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You are an expert at summarizing long articles. Proceed to summarize the following text: in recent years a significant effort has been devoted to the understanding of glass - forming systems . recent theoretical and numerical results clearly show that the slowing down of the dynamics near the glass transition is strongly connected to the potential energy landscape geometry . the trajectory of the representative point in the configuration space can be viewed as a path in a multidimensional potential energy surface @xcite . the dynamics is therefore strongly influenced by the topography of the potential energy landscape : local minima , barriers heights , basins of attraction an other topological properties all influence the dynamics . the potential energy surface of a super - cooled liquid contains a large number of local minima , called _ inherent structures _ ( is ) by stillinger @xcite . all states that under local energy minimization will flow into the same is define the _ basin _ of the is ( valley ) . with this pictures in mind the time evolution of the system can be seen as the result of two different processes : thermal relaxation into basins ( _ intra - basin _ motion ) and thermally activated potential energy barrier crossing between different basins ( _ inter - basin _ motion ) . when the temperature is lowered down to the order of the critical mode coupling theory ( mct ) temperature @xmath1 the inter - basin motion slows down and the relaxation dynamics is dominated by the slow thermally activated crossing of potential energy barriers @xcite . if the temperature is further reduced the relaxation time eventually becomes of the same order of the observation time and the system falls out of equilibrium since there is not enough time to cross barriers and equilibrate . this define the `` experimental''glass transition temperature @xmath2 . the regime between @xmath1 and @xmath2 can not be described by the mct since it neglects activated processes responsible for barrier crossing . in mct the relaxation time diverges at @xmath1 , leading to @xmath3 , and the dynamics remains confined into a single basin forever . the essential features of mct for glass - forming systems are also common to the high temperature phase of some fully connected spin glass models @xcite , the most well known being the spherical @xmath0-spin spin glass model @xcite . we shall call these models _ mean - field @xmath0-spin - like _ glass models . as a consequence at the critical temperature @xmath1 , called @xmath4 in @xmath0-spin language , an ergodic to non - ergodic transition takes place . below this temperature the system is dynamically confined to a metastable state ( a basin ) @xcite since relaxation to true equilibrium can only take place via activated processes , absent in mean - field models . for these systems , nevertheless , it is known that the true equilibrium transition to a low temperature phase occurs below @xmath4 at the static critical temperature @xmath5 , also denoted by @xmath6 @xcite . this is the analogous of the kauzmann temperature @xmath7 for liquids . the glass transition temperature @xmath2 of real systems sits somewhere in between @xmath5 and @xmath4 . this transition , obviously , can not be reached even on infinite time in mean - field models . despite these difficulties mean - field models , having the clear advantage of being analytically tractable , have been largely used to study the properties of fragile glassy systems , especially between the dynamical temperature @xmath4 and the static temperature @xmath5 . the picture that emerges is however not complete since activated process can not be captured by mean - field models . therefore the relevance of mean - field results for real systems can not be considered completely stated . only recently activated processes in mean - field - like models have been invesigated in extended numerical investigation of _ finite - size _ fully - connected @xmath0-spin - like models @xcite . comparing the results with the observed behavior of super - cooled liquids near @xmath1 we can conclude that , once activated process are allowed , mean - field @xmath0-spin - like models are highly valuable for a deep understanding of the glass transition in real systems . we report the main results obtained for the ising - spin random orthogonal model ( rom ) @xcite , defined by the hamiltonian @xcite , @xmath8 where @xmath9 are @xmath10 ising spin variables , and @xmath11 is a @xmath12 random symmetric orthogonal matrix with @xmath13 . for @xmath14 this model has the same thermodynamic properties of the @xmath0-spin model : a dynamical transition at @xmath15 , with threshold energy per spin @xmath16 , and a static transition at @xmath17 , with critical energy per spin @xmath18 @xcite . the free energy analysis ( tap ) @xcite reveals that the phase space is composed by an exponentially large ( in @xmath10 ) number of different basins , separated by infinitely large ( for @xmath14 ) barriers . each basin is unambiguously labelled by the value of the energy density @xmath19 of the local minimum contained within it , i.e. the is of the system . in this picture the dynamical transition is associated with is having the largest basin of attraction for @xmath14 , while the static transition with is with the lowest accessible free energy ( vanishing configurational entropy ) @xcite . in the mean - field limit , the allowed values of @xmath19 are between @xmath20 and @xmath21 . solutions with @xmath19 larger than @xmath21 are unstable ( saddles ) , while solutions with @xmath19 smaller than @xmath20 have negligible statistical weight . moreover in the @xmath14 limit is with @xmath22 attract most ( exponentially in @xmath10 ) of the states and dominate the behavior of the system . other is are irrelevant for @xmath14 . for finite @xmath10 the scenario is different since not only the basins of is with @xmath23 acquire statistical weight , but it may happen that solutions with @xmath24 and few negative directions ( saddles with few downhill directions ) become stable , simply because there are not enough degrees of freedom to hit them . to get more insight the is - structure of finite systems we follow stillinger and weber @xcite and decompose the partition sum into a sum over basins of different is and a sum within each basin . collecting all is with the same energy @xmath19 , denoting with @xmath25\,de$ ] the number of is with energy between @xmath19 and @xmath26 , and shifting the energy of each basin with that of the associated is , the partition sum can be rewritten as @xcite @xmath27\ ] ] where @xmath28 can be seen as the free energy density of the system when confined in one of the basin associated with is of energy @xmath19 . the function @xmath29 is the _ configurational entropy density _ also called _ complexity_. from the partition function we can compute the average internal energy density @xmath30 @xmath31 . the first term is the average energy of the is relevant for the thermodynamics at temperature @xmath32 , while the second is the contribution from fluctuations inside the associated basins . in the limit @xmath14 only is with @xmath22 contribute and @xmath33 for any @xmath34 . for finite @xmath10 , and @xmath32 not too close to @xmath4 , the thermodynamics is dominated by is with @xmath24 and @xmath35 @xcite . this is more evident from the ( equilibrium ) probability distribution of @xmath19 since it is centered about @xmath36 indicating that is with @xmath37 have the largest basins . this scenario has been also observed in real glass - forming systems@xcite . from the knowledge of is - energy distribution we can reconstruct the complexity @xmath29 since from eq . ( [ eq : part ] ) the probability that an equilibrium configuration at temperature @xmath38 lies in a basin associated with is of energy between @xmath19 and @xmath26 is : @xmath39 / z_n(t ) . $ ] in the temperature range where this applies , the curves @xmath40 are equal , except for a temperature dependent factor @xmath41 , to @xmath42 . if the @xmath19-dependence of @xmath28 can be neglected , then it is possible to superimpose the curves for different temperatures , see fig . [ fig : f2 ] ( a ) . the data collapse is rather good for @xmath43 . above the curves can not be superimposed anymore indicating that the @xmath19-dependence of @xmath28 can not be neglected . in liquid this is called the anharmonic threshold @xcite . direct consequence of @xmath44 for @xmath45 is that in this range the partition function can be written as the product of an intra - basin contribution [ @xmath46 ) ] and of a configurational contribution which depends only on the is energy densities distribution . the system can then be considered as composed by two independent subsystems : the intra - basin subsystem describing the equilibrium when confined within basins , and the is subsystem describing equilibrium via activated processes between different basins . as the temperature is lowered and/or @xmath10 increased the two processes get more separated in time and the separation becomes more and more accurate . a scenario typical of super - cooled liquids near the mct transition @xcite . more informations on the is structure can be obtained from non - equilibrium relaxation processes . to study the non - equilibrium dynamics we quench at time zero the system from an initial equilibrium configuration at temperature @xmath47 to a final temperature @xmath48 and study the evolution of the average is energy per spin @xmath49 as function of time , see fig.[fig : e - aging ] ( b ) . two different relaxation processes are clear seen . a first regime independent of @xmath50 , and a second regime independent of both @xmath51 and @xmath50 . the final temperature @xmath50 controls the cross - over between the two regimes . a similar behavior has been observed in molecular dynamics simulations of super - cooled liquids @xcite . the two regimes are associated with different relaxation processes . in the first part the system has enough energy and relaxation is mainly due to _ path search _ out of basins through saddles of energy lower than @xmath52 . this part depends only on the initial equilibrium temperature @xmath51 since it sets the initial phase space region . different @xmath51 leads to different power law . in particular relaxation must slow down as @xmath51 decreases since we expect that lower states are surrounded by higher barriers , in agreement with numerical data @xcite . during this process the system explores deeper and deeper valleys ( basins ) while decreasing its energy . the process stops when all barrier heights become of @xmath53 . from now on the relaxation proceeds only via activated process . a first consequence is that lower the final temperature @xmath50 shorter the first relaxation , in agreement with our findings [ see figures [ fig : e - aging ] ] . the analysis of the distance between the instantaneous system state and the corresponding is , counting the number of single spin flip needed to reach the is , reveals that for all times the systems stays in configurations few spin flips away from an is . a similar study starting from equilibrium configurations at temperature @xmath54 evaluated comparing panels ( a ) and ( b ) of figure [ fig : e - aging ] @xcite leads to similar numbers . we then conclude that during relaxation the aging system explores the same type of minima ( and basins ) visited in equilibrium at temperature @xmath55 . direct consequence is that once the system has reached the activated regime there can not be memory of the initial @xmath51 , and all curves with different @xmath51 but same @xmath50 should collapse for large time @xcite . to summarize , we have shown that _ finite - size _ mean - field @xmath0-spin - like models are good candidates for studying the glass transition . the key point is that near the glass transition the thermodynamics of the systems is dominated by the is distributions , therefore all systems with similar is distributions should have similar behavior . finite - size mean - field @xmath0-spin - like models have the double advantage of being analytically tractable for @xmath14 and easily simulated numerically for finite @xmath10 , offering good models to analyze the glass transition .

we analyze the properties of the energy landscape of _ finite - size _ fully connected @xmath0-spin - like models whose high temperature phase is described , in the thermodynamic limit , by the schematic mode coupling theory of super - cooled liquids . we show that _ finite - size _ fully connected @xmath0-spin - like models , where activated processes are possible , do exhibit properties typical of real super - cooled liquid when both are near the critical glass transition . our results support the conclusion that fully - connected @xmath0-spin - like models are the natural statistical mechanical models for studying the glass transition in super - cooled liquids . and glass transition , spin - glass , random models pacs : 64.70.pf , 75.10.nr , 61.20.gy , 82.20.wt

You are an expert at summarizing long articles. Proceed to summarize the following text: let us consider the following nonlinear oscillator described by the so called modified emden equation with linear forcing term @xcite , @xmath1 here @xmath2 is a parameter . equation ( 1 ) can be considered as the cubic anharmonic oscillator with additional position dependent damping type nonlinear force @xmath3 . this type of equation has been well studied in the literature . for example , eq . ( 1 ) with @xmath4 arises in a wide range of physical problems : it occurs in the study of equilibrium configurations of a spherical gas cloud acting under the mutual attraction of its molecules and subject to the laws of thermodynamics @xcite and in the modelling of the fusion of pellets @xcite . it also governs spherically symmetric expansion or collapse of a relativistically gravitating mass @xcite . this equation can also be thought of as a one - dimensional analog of the boson ` gauge - theory ' equations @xcite . equation ( [ mod01a ] ) has been shown to posses an unusual property which is not a general characteristic of a nonlinear equation : the frequency of oscillation of the oscillator is independent of the amplitude similar to that of a linear harmonic oscillator @xcite . an oscillator which possesses this property is also known as an isochronous oscillator @xcite . for a detailed study about isochronous orbits and isochronous oscillators one may refer to refs . @xcite . equation ( [ mod01a ] ) admits the following nonsingular , periodic solution : @xmath5 here @xmath6 and @xmath7 are arbitrary constants , expressible in terms of the two integrals of motion or integration constants obtained by solving ( [ mod01a ] ) ( for details see ref . @xcite ) . note that the angular frequency of oscillation @xmath2 continues to be the same as that of the linear oscillation . from this solution it is obvious that for @xmath8 , equation ( [ mod01a ] ) exhibits the property of amplitude independence of the frequency of oscillation . one can starightforwardly write down the solution of the initial value problem from the general solution ( [ mod02a ] ) . for example , for the initial condition @xmath9 , @xmath10 , from ( [ mod02a ] ) we have the solution as @xmath11}{\sqrt{b^2+\omega^2}-b\cos\left[\omega t+\cos^{-1}\left(\frac{b}{\sqrt{b^2+\omega^2}}\right)\right]}.\end{aligned}\ ] ] note that @xmath12 is the amplitude of oscillation . figure [ fig1 ] shows the periodic oscillations admitted by eq . ( [ mod01a ] ) for three different sets of initial conditions @xmath13 and @xmath14 with @xmath15 in terms of three different colours . we note here that the frequency of the oscillations is independent of the initial conditions as in the case of the linear harmonic oscillator . ) exhibiting periodic oscillation for three different initial conditions ( three different colours ) and @xmath15 ( b ) phase space portrait of eq . ( [ mod01a]),width=529 ] one can trace the origin of this property of equation ( [ mod01a ] ) to the fact that it can be transformed to the linear harmonic oscillator equation , @xmath16 through a nonlocal transformation , @xmath17 the solution ( [ mod02a ] ) can be obtained ( see below , equation ( [ nld05 ] ) ) from the solution of ( [ horm1 ] ) , @xmath18 , where @xmath6 and @xmath7 are arbitrary constants and the frequency , @xmath2 , is independent of the amplitude . such a linearization property is one of the fascinating features associated with a class of nonlinear equations exhibiting large number of symmetries and extensive search for such linearizing transformations is being made in the recent literature @xcite . in fact , there exists a class of nonlinear oscillators which are connected to the linear oscillator equation ( [ horm1 ] ) through the following nonlocal transformation @xcite @xmath19 where @xmath20 is an arbitrary function of @xmath21 . now substituting ( [ int02 ] ) into ( [ horm1 ] ) we get a nonlinear ordinary differential equation ( ode ) of the form @xmath22 where prime denotes differentiation with respect to @xmath23 . equation ( [ int03 ] ) is a special case of the well known lienard equation ( le ) @xcite @xmath24 one can also consider a more general nonlocal transformation of the form @xmath25 and substituting this in ( [ horm1 ] ) we get @xmath26 we find the above equation reduces to a linard type equation only for the choice @xmath27 . interestingly for @xmath28 , equation ( [ int03 ] ) becomes the well known isotonic oscillator @xcite equation , @xmath29 the solution of the nonlinear equation ( [ int03 ] ) is obtained by using the identity @xmath30 since @xmath31 , where @xmath6 and @xmath7 are integration constants , is the solution of the linear harmonic oscillator ( [ horm1 ] ) , equation ( [ nld05 ] ) can be rewritten as the first order nonlinear differential equation of form @xmath32 now one can get the solution of ( [ int03 ] ) by solving ( [ mod07aa ] ) . in particular , for the specific case @xmath33 equation ( [ mod07aa ] ) becomes a bernoulli equation of the form @xmath34 the corresponding ode ( [ int03 ] ) becomes @xmath35 and equation ( [ mod01a ] ) is the special case corresponding to @xmath36 . upon integrating ( [ mod07b ] ) we get the periodic solution of ( [ mod01 ] ) as @xmath37^{\frac{1}{(2m+1)}}},\end{aligned}\ ] ] where @xmath38 , @xmath39 , @xmath40 , @xmath41 and @xmath7 are arbitrary constants . here @xmath42 is a non - negative integer and @xmath2 is the angular frequency . one can note that solution ( [ mod02 ] ) is also isochronous . this has indeed been reported recently by iacono and russo @xcite using a different procedure . in figure [ fig2 ] we show the periodicity of the solution for the case @xmath43 and with the initial conditions @xmath44 and @xmath10 . we additionally remark here that the case @xmath45 , @xmath46 of equation ( [ mod01 ] ) is also exactly solvable but the solutions are of damped oscillatory type as will be proved later in this article ( sec . [ sec2 ] ) even for coupled systems . ) for @xmath43 exhibiting periodic oscillations with the initial condition @xmath44 and @xmath10 for @xmath47 ( b ) phase space portrait of eq . ( [ mod01]),width=529 ] in this paper we will show that this unusual ( amplitude independent frequency ) property possessed by the class of equations ( [ int03 ] ) is not limited to scalar les alone but is also possessed by a class of @xmath0 coupled nonlinear oscillator equations . in order to demonstrate the existence of periodic and quasiperiodic solutions for such systems , we extend the above mentioned procedure to @xmath0 coupled les . we also derive the integrals of motion and the general solution to the system of @xmath0 coupled les . we also point out that there exists another class of @xmath0 coupled nonlinear oscillators which is almost integrable in the sense that it admits @xmath48 independent integrals and reduces to a single first order nonautonomous nonlinear differential equation that can be solved numerically . the frequency of oscillation of its periodic solutions is again independent of amplitude . in the following section we show that a system of @xmath0 coupled les can be explicitly integrated to find periodic and quasiperiodic solutions whose frequencies are identical to a system of uncoupled linear oscillators . in sec . [ sec3 ] we deduce the underlying @xmath49 integrals of motion for the system and analyze their structure . in sec . [ nearlyintegrable ] we numerically solve the system of @xmath0 coupled les for which the explicit general solution is not known , but @xmath48 integrals are known . we finally summarize our results in sec . generalizing the above results of the scalar system ( [ int03 ] ) to a system of coupled nonlinear oscillators of lienard type , we relate them to a system of uncoupled @xmath0-dimensional linear harmonic oscillators . for this purpose , we consider the @xmath0-dimensional anisotropic harmonic oscillator equation of the form @xmath50 where the freqencies @xmath51 are in general different . let us now introduce the nonlocal transformation @xmath52 where @xmath53 are arbitrary functions of the variables . subsituting ( [ nhorm2 ] ) into ( [ nhorm1 ] ) we get a system of @xmath0 coupled lienard type second order nonlinear oscillator equations of the form @xmath54 where @xmath55 , @xmath56 . in the following we shall demonstrate the existence of periodic and quasiperiodic solutions with amplitude independent frequencies , @xmath51 ( isochronous property ) for the above mentioned nonlinear oscillator equation with specific forms of @xmath57 s . to obtain the desired results , we make use of the following identities , @xmath58 derived from equation ( [ nhorm2 ] ) . obviously the solution of the system of second order linear odes ( [ nhorm1 ] ) is @xmath59 where @xmath60 s and @xmath61 s , @xmath62 , are @xmath49 integration constants . now substituting the solution ( [ ham_sol ] ) in equation ( [ nhorm4 ] ) we find the following system of coupled first order odes to represent ( [ nhorm3 ] ) , @xmath63 it may be noted that @xmath0 integration constants , @xmath61 s , @xmath62 , of the @xmath0 coupled second order odes ( [ nhorm3 ] ) appear explicitly in ( [ mod07 ] ) . in order to find the general solution of ( [ nhorm3 ] ) we need @xmath0 more integration constants which are to be obtained by integrating ( [ mod07 ] ) . for general forms of @xmath64 in ( [ nhorm3 ] ) this can not be done . however for the special choice , @xmath65 equation ( [ mod07 ] ) can be integrated . in order to perform this integration we multiply each equation of the system ( [ mod07 ] ) by @xmath66 for @xmath67 and the last equation by @xmath68 and subtract them to get @xmath69 dividing eq . ( [ int1 ] ) throughout by @xmath70 , we get @xmath71 integrating the above equation , we get @xmath72 rewriting the above equation , we get @xmath73 where @xmath74 s , @xmath75 , are @xmath76 integration constants . we are also left with a first order ode of the form @xmath77 where @xmath78 and @xmath79 s are given by equation ( [ mod08 ] ) . now the problem of solving the set of @xmath0 coupled autonomous second order odes ( [ nhorm3 ] ) is reduced to the problem of solving a single non - autonomous first order ode ( [ mod07a ] ) . therefore one can get the general solution for equation ( [ nhorm3 ] ) for the case ( [ mod07e ] ) by solving equation ( [ mod07a ] ) . equation ( [ mod07a ] ) again can not be in general solved explicitly for arbitrary form of the function @xmath80 @xcite . hence , in order to solve ( [ mod07a ] ) we assume that the function @xmath80 has a symmetry @xmath81 , where @xmath82 , @xmath83 and @xmath84 are arbitrary parameters . this implies that @xmath80 is a homogeneous polynomial and we assume the following form of @xmath80 , @xmath85 where @xmath86 s are arbitrary functions of @xmath87 . however , in the present case we assume @xmath88 s to be constants only for simplicity . even when they are functions of @xmath87 the following integration procedure holds good . with the above choice of @xmath80 , equation ( [ nhorm3 ] ) reduces to the system of coupled nonlinear oscillator equations , @xmath89 the solution of this system of coupled nonlinear oscillators can be obtained by solving the following first order nonlinear differential equation obtained by substituting ( [ mod10e ] ) into ( [ mod07a ] ) along with ( [ mod08 ] ) : @xmath90 where @xmath91 , @xmath92 , and @xmath93 . the above equation is of the first order bernoulli equation type @xcite , namely @xmath94 where @xmath95 and @xmath96 are arbitrary functions of the independent variable @xmath97 . with the substitution @xmath98 in ( [ bern - eq1 ] ) we get the following first order linear inhomogenous ode @xmath99 the general solution of ( [ bern1 ] ) is obviously @xmath100},\end{aligned}\ ] ] where @xmath101 is the integration constant . rewriting the above in terms of @xmath66 , we get @xmath102^{1/q}},\end{aligned}\ ] ] where @xmath101 is the @xmath103 integration constant which we are looking for . the integral appearing in the denominator of the above expression can be integrated explicitly for arbitrary values of @xmath104 . however , we find that the system admits oscillatory solutions only when @xmath104 is a positive integer and for this choice we find the above expression reduces to either of the following forms depending on the value of @xmath104 @xcite . * ( i ) case 1 - @xmath104 odd :* we take @xmath105 , @xmath106 . then equation ( [ mod11 ] ) reduces to the form [ sol_odd ] @xmath107^{\frac{1}{2m+1}}},\end{aligned}\ ] ] where @xmath108 , @xmath109 and @xmath110 . using ( [ mod08 ] ) and ( [ mod12 ] ) we get the remaining @xmath76 variables as @xmath111^{\frac{1}{2m+1}}},\end{aligned}\ ] ] where @xmath75 . note that for the solution ( [ sol_odd ] ) to be nonsingular periodic , we require the condition @xmath112 . we note here that for the special case @xmath113 the solutions ( [ mod12 ] ) and ( [ mod13 ] ) become the periodic / quasiperiodic solution @xmath114 which exactly matches with the solution given in @xcite for the so called coupled modified emden equation . * ( ii ) case 2 - @xmath104 even * , here we take @xmath45 , @xmath115 . then equation ( [ mod11 ] ) reduces to the form [ sol_even ] @xmath116^{\frac{1}{2m+1}}},\end{aligned}\ ] ] where @xmath117 , @xmath118 , @xmath119 . using ( [ mod08 ] ) and ( [ mod12b ] ) we get the remaining @xmath76 variables as @xmath120^{\frac{1}{2m+1}}}\,\,.\end{aligned}\ ] ] from ( [ mod12b ] ) and ( [ mod12c ] ) we find that equation ( [ nhorm3a ] ) admits only oscillatory dissipative type solution for the choice @xmath45 , @xmath115 . note that for @xmath104 odd positive integer in ( [ nhorm3a ] ) , one can have either periodic or quasiperiodic solutions , depending on whether the uncoupled frequencies @xmath121 s are commensurate or not . in fig . 1 we have presented quasiperiodic and periodic solutions for suitable choices of the uncoupled frequencies @xmath122 with @xmath123 in the form of projected phase space plots in the @xmath124 plane . we find that for the choice @xmath125 and @xmath126 , the system ( [ nhorm3a ] ) exhibits @xmath127 period oscillations . similarly for the choice @xmath128 and @xmath126 , the system ( [ nhorm3a ] ) is found to exhibit quasiperiodic oscillations . one can note that equation ( [ nhorm3a ] ) can also be rewritten in the following first order form as was done by iacono and russo @xcite for the scalar case as @xmath129 [ mod05 ] multiplying eq . ( [ mod05aa ] ) by @xmath130 and eq . ( [ mod05ab ] ) by @xmath68 and subtracting the resulting equations we get @xmath131 dividing throughout by @xmath132 and rewriting we get @xmath133 upon introducing the angle variable @xmath134 , equation ( [ mod06 ] ) becomes @xmath135 from ( [ mod06c ] ) it is obvious that the angle and hence the frequency ( which is similar to that of the linear harmonic oscillator ) are independent of the amplitude of oscillation , irrespective of the form of @xmath136 . however , one may note that this does not always imply isochronocity as the amplitude of oscillation may decay with time , as shown in equation ( [ sol_even ] ) . ) in the @xmath124 plane with @xmath123 , for two different values @xmath43 ( figures ( a , c ) ) and @xmath137 ( figures ( b , d ) ) , respectively . figures ( a ) and ( b ) describe the @xmath127 period oscillations for the choice @xmath125 and @xmath138 . figures ( c ) and ( d ) describe the quasiperiodic oscillations for the choice @xmath128 and @xmath138 . , in this section we show the existence of @xmath49 independent integrals of motion for equation ( [ nhorm3 ] ) with @xmath139 , @xmath140 , being a homogeneous polynomial ( [ mod10e ] ) . in order to show that there exists @xmath0 time dependent integrals , let us consider the equivalent form of ( [ mod07 ] ) , that is , @xmath141 rewriting the above equation ( the first and the last expressions ) we get @xmath0 time dependent integrals as @xmath142-\omega_it , \ ; i=1,2 , \ldots , n,\end{aligned}\ ] ] where @xmath143 . now to find the remaining @xmath0 integrals for @xmath104 odd or even integer in ( [ mod10e ] ) we proceed as follows . from ( [ mod16 ] ) we get @xmath144 . using this expression in the well known trignometric identity @xmath145 and @xmath146 , we get @xmath147 [ mod17 ] rewriting now ( [ mod08 ] ) in the form @xmath148 we obtain @xmath149 now by using ( [ mod17a ] ) we can write @xmath150 substituting the above in ( [ rev2 ] ) we obtain the first @xmath76 time independent integrals as @xmath151 for @xmath104 odd positive integer , substituting ( [ mod17 ] ) and ( [ mod15 ] ) into ( [ mod12 ] ) and rearranging we arrive at the following form for the integral @xmath152 , @xmath153 which is the @xmath0th time independent integral . for @xmath113 the integral ( [ mod14 ] ) becomes @xmath154 one can note that for @xmath113 the integrals ( [ mod16a ] ) , ( [ mod15 ] ) and ( [ mod14a ] ) exactly match with those presented in @xcite . for instance , equation ( [ mod14 ] ) exactly reduces to the corresponding form given in @xcite for the scalar case ( @xmath155 ) . similarly , for the case where @xmath104 is even positive integer , we substitute ( [ mod17 ] ) and ( [ mod15 ] ) into ( [ mod12b ] ) and rearranging we arrive at the following form for the integral @xmath156 , @xmath157 we note here that the above first integral is a time dependent one . in sec . [ sec3 ] we obtained the general solution of ( [ nhorm3 ] ) for the choice @xmath158 , @xmath140 and @xmath80 is a homogeneous polynomial as in ( [ mod10e ] ) . however , we wish to point out that the system ( [ nhorm3 ] ) for any arbitrary choice of ( [ mod07e ] ) is almost integrable as there always exist @xmath0 time dependent integrals ( [ mod16a ] ) and @xmath159 time independent integrals ( [ mod15 ] ) . for complete integrability only the first order nonautonomous differential equation ( [ mod07a ] ) needs to be integrated . for those forms of @xmath80 for which this can not be done explicitly , one can always carry out a numerical integration of ( [ mod07a ] ) or apply a suitable approximation method to find @xmath160 . the remaining @xmath161 s , @xmath162 , can be obtained readily using ( [ mod08 ] ) and @xmath160 . equation ( [ mod06c ] ) ensures that if the solutions are periodic or quasiperiodic , the frequency is independent of amplitude . ) in the @xmath163 plane for the choices ( a ) @xmath164 exhibiting @xmath165 period oscillation , ( b ) @xmath166 exhibiting quasiperiodic oscillation ] in order to demonstrate the above , let us consider the special case @xmath167 in ( [ nhorm3a ] ) and @xmath168 . this choice reduces equation ( [ nhorm3a ] ) to the following system of coupled second order nonlinear odes , [ nonintegrable ] @xmath169 the solution of the above system of equations can be deduced by solving the following first order ode obtained by the procedure discussed in the previous section , @xmath170 however , we find that the explicit general solution of the above first order ode is not known @xcite . one can apply suitable numerical methods to solve this equation . figure [ fig4 ] is plotted by solving eq . ( [ first ] ) using a variable step size runge - kutta fourth order method . we find the system ( [ nonintegrable ] ) exhibits periodic and quasiperiodic oscillations which is shown in fig [ fig4 ] . here the projection of the phase space of ( [ nonintegrable ] ) in the @xmath163 plane for the set @xmath171 and @xmath172 are shown for @xmath165 periodic behaviour and quasiperiodic behaviour , respectively . .[table1]a comparison of the solution of eq . ( [ first ] ) obtained using the variable step size runge - kutta fourth order ( rk4 ) numerical procedure and the homotopy perturbation method ( hpm ) . [ cols= " < , < , < " , ] one can also apply perturbation techniques such as the homotopy analysis @xcite to find approximate solution of high accuracy for eq . ( [ first ] ) and compare the results with the numerical analysis . the homotopy perturbation method @xcite involves the introduction of an artificial parameter , say @xmath173 , into the original nonlinear equation @xmath174 as @xmath175 where @xmath176 is the linear operator corresponding to the linear part of the given equation and the @xmath6 is the operator corresponding to the given nonlinear equation . here @xmath177 is the lowest order approximate solution . expressing the solution @xmath97 as a power series in @xmath173 , i.e. @xmath178 where @xmath179 , @xmath180 are the higher order approximations , one can substitute this series solution in ( [ first ] ) to obtain a system of linear first order ordinary differential equations . solving this system with suitable initial conditions one can obtain the approximate solution to the given equation in the limit @xmath181 . for further details on this procedure one can refer to refs . @xcite . in table [ table1 ] we compare the solution of eq . ( [ first ] ) obtained through this procedure with the numerical solution for the parametric choice @xmath182 . the values listed in table [ table1 ] for the homotopy perturbation method are calculated up to third order approximation . we note here that the accuracy of the perturbation solution will improve if further higher order approximations are taken into the calculation @xcite , which we do not pursue here . in this paper , we have shown that a system of @xmath0 coupled nonlinear lienard type oscillators admits periodic and quasiperiodic solutions or damped oscillatory periodic solutions with amplitude independent frequency of oscillations . we have derived explicit general solution and @xmath49 integrals of motion for this system . thus we prove this system to be completely integrable . we have also shown that another system of @xmath0 coupled lienard type oscillators is almost integrable in the sense that it admits @xmath48 independent integrals and reduces to a single nonautonomous first order nonlinear differential equation . we have also shown that this almost integrable system also exhibits periodic and quasiperiodic oscillations for suitable parametric choices . for the general system of @xmath0 coupled nonlinear oscillators ( [ nhorm3 ] ) with arbitrary form of nonlinearity , the nonlocal transformations reduce it to a system of @xmath0 coupled first order odes ( [ mod07e ] ) . it will be interesting to investigate further whether other forms of nonlinearity ( different from ( [ mod10e ] ) ) are also amenable to analysis . the work is supported by a department of science and technology ( dst)ramanna fellowship project and a dst irhpa research project of m. l. , who is also supported by a dae raja ramanna fellowship . jhs is supported by a dst fast track young scientist research project . v. j. erwin , w. f. ames and e. adams _ wave phenomena : modern theory and applications _ ed c rogers and j b moodie ( amsterdam : north - holland ) g. c. mcvittie , _ mon . not . r. astron . soc . _ * 93 * 325 ( 1933 ) ; _ ann . inst . h poincar _ * 6 * 1 ( 1967 ) ; * 40 * 231 ( 1984 ) a. kimiaeifar , a. r. saidi , g. h. bagheri , m. rahimpour and d. g. domairry , _ chaos , solitons & fractals _ * 42 * 2660 ( 2009 ) ; m. bayat , m. shahidi , a. barari and g. domairry , _ int . j. phys . sci . _ * 5 * 1074 ( 2010 )

existence of amplitude independent frequencies of oscillation is an unusual property for a nonlinear oscillator . we find that a class of n coupled nonlinear linard type oscillators exhibit this interesting property . we show that a specific subset can be explicitly solved from which we demonstrate the existence of periodic and quasiperiodic solutions . another set of @xmath0-coupled nonlinear oscillators , possessing the amplitude independent nature of frequencies , is almost integrable in the sense that the system can be reduced to a single nonautonomous first order scalar differential equation which can be easily integrated numerically . nonlinear oscillators , coupled ordinary differential equations , complete integrability , isochronous systems

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Proceed to summarize the following text: ever since the seminal work of kramers on the diffusion model of chemical reactions was published about half a century ago @xcite , the theory of activated processes has become a central issue in many areas of science @xcite , notably in chemical physics , nonlinear optics and condensed matter physics . kramers considered a model brownian particle trapped in a one - dimensional well representing the reactant state which is separated by a barrier of finite height from a deeper well signifying the product state . the particle was supposed to be immersed in a medium such that the medium exerts a frictional force on the particle but at the same time thermally activates it so that the particle may gain enough energy to cross the barrier . over several decades the model and many of its variants have served as standard paradigms in various problems of physical and chemical kinetics to understand the rate in multidimensional systems in the overdamped and underdamped limits @xcite , effect of anharmonicities @xcite , rate enhancement by parametric fluctuations @xcite , the role of non - gaussian white noise @xcite , role of a relaxing bath @xcite , quantum and semiclassical corrections @xcite to classical rate and related similar aspects . the vast body of literature has been the subject of several reviews @xcite and monograph @xcite . the common feature of overwhelming majority of the aforesaid treatments is that the system is thermodynamically closed which means that the noise of the medium is of _ internal _ origin so that the dissipation and fluctuations get related through the fluctuation - dissipation relation @xcite . however , in a number of situations the system is thermodynamically open , i.e. , when the system is driven by an _ external _ noise which is _ independent _ of system s characteristic damping @xcite . the distinctive feature of the dynamics in this case is the absence of any fluctuation - dissipation relation . while in the former case a zero current steady state situation is characterized by an equilibrium boltzmann distribution , the corresponding situation in the latter case is defined only by a steady state condition , if attainable . it may therefore be anticipated @xcite that the independence of fluctuations and dissipation tends to make the steady state distribution function depend on the strength and correlation time of external noise as well as on the dissipation of the system . the elucidation of the role of this steady state distribution in rate theory is worth - pursuing . our aim in this paper is to generalize kramers theory of activated processes for external noise in this context . we thus allow the brownian particle in a potential field to be driven by both external and internal stationary and gaussian noise fluctuations with arbitrary decaying correlation functions . the external noise may be of thermal or non - thermal type . we consider the stochastic motion to be spatial - diffusion - limited and calculate the rate of escape over the barrier in the intermediate to strong damping regime within an unified description . the theory we develop here follows closely the original flux over population method of farkas @xcite . the distinctive aspect , however , is the consideration of a steady state distribution instead of the equilibrium boltzmann distribution for determination of quasi - stationary population in the source well . this affects the generalized rate expression significantly in two ways . first , the dynamics around the bottom of the source well exhibits the dependence of steady state distribution on the dissipation . second , the rate expression remains valid even in absence of any internal thermal noise . we mention , in passing , that the former point had earlier been rightly emphasized by melnikov @xcite as a specific requirement for a general theory . some pertinent points regarding the rate theory for nonequilibrium systems may be in order . it is wellknown that though thermodynamically closed systems with homogenous boundary conditions possess in general time - independent solutions , the driven or open systems may settle down to complicated multiple steady states @xcite when one takes into account of nonlinearity of the system in full . secondly in most nonequilibrium systems the lack of detailed balance symmetry gives rise to severe problem in determination of stationary probability density for multidimensional problem @xcite . because of its one - dimensional and linearized description the present treatment is free from these difficulties . it is important to point out that the externally generated nonequilibrium fluctuations can bias the brownian motion of a particle in an anisotropic medium and may used for design of molecular motors and pumps @xcite . the nonequilibrium , non - thermal systems has also been investigated by a number of worker in different contexts , e.g. , for examining the role of colour noise in stationary probabilities @xcite , the properties of nonlinear systems @xcite , the nature of cross - over @xcite , the effect of monochromatic noise @xcite , the rate of diffusion - limited coagulation processes @xcite , etc . the outlay of the paper is as follows : in sec.ii we generalize kramers theory of reaction rate for external noise . the stationary , gaussian noise processes are of both external and internal type with arbitrary decaying correlation functions . a general form of steady state distribution function in the source well and a rate expression for barrier crossing dynamics for the nonequilibrium open system have been pointed out . in sec.iii we explicitly calculate the detailed form of the rate expressions for the specific cases . the paper is concluded in sec.iv . we consider the motion of a particle of unit mass moving in a kramers type potential @xmath0 such that it is acted upon by random forces @xmath1 and @xmath2 of both internal and external origin , respectively , in terms of the following generalized langevin equation @xmath3 where the friction kernel @xmath4 is connected to internal noise @xmath1 by the wellknown fluctuation - dissipation relationship @xcite @xmath5 we assume that both the noises @xmath1 and @xmath2 are stationary and gaussian . their correlation times may be of arbitrary decaying type . the external noise is independent of the memory kernel and there is no corresponding fluctuation - dissipation relation . we further assume , without any loss of generality , that @xmath1 is independent of @xmath2 so that we have @xmath6 the external noise modifies the dynamics of activation in two ways . first , it influences the dynamics in the region around the barrier top so that the effective stationary flux across it gets modified . second , in presence of this noise the equilibrium distribution of the source well is disturbed so that one has to consider a new stationary distribution , if any , instead of the standard boltzmann distribution . this new stationary distribution must be a solution of the generalized fokker - planck equation around the bottom of the source well region and serve as an appropriate boundary condition analogous to kramers problem . we consider these two aspects separately in the next two subsections . we consider the potential @xmath0 as shown in fig.1 . linearizing the potential around the barrier top at @xmath7 we write @xmath8 thus the langevin equation takes the following form @xmath9 where @xmath10 the general solution of eq.([eq5 ] ) is given by , @xmath11 where @xmath12 with @xmath13 and @xmath14 being the initial position and velocity of the brownian particle that are assumed to be nonrandom , and @xmath15 the kernel @xmath16 is the laplace inversion of , @xmath17 with @xmath18 the time derivative of eq.([eq7 ] ) gives @xmath19 with @xmath20 and @xmath21 now using the symmetry of the correlation function , @xmath22 we compute the explicit expressions of the variances in terms of @xmath16 and @xmath23 as , @xmath24 ^ 2 \rangle \nonumber \\ & = & 2 \int_0^t m_b(t_1 ) \ ; dt_1 \int_0^{t_1 } m_b(t_2 ) \ ; c(t_1 - t_2 ) \ ; dt_2 \ ; \ ; , \\ \sigma_{vv}^2 ( t ) & \equiv & \langle [ v(t ) - \langle v(t ) \rangle ] ^2 \rangle \nonumber \\ & = & 2 \int_0^t m_b(t_1 ) \ ; dt_1 \int_0^{t_1 } m_b(t_2 ) \ ; c(t_1 - t_2 ) \ ; dt_2 \ ; \ ; , \\ \sigma_{xv}^2 ( t ) & \equiv & \langle [ x(t ) - \langle x(t ) \rangle ] \ ; [ v(t ) - \langle v(t ) \rangle ] \rangle \nonumber \\ & = & \int_0^t m_b(t_1 ) \ ; dt_1 \int_0^t m_b(t_2 ) \ ; c(t_1 - t_2 ) \ ; dt_2\end{aligned}\ ] ] and from ( 14a ) and ( 14c ) we see that @xmath25 while calculating the variances it should be remembered that by virtue of eq.([eq6 ] ) @xmath26 since , in principle we know all the average quantities and variances of the linear system driven by gaussian noise one can make use of the characteristic function method to write down the fokker - planck equation for phase space distribution function @xmath27 near the barrier top @xcite @xmath28 p(x , v , t ) & = & \bar{\gamma}_b ( t ) \frac{\partial}{\partial v } v p(x , v , t ) \nonumber \\ & & + \phi_b(t ) \frac{\partial^2}{\partial v^2 } p(x , v , t ) + \psi_b(t ) \frac{\partial^2}{\partial v \partial x } p(x , v , t)\end{aligned}\ ] ] with @xmath29 regarding the fokker - planck equation ( [ eq16 ] ) three points are to be noted . first , although bounded the time dependent functions @xmath30 , @xmath31 and @xmath32 may not always provide long time limits . these play a decisive role in the calculation of non - markovian kramers rate . therefore , in general , one has to work out frequency @xmath33 and friction @xmath34 functions for analytically tractable models @xcite . second , when the noise is purely internal ( i.e. , there exist a fluctuation - dissipation relation ) we have @xcite @xmath35 \ ; \ ; .\ ] ] third , for pure external noise with markovian relaxation , i.e. , @xmath36 we have @xmath37 in order to calculate the stationary distribution near the bottom of the left well we now linearize the potential @xmath0 around @xmath38 . the corresponding fokker - planck equation can be constructed using the above - mentioned technique to obtain @xmath39 p(x , v , t ) & = & \bar{\gamma}_0 ( t ) \frac{\partial}{\partial v } v p(x , v , t ) \nonumber \\ & & + \phi_0(t ) \frac{\partial^2}{\partial v^2 } p(x , v , t ) + \psi_0 ( t ) \frac{\partial^2}{\partial v \partial x } p(x , v , t)\end{aligned}\ ] ] with @xmath40 here the subscripts ` 0 ' signifies the dynamical quantities corresponding to the bottom of the left well . it may be easily checked that the stationary solution of eq.([eq20 ] ) is given by @xmath41\ ] ] where , @xmath42 ; @xmath43 , @xmath44 and @xmath45 are the values at long time limit and @xmath46 is the normalization constant . here @xmath47 is the renormalized linearized potential with a renormalization in its frequency . it must be emphasized that the distribution ( [ eq22 ] ) is not an equilibrium distribution . this stationary distribution for the open system plays the role of an equilibrium distribution for the closed system which may be however recovered in the absence of external noise terms . we also point out in passing that because of the linearized potential @xmath48 the steady state is unique and the question of multiple steady states does not arise . in the spirit of kramers celebrated ansatz @xcite we now demand a solution of the eq.([eq16 ] ) at the stationary limit of the type @xmath49 \ ; \xi ( x , v)\ ] ] with @xmath50 and @xmath51 are the long time limits of the corresponding time dependent quantities specific for the barrier top region . the notable difference from the kramers ansatz is that the exponential factor in ( [ eq23 ] ) is not the boltzmann factor but pertains to the dynamics at the barrier top . the ansatz of the form ( [ eq23 ] ) denoting the steady state distribution is motivated by the local analysis near the bottom and top of the barrier in the kramers sense . for a nonequilibrium system , as in the present problem of external time - dependent potential field , the relative population of the two regions , in general , depends on the global properties of the potential . thus although at equilibrium the probability density is given by a boltzmann distribution , the external modulation of the potential requires energy input and drives the system away from equilibrium , disturbing the boltzmann distribution . at this point one may anticipate the signature of dynamics in the kramers-like ansatz ( [ eq23 ] ) compared to the standard kramers ansatz for closed system ( i.e. , when the external field is absent ) . thus while in the latter case one considers a complete factorization of the equilibrium part ( boltzmann ) and the dynamical part , @xmath52 , the ansatz ( [ eq23 ] ) incorporates the additional dynamical contribution through dissipation and the strength of the noise into the exponential part . this explicit dynamical modification of kramers ansatz in the form of ( [ eq23 ] ) is valid so long as the extra dynamical contribution in the exponential factor in ( [ eq23 ] ) does not become too severe , i.e. , the amplitude of the external noise field is not too strong . to put it in a more quantitative way , this implies ( assuming for simplicity @xmath53 , @xmath54 ) that the thermal length scale , i.e. , the maximum value of @xmath55 on which the velocity of the particle is thermalized , should be shorter than the other characteristic length scales of the system , e.g. , @xmath56 these considerations are necessary for making spatial diffusion regime and quasi - stationary condition meaningful in the present context . now inserting ( [ eq23 ] ) in ( [ eq16 ] ) in the steady state we get @xmath57 \frac{\partial \xi}{\partial v } + \phi_b \frac{\partial^2 \xi}{\partial v^2 } + \psi_b \frac{\partial^2 \xi}{\partial v \partial x } = 0 \ ; \ ; .\ ] ] at this point we set @xmath58 and with the help of the transformation ( [ eq25 ] ) , eq.([eq24 ] ) is reduced to the following form @xmath59 \frac{d \xi}{d u } = 0 \ ; \ ; .\ ] ] now , let @xmath60 where @xmath61 is a constant to be determined later . from ( [ eq25 ] ) and ( [ eq27 ] ) we have @xmath62 with @xmath63 by virtue of the relation ( [ eq27 ] ) , eq.([eq26 ] ) becomes @xmath64 where @xmath65 the general solution of the homogenous differential equation ( [ eq30 ] ) is @xmath66 where @xmath67 and @xmath68 are the constants of integration . the integral in the eq.([eq32 ] ) converges for @xmath69 if only @xmath70 is positive . the positivity of @xmath70 depends on the sign of @xmath71 ; so by virtue of eqs.([eq25 ] ) and ( [ eq27 ] ) we find that the negative root of @xmath71 , i.e. , @xmath72 guarantees the positivity of @xmath70 since @xmath73 to determine the value of @xmath67 and @xmath68 we impose the first boundary condition on @xmath74 @xmath75 this condition yields @xmath76 inserting ( [ eq35 ] ) into ( [ eq32 ] ) we have as usual @xmath77 \ ; \ ; .\ ] ] since we are to calculate the current around the barrier top , we expand the renormalized potential @xmath47 around @xmath78 @xmath79 thus with the help of ( [ eq36 ] ) and ( [ eq37 ] ) , eq.([eq23 ] ) becomes @xmath80\ ] ] with @xmath81 now defining the steady state current @xmath82 across the barrier by @xmath83 we have using eq.([eq38 ] ) @xmath84 \ ; \ ; .\ ] ] having obtained the steady state current over the barrier top we now look for the value of the undetermined constant @xmath68 in eq.([eq41 ] ) in terms of the population in the left well . we show that this may be done by matching two appropriate _ reduced _ probability distributions at the bottom of the left well . to do so we return to the eq.([eq23 ] ) which describes the steady state distribution at the barrier top . again with the help of ( [ eq36 ] ) we have @xmath85 \ ; \exp \left [ -\frac{v^2}{2d_b } - \frac { \tilde{v}(x)}{d_b+\psi_b } \right ] \ ; \ ; .\ ] ] we first note that , as @xmath86 , the pre - exponential factor in @xmath87 reduces to the following form @xmath88 = f_2 \ ; \left ( \frac{2\pi}{\lambda } \right ) ^{1/2 } \ ; \ ; .\ ] ] we now define a reduced distribution function in @xmath89 @xmath90 hence we have from ( [ eq44 ] ) and ( [ eq45 ] ) @xmath91 \ ; \ ; .\ ] ] similarly we derive the reduced distribution function in the left well , around @xmath92 using ( [ eq22 ] ) as @xmath93\ ] ] where we have employed , the expansion of @xmath48 as @xmath94 and @xmath46 as the normalization constant . at this juncture we impose the second boundary condition that , at @xmath95 the reduced distribution function ( [ eq46 ] ) must go over to stationary reduced distribution function ( [ eq47 ] ) at the bottom of the left well . thus we have @xmath96 the above condition is used to determine the undetermined constant @xmath68 in terms of the normalization constant @xmath46 of eq.([eq22 ] ) @xmath97 \ ; \exp \left [ \frac { \tilde{v}(0 ) -\frac{1}{2}\bar{\omega}_b^2 x_a^2 } { d_0+\psi_0 } \right ] \ ; \ ; .\ ] ] evaluating the normalization constant by explicitly using the relation @xmath98 and then inserting its value in ( [ eq50 ] ) we obtain @xmath99 \ ; \ ; .\ ] ] making use of the relation @xmath100 in ( [ eq52 ] ) and then the value of @xmath68 in eq.([eq41 ] ) we arrive at the expression for the normalized current or barrier crossing rate @xmath101 \ ; \ ; .\ ] ] where the activation energy @xmath102 is defined as @xmath103 as shown in fig.1 . since the temperature due to internal thermal noise , the strength of the external noise and damping constant are buried in the parameters @xmath104 , @xmath105 , @xmath43 , @xmath51 and @xmath70 the general expression ( [ eq53 ] ) looks somewhat cumbersome . we note that the subscripts ` 0 ' and ` b ' in @xmath106 or @xmath107 refer to the well or the barrier top region , respectively . we discuss it in greater detail in the next section . eq.([eq53 ] ) is the central result of this paper . this generalizes the kramers expression for rate of the activated processes for the nonequilibrium open systems . both the internal and the external noises may be of arbitrary long correlation time . it is important to note that the pre - exponential dynamical factors as well as the exponential factor are modified due to the openness of the system . the modification of the exponential factor is due to the fact that depending on the strength of the external noise @xmath2 the system settles down to a stationary distribution which does not coincide with the usual equilibrium boltzmann distribution . the system therefore attains the steady state at a different ` effective ' temperature . this aspect will be clarified in greater detail when we consider the limiting case in subsection d. in general , both the factors in the rate depend on the strength of the noise , correlation time of fluctuations of both external and internal noise processes and dissipation . the rate is spatial - diffusion - limited and is valid for intermediate to strong damping regime . this validity must be appreciated in the present context of driven system in the sense that while on the one hand thermal length scale of the system must be short compared to other characteristic length scales of the sytem corresponding to the inequality ( [ eq71 ] ) , dissipation should also obey the restriction that during one round trip of the particle in phase space ( in action , angle space ) under purely deterministic motion corresponding to ( [ eq1 ] ) , the energy dissipated is greater than the thermal energy , i.e. , @xmath108 where @xmath109 , the action , is equivalent to unperturbed energy @xmath102 in the weak friction limit . both the inequalities ( [ eq71 ] ) and ( [ eq72 ] ) are therefore relevant for quantifying the spatial - diffusion - limited intermediate to strong damping regime . in what follows we shall be concerned with several limiting situations to illustrate the general result ( [ eq53 ] ) systematically for both thermal and non - thermal activated processes . we first consider the case with no external noise and the internal thermal noise is @xmath110-correlated . to this end we set @xmath111 making use of the abbreviations in eqs.(17 ) and ( 21 ) it follows after some algebra that @xmath112 the above relations reduce the general expression ( [ eq53 ] ) to classical expression for kramers rate @xcite @xmath113 \ ; e^{-e / k_bt } \ ; \ ; .\ ] ] next we consider the case with no external noise but the internal noise is of ornstein - uhlenbeck type @xcite . thus we have @xmath114 here @xmath115 denotes the strength while @xmath116 refers to the correlation time of the noise . again from eqs.(17 ) , ( 18 ) and ( 21 ) along with ( [ eq56 ] ) we derive the following relations @xmath117 \ ; \ ; { \rm and } \\ a_\pm = \frac{1}{1+d_b } \ ; \left [ -\frac { \bar{\gamma}_b } { 2 } \pm \sqrt { \frac { \bar{\gamma}_b^2}{4 } + \bar{\omega}_b^2 } \right ] \ ; \ ; .\end{aligned}\ ] ] and hence the rate becomes @xmath118 \ ; e^{-e / k_bt } \ ; \ ; .\ ] ] whereby we recover the result of grote - hynes @xcite and hnggi - mojtabai @xcite obtained several years ago . next we consider the case where the noise is completely due to of external source and the external noise is of ornstein - uhlenbeck type @xcite so that we set @xmath119 note that since in this case the dissipation is independent of fluctuations we may assume markovian relaxation so that @xmath36 ( see also eqs.(18 ) and ( 19 ) ) . the above condition ( [ eq58 ] ) when used in eqs.(17 ) , ( 19 ) and ( 21 ) we obtain after some lengthy algebra @xmath120 \ ; \ ; { \rm and } \\ a_\pm = \frac{1}{1+\gamma \tau_c } \ ; \left [ - \frac{\gamma}{2 } \pm \sqrt { \frac{\gamma^2}{4 } + \omega_b^2 } \right ] \ ; \ ; .\end{aligned}\ ] ] and the rate becomes @xmath121 \ ; \exp \left [ - \frac { \gamma ( 1+\gamma \tau_c - \omega_b^2 \tau_c^2 ) } { { \cal d } ( 1 + \gamma \tau_c ) } e \right ] \ ; .\ ] ] it is interesting to note that the expression ( [ eq59 ] ) denotes the external noise - induced barrier crossing rate which crucially depends on the strength @xmath115 and correlation time @xmath116 of the coloured noise . the absence of temperature and the appearance of dissipation @xmath122 explicitly demonstrates the non - thermal origin of the noise processes as well as the absence of fluctuation - dissipation relation . we finally consider both the internal and external noise to be @xmath110-correlated , i.e. , @xmath123 @xmath124 being the strength of the external white noise . hence , by virtue of ( 15 ) , ( 17 ) and ( 21 ) we have @xmath125 hence the rate becomes @xmath126 \ ; \exp \left [ \frac{e } { k_bt + ( \alpha/\gamma ) } \right ] \ ; \ ; .\ ] ] in the limit @xmath127 we recover the kramers original result ( [ eq55 ] ) for pure internal white noise . we note here that @xmath128 defines a new ` effective ' temperature due to external noise . the effective temperature which depends on the strength of the external noise had been discussed earlier by bravo et . @xcite in a somewhat different context . we note that while in the latter case the bath is driven by external fluctuations , the present treatment concerns the direct driving of the reaction co - ordinate by external noise . in this paper we have generalized kramers theory of activated processes for nonequilibrium open systems . the theory takes into account of both internal and external gaussian noise fluctuations with arbitrary decaying correlation functions in an unified way . the treatment is valid for intermediate to strong damping regime for spatial diffusive processes . \(i ) we have shown that not only the motion at the barrier top is influenced by the dynamics , it has an important role to play in establishing the stationary state near the bottom of the source well for the open systems . thus the stationary distribution function in the well depends crucially on the correlation time of the external noise processes as well as on damping . this is distinctly a different situation ( but analogous ) as compared to an equilibrium boltzmann distribution in the source well for standard kramers theory for closed systems . \(iii ) we have checked and examined the various limits of the generalized rate expression to obtain kramers rate , its non - markovian counterpart as well as the other cases for specific external noise processes in presence and absence of the internal noise . we conclude by noting that since the validity of the rate expression derived in the paper depends on the existence of long time limit of the moments for the stochastic processes , the theory can not be directly extended to , say , fractal noise processes . these and the related noise processes remain outside the scope of the present treatment . suitable extension of the kramers theory in this direction is worth - pursuing . j. ray chaudhuri , g. gangopadhyay and d. s. ray , j. chem . phys . * 109 * , 5565 ( 1998 ) ; j. r. chaudhuri , b. deb , g. gangopadhyay and d. s. ray , j. phys . b * 31 * , 3859 ( 1998 ) ; j. r. chaudhuri , s. k. banik , b. deb and d. s. ray , eur . j. d * 6 * , 415 ( 1999 ) .

the kramers theory of activated processes is generalized for nonequilibrium open one - dimensional systems . we consider both the internal noise due to thermal bath and the external noise which are stationary , gaussian and are characterized by arbitrary decaying correlation functions . we stress the role of a nonequilibrium stationary state distribution for this open system which is reminiscent of an equilibrium boltzmann distribution in calculation of rate . the generalized rate expression we derive here reduces to the specific limiting cases pertaining to the closed and open systems for thermal and non - thermal steady state activation processes .

You are an expert at summarizing long articles. Proceed to summarize the following text: about ten years ago , a peculiar dynamical phenomenon was discovered in populations of identical phase oscillators : under nonlocal symmetric coupling , the coexistence of coherent ( synchronized ) and incoherent oscillators was observed @xcite . this highly counterintuitive phenomenon was given the name chimera state after the greek mythological creature made up of different animals @xcite . since then the study of chimera states has been the focus of extensive research in a wide number of models , from kuramoto phase oscillators @xcite to periodic and chaotic maps @xcite , as well as stuart - landau oscillators @xcite . the first experimental evidence of chimera states was found in populations of coupled chemical oscillators as well as in optical coupled - map lattices realized by liquid - crystal light modulators @xcite . recently , moreover , martens and coauthors showed that chimeras emerge naturally from a competition between two antagonistic synchronization patterns in a mechanical experiment involving two subpopulations of identical metronomes coupled in a hierarchical network @xcite . in the context of neuroscience , a similar effort has been undertaken by several groups , since it is believed that chimera states might explain the phenomenon of unihemispheric sleep observed in many birds and dolphins which sleep with one eye open , meaning that one hemisphere of the brain is synchronouns whereas the other is asynchronous @xcite . the purpose of this paper is to make a contribution in this direction , by identifying for the first time a variety of single and multi - chimera states in networks of non - locally coupled neurons represented by hindmarsh rose oscillators . recently , multi - chimera states were discovered on a ring of nonlocally coupled fitzhugh - nagumo ( fhn ) oscillators @xcite . the fhn model is a 2dimensional ( 2d ) simplification of the physiologically realistic hodgkin - huxley model @xcite and is therefore computationally a lot simpler to handle . however , it fails to reproduce several important dynamical behaviors shown by real neurons , like rapid firing or regular and chaotic bursting . this can be overcome by replacing the fhn with another well - known more realistic model for single neuron dynamics , the hindmarsh - rose ( hr ) model @xcite , which we will be used throughout this work both in its 2d and 3d versions . in section 2 of this paper , we first treat the case of 2d - hr oscillators represented by two first order ordinary differential equations ( odes ) describing the interaction of a membrane potential and a single variable related to ionic currents across the membrane under periodic boundary conditions . we review the dynamics in the 2d plane in terms of its fixed points and limit cycles , coupling each of the @xmath0 oscillators to @xmath1 nearest neighbors symmetrically on both sides , in the manner adopted in @xcite , through which chimeras were discovered in fhn oscillators . we identify parameter values for which chimeras appear in this setting and note the variety of oscillating patterns that are possible due to the bistability features of the 2d model . in particular , we identify a new `` mixed oscillatory state '' ( mos ) , in which the desynchronized neurons are uniformly distributed among those attracted by a stable stationary state . furthermore , we also discover chimera like patterns in the more natural setting where only the membrane potential variables are coupled with @xmath2 of the same type . next , we turn in section 3 to the more realistic 3d - hr model where a third variable is added representing a slowly varying current , through which the system can also exhibit bursting modes . here , we choose a different type of coupling where the two first variables are coupled symmetrically to @xmath2 of their own kind and observe only states of complete synchronization as well as mos in which desynchronized oscillators are interspersed among neurons that oscillate in synchrony . however , when coupling is allowed only among the first ( membrane potential ) variables chimera states are discovered in cases where spiking occurs within sufficiently long bursting intervals . finally , the paper ends with our conclusions in section 4 . following the particular type of setting proposed in @xcite we consider @xmath0 nonlocally coupled hindmarsh - rose oscillators , where the interconnections between neurons exist with @xmath3 nearest neighbors only on either side as follows : @xmath4 \label{eq:01 } \\ \dot y_k & = & 1 - 5x_k^2-y_k+ \frac{\sigma_y}{2r}\sum_{j = k - r}^{j = k+r } [ b_{yx}(x_j - x_k)+b_{yy}(y_j - y_k ) ] . \label{eq:02 } \end{aligned}\ ] ] in the above equations @xmath5 is the membrane potential of the @xmath6-th neuron , @xmath7 represents various physical quantities related to electrical conductances of the relevant ion currents across the membrane , @xmath8 , @xmath9 and @xmath10 are constant parameters , and @xmath11 is the external stimulus current . each oscillator is coupled with its @xmath12 nearest neighbors on both sides with coupling strengths @xmath13 . this induces nonlocality in the form of a ring topology established by considering periodic boundary conditions for our systems of odes . as in @xcite , our system contains not only direct @xmath14 and @xmath15 coupling , but also cross - coupling between variables @xmath16 and @xmath17 . this feature is modeled by a rotational coupling matrix : @xmath18 depending on a coupling phase @xmath19 . in what follows , we study the collective behavior of the above system and investigate , in particular , the existence of chimera states in relation to the values of all network parameters : @xmath0 , @xmath3 , @xmath19 , @xmath20 and @xmath21 . more specifically , we consider two cases : ( @xmath22 ) direct and cross - coupling of equal strength in both variables ( @xmath23 ) and ( @xmath24 ) direct coupling in the @xmath16 variable only ( @xmath25 , @xmath26 ) . similarly to @xcite we shall use initial conditions randomly distributed on the unit circle ( @xmath27 ) . at @xmath28 of eqs . ( [ eq:01]),([eq:02 ] ) ( left ) and selected time series ( right ) for @xmath29 . ( a ) @xmath30 , ( b ) @xmath31 and ( c ) @xmath32 . @xmath33 and @xmath34.,scaledwidth=70.0% ] typical spatial patterns for case ( @xmath22 ) are shown on the left panel of fig . [ fig : fig1 ] , where the @xmath16 variable is plotted over the index number @xmath6 at a time snapshot chosen after a sufficiently long simulation of the system . in this figure the effect of different values of the phase @xmath19 is demonstrated while the number of oscillators @xmath0 and their nearest neighbors @xmath3 , as well as the coupling strengths @xmath23 are kept constant . for example , for @xmath30 ( fig . [ fig : fig1](a ) ) an interesting novel type of dynamics is observed that we shall call `` mixed oscillatory state '' ( mos ) , whereby nearly half of the @xmath5 are attracted to a fixed point ( at this snapshot they are all at a value near @xmath35 ) , while the other half are oscillating interspersed among the stationary ones . from the respective time series ( fig . [ fig : fig1](a ) , right ) it is clear that the former correspond to spiking neurons whereas the latter to quiescent ones . this interesting mos phenomenon is due to the fact that , in the standard parameter values we have chosen , the uncoupled hr oscillators are characterized by the property of _ bistability_. clearly , as shown in the phase portrait of fig . [ fig : fig2 ] , each oscillator possesses three fixed points : the leftmost fixed point is a stable node corresponding to the resting state of the neuron while the other two correspond to a saddle point and an unstable node and are therefore repelling . for @xmath11 ( which is the case here ) a stable limit cycle also exists which attracts many of the neurons into oscillatory motion , rendering the system bistable and producing the dynamics observed in fig . [ fig : fig1](a ) . now , when a positive current @xmath36 is applied , the @xmath16-nullcline is lowered such that the saddle point and the stable node collide and finally vanish . in this case the full system enters a stable limit cycle associated with typical spiking behaviour . similar complex patterns including mos and chimeras have been observed in this regime as well . . three fixed points coexist with a stable limit cycle.,scaledwidth=40.0% ] for @xmath31 , on the other hand , there is a `` shift '' in the dynamics of the individual neurons into the spiking regime , as seen in fig . [ fig : fig1](b ) ( right ) . the corresponding spatial pattern has a wave - like form of period 2 . then , for @xmath32 a classical chimera state with two incoherent domains is observed ( ( fig . [ fig : fig1](c ) , left ) . diagonal coupling ( @xmath37 , @xmath38 ) is , therefore , identified as the necessary condition to achieve chimera states . by contrast , it is interesting to note that in nonlocally coupled fitz - hugh nagumo oscillators @xcite it has been shown both analytically and numerically that chimera states occur for _ off - diagonal _ coupling . by decreasing the range of coupling @xmath3 and increasing the system size @xmath0 , chimera states occur with multiple domains of incoherence and coherence for @xmath32 ( fig . [ fig : fig3](c , d ) ) , and , accordingly , periodic spatial patterns with larger wave numbers arise for @xmath31 , as seen in fig . [ fig : fig3](a , b ) . this is in agreement with previous works reported in @xcite . next we consider the case ( @xmath24 ) where the coupling term is restricted to the @xmath16 variable . this case is important since incorporating the coupling in the voltage membrane ( @xmath16 ) alone is more realistic from a biophysiological point of view . in fig . [ fig : fig4 ] spatial plots ( left ) and the corresponding @xmath39-plane ( right ) for increasing coupling strength are shown . chimera states ( fig . [ fig : fig4 ] ( b , c ) ) are observed as an intermediate pattern between desynchronization ( fig . [ fig : fig4 ] ( a ) ) and complete synchronization ( fig . [ fig : fig4 ] ( d ) ) . of eqs . ( [ eq:01]),([eq:02 ] ) at @xmath28 for @xmath23 . @xmath40 ( top ) and @xmath41 ( bottom ) whereas @xmath31 ( left ) and @xmath32 ( right ) . in ( a ) and ( b ) @xmath42 , in ( c ) @xmath43 , and in ( d ) @xmath44.,scaledwidth=50.0% ] ( left ) and in the @xmath45-plane ( right ) of eqs . ( [ eq:01]),([eq:02 ] ) at @xmath28 for @xmath25 , @xmath33 and @xmath34 . ( a ) @xmath46 , ( b ) @xmath47 , ( c ) @xmath48 and ( d ) @xmath49.,scaledwidth=40.0% ] in order to complete our study of the hindmarsh - rose model we shall consider , in this section , its three - dimensional version . the corresponding equations read : @xmath50 the extra variable @xmath51 represents a slowly varying current , which changes the applied current @xmath52 to @xmath53 and guarantees firing frequency adaptation ( governed by the parameter @xmath54 ) as well as the ability to produce typical bursting modes , which the 2d model can not reproduce . parameter @xmath55 controls the transition between spiking and bursting , parameter @xmath56 determines the spiking frequency ( in the spiking regime ) and the number of spikes per bursting ( in the bursting regime ) , while @xmath57 sets the resting potential of the system . the parameters of the fast @xmath58 system are set to the same values used in the two - dimensional version ( @xmath8 , @xmath10 ) and typical values are used for the parameters of the @xmath51-equation ( @xmath59 , @xmath60 , @xmath61 ) . the 3d hindmarsh - rose model exhibits a rich variety of bifurcation scenarios in the @xmath62 parameter plane @xcite . thus , we prepare all nodes in the spiking regime ( with corresponding parameter values @xmath9 and @xmath63 ) and , as in section 2 , we use initial conditions randomly distributed on the unit sphere ( @xmath64 ) . first let us consider direct coupling in both variables @xmath16 and @xmath17 and vary the value of the equal coupling strengths @xmath23 , while @xmath0 and @xmath3 are kept constant . naturally , the interaction of @xmath0 spiking neurons will lead to a change in their dynamics , as we discuss in what follows . for low values of @xmath65 we observe the occurrence of a type of mos where nearly half of the neurons spike regularly in a synchronous fashion , while the rest are unsynchronized and spike in an irregular fashion . this is illustrated in the respective time series in the right panel of fig . [ fig : fig5](a ) at higher values of the coupling strength the the system is fully synchronized ( fig . [ fig : fig5](b ) ) . of eqs . ( [ eq:03])-([eq:05 ] ) at @xmath66 ( left ) and selected time series ( right ) for : ( a ) @xmath67 and ( b ) @xmath68 . @xmath33 and @xmath34.,scaledwidth=50.0% ] next we check the case of coupling in the @xmath16 variable alone ( @xmath69 , @xmath25 ) . figure [ fig : fig6 ] displays characteristic synchronization patterns obtained when we increase @xmath20 ( left panel ) and selected time series ( right panel ) . at low values of the coupling strength all neurons remain in the regular spiking regime and desynchronization alternates with complete synchronization as @xmath20 increases ( fig . [ fig : fig6](a , b , c ) ) . for intermediate values of the coupling strength , chimera states with one incoherent domain are to be observed . these are associated with a change in the dynamics of the individual neurons , which now produce irregular bursts ( fig . [ fig : fig6](d ) ) . the number of spikes in each burst increases at higher @xmath20 values and the system is again fully synchronized ( fig . [ fig : fig6](e ) ) . extensive simulations show that the chimera states disappear and reappear again by varying @xmath20 which is most likely due to the system s multistability and sensitive dependence on initial conditions . of eqs . ( [ eq:03])-([eq:05 ] ) at @xmath66 ( left ) and selected time series for @xmath25 , @xmath33 and @xmath34 . ( a ) @xmath70 , ( b ) @xmath71 , @xmath72 , ( d ) @xmath73 , and ( e ) @xmath74 . , in this paper , we have identified the occurrence of chimera states for various coupling schemes in networks of 2d and 3d hindmarsh - rose models . this , together with recent reports on multiple chimera states in nonlocally coupled fitzhugh - nagumo oscillators , provide strong evidence that this counterintuitive phenomenon is very relevant as far as neuroscience applications are concerned . chimera states are strongly related to the phenomenon of synchronization . during the last years , synchronization phenomena have been intensely studied in the framework of complex systems @xcite . moreover , it is a well - established fact the key ingredient for the occurrence of chimera states is nonlocal coupling . the human brain is an excellent example of a complex system where nonlocal connectivity is compatible with reality . therefore , the study of chimera states in systems modelling neuron dynamics is both significant and relevant as far as applications are concerned . moreover , the present work is also important from a theoretical point of view , since it verifies the existence of chimera states in coupled bistable elements , while , up to now , it was known to arise in oscillator models possessing a single attracting state of the limit cycle type . finally , we have identified a novel type of mixed oscillatory state ( mos ) , in which desynchronized neurons are interspersed among those that are either stationary or oscillate in synchrony . as a continuation of this work , it is very interesting to see whether chimeras and mos states appear also in networks of _ populations _ of hindmarsh - rose oscillators , which are currently under investigation . the authors acknowledge support by the european union ( european social fund esf ) and greek national funds through the operational program `` education and lifelong learning '' of the national strategic reference framework ( nsrf ) - research funding program : thales . investing in knowledge society through the european social fund . funding was also provided by ninds r01 - 40596 . hagerstrom , a. m. , thomas , e. , roy , r. , hvel , p. , omelchenko , i. & schll , e. [ 2012 ] `` chimeras in coupled - map lattices : experiments on a liquid crystal spatial light modulator system '' , _ nature physics _ * 8 * , 658 . omelchenko , i. , omelchenko , o. e. , hvel , p. & schll , e. [ 2013 ] `` when nonlocal coupling between oscillators becomes stronger : patched synchrony or multichimera states '' , _ phys . lett . _ * 110 * , 224101 . rattenborg , n. c. , amlaner , c. j. & lim , s. l. [ 2000 ] `` behavioral , neurophysiological and evolutionary perspectives on unihemispheric sleep , '' _ neuroscience and biobehavioral reviews _ * 24 * , pp . 817-842 .

we have identified the occurrence of chimera states for various coupling schemes in networks of two - dimensional and three - dimensional hindmarsh - rose oscillators , which represent realistic models of neuronal ensembles . this result , together with recent studies on multiple chimera states in nonlocally coupled fitzhugh - nagumo oscillators , provide strong evidence that the phenomenon of chimeras may indeed be relevant in neuroscience applications . moreover , our work verifies the existence of chimera states in coupled bistable elements , whereas to date chimeras were known to arise in models possessing a single stable limit cycle . finally , we have identified an interesting class of mixed oscillatory states , in which desynchronized neurons are uniformly interspersed among the remaining ones that are either stationary or oscillate in synchronized motion .

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Proceed to summarize the following text: white dwarfs ( wds ) are the final stage for the the evolution of majority of low and medium mass stars with initial masses @xmath2 @xmath3 . since there are no fusions reaction , the evolution of wds is primarily determined by a well understood cooling process ( fontaine et al . 2001 ; salaris et al . 2000 ) . thus , they can be used for cosmochronology , an independent age - dating method . also , the luminosity function of wds provides firm constraints on the local star formation rate and history of the galactic disk ( krzesinski et al . 2009 ) . mccook @xmath4 sion ( 1999 ) present a catalog of 2249 wds which have been identified spectroscopically . in addition , the sloan digital sky survey ( sdss ; york et al . 2000 ) has greatly expanded the number of spectroscopically confirmed wd stars ( harris et al . 2003 ; kleinman et al . 2004 ; eisenstein et al . 2006 ; kleinman et al . the latter presented a catalog of 20,407 spectroscopically confirmed white dwarfs from the sdss data release 4 ( dr4 ) , roughly doubling the number of spectroscopically confirmed white dwarfs . large sky area multi - object fiber spectroscopic telescope ( lamost , so called the guoshoujing telescope ) is a national major scientific project undertaken by the chinese academy of science ( wang et al . 1996 ; cui et al . lamost has recently completed the pilot survey from october 2011 to may 2012 , which obtained several hundred thousand spectra ( luo et al . 2012 ) . from september of 2012 , lamost has undertaken the general survey and will observe about 1 million stars per year . lamost has the capability to observe large , deep and dense regions in the milky way galaxy , which will enable a number of research topics to explore the evolution and the structure of the milky way . therefore , it will definitely yield a large sample of wds . wds whose primary spectral classification is da have hydrogen - dominated atmospheres . they make up the majority ( approximately 75% ) of all observed wds ( fontaine @xmath4 wesemael 2001 ) . such wds are easy to identify using optical spectra . here we present a catalog of da wds from the lamost pilot survey ( luo et al . * we do not expect the completeness of this sample . * in section 2 we describe the spectra obtained . section 3 discusses how the @xmath5 , log @xmath6 , mass and distance of the wds were estimated . the kinematics of these wds are illustrated in section 4 . a summary of our pilot study results is given in section 5 . the lamost spectra have a resolving power of r @xmath0 2000 spanning 3700@xmath7 9000@xmath8 . two arms of each spectrograph cover this wavelength range with overlap of 200 @xmath9 . the blue spectral coverage is 3700@xmath10 5900@xmath8 while that in the red is 5700@xmath10 9000 @xmath8 . the raw data were reduced with lamost 2d and 1d pipelines ( luo et al . these pipelines include bias subtraction , cosmic - ray removal , spectral trace and extraction , flat - fielding , wavelength calibration , sky subtraction , and combination . the throughput in red is higher than the blue band . the pilot survey obtained spectra of stars in the milky way , which included fainter objects on dark nights ( yang et al . 2012 ; carlin et al . 2012 ) , brighter objects on bright nights ( zhang et al . 2012 ) , objects in the disk of the galaxy with low latitude ( chen et al . 2012 ) and objects in the region of the galactic anti - center . it also targets extragalactic objects located in two regions , i.e. , the south galactic cap and the north galactic cap . we found twenty wd spectra in both sdss and lamost pilot survey catalogs . 1 shows a portion of a typical spectrum . the top panel compares the sdss dr7 and lamost spectra for the object j100316.35 - 002336.95 . the solid line is the sdss spectrum . the dotted line is the lamost spectrum . the bottom panel shows the residual between two spectra . the mean difference between two spectra is less than 10 % . the initial wd candidates we selected are from two sources . one is the lamost pipeline ( luo et al . 2012 ) which yielded about 2000 candidates using the `` pcaz '' method . for stars with sdss photometry , we used the formulas 1 - 4 of elsenstein et al . ( 2006 ) to idendify candidates . next , each of these spectra was inspected by eye . stars with signal - to - noise ratio ( s / n ) smaller than 10 were excluded . finally , if the balmer line profiles of the star were a little too narrow ( log @xmath6 @xmath2 7.0 ) , the spectrum was rejected even if selected by the pipeline . after these filters , 72 da wds were left . table 1 presents the physical data for these wds . column 1 is an i d number . columns 2 - 5 list the name , ra and dec . the estimated @xmath5 , log @xmath6 , mass and the cooling time are given in columns 5 - 8 . columns 9 - 13 list the apparent magnitudes of each wds . column 14 indicates the source of the magnitudes . the last two columns are estimates of the color excess ( b - v ) and distance . the e(b - v ) is estimated from schlegel et al . for our da wd candidates , the @xmath11@xmath12 and log @xmath6 were derived via simultaneous fitting of the h@xmath13 to h8 balmer line profiles using the procedure outlined by bergeron et al . the line profiles in both observed spectra and model spectra were first normalized using two points at the continuum level on either side of each absorption line . thus , the fit should not be affected by the flux calibration . model atmospheres used for this fitting were derived from model grids provided by koester ( 2010 ) . details of the input physics and methods can be found in that reference . fitting of the line profiles was carried out using the idl package mpfit ( markwardt 2008 ) , which is based on @xmath14 minimization using levenberg - marquardt method . this package can be downloaded from the project website . errors in the @xmath11@xmath12 and log @xmath6 were calculated by stepping the parameter in question away from their optimum values and redetermining minimum @xmath14 until the difference between this and the true minimum @xmath14 corresponded to 1@xmath15 for a given number of free model parameters . 2 - 3 show examples of @xmath11@xmath12 and log @xmath6 determinations for j150156.26 + 302300.13 . fig.2 is the contour plot of the @xmath14 residual and the rough @xmath11@xmath12 and log @xmath6 implied by these error eclipses . 3 shows the actual fits of the observed balmer lines for j150156.26 + 302300.13 . the black solid lines are the observed profiles of balmer lines from h@xmath13 to h8 . the red dashed lines are the model spectra . the derived @xmath11@xmath12 , log @xmath6 and uncertainties for all the wds are shown in columns 5 - 6 of table 1 . estimated @xmath11@xmath12 and log @xmath6 values for 14 das were also available in the literature , allowing the comparisons shown in fig . the solid line represents the unit slope relation . plus ( + ) symbols represent the wds with high s / n spectra while squares represent wds with low s / n spectra . the three spectra of lowest s / n are outliers in the log @xmath6 comparison plot - suggesting the importance of s / n in determining this parameter . for most of other wds , the mean differences between our and the literature @xmath11@xmath12 values are less than 1000 k and the log @xmath6 difference is less than 0.2 dex . within this scatter , our results are consistent with those in the literatures . one of our candidates , j104311.45 + 490224.35 has also been identified as da wd by mccook @xmath4 sion ( 1999 ) . however , we were unable to determine its @xmath11@xmath12 and log @xmath6 because h@xmath13 was not included in the spectrum we obtained . from the @xmath11@xmath16 and log @xmath6 of each wd , its mass ( m@xmath17 ) and cooling time ( t@xmath18 ) were estimated from bergeron s cooling sequences . for the model atmospheres above @xmath11@xmath16 = 30,000 k we used the carbon - core cooling models of wood ( 1995 ) , with thick hydrogen layers of q@xmath19 = m@xmath19/m@xmath20 = 10@xmath21 . for @xmath11@xmath16 below 30,000 k we used cooling models similar to those described in fontaine , brassard @xmath4 bergeron ( 2001 ) but with carbon - oxygen cores and q@xmath19 = 10@xmath21 ( see bergeron , leggett @xmath4 ruiz 2001 ) . 5 is the mass distribution of our sample resulting from the above procedure . masses are found to range from 0.4 @xmath1 to 1.2 @xmath1 . the curve is a gaussian fit with a peak at about 0.61 @xmath1 , which is consistent with the mean mass 0.613 @xmath1 from tremblay et al . ( 2011 ) derived from sdss da wds sample . the determination of distances for wds is very difficult because of their low luminosity . currently only about 300 wds have trigonometric parallaxes . in the absence of parallaxes , color - magnitude relations and empirical photometric methods are often used . holberg et al . ( 2008 ) provided improved distance estimates for da wds using multi - band synthetic photometry tied to spectroscopic temperatures and gravities . this method was called synthetic spectral distances ( ssd ) . the unique aspect of ssd is the systematic use of calibrated multi - channel synthetic absolute magnitudes , interpolated within the grid by the @xmath5 and log @xmath6 . @xmath22 in this paper , the distances of wds in our sample were estimated using equation 1 . here , @xmath23 are the photometric magnitudes of the wds . most of our has _ u _ , _ g _ , _ r _ , _ i _ , _ z _ magnitudes . almost all have at least _ g _ , _ r _ and _ i _ magnitudes . a few wds still only have _ v _ magnitude . @xmath24 is the model absolute magnitudes calculated by interpolations in the atmospheric models provided by bergeron . @xmath25 is the reddening and d is the distance in parsecs . in general for each magnitude a corresponding distance can be calculated . the final distance is estimated by using weighted average . the weights adopted are the errors in the magnitude . here , we only calculated the distances for wds having _ u _ , _ g _ , _ r _ , _ i _ , _ z _ or _ v _ magnitude data . distances for two wds in table 1 could not be estimated . oppenheimer et al . ( 2001 ) suggested that halo wds could provide a significant contribution to the galactic dark matters component , which prompted much interest in wd kinematics . in a related study , silvestri et al . ( 2002 ) observed 116 common proper - motion binaries consisting of a wd plus m dwarf component . they determined full space motions of their wds from the companion m dwarfs . most of their wds were found to be members of the disk ; only one potential halo wd was identified . even the much larger samples of wds such as the pauli et al . ( 2003 , 2006 ) sn ia progenitor survey ( spy ) have found relatively few genuine halo and thick disk candidates . in their magnitude - limited sample of 398 wds , they examined both the uvw space motions and the galactic orbits of their stars . they found only 2% of their sample kinematically belonged to the halo and 7% to the thick disk . sion et al . ( 2009 ) presented the kinematical properties of the wds within 20 pc of the sun . in their nearby sample , they found no convincing evidence of halo members among 129 wds , nor was there convincing evidence of genuine thick disk subcomponent members within 20 parsecs . the entire 20 pc sample likely belongs to the thin disk . the proper motions of our sample were derived by the cross - correlating with ppmxl catalog ( roeser et al . silvestri et al . ( 2002 ) ; pauli et al . ( 2003 , 2006 ) , sion et al . ( 2009 ) found relatively little kinematical difference among the samples whether they used radial velocity ( rv ) to compute full space motions or used the simple zero rv for simple wds . we have assumed zero rvs in the analysis of our sample . u is measured positive in the direction of the galactic anti - center , v is measured positive in the direction of the galactic rotation , and w is measured positive in the direction of the north galactic pole . the u , v and w velocities were corrected for the peculiar solar motion ( u , v , w ) = ( -9 , + 12 , + 7 ) km s@xmath26 ( wielen 1982 ) . the space motions of 59 wds with sufficient kinematical information ( photometric or trigonometric parallax , proper motion ) in our sample were calculated . the top panel of fig . 6 shows contours , centered at ( u , v ) = ( 0 , -220 ) km s@xmath26 , that represent 1@xmath15 and 2@xmath15 velocity ellipsoids for stars in the galactic stellar halo as defined by chiba @xmath4 beers ( 2000 ) . only one of our candidate wds lies outside the 2@xmath15 velocity contour centered on ( u , v ) = ( 0 , -35 ) km s@xmath26 defined for disk stars ( chiba @xmath4 beers 2000 ) . the bottom of fig . 6 shows a toomre diagram for our stars . venn et al . ( 2004 ) suggested stars with v@xmath27 @xmath28 180 km s@xmath26 are possible halo members . none of our stars meet this criterion . we conclude that our sample consistes entirely of disk stars . wegg @xmath4 phinney ( 2012 ) concluded that kinematical dispersion decreases with increasing wd mass among young wds whose cooling time is smaller than 3@xmath2910@xmath30 years . progenitors of high mass wds have shorter lifetimes , hence they should be closer to the galactic plane and have small kinematical dispersion in accord with the disk ` heating ' theory . since most wds in our sample are relative young , we investigated the relation between mass and w , as well as mass and vertical distance of galactic plane @xmath31z@xmath31 ( see fig . 7 ) . in the top panel of fig . 7 , wds with mass larger than 0.8 @xmath32 are seen to have smaller w. also , the vertical distances from the galactic plane of wds with larger mass are relative small . although there is no strict relation such as seen in wegg @xmath4 phinney ( 2012 ) , our sample support the general expectation that high mass wds tend to have lower w and @xmath31z@xmath31 . from the lamost pilot survey data , 72 da wds were detected with s / n @xmath28 10 . @xmath11@xmath12 , log @xmath6 , cooling time , mass and distance of these wds were determined from their spectra . the @xmath5 of most wds range from 12000 k to 35000 k and the cooling times of all the wds are younger than 300 myr . all these wds were found to be members of galactic disk . wds with higher mass tend to have smaller vertical distance from the galactic disk , which partly supports the conclusions of wegg et al . the da wd catalogue of the lamost pilot survey provides a first glimpse of how useful the survey will be to search for nearby wds . the upcoming formal lamost survey will enlarge the sample of wds rapidly , perhaps providing the largest sample of wds available . this large sample will open the door to much more detailed investigation of the physical @xmath4 kinematic properties of wds in the solar neighborhood as well as the local structure and evolution of the galaxy . many thanks to d. koester for providing his wd models . balmer / lyman lines in the models were calculated with the modified stark broadening profiles of tremblay @xmath4 bergeron ( 2009 ) , kindly made available by the authors . this study is supported by the national natural science foundation of china under grant no . 11233004 , 11078019 and 10973021 . t.d.o . acknowledges support from nsf grant ast-0807919 to florida institute of technology . guoshoujing telescope ( the large sky area multi - object fiber spectroscopic telescope lamost ) is a national major scientific project built by the chinese academy of sciences . funding for the project has been provided by the national development and reform commission . lamost is operated and managed by the national astronomical observatories , chinese academy of sciences . carlin , j. l. , lpine , s. , 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astrophysics , vol . 2c , sec . 8.4 ( berlin : springer ) , 202 yang , f. , carlin , j. l. , newberg , h. j. , et al . 2012 , raa ( research in astronomy and astrophysics ) , 12 , 781 york , d. g. , et al . 2000 , , 120 , 1579 zhang , y. , carlin , j. , liu , c. , et al . 2012 , raa ( research in astronomy and astrophysics ) , 12 , 792 ccccccccccccccccc 0 & j220522.86 + 021837.56 & 331.345250 & 2.310432 & 15377 @xmath33 493 & 8.02 @xmath33 0.10 & 0.63 @xmath33 0.06 & 190 @xmath33 43 & 17.35 & 17.00 & 17.25 & 17.45 & 17.71 & & 1 & 0.05 & 135 + 1&j025737.25 + 264047.89 & 44.405201 & 26.679970 & 19008 @xmath33 669 & 7.87 @xmath33 0.12 & 0.55 @xmath33 0.06 & 66 @xmath33 21 & & 16.91 & 17.00 & 17.16 & & & 2 & 0.16 & 139 + 2&j030214.72 + 285707.41 & 45.561340 & 28.952057 & 21894 @xmath33 1406 & 8.01 @xmath33 0.23 & 0.64 @xmath33 0.13 & 46 @xmath33 37 & & 17.21 & 17.60 & 17.80 & & & 2 & 0.18 & 173 + 3&j040449.34 + 280023.65 & 61.205600 & 28.006570 & 29302 @xmath33 2525 & 8.25 @xmath33 0.55 & 0.79 @xmath33 0.33 & 20 @xmath33 32 & & 15.96 & 15.84 & 16.00 & & & 2 & 0.21 & 87 + 4&j004036.79 + 413138.79 & 10.153296 & 41.527443 & 13000 @xmath33 651 & 7.75 @xmath33 0.08 & 0.48 @xmath33 0.04 & 216 @xmath33 47 & & 15.90 & 16.21 & 16.40 & & & 2 & 0.07 & 83 + 5&j003956.55 + 422929.55 & 9.985629 & 42.491542 & 18053 @xmath33 816 & 7.32 @xmath33 0.15 & 0.34 @xmath33 0.05 & 50 @xmath33 10 & & 16.43 & 16.58 & 16.72 & & & 2 & 0.06 & 181 + 6&j004128.67 + 402324.09 & 10.369458 & 40.390026 & 25996 @xmath33 733 . & 7.92 @xmath33 0.10 & 0.59 @xmath33 0.05 & 15 . @xmath33 3 . & & & & & & 17.14 & 3 & 0.08 & 68 + 7&j005340.53 + 360116.89 & 13.418857 & 36.021358 & 29772 @xmath33 158 & 7.96 @xmath33 0.04 & 0.63 @xmath33 0.02 & 9 @xmath33 0 & & 14.10 & 14.58 & 14.91 & & & 2 & 0.05 & 72 + 8&j100551.51 - 023417.87 & 151.464628 & -2.571630 & 22072 @xmath33 477 & 8.22 @xmath33 0.07 & 0.76 @xmath33 0.04 & 78 @xmath33 16 & 15.15 & 15.10 & 15.46 & 15.76 & 16.08 & & 1 & 0.05 & 68 + 9&j100316.35 - 002336.95 & 150.818141 & -0.393597 & 22249 @xmath33 330 & 7.92 @xmath33 0.05 & 0.59 @xmath33 0.03 & 33 @xmath33 4 & 15.97 & 15.93 & 16.25 & 16.56 & 16.85 & & 1 & 0.05 & 123 + 10 & j100941.45 - 004404.55 & 152.422705 & -0.734597 & 16489 @xmath33 601 & 7.98 @xmath33 0.13 & 0.60 @xmath33 0.08 & 140 @xmath33 45 & 17.36 & 16.98 & 17.24 & 17.44 & 17.74 & & 1 & 0.04 & 148 + 11 & j054613.53 + 255031.70 & 86.556364 & 25.842139 & 22935 @xmath33 498 & 7.99 @xmath33 0.07 & 0.63 @xmath33 0.04 & 34 @xmath33 10 & & 17.33 & 17.62 & 17.78 & & & 2 & 1.72 & 27 + 12 & j090734.26 + 273903.32 & 136.892757 & 27.650923 & 18619 @xmath33 386 & 8.56 @xmath33 0.07 & 0.97 @xmath33 0.04 & 272 @xmath33 39 & 16.31 & 16.08 & 16.37 & 16.64 & 16.89 & & 1 & 0.03 & 72 + 13 & j004628.31 + 343319.90 & 11.617971 & 34.555527 & 14644 @xmath33 808 & 7.60 @xmath33 0.18 & 0.41 @xmath33 0.08 & 120 @xmath33 47 & 16.83 & 16.33 & 16.40 & 16.53 & 16.75 & & 1 & 0.08 & 112 + 14 & j005340.53 + 360116.89 & 13.418857 & 36.021358 & 26534 @xmath33 394 & 7.88 @xmath33 0.06 & 0.58 @xmath33 0.03 & 13 @xmath33 1 & & 14.10 & 14.58 & 14.91 & & & 2 & 0.05 & 67 + 15 & j052038.36 + 304822.65 & 80.159836 & 30.806293 & 15924 @xmath33 348 & 8.00 @xmath33 0.07 & 0.61 @xmath33 0.04 & 164 @xmath33 28 & & 15.38 & 15.68 & 15.88 & & & 2 & 0.85 & 24 + 16 & j031236.50 + 515511.74 & 48.152099 & 51.919927 & 23558 @xmath33 1966 & 7.93 @xmath33 0.29 & 0.59 @xmath33 0.14 & 25 @xmath33 13 & & & & & & 15.44 & 3 & 0.84 & 103 + 17 & j055046.51 + 261220.27 & 87.693772 & 26.205631 & 28000 @xmath33 1916 & 8.34 @xmath33 0.39 & 0.84 @xmath33 0.24 & 37 @xmath33 57 & & 15.13 & 15.64 & 15.91 & & & 2 & 1.50 & 13 + 18 & j013938.94 + 291859.80 & 24.912266 & 29.316611 & 20934 @xmath33 515 & 8.13 @xmath33 0.08 & 0.70 @xmath33 0.05 & 77 @xmath33 22 & & 17.53 & 17.94 & 18.19 & & & 2 & 0.05 & 213 + 19&j105811.27 + 475752.75 & 164.546942 & 47.964653 & 29532 @xmath33 490 & 7.84 @xmath33 0.11 & 0.56 @xmath33 0.05 & 9 @xmath33 0 & 17.09 & 17.29 & 17.75 & 18.10 & 18.35 & & 1 & 0.01 & 353 + 20 & j104311.45 + 490224.35 & 160.797708 & 49.040097 & & & & & 15.47 & 15.84 & 16.40 & 16.76 & 17.18 & & 1 & 0.01 & + 21 & j053931.86 + 285456.66 & 84.882770 & 28.915740 & 23865 @xmath33 1774 & 8.63 @xmath33 0.26 & 1.01 @xmath33 0.15 & 147 @xmath33 97 & & 17.39 & 16.64 & 16.17 & & & 2 & 1.43 & 15 + 22 & j094104.43 + 282224.58 & 145.268457 & 28.373495 & 16713 @xmath33 438 & 7.86 @xmath33 0.09 & 0.54 @xmath33 0.05 & 109 @xmath33 24 & 15.70 & 15.42 & 15.70 & 15.94 & 16.25 & & 1 & 0.02 & 82 + 23 & j081845.28 + 121952.45 & 124.688667 & 12.331236 & 22271 @xmath33 531 & 8.34 @xmath33 0.08 & 0.83 @xmath33 0.05 & 100 @xmath33 22 & 16.32 & 16.18 & 16.57 & 16.88 & 17.19 & & 1 & 0.03 & 107 + 24 & j014147.59 + 302135.45 & 25.448307 & 30.359846 & 17520 @xmath33 367 & 8.17 @xmath33 0.07 & 0.72 @xmath33 0.04 & 162 @xmath33 27 & & 16.96 & 17.39 & 17.54 & & & 2 & 0.05 & 138 + 25 & j014933.76 + 285610.60 & 27.390679 & 28.936279 & 32200 @xmath33 631 & 8.33 @xmath33 0.12 & 0.84 @xmath33 0.07 & 17 @xmath33 10 & & 16.88 & 17.47 & & & & 2 & 0.06 & 140 + 26 & j074742.05 + 280945.57 & 116.925192 & 28.162658 & 15085 @xmath33 596 & 7.66 @xmath33 0.13 & 0.44 @xmath33 0.06 & 117 @xmath33 31 & 17.83 & 17.43 & 17.69 & 17.88 & 18.13 & & 1 & 0.04 & 209 + 27 & j075251.35 + 271513.85 & 118.213962 & 27.253847 & 25134 @xmath33 711 & 7.94 @xmath33 0.10 & 0.60 @xmath33 0.06 & 19 @xmath33 9 & 16.72 & 16.73 & 17.15 & 17.46 & 17.79 & & 1 & 0.03 & 206 + 28 & j075106.48 + 301726.96 & 117.776979 & 30.290822 & 34418 @xmath33 580 & 8.21 @xmath33 0.10 & 0.78 @xmath33 0.06 & 9 @xmath33 1 & 15.65 & 15.92 & 16.39 & 16.72 & 17.05 & & 1 & 0.05 & 156 + 29&j113614.04 + 290130.26 & 174.058504 & 29.025072 & 24106 @xmath33 255 & 7.80 @xmath33 0.03 & 0.53 @xmath33 0.02 & 20 @xmath33 1 & 14.64 & 14.68 & 15.13 & 15.44 & 15.75 & & 1 & 0.02 & 87 + 30 & j113705.17 + 294757.54 & 174.271529 & 29.799317 & 21786 @xmath33 160 & 8.58 @xmath33 0.03 & 0.98 @xmath33 0.02 & 174 @xmath33 11 & 12.29 & 12.31 & 12.69 & 12.99 & 13.31 & & 1 & 0.02 & 15 + 31 & j113423.35 + 314606.58 & 173.597300 & 31.768494 & 14683 @xmath33 832 & 8.02 @xmath33 0.14 & 0.62 @xmath33 0.08 & 219 @xmath33 73 & 15.53 & 15.17 & 15.44 & 15.68 & 15.95 & & 1 & 0.03 & 58 + 32 & j093903.33 + 114418.62 & 144.763879 & 11.738506 & 16673 @xmath33 815 & 8.75 @xmath33 0.09 & 1.08 @xmath33 0.05 & 513 @xmath33 116 & 17.37 & 17.01 & 17.21 & 17.41 & 17.67 & & 1 & 0.03 & 82 + 33 & j070755.01 + 265102.94 & 106.979210 & 26.850817 & 17854 @xmath33 893 & 8.87 @xmath33 0.12 & 1.14 @xmath33 0.06 & 554 @xmath33 202 & & 15.53 & 15.86 & 16.01 & & & 2 & 0.07 & 39 + 34 & j104946.47 + 003635.81 & 162.443625 & 0.609947 & 19832 @xmath33 550 & 8.08 @xmath33 0.10 & 0.67 @xmath33 0.06 & 87 @xmath33 23 & 17.25 & 17.27 & 17.67 & 17.97 & 18.30 & & 1 & 0.05 & 191 + 35 & j104623.28 + 024236.57 & 161.596987 & 2.710158 & 13000 @xmath33 728 & 7.73 @xmath33 0.08 & 0.47 @xmath33 0.04 & 211 @xmath33 52 & 16.43 & 16.03 & 16.26 & 16.48 & 16.72 & & 1 & 0.04 & 92 + 36 & j104928.89 + 275423.77 & 162.370375 & 27.906603 & 14212 @xmath33 681 & 7.68 @xmath33 0.15 & 0.45 @xmath33 0.07 & 148 @xmath33 44 & 15.74 & 15.32 & 15.51 & 15.75 & 15.98 & & 1 & 0.02 & 75 + 37 & j115506.22 + 264924.59 & 178.775929 & 26.823497 & 17291 @xmath33 679 & 8.47 @xmath33 0.13 & 0.91 @xmath33 0.08 & 285 @xmath33 88 & 17.03 & 16.69 & 16.97 & 17.23 & 17.50 & & 1 & 0.02 & 97 + 38 & j094627.81 + 313211.08 & 146.615867 & 31.536411 & 15000 @xmath33 2362 & 8.34 @xmath33 0.20 & 0.82 @xmath33 0.13 & 342 @xmath33 218 & 17.30 & 16.91 & 17.12 & 17.34 & 17.55 & & 1 & 0.02 & 103 + 39&j070057.53 + 284310.06 & 105.239692 & 28.719461 & 16000 @xmath33 735 & 8.15 @xmath33 0.13 & 0.70 @xmath33 0.08 & 207 @xmath33 66 & 17.37 & 16.98 & 17.23 & 17.45 & 17.73 & & 1 & 0.08 & 120 + 40 & j040613.25 + 465133.66 & 61.555205 & 46.859349 & 33026 @xmath33 436 & 7.50 @xmath33 0.10 & 0.45 @xmath33 0.03 & 6 @xmath33 1 & & & & & & 14.77 & 3 & 0.82 & 145 + 41 & j103535.22 + 395502.27 & 158.896764 & 39.917298 & 16652 @xmath33 550 & 8.05 @xmath33 0.11 & 0.65 @xmath33 0.07 & 155 @xmath33 39 & 17.43 & 17.12 & 17.33 & 17.55 & 17.84 & & 1 & 0.01 & 154 + 42 & j105443.36 + 270658.42 & 163.680650 & 27.116228 & 24915 @xmath33 131 & 8.38 @xmath33 0.02 & 0.86 @xmath33 0.02 & 74 @xmath33 4 & 13.86 & 13.98 & 14.34 & 14.64 & 14.97 & & 1 & 0.02 & 41 + 43 & j064452.84 + 260947.75 & 101.220170 & 26.163263 & 16835 @xmath33 598 & 7.78 @xmath33 0.13 & 0.50 @xmath33 0.06 & 90 @xmath33 20 & & 15.48 & 15.98 & 16.22 & & & 2 & 0.10 & 86 + 44 & j065601.55 + 115745.85 & 104.006460 & 11.962736 & 31347 @xmath33 603 & 7.42 @xmath33 0.15 & 0.41 @xmath33 0.05 & 8 @xmath33 1 & & 14.31 & 13.48 & 13.13 & 11.94 & & 1 & 0.22 & 63 + 45 & j013914.45 + 290057.61 & 24.810197 & 29.016003 & 16808 @xmath33 478 & 8.06 @xmath33 0.10 & 0.65 @xmath33 0.06 & 153 @xmath33 34 & & 16.20 & 16.53 & 16.68 & & & 2 & 0.06 & 97 + 46 & j094126.79 + 294503.39 & 145.361630 & 29.750942 & 21798 @xmath33 267 & 8.15 @xmath33 0.04 & 0.72 @xmath33 0.03 & 68 @xmath33 10 & 15.88 & 15.91 & 16.25 & 16.55 & 16.87 & & 1 & 0.02 & 106 + 47 & j100549.01 + 424804.68 & 151.454200 & 42.801300 & 23923 @xmath33 812 & 8.11 @xmath33 0.12 & 0.70 @xmath33 0.07 & 38 @xmath33 13 & 16.04 & 16.00 & 16.39 & 16.70 & 16.98 & & 1 & 0.01 & 127 + 48 & j093047.11 + 160012.98 & 142.696300 & 16.003606 & 32492 @xmath33 634 & 8.00 @xmath33 0.13 & 0.66 @xmath33 0.07 & 7 @xmath33 1 & 16.35 & 16.64 & 17.12 & 17.50 & 17.84 & & 1 & 0.04 & 249 + 49&j093451.69 + 171814.00 & 143.715358 & 17.303889 & 14645 @xmath33 566 & 7.80 @xmath33 0.12 & 0.50 @xmath33 0.06 & 156 @xmath33 45 & 17.10 & 16.79 & 17.09 & 17.36 & 17.66 & & 1 & 0.03 & 143 + 50 & j092518.36 + 180534.20 & 141.326500 & 18.092833 & 26274 @xmath33 324 & 8.29 @xmath33 0.06 & 0.81 @xmath33 0.04 & 43 @xmath33 11 & 16.07 & 16.17 & 16.61 & 16.94 & 17.23 & & 1 & 0.05 & 127 + 51 & j052147.24 + 283532.50 & 80.446823 & 28.592361 & 18917 @xmath33 466 & 7.81 @xmath33 0.09 & 0.52 @xmath33 0.05 & 59 @xmath33 16 & & 17.50 & 17.73 & 17.86 & & & 2 & 0.62 & 108 + 52 & j071223.81 + 260933.41 & 108.099190 & 26.159281 & 14278 @xmath33 632 & 7.75 @xmath33 0.14 & 0.48 @xmath33 0.06 & 159 @xmath33 40 & & 16.85 & 17.24 & 17.41 & & & 2 & 0.08 & 141 + 53 & j102521.36 + 455553.91 & 156.338987 & 45.931643 & 23547 @xmath33 908 & 7.51 @xmath33 0.13 & 0.41 @xmath33 0.05 & 19 @xmath33 3 & 18.12 & 18.27 & 18.60 & 18.89 & 19.26 & & 1 & 0.02 & 530 + 54 & j101806.60 + 455830.36 & 154.527482 & 45.975101 & 22475 @xmath33 1541 & 8.36 @xmath33 0.22 & 0.85 @xmath33 0.14 & 101 @xmath33 59 & 17.61 & 17.55 & 17.90 & 18.19 & 18.47 & & 1 & 0.01 & 201 + 55 & j033149.69 + 305944.92 & 52.957023 & 30.995811 & 19435 @xmath33 332 & 8.64 @xmath33 0.06 & 1.02 @xmath33 0.03 & 277 @xmath33 37 & & 16.89 & 17.41 & 17.67 & & & 2 & 1.18 & 25 + 56 & j033253.91 + 284006.91 & 53.224625 & 28.668586 & 19000 @xmath33 1400 & 9.84 @xmath33 0.09 & 1.20 @xmath33 0.15 & 155 @xmath33 0 & & 17.06 & 17.41 & 17.58 & & & 2 & 0.25 & 27 + 57 & j090918.99 + 292929.61 & 137.329125 & 29.491558 & 22588 @xmath33 346 & 8.04 @xmath33 0.05 & 0.65 @xmath33 0.03 & 43 @xmath33 8 & 15.74 & 15.68 & 16.04 & 16.34 & 16.59 & & 1 & 0.02 & 107 + 58 & j102155.50 + 405014.85 & 155.481261 & 40.837458 & 23364 @xmath33 999 & 7.96 @xmath33 0.15 & 0.61 @xmath33 0.09 & 28 @xmath33 18 & 16.19 & 16.25 & 16.61 & 16.91 & 17.23 & & 1 & 0.01 & 153 + 59&j121336.54 + 314808.77 & 183.402250 & 31.802436 & 13308 @xmath33 405 & 8.26 @xmath33 0.08 & 0.77 @xmath33 0.05 & 418 @xmath33 67 & 16.21 & 15.79 & 16.00 & 16.21 & 16.49 & & 1 & 0.01 & 60 + 60 & j134922.51 - 003503.15 & 207.343783 & -0.584208 & 16401 @xmath33 1151 & 8.52 @xmath33 0.19 & 0.94 @xmath33 0.12 & 363 @xmath33 153 & 17.24 & 16.91 & 17.25 & 17.46 & 17.73 & & 1 & 0.03 & 99 + 61 & j144433.83 - 005958.83 & 221.140967 & -0.999675 & 12165 @xmath33 856 & 8.01 @xmath33 0.21 & 0.61 @xmath33 0.13 & 367 @xmath33 154 & 16.58 & 16.20 & 16.38 & 16.58 & 16.85 & & 1 & 0.04 & 77 + 62 & j112518.85 + 541936.65 & 171.328550 & 54.326847 & 15272 @xmath33 209 & 7.85 @xmath33 0.04 & 0.53 @xmath33 0.02 & 147 @xmath33 14 & 15.62 & 15.28 & 15.57 & 15.83 & 16.08 & & 1 & 0.01 & 73 + 63 & j113203.47 + 065509.52 & 173.014441 & 6.919311 & 12455 @xmath33 1141 & 7.93 @xmath33 0.45 & 0.57 @xmath33 0.25 & 310 @xmath33 201 & 15.21 & 14.89 & 15.13 & 15.35 & 15.64 & & 1 & 0.04 & 46 + 64 & j084107.69 + 163221.71 & 130.282053 & 16.539363 & 16626 @xmath33 580 & 8.30 @xmath33 0.11 & 0.80 @xmath33 0.07 & 237 @xmath33 64 & 17.52 & 17.22 & 17.45 & 17.67 & 17.97 & & 1 & 0.02 & 134 + 65 & j150156.26 + 302300.13 & 225.484400 & 30.383369 & 27051 @xmath33 339 & 7.84 @xmath33 0.05 & 0.56 @xmath33 0.03 & 12 @xmath33 1 & 14.13 & 14.24 & 14.71 & 14.96 & 15.32 & & 1 & 0.02 & 77 + 66 & j063406.26 + 025401.30 & 98.526074 & 2.900361 & 31607 @xmath33 1274 & 7.82 @xmath33 0.31 & 0.56 @xmath33 0.14 & 7 @xmath33 1 & & & & & & & 4 & 1.60 & + 67 & j064438.16 + 030704.39 & 101.159020 & 3.117885 & 12067 @xmath33 2288 & 8.37 @xmath33 0.65 & 0.84 @xmath33 0.42 & 650 @xmath33 791 & & 14.10 & 13.66 & 13.51 & & & 2 & 1.02 & 5 + 68 & j063517.47 + 054917.94 & 98.822796 & 5.821650 & 22885 . @xmath33 3739 & 7.48 @xmath33 0.54 & 0.40 @xmath33 0.14 & 21 @xmath33 13 & & 14.19 & 13.96 & 13.84 & & & 2 & 0.41 & 38 + 69&j152130.83 - 003055.70 & 230.378443 & -0.515472 & 13000 @xmath33 1056 & 7.63 @xmath33 0.10 & 0.42 @xmath33 0.05 & 186 . @xmath33 71 . & 15.52 & 15.24 & 15.53 & 15.78 & 16.09 & & 1 & 0.07 & 67 + 70 & j113705.14 + 294757.77 & 174.271408 & 29.799381 & 23829 @xmath33 127 & 8.49 @xmath33 0.02 & 0.93 @xmath33 0.02 & 111 @xmath33 6 & 13.54 & 12.45 & 12.88 & 13.16 & 13.69 & & 1 & 0.02 & 18 + 71 & j191927.67 + 395839.30 & 289.865292 & 39.977583 & 20376 @xmath33 345 & 7.93 @xmath33 0.06 & 0.59 @xmath33 0.03 & 54 @xmath33 8 & & & & & & & 4 & 0.14 & +

we present a spectroscopically identified catalogue of 72 da white dwarfs from the lamost pilot survey . 35 are found to be new identifications after cross - correlation with the eisenstein et al . and villanova catalogues . the effective temperature and gravity of these white dwarfs are estimated by balmer lines fitting . most of them are hot white dwarfs . the cooling times and masses of these white dwarfs are estimated by interpolation in theoretical evolution tracks . the peak of mass distribution is found to be @xmath0 0.6 @xmath1 which is consistent with prior work in the literature . the distances of these white dwarfs are estimated using the method of synthetic spectral distances . all of these wds are found to be in the galactic disk from our analysis of space motions . our sample supports the expectation white dwarfs with high mass are concentrated near the plane of galactic disk .

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Proceed to summarize the following text: one of the factors of the enormous success of our societies is our ability to cooperate @xcite . while in most animal species cooperation is observed only among kin or in very small groups , where future interactions are likely , cooperation among people goes far beyond the five rules of cooperation @xcite : recent experiments have shown that people cooperate also in one - shot anonymous interactions @xcite and even in large groups @xcite . this poses an evolutionary puzzle : why are people willing to pay costs to help strangers when no future rewards seem to be at stake ? a growing body of experimental research suggests that cooperative decision - making in one - shot interactions is most likely a history - dependent dynamic process . _ dynamic _ because time pressure @xcite , cognitive load @xcite , conceptual priming of intuition @xcite , and disruption of the right lateral prefrontal cortex @xcite have all been shown to promote cooperation , providing direct evidence that automatic actions are , on average , more cooperative than deliberate actions . _ history - dependent _ because it has been found that previous experience with economic games on cooperation and intuition interact such that experienced subjects are less cooperative than inexperienced subjects , but only under time pressure @xcite and that intuition promotes cooperative behavior only among inexperienced subjects with above median trust in the setting where they live @xcite . while this latter paper also shows that promoting intuition versus reflection has no effect among experienced subjects , its results are inconclusive with regard to people with little trust in their environment , due to the limited number of observations . more generally , the limitation of previous studies is that they have all been conducted in developed countries and so they do not allow to draw any conclusions about what happens among people from a societal background in which they are exposed to frequent non - cooperative acts . two fundamental questions remain then unsolved . what is the effect of promoting intuition versus deliberation among people living in a non - cooperative setting ? how does this interact with previous experience with economic games on cooperative decision - making ? the first question is particularly intriguing since , based on existing theories , several alternatives are possible . the social heuristics hypothesis ( shh ) , introduced by rand and colleagues @xcite to explain the intuitive predisposition towards cooperation described above `` posits that cooperative decision making is guided by heuristic strategies that have generally been successful in one s previous social interactions and have , over time , become internalized and automatically applied to social interactions that resemble situations one has encountered in the past . when one encounters a new or atypical social situation that is unlike previous experience , one generally tends to rely on these heuristics as an intuitive default response . however , through additional deliberation about the details of the situation , one can override this heuristic response and arrive at a response that is more tailored to the current interaction '' @xcite . then , according to the shh , inexperienced subjects living in a non - cooperative setting should bring their non - cooperative strategy ( learned in the setting where they live ) in the lab as a default strategy . these subjects are then predicted to act non - cooperatively both under time pressure , because they use their non - cooperative default strategy , and under time delay , because defection is optimal in one - shot interactions . however , this is not the only possibility . several studies have shown that patients who suffered ventromedial prefrontal cortex damage , which causes the loss of emotional responsiveness , are more likely to display anti - social behavior @xcite . these findings support the interpretation that intuitive emotions play an important role in pro - social behavior and form the basis of haidt s social intuition model ( sim ) according to which moral judgment is caused by quick moral intuitions and is followed ( when needed ) by slow , ex post facto , moral reasoning @xcite . while the sim does not make any prediction on what happens in the specific domain of cooperation , it would certainly be consistent with a general intuitive predisposition towards cooperation , mediated by positive emotions , and independent of the social setting in which an individual is embedded . a third alternative is yet possible . motivated by work suggesting that people whose self - control resources have been taxed tend to cheat more @xcite and be less altruistic @xcite , it has been argued that self - control plays an important role in overriding selfish impulses and bringing behavior in line with moral standards . this is consistent with kohlberg s rationalist approach @xcite , which assumes that moral choices are guided by reason and cognition : as their cognitive capabilities increase , people learn how to take the other s perspective , which is fundamental for pro - social behavior . this rationalist approach makes the explicit prediction that promoting intuition always undermine cooperation . in sum , the question of how promoting intuition versus reflection affects cooperative behavior among people living in a non - cooperative setting is far from being trivial and , based on existing theories , all three possibilities ( positive effect , negative effect , no effect ) are , a priori , possible . concerning previous experience on economic games , while the sim and the rationalist approach do not make any prediction about its role on cooperative decision - making among people living in a non - cooperative setting , the shh predicts that it has either a null or a positive effect driven by intuitive responses . this because experienced participants , despite their living in a non - cooperative setting , _ might _ have internalized a cooperative strategy to be used only in experiments . of course , the shh does not predict that a substantial proportion of subjects have _ in fact _ developed such a context - dependent cooperative intuition - and this is why the predicted effect is _ either _ positive _ or _ null . in the former case , however , the shh predicts that the positive effect should be driven by intuitive responses , since the shh assumes that experience operates primarily through the channel of intuition . here we report on an experiment aimed at clarifying these points . we provide evidence of two major results : ( i ) promoting intuition versus reflection has no effect on cooperation among subjects living in a non - cooperative setting and with no previous experience with economic games on cooperation ; ( ii ) experienced subjects are more cooperative than inexperienced subjects , but only when acting under time pressure . taken together , these results suggest that cooperation is a learning process , rather than an instinctive impulse or a self - controlled choice , and that experience operates primarily via the channel of intuition . in doing so , they shed further light on human cooperative decision - making and provide further support for the social heuristics hypothesis . we have conducted an experiment using the online labor market amazon mechanical turk ( amt ) @xcite recruiting participants only from india . india is a particularly suited country to hire people from for our purpose : if , as many studies have confirmed @xcite , good institutions are crucial for the evolution of cooperation , and if , as many scholars have argued @xcite , corruption and cronyism are endemic in indian society , then residents in india are likely to have very little trust on strangers and so they are likely to have internalized non - cooperative strategies in their every - day life . one study confirms this hypothesis , by showing that spiteful preferences are widespread in the village of uttar pradesh and this ultimately implies residents inability to cooperate @xcite . at the same time , according to demographic studies on amt population @xcite , india is the second most active country on amt after the us , which facilitates the procedure of collecting data . participants were randomly assigned to either of two conditions : in the _ time pressure _ condition we measured intuitive cooperation ; in the _ time delay _ condition we measured deliberate cooperation . as a measure of cooperation , we adopted a standard two - person prisoner s dilemma ( pd ) with a continuous set of strategies . specifically , participants were given an endowment of @xmath2 , and asked to decide how much , if any , to transfer to the other participant . the amount transferred would be multiplied by 2 and earned by the other participant ; the remainder would be earned by themselves , but without being multiplied by any factor . each participant was informed that the other participant was facing the same decision problem . participants in the time pressure condition were asked to make a decision within 10 seconds and those in the time delay condition were asked to wait for at least 30 seconds before making their choice . after making their decision , participants had to answer four comprehension questions , after which they entered the demographic questionnaire , where , along with the usual questions , we also asked `` to what extent have you previously participated in other studies like to this one ( e.g. , exchanging money with strangers ) ? '' using a 5 point likert - scale from `` never '' to `` several times '' . as in previous studies @xcite , we used the answer to this question as a measure of participant s previous experience with economic games on cooperative decision - making . as in these studies , we say that a subject is _ inexperienced _ if he or she answered `` never '' to the above question . in the supplementary online material we also report the results of a pilot , in which we measured participants level of experience by asking them to report the extent to which they had participated in _ exactly _ the same task before . although the use of the word `` exactly '' may lead to confusion , with some minor differences in the details , our main results are robust to the use of this measure ( see supplementary online material for more details ) . after collecting the results , bonuses were computed and paid on top of the participation fee ( @xmath3 ) . no deception was used . a total of 949 subjects participated in our experiment . taken globally , results contain a lot of noise , since only 449 subjects passed the comprehension questions . here we restrict our analysis to subjects who passed all comprehension questions and we refer the reader to the supplementary online material for the analysis of those subjects who failed the attention check . we include in our analysis also subjects who did not obey the time constraint in order to avoid selection problems that impair causal inference @xcite . first we ascertain that our time manipulation effectively worked . analyzing participants decision times , we find that those in the time delay condition ( @xmath4 ) took , on average , 45.64 seconds to make their decision , while those under time pressure took , on average , only 20.04 seconds . thus , although many subjects under time pressure did not obey the time constraint , the time manipulation still had a substantial effect . participants under time pressure transferred , on average , 27.93% of their endowment , and those under time delay transferred , on average , 28.57% of their endowment . linear regression using time manipulation as a dummy variable confirms that the difference is not statistically significant ( coeff @xmath5 , @xmath6 ) , even after controlling for age , sex , and level of education ( coeff @xmath7 @xmath8 ) . we note that restricting the analysis to subjects who obeyed the time constraint leads to qualitatively equivalent results ( 26.11% under time pressure vs 29.02% under time delay , coeff @xmath9 , @xmath10 ) . thus , promoting intuition versus reflection does not have any effect on cooperative decision - making . _ en passant _ , we note that these percentages are far below those observed among us residents in a very similar experiment @xcite . more precisely , in this latter paper , us residents started out with a @xmath11 endowment and were asked to decide how much , if any , to give to the other person . as in the current study , the amount transferred would be multiplied by 2 and earned by the other player . strictly speaking , these two experiments are not comparable for three reasons . first , in @xcite there was no time manipulation ; second , the initial endowments were different ; third , stakes used in the experiment in the us did not correspond to the same stakes in indian currency . according to previous research , these differences are minor . indeed , recent studies have argued that stakes do not matter as long as they are not too high @xcite and that neutrally framed pds give rise to a percentage of cooperation sitting between that obtained in the time pressure condition and that obtained in the time delay condition @xcite . thus comparing the percentage of cooperation in the current study ( 28% ) with that reported in @xcite ( 52% ) supports our assumption that the average indian sample is particularly non - cooperative or , at least , less cooperative than the average us sample . next we investigate our main research questions . figure 1 summarizes our results , providing visual evidence that ( i ) promoting intuition versus reflection has no significant effect on cooperation among inexperienced subjects ; that ( ii ) experienced subjects cooperate more than inexperienced subjects , but only when acting under time pressure . more specifically , we find that inexperienced subjects under time pressure ( @xmath12 ) transferred , on average , @xmath13 of their endowment while those under time delay ( @xmath14 ) transferred , on average , @xmath15 of their endowment . the difference is not significant ( coeff @xmath16 , @xmath17 ) , even after controlling for all socio - demographic variables ( coeff @xmath18 , @xmath19 ) . thus , promoting intuition versus reflection has no effect on cooperation among inexperienced subjects living in a non - cooperative setting . this finding is robust to controlling for people who did not obey the time constraint ( coeff @xmath20 , @xmath21 ) and to controlling for people who obeyed the time contraint ( coeff @xmath22 , @xmath23 ) . to explore the interaction between experience and cooperative behavior , as in previous studies @xcite , we separate subjects into experienced and inexperienced . this procedure comes from the observation that , although the level of experience is a categorical variable , the association between participant s objective level of experience and their answer to our question is objective only in case of inexperienced subjects . linear regression predicting cooperation using experience as a dummy variable confirms that experience with economic games on cooperation favors the emergence of cooperative choices , but only among people in the time pressure condition ( time pressure : coeff @xmath24 , @xmath25 ; time delay : coeff @xmath26 , @xmath27 ) . these results are robust to including control on the socio - demographic variables ( time pressure : coeff @xmath28 , @xmath29 ; time delay : coeff @xmath30 , @xmath31 ) . our main results are also robust to using non - parametric tests , such as wilcoxon rank - sum : the rate of cooperation of inexperienced subjects is not statistically distinguishable from the rate of cooperation of inexperienced subjects acting under time delay both when they act under time pressure ( @xmath32 ) and under time delay ( @xmath33 ) ; and the rate of cooperation of inexperienced subjects is significantly smaller than that of experienced subjects , but only among those acting under time pressure ( time pressure : @xmath34 ; time delay : @xmath35 ) . for completeness , we also report the results of linear regression predicting cooperation using level of experience as independent variable . we find that level of experience has a marginally significant positive effect on cooperation among subjects acting under time pressure ( coeff @xmath36 , @xmath37 ) and has no effect on cooperation among subjects acting under time delay ( coeff = @xmath38 , @xmath39 ) . also these results are robust to including control on all the socio - demographic variables ( time pressure : coeff @xmath40 , @xmath41 ; time delay : coeff @xmath42 , @xmath43 ) . the increase of cooperation from inexperienced subjects to experienced subjects seems to be driven by participants under time pressure who did _ not _ obey the time constraint . specifically , linear regression predicting cooperation using experience as a dummy variable yields non - significant results in case of participants who obeyed the time pressure condition ( without control : coeff @xmath44 , @xmath45 ; with control : coeff @xmath46 , @xmath47 ) and significant results in case of participants who did not obey it ( without control : coeff @xmath48 , @xmath49 ; with control : coeff @xmath50 , @xmath51 ) . this is not surprising and it is most probably due to noise generated by a combination of two factors : the set of people who obeyed the time pressure contraint is very small ( for instance , only 14 inexperienced people obeyed the time constraint ) and it is more likely to contain people who did not understand the decision problem but passed the comprehension questions by chance ( which we estimated to be 5% of the total . see supplementary online material ) . ) . previous experience with economic games on cooperation has a positive effect on cooperation , but only among participants in the time pressure condition ( @xmath25 ) . _ _ ] we have shown that ( i ) promoting intuition via time pressure versus promoting deliberation via time delay has no effect on cooperative behavior among subjects residents in india with no previous experience with economic games on cooperation , and that ( ii ) experience has a positive effect on cooperation , but this effect is significant only among subjects acting under time pressure . our results have several major implications , the first of which is providing further support for the social heuristics hypothesis ( shh ) @xcite . introduced in order to organize the growing body of literature providing direct @xcite and indirect @xcite evidence that , on average , intuitive responses are more cooperative than reflective responses , the shh contends that people internalize strategies that are successful in their everyday social interactions and then apply them to social interactions that resemble situations they have encountered in the past . thus , when they encounter a new or atypical situation , people tend to rely on these heuristics and use them as intuitive responses . deliberation can override these heuristics and adjust the behavior towards one that is more tailored to the current interaction . as such , the shh makes a prediction that has not been tested so far : inexperienced subjects living in a non - cooperative setting should act non - cooperatively both under time pressure , because they use their non - cooperative default strategy ( learned in the setting where they live ) , and under time delay , because defection is optimal in one - shot interactions . our results support this prediction . besides this prediction , the shh is also consistent with an interaction between level of previous experience with economic games on cooperation , time pressure , and cooperation in one - shot interactions : experienced people , despite their living in a non - cooperative setting , _ might _ have internalized a cooperative strategy , to be used only in amt . the shh does not predict that a substantial proportion of experienced people have _ in fact _ developed this context - dependent intuition for cooperation , but it is certainly consistent with a positive effect of experience on cooperation driven by intuitive responses . our results provide evidence for this phenomenon . as mentioned in the introduction , kohlberg s rationalist approach makes the explicit prediction that promoting intuition should always undermine cooperation . thus our results support the shh versus kohlberg s rationalistic approach . of course , this does _ not _ imply that the rationalist approach should be completely rejected : it is indeed supported by many experimental studies involving pro - social behaviors other than cooperation . if anything , our results point out that different pro - social behaviors may emerge from different cognitive processes . classifying pro - social behaviors in terms of the processes involved is an important direction for future research towards which , to the best of our knowledge , only one recent study has attempted a first step @xcite . supporting the shh , our results suggest that economic models of human cooperation should start taking dual processes and individual history into account . indeed , virtually all major models of human cooperation are static and decontextualized and only a handful of papers have recently attempted a first step in the direction of taking dual processes into account @xcite . we believe that extending these approaches to incorporate also individual history could be a promising direction for future research . our findings go beyond the mere support of the shh . our cross cultural analysis , although it is formally not correct , shows that residents in india are , on average , less cooperative than us residents . the difference is so large ( 28% vs 52% ) that it is hard to explain it by appealing to minor differences in the experimental designs and so it deserves to be commented . one possibility , supported by the experimental evidence that good institutions are crucial in promoting cooperation @xcite and the evidence that india struggles on a daily basis to fight corruption in politics at both the national and local levels @xcite , is that residents in india may have internalized non cooperative behavior in their everyday life ( because cooperation is not promoted by their institutions ) and they tend to apply it also to the new situation of a lab experiment . one far - reaching consequence of this interpretation is that the role of local institutions may go far beyond regularizing behavior . if institutions do not support cooperative behavior , selfishness may even get internalized and applied to atypical situations where people rely on heuristics . while this interpretation is supported by a recent study @xcite showing that norms of cooperation learned in one experiment spill over to subsequent experiments where there are no norms , we recommend caution on our interpretation , since our results do _ not _ show directly that inexperienced residents in india are less cooperative than us residents _ because _ they are embedded into a society whose institutions do not promote cooperative behavior . however , we believe that this is a fundamental point that deserves to be rigorously addressed in further research . interestingly , we have shown that experienced residents in india are significantly more cooperative than inexperienced ones . this correlation appears to be even more surprising if seen in light of recent studies reporting that experience has a _ negative _ effect on cooperation among residents in the us @xcite . although the sign of these effects are different , they share the property that they are driven by intuitive responses . thus they are in line with the shh , which assumes that experience operates mainly through the channel of intuition , but it does not make any prediction about the sign of the effect of experience , which may ultimately depend on a number of factors . while it is relatively easy to explain a negative effect of experience with economic games on cooperation , by appealing to learning of the payoff maximizing strategy , explaining a positive effect is harder . one possibility is that experienced subjects have learned cooperation in iterated games , where it might be strategically advantageous , and tend to apply it also in one - 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demographics of participants who passed the comprehension questions . data are summarized in the table . we remind that participants level of experience was measured using a 5-point likert - scale from 1=never to 5=several times . the socio - demographic statistics show that the majority of subjects in the time pressure condition did not obey the time constraint . this is likely due to the fact that reading the instructions of the decision problem takes about six seconds and so participants had only 4 seconds to understand the problem and make their decision . however , the mean decision time of participants in the time pressure condition was much smaller than the mean decision time of participants in the time delay condition , providing evidence that the time manipulation still had a substantial effect . [ cols="<,^,^",options="header " , ] finally we analyze subjects who failed the attention test . indeed , the large number of participants who failed the comprehension questions ( about half of the total number ) is worrisome that a substantial fraction of those who passed the comprehension questions may have passed them just by chance . although this rate of passing is not much lower than that in similar studies conducted in the us ( in a very similar experiment , conducted with american subjects and published in @xcite , 32% of subjects did not pass the comprehension questions ) it could potentially generate noise in our data . indeed , subjects who failed the attention test played essentially at random ( time pressure : average transfer = 47.20% ; time delay : average transfer = 46.35% ) . to exclude this possibility , we analyse failers responses to the comprehension questions . we start with the first comprehension question , which asked `` what is the choice by you that maximizes your outcome ? '' . participants could choose any even amount of money from 0c to 20c , for a total of 11 possible choices . figure 1 reports the distribution of responses of people who failed at least one comprehension questions . the distribution is clearly tri - modal , with 60% of responses equally distributed among the two extreme choices ( full cooperation and full defection ) and the mid - point ( transfer half ) . the remaining 40% is equally distributed among all other choices . consequently , assuming that confused participants respond to the first comprehension question according to this distribution , the probability that a confused participant pass the first comprehension question by chance is equal to 1/5 . next we analyze the responses to the second comprehension question , which asked `` what is the choice by you that maximizes the other participant s outcome ? '' . figure 2 reports the distribution of answers . also in this case we find a tri - modal distribution , although this time the correct answer appeared with higher frequency ( about 50% ) . assuming that a confused participant answered this question according to this distribution , we conclude that a confused participant had probability about 1/10 to pass the first two comprehension questions by chance . then we analyze the responses to the third comprehension question , which asked `` what is the choice by the other participant that maximizes your outcome ? '' figure 3 reports the distribution of answers . this time the distribution is essentially bi - modal , with about 3/5 of people answering correctly . assuming that confused participants answered according to this distribution , we conclude that a confused participant had probability about 3/50 to pass the first three comprehension questions by chance . now , it is impossible to have a clue of what the proportion of confused participants who passed the fourth comprehension question by chance is . the analysis above suggests that 3/50 is an upper bound of the probability that a confusing subject passed all comprehension questions by chance . interestingly , 88% of subjects who answered 20c in the third comprehension and failed the fourth comprehension question , answered 20c also to the fourth comprehension question , which asked `` what is the choice by the other participant that maximizes the other participant s outcome ? '' . this suggests that the actual probability of passing all comprehension questions by chance is much lower that 3/50 . in any case , although our results do not allow to make a precise estimation of noise , the proportion of people who passed the comprehension questions by chance is likely below 5% of the total , suggesting that noise is a minor problem in our data . participants were randomly assigned to either the time pressure condition or the time delay condition . in both conditions , after entering their worker i d , participants were informed that they would be asked to make a choice in a decision problem to be presented later and that comprehension questions would be asked . participants were also informed that the survey ( which was made using the software qualtrics ) contained a skip logic which would automatically exclude all participants failing any of the comprehension questions . specifically , this screen was as follows : _ welcome to this hit . _ _ this hit will take about five minutes . for the participation to this hit , you will earn 0.50 us dollars , that is , about 31 inr . you can also earn additional money depending on the decisions that you and the other participants will make . _ you will be asked to make one decision . there is no incorrect answer . however : _ _ important : after making the decision , to make sure you understood the decision problem , we will ask some simple questions , each of which has only one correct answer . if you fail to correctly answer any of those questions , the survey will automatically end and you will not receive any redemption code and consequently you will not get any payment . _ _ with this in mind , do you wish to continue ? _ at this stage , they could either leave the study or continue . those who decided to continue were redirected to an introductory screen where we gave them all the necessary information about the decision problem , but without telling exactly which one it is . this is important in order to have the time pressure and time delay conditions work properly in the next screen . this introductory screen for the participants in the time pressure condition was the following : _ you have been paired with another participant . you can earn additional money depending on the decision you will make in the next screen . you will be asked to make a choice that can affect your and the other participant s outcome . the decision problem is symmetric : also the other participant is facing the same decision problem . after the survey is completed , you will be paid according to your and the other participant s choices . _ _ you will have only 10 seconds to make the choice . _ _ this is the only interaction you have with the other participant . he or she will not have the opportunity to influence your gain in later parts of the hit . if you are ready , go to the next page . _ the introductory screen for the participants in the time delay condition was identical , a part from the fact that the sentence ` you will have only 10 seconds to make the choice ' was replaced by the sentence ` you will be asked to think for at least 30 seconds before making your choice . use this time to think carefully about the decision problem ' . the decision screen was the same in both conditions : _ you and the other participant are both given @xmath2 us dollars . you and the other participant can transfer , independently , money to the each other . every cent you transfer , will be multiplied by @xmath52 and earned by the other participant . every cent you do not transfer , will be earned by you . _ _ how much do you want to transfer ? _ by using appropriate buttons , participants could transfer any even amount of money from @xmath53 to @xmath2 . en passant , we observe that reading the decision screen takes about six seconds and thus participants under time pressure had only about four seconds to make their choice . to assure that time pressure and time delay work properly , it is necessary that comprehension questions are asked after the decision has been made . thus , right after the decision screen , participants faced the following four comprehension questions . _ what is the choice by you that maximizes your outcome ? _ _ what is the choice by you that maximizes the other participant s outcome ? _ _ what is the choice by the other participant that maximizes your outcome ? _ _ what is the choice by the other participant that maximizes the other participant s outcome ? _ by using appropriate buttons , participants could select any even amount of money from @xmath53 to @xmath2 . participants who failed any of the comprehension questions were automatically excluded from the survey . those who answered all questions correctly entered the demographic questionnaire , where we asked for their age , sex , reason for their choice , and , most importantly , level of experience in these games . specifically , we asked the following question : _ to what extent have you previously participated in other studies like to this one ( e.g. , exchanging money with strangers ) ? _ answers were collected using a 5 point likert - scale from `` 1=never '' to `` 5=several times '' . our pilot experiment was identical to our main experiment , except for the fact that , as a measure of experience , we asked participants to self - report the extent to which they had participated in `` exactly '' the same task before . participants could choose between : never , once or twice , and several times . the use of the word `` exactly '' is problematic , since it might lead to confusion : what does the answer `` exactly the same task '' imply ? does it imply that participants have participated in a task with exactly the same instructions ( including time constrains ) or does it imply that the participant have participated in a task containing the same economic game ? figure 4 reports the results of the pilot . we observe that , indeed , the details are different : level of experience seem to have a inverted - u effect on cooperation , which , since it has not been replicated in the main experiment , is probably due to confusion regarding the interpretation of the word `` exactly '' . however , as in our main experiment , we find that experienced subjects are significantly more cooperative than little or no experienced subjects and that this behavioral change is mainly driven by intuitive responses ( see figure 5 ) . specifically , linear regression confirms that little experienced subjects are significantly less cooperative than inexperienced subjects both under time pressure ( coeff @xmath54 , @xmath55 ) and under time delay ( coeff @xmath56 , @xmath57 ) ; and confirms that experienced subjects are significantly more cooperative than little experienced subjects both under time pressure ( coeff @xmath58 , @xmath59 ) and under time delay ( coeff @xmath60 , @xmath61 ) . moreover , experienced subjects were also significantly more cooperative than inexperienced subjects , both under time pressure ( coeff @xmath62 , p @xmath63 ) and under time delay ( coeff @xmath64 , p @xmath63 ) . thus experience has a significant inverted - u effect on cooperation , where little experienced subjects cooperate the least and experienced subjects the most . the coefficients of the previous regressions suggest that the motivations behind the initial decrease of cooperation , which affects subjects under time pressure and those under time delay to exactly the same extent , are different from the motivations behind the subsequent flourishing of cooperation , which seems to affect subjects under time pressure to a larger extent than those under time delay . to confirm this , we use linear regression to predict decision among experienced subjects using time pressure as a dummy variable . we find that experienced subjects under time pressure are nearly significantly more cooperative than experienced subjects under time delay ( coeff @xmath65 , @xmath66 ) .

recent studies suggest that cooperative decision - making in one - shot interactions is a history - dependent dynamic process : promoting intuition versus deliberation has typically a positive effect on cooperation ( dynamism ) among people living in a cooperative setting and with no previous experience in economic games on cooperation ( history - dependence ) . here we report on a lab experiment exploring how these findings transfer to a non - cooperative setting . we find two major results : ( i ) promoting intuition versus deliberation has no effect on cooperative behavior among inexperienced subjects living in a non - cooperative setting ; ( ii ) experienced subjects cooperate more than inexperienced subjects , but only under time pressure . these results suggest that cooperation is a learning process , rather than an instinctive impulse or a self - controlled choice , and that experience operates primarily via the channel of intuition . in doing so , our findings shed further light on the cognitive basis of human cooperative decision - making and provide further support for the recently proposed social heuristics hypothesis . @xmath0center for mathematics and computer science ( cwi ) , 1098 xg , amsterdam , the netherlands . @xmath1department of political sciences , luiss guido carli , 00197 , roma , italy . contact author : v.capraro@cwi.nl__forthcoming in proceedings of the royal society b : biological sciences _ _

You are an expert at summarizing long articles. Proceed to summarize the following text: photonic crystal fibers ( pcf ) are a new class of optical fibers which has revealed many surprising phenomena and also holds a big promise for future applications ( see _ e.g. _ these pcfs are made from pure silica with a cladding consisting of a regular lattice of air - holes running along the fiber axis . depending on the arrangement of the air - holes the guiding of light can be provided by either modified total internal reflection @xcite or by the photonic band - gap effect @xcite and pcfs can even be endlessly single - mode @xcite because of the wavelength dependence of the cladding index . for the basic operation we refer to the review of broeng _ et al . _ @xcite . understanding the shape and radiation pattern , as illustrated in fig . [ fig1 ] , of the mode in the endlessly single - mode pcf is very important . e.g. in tests and applications this is essential for estimations of coupling efficiencies and for determining the mode field diameter from the far - field distribution . furthermore , it is fundamentally the simplest structure with a hexagonal cladding , and hence the understanding of this structure will be a natural basis for understanding the modes of more sophisticated pcf structures . in this paper we present a semi - empirical model which is capable of explaining both the near and far - field distribution of the mode , but most importantly also accounts for the fine structure in the transition from the near to the far field . the simplicity of the model allows for a phenomenological interpretation of the shapes of the near and far - field patterns . the measurements reported are for a pcf with a triangular air - hole lattice with pitch of @xmath1 and air holes of diameter @xmath2 . the measurements reported here were performed at a free - space wavelength of @xmath3 , where the light is guided in a single mode in the silica core of the fiber formed by a `` missing '' air hole . in panel a of fig . [ fig2 ] a micro - graph of the fiber structure can be seen . the near - field distribution was measured using a microscope objective to magnify the mode onto a si - based ccd camera . in fig . [ fig2]b the intensity distribution is shown at focus . by translating the fiber away from the focal plane , the intensity distribution may be imaged at different distances between the near and the far field . this is shown in panels b to h in fig . as expected the mode at focus has a hexagonal shape , that extends in the six regions between the inner holes and is sharply confined at the six silica - hole interfaces . however , when the image is defocused , the shape at first transforms into a nearly circular shape ( panel c ) followed by a hexagonal shape rotated by an angle of @xmath0 with respect to the focus ( panel d ) . after this the shape again becomes close to circular ( panel e ) , and finally transforms into the original hexagonal orientation ( panel f ) with six satellites emerging from the distribution ( panels g and h ) . it is noted that the orientation of the satellites is rotated by @xmath0 with respect to the six inner holes surrounding the core . in fig . [ fig3 ] ( right ) the intensity distribution in the far - field limit is shown ( several centimeters from the fiber end - facet ) , obtained using a commercial far - field profiler . here , the satellites have fully developed and as shown in the cross sectional plot in fig . [ fig3 ] ( left ) the peak intensities of the satellites are more than two orders of magnitude lower than the main peak . hence , a reasonably accurate analysis of the far field may be performed considering only the main peak . apart from being a fascinating and intriguing evolution of the mode shape from the near to the far field , it is important to be aware of these transitions in any application that involves imaging of the modes . _ e.g. _ for estimations of the mode field diameter and effective area based on near - field analysis , it is important to focus the mode correctly , and the positions corresponding to panel b and panel d in fig . [ fig2 ] may easily be confused . they both show the hexagonal shape and have only slightly different mode sizes . hence , as a measurement procedure for determining the mode field diameter , a direct measurement of the near field may be even more tricky than it is for `` standard technology fibers '' with circular symmetry . in panel a of fig . [ fig4 ] two cross - sections of the measured near - field distribution are shown , one taken along a line passing through opposite hole centers ( 1 ) and the second taken along a line passing between the holes ( 2 ) ( rotated by an angle @xmath0 with respect to the first ) . it is noted that a gaussian distribution is a significantly better fit to the intensity along line ( 2 ) , motivating a simple interpretation of the mode shape : the mode is a circular gaussian distribution from which a narrow distribution localized at each of the centers of the six inner holes is subtracted . this simple interpretation is theoretically modeled in the following . in order to simulate the radiated field we start from the fully - vectorial fields in the single - mode all - dielectric pcf @xmath4 where @xmath5 and @xmath6 are the transverse fields and the propagation constant , respectively . these we calculate numerically by a fully - vectorial plane - wave method @xcite . substantial insight in the physics of the radiation problem can be gained by expanding @xmath5 in gaussians . introducing the notation @xmath7 and using that the components of @xmath5 can be chosen either real or imaginary we consider @xmath8 for the radiation into free space this gives a linear combination of expanding gaussian beams and this is a well - studied problem , see _ e.g. _ @xcite . neglecting the small back - scattering from the end - facet , the gaussian @xmath9 at finite @xmath10 transforms as @xmath11,\ ] ] where @xmath12 and @xmath13 . in the following we consider a particular simple linear combination in the pcf ; @xmath14 where @xmath15 with @xmath16 being the center position of the six air holes nearest to the core . here , @xmath17 ( the radius of the silica core ) , @xmath18 ( the mode - field radius ) , and @xmath19 ( the radius of the air holes ) . the first term gives the over - all gaussian intensity profile of the mode and with @xmath20 the additional six terms of opposite sign suppress the intensity at the six air - holes nearest to the core . for finite @xmath10 the intensity transforms as @xmath21 in panel b of fig . [ fig4 ] we show an example of the intensity distribution in an ideal pcf with @xmath22 and @xmath23 at @xmath3 corresponding to experimental situation . for the dielectric function we have used @xmath24 for the air holes and for the silica we have used @xmath25 based on the sellmeier formula . while eq . ( [ h_gaussian ] ) may seem too simplistic the good fits to gaussians strongly justify it and as we shall see it reproduces the physics observed experimentally . in fig . [ fig5 ] we show the corresponding near field based on eq . ( [ h_gaussian ] ) . the profile at the end - facet ( panel a ) first transforms into a close - to - circular profile ( panel c ) followed by a hexagonal shape rotated by @xmath0 ( panels d to f ) , a close - to - circular profile ( panel g ) , and finally a hexagonal shape ( panels h and i ) with the same orientation as at the end - facet ( panel a ) . comparing with fig . [ fig2 ] this is qualitatively in excellent agreement with the experimental observations . the fact that the fully coherent scattering description qualitatively reproduces the experimentally observed @xmath0 rotation gives strong indications of its nature ; it is a phenomena caused by an interference between the different gaussian elements used in the decomposition of the fundamental mode in the pcf . in fig . [ fig6 ] we show the corresponding intensity distribution in the far - field limit which is in a very good agreement with the experiments , see fig . it is seen that the satellites are reproduced and are in fact oriented in the same way as in the experiment . moreover the relative intensities between the satellites and the main peak in fig . [ fig6 ] ( left ) are very similar to the ones in fig . [ fig3 ] ( left ) . finally , for fibers where the air holes modify the overall gaussian profile sufficiently ( not shown ) we find indication of additional higher - order spots further away from the center of the intensity distribution which can also be seen experimentally . the evolution of the mode shape of a pcf with a triangular cladding has been investigated in the transition from the near to the far field . when moving away from the near field at the focus of the fiber end - facet , it has been observed that the hexagonal orientation is rotated two times by @xmath0 after which six satellites emerge in the radiation pattern . in the far - field limit the satellites remain in the pattern , having a relative peak intensity more than two orders of magnitude less than the main peak and with an orientation rotated by @xmath0 relative to the six inner holes around the fiber core . all these observations have been reproduced theoretically , by approximating the near - field distribution by a main gaussian peak from which six narrow gaussians located near the center of the six inner holes have been subtracted . from the simulations it is concluded that the changes of shape in the radiation pattern are caused by an interference between the different gaussian elements used in the decomposition of the fundamental mode in the pcf . the results presented here are very important for understanding and analysing the behavior of the mode in many optical systems based on photonic crystal fibers especially those involving imaging and focusing the mode . furthermore , the successful idea of decomposing the near field of the mode into seven localized distributions can be adapted in future work aimed at a simple quantitative description of the near and far - field distributions , particularly for relating the measured far field to the physical structure around the fiber core that influences the near field . the latter is very interesting in the field of fiber measurement procedures , since the far - field analysis of conventional optical fibers can not directly be adapted to pcfs because of the lack of cylindrical symmetry . we thank j. riishede and t. p. hansen ( com , technical university of denmark ) , p. m. w. skovgaard and j. broeng ( crystal fibre a / s ) for technical assistance and useful discussions .

the transition from the near to the far field of the fundamental mode radiating out of a photonic crystal fiber is investigated experimentally and theoretically . it is observed that the hexagonal shape of the near field rotates two times by @xmath0 when moving into the far field , and eventually six satellites form around a nearly gaussian far - field pattern . a semi - empirical model is proposed , based on describing the near field as a sum of seven gaussian distributions , which qualitatively explains all the observed phenomena and quantitatively predicts the relative intensity of the six satellites in the far field . 10 opt . express * 9 * , 674779 ( 2001 ) , http://www.opticsexpress.org/issue.cfm?issue id=124 . j. opt . a : pure appl . opt . * 3 * , s103s207 ( 2001 ) . j. c. knight and p. s. j. russell , `` applied optics : new ways to guide light , '' _ science _ * 296 * , 276277 ( 2002 ) . j. c. knight , t. a. birks , p. s. j. russell , and d. m. atkin , `` all - silica single - mode optical fiber with photonic crystal cladding , '' _ opt . lett . _ * 21 * , 15471549 ( 1996 ) . j. c. knight , t. a. birks , p. s. j. russell , and d. m. atkin , `` all - silica single - mode optical fiber with photonic crystal cladding : errata , '' _ opt . lett . _ * 22 * , 484485 ( 1997 ) . j. c. knight , j. broeng , t. a. birks , and p. s. j. russell , `` photonic band gap guidance in optical fibers , '' _ science _ * 282 * , 14761478 ( 1998 ) . r. f. cregan , b. j. mangan , j. c. knight , t. a. birks , p. s. j. russell , p. j. roberts , and d. c. allan , `` single - mode photonic band gap guidance of light in air , '' _ science _ * 285 * , 15371539 ( 1999 ) . t. a. birks , j. c. knight , and p. s. j. russell , `` endlessly single mode photonic crystal fibre , '' _ opt . lett . _ * 22 * , 961963 ( 1997 ) . j. broeng , d. mogilevstev , s. e. barkou , and a. bjarklev , `` photonic crystal fibers : a new class of optical waveguides , '' _ opt . fiber technol . _ * 5 * , 305330 ( 1999 ) . s. g. johnson and j. d. joannopoulos , `` block - iterative frequency - domain methods for axwell s equations in a planewave basis , '' _ opt . express _ * 8 * , 173190 ( 2000 ) , http://www.opticsexpress.org/abstract.cfm?uri=opex-8-3-173 . a. k. ghatak and k. thyagarajan , _ introduction to fiber optics _ ( cambridge , cambridge university press , 1998 ) . a. k. ghatak and k. thyagarajan , _ optical electronics _ ( cambridge , cambridge university press , 1989 ) .

You are an expert at summarizing long articles. Proceed to summarize the following text: a precise measurement of the stellar initial mass function ( imf ) and its functional dependence on environmental conditions would impact astronomy over a wide range of physical scales . it would be of great help to theorists in untangling the mysteries of star formation and it is a key input in spectral synthesis models used to interpret the observed properties of galaxies both nearby and in the early universe . the current question is whether the imf is _ universal_ the same regardless of time and environmental conditions . @xcite concisely states the current understanding of imf universality . it is difficult to believe that the imf is universal given the diversity of galaxy types , environments , star formation rates , and populations within galaxies over the range of observable lookback times . on the other hand , while imf measurements do vary they are all consistent with a universal imf within measurement errors and sampling statistics . the only way to proceed then is to strive for smaller measurement errors and improving sample sizes . a definitive theoretical derivation of the imf does not yet exist . theoretical approaches to the imf usually center around the jeans mass , @xmath9 , the mass at which a homogeneous gas cloud becomes unstable . at first the collapse of a cloud is isothermal and the jeans mass decreases which leads to fragmentation of the cloud @xcite . both @xcite and @xcite suggested that at some point during the cloud collapse the line cooling opacity becomes high enough that the collapse is no longer isothermal . at this point the jeans mass increases and fragmentation stops . the minimum jeans mass is the smallest fragment size at this point and it provides a lower limit to size of the stars formed . authors have calculated jeans masses and minimum jeans masses using a variety of methods . in the classical derivation of the jeans mass @xmath10 . @xcite finds that @xmath11 where @xmath12 is the mass of gas atoms or molecules and @xmath13 is the opacity at the final fragmentation . more recently turbulence in clouds has been studied . @xcite found @xmath14 where @xmath15 is the number density and @xmath16 is the velocity dispersion of the gas . other investigators have looked at the hierarchical fractal geometry of molecular clouds , thought to arise from turbulence , as a generator of the imf ( e.g. @xcite ) . on a related note @xcite point out that molecular clouds exhibit structure on all resolvable spacial scales suggesting that as no characteristic density exists for the clouds neither does a single jeans mass . they develop a semi - empirical model for determining the final masses of stars from the initial conditions of molecular clouds without invoking jeans mass arguments and use it to construct imf models . the key components of their model are sound speed and rotation rate of cloud cores and the idea that stars help determine their final masses through winds and outflows . but from the beginning the study of the imf has been driven by measurements . in 1955 salpeter was the first to make a measurement of the imf inferring it from his observed stellar luminosity function @xcite . we parameterize the imf by : @xmath17 following @xcite . salpeter found that @xmath18 . it is often overlooked that his measurement only covered masses for which @xmath19 . nonetheless his original measurement is surprisingly consistent with modern values over a wide range of masses . the imf in equation [ eq : imfbaldry ] is similar to the salpeter imf for @xmath20 . the difference is that there are fewer stars with masses less than 0.5 @xmath1 . we adopt a two part power law as there is agreement amongst several authors that there is a change in the imf slope near 0.5 @xmath21 @xcite . the technique we will use is not sensitive to the imf at low masses so we assume a constant value in that regime . salpeter s idea continues to be used today in imf measurements of resolved stellar populations . the technique can be applied to field stars as well as clusters . however salpeter s method has several inherent limitations . the nature of stars presents a challenge . on the high mass end stars are very luminous , but live only a few million years , while on the low end stars are faint but have lifetimes many times longer than the current age of the universe . there are very few star clusters which are both young and close enough to allow us access to the imf over the full mass range . in addition , the main sequence mass - luminosity relationship is a function of age , metallicity , and speed of rotation in addition to mass . it is not yet well - known at the low and high mass extremes . unresolved binaries can also affect the measured imf @xcite . the light from unresolved binaries is dominated by the more massive of the pair . as a result the less massive star is typically not detected which leads to a systematic under - counting of low mass stars . @xmath22 of main sequence m stars @xcite and 43% of main sequence g stars @xcite are primary stars in multiple star systems . these are both lower limits as some companions may have eluded detection . as roughly half of stars are in multiple systems it has a potentially large effect on the observed luminosity function . except for at the high mass end field stars in the solar neighborhood offer the best statistics for imf measurements . however the solar neighborhood imf is found to be deficient in massive stars when compared to other galaxies . for example the @xcite and @xcite solar neighborhood imfs are rejected by integrated light approaches , i.e. @xcite , hereafter k83 , @xcite ( ktc94 ) , and @xcite . analysis of individual star clusters can be used to detect imf variations . as methods and data quality can vary between authors comparisons between individual clusters are difficult . however @xcite studied eight young open clusters with the same technique . on the extremes they measured @xmath23 for ngc 663 and @xmath24 for ngc 581 over masses from around 1 to 12 m@xmath25 . @xcite considered the case in which one could have perfect knowledge of the masses of all stars in a cluster . clusters have a finite size so even with no measurement errors uncertainty arises from sampling the underlying imf . he shows that the observed scatter in imf power law values above 1 m@xmath25 can be accounted for by sampling bias for clusters with between @xmath26 and @xmath27 members ( the @xcite clusters have memberships in this range ) . furthermore , dynamical evolution of clusters can affect measurements of the imf . this can happen by preferentially expelling lower mass stars from the cluster and by breaking up binary systems within the cluster . in total @xcite finds that for stars with @xmath28 the spread in the observed imf power - law slopes in clusters is around 1 when both binary stars and sampling bias are considered even when the underlying imfs are identical . stochastic processes can also influence the ability to detect systematic imf variations . o and b stars produce ionizing photons and cosmic rays which affect the surrounding nebula . however the probability of creating one of these massive stars is comparatively small . if one of these massive stars happens by chance to form first it may drive up the nebular temperature and depress the formation of less massive stars compared to regions without massive stars @xcite . an alternative to imf measurements based on the stellar luminosity functions of resolved stellar populations is to infer the characteristics of stellar populations from the integrated light of galaxies using spectral synthesis models . the advantage of using integrated light techniques is that many of the problems plaguing imf investigations of resolved stellar populations are avoided . stochastic effects are washed out over a whole galaxy . unresolved multiple star systems are irrelevant . the number of observable galaxies is large and their environments span a much larger range than those of milky way clusters . integrated light techniques can be applied to the high redshift universe . this creates a strong motivation to develop imf techniques and test them for galaxies in the local universe which can later be used to probe earlier stages of galaxy formation . the assumption of a universal imf has a huge influence on the interpretation of the high redshift universe . the reionization of the universe and the madau plot the global star formation rate of the universe over cosmic history are two areas where imf variations could impact the current picture of galaxy formation and evolution . on the downside conclusions are dependent on the stellar evolution models used , which are not well constrained at high masses or with horizontal branch stars at low metallicities . the biggest problem is that changes in the imf , star formation history ( sfh ) or age of a galaxy model can have similar effects in the resulting spectral energy distribution ( sed ) . any integrated light technique needs to address these degeneracies . it is also difficult to probe the imf at sub - solar masses using integrated light . on the level of galaxies the concept of a universal imf has recently become more complicated . a number of recent studies have investigated the effect of summing the imf in individual clusters over a galaxy in the presence of power law star cluster mass functions . @xcite argue that the integrated galaxial imf will appear to vary as function of galactic stellar mass even if the stellar imf is universal . however @xcite argues that the galaxy wide imf should not differ from the imf in individual clusters based on analytical arguments and monte carlo simulations . our approach can only measure the imf averaged over whole galaxies and can not address this distinction . even so systematic variations of any kind from the salpeter slope have not been measured outside of some evidence for non - standard imfs in low surface brightness galaxies ( lsb ) and galaxies experiencing powerful bursts of star formation @xcite . either way , observational evidence for systematic variations of the imf in galaxies would be highly valuable . the plan of this paper is as follows : 2 explains our method for constraining the imf . in 3 we describe the sdss data and our sample selection . 4 details our modeling scheme . in 5 we discuss our statistical techniques . 6 reports our results . in 7 we check our results against the h@xmath29 distribution of the data and 8 presents our conclusions . this paper revisits the `` classic '' method of k83 ( and the subsequent extension ktc94 ) to constrain the imf of integrated stellar populations . the method takes advantage of the sensitivity of the h@xmath0 equivalent width ( ew ) to the imf . k83 showed that model imf tracks can be differentiated in the @xmath30 plane . the total flux of a galaxy at 6565 is the combination of the underlying continuum flux plus the flux contained in the h@xmath0 emission line . the h@xmath0 flux and the continuum flux have different physical origins , both of which can be used to gain physical insights into galaxies . in the absence of agn activity the h@xmath0 flux is predominantly caused by massive o and b stars which emit ionizing photons in the ultraviolet . o and b stars are young and found in the regions of neutral hydrogen in which they formed . in case b recombination , where it is assumed that these clouds are optically thick , any emitted lyman photons are immediately reabsorbed . after several scattering events the lyman photons are converted into lower series photons ( including h@xmath0 ) and two photon emission in the @xmath31 continuum . these photons experience smaller optical depths and can escape the cloud . the transition probabilities are weakly dependent on electron density and temperature and can be calculated . through this process the measured h@xmath0 flux can be converted into the number of o and b stars currently burning in an integrated stellar population . however case b recombination is an idealized condition and it is possible that ionizing photons can escape the cloud without this processing , a situation known as lyman leakage . as such the h@xmath0 flux is a lower limit on the number of o and b stars present . the continuum flux of a galaxy is due to the underlying stellar population . at 6565 the continuum is dominated by red giant stars in the 0.7 - 3 m@xmath25 range while the h@xmath0 flux comes from stars more massive than 10 m@xmath25 . the ew is defined as the width in angstroms of an imaginary box with a height equal to the continuum flux level surrounding an emission or absorption line which contains an area equal to the area contained in the line . this is effectively the ratio of the strength of a emission or absorption line to the strength of the continuum at the same wavelength . given the physical origins of the h@xmath0 flux and the continuum at 6565 the h@xmath0 ew is the ratio of massive o and b stars to stars around a solar mass . therefore the h@xmath0 ew is sensitive to the imf slope above around 1 m@xmath32 and can be used to probe the imf in galaxies . as mentioned in the introduction several degeneracies plague the study of the imf from the integrated light properties of galaxies . variations in the imf , age , metallicity , and sfh of galaxy models can all yield similar effects in the resulting spectra . for example , increasing the fraction of massive stars , reducing the age of a galaxy , lowering the metallicity , and a recent increase in the star formation rate will all make a galaxy bluer . metallicity effects were not discussed in either k83 or ktc94 and galaxy ages were assumed , 15 gyr for k83 and 10 gyr for ktc94 . the sfh in k83 is addressed by calculating models with exponentially decreasing sfhs for a range of e - folding times , as well as a constant and a linearly increasing sfh . in the @xmath30 plane the effect of varying the sfh e - folding time is orthogonal to imf variations . however this is only true for smoothly varying exponential and linear sfhs . discontinuities , either increases ( bursts ) or decreases ( gasps ) , in the star formation rate can affect the h@xmath0 ew relative to the color in ways similar to a change in the imf . along with the exponential sfhs ktc94 also uses models with instantaneous bursts on top of constant sfhs . however this was done to access high ews at an age of 10 gyr rather than to fully flesh out the effects of sfh discontinuities . the assumption of smoothly varying sfhs is a key assumption in our analysis . most late - type galaxies are thought to form stars at a fairly steady rate over much of recent time although bursts of star formation may play a significant role in low mass galaxies @xcite . for these galaxies smoothly varying sfhs are justified . however there are other galaxies clearly in the midst of a strong burst of star formation ( e.g. m82 ) and dwarf galaxies with complex sfhs , e.g. ngc 1569 @xcite , for which this assumption is a poor one . the effects of violations of our sfh assumptions are described in detail in the results section . k83 cites four major sources of error all of which are improved upon or eliminated by the high , uniform quality of sdss spectroscopic and photometric data . the h@xmath0 fluxes in his sample can be contaminated by nonthermal nuclear emission . in the updated investigation , ktc94 , this problem is addressed by removing objects with known seyfert or liner activity and luminous agn . the sdss spectra allow measurements of emission line ratios which can be used to separate star forming galaxies from agn @xcite . the second problem is that the h@xmath0 emission flux will be underestimated if the underlying stellar absorption of h@xmath0 is not taken into consideration . the narrow band filter photometry of @xcite could not measure this effect for individual galaxies so a fixed ratio was assumed for all galaxies . the sdss spectroscopic pipeline does not take this into account either . we use the h@xmath0 fluxes measured from the sdss spectra by @xcite which fit the continua with stellar population models to more accurately measure the h@xmath0 emission . while the sdss pipeline method is sufficient for strong emission lines the h@xmath0 absorption ew can be as large as 5 which is significant for weaker emission lines . thirdly , their narrow band h@xmath0 imaging includes [ ] emission which is corrected for by assuming a constant [ ] /h@xmath0 ratio . the h@xmath0 and [ ] emission lines are resolved in the sdss spectra so there is no need for a correction . this is a significant improvement . @xcite found from a literature survey that the mean value of the h@xmath0/(h@xmath0 + [ ] ) ratio is @xmath33 for spiral galaxies and @xmath34 in irregular galaxies . these mean corrections were applied uniformly to the k83 data . in ktc94 a uniform correction was applied using [ ] /h@xmath0 = 0.5 . for comparison in our sample the mean value of the h@xmath0/(h@xmath0 + [ ] ) ratio is a strikingly similar 0.752 and the mean [ ] /h@xmath0 ratio is 0.340 . however [ ] /h@xmath0 ranges from 0.0 to 0.6 . if our h@xmath0 and [ ] lines were blended , applying a fixed correction would introduce errors of as much as 25% in the h@xmath0 ews of individual galaxies . lastly , in both k83 and ktc94 extinction corrections were addressed by plotting data alongside models which were either assumed an average value for the extinction or were unextincted . the balmer decrement ( h@xmath0/h@xmath35 ) can be measured from sdss spectra which allows for extinction corrections for individual galaxies . another major advantage of this study is that the sample size is much larger than that of k83 and ktc94 . the ktc94 sample contains 210 galaxies , whereas ours has @xmath36 . the large sample size allows us to investigate imf trends as functions of galaxy luminosity , redshift and aperture fraction with subsamples larger than the entire ktc94 sample . there is a key disadvantage to this method as well . k83 and ktc94 were able to adjust the sizes of their photometric apertures to contain the entire disk of individual galaxies to a limiting isophote of 25@xmath37 given by the rc2 catalog @xcite . the advantage of narrow band measurements of h@xmath0 ew is that they can cover a much larger aperture and match the broadband measurements set to match the physical size of individual galaxies . the sdss has fixed 3 `` spectroscopic apertures . this problem is partly offset by using matching 3 '' photometry apertures from the sdss . however this introduces aperture effects as observed galaxies have a wide range of angular sizes due to the range of physical sizes and distances present in the local universe . this is significant as radial metallicity gradients ( e.g. @xcite ) are observed in spiral galaxies which could incorrectly be interpreted as radial imf gradients . even so , this method can constrain the imf within the sdss apertures . for our program galaxies 23% of the total light falls in the sdss apertures . the fact that the more distant galaxies are more luminous and tend to be larger helps to balance out the larger physical scales of the fixed aperture size at greater distances . on average 17% of the light falls in the aperture for the faintest galaxies while it is 25% for the brightest bin . in spite of their limited size the sdss apertures still contain a great diversity of stellar populations which make this data set an excellent test bed for imf universality . we will present extensive tests of aperture effects below . the sample is selected from sloan digital sky survey data . the project goal of the sdss is to image a quarter of the sky in five optical bands with a dedicated 2.5 m telescope @xcite . from the imaging @xmath38 galaxies and @xmath39 quasars will be selected for spectroscopic followup . photometry is done in the @xmath40 filter system described by @xcite . magnitudes are on the ` arcsinh ` system @xcite which approaches the ab system with increasing brightness . spectra are taken with a multi - object fiber spectrograph with wavelength coverage from 3800 to 9200 and @xmath41 @xcite . our sample is a sub - sample of the main galaxy sample from sdss dr4 @xcite . the main galaxy sample ( mgs ) targets galaxies with @xmath42 in petrosian magnitudes @xcite . all galaxies in the mgs are strong detections so the differences between luptitudes and the ab system can be ignored . in order to avoid fiber crosstalk in the camera an upper brightness limit of @xmath43 , @xmath44 and @xmath45 is imposed . targets are selected as galaxies from the imaging by comparing their psf magnitudes to their de vaccouleur s and exponential profile magnitudes . exposure times for spectroscopy are set so that the cumulative median signal - to - noise satisfies @xmath46 at @xmath47 and @xmath48 fiber magnitudes . the time to achieve this depends on observing conditions but always involves at minimum three 15 minute exposures . due to the construction of the spectrograph fibers can not be placed closer than 55 " to each other . this may be a source of bias in the sample . cluster galaxies may be preferentially excluded from the sample . the sdss collaboration has plans to quantify this effect in the near future @xcite . lsbs are excluded from the mgs with a surface brightness cut which may also bias the sample @xcite . there is some evidence that lsbs may have imfs which differ from a universal imf . @xcite find that the comparatively high mass - to - light ratios of lsbs can be explained with an imf deficient in massive stars relative to normal galaxies . our sample begins with the fourth data release ( dr4 ) of the sdss @xcite . dr4 covers 4783 deg@xmath49 in spectroscopy for a total of 673,280 spectra , 567,486 of which are galaxy spectra . 429,748 of these have flags set indicating they are part of the mgs . dr4 also includes special spectroscopic observations of the southern stripe which were not selected by the standard algorithm but which nonetheless have the ` target_galaxy ` flag set in ` primtarget ` which usually indicates membership in the mgs . these objects are identified by comparing their spectroscopic plate number to the list of special plates in @xcite and rejected . this leaves 423,285 spectra . the first round of cuts to our sample address general data quality . while the overall quality of sdss data is high there are a handful of objects with pathological values for one or more parameters . first we require that all parameters of interest have reasonable , real values . this means that the petrosian and fiber magnitudes must be between 0 and 25 and have errors smaller than 0.5 in all five @xmath40 bands . line flux errors are capped at @xmath50 @xmath51 and equivalent widths at @xmath39 . for most of the parameters less than 1% of objects fail this loose requirement . however 6.6% of the objects fail the petrosian magnitude error requirement . this is most likely due to the photometric pipeline having a difficult time defining the petrosian radius . as such this constraint is potentially biased against lsbs or galaxies with unusual morphologies . in defense of this cut we later bin our data by luminosity and aperture fraction both of which are determined in part by the petrosian magnitudes and also by @xmath52-corrections determined from them . in addition , limiting flux errors to 50% is hardly unreasonable . altogether 391,160 galaxies pass these combined requirements , which is 92.4% of the mgs . galaxies from photometry run 1659 are removed because of a known problem with the photometry . this excludes 4,485 galaxies ( 1.1% ) from a continuous strip on the sky and should not be a source of bias . we place a further constraint on the @xmath53 band fiber magnitude requiring that the error be less than 0.15 . the @xmath53 band generally suffers from the most noise so this requirement ensures that the fiber magnitude quality is good enough to minimize the chance of erroneous @xmath52-corrections which could affect our colors . only 2,866 ( 0.7% ) mgs galaxies fail this test . combining the general data quality requirements , the run 1659 rejection and the fiber @xmath53 band error limit leaves 386,647 galaxies ( 91.3% of the mgs ) . the next round of cuts to our sample , while necessary , have clear astrophysical implications . many of the objects in the mgs have agn components . as we are interested in studying only the underlying stellar populations of these objects agn must be removed . this is done using the classical @xcite diagram comparing the logarithms of the [ @xmath54/h@xmath35 and [ @xmath55/h@xmath0 emission line ratios . we used the criterion of @xcite where objects for which @xmath56/\rm h\beta ) > \frac{0.61}{\log([\rm{n\ ii}]/h\alpha)-0.05}+1.3 \label{eq : kbpt}\ ] ] are classified as agn and rejected . following @xcite we require the s / n of the h@xmath0 , h@xmath35 , [ ] and [ ] lines to be at least 3 to properly classify a galaxy as a star forming one . 131,807 galaxies ( 34.1% of mgs objects surviving our first round of cuts ) survive this cut . the above cut automatically rejects any galaxies with weak [ ] and [ ] lines . this excludes galaxies with weak star formation . to a lesser extent metal poor galaxies which also have weak [ ] emission are rejected as well . @xcite also define a low s / n star forming class of galaxies which we identify and keep in our sample . these are the galaxies which have not already been classified as star forming or agn by strong lines and equation [ eq : kbpt ] , nor have been identified as low s / n agn by [ @xmath55/h@xmath0 @xmath57 0.6 with s / n @xmath57 3 in both lines , yet still have h@xmath0 with s / n at least 2 . 79,548 ( 20.6% ) of the sample falls into the low s / n star forming galaxy category . combining the two classes 211,355 ( 54.7% ) of the galaxies survive the agn cut . the agn cut also has a strong luminosity bias for two reasons . galaxies with agn components tend to be brighter . the bimodal distribution of galaxies in color - magnitude space @xcite also plays a role . the most luminous galaxies are predominantly red with minimal star formation and thus weak emission lines . luminous galaxies are rejected both for having agn components and for having low s / n emission lines . over 95% of galaxies fainter than @xmath58 meet this criteria , but by @xmath59 the fraction is only 38.9% . a color bias is also introduced by the agn cut . over 95% of galaxies bluer than @xmath60 pass , which drops to 24% by @xmath61 . this is mainly due to the s / n requirement for the emission lines . redder galaxies tend to have weak emission lines and are rejected . the balmer decrement is used for the extinction correction so h@xmath0 and h@xmath35 s / n are required to be at least 5 to reduce errors . 214,912 galaxies ( 55.6% ) have h@xmath35 s / n @xmath62 which is the more restrictive of the two criteria . this cut has a clear luminosity bias . roughly 85% of galaxies with @xmath63 satisfy this requirement , but this fraction decreases with increasing luminosity until only 12.9% survive at @xmath64 . this is again due to the bimodal distribution of galaxies . the luminous red galaxies with weak emission lines are rejected . there is also a color bias . over 93% of the bluest galaxies blueward of @xmath65 survive the cut while only 23% of the reddest pass this requirement . as previously mentioned , by nature the reddest galaxies have weak balmer lines as a result of their low sfrs and are preferentially rejected . a redshift cut of @xmath66 is applied to ensure that peculiar velocities do not dominate at low redshift and to limit the range of galaxy ages . 384,349 galaxies ( 99.4% ) meet this criteria . nearly all galaxies from @xmath67 to @xmath68 survive this cut . on the low luminosity end only 27.4% of @xmath69 galaxies are distant enough to pass and on the high end 93.1% of @xmath70 galaxies are close enough to survive . only 63% of the bluest galaxies pass . many of the blue galaxies that fail are actually regions of local group galaxies which are treated as their own objects by the sdss pipeline so removing them actually improves the integrity of our sample . the stellar populations of galactic bulges can be significantly different from those in the spiral arms . the sdss fibers have a fixed aperture of 3 so over the large range of luminosities and distances in the mgs aperture affects can become very important . to remove outliers we require that at least 10% of the light from a galaxy falls within the spectroscopic aperture . this is done by comparing the petrosian magnitude to a fixed 3 aperture fiber magnitude . both of these quantities are calculated for all objects in the sdss by the photometric pipeline . 371,777 ( 96.2% ) galaxies survive this cut . this cut rejects proportionally more faint galaxies ; 99.2% pass at @xmath70 compared to 50.9% at @xmath69 . the aperture cut has low sensitivity to color . the intersection of the agn , balmer line s / n , redshift and aperture fraction cuts leaves 140,598 galaxies 36.4% of the high quality mgs data defined by our first round of cuts and 33.2% of the mgs as a whole . at this point three final cuts are applied to the sample . one galaxy is removed for surviving all criteria , but having a negative h@xmath0 ew . the extinction of individual galaxies is estimated using the balmer decrement ( h@xmath0/h@xmath35 emission flux ratio ) for each galaxy . given case b recombination , a gas temperature of 10,000k and density of 100 @xmath71 the balmer decrement is predicted to be 2.86 @xcite . this ratio is weakly dependent on nebular temperature and density . @xcite lists values down to 2.74 for case b recombination in environments where both the temperature and electron density are high . 3.2% of the galaxies suffer from the problem that the balmer decrement is less than 2.86 and 2.1% have a balmer decrement below 2.74 . 538 galaxies ( 0.4% ) have balmer decrements more than 3@xmath72 below 2.74 , which is around 6 times more than expected . this is does not suggest a problem with the case b assumption as the predicted balmer decrements for case a recombination are nearly identical in each temperature regime . to understand the reason behind this problem around 100 of the offending spectra were inspected revealing a few different causes for the problem . around 80 galaxies are at redshifts where the telluric @xmath73 line affects the measurement of h@xmath35 . many of these galaxies have very strong emission lines with extremely weak stellar absorption . using the sdss pipeline values instead of the @xcite values yields acceptable balmer decrements . the rest are galaxies with low flux where the balmer lines are in absorption . this shows there are a few cases where attempting to fit the underlying stellar absorption lines fails and this failure is not reflected in the error values . these galaxies are rejected without any apparent introduction of bias . all of these cuts combined leave 140,060 galaxies , which is 33% of the mgs and 36% of the high quality mgs data . after removing duplicate observations there are 130,602 galaxies in our sample . of these objects 1.7% overlap with the luminous red galaxy sample . overall the bulk of the galaxies are removed by agn rejection and the h@xmath35 s / n requirement , with the rest of the cuts having little effect . both of these cuts are necessary . agn must be removed to ensure that the h@xmath0 emission represents the underlying stellar population and not an accretion disk . our method requires that the galaxies have measurable balmer emission lines . this coupled with the need for accurate extinction corrections justifies the balmer line s / n requirement . our cuts bias our sample by preferentially excluding galaxies at both luminosity and both color extremes . the faintest galaxies are most affected by the redshift and aperture cuts while the luminous galaxies succumb to the h@xmath35 s / n requirement . at the red extremes it is the h@xmath35 s / n and agn requirements that play equally large roles , while the bluest objects are primarily rejected by the hubble flow redshift requirement . although our sample is biased by our cuts each one is a necessary evil . we do not attempt to correct this bias , but we remind the reader that the following results are only representative of actively star forming galaxies without any agn activity . the aim of this paper , however , is to test imf _ universality_. if the imf is truly universal it should be universal in any subsample of galaxies . the fact that our sample is slightly biased with respect to luminosity and color is not a significant barrier to achieving our goal . the sdss includes a number of different calculated magnitudes . we use the _ fiber _ magnitudes which are 3 fixed aperture magnitudes . although it was not the case in earlier versions of the photometric pipeline , fiber magnitudes are now seeing corrected @xcite . the fiber magnitudes were not originally intended for science purposes but rather to get an idea of how bright an object will appear in the spectrograph . we use the fiber magnitudes to reduce the aperture effects arising from comparing a 3 spectroscopic aperture to petrosian magnitudes . originally the sdss used `` smear '' exposures to correct spectra for light falling outside the 3 aperture due seeing , guiding errors and atmospheric refraction @xcite . the smear technique was later found to be an improvement only for high s / n point sources and its use was discontinued @xcite . the spectra here are not seeing corrected . after paring the sample to its final size a number of corrections must be made to both the photometric and spectroscopic data . galactic reddening from the milky way must be corrected for . sdss database includes the @xcite dust map values for each photometry object . the extinction of individual galaxies is estimated using the balmer decrement for each galaxy . the data is corrected assuming that 2.86 is the true value of the balmer decrement using the milky way dust models of @xcite . the assumption of milky way dust is not significant as models of the dust attenuation in the milky way , smc and lmc are nearly identical in the @xmath74 band and redward . as aforementioned a few percent of our galaxies have balmer decrements below 2.86 . our solution is to set the emission line extinction to @xmath75 magnitudes for these galaxies . massive young stars and their surrounding ionized nebulae tend to be embedded in their star forming regions more so than older , lower mass stars which have had time to migrate from their birth regions . as such nebular emission lines will experience more extinction than the continuum . @xcite find the ratio of emission to continuum line extinction is @xmath76 . we assume this value to be 2.0 and correct the continuum and emission lines separately . this is the same value used by k83 and ktc94 in their extinction corrected models . we note that the spatial geometry of the dust can influence the extinction law , but this complication is beyond the scope of this paper . galaxy photometry is @xmath52-corrected to @xmath77 using version 4.1.4 of the code of @xcite . this redshift is roughly the median of the sample and is selected to minimize errors introduced by the @xmath52-corrections . the @xmath78 colors we use are the @xmath79 colors we would observe if the galaxies were all located at @xmath80 . stated explicitly the @xmath78 color is @xmath81 where @xmath82 and @xmath83 are @xmath52-corrections , @xmath84 and @xmath85 are milky way reddening values , and 1.153 and 0.834 relate the v band extinction to the @xmath74 and @xmath86 bands assuming a milky way dust model . the corrected equivalent width is obtained as follows @xmath87^{-1 } \label{eq : width}\ ] ] where @xmath88 is the measured , uncorrected equivalent width . the @xmath89 arises from the fact that the total flux in the h@xmath0 line is not affected by cosmological expansion of the universe but the flux per unit wavelength of the continuum is depressed by a factor of @xmath89 . the following term is the extinction correction . the 0.775 relates the v band extinction to the h@xmath0 line assuming a milky way dust model and the @xmath90 is due to the fact that the emission line and continuum experience different amounts of extinction as previously explained . figure [ fig : grha_data ] shows the distribution of the galaxies in the color vs. h@xmath0 ew plane . in order to conduct a likelihood analysis we need error estimates which take into account both the errors induced by the photometry and spectroscopy and those by the aforementioned corrections . the error in the corrected color , @xmath91 , is given by @xmath92 where @xmath93 and @xmath94 are the poisson errors in the observed @xmath74 and @xmath86 band photometry , @xmath95 is the ratio of the emission line to continuum extinction , and @xmath96 is the error introduced by the @xmath52-corrections . the terms inside the square root in equation [ eq : sigmac ] are obtained through a straight error propagation of equation [ eq : color ] . the 0.03 outside the square root is due to the systematic zero point errors of the sdss filter system . following @xcite @xmath95 is fixed at 2.0 and @xmath97 is set to 0.4 . the error in emission line @xmath98 is given by @xmath99 which is dependent on the fractional uncertainty of the h@xmath0 and h@xmath35 emission line fluxes . @xcite reports a 16% error in their milky way reddening values . because a dust model is assumed reddening in the @xmath74 and @xmath86 bands are linearly related , so the error in @xmath100 is a function of @xmath84 . this relationship combined with the 16% error yields the 0.0440 in equation [ eq : sigmac ] . the median value of @xmath74 band reddening is 0.10 so the errors introduced by the mw reddening correction are insignificant for the majority of objects . errors in the redshift determination are negligible , typically 0.01% . the value of @xmath96 is estimated to be 0.02 by visual inspection of a plot of @xmath101 as a function of redshift . for a typical galaxy the term involving @xmath97 is the largest contributor to the extinction corrected color error . the median poisson error from the photometry is 0.01 in both bands . the median value of @xmath91 is 0.085 . the error in the corrected h@xmath0 equivalent width , @xmath102 , is given by @xmath103 \label{eq : sigmaw}\ ] ] equation [ eq : sigmaw ] is the result of propagating the errors in equation [ eq : width ] , neglecting the insignificant redshift errors . again , the term involving @xmath97 is the largest contributor to the error for typical galaxies . the median error in the equivalent width is 17% . the median error bars for the sample are shown in figure [ fig : grha_data ] . for comparison , k83 reports equivalent width errors of around 10% , but the extinction is uncertain at the 20 - 30% level . model galaxy spectra were calculated using the publicly available pegase.2 spectral synthesis code @xcite . models are calculated for ages from 1 myr to 13 gyr . 25 smoothly varying sfhs generated by analytic formulae are considered . the sfhs range from 19 exponentially decaying sfhs with time constants from 1.1 to 35 gyr , a constant sfr , and four increasing sfhs which are proportional to @xmath104 where @xmath105 is the time constant . the precise values of the time constants were selected to smoothly sample the h@xmath0 ew vs. @xmath78 plane . the metallicity of the stars is assumed to be constant with respect to time and calculated for @xmath106 = 0.005 , 0.010 , 0.020 and 0.025 . galactic winds , galactic infall and dust extinction are turned off . the dust extinction is not modeled because we have applied an extinction correction to the data . nebular emission is calculated from the strength of the lyman continuum . emission line ratios are fixed . the model spectra are redshifted to @xmath80 to match the redshift range of the data . the model parameter of interest is the imf . implicit in equation [ eq : imfbaldry ] is the assumption that the imf does not vary as a function of time . the continua of galaxies are weakly influenced by low mass stars in the optical . this method is sensitive to the imf for masses above around 1 m@xmath25 so the slope is fixed below 0.5 m@xmath25 . above 0.5 m@xmath25 models are calculated for @xmath107 , where @xmath108 is incremented by 0.05 between models . we treat our imf model as though it has only one degree of freedom @xmath108 above 0.5 m@xmath25 . in truth it has three more as written : the lower and upper mass cutoffs and the point at which the slope changes . so how well are our assumptions justified ? we have parameterized the imf as a piecewise power law with two components . piecewise power laws are motivated by empirical fits to data starting with @xcite , which had only one component . by contrast the power law formulation of the @xcite imf has 24 components . the log - normal distribution is a more physical choice as it can arise from stochastic processes . @xcite were the first to fit an observational measurement of the imf with a log - normal distribution . the log - normal distribution is normalizable as it goes to zero smoothly at both extremes without any awkward truncation . its main drawback is that it can not fit any structure in the imf over small mass ranges . log - normal distributions have three degrees of freedom . this is less than the four our model has . however , we are not sensitive to imf over the full range of masses which makes it much more difficult to fit the parameters of the log - normal distribution . instead we use this piecewise model and lower the degrees of freedom through physical arguments . many investigators find a change in slope in the imf around 0.5 m@xmath25 . our fixed lower end of the imf is designed to be consistent to this . as our technique is not sensitive to this regime this assumption does not impact the results . the imf must be normalizable because the total mass of a stellar population is finite . in our parameterization this is achieved by truncation at 0.1 and 120 m@xmath25 . this seems unphysical as the existence of brown dwarfs suggests that the imf should continue below the hydrogen burning limit . however , stars at 0.1 m@xmath25 do not contribute much to the integrated light of galaxies . as we are not sensitive to stars in this mass range this choice is not unreasonable . in fact , truncating the imf at 0.5 m@xmath25 yields models which are at worse differ by 0.002 in @xmath78 and 3% in h@xmath0 ew from those with low mass stars . the slope below 0.5 m@xmath25 has essentially no effect on our results . only when the imf is truncated at 0.9 m@xmath25 do the models differ at the level of the errors in the data . on the high mass end the choice of limit does matter . there is a physical upper limit to the size of stars associated with the eddington limit . the value of this theoretical limit is not widely agreed upon . the largest stellar mass measured reliably , via analysis of a binary system , is @xmath109 m@xmath25 @xcite . @xcite argue that given the large mass and youth of the star forming cluster r136 in the large magellanic cloud stars in excess of 750 m@xmath25 should be present given a salpeter imf with no upper mass limit to stars , whereas no stars above 150 m@xmath25 are observed . an analysis of the arches cluster , the youngest observable cluster , gives an upper limit of 150 m@xmath25 based on monte carlo simulations although stars above 130 m@xmath25 are not detected @xcite . the pegase model tracks only extend up to 120 m@xmath25 so this is the cutoff used . another issue is that the physics and evolution of such high mass stars is not well known so the models themselves may be a significant source of error in this regime . the left half of figure [ fig : oddimf ] shows the effects of varying the high mass cutoff in the imf . the effect in the @xmath78-h@xmath0 ew plane is seen to be very similar to increasing the value of @xmath108 . lowering the upper mass cutoff from 120 to 90 m@xmath25 has roughly the same effect as increasing @xmath108 from 1.35 to 1.45 . the relationship between the change in the upper mass cutoff ( from 120 m@xmath25 ) and the apparent change in @xmath108 is @xmath110 and is roughly linear for upper mass cutoffs down to 50 m@xmath25 . the coefficient in the relationship is a week function of the age of the population ranging from 0.004 for 13 gyr old populations to 0.006 for 100 myr old populations . the right half of figure [ fig : oddimf ] shows the affect of adding a second break in the imf at 10 m@xmath25 . reducing the value of @xmath108 over the 0.5 - 10 m@xmath25 range while keeping it fixed at @xmath111 above 10 m@xmath25 has a similar effect to decreasing @xmath108 in a two component model . this illustrates one of the limitations of this model . at this point it is not possible to detect fine structure in the imf slope or to state precise values for the imf slope . in this limited space of observables the imf models themselves are degenerate . what it does provide is a framework with which to detect variations in the imf . although we can construct similar tracks from different imf models , we can still detect the differences between two groups of galaxies . while we will report our results as a function of @xmath108 it must always be kept in mind that it is degenerate with the upper mass cutoff and other fine structure in the imf at high stellar masses . the assumption of smoothly varying sfhs is of great consequence . in the event that a galaxy is experiencing or has recently experienced a burst our sfh assumption can lead to measured @xmath108 values that are off by as much as 0.5 . the effects of bursts are more closely examined in a later section . within the assumption of smoothly varying sfhs much can be said about the effects of the imf , metallicity , age , and sfh in the color - h@xmath0 ew plane . figure [ fig : models ] demonstrates these relationships . in both panels the ages of the models decrease from the upper left to lower right . the effects of the age of a stellar population are largely orthogonal to those of imf variations . in figure [ fig : models]a the effects of changing the functional form of the smoothly varying sfh with fixed metallicity and @xmath108 are shown . sfh variation is degenerate with the imf . however the effect is relatively small over a wide range of sfhs . the solid lines are exponentially decreasing sfhs with @xmath112 gyr where the bulk of the star formation occurs early in the galaxy s life . the dashed lines have sfhs that are increasing with time where most star formation occurs at late ages . the effect of variations in the form of smooth sfhs is larger at later ages and higher values of @xmath108 but does not dominate the effects of imf variations . with all other parameters fixed the range of smooth sfhs cause systematic uncertainties at the level of @xmath4 in @xmath108 . in figure [ fig : models]b the effects of metallicity variations with fixed sfh and @xmath108 are shown . the metallicity variations are also degenerate with imf variations . with all other parameters fixed between colors of @xmath113 the systematic uncertainty due to metallicity is less than 0.05 in @xmath108 . this uncertainty increases to 0.35 at @xmath114 and 0.7 . as aforementioned , the extinction correction is another potential problem . the arrows in figure [ fig : models ] show the length and direction of the extinction correction for typical galaxies in our sample . it is assumed that @xmath115 , but @xmath116 and @xmath117 are also plotted to show the potential effect of variations in @xmath95 . the reddening vectors for @xmath115 and 4 are fortuitously orthogonal to the imf variations . only when the continuum and emission extinctions are equal , when @xmath116 , do the extinction correction and variations in @xmath95 become a larger concern than metallicity and sfh . however such low @xmath95 ratios are not observed in galaxies ( @xcite and section [ sec : fratio ] ) . figure [ fig : all135 ] shows all 18,480 model points with @xmath111 . models are interpolated in sfh history for fuller coverage of the color - h@xmath0 ew plane . for each value of @xmath108 the models cover a stripe rather than a single line . it can be seen in the lower right of figure [ fig : all135 ] that the model become degenerate in @xmath108 for old , red galaxies with weak current star formation . the data and models are compared using a pseudo-@xmath118 " minimization . for various reasons ( detailed below ) the classical @xmath118 estimator assumptions are violated so we can not use traditional tables for error estimates but we can still use the @xmath118 as a statistical estimator as long as the confidence regions are calibrated by monte carlo ( mc ) techniques as we will do . we proceed as follows : for each galaxy @xmath119 we have a measured @xmath78 color , @xmath120 , and an h@xmath0 ew , @xmath121 , and measurement errors @xmath122 and @xmath123 , given by equations [ eq : sigmac ] and [ eq : sigmaw ] . we also have model values @xmath124 and @xmath125 for a range of imf slopes @xmath108 , metallicities @xmath106 , ages @xmath126 and sfhs @xmath127 . we can then construct a @xmath118 value as @xmath128 which is calculated by brute force . the goal of this paper however is to investigate the imf with relatively simple measurements of the h@xmath0 ew and a broadband color . while making crude measurements of the mean stellar metallicities of individual galaxies is possible , disentangling age and sfh effects on an individual basis is a daunting task . assuming that it is possible to do , it does not scale up well to the high redshift universe where observations will be of lower quality . it does not make sense to minimize @xmath118 over all galaxies for a particular set of @xmath129 because we have no a priori reason to think that all of the galaxies should have the same metallicity or sfh . in fact we expect that they would not . the solution is to marginalize @xmath118 over metallicity , age and sfh for each galaxy such that @xmath130 _ { z , t,\psi } \label{eq : chisquare2}\ ] ] this is somewhat unorthodox , because for some galaxies the data points are over - fitted , i.e. there will be a stripe in ( @xmath131 ) space corresponding to a given @xmath108 and we can get @xmath132 values very close to zero ( but not exactly because of the discrete nature of the model grid ) for galaxies within the stripe . we note we also have partial degeneracy between parameters such as age and metallicity they both shift the tracks in similar directions largely orthogonal to @xmath108 ( though not completely which is why we have a stripe in parameter space not a line ) . this makes it difficult to calculate the traditional number of degrees of freedom . " despite these limitations it is clear that galaxies inconsistent with a particular @xmath108 will _ still _ have large values of @xmath132 for example a very blue galaxy with a low equivalent width in figure [ fig : all135 ] . the complication is that the stripes for similar @xmath108 values overlap , and for red galaxies with low equivalent widths the stripes for vastly different @xmath108 values overlap . as such , the imf for an individual galaxy is only broadly constrained . measuring a precise best imf for an individual galaxy boils down to random chance and the discrete nature of the models . however , by summing @xmath133 over many galaxies the imf is narrowly constrained for the sample being summed over as long as we are careful in our confidence region analysis . because of this over - fitting and partial degeneracy we can not apply the textbook notions of the @xmath118 distribution , calculate degrees of freedom and choose @xmath134 contours for different confidence regions . further to this @xmath120 and @xmath121 are not truly independent variables . both the colors and ews are subject to the same extinction and reddening corrections which tie the errors together . for galaxies with @xmath135 the h@xmath0 line is in the observed @xmath86 band , although this only affects a relatively small number of galaxies in the sample almost all of which have @xmath136 . also the direct statistical interpretation of @xmath118 is predicated on the assumption of normal errors . equations [ eq : sigmac ] , [ eq : sigmaav ] and [ eq : sigmaw ] reveal that our errors are complicated mixtures of individual measurements which are most likely poisson distributed . thus @xmath91 and @xmath102 are unlikely to be normally distributed . bursty sfhs can potentially create outliers which are statistically significant due to the fact that neither @xmath91 or @xmath102 include a term for this difficult to quantify effect . the problem is even worse if the errors are non - symmetric which could potentially arise from the aforementioned bursty sfhs . in the case of non - symmetric errors the best value of @xmath108 could erroneously be pulled away from the true value . given all this we abandon the direct statistical interpretation of @xmath118 and regard it as an estimator of the goodness of fit whose confidence regions have to be calibrated empirically . we do this via mc simulations ( as recommended by @xcite ) where we simulate data points for a given @xmath108 with the correct error distributions and propagate everything through the analysis in the same way as for the actual data . for each of our mc simulations we add poisson errors to the @xmath40 fiber magnitudes , h@xmath0 and h@xmath35 fluxes and the observed h@xmath0 ew . we assume that these observed quantities have poisson dominated errors as members of the mgs they are high s / n measurements . the entire analysis described above is repeated , including a new extinction measurement and a recalculation of the @xmath52-corrections . for each value of @xmath108 we run 100 mc simulations to estimate the 95% confidence interval . setting up the mc architecture in this way has the further advantage that we can use the same machinery to test the effect of systematic errors such as the violation of our smooth sfh assumptions , as we will do later . the main downside of course is that this approach is computationally intense . run times for the 100 mc simulations are typically 18 days on a 2 ghz desktop pc for the samples considered here . in practice it turns out that @xmath137 is still a smooth well - behaved function with , not surprisingly , a quadratic minimum which has the advantage that we can then interpolate it to increase the resolution in the best - fitting @xmath108 without incurring the additional computational expense . this arises of course from the fact that our estimator is similar to a traditional @xmath118 and is a good reason to stick with this similarity over some more exotic goodness of fit measure . an estimate of the systematic errors is discussed later . regardless of how poorly a sample is modeled by a universal imf the above method will still find a best fitting @xmath108 and corresponding confidence region . we still expect @xmath118 to be small for a model that is a good fit and large for one that is not . one nuance in comparing @xmath118 between different sub - samples , as we will do , is that the samples are often of considerably different sizes . because of this we choose instead to use the _ mean _ @xmath118 , @xmath138 instead , as a sample metric this has the advantage that absolute @xmath138 values and confidence regions are more similar between the sub - samples , though we note that the confidence regions on @xmath138 are still determined directly from our mc simulations . figure [ fig : all - like ] shows the results of our analysis for the full sample of galaxies using just the observed data set . the `` x '' marks the best fitting imf , where @xmath139 and @xmath140 with @xmath141 . at @xmath142 @xmath143 while steeper imfs are more heavily rejected with @xmath144 at @xmath145 . for comparison several `` classic '' imfs are also plotted in figure [ fig : all - like ] at their approximate equivalent values of @xmath108 . with @xmath146 the @xcite solar neighborhood imf is a particularly bad fit . two more recent solar neighborhood imfs , @xcite and @xcite , yield @xmath147 and 1.98 respectively . these results reinforce the conclusions of k83 , ktc94 and @xcite that the solar neighborhood is not representative of galaxies on the whole as far as the imf is concerned . on the other hand the @xcite imf , established from a review of star cluster imf studies in the literature , is a better fit than the best @xmath108 value in our parameterization with @xmath148 . this result highlights the degeneracy of the imf models themselves in the color - h@xmath0 ew plane two considerably different imfs ( one with one break and the other with two ) fit nearly equally as well . the results of the mc simulation show that the 95% confidence region is @xmath149 for the data set as a whole . the mc simulation shows that additional data will not improve the overall results as the random errors are already small . clearly and not surprisingly systematic errors , which are discussed later , dominate . the overall result of @xmath150 is steeper than the original salpeter value of @xmath111 . it is also steeper than the @xcite value of @xmath151 derived from galaxy luminosity densities in the uv to nir . it is however well within their 95% confidence limit of @xmath152 as well as their measurement of @xmath153 based on the h@xmath0 luminosity density . the difference between their two results suggests that the h@xmath0 and mid - uv to optical fluxes may have different sensitivities to massive stars . @xcite estimated the uncertainty in @xmath108 , either due to measurement uncertainties , real imf variations or both , in his star cluster imf based on the spread of results in the literature . our result is well within his range of uncertainty in both mass regimes @xmath154 for @xmath155 and @xmath156 for @xmath157 . the luminosity of a galaxy could potentially have an effect on the imf within it . for one the ambient radiation field is likely higher in more luminous galaxies . figure [ fig : allmagr ] shows the best fitting imf and @xmath138 values as a function of @xmath158 for all 130,602 galaxies . the galaxies have been binned in @xmath158 such that there are 500 objects in each bin . the bin size was chosen to maximize coverage in @xmath158 yet still keep the random errors in each bin small . the solid lines represent the lower and upper 95% confidence region determined from the mc simulation . figure [ fig : allmagr ] reveals a constant value of @xmath7 for galaxies with @xmath158 between @xmath159 and @xmath160 with linear increases in @xmath108 for both brighter and fainter galaxies . there is also a sudden downturn in @xmath108 values for galaxies fainter than @xmath161 . given the sizes of the random errors the differences in @xmath108 between @xmath162 and @xmath160 are substantial , from 1.59 to 1.41 , and statistically significant . the agreement with the salpeter slope is the best for galaxies between @xmath159 and @xmath160 in @xmath158 . in many ways it is not surprising that previous investigators have not found this trend . the milky way is thought to have a luminosity of @xmath163 @xcite ; the @xmath164 and @xmath165 filter curves cover roughly the same wavelengths . at comparable luminosities our results are similar to salpeter . the galaxies in the k83 sample have a median @xmath166 with only 16% ( 18 objects ) fainter than @xmath167 @xcite . this is a significant bias toward more luminous galaxies where our results are in agreement with a universal imf . by contrast 30% of our sample is fainter than @xmath168 @xcite . we have a sample of 39,350 galaxies fainter than @xmath169 . the lower panel of figure [ fig : allmagr ] shows that the relative quality of the fits rapidly deteriorates as the luminosity of the galaxies decrease . for the brightest galaxies @xmath138 floats around 0.15 , while in the faintest bin it is over 6 . for comparison @xmath170 for the large magellanic cloud and @xmath171 for the small magellanic cloud @xcite this trend could indicate that a universal imf is a good fit to the most luminous galaxies , but dwarf galaxies can not be described by a universal imf , even if a different universal slope is allowed . however it could have a more mundane explanation . it could be that errors are over or underestimated as a function of luminosity . it also could arise from deviations from our assumption of smoothly varying sfhs . we can not bin our data by stellar mass without assuming an imf which is contrary to the goals of the project . we can repeat the analysis of figure [ fig : allmagr ] using @xmath172 in the place of @xmath158 . @xmath172 , being redder , is a better proxy to stellar mass . the resulting plot is nearly identical to figure [ fig : allmagr ] which shows that the relationship persists across several wavebands . figure [ fig : allmagr ] reveals a clear , statistically significant trend in @xmath108 and @xmath138 with respect to luminosity . the rest of this section focuses whether this trend is a manifestation of true imf variations or if it is the result of sample biases or poor assumptions . if the imf is truly universal and our method successfully probes the imf any subsample of galaxies that we could choose should yield the same @xmath173 value as any other in spite of any selection biases or aperture effects . figure [ fig : allmagr ] clearly shows that the preceding statement is false . in this section we set aside the possibility of imf variations and search for biases in our sample . figure [ fig : allmagr ] shows that the overall result of @xmath174 is really a weighted average . the sdss mgs is a magnitude limited sample , one defined by flux limits , with both upper and lower limits . table [ tab : lumbins ] gives the number of objects in each luminosity bin . there are 41,411 galaxies with @xmath175 but only 28 for which @xmath176 . as such the overall result is heavily biased by more luminous galaxies . malmquist bias will affect any magnitude limited sample . because brighter objects can be seen at greater distances a magnitude limited sample contains bright objects from a greater volume of space than fainter objects . the result is that the ratio objects by luminosity in a magnitude limited sample differs from the true ratio in nature ; brighter objects are over - represented . to eliminate malmquist bias volume limited bins where subsamples are complete for a range of luminosities are constructed . figure [ fig : vl - def ] details the construction of these bins . given both the upper and lower flux limits of the mgs ( @xmath177 ) only a factor of 13 in luminosity falls in the sample at any given redshift . the redshift limits of each volume limited bin are defined such that no galaxies within the magnitude limits of the bin are affected by the flux limits of the mgs . within each box in figure [ fig : vl - def ] the true ratio of galaxy luminosities is preserved and is thus free of malmquist bias . figure [ fig : vl - plot ] shows the results for volume limited magnitude bins . most error bars are smaller than the plotting symbols due to the larger number of galaxies , 329 to 29,701 as given in table [ tab : lumbins ] , in each bin . figures [ fig : allmagr ] and [ fig : vl - plot ] show the exact same trends . the largest difference in @xmath108 between the whole and volume limited samples is 0.0116 in the @xmath67 bin . the other notable difference is that the fainter galaxies have larger @xmath138 values in the volume limited case . however malmquist bias across bins is not responsible for the luminosity trends in figure [ fig : allmagr ] . another effect of magnitude limited samples is that the faintest galaxies are much closer than the most luminous ones . the mean redshifts of the volume limited magnitude bins range from @xmath178 for @xmath67 to @xmath179 for @xmath180 . this corresponds to a difference in age of around 1.8 gyr . as aforementioned , model tracks reveal that age is largely orthogonal to the imf in our parameter space , but there could be other effects tied to age and distance . in addition the imf could evolve with time . the large number of galaxies in our sample affords us the luxury of investigating the effects of luminosity and redshift simultaneously to obtain a better understanding of what role , if any , the redshift plays in our analysis . figure [ fig : redshift ] shows our fitted parameters for all 130,602 galaxies in bins that are 0.25 magnitudes wide in luminosity and 0.005 wide in redshift . the upper left panel shows the best fitting @xmath108 for each two dimensional bin . on the upper right the width of the 95% confidence region in @xmath108 for each bin is shown . at bottom left is the @xmath181 and at bottom right is the log of the number of galaxies in each bin . the white contour demarcates the region in which each bin contains at least 50 galaxies . the number 50 is arbitrary but it shows the region where poisson errors are expected to be small . the black areas are regions where there are no galaxies with the given parameters . using the plot of @xmath108 at the upper left we can look for potential redshift biases . this is complicated by the fact that at a fixed luminosity there is a limit to the range of redshifts in the sample due to the flux limits of the sample described earlier . looking at vertical slices through the plot at any fixed luminosity there is a trend towards larger values of @xmath108 with increasing redshift . however , for horizontal slices of fixed redshift the same relationship between @xmath108 and luminosity that is present for the whole sample is seen modulo a normalization factor . the right half of figure [ fig : redshift ] shows a strong relationship between the number of galaxies per bin and the width of the 95% confidence region in @xmath108 . this simply reflects the fact that larger samples are less affected by poisson errors . the lower left panel of figure [ fig : redshift ] provides an excellent example of why our metric of fit quality , @xmath138 , is so important . bins with similar numbers of galaxies and @xmath182 values can have vastly different values of @xmath138 . it is worth reminding that the contours in @xmath138 are logarithmic . at fixed luminosity the galaxies are better fit by a universal imf at higher redshift . similar to the sample as a whole the quality of fit improves with luminosity . while there does appear to be some weak trending of @xmath108 and @xmath138 with redshift , redshift effects are not driving the relationship seen between imf and luminosity as it persists at fixed redshifts . one explanation for the trend in @xmath108 with redshift is aperture effects . again , if the imf is truly universal aperture effects should not exist . the sdss spectra have a fixed aperture of 3 for all galaxies . depending on the angular extent and distance to a galaxy a different fraction of the total light of the galaxy will fall into the aperture . the problem is mitigated by the fact that the most distant galaxies are the most luminous and more likely to have a larger physical size . as the physical area contained in the aperture increases with distance , so too does the size of the galaxies being observed . however , the two effects do not exactly balance out . table [ tab : lumbins ] shows that the mean aperture fraction for the @xmath183 bin is 0.20 and increases to 0.27 at @xmath184 . on average 35% more of the most luminous galaxies fall within the aperture compared to the faintest . figure [ fig : aperture ] shows the behavior of our fitted parameters for two dimensional bins of luminosity and aperture fraction in the same manner as figure [ fig : redshift ] did for luminosity and redshift . for fixed luminosities increasing aperture fraction leads to decreasing values of @xmath108 . however at fixed aperture fraction the qualitative imf - luminosity trend remains . the @xmath138 values are a strong function of luminosity , but @xmath138 does increase with aperture fraction at each fixed luminosity . the trend with aperture fraction is the exact opposite of what would be expected in the presence of a systematic effect operating given the redshift result in figure [ fig : redshift ] . the nearest galaxies should have the smallest aperture fractions in a particular luminosity bin . the nearest galaxies in figure [ fig : redshift ] have the smallest values of @xmath108 while the smallest aperture fractions in figure [ fig : aperture ] have the largest values of @xmath108 . figure [ fig : aperture ] suggests that the measured imf is more dependent on the aperture fraction than the redshift . there are several possible physical explanations for imf trends with the aperture fraction , all of which are related to radial gradients in disk galaxies . @xcite make the theory based claim that the imf should be a function of the original local temperature of the star - forming molecular clouds . metallicity gradients are also known to exist in disk galaxies , including the milky way @xcite . @xcite measure a linear , radial light metal ( c , o , mg & si ) abundance gradient of @xmath185 in the disk of the milky way . given the increased efficiency of cooling with metal lines we would expect the most low mass stars where metallicity is the highest on average towards the center of galaxies . the trend in @xmath108 in figure [ fig : aperture ] is qualitatively consistent with this idea . if there are radial imf gradients in galaxies one would expect the fits to decrease in quality with increasing aperture fraction . a blend of imfs will not be fit as well as a universal one given our technique . this idea is consistent with the results in figure [ fig : aperture ] . however , in well - resolved stellar populations there is no evidence for a relationship between the imf and metallicity , except perhaps at masses lower than those probed by our method @xcite . if metallicity plays a role in determining the imf the effects are only being revealed as a global trend in our large sample of integrated stellar populations . for individual clusters metallicity must play a secondary role to stochastic effects . the extinction correction is another potential source of bias . it is possible that there is a second order correction that our fairly simple extinction correction fails to take into account . this could potentially lead to an erroneous imf trend with extinction correction . this affects the luminosity results because more luminous galaxies tend to be dustier , as evidenced in table [ tab : lumbins ] . the problem is further complicated by the fact that dust is thought to play an integral part in star formation so it is not unreasonable that an observed imf trend with extinction may be real . figure [ fig : extinction ] , similar to figures [ fig : redshift ] and [ fig : aperture ] , shows the results of our analysis for two dimensional bins of luminosity and the extinction correction that was measured and applied . vertical slices through the upper left panel of figure [ fig : extinction ] show that @xmath108 does depend on @xmath186 , trending towards lower @xmath108 values with increased extinction over the region where the poisson error in @xmath108 is reasonable . yet again , horizontal cuts of fixed extinction show the imf - luminosity relationship . the decreasing @xmath108 values with increasing extinction is counter - intuitive . dustier regions tend to be more metal rich . if metal cooling plays a significant role in the imf the dustiest regions should have the steepest imfs . at fixed luminosity @xmath138 increases with extinction . as aforementioned our calculated errors in color and ew ( equations [ eq : sigmac ] and [ eq : sigmaw ] ) have a functional dependence on the observed emission line extinction . in both cases it is the term proportional to @xmath187 which is on average the major contributor to the calculated error . because luminous galaxies tend to be more heavily extincted they will also be more likely to have larger errors . this is potentially problematic for our observed imf trend with luminosity . if we are unknowingly underestimating the errors for faint galaxies with low extinction the source of the poor fit qualities of these galaxies could be systematic instead of astrophysical . however the lower left panel of figure [ fig : extinction ] shows that the most extincted galaxies have the poorest fits where such a bias would suggest that they should fit the best due to the large accommodating errors . it is also possible that biases in our @xmath108 measurements could depend on two parameters simultaneously . figure [ fig : avap ] shows the measured galaxy of @xmath108 as a function of both aperture fraction and measured emission line extinction for six volume limited luminosity bins . when holding all other parameters fixed , increasing the aperture fraction leads to lower values of @xmath108 in all statistically significant areas of figure [ fig : avap ] . this is the same relationship found in the earlier section on aperture fraction . decreasing values of @xmath108 are also seen for increasing extinction when all other parameters are constant . a notable exception to this is that galaxies with large extinction and small aperture fractions favor higher @xmath108 values . most importantly when looking at a particular combination of aperture fraction and extinction the imf becomes shallower with increasing luminosity until the highest luminosities where it becomes steeper again . even in the narrowest slices of the data set the same imf - luminosity trend is seen , albeit with slightly different absolute values of @xmath108 . as discussed previously we have allowed two classes of star forming galaxies into our sample . 111,806 galaxies ( 86% ) fall in the star forming class and the other 18,796 ( 14% ) belong to the low s / n star forming class where the or lines are weak , but the h@xmath0 and h@xmath35 lines still have s / n @xmath188 . by comparing the results from these two subsamples we can investigate a possible bias of the results with respect to the level of star formation . figure [ fig : sfclass ] shows the results for both classes as a function of luminosity . as it comprises 86% of the total sample it is not surprising that the results for the star forming class are similar to those of the sample as a whole in figure [ fig : allmagr ] . the low s / n class exhibits a similar qualitative behavior to the set as a whole with a few notable differences . the @xmath108 values are offset by at least 0.08 towards larger @xmath108 . the measured imf turns toward steeper values at lower luminosities than for the sample as a whole . the @xmath138 are several times lower as well . while the galaxies in low s / n class meet the same requirement of s / n @xmath57 5 in the balmer lines as the star forming class they are biased towards noisier h@xmath0 line measurements . this corresponds to lines that are either weak ( low sfr ) or weak compared to the continuum ( low present sfr compared to the past ) both of which lead to low ews . another issue at play is that the relationship between the h@xmath0 line flux and the sfr is dependent on the imf . at a fixed metallicity and sfr increasing @xmath108 by 0.05 reduces the h@xmath0 flux by 20% . in fact the h@xmath0 flux of a galaxy with @xmath142 will be 33 times larger than a galaxy with the same sfr and @xmath145 . in the presence of real imf variations at any fixed luminosity the low s / n class will be biased towards galaxies with steeper imfs . both low sfrs and steep imfs potentially lead to low s / n h@xmath0 flux . however it is difficult to determine the level of influence of each effect . as shown in figure [ fig : grha_data ] low h@xmath0 ews lead to larger values of @xmath108 for any fixed color . as the low s / n class tends toward noisier h@xmath0 fluxes and therefore ews it is easy to see from equations [ eq : sigmac ] , [ eq : sigmaav ] and [ eq : sigmaw ] that the errors for this class will tend to be larger . this in turn leads to lower @xmath118 values . as the qualitative imf - luminosity trend occurs in both star forming classes the strength of star formation is unlikely to be a significant bias on our results . as mentioned in the data corrections section , the @xmath95 ratio is ratio of the extinction experienced by the nebular emission lines to that experienced by the stellar continuum . the assumption of a value for @xmath95 could potentially bias our results . an alternative way of looking at the same problem is that our data in the color - h@xmath0 ew plane can be used to constrain the @xmath95 ratio by assuming a universal salpeter imf . figure [ fig : f - all ] gives the results of this analysis for the data set as a whole . a value of @xmath189 is found with @xmath190 . this is in good agreement with to the @xcite value of @xmath191 . the quality of the fit in the best case is worse than in figure [ fig : all - like ] . part of the reason for this is that the errors used were slightly smaller as the @xmath97 terms in equations [ eq : sigmac ] and [ eq : sigmaw ] are set equal to 0 . the quality of the fit drops sharply below @xmath115 and more gradually for @xmath192 . values of @xmath95 near 1 are heavily rejected . however in this particular plot the results are dominated by luminous galaxies . the values of @xmath95 as a function of luminosity are shown in figure [ fig : f - vl ] . for galaxies @xmath193 and brighter the best value of @xmath95 is consistent with @xmath115 . the faintest two bins the prefer an @xmath95 ratio closer to 2.5 . however the lower panel shows that this new @xmath95 value does not translate to improved fit quality . in fact the faintest galaxies have in general smaller measured extinctions and are therefore less susceptible to changes in @xmath95 . the same qualitative trend of worsening fits with decreasing luminosity seen when allowing @xmath108 to float is seen with a varying @xmath95 value . together these two @xmath95 ratio plots provide a number of insights . for one it shows that our choice of the @xmath95 ratio is very sensible and provides an independent confirmation of other @xmath95 ratio measurements . the fact that our best fitting @xmath108 values are at least 0.05 above the salpeter value can not be reconciled by changing the geometry of the dust screen . it provides further evidence that the relationship between @xmath108 and luminosity is not a function of extinction or a byproduct of our extinction correction . in this section we have investigated several possible sources of bias to account for our observed trend between the imf and luminosity . relationships between the imf and redshift , aperture fraction , extinction and star formation strength have been uncovered . two parameter biases were also found . in all cases in narrow slices through the data where potential biases are held fixed the qualitative imf - luminosity relationship appears . the parameters primarily act to offset the value of @xmath108 at a particular luminosity . the ratio of continuum to emission line extinction , @xmath95 , was found to be a sensible choice and the results are not sensitive to small changes in this value . there are two possible interpretations to the relationships between the imf and potential biases . one is that they are systematic effects due to some problem with our measurement of @xmath108 . the second is that they are real physical effects . it is not clear from the data which of these statements is more correct . in the previous section several possible sources of bias were investigated , but none were able to account for our observed trend in @xmath108 with luminosity or the inability of a universal imf to fit low luminosity galaxies . figure [ fig : hagr1723 ] shows the distribution in color - h@xmath0 ew space for the least and most luminous bins , @xmath67 and @xmath68 . from this figure it is apparent that the most luminous galaxies lie roughly parallel to the imf tracks while the faintest galaxies are more perpendicular to the tracks . in the low luminosity bin there are galaxies which are simultaneously blue and have low ews . these galaxies are not consistent with a universal imf with @xmath111 and as mentioned before are not consistent with a universal imf with a different slope . in addition the faintest galaxies have the lowest extinctions they are the least sensitive to dust and @xmath95 ratio issues . before concluding that this is evidence for imf variations we must first consider whether our model assumption of smooth sfhs is justified . the sfh of individual galaxies is the most problematic aspect of the k83 analysis . a sudden burst on top of a smoothly varying background will immediately increase the h@xmath0 ew . this is due to the formation of o and b stars which indirectly increase the h@xmath0 flux through processing of their ionizing photons . the new presence of o and b stars also makes the color of the galaxy bluer . both of these effects are proportional to the size of the burst . after the burst is over the h@xmath0 ew is smaller and the colors are redder than they would be if the burst had not occurred . the h@xmath0 ew drops because there is no longer an excess of o and b stars and their ionizing photons which reduces the h@xmath0 flux to pre - burst levels . however there is now an excess of red giants due to the less massive stars from the burst leaving the main sequence . this increases the continuum around the h@xmath0 line which further drops the ew in addition to making the galaxy colors redder . after enough time has elapsed after the burst the galaxy returns to the same position in the color - h@xmath0 ew plane it would have occupied had no burst occurred , although it will have taken longer to get there . figure [ fig : burst ] gives one example of this cycle . a solar metallicity galaxy with @xmath111 and an exponentially decreasing sfh with @xmath194 gyr experiences a burst of star formation at an age of 4.113 gyr which lasts 250 myr and forms 10% of the stellar mass . the black dots , spaced at 100 myr intervals , show that comparatively more time is spent below the nominal track than above it . the peak h@xmath0 ew is reached just 5 myr after the start of the burst . if you happen to be observing the galaxy during the burst a shallower imf will be measured ( assuming a burst - free sfh ) , after the burst for 1 gyr a steeper imf will be measured and after that the effects of the burst largely disappear , although the galaxy will appear younger than it actually is . figure [ fig : burstfit ] gives the best fitting imf slope from our analysis , which assumes no bursts , as a function of age for the galaxy in figure [ fig : burst ] . the jitter in the best fit @xmath108 values is due to the discrete nature of our model grid , and the fact that the model tracks for different imfs run together at large ages . within a 300 myr period during and just after the burst the best fitting imf slope is anywhere from @xmath195 to 1.95 . this shows that even if a universal imf exists the sfh can mimic a huge range of imf models . roughly 1 gyr after the burst the measured imf is back to its true value . a galaxy with a bursty sfh viewed at a random time will be biased toward a steeper imf than what is the true imf . while one model is not an exhaustive study of the effects of the sfh on imf measurements it does give a good sense of what issues arise . to eliminate this uncertainty we investigated cutting the sample on sfh . in order to detect a relative lack or excess of present star formation it is necessary to measure both the present and past star formation rates or at least be able to compare the two in some way . the problem lies in the fact that conversions of observables into star formation rates assume an imf to do so . our aim is to measure the imf so we can not make strong a priori assumptions about it . instead of biasing or results from the start we fit all galaxies and then try to determine the affect of sfhs on our conclusions . the simplest burst model is that of a single burst at a random time on top of our smooth exponential sfhs . a grid of 1000 sfhs was constructed by first selecting one of the 24 smoothly varying sfhs at random . a burst lasting 200 myr was superposed on the sfh at a time selected uniformly at random over a range of 12.5 gyr . the strength of the burst was randomly selected up to 40% of the total stellar mass , with preference given to smaller bursts . the colors and ews of these sfhs were calculated at 1 myr intervals over 12.5 gyr for the imfs @xmath18 and 1.80 and a fixed metallicity of @xmath196 . this metallicity choice is based on the luminosity - metallicity relationship in @xcite for @xmath197 galaxies . plotting these models in the color - h@xmath0 ew plane shows that all observed data points are covered by either imf . to test whether the observed distribution of points in the @xmath162 bin can be explained by bursting sfhs the mc techniques of 5 were used . the models of 4 were replaced with the grid of single burst models . the 100 mc simulations were constructed as described earlier , but using only the 329 @xmath162 galaxies as a basis . the analysis yielded @xmath198 for @xmath111 and @xmath199 for @xmath200 , both of which are over fits . this shows than an individual galaxy can be fit with an arbitrary imf given the freedom to choose a sfh . however , our advantage is that we have _ many _ galaxies and the distribution of the properties of the best fit models can be shown to be implausible . figure [ fig : chibursts ] demonstrates the problem with the single burst model . on the left of the figure the distribution of the best fitting burst strengths are plotted as a fraction of the total stellar mass formed . at right is the distribution of the best fitting times from the burst onset . for example , a galaxy which best fits a model with a burst at 1.000 gyr at an age of 1.211 gyr has a time from burst onset of 211 myr . this measure is used because in an investigation of the effects of bursts the age relative to the burst is more important than the age given that the bursts occur at different , random times across the models . for both @xmath111 and @xmath200 the number of objects best fit by a model prior to the burst is @xmath201 , or 3% , and are not plotted . in both cases the distributions of the best fitting ages and ages at which the burst begins are roughly uniform . given this fact it is expected that half of the galaxies should be best fit by a pre - burst model . furthermore the right hand panel shows sharp , significant discontinuities in the distribution of best fitting time from burst onset . again , viewed at random times this distribution should be uniform but is highly peaked in the 25 myr at the start of the burst and the 25 myr just after the burst ends . in both cases the errors bars show that the discontinuities are significant . in the case of @xmath111 , 3.5 times as many galaxies are in the 200 myr after the burst ends than the 200 myr during it and this 400 myr accounts for 57% of all galaxies . although our sample is @xmath86-band selected the stars in the 0.7 to 3 m@xmath25 range which dominate the red continuum in the red giant phase do not start to leave the main sequence for 300 myr . the sharp increase in galaxies fit at 200 myr after the burst can not be due to a selection effect . assuming a universal imf this points to a strong coordination of sfhs across a population of galaxies unrelated in space . these arguments show that while a single burst model can fit the data extremely well , it does not do so in a physically self - consistent fashion . to find a physical motivation for sfh models for low luminosity galaxies we look towards the local group . there have been a number of recent studies of the sfhs of local dwarf galaxies which use hst to get color - magnitude diagrams ( cmd ) of resolved stellar populations . the sfh is determined by fitting isochrones to the cmd . the sample here is biased by local group membership and by what galaxies have been observed to date . the galaxies mentioned here give a point of reference rather than a well - defined distribution of sfhs . the blue compact dwarf ( bcd ) ugca 290 was found to quiescently form stars over the past gyr up until a ten - fold increase in sfr from 15 to 10 myr ago which more recently has decreased to a quarter of its peak value @xcite . the dwarf irregular ic 1613 , which is relatively isolated and non - interacting , was found to have sfr enhanced by a factor of 3 from 3 to 6 gyr ago without evidence of strong bursts @xcite . the dwarf irregular ngc 6822 , also relatively isolated , is found to have a roughly constant sfh @xcite . the bcd ngc 1705 is found to be _ gasping_- a sfh marked by moderate activity punctuated by short periods of decreased star formation @xcite . the authors also note that ngc 1705 is best fit by an imf with @xmath202 . ngc 1569 likely experienced three strong bursts in the last gyr as well as a quiescent phase from 150 to 300 myr ago @xcite . inspired by the preceding local group sfhs we constructed six sfh classes with multiple bursts . these sfh classes are described in table [ tab : sfhs ] . each class starts with an underlying smooth sfh . sfhs # 1 & 2 have no star formation , # 3 & 4 have a constant sfh and # 5 & 6 have exponentially decreasing sfhs like those previously described . star formation discontinuities are then superimposed on top of the smooth sfhs . these discontinuities are in the form of increased ( bursts ) or decreased ( gasps ) star formation for periods of 200 myr . the time and spacing of the discontinuities is random with the mean interval between bursts listed in table [ tab : sfhs ] . for each sfh class described above we randomly generated 1,000 sfhs . colors and ew widths were calculated for each sfh using @xmath111 and @xmath196 . according to the sdss mass - metallicity relationship @xcite galaxies at @xmath67 will on average have @xmath196 . we then repeated our @xmath118 analysis with the 100 mc simulations of the @xmath162 galaxies in the same manner as for the single burst models . the results of our analysis for each of the six sfh classes are shown in figure [ fig:6sfh ] . sfh 5 , the gasps on top of exponential sfhs , is the best fit with @xmath203 . extended periods of no star formation punctuated by bursts ( sfhs # 1 & 2 ) do not fit the data . as was the case for the single burst models an unreasonable fraction of galaxies are best fit by sfhs in the 20 myr immediately following a burst or the first 20 myr of a gasp . if the sfh models are reasonable we should see roughly equal numbers of galaxies in each time bin . there is no reason why all of the low luminosity galaxies across the large volume of space in the sdss footprint should have experienced coordinated bursts . however each panel of figure [ fig:6sfh ] has at least 40% of the galaxies in one 20 myr bin . one explanation for this is that it is an artifact of our sample being selected in the @xmath86 band . however spectral synthesis models show that for instantaneous bursts of star formation the @xmath165 magnitude is brightest at the burst time and decays smoothly for a range of @xmath108 . if anything it is more likely to catch galaxies during a burst rather than after or after a gasp instead of during one . regardless of the sfh model the presence of blue galaxies with low h@xmath0 ew requires a recent discontinuity in the sfr for @xmath111 . based on the evolution of the @xmath165 band luminosity we expect to see a similar number of galaxies with excess h@xmath0 ews . the fact that these galaxies are missing shows that the discontinuous sfh models do not match our observations . therefore imf variations are a more likely explanation for the observed distribution of @xmath67 galaxies . as a last exercise the best fitting sfh models from the previous section can be run forward to see if the correct imf can be recovered . for each sfh model grids 10,000 data points were chosen by selecting a random sfh and a _ uniformly distributed _ age . normal errors were added using the error characteristics of the @xmath67 bin . this synthetic data was analyzed in the same fashion as the real data in the earlier sections . for the single burst models the recovered imf models for @xmath204 and 1.80 were 1.34 and 1.79 respectively with best fitting @xmath205 0.80 and 0.65 . for sfh 5 the recovered imf was also @xmath206 with @xmath207 . in all three cases the correct imfs were recovered although the fit was worsened by by the burst activity . this reinforces the difficulty in producing enough blue , low h@xmath0 ew galaxies to match the observed data with simple sfh models . in the previous sections we have expanded on the k83 method and exploited the h@xmath0 and color information as much as possible . in the bias section we found that various possible biases do not fully explain either the increased values of @xmath108 or the poor fit to a single imf in the lowest luminosity bins . in the sfh section we found that an arbitrary @xmath108 value coupled with a plausible sfh with bursts or gasps can account for the position of any individual galaxy in the color - h@xmath0 ew plane . however taking the population of @xmath67 galaxies together necessitates an incredibly unlikely coordination of sfhs across the disparate group of objects . this points to the extraordinary conclusion that while the imf may be universal across luminous galaxies , it is not in fact universal in low luminosity galaxies . such an extraordinary claim would ideally be backed by extraordinary evidence . in this section we take a look beyond the k83 method for some reinforcement of our result . the h@xmath208 absorption feature can be used to gain additional insight into the nature of stellar populations . absorption is due to absorption lines form stellar photospheres . the balmer absorption lines are most prominent in a stars and weaken due to the saha equation for both hotter and cooler stars . as such the h@xmath208 absorption is a proxy for the fraction of light of a stellar population being supplied by a stars , and to a lesser extent b and f type stars . in a stellar population of a uniform age the h@xmath208 absorption will peak after the o and b stars burn out , but before the a stars leave the main sequence . for this reason the strength of h@xmath208 can be used help determine the age of a population or to detect bursts of star formation which occurred around 1 gyr in the past . @xcite describes two different methods for measuring h@xmath208 absorption . the h@xmath209 definition is tuned to most accurately measure the h@xmath208 absorption from f stars . the h@xmath29 definition has a wider central bandpass to match the line profiles of a stars . they state that the h@xmath29 definition is better to use for galaxies because it is less noisy in low s / n galaxies and velocity dispersion acts to widen absorption features . on the downside the narrower h@xmath209 definition is much more sensitive to population age where it can be used . @xcite observationally determines the range of h@xmath29 values to be from 13 for a4 dwarf stars to -9 for m - type giant stars . h@xmath29 values can be measured from the sdss spectra . like the h@xmath0 ew values our h@xmath29 values come from @xcite instead of the sdss pipeline . figure [ fig : hdfig ] compares the distribution of the h@xmath29 values for the @xmath67 and @xmath180 luminosity bins . for reference recall that figure [ fig : hagr1723 ] plots the color and h@xmath0 ews for these two bins . the difference between the high and low luminosity bins is clear . gaussian profiles can be fit to both distributions . for the @xmath180 bin the standard deviation of the profile is twice as large as the measurement error in h@xmath29 suggesting that the true distribution has a range of values . assuming both the errors and underlying distribution are gaussian the distribution of h@xmath29 for the luminous galaxies is centered at h@xmath210 with a standard deviation of 1.5 . by contrast , the gaussian fit to the @xmath67 bin has the same standard deviation as the median error in the galaxies . this is consistent with nearly all of the galaxies having h@xmath211 . this shows that the fainter galaxies on average have significantly larger fractions of a - type stars amongst their stellar populations . it also shows a seemingly unlikely coordination of h@xmath29 in the low luminosity galaxies , reminiscent of the earlier suggestion of coordinated sfhs . the question then becomes why ? for a possible explanation we look again to the models . modeling the behavior of h@xmath29 requires an extra step . the standard pegase.2 models do not have the required resolution to accurately measure the h@xmath29 index . this is remedied with the use of the pegase - hr code @xcite which uses a library of echelle spectra of 1503 stars to calculate spectral synthesis models with r=10,000 over the range of 4000 to 6800 . using pegase - hr in the low resolution mode yields the same results as pegase.2 , and the same input parameters are used for both codes . the models here are the same as those described earlier in the models section , but have been recalculated using pegase - hr to allow h@xmath29 measurements . figure [ fig : hdmodplot ] shows the behavior of the h@xmath29 index as a function of age for four different imf models , @xmath142 , 1.40 , 1.70 and 2.00 . for each imf models of all metallicities and smooth sfhs are plotted for each age . the qualitative behavior of all models is the same . the h@xmath29 holds steady for the first 20 to 40 myr before increasing to a peak value at 700 myr to 1 gyr and then falls off . prior to reaching the peak value for each imf the metallicity has the strongest effect on the h@xmath29 index . at this point the lowest metallicity galaxies have the highest h@xmath29 . after the peak the sfh has the strongest effect with the constant and increasing sfhs maintaining higher h@xmath212 values . after a few gyr the differences in h@xmath29 between models with different imfs disappear . prior to that there are three main differences . first , the peak h@xmath29 values are higher for larger values of @xmath108 . for @xmath145 h@xmath29 reaches a maximum value of nearly 8 and the maximum value is similar for all metallicities . for @xmath142 the maximum value ranges from 4.5 to 6 depending on the metallicity . this is due to the fact that the steeper imfs have fewer luminous massive stars to dilute the h@xmath208 absorption features from the a star population . secondly , the low @xmath108 models have h@xmath29 values that start their initial increases at a later time . lastly the low @xmath108 models reach their peak values later in time than those with fewer massive stars . to compare the h@xmath29 values for the @xmath180 and @xmath213 bins to the models the range of the middle 90% values from figure [ fig : hdfig ] for each bin are overlaid on figure [ fig : hdmodplot ] . once again the @xmath180 bin is in good agreement with our assumption of a universal imf and smooth sfh . the range of h@xmath29 values can be accomplished with a single salpeter - like @xmath214 imf with only the proviso that most galaxies be older than a few gyr or younger than 300 myr . however the exact same statement can be made for @xmath215 so h@xmath29 provides only a constraint on the age of the most luminous galaxies , but not the imf . for the @xmath67 galaxies the distribution of h@xmath29 can not be achieved with the shallowest imfs investigated under the assumption of smooth sfhs . however salpeter and steeper imfs can be accomplished . what changes is the range of ages over which the models have the correct h@xmath29 . steeper imfs require that the galaxies be either older than a few gyr or 200 myr old to accommodate the observed h@xmath29 values . most troubling is that there does not appear to be any reason why h@xmath29 should stack up at 6.1 for the @xmath67 galaxies . h@xmath29 values can also be calculated for the same sfh class models used earlier . unfortunately this does not provide any added constraints . values remain elevated for several 100 myr after the start of a burst or the end of a gasp of sfh . the behavior of h@xmath29 in the presence of sfh discontinuities provides no need for galaxies to stack up in the narrow 20 myr intervals seen in the earlier multiple burst section , nor do these models suggest why the low luminosity galaxies are consistent with a single value of h@xmath29 . there is no satisfactory model to account for the h@xmath29 distribution of the low luminosity galaxies . however the sfh results from the previous section strongly suggest that the incredible coordination of discontinuities is highly unlikely . steeper imfs do allow for higher h@xmath29 values for longer periods of time , thus relaxing the sfh coordination requirement . this agrees with our earlier results for faint galaxies . the low luminosity galaxies are most likely the result of a mix of imfs which are on average steeper than salpeter . the goal of this paper was to revisit the k83 method for inferring the imf from integrated stellar populations and to harness the richness of the sdss data , improved spectral synthesis models and greater computational power available today to make a state - of - the - art measurement of the imf . the quality of the sdss spectroscopy allowed us to address several of the limitations of k83 and ktc94 we resolve the [ ] lines ( a significant improvement in the accuracy of the h@xmath0 ews of individual galaxies ) , eliminate contamination from agn , make extinction corrections for individual galaxies and fit underlying stellar absorption of h@xmath0 . we succeeded in achieving more accurate ews for individual galaxies . the median total ew error for our sample is 17% compared to a 10% uncertainty in ews combined with a 20 - 30% uncertainty in the extinction correction for k83 . we expanded the grid of models to allow for a range of ages and metallicities . we used @xmath118 minimization to go beyond differentiating between two or three imf models to actual fitting for the best imf slope . the vast size of the sdss sample allowed us to both drive down random errors and to cut the data into narrow parameter ranges which were still statistically viable . the size of the dr4 sample yielded @xmath216 95% confidence region due to random error for the sample as a whole . even the volume limited @xmath67 luminosity bin with only 329 objects has a random error of @xmath217 . only in bins with fewer than 10 objects do the random errors become significant . our imf fitting is therefore dominated by systematics . originally we believed that our systematics would be dominated by the effects of sfh discontinuities . however we conducted several experiments where we selected populations of galaxies from models with bursting or gasping sfhs and gave them measurement errors consistent with those in the @xmath162 luminosity bin . to our surprise our @xmath118 minimization revealed the true imfs with @xmath218 . the main effect was to reduce the quality of the fits . this is due to the fact that h@xmath0 ews return to nominal levels in a relatively short time after sfh discontinuities . another way to estimate the size of the systematic errors is to look at the trends of the @xmath219 and @xmath160 luminosity bins , because they have the largest membership , in figures [ fig : redshift ] , [ fig : aperture ] and [ fig : extinction ] . assuming that the imf is universal and that our method is perfect we should get the same answer for any subset of the data we might choose . the largest ranges are @xmath220 for redshift binning , @xmath221 by aperture and @xmath221 by extinction . conservatively then the systematic error is @xmath4 . there are two points to be kept in mind about this estimate of the systematic error . for one it is the systematic error in the exact value of @xmath108 . even in figure [ fig : avap ] where more narrow bands of measured extinction and aperture fraction are considered the same trends with luminosity are seen as with the sample as a whole . the relative systematics between luminosity bins in these narrow slices is much smaller . the second thing to remember is the way in which we empirically defined our systematic error discounts the possibility of imf variations . what we have called systematics could actually be science . if galaxies have radial imf gradients or if dust content plays a strong role in star formation the systematic error could be much smaller . the main area in which we were unable to improve upon the k83 and ktc94 studies is that they were able to match the aperture size to the galaxies which avoids the issue of aperture effects . first , for galaxies brighter than @xmath222 the best fitting imfs are salpeter - like ( @xmath7 ) . in addition the assumption of a universal imf and smoothly varying sfhs is a good fit . this is reassuring as it follows the conventional wisdom and provides confidence that the method works . secondly , galaxies fainter than @xmath222 are best fit by steeper imfs with larger fractions of low mass stars . for these galaxies a universal imf and smooth sfh is a poor assumption . this result is in qualitative agreement with evidence that lsbs have bottom - heavy imfs @xcite . it is worth mentioning the main caveat of our @xmath108 values again . as illustrated in figure [ fig : oddimf ] imf parameterizations are themselves degenerate in our parameter space . increasing the imf slope has a similar effect to lowering the highest mass stars that are formed or increasing the fraction of intermediate mass stars . this method can not explicitly determine if two populations have the same underlying imf . figure [ fig : all - like ] shows that for the sample as a whole the @xcite three part power law yields nearly the same result as our two part power law . however our method is sensitive in many cases if the imfs are different . in terms of star forming cloud temperatures the harsher ambient radiation and larger number of sources of cosmic rays present in more luminous galaxies agree qualitatively with our results . with the extra energy hitting the star forming clouds larger masses may be needed for contraction and fractionization may end sooner , suppressing the formation of less massive stars @xcite . @xcite find that while the regions of the luminous grand design spiral ngc 5457 ( m 31 ) can be reproduced by a single salpeter imf , for the low luminosity flocculent galaxy ngc 4395 a blend of two imfs is required . however , such trends are not seen in studies of well - resolved stellar populations @xcite . another explanation for the absence of massive stars is that the massive stars are there , but are not visible . extinction to the center of star forming regions , where massive stars preferentially exist , can reach @xmath223 @xcite . however the low luminosity galaxies have the lowest observed extinctions ( see table [ tab : lumbins ] ) which is the opposite of what would be expected given our imf results . it is also possible that the imf is in fact universal , but the way in which it is sampled in embedded star clusters leads to an integrated galaxial imf which varies from the true imf . @xcite use a universal imf with the assumption that stars are born in clusters where the maximum cluster mass is related to the star formation rate . for a range of models this leads to a narrow range for the apparent imf in high mass galaxies . for low mass galaxies the imf is steeper with a wider range of slopes . the results here are in qualitative agreement for some of the integrated galactic imf scenarios in @xcite given that there should be a rough correlation between galaxy luminosity and mass . once again , @xcite argues that the galaxy wide imf should be the same as the imf in clusters regardless . in light of the theory of @xcite , whether the results of this paper speak to a relationship between environment and the formation of individual stars is open to interpretation . however the impact on the modeling and interpretation of the properties of galaxies is clear . @xcite note that the integrated galaxial imf is the correct imf to use when studying global properties of galaxies . even if the imf of stars is in truth universal it may currently be misused in the modeling of galaxies . furthermore a varying integrated galaxial imf could open the door to new insights in galaxy evolution . for instance , @xcite suggest that the observed mass - metallicity relationship in galaxies naturally arises from a variable integrated galaxial imf similar to the results of this paper . future work will expand in several directions . the success constraining the imf with only two observed parameters ( albeit carefully chosen to be orthogonal to systematic errors ) motivates a more expansive analysis with more parameters . information from wavelength regimes beyond the sdss can be used . for instance the absorption strength and p - cygni profile shape of the @xmath224 line due to massive stars is sensitive to the imf slope and upper mass cutoff @xcite . however it could be contaminated by absorption from the interstellar medium and would need to be disentangled @xcite . a full markov chain monte carlo analysis could be implemented fitting to multiple spectral features and marginalizing over a range of sfhs . in addition to luminosity , surface brightness and gas phase metallicity can be tested for systematic imf variations . while the results for luminous galaxies are already dominated by systematics , the continued progress of the sdss can provide better statistics for a more detailed analysis of what physical processes are behind the imf variations in faint galaxies . e.a.h . and k.g . acknowledge generous funding from the david and lucile packard foundation . we would also like to thank c. tremonti , g. kauffmann and t. heckman for an early look at their sdss spectral line catalogs . e.a.h . would like to thank johns hopkins for funding from various sources as well as c. tremonti , e. peng and a. pope for invaluable assistance with the nuances of the sdss . several plots were created with the use of publicly available idl routines written by d. schlegel . funding for the creation and distribution of the sdss archive has been provided by the alfred p. sloan foundation , the participating institutions , the national aeronautics and space administration , the national science foundation , the u.s . department of energy , the japanese monbukagakusho , and the max planck society . the sdss web site is http://www.sdss.org/. the participating institutions are the university of chicago , fermilab , the institute for advanced study , the japan participation group , the johns hopkins university , the max - planck - institute for astronomy ( mpia ) , the max - planck - institute for astrophysics ( mpa ) , new mexico state university , princeton university , the united states naval observatory , and the university of washington . abazajian , k. , et al . 2004 , , 128 , 502 adams , f. c. , & fatuzzo , m. 1996 , 464 , 256 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crcccccccrcc -14 & - & - & - & - & - & - & - & - & 28 & 1.3835 & 7.43 + -15 & - & - & - & - & - & - & - & - & 188 & 1.3892 & 9.86 + -16 & - & - & - & - & - & - & - & - & 406 & 1.5461 & 4.64 + -17 & 329 & 1.6000 & 1.6045 & 1.6086 & 3.41 & 0.013 & 0.38 & 0.20 & 1,304 & 1.5879 & 2.57 + -18 & 1,555 & 1.5338 & 1.5370 & 1.5424 & 2.05 & 0.022 & 0.50 & 0.21 & 4,327 & 1.5326 & 1.57 + -19 & 4,935 & 1.4772 & 1.4788 & 1.4813 & 1.17 & 0.032 & 0.70 & 0.21 & 10,375 & 1.4813 & 0.96 + -20 & 12,951 & 1.4306 & 1.4320 & 1.4330 & 0.63 & 0.050 & 0.91 & 0.22 & 24,851 & 1.4436 & 0.56 + -21 & 28,633 & 1.4051 & 1.4057 & 1.4063 & 0.32 & 0.077 & 1.15 & 0.24 & 41,411 & 1.4064 & 0.31 + -22 & 29,701 & 1.4036 & 1.4042 & 1.4050 & 0.19 & 0.116 & 1.32 & 0.25 & 38,406 & 1.4084 & 0.20 + -23 & 8,049 & 1.4545 & 1.4556 & 1.4568 & 0.15 & 0.168 & 1.55 & 0.27 & 9,106 & 1.4505 & 0.16 + -24 & - & - & - & - & - & - & - & - & 192 & 1.5329 & 0.12 + all * & 130,602 & 1.4432 & 1.4437 & 1.4443 & 0.50 & 0.090 & 1.05 & 0.25 & - & - & - + [ tab : lumbins ] sfh 1 & burst & 200 & - & 3 & none + sfh 2 & burst & 200 & - & 1 & none + sfh 3 & burst & 200 & 4.0 & 3 & constant + sfh 4 & gasp & 200 & 0.0 & 1.5 & constant + sfh 5 & gasp & 200 & 0.1 & 1.5 & exponential + sfh 6 & burst & 200 & 5.0 & 1.5 & exponential + [ tab : sfhs ]

this paper revisits the classical kennicutt method for inferring the stellar initial mass function ( imf ) from the integrated light properties of galaxies . the large size , uniform high quality data set from the sloan digital sky survey dr4 is combined with more in depth modeling and quantitative statistical analysis to search for systematic imf variations as a function of galaxy luminosity . galaxy h@xmath0 equivalent widths are compared to a broadband color index to constrain the imf . this parameter space is useful for breaking degeneracies which are traditionally problematic . age and dust corrections are largely orthogonal to imf variations . in addition the effects of metallicity and smooth star formation history e - folding times are small compared to imf variations . we find that for the sample as a whole the best fitting imf slope above 0.5 @xmath1 is @xmath2 with a negligible random error of @xmath3 and a systematic error of @xmath4 . galaxies brighter than around @xmath5 ( including galaxies like the milky way which has @xmath6 ) are well fit by a universal @xmath7 imf , similar to salpeter , and smooth , exponential star formation histories ( sfh ) . fainter galaxies prefer steeper imfs and the quality of the fits reveal that for these galaxies a universal imf with smooth sfhs is actually a poor assumption . several sources of sample bias are ruled out as the cause of these luminosity dependent imf variations . analysis of bursting sfh models shows that an implausible coordination of burst times is required to fit a universal imf to the @xmath8 galaxies . this leads to the conclusions that the imf in low luminosity galaxies has fewer massive stars , either by steeper slope or lower upper mass cutoff , and is not universal .

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Proceed to summarize the following text: over the last several years , there has been an enormous amount of interest in the physics of dilute fermi gases confined in magneto - optic traps@xcite . with the possibility of tuning the atomic scattering lengths from the repulsive regime to an attractive one using the feshbach resonance technique , there has been considerable experimental activity in looking for phenomenon such as superfluidity , and other phase transitions in these systems@xcite . this has led to equally vigorous theoretical activity starting from the studies of so - called bec - bcs crossover physics@xcite , search for shell - structure in these systems@xcite , to the study of more complex phases@xcite . as far as the spin of the fermions is concerned , most attention has been given to the cases of two - component gases which can be mapped to a system of spin-@xmath0 atoms@xcite . therefore , in our opinion , a quantum - mechanical study of spin-@xmath0 fermions moving in a harmonic oscillator potential , and interacting via a pair - wise delta function potential , can help us achieve insights into the physics of dilute gases of trapped fermionic atoms . with the aforesaid aims in mind , the purpose of this paper is to describe a fortran 90 computer program developed by us which can solve the hartree - fock equations for spin-@xmath0 fermions moving in a three - dimensional ( 3d ) harmonic oscillator potential , and interacting via delta - function potential . a basis set approach has been utilized in the program , in which the single - particle orbitals are expanded as a linear combination of the 3d simple harmonic oscillator basis functions , expressed in terms of cartesian coordinates . the program can solve both the restricted - hartree - fock ( rhf ) , and the unrestricted hartree - fock ( uhf ) equations , the latter being useful for fermi gases with imbalanced populations . we would like to clarify , that as far as the applications of this approach to dilute fermi gases is concerned , at present it is not possible to reach the thermodynamic limit of very large @xmath4 , where @xmath4 is the total number of atoms in the trap . however , we believe that by solving the hf equations for a few tens of atoms , one may be able to achieve insights into the microscopic aspects such as the nature of pairing in such systems . this program is an extension of an earlier program developed in our group , aimed at solving the time - independent gross - pitaevskii equation ( gpe ) for harmonically trapped bose gases@xcite . thus the combined total program accompanying this paper can now solve for both bose and fermi systems , confined to move in a harmonic oscillator potential , with mutual interactions of the delta - function form . as with our earlier boson program , because of the use of a cartesian harmonic oscillator basis set , the new program can handle trap geometries ranging from spherical to completely anisotropic , and it can also account for those trap anharmonicities which can be expressed as polynomials in the cartesian coordinates . the nature of interparticle interactions , i.e. , whether they are attractive or repulsive , also imposes no restrictions on the program . we note that yu _ _ et al.__@xcite _ _ have recently described a hartree - fock approach for dealing with two - component fermions confined in harmonic traps with spherical symmetry , employing a finite - difference - based numerical approach . however , we would like to emphasize that , as mentioned earlier , our approach is more general in that it is not restricted to any particular trap symmetry . apart from describing the program , we also present and discuss several of its applications . with the aim of exploring the shell - structure in trapped fermionic atoms , using our uhf approach we compute the addition energy for spherically trapped fermions for various particle numbers , and obtain results consistent with a shell - structure and hund s rule . the remainder of the paper is organized as follows . in the next section we discuss the basic theoretical aspects of our approach . in section [ sec - program ] , we briefly describe the most important subroutines that comprise the new enlarged program . section [ sec : install ] contains a brief note on how to install the program and prepare the input files . in section [ sec - results ] we discuss results of several example runs of our program for different geometries . in the same section , we also discuss issues related to the convergence of the procedure . finally , in section [ sec - conclusions ] , we end this paper with a few concluding remarks . we consider a system of @xmath4 identical spin-@xmath0 particles of mass @xmath5 , moving in a 3d potential with harmonic and anharmonic terms , interacting with each other via a pair - wise delta function potential . the hamiltonian for such a system can be written as @xmath6 where @xmath7 represents the position vector of @xmath8th particle , @xmath9 represents the strength of the delta - function interaction , and @xmath10 denotes the one - particle terms of the hamiltonian@xmath11 where @xmath12 , @xmath13 and @xmath14 are the angular frequencies of the external harmonic potential in the @xmath15 , @xmath2 and @xmath3 directions , respectively , and @xmath16 represents any anharmonicity in the potential . in order to parametrize the strength of the delta - function interactions , we use the formula @xmath17 in our program , where @xmath18 is the @xmath19-wave scattering length for the atoms . next we will obtain the rhf and the uhf equations for the system . assuming that @xmath20 , and that the many - particle wave function of the system can be represented by a single closed - shell slater determinant , the rhf equations for the @xmath21 doubly occupied orbitals @xmath22 of the system are obtained to be@xcite@xmath23 similarly , for a system with @xmath24 up - spin ( @xmath25 ) fermions , and @xmath26 down - spin ( @xmath27 ) fermions @xmath28 , the uhf equations for the up - spin orbitals can be written as@xcite@xmath29 where @xmath30 and @xmath31 , represent the occupied orbitals corresponding to the up and the down spins , respectively . similar to the the uhf equations for the down - spin orbitals can be deduced easily from eq . ( [ eq : uhf ] ) . as in our earlier work on the bosonic systems@xcite , we adopt a basis - set approach and expand the hf orbitals in terms of the 3d harmonic oscillator basis functions . this approach is fairly standard , and is well - known as the hartree - fock - roothan procedure in the quantum chemistry community@xcite . thus , for the rhf case , the orbitals are expressed as@xmath32 where @xmath33 represents the coefficient corresponding to the @xmath34-th 3d harmonic oscillator basis function @xmath35 , in the expansion of the @xmath36-th occupied orbital @xmath37 , and @xmath38 is the total number of basis functions used . note that @xmath35 is itself a product of three linear harmonic oscillator eigenfunctions of quantum numbers @xmath39 @xmath40 , and @xmath41 . therefore , a set of functions @xmath35 , for different values of @xmath39 @xmath40 , and @xmath41 , will constitute an orthonormal basis set , leading to an overlap matrix which is identity matrix . for the uhf case , the corresponding expansion for up - spin particles is@xmath42 from which the expansion for the down - spin particles can be easily deduced . upon substituting eqs . ( [ eq : basis - rhf ] ) and ( [ eq : basis - uhf ] ) , in eqs . ( [ eq : rhf ] ) and ( [ eq : uhf ] ) , respectively , one can obtain the matrix forms of the rhf / uhf equations@xcite . as outlined in our earlier work@xcite , numerical implementation of the approach is carried out in the so - called harmonic oscillator units , in which the unit of length is the quantity @xmath43 , and that of energy is @xmath44 . the resulting matrix equation for the rhf case is @xmath45 where @xmath46 represents the column vector containing expansion coefficients @xmath47 of @xmath48 , @xmath49 is the corresponding energy eigenvalue , and the elements of the fock matrix @xmath50 are given by@xmath51 above@xmath52 expressed in terms of aspect ratios @xmath53 and @xmath54 , @xmath55 are the matrix elements of the anharmonic term in the confining potential , @xmath56 is a density - matrix element , and @xmath57 represents the 3d two - fermion repulsion matrix defined as@xmath58 each one of the @xmath59 matrices in eq . ( [ eq - jtilde ] ) , corresponding to the three cartesian directions , can be written in the form@xmath60 where @xmath61 is the corresponding cartesian coordinate in the harmonic oscillator units . an analytical expression for @xmath62 can be found in our earlier work@xcite . in the uhf case , one obtains two matrix equations for the up / down - spin particles of the form@xmath63 where the @xmath64 represents the fock matrix for the up - spin particles given by @xmath65 , @xmath66 is the energy eigenvalue , and @xmath67 , are the elements of the down - spin density matrix . we can easily deduce the form of the fock equation for the down - spin particles from eq . ( [ eq - eigval - uhf ] ) . in our program , ( [ eq - eigval - rhf ] ) and ( [ eq - eigval - uhf ] ) are solved employing the self - consistent field ( scf ) procedure , which requires the iterative diagonalization of the fock equations@xcite . in this section we briefly describe the main program and various subroutines which constitute the entire module . as mentioned in the introduction , the present program is an extension of our earlier program for bosons@xcite . thus the new program , which compiles as _ trap.x _ , can solve for : ( a ) time - independent gross - pitaevskii equation for bosons , and ( b ) hartree - fock equations for fermions , confined in a trap . therefore , most of the changes in the present program , as compared the earlier bosonic program , are related to its added fermionic hf capabilities . however , we have also tried to optimize the earlier bosonic module of the program wherever possible . a readme file associated with this program lists all its subroutines . thus , in what follows , we will describe only those subroutines which are either new ( fermion related ) , or modified , as compared to the older bosonic code@xcite . for an account of the older subroutines not described here , we refer the reader to our earlier work@xcite . additionally , with the aim of making the calculations faster , in the present code , we use the diagonalization routines of lapack library@xcite , which requires the linking of our code to that library . therefore , for this program to work , the user must have the lapack / blas program libraries installed on his / her computer system . the letter f or b has been included in parenthesis after the name of each subroutine to show whether the subroutine is useful for fermionic or bosonic calculations . if it is applicable for both , we denote this by writing bf . this is the main program of our package which reads the input data , dynamically allocates relevant arrays , and then calls other subroutines to perform tasks related to the remainder of the calculations . the main modification in this program , as compared to its earlier version@xcite , is that it now allows for input related to fermionic hf calculations . thus , the user now has to specify whether the particles considered are bosons or fermions . if the particles considered are fermions , one has to further specify whether the rhf or the uhf calculations are desired . for the case of uhf calculations , the user also needs to specify the number of up- and down - spin orbitals . because of the dynamic array allocation throughout , no data as to the size of the arrays is needed from the user . the program will stop only if it exhausts all the available memory on the computer . there is one major departure in the storage philosophy in the present version of the code as compared to the previous one@xcite in that now only the lower / upper triangles of most of the real - symmetric matrices ( such as the fock matrix ) are stored in the linear arrays in the packed format . this not only reduces the memory requirements roughly by a factor of two , but also leads to faster execution of the code . this is the modified version of the old subroutine bec_drv , and is called from the main program oscl . as its name suggests , it is the driver routine for performing : ( a ) calculations of the bose condensate wave function for bosons , or ( b ) solving the rhf / uhf equations for fermions . apart from allocating a few arrays , the main task of this routine is to call either : ( a ) routines bose_scf or bose_steep depending upon whether the user wants to use the scf or the steepest - descent approach meant for solving the gpe@xcite , or ( b ) routines fermi_rhf or fermi_uhf depending on whether the rhf or uhf calculations are to be performed . this subroutine solves the rhf equations for the fermions in a trap using the scf procedure , mentioned earlier . its main tasks are as follows : 1 . allocate various arrays needed for the scf calculations 2 . setup the starting orbitals . this is achieved by diagonalizing the one - particle part of the hamiltonian . 3 . perform the scf calculations . for this purpose , the two - particle integrals @xmath68 ( cf . ( [ eq - jtilde ] ) ) are calculated during each iteration@xcite . if the user has opted for fock matrix / orbital mixing , it is implemented using the formula @xmath69 where @xmath70 is the quantity under consideration in the @xmath36-th iteration , and parameter @xmath71 quantifying the mixing is user specified . thus , if fock matrix mixing has been opted , @xmath71 specifies the fraction of the new fock matrix in the total fock matrix in the @xmath36-th iteration . if the user has opted for the orbital mixing , then each occupied orbital is mixed as per the formula above . the fock matrix constructed in each iteration is diagonalized using the lapack routine dspevx@xcite , which can obtain a selected number of eigenvalues / eigenvectors of a real - symmetric matrix , as against traditional diagonalizers which calculate the entire spectrum of such matrices . we use dspevx during the scf iterations to obtain only the occupied orbitals and their energies , thereby , leading to a much faster completion of the scf process in comparison to using a diagonalizer which computes all the eigenvalues / vectors of the fock matrix . the occupied orbitals are identified according to the _ aufbau _ principle . the total energy and the wave function obtained after every iteration are written in various data files so that the progress of the calculation can be monitored . this process continues until the required precision ( user specified ) in the total hf energy is obtained . in structure and philosophy this subroutine is similar to fermi_rhf , except that its purpose is to solve the uhf equations for interacting spin-@xmath0 fermions confined in a harmonic potential . because there are two separate fock equations corresponding to the up- and the down - spin fermions , the computational effort associated with this subroutine is roughly twice that of routine fermi_rhf . this subroutine aims at solving the time - independent gpe for bosons using the iterative diagonalization approach , and was described in our earlier paper@xcite . the diagonalizing routine which was being used for the purpose obtained all the eigenvalues and eigenvectors of the gpe , which is quite time consuming for calculations involving large basis sets . since the condensate corresponds to the lowest - energy solution of the gpe , using diagonalizing routines which obtain all its eigenvalues and eigenvectors is wasteful . therefore , in the new version of bose_scf we now use the lapack@xcite routine dspevx to obtain the lowest eigenvalue and the eigenvector of the hamiltonian during the scf cycles , leading to substantial improvements in speed . this subroutine aims at solving the time - independent gpe for bosons using the steepest - descent method , and was also described in our earlier paper@xcite . in this routine , the main computational step is multiplication of a trial vector by the matrix representation of the hamiltonian . in the earlier version of the code , because the entire hamiltonian was being stored in a two - dimensional array , we used the fortran 90 intrinsic subroutine matmul for the purpose . however , now that we only store the upper triangle of the hamiltonian in a linear array , it is fruitful to use an algorithm which utilizes this aspect . therefore , we have replaced the call to matmul by a call to a routine called matmul_ut written by us . this has also lead to significant speed improvements . as mentioned in the previous section , the aim of this subroutine is to multiply a vector by a real - symmetric matrix , whose upper triangle is stored in a linear array . this routine is called from the subroutine bose_steep , and it utilizes a straightforward algorithm for achieving its goals by calling two blas@xcite functions ddot and daxpy . we have also significantly improved the capabilities of the program as far as plotting of the orbitals and the associated densities is concerned . now the orbitals , or corresponding densities , can be computed both on one - dimensional and two - dimensional spatial grids , along user - specified directions , or planes . the driver subroutine for the purpose is called plot_drv , which in turn calls the specific subroutines suited for the calculations . these subroutines are plot_1d , and plot_1d_uhf for the one - dimensional plots , and plot_2d and plot_2d_uhf for the planar plots . the output of this module is written in a file called ` orb_plot.dat ` , which can be directly used in plotting programs such as ` gnuplot ` or ` xmgrace ` . in our earlier paper , we had described in detail how to install , compile , and run our program on various computer systems@xcite . additionally , we had explained in a step - by - step manner how to prepare the input file meant for running the code , and also the contents of a typical output file@xcite . because , various aspects associated with the installation and running of the program remain unchanged , except for some minor details , we prefer not to repeat the same discussion . instead , we refer the reader to the readme file in connection with various details related to the installation and execution of the program . additionally , the file `input_prep.pdf` explains how to prepare a sample input file . several sample input and output files corresponding to various example runs are also provided with the package . in this section we report results of some of the calculations performed by our code on fermionic systems . we present both rhf and uhf calculations for various types of traps . further , we discuss some relevant issues related to the convergence of the calculations . in this section our aim is to investigate the convergence properties of the total hf energy of our program with respect to : ( a ) number of particles in the trap , ( b ) symmetry of the confining potential , ( c ) nature and strength of interactions , and ( d ) number of basis functions employed in the calculations . as far as the number of particles is concerned , we have considered two closed - shell systems namely with two particles ( @xmath72 ) , and with eight particles ( @xmath73 ) . for @xmath72 case , calculations have been performed for all possible trap geometries ranging from a spherical trap to a completely anisotropic trap . during these calculations , we have considered both attractive and repulsive interactions , corresponding to negative and positive scattering lengths , respectively . the magnitude of the scattering length ( @xmath74 ) employed in these calculations ranges from @xmath75 to @xmath76 . to put these numbers in perspective , we recall that in most of the atomic traps , @xmath77 @xmath78 m , and for a two - component @xmath79li trapped gas , the estimated value of the scattering length is anomalously large @xmath80@xcite , where @xmath81 is the bohr radius . thus , for this very strongly interacting system , the scattering length @xmath82 , is well within the range of the scattering lengths considered in these calculations . therefore , the systems considered here ranging from weakly interacting ones to very strongly interacting ones truly test our numerical methods . .convergence of total hf energy ( @xmath83 ) for a spherically symmetric trap containing two particles , with respect to the size of the basis set , for various positive values ( repulsive interactions ) of the scattering length . above , @xmath84 is the maximum value of the quantum number of the sho basis function in a given direction , and @xmath38 is the total number of basis functions corresponding to a given value of @xmath84 . in some cases , fock matrix mixing approach was used to achieve convergence . [ cols="^,^,^,^,^,^",options="header " , ] finally , in fig . [ fig : den - plot ] we present the orbital density plots for the @xmath72 case with both attractive and repulsive interactions , corresponding to @xmath85 . the noteworthy point in the graph is the accumulation of the density at the center of the trap in case of attractive interactions , as compared to when the interactions are repulsive . with increasingly attractive interactions , this phenomenon becomes even more pronounced , possibly causing the instabilities in the hf approach . plotted along the @xmath15-axis , obtained from rhf calculations on a two - particle system in an isotropic trap with @xmath86 ( solid lines ) , and @xmath87 ( dashed lines ) . distance @xmath88 is in harmonic oscillator units . , width=321 ] in this section we describe the results of our uhf calculations . if one performs a uhf calculation on a closed - shell system , one must get the same results as obtained by an rhf calculation . similarly , the total energy and orbitals of a system with @xmath5 up - spin and @xmath21 down - spin particles should be the same as that of a system with @xmath21 up - spin and @xmath5 down - spin particles . these properties of the uhf calculations can be used to check the correctness of the underlying algorithm . we verified these properties explicitly by : ( a ) performing uhf calculations on closed - shell systems with various scattering lengths and geometries , and found that the results always agreed with the corresponding rhf calculations , and ( b ) by performing uhf calculations on various open - shell systems with interchanged spin configurations and found the results to be identical . therefore , we are confident of the essential correctness of our uhf program , and in what follows , we describe its applications in calculating the addition energy of fermionic atoms confined in a spherical trap . the aim behind this calculation is to explore whether such a system follows : ( a ) shell - structure , and ( b ) hund s rule , in analogy with harmonically trapped electrons confined in a quantum dot . we also note that a study of hund s rule for fermionic atoms confined in an optical lattice was carried out recently by krkkinen _ _ et al.__@xcite . the addition energy , _ i.e _ , the energy required to add an extra atom , to an @xmath4-atom trap is defined as @xmath89 , where @xmath90/@xmath91 represents the chemical potential of an @xmath92 particle system . the chemical potentials , in turn , are defined as @xmath93 , where @xmath94/@xmath95 represents the total energy of an @xmath92 particle system . in our calculations , the total energies were calculated using the uhf approach for various values of the scattering length and our results for the addition energy for an @xmath96 spherical trap are presented in fig . [ fig : uhf - delmu ] , for the values from @xmath97 to @xmath98 . ) of a spherical trap ( in the units of @xmath44 ) with scattering length @xmath96 , plotted as a function of the particle number @xmath4 , ranging from @xmath97 to @xmath98 . [ fig : uhf - delmu],width=453 ] for the range of @xmath4 values studied here , in a noninteracting model the charging energy acquires nonzero values @xmath99 , only for @xmath100 @xmath101 , and @xmath102 , corresponding to filled - shell configurations . in an interacting model , however , @xmath103 should additionally exhibit smaller peaks at @xmath104 , @xmath105 , corresponding to the half - filled shells . if the inter - particle repulsion is strong enough to split @xmath106 and @xmath107 shells significantly , we will additionally obtain a peak at @xmath108 corresponding to the filled @xmath107 shell , while the peaks corresponding to the half - filled shells will occur at @xmath109 , and @xmath110 , instead of @xmath105 . moreover , it is of considerable interest to examine whether the hund s rule is also satisfied for open - shell configurations of such spherically trapped fermionic atoms , as is the case , _ e.g. _ , for electrons in quantum dots@xcite . from fig . [ fig : uhf - delmu ] it is obvious that major peaks are located at @xmath100 @xmath101 , and @xmath102 , while the minor ones are at @xmath104 , and @xmath111 , with no peaks at @xmath109 , @xmath112 , or @xmath113 . the heights of the major peaks are in the descending order with increasing @xmath4 , ranging from @xmath114 ( @xmath115 to @xmath116 ( @xmath117 ) . additionally , for all the open - shell cases , the lowest - energy configurations were consistent with the hund s rule in that , a given shell is first filled with fermions of one ( say up ) spin - orientation , and upon completion , followed by the fermions of other ( down ) spin orientation . we note that these results are qualitatively similar to the results obtained for spherical quantum dots@xcite . thus , we conclude that for the small number of particles considered by us , the shell structure and the hund s rule are also followed by atoms confined in harmonic traps where the mutual repulsion is through short - range the contact interaction . we have performed a number of uhf calculations on traps of different geometries , and scattering lengths , whose results will be published elsewhere . however , we would like to briefly state that as the scattering length is increased , in several cases the ferromagnetic configurations violating the hund s rule become energetically more stable . this implies that for large scattering lengths the uhf mean - field approach may not be representative of the true state , and inclusion of correlation effects may be necessary . in this paper we reported a fortran 90 implementation of a harmonic oscillator basis set based approach towards obtaining the numerical solutions of both the restricted , as well as the unrestricted hartree - fock equations for spin-@xmath0 fermions confined by a harmonic potential , and interacting via pair - wise delta - function potential . the spin-@xmath0 fermions under consideration could represent a two - component fermi gas composed of atoms confined in harmonic traps . we performed a number of calculations assuming both attractive , and repulsive , inter - particle interactions . as expected , the hartree - fock method becomes unstable with the increasing scattering length for attractive interactions , while no such problem is encountered for the repulsive interactions . additionally , we performed a uhf study of atoms confined in a spherical harmonic trap and verified the existence of a shell structure , and that the hund s rule is followed . these results are in good qualitative agreement with similar studies performed on harmonically confined electrons in quantum dots , interacting via coulomb interaction . in future , we intend to extend and improve the fermionic aspects of the present computer program in several possible ways . as far as problems related to fermionic gases in a trap are concerned , we would like to implement the hartree - fock - bogoliubov approach to allow us to study such systems in the thermodynamic limit , and at finite temperatures . with the aim of studying the electronic structure of quantum dots , we plan to introduce the option of using the coulomb - repulsion for interparticle interactions , a step which will require significant code writing for the two - electron matrix elements . additionally , we also aim to introduce the option of studying the dynamics of electrons in the presence of an external magnetic field , which will also allow us to study fermionic gases in rotating traps . finally , we plan to implement the option of including spin - orbit coupling in our approach , which , at present , is a very active area of research . we will report results along these lines in the future , as and when they become available . e.g. _ , k. m. ohara , s. l. hemmer , m. e. gehm , s. r. granade , and , j. e. thomas , science 298 ( 2002 ) 2179 ; c. a. regal , c. ticknor , j. l. bohn , and d. s. jin , nature 424 ( 2003 ) 47 ; m. greiner , c. a. regal , and d. s. jin , 426 ( 2003 ) 537 ; s. jochim , m. bartenstein , a. altmeyer , g. hendl , s. riedl , c. chin , j. hecker denschlag , and r. grimm , science 302 ( 2003 ) 2101 ; m. w. zwierlin , c. a. stan , c. h. schunk , s. m. f. raupach , s. gupta , z. hadzibabic , and w. ketterle , phys . rev . lett . 91 ( 2003 ) 250401 ; m. w. zwierlin , a. schirotzek , c. h. schunk , and w. ketterle , science 311 ( 2006 ) 492 ; g. b. patridge , w. li , r. i. kamar , y. liao , r. g. hulet , science 311 ( 2006 ) 503 ; m. w. zwierlin , c. h. schunk , a. schirotzek , and w. ketterle , nature 442 ( 2006 ) 54 . e. anderson , z. bai , c. bischof , s. blackford , j. demmel , j. dongarra , j. du croz , a. greenbaum , s. hammarling , a. mckenney , and d. sorensen , lapack users guide , 3rd edn . , ( 2002 ) , siam , philadelphia ( usa ) .

a set of weakly interacting spin-@xmath0 fermions , confined by a harmonic oscillator potential , and interacting with each other via a contact potential , is a model system which closely represents the physics of a dilute gas of two - component fermionic atoms confined in a magneto - optic trap . in the present work , our aim is to present a fortran 90 computer program which , using a basis set expansion technique , solves the hartree - fock ( hf ) equations for spin-@xmath0 fermions confined by a three - dimensional harmonic oscillator potential , and interacting with each other via pair - wise delta - function potentials . additionally , the program can also account for those anharmonic potentials which can be expressed as a polynomial in the position operators @xmath1 @xmath2 , and @xmath3 . both the restricted - hf ( rhf ) , and the unrestricted - hf ( uhf ) equations can be solved for a given number of fermions , with either repulsive or attractive interactions among them . the option of uhf solutions for such systems also allows us to study possible magnetic properties of the physics of two - component confined atomic fermi gases , with imbalanced populations . using our code we also demonstrate that such a system exhibits shell structure , and follows hund s rule . trapped fermi gases , hartree - fock equation numerical solutions 02.70.-c , 02.70.hm , 03.75.ss , 73.21.la * program summary * + _ title of program : _ trap.x + _ catalogue identifier : _ + _ program summary url : _ + _ program obtainable from : _ cpc program library , queen s university of belfast , n. ireland + _ distribution format : _ tar.gz + _ computers : _ pcs / linux , sun ultra 10/solaris , hp alpha / tru64 , ibm / aix + _ programming language used : _ mostly fortran 90 + _ number of bytes in distributed program , including test data , etc . : _ size of the gzipped tar file 371074 bytes + _ card punching code : _ ascii + _ nature of physical problem : _ the simplest description of a spin @xmath0 trapped system at the mean field level is given by the hartree - fock method . this program presents an efficient approach of solving these equations . additionally , this program can solve for time - independent gross - pitaevskii and hartree - fock equations for bosonic atoms confined in a harmonic trap . thus the combined program can handle mean - field equations for both the fermi and the bose particles . + _ method of solution : _ the solutions of the hartree - fock equation corresponding to the fermi systems in atomic traps are expanded as linear combinations of simple - harmonic oscillator eigenfunctions . thus , the hartree - fock equations which comprises of a set of nonlinear integro - differential equation , is transformed into a matrix eigenvalue problem . thereby , its solutions are obtained in a self - consistent manner , using methods of computational linear algebra . + _ unusual features of the program : _ none

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Proceed to summarize the following text: analog to analog ( a2a ) compression of signals has recently gathered interest in information theory @xcite . in a2a compression , a high dimensional analog signal @xmath0 is encoded into a lower dimensional analog signal @xmath1 . the goal is to design the encoding so as to preserve in @xmath2 all the information about @xmath3 , and to obtain successful decoding for a given distortion measure like mse or error probability . in particular , the encoding may be corrupted by noise . it is worth mentioning that when the alphabet of @xmath4 and @xmath5 is finite , this framework falls into traditional topics of information theory such as lossless and lossy data compression , or joint source - channel coding . the novelty of a2a compression is to consider @xmath4 and @xmath5 to be real valued and to impose regularity constrains on the encoder , in particular linearity , as motivated by compressed sensing @xcite . the challenge and practicality of a2a compression is to obtain dimensionality reduction , i.e. , @xmath6 , by exploiting a prior knowledge on the signal . this may be sparsity as in compressed sensing . for @xmath7-sparse signals , and without any stability or complexity considerations , it is not hard to see that the dimensionality reduction can be of order @xmath8 . a measurement rate of order @xmath9 has been shown to be sufficient to obtain stable recovery by solving tractable optimization algorithms like convex programming ( @xmath10 minimization ) . this remarkable achievement has gathered tremendous amount of attention with a large variety of algorithmic solutions deployed over the past years . the vast majority of the research has however capitalized on a common sparsity model . several works have explored connections between information theory and compressed sensing , in particular @xcite , however it is only recently @xcite that a foundation of a2a compression has been developed , shifting the attention to probabilistic signal models beyond the sparsity structure . it is shown in @xcite that under linear encoding and lipschitz - continuous decoding , the fundamental limit of a2a compression is the rnyi information dimension ( rid ) , a measure whose operational meaning had remained marginal in information theory until @xcite . in the case of a nonsingular mixture distribution , the rid is given by the mass on the continuous part , and for the specific case of sparse mixture distributions , this gives a dimensionality reduction of order @xmath8 . it is natural to ask whether this improvement on compressed sensing is due to potentially complex or non - robust coding strategies . @xcite shows that robustness to noise is not a limitation of the framework in @xcite . two other works @xcite have corroborated the fact that complexity may not be a limitation either . in @xcite spatially - coupled matrices are used for the encoding of the signal , leveraging on the analytical ground of spatially - coupled codes and predictions of @xcite . in particular , @xcite shows that the rid is achieved using approximate message passing algorithm with block diagonal gaussian measurement matrices measurement matrices . however , the size of the blocks are increasing as the measurement rate approaches the rid . in @xcite , using a new entropy power inequality ( epi ) for integer - valued random variables that was further developed in @xcite , the polarization technique was used to deterministically construct partial hadamard matrices for encoding discrete signals over the reals . this provides a way to achieve a measurement rate of @xmath11 for signals with a zero rid along with a stable low complexity recovery algorithm . the case of mixture distributions was however left open in @xcite . this paper proposes a new approach to a2a compression by means of a polarization theory over the reals . the use of polarization techniques for sparse recovery was proposed in @xcite for discrete signals , relying on coding strategies over finite fields . in this paper , it is shown that using the rid , one obtains a natural counter - part over the reals of the entropy polarization phenomenon @xcite . specifically , the entropy ( or source ) polarization phenomenon @xcite shows that transforming an i.i.d . sequence of discrete random variables using an hadamard matrix polarizes the conditional entropies to the extreme values of 0 and 1 ( deterministic and maximally random distributions ) . we show in this paper that the rid of an i.i.d . sequence of mixture random variables also polarizes to the two extreme values @xmath12 and @xmath13 ( discrete and continuous distributions ) . to get to this result , properties of the rid in vector settings and related information measures are first developed . it is then shown that the rid polarization is , as opposed to the entropy polarization , obtained with an analytical pattern . in other words , there is no need to rely on algorithms to compute the set of components which tend to 0 or 1 , as this is given by a known pattern equivalent to the bec channel polarization @xcite . this is then used to obtain universal a2a compression schemes based on explicit partial hadamard matrices . the current paper focuses on the encoding strategies and on extracting the rid without specifying the decoding strategy . numerical simulations provide evidence that efficient message passing algorithms may be used in conjunction to the obtained encoders . finally , the paper extends the realm of a2a compression to a multi signal settings . techniques of distributed compressed sensing were introduced in @xcite for specific classes of sparse signal models . we provide here an information theoretic framework for general multi signal a2a compression , as a counter part of the slepian&wolf coding problem in source compression @xcite . a measurement rate region to extract the rid of correlated signals is obtained and is shown to be tight . the set of reals , integers and positive integers will be denoted by @xmath14 , @xmath15 and @xmath16 respectively . @xmath17 will denote the set of strictly positive integers . for @xmath18 , @xmath19=\{1,2,\dots , n\}$ ] denotes the sequence of integers from @xmath13 to @xmath20 . for a set @xmath21 , the cardinality of the set will be denoted by @xmath22 , thus @xmath23|=n$ ] . all random variables are denoted by capital letters and their realization by lower case letter ( @xmath4 is a realization of the random variable @xmath24 ) . the expected value and the variance of a random variable @xmath24 are denoted by @xmath25 and @xmath26 . for @xmath27 , @xmath28 is a column vector consisting of the random variables @xmath29 and for @xmath30 , we set @xmath28 equal to null . for a discrete random variable @xmath24 with a distribution @xmath31 , @xmath32 denotes the discrete entropy of @xmath24 . for the continuous case , @xmath33 denotes the differential entropy of @xmath24 . throughout the paper , we assume that all of discrete and continuous random variables have well - defined discrete entropy and differential entropy respectively . for random elements @xmath24 , @xmath34 and @xmath35 , @xmath36 and @xmath37 denote the mutual information of @xmath24 and @xmath34 and the conditional mutual information of @xmath24 and @xmath34 given @xmath35 . @xmath38 denotes the mutual information of @xmath24 and @xmath34 given a specific realization @xmath39 . hence , @xmath40 . for simplicity , we also assume that all of the random variables ( discrete , continuous or mixture ) have finite second order moments . all probability distributions are assumed to be nonsingular . hence , in the general case for a random variable @xmath24 , the distribution of @xmath24 can be decomposed as @xmath41 , where @xmath42 and @xmath43 are the continuous and the discrete part of the distribution and @xmath44 is the weight of the continuous part . thus , @xmath45 and @xmath46 corresponds to the fully discrete and fully continuous case respectively . for such a probability distribution , the rnyi information dimension is interchangeably denoted by @xmath47 or @xmath48 and is equal to the weight of the continuous part @xmath49 . there is another representation for a random variable @xmath24 that we will repeatedly use in the paper . assume @xmath50 is a continuous random variable with probability distribution @xmath42 and @xmath51 is a discrete random variable with probability distribution @xmath43 and @xmath50 and @xmath51 are independent . let @xmath52 be a binary valued random variable , independent of @xmath50 and @xmath51 with @xmath53 . it is easy to see that we can represent @xmath24 as @xmath54 , where @xmath55 . in this case , the random variable @xmath24 will have the distribution @xmath41 . also , if @xmath56 is a sequence of such random variables with the corresponding binary random variables @xmath57 , @xmath58 : \theta_i=1\}$ ] is a random set consisting of the position of the continuous components of the signal . similarly , @xmath59\backslash c_\theta$ ] is defined to be the position of the discrete components . for a matrix @xmath60 of a given dimension @xmath61 and a set @xmath62 $ ] , @xmath63 is a sub - matrix of dimension @xmath64 consisting of those columns of @xmath60 having index in @xmath21 . similarly , for a vector of random variables @xmath56 , the vector @xmath65 is a sub - vector of @xmath56 consisting of those random variables having index in @xmath21 . for two matrices @xmath66 and @xmath67 of dimensions @xmath68 and @xmath69 , @xmath70 $ ] denotes the @xmath71 matrix obtained by vertically concatenating @xmath66 and @xmath67 . for an @xmath72 and a @xmath73 , @xmath74_q=\frac { \lfloor q x \rfloor } { q}$ ] denotes the uniform quantization of @xmath4 by interspacing @xmath75 . similarly , for a vector of random variables @xmath56 , @xmath76_q$ ] will denote the component - wise uniform quantization of @xmath56 . for @xmath77 and @xmath78 two functions of @xmath79 , @xmath80 or equivalently @xmath81 will be used for @xmath82 similarly , @xmath83 is equivalent to @xmath84 . an ensemble of single terminal measurement matrices will be denoted by @xmath85 , where @xmath86 is the labeling sequence and can be any subsequence of @xmath87 . the dimension of the family will be denoted by @xmath88 , where @xmath89 is the number of measurements taken by @xmath90 . the asymptotic measurement rate of the ensemble is defined by @xmath91 . we will also work with an ensemble of multi terminal measurement matrices . we will focus to the two terminal case and the extension to more than two terminals will be straightforward . we will denote these two terminals by @xmath92 and the corresponding ensemble by @xmath93 with the corresponding dimension @xmath94 and @xmath95 . the measurement rate vector for this ensemble will be denoted by @xmath96 , where @xmath97 . let @xmath24 be a random variable with a probability distribution @xmath31 over @xmath14 . the upper and the lower rid of this random variable are defined as follows : @xmath98_q)}{\log_2(q)},\\ \underline{d}(x)&=\liminf_{q\to \infty } \frac{h([x]_q)}{\log_2(q)}.\end{aligned}\ ] ] by lebesque decomposition or jordan decomposition theorem , any probability distribution over @xmath14 like @xmath31 can be written as a convex combination of a discrete part , a continuous part and a singular part , namely , @xmath99 where @xmath43 , @xmath42 and @xmath100 denote the discrete , continuous and the singular part of the distribution and @xmath101 and @xmath102 . in @xcite , rnyi showed that if @xmath103 , namely , there is no singular part in the distribution and @xmath104 for some @xmath105 $ ] , then the rid is well - defined and @xmath106 . moreover , he proved that if @xmath56 is a continuous random vector then @xmath107_q)}{\log_2(q)}=n$ ] , implying the rid of @xmath20 for the @xmath20-dimensional continuous random vector . our objective is to extend the definition of rid for arbitrary vector random variables , which are not necessarily continuous . to do so , we first restrict ourselves to a rich space of random variables with well - defined rid . over this space , it will be possible to give a full characterization of the rid as we will see in a moment . let @xmath108 be a standard probability space . the space @xmath109 is defined as @xmath110 , where @xmath111 is the set of all nonsingular random variables and for @xmath112 , @xmath113 is the space of @xmath20-dimensional random vectors defined as @xmath114 it is not difficult to see that all @xmath20-dimensional vector random variables , singular or nonsingular , can be well approximated in the space @xmath115 , for example in @xmath116-sense . however , this is not sufficient to fully characterize the rid . specially , the rid is discontinuous in @xmath117 topology , @xmath118 . for example , we can construct a sequence of fully discrete random variables in @xmath115 converging to a fully continuous random variable in @xmath119 , whereas the rid of the sequence is @xmath12 and does not converge to @xmath13 . although we have such a mathematical difficulty in giving a characterization of the rid , we think that the space @xmath115 is rich enough for modeling most of the cases that we encounter in applications . over @xmath115 , we will generalize the definition of the rid to include joint rid , conditional rid and rnyi information defined as follows . let @xmath56 be a random vector in @xmath115 . the joint rid of @xmath56 provided that it exists , is defined as @xmath120_q)}{\log_2(q)}.\ ] ] let @xmath121 be a random vector in @xmath115 . the conditional rid of @xmath56 given @xmath122 and rnyi information of @xmath123 about @xmath56 , provided they exist , are defined as follows : @xmath124_q|y_1^m)}{\log_2(q)}\\ & i_r(x_1^n;y_1^n)=d(x_1^n)-d(x_1^n|y_1^m).\end{aligned}\ ] ] generally , it is difficult to give a characterization of rid for a general multi - dimensional distribution because it can contain probability mass over complicated subsets or sub - manifolds of lower dimension . however , we will show that the vector rnyi information dimension is well - defined for the space @xmath115 . in order to give the characterization of rid over @xmath115 , we also need to define some concepts from linear algebra of matrices , namely , for two matrices of appropriate dimensions , we propose the following definition of the influence " of one matrix on another matrix and residual " of one matrix given another matrix . let @xmath66 and @xmath67 be two arbitrary matrices of dimension @xmath68 and @xmath69 . also let @xmath125 $ ] . the influence of the matrix @xmath67 on the matrix @xmath66 and the residual of the matrix @xmath66 given @xmath67 over the column set @xmath126 are defined to be @xmath127&={\mathrm{rank}}([a;b]_k)-{\mathrm{rank}}(a_k),\\ r(a;b)[k]&={\mathrm{rank}}([a;b]_k)-{\mathrm{rank}}(b_k).\end{aligned}\ ] ] it is easy to check that @xmath128 $ ] is the amount of increase of the rank of the matrix @xmath129 by adding rows of the matrix @xmath130 and @xmath131 $ ] is the residual rank of the matrix @xmath129 knowing the rows of the matrix @xmath130 . moreover , one can easily check that @xmath128=r(b;a)[k]$ ] . [ rid_maintheorem ] let @xmath121 be a random vectors in the space @xmath115 , namely , there are i.i.d . nonsingular random variables @xmath132 and two matrices @xmath66 and @xmath67 of dimension @xmath133 and @xmath134 such that @xmath135 and @xmath136 . let @xmath137 be the representation for @xmath138 $ ] . then , we have 1 . @xmath139 , 2 . @xmath140\}$ ] , where @xmath141 : \theta_i=1\}$ ] is the random set consisting of the position of continuous components . notice that the results intuitively make sense , namely , for a specific realization @xmath142 if @xmath143 we can neglect @xmath144 because it is fully discrete and does not affect the rid . moreover , over the continuous components the resulting contribution to the rid is equal to the rank of the matrix @xmath145 , which is the effective dimension of the space over which the continuous random variable @xmath146 is distributed . finally , all of these contributions are averaged over all possible realizations of @xmath147 . using theorem [ rid_maintheorem ] , it is possible to prove a list of properties of the rid . [ rid_extensions ] let @xmath121 be a random vector in @xmath115 as in theorem [ rid_maintheorem ] . then , we have the following properties : 1 . @xmath148 for any arbitrary invertible matrix @xmath149 of dimension @xmath150 . 2 . @xmath151 . 3 . @xmath152 . 4 . @xmath153 and @xmath154 if and only if @xmath56 and @xmath122 are independent after removing discrete common parts , namely , those @xmath155 $ ] that are fully discrete . further investigation also shows that we have a very nice duality between the discrete entropy and the rid as depicted in table [ tab : duality1 ] . as we will see in subsection [ results : single ] and [ results : multi ] , this duality can be generalized to include some of the theorems in classical information theory like single terminal and multi terminal ( slepian & wolf ) source coding problems . .duality between @xmath156 and @xmath157 [ cols="^,^ " , ] in this section , we will give a brief overview of the results proved in the paper . subsection [ results : rid_polarize ] is devoted to the results obtained for the polarization of the rnyi information dimension . these results are used in subsections [ results : single ] and [ results : multi ] to study _ compression _ problem from an information theoretic point of view . subsection [ results : single ] considers the single terminal case whereas subsection [ results : multi ] is devoted to the multi terminal case . before stating the polarization result for the rid , we define the @xmath158-dimensional erasure process as follows . let @xmath159 $ ] . an `` erasure process '' with initial value @xmath160 is defined as follows . 1 . @xmath161 . @xmath162 and @xmath163 . 2 . let @xmath164 , for some arbitrary @xmath165-valued sequence @xmath166 . define @xmath167 notice that using the @xmath165 labeling , we can construct a binary tree where each leaf of the tree is assigned a specific @xmath165-valued sequence . let @xmath168 be a sequence of i.i.d . uniform @xmath165-valued random variables . by replacing @xmath169 for @xmath165-labeling @xmath166 in the definition of the erasure process , we obtain a stochastic process @xmath170 . let @xmath171 be the @xmath172-field generated by @xmath169 . using the bec polarization @xcite , we have the following results : 1 . @xmath173 is a positive bounded martingale . @xmath174 converges to @xmath175 with @xmath176 . 3 . for any @xmath177 , @xmath178 , where @xmath179 is the number of all possible cases that @xmath174 can take . let @xmath18 and @xmath179 . assume that @xmath180 is a sequence of i.i.d . nonsingular random variables with a rid equal to @xmath48 and let @xmath181 , where @xmath182 is the hadamard matrix of order @xmath86 . for @xmath183 $ ] , let us define @xmath184 . assume that @xmath166 is the binary expansion of @xmath185 . by replacing @xmath12 by @xmath186 and @xmath13 by @xmath187 , we can equivalently represent @xmath188 be a sequence of @xmath165 values , namely , @xmath189 . similar to the erasure process , we can convert @xmath190 to a stochastic process @xmath191 by using i.i.d . uniform @xmath165-valued random variables @xmath169 . we have the following theorem . [ single_rid_polarization ] @xmath192 is an erasure stochastic process with initial value @xmath48 polarizing to @xmath193 . for @xmath18 and @xmath179 , let @xmath194 be a sequences of random vectors in the space @xmath115 , with joint and conditional rid @xmath195 , @xmath196 and @xmath197 . let @xmath181 and assume that @xmath198 . let us define two processes @xmath190 and @xmath199 as follows . @xmath200,\\ j_n(i)&=d(w_i|w_1^{i-1},z_1^n ) , i\in [ n].\end{aligned}\ ] ] similarly , we can label @xmath190 and @xmath199 by a sequence of @xmath166 and convert them to stochastic processes @xmath191 and @xmath201 . by this definition , we have the following theorem . [ multi_rid_polarization ] @xmath192 and @xmath202 are erasure stochastic processes with initial value @xmath48 and @xmath197 , both polarizing to @xmath193 . in the @xmath203 terminal case @xmath204 for a @xmath203 terminal source @xmath205 , using a similar method it is possible to construct erasure processes with initial values @xmath206 , polarizing to @xmath193 . in this subsection , we will use the properties of the rid developed in section [ section : rid ] to study the a2a compression of memoryless sources . we assume that we have a memoryless source with some given probability distribution . the idea is to capture the information of the source , to be made clearer in a moment , by taking some linear measurements . as is usual in information theory , we are mostly interested in asymptotic regime for large block lengths . to do so , we will use an ensemble of measurement matrices to analyze the asymptotic behavior . we will also define the notion of rep ( restricted iso - entropy property ) for an ensemble of measurement matrices . this subsection is devoted to the single terminal case . the results for the multi terminal case will be given in subsection [ results : multi ] . we are mostly interested to the the measurement rate region of the problem in order to successfully capture the source . let @xmath180 be a sequence of i.i.d . random variables with a probability distribution @xmath31 ( discrete , mixture or continuous ) over @xmath14 , and let @xmath207_q$ ] for @xmath208 . the family of measurement matrices @xmath209 , indexed with a subsequence of @xmath87 and with dimension @xmath88 , is @xmath210-rep(@xmath31 ) with the measurement rate @xmath211 if @xmath212 to give some intuitive justification for the rep definition , let us assume that all of the measurements are captured with a device with finite precision @xmath213 for some @xmath214 . in that case , although the potential information of the signal , in terms of bits , can be very large , but what we effectively observe through the finite precision device is only @xmath215_{q_0})$ ] . in such a setting , the ratio of the information we lose after taking the measurements , assuming that some genie gives us the infinite precision measurement captured from the signal , is exactly what we have in the definition of rep , namely , @xmath216 where we assume that @xmath207_{q_0}$ ] . this might be a reasonable model for application because pretty much this is what happens in reality . the problem with this model is that it is not invariant under some obvious transformations like scaling . for example , assume that we are scaling the signal by some real number . in this case , through some simple examples it is possible to show that the ratio in ( [ information_ratio ] ) can change considerably . there are two approaches to cope with this problem . one is to scale the signal with a desired factor to match it to the finite precision quantizer , which in its own can be very interesting to analyze but probably will be two complicated . the other way , is to take our approach and develop a theory for the case in which the resolution is high enough so that the quality measure proposed in ( [ information_ratio ] ) is not affected by the shape of the distribution of the signal . notice that in the fully discrete case , the rep definition is simplified to the equivalent form @xmath217 for a non discrete source with strictly positive rid , @xmath218 , if we divide the numerator and the denumerator in the expression ( [ rep_definition ] ) by @xmath219 , take the limit as @xmath79 tends to infinity and use the definition of the rid , we get the equivalent form @xmath220 interestingly , this implies that in the high resolution regime that we are considering for analysis , the information isometry ( keeping more than @xmath221 ratio of the information of the signal ) is equivalent to the rnyi isometry . moreover , from the properties of the rid , it is easy to see that this rep measure meets some of the invariance requirements that we expect . for example , it is scale invariant and any invertible linear transformation of the input signal @xmath180 keeps the @xmath210-rep measure unchanged . we can also extend the definition when the probability distribution of the source is not known exactly but it is known to belong to a given collection of distributions @xmath222 . assume @xmath223 is a class of nonsingular probability distributions over @xmath14 . the family of measurement matrices @xmath209 , indexed with a subsequence of @xmath87 and with dimension @xmath88 , is @xmath210-rep ( @xmath222 ) for measurement rate @xmath211 if it is @xmath210-rep ( @xmath224 ) for every @xmath225 . now that we have the required tools and definitions , we give a characterization of the required measurement rate in order to keep the information isometry . similar to all theorems in information theory , we do this using the converse " and achievability " parts . [ mixture_converse ] let @xmath180 be a sequence of i.i.d . random variables in @xmath115 . suppose @xmath209 is a family of @xmath210-rep(@xmath31 ) measurement matrices of dimension @xmath88 , then @xmath226 . this result implies that to capture the information of the signal the asymptotic measurement rate must be approximately greater then the rid of the source . this in some sense is similar to the single terminal source coding problem in which the encoding rate must be grater then the entropy of the source . this again the emphasizes the analogy between @xmath156 and @xmath157 . moreover , in the discrete case , @xmath227 , the result is trivial . it was proved in @xcite that under linear encoding and block error probability distortion condition , the measurement rate must be higher than the rid of the source , @xmath228 . theorem [ mixture_converse ] strengthen this result stating that @xmath229 must hold even under the milder @xmath210-rep restriction on the measurement ensemble . theorem [ mixture_converse ] puts a lower bound on the measurement rate in order to keep the @xmath210-rep property . however , it might happen that there is no measurement family to achieve this bound . fortunately , as we will see , it is possible to deterministically truncate the family of hadamard matrices to obtain a measurement family with @xmath210-rep property and measurement rate @xmath48 . this is summarized in the following two theorems . notice that in the fully continuous case as theorem [ mixture_converse ] implies , the feasible measurement rate is approximately @xmath13 which for example can be achieved with any complete orthonormal family , thus no explicit construction is necessary . for the noncontinuous case , we will distinguish between the fully discrete case and the mixture case because they need different proof techniques . theorem [ achievability_discrete ] and [ achievability_hadamard ] summarize the results . [ achievability_discrete ] let @xmath180 be a sequence of i.i.d . discrete integer - valued random variables . then , for any @xmath230 , there is a family of @xmath210-rep(@xmath31 ) partial hadamard matrices of dimension @xmath88 , for @xmath179 with @xmath231 . [ achievability_hadamard ] let @xmath180 be a sequence of i.i.d . random variables in @xmath115 . then , for any @xmath230 , there is a family of @xmath210-rep(@xmath31 ) partial hadamard matrices of dimension @xmath88 , for @xmath179 with @xmath232 . we have also the general result in theorem [ universal_mixture ] which implies that we can construct a family of truncated hadamard matrices which is @xmath210-rep for a class of distributions . [ universal_mixture ] let @xmath222 be a family of probability distributions with strictly positive rid . then , for any @xmath230 , there is a family of @xmath210-rep ( @xmath222 ) partial hadamard matrices of dimension @xmath88 , for @xmath179 , with @xmath233 . theorem [ universal_mixture ] implies that there is a fixed ensemble of measurement matrices capable of capturing the information of the all of the distributions in the family @xmath222 . this is very useful in applications because usually taking the measurements is costly and most of the time we do not have the exact distribution of the signal . if each distribution needs its own specific measurement matrix , we have to do several rounds of the measurement each time taking the measurements compatible with one specific distribution and do the recovery process for that specific distribution . the benefit of theorem [ universal_mixture ] is that one measurement ensemble works for all of distributions . it is also good to notice that although the measurement ensemble is fixed , the recovery ( decoding ) process might need to know the exact distribution of the signal in order to have successful recovery . in this section , our goal is to extend the a2a compression theory from the single terminal case to the multi terminal case . in the multi terminal setting , we have a memoryless source which is distributed in more than one terminal and we are going to take linear measurements from different terminals in order to capture the information of the source . we are again interested in an asymptotic regime for large block lengths . to do so , we will use an ensemble of distributed measurement matrices that we will introduce in a moment . similar to the single terminal case , we are interested in the measurement rate region of the problem , namely , the number of measurements that we need from different terminals in order to capture the signal faithfully . we will analyze the problem for two terminal case . the extension to more than two terminals is straightforward . let @xmath234 be a two terminal memoryless source with @xmath235 being in @xmath115 . the family of distributed measurement matrices @xmath236 , indexed with a subsequence of @xmath87 , is @xmath210-rep @xmath237 for the measurement rate @xmath96 if @xmath238_q,[y_1^n]_q|\phi^x_n x_1^n , \phi^y_n y_1^n)}{h([x_1^n]_q,[y_1^n]_q ) } \leq \epsilon,\\ & \limsup _ { n \to \infty } \frac{m^x_n}{n}\leq\rho_x,\ \ \limsup_{n \to \infty } \frac{m^y_n}{n}\leq \rho_y.\nonumber\end{aligned}\ ] ] if @xmath239 is a random vector in @xmath115 with @xmath240 , similar to what did in the single terminal case , dividing the numerator and the denumerator in the expression ( [ multiterminal_formula ] ) by @xmath219 and taking the limit as @xmath79 tends to infinity , we get the equivalent definition @xmath241 which implies the equivalence of the information isometry and the rnyi isometry . notice that in the fully discrete case , the definition above is simplified to the equivalent form @xmath242 we can also extend the definition to a class of probability distributions . assume that @xmath223 is a class of nonsingular probability distributions in @xmath115 . the family of measurement matrices @xmath243 is @xmath210-rep ( @xmath222 ) for measurement rate @xmath96 if it is @xmath210-rep ( @xmath224 ) for every @xmath225 . let @xmath239 be a two dimensional random vector in @xmath115 with a distribution @xmath244 . the rnyi information region of @xmath244 is the set of all @xmath245 ^ 2 $ ] satisfying @xmath246 assume that @xmath222 is a class of two dimensional random vectors from @xmath115 . the rnyi information region of the class @xmath222 is the intersection of the rnyi information regions of the distributions in @xmath222 . similar to the single terminal case , we are interested in the rate region of the problem . we have the following converse and achievability results . [ converse_theorem_multi ] let @xmath234 be a two - terminal memoryless source with @xmath235 being in @xmath115 . assume that the distributed family of measurement matrices @xmath243 is @xmath210-rep with a measurement rate @xmath96 . then , @xmath247 this rate region is very similar to the rate region of the distributed source coding ( slepian&wolf ) problem with the only difference that the discrete entropy has been replaced by the rid , which again emphasizes the analogy between the discrete entropy and the rid . similar to the slepian&wolf problem , we call @xmath248 the dominant face of the measurement rate region . [ achievability_discrete_multi ] let @xmath234 be a discrete two - terminal memoryless source . then there is a family of @xmath210-rep partial hadamard matrices @xmath243 with @xmath249 . [ achievability_hadamard_multi ] let @xmath234 be a two - terminal memoryless source with @xmath235 belonging to @xmath115 . given any @xmath96 satisfying @xmath250 there is a family of @xmath210-rep partial hadamard matrices with measurement rate @xmath96 . we have also the general result in theorem [ universal_mixture_multi ] which implies that we can construct a family of truncated hadamard matrices which is @xmath210-rep for a class of distributions . [ universal_mixture_multi ] let @xmath222 be a family of two dimensional probability distributions in @xmath115 . then , for any @xmath96 in the measurement region of @xmath222 , there is a family of partial hadamard matrices which is @xmath210-rep ( @xmath222 ) with a measurement rate @xmath96 . in this section , we will give a brief overview of the techniques used to prove the results . we will divide this section into three subsections . in subsection [ prooftech : rid ] , we will overview the proof techniques for the rid . subsection [ prooftech : single ] and [ prooftech : multi ] will be devoted to proof ideas and intuitions about the a2a compression problem in the single and multi terminal case . in this section we will prove theorem [ rid_maintheorem ] and [ rid_extensions ] and we will give further intuitions about the rid over the space @xmath115 . * proof of theorem [ rid_maintheorem ] : * to prove the first part of the theorem , notice that @xmath251_q ) \doteq h([x_1^n]_q , \theta_1^k ) \doteq h([x_1^n]_q|\theta_1^k),\ ] ] because @xmath252 . as @xmath253 and takes finitely many values , it is sufficient to show that for any realization @xmath142 , @xmath254_q|\theta_1^k)}{\log_2(q)}={\mathrm{rank}}(a_{{c_\theta } } ) . \end{aligned}\ ] ] taking the expectation over @xmath147 , we will get the result . to prove ( [ rid_maintheorem_formula1 ] ) , notice that @xmath255_q|\theta_1^k)&= h([a_{{c_\theta}}u_{{c_\theta}}+ a_{{\bar{c}_\theta}}v_{{\bar{c}_\theta}}]_q)\nonumber \\ & \doteq h([a_{{c_\theta}}u_{{c_\theta}}+ a_{{\bar{c}_\theta}}v_{{\bar{c}_\theta}}]_q|v_{{\bar{c}_\theta}})\label{rid_maintheorem_formula2}\\ & \doteq h([a_{{c_\theta}}u_{{c_\theta}}]_q),\label{rid_maintheorem_formula3}\end{aligned}\ ] ] where we used @xmath256 . we also used the fact that knowing @xmath257 , @xmath258_q$ ] and @xmath259_q$ ] are equal up to finite uncertainty . specifically , suppose @xmath260 is the minimum number of lattices of size @xmath75 required to cover @xmath261^{|{{\bar{c}_\theta}}|}$ ] , which is a finite number . then @xmath262_q | v_{{\bar{c}_\theta } } , [ a_{{c_\theta}}u_{{c_\theta}}+ a_{{\bar{c}_\theta}}v_{{\bar{c}_\theta}}]_q ) \leq \log_2(l),\ ] ] which implies ( [ rid_maintheorem_formula2 ] ) and ( [ rid_maintheorem_formula3 ] ) . generally @xmath145 is not full rank . assume that the rank of @xmath145 is equal to @xmath158 and let @xmath263 be a subset of linearly independent rows . it is not difficult to see that knowing @xmath264_q$ ] there is only finite uncertainty in the remaining components of @xmath258_q$ ] , which is negligible compared with @xmath219 as @xmath79 tends to infinity . therefore , we obtain @xmath255_q|\theta_1^k)&\doteq h([a_{{c_\theta}}u_{{c_\theta}}]_q)\\ & \doteq h([a_m u_{{c_\theta}}]_q)\\ & \doteq m \log_2(q).\end{aligned}\ ] ] thus , taking the limit as @xmath79 tends to infinity , we obtain @xmath265|\theta_1^k)}{\log_2(q)}={\mathrm{rank}}(a_{{c_\theta}}).\ ] ] also , taking the expectation with respect to @xmath147 , we obtain @xmath266 , which is the desired result . to prove the second part of the theorem , notice that @xmath251_q | y_1^m ) \doteq h([x_1^n]_q | y_1^m , \theta_1^k).\ ] ] for a specific realization @xmath142 we have @xmath267_q | y_1^m,\theta_1^k)\\ & = h([a_{{c_\theta}}u_{{c_\theta}}+ a_{{\bar{c}_\theta}}v_{{\bar{c}_\theta}}]_q | b_{{c_\theta}}u_{{c_\theta}}+ b_{{\bar{c}_\theta}}v_{{\bar{c}_\theta}})\\ & \doteq h([a_{{c_\theta}}u_{{c_\theta}}+ a_{{\bar{c}_\theta}}v_{{\bar{c}_\theta}}]_q | b_{{c_\theta}}u_{{c_\theta}}+ b_{{\bar{c}_\theta}}v_{{\bar{c}_\theta } } , v_{{\bar{c}_\theta}})\\ & \doteq h([a_{{c_\theta}}u_{{c_\theta}}]_q | b_{{c_\theta}}u_{{c_\theta}}).\end{aligned}\ ] ] generally , @xmath145 is not full - rank . let @xmath263 be the set of all linearly independent rows of @xmath145 of size @xmath158 . then @xmath262_q | b_{{c_\theta}}u_{{c_\theta}})\doteq h([a_m u_{{c_\theta}}]_q|b_{{c_\theta}}u_{{c_\theta}}).\ ] ] it may happen that some of the rows of @xmath263 can be written as a linear combination of rows of @xmath268 . let @xmath269 be the remaining matrix after dropping @xmath270 predictable rows of @xmath263 . given , @xmath271 , @xmath272 has a continuous distribution thus @xmath273_q|b_{{c_\theta}}u_{{c_\theta } } ) \doteq r \log_2(q).\ ] ] it is easy to check that @xmath274 is exactly @xmath275 $ ] . therefore , taking the expectation with respect to @xmath147 , we get @xmath276\}.\ ] ] we also get the following corollary , which shows the additive property of the rid for the independent random variables from @xmath115 . let @xmath56 be independent random variables from @xmath115 . then @xmath277 . notice that we can simply write @xmath278 , where @xmath279 is the identity matrix of order @xmath86 . therefore , by the rank characterization for the rid , we have @xmath280)\}={\mathbb{e}}\{\sum_{i=1}^n \theta_i \}=\sum_{i=1}^n d(x_i),\end{aligned}\ ] ] where we used the fact that the columns of @xmath279 are linearly independent thus adding a column increases the rank by @xmath13 . therefore , the rank of @xmath281 is equal to the number of @xmath13 s is @xmath282 , namely , @xmath283 . using the results of theorem [ rid_maintheorem ] , we can prove theorem [ rid_extensions ] . * proof of theorem [ rid_extensions ] : * for part @xmath13 , the proof is simple by considering the rank characterization . we know that @xmath135 and @xmath284 . moreover , @xmath285 thus @xmath286 . as @xmath149 is invertible @xmath287 , thus we get the result . for part @xmath288 , notice that for any realization @xmath142 and the corresponding set @xmath289 , @xmath290_{{c_\theta}})&={\mathrm{rank}}(a_{{c_\theta } } ) + r(b;a)[{{c_\theta}}]\\ & = { \mathrm{rank}}(b_{{c_\theta } } ) + r(a;b)[{{c_\theta}}].\end{aligned}\ ] ] taking the expectation over @xmath147 , we get the desired result @xmath291 for part @xmath292 , using the chain rule result from part @xmath288 and applying the definition of @xmath293 , we get @xmath294 which shows the symmetry of @xmath295 with respect to @xmath56 and @xmath122 . for part @xmath296 , notice that for a specific realization @xmath142 , a simple rank check shows that @xmath275 \leq { \mathrm{rank}}(a_{{c_\theta}})$ ] . taking the expectation over @xmath147 , we get @xmath297 . if @xmath56 and @xmath122 are independent , the equality follows from the definition . for the converse part , notice that if @xmath298 is fully discrete then @xmath299 . similarly , if @xmath122 is fully discrete then @xmath300 and using the identity @xmath301 , we get the equality . this case is fine because after removing the discrete @xmath155 $ ] , either @xmath56 or @xmath122 is equal to @xmath12 , namely , a deterministic value , and the independence holds . assume that none of @xmath56 or @xmath122 is fully discrete . without loss of generality , let @xmath302 be the non - discrete random variables among @xmath132 and let @xmath303 and @xmath304 be the resulting random vectors after dropping the discrete constituents , namely , we have @xmath305 and @xmath306 , where @xmath269 and @xmath307 are the matrices consisting of the first @xmath274 columns of @xmath66 and @xmath67 respectively . it is easy to check that @xmath308 and @xmath309 . thus it remains to show that @xmath303 and @xmath304 are independent . as we have dropped all of the discrete components , the resulting @xmath310 $ ] are @xmath13 with strictly positive probability . this implies that for any realization of @xmath311 and the corresponding @xmath312 , @xmath313={\mathrm{rank}}(a_{r,{{c_\theta}}})$ ] . in particular , this holds for any @xmath312 of size @xmath13 , namely , for any column of @xmath269 and @xmath307 , which implies that if @xmath269 has a non - zero column the corresponding column in @xmath307 must be zero and if @xmath307 has a non - zero column then the corresponding column in @xmath269 must be zero . this implies that @xmath303 and @xmath304 depend on disjoint subsets of the random variables @xmath302 . therefore , they must be independent . in this section , we will prove the polarization of the rid in the single and multi terminal case as stated in theorem [ single_rid_polarization ] and theorem [ multi_rid_polarization ] . the main idea is to use the recursive structure of the hadamard matrices and the rank characterization of the rid in the space @xmath115 . for the initial value , we have @xmath314 . let @xmath18 and @xmath179 . to simplify the proof , instead of the hadamard matrices , @xmath156 , we will use shuffled hadamard matrices , @xmath315 , constructed as follows : @xmath316 and @xmath317 is constructed from @xmath318 as follows @xmath319 where @xmath320 , @xmath183 $ ] denotes the @xmath321-th row of the @xmath318 . let @xmath56 be as in theorem [ single_rid_polarization ] and let @xmath322 , where @xmath182 is replaced by @xmath318 . also , let @xmath323 , @xmath183 $ ] . we first prove that @xmath324 is also an erasure process with initial value @xmath325 and evolves as follows @xmath326 where @xmath327 $ ] with the corresponding @xmath165-labeling @xmath166 . also , let @xmath328 and @xmath329 denote the first @xmath185 and the first @xmath321 rows of @xmath318 . also , let @xmath320 denote the @xmath321-th row of @xmath318 . thus , we have @xmath330 and @xmath331 . as @xmath180 are i.i.d . nonsingular random variables , it results that @xmath332 belong to the space @xmath115 generated by the @xmath180 random variables . notice that using the rank characterization for the rid over @xmath115 , we have @xmath333\},\end{aligned}\ ] ] where @xmath334 \in \{0,1\}$ ] is the amount of increase of rank of @xmath335 by adding @xmath320 . now , consider the stage @xmath336 , where we have the shuffled hadamard matrix @xmath317 . consider the row @xmath337 which corresponds to the row @xmath338 of @xmath317 . now , if we look at the first block of the new matrix , we simply notice that adding @xmath320 has the same effect in increasing the rank of this block as it had in @xmath318 . a similar argument holds for the second block . moreover , adding @xmath320 increases the rank of the matrix if it increases the rank of either the first or the second block or both . let @xmath339 denote the random rank increase in @xmath328 by adding @xmath320 , then we have @xmath340 @xmath282 and @xmath341 are i.i.d . random variables and a simple check shows that @xmath342 and @xmath343 are also i.i.d .. taking the expectation value , we obtain @xmath344 moreover , if we denote @xmath345 , then by the structure of @xmath318 it is easy to see that @xmath346 and @xmath347 can be written as follows : @xmath348 using the chain rule for the rid , we have @xmath349 which along with ( [ plus_part ] ) , implies that @xmath350 therefore , @xmath324 evolves like an erasure process with initial value @xmath48 . now , notice that the only difference between @xmath182 and @xmath318 is the permutation of the rows , namely , there is a row shuffling matrix @xmath351 such that @xmath352 . it was proved in @xcite that @xmath351 and @xmath182 commute , which implies that @xmath353 . however , notice that @xmath180 is an i.i.d . sequence and @xmath354 is again an i.i.d . sequence with the same distribution as @xmath180 . in particular , adding or removing @xmath351 does not change the rid values , which implies that for @xmath181 and @xmath184 , @xmath355 . therefore , @xmath356 is also be an erasure process with initial value @xmath48 , which polarizes to @xmath193 . using a similar technique , we can prove theorem [ multi_rid_polarization ] . the main idea is that @xmath239 are correlated random variables in the space @xmath115 and they can be written as a linear combination of i.i.d . nonsingular random variables . for the initial value , we have @xmath314 and @xmath357 . as @xmath234 is a memoryless source , similar to the single terminal case , it is easy to see that @xmath356 is an erasure process with initial value @xmath325 and it remains to show that @xmath358 is also an erasure process but with initial value @xmath359 . let @xmath328 , @xmath329 and @xmath320 denote the first @xmath185 rows , the first @xmath321 rows , and the @xmath321-th row @xmath318 . as @xmath360 there is a sequence of i.i.d . random variables @xmath361 and two vectors @xmath362 and @xmath363 such that @xmath364 and @xmath365 . as @xmath234 is memoryless , there is a concatenation of sequence of i.i.d . copies of @xmath361 , @xmath366 , such that @xmath367 e,\\ w_{1}^{n}&=\tilde{h}_n y_1^n=[(b_n h_n ) \otimes ( b_1^k)^t ] e,\end{aligned}\ ] ] where @xmath368 denotes the kronecker product and @xmath369 are the transpose of the column vectors @xmath362 and @xmath363 . let @xmath370 be the random element corresponding to the @xmath371 pattern of @xmath372 $ ] , where @xmath373 $ ] . using the rank result developed for the rid , it is easy to see that for every @xmath374 $ ] @xmath375;h_j\otimes ( b_1^k)^t)[c_\gamma]\}.\end{aligned}\ ] ] for @xmath183 $ ] , let @xmath339 denote the random increase of rank of @xmath376_{c_{\gamma}}$ ] by adding @xmath377 . now , consider the stage @xmath336 , where we are going to combine two copies of @xmath318 to construct the matrix @xmath317 . the the row @xmath321 corresponding to @xmath378 is split into two new rows @xmath337 and @xmath379 which correspond to the row number @xmath338 and the row number @xmath380 of @xmath317 . @xmath381 similar to the single terminal case , we see that adding @xmath382 increases the rank of the matrix if it increases the rank of the either the first or the second block . in other words , @xmath383 where @xmath384 are the corresponding amount of increase of the rank of the first and second block by adding the @xmath321-the row . in particular , @xmath282 and @xmath385 are i.i.d . so are @xmath342 and @xmath343 . taking the expectation , similar to what did in the single terminal case , we obtain that @xmath386 moreover , one can also show that for @xmath327 $ ] , @xmath387 which together with ( [ plus_part_multi ] ) , implies that @xmath388 . therefore , @xmath358 is also an erasure process with initial value @xmath197 . similar to the single terminal case , one can also show that the permutation matrix @xmath351 is not necessary , thus the proof is complete . in this part , we will overview the techniques used to prove the achievability part . the converse part , given in theorem [ mixture_converse ] , has been proved in appendix [ converse_proof_single ] . we will give separate constructions for the fully discrete case and the mixture case although the proof techniques used are very similar . * achievability proof for the mixture case : * we will give an explicit construct of the the measurement ensemble as follows . let @xmath18 and let @xmath179 . assume that @xmath180 is a sequence of i.i.d . nonsingular random variables with rid equal to @xmath48 . let @xmath181 , where @xmath182 is the hadamard matrix of order @xmath86 . also assume that @xmath184 , @xmath183 $ ] . as we proved in theorem [ single_rid_polarization ] , @xmath356 is an erasure process with initial value @xmath48 . we will construct the measurement matrix @xmath90 by selecting all of the rows of @xmath182 with the corresponding @xmath190 value greater than @xmath389 . therefore , we can construct the measurement ensemble @xmath209 labelled with all @xmath86 that are a power of @xmath288 . assume that the dimension of @xmath90 is @xmath88 . it remains to prove that the ensemble @xmath209 is @xmath210-rep with measurement rate @xmath48 . this will complete the proof of theorem [ achievability_hadamard ] . we first show that the family @xmath209 has measurement rate @xmath48 . notice that the process @xmath190 converges almost surely . thus , it also converges in probability . specifically , considering the uniform probability assumption , this implies that @xmath390 : i_n(i)\geq \epsilon\ , d(x ) \}}{n}\\ & = \limsup_{n \to \infty } { \mathbb{p}}(i_n \geq \epsilon\ , d(x))\\ & = { \mathbb{p}}(i_\infty \geq \epsilon\ , d(x))=d(x).\end{aligned}\ ] ] it remains to prove that @xmath209 is @xmath210-rep . let @xmath391 : i_n(i ) \geq \epsilon \ , d(x)\}$ ] denote the selected rows to construct @xmath90 and let @xmath181 be the full measurements . it is easy to check that @xmath392 . also let @xmath393 $ ] denote all of the indices in @xmath21 before @xmath321 . we have @xmath394 which shows the @xmath210-rep property for @xmath209 . * achievability proof for the discrete case : * for the discrete case , the construction of the measurement family is very similar to the mixture case with the only difference that instead of using the erasure process corresponding to the rid , we use the discrete entropy function . more exactly , in the discrete case , assuming that @xmath181 , we define the following process for @xmath395 $ ] , @xmath396 . in @xcite , using the conditional epi result @xcite , the following was proved . @xmath397 is a positive martingale converging to @xmath12 almost surely . similar to the mixture case , we again construct the family @xmath209 by selecting those rows of the shuffled hadamard matrix with @xmath356 value greater than @xmath398 . by a similar procedure , it is easy to show that @xmath209 has zero measurement rate . @xmath399 moreover , assuming that @xmath391 : i_n(i ) \geq \epsilon\ , h(x_1 ) \}$ ] and @xmath393 $ ] , we have @xmath400 which show the @xmath210-rep property for @xmath209 . the last step is to prove theorem [ universal_mixture ] , namely , to show that for a family of mixture distributions @xmath222 with strictly positive rid , there is a fixed measurement family @xmath209 which is @xmath210-rep for all of the distributions in @xmath222 with a measurement rate vector lying in the rnyi information region of of the family . the proof is simple considering the fact that the construction of the family @xmath209 in the proof of theorem [ achievability_hadamard ] depends only on the erasure pattern . also , the erasure pattern is independent of the shape of the distribution and only depends on its rid . moreover , it can be shown that the erasure patterns for different value of @xmath49 are embedded in one another , namely , for @xmath401 , @xmath402 $ ] . considering the method we use to construct the family @xmath209 , this implies that an @xmath210-rep measurement family designed for a specific rid @xmath49 is @xmath210-rep for any distribution with rid less than @xmath49 . thus , if we design @xmath209 for @xmath403 , it will be @xmath210-rep for any distribution in the family . figure [ absorption ] shows the _ absorption phenomenon _ for a binary random variable with @xmath404 . figure [ polarization ] shows the polarization of the rid for a random variable with rid @xmath405 . , width=2 ] , width=2 ] in this section , we will give a brief overview of the techniques used to prove the achievability part . the proof of the converse part is given in appendix [ converse_proof_multi ] . * acievability proof for the mixture case : * the proof technique is very similar to the single terminal case . we will define the suitable erasure process and we will use it to construct the desired @xmath210-rep measurement matrices for the multi terminal case . let @xmath234 , @xmath327 $ ] , be a two - terminal memoryless source , where @xmath86 is a power of two . let @xmath181 and @xmath198 . for @xmath327 $ ] , let us define @xmath184 and @xmath406 . using theorem [ multi_rid_polarization ] , we can show that @xmath190 and @xmath199 are erasure processes with initial values @xmath48 and @xmath197 polarizing to @xmath193 . the next step is to construct the two terminal measurement ensemble . let @xmath18 and @xmath179 . we will construct @xmath407 by selecting those rows of the hadamard matrix , @xmath182 , with @xmath408 . similarly , @xmath409 is constructed by selecting those rows of @xmath182 with @xmath410 . it remains to prove that the family @xmath243 labeled with @xmath86 , a power of @xmath288 , and of dimension @xmath411 and @xmath412 is @xmath210-rep with measurement rate @xmath413 . by this construction , we can achieve one of the corner points of the dominant face of the rate region . if we switch the role of @xmath24 and @xmath34 we will get the other corner point @xmath414 . one way to obtain any point on the dominant face is to use time sharing for the two family . however , it is also possible to use an explicit construction proposed in @xcite , which directly gives any point on the dominant face of the measurement rate region without any need to time sharing . we will just prove the achievability for the corner point @xmath413 . we first show that the family @xmath243 has measurement rate @xmath413 . notice that the processes @xmath415 converge almost surely thus , thay converge in probability . specifically , considering the uniform probability assumption and using a similar technique as we used in the single terminal case , we get the following : @xmath416 : i^x_n(i)\geq \epsilon\ , d(x ) \}}{n}\\ & = \limsup_{n \to \infty } { \mathbb{p}}(i^x_n \geq \epsilon\ , d(x))\\ & = { \mathbb{p}}(i^x_\infty \geq \epsilon\ , d(x))=d(x).\end{aligned}\ ] ] similarly , we can show that @xmath417 . it remains to prove that @xmath243 is @xmath210-rep . let @xmath418 : i_n(i ) \geq \epsilon \ , d(x)\}$ ] and @xmath419 : j_n(i ) \geq \epsilon \ , d(y|x)\}$ ] denote the selected rows to construct @xmath243 and let @xmath420 and @xmath421 be the full measurements for the @xmath4 and the @xmath5 terminal . let @xmath422 $ ] and @xmath423 $ ] be the set of all indices in @xmath424 and @xmath425 less than @xmath321 . we have @xmath426 which shows the @xmath210-rep property for the two terminal measurement family @xmath243 . * achievability proof for the discrete case : * in the fully discrete case , the construction is very similar to the mixture case with the only difference that instead of using the rid , we will use the entropy . similar to the single terminal case , we can prove the following . @xmath192 and @xmath427 are positive martingale converging to @xmath12 almost surely . we again construct the family @xmath243 by selecting those rows of @xmath182 with @xmath428 and @xmath429 . similar to the single terminal case , it is easy to show that @xmath243 has measurement rate @xmath430 . it remains to prove that @xmath243 is @xmath210-rep . let @xmath418 : i_n(i ) \geq \epsilon \ , h(x)\}$ ] and @xmath419 : j_n(i ) \geq \epsilon \ , h(y|x)\}$ ] denote the selected rows to construct @xmath243 and let @xmath420 and @xmath421 be the full measurements for the @xmath24 and the @xmath34 terminal . let @xmath422 $ ] and @xmath423 $ ] be the set of all indices in @xmath424 and @xmath425 less than @xmath321 . we have the following : @xmath431 which shows the @xmath210-rep property for the two terminal measurement family @xmath243 . the last step is to prove theorem [ universal_mixture ] , namely , to show for a family of mixture distributions @xmath222 , there is a fixed measurement family @xmath243 , which is @xmath210-rep for all of the distributions in @xmath222 with a measurement rate in the ryi information region of the family . the proof is simple considering the fact that the construction of the family @xmath236 in the proof of theorem [ achievability_hadamard_multi ] depends only on the erasure pattern which is independent of the shape of the distribution and only depends on its rid . this implies that for any @xmath96 in the rnyi information region of @xmath222 , the designed measurement family @xmath243 is @xmath210-rep @xmath432 . up to now , we defined the notion of @xmath210-rep for an ensemble of measurement matrices . this definition is what we call an `` informational '' characterization , in the sense that taking measurements by the ensemble potentially keeps more than @xmath221 ratio of the information of the source . now , we can ask the natural question that weather this has some `` operational '' implication , in the sense that after having the linear measurements , is it possible to recover the source up to an acceptable distortion ? in particular , is there a computationally feasible algorithm to do that ? to explain the operational view more , let us give an example from polar codes for binary source compression which has lots of similarities with what we have done . as shown in @xcite , for a binary memoryless source with @xmath433 , for a large block length @xmath20 , there is a matrix @xmath434 , of dimension approximately equal to @xmath435 such that the linear measurement of the source by this matrix over @xmath436 faithfully captures all of the randomness of the source . this in its own only solves the encoding part of problem without directly addressing the decoding part , namely , it does not imply the existence of a decoder to recover the source from the measurements up to negligible distortion ( error probability ) . therefore , the operational picture is not complete yet . fortunately , in the case of polar codes , the successive cancellation decoder ( or other decoders proposed ) fills up the gap and shows that the _ informational _ characterization implies the _ operational _ characterization . for simulations , we use a unit variance sparse distribution @xmath437 , where @xmath438 is the unit delta measure at point zero , @xmath42 is the distribution of the continuous part and @xmath439 is the rid of the signal . we use the mse ( mean square error ) as distortion measure . the simulations are done with the hadamard matrix of order @xmath440 . to build the measurement matrix @xmath66 , we select all of the rows of @xmath182 with highest conditional rid , as stated in [ prooftech : single ] , until we get acceptable recovery distortion . figure [ ell_1_pt ] shows the phase transition ( pt ) diagram for the @xmath441-minimization algorithm . the simulations are done with @xmath292 different distributions for @xmath42 : gaussian , laplacian and uniform . the acceptable recovery distortion is set to @xmath442 . the recovery is successful for the measurement rates above the plotted curves . the results show the insensitivity of the pt region to the distribution of the continuous components . -minimization , width=3 ] -minimization , width=3 ] we also used the amp algorithm to recover the signal , where for simplicity , we only did the simulations for the gaussian case for @xmath42 . the amp iteration is as follows : @xmath443 where @xmath444 is the linear measurements taken by @xmath66 , @xmath445 is the measurement rate , @xmath446 , @xmath447 and where @xmath448 , with @xmath449 independent of the signal @xmath24 and @xmath450 given by the state evolution equation for amp , is the soft - thresholding function designed for the known distribution of @xmath24 . for initialization , we use @xmath451 and @xmath452 . 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pp . 42034215 , dec . 2005 . e. j. cands , t. tao , `` near - optimal signal recovery from random projections : universal encoding strategies '' , ieee transaction on information theory , vol . 54065425 , dec . 2006 . a. rnyi,on the dimension and entropy of probability distributions , " _ acta mathematica hungarica _ , 1 - 2 , mar . in this section , we will prove theorem [ mixture_converse ] , which constitutes the converse part and puts a lower bound on the minimum number of linear measurements in order to to keep @xmath210-rep property . we will prove the following lemmas which will be used repeatedly for other parts . [ cauchy - binnet ] assume that @xmath60 is a full - rank matrix of dimension @xmath61 , for @xmath454 , and @xmath455 . then , there exists @xmath456 , |s|=m$ ] such that @xmath457 . as @xmath454 from cauchy - binnet formula we have @xmath458 , |s|=m } \det(\phi_s \phi_s^t)\\ & = \sum _ { s\subset [ n ] , |s|=m } \det(\phi_s)^2.\end{aligned}\ ] ] as all of the terms are positive , there must be a @xmath62 $ ] of size @xmath158 such that @xmath459 which implies that @xmath460 . [ hqxd1 ] let @xmath24 be a continuous random variable with finite differential entropy and let @xmath461_q$ ] . suppose @xmath462 is a random element for which the differential entropy and the rid of @xmath24 given @xmath462 are well - defined . then , we have @xmath463 we have @xmath464 we know that @xmath465 , which implies that @xmath466 has a bounded support at most @xmath467 $ ] . as the uniform distribution maximizes the differential entropy for a fixed support , we have @xmath468)=0 $ ] . we also have @xmath469 given @xmath462 , @xmath24 has a well - defined differential entropy , which implies that almost surely for all @xmath470 , @xmath24 conditioned on @xmath462 is a continuous random variable . therefore , @xmath471 taking the limit as @xmath79 tends to the infinity we get the result . putting @xmath462 equal to null in the lemma [ hqxd1 ] , we get the following corollary . [ hqxd2 ] let @xmath24 be a continuous random variable with a well - defined differential entropy and let @xmath461_q$ ] . then @xmath472 and @xmath473 . [ hqpxd_lb ] let @xmath56 be a sequence of i.i.d . continuous random variables and let @xmath474_q$ ] . assume that @xmath60 is a full - rank matrix of dimension @xmath61 where @xmath454 and @xmath475 . suppose @xmath462 is a random element such that the differential entropy of @xmath476 given @xmath462 is well - defined . then @xmath477 . by lemma [ cauchy - binnet ] , there is a @xmath62 $ ] of size @xmath158 such that @xmath478 . hence , we have @xmath479 where @xmath480 $ ] and @xmath481 . the final result follows by applying lemma [ hqxd1 ] . without any loss of generality , we can assume that @xmath209 is a full - rank family , otherwise , we can drop some of the rows of @xmath90 and obtain an equivalent family with lower measurement rate . also , we can assume that the rows of @xmath90 are orthonormal . otherwise , by gram - schmidt procedure , we can obtain an equivalent family with orthonormal rows . in other words , there is a lower triangular and invertible @xmath482 matrix @xmath483 such that @xmath484 has orthonormal rows . as @xmath483 is invertible , it results that @xmath485 thus the equivalent family @xmath486 is also @xmath210-rep and has orthonormal rows , namely , @xmath487 , where we again dropped the dependence of @xmath158 on @xmath86 . we also represent each @xmath488 $ ] as @xmath489 . from @xmath210-rep assumption , for any @xmath490 there is a @xmath491 such that for @xmath492 @xmath493 where @xmath207_q$ ] . as we are going to take the limit as @xmath79 tends to infinity , we can drop the negligible terms . in other words , we have @xmath494 where we used the fact that @xmath495 . for a specific realization @xmath496 , let @xmath497 : \theta_i=1\}$ ] and @xmath498\backslash { { c_\theta}}$ ] as introduced before . then , we obtain @xmath499 where @xmath500_q$ ] and @xmath501_q$ ] denote the component - wise quantization of @xmath502 and @xmath257 . we also used the fact that @xmath503 let @xmath504_q$ ] and @xmath505_q$ ] . we consider two cases : first , if @xmath506 , using ( [ mixture4 ] ) , we have @xmath507 second , if @xmath508 , generally , @xmath509 is neither full - rank nor orthonormal . however , we can drop the redundant rows and by using the gram - scmidt procedure , we can create an equivalent orthonormal matrix @xmath510 of dimension @xmath511 with @xmath512 . therefore , for this case we obtain @xmath513 where @xmath514 is the variance of @xmath515 . we also used lemma [ hqpxd_lb ] , @xmath516 , and the fact that the gaussian distribution maximizes the differential entropy for a given covariance matrix . combining ( [ mixture5 ] ) and ( [ mixture6 ] ) , and @xmath517 we obtain @xmath518 which implies that @xmath519 moreover , from ( [ mixture1 ] ) , ( [ mixture2 ] ) and ( [ mixture7 ] ) , we get @xmath520 taking the limit as @xmath79 tends to infinity , we obtain @xmath521 which implies that @xmath522 as @xmath490 is arbitrary , we get the result . this section is devoted to the proof of theorem [ converse_theorem_multi ] . this theorem puts constraints on the number of linear measurements we should take from different terminals in order to keep @xmath210-rep property . from @xmath210-rep property , we have @xmath523_q,[y_1^n]_q ; & \phi^x_n x_1^n , \phi^y_n y_1^n)\nonumber\\ & \geq ( 1-\epsilon ) h([x_1^n]_q,[y_1^n]_q).\label{first_formula}\end{aligned}\ ] ] similar to the @xmath524 notation that we used in ( [ gamma_notation ] ) for the representation for @xmath180 and @xmath525 , we have @xmath523_q&,[y_1^n]_q ; \phi^x_n x_1^n , \phi^y_n y_1^n ) \\ & \doteq i([x_1^n]_q,[y_1^n]_q ; \phi^x_n x_1^n , \phi^y_n y_1^n,\gamma_1^n)\\ & \doteq i([x_1^n]_q,[y_1^n]_q ; \phi^x_n x_1^n , \phi^y_n y_1^n|\gamma_1^n)\end{aligned}\ ] ] as @xmath526 takes finitely many values , we can obtain the result for a specific realization @xmath527 and then take expectation over all possible realizations . for a specif realization @xmath527 , if some of the components of @xmath528 and @xmath529 are discrete or they are linearly dependent we can drop them . with some abuse of notation , let @xmath530 denote the remaining components which have will have dimension @xmath531 and @xmath532 , where @xmath533 and @xmath534 depend on the specific realization @xmath527 . @xmath535_q,[y_1^n]_q ; \phi^x_n x_1^n , \phi^y_n y_1^n)\nonumber\\ & = h_\gamma(\phi^x_n x_1^n , \phi^y_n y_1^n ) \label{form1}\\ & - h_\gamma(\phi^x_n x_1^n , \phi^y_n y_1^n|[x_1^n]_q,[y_1^n]_q)\nonumber\\ & \preceq - h_\gamma(\phi^x_n x_1^n , \phi^y_n y_1^n|[x_1^n]_q,[y_1^n]_q)\nonumber\\ & \doteq - h_\gamma(q\phi^x_n x_1^n , q\phi^y_n y_1^n|[x_1^n]_q,[y_1^n]_q)\label{form2}\\ & + ( r^x_n + r^y_n ) \log_2(q)\nonumber\\ & \preceq ( m^x_n+m^y_n ) \log_2(q)\nonumber,\end{aligned}\ ] ] where in ( [ form1 ] ) , we used the fact that @xmath536 is upper bounded by the differential entropy of a gaussian random vector with appropriate covariance matrix which vanishes in the limit as @xmath79 tends to infinity . also , in ( [ form2 ] ) , we used lemma [ hqpxd_lb ] to show that @xmath537_q,[y_1^n]_q)\doteq0 $ ] . therefore , taking the expectation over @xmath526 we obtain that @xmath523_q,[y_1^n]_q ; \phi^x_n x_1^n , \phi^y_n y_1^n)\preceq ( m^x_n+m^y_n)\log_2(q).\end{aligned}\ ] ] we also have @xmath215_q,[y_1^n]_q ) \doteq n d(x , y ) \log_2(q)$ ] . therefore , using ( [ first_formula ] ) and taking the limit as @xmath79 tends to infinity , we obtain @xmath538 to prove the other two inequalities , notice that @xmath539_q&,[y_1^n]_q ; \phi^x_n x_1^n , \phi^y_n y_1^n)\nonumber\\ & = i_\gamma([y_1^n]_q ; \phi^x_n x_1^n , \phi^y_n y_1^n ) \nonumber\\ & + i_\gamma([x_1^n]_q ; \phi^x_n x_1^n , \phi^y_n y_1^n | [ y_1^n]_q)\nonumber\\ & \leq h_\gamma([y_1^n]_q ) + i_\gamma([x_1^n]_q ; \phi^x_n x_1^n|[y_1^n]_q)\nonumber \\ & + i_\gamma([x_1^n]_q ; \phi^y_n y_1^n | \phi^x_n x_1^n,[y_1^n]_q).\label{formula_for_xy}\end{aligned}\ ] ] for the last term , @xmath540_q ; \phi^y_n y_1^n | \phi^x_n x_1^n,[y_1^n]_q)$ ] , we can again assume that we have dropped all of discrete and linearly dependent terms from @xmath529 so that it has a well - defined differential entropy . thus , we obtain @xmath539_q & ; \phi^y_n y_1^n | \phi^x_n x_1^n,[y_1^n]_q)\nonumber\\ & = i_\gamma([x_1^n]_q ; q\phi^y_n y_1^n | \phi^x_n x_1^n,[y_1^n]_q)\nonumber\\ & = h_\gamma(q\phi^y_n y_1^n| \phi^x_n x_1^n,[y_1^n]_q)\label{formula_for_y1}\\ & - h_\gamma(q\phi^y_n y_1^n| \phi^x_n x_1^n,[y_1^n]_q,[x_1^n]_q)\label{formula_for_y2}.\end{aligned}\ ] ] notice that , for the first term in ( [ formula_for_y1 ] ) , @xmath541_q)&\leq h_\gamma(q \phi^y_n(y_1^n - [ y_1^n]_q)).\end{aligned}\ ] ] it is easy to see that the random vector @xmath542 ) \in [ 0,\frac{1}{q}]^{r^y_n}$ ] has a bounded support independent of @xmath79 thus @xmath543_q)$ ] has an upper bound independent of @xmath79 . therefore , @xmath544_q ) \preceq 0.\ ] ] using a similar argument for ( [ formula_for_y2 ] ) , we have @xmath544_q,[x_1^n]_q ) \preceq 0.\ ] ] assume @xmath260 is a lower triangular invertible matrix , obtained through the gram - scmidt procedure , such that @xmath545 is an orthonormal matrix . then applying lemma [ hqpxd_lb ] , we obtain that @xmath546_q ) \succeq - \log_2(|\det(l)| ) \succeq 0,\\ & h_\gamma(q\phi^y_n y_1^n| \phi^x_n x_1^n,[y_1^n]_q,[x_1^n]_q ) \succeq - \log_2(|\det(l)| ) \succeq 0.\end{aligned}\ ] ] this implies that @xmath540_q ; \phi^y_n y_1^n | \phi^x_n x_1^n,[y_1^n]_q)\doteq 0 $ ] . thus , from ( [ formula_for_xy ] ) , we obtain @xmath547_q,[y_1^n]_q ; \phi^x_n x_1^n , \phi^y_n y_1^n)\\ & \preceq h_\gamma([y_1^n]_q ) + i_\gamma([x_1^n]_q ; \phi^x_n x_1^n|[y_1^n]_q ) . \end{aligned}\ ] ] again if @xmath528 has discrete components or if some of the components are linearly dependent or can be predicted from @xmath548_q$ ] we can drop them . with some abuse of notation let @xmath528 denote the resulting random vector of dimension @xmath533 . we have @xmath539_q & ; \phi^x_n x_1^n|[y_1^n]_q)\nonumber\\ & = h_\gamma(\phi^x_n x_1^n|[y_1^n]_q)-h_\gamma(\phi^x_n x_1^n|[y_1^n]_q,[x_1^n]_q)\nonumber\\ & \doteq -h_\gamma(\phi^x_n x_1^n|[y_1^n]_q,[x_1^n]_q)\nonumber\\ & \doteq -h_\gamma(q\phi^x_n x_1^n|[y_1^n]_q,[x_1^n]_q ) + r^x_n \log_2(q)\nonumber\\ & \doteq r^x_n \log_2(q ) \preceq m^x_n \log_2(q),\label{last_formula}\end{aligned}\ ] ] where we used the fact that @xmath549_q)\doteq0 $ ] and from lemma [ hqpxd_lb ] , @xmath550_q,[x_1^n]_q ) \doteq 0 $ ] . therefore , taking the expectation over @xmath526 and using ( [ first_formula ] ) and ( [ last_formula ] ) , we obtain @xmath551_q,[y_1^n]_q|\gamma_1^n)-h([y_1^n]_q|\gamma_1^n),\\ & \doteq ( 1-\epsilon ) h([x_1^n]_q,[y_1^n]_q ) - h([y_1^n]_q),\end{aligned}\ ] ] which implies that @xmath552 . therefore , taking the limit as @xmath86 tends to infinity , we get @xmath553 the last inequality in the theorem follows by symmetry .

this paper shows that the rnyi information dimension ( rid ) of an i.i.d . sequence of mixture random variables polarizes to the extremal values of 0 and 1 ( fully discrete and continuous distributions ) when transformed by an hadamard matrix . this provides a natural counter - part over the reals of the entropy polarization phenomenon over finite fields . it is further shown that the polarization pattern of the rid is equivalent to the bec polarization pattern , which admits a closed form expression . these results are used to construct universal and deterministic partial hadamard matrices for analog to analog ( a2a ) compression of random i.i.d . signals . in addition , a framework for the a2a compression of multiple correlated signals is developed , providing a first counter - part of the slepian - wolf coding problem in the a2a setting . rnyi information dimension , polarization , information preserving matrices , analog compression , distributed analog compression , compressed sensing .

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Proceed to summarize the following text: electromagnetic fields are the main tool in the control and manipulation of the motion of polarizable dipoles , such as atoms and molecules , which are key objects in physics and chemistry . optical traps @xcite and laser cooling @xcite are widely used to induce external mechanical forces on individual dipoles , whereas interactions between atoms are often tuned with the help of external magnetic or electric fields @xcite . in this work , we address laser - induced dipole - dipole interactions ( liddi ) by establishing their relation to resonant inter - dipolar interaction ( excitation exchange ) in any confined geometry . polarizable dipoles subject to static electric fields interact via the electrostatic dipole - dipole interaction that scales in free space as @xmath0 , @xmath1 being the inter - dipolar distance . a dynamic analog is obtained when a laser field , far - detuned from the dipolar resonant frequency , illuminates the ( induced ) dipoles . in the retarded regime , where @xmath2 with @xmath3 the laser wavenumber , the resulting liddi scales like @xmath4 , whereas in the non - retarded quasistatic case , where @xmath5 , the electrostatic scaling @xmath0 is restored @xcite . the retarded long - range liddi was shown to be associated with peculiar many - body effects , such as self trapping @xcite , density modulations @xcite and the existence of a roton - like collective excitations @xcite , in an atomic bose - einstein condensate . recently , the possibility to control and shape the space - dependence of the inter - dipolar liddi potential was considered . one option is to tune the laser parameters @xcite or the radiation spectrum @xcite . another , more recent approach , is to consider dipoles coupled to structures that support confined photon modes @xcite . once illuminated by off - resonant light and virtually excited , these dipoles interact via confined virtual photons whose spatial - mode structure determines the resulting liddi space - dependence . for example , in the case of many atoms that are trapped in the vicinity of an optical fiber @xcite and free to move along its axis as in @xcite , the fiber - mediated liddi can effectively become one - dimensional ( 1d ) , such that it extends to any range and the atoms may self - organize @xcite . when the fiber incorporates a bragg grating ( 1d photonic crystal ) , the relaxation dynamics of the resulting many - atom system was shown to reveal the inequivalence of statistical ensembles typical of non - additive systems @xcite . self organization and dynamics of laser - illuminated atoms inside a cavity were also studied theoretically and experimentally @xcite . a similar situation arises when considering the spatial dependence of the dispersive interaction between the internal degrees of freedom of the dipoles , the so - called resonant dipole - dipole interaction ( rddi ) , where a dipole , e.g. an atom initially in the excited state , periodically exchanges its excitation with another dipole , e.g. an atom initially in the ground state @xcite . rddi is mediated by the exchange of virtual photons between the atoms , hence its space - dependence is determined by the spatial structure / propagation of the mediating photon modes . in free space and for @xmath6 , @xmath7 being the typical dipolar transition wavelength , the rddi exchange frequency scales as @xmath0 whereas in the retarded regime @xmath8 it takes the form @xmath9 . such rddi retardation effects are of interest for quantum information processing @xcite . studies of geometries where rddi is mediated by confined photon modes include , for example , rddi via surface - plasmon - polariton and coplanar waveguide modes in one dimension @xcite or in geometries that create cutoffs or bandgaps in the photonic mode spectrum @xcite , where spontaneous emission is absent and enhanced long - range rddi may become dominant @xcite . the extensive experimental progress in the ability to couple atoms / dipoles to various confined photonic structures @xcite , makes the discussion of liddi in such structures ever more relevant . in this study we derive a general formalism for liddi in _ any geometry _ by relating it to the underlying rddi process , thus providing a simple and intuitive recipe for calculating liddi through the less complex rddi . we also show that the same holds for the relation between scattering of laser photons and spontaneous emission , respectively . our formalism has several appealing features : ( 1 ) it is fully quantum mechanical , with the laser radiation taken to be in a coherent state ( in contrast to the less realistic number state @xcite ) . ( 2 ) it interprets _ liddi as rddi between dressed dipoles _ , thus allowing to account for liddi in strong radiation fields . this considerably simplifies the calculation of liddi in a general geometry to merely calculating the rddi in that geometry , thereby avoiding the need for the rather cumbersome fourth - order perturbative qed calculation which contains 24 diagrams @xcite . ( 3 ) it enables the analysis of _ nonlinear effects _ in liddi , resulting from the nonlinear response of the dipoles to the laser , unaccounted for previously . ( 4 ) for the case of far - detuned laser field , it reveals a simple relation between the pairwise rddi energy @xmath10 and the liddi potential @xmath11 , where @xmath12 is the laser rabi frequency and @xmath13 the dipole - laser detuning . this provides an intuitive picture for liddi : the dipole / atom is virtually excited by the off - resonant laser with probability @xmath14 and , once excited , interacts with the other dipole / atom via rddi . the paper is organized as the following . section ii is dedicated to the derivation of the general formalism that relates liddi and scattering to their underlying rddi and spontaneous emission , respectively , arriving at our central , general result for the liddi potential , eq . ( [ liddi1 ] ) . the large - detuning limit is discussed in section iii , whereas the application of our formalism to the analysis of liddi in a cavity is illustrated in section iv . in section v we put forward intuitive and simple arguments that explain the origin of liddi effects caused by nonlinear dipole response to the laser . our conclusions are presented in section vi . we adopt a two - level atom model for the polarizable dipole . thus , while still capturing the essence of nonlinear response of the dipoles , the discussion is kept simple and intuitive . the two - level atom approximation is justified for the typical situation where a laser frequency is close to a specific dipole - allowed transition of a system with discrete energy levels , such as atoms , molecules , quantum dots or even superconducting qubits . the atoms , with excited state @xmath15 and ground state @xmath16 interact with a laser ( @xmath17 below ) . considering a laser mode in a coherent state , an _ exact _ transformation due to mollow @xcite allows to treat the interacting system of atoms + single field mode in a coherent state ( laser ) + vacuum in the rest of the modes , as a system of atoms + single - mode external field ( laser ) + vacuum in all field modes @xcite . the atoms are coupled to the vacuum modes @xmath18 in the considered geometry ( @xmath19 below ) via dipole couplings ( @xmath20 below ) which ultimately lead to the inter - atomic interaction . the hamiltonian @xmath21 reads @xmath22 , \nonumber\\ h_v&=&\sum_k\hbar\omega_k\hat{a}^{\dag}_k\hat{a}_k , \nonumber \\ h_{av}&=&\sum_{\nu=1}^2\sum_k\hbar\left[i g_{k\nu } \hat{a}_k - i g^{\ast}_{k\nu } \hat{a}^{\dag}_k \right]\left[\hat{\sigma}_{\nu}^{-}+\hat{\sigma}_{\nu}^{+}\right ] . \nonumber \\ \label{h}\end{aligned}\ ] ] here @xmath23 is the electromagnetic vacuum mode @xmath24 lowering operator , whereas @xmath25 and @xmath26 are the laser frequency and wavevector with rabi frequency @xmath27 , @xmath28 and @xmath29 being its electric field amplitude and polarization vector , respectively , and @xmath30 denoting the atomic dipole matrix element . in order to find the pairwise - liddi potential it is enough to focus on a single pair of atoms with indices @xmath31 and corresponding atomic operators @xmath32 , @xmath33 . the atom - vacuum dipolar couplings are @xmath34 , @xmath35 being the location of atom @xmath36 , and @xmath37 the spatial function of mode @xmath24 . moving to the interaction picture with respect to ( w.r.t ) @xmath38 the hamiltonian becomes @xmath39 . \label{hi1}\ ] ] here the rotating - wave approximation @xmath40 is taken , and the interaction - picture atomic operators are @xmath41 with @xmath42 , where @xmath43 denotes time ordering . we are now interested in finding the interaction - picture operator @xmath44 of the coupled atom - laser system . this is done for each atom @xmath31 separately , hence the index @xmath36 is suppressed here . the operator @xmath44 can be viewed as a heisenberg - picture operator w.r.t the hamiltonian @xmath17 from eq . ( [ h ] ) . then , dividing @xmath17 into @xmath45 and @xmath46 , we move to the so - called laser - rotated frame , which is an interaction picture w.r.t @xmath47 , where the hamiltonian @xmath17 becomes @xmath48 with @xmath49 and where @xmath50 is the atom - laser detuning . the eigenstates @xmath51 and eigenvalues @xmath52 of @xmath53 are found to be ( see also @xcite ) @xmath54 where @xmath55 is the effective rabi frequency . considering the relation between operators in the heisenberg and interaction pictures we find @xmath56 where @xmath57 and @xmath58 , such that @xmath59 . then , recalling @xmath60 , @xmath61 , we can use the transformation ( [ tran ] ) to express the operators @xmath62 in terms of the @xmath51-dressed - basis projection operators and easily apply @xmath63 on these operators , obtaining @xmath64 where @xmath65 and @xmath66 are the spin operators in the dressed - basis . this result can be written in a more compact form by absorbing the time - independent coefficients into the operators , as @xmath67 eqs . ( [ sig1 ] ) and ( [ sig2 ] ) describe the dynamics of the atom operator @xmath68 , dressed by the laser , as composed of a linear combination of the three spin operators in the dressed - basis @xmath69 , each oscillating in time with a distinct frequency @xmath70 , hence allowing all possible transitions in this basis . inserting @xmath71 from eq . ( [ sig2 ] ) into eq . ( [ hi1 ] ) , we obtain the interaction picture hamiltonian in the form @xmath72 . \label{hi2}\ ] ] this hamiltonian describes the interaction of atoms , dressed by the laser ( operators @xmath73 ) , with the electromagnetic vacuum ( operators @xmath23 ) . we are interested to use lowest order perturbation theory in order to find the effective interaction between the dressed atoms , mediated by the vacuum , similar to the case of rddi @xcite . equivalently , this can be performed by the derivation of the master equation for the density operator of the two - atom system @xmath74 when it interacts with the vacuum reservoir in a stationary vacuum state @xmath75 . to lowest order ( born approximation ) the master equation is given by @xcite @xmath76\right]\right\}. \label{me1}\ ] ] inserting the hamiltonian from eq . ( [ hi2 ] ) into eq . ( [ me1 ] ) and using standard methods ( see appendix a ) , we obtain the markovian master equation @xmath77+\sum_{\nu,\nu'=1}^2\sum_{i=\pm , z}\left[\gamma_{\nu \nu'}^i\tilde{s}_{i\nu}^{\dag}\rho\tilde{s}_{i\nu ' } \right . \nonumber \\ & & \left . -\frac{1}{2}\gamma_{\nu \nu'}^i\left(\tilde{s}_{i\nu}\tilde{s}_{i\nu'}^{\dag}\rho+\rho\tilde{s}_{i\nu}\tilde{s}_{i\nu'}^{\dag}\right)\right ] , \label{me2}\end{aligned}\ ] ] with the effective dipole - dipole hamiltonian @xmath78 the dispersive , virtual - photon - mediated , rddi energy @xmath79 and the corresponding cooperative emission rate @xmath80 , are related by kramers - kronig relation and given by @xmath81 where @xmath82 is the vacuum - reservoir two - point ( autocorrelation ) spectrum defined in the continuum limit @xmath83 . the geometry dependence of the liddi and related effects discussed below is thus encoded in the function @xmath82 , since this function depends on the dipole couplings @xmath84 , which in turn depend on the spatial photon modes of the confined geometry @xmath85 . two important assumptions regarding the temporal resolution of interest ( coarse - graining time ) @xmath86 were made in the derivation of eqs . ( [ me2 ] ) , ( [ hdd ] ) and ( [ d ] ) ( appendix a ) : ( 1 ) @xmath87 , i.e. @xmath88 . this allows to take terms oscillating as @xmath89 to be kronecker deltas @xmath90 . ( 2 ) @xmath91 , where @xmath92 is the correlation time of the reservoir , namely @xmath93 is the width of the functions @xmath94 around @xmath70 . this allows to take the markov approximation and obtain a master equation which is local in time ( see appendix a and refs . @xcite ) . in the absence of the external laser light , i.e. taking the limit @xmath95 in eqs . ( [ sig2 ] ) and ( [ tran ] ) , we obtain @xmath96 , @xmath97 , @xmath98 and @xmath99 , such that the effective dipole - dipole hamiltonian , eq . ( [ hdd ] ) , reduces to that of rddi , @xmath100 then , once one of the atoms happens to be excited , it coherently and reversibly exchanges its excitation with the other , initially unexcited , atom . the complementary incoherent and radiative cooperative process , namely that of irreversible excitation exchange between the atoms via the emission of a real photon at the atomic frequency @xmath101 @xcite , is described in the non - hamiltonian part of the master equation ( [ me2 ] ) by terms proportional to @xmath102 , e.g. @xmath103 . these bare - atom processes are depicted by the diagram in fig . 1(a ) , where the atoms exchange their excitation via a photon at frequency @xmath104 . in the radiative process , the photon is real and hence has the frequency of the atomic transition @xmath105 , whereas the dispersive effect involves the summation over all diagrams with virtual photons @xmath104 around @xmath101 , as can be seen by the sampling of the functions @xmath106 and @xmath107 in eq . ( [ d ] ) , at @xmath99 for @xmath108 and @xmath109 , respectively . both rddi and its irreversible counterpart rely on the fact that one of the atoms is excited , and are absent when both atoms are in their ground states . as we shall see below , the essence of liddi revealed by our formalism is the possibility to excite the atoms by the laser , thus allowing them to interact via rddi . in the more general case , where the laser is turned on , we write the effective dipole - dipole hamiltonian ( [ hdd ] ) explicitly using eq . ( [ sig2 ] ) , as @xmath110 , \nonumber \\ & -&\hbar\delta^-_{12}\frac{(\delta+\bar{\omega})^2}{4\bar{\omega}^2}\left[e^{-i\mathbf{k}_l\cdot\mathbf{r}_{12}}\hat{s}^-_1\hat{s}^+_2+e^{i\mathbf{k}_l\cdot\mathbf{r}_{12}}\hat{s}^+_1\hat{s}^-_2\right ] , \nonumber \\ \label{hdd_d}\end{aligned}\ ] ] where @xmath111 . we also note that here the single - atom @xmath112 ( @xmath113 ) terms are not considered , since they are not important for the interaction between the dipoles , and that we have assumed the typical situation where @xmath114 is real . the complementary dissipative effects include terms of similar form , e.g. with @xmath115 and @xmath116 replacing the corresponding @xmath117 terms in ( [ hdd_d ] ) . in analogy to the bare - atom case , the above hamiltonian and its dissipative counterparts describe the rddi and cooperative emission , respectively , between two dressed atoms , whose level scheme , consisting of two levels separated by an energy @xmath118 [ eq . ( [ tran ] ) ] , is presented in fig . however , as opposed to the bare - atom case , where only one exchange process exists [ see eq . ( [ rddi ] ) and fig . 1(a ) ] , here several processes can occur : + ( i ) the first line in eq . ( [ hdd_d ] ) describes a process where the internal states of the dressed atoms do not change , giving rise to a cooperative energy shift to the dressed states . + ( ii ) the process described in the second line , shown in fig . 1(c ) , is analogous to the bare - atom rddi in eq . ( [ rddi ] ) and fig . 1(a ) , where the dressed states @xmath119 and @xmath120 take the role of the excited and ground states @xmath15 and @xmath16 , respectively . the relevant atomic transition is then @xmath121 , which in fig . 1(b ) is shown to be resonant with @xmath118 . however , since the dressed - states ( [ tran ] ) are written in a frame rotating with frequency @xmath25 , the actual transition frequency is @xmath122 . then , a real - photon , dissipative effect involves a photon exchange with frequency @xmath123 , whereas the dispersive rddi results from the equivalent virtual - photon effect , just as seen by the sampling of @xmath124 and @xmath125 at @xmath123 in eq . ( [ d ] ) for @xmath126 and @xmath127 , respectively . + ( iii ) the last process , described by the third line of ( [ hdd_d ] ) is that of the atomic transition @xmath128 , shown in fig . namely , atom 1 , initially in state @xmath120 , transfers a photon to atom 2 which is initially in state @xmath119 , and as a result both of them flip their states . the emission of a photon from the state @xmath120 and its absorption by the state @xmath119 underlying this process , is enabled by the fact that the dressed state @xmath120 contains a component of the excited state and likewise the dressed state @xmath119 contains a ground - state component . the transition frequency in the dressed - state basis is then @xmath129 , so that the actual frequency is @xmath130 . the three processes surveyed above , describe the interaction between the atoms mediated by photon modes at frequencies @xmath131 and @xmath132 , respectively , in analogy with the three spectral components of the well - known mollow - triplet of resonance fluorescence @xcite . the distinction between these three processes and their physical meaning are further discussed in sec . v c below . we now present a central insight of this paper , namely , that liddi ( and scattering ) originate from rddi ( and emission ) of dressed atoms . the average interaction energy between the dressed atoms , interpreted as the liddi potential , is obtained by averaging quantum mechanically over the hamiltonian @xmath133 , eq . ( [ hdd_d ] ) , @xmath134 , \label{liddi}\ ] ] where @xmath135 is the density matrix of the two - atom state . we note that the hamiltonian @xmath133 is assumed to be time - independent in this formalism . in fact , @xmath133 depends on the external degrees of freedom of the atoms , namely their positions @xmath136 , and hence becomes time - dependent due to their dynamics which is driven by the liddi potential @xmath137 . however , assuming this dynamics is slow enough , namely , @xmath138 is much smaller than any relevant energy scale such as @xmath118 , a description using the potential @xmath137 from eq . ( [ liddi ] ) can be viewed as a valid adiabatic approximation . ( [ hdd_d ] ) and ( [ liddi ] ) allow us to write the liddi potential for a general two - atom density operator @xmath135 . using the two - atom basis , @xmath139 and denoting the corresponding density matrix elements @xmath140 , we find @xmath141=1 - 2(\rho_{22}+\rho_{33})$ ] and @xmath142=\rho_{23}$ ] . the liddi potential between a pair of atoms is then , @xmath143 , \nonumber \\ u^{\pm}(\mathbf{r}_{12})&=&-\hbar\delta^{\pm}_{12}\frac{(\delta\mp\bar{\omega})^2}{4\bar{\omega}^2}\left[e^{\pm i\mathbf{k}_l\cdot\mathbf{r}_{12}}\rho_{23}+e^{\mp i\mathbf{k}_l\cdot\mathbf{r}_{12}}\rho^{\ast}_{23}\right ] . \nonumber \\ \label{liddi1}\end{aligned}\ ] ] here we have separated the potential @xmath144 into its components @xmath145 according to its contributions from processes mediated by different photon mode frequencies @xmath70 ( @xmath69 ) . for example , if @xmath135 does not allow for a superposition of @xmath51 states in each atom , then @xmath146 vanishes and the only contribution to the liddi originates from the @xmath132 mediated process , later interpreted here as the linear component of liddi . we recall that @xmath12 and @xmath147 contain the atomic - dipole matrix element @xmath30 which , in spherically - symmetric cases , is determined by the laser polarization @xmath29 , the only preferred direction ( see appendix b ) . the two - atom density matrix @xmath135 , to be used in eq . ( [ liddi1 ] ) for the liddi potential , is determined by the solution of the master equation ( [ me2 ] ) . in general , this yields a time - dependent potential @xmath137 obtained by averaging in eq . ( [ liddi ] ) with the time - dependent solution @xmath74 . if however the time - resolution of interest is longer than the typical decay times @xmath148^{-1}$ ] and @xmath149^{-1}$ ] , then the steady - state solution of the master equation , found by setting @xmath150 in ( [ me2 ] ) , can be used to obtain the stationary liddi potential . on the contrary , if the dynamics of interest occurs at times much shorter than the decay times @xmath151 ( @xmath69 ) , then @xmath74 and hence @xmath137 , may be obtained by simply solving eq . ( [ me2 ] ) without the dissipative terms . these two limiting cases are illustrated in sec . iv below for atoms in a cavity . turning to the dissipative counterpart of the liddi potential , the dressed atoms can scatter laser photons via spontaneous emission , resulting in a random diffusive motion for the atoms , on top of that driven by the liddi potential . upon ignoring ( for the time being ) the cooperative scattering terms , the scattering rate from a single atom ( e.g. atom 1 without loss of generality ) can be obtained by averaging the dissipative , imaginary - hamiltonian - like , terms of the master equation for @xmath152 . in analogy to eq . ( [ liddi1 ] ) , we obtain the single atom scattering rate @xmath153 as @xmath154 where @xmath155 $ ] is the density operator of atom 1 , obtained by tracing over atom 2 , and @xmath51 are the states of atom 1 . here we used @xmath156=1 $ ] and @xmath157=\langle \pm|\rho^{(1 ) } |\pm\rangle$ ] . the scattering rate given above is composed of scattering rates of photons at three distinct frequencies : @xmath132 , the frequency of the incoming laser light , and two sidebands at frequencies @xmath158 , which reproduce the mollow - triplet fluorescence spectrum . the amplitudes of the sidebands are determined by their corresponding scattering processes : an atom that scatters a @xmath159 photon is initially in state @xmath51 and finally in state @xmath160 . hence , the amplitude is related to the probability of the atom to be in the initial state , i.e. @xmath161 that appears in the expressions for @xmath162 . likewise , scattering at @xmath25 does not change the internal state , which explains its independence of the atomic state @xmath135 . our liddi result of eq . ( [ liddi1 ] ) reveals that liddi is mediated by virtual photons centered around three frequencies ; namely , that of the incident laser , @xmath25 , and those of the sidebands @xmath159 . the former , linear , process gives rise to the liddi term proportional to @xmath163 , whereas the latter give rise to a nonlinear liddi process manifest by @xmath164 . here , the term `` linear '' is more easily understood when we consider the scattering rate @xmath153 [ see eq . ( [ r ] ) ] : if the scattered real photons are at the same frequency as the incident light , as in the term @xmath165 , then the process is linear , as in elastic scattering . however , if the scattered photons are at a different frequency , here @xmath131 , then the scattering is inelastic and we call it a nonlinear process . likewise , viewing liddi as the virtual - photon ( and cooperative ) counterpart of scattering , we call the process mediated by virtual photons around @xmath132 `` linear '' and that mediated by @xmath159 `` nonlinear '' . as discussed below , previous treatments of liddi @xcite obtained only the linear contribution , whereas our formalism accounts also for the nonlinear response of the atoms to the laser light . most of the previous results of liddi , e.g. those of refs . @xcite , are obtained in the limit of weak laser amplitude @xmath12 compared to atom - laser detuning @xmath13 , assuming also that the atoms are in the ground state . navely , this result can be reproduced by our formalism in this large - detuning limit @xmath166 as follows : we first note that @xmath167 and further approximate @xmath168 for each atom , such that @xmath169 . we then have @xmath170 , such that we find that the nonlinear liddi vanishes , @xmath171 , and only the linear part remains , @xmath172 by recalling the dynamical polarizability of atoms in their ground - state and in the limit of far detuned laser , @xmath173 , and considering the known solution for the rddi @xmath163 in free space @xcite , we recover the free - space liddi result from @xcite . this result fails to capture the nonlinear part of liddi which is clearly important beyond the weak laser ( i.e. large detuning ) approximation . interestingly enough however , the above result , eq . ( [ ul ] ) , appears to be inconsistent even within the @xmath166 approximation , from two main reasons : ( 1 ) the assumption that the atoms always stay in the ground state , which although seems reasonable for large detuning , is not always true , especially for long times ( see sec . iv b below ) , and generally the state of the atoms @xmath135 has to be found by solving the master equation ( [ me2 ] ) . ( 2 ) even if we assume that the atoms are in the ground state , i.e. @xmath174 , and take the approximate @xmath16 from ( [ g ] ) instead of just taking @xmath168 , we find @xmath175 . then , to second order in @xmath176 , there exists also a nonlinear term , describing liddi mediated at ac - stark shifted atomic frequency ( for e.g. @xmath177 ) @xmath178 , @xmath179 which is totally missed by previous treatments and eq . ( [ ul ] ) . this is because in previous treatments linearity , namely , that the mediating virtual photons are at the same frequency as the incident laser photons , was imposed by the method of calculation . in appendix c , we briefly review two such previous approaches and show how linearity is imposed by them , thus clarifying the reason for the absence of the nonlinear term therein . in fact , as shown in sec . iv below , the above results , ( [ ul ] ) and ( [ unl ] ) , are obtained for liddi in the large - detuning and transient regime , for atoms initially in their ground states , whereas in steady - state only the linear part ( [ ul ] ) survives . moreover , eqs . ( [ ul ] ) and ( [ unl ] ) provide an interesting insight into the nature of liddi in the large - detuning limit : looking at the coefficient @xmath180 in front of the rddi rate @xmath147 in both equations , and recalling that it is proportional to the probability to excite an initially ground - state atom in this regime , we can interpret liddi as the excitation of an atom by the laser at probability @xmath180 thus allowing it to interact with another atom via rddi at rate @xmath147 . let us illustrate our formalism by considering the liddi potential for atoms inside a cavity geometry . such an analysis is very relevant for current experimental and theoretical research in many - atom systems inside cavities @xcite . we treat this example in two cases , the first case where dissipation ( emission / scattering ) is significant such that the relevant liddi is that in the steady state , and the second case in the transient regime where dissipation is negligible . for both cases however , we first have to find the underlying effects of rddi and emission . for simplicity , we assume a perfect cavity of length @xmath181 and effective area @xmath182 , and consider a single transverse mode , such that the relevant photon modes are those in the longitudinal direction @xmath183 , @xmath184 with frequencies @xmath185 and where @xmath186 is the polarization index , e.g. @xmath187 . assuming all three @xmath70 s ( @xmath69 ) are between and do not coincide with the cavity modes @xmath188 , there exists no atom dissipation to the cavity modes , i.e. they do not contribute to @xmath80 , whereas their contribution to the rddi , given by eqs . ( [ d ] ) , ( [ un ] ) , is @xmath189 with @xmath190 ( for a more complete treatment of rddi in a cavity see @xcite ) . since the transverse ( @xmath191 ) area of the cavity mirrors is finite , the atoms are also coupled to non - confined photon modes . we then expect that their spatial scaling is similar to that of free - space modes , such that their contribution to the cooperative effects @xmath192 and @xmath117 falls off initially like @xmath193 and is negligible for typical atomic distances larger than the atomic wavelength . then these modes only contribute to the single - atom spontaneous emission rate , which is approximated as that of free - space . finally , we have @xmath194 and @xmath195 from eq . ( [ dc ] ) . assuming the dissipation rate is larger or comparable to the rddi rate , i.e. @xmath196 , there are no interesting dynamics at time - scales much shorter than @xmath197^{-1}$ ] , such that dissipation is important . then , the relevant liddi is that in the steady - state , which is calculated by solving the master equation ( [ me2 ] ) for @xmath150 with the dissipation and rddi parameters from eqs . ( [ dc ] ) and ( [ gfs ] ) . for the density matrix elements of interest , we find the steady - state equation , @xmath198 \\ \mathrm{im}[\rho_{23 } ] \\ \end{array } \right ) = \left ( \begin{array}{c } 0 \\ -\tilde{\gamma}^- \\ -\tilde{\gamma}^- \\ 0 \\ 0\\ \end{array } \right ) , \label{mest}\ ] ] with @xmath199 , @xmath200 and @xmath201 . the steady - state solution is then @xmath202 and @xmath203 . inserting this solution into the state - dependent liddi potential in eq . ( [ liddi1 ] ) we find only linear liddi in steady - state , @xmath204 ^ 2}\right ] \nonumber \\ & & \times \hbar\delta^z_{12}\cos(\mathbf{k}_l\cdot\mathbf{r}_{12 } ) . \label{ust}\end{aligned}\ ] ] for the large detuning limit , @xmath166 , we expand the above result to lowest order and find , @xmath205 reproducing the linear - analysis result shown in sec . ( [ ul ] ) . our analysis shows that while rddi is mediated by confined cavity photon modes , the dissipation is driven by the non - confined free - space - like modes . for @xmath25 ( and hence @xmath70 s ) close to a cavity mode @xmath188 , this leads to the possibility of rddi much stronger than dissipation , as revealed by eqs . ( [ dc ] ) and ( [ gfs ] ) . therefore , in such a regime , it is relevant to consider the dynamics , and hence liddi , at time - scales much shorter than the dissipation time @xmath206^{-1}$ ] . this allows to set @xmath207 in the master equation ( [ me2 ] ) for @xmath135 , obtaining , @xmath208 \\ \end{array } \right ) = \left ( \begin{array}{ccc } 0 & 0 & 2\delta \\ 0 & 0 & -2\delta \\ -\delta & \delta & 0 \\ \end{array } \right ) \left ( \begin{array}{c } \rho_{22 } \\ \rho_{33 } \\ \mathrm{im}[\rho_{23 } ] \\ \end{array } \right ) , \label{met}\ ] ] and @xmath209=0 $ ] , where @xmath210 . taking the initial state where both atoms are in the ground state , @xmath211 , we find @xmath212 . the liddi potential in the transient regime @xmath213^{-1}$ ] then becomes @xmath214 . \nonumber\\ \label{ut}\end{aligned}\ ] ] here we divided the liddi into its linear and nonlinear contributions , @xmath215 and @xmath216 , respectively . in the large - detuning approximation @xmath166 we find @xmath217 while the linear part of this result is the same as that of the steady - state result , here there is an additional non - linear contribution , which in the large - detuning regime is equivalent to that discussed in sec . ( [ unl ] ) . since previous treatments @xcite have only yielded linear liddi processes , it is interesting to focus on the mechanism that leads to nonlinear effects in our more general approach . these effects are manifest in the probing of @xmath125 at frequencies different than that of the incident laser @xmath25 in the expression for liddi , eq . ( [ liddi1 ] ) , which is relevant at time resolution @xmath218 , with @xmath55 , as revealed by our dynamic master - equation - based formulation ( sec . ii b ) . in what follows , we discuss , from different perspectives , the origin of the nonlinear effects and when they are expected to be significant . first , let us take an intuitive dynamic approach towards the fluorescence of an illuminated atom described by hamiltonian ( [ ha ] ) . the solution for the dynamics of the atomic excitation - probability @xmath219 is known to be @xcite @xmath220.\ ] ] hence , the excitation probability @xmath219 , and consequently the scattering rate from a single atom , is modulated . since this scattered radiation is centered around @xmath25 [ recalling that hamiltonian ( [ ha ] ) is written in a frame rotated by @xmath25 ] , this expression implies an amplitude - modulated scattered signal , whose spectrum contains peaks around @xmath132 and @xmath131 , as found above . the distance between the peaks is @xmath118 , hence they can be resolved only at times larger than @xmath222 . this provides an intuitive interpretation for the scattering rate in eq . ( [ r ] ) and the liddi related to it , eq . ( [ liddi1 ] ) : at times @xmath218 it is possible to distinguish between scattered or liddi - mediating photons at three frequencies @xmath70 ( @xmath69 ) ; hence , @xmath223 or @xmath125 , respectively , are probed at these frequencies for such a time - resolution . let us recall the master - equation approach used to derive the liddi and scattering rate expressions above : in sec . ii a we first solved for the relevant operator of the combined atom - laser , or dressed - atom , system that couples it to the reservoir , i.e. @xmath224 . in analogy to the bare - atom case , where @xmath225 , we then proceeded to calculate the rddi between atoms , however this time they are dressed , with the time - dependent @xmath226 replacing the bare time - independent @xmath227 . then , from the point of view of the reservoir , it interacts with a dipole that oscillates at all frequencies contained in @xmath228 ( around @xmath101 ) , and hence responds resonantly with all these frequencies , which are the same @xmath70 ( @xmath69 ) as before . this explains why @xmath223 or @xmath125 are probed at these ( resonant ) frequencies . moreover , it provides a frequency - domain picture for the time - scales at which the nonlinear effect , namely , the probing of @xmath223 and @xmath125 at frequencies other than @xmath132 , becomes significant : the overlap integral in eq . ( [ a1f ] ) of appendix a gives distinct results for different @xmath70 ( @xmath69 ) when the sinc function @xmath229 is narrower than the differences between different @xmath70 , @xmath218 and as long as the widths of @xmath223 and @xmath125 around @xmath70 , @xmath93 , are not larger than their difference , @xmath230 . this approach to our results is related to known theories for the interaction between modulated systems and a reservoir @xcite : there , modulation of system parameters , such as atomic energy levels , by external fields , yields the replacement of the bare @xmath231 operator by @xmath232 in the system - reservoir coupling , @xmath233 being a time - dependent function determined by the modulation , in analogy to @xmath234 in our case . by writing @xmath235 , one then obtains that the reservoir response , e.g. @xmath223 , becomes @xmath236 instead of the bare - system @xmath237 @xcite , in analogy with our liddi and bare - atom - rddi results , respectively . in section ii c we have treated the effects that underlie liddi and scattering as dipolar interactions driven by three distinct processes , each involving a different transition between the dressed states of fig . 1(b ) . equations ( [ hdd_d ] ) , ( [ liddi1 ] ) show that each of these processes has a different amplitude : @xmath238 for the process mediated by @xmath132 and @xmath239 for the process mediated by @xmath131 . in the following we show that these amplitudes are directly related to those of corresponding transitions between laser - quantized dressed states @xcite . in the spirit of chapter vi in ref . @xcite , we consider the coupling of a quantized laser mode @xmath240 to an atom via @xmath241 and assume the laser to be in a coherent state with average photon occupation @xmath242 . for large enough @xmath242 , we can ignore the laser photon fluctuations around the average @xmath243 compared to the average @xmath242 , such that in the @xmath242-manifold of the atom - laser space , @xmath244 , the relevant rabi frequency is @xmath245 and the corresponding dressed states are @xmath246 with eigenenergies @xmath247 , respectively . the energy level diagram of two adjacent manifolds is plotted in fig . 2 . radiative transitions , i.e. emission of a photon from the atom to the vacuum , do not change the number of laser photons but take the atom from the excited to the ground state @xmath248 , i.e , involve only a @xmath249 operation . hence , such transitions can only occur between adjacent manifolds . there are four possible transitions , marked by dashed lines in fig . 2 : the two at the center represent the linear process which involves an emission of a photon at the laser frequency @xmath132 and the other two transitions at frequencies @xmath131 represent the nonlinear processes . hence , the strength of any process that involves an emission or exchange of a photon at frequency @xmath70 ( @xmath69 ) as in the scattering and liddi of eqs . ( [ r ] ) and ( [ liddi1 ] ) should be proportional to the probability of the transition between dressed states with energy difference @xmath250 in the emitting atom . figure 2 shows that a nonlinear process with a @xmath122 photon involves the transition from @xmath251 to @xmath252 with transition amplitude @xmath253 [ inferred from eq . ( [ drs ] ) ] . the associated probability is then @xmath254 , just as in the prefactor that appears in the @xmath255 scattering and liddi of eqs . ( [ r ] ) and ( [ liddi1 ] ) . likewise , the relevant nonlinear @xmath256 process seen in fig . 2 is that of the @xmath257 transition , with amplitude @xmath258 and probability @xmath259 , as in the prefactor of the @xmath260 terms of eqs . ( [ liddi1]),([r ] ) . finally , there are two different processes that involve a @xmath25 photon , i.e. @xmath261 and @xmath262 with identical transition probabilities @xmath263 . summing the probabilities of both processes we obtain @xmath238 as in the prefactor of the @xmath264 terms of eqs . ( [ liddi1]),([r ] ) . this study has addressed laser - induced interactions between polarizable dipoles ( liddi ) , e.g. atoms , molecules or nanoparticles , in a general geometry . it is expected to prove useful to the understanding and design of liddi between polarizable dipoles in confined electromagnetic environments , encountered in various experimental systems @xcite . we have analyzed in detail liddi in a general geometry without assuming that the dipoles respond linearly to the applied laser field , and have found that it can be described as the resonant dipole - dipole interaction ( rddi ) between laser - dressed atoms in the same geometry . this description has allowed us to obtain general formulae for both liddi and scattering in terms of the corresponding rddi and spontaneous emission , respectively [ eqs . ( [ liddi1 ] ) and ( [ r ] ) ] . the liddi and scattering revealed by our formulae contain contributions due to the _ nonlinear _ response of the dipoles to the laser light , which are missing in previous treatments , e.g. those of ref . @xcite . by reviewing the principles of these previous treatments , we have demonstrated their imposed linearity as the source of the absence of nonlinear liddi / scattering effects in these treatments . we have then explained , using several simple and intuitive approaches , the origin of the nonlinear liddi and scattering terms in our expressions . considerable progress has been reported in the ability to couple atoms to cavities @xcite and nano - fibers @xcite . this promising direction has already lead to exciting predictions @xcite and experiments @xcite concerning the many - body physics of illuminated atoms in confined geometries . in this context , the generality of our formalism and its illustration for the analysis of liddi in a cavity , suggest its applicability for further exploration of this timely line of research . we appreciate useful discussions with ilya averbukh . the support of isf and bsf is acknowledged . here we present the derivation of the master equation ( me ) , eq . ( [ me2 ] ) . beginning with the double commutator in the me ( [ me1 ] ) , @xmath265 $ ] , it includes four terms , @xmath266 . \label{com}\end{aligned}\ ] ] the rest of the derivation is similar for these four terms , hence in the following we present in detail only that of the first term . inserting @xmath267 from eq . ( [ hi2 ] ) into the first term in ( [ com ] ) we obtain @xmath268 . \nonumber\\ \label{a1a}\end{aligned}\ ] ] taking the trace @xmath269 in eq . ( [ me1 ] ) over the reservoir in the vacuum state , we have @xmath270=\langle \hat{a}_k \hat{a}_{k'}^{\dag } \rangle=\delta_{kk ' } , \label{tr}\ ] ] and similarly @xmath271=\mathrm{tr}_v[\rho_v\hat{a}_k^{\dag } \hat{a}_{k'}]=\mathrm{tr}_v[\rho_v\hat{a}_k^{\dag } \hat{a}_{k'}^{\dag}]=0 $ ] , such that only the third term in ( [ a1a ] ) contributes , and the trace over ( [ a1a ] ) becomes @xmath272 assuming that the relevant time - scale for the effects we are interested to resolve ( coarse graining resolution ) , @xmath86 , is much larger than @xmath273 for any @xmath274 , or simply that @xmath275 , we can view the exponent on the right hand side of ( [ a1b ] ) @xmath276 as a fast oscillation that averages out for @xmath274 and hence replace it with a kronecker delta @xmath90 , yielding for ( [ a1b ] ) @xmath277 taking the continuum limit @xmath278 , where @xmath279 is the density of photon modes @xmath18 in the considered geometry [ e.g. @xmath280 in free - space or @xmath281 in a waveguide with on - axis wavenumber @xmath282 , we identify the vacuum reservoir two - point ( autocorrelation ) spectrum as @xmath283 then , performing the integration @xmath284 in the me ( [ me1 ] ) and using ( [ g ] ) , the term in ( [ a1c ] ) becomes @xmath285 such that the me has @xmath286 in its left - hand side ( lhs ) and the above term ( [ a1d ] ) , along with 4 similar terms , in the right - hand side ( rhs ) . _ markov approximation_. assume now that we are interested to solve the me for very short times @xmath287 , much shorter than the typical time - scale of variation for @xmath74 . namely , we assume that @xmath288 hardly changes within the integral from @xmath289 to @xmath290 , such that it can be taken out of the integral and approximated as @xmath291 , which is the so - called markov approximation . at first , this seems like a strange assumption , considering the fact that eventually we are interested to find an equation for the dynamics of @xmath74 on time - scales @xmath290 where @xmath74 does change appreciably , but this will become clearer in the following , where @xmath86 is interpreted as a coarse - graining time . next , we need to solve the resulting integral over @xmath292 , @xmath293 where the coordinate transformation @xmath294 was used and @xmath295 is the heaviside step function . using also the relations @xmath296 where @xmath297 denotes the principal value upon integration , the me term ( [ a1d ] ) becomes @xmath298 . \label{a1e}\end{aligned}\ ] ] denoting the sinc function @xmath299 , ( [ a1e ] ) becomes @xmath300 , \nonumber\\ \label{a1f}\end{aligned}\ ] ] with @xmath106 and @xmath107 from eq . ( [ d ] ) in the main text . this expression describes an overlap integral between the complex linear response ( spectrum ) of the vacuum reservoir to the two - atom system @xmath301 and the sinc function @xmath229 of width @xmath302 around @xmath303 . denoting the typical width of the vacuum spectral response functions @xmath106 and @xmath107 around @xmath70 by @xmath93 , and assuming @xmath91 , we can approximate @xmath304 and finally get @xmath305\tilde{s}_{i\nu}\tilde{s}_{i\nu'}^{\dag}\rho(t ) . \label{a1g}\ ] ] the rest of the three terms that appear on the rhs of the me can be treated similarly , thus obtaining the me in eq . ( [ me2 ] ) . however , here it is written for short times @xmath287 w.r.t the typical time - variation of @xmath74 . nevertheless , since the coefficients of the me are constant , it has the form @xmath306 with @xmath307 a time - independent superoperator . then , its formal solution for @xmath287 is @xmath308 , and similarly @xmath309 for @xmath310 . this procedure can be repeated successively for any time duration @xmath290 , so that the me is in fact valid for arbitrary long time @xmath290 , including time durations where the system state @xmath74 changes considerably . _ separation of time - scales_. the coefficients of the me terms become time - independent in the markov approximation , namely by assuming that the time - resolution of interest , @xmath86 , is much larger than the so - called correlation time of the reservoir @xmath92 . on the other hand , in order to have sufficient resolution in probing the system dynamics , recall that we assumed @xmath86 to be much shorter than the typical time of variation for the system , that can be directly read from the me to be @xmath311 , i.e. the inverse of the rates @xmath312 . this implies , @xmath313 namely , that the time - scales for the reservoir memory / correlation , @xmath92 , and the system dynamics , @xmath314 , are well separated . let us consider the dependence of the liddi potential , eq . ( [ liddi1 ] ) , on the orientation of the dipolar matrix element @xmath30 and laser polarization . we note that both @xmath12 and @xmath147 ( via @xmath315 ) depend on @xmath30 , which we recall to be @xmath316 , where @xmath317 is the dipole operator ( e.g. the electron charge times electron position operator in an atom ) . for the case of polarizable dipoles with a fixed orientation , @xmath30 is fixed , and so are the products @xmath318 and @xmath319 , such that @xmath12 , @xmath147 and hence the liddi @xmath144 can be calculated . however , for cases where the atomic polarizability is effectively isotropic , the only preferred direction is that of the laser polarization , such that it effectively determines the orientation of @xmath30 . in the following we illustrate this idea for two typical scenarios of atomic and molecular dipoles _ isotropic atoms_. consider the ground state @xmath16 to have a spherical symmetric electronic wavefunction , i.e. with spherical harmonic quantum numbers @xmath320 : @xmath321 . then , the quantization axis @xmath183 that fixes the quantum number @xmath322 is arbitrary and , assuming the laser is linearly polarized , we can choose it to be that of the laser polarization , i.e. @xmath323 . consider now the possible dipole - allowed excited states in a spherically symmetric potential for the electron , @xmath324 with @xmath325 . then , by dipole selection rules , the dipole operator is generally written as @xmath326 where @xmath327 is a right / left circular polarization unit vector , respectively . then , the product @xmath328 in @xmath12 implies that the only state to be excited by the laser is that of @xmath329 , namely , @xmath330 , such that the relevant dipolar matrix element that appears in the liddi potential @xmath144 is @xmath331 . considering a circular laser polarization , @xmath332 , the arbitrary quantization axis @xmath183 is now chosen to be perpendicular to both circular polarizations , and accordingly the product @xmath319 in @xmath12 imposes @xmath333 and @xmath334 , again parallel to the laser polarization . _ randomly oriented anisotropic molecules_. consider now molecules with an anisotropic polarizability , e.g. where for a given quantization axis not all possible @xmath325 excited states exist , and the dipole matrix element is some fixed @xmath30 . however , by allowing for random molecule orientation , the quantization axis of each molecule ( with index @xmath36 ) becomes random and so does the orientation of its corresponding dipole matrix element @xmath335 . we recall , @xmath336 where @xmath337 is the unit vector of @xmath338 and where we considered for simplicity @xmath339 in the large - detuning approximation as in eqs . ( [ ul ] ) and ( [ unl ] ) . the , choosing the spherical coordinates of the vector @xmath335 with angles @xmath340 and @xmath341 , setting @xmath323 , we have @xmath342 \nonumber \\ \mathbf{d}_{\nu}\cdot \mathbf{e}_l&=&|\mathbf{d}|\cos(\theta_{\nu}),\end{aligned}\ ] ] with @xmath343 ( @xmath344 ) . inserting the above products into @xmath144 in ( [ u12 ] ) , and assuming no correlation between the random orientation of different molecules , we obtain @xmath345,\end{aligned}\ ] ] where @xmath346 denotes averaging over orientation with an isotropic distribution . performing the averaging we finally get @xmath347 this result is equivalent to taking @xmath348 in @xmath147 and @xmath12 . therefore , again , yet for a different spherically symmetric system , we effectively obtained @xmath349 . in section iii we have mentioned that the apparent discrepancy between our liddi results and those obtained by previous treatments ( e.g. @xcite ) is due to the linearity imposed in those treatments . here we would like to briefly review the essence of two typical approaches used previously , and highlight how linearity is imposed in them , thus preventing them to capture the nonlinear terms of liddi revealed by our formalism . consider the following description of liddi depicted in fig . 3(a ) : a strong laser beam illuminates two atoms , such that atom 1 is subject to the laser electric field @xmath350 and atom 2 to the field @xmath351 . focusing on atom 1 , the laser induces its polarization , which to first order in @xmath28 is @xmath352 , @xmath353 being the atomic linear polarizability at the laser frequency @xmath25 . then , since the polarized atom 1 is an oscillating dipole , it scatters a field @xmath354 , which then arrives at atom 2 where it is given to lowest order ( born approximation ) by @xmath355 . here @xmath356 is proportional to the green s function of the electromagnetic - field propagation at frequency @xmath25 from @xmath357 to @xmath358 in the considered geometry . since the electromagnetic energy of atom 2 ( like any dipole ) is @xmath359 , @xmath360 being the total electric field at @xmath358 , the lowest - order interaction energy @xmath144 , interpreted as liddi , becomes , @xmath361 where the last step ( right - hand side ) is valid for atoms in their ground state and for large detuning . the result and description above render the essence of treatments used in refs . . its analogy to the linear part of our result ( [ ul ] ) , @xmath362 , becomes transparent upon recalling that the rddi term @xmath125 is directly related to the green s function @xmath363 @xcite . the origin of linearity in this treatment is clear : the atoms are modeled as polarizable dipoles by using their linear response to electric fields , @xmath364 , at the same frequency as the incident radiation . this leads to the sampling of the green s function at frequency @xmath25 , thereby obtaining @xmath365 . clearly , by imposing a description where the highly nonlinear two - level atoms linearly respond to the strong laser , effects due to real or virtual photons at frequencies other than @xmath25 can not be revealed . a different approach , applied in e.g. refs.@xcite , uses a qed time - independent perturbative treatment similar to that used to calculate the van der waals and casimir - polder forces . taking the state of the laser as a number state for the time - being , the laser and the vacuum are treated as a single system ( the photons ) , where there are @xmath242 photons in the mode with @xmath25 and vacuum in the rest of the modes . then , the liddi potential is interpreted as the lowest ( fourth ) order correction ( in the atom - photon coupling ) of the state @xmath366 , namely , both atoms in the ground state , @xmath242 photons in the laser mode @xmath181 and vacuum ( @xmath289 ) in the rest of the modes . this energy correction is then given by @xcite @xmath367 with @xmath20 from eq . ( [ h ] ) . this is a sum over intermediate / virtual states , @xmath368 , @xmath369 and @xmath370 , where @xmath371 is the energy of state @xmath372 without the interaction @xmath20 , e.g. @xmath373 ( taking the atomic ground - state energy to zero here ) . the possible sets of virtual states can be represented by diagrams , such as that in fig . 3(b ) , with a total of 24 diagrams @xcite . in order to illustrate the essence of this approach , it is enough to focus on the diagram in fig . 3(b ) : it describes the intermediate states @xmath374 , @xmath375 and @xmath376 with energies @xmath377 , @xmath378 and @xmath379 , respectively , where @xmath104 denotes the mode of the virtual photon that mediates the interaction between vertices ii and iii , with @xmath380 . by identifying the laser rabi frequency as @xmath381 , @xmath382 being the dipole coupling to the laser mode , the resulting energy correction for this diagram becomes @xmath383 yielding again the linear liddi . the crucial step that leads to linearity here is the treatment of atom - laser interaction in vertices i and iv : these vertices give rise to the factor @xmath384 which is indeed small for large detuning . however , such a treatment , where the interaction with the strong laser is taken only in two bare atom - laser vertices , does not allow to account for the nonlinear effect . c. j. pethick and h. smith , _ bose - einstein condensation in dilute gases _ ( cabridge university press , 2002 ) . c. cohen - tannoudji , _ atomic motion in laser light _ , in _ fundamental systems in quantum optics _ , les houches , session liii , 1990 , pp . 1 - 164 ( elsevier science publisher b.v . , 1992 ) . g. kurizki , i. e. mazets , d. h. odell and w. p. schleich , int . b * 18 * , 961 ( 2004 ) . d. p. craig and t. thirunamachandran , _ molecular quantum electrodynamics _ ( academic , london , 1984 ) . t. thirunamachandran , mol . phys . * 40 * , 393 ( 1980 ) . a. salam , advances in quantum chemistry * 62 * , 1 ( 2011 ) . d. odell , s. giovanazzi , g. kurizki and v. m. akulin , phys . * 84 * , 5687 ( 2000 ) . s. giovanazzi , d. odell and g. kurizki , phys . rev . lett . * 88 * , 130402 ( 2002 ) . d. h. j. odell , s. giovanazzi and g. kurizki , phys . lett . * 90 * , 110402 ( 2002 ) . m. lemeshko , phys . a * 83 * , 051402(r ) ( 2011 ) . j. rodrguez and d. l. andrews , opt . com . * 282 * , 2267 ( 2009 ) . d. e. chang , j. i. cirac and h. j. kimble , phys . * 110 * , 113606 ( 2013 ) . e. shahmoon , i. mazets and g. kurizki , arxiv:1309.0555 ( 2013 ) . a. goban , k. s. choi , d. j. alton , d. ding , c. lacroute , m. pototschnig , t. thiele , n. p. stern and h. j. kimble , phys . lett . * 109 * , 033603 ( 2012 ) . e. vetsch , d. reitz , g. sagu , r. schmidt , s. t. dawkins , and a. rauschenbeutel , phys . lett . * 104 * , 203603 ( 2010 ) . h. ritsch , p. domokos , f. brennecke and t. esslinger , rev . phys . * 85 * , 553 ( 2013 ) . r. h. lehmberg , phys . rev . a * 2 * , 883 ( 1970 ) . d. petrosyan and g. kurizki , phys . lett . * 89 * , 207902 ( 2002 ) ; 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polarizable dipoles , such as atoms , molecules or nanoparticles , subject to laser radiation , may attract or repel each other . we derive a general formalism in which such laser - induced dipole - dipole interactions ( liddi ) in any geometry and for any laser strength are described in terms of the resonant dipole - dipole interaction ( rddi ) between dipoles dressed by the laser . this approach provides a simple route towards the analysis of liddi in a general geometry . our general results reveal liddi effects due to nonlinear dipolar response to the laser , previously unaccounted for . the origin of these nonlinear effects is discussed . our general formalism is illustrated for liddi between atoms in a cavity .

You are an expert at summarizing long articles. Proceed to summarize the following text: the fossil record of star formation and galaxy evolution is imprinted on the spatial distribution , ages and metallicities of galactic stellar populations . surprisingly little is known about the old and intermediate - age stellar populations in massive galaxies outside our own milky way . located at @xmath1780 kpc , m31 provides our best opportunity to explore the stellar populations across the face of a large disk galaxy . furthermore , its favourable inclination means that the disk and halo components at large radii can be easily distinguished and independently studied . most previous studies of m31 have either been based on observations of single fields ( eg . holland et al 1996 , durrell et al 2001 ) or on large - area surveys limited to the bright disk ( eg . walterbos & kennicutt 1988 ) . it is generally thought that m31 resembles the milky way in many ways , although it is somewhat larger , more luminous and has a denser stellar halo ( see freeman 1999 for a review ) . the disparity between the mean metallicity of m31 s field halo and that of the milky way ( @xmath2 factor of 10 lower ) was first recognized by mould & kristian ( 1986 ) but remains to this day poorly understood . we searched the hst archive for all deep ( t@xmath34000s ) wfpc2 pointings towards m31 in the f555w and f814w filters . we focus here on those fields which sample the disk and halo at large radius ( r@xmath4kpc ) . @xmath5 ) , 47 tuc ( [ fe / h]@xmath6 ) and ngc 6553 ( [ fe / h]@xmath7 ) . also indicated is the mean v magnitude of the blue horizontal branch detected in m31 halo by holland et al ( 1996).,title="fig : " ] @xmath5 ) , 47 tuc ( [ fe / h]@xmath6 ) and ngc 6553 ( [ fe / h]@xmath7 ) . also indicated is the mean v magnitude of the blue horizontal branch detected in m31 halo by holland et al ( 1996).,title="fig : " ] figure 1 shows the wfpc2 cmd of a field at @xmath8 kpc ( or 5 disk scale lengths ) along the major axis . based on extrapolation of the structural parameters of walterbos & kennicutt ( 1988 ) , we expect 95% of the stars in this field to belong to the disk . the prominence and morphology of the red giant branch ( rgb ) and red clump ( rc ) indicate an old - to - intermediate age , fairly metal - rich population . comparison of the outer disk rgb with globular cluster fiducials indicates a mean metallicity comparable to 47 tuc ( [ fe / h]@xmath6 ) . the significant width of the rgb exceeds that of photometric errors , and is most easily explained by an intrinsic dispersion ( @xmath9 dex ) in the metallicity of the stellar population . while these properties have been noted before for the m31 halo ( eg . holland et al 1996 ) , we find here that they also characterise the outer disk . see ferguson & johnson ( 2001 ) for details . the rc properties , when coupled with the mean metallicity from the rgb colour , suggest a mean age for the population of @xmath10 gyr ( cole 1999 , girardi & salaris 2001 ) . this is also consistent with the apparent lack of asymptotic giant branch stars above the tip of the rgb ( i@xmath120.5 ) , which would represent young - to - intermediate age ( 26 gyr ) shell he - burning stars . finally , we note the marginal detection of horizontal feature in the cmd connecting the blue plume at v@xmath11 to the rc . while this could represent the subgiant branch of a @xmath12 gyr population , it is more likely to be an old ( @xmath13 gyr ) , metal - poor ( [ fe / h]@xmath14 ) horizontal branch ( see figure 1 ) . figure 2 shows cmds for a representative halo - dominated field and a field in which disk and halo are expected to contribute in roughly equal amounts . apart from stellar density ( which reflects both intrinsic variations as well how much of the wf area was useable in our analysis ) , the dominant stellar population in these fields appears strikingly similar ( and indeed , both resemble the outer disk cmd in figure 1 ) . the homogeneity of halo and outer disk populations was first remarked upon by morris et al ( 1994 ) from shallow ground - based data , but we show here the similarity holds to well below the horizontal branch . while providing detailed information on the stellar populations , the hst pointings tell us nothing about the large - scale structure of m31 ( wfpc2 fov @xmath15 kpc@xmath16 ) . we have therefore embarked upon a ground - based , moderate - depth panoramic imaging survey of the outskirts of m31 with the int wide field camera . to date , 58 contiguous fields ( 0.3 @xmath17 per pointing ) have been observed in the se half of the galaxy , mapping out to a projected radius of @xmath0 kpc . an exciting first result has been the discovery of a giant stream of stars ( overdensity a factor of 2 ) in the halo of m31 near the southern minor axis ( ibata et al 2001 ) . the stream extends beyond the limit of our imagery , or @xmath18 kpc at this position angle , and has an average v - band surface brightness of @xmath19 . if an old coeval population is assumed , the excess stream population is found to be of similar or slightly higher metallicity than the m31 field halo and outer disk . the stream appears to lie along a line which connects m32 and ngc 205 , the two dwarf elliptical companions of m31 . both these satellites display rather odd properties for their morphological class including young and/or intermediate - age stars and in the case of ngc 205 , cold gas and both exhibit distorted isophotes in the outer regions , suggestive of tidal distortion and possibly disruption . the similarity between the metallicities of these dwarfs and that of the stream suggests that one or both of them may be the origin of the feature . we are exploring the stellar populations in the outskirts of m31 using deep hst archival pointings in combination with a ground - based panoramic imaging survey . we summarise our first results as follows : + @xmath20 the disk of m31 appears to have a significant mean age ( @xmath21 gyr ) . for current cosmologies , this lookback time corresponds to a redshift of @xmath22 by which about half the stellar disk was in place at 30 kpc . this may be problematic for theories which invoke delayed disk formation as a way to circumvent the angular momentum problem seen in numerical simulations of galaxy formation ( eg . weil et al 1998 ) . equally puzzling is our finding that the dominant populations in the outer disk and halo are remarkably similar , suggesting these components have experienced similar formation epochs and evolutionary histories . + @xmath20 the discovery of a giant stellar stream in the outer halo of m31 attests to the fact that the hierarchical process of galaxy formation continues to the present - day and that at least some fraction of the m31 field halo has not formed _ in - situ _ , but has been accreted from lower mass objects . could this interaction have polluted the outer regions of m31 to such an extent that it explains that apparent uniformity of the stellar populations in these parts ? future observations will help us assess this possibility more throughly . i gratefully acknowledge the input of my collaborators on the work discussed here specifically , rachel johnson and nial tanvir on the hst studies and mike irwin , rodrigo ibata , geraint lewis and nial tanvir on the int / wfc survey . thanks also to the leids kerkhoven - bosscha fonds for financial assistance to attend this conference .

we discuss the first results from ongoing studies of the resolved stellar populations in the outskirts of our nearest large neighbour , m31 . deep hst / wfpc2 archival observations are used to construct colour - magnitude - diagrams which reach well below the horizontal branch at selected locations in the outer disk and halo , while a panoramic ground - based imaging survey maps spatial density variations through resolved star counts to a projected radius of @xmath0kpc .

You are an expert at summarizing long articles. Proceed to summarize the following text: that the global properties of an extended system may be mapped to the boundaries is an idea that has found success in holographic theories of general relativistic systems @xcite , and in magnetically confined plasmas @xcite . we report on a similar behavior observed in incompressible hydrodynamic flows in a taylor - couette apparatus where it is observed that certain characteristics of the global flow are largely dictated by the boundaries . this finding is particularly relevant for experiments that examine quasi - keplerian ( qk ) flows , that is , rotation satisfying @xmath0 where @xmath1 , @xmath2 is the fluid angular velocity and @xmath3 is the radial coordinate , as models of astrophysical systems , namely accretion disks . numerous recent studies have commented on the hydrodynamic stability of such systems @xcite , with extensions to magnetohydrodynamics in electrically conducting fluids @xcite . while there is some disagreement between studies as to whether hydrodynamic turbulence can be induced in qk flows , the balance seems to lean toward the negative , at least insofar as incompressible turbulence is considered , and points to the important role of magnetohydrodynamic effects in astrophysical systems . however , while it is known that qk flows are linearly stable it remains unknown whether there exists a nonlinear transition to turbulence , even for incompressible hydrodynamic systems . some experiments @xcite and simulations @xcite indicate that such a transition is not likely , while others present evidence that suggests that a subcritical transition may exist @xcite and some simulations find significant transient growth of perturbations that may allow for nonlinear effects to enter @xcite . fluid experiments in other regimes of operation that are not astrophysically- elevant have observed bi - stability @xcite , suggesting that should a similar mechanism exist for qk systems then a subcritical pathway to turbulence may explain angular momentum transport in accretion disks @xcite . we show in this work that the influence of the boundaries is intimately connected to the global structure of flows in taylor- ouette experiments and , by extension , is also related to the tendency of these systems to generate and sustain turbulence . one of the long - standing challenges of taylor - couette experiments in the quest to understand angular momentum transport in astrophysically - relevant flows has been the parasitic presence of ekman circulation ( secondary circulation ) induced by the mismatch between the fluid velocity and the solid body rotation of the axial boundaries . a significant reduction in ekman circulation has been realized in experiment by using axial boundaries that are split into multiple rings capable of differential rotation . under particular boundary conditions , azimuthal velocity profiles of the fluid can be generated that very nearly match that of ideal couette rotation @xcite , the rotation profile that is expected in the absence of axial influences for a constant radial flux of angular momentum , and has been observed to hold over a wide range of reynolds numbers @xcite . in contrast , studies in the `` classical '' configuration where the axial boundaries rotate with the outer cylinder have shown performance that further deviates from ideal couette as the reynolds number is increased @xcite . such trends are revealing of whether these systems are dominated by boundary interactions or internal dynamics , a distinction with important consequences for the applicability of such experiments to interpretation of astrophysical systems , especially at large reynolds numbers . first , through the experiments reported here we identify two necessary criteria that define constraints on the boundary configurations that allow near - ideal flows to develop . we then discuss the competing roles of radial ( stewartson ) boundary layers and axial ( ekman ) boundary layers , from which we develop a model that describes the quantitative departure of the rotation profiles from ideal couette flow as a function of the angular momentum fluxes through the boundaries . a taylor - couette ( tc ) device is a system of coaxial cylinders that rotate independently of each other with the experimental fluid region between . the tc apparatus used in these studies , called the hydrodynamic turbulence experiment ( htx ) , is a modified version of the classical device in that the axial boundaries in htx are segmented to allow differential rotation across the boundaries @xcite . the inner cylinder radius is @xmath4 cm and the outer cylinder radius is @xmath5 cm . the inner radius and outer radius of the independent rings are defined by the parameters @xmath6 and @xmath7 , respectively . the axial length of the experimental volume is @xmath8 cm , giving an aspect ratio of @xmath9 ( see fig . [ fig1 ] ) . corresponding components on the top and bottom are driven by the same motor so that the system is up - down symmetric . the angular velocities of the inner cylinder , outer cylinder and rings are identified by @xmath10 , @xmath11 and @xmath12 , respectively . rotary encoders report the speed of the motors to the control system . a laser doppler velocimeter ( ldv ) diagnostic system is used to measure the local , azimuthal velocity ( @xmath13 ) , which in the experiments reported here were measured at the midplane of the device . the ldv system is calibrated by measuring fluid flow in solid body for which , after spin - up , the only velocity component is @xmath13 , which is a unique function of the motor speeds . for these studies we define the global shear reynolds number as @xmath14 , where @xmath15 is the geometric- ean radius , @xmath16 and @xmath17 is the kinematic viscosity , approximately @xmath18 m@xmath19/s for water . ( color online ) illustration of the htx device at pppl showing the segmented axial boundaries ( yellow , blue and green segments ) , the inner cylinder in black , the outer cylinder in green and the working fluid in dark blue.,width=377 ] the core observation of this work is summarized in fig . [ fig2 ] , presenting flows under two different values of normalized shear ( @xmath20 and @xmath21 ) and three different boundary conditions : ` split ` , ` optimized ` , and ` ekman ` . for the @xmath20 cases these configuration speeds , reported as @xmath10-@xmath12-@xmath11 , are multiples of 350 - 350 - 50 rpm ( ` split ` ) , 350 - 185 - 50 rpm ( ` optimized ` ) , and 350 - 50 - 50 rpm ( ` ekman ` ) . for the @xmath21 these are 250 - 250 - 50 rpm ( ` split ` ) , 250 - 140 - 50 rpm ( ` optimized ` ) , and 250 - 50 - 50 rpm ( ` ekman ` ) . the reference profile for these studies is described by ideal couette rotation , defined in terms of angular velocity as @xmath22 , where @xmath23 , @xmath24 . as a function of azimuthal velocity the couette solution is @xmath25 . it is interesting that while the ` ekman ` and ` split ` configurations exhibit progressive departure from ideal couette as the reynolds number is increased , the shape of the ` optimized ` cases is nearly invariant with respect to scaling of the reynolds number . ( color online ) scaled measurements of @xmath13 at various @xmath26 for ( a ) @xmath20 and ( b ) @xmath21.,width=377 ] recognizing that the axial boundaries in the ` split ` cases will tend to increase the angular momentum flux to the bulk , and conversely for the ` ekman ` cases , we begin by considering the ansatz that the balance of angular momentum fluxes across the axial boundaries determines the deviation from ideal couette . a continuous variation of @xmath12 from @xmath11 to @xmath10 would then suggest that there is some intermediate state for which the profile must pass close to ideal couette rotation . we now address whether there is a method for predicting this optimal value of @xmath12 . the transition between the bulk flow and the walls of the tc device , which may move at very different speeds , occurs over thin boundary layers . the structure of these boundary layers depends on whether we are considering the balance of forces in the axial or radial directions . the bulk flow transitions to the axial boundary speeds over ekman boundary layers whose thickness scales like @xmath27 , and in the ideal model of radial boundaries , over stewartson boundary layers that scale like @xmath28 , where @xmath29 is a local reynolds number particular to the boundary location @xmath3 with boundary speed @xmath30 @xcite . the fluxes of angular momentum across these boundary layers , being inversely proportional to the boundary layer thickness , do not scale proportionally , and hence we anticipate that self - similarity of the global properties need not be preserved as the reynolds number is scaled . thus , while the observed variation in the shape of the ` ekman ` and ` split ` profiles is expected , it comes as some surprise that the ` optimized ` profiles are effectively independent of the reynolds number . the experiments reported here showed no significant temporal variation in the mean values , hence , it can be stated that in steady - state the net flux of angular momentum into these flows must sum to zero , that is , @xmath31 where @xmath32 and @xmath33 are the fluxes integrated over the inner and outer cylinders , respectively , and @xmath34 is the axial flux integrated over each axial boundary ( with inward towards the fluid defined as a positive flux ) . the ideal couette profile is the response to a constant radial flux of angular momentum , implying that @xmath35 and that the axial fluxes of angular momentum are everywhere zero . such a state can be imagined in a tc system with free - slip conditions on the axial boundaries , or with a continuously variable boundary that can perfectly match the ideal couette profile , conditions that will result in vanishing stress at the axial boundaries . the radial flux of angular momentum under these conditions ( which we call @xmath36 , the couette flux ) has a magnitude equal to @xmath37 where @xmath38 . as an aside , it is interesting to note that @xmath39 when the radius ratio ( @xmath40 ) is equal to the golden ratio . even though there is , in general , a large mismatch between fluid and axial boundary speeds , we find that the flows under the ` optimized ` boundary conditions very closely approximate the ideal couette profile , and therefore have a nearly constant radial flux of angular momentum . we now show that rather than satisfying the condition that the axial flux of angular momentum everywhere vanishes , these flows satisfy a much weaker constraint : it is the surface integral of the axial angular momentum flux that vanishes , that is , @xmath41 . to calculate the axial flux of angular momentum , being proportional to @xmath42 , we make the assumption that @xmath43 , where @xmath44 is the difference of the boundary angular velocity ( @xmath30 ) and the fluid angular velocity just inside the ekman boundary layer ( @xmath2 ) , and that @xmath45 represents the thickness of an ekman boundary layer with an unknown numerical constant @xmath46 . defining the normalized quantities @xmath47 ( recalling , @xmath16 ) , @xmath48 , @xmath49 and @xmath50 , we have @xmath51 that the radial flux of angular momentum under ideal couette rotation [ eq . [ phic ] ] scales like @xmath26 , whereas @xmath34 scales like @xmath52 means that unless steps are taken to force the integral in eq . [ phiz ] to zero , there will always exist a reynolds number beyond which the axial flux will overwhelm the couette flux and cause the flow to depart from ideal rotation , regardless of aspect ratio . from eq . [ phiz ] we can immediately conclude that tc devices of the `` classical '' geometry with the axial boundaries co - rotating with either the inner cylinder or outer cylinder will always have non - zero @xmath34 and can therefore never be a good model for astrophysical systems at sufficiently high reynolds numbers to be of interest . in looking for the conditions under which ideal couette flow may be generated , we set @xmath53 and search for the boundary conditions ( the @xmath54 ) that cause the integral of eq . [ phiz ] to vanish . ( color online ) experiments conducted in @xmath20 flows showing @xmath13 as a function of @xmath12 for the ( a ) ` htx ` and ( b ) ` wide - ring ` geometries , progressing from low ring speeds ( bottom curves ) to higher ring speeds ( upper curves ) . in panels ( c ) and ( d ) the experimental @xmath55 ( dashed green ) , for the measured profiles relative to the ideal coutte profile for these boundary conditions , is plotted against the calculated axial flux of angular momentum from eq . [ phiz ] ( red , increasing function ) and the pressure differential from eq . [ dp ] ( blue , decreasing function ) for the ` htx ` and ` wide - ring ` geometries , respectively . @xmath34 is normalized by @xmath56 and @xmath57 is normalized by the kinetic energy density for ideal couette flow . the vertical dotted lines indicate the zeros of @xmath57 and @xmath34.,width=529 ] figure [ fig3 ] summarizes fluid velocity measurements over a scan of @xmath12 for @xmath20 flows for two axial boundary configurations : the ` htx ` configuration with @xmath58 cm and @xmath59 cm , and a ` wide - ring ` configuration with rings that span the entire radial gap . these experiments show that the ` htx ` configuration produces an exceptional match to the ideal couette profile for a narrow range of ring speeds centered about @xmath60 rpm , with very low fluctuation levels spanning the entire gap @xcite . importantly , the boundary conditions that generate near - ideal flows for the ` htx ` configuration include the zero crossing of @xmath34 [ fig . [ fig3](c ) ] . however , the ` wide - ring ` case also has a zero in @xmath34 , near @xmath61 rpm [ fig . [ fig3](d ) ] , yet its flows never resemble ideal couette , indicating that the vanishing of @xmath34 is a necessary but not sufficient condition for achieving near - ideal flows . ( color online ) same as fig . [ fig3 ] except for @xmath21.,width=529 ] the role of the axial boundaries in dictating global performance has also been interpreted through the competition of pressures from the bulk flow and from the boundary flow that is viscously coupled to the axial boundaries @xcite . the tendency to drive secondary flows was shown to be consistent with whether the bulk pressure is larger than ( ekman circulation ) or smaller than ( anti - ekman circulation ) the boundary pressure . following the intuition motivated by these simulations , we define a function @xmath57 that characterizes the average pressure difference between ideal couette rotation and boundary rotation , @xmath62 for the ` wide - ring ` configuration the zeros of @xmath57 and @xmath34 are widely separated , meaning that no circumstance exists where pressure balance and zero net axial flux can be simultaneously satisfied . in contrast , the ` htx ` configuration has zeros of these functions that are nearly coincident and fall within the operating range in which small fluctuations were observed in ref . other boundary configurations with @xmath21 also have nearly coincident zeros of @xmath34 and @xmath57 , in remarkably good agreement with experiments ; see fig . note that the requirement of coincident or nearly - coincident zeros represents an effective third constraint . these studies suggest that the requirements of having nearly coincident zeros of @xmath34 and @xmath57 for the generation of ideal flows may represent necessary and sufficient conditions . additional research conducted over a wider range of geometries and shear conditions will provide a stronger test of this hypothesis . while the coincident vanishing of @xmath34 and @xmath57 define what may be necessary and sufficient conditions for ideal flows to develop , they do not reveal how qk systems should behave under non - optimized conditions . a general model of the fluid response to forcing by the boundaries must account for the fluxes across both axial and radial boundary layers . as there does not exist a theory of stewartson boundary layers under the conditions of qk rotation at large reynolds numbers , we can not rely on results derived from a linear analysis of perturbative differences in rotational speeds @xcite , especially for experimental conditions where the stewartson boundary layers are turbulent @xcite . instead , we assume a generalized scaling of the stewartson boundary layer thickness of the form @xmath63 , taking the exponent of the reynolds number ( @xmath64 ) as a free - parameter to be determined from the observed scalings in fig . [ fig2 ] . while the numerical factors @xmath65 and @xmath66 are introduced through the definitions of the boundary layer thicknesses we can not measure the boundary layers directly , and therefore we must interpret their meaning as a measure of the effectiveness of angular momentum flux . we begin this analysis by considering the ` ekman ` configurations from fig . [ fig2 ] where we note that @xmath13 transitions to solid body rotation at @xmath11 near the outer cylinder , implying that @xmath67 . with a jump in azimuthal velocity ( a negative @xmath68 ) occurring over a stewartson layer at the inner cylinder , the appropriate form for @xmath32 is @xmath69 where @xmath70 is a result of converting from a representation of the boundary layer thickness that depends on the local @xmath71 to a global @xmath26 . we approximate these flows with a piecewise function of the form @xmath72 and @xmath73 , where @xmath74 is the transition radius , similar to the identification of separate flow regions in the recent work of nordsiek _ et al . _ @xcite . substituting this rotation profile into eq . [ phiz ] ( using @xmath75 ) and using eq . [ phi1 ] for the radial flux at @xmath76 , we solve the global balance of angular momentum , @xmath77 , for the dimensionless @xmath78 as a function of @xmath26 . noting that @xmath34 has two parts given this representation , our solution has three terms and is of the form @xmath79 where @xmath80 , and the integral expressions @xmath81 and @xmath82 are defined as @xmath83 @xmath84 ( color online ) scaling of the velocity differential @xmath85 for the three groups of boundary conditions from fig . [ fig2 ] for ( a ) @xmath20 and ( b ) @xmath21 . the dashed lines in the figure are the solutions to eqs . [ dv1 ] and [ dv2 ] with @xmath86 and @xmath87.,width=377 ] equation [ dv1 ] is a transcendental expression in @xmath68 since the limits of the integrals in @xmath81 and @xmath82 are functions of @xmath74 which itself depends on @xmath68 . equation [ dv1 ] has only two free parameters : @xmath64 and @xmath88 . comparison of eq . [ dv1 ] with experimental measurements of @xmath68 is presented in fig . [ fig5 ] for the @xmath20 and @xmath21 cases , where values of @xmath89 and @xmath90 provide the best fit to the experimental measurements . this analysis can be extended to the case of the ` split ` configuration with positive @xmath68 by accounting for the different boundary conditions and a slightly more complex flow structure . the most important feature of the positive @xmath68 cases is that there is a nearly linear decrease in @xmath68 across the gap , where @xmath68 at @xmath91 is defined to be equal to @xmath92 . in the @xmath20 cases @xmath68 decreases by about @xmath93 across the gap ( @xmath94 ) , and by @xmath95 for @xmath21 ( @xmath96 ) , nearly independent of reynolds number . the radial flux at @xmath91 in terms of the @xmath97 reduced @xmath68 is @xmath98 where @xmath99 . the solution for positive , normalized @xmath68 is @xmath100 where , similar to eq . [ dv1 ] , we have @xmath101 @xmath102 \omega_b^{1/2 } s^2 ds . \label{f4}\ ] ] for the ` split ` cases , the term @xmath103 defines the transition point from increasing to decreasing @xmath68 , and the terms in square brackets in eq.[f4 ] derive from the variation of @xmath68 with radius ( note that if @xmath68 were constant across the gap then @xmath104 and the term in brackets reduces to unity , recovering eq . equation [ dv2 ] is compared against the measured @xmath105 in fig . [ fig5 ] and is also found also to be in good agreement with experiment when using the same values of @xmath106 and @xmath64 derived from the analysis of the ` ekman ` cases . the success of this model in reproducing the general features of the departures from ideal couette is remarkable given that it represents a simple accounting of the angular momentum fluxes across the boundaries , ignoring completely the complex internal dynamics of these flows , especially at large reynolds number where a substantial fraction of the volume exhibits turbulent fluctuations @xcite . what disagreement exists between experiment and theory should not overshadow the success of this model in reproducing the general trend of the velocity deviations , the relative amplitude of the positive and negative @xmath68 cases , and the change in the amplitude of the positive @xmath68 cases between @xmath20 and @xmath21 . while the model presented here uses a single set of parameters for both values of @xmath107 , better fits to the data can be found if we let these parameters depend on @xmath107 . it is also possible the these parameters may depend on geometry and reynolds number in a way that is not captured by the power - law scaling used here . studies of the measured fluctuation levels arising from the stewartson boundary layers in ref . @xcite found that the transition to turbulent stewartson boundary layers was consistent with the taylor model of low reynolds number centrifugal instability . so while can not offer a precise explanation for why @xmath64 should take a value close to @xmath108 , we reiterate that the inferred scaling applies to the angular momentum flux which is a combination of variations in the boundary layer thickness and an effective viscosity within these layers that is modified by turbulent fluctuations . it is beyond the scope and ability of these experiments to identify the separate scalings of the turbulent viscosity and the thickness of the stewartson boundary layers , though perhaps future experiments will be able to provide greater insight into this problem . we conclude with an outlook to future physical and numerical experiments . the ratio of @xmath34 and @xmath36 explored earlier can be recast as a constraint on the aspect ratio @xmath109 as a function of the reynolds number . the relative contribution of @xmath34 can be made arbitrarily small by increasing the axial size of the system , so that to make @xmath110 for an ` ekman ` configuration , for example , one would need an aspect ratio of order @xmath111 at a reynolds number of @xmath112 . outside of using very large aspect ratios , one can employ mechanical advantages as we have done in htx , such as extensions of the inner and outer cylinder or independent rings . despite the desire to simplify the mechanical design of such modified tc devices , further analysis shows that the only way to achieve simultaneous zeros in @xmath34 and @xmath57 is through a design with at least one independent ring . greater insight into the balance of forces in quasi- eplerian flows could be explored in future experiments in modified taylor couette devices like htx . in particular , further exploration of the reynolds number dependence on these deviations , the dependence of the model parameters on @xmath107 , and the structure of the ekman cells would tell us much more about the influence of the boundaries on global behavior . we have shown that nearly ideal flows exhibit profile shape invariance under scaling of the reynolds number , an effect we interpret through the dual conditions of vanishing axial angular momentum flux and vanishing pressure differential that are nearly simultaneously satisfied , offering predictive capability for selecting optimized boundary conditions and in experimental design . the strongest piece of evidence in support of this model is the prediction of self - similarity of the profiles with respect to scaling of the reynolds number only for cases in which the axial flux of angular momentum and the pressure differential vanish nearly coincidentally , a prediction in excellent agreement with the observations presented in fig . it should be reiterated that while the experiments for ` optimized ` flows show very small departures from ideal couette , and whose interpretation is congruent with the coincident zeros in @xmath57 and @xmath34 , this does not strictly prove that these criteria represent true sufficient conditions . an interesting problem for future studies would be to measure how the departures from ideal couette depend on the `` distance '' between these zeros . it is interesting to note that in all cases presented here the local axial fluxes of angular momentum may be quite large given the substantial difference in speed between the bulk fluid and the axial boundaries , and that it is only on summation over the axial boundaries that zero net axial flux is realized for the ` optimized ` cases . recalling that multiple experiments @xcite and simulations @xcite have observed a nearly uniform axial structure through the bulk of the fluid volume , the existence of the large axial fluxes naturally raises the question : what allows the bulk flows to depart from the solid body rotation forced by the boundaries ? intuition based on the taylor - proudman theorem for rayleigh - stable flows , that is @xmath113 , would suggest that the bulk should tend to follow the boundary . however , it should be recalled that the ekman boundary layers do not need to satisfy the taylor- roudman theorem because the axial gradient of @xmath13 is balanced with a viscous diffusion of vorticity over the scale of the boundary layer thickness . thus , the nearly uniform axial structure suggests that the large fluxes of angular momentum that must be present due to the existence of the the ekman boundary layers are redistributed locally , perhaps by small ekman cells that do not extend far into the bulk of the fluid or even in the boundary layers themselves . as has been shown in prior studies , the free - shear layer that develops at the interface between the differentially rotating rings on the axial boundaries is expected to be centrifugally unstable and may help to enhance the redistribution of angular momentum locally @xcite . another problem for future experiments , both physical and numerical , will be to explore in greater detail how the very large , local fluxes of angular momentum are redistributed near the axial boundaries with only minor effect on the bulk flow in the ` optimized ` configurations . non - optimized boundaries , like the ` split ` and ` ekman ` configurations , show progressive departure from the ideal couette flow as the reynolds number is increased , in agreement with the expectations of a dominant axial flux of angular momentum . a good test for numerical experiments will be to accurately model the global behavior of the mean flows in tc experiments by properly accounting for the angular momentum fluxes from the boundaries . we believe that the boundary layer scalings presented here may enable simulations to bootstrap to larger effective reynolds numbers by using specified boundary flux models , thus bypassing the need for very fine grids to resolve the boundary layer structure directly . with the combination of recent simulations of qk flows that can attain reynolds numbers of the order of @xmath114 @xcite , though with axially - periodic boundary conditions , a specified - flux boundary model may allow simulations to reach something resembling experimental conditions of large reynolds number tc flows . we thank j. goodman for sharing his thoughts on angular momentum transport and boundary layers in our studies , and e. schartman , e. gilson , and p. sloboda in helping to keep the experiments running smoothly . this work was supported by the u.s . department of energy , fusion energy sciences under contract de - ac02 - 09ch11466 through the center for momentum transport & flow organization in plasmas and magentofluids ( cmtfo ) . j. e. rice , j. w. hughes , p. h. diamond , y. kosuga , y. a. podpaly , m. l. reinke , m. j. greenwald , . d. grcan , t. s. hahm , a. e. hubbard , e. s. marmar , c. j. mcdevitt , and d. g. whyte . edge temperature gradient as intrinsic rotation drive in alcator @xmath116-mod tokamak plasmas . , 106:215001 , 2011 . m. seilmeyer , v. galindo , g. gerbeth , t. gundrum , f. stefani , m. gellert , g. rudiger , m. schultz , and r. hollerbach . experimental evidence for magnetorotational instability in a taylor - couette flow under the influence of a helical magnetic field . , 113:024505 , 2014s . f. stefani , t. gundrum , g. gerbeth , g. rdiger , m. schultz , j. szklarski , and r. hollerbach . experimental evidence for magnetorotational instability in a taylor - couette flow under the influence of a helical magnetic field . , 97:184502 , 2006 . d. r. sisan , n. mujica , w. a. tillotson , y. huang , w. dorland , a. b. hassam , t. m. antonsen , and d. p. lathrop . experimental observation and characterization of the magnetorotational instability . , 93:114502 , 2004 .

we present measurements of quasi - keplerian flows in a taylor - couette device that identify the boundary conditions required to generate near - ideal flows that exhibit self - similarity under scaling of the reynolds number . these experiments are contrasted with alternate boundary configurations that result in flows that progressively deviate from ideal couette rotation as the reynolds number is increased . these behaviors are quantitatively explained in terms of the tendency to generate global ekman circulation and the balance of angular momentum fluxes through the axial and radial boundary layers . + eric m. edlund * and hantao ji + princeton plasma physics laboratory + september 1 , 2015

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Proceed to summarize the following text: the complex relationships among the nearby dwarf spheroidal ( dsph ) galaxies , the globular clusters and the general halo of our galaxy are far from clear , though all belong to baade s ( 1944 ) original population ii . it is of interest to discuss the similarities and differences of their stellar populations in terms of possible scenarios for the origin of the galactic halo . according to eggen , lynden - bell & sandage ( 1962 ) the galactic halo , including the globular clusters , formed in a single monolithic event of gravitational collapse lasting about @xmath0 years . during that time , the vast majority of halo stars and stars in the globulars formed to provide the present population ii . an alternative scenario was suggested by searle & zinn ( 1978 ) in which the halo was accumulated by the capture of many small systems such as dsphs over a timescale at least an order of magnitude longer . many current versions of hierarchical galaxy formation theories invoke similar scenarios ( e.g. klypin et al . 1999 , moore et al . strong observational evidence in favor of at least some contribution of dsph systems to the halo was first provided by ibata , gilmore & irwin ( 1994 ) with the discovery that the galaxy is currently capturing the sagittarius system ( sgr ) with its attendant globular clusters . more recent evidence ( e.g. yanny et al . 2003 , martin et al . 2004 ) indicates that the galaxy may well be absorbing or has absorbed additional systems . in addition , the complicated history of gradual metal growth in @xmath1 centauri indicates that it probably orbited our galaxy for several gyr before being captured ( hughes & wallerstein 2000 ; hilker & richtler 2000 ) . that the galaxy , as well as m31 and presumably other similar spirals , are intimately surrounded by a number of dsphs is made particularly clear in the graphic 3-dimensional galaxy distribution map of the local group given in grebel ( 1999 ) . a full review of current ideas regarding the formation of the galaxy can be found in freeman & bland - hawthorn ( 2002 ) . one approach that helps to constrain formation scenarios is to compare the populations of the surviving dsph systems with that of the halo . if the halo is indeed made up in large part by dissolved systems initially like the dsphs we see today , one would expect to find many similarities in their stellar . of various methods of comparison , three stand out as potentially the most viable . the first is a comparison of the types and period distributions of variable stars ( renzini 1980 ) . this method is not currently useable because of selection effects that plague any census of variables in the halo , although various surveys such as the sloan digital sky survey should help pin these down . the second is to compare the cmds in detail and try to set limits on the percentage of present day dsph populations that may have contributed to the halo . by comparing the turnoff colors in these systems , unavane , wyse & gilmore ( 1996 ) have set an upper limit of @xmath2 on this contribution , as the intermediate - age stars generally found in dsphs are lacking in the halo , hinting that dsphs may not be the generic galactic building blocks they are often imagined to be . the third approach is a direct comparison of the detailed chemical compositions of stars from the two environments , based on high resolution spectroscopy . a large sample of galactic field stars with detailed abundance analyses is now available ( e.g. mcwilliam 1997 ; burris et al . 2000 ; ryan , norris & beers 1996 , carretta et al . 2002 , nissen & schuster 1997 , fulbright 2002 , johnson 2002 , gratton et al . 2003a , stephens and boesgaard 2002 ) . a kinematic analysis of these and other galactic field stars shows most belong to the halo below [ fe / h ] = -1 , although several thick disk stars have metallicities that extend below [ fe / h ] = -2 ( venn et al . 2004 ) . here , we assume that stars with [ fe / h ] below -1 are representative of the galactic halo . the complementary studies of stars in dsph systems have only recently begun . in a pioneering study , shetrone , bolte & stetson ( 1998 ) investigated four stars in the draco dsph . they found a metallicity range from [ fe / h ] = @xmath3 to @xmath4 with a mean value of [ @xmath5/fe ] of + 0.2 and a spread in [ o / fe ] from + 0.38 to @xmath6 . subsequently , shetrone , ct , & sargent ( 2001 - hereafter s01 ) published results for an additional two stars in draco , six in ursa minor and 5 in sextans , and shetrone et al . ( 2003 - hereafter s03 ) added five giants each in carina and sculptor ( hereafter scl ) , three in fornax and two in leo i. combining their studies , they find that at a given the dsph giants exhibit significantly lower [ /fe ] abundance ratios than stars in the galactic halo . they conclude that the general galactic halo could not be built up of stars like those seen in their dsph samples . however , they find that a small subset of stars , represented by the , high @xmath7 , high @xmath8 halo stars studied by nissen & schuster ( 1997 ) , do mimic the dsph stars of similar in their detailed chemical composition and support nissen & schuster s claim that up to 50% of the halo could be explained by dsph accretion , although this relied on only two dsph stars of the appropriate . however , venn et al . ( 2004 ) further analysed the nissen & schuster ( 1997 ) stars and find that the ni - na relationship , the basis for the s03 claim about 50% of the metal - rich halo possibly being stripped dsphs , is a general nucleosynthetic signature and not relevant to the discussion of merged galaxies . finally , bonifacio et al . ( 2000 ) , bonifacio and caffau ( 2003 ) , bonifacio et al . ( 2004 ) and smecker - hane & mcwilliam ( 2002 ) have studied a large sample of stars in the sgr dsph . the latter find significant differences between sgr and galactic field stars of comparable , in particular with regards to al , na and elements , in the same sense as for the samples of shetrone and collaborators . bonifacio et al . ( 2004 ) suggest that the chemical similarities of dsphs and damped lysystems , particularly in regard to their depressed , may demonstrate a common evolutionary history and nature . gratton et al ( 2003a ) have used kinematics to divide field subdwarfs and early subgiants into two subpopulations , one which they ascribe to a dissipational collapse and one that is likely to represent accreted stars . in particular , they have compared the ratios of alpha - elements to iron in the two subpopulations . they find that on average the supposed accreted population has lower [ /fe ] than their dissipative collapse counterparts . if the accreted stars have been accumulated by the capture of systems like scl , their compositions should be similar to those of scl . to expand the data base for dsph galaxies and to further test the hypothesis that the halo of the milky way may have been sculpted " from galaxies like scl , we have observed 4 red giants in the scl galaxy . scl was one of the first two dsph companions to the milky way to be discovered ( shapley 1938 ) . indeed , as the prototype , dsph galaxies were originally referred to as scl - like systems " ( e.g. shapley 1943 ) . photometry in scl ( e.g. da costa 1984 , schweitzer et al . 1995 ) has shown that the red giant branch is broad , indicating a spread in metallicity ( as is seen in all dsphs and in @xmath1 cen ) . in a survey of 37 red giants in scl , tolstoy et al . ( 2001 ) used the ir caii triplet to derive metallicities rather than [ ca / h ] which was actually measured . the translation from [ ca / h ] to can be problematical ( e.g. cole et al . 2000 ) , especially when [ ca / fe ] may vary between program objects and abundance calibrators . ] ranging from @xmath9 down to @xmath10 dex with most of the stars distributed between @xmath11 and @xmath12 . the distribution is asymmetric with a mean of @xmath13 and a peak near @xmath11 . both the mean and spread are very similar to @xmath1 cen ( norris , freeman & mighell 1996 ; sunzeff & kraft 1996 ) but the asymmetry in @xmath1 cen is opposite to that of scl with the peak of the distribution on the metal - poor side of the mean . apparently the two systems had slightly different rates of self - enrichment and star formation . scl provides an excellent opportunity to compare the compositions of dsph stars with those in the halo . at a distance of 87 kpc ( mateo 1998 ) , it is one of the nearest dsphs . its brightest giants , at @xmath14 , are therefore accessible to 8m - class telescopes equipped with high resolution spectrographs . since scl shows a spread in metallicity , there is the possibility to compare the ratios of various species to iron as a function of metallicity with the same elemental ratios in the halo field and the globular clusters . additionally , detailed abundances are required to disentangle age from effects when attempting to derive the star formation and chemical enrichment history of scl , as discussed by tolstoy et al . ( 2003 ) for their age determinations . photometric data alone are not sufficient to break the age - metallicity degeneracy present in old intermediate age stellar systems . accurate age determinations require the knowledge of the element as well as . scl also is unusual or even unique in several respects . it appears to be composed mainly or almost exclusively of an old , globular cluster- aged population ( e.g. da costa 1984 , hurley - keller et al . 1999 , monkiewicz et al . 1999 , dolphin 2002 ) . however , there are some blue stars brighter than the turnoff ( demers & battinelli 1998 ) and it may contain neutral h gas ( carignan et al . 1998 , bouchard et al . 2003 ) . walcher et al . ( 2002 ) find evidence for tidal tails . hurley - keller et al . ( 1999 ) , majewski et al . ( 1999 ) and harbeck et al . ( 2001 ) found a gradient in the morphology of the horizontal branch , with a much higher percentage of red hb stars in the inner regions than in the outer regions . harbeck et al . ( 2001 ) found scl to have the most significant hb morphology gradient of any of their sample of 9 dsphs . majewski et al . even suggested the possibility of a bimodal distribution based on their cmd . however , tolstoy et al . ( 2001 ) found no indication for either a gradient or bimodality in their low resolution spectroscopic for a large sample of stars . because of its importance , several groups independently began high resolution studies of scl stars . after we began our study we became aware of the other groups . the results of one of these groups have subsequently been published ( s03 ) and we have combined our results with theirs . this allows for a substantial increase in sample size and coverage , allowing us to investigate any abundance trends with much greater confidence . an additional aspect of a detailed analysis of scl s red giants is the internal evolution of the stars themselves . while all highly evolved , metal - poor , low - mass red giants show evidence of extra - mixing beyond the canonical first dredge - up ( see references in , e.g. , weiss & charbonnel 2003 ) , the degree of mixing may depend on the initial stellar metallicity . in both field and globular cluster stars near the red giant tip , the very low @xmath15c/@xmath16c ratio , ranging from @xmath174 to 8 , can be unambiguously attributed to an in situ ( evolutionary ) mixing mechanism . we can test the universality of this phenomenon by deriving the carbon isotopic ratios for our two most metal - rich stars . on the other hand , the o / fe , na / fe , and al / fe ratios vary greatly from star to star in globular clusters . the abundance anomalies of these isotopes from higher p - burning cycles , which are not seen in field halo stars , seem to be mostly of primordial origin according to the preponderance of observational evidence ( e.g .. , gratton et al . 2001 ; grundhal et al . 2002 ; yong et al . massive agb stars have been claimed to be the favorite candidates to have polluted the intracluster gas or the surface of cluster stars ( e.g. , cottrell & da costa 1981 ; ventura et al . however fenner et al . ( 2004 ) recently showed that the abundance patterns observed in a classical " globular cluster like ngc 6752 could not be matched by a model of chemical evolution incorporating self - consistently the detailed nucleosynthesis yields from agb stars . in particular , neither the o - na nor the mg - al anticorrelations could be reproduced , in agreement with the previous findings by denissenkov & herwig ( 2003 ) and herwig ( 2004 ) . so the details of this primordial ( or pollution ) scenario still await clarification . by investigating the relative abundances of o , na and al we can see how the presence of these chemical variations depends on environment and probe the connection between dsphs and globular clusters . our paper is arranged as follows : in section 2 we present the observations and reductions and in 3 the details of the abundance analysis . in 4 we present the abundance results . the heavy element star we discovered is discussed in detail in section 5 . in section 6 we summarize our major findings . echelle spectra of four giants in scl were obtained on the nights of september 18 and 19 , 2000 with the uves instrument on the 8.2 m vlt ut2 telescope ( kueyen ) of the european southern observatory . the target stars were selected from the study of schweitzer et al . ( 1995 ) as being amongst the brightest giant members ( with proper motion membership probabilities @xmath18 ) and covering the full color width near the tip of the giant branch . this latter should reflect to first order as complete coverage of the full range as possible . each star was observed for a total of four hours , divided into one - hour exposures . the stars were observed simultaneously through the blue and red arms of uves using a dichroic beam splitter . this yields complete coverage from @xmath19 in the red except for loss of a single order near the center where there is a gap between the two ccds , and complete coverage from @xmath20 in the blue . the resolving power with a @xmath21 slit is about 22,000 in the red and 16,000 in the blue . the seeing was generally @xmath22 . spectra of hot , rapidly rotating stars were also obtained in order to divide out telluric absorption lines . the data were sky subtracted , reduced to one - dimensional wavelength calibrated spectra , and then the individual spectra for each star were co - added , using the standard software packages available in iraf . typical s / n ratios for the final combined spectra are about 120 per pixel at 6700 and 65 per pixel at 4500 . radial velocities of all 4 stars , given in table 1 , show that they are indeed members , as the mean radial velocity of scl stars is @xmath23 ( mateo 1998 ) . combining our four velocities with five velocities in s03 , we find a mean of + 110.0 km - s@xmath24 , with a dispersion ( standard deviation ) of 6.9 km s@xmath24 . a small segment of spectrum is shown for a metal - poor star ( 195 , with [ fe / h]= -2.1 ) and for a more metal - rich star ( 1446 , with [ fe / h]= -1.2 ) in figure 1 . the difference in the line strengths is obvious ; as the stars have rather similar effective temperatures and gravities which would lead to only very modest differences in line absorption for the same abundances , most of the differences in the observed line strengths are caused by abundance differences . this is the most graphic evidence for a real abundance spread in this dsph . one of the fundamental parameters needed in stellar abundance analyses is the effective temperature of the star in question . the procedure in this study is to base effective temperatures on two broadband color ( v k and j k ) calibrations of t@xmath25 . table 1 lists the stars observed along with various apparent and absolute magnitudes and colors and the derived radial velocities . the star designations , v - magnitudes and ( b v ) colors are taken from schweitzer et al . ( the two - micron all sky survey ( 2mass ) database , accessed via `` http://irsa.ipac.caltech.edu '' , is the source of the k - magnitudes and ( j k ) colors . the 2mass magnitudes and colors have been transformed to the system defined by bessell & brett ( 1988 ) , as we use t@xmath25-calibrations from bessell , castelli & plez ( 1998 ) , who use the color system defined in bessell & brett ( 1988 ) . the 2mass corrections are those defined by carpenter ( 2001 ) in his appendix a and are fairly small : a constant 0.04 magnitude offset in k , and a small color term that is about 0.03 magnitude in ( j k ) . the v - magnitudes used to compute ( v k ) are corrected for reddening based on the ( b v ) color excess of @xmath26 ( mateo 1998 ) and a@xmath27= 3.3e(b v ) . no absorption correction is applied to k , since the overall reddening to scl is quite small and @xmath28 . in addition , in table 1 are the absolute k - magnitudes ( for a true distance modulus of @xmath29 - mateo 1998 ) , along with k - band bolometric corrections from bessell et al . ( 1998 ) and the subsequent absolute bolometric magnitudes and luminosities . effective temperatures are derived for the program stars using the ( v k ) and ( j k ) colors from table 1 , along with calibrations discussed and defined in bessell et al . these authors point out that ( v k ) versus t@xmath25 has almost no sensitivity to metallicity ( hereafter taken as [ fe / h ] ) so long as t@xmath25 is greater than @xmath174000k , while ( j k ) has only a small dependency ; their near independence from [ fe / h ] is why these two colors are used . table 2 lists effective temperatures for each star as defined by ( v k ) and ( j k ) colors . with just the four program stars in question , there is no systematic offset between the two sets of t@xmath25 s , and their differences scatter around 100k with no systematic trend . for the final t@xmath25 to be used in the abundance analyses , we adopt the average of the two values and round this to the nearest 25k in defining the model atmosphere effective temperature ( shown in column 4 of table 2 ) . other color effective temperature relations are available in the literature and we compare one other source ( mcwilliam 1990 ) to the calibrations used here . using mcwilliam s ( v k ) calibration and comparing it to the values of colors and t@xmath25 s in tables 1 and 2 , the mean difference ( in the sense of bessell et al . ( 1998 ) minus mcwilliam ) and standard deviation is found to be + 7@xmath3019k : very good agreement . using ( j k ) , the mean and standard deviation are + 90@xmath3051k . this comparison suggests that these various temperature scales for the types of red giants analyzed here are in good agreement with differences of less than about 100k . with an effective temperature scale defined by broadband colors , the remaining global stellar parameters of surface gravity ( parameterized as log g ) , microturbulent velocity ( @xmath31 ) , and overall metallicity ( defined by the fe abundance ) are set by the fe i and fe ii lines . the spectroscopic analysis of the fe lines , as well as all of the other elements to be discussed , uses a recent version of the lte spectrum synthesis code moog , first described by sneden ( 1973 ) . the model atmospheres adopted are those generated from a version of the marcs code as discussed by gustafsson et al . the combination of lte analysis from moog using marcs model atmospheres has been used extensively in abundance analyses of k - giants , and yields accurate chemical abundances of many species . recent examples of similar analyses of red giants includes ivans et al . ( 1999 , 2001 ) , ramirez et al . ( 2001 ) , and s01 and s03 . the microturbulent velocity , at a given log g , is found by forcing all fe i lines to yield the same iron abundance , i.e. , with no significant slope of a(fe)+ 12.0 . ] versus reduced equivalent width ( log ( w/@xmath32 ) . once the microturbulent velocity is defined , the surface gravity is tested for consistency by comparing the fe i and fe ii abundances , with ionization equilibrium demanding both neutral and singly ionized fe to yield the same abundances . the now determined stellar parameters ( t@xmath25 , log g , and @xmath31 ) along with the fe i and fe ii lines are then used to derive the overall metallicity of the stellar atmosphere . the entire process of defining log g , @xmath31 , and [ fe / h ] is iterated until a consistent set of stellar and model atmosphere parameters is found , and this atmosphere is finally used in the derivation of the other elemental abundances . when enforcing the ionization equilibrium of fe i and fe ii to derive surface gravity in metal - poor red giants , some care should be taken to investigate the possible influence of over - ionization on fe i , as discussed by thevenin & idiart ( 1999 ) . data from three recent analyses of globular cluster giants can be used to partially address this question : ivans et al . ( 1999 ) for m4 , ramirez et al . ( 2001 ) for m71 , and ivans et al . ( 2001 ) for m5 . in all these studies , `` evolutionary '' gravities are calculated from stellar model tracks ( with known masses ) coupled to the derived red - giant effective temperatures and luminosities . iron abundances are then computed and can be compared between fe i and fe ii . from the three studies noted above , no significant effect is seen over the metallicity range from [ fe / h]= -0.84 ( for m71 with @xmath33(fe i fe ii)= + 0.14@xmath300.17 ) , to [ fe / h]= -1.15 ( for m4 with @xmath33(fe i fe ii)= -0.01@xmath300.09 ) , and [ fe / h]= -1.21 ( for m5 with @xmath33(fe i fe ii)= -0.13@xmath300.07 ) . based on the above observational constraints , overionization may affect derived gravities by @xmath17 0.1 dex , and possibly lead to small over / underestimates of the iron abundance by @xmath17 0.1 dex . such a possibility will not , however , have a significant effect on the derived abundances and abundance trends to be discussed here . the derived effective temperatures , gravities , microturbulent velocities , and iron abundances are listed in table 2 . the iron abundances are derived using the accurate sets of gf - values from martin , fuhr , & wiese ( 1988 ) , bard , kock , & kock ( 1991 ) , holweger et al . ( 1991 ) , and obrian et al . it is worth noting that the absolute accuracy of these fe gf - values is now at the few percent level , and an analysis of solar fe i and fe ii lines yields photospheric iron abundances with a scatter of @xmath17 0.05 dex and essentially perfect agreement with the meteoritic abundance ( a(fe)= 7.50 - grevesse & sauval ( 1999 ) ) . table 3 gives relevant parameters for all of our measured fe lines . we will later combine our abundances to those derived by s03 for five other sculptor red giants in order to create a larger database . abundances will be given as values of [ x / fe ] so we must check for any offsets in the two abundance scales caused by either gf - values or adopted solar abundances . in the case here for iron , we note first that s03 used a(fe)= 7.52 for the sun , whereas we adopt 7.50 ; to strictly compare our respective fe abundances , we should add + 0.02 dex to the s03 values to bring them onto our scale . in addition , however , a comparison of gf values for fe lines reveals that for 15 lines in common , there is a mean offset of + 0.04 dex in log gf ( in the sense of us s03 ) . based on this difference ( if it is indicative of a general trend for all fe lines ) , we should then subtract 0.04 dex from the s03 [ fe / h ] values to bring them onto our gf - scale . the net result of considering the adopted solar fe and gf - value scales would be to shift by -0.02 dex the s03 values of [ fe / h ] to bring them into agreement with our scale . this is such a small offset , well within the uncertainties of the gf - values themselves and even the solar fe abundance , that we consider such a difference to be insignificant and apply no corrections to the s03 values of [ fe / h ] : both studies are effectively on the same abundance scale . of the spectral species studied here in common with smith et al . ( 2000 ) , we have adopted the gf - values from that study , and their sources are discussed in detail in that paper . these elements are o i , na i , mg i , al i , si i , ca i , sc ii , ti i , ti ii , fe i , fe ii , ni i , y i , y ii , zr i , zr ii , ba ii , la ii , and eu ii . smith et al . ( 2000 ) analyzed both the sun and arcturus using this linelist and gf - values and derived expected abundances , indicating that these gf - values can be used to derive accurate abundances . in addition , we have added the species mn i and zn i in this study , with the mn i gf - values taken from prochaska & mcwilliam ( 2000 ) . neutral zinc gf - values were adjusted to yield a solar abundance of a(zn)=4.60 using a 1-d marcs model ; this resulted in log gf= -0.44 for the 4722 line ( in excellent agreement with the value of -0.39 from s03 ) , and log gf= -0.24 for the 4810 line ( again in excellent agreement with s03 who used -0.17 ) . a comparison of the other elemental gf - values with the same lines used by s03 finds differences of less than 0.05 dex in log gf for all cases except al i. in the case of differences having less than 0.05 dex , no offsets will be applied to the s03 abundances when adding them to our dataset . for al i the offset is + 0.26 dex ( in the sense of us s03 ) , this difference was applied in order to combine our sculptor aluminium abundances with those from s03 . there are also a few other points concerning gf - values that should be noted . two ca i lines have differing log - gf values greater than 0.05 dex ; the 6439.08 gf - value used here is 0.08 dex larger than in shetrone et al . ( 2003 ) , but this is only marginally larger than 0.05 dex . the gf - value for the 6161.30 line ( smith et al . 2000 ) is 0.24 dex larger than that adopted by shetrone et al . ; however the particular value used here , along with a marcs solar model , yields a solar calcium abundance of a(ca)=6.25 . this is acceptably close to the recommended value of a(ca)=6.34 from lodders ( 2003 ) . as calcium is represented by 8 lines in this study , and 9 lines from shetrone et al . , with the other gf - values in close agreement , the differences discussed above will not affect significantly the average ca abundances . the mg i lines used here at 8717.83 and 8736.04 were not in smith et al . ( 2000 ) and the gf - values here are taken from the kurucz ( 1991 ) compilation . analysis of these lines in the solar flux spectrum results in respective magnesium abundances of a(mg)=7.51 and 7.54 , close to the lodders ( 2003 ) recommended value of 7.55 . there are no lines in common between this study and that of shetrone et al . ( 2003 ) for ti i , ti ii , y i , and y ii , but later inspections of figure 6 ( for ti ) and figure 8 ( for y ) will find no significant differences in the respective behaviors of ti and y with metallicity between the two studies . hyperfine splitting ( hfs ) was included for the species sc ii ( with the hfs data taken from prochaska & mcwilliam 2000 ) , mn i ( hfs data also taken from prochaska & mcwilliam 2000 ) , y i ( with hfs data taken from biehl 1976 ) , la ii ( with hfs data taken from lawler , bonvallet , & sneden 2001 ) , and eu ii ( with hfs data taken from biehl 1976 ) . for those species with multiple isotopes , solar isotopic ratios were assumed . table 4 provides the relevant parameters for all of our measured non - fe lines , as well as their equivalent widths in the sculptor red giants . although we present the equivalent widths , it must be noted that all abundances were derived via spectrum synthesis . the final abundances are given in table 5 , in the form of [ x / h ] values , as well as the adopted solar abundances in the form of a(x ) . as discussed in section 3.1 , various comparisons between ( v - k ) and ( j - k ) temperature calibrations can reveal systematic differences of up to 90k , with a scatter of 50k , thus an expected uncertainty ( @xmath171@xmath34 ) of about 100k for t@xmath25 in red giants is a reasonable value . in addition , the 1@xmath34 scatter set by the fe i lines in defining a mean iron abundance is about 0.15 to 0.20 dex ; this scatter is carried into the determination of the surface gravity from the fe ii lines , and leads to an uncertainty here of about 0.3 dex in log g. finally , the microturbulence is defined by using the fe i lines ( with the criterion of no trend in fe abundance with reduced equivalent width ) and the minimum scatter in the fe i abundances leads to an uncertainty in @xmath31 of about 0.3 km s@xmath24 . the uncertainties of @xmath30100k in t@xmath25 , @xmath300.3 dex in log g , and @xmath300.3 km s@xmath24 represent approximate 1@xmath34 values for these fundamental stellar parameters . all of these elemental species present different sensitivities in their derived abundances to the primary stellar parameters of t@xmath25 , log g , and @xmath31 . table 6 quantifies these sensitivities for star 770 , which is near the middle of our sample in terms of effective temperature , gravity , and micorturbulence ; the other stars will exhibit very similar sensitivities to changes in stellar parmaeters . the differences for each species are tabulated for a change of + 100k in t@xmath25 , + 0.3 in log g , and + 0.3 km - s@xmath24 in @xmath31 , with the final column showing the quadratic sum of these uncertainties . this final value is a fair estimate of the uncertainty in the derived abundances caused by realisitic uncertainties in defining the fundamental stellar parameters . we first combine our data with that of s03 obtained with the same telescope and instrument . this gives us a total of 9 stars in scl , the second largest sample of high resolution abundances yet obtained for a dsph , surpassed only by several sagittarius studies , with a range in [ fe / h ] from 2.10 to 0.97 . note that this substantially extends the range of s03 s sample , which covered 1.95 to 1.2 . we also cover most of the known range in this galaxy , based both on the extreme colors of our sample in observed cmds and the large sample of ca - triplet abundances derived by tolstoy et al . ( 2001 ) for 37 stars . other than 2 stars with of 2.2 to 2.3 , their next most star has @xmath35 and at the end their distribution stops at 1 except for a single star with 0.8 . in particular , our combined sample allows us to see abundance trends with more clearly . also note that our spectra are of significantly higher s / n than those of s03 s scl observations . our mean @xmath36 ( standard error of the mean ) . this compares very well with the mean of @xmath13 found by tolstoy et al . ( 2001 ) and the mean of 1.5 found by dolphin ( 2002 ) . we do not find any indication of a gradient , although our sample is small and only covers a limited radial range , from 0.1 1.1 core radii . tolstoy et al . ( 2001 ) have a much larger sample and radial extent and found no gradient . sodium is produced in the ne - na cycle in which protons are captured by the ne isotopes accompanied by the necessary @xmath37-decays . the stellar environment for the ne - na cycle is uncertain but it can occur at relatively low temperatures , near 30 million k , in evolved stable stars . in addition na can be produced by carbon burning which requires temperatures that can be realized only in advanced stages of stellar evolution and in massive stars . in our nine scl stars the mean [ na / fe ] ratio is @xmath38 , with no trend with . this ratio differs from recent analyses of globular clusters in which an excess of na is usually seen . in m15 and m92 sneden , pilachowski and kraft ( 2000 ) find [ na / fe]=+0.2 on average . in m5 ivans et al ( 2001 ) find [ na / fe]@xmath17 -0.2 in oxygen - rich stars , rising to + 0.4 in oxygen - poor stars . in m4 ivans et al . ( 1999 ) find [ na / fe]@xmath17 0.0 for stars that do not show evidence of a deficiency of oxygen . clearly the na deficiency seen in scl differs from the situation in the globulars . in the halo field , fulbright ( 2002 ) found that [ na / fe ] scattered around 0.0 all the way down to [ fe / h]= 4.0 with some stars showing [ na / fe ] as large as + 0.5 between [ fe / h]= 2.0 to 3.5 and some scattering down to [ na / fe ] = 0.5 between [ fe / h]= 1.5 to 2.5 . indeed the stars in his high velocity " bin have [ na / fe ] very similar to our sample . even if a correction of + 0.2 is added to our [ na / fe ] values to compensate for potential non - lte effects ( e.g. tautvaisiene et al . 2004 ) , our na abundances are still @xmath39 dex lower than typical galactic stars of similar metallicity . our oxygen abundances were derived from the single 6300 line of [ oi ] , while s03 used both this line ( mainly ) as well as the 6363 line when available ( 2 stars ) . although generally weak and only a single line , we feel that our o abundances are well - determined via the spectrum synthesis technique . a comparison of observed and synthetic spectra covering the [ o i ] line in star 1446 is shown in figure 2 to illustrate the quality of the spectra and corresponding synthetic matches . in figure 3 we show the correlation of [ o / fe ] with [ fe / h ] for both the sculptor giants ( 8 stars sampled , with 4 from this study and 4 from s03 ) , and a sample of galactic field stars from a number of studies ( noted in the figure caption ) . the galactic studies shown in figure 3 include those that rely either on the [ o i ] 6300 line , or the infrared vibration - rotation lines from oh . at the metal - poor end , the [ o / fe ] value is similar to , albeit a bit lower in the mean than , that which is seen in metal - poor globular clusters and the halo field ( mcwilliam 1997 ) . however , the ratio of o / fe decreases steadily and rapidly as increases above 1.5 , reaching [ o / fe]@xmath17 0.0 at [ fe / h]= 1.2 , and the most star at @xmath40 has a very low [ o / fe ] abundance of 0.3 . this is distinctly different from the globulars and the halo and disk field where [ o / fe ] only begins to fall dramatically at [ fe / h]=1 and does not reach zero until [ fe / h ] is near zero . the o abundance of star 982 is some 0.6 dex lower than the mean for the at this . note that this trend is seen much more clearly here than in s03 due to the extended range , or for that matter from our data alone , which suggest a strong monotonic decline . as noted in the introduction , it is very important to check for the existence of any na - o ( anti)correlation in order to probe the connection between dsphs , the galactic halo and globular clusters . in figure 4 we present [ na / fe ] vs. [ o / fe ] for the sculptor red giants along with the data for galactic field stars . there is no trend in sculptor , although note that even the low - o star 982 , the heavy element star ( see section 5 ) , shows [ na / fe]= -0.2 . where the anti - correlation is seen , it has often been ascribed to deep mixing within the star which has carried material to the stellar surface that had been processed by the on , nena and mgal cycles ( see dennisenkov and weiss 2001 and references therein ) . however recent observational evidence and theoretical developments ( see charbonnel & palacios 2003 and references therein ) now point toward an alternative explanation . namely , these anomalies may be due to stochastic enrichment by an earlier generation of stars followed by scattered incorporation into some , but not all , of the stars now seen in globular clusters ( cottrell & da costa 1981 ; dantona et al . 1983 ; jehin et al . 1998 ; parmentier et al . 1999 ) . however the major source of pollution for gcs still remains to be determined . indeed , based on current theories of intermediate mass stellar evolution and nucleosynthesis , it now appears that massive agbs , which seemed to be the most plausible candidates for this pollution , are not responsible for the observed globular cluster abundance anomalies ( denissenkov & herwig 2003 ; herwig 2004 ; fenner et al . 2004 ) . since we see neither the o / na nor the o / al anti - correlation in the sculptor stars it appears that neither of the above scenarios ( in situ extra - mixing of nena and mgal cycle material or pollution ) that induced the anomalous chemical patterns in globular clusters has occurred in this dwarf galaxy . the anti - correlation is never seen in field stars of the galactic halo . these facts tend to associate sculptor with field stars rather than with globular cluster stars . we note , however , that mcwilliam et al . ( 2003 ) have found one star in sgr with a mild excess of [ na / o ] . the other light odd element is al for which our fragmentary data ( we only derive al abundances for the two most stars ) shows a deficiency similar to that of na , while in globulars the [ al / fe ] value varies , reaching as much as + 0.6 in m4 in stars that show no evidence for the depletion of oxygen ( ivans et al . 1999 ) . while the na / o ratios in scl do not provide evidence for deep mixing , the ratio of @xmath15c/@xmath16c in two stars shows that moderately deep mixing has indeed occurred . for our two most metal - rich stars it was possible to derive the carbon isotope ratios from the 8005 @xmath16cn feature . for star 1446 the derived ratio is 3 with a substantial uncertainty . for the heavy element star , 982 , the @xmath16cn feature is strong enough for an estimate of the uncertainty and we find a ratio of @xmath42 . we show the fit for star 982 in figure 5 . these low ratios of @xmath15c/@xmath16c are almost exactly the cno equilibrium ratio and show that the vast majority of the material presently in the stellar atmosphere has been subject to proton capture at temperatures of at least 10@xmath43k . low @xmath15c/@xmath16c ratios are the general rule for stars near the red giant tip in the field ( charbonnel , brown & wallerstein 1998 ; gratton et al . 2000 ) , and in galactic open ( gilroy 1989 ; gilroy & brown 1991 ) and globular clusters ( see charbonnel & do nascimento 1998 for early references ; smith et al . 2000 ; shetrone 2003 ) . the same is true for rgb stars in the lmc ( smith et al . 2002 ) and in the smc ( hill et al . our results thus confirm the universality of an extra - mixing mechanism which transports matter between the outer layers of the hydrogen burning shell and the convective envelope in all low - mass red giants , and which is still not part of the standard evolution theory . this extra - mixing is independent of the stellar environment and metallicity , although it has been shown to depend on the initial stellar mass . for the purposes of this section , we will include here as alpha elements mg , si , ca and ti . see s03 for a more thorough discussion of the various nucleosynthetic origins for the alpha elements . silicon scatters about a mean near solar while magnesium , calcium , and titanium show interesting and similar trends albeit at different significance levels , as shown in figure 6 . all three elements show trends generally similar to that of o , with solar or enhanced and relatively constant values for the stars with @xmath44 that are generally less than those of normal galactic field stars at similar , but the more stars have significantly lower that are generally much lower than those of galactic stars of the same . the deficit with respect to galactic field stars is in fact true for si at all as well . in the case of mg , the lowest star has the most enhanced and the trend of decreasing over the full range is the most pronounced . the decrease in for the most stars is smallest for ti and indeed the difference between these and the more stars is not very significant . [ ti / fe ] is nearly constant at about solar with only a small decrease to 0.2 for the three stars near 1 . this observed behavior in the scl stars is different from the trends in the globulars and the halo where [ ti / fe ] is usually near + 0.3 over the range of [ fe / h ] from -2 to -1 ; though some metal - poor globulars show nearly the solar ratios ( lee and carney , 2002 ) . in fact , models of type ii supernovae usually place ti as an iron - peak element rather than associate it with the alpha - elements ( arnett 1996 ) . it appears that in scl ti follows snii models better than do most of the globulars and field stars of low metallicty . but models comparing the relative yields of snii and ia ( e.g. lee and carney 2002 , fig . 9 , based on woosley and weaver 1995 models ) show that ti is _ the _ element most produced in snii s compared to snia s . shetrone ( 2004 ) suggested that the light " ( o and mg ) and heavy " ( ca and ti ) elements may show differences in their behavior due to potential differences in their nucleosynthetic origins . we find that the mean enhancements of mg and ca in scl ( 0.08 and 0.11 dex ) are very similar but o ( 0.22 dex ) is significantly enhanced with respect to ti ( -0.08 dex ) and thus there is no clear trend . all of the elements from sc to zn are usually ascribed to the fe - peak though a few of the odd elements may be enhanced by small neutron - capture processes . the relative abundances of sc , mn , cu , and zn are shown in figure 7 . in galactic metal - poor stars sc follows fe very uniformly ( mcwilliam 1997 ) . in scl , between [ fe / h]=-2.1 and -1.5 , sc follows fe just as in the galaxy , although the mean [ sc / fe ] is slightly less than in the galaxy . but the higher scl stars deviate significantly from their galactic counterparts : [ sc / fe ] begins to decrease at a of 1.5 and drops rapidly to about 0.5 by [ fe / h]=-1.0 , very reminiscent of the general behavior . we know of no other object in which this phenomenon occurs . for cr we must rely on the data of s03 whose cr / fe ratios for 5 stars lie close to -0.15 dex which is very similar to field stars in the halo ( mcwilliam 1997 , fig . the mean [ mn / fe ] ratio is near -0.4 which is similar to that in halo field stars of the same metallicity interval . however the [ mn / fe ] values for the largest and smallest deficiencies of fe suggest a small downward trend in [ mn / fe ] with [ fe / h ] . while the reality of this trend , which only depends on the two endpoints , is uncertain , especially when the error bars are taken seriously , the trend appears to be real . if so , this trend is contrary to the rise of mn with fe seen in the galactic halo , thick disk and bulge ( e.g. mcwilliam et al . 2003 ) . for co / fe the 4 stars analysed by s03 also track iron as do the halo field stars with [ fe / h ] @xmath45 . the same holds for the ni / fe ratio . for cu the story is very different . in the 5 stars for which we have derived copper abundances , with [ fe / h ] between -1.2 and -1.8 , the values of [ cu / fe ] are very near -1.0 with a hint of a downward trend with , significantly different from the field halo stars and stars in @xmath1 cen , where it is close to -0.5 in the same interval of [ fe / h ] ( mcwilliam 1997 , cunha et al . the field stars show a significant slope , reaching -0.8 dex in [ cu / fe ] versus [ fe / h ] at [ fe / h]=-2.5 . mishenina et al . ( 2002 ) find a steady decrease in [ cu / fe ] with decreasing [ fe / h ] . however , over the interval of [ fe / h]= 1.2 to 1.8 , their data are consistent with [ cu / fe]=0.5 . finally for zn there appears to be no mean trend away from [ zn / fe]@xmath17 0.0 , as seen in the halo field stars , but it appears that there may be a bifurcation , with one group of scl stars with [ zn / fe]@xmath46 and another with [ zn / fe]@xmath47 . elements heavier than zn are known to be produced by neutron capture . the two neutron capture phenomena are the slow ( s - process ) capture sequence in which the time scale for beta - decay is shorter than the time scale between neutron captures and the rapid ( r - process ) capture sequence in which a flood of neutrons ( or a burst of nuclear reactions whose results mimic neutron captures ) drives the nuclei to extremely neutron - heavy isotopes which finally decay to the valley of stability . the readily observable s - process species may be divided into the light ( ls ) group which consists of rb , sr , y , and zr ; and the heavy ( hs ) group which consists of ba , la and the light rare earths . the only observable r - process element in our stars is eu . the heavier r - process species have lines that are too weak to be measurable on our spectra . starting with the ls elements , either y , zr , or both are available in 8 stars of the combined data . ignoring for the moment the most star ( 982 ) , the mean ls ( using the average of y and zr when available ) is 0.26 with a large @xmath48 scatter of 0.4 dex and no trend . this value is very low compared to galactic field stars ( e.g. fulbright 2002 ) . as shown in figure 8 , star 982 is unique , standing far above all the other stars with [ y / fe]= 1.1 . this is a heavy element star ! we discuss this star in detail in section 5 . the [ hs / fe ] data ( taking the mean of ba and la ) scatter around a value of 0.1 with no trend , as shown in figure 9 ( omitting the heavy element star 982 which again is remarkable , as seen from this figure ) . with two exceptions , the scl stars mimic their counterparts in [ hs / fe ] . [ eu / fe ] ( figure 10 ) appears to decrease from near + 0.7 at [ fe / h]@xmath49 to @xmath17 solar at [ fe / h]=1.2 , although the trend again is mainly determined by only two stars . once again we omit the heavy element star 982 in which there is a huge excess of eu . with the exception of these three stars , the remainder are in good agreement with the mean for similar stars . perhaps the best discriminator of the relative importance of s - process to r - process enhancement is the behavior of [ ba / eu ] as a function of . in figure 11 we present our scl results . the long dashed line below represents the pure r - process ratio from arlandini et al . ( 1999 ) , while the long dashed line above represents the pure s - process ratio . in the galaxy , this ratio is @xmath50 constant at @xmath51 throughout the halo regime but begins to rise at @xmath52 and reaches a solar value near = 0.4 , significantly above the r - process line but well below the s - process value . scl has a unique behavior in this diagram . the most stars follow the . however , the stars more than = -1.5 follow an upward trend reaching to [ ba / eu]=+0.5 at = 1 , some 0.8 dex above the value . note that the heavy element star 982 does not stand out in this diagram as being unusual but only extends the trend begun by its nearest companions in . the rise in [ ba / eu ] for scl from figure 11 occurs near [ fe / h]= -1.5 , while in field stars no such rise is seen until [ fe / h]@xmath53 . in @xmath1 cen , however , there is a much steeper increase between [ fe / h]= -2.0 and -1.4 , as shown in figure 11 , plotted as the filled squares . other s - process species show a similar rise in @xmath1 cen ( e.g. vanture , wallerstein , & brown 1994 , norris & da costa 1995 , smith et al . 2000 ) . in the galaxy , note that the metal - poor stars have an @xmath50 r - process ratio , which gradually increases towards the s - process value as [ fe / h ] increases ; this reflects the increasingly important contribution to the heavy elements from agb stars as the galaxy evolves chemically . @xmath1 cen is known to be heavily influenced in its chemical evolution by agb stars and this is shown by the rapid increase ( in terms of an increase in [ fe / h ] ) to a pure s - process ratio in ba / eu as fe increases . the s / r - process chemical evolution in scl appears to have been intermediate to that experienced by the milky way and @xmath1 cen . however , note that johnson and bolte ( 2001 ) found that the interpretation of ba in this metallicity range is complicated . galactic ba abundances from the 4554 line can show a rise in ba / eu occuring at very low metallicities ( around 2.0 ) , which is very unlikely to be due to s - process enrichments since they are not matched by la / eu enhancement . this line is not used in our analysis nor that of s03 . venn et al . ( 2004 ) showed that this ba / eu early rise is also seen in other dsph stars . figure 12 compares a small portion of one of the orders from the spectrum of stars 1446 ( top ) and 982 ( bottom ) . note that these stars have similar effective temperatures and fe abundances , as evidenced by the similarity in the strengths of the fei lines . however , note the immense strength of lines due to the s - process elements zirconium and barium , as represented by zr i and ba ii , in the spectrum of star 982 . the unusually large ratio of all species from y to eu relative to fe mark sc982 as a heavy element star . such stars are extremely rare in our halo but several have been found now in dsphs ( s01 , s03 ) . there are three types of heavy element stars . lloyd evans ( 1983 ) suggested that the heavy element stars in @xmath1 cen were formed with their observed heavy - element excess . many galactic heavy - element stars , referred to as intrinsic " , have generated their own excess heavies . the best example of such objects are the s stars which contain technetium . a third type of heavy - element star has received a dose of heavies from a now defunct companion . they are referred to as extrinsic " , and are identified by the fact that they are spectroscopic binaries with periods near a year or somewhat greater . we have searched for the tc i lines in the 4238 - 4297 region in scl 982 and have not found them . due to the heavy blending and modest s / n of the spectrum , this test is not definitive , but is indicative that we are not dealing with a recently self - polluted s or sc star . we have no information on a possible variable radial velocity of 982 , but its observed velocity falls within the spread of the other stars . nevertheless we can derive some useful information about the star or stars that produced the heavy elements seen in sc982 . a useful spectroscopic criterion is the hs / ls ratio with the ls species represented by y and zr and the hs represented by ba and la . in addition we have measured the rb abundance which is sensitive to the neutron density during the neutron capture events that added to the heavy elements . using our observed value of 0.6 for [ hs / ls ] and figures 17 and 18 of smith ( 1997 ) , we find a neutron exposure of tau = 1.1(mb@xmath54 ) and a log n(n ) = 8.6 ( @xmath55 ) . in addition , the ratio of 0.6 for [ hs / ls ] combined with the metallicity of [ fe / h]=-1 places sc982 among the ch stars ( vanture 1992 ) though it does not appear to have the enhanced ch and c@xmath56 shown by ch stars . hence it is possible that the star that produced the heavies was a ch star . one phenomenon seen in many red giants in globular clusters ( but not in field stars ) is a deficiency of oxygen combined with an excess of na , as discussed above . the enhanced na / o ratio has been explained with proton captures by both @xmath57o and the nena cycle , either within the observed red giant or by agb stars that enriched the presently observed star . a significant enhancement of the na / o ratio requires a temperature near 30 x 10@xmath58k for the required proton captures to be effective on a reasonable timescale . the abundances of star 982 as shown in table 5 indicate a low o / fe when compared to other sculptor red giants : [ o / fe]= -0.30 , while the mean for the other stars is + 0.30@xmath300.20 . the plot of [ na / fe ] versus [ o / fe ] shown in figure 4 shows no strong trend , although star 982 has the lowest [ o / fe ] and largest [ na / fe ] values . in this paper we have combined our new vlt plus uves high resolution abundance data for 4 stars in the sculptor dwarf spheroidal galaxy with similar data obtained with the same instrument for 5 stars by shetrone et al . this extends the range of [ fe / h ] covered from -2.1 to -0.97 and allows us to distinguish a number of interesting trends of various elemental abundances with fe that were either not visible or only hinted at in the more limited dataset of s03 . the most important single fact that emerges is that from oxygen to manganese many elements show a relatively constant elemental ratio [ x / fe ] at the end and then declining rapidly for @xmath59 , or a steadily decreasing value of [ x / fe ] as [ fe / h ] rises from -2 to -1 . the elements showing this behavior include o , mg , ca , ti , sc and mn ( while the ba / eu ratio increases with above = 1.5 ) . it is unique to see the same pattern for all of these elements . in particular the ratios of [ o , mg , ca / fe ] near [ fe / h ] = -2 are similar to but slightly less than their values in the galactic halo at similar , but their decline to @xmath60 solar or less near [ fe / h ] = -1.0 rather than at [ fe / h ] = 0.0 is unique . [ sc / fe ] and [ ti / fe ] are near solar at [ fe / h ] = -2 and become substantially negative at [ fe / h ] = -1 . one of our prime motivations for undertaking this study was to further test the hypothesis that the halo of our may have been accreted from dsph - like objects such as scl , as first proposed by searle and zinn ( 1978 ) . our derived composition of the scl stars does not support the suggestion that the halo of our galaxy was formed from stars such as those now seen in scl . this point has already been made by s01 , f02 , s03 and tolstoy et al . we find that scl stars are significantly underabundant in [ /fe ] at all with respect to typical galactic field stars . agb stars in scl were more important in the chemical evolution of scl than in the in causing the high s - process / r - process ratios ocurring in the most stars . finally , we find a heavy element star , with very strong enhancement of s - process elements . we thank paranal observatory for the excellent support received during our observing run , especially from t. szeifert and g. marconi . we would like to gratefully acknowledge m. shetrone , e. tolstoy , v. hill , k. venn , a. kaufer and f. primas for allowing us to access their results prior to publication . t. richtler made valuable comments on an earlier draft . special thanks to the referee , k. venn , for a very helpful , thorough and constructive job which significantly improved this paper . this work is supported in part by the national science foundation through ast99 - 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ph/0207467 + weiss , a . & charbonnel , c . 2003 , in proceedings of jd 04 at the xxiii iau general assembly , ed . f. dantona and g. da costa , mem.soc.ast.it . , 75 , 347 + woolf , v.m . , tomkin , j. , & lambert , d.l . 1995 , apj , 453 , 660 + woosley , s.e . , & weaver , t.a . 1995 , apjs , 101 , 181 + yanny , b. , et al . , 2003 , apj , 588 , 824 + yong , d. , grundhal , f. , lambert , d.l . , nissen , p.e . , & shetrone , m. , 2003 , a&a , 402 , 985 + versus [ o / fe ] for the sculptor red giants ( large magenta filled circles - this paper , large magenta filled squares - s03 ) , along with results from galactic field stars ( small blue symbols ) . due to the large numbers of galactic stars , their symbols must be kept small , so the different symbols are not apparent , but these studies include edvardsson et al . ( 1993 - filled squares ) , nissen et al . ( 1997 - filled triangles ) , prochaska et al . ( 2000 - 6-pointed stars ) , or reddy et al . ( 2003 - filled circles ) . , width=529 ] , [ si / fe ] , [ ca / fe ] , and [ ti / fe ] versus [ fe / h ] for the sculptor red giants ( large magenta filled circles - this paper , large magenta filled squares - s03 ) and samples of galactic field stars ( small blue symbols ) . due to the large numbers of galactic stars , their symbols must be kept small , so the different symbols are not apparent , but these studies include gratton & sneden ( 1988 ) , edvardsson et al . ( 1993 ) , mcwilliam et al . ( 1995 ) , nissen et al . ( 1997 ) , prochaska et al . ( 2000 ) , carretta et al . ( 2002 ) , fulbright ( 2002 ) , johnson ( 2002 ) , and reddy et al . ( 2003 ) . , width=529 ] , [ mn / fe ] , [ cu / fe ] and [ zn / fe ] versus [ fe / h ] for the sculptor red giants ( large magenta filled circles - this paper , large magenta filled squares - s03 ) and samples of galactic field stars ( small blue symbols ) . due to the large number of galactic points , these symbols are kept small and the different symbols are not apparent , but these studies include sneden & crocker ( 1988 ) , sneden et al . ( 1991 ) , gratton & sneden ( 1991 ) , mcwilliam et al . ( 1995 ) , nissen et al . ( 2000 ) , mishenina et al . ( 2002 ) , johnson ( 2002 ) , and reddy et al . , width=529 ] versus [ fe / h ] for the sculptor red giants ( large magenta filled circles - this paper , large magenta filled squares - s03 ) and samples of galactic field stars ( small blue symbols ) . yttrium is used as a surrogate for the light s - process elements ( often the average of y and zr are used , but most of the galactic studies of metal - poor stars do not include zr ; the addition of zr to the sculptor stars would not change their positions significantly ) . the galactic studies include gratton & sneden ( 1994 - filled triangles ) , fulbright ( 2002 - 6-pointed stars ) , johnson ( 2002 - 5-pointed stars ) , and reddy et al . ( 2003 - filled circles ) . note the extreme [ y / fe ] enhancement in star 982 . , width=529 ] versus [ fe / h ] , where hs " is the mean of the heavy s - process elements ba and la , for the sculptor red giants ( large magenta filled circles - this paper , large magenta filled squares - s03 ) and galactic field stars ( small blue symbols ) . the galactic studies consist of gratton & sneden ( 1994 - filled triangles ) , fulbright ( 2002 - 6-pointed stars ) , and reddy et al . ( 2003 - filled circles ) . note the extreme heavy s - process enhancement in star 982 . , width=604 ] versus [ fe / h ] for the sculptor red giants ( large magenta filled circles - this paper , large magenta filled squares - s03 ) and samples of galactic field stars ( small blue symbols ) . the galactic studies include gratton & sneden ( 1994 - filled triangles ) , woolf et al . ( 1995 - filled triangles ) , mcwilliam et al . ( 1995 - 4-pointed crosses ) , fulbright ( 2002 - filled squares ) , johnson ( 2002 - 6-pointed stars ) , and reddy et al . ( 2003 - filled circles ) . note the extreme enhancement of star 982 . , width=604 ] versus [ fe / h ] for the sculptor red giants ( large magenta filled circles - this paper , large magenta filled squares - s03 ) , galactic field stars ( small blue symbols ) and red giants from the peculiar globular cluster @xmath1 cen ( large red squares ) . the long dashed line below represents the pure r - process abundance ratio and the long dashed line above represents the pure s - process abundance ratio from arlandini et al . the galactic studies include gratton & sneden ( 1994 - filled squares ) , mcwilliam et al . ( 1995 - filled squares ) , burris et al . ( 2000 - 6-pointed stars ) , fulbright ( 2002 - filled pentagons ) , johnson ( 2002 - filled hexagons ) , and reddy et al . ( 2003 - filled circles ) . the points for @xmath1 cen are from smith et al . , width=529 ]

we have used high - resolution , high signal - to - noise spectra obtained with the vlt and uves to determine abundances of 17 elements in 4 red giants in the sculptor dwarf spheroidal galaxy . our [ fe / h ] values range from 2.10 to 0.97 , confirming previous findings of a large metallicity spread . we have combined our data with similar data for five sculptor giants studied recently by shetrone et al . to form one of the largest samples of high resolution abundances yet obtained for a dwarf spheroidal galaxy , covering essentially the full known metallicity range in this galaxy . these properties allow us to establish trends of [ x / fe ] with [ fe / h ] for many elements , x. the trends are significantly different from the trends seen in galactic halo and globular cluster stars . this conclusion is evident for most of the elements from oxygen to manganese . we compare our sculptor sample to their most similar galactic counterparts and find substantial differences remain even with these stars . the many discrepancies in the relationships between [ x / fe ] as seen in sculptor compared with galactic field stars indicates that our halo can not be made up in bulk of stars similar to those presently seen in dwarf spheroidal galaxies like sculptor , corroborating similar conclusions reached by shetrone et al . , fulbright and tolstoy et al . these results have serious implications for the searle - zinn and hierarchical galaxy formation scenarios . we also find that the most metal - rich star in our sample is a heavy element - rich star . this star and the [ ba / eu ] trend we see indicates that agb stars must have played an important role in the evolution of the s - process elements in sculptor . a very high percentage of such heavy element stars are now known in dwarf spheroidals compared to the halo , further mitigating against the formation of the halo from such objects .

You are an expert at summarizing long articles. Proceed to summarize the following text: the specific curve pairs are the most popular subjects in curve and surface theory and involute - evolute pair is one of them . we can see in most textbooks various applications not only in curve theory but also in surface theory and mechanic . in this study , the spherical indicatrices of involute of a space curve are given . in order to make a involute of a space curve and its evolute curve slant helix , the feature that spherical indicatrices curve s need to have are examined . let @xmath0 be a curve with @xmath1 , where @xmath2 @xmath3 . the arc - lenght @xmath4 of a curve @xmath5 is determined such that @xmath6 let us denote @xmath7 and we call @xmath8 a tangent vector of @xmath9 at @xmath10 . we define the curvature of @xmath11 by @xmath12 . if @xmath13 then the unit principal normal vector @xmath14 of the curve at @xmath10 is given by @xmath15 . the unit vector @xmath16 is called the unit binormal vector of @xmath11 at @xmath17 . then we have the frenet - serret formulae @xmath18where @xmath19 is the torsion of @xmath11 at @xmath10 @xcite . the curve @xmath11 is called evolute of @xmath20 if the tangent vectors are orthogonal at the corresponding points for each @xmath21 in this case , @xmath20 is called involute of the curve @xmath11 and there exists a relationship between the position vectors as@xmath22where @xmath23 is the distance between the curves @xmath11 and @xmath24 at the corresponding points for each @xmath25 the pair of ( @xmath24 , @xmath11 ) is called a involute - evolute pair . @xmath26 is not a constant for involute - evolute pairs@xcite . on the other hand , izumiya and takeuchi have introduced the concept of slant helix by saying that the normal lines make a constant angle with a fixed straight line . they characterize a slant helix if and only if the geodesic curvature of the principal image of the principal normal indicatrix@xmath27is a constant function , where @xmath28@xcite . in this study , we denote @xmath29 , @xmath30 , @xmath31 @xmath32 , @xmath33 and @xmath34 , @xmath35 , @xmath36 @xmath37 , @xmath38 are the frenet equipments of @xmath11 and @xmath39 respectively . tangent , principal normal and binormal vectors are described for the spherical curves which are called tangent , principal normal and binormal indicatrices both the curves @xmath11 and @xmath39 respectively . throughout this study , both involute and evolute curves are regular . in this section , we introduced the spherical indicatrices of involute curve of a curve in euclidean 3-space and gave considerable results by using the properties of the curves , similar to the previous section . let @xmath11 be a curve with its involute curve @xmath20 then @xmath40where @xmath41and @xmath42 is definitely positive . let @xmath43 be the sign of @xmath42 such that if @xmath44 , @xmath45 and if @xmath46 , @xmath47 we differentiate the equation ( [ 2 ] ) with respect to @xmath4 , we get@xmath48since @xmath29 and @xmath49 are orthogonal , there is no any component of @xmath50 on @xmath29 . thus @xmath43 has to be @xmath51 . [ t1]let @xmath20 be involute of a space curve , then we have frenet formula:@xmath52where @xmath53with the parametrization @xmath54@xmath55and the curvature and torsion of @xmath56 are @xmath57the geodesic curvature of the the principal image of the principal normal indicatrix of involute curve is@xmath58 from ( [ 5 ] ) , it is obvious that involute of @xmath11 is a planar curve if and only if @xmath11 is a generalized helix . for further usage we denote @xmath59 as @xmath60 . by using ( [ 1 ] ) and ( [ 5 ] ) we obtained the relation@xmath61and so we have @xmath62thus we have the following theorem . [ t2]if the frenet frame of the tangent indicatrix @xmath68 of involute of @xmath10 is @xmath69 , we have frenet formula:@xmath70where@xmath71with the parametrization@xmath72and the curvature and torsion of @xmath73 are@xmath74the geodesic curvature of the principal image of the principal normal indicatrix of @xmath73 is@xmath75 let @xmath20 be involute of a space curve @xmath11 then spherical image of the tangent indicatrix of @xmath20 is a spherical helix if and only if involute of @xmath11 is a slant helix . in this case , spherical image of the tangent indicatrix of @xmath20 is a slant helix on unit sphere too . if the frenet frame of the principal normal indicatrix @xmath82 of involute of the curve @xmath10 is @xmath83 , we have frenet formula:@xmath84where@xmath85with the parametrization@xmath86and the curvature and torsion of @xmath87 are@xmath88 + \left [ \left ( \tfrac{-% \widetilde{f}^{^{\prime } } \left ( 1+\widetilde{f}^{2}\right ) ^{\frac{3}{2}}}{% \rho } \right ) \left ( \tfrac{\widetilde{\kappa } ^{2}\left ( 1+\widetilde{f}% ^{2}\right ) ^{\frac{5}{2}}}{\rho } \right ) ^{^{\prime } } \right ] + \left [ \left ( \tfrac{\widetilde{\kappa } \widetilde{f}^{^{\prime } } \left ( 1+% \widetilde{f}^{2}\right ) ^{\frac{3}{2}}}{\rho } \right ) ^{^{\prime } } \left ( \tfrac{\widetilde{\kappa } \left ( 1+\widetilde{f}^{2}\right ) ^{\frac{5}{2}}}{% \rho } \right ) \right ] \right\ } \notag\end{aligned}\ ] ] where @xmath89@xmath90the geodesic curvature of the principal image of the principal normal indicatrix of @xmath87 is@xmath91where if the frenet frame of the binormal indicatrix @xmath96 of involute of the curve @xmath10 is @xmath97 , we have frenet formula:@xmath98where @xmath99with the parametrization @xmath100and the curvature and torsion of @xmath101 are@xmath102the geodesic curvature of the principal image of the principal normal indicatrix of @xmath101 is@xmath103 let @xmath11 be a space curve and @xmath20 be its involute with nonzero torsion then spherical image of binormal indicatrix of @xmath24 is a circle on unit sphere if and only if @xmath104 is a generalized helix . let @xmath112 and @xmath113be two regular curves in @xmath114then @xmath115 and @xmath116 are similar curves with variable transformation if and only if the principal normal vectors are the same for all curves @xmath117 under the particular variable transformation @xmath118@xmath119of the arc - lengths . let @xmath120 and @xmath121be two regular curves in @xmath114then @xmath73 and @xmath122 are similar curves with variable transformation if and only if the principal normal vectors are the same for all curves @xmath123 under the particular variable transformation @xmath118@xmath124of the arc - lengths . in @xcite , the general equation of spherical helix family is obtained by monterde which is , @xmath125where @xmath126 @xmath127 in 4b , kula et al . obtained the general equation of a slant helix family similar to following@xmath128where @xmath129}{% 2w\left ( w+1\right ) } + \frac{\left ( w+1\right ) \sin [ \left ( w-1\right ) t]}{% 2w\left ( w-1\right ) } \\ \gamma _ { \mu } ^{2}(s ) & = & \frac{\left ( w+1\right ) \cos [ \left ( w-1\right ) t]}{% 2w\left ( w-1\right ) } + \frac{\left ( w-1\right ) \cos [ \left ( w+1\right ) t]}{% 2w\left ( w+1\right ) } \\ \gamma _ { \mu } ^{3}(s ) & = & -\frac{\cos \left ( s\right ) } { \mu w}.\end{aligned}\]]and @xmath130 @xmath131 by using the theorem [ c5 ] , we can obtained the general equation of general helix family in euclidean 3-space according to the non - zero constant @xmath132 as follows@xmath133where@xmath134+\left ( w-1\right ) \cos [ \left ( w+1\right ) t]\right\ } \\ & & + \frac{1}{2w\left ( w^{2}-1\right ) } \left\ { \left ( w+1\right ) ^{2}\sin [ \left ( w-1\right ) t]+\left ( w-1\right ) ^{2}\sin [ \left ( w+1\right ) t]\right\ } \\ \widetilde{\gamma } _ { \mu } ^{2}(s ) & = & \frac{-\left ( c - s\right ) } { 2w}\left\ { \left ( w+1\right ) \sin [ \left ( w-1\right ) t]+\left ( w-1\right ) \sin [ \left ( w+1\right ) t]\right\ } \\ & & + \frac{1}{2w\left ( w^{2}-1\right ) } \left\ { \left ( w+1\right ) ^{2}\cos [ \left ( w-1\right ) t]+\left ( w-1\right ) ^{2}\cos [ \left ( w+1\right ) t]\right\ } \\ \widetilde{\gamma } _ { \mu } ^{3}(s ) & = & \frac{\left ( c - s\right ) \sin \left ( s\right ) -\cos \left ( s\right ) } { \mu w}\end{aligned}\]]and @xmath130 @xmath135 with the parametrization@xmath136and the curvature and torsion of curve @xmath137 is @xmath138

in this work , we studied the properties of the spherical indicatrices of involute curve of a space curve and presented some characteristic properties in the cases that involute curve and evolute curve are slant helices and helices , spherical indicatrices are slant helices and helices and we introduced new representations of spherical indicatrices .

You are an expert at summarizing long articles. Proceed to summarize the following text: the fast rise , slow decay of subpulses in grb is a common feature . there could be many ways to explain it ( e.g. impulsive energy infusion followed by slower cooling or light echoing ) . it is therefore desirable to discriminate among the different models with quantitative tests and predictions whenever possible . in a previous paper ( eichler and manis 2007 , hereafter em07 ) , it was suggested that fast rise , slow decay subpulses constitute a qualitative manifestation of baryons being accelerated by radiation pressure . more generally , the basic idea can apply to any flow in which a light , fast fluid imparts energy to a clumpy , denser component of the flow by overtaking the clumps from the rear , but for convenience in this discussion we refer to the fast light component as photons that scatter off the clumps . it was proposed that the fast rise of a subpulse is the stage where a cloud of baryons scatters photons into a progressively narrowing beaming cone of width @xmath3 , where @xmath4 is the bulk lorentz factor of the accelerating cloud . this narrowing of the @xmath3 cone causes brightening as long as @xmath4 remains below @xmath5 , where @xmath6 is the viewing angle offset between the observer s line of sight and the velocity vector of the scattering cloud . once the scattering cloud accelerates to a lorentz factor exceeding @xmath5 , the viewer is no longer inside the beaming cone and apparent luminosity begins to decline . if the cloud accelerates with roughly constant radiative force , as is reasonable to suppose over timescales that are short compared to the hydrodynamic expansion time , then the decline in luminosity is considerably slower than the rise time , because the acceleration time increases so dramatically as the velocity approaches c. it was shown in em07 that the spectral peak frequency as seen by the observer remains roughly constant during the rising phase and , well into the declining phase , softens as @xmath7 , as reported by ryde ( 2004 ) . the spectral softening of the pulse has been carefully studied by norris and coworkers , who have noted that the asymmetry of the subpulse increases with decreasing frequency and that the width of the subpulse scales roughly as the frequency to the power -0.4 ( fenimore et al 1995 ) in the batse energy range . this represents additional information , as the result of ryde is in principle consistent with symmetric pulses . in this letter , we derive the light curves as a function of both time and frequency . we show that the asymmetry of the subpulse decreases with frequency and that the hypothesis of em07 is quantitatively consistent with the formulation of fenimore et al ( 1995 ) . the basic assumption in our hypothesis - that a scattering screen can _ enhance _ the detected signal - presupposes that the unscattered radiation is beamed and directed slightly away from the observer s line of sight , so that the scattering of photons into the line of sight creates a `` flash - in - the - pan '' type brightening . this assumption is non - trivial , but has been suggested as being an explanation for the amati relation ( 2002 ) in earlier papers ( eichler and levinson 2004 , 2006 ; levinson and eichler 2005 ) . in this series of papers , it was suggested that a significant fraction of all grb are actually brighter and harder in spectrum than they appear to be , and that they appear dimmer and softer because we , the observers , are viewing the burst from a slightly offset angle relative to the direction of the fireball . the interpretation of the subpulses given here and in em07 is thus in accord with this picture . the equations describing matter that is being accelerated by highly collimated radiation pressure were presented in em07 . here we apply the solutions described in em07 to calculate the profile of a subpulse as a function of photon energy . we assume that the differential primary photon spectrum @xmath8 has the form @xmath9exp(@xmath10 ) , where @xmath11 is the photon energy in the frame of the central engine . this form is consistent with a comptonized thermal component . it does not , however , exclude a power law photon spectrum produced further downstream by internal shocks . after scattering off a baryon clump that moves with velocity @xmath12 , the photon energy as seen by an observer at angle @xmath6 is @xmath13=e_o(1-\beta)/(1-\beta cos\theta).\ ] ] together with the solution for the accelerating trajectory @xmath14 given in em07 , the source / observer frame invariance of the number of photons @xmath15 scattered within energy interval de and time interval dt , and solid angle @xmath16 , equation ( 1 ) determines the light curve n(e , t ) as a function of observed photon energy e and observer time t. in figure 1 the subpulse light curves are plotted for three different frequencies . it is clear that the pulse width is larger and the rise - fall asymmetry is more pronounced at lower frequencies , as reported by fenimore et al . ( 1995 ) and references therein . in figure 2 the width is plotted as a function of photon energy . at high energies , which correspond to the batse measurements used by these authors , the width is seen to scale approximately as the photon energy to the power @xmath17 , as reported by fenimore et al . , above @xmath18 kev . similar calculations with varying values for the low energy power law index , @xmath19 , of the primary spectrum show that this dependence is weakly dependent on @xmath19 and on viewing angle . for a viewing offset angle of 10 degrees , the width depends on @xmath20 , with @xmath21 when @xmath22 with the sensitivity @xmath23 at @xmath24 . for viewing offset of 15 degrees , the value of @xmath25 is increased by about 0.033 so that a given range of @xmath25 is occupied by a somewhat lower ( i.e. more negative ) range of @xmath19 than for smaller viewing offsets . for an extended beam , some contribution from larger offsets is inevitable , but a synthesis of light curves from extended beams is deferred for future work . it can be seen from figure 2 that the value of @xmath25 increases with @xmath26 , and the range of @xmath26 that corresponds to batse sensitivity depends on cosmological redshift , larger z implies larger intrinsic values of @xmath26 , hence steeper e dependence of the pulse width , over a given range of observed photon energies . finally , the primary source spectrum , which we argue is not a direct observable , is somewhat uncertain . altogether , the range of @xmath27 is consistent with the ranges @xmath28 , @xmath29 , @xmath30 , and @xmath31 . it is predicted that the dependence of width on e weakens ( i.e. @xmath25 decreases ) at lower photon energies , and this should be testable with detectors that are more sensitive at lower energies , such as the gamma ray burst monitor . as the acceleration time is inversely proportional to the radiation flux on the scatterer , it is clear , all other things being equal , that the rise time of the pulse and spectral lag are inversely proportional to source luminosity , as observed ( e.g. gehrels et al . , 2006 ) . of course , scatter in other variables , such as the distance from the source of illumination , optical depth of the scatterer etc . , creates scatter in the constant of proportionality . if the scattering is isotropic ( or backwardly biased due to high optical depth ) in the scattering frame , it follows from equation ( 1 ) that the scattered radiation , averaged over angle , is a factor of 2 ( or more ) softer than the primary emission . as explained in em07 , the other half of the energy goes into the acceleration of the scatterer . on the other hand , at most viewing angles @xmath6 , the scattered radiation is harder than scattered radiation after the scatterer has reached terminal lorentz factor @xmath32 if @xmath33 . as the scattered radiation during the acceleration phase of the scatterer is likely to be the most time dependent , it may be possible to separate out this component from the other two . the extent to which the scattering affects the spectrum depends , of course , on the fraction of primary radiation that is scattered . equating the scattered photon energy with the baryon afterglow energy , and applying the results of eichler and jontof - hutter ( 2005 ) , which estimated the afterglow efficiency , we may tentatively estimate that about 30 percent of the primary emission is scattered , about half of that 30 percent going into baryons and the other half ending up in a scattered subpulse component . clearly there is variation in the scattered fraction as well as uncertainty in theoretical inferences of the baryon energy from afterglow calorimetry , so this estimate should be considered rough and preliminary . the time - integrated spectrum at a _ given _ viewing angle can be different from the average , because the scattered radiation is not isotropic but , rather , beamed in an ever narrowing cone as the scatterer accelerates . consider a primary emission spectrum that is a delta function @xmath34 . at a given @xmath35 and a given observed photon energy @xmath36 , a monochromatic primary spectrum @xmath37 is , after scattering , monochromatic at photon energy @xmath38 $ ] given by equation ( 1 ) , so the contribution to the emitted power at energy e comes only at @xmath39 the time integrated energy @xmath40 of the scattered radiation at observed photon energy e and viewing angle @xmath6 is @xmath41 = @xmath42 where @xmath43 is the power of the scattered radiation as observed at photon energy e in the frame of the primary source , is to make it a positive quantity . ] t is the elapsed time in the frame of the source ( and in any case the variable of integration ) , @xmath44 is has front - back symmetry in the frame of the scatterer . a high , time varying optical depth would be more complicated . ] @xmath45 ( em07 ) . making the simplifying assumption that the scattered radiation is isotropic in the frame of the scatterer , i.e. that @xmath46 , using the transformation for emitted power @xmath47 ( equation 4.97a in rybicki and lightman , 1979 ) and evaluating @xmath48 in units of @xmath11 from equation ( 1 ) , it follows that @xmath49^{1/2}/e_o[1-\mu]^{3/2}.\ ] ] in the limit that @xmath50 is sufficiently below unity that @xmath51 does not depend significantly on e , @xmath52 is proportional to @xmath53 , i.e. @xmath54 . that this is softer than the average over all viewing angles - for which @xmath55 - can be understood as the result of most of the emission at large t going into a narrower cone than the one the observer is on , so that the observer sees only the soft fringes of this dominant component . also note that this result assumes that the scatterer s acceleration proceeds indefinitely . if the scatterer reaches a terminal velocity @xmath56 , then the observer would not see any of the primary radiation originally at @xmath11 scattered to an energy below @xmath57 . while the scattered radiation is not the only observable component , the hypothesis that it comprises a significant fraction of the total fluence of many grb is broadly consistent with the tendency of the low energy spectral index to not exceed 0 . the question of how much radiation is scattered , before and after the scatterers have reached terminal velocity , remains somewhat open at the quantitative level , but the gbm data may provide the opportunity to address these questions quantitatively as well as qualitatively . we have presented evidence that subpulses in @xmath0-ray bursts are photons that are scattered into our line of sight by scatterers with lower lorentz factors than the frame in which the prescattered photons had zero net momentum . because scattering can never increase the intensity of a beam of photons , the hypothesis presupposes that the observed intensity is lower in the observer s direction than in the direction of the beam , and that the scattering by the slower baryons broadens the beam enough that it engulfs the observer s line of sight . it is this widening of the beam that allows the observer to see enhanced flux . this fits the picture already put forth to explain the amati relation . as most grb with known redshifts have spectral peaks and energies that are both below those of the hardest , brightest grb , it would follow , according to our interpretation , that a large fraction of , perhaps most , grb are observed from an offset viewing angle . the question then arises as to why there are so many offset observers relative to those that are within the @xmath58 beaming cone of the primary radiation . in an earlier paper ( eichler and levinson 2004 ) , it was proposed that the complex shape of the primary beam - e.g. an annular shape - would allow a comparable number of viewers just off ( i.e. within several times @xmath58 ) of the periphery of the primary beam to the number of viewers within this periphery . it was shown that a thick annulus , in which the separation between the inner and outer radii is comparable to half the outer radius , gives a distribution of offsets that is consistent with the observed distribution of spectral peaks . there may be several reasons for grb jets to have an annular or otherwise complex morphology : it could be that baryons are entrained in the flow from ( or are fed neutrons by ) the confining walls ( levinson and eichler 2003 ) and that much of the liberated energy is due to dissipation associated with this entrainment . or , it could be that dissipation from shocks associated with wall impact could preferentially liberate energy from near the confining walls ( begelman , private communication ) . here we suggest another simple mechanism that would give the scattered @xmath0-radiation some measure of annular bias , depending on the fraction scattered : consider the region of flow where the @xmath0-rays make their last scattering off baryons within the flow . if , as seems more likely than not , the baryons are clumpy , it is likely that the clumps are optically thick , while the interclump medium is optically thin . in this case the photons are most likely to make their last scattering off the surface of a clump . if the clumps are moving more slowly than the primary fireball , then the photons are most likely to overtake them from the rear , and , because the clumps are optically thick , the photon is likely to emerge from its last scattering from the rear end of the clump . it is then obscured from a viewer who lies along the velocity vector of the clump , just as sunlight scattering off the moon is obscured to a viewer on the dark side of the moon . the viewers best positioned to see the back side illumination are those who see photons that are emitted backward in the frame of the cloud - i.e. those that are offset by more than @xmath3 from the velocity vector of the clump . this , we suggest , could be a reason so many grb are observed from an offset angle of more than @xmath3 . some subpulses have such fast rises that they can be interpreted as the @xmath3 shadow of an optically thick cloud narrowing from above to below the offset angle of the observer , @xmath6 , as the cloud accelerates ( em07 ) . the very sudden rise is then attributed to the observer emerging from the shadow of the clump . to summarize , we suggest that spectral lags from long bursts are the result of high @xmath4 radiation ( where @xmath4 is the lorentz factor of the frame in which the radiation is isotropic ) impacting slower baryons from behind , and scattering off them while accelerating them . as in ( em07 ) , the quantitative agreement with the data is excellent . the inverse correlation between lag and luminosity ( e.g. gehrels et al 2006 ) follows from that between acceleration time and luminosity . the assumptions needed to make the general scheme work are minimal . fluid in the frame of the clump . however , in the simplest internal shock model , where the average electron energy , magnetic field , and blue shift all scale with @xmath4 , the spectral peak varies as a high power of @xmath4 . ] in any case , the minimal conclusion to be drawn from the quantitative success of this model in explaining spectral lags of long grb is that baryons are still undergoing considerable acceleration by the time the fireball as a whole is becoming optically thin . were energy systematically removed from baryons beyond the photosphere ( e.g. because they collided with other scatterers in their shadow ) , one would expect negative spectral lags . it is predicted that the time integrated spectra of the subpulses should be slightly softer than the primary emission , and harder than the emission that is scattered at terminal lorentz factor . the glast / fermi gbm monitor offers the potential opportunity for unraveling these three components . for short hard bursts ( shb ) , the subpulses are about a factor of 10 to 30 shorter than for long ones and the spectral lags are much smaller . this difference can perhaps be attributed to the difference in timescale over which the baryons are undergoing acceleration . for example shb are likely to be observable somewhat closer to the central engine , being unobscured by the envelope of a massive host star , and at a wider angle ( eichler , guetta , and manis 2008 ) . if this is indeed the case , then we may be able to see baryonic clumps at an earlier stage of their acceleration , when the acceleration time is considerably shorter . there are many unknowns in this hypothesis - e.g. the optical depth of the baryons , their point of injection and their covering factor - on which the qualitative nature of the subpulses may depend , and future work will focus on the question of whether reasonable ranges for these parameters can explain the wide diversity of grb light curves and spectra . this work was supported by the joan and robert arnow chair of theoretical astrophysics , the us - israel binational science foundation and the israeli science foundation s center of excellence program . 99 amati , l. , et al . , 2002 , a & a , 390 , 81 eichler , d. & jontof - hutter , d. 2005 , apj , 635 , 1182 eichler , d. & levinson , a. , 2004 , apj . 614 , l13 eichler , d. & levinson , a. , 2006 , apj , 649 , l5 eichler , d. & manis , h. , 2007 , apj , 669 , l65 eichler , d. guetta , d. & manis , h. , 2008 , apj ( submitted ) fenimore , e. e. , in t zand , j. j. m. , norris , j. p. , bonnell , j. t. & nemiroff , r. j. , 1995 , apj , 448 , l101 gehrels , n. et al . 2006 , nature 444 , 1044 levinson , a. & eichler , d. , 2003 , apj , 594 , l19 levinson , a. & eichler , d. , 2005 , apj , 629 , l13 ryde , f. , 2004 , apj , 611 , l41

a quantitative theory of spectral lags for @xmath0-ray bursts ( grbs ) is given . the underlying hypothesis is that grb subpulses are photons that are scattered into our line of sight by local concentrations of baryons that are accelerated by radiation pressure . for primary spectra that are power laws with exponential cutoffs , the width of the pulse and its fast rise , slow decay asymmetry is found to increase with decreasing photon energy , and the width near the exponential cutoff scales approximately as @xmath1 , with @xmath2 , as observed . the spectral lag time is naturally inversely proportional to luminosity , all else being equal , also as observed .

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Proceed to summarize the following text: nonlinear conduction in correlated electron systems such as one - dimensional mott insulators@xcite and two - dimensional charge - ordered materials@xcite has been of great interest in the past few decades . they offer intriguing subjects of nonequilibrium physics in condensed matter and possibility for novel functions of electronic devices . for example , in a typical quasi - one - dimensional mott insulator , sr@xmath9cuo@xmath10,@xcite a dielectric breakdown has been observed experimentally by applying a strong electric field . a dielectric breakdown has been reported also in an organic spin - peierls insulator , k - tcnq@xcite [ tcnq = tetracyanoquinodimethane ] . for another organic compound , ( bedt - ttf)(f@xmath9tcnq ) [ bedt - ttf = bis(ethylenedithio)tetrathiafulvalene ] , which is a quasi - one - dimensional mott insulator , metal - insulator - semiconductor field - effect transistor device structures have been reported,@xcite where its field - effect characteristics are different from those of band insulators.@xcite so far , theoretical investigations on nonlinear conduction for interacting electron systems that are initially insulating in their equilibrium states have been done by several authors.@xcite in general , these studies are classified into two approaches depending on whether an external force is written as an electric field@xcite or a bias voltage.@xcite the former approach is to consider electron systems without electrodes . the electric field is usually applied with open boundary condition,@xcite or equivalently with periodic boundary condition by using a time - dependent magnetic flux.@xcite in one dimension , oka and aoki studied the hubbard model under a strong electric field by the time - dependent density - matrix - renormalization - group method.@xcite one of their important results is that the dielectric breakdown of mott insulators is interpreted as a many - body counterpart of the landau - zener ( lz ) breakdown.@xcite this is known to describe the breakdown of band insulators where the one - particle picture holds . the threshold is given as @xmath11 with @xmath5 being an energy gap . a mean - field approach to electric - field - induced insulator - to - metal transitions by using keldysh green s functions has been reported in ref . . the latter approach is to consider an insulator attached to electrodes . interface structures must be explicitly taken into account in real electric devices . along this line , okamoto investigated the current - voltage ( @xmath12-@xmath13 ) characteristics of heterostructures that consist of mott - insulator layers sandwiched by metallic leads by combining the dynamical mean - field theory with the keldysh green s function technique.@xcite ajisaka _ et al . _ studied the @xmath12-@xmath13 characteristics of an electron - phonon system coupled with two reservoirs by a field - theoretical method.@xcite in the studies listed above , okamoto discussed the @xmath12-@xmath13 characteristics in the framework of the lz breakdown , while ajisaka _ proposed a different mechanism with the threshold bias voltage @xmath2 . these results seem to be inconsistent with each other . when we consider a nanostructure in which some material is attached to left and right metallic electrodes , the bias voltage @xmath13 applied to the material is described as @xmath14 with @xmath15 and @xmath16 being the chemical potentials of the left and right electrodes , respectively . here we assume that the work - function difference at the interfaces is absent for simplicity . in this case , one might expect that the current flows when some energy levels of the material appear in the region between @xmath15 and @xmath16 , indicating the threshold governed by the applied bias voltage @xmath2 . in fact , for the transport in a field - effect transistor with a small channel , such an explanation has been used frequently.@xcite however , for a bulk insulator with an energy gap @xmath5 , this picture does not hold and should be replaced by the lz mechanism where the threshold is determined by the electric field , @xmath11 . this consideration poses us a question about which parameter determines the mechanism of the breakdown . in particular , we address the condition for the realization of the lz breakdown in a structure with electrodes . it is also important to know the relation between the approaches using the structure with electrodes and those which do not include them explicitly . we point out that the size of an insulator as well as the potential distribution inside it determine the nature of the breakdown . in this paper , we study one - dimensional band and mott insulators attached to two electrodes ( see fig . [ fig : model ] ) , using the nonequilibrium green s function approach that has previously been used to discuss the suppression of rectification at metal - mott - insulator interfaces.@xcite this approach more naturally describes nonequilibrium steady states than the approach based on the time - dependent schr@xmath17dinger equation because a current oscillation is inevitable in the latter owing to finite - size effects . the present method can be easily applied to higher - dimensional systems . preliminary results for the @xmath12-@xmath13 characteristics of two - dimensional charge - ordered systems are reported in ref . . we show that the applied bias voltage @xmath13 induces a breakdown of band and mott insulators at zero temperature . for both insulators , the threshold shows a crossover as a function of the size of the insulating region @xmath1 . for systems with @xmath1 smaller than the correlation length @xmath18 , i.e. , the characteristic decay length of the wave function in the insulator , the breakdown takes place when the bias @xmath13 exceeds the energy gap @xmath5 . for @xmath6 , it is governed by the electric field , @xmath19 , which is consistent with the lz tunneling mechanism . whether the charge gap is produced by the band structure or by the electron - electron interaction is irrelevant to the crossover phenomenon . we will focus on the spatial modulation of the wave functions . for the crossover behavior and the deformation of the wave functions , the spatial dependence of the scalar potential inside the band or mott insulator is important . this is demonstrated in the appendix by showing the case without the scalar potential , where the bias - induced transition occurs at @xmath20 regardless of the length @xmath1 . we consider a one - dimensional insulator that is attached to semi - infinite metallic electrodes on the left and right sides as shown in fig . [ fig : model ] . the insulator is referred to as the central part , and described by the hubbard model for a mott insulator and a tight - binding model with alternating transfer integrals for a band insulator , both at half filling : @xmath21 ( c^{\dagger}_{i\sigma}c_{i+1\sigma}+h.c.)\nonumber \\ & + & u\sum_{i}(n_{i\uparrow}-\frac{1}{2 } ) ( n_{i\downarrow}-\frac{1}{2})\nonumber \\ & + & \sum_{\langle\langle ij\rangle\rangle}v_{ij}(n_i-1)(n_j-1 ) , \label{eq : ham}\end{aligned}\ ] ] where @xmath22 denotes the creation ( annihilation ) operator for an electron with spin @xmath23 at the @xmath24th site , @xmath25 , and @xmath26 . @xmath1 is the total number of sites in the central part . the parameter @xmath27 denotes the transfer integral , @xmath28 its modulation , and @xmath29 the on - site interaction . we use @xmath27 as a unit of energy . for @xmath30 and @xmath31 , the first and second terms in eq . ( [ eq : ham ] ) become the one - dimensional repulsive hubbard model , whereas they describe a band insulator for @xmath32 and @xmath33 . the long - range coulomb interaction term with @xmath34 is introduced because it is responsible for the potential modulation near the metal - insulator interfaces ( @xmath35 and @xmath36 ) . the @xmath37 term is treated by the hartree approximation , which is equivalent to the introduction of a scalar potential that satisfies the poisson equation . here @xmath38 means the summation over pairs of the @xmath24th and @xmath39th sites with @xmath40 in the central part ( @xmath41 ) . we briefly review our formulation@xcite below to treat steady states with a finite voltage . for the metallic electrodes , we consider noninteracting electrons . the effects of the left and right @xmath42 electrodes on the central part are then described by the retarded self - energies.@xcite for simplicity , we take the wide - band limit so that the self - energies are independent of energy . in the present case , they become diagonal matrices@xcite @xmath43 where @xmath44 and @xmath45 are the kronecker delta and @xmath46 ( @xmath47 ) denotes the site connected with the left ( right ) electrode . we consider the case of @xmath48 . within the hartree - fock approximation for the on - site term in eq . ( [ eq : ham ] ) , the retarded green s function for spin @xmath23 is written as @xmath49_{ij}=\epsilon \delta_{ij}-(h^{r}_{\rm hf\sigma})_{ij},\ ] ] with @xmath50 the off - diagonal elements of @xmath51 come from the first term of eq . ( [ eq : ham ] ) and the diagonal elements are written as @xmath52 with @xmath53 . here , the scalar potential @xmath54 is defined by the hartree approximation to the long - range coulomb interaction as @xmath55 with @xmath56 and @xmath57 being constants . these constants are so determined that @xmath58 satisfies the boundary conditions : @xmath59 for the bias voltage @xmath60 . when @xmath13 is positive , the left electrode has a higher potential for the electrons and the current flows from left to right.@xcite here we assume that the work - function difference is absent at the interfaces . by diagonalizing the complex symmetric matrix @xmath61 , the retarded green s function is obtained as @xmath62_{ij}=\sum_{m}\frac{u^{\sigma}_{m}(i)u^{\sigma}_{m}(j)}{\epsilon -e^{\sigma}_m } , \label{eq : gr}\ ] ] where @xmath63 ( @xmath64 and @xmath65 are real ) is the eigenvalue of @xmath61 and @xmath66 is the corresponding right eigenvector . the electron density is calculated by decomposing it into the `` equilibrium '' and `` nonequilibrium '' parts@xcite as @xmath67 the `` equilibrium '' part is defined by integrating the local density of states as @xmath68_{ii } f_c(\epsilon ) , \label{eq : neq}\ ] ] where @xmath69 is the fermi distribution function with the chemical potential @xmath70 at the midpoint of the right and left chemical potentials , @xmath71 . since @xmath72 and @xmath73 , we have @xmath74 . for @xmath75 , where the left chemical potential is higher than the right , @xmath76 ( @xmath77 ) is interpreted as the inflow ( outflow ) . the `` nonequilibrium '' part of the density @xmath78 is obtained from the `` nonequilibrium '' part of the lesser green s function : @xmath79_{ii}. \label{eq : nneq}\ ] ] in order to obtain @xmath80 , we first decompose the lesser self - energy in the wide - band limit@xcite as in eq . ( [ eq : n_parts ] ) : @xmath81 with @xmath82 and @xmath83,\ ] ] where @xmath84 . then , we employ the keldysh equation @xmath85 where @xmath86 is the hermitian conjugate of @xmath87 . the expressions for @xmath88 and @xmath78 , with which the numerical calculations are carried out , are obtained by substituting eq . ( [ eq : gr ] ) into eqs . ( [ eq : neq ] ) and ( [ eq : nneq ] ) . the results are @xmath89 ^ 2\bigl[\frac{1}{\pi}\tan^{-1}\frac{2(\mu_c-\epsilon^{\sigma}_m ) } { \gamma^{\sigma}_{m}}+\frac{1}{2}\bigr ] , \label{eq : neq2}\ ] ] and @xmath90\nonumber \\ & = & \frac{\gamma_{\alpha}}{2\pi}\sum_{n , m}\bigl\ { { \rm i m } \bigl [ \frac{u^{\sigma}_m(i)u^{\sigma}_m(i_{\alpha})u^{\sigma * } _ n(i)u^{\sigma * } _ n(i_{\alpha } ) } { \epsilon^{\sigma}_m-\epsilon^{\sigma}_n - i\gamma^{\sigma}_m/2-i\gamma^{\sigma}_n/2 } \bigr ] \bigr\}\nonumber \\ & \times & \bigl [ \tan^{-1}\frac{2(\mu_{\alpha}-\epsilon^{\sigma}_m)}{\gamma^{\sigma}_m } -\tan^{-1}\frac{2(\mu_{c}-\epsilon^{\sigma}_m)}{\gamma^{\sigma}_m}\nonumber \\ & + & \tan^{-1}\frac{2(\mu_{\alpha}-\epsilon^{\sigma}_n)}{\gamma^{\sigma}_n } -\tan^{-1}\frac{2(\mu_{c}-\epsilon^{\sigma}_n)}{\gamma^{\sigma}_n } \bigr]\nonumber \\ & + & { \rm re } \bigl [ \frac{u^{\sigma}_m(i)u^{\sigma}_m(i_{\alpha})u^{\sigma * } _ n(i)u^{\sigma * } _ n(i_{\alpha } ) } { \epsilon^{\sigma}_m-\epsilon^{\sigma}_n - i\gamma^{\sigma}_m/2-i\gamma^{\sigma}_n/2 } \bigr]\nonumber \\ & \times & \bigl [ \frac{1}{2 } \ln \frac{(\mu_{\alpha}-\epsilon^{\sigma}_m)^2+(\gamma^{\sigma}_m/2)^2 } { ( \mu_c-\epsilon^{\sigma}_m)^2+(\gamma^{\sigma}_m/2)^2}\nonumber \\ & - & \frac{1}{2 } \ln \frac{(\mu_{\alpha}-\epsilon^{\sigma}_n)^2+(\gamma^{\sigma}_n/2)^2 } { ( \mu_c-\epsilon^{\sigma}_n)^2+(\gamma^{\sigma}_n/2)^2 } \bigr ] . \label{eq : nneq2}\end{aligned}\ ] ] in the above equations , we recover the electron density in the equilibrium state without the electrodes if @xmath91 , since the bracket in eq . ( [ eq : neq2 ] ) is reduced to the step function and @xmath92 . the current from the left electrode is expressed by the `` nonequilibrium '' part of the density as@xcite @xmath93 [ f_l(\epsilon)-f_r(\epsilon)]\nonumber \\ & = & \frac{\gamma_l\gamma_r}{2\pi}\int^{\infty}_{-\infty}d\epsilon \sum_{\sigma}|[g^{r}_{\sigma}(\epsilon)]_{i_li_r}|^2[f_l(\epsilon)-f_r(\epsilon)]\nonumber \\ & = & \gamma_{r}\sum_{\sigma}\delta n^{l}_{i_r\sigma}-\gamma_{l}\sum_{\sigma}\delta n^{r}_{i_l\sigma } , \label{eq : cur}\end{aligned}\ ] ] where we set @xmath94 . in this section , we show the results of @xmath12-@xmath13 characteristics , charge densities , and the spatial dependence of wave functions for band and mott insulators . for both models , a breakdown of the insulating state takes place when the bias @xmath13 becomes sufficiently large . the threshold shows a crossover behavior as a function of the size of the central part @xmath1 , which indicates the mechanism of the breakdown changes according to @xmath1 . the profile of the scalar potential @xmath54 has crucial importance on the way of the breakdown . this is demonstrated in the appendix by showing that , if @xmath54 is artificially set to zero for all @xmath24 , the crossover phenomenon disappears . figure [ fig : band_iv1 ] shows the @xmath12-@xmath13 characteristics for band insulators with @xmath95 and @xmath96 . the other parameters are @xmath97 , @xmath33 , @xmath98 , and @xmath99 . for comparison , we show the results for the regular transfer integrals ( @xmath30 ) with @xmath97 , @xmath33 , @xmath100 , and @xmath99 . for @xmath30 , the current @xmath101 becomes nonzero for @xmath102 since the central part is metallic . the @xmath12-@xmath13 curve has stepwise structures owing to the finite - size effect . for @xmath103 and @xmath104 , the central part is a band insulator with the energy gap @xmath105 . because of the gap , @xmath101 is suppressed near @xmath103 . the @xmath12-@xmath13 curves for finite @xmath28 show more complex structures than that for @xmath30 . apart from the fine structures , @xmath101 increases almost linearly for large @xmath13 , which indicates a breakdown of the band insulator . -@xmath13 characteristics of one - dimensional band insulators for @xmath95 and @xmath106 . the other parameters are @xmath97 , @xmath33 , @xmath98 , and @xmath107 . results for @xmath30 are also shown where we set @xmath97 , @xmath33 , @xmath100 , and @xmath108.,height=215 ] , `` equilibrium '' part , @xmath109 , and ( b ) `` nonequilibrium '' parts @xmath110 and @xmath111 for @xmath95 , @xmath97 , @xmath33 , @xmath98 , @xmath108 , and @xmath112 . the scalar potential @xmath54 is shown in the inset of ( a).,title="fig:",height=200 ] , `` equilibrium '' part , @xmath109 , and ( b ) `` nonequilibrium '' parts @xmath110 and @xmath111 for @xmath95 , @xmath97 , @xmath33 , @xmath98 , @xmath108 , and @xmath112 . the scalar potential @xmath54 is shown in the inset of ( a).,title="fig:",height=200 ] , `` equilibrium '' part , @xmath109 , and ( b ) `` nonequilibrium '' parts @xmath110 and @xmath111 for @xmath95 , @xmath97 , @xmath33 , @xmath98 , @xmath113 , and @xmath114 . the scalar potential @xmath54 is shown in the inset of ( a).,title="fig:",height=200 ] , `` equilibrium '' part , @xmath109 , and ( b ) `` nonequilibrium '' parts @xmath110 and @xmath111 for @xmath95 , @xmath97 , @xmath33 , @xmath98 , @xmath113 , and @xmath114 . the scalar potential @xmath54 is shown in the inset of ( a).,title="fig:",height=200 ] for finite voltages applied , the charge distributions in resistive and conductive states for @xmath95 are shown in figs . [ fig : band_res ] and [ fig : band_con ] , respectively . in fig . [ fig : band_res](a ) , the electron density @xmath115 and its `` equilibrium '' part @xmath116 for @xmath112 are shown . a @xmath117 oscillation in the charge distribution is induced by the boundaries.@xcite for all @xmath24 , @xmath118 is almost unity as in the equilibrium case ( @xmath119 for @xmath103 ) . the electron density @xmath120 is basically unchanged by the bias voltage @xmath13 when @xmath101 is small . the scalar potential @xmath54 has a linear dependence on @xmath24 as shown in the inset of fig . [ fig : band_res](a ) . this is because the long - range interaction term in eq . ( [ eq : psi ] ) is small for @xmath121 , so that @xmath54 is determined only by the boundary conditions . the `` equilibrium '' part @xmath109 , on the other hand , deviates from unity near the left and right electrodes , where the deviation is canceled by the `` nonequilibrium '' parts @xmath122 , as shown in fig . [ fig : band_res](b ) . the quantity @xmath123 have nonnegative values for all @xmath24 because electrons come in from the left electrode . although @xmath123 is large near the left electrode , it decays as @xmath24 increases . on the other hand , @xmath124 have nonpositive values for all @xmath24 because electrons go out to the right electrode . note that @xmath125 for @xmath48 , no work - function differences , and at half filling.@xcite the behaviors of @xmath123 and @xmath124 indicate that electrons and holes hardly penetrate into the central part and the resistive state is maintained . the current hardly flows through the central part , because @xmath101 in eq . ( [ eq : cur ] ) is determined by the difference between the density modulation by the left electrode at the right boundary , @xmath126 , and that by the right electrode at the left boundary , @xmath127 . both terms @xmath126 and @xmath127 are vanishingly small , as shown in fig . [ fig : band_res](b ) . the charge distribution for @xmath114 , where the system is conductive , is qualitatively different from that for @xmath112 as shown in fig . [ fig : band_con ] . the spatial dependences of @xmath109 and @xmath118 are nearly the same . they increase almost linearly from left to right except in the vicinities of the electrodes , where some oscillatory structure appears . the distributions of @xmath109 and @xmath118 are understood by that of @xmath54 shown in the inset of fig . [ fig : band_con](a ) . the electron density is higher ( lower ) on the right ( left ) half where @xmath54 is low ( high ) . this behavior is caused by the electrons that move through the system in the conductive phase . the profile of @xmath54 shows almost a linear dependence on @xmath24 although a small deviation from the linearity near the electrodes is visible in contrast to the resistive phase , which is because the charge redistribution ( @xmath128 ) is easier in the conductive phase . as for the `` nonequilibrium '' parts of the density shown in fig . 4(b ) , @xmath123 have positive values for all @xmath24 , while @xmath124 are negative for all @xmath24 because the electrons come in from the left electrode and go out to the right electrode . a finite current flows through the central part : @xmath126 and @xmath127 are finite and have the opposite signs . -@xmath13 characteristics of one - dimensional band insulators for several values of @xmath1 with @xmath95 , @xmath33 , @xmath98 , and @xmath108 . the solid lines show the function @xmath129 which fits to the results.,height=215 ] and the threshold electric field @xmath130 on the size of the central part @xmath1 . the other parameters are the same as in fig . [ fig : band_iv2 ] . the error bars in the fitting are also shown.,height=215 ] next , we discuss the breakdown mechanism of band insulators . in fig . [ fig : band_iv2 ] , we show the @xmath12-@xmath13 curves for different sizes of the central parts @xmath1 with @xmath95 . to the numerical results , the function , @xmath131 is well fitted , where @xmath56 and @xmath132 are parameters . this expression originates from the lz tunneling mechanism through which the insulator breaks down with the threshold voltage @xmath132.@xcite for @xmath133 , the current @xmath101 is exponentially suppressed due to the energy gap , while it increases linearly for @xmath134 . when the central part is large , the fitting works well as shown in fig . [ fig : band_iv2 ] , so that the breakdown is consistent with the lz tunneling picture , although there exist fine structures in the @xmath12-@xmath13 characteristics which come from the discreteness of the energy spectrum of the central part . as @xmath1 decreases , the structure becomes more prominent . for @xmath135 , for example , a deviation from the fitting curve due to the stepwise structure becomes large , which indicates the lz mechanism is no longer applicable to small-@xmath1 systems . figure [ fig : band_cross ] shows @xmath132 determined by fitting eq . ( [ eq : lz_iv ] ) to the data for each @xmath1 , together with the corresponding electric field @xmath136 . for large @xmath1 , @xmath132 is proportional to @xmath1 , so that @xmath130 becomes a constant . in band insulators , the lz breakdown is known to be induced by the applied electric field.@xcite since the one - particle picture holds in band insulators , this breakdown can be analyzed as a usual interband tunneling problem and the threshold electric field becomes @xmath137.@xcite the breakdown occurs when the energy gain by displacing an electron with charge @xmath138 in an electric field @xmath139 by the distance @xmath3 , @xmath140 , overcomes the energy gap @xmath5 . here , @xmath141 is the bandwidth and @xmath142 . we have obtained the threshold @xmath143 , which is comparable with the value obtained by the lz formula,@xcite @xmath144 with @xmath145 . in short , the threshold is governed by the electric field . when the central part is small , the fitting to the @xmath12-@xmath13 curve becomes worse because the finite - size effect becomes severe . the lz mechanism is not suitable for understanding this breakdown . in this case , another mechanism , in which the threshold is determined by the bias voltage , is more appropriate for the following reason . as @xmath1 decreases , it eventually becomes smaller than the correlation length @xmath18 . the tunneling occurs when the energy gain by displacing an electron by the distance @xmath1 , @xmath146 , overcomes the energy gap @xmath5 . this indicates that the mechanism of the breakdown continuously changes around @xmath147 as a function of @xmath1 . when @xmath13 exceeds @xmath5 , some energy levels of the central part come in between @xmath15 and @xmath16 . for @xmath148 , the wave functions do not fully decay in the system : the electron injected from the left electrode with energy higher than @xmath149 can reach the right electrode through these levels so that the current flows . this can be clearly seen in fig . [ fig : band_iv2 ] for @xmath135 where the gap is @xmath150 due to the finite - size effect . the @xmath12-@xmath13 curve shows an abrupt increase at @xmath151 because @xmath15 exceeds the lowest unoccupied energy level of the central part . each stepwise increase in the @xmath12-@xmath13 characteristics corresponds to the increase in the number of energy levels located between @xmath15 and @xmath16 . we have numerically confirmed that the results are qualitatively unchanged even if the long - range coulomb interaction strength @xmath152 and the system - electrode coupling strength @xmath153 ( @xmath154 ) are varied . thus , the threshold shows a crossover as a function of @xmath1 . when @xmath155 , the lz - type breakdown occurs and the threshold is governed by the electric field . for @xmath148 , on the other hand , the current flows when @xmath13 exceeds the energy gap @xmath5 . it is noted that the spatial dependence of @xmath54 is important for the realization of the field - induced breakdown as well as the spatial modulation of the wave functions as discussed below . for small @xmath13 , @xmath54 has a linear dependence on @xmath24 throughout the central part because the electrons are localized , @xmath156 , so that the effect of the long - range interaction on @xmath54 is small . when the system is conductive , the charge redistribution occurs near the interfaces where a small deviation from the linearity is seen in @xmath54 . this charge redistribution weakens the electric field on the sites away from the interfaces in the central part . for comparison , we show in the appendix the @xmath12-@xmath13 characteristics that are obtained by artificially setting @xmath157 for all @xmath24 . this corresponds to a hypothetical case where a sufficiently large charge redistribution occurs near the interfaces . there is no electric field in the central part : a voltage drop occurs only at the interfaces . in this extreme case , we obtain the threshold bias voltage @xmath2 regardless of the length @xmath1 , which is in contrast to the lz - type behavior for large @xmath1 in fig . [ fig : band_cross ] . such a situation never occurs in our calculations with @xmath54 and for realistic parameters . as we will discuss in the next section and also in the appendix , the effect of the spatial profile of @xmath54 on the breakdown mechanism of mott insulators is basically the same as in the case of band insulators . thus , the model without @xmath54 is inappropriate for realistic insulators . with @xmath158 plotted against @xmath159 for @xmath103 , @xmath160 , @xmath161 , and @xmath162 in the case of @xmath95 , @xmath163 , @xmath33 , @xmath98 , and @xmath164 . for @xmath165 , and @xmath162 , the lines are shifted upward by @xmath166 , @xmath162 , and @xmath167 , respectively.,height=249 ] in discussing the breakdown for @xmath6 , the spatial dependences of the wave functions @xmath168 are crucial as explained above . in fig . [ fig : band_uiruil ] , we show @xmath169 as a function of the real part of the one - particle energy @xmath159 for several values of @xmath13 in the case of @xmath163 . the behavior of this quantity for @xmath170 is the same . it shows whether a given one - particle state contributes to the current @xmath101 . note that @xmath101 is obtained by integrating @xmath171_{i_li_r}|^2 $ ] over @xmath172 . since @xmath173 appears in the denominator for @xmath174_{ij}$ ] [ eq . ( [ eq : gr ] ) ] , the one - particle state @xmath175 with finite @xmath176 in the interval @xmath177 gives a large contribution to @xmath101 . this quantity directly shows whether the one - particle state @xmath175 is localized or delocalized because it comes from the product of the amplitudes of the wave function at the two interfaces @xmath178 and @xmath179 . if @xmath176 is large , the state has finite amplitudes at both sides of the central part , so that it is delocalized . if @xmath176 is small , on the other hand , the state has a small amplitude at either of the interfaces . for @xmath103 , @xmath169 shows two bands that correspond to the conduction and valence bands in the band insulator . because of the energy gap , no state exists in the region @xmath180 for @xmath103 so that the current does not flow at least for @xmath181 . when @xmath182 , several states appear in the region @xmath183 , corresponding to the leakage of one - particle states from the electrodes to the central part . however , these states do not contribute to the current because their @xmath169 are vanishingly small as shown in fig . [ fig : band_uiruil ] . note that the line for each @xmath102 is shifted upward by @xmath184 . for @xmath185 , the number of states around @xmath186 with small @xmath169 increases . the energy range where these localized states appear becomes wider as @xmath13 increases . consequently , the delocalized states that contribute to the current depart from the region @xmath172 . therefore , the current does not flow even if @xmath13 barely exceeds the gap . as we show in the appendix , the localized states do not appear if we set @xmath157 for all @xmath24 . it is crucial to take the spatial dependence of @xmath54 into account to obtain the modulation of the wave functions . for ( a ) @xmath187 , ( b ) @xmath188 , and ( c ) @xmath189 in the case of @xmath185 . the other parameters are the same as in fig . [ fig : band_uiruil ] . the corresponding @xmath159 are indicated by the arrows in fig . [ fig : band_uiruil].,height=340 ] figure [ fig : band_wf ] shows the spatial dependences of the squares of the absolute values of the wave functions @xmath190 for @xmath185 and several @xmath175 whose @xmath159 are located at the positions indicated by the arrows in fig . [ fig : band_uiruil ] . here @xmath175 is so labeled that @xmath191 for @xmath192 . figure [ fig : band_wf](a ) shows the one - particle state in the lower band with @xmath193 ( @xmath187 ) which is inside the gap for @xmath103 . this state is localized on the left half of the central part . the reason is as follows . the scalar potential is high ( low ) near the left ( right ) electrode . within each of the conduction and valence bands , the state whose weight is large near the left ( right ) electrode has a higher ( lower ) energy than others . in the present case , the indexes @xmath175 for the valence band are @xmath194 , @xmath195 , @xmath196 , @xmath197 , @xmath198 , @xmath199 , @xmath200 , while those for the conduction band are @xmath201 , @xmath202 , @xmath203 , @xmath204 , @xmath205 , @xmath206 , @xmath207 . as @xmath175 is lowered , the wave function of the one - particle state is generally extended to a wider region and its largest amplitude is shifted to the right , as shown in fig . [ fig : band_wf](b ) for the case of @xmath208 ( @xmath188 ) . as @xmath175 is lowered further , e.g. , for @xmath209 ( @xmath189 ) in fig . [ fig : band_wf](c ) , the one - particle state is delocalized to reach the right electrode . then , its wave function has large amplitudes near both electrodes . in order to overview the behaviors of the one - particle states , we show the contour map of one - particle states @xmath190 on the @xmath210 plane for @xmath185 in fig . [ fig : band_map ] . for @xmath185 , the states @xmath211 are localized near the left electrode , and the states below are delocalized . at the bottom of the lower band , the states are localized near the right electrode because of the low scalar potential near the right electrode . in @xmath210 plane for @xmath185 . the other parameters are the same as in fig . [ fig : band_uiruil].,height=302 ] in this section , we consider the case where the central part is described by the hubbard model . figure [ fig : mott_iv ] shows the @xmath12-@xmath13 characteristics for @xmath30 , @xmath97 , @xmath212 , @xmath98 , and @xmath99 . when @xmath103 , the system is an antiferromagnetic insulator owing to the hartree - fock approximation . the energy gap @xmath5 is then 0.25 . in the previous studies on the @xmath12-@xmath13 characteristics of metal - mott - insulator interfaces,@xcite the current was calculated by solving the time - dependent sch@xmath17dinger equation . it is argued that the results obtained by the time - dependent hartree - fock approximation for the electron - electron interaction are consistent with those obtained by exact many - electron wave functions on small systems . for example , the suppression of rectification at metal - mott - insulator interfaces is described by both methods.@xcite although the present time - independent hartree - fock approximation is worse , we expect the present approach captures the essential features of nonequilibrium steady states under the bias voltage . as we increase @xmath13 , the current begins to flow at @xmath213 . the breakdown becomes a first - order transition due to the hartree - fock approximation , which is in contrast to the case of band insulators in the previous section . -@xmath13 characteristics of one - dimensional hubbard model for @xmath30 , @xmath97 , @xmath212 , @xmath98 , and @xmath108.,height=215 ] , `` equilibrium '' part , @xmath109 , and ( b ) `` nonequilibrium '' parts @xmath110 and @xmath111 for @xmath30 , @xmath97 , @xmath212 , @xmath98 , @xmath108 , and @xmath214 . the scalar potential @xmath54 is shown in the inset of ( a).,title="fig:",height=200 ] , `` equilibrium '' part , @xmath109 , and ( b ) `` nonequilibrium '' parts @xmath110 and @xmath111 for @xmath30 , @xmath97 , @xmath212 , @xmath98 , @xmath108 , and @xmath214 . the scalar potential @xmath54 is shown in the inset of ( a).,title="fig:",height=200 ] , `` equilibrium '' part , @xmath109 , and ( b ) `` nonequilibrium '' parts @xmath110 and @xmath111 for @xmath30 , @xmath97 , @xmath212 , @xmath98 , @xmath113 , and @xmath114 . the scalar potential @xmath54 is shown in the inset of ( a).,title="fig:",height=200 ] , `` equilibrium '' part , @xmath109 , and ( b ) `` nonequilibrium '' parts @xmath110 and @xmath111 for @xmath30 , @xmath97 , @xmath212 , @xmath98 , @xmath113 , and @xmath114 . the scalar potential @xmath54 is shown in the inset of ( a).,title="fig:",height=200 ] the charge distributions in resistive and conductive phases at finite @xmath13 are shown in figs . [ fig : mott_res ] and [ fig : mott_con ] , respectively . their overall features are similar to those in band insulators . for all @xmath24 and @xmath133 , the electron density @xmath118 is almost unity , which is basically the same as in the equilibrium case ( @xmath119 for @xmath103 ) . the scalar potential @xmath54 has a linear dependence on @xmath24 as shown in the inset of fig . [ fig : mott_res](a ) . as for the `` nonequilibrium '' parts , @xmath123 ( @xmath124 ) is large near the left ( right ) electrode and decays as @xmath24 increases ( decreases ) [ fig . 11(b ) ] . electrons and holes do not penetrate into the central part so that the current does not flow . for @xmath215 , the spatial dependences of @xmath109 and @xmath118 are shown in fig . [ fig : mott_con](a ) . they increase almost linearly from left to right . this reflects the profile of the scalar potential @xmath54 that is higher ( lower ) on the left ( right ) half . the `` nonequilibrium '' parts of the densities , @xmath123 and @xmath124 are extended over the whole system with small spatial dependence [ fig . 12(b)],@xcite which is in contrast to the resistive phase . since @xmath126 and @xmath127 are finite with opposite signs , a finite current flows through the central part . and the threshold electric field @xmath130 on the size of the central part @xmath1 . the other parameters are the same as in fig . [ fig : mott_iv].,height=207 ] in fig . [ fig : mott_cross ] , we show the threshold bias voltage @xmath132 and the corresponding electric field @xmath216 for the first - order transition as a function of @xmath1 . when the central part is small , i.e. , @xmath217 , @xmath132 is almost a constant near the energy gap @xmath218 . for small @xmath1 , the electron injected from the left electrode with energy higher than @xmath149 can reach the right electrode since the correlation length @xmath219 is comparable to @xmath1 . thus , the threshold for @xmath148 is determined by the bias voltage . for @xmath6 , on the other hand , @xmath132 is proportional to @xmath1 , so that the threshold is governed by the electric field . in recent theoretical studies,@xcite oka and aoki have proposed that the lz breakdown occurs also in mott insulators by applying the time - dependent density - matrix - renormalization - group method to the one - dimensional hubbard model under an electric field with open boundary condition . in our calculations , the scalar potential @xmath54 is linearly increasing with @xmath24 in the resistive phase as shown in fig . [ fig : mott_res ] ( a ) , which means that the electrons feel a uniform electric field in the central part . therefore , our model describes the lz breakdown as in the open hubbard chain as long as the electrodes do not affect the nature of the breakdown for @xmath6 . in fact , the threshold @xmath130 is about 0.0033 , which is comparable to the lz value,@xcite @xmath220 , with @xmath145 . thus , the threshold shows a crossover as a function of @xmath1 as in the case of band insulators . the lz breakdown is explained as before by comparing the charge gap @xmath5 and the work which is done by the electric field on an electron moving over the correlation length @xmath18 . if the work @xmath221 exceeds @xmath5 , the electron in the lower band may go over to the upper band so that the current flows . according to the results by oka and aoki,@xcite this consideration is applicable to mott insulators where the correlation effects are important . thus , we expect that the results obtained by the hartree - fock approximation are qualitatively unchanged even if we take account of the electron correlation . it is well known that the hartree - fock theory overestimates the charge gap @xmath5 . it predicts the antiferromagnetic spin ordering which is actually destroyed if quantum fluctuations are appropriately taken into account . however , the overestimated @xmath5 will alter the threshold only quantitatively . we also expect that the spin ordering does not essentially affect the breakdown itself since only the charge degrees of freedom are relevant to the mechanism . we note that the time - dependent density - matrix - renormalization - group method has been applied to a mott insulator with electrodes very recently,@xcite and that the @xmath12-@xmath13 characteristics have been consistently explained by the lz tunneling mechanism . however , it is also shown that physical quantities such as the spin structure factor and the double occupancy do not reach a stationary state in the accessible time window . therefore , a direct description of nonequilibrium steady states as in the present study is important to deepen our understanding of the breakdown . since the crossover behavior is obtained for both band and mott insulators , the electron - electron interaction is not responsible for the phenomenon . we emphasize that the spatial profile of @xmath54 is important for the realization of the lz breakdown . in the appendix , this is demonstrated for the mott insulator by showing the @xmath12-@xmath13 characteristics that are obtained by artificially setting @xmath157 for all @xmath24 . in this case , no electric field exists in the central part so that the modification of the wave functions does not occur . the @xmath12-@xmath13 curves do not show any @xmath1 dependence apart from the fine structures coming from the discreteness of the energy spectrum . we obtain the threshold bias voltage @xmath2 regardless of the length @xmath1 as in band insulators . in fig . [ fig : mott_uiruil ] , we show @xmath169 as a function of the real part of the one - particle energy @xmath159 for several values of @xmath13 , where each line for @xmath102 is shifted upward by @xmath13 . the behavior of this quantity for @xmath170 is the same . with @xmath158 plotted against @xmath159 for @xmath103 , @xmath166 , @xmath162 , and @xmath167 in the case of @xmath222 , @xmath97 , @xmath212 , @xmath98 , and @xmath164 . for @xmath112 , @xmath162 , and @xmath167 , the lines are shifted upward by @xmath166 , @xmath162 , and @xmath167 , respectively.,height=249 ] for ( a ) @xmath223 , ( b ) @xmath224 , and ( c ) @xmath225 in the case of @xmath226 . the other parameters are the same as in fig . [ fig : mott_uiruil ] . the corresponding @xmath159 are indicated by the arrows in fig . [ fig : mott_uiruil].,height=340 ] for @xmath103 , no state exists in the region @xmath180 since the energy gap opens . @xmath169 shows two bands that correspond to the upper and lower hubbard bands . when @xmath112 , one - particle states leak from the electrodes to the central part so that several states appear in the region @xmath183 . as in band insulators , these states do not have any contributions to the current because their @xmath169 are vanishingly small . as we increase @xmath13 further , e.g. , @xmath226 , the states with vanishingly small @xmath169 appear in a wider range around @xmath186 . the number of these localized states also increases . the appearance of the localized states keeps the central part resistive until @xmath19 reaches the threshold electric field @xmath130 even if @xmath227 holds . when the system is conductive ( @xmath134 ) , the upper and lower hubbard bands are merged into a single metallic band . in this case , all the states around @xmath228 are delocalized and contribute to the current . figure [ fig : mott_wf ] shows @xmath190 as a function of @xmath24 for @xmath226 . here @xmath175 is chosen at @xmath229 , @xmath230 , and @xmath231 , whose @xmath159 are located at the positions indicated by the arrows in fig . [ fig : mott_uiruil ] . in fig . [ fig : mott_wf](a ) , we show the wave function with @xmath232 ( @xmath223 ) , which is localized on the left half of the central part . this state belongs to the lower hubbard band . since the scalar potential is high ( low ) near the left ( right ) electrode , the state is located near the top of the lower hubbard band . the one - particle states in the band gradually lose their localized nature as @xmath175 is lowered . this can be seen in fig . [ fig : mott_wf](b ) for the case of @xmath233 ( @xmath224 ) , where its largest amplitude is shifted to the right compared to that of @xmath223 . figure [ fig : mott_wf](c ) shows the one - particle state for @xmath234 ( @xmath225 ) , which is completely delocalized . its wave function has large amplitudes near both electrodes . the spatial dependences of the one - particle states in the resistive phase are similar to those in band insulators . we have investigated the @xmath12-@xmath13 characteristics of the one - dimensional band and mott insulators attached to electrodes . a tight binding model with alternating transfer integrals for the band insulator and the hubbard model for the mott insulator are studied by using the nonequilibrium green s function method . the applied bias voltage induces a breakdown of the insulating state to convert into a conductive state for both models . the threshold shows a crossover as a function of the size @xmath1 of the insulators . for @xmath235 , the breakdown occurs at @xmath2 so that the threshold is governed by the bias voltage . for @xmath6 , the electric field determines the threshold , @xmath8 , which is consistent with the lz breakdown reported previously.@xcite since the crossover is obtained for both band and mott insulators , the electron - electron interaction is not responsible for the phenomenon . the profile of the scalar potential @xmath54 , which is linearly increasing with @xmath24 in the resistive phase so that the electrons in the central part feel an almost uniform electric field , is important for the realization of the lz breakdown and the crossover behavior . this work was supported by grants - in - aid for scientific research ( c ) ( grant no . 19540381 ) and scientific research ( b ) ( grant no . 20340101 ) , and by `` grand challenges in next - generation integrated nanoscience '' from the ministry of education , culture , sports , science and technology of japan . in this appendix , we show the results when we artificially set @xmath157 for all @xmath24 . the @xmath12-@xmath13 curves of the band insulator with @xmath106 , @xmath33 , @xmath164 , and @xmath157 for @xmath236 , @xmath203 , and @xmath207 are shown in fig . [ fig : band_psi0 ] . it is apparent that the breakdown occurs at @xmath2 regardless of the length @xmath1 , which is consistent with ajisaka _ et al._@xcite this is in contrast to the results in fig . [ fig : band_cross ] where @xmath132 is proportional to @xmath1 for @xmath237 . when we fix @xmath157 , one - particle states do not leak from the electrodes to the central part since the electric field is absent from the central part . in fig . [ fig : band_app_uiruil ] , we plot @xmath169 as a function of @xmath159 for several values of @xmath13 . the results indicate that the one - particle energies and the wave functions are not affected by @xmath13 . this comes from the fact that the hamiltonian in eq . ( 1 ) does not depend on @xmath118 for band insulators if we set @xmath157 for all @xmath24 . therefore , no localized state appears inside the gap for @xmath103 . in this case , the current begins to flow when @xmath13 merely exceeds @xmath2 since one - particle states with finite @xmath169 appear in the region @xmath238 . in fig . [ fig : mott_psi0 ] , we show the @xmath12-@xmath13 curves of the mott insulator with @xmath30 , @xmath212 , @xmath164 , and @xmath157 for @xmath236 , @xmath203 , and @xmath207 . as in the case of band - insulators , the breakdown occurs at @xmath2 for all @xmath1 . although the breakdown seems to be continuous for @xmath236 and @xmath203 , a small discontinuity is evident for @xmath163 , which indicates a first - order transition . the discontinuity is more obvious for large @xmath29 as shown in the inset of fig . [ fig : mott_psi0 ] for @xmath239 with the gap @xmath240 . -@xmath13 characteristics in the case of @xmath242 , @xmath212 , @xmath164 , and @xmath157 for several values of @xmath1 . the results for @xmath239 and @xmath157 are also shown in the inset . the arrow indicates the location of the gap @xmath5.,height=215 ] 99 y. taguchi , t. matsumoto , and y. tokura , phys . b * 62 * , 7015 ( 2000 ) . r. kumai , y. okimoto , and y. tokura , science * 284 * , 1645 ( 1999 ) . s. yamanouchi , y. taguchi , and y. tokura , phys . lett . * 83 * , 5555 ( 1999 ) . f. sawano , i. terasaki , h. mori , t. mori , m. watanabe , n. ikeda , y. nogami , and y. noda , nature * 437 * , 522 ( 2005 ) . r. kondo , m. higa , and s. kagoshima , j. phys . * 76 * , 033703 ( 2007 ) . s. niizeki , f. yoshikane , k. kohno , k. takahashi , h. mori , y. bando , t. kawamoto , and t. mori , j. phys . jpn . * 77 * , 073710 ( 2008 ) . f. sawano , t. suko , t. s. inada , s. tasaki , i. terasaki , h. mori , t. mori , y. nogami , n. ikeda , m. watanabe , and y. noda , j. phys . . jpn . * 78 * , 024714 ( 2009 ) . t. s. inada , i. terasaki , h. mori , and t. mori , phys . b * 79 * , 165102 ( 2009 ) . t. hasegawa , k. mattenberger , j. takeya , and b. batlogg , phys . b * 69 * , 245115 ( 2004 ) . k. yonemitsu , j. phys . . jpn . * 74 * , 2544 ( 2005 ) . k. yonemitsu , n. maeshima , and t. hasegawa , phys . b * 76 * , 235118 ( 2007 ) . k. yonemitsu , j. phys . 78 * , 054705 ( 2009 ) . t. oka , r. arita , and h. aoki , phys . lett * 91 * , 066406 ( 2003 ) . t. oka and h. aoki , phys . lett * 95 * , 137601 ( 2005 ) . t. oka and n. nagaosa , phys . lett * 95 * , 266403 ( 2005 ) . t. oka and h. aoki , phys . b * 81 * , 033103 ( 2010 ) . s. onoda , n. sugimoto , and n. nagaosa , prog . phys . * 116 * , 61 ( 2006 ) . n. sugimoto , s. onoda , and n. nagaosa , prog . . phys . * 117 * , 415 ( 2007 ) . n. sugimoto , s. onoda , and n. nagaosa , phys . b * 78 * , 155104 ( 2008 ) . s. okamoto , phys . b * 76 * , 035105 ( 2007 ) . s. okamoto , phys . * 101 * , 116807 ( 2008 ) . s. ajisaka , h. nishimura , s. tasaki , and i. terasaki , prog . . phys . * 121 * , 1289 ( 2009 ) . l. d. landau , phys . z. sowjetunion * 2 * , 46 ( 1932 ) . c. zener , proc . london , ser . a * 137 * , 696 ( 1932 ) . s. datta , _ quantum transport : atom to transistor _ ( cambridge university press , cambridge , 2005 ) y. tanaka and k. yonemitsu , physica b * 405 * , s211 ( 2010 ) . y. meir and n. s. wingreen , phys . * 68 * , 2512 ( 1992 ) . n. s. wingreen , a .- p . jauho , and y. meir , phys . b * 48 * , 8487 ( 1993 ) . a .- jauho , n. s. wingreen , and y. meir , phys . b * 50 * , 5528 ( 1994 ) . h. haug and a .- yauho , _ quantum kinetics in transport and optics of semiconductors _ ( springer , berlin , 2008 ) 2nd ed . j. m. ziman , _ principles of the theory of solids _ ( cambridge university press , cambridge , 1979 ) f. heidrich - meisner , i. gonz@xmath243lez , k. a. al - hassanieh , a. e. feiguin , m. j. rozenberg , and e. dagotto , phys . b * 82 * , 205110 ( 2010 ) .

nonequilibrium states induced by an applied bias voltage @xmath0 and the corresponding current - voltage characteristics of one - dimensional models describing band and mott insulators are investigated theoretically by using nonequilibrium green s functions . we attach the models to metallic electrodes whose effects are incorporated into the self - energy . modulation of the electron density and the scalar potential coming from the additional long - range interaction are calculated self - consistently within the hartree approximation . for both models of band and mott insulators with length @xmath1 , the bias voltage induces a breakdown of the insulating state , whose threshold shows a crossover depending on @xmath1 . it is determined basically by the bias @xmath2 for @xmath1 smaller than the correlation length @xmath3 where @xmath4 denotes the bandwidth and @xmath5 the energy gap . for systems with @xmath6 , the threshold is governed by the electric field , @xmath7 , which is consistent with a landau - zener - type breakdown , @xmath8 . we demonstrate that the spatial dependence of the scalar potential is crucially important for this crossover by showing the case without the scalar potential , where the breakdown occurs at @xmath2 regardless of the length @xmath1 .

You are an expert at summarizing long articles. Proceed to summarize the following text: our understanding of distant galaxies and the history of galaxy formation has undergone a revolution in the past decade . galaxies are now routinely discovered and studied out to redshifts @xmath8 ( e.g. , dickinson et al . 2004 ; yan et al . 2005 ; bouwens et al . 2007 ; bouwens et al . 2010 ) . samples of a few dozen objects have been found at even higher redshift , back to the era of reionization ( @xmath9 ) , and perhaps some galaxies have been discovered at even higher redshifts , @xmath10 ( e.g. , bouwens et al . 2010 ; finkelstein et al . this relatively rapid advance in our discovery of the earliest galaxies is the direct result of technical advances in spectroscopy and imaging over the past decade , in which the _ hubble space telescope _ ( _ hst _ ) has played a leading role . historically , distant galaxies are found within deep optical imaging surveys , and are confirmed as high redshift galaxies with large multi - object spectrographs on 8 - 10 meter telescopes , which came online in the mid-1990s . it can be argued however that some of the most important advances in our understanding of galaxies has come about from very deep imaging , especially from _ hst_. the _ hubble _ space telescope has played a key role in high - redshift discoveries and our understanding of galaxy evolution through large blank field and targeted programs such as the _ hubble _ deep and ultra deep fields , goods , egs , and cosmos , among others ( e.g. , williams et al . 1996 ; giavalisco et al . 2004 ; beckwith et al . 2006 ; davis et al . 2007 ; scoville et al . 2007 ) . this _ hubble _ imaging has proven invaluable for two primary reasons . one is simply due to the depth that can be achieved with a high photometric fidelity , ensuring that exquisite photometry of distant galaxies can be obtained . whilst ground based telescopes can reach the depths of _ hst _ at optical wavelengths , in principle the accuracy and precision of this photometry is not nearly as good , due to a higher background , and importantly , the large and variable psf . this makes accurate measurements of light difficult , particularly for colours which require exact apertures for accurate measures . furthermore , _ hst _ data have proven important for the discovery of the most distant galaxies in the universe through the use of the lyman - break method of looking for drop - out galaxies in bluer bands . many filter choices within multi - colour deep imaging programmes were in fact selected to facilitate optimal drop - out searches ( e.g. , giavalisco et al . 2004 ) . _ hubble _ imaging furthermore has facilitated a renaissance in the study of galaxy structure in the distant universe , which provides a key observable for understanding how distant galaxies form and evolve ( e.g. , conselice et al . 2003 ; conselice et al . 2008 , 2009 ; lotz et al . 2008 ; buitrago et al . 2008 ; jogee et al . 2009 ; bluck et al . 2009 ; casatta et al . these structural measurements have proven critical for determining how galaxy morphologies , sizes , and merger / kinetic states have evolved through time ( e.g. , trujillo et al . 2007 ; ravindranath et al . 2004 ; conselice et al . this allows us to examine how the merger history of galaxies has changed ( e.g. , conselice et al . 2003 , 2008 ; lotz et al . 2008 ; jogee et al . 2009 ) , and thus we can begin to derive how galaxies form , as opposed to simply when . it is not currently straightforward to measure the structures of distant galaxies with ground based imaging even with adaptive optics , and thus _ hubble _ has and continues to provide a key aspect for tracing evolution using these methods . however , one key aspect of parameter space that has not yet been explored with _ hst _ , or other space - based telescopes in any depth over large areas , is deep infrared imaging over a relatively large area . previously there exists deep nic3 imaging over the _ hubble _ deep field ( dickinson et al . 2000 ) and hubble ultra - deep field ( thompson et al . 2004 ) , as well as very deep nicmos imaging over a small area of the hdf - n ( thompson et al . 1999 ) . these areas are however very small , and while nic3 parallel data exists over the cosmos and egs fields , it is quite shallow at @xmath11 orbit depth . _ hst _ imaging data however has a distinct advantage over ground - based imaging not only in terms of the higher quality photometric fidelity and higher resolution , but also the depth which can be achieved in the near infrared ( nir ) with _ hst _ as opposed to the ground - based optical where comparable depths to _ hst _ can be reached . within one or two orbits , the _ hst _ can reach a depth in the nir which is difficult to obtain from the ground even with an 8 - 10-m class telescope , and which will not have the same photometric quality , nor resolution as the _ hubble _ data . we thus designed and carried out the goods nicmos survey ( gns ) , which is a large _ hst _ programme intended to remedy this situation by providing through an initial 180-orbit program of nic3 imaging in the goods fields , a data set designed to examine a host of problems requiring very deep nir data . the gns data consist of 60 nicmos nic3 pointings , centred on the most massive ( @xmath12@xmath13 ) galaxies at @xmath14 . the depth of each image is 3 orbits / pointing within the @xmath15-bandpass over a total area of @xmath16 arcmin@xmath17 ( buitrago et al . 2008 ; bluck et al . 2009 , casey et al . 2009 , bouwens et al . 2009 , 2010 present results using this data ) . with these nicmos data we are able to explore the rest - frame optical features of galaxies at @xmath18 in detail . this allows a few measurements to be made that can not be easily reproduced with optical imagining and/or deep nir imaging from the ground . this includes : filling in the important near - infrared gap in galaxy spectral energy distributions ( seds ) ; sampling the rest - frame optical structures and sizes of galaxies out to @xmath19 ( buitrago et al . 2008 ) ; and the detection and characterization of the population of massive @xmath20 galaxies and agn , and determining the relation of agn evolution to that of massive galaxies ( e.g. , bluck et al . 2010 ) . in this paper we present the basic outline , background , and results from this survey . we discuss the design of the observations , our field selection , as well as the selection for our initial massive galaxy sample which has guided the centres for our nicmos nic3 pointings . we also discuss the various methods for locating the massive galaxy population at higher redshifts , and the connection of these massive galaxies to those at @xmath4 . we show that no one colour method is able to identify the massive galaxy population at high redshifts , and that a combination of methods and photometric redshifts are needed to construct a semi - complete massive galaxy sample at higher redshifts . in this paper we construct as complete as possible sample of massive galaxies within our fields , and discuss the properties of these galaxies , as well as some features of lower mass galaxies . this paper is organised as follows : 2 gives a summary of our observations and the design of the gns , including how the initial sample of galaxies was selected . 3 gives a description of the derived parameters from the @xmath21band imaging , including photometric redshifts and catalogue matching . 4 describes our initial analysis of the survey data , including how the various selections for massive galaxies at high redshifts compare , while finally 5 is our summary . we use a standard cosmology of _ km s@xmath23 mpc@xmath23 , and @xmath24 = 0.3 throughout . the gns selection and field coverage is based on the previous optical acs and ground - based imaging from the original goods program ( giavalisco et al . the goods programme is a multi - wavelength campaign to obtain a coherent collection of deep imaging and spectroscopy in two 150 arcmin@xmath17 areas in the northern and southern hemispheres ( goods - n and goods - s ) . these two fields are centred around the _ hubble _ deep field - north ( hdf - n ) and _ chandra _ deep field - south ( cdf - s ) , which are areas of very low dust extinction , and minimal stellar and radio contamination . the existing goods / acs fields match the coverage of the goods _ spitzer _ program and cover the 2 msec exposure of the _ chandra _ deep field south and the 2 msec exposure of the _ chandra _ deep field north ( luo et al . large ongoing campaigns to obtain spectroscopy for the goods fields have also been carried out , including 3000 spectra as part of the keck treasury redshift survey ( wirth et al . another @xmath25 redshifts in goods - s have been measured from various eso programs ( e.g. , vanzella et al . 2008 ; le fevre et al . 2005 ; popesso et al . 2009 ; balestra et al . 2010 ) . the co - moving volume probed by goods at high redshifts , @xmath26 , is similar to the co - moving volume covered by the cosmos field ( e.g. , scoville et al . 2007 ) at @xmath27 . furthermore , due to its depth at all wavelengths the goods fields are thus an ideal location for examining the formation and evolution of early galaxies . deep nir imaging of these fields however is lacking , although some deep nir imaging has been obtained with eso telescopes using sofi and isaac for the goods - south , as well as deep cfht wircam imaging , subaru imaging , and some keck imaging over the goods north ( e.g. , kajisawa et al . 2009 ; retzlaff et al . 2010 ; wang et al . however , these data only reach modest depths of @xmath28 compared to our _ hst _ imaging . the depth of our suvey is only comparable to previous nicmos deep programmes covering the hdf - north and hdf - south fields , as well as new near - ir data obtained with wfc3 ( cassata et al . 2010 ) . ideally , one would want to cover both goods north fields completely , yet given the small nicmos field of view it is not practical to cover the entire goods fields any deeper than one orbit with nicmos . the wfc3 camera will , however , soon cover these fields to an even great depth with the candels programme . our strategy is not to map out a continuous area , but to collect 60 pointed observations directed towards the most massive galaxies at @xmath29 found in the goods fields ( 2.2 ) , maximised to obtain the largest number of galaxies based on our selection methods . to obtain the most unique and useful science we therefore constructed a program which covers a sixth of the area of a single goods field ( in total 43.7 arcmin@xmath17 ) in three orbits depth in the @xmath15 band , in areas of the deepest _ spitzer _ , _ chandra _ , and acs imaging , and where the greatest amount of spectroscopy already exists . some of these fields were then observed in the @xmath30-band ( @xmath31 ) with nicmos or wfc3 as part of a follow up programme to obtain near infrared seds to look for high redshift drop - out galaxies ( bouwens et al . 2010 ) . our nicmos pointings were chosen to target a set of objects selected to be known massive galaxies at high redshift , identified using a variety of color selection methods . these include `` distant red galaxies '' ( drg : franx et al . 2003 ; papovich et al . 2006 ) , irac extremely red objects ( yan et al . 2006 ) , and bzk color - selected galaxies ( daddi et al . 2004 , 2007 ) . all of these methods are designed to find red , dusty or passively evolving older galaxies at @xmath32 . in practice , we utilised all three of these colour - selections separately , in order to obtain as much as possible a complete sample of massive galaxies at @xmath33 . to optimise our field placement , we also used catalogues of lyman - break selected bm / bx objects ( reddy et al . 2008 ) , as well as high redshift drop - outs and sub - mm galaxies . however , the primary field selection was done in terms of the massive galaxy selection through the three primary colour criteria as described further below . colour selection of distant galaxies has a long history dating back to the early work of finding lyman - break galaxies through image drop - outs in blue bands ( e.g. , guhathakurta , tyson , majewski 1990 ; steidel & hamilton 1992 ) . it is generally accepted that no single method can find all galaxies at a given redshift , and some of these methods are better at finding star - forming objects , as opposed to those which are more passive and evolved . in fact , it is generally agreed that no method or combination of methods can identify an obviously complete sample of high - z galaxies . one of the methods we use for finding likely passively evolving and dusty red galaxies is to find distant red galaxies ( drgs ) defined by a nir colour cut ( e.g. , franx et al . 2003 ; papovich et al . 2006 ; conselice et al . the selection we use to find drgs , and to be included within our sample , is galaxies at @xmath29 with ( @xmath34 ) @xmath35 mag in vega magnitudes ( or @xmath36 in ab mags ) . the selection for these galaxies is based on ground - based imaging from isaac on the very large telescope ( vlt ) . this selection is only used for choosing systems in goods - s , as deep nir imaging over the entire goods - n field was not available when the target selection was carried out . this goods - s drg sample is approximately complete for @xmath0 drg selected galaxies at @xmath37 ( papovich et al . 2006 ) . another selection we use to construct our initial massive sample is the _ spitzer _ selected extremely red objects ( eros ) , otherwise known as infrared eros ( ieros ) . these were first described in yan et al . ( 2004 ) , based on nir and _ spitzer _ data within the goods fields . the selection for these objects is s@xmath38(3.6 @xmath39m)/s@xmath40 . these objects were found by yan et al . ( 2004 ) , based on sed fits , to have a mixture of old and younger populations . note that selecting galaxies in this way ensures that they are massive given their brightness in the ir . however , because they are selected with _ spitzer _ imaging , which has a large psf , resulting in potential confusion from neighboring objects , this selection can have issues with contamination from other galaxies . hence any galaxies which would satisfy the criteria but are too close to another galaxy will not be included simply due to the problem of confusion . another method we use to select distant galaxies is through the bzk approach , which is described in daddi et al . ( 2007 ) in terms of selection within the goods fields . the selection for these objects is slightly more complicated than that of the drgs or ieros , since they are selected through colours using the @xmath41 , @xmath42 and @xmath43-bands together . this method proposes to separate evolved galaxies or passive pbzks , and those which are star forming , or sbzks . the selection for these galaxies is done through the quantity @xmath44 , defined using these three bands by : star forming galaxies at @xmath46 are proposed to have @xmath47 . the redder , possibly more evolved galaxies , are found through the selection @xmath48 and @xmath49 . for the bzk sample we use , the selection is somewhat more limited than for the other colour selections as these sources were selected down to @xmath50 vega in the north and @xmath51 in the south . we utilise photometric redshifts and stellar masses of the galaxies selected through these methods to identify and study these colour selected populations taken directly from papovich et al . ( 2006 ) , yan et al . ( 2004 ) and daddi et al . ( 2007 ) . our initial massive galaxy sample from which we optimise our nicmos pointings are selected through these three methods , with a further photometric redshift cut of @xmath14 , and with a stellar mass cut of @xmath0 . in practice our final pointings were chosen by finding the locations within the goods fields where the number of these massive galaxies was maximised within the nic3 fields . in total we imaged 45 pre - selected massive @xmath0 galaxies at @xmath14 in the goods - n , and 35 in the goods - s . galaxies selected in other ways were also used to optimise the number of galaxies in each nic3 pointing , although each pointing was designed to have at least one massive galaxy with the properties above . these ` additional ' galaxies are selected through the lyman - break drop - out method utilising @xmath41 , @xmath52 and @xmath53 drop - outs , the bx / bm selection , as well as sub - mm galaxies from greve et al . each nic3 pointing contained between four to 19 of each of these galaxy types . figure 1 shows our field layout within the goods fields with the different galaxy types shown as different colours and symbols , and figure 2 shows a typical nicmos nic3 pointing of one of our fields . tables 1 and 2 list the statistics and positions of our 60 pointings , with 30 nic3 pointings in the goods - north field , and 30 in goods - south . we also list the number of various other types of galaxies within each of these fields . tables 3 and 4 list the initial massive galaxies for which we picked our fields , along with basic information such as their photometric redshifts , stellar masses , and information on the optical and @xmath21band magnitudes for these systems . nicmos and acs images of ten of these massive galaxies are shown in figure 3 . this data , including catalogs of sources , redshifts and stellar masses , as well as the original reduced nic3 imaging itself is available at * http://www.nottingham.ac.uk / astronomy / gns/*. + notes . ( a ) the values of the errors on @xmath54 and sersic @xmath55 are representative of the 1 @xmath56 model errors from galfit ( see buitrago et al . this does not take into account many possible sources of error that may bias these measurements , including magnitude of galaxy , concentration of its light profile , etc . the uncertainty in these structural parameters increase by on order of 10 percent for @xmath54 and 20 percent for @xmath55 due to changes in the psf across the nicmos nic3 field of view . also listed is the fitted axis ratios for these galaxies ( b / a ) , and position angles ( p.a . ) . the effective surface brightness ( @xmath57 ) is listed , as is the spectroscopic redshift ( @xmath58 ) , if available . the value of ` sep ' is the difference between the position of an object and the identification of the spectroscopic target , in arcsec . finally , the calculated photometric redshift , @xmath59 , is shown . balestra , i. et al . 2010 , a&a , 512 , 12 barger , a. et al . , 2008 , apj 689 , 687 beckwith , s. , et al . 2006 , aj , 132 , 1729 benitez , n. 2000 , apj 536 , 571 benitez , n. 2000 , apj , 536 , 571 bertone , s. , conselice , c.j . 2009 , mnras , 396 , 2345 bluck , a.f.l . , conselice , c.j . , bouwens , r.j . , daddi , e. , dickinson , m. , papovich , c. , yan , h. 2009 , mnras , 394 , 51l bluck , a.f.l . , conselice , c.j . , almaini , o. , laird , e.s . , nandra , k. , gruetzbauch , r. mnras in press , arxiv:1008.2162 bohlin , r.c . , mack , j. , hartig , g. , sirianni , m. 2005 , instrument science report acs , 2005 - 12 bolzonella , m. et al . 2000 , a&a 363 , 476 bouwens , r.j . , illingworth , g.d . , franx , m. , ford , h. 2007 , apj , 670 , 928 bouwens , r.j . , 2010 , arxiv:1003.1706 bouwens , r.j . , 2009 , apj , 705 , 936 bruzual , g. , charlot , s. 2003 , mnras , 344 , 1000 buitrago , f. , trujillo , i. , conselice , c.j . , bouwens , r.j . , dickinson , m. , yan , h. 2008 , apj , 687 , 61l bundy , k. , et al . 2006 , apj , 651 , 120 bundy , k. , et al . 2008 , apj , 681 , 931 calzetti et al . 2000 , apj 533 , 682 casey , c.m . , chapman , s.c . , muxlow , t.w.b . , beswick , r.j . , alexander , d.m . , conselice , c.j . 2009 , mnras , 395 , 1249 cassata , p. , et al . 2010 , apj , 714 , 79 conselice , c.j . , bershady , m.a . , dickinson , m. , papovich , c. 2003 , aj , 126 , 1183 conselice , c.j . bundy , k. , ellis , r.s . , brichmann , j. , vogt , n.p . , philips , a.c . 2005 , apj , 628 , 160 conselice , c.j . , et al . 2007a , mnras , 381 , 962 conselice , c.j . , 2007b , apj , 660 , 55l conselice , c.j . , rajgor , s. , myers , r. 2008 , mnras , 386 , 909 conselice , c.j . , bundy , k. , u , v. , eisenhardt , p. , lotz , j. , newman , j. 2008 , mnras , 383 , 1366 conselice , c.j . , yang , c. , bluck , a. , 2009 , mnras , 394 , 1956 daddi , e. , et al . 2004 , apj , 617 , 746 daddi , e. , et al . 2007 , apj , 670 , 156 davis , m. , et al . 2007 , apj , 660 , 1l de jong , r.s . 2006 , instrument science report nicmos 2006 - 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we present the details and early results from a deep near - infrared survey utilising the nicmos instrument on the _ hubble space telescope _ centred around massive @xmath0 galaxies at @xmath1 found within the great observatories origins deep survey ( goods ) fields north and south . the goods nicmos survey ( gns ) was designed to obtain deep f160w ( h - band ) imaging of 80 of these massive galaxies , as well as other colour selected objects such as lyman - break drop - outs , bzk objects , distant red galaxies ( drgs ) , extremely red objects ( eros ) , _ spitzer _ selected eros , bx / bm galaxies , as well as flux selected sub - mm galaxies . we present in this paper details of the observations , our sample selection , as well as a description of features of the massive galaxies found within our survey fields . this includes : photometric redshifts , rest - frame colours , and stellar masses . we furthermore provide an analysis of the selection methods for finding massive galaxies at high redshifts , including colour selection methods and how galaxy populations selected through these colour methods overlap . we find that a single colour selection method can not locate all of the massive galaxies , with no one method finding more than 70 percent we however find that the combination of these colour methods finds nearly all the massive galaxies , as selected by photometric redshifts with the exception of apparently rare blue massive galaxies . by investigating the rest - frame @xmath2 vs. m@xmath3 diagram for these galaxies we furthermore show that there exists a bimodality in colour - magnitude space at @xmath4 , driven by stellar mass , such that the most massive galaxies are systematically red up to @xmath5 , while lower mass galaxies tend to be blue . we also discuss the number densities for galaxies with stellar masses @xmath0 , whereby we find an increase of a factor of eight between @xmath6 and @xmath7 , demonstrating that this is an epoch when massive galaxies establish most of their mass . we also provide an overview of the evolutionary properties of these galaxies , such as their merger histories , and size evolution . [ firstpage ] galaxies : evolution , formation , structure , morphology , classification

You are an expert at summarizing long articles. Proceed to summarize the following text: the baker - campbell - hausdorff formula is the expansion of @xmath3 in terms of nested commutators for the non - commuting variables @xmath1 and @xmath2 , where the commutator of @xmath4 and @xmath5 is defined as @xmath6 : = ab - ba$ ] . the explicit combinatorial form of it was given by dynkin in his 1947 paper @xcite . by considering the linear extension of the map @xmath7 defined by @xmath8 , @xmath9 , @xmath10 , @xmath11 $ ] and @xmath12 $ ] he proved that @xmath13 this series is related to lie s third theorem and the history around it is too rich to be retold here , so we refer the reader to the recent monograph @xcite and references therein for a historical account . the baker - campbell - hausdorff formula , as well as many other results in lie theory , firmly belongs to associative algebra . however , after the work of mikheev and sabinin on local analytic loops @xcite and the description of the primitive operations in non - associative algebras by shestakov and umirbaev @xcite , associativity does not seem to be as essential for the lie theory as previously thought @xcite . in this paper we address the problem of determining @xmath14 in @xmath15 where @xmath16 in terms of shestakov - umirbaev operations for the primitive elements of the non - associative algebra freely generated by @xmath1 and @xmath2 . our approach uses a generalization of the magnus expansion ( see @xcite for a readable survey ) , that is , we will study the differential equation @xmath17 where @xmath18 stands for @xmath19 and both @xmath20 and @xmath21 belong to a non - associative algebra . the differential equation @xmath22 satisfied by @xmath21 ( corollary [ cor : magnus ] ) is obtained with the help of a non - associative version of the dynkin - specht - wever lemma ( lemma [ lem : dsw ] ) . this equation leads to a recursive formula for computing the expansion of @xmath14 , which gives , in degrees smaller than 5 , the following expression : @xmath23\\ & \quad + \frac{1}{12}[x,[x , y ] ] - \frac{1}{3}\langle x;x , y\rangle -\frac{1}{12 } [ y,[x , y ] ] - \frac{1}{6 } \langle y ; x , y \rangle -\frac{1}{2}\phi(x;y , y ) \\ & \quad -\frac{1}{24}\langle x ; x , [ x , y]\rangle - \frac{1}{12 } [ x,\langle x;x , y\rangle ] - \frac{1}{8}\langle x , x;x , y\rangle \\ & \quad + \frac{1}{24}[[x,[x , y]],y ] - \frac{1}{24}[x,\langle y;x , y\rangle ] -\frac{1}{4}\phi(x , x;y , y ) - \frac{1}{4}[x,\phi(x;y , y ) ] \\ & \quad -\frac{1}{24}[\langle x;x , y\rangle , y ] -\frac{1}{24}\langle x ; [ x , y],y\rangle - \frac{1}{6}\langle x , y;x , y\rangle + \frac{1}{24}\langle y , x;x , y\rangle \\ & \quad + \frac{1}{12 } [ \phi(x;y , y),y ] + \frac{1}{24}\langle y ; y,[x , y]\rangle - \frac{1}{24 } \langle y , y;x , y\rangle - \frac{1}{6}\phi(x;y , y , y ) \\ & \quad + \dots\end{aligned}\ ] ] when all the operations apart from @xmath24 $ ] vanish we recover the usual baker - campbell - hausdorff formula . a different approach to the non - associative baker - campbell - hausdorff formula has appeared in @xcite ; it does not explicitly use the dynkin - specht - wever lemma or the magnus expansion . for the treatment of the subject from the point of view of differential geometry see @xcite ; actually , geometric considerations also motivate a different type of a baker - campbell - hausdorff formula , see @xcite ; although it is of importance for the non - associative lie theory , we shall not consider it here . our results are presented for the unital @xmath25-algebra of formal power series @xmath26 in two non - associative variables @xmath1 and @xmath2 . readers with background in non - associative structures will realize that a more natural context for the baker - campbell - hausdorff formula is the completion of the universal enveloping algebra of a relatively free sabinin algebra on two generators . the extension of our results to that context is rather straightforward . readers familiar with free lie algebras might wonder about the existence , behind of the scenes , of certain non - associative lie idempotents responsible for some of our formulas . the answer is affirmative ; however , this topic is not discussed in the present paper since it requires some knowledge of sabinin algebras and treating it would significantly increase the length of text . very briefly , the context for the non - associative lie idempotents is as follows . one can start with a variety @xmath27 of loops containing all the abelian groups and define a relatively free sabinin algebra @xmath28 associated to the variety @xmath27 and freely generated by @xmath29 . let @xmath30 be the universal enveloping algebra of @xmath28 . this algebra is a non - associative graded hopf algebra @xmath31 once we set @xmath32 . the convolution @xmath33 defines a non - associative product on the space of @xmath25-linear maps @xmath34 . the subalgebra generated by the projections @xmath35 of @xmath30 onto @xmath36 ( where @xmath37 ) with respect to this convolution product , which in the associative setting is anti - isomorphic to solomon s descent algebra , is isomorphic as a graded hopf algebra to @xmath30 . therefore , @xmath30 is a non - associative hopf algebra with an extra associative ( inner ) product inherited from the composition in @xmath34 . in the associative case , this is the subject of study in the theory of non - commutative symmetric functions , so a similar theory seems possible in the non - associative setting . eulerian , dynkin and klyachko idempotents , among others , are easily understood in @xmath30 as particular examples of primitive elements with respect to the comultiplication that , in addition , are idempotent with respect to the associative inner product , and they ultimately explain some of the formulas in this paper . throughout this paper the characteristic of the base field @xmath25 is zero . the unital associative @xmath25-algebra freely generated by a set of generators @xmath38 will be denoted by @xmath39 while @xmath40 will stand for the unital associative algebra of formal power series on @xmath38 with coefficients in @xmath25 . their non - associative counterparts , namely , the unital non - associative @xmath25-algebra freely generated by @xmath38 and the unital non - associative @xmath25-algebra of formal power series on @xmath38 with coefficients in @xmath25 , will be denoted by @xmath41 and @xmath42 respectively . for any algebra @xmath43 , @xmath44 $ ] will denote the algebra of formal power series in @xmath45 with coefficients in @xmath43 . the parameter @xmath45 commutes and associates with all the elements in @xmath44 $ ] . finally , we will stick to the following order of parentheses for powers : @xmath46 ( @xmath47 times ) . a coalgebra @xmath48 is a vector space equipped with two linear maps @xmath49 ( _ comultiplicaton _ ) and @xmath50 ( _ counit _ ) such that @xmath51 where @xmath52 stands for @xmath53 ( _ sweedler notation _ ) . coassociative and cocommutative coalgebras are those coalgebras @xmath48 that , in addition , satisfy @xmath54 ( _ coassociativity _ ) and @xmath55 ( _ cocommutativity _ ) where @xmath56 . coassociativity ensures that @xmath57 so we can safely write @xmath58 for any of the sides of this equality . for coassociative coalgebras the result of the iterated application @xmath47 times of @xmath59 to @xmath1 does not depend on the selected factors and it is denoted by @xmath60 . cocommutativity ensures that we can freely permute the factors of @xmath60 without altering the value of this expression @xcite . in this paper , by a _ ( non - associative ) hopf algebra _ @xmath61 we shall mean a cocommutative and coassociative coalgebra @xmath62 endowed with the following linear maps : a _ product _ @xmath63 , a _ unit _ @xmath64 , a _ left division _ @xmath65 and a _ right division _ @xmath66 so that @xmath67 , @xmath68 , @xmath69 , @xmath70 and @xmath71 where @xmath72 and @xmath73 is the unit element ( see @xcite for a survey on non - associative hopf algebras ) . in case that @xmath43 is associative then the left and right divisions can be written as @xmath74 and @xmath75 where @xmath76 is the antipode . however , non - associative hopf algebras lack antipodes in general . the most important example of a non - associative hopf algebra in this paper is the unital non - associative algebra @xmath77 freely generated by @xmath78 . the maps @xmath79 and @xmath80 ( @xmath81 ) induce homomorphisms of unital algebras @xmath82 and @xmath83 so that @xmath84 is a coassociative and cocommutative coalgebra . by induction on the degree of @xmath1 , the formulas ( [ eq : left_division ] ) and ( [ eq : right_division ] ) uniquely determine the left and the right division in @xmath77 . for instance , @xmath85 implies @xmath86 and @xmath87 etc . the operations @xmath59 , @xmath88 , @xmath89 and @xmath90 , together with the product and the unit , provide @xmath77 with the structure of a non - associative hopf algebra . far from being a fancy feature , the divisions are a valuable tool for computations . an element @xmath4 in a hopf algebra @xmath43 such that @xmath91 is called _ primitive _ ; the subspace of all such elements is denoted by @xmath92 . while for associative hopf algebras this subspace is a lie algebra with the commutator product @xmath93:= xy - yx$ ] , shestakov and umirbaev @xcite realized that if @xmath43 is non - associative , many more operations are required to describe its algebraic structure completely . let @xmath94 , @xmath95 and @xmath96 be disjoint sets of symbols that we take to be the free generators of @xmath97 . write @xmath98 , @xmath99 and define @xmath100 in @xmath97 , where @xmath101 denotes the associator @xmath102 of @xmath103 and @xmath104 . each of the elements @xmath105 is primitive . considered as non - associative polynomials , @xmath106 can be evaluated in any algebra @xmath107 so we can think of them as of new multilinear operations derived from the binary product of @xmath107 . define @xmath108 & : = xy - yx \\ \langle x_1,\dots , x_m ; y , z \rangle & : = -p(x_1,\dots , x_m;y;z ) + p(x_1,\dots , x_m ; z;y)\\ \phi(x_1,\dots , x_m ; y_1,\dots , y_n ; y_{n+1 } ) & : = \\ & \hskip -2.5 cm \frac{1}{m!(n+1 ) ! } \sum_{\sigma \in s_n , \tau \in s_{m+1 } } p(x_{\sigma(1)},\dots , x_{\sigma(m ) } ; y_{\tau(1)},\dots , y_{\tau(n ) } ; y_{\tau(n+1)})\end{aligned}\ ] ] where @xmath109 and @xmath110 stands for the symmetric group on @xmath111 . in order to simplify the notation , for @xmath112 we write @xmath113.\ ] ] with this convention , ( [ eq : p ] ) gives @xmath114 shestakov and umirbaev proved that @xmath115 thus , while ( [ eq : dynkin_formula ] ) can be written in terms of commutators , the natural language to write its non - associative counterpart uses @xmath116 and @xmath117 . the algebra @xmath118 ( respectively , @xmath119 ) of formal power series in @xmath1 with coefficients in @xmath25 is a topological hopf algebra with the continuous extension of the operations of the hopf algebra @xmath120 ( respectively , @xmath121 ) . since @xmath122 , the _ group - like _ elements of @xmath119 , that is , the elements @xmath123 such that @xmath124 and @xmath125 , are of the form @xmath126 with @xmath127 . therefore , @xmath128 is , in a sense , canonical among all of them . however , @xmath129 is infinite - dimensional and @xmath130 has an infinite number of group - like elements that could rightfully be considered as the non - associative analogs of the exponential series . apart from the most obvious non - associative versions of the exponential @xmath131 other series have been proposed as non - associative analogs of @xmath132 , each leading to a different logarithm @xcite . a group - like element @xmath133 is a _ base for logarithms _ if its homogeneous component @xmath134 of degree one in @xmath1 is not zero . we say that the base for logarithms @xmath135 is _ normalized _ if @xmath136 . associated with any base for logarithms @xmath135 there exists a primitive element @xmath137 determined by @xmath138 the _ exponentiation _ on @xmath42 _ with base _ @xmath135 and the _ logarithm on @xmath139 to the base @xmath135 _ are the maps @xmath140 where @xmath141 denotes the space of formal power series with zero constant term . both maps are inverse to each other and give a bijection between the primitive and the group - like elements in @xmath139 . the logarithms to the bases @xmath142 and @xmath143 will be denoted by @xmath144 and @xmath145 , respectively . any base for logarithms @xmath135 determines a baker - campbell - hausdorff series in @xmath26 : @xmath146 the element @xmath147 is primitive so it can be written in terms of the shestakov - umirbaev operations @xmath116 and @xmath148 . since these operations are defined via the left - normed products @xmath149 and @xmath150 , the base @xmath142 is better adapted to recursive computations . in @xcite @xmath151 has been described as follows . for @xmath152 set @xmath153 . if @xmath154 is a non - associative monomial in @xmath1 , there is only one way of writing @xmath155 as a product @xmath156 . set @xmath157 and @xmath158 where @xmath159 is the @xmath160th bernoulli number . with this notation we have @xmath161 the baker - campbell - hausdorff series for different bases are related in a straightforward manner . if @xmath162 and @xmath163 are two bases for logarithms , the series @xmath164 is a primitive element of @xmath130 whose term of degree 1 is non - zero . in particular , it has a composition inverse @xmath165 such that @xmath166 . it is then clear that @xmath167 moreover , for any baker - campbell - hausdorff series @xmath168 and any primitive @xmath169 with @xmath170 , the series @xmath171 is also a baker - campbell - hausdorff series for some base . let @xmath172 be a derivation of @xmath77 that preserves @xmath173 , that is @xmath174 define @xmath175 ; thus , @xmath176 for all @xmath177 . the proof of the following result was inspired by @xcite . [ lem : dsw ] let @xmath172 be a derivation of @xmath77 that preserves @xmath173 , @xmath177 and @xmath178 . we have @xmath179 let us compute @xmath180 in two ways : @xmath181 so that by ( [ eq : brackets ] ) @xmath182 using ( [ eq : left_division ] ) , divide by @xmath183 to get the result . let us compute the expansion @xmath184 up to degree @xmath185 in terms of the shestakov - umirbaev operations with the help of the dynkin - specht - wever lemma . since , up to the summands of degree @xmath186 , we have @xmath187 the expansion of @xmath188 up to degree @xmath185 is @xmath189 + \frac{1}{3}x^2y - \frac{1}{4}x(xy)+\frac{1}{4 } x(yx ) -\frac{5}{12 } ( xy)x + \frac{1}{12}(yx)x\\ + \frac{1}{2 } xy^2 -\frac{5}{12}(xy)y + \frac{1}{12}(yx)y - \frac{1}{4}y(xy ) - \frac{1}{6}y^2x + \frac{1}{4}y(yx ) + \cdots\end{gathered}\ ] ] now , apply lemma [ lem : dsw ] with @xmath190 , where @xmath191 denotes the degree of @xmath192 , for homogeneous @xmath193 . first , observe that @xmath194 , @xmath195 and @xmath196 applying @xmath197 to the homogeneous summands in ( [ eq : bch3 ] ) and dividing by their degree , we can write ( [ eq : bch3 ] ) as @xmath198 + \frac{1}{12}[x,[x , y ] ] - \frac{1}{3}\langle x;x , y\rangle -\frac{1}{12 } [ y,[x , y ] ] + \frac{1}{6 } \langle y ; y , x \rangle -\frac{1}{2}\phi(x;y , y ) + \cdots\ ] ] the differential equation @xmath199 when @xmath18 and @xmath20 do not necessarily commute ( for instance , @xmath18 may belong to a matrix lie group and @xmath20 to the corresponding lie algebra ) has been studied since long ago @xcite . a fruitful approach is to look for solutions of the form @xmath200 for some @xmath21 , where @xmath128 denotes the usual exponential . the solution @xmath21 is determined by the initial condition and by the differential equation @xmath201 where @xmath202 denotes the @xmath47-th bernoulli number . take @xmath203 ; then @xmath204 so we can use ( [ eq : magnus_recursion ] ) in order to study @xmath205 . however , in a non - associative setting there are some details to be taken care of , since , for instance , equality @xmath2061@xmath207 above requires the associativity . let @xmath43 be a unital algebra , @xmath133 a base for logarithms and @xmath208 with @xmath209 $ ] such that @xmath210 . for any @xmath211 $ ] the solution @xmath21 to the equation @xmath212 satisfies @xmath213 where @xmath214 is defined by @xmath215 evaluating at @xmath216 and @xmath217 we get @xmath218 if @xmath219 then @xmath220 for some @xmath221 and there exists @xmath222 such that @xmath223 . therefore @xmath224 . in order to compute @xmath225 in terms of the shestakov - umirbaev operations , we will use the dynkin - specht - wever lemma . consider the derivation @xmath226 of @xmath26 determined by @xmath227 by induction on the degree @xmath228 of @xmath192 we can check that @xmath229 so that @xmath230 preserves @xmath231 and it is related to @xmath232 via @xmath233 now , in order to apply the dynkin - specht - wever lemma recursively @xmath135 should be a linear combination of left - normed products of primitive elements . this is the main reason for restricting ourselves to @xmath142 . [ lem : tau ] the component @xmath234 of degree @xmath47 in @xmath1 of @xmath235 is @xmath236 where @xmath237 . the expansion of @xmath238 can be easily obtained from the expansion of @xmath235 . given a tuple @xmath239 with @xmath240 define @xmath241 the concatenation @xmath242 of @xmath243 and @xmath244 will be denoted by @xmath245 . [ thm : degoneexpansion ] in @xmath26 we have @xmath246 where @xmath247 runs over all possible tuples with entries @xmath248 and @xmath249 let @xmath226 be the derivation of @xmath26 determined by ( [ eq : derivation ydeltax ] ) . the dynkin - specht - wever lemma implies @xmath250 so @xmath251 and @xmath252 since @xmath253 , the result follows . [ cor : magnus ] let @xmath43 be a unital algebra and @xmath211 $ ] . the solution @xmath209 $ ] of the equation @xmath212 with @xmath254 and @xmath210 satisfies @xmath255 where @xmath247 runs over all possible tuples with the components @xmath248 . we will use the formula for @xmath256 in theorem [ thm : degoneexpansion ] to describe , in terms of the shestakov - umirbaev operations , the differential equation satisfied by @xmath257 . [ prop : bch ] let @xmath135 be a normalized base for logarithms . in @xmath26 we have @xmath258 consider @xmath259 . since @xmath260 evaluating at @xmath261 , we get @xmath262 so @xmath263 . in the case when @xmath135 is @xmath142 or @xmath143 , proposition [ prop : bch ] was proved in @xcite . the components of degree @xmath264 and @xmath185 of @xmath235 are @xmath265 , @xmath266 , @xmath267 and @xmath268 . thus , the component of degree one in @xmath2 in @xmath184 is @xmath269 we can compute directly the coefficient of @xmath270 in @xmath184 , for instance . since @xmath271 then theorem [ thm : degoneexpansion ] ensures that this coefficient equals @xmath272 . @xmath273 [ prop : bchrecurr ] let @xmath274 . in @xmath275 $ ] we have @xmath276 we have @xmath277 so @xmath278 . the result follows from theorem [ thm : degoneexpansion ] . the component of @xmath184 of degree @xmath279 in @xmath1 and degree @xmath280 in @xmath2 is @xmath281 + \frac{1}{6}\langle y;y , x\rangle - \frac{1}{2}\phi(x;y , y).\end{aligned}\ ] ] based on proposition [ prop : bchrecurr ] we can compute the initial terms of the expansion of @xmath184 . [ thm : bch ] the expansion of @xmath184 in @xmath26 is @xmath282\\ & \quad + \frac{1}{12}[x,[x , y ] ] - \frac{1}{3}\langle x;x , y\rangle -\frac{1}{12 } [ y,[x , y ] ] - \frac{1}{6 } \langle y ; x , y \rangle -\frac{1}{2}\phi(x;y , y ) \\ & \quad -\frac{1}{24}\langle x ; x , [ x , y]\rangle - \frac{1}{12 } [ x,\langle x;x , y\rangle ] - \frac{1}{8}\langle x , x;x , y\rangle \\ & \quad + \frac{1}{24}[[x,[x , y]],y ] - \frac{1}{24}[x,\langle y;x , y\rangle ] -\frac{1}{4}\phi(x , x;y , y ) - \frac{1}{4}[x,\phi(x;y , y ) ] \\ & \quad -\frac{1}{24}[\langle x;x , y\rangle , y ] -\frac{1}{24}\langle x ; [ x , y],y\rangle - \frac{1}{6}\langle x , y;x , y\rangle + \frac{1}{24}\langle y , x;x , y\rangle \\ & \quad + \frac{1}{12 } [ \phi(x;y , y),y ] + \frac{1}{24}\langle y ; y,[x , y]\rangle - \frac{1}{24 } \langle y , y;x , y\rangle - \frac{1}{6}\phi(x;y , y , y)\end{aligned}\ ] ] plus terms of degree @xmath186 . formulas ( [ eq : magnus_recursion ] ) and ( [ eq : nonassociative_magnus_recursion ] ) give an alternative point of view on the relation between the numbers @xmath283 and @xmath284 which has been established in @xcite . we have @xmath285 by definition , @xmath286 and @xmath287 vanish in any associative algebra , with the only exception of @xmath288 . thus , after projecting from @xmath26 , in @xmath289 we get @xmath290 since it is well - known that @xmath291 holds in @xmath289 we get the result ; the sign @xmath292 in the latter formula comes from our choice @xmath293 instead of @xmath294 . in @xcite woon gave an algorithm to compute @xmath295 with the help of the binary tree ( v1 ) at ( -1,3 ) @xmath296 $ ] ; ( v2 ) at ( -3,2 ) @xmath297 $ ] ; ( v3 ) at ( 1,2 ) @xmath298 $ ] ; at ( 5,3 ) level 1 ; at ( 5,2 ) level 2 ; ( v4 ) at ( -4,1 ) @xmath299 $ ] ; ( v5 ) at ( -2,1 ) @xmath300 $ ] ; ( v6 ) at ( 0,1 ) @xmath301 $ ] ; ( v7 ) at ( 2,1 ) @xmath302 $ ] ; ( v1 ) edge ( v2 ) ; ( v1 ) edge ( v3 ) ; ( v2 ) edge ( v4 ) ; ( v2 ) edge ( v5 ) ; ( v3 ) edge ( v6 ) ; ( v3 ) edge ( v7 ) ; at ( -1,1 ) @xmath303 ; at ( 5,1 ) @xmath303 ; here , the nodes are labeled by @xmath304 $ ] ; the root is @xmath296 $ ] and at any node we have ( v1 ) at ( 0,2 ) @xmath304 $ ] ; ( v2 ) at ( -2,1 ) @xmath305 $ ] ; ( v3 ) at ( 2,1 ) @xmath306 $ ] ; ( v1 ) edge ( v2 ) ; ( v1 ) edge ( v3 ) ; the factorial of the node @xmath307 $ ] is @xmath308 . woon proved the equality @xmath309 for @xmath310 , where @xmath311 runs over the nodes in the level @xmath160 . in @xcite fuchs extended this construction as follows . consider the _ general pi binary tree _ ( v1 ) at ( -1,3 ) @xmath312 ; ( v2 ) at ( -3,2 ) @xmath313 ; ( v3 ) at ( 1,2 ) @xmath314 ; at ( 5,3 ) level 1 ; at ( 5,2 ) level 2 ; at ( 5,1 ) @xmath303 ; ( v4 ) at ( -4,1 ) @xmath315 ; ( v5 ) at ( -2,1 ) @xmath316 ; ( v6 ) at ( 0,1 ) @xmath317 ; ( v7 ) at ( 2,1 ) @xmath318 ; ( v1 ) edge ( v2 ) ; ( v1 ) edge ( v3 ) ; ( v2 ) edge ( v4 ) ; ( v2 ) edge ( v5 ) ; ( v3 ) edge ( v6 ) ; ( v3 ) edge ( v7 ) ; at ( -1,1 ) @xmath303 ; with root @xmath312 and at each node @xmath319 for any sequence @xmath320 of complex numbers change the node @xmath321 by @xmath322 . then define @xmath323 to be the sum of the nodes in the @xmath160-th level . this value depends on the sequence @xmath320 . in the case when @xmath324 we get the tree ( v1 ) at ( -1,3 ) @xmath325 ; ( v2 ) at ( -3,2 ) @xmath326 ; ( v3 ) at ( 1,2 ) @xmath327 ; ( v4 ) at ( -4,1 ) @xmath328 ; ( v5 ) at ( -2,1 ) @xmath329 ; ( v6 ) at ( 0,1 ) @xmath330 ; ( v7 ) at ( 2,1 ) @xmath331 ; ( v1 ) edge ( v2 ) ; ( v1 ) edge ( v3 ) ; ( v2 ) edge ( v4 ) ; ( v2 ) edge ( v5 ) ; ( v3 ) edge ( v6 ) ; ( v3 ) edge ( v7 ) ; at ( -1,1 ) @xmath303 ; and for each @xmath160 we have @xmath332 . to relate these constructions to the numbers @xmath333 in theorem [ thm : degoneexpansion ] we use a binary tree to collect the summands involved in @xmath334 . consider associative but non - commutative indeterminates @xmath335 and the tree ( v1 ) at ( -1,3 ) @xmath336 ; ( v2 ) at ( -3,2 ) @xmath337 ; ( v3 ) at ( 1,2 ) @xmath338 ; ( v4 ) at ( -4,1 ) @xmath339 ; ( v5 ) at ( -2,1 ) @xmath340 ; ( v6 ) at ( 0,1 ) @xmath341 ; ( v7 ) at ( 2,1 ) @xmath342 ; ( v1 ) edge ( v2 ) ; ( v1 ) edge ( v3 ) ; ( v2 ) edge ( v4 ) ; ( v2 ) edge ( v5 ) ; ( v3 ) edge ( v6 ) ; ( v3 ) edge ( v7 ) ; at ( -1,1 ) @xmath303 ; at ( 5,3 ) level 1 ; at ( 5,2 ) level 2 ; at ( 5,1 ) @xmath303 ; where at any node on the level @xmath343 we have ( v1 ) at ( 0,2 ) @xmath344 ; ( v2 ) at ( -2,1 ) @xmath345 ; ( v3 ) at ( 2,1 ) @xmath346 ; ( v1 ) edge ( v2 ) ; ( v1 ) edge ( v3 ) ; consider a sequence of numbers @xmath347 . define for @xmath348 the number @xmath349 and replace any node @xmath344 with @xmath350 . the sum of the nodes in the level @xmath47 of the resulting tree is @xmath351 . in case that @xmath352 , in the previous construction we can replace the label @xmath353 by @xmath354 without losing information . with these new labels , at any node on the level @xmath343 of the tree we have ( v1 ) at ( 0,2 ) @xmath321 ; ( v2 ) at ( -2,1 ) @xmath355 ; ( v3 ) at ( 2,1 ) @xmath356 ; ( v1 ) edge ( v2 ) ; ( v1 ) edge ( v3 ) ; which essentially gives the general pi binary tree . the number that we attach to the node @xmath357 is @xmath358 , so we recover the construction of fuchs . in this section we briefly study the coefficients of the non - associative monomials in the series @xmath184 . unfortunately , these monomials are not left - normed so we can not directly apply the dynkin - specht - wever lemma to them to get a closed form of the baker - campbell - hausdorff formula similar to ( [ eq : dynkin_formula ] ) . given a monomial @xmath359 , that we can identify with a binary planar rooted tree , any tuple @xmath360 of monomials satisfying @xmath361 with @xmath362 will be called a _ cut _ of @xmath363 recall that @xmath364 denotes the degree of @xmath365 in @xmath1 . attached to @xmath366 there is the set @xmath367 . the pair @xmath368 is a _ branch _ of @xmath363 . sometimes we will refer to @xmath369 as a branch of @xmath363 , although this is an abuse of notation since we should specify the positions that the branch occupies inside @xmath363 . monomials @xmath370 are represented by binary planar rooted trees with leaves decorated with @xmath1 or @xmath2 . the element @xmath375 in @xmath26 is expanded in terms of monomials @xmath376 with @xmath377 . unfortunately , monomials @xmath376 might represent the same monomial @xmath378 for different values of @xmath155 and @xmath379 one way of computing the coefficient of @xmath378 in the series @xmath184 is to determine the cuts of @xmath378 where every branch is of the form @xmath380 ( _ bch - cuts _ ) . if we denote by @xmath381 the set of all bch - cuts of @xmath378 , the coefficient in @xmath184 of the monomial @xmath378 is @xmath382 where @xmath383 is the coefficient of @xmath155 in @xmath151 . the set @xmath381 can be easily determined since lemma [ lem : branches ] implies that there exists a unique @xmath384 with minimal @xmath385 . the branches of any @xmath386 can be obtained as the branches in a bch - cut of the monomials @xmath387 , , @xmath388 . each monomial @xmath389 produces @xmath390 ( @xmath391 ) , @xmath392 ( @xmath393 ) or @xmath394 ( @xmath395 ) bch - cuts . using @xmath396 $ ] to delimit branches and writing @xmath397,\dots,[x^{i_{\vert\tau\vert } } y^{j_{\vert\tau\vert}}])$ ] instead of @xmath398 , some bch - cuts are @xmath399 , [ x^2][y ] , ( [ x][x])[y ] \\ \hline x(xy ) & [ x][xy ] , [ x]([x][y ] ) \\ \hline x(yx ) & [ x]([y][x ] ) \\ \hline ( xy)x & [ xy][x ] , ( [ x][y])[x ] \\ \hline \vdots & \vdots \\ \hline \end{array}\ ] ] thus , for instance , the coefficient of @xmath400 in @xmath184 is @xmath401 while the coefficient of @xmath402 is @xmath403 in a similar way , the coefficient of @xmath404 is @xmath405 finally , let us compute the coefficient of @xmath406 with @xmath407 . the bch - cuts are @xmath408 $ ] and @xmath409[x])\cdots [ x])(([y^j][y])\cdots[y])$ ] @xmath410 and @xmath411 . hence , the coefficient is @xmath412 however , we observe that the coefficients of the monomials in the expansion of @xmath413 agree with those in @xmath184 ( take @xmath414 ) . the coefficient in @xmath413 of @xmath415 ( @xmath416 ) is @xmath417 however , @xmath418 so this coefficient is @xmath419 . therefore , the coefficient of @xmath406 in @xmath184 is @xmath420

we address the problem of constructing the non - associative version of the dynkin form of the baker - campbell - hausdorff formula ; that is , expressing @xmath0 , where @xmath1 and @xmath2 are non - associative variables , in terms of the shestakov - umirbaev primitive operations . in particular , we obtain a recursive expression for the magnus expansion of the baker - campbell - hausdorff series and an explicit formula in degrees smaller than 5 . our main tool is a non - associative version of the dynkin - specht - wever lemma . a construction of bernouilli numbers in terms of binary trees is also recovered .

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Proceed to summarize the following text: the phase diagram structure of spin - glasses remains an open field of inquiry , since most approaches to the problem rely on approximations . any exact analytical result in this area is thus very valuable , both for the direct information it provides and as a test for approximation methods . over the last few years striking progress has been made combining the replica method , duality , and symmetry arguments @xcite , an approach which has yielded the exact locations of the multicritical points in the ising and potts spin - glasses on the square lattice and in the four - dimensional random - plaquette gauge model . the most recent result in this series @xcite is a general conjecture relating the multicritical point locations of any spin - glasses on a pair of mutually dual lattices . in support of the conjecture , estimates based on monte carlo simulations were given for ising spin - glasses , in @xmath2 , on the dual pairs of triangular and hexagonal lattices and , in @xmath3 , on the dual pairs of bilinear and lattice - gauge interactions on the cubic lattice . in both cases , within the numerical limitations , the conjecture is approximately satisfied . we propose here to extensively test the conjecture in an alternative fashion using hierarchical lattices @xcite , by looking at ising spin - glasses on mutually dual pairs @xcite of such lattices . these constitute ideal testing grounds , since an exact renormalization - group transformation for the quenched bond probability distribution can be constructed for such lattices , yielding global phase diagrams and critical properties . accordingly , the location of the phase boundaries and of the multicritical points are precisely determined . we thus investigate three pairs of hierarchical lattices , and in the end find that the conjecture is very nearly satisfied for all of them . the ising spin - glass is given by the hamiltonian @xmath4 where @xmath5 at each site @xmath6 , @xmath7 denotes a sum over nearest - neighbor pairs of sites , and the bond strengths @xmath8 are equal to @xmath9 with probability @xmath10 and @xmath11 with probability @xmath12 . the limits @xmath13 and @xmath14 correspond to purely ferromagnetic and purely antiferromagnetic systems respectively . to give a brief overview of the conjecture , let us consider the model on an arbitrary lattice , and treat the randomness through the replica method , where the system is replicated @xmath15 times and the @xmath16 limit is eventually taken , in order to get results for the physical system . the partition function of the @xmath15-replicated system after averaging over randomness , @xmath17 , can be expressed entirely as a function of @xmath18 `` averaged '' combinations of edge boltzmann factors , @xmath19 , associated with nearest - neighbor bonds @xcite . these averaged boltzmann factors , @xmath20 , @xmath21 , have the form @xmath22 where the @xmath23th factor corresponds to a configuration with a parallel - spin bond in @xmath24 replicas and an antiparallel - spin bond in @xmath23 replicas @xcite . thus , @xmath25 the partition function on the dual lattice , @xmath26 , can be expressed in a similar form , @xmath27 with the dual counterparts to the averaged boltzmann factors given by @xmath28 for @xmath29 . @xmath17 and @xmath30 are related as @xcite @xmath31 where @xmath32 is a constant , which can be eliminated by using eq . evaluated at two different sets of parameters , @xmath33 and @xmath34 , giving a relationship of the form @xmath35 the individual partition functions @xmath17 can be rewritten by extracting @xmath36 , the averaged boltzmann factor corresponding to an all - parallel spin state , thus effectively measuring the energy of the system relative to this state @xcite : @xmath37 where @xmath38 is the number of bonds in the lattice , and the reduced variables are @xmath39 . . becomes @xmath40^{n_b } { \cal z}_n(u_1(p_1,j_1),\,\ldots,\,u_n(p_1,j_1))\\ & \qquad \cdot { \cal z}_n^\ast(u_1(p_2,j_2),\,\ldots,\,u_n(p_2,j_2))\\ & = [ x^\ast_0(p_1,j_1)x^\ast_0(p_2,j_2 ) ] ^{n_b } { \cal z}_n^\ast(u^\ast_1(p_1,j_1),\,\ldots,\,u^\ast_n(p_1,j_1))\\ & \qquad \cdot { \cal z}_n(u^\ast_1(p_2,j_2),\,\ldots,\,u^\ast_n(p_2,j_2))\ , . \end{split}\ ] ] in general , the form of eq . is too complicated to yield useful information relating the locations of phase transitions . however , the multicritical points in both original and dual systems are expected to lie @xcite on the nishimori line @xcite , which simplifies the relation . furthermore , the conjecture advanced in ref . @xcite states that , for the multicritical points @xmath41 of the original system and @xmath42 of its dual , eq . is satisfied when the leading boltzmann factors @xmath36 from each side are equal , @xmath43 since @xmath41 and @xmath42 lie on the nishimori line , @xmath44 from eqs . and , eq . gives @xmath45 finally taking the limit , @xmath46 , one obtains the condition @xmath47 where @xmath48 . as expressed in eq . , the conjecture is asserted to hold for multicritical points of ising spin - glasses on any pair of mutually dual lattices @xcite . hierarchical lattices @xcite are constructed by replacing every single bond , in a connected cluster of bonds , with the connected cluster of bonds itself , and repeating this step an infinite number of times . these provide models exactly solvable by renormalization group , with which complex problems have been studied and understood . for example , frustrated @xcite , spin - glass @xcite , random - bond @xcite and random - field @xcite , schrdinger equation @xcite , lattice - vibration @xcite , dynamic scaling @xcite , aperiodic magnet @xcite , complex phase diagram @xcite , and directed - path @xcite systems , etc . , have been solved on hierarchical lattices . to test the conjecture of eq . , we study ising spin - glasses on the dual pairs of hierarchical lattices , depicted in figs . [ lfig1 ] , [ lfig2 ] , and [ lfig3 ] . each lattice in a given pair is the dual of the other . these particular choice of lattices are motivated by their properties under renormalization - group transformation as related to physical lattices . the hierarchical lattices of fig . 1(a ) and ( b ) yield the two variants of the migdal - kadanoff recursion relations @xcite for dimension @xmath2 with length rescaling factor @xmath49 . similarly , the lattice in fig . 2(a ) yields a migdal - kadanoff recursion relation for @xmath3 , @xmath49 . its dual lattice in fig . 2(b ) has @xmath50 . ( the two variants of the migdal - kadanoff recursion relations correspond to mutually dual hierarchical lattices only in @xmath2 . ) lastly , the hybrid lattice in fig . 3(a ) is interesting because it has been shown to give very accurate results for the critical temperatures of the @xmath3 isotropic and anisotropic ising model @xcite . this lattice has @xmath51 , while its dual in fig . 3(b ) has @xmath50 . , @xmath49 migdal - kadanoff recursion relations are exact . ] , @xmath49 . its dual lattice , in ( b ) , has @xmath52 , @xmath53 . ] , @xmath49 and @xmath52 , @xmath53 respectively . ] for a pure system , the renormalization - group transformation on a hierarchical lattice consists of a decimation by summing over the internal sites in each of the connected clusters making up the lattice ( the right - hand sides of figs . [ lfig1]-[lfig3 ] ) . thus , the hierarchical lattice construction process is reversed , as each connected cluster is replaced by a single renormalized bond . the decimation can be expressed as a mapping , @xmath54 where the set @xmath55 are all the bonds within the connected cluster of the original system and @xmath56 is the renormalized bond between sites @xmath57 and @xmath58 of the rescaled system . in the pure case , all @xmath8 bonds are independent for @xmath59 , and the implementation of eq . is straightforward . when quenched randomness is added to the system , the renormalization - group transformation is expressed in terms of quenched probability distributions @xcite , where the quenched probability distribution @xmath60 in the rescaled system is calculated from @xmath61 in the original system through the convolution @xmath62 \delta\left(j^\prime_{i^\prime j^\prime } - r(\{j_{ij}\})\right)\,.\ ] ] here the product runs over all the bonds @xmath59 in the connected cluster of the original system between sites @xmath57 and @xmath58 . the recursion of the quenched probability distribution , eq . , is implemented numerically . the probability distribution is represented by histograms , each histogram being specified by a bond strength and an associated probability . thus , for the spin - glass problem , the starting distribution consists of two histograms , one at @xmath63 with probability @xmath10 , and one at @xmath11 with probability @xmath12 . . dictates the convolution of 9 probability distributions for the lattices of fig . [ lfig1 ] , and the convolution of 27 distributions for the lattices of figs . [ lfig2 ] and [ lfig3 ] . in this task , computational storage limits can be maximally exploited by factorizing eq . into an equivalent series of pairwise convolutions , each of which involves only two distributions convoluted using an appropriate @xmath64 function . the types of pairwise convolutions needed are a `` bond - moving '' convolution , with @xmath65 and a decimation convolution , with @xmath66\,,\ ] ] which is just the standard decimation transformation for a two - bond ising segment . consider the hierarchical lattice in fig . [ lfig1](a ) . if @xmath67 is the initial probability distribution , a series of pairwise convolutions which yields the total convolution of eq . for this lattice is : ( i ) a bond - moving convolution of @xmath68 with itself , yielding @xmath69 ; ( ii ) a bond - moving convolution of @xmath69 with @xmath68 , yielding @xmath70 ; ( iii ) a decimation convolution of @xmath71 with itself , yielding @xmath72 ; ( iv ) a decimation convolution of @xmath72 with @xmath70 , yielding @xmath73 . for the lattice in fig . [ lfig2](a ) , the series is : ( i ) a decimation convolution of @xmath68 with itself , yielding @xmath69 ; ( ii ) a decimation convolution of @xmath69 with @xmath68 , yielding @xmath70 ; ( iii ) a bond - moving convolution of @xmath70 with itself , yielding @xmath72 ; ( iv ) a bond - moving convolution of @xmath74 with itself , yielding @xmath75 ; ( v ) a bond - moving convolution of @xmath75 with itself , yielding @xmath76 ; ( vi ) a bond - moving convolution of @xmath76 with @xmath70 , yielding @xmath77 . for the lattice in fig . [ lfig3](a ) , the series is : ( i ) a bond - moving convolution of @xmath68 with itself , yielding @xmath69 ; ( ii ) a decimation convolution of @xmath69 with itself , yielding @xmath71 ; ( iii ) a decimation convolution of @xmath70 with @xmath78 , yielding @xmath72 ; ( iv ) a bond - moving convolution of @xmath72 with itself , yielding @xmath75 ; ( v ) a bond - moving convolution of @xmath75 with itself , yielding @xmath76 ; ( vi ) a decimation convolution of @xmath68 with itself , yielding @xmath79 ; ( vii ) a decimation convolution of @xmath80 with @xmath68 , yielding @xmath81 ; ( viii ) a bond - moving convolution of @xmath81 with @xmath76 , yielding @xmath77 . as for the dual lattices in figs . [ lfig1](b ) , [ lfig2](b ) , and [ lfig3](b ) , the series of pairwise convolutions are identical to their counterparts above , except that each bond - moving is replaced by a decimation , and vice versa . since the number of histograms that constitute the probability distribution increases rapidly with each renormalization iteration , a binning procedure is used when the desired ( large , namely up to @xmath0 ) number of histograms is reached : before every pairwise convolution , the histograms are placed on a grid , and all histograms falling into the same grid cell are combined into a single histogram in such a way that the average and the standard deviation of the probability distribution are preserved . histograms falling outside the grid , representing a negligible part of the total probability , are similarly combined into a single histogram . any histogram within a small neighborhood of a cell boundary is proportionately shared between the adjacent cells . in the current study , the binning procedure is done separately for @xmath82 and @xmath83 . after the convolution , the original number of histograms is reattained . in the current study , 40,000 bins are generally used , representing the renormalization - group flows of 80,000 variables , requiring the calculation of 40,000 local renormalization - group transformations at each renormalization - group iteration . the numerical results converge rapidly with increasing bin number . for maximal accuracy in determining the exact locations of the multicritical points , in the immediate vicinity of these points we used at least 1,000,000 histograms , representing the renormalization - group flows of 2,000,000 variables , requiring the calculation of 1,000,000 local renormalization - group transformations at each renormalization - group iteration . it should thus be noted that our analysis is an exact numerical solution of ising spin - glasses on hierarchical lattices . and @xmath84 probability bins were used in ( a ) and ( b ) respectively . ] global phase diagrams for the various hierarchical lattices are obtained from the renormalization - group flows of the probability distributions . each phase has a corresponding sink , namely a completely stable fixed distribution . the boundaries between phases flow to unstable fixed distributions , analysis of which yields the order of the phase transition and the values of the critical exponents of second- and higher - order transitions . all the phase diagrams are plotted in terms antiferromagnetic bond concentration @xmath12 versus temperature @xmath86 . the diagrams are symmetric around @xmath87 , with the ferromagnetic phase in the @xmath88 half - space mapping onto the antiferromagnetic phase in the @xmath89 half - space . thus in the figures only the @xmath88 portions are shown . 4(a ) and ( b ) show the phase diagrams for the dual pair of hierarchical lattices in fig . 1(a ) and ( b ) respectively . the phase structure of both diagrams is topologically identical to that of the @xmath2 ising spin - glass on a square lattice , which is only natural considering that the @xmath2 , @xmath49 migdal - kadanoff recursion relations are exact on these hierarchical lattices.@xcite the @xmath13 transition temperatures of the two models are related by the duality algebra @xcite which is also true for the two other pairs of mutually dual hierarchical models . furthermore , the @xmath13 transition temperatures in fig . 4(a ) and ( b ) are related @xcite by @xmath91 since the mappings of the interaction constant in the repetition of renormalization - group transformations differs only by an initial bond strengthening by a factor of @xmath92 ; note that eq . does not apply to @xmath93 , since there the bond - moving is not a mere multiplicative strengthening , but a @xmath94-fold convolution of the probability distributions that alters this distribution in a non - simple way . . is also not applicable to the two other pairs of mutually dual models , since the repetition of renormalization - group transformations are not differentiated by only a preliminary bond - moving . in each of fig . 4(a ) and ( b ) , a ferromagnetic phase at low temperatures and low @xmath12 is separated from the disordered paramagnetic phase by two second - order phase boundaries , meeting at a multicritical point . ( in a narrow neighborhood of all multicritical points in our results , reentrance is observed : paramagnetic , then ferromagnetic , then paramagnetic or spin - glass phases are encountered as temperature is lowered at fixed @xmath12 . ) the two second - order boundaries flow to distinct unstable probability distributions with different critical exponents , constituting a strong violation of universality @xcite and consistent with the prediction , generally , of the absence of first - order transition under quenched randomness in @xmath2 . @xcite as expected from symmetry considerations , the multicritical points fall @xcite precisely on the nishimori line @xcite as seen in table i. as also seen in table i , @xmath95 , so that the conjecture is realized to a very good approximation . 5 shows the phase diagrams for the dual pair of hierarchical lattices in fig . 2 . while fig . 5(b ) has the same phase topology as the diagrams in fig . 4 , being at @xmath52 below the spin - glass lower - critical dimension , a different structure occurs in fig . 5(a ) . here the @xmath3 , @xmath49 migdal - kadanoff relations are exact on the hierarchical lattice , and for low temperatures in the vicinity of @xmath96 there exists a spin - glass phase . the multicritical point occurs where the ferromagnetic , paramagnetic , and spin - glass phases meet . as expected both multicritical points lie directly on the nishimori line . from table i we see that @xmath97 , so that the conjecture is realized to a very good approximation , even when the mutually dual models belong to different dimensionalities @xmath1 and have different phase diagram topologies at the multicritical points of the conjecture . the phase diagram structures in fig . 6 , corresponding to the dual pair of hierarchical lattices in fig . 3 , are similar to those of fig . 5 , illustrating dimensions above and below the spin - glass lower - critical dimension . again the multicritical points for both cases lie directly on the nishimori line . in this case @xmath98 , and the conjecture is realized to a very good approximation , again for mutually dual models belonging to different dimensionalities @xmath1 and having different phase diagram topologies at the multicritical points of the conjecture . thus , we find that for all three mutually dual pairs of hierarchical lattices , the conjecture relating the locations of the multicritical points is satisfied to a very good approximation . this is all the more remarkable , since , as seen in table i , the contributions of @xmath99 and @xmath100 to the conjecture are strongly asymmetric . however , it should be noted that ( 1.0172,0.9829,0.9911 ) , while being very close to 1 , are different from integer 1 . in our numerical implementation of the convolutions of the probability distributions , the results have converged to the precision of the digits shown in table i. further increase of the already very large number of probability bins does not change the entries in the table . further tests of the conjecture , using other systems , would be very useful . similar to our current study , the use of hierarchical lattices to study phenomena linked to mutually dual lattices , e.g. , ref . @xcite , would also be very useful . we thank h. nishimori and k. takeda for useful correspondance . this research was supported by the scientific and technical research council of turkey ( tbitak ) and by the academy of sciences of turkey . mh gratefully acknowledges the hospitality of the feza grsey research institute and of the physics department of istanbul technical university . h. nishimori , prog . . phys . * 66 * , 1169 ( 1981 ) . h. nishimori and k. nemoto , j. phys . japan * 71 * , 1198 ( 2002 ) . maillard , k. nemoto , and h. nishimori , j. phys . a : math . gen . * 36 * , 9799 ( 2003 ) . k. takeda and h. nishimori , nucl . b * 686 * , 377 ( 2004 ) . k. takeda , t. sasamoto , and h. nishimori , j. phys . gen . * 38 * , 3751 ( 2005 ) . a.n . berker and s. ostlund , j. phys . c * 12 * , 4961 ( 1979 ) . m. kaufman and r.b . griffiths , phys . b * 24 * , 496 ( 1981 ) . m. kaufman and r.b . griffiths , phys . b * 30 * , 244 ( 1984 ) . m. kaufman and d. andelman , phys . rev . b * 29 * , 4010 ( 1984 ) . m. kaufman , phys . b * 30 * , 413 ( 1984 ) . c. itzykson and j.m . luck , proceedings of the brasov international summer school ( 1984 ) . h. ottavi and g. albinet , j. phys . a * 20 * , 2961 ( 1987 ) . le doussal and a. georges , yale university report no . yctp - 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the ising spin - glasses are investigated on three dual pairs of hierarchical lattices , using exact renormalization - group transformation of the quenched bond probability distribution . the goal is to investigate a recent conjecture which relates , on such pairs of dual lattices , the locations of the multicritical points , which occur on the nishimori symmetry line . towards this end we precisely determine the global phase diagrams for these six hierarchical spin - glasses , using up to @xmath0 probability bins to represent the quenched distribution subjected to an exact renormalization - group transformation . we find in all three cases that the conjecture is realized to a very good approximation , even when the mutually dual models belong to different spatial dimensionalities @xmath1 and have different phase diagram topologies at the multicritical points of the conjecture and even though the contributions to the conjecture from each lattice of the dual pair are strongly asymmetric . in all six phase diagrams , we find reentrance near the multicritical point . in the models with @xmath2 or 1.5 , the spin - glass phase does not occur and the phase boundary between the ferromagnetic and paramagnetic phases is second order with a strong violation of universality . pacs numbers : 75.10.nr , 64.60.kw , 05.45.df , 05.10.cc

You are an expert at summarizing long articles. Proceed to summarize the following text: one of the most pervasive tendencies of humans is putting things in ranking order . in human societies these tendencies are reflected in their social interactions and networks being hierarchical in many respects . hierarchies and ranks emerge due to individuals subjective perceptions that some other individuals are in some respect better . then a relevant research question is whether or not the formation and structure of hierarchies in human societies can be understood by making the assumption that the dominant driving force of people in social interactions is to enhance their own `` value '' or `` status '' relative to others . we call this assumption `` better than - hypothesis '' ( bth ) and note that it is closely related to the thinking of the school of individual psychology founded by adler in the early 1900s @xcite , which , while starting with the assumption that human individuals universally strive for `` superiority '' over others , emphasizes inferiority avoidance as a motive for many human actions . further studies of this kind of individuals status - seeking behaviour , especially concerning consumer behaviour and economics , include the canonical references by veblen @xcite , duesenberry @xcite and packard @xcite ( see also refs @xcite ) . in addition there is a closely related sociological model called social dominance theory @xcite , which proposes that the construction and preservation of social hierarchies is one of the main motivations of humans in their social interactions and networks . however , the most relevant observational facts concerning bth come from the field of experimental economics , especially from the results of experiments on the so - called `` ultimatum game '' @xcite , where the human players have been shown to reject too unequal distributions of money . the concept of _ inequity aversion _ , that is the observed social phenomenon of humans preferring equal treatment in their societies , is often invoked to explain these observations . recently some models featuring inequity aversion have been proposed in refs . @xcite . all of these models , although from different fields of study , have something to do with the relative standings between different human individuals and groups , and so they could all be considered to emerge from or be based on a single principle such as bth . it is this generality which makes bth an intriguing and interesting object of study . there are even some studies on economic data , such as @xcite , that suggest a link between relative social standings and human well - being , and considerations of social status have measurable effects on brain functions , as shown in e.g. @xcite . these studies imply that bth could well be something fundamental to human nature . the competition for a better hierarchical position among humans can be intense and sometimes even violent . however , humans have other characteristics including egalitarianism as well as striving for fairness . these traits could be interpreted in the context of bth by remarking that people need to live in societies and make diverse social bonds , which in turn would contribute to their social status . this means that the members of society when they make decisions , need to take the feelings of others into account . hence the behavioral patterns of individuals in social networks should then be characterised by sensitivity to the status of the other individuals in the network . this sensitivity manifests itself as inequity aversion and treating others fairly . to find out what in this context are the plausible and relevant mechanisms of human sociality driving societal level community formation we will focus on improving the bth - based approach by using the frame of agent - based models and studying the emergence of social norms in such social systems , following the tradition presented in refs . @xcite . in this study we use an agent - based network model applying bth - based approach to simulate social interactions dependent on societal values and rank , to get insight to their global effects on the structure of society . we find that in such a model society with a given constant ranking system the social network forms a degree hierarchy on top of the ranking system under bth , such that the agents degrees tend to increase , the further away their rank is from the average . the structure of the paper is as follows . in section [ model ] we motivate the basics of bth using the simple and well - researched ultimatum game as an example , and in section [ modelv1 ] we show how the findings from this can be utilised as a part of agent - based models . in section [ nc ] we present the numerical results of the simulations from the model , and in section [ meanfield ] we analyse them . the two final sections discuss the possible interpretations of the results and present the conclusions . in this section we describe the theoretical basis for our model . we start by analysing the ultimatum game first proposed in @xcite , as it allows us to derive a basic form for the social gain function in our model . the ultimatum game is a game with two players , where one player has the task to make a proposal to the other player about how a given sum of money should be divided between them . the second player then gets to choose if the proposal is acceptable or not ; if it is , the money is divided as proposed . if not , neither player gets anything . experiments show that humans playing this game normally do not accept deals that are perceived to be unfair , i.e. in situations in which the proposer gets too large a share of the money ( see , e.g. refs . this is a classic problem in the mainstream economics , where humans are assumed to be rational and , therefore , accept something rather than nothing . we implement bth in the ultimatum game by interpreting the money used in a deal as a way of comparing the status between the one who accepts the proposal ( called from now on the accepter ) and its proposer . we denote the change of `` status '' of the accepter as @xmath0 , which takes into account it s own monetary gain , and the gain in relation to the proposer . therefore , the simplest expression for @xmath0 is , @xmath1\\ & -[r_a(t_0 ) - r_p(t_0 ) ] , \end{aligned } \label{perusta}\ ] ] where @xmath2 and @xmath3 stand for the monetary reserves ( in the context of the game ) of the accepter and proposer , respectively , at time @xmath4 , with @xmath5 being the time before the deal and @xmath6 the time after the deal . in terms of economic theory , @xmath0 would be called the accepter s change of utility , which is ordinarily assumed to consist of the term @xmath7 or the absolute payoff of the accepter . the additional terms @xmath8-[r_a(t_0 ) - r_p(t_0)]$ ] that stem from the bth measuring the change in relative standings of the accepter and proposer . the actual bth utility function for the accepter in the ultimatum game takes the form @xmath9.\ ] ] according to eq . ( [ perusta ] ) , the accepter will refuse the deal for @xmath10 , and will accept it for @xmath11 , with @xmath12 being the borderline case . should the deal be rejected , @xmath13 and @xmath14 , and , consequently , @xmath15 . if we denote by @xmath16 the total amount of money to be shared and by @xmath17 the actual amount money that the proposer has reserved for the accepter , then in the case where the transaction does take place we have @xmath18 and @xmath19 . if we further assume that @xmath20 ( i.e. the players start on equal footing , which may very well be the case in the context of the game at least ) , it follows from eq.([perusta ] ) that the smallest offer that the accepter expects from the proposer is one third of the maximum @xmath16 , i.e. the condition @xmath21 must hold for the proposal to be acceptable . previous literature shows that the minimum offers that people are usually willing to accept are around @xmath22 of a given quantity @xcite , in close agreement with the calculation above . moreover , we note that if the term @xmath7 in eq . ( [ perusta ] ) is neglected , then the accepter will never settle for less than half of the total amount . next we use eq . ( [ perusta ] ) to illustrate how the bth - based approach can be implemented in the context of agent based social simulations . the trick is to generalise this equation to the cases of many players , and with multiple and different kinds of items being exchanged . if there are @xmath23 players ( denoted by @xmath24 ) , the change of status @xmath25 of the individual @xmath26 may be written as follows @xmath27 \nonumber \\ & & - \sum_{j \neq i } [ r_i(t_0 ) - r_j(t_0 ) ] . \label{glin}\end{aligned}\ ] ] in the case that players have several ways to measure their status , we may introduce normalisation factors to compare the relative value of the exchanged items , and write the change of status as , @xmath28 \nonumber \\ & & - \bigg . \sum_{j \neq i } \left [ r^{\alpha}_i(t_0 ) - r^{\alpha}_j(t_0 ) \right ] \bigg\ } \label{ydelta}\end{aligned}\ ] ] where the index @xmath29 runs over the various items determining the status , and @xmath30 is the normalisation factor of each item , which may vary from one player to another . the utility function associated with eq . ( [ ydelta ] ) reads then @xmath31 \bigg\}.\ ] ] in this section we present an agent - based model of a ranked social system of @xmath23 agents , in which the agents exchange their views of the ranking system itself . to each agent @xmath26 we assign a parameter @xmath32 to describe the rank of the agent , and a state variable @xmath33 to denote the opinion of the agent @xmath26 of the social value attached to parameter @xmath34 . the social value is then a relative quantity in the minds of the agents , and they value each other in either ascending or descending order according to the `` ranking parameter '' , we call @xmath34 , and @xmath33 determines which order a given agent @xmath26 prefers and how strongly . generally speaking , the sign of @xmath33 represents the chosen order , @xmath35 for descending and @xmath36 for ascending order , while its magnitude represents the strength of conviction : with @xmath37 the agent can be said to support equality of all the agents irrespective of the ranking parameter , with @xmath38 the agent thinks that @xmath34 should directly define the hierarchy of the society , and for the case @xmath39 or @xmath40 correspondinly downplaying or emphasizing the significance of @xmath34 , respectively . here , we adopt the maximum value for @xmath41 to be @xmath42 . to put the relation of the social value and the opinion parameter into more precise terms , we adopt the following expression for the term we call `` ranking pressure '' : @xmath43 where the summation is over the whole network . now , we define the social value of the agent @xmath26 in the eyes of agent @xmath44 as @xmath45 , and assume that the agents take into account the views of their neighbours in addition to their own when evaluating their total social value @xmath46 . thus , we write @xmath47 where @xmath48 denotes agents that are one step away from agent @xmath26 . the first term on the right could be considered as the agent s `` self esteem '' and the second the `` social value '' given to it by its first neighbours . it should be noted at this point that in defining @xmath46 in terms of @xmath49 we have assumed that @xmath32 does not confer direct advantages or disadvantages for the agents . therefore , the most natural interpretation for @xmath32 is that it represents the ownership of pure status symbols or veblen goods , i.e. goods that are only , or mostly , desirable due to their status - enhancing properties , such as luxury items . if @xmath32 would give some advantages or disadvantages for the agents , eq . ( [ totsosval ] ) would have to be revised accordingly . the agents in our simulations attempt to gain as much social value as possible , both in absolute and relative terms , and do this either by changing their opinion variables or adjusting their relations to other agents . the system we use here is purely reactive , with agents reacting to the changes in their social environment in accordance with bth . the social gain function could then be written as @xmath50 where @xmath5 and @xmath6 are the two consecutive time steps . the sign of this function determines the direction of the changes in @xmath33 . the decision making method employed by the agents is thus a simple hill climbing algorithm : at a given time step @xmath4 , first an agent observes the quantity @xmath51 , and then changes its variable @xmath52 , which leads to a recurrence relation of the form @xmath53 where @xmath54 $ ] and @xmath55 is a small increment . in the spirit of simulated annealing techniques , the magnitude of the change is larger at the beginning of the dynamics and falls linearly with time to a minimum value , the maximum and minimum values being @xmath56 and @xmath57 , respectively , and the time period to reach the minimum is 1000 time steps . in general , the links between the agents in the social network may change in time for which purpose we use the following rewiring scheme of ref . the social network of the agents is initially random , but will change periodically , i.e. at every @xmath58 time steps of the dynamical eq . ( [ dm ] ) . given the definition of the total social value of an agent in eq . ( [ totsosval ] ) , the gain function eq.([altdel ] ) can be used to calculate the loss or gain in total social status when forming or breaking new social bonds . in this study , we take any positive gain as sufficient to justify the rearrangement of social relations between the agents . when agent @xmath26 considers cutting an existing bond with agent @xmath44 , the gain function has the form @xmath59 where @xmath60 is the current number of neighbours of agent @xmath26 . similarly , when agent @xmath26 considers forming a new bond with agent @xmath44 , the social gain function reads @xmath61 since any positive change indicated by the functions above leads to rewiring , it is the sign of these functions that determines whether or not links between agents are broken or created . for instance , if @xmath62 , the link between agents @xmath26 and @xmath44 will be cut , and preserved if @xmath63 . in the same vein , a link between agents @xmath26 and @xmath44 will be created if @xmath64 , and not created otherwise . it should be noted that when forming links the opinions of both agents are taken into account : the relation formation only succeeds if @xmath65 and @xmath66 are both positive . the agents will form all the relationships they can in a rewiring cycle . the numerical simulations of the models of social system described in the previous sections are performed as follows . first , the initial state of the system is set at random with the agents given a relatively small initial opinion @xmath33 between @xmath67 and @xmath68 , a ranking parameter @xmath32 between @xmath69 and maximum value of @xmath70 and initial connections to other agents , with initial average degree of @xmath71 . the opinion and ranking parameters are chosen using a random number generator , which returns a flat distribution . the dynamics are then run for @xmath72 time steps , which is , according to our test runs , sufficient for the general structure of the network to settle . however , the dynamics of the opinion variables do not have a set stopping point , so they may experience fluctuations even when such fluctuations do not have an effect on the network structure anymore . to obtain reliable statistics , the same simulations are repeated @xmath73 times with random initial values , and averages are calculated from these repeated tests for the quantities under study . the rewiring timescale @xmath58 is fixed to @xmath73 in our simulations , since this value lies in the range where communities are formed in the opinion formation model of ref . the main parameter whose effect is studied here is the number of simulated agents , @xmath23 . the main objective of this research is to study the structure of the social networks created under bth assumption in the case of a rigid ranking system , which we perform using the model explained in section [ modelv1 ] . the most interesting properties of the system are then associated with assortativity , or the tendency of agents with high degrees connecting to other highly connected agents , and homophily , or the inclination of similar agents forming connections between each other . in the context of this study , homophily refers to agents with similar ranking parameters forming connections with each other . the averaged numerical results extracted from the simulations consist then of the standard network properties , i.e. degree @xmath74 the shortest path @xmath75 , the average clustering coefficient @xmath76 , the mean number of second neighbours @xmath77 , susceptibility @xmath78 and average assortativity coefficient @xmath79 , and a homophily coefficient @xmath80 . susceptibility here refers to average cluster size , which is calculated as the second moment of the number of @xmath81 sized clusters , @xmath82 : @xmath83 as customary in percolation theory , the largest connected component of the network is not counted in calculating @xmath81 . for the assortativity coefficient we use the definition given in @xcite , and the homophily coefficient is defined using pearson s product moment coefficient , which measures the goodness of a linear fit to a given data . for a sample it can be defined as @xmath84 where @xmath85 and @xmath86 are vectors containing the value parameters of agents linked by link @xmath26 , @xmath87 and @xmath88 are the mean values of these vectors , respectively , and @xmath89 is the total number of links . more specifically , if agents @xmath29 and @xmath90 are connected by link @xmath26 , then @xmath91 and @xmath92 . the links are indexed as follows : the links involving the first agent are given the first indices , then follow the links involving the second agent but not the first , and so on , without repeating links that have already been indexed . it should be noted , however , that @xmath93 only measures linear correlation between the ranking parameters of linked agents , it does not indicate how steep these trends are . to check whether the system is truly homophilic , then , one needs to make a linear fit to the data : the closer the obtained linear coefficients are to @xmath68 , the greater the homophily . = 0.85 the average network properties of the system , with graphs illustrating the behaviour of the system are shown in fig . [ fig : np100r ] as functions of the population size , which is varied between @xmath94 and @xmath95 . the main observations that can be made about the graphs in fig.[fig : np100r](a ) are that at lower population levels they show a tendency of breaking apart into many subcomponents of different sizes , while for larger population sizes they tend to consist of a single large component and possibly some smaller separate clusters . a noteworthy fact about these clusters is that they consist of agents with similar values of the ranking pressure @xmath49 , which means that the network exhibits homophily in this case . the largest clusters are found at extreme values , and they become smaller when one approaches @xmath69 , which also corresponds to average ranking parameters . in the high population case the picture becomes more complicated due to the emergence of clusters that contain agents with opposing opinions as well . these new clusters tend to be less connected than the previously described homogeneous ones , and they tend to connect to the large subgraphs , thus forming a single giant graph . a closer look reveals that these agents generally have opinion variables and ranking pressures with opposite signs and are depicted as triangles in fig . [ fig : np100r ] and named ` contrarians' from now on . a naive analysis would indicate that the agents with positive ranking pressures should always support the ascending hierarchy , and the agents with negative ranking pressures should always support the descending hierarchy . however , the contrarian agents exhibit opposite preferences . the reason why this behaviour is status - wise profitable can be found by looking into the connections of the contrarian agents , details of which are shown in figs . [ fig:1bis ] and [ fig : scatter2 ] . as it turns out , most of the connections they form are to other similar agents but with opposite `` polarity '' to theirs , i.e. the contrarian agent with negative ranking pressure forms connections mostly with contrarian agents of positive ranking pressure , and vice versa . an important quantity in the model is the social value @xmath46 , which is a product of the opinion and ranking pressure , as it is seen in eq . [ totsosval ] . if @xmath33 and @xmath49 have opposite signs then the `` self esteem '' part of @xmath46 is negative , which in itself does not mean that the agent can not develop a contrarian opinion , since the second sum in the equation could be positive because it depends on the opinions of the neighbours @xmath44 . this allows the contrarian to be able to make connections with agents of the same or opposite ranking pressure . additionally , by looking at all connections among contrarians we find that they mostly have @xmath49 of the same sign as @xmath96 , as illustrated in the example of fig . [ fig:1bis ] . = 0.85 this situation can be status - wise beneficial to all parties involved , since the small penalty to an agent s self - esteem is more than compensated by the respect that the agent will gain in this case from other agents . the fact that the agents could find this strategy using as primitive an intelligence setup as hill climbing is astounding . another interesting thing about the contrarians is that they appear mostly as connections between clusters that are defined as communities of `` normal '' agents . the various kinds of behaviour exhibited by the social networks have a marked effect on the network properties also shown in fig . [ fig : np100r](b ) . the most obvious is the gradually rising normalised maximum cluster size ( @xmath97 ) , which is about 40% of total population size for @xmath98 , and over 95% for @xmath99 . from the figure it seems that the maximum cluster size reaches 50% of the population size for approximately @xmath100 , after which point we may assume that the contrarian behavioural patterns start to become progressively more pronounced . the susceptibility @xmath101 at first rises pretty much linearly , which is not too surprising because of the tendency of network to break into smaller subgraphs at low population sizes . however , once the population size reaches about @xmath102 , the susceptibility starts to decay , most likely due to the main component of the network becoming more prominent , with a decay pattern that is almost piecewise linear itself , apart from fairly large fluctuations . while fairly high throughout , the homophily @xmath103 and clustering coefficients @xmath104 gradually fall as functions of population size , almost certainly due to the proliferation of contrarians . there are no great changes in other network properties , as in the average assortativity coefficient ( @xmath105 ) , although some faint systematic tendencies can be discerned , a slight rising of the average path length ( @xmath106 ) , as well as slightly decreasing average number of clusters ( @xmath107 ) . the rest of the properties per agent are nearly constant , a slight rise of the average number number of second neighbours ( @xmath108 ) , and a just perceptible decrease of the average cluster size ( @xmath109 ) and average degree ( @xmath110 ) . a way to illustrate the homophily of the system is to make a scatterplot of the ranking parameters of linked agents . as it is seen in fig.[fig : scatter2 ] , the correlation turns out to be very homophilous , as the ranking parameters of linked agents correspond very closely to one other . the emergence of the contrarians is also clearly seen : for @xmath98 , the percentage of contrarians in 10 realisations is 5.4% , while for @xmath111 is 13.8% , for @xmath112 is 24.9% and for one realisation in in a network of 1000 agents is 38.5% . there is also a clear decreasing linear trend due to the contrarians . the rising trend is without doubt caused by the normal agents , who tend to associate with agents of similar rank , and the decreasing trend is likewise due to the contrarians . both trends have a similar tendency to form square - like patterns along the diagonals , with each `` square '' corresponding to some of the many visible communities of the graphs . = 0.85 + as explained above , it is necessary to check whether the correspondence of the rankings is truly homophilic . in table [ tab:1 ] we show the value of the homophily coefficient of normal and contrarian agents for networks of various sizes . the data were taken from 10 different numerical realisations in each case . observe that normal agents have values very near one , and contrarians are around one half . also in the table we show the slope of the regression of @xmath32 vs. @xmath113 . normal agents are very close to one , indicating high degree of homophily , and the contrarians are negative and around 0.6 , indicating that they mostly form connection with agents that have opposite signs of the ranking pressure , and are much less homophilic . .homophily measurement from 10 numerical realisations of networks with different sizes . [ cols="<,^,^,^,^,^,^,^,^ " , ] from condition([expla4 ] ) it follows that the largest community in @xmath114 comprises of those agents with ranking parameters over @xmath115 , which means that the community will have approximately @xmath116 members , when one takes into account the fact that the ranking parameters are uniformly distributed . the second largest community , likewise , consists of those agents with ranking parameters between @xmath117 and @xmath118 , and has about @xmath119 members . the nth ( @xmath120 ) largest group will have ranking parameters between @xmath121 and @xmath122 , and have @xmath123 members . while ranking parameters naturally vary from agent to agent within the communities , the average value of the ranking parameters of each group @xmath124 falls approximately to the middle point of each ranking range due to the uniform distribution of the parameters : @xmath125 from figs . [ fig : scatter2 ] and [ fig : p1000 ] we see that only a maximum of four to five of these communities exist in practice at any one time , so we limit our approximation to these groups . by assuming the groups to be fully connected , as they seem to be in the graphs , and approximating sums of the products of the degrees and ranking parameters with the products of their average values we get @xmath126 where @xmath127 denotes the average ranking parameter and @xmath128 the number of members of the @xmath129th largest group . substituting the values given in table [ t:1 ] we get @xmath130 which in turn can be inserted into ( [ cgsimple ] ) : @xmath131 finally , using the definition of @xmath49 the condition ( [ cgsimple2 ] ) can be written in the form @xmath132 where @xmath133 . from condition ( [ cgsimple3 ] ) we can see that the probability of agent @xmath26 following the conventional wisdom diminishes with rising @xmath23 and decreasing @xmath60 , which is what we saw happening in our simulations judging from the results shown in the previous section . to take an example , for the @xmath99 case the largest community of agents with positive ranking pressures comprises of about @xmath134 agents . this means , according to the inequality ( [ cgsimple3 ] ) , that the agents with @xmath135 could definitively be expected to always choose to have positive opinion variables . from fig . [ fig : scatter2 ] we can tell that the real threshold is closer to @xmath136 , which is , however , in remarkably good agreement with the approximate value of @xmath137 when one takes into the consideration the fact that the appearance of the contrarians themselves was not taken into account in the derivation of ( [ cgsimple3 ] ) , and that for @xmath99 they are already very prominent . if one were to derive the condition equivalent to ( [ cgsimple3 ] ) with contrarian strategies taken into account , one would need to consider the effect that the contrarians have on their neighbours total social value @xmath138 . thus , condition ( [ cgsimple3 ] ) will most likely not hold for networks with larger @xmath23 . the last question we need to address as regards to the contrarians is the fact that they are often embedded in the groups of normal agents . so why would it be status - wise beneficial for an agent using normal strategy to retain , let alone form , a link with a contrarian agent ? let us consider a situation where an agent pursues contrarian strategies in a group of normal agents . returning to inequality ( [ expla1 ] ) , we see that it is acceptable for a normal agent @xmath26 with @xmath139 and maximal @xmath140 to form a link with a contrarian agent @xmath44 with @xmath141 and minimal @xmath142 if @xmath143 and we assume that all the neighbours of agents @xmath26 and @xmath44 also have the maximal opinion parameters . the striking fact about this relation is that it is always fulfilled in this case , meaning that @xmath26 would always find formation of links with contrarians acceptable . from the point of view of agent @xmath44 , however , linking to @xmath26 is only acceptable if @xmath144 which leads to the very same result as before . i.e. @xmath44 will not form a link with @xmath26 unless @xmath145 , if @xmath26 and @xmath44 are to belong to the same group . thus the contrarian agents would behave and be treated as normal agents when forming relations , which is surprising considering that their contribution to the total social value of other agents is negative . the latter fact is demonstrated in the simulations with some of the most counterintuitive behaviours of the model , namely , relations between agents being first broken and immediately reinstated . let us use eq . ( [ cutgainij ] ) to determine , whether the agent @xmath26 from the previous calculations would benefit from cutting the link with agent @xmath44 , even when expression ( [ cnrel ] ) says that @xmath26 would also form a link with @xmath44 in the event that such a link did not exist . with the previously stated assumptions , we arrive to the following condition @xmath146 for the link to be cut . from ( [ cncut ] ) we see that unless @xmath147 , agent @xmath26 will cut its ties with the contrarians . since @xmath148 for the largest groups , the inequality ( [ cncut ] ) is likely to be true most of the time , leading to links between agents @xmath26 and @xmath44 being cut and immediately reformed repeatedly , since inequality ( [ cnrel ] ) also holds . as suggested above , we have observed this behavioural pattern in our simulations , and to some extent it can be observed in fig . [ fig:1bis ] , in which it is seen that only the connections between contrarian and normal agents @xmath26 and @xmath44 can the signs of @xmath33 and @xmath149 being of opposite signs . it should again be stressed , however , that the calculations above do not take into account the existence of more than one contrarian . having more contrarians in the system allows them to form links between each other , which has a sizeable effect on the overall structure of the network . in summary , it could be said that while conditions ( [ expla3 ] ) and ( [ cnrel2 ] ) provide surprisingly well fitting approximations as to how a given agent chooses to link with other agents , the conditions ( [ cgsimple3 ] ) , ( [ cnrel ] ) and ( [ cncut ] ) ( though pointing to the right direction ) only give vague qualitative explanations for the behaviour of contrarians and can not be expected to yield precise numerical predictions . the interpretation of the ranking parameter @xmath32 serves as the key to find possible parallels between our model and the real world . as it describes a single property of an agent , the links between the agents only correspond to exchanges of opinion on whether the agents with larger @xmath32 are `` better '' than the agents with smaller @xmath32 , or vice versa . agents could be considered as being embedded a larger social context , and in this context they could , in principle , have other social connections . in this case , the results presented in the previous section are best interpreted in terms of echo chambers , which means that agents prefer such a social hierarchy in which they have better relative rank , and seek to communicate their opinion to others . the agents whose ranking parameters are further away from the average , are more vocal in broadcasting their views and gather supporters , since they rank highly in their chosen hierarchy and , therefore , would benefit from their hierarchy becoming more widely accepted . on the other , the agents with average ranking parameters are much more reluctant to take part in the conversation at all , since they do not rank highly in either of the hierarchies . then the end result for small system sizes is that agents divide themselves according to their ranking pressure into two or more distinct communities supporting opposing hierarchies , in which the agents with similar rankings lump together and refuse to communicate with those that disagree . it is the shutting out of the opposing point of view that makes this system s behaviour reminiscent of echo chambers found in reality . however , with increasing system size the agents develop more nuanced positions on their preferred hierarchies due to mounting social competition , as is seen in the emergence of the contrarians . there is , however , an alternative way to interpret the ranking parameter . it could be taken to represent an aggregate of all the social properties of an agent , thereby representing its total standing in the societal status measures . in this case the connections could represent the totality of the agents social interactions , and the opinion variables the agents attitude to the ( current ) state society at large . with this interpretation the rupture between the different communities observed for smaller system sizes would actually represent a real disintegration of the society . this might have implications concerning early human migrations , as they could easily have been influenced by social pressures as well as material needs . if the environmental pressures define a minimum group size necessary for a comfortable life for a tribe , and this minimum is smaller than the limit at which the tribe is forced to be adopting more advanced strategies to enhance social stability , as exemplified by the contrarians in our simulations , the tribe may well split , with splinter groups migrating elsewhere . other than the different economic games , bth can also shed some light into the well known paradox of value , also known as diamond - water paradox , which refers to the fact that diamonds are far more valued in monetary terms than water , even though water is necessary for life and diamonds not . from the bth view point the solution to this paradox is obvious : water , being necessary requirement for life , has to be available in sufficient quantities to all living humans , which means that owning water or its source does not set an individual apart from others , that is , an individual can not really compare favorably to others on grounds of having water . diamonds , on the other hand , are relatively rare , and thus can not be owned by everyone . therefore , an individual possessing diamonds is compared favourably to others , and so diamonds acquire a relatively high value in comparison to water in the minds of humans , in a very similar manner with which the veblen goods become valuable . then bth , in a sense , contains in itself a natural definition of value , although further work is needed to determine how exactly this status - value relates to other forms of value , such as value derived from usefulness or necessity . in section [ model ] we only analysed the behaviour of the accepter , since this is straightforward in comparison to predicting the behaviour of the proposer . the experiments on the ultimatum game often find that the proposers tend to offer fair shares to accepters , which is easily explained in the context of bth by the desire of the proposer to have the proposal accepted : the proposers only offer shares that they would accept themselves , and in this way eq.([accepted ] ) also restricts the proposers offers , although it can not tell the exact amount of money offered . to be able to give a better estimate for the offers one would need to study the learning processes that shape the proposers experience on how uneven treatment people are usually willing to tolerate . this is , however , outside the scope of this paper . the behaviour of dictators in the dictator game @xcite is somewhat more difficult to analyse using bth . the dictator game is similar to the ultimatum game , the only difference being that the other player does not even get to make a choice , and only receives what the first player , or dictator , endows . it has been observed that @xcite in this game the dictators tend to be rather generous , which is difficult but not impossible to explain in the context of bth , if one takes into account the effect of reputation and other `` social goods '' . the nature of such influence on the behaviour of the dictator will be studied in a later work . however , there are some indications that bth could very well be applied to the dictator game when all the social effects are taken into account . it has been reported @xcite that when the rules of the dictator game are modified so that instead of giving money to the other players , the dictator gets to take some or all of the money given to the other players ( thus turning the game into a `` taking game '' ) , the dictator s behaviour changes from egalitarian to self serving , i.e. taking often the majority or even all of the available money . from the bth point of view , the dictator s observed behaviour change can potentially be explained in terms of social norms . in the ordinary dictator game the dictator may still feel bound by the usual norms of the society , while in the `` taking game '' it is encouraged to go against these norms . this sets the `` taker '' apart from the other player in particular , and other members of the society in general . hence the dictator feels `` better '' than the others when breaking the norms with impunity , and act on this feeling by taking money from the other players . the fact that bth can possibly lead to formation of norms as well as rebellion against these norms is well worth of further studies . relating to the known results of the ultimatum game , we have formulated a hypothesis explaining the observed behaviour of humans in terms of superiority maximization , or `` better than''-hypothesis , and presented a simple agent - based model to implement this hypothesis . the model describes agents with constant ranking parameters and raises the question whether the agents with larger ranks are `` better '' than agents with smaller ranks or the other way around . we have found that the social system produced by our model , features homophily , meaning that agents forming social ties with other agents with similar ranking parameters , and assortativity , describing the tendency of highly / lowly connected agents forming links with other highly / lowly connected agents . in addition we find community formation , both in terms of there being communities with opposing opinions and in terms of the communities with the same opinion fracturing into smaller ones according to their ranking parameters . furthermore , we have observed the formation of a hierarchy , in the sense of a connectivity hierarchy being formed on top of the one defined by the ranking parameters , with the agents with extreme ranking parameters presenting higher connectivity than the agents with average ranking parameters . moreover , we have found that the resulting social networks tend to be disconnected for small system sizes , but mostly connected for larger system sizes . this fact may have some relevance for research of early human migrations , hinting of the effects of social pressure in shaping the social network . j.e.s . acknowledges financial support from niilo helander s foundation , g.i . acknowledges a visiting fellowship from the aalto science institute , and k.k . acknowledges financial support by the academy of finland research project ( cosdyn ) no . 276439 and eu horizon 2020 fet open ria project ( ibsen ) no . r.a.b . wants to thank aalto university for kind hospitality during the development of this work . rab acknowledges financial support from conacyt through project 799616 . we acknowledge the computational resources provided by the aalto science - it project . fagundes , m. s. , ossowski , s. , cerquides , j. & noriega , p. ( 2016 ) design and evaluation of norm - aware agents based on normative markov decision processes . _ international journal of approximate reasoning _ , * 78 * , 3361 . j. , h. , boyd , r. , bowles , s. , camerer , c. , fehr , e. & gintis , h. ( 2004 ) _ foundations of human sociality : economic experiments and ethnographic evidence from fifteen small - scale societies_. oxford university press

in human societies , people s willingness to compete and strive for better social status as well as being envious of those perceived in some way superior lead to social structures that are intrinsically hierarchical . here we propose an agent - based , network model to mimic the ranking behaviour of individuals and its possible repercussions in human society . the main ingredient of the model is the assumption that the relevant feature of social interactions is each individual s keenness to maximise his or her status relative to others . the social networks produced by the model are homophilous and assortative , as frequently observed in human communities and most of the network properties seem quite independent of its size . however , it is seen that for small number of agents the resulting network consists of disjoint weakly connected communities while being highly assortative and homophilic . on the other hand larger networks turn out to be more cohesive with larger communities but less homophilic . we find that the reason for these changes is that larger network size allows agents to use new strategies for maximizing their social status allowing for more diverse links between them . community formation , opinion formation , social hierarchy

You are an expert at summarizing long articles. Proceed to summarize the following text: the bilayer heisenberg antiferromagnet , @xmath4 where the intralayer couplings run over nearest neighbors on square lattices and the @xmath5 are @xmath3 quantum spins , has turned out to be an excellent testing ground for notions of quantum criticality . a variety of controlled calculations , such as strong - coupling expansions at zero temperature ( starting with the work of hida@xcite and extended by two of the present authors@xcite ) , finite - temperature quantum monte carlo calculations@xcite and high - temperature expansions@xcite fit extremely well into the general field - theoretic framework for zero - temperature order - disorder critical points in unfrustrated two - dimensional quantum antiferromagnets.@xcite the critical point in the @xmath3 bilayer heisenberg model is between a nel ordered phase when @xmath6 is sufficiently large ( greater than the critical value @xmath7 ) and a magnetically disordered `` dimer '' phase which is the adiabatic continuation of the @xmath8 ground state ( which is a product of interlayer singlet pair wavefunctions ) . the universality class of the phase transition is the same as for finite - temperature transitions in @xmath9 , @xmath10 symmetric models . it seemed most natural to us that for larger values of the quantum spin @xmath0 the same scenario would play out : the value of @xmath11 would change , but qualitatively nothing would be different . it therefore came as a surprise when ng , zhang , and ma@xcite described the results of schwinger boson mean - field ( sbmf ) calculations , which indicate that for @xmath12 the transition between magnetically ordered and disordered phases is _ first - order_. the calculation leads to @xmath13 , and hence indicates that for all physical values of @xmath0 the transition should be first - order ; but it is reasonable to accept the suggestion of ng _ et al . _ that sbmf theory leads to phase diagram of @xmath0 versus @xmath14 which is qualitatively valid even if @xmath15 is underestimated . two more salient points regarding the sbmf results should be noted . at large @xmath0 , the value of @xmath14 at the dimer - nel phase boundary scales as @xmath16 ; and at sufficiently large @xmath0 the ground state on the dimer side of the boundary is the pure interlayer singlet pair state ( _ i.e. , _ identical to the ground state at @xmath8 ) . here we will argue that the results of sbmf theory for phase transitions in the bilayer heisenberg model are qualitatively incorrect , even in the limit of large @xmath0 where one might expect such an approximation to be reliable . ng _ et al . _ offer an elementary argument in support of their proposed scenario , which it is informative to reconsider . they note that the ideal nel state and the interlayer singlet state become degenerate for @xmath17 ( with the interlayer singlet state energetically favored for @xmath18 ) , while linear spin - wave theory indicates that the magnetization should vanish at @xmath19 ( with nel order existing for @xmath20 ) . at large @xmath0 , it is clear that @xmath21 . _ suggest that , on decreasing @xmath14 from infinity , the transition that one might suspect should take place at @xmath22 ( presumably a continuous transition from the nel to dimer phase ) is pre - empted by a first - order transition at @xmath23 . this is certainly consistent with their sbmf calculations . however , we suggest an alternative interpretation . we believe that the argument in favor of a first - order transition at @xmath23 is too simplistic : using simple trial states to represent the ground states in the respective phases is dangerous , particularly since it is a comparison of subleading ( in powers of @xmath0 ) terms in the variational energies for each phase which leads to the estimate for @xmath23 . furthermore , there is an elementary calculation which strongly suggests that the dimer phase is unstable with respect to the development of nel order for @xmath24 . consider a single dimer , subject to a staggered magnetic field which is produced self - consistently by the staggered magnetization of the system . let @xmath25 denote the staggered susceptibility of a single dimer . then , within this `` dimer mean - field approximation '' one has @xmath26 . one can calculate @xmath27 directly from second - order perturbation theory , based on explicit formulae for 3-@xmath28 symbols,@xcite to find that @xmath29 . as promised , the domain of stability of the dimer phase is @xmath30 , in agreement with the spin - wave estimate for the stability of the nel phase . it should be clear that the dimer mean - field approximation underestimates the stability of the dimer phase , just as the standard weiss theory overestimates curie temperatures . one could reasonably object to our argument , then , by suggesting that the corrections to this mean - field theory could be large : in particular , @xmath31 might be pushed up to a value of order @xmath16 . in the following section , we describe high - order perturbation expansions about @xmath8 for bilayers with @xmath32 . combined with the existing results for @xmath3 , they strongly suggest that the corrections to dimer mean - field theory are order unity rather than order @xmath0 . in addition , for @xmath2 bilayers we present a wide variety of results based on expansions about @xmath8 and about ising models . they are all qualitatively similar to what is found for @xmath3 bilayers , providing additional confirmation that the phase diagram is unchanged with increasing @xmath0 ( in particular , there is no evidence for any phase besides dimer or nel ) and offering accurate results which could prove useful in interpreting data from experimental realizations of @xmath2 bilayers . perturbation expansions@xcite for the ground - state energy @xmath33 , the triplet excitation spectrum , and the antiferromagnetic susceptibility @xmath34 have been carried out for bilayers with all integer and half - integer values of @xmath0 from 1 to 4 . the series coefficients for integer @xmath0 bilayers for the minimum gap @xmath35 , corresponding to wave vector @xmath36 , and @xmath34 are presented in table [ table : gaps ] . for @xmath3 bilayers such expansions have already been presented by zheng@xcite to @xmath37 . [ gapapprox ] is a set of `` scatter plots '' of estimated critical @xmath14 values versus estimated critical exponents derived from unbiased @xmath38log - pad approximants to the gap series for @xmath2 , 2 , 3 and 4 bilayers . an analogous plot for @xmath3 is presented in ref . . in every case they have very similar character , with nearly all exponent estimates lying slightly above the anticipated value of 0.71 , and with clear correlations between critical point and exponent values . ( scatter plots for the antiferromagnetic susceptibility have very similar character ; likewise for the gap and susceptibility of half - integer @xmath0 bilayers . ) as @xmath0 increases the series actually become better behaved , in that the approximants derived from a fixed number of terms are more tightly clustered ( neglecting the outliers ) and estimated exponents are closer to the expected value . biasing the exponent one can obtain more precise estimates of the critical points @xmath11 , as listed in table [ table : xc ] . the trend is clear : with increasing @xmath0 , the critical value of @xmath14 is approaching a constant multiple of the dimer mean - field value @xmath39 . if we plot @xmath40 as a function of @xmath41 ( shown in fig . [ lambda_c ] ) , we can see that the results can be remarkably well fitted by a straight line @xmath42 with an intercept at @xmath43 slightly above @xmath44 . the constants @xmath45 and @xmath46 determined by a linear least squares fit are : @xmath47 for general @xmath0 we have found empirically , based on the ground state energy @xmath33 , the minimum triplet gap @xmath35 , and the antiferromagnetic susceptibility @xmath34 series for the eight values of @xmath0 that have been explicitly calculated , that the coefficients of @xmath48 can be expressed as polynomials of order @xmath49 in the variable @xmath50 . for example , to third order @xmath51\lambda^2 \label{delta}\\ & & + [ ( 56/135)r-(116/135)r^2-(3824/1215)r^3]\lambda^3\ .\nonumber\end{aligned}\ ] ] further coefficients are listed in the appendix . one can then define @xmath52 and consider the @xmath53 , @xmath54 limit . the resulting series in @xmath55 corresponds to the terms in the double - series ( in @xmath50 and @xmath14 ) of the form @xmath56 . its @xmath38log - pad approximants are as well behaved as the series for the larger values of @xmath0 shown above , with unbiased approximants clustering and exponent estimates just above the anticipated value . biasing the critical point estimates leads to a critical @xmath55 of 0.2691(7 ) ( this is consistent with the value of @xmath45 in eq . ( [ r_c ] ) ) , so the ratio of the exact critical point value in the large-@xmath0 limit to that coming from the dimer mean - field theory is 1.435(4 ) . we can also try to use the double - expansions in @xmath50 and @xmath14 to analytically continue to unphysically small values of @xmath0 . an interesting feature of the general-@xmath0 double - series ( both for the @xmath35 and @xmath34 ) is that all terms of the form @xmath57 appear to have vanishing coefficients . consequently , for @xmath58 there is no phase transition to a nel - ordered state . presumably there is a critical spin @xmath15 , less than @xmath59 , at and below which the dimer state is stable for all values of @xmath14 . it has not been possible to obtain a reliable estimate of @xmath15 by constructing @xmath38log - pad approximants at fixed values of @xmath0 : for @xmath60 the approximants depend strongly on the number of terms used . however , the extrapolation from larger @xmath0 , summarized in eqs . ( [ fit_lambdac ] ) and ( [ r_c ] ) , suggests that @xmath61 . although we have presented evidence above that the dimer phase is unstable for @xmath14 larger than a critical value of @xmath30 , we did not firmly establish that the structure of the phase diagram is simply dimer phase critical point nel phase . the fact that the unbiased exponent values are consistent with a lorentz - invariant , @xmath9 , @xmath10 universality class is certainly suggestive . for the @xmath2 bilayers we have done much more . in addition to expansions about @xmath8 we have constructed expansions about ising models , generating a set of results analogous to those presented by zheng@xcite for @xmath3 bilayers . taken together , these leave little room for doubt about the phase diagram . because the calculations follow those presented in ref . so closely we refer the reader to sec . iii of that paper for a description of the calculation ( note that the parameter @xmath63 in that paper corresponds to @xmath64 ) . expansions were obtained to @xmath65 ( with @xmath66 denoting the ratio of transverse to longitudinal exchange strength ) for the sublattice magnetization @xmath67 , uniform transverse susceptibility @xmath68 , and triplet excitation spectrum . from these we could derive the gap to the optical branch of the spin - wave spectrum at wave vectors @xmath69 ( @xmath70 , the minimum gap ) and @xmath71 ( @xmath72 ) , spin - wave velocity @xmath73 , and spin - wave stiffness @xmath74 . the expansion coefficients will not be presented here but are available from the authors on request . results of series extrapolations are shown in figs . [ optical_gap ] , [ m_chi ] , and [ c_rho ] . they demonstrate that expansions about the nel state lead to estimates of the domain of stability of the nel phase which are completely consistent with the critical point found by expanding about @xmath8 . this is nicely confirmed by fig . [ e0 ] , which displays the ground - state energy per site . the values from the dimer expansion match very smoothly on to those from the ising expansion around the critical point , whereas for a first - order transition there would be a discontinuity in slope at the transition . all of the results presented in this subsection represent the best available theoretical values for experimentally accessible properties of nel - ordered @xmath2 bilayers , which should prove useful in the interpretation of experimental data if any such systems are studied . ( in real compounds single - ion anisotropy is always present to some degree . the present calculations could readily be extended to include such terms in the hamiltonian . ) finally , it is amusing to note that in the dimer phase for @xmath75 bilayers , sufficiently close to @xmath8 , there exist spin-2 elementary excitations . let us consider @xmath2 bilayers specifically . triplet spectra for various values of @xmath14 in the dimer phase are shown in fig . [ mk_triplet ] while _ quintuplet _ spectra are shown in fig . [ mk_quintuplet ] . ( these spectra are obtained by direct sums of the terms in the series to the maximum order available . ) the latter spectra lose physical significance when the quintuplet excitations can decay into multiple triplet excitations . one might imagine this happens for arbitrarily small values of @xmath14 , since at @xmath8 the quintuplet gap is three times the triplet gap . however , the parity , with respect to interchange of layers in the bilayer system , of the quintuplet excitations is opposite to that of the triplet excitations , and symmetry forbids decay of the quintuplet excitations into an odd number of triplet excitations . hence stable spin-2 excitations lie between the 2-triplet and 4-triplet continua for sufficiently small @xmath14 . the order - disorder transition in bilayer heisenberg antiferromagnets appears to be a problem for which schwinger boson mean - field theory leads to qualitatively incorrect results even in the limit of large @xmath0 . we have shown by high order perturbation expansions about the limit of uncoupled interlayer singlets that there is strong evidence for continuous transitions between dimer and nel phases with critical values of @xmath76 scaling as @xmath77 . there seems to be no evidence in favor of the scenario described by ng _ et al._,@xcite in which the transition would become first - order for sufficiently large @xmath0 . of course this does not imply that the phase diagram for bilayer heisenberg antiferromagnets is completely universal . if further - neighbor interactions or higher - order exchange interactions ( for example , terms in the hamiltonian of the form @xmath78 ) were allowed then a wide variety of new phases could be stabilized . however , for the simplest bilayer models it appears that simple arguments based on the instability of the dimer phase to nel order ( using dimer mean - field theory ) and the instability of the nel phase to a spin - disordered phase ( using linear spin - wave theory to determine where the staggered magnetization vanishes ) , which both suggest a critical point at @xmath79 , are correct . the double expansions in @xmath14 and @xmath41 for the ground - state energy per site @xmath80 , triplet minimum energy gap @xmath35 , and antiferromagnetic susceptibility @xmath34 for spin-@xmath0 bilayers are presented in full here . @xmath81\lambda^2\\ % & & + [ ( 56/135)r-(116/135)r^2-(3824/1215)r^3]\lambda^3+\cdots\ .\nonumber \chi & & = { 4r\over 3 } { \big [ } 1 + { { 16\,\lambda \,r}\over 3 } + { { { { \lambda } ^2}\,r\,\left ( -1 + 140\,r \right ) } \over 6 } \nonumber \\ & & + { { \lambda } ^3}\,r\,\left ( -{1\over 8 } - { { 143\,r}\over { 36 } } + { { 2677\,{r^2}}\over { 27 } } \right ) \nonumber \\ & & + { { \lambda } ^4}\,r\ , { \big ( } -0.10671296 - 1.45847442\,r \nonumber \\ & & - 33.2428954\,{r^2 } + 403.712863\,{r^3 } { \big ) } \nonumber \\ & & + { { \lambda } ^5}\,r\ , { \big ( } -0.090168701 - 0.58931360\,r \nonumber \\ & & - 3.49146301\,{r^2 } - 221.1724119\,{r^3 } + 1622.218477\,{r^4 } { \big ) } \nonumber \\ & & + \ ! { { \lambda } ^6}r { \big ( } \ ! -0.076292486 + 0.12455955 r + 2.92215451 { r^2 } \nonumber \\ & & + 8.15178055\,{r^3 } - 1231.683801\,{r^4 } + 6416.371180\,{r^5 } { \big ) } \nonumber \\ & & + o(\lambda^7 ) { \big ] } \end{aligned}\ ] ] k .- k . ng , f. c. zhang , and m. ma , phys . b * 53 * , 12196 ( 1996 ) . m. rotenberg , r. bivins , n. metropolis , and john k. wooten , jr . , _ the 3-j and 6-j symbols _ ( cambridge , massachusetts : the technology press , 1959 ) . see especially pp . 4 , 12 .

spin-@xmath0 bilayer heisenberg models ( nearest - neighbor square lattice antiferromagnets in each layer , with antiferromagnetic interlayer couplings ) are treated using dimer mean - field theory for general @xmath0 and high - order expansions about the dimer limit for @xmath1 . we suggest that the transition between the dimer phase at weak intraplane coupling and the nel phase at strong intraplane coupling is continuous for all @xmath0 , contrary to a recent suggestion based on schwinger boson mean - field theory . we also present results for @xmath2 layers based on expansions about the ising limit : in every respect the @xmath2 bilayers appear to behave like @xmath3 bilayers , further supporting our picture for the nature of the order - disorder phase transition .

You are an expert at summarizing long articles. Proceed to summarize the following text: theoretical descriptions of photonic crystal fibers ( pcfs ) have traditionally been restricted to numerical evaluation of maxwell s equations . in the most general case , a plane wave expansion method with periodic boundary conditions is employed @xcite while other methods , such as the multipole method @xcite , take advantage of the localized nature of the guided modes and to some extend the circular shape of the air - holes . the reason for the application of these methods is the relatively complex dielectric cross section of a pcf for which rotational symmetry is absent . the aim of this work is to provide a set of numerically based empirical expressions describing the basic properties such as cutoff and mode - field radius of a pcf based on the fundamental geometrical parameters only . we consider the fiber structure first studied by knight _ _ @xcite and restrict our study to fibers that consist of pure silica with a refractive index of 1.444 . the air holes of diameter @xmath1 are arranged on a triangular grid with a pitch , @xmath2 . in the center an air hole is omitted creating a central high index defect serving as the fiber core . a schematic drawing of such a structure is shown in the inset of the right panel in fig . [ fig1 ] . depending on the dimensions , the structure comprises both single- and multi - mode fibers with large mode area as well as nonlinear fibers . the results presented here cover relative air hole sizes , @xmath3 , from 0.2 to 0.9 and normalized wavelengths , @xmath4 , from around 0.05 to 2 . the modeling is based on the plane - wave expansion method with periodic boundary conditions @xcite . for the calculations of guided modes presented the size of the super cell was @xmath5 resolved by @xmath6 plane waves while for calculations on the cladding structure only , the super cell was reduced to a simple cell resolved by @xmath7 planes waves . when attempting to establish a simple formalism for the pcf it is natural to strive for a result similar to the @xmath0parameter known from standard fibers @xcite . however , a simple translation is not straight forward since no wavelength - independent core- or cladding index can be defined . recently , we instead proposed a formulation of the @xmath0parameter for a pcf given by @xcite @xmath8 although this expression has the same overall mathematical form as known from standard fibers , the unique nature of the pcf is taken into account . in eq . ( [ vpcf ] ) , @xmath9 is the wavelength dependent effective index of the fundamental mode ( fm ) and @xmath10 is the corresponding effective index of the first cladding mode in the infinite periodic cladding structure often denoted the fundamental space filling mode ( fsm ) . for a more detailed discussion of this expression and its relation to previous work we refer to ref . @xcite and references therein . we have recently argued that the higher - order mode cut - off can be associated with a value of @xmath11 @xcite and showed that this criterion is indeed identical to the single - mode boundary calculated from the multipole method @xcite . recently the cut off results have also been confirmed experimentally @xcite . further supporting the definition of @xmath12 is the recent observation @xcite that the relative equivalent mode field radius of the fundamental mode , @xmath13 as function of @xmath12 fold over a single curve independent of @xmath3 . the mode field radius @xmath14 is defined as @xmath15 and corresponds to the @xmath16 width of a gaussian intensity distribution with the same effective area , @xmath17 , as the fundamental mode itself @xcite . in the left panel of fig . [ fig1 ] , calculated curves of @xmath12 as function of @xmath18 are shown for @xmath3 ranging from 0.20 to 0.70 in steps of 0.05 . in general , all curves are seen to approach constant levels dependent on @xmath3 . the horizontal dashed line indicates the single - mode boundary @xmath11 . in the right panel , @xmath13 is plotted as function of @xmath12 for each of the 9 curves in the left panel and as seen all curves fold over a single curve . an empirical expression for @xmath13 can be found in ref . the mode is seen to expand rapidly for small values of @xmath12 and the mode - field radius saturates toward a constant value when @xmath12 becomes large . in fact , it turns out that @xmath19 for @xmath20 and @xmath21 for @xmath11 . in the left panel of fig . [ fig2 ] , curves corresponding to constant values of @xmath12 are shown in a @xmath4 versus @xmath3 plot . in the right panel , curves of constant @xmath13 is shown , also in a @xmath4 versus @xmath3 plot . since there is a unique relation between @xmath13 and @xmath12 @xcite the curves naturally have the same shape . when designing a pcf any combination of @xmath1 and @xmath2 is in principle possible . however , in some cases the guiding will be weak causing the mode to expand beyond the core and into the cladding region @xcite corresponding to a low value of @xmath12 . in the other extreme , the confinement will be too strong allowing for the guiding of higher - order modes @xcite . since both situations are governed by @xmath12 the design relevant region in a @xmath4 versus @xmath3 plot can be defined . this is done in fig . [ fig3 ] where the low limit is chosen to be @xmath22 where @xmath23 . how large a mode that can be tolerated is of course not unambiguous . however , for @xmath24 leakage - loss typically becomes a potential problem in pcfs with a finite cladding structure . in non - linear pcfs it is for dispersion reasons often advantageous operating the pcf at @xmath25 and then a high number of air - hole rings is needed to achieve an acceptable level of leakage loss @xcite . finally , we note that the practical operational regime is also limited from the low wavelength side . in ref . @xcite a low - loss criterion was formulated in terms of the coupling length @xmath26 $ ] between the fm and the fsm . in general scattering - loss due to longitudinal non - uniformities increases when @xmath27 increases and a pcf with a low @xmath27 will in general be more stable compared to one with a larger @xmath27 . using @xmath28 we can rewrite eq . ( [ vpcf ] ) as @xmath29 from which it is seen that a high value of the @xmath0parameter is preferred over a smaller value . in fig . ( [ fig3 ] ) it is thus preferable to stay close to the single - mode boundary ( @xmath30 ) but in general there is a practical lower limit to the value of @xmath4 which can be realized because when @xmath31 one generally has that @xmath32 @xcite . although the @xmath0parameter offers a simple way to design a pcf , a limiting factor for using eq . ( [ vpcf ] ) is that a numerical method is still required for obtaining the effective indices . in analogy with expressions for standard fibers @xcite it would therefore be convenient to have an alternative expression only dependent on the wavelength , @xmath33 , and the structural parameters @xmath1 and @xmath2 . in fig . [ fig4 ] , we show @xmath12 as function of @xmath4 ( data are shown by open circles ) for @xmath3 ranging from 0.20 to 0.80 in steps of 0.05 . each data set in fig . [ fig4 ] is fitted to a function of the form [ vpcf_fit ] @xmath34 + 1}\ ] ] and the result is indicated by the full red lines . ( [ vpcf_fit_v ] ) is not based on considerations of the physics of the v - parameter but merely obtained by trial and error in order to obtain the best representation of calculated data with the lowest possible number of free parameters . prior to the fit , the data sets are truncated at @xmath35 since @xmath36 in this region ( see left panel in fig . [ fig1 ] ) and the data is thus not practically relevant . in eq . ( [ vpcf_fit_v ] ) the fitting parameters @xmath37 , @xmath38 , and @xmath39 depend on @xmath3 only . in order to extract this dependency , suitable functions ( again obtained by trial and error ) are fitted to the data sets for @xmath37 , @xmath38 , and @xmath39 . we find that the data are well described by the following expressions @xmath40 @xmath41 @xmath42 the above set of expressions , eqs . ( [ vpcf_fit ] ) , constitute our empirical expression for the @xmath0parameter in a pcf with @xmath4 and @xmath3 being the only input parameters . for @xmath43 and @xmath44 the expression gives values of @xmath12 which deviates less than @xmath45 from the correct values obtained from eq . ( [ vpcf ] ) . the term endlessly single - mode ( esm ) refers to pcfs which regardless of wavelength only support the two degenerate polarization states of the fundamental mode @xcite . in the framework of the @xmath0parameter this corresponds to structures for which @xmath46 for any @xmath4 @xcite . as seen in the left panel of fig . [ fig1 ] this corresponds to sufficiently small air holes . however , from the plot in fig . [ fig1 ] it is quite difficult to determine the exact @xmath3 value for which @xmath11 for @xmath33 approaching 0 . from eq . ( [ vpcf_fit ] ) it is easily seen that the value may be obtained from @xmath47 fig . [ fig5 ] illustrates this equation graphically where we have extrapolated the data in fig . [ fig4 ] to @xmath48 . from the intersection of the full line with the dashed line we find that @xmath49 bounds the esm regime . solving eq . ( [ d_esm ] ) we get @xmath50 and the deviation from the numerically obtained value is within the accuracy of the empirical expression . there are several issues to consider when designing a pcf . in this work we have addressed the single / multi - mode issue as well as those related to mode - field radius / field - confinement , and mode - spacing . we have shown how these properties can be quantified via the @xmath0parameter . based on extensive numerics we have established an empirical expression which facilitate an easy evaluation of the @xmath0-parameter with the normalized wavelength and hole - size as the only input parameters . we believe that this expression provides a major step away from the need of heavy numerical computations in design of solid core pcfs with triangular air - hole cladding . we thank j. r. folkenberg for stimulating discussion and m. d. nielsen acknowledges financial support by the danish academy of technical sciences .

based on a recent formulation of the @xmath0parameter of a photonic crystal fiber we provide numerically based empirical expressions for this quantity only dependent on the two structural parameters the air hole diameter and the hole - to - hole center spacing . based on the unique relation between the @xmath0parameter and the equivalent mode field radius we identify how the parameter space for these fibers is restricted in order for the fibers to remain single mode while still having a guided mode confined to the core region . 10 s. g. johnson and j. d. joannopoulos , `` block - iterative frequency - domain methods for axwell s equations in a planewave basis , '' opt . express * 8 * , 173 ( 2001 ) , + http://www.opticsexpress.org/abstract.cfm?uri=opex-8-3-173 . t. p. white , b. t. kuhlmey , r. c. mcphedran , d. maystre , g. renversez , c. m. de sterke , and l. c. botton , `` multipole method for microstructured optical fibers . i. formulation , '' j. opt . soc . am . b * 19 * , 2322 ( 2002 ) . j. c. knight , t. a. birks , p. s. j. russell , and d. m. atkin , `` all - silica single - mode optical fiber with photonic crystal cladding , '' opt . lett . * 21 * , 1547 ( 1996 ) . a. w. snyder and j. d. love , _ optical waveguide theory _ ( chapman & hall , new york , 1983 ) . d. marcuse , `` gaussian approximation of the fundamental modes of graded - index fibers , '' j. opt . . am . * 68 * , 103 ( 1978 ) . n. a. mortensen , j. r. folkenberg , m. d. nielsen , and k. p. hansen , `` modal cut - off and the @xmath0parameter in photonic crystal fibers , '' opt . lett . * 28 * , 1879 ( 2003 ) . b. t. kuhlmey , r. c. mcphedran , and c. m. de sterke , `` modal cutoff in microstructured optical fibers , '' opt . lett . * 27 * , 1684 ( 2002 ) . j. r. folkenberg , n. a. mortensen , k. p. hansen , t. p. hansen , h. r. simonsen , and c. jakobsen , `` experimental investigation of cut - off phenomena in non - linear photonic crystal fibers , '' opt . lett . * 28 * , 1882 ( 2003 ) . m. d. nielsen , n. a. mortensen , j. r. folkenberg , and a. bjarklev , `` mode - field radius of photonic crystal fibers expressed by the @xmath0parameter , '' opt . lett . * 28 * , in press ( 2003 ) , + http://arxiv.org/abs/physics/0309030 . t. p. white , r. c. mcphedran , c. m. de sterke , l. c. botton , and m. j. steel , `` confinement losses in microstructured optical fibers , '' opt . lett . * 26 * , 1660 ( 2001 ) . b. t. kuhlmey , r. c. mcphedran , c. m. de sterke , p. a. robinson , g. renversez , and d. maystre , `` microstructured optical fibers : where s the edge ? , '' opt . express * 10 * , 1285 ( 2002 ) , + http://www.opticsexpress.org/abstract.cfm?uri=opex-10-22-1285 . w. h. reeves , j. c. knight , p. s. j. russell , and p. j. roberts , `` demonstration of ultra - flattened dispersion in photonic crystal fibers , '' opt . express * 10 * , 609 ( 2002 ) , + http://www.opticsexpress.org/abstract.cfm?uri=opex-10-14-609 . n. a. mortensen and j. r. folkenberg , `` low - loss criterion and effective area considerations for photonic crystal fibers , '' j. opt . a : pure appl . opt . * 5 * , 163 ( 2003 ) . t. a. birks , j. c. knight , and p. s. j. russell , `` endlessly single mode photonic crystal fibre , '' opt . lett . * 22 * , 961 ( 1997 ) .

You are an expert at summarizing long articles. Proceed to summarize the following text: in continuum non abelian field theories , most popular choices of fixing the gauge ( e.g. landau , coulomb ) suffer from the gribov ambiguity @xcite . it is now well established that this problem also affects the lattice formulation of these theories @xcite-@xcite . this problem has been neglected for a long time because , in principle , the computation of gauge invariant operators in compact lattice theories does not require gauge fixing . fixing the gauge is , however , necessary in several cases . monopole studies in su(2 ) pure gauge theory have been done in the unitary gauge and the effect of the gribov ambiguity on the number of su(2 ) monopoles has been investigated @xcite . the authors conclude that , in their case , the gribov noise does not exceed the statistical uncertainty . in su(3 ) gauge theory , gauge fixing is essential in the the computation of gauge dependent quantities , such as gluon and quark propagators . there are now several studies of lattice propagators . the gluon propagator has been calculated in @xcite-@xcite with the aim of studying the mechanism through which the gluon may become massive at long distances . more recent attempts have investigated its behaviour as a function of momentum @xcite . analogous studies have also been performed on the quark propagator ( see , for example @xcite ) . in practice , there are also cases in which it is convenient to implement a gauge dependent procedure for the computation of gauge invariant quantities @xcite-@xcite . for example , smeared fermionic interpolating operators are widely being used in lattice qcd spectroscopy and phenomenology , in order to optimise the overlap of the lower - lying physical state with the operator . the point - splitted smeared operators are gauge dependent , and therefore the gauge must be fixed before they are calculated . in particular , the calculation of the decay constant of the @xmath5 meson in the static approximation , in which the @xmath6-quark has infinite mass , requires the computation of the two point correlation function of the axial current . the isolation of the lightest state at large times is not possible if local ( gauge invariant ) operators are used . a nice way out consists in smearing the bilocal operator over a small cube and extracting @xmath7 by forming suitable ratios of smeared and local correlation functions @xcite . this is an explicitly gauge dependent procedure which is most naturally carried out in the coulomb gauge . in ref.@xcite the smeared - smeared correlation functions on a few individual configurations were computed . two gribov copies were produced per configuration . the gribov noise on individual configurations was found to vary from @xmath8 to @xmath9 depending on the time - slice , which implies that it may still be a considerable effect after averaging over configurations . however , it was not possible to estimate its effect beyond individual configurations . the reason is that in such a study other sources of error dominate , such as the systematic error arising from fitting the exponential decay of the correlation function with time . thus the isolation of the gribov noise is difficult . in the static limit @xcite uses a different method for constructing ratios of smeared and local correlators which avoids fitting . this method , however , requires a large temporal extention of the lattice . ] in this paper we study a different physical quantity , namely the renormalisation constant @xmath0 of the lattice axial current . a knowledge of these renormalisation constants is necessary for matching the matrix elements computed using lattice simulations to those required in a definite continuum renormalisation scheme . provided that the lattice spacing is sufficiently small it is possible to calculate these renormalisation constants in perturbation theory . for a more reliable determination of these constants it has been suggested to impose the chiral ward identities of @xmath10 non - perturbatively @xcite . here we focus our attention on the determination of the r@xmath11le of the gribov ambiguity in the calculation of @xmath0 , obtained from quark state correlation functions . a recently proposed method to determine @xmath0 and other renormalisation constants , based on truncated quark green functions in momentum space @xcite can also in principle be afflicted by gribov fluctuations . since reasonably small errors are expected , in this kind of calculations , it is crucial to investigate the r@xmath12le of the gribov noise . moreover , the renormalisation constant @xmath0 of the axial current is particularly well suited to the study of the gribov fluctuations , mainly for two reasons . firstly , @xmath0 can be obtained from chiral ward identities in two distinct ways : a gauge independent one , which consists in taking the matrix elements between hadronic states , and a gauge dependent one , which consists in taking the matrix elements between quark states . hence , there is an explicitly gauge invariant estimate of @xmath0 which is free of gribov noise and which can be directly compared to the gauge dependent , gribov affected , estimate . the second advantage is that @xmath0 is obtained by solving a first degree algebraic equation for each lattice time slice , thus avoiding the usual systematic errors arising from fitting exponentially decaying signals in time . the theoretical framework for the non - perturbative evaluation of @xmath0 for wilson fermions , has been developed in @xcite . the renormalisation constant is obtained through ward identities generated by axial transformations . a first application of these techniques in numerical simulations using the wilson action was attempted in @xcite . the extension of these methods to the @xmath13 improved clover action @xcite ( @xmath14 is the lattice spacing ) was presented in @xcite , which we follow most closely . here we only give a brief outline of the results which are essential to our work . in this study terms that , close to the continuum limit , are effectively of @xmath13 are eliminated by using the clover action @xcite and rotating all quark fields of the matrix elements according to the `` improved improvement '' prescription of @xcite : @xmath15 ( @xmath16 is the wilson term parameter ; in this work @xmath17 ) . the two fermion local operators considered in the following are the axial and vector currents and the pseudoscalar density : @xmath18 ( @xmath19 is a flavour label and the notation is generic for any quark fields @xmath20 and @xmath21 ) . in order to ensure that the lattice axial current @xmath22 has the correct chiral properties , it is normalised by a renormalisation constant @xmath0 @xcite ; this implies that @xmath23 ( m is the bare quark mass and @xmath24 the pseudoscalar state ) . as can be seen from the above equation , @xmath25 is gauge invariant . the gauge dependent calculation of @xmath0 relies on the following ward identity for quark green functions @xcite @xmath26 \nonumber \\ = \left ( \frac{1}{z_{a}}-{\rho ra}\right ) tr \left [ \int d^3\vec y < ( \gamma_5 d(y)\bar d(0)\ , + \ , u(y)\bar u ( 0)\gamma _ 5 ) > _ { \alpha,\beta } \right ] \label{eq : wiqp}\end{aligned}\ ] ] in eq.([eq : wiqp ] ) we work explicitly with up and down quark fields with spinor labels @xmath27 and @xmath28 . the trace is over colour indices . the expectation values on both sides of ( [ eq : wiqp ] ) are evaluated as functions of @xmath29 . taking the value of @xmath30 obtained using eq.([eq : twomzal ] ) , @xmath31 can then be determined . in order to enhance the signal , we add in both sides of ( [ eq : wiqp ] ) the four contributions @xmath32=(1,3 ) , ( 3,1 ) , ( 2,4 ) and ( 4,2 ) , which were found to give the clearest signal @xcite . a plateau in @xmath29 is typically obtained and @xmath0 is estimated from it . the crucial point is that both sides of eq.([eq : wiqp ] ) are gauge dependent , and thus this determination of @xmath0 is in principle sensitive to the gribov noise . a gauge invariant determination of @xmath0 is obtained through the ward identity @xmath33 in the above equation , the vector current renormalisation constant @xmath34 is also needed . for the clover action , @xmath34 is calculated with the aid of the so - called conserved and improved vector current @xmath35 \nonumber\\ & & \ \ \ \ \ \ \ \ + ( x\rightarrow x-\hat\mu ) + \frac{r}{2 } \sum_{\rho}\partial_\rho\left(\bar\psi(x)\sigma _ { \rho\mu}\psi(x)\right ) \label{eq : vci}\end{aligned}\ ] ] since the current is conserved , its renormalisation constant is precisely 1 , and , since it is improved " , its matrix elements have no corrections of @xmath13 . the normalisation constant @xmath34 of the local vector current @xmath36 is determined through the ratio of the vacuum to vector state matrix elements of the two vector currents @xcite : @xmath37 ( here @xmath38 denotes the vector state ) . by calculating all correlation functions of eq.([eq : wivaint ] ) and @xmath34 from eq.([eq : zvrat ] ) and by requiring that eq.([eq : wivaint ] ) holds at each @xmath29 , we can determine @xmath0 in an explicitly gauge invariant fashion for which the gribov ambiguity is irrelevant . one more comment is in place here : the terms proportional to @xmath39 on the right - hand - side of eqs.([eq : wiqp ] ) and ( [ eq : wivaint ] ) arise from the rotations of the fermion fields defined in eq.([rot ] ) , which are inherent to clover action improvement @xcite . the @xmath40 rotations , combined with the equations of motion , generate contact terms which , to @xmath13 , give rise to the terms proportional to @xmath39 @xcite . gauge fixing and the generation of gribov copies on the lattice is by now a standard procedure . given a thermalised configuration generated by a monte carlo simulation , the landau gauge is fixed by minimising the functional @xcite @xmath41 \equiv - \frac{1}{v } re \ tr \sum_{\mu=1}^{4 } \sum_{n } u_{\mu}^{g}(n ) \label{eq : bigf}\ ] ] with respect to @xmath42 . here @xmath43 is the lattice volume and @xmath44 is the compact @xmath1 gauge field , gauge transformed by a local gauge transformation @xmath45 . the extrema of @xmath46 correspond to configurations that satisfy the gauge condition @xmath47 in discretised form . the minimisation of @xmath46 is obtained through iteration : each lattice site is visited and @xmath46 is minimized locally . after several lattice sweeps @xmath46 becomes constant , and the gauge is fixed . this is nothing else but a discretised analogue of the continuum formulation @xcite , according to which the local minima of the functional @xmath48 = - \sum_{\mu=1}^4 tr \int d^4x ( a^g_\mu)^2 $ ] with respect to @xmath49 correspond to configurations in the landau gauge . even if there is a straightforward correspondence between the lattice and continuum gauge fixing procedures , it is interesting to note that on the lattice , because of discretisation , there can be more minima than in the continuum @xcite . the production of gribov copies consists in generating gauge equivalent configurations , by applying random gauge transformations to our thermalised link configuration @xcite , @xcite . the gauge is then fixed and @xmath46 is measured . since it is a gauge dependent quantity , a different value of @xmath46 for two gauge equivalent , gauge fixed configurations means that they are gribov copies . the over - relaxation technique of @xcite ( which consists in accelerating the gauge fixing algorithm by raising the gauge transformation @xmath49 to a real tunable power @xmath50 at every iteration ) has been implemented at fixed @xmath50 . the over - relaxation itself can be used for the generation of gribov copies , by varying the value of @xmath50 , as proposed in @xcite . in this work we have opted for the random gauge transformation method . we work in the framework of the quenched approximation with the clover action of su(3 ) gauge theory . the lattice volume is @xmath51 and @xmath3 . after @xmath52 thermalising sweeps , 36 configurations were generated , separated by @xmath53 sweeps . an @xmath54 hit metropolis algorithm was used . for each thermalised configuration , we generated @xmath55 gribov copies . this was done by fixing the gauge both on the original configuration and on @xmath56 gauge equivalent replicas , obtained by applying random gauge transformations . it is remarkable that , in our case , each random gauge transformation produced a gribov copy . this high probability to find gribov copies is a characteristic of large volume lattices in the confined region , as discussed in @xcite . we fix the landau gauge , using the over - relaxation algorithm suggested in @xcite at fourth order in the over - relaxation parameter @xmath57 . the stopping condition we have imposed is that the relative variation @xmath58 , of the minimised functional @xmath46 , between two consecutive gauge fixing sweeps be less than @xmath59 . this value guarantees a good quality of the gauge fixing , allows us to distinguish gribov copies , and it is typically reached after a number of sweeps which varies between 500 and 1500 . the gauge fixing was done on an ibm risc 6000/550 equipped with 128 mbyte of ram memory and with a cpu working at 42 mhz ; with this machine a single gauge fixing sweep takes about 40 s. the quark propagators , rotated as indicated by eq.([rot ] ) were obtained at a wilson hopping parameter value of @xmath60 , which , for the clover action , corresponds to a pion of roughly 900 mev . before passing to a detailed discussion of the gribov noise , it is appropriate to present a comparison of the results for @xmath0 , obtained in a gauge invariant way ( see eq.([eq : wivaint ] ) ) , to those based on the gauge dependent identity of eq.([eq : wiqp ] ) . this calculation has been already presented in @xcite , on the first 18 configurations of our ensemble ; here we have doubled the statistics . moreover , we have re - gauged the first 18 configurations in order to reach the better quality of gauge fixing required for this study . in fig.(1 ) we show the behaviour of the two estimates of @xmath0 , as a function of @xmath29 , calculated on the same ensemble of 36 configurations . the gauge invariant values of @xmath0 show a flat behaviour already at @xmath61 and with small error bars . the new value of @xmath0 , obtained with the gauge independent technique , over 36 configurations , is @xmath62 to be compared with the old one obtained over 18 configurations : @xmath63 ( see ref.@xcite ) . in the gauge dependent case , we have taken the average over the gribov copies , in the way that will be discussed below . in this case , the @xmath0 behaviour becomes flat only at @xmath64 showing a large sensitivity to the contact terms of the ward identity at small @xmath29 values . the new value of @xmath0 , obtained from the gauge dependent ward identity , as can be seen from fig.(1 ) is @xmath65 ; to be compared to the value obtained from 18 configurations , @xmath66 ( see ref.@xcite ) . as already stressed in @xcite , this method gives results that are compatible within the errors with the gauge independent ones . moreover , the error of the gauge invariant calculation of @xmath0 is always smaller than the error of its gauge dependent counterpart . this is due to the fact that the quark state correlation functions fluctuate more than the gauge invariant correlation functions , but it may also indicate the presence of another effect , which is probably the gribov ambiguity . in the hypothetical case of two determinations of @xmath0 , affected by the same statistical error , the gauge dependent estimate should fluctuate more due to the gribov noise . then the amount of gribov noise could be estimated as the difference ( in quadrature ) of the two errors . normally , in a standard simulation of gauge dependent quantities , one does not generate gribov copies . consequently , one measures a given quantity by taking the average and error over the ensemble of the gauge fixed configurations that have been generated . this implies a particular and arbitrary choice of gribov copies . the error estimated , for example , by a jacknife method , is not purely statistical as it implicitly contains the uncertainty due to the particular choice of a gribov copy . in our case , having generated @xmath67 copies for each of the @xmath68 thermalised configurations , there are @xmath69 possible combinations that we may consider when forming a particular ensemble . to the best of our knowledge , the distribution of the gribov copies of a given configuration is unknown ; thus the weight to be associated to it is arbitrary . moreover , any technique used to generate different gribov copies selects a particular copy in a completely uncontrolled way . hence , we assume that the particular choice of different combinations of gribov copies when forming a statistical ensemble is arbitrary . in order to exhibit the effect of such arbitrariness , we show in fig.(2 ) the behaviour of @xmath70 for @xmath71 arbitrary choices of copies . the @xmath71 different behaviours are compatible , and the same is true for the jacknife errors . it is clear , however , that the presence of gribov copies is a visible effect ; each of the six estimates of @xmath0 shown has a slightly different profile as a function of @xmath29 . we now expose the procedure we implemented for taking into account the gribov ambiguity . the gauge dependent traces of the two and three point correlation functions appearing in eq.([eq : wiqp ] ) , calculated on a single configuration @xmath72 and for a particular gribov copy @xmath42 are denoted by : @xmath73 \nonumber \\ t_3(t_y ; c , g ) = tr \left [ \int d^4x \int d^3\vec y \ , u(y)\,\left(\bar u(x)\gamma_5 d(x)\right ) \,\bar d(0)\ \right ] \label{eq : t2t3a}\end{aligned}\ ] ] ( in the above equations the dirac indices have been implicitly averaged over , as explained in sect.(2 ) ) . then , for a given configuration @xmath72 , we consider our `` best estimate '' of these matrix elements to be their average over the @xmath67 gribov copies : @xmath74 the average of the above @xmath75 s over the @xmath76 configurations will be taken as our `` best estimate '' @xmath77 for the gauge dependent traces . on the other hand , @xmath25 , being gauge invariant , does not depend on a particular choice of gribov copies , but only on the configuration ensemble . then @xmath70 is obtained by applying eq.([eq : twomzal ] ) as follows @xmath78 the error is obtained by a standard jacknife method performed on the quantities @xmath79,@xmath80 and @xmath81 , by decimating one configuration at a time . this is the gauge dependent @xmath70 result shown in fig.(1 ) . we want to stress that the values of @xmath82 and @xmath83 fluctuate more than their ratio @xmath84 . the latter quantity is gauge invariant , according to eq.([eq : zam1 ] ) . for example , at @xmath85 , @xmath86 , @xmath87 and @xmath88 . the strong reduction of the relative error indicates a great sensitivity of @xmath89 to the gribov fluctuation , as opposed to a relative stability of the gauge invariant quantities . in order to estimate the uncertainty arising from a particular choice of copies , out of the @xmath90 possible ones , we have applied a procedure which takes this arbitrariness into account we have chosen randomly @xmath91 combinations of copies of our ensemble and have calculated @xmath70 for each one of these combinations at fixed @xmath29 . the histogram of the values obtained for @xmath70 is well fitted by a gaussian , the r.m.s . width of which is taken as an estimate of the fluctuation . in fig.(3 a ) we compare the jacknife error of our `` best estimate '' to the width of the gaussian . we see that for all @xmath29 the width is smaller than the jacknife error . this implies that the fluctuations induced by the particular choice of gribov copy when forming the ensemble are small and do not overcome the statistical uncertainty . as it is also important to understand how the above behaviour scales with increasing the number of configurations , we have performed the same analysis for the first @xmath92 configurations ( half of our ensemble ) . the result is shown in fig.(3 b ) . comparing fig.(3 a ) to fig.(3 b ) , we note that both errors scale at least as @xmath93 . thus , within our moderate statistics , we find that the error , even if affected by the gribov ambiguity , decreases with increasing configuration number . in conclusion , our investigation , even within its limitations , shows that , on the lattice , the residual gauge dependence associated with the gribov copies does not generate an overwhelming fluactuation of the @xmath0 measurements performed by the gauge dependent method . nevertheless the arbitrariness associated to a particular choice of gribov copies is a visible effect which manifests itself , especially in the behaviour of @xmath0 as a function of @xmath29 . the jacknife error is greater in the gauge dependent determination than in the gauge independent one . however , in the former case , even though the error is afflicted by the gribov uncertainty , it is still decreasing with increasing statistics . this implies that the uncertainty arising from the gribov ambiguity may be a secondary effect . analogous studies on different physical quantities and renormalisation constants could further support this belief . we warmly thank g.martinelli , c.t.sachrajda and m. testa for many discussions and partecipation throughout this work . useful discussions with v.n . gribov and a.j . van der sijs are also greatfully acknowledged . we also thank the ibm - semea for providing us with 32 mbytes of memory . figure 1 : comparison of the gauge independent calculation of @xmath70 ( diamonds ) to the gauge dependent one ( crosses ) . the errors are jacknife . the crosses have been slightly displaced to help the eye . + figure 2 : the gauge dependent calculation of @xmath70 for 4 arbitrarily chosen gribov copies . the errors are jacknife . + figure 3 : comparison of the jacknife error ( crosses ) to the r.m.s . gaussian width ( diamonds ) due to the gribov ambiguity ( see text ) . the crosses have been slightly displaced to help the eye . case ( a ) is with 36 configurations ; case ( b ) with the first 18 configurations

we study the influence of gribov copies , in the landau gauge , on the lattice renormalisation constant @xmath0 of the axial current , obtained from a ward identity on quark state correlation functions , with the clover action , in quenched @xmath1 gauge theory . a comparison between the gauge invariant determination of @xmath0 and the gauge dependent one is discussed . on a @xmath2 lattice at @xmath3 and with @xmath4 , the values , on a sample of 36 configurations , are : @xmath0 = 1.08(5 ) ( gauge dependent calculation ) and @xmath0 = 1.06(2 ) ( gauge independent calculation ) . we find that the residual gauge freedom associated to gribov copies induces observable effects , which , at the level of numerical accuracy of our simulation , are included in the statistical uncertainty inherent in a monte carlo simulation . doubling the statistics suggests that the fluctuation due to the lattice gribov ambiguity scales down at least as fast as a pure statistical error .

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Proceed to summarize the following text: for decades , galaxy merging has been a popular explanation for the observed evolution in galaxy properties . galaxy mergers were first invoked to explain the morphological transformation of galaxies @xcite . merging remains the backbone in cosmological simulations in building up large galaxies ( e.g. * ? ? ? * ; * ? ? ? gas - rich major mergers at high redshifts ( @xmath10 ) are thought to trigger starburst and active galactic nuclei ( agn ) episodes , quench star formation , and lead to bulge formation , thereby building the massive ellipticals in the local universe @xcite . an alternative scenario has been proposed more recently , in which massive galaxies at high redshift are clumpy disks which are very efficient in turning incoming cold gas into stars @xcite . the most luminous agns and ultra - luminous infrared galaxies ( ulirgs ) take place in major galaxy mergers @xcite . merging galaxies have enhanced star formation activity compared to isolated ones ( @xcite ; but also see @xcite ) . as galaxy merging may have profound influence on how the galaxy population evolved to this day , quantifying its rate of occurrence is essential to judge whether it explains any of the observed evolutionary trends . as the timescale for galaxy mergers is on the order of a gyr ( e.g. * ? ? ? * ) , the conventional way to measure galaxy merger rate is to divide the observed fraction of galaxies undergoing mergers by a typical merging ( observability ) timescale at different redshift bins . merging galaxies can be identified as close galaxy pairs or galaxies displaying disturbed morphologies , and the timescale required to convert the merger fraction to merger rate depends on the specific selection technique . in this work we use the pair selection method , as the merger fraction measured from morphological selection ( e.g. * ? ? ? * ; * ? ? ? * ) and the merging observability timescale are dependent on the imaging depth and resolution . the advent of multi - wavelength blank field observations in the past decade have enabled many improvements in the measurement of merger fractions , including the following : ( 1 ) the merger fraction of massive galaxies can be measured beyond @xmath12 ; ( 2 ) the photometric redshifts allow more accurate removal of the pairs projected along the line - of - sight ; ( 3 ) the stellar masses derived from the spectral energy distribution ( sed ) fitting provide the stellar mass ratio of galaxy pairs , which is a more physically meaningful proxy for the dynamical interaction than a single - band flux ratio ; ( 4 ) deeper and wider area surveys provide larger samples , which in turn allow the dependence of merger fractions on different parameters to be explored . multiple authors have measured the merger fraction at @xmath13 , presenting somewhat conflicting results : does the merger fraction increase with redshift @xcite , remain constant , or even diminish @xcite ? as shown in @xcite , the variation of the parent galaxy selection and and mass ratio limits can contribute to some of the discrepancies across studies . on the other hand , the average merging observability timescale is hard to estimate due to the large possible variety of orbital parameters and viewing angles , as well as the lack of observed dynamical information on a galaxy - to - galaxy basis . the uncertainties in the implied merger rates are discussed thoroughly in @xcite . in this work , we present the largest sample of photometrically selected mergers at @xmath14=0.1 - 3 to date from stellar mass complete catalogs . the _ ks_-band selected catalog from the ultravista / cosmos survey @xcite covers a large area , allowing us to expand our merger sample to more than five times times larger than previous studies . we complement the ground - based ultravista catalog with the space - based 3dhst+candels @xcite catalog , which is deeper and has higher spatial resolution , to study possible systematic effects in measuring merger fractions . the remainder of the paper is structured as follows : section [ sec : data ] describes the ultravista and the 3dhst+candels catalogs used in our study . we present the criteria for selecting massive galaxies and mergers , as well as the completeness of the catalogs . in section [ sec : method ] we present the method of measuring the merger fractions as a function of redshift . we compare the merger fractions measured using the two catalogs , as well as the selection using the stellar mass ratio and @xmath1-band flux ratio . we examine the stellar mass ratio distribution of the selected mergers . we discuss the two main sources of uncertainties in the merger fraction measurements . we show that we are complete to detecting minor mergers up to @xmath15 . finally we convert the merger fractions to merger rates , and infer the merger contribution in the stellar mass , size , velocity dispersion and number density evolution of massive galaxies . based on our findings , we address some broader questions in the context of galaxy evolution in section [ sec : discussion ] : what do the merger rates imply for the evolution of massive quiescent galaxies ? is merging an influential process in the cosmic star formation history or not ? we also discuss the future prospects of merger fraction studies . the conclusions of this work is summarised in section [ sec : conclusions ] . in appendix [ sec : missing_mergers ] we present the simulations we perform to test for the completeness limits of the faintest possible satellites . appendix [ sec : pf_lit ] provides an in - depth comparison to similar merger fraction measurements in the literature . all magnitudes are quoted in the ab system . a cosmology of @xmath16 = 70 km s@xmath17 mpc@xmath17 , @xmath18 = 0.3 and @xmath19 = 0.7 is adopted throughout this work . we use the _ ks_-band selected catalog for the ultravista survey compiled by @xcite . the ultravista survey targets the cosmos field @xcite with the eso vista survey telescope . the effective survey area of ultravista is 1.62 deg@xmath20 . the catalog contains psf - matched photometry in 30 photometric bands covering the wavelength range 0.15 - 24@xmath21 and includes the _ galex _ @xcite , cfht / subaru @xcite , ultravista @xcite , s - cosmos @xcite , and zcosmos @xcite datasets . the ultravista source detection is performed on the _ ks_-band image with a @xmath22 aperture , which has a limiting magnitude of @xmath23 ( @xmath24-aperture ) . in total there are 154 803 detected sources with reliable photometry having _ _ ks__@xmath25 , which is the 90% completeness limit and the adopted luminosity limit in this work . the stellar masses quoted in this paper are derived assuming a chabrier imf . further details regarding the photometric redshifts ( photo-@xmath14 s ) and sed fitting can be found in @xcite . to complement the ground - based _ yjhk@xmath26 _ imaging from vista , we use the 3dhst catalog presented in @xcite and @xcite , which includes _ hst _ imaging from the candels survey @xcite over five fields : cosmos , goods - north and south , aegis , and uds with a combined usable area of @xmath27 deg@xmath20 . @xcite performed photometry ( aperture of @xmath28 ) on the psf matched images and compiled a photometric catalog with photo-@xmath14 s and sed best fits . we only use the objects marked with good photometry to ensure reliable photo-@xmath14 s and stellar masses . we use close galaxy pairs as a probe for galaxy mergers following similar criteria used in the literature @xcite . in the ultravista catalog , there are 9829 massive ( log@xmath4 ) galaxies in the redshift range of @xmath29 , and 380 ( @xmath30 ) of them are covered by the _ h_-band imaging from the candels and 3dhst cosmos surveys . around these massive galaxies , we search for galaxy satellites fulfiling the following criteria : 1 . within a projected separation of @xmath31 kpc @xmath32 . stellar mass ratio @xmath33 of 1:1 - 4:1 as major merger , 4:1 - 10:1 as minor merger . 3 . the 1@xmath34 confidence intervals of the photo-@xmath14 s of the pair overlap . we calculate @xmath35 using the angular scale based on the photo-@xmath14 s of the more massive galaxy . as the fwhm of the ground - based ultravista @xmath36-band image is @xmath37 , corresponding to a maximum of 9.7 kpc @xmath32 at @xmath38 , we use 10 kpc @xmath32 as the lower limit of @xmath35 to ensure that no close pairs are missed due to blending . in section [ sec : compare_merger_rates ] we explore the use of different @xmath35 bins up to 100 kpc @xmath32 . we explore the use of the @xmath7-band flux ratio as a probe for the stellar mass ratio in section [ sec : pf_ratios ] , which we demonstrate to have a profound impact on the merger fraction evolution at @xmath39 . the redshift distribution of massive galaxies and pairs are listed on table [ table : pf_massratio ] . we assess the completeness limit of the massive galaxies and their 4:1 and 10:1 satellites in two aspects : the stellar mass completeness and the surface brightness limits . we detail our analysis in appendix [ sec : missing_mergers ] and give the summary as follows . we find that the surface brightness limit is the constraining factor for detecting the satellites of massive galaxies . if completeness is only estimated by comparing the magnitude - redshift distribution to deeper catalogs , the completeness limits may be overstated . we find that ultravista ( 3dhst+candels ) is complete to @xmath40 and @xmath41 ( @xmath42 and @xmath15 ) for major and minor mergers respectively . in this work , the data points at redshift bins which are mass incomplete are either omitted or plotted as lower limits , to ensure that incompleteness does not affect our conclusions . despite the fact that 3dhst+candels is deeper than ultravista and can probe the merger fractions to higher redshifts , we demonstrate in section [ sec : pf_uvista ] that we do not get a higher merger fraction , both major and minor , with 3dhst+candels compared to ultravista , suggesting that there is not a significant population of mergers that have faint quiescent satellites only detectable in the 3dhst+candels catalog . [ fig : pf_massratio ] [ fig : pf_fluxratio ] the relation between the number of observed galaxy pairs ( @xmath43 ) and the number of ongoing physical galaxy mergers ( @xmath44 ) can be described as @xmath45 . the quantity @xmath43 is defined as the number of galaxy pairs observed that satisfy a projected separation and mass ( or flux ) ratio criteria , e.g. pairs fulfiling the first two criteria listed in [ sec : select_pairs ] . among the observed pairs , some are galaxy pairs of physical proximity , while some pairs are galaxies projected along a similar line - of - sight . the line - of - sight projected galaxy pairs can be corrected for using redshift measurements ( photometric or spectroscopic ) or statistical arguments based on the galaxy mass or luminosity function . in this work we apply a photo-@xmath14 criterion as listed in section [ sec : select_pairs ] to correct for projected pairs ( @xmath46 ) . we have demonstrated in @xcite that using the photo-@xmath14 s to correct for chance alignments yield results consistent with statistical corrections at @xmath47 . in this work we do not correct for physical galaxy pairs at matching redshifts that are not energetically bound to merge , i.e. we assume @xmath48 = 0 . cosmological simulations can provide a statistical estimate of @xmath48 to account for the unbound galaxy pairs in cluster environments with high relative velocities . however , the interpretation may be complicated by the presence of a third neighbor which is not uncommon @xcite , or these pairs simply require more time before the eventual coalescence @xcite . galaxy fly - bys may be frequent @xcite but it remains unexplored how high - speed encounters may impact the mass distribution and light profiles of galaxies . even if the cores do not coalesce , mass from the satellite may still be deposited onto the host galaxy , and the energy exchange can lead to size growth akin to a `` real '' merger @xcite . it is not well understood how @xmath48 evolves with the environment and redshift . at higher redshift , massive galaxies are expected to be less clustered than at the present day , so the effect is likely more dominant at low redshift . future studies of the dynamical properties of galaxy pairs at different redshifts and environments may provide new insights into this effect , but for now we do not have enough information to correct for it . we note that by including non - energetically bound pairs in our selection , the merger fractions derived in this paper are formally upper limits . hereafter we refer to @xmath44 as @xmath49 for simplicity . we define the merger fraction as the fraction of massive galaxies that are merging with a less massive companion , i.e. @xmath50 . the major and minor merger fractions ( @xmath51 and @xmath52 ) in redshift bins are listed on table [ table : pf_massratio ] and plotted on figure [ fig : pf_massratio ] ( left ) . we parameterise the merger fractions within the completeness limits by a power law using least squares fitting . in the case of @xmath51 declining beyond @xmath38 in ultravista , the reduced @xmath53 value for the power law fit exceeds 10 indicating a bad fit so we fit the data points with a quadratic function instead . we list the best fitting parameters in table [ table : bestfit ] . using the stellar mass ratio selection , we find that @xmath51 ( @xmath52 ) increases from @xmath54 to reach a peak at @xmath55 , remains relatively constant to @xmath56 ( @xmath57 ) and then diminishes towards higher redshift . a comparison between the merger fractions derived from the ground - based ultravista and the deeper , higher resolution 3dhst+candels reveals very similar @xmath51 and @xmath52 in both samples . in fact , @xmath51 is slightly lower in 3dhst+candels than in ultravista at @xmath58 . if we include the pairs without photo-@xmath14 information ( columns 3 and 7 on table [ table : pf_massratio ] ) in our merger sample , the @xmath51 of 3dhst+candels at this redshift bin becomes consistent with the one from ultravista . this illustrates that space - based data is not required for measuring the galaxy merger fraction . in fact , ground - based data with a large survey volume such as ultravista provides the optimal dataset , as the sample is adequately large to measure the redshift dependence of the merger fractions in finer redshift bins . we elaborate on the uncertainties of merger fraction measurements in section [ sec : cv ] . cccc + ultravista & stellar mass ratio & @xmath59 & @xmath60 + ultravista & @xmath1 flux ratio & @xmath61 & @xmath62 + 3dhst & stellar mass ratio & @xmath63 & @xmath64 + 3dhst & @xmath1 flux ratio & @xmath65 & @xmath66 + + ultravista & stellar mass ratio & @xmath67 & @xmath68 + ultravista & @xmath1 flux ratio & @xmath69 & @xmath70 + 3dhst & stellar mass ratio & @xmath71 & @xmath72 + 3dhst & @xmath1 flux ratio & @xmath73 & @xmath74 + [ table : bestfit ] ccccccccccc + @xmath75 & 628 & 23 & 12 & 201 & 3.66@xmath76 & & 29 & 6 & 170 & 4.62@xmath77 + @xmath78 & 772 & 40 & 4 & 99 & 5.18@xmath79 & & 52 & 3 & 117 & 6.74@xmath80 + @xmath81 & 1618 & 158 & 2 & 170 & 9.77@xmath82 & & 146 & 6 & 179 & 9.02@xmath83 + @xmath84 & 1692 & 184 & 6 & 169 & 10.87@xmath85 & & 140 & 9 & 140 & 8.27@xmath86 + @xmath87 & 1426 & 142 & 5 & 133 & 9.96@xmath88 & & 134 & 10 & 143 & 9.4@xmath89 + @xmath90 & 1163 & 133 & 8 & 99 & 11.44@xmath91 & & 81 & 13 & 102 & 6.96@xmath92 + @xmath93 & 1087 & 99 & 9 & 97 & 9.11@xmath94 & & 42 & 20 & 125 & 3.86@xmath95 + @xmath96 & 560 & 40 & 9 & 63 & 7.14@xmath97 & & 15 & 4 & 63 & 2.68@xmath98 + @xmath99 & 536 & 28 & 13 & 56 & 5.22@xmath100 & & 18 & 9 & 73 & 3.36@xmath101 + @xmath102 & 347 & 20 & 3 & 31 & 5.76@xmath103 & & 10 & 4 & 46 & 2.88@xmath104 + + @xmath105 & 123 & 10 & 1 & 43 & 8.13@xmath106 & & 7 & 1 & 44 & 5.69@xmath107 + @xmath108 & 61 & 3 & 0 & 9 & 4.92@xmath109 & & 5 & 0 & 14 & 8.2@xmath110 + @xmath111 & 88 & 5 & 0 & 14 & 5.68@xmath112 & & 9 & 1 & 16 & 10.23@xmath113 + @xmath114 & 63 & 6 & 0 & 20 & 9.52@xmath115 & & 5 & 1 & 18 & 7.94@xmath116 + @xmath117 & 45 & 0 & 0 & 7 & 0.0@xmath118 & & 3 & 1 & 7 & 6.67@xmath119 + + @xmath105 & 84 & 3 & 2 & 22 & 3.57@xmath120 & & 6 & 5 & 35 & 7.14@xmath121 + @xmath108 & 72 & 6 & 2 & 13 & 8.33@xmath122 & & 7 & 2 & 13 & 9.72@xmath123 + @xmath111 & 57 & 1 & 2 & 7 & 1.75@xmath124 & & 3 & 2 & 4 & 5.26@xmath125 + @xmath114 & 65 & 1 & 0 & 6 & 1.54@xmath126 & & 1 & 3 & 13 & 1.54@xmath127 + @xmath117 & 37 & 0 & 0 & 6 & 0.0@xmath118 & & 1 & 0 & 9 & 2.7@xmath128 + + @xmath105 & 66 & 4 & 0 & 11 & 6.06@xmath129 & & 6 & 2 & 25 & 9.09@xmath130 + @xmath108 & 77 & 6 & 1 & 9 & 7.79@xmath131 & & 4 & 1 & 11 & 5.19@xmath132 + @xmath111 & 74 & 3 & 0 & 9 & 4.05@xmath133 & & 6 & 5 & 12 & 8.11@xmath134 + @xmath114 & 47 & 2 & 0 & 7 & 4.26@xmath135 & & 2 & 0 & 8 & 4.26@xmath136 + @xmath117 & 39 & 4 & 0 & 6 & 10.26@xmath137 & & 2 & 1 & 7 & 5.13@xmath138 + + @xmath105 & 102 & 8 & 3 & 54 & 7.84@xmath139 & & 3 & 5 & 76 & 2.94@xmath140 + @xmath108 & 124 & 7 & 4 & 23 & 5.65@xmath141 & & 11 & 10 & 38 & 8.87@xmath142 + @xmath111 & 141 & 19 & 1 & 31 & 13.48@xmath143 & & 9 & 5 & 22 & 6.38@xmath144 + @xmath114 & 86 & 5 & 3 & 16 & 5.81@xmath145 & & 3 & 1 & 21 & 3.49@xmath146 + @xmath117 & 54 & 4 & 1 & 8 & 7.41@xmath147 & & 1 & 3 & 14 & 1.85@xmath148 + + @xmath105 & 87 & 11 & 6 & 29 & 12.64@xmath149 & & 9 & 6 & 19 & 10.34@xmath150 + @xmath108 & 103 & 5 & 2 & 13 & 4.85@xmath151 & & 8 & 4 & 17 & 7.77@xmath152 + @xmath111 & 162 & 11 & 2 & 15 & 6.79@xmath153 & & 11 & 2 & 38 & 6.79@xmath154 + @xmath114 & 98 & 3 & 0 & 12 & 3.06@xmath155 & & 2 & 2 & 10 & 2.04@xmath156 + @xmath117 & 65 & 4 & 0 & 11 & 6.15@xmath157 & & 0 & 2 & 16 & 0.0@xmath158 + [ table : pf_massratio ] [ fig : mtol_uvista ] [ fig : mtol_3dhst ] selected by stellar mass ratio ( blue ) or @xmath1 flux ratio ( red ) from the 3dhst+candels catalog . the ratios are defined such that the mass- ( flux-)ratio selected mergers will have ratios of 1 - 10 . on the top panel we show the histogram of the stellar mass ratios , and at bottom the histogram of the @xmath1 flux ratios . the solid , dashed and dotted gray lines represent the 1:1 , 1:4 & 4:1 , 1:10 & 10:1 ratios respectively . from the top panel , we observe that a large excess of flux ratio selected mergers in 3dhst+candels have @xmath1 flux ratios between 1 and 10 , but have stellar mass ratios between 10 and 100 . this explains the rising merger fractions observed in figure [ fig : pf_fluxratio ] ( right ) due to bright satellites with log@xmath159 being included in the flux ratio selected sample . [ fig : mr_fr_histogram ] , scaledwidth=50.0% ] merger fraction measurements have led to conflicting conclusions regarding whether it increases with redshift at @xmath39 @xcite or not @xcite . the former studies use the single band flux ratio from _ hst _ _ h_-band imaging to estimate the mass ratio , rather than full the stellar mass ratio from sed fits used in the latter studies . we explore the possibility of a systematic effect regarding the ratio used in the merger selection . we repeat the selection of mergers with the @xmath1-band flux ratio instead of using the stellar mass ratio on the same dataset presented in section [ sec : data ] , namely the ultravista and 3dhst+candels catalogs . the results are presented in figure [ fig : pf_fluxratio ] ( right ) . it is apparent that the combination of using the flux ratio to select mergers and the 3dhst+candels catalog leads to an increasing redshift trend of @xmath51 ( @xmath52 ) up to @xmath160 ( @xmath15 ) where the catalog is complete for major ( minor ) satellites . this is in contrast to the flat or even diminishing evolution found when mergers are selected by the stellar mass ratio ( figure [ fig : pf_massratio ] , left ) , as well as using flux ratio to select mergers from ultravista ( filled circles in figure [ fig : pf_fluxratio ] , right ) . our results are in good agreement with the trends found in literature ( see appendix [ sec : pf_lit ] for details of the comparison ) meaning that we are able to reproduce the increasing redshift trend of the merger fraction if mergers are selected by flux ratio . by comparing the mergers selected in the overlapping area of the ultravista and candels - cosmos surveys , we find that the flux - ratio selected satellites at @xmath10 are close to the survey depth of ultravista dr1 ( @xmath161 , @xcite ) , and therefore fainter satellites are missed due to low surface brightness . we interpret the difference between the flux ratio selected merger fraction between the ultravista and the 3dhst - candels samples as being due to the observation limit of the ultravista dr1 data . this is expected to improve for the forthcoming data release of ultravista in which the survey depth of the four ultra - deep stripes will be @xmath162 mag deeper . in order to explain the difference between the flux and stellar mass ratio selections using the 3dhst+candels catalog , we compare the stellar mass ratio and flux ratio distribution of the mergers using both selection techniques in figure [ fig : mr_fr_histogram ] . we display the results for the redshift bin @xmath163 where the discrepancy in the merger fraction is most significant between the two selection techniques . we find that almost all of the stellar mass ratio ( 1:1 - 10:1 ) selected mergers have @xmath7-band flux ratio in the same range . on the other hand , flux ratio selected mergers ( 1:1 - 10:1 ) include mergers with stellar mass ratios in the same range , as well as mergers with more extreme stellar mass ratios ( @xmath16410:1 ) . among the major _ flux ratio _ pairs at @xmath163 in 3dhst+candels , only @xmath165 have major _ stellar mass _ ratios . the remaining pairs consist of minor _ stellar mass _ ratio ( @xmath166 ) and mostly very minor _ stellar mass _ ratio ( @xmath167 ) with @xmath16810:1 . this demonstrates that the observed @xmath7-band flux is a biased tracer of the stellar mass at @xmath10 . using the @xmath7-band flux ratio as a probe for the stellar mass ratio leads to the inclusion of bluer , less massive galaxies as satellites . in another words , at @xmath10 most of the satellites are star - forming blue galaxies that are bright in the rest - frame optical b- or v - bands . we conclude that the flux ratio selection yields a higher merger fraction than mass ratio selection at all redshifts for two reasons : ( 1 ) the observed @xmath7-band probes bluer rest - frame bands at higher @xmath14 ; ( 2 ) lower @xmath169 satellites enter the sample @xcite , where @xmath169 is the ratio of the stellar mass to the rest - frame @xmath170-band luminosity . we illustrate the redshift dependence of @xmath169 in figure [ fig : mtol_z ] . there is overall @xmath169 redshift evolution in both the massive galaxies and their satellites , in which the ratio increases over cosmic time . both catalogs show a similar @xmath169 evolution except for the @xmath7-band flux ratio selected pairs in the candels+3dhst sample , where the evolution is steeper implying the inclusion of lower @xmath169 at @xmath10 than for the stellar mass ratio selection . at @xmath171 the observed @xmath1-band roughly corresponds to the rest - frame @xmath172 and @xmath170 bands . our simulations in appendix [ sec : sb_limit ] indicate that we are complete to @xmath173 for major ( minor ) mergers in 3dhst+candels , therefore the @xmath169 evolution can not be explained by observational effects and is intrinsic . the @xmath169 evolution reflects the higher star formation activity at @xmath174 compared to that of the present day ( e.g. * ? ? ? having shown that the use of the flux and stellar mass ratio can reproduce the discrepancy in merger fraction in literature , we proceed to find the ratio that best describes the dynamics and future evolution of the merging galaxies . although using the @xmath7-band flux ratio selection is biased towards star - forming but low stellar mass satellites , the use of the stellar mass ratio may be biased against gas - rich satellites at @xmath13 . galaxies appear to be more gas - rich at higher redshift and at lower masses @xcite . such a dependence implies that the baryon mass ratio is closer to unity than the stellar mass ratio , since cold gas mass is included into the baryon mass calculation . the baryon mass of a galaxy is a better probe of its total mass ( which also includes dark matter ) than the stellar mass alone , as shown in cosmological simulations @xcite . a merger can be _ major _ or _ minor _ depending on whether the stellar mass , baryon mass or total mass is considered for the mass ratio @xcite . intermediate mass galaxies of log@xmath175 are the satellites to the massive galaxies studied here , and their molecular gas mass may not be negligible in the total mass budget that governs the dynamics of the galaxies , especially at @xmath174 . if the cold gas fraction increases with redshift and decreases with stellar mass as previously claimed @xcite , there is a redshift - dependent underestimation if we use the stellar mass to trace the baryon mass . the correction is likely larger at higher redshift due to the higher gas fraction . therefore merging with these gas - rich satellites with stellar mass ratios more extreme than 10:1 may contribute to the star formation budget of the massive galaxies @xcite , in the form of gas accretion or very minor mergers if characterised by the stellar mass ratio . we note that gas - rich satellites are not equivalent to gas - rich mergers ( e.g. * ? ? ? * ) , which is usually defined as the average gas fraction of both galaxies . despite the importance of the gas content in the merger definition as well as its contribution to star formation activity , direct measurements of the molecular gas mass are so far only available for limited samples of galaxies @xcite , mostly starbursting sub - millimeter galaxies . alma surveys of large samples of normal " star - forming galaxies will shed light on this topic in the future @xcite . ccccccccc @xmath176 & 7.8 & 0.18 & 0.39 & 0.43 & 6.7 & 0.20 & 0.39 & 0.44 + @xmath177 & 6.2 & 0.21 & 0.16 & 0.27 & 8.0 & 0.18 & 0.11 & 0.21 + @xmath178 & 7.5 & 0.17 & 0.59 & 0.61 & 7.3 & 0.18 & 0.19 & 0.26 + @xmath179 & 4.7 & 0.27 & 0.58 & 0.64 & 3.6 & 0.32 & 0.63 & 0.70 + @xmath180 & 5.0 & 0.33 & 0.86 & 0.92 & 2.9 & 0.45 & 0.80 & 0.92 + [ table:3dhst_cv ] it is apparent from figure [ fig : pf_3dhst ] that a considerable scatter exists for the merger fractions measured in the individual fields of the 3dhst+candels . the small survey area ( @xmath20.05deg@xmath20 for each of the five fields ) could lead to systematic uncertainties comparable to or larger than the poisson uncertainties . we list the fractional errors ( @xmath181 ) of the merger fraction measurements of the candels+3dhst sample in table [ table:3dhst_cv ] . the poisson uncertainties of the merger fractions are calculated as @xmath182 . we compute the standard deviation of the merger fraction in each field from the combined mean as @xmath183 , where @xmath184 represents the measurement of each of the five fields . the cosmic variance is simply the observed variance in excess of the poisson random noise , given by @xmath185 . the cosmic variance is a comparable or sometimes larger contributor to the total error budget of the merger fraction measurements than the poisson uncertainty , as visualised in figure [ fig : pf_3dhst ] . more specifically , in the redshift range of @xmath186 the @xmath51 measured from aegis is @xmath187 , whereas the same quantity is measured to be @xmath188 in goods - n . while each of these quantities are @xmath189 from the @xmath190 averaged over the five candels fields , if the individual measurements are taken at face value without including the cosmic variance in the error budget , the results can differ by a maximum of @xmath191 depending on the field used . combining the measurements from the five candels fields is crucial to mitigate cosmic variance , also known as the field - to - field variance @xcite . the cosmic variance affecting the merger fraction measurements depends primarily on the number densities of the massive galaxies and their satellites , as well as the cosmic volume probed , as shown by @xcite . here we use their parametrisation to estimate the relative cosmic variance for the ultravista and 3dhst+candels samples . if we assume that the number densities of the massive galaxies and their satellites are not different in ultravista than in the combined five fields of 3dhst+candels , the cosmic variance has a dependence on the comoving volume as @xmath192 . since the comoving volume is proportional to the survey area , and ultravista covers @xmath193 larger area than the fields of 3dhst+candels combined , we expect the @xmath194 of ultravista to be @xmath195 that of 3dhst+candels . another prominent error of the merger fraction is the poisson number count of pairs . as @xmath196 is proportional to @xmath197 , and again assuming similar number densities of satellites in both fields , we expect @xmath198 and therefore the poisson errors should be @xmath199 smaller in ultravista than that in 3dhst+candels . this implies that the total fractional error of merger fraction measured from ultravista to be @xmath200 that of 3dhst+candels . to summarise , we caution against drawing conclusions from merger fraction measurements based on individual candels - sized fields . the merger fraction measurements from the five 3dhst+candels fields combined are comparable to those from ultravista which covers @xmath201 larger area , albeit with larger poisson uncertainties and in coarser redshift bins . we call for including cosmic variance as a systematic uncertainty for pencil beam surveys such as 3dhst+candels for merger fraction measurements @xcite . minor dry mergers are often invoked as the primary driver of the observed size evolution of quiescent massive galaxies from @xmath174 to 0 . predictions from numerical simulations and virial arguments @xcite suggest that they are more efficient than major dry mergers in puffing up the sizes of quiescent galaxies per unit mass added . from previous minor merger fraction measurements @xcite and this work ( see section [ sec : merger_rates ] ) it is inferred that massive galaxies undergo less than one minor merger since @xmath174 . however , if the sole explanation of the observed size evolution is minor merging , multiple minor mergers are required ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? here we investigate the possibilities of missing faint satellites to massive galaxies at @xmath39 . as discussed in section [ sec : pf_ratios ] , we find that neither the major nor minor merger fractions in the candels deep fields are higher than those in the candels wide fields , although measurements from individual fields are subject to high cosmic variance ( see section [ sec : cv ] ) . additionally , the merger fractions from stellar mass ratio selected mergers of ultravista and 3dhst+candels are remarkably consistent ( figure [ fig : pf_massratio ] , left ) , even in the redshift bins where ultravista is incomplete for low surface brightness galaxies . even though the candels _ h_-band imaging is @xmath1643 magnitudes deeper and has @xmath202 smaller psf compared to ultravista , ultravista has the advantage that it probes a redder band ( _ ks _ ) where high redshift galaxies are brighter . to make a robust claim that we do not miss minor mergers lying just below the surface brightness limits ( sb ) of our surveys , we refer to the simulation performed for the completeness limits as introduced in appendix [ sec : sb_limit ] . in short , we confirm that we do not miss minor mergers up to @xmath15 in 3dhst+candels . we arrive at this conclusion by making the most conservative assumption that the faintest possible satellite is a maximally old , dust - free galaxy of log@xmath203 for a range of light profiles . the completeness limits hold except for the extreme cases not simulated : ( 1 ) they have very compact sizes ( @xmath204 kpc ) and sersic index @xmath205 so that they have insufficient contiguous pixels above the detection threshold ; ( 2 ) they have very large sizes ( @xmath206 kpc ) and low @xmath207 so they have low sb ; ( 3 ) their dust extinction causes them to be fainter than a dust - free maximally old galaxy . these size limits are motivated by the scaling relations for quiescent or early - type galaxies @xcite and simulation assumptions regarding the size of the stellar halo @xcite . unless these intermediate mass galaxies have light profiles very different from the more massive galaxies at similar redshift and similar mass galaxies at lower redshifts , ( 1 ) and ( 2 ) are not likely explanations . the rest - frame optical faintest galaxies at @xmath10 should be quiescent and therefore should be dust - free , therefore ( 3 ) is not a likely explanation either . from binary merger simulations @xcite , the observability timescales of major and minor mergers are very short at @xmath208 kpc @xmath32 ( @xmath209 gyr ) and therefore we do not expect many close pairs blended by the psf . as long as the lower @xmath35 limit for the close pair search is set according to the seeing and sb limit of the data , the resolution is not expected to cause a bias in the merger fraction . as lower mass galaxies are more abundant than massive galaxies , one may expect that minor mergers are more frequent than major mergers from a statistical argument . minor mergers are expected to be visible as pairs for longer than major mergers , according to dynamical friction timescales arguments and binary simulations @xcite . therefore one intuitively expects the minor merger fraction and rate to be higher than the major ones . however , cosmological simulations indicate that the major and minor merger rates are comparable in the stellar mass range probed in this work @xcite due to the stellar mass dependence on the @xmath210 relation . with our large complete sample of mergers , we can study the relative fractions of mergers of different stellar mass ratios ( @xmath211 ) . we present our merger fractions in various @xmath211 bins in figure [ fig : pf_mu ] . the merger fraction decreases as the @xmath211 gets more extreme . the minor ( @xmath212 ) merger fractions are comparable to the major merger ( @xmath213 ) fractions at all redshifts . this is in qualitative agreement with previous observations @xcite . for our sample of stellar mass ratio selected mergers from both datasets , the geometric number - weighted mean stellar mass ratio is @xmath214 4:1 - 5:1 and the mass - weighted mean stellar mass ratio is @xmath215 3:1 - 4:1 . this is in consistency with various model predictions @xcite except @xcite , who find @xmath215 5:1 but @xmath216 16:1 . their simulation is able to resolve down to 100:1 mergers , whereas we impose a cut at 10:1 mergers . we attribute the discrepancy to a higher minor merger rate of their simulated massive galaxies , as well as our imposed cutoff at @xmath211=10:1 . -band flux ratio ) , except for a declining tail towards lower stellar mass ratios for the @xmath7-band flux ratio selection as discussed in section [ sec : pf_ratios ] . on this plot we display the stellar mass selected ratio mergers from 3dhst+candels for illustration . only the data points in which they are complete in stellar mass and surface brightness are shown ( see table [ table : completeness_all ] ) . the major merger fractions appear to be comparable to the minor merger fractions at all redshifts . , scaledwidth=50.0% ] the goal of measuring the galaxy merger fraction is to determine the time integral of the merger rate , defined as the number of mergers ( @xmath217 ) that a massive galaxy experiences on average over a time span . the merger rate can be compared to the observed evolution of the galaxy population , such as in numbers , mass , size , etc . , so that we can infer if galaxy merging is likely a driver . merger rates scale as the number of mergers ( @xmath218 ) occurred during the time span ( @xmath219 ) defined by the redshift bin , divided by the time span , ( rate @xmath220 ) . we measure @xmath221 as the number of observed merging galaxies ( @xmath222 ) divided by the observability timescale of mergers ( @xmath223 ) , i.e. rate @xmath224 . the two common definitions of merger rates can be generalized as follows ( * ? ? ? * and references therein ) : \(1 ) the number of merger events per unit time per unit volume ( @xmath221 ) : @xmath225 = \frac{n_{merge , obs}(z ) / \tau_\mathrm{obs}}{v_\mathrm{comoving}(z ) } = \frac{n_\mathrm{merge}}{\tau_\mathrm{obs } } \ ] $ $ \ ] ] where @xmath226 refers to the number of major ( or minor ) satellites around massive galaxies in that redshift , @xmath227 is the average observable timescale for the mergers of the mass ratio range observed to be within @xmath35 , and @xmath228 is the comoving volume projected by the survey area within the concerned redshift interval . \(2 ) the number of merger events per galaxy per unit time ( @xmath229 ) is defined as : @xmath230 = \frac{\gamma ( z)}{n_\mathrm{massive}(z ) } = \frac{n_\mathrm{merge } } { n_\mathrm{massive } \tau_\mathrm{obs } } = \frac{f_\mathrm{merge}}{\tau_\mathrm{obs } } \ ] $ $ \ ] ] where @xmath231 is the number density of massive galaxies per unit volume . the number of mergers a massive galaxy undergoes on average ( @xmath232 ) is simply the time integral of the merger rate per galaxy : @xmath233 $ $ \ ] ] where @xmath234 is the hubble time , and @xmath235^{1/2}$ ] @xcite with the @xmath236 s denoting the density parameters . merger rates can be inferred by observing the merger fraction as a function of redshift , and then a merging timescale is assumed to convert the fraction to a rate . the assumed merging timescale either comes from binary merger simulations @xcite , cosmological simulations @xcite , or approximation using the dynamical friction timescale . here we briefly discuss the various options and justify the merger timescales used in this work . the dynamical friction timescale @xcite is a suitable approximation for dark matter halo mergers of large mass ratios ( i.e. minor mergers ) . however , it remains uncertain whether it can describe mergers with baryons or major mergers in which violent relaxation is the dominant mechanism determining the duration of the merger . the timescales from binary simulations and cosmological simulations are conceptually distinct . in binary merger simulations ( e.g. * ? ? ? * ) , two galaxies are set on approaching orbits , and the observability timescale ( @xmath223 ) samples the distribution of pre - coalescence pairs as a function of @xmath35 . the timescale @xmath223 is a well - defined quantity which is directly applicable to the merger fraction to rate conversion . this direct simulation method provides an accurate and comprehensible description of merging for the assumed conditions of relative velocity , gas fraction , morphology , etc . on the other hand , merging timescale ( @xmath237 ) defined in cosmological simulations @xcite depends on how the start and end of merging are defined , for example whether the end is the final coalescence of the two galaxy cores or when most of the mass of the satellite galaxy is deposited onto the massive one . another complication is that there are different treatments of mapping stellar masses to the dm halos in cosmological simulations ( e.g. * ? ? ? * ; * ? ? ? we note that merging timescales for major mergers derived using cosmological simulations are shown to be @xmath238gyr longer compared with simulations that include baryons @xcite . most importantly , @xmath223 instead of @xmath237 should be used to convert the observed fractions into rates . therefore in this work we use the @xmath223 from @xcite . the cosmological simulations are useful to weigh the timescales of mergers from binary simulations with different assumptions , such as gas fraction , orbital parameters , as discussed in details in @xcite . due to the systematic uncertainties in these assumptions , as well as random uncertainties due to viewing angles of pairs projected in 2d , the merging ( observability ) timescale can only be determined at best to @xmath239 accuracy ( * ? ? ? * and references therein ) . the merger rates derived using equations [ eqt : mergerrate_vol ] and [ eqt : nmerger ] normalised to timescales of 1 gyr are shown in table [ table : merger_rates_normalized ] . we plot the inferred merger rates on figure [ fig : merger_rates ] . as expected from the merger fractions , we find the merger rates from ultravista are consistent with those from 3dhst+candels within the completeness range , and that the flux ratio selection method gives an increasing trend while the stellar mass ratio selection method gives a flat or diminishing trend for the 3dhst+candels catalog . we list the best fitting parameters for the observed merger rates to a power law in table [ table : bestfit ] for easy comparison to literature . as the merger rate uncertainties are considerably larger than the measured merger fractions due to the @xmath239 uncertainty in @xmath223 , the redshift dependence is weaker and we therefore deem a quadratic fit which has one more degree of freedom than the power law unnecessary . we show the integrated number of major and minor mergers in table [ table : nmerger ] for the two catalogs and selection methods . we find that at @xmath10 the observed merger rates using the stellar mass ratio selection are lower than predicted from the semi - analytical models ( sams ) of @xcite as shown in figure [ fig : merger_rates ] , but are consistent with the gas - poor merger rate ( @xmath240 , where the gas fraction @xmath241 is defined as the ratio of the total gas mass to the total baryon mass of the merging galaxies ) . in general the sams predict that the galaxy merger rates increase monotonically with redshift . our measurements using the @xmath7-band flux ratio selection show an increasing trend similar to the gas - rich merger rate of @xcite ( @xmath242 ) , even though the @xmath7-band flux is not a direct tracer of cold gas mass or star formation rate . this lends support to our claim in section [ sec : pf_ratios ] that using the stellar mass ratio as a probe for the baryon mass ratio may be subject to a bias against gas - rich mergers at @xmath10 , an epoch at which cold gas fraction is non - negligible especially for intermediate mass galaxies @xcite . we also compare the merger rates inferred from the merger fractions of various @xmath35 bins in figure [ fig : merger_rates_difft ] . we only show results for the stellar mass ratio selection , but the following conclusions also hold for the @xmath7-band flux ratio selection . we find that the merger rates are consistent for different @xmath35 bins once the suitable observability timescales from @xcite are applied . on average , the merger rates derived from mergers with @xmath35 = 10 - 100 kpc @xmath32 kpc @xmath32 is still small compared to the typical photo-@xmath14 uncertainty . the typical photo-@xmath14 error at @xmath243 is @xmath244 @xcite , corresponding to 84 mpc / h at @xmath41 . therefore we do not expect the photo-@xmath14 uncertainty to constrain widely separated pairs . ] are up to @xmath245 higher than for smaller @xmath35 bins , although still consistent within the large uncertainties due to the 50% uncertainty in the merger observability timescale . this implies that there are more widely separated mergers ( @xmath246 kpc @xmath32 ) than expected from the timescales of binary merger simulations . possible explanations could be : ( 1 ) the large scale environment of galaxies are probed at separations of @xmath247 kpc @xmath32 , therefore we may include galaxies in the same over - densities that are not bound to merge ; ( 2 ) the merging observability timescales for wide pairs may be systematically longer than the assumed tilted polar orbit for close pairs , e.g. relative velocities of merging pairs are higher than assumed in the binary simulations ( typically @xmath248 km s@xmath17 ) which may be true in over - densities , or if the merger orbit is more like a circular orbit the merging timescale can be up to @xmath249 longer @xcite . we note that the discrepancy is larger at lower redshift , hinting that the effect could be related to large - scale structure formation . cosmological simulations may provide estimates of these effects . although we do not use the timescale of @xcite for galaxy merger fraction measurements for the reasons explained in section [ sec : compare_timescales ] , for comparison we note that using it leads to lower merger rates than those derived using the shorter timescales of @xcite as expected from the inverse scaling between timescale and rate . cccccc + @xmath75 & 0.111@xmath250 & 0.109@xmath251 & & 0.14@xmath252 & 0.138@xmath253 + @xmath78 & 0.076@xmath254 & 0.09@xmath255 & & 0.098@xmath255 & 0.117@xmath256 + @xmath81 & 0.182@xmath255 & 0.125@xmath257 & & 0.169@xmath258 & 0.116@xmath257 + @xmath84 & 0.131@xmath257 & 0.123@xmath259 & & 0.1@xmath260 & 0.094@xmath260 + @xmath87 & 0.084@xmath261 & 0.083@xmath261 & & 0.08@xmath261 & 0.079@xmath261 + @xmath90 & 0.072@xmath262 & 0.072@xmath262 & & 0.044@xmath263 & 0.044@xmath263 + @xmath93 & 0.051@xmath263 & 0.045@xmath263 & & 0.021@xmath264 & 0.019@xmath265 + @xmath96 & 0.02@xmath265 & 0.028@xmath263 & & 0.008@xmath266 & 0.01@xmath265 + @xmath99 & 0.014@xmath265 & 0.016@xmath265 & & 0.009@xmath266 & 0.011@xmath265 + @xmath102 & 0.01@xmath265 & 0.015@xmath264 & & 0.005@xmath266 & 0.007@xmath265 + + + @xmath105 & 0.115@xmath267 & 0.5@xmath268 & & 0.099@xmath269 & 0.43@xmath270 + @xmath108 & 0.066@xmath258 & 0.096@xmath271 & & 0.086@xmath272 & 0.124@xmath273 + @xmath111 & 0.08@xmath258 & 0.073@xmath254 & & 0.078@xmath258 & 0.071@xmath274 + @xmath114 & 0.033@xmath259 & 0.031@xmath260 & & 0.025@xmath260 & 0.024@xmath260 + @xmath117 & 0.024@xmath260 & 0.023@xmath260 & & 0.014@xmath262 & 0.013@xmath262 + [ table : merger_rates_normalized ] cccccc + 10 - 30 kpc @xmath32 & @xmath275 & @xmath276 & & @xmath277 & @xmath275 + 10 - 100 kpc @xmath32 & @xmath278 & @xmath279 & & @xmath280 & @xmath281 + + + 10 - 30 kpc @xmath32 & @xmath282 & @xmath283 & & @xmath284 & @xmath285 + 10 - 100 kpc @xmath32 & @xmath281 & @xmath286 & & @xmath287 & @xmath288 + [ table : nmerger ] we compute the merger - driven stellar mass accretion rate as @xmath289 [ @xmath290 / gyr / galaxy ] = @xmath291 , where @xmath292 is the median stellar mass of the massive galaxies , @xmath229 is the major ( minor ) merger rate , and @xmath293 is the median stellar mass ratio of the major ( minor ) mergers . all these quantities are redshift dependent so we are able to calculate the merger - driven stellar mass growth as a function of time . there is controversy regarding whether merging triggers significant star formation episodes compared to isolated galaxies ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? @xcite study the evolution of the age- , mass - metallicity relation of massive galaxies since @xmath294 to @xmath295 , and report that neither new star formation nor chemical enrichment is needed for the evolution of massive quiescent galaxies . additionally , we do not have measurements of the gas fraction of our merger sample . therefore we note that our analysis only accounts for the accretion of existing stars and ignores stars formed during mergers , setting the lower limit on the merger contribution to the stellar mass growth . we show the stellar mass accretion rate as a function of redshift in figure [ fig : mass_growth ] . for the average massive galaxy of log@xmath4 , we find that major ( minor ) merging leads to an average stellar mass growth of and @xmath296 during @xmath5 . this amounts to a total of @xmath297 being accreted via 1:1 - 10:1 mergers , implying that the average @xmath298 galaxies increase their stellar masses by at least @xmath299 through accreting existing stars from satellite galaxies from @xmath15 to 0.1 . our results are in agreement with similar observations for bright central galaxies in galaxy clusters @xcite and field galaxies @xcite up to @xmath12 , showing that major merging plays a significant role in the mass assembly of massive galaxies ( and therefore its number density evolution ) independent of the environment . our stellar mass accretion rates are also consistent with simulation predictions @xcite with the exception of @xcite . @xcite follow the history of simulated massive galaxies and find that by @xmath300 , 80% of the stars in massive galaxies are formed at @xmath301 ex - situ of the original halo at @xmath302 , and are accreted at @xmath303 with an average rate of @xmath304/yr . their average mass accretion rate stays relatively flat at @xmath10 and decreases at lower redshift , which is qualitatively similar to our observed trends but on average @xmath305 higher , as seen in fig . [ fig : mass_growth ] . as we discussed in section [ sec : expect_minor_merger ] , this is explained by the higher minor merger rates in their simulations compared to the observations of this works and others . we emphasise that the stellar mass accretion rate presented here does not include new stars formed due to merger - triggered star formation episodes , and therefore represents a lower limit of the true merger - driven stellar mass growth rate ( see also the discussion in section [ sec : pf_ratios ] ) . dry merging provides a channel to increase the sizes of compact ( @xmath306 kpc ) massive quiescent galaxies ( qgs ) at @xmath10 by a few factors to @xmath295 ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) , as discussed in section [ sec : few_minor ] . we use our measured stellar mass accretion rate to infer an upper limit on the size evolution due to `` dry '' dissipationless merging . since qgs are expected to remain quiescent for the build - up of the red sequence , and the dissipation from gas in merging galaxies can reduce the efficiency of puffing up sizes of galaxies , for this exercise we make the simplistic assumption that all observed mergers are dissipationless . the aim of the test is to investigate to what extend the observed frequency of galaxy merging can explain the size evolution of qgs . we find that the merger fractions of massive galaxies and the quiescent subset are consistent within their uncertainties , therefore we simply use the merger fractions of the overall massive galaxy population in the following analysis . the virial theorem and more sophisticated merger simulations have been used to predict the size evolution due to dry merging . the size evolution can be parameterised as @xmath307 , where @xmath308 for major merging and @xmath309 for minor merging predicted using the virial theorem @xcite , or alternatively @xmath310 for major merging and @xmath311 for minor merging according to the simulations of @xcite . the high value of @xmath312 for minor merging in @xcite implies that it is very efficient in increasing the sizes of galaxies , and likely represents an upper limit due to the high dark matter content and extended stellar haloes of the satellites assumed in their simulation . for each redshift bin , we multiply the average stellar mass accretion rate ( see section [ sec : mass_growth ] ) with the time elapsed in the redshift bin to get the stellar mass accreted , and scale the predicted size growth to the stellar mass accretion using the @xmath312 values as discussed above . the maximum merger - driven size growth using both catalogs are plotted in figure [ fig : size_evolution ] . we observe that the total amount of merging can only increase the size of massive qgs by a factor of two , from 1.5 kpc at @xmath15 to @xmath313 kpc at @xmath295 . this result is insensitive to the size growth model used , meaning that the virial theorem provides a good approximation of the size evolution due to dissipationless merging . the observed size evolution of massive qgs ( or early - type galaxies ) has been presented in various works . on figure [ fig : size_evolution ] we compare our predicted merger - driven size evolution to two recent measurements using candels . @xcite report an average size growth of @xmath314 from @xmath15 to 0 , with a redshift dependence of @xmath315 , consistent with previous works including @xcite and @xcite . on the other hand , @xcite report a consistent but slightly stronger size growth of @xmath316 times in the same redshift range , with a redshift dependence of @xmath317 , similar to the finding of @xcite . both works report the scatter of the stellar - mass size relation to be consistent with being constant . the difference of the observed size evolution may be due to the stellar mass threshold , as well as the size measurement technique . as the primary focus of this paper is not the observed size evolution , we can only conclude that merging increases the sizes of a @xmath298 qg by a factor of two at most from @xmath318 to 0 . while this is insufficient to explain the observed average size growth of a factor of 3 - 5 , it is enough to bring the average sizes of massive qgs to @xmath11 below the local mean stellar mass - size relation if the redshift dependence is on the milder end of the observations ( @xmath315 ) like in @xcite . if the sizes follow a normal distribution , the massive qgs already formed and quenched since @xmath318 evolve through merging to form the smallest 16% ( 2% ) of local massive qgs since they lie at @xmath11 ( @xmath319 ) below the mean . if the sizes follow a skewed distribution instead , as shown by @xcite , the fraction can be even higher ( e.g. up to the smallest 12.5% for @xmath319 below mean following chebyshev s inequality ) . this may be a more relevant representation if these compact qgs end up to lie below the local mass - size relation , while the majority of later quenched qgs occupy the upper part of the relation . recent measurements of compact massive qgs reveal that their number densities peak at @xmath320 , and decrease at lower redshifts @xcite , therefore they must undergo structural changes . incidentally this is the same redshift range in which our merger rate peaks ( major : @xmath321 , minor : @xmath322 , see fig . [ fig : merger_rates ] ) . we will further the discussion on the observed size evolution in section [ sec : etg_evolution ] . even though there may be a significant number of minor mergers rejected by the stellar mass ratio criterion ( flux ratio between 1:1 and 10:1 , but stellar mass ratio more extreme than 10:1 ) , these mergers are more likely to have non - negligible gas mass and more dissipation so it does not help to solve the problem of the observed size evolution . the gas content of merging galaxies may explain the scatter of the redshift - size evolution @xcite . however without gas measurements we are not able to test this hypothesis at this point . the virial theorem predicts that equal - mass mergers do not change the stellar velocity dispersion @xmath323 , and minor mergers reduces the @xmath323 by @xmath324 if the satellite has a @xmath323 much lower than the massive galaxy it is merging with @xcite . using the stellar mass accretion rate we estimate that 4:1 - 10:1 minor mergers can only reduce the @xmath323 of massive galaxies by 6% from @xmath15 to 0.1 . if we relax the assumption and allow 1:1 - 4:1 mergers to be equally efficient in reducing @xmath323 , the total stellar mass accreted implies that the @xmath323 decreases by maximum 25% from @xmath15 to 0.1 . from this we conclude that merging is insufficient to reduce the high @xmath323 ( @xmath325 km s@xmath17 ) observed in @xmath174 qgs @xcite by @xmath326 to match the average of the local population . this is consistent with claims that the addition of lower @xmath323 galaxies to the quiescent population at later times contribute to the decreasing average @xmath323 of the overall massive qg population @xcite . we note that if a significant amount of dark matter is accreted by these massive qgs , the total mass increases and therefore the velocity dispersion and the sizes may change without any observable stellar mass growth . ) above two stellar mass thresholds . the blue ( red ) filled circles represent the observed @xmath327 of massive galaxies of log@xmath8 ( 11.4 ) , and the error bars represent the poisson error of the number counts . the triangles represent the major - merger driven @xmath327 growth using two merger observability timescales ( @xmath3280.5 gyr : downward triangles , dashed lines and yellow shades ; 1.0 gyr : upward triangles , solid lines , purple shades ) . the colored shades show the uncertainty on @xmath327 propagated from the poisson errors of the number of mergers . the predicted major - merger driven @xmath327 growth accounts for the formation of `` new '' massive galaxies above the threshold due to major merging , as well as the reduction in numbers of massive galaxies that merge with each other ( a minor effect as observed ) . the predicted growth is normalised to the observed @xmath327 of massive galaxies @xmath329 to which we are complete for major mergers . we only perform this exercise on the ultravista catalog , because the 3dhst+candels contain too few galaxies above these stellar mass thresholds for meaningful @xmath327 constraints . we find that the slope of the observed @xmath327 evolution of the most massive galaxies follows the predicted slope due to major merging , if the @xmath330 gyr @xcite for major merging which is roughly the average of the two timescales shown . to keep the slope consistent with the observed number densities , a maximum of @xmath331 stellar mass can be added in addition to major merging , implying @xmath332 for mechanisms other than major and minor merging . , scaledwidth=50.0% ] to understand what the merger rates from section [ sec : compare_merger_rates ] imply for the overall galaxy evolution , in this section we aim to quantify the contribution of merging to the observed increase in the number density ( @xmath327 ) of massive galaxies in the redshift range @xmath333 . as shown in section [ sec : mass_growth ] , most of the stellar mass accreted is through major merging , so in this section we only consider major merging for which our samples are complete to higher redshifts . merging can affect the number counts of massive galaxies in two counteracting ways . on one hand , merging among lower mass galaxies can increase the number of massive galaxies above a stellar mass threshold . on the other hand , merging among massive galaxies already above the mass threshold will lead to a decreased number count . we denote @xmath334 as the number of mergers with individual stellar masses lower than a given threshold , but with the sum of their stellar masses above the threshold @xcite , and @xmath335 as the number of mergers with the individual stellar masses of both galaxies above the threshold . the net change of @xmath327 due to major merging is @xmath336 , where @xmath337 and @xmath338 are the comoving volume and the elapsed time of the redshift range , and @xmath223 is the merger observability timescale given the projected separation ( @xmath35 ) range . the @xmath223 for major mergers with @xmath35 = 10 - 30kpc @xmath32 is about 0.6 - 0.7 gyr @xcite with an error of @xmath339 gyr . in this exercise we show the results of two values of @xmath223 ( 0.5 and 1.0 gyr ) . since we assume that no new stars are formed during mergers for the reasons discussed in section [ sec : mass_growth ] , the presented quantities mark the minimum merger contribution to the formation of new massive galaxies . we present the results in figure [ fig : numdens_growth ] . we find that major merging alone can explain the @xmath327 evolution of galaxies more massive than @xmath340 if @xmath223 lies between 0.5 - 1 gyr . if @xmath223 was systematically much longer than 1 gyr , then additional mechanisms may be required to explain the @xmath327 evolution of these very massive galaxies . we note that 3dhst+candels is inadequate for tracing the @xmath327 growth of the most massive galaxies . the volume probed is too small leading to large cosmic variance on the observed number density and therefore is not shown . taking our results further , we use the observed @xmath327 evolution of the most massive galaxies to constrain the upper limit of the stellar masses that can be added in addition to major merging . we increase the stellar masses of all the galaxies by an arbitrary factor , and count the number of galaxies @xmath341 that cross the given mass thresholds . its contribution to the @xmath327 evolution is @xmath342 . we find that the observed @xmath327 evolution is marginally consistent with a maximum 15% of stellar mass growth of the overall massive galaxy population in addition to major merging since @xmath318 . any non - major merging stellar mass growth beyond @xmath331 would overproduce the number of the most massive galaxies . as shown in section [ sec : mass_growth ] , minor merging accounts for @xmath343 of the stellar mass accreted . therefore we conclude that there remains little room ( @xmath332 ) for the most massive galaxies to increase their stellar masses by mechanisms other than major and minor merging , such as star formation or very minor mergers ( @xmath34410:1 ) . there are comparative studies of the possible mechanisms that can explain the size evolution @xcite . merging , in particular dry minor merging , appears to be a viable means to explain the observed size and velocity dispersion evolution . however , even when we assume that all mergers were dry ( dissipationless ) , the size evolution inferred from our merger fraction can only account for a factor of two of size increase from @xmath3452.5 to 0.1 . this is marginally consistent with being @xmath11 below the mean stellar - mass size relation of the measurement of @xcite , but @xmath346 compared to that of @xcite . this necessitates additional mechanisms to explain the observed size increase for the bulk of the population . the apparent strong size evolution may be in part due to observational effects . our observations indicate that massive galaxies tend to merge with galaxies with lower stellar mass - to - light ratios ( see figure [ fig : mtol_z ] and section [ sec : pf_ratios ] ) . if the younger , bluer stars of the companion are added to the outskirts of massive galaxies consisting of older stellar populations @xcite , then the half - light radius ( @xmath347 ) measured in rest - frame optical bands increases . this scenario is supported by the observed negative colour gradients @xcite , and is consistent with the observation of @xcite that the @xmath347 of massive galaxies are smaller when measured at longer wavelengths . @xcite show that the half - mass radii of massive qgs are on average @xmath348 smaller than the half - light radii measured from the rest - frame @xmath349-band . therefore the observed size evolution is perhaps in part due to the radial dependence of the @xmath350 . since the number- and mass - weighted average stellar mass ratio is @xmath2 4:1 for the mergers in this work , the satellites may strip off their stars at the outskirts like the 5:1 intermediate mass ratio merger simulated by @xcite , lending support to merging as a viable explanation for the observed size evolution and color gradients . it is important to distinguish between the growth of individual galaxies and the evolution of the overall population . the number density of the massive qgs evolves with redshift , for instance massive ( @xmath298 ) galaxies are 30 times more abundant at @xmath351 than @xmath352 ( e.g. @xcite and references therein , also see section [ sec : numdens_evolution ] ) . therefore if larger , later quenched galaxies are continuously added to the qg population , it may be sufficient to increase the average sizes of qgs ( more details about the so - called `` progenitor bias '' in @xcite ) . this assumes that the sizes of qgs are correlated with their age or time since being quenched , a trend which is observed in some works @xcite but not in others @xcite . another implication is that the scatter of the size evolution is expected to increase if the progenitor bias is the sole explanation for the observed size evolution , which contradicts the constant scatter observed @xcite . additionally , the progenitor bias alone does not explain the disappearance of compact qgs observed at @xmath10 @xcite . the number density of compact qgs peaks at @xmath353 and decreases towards lower and higher redshifts . our merger fractions ( stellar mass ratio selected ) peak at @xmath354 , and one may speculate on a causal relation between the two observations . a fixed number density selection may provide a more direct comparison between massive qgs at @xmath174 and their descendants at lower redshifts ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? if the descendants of compact massive qgs at @xmath355 are the most compact qgs in clusters in the local universe , the sizes of individual qgs will only need to increase by a factor of @xmath356 @xcite , which is in good agreement with the size evolution inferred from our merger rates . apart from the observational effects and the progenitor bias discussed above , alternative means to increase the sizes of individual qgs have been proposed . some examples include agn and/or supernova feedback @xcite , adiabatic cooling via the mass loss of old stars @xcite , and halo size evolution @xcite . it is beyond the scope of this work to draw conclusions on the relative contributions of the possible options in explaining the size evolution . we emphasise that our results provide a strong constraint : whichever mechanisms are responsible for the observed size evolution , there is little room for further stellar mass to be created or added ( @xmath9 at most for @xmath357 ) for the most massive galaxies ( @xmath358 ) in order not to over - produce the observed numbers at different redshifts . the open question of whether merging is a major contributor to the cosmic star formation history ( sfh ) has been tackled in different ways : do merging galaxies have higher star formation rates compared to isolated ones @xcite ? at each epoch , are star - forming galaxies primarily mergers or isolated disks @xcite ? these different perspectives can lead to seemingly contradictory conclusions . despite the apparent offset of visually identified mergers from the sfr-@xmath359 relation ( dubbed main - sequence , @xcite ) , merging galaxies only show disturbed morphologies for a limited time ( @xmath360 gyr , e.g. @xcite ) . if the duty cycle of mergers is interpreted as the cause for the scatter of the sfr-@xmath359 relation , major mergers account for a majority of the total sf at @xmath361 @xcite . @xcite have shown that mergers can enhance sfr to @xmath362 kpc , and such widely separated merging galaxies are likely not identified in morphological selected samples which probe later - stage mergers . on the other hand , the existence of isolated star - forming disks has been used as evidence against mergers being a contributor of cosmic sf budget based on the assumption that mergers destroy disks ( e.g. * ? ? ? while mergers _ can _ destroy disks and remain a popular explanation for bulge formation @xcite , various works have shown that disks can reform after gas - rich mergers @xcite . even though galaxy merging may not increase the total amount of stars formed from the available cold gas reservoir , it can trigger starburst episodes by temporarily enhancing the star formation efficiency , leading to faster cold gas depletion @xcite . detailed studies of the sfh of individual galaxies can provide an answer to whether most stars in galaxies are formed during merging or isolated phases ( continuous vs bursty sfh ) . in section [ sec : pf_ratios ] we have shown that using the @xmath7-band flux ratio to select mergers leads to an increasing merger fraction evolution , as opposed to the flat or diminishing trend seen using stellar mass ratio selected pairs . the former merger fraction share a similar redshift evolution as the cosmic star formation rate density ( e.g. @xcite and references therein ) albeit with considerable uncertainties : both rise from @xmath295 to @xmath12 and reach a plateau or increase mildly from @xmath12 to @xmath318 . this may be a hint that at @xmath363 , massive galaxies are primarily merging with low stellar mass ( @xmath364 10:1 ) but gas - rich satellites . these mergers are classified as major or minor depending on whether the baryon mass or stellar mass ratio is used . when inferring the merger contribution to the cosmic star formation budget , we need to account for these `` missing '' mergers @xcite that did not enter the stellar mass ratio selection . future surveys of the molecular gas mass of high-@xmath14 galaxies are needed to make progress on this issue . the merger fraction of massive galaxies is @xmath365 , resulting in low number densities of mergers ( @xmath366 mpc@xmath367 ) at @xmath10 . as we show in section [ sec : cv ] , cosmic variance is the dominant source of uncertainty in merger fraction measurement with candels - sized surveys , due to the small survey area and low source number density . we note that the merger fractions measured from ultravista and 3dhst+candels yield very consistent results ( see figure [ fig : pf_3dhst ] ) , even at the redshifts where ultravista is expected to be incomplete for low surface brightness satellites . this is due to the fact that most satellites have lower @xmath350 ratios ( see section [ sec : pf_ratios ] and figure [ fig : mtol_z ] ) . as long as the lower limit of @xmath35 is set so that no close pairs are missed due to blending , and the relevant observability timescales are applied for the @xmath35 range @xcite , deep ground - based nir surveys like ultravista and uds provide as accurate results as _ hst _ surveys . ground - based surveys have the additional advantage of larger sample sizes , so that the evolution can be probed in finer redshift bins with small poisson uncertainties . put another way , large area surveys are crucial to mitigate cosmic variance and poisson uncertainties in galaxy merger fraction measurements . a limitation of the pair selection is that a minimum @xmath35 must be imposed to match the resolution of the imaging data , for example 10 kpc @xmath32 in this work . if the scientific interest is on the incidence of late stage mergers of @xmath368kpc @xmath32 among agns or ulirgs ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) , alternative merger identifications may be a more appropriate choice ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? photometric redshifts ( photo-@xmath14 s ) are essential in removing line - of - sight projected pairs from merger samples . the projected pair fraction is redshift dependent and can reach @xmath369 at @xmath370 ( see table [ table : pf_massratio ] ) . statistical simulations can provide an estimate for the number of projected pairs , however photo-@xmath14 s are crucial for selecting real mergers for spectroscopic follow - up . one may expect photometric samples of mergers to include more mergers due to the larger uncertainties of photo-@xmath14 s than spec-@xmath14 s , however the merger fractions presented in this work using photometrically selected mergers are in agreement or even lower than those using spectroscopic selected mergers @xcite . aside from the variations of the parent sample as discussed in @xcite , this may be an indication that the selection effects associated with the spectroscopic merger samples outweigh the uncertainties of photo-@xmath14 s in photometric merger samples , e.g. mass - incompleteness ( due to flux - limited selection ) , slit / fiber placement incompleteness , limited sample sizes and so on . therefore we argue that large - area ( @xmath371 deg@xmath20 ) surveys with accurate photo-@xmath14 s currently provide the most time - efficient datasets for measuring galaxy merger fractions . on the theoretical front , the merging probability of galaxy pairs in close physical separations need to be quantified as a function of redshift and environment , as discussed in section [ sec : method ] . it is also important to understand how galaxy fly - bys can impact the structure and dynamics of massive galaxies . these are subtle yet crucial quantities that fold into the interpretation of the inferred galaxy merger rates , which are paramount in determining whether galaxy merging is a significant driver of its evolution . we present the largest sample of photometrically selected mergers at @xmath333 from mass - complete catalogs , using complementary datasets of a large area ground - based survey ( ultravista ) and a deep spaced - based survey ( 3dhst+candels ) . we measure the galaxy major and minor merger fractions ( @xmath51 and @xmath52 ) . applying the merging observability timescale ( @xmath223 ) from @xcite , we infer the merger rates , as well as the evolution in stellar mass , size and number density for massive galaxies . we summarise our findings as follows : 1 . the merger fraction shows a steep increase from @xmath295 to 1 , with @xmath51 showing a stronger evolution than @xmath52 . using the stellar mass ratio selection ( figure [ fig : pf_massratio ] , left ) , @xmath51 and @xmath52 show a plateau at @xmath372 and diminishes beyond @xmath320 . if the observed @xmath7-band flux ratio selection is used instead ( figure [ fig : pf_fluxratio ] , right ) , @xmath51 and @xmath52 increase monotonically with redshift . the ultravista and 3dhst+candels show discrepant results at @xmath39 due to the magnitude limit of the ultravista dr1 survey . 2 . selecting mergers by the observed @xmath7-band flux ratio leads to an increasing merger fraction with redshift , while selecting mergers by stellar mass ratio shows a diminishing redshift dependence . this variation in merger selection technique is the cause of the discrepant merger fraction measurements at @xmath39 in the literature @xcite . the discrepancy is a consequence of the @xmath350 evolution of galaxies with redshift : at high redshifts and lower @xmath359 , galaxies have higher star formation rates and lower @xmath350 ratios . the two selections produce consistent merger fractions at @xmath373 , but the fractions diverge at @xmath374 the @xmath7-band flux ratio selection is biased towards bright , star - forming low - mass satellites at @xmath363 , and the stellar mass ratio selection is biased against low - mass satellites which have significant cold gas mass . cold gas measurements for massive galaxies and their satellites are required to refine the merger definition using the baryon mass ratio . our inferred merger rates using the stellar mass ratio selection is consistent with the gas - poor ( @xmath375 ) merger rates of the simulations of @xcite . on the other hand , our inferred merger rates using the @xmath7-band flux ratio selection is consistent with their predicted gas - rich ( @xmath376 ) ones . we get consistent merger rates when mergers are selected from different @xmath35 bins ( 5 - 20 , 10 - 30 , 10 - 50 , 10 - 100 kpc @xmath32 ) when the relevant @xmath223 from @xcite are applied . however , we note that the widest @xmath35 are systematically higher than the other bins , with a more noticeable discrepancy at lower redshift . this is consistent with the pairs at 50 - 100 kpc @xmath32 probing large - scale structure formation . the results imply that an average massive ( @xmath377 ) galaxy experiences @xmath378 major and @xmath379 minor mergers over the redshift range of @xmath5 , if mergers are selected by stellar mass ratio . there may be an additional @xmath380 major merger and @xmath360 minor merger if mergers are selected by the @xmath7-band flux ratio . the mass - weighted average stellar mass ratio is @xmath2 3:1 - 4:1 , implying that the inferred stellar mass accretion rate is primarily driven by intermediate mass ratio mergers up to @xmath318 . this work extend the expectations from @xmath381 to @xmath318 that major merging is the dominant process for stellar mass accretion for massive galaxies . 7 . major and minor merging combined can at most increase the sizes by a factor of two from @xmath15 to 0.1 for an average @xmath382 quiescent galaxy , if we assume that all mergers are dry . additional mechanisms are thus required to explain the strong observed size evolution ( factor of @xmath383 ) . the observed amount of major merging is sufficient to explain the evolution of the formation of new massive ( @xmath358 ) galaxies by number density arguments . these very massive galaxies can only increase their stellar masses by at most @xmath384 during @xmath357 by processes in addition to major and minor merging , in order to match the observed number density evolution . this hints that star formation and very minor merging are unlikely mechanisms responsible for the observed size evolution . we are grateful for the contributions of the ultravista , comsos , candels and 3dhst collaborations for making the catalogs available for public use . am acknowledges tomo goto , knud jahnke , and jennifer lotz for helpful conversations at the early phase of this project . am also thanks bo milvang - jensen for clarification of the ultravista data , and anna gallazzi for clarifying the stellar population models . the dark cosmology centre is funded by the danish national research foundation . we acknowledge the hpc facility at the university of copenhagen for providing the computing resources used in this work . st and az gratefully acknowledge support from the lundbeck foundation . this work has made use of the ultravista catalog , which is based on data products from observations made with eso telescopes at the la silla paranal observatory under eso programme i d 179.a-2005 and on data products produced by terapix and the cambridge astronomy survey unit on behalf of the ultravista consortium . this work is in part based on observations taken by the 3d - hst treasury program ( go 12177 and 12328 ) with the nasa / esa hst , which is operated by the association of universities for research in astronomy , inc . , under nasa contract nas5 - 26555 . the 3dhst+candels catalog is compiled using the datasets in these papers : @xcite ; almaini / foucaud in prep and dunlop et al . in prep . in order to measure the merger fraction evolution robustly , it is essential to ensure completeness in the identification of merging satellites especially at high redshifts . we assess the completeness of faint satellites in two aspects : 1 . stellar mass completeness : is ultravista mass complete at high-@xmath14 for the 10:1 satellites ? 2 . surface brightness ( sb ) : do we miss low sb faint satellites ? we present our analysis in the following subsections . we estimate the stellar mass ( @xmath359 ) completeness of the ultravista catalog by comparing the @xmath385-band magnitudes and photo-@xmath14 s of the detected galaxies with those of the deeper @xmath385-band selected fireworks catalog ( @xmath386 at 5@xmath34 depth , @xcite ) in the chandra deep field south . assuming that the fireworks catalog is 100% complete , we take the fractions of massive galaxies in fireworks above different @xmath359 in different redshift bins which are fainter than the ultravista survey magnitude limit as the mass completeness limits . the results are shown in figure [ fig : mass_completeness ] . from this comparison we estimate that for the ultravista sample , massive galaxies of log@xmath4 are @xmath387 complete at @xmath388 . major ( @xmath389 4:1 ) satellites of log@xmath390 log@xmath391 are above 80% complete for @xmath392 . minor satellites ( 4:1 @xmath393 10:1 ) of log@xmath394 log@xmath395 are above 80% complete for @xmath396 . we list the @xmath387 limits in table [ table : completeness_all ] . the candels survey is sensitive to faint objects ( @xmath397 at 5@xmath34 depth , @xcite ) . for example quiescent galaxies with @xmath398 are 50% complete at @xmath399 ( 3.2 ) for wide and deep regions @xcite , therefore we expect the stellar mass completeness not to be an issue . -band magnitude distribution of ultravista to the deeper fireworks catalog . the stellar mass bins of the massive galaxies of log@xmath4 , as well as their 4:1 and 10:1 satellites are shown in different colors as indicated in the legend . the dashed line shows the @xmath400 completeness limit . , scaledwidth=50.0% ] ccc ultravista & @xmath401 & @xmath402 + candels & & @xmath402 + ultravista & @xmath403 & @xmath40 + candels & & @xmath42 + ultravista & @xmath40 & @xmath41 + candels & & @xmath15 + [ table : completeness_all ] the detection of objects at faint magnitudes is sensitive to their surface brightness ( sb ) profiles and the source extraction thresholds . in order to test the redshift limit up to which we are complete to detecting the faintest possible satellites , we simulate the source detection by simulating galaxies with a range of sersic profiles with magnitudes determined by a dust - free , maximally old stellar population at given @xmath359 and redshifts . the effective half - light radii ( @xmath347 ) assumed are the extrema calculated from the observed scaling relations and/or simulations , as described in detail below . to emulate the actual observations of ultravista and candels , the sersic profiles are smoothed to the instrument psf and added to images with blank patches of sky , and sextractor is run with the object detection settings of the respective catalogs . structural measurements of intermediate mass ( @xmath404 ) galaxies at @xmath355 are sparse due to their faintness . we list the possibilities here and select the extreme sizes for our simulations . 1 . observationally , the sizes of local elliptical or early - type galaxies scale with stellar mass as @xmath405 for @xmath406 . the observed @xmath174 stellar mass - size relation has a similar slope @xcite . if intermediate mass galaxies have the same stellar density as massive galaxies , then the radius scales with stellar mass as @xmath407 . lastly , numerical simulations for merger - driven size evolution have shown that a hernquist profile in projection can be described by a sersic index of @xmath408 @xcite . these simulations use the same scale radius for the stellar halos of the host galaxy and the satellite which has only a tenth of the host stellar mass for the `` diffuse '' case . considering the above mentioned possibilities , we simulate the two extreme sizes of a @xmath409 quiescent galaxy : the most compact ( constant stellar density : @xmath407 ) and the most extended ( simulation : @xmath410 ) . observations show that a @xmath411 quiescent galaxy has log@xmath412 at @xmath413 and log@xmath414 at @xmath415 @xcite , with a scatter of @xmath416 . we scale the sizes to one - tenth of the stellar mass with the extreme scenarios , e.g. our simulated @xmath409 galaxy at @xmath15 has @xmath347 of 0.29 kpc ( compact ) to 1.95 kpc ( extended ) , equivalent to 0.035@xmath417 and 0.248@xmath417 . we assume a maximally old , dust - free stellar population with a single burst and highest metallicity ( @xmath418=0.03 ) to compute the faintest possible magnitudes for these intermediate mass galaxies using the updated version ( 2012 ) of the stellar population synthesis model library of @xcite . this corresponds to magnitude limits of @xmath419 and @xmath420 for a @xmath409 maximally old galaxy at @xmath15 . we simulate different light profiles using three sersic indices ( @xmath421[0.5 , 1 , 4 ] ) , in which the latter two represent the exponential disk profile and the de vaucouleurs profile respectively . we assume two axial ratios of @xmath422 $ ] , though we note that lower axial ratios are easier to detect when the source is closer to the sb limit . with the assumed parameters we generate sersic models according to the @xmath7 and @xmath385 limits . we smooth the images with a gaussian beam corresponding to the psf size of the imaging surveys . then we add them to blank regions on the candels - wide @xmath7-band and the ultravista @xmath385-band images , and we extract sources from the simulated images with the corresponding sextractor settings of the two surveys . we outline the results of our simulation for both catalogs . for the ultravista dr1 catalog , as long as the source is brighter than @xmath385=24.2 - 24.3 mag arcsec@xmath20 , we are able to extract the sources for all the sersic models simulated . this corresponds to @xmath40 ( 1.5 ) for using ultravisata dr1 to detect major ( minor ) satellites . the limit for the candels wide catalog is @xmath7=26.45 mag arcsec@xmath20 , corresponding to @xmath160 ( 2.5 ) for major ( minor ) satellites . we note that these limits are more constraining that those derived from a simple stellar mass completeness argument ( appendix [ sec : mass_completeness ] ) . from this test we observe that the source detection for faint objects close to the sb limit depends on the following structural parameters : ( 1 ) @xmath347 : for a given integrated magnitude , the larger the @xmath347 the lower the sb per pixel . sufficient pixels ( 10 pixels following ultravista and candels settings ) above the sb threshold are required for a detection ; ( 2 ) @xmath423 : for a given integrated magnitude , the combination of a very low @xmath423 and very extended @xmath347 may lead to too low sb / pix for detection . on the other hand , for a very high @xmath423 and very compact @xmath347 a non - detection may result due to the insufficient number of pixels above the detection threshold ; ( 3 ) @xmath424 : if the axis ratio is close to 1 , the flux densities are divided over more pixels than the case of a lower @xmath424 , resulting in an insufficient number of pixels above the detection threshold . we note that our derived limits may be subject to change , if there are systematic uncertainties in the magnitudes and/or the stellar mass . namely , the magnitude limits are derived from dust - free models , which may be reasonable assumptions given that the faintest possible galaxies at @xmath15 are not actively star - forming . on the other hand , there are known systematic uncertainties in stellar masses ( @xmath425 dex ) and ages from sed fitting due to different assumptions of imf or stellar population synthesis model . if the modeled magnitudes are actually fainter or if the stellar masses are underestimated , then our sb completeness limit may be lower than the numbers quoted here . we only compare our results with previous merger fraction measurements using the close pair selection but not the morphological selection ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? as the morphological selection is sensitive to the imaging quality , merger fraction measurements may be subject to large systematic uncertainties beyond @xmath12 . we refer readers to @xcite for a comprehensive review on the two methods , and focus on comparing our results with works that use the close pair method to identify mergers . we note that for the few studies which cover a different @xmath35 range than our data points shown on figure [ fig : pf_lit ] , we use the observability timescales of @xcite to correct the merger fractions for a fair comparison . we compare our merger fractions with @xmath426 studies using the close pair selection . as the selection criteria vary slightly across studies , we re - run our selection according to the published studies for a fair comparison . we compare our merger fractions with similar studies that select mergers using the stellar mass ratio @xcite . in these studies , the projected separation limits are @xmath35 = 13 - 30 kpc @xmath32 and 10 - 30 kpc @xmath32 respectively . we replicate the selections by slightly modifying our criteria : we search for satellites around massive quiescent galaxies ( @xmath427 and ssfr @xmath428 ) , using a limit of @xmath35=10 - 30 kpc @xmath32 . we note that the results of @xcite are based on satellites around quiescent galaxies at lower stellar masses ( @xmath429 ) . we check that lowering the stellar mass cut by 0.3 dex gives consistent merger fractions within the large poisson uncertainties , as is also shown in ( * ? ? ? * table 3 ) . the comparison is shown in figure [ fig : pf_lit_massratio ] ( left ) . we find our @xmath51 to be consistent with that of @xcite , and the one measured from ultravista is @xmath430 higher than that from @xcite at @xmath12 and 1.8 . we note that in these redshift bins , @xcite show slightly higher @xmath52 than in other fields . therefore we conclude that the combined @xmath51 and @xmath52 measured in our data and in @xcite are in good agreement . the discrepancy of @xmath431 in the @xmath51 can be explained by the separation of major and minor mergers , as well as cosmic variance and photo-@xmath14 criterion variation . this discrepancy does not affect the conclusions made in this work . @xcite and @xcite search for satellites of @xmath7-band flux ratios down to 4:1 around galaxies more massive than @xmath432 , within projected separations of @xmath433 kpc , i.e. 21 kpc @xmath32 . in particular , @xcite impose a lower limit of @xmath434 kpc to screen out confused pairs which are likely unresolved with nicmos . this comparison is illustrated in figure [ fig : pf_lit_fluxratio ] ( right ) . our @xmath51 is consistent with these studies . @xcite present the first measurement of the @xmath51 at @xmath13 in the hudf using the stellar mass ratio selection . they use a smaller @xmath435 20 kpc @xmath32 and search for satellites around galaxies of @xmath436 , which is six times lower than our mass criteria . this may explain why their @xmath51 to be @xmath239 higher than ours . as discussed in section [ sec : future ] , flux - limited spectroscopic surveys may lead to biased merger fractions due to mass incompleteness , slit / fiber collision , etc . bearing in mind the difference in the merger selection , we compare our results using photometric mergers with those using spectroscopic mergers . our results are consistent with @xcite who measure the @xmath51 and @xmath52 of @xmath437 galaxies from the spectroscopic survey of vvds up to @xmath12 using the @xmath172-band flux ratio selection . the observed @xmath7-band corresponds approximately to the rest - frame @xmath172-band at @xmath318 and therefore our results using the flux ratio selection are directly comparable to their work . @xcite and @xcite extend measurements of spectroscopic merger fractions to @xmath438 , in which the former use a flux ratio selection for star - forming galaxies and the latter a stellar mass ratio selection . both works report a @xmath51 of @xmath439 . our major merger fraction are marginally consistent with that of @xcite although we note that their primary sample consists of star - forming galaxies only , and may include more mergers if merging does trigger star formation activity . our merger fractions are @xmath440 lower than that of @xcite . both of these studies sample the mergers around less massive galaxies ( 0.8 - 1.7 dex lower than our mass limit ) , and we speculate that it may account for the higher fractions . @xcite select mergers photometrically with the @xmath385-band flux ratio , and report a mildly increasing @xmath51 for massive ( @xmath442 ) galaxies from @xmath300 to 1.2 . when compared to our @xmath51 using the @xmath7-band flux ratio for the similar @xmath359 and @xmath35 range ( figure [ fig : pf_lit_fluxratio ] , right ) , our results are in good agreement with theirs . @xcite measure the @xmath51 and @xmath52 of massive ( @xmath442 ) galaxies in zcosmos at @xmath443 , selecting mergers by stellar mass ratio and relative velocity @xmath444 km s@xmath445 . they find a redshift dependence of the @xmath51 as @xmath446 , and a redshift - constant @xmath52 in this redshift range . @xcite present results for @xmath51 at @xmath447 for cosmos with similar selection criteria . we compare to their @xmath51 for galaxies with log(@xmath448)=11 - 11.4 . our results are consistent to these two works , as shown in figure [ fig : pf_lit_massratio ] ( left ) . @xcite demonstrate that the variation in selecting the parent galaxy sample and the mass ratio probe leads to different redshift trends in the merger fraction . therefore we do not compare our results directly with the pair fraction measurements at @xmath441 with different selection criteria ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? we note that once the selection differences are accounted for , the merger rate per galaxy presented in section [ sec : compare_merger_rates ] of this work is consistent with those inferred from these works as presented in @xcite : both follow a monotonically increasing trend from @xmath295 to @xmath449 . , h. , kosugi , g. , akiyama , m. , takata , t. , sekiguchi , k. , & furusawa , j. 2008 , in astronomical society of the pacific conference series , vol . 399 , panoramic views of galaxy formation and evolution , ed . t. kodama , t. yamada , & k. aoki , 131

we measure the merger fraction of massive galaxies using the ultravista / cosmos _ ks_-band selected catalog , complemented with the deeper , higher resolution 3dhst+candels catalog selected in the _ hst_/wfc3 _ h_-band , presenting the largest mass - complete photometric merger sample up to @xmath0 . we find that selecting mergers using the @xmath1-band flux ratio leads to an increasing merger fraction with redshift , while selecting mergers using the stellar mass ratio causes a diminishing redshift dependence . defining major and minor mergers as having stellar mass ratios of 1:1 - 4:1 and 4:1 - 10:1 respectively , the results imply @xmath21 major and @xmath31 minor merger for an average massive ( log@xmath4 ) galaxy during @xmath5 . there may be an additional @xmath6 major ( minor ) merger if we use the @xmath7-band flux ratio selection . the observed amount of major merging alone is sufficient to explain the observed number density evolution for the very massive ( log@xmath8 ) galaxies . we argue that these very massive galaxies can put on a maximum of @xmath9 of stellar mass in addition to major and minor merging , so that their number density evolution remains consistent with observations . the observed number of major and minor mergers can increase the size of a massive quiescent galaxy by a factor of two at most . this amount of merging is enough to bring the compact quiescent galaxies formed at @xmath10 to lie at @xmath11 below the mean of the stellar mass - size relation as measured in some works ( e.g. * ? ? ? * ) , but additional mechanisms are needed to fully explain the evolution , and to be consistent with works suggesting stronger evolution ( e.g. * ? ? ? * ) .

You are an expert at summarizing long articles. Proceed to summarize the following text: it is widely believed that most millisecond pulsars ( msps ) are spun up by accretion from a close binary companion . this recycling ( radhakrishnan & srinivasan , 1981 ) can only take place when the neutron star magnetic field has decayed to a value @xmath4 g. if accretion subsequently stops for some reason , the neutron star appears as a millisecond pulsar with a very low spindown rate , as its low magnetic field makes dipole radiation very weak . in line with these ideas a large proportion of millisecond pulsars are members of binary systems . often these pulsars undergo very wide eclipses , with an obscuring object much larger than the companion star s roche lobe . this must be an intense wind from the companion star , driven in some way by the pulsar emission ( fruchter et al . , 1988 ) . whenever such eclipses are seen the binary eccentricity and the pulsar mass function are extremely small , implying companion masses @xmath5 . there is another group of binary millisecond pulsars with systematically lower mass functions , and it is natural to assume that these are also evaporating systems with orbital inclinations which prevent us seeing the eclipses ( freire et al . , 2001 ) . in a recent paper ( king et al . 2003 ; hereafter kdb ) we pointed out that the incidence of black widow pulsars ( bwps ) is far higher in globular clusters than in the field . we identified a favoured formation mechanism for bwps in globulars in which turnoff mass stars exchange into wide binaries containing recycled millisecond pulsars ( msps ) and eject their helium white dwarf companions . once angular momentum loss or nuclear expansion bring the companion into contact with its roche lobe the pulsar is able to expel the matter issuing through the inner lagrange point @xmath6 . thus mass is lost on the binary evolution timescale . bwps are observable only when this evolution and thus the mass loss are slow . here we consider the consequences of this picture for the subsequent evolution of bwps . our main aim is to understand the distribution of bwps in the plane of minimum companion mass @xmath7 and orbital period @xmath8 ( fig . [ figmsp ] ) . bwps appear to fall into two distinct groups which we shall call high mass and low mass , depending on whether @xmath9 or @xmath10 . the high mass group have noticeably more irregular eclipses than the low mass group ( e.g. scott ransom , talk at the aspen meeting on binary millisecond pulsars in january 2004 ) . in the standard way we may associate with these two groups other binary msps which do not eclipse but whose values of @xmath11 are even smaller than those of the eclipsing systems . the interpretation is that these are bwps seen at low orbital inclination . with this association the distribution of fig . [ figmsp ] puts the dividing line between high and low mass bwps at @xmath12 . we consider the low mass group first . the small secondary mass ( close to @xmath7 for eclipsing systems ) implies a thermal time much longer than the binary evolution timescale for mass loss causing bwp behaviour . the star therefore reacts adiabatically . as it is either fully convective or degenerate we can model approximately it as an @xmath13 polytrope , with radius @xmath14 ( cf e.g. king 1988 ; for more accurate representations see nelson & rappaport , 2003 ) . here @xmath15 is the fractional hydrogen content , and @xmath16 measures the deviation from the fully degenerate radius given by @xmath17 . we assume that low mass bwps have a range @xmath18 few for reasons we will discuss in the next section . assuming a roche - lobe filling donor gives the mass - period relation @xmath19 where @xmath20 is the orbital period measured in hours . we assume further that loss of orbital angular momentum via gravitational radiation ( gr ) drives orbital evolution , so that @xmath21 where @xmath22 , @xmath23 is the pulsar mass , and @xmath24 is the specific angular momentum of the lost mass relative to that of the secondary ( van teeseling & king 1998 ) . thus @xmath25 where @xmath26 is the binary separation and @xmath27 is the distance from the centre of mass at which the matter is ejected from the binary . this is between the circularization radius of the infalling matter and the @xmath6 point . as all bwps have @xmath28 , @xmath29 except for very small @xmath30 ( see below ) the first term on the right hand side of ( [ dotm ] ) is @xmath31 and we can rewrite ( [ dotm ] ) as @xmath32 where @xmath33 and @xmath34 . according to kdb the system is only visible as a bwp if this rate is less than the critical value @xmath35 for free free absorption at typical ( 400 1700 mhz ) observing frequencies , where @xmath36 is the temperature of the lost mass near @xmath6 in units of @xmath37 k. the resulting constraint @xmath38 is plotted on fig . [ figbwp ] as ` visibility line ' . in a similar way we can plot various other constraints on this figure . for very small mass ratios @xmath39 the @xmath24 term in the denominator of ( 3 ) becomes significant . this signals dynamical instability , as the roche lobe moves inwards wrt the stellar surface ( cf stevens , rees & podsiadlowski , 1992 ) . at such masses the companion must be broken up and sheared into a large disc surrounding the pulsar . this is presumably the origin of the planets observed around psr b1257 + 12 ( wolszczan & frail 1992 ; konacki & wolszczan 2003 ) . direct numerical calculation of @xmath27 from roche geometry shows that dynamical instability occurs when @xmath40 if matter is ejected from the @xmath41 point . however , matter can be ejected from anywhere between the @xmath41 point and the circularization radius ( which even for small mass ratios is not comparable with the distance to the @xmath41 point ) . if matter reaches in 5 per cent of the distance from the centre of mass to the @xmath41 point this gives @xmath42 i.e. @xmath43 . this line is labelled ` dynamical instability ' on fig . [ figbwp ] . we also plot the line ( ` hubble line ' ) on which the binary evolution has a characteristic timescale longer than a hubble time . finally we plot the binary evolution of a system whose secondary is fully degenerate ( ` degenerate ' ) and one whose radius is 3 times as large for the same mass ( @xmath44 ) as given by equation ( [ mp ] ) . arrows on the degenerate sequences indicate the direction of evolution . if this is a viable picture of the evolution of low mass bwps , we expect them to have combinations of @xmath45 lying inside the various constraints shown in fig . [ figbwp ] . given that eclipsing systems ( solid circles ) have @xmath46 , while non eclipsing systems have @xmath47@xmath48@xmath7 , we see that the data plotted on fig . [ figbwp ] are indeed reasonably consistent with adiabatic evolution under mass loss driven by gravitational radiation . we now consider the remaining systems on fig . [ figbwp ] , i.e. those with @xmath49 . nelson ( talk at the aspen meeting on binary millisecond pulsars in january 2004 ) has studied the evolution of the long period system j17405340 , which has a subgiant companion ( damico et al . 2001 ; ferraro et al . 2001 ) and shown that it is consistent with a location near a bifurcation point , at which nuclear evolution and angular momentum loss are comparable . the remaining 4 systems have @xmath50hr and @xmath51 . except for implausibly small orbital inclinations these companions have main sequence radii far too small to fill their roche lobes . there are various ways around this difficulty , including competition between nuclear and orbital evolution , or attributing the eclipses to strong stellar winds of detached companions . however the most likely explanation appears to us to follow from the turbulent history these binaries must have had . the process of gaining a new companion must involve considerable disturbance to that star , probably leading to extensive mass loss . this is clearly indicated for tidal capture ( podsiadlowski 1996 ) , but is probably true in any picture . for an exchange encounter to give a short period system a large post - exchange eccentricity is required ( see section [ discuss ] ) . when this eccentric system circularizes the tidal effects on the secondary can in some cases be similar to tidal capture if the periastron distance is comparable to the stellar radii . the globular cluster x ray binary ac211 ( van zyl et al . 2004 ; charles , clarkson & van zyl 2002 , and references therein ) may be an example where the companion star is oversized because of the capture process , the only difference being that the neutron star in this system accretes the overflowing matter rather than expelling it , presumably because no previous partner recycled it . for an exchange encounter the lowest mass star is ejected and so the captured star , in a high - mass bwp formed through this route , must have lost a significant portion of its initial mass . this mass loss , however , occurs during the tidal circularization ( not afterwards ) and so the companion could still be thermally bloated . after the tidal disturbance and consequent mass loss , the companion attempts to reach its new main sequence radius on a thermal timescale . this process competes with orbital shrinkage via gr and reduces the mass loss ( already weak cf equation ( [ dotm2 ] ) for the relatively long orbital periods of most of this group ) . it is therefore plausible that bwps ultimately emerge from the tidally induced mass loss with the parameters of the high mass bwps . the thermal timescale of some of these bwps may be less than the gr timescale and so eventually these would shrink from contact with their roche lobes . however , they would still be observed as high - mass bwps for the length of their thermal timescale . if the companion has a sufficiently strong stellar wind , this can produce orbital eclipses through free free absorption . this appears to happen in psr 171819 ( wijers & paczyski 1993 ; burderi & king 1994 ) . here @xmath52hr , and modelling of the absorption light curve ( burderi & king 1994 ) shows that @xmath53 . we note that psr 171819 is probably a cluster member ( wijers & paczyski 1993 ) and that the stellar wind must be the eclipsing agent as the pulsar is not an msp , and thus incapable of driving mass loss . although the companion must relax towards the main sequence , its radius ultimately shrinks only slowly , whereas orbital shrinkage via gr accelerates . depending on the initial separation , the binary reaches contact with the companion somewhat oversized compared with its thermal equilibrium radius . this is probably the origin of the range of radii ( @xmath54values ) inferred for low mass bwps above . grindlay ( talk at aspen meeting on binary millisecond pulsars in january 2004 ) has found 108 x - ray sources in the globular cluster 47 tuc . a number of these are claimed to be quiescent low - mass x - ray binaries ( lmxbs ) and have measured periods from x - ray dips / eclipses ( eg w37 at 3hrs ) , power law components ( possibly from a wind ) and variable absorption . these may be bwps , rather than quiescent lmxbs , where the absorption is too large to observe the radio emission and the stellar wind is responsible for the x - ray emission . fig . [ figmsp ] reveals the striking fact that _ all binary msps with orbital periods _ @xmath3hr _ are bwps : 10 out of 16 actually eclipse . _ there are two obvious possible explanations for this : \(a ) fan beam . the pulsar beam of a recycled msp is so wide that it always includes the orbital plane , whatever the relative orientation of spin and orbit . for a wide fan beam such systems are detectable at any spin inclination . then all binary msps become bwps as soon as their companions fill their roche lobes . \(b ) pencil beam . the beam of a recycled msp is narrowly confined . clearly the only plausible geometry for making bwps has the beam axis orthogonal to the spin , with the latter roughly aligned with the binary orbit . it is quite difficult to break the degeneracy between these two possibilities . however the pencil beam requires alignment , and thus that the pulsar has accreted @xmath55 from its _ current _ companion . this requires it to have been an lmxb and then somehow broken contact , and seems harder to reconcile with the picture of high mass bwp evolution we have sketched above . we thus tentatively conclude that a fan beam offers a simple explanation for the universality of eclipsing behaviour in msps with short orbital periods . it is generally agreed that dynamical encounters must be invoked to explain the overproduction of bwps in globular clusters ( kdb ; rasio et al . rappaport , putney & verbunt ( 1989 ) considered the problem of exchanging out the white dwarf companion of the neutron star in a binary msp . their fig . 1 shows that the timescale ( in 47 tuc ) for ejecting the white dwarf from systems with periods of 10 , 100 and 1000 days is @xmath5610 , 3 , and 1 gyr , respectively . thus an exchange encounter is likely only for orbital periods @xmath57 days . the post exchange period is generally not substantially shorter , and most of these systems would not become bwps . however the post exchange eccentricity @xmath58 is evenly distributed in @xmath59 ( heggie 1975 ) . for large @xmath59 tides circularize the orbit at much tighter separations than the initial post - exchange value . the post - exchange period after tidal circularization can thus be much shorter ( hours ) in a few cases , producing a bwp . the need for the exchange into the binary msp is set by the assumption that radio pulsars do not turn on as long as there is a binary companion in near contact ; thus the need for removing it altogether after the initial recycling . burgay et al . ( 2003 ) , however , claim that the lack of detection of radio pulsations from quiescent lmxbs could result from pulsar radiation driving a large outflow from the system which then attenuates the radio emission . a problem with this scenario is that the pulsar must remove the accretion disc before it can blow away matter flowing through the inner lagrangian point . burderi et al . ( 2001 ) show that this can only occur in some wide systems , and only then when the magnetic pressure due to the neutron star overcomes the disc s internal pressure . consequently it is unlikely that the companion in a bwp is the one that spun the pulsar up , unless the latter has somehow contracted well inside its roche lobe . if this occurs through evolutionary loss of the envelope one then has the problem of bringing the system to short periods in a reasonable time . converting wide orbits to narrow ones was the main motivation for rasio et al . ( 2000 ) in considering an alternative scenario . this invokes binary formation through exchange interactions , producing neutron star systems with companions which are more massive . this must occur in a relatively short interval of @xmath60 gyr before all the potential extended companions with masses larger than that of the neutron star have finished their evolution . given such a system , a common envelope phase results once the companion evolves and overfills its roche lobe . the envelope of the red giant is ejected and the core spirals in , leaving a he rich white dwarf in a relatively close orbit with the neutron star . if contact is established , the pulsar could presumably begin to evaporate the companion . however because the donor star has no hydrogen , the orbital periods of systems made this way are much shorter than those of observed bwps ( see equation [ r2 ] and fig . [ figbwp ] ) . the evolutionary outcome for bwps depends on whether they have evolved to a period where they have become dynamically unstable . a number of bwps exist which have reached or are evolving towards the ` hubble line ' and so are not yet old enough to have been dynamically disrupted . the systems which have undergone dynamical instability are observed as isolated msps . in the globular 47 tuc 7 out of 21 of msps are isolated ( see http://www.naic.edu/~pfreire/gcpsr.html[http://www.naic.edu/@xmath56pfreire/gcpsr.html ] for an up to date list of pulsars in globular clusters ) while 5 out of 21 are low - mass bwps . objects may undergo evolution through the low - mass evolutionary route and disruption of the companion . however , the number of resulting isolated msps should not be significantly greater than the number of bwps unless there is a process which strongly favours short periods ( less bloated secondaries ) . the similar number of low - mass bwps and isolated pulsars in 47 tuc supports this conclusion . we thank the referee saul rappaport for comments which have improved this paper . we are also grateful to the organisers of the aspen meeting on binary millisecond pulsars in january 2004 which provided much stimulus for this work . dr and ks are supported by the leicester pparc rolling grant for theoretical astrophysics , and js by a pparc studentship . ark gratefully acknowledges a royal society wolfson research merit award . meb acknowledges the support of a ukaff fellowship . burderi l. , king a.r . , 1994 , apj , 430 , l57 burderi l. et al . , 2001 , apj , 560 , l71 burgay m. , burderi l. , possenti a. , damico n. , manchester r.n . , lyne a.g . , camilo f. , campana s. , 2003 , apj , 589 , 902 charles p.a . , clarkson w.i . , van zyl l. , 2002 , new ast , 7 , 21 damico n. , possenti a. , manchester r.n . , sarkissian j. , lyne a.g . , camilo f. , 2001 , apj , 548 , l171 ferraro f.r . , possenti a. , damico n. , sabbi e. , 2001 , apj , 561 , l93 freire , p. c. ; camilo , f. c. ; kramer , m. ; lorimer , d. r. ; lyne , a. g. ; manchester , r. n. ; damico , n. , 2001 , aas abstract 199 , # 159.04 fruchter , a.s . , stinebring , d.r . , taylor , j.h . , 1988 , 312 , 255 grindlay j.e . , 2004 , in rasio f.a . , stairs i.h . , eds , binary radio pulsars , asp conf . ser . , san francisco heggie d.c . , 1975 , mnras , 173 , 729 king a.r . , 1988 , qjras , 29 , 1 king a.r . , davies m.b . , beer m.e . , 2003 , mnras , 345 , 678 konacki m. , wolszczan a. , 2003 , apj , 591 , l147 nelson l.a . , 2004 , in rasio f.a . , stairs i.h . , eds , binary radio pulsars , asp conf . , san francisco nelson , l.a . , rappaport , s.a . 2003 , apj 598 , 431 podsiadlowski ph . , 1996 , mnras , 279 , 1104 radhakrishnan , v. & srinivasan , g. , 1981 , paper presented at 2nd asian pacific regional iau meeting , bandung ransom s.m . , 2004 , in rasio f.a . , stairs i.h . , eds , binary radio pulsars , asp conf . , san francisco rappaport s. , putney a. , verbunt f. , 1989 , apj , 345 , 210 rasio f.a . , pfahl e.d . , rappaport s. , 2000 , apj , 532 , l47 stevens i.r . , rees m.j . , podsiadlowski ph . , 1992 , mnras , 254 , 19 van zyl l. , charles p.a . , arribas s. , naylor t. , mediavilla e. , hellier c. , 2004 , mnras , 350 , 649 van teeseling a. , king a.r . , 1998 , a&a , 338 , 957 wijers r.a.m.j . , paczyski b. , 1993 , apj , 415 , l115 wolszczan a. , frail d.a . , 1992 , nat , 355 , 145

we consider the population of black widow pulsars ( bwps ) . the large majority of these are members of globular clusters . for minimum companion masses @xmath00.1m@xmath1 , adiabatic evolution and consequent mass loss under gravitational radiation appear to provide a coherent explanation of all observable properties . we suggest that the group of bwps with minimum companion masses @xmath2 are systems relaxing to equilibrium after a relatively recent capture event . we point out that all binary millisecond pulsars ( msps ) with orbital periods @xmath3hr are bwps ( our line of sight allows us to see the eclipses in 10 out of 16 cases ) . this implies that recycled msps emit either in a wide fan beam or a pencil beam close to the spin plane . simple evolutionary ideas favour a fan beam . [ firstpage ] binaries : close

You are an expert at summarizing long articles. Proceed to summarize the following text: the discovery by donati et al . ( 2002 ) of a magnetic field in the periodic variable @xmath0orionisc ( hd37022 ; o67 v : morgan & keenan 1973 ; o7 : stahl et al . 1996 ) solidified the expectation that it is a hot analog of the magnetic b - type variables , the so called he - anomalous bp stars . in contrast to the bp stars , the modulation of soft x - ray flux in @xmath0 ori c reveals the presence of a hot , co - rotating magnetosphere ( gagn et al . 1997 ) . the magnetic and x - ray periods are presumed to correspond to the rotational period of 15.422 days , as determined from ultraviolet and optical line variations ( stahl et al . 1993 , stahl et al . 1996 , reiners et al . nonetheless , questions have lingered as to how to reconcile the phased variations of strong optical and uv lines with the geometry of the bp magnetosphere model ( donati et al . 2002 ) . the magnetic bp variables can be classified as a subgroup of a larger class of early - type stars that includes at least two pulsating @xmath3cephei stars ( henrichs et al . 1998 ) , the sn " stars4267 lines . ] recently examined by shore et al . ( 2004 ) , and other related objects ( e.g. , smith & groote 2001 ; hereafter sg01 ) . according to the usual picture , winds in magnetic bp stars are prevented from escaping from the magnetic poles , but are channeled outwards along dipolar field loops . the wind is constrained to flow along these loops until it encounters streams from the opposite pole . when the two streams collide , they shock , cool , and either circulate back to the star or leak out through the outer disk edge where they are lost to the system . the initial formulation of this picture by shore and collaborators ( e.g. , shore & brown 1990 ; hereafter sb90 ) envisaged polar outflows ( jets " ) and a cool , toroidal disk . subsequent magnetostatic modeling of the wind flows of bp stars by babel & montmerle ( 1997 ) has dealt with the post - shock cooling of the gas in the magnetospheric disk as the principal site of the uv and h@xmath4 line variations . in this formulation , rotation is so small as to be negligible . the frozen - in magnetosphere then resembles a wobbly inner tube that co - rotates with the bp star . if the disk is extended enough in radius , it can contribute substantial line emission when it is viewed face - on . moreover , disk regions passing in front of the star generally have a lower temperature than the photosphere and therefore remove additional flux from photospheric absorption lines . in the case of resonance lines , scattering of line flux has the same effect . thus , over the cycle various viewing aspects of the wind / disk complex can produce a variety of emissions and absorptions in the uv resonance lines of , , and . in addition , modulations of h@xmath4 emission ( short & bolton 1994 ) and high - level balmer line absorptions ( groote & hunger 1976 ) provide excellent diagnostics for well - developed magnetospheres , such as the one surrounding the prototypical he - strong star @xmath5orionise . along the hot edge of the classical bp domain , dense winds insure that the wind particles do not fractionate chemically , with the result that the surface abundances of these stars remain normal . the study of the wind geometry of @xmath0 ori c is compelling because it is the only o star currently known to possess a magnetic field ( henrichs 2005 ) . the presence of a strong field in a dense , high - velocity wind make its circumstellar environment a unique laboratory to study high - energy magnetohydrodynamic plasma ( babel & montmerle 1997 ) . the interaction of various components of the wind produces the excitation of hot gas visible in x - ray spectra ( gagn et al . 2005 ) and potentially in the far - uv . also , despite the inhibition caused by the strong magnetic field , the mass - loss rate of @xmath0 ori c is several hundred times larger than then mass - loss rates typical of hot bp stars ( donati et al . 2002 ) . the strong stellar wind , together with the star s slow rotation rate , permit spectral features to be doppler shifted outside the photospheric line profile , which provides an opportunity to map the geometry of the wind over a wide range of stellar distances and azimuths . in this study we adopt the values for the rotational and magnetic tilt angles @xmath6 and @xmath3 from the analysis of the longitudinal magnetic field curves by donati et al . ( 2002 ) : both parameters are close to 45 . these values are the same as those adopted by stahl et al . ( 1996 ; hereafter s96 ) and nearly the same to those of reiners et al . ( 2000 ; hereafter r00 ) . however , as noted by donati et al . ( 2002 ) , the mapping between line variations and magnetic phase adopted by s96 implies that the magnetic pole is viewed face - on at @xmath1 = 0.5 , which is difficult to reconcile with the observed h@xmath4 absorption at this phase . instead , the magnetic - field measurements of donati et al . ( 2002 ) show conclusively that phase @xmath1 = 0.0 corresponds to the passage of the north magnetic pole across the projected center of the star . at this phase , material in the equatorial plane face - on " by a distant observer . at @xmath1 = 0.5 , the north magnetic pole has rotated to a point on the star s upper " limb and the south magnetic pole makes its only brief appearance at the star s lower " limb . at this time the observer sees the post - shock magnetosphere ( i.e. , the material near the magnetic equator ) edge - on . if the bp paradigm of sb90 and sg01 were applicable , maximum h@xmath4 emission and excess absorption in uv resonance lines should be observed at @xmath1 = 0 and 0.5 , respectively . as we will see below , this is not the case . furthermore , donati et al . ( 2002 ) were unable to explain the 50 kms@xmath7 blueshift in the centroid of the h@xmath4 emission component in the context of the bp picture . the configuration described in this paper , shown as a sketch in figure[cartoon ] , is much different from this standard bp paradigm . our geometry attributes the blueshifted absorption in the resonance lines at @xmath1 = 0.5 to a radially outflowing , high - velocity wind . the point marked x " designates schematically the alfven radius at the magnetic equator . for a very slowly rotating star this is where a bifurcation occurs in wind stream trajectories ( e.g. , ud - doula & owocki 2002 ) . all streams sketched in fig.[cartoon ] are diverted toward the magnetic equator because the particles are constrained at first to flow along field lines by the dominant magnetic forces . however , wind streams originating from high magnetic latitudes follow less confined loops that eventually extend beyond the alfven radius . as these loops traverse the alfven radius , radiation pressure ( which falls off less quickly with distance from the star ) overwhelms the magnetic forces and drags the field radially outward . thus , these streams evolve to become a concentrated outflow confined to low magnetic latitudes . in contrast , streams in the inner wind zone that originate from intermediate magnetic latitudes on the star remain confined and contribute or remove line flux only when they collide with one another , shock , and return to the star ( [ infall ] ) . the fate of this wind component contrasts strongly with the formation of the quasi - static , corotating disk typical of bp stars , and is the subject of much of the rest of this paper . in their seminal paper , s96 demonstrated that the emissions in h@xmath4 , @xmath84686 , and the absorptions of the and resonance doublets of @xmath0 ori c are regularly modulated over a period of 15.422 days . this value is assumed to be the stellar rotational period . phases in magnetic bp variables are traditionally reckoned from the time of magnetic maximum , i.e. , from the transit of a magnetic pole across the observer s line - of - sight . typically for these stars , if magnetic ephemerides are not available , the zero point is taken as the epoch of h@xmath4 emission maximum ( equivalent width minimum ) . because the central axis of the magnetosphere coincides with the magnetic plane , these zero points coincide . using this zero point and period , s96 and r00 measured the equivalent - width variations with phase for several strong uv resonance and optical excited lines . the variations exhibit a maximum at @xmath1 = 0 , and both groups of authors interpreted this as excess absorptions in the line cores . the variations in the uv resonance , h@xmath4 , h@xmath3 ( g. wade , private communication ) , and @xmath84686 lines are also highly nonsinusoidal . as measured across the entire profile , the phased equivalent width variations have an m "- shape centered at @xmath1 = 0.5 . the second maximum of the m " is centered at @xmath1 @xmath9 0.60.7 and is generally stronger than the first prong at @xmath1 = 0.4 . in the case of the h@xmath4 line , the local minimum at @xmath1 = 0.5 arises at least partly from the brief appearance of a red emission component ( stahl et al . 1993 ) . to reinvestigate the variations exhibited by the uv lines , particularly the and resonance doublets , we retrieved the 22 high - dispersion _ iue _ spectra of @xmath0 ori c obtained through the large aperture with the swp camera from the multi - mission archive at space telescope . the basic features of the variations are shown in figure [ grysc ] , which llustrates the fluctuations in as a function phase in in the form of a grayscale ( dynamic " ) spectrum . the format of fig . [ grysc ] is similar to figure 10 of s96 , except that the mean profile has been subtracted from individual spectra in the time series rather than the profile near @xmath10 , and the difference spectra illustrated in the image have been smoothed over 1 intervals . [ grysc ] shows that the variations in the resonance doublet are dominated by enhanced absorption ( darker shades ) in the high - velocity region of the unusually shallow p cygni absorption trough . however , weaker variations are also present throughout the absorption trough and even extend into the redshifted part of the profile . the red - wing variations of the @xmath81548 line are confused by blending between the doublet components , but the corresponding variations of the @xmath81550 can be seen clearly in window e " in the figure . as discussed below , the weaker variations exhibit different behavior as a function of phase than the high - velocity absorption excess . the equivalent width - phase curves ( e.g. , figure8 in s96 ) show some interesting features , and we have already mentioned the m - shaped morphology . these features , particularly those pertaining to the different dependences of fluxes across the line profile , can be investigated further by comparing the individual spectra at strategic phases . a prime example of this is the different manner in which the fluxes across the line profiles contribute to the equivalent width of the full profile . since the maximum swing in equivalent widths occurs over the comparatively narrow phase range between 0.6 and 0.9 , it is illuminating to compare the three available spectra at these phases . these spectra are shown in figure [ c4mnmx ] . the large flux variations at wavelengths shorter than 1543 are the most obvious differences . however , we also note that the fluxes in the blue and red regions of the doublet vary in opposite senses . notice that while profiles at @xmath1 = 0.50.6 exhibit increased blue absorption relative to the profiles observed near @xmath1 = 0.0 , the red - wing flux of the @xmath81550 line ( the region labeled `` e '' ) is high relative to the wing in the observation at @xmath1= 0.94 . put another way , the flux of the red wing of @xmath81550 is high when the fluxes of the ( blended ) blue wing of the doublet is low , and vice versa , over the rotational cycle . the blue wing variations overwhelms the anticorrelated red - wing ones for the lines , which is why a depiction of the variation of the equivalent width extracted from the full profile conceals the red wing variation . figure [ n5mnmx ] likewise shows observations of the @xmath111238 , 1242 at the same phases as the previous figure . the blue - wing flux variations are again present , but in this case they are much smaller than for the doublet . in fact , it is difficult to see a variation at all between phases 0.6 and 0.9 . however , the red - wing variation , ( unblended in the two components ) , as indicated also by the smoothed difference spectrum at the bottom of the figure , is clearly visible . another interesting feature of figures[c4mnmx ] and [ n5mnmx ] is that when allowance is made for the different local continuum levels , the amplitude of the red - wing variations is virtually the same for both components of the and doublets . although there are additional complications due to blending , the implication is that the variable component is formed in an optically thick medium . the phase - resolved fluctuations in different regions of the resonance doublet just discussed can be compared by plotting pseudo - equivalent widths extracted from different wavelength intervals . to provide quantitative measures of the line absorptions , we computed equivalent width indices , " which are summed net " fluxes in a defined series of windows that span the line . we used the net fluxes to avoid the errors introduced by the ripple correction for high - dispersion _ iue _ data processed with newsips . " the equivalent width index in a given window was computed as the ratio of the total flux in it to the total flux in the particular order of the echelle spectrum . like genuine equivalent widths , larger , positive values of these indices indicate greater absorption . they can be measured with precision since they are independent of continuum placement errors , but they can not be converted to true equivalent widths in angstroms . error bars were determined from the median point - to - point fluctuations in the values of the indices . this criterion likely overestimates the uncertainties because of the slow variations of the index through the cycle . when summed over the entire profile , our plots of the indices mimic the equivalent - width curves presented by s96 . however , the behavior in individual wavelength bins distributed across the weak p cygni profile is markedly different . for example , the shortest wavelengths are dominated by a single , large - amplitude variation with maximum absorption near @xmath1 = 0.5 . in contrast , the indices formed from bins that sample the core and red wing reflect systematic , small variations in the red wing , as suggested by fig . [ c4mnmx ] . these red - wing variations are illustrated in figure [ c4n5phas ] for and . from this figure it is apparent that a sinusoidal variation ( shown as a dashed line for reference ) is not a good fit to the data . [ c4n5phas ] also shows that the red - wing variations of the @xmath81242 profile ( 0+250 kms@xmath7 ) track the red - wing variations of the doublet . we also find that a simple blue / red - profile dichotomy is an inadequate description of the variations of different regions of the doublet with phase . in figure[c4wng ] we represent fluxes in four additional 1-angstrom bins at increasingly blueshifted portions of the absorption trough . at the bottom of the figure we replot the curve from the red - wing extraction of 1550 in shown in fig.[c4n5phas ] . them "- pattern is present in curves _ b _ and _ c _ , but in curve _ a , _ representing the most negative velocities , the two prongs have merged together . the same behavior is evident in the dynamic spectrum ( fig . [ grysc ] ) . according to the revised geometry of the wind flow , spectroscopic observers in the right - hand section of fig . [ cartoon ] will see the effects on the line profiles of a collimated , high - velocity wind rotating into their line of sight as phase @xmath1 = 0.5 approaches . thus , the equivalent width indices extracted from the far - blue wing exhibit maximum absorption strength at this phase . the m "- shaped curves for the near - blue wing and smaller velocities can also be explained by this same flow geometry . between phases @xmath1 = 0.30.4 and 0.60.7 , the absorptions occur at positions in the line that correspond approximately to the wind terminal velocity multiplied by the cosine of the angle between the line - of - sight to the center of the star and the bisector line in the equatorial plane . since the column responsible for the high velocity absorption at @xmath1 = 0.5 is shifted to intermediate velocities just before and after this phase , a distant observer sees the maximum absorption at these times . the wind flows seen against the rest of the stellar disk at these times are changing their direction and are also accelerating rapidly . consequently , the absorptions these flows impose on the line profile are distributed over a broader velocity range and thus have a smaller influence on the equivalent widths extracted from any one narrow wavelength window in the absorption trough . in table1 we summarize the relevant properties of a selection of optical and uv lines of @xmath12@xmath13oric reported in the recent literature . all of these lines show a phase modulation . we include the phases at which absorption equivalent widths of various lines attain their maximum absorption values . those lines for which the m "- morphology is most prominent are denoted by phases 0.40.6 . " we also show two entries for the @xmath111548,1550 doublet . following s96 , the first of these refers to the equivalent width across the full profiles , while the second describes the results for the red half of the profile only . a clear division can be made between the lines predominantly formed in the wind ( i.e. , the uv resonance lines , h@xmath4 , the emission in h@xmath3 , and @xmath84686 ) and the lines largely formed in the stellar photosphere ( e.g. , the excited transitions of and ) . as already noted , h@xmath4 and @xmath84686 exhibit complicated emissions . at @xmath1 = 0.0 , a strong emission appears in the blue wing , and subtle variations interpreted by stahl et al . ( 1993 ) as emission appear at @xmath1 = 0.5 to the red of line center . according to the sketch in figure[cartoon ] , blueshifted emissions might be observed as the magnetic pole crosses the projected center of the star at @xmath1 = 0 ; this possibility is addressed below . for now we note that the appearance of strong h@xmath4 emission ( slightly blueshifted or otherwise , and discounting a nebular contribution ) is unusual in the spectra of mid - o dwarfs . its presence in spectra of @xmath0 ori c is certainly attributable to the unusual conditions in the magnetically - channeled wind . s96 and r00 illustrated the modulations of several excited lines of , , , and . depending on the data quality , the absorption curves of these lines exhibit either a broad plateau centered at @xmath1 = 0.5 , or a shallow local minimum at this phase . although these authors constructed these curves by measuring the whole profile , differentiation with velocity can be found in the dynamic spectra of @xmath85592 and @xmath85812 presented in these papers . these depictions show the blue - to - red migration of an apparent absorption , with line center crossing at @xmath14 . the traditional interpretations for the origins of migrating subfeatures in line profiles of hot stars include surface spots due to chemical inhomogeneities , photospheric velocity fields due to nonradial pulsations , or circumstellar clouds forced into co - rotation . the first of these alternatives is improbable in an o star having a chemically homogeneous wind . the second is problematical for several reasons , but in particular because the migrating features are present during only half the cycle . we will return presently to the third alternative , co - rotating clouds , " in the modified form of dense , infalling material . s96 noted that the strength of the excited @xmath81371 line exhibits phase - dependent variations , but they did not discuss its behavior in detail . figure [ o5phas ] shows the equivalent width indices from fluxes binned over the ranges from @xmath15300 to @xmath15100 kms@xmath7 and from @xmath15100 to @xmath16100 kms@xmath7 in the line profile . these curves demonstrate that blue " and central - red " indices follow the morphology of the variations in the resonance lines ( fig . [ c4n5phas ] ) ; i.e. , the two halves of the line profiles show anticorrelated behavior . thus , the blue half exhibits excess absorption at @xmath1 = 0.5 , probably because of the importance of the accelerating wind seen at this orientation . in contrast , the red half shows either less absorption or more emission at this phase . we have repeated these extractions for the excited @xmath81718 line . although the curves are noisier , they show the same anticorrelated blue / red behavior . we now return to the more complicated issue of the origin of the red - wing variations , e.g. , in the and resonance lines . within the limits imposed by the data quality , these and other optical lines listed in table1 show the same variations in the phase range 0.30.7 . this fact suggests that the fluctuations are formed in the same geometrical structures in the wind . because the behavior of this ensemble of lines is similar regardless of their excitation and ionization potentials , we expect that they are formed in a region of thermal equilibrium that is denser than an o - star wind . the variations are probably optically thick , though further modeling is required to demonstrate this explicitly . there are two basic interpretations for the red - wing variations : either they represent _ excess absorption _ at @xmath1 @xmath9 0.90.0 or they represent _ additional emission _ at @xmath1 @xmath9 0.50.6 . the distribution of the circumstellar material is implicit in these interpretations . in particular , if the red - wind variations are interpreted as excess emission , the material must be centered ( or nearly centered ) on either the star s magnetic pole or its magnetic equator in order to produce equivalent - width curves that are symmetric about the pole - on and equator - on phases . we have identified three scenarios to address this issue : _ option i : _ : : central and red - wing _ absorption _ observed at @xmath1 = 0.0 in closed loops projected against the limb of the star . the absorbing wind streams have a net positive velocity component relative to the observer . _ option ii : _ : : central and red - wing _ emission _ seen in the plane of the sky due to columns of material emanating from the magnetic poles . at @xmath1 = 0.5 these columns have rotated to opposite limbs of the star . _ option iii : _ : : central and red - wing _ emission _ from infalling material viewed edge - on in the magnetic equatorial plane . _ option i _ suggests that the variations are due to absorption that is redshifted with respect to an observer situated above the magnetic pole in fig.[cartoon ] . thus , the gas responsible for it must be moving with a vector component directed away from the observer , even though the wind is predominantly flowing outward along the magnetic field lines in this region . this configuration is difficult to achieve , though it might arise from wind streams moving along closed loops near the star . in this case , the line absorption would be visible at phases when the flow appears to cross the edge of the stellar disk . to explore this idea , we can estimate the fraction of the stellar disk occulted by such a wind stream in the most ideal circumstances . we have computed plasma absorptions using the hubeny atmosphere and spectral synthesis and cloud " transfer codes _ synspec _ and _ circus _ ( see sg01 and smith 2001 ) for a wind having a temperature of 35,00040,000k . we find that in the best " possible case for _ option i , _ ( i.e. , for an optically thick wind with no background stellar - limb darkening ) , an absorption component formed at @xmath15200 kms@xmath7 would have to be produced by streams covering at least 9% of the stellar disk with a column density in h of at least 10@xmath17 @xmath18 in order to produce the redshifted variations observed in and . even so , it is difficult to cover this much of the stellar disk with material that is redshifted by as much as @xmath19 kms@xmath7 ; see fig.[c4mnmx ] . since the wind streams near the stellar limb are moving nearly parallel to the stellar surface , speeds of @xmath21500 kms@xmath7 are required to account for a projected component of 400 kms@xmath7 . however , the terminal velocity of the wind in this star is no more than 2000 kms@xmath7 ( walborn & nichols 1994 ) : for a line - driven stellar wind characterized by a @xmath3=1 velocity law , a velocity as large as 1500 kms@xmath7 would only be reached at distances of @xmath24@xmath20 ; i.e. , far from the stellar surface . in view of the implausible degree of fine - tuning required by _ option i _ , we do not think it is likely . _ option ii _ has the advantage that doppler - shifted emission is known to be present in some h and he lines at other phases , particularly @xmath1 = 0 , where it extends @xmath2300 kms@xmath7 redward of line center ( stahl et al . 1993 ; s96 ; g. wade , private communication ) . since the behavior of the uv resonance lines is similar , it is reasonable to conclude that their variations are also due to changes in emission . according to _ option ii , _ scattering brings flux into our line of sight when the north magnetic pole rotates to the stellar limb . an observer to the right in fig.[cartoon ] observes a glowing column of particles moving through the plane of the sky . however , as the column ( jet " ) fans out from the plane of the sky , we expect to see primarily blueshifted h@xmath4 emission unless the line is optically thin . thus , this picture fails to explain the exclusively redshifted emission observed in and . perhaps the idea could be saved by invoking a significant departure from magnetic axi - symmetry , e.g. , with a decentered dipole and a jet that is compressed in one 180 azimuthal sector . in such a scheme the observer might see more gas receding over a large area of the sky than gas exhibiting blueshifts in the foreground . however , even this geometry would produce significant blueshifted emission , which is not indicated in figs.[grysc ] , [ c4mnmx ] or [ n5mnmx ] . thus , _ option ii _ is probably not viable . _ option iii _ associates the emission maximum at @xmath1 = 0.5 with infalling material near the magnetic equator rather than the poles . since the lines of sight to this emission are also those that intersect the high - velocity wind moving toward the observer at this phase , this idea seems at first counterintuitive . how can the wind along the central line of sight to the star be approaching and receding at the same time ? furthermore , this scenario must explain the appearance of redshifted in many strong lines , including h@xmath4 ( stahl et al . 1993 ) . in the resonance lines , the emission is probably optically thick . moreover , since both low- and high - excitation lines show emission , it is most likely formed in a dense circumstellar medium and thus probably close to the star . the geometry illustrated in fig.[cartoon ] addresses these issues , in particular by showing how rapidly moving , red- and blue - shifted material can be found between between the star and a distant observer at @xmath1 = 0.5 . the presence of the infalling material required by _ option iii _ is is largely confirmed by recent magnetohydrodynamic ( mhd ) simulations by dr . a. ud - doula ( private communication ; see gagn et al . 2005 for a description of this work ) , which also provide a physical basis for the geometry sketched in fig.[cartoon ] . as with previous studies of bp stars , the circulation results from the collision of wind streams that originate in opposite magnetic hemispheres and are guided by closed loops toward the magnetic equator . in the case of @xmath0 ori c , which has a much denser wind than a typical bp star , the collisions at the magnetic equator result in the formation of dense , optically thick blobs . these structures can not be supported by radiation pressure from the star , and fall back toward the surface . since the blobs are still constrained to move along magnetic fields lines , they fall obliquely back toward the star along the lines that originally carried them to the magnetic equator . observers viewing the system edge - on ( @xmath1 = 0.5 ) will see the infall as a redshifted , optically thick feature ; in spectra with sufficiently high signal - to - noise ratio , this emission might be variable from one cycle to the next . ud - doula s models also include effects of radiative cooling . his models suggest that the condensations are part of the shocks . these structures can be expected to contribute to line emission observed in the uv and optical regions . since @xmath0 ori c rotates so slowly , it is appropriate to think of this circulation as a rigid , axisymmetric structure comprised of dense blobs . we expect that the circulation we have described is related to the h@xmath4 flux and radial - velocity variations ( s96 ; gagn et al . an additional consistency check for this infall picture is the _ absence _ of evidence for a cool , static disk in the magnetic equatorial plane . such disks are the hallmarks of uv - variable bp stars . our modeling of disks for early and late - type bp stars using _ synspec _ and _ circus _ ( e.g. , sg01 ; smith 2003 ) suggests that the disk temperature is typically 0.6@xmath21 , which for @xmath0 ori c implies a disk temperature near 25,000k . the ultraviolet line opacity tends to decrease with temperature in this temperature range . yet even for temperatures as high as 30,000k , our modeling experiments suggest that a geometrically thin disk with a column density of 10@xmath22 @xmath18 in h should be detectable as an absorption feature near rest velocity in the resonance doublet . the fact that a static disk is _ not _ visible implies that the circulating component of the wind interior to the alfven radius must be transported somewhere else . we believe the redshifted emission at @xmath1=0.5 is evidence of this transport . donati et al . ( 2002 ) noted that they were unable to duplicate the significantly blueshifted ( @xmath23 kms@xmath7 ) h@xmath4 emission in their simulation at the pole - on phase ( @xmath1 = 0.0 ) . they found that the profile was formed mainly from a static disk , which in their model has a radius of @xmath23@xmath20 . it is not surprising that the contribution from such a large disk would dominate the contribution from the polar wind , but the disk can not produce a blueshifted feature at @xmath1 = 0 in this geometry . perhaps the blueshifts are caused by the acceleration of jet - like wind from the pole ? we have investigated this possibility with _ circus . _ we find that even with generous values for relevant parameters the contribution of a jet produces negligible h@xmath4 emission relative to the photospheric flux . moreover , it is even more difficult for a jet - emission model to work for @xmath84686 . in contrast , our same modeling experiments demonstrate that for similar temperatures and densities , a much smaller volume viewed beyond the projected face of the star could easily produce the observed h@xmath4 emission . from these results we speculate that these blueshifts at @xmath1 = 0 have the same cause as the uv resonance line redshifts at @xmath1 = 0.5 : i.e. , they arise from the projection of the component of post - shock matter falling obliquely toward the star , as indicated by the closed arrowheads in fig.[cartoon ] . from archival _ iue _ spectra of the magnetic o - type star @xmath0 ori c , we have discovered new variations of the and resonance doublets , as well as the strong , excited lines of @xmath81371 and @xmath81718 . we found strong similarities between these variations and those exhibited by several prominent optical lines reported in the literature , including h@xmath4 and @xmath84686 . lines with the largest variations exhibit an anticorrelation in blue- and red - wing fluxes at @xmath24 : as the red wing weakens , the blue wing strengthens . in addition to this dichotomy , extractions of the profiles over different velocity bins show that the m "- shaped wave form present in optical lines is also present in the lines . this morphology disappears at the highest wind velocities sampled ( i.e. , at the most blueshifted wavelengths ) , where a single - peaked absorption at @xmath1 = 0.5 replaces the m "- pattern . the high - velocity strengthenings of the blue wing of at @xmath1 = 0.5 are explained by the flow geometry of ud - doula s mhd simulations . these predict that radiative forces open the field lines and carry them away from the star once they extend beyond the alfven radius ( i.e. , several tenths of a stellar radius above the surface ; gagn et al . 2005 ) . at @xmath1 = 0.5 , the enhanced blue wings of these lines represent material accelerating away from the star along the magnetic plane . we also argued that at @xmath25 and 0.7 the observer looks along a smaller projected path length through flows that have moderate outward velocities . this means that the excess absorptions at these phases are weaker than the absorptions formed over the longer path length of the open , high - velocity flow viewed at @xmath1 = 0.5 ( fig.[cartoon ] ) . the red variations at the edge - on phases ( @xmath26 ) , which we interpret as emission , require a more complicated geometry in which post - shock material flows towards the star , likely as dense clumps . the presence of this material is also predicted by ud - doula s simulations , though in the current generation of models the infalling material only occurs episodically . finally , _ in contrast to bona fide bp stars , there is no evidence for a cool , static disk . _ if post - shock material were to accumulate , given reasonable extrapolations from models of bp - star disks , a disk ought to be visible in spectra of @xmath0 ori c. we conclude that such accumulation does not happen in the circumstellar environment of @xmath0 ori c. the authors are grateful to drs . marc gagn , richard townsend , asif ud - doula , and stan owocki for discussions of their mhd simulations which have led to a clearer description of our geometry . we also thank both dr . detlef groote and the referee for their careful readings of this paper and helpful comments to improve its clarity . finally , we thank dr . gregg wade for permission to refer to his new optical data in advance of publication . this work was supported in part by nasa grant # nng04ge75 g . figure [ cartoon ] : : a sketch ( not to scale ) of the suggested wind / dipolar magnetic - field orientation for @xmath0 ori c at @xmath1 = 0.0 ( from top ) , 0.3 and 0.7 ( from 2 oclock position ) , and 0.5 ( from right ) , assuming angles @xmath6 = @xmath3 = 45 . in this planar projection we have collapsed the observer s positions at @xmath1 = 0.3 and 0.7 , since they are at equal angles in and out of page , respectively . the stellar wind expands freely only from the regions near the magnetic poles ( top and bottom of the star ) . the magnetic field causes radiatively - driven material from other regions of the stellar surface to follow flux lines toward the magnetic equator . the wind is divided into two zones according to its position relative to the star s alfven radius , which is designated by the dotted locus and the x " on the line - of - sight from the center of the star to the observer ( dashed line ) . beyond this radius radiative forces dominate over magnetic forces and cause the wind to accelerate away from the star , approximately parallel to the magnetic equatorial plane . within the alfven radius , wind particles emerge from near the magnetic poles , follow closed loop lines ( open arrowheads ) , and encounter streams from the opposite hemisphere at the magnetic equator , where they shock and accumulate . periodically , the particles formerly in the wind condense into optically thick blobs , which return to the star along the original field lines ( full arrowheads ) . the redshifted variations suggest that the post - shock gas falls at speeds of 300400 kms@xmath7 . figure [ grysc ] : : : dynamic spectrum of the 22 high - dispersion _ iue _ echellograms of @xmath0 ori c obtained through the large - aperture with the swp camera showing variations in the @xmath111548 , 1551 doublet as a function of magnetic / rotational phase . the shading represents differences with respect to the time - averaged mean spectrum ( lower panel ) : darker shades denote phases and wavelengths where the profile is deeper ( i.e. , less flux ) than its time - averaged value . the difference spectra have been smoothed over @xmath21 . phase gaps in this spectrum are due to the paucity of observations . brackets in the lower panel indicate the extent of the blue and red components of the doublet for an assumed terminal velocity of 2500 kms@xmath7 ( s96 ) . although the dominant variations occur in the blue wing of the absorption trough ( window a " ) , significant variations occur throughout the profile . the variations in window e " are anti - correlated with those in window a. " figure [ c4mnmx ] : : : three high - resolution _ iue _ spectra in the region of the resonance lines for phases @xmath1 = 0.51 , 0.60 , and 0.94 ( swp54040 , swp07481 , and swp14665 ) . the units for the flux are ergs@xmath7@xmath18@xmath7 . all spectra are binned over 4 pixels , which undersamples the line cores and give them a sharp appearance . the spectra obtained at @xmath1 = 0.60 and 0.94 ( solid and dashed lines , respectively ) exhibit the maximum difference in blue- and red - wing flux . wavelength windows _ a e _ are referenced in figs.[grysc ] and [ c4wng ] . following s96 , phase zero is defined by the epoch hjd=2448833.0 ( mjd=48832.5 ) . figure [ n5mnmx ] : : : _ iue _ spectra for phases 0.60 and 0.64 ( swp07481 and 54058 ; solid line is their mean ) and 0.90 and 0.94 ( swp54094 and 14665 ; dashed line is their mean ) are compared in the region of the doublet of @xmath0 ori c. the spectra have been binned over 4 points . the computed difference spectrum , binned over 8 points , is shown at the bottom in units of ergs@xmath7@xmath18@xmath7 . figure [ c4n5phas ] : : : relative equivalent width index ( normalized to the maximum value ) versus phase for the red members of the and and doublets , as extracted over the velocity range between 0 and @xmath16250 kms@xmath7(01.3 ) . the dotted line is a reference sinusoid . figure [ c4wng ] : : : equivalent width index versus phase extracted from four velocity ranges in the blue wing of @xmath81548 ; see windows _ a e " _ in fig . [ grysc ] . the curves have been offset vertically for clarity . the corresponding extraction from 1548 was not used due to interference with the wind absorption of the 1550 line . note the progression from no variation , to an m "- like pattern , and finally to a morphology featuring a single peak at @xmath1 @xmath9 0.5 . figure [ o5phas ] : : : equivalent width index variations with phase of the excited @xmath81371 line , as extracted from negative and positive velocities of the line profile . lccccl & 6563 & 10.2 & 13.2 & 0.40.6 & s96 - 4 + & 4686 & 48.4 & 54.4 & 0.40.6 & s96 - 5 + & 4471 & 21.0 & 24.6 & 0.0 & s96 - 13 + & 4541 & 51.0 & 54.4 & 0.0 & s96 - 14 + & 5801 & 37.5 & 64.5 & 0.0 & s96 - 15 + & 5812 & 37.5 & 64.5 & 0.0 & r00 - 7 + & 1548 , 1550 & 0.0 & 64.5 & 0.40.6 & s96 - 8 + & 1548 , 1550 & 0.0 & 64.5 & 0.0 & fig . [ c4n5phas ] + & 1718 & 16.2 & 77.4 & 0.0 & [ uvexc ] + & 1238 , 1242 & 0.0 & 97.9 & 0.0 & fig . [ c4n5phas ] + & 5592 & 33.9 & 55.0 & 0.0 & s96 - 16 + & 1371 & 19.7 & 113.9 & 0.0 & fig . [ o5phas ] + & 1394 , 1402 & 0.0 & 45.1 & 0.40.6 & s96 - 9 +

the star @xmath0orionisc ( o67v ) is often cited as a hot analog of bp variables because its optical and uv line and x - ray continuum fluxes modulate over the magnetic / rotational period . in this circumstance , one expects emission and absorption components of the uv resonance lines to vary as a flattened magnetosphere co - rotates with the star . in this paper we re - examine the detailed velocity behavior of several strong uv lines . whereas past work has focused on variations of the full profiles , we find that the blue and red wings of the and resonance lines exhibit anticorrelated modulations . these appear as absorption excesses at large blueshifts , and flux elevations at moderate redshifts at the edge - on phase @xmath1=0.5 . no rest - frame absorption features , which are the typical signatures of cool , static disks surrounding bp stars , can be detected at any phase . we suggest that this behavior is caused by two geometrically distinct components of the wind , which are defined by the relationship between the extent of a magnetic loop and the local alfven radius . streams on field lines opening outside this radius are first channeled toward the magnetic equator , but after reaching the alfven radius they are forced outward by radiative forces , eventually to become an expanding radial outflow . this wind component causes blueshifted absorption as the co - rotating magnetic equatorial plane crosses the observer s line of sight ( @xmath1= 0.5 ) . the geometry of the inner component requires a more complicated interpretation . wind streams first follow closed loops and collide at the magnetic equator with counterpart streams from the opposite pole . there they coalesce and fall back to the star along their original field lines . the high temperatures in these falling condensations cause the redshifted emission . the rapid circulation of these flows is likely the reason for the absence of signatures of a cool disk ( e.g. , zero - velocity absorptions at @xmath1 @xmath2 0.5 ) in the strong uv lines .

You are an expert at summarizing long articles. Proceed to summarize the following text: core - collapse supernovae ( ccsne ) are the signposts of recent massive star formation . since much of the massive star formation is embedded in dust , a substantial fraction of the ccsne in the universe will remain hidden in optical searches @xcite . this is particularly true in the case of star formation taking place in the dusty environments of luminous ( @xmath7)@xmath8 $ ] . ] , and ultra - luminous ( @xmath9 ) infrared ( ir ) galaxies ( lirgs and ulirgs , respectively ) , which dominates the star formation rate density at @xmath10 ( e.g. , * ? ? ? in many lirgs , the bulk of star formation occurs within their circumnuclear regions ( @xmath11kpc ) , so that the need for high - resolution observations becomes crucial in the detection and study of ccsne therein . the synergy between high resolution radio and near - infrared ( nir ) _ k_-band ( where the extinction is 10 times lower than in the optical ) observations is currently being used to help build a complete picture of the supernova ( sn ) activity in dusty starbursts and lirgs . outstanding examples are sne 2000 ft ( in ngc7469 ; * ? ? ? * and references therein ) , 2004ip ( in iras18293@xmath123413 ; * ? ? ? * ; * ? ? ? * ) , 2008cs ( in iras17138@xmath121017 ; * ? ? ? * ) , and 2008iz ( in m82 ; * ? ? ? * ; * ? ? ? * ) where both high resolution ( to disentangle the emission of the sn from its host ) and reduced extinction measurements ( radio and nir ) , were essential in their detection and characterisation . arp299 is a lirg with an ir luminosity ( @xmath13l@xmath14 @xcite , at an adopted luminosity distance of 44.8mpc , assuming @xmath15kms@xmath16mpc@xmath16 . the system is composed of two interacting galaxies , whose major nuclei ( a and b1 ) and core components in the interacting region ( c@xmath17 and c ) are bright radio and nir emitters @xcite . arp299 is a very prolific sn factory as proved by the detection of several radio sne and sn remnants ( snrs ) in the innermost nuclear regions of arp299a and arp299b @xcite , and the detection within the last 20 years of seven optical / nir sne in the circumnuclear regions of the system ( see * ? ? ? * ; * ? ? ? many attempts have been made to detect at radio wavelengths the sne occurring in the circumnuclear regions of arp299 , mostly under observing programmes with the very large array ( vla ) : as333 carried out from 1990 may to 1993 december ; as525 on 1994 february to detect sn1993 g ; as568 on 1999 january , february , april and october to detect sn1999d ; aw641 on 2005 february , june and august to detect sn2005u . we are currently monitoring arp299 at radio wavelengths under programmes ap592 and ap614 , `` unveiling the hidden population of sne in local luminous infrared galaxies '' ( pi : m. a. prez - torres ) , as part of a combined radio and nir sn search in a sample of local lirgs . in this paper we study the nature of the most recently detected sne in the system , 2010o and 2010p , by means of their late - time radio emission . sn2010o was discovered at optical wavelengths on 2010 january 24 @xcite . the object exploded on 2010 january 7 and lies on a location with intermediate extinction , @xmath18mag ( kankare et al . it was classified as a type ib sn by @xcite based on a low - resolution optical spectrum obtained with the nordic optical telescope ( not ) . @xcite reported an x - ray transient at the position of sn2010o prior to its explosion , and suggested that the progenitor of this sn was part of a wolf - rayet x - ray binary system , similar to those found in our galaxy . sn2010p was discovered at nir wavelengths with the not on 2010 january 18 @xcite , a few days after explosion ( 2010 january 10 , kankare et al . the spectrum of sn2010p obtained with the gemini - north telescope revealed a deficiency of hydrogen and matched with spectra of type ib / iib sne @xcite , and the absolute magnitude and colours from optical and nir observations with the not and the gemini - north telescope yielded a likely high host galaxy extinction of @xmath19 ( kankare et al . 2013 ) . this paper accompanies @xcite by kankare et al . ( 2013 ) which presents a study of the early optical and nir emission of sne 2010o and 2010p . here we analyse long - term follow - up radio observations of arp299 carried out by us and obtained from archives , with particular emphasis on characterising sne 2010o and 2010p . in section [ sec : radio ] we give details on the data reduction and analysis . we describe our results in section [ sec : results ] , which we then discuss in section [ sec : discussion ] . in section [ sec : concl ] we summarize the conclusions of our work . we have collected observations of arp299 obtained with the multi - element radio linked interferometry network ( merlin ) , the electronic multi - element remotely linked interferometer network ( e - merlin ) , the vla of the national radio astronomy observatory ( nrao ) , and the european very long baseline interferometry ( vlbi ) network ( evn ) . in table [ tab : radobs ] we show basic information for these observations ( all of which have an assigned label ) including the range of frequencies observed , the total time on target ( @xmath20 ) , and the peak intensity in each epoch of j1128@xmath215925 , which was used as the phase calibrator in all the observations we report here .

we report radio observations of two stripped - envelope supernovae ( sne ) , 2010o and 2010p , which exploded within a few days of each other in the luminous infrared galaxy arp299 . whilst sn2010o remains undetected at radio frequencies , sn2010p was detected ( with an astrometric accuracy better than 1milli arcsec in position ) in its optically thin phase in epochs ranging from @xmath0 to @xmath1yr after its explosion date , indicating a very slow radio evolution and a strong interaction of the sn ejecta with the circumstellar medium . our late - time radio observations toward sn2010p probe the dense circumstellar envelope of this sn , and imply @xmath2 / v_{\rmn{wind } } [ 10\,{\,\mbox{\ensuremath{\rmn{km\,s^{-1 } } } } } { } ] = ( 3.0$]5.1)@xmath3 , with a 5ghz peak luminosity of @xmath4@xmath5 on day @xmath6464 after explosion . this is consistent with a type iib classification for sn2010p , making it the most distant and most slowly evolving type iib radio sn detected to date . [ firstpage ] supernovae : general supernovae : individual : sn2010o supernovae : individual : sn2010p

You are an expert at summarizing long articles. Proceed to summarize the following text: entanglement , first recognized as the characteristic trait of quantum mechanics @xcite , has been used for a long time as the main indicator of the quantumness of correlations . indeed , as shown in ref . @xcite , for pure - state computation , exponential speed - up occurs only if entanglement grows with the size of the system . however , the role played by entanglement in mixed - state computation is less clear . for instance , in the so - called deterministic quantum computation with one qubit ( dqc1 ) protocol @xcite , quantum speed - up can be achieved using factorized states . as shown in ref . @xcite , speed - up could be due to the presence of another quantifier , the so called quantum discord @xcite , which is defined as the difference between two quantum analogs of the classical mutual information . the relationship between entanglement and quantum discord has not been completely understood , since they seem to capture different properties of the states . in ref . @xcite , it is shown that even if quantum discord and entanglement are equal for pure states , mixed states maximizing discord in a given range of classical correlations are actually separable . the relation between discord and entanglement has been discussed in refs . @xcite , and an operational meaning in terms of state merging has been proposed in @xcite . recently , the use of quantum discord has been extended to multipartite states . a measure of genuinely multipartite quantum correlations has been introduced in @xcite . in ref . @xcite , an attempt to generalize the definition of quantum discord in multipartite systems based on a collective measure has been proposed . in ref . @xcite , the authors proposed different generalizations of quantum discord depending on the measurement protocol performed . entanglement in multipartite systems has been shown to obey monogamy in the case of qubits @xcite and continuous variables @xcite . monogamy means that if two subsystems are highly correlated , the correlation between them and other parties is bounded . _ proved that , unlike entanglement , quantum discord is in general not monogamous @xcite . they also suggested , based on numerical results , that w states are likely to violate monogamy . in this brief report , using the koashi - winter formula @xcite , we prove that , for pure states , quantum discord and entanglement of formation obey the same monogamy relationship , while , for mixed states , distributed discord exceeds distributed entanglement . then , we give an analytical proof of the violation of monogamy by all w states . furthermore , we suggest the use of the interaction information @xcite as a measure of monogamy for the mutual information . finally , as a further application of koashi - winter equality , we prove , the conjecture on upper bounds of quantum discord and classical correlations formulated by luo _ et al . _ @xcite for rank 2 states of two qubits . in classical information theory , mutual information between parties @xmath0 and @xmath1 is defined as @xmath2 , where @xmath3 is the shannon entropy and @xmath4 is the conditional shannon entropy of @xmath0 after @xmath1 has been measured . an equivalent formulation [ @xmath5 can be obtained using bayes rules , because of which @xmath6 . on the other hand , if we try to quantize these quantities , replacing probabilities with density matrices and the shannon entropy with the von neumann entropy , their counterparts differ substantially @xcite . the quantum mutual information is defined as @xmath7 where @xmath8 is the von neumann entropy and @xmath9 are the reduced states after tracing out party @xmath10 , while the quantized version of @xmath11 measures the classical part of the correlations @xcite and it is given by @xmath12,\label{clas}\ ] ] with the conditional entropy defined as @xmath13 , @xmath14 and where @xmath15 is the density matrix after a positive operator valued measure ( povm ) @xmath16 has been performed on @xmath1 . in some cases , orthogonal measurements are enough to find the maximum in eq . ( [ clas ] ) @xcite . quantum discord is thus defined as the difference between @xmath17 and @xmath18 : @xmath19.\ ] ] quantum discord can be considered as a measure of how much disturbance is caused when trying to learn about party @xmath0 when measuring party @xmath1 , and has been shown to be null only for a set of states with measure zero @xcite . both classical correlations and quantum discord are asymmetric under the exchange of the two sub - parties ( i.e. , @xmath20 and @xmath21 ) . while @xmath22 is invariant under local unitary transformations and can not increase under local operations and classical communication , @xmath23 is not monotonic under local operations . for instance , in @xcite , it is shown how to create quantum correlations under the action of local noise . given a measure of correlation @xmath24 , monogamy implies a tradeoff on bipartite correlations distributed along all the partitions @xmath25 ( @xmath26 ) : @xmath27 coffman , kundu , and wootters @xcite showed that this property applies to three - qubit states once the square of the concurrence @xcite ( @xmath28 ) plays the role of @xmath24 . the extension of the proof to @xmath29-partite ( @xmath30 ) qubit systems has been given in ref . as pointed out in @xcite , however , entanglement of formation does not satisfy the criterion given in eq . ( [ monog ] ) . then , even if people usually refers to entanglement as a monogamous quantity , it would be worth paying attention to the entanglement monotone in use . in trying to apply this property to quantum discord , prabhu _ et al . _ showed that monogamy is obeyed if and only if the interrogated interaction information is less than or equal to the unmeasured interaction information @xcite . then , the authors found through numerical simulations that the subset of w states are not monogamous , in contrast with greenberger - horne - zeilinger ( ghz ) states , which can be monogamous or not . here , we prove that , for pure states , the monogamy equations for quantum discord and for entanglement of formation coincide . let us consider the pure tripartite state @xmath31 . quantum discord of any of the couples of sub - parties is given by @xmath32 , where @xmath33 , while @xmath34 . as shown in ref . @xcite , the following relationships between conditional entropies and entanglement of formation ( @xmath35 ) holds : @xmath36 this formula allows one to write @xmath37 using eq . ( [ kw ] ) , monogamy equation @xmath38 is then equivalent to @xmath39 where @xmath40 has been employed . the equality of conservation law for distributed entanglement of formation and quantum discord , even if not associated to monogamy , was already noticed by fanchini _ et al . _ because of this equivalence , the violation of eq . ( [ entmon ] ) by w states , whose numerical evidence has been given in ref . @xcite , admits an analytical proof . let us recall that , apart from local operations , a generic pure state of three qubits , belonging to the ghz class , can be written as @xmath41 @xcite . the family of w states is obtained fixing @xmath42 . as shown by coffman , kundu , and wootters @xcite , w states have zero three - tangle ; that is , they obey @xmath43 to show that eq . ( [ concw ] ) implies @xmath44 , it is enough to note that @xmath35 is a concave function of @xmath28 , since @xmath45 $ ] , where @xmath46 is the binary entropy @xmath47 , and both @xmath35 and @xmath48 admit values between @xmath49 and @xmath50 . then , if we apply the mapping from @xmath28 to @xmath35 to the three elements of eq . ( [ concw ] ) , we find @xmath51 if @xmath52 ( i.e. for biseparable states ) and @xmath53 otherwise . as noticed in ref . @xcite , ghz states can be monogamous or not . actually , a numerical analysis shows that about half of them do not respect monogamy . to see a transition from observation to violation of monogamy , we consider the family of states @xmath54 . note that @xmath55 is the maximally entangled ghz state @xmath56 , while @xmath57 coincides with the maximally entangled w state @xmath58 . for @xmath59 , qubit @xmath0 is factorized , and ( [ discmon ] ) becomes an equality . in fig . [ figure ] , @xmath60 is plotted as a function of @xmath61 for different values of @xmath62 . as expected , for any @xmath63 , there is a threshold for @xmath61 above which the states are monogamous . , as quantified by @xmath64 , as a function of @xmath61 for different values of @xmath62 ( see the main text ) . states are monogamous where the respective curves are positive . black solid line is for @xmath65 , black dashed line is for @xmath66 , orange ( gray ) solid line is for @xmath67 , and orange ( gray ) dashed line is for @xmath68 . according to the analytical proof , for w states ( @xmath65 ) @xmath60 is always positive ( i.e. , these states are never monogamous ) . for ghz states , there exists a threshold value of @xmath61 above which monogamy is satisfied . this threshold goes to zero for vanishing @xmath61 , since in this case qubit @xmath0 becomes factorized and all the related entanglement quantifiers vanish as well.,title="fig:",width=302 ] + once the assumption of pure state is relaxed , eq . ( [ kw ] ) becomes an inequality : @xmath69 . then , @xmath70 , or , using the subadditivity of von neumann entropy , @xmath71 thus , for mixed states , monogamy of quantum discord has a stricter bound than monogamy of entanglement . the search for monogamy of correlations can be extended to @xmath72 . it is actually easy to note that , for pure tripartite states , monogamies of quantum discord and classical correlations are complementary , that is @xmath73 . to prove it , it is sufficient to observe that mutual information obeys @xmath74 the generalization of eq . ( [ minfmon ] ) to mixed states presents some interesting aspects . we have @xmath75 where @xmath76 is called correlation information @xcite . in the language of density matrices , it can be defined as @xmath77 where @xmath78 are all the possible strings containing integer numbers between @xmath50 and @xmath29 , with @xmath79 for any @xmath80 , and @xmath81 counts the length of each string . for instance , for a bipartite system , @xmath82 coincides with the ordinary mutual information , and in the tripartite case @xmath83 . it can be checked that , for @xmath29 odd , @xmath84 for any pure state . in classical information theory , the interaction information has been introduced with the aim of measuring the information that is contained in a given set of variables and that can not be accounted for considering any possible subset of them . it should then measure genuine @xmath29-partite correlations . actually , @xmath85 can be negative . thus , according to the criteria given , for instance , in ref . @xcite , it can not be used as a correlation measure . then , its meaning is widely debated . equation ( [ minfxmon ] ) suggests that it plays the role played by the tangle in the distribution of @xmath28 @xcite , since it is invariant under index permutation , and it can be called a `` mutual information tangle . '' when @xmath86 is negative , it quantifies the lack of monogamy of the mutual information . as shown by prabhu _ , monogamy of discord relies on the relationship between @xmath86 and its interrogated version @xcite . @xcite , it has been conjectured that , given a bipartite state @xmath87 , defined in the hilbert space @xmath88 , the following upper bounds for quantum discord and classical correlations could exist : @xmath89 , \\{\cal j}_{a , b}&\le & \min [ s(\varrho_{a}),s(\varrho_{b})].\end{aligned}\ ] ] it is trivial to prove the existence of such an upper bound for entanglement of formation , its definition being based on the convex roof construction . if inequality ( [ discmix ] ) were an equality , it would be easy to extend the proof to discord . actually , inequality ( [ discmix ] ) is telling us that distributed discord could exceed distributed entanglement , and these upper bounds could be violated . while a a partial proof of the conjecture has been given in ref . @xcite using the language of quantum operations , a full proof for the case of rank 2 states of two qubits can be given using the koashi - winter formula . by applying a purification procedure , we add an ancillary hilbert space @xmath90 and write a pure tripartite state @xmath91 such that @xmath92 . since @xmath87 has rank 2 , @xmath91 is a three - qubit state . as a consequence of eq . ( [ kw ] ) , we have @xmath93 then , inequalities @xmath94 and @xmath95 are immediately verified . let us now separately discuss the cases @xmath96 and @xmath97 . in the first case , we only need to prove @xmath98 . in ref . @xcite , using the invariance under index permutation of the three tangle introduced by coffman , kundu , and wootters @xcite , we proved that , for the case of three qubits , if @xmath96 , then @xmath99 . this chain rule implies @xmath100 , and then @xmath101 as we wanted to prove . assuming now @xmath97 , we are left to show that @xmath102 . writing explicitly @xmath103 , we use the chain rule to write @xmath104 and to obtain @xmath105 this ends the proof . we have studied the monogamy properties of pure tripartite state . we have shown that quantum discord and entanglement of formation obey the same monogamy relationship . applying this equivalence to the case of three qubits , we have shown , by analytical demonstration , that , for all the w states , quantum discord is not monogamous , in contrast with ghz states , where discord can be monogamous or not . in an example , we have shown the transition from monogamy to absence of monogamy for a subfamily of ghz states . the equivalence between quantum discord and entanglement of formation concerning monogamy raises a subtle question that it is worth considering . while people usually claim that , for qubits , entanglement is monogamous , all we know is that there exists an entanglement monotone ( the square of the concurrence ) that is in fact monogamous . by analogy , we can say that the results of ref . @xcite do not exhaust the search for monogamy of quantum discord and other correlations , where monogamous monotone indicators could be found . using the connection between discord end entanglement of formation , we have also shown that in the case of rank 2 states of two qubits , as conjectured by luo _ et al . _ @xcite , quantum discord and classical correlations are bounded from above by the single - qubit von neumann entropies . a full proof can not be given because , for mixed states , the equality of conservation law for distributed entanglement of formation and quantum discord is broken . g. adesso and f. illuminati , new j. phys . * 8 * , 15 ( 2006 ) ; g. adesso , a. serafini , and f. illuminati , phys . rev . a * 73 * , 032345 ( 2006 ) ; t. hiroshima , g. adesso , and f. illuminati , phys . rev . lett . * 98 * , 050503 ( 2007 ) . s. campbell , t. j. g. apollaro , c. di franco , l. banchi , a. cuccoli , r. vaia , f. plastina , and m. paternostro , arxiv:1105.5548 ; f. ciccarello and v. giovannetti , arxiv:1105.5551 ; a. streltsov , h. kampermann , and d. bruss , phys . lett . * 107 * , 170502 ( 2011 ) . a. acn , a. andrianov , l. costa , e. jan , j. i. latorre , and r. tarrach , phys . lett . * 85 * , 1560 ( 2000 ) ; w. dr , g. vidal , and j. i. cirac , phys . a * 62 * , 062314 ( 2000 ) ; a. acn , d. bruss , m. lewenstein , and a. sanpera , phys . lett . * 87 * , 040401 ( 2001 ) .

in contrast with entanglement , as measured by concurrence , in general , quantum discord does not possess the property of monogamy , that is , there is no tradeoff between the quantum discord shared by a pair of subsystems and the quantum discord that both of them can share with a third party . here , we show that , as far as monogamy is considered , quantum discord of pure states is equivalent to the entanglement of formation . this result allows one to analytically prove that none of the pure three - qubit states belonging to the subclass of w states is monogamous . a suitable physical interpretation of the meaning of the correlation information as a quantifier of monogamy for the total information is also given . finally , we prove that , for rank 2 two - qubit states , discord and classical correlations are bounded from above by single - qubit von neumann entropies .

You are an expert at summarizing long articles. Proceed to summarize the following text: the largest source of information today is the world wide web . the estimated number of documents nears 10 billion . similarly , the number of documents changing on a daily basis is also enormous . the ever - increasing growth of the web presents a considerable challenge in finding novel information on the web . in addition , properties of the web , like scale - free small world ( sfsw ) structure @xcite may create additional challenges . for example the direct consequence of the scale - free small world property is that there are numerous urls or sets of interlinked urls , which have a large number of incoming links . intelligent web crawlers can be easily trapped at the neighborhood of such junctions as it has been shown previously @xcite . we have developed a novel artificial life ( a - life ) method with intelligent individuals , crawlers , to detect new information on a news web site . we define a - life as a population of individuals having both static structural properties , and structural properties which may undergo continuous changes , i.e. , adaptation . our algorithms are based on methods developed for different areas of artificial intelligence , such as evolutionary computing , artificial neural networks and reinforcement learning . all efforts were made to keep the applied algorithms as simple as possible subject to the constraints of the internet search . evolutionary computing deals with properties that may be modified during the creation of new individuals , called multiplication. descendants may exhibit variations of population , and differ in performance from the others . individuals may also terminate . multiplication and selection is subject to the fitness of individuals , where fitness is typically defined by the modeler . for a recent review on evolutionary computing , see @xcite . for reviews on related evolutionary theories and the dynamics of self - modifying systems see @xcite and @xcite , respectively . similar concepts have been studied in other evolutionary systems where organisms compete for space and resources and cooperate through direct interaction ( see , e.g. , @xcite and references therein . ) selection , however , is a very slow process and individual adaptation may be necessary in environments subject to quick changes . the typical form of adaptive learning is the connectionist architecture , such as artificial neural networks . multilayer perceptrons ( mlps ) , which are universal function approximators have been used widely in diverse applications . evolutionary selection of adapting mlps has been in the focus of extensive research @xcite . in a typical reinforcement learning ( rl ) problem the learning process @xcite is motivated by the expected value of long - term cumulated profit . a well - known example of reinforcement learning is the td - gammon program of tesauro @xcite . the author applied mlp function approximators for value estimation . reinforcement learning has also been used in concurrent multi - robot learning , where robots had to learn to forage together via direct interaction @xcite . evolutionary learning has been used within the framework of reinforcement learning to improve decision making , i.e. , the state - action mapping called policy @xcite . in this paper we present a selection based algorithm and compare it to the well - known reinforcement learning algorithm in terms of their efficiency and behavior . in our problem , fitness is not determined by us , but fitness is implicit . fitness is jointly determined by the ever changing external world and by the competing individuals together . selection and multiplication of individuals are based on their fitness value . communication and competition among our crawlers are indirect . only the first submitter of a document may receive positive reinforcement . our work is different from other studies using combinations of genetic , evolutionary , function approximation , and reinforcement learning algorithms , in that i ) it does not require explicit fitness function , ii ) we do not have control over the environment , iii ) collaborating individuals use value estimation under ` evolutionary pressure ' , and iv ) individuals work without direct interaction with each other . we performed realistic simulations based on data collected during an 18 days long crawl on the web . we have found that our selection based weblog update algorithm performs better in scale - free small world environment than the rl algorithm , eventhough the reinforcement learning algorithm has been shown to be efficient in finding relevant information @xcite . we explain our results based on the different behaviors of the algorithms . that is , the weblog update algorithm finds the good relevant document sources and remains at these regions until better places are found by chance . individuals using this selection algorithm are able to quickly collect the new relevant documents from the already known places because they monitor these places continuously . the reinforcement learning algorithm explores new territories for relevant documents and if it finds a good place then it collects the existing relevant documents from there . the continuous exploration of rl causes that it finds relevant documents slower than the weblog update algorithm . also , crawlers using weblog update algorithm submit more different documents than crawlers using the rl algorithm . therefore there are more relevant new information among documents submitted by former than latter crawlers . the paper is organized as follows . in section [ s : related ] we review recent works in the field of web crawling . then we describe our algorithms and the forager architecture in section [ s : architecture ] . after that in section [ s : experiments ] we present our experiment on the web and the conducted simulations with the results . in section [ s : discussion ] we discuss our results on the found different behaviors of the selection and reinforcement learning algorithms . section [ s : conclusion ] concludes our paper . our work concerns a realistic web environment and search algorithms over this environment . we compare selective / evolutionary and reinforcement learning methods . it seems to us that such studies should be conducted in ever changing , buzzling , wabbling environments , which justifies our choice of the environment . we shall review several of the known search tools including those @xcite that our work is based upon . readers familiar with search tools utilized on the web may wish to skip this section . there are three main problems that have been studied in the context of crawlers . rungsawang et al . @xcite and references therein and menczer @xcite studied the topic specific crawlers . risvik et al . @xcite and references therein address research issues related to the exponential growth of the web . cho and gracia - molina @xcite , menczer @xcite and edwards et . al @xcite and references therein studies the problem of different refresh rates of urls ( possibly as high as hourly or as low as yearly ) . rungsawang and angkawattanawit @xcite provide an introduction to and a broad overview of topic specific crawlers ( see citations in the paper ) . they propose to learn starting urls , topic keywords and url ordering through consecutive crawling attempts . they show that the learning of starting urls and the use of consecutive crawling attempts can increase the efficiency of the crawlers . the used heuristic is similar to the weblog algorithm @xcite , which also finds good starting urls and periodically restarts the crawling from the newly learned ones . the main limitation of this work is that it is incapable of addressing the freshness ( i.e. , modification ) of already visited web pages . menczer @xcite describes some disadvantages of current web search engines on the dynamic web , e.g. , the low ratio of fresh or relevant documents . he proposes to complement the search engines with intelligent crawlers , or web mining agents to overcome those disadvantages . search engines take static snapshots of the web with relatively large time intervals between two snapshots . intelligent web mining agents are different : they can find online the required recent information and may evolve intelligent behavior by exploiting the web linkage and textual information . he introduces the infospider architecture that uses genetic algorithm and reinforcement learning , also describes the myspider implementation of it . menczer discusses the difficulties of evaluating online query driven crawler agents . the main problem is that the whole set of relevant documents for any given query are unknown , only a subset of the relevant documents may be known . to solve this problem he introduces two new metrics that estimate the real recall and precision based on an available subset of the relevant documents . with these metrics search engine and online crawler performances can be compared . starting the myspider agent from the 100 top pages of altavista the agent s precision is better than altavista s precision even during the first few steps of the agent . the fact that the myspider agent finds relevant pages in the first few steps may make it deployable on users computers . some problems may arise from this kind of agent usage . first of all there are security issues , like which files or information sources are allowed to read and write for the agent . the run time of the agents should be controlled carefully because there can be many users ( google answered more than 100 million searches per day in january - february 2001 ) using these agents , thus creating huge traffic overhead on the internet . our weblog algorithm uses local selection for finding good starting urls for searches , thus not depending on any search engines . dependence on a search engine can be a suffer limitation of most existing search agents , like myspiders . note however , that it is an easy matter to combine the present algorithm with urls offered by search engines . also our algorithm should not run on individual users s computers . rather it should run for different topics near to the source of the documents in the given topic e.g. , may run at the actual site where relevant information is stored . risvik and michelsen @xcite mention that because of the exponential growth of the web there is an ever increasing need for more intelligent , ( topic-)specific algorithms for crawling , like focused crawling and document classification . with these algorithms crawlers and search engines can operate more efficiently in a topically limited document space . the authors also state that in such vertical regions the dynamics of the web pages is more homogenous . they overview different dimensions of web dynamics and show the arising problems in a search engine model . they show that the problem of rapid growth of web and frequent document updates creates new challenges for developing more and more efficient web search engines . the authors define a reference search engine model having three main components : ( 1 ) crawler , ( 2 ) indexer , ( 3 ) searcher . the main part of the paper focuses on the problems that crawlers need to overcome on the dynamic web . as a possible solution the authors propose a heterogenous crawling architecture . they also present an extensible indexer and searcher architecture . the crawling architecture has a central distributor that knows which crawler has to crawl which part of the web . special crawlers with low storage and high processing capacity are dedicated to web regions where content changes rapidly ( like news sites ) . these crawlers maintain up - to - date information on these rapidly changing web pages . the main limitation of their crawling architecture is that they must divide the web to be crawled into distinct portions manually before the crawling starts . a weblog like distributed algorithm as suggested here my be used in that architecture to overcome this limitation . cho and garcia - molina @xcite define mathematically the freshness and age of documents of search engines . they propose the poisson process as a model for page refreshment . the authors also propose various refresh policies and study their effectiveness both theoretically and on real data . they present the optimal refresh policies for their freshness and age metrics under the poisson page refresh model . the authors show that these policies are superior to others on real data , too . they collected about 720000 documents from 270 sites . although they show that in their database more than 20 percent of the documents are changed each day , they disclosed these documents from their studies . their crawler visited the documents once each day for 5 months , thus can not measure the exact change rate of those documents . while in our work we definitely concentrate on these frequently changing documents . the proposed refresh policies require good estimation of the refresh rate for each document . the estimation influences the revisit frequency while the revisit frequency influences the estimation . our algorithm does not need explicit frequency estimations . the more valuable urls ( e.g. , more frequently changing ) will be visited more often and if a crawler does not find valuable information around an url being in it s weblog then that url finally will fall out from the weblog of the crawler . however frequency estimations and refresh policies can be easily integrated into the weblog algorithm selecting the starting url from the weblog according to the refresh policy and weighting each url in the weblog according to their change frequency estimations . menczer @xcite also introduces a recency metric which is 1 if all of the documents are recent ( i.e. , not changed after the last download ) and goes to 0 as downloaded documents are getting more and more obsolete . trivially immediately after a few minutes run of an online crawler the value of this metric will be 1 , while the value for the search engine will be lower . edwards et al . @xcite present a mathematical crawler model in which the number of obsolete pages can be minimized with a nonlinear equation system . they solved the nonlinear equations with different parameter settings on realistic model data . their model uses different buckets for documents having different change rates therefore does not need any theoretical model about the change rate of pages . the main limitations of this work are the following : * by solving the nonlinear equations the content of web pages can not be taken into consideration . the model can not be extended easily to ( topic-)specific crawlers , which would be highly advantageous on the exponentially growing web @xcite , @xcite , @xcite . * the rapidly changing documents ( like on news sites ) are not considered to be in any bucket , therefore increasingly important parts of the web are disclosed from the searches . however the main conclusion of the paper is that there may exist some efficient strategy for incremental crawlers for reducing the number of obsolete pages without the need for any theoretical model about the change rate of pages . there are two different kinds of agents : the foragers and the reinforcing agent ( ra ) . the fleet of foragers crawl the web and send the urls of the selected documents to the reinforcing agent . the ra determines which forager should work for the ra and how long a forager should work . the ra sends reinforcements to the foragers based on the received urls . we employ a fleet of foragers to study the competition among individual foragers . the fleet of foragers allows to distribute the load of the searching task among different computers . a forager has simple , limited capabilities , like limited number of starting urls and a simple , content based url ordering . the foragers compete with each other for finding the most relevant documents . in this way they efficiently and quickly collect new relevant documents without direct interaction . at first the basic algorithms are presented . after that the reinforcing agent and the foragers are detailed . a forager periodically restarts from a url randomly selected from the list of starting urls . the sequence of visited urls between two restarts forms a path . the starting url list is formed from the @xmath0 first urls of the weblog . in the weblog there are @xmath1 urls with their associated weblog values in descending order . the weblog value of a url estimates the expected sum of rewards during a path after visiting that url . the weblog update algorithm modifies the weblog before a new path is started ( algorithm [ t : weblog_pseudo ] ) . the weblog value of a url already in the weblog is modified toward the sum of rewards in the remaining part of the path after that url . a new url has the value of actual sum of rewards in the remaining part of the path . if a url has a high weblog value it means that around that url there are many relevant documents . therefore it may worth it to start a search from that url . ' '' '' ' '' '' [ t : weblog_pseudo]*weblog update*. @xmath2 was set to 0.3 ' '' '' xxx = xx = xx = xx = xx = xx = xx = xx = xx = ` input ` + @xmath3 the steps of the given path + @xmath4 the sum of rewards for each step in the given path + ` output ` + starting url list + ` method ` + @xmath5 cumulated sum of @xmath6 in reverse order + @xmath7 not having value in @xmath8 + @xmath9 having value in @xmath8 + ` for each ` @xmath10 + @xmath11 + ` endfor ` + ` for each ` @xmath12 + @xmath13 + @xmath14 + ` endfor ` + @xmath15 descending order of values in @xmath8 + @xmath15 truncate @xmath8 after the @xmath16 + element + starting url list @xmath17 first @xmath18 elements of @xmath8 ' '' '' ' '' '' without the weblog algorithm the weblog and thus the starting url list remains the same throughout the searches . the weblog algorithm is a very simple version of evolutionary algorithms . here , evolution may occur at two different levels : the list of urls of the forager is evolving by the reordering of the weblog . also , a forager may multiply , and its weblog , or part of it may spread through inheritance . this way , the weblog algorithm incorporates most basic features of evolutionary algorithms . this simple form shall be satisfactory to demonstrate our statements . a forager can modify its url ordering based on the received reinforcements of the sent urls . the ( immediate ) profit is the difference of received rewards and penalties at any given step . immediate profit is a myopic characterization of a step to a url . foragers have an adaptive continuous value estimator and follow the _ policy _ that maximizes the expected long term cumulated profit ( ltp ) instead of the immediate profit . such estimators can be easily realized in neural systems @xcite . policy and profit estimation are interlinked concepts : profit estimation determines the policy , whereas policy influences choices and , in turn , the expected ltp . ( for a review , see @xcite . ) here , choices are based on the greedy ltp policy : the forager visits the url , which belongs to the _ frontier _ ( the list of linked but not yet visited urls , see later ) and has the highest estimated ltp . in the particular simulation each forager has a @xmath19 dimensional probabilistic term - frequency inverse document - frequency ( prtfidf ) text classifier @xcite , generated on a previously downloaded portion of the geocities database . fifty clusters were created by boley s clustering algorithm @xcite from the downloaded documents . the prtfidf classifiers were trained on these clusters plus an additional one , the @xmath20 , representing general texts from the internet . the prtfidf outputs were non - linearly mapped to the interval [ -1,+1 ] by a hyperbolic - tangent function . the classifier was applied to reduce the texts to a small dimensional representation . the output vector of the classifier for the page of url @xmath21 is @xmath22 . ( the @xmath20 output was dismissed . ) this output vector is stored for each url ( algorithm [ t : pageinfo_urlordering_pseudo ] ) . ' '' '' ' '' '' [ t : pageinfo_urlordering_pseudo]*page information storage * ' '' '' xxx = xx = xx = xx = xx = xx = xx = xx = xx = ` input ` + @xmath23 urls of pages to be stored + ` output ` + @xmath24 the classifier output vectors for pages of @xmath25 + ` method ` + ` for each ` @xmath26 + @xmath27 text of page of @xmath28 + @xmath29 classifier output vector for @xmath30 + ` endfor ` ' '' '' ' '' '' a linear function approximator is used for ltp estimation . it encompasses @xmath31 parameters , the _ weight vector _ @xmath32 . the ltp of document of url @xmath21 is estimated as the scalar product of @xmath33 and @xmath34 : @xmath35 . during url ordering the url with highest ltp estimation is selected . the url ordering algorithm is shown in algorithm [ t : urlordering_pseudo ] . ' '' '' ' '' '' [ t : urlordering_pseudo]*url ordering * ' '' '' xxx = xx = xx = xx = xx = xx = xx = xx = xx = ` input ` + @xmath36 the set of available urls + @xmath24 the stored vector representation of the urls + ` output ` + @xmath37 url with maximum ltp value + ` method ` + ` for each ` @xmath38 + @xmath39 + ` endfor ` + @xmath37 url with maximal ltp @xmath40 ' '' '' ' '' '' the weight vector of each forager is tuned by temporal difference learning @xcite . let us denote the current url by @xmath41 , the next url to be visited by @xmath42 , the output of the classifier for @xmath43 by @xmath44 and the estimated ltp of a url @xmath43 by @xmath45 . assume that leaving @xmath41 to @xmath42 the immediate profit is @xmath46 . our estimation is perfect if @xmath47 . future profits are typically discounted in such estimations as @xmath48 , where @xmath49 . the error of value estimation is @xmath50 we used throughout the simulations @xmath51 . for each step @xmath52 the weights of the value function were tuned to decrease the error of value estimation based on the received immediate profit @xmath46 . the @xmath53 estimation error was used to correct the parameters . the @xmath54 component of the weight vector , @xmath55 , was corrected by @xmath56 with @xmath57 and @xmath58 . these modified weights in a stationary environment would improve value estimation ( see , e.g , @xcite and references therein ) . the url ordering update is given in algorithm [ t : urlordering_update_pseudo ] . ' '' '' ' '' '' [ t : urlordering_update_pseudo]*url ordering update * ' '' '' xxx = xx = xx = xx = xx = xx = xx = xx = xx = ` input ` + @xmath59 the step for which the reinforcement is received + @xmath60 the previous step before @xmath42 + @xmath61 reinforcement for visiting @xmath42 + ` output ` + @xmath62 the updated weight vector + ` method ` + @xmath63 + @xmath64 ' '' '' ' '' '' without the update algorithm the weight vector remains the same throughout the search . a document or page is possibly relevant for a forager if it is not older than 24 hours and the forager has not marked it previously . algorithm [ t : relevant_pseudo ] shows the procedure of selecting such documents . the selected documents are sent to the ra for further evaluation . ' '' '' ' '' '' [ t : relevant_pseudo]*document relevancy at a forager * ' '' '' xxx = xx = xx = xx = xx = xx = xx = xx = xx = ` input ` + @xmath65 the pages to be examined + ` output ` + @xmath66 the selected pages + ` method ` + @xmath67 previously selected relevant pages + @xmath66 all pages from @xmath68 which are + not older than 24 hours and + not contained in @xmath69 + @xmath67 add @xmath70 to @xmath69 ' '' '' ' '' '' during multiplication the weblog is randomly divided into two equal sized parts ( one for the original and one for the new forager ) . the parameters of the url ordering algorithm ( the weight vector of the value estimation ) are either copied or new random parameters are generated . if the forager has a url ordering update algorithm then the parameters are copied . if the forager does not have any url ordering update algorithm then new random parameters are generated , as shown in algorithm [ t : multiplication_pseudo ] . ' '' '' ' '' '' [ t : multiplication_pseudo]*multiplication * ' '' '' xxx = xx = xx = xx = xx = xx = xx = xx = xx = ` input ` + @xmath8 + weight vector of url ordering + ` output ` + @xmath71 + @xmath72 + ` method ` + @xmath73 randomly selected + urls and values from @xmath8 + @xmath15 delete @xmath71 from @xmath8 + ` if ` forager has url ordering update algorithm + @xmath74 copy the weight vector of url ordering + ` else ` + @xmath74 generate a new random weight vector + ` endif ` ' '' '' ' '' '' a reinforcing agent controls the `` life '' of foragers . it can start , stop , multiply or delete foragers . ra receives the urls of documents selected by the foragers , and responds with reinforcements for the received urls . the response is @xmath75 ( a.u . ) for a relevant document and @xmath76 ( a.u . ) for a not relevant document . a document is relevant if it is not yet seen by the reinforcing agent and it is not older than 24 hours . the reinforcing agent maintains the score of each forager working for it . initially each forager has @xmath77 score . when a forager sends a url to the ra , the forager s score is decreased by @xmath78 . after each relevant page sent by the forager , the forager s score is increased by @xmath79 ( algorithm [ t : manageurl_pseudo ] ) . ' '' '' ' '' '' [ t : manageurl_pseudo]*manage received url * ' '' '' xxx = xx = xx = xx = xx = xx = xx = xxxxx = ` input ` + @xmath80 received url from forager + ` output ` + reinforcement to forager + updated forager score + ` method ` + @xmath81 relevant pages seen by the ra + @xmath27 get page of @xmath28 + decrease forager s score with @xmath82 + ` if ` @xmath83 or page date is older than 24 hours + send @xmath84 to forager + ` else ` + @xmath81 add @xmath30 to @xmath85 + send @xmath86 to forager + increase forager s score with @xmath87 + ` endif ` ' '' '' ' '' '' when the forager s score reaches @xmath88 and the number of foragers is smaller than @xmath89 then the forager is multiplied . that is a new forager is created with the same algorithms as the original one has , but with slightly different parameters . when the forager s score goes below @xmath90 and the number of foragers is larger than @xmath91 then the forager is deleted ( algorithm [ t : manageforager_pseudo ] ) . note that a forager can be multiplied or deleted immediately after it has been stopped by the ra and before the next forager is activated . ' '' '' ' '' '' [ t : manageforager_pseudo ] * : manage forager * ' '' '' xxx = xx = xx = xx = xx = xx = xx = xxxxx = ` input ` + @xmath92 the forager to be multiplied or deleted + ` output ` + possibly modified list of foragers + ` method ` + ` if ` ( @xmath93 s score @xmath94 @xmath95 ` and ` + number of foragers @xmath96 @xmath97 ) + @xmath98 call @xmath93 s + * multiplication , alg . [ t : multiplication_pseudo ] * + @xmath93 may modify it s own weblog + @xmath99 create a new forager with the received + @xmath8 and @xmath100 + set the two foragers score to @xmath101 + ` else if ` ( @xmath93 s score @xmath102 @xmath103 ` and ` + number of foragers @xmath104 @xmath105 ) + delete @xmath93 + ` endif ` ' '' '' ' '' '' foragers on the same computer are working in time slices one after each other . each forager works for some amount of time determined by the ra . then the ra stops that forager and starts the next one selected by the ra . the pseudo - code of the reinforcing agent is given in algorithm [ t : reinforcing_pseudo ] . ' '' '' ' '' '' [ t : reinforcing_pseudo ] * : reinforcing agent * ' '' '' xxx = xx = xx = xx = xx = xx = xx = xxxxx = ` input ` + seed urls + ` output ` + @xmath81 found relevant documents + ` method ` + @xmath81 empty set /*set of all observed relevant pages + initialize @xmath105 foragers with the seed urls + set one of them to be the next + ` repeat ` + start next forager + receive possibly relevant url + call * manage received url , alg . [ t : manageurl_pseudo ] * with url + stop forager if its time period is over + call * manage forager , alg . [ t : manageforager_pseudo ] * with this forager + choose next forager + ` until ` time is over ' '' '' ' '' '' a forager is initialized with parameters defining the url ordering , and either with a weblog or with a seed of urls ( algorithm [ t : initforager_pseudo ] ) . after its initialization a forager crawls in search paths , that is after a given number of steps the search restarts and the steps between two restarts form a path . during each path the forager takes @xmath106 number of steps , i.e. , selects the next url to be visited with a url ordering algorithm . at the beginning of a path a url is selected randomly from the starting url list . this list is formed from the 10 first urls of the weblog . the weblog contains the possibly good starting urls with their associated weblog values in descending order . the weblog algorithm modifies the weblog and so thus the starting url list before a new path is started . when a forager is restarted by the ra , after the ra has stopped it , the forager continues from the internal state in which it was stopped . the pseudo code of step selection is given in algorithm [ t : stepselection_pseudo ] . ' '' '' ' '' '' [ t : initforager_pseudo]*initialization of the forager * ' '' '' xxx = xx = xx = xx = xx = xx = xx = xx = xx = ` input ` + weblog or seed urls + url ordering parameters + ` output ` + initialized forager + ` method ` + set path step number to @xmath107 /*start new path + set the weblog + either with the input weblog + or put the seed urls into the weblog with 0 weblog value + set the url ordering parameters in url ordering algorithm ' '' '' ' '' '' ' '' '' ' '' '' [ t : stepselection_pseudo]*url selection * ' '' '' xxx = xx = xx = xx = xx = xx = xx = xx = xx = ` input ` + @xmath36 set of urls available in this step + @xmath108 set of visited urls in this path + ` output ` + @xmath109 selected url to be visited next + ` method ` + ` if ` path step number @xmath110 + @xmath109 selected url by * url ordering , alg . [ t : urlordering_pseudo ] * + increase path step number + ` else ` + call the * weblog update , alg . [ t : weblog_pseudo ] * to update the weblog + @xmath109 select a random url from the starting url list + set path step number to 1 + @xmath36 empty set + @xmath108 empty set + ` endif ` ' '' '' ' '' '' the url ordering algorithm selects a url to be the next step from the frontier url set . the selected url is removed from the frontier and added to the visited url set to avoid loops . after downloading the pages , only those urls ( linked from the visited url ) are added to the frontier which are not in the visited set . in each step the forager downloads the page of the selected url and all of the pages linked from the page of selected url . it sends the urls of the possibly relevant pages to the reinforcing agent . the forager receives reinforcements on any previously sent but not yet reinforced urls and calls the url ordering update algorithm with the received reinforcements . the pseudo code of a forager is shown in algorithm [ t : forager_pseudo ] . ' '' '' ' '' '' [ t : forager_pseudo]*forager * ' '' '' xxx = xx = xx = xx = xx = xx = xx = xx = xx = ` input ` + @xmath36 set of urls available in the next step + @xmath108 set of visited urls in the current path + ` output ` + sent documents to the ra + modified @xmath111 and @xmath112 + modified @xmath8 and url ordering weight vector + ` method ` + ` repeat ` + @xmath109 call * url selection , alg . [ t : stepselection_pseudo ] * + @xmath36 remove @xmath113 from @xmath111 + @xmath108 add @xmath113 to @xmath112 + @xmath27 download the page of @xmath113 + @xmath114 links of @xmath30 + @xmath115 @xmath116 which are not @xmath112 + @xmath36 add @xmath117 to @xmath111 + download pages of @xmath116 + call * page information storage , alg . [ t : pageinfo_urlordering_pseudo ] * with @xmath117 + @xmath66 call * document relevancy , alg . [ t : relevant_pseudo ] * for + all pages + send @xmath70 to reinforcing agent + receive reinforcements for sent but not yet reinforced pages + call * url ordering update , alg . [ t : urlordering_update_pseudo ] * with + the received reinforcements + ` until ` time is over ' '' '' ' '' '' we conducted an 18 day long experiment on the web to gather realistic data . we used the gathered data in simulations to compare the weblog update ( section [ sss : weblog ] ) and reinforcement learning algorithms ( section [ sss : rl ] ) . in web experiment we used a fleet of foragers using combination of reinforcement learning and weblog update algorithms to eliminate any biases on the gathered data . first we describe the experiment on the web then the simulations . we analyze our results at the end of this section . we ran the experiment on the web on a single personal computer with celeron 1000 mhz processor and 512 mb ram . we implemented the forager architecture ( described in section [ s : architecture ] ) in java programming language . in this experiment a fixed number of foragers were competing with each other to collect news at the cnn web site . the foragers were running in equal time intervals in a predefined order . each forager had a 3 minute time interval and after that interval the forager was allowed to finish the step started before the end of the time interval . we deployed 8 foragers using the weblog update and the reinforcement learning based url ordering update algorithms ( 8 wlrl foragers ) . we also deployed 8 other foragers using the weblog update algorithm but without reinforcement learning ( 8 wl foragers ) . the predefined order of foragers was the following : 8 wlrl foragers were followed by the 8 wl foragers . we investigated the link structure of the gathered web pages . as it is shown in fig . [ f : sf ] the links have a power - law distribution ( @xmath118 ) with @xmath119 for outgoing links and @xmath120 for incoming links . that is the link structure has the scale - free property . the clustering coefficient @xcite of the link structure is 0.02 and the diameter of the graph is 7.2893 . we applied two different random permutations to the origin and to the endpoint of the links , keeping the edge distribution unchanged but randomly rewiring the links . the new graph has 0.003 clustering coefficient and 8.2163 diameter . that is the clustering coefficient is smaller than the original value by an order of magnitude , but the diameter is almost the same . therefore we can conclude that the links of gathered pages form small world structure . ) . vertical axis : relative frequency of number of edges at different urls ( @xmath121 ) . dots and dark line correspond to outgoing links , crosses and gray line correspond to incoming links . , width=240 ] the data storage for simulation is a centralized component . the pages are stored with 2 indices ( and time stamps ) . one index is the url index , the other is the page index . multiple pages can have the same url index if they were downloaded from the same url . the page index uniquely identifies a page content and the url from where the page was download . at each page download of any foragers we stored the followings ( with a time stamp containing the time of page download ) : 1 . if the page is relevant according to the ra then store `` relevant '' 2 . if the page is from a new url then store the new url with a new url index and the page s state vector with a new page index 3 . if the content of the page is changed since the last download then store the page s state vector with a new page index but keep the url index 4 . in both previous cases store the links of the page as links to page indices of the linked pages 1 . if a linked page is from a new url then store the new url with a new url index and the linked page s state vector with a new page index 2 . if the content of the linked page is changed since the last check then store the page s state vector with a new page index but same url index for the simulations we implemented the forager architecture in matlab . the foragers were simulated as if they were running on one computer as described in the previous section . during simulations we used the web pages that we gathered previously to generate a realistic environment ( note that the links of pages point to local pages ( not to pages on the web ) since a link was stored as a link to a local page index ) : * simulated documents had the same state vector representation for url ordering as the real pages had * simulated relevant documents were the same as the relevant documents on the web * pages and links appeared at the same ( relative ) time when they were found in the web experiment - using the new url indices and their time stamps * pages and links are refreshed or changed at the same relative time as the changes were detected in the web experiment using the new page indices for existing url indices and their time stamps * simulated time of a page download was the average download time of a real page during the web experiment . we conducted simulations with two different kinds of foragers . the first case is when foragers used only the weblog update algorithm without url ordering update ( wl foragers ) . the second case is when foragers used only the reinforcement learning based url ordering update algorithm without the weblog update algorithm ( rl foragers ) . each wl forager had a different weight vector for url value estimation during multiplication the new forager got a new random weight vector . rl foragers had the same weblog with the first 10 urls of the gathered pages that is the starting url of the web experiment and the first 9 visited urls during that experiment . in both cases initially there were 2 foragers and they were allowed to multiply until reaching the population of 16 foragers . the simulation for each type of foragers were repeated 3 times with different initial weight vectors for each forager . the variance of the results show that there is only a small difference between simulations using the same kind of foragers , even if the foragers were started with different random weight vectors in each simulation . table [ t : params ] shows the investigated parameters during simulations . [ cols= " < , < " , ] from table [ t : data ] we can conclude the followings : * rl and wl foragers have similar download efficiency , i.e. , the efficiencies from the point of view of the news site are about the same . * wl foragers have higher sent efficiencies than rl foragers , i.e. , the efficiency from the point of view of the ra is higher . this shows that wl foragers divide the search area better among each other than rl foragers . sent efficiency would be 1 if none of two foragers have sent the same document to the ra . * rl foragers have higher relative found url value than wl foragers . rl foragers explore more than wl foragers and rl found more urls than wl foragers did per downloaded page . * wl foragers find faster the new relevant documents in the already found clusters . that is freshness is higher and age is lower than in the case of rl foragers . [ f : efficiency ] shows other aspects of the different behaviors of rl and wl foragers . download efficiency of rl foragers has more , higher , and sharper peaks than the download efficiency of wl foragers has . that is wl foragers are more balanced in finding new relevant documents than rl foragers . the reason is that while the wl foragers remain in the found good clusters , the rl foragers continuously explore the new promising territories . the sharp peaks in the efficiency show that rl foragers _ find and recognize _ new good territories and then _ quickly collect _ the current relevant documents from there . the foragers can recognize these places by receiving more rewards from the ra if they send urls from these places . the predefined order did not influence the working of foragers during the web experiment . from fig . [ f : efficiency ] it can be seen that foragers during the 3 independent experiments did not have very different efficiencies . on fig . [ f : freshness ] we show that the foragers in each run had a very similar behavior in terms of age and freshness , that is the values remains close to each other throughout the experiments . also the results for individual runs were close to the average values in table [ t : data ] ( see the standard deviations ) . in each individual run the foragers were started with different weight vectors , but they reached similar efficiencies and behavior . this means that the initial conditions of the foragers did not influence the later behavior of them during the simulations . furthermore foragers could not change their environment drastically ( in terms of the found relevant documents ) during a single 3 minute run time because of the short run time intervals and the fast change of environment large number of new pages and often updated pages in the new site . during the web experiment foragers were running in 8 wlrl , 8 wl , 8 wlrl , 8 wl , temporal order . because of the fact that initial conditions does not influence the long term performance of foragers and the fact that the foragers can not change their environment fully we can start to examine them after the first run of wlrl foragers . then we got the other extreme order of foragers , that is the 8 wl , 8 wlrl , 8 wl , 8 wlrl , temporal ordering . for the overall efficiency and behavior of foragers it did not really matter if wlrl or wl foragers run first and one could use mixed order in which after a wlrl forager a wl forager runs and after a wl forager a wlrl forager comes . however , for higher bandwidths and for faster computers , random ordering may be needed for such comparisons . our first conjecture is that selection is efficient on scale - free small world structures . lrincz and kkai @xcite and rennie et al . @xcite showed that rl is efficient in the task of finding relevant information on the web . here we have shown experimentally that the weblog update algorithm , selection among starting urls , is at least as efficient as the rl algorithm . the weblog update algorithm finds as many relevant documents as rl does if they download the same amount of pages . wl foragers in their fleet select more different urls to send to the ra than rl foragers do in their fleet , therefore there are more relevant documents among those selected by wl foragers then among those selected by rl foragers . also the freshness and age of found relevant documents are better for wl foragers than for rl foragers . for the weblog update algorithm , the selection among starting urls has no fine tuning mechanism . throughout its life a forager searches for the same kind of documents goes into the same ` direction ' in the state space of document states determined by its fixed weight vector . the only adaptation allowed for a wl forager is to select starting urls from the already seen urls . the wl forager can not modify its ( ` directional ' ) preferences according goes newly found relevant document supply , where relevant documents are abundant . but a wl forager finds good relevant document sources in its own direction and forces its search to stay at those places . by chance the forager can find better sources in its own direction if the search path from a starting url is long enough . on fig . [ f : efficiency ] it is shown that the download efficiency of the foragers does not decrease with the multiplication of the foragers . therefore the new foragers must found new and good relevant document sources quickly after their appearances . the reinforcement learning based url ordering update algorithm is capable to fine tune the search of a forager by adapting the forager s weight vector . this feature has been shown to be crucial to adapt crawling in novel environments @xcite . an rl forager goes into the direction ( in the state space of document states ) where the estimated long term cumulated profit is the highest . because the local environment of the foragers may changes rapidly during crawling , it seems desirable that foragers can quickly adapt to the found new relevant documents . relevant documents may appear lonely , not creating a good relevant document source , or do not appear at the right url by a mistake . this noise of the web can derail the rl foragers from good regions . the forager may `` turn '' into less valuable directions , because of the fast adaptation capabilities of rl foragers . our second conjecture is that selection fits sfsw better than rl . we have shown in our experiments that selection and rl have different behaviors . selection selects good information sources , which are worth to revisit , and stays at those sources as long as better sources are not found by chance . rl explores new territories , and adapts to those . this adaptation can be a disadvantage when compared with the more rigid selection algorithm , which sticks to good places until ` provably ' better places are discovered . therefore wl foragers , which can not be derailed and stay in their found ` niches ' can find new relevant documents faster in such already known terrains than rl foragers can . that is , freshness is higher and age is lower for relevant documents found by wl foragers than for relevant documents found by rl foragers . also , by finding good sources and staying there , wl foragers divide the search task better than rl foragers do , this is the reason for the higher sent efficiency of wl foragers than of rl foragers . we have rewired the network as it was described in section [ ss : real ] . this way a scale - free ( sf ) but not so small world was created . intriguingly , in this sf structure , rl foragers performed better than wl ones . clearly , further work is needed to compare the behavior of the selective and the reinforcement learning algorithms in other then sfsw environments . such findings should be of relevance in the deployment of machine learning methods in different problem domains . from the practical point of view , we note that it is an easy matter to combine the present algorithm with urls offered by search engines . also , the values reported by the crawlers about certain environments , e.g. , the environment of the url offered by search engines represent the neighborhood of that url and can serve adaptive filtering . this procedure is , indeed , promising to guide individual searches as it has been shown elsewhere @xcite . we presented and compared our selection algorithm to the well - known reinforcement learning algorithm . our comparison was based on finding new relevant documents on the web , that is in a dynamic scale - free small world environment . we have found that the weblog update selection algorithm performs better in this environment than the reinforcement learning algorithm , eventhough the reinforcement learning algorithm has been shown to be efficient in finding relevant information @xcite . we explain our results based on the different behaviors of the algorithms . that is the weblog update algorithm finds the good relevant document sources and remains at these regions until better places are found by chance . individuals using this selection algorithm are able to quickly collect the new relevant documents from the already known places because they monitor these places continuously . the reinforcement learning algorithm explores new territories for relevant documents and if it finds a good place then it collects the existing relevant documents from there . the continuous exploration and the fine tuning property of rl causes that rl finds relevant documents slower than the weblog update algorithm . in our future work we will study the combination of the weblog update and the rl algorithms . this combination uses the wl foragers ability to stay at good regions with the rl foragers fine tuning capability . in this way foragers will be able to go to new sources with the rl algorithm and monitor the already found good regions with the weblog update algorithm . we will also study the foragers in a simulated environment which is not a small world . the clusters of small world environment makes it easier for wl foragers to stay at good regions . the small diameter due to the long distance links of small world environment makes it easier for rl foragers to explore different regions . this work will measure the extent at which the different foragers rely on the small world property of their environment . this material is based upon work supported by the european office of aerospace research and development , air force office of scientific research , air force research laboratory , under contract no . fa8655 - 03 - 1 - 3036 . this work is also supported by the national science foundation under grants no . int-0304904 and no . any opinions , findings and conclusions or recommendations expressed in this material are those of the author(s ) and do not necessarily reflect the views of the european office of aerospace research and development , air force office of scientific research , air force research laboratory . j. edwards , k. mccurley , and j. tomlin , _ an adaptive model for optimizing performance of an incremental web crawler _ , proceedings of the tenth international conference on world wide web , 2001 , pp . 106113 . b. gbor , zs . palotai , and a. lrincz , _ value estimation based computer - assisted data mining for surfing the internet _ , int . joint conf . on neural networks ( piscataway , nj 08855 - 1331 ) , ieee operations center , 26 - 29 july , budapest , hungary 2004 , pp . 1035 . , ieee catalog number : 04ch37541c , ijcnn2004 cd rom conference proceedings . thorsten joachims , _ a probabilistic analysis of the rocchio algorithm with tfidf for text categorization _ , proceedings of icml-97 , 14th international conference on machine learning ( nashville , us ) ( douglas h. fisher , ed . ) , morgan kaufmann publishers , san francisco , us , 1997 , pp . 143151 . k. tuyls , d. heytens , a. nowe , and b. manderick , _ extended replicator dynamics as a key to reinforcement learning in multi - agent systems _ , ecml 2003 , lnai 2837 ( n. lavrac et al . , ed . ) , springer - verlag , berlin , 2003 , pp .

in this paper we compare the performance characteristics of our selection based learning algorithm for web crawlers with the characteristics of the reinforcement learning algorithm . the task of the crawlers is to find new information on the web . the selection algorithm , called weblog update , modifies the starting url lists of our crawlers based on the found urls containing new information . the reinforcement learning algorithm modifies the url orderings of the crawlers based on the received reinforcements for submitted documents . we performed simulations based on data collected from the web . the collected portion of the web is typical and exhibits scale - free small world ( sfsw ) structure . we have found that on this sfsw , the weblog update algorithm performs better than the reinforcement learning algorithm . it finds the new information faster than the reinforcement learning algorithm and has better new information / all submitted documents ratio . we believe that the advantages of the selection algorithm over reinforcement learning algorithm is due to the small world property of the web .

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Proceed to summarize the following text: there is the classical weierstrass representation for minimal surfaces in euclidean @xmath0-space @xmath3 , which gives a local conformal parametrization for any minimal surface . it involves choosing two holomorphic functions ( or perhaps meromorphic functions when considering the surfaces more globally ) on a riemann surface . if one restricts to isothermic parametrizations , that is , conformal parametrizations that are also curvature lines coordinate systems , then the representation reduces to the choice of just one holomorphic function . since every minimal surface has local isothermic coordinates ( away from umbilics ) , this reduction does not involve any loss of generality beyond avoiding umbilic points . once one restricts to isothermic parametrizations , it becomes possible to give a definition for discrete analogs of minimal surfaces @xcite . these analogs are `` discrete isothermic '' meshes ( a definition of this is given later in this paper ) . they are comprized of planar quadrilaterals , which in particular have concircular vertices . by a transformation called the lawson correspondence or @xmath4-transformation or calapso transformation @xcite , one can produce all constant mean curvature ( cmc ) @xmath2 surfaces in hyperbolic @xmath0-space @xmath1 from minimal surfaces in @xmath3 . there is a corresponding holomorphic representation for those surfaces as well , first given by bryant @xcite . correspondingly , without loss of generality beyond avoiding umbilics , one can restrict to isothermic coordinates in this case also , and one has a discrete analog of cmc @xmath2 surfaces in @xmath1 , first found by hertrich - jeromin @xcite . in the case of smooth surfaces there is also a holomorphic representation for flat ( i.e. intrinsic curvature zero ) surfaces in @xmath1 @xcite and this also ties in to the above - mentioned bryant representation , as there are deformations from cmc @xmath2 surfaces in @xmath1 to flat surfaces via a family of linear weingarten surfaces in @xmath1 @xcite . these do not include all linear weingarten surfaces , but rather a certain special subclass called linear weingarten surfaces of _ bryant type _ @xcite @xcite , so named because they have bryant - type representations . thus it is natural to wonder if flat surfaces also have a discrete analog , and we will see here that they do . once this discrete analog is found , a new question about `` singularities on discrete flat surfaces '' naturally presents itself , in this way : unlike the smooth isothermic minimal surfaces in @xmath3 and cmc @xmath2 surfaces in @xmath1 , smooth flat fronts have certain types of singularities , such as cuspidal edges and swallowtails ( in fact , indirectly , this is what the naming `` fronts '' rather than `` surfaces '' indicates ) . the means for recognizing where the singularities are on smooth flat fronts are clear , and one can make classifications of those surfaces most generic types of singularities just from looking at the choices of holomorphic functions used in their representation @xcite . however , in the case of discrete flat surfaces , it is not apriori clear where the singularities are , nor even what such a notion would mean . since one does not have first and second fundamental forms at one s disposal in the discrete case , one must find an alternate way of defining singularities . we aim towards this by defining and using a discrete analog of caustics , also called focal surfaces , for smooth flat fronts . for a smooth flat front , the caustic is the surface comprised of all the singular points on all parallel surfaces of that flat front . ( the parallel surfaces are also flat . ) thus the singular set of the flat front can be retrieved by taking its intersection with its caustic . in the case of a smooth flat front , the caustic is again a flat surface , but this will not quite be the case for discrete flat surfaces . we will also present a number of examples of these discrete flat surfaces . in addition to the rather simple examples of discrete cylinders and discrete surfaces of revolution , we will also discuss a discrete flat surface based on the airy equation . this example exhibits swallowtail singularities and a stokes phenomenon , similar to that of the analogous surface in the smooth case , as shown by two of the authors in @xcite . this last example hints at existence of a robust collection of discrete flat surfaces with interesting geometric properties yet to be explored . thus , the purpose of this paper is to : 1 . provide a definition for discrete flat surfaces and discrete linear weingarten surfaces of bryant type in hyperbolic @xmath0-space @xmath1 ; 2 . give properties of these surfaces that justify our choice of definitions ( in particular , as smooth flat fronts have extrinsic curvature @xmath2 , we identify notions of discrete extrinsic curvature of discrete flat surfaces which do indeed attain the value @xmath2 ) ; 3 . show that these surfaces have concircular quadrilaterals ; 4 . study examples of these surfaces , and in particular look at swallowtail singularities and global properties of an example related to the airy equation ; 5 . give a definition of discrete caustics for discrete flat surfaces ; 6 . show that the caustics also have concircular quadrilaterals and that they provide a means for identifying a notion of singularities on discrete flat surfaces . in section [ section1 ] we describe smooth and discrete minimal surfaces in euclidean @xmath0-space @xmath3 , to help motivate later definitions , and we also give the definition of a discrete holomorphic function , which will be essential to everything that follows . in section [ section2 ] we describe smooth cmc @xmath2 surfaces , and flat surfaces and linear weingarten surfaces of bryant type in @xmath1 , again as motivational material for the definitions of the corresponding discrete surfaces in section [ section3 ] . we prove in section [ section3 ] that discrete flat surfaces and linear weingarten surfaces of bryant type have concircular quadrilaterals . also , section [ section3 ] provides a natural representation for discrete flat surfaces which gives the mapping of the surfaces as products of @xmath5 by @xmath5 matrices times their conjugate transposes , and we show that this representation applies to the case of discrete cmc @xmath2 surfaces as well . the definition for discrete cmc @xmath2 surfaces is already known @xcite , but the representation here for those surfaces is new . in section [ section4 ] we look at a specific discrete example whose smooth analog is equivalent to solutions of the airy equation , and we look at the asymptotic behavior of that surface , which exhibits swallowtail singularities and a stokes phenomenon . in section [ section5 ] , we look at normal lines to discrete flat surfaces . with this we can do several things . for example , we look at parallel surfaces ( which are also discrete flat ) and show that the area of corresponding quadrilaterals of the normal map equals the area of the quadrilaterals of the surface itself , as should be expected , since in some sense the extrinsic curvature of the surface is identically equal to @xmath2 ( note that the analogous statement is true for smooth surfaces with extrinsic curvature @xmath2 , infinitesimally ) . then , using distances from the surface s vertices to the intersection points of the normal lines , we consider a discrete analog of the extrinsic curvature and see that it is @xmath2 in the discrete case as well . furthermore , those intersections give us a means to define discrete caustics , and , as mentioned above , we use those caustics to study the nature of `` singularities '' on discrete flat surfaces , in the final section [ section6 ] . the authors thank udo hertrich - jeromin for fruitful discussions and valuable comments . a useful choice of coordinates for a surface is isothermic coordinates . not all surfaces have such coordinates , but cmc surfaces in space forms such as @xmath3 and @xmath1 do have them , away from umbilic points . isothermic coordinates will be of central importance in this paper . another useful tool in the study of surfaces in space forms is the hopf differential , which is defined as @xmath6 , where the surface @xmath7 is a map from points @xmath8 in a portion of the complex plane @xmath9 , @xmath10 is the bilinear extension of the metric for the ambient space form to complex vectors , and @xmath11 is the unit normal to the surface . when the coordinate @xmath8 is conformal and the surface is cmc , then @xmath12 will be holomorphic in @xmath8 . umbilic points of the surface occur precisely at the zeros of the hopf differential . locally , away from umbilics , we can always take a smooth minimal immersion @xmath13 into @xmath3 to have isothermic coordinates @xmath14 in a domain of @xmath15 . let @xmath11 denote the unit normal vector to @xmath7 . then , setting @xmath16 , the hopf differential becomes @xmath17 for some real constant @xmath18 , and rescaling the coordinate @xmath8 , we may assume @xmath19 . let @xmath20 be the stereographic projection of the gauss map @xmath11 to the complex plane , and set @xmath21 . as we are only concerned with the local behavior of the surface , and we are allowed to replace the surface with any rigid motion of it , we may ignore the possibility that @xmath20 has poles or other singularities , and so the map @xmath22 is holomorphic . because we avoid umbilic points of @xmath7 , we also know that @xmath23 is never zero . thus the weierstrass representation is ( with @xmath24 regarded as lying in the complex plane @xmath9 ) @xmath25 associating @xmath26 , @xmath27 and @xmath28 with the quaternions @xmath29 , @xmath30 and @xmath31 , respectively , we have @xmath32 we have converted to a formulation using quaternions here , because this type of formulation has been used to define discrete minimal surfaces in @xmath3 and discrete cmc @xmath2 surfaces in @xmath1 , and we wish to make comparisons to those formulations . note that by restricting to isothermic coordinates , we can then determine minimal surfaces by choosing just one holomorphic function @xmath20 . to define discrete minimal surfaces , we use discrete holomorphic functions @xmath33 , where @xmath34 is the square integer lattice @xmath35 , or a subdomain of it . discrete holomorphic functions are defined as follows : defining the cross ratio of @xmath20 to be @xmath36 we say that @xmath20 is _ discrete holomorphic _ if there exists a discrete mapping @xmath37 to @xmath38 such that @xmath39 with @xmath40 and @xmath41 for all quadrilaterals ( squares with edge length @xmath2 and vertices in @xmath34 . ) see @xcite . we call the discrete map @xmath37 a _ cross ratio factorizing function _ for @xmath20 . note that @xmath37 is defined on edges of @xmath34 , not vertices . note also that @xmath37 is symmetric , that is , @xmath42 and @xmath43 . there is a freedom of a single real factor in the choice of these @xmath44 and @xmath45 , since we could replace all of them with @xmath46 and @xmath47 for any nonzero real constant @xmath48 , and all relevant properties would still hold . throughout this paper we use @xmath48 to denote that free factor . in the above definition of the cross ratio , we have a product of four terms . since @xmath49 , these terms all commute , and so we could have written this cross ratio simply as a product of two fractions . however , when we later consider the cross ratio for quaternionic - valued objects or matrix - valued objects , commutativity no longer holds and the order of the product in the cross ratio becomes vital . so , for later reference , we have chosen to write the cross ratio in the somewhat cumbersome way above . the definition above for discrete holomorphic functions is in the `` broad '' sense . the definition in the `` narrow '' sense would be that @xmath50 is identically @xmath51 on @xmath34 ( see @xcite , @xcite ) . furthermore , note that , unlike the case of smooth holomorphic functions , the discrete derivative or discrete integral of a discrete holomorphic function is generally not another discrete holomorphic function . let us exhibit some examples of discrete holomorphic functions : 1 . let @xmath52 , and set @xmath53 for @xmath54 a complex constant . 2 . let @xmath55 , and set @xmath56 for @xmath54 a real or pure imaginary constant . one could also take the function @xmath57 for choices of real constants @xmath58 and @xmath59 so that the cross ratio is identically @xmath51 , giving a discrete holomorphic function in the narrow sense . [ item5 ] in section [ section4 ] we will describe a discrete flat surface based on a discrete version of the power function @xmath60 ( @xmath61 ) , which we define here . this function is discrete holomorphic in the narrow sense . it is defined by the recursion @xmath62 we start with @xmath63 . for @xmath64 , the initial conditions should be @xmath65 we can then use to propagate along the positive axes @xmath66 and @xmath67 with @xmath68 and @xmath69 , respectively . we can then compute general @xmath70 ( for both @xmath71 and @xmath72 ) by using that the cross ratio is always @xmath51 . the @xmath70 will then automatically satisfy the recursion relation . this definition of the discrete power function can be found in bobenko @xcite . ( it is also found in a recently published textbook @xcite . ) agafonov @xcite showed that these discrete power functions are embedded in wedges ( see figure [ fct43 ] ) , and are schramm circle packings ( see @xcite ) . note that , for @xmath73 and @xmath74 , @xmath75 where @xmath76 denotes the pochhammer symbol , and a closed expression for general @xmath70 is still unknown . we explore this difference equation in more detail in appendix [ appendix ] at the end of this paper . [ cols="^,^ " , ] in this and the next section ( and in appendix [ appendix9 ] ) , since we will consider flat surfaces and their caustics exclusively , we will abbreviate the notation `` @xmath77 '' to `` @xmath7 '' . let @xmath78 be the collection of focal points , i.e. the _ focal surface _ , also called the _ caustic _ , of a smooth flat surface @xmath7 in @xmath1 ( note that we should assume @xmath7 is a flat front , so that the caustic will exist ) . if @xmath79 is the lift of @xmath80 ( determined from @xmath20 ) , then ( see @xcite ) @xmath81 is a lift of @xmath82 . although we have just described the caustic in terms of weierstrass data , it is independent of the choice of that data . we know that @xmath78 is a flat surface ( see @xcite , @xcite ) , because its lift @xmath83 satisfies the following equation : @xmath84 for the case of a discrete flat surface @xmath7 in @xmath1 with discrete lift @xmath85 , we must first consider how to define the caustic @xmath78 . we can define the normal @xmath86 as in at each vertex @xmath87 of @xmath7 , so we have normal geodesics emanating from each vertex , and we can consider when normal geodesics of adjacent vertices will intersect . once we have those intersection points , we will see that we can consider them as vertices of @xmath78 , giving us a definition for @xmath78 . [ single - caustic - lemma ] let @xmath7 be a discrete flat surface in @xmath1 with lift @xmath85 , as in , constructed using the discrete holomorphic function @xmath20 . let @xmath88 be a cross ratio factorizing function for @xmath20 . then the normal geodesics in @xmath1 emanating from two adjacent vertices @xmath87 and @xmath89 will intersect if and only if @xmath90 , in which case the intersection point is unique and is equidistant from @xmath87 and @xmath89 . furthermore , even when the two normal geodesics do not intersect , they still lie in a single common geodesic plane . consider one edge @xmath91 . applying an isometry of @xmath1 if necessary , we may assume without loss of generality that @xmath92 then , with @xmath93 , @xmath94 thus @xmath95 the condition for the two normal geodesics to intersect is that there exist reals @xmath96 and @xmath97 so that @xmath98 in other words ( we now abbreviate @xmath99 to @xmath100 , and @xmath88 to @xmath37 ) , @xmath101 @xmath102 there exists a @xmath97 so that this last sum on the right - hand side is a diagonal matrix if and only if latexmath:[\[\left| \frac{\lambda \alpha - |dg|^2}{\lambda \alpha + @xmath1 if and only if @xmath104 this is equivalent to @xmath105 and then @xmath97 satisfies @xmath106 now , to get the diagonal terms in the above matrix equation to match , we want @xmath96 such that @xmath107\ ; , \ ] ] @xmath108\ ; .\ ] ] such a @xmath96 does exist , and in fact a computation shows that this @xmath96 is equal to @xmath97 . now the sign of @xmath109 determines which side of the quadrilateral the intersection lies on , and @xmath110 if and only if @xmath111 . note that because @xmath112 , the distance from the intersection point of the normal lines to either of @xmath87 and @xmath89 is the same . by a further isometry of @xmath1 that preserves @xmath87 and @xmath86 , we may change @xmath20 to @xmath113 for some constant @xmath114 . thus without loss of generality we may assume @xmath115 . it is then clear from the above equations that the two geodesics emanating in the normal directions from @xmath87 and @xmath89 both lie in the geodesic plane @xmath116 of @xmath1 , with @xmath1 represented as in section [ section3pt1 ] . this proves the last claim of the lemma . [ single - caustic - lemma - but - in - r31 ] by an argument similar to that of the proof of lemma [ single - caustic - lemma ] , but simpler , we also have the following statements : let @xmath7 be a discrete flat surface in @xmath1 with lift @xmath85 , as in , constructed using the discrete holomorphic function @xmath20 . let @xmath88 be a cross ratio factorizing function for @xmath20 . let @xmath86 denote the normal as in at @xmath87 . then the lines in @xmath117 ( not @xmath1 ) emanating from two adjacent vertices @xmath87 and @xmath89 in the directions of @xmath86 and @xmath118 ( respectively ) will either be parallel or will intersect at a unique point that is equidistant from @xmath87 and @xmath89 . the distance from either @xmath87 or @xmath89 to that intersection point is @xmath119 this is true on each edge @xmath91 , regardless of the sign of @xmath120 . furthermore , these two lines stemming from @xmath87 and @xmath89 will not be parallel if @xmath121 now , for all that follows , we introduce the following assumption . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ * assumption : * for @xmath20 the discrete holomorphic map used to construct a discrete flat surface , assume that all quadrilaterals in the image of @xmath20 in the complex plane are properly embedded . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ this implies the following properties : 1 . @xmath99 is never zero for any edge @xmath122 between adjacent vertices @xmath123 and @xmath124 in the domain @xmath34 . note that this was already assumed in section [ sect:2pt3 ] . 2 . @xmath99 and @xmath125 are never parallel for any square of edge - length @xmath2 with vertices @xmath126 in @xmath34 . ( if @xmath99 and @xmath125 are parallel , then @xmath127 all lie in one line , which implies that the interior of the quadrilateral in the complex plane is a half - plane , which is not properly embedded . ) because the cross ratio for @xmath20 is always negative , the term @xmath120 is negative on exactly all of the horizontal edges in the domain @xmath128 of @xmath7 , or exactly all of the vertical edges . applying a @xmath129 degree rotation to the domain if necessary , we can , and now do , assume without loss of generality that @xmath120 is negative if and only if the edge @xmath91 is vertical in @xmath34 . thus every vertical edge @xmath91 provides a unique intersection point of the normal geodesics , which we denote by @xmath130 . the domain @xmath131 of this new mesh is now the collection of all vertical edges of @xmath34 . if we let the vertical edges be represented by their midpoints , then we can consider @xmath78 to also be defined on a square grid , but now a shifted one , i.e. @xmath132 . we call the discrete surface @xmath133 the _ caustic _ , or _ focal surface _ , of @xmath7 . this new discrete surface @xmath134 is generally not a discrete flat surface , as the vertices in the quadrilaterals of the caustic will generally not be concircular . however , it has a number of interesting properties closely related to discrete flat surfaces , which we will describe . the first property is stated in this lemma : [ lem6pt6 ] the quadrilaterals of @xmath130 lie in geodesic planes of @xmath1 . this is clear from the geometry of the construction , and from the fact that adjacent normal geodesics always lie in a common geodesic plane of @xmath1 , by lemma [ single - caustic - lemma ] . now that we have seen the proof of lemma [ single - caustic - lemma ] , we are able to give one geometric justification for why we can say the discrete surfaces in section [ section4pt3 ] are `` flat '' . we will give an argument here showing the `` discrete extrinsic curvature '' is identically @xmath2 . for a smooth flat surface given by a frame @xmath85 solving , we find that the two functions @xmath135 give the principle curvatures of the surface , and also give the inverses of the distances to the focal points in @xmath117 of the curves in the surface along which either @xmath136 or @xmath137 is constant . here we are considering the focal points found in @xmath117 ( not @xmath1 ) , but we are finding those focal points with respect to the normal directions to the surface in @xmath1 given by the normal vectors , which are actually tangent vectors to @xmath1 itself . since the extrinsic curvature @xmath138 is exactly @xmath2 , if @xmath139 , we have @xmath140 and @xmath141 , and we have that @xmath142 this last right - hand equality , of course , is clear from the fact that @xmath143 . however , the right - hand equality encodes that the extrinsic curvature is exactly @xmath2 in a way that can be applied to the discrete case , as follows : the corresponding equation in the discrete case is given by the corresponding summation about the four edges ( assume @xmath144 and @xmath145 the other case can be handled similarly ) latexmath:[\[\label{artanhguydiscrete } \text{arctanh}\left(\frac{-\lambda \alpha_{pq}+ + |g_q - g_p|^2}\right ) - \text{arctanh}\left(\frac{\lambda \alpha_{qr}+ + |g_s - g_r|^2}\right ) - \text{arctanh}\left(\frac{\lambda \alpha_{sp}+ quadrilateral with vertices associated to @xmath126 ( given in counterclockwise order about the quadrilateral in @xmath34 ) in the discrete surface . the analogous geometric meaning of @xmath147 is preserved in the discrete case , as it is the inverse of the ( oriented ) distance from either @xmath87 or @xmath89 to the intersection point of the geodesics in @xmath117 stemming off of @xmath87 and @xmath89 in the directions of @xmath86 and @xmath118 , respectively ( see remark [ single - caustic - lemma - but - in - r31 ] ) . furthermore , equation follows immediately from the definition of the cross ratio factorizing function @xmath37 . in this sense , we can say that the `` discrete extrinsic curvature '' is identically @xmath2 . we now consider a second approach to discrete extrinsic curvature . for a smooth surface of constant extrinsic curvature @xmath2 , the infinitesimal ratio of the area of the gauss map to the area of the surface is exactly @xmath2 . so another way to give a notion that the `` discrete extrinsic curvature '' be identically @xmath2 for a discrete flat surface is to show the analogous property in the discrete case . that is the purpose of the following lemma . let @xmath7 be a discrete flat surface in @xmath1 with normal map @xmath11 . consider the vertices @xmath148 of one quadrilateral ( in @xmath1 ) of the surface associated with the quadrilateral with vertices @xmath123 , @xmath124 , @xmath18 , @xmath149 ( given in counterclockwise order ) in @xmath34 . these four vertices @xmath148 also determine another quadrilateral @xmath150 , now in @xmath117 , again with vertices @xmath148 , but now with geodesic edges @xmath151 , @xmath152 , @xmath153 , @xmath154 in @xmath117 , and which is planar in @xmath117 . likewise , the normals @xmath155 determine a planar quadrilateral @xmath156 in @xmath117 with vertices @xmath155 and with geodesic edges @xmath157 , @xmath158 , @xmath159 , @xmath160 in @xmath117 . these two quadrilaterals @xmath150 and @xmath156 lie in parallel spacelike planes of @xmath117 and have the same area . because @xmath148 lie in a circle @xmath161 in @xmath1 , there exists a planar quadrilateral @xmath150 in @xmath117 with edges that are geodesics in @xmath117 , and with vertices @xmath148 . since @xmath86 and @xmath118 have reflective symmetry with respect to the edge of @xmath150 from @xmath87 and @xmath89 ( see remark [ single - caustic - lemma - but - in - r31 ] ) , and since similar symmetry holds on the other three edges of @xmath150 , we know that @xmath162 are the vertices of a planar quadrilateral @xmath163 with geodesic edges in @xmath117 . it follows that @xmath155 are then the vertices of a planar quadrilateral @xmath156 with geodesic edges in @xmath117 . in fact , @xmath150 , @xmath163 and @xmath156 all lie in parallel spacelike planes . the goal is to show that @xmath150 and @xmath156 have the same area . since @xmath150 and @xmath156 are parallel , it is allowable to replace the metric of @xmath117 with the standard positive - definite euclidean metric for @xmath164 and simply prove that @xmath150 and @xmath156 have the same area with respect to that metric . the advantage of this is that it allows us to use known computational methods involving mixed areas . see @xcite , for example , for an explanation of mixed areas . noting that @xmath165 lies in the @xmath0-dimensional light cone of @xmath117 , i.e. @xmath166 , we define the lightlike vectors @xmath167 ( in fact , the normal geodesic at each vertex of @xmath7 in @xmath1 is asymptotic to the lines in the light cone determined by @xmath168 and @xmath169 . ) the two concircular sets @xmath170 and @xmath171 have the same real cross ratio , so the two quadrilaterals in @xmath117 that they determine are either congruent or dual to each other . ( in fact , it is generally the second of these two cases that occurs . ) it follows that the mixed area of @xmath168 and @xmath169 is zero @xcite . we then have , with `` @xmath172 '' denoting area and `` @xmath173 '' denoting mixed area , @xmath174 @xmath175 hence @xmath176 . the proof of lemma [ single - caustic - lemma ] shows that for a vertical edge @xmath122 ( of length @xmath2 ) of @xmath34 , the geodesic edge @xmath151 , the geodesic through @xmath87 in the direction of @xmath86 and the geodesic through @xmath89 in the direction of @xmath118 form the boundary of a planar equilateral triangle in @xmath1 . in particular , it follows that @xmath87 , @xmath89 , @xmath177 and @xmath178 are concircular , for any value of @xmath179 . one can also show that @xmath87 , @xmath89 , @xmath177 and @xmath178 are concircular even when @xmath122 is a horizontal edge in @xmath34 . this shows that the quadrilaterals formed by the two points of an edge and the two points of the corresponding edge of a parallel flat surface are always concircular . this in turn implies that a quadrilateral of the surface and the corresponding quadrilateral on a parallel surface have a total of eight vertices all lying on a common sphere - forming a cubical object with concircular sides . this gives ( see @xcite ) a discrete version of a triply orthogonal system , that is , a map from @xmath180 or a subdomain of @xmath180 to @xmath3 where all quadrilaterals are concircular . in lemma [ lem6pt6 ] we gave one property of caustics that is closely related to discrete flat surfaces . here we give a second such type of property , as seen in theorem [ anotherthm - forcaustic ] below . the equation for the lift @xmath85 of @xmath7 is @xmath181 however , since the formula for the surface has a mitigating scalar factor @xmath182 , we can change the equation above so that the potential matrix has determinant one , without changing the resulting surface , so let us instead use : @xmath183 @xmath184 we also now assume that @xmath48 is sufficiently close to zero so that @xmath185 for all edges @xmath91 . we now define , for each vertical edge @xmath91 , @xmath186{\lambda \alpha_{pq } } } & 0 \\ 0 & \frac{\sqrt[4]{\lambda \alpha_{pq}}}{\sqrt{dg_{pq } } } \end{pmatrix } \cdot p \ ; , \ ] ] where @xmath187 and @xmath188 are any choice of nonnegative reals such that @xmath189 ( recall the definition of @xmath190 in ) . this is a natural discretization of the @xmath83 for the case of smooth surfaces ( see ) , where we now must take a weighted average of @xmath191 and @xmath192 , and we allow any choice of weighting @xmath193 . for the smooth case in equation , the fourth root of the upper right term of @xmath194 divided by the lower left term gives the @xmath195 appearing in equation . for the discrete case in equation , the fourth root of the upper right term of @xmath196 divided by the lower left term gives the @xmath197{\lambda \alpha_{pq}}$ ] appearing in equation here . this explains why we insert the @xmath198{\lambda \alpha_{pq}}$ ] factors here . note that @xmath198{\lambda \alpha_{pq}}$ ] is not real , because @xmath90 . [ anotherthm - forcaustic ] the formula @xmath199 for the discrete caustic holds for all vertical edges @xmath91 , and this formula does not depend on the choice of @xmath187 and @xmath200 . a computation gives @xmath201@xmath202 the scalar factor @xmath203 is the only part of @xmath204 that depends on @xmath187 and @xmath188 , but this scalar factor is irrelevant in the formula , so we see independence from the choice of @xmath187 and @xmath188 . the result now follows from the proof of lemma [ single - caustic - lemma ] . however , the choice of normal direction at the vertices of @xmath134 does depend on the choice of @xmath187 and @xmath188 , as the following equation shows : @xmath205 where @xmath206 is @xmath207 equation implies that , for both horizontal and vertical edges , the adjacent normal geodesics of the caustic typically do not intersect , for any generic choice of @xmath187 and @xmath188 . the purpose of the next results is to show that the discrete caustics in section [ section5 ] have properties similar to the caustics in the smooth case . in what follows , we will regard both the discrete flat surface @xmath7 and its caustic @xmath134 as discrete surfaces that have edges and faces in @xmath1 ( not just vertices ) . since there is a unique geodesic line segment between any two points in @xmath1 , the edge between any two adjacent vertices of @xmath7 is uniquely determined . the same is true of @xmath134 . then , since the image of the four vertices of any given fundamental quadrilateral in @xmath34 ( resp . in @xmath131 ) under @xmath7 ( resp . @xmath134 ) has image lying in a single geodesic plane ( see theorem [ thm : concirc - quads ] and lemma [ lem6pt6 ] ) , the image of the fundamental quadrilateral in @xmath34 ( resp . @xmath131 ) can be regarded as a quadrilateral in a geodesic plane of @xmath1 bounded by four edges of @xmath7 ( resp . @xmath134 ) . this is the setting for the results given in this section . in the case of a smooth flat surface ( front ) @xmath7 in @xmath1 and its smooth flat caustic @xmath78 , every point in @xmath78 is a point in the singular set of one of the parallel flat surfaces of @xmath7 . in lemma [ finalprop1 ] , we are stating that every point in the discrete caustic @xmath78 of a discrete flat surface @xmath7 is a point in the edge set of some parallel flat surface of @xmath7 . this , in conjuction with theorem [ finalprop2 ] , suggests a natural candidate for the definition of the singular set of a discrete flat surface . the proof of lemma [ finalprop1 ] is immediate from the definitions of parallel surfaces and caustics . [ finalprop1 ] let @xmath7 be a discrete flat surface defined on a domain @xmath208 determined by a discrete holomorphic function @xmath209 with properly embedded quadrilaterals , and let @xmath78 be its caustic . let @xmath210 be any point in @xmath78 , so @xmath190 lies in the quadrilateral @xmath211 of @xmath78 that is determined by two adjacent vertices @xmath87 , @xmath89 of @xmath7 and the normal geodesics ( which contain two opposite edges of @xmath211 ) in the directions @xmath86 , @xmath118 at @xmath87 , @xmath89 , respectively . ( thus @xmath122 will be a horizontal edge of @xmath34 . ) @xmath190 can lie in either the interior of @xmath211 , or an edge of @xmath211 , or could be a vertex of @xmath211 . then @xmath190 lies in the edge @xmath212 of some parallel surface @xmath213 of @xmath7 . the next proposition will be used in the proof of theorem [ finalprop2 ] . [ finalprop3 ] let @xmath7 be a discrete flat surface with normal @xmath11 produced from a discrete holomorphic function @xmath20 with properly embedded quadrilaterals . then for all vertices @xmath123 , @xmath86 is not tangent to any quadrilateral of @xmath7 having vertex @xmath87 . take a quadrilateral with vertices @xmath87 , @xmath89 , @xmath214 and @xmath215 of @xmath7 so that @xmath122 is a horizontal edge of @xmath34 . we may make all of the assumptions in the proof of lemma [ single - caustic - lemma ] , including the assumption that @xmath99 is real . of course , the normal @xmath216 lies in the hyperplane @xmath217 of @xmath117 ( here we regard points of @xmath117 as hermitean matrices as in section [ section3pt1 ] ) , and so does the point @xmath89 , since @xmath115 . furthermore , @xmath122 is a horizontal edge of @xmath34 , so @xmath218 , which implies that @xmath89 does not lie in the geodesic in @xmath1 containing @xmath87 and tangent to @xmath86 . however , @xmath20 has properly embedded quadrilaterals , so both @xmath125 and @xmath219 will not lie in @xmath38 , and thus @xmath215 will not lie in the hyperplane @xmath217 . it follows that @xmath86 will not be parallel to the geodesic plane in @xmath1 containing the two geodesics from @xmath87 to @xmath89 and from @xmath87 to @xmath215 . for a smooth flat surface ( front ) @xmath7 and its caustic @xmath78 , a parallel flat surface to @xmath7 , including @xmath7 itself , will meet @xmath78 along its singular set , and that singular set is generally a graph in the combinatorial sense ( whose edges consist of immersable curves ) . furthermore , all vertices of that combinatorial graph have valence at least two . for example , cuspidal edges form the edges of this graph , and swallowtails give vertices of this graph with valence two . in particular , no cuspidal edge can simply stop at some point without continuing on to at least one other cuspidal edge ( as this would give a vertex of valence one ) . the following theorem shows that an analogous property holds in the discrete case . note that when we are speaking of the vertices of this combinatorial graph in the theorem below , these vertices are _ not _ the same as the vertices of the discrete flat surface , nor its discrete caustic , in general . [ finalprop2 ] let @xmath7 be a discrete flat surface defined on a domain @xmath208 determined by a discrete holomorphic function @xmath209 with properly embedded quadrilaterals , and let @xmath78 be its caustic . let @xmath213 be a parallel surface , and let @xmath220 be the set of all @xmath210 as in lemma [ finalprop1 ] , for that value of @xmath179 , and for any adjacent endpoints @xmath123 and @xmath124 of a horizontal edge of @xmath34 . assume that no two adjacent vertices of @xmath213 are ever equal , and that the faces of @xmath78 are embedded . then @xmath220 is a graph ( in the combinatorial sense ) with edges composed of geodesic segments lying in the image in @xmath1 of the horizontal edges of @xmath34 under @xmath213 , and with all vertices of @xmath220 having valence at least two . the snowman shown on the right - hand side of figure [ uglyfigure8 ] ( see example [ example4pt12 ] ) provides an example to which theorem [ finalprop2 ] applies . the airy example in section [ section4 ] also satisfies the conclusion of this theorem ( see figure [ uglyfigure6 ] ) , although it does not actually satisfy the condition in the theorem that the faces of the caustic be embedded . at least one of the assumptions in theorem [ finalprop2 ] that the quadrilaterals of @xmath78 are embedded and that no two adjacent vertices of @xmath213 coincide is necessary . without them , the discrete hourglass , as seen in figure [ uglyfigure8 ] ( see example [ example4pt12 ] ) , would provide a counterexample to the result . however , it is still an open question whether both of those conditions are really needed . there are reasons why it is not obvious that we can remove one of those two conditions . we explore those reasons in appendix [ appendix9 ] . . ] we must show that all vertices of @xmath220 have valence at least two . there are essentially only two situations for which we need to show this , one obvious and one not obvious . the obvious case is when an entire edge @xmath221 lies in @xmath220 , and then the result is clear . the non - obvious case that we now describe , where only a part of @xmath221 lies in @xmath220 , is essentially only one situtation , since any other non - obvious situation can be reformulated in terms of the notation given below . without loss of generality , replacing @xmath213 by @xmath7 if necessary , we may assume that @xmath222 . let @xmath34 be a domain in @xmath35 containing @xmath223 , @xmath224 , @xmath225 , @xmath226 , @xmath227 , @xmath228 , @xmath229 , @xmath230 , and let @xmath20 be a discrete holomorphic function defined on @xmath34 . let @xmath231 be the resulting discrete flat surface . we have a normal direction defined at each vertex of @xmath7 , which determines normal geodesics @xmath232 , @xmath233 at the vertices @xmath234 , @xmath235 , respectively . note that @xmath232 and @xmath233 never intersect ( and thus @xmath234 and @xmath235 are never equal ) , although they do lie in the same geodesic plane , and that @xmath232 and @xmath236 ( resp . @xmath233 and @xmath237 ) intersect at a single point that we call @xmath238 ( resp . @xmath239 ) , by lemma [ single - caustic - lemma ] . the point @xmath238 ( resp . @xmath239 ) is equidistant from the two vertices @xmath234 and @xmath240 ( resp . @xmath235 and @xmath241 ) , as in lemma [ single - caustic - lemma ] . now the caustic @xmath134 has two quadrilaterals , described here by listing their vertices in order about each quadrilateral : @xmath242 let @xmath137 be a point in the geodesic edge @xmath243 so that @xmath137 also lies in the edge @xmath244 . since no two adjacent vertices of @xmath7 are ever equal , it follows that @xmath137 lies strictly in the interior of @xmath243 . ( see the left - hand side of figure [ uglyfigure1 ] . ) thus , since the face @xmath245 of the caustic is embedded , there exists a half - open interval @xmath246 $ ] or @xmath247 contained entirely in the interior of @xmath248 so that @xmath249 lies in @xmath245 . thus @xmath137 can become a vertex of the graph @xmath250 . we wish to show that the valence at @xmath137 is at least two . this would mean that the visual representation is more like in the right - hand side of figure [ uglyfigure1 ] , where the quadrilateral of @xmath7 with vertices @xmath251 is nonembedded . it suffices to show that there exists a half - open interval @xmath252 $ ] or @xmath253 contained entirely in the interior of the geodesic edge @xmath254 so that : 1 . @xmath255 lies in the face @xmath256 of the caustic , and 2 . @xmath257 . because @xmath137 lies in both the geodesic plane determined by @xmath251 and the geodesic plane determined by @xmath258 , and because proposition [ finalprop3 ] implies these two geodesic planes are not equal , @xmath137 must also lie in the line determined by @xmath259 . then , because @xmath137 lies in the embedded face @xmath245 , and because both @xmath244 and @xmath260 lie in the geodesic planar region between @xmath261 and @xmath262 , @xmath137 must lie in the edge @xmath259 itself . ( note that we have now proven that the quadrilateral with vertices @xmath263 , @xmath264 , @xmath265 and @xmath266 is not embedded , so in fact the visual representation must be more like in the right - hand side of figure [ uglyfigure1 ] . ) keeping in mind that @xmath137 also lies in the edge @xmath267 , then since both edges @xmath259 and @xmath267 lie in the plane determined by @xmath256 , and since @xmath256 is embedded , we conclude existence of such an interval @xmath255 with the required properties . the above lemma [ finalprop1 ] and theorem [ finalprop2 ] suggest that @xmath220 has many of the right properties to make it a natural candidate for the singular set of any discrete surface @xmath213 in the parallel family of @xmath7 . we have chosen to consider the set @xmath220 in the image @xmath268 of a discrete flat surface , rather than in the domain @xmath34 itself ( as is usually done for the singular set in the smooth case ) , because @xmath220 becomes a collection of connected curves in the image , while in the domain @xmath34 it would jump discontinuously between points in the lower and upper horizontal edges of quadrilaterals of @xmath34 ( as we have seen in the above proof ) . however , one could remedy this by inserting vertical lines between those lower and upper edge points in quadrilaterals of @xmath34 , and then consider the set in the domain . in figure [ discairycmc ] , we have drawn some of the discrete surfaces in the linear weingarten family associated with the discretization of the airy equation . for making these graphics , we used the discrete holomorphic power function . we explain here how to solve the difference equation for determining the discrete power function , as follows : @xmath269 with the initial conditions @xmath270 once we know @xmath271 and @xmath272 , the full solution is given by solving the first equation for the cross ratio . we fix @xmath273 and set @xmath274 then , it is seen from the cross ratio condition that @xmath275 if we set @xmath276 then @xmath277 satisfies @xmath278 we define the recurrence relations : @xmath279 so that @xmath280 the initial conditions are @xmath281 the relation is written in the form @xmath282 and the eigenvalues of the @xmath283 matrix just above are @xmath284 . the difference equation satisfied by @xmath285 is @xmath286 where @xmath287 and @xmath288 . once we have determined @xmath289 , then @xmath290 determine @xmath291 then , using equation , we can determine @xmath70 for all nonegative @xmath292 and @xmath273 . we have the following fact , which was also stated in @xcite : [ lem - app8 ] suppose that @xmath70 is the discrete holomorphic function solving and for one choice of @xmath293 , and suppose @xmath294 is the same , but with @xmath293 replaced by @xmath295 . then @xmath70 and @xmath296 satisfy equation with @xmath297 ( resp . @xmath298 ) on horizontal ( resp . vertical ) edges . that holds on the edges @xmath299 and @xmath300 can be easily confirmed from equation . then an induction argument proves the result on all other edges as well . if @xmath301 is a smooth nonconstant holomorphic function with respect to the usual complex coordinate @xmath8 for @xmath9 , then @xmath302 is a harmonic function , and the maximum principle for harmonic functions tells us that @xmath302 can not have a local finite minimum at an interior point of the domain . thus , if @xmath303 has a local minimum at an interior point @xmath304 , it must be that @xmath305 . there are various ways to discretize the notions of holomorphicity and harmonicity . see @xcite , @xcite , @xcite , @xcite , @xcite , @xcite , @xcite , @xcite , to name just a few of the possible references however , the history of this topic goes back much further than just the references mentioned here . these ways do provide for discrete versions of the maximum principle . the definition we have chosen here for discrete holomorphic functions based on cross ratios , however , does not satisfy a particular simple - minded discrete version of the maximum principle , as we can see by the first explicit example below . a more sophisticated consideration is needed to produce a proper discrete version of the maximum principle , but we do not discuss that here , as the simplest questions are what are relevant to theorem [ finalprop2 ] . [ exa9pt1 ] set @xmath306 . then , with @xmath307 , set @xmath308@xmath309@xmath310 and extend @xmath20 to all of @xmath34 so that all cross ratios of @xmath20 on @xmath34 are @xmath51 . that is , take @xmath20 so that all @xmath311 and all @xmath312 . then we have the following two properties : 1 . amongst all vertices of @xmath34 , @xmath303 has a strict minimum of @xmath313 at the interior vertex @xmath314 . 2 . amongst all edges of @xmath34 between adjacent vertices @xmath123 and @xmath124 ( both horizontal and vertical ) , @xmath315 has a strict minimum of @xmath2 at the interior edge from @xmath314 to @xmath316 . which meet @xmath213 for some @xmath179 at a single point , with @xmath20 taken as in examples [ exa9pt1 ] and [ exa9pt2 ] . the caustics are shown in red , and the @xmath213 are shown in blue.,title="fig : " ] which meet @xmath213 for some @xmath179 at a single point , with @xmath20 taken as in examples [ exa9pt1 ] and [ exa9pt2 ] . the caustics are shown in red , and the @xmath213 are shown in blue.,title="fig : " ] taking @xmath20 as in the previous example , and taking @xmath317 , we can produce the flat surfaces @xmath213 and the caustic @xmath78 . it turns out that the surface @xmath213 and the caustic @xmath78 intersect at just one point if @xmath318 is chosen correctly . see the left - hand side of figure [ fig - appendix9 ] . in this case , two adjacent vertices ( coming from @xmath314 and @xmath316 ) of @xmath213 do coincide . however , the quadrilaterals of the caustic @xmath78 are not embedded in this case , so this example does not suffice to show that both assumptions at the end of the first paragraph of theorem [ finalprop2 ] are truly needed . in light of the equations in the proof of lemma [ single - caustic - lemma ] , a natural next step toward understanding the role of coincidence of vertices of @xmath213 in theorem [ finalprop2 ] ( and thus toward understanding which assumptions the theorem really needs ) is to consider a discrete holomorphic function with the properties as in the next explicit example . [ exa9pt2 ] taking the same domain @xmath34 as in the previous example , we now set @xmath319@xmath320@xmath321 and , like in the previous example , we extend @xmath20 to all of @xmath34 so that all @xmath311 and all @xmath312 . then we have the following two properties : 1 . amongst all vertical edges of @xmath34 between adjacent vertices @xmath123 and @xmath124 , @xmath315 has a strict minimum of @xmath2 at the interior edge from @xmath314 to @xmath316 . 2 . amongst any three vertical edges @xmath322 and @xmath323 and @xmath324 at the same height in @xmath34 , @xmath315 is ( strictly ) minimized at the central edge @xmath300 . using the function @xmath20 in example [ exa9pt2 ] , one might hope that the resulting surface @xmath213 would show the necessity of the assumption in theorem [ finalprop2 ] that the adjacent vertices of @xmath213 do not coincide . however , it turns out that the quadrilaterals of @xmath78 are not embedded in this case as well . see the right - hand side of figure [ fig - appendix9 ] , where again @xmath325 and @xmath179 ( @xmath326 ) is taken so that the two vertices of @xmath213 coming from @xmath314 and @xmath316 coincide . a. i. bobenko , c. mercat and y. b. suris , _ linear and nonlinear theories of discrete analytic functions . integrable structure and isomonodromic green s function _ , j. reine und angew . math . 583 ( 2005 ) , 117 - 161 .

we define discrete flat surfaces in hyperbolic @xmath0-space @xmath1 from the perspective of discrete integrable systems and prove properties that justify the definition . we show how these surfaces correspond to previously defined discrete constant mean curvature @xmath2 surfaces in @xmath1 , and we also describe discrete focal surfaces ( discrete caustics ) that can be used to define singularities on discrete flat surfaces . along the way , we also examine discrete linear weingarten surfaces of bryant type in @xmath1 , and consider an example of a discrete flat surface related to the airy equation that exhibits swallowtail singularities and a stokes phenomenon .

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Proceed to summarize the following text: is a novel transient radio source @xcite , whose notable properties have included `` bursts '' with approximately 1 jy peak flux density lasting approximately 10 min . each and occurring at apparently regular 77 min.intervals . we first identified from archival 330 mhz ( 90 cm ) observations taken with the very large array ( vla ) on 2002 september 30 . we then detected a single burst from the source with the giant metrewave radio telescope ( gmrt ) in archival observations from 2003 september 28 at the same frequency . is located about 1.25 south of the galactic center just outside ( in angular distance ) the shell - type supernova remnant @xcite . the environment of the source is discussed further in @xcite . the source is notable because it is one of a small number of _ radio - selected _ transients . moreover , with only a few exceptions @xcite such as electron cyclotron masers from flare stars and the planets , plasma emission from solar radio flares , pulsar radio emission , and molecular - line masers , most radio transients are incoherent synchrotron emitters . for an incoherent synchrotron emitter , the energy density within the source is limited to an effective brightness temperature of roughly @xmath3 k by the inverse compton catastrophe @xcite . the properties of suggest strongly that its brightness temperature exceeds @xmath3 k by a large factor and that it is a member of a new class of coherent emitters @xcite . the two previous detections of were based on vla and gmrt 330 mhz observations from two different epochs , but from which similar source properties were observed . this _ letter _ reports on a third 330 mhz archival observation of , made with the gmrt in 2004 , in which a single , much fainter and shorter burst is detected in contrast to the burst properties observed in 2002 and 2003 . in addition , the burst is found to have a very steep spectrum , as expected for a coherent emitter , providing another important clue to understanding the nature of this enigmatic source . possible models for include nearby objects such as a flaring brown dwarf , flare star , or extra - solar planet , although , as discussed in @xcite , the properties of the source do not easily fit these classes . alternative models proposed for thus far are a nulling pulsar ( @xcite ) , a double pulsar ( @xcite ) , a transient white dwarf pulsar ( @xcite ) , and a precessing radio pulsar ( @xcite ) . as noted in the latter paper , the double pulsar model predicts a @xmath03-yr bursting activity period which is not supported by the redetections in 2003 and 2004 . furthermore , in @xcite we presented tentative evidence suggesting that the 2003 burst was an isolated one , and not one of several bursts emitted at 77 min . intervals , as observed in 2002 . this evidence could be problematic for some or all of these models . for example , the proposed double pulsar would link the radio emission to a 77 min . orbital period . in this model , coherent radio emission is triggered by the shock formed through the interaction of the wind of the more enigmatic pulsar with the magnetosphere of the companion pulsar . on the other hand , in analogy to the psr b1259 - 63 system which consists of a pulsar and be star companion , it is also possible that the magnetosphere of the companion is not constant , and therefore that the radio bursts are not always triggered every orbit . indeed , the much fainter detection from reported in this _ letter _ could be evidence of variable conditions in the environment around a companion star . similarly , the precessing radio pulsar and transient white dwarf pulsar models would also require one or more types of nulling effects to explain the occurrence of isolated bursts in the short term , as well as the low duty cycle in the long term . a few radio pulsars are known to have a very large nulling fraction . psr b1931 + 24 remains in an off state for @xmath090% of the time , and it emits bursts quasi - periodically at @xmath040 per day @xcite . such a high nulling fraction may be consistent with the measured duty cycle estimated for . the new , serendipitous detection reported in this paper is derived from 330 mhz gmrt galactic center observations obtained by two of us ( s. roy and s. bhatnagar ) as part of an unrelated project and not included in @xcite . one of the observations , from 2004 march 20 - 21 , is pointed 0.5 from and consists of eleven 10 min . scans spread over six hours . the observations were carried out using the default observing mode with a bandwidth of 16 mhz in each of the two available sidebands . the sources b1822 - 096 and b1714 - 25 were used as secondary calibrators . the gmrt does not measure the system temperature ( @xmath4 ) , and the increase in @xmath4 from the calibrator field to the target source affects the source visibility amplitudes in the default observing mode ( i.e. , the automatic level control [ alc ] in the system is turned on ) . we employed the following method to correct for the @xmath4 variation . as the system gain does not change with the alc off , we observed 3c48 and b1822 - 096 once with the alc off and determined the flux density of b1822 - 096 to be 10.8 jy using the known flux density of 3c48 from @xcite . also with the alc off , we determined the ratio of the total power on the target source to that of b1822 - 096 to be 1.8 . since this ratio was quite similar ( within 10% ) for almost all the antennas , rather than correcting the antenna - based gains , we multiplied the final map of the source intensity distribution by this value . we estimate the overall calibration uncertainty to be within @xmath515% . the initial images were improved by phase and later amplitude and phase self - calibration . to produce the final image , separately self - calibrated data from both the sidebands were combined to improve the uv - coverage . we detect a @xmath010@xmath6 burst from in the middle of the first scan beginning at approximately 21:31:00 ( iat ) on march 20 and lasting for approximately 2 min . the source is unresolved and has a flux density of 57.9@xmath56.6 mjy . as a check on the calibration , we find the flux densities of several bright sources in the field - of - view to be consistent with those obtained in the 330 mhz vla survey of the galactic center by @xcite . over the past several years , we have conducted a 330 mhz search for radio transients in the galactic center @xcite using archival observations made between 1989 and the present , and monitoring observations beginning in 2002 . @xcite summarize the vla and gmrt observations and data reduction through 2005 . most of these observations are pointed towards sgr a * where the large primary beam at 330 mhz ( 2.2and 1.4 ) and high stellar density subtended optimize the likelihood of detecting a transient , including . our 2006 and 2007 monitoring program includes observations at 330 mhz ( vla ) and simultaneous observations at 235 and 610 mhz ( gmrt ) ; these will be described in a subsequent paper . the flux density of the detected 2004 burst is much fainter than the @xmath01 jy peak flux density measured for the five bursts detected in 2002 @xcite and for the one burst detected in 2003 @xcite . due to the relatively short and infrequent scans in the 2004 observations , we detect only a single burst , as in the 2003 epoch . we can not rule out the possibility that additional bursts were emitted at the same 77 min . interval observed in 2002 . we note , however , that nondetections in additional scans made at multiples of 77 min . after the detected 2003 burst , but in which is located well beyond the half power point of the primary beam , suggest that the 2003 detected burst is actually an isolated one @xcite . figure [ fig : lightcurve04 ] shows the light curves for all seven bursts detected to date . while only the @xmath02 min . decay portion of the 2003 burst was detected due to its coincidence with the beginning of a scan , the 2004 burst is sampled completely within one scan . it is detected for only @xmath02 min . compared with the @xmath010 min . duration observed for the bursts in 2002 . is unresolved in all three epochs . the resolution for the 2004 epoch is approximately 20 @xmath1 10 , comparable to the 2003 epoch , and approximately a factor of 2 better than in the 2002 epoch . a gaussian fit to the 2004 detection yields a position of ( j2000 ) right ascension @xmath7 @xmath8 509 ( @xmath9 ) , declination @xmath10 09 56.4 ( @xmath11 ) , which is consistent with the 2003 and 2002 positions and is approximately 2.5@xmath1 and 5@xmath1 more accurate , respectively . the source position and uncertainty cited above include a correction for ionospheric refraction which is prevalent in low frequency observations and discussed in @xcite and @xcite . separate images were made for the upper ( 333 mhz ) and lower ( 317 mhz ) sidebands of the observations and yield flux densities of 42.1@xmath57.2 mjy and 72.5@xmath59.5 mjy , respectively . no significant differences are found in the shapes of the separate light curves generated for each sideband . figure [ fig : spectrum04 ] shows the spectrum of obtained by imaging pairs of adjacent frequency channels across the two sidebands . a power - law fit yields a very steep spectrum of @xmath12 for . an identical analysis of the data for the nearby strong source g358.638@xmath131.160 yields a spectral index of @xmath14 , consistent with the determination of @xcite who found a spectral index of @xmath15 between 330 and 1400 mhz . a monte - carlo simulation was conducted to assess the confidence level of the steep spectrum obtained for the 2004 detection of . first , a spectrum was generated based on the fitted spectral index of @xmath16 and normalized to the observed flux density at a particular channel . one thousand spectra were simulated by randomly adding gaussian noise to the flux densities of the normalized spectrum , based on the observed noise level obtained for the individual channel images . a flat spectrum fit to the data is ruled out with a confidence level of 99.7% . unfortunately , however , due to limitations in the analysis of the 2002 and 2003 observations , we were not able to detect reliable evidence of a steep spectrum for those bursts . no emission is detected from when imaging the 2004 observation at times when the burst is not occurring . from these 2004 observations , we are able to improve the ( 5@xmath6 ) upper limit for quiescent 330 mhz emission between bursts to 6 mjy , as compared to earlier limits of 75 mjy and 25 mjy from the 2002 and 2003 observations , respectively . the upper limit on quiescent emission during periods of no burst activity is 15 mjy at 330 mhz @xcite . in addition , we find an upper limit of 50% circular polarization for the 2004 burst , while an upper limit of 15% was previously determined for both the 2002 and 2003 bursts . in @xcite we crudely estimated the activity of the bursting behavior of by comparing the time during which the source is observed to be active to the total amount of observing time . with the addition of @xmath040 hr of 330 mhz vla observations analyzed in 2006 , with no detection , the total observing time is @xmath0120 hr . the 2002 september 30 - october 1 bursts lasted for at least 6 hr , and since only a single burst was detected on 2003 september 28 and on 2004 march 20 , we assume that the source was active for @xmath01 hr in each of these epochs . thus , has exhibited bursting activity approximately 7% of the time it was observed . the 2002 and 2003 detections of occured in late september and the 2004 detection occurred in late march , suggesting a roughly 6-month interval between active periods . although we find no detections in our 1998 , 2002 , 2005 , and 2006 observations in march and september , it is still possible that bursting activity did occur in these months but was either too faint or lasted too short a time to be detected . similarly , while our database has many observations in 2002 , 2003 , and 2006 , we did not monitor for transients nearly as often in 2004 and 2005 , and useful archival data is very limited in those years . since we have now detected in three significantly different observed states ( @xmath01 jy versus @xmath050 mjy burst strengths , @xmath010 min . versus @xmath02 min . burst durations , and regularly repeating versus isolated bursts ) , future detections may yet exhibit additional properties ( e.g. , emission in other frequency bands and polarized emission ) that will lead to a definitive understanding of the nature of . given the high variability in flux density already detected , and the very steep spectrum of the 2004 burst , it is even conceivable that bursts much stronger than @xmath01 jy will be detected at 330 mhz and lower frequencies , although it is also possible that the bursts are now continually decreasing in strength at all frequencies . we note that all three detections suggest strongly that is a coherent emitter , as first indicated in @xcite , and now even more so by the very steep spectrum reported here . although the 2004 burst is much weaker than the previous bursts it also appears to have much shorter rise and decay times , @xmath17 , as seen in figure [ fig : lightcurve04 ] . if we constrain the emitting region to be less than c@xmath17 , with @xmath17 @xmath01-min , then the brightness temperature of is @xmath18 , where @xmath19 is the distance to the source . if the source is at the galactic center , @xmath08.5 kpc distant , its brightness temperature then still far exceeds the @xmath3k upper limit for an incoherent synchrotron emitter . in summary , the new detection of consists of a single @xmath050 mjy burst lasting only 2-min and exhibiting a very steep spectrum ( @xmath20 ) . the burst is significantly weaker and shorter than the @xmath01 jy and 10-min . bursts detected in 2002 and 2003 . like the 2003 detection , the single burst detected in 2004 appears to be an isolated one , although the sparse sampling of the observation does not rule out the possibility that additional bursts were emitted at the same 77 min . period observed in 2002 @xcite . given the @xmath0120 hrs of 330 mhz observations searched , we estimate that the source is in a detectable bursting state @xmath07% of the time . further , multi - wavelength observations are needed to better constrain the physical nature of . together with continued radio monitoring for transient or weaker pulsed emission ( e.g. pulsar - like ) , additional desired observations include searches for quiescent or variable infrared or x - ray counterparts . the national radio astronomy observatory is a facility of the national science foundation operated under cooperative agreement by associated universities , inc . we thank the staff of the gmrt that made these observations possible . gmrt is run by the national centre for radio astrophysics of the tata institute of fundamental research . is supported by funding from research corporation and sao _ chandra _ grants go6 - 7135f and go6 - 7038b . basic research in radio astronomy at the naval research laboratory is supported by 6.1 base funding .

is a transient bursting radio source located in the direction of the galactic center . it was discovered in a 330 mhz vla observation from 2002 september 30october 1 and subsequently rediscovered in a 330 mhz gmrt observation from 2003 september 28 by hyman et al . here we report a new radio detection of the source in 330 mhz gmrt data taken on 2004 march 20 . the observed properties of the single burst detected differ significantly from those measured previously , suggesting that was detected in a new physical state . the 2004 flux density was @xmath00.05 jy , @xmath010@xmath1 weaker than the single 2003 burst and @xmath030@xmath1 weaker than the five bursts detected in 2002 . we derive a very steep spectral index , @xmath2 , across the bandpass , a new result previously not detectable due to limitations in the analysis of the 2002 and 2003 observations . also , the burst was detected for only @xmath02 min . , in contrast to the 10 min . duration observed in the earlier bursts . due to sparse sampling , only the single burst was detected in 2004 , as in the 2003 epoch , and we can not rule out additional undetected bursts that may have occurred with the same @xmath077 min . periodicity observed in 2002 or with a different periodicity . considering our total time on source throughout both our archival and active monitoring campaigns , we estimate the source exhibits detectable bursting activity @xmath07% of the time .

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Proceed to summarize the following text: after the first successful space flight use of the x - ray charge coupled device ( ccd ) of the sis ( @xcite ) on board asca , the ccd has been playing a major role in imaging spectroscopy in the field of x - ray astronomy . however , the charge transfer inefficiency ( cti ) of x - ray ccds increases in orbit due to the radiation damage ; the cti is defined as the fraction of electrons that are not successfully moved from one ccd pixel to the next during the readout . since the amount of charge loss depends on the number of the transfers , the energy scale of x - ray ccds depends on the location of an x - ray event . furthermore , there is a fluctuation in the amount of the lost charge . therefore , without any correction , the energy resolution of x - ray ccds in orbit gradually degrades . in the case of the x - ray imaging spectrometer ( xis ) @xcite on board the suzaku satellite @xcite launched on july 10 , 2005 , the energy resolution in full width at half maximum ( fwhm ) at 5.9 kev was @xmath0140 ev in august 2005 , but had degraded to @xmath0230 ev in december 2006 . the increase of the cti is due to an increase in the number of charge traps at defects in the lattice structure of silicon made by the radiation . since the trap distribution is not uniform , it would be best if we could measure the cti of each pixel as chandra acis @xcite . in the case of the xis , however , it is impossible to measure the cti values of all the pixels , mainly because the onboard calibration sources do not cover the entire field of view of the xis . therefore , we use the cti of each column to correct the positional dependence of the energy scale . the xis is equipped with a charge injection structure @xcite which can inject an arbitrary amount of charge in arbitrary positions . using this capability , we can precisely measure the cti of each column @xcite . by applying the column - to - column cti correction , the positional dependence of the cti corrected energy scale is greatly reduced , and the over - all energy resolution is also improved @xcite . in @xcite , the results of the cti correction was mainly based on the ground - based charge injection experiments . in - orbit measurements were limited within one year after the launch . this paper reports more comprehensive and extended in - orbit experiments up to two years after the launch . the results are based on the data with the normal full window mode @xcite without a spaced - row charge injection @xcite , and have been implemented to the suzaku calibration database and applied to all the data obtained with the same mode . all the errors are at the 1@xmath1 confidence level throughout this paper unless mentioned . the xis is the set of four x - ray ccd camera systems . three sensors ( xis 0 , 2 , and 3 ) contain front - illuminated ( fi ) ccds and the other ( xis 1 ) contains back illuminated ( bi ) ccd . the xis 2 sensor became unusable on november 9 , 2006 . therefore there are no data for xis 2 after that day . the detailed structure of the ccd has been provided in @xcite . in this paper , we define a `` row '' and a `` column '' as a ccd line along the @xmath2 and @xmath3 axes , respectively ( see figure 3 in @xcite ) . in the imaging area , the _ actx _ value runs 0 to 1023 from the segment a to d , while the _ acty _ value runs from 0 to 1023 from the readout node to the charge injection structure . the charge injection structure lies adjacent to the top row ( _ acty _ = 1023 ) in the imaging area . we can inject charges from @xmath0 50 e@xmath4 to @xmath0 4000 e@xmath4 per pixel ; the equivalent x - ray energy ranges from @xmath00.2 kev to @xmath015 kev . a charge packet generated by an incident x - ray is transferred to the readout node , then is converted to a pulse - height value . we define @xmath5 to be the original pulse height generated by the x - ray . in the real case , the readout pulse height of the packet ( @xmath6 ) is smaller than @xmath5 , because some amount of charges is lost during the transfer . to measure the charge loss , we have to know both @xmath5 and @xmath7 . however , we can usually measure only @xmath6 , and hence it is difficult to obtain @xmath5 . @xcite and @xcite reported a technique to solve this problem by the charge injection method , and @xcite applied this technique to the xis . we briefly repeat by referring figure 3 in @xcite . first , we inject a `` test '' charge packet into the top ccd row ( _ acty _ then , after the gap of a few rows , five continuous packets are injected with the same amount of charge of the test packet . the former four packets are called `` sacrificial '' charge packets , while the last one is called a `` reference '' charge packet . the test packet loses its charge by the charge traps . on the other hand , the reference packet does not suffer from the charge loss , because the traps are already filled by the preceding sacrificial packets . thus we can measure the charge loss by comparing the pulse - height values of the reference charge ( @xmath8 ) and the test charge ( @xmath9 ) . the relation between sacrificial charge packets and reference charge packets is described in gendreau ( 1995 ) . we can obtain a checker flag pattern by these injected packets in the x - ray image ( right panel of figure 3 in @xcite ) because of the onboard event - detection algorithm @xcite . therefore in this paper , we call this technique a `` checker flag charge injection ( cfci ) . '' a charge packet in the xis loses its charge during ( a ) the fast transfer ( 24 @xmath10s pixel@xmath11 ) along the _ acty _ axis in the imaging area , ( b ) the fast transfer along the _ acty _ axis in the frame - store region , ( c ) the slow transfer ( 6.7 ms pixel@xmath11 ) along the _ acty _ axis in the frame - store region , ( d ) the fast transfer to the readout node along the _ actx _ axis . the cti depends on many parameters such as the transfer speed and the number density of the charge traps @xcite . the frame - store region is covered by the shield and is not exposed to the radiation directly . furthermore , the pixel size of the frame - store region ( 21 @xmath10m@xmath1213.5 @xmath10 m ) is different from that of the imaging area ( 24 @xmath10m@xmath1224 @xmath10 m ) . thus the number of traps per pixel may be different between the imaging area and the frame - store region . then we assumed that the four transfers have different cti values . we examined the transfer ( d ) by using the calibration source data taken in april 2007 , and found no significant decrease of the pulse height along the _ actx _ axis . we , therefore , ignore the charge loss in the transfer ( d ) . we define that @xmath13 is the transfer number in the imaging area ( @xmath13=@xmath3 + 1 ; here , @xmath3 is a coordinate value where an incident x - ray generates a charge packet ) . then the relation between @xmath6 and @xmath5 is expressed as , @xmath14 where @xmath15 , @xmath16 and @xmath17 are the cti values in the transfers ( a ) , ( b ) , and ( c ) , respectively . here we used the fact that the cti values are much smaller than 1 . thus we can separate the charge loss into @xmath13-dependent component ( the second term in the right - hand side of equation [ eq : cti_def_0 ] ) and constant component ( the third term ) . we therefore substitute the cti with cti1 ( the former component ) and cti2 ( the latter component ) , which have the cti values of @xmath18 and @xmath19 , respectively . then equation [ eq : cti_def_0 ] can be written as @xmath20 since the cti values depend on the amount of transfer charge which is proportional to the pulse height , we assume the cti is described by a power function of the pulse height ( prigozhin et al . 2004 ) and expressed as @xmath21 where @xmath22 and @xmath23 are scale factors for the cti1 and cti2 , and the index @xmath24 is common to the cti1 and cti2 . we have conducted the cfci experiments six times in orbit . effective exposure time for each experiment ranges from a few to @xmath020 ks . the equivalent x - ray energy of the injected charge packets ranges from @xmath00.3 kev to @xmath08 kev . since june 2006 , we injected various amounts of charge in one experiment . the log is summarized in table 1 . in the cfci experiments , the test charge is injected to the row at @xmath25 ( @xmath26 ) , and hence @xmath27 . the reference charge should be equal to the original charge which does not suffer from the charge loss , and hence @xmath28 . then equation [ eq : cti_def ] can be written as @xmath29 we determined @xmath30 by measuring the ratio @xmath31 for each column . from equation 3 , we can obtain the relation in the cfci experiments as , @xmath32 the index @xmath24 and @xmath33 were derived by fitting equation [ eq : cfci_cti2 ] to the values of @xmath30 obtained with the cfci experiments with multiple amount of charge injections ( multiple @xmath8s ; log number 36 in table 1 ) . the mean and standard deviation of the best - fit @xmath24 of equation [ eq : cfci_cti2 ] averaged over each sensor are shown in figure [ fig : ctipow ] . the mean value of @xmath24 shows no time variation , and the time averaged values of xis 0 , 1 , and 3 are 0.31 , 0.22 , 0.15 for xis 0 , 1 , and 3 , respectively . as for xis 2 , there was only one data point , and we obtained @xmath34 . if a charge packet has a volume proportional to the number of electrons and is spherically symmetric , the probability that one electron encounters a charge trap is proportional to the cross section of the charge packet . in this case , we can expect @xmath35 . from equation [ eq : cfci_cti ] , the cti value is proportional to @xmath36 , and hence @xmath37 . thus the simple model is roughly consistent with the observed @xmath24 values . using the results of all the cfci experiments ( log number 16 in table 1 ) and above determined @xmath24 values , we then re - estimated @xmath22 and @xmath23 separately . in this process , we assumed @xmath38 is equal to @xmath39 which was estimated by the 6.4 kev line from the sgr c region to be 0.67 and 1.5 for the fi and bi ccd , respectively ( @xcite ) . from this @xmath22 , we can obtain the final value of @xmath40 . figure [ fig : baratuki ] shows an example of the distribution of @xmath40 in july 2006 . we can see significant column - to - column dispersion . figure [ fig : ctihenka ] shows the change of @xmath40 from july 2006 to september 2007 . we can see that the cti values of all columns increased , but the increasing rate was different from column to column . the results of figures 2 and 3 indicate that the cti correction at the column level is strongly required . in figure [ fig : ctijikanhatten ] , we show the column - averaged @xmath40 value as a function of time . since the cfci experiments were only sparsely conducted ( see table 1 ) , we interpolate the @xmath40 and @xmath41 values for the observations of inter - cfci epochs . as for the determination of the cti values before the first cfci experiment , see appendix . a cti correction , which is the conversion of @xmath42 to @xmath5 , is made with equation [ eq : cti_def ] , where @xmath40 and @xmath41 are calculated from equation 3 by using the @xmath22 , @xmath23 , and @xmath24 values determined in section 3 . we used the emission lines from the onboard calibration sources , the perseus cluster of galaxies , and the supernova remnant 1e0102.2@xmath437219 . we retrieved the data from the data archives and transmission system of isas / jaxa . all data were acquired with the normal full window mode and the 3@xmath123 or 5@xmath125 editing mode @xcite . we used the data with the asca grades of 0 , 2 , 3 , 4 , and 6 . as is mentioned in @xcite , a small fraction of the charge in a pixel is left behind ( trailed ) to the next pixel in the same column during the transfer . all data used in this paper were corrected for the trail phenomenon . the observations are summarized in table [ tab : objlog ] . the calibration source @xmath44fe produces the mn k@xmath45 line . the theoretical line center energy is 5895 ev @xcite . we used the data from august 2005 to april 2007 . this is one of the x - ray brightest clusters of galaxies in the sky . the x - ray spectrum is that of a thin thermal plasma with the strong k@xmath45 line of fe . the plasma temperature changes smoothly from @xmath46 4 kev to @xmath0 7 kev toward the outer region @xcite , and the center energy of the fe k@xmath45 triplet is almost constant ( @xmath06.56 kev at @xmath47 ) within this temperature range . its radius of @xmath48 can cover the entire field of view of the xis ( @xmath49 ) . thus this source is suitable for measuring the positional dependence of the energy scale . this is one of the brightest supernova remnants in the small magellanic cloud . with the spatial resolution of suzaku , it can be regarded as a point source . there are many bright emission lines originated from thermal plasma in the x - ray spectrum below 2 kev . these lines are resolved with the xmm - newton rgs , and the accurate energies of the line centroids are known @xcite . this object has been used by many instruments for the calibration in the low - energy band , and an empirical model to describe the spectrum has been established . we used this source as the energy - scale calibrator in the low - energy band . for the data of the perseus cluster of galaxies , we divided the imaging area into four regions along the _ acty _ axis , and extracted a spectrum from each region . then we fitted the spectra in the 57.3 kev band with a power - law model and a gaussian function , and obtained the center pulse height of the fe k@xmath45 line . figure [ fig : perseus ] shows the center pulse height as a function of @xmath13 . triangles and circles indicate the data before and after the cti correction , respectively . we can see no significant @xmath13 dependence after the cti correction , and this supports the validity of our correction . the goal is to determine a relation of @xmath5 and x - ray energy @xmath51 . from the ground experiments , we found that the @xmath5-@xmath51 relation can be expressed as a broken - linear function linked at the si - k edge energy of 1839 ev @xcite . we then determined the @xmath5-@xmath51 relation of each segment by using the lines of the calibration sources ( mn k@xmath45 line at 5895 ev ) and 1e@xmath50 ( k@xmath45 lines of o , ne and ne around 6501020 ev ) . we show the results after the cti correction and the @xmath5-@xmath51 conversion . figure [ fig : calpeak ] shows measured center energies of the mn k@xmath45 line as a function of time . each mark in the plot has an effective exposure of more than 60 ks . the mean values of the center energy are 5896.2 , 5895.4 , 5895.0 , and 5895.4 ev for xis 0 , 1 , 2 , and 3 , respectively . the deviation around the theoretical center energy ( 5895 ev ) is 7.8 , 4.4 , 6.6 , and 7.8 ev for xis 0 , 1 , 2 , and 3 , respectively . therefore , the time - averaged uncertainty of the absolute energy is @xmath52 0.1 % for the mn k@xmath45 line of the calibration sources . we also studied the time evolution of the deviation around the theoretical center energy , and the results are shown in figure [ fig : peakbunsanhenka ] . we can see that the deviation gradually increases with time . figure [ fig : sciofflowpeak ] shows the center energy of the o k@xmath45 line from the 1e@xmath50 data . the mean values of the center energy are 652.6 , 653.8 , 652.7 , and 652.8 ev for xis 0 , 1 , 2 , and 3 , respectively . the deviation around the center energy of the empirical model ( 653 ev ) is 1.4 , 1.4 , 2.3 , and 1.1 ev for xis 0 , 1 , 2 , and 3 . therefore , the uncertainty of the absolute energy is @xmath53 0.2% for the o k@xmath45 line of 1e@xmath50 . we examined the energy resolution in fwhm ( @xmath54 ) @xcite for each sensor ; @xmath54 is common to all segments . we expressed @xmath54 as @xmath55 where @xmath56 and @xmath57 are time dependent parameters and @xmath58 is the energy resolution determined by the ground experiments and obtained using equation 1 in @xcite . we determined @xmath56 and @xmath57 by using the time history of the calibration sources and 1e@xmath50 . the @xmath54 values obtained in this way is incorporated into the redistribution matrix file ( rmf ) . figure [ fig : segcolumn ] shows the energy resolution of the mn k@xmath45 line after the column - to - column cti correction . we also plot the results of the cti correction , where we used the cti values averaged over a segment ( the column - averaged cti correction ) . we can see that the energy resolution is greatly improved by the column - to - column cti correction . for example , the energy resolution in december 2006 was greatly improved from @xmath0230 ev to @xmath0 190 ev . on the other hand , with the column - averaged cti correction , the energy resolution is @xmath0230 ev and is not significantly improved . in figure [ fig : calsig ] , we compared the energy resolution of the mn k@xmath45 line with our rmf model . the deviation of the data points around our model is 5.6 , 4.9 , 3.4 , and 6.3 ev for xis 0 , 1 , 2 , and 3 , respectively . we have conducted the cfci experiments six times in orbit . the cti correction has been done with the cfci results . we calibrated the energy scale of the xis precisely using the onboard calibration sources and 1e@xmath50 . our calibration results have been applied to all the data obtained with the normal full window mode without the spaced - row charge injection . the results of the cfci experiments and the current calibration status are summarized as follows : 1 . we determined the cti1 and cti2 values of each column precisely based on the data of the cfci experiments . we also found that the pulse height dependence of the cti does not change with time . 2 . after the column - to - column cti correction , we determined the @xmath5-@xmath51 relation . we also modeled the time - dependent energy resolution . the uncertainty of the energy scale is @xmath59 0.2 % for the o k@xmath45 line ( @xmath0 0.65 kev ) of 1e@xmath50 , and @xmath59 0.1 % for the mn k@xmath45 line ( @xmath605.9 kev ) of the calibration sources . 4 . with the column - to - column cti correction , the energy resolution at 5.9 kev in december 2006 was greatly improved from @xmath0230 ev to @xmath0 190 ev . the authors thank all the xis members for their support and useful information . this work was 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 . m.o . , h.u . , and h.n . are financially supported by the japan society for the promotion of science . is also supported by the mext , grant - in - aid for young scientists ( b ) , 18740105 , 2008 , and by the sumitomo foundation , grant for basic science research projects , 071251 , 2007 . h.t . and k.h . were supported by the mext , grant - in - aid 16002004 . first , we determined the cti values of the segments a and d in august 2005 . we combined the data of the calibration sources from august 11 to 31 , 2005 , and obtained the @xmath6 of the mn k@xmath45 line . we also estimated @xmath5 at 5895 ev from the @xmath5-@xmath51 relation determined by the ground experiments , and obtained the ratio @xmath61 . from equation [ eq : cti_def ] , the ratio can be expressed approximately as @xmath62 , where @xmath63 is the mean transfer number of the calibration events ( typically @xmath0900 ) . we determined @xmath40 and @xmath41 at 5895 ev from @xmath61 with the @xmath39 ratio fixed to the values shown in section 3 . the @xmath40 and @xmath41 for other pulse - height values were calculated from equation [ eq : cti_ph ] , where we used the column - averaged and time - averaged @xmath24 values determined in section 3 . then for segments b and c , we took the average cti values of the segments a and d. we regard the cti values obtained in this procedure as those on august 11 , 2005 ( the day of the xis first light ) . note that these values are determined for each segment , not for each column .

the x - ray imaging spectrometer ( xis ) on board the suzaku satellite is an x - ray ccd camera system that has superior performance such as a low background , high quantum efficiency , and good energy resolution in the 0.212 kev band . because of the radiation damage in orbit , however , the charge transfer inefficiency ( cti ) has increased , and hence the energy scale and resolution of the xis has been degraded since the launch of july 2005 . the ccd has a charge injection structure , and the cti of each column and the pulse - height dependence of the cti are precisely measured by a checker flag charge injection ( cfci ) technique . our precise cti correction improved the energy resolution from 230 ev to 190 ev at 5.9 kev in december 2006 . this paper reports the cti measurements with the cfci experiments in orbit . using the cfci results , we have implemented the time - dependent energy scale and resolution to the suzaku calibration database .

You are an expert at summarizing long articles. Proceed to summarize the following text: the vela pulsar ( b0833 - 45 ) , with a period of 89 millisecond and a characteristic age of 11,000 years , it is the prototype of the young `` vela - like '' pulsars . the excellent angular resolution of _ chandra _ allows us to separate the pulsar from its synchrotron nebula and study its x - ray properties . we extracted a total of 16000 counts in the range 0.252.0 kev from two hrc - s / letg observations of 25 ks each . no deviations of counts in the individual bins from the mean value of neighboring 1020 bins were found with statistical significance higher than 2.7@xmath7 . this may indicate that there are no elements heavier than h present on the ns surface . the spectral fits with thermal models show excess counts at the high energy end . to get a handle on the harder component , we used the publicly available acis / hetg data obtained in the continuous clocking ( cc ) mode . we used only the zero order image , which is equivalent to about 3 ks observation without the grating . we applied manual corrections for the dither and the sim motion for both timing and the event position in sky plane ( see zavlin et al . 2000 for details ) . the background was estimated by interpolating between the neighboring regions in the 1d image of the pulsar and its nebula . since the acis response is poorly known at e @xmath8 0.5 kev , and the background dominates above 8.0 kev in the cc mode , we use the events in the energy range 0.58.0 kev . lcccccc thermal & @xmath9 & t@xmath10 & r@xmath10 & l@xmath11 & @xmath12 & l@xmath13 + model & ( @xmath14 @xmath15 ) & ( mk ) & ( km ) & ( erg s@xmath16 ) & & ( erg s@xmath16 ) + blackbody & 2 & 1.5 & 2 & 1.5@xmath17 & 2.7 & @xmath18 + h atmosphere & 3 & 0.7 & 12 & 3.8@xmath17 & 1.5 & 1.5@xmath19 + the combined spectral fit to the hrc - s / letg and acis / hetg - cc data clearly shows two components a soft thermal component , which fits equally well with a blackbody or a magnetic neutron star atmosphere model , and a harder component , which we interpret as a power - law ( pl ) . the parameters of the fit depend on the model chosen for the soft thermal component . table 1 shows the parameters of fits to the combined data ( see pavlov et al . 2001 for details ) . figure 1 shows the broadband spectrum of the vela pulsar . the soft x - ray spectral parameters are for the atmosphere model . the extrapolation of the pl component matches both the optical and the hard x - ray/@xmath12-ray flux . this is the first conclusive observation of both the thermal and the non - thermal radiation from the vela pulsar . the vela pulsar was observed twice with the hrc - i and twice with the hrc - s / letg . both of the hrc - i observations and the first letg observation suffered from the hrc timing problem . we corrected the times by shifting to the time of the next event in the level 1 event list taking only the events with time correction less than 4 ms . the second letg observation was in the timing mode , so we could recover all the event times . [ fig2 ] we used the radio ephemeris of the vela pulsar based on the observations at the hobart and parkes radio observatories . the phase of each event was determined and folded to get the pulse profiles . the left panel of figure 2 shows the observed pulse profiles of the hrc observations . the pulse profiles show three peaks separated by about 1/3 of phase , with a total pulsed fraction of @xmath20 ( see also helfand et al . 2001 ) . the pulsed fluxes in the three peaks are shown in figure 1 . for the acis / hetg - cc observation , the recorded times are the event readout times . we corrected the times for the dither and sim motion , and the delay from the event to its readout . the correction for absolute time depends on the actual position of the source on the ccd , which is poorly known at present . therefore , we are unable to get the absolute times for the cc observation . the relative times , on the other hand , are accurate . the right panel of figure 2 shows the acis pulse profiles in two energy bands ( @xmath21 kev and @xmath6 kev ) . the difference in the profile shapes and the pulsed fraction is striking . we estimate the intrinsic pulsed fraction of the pulsar to be @xmath22 and @xmath23 in the @xmath21 kev and @xmath6 kev bands , respectively . * the hrc - s / letg spectrum shows no conclusive evidence for lines in the 0.252.0 kev range . the spectrum shows two components ( thermal + non - thermal ) , similar to middle - aged pulsars . the thermal component is consistent with the emission from a magnetic hydrogen atmosphere . extrapolation of the pl component matches the optical and the hard x - ray points . * the thermal luminosity of the vela pulsar is lower than that predicted by the `` standard '' neutron star cooling models ( tsuruta 1998 ) . * hrc data show three peaks with total pulsed fraction of about @xmath20 . energy - resolved acis pulse profiles show energy - dependent shape and pulsed fraction . the pulse profile at @xmath21 kev is similar to that observed with hrc . at @xmath6 kev , the pulse profile shows only two peaks with estimated intrinsic pulsed fraction of @xmath23 .

we report the results of the spectral and timing analysis of observations of the vela pulsar with the _ chandra _ x - ray observatory . the spectrum shows no statistically significant spectral lines in the observed 0.258.0 kev band . it consists of two distinct continuum components . the softer component can be modeled as either a magnetic hydrogen atmosphere spectrum with @xmath0 ev , @xmath1 km , or a standard blackbody with @xmath2 ev , @xmath3 km ( the radii are for a distance of 250 pc ) . the harder component , modeled as a power - law spectrum , gives photon indices depending on the model adopted for the soft component : @xmath4 for the magnetic atmosphere soft component , and @xmath5 for the blackbody soft component . timing analysis shows three peaks in the pulse profile , separated by about 0.3 in phase . energy - resolved timing provides evidence for pulse profile variation with energy . the higher energy ( @xmath6 kev ) profile shows significantly higher pulsed fraction . # 1_#1 _ # 1_#1 _ = # 1 1.25 in .125 in .25 in

You are an expert at summarizing long articles. Proceed to summarize the following text: despite of the increasing success of qcd in describing a large variety of phenomena , both in the perturbative as well as in the non - perturbative regimes , some fundamental questions remain unsolved . prominent examples are the very nature and detailed properties of the strongly coupled quark gluon plasma which is the conjectured state of qcd matter at temperatures comparable and larger than the qcd energy scale @xmath0 . furthermore , the nature and properties of the chiral and deconfinement phase transition as well as the position of a conjectured critical point ( cp ) in the qcd phase diagram are among the still challenging issues @xcite . to answer these questions experimentally , a number of large scale experiments is currently running ( alice , atlas and cms at the lhc , and star and phenix at rhic ) , planned ( mpd at nica ) or under construction ( cbm and hades at fair ) . on the theory side , lattice qcd yields a smooth crossover from the hadronic phase to the quark - gluon phase for small chemical potential at temperatures of about @xmath1 . at sufficiently large net baryon density the crossover may turn into a first order phase transition at a cp . at non - zero net densities ( _ i.e._non - zero quark chemical potential ) there is no first - principle approach to the phase diagram . therefore , one has to rely on effective models or on truncation schemes . nevertheless , many of these approaches seem to point to a first order phase transition connected to the spontaneous breaking of chiral symmetry at densities a few times the nuclear density . the end point of this transition line has interesting properties on its own . from macroscopic examples , the phenomenon of critical opalescence , _ i.e._the diverging scattering strength of transparent media in the vicinity of a critical point , has been known for a long time @xcite . quite common also is the phenomenon of critical slowing down , _ i.e._the diverging relaxation time at criticality @xcite . these two examples as well as most of the special properties of critical points have their reason in the diverging correlation length making the system scale free . to understand the mass generation connected to chiral symmetry breaking , several effective models have been constructed with the nambu - jona - lasinio ( njl ) model @xcite and the linear sigma model ( l@xmath2 m ) @xcite being the most prominent ones . our present investigation is motivated by the question whether penetrating probes reflect directly the phase structure of strongly interacting matter . we focus here on real photons and select the @xmath3 to mimic the above anticipated phase structure . the @xmath3 contains quark and meson ( pion and sigma ) fields as basic degrees of freedom , where the fluctuations of the latter ones are accounted for in linear approximation , as in @xcite and the photon field is minimally coupled to the strongly interacting components of the @xmath3 . there is a large difference in the time scales concerning the strong and the electromagnetic interactions , respectively . this makes possible separating the two interactions involved . the strong interaction is responsible for the relaxation towards a local thermal equilibrium as well as to the mass generation via the spontaneous breaking of chiral symmetry . the electromagnetic interaction with a perturbative radiation field contributes little to this , because its effects are @xmath4 suppressed . therefore we might calculate the thermodynamics without regarding electromagnetism and use the thermodynamic properties as well as the effective masses of the dressed quarks and mesons later on in the photon emission calculations . the @xmath3 is a widely used effective model of qcd and has been applied often for studying various aspects of thermodynamics of strongly interacting matter . it was suggested by gell - mann and levy in 1960 @xcite for studying chiral symmetry breaking . in absence of an explicit symmetry breaking term , the model has a symmetry and therefore belonging to the same universality class as @xmath5 qcd in the chiral limit @xcite . this symmetry present at high temperatures is spontaneously broken to a residual @xmath6 symmetry with the three pseudoscalar @xmath7 mesons being the goldstone modes . breaking chiral symmetry explicitly , the pions acquire non - zero masses . besides these satisfying properties there is a close connection to the non - linear @xmath2 model , which in turn is equivalent to leading order chiral effective field theory of qcd . compared to the njl model the @xmath3 has the advantage of including the mesons directly as dynamic field quanta , making it easier to address their properties . the @xmath3 lagrangian reads @xmath8 where the dirac field @xmath9 describes a doublet of quarks , @xmath2 corresponds to an iso - scalar and lorentz - scalar field , and @xmath10 describes an iso - vector and lorentz - pseudoscalar field , the latter ones conveniently interpreted as the @xmath2 and @xmath7 mesons . from the lagrangian the thermodynamic potential @xmath11 is constructed via the path integral of the exponential of the euclidean action and evaluated following the procedure described in @xcite for including linearized fluctuations . first , one integrates over the fermionic fields @xmath9 and @xmath12 . the remaining path integral corresponds to a purely mesonic theory with a complicated interaction potential , which is approximated by a quadratic one to account for small fluctuations . the parameters of this quadratic potential are identified with the masses and thermodynamic averages of the meson fields . this leads to self consistency relations for the masses . the parameters are fixed by the following requirements : the mass of the pions is set to @xmath13 in vacuum ( @xmath14 ) and the sigma meson mass to @xmath15 . the effective quark mass in the vacuum is fixed to one third of the nucleon mass @xmath16 , and the parameter @xmath17 is identified with the pion decay constant in vacuum , @xmath18 . with these parameters one obtains the results depicted in fig . [ fig_thermodyn ] . + figs . [ subfig_m_pi_lf]-[subfig_m_q_lf ] show contour plots of the masses over the phase diagram . one notes that the pion mass ( fig . [ subfig_m_pi_lf ] ) increases with temperature and chemical potential with the strongest change at the phase boundary . the sigma meson mass ( fig . [ subfig_m_si_lf ] ) on the other hand exhibits a valley of low mass values around the phase boundary and with a global minimum at the critical point . the quark mass plotted in fig . [ subfig_m_q_lf ] drops from its vacuum value to about @xmath19 . the most drastic change , again , is at the phase boundary , signaling that the mechanism for mass generation is indeed the spontaneous breaking of chiral symmetry within the @xmath3 . because the chiral symmetry is also explicitly broken by a nonzero @xmath20 in the lagrangian , the quark mass does not drop to zero , but stays finite in the high temperature phase . comparing the meson masses ( _ cf._figs . [ subfig_m_pi_lf ] and [ subfig_m_si_lf ] ) , one realizes that they are degenerate above the 1st order phase transition curve and the crossover region , respectively , but very different below . this behavior of the mass difference of these chiral partners is another sign of the chiral symmetry breaking and restoration . for quantifying the size of the critical region the quark number susceptibility is chosen , since the susceptibility scales with the correlation length whose divergence causes many of the special features of a cp . in fig . [ subfig_chi_lf ] , @xmath21 is normalized to the susceptibility @xmath22 of a massless ideal fermion gas to scale out trivial contributions . for calculating photon emission rates , the @xmath3 lagrangian is extended by an electromagnetic sector coupled minimally ( _ cf._@xcite ) to the strongly interacting part . @xmath23 where @xmath24 is the free photon lagrangian and @xmath25 denotes the photon field . photon emission rates are , in a kinetic theory approach , convolutions of squared matrix elements @xmath26 and phase space distribution functions @xmath27 , the latter ones explicitly depending on @xmath28 and @xmath29 . superimposed are implicit @xmath28 and @xmath29 dependencies from the effective masses of the involved fields , as displayed in . given the marked variations of these masses one can expect an pronounced impact on the emission rates @xmath30 owing to the weakness of the electromagnetic interaction we restrict the calculations to first order in the electromagnetic coupling . since we expect to have captured the dominant part of the strong interaction in the calculation of the thermodynamic potential and the effective masses , the residual interaction is expected to be relatively weak . therefore we restrict our calculation to 1st order processes in the quark - meson coupling . within this approximation the contributing processes are the tree - level processes in the @xmath31 , @xmath32 and @xmath33 channels . in , four of the nine integrations can be carried out exactly applying the delta distribution . another ( angular ) integration drops out by symmetry reasons , so one is left with four integrals , which have to be executed numerically resulting in the rates depicted in fig . [ fig_rates ] . + when photon energies @xmath34 are much larger than the respective masses , it is not expected to see much of the details of the phase structure . contrary , at lower energies there are huge differences in available phase space and matrix elements squared leading to pronounced patterns which reflect phase diagram features , in particular the effective masses . for this reason @xmath35 is chosen . figure [ fig_rates ] shows contour plots of the photon rates for the different contributing processes over the phase diagram . in figs . [ subfig_rate_qp_gq_omega=0010 ] and [ subfig_rate_qs_gq_omega=0010 ] we see an enhancement in the crossover region and in fig . [ subfig_rate_qs_gq_omega=0010 ] a global maximum in the critical region . in figs . [ subfig_rate_qq_gp_omega=0010 ] and [ subfig_rate_qq_gs_omega=0010 ] one notices large rates in the chirally restored phase and much less photon emission in the chirally broken phase , which in case of [ subfig_rate_qq_gs_omega=0010 ] is superimposed by an island of enhanced rates for @xmath36 and @xmath37 . figures [ subfig_rate_qq_gp_omega=0010 ] and [ subfig_rate_qp_gq_omega=0010 ] show photon rates from processes involving pions . pions exhibit a large mass difference between the two phases , but contrary to the sigma meson whose mass has a global minimum at the cp the pion mass does not show special features at this point . this leads to a large difference in the emissivity between the phases but no features characteristic for the cp itself . for the pion - involving compton process ( fig . [ subfig_rate_qp_gq_omega=0010 ] ) there is an enhancement in the crossover region . this is probably due to a combination of phase space effects and the ( comparatively ) large probability for the internally propagating pion to get on - shell . a better channel for obtaining signatures of a cp are sigma involving processes . this is expected , since the sigma meson is precisely the mode getting massless at the cp making long range interactions possible and thus driving the critical processes . unfortunately , the inclusion of linearized fluctuations increases the sigma mass , so it is not clear whether the endpoint of the 1st order phase transition shows correctly the critical behavior . but linearizing fluctuations anyhow restricts to small fluctuations making it not adequate very near the cp . nevertheless the sigma mass drops to small values in the critical region , which has a notable effect on the corresponding processes , _ e.g._the excess of the photon rate in the critical region in fig . [ subfig_rate_qs_gq_omega=0010 ] . there is a large difference in the rates for the processes under consideration , even between the corresponding compton ( figs . [ fig_rates](b ) and [ fig_rates](d ) ) and annihilation ( figs . [ fig_rates](a ) and [ fig_rates](c ) ) processes . these can be understood in terms of available phase space in combination with thermal suppression . within a boltzmann approximation two of the remaining integrals in can be solved to obtain @xmath38 the difference between the minimal kinematically allowed value of the center of mass energy @xmath39 for the different processes , together with a small value of @xmath34 leads to the huge thermal suppression at small @xmath28 seen in figs . [ fig_rates ] ( a ) and ( d ) . focusing on soft - photon emission rates we demonstrate that some features of the phase diagram provided by the linear sigma model are nicely mapped out . being aware of some limitations , such as the restriction to linearized fluctuations ( _ cf._@xcite for a proper account of fluctuations ) and the need to implement more complete rates in a model of space - time evolution of the matter , we hope that improved calculations can provide useful complementary information on strongly interacting matter produced in the course of relativistic heavy - ion collisions at various energies , system sizes and centralities .

soft - photon emission rates are calculated within the linear sigma model . the investigation is aimed at answering the question to which extent the emissivities map out the phase structure of this particular effective model of strongly interacting matter .

You are an expert at summarizing long articles. Proceed to summarize the following text: it is well known that the density of states in a pure one dimensional interacting electron gas ( luttinger liquid ) vanishes as a power law in the vicinity of the fermi level @xcite : @xmath9 , where the exponent is determined by the interaction strength . such a behavior should reveal itself in the dependence of the differential tunnel conductance on the applied bias : @xmath10 . it is obvious that the presence of a disorder would perturb the local value of the density of states and , thus , cause some random correction , @xmath3 , to the conductance . then the relevant questions are : \i ) what is the typical magnitude of @xmath7 for a given realization of the disorder ? \ii ) how the values of @xmath7 at different voltages are correlated ? \iii ) how the correlation between the values of @xmath7 at the same voltage , but at different points of the liquid , falls off with increasing separation between the points ? these questions are addressed in the present paper . we will consider the case of a smooth disorder . namely , we will assume that the correlation radius is much larger than the fermi wavelength . this assumption simplifies the problem drastically , since it permits one to neglect the backward scattering of electrons and , hence , to view the disorder as a random variation of the electron concentration with coordinate . the key to understanding the role of the disorder is provided by the bosonization procedure @xcite , which allows to describe the low energy excitations of the system in terms of bosonic excitations ( plasmons ) , propagating along the liquid . since the velocity of a plasmon depends on the concentration of electrons , the spatial variation of the concentration would give rise to the backscattering of plasmons . in other words , in the presense of a disorder , a plasmon with frequency @xmath1 acquires a finite mean free path @xmath0 . it is important to note that a smooth disorder , for which the backscattering of electrons at the fermi level is suppressed , might , ultimately , be quite efficient in backscattering of plasmons with wavelengths of the order of the correlation radius . for non interacting electrons in one dimension it is established that even a weak disorder leads to the localization of all eigenstates @xcite . then the mean free path acquires the meaning of the localization radius . the same conclusion applies , certainly , to plasmons . the difference is , however , that an electron becomes more and more localized as its energy decreases , whereas for plasmons the situation is the opposite : the lower is the frequency , the weaker is the localization . since the dispersion law of a plasmon is linear : @xmath11 , where @xmath12 is the sound velocity , the density of the plasmon states is frequency independent . the reason for suppression of the localization for the low frequency plasmons is that the matrix element of the baskscattering vanishes in the limit of long wavelengths . in particular , we show that for small @xmath1 the frequency dependence of the mean free path is @xmath2 and , correspondingly , the product @xmath13 , which determines the localization strength , increases as @xmath14 when @xmath1 goes to zero . we also show that @xmath0 turns to infinity when the interactions are switched off , thus revealing that the backscattering of plasmons is possible only due to interactions . if the relative variation of the concentration is small , the disorder induced correction to @xmath15 can be expressed in terms of the spectral characteristics of the localized plasmons ( such as local density of states ) . on the other hand , these characteristics were the subject of detailed studies in application to the localized electrons @xcite . by utilizing the approach developed in refs . , we calculate the variance @xmath16 , and the two point correlator @xmath17 . we find that the ratio @xmath18 increases with voltage as @xmath19 and that the characteristic `` period '' of change of @xmath3 with voltage is of the order of @xmath4 . we also find that the two point correlator falls off with @xmath20 as a power law and oscillates with the period @xmath21 , which is one half of the wavelength of a plasmon with frequency @xmath8 . the paper is organized as follows . in the next section the formula for the mean free path of a plasmon is derived . in section [ ctc ] we calculate the correlator of fluctuations of the tunnel conductance . section [ con ] concludes the paper . the hamiltonian of a luttinger liquid with concentration of electrons , @xmath22 , being a function of coordinate has the form @xcite @xmath23 , \ ] ] where @xmath24 and @xmath25 are , correspondingly , the displacement and conjugate momentum ( @xmath26=i\hbar \delta(x - x')$ ] ) ; @xmath27 is the effective interaction strength : @xmath28 . the hamiltonian can be reduced to a system of harmonic oscillators @xmath29 , \ ] ] by means of the following transformation @xmath30 here @xmath31 are the eigenfunctions of the operator @xmath32 which is defined as @xmath33= \omega_{\mu}^2 \phi_{\mu}.\ ] ] the eigenvalues of @xmath32 determine the frequencies , @xmath34 , of the oscillators . if the concentration is constant ( @xmath35 ) , the solutions of ( [ d ] ) are the plane waves @xmath36 with a linear spectrum @xmath37 ( @xmath38 is the normalization length ) . substituting ( [ pw ] ) into ( [ d ] ) , we get the standard expression for the sound velocity @xcite @xmath39 where @xmath40 is the fermi velocity . assuming that the relative variation of the concentration is small , @xmath41 , the expression for the mean free time , @xmath42 , for a plasmon with frequency @xmath1 is given by the golden rule @xmath43 where @xmath44 is the plane wave with @xmath45 and @xmath46 stands for the averaging over the fluctuations of @xmath22 . the factor @xmath14 in ( [ gr ] ) appears since we define the mean free time as @xmath47 . keeping only the first order terms in the difference @xmath48 , we get the following expression for the matrix element @xmath49 .\end{aligned}\ ] ] the energy conservation , insured by the @xmath50-function in ( [ gr ] ) , requires that @xmath51 ( backscattering ) . then we have @xmath52 and , as it can be easily seen , the second term in the matrix element vanishes identically . as a result , the matrix element takes the form @xmath53 we see that the matrix element for backscattering is proportional to the interaction strength , which reflects the fact that this process is possible only due to interactions . the mean free path , @xmath0 , defined as @xmath54 , can be found after substituting ( [ fme ] ) into ( [ gr ] ) @xmath55 where we have introduced the correlator of the fluctuations of @xmath22 @xmath56 here @xmath57 is the correlation radius and @xmath58 is the mean square fluctuation of @xmath22 , so that @xmath59 . it is instructive to rewrite ( [ mf ] ) for the product @xmath60 , which measures the effective scattering strength . assuming the gaussian form of the correlator , @xmath61 , we obtain @xmath62 note that the last factor in ( [ mfg ] ) is a function of the argument @xmath63 and this function does not exceed unity . thus , the product @xmath13 is large for any @xmath1 . obviously , the backscattering is ineffective when @xmath64 , i.e. when the correlation radius @xmath57 exceeds the wave length of the plasmon . in the opposite limit , @xmath65 , the mean free path behaves as @xmath66 . as it was mentioned in the introduction , the real meaning of @xmath0 is the localization length of a plasmon with frequency @xmath1 . the fact that @xmath67 allows one to apply to localized plasmons , the description developed earlier for localized electrons @xcite just in this limit . this is done in the next section . once the hamiltonian is diagonalized , the derivation of the formula for the density of states becomes standard @xcite . the operator @xmath68 , creating an electron at point @xmath69 , can be presented in the form @xcite @xmath70 where the coefficients @xmath71 and @xmath72 are defined as @xmath73 the calculation of the green function @xmath74 does not differ from that for a pure luttinger liquid and leads to the following expression for the density of states @xmath75 where @xmath76 is the sum over eigenmodes @xmath77 if @xmath22 is constant so that the eigenfunctions are given by ( [ pw ] ) , eq.([dos ] ) reproduces the known result @xcite for the density of states : @xmath78 , with @xmath79 indeed , for plane waves we have latexmath:[$|\alpha_{\mu}|^2=1/k_{\mu}^2ln_0 , the function @xmath76 @xmath81 where @xmath82 is the cutoff parameter . substituting this expression into ( [ dos ] ) leads to ( [ kappa ] ) . to find the disorder - induced correction to @xmath83 , we treat the difference @xmath84 as a perturbation and expand the exponent in ( [ dos ] ) to the first order . this gives @xmath85.\ ] ] to calculate the integral over @xmath86 , it is convenient to present the denominator in ( [ ddos ] ) in the form @xmath87 after substituting ( [ ga ] ) into ( [ ddos ] ) , both integrations , over @xmath88 and @xmath86 can be easily carried out , and one obtains @xmath89 ( \omega-\omega_{\mu})^{\kappa } \right . \nonumber \\ \left.- ( \kappa+1 ) \int_0^{\omega / s}\frac{dq}{q}(\omega - sq)^{\kappa } \right\ } .\end{aligned}\ ] ] note that the constant @xmath90 can be expressed through the derivative @xmath91 by integrating the equation ( [ d ] ) from @xmath69 to @xmath92 . then , one gets @xmath93 to study the correlation properties of @xmath94 , we introduce the following local densities @xmath95 the average values of these densities are equal @xmath96 then the correction @xmath97 can be rewritten in terms of the fluctuations @xmath98 and @xmath99 @xmath100 .\ ] ] correspondingly , the correlator of @xmath97 at different points is expressed through the correlators of the fluctuations @xmath101 @xmath102 ^ 2 \times \nonumber \\ & & \int_0^{\omega } \frac{d\omega_1}{\omega_1 } \int_0^{\omega } \frac{d\omega_2}{\omega_2 } ( \omega - \omega_1)^{\kappa } ( \omega - \omega_2)^{\kappa } \left(\frac{v_f^2}{s^2 } f_1 + \frac{s^2}{v_f^2 } f_2 + 2 f_{12 } \right ) \ ; , \end{aligned}\ ] ] where the correlators @xmath103 and @xmath104 are defined as @xmath105 the correlator @xmath106 is , in fact , the correlator of the fluctuations of the local density . in application to localized electrons , it was studied in refs . using , correspondingly , the berezinski technique @xcite and the technique developed by berezinski and gorkov @xcite . in the limit @xmath107 , the correlator is nonzero only if the difference @xmath108 is small enough : @xmath109 . for distances @xmath110 , the expression for @xmath106 , obtained in ref . , reads @xmath111 \ ; , \ ] ] where @xmath112 is the difference between the two frequencies and the function @xmath113 is given by @xmath114 as it was noted in refs . , at @xmath115 and @xmath116 , we have @xmath117 , i.e. the correlation is absent . it is easy to establish that the correlator @xmath118 is equal to @xmath106 . concerning the correlator @xmath104 , we did not find the expression for this correlator in the literature . so , we have calculated it using the berezinski - gorkov technique and obtained the following expression @xmath119 \ ; .\ ] ] it is seen that in contrast to @xmath106 , the correlator @xmath104 is non - zero at @xmath115 and @xmath120 @xcite . for finite @xmath121 , both correlators are proportional to @xmath122 and decay with increasing @xmath121 as @xmath123 . to calculate the double integral in ( [ nucor ] ) , we make use of the fact that the correlators @xmath124 and @xmath104 are sharp functions of @xmath121 , i.e. the major contribution to the integral comes from the domain @xmath125 . this allows to put @xmath126 in all other factors and to extend the integration over @xmath127 to @xmath128 . noting that the integral of @xmath129 is equal to @xmath130 , we obtain @xmath131 note that the coefficients in front of @xmath124 and @xmath104 in ( [ nucor ] ) have combined into the factor @xmath132 which is proportional to @xmath133 at small @xmath27 . using ( [ int_e ] ) , the expression for the correlator @xmath134 takes the form @xmath135 ^ 2 \int_0^{\omega } \frac{d \omega_1}{\omega_1 ^ 2 \tau_{\omega_1 } } ( \omega -\omega_1)^{2 \kappa } \cos \left(\frac{2 \omega_1 z}{s } \right ) \ ; .\ ] ] apparently , the factor @xmath136 in ( [ nnucor ] ) diverges at small @xmath137 . however , this divergency is compensated by the frequency dependence of the mean free time , which at small @xmath137 behaves as @xmath136 . substituting @xmath138 in ( [ nnucor ] ) and introducing the new variable @xmath139 , we get the final result @xmath140 ^ 2 \left(1-\frac{v_f^2}{s^2}\right)^4 \frac{\overline{(\delta n)^2}}{n_0 ^ 2 } \frac{s r_c \omega^{2\kappa+1}}{v_f^2 } r(\tilde{z } ) \ ; , \ ] ] where the function @xmath141 is defined as @xmath142 here , @xmath143 is the dimensionless distance @xmath144 the function @xmath141 is defined in such a way that , in the only interesting limit @xmath145 , it turns to @xmath146 at @xmath147 . this function determines the coordinate dependence of the correlator ( [ nucor ] ) and , consequently , the coordinate dependence of the correlator of the fluctuations of the tunnel conductance @xmath148 the function @xmath141 is ploted in fig.1 . with increasing distance , it falls off and oscillates . the asymptotic behavior of @xmath149 at @xmath150 is as follows @xmath151 the spatial period of the oscillations of @xmath141 is @xmath152 and it decreases with increasing voltage . it is to be emphasized that such a behavior of the correlator of @xmath7 ( slow decay and oscillations ) is entirely due to interactions . in the absence of the interactions , the correlator ( [ tcf ] ) would simply reproduce the correlator ( [ dc ] ) of the fluctuations of @xmath22 , i.e. it would decay monotonously at distance @xmath153 which is much smaller than @xmath154 . at @xmath147 , the formula ( [ nucor ] ) defines the variance of @xmath97 . it is convenient to rewrite it for the relative magnitude of the fluctuations of @xmath7 @xmath155 naturally , the variance @xmath156 is proportional to the magnitude of the fluctuations of the density . less trivial is that ( [ tcv ] ) contains two additional factors : @xmath157 which is proportional to @xmath133 at small @xmath27 , and a small factor @xmath158 ( note that ( [ tcv ] ) is written in the limit @xmath159 ) . this factor determines the voltage dependence : @xmath160 . qualitatively , this dependence can be interpreted as follows . formula ( [ fddos ] ) shows that all plasmon modes with frequencies smaller than @xmath1 contribute to @xmath161 . obviously , the major contribution comes from plasmons for which the center of localization lies within the localization radius from the point @xmath69 . let us consider some frequency strip @xmath162 centered at @xmath163 and with the width @xmath164 . then , the localization radius for all modes within this strip is approximately @xmath165 . for a spatial interval of length @xmath0 , the typical frequency spacing between modes is @xmath166 . correspondingly , the average number of modes within the strip is @xmath167 and , hence , the relative fluctuation of this number is @xmath168 . the final estimate for the fluctuation of @xmath94 emergies if we set @xmath169 . then , one gets @xmath170 . substituting ( [ mfg ] ) for @xmath171 , we reproduce the voltage dependence in ( [ tcv ] ) . however , the estimate and the result of the calculation still differ by a factor @xmath172 . this factor originates from the specific details of the strucure of the eigenfunctions ( see the comment after eq . ( [ int_e ] ) ) , and we can not interpret it qualitatively . note in conclusion of the section , that since all frequencies from @xmath173 to @xmath1 give , roughly , comparable contributions to @xmath161 , it will change significantly only when the frequency doubles . this means that at fixed @xmath69 the characteristic period of fluctuations in @xmath7 as a function of voltage is of the order of @xmath4 . the main result of the present paper is that the randomness in the concentration of electrons in the luttinger liquid causes a random component in the tunnel conductance which changes semiperiodically along the liquid , with the period depending on the applied bias . this behavior results from the fact that at a fixed bias , v , the frequencies of plasmons , responsible for the correction to the density of states , are strictly limited by @xmath174 . correspondingly , the oscillating behavior of the correlator of the fluctuations reflects the distribution of density in a plasmon with maximal frequency . if the conductance is studied as a function of bias , the disorder would cause fluctuations with characteristic period of the order of @xmath4 . the fluctuations disappear as the wavelength of the plasmon with frequency @xmath174 becomes smaller than the spatial scale of the change of the concentration , @xmath57 . there is also a limitation from low voltages , imposed by the finite length of the liquid , @xmath38 . namely , our calculation applies when the mean free path of a plasmon , @xmath0 , is smaller than @xmath38 . at low enough frequencies @xmath0 exceeds @xmath38 and the disorder does not play any role . for such frequencies the oscillations of the tunnel conductance with voltage have their origin in the size quantization of plasmons , the period of oscillations being @xmath175 . these oscillations were studied in ref . . note finally , that the approach developed in the present paper can be extended to the case of the multichannel luttinger liquid @xcite . in the latter case , the scattering of plasmons between the channels , caused by a disorder , should be taken into account . y. oreg and a. m. finkelstein , phys . lett . * 76 * , 4230 ( 1996 ) . d. c. mattis and e. h. lieb , j. math . * 6 * , 304 ( 1965 ) . f. d. m. haldane , j. phys . c * 14 * , 2585 ( 1981 ) . n. f. mott and w. d. twose , adv . * 10 * , 107 ( 1961 ) . v. l. berezinski , zh . * 65 * , 1251 ( 1973 ) [ sov . jetp * 38 * , 620 ( 1974 ) ] . v. l. berezinski and l. p. gorkov , zh . . fiz . * 77 * , 2489 ( 1979 ) [ sov . jetp * 50 * , 1209 ( 1979 ) ] . e. p. nakhmedov , v. n. prigodin , and yu . a. firsov , zh . * 92 * , 2133 ( 1987 ) [ sov . jetp * 65 * , 1202 ( 1987 ) ] . l. p. gorkov , o. n. dorokhov , and f. v. prigara , zh . . fiz . * 84 * , 1440 ( 1983 ) [ sov . jetp * 57 * , 838 ( 1983 ) ] . yu . v. nazarov , a. a. odintsov , and d. v. averin , bull . . soc . * 41 * , 376 ( 1996 ) . a. gramada , m. e. raikh , preprint cond - mat/9604090 g. mahan _ many - particle physics _ , 2nd ed . ( plenum , new york , 1990 ) . b. l. altshuler and v. n. prigodin , zh . eksp . fiz . * 95 * , 348 ( 1989 ) [ sov . jetp * 68 * , 198 ( 1989 ) ] . actually there is no qualitative understanding by now , why @xmath106 turns to zero at coinciding points @xcite . k. a. matveev and l. i. glazman , phys . * 70 * , 990 , ( 1993 ) .

the luttinger liquid in which the concentration of electrons varies randomly with coordinate is considered . we study the fluctuations of the tunnel conductance , caused by the randomness in the concentration . if the concentration changes slowly on the scale of the fermi wavelength , its prime role reduces to the scattering of the plasmon waves , propagating along the system . as a result of such a scattering , plasmons get localized . we show that the localization length , @xmath0 , of a plasmon with frequency @xmath1 is inverse proportional to the square of the interaction strength and changes with frequency as @xmath2 . if the relative variation of the concentration is small , the randomness induced correction to the tunnel conductance , @xmath3 , where @xmath4 is the applied bias , can be expressed through the spectral characteristics of the localized plasmons . the magnitude of the correction , @xmath5 , increases with @xmath4 as @xmath6 . the typical period of the fluctuations in @xmath3 is of the order of @xmath4 . at a fixed @xmath4 , the correlator of @xmath7 at different points of the liquid falls off with distance as a power law and oscillates with the period which is one half of the wavelength of a plasmon with frequency @xmath8 .

You are an expert at summarizing long articles. Proceed to summarize the following text: the increasing ability of the direct @xmath0-body method to provide reliable models of the dynamical evolution of star clusters has closely mirrored increases in computing power ( heggie 2011 ) . the community has progressed from small-@xmath0 models performed on workstations ( e.g. von hoerner 1963 ; mcmillan , hut & makino 1990 ; giersz & heggie 1997 ) to models of old open clusters and into the @xmath9 regime ( baumgardt & makino 2003 ) by making use of special - purpose grape hardware ( makino 2002 ) . software advances over a similar timeframe have produced sophisticated codes ( aarseth 1999 ; portegies zwart et al . 2001 ) that increase the realism of the models by incorporating stellar and binary evolution , binary formation , three - body effects and external potentials . as a result , @xmath0-body models have been used in numerous ways to understand the evolution of globular clusters ( gcs : vesperini & heggie 1997 ; baumgardt & makino 2003 ; zonoozi et al . 2011 ) , even though the best models still only touch the lower end of the gc mass - function ( see aarseth 2003 and heggie & hut 2003 for a more detailed review of previous work ) . at the other end of the spectrum , monte carlo ( mc ) models have proven effective at producing dynamical models of @xmath10particles ( giersz & heggie 2011 ) . these models have shown that clusters previously defined as non - core - collapse can actually be in a fluctuating post - core - collapse phase ( heggie & giersz 2008 ) . in practice the two methods are complimentary with mc informing the more laborious @xmath0-body approach ( such as refining initial conditions ) and @xmath0-body calibrating aspects of mc . in this paper we present an @xmath0-body simulation of star cluster evolution that begins with @xmath1 stars and binaries . this extends the @xmath0 parameter space covered by direct @xmath0-body models and performs two important functions . firstly it provides a new calibration point for the mc method this statistical method is increasingly valid for increasing @xmath0 so calibrations at higher @xmath0 are more reliable . it also allows us to further develop our theoretical understanding of star cluster evolution and investigate how well inferences drawn from models of smaller @xmath0 scale to larger values . the latter is the focus of this current paper . a good example of the small-@xmath0 models that we wish to compare with is the comprehensive study of star cluster evolution presented by giersz & heggie ( 1997 ) using models that included a mass function , stellar evolution and the tidal field of a point - mass galaxy , albeit starting with @xmath7 stars instead of @xmath11 . more recent examples for comparison include baumgardt & makino ( 2003 ) and kpper et al . we were also motivated to produce a model that exhibited core - collapse close to a hubble time without dissolving by that time . what we find when interpreting this model is that much of the behaviour reported previously for smaller @xmath0-body models stands up well in comparison but that the actions of a binary comprised of two black holes ( bhs ) provides a late twist to the evolution of the cluster core . in section 2 we describe the setup of the model . this is followed by a presentation of the results in sections 3 to 7 focussing on general evolution ( cluster mass and structure ) , the impact of the bh - bh binary , mass segregation , velocity distributions and binaries ( binary fraction and binding energies ) . throughout these sections the results are discussed and compared to previous work where applicable . then in section 8 we specifically look at how the evolution timescale of the new model compares to findings presented in the past . for our simulation we used the nbody4 code ( aarseth 1999 ) on a grape-6 board ( makino 2002 ) located at the american museum of natural history . nbody4 uses the 4th - order hermite integration scheme and an individual timestep algorithm to follow the orbits of cluster members and invokes regularization schemes to deal with the internal evolution of small-@xmath0 subsystems ( see aarseth 2003 for details ) . stellar and binary evolution of the cluster stars are performed in concert with the dynamical integration as described in hurley et al . ( 2001 ) . the simulation started with @xmath2 single stars and @xmath3 binaries . we will refer to this as the k200 model . the binary fraction of 0.025 is guided by the findings of davis et al . ( 2008 ) which indicated a present day binary fraction of @xmath12 for the globular cluster ngc@xmath13 , measured near the half - light radius of the cluster . as shown in hurley , aarseth & shara ( 2007 ) and discussed in hurley et al . ( 2008 ) , this can be taken as representative of the initial binary fraction of the cluster . thus we adopted this value for our model . validation of the binary fraction approach will be provided in section 7 . masses for the single stars were drawn from the initial mass function ( imf ) of kroupa , tout & gilmore ( 1993 ) between the mass limits of 0.1 and @xmath14 . each binary mass was chosen from the imf of kroupa , tout & gilmore ( 1991 ) , as this had not been corrected for the effect of binaries , and the component masses were set by choosing a mass - ratio from a uniform distribution . in nbody4 we assume that all stars are on the zero - age main sequence when the simulation begins and that any residual gas from the star formation process has been removed . a metallicity of @xmath15 was set for all stars . the orbital separations of the @xmath3 primordial binaries were drawn from the log - normal distribution suggested by eggleton , fitchett & tout ( 1989 ) with a peak at @xmath16au and a maximum of @xmath17au . orbital eccentricities of the primordial binaries were assumed to follow a thermal distribution ( heggie 1975 ) . for the tidal field of the parent galaxy we have used the point - mass galaxy approach with the model cluster on a circular orbit at @xmath18kpc with an orbital velocity of @xmath19 . in setting @xmath20 we have been primarily guided by a desire for the cluster to have its moment of core - collapse between @xmath21gyr . previous experience suggested that @xmath22kpc would provide this for a model starting with @xmath1 . we used a plummer density profile ( plummer 1911 ; aarseth , hnon & wielen 1974 ) and assumed the stars and binaries are in virial equilibrium when assigning the initial positions and velocities . the plummer profile formally extends to infinite radius so in practice a cut - off at a radius of @xmath23 is applied , where @xmath24 is the half - mass radius . this is to avoid rare cases of large distance in the initial distribution . the tidal field sets a tidal radius according to : @xmath25 where @xmath26 is the gravitational constant and @xmath27 is the cluster mass ( see giersz & heggie 1997 ) . we chose the @xmath0-body length - scale of our model so that the outermost star of the initial model sits at @xmath28 . this reflects the expansion expected when gas leftover from star formation is removed from the potential well ( which we do not model ) , hence our initial model should be more radially extended than a compact protocluster . stars were removed from the simulation when their distance from the density centre exceeds twice that of the tidal radius of the cluster . with these choices the initial parameters of the k200 model were @xmath29 , @xmath30pc and @xmath31pc . the half - mass relaxation timescale of the initial model was @xmath32myr . much of the behaviour of this model will be compared to that reported by hurley et al . ( 2008 ) for a model that started with @xmath33 single stars and @xmath3 binaries . this will be referred to as the k100 model . the k100 model was placed on a circular orbit about a point - mass galaxy at a radial distance of @xmath34kpc . it had the same number of binaries as the k200 model but twice the binary fraction . the parameters of the stars and binaries were set up in the same manner as described above for the k200 model . model orbiting at @xmath18kpc ( solid line ) , the hurley et al . ( 2008 ) @xmath35 ( k100 ) model orbiting at @xmath34kpc ( dashed line ) and a @xmath35 model orbiting at @xmath36kpc ( dotted line ) . [ f : fig1],width=317 ] in figure [ f : fig1 ] we look at the evolution of the total cluster mass with time for the k200 model . this simulation was stopped for analysis at @xmath4gyr having satisfied the goal of providing a post - core - collapse model ( see below ) at the approximate age of a milky way globular cluster . at this point the model cluster has lost @xmath37 of its initial mass . in terms of stars remaining at @xmath4gyr the k200 model has @xmath38 comprised of @xmath39 single stars and @xmath40 binaries . we include the evolution of the k100 model from hurley et al . ( 2008 ) in figure [ f : fig1 ] and see that at @xmath4gyr the two model clusters have the same amount of mass remaining ( after starting with a factor of two difference ) . for comparison we also show in figure [ f : fig1 ] a model that started with @xmath8 stars on the same orbit as the k200 model . as expected this model does not last for a hubble time and is completely dissolved after about @xmath41gyr . the slope in the mass - age plane is similar to that of the k200 model and distinct from that of the k100 model with @xmath34kpc . an investigation of the mass - loss rates and dissolution times of star clusters as a function of orbit within the galaxy will be the subject of another paper ( madrid et al . 2012 ) . figure [ f : fig2 ] shows the behaviour of the core radius , the half - mass radius and the tidal radius as the k200 model evolves . both the half - mass and core radii show an initial increase corresponding to stellar evolution mass - loss from massive stars , which are mostly found in the inner regions of the cluster . the half - mass radius then plateaus before appearing to follow the decreasing trend of the tidal radius at later times . at @xmath4gyr we have @xmath42pc which is comparable to the average effective radius found for globular clusters ( e.g. jordn et al . 2005 ) . the core - radius shows a deep minimum at @xmath5gyr which we identify as the moment that the initial core - collapse phase ends . this is based purely on inspection of figure [ f : fig2 ] , noting that the same method looking for the first deep minimum of the density- or mass - dependent central radius has been commonly employed in the past ( e.g. baumgardt & makino 2003 ; hurley et al . 2004 ; kpper et al . 2008 ) . at this point the core density is @xmath43 , increased from an initial value of @xmath44 . subsequently the core fluctuates markedly , corresponding to post - core - collapse oscillations highlighted by heggie & giersz ( 2009 ) . however , we note that the core exhibits fluctuating behaviour leading up to the moment of core - collapse as well . -body core radius ( + symbols ) , the half - mass radius ( lower solid line ) and tidal radius ( upper solid line ) for the @xmath1 model . all radii are three - dimensional . the vertical dotted line at @xmath5gyr denotes the time we have identified with the end of the initial core - collapse phase . [ f : fig2],width=317 ] the core radius in figure [ f : fig2 ] is the density radius commonly used in @xmath0-body simulations ( casertano & hut 1985 ) , calculated from the density weighted average of the distance of each star from the density centre ( aarseth 2003 ) . following heggie & giersz ( 2009 ) we have also looked at the dynamical core radius used in their monte carlo models , calculated as @xmath45 where @xmath46 is the mass - weighted central velocity dispersion and @xmath47 is the central density , both calculated from the innermost 20 stars . we find that @xmath48 and @xmath49 cover a similar range at all times . in particular , for the period @xmath50gyr , i.e. @xmath6gyr after core - collapse , both oscillate between 0.1 to @xmath51pc for the majority of the time . for reference , the radius containing the inner 1% of the cluster mass is @xmath52pc over this period and is thus a good proxy for the average core radius . also following heggie & giersz ( 2009 ) we have used the autocorrelation method to determine if there is any clear periodicity in the fluctuations of the @xmath0-body core radius in the @xmath6gyr subsequent to core - collapse . this showed a period of about @xmath53myr : greater than the crossing time ( few myr ) and less than the relaxation time ( @xmath54myr ) . in comparison , heggie & giersz ( 2009 ) reported an oscillation period of @xmath55myr for their @xmath0-body model - body model was evolved for @xmath6gyr starting from a post - collapse monte carlo model at an age of @xmath4gyr . ] , although this contained more @xmath0 than our post - collapse k200 model and correspondingly had a higher half - mass relaxation timescale of @xmath56myr . it is of interest to examine how previous @xmath0-body results reported for smaller @xmath0 models hold up in comparison to our new model . the k100 model of hurley et al . ( 2008 ) reached a similar core radius at the end of core - collapse as did our k200 model ( @xmath57pc ) , albeit at a later time ( @xmath58gyr compared to @xmath5gyr ) . as noted above , the average value of @xmath48 near core - collapse is similar to the 1% lagrangian radius for the k200 model . for the k100 model @xmath48 evolves similarly to the 2% lagrangian radius ( although it does dip down to the 1% radius on occasion ) and for the @xmath59 models of giersz & heggie ( 1997 ) @xmath48 is at the 5% lagrangian radius or greater . thus it seems that with increasing @xmath0 the depth of core - collapse increases relative to the cluster mass distribution . if we look instead at scaled quantities , particularly the evolution of the ratio @xmath60 as a function of age scaled by the half - mass relaxation timescale , we find that the k200 and k100 simulations track each other very well . the ratio starts at @xmath61 and steadily decreases to an average value of @xmath62 at the end of core - collapse . giersz & heggie ( 1997 ) find @xmath63 for their models at core - collapse , noting that they use @xmath49 rather than the casertano & hut ( 1985 ) definition and that latter is about twice as large in the post - collapse phase for their models ( we found that the two gave similar average values ) . mcmillan ( 1993 ) performed models with @xmath64 , primordial binaries and a tidal field . for these models @xmath60 stabilized at about 0.1 in good agreement with prior models of isolated clusters ( see mcmillan 1993 for details ) . it therefore seems that this can be taken as a reliable value for clusters at the end of core - collapse ( although see the next section for mention of some unusual cases ) . a remarkable feature in figure [ f : fig2 ] is the sharp change in the behaviour of the core at @xmath65gyr , the last deep minimum , when the size of the core suddenly increases and evolves steadily from that point onwards . this change is related to an interaction within the core involving a binary comprised of two bhs . the binary in question is non - primordial . each bh formed from a massive single star within the first @xmath66myr of evolution with masses of @xmath67 and @xmath68 , respectively . the two bhs formed a binary at @xmath69myr in a four - body interaction , initially with a very high eccentricity and long orbital period of @xmath70d . it resided in the core for the majority of its lifetime and suffered a series of perturbations and interactions that saw the eccentricity vary between 0.2 to 0.95 and the orbital period reduced to @xmath71d at @xmath72myr . at that time the bh - binary becomes embroiled in a strong interaction with a binary comprised of two main - sequence stars ( masses of @xmath73 and @xmath74 ) . the second binary is broken - up and the two main - sequence stars are ejected rapidly from the cluster ( velocities of @xmath75 and @xmath76 ) . this causes a recoil of the bh - binary which leaves the core and then the cluster entirely ( @xmath66myr later with a velocity of @xmath77 ) . the domination of the central region of the cluster by this bh - binary and its subsequent ejection are similar to the processes described by aarseth ( 2012 ) . the sudden loss of mass from the core the average mass drops by 30% ( see next section ) combined with the rapid ejection of the two main - sequence stars causes the core to expand . we see that after this event the core radius does start to decrease once more but without fluctuations . thus , the influence of one strong interaction involving a massive binary has halted the core oscillation process . compared to the point that we identified as the end of the initial core - collapse phase the core radius has increased by a factor of about six . the structure of globular clusters is often quantified by the concentration parameter @xmath78 ( king 1966 ) . milky way gcs exhibit a range of @xmath79 values ( harris 1996 ) with the most obvious core collapse examples having @xmath80 but with @xmath81 generally taken as indicative of a high - density cluster or a possible core - collapse cluster ( mateo 1987 ) . at the end of the core - collapse phase our k200 model has @xmath82 and this decreases to @xmath83 after the bh - binary is ejected from the core . thus , the cluster would not be expected to appear as a core - collapse cluster if observed at this point . hurley ( 2007 ) showed that the presence of a long - lived bh - bh binary in the core , with both bhs being of stellar mass , could significantly increase the @xmath60 ratio of a model with @xmath8 stars . the bh - bh binary in our @xmath1 model has not produced a similar inflation of the ratio . mackey et al . ( 2007 ) performed @xmath0-body simulations with @xmath9 in which they retained @xmath84 stellar mass bhs . they found that the bhs formed a dense core in which interactions were common and bhs could be ejected from the cluster , leading to a significantly inflated core radius . it is our intention in the near future to look at a wide range of @xmath0-body simulations and document in detail the statistics and outcomes of bh - bh binaries in the cores of model clusters . this will include fitting king ( 1966 ) models to the density profiles of the model clusters so as to properly calculate the concentration parameter rather than using @xmath0-body values as we have done here in our preliminary analysis . model . the vertical dotted line marks the end of the initial core - collapse phase as in figure [ f : fig2 ] . [ f : fig3],width=317 ] figure [ f : fig3 ] looks at how the average stellar mass behaves for the k200 model , focussing on four different lagrangian regions : a central volume that encompasses the inner 1% of the cluster mass , a central shell that lies between radii enclosing 1% and 10% of the cluster mass , an intermediate shell that lies between the 10% and 50% lagrangian radii , and an outer shell that includes all stars beyond the 50% lagrangian radius . note that binaries are included and are assumed to be unresolved . the average stellar mass throughout the entire cluster is @xmath85 at the start of the simulation there is no primordial mass segregation and drops initially in all regions owing to stellar evolution mass - loss of massive stars for the first @xmath86myr . we then see that the effect of mass segregation driven by two - body encounters takes over , causing an increase in the average stellar mass in the inner regions and a corresponding decrease in the outer region . by the time that one half - mass relaxation timescale has elapsed ( @xmath87myr ) there is a clear distinction between the average mass in each of the regions . in the central regions the average stellar mass continues to increase up to the end of core - collapse and then flattens post - collapse ( until the ejection of the bh - bh binary ) . the value in the very centre is noisy owing to a smaller number of objects and the cycling of these objects in and out of the region . we see a marked increase in this central value as the cluster gets closer to the end of core - collapse ( note the correlation with @xmath48 in figure [ f : fig2 ] ) . the decreasing average mass in the outer region is gradually arrested by the effect of the external tide which preferentially removes the lower - mass stars that have been pushed out to this region . we see that from @xmath88myr onwards the average mass in the outer region is now increasing and that this effect is also felt in the intermediate region . but now showing the velocity dispersion . [ f : fig4],width=317 ] if we instead look at the behaviour in two - dimensional projected regions we find that the average mass is the same in the two outermost regions but drops by about 5% in the 1 - 10% region and 10 - 15% in the central region ( after the first gyr ) . giersz & heggie ( 1997 ) look at the evolution of average mass in different regions , as we have done in figure [ f : fig3 ] . they see very similar behaviour which can be summarised as : ( i ) a sharp increase of the average mass within the 1% lagrangian region towards core - collapse ; ( ii ) a decrease in the outer regions that flattens out with time and then increases at late times ; and ( iii ) similar trends in the intermediate regions . however , in their @xmath59 models they find that the timescale for the initial phase of mass segregation is about the same as the core - collapse timescale , whereas we find that mass segregation is fully established well before core - collapse . the @xmath89 models of baumgardt & makino ( 2003 ) agree with our k200 model in that respect and with the general behaviour , including the flattening of the average mass in the central regions post - collapse . but now showing the anisotropy parameter . [ f : fig5],width=317 ] in figure [ f : fig4 ] we show the velocity dispersion for the same lagrangian regions as in figure [ f : fig3 ] . all regions show a rapid initial decrease owing to an overall expansion of the cluster . this is followed by a more gradual decrease as the cluster evolves towards core - collapse , with all regions declining in an almost homologous manner . as the model cluster nears the end of core - collapse there is a pronounced upturn in the velocity dispersion within the 1% radius . this is also seen out to the 10% radius and even at the half - mass radius , although to a much smaller extent . note that the behaviour for two - dimensional velocities in projected lagrangian regions is the same but with values that are typically 20% less . the main features of our new model mirror those in the smaller-@xmath0 models of giersz & heggie ( 1997 ) . these features are the following : the values within the 1% and 10% regions overlap ; the velocity dispersion in the outer regions is clearly lower than for these inner regions ; and the values for the 50% region are much closer to those of the inner regions than the outer regions . another aspect of velocity to consider is the anisotropy . we have chosen to define this as the ratio of the mean square transverse to radial velocity components , @xmath90 ( as in giersz & heggie 1997 ) and the result is shown in figure [ f : fig5 ] . in three dimensions this is equal to two for isotropy , greater than two for a tangentially anisotropic distribution and less than two for a radially anisotropic distribution . an alternative is to use the anisotropy parameter @xmath91 ( binney & tremaine 1987 ; baumgardt & makino 2003 ; wilkinson et al . 2003 ) where @xmath92 is isotropic , @xmath93 is tangentially anisotropic and @xmath94 is radially anisotropic . we see from figure [ f : fig5 ] that the inner regions of the cluster remain close to isotropic throughout the evolution while the outer region develops an increasing tangential anisotropy over the first @xmath95gyr and then flattens out to a constant value for the remainder of the evolution . this overall behaviour is similar to that observed by baumgardt & makino ( 2003 ) in their models . it is also similar to the behaviour for the k100 model , although the degree of anisotropy in the outer region is about 30% less than in the k200 model . also , because the k100 model is evolved well past core - collapse it is possible to see that the anisotropy is reduced post - collapse and tends towards isotropy in the final stages of evolution ( as was noted by baumgardt & makino 2003 ) . the tangential anisotropy in the outer region is contrary to previous findings of radial anisotropy for smaller-@xmath0 models ( giersz & heggie 1997 ; wilkinson et al . tangential anisotropy can be explained by stars expelled from the central regions on radial orbits preferentially escaping from the cluster at the expense of stars on tangential orbits which find it harder to escape . in the very central region ( within the 1% lagrangian radius ) the anisotropy parameter is very noisy and fluctuates between radial and tangential anisotropy . however , the average behaviour is isotropy . model . shown are the binary fraction of the entire cluster ( dashed line ) , within the 10% lagrangian radius ( black dotted line ) and within the core ( red solid line ) . also shown is the binary fraction of main - sequence stars and binaries near the half - mass radius ( red dotted line : which closely follows the binary fraction of the entire cluster ) . compare with figure 3 of hurley , aarseth & shara ( 2007 ) . the vertical dotted line once again marks the end of the initial core - collapse phase as in figure [ f : fig2 ] . [ f : fig6],width=317 ] hurley , aarseth & shara ( 2007 ) documented the evolution of binary fraction with time for a range of @xmath0-body models covering @xmath96 to @xmath8 and primordial binary fractions from 0.05 to 0.5 . in all cases they found that the binary fraction within the cluster core and the 10% lagrangian radius increases as the cluster evolves towards core - collapse while the overall binary fraction of the cluster stays close to the primordial value . figure [ f : fig6 ] shows the same evolution for the k200 model . similar results are seen in that the core binary fraction increases markedly as the cluster evolves , particularly towards core - collapse when it becomes a factor of six or more greater than the primordial value , while the overall binary fraction does stay close to the primordial value ( although the inclusion of a large proportion of initially wide binaries would lead to an initial decrease ) . this gives added confidence in the results of hurley , aarseth & shara ( 2007 ) although models of greater @xmath0 ( and density ) are still required before the situation for the majority of the milky way globular clusters can be firmly established . davis et al . ( 2008 ) measured the binary fraction amongst main - sequence stars near the half - mass radius of ngc@xmath13 and found it to be a few per cent . this was taken as representative of the primordial binary fraction of ngc@xmath13 by leaning on the result from hurley et al . ( 2008 ) showing that the binary fraction measured near the half - mass radius of a cluster can be taken as a good indication of the primordial binary fraction . this is reinforced in figure [ f : fig6 ] for our k200 model where we show that the binary fraction of main - sequence stars and binaries near the half - mass radius of the cluster stays at roughly the same value throughout the evolution . the vertical dotted line denotes the time identified with the end of the initial core - collapse phase for this model . [ f : fig7],width=317 ] but now for the k100 model . the vertical dotted line denotes the time identified with the end of the initial core - collapse phase for this model . [ f : fig8],width=317 ] next we look at the energy in binaries for the k200 simulation in figure [ f : fig7 ] and for the k100 simulation in figure [ f : fig8 ] . here we see that for the @xmath97% lagrangian mass regions the values and behaviour for the two simulations are very similar . as expected the average energy per binary increases as we move inwards towards the cluster centre , although the values within the 1 - 10% and 10 - 50% regions are converging towards the end of the simulation . we note that the binary binding energy is calculated in arbitrary yet physical units of @xmath98 to enable direct comparison between the internal energies of the two binary populations . conversion to cluster units of @xmath99 t based on the average kinetic energy of the cluster stars ( e.g. mcmillan 1993 ) increases the k100 values by a factor of three compared to the k200 values , i.e. the binaries have relatively more internal energy in the simulation with less stars . sharp dips in the binding energy occur throughout the simulations when binaries escape ( as documented by kpper , kroupa & baumgardt 2008 ) although these are less evident in figure [ f : fig8 ] owing to less frequent sampling . the energy per binary in the inner 1% region by mass starts off similarly for both simulations , oscillating about the 1 - 10% values . however , the influence of the bh - binary in the k200 simulation from @xmath100gyr onwards is clear to see , increasing the average energy by almost two orders of magnitude before a sharp drop when the binary is removed from the core . noticeable increases in binding energy are also evident at other times and arise from the creation of tight binaries by various means . for example , the spike in the 50 - 100% lagrangian region at @xmath101myr in figure [ f : fig7 ] results from a primordial binary in which common - envelope evolution creates a short - period binary comprised of two white dwarfs which subsequently come into contact and merge . thus the binary dominates the energy in this region for a short time and then disappears . we see a more persistent increase in the average energy within the 1% region at @xmath102myr in figure [ f : fig7 ] . this is caused when a main - sequence star and a black hole form a short - period binary via an exchange interaction . the binary survives for about @xmath103myr in the cluster core before the main - sequence star is consumed by the black hole . it is worthwhile to ask if the signatures of these short - lived energetic binaries be observed ? the binary that resulted in the merger of two wds did not produce a wd with a mass in excess of the chandrasekhar limit so could not be a potential type ia supernova ( see shara & hurley 2002 ) . however , the resulting wd will be relatively massive , hot and _ young_. thus it could be expected to remain one of the brightest wds in the cluster for several gyrs and would be easily observed . the other binary mentioned resulted in a main - sequence star being tidally disrupted and swallowed by a black - hole of @xmath104 . this could be expected to give rise to a burst of x - rays and perhaps gamma - rays , along the lines of burrows et al . ( 2011 : albeit for a supermassive black hole ) . . comparison of core - collapse , @xmath105 , and relaxation timescales , @xmath106 , for the k200 and k100 simulations . note that the averaged @xmath106 is calculated to @xmath107gyr and @xmath108gyr for the k200 and k100 simulations , respectively . also shown is the time for the cluster to lose half of the initial mass , @xmath109 . [ t : table1 ] [ cols="<,>,>",options="header " , ] timescales for star cluster evolution are important to understand , with the time until core - collapse and the time until dissolution being quantities of interest ( e.g. gnedin & ostriker 1997 ) . furthermore , the scaling of these with @xmath0 or related properties of clusters / models is necessary ( baumgardt 2001 ) , particularly while direct models of globular clusters remain out of reach . in table [ t : table1 ] we summarize the significant timescales for the k200 and k100 simulations : the time for half of the initial mass to be lost , the time until the end of the core - collapse phase and the half - mass relaxation time at various key points . kpper et al . ( 2008 ) presented a range of open cluster models in a steady tidal field and found that the core - collapse time scaled by the initial half - mass relaxation time ranged from 17 for clusters starting well within their tidal radii ( smaller @xmath110 ) to 9 for clusters that fill their tidal radii from the start ( larger @xmath110 ) . our k200 model and the k100 model of hurley et al . ( 2008 ) start by filling their tidal radius and have @xmath111 which is in good agreement with kpper et al . ( 2008 ) . baumgardt ( 2001 ) looked at the scaling of @xmath0-body models using simulations of up to @xmath112 equal - mass stars . in this work the time for a cluster to lose half of its mass , @xmath109 , was taken as an indication of the cluster lifetime with @xmath113 shown to be the appropriate scaling . baumgardt & makino ( 2003 ) subsequently used their larger-@xmath0 models ( up to @xmath114 stars ) to once again show that scaling the dissolution time as @xmath115 is appropriate , this time for multi - mass models that included stellar evolution . however , we see from table [ t : table1 ] that the half - mass relaxation time varies considerably across the lifetime of a cluster and it is not immediately clear which value to use when scaling timescales . for an observed cluster it will be the value at the current age of the cluster . a fairer comparison would be to use the average half - mass relaxation time across the lifetime of the cluster , @xmath116 of course this is not known for an observed cluster . for the k200 and k100 models we find that @xmath117 is the same for both simulations , so the agreement for this key timescale is in excellent agreement with the previous suggestion . we also find that @xmath109 scales quite well with @xmath118 . however , if we instead use @xmath119 then we do not find good agreement for either the core - collapse or half - life timescales . thus our agreement with the scalings found in previous works is dependent on which @xmath106 is used . we have presented an @xmath0-body model that started with @xmath11 stars and binaries , evolves to the moment of core - collapse at @xmath5gyr and has @xmath120 stars remaining at @xmath4gyr . we have used our direct @xmath0-body model to confirm the post - core - collapse fluctuations described in the monte carlo model of heggie & giersz ( 2008 ) and the hybrid @xmath0-body / mc approach of heggie & giersz ( 2009 ) . we have also shown that these fluctuations can be halted by the ejection of a dominant bh - binary from the core . this produces a core that shows no sign that it has previously evolved through core - collapse . we have looked at how the results of previous works compare to a model of larger @xmath0 and find good agreement provided that appropriate scalings are used ( such as the core - radius to half - mass radius ratio at core - collapse ) . in terms of raw values some variations exist : the core radius at core - collapse reaches deeper in to the mass distribution for larger @xmath0 , for example . the behaviour of quantities such as average stellar mass and velocity dispersion have been documented and the general behaviour matches expectations from earlier models . looking at time scales such as the time to core - collapse and the dissolution time we also find agreement with scaling relations previously reported in the literature , however this is dependent on which value of the half - mass relaxation timescale is used . in particular , the scaling of dissolution time with @xmath115 reported by baumgardt ( 2001 ) could be reproduced provided that the average @xmath106 was used and not the initial @xmath106 . the @xmath1 simulation reported in this work took the best part of a year on a grape-6 board to complete . it continues the gradual increase of @xmath0 used in realistic @xmath0-body models from the @xmath59 model of giersz & heggie ( 1997 ) to the @xmath89 model of baumgardt & makino ( 2003 ) . however , with only @xmath121 remaining in our model at an age of @xmath107gyr we are still only touching the lower end of the globular cluster mass - function . this is after considerable effort . in particular , we are still some way from the goal of a full million - body model of a globular cluster ( heggie & hut 2003 ) . how can we push forward to reach that goal ? the shift towards graphics processing units ( gpus ) as the central computing engine for @xmath0-body codes , combined with sophisticated software development , offers hope ( nitadori & aarseth 2012 ) . simulations of @xmath8 stars can be performed comfortably on a single - gpu ( hurley & mackey 2010 ; zonoozi et al . 2011 ) and the introduction of multiple - gpu support will likely make simulations of the type presented here commonplace in the near future . further hardware advances and a revisiting of efforts to parallelize direct @xmath0-body codes ( spurzem 1999 ) will also aid the push towards greater @xmath0 . in a follow - up paper we will conduct a full investigation of the stellar and binary populations of our model . it is also our intention to make model snapshots saved at frequent intervals across the lifetime of the simulation available for others to _ observe _ and analyse . these can be obtained by contacting the authors . at the beginning of this paper we indicated that an important function of a large-@xmath0 model would be to aid in the calibration of the monte carlo technique . this is currently underway ( giersz et al . 2012 ) and will include a direct comparison of @xmath0-body and mc models starting from the same initial conditions . we acknowledge the generous support of the cordelia corporation and that of edward norton which has enabled amnh to purchase grape-6 boards and supporting hardware . we thank harvey richer for helping to provide the motivation for this model and ivan king for many helpful suggestions . aarseth s. j.,1999 , pasp , 111 , 1333 aarseth s.j . , 2003 , gravitational n - body simulations : tools and algorithms ( cambridge monographs on mathematical physics ) . cambridge university press , cambridge aarseth s. j. , 2012 , mnras , 422 , 841 aarseth s. , hnon m. , wielen r. , 1974 , a&a , 37 , 183 baumgardt h. , 2001 , mnras , 325 , 1323 baumgardt h. , makino j. , 2003 , mnras , 340 , 227 binney j. , tremain s. , 1987 , galactic dynamics . princeton university press , princeton burrows d.n . , et al . , 2011 , nature , 476 , 421 casertano s. , hut p. , 1985 , apj , 298 , 80 davis d.s . , richer h.b . , anderson j. , brewer j. , hurley j. , kalirai j. , rich r.m . , stetson , p.b . , 2008 , aj , 135 , 2155 eggleton p.p . , fitchett m. , tout c.a . , 1989 , apj , 347 , 998 giersz m. , heggie d.c . , 1997 , mnras , 286 , 709 giersz m. , heggie d. c. , 2011 , mnras , 410 , 2698 giersz m. , heggie d.c . , hurley j. , hypki a. , 2012 , mnras ( arxiv:1112.6246 ) gnedin o.y . , ostriker j.p . , 1997 , apj , 474 , 223 harris w.e . , 1996 , aj , 112 , 1487 heggie d.c . , 1975 , mnras , 173 , 729 heggie d.c . , 2011 , bulletin of the astronomical society of india , 39 , 69 heggie d.c . , giersz m. , 2008 , mnras , 389 , 1858 heggie d.c . , giersz m. , 2009 , mnras , 397 , l46 heggie d.c . , hut p. , 2003 , the gravitational million body problem . cambridge university press , cambridge hurley j. r. , tout c. a. , aarseth s. j. , pols o.r . , 2001 , mnras , 323 , 630 hurley j. r. , tout c. a. , aarseth s. j. , pols o.r . , 2004 , mnras , 355 , 1207 hurley j.r . , 2007 , mnras , 379 , 93 hurley j.r . , aarseth s.j . , shara m.m . , 2007 , mnras , 665 , 707 hurley j.r . , shara m.m . , richer h.b . , king i.r . , davis s.d . , kalirai j.s . , hansen b.m.s . , dotter a. , anderson j. , fahlman g.g . , rich r.m . , 2008 , aj , 135 , 2129 hurley j.r . , mackey a.d . 2010 , mnras , 408 , 2353 jordn a. , et al . , 2005 , apj , 634 , 1002 king i.r . , 1966 , aj , 71 , 64 kroupa p. , tout c. a. , gilmore g. , 1991 , mnras , 251 , 293 kroupa p. , tout c. a. , gilmore g. , 1993 , mnras , 262 , 545 kpper a.h.w . , kroupa p. , baumgardt h. , 2008 , mnras , 389 , 889 mackey a.d . , wilkinson m.i . , davies m.b . , gilmore g.f . , 2007 , mnras , 379 , l40 madrid j.p . , hurley j.r . , sippel a.c . , 2012 , apj , accepted ( arxiv:1208.0340 ) makino j. , 2002 , in shara m.m . , ed , asp conference series 263 , stellar collisions , mergers and their consequences . asp , san francisco , p. 389 mateo m. , 1987 , apj , 323 , l41 mcmillan s.l.w . , 1993 , in djorgovski s. , meylan g. , eds , asp conference series 50 , dynamics of globular clusters . asp , san francisco , p. 171 mcmillan s. , hut p. , makino j. , 1990 , apj , 362 , 522 nitadori k. , aarseth s. j. , 2012 , mnras , 424 , 545 plummer h.c . , 1911 , mnras , 71 , 460 portegies zwart s.f . , mcmillan s.l.w . , hut p. , makino j. , 2001 , mnras , 321 , 199 shara m.m . , hurley j.r . , 2002 , apj , 571 , 830 spurzem r. , 1999 , journal of computational and applied mathematics , 109 , 407 vesperini e. , heggie d.c . , 1997 , mnras , 289 , 898 von hoerner s. , 1963 , z. astrophys . , 57 , 47 wilkinson m.i . , hurley j.r . , mackey a.d . , gilmore g.f . , tout c.a . , 2003 , mnras , 343 , 1025 zonoozi a.h . , kpper a.h.w . , baumgardt h. , haghi h. , kroupa p. , hilker , m. , 2011 , mnras , 411 , 1989

we report on the results of a direct @xmath0-body simulation of a star cluster that started with @xmath1 , comprising @xmath2 single stars and @xmath3 primordial binaries . the code used for the simulation includes stellar evolution , binary evolution , an external tidal field and the effects of two - body relaxation . the model cluster is evolved to @xmath4gyr , losing more than 80% of its stars in the process . it reaches the end of the main core - collapse phase at @xmath5gyr and experiences core oscillations from that point onwards direct numerical confirmation of this phenomenon . however , we find that after a further @xmath6gyr the core oscillations are halted by the ejection of a massive binary comprised of two black holes from the core , producing a core that shows no signature of the prior core - collapse . we also show that the results of previous studies with @xmath0 ranging from @xmath7 to @xmath8 scale well to this new model with larger @xmath0 . in particular , the timescale to core - collapse ( in units of the relaxation timescale ) , mass segregation , velocity dispersion , and the energies of the binary population all show similar behaviour at different @xmath0 . [ firstpage ] stars : evolution globular clusters : general galaxies : star clusters : general methods : numerical binaries : close stars : kinematics and dynamics

You are an expert at summarizing long articles. Proceed to summarize the following text: observations by @xmath3 of binary systems have so far yielded 11 transiting circumbinary planets . until recently , all discovered circumbinary planets have resided near to or outside the dynamical stability limit characterized by @xcite . this finding has prompted many to study how planets form in such systems and why they seem biased towards lying at the brink of dynamical instability @xcite . simulations and theoretical arguments by @xcite and @xcite rule out in situ formation in the inner edge of the disc near where several circumbinary planets have been observed . currently , one of the most successful models in explaining the circumbinary planet population is planetary migration . in this model , a planet , having formed farther out in the disc , migrates inward through viscous interactions with a gaseous disc , potentially undergoing planet - planet scattering , until it reaches near its observed location . several previous studies applied migration to observed circumbinary planetary systems and have been able to show that for certain disc and viscous drag models , the observed planet orbits are nearly recovered ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? however , the recent discovery by @xcite of the first long - period transiting circumbinary planet complicates this picture . the newly discovered planet candidate , koi-2939b , is a jupiter - sized exoplanet on a roughly 3 year orbit that suggestively lies in the conservative habitable zone of two short - period g dwarfs @xcite . the existence of a such a system shows that not all circumbinary planets are driven , either through migration or other mechanisms , inward towards the dynamical stability limit . this has important consequences for not only how planets migrate and evolve in a viscous circumbinary disc , but also for how and where these planets form within the protoplanetary disc . clearly , characterizing the influence exerted on the planet by the protoplanetary disc is important for understanding how such systems form . additionally , the influence of the central binary on the external disc causes the disc to evolve and potentially undergo large scale changes . previous studies of both protoplanetary discs and accretion discs around binary supermassive black holes ( smbhs ) found that circumbinary discs can become eccentric , precess , and have density waves excited from resonances @xcite . specifically in the context of planetary systems , a hydrodynamic theory presented by @xcite showed that nonlinear coupling mediated by density waves launched at eccentric lindblad resonances causes disc eccentricity growth . much theoretical work in this area has focused on how planetesimals can grow and evolve in circumbinary discs . @xcite found that for planetesimals on circumbinary orbits , in situ formation proves quite difficult , suggesting that planets form far out in the disc and subsequently migrate inward . more recently , @xcite showed that planetesimals on the most circular orbit about the central binary can attain small relative velocities , facilitating their growth . with a model that considers both the gravity and gas drag of a precessing , eccentric circumbinary disc , @xcite found that the inner radius for @xmath4 m planetesimal growth depends on the disc eccentricity , density and precession rate . the binary s influence on the disc is not without a cost , however , as the disc also drives changes in the binary orbital elements , which as they evolve , can change how the binary influences the disc . early work on how binary systems interact with accretion discs by @xcite studied jupiter s interaction with the sun s protoplanetary disc . this study showed that a satellite s orbital eccentricity could be increased through energy and momentum transfers with the disc at lindblad resonances , causing significant changes over a few thousand years . the case of accretion discs around two objects was explored by @xcite and later by @xcite . @xcite found that a central binary s interaction with an external accretion disc can decrease binary separation and change binary eccentricity , depending on disc structure . in the context of a jupiter to brown dwarf mass companion orbiting a central star , @xcite found that for sufficiently massive companions , a coupling between small initial disc eccentricity and the companion s tidal potential excited an @xmath5 wave from the 1:3 eolr causing further disc eccentricity growth . wave excitation at the 1:3 eolr can also couple with the interaction between the eccentric disc and the companion to induce orbital eccentricity growth of the companion . many recent and past works explored the more general case of binary stars evolving under the influence of a gaseous circumbinary disc . simulations of unequal mass binaries embedded in a protoplanetary disc by @xcite show rapid binary eccentricity growth and semimajor axis decay due to interactions with the 1:3 eolr and the viscous disc . subsequent theoretical work by @xcite and @xcite derived relations to show how resonant and viscous interactions drive @xmath6 and @xmath7 . binary orbital element and disc evolution has also been explored in systems with circumbinary gaseous discs containing migrating planets in simulations by @xcite who also found binary and disc eccentricity growth . characterizing disc - binary interactions are important on much larger scales , as well . simulations by @xcite show that eccentric supermassive black hole binaries can rapidly form from the merger between two spiral galaxies . an external disc forms exterior to the smbh binary as interactions with the disc and external gas clouds can cause the black hole separation to decrease @xcite . numerous studies have been conducted to explore how binary smbh - disc interactions cause binary smbh eccentricity growth and semimajor axis decay , possibly explaining the last parsec " problem allowing smbh binaries to coalesce ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? extensive work has also been made to explore accretion onto binary smbhs , how it varies with both the binary and disc properties and what impact accretion has on the dynamical properties of the system ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) . for circumbinary systems , the binary and its disc are intertwined in non - trivial ways . the coevolution of such systems depends strongly on resonant interactions that can impart significant changes on both the binary and disc . previous simulations focused on studying either disc evolution or binary evolution , often making approximations such as holding the binary orbital elements fixed . in this work we seek to explore how the coevolution of the disc and binary proceeds by allowing all particles to gravitationally interact . using the kepler 38 binary as our model system , we present the results of n - body sph simulations of unequal mass binaries of variable initial eccentricity that probe the dynamical coupling of binary stars and a circumbinary disc . we show that the initial eccentricity of the binary dictates how strongly the disc - binary system resonantly couples . the strength of this coupling in turn dictates how eccentricity grows in either the disc , the binary or both and how structure forms within the disc . the simulations described in this paper were performed in 3d using the massively parallel n - body and smooth particle hydrodynamics ( sph ) code , changa @xcite . changa , implemented in charm++ @xcite , uses a modified version of the barnes - hut tree algorithm with hexadecapole order multiples and a node opening criterion of @xmath8 for fast and accurate calculation of gravitational forces . the euler equations which describe the gas dynamics of the simulated circumbinary disc were solved using an sph implementation based on @xcite . changa uses a multistepping algorithm that gives each particle its own timestep to ensure sufficient dynamical resolution @xcite . artificial viscosity was implemented using monaghan viscosity @xcite . the viscosity @xmath9 and @xmath10 parameters were set to 1 and 2 , respectively . the balsara switch was used to limit shear viscosity @xcite . for additional information about the implementation and performance of changa see @xcite . .kepler 38 parameters adapted from @xcite . [ cols="^,^,^,^",options="header " , ] [ tab : table_2 ] as the simulation evolves , the time varying gravitational force of the binary truncates the inner edge of the circumbinary disc and excites various lindblad resonances within the disc . the gap , cleared quickly in about 100 binary orbits , is preserved by a balance of resonant and viscous torques within the disc @xcite . for larger gaps , corotation and lindblad resonances can fall within the evacuated region removing their influence from the system . these resonances , especially those closest to the inner edge of the disc , can drive evolution in the binary s orbital elements @xcite and pump eccentricity in the disc @xcite . therefore , in order to understand the subsequent dynamical evolution of both the binary and the disc , the inner disc edge structure must be understood to see which resonances may play a role . the approximate size of the gaps found in these simulations is @xmath11 with more eccentric binaries producing larger gaps , in good agreement with the results of @xcite . our results are also consistent with the findings of @xcite who show that for binary mass ratios above @xmath12 as is the case for our simulations , a hollow central cavity forms around the secondary within the circumbinary disc . gaps of this size tend to remove the eccentricity - damping 1:2 corotation resonance while leaving the 1:3 eolr as the closest to the inner disc edge at a radius of @xmath13 suggesting that this resonance dominates the evolution of the binary s eccentricity and semimajor axis @xcite . to verify the absence of the 1:2 corotation resonance and prominence of the 1:3 eolr , the surface density profiles for simulations @xmath14 are shown in fig . [ fig : figure1 ] after 200 years of evolution . for all @xmath15 , the surface density at the 1:3 eolr is at least an order of magnitude larger than at the 1:2 corotation resonance when it is present within the disc . since the gap opens rather quickly , in of order 5 years for the kepler 38 binary , this finding supports the hypothesis that the 1:3 eolr will be the dominant resonance within the disc that will drive subsequent evolution within the system . the structure and eccentricity of the protoplanetary disc was examined over @xmath1 binary periods in over 230 snapshots for each simulation to track how it evolves with the central binary . a central binary excites eccentricity in the surrounding circumbinary disc via resonant gravitational interactions @xcite . previous simulations of circumbinary discs @xcite found that disc eccentricity increases due to interactions that occur at the 1:3 eolr . to explain why the disc becomes eccentric , @xcite showed that disc eccentricity growth occurs via a parametric instability driven by coupling between the binary s tidal potential and a disc @xmath2 mode due to a small initial disc eccentricity . this coupling excites an @xmath5 spiral wave from the 1:3 eolr that removes angular momentum from the disc at constant energy making the gas orbits eccentric . the material at the 1:3 eolr rotates more slowly that the orbital pattern speed allowing the resonant torques to grow eccentricity in the system through the @xmath5 wave . we therefore expect our circumbinary discs to become eccentric as well . to explore this effect in our simulations , we computed the disc eccentricity via a mass average following the prescription of @xcite @xmath16 where @xmath17 is the local surface density and the integral was evaluated out to a radius of 3 au over 50 radial bins . the disc eccentricity in each radial bin was taken to be the mass - weighted average of the eccentricity of all gas particles within the bin assuming the particles orbit the system s barycenter . we neglected the influence of gas pressure in this calculation . for all simulations , disc eccentricity growth is observed and consistent with the results of the similar gaseous circumbinary disc simulations of @xcite , @xcite and @xcite . the disc eccentricity change over time for each simulation is shown in fig . [ fig : figure2 ] . for @xmath18 , significant disc eccentricity growth occurs . after about 500 years , the disc reaches eccentricities of about 0.1 while continuing to grow linearly . for larger initial @xmath15 up to @xmath19 , less eccentricity growth occurs indicating that more eccentric binaries tend to produce less eccentric discs . for non - zero @xmath15 , the disc eccentricity grows linearly with a periodic modulation . the period of this disc eccentricity oscillation is similar to the inner disc edge clump precession timescale discussed in section [ discstructevolution ] below suggesting that the clump impacts the disc s eccentricity modulation , but only when the binary is sufficiently eccentric . this finding is consistent with the work of @xcite who explain that the inner disc edge precesses with a period of @xmath20 binary periods when @xmath21 . we find that the global disc eccentricity growth is predominantly due to the inner edge of the disc becoming eccentric . [ fig : figure3 ] shows the disc eccentricity versus radius for simulations 1 - 5 after about 520 years of evolution . for binaries with @xmath18 , disc inner edge eccentricities are of order 0.4 while more eccentric binaries tend to produce less eccentric inner edges . to understand the eccentricity growth of our circumbinary discs , specifically why less eccentric binaries tend to produce more eccentric discs , we turn to the theory of @xcite . @xcite explains that a nonlinear coupling between the binary and a small initial disc eccentricity excites an @xmath5 wave from the 1:3 eolr within the disc with a resonant forcing pattern speed @xmath22 for binary orbital frequency @xmath23 . this wave transports angular momentum outwards , driving eccentricity growth in the system . we confirmed the presence of the @xmath5 spiral wave originating from the 1:3 eolr in our simulations via a fourier transform over azimuthal angle of the disc surface density . this wave removes angular momentum from the disc , accounting for the increase in disc eccentricity . disc eccentricity growth as a function of @xmath15 depends on how strongly the binary couples to the disc . when strong coupling occurs , both the binary and inner disc edge grow to similar eccentricities while weak coupling results in the less massive of the binary or the disc inner edge developing appreciable eccentricity . for initially circular binaries , @xcite described strong coupling as occurring when the disc mass within the gap radius is comparable to the mass of the secondary . for our simulations , the mass of the secondary is roughly an order of magnitude larger than the mass of the entire disc , so we expect the binary to be weakly coupled to the disc resulting in significant disc eccentricity growth as is observed in our simulations ( see fig . [ fig : figure2 ] and fig . [ fig : figure3 ] ) . for initially eccentric binaries , the @xcite strong coupling criterion does not apply . instead , we note that the time - averaged orbit of an eccentric binary corresponds to an azimuthal @xmath2 mode perturbation to a circular orbit . this @xmath2 mode couples to the @xmath2 mode of the eccentric inner disc edge , placing the system in the strong coupling regime causing the disc eccentricity and @xmath15 to grow to similar magnitudes , as is observed in our simulations ( see fig . [ fig : figure2 ] and fig . [ fig : figure4 ] ) . we discuss these coupling mechanisms and their implications in more detail in sections [ discstructevolution ] and [ binaryevolution ] . the @xmath24 and @xmath25 simulations show larger disc eccentricity than the @xmath26 case in contrast to expected behavior . for @xmath25 , this can be understood as intermediate coupling . the initial binary eccentricity is not low enough to conform exactly to the @xcite criterion and is not large enough to couple strongly to the disc inner edge resulting in an intermediate coupling with larger disc eccentricity growth than the @xmath26 case and also appreciable @xmath15 growth that is still less than the @xmath26 case ( see section [ binaryevolution ] ) . for the @xmath24 case , the binary eccentricity may be large enough that higher order resonances in the disc begin to impact the evolution @xcite , potentially accounting for the system s departure from expected behavior . the gravitational influence of the binary forces several major changes within the structure of the circumbinary disc . to explore how the structure of the disc changes with time and @xmath15 , we examined the orbits of gas particles and how they vary with distance from the binary . to accomplish this , two - dimensional histograms of all gas particles within a radial distance of 3 au from the barycenter were made for each snapshot . we computed the histograms over semimajor axis , @xmath27 , and the longitude of periastron , @xmath28 , defined in this work as the sum of the argument of periastron , @xmath23 , and the longitude of the ascending node , @xmath29 , relative to that of the binary , @xmath30 , for each gas particle . [ fig : figure5 ] and fig . [ fig : figure6 ] shows these histograms for all particles in the disc out to 3 au for several of our simulations . in all simulations , a precessing overdense knot was found just outside inner edge of the disc . the knot corresponds to a coherent precession of eccentric gas particle orbits at the inner disc edge . the knot depicted in fig . [ fig : figure5 ] precesses relative to the binary in the prograde sense with a period of about 20 years , or 400 binary periods , for @xmath18 . fig . [ fig : figure5 ] shows the @xmath31 histogram for two simulations with initial @xmath15 of 0 and 0.1032 , respectively after 200 years of evolution . precession of orbits near the binary at the inner disc edge are expected due to the binary s time - varying potential as shown in the simulations and analytic theory of @xcite . the existence of this knot is consistent with the identification of a similar overdense lump located at the inner edge of gaseous circumbinary accretion discs about binary black holes from 2d simulations by @xcite . an additional structure identified in the disc is a single arm ( @xmath2 ) spiral wave launched from near the inner edge of the disc as shown in fig . [ fig : figure5 ] . the spiral wave is an alignment of gas particle longitude of periastrons relative to the binary s . for the initial @xmath26 simulation , the wave develops rapidly within the first 50 years , or about @xmath32 binary periods . recent n - body simulations of circumbinary planetesimal discs by @xcite confirmed the presence of an @xmath2 wave discs about eccentric binaries . when under the influence of an asymmetric gaseous disc potential , @xcite identified the wave as a preferential alignment of planetesimal longitude of periastrons as a function of orbital radius around the eccentric binary of the kepler 16 system , similar to the spiral wave found in this work . in our simulations , the spiral wave s orientation remains locked to the binary s slow prograde @xmath30 precession throughout the entire simulation , although a slight drift of a degree or so does occur . the wave s fixed orientation relative to the binary is an important effect whose consequences will be examined more carefully in section [ binaryevolution ] . simply put , if the wave did circulate relative to the binary , it s long term effect on the system , if any , would average out to zero , so the fixed orientation could indeed dynamically impact the system . note that in the simulation with an initial @xmath33 ( left panel ) , no spiral @xmath2 arm exists in contrast to the @xmath26 simulation ( right panel ) which shows a prominent spiral arm . the arm is also observed in the simulation with initial @xmath24 but not with initial @xmath34 . since the spiral arm is only observed when the binary has an appreciable eccentricity , we can infer that a coupling between binary eccentricity and the inner disc edge impacts its formation . to investigate the role of binary eccentricity in exciting the wave , simulation 3 was ran with initial @xmath25 to see if the spiral wave could be excited with an intermediate @xmath15 between the two regimes identified above . in the intermediate regime as explained in section [ disceccevolution ] , @xmath15 is not large enough to be strongly coupled to the disc while also not low enough to weakly couple to the disc to drive disc eccentricity . from the onset of the initial @xmath25 simulation , a faint @xmath2 spiral wave appeared and gradually strengthened as shown in fig . [ fig : figure6 ] . the wave , initially weak , became more apparent after about 900 years . the @xmath2 spiral arm in this simulation does not become as pronounced as the one seen in the initial @xmath26 simulation , suggesting that the strength of the arm depends on @xmath15 and supporting the notion that this disc - binary system undergoes an intermediate coupling . we again apply the theory of @xcite to understand the origin and behavior of the spiral wave . we know from section [ disceccevolution ] that a nonlinear coupling between non - zero disc eccentricity and the binary s tidal potential excites an @xmath5 spiral density wave from the 1:3 eolr that mediates angular momentum transfer in the system . also , we have shown that the strength of the coupling between the binary and disc , which depends on @xmath15 , determines the magnitude of the disc inner edge eccentricity . additional structural changes within the disc proceed via a higher order coupling . in @xcite , the authors show that the @xmath35 density wave emitted at the 1:3 eolr can couple back through the binary tidal potential . this additional coupling produces a time independent @xmath2 wave and an associated potential . the extra potential from the @xmath2 wave can allow for the removal of angular momentum from the system via resonant torques . the @xmath2 wave produced via the recoupling mechanism is precisely the @xmath2 spiral wave identified in this work . the presence of the @xmath2 wave was reconfirmed via a fourier decomposition of the disc surface density . since we only observe the @xmath2 spiral wave in discs around eccentric binaries , we infer that this recoupling mechanism only occurs when the disc and binary are strongly coupled . @xcite s observation of a @xmath2 spiral wave present in their simulation of a planetesimal disc surrounding the eccentric kepler 16 binary support this argument . since the orientation of the @xmath2 wave in our simulations remains locked to that of the binary s , it does not circulate and hence is independent of time . also , as we will explore in section [ binaryevolution ] , the additional potential from the @xmath2 resonantly torques on the binary causing evolution in its orbital eccentricity . next we examine why the orientation of the spiral wave remains fixed relative to the binary . we apply the analytic theory for circumbinary orbits of @xcite to partially explain this effect . the theory , accurate to first order in @xmath15 , decomposes the orbit of a test particle about two stars into a superposition of the circular motion of a guiding center and the radial and vertical epicyclic motion due to the non - axisymmetric components of the binary s potential . @xcite give the equations for the precession rate of the argument of periastron , @xmath23 , and the longitude of the ascending node , @xmath29 , respectfully , to be @xmath36 @xmath37 where @xmath38 and @xmath39 are the masses of the primary and secondary stars , respectively , @xmath40 is the binary semimajor axis and @xmath41 is the radial distance from the barycenter . since these rates are approximately equal and opposite , one expects a gas particle s longitude of periastron , as defined earlier , to remain fixed , as is observed for the spiral waves in our simulations about sufficiently eccentric binaries . in the context of gaseous circumbinary discs , this interpretation has a few potential shortcomings . first , a given gas particle does not live in isolation since it feels the effects of disc self gravity and pressure gradients within the disc that can impact its orbit . also since this theory is only linear in @xmath15 , its applicability could lessen as the binary becomes more eccentric due to interactions with the disc . however , the binaries considered have low to moderate eccentricities and the discs are rather low - mass such that disc self gravity is negligible so the gravitational influence of the binary should dominate . therefore , we expect this theory to still provide a decent approximate explanation for why the spiral arm remains fixed relative to the binary . secular theory @xcite and previous simulations @xcite show that angular momentum losses to a disc change the central binary s orbital elements . angular momentum loss occurs mainly through resonant gravitational torques at the lindblad and corotation resonances . in the case of binary stars embedded in an external disc , loss of angular momentum can result in changes to the binary eccentricity and semimajor axis . for unequal mass binary stars with low to moderate eccentricity , the majority of eccentricity growth is due to resonant torques at the 1:3 eolr . this resonance dominates since these systems tend to open a gap in the disc such that the 1:3 eolr resides nearest to the inner edge of the disc while the eccentricity damping 1:2 corotation resonance lies within the evacuated region @xcite . simulations of binaries embedded in circumbinary discs by @xcite agree with this interpretation as they identified gravitational torque density peaks responsible for binary eccentricity evolution located at the 1:3 eolr in the disc . for the simulations presented in this work , we expect a secular increase in binary eccentricity and decrease in semimajor axis as the binary loses angular momentum to circumbinary disc . as shown in fig . [ fig : figure4 ] , binary eccentricity for initially eccentric binaries grows over the duration of the simulation . the eccentricity growth rate , @xmath6 , seems to scale with initial @xmath15 . in the simulation with initial @xmath24 , however , @xmath15 increases more slowly than the initial @xmath26 case . when the initial @xmath42 , no binary eccentricity growth occurs in contrast to the results of similar simulations of binaries embedded in an external disc by @xcite who find significant eccentricity growth with an initial @xmath18 , a discrepancy we will address later . in all cases , @xmath15 oscillates as the simulations progress . this oscillation is due to forcing by the @xmath2 potential of the eccentric external disc @xcite . for all simulations , the binary semimajor axis secularly decreases where the rate of decline is lower for increasingly eccentric binaries as shown in fig . [ fig : figure7 ] . as before , the initial @xmath24 case defies this trend as it shows a greater semimajor axis decline than the initial @xmath26 case instead of the expected lesser decline . additionally , the binary s longitude of periastron slowly precesses in the prograde sense less than @xmath43yr@xmath44 over the duration of the simulation similar to the results of comparable simulations by @xcite . any changes in the binary orbital elements will be depend on the details of the binary s interactions with the disc . as discussed in section [ disceccevolution ] , we applied the secular theory of @xcite to show that the strength of the disc - binary coupling dictates how eccentricity grows within the system . we apply the same arguments used above to understand disc eccentricity growth to binary eccentricity evolution . for circular binaries , we argued that the binary and disc are weakly coupled since the mass of the secondary is much greater than the mass of the entire circumbinary disc . in this weak coupling regime , the eccentricity of the less massive of the binary - disc system grows , which in this case is the disc . as expected , the disc develops appreciable eccentricity . from these arguments , we expect the binary to develop very little , if any , eccentricity . this behavior is exactly what we observe in our simulations ( see fig . [ fig : figure4 ] ) . as a function of @xmath40 from simulations with initial @xmath15 of 0.05 , 0.1032 , and 0.25 over - plotted with the analytic fit of equation [ eqn : total_deda ] with @xmath45 . as the system evolves , time advances towards the left in this depiction . ] for simulations of discs around eccentric binaries , we argued that the disc and binary are strongly coupled through the @xmath2 modes of the eccentric binary orbit and inner disc edge orbits . in this regime , both the binary and disc eccentricities grow together and further coupling between the disc and binary can occur . additional disc - binary coupling discussed at length in section [ discstructevolution ] lead to the excitation of a time - independent @xmath2 spiral wave and associated potential from the 1:3 eolr whose orientation remains locked to that of the binary s orbit ( see fig . [ fig : figure5 ] ) . since this wave remains fixed relative to the binary , it resonantly torques the binary through the 1:3 eolr , removing angular momentum from the binary s orbit , increasing @xmath15 . if the wave instead circulated over time , its potential would time - average to zero and have no effect on the binary orbital elements . the presence of the @xmath2 spiral wave leads to the qualitatively different binary eccentricity evolution for the @xmath26 case relative to the @xmath18 case displayed in fig . [ fig : figure4 ] . in all simulations , the binary semimajor axis decreases due to energy dissipation from the viscous disc . similar n - body sph simulations of binaries embedded in a circumbinary disc by @xcite showed that for binaries with mass ratio @xmath46 and @xmath19 similar to the systems examined in this paper , resonant interactions with the surrounding disc at the 1:3 eolr drive eccentricity growth and semimajor axis decay . in this regime , we expect the binary eccentricity growth observed by @xcite since the binary and disc are strongly coupled . with the origin of binary eccentricity evolution understood using the theory of @xcite , we seek to quantify binary orbital evolution . following the analysis of @xcite , we applied the theory of @xcite to quantify the eccentricity and semimajor axis evolution of a binary with initial @xmath19 embedded in an external gaseous disc due to resonant interactions with the 1:3 eolr using the following relation @xmath47 where @xmath48 is the potential component corresponding to the 1:3 eolr @xcite and @xmath49 and @xmath27 are the binary eccentricity and semimajor axis , respectively . for less eccentric binaries , the semimajor axis and eccentricty evolution is well - described by @xmath50 where @xmath51 is the effective standard viscosity parameter @xcite . note that @xmath51 in equation [ eqn : deda1996 ] is not in general the same as the @xmath9 viscosity parameter discussed in section [ methods_section ] . to relate @xmath51 and @xmath9 , we use the following relation from @xcite and @xcite @xmath52 where @xmath53 is the smoothing length , @xmath54 is the disc aspect ratio and the factor of @xmath55 comes from the @xcite derivation of the coefficient for the @xcite viscosity implementation used in changa @xcite . the @xmath56 factor arises from our use of the balsara switch which limits shear viscosity by scaling both @xmath9 and @xmath10 @xcite . the range of @xmath56 is [ 0,1 ] . since both @xmath53 and @xmath56 can vary between gas particles and @xmath54 can vary radially as the disc evolves , we average over the disc to get @xmath57 and @xmath58 = 0.4 . given these values , we set @xmath45 as the approximate value for our simulations with @xmath59 . equations [ eqn : deda ] and [ eqn : deda1996 ] , derived by @xcite via examining the balance between viscous and resonant interactions at the inner disc edge , apply in separate regimes that depend sensitively on @xmath15 . @xcite estimate that once @xmath60 , the 1:3 eolr dominates binary eccentricity growth while below this threshold for circular binaries , no eccentricity growth occurs . these separate regimes correspond to the weak and strong disc - binary coupling for circular and eccentric binaries , respectively , discussed in previous sections . for nearly circular binaries , @xmath61 and @xmath62 as expected for weak disc - binary coupling . conversely for eccentric binaries , @xmath63 and @xmath64 as demonstrated above for strong disc - binary coupling . we combine equations [ eqn : deda1996 ] and [ eqn : deda ] to model how @xmath15 and @xmath40 should evolve under the influence of an external disc following the models of @xcite and @xcite @xmath65 hence , eccentricity increases while the semimajor axis decreases . to verify that equation [ eqn : total_deda ] is a proper model for our simulations , the observed evolution of @xmath15 and @xmath40 for simulations with initial @xmath66 0.05 , 0.1032 , and 0.25 were compared with the theoretical result of equation [ eqn : total_deda ] assuming @xmath45 . the comparison is shown in fig . [ fig : figure8 ] . the results of both the simulations with initial @xmath26 and @xmath24 are in good agreement with the theoretical expectations of equation [ eqn : deda ] and also in accordance with the simulation of a similar system with initial @xmath67 by @xcite . equation [ eqn : total_deda ] can also be applied to the initial @xmath18 cases . for these simulations , the binary semimajor axis decreases via energy dissipation through the viscous disc while no eccentricity growth occurs as seen in fig . [ fig : figure4 ] and fig . [ fig : figure7 ] . our results are consistent with the prediction of equation [ eqn : total_deda ] . these findings indicate that equation [ eqn : total_deda ] successfully quantifies how the binary evolves in the different disc - binary coupling regimes . the initial @xmath15 = 0.05 case proves troublesome . equation [ eqn : total_deda ] does a poor job fitting the binary eccentricity and semimajor axis evolution . the poor fit can be understood in the context of how the disc and binary undergo an intermediate coupling in between the strong and weak regimes . as discussed in sections [ disceccevolution ] and [ discstructevolution ] , the binary eccentricity is not large enough to launch a prominent @xmath2 spiral wave and drive the eccentricity growth for more eccentric binaries . in between regimes , we expect the binary eccentricity to grow weakly as we observe in our simulations . equation [ eqn : total_deda ] succeeds for systems firmly in the weakly or strongly coupled regime but does not perform well for intermediate coupling . this behavior has interesting consequences for the subsequent evolution of a system . for the intermediate case as the binary eccentricity grows with time , it will eventually reach @xmath19 and will then begin to strongly couple to the gaseous disc . as discussed previously , strong coupling launches a @xmath2 spiral wave in the disc and increases the growth rate of binary eccentricity . also in the strongly coupled regime , we expect the disc eccentricity to decrease from higher values and be similar in magnitude to the binary eccentricity . in a similar vein for nearly circular binaries , we expect them to remain circular . other simulations of binaries embedded in circumbinary discs such as those by @xcite and @xcite have found that initially circular binaries eventually develop appreciable non - zero eccentricities . this behavior can be understood by examining equation [ eqn : total_deda ] . if the binary is perturbed and some non - zero eccentricity develops , we would expect the binary eccentricity to grow very slowly , gradually strengthening the coupling between the disc and binary until intermediate coupling is reached and the system progresses as described above . this picture is consistent with the results of @xcite who found the binary eccentricity began to grow on timescales longer than those explored in this work . therefore over timescales much longer than simulated here , we would expect our initial @xmath68 case to become appreciably eccentric . we extrapolate the results of our simulations to longer timescales and consider the consequences below . we ran additional simulations to ensure that the 1:3 eolr did indeed dominate binary evolution and no other effect played a major role . to do this , a shorter simulation with an initial binary eccentricity of 0.1032 was performed with the initial disc gap radius located outside of the 1:3 eolr . minimal binary eccentricity growth and effectively no binary semimajor axis decay occurred until the disc viscously spread inward . additionally , the @xmath2 spiral wave observed in other simulations of sufficiently eccentric binaries also did not exist until mass was able to drift inward and accumulate at the 1:3 eolr at which point the @xmath15 and @xmath40 began to evolve . these findings support the supposition that interactions with the 1:3 eolr drives the binary evolution as anticipated . to study what effect , if any , accretion has on the how the binary stars orbital elements vary , a procedure similar to that used by @xcite was performed . for a given simulation each accretion event was tracked such that the accreted gas particle s mass and velocity components were outputted . using these events , @xmath15 and @xmath40 were evolved by adding each accreted particle to the binary imposing linear momentum and mass conservation as is done natively in changa for sink particles . this test demonstrated that accretion had a negligible effect on @xmath15 and @xmath40 . therefore , it is safe to assume that the evolution of the binary orbital elements is primarily driven by interactions with the external disc , in particular at the 1:3 eolr for the systems considered in this work . previous studies have examined how varying disc properties can change how a circumbinary disc evolves . for example , @xcite showed that disc eccentricity is sensitive to the initial disc surface density gradient and aspect ratio . here , we analyse the results of simulations that vary disc mass , gas resolution , and aspect ratio in order to examine how disc properties impact the disc - binary coevolution . to study how our results vary with disc mass , three additional simulations were run with @xmath69 to see if varying disc mass changes binary evolution . simulation 6 with @xmath70 , simulation 7 with @xmath71 , and simulation 8 with @xmath72 were run . for these additional simulations , we expect the disc to be strongly coupled to the eccentric binary since the coupling only depends on the magnitude of binary eccentricity . we do expect , however , that eccentricity growth to occur more quickly for systems with more massive discs since torque scales with the disc mass . the results of the simulations are shown in fig . [ fig : figure9 ] . the disc eccentricity of simulation 7 was similar to that of simulation 4 while simulation 6 showed larger disc eccentricity values . as expected , more massive discs became more eccentric . not depicted is the spiral arm development . similar to the standard initial @xmath26 case , a prominent @xmath2 spiral wave quickly forms in simulations 6 , 7 , and 8 consistent with the picture that binary eccentricity determines how strongly the disc and binary couple . in all three simulations , the wave has the same shape and remains fixed relative to the binary except for simulation 6 which showed slight prograde precession of the spiral arm . for more massive discs , binary eccentricity grew more quickly and the binary underwent more semimajor axis decay . although not plotted , both simulations 6 , 7 and 8 are still well - described by equation [ eqn : total_deda ] and hence correspond to either faster or slower binary evolution timescales . to ensure that our simulations were sufficiently resolved , we ran two additional simulations , simulations 9 and 10 , which has decreased and increased the initial number of gas particles by a factor of 2 to @xmath73 and @xmath74 , respectively . the results of these simulations are shown in fig . [ fig : figure9 ] . in both simulations , we find the general trend of eccentricity growth and semimajor axis decay holds . the lower resolution simulation 9 eccentricity growth is less than the fiducial simulation 4 . in addition , the disc eccentricity does not oscillate as seen in other simulations . this suggests that the clump which dominates the disc eccentricity does not form into a coherent structure , indicating that n@xmath75 might not be large enough to properly resolve all the physics at the disc inner edge . we find that the @xmath2 spiral wave appears in the disc indicating that the binary strongly couples to the disc as we expect from our previous simulations . the higher resolution simulation 10 exhibits both binary and disc eccentricity evolution that is in good agreement with the fiducial simulation 4 . additionally , we again observed a prominent @xmath2 spiral wave within the disc that behaved identically to its simulation 4 counterpart . one disagreement between simulation 10 and simulation 4 is that the higher resolution simulation displayed less binary semimajor axis decay . this result is expected , however , since higher resolution n - body sph simulations will have smaller gas softening lengths , @xmath53 . as shown in equation [ eqn : alphas ] , the effective standard viscosity parameter @xmath76 . the theoretical work of @xcite estimate that @xmath77 . we therefore expect a higher resolution run with smaller @xmath53 and hence smaller @xmath51 to exhibit less binary semimajor axis decay due to dissipation from the viscous disc . since the main effects explored in this work , the binary eccentricity evolution and the accompanying development of disc eccentricity and structure , are in good agreement between the standard and higher resolution runs , we find that our nominal resolution of n@xmath78 is sufficient . simulations of accretion discs around binary black holes have examined the effects of larger aspect ratio discs , mainly focusing on accretion rates . two dimensional sph simulations of gas accretion onto binaries embedded in a circumbinary disc by @xcite found that increasing gas temperature leads to increased accretion rates onto the primary star and growth in the binary mass ratio . simulations of discs about black hole binaries by @xcite showed that discs with aspect ratios h / r @xmath79 have enhanced accretion rates as the inspiralling gas is not supressed by the binary s gravitational torque . to examine disc aspect ratio s effect on disc - binary evolution in our simulations , we ran simulation 11 with a disc aspect ratio of @xmath80 . to initialize the disc with a larger aspect ratio , we increased the disc temperature by setting t@xmath81 in equation [ eqn : disc_temp_profile ] to 2500 k giving us about a factor of @xmath82 larger aspect ratio relative to the fiducial simulation 4 . the result of simulation 11 is presented in fig . [ fig : figure9 ] . we found that the thicker disc resulted in greater binary semimajor axis decay and eccentricity growth compared to the fiducial simulation 4 . the increased binary semimajor axis decay agrees with the theoretical expectation of @xcite who estimate that @xmath83 . the enhanced binary eccentricity growth follows as a consequence of equation [ eqn : deda ] . in simulation 11 , the binary accretion rate was enhanced by about a factor of @xmath84 relative to the fiducial simulation 4 in agreement with the general findings of both @xcite and @xcite . we found that accretion had little impact on the binary orbital element evolution . as expected from arguments presented in section [ discstructevolution ] , the disc and binary were strongly coupled , producing a prominent @xmath2 spiral wave similar to the one seen in fig . [ fig : figure5 ] . the disc also displayed larger eccentricity initially but it did not grow appreciably over the course of the simulation . the coevolution of a binary with a gaseous circumbinary disc , primarily driven by resonant interactions at the 1:3 eolr , has several important consequences for the subsequent dynamical evolution of the system . as shown above , an eccentric binary system tends to gain eccentricity and experience a secular decay in semimajor axis due to viscous and resonant interactions with the disc . this evolution not only changes due to additional feedback with the disc , but also impacts regions in the disc where planets form and migrate . n - body simulations of an unequal mass binary embedded in a protoplanetary disc by @xcite found rapid semimajor axis decay leading the authors to suggest that the binary separation may become small enough that tidal effects or even stellar coalescence may occur for such systems . tides between stellar companions tend to circularize the orbit over long timescales once the stellar separation becomes sufficiently small . detailed studies of companions to sun - like stars by @xcite and measurements of solar - type spectroscopic binaries in m35 by @xcite both found that binaries from these populations with periods less than about 10 days tend to be circularized . theoretical work on the premain - sequence evolution of @xmath85 @xmath86 binaries by @xcite demonstrate that binaries with orbital periods of about 8 days or less are tidally circularized with effectively all of the circularization occurring before the stars reach the main - sequence . for binaries with an initial period slightly greater than the @xmath87 day tidal circularization boundary embedded in a circumbinary disc , binary - disc interactions could potentially decrease @xmath40 enough to make tidal effects important for subsequent evolution given that the lifetimes of protoplanetary discs are of order 1 myr @xcite . in addition , we would expect some longer period binaries to develop appreciable eccentricity through this mechanism . observations of spectroscopic binaries discussed by @xcite show that a large number of such binaries have large eccentricities , some up to @xmath88 , suggesting that disc - binary interactions may in fact be an important mechanism in pumping binary eccentricity . ideally , additional observations of binaries with circumbinary planets , systems guaranteed to have had protoplanetary discs , will allow us to better constrain and model this effect . the extent to which disc - binary interactions impact astrophysical systems over the disc s lifetime is difficult to measure . over the course of the disc s lifetime , what may occur is some process that removes the 1:3 eolr from the disc . as @xmath40 decays through disc - binary interactions , the location of the 1:3 eolr shifts inwards . also as @xmath15 grows , the central gap size increases @xcite . the combined @xmath40 and @xmath15 evolution could result in the 1:3 eolr moving into the evacuated disc gap , removing its influence from the system , leaving higher order resonances to influence the binary . for binaries with large @xmath15 , @xcite speculates that the combination of higher order inner and outer lindblad resonances and corotation resonances should combine to reduce the magnitude of @xmath6 and @xmath7 , potentially preventing subsequent evolution . this picture is not so simple , however , as simulations of binary smbhs embedded in gaseous discs by @xcite and @xcite both find that binary eccentricity growth continues to @xmath89 where this growth did not slow until @xmath90 . we note that the simulations of @xcite assumed a fixed @xmath40 which neglects the inward motion of the resonances as @xmath40 decays , potentially leaving them in the evacuated region , removing their effects from the system . the impact of higher order resonances on binary evolution is a complicated matter that requires proper treatment in which both the binary and disc are allowed to coevolve together and likely depends on disc structure and artificial viscosity implementation . additionally , findings by @xcite show that in principle , there is no limit to the amount of angular momentum that can be lost by a central binary to an external disc suggesting that binary coalescence is not as unrealistic as it sounds . we caution that when performing simulations of binaries embedded in a gaseous disc that explore the role of semimajor axis decay , one should ensure that their observed semimajor axis decay has converged as both resolution and non - trivial effects such as accretion ( e.g. * ? ? ? * ) can have a substantial impact . one effect not explored in this work is the possibility of kozai - lidov ( kl ) oscillations for the general case of a misalign discs in binary systems . for an inclined test particle orbiting one component of a binary , periodic kl oscillations allow for the particle s eccentricity to grow at the expense of its inclination @xcite . for the case of an inclined circumstellar disc about one component of the binary , @xcite found that the disc can exhibit kl cycles with the periodic disc eccentricity maxima approaching @xmath91 . a later study of similar systems by @xcite demonstrated that misaligned discs can become much more extended than coplanar discs and potentially could overflow the roche lobe of the star . simulations of misaligned circumbinary discs by @xcite showed that discs of almost all inclinations can tear leading to massive accretion and potentially a merger of the central binary . given these results in the general case of systems with misaligned circumbinary discs , the binary eccentricity evolution is likely significantly impacted by the disc evolution and depart from the results presented here for thin , coplanar discs . the disc , if misaligned , could reach large eccentricities due to kl oscillations and via interactions with the binary if it does not tear . if the disc does in fact tear , the binary would likely not couple with the disc at all but could in fact coalesce as demonstrated by @xcite . the general case of a binary coupling with an inclined circumbinary disc is greatly complicated by kl oscillations , torque scaling with inclination and the potential for tearing and warrants a more robust future study . the observed orbital elements of binary stars that host a circumbinary planet are the product of a complex evolutionary history . from fig . [ fig : figure4 ] and fig . [ fig : figure7 ] , we see that for systems similar to the ones considered in this work , appreciable changes can occur on order @xmath1 binary orbits . as shown in section [ varyingdiscmass ] , the mass of the disc strongly influences the binary evolution . more massive discs , for example , drive much faster @xmath15 growth and @xmath40 decay . faster dynamical binary evolution due to massive discs could be particularly relevant for _ kepler _ circumbinary planets as the work of @xcite suggests that these circumbinary planets formed and migrated in massive discs . additionally , disc - binary interactions can make planet formation more difficult . simulations by @xcite identified an @xmath2 spiral wave in the circumbinary disc that corresponds to an alignment of planetesimal longitudes of periastron . this wave , whose origin was explained in this work , caused an increase in erosive planetesimal collisions making in - situ formation difficult in circumbinary protoplanetary discs . the decay of @xmath40 via disc - binary interactions also causes the inward shift of mean motion and lindblad resonances . these resonances can significantly impact the orbital stability of local objects in the disc in several important ways . for the restricted three body problem , resonance overlapping can lead to stochastic orbital evolution as shown from the criterion derived by @xcite . for the case of binary orbital evolution driven by tides , @xcite point out that evolving binary eccentricity and semimajor axis changes the location of critical resonances and hence where they overlap , potentially making stable systems unstable over time . the location of mean motion resonances also dictate where circumbinary planets may reside . the numerical integrations of both @xcite and @xcite show that many circumbinary planets lie in a stable region shepherded by unstable mean motion resonances . if @xmath40 evolves significantly on short enough timescales , so too do the locations of the resonances , sweeping inward and potentially destabilizing orbits . we note , however , that @xmath40 evolution appears to be a resolution dependent effect which future work should address . this behavior is of particular importance for studies of planetary migration in circumbinary discs . studies of circumbinary planetary migration in a viscous , eccentric disc find that planets tend to migrate inwards until they are trapped in or near the 4:1 mean motion resonance in the region of stability identified by @xcite @xcite . since the resonances and the region of stability move as binary eccentricity and semimajor axis evolve , the final location and stability of migrating planets in circumbinary discs is sensitive to binary evolution . simulations of circumbinary systems , especially those using n - body sph methods like the ones presented in this work , must ensure that they properly account for the disc - binary interactions . in this work , we showed that unequal mass binary stars embedded in a circumbinary gaseous disc carved out a gap in the disc and caused structural changes within the disc . resonant interactions with the binary at the 1:3 eolr excited disc eccentricity . sufficiently eccentric binaries excited a @xmath2 spiral wave within the disc . this wave corresponded to an alignment of gas particle longitude of periastrons that varied with radius . the spiral wave formed within 50 years for discs about sufficiently eccentric binaries but took longer to strengthen for less eccentric binaries ( see fig . [ fig : figure5 ] ) . eccentric binary stars became more eccentric and experienced a secular decrease in semimajor axis while initially nearly circular binaries underwent no eccentricity growth over the timescales considered . eccentricity growth within the system was understood in the context of the theory of @xcite in which nonlinear coupling between non - zero disc eccentricity and the binary s tidal potential excites an @xmath5 spiral density wave from the 1:3 eolr that mediates angular momentum transfer in the system . nearly circular binaries weakly couple to the external disc and drive the inner disc edge to become very eccentric . eccentric binaries , however , strongly couple to the disc leading to eccentricity growth for both the disc and binary . the origin of the @xmath2 wave within the disc is understood as a recoupling of the @xmath92 spiral density wave with the binary tidal potential . this model does have limited applicability as disc gap size scales with @xmath15 , so the 1:3 eolr could fall within the evacuated region removing its effect from the system , potentially slowing down binary evolution . for simulations of gaseous circumbinary discs , we caution that the disc - binary interaction must be sufficiently accounted for to properly model the system . we leave the characterization of the long - term impact of disc - binary interactions to future work . limitations of this work include the difficulty in integrating the binary orbit . since the binary feels the force of every other sph particle in our simulations and is integrated using changa s native leapfrog integrator , very conservative timestepping was employed to ensure that the binary orbit was well - resolved and physically accurate . the conservative timestepping scheme significantly slowed our simulations . in light of this limitation , potential future work could include running a long - term higher resolution simulation over at least @xmath93 binary orbits for small yet non - zero @xmath15 in order to better characterize how the disc and binary coevolve . additional future work could involve examining equal mass binaries or binaries with larger eccentricities than those explored in this work . since binaries with large eccentricities excite higher order resonances within the disc ( e.g. * ? ? ? * ; * ? ? ? * ) and carve out gaps that could remove the 1:3 eolr from the disc @xcite , it would be interesting to examine how these other resonances can impact binary evolution . a study on how different numerical viscosity implementations impact binary evolution would also prove fruitful to examine its influence on disc - binary coevolution , specifically binary semimajor axis decay . we thank the anonymous referee for helpful comments and suggestions that improved the quality of the manuscript . we would also like to thank isaac backus for a careful reading of the manuscript and jacob lustig - yaeger for helpful feedback . this work was facilitated through the use of advanced computational , storage , and networking infrastructure provided by the hyak supercomputer system at the university of washington . we made use of _ pynbody _ ( https://github.com/pynbody/pynbody ) in our analysis for this paper . this work was performed as part of the nasa astrobiology institute s virtual planetary laboratory , supported by the national aeronautics and space administration through the nasa astrobiology institute under solicitation nnh12zda002c and cooperative agreement number nna13aa93a . david fleming is supported by an nsf igert dge-1258485 fellowship . thomas quinn is supported by nasa grant nnx15ae18 g .

the recent discoveries of circumbinary planets by @xmath0 raise questions for contemporary planet formation models . understanding how these planets form requires characterizing their formation environment , the circumbinary protoplanetary disc , and how the disc and binary interact and change as a result . the central binary excites resonances in the surrounding protoplanetary disc that drive evolution in both the binary orbital elements and in the disc . to probe how these interactions impact binary eccentricity and disk structure evolution , n - body smooth particle hydrodynamics ( sph ) simulations of gaseous protoplanetary discs surrounding binaries based on kepler 38 were run for @xmath1 binary periods for several initial binary eccentricities . we find that nearly circular binaries weakly couple to the disc via a parametric instability and excite disc eccentricity growth . eccentric binaries strongly couple to the disc causing eccentricity growth for both the disc and binary . discs around sufficiently eccentric binaries that strongly couple to the disc develop an @xmath2 spiral wave launched from the 1:3 eccentric outer lindblad resonance ( eolr ) that corresponds to an alignment of gas particle longitude of periastrons . all systems display binary semimajor axis decay due to dissipation from the viscous disc . = 1 [ firstpage ] binaries : general hydrodynamics protoplanetary discs

You are an expert at summarizing long articles. Proceed to summarize the following text: one of the crucial properties of the _ dark _ matter ( dm ) is the feebleness of its coupling to the electromagnetic field . the early decoupling of dm from the baryon - photon fluid is a basic ingredient of the current picture of structure formation , and various direct dm detection experiments set stringent limits on the coupling of dm with ordinary matter . the phenomenological possibilities of a charged @xcite , or a milli - charged @xcite dm species , or that of dm featuring an electric or magnetic dipole moment @xcite were considered in several recent studies , all pointing towards a severe suppression of any coupling of the dm with photons . significant absorption or scattering of photons by dm appears to ruled out , perhaps implying that _ dm does not cast shadows_. in this analysis we investigate the possibility that , while the typical scattering cross section of dm with photons is very small , photons with the right energy can resonantly scatter off dm particles . we show that this resonant scattering might result in peculiar absorption features in the spectrum of distant sources . this effect can occur if the extension of the standard model of particle physics required to accommodate a ( neutral ) dm particle candidate @xmath0 also encompasses ( 1 ) a second , heavier neutral particle @xmath1 and ( 2 ) an electric and/or magnetic transition dipole moment which couples the electromagnetic field to @xmath0 and @xmath1 . we also assume , for definiteness , that @xmath0 and @xmath1 are fermionic fields . in this setting , there exists a special photon energy @xmath2 where the scattering cross - section of photons by dm is resonantly enhanced to the unitarity limit . if the resonance is broad enough , and the cross section and dm column number density are large enough , the spectrum of distant photon sources might in principle feature a series of absorption lines corresponding to dm halos at different redshifts . if such anomalous absorption features exist , not only would they provide a smoking gun for the particle nature of dm , but they could also potentially give information about the distribution of dm in the universe . we adopt here a completely model - independent setting , where we indicate with @xmath4 the masses of @xmath5 , and consider the effective interaction lagrangian @xmath6 in the rest frame of @xmath0 , the photon - dm scattering mediated through an @xmath7-channel @xmath1 exchange [ see fig . [ fig : feyn](a ) ] is resonant at the photon energy @xmath8 for @xmath9 , the @xmath10-dm scattering cross section can be approximated with the relativistic breit - wigner ( bw ) formula @xmath11 where @xmath12 indicates the modulus of the center - of - mass momentum , @xmath7 is the center - of - mass energy squared , @xmath13 is the total decay width of @xmath1 , and @xmath14 is the decay width of @xmath1 into @xmath0 and a photon . the value of @xmath15 at @xmath16 saturates the unitarity limit provided @xmath17 . under this assumption , even if @xmath0 and @xmath1 featured interactions different in their detailed microscopic nature from those described by eq . ( [ eq : lagr ] ) ( such as a transition milli - charge , or fermion - sfermion loops in neutralino dm models ) , the maximal resonant @xmath10-dm scattering cross section would always be given by @xmath18 for @xmath19 . from eq . ( [ eq : lagr ] ) , we compute @xmath20 in the remainder of this study , for conciseness , we shall denote @xmath21 and , in order to maximize the scattering rate of photons by dm , we will assume a model with @xmath22 . all the quantities above can be trivially rephrased in terms of @xmath23 and of the two ratios @xmath24 and @xmath25 as @xmath26 & \nonumber & e^{\rm res}_{\gamma } = m_{2}\ \frac{1-r^2}{2r } \quad \sigma_{\gamma\chi_1}(e^{\rm res}_{\gamma } ) = \frac{8\pi}{\left(1-r^2\right)^2}\frac{1}{m_{2}^2}\\[0.3 cm ] & & \nonumber \sigma_{\gamma\chi_1}(\tilde e\equiv \frac{e_\gamma}{m_{2}})=\frac{2\pi}{m_{2}^2}\frac{r+2\tilde e}{r\tilde e^2}\frac{\tilde\gamma^2}{\left(r^2 + 2r\tilde e-1\right)^2+\tilde\gamma^2}\end{aligned}\ ] ] where @xmath27 . let us now turn to the effects of the resonant scattering of photons emitted by a distant source . the mean specific intensity at the observed frequency @xmath28 as seen by an observer at redshift @xmath29 from the direction @xmath30 is given by @xmath31\epsilon(\nu , z,\psi)e^{-\tau_{\rm eff}},\ ] ] where @xmath32 @xmath33 is the emissivity per unit comoving volume , and @xmath34 is the effective opacity . the latter can be cast as @xmath35 \ , , \ ] ] where @xmath36 is the dm density . to get a numerical feeling of whether the resonant scattering of photons leads to a sizable effect , we define @xmath37 where @xmath38 indicates an effective dm _ surface density _ , associated with the integral along the line of sight of the dm density . when the quantity @xmath39 in eq . ( [ eq : tau ] ) for @xmath40 we expect a significant absorption for photon energies @xmath19 . once a photon from a background source scatters off an intervening dm particle , the flux from the source itself is attenuated as long as the photon is diffused into an angle larger than the angular resolution of the instrument . the kinematics of the process closely resembles that of the relativistic compton scattering @xcite or thompson scattering at lower energies . roughly speaking , the relevant quantity can be cast as the fraction @xmath41 of scattered photons which end up being scattered into an angle smaller than the instrumental angular resolution @xmath42 , over the total number of scattered photons . for an order of magnitude estimate it is easy to show that , apart from the details of the dm distribution geometry , @xmath41 depends on the two variables @xmath42 and @xmath43 . making simple assumptions , we estimate the values of @xmath41 for an instrument featuring an angular resolution of one degree , over the range @xmath44 , fall within @xmath45 . we therefore can safely assume that if a photon scatters off dm , it is effectively lost ( _ i.e. _ the flux of photons from the background source is effectively depleted by photon - dm scattering processes ) . since the lagrangian ( [ eq : lagr ] ) effectively couples the electromagnetic field to @xmath0 and @xmath1 , depending upon the size of the coupling and the mass of the two particles @xmath5 , the constraints that apply to a milli - charged particle @xmath30 ( _ e.g. _ neutrinos featuring a small electric charge @xmath46 @xcite , or the paratons of ref . @xcite ) will also be relevant for the present setting . the parameter space of the model we consider here consists of the parameters @xmath23 , @xmath47 and @xmath48 ; for future convenience , we choose to represent the viable range of parameters on the ( @xmath49 ) plane , at fixed representative values of @xmath50 and 0.99 [ fig . [ fig : fig2 ] ] . to translate the constraints from the milli - charged scenario in the present language , we need to compute the cross section @xmath51 , and compare it to the standard @xmath52 cross section , where @xmath53 . we find @xmath54 a first simple astrophysical constraint on the model is based on avoiding excessive energy losses in stars that can produce @xmath55 pairs by various reactions , in particular through plasma decay processes . the most stringent limits come from avoiding an unacceptable delay of helium ignition in low - mass red giants . the relevant energy scale in the process is the plasma frequency @xmath56 kev , and the limit applies , roughly , to masses @xmath57 , constraining @xcite @xmath58}.\ ] ] while the lower limit stems from the energy losses argument , the upper limit comes from the requirement that the mean free path of @xmath0 is smaller than the physical size of the stellar core : if the @xmath0 particles get trapped , the impact on the stellar evolution through energy transfer would in any case be negligible compared to other mechanisms @xcite . at such low masses , however , constraints from large scale structure , and namely from lyman-@xmath3 forest data , on the smallest possible mass for the dm particle force @xmath59 kev @xcite . this bound corresponds to the left - most horizontal lines in fig . [ fig : fig2 ] . for a narrow range of effective @xmath60 couplings , the cooling limit discussed above can be applied to the sn 1987a data . for a sn core plasma frequency @xmath61 mev , values of @xmath57 can be ruled out in the range of couplings @xcite @xmath62}.\ ] ] the limits from sn 1987a show up on the ( @xmath49 ) plane as the rectangular regions of parameter space shown in fig . [ fig : fig2 ] . if @xmath5 reach thermal equilibrium in the early universe before big bang nucleosynthesis ( bbn ) , they contribute to the energy density and thus to the expansion rate . translating in the present language the constraints from bbn found in ref . @xcite , if @xmath63 then @xmath64}.\ ] ] the bbn limit corresponds to the central horizontal lines shown in fig . [ fig : fig2 ] . as pointed out in @xcite , a strong constraint on the size of dm magnetic or electric dipole moments is related to the size of the photon transverse vacuum - polarization tensor [ see fig . [ fig : feyn ] , ( b ) ] , @xmath65 the strongest constraint derived from ( [ eq : vacu ] ) comes from the effect of the running of the fine - structure constant , for momenta ranging up to the @xmath66 mass , on the relationship between @xmath67 , @xmath68 and @xmath69 . using ( [ eq : lagr ] ) , we computed @xmath70 finding @xmath71.\end{aligned}\ ] ] the theoretically computed standard model values and the experimental inputs yield a limit on extra contributions to the running of @xmath3 , namely @xmath72 at 95% c.l . @xcite . with particle masses @xmath73 , eq . ( [ eq : loop ] ) reduces to @xmath74 implying @xmath75 for consistency with electroweak precision observables . the limits from eq . ( [ eq : loop ] ) rule out the region below the line labeled as `` vac.pol.-ew precision '' in fig . [ fig : fig2 ] . lastly , high energy accelerator experiments also put constraints on particles with an effective coupling to photons . such particles could have been seen in free quark searches @xcite , at the anomalous single photon ( asp ) detector at the slac storage ring pep @xcite ( designed to look for events in the form @xmath76 weakly interacting particles ) and in beam - dump experiments from vector - meson decays and direct drell - yan production @xcite . the combination of all accelerator constraints rules out the relatively massive and strongly coupled models lying below the upper - right curvy lines on the ( @xmath49 ) plane shown , for three values of @xmath48 , in fig . [ fig : fig2 ] . this completes our discussion of the constraint on the parameter space of the model under consideration here : the viable parameter space , for a given @xmath48 , lies _ above _ the lines shown in fig . [ fig : fig2 ] , while the portions of parameter space that are ruled out correspond to the regions of the plot _ below _ the various constraint lines . since dm particles live in halos characterized by a velocity dispersion @xmath77 , which depending upon the mass of the dm halo can take values from roughly @xmath78km / s to over @xmath79km / s , the momentum distribution of the dm particles approximately follows a maxwell - boltzmann distribution , @xmath80 an incoming photon will therefore scatter off dm particles with the above momentum distribution , and the `` _ effective _ '' scattering cross section will be given by the following average : @xmath81 where @xmath82 where @xmath83 is the cosine of the incident dm-@xmath10 angle , and where the center of mass energy and momentum squared read @xmath84 and @xmath85 the integral in eq . ( [ eq : anginte ] ) can be solved analytically , and we report the result in the appendix . as a result of the averaging procedure of eq . ( [ eq : convolution ] ) , the maximum of the effective cross section is no longer the peak value @xmath86 , but will be a non - trivial combination of the latter , @xmath13 , @xmath77 , and @xmath87 . we illustrate an instance of the result of the broadening of the bw cross section in fig . [ fig : resonance ] . given an instrument with an energy resolution @xmath88 , defined as the relative energy resolution ( _ i.e. _ the ratio of the energy resolution at the energy @xmath89 over the energy @xmath89 itself ) , we require the width @xmath90 of the resonance ( which we define , for convenience , to be the range of values of @xmath91 where @xmath92 ) to be at least as large as @xmath93 . to a good approximation , the solution to the equation @xmath94 is independent of @xmath47 , since @xmath95 for @xmath96 . also , since @xmath97 [ see eq . ( [ eq : distribution ] ) ] , at fixed @xmath48 and small @xmath13 , the ratio @xmath98 is independent of @xmath23 as well . we therefore plot , in fig . [ fig : sigmat ] , curves at constant values of @xmath98 on the @xmath99 plane . as clear from eq . ( [ eq : distribution ] ) , the larger the value of the velocity dispersion @xmath77 , the larger @xmath90 . from eq . ( [ eq : array ] ) we also understand that , as @xmath100 , @xmath101 , explaining why arbitrarily large values of @xmath98 can be obtained for large @xmath48 [ see the upper part of fig . [ fig : sigmat ] ] . how would the spectrum of a background source look like after photons have resonantly scattered off dm ? we address this question in fig . [ fig : flux ] . we assume for definiteness ( our results do nt critically depend upon the particular spectral shape ) a power - law spectrum of the form @xmath102 we consider a setup where @xmath103 mev and @xmath104 mev , and as an example we focus on the case of a source located behind ( or at the center of ) a cluster with features similar to those of the coma cluster . making use of the estimates provided in ref . @xcite , we consider a dm surface density ( integrating the d05 dm profile @xcite along the direction of the center of the cluster , within one virial radius of the cluster center ) of @xmath105 . also , we assume a velocity dispersion of @xmath106km / s . notice that the redshift of the coma cluster , @xmath107 , is small enough that the effect of photon redshift on the shape and location of the absorption feature is completely negligible . making use of these estimates , the effect on the background source spectrum depends entirely upon the value of @xmath13 : for large values of the latter quantity the dm halo is opaque to photons with energies around @xmath2 . we show in fig . [ fig : flux ] how the spectrum defined in eq . ( [ eq : spectru ] ) is affected by setups with various different values of @xmath13 . for @xmath108 ev ( orange line ) , the absorption is almost complete around @xmath2 . smaller values of @xmath13 imply only a partial deformation of the spectrum , and a reduced energy range where absorption effectively takes place . for @xmath109 ev the absorption feature would be almost invisible . we summarize our results on the @xmath110 plane in fig . [ fig : fig6 ] , for the same reference values we employed in fig . [ fig : sigmat ] , _ i.e. _ @xmath111 and @xmath106km / s . for this choice of parameters , @xmath112 . the area shaded in yellow at the bottom right of the plot is ruled out by the various constraints discussed in the previous section . the green dashed lines correspond to fixed values for @xmath38 such that @xmath39 in eq . ( [ eq : tau ] ) , in units of @xmath113 ( blue solid line ) . for dm surface densities @xmath114 , absorption is possible for dm particle masses @xmath115@xmath116 mev . the absorption feature , in this plot , is predicted according to eq . ( [ eq : eres ] ) to occur at a photon energy @xmath117 ; henceforth , in the range above , we predict @xmath118@xmath119 mev . the analogue of fig . [ fig : fig6 ] for other values of @xmath77 and @xmath48 can be directly read out of our results shown in fig . [ fig : sigmat ] taking into account the constraints shown in fig . [ fig : fig2 ] , and the fact that the values of @xmath38 such that @xmath39 in eq . ( [ eq : tau ] ) scale approximately linearly with @xmath48 . for instance , again for dm surface densities @xmath120 , we would predict a dm particle mass @xmath121 mev for @xmath122 , and around 150 mev for @xmath123 . analogously , the location of the absorption feature is predicted in the range @xmath124 1 mev to 150 gev for @xmath122 , and at @xmath124 10500 kev for @xmath123 . in the present setup , therefore , for reasonable dm surface densities , the location of the absorption feature varies in a wide range of photon energies , from tens of kev up to several gev . photons from background sources will in general pass through various dm halos at all intermediate redshifts , resulting in a cumulative cosmological effect leading , in principle , to a broadening and modulation of the absorption feature described above . in ref . @xcite , for instance , an analogous computation was carried out for the monochromatic photon emission from dm pair annihilations into two photons ; the detailed setup here is however different as the effect depends linearly rather than quadratically on the dark matter density distribution . a similar cosmological broadening was also discussed for the case of resonant @xmath125 high energy neutrino absorption _ e.g. _ in ref . the spatial homogeneity of the cosmic neutrino background results however in a completely different column density structure than in the present case . the detailed computation of this cumulative cosmological effect depends on several assumptions about the distribution and nature of dm structures in the universe , on the presence of dm clumps or other substructures @xcite , and on the assumed halo density profiles and velocity distributions @xcite . we leave the detailed analysis of this effect to a future study . in passing we note that thermally - produced dm candidates with masses in the tens of kev to the mev range , often referred to as _ warm _ dm candidates , exhibit potentially interesting features in structure formation , suppressing , through free - streaming , small scale structures , and partly alleviating the cusp problem of cold dm models ( see _ e.g_. ref . @xcite and references therein ) . depending on the details of the particle physics model constraints on such warm dm candidates might be used to constrain our scenario . closing the photon line in fig . [ fig : feyn ] * ( a ) * into a loop generates radiative corrections to @xmath126 . if the latter are too large , the values we employed must be corrected accordingly , and small values of @xmath126 might not be theoretically allowed . we can estimate the size of these corrections as @xmath127 radiative corrections are therefore smaller than @xmath128 provided @xmath129 , a condition which is always widely satisfied in the parameter space under consideration here . a mass mixing term would also be generated by the interaction responsible for the effective lagrangian ( [ eq : lagr ] ) ; in principle , one should then rotate eq . ( [ eq : lagr ] ) to the proper mass - eigenstate basis . however , the relative size of the induced @xmath130 mixing is also very suppressed , as it roughly scales as @xmath131 , and can be thus safely neglected here . in the scenario we are discussing here , the @xmath0 particles can also pair annihilate into two photons through a @xmath1 @xmath132- or @xmath133- channel exchange . the resulting cross section can be estimated as @xmath134 pair annihilation of @xmath0 s into photons can _ a priori _ be the process through which dm annihilates in the early universe and potentially this could thermally produce the amount of dm inferred in the current cosmological standard model . in the range of couplings and masses we obtain here , the above mentioned annihilation channel is insufficient to produce a large enough pair annihilation rate in the early universe in order to get the required dm abundance @xmath135 . other channels , otherwise irrelevant for the present discussion , and compatible with the present setting , can however contribute to give the @xmath0 particles the right pair annihilation rate the same diagram discussed above , and the same pair annihilation cross section , intervene in the pair annihilation rate of @xmath0 s today into monochromatic photons of energy @xmath136 . the flux of photons per unit solid angle from monochromatic pair annihilations of @xmath0 s can be written as @xmath137 where , in this instance , the quantity @xmath138 refers to the following line - of - sight integral along the direction @xmath139 averaged over the solid angle @xmath140 @xmath141 when @xmath142 , we can derive an upper limit to the monochromatic photon flux which is independent of @xmath4 , namely @xmath143 taking an angular region @xmath144 sr in the direction of the galactic center the range of values which @xmath138 can take for various viable dm halo models is @xmath145@xmath146 . this means that one expects a flux of monochromatic photons in the range @xmath147@xmath148 . the diffuse gamma - ray flux in the galactic center region as measured by comptel and egret @xcite is at the level of 0.01 @xmath149 at a gamma - ray energy of 1 - 3 mev . extrapolating to smaller energies we expect an even larger flux at energies around or smaller than 100 kev . this makes it extremely hard to reconstruct a would be annihilation signal from the galactic background . dedicated searches for line emissions show that instruments such as integral - spi also fail to achieve the sensitivity required here @xcite . on the other hand , this also means that the class of models discussed above is not currently constrained by monochromatic photon emissions . furthermore , observations of objects where the diffuse gamma - ray background is expected to be suppressed , such as the nearby dwarf galaxies @xcite , can potentially lead to constraints or even to the detection of the monochromatic emission line predicted here . if , as we describe here , photons scatter off dm at significant rates , one might also expect other associated features besides the absorption lines and the monochromatic emissions described above . scattering off dm might generate an effective `` _ index of refraction _ '' in the photon propagation , possibly inducing _ e.g. _ time delays in transient sources at different frequencies , or frequency - dependent distortions of the photon paths for steady sources . a detailed discussion of these effects lies , however , beyond the scopes of the present analysis . neutralino dm in the context of the minimal supersymmetric extension of the standard model ( mssm ) can in principle produce an effective lagrangian setup as that in eq . ( [ eq : lagr ] ) , for instance through fermion - sfermion loops coupling two different neutralinos @xmath150 and @xmath151 , @xmath152 . from the discussion above , however , it is clear that supersymmetric dm can not produce any sizable photon absorption . first , the lightest supersymmetric particle ( lsp ) in any viable low energy supersymmetry setup is typically heavier than at least a few gev ( for exceptions , _ e.g. _ in the next - to - mssm , see @xcite ) . this implies , as can be read off fig . [ fig : fig6 ] , very large values of @xmath38 to get @xmath39 in eq . ( [ eq : tau ] ) . secondly , the assumptions we made at the beginning that @xmath22 does not hold in general in the mssm : the radiative @xmath153 decay can be the dominant mode only in restricted regions of parameter space , _ e.g. _ when phase space suppresses other three - body decay modes . the resulting effective @xmath47 , in the notation set above , is in any case limited from above by @xmath154 requiring @xmath155 implies @xmath156 , and typically @xmath157 . furthermore , since in the mssm when two neutralinos are quasi degenerate the lightest chargino is also quasi degenerate with them , lep2 limits on the chargino mass @xcite force @xmath158 gev . @xmath157 also implies @xmath159 gev . these values for the model parameters imply ( _ a _ ) small relative widths and ( _ b _ ) too large dm surface densities for the absorption feature to be detectable . relaxing the requirement that @xmath22 would not help anyway , since the cross section ( [ eq : xsec ] ) receives the large suppression factor @xmath160 , and the photon absorption process is again suppressed . one can envision , however , various particle physics scenarios where the phenomenology described above can take place . for instance , a concrete particle physics setup which can explain at once the dm abundance , neutrino masses and mixing , the baryon asymmetry of the universe and , potentially , inflation , is the so - called @xmath161msm @xcite , or one of its extensions @xcite . these models feature a light quasi - stable sterile neutrino with a mass in the tens of kev @xcite up to @xmath162(10 ) mev @xcite range , and heavier majorana neutrinos with a mass at the gev scale . extending this class of models with an effective interaction of the form of our eq . ( [ eq : lagr ] ) gives rise to the phenomenology described above and , hence , to possible resonant photon scattering . we have shown that photons can , in principle , resonantly scatter off dm , through an effective lagrangian featuring a dipole transition moment coupling photons , the dm particle @xmath0 and a heavier neutral particle @xmath1 . we discussed the constraints on the model from stellar energy losses , data from sn 1987a , the lyman-@xmath3 forest , big bang nucleosynthesis , electro - weak precision measurements and accelerator searches . the effective resonant `` absorption '' cross section is broadened by the effect of the momentum distribution of dm particles in dm halos . we showed that dm particles in the tens of kev to a few mev range can lead to resonant photon scattering ( resulting in absorption lines which can lie between tens of kev up to tens of gev ) provided the dm surface mass density is at least of @xmath163 . we also pointed out that typical supersymmetric dm ( the weak - scale neutralino ) does not cast any shadows ( _ i.e. _ it does not `` absorb '' photons ) , while photon absorption can take place in other particle physics setups which can explain various pieces of physics beyond the standard model . we thank john beacom and christopher hirata for insightful comments on an earlier draft of this manuscript . we thank vincenzo cirigliano , shane davis , mikhail gorshteyn , tesla jeltema , marc kamionkowski and enrico ramirez - ruiz for related discussions . sp is supported in part by doe grants de - fg03 - 92-er40701 and de - fg02 - 05er41361 , and nasa grant nng05gf69 g . ks is supported by nasa through hubble fellowship grant hst - hf-01191.01-a awarded by the space telescope science institute , which is operated by the association of universities for research in astronomy , inc . , for nasa , under contract nas 5 - 26555 . the angular integral @xmath164 can be computed analytically with the result @xmath165}\times\\[0.3 cm ] & & \nonumber\left\ { 2 + \frac{e_{\gamma}}{p}\left[c_f(f_1-f_2)+c_g g\right]\right\ } \,\end{aligned}\ ] ] where @xmath166 @xmath167 @xmath168\\[0.3 cm ] & & \nonumber-\frac { ( 2m_{1}^2 - \delta m^2)m_{2}\gamma_{\chi_2}}{(\delta m^2)^2 + m_{2}^2\gamma_{\chi_2}^2 } \,\end{aligned}\ ] ] @xmath169 \ , , \end{aligned}\ ] ] @xmath170 \ , , \end{aligned}\ ] ] and @xmath171\\[0.3 cm ] & & \nonumber - \arctan\left[\frac{\delta m^2 - 2e_\gamma \left(p+\sqrt{m_1 ^ 2+p^2}\right)}{m_2\gamma_{\chi_2}}\right].\end{aligned}\ ] ] s. davidson , s. hannestad and g. raffelt , jhep * 0005 * ( 2000 ) 003 [ arxiv : hep - ph/0001179 ] ; s. l. dubovsky , d. s. gorbunov and g. i. rubtsov , jetp lett . * 79 * ( 2004 ) 1 [ pisma zh . * 79 * ( 2004 ) 3 ] [ arxiv : hep - ph/0311189 ] . s. colafrancesco , s. profumo and p. ullio , astron . & astrophys , * 455 * ( 2006 ) 21 [ arxiv : astro - ph/0507575 ] . j. diemand , m. zemp , b. moore , j. stadel and m. carollo , mon . not . * 364 * ( 2005 ) 665 [ arxiv : astro - ph/0504215 ] . s. profumo , k. sigurdson and m. kamionkowski , phys . * 97 * ( 2006 ) 031301 [ arxiv : astro - ph/0603373 ] . p. bode , j. p. ostriker and n. turok , astrophys . j. * 556 * ( 2001 ) 93 [ arxiv : astro - ph/0010389 ] ; b. moore , t. quinn , f. governato , j. stadel and g. lake , mon . not . soc . * 310 * , 1147 ( 1999 ) [ arxiv : astro - ph/9903164 ] ; v. avila - reese , p. colin , o. valenzuela , e. donghia and c. firmani , astrophys . j. * 559 * ( 2001 ) 516 [ arxiv : astro - ph/0010525 ] . a. w. strong , h. bloemen , r. diehl , w. hermsen and v. schoenfelder , arxiv : astro - ph/9811211 ; a. w. strong , i. v. moskalenko and o. reimer , astrophys . j. * 537 * ( 2000 ) 763 [ erratum - ibid . * 541 * ( 2000 ) 1109 ] [ arxiv : astro - ph/9811296 ] ; a. w. strong , i. v. moskalenko and o. reimer , astrophys . j. * 613 * ( 2004 ) 962 [ arxiv : astro - ph/0406254 ] .

we carry out a model independent study of resonant photon scattering off dark matter ( dm ) particles . the dm particle @xmath0 can feature an electric or magnetic transition dipole moment which couples it with photons and a heavier neutral particle @xmath1 . resonant photon scattering then takes place at a special energy @xmath2 set by the masses of @xmath0 and @xmath1 , with the width of the resonance set by the size of the transition dipole moment . we compute the constraints on the parameter space of the model from stellar energy losses , data from sn 1987a , the lyman-@xmath3 forest , big bang nucleosynthesis , electro - weak precision measurements and accelerator searches . we show that the velocity broadening of the resonance plays an essential role for the possibility of the detection of a spectral feature originating from resonant photon - dm scattering . depending upon the particle setup and the dm surface mass density , the favored range of dm particle masses lies between tens of kev and a few mev , while the resonant photon absorption energy is predicted to be between tens of kev and few gev .

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Proceed to summarize the following text: the kinetics of surface particle adsorption and of transport through interfaces play a key role in surfactant phenomena @xcite , membrane biology and cell signaling @xcite , marine layer oceanography @xcite , and other biological and chemical processes . particle adsorption may fundamentally alter the physical and chemical properties of the interface , and it is crucial to understand both equilibrium and dynamical properties of the adsorbed layers @xcite . in the seminal work of ward and tordai @xcite , a bulk phase acting as a reservoir of particles is physically limited by an empty surface onto which the particles can adsorb . particles are assumed to lower their free energy with respect to the bulk phase by irreversibly and instantaneously adsorbing onto the interface . under these conditions , the total concentration of adsorbed particles may be estimated in relation to measurable interfacial properties , such as the dynamic surface tension . several applications , extensions and alternate approaches to this work have been proposed @xcite . in particular , adsorption dynamics in the ward - tordai setting can be rederived through a free energy approach @xcite , allowing for the inclusion of ionic surfactant effects and electrostatic interactions . in many biochemical systems , the complete adsorption of a particle arriving from the bulk requires a series of auxiliary transformations at the surface before the particle can be successfully incorporated , or ` fused ' into the surface . these intermediate steps gives rise to a lag - time in the complete adsorption process . for example , the incorporation of emulsifying proteins onto an air - water interface may be delayed by the unfolding of the polypeptide at the interface @xcite . adsorption of proteins on polymer - grafted interfaces , such as the glycocalyx layer of vascular endothelial cells , is also delayed due to the progressive insertion of the protein through the polymer brush@xcite . kinetic delays have also been observed in the adsorption of the hemagglutinin glycoprotein ( ha ) of the influenza virus as it enters target host cellular membranes @xcite . the mechanisms underlying this delay are not known in detail but are believed to involve conformational changes of ha molecules into fusion enabling complexes , mediated by the presence of binding receptors and coreceptors on the target cell membrane @xcite . similarly , the incorporation of an hiv particle into a t - cell or a macrophage is possible only after the gp120 glycoprotein of the hiv virus membrane recognizes and binds to the target cell surface receptor cd4 , and subsequently to other coreceptors such as ccr5 or cxcr4 . as in the case of ha and influenza , the exact number of gp120-bound receptors and coreceptors required for hiv particle fusion is yet unknown and might depend on gp120 conformations and receptor / coreceptor binding cooperativity @xcite . the complex nature of surface biochemistry makes quantitative kinetic measurements challenging . recently , the binding kinetics of the cd4 cellular receptor to the gp120 hiv ligand have been measured under different experimental conditions yielding widely different dissociation rates @xcite . in this work , we will provide a quantitative framework that can be used to better understand the experimentally observed lag - times in surface kinetics phenomena that involve multistage surface chemistry . . particles such as viruses are spread over the cell layer in a thin supernatant film . inset : after initial nonspecific viral adsorption on the supernatant - cell interface , cellular receptors and coreceptors bind to the virus via a certain stoichiometry , forming fusion intermediates @xmath1 . the subsurface layer of thickness @xmath2 , the subsurface concentration @xmath3 , and the adsorbed species @xmath4 are discussed in the text.,height=201 ] in particular , we will explicitly consider intermediate , reversible steps for surface binding in the ward - tordai formalism , deriving an effective boundary condition to complement the bulk diffusion process . chemical transitions among the surface species will introduce memory terms in the boundary conditions for the bulk concentration . our analysis can be readily applied to the titration of replication - incompetent virus via a colony formation assay @xcite as shown in fig.[fig1 ] . in this section , we motivate and derive the equations coupling bulk diffusion to surface layer evolution . we consider a general , linear reaction scheme to describe the multistep surface reaction dynamics . effective boundary conditions for diffusion from the bulk are derived in sect.3 . as we shall discuss in detail , we are able to embody the response of the adsorbing particle system to the existence of intermediate chemical steps at the surface , into a unique delay kernel regulating the boundary dynamics . all microscopic details stemming from the surface dynamics , no matter how complicated , are contained in the derived memory kernel . our approach includes ligand rebinding to surfaces , found to be important for analyzing surface plasmon resonance assays of biochemical systems @xcite . in sect.4 we particularize our surface reaction scheme to a specific markov process chain and evaluate all physically relevant quantities . in the continuum limit , the density of particles @xmath5 in the bulk phase obeys the convection - diffusion equation @xmath6 + \frac 1 { k_b t}\nabla\cdot\left[{d}\,n\nabla u \right ] , \label{convectiondiffusion}\ ] ] where @xmath7 and @xmath8 are the local diffusion coefficient and potential of mean force , respectively , and @xmath9 is the thermal energy . spatial variation of @xmath7 and @xmath8 may arise from interactions with the interface as shown in fig.[layer ] . boundary conditions are typically applied at the mathematical surface onto which the particles adsorb or reflect . by balancing the diffusive flux just above this mathematical interface with the particle rate of insertion into the interface , a mixed boundary condition arises @xmath10 here , @xmath11 denotes the substrate ; its normal direction is @xmath12 . the parameter @xmath13 , which has the physical units of speed , is proportional the probability @xmath14 ( often called the accommodation coefficient @xcite or sticking probability @xcite ) that a particle is adsorbed into the mathematical interface upon collision . we define @xmath15 such that in the limit @xmath16 and @xmath17 , eq.[bc0 ] is equivalent to @xmath18 , an absorbing boundary condition . a reflecting boundary condition , @xmath19 , arises when @xmath20 . equations [ convectiondiffusion ] and [ bc0 ] are commonly used to model simple diffusion - adsorption processes at surfaces . in many applications , particles at an interface undergo chemical or physical modifications that control for example , surface reactivity , surface tension @xcite , and conductivity @xcite . biological examples include tissue factor initiated coagulation reactions and viral entry . coagulation factors must work their way through the glycocalyx layer before they can be enzymatically primed by the membrane - bound tissue factors @xcite . entry of viruses , such as hiv , into cells require the binding of membrane - bound receptors and coreceptors before fusion with the target cell can occur . all of these processes can be thought of as reactions at the membrane surface . immediately after adsorption from the bulk , the surface particle concentration , whether of coagulation factors or of virus particles , is denoted by @xmath21 . for example , in the case of viruses , we can identify the @xmath22 state as being that of a virus bound to @xmath23 cd4 surface receptor . the initially adsorbed species can then kinetically evolve into the other species @xmath24 representing virus particles with @xmath25 bound receptors or coreceptors . the kinetics among the @xmath26 surface species follows the linear rate equation @xmath27 , where @xmath28 , @xmath29 is the transition matrix among the @xmath26 surface states , and @xmath30 is the source of the first , originating source species @xmath21 coming from the bulk . . the source species @xmath21 is supplied by the bulk surface concentration at the interface , @xmath31 . in the context of virus recognition and infection , the intermediate steps label various numbers of receptors or coreceptors associated with the surface bound virus particle . for example , @xmath24 may denote the surface concentration of hiv particles with @xmath32 receptors and coreceptors attached . this catenary model could also represent successive degrees of insertion of an absorbing species through a polymer brush or glycocalyx coated interface.,height=33 ] figure [ chain ] illustrates a simple example of a linear surface reaction scheme that can be described by the above linear rate equation . in this case the reaction matrix @xmath29 is tridiagonal . general reaction matrices can also be analyzed since our results depend only on the eigenvalues and eigenvectors of @xmath29 . because the surface densities @xmath33 carry units of number per area , and the bulk densities are expressed by number per volume , any kinetic parameter linking bulk source concentrations to those at the interface must introduce a physical length scale . diamant and andelman @xcite have introduced the sublayer thickness as a mathematical step coupling the bulk density to surface density . here , we physically motivate this `` surface layer '' and the associated transport . let us thus introduce a thin layer of thickness @xmath2 near the surface , in which the particle density is denoted @xmath3 and is still expressed in units of number per volume . the continuum approximation eq.[convectiondiffusion ] breaks down when resolving the transport within distances of a few mean free paths . if we identify the sublayer thickness @xmath2 with the mean free path @xmath34 , as shown in fig.[layer ] , we must solve eq.[convectiondiffusion ] with a nonuniform @xmath8 , and possibly a nonuniform @xmath7 , to within a distance @xmath35 of the interface . for the choice @xmath36 , the adsorption velocity @xmath37 can be approximated by the thermal velocity @xmath38 such that @xmath39 . the value of the bulk density at the boundary @xmath40 is defined as the sublayer density : @xmath41 . the equation for the rate of change of the number per area of molecules in the thin layer , @xmath42 can be obtained by balancing the latter with the diffusive flux into the layer @xmath43 , the adsorption into the surface concentration @xmath21 , and the spontaneous desorption from the initially adsorbed species @xmath21 occurring at rate @xmath44 . the complete set of equations coupling the variables @xmath45 and @xmath24 is thus @xmath46 + \frac 1 { k_b t } \nabla\cdot\left[{d}\,n \ , \nabla u \right ] , \label{convectiondiffusion2 } \\ \displaystyle \l{\dd n_{s } \over \dd t } & = & -f + d \,{\bf \hat n}\cdot\nabla n \ , \big|_{\r = \s+\l{\bf n } } + q_{1}\ , \g_{1},\label{ns } \\ { \dd { \bf \g } \over \dd t } & = & { \bf m}{\bf \g } + { \bf f } , \quad \ , { \bf f } = ( f,0,0,\ldots,0 ) . \label{gamma}\end{aligned}\ ] ] here , @xmath47 , and @xmath48 $ ] is the flux of the surface concentration @xmath31 into the incipiently adsorbed species @xmath21 . this functional may depend on interactions among the adsorbed species @xmath33 , including cooperative or crowding effects , and may be modeled using free energies and chemical potential differences between the bulk and surface @xcite . , or the mean free path @xmath34 , depending upon which is larger.,height=192 ] a further simplification can be introduced by defining a different sublayer thickness @xmath49 , where @xmath50 is the typical range of the particle - surface interaction as shown in fig.[layer ] . in this case , at least at a distance @xmath50 from the interface , @xmath7 is constant , @xmath8 is zero and eq.[convectiondiffusion ] is approximated by the standard diffusion equation @xmath51 all effects of the potential of mean force @xmath8 and spatially varying @xmath7 are now subsumed into an effective source @xmath52 $ ] . this is consistent with all previous treatments @xcite in which transport in the bulk phase was described by simple diffusion with uniform @xmath53 and @xmath54 . provided @xmath55 $ ] is independent of @xmath24 , equations [ convectiondiffusion]-[gamma ] can be explicitly solved in simple geometries . for low surface densities @xmath24 such that additional adsorption is not hindered by steric exclusion , one can assume a form @xmath56 $ ] independent of surface concentrations @xmath33 . for @xmath57 , @xmath13 is now interpreted as an effective adsorption coefficient allowing eq.[convectiondiffusion ] to be replaced by eq.[diffusion0 ] , and simplifying the bulk concentration equation . following the original work of ward and tordai , subsequent studies on adsorption and dynamic surface tension measurements @xcite eliminate the bulk density at the interface in eq.[ns ] to yield two coupled integro - differential equations for @xmath31 and @xmath21 which must be numerically self - consistently solved . here , we solve the linear eqs.[gamma ] independently from the bulk densities , but with a source term @xmath56 $ ] that connects the surface concentrations @xmath24 with the bulk concentration @xmath5 . if @xmath56 $ ] is independent of @xmath24 , the explicit solution to eqs . [ gamma ] can be found by evaluating the eigenvalues @xmath58 and corresponding eigenvectors @xmath59 of the chemical transition matrix @xmath29 . denoting the similarity transform matrix @xmath60 such that @xmath61diag(@xmath58 ) , the surface densities are @xmath62\delta_{1 m } \dd t'\right ] \\[13pt ] \gamma_{k}(t ) & \displaystyle = \sum_{j , m = 1}^{n } v^{-1}_{kj}v_{jm } \gamma_{m}(0)e^{\lambda_{j}t } \\[13pt ] \ : & \hspace{6 mm } \displaystyle + \sum_{j=1}^{n}v^{-1}_{kj}v_{j1}\int_{0}^{t}\!e^{\lambda_{j}(t - t')}f[n_{s}(t')]\dd t ' , \label{gammasoln } \end{array}\ ] ] where @xmath63 are the intermediate surface concentrations at @xmath64 . if there are no spontaneous sources of the surface cell intermediates , all eigenvalues @xmath65 . from eq.[gammasoln ] , in the case @xmath66 , @xmath67 is proportional to @xmath68 $ ] . upon substituting @xmath69 by setting @xmath70 in eq . [ gammasoln ] , we substitute @xmath69 into eq.[ns ] , and find a concise description of the diffusion - adsorption process : @xmath71 @xmath72 \dd t ' , \label{effectiveboundary}\ ] ] @xmath73 where @xmath74 is the kernel constructed from the eigenvalues and eigenvectors of @xmath29 . it is composed of an instantaneous response the immediate depletion of @xmath31 due to adsorption into the @xmath21 surface species and delay terms arising from the surface kinetics of eq.[gamma ] . the complete set of equations [ diffusion1]-[kernel ] is one of our main findings . this result explicitly shows how multistage adsorption is modeled by a bulk diffusion equation with an nonlinear integro - differential boundary condition that incorporates the delay arising from the multistep kinetics . under this scheme , all effects of surface reactions are incorporated in the kernel @xmath74 . our analysis can be carried further by specifying a linear form for the @xmath33-independent source term @xmath75 = \gamma n_{s}(t ) , \label{linearapprox}\ ] ] which simply takes the source for the surface concentration @xmath76 of the first species to be proportional to the subsurface concentration . the surfaces densities @xmath4 can be found by substituting @xmath3 , derived from eq.[diffusion1]-[kernel ] , into the expression for @xmath68 $ ] in eq.[gammasoln ] . note that the boundary condition eq.[effectiveboundary ] contains a singular perturbation , and that for times @xmath77 , the `` outer solution '' approximation @xmath78 yields the standard mixed boundary condition eq.[bc0 ] with an additional memory kernel . moreover , in the linear approximation of eq . [ linearapprox ] , the convolution of the delay term in the effective boundary condition [ effectiveboundary ] is amenable to analysis by laplace transforms . for simplicity , we will assume a simple one - dimensional problem where all quantities vary spatially only in the direction normal to an infinite , flat interface at @xmath79 . we nondimensionalize all quantities by using @xmath2 as the unit of length , and @xmath80 as the unit of time . henceforth , in all equations , we make the replacements @xmath81 and @xmath82 . to render the notation less cumbersome we omit the bars from the redefined quantities . in the discussion that follows , the @xmath83 parameters are intended as nondimensional . upon taking the laplace transform in time of the dimensionless forms of eqs.[diffusion1 ] , [ effectiveboundary ] , and [ kernel ] , we obtain @xmath84 @xmath85 where @xmath86 and @xmath87 is the initial , dimensionless constant bulk and sublayer concentration . the general solution to the bulk density @xmath88 from eq.[lpdiffusion1 ] is @xmath89 once the bulk density is derived , all other quantities can be found by inverse laplace transforming @xmath90 . in the absence of spontaneous sources of the surface intermediates , @xmath91 . in this case , it is possible to show that @xmath92 only has a simple pole at @xmath93 and a branch cut on @xmath94 $ ] . performing the integral along the latter , we find the exact results @xmath95 and @xmath96 where @xmath97 equations [ nzt]-[l ] are used to numerically compute all of our results in the next section . for completeness , analytic expressions for asymptotically short and long time limits are derived in the appendix . we now specify a surface reaction scheme and construct its delay kernel by using its associated eigenvalues and eigenvectors . for applications such as multiple receptor binding of the adsorbed species , we consider a reversible sequential markov process among @xmath26 chemical intermediates @xmath98 . as shown in fig.[chain ] , formation of state @xmath99 from state @xmath24 occurs at rate @xmath100 , while the reverse step occurs at rate @xmath101 . the final state @xmath102 is irreversibly annihilated , by transport across the membrane , or by irreversible reaction , with rate @xmath103 . we can then explicitly write the dimensional form of eq.[gamma ] , where @xmath29 is a tridiagonal transition matrix , as @xmath104 - ( p_{1}+q_{1 } ) \g_{1}+q_{2}\g_{2 } , \\[13pt ] \displaystyle { \dd \g_{i } \over \dd t } & = & \displaystyle p_{i-1}\g_{i-1}-(q_{i}+p_{i})\g_{i}+q_{i+1}\g_{i+1 } \quad 2 \leq i \leq n-1 \\[13pt ] \displaystyle { \dd \g_{n } \over \dd t } & = & \displaystyle -(p^{*}+q_{n})\g_{n}+p_{n-1}\g_{n-1}. \\ \nonumber \end{array}\ ] ] in the simplest case where there is only one surface intermediate before transport across the interface , @xmath105 and the dimensionless ( @xmath106 ) delay kernel is simply @xmath107 . the sublayer concentration @xmath3 derived from eq.[nzt ] , and the surface concentration evaluated from eq.[gamma2 ] , are shown in figs.[plot1 ] for various values of @xmath13 . let us estimate typical parameter values for viral fusion or molecular binding processes . typical diffusion constants for viruses of diameter @xmath108 nm and in aqueous environments , are @xmath109@xmath110/s . using the typical screened electrostatic interaction potential , @xmath111 cm , we estimate the dimensionless diffusion coefficient @xmath112s@xmath113 . on the other hand , typical diameters of small ligand molecules are of the order of 1 nm yielding a nondimensional diffusion constant @xmath114s@xmath113 . the nondimensional @xmath13 estimated using the thermal velocity @xmath38 is now @xmath115 . for virus particles @xmath116s@xmath117 , while for molecular ligands @xmath118s@xmath117 . the dissociation rate @xmath44 is highly variable and typically falls in the broad range @xmath119 s@xmath120 to @xmath121s@xmath120 . for the gp120-cd4 interaction , the dissociation has been estimated in model systems @xcite to be @xmath122s@xmath120 , while the detachment rate for mutant viral species @xcite can be as high as @xmath123s@xmath120 . lower dissociation rates are possible in tighter binding ligand receptor pairs such as egf - receptor @xcite where @xmath124s@xmath120 . for other pairs such as p - selectin and its receptors @xcite , @xmath125s@xmath120 to @xmath126 s@xmath120 . the sticking probability @xmath14 is proportional to the binding probability of upon ligand - receptor contact , multiplied by the receptor area fraction at the interface . the factor @xmath14 depends on the receptor density , but is typically of the order @xmath127 . in fig.[plot1](a ) we plot the sublayer density @xmath128 as a function of time . for @xmath129 , fig.[plot1](a ) shows that the sublayer density @xmath3 starts at its initial value @xmath87 and decreases with a nondimensional rate proportional to @xmath13 , eventually monotonically reaching @xmath130 . if the annihilation rate @xmath103 is decreased , @xmath31 may no longer be monotonic . the observed increase in the surface concentration is due to the slow consumption of material at the interface , allowing some of the material to desorb after being delayed at the interface , rather than irreversibly reaching the final annihilated or fused state . for example , when @xmath131 , the surface concentration @xmath3 first decreases but recovers slightly at longer times , before ultimately decaying to zero as shown in fig.[plot1](b ) . for various @xmath132 with @xmath133 and @xmath134 . ( b ) the bulk density profiles @xmath135 as a function of position @xmath136 at times @xmath137 , and @xmath138.,height=412 ] note that the curves for @xmath3 exhibit a transient at short times determined by @xmath139 . beyond this transient , the full solution we plot in fig.[plot1](b ) reduces to the outer solution , corresponding to setting @xmath140 on the left hand side of eq.[effectiveboundary ] . fig.[plot1](c ) explicitly shows different behaviors of the full and outer solutions during the transient time . the effect of the @xmath141 term is to slow down the initial decrease of @xmath31 , particularly for short times within the transient defined by @xmath139 . the effects of neglecting the boundary layer are less pronounced for larger bulk diffusivities @xmath53 . the temporal evolution of @xmath142 is strongly dependent on @xmath143 . in fact , the nonmonotonicity of @xmath142 for small @xmath103 shown in fig.[plot1](b ) arises from the build - up of @xmath76 indicated in fig.[plot2](a ) which can get released back into the subsurface layer . for smaller @xmath103 , @xmath76 reaches larger values . as long as @xmath144 , both @xmath31 and @xmath76 vanish at sufficiently long times . complete particle depletion near the surface occurs in dimensions less than two because there is no bounded steady - state solution to the diffusion equation and the depletion zone moves away from the interface for all times as shown in fig.[plot2](b ) . despite free diffusion , the bulk is unable to sustain a particle source near the surface as is known from classic diffusion theory @xcite . the replenishment at small annihilation rates @xmath132 also manifests itself in the bulk . in the case shown in fig.[plot2](b ) , as time increases from @xmath145 to @xmath146 , the bulk concentration near the interface recovers before ultimately decreasing according to eq.[longtime ] . in the surface reaction scheme . ( a ) the sublayer density @xmath31 as @xmath26 is increased . the initial rapid fall from @xmath147 is imperceptible on this scale . ( b ) the surface concentrations @xmath33 as a function of time for @xmath148.,height=412 ] for general @xmath26 , the eigenvectors and eigenvalues must be explicitly computed . in fig . [ plot3](a ) , we plot @xmath31 as a function of @xmath26 for uniform @xmath149 and uniform @xmath150 . for small @xmath150 , the surface kinetics is a highly biased random walk away from @xmath151 toward @xmath76 , resulting in a larger @xmath31 . both small @xmath150 and large @xmath26 , hinder the annihilation process and impart a more reflective character to the interface . after initial transients , both @xmath31 and the surface concentrations @xmath33 maintain a high level for a long time before dissipating . larger @xmath26 also effectively trap surface material in the surface reservoir @xmath33 . the relative amounts of @xmath33 for @xmath148 are shown in fig.[plot3](b ) . for the small @xmath152 used , most of the surface density lies in the initial species @xmath76 , decreasing in the latter species . we derived an effective , integro - differential equation for the boundary condition of a simple diffusion process . the approach presented differs from the typical ward and tordai treatments since we use a linear model for the time rate of change of the initially adsorbed species @xmath21 , rather than eliminating the bulk diffusion equation . the effects of intermediate chemical steps at the boundary are described by a delay kernel that can be decomposed using laplace transforms . this kernel is an explicit function of the eigenvalues and eigenvectors of the surface reaction transition matrix . our results suggest that measurement of a few quantities , such as fluorescence monitoring of the sublayer density @xcite , can be used to reconstruct the principal components of @xmath74 . this approach can be used to probe qualitative features of the surface kinetics important in modeling cell membrane signaling and viral infection , where a sequence of chemical steps at the surface are required before initiation of signaling or viral fusion . in hiv infection , the initial adsorption rate would be proportional to the surface cd4 concentration , and the subsequent rates in the reaction scheme in fig.[chain ] would depend on the coreceptor concentrations , their surface mobilities , as well as the effects of cooperative binding @xcite . all of these physical attributes are encoded in the distribution of eigenvalues and eigenvectors of @xmath29 . for simple linear reaction schemes on a flat surface , we find explicit dependences of the surface concentrations @xmath33 and sublayer concentrations on the eigenvalues and eigenvectors of the transition matrix . for smaller annihilation rates @xmath132 , and at least one ( @xmath153 ) surface intermediate , we find that the surface concentration persists and can replenish the bulk concentrations @xmath31 after its initial decay . the depletion zone in the bulk can also recover . delays that induce instabilities in dynamical systems have been well established @xcite . here , although the delay occurs in a boundary condition , we observe nonmonotonic behavior arising in the bulk concentrations as well . this rebounding effect is also apparent if one differentiates eq.[effectiveboundary ] with respect to time , giving a second order , harmonic oscillator - like equation , plus a dissipative coupling to the bulk concentration . whether the surface concentration or the bulk concentration near the surface vanishes at long times depends on the surface kinetics as well as geometry . if the combined surface kinetics towards annihilation is slow relative to bulk diffusion , the decay of the sublayer concentration @xmath31 can be extremely slow . similarly , if the number of surface states is large , there is an effective delay to annihilation and a higher probability that a surface species can detach and replenish the sublayer concentration . this effect is very sensitive to the annihilation rate @xmath132 and the size of the reaction @xmath26 and can keep the subsurface concentration high for essentially all times . a number of extensions and related approaches to this and related systems can be readily investigated . for example , in applications such as surfactant adsorption , the surface concentration @xmath154 can be appreciable and suppress additional adsorption . if surface species @xmath24 has molecular area @xmath155 , an adsorption term including steric exclusion would be @xmath156 . the surface rate equations remain linear in @xmath157 , but with a time - dependent transition matrix @xmath29 . the effective boundary condition eq.[effectiveboundary ] is now nonlinear in @xmath3 through @xmath56 $ ] . however , for many biochemical applications ( such as cell signaling and virus adsorption and entry ) the total surface concentration is low such that @xmath158 and the adsorption term can be linearized . in our one - dimensional analysis , as long as @xmath26 is not too large and there is an appreciable annihilation process , the surface concentrations all vanish in time . the effects of multistage adsorption can also be explored on surfaces of arbitrary shape , particularly for cylinders and spheres . for multistage processes on a sphere , the sublayer concentration approaches a positive value @xmath159 . we also expect positive eigenvalues @xmath160 of @xmath29 to have a striking effect on the transport . finally , although we have only considered simple linearized surface reaction schemes with negative eigenvalues , systems that support oscillations , such as those involved in surface - mediated cell signalling , could also be treated within our framework . features of the surface reactions and the bulk concentrations near the reacting surface remain coupled through the kernel @xmath74 . under certain conditions , nonlinear surface reaction schemes may also be linearized . one example is in the _ stochastic _ representation of the surface reactions . if we write the surface quantities in terms of the probability distribution function @xmath161 that there are @xmath162 molecules of of type 1 , @xmath163 of type 2 , etc . , the surface reactions can be written as a linear master equation . this allows our approach to be applied when @xmath76 in the last term of eq . [ ns ] is interpreted as @xmath164 , the ensemble average @xmath165 . using this interpretation , the full problem can be solved using linear methods similar to those presented , albeit for extremely large matrix dimension . the authors thank benhur lee for stimulating discussions . tc acknowledges support from the nsf ( dms-0349195 ) , and the nih ( k25ai41935 ) . part of this work was done during the cells and materials workshop at ipam ucla . here , we derive asymptotic expression for bulk and surface densities . the trivial short time behavior of the subsurface density is @xmath166 , independent of the surface reactions since the first physical phenomenon to occur is particle adsorption from the bulk to the interface , at rate @xmath13 . the condition @xmath170 can be interpreted as a comparison between two typical velocities . the usual diffusive velocity , @xmath171 , is compared to an effective reaction velocity expressed by @xmath13 modulated by surface effects through the kernel @xmath172 . we may thus define an effective damkhler number @xmath173 , so that eqn.[largez ] is valid only at large distances and for large values of @xmath174 . the leading term on the right - hand - side above is independent of the surface kinetics : the first information to have traveled away from the interface is the initial depletion of the @xmath31 layer into the surface and interfacial effects emerge as first order corrections . in the @xmath175 limit , the dominant contribution to @xmath135 comes from small values of @xmath176 in eq.[nzt ] . approximating @xmath177 with its @xmath178 limit , we find the asymptotic long time limit this expression is valid only if the surface dynamics include a net sink of material . as long as there is some annihilation , @xmath180 and eq.[longtime ] holds . a similar consideration of the small-@xmath176 dominated integration in eq.[gamma2 ] yields for the surface concentrations where @xmath182 is the largest eigenvalue of the chemical transition matrix @xmath29 . both eq . [ longtime ] and [ gammaasymp ] exhibit diffusion - limited @xmath183 behavior . these general results rely only on the linearity of @xmath56 $ ] and are valid for _ any _ surface reaction scheme through the eigenvalues and eigenvectors of the transition matrix @xmath29 and the resulting function @xmath184 .

we derive the equations that describe adsorption of diffusing particles onto a surface followed by additional surface kinetic steps before being transported across the interface . multistage surface kinetics occurs during membrane protein insertion , cell signaling , and the infection of cells by virus particles . for example , viral entry into healthy cells is possible only after a series of receptor and coreceptor binding events occur at the cellular surface . we couple the diffusion of particles in the bulk phase with the multistage surface kinetics and derive an effective , integro - differential boundary condition that contains a memory kernel embodying the delay induced by the surface reactions . this boundary condition takes the form of a singular perturbation problem in the limit where particle - surface interactions are short - ranged . moreover , depending on the surface kinetics , the delay kernel induces a nonmonotonic , transient replenishment of the bulk particle concentration near the interface . the approach generalizes that of ward and tordai @xcite and diamant and andelman @xcite to include surface kinetics , giving rise to qualitatively new behaviors . our analysis suggests a simple scheme by which stochastic surface reactions may be coupled to deterministic bulk diffusion .

You are an expert at summarizing long articles. Proceed to summarize the following text: it is common accepted that braking of pulsars is caused by the magneto - dipole radiation of the rotating magnetic star . in this case the rate of losses of the neutron star rotation energy can be equated to the power of its magneto - dipole radiation : @xmath1 + where _ i _ is the moment of inertia of the neutron star , @xmath2 - the angular speed of its rotation , @xmath3 - its magnetic moment , @xmath0 - the angle between the rotation axis and the magnetic moment , _ c _ - speed of light . for standard parameters of neutron stars : masses of order of the solar mass ( @xmath4 ) and radii _ r _ of order of @xmath5 cm we can put _ i _ = @xmath6 . for the magnetic moment we have @xmath7 + here @xmath8 is the magnetic induction at the magnetic pole , @xmath9 ? the induction at the magnetic equator . instead of @xmath2 the rotation period @xmath10 is usually measured and we can obtain from ( 1 ) and ( 2 ) : @xmath11 + this equality is used usually to calculate magnetic inductions of pulsars assuming that @xmath12 for all objects . the known catalogs ( see , for example manchester et al . , 2005 ) contain as a rule @xmath9 instead of @xmath8 . here we propose to decline the assumption on the constancy of @xmath13 and use some estimations of this parameter to calculate more accurate values of pulsar magnetic inductions . in a number of our works ( malov & nikitina , 2011a , b , 2013 ) some methods for calculations of the angle @xmath0 have been put forward and applied to some catalogs of pulsars ( keith et al . , 2010 ; van ommen et al . , 1997 ; weltevrede & johnston , 2008 ) at approximately 10 , 20 and 30 cm . basic equations for this aim are ( manchester & taylor , 1977 ) : @xmath14 @xmath15 + here @xmath16 is the angle between the line of sight and the rotation axis , @xmath17 - the angular radius of the emission cone , @xmath18 - a half of the angular width of the observed pulse , @xmath19 - the position angle of the linear polarization , @xmath20 - longitude . the simplest case for the calculations of the angle @xmath0 is realized when the line of sight passes through the center of the emission cone , i.e. @xmath21 + in this case we can use the dependence of the observed pulse width @xmath22 at the @xmath23 level on the rotation period and determine the lower boundary in the corresponding diagram to obtain @xmath24 + as the result we have from ( 4 ) , ( 5 ) and ( 7 ) ( malov & nikitina , 2011a ) : @xmath25 + the values of angles calculated by this method are denoted as @xmath26 and given in the table 1 . usually polarization measurements are made inside the pulse longitudes only . in this case we can use the maximal derivative of the position angle . from ( 5 ) we have @xmath27 we can obtain from the dependence of @xmath22 on _ p _ by the least squares method @xmath28 + the third equation for the calculations of the angle @xmath0 is ( 4 ) . from these three equations we obtain @xmath29y^2 + 2c(d - b^2)y+c^2d^2-b^2(1+c^2)=0.\\\ ] ] + here @xmath30 + we can transform the equation ( 9 ) to the following form @xmath31 + then finding the value of y from the equation ( 11 ) we can calculate @xmath0 from ( 13 ) . we have calculated values of @xmath0 by this method and list them in the table 1 as @xmath32 . here we correct the misprint in the equation ( 11 ) made in our papers ( malov & nikitina , 2011a , b , 2013 ) . there is an additional way to calculate angles @xmath0 . this way uses observable values of position angles and shapes of average profiles for individual pulsars . in this case , original equations form the closed system for calculations of the angles @xmath17 , @xmath16 and @xmath0 : @xmath33 as the observed pulsar profiles have various forms , the coefficient _ n _ has a different value depending on a profile structure . we put arbitrary the following values of _ n _ ( fig.1 ) . if the ratio of the intensity @xmath34 in the center of the pulse to the maximal intensity @xmath35 is zero then @xmath36 . for @xmath37 @xmath38 , @xmath39 @xmath40 , @xmath41 @xmath42 , and for @xmath43 @xmath44 . it is worth noting that the solution of the system ( 14 ) can be obtained numerically for any value of _ n_. for example , if @xmath45 , the solution for @xmath46 can be obtained from the equation : @xmath47 at n = 2 : @xmath48 y^4 + 2c \left [ c^2 ( 1 + d - 2d^2 ) - 2 - d \right ] y^3 + \left [ 2dc^4 ( 1 - d ) - \right . . - c^2 ( 2d^2 - 6d + 7 ) + 5 \right ] y^2 + 2c \left [ c^2 d^2 + d(1 + c^2 ) - 2 ( c^2 - 1 ) \right ] y + c^2 d^2 ( 1 + c^2 ) - ( c^2 - 1)^2 = 0;\\ \end{array}\ ] ] at n = 3/2 : @xmath49 \sqrt{\frac{1 + \frac{c + y}{\sqrt{c^2 + 2cy + 1}}}{2 } } - c y^2 ( 1 - d ) - y - cd = 0;\ ] ] at n = 5/4 : @xmath50 this method gives angles @xmath51 ( see the table 1 ) . for some pulsars calculations were made by one method only . when it was possible we used two or all three methods . in these cases , the mean value of the angle @xmath0 has been calculated . the resulting values @xmath52 are listed in the table 1 . some other authors ( for example , kuzmin & dagkesamanskaya , 1983 ; kuzmin et al . , 1984 ; lyne & manchester , 1988 ) carried out calculations of the angle @xmath0 earlier for the shorter samples of pulsars using some additional assumptions . we will use further our estimations to calculate magnetic inductions at the surface of the neutron stars . the distribution of the angles @xmath0 from the table 1 ( fig.2 ) shows that the majority of pulsars have rather small inclinations of the magnetic moments . these pulsars are old enough , and we can conclude that they evolve to the aligned geometry . the average characteristic age for our sample of pulsars is @xmath53 years . we must note however that the angles calculated by the method * _ 1 ) _ * are the lower limits of this parameter . this explains partly the predominance of the small values of @xmath0 . from the table 1.,width=453 ] .values of the angle @xmath0 ( deg ) . [ cols="^,^,^,^,^,^,^,^,^,^,^ " , ] 1 . some methods for calculations of the angle @xmath0 between rotation and magnetic axes were applied to obtain the values of @xmath0 for 376 radio pulsars . the distribution of these values shows the predominance of small inclinations of the magnetic axes . 2 . magnetic inductions at the surface of 375 pulsars considered were calculated . there is no the measured derivative @xmath54 for the pulsar j1713 - 3949 and it is excluded from the consideration . the distribution of the calculated magnetic inductions can be described by the gaussian with the maximal value of @xmath55 and the width in the logarithmic scale nearly 1 . the calculated inductions are higher than the catalog equatorial inductions with the mean value of the ratio of these quantities of 5 . for the pulsar j1410 - 7404 @xmath56 . the maximal value of the ratio @xmath57 for the pulsar j2007 + 0809 . this work has been carried out with the financial support of basic research program of the presidium of the russian academy of sciences * _ transitional and explosive processes in astrophysics _ * ( p-41 ) . we thank a.v.biryukov for very useful comments and discussions . 99 keith m.j . , johnston s. , weltevrede p. and kramer m. , 2010 , mnras , 402 , 745 kuzmin a.d . , dagkesamanskaya i.m . , 1983 , soviet astron . letters , 9 , 80 kuzmin a.d . , dagkesamanskaya i.m . , pugachev v.d . , 1984 , soviet astron . letters , 10 , 357 lyne a.g . , manchester r.n . , 1988 , mnras , 243 , 477 manchester r.n . , taylor j.h . , 1977 , pulsars . w.h.freeman and company , san francisco manchester r.n . et al . , 2005 , j. , 129 , 1993 . malov i.f . , nikitina e.b . , 2011a , astron.rep . , 55 , 19 malov i.f . , nikitina e.b . , 2011b , astron.rep . , 55 , 878 malov i.f . , nikitina e.b , 2013 , astron.rep . , 57 , 833 van ommen t.d . et al . , 1997 , mnras , 287 , 1210 weltevrede p. , johnston s. , 2008 , mnras , 391 , 1210

we used the magneto - dipole radiation mechanism for the braking of radio pulsars to calculate the new values of magnetic inductions at the surfaces of neutron stars . for this aim we estimated the angles @xmath0 between the rotation axis and the magnetic moment of the neutron star for 376 radio pulsars using three different methods . it was shown that there was the predominance of small inclinations of the magnetic axes . using the obtained values of the angle @xmath0 we calculated the equatorial magnetic inductions for pulsars considered . these inductions are several times higher as a rule than corresponding values in the known catalogs . * keywords * magnetic fields ; methods : data analysis ; methods : statistical ; _ ( stars : ) _ pulsars : general

You are an expert at summarizing long articles. Proceed to summarize the following text: supergiant fast x ray transients ( hereafter sfxts ; smith et al . 2006a ) are a new class of hard x ray sources mostly discovered by the @xmath2 satellite ( negueruela et al . 2005a , sguera et al . 2005 ) . they are transient sources which seem to emit x rays only during `` short '' outbursts ( few hours , as observed with @xmath2 or @xmath3 ) and their optical counterparts are blue supergiant stars . their x ray spectra are similar to those of accreting pulsars , thus it is likely that they are high mass x ray binaries ( hmxbs ) hosting neutron stars . in two sfxts ( among about twenty sources , comprising candidate sfxts ) x ray pulsations have been indeed observed ( igr j18410 - 0535/ax j1841.0 - 0536 , p@xmath4=4.74 s , bamba et al . 2001 ; igr j112155952 , p@xmath5s , smith et al . 2006b , swank et al . 2007 ) . sfxts reach x ray luminosities up to a few 10@xmath6 erg s@xmath7 , while the quiescent level ( @xmath810@xmath9 erg s@xmath7 ) has been observed only in igr j17391@xmath13021/xte j1739@xmath1302 ( smith et al . 2006a ; sakano et al . 2002 ) , igr j17544@xmath12619 ( int zand 2005 ) and probably igr j18410@xmath10535/ax j1841.0@xmath10536 ( halpern et al . 2004 ; bamba et al . 2001 ) . it is important to note that none of the sfxt sources has ever been caught to undergo a transition from quiescence to outburst and then back to quiescence in a few hours . the quiescent emission had always been observed well far away from the outbursts , except in one case : only int zand ( 2005 ) did observe the transition from quiescence to outburst with @xmath10 ( in igr j17544@xmath12619 ) , but the observation finished before the start of the declining phase to quiescence . thus the real duration of this outburst could not be measured . the so - called `` short '' duration ( a few hours ) of the outbursts from sfxts is indeed based on observations with instruments not sensitive enough to detect the quiescence level . the instruments onboard @xmath3 and @xmath2 could only observe the brightest fast flaring activity ( lasting a few hours , less than one day ) reaching a few 10@xmath6 erg s@xmath7 . hence , the definition of sfxts as transient sources displaying `` short '' x ray outbursts lasting only a few hours is strongly biased . this has been observationally demonstrated by our recent deep campaign with @xmath0 ( romano et al . 2007 , hereafter paper ii ) of the outburst from the unique sfxt displaying `` periodic outbursts '' , igr j112155952 ( sidoli et al . 2006 , hereafter paper i ) . these very sensitive observations showed that the accretion phase in sfxts lasts longer than what previously thought : a few days instead of only hours . with these new observations at hand , we report on an alternative model to explain the outbursts from this new class of sources , based on @xmath0 monitoring observations of igr j112155952 during the last two outbursts ( starting on february 9 and july 24 , 2007 ; sidoli et al . 2007 , hereafter paper iii ) . is an x ray transient discovered by during a fast outburst in april 2005 ( lubinski et al . the optical counterpart is a b1 supergiant , hd 306414 , located at a distance of 68 kpc ( negueruela et al . 2005b , masetti et al . 2006 , steeghs et al . 2006 ) . from the analysis of observations of the source field , we discovered ( paper i ) that the outbursts are equally spaced by @xmath8330 days ( although a half of this period could not be excluded , due to a lack of observations ) . this periodicity was later confirmed in march 2006 ( outburst after 329 days ; smith et al . 2006c ) during a monitoring with @xmath11 , and was related in a natural way to the orbital period of the system , with the outbursts triggered at ( or near to ) the periastron passage ( paper i ) . based on this known periodicity , a new outburst was expected for 2007 february 9 and we planned a monitoring campaign with @xmath0 , starting on 2007 february 4 ( romano et al . a second monitoring campaign was performed with @xmath0 in july 2007 , in order to monitor the quiescent level and the epoch of the supposed apastron passage ( based on the 329 days period ; romano et al . these observations led to the detection of a new unexpected outburst starting on 2007 , july 24 , which reached roughly the same flux as during the february 2007 outburst ( paper iii ) . details of the @xmath0 data analysis and spectral / timing results are reported in paper ii and paper iii . here we concentrate on the shape of the x ray lightcurve in order to understand the physical mechanism which produces the outbursts . the igr j112155952 lightcurve observed during the february 2007 outburst represents the most complete set of observations of a sfxt outburst ( fig . [ fig : ecc ] , black curve ) . the first important result of these observations is that the whole outburst phase lasts longer than what previously thought , based on less sensitive instruments : a few days , instead of a few hours . only the brightest part of the outburst is short ( lasts less than 1 day ) and would have been seen by the integral instruments . intense flaring activity is also present , both during the bright peak and the declining phase of the outburst , with each single flare lasting minutes or a few hours . it is natural to associate the clock responsible for the outbursts with the orbital periodicity of the binary system . since displayed a new outburst after about a half of the 329 days period ( paper iii ; romano et al . 2007c ) , it is possible that 164.5 days is indeed the real orbital period which escaped detection up to now . in both cases ( p@xmath12=329 days or 164.5 days ) , the system is a wide binary where the blue supergiant does not fill its roche lobe , and the system is very likely wind - fed . applying the bondi - hoyle wind accretion scenario , where the neutron star accretes from the wind of the supergiant at different rates depending on the wind density and relative velocity along the orbit , and assuming reasonable parameters for the b - supergiant , we obtain that the observed x ray lightcurve is always too narrow and steep to be explained with accretion from a spherically symmetric wind , even adopting extreme eccentricities for the binary system ( see fig . [ fig : ecc ] ) . this result led us to suggest that the wind from the b supergiant is not spherically symmetric . the alternative viable explanation we propose for the sharpness of the observed x ray lightcurve is that in the supergiant wind has a second component ( besides the polar spherically symmetric one ) , in the form of an `` equatorial disk '' , inclined with respect to the orbital plane ( see fig . [ fig : geom ] for an artistic view of the geometry of the system ) . the short outburst is then produced when the neutron star crosses this equatorial wind component , denser and slower than the polar one . deviations from spherical symmetry in hot massive star winds are also suggested by optical observations ( e.g. prinja 1990 , prinja et al . 2002 ) and the presence of equatorial disk components , denser and slower with respect to the polar wind , also results from simulations ( ud - doula et al . 2006 ) . the thickness _ h _ of the densest part of this supergiant equatorial wind can be calculated from the duration of the brightest part of the outburst ( which lasts less than 1 day , time needed for the neutron star to cross it ) and the neutron star velocity , 100200 km s@xmath7 : _ cm , where @xmath16 is the inclination angle of the equatorial wind with respect to the orbital plane ( with @xmath16=90@xmath17 if the disk is perpendicular to the orbital plane ) . the model we are proposing can explain also the short flares from all the other sfxts where a clear periodicity in the outbursts recurrence has not been found yet , if a different geometry of the equatorial wind component with respect to the orbital plane is assumed : in , where the outbursts are equally spaced and occur with a fixed periodicity , the inclined equatorial disk wind component should intersect the neutron star at the periastron ( or very close to it , see the left panel in fig . [ fig : geom ] ) and can intersect the neutron star orbit once or twice depending both on the extension of the wind disk and on the orbital eccentricity . instead , it is possible that in the other sfxts the inclined disk wind intersects twice a wide and highly eccentric orbit , not at the periastron ( see the right panel in fig . [ fig : geom ] ) , leading to a double periodicity ( one shorter than the other ) which has not been found yet _ only _ because of a lack of a continuous monitoring . this model can also explain the x ray emission from the persistently accreting hmxbs , if we admit that in this case the neutron star is always moving inside the equatorial wind component which lies on the orbital plane . in this framework , the sharp x ray lightcurve observed from can be modelled with different wind parameters ( for both polar and equatorial components ) depending on the orbital period ( 164.5 days or 329 days ) and the eccentricity of the binary . we assume a blue supergiant with a mass of 39 m@xmath18 and radius of 42 r@xmath18 , and a polar wind component with a terminal velocity of 1800 km s@xmath7 . the x ray lightcurve observed with @xmath0 is better reproduced assuming a `` polar wind '' mass loss rate of 5@xmath1310@xmath19 m@xmath18 yrs@xmath7 ( for a p@xmath12 of 164.5 days and an eccentricity of 0.4 ) and 9@xmath1310@xmath20 m@xmath18 yrs@xmath7 ( for a p@xmath12 of 329 days and a circular orbit , which is required by the fact that the two consecutive outbursts from reached roughly the same peak flux ) . the equatorial wind component should have a variable velocity ranging from 750 km s@xmath7 to 1400 km s@xmath7 ( for p@xmath12=164.5 days ) , and from 850 km s@xmath7 to 1600 km s@xmath7 ( for p@xmath12=329 days ) , and a density about 100 times higher than the polar wind component . note however that since the x ray luminosity expected for the wind accretion is proportional to @xmath21@xmath22@xmath23 ( where @xmath21 is the wind mass loss rate , and @xmath22 is the relative velocity of the wind with respect to the neutron star ) , different combinations of wind density and velocity in the equatorial component can reproduce the x ray lightcurve as well . in conclusion , in our model we explain the short recurrent flares if the neutron star intersects an inclined equatorial wind component ( once or twice ) during its orbit . a different particular geometry and inclination of this equatorial wind with respect to the orbital plane can account for the whole phoenomenology of both sfxts and persistently accreting hmxbs in general . both the orbital eccentricity and no - coplanarity can be explained by a substantial supernova kick at birth . this could indicate that sfxts are likely young systems , probably younger than persistent hmxbs . based on observations obtained with xmm - newton , an esa science mission with instruments and contributions directly funded by esa member states and the usa ( nasa ) . based on observations with integral , an esa project with instruments and the science data centre funded by esa member states ( especially the pi countries : denmark , france , germany , italy , switzerland , spain ) , czech republic and poland , and with the participation of russia and the usa . we thank the , , and @xmath24 teams for making these observations possible , in particular the duty scientists and science planners . pr thanks inaf - iasfmi for their kind hospitality . dg acknowledges the french space agency ( cnes ) for financial support . this work was supported by contract asi / inaf i/023/05/0 .

the physical mechanism responsible for the short outbursts in a recently recognized class of high mass x - ray binaries , the supergiant fast x - ray transients ( sfxts ) , is still unknown . recent observations performed with @xmath0 , and of the 2007 outburst from igrj11215@xmath15952 , the only sfxt known to exhibit periodic outbursts , suggest a new explanation for the outburst mechanism in this class of transients ; the outbursts could be linked to the possible presence of a second wind component in the supergiant companion , in the form of an equatorial wind . the applicability of the model to the short outburst durations of all other supergiant fast x - ray transients , where a clear periodicity in the outbursts has not been found yet , is discussed . the scenario we are proposing also includes the persistently accreting supergiant high mass x ray binaries .

You are an expert at summarizing long articles. Proceed to summarize the following text: several theories of particle physics as well as cosmology predict the existence of at least one sub ev scalar , that is , spin - zero , boson @xcite . correspondingly , many theories of physics beyond the standard model ( sm ) can accommodate scalars with very small masses and feeble couplings to sm fields @xcite . an intriguing possibility in astrophysics and cosmology is that these weakly interacting , sub ev particles ( wisps ) may constitute at least some component of the cold dark matter in the universe @xcite . it has been shown that these arguments apply for both pseudoscalar , namely axion @xcite , and scalar @xcite wisps , e.g. axion like particles ( alps ) that couple to two photons . the current experimental programs that explore the parameter space of weakly interacting , light , spin - zero bosons include sensitive searches that make use of resonant cavities @xcite . scalar alps may be detected by stimulation of the conversion to photons in a magnetic field , similar to the way that axion conversion may take place @xcite . this conversion can occur in a suitable cavity that is properly tuned to the mass of the alp . while it should be noted that scalar couplings to two photons are strongly excluded by fifth force experiments @xcite , it is possible that a low energy form factor could relax these constraints in some models @xcite . the lagrangian that describes the coupling of a light scalar particle to two photons in the presence of an external magnetic field takes the form : @xmath3 where @xmath4 is the electromagnetic field strength tensor , @xmath5 is the strength of the coupling between the scalar particle @xmath6 and two photons , @xmath7 is the magnetic field of the incident photon , and @xmath8 is the external magnetic field . from this lagrangian , the equations of motion that result may be used to write the power @xmath9 that results from axion to photon conversion on resonance in the microwave cavity as @xcite @xmath10 here , @xmath11 is the cavity volume , @xmath12 is the form factor associated with the cavity mode , @xmath13 is the quality factor of the cavity , and @xmath14 is the assumed scalar alp density . the results presented here are the first from a search for primordial scalar particles coupling to a strong external magnetic field using resonant cavities in the mass range of 0.14 mev corresponding to 34 ghz frequency . the apparatus @xcite consists of a tunable , 34 ghz resonant cavity ( q@xmath1510@xmath16 ) made from oxygen - free copper and a high electron mobility transistor ( hemt ) cryogenic amplifier @xcite located at the bottom of a cold gas cryostat that is oriented vertically and cooled to approximately 4 k. the cavity and cryogenic amplifier are coupled by approximately 10 cm of wr28 waveguide . the cryostat rests inside the vertical bore of a 7 t cryomagnet . the temperature inside the cryostat is monitored at multiple locations with cryogenic thin film resistance temperature sensors . the resonant cavity has one critically coupled and one weakly coupled port , each connected to wr28 waveguide . the critically coupled waveguide terminates at the input to the hemt amplifier , after which both waveguides feed out of the cryostat . the air in the cavity is pumped out through the weakly coupled waveguide . the q of the cavity is measured prior to each data run using a network analyzer connected to both waveguides . the cavity is tuned with an adjustable plunger that is vacuum tight at 4 k. a fiberglass g-10 rod is threaded through the top of the cryostat and fastened to the tuning plunger with a horizontal lever . when the g-10 rod is turned , the plunger moves vertically . the range of tuning in the cavity is @xmath175 mm which corresponds to approximately @xmath170.8 ghz . the signal from the cryogenic amplifier terminates outside the cryostat at a waveguide to coaxial adapter and triple heterodyne microwave receiver @xcite . after the receiver the voltages from the in phase and quadrature components are digitized for the complex fast fourier transform ( fft ) and further analysis offline . the sensitivity of the experiment is limited by the system noise temperature @xmath18 according to the dicke radiometer equation @xcite . @xmath19 typical power spectra measured with the cryogenic amplifier held at 6 k are shown in the left panel of figure [ fig : results ] . the resonance of the cavity appears as a dip in the spectrum . this dip is qualitatively consistent with the combination of two effects : a frequency - dependent reflection of noise power coming from the input of the hemt , and a change in the gain of the hemt as the impedance of the source decreases ( e.g. @xcite ) . the width of the dip is approximately 3 - 4 mhz which is in accordance with the cavity s q of 10@xmath16 . coupling from data similar to those shown in the left panel . in this initial result the measurements are centered at 34.29 ghz ( 0.142 mev ) and span across the width of the last bandpass filter ( 30 mhz ) . the gap in the center of the plot corresponds to frequencies at or near 0 hz in the baseband and have been excluded from this analysis.[fig : results],title="fig:",width=288 ] coupling from data similar to those shown in the left panel . in this initial result the measurements are centered at 34.29 ghz ( 0.142 mev ) and span across the width of the last bandpass filter ( 30 mhz ) . the gap in the center of the plot corresponds to frequencies at or near 0 hz in the baseband and have been excluded from this analysis.[fig : results],title="fig:",width=288 ] the expected power from couplings between a scalar alp and 2 photons is given in equation [ eq : power ] @xcite . the cavity form factor @xmath12 , adapted from @xcite for the case of a scalar alp , is @xmath20 where @xmath21 is the oscillating magnetic field in the cavity , @xmath22 is the static magnetic field , and @xmath23 is the magnetic permeability . for the configuration used in these measurements @xmath12=10@xmath24 is lower than would be desirable , but the cavity geometry was dictated by the constraints of a different experiment @xcite . using a limited set of measurements similar to those in the left panel of figure [ fig : results ] and assuming a primordial alp density @xmath14 of 10@xmath25/@xmath26 @xcite , the right panel shows that we exclude couplings with g@xmath010@xmath1/gev between 0.14 mev scalar particles and two photons with 5@xmath2 confidence . although the sensitivity of the present measurement does not exceed model - dependent limits on g@xmath27 set in previous searches for solar alps @xcite , by astrophysical observations @xcite , and by fifth force experiments @xcite , it is the first glimpse into this energy regime with a technique that has mass resolution . it is also the first direct search for alps as cold dark matter at 0.1 mev . immediate plans for the experiment include a cavity in a transverse magnetic mode which will allow coupling to pseudoscalar alps and will be 3 orders of magnitude more sensitive than the present measurement . a wider mass range will also be searched by tuning the receiver . in conclusion although this measurement is primarily a first step toward the goal of a more sensitive experiment , it is still an unprecedented , narrow band test of @xmath28 coupling limits that are otherwise model - dependent . this work was supported by the office of naval research award n000140910481 . 99 m. ahlers et . al . , phys . rev . d77 ( 2008 ) 095001 . j. jaeckel and a. ringwald , phys . lett . b659 ( 2008 ) 509 . b. holdom , phys . b166 ( 1986 ) 196 . r. foot and x.g . he , phys . b267 ( 1991 ) 509 . j. jaeckel , j. redondo , and a. ringwald , phys . ( 2008 ) 131801 . k. choi , phys . d 62 ( 2000 ) 043509 . i. waga and j. frieman , phys . d 62 ( 2000 ) 043521 . p. sikivie , invited talk at the idm2010 international conference on the identification of dark matter , montpellier , france , july 26 - 30 , ( 2010 ) . l. visinelli and p. gondolo , phys . d81 ( 2010 ) 063508 . p. arias , d. cadamuro , m. goodsell , et al . , desy-11 - 226 , mpp-2011 - 140 , cern - phth- 2011 - 323 , ippp-11 - 80 , dcpt-11 - 160 ( 2012 ) . p. sikivie , phys . rev . d32 ( 1985 ) 2988 . r. bradley et . al . , rev . phys . 75 ( 2003 ) 777 . f. caspers , j. jaeckel , and a. ringwald jinst 4 ( 2009 ) p11013 . j. jaeckel and a. ringwald , phys . b659 ( 2008 ) 509 . p. slocum et . al . , proceedings of the 6th patras workshop on axions , wimps and wisps , zurich , switzerland ( 2010 ) . e. g. adelberger , b. r. heckel , s. hoedl , et al . , 98 ( 2007 ) 131104 . s. j. asztalos , g. carosi , c. hagmann , et al . 104 ( 2010 ) 104 . p. slocum et . , proceedings of the 5th patras workshop on axions , wimps and wisps , durham , u.k . a. martin et . al . , proceedings of the 7th patras workshop on axions , wimps and wisps , mykonos , greece ( 2011 ) . s. weinreb , m. w. pospieszalski , r. norrod , proc . 1988 ieee mit - s international microwave symp . ( 1988 ) 945 . r. h. dicke , rev . sci instrum . 17 ( 1946 ) 268 . s. w. wedge and d. b. rutledge , ieee trans . on microwave theory and tech . 40:11 ( 1992 ) 2004 . r. p. meys , ieee trans . on microwave theory and tech . 26 ( 1978 ) 34 . e. i. gates , g. gyuk , m. s. turner , astrophys . j. lett . l123 ( 1995 ) 449 . k. zioutas , phys . 94 ( 2005 ) 121301 . g. raffelt , stars as laboratories for fundamental physics : the astrophysics of neutrinos , axions , and other weakly interacting particles , chicago university press ( 1996 ) .

light axion like particles ( alps ) that couple to two photons are allowed in a number of proposed extensions to the standard model of elementary particles . of particular interest from a theoretical and observational standpoint is the energy regime near 0.1 mev . we present results from a pilot experiment to search for a signal from a 0.14 mev scalar alp by way of its coupling to two photons . using a copper resonant cavity cooled to four degrees kelvin while immersed in a seven tesla magnetic field , and coupled to a low noise cryogenic amplifier and room temperature receiver , we exclude an alp driven excess of 34 ghz photons with g @xmath010@xmath1/gev with 5@xmath2 confidence . we discuss the ramifications of this initial measurement as well as planned modifications to the experiment for increased sensitivity .

You are an expert at summarizing long articles. Proceed to summarize the following text: the ground state ( `` vacuum '' ) of non - abelian gauge theories like qcd is known to be very rich . it includes topologically non - trivial fluctuations of the gauge fields , carrying an integer topological charge @xmath1 . the simplest building blocks of topological structure in the vacuum are @xcite _ instantons _ with @xmath2 and _ anti - instantons _ with @xmath3 . instantons represent gluon field configurations that are localized ( `` instantaneous '' ) in euclidean time and space . while they are believed to play an important role in various long - distance aspects of qcd , there are also important short - distance implications . in qcd with @xmath4 ( light ) flavours , instantons induce hard processes violating _ chirality _ in accord @xcite with the selection rule @xmath5 _ chirality _ @xmath6 , due to the general chiral anomaly . while in ordinary perturbative qcd ( @xmath7 ) these processes are forbidden , their experimental discovery would clearly be of basic significance . the deep - inelastic scattering regime is strongly favoured in this respect , since hard instanton - induced processes are both calculable @xcite within instanton - pertubation theory and have good prospects for experimental detection at hera @xcite . qcdins @xcite is a monte carlo package for simulating qcd - instanton induced scattering processes in deep - inelastic scattering ( hera ) . it is designed as an `` add - on '' hard process generator interfaced by default to the monte carlo generator herwig @xcite . it incorporates the theoretically predicted production rate for instanton - induced events as well as the essential characteristics that have been derived theoretically for the partonic final state of instanton - induced processes : notably , the flavour democratic @xcite and isotropic @xcite production of the final state partons , energy weight factors different for gluons and quarks @xcite , and a high average multiplicity @xmath8 of produced partons with a ( approximate ) poisson distribution of the gluon multiplicity @xcite . earlier versions of qcdins have been used already to establish first experimental bounds on the rate of instanton - induced events at hera @xcite and to develop instanton search strategies @xcite . in the present report a comprehensive description is given of the theoretical framework built into the program ( section [ theorie ] ) as well as of the various program components ( section [ qcdins ] ) and of their usage ( section [ usage ] ) . let us briefly summarize in this section the underlying physics picture , some relevant formulae and the main stages involved in qcdins to generate the complete instanton - induced partonic final state . the remaining formulae may be found under the corresponding descriptions of qcdins routines in section [ qcdins ] . ll & l + dis variables : + @xmath9 + @xmath10 + @xmath11 + @xmath12 + @xmath13 + @xmath14 + + variables of instanton - subprocess : + @xmath15 + @xmath16 + @xmath17 + in deep - inelastic @xmath18 scattering , instanton - induced events are predominantly associated @xcite with a process structure as sketched in fig . [ kin - var ] : a photon splitting into a @xmath19-pair fuses with a gluon from the proton in the background of an instanton ( @xmath20 ) or an anti - instanton ( @xmath21 ) . for each ( light ) flavour , @xmath22 , a violation of _ chirality _ is induced , @xmath23=\pm\ , 2 \ \mbox{\ for an } \left\{\begin{array}{l}i\\{{\overline{i}}}\end{array } \right . \,,\ ] ] in agreement with the general chiral anomaly relation @xcite . correspondingly , the partonic final state exhibits `` flavour democracy '' , i.e. @xmath24 ( @xmath25 ) pairs of _ all _ @xmath4 light flavours occur precisely once in case of an instanton ( anti - instanton ) , @xmath26 + n_g\,g \,.\ ] ] as illustrated in fig . [ kin - var ] , one of those partons acts as a current - quark ( jet ) @xmath27 , whereas the other @xmath28 ( anti-)quarks and some number @xmath29 of gluons are directly emitted from the instanton ( anti - instanton ) `` blob '' . instanton - induced processes initiated by a quark from the proton are suppressed by a factor of @xmath30 with respect to the gluon initiated process @xcite . this fact , together with the high gluon density in the relevant kinematical domain at hera , justifies to neglect quark initiated processes . in instanton - perturbation theory , the dominant instanton - induced contribution to the inclusive @xmath31 cross section at cross sections . ] , subject to appropriate kinematical restrictions and ( theoretical ) fiducial cuts , has a convolution - like structure @xcite , @xmath32 \label{evwgt } & & \times \int\limits_{{\rm max}\left(\frac{q^{\prime 2}}{sx^\prime y_{\rm bj\ , max}},\frac{x_{\rm bj\,min}}{x^\prime } \right)}^{z_{\rm max } } \frac{dz}{z}\,f_g ( z ) \int\limits_{x_{\rm bj\,min}}^{x^\prime z - \frac{m_{k}^2}{s } \frac{1}{y_{\rm bj\,max}-\frac{q^{\prime 2}}{sx^\prime z } } } \frac{dx_{\rm bj}}{x_{\rm bj}}\ , \\[2ex ] \nonumber & & \times \int\limits_{{\rm max}\left ( \frac{q^{\prime 2}}{sx^\prime z}+\frac{m_{k}^2}{s } \frac{1}{x^\prime z -x_{\rm bj}},y_{\rm bj\,min } \right)}^{y_{\rm bj\,max } } \frac{dy_{\rm bj}}{y_{\rm bj}}\ , \theta ( s x_{\rm bj } y_{\rm bj } - q^2_{\rm min})\ , \\[2ex ] \nonumber & & \times \frac{1+(1-y_{\rm bj})^2}{y_{\rm bj}}\ p_{q^\prime}^ { { ( i)}}\ .\end{aligned}\ ] ] it involves integrations over the gluon density @xmath33 , the virtual photon flux and the flux of virtual ( anti-)quarks @xmath34 in the instanton - background @xcite , @xmath35 all relevant kinematical variables in eq . ( [ evwgt ] ) are defined in fig . [ kin - var ] , and @xmath36 denotes the electric charge squared of the virtual ( anti-)quark @xmath34 in units of the electric charge squared @xmath37 . [ ht ] in eq . ( [ evwgt ] ) , @xmath38 denotes the instanton - induced total cross section of the @xmath39-subprocess ( c.f . [ kin - var ] ) and contains the essential instanton dynamics . its analytical form @xcite used in qcdins may be found in eq . ( [ qgcross - vxi ] ) . as illustrated in fig . [ isorho ] , @xmath38 is very steeply growing for decreasing values of @xmath40 and @xmath41 , respectively . eventually , our theoretical predictions based on instanton - perturbation theory @xcite will cease to hold . therefore , the following cuts inferred from a high - quality lattice simulation of qcd @xcite are implemented by default in qcdins 2.0 ( table [ floating ] ) , @xmath42 a further cut on the photon virtuality , @xmath43 is applied in order to warrant sufficient suppression of non - planar contributions @xcite , which are hard to calculate and may spoil the validity of eq . ( [ evwgt ] ) . the cross section @xmath38 exhibits a rather weak residual dependence on the renormalization scale @xmath44 . as an `` optimal '' choice , the value @xmath45 , corresponding to the minimum @xcite , @xmath46 , is taken by default ( table [ floating ] ) . however , @xmath38 depends strongly on the qcd scale @xmath47 . since , strictly speaking , the underlying theoretical framework refers to massless quarks , the ( default ) number of flavours is set to @xmath48 ( table [ floating ] ) . the respective value @xmath49 is obtained by a standard 3-loop perturbative flavour reduction ( eq . ( 9.7 ) of ref . @xcite ) from the 1998 world - average of the running qcd coupling at the z - boson mass @xcite , @xmath50 the central values of these parameters are taken as default in qcdins 2.0 ( table [ floating ] ) . this upgrade of @xmath47 , together with the modified cuts ( [ fiducial ] ) and ( [ fiducialq ] ) , represents an improved understanding of the input parameters and a considerable reduction of uncertainties , as compared to the original publication @xcite and earlier versions of qcdins . note that it also implies a significant change in the predicted rate . next , let us summarize the various stages of event generation by means of qcdins . in a first stage , the various bjorken variables @xmath51 of the instanton - induced process ( c.f . [ kin - var ] ) are generated , with a distribution according to the normalized differential cross section from eq . ( [ evwgt ] ) , @xmath52 in the second stage of the event generation , the 4-momenta @xmath53 of the incoming gluon @xmath54 , the virtual photon @xmath55 , the virtual quark @xmath34 and the current quark @xmath27 , respectively , are filled . sudakov decompositions of these momenta are used to incorporate various constraints , e.g. on the momenta squared . the 4-momentum @xmath56 of the outgoing lepton is calculated subsequently . in the third stage , the partonic final state of the instanton - induced @xmath57-subprocess is generated in its centre - of - mass system ( cms ) as follows . the number @xmath29 of produced gluons is generated according to a poisson distribution with mean @xmath58 ( for the cuts ( [ fiducial ] ) ) , as calculated theoretically ( eq . ( [ ng ] ) ) in instanton - perturbation theory @xcite . next , @xmath59 @xmath60$]-``strings '' of partons are set up , each beginning with a quark , followed by a random number @xmath61 of gluons and ending with an anti - quark of randomly chosen flavour . there are @xmath62 gluons in total and , due to the required flavour democracy ( [ flavour - democr ] ) , @xmath19-pairs of all @xmath4 flavours occur precisely once . a quark and a gluon among these @xmath63 partons are ( randomly ) marked as incoming . the momenta @xmath64 of the @xmath65 outgoing partons are then generated in the cms of the instanton subprocess , according to the energy - weighted phase - space @xmath66 these different energy weights for quarks and gluons @xcite , along with the angular isotropy @xcite , are characteristic features of the leading - order partonic final state ( after averaging over colour ) . next , the colour and flavour connections of the partons are set up . the colour flow is obtained simply by connecting the colour lines of adjacent partons within each of the above - mentioned @xmath4 @xmath60$]-``strings '' in a planar manner ( consistent with the leading order @xmath67 expectation ) . this choice is inspired by the leading - order partonic final state ( after averaging over colour ) @xcite , but may well deserve further research . the flavour flow is constructed by connecting the flavour lines of the quark at the beginning of a string with the flavour line of the anti - quark at the end of a string . the hard subprocess generation ends by boosting the momenta of the final state partons to the laboratory frame . while the subsequent perturbative evolution of the generated partons is always handled by the herwig @xcite package , the final hadronization step may optionally be performed also by means of jetset @xcite . this section is devoted to a systematic description of the various routines of the qcdins package that is designed as an `` add - on '' hard process generator , interfaced to the monte carlo generator herwig @xcite . this reference section is organized as follows : while all subroutines and functions of the qcdins package are described in _ alphabetical order _ in section [ routines ] , a _ logical flow - chart _ is provided in form of tables [ flow - chart ] and [ flow - qihgen ] below . they should always be used as the main guide through the description of the package . a routine listed in the n - th column and the m - th row of these tables progressively calls all routines in the ( n+1)-th column starting in the ( m+1)-th row . all routines called at the level of the main hard process generator * qihgen * and below are documented in table [ flow - qihgen ] . a specific application requires the writing of a steering program by the user ( c.f . section [ usage ] and appendix a ) . it must contain the standard herwig - initialization calls as well as the calls to various initialization routines for qcdins . the latter comprise essentially the four routines listed in the second column of table [ flow - chart ] . by calling the last one of these ( * qcloop * ) , the user starts the proper simulation which comprises the chain of internally called qcdins routines as documented in tables [ flow - chart ] and [ flow - qihgen ] . a specific and quite extensive example is provided with the qcdins distribution ( qtesthz.f ) and may be found in the directory qcdtest. it also illustrates the use of the event analysis routine * hwanal * called by herwig after each processed event . meters ( tables [ floating ] , [ flags ] ) & [ -1.5ex]*qiinit * & & & & + flavour reduction : & & & & & + @xmath68 & & [ -1.5ex]*lamnf * & & & + initialize particle " & & & & & + inst in event record & & [ -1.5ex]*qiinih * & & & + print input parame- & & & & & + ters and warnings & [ -1.5ex]*qistat * & & & & + initialize jetset & & & & & + common block data & [ -1.5ex]*gjeini * & & & & + loop over desired & & & & & + number of events & [ -1.5ex]*qcloop * & & & & + generate one event & & * qcdgen * & & & + assign hard process & & & & & + variables ( herwig ) & & & [ -1.5ex]_hwepro _ & & + no action ( modified & & & & & + herwig routine ) & & & & [ -1.5ex]*hwegam * & + call main instanton & & & & & + process generator & & & & [ -1.5ex]*hvhbvi * & + main ( hard ) instanton & & & & & + process generator & & & & & ' '' '' [ -1.5ex ] + generate parton & & & & & + cascades ( herwig ) & & & [ -1.5ex]_hwbgen _ & & + combine jets with & & & & & + correct kinematics & & & & [ -1.5ex]*hwbjco * & + convert herwig to & & & & & + jetset block data & & & [ -1.5ex]*herlund * & & + jetset event record & & & & & + to hepevt common & & & [ -1.5ex]*luhepc * & & + radiation from lepton & [ -1.5ex]*exfrac * & & + check kinematical boundaries & * qicalc & & + generate identity code of current & & & + quark @xmath27 and virtual quark @xmath69 & [ -1.5ex]*qihpar * & & + generate @xmath70 and associated weight & * qihins & & + generate @xmath71 as @xmath72 & & * qirdis & + calculate total cross section of & & & + instanton - induced subprocess @xmath73 & & [ -1.5ex]*q2sig * & + calculate @xmath74-valley action @xmath75 & & & * action + calculate fermionic overlap @xmath76 & & & * omega + calculate lambert w - function & & & * lambertw + calculate saddle - point value of & & & + conformally invariant @xmath74-distance @xmath77 & & & [ -1.5ex]*xi * + * * * * * * ' '' '' calculate inverse running coupling @xmath78 & & & ' '' '' * xqs * + generate number @xmath29 of emitted gluons & * qigmul & & + calculate gluon multiplicity @xmath79 & & * gmult & + * * ' '' '' calculate inverse running coupling @xmath78 & & & ' '' '' * xqs * + calculate @xmath74-valley action @xmath75 & & & * action + calculate lambert w - function & & & * lambertw + calculate flux of virtual quark @xmath69 & * qisplt & & + calculate remaining weight & * qipvwt & & + calculate momentum of incoming gluon & * qikpar & & + generate momentum of virtual photon & * qikgam & & + generate momenta of virtual quark @xmath69 & & & + and current quark @xmath27 & [ -1.5ex]*qikgsp * & & + generate partonic final state & * qistid & & + generate the partons of the instanton & & & + subprocess in form of @xmath80$]-strings & & [ -1.5ex]*qiglst * & + find the incoming partons & & & + in the @xmath80$]-strings & & [ -1.5ex]*qigpar * & + assign masses of outgoing partons & & * qiplst & + generate 4-momenta of outgoing partons & & * qipsgn & + calculate relative energy weight & & & + of outgoing partons & & & [ -1.5ex]*qipswt * + store 4-momenta of outgoing partons & & & + into phep common block of herwig & & [ -1.5ex]*qipsto * & + colour / flavour connections for each string & & * qiccon & + * * * * * * * * * * subroutine * action*(xi4,s , ds , dds ) abcdefghikl calculation of the @xmath74-action as well as its 1st and 2nd derivatives , as function of the conformally invariant @xmath74-distance . + abcdef conformally invariant @xmath74-distance @xmath77 . @xmath74-action @xmath75 , eq . ( [ action ] ) . @xmath81 , @xmath82 the action is calculated according to the exact valley form @xcite , @xmath83 \ln\left [ \frac{1}{2\xi } \bigl ( f(\xi ) + 4\bigr)\right ] \ , , \\[2.4ex ] f(\xi ) & = & \xi^2+\sqrt{\xi^2 - 4}\xi-4 \ , . \end{aligned}\ ] ] subroutine * exfrac*(a ) abcdefghikl optional account of initial state radiation from the lepton . + ab abcdef dummy rescaling factor of the incoming lepton momentum ; double precision output variable . the actual routine has to be provided by the user . subroutine * gjeini * abcdefghikl initialization of the jetset @xcite parameter common blocks ludat1 , ludat2 , ludat3 , ludat4 and ludatr . + * gjeini * has to be called by the user before any other jetset routine . * gjeini * is from ref . @xcite . function * gmult*(xpr , xi_min , xi_max , qlam , kappa , nf , loopfl ) abcdefghikl calculation of the average gluon multiplicity @xmath84 depending on @xmath41 , @xmath85 , @xmath86 , @xmath4 and loop - order . here , @xmath44 and @xmath4 denote the renormalization scale and the number of light flavours , respectively . + abcdefghi @xmath41 @xmath87 ; lower boundary of @xmath77 used for interpolation . @xmath88 ; upper boundary of @xmath77 used for interpolation . @xmath85 @xmath86 @xmath4 ; number of light flavours . abcd 1-loop renormalization group ( rg ) invariance @xcite along with 1-loop form of @xmath89 . 2-loop rg invariance @xcite along with 2-loop form of @xmath90 . ( default ) 2-loop rg invariance along with 3-loop form of @xmath90 . from an analysis based on the generalized ( mueller @xcite ) optical theorem for the @xmath91 forward scattering amplitude and the @xmath74-valley method , one infers @xcite the differential one - gluon inclusive @xmath92 cross section , normalized by the total cross section @xmath38 . the mean gluon multiplicity @xcite is then found by phase space integration , @xmath93 the function * gmult * calculates and returns the average gluon multiplicity ( [ ng ] ) . the stars @xmath94 in eq . ( [ ng ] ) denote the saddle point values of the @xmath74 collective coordinates @xmath95 , @xmath96 and @xmath77 . their computation proceeds as in the descriptions of the functions * q2sig * and * xi*. the required values of @xmath97 are calculated and returned by the function * xqs*. the @xmath74-action @xmath75 and its @xmath77-derivatives are provided by the subroutine * action*. subroutine * herlund * abcdefghikl conversion of the herwig @xcite event record in the hepevt common block to the respective jetset @xcite common block . + * herlund * is a modified @xcite copy from the jetset subroutine luhepc . subroutine * hvhbvi * abcdefghikl call of the main ( hard ) instanton process generator * qihgen*. + replaces a dummy stub in herwig @xcite that was originally used as event generation interface for the monte carlo generator herbvi @xcite for baryon number violating interactions . used in the qcdins package to select qcd - instanton induced processes via mod(iproc/100,100 ) @xmath98 75 . the process code iproc (= 17600 ) has to be set in the user s steering program ( c.f . appendix a ) . furthermore , the qcdins program header is printed . subroutine * hwbjco * abcdefghikl modification of herwig @xcite routine to account for instanton - induced scattering . + the modifications are @xcite : the logical flag ( dispro ) for keeping the lepton momenta fixed in herwig 5.9 is modified to include also instanton - induced dis , ipro = 76 . furthermore , a bug concerning energy - momentum conservation in the original routine from herwig 5.9 has been fixed . subroutine * hwegam*(ihep , zmi , zma , wwa ) abcdefghikl modification to avoid standard generation of the ( virtual ) photon at this stage . + c.f . ref . @xcite this is a modified routine from herwig 5.9 ( c.f . ref . usually , * hwegam * generates an incoming photon from the incoming @xmath99 . within qcdins , however , the photon is generated at a later stage in the subroutine * qikgam*. thus , the herwig routine * hwegam * has been modified to immediately return for instanton - induced processes ( ipro=76 ) . function * lambertw*(x ) abcdefghikl calculation of the principal branch of the lambert w - function @xmath100 for @xmath101 . + x : @xmath102 ; argument of the lambert w - function . @xmath100 is the ( real ) solution of @xmath103 , analytic at @xmath104 . the following simple , but accurate approximation is used and returned by * lambertw * : @xmath105 \ln(x-4)-(1-\frac{1}{\ln(x)})\cdot\ln(\ln(x ) ) ; & { \rm\ for\ } x > 500\ , . \end{array } \right.\ ] ] function * lamnf*(nf , lambda5 ) abcdefghikl calculation of @xmath106 from @xmath107 to order @xmath108 . + abcdefghijklm number of ( light ) flavours , @xmath4 . input value @xmath107 . the flavour reduction of @xmath107 to the desired number of light flavours is performed by using eq . ( 9.7 ) of ref . @xcite . subroutine * luhepc*(mconv ) abcdefghikl conversion of the jetset @xcite event record contents back to the hepevt common block . + mconv = 1 the present routine is a modified @xcite version of the jetset routine * luhepc*. function * omega*(xi4 ) abcdefghikl calculation of the fermionic overlap , as function of the conformally invariant @xmath74-distance . + xi4 : conformally invariant @xmath74-distance @xmath77 . the following simple , but accurate approximation for the fermionic overlap @xcite @xmath109 is used and returned by * omega * : @xmath110 function * q2sig*(xprime , qlam , kappa , loopfl , nf ) abcdefghikl calculation of the total cross section @xmath111\ ] ] for the instanton - induced subprocess , depending on @xmath41 , @xmath85 , @xmath86 , loop - order and @xmath4 . here , @xmath44 and @xmath4 denote the renormalization scale and the number of light flavours , respectively . + abcdefghi @xmath41 @xmath85 @xmath86 abcd 1-loop renormalization group ( rg ) invariance @xcite along with 1-loop form of @xmath89 . 2-loop rg invariance @xcite along with 2-loop form of @xmath90 . ( default ) 2-loop rg invariance along with 3-loop form of @xmath90 . @xmath4 ; number of light flavours . the function * q2sig * calculates and returns the cross section ( [ cs ] ) as derived in ref . @xcite , @xmath112 & & \times \frac{\omega ( \xi_\ast ) ^{2n_f-1}(\xi_\ast -2)^3 v^{\ast\,5 } } { \sqrt{\frac{1}{2}({\tilde{s}}-v_\ast -2 { d(\tilde{s})})^2 + { \tilde{s}}({\tilde{s}}-v_\ast ) { d\left ( \ln \left ( \frac{{d(\tilde{s})}}{\sqrt{\xi_\ast -2 } } \right)\right ) } } } \\[1.6ex ] & & \times \left ( \frac{4\pi } { \alpha_{\overline{\rm ms}}\left(\mu _ r \right)}\right)^{19/2 } \exp \left [ -\frac{4\pi } { \alpha_{\overline{\rm ms}}\left(\mu _ r \right ) } s^{({i\overline{i}})}\left(\xi_\ast \right ) - 2 \left(1-\ln\left(\frac{v_\ast\mu_r}{q^\prime}\right)\right ) \,{\tilde{s}}\right ] \ , . \nonumber\end{aligned}\ ] ] it is expressed entirely in terms of the saddle point values for the @xmath74 collective coordinates , @xmath113 ( conformally invariant distance ) and @xmath114 ( scaled size ) . for given @xmath115 , @xmath85 and ( scaled ) renormalization scale @xmath86 , these are in turn unique solutions of the saddle point equations @xcite @xmath116 \label{eq1 } v_\ast & = & 2\ , d({\tilde{s}}(\xi_\ast))\ , w\left ( \frac{q^\prime}{\mu_r } \frac{\exp\left\{\frac{1}{2 } \left[\frac{4\pi } { \alpha_{\overline{\rm ms}}\left(\mu _ r \right)}\frac{1}{\delta_1\beta_0 } + \frac{{\tilde{s}}(\xi_\ast)}{d({\tilde{s}}(\xi_\ast ) ) } \right ] \right\ } } { 2\,d({\tilde{s}}(\xi_\ast ) ) } \right),\end{aligned}\ ] ] with @xmath117 and @xmath118 the @xmath74-action @xmath75 as well its @xmath77-derivatives , entering the cross section ( [ qgcross - vxi ] ) and eqs . ( [ eq2 ] ) , ( [ eq1 ] ) through @xmath119 are calculated in the subroutine * action*. the fermionic overlap @xmath120 is calculated and returned by the function * omega*. in eq . ( [ eq1 ] ) , @xmath121 denotes the principal branch of the lambert @xmath121-function , i.e. the ( real ) solution of @xmath103 , analytic at @xmath104 . the latter is calculated and returned by the function * lambertw*. the first step in the solution of the saddle - point equations ( [ eq2 ] ) , ( [ eq1 ] ) consists in eliminating @xmath122 in eq . ( [ eq2 ] ) by inserting eq . ( [ eq1 ] ) . next , for given @xmath115 , @xmath123 and @xmath86 , the resulting implicit equation is solved numerically for @xmath113 . this is done by the function * xi * which provides @xmath124 on return . the latter is then inserted into eq . ( [ eq1 ] ) providing @xmath125 . the values of @xmath97 are calculated and returned by the function * xqs*. subroutine * qcdgen * abcdefghikl interface for the generation of one instanton - induced event , including calls to event initialization , hadronization and event termination routines . + * an instanton - induced event is initialized by a call to the herwig @xcite subroutine * hwuine*. * the partonic instanton subprocess is generated by a call to the herwig subroutine * hwepro*. * if the hadronization flag qicont(21 ) is set .true . ( default ) , the event is fully hadronized . else , the event is finalized ( * hwufne * ) immediately after the call of * hwepro*. depending on the control flag for the hadronization model , qicont(18)=.true./.false . , the hadronization is either performed by appropriate herwig @xcite or ( modified @xcite ) jetset @xcite routines , respectively ( see table [ hadroutines ] ) . * furthermore , in this routine energy - momentum conservation in the generated event is checked . .[hadroutines ] depending on the flag qicont(18 ) = .true./.false . , herwig @xcite and jetset @xcite hadronization may be selected . the respective calls to herwig and ( modified ) jetset routines used in * qcdgen * are displayed . [ cols="^,<,<",options="header " , ] subroutine * qiusps(wt ) * abcdefghikl dummy routine for optional modification of relative phase space weight as calculated in * qipswt*. + wt : modified relative phase space weight . called from * qipswt * if qicont(2 ) = .false .. calculation of wt has to be provided by the user . function * xi * ( xpr , xi_min , xi_max , xq , xmu , nf , loopfl ) abcdefghikl the saddle point value of the conformally invariant @xmath74-separation @xmath77 is calculated as function of @xmath115 , @xmath126 , @xmath127 and @xmath4 . + + abcdefghi @xmath115 @xmath87 ; lower boundary of @xmath77 used for interpolation . @xmath88 ; upper boundary of @xmath77 used for interpolation . @xmath126 @xmath127 @xmath4 ; number of light flavours . abcd 1-loop renormalization group ( rg ) invariance @xcite along with 1-loop form of @xmath89 . 2-loop rg invariance @xcite along with 2-loop form of @xmath90 . ( default ) 2-loop rg invariance along with 3-loop form of @xmath90 . first , the saddle point equation ( [ eq2 ] ) is solved analytically for @xmath41 , and the @xmath128-values corresponding to a discrete set of @xmath129 values are calculated as @xmath130 with @xmath131 and @xmath132 from eqs . ( [ eq1])([dadada ] ) inserted : @xmath133 \nonumber & & \hspace{-5ex } w\left ( \frac { \exp\left\ { \frac{1}{2}\frac{{\tilde{s}}}{{d(\tilde{s } ) } } + \frac{1}{2 } \frac{\delta_1 - 1}{\delta_1\beta_0 } \left [ \delta_1 \ln \left ( \frac{\delta_1 x(\mu_r ) } { x(q^\prime ) + ( \delta_1 - 1)x(\mu_r ) } \right ) -1 \right ] x(\mu_r ) + \frac{1}{2}\frac{x(q^\prime ) } { \beta_0 } \right\ } } { 2\,{d(\tilde{s } ) } } \right ) .\end{aligned}\ ] ] in eq . ( [ vastxq ] ) , the explicit dependence of @xmath132 on @xmath134 ( c.f . ( [ eq1 ] ) has been eliminated in favour of an @xmath135 , @xmath136 dependence by means of the standard scale transformation of @xmath90 . here , we have introduced the shorthand @xmath137 the desired continuous inversion @xmath138 is then achieved by means of numerical interpolation based on the above exact supporting points ( @xmath139 ) . function * xqs*(qlam , loopfl , nf ) abcdefghikl calculation of @xmath140 as function of @xmath141 , where @xmath142 is a generic mass scale . + abcdefghik @xmath141 abcd 1-loop form of @xmath89 . 2-loop form of @xmath90 . ( default ) 3-loop form of @xmath90 . @xmath4 ; number of light flavours . the running coupling @xmath90 is calculated according to the explicit formula eq . ( 9.5a ) in ref . @xcite , which is accurate to 3-loop . the quantity @xmath97 is returned . the integer variable loopfl = @xmath143 controls the loop - order at which eq . ( 9.5a ) in ref . @xcite is truncated . qcdins 2.0 should be loaded together with herwig @xcite version 5.9 and jetset @xcite version 7.4 , that are part of the cernlib distribution . the program is a slave system , which the user must call from his own steering program . a very simple example is provided in appendix a. by default , qcdins is compiled in form of a library , libqcdins.a , that may be linked together with libherwig59.a , libjetset74.a and the library libpdflib.a of parton distribution functions to the steering program . an extensive demonstration program , including an interface to the hztool @xcite package , and a detailed installation instruction are included in the distribution . information about the distribution , its update history , an interactive manual , the source code , pictures of typical events etc . can be accessed via the qcdins www site , http://www.desy.de/~t00fri/qcdins/qcdins.html . first of all , we are grateful to m. gibbs who left physics in may 1995 . without his early contributions the qcdins package would presumably not exist . a number of people helped improving the code . in particular , let us mention t. carli , g. grindhammer , h. jung and m. seymour . specifically , we thank h. jung for his efforts to incorporate an interface to the jetset package . moreover , we are grateful to m. seymour for an evaluation of the qcdins package during the desy workshop 1998/99 on monte carlo generators for hera physics . finally , we thank g. ingelman for a careful reading of the manuscript . the following code may serve as the simplest illustration of a steering program for the qcdins / herwig package . note that the two subroutines * hwanal * and * hwaend * must exist . for an example of an interface with the jetset hadronization , we refer to the extensive steering program qtesthz included in the qcdins distribution . .... program qcdins # include " herwig.inc " c force inclusion of block data external hwudat c initialize process number iproc = 17600 c maximal number of events in this run maxev = 1000 c beam particles pbeam1 = 27.5d0 pbeam2 = 820.0d0 part1 = ' e+ ' part2 = ' p ' c initialize common blocks call hwigin c user can reset parameters at this point , otherwise values c set in hwigin will be used . c no vertex information in event printout prvtx=.false . c reset number of shots for initial max weight search ibsh = 5000 lrsud=0 lwsud=77 c seeds nrn(1)=106645412 nrn(2)=135135176 c use laboratory frame usecmf = .false . c compute parameter - dependent constants call hwuinc c number of herwig events to print out maxpr = 1 c call hwusta to make any particle stable call hwusta('pi0 ' ) call hwusta('k_s0 ' ) c initialize default qcdins input parameters call qiinit c print input parameters call qistat c loop over events call qcloop c user 's terminal calculations call hwaend stop end subroutine hwanal return end subroutine hwaend return end .... below , we display the essential output from a test run of the very simple steering program from appendix a. this test run simulates 1000 complete instanton - induced events in deep - inelastic @xmath144 scattering ( hera ) , with @xmath145 gev and @xmath146 gev , in the laboratory frame . all the qcdins parameters correspond to the default values as set in the initialization routine * qiinit * ( c.f . tables [ floating ] and [ flags ] ) . .... = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = qcd instanton monte carlo information version 2.0 = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = parameter value = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = default gluon mass ( gev ) 0.7500e+00 minimum allowed value of x_bj 0.1000e-02 maximum allowed value of z ( proton mom.frac . ) 0.1000e+01 maximum allowed value of y_bj 0.1000e+01 minimum allowed value of y_bj 0.1000e+00 maximum allowed value of x prime 0.9000e+00 minimum allowed value of x prime 0.3500e+00 maximum allowed value of q prime * * 2 ( gev**2 ) 0.1652e+04 minimum allowed value of q prime * * 2 ( gev**2 ) 0.1134e+03 lower cut for me calculation on x prime 0.3500e+00 lowest allowed weight efficiency cut -.1000e-09 minimum instanton invariant mass 0.0000e+00 lambda - ms - bar(nf ) [ gev ] from pdg 1998 0.3459e+00 minimum total k.e . of outgoing partons ( gev ) 0.0000e+00 renormalization point kappa = mu_r / qprime : 0.1500e+00 minimum allowed value of q * * 2 ( gev**2 ) 0.1134e+03 factorization scale ( gev ) 0.1065e+02 = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = i d code assumed for instanton 206 maximum number of mambo iterations per ps wt . 100 maximum number of ps wt . rejections per event 300 number nf of ( light ) flavours 3 maximum average gluon multiplicity in distbn . 10 maximum number of iterations for me generation 20 maximum number of qigmul iterations 40 number of loops in rg - invariance / alpha_s 3 = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = control flag option setting = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = reweight phase space configurations true use default phase space reweighting true use matrix element weight true disregard instanton minimum mass requirement true generate q prime before x prime true enforce mass of current quark in kinematics true enforce limit on maximum number of gluons true ensure mass less than subprocess energy true kill events with insufficient instanton mass true use z generation as dz / z true use x prime * * -n generation for efficiency true use q prime * * -n generation for efficiency true use herwig rather than jetset hadronization true use random azimuth angle for scattered electron true use full hadronization true = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = process code iproc = 17600 random no . seed 1 = 1246579 seed 2 = 8447766 number of shots = 5000 * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * qcdins version 2.0 * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * .... .... new maximum weight = 2.4520378923431534e-05 new maximum weight = 6.0858601240747906e-02 new maximum weight = 0.1289870599803515 new maximum weight = 0.3662942637479250 new maximum weight = 0.5705336816482935 new maximum weight = 0.8061467165013312 number of events = 0 number of weights = 5000 mean value of wgt = 2.8721e-02 rms spread in wgt = 6.4377e-02 actual max weight = 7.3286e-01 assumed max weight = 8.0615e-01 ihep i d idpdg ist mo1 mo2 da1 da2 p - x p - y p - z energy mass 1 e+ -11 101 0 0 4 0 0.00 0.00 27.50 27.50 0.00 2 p 2212 102 0 0 0 0 0.00 0.00 -820.00 820.00 0.94 3 cmf 0 103 1 2 0 0 0.00 0.00 -792.50 847.50 300.33 ihep i d idpdg ist mo1 mo2 da1 da2 p - x p - y p - z energy mass 5 e+ -11 121 7 9 21 9 0.00 0.00 27.50 27.50 0.00 6 gluon 21 122 7 20 22 19 0.00 0.00 -50.37 50.38 0.75 7 hard 0 120 5 6 9 20 0.53 0.99 -22.98 77.98 74.51 8 inst 0 3 7 0 0 0 0.23 9.33 -43.54 47.01 15.07 9 e+ -11 123 7 5 26 5 8.20 -7.47 3.48 11.63 0.00 10 ubar -2 124 7 14 27 17 -8.43 -1.86 17.19 19.24 0.31 11 dqrk 1 124 7 12 31 13 -0.07 0.52 -7.65 7.67 0.32 12 gluon 21 124 7 13 33 11 0.16 -0.28 -0.64 1.04 0.75 13 dbar -1 124 7 11 35 12 -0.67 -0.12 -2.65 2.75 0.32 14 uqrk 2 124 7 15 37 10 0.01 4.97 -11.73 12.75 0.32 15 gluon 21 124 7 16 39 14 0.53 0.62 -2.17 2.44 0.75 16 gluon 21 124 7 17 41 15 0.43 -0.43 -8.44 8.49 0.75 17 gluon 21 124 7 10 43 16 -0.72 0.09 -2.40 2.62 0.75 18 sqrk 3 124 7 19 45 20 -0.11 2.50 -4.93 5.55 0.50 19 gluon 21 124 7 6 47 18 0.81 0.93 -1.46 2.06 0.75 20 sbar -3 124 7 18 51 6 -0.14 0.52 -1.47 1.64 0.50 21 z0/gama * 23 3 5 7 0 0 -8.20 7.47 24.02 15.87 -21.17 .... .... check of energy - momentum conservation in the event : = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = sum_i(p_i)[gev ] = ( 4.87e-13 -5.49e-13 1.59e-12 -4.20e-12 ) with 60 stable particles in final state contributing . number of events = 1000 number of weights = 27258 mean value of wgt = 2.8837e-02 rms spread in wgt = 6.3371e-02 actual max weight = 7.0199e-01 assumed max weight = 8.0615e-01 9 a. belavin , a. polyakov , a. schwarz and yu . tyupkin , . t hooft , ; ; ( erratum ) ; . s. moch , a. ringwald and f. schrempp , . a. ringwald and f. schrempp , . a. ringwald and f. schrempp , hep - ph/9411217 , in : quarks 94 , eds . d. grigoriev , v. matveev , v. rubakov , d. son and a. tavkhelidze ( world scientific , singapore , 1995 ) p. 170 . m. gibbs , a. ringwald and f. schrempp , hep - ph/9506392 , in : workshop on deep inelastic scattering and qcd ( dis 95 ) , eds . f . laporte and y. sirois ( ecole polytechnique , paris , 1995 ) p. 341 . t. carli , j. gerigk , a. ringwald and f. schrempp , hep - ph/9906441 , in : monte carlo generators for hera physics , eds . a.t . doyle , g. grindhammer , g. ingelman and h. jung , desy - proc-1999 - 02 ( desy , hamburg , 1999 ) p. 329 . g. marchesini et al . , . s. moch , a. ringwald and f. schrempp , hep - ph/9706400 , in : deep inelastic scattering and qcd ( dis 97 ) , eds . j. repond and d. krakauer ( american institute of physics , woodbury , new york , 1997 ) p. 1007 . a. ringwald and f. schrempp , hep - ph/9706399 , in : deep inelastic scattering and qcd ( dis 97 ) , eds . j. repond and d. krakauer ( american institute of physics , woodbury , new york , 1997 ) p. 781 . s. aid et al . , h1 collaboration , . s. aid et al . , h1 collaboration , . t. carli and m. kuhlen , . s. moch , a. ringwald and f. schrempp : in preparation a. ringwald and f. schrempp , . c. caso et al . , particle data group , . h. jung , private communication . t. sjstrand , . v.v . khoze and a. ringwald , . j. verbaarschot , . g. grindhammer , private communication . a. mueller , . m. gibbs and b. webber , . t. carli , private communication . j. owens , . r. kleiss and w. stirling , . m. seymour , private communication . j. bromley et al . , in : future physics at hera , eds . g. ingelman , a. de roeck and r. klanner ( desy , hamburg , 1996 ) vol . 1 ,

we describe a monte carlo event generator for the simulation of qcd - instanton induced processes in deep - inelastic scattering ( hera ) . the qcdins package is designed as an `` add - on '' hard process generator interfaced to the general hadronic event simulation package herwig . it incorporates the theoretically predicted production rate for instanton - induced events as well as the essential characteristics that have been derived theoretically for the partonic final state of instanton - induced processes : notably , the flavour democratic and isotropic production of the partonic final state , energy weight factors different for gluons and quarks , and a high average multiplicity @xmath0 of produced partons with a poisson distribution of the gluon multiplicity . while the subsequent perturbative evolution of the generated partons is always handled by the herwig package , the final hadronization step may optionally be performed also by means of the general hadronic event simulation package jetset . and qcd ; instanton ; deep - inelastic scattering ; monte carlo simulation 11.15.kc ; 12.38.lg ; 13.60.hb * program summary * + _ title of program : _ qcdins 2.0 + _ catalogue identifier : _ + _ program obtainable from : _ http://www.desy.de/~t00fri/qcdins/qcdins.html + _ computer for which the program is designed and others on which it has been tested : _ any computer with a fortran 77 compiler + _ operating systems under which the program has been tested : _ linux 2.0.x ; hp - ux 10.2 + _ programming language used : _ fortran 77 + _ memory required to execute with typical data : _ size of executable program is approximately 2.6 mb . the size of the qcdins library itself is about 200 kb ; the required routines from the herwig and jetset libraries constitute the dominant portion of the needed memory . + _ no . of processors used : _ 1 + _ has the code been vectorised or parallelized ? : _ no + _ no . of bytes in distributed program , including test data , etc . : _ 1071106 + _ distribution format : _ ascii + _ cpc program library subprograms used : _ herwig [ 1 ] version 5.9 ; jetset 7.4 [ 2 ] + _ keywords : _ qcd ; instanton ; deep - inelastic scattering ; monte carlo simulation + _ nature of physical problem _ + instantons are a basic aspect of quantum chromodynamics . being non - perturbative fluctuations of the gauge fields , they induce hard processes absent in conventional perturbation theory . deep - inelastic lepton - nucleon scattering at hera offers a unique possible discovery window for such processes induced by qcd - instantons through their characteristic final - state signature and a sizable rate , calculable within instanton - perturbation theory . an experimental discovery of such a novel , non - perturbative manifestation of non - abelian gauge theories would be of fundamental significance . however , instanton - induced events are expected to make up only a small fraction of all deep - inelastic events . therefore , a detailed knowledge of the resulting hadronic final state , along with a multi - observable analysis of experimental data by means of monte carlo techniques , is necessary . _ method of solution _ + the qcdins package is designed as an `` add - on '' hard process generator interfaced to the general hadronic event simulation package herwig . it incorporates the theoretically predicted production rate for instanton - induced events as well as the essential characteristics that have been derived theoretically for the partonic final state of instanton - induced processes : notably , the flavour democratic and isotropic production of the partonic final state , energy weight factors different for gluons and quarks , and a high average multiplicity @xmath0 of produced partons with a poisson distribution of the gluon multiplicity . while the subsequent perturbative evolution of the generated partons is always handled by the herwig package , the final hadronization step may optionally be performed also by means of the general hadronic event simulation package jetset . _ restrictions on the complexity of the problem _ + the default values of the implemented kinematical cuts represent the state of the art limits for the reliability of the generated instanton - induced event rate and event topology . _ typical running time _ + 10 - 100 events per second for a pc with pentium cpu , depending on its clock frequency . on a hp 9000/735 ( 99 mhz ) workstation , 6 events per second are generated . _ unusual features of the program _ + none _ references _ + [ 1 ] g. marchesini et al . , comput . phys . commun . 67 ( 1992 ) 465 . + [ 2 ] t. sjstrand , comput . phys . commun . 82 ( 1994 ) 74 . * long write - up * +

You are an expert at summarizing long articles. Proceed to summarize the following text: the apparent metallic - like temperature dependence of the resistivity is observed in various two - dimensional ( 2d ) carrier systems @xcite and remains a focus of a broad interest because it challenges the conventional theory of the metallic conduction . the effect manifests itself in a strong drop of the resistivity over a limited range of temperatures , @xmath0 , from a high temperature value ( call it ` drude ' value ) @xmath1 to a low temperature one @xmath2 , here @xmath3 is the fermi temperature . upon lowering the temperature further , a strong ` metallic ' drop in @xmath4 was found to cross over to the conventional weak localization type dependence @xcite . recently , it was demonstrated @xcite that the ` metallic ' drop is not related to quantum interference and in this sense should have a semiclassical origin . magnetic field applied in the plane of the 2d system causes a dramatic increase in resistivity @xcite . it was proven experimentally @xcite that the magnetoresistance ( mr ) in si - mos samples is mainly caused by spin - effects , though certain contribution of orbital effects @xcite is noticeable at very large fields @xcite , bigger than the field of the spin polarization . the two effects , _ strong metallic drop in resistivity _ and _ parallel field magnetoresistance _ remain _ puzzling_. in this paper we demonstrate that ( i ) the strength of both effects is well matched and ( ii ) the action of magnetic field on conduction is similar to that of disorder and to some extent to that of temperature . in particular , in the presence of in - plane magnetic field the strong temperature dependence of the resistivity for a high mobility sample transforms into a weaker one and shifts to higher temperatures ; both these features are typical for low mobility samples . next , ( iii ) the effect of the magnetic field on the resistivity is continuous and shows no signatures of a threshold . finally , ( iv ) increase in either temperature or parallel magnetic field restores the same high - temperature ( ` drude ' ) resistivity value while transport remains non - activated . matching of the actions of the above controlling parameters sets substantial constraints on the choice of theoretical models . we performed measurements on a number of different mobility ( 100 ) @xmath5si - mos samples , because this system demonstrates the most dramatic appearance of the discussed effects . table [ samples ] below shows the relevant parameters for the four most intensively studied samples whose mobilities differ by a factor of 25 ; parameters for other samples were reported in refs . @xcite . .relevant parameters of the five samples . @xmath6 [ m@xmath7/vs ] is the peak mobility at @xmath8k , @xmath9 is in units of @xmath10 , and @xmath11 is in @xmath12@xmath13 . [ cols="^,^,^,^",options="header " , ] for the curve _ 3 _ , the fit is not perfect because it oscillates vs temperature , which is beyond the frameworks of the model . the only steep feature may be found in the field dependence of @xmath14 at the transition from insulating to metallic behavior ; this however may be caused by a minor mismatching of the oversimplified model curves in the metallic and insulating ranges . the ` activation energy ' @xmath15 depends on the choice of the model and on average is 2 - 3 times larger than the values estimated from fig . the right columns in table 2 ) . this discrepancy is presumably caused by the admixture to the data of a non - exponential temperature dependence which is essentially strong for the considered sample and which can not be separated at low temperatures from the exponential one . for example , we were able to achieve a reasonable fit of the data with eq . ( 2 ) , using the _ calculated _ in table 2 values @xmath16 when we modeled the ` tilted separatrix ' in eq . ( [ tilted ] ) by a power low factor , @xmath17 , rather than by the polynomial factor . in ref . @xcite it was stated that `` the effect of magnetic field can not be ascribed solely to a field - induced change in the critical electron density '' . we verified this possibility and arrive at the opposite conclusion . the success of our fitting confirms that the action of the parallel field , at least , to the first approximation consists in progressive increase of disorder . this is described by increasing the @xmath18-independent scattering rate @xmath19 , decreasing the magnitude of the resistivity drop , @xmath20 , and increasing the critical carrier density @xmath21 ; the three parameters are not independent and within the same material system may be reduced only to the single parameter , e.g. @xmath21 . to summarize , the results reported in this paper demonstrate that the effect of the parallel magnetic field on resistivity in high mobility si - mos samples ( though via zeeman coupling ) is similar to that of disorder and , to some extent , to that of temperature . in other words , _ parallel magnetic field increases disorder_. the temperature dependence of resistivity for various disorder and magnetic fields may be reduced to the dependences of the ` critical ' density @xmath11 on magnetic field and on disorder ; the changes in the resistivity drop with temperature , @xmath22 , caused by disorder and magnetic field may be also mapped onto each other . we find a similarity between the action of the parallel field and temperature for the region of carrier density and field where conduction remains ` non - activated ' . these findings set constraints on the choice of the developed microscopic models . the analogy between the parallel field and disorder points to the existence of a sub - band of localized carriers . such band of localized carriers should lift in energy as a function of the parallel field , gets ` undressed ' after passing through the fermi energy and cause the increase in the scattering rate . the strong evidence for this mechanism is the variation in the hall voltage observed in the vicinity of the metal - insulator transition . we can not exclude also the contribution of the inter spin - subband scattering , as a complementary feature to the above mechanism . however , our shubnikov - de haas data @xcite indicates that the mobility in different spin and valley sub - bands of conducting electrons are almost equal and the intersubband scattering does not play a major role in the themperature- and parallel field- dependence of the resistance . applying the model of a field dependent disorder , we find a qualitative explanation of the whole set of presented results . one of the authors ( v.p . ) acknowledges hospitality of the lorenz centre for theoretical physics at leiden , where part of this work was done and reported at the workshop in june , 2000 . the work was supported by nsf dmr-0077825 , austrian science fund ( fwf p13439 ) , intas ( 99 - 1070 ) , rfbr ( 00 - 02 - 17706 ) , the programs physics of solid state nanostructures , statistical physics , integration and ` the state support of the leading scientific schools ' . s. v. kravchenko , g. v. kravchenko , j. e. furneaux , v. m. pudalov , and m. diorio , phys . b * 50 * , 8039 ( 1994 ) ; s. v. kravchenko , g. e. bowler , j. e. furneaux , v. m. pudalov , and m. diorio , phys . rev . b * 51 * , 7038 ( 1995 ) . d. popovi , a. b. fowler , s. washburn , phys . lett . * 79 * , 1543 ( 1997 ) . t. coleridge , r. l. williams , y.feng , p. zawadszki , phys . b. * 56 * , r12764 ( 1997 ) . a. r. hamilton , m. y. simmons , m. pepper , e. h. linfield , p. h. rose , and d. a. ritchie , phys . * 82 * , 1542 ( 1999 ) . the magnetic field driven transition from non - activated to activated conduction regime causes a strong exponential magnetoresistance @xmath23 , where @xmath24 ; therefore , @xmath25 scales as @xmath26 . a theoretical consideration see in ref .

we compare the effects of temperature , disorder and parallel magnetic field on the strength of the metallic - like temperature dependence of the resistivity . we found a similarity between the effects of disorder and parallel field : the parallel field weakens the metallic - like conduction in high mobility samples and make it similar to that for low - mobility samples . we found a smooth continuous effect of the in - plane field on conduction , without any threshold . while conduction remains non - activated , the parallel magnetic field restores the same resistivity value as the high temperature does . this matching sets substantial constraints on the choice of the theoretical models developed to explain the mechanism of the metallic conduction and parallel field magnetoresistance in 2d carrier systems . we demonstrate that the data for magneto- and temperature dependence of the resistivity of si - mos samples in parallel field may be well described by a simple mechanism of the magnetic field dependent disorder . 2

You are an expert at summarizing long articles. Proceed to summarize the following text: algol ( @xmath5 perseus ) is the eponymous eclipsing binary system which consists of a primary early - type main sequence star and a roche lobe - filling secondary late - type giant or subgiant star . these systems have undergone a period of mass transfer during which material was transferred from the initially more massive present day late - type star to its initially less massive early - type companion . algol is the brightest and nearest example of this type . the primary star ( algol a ) is a b8 v main sequence star , while the secondary ( algol b ) is a k2 iv subgiant that has lost about half of its original mass to the present day primary ( e.g. , * ? ? ? * ) . since algol has a short period orbit of 2.87 days ( e.g. , * ? ? ? * ) the two stars are tidally locked , and their orbital and rotational periods are synchronized . algol was first confirmed as an x - ray source by @xcite , who suggested that the x - ray emission might be explained by a mass - transfer model , where the more massive b star accretes material from the less massive k star , either through roche lobe overflow or a stellar wind . in both cases , the infalling material is shock - heated to x - ray temperatures when it hits the b star . according to this mass - transfer model , we would expect to observe an x - ray eclipse during the optical primary eclipse . however , observations made with the solid state spectrometer ( sss ) of the _ einstein _ observatory revealed no sign of an x - ray eclipse at the time of optical primary eclipse @xcite . at present , it is widely believed that the x - ray emission from algol arises mostly , if not completely , from the corona of algol b @xcite . the two stars are tidally locked and rapidly rotating , and within the rotationally - excited dynamo paradigm , it is expected that the convection zone of the k star would experience increased dynamo activity whose resulting magnetic energy is subsequently dissipated at the stellar surface in the form of a hot , x - ray emitting corona @xcite . however , the nature and structure of this corona remains a topic of debate . more than four orders of magnitude in x - ray luminosity and an order of magnitude in plasma temperature separate the solar corona from coronae of most active stars such as the algol - type secondaries ( e.g. * ? ? ? we currently have very little idea of the physical and spatial characteristics and appearance of these much more active coronae , and whether or not magnetic structures are restricted to the stellar surface or whether significant x - ray emitting plasma resides in larger structures , including inter - binary loops , with magnetic structures linking the two stars @xcite . @xcite analyzed exosat observations of algol and found no clear x - ray eclipse during the optical secondary minimum . if this x - ray emission was indeed from the secondary star , his study suggests that the coronal extent of algol b must be comparable to or greater than the size of the star itself . however , @xcite did detect a secondary x - ray eclipse in rosat observations , and concluded that the scale height of the algol b corona is @xmath6 . similarly , a shallow eclipse at optical secondary minimum was seen by asca @xcite . a dramatic eclipse of a large flare observed by bepposax enabled the size of the flaring structure to be estimated as being less than 0.6 stellar radii and pinpointed the location of the structure to polar regions @xcite . based on these deductions and on observations of flares on algol seen by ginga , exosat , rosat , and xmm - newton , @xcite suggest that the corona is essentially concentrated onto the polar regions of the k star , with a more compact ( smaller than the star ) flaring component and a perhaps somewhat more extended ( comparable in size to the star ) quiescent corona . such a picture of polar - dominated activity has long been suspected based on optical doppler imaging techniques , which consistently see dark polar spots on active stars ( e.g. * ? ? ? @xcite also suggested polar emission was responsible for the lack of rotational modulation in the observed flux in a _ low energy transmission grating spectrograph ( letgs ) observation of algol . however , an xmm - newton observation of an algol flare studied by @xcite was interpreted as lying at lower latitudes . while this interpretation is somewhat subjective , it does suggest that even if polar emission dominates , significant activity at all latitudes is likely present . similar conclusions regarding the corona of the k star might be drawn based on radio detections of algol . in particular , @xcite presents results from 8.4 and 15 ghz observations of algol obtained using the very long baseline array ( vlba ) in 1995 . the vlba maps show a double - lobed structure in the radio corona of the k star , which appear to originate in or near the polar caps . the quiescent radio emission does not show signs of orbital modulation , again suggestive of a corona concentrated on the poles of the k star . caution is warranted in the interpretation of the spatial distributions of radio and x - ray emission , however , since the former is produced by gyrosynchrotron emission from relativistic electrons , and this population might not be co - spatial with the thermal electrons responsible for x - ray emission . direct x - ray observations present the only means to probe the spatial distribution of this thermal population . in this paper , we primarily use the _ chandra _ high energy transmission grating spectrograph ( hetgs ) x - ray spectrum of algol to test this emerging picture of coronal activity . we compare observed doppler shifts with the theoretical orbital velocities of the primary and secondary stars of algol , and compare the observed line widths with theoretical expectations based on instrumental , thermal , and rotational broadening . based on these comparisons , we are able to place the first observational limits on the contribution of the b8 dwarf to the x - ray emission of algol , and to place constraints on the scale height , and any non - thermal motions , characterizing the corona of algol b. finally , we show that lines of o vii seen from quadrature to primary eclipse in a low energy transmission grating spectrograph ( letgs ) observation of algol support our conclusions that the corona of algol b at temperatures of a few @xmath7 k must be radially extended to at least one stellar radius . the _ chandra _ hetgs employed for the observations used in this analysis consists of two gratings the high energy grating ( heg ) and the medium energy grating ( meg ) . the heg covers a wavelength range of 1.2 - 15 ( 10.0 - 0.8 kev ) , with a typical line width of @xmath80.012 . the meg covers a wider wavelength range of 2.5 - 31 ( 5.0 - 0.4 kev ) , with a typical line width of @xmath80.023 . a more thorough description of this instrumentation can be found in @xcite . we made use of both heg and meg spectra . the hetgs observation of algol ( obsid 604 ) studied here was obtained as part of the guest observer ( go ) program and was undertaken in the standard instrument configuration using the acis - s detector between ut 02:20 and 17:31 on 2000 april 1 , for a total effective exposure time of 54660 s. in order to investigate the accuracy of the instrument line response function , we also analyzed an observation of capella ( obsid 2583 ) obtained for routine calibration . this observation was also taken in the standard instrument configuration using the acis - s detector between ut 17:46 and 02:03 on 2002 april 29 to april 30 , for a total exposure time of 29700 s. we chose this particular observation of capella because it was obtained at an orbital phase of @xmath8 0.5 , when the two giants were near conjunction and spectral features were minimally broadened by orbital motion . we also analyzed an letgs observation of algol ( obsid 2 ) , in order to examine the ratio of the o vii forbidden to intercombination lines as a function of orbital phase . the data were observed between ut 18:22 and 17:11 on 2000 march 12 to march 13 , for a total exposure time of 82200 s. for details of the observation itself we refer to @xcite . the standard ciao pipeline - processed ( version 2.2 ) data were downloaded from the chandra public data archive . subsequent analysis was done using the idl software package pintofale @xcite . the first order meg and heg spectra are illustrated in figure [ f : spec ] , together with identifications of prominent spectral lines . emission lines used in our analyses are labeled with a larger font . prior to spectral analysis , we examined the lightcurves of both algol and capella in order to determine their level of variability and whether any significant events ( such as large flares ) occurred during the observations that may be relevant for subsequent interpretation of doppler shifts and line broadening . the lightcurves were derived by extracting all _ dispersed _ events in the standard ciao spectral extraction region , then binning the events at 100 second time intervals . we emphasize here that we did not use the 0th order events since these are strongly affected by pile - up . the observed lightcurve of capella is relatively flat , with no obvious signs of flaring , or other variability . the lightcurve of algol on the other hand , shows a definite and significant flare in the beginning of the observation ( figure [ f : lc ] ) . note that the orbital phase of the algol observation begins at @xmath9 = 0.48 , just before algol b starts to come out of eclipse . it is possible that the impulsive phase of the flare has been affected by geometric occultation and we return to this in [ s : results ] . the flare has decayed by @xmath10 , soon after algol b passes conjunction . algol b appears to be quiescent for the remainder of the observation , which ends at @xmath11 . in this section , we investigate the algol spectra for evidence of doppler shifts resulting from orbital motion . the following analysis is restricted to the emission lines listed in table [ t : whichlines ] . we have chosen these particular lines because they have high s / n ratios and are not significantly blended with lines of other atomic species . [ h ] c c c c wvl [ ] & ion & grating & diffraction order + 8.42 & mg xii & meg & 1,3 + 12.13 & ne x & meg , heg & 1 + 15.01 & fe xvii & meg & 1 + 16.78 & fe xvii & meg & 1 + 18.98 & o viii & meg & 1 + 24.78 & n vii & meg & 1 + [ t : whichlines ] orbital velocity as a function of orbital phase is derived separately for each emission line listed in table [ t : whichlines ] . for each line , we first bin the events into time intervals ( which are later converted to orbital phase ) , where the bin size is proportional to the total observed counts of the emission line , thereby maintaining an approximately constant s / n ratio for each bin . negative and positive order events ( in both meg and heg ) were combined to obtain higher signal to noise ( s / n ) ratios . since we are trying to accurately measure the wavelength centroids of emission lines , what is important here is the s / n ratio , rather than the resolution . although the heg has better resolution than the meg , most of even the brightest lines proved too faint for useful analysis in heg spectra . most of the lines we analyze here are therefore meg lines . a modified lorentzian function ( ` beta - profile ' ) described by the relation @xmath12 where @xmath13 is the amplitude and @xmath14 is a characteristic line width , is then fit to the events along the wavelength axis for each time interval . with @xmath15 , this function has been found to be a good match to observed hetgs line profiles , better than gaussians ( * ? ? ? * in preparation ) . in this way , we obtain the observed wavelength centroids as a function of time and orbital phase . the line - of - sight orbital velocities were derived for all emission lines by subtracting the rest wavelengths of lines from their observed centroids . we expect the dominant component of the doppler shifts to result from the orbital motion of algol b , which amounts to @xmath16 km s@xmath2 at quadrature . the predicted orbital velocity is described by @xmath17 where @xmath18 is the radius of orbit , @xmath19 is the inclination , @xmath9 is the orbital phase , and @xmath20 is the orbital period . we used an inclination of 81.4 degrees @xcite and a period of 2.87 days . we fitted a sine model to the observed line wavelengths using the ciao fitting engine sherpa . the model used to fit the data is given by @xmath21 ) \label{e : sinemodel}\ ] ] where @xmath22 is a constant y - offset [ ] , @xmath23 is the rest wavelength of the emission lines [ ] , @xmath24 is the speed of light [ km s@xmath2 ] , @xmath26 is the amplitude of the final fit which represents the line - of - sight orbital velocity [ km s@xmath2 ] ( given by 2@xmath27r @xmath28/p ) , @xmath9 is the orbital phase , r is the effective orbital radius of x - ray emitting material [ km ] , @xmath19 is the inclination of orbit [ radians ] , and p is the orbital period [ s ] . the parameters for all emission lines were forced to be the same , and the orbital period of algol b was fixed at the value noted above . the only parameters allowed to vary were @xmath22 and @xmath26 . we allowed @xmath22 to be a free parameter because the absolute wavelength calibration of hetg+acis - s observations is @xmath8 0.011 for meg and @xmath8 0.006 for heg . therefore , each emission line may need a different @xmath29 offset in order to account for wavelength calibration uncertainties . because @xmath22 was a free parameter , errors on the reference wavelengths , though extremely small for the h- and he - like ions , are not relevant for our analysis . the data for all spectral lines were fit simultaneously . the amplitude of the best fit was a=187.5 km s @xmath2 . from the amplitude , we can compute the effective orbital radius of x - ray emitting material , @xmath18 ( equation [ e : vorb ] ) . we obtain an effective orbital radius of 10.74 @xmath30 0.93 r@xmath31 . this result will be further discussed in [ s : results ] . figure [ f : orbit_all ] illustrates the observed line of sight orbital velocity as a function of orbital phase , derived from various emission lines . figure [ f : orbit_all ] also shows the theoretical orbital velocities of algol a and algol b , calculated under the assumption that the x - ray emission is centered on one of the two stars ( i.e.the expected curve if _ all _ of the x - ray emission had been located at the poles of algol b or algol a ) . in addition to measuring individual emission lines , we also utilized a cross - correlation technique to obtain doppler shifts as a function of orbital phase . events were binned into 5 time intervals corresponding to different orbital phases , and spectra were extracted for each phase bin . spectra from each phase bin were cross - correlated with a reference spectrum whose orbital velocity is known . we used capella ( obsid 2583 ) as our reference spectrum because , as noted in [ s : observations ] , this particular observation was taken when the two giants were near conjunction in orbital phase . this ensures that the orbital velocity of the spectrum is nearly zero . other systematic velocities involved can all be determined with high precision and subtracted : these are the radial velocities of the capella and algol systems , and the velocity of the _ chandra _ satellite in the direction of these objects during the observations . algol has a radial velocity of only 4 km s@xmath2 @xcite , while capella has a radial velocity of 30 km s@xmath2 @xcite , therefore we must redshift the algol velocities by 26 km s@xmath2 . the orbital velocity of _ chandra _ during these observations was less than @xmath82 km s@xmath2 . since this velocity is small compared to the errors incurred during the cross - correlation analysis , we ignored this orbital motion . the algol and capella spectra were cross - correlated by shifting the algol spectrum with respect to the capella spectrum in velocity steps of 25 km s@xmath2 , ranging from -200 to 500 km s@xmath2 . at each velocity shift , we computed the @xmath32 value as follows : @xmath33 ^ 2}{\sigma_{ref}(\lambda)^{2}+\sigma_{2}(\lambda(1+\frac{v}{c}))^2 } ; \label{e : chi2}\ ] ] where @xmath34 and @xmath35 are the two spectra being compared ( capella and velocity shifted algol ) , and @xmath36 and @xmath37 are their respective errors . these quantities are then summed over the entire spectral region . the velocity shift at which minimum @xmath32 is achieved represents the doppler shift in that particular phase bin of the algol spectrum relative to the capella spectrum . in order to better understand the errors incurred from cross - correlating the two spectra and computing @xmath32 , we utilized a monte carlo method by randomizing the counts within errors on the algol spectrum 25 times , and repeating cross - correlation for each randomized data set . parabolae were then fit to @xmath32 versus velocity shift for each of the 25 simulations in order to find the velocity that corresponds to the minimum @xmath32 of each simulation . the mean velocity is then adopted as the final doppler shift for that given interval of phase , and the standard deviation of the mean is the @xmath38 error on the final doppler shift . figure [ f : orbit_crosscor ] illustrates the line - of - sight velocity of algol as a function of orbital phase , as obtained by the cross - correlation / monte - carlo analysis method described above . also illustrated are the theoretical orbital velocities of algol a and algol b. once again , we used the sherpa fitting engine to fit a sine model to the observed doppler shifts obtained via the cross - correlation analysis . doppler shifts for the positive and negative orders , as well as both meg and heg , were fit simultaneously with the model described by equation [ e : sinemodel ] . the amplitude of the sine model is a=173.4 @xmath30 5.1 km s@xmath2 . assuming an orbital period of 2.87 days , and an inclination angle of 81.4 degrees , this amplitude corresponds to an orbital radius of 9.93 @xmath30 0.29@xmath39 . this radius is consistent , within 1@xmath40 uncertainties , with our previous result of 10.74 @xmath30 0.93@xmath39 obtained by measuring wavelength centroids of individual emission lines . we also applied the cross - correlation / monte - carlo analysis to the algol spectrum for regions greater than and less than 10 , in order to compare the line - of - sight velocities for lines formed at cooler versus hotter temperatures . doppler shifts for both regions of the spectrum were consistent , though uncertainties for the hotter region of the spectrum were considerably larger ( @xmath850 km s@xmath2 ) than the cooler region ( @xmath810 km s@xmath2 ) , since most of the stronger emission lines lie at wavelengths greater than 10 . in order to test for the presence of non - thermal broadening in the observed algol line profiles , we compare these with carefully - computed theoretical profiles . since the dominant source of line - broadening in the observed spectra is the instrumental profile , it is important that we confirm that our understanding of the instrumental broadening is correct , before proceeding with the comparison . in a far ultra - violet study of capella , @xcite found that the observed profile of fe xviii originated largely from the g8 component , and that only instrumental , rotational ( 3 km s@xmath2 ) , and thermal broadening were required to match the observed profile . this gives us some confidence that the coronal spectra of the capella giants are free from significant additional sources of non - thermal broadening . it is not certain which of the two capella stars dominated during the time of the observation analyzed here ; for example , @xcite found that , in 1999 september 12 , the g1 component dominated in the light of fe xxi 1354 seen by the space telescope imaging spectrograph ( stis ) . the dominance of one component over the other is not important here . by constructing two theoretical profiles ( described below ) , one with rotational broadening of 3 km s@xmath2 ( g8 component ) and another with rotational broadening of 36 km s@xmath2 ( g1 component ; @xcite ) , we find that the difference in fwhm of the two profiles are negligible of order 0.0001 and two orders of magnitude smaller than typical observed line widths . theoretical line profiles were synthesized by convolving the predicted rotational , thermal , and instrumental line broadening , where rotational profiles were derived by assuming solid body rotation at the stellar surface ( e.g. * ? ? ? in our final synthesis of capella theoretical line profiles , we have adopted a rotational velocity value of the g1 star ( 36 km s@xmath2 ; @xcite ) . the width of the thermal profile is calculated as the weighted average of the thermal widths computed at various temperatures where the weighting function is given by the product of the emission measure and the line emissivity at each temperature . the fwhm of the thermal width for a given ion , at a given temperature , is then described by : @xmath41 where @xmath42 is the emission measure , @xmath43 is the emissivity , @xmath44 is the electron temperature , @xmath45 is the boltzmann constant , and @xmath46 is the ion mass . we derived an emission measure @xmath42 for capella based on the measured line strength ratios of h - like and he - like ions , as described by ( * ? ? ? * in preparation ) . this method of using line strength ratios to derive an emission measure is similar to what is described in ( * ? ? ? * in press ) . we defer a detailed discussion of this analysis to a future paper , but emphasize that the final thermal profile does not depend critically on the exact details of the emission measure distribution : choosing a single isothermal emission measure at a temperature of @xmath47 produced very similar results . line profiles for the h - like lyman-@xmath48 doublets of o viii , ne x , and mg xii were each modeled with the doublet wavelength separation fixed at @xmath49 for ne x and mg xii , and @xmath50 for o viii @xcite . both components of a given line were forced to have the same line width , and relative fluxes of the two components were fixed at the theoretical ratio of 2:1 . the final theoretical line profiles are then created by convolving the rotational and thermal profiles with the expected instrumental broadening . the instrumental line profile was derived from high - fidelity ray - traces of the _ chandra _ instruments ( * ? ? ? * in press ) . the predicted and observed line widths are compared in table [ t : linewidths ] and figure [ f : order_comp ] . these comparisons show that the theoretical capella line widths are in good agreement with the observed line widths of the data . we are therefore confident that the current model of the instrumental line broadening is sufficient to yield theoretical profiles that are accurate to within errors of the observed algol line widths . prior to measuring the algol line widths , we eliminated line broadening due to the orbital velocity of algol b by shifting the event positions by the predicted amount ( @xmath51 ) . here , we adopt the true radius of orbit of algol b , @xmath52 ; we also performed the same line width analysis assuming the smaller of the two effective radii , @xmath53 found from the doppler shifts above , but found no significant differences between the two . the negative and positive order spectra were then coadded . great care has to be taken here . bright sources such as algol suffer from significant pile - up in the 0th order , resulting in a distorted source profile . this distortion can confuse the standard pipeline 0th order location algorithm , resulting in a computed centroid that does not correspond exactly to the true centroid . if the 0th order centroid location is not precise , there will be a systematic wavelength shift between opposite orders . co - addition of the opposite orders will then produce an artificially broadened line profile . in order to avoid this problem , we therefore ensured that the line centroids in opposite spectral orders were identical prior to co - addition , on a line by line basis . the widths of lines listed in table [ t : whichlines ] were then measured by fitting with a modified lorentzian function ( equation [ e : lorentz ] ) . since the measured line widths are sensitive to the adopted level of the continuum , we have computed a model continuum based on the derived emission measure distribution of algol and have used this to guide continuum placement . in this way , the continuum level adopted makes use of the signal in the spectrum over a broad wavelength range and accurate placement can be achieved such that final uncertainties in measured line quantities do not have a significant contribution from continuum placement errors . the final continuum placement was done by eye , using effectively `` line free '' regions as a guide . in order to verify that this continuum placement did not introduce significant errors in our line parameters , we performed a sensitivity test by changing the adopted continuum level by different amounts that stretched plausibility regarding the true continuum level , and re - measuring lines for the different levels . although the continuum placement is partly subjective because of the pseudo - continuum due to the presence of weak lines and broad wings of closely spaced lines , in the worst case the uncertainty of the fwhm of any of our lines resulting from uncertainties in continuum placement is not more than @xmath80.002 . for some of the stronger lines , such as ne x , it is unlikely that the line width uncertainty is at most @xmath80.001 . theoretical profiles for algol were generated in the same way as described for capella in [ s : capella ] . the theoretical fwhm were then compared to the measured line widths of the observed data ; these comparisons are listed in table [ t : linewidths ] and illustrated in figure [ f : order_comp ] . we find some evidence for moderate excess line broadening in several of the emission lines . in the meg spectra , all lines but n vii , o viii , and mg xii show signs of significant excess line widths . the two fe xvii lines , ne x , and mg xii show significant excess line widths , ranging from 3@xmath40 to almost 6@xmath40 . in the heg data , ne x shows an excess width of 3@xmath40 , though mg xii shows no sign of excess line broadening . in order to determine whether any of the observed excess broadening could be due to the flare activity in the first part of the observation we also performed the analysis described above for flare and quiescent periods . line widths for both periods were found to be consistent within statistical uncertainties . the excess widths we have detected in algol may be attributed to either or both of two plausible sources of non - thermal velocities : turbulence or flows within coronal structures , and `` excess '' rotational broadening above what is expected from surface emission . figure [ f : profile ] is an example comparison between observed and theoretical line profiles . this figure shows that a theoretical profile with an additional gaussian velocity of 125 km s@xmath2 is a reasonably good match to the fe xvii line , while an additional velocity of 300 km s@xmath2 is too wide , and a velocity of 0 km s@xmath2 is too narrow . we obtain a crude estimate of the turbulent velocities involved by comparison of observed and theoretical line profiles , computed as described in the previous section . we convolved the latter with an additional gaussian broadening corresponding to a range of velocities from 0 to 300 km s@xmath2 . figure [ f : turbulent ] illustrates the fwhm of these synthesized profiles as a function of the additional non - thermal velocity , and also indicates where the observed fwhm fall on these curves . most of the lines we have analyzed show observed fwhm that correspond to additional velocities of @xmath8 50 - 150 km s@xmath2 . we are able to place an upper - limit on additional turbulent velocities at 170 km s@xmath2 . excess rotational broadening can occur if the corona on algol b is significantly radially extended . line widths can therefore be used to place direct spectroscopic constraints on this radial extension . limits on the radial extent of the corona of algol b were determined by comparison of observed and theoretical profiles computed for different scale heights . we varied the scale height of the theoretical rotational profiles from within the stellar surface ( zero rotation ) to 8.75@xmath54 , where the scale height is defined as zero at the stellar surface . then we convolved these with the thermal and instrumental broadening , as before . the final profiles are compared with the fe xvii 15.01 resonance line in figure [ f : profile ] . this figure shows good qualitative agreement with a profile computed for coronal scale height of 3@xmath54 ; a scale height of 8@xmath54 is clearly too broad , while purely surface rotational broadening is too narrow . the upper two panels of figure [ f : scale ] show the fwhm of the theoretical profiles as a function of coronal scale height , together with the observed fwhm overplotted in thicker lines . more than half of the observed fwhm fall approximately at a scale height of @xmath8 2 to 3@xmath54 , which corresponds to an excess rotational velocity of @xmath8 125 to 185 km s@xmath2 , in addition to the rotational velocity at the stellar surface ( @xmath8 62 km s@xmath2 ) . the bottom panel of figure [ f : scale ] shows a distribution of scale heights which were derived by randomly generating a set of line widths that are distributed in a gaussian manner , centered on the measured line width of each emission line . the number of monte - carlo realizations for each emission line was weighted by the measured flux of that line . we generated a total of 30087 realizations for all lines combined . we then obtained the scale height that corresponds to each realization of a line width by interpolating along the theoretical curves shown in the top two panels of figure [ f : scale ] . the lowest panel of figure [ f : scale ] shows the distribution of the scale heights for all realizations of line widths , for both meg and heg . vertical lines indicate the median scale height , and the 1@xmath40 and 3@xmath40 limits . the median indicates a scale height of 3.1@xmath54 , with 1@xmath40 and 3@xmath40 upper limits occurring at scale heights of 3.8 and 4.6 stellar radii , respectively . the lower 1@xmath40 limit occurs at 0.9@xmath54 . in the case of line width realizations that are smaller than what is physically possible , these scale heights have been set equal to -1@xmath54 , which accounts for the large peak at -1@xmath54 . the uv excitation of the @xmath56 term by algol a provides a possible test of coronal geometric extent . in the spherically - symmetric case , a very compact corona should exhibit modulation of the ratio of the forbidden and intercombination lines , @xmath55 , with orbital phase : at phase @xmath57primary optical eclipse the visible hemisphere of algol b is not illuminated by algol a and the @xmath55 ratio should revert very closely to its pressure - dominated value . the active star survey of ( * ? ? ? * in preparation ) indicates that this ratio should be @xmath58 - 3 , instead of 0.94 , as observed by @xcite . there is no evidence for orbital modulation of the @xmath55 ratio in the hetgs observation ; this is not surprising because the observation covered only those phases during which the visible k star corona was substantially illuminated by algol a. we have therefore re - examined the letgs observation ( obsid 2 ) that covers phases @xmath59 - 1.03 to search for evidence of orbital - modulated changes in the @xmath55 ratio . the end of this observation corresponds to primary eclipse when the visible hemisphere of the k star is not irradiated by algol a. we split the algol letgs data into three phase bins ( lacking the signal in the o vii lines to make more fine divisions ) and examined the relative strengths of the intercombination and forbidden lines at 21.80 and 22.10 . the spectral region containing these lines is illustrated for the three phase bins in figure [ f : o7letg ] : there is no perceptible change in the ratio between these different bins . by modeling observed doppler shifts in terms of orbital motion , we were able to estimate the effective orbital radius of the x - ray emitting material . we found this effective radius to be @xmath60 @xmath30 0.93@xmath39 from the analysis of individual emission lines , and @xmath61 @xmath30 0.29@xmath39 from the cross - correlation analysis ; these results are consistent within @xmath62 uncertainties . the statistical uncertainties in the cross - correlation method are considerably lower because all the information in the spectrum is used , rather than just information in bright lines . the 1@xmath40 error bar of the line - of - sight velocities obtained via the cross - correlation analysis is @xmath88 km s@xmath2 for the meg data and @xmath811 km s@xmath2 for the heg data . however , limitations due to uncertainties in the calibration of the _ chandra _ instruments cause systematic uncertainties which appear somewhat larger than what is implied by the statistical uncertainties . for example , a few of the data points in figure [ f : orbit_crosscor ] show discrepancies between the meg and heg results , by @xmath82@xmath40 . in comparison with the x - ray doppler radius , the orbital radius of algol b around the system center of mass is @xmath63 @xcite . since the effective radius we have derived is smaller than @xmath64 , we can infer that the x - ray emitting material is in fact not perfectly centered on algol b , but is shifted slightly inward toward the primary star . the effective orbital radius of the x - ray emission allows us to place a constraint on the possible contribution from algol a emission that would skew the apparent x - ray doppler radius toward the center of gravity . as mentioned in [ s : intro ] , it has been argued that accretion onto the primary star could be a source of x - rays . if this were true , a small x - ray contribution from the accretion activity of algol a could explain the inward shift of the effective radius of x - ray emitting material . assuming that the emission of the k star corona is centered on the star itself , an effective radius of @xmath65 for the x - ray emission would indicate that @xmath885% of the total x - ray emission is from algol b , and the remaining 15% is from algol a. @xcite considered the issue of accretion - driven x - ray emission in algol - type binaries in their comparison between algols and rs cvn - type binaries . the latter are comprised of two late - type stars in which neither component filled their roche lobes . @xcite found that the algol - type binaries are in fact slightly x - ray deficient relative to their rs cvn cousins , suggesting strongly that the accretion activity of algol - type binaries is not a significant source of x - rays . the 15% effect we are seeking here would , however , be quite inconspicuous in this type of statistical study . the possibility of accretion giving rise to significant x - ray flux in algol was reviewed by @xcite . two key parameters are the shock temperature of the accreting gas , and the mass transfer rate . it seems unlikely that the accretion shock can exceed @xmath66 k , and pustylnik suggests a maximum x - ray luminosity from accretion of about @xmath67 erg s@xmath2 based on a mass transfer estimate of order @xmath68 yr@xmath2 . the x - ray luminosity of algol as measured from the hetg observation analyzed here is about @xmath69 erg s@xmath2 ( * ? ? ? * in preparation ) ; accretion does therefore appear sufficient to account for @xmath70% of the total luminosity . one test of this would be whether or not lines formed only above @xmath71 k lines that could only plausibly originate from coronal emission show the same center of gravity as lines formed at cooler temperatures . unfortunately , our data are of insufficient quality to perform this test since only weaker lines in the spectrum exclusively originate from such hot plasma . the brightest line formed at these higher temperatures is h - like si xiv @xmath726.18 . the line - of - sight velocity as a function of orbital phase indicated by this line is consistent within statistical uncertainties with the velocity seen in the other , cooler lines ( figure [ f : orbit_all ] ) . however , measurement errors are sufficiently large that we can not rule out differences at the 15% level . evidence for plasma associated with accretion and with temperatures of at least @xmath73 k has , however , been found in the algol systems v356 sgr and tt hya from recent fuse observations ( * ? ? ? * ; * ? ? ? * respectively ) . emission detected from o vi appeared to be associated with a bipolar flow that makes a large angle with the orbital plane . however , in order for this plasma to contribute significantly at x - ray wavelengths , it must be comprised of components at least an order of magnitude hotter than the formation temperature of o vi . another possible , and we suggest more likely , explanation for the inward shift of the effective radius is that the corona of algol b is not spherically symmetric or exactly centered on the center of mass of algol b , but rather has some asymmetry and structure on the side facing inward , toward algol a. the surface of algol b itself will be severely distended in this direction by the gravitational field of algol a the roche lobe filling factor for algol b is expected to be very close to unity , such that any equatorial corona could well lie beyond the l1 point and would not be gravitationally bound to algol b. we illustrate the approximate geometry of the system in figure [ f : geom ] . this figure was produced using the _ nightfall _ program by r. wichmann , which accounts for the relative masses of the two stars , the roche lobe filling factors , and inclination of the orbit . we also note that , in the x - ray lightcurve ( figure [ f : lc ] ) , we see a flare just as algol b comes out of eclipse . this flare does not have a sharp rise , but a more gradual one that appears to start before eclipse egress . it is possible that this flaring plasma is located on the hemisphere facing algol a , and that the characteristics of its intensity evolution over time are modulated by the eclipse . if the flaring plasma is indeed located on the hemisphere facing algol a , this could also explain why the effective radius of the x - ray emission during this observation is shifted toward algol a. by comparing observed with theoretical line widths , we have some found evidence for moderate excess line widths that we attribute to the possible presence of non - thermal broadening . while not definitive , this result is supported by recent analyses of far ultraviolet spectra obtained by the hubble space telescope @xcite and the far ultraviolet explorer @xcite . both studies report the need for excess line broadening to understand the profiles of the forbidden lines of fe xxi at 1354 and fe xviii at 976 in the more active stars of their sample with the largest projected rotational velocities ( @xmath74 ) . one source of excess broadening might be turbulence or `` explosive events '' . our line profile analysis ( e.g. , figures [ f : turbulent ] & [ f : profile ] ) suggests that a random velocity component of @xmath75 km s@xmath2 could explain the observed line widths . this is similar to the sound speed at @xmath76 k. such a velocity is reminiscent of the non - thermal broadening of over 100 km s@xmath2 seen in transition region lines of stars of different activity as reviewed by @xcite . it was suggested in this body of work that the broadening might be caused by the acceleration of plasma in magnetic reconnection events associated with microflaring . it is possible that the broadening we see in hot x - ray lines is related to this . other types of flows are also possible . @xcite observed flows with velocities up to 40 km s@xmath2 in solar coronal loops based on detailed transition region and coronal explorer ( _ trace _ ) time - resolved imaging and simultaneous solar and heliospheric observatory ( _ soho _ ) sumer spectra . while the velocities inferred here are three times this , it is not difficult to envisage faster flows when considering that the coronal energy deposition rate of algol b is @xmath77 times that of the sun . however , as pointed out by @xcite , in both the above cases the broadening mechanisms should be ubiquitous among the more active stars and should not be seen only in those with the largest @xmath74 . another possible source of excess line widths is rotational broadening originating in a corona with significantly radially - extended structure . as noted in [ s : intro ] , the evidence from observations of algol during eclipse does not provide an unambiguous picture , with some observations requiring apparently extended coronae @xcite , and others being consistent with scale heights of a stellar radius or less @xcite . the scale height for plasma of mean molecular weight @xmath78 at a temperature @xmath44 on a star of mass @xmath79 and radius @xmath80 is @xmath81 . the parameters for algol b were summarized recently by @xcite , who adopted @xmath82 and @xmath83 ( see also * ? ? ? * ; * ? ? ? * ) . for a coronal temperature @xmath84 k on algol b , the scale height is then about @xmath85 in the absence of strong centrifugal forces . our median scale height and 1@xmath40 upper limit are 3.1@xmath54 and 3.8@xmath54 , respectively . while this is slightly larger than the thermal scale height in the absence of strong centrifugal forces arising from rapid rotation , it is important to keep in mind the following things . firstly , the line profile analysis depends on subtraction of the dominant instrumental and thermal broadening components and is inherently prone to additional systematic error arising as a result of imperfections in the description of these dominant effects . secondly , for a corona on algol b with a radial extent comparable to the stellar radius , the competition between gravitational and centrifugal forces becomes significant : centrifugal acceleration on a single star with the same rotation rate is equivalent to gravitational acceleration at a height of @xmath86 . while the binary nature of algol complicates the picture , it is possible that some regions of an extended corona comprise plasma bound by magnetic fields , as has been discussed in connection with other rapidly rotating stars , such as ab dor ( e.g. * ? ? ? ( see also @xcite who concludes that the 1032 o vi profile of ab dor , observed with orpheus , is not rotationally broadened beyond the photospheric surface @xmath74 value . ) we conclude that , in the interpretation that the tentative evidence for excess line widths is the result of rotational broadening , the extension of the corona would be similar to the expected thermal scale height and comparable to the stellar radius . this interpretation is consistent with that of @xcite based on their fuse analysis of active stars . these authors suggested that their excess line widths could result from coronal structures with heights up to @xmath87 . such a picture would also be consistent with the eclipse studies of @xcite and @xcite . with regard to eclipses , it is important to note that extended structure at the stellar poles would give rise to no obvious x - ray eclipse . however , pole - dominated emission also would not give rise to strong rotational broadening . our results suggest that , if rotational broadening is the correct interpretation for our observed line widths , this emission must be distributed around the star and not concentrated at the poles . the hetg observation does cover the secondary optical eclipse , and there is some evidence that we do indeed see some obscuration of the x - ray corona , despite the complications presented by the flare . we have computed theoretical x - ray lightcurves for algol using a spherically - symmetric , optically - thin emitting shell coronal model . the model assumes that algol a is 100% x - ray dark , and uses the following physical and orbital parameters : @xmath88 , @xmath89 , @xmath90 , @xmath91 , @xmath92 days , and @xmath93 degrees @xcite . theoretical lightcurves were calculated for an array of algol b coronal scale heights , ranging from 0.1 to 4.0 stellar radii . figure [ f : lc ] shows the observed lightcurve of algol with three theoretical lightcurves of various scale heights overplotted . also plotted is a similar curve computed for polar emission restricted to high latitudes corresponding to the scale height predicted by quasi - static loop models ( see below ) . if we assume that the count rate at phase @xmath8 0.5 represents the quiescent level of algol b , as does the level toward the end of the observation , we can see from the overplotted models that scale heights of @xmath94-@xmath95 represent the data reasonably well . the polar emission model clearly does not represent the observed eclipse minimum ; we discuss this further below . a scale height commensurate with the thermal scale height would appear to clash with the recent interpretation of density - sensitive line ratios seen in _ letgs observations of algol by @xcite . the interpretation of helium - like line ratios in terms of plasma density is complicated in the case of algol by the radiation field of algol a , which is sufficiently strong to cause significant radiative excitation of electrons from the upper level of the forbidden line to the upper level of the intercombination line through the transition @xmath4 in the ions c , n and o. nevertheless , @xcite pointed out that , even with densities of order @xmath96 @xmath97similar to those found for other active stars in the extensive survey of ( * ? ? ? * in preparation)simple quasi - static coronal loop model scaling laws ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) that predict that loop peak temperature @xmath44 , pressure @xmath20 and length @xmath98 are related by @xmath99 , suggest maximum loop lengths of order a few @xmath96 cm . this is only a fraction of the radius of algol b. as mentioned in [ s : f / i_analysis ] , there was no detectable orbital modulation of the o vii @xmath55 ratio in the algol letgs data , which was observed during orbital phase @xmath59 - 1.03 . @xcite also found no differences in the @xmath55 ratio of algol spectra which was extracted during the quiescent and flaring phases of an xmm - newton observation . this result has important implications . if we assume that the corona of algol is indeed similar to that of other active stars , such as its `` coronal twin '' hr 1099 @xcite , then we must conclude that _ the bulk of the o vii emission is always illuminated by algol a_. this can be achieved by having a fairly extended corona with a scale height @xmath100 , as our line profile analysis indicates . instead , _ if the o vii emitting coronal structures are compact compared to the stellar radius , then they must reside predominantly at the poles _ where the algol a uv radiation field is not significantly shadowed at primary eclipse . if scale heights are fairly large ( @xmath101 ) , the classical quasi - static loop scaling laws must then be quite inapplicable to these coronal structures . in the case of compact ( by necessity ) quasi - static loops with a plasma density of a few @xmath96 @xmath97 , @xcite estimate a coronal volumetric filling factor of up to 0.3unreasonably large for emission confined to polar regions . as noted earlier in this section , the synthetic lightcurve computed for the case of polar emission with a scale height of @xmath102 , similar to that predicted by quasi - static models , is a very poor match to the observed eclipse minimum light ( figure [ f : lc ] ) . this model predicts very sharp eclipses which have never been observed in x - ray observations of algol . in summary , the combined evidence of a tentative detection of excess line widths , the shape of the x - ray lightcurve , and the lack of orbital modulation of the o vii @xmath55 ratio going into primary eclipse , all point to the plasma source at temperatures up to several @xmath7 k having a scale height of the same order as the stellar radius , and being distributed around the k star . such distributed coronal structure could be responsible for , e.g. , the flaring at lower latitudes , which was inferred by @xcite based on an xmm - newton x - ray lightcurve . unfortunately , the x - ray data alone can not rule out other scenarios involving compact structures , though these would require special and fortuitous placement of dominant active regions in order to remain consistent with past and present observations of the coronal emission of algol . @xcite drew similar conclusions based on the failure to observe a strong x - ray eclipse in the exosat observation , though with the benefit of not having seen data from subsequent observations that have led to strong ambiguities in interpretation . our favored picture of the x - ray emission of algol is also consistent with the picture painted by @xcite , and with the interpretation of excess line widths seen in uv and fuv coronal forbidden lines by @xcite . what we have not tackled in this article , however , is the very hot ( @xmath103 k ) plasma that often flares and seems to be located at the poles . the recent plasma density survey of ( * ? ? ? * in preparation ) indicates that this very hot material is at much higher densities of @xmath104 @xmath97 , and so occupies quite different structures . these are likely to be the structures responsible for pole - dominated flaring activity , such as that highlighted by @xcite . a detailed analysis of high quality _ chandra _ hetgs spectra of the algol system provides new insights into the geometry of the x - ray emission in this system . based on this analysis we draw the following conclusions : 1 . our study clearly reveals doppler shifts of the x - ray emitting plasma corresponding to orbital motion with a velocity at quadrature of 150 km s@xmath2 . these data thus provide the first definitive proof that the coronal activity of algol b dominates the x - ray emission of the system , as has long been suspected ( e.g. , * ? ? ? the observed doppler motion of the x - ray plasma on algol b appears to be off - center relative to the stellar center of mass , and shifted toward algol a. we suggest that this occurs as a result of the tidal distortion of the surface of algol b. the hetgs light curve exhibits a flare near secondary eclipse which , if located on the hemisphere facing the primary b8 star as expected , might also suggest the presence of a dominant active region that could bias the emission toward a slightly smaller orbital radius . 3 . alternatively , x - ray activity of algol a , possibly as a result of accretion from algol b , could be responsible for the smaller apparent orbital radius of the x - ray emission . in this case , we estimate that such a contribution amounts to no more than 15% of the total emission . such an x - ray flux would be consistent with current ideas as to the plausible range of accretion activity in algol . we have found some evidence for excess line broadening in bright x - ray emission lines , above that expected from surface rotation and thermal motions . if this effect is real , it suggests broadening by rotation of a radially - extended corona . the observed widths would require a coronal scale height of at least one stellar radius this would be consistent with recent observations that detected excess broadening in the forbidden fe xviii 974 line , seen in fuse fuv spectra of rapidly rotating active stars . 5 . while turbulence , microflares or flows could also produce excess line broadening , the idea of a coronal plasma with significant radial extent is also supported by uv - sensitive lines of o vii seen in an letgs observation of algol . no change in the uv - sensitive lines was seen going into primary eclipse , when the majority of the visible hemisphere of algol b was shadowed from the radiation field of algol a. 6 . while it is possible that other physical effects are responsible for excess line widths and lack of modulation of the uv - sensitive o vii lines , coronal extension would appear to be the most likely . a consistently high plasma density of a few @xmath105 @xmath97 , as would be indicated by the o vii lines in the absence of uv excitation , would be unlike _ any _ of the densities seen at o vii temperatures in the sample of active stars surveyed by ( * ? ? ? * in preparation ) . this picture also appears to be consistent with conclusions drawn from recent lightcurve analyses which were summarized in [ s : intro ] . + coupled with the results of the coronal density survey of ( * ? ? ? * in preparation ) , we can summarize as follows . coronal plasma with temperatures of up to several @xmath7 k appears to have a significant component lying in extended structures with a scale size similar to the stellar radius and densities typically of a few @xmath96 @xmath97similar to solar active regions . very hot coronal plasma appears more exclusively in much more compact regions with densities of order @xmath104 @xmath97 , and in very active , rapidly rotating stars appears to be concentrated more toward stellar poles . further spectroscopic studies of these apparently different coronal regimes , especially with regard to possible abundance anomaly differences between them , should prove very interesting . 24.78 & 0.0306 @xmath30 0.0033 & 0.0217 @xmath30 0.0055 & 0.0285 & 0.0267 & 0.0021 + 18.97 & 0.0256 @xmath30 0.0013 & 0.0216 @xmath30 0.0009 & 0.0249 & 0.0238 & 0.0006 + 16.78 & 0.0284 @xmath30 0.0025 & 0.0190 @xmath30 0.0007 & 0.0212 & 0.0211 & 0.0072 + 15.01 & 0.0266 @xmath30 0.0017 & 0.0196 @xmath30 0.0005 & 0.0199 & 0.0199 & 0.0067 + 12.13 & 0.0266 @xmath30 0.0008 & 0.0205 @xmath30 0.0006 & 0.0217 & 0.0212 & 0.0049 + 8.42 & 0.0211 @xmath30 0.0009 & 0.0191 @xmath30 0.0008 & 0.0197 & 0.0195 & 0.0014 + , t. h. , canizares , c. r. , dewey , d. , mcguirk , m. , pak , c. s. , & schattenburg , m. l. 1994 , in proc . 2280 , p. 168 - 180 , euv , x - ray , and gamma - ray instrumentation for astronomy v , oswald h. siegmund ; john v. vallerga ; eds . , 168180

in a study of _ chandra _ high energy transmission grating spectra of algol , we clearly detect doppler shifts caused by the orbital motion of algol b. these data provide the first definitive proof that the x - ray emission of algol is dominated by the secondary , in concordance with expectations that the primary b8 component should be x - ray dark . however , the measured doppler shifts are slightly smaller than might be expected , implying an effective orbital semi - major axis of about @xmath0 instead of @xmath1 for the algol b center of mass . this could be caused by a small contribution of algol a , possibly through accretion , to the observed x - ray flux , in which case such a contribution does not exceed 10 - 15% . we suggest the more likely explanation is an asymmetric corona biased toward the system center of mass by the tidal distortion of the surface of algol b. a detailed analysis of the profiles of the strongest lines indicates the presence of excess line broadening amounting to approximately 150 km s@xmath2 above that expected from thermal motion and surface rotation . possible explanations for this additional broadening include turbulence , flows or explosive events , or rotational broadening from a radially extended corona . we favor the latter scenario and infer that a significant component of the corona at temperatures @xmath3 k has a scale height of order the stellar radius . this interpretation is supported by the shape of the x - ray lightcurve and tentative detection of a shallow dip at secondary eclipse . we also examine the o vii intercombination and forbidden lines in a low energy transmission grating spectrograph observation and find no change in their relative line fluxes as the system goes from quadrature to primary eclipse . since these lines appear to be strongly affected by uv irradiation from algol a through radiative excitation of the @xmath4 transition , this supports the conjecture that the corona of algol b at temperatures of several million k must be significantly extended and/or located toward the poles to avoid being shadowed from algol a during primary eclipse .

You are an expert at summarizing long articles. Proceed to summarize the following text: antiferromagnetism ( af ) appears in the hubbard and related models for an arbitrary interaction strength @xmath0 provided we are close to the half - filled - band situation ( @xmath1 in the orbitally nondegenerate case ) @xcite . this is easy to understand qualitatively , since the intraatomic hubbard interaction @xmath2 is diminished by keeping apart the electrons with the opposite spins . in the strong - correlation limit @xcite this interaction leads to an antiferromagnetic kinetic exchange for an arbitrary band filling @xmath3 . at the same time , the band energy is not increased because the concomitant nesting condition for the quasiparticle states achievable for bipartite lattices only leads to the energy decrease of the occupied states , even when going beyond the hartree - fock picture . in effect , the regime of the band filling @xmath4 for which the af state is stable at given @xmath0 has been determined for variety of theoretical approaches @xcite - @xcite . the reliability of the results for the half - filled case is not in question as long as they reduce to those in the hartree - fock and to the mean - field ( heisenberg ) approximations in the weak- and strong - correlation limits , respectively , as we will discuss in the following . the theoretical results are in accord with the fact that _ all _ known mott insulators with the half - filled band configuration are also antiferromagnetic insulators . the main purpose of this paper is to overview the situation by discussing the crossover behavior from the hartree - fock to the mean - field approximation for the heisenberg antiferromagnet . similar , though not equivalent results can be reached within the dynamic mean - field theory ( dmft , see below ) . in the orbitally degenerate situation ( with the degeneracy @xmath5 ) the same type of magnetic(slater ) gap is generated by an alternant orbital ordering @xcite , as is also discussed at the end of the paper . explicitly , we concentrate our attention to two specific features of quasiparticle states not elaborated in detail so far , namely , ( i ) an evolution of the magnetic gap ( renormalized by the electronic correlations ) into the mott - hubbard gap , and ( ii ) a rather weak renormalization of the effective mass for the half - filled - band case , which is in contrast with that calculated in the paramagnetic ( para ) case @xcite . a rather strong mass enhancement is retained in the @xmath6 case , in direct analogy to the paramagnetic case @xcite . these results are obtained within the slave - boson approach in the saddle - point approximation , which we compare with the corresponding analysis in the infinite - dimension limit @xcite . in particular , we introduce the concept of a nonlinear staggered molecular field , which shows up as the effective ( nonlinear - in - magnetization ) magnetic gap , evolving at temperature @xmath7 continuously with increasing @xmath0 from the slater ( hartree - fock ) gap @xmath8 into the hubbard gap . in connection with this evolution , we single out the magnetic and coulomb parts of the localization energy . these particular features resolve explicitly the old question about the difference between the slater and mott - hubbard insulators in the sense that only the mott - hubbard gap survives when antiferromagnetism disappears ( i.e. above the nel temperature ) . most of the results are contained in the mathematical formulation established earlier @xcite - @xcite . here , we only discuss those points in an explicit manner . we believe that these points are relevant to the general physics of correlated systems . this reason is also behind publishing , perhaps belatedly , such a simple paper . additionally , since the effective mass is only weakly renormalized for the @xmath9 case and it never reaches zero for @xmath6 , the role of the quantum fluctuations is relatively not as crucial as for the discussion of the paramagnetic state . the structure of the paper is as follows . in section ii we provide the analytic details of the af solution , which are followed by the numerical analysis ( section iii ) and the discussion in physical terms in section iv , where we also compare the results with those for a doubly degenerate - band case for @xmath9 . to stress the role of the molecular field coming from the electronic correlations , we start from the extended hubbard model , with intersite exchange interactions included , i.e. write @xmath10 where the labels @xmath11 and @xmath12 represent two interpenetrable sublattices , each containing half @xmath13 of available atomic sites . the first term represents single - particle hopping of electrons between the sublattices ( nearest neighbors ) , the second and the third express the intraatomic interaction of the same magnitude on _ all _ sites , the fourth includes the heisenberg exchange between the sublattices , and @xmath14 is the reference energy with @xmath15 being the chemical potential , and @xmath16the total number of fermions . the heisenberg term has been added only to provide an illustration of the concept of the exchange field coming from the electron correlations ( they will add to one another ) . in the mean - field ( saddle - point ) approximation for the slave bosons , the rotationally invariant approach of li et al . @xcite and the original kotliar - ruckenstein @xcite formulations can be brought to an equivalent form @xcite . explicitly , the six bosons @xmath17 and @xmath18 appearing in the approach fulfill the following five constrains in the mean - field approximation @xmath19 @xmath20 where the subscript @xmath21 characterizes the sublattices , and fermion operators @xmath22and @xmath23 represent the new ( quasiparticle ) fermion operators appearing in the theory . the effective hamiltonian in the saddle - point approximation takes the form in reciprocal ( @xmath24 ) * * * * space @xmath25 where @xmath26 contains the constraints appearing in the theory in the form of lagrange multipliers @xmath27 + \notag \\ & \frac{n}{2}\lambda ^{(1b)}\left ( e^{2}+d^{2}+p_{b\uparrow } ^{2}+p_{b\downarrow } ^{2}-1\right ) - \notag \\ & \frac{n}{2}\left [ \lambda _ { \uparrow } ^{(2b)}\left ( p_{b\uparrow } ^{2}+d^{2}\right ) + \lambda _ { \downarrow } ^{(2b)}\left ( p_{b\downarrow } ^{2}+d^{2}\right ) \right ] \notag\end{aligned}\ ] ] the summation over ( @xmath24 * * ) * * comprises the states in the halved brillouin zone and @xmath28 is the number of nearest neighbors . the lagrange multipliers @xmath29 and @xmath30 correspond to the constraint ( [ con1 ] ) and , since they are spin symmetric we can put @xmath31 the spin - dependent lagrange multipliers @xmath32 and @xmath33 represent spin - resolved constraint ( [ con2 ] ) . in the antiferromagnetic ( af ) case the magnetic moments @xmath34 @xmath35 have the opposite sign , i.e. @xmath36 this means that the constraints @xmath37 obey the relations @xmath38 the narrowing factor @xmath39 renormalizing the bare band energy @xmath40 assumes the form @xmath41 the hamiltonian ( [ hf ] ) can be diagonalized with the help of the bogolyubov transformation @xcite . in effect , the free energy functional of the system ( per atom ) takes the form @xmath42 -\frac{k_{b}t}{n}% \underset{\mathbf{k}\sigma}{\sum}\ln\left [ 1+e^{-\beta\left ( -e_{\mathbf{k}% } -\mu_{eff}\right ) } \right ] \quad \label{hb } \\ & + \frac{1}{2}jzm^{2}+ud^{2}+\mu\frac{n_{e}}{n}+h_{con } , \notag\end{aligned}\ ] ] where now @xmath43 here the quasiparticle energies are @xmath44 , with the magnetic gap @xmath45 , and the effective chemical potential @xmath46 with @xmath47 . the quantity @xmath48 plays a role of the correlation induced molecular field , since it adds to the effective heisenberg field @xmath49 in the case @xmath50 ( taken in the numerical analysis ) @xmath51 constitutes the entire gap induced by the magnetic ordering ( it is the _ magnetic _ gap ) . on the whole , the first two terms in ( [ hb ] ) provide the contribution to the thermodynamics coming from the single particle excitations in the magnetic ( slater ) subbands caused by af ordering and having energies @xmath52 these quasiparticle energies comprise the effective band narrowing or the mass renormalization @xmath53 and the molecular field @xmath54 both to be determined in a self - consistent manner detailed below . the field @xmath51 arises from the local constraint ( [ con2 ] ) . thus , one can say that the molecular field is induced by the correlations . the functional ( [ hb ] ) must be minimized with respect to all appearing bose fields and the chemical potential @xmath55 effectively , one can reduce ( [ hb ] ) to the form with two variables @xmath56 and @xmath18 . for the purpose of simplicity and clarity of our points we take constant density of states @xmath57 for @xmath58 for which the ground state energy takes the form @xmath59 where @xmath60 is the ground - state energy , and @xmath61 in this expression we have already connected @xmath62 to @xmath63via the relation @xmath64 for the sake of completness , we list also the explicit form of the band narrowing factor for @xmath9 : @xmath65 .\ ] ] note that the variable @xmath66 has a physical meaning of the ratio of the slater gap to the renormalized band energy . in other words , it provides a relative strength of the effective local field with respect to the renormalized kinetic energy . the growing ratio @xmath67 drives the system towards localization induced by the formation of staggered magnetic moments arranged in two sublattices , whereas the growing ratio @xmath68 drives the system towards localization independent of magnetic ordering . therefore , the present formulation will allow us to single out the contributions coming from the two factors . the magnetic energy is thus measured with respect to the band energy @xmath69 renormalized by the coulomb interaction . we now show that the results obtained above provide correctly a mean - field solution in the @xmath72 limit . for the sake of simplicity consider the half - filled band case with @xmath50 and for symmetric form of the density of states , @xmath73 . we can then write the ground state energy ( per site ) in the form @xmath74 for large @xmath0 , the gap @xmath75 is also large . in that limit the energy has the form @xmath76 where @xmath77 . minimizing this expression with respect to @xmath75 we obtain that @xmath78 minimization with respect to @xmath62 and @xmath79 gives the relations @xmath80 and @xmath81 where @xmath82 . combining eqs.([m1])-([m3 ] ) we obtain that @xmath83 , and @xmath84 . explicitly , from the fact that @xmath85 and that numerically @xmath86 , we obtain that @xmath87 , and @xmath88 . thus finally , for the featureless form of the density of states we have @xmath89 and @xmath90 in other words , in the @xmath72 limit the hubbard model reduces to the heisenberg model @xcite with the hubbard gap @xmath91 , since in the mean - field approximation the ground - state energy is then given by the kinetic exchange contribution @xcite @xmath92 with @xmath93 . also , the magnetic gap reduces to the atomic value of the hubbard gap , as @xmath94 with @xmath95 . the hartree - fock @xmath96 limit is recovered once one notices that the method has been constructed to obtain @xmath97 in the weak coupling limit @xcite , @xcite . under that circumstance eq.([eg ] ) reduce to the usual hartree - fock form if we assume that now @xmath98 . this limit was checked out also numerically , but the results are not reproduced here . in fig.1 we have displayed both the effective magnetic ( slater - type ) gap @xmath99 and the mott - hubbard gap for the paramagnetic phase , both for @xmath9 [ ptb ] those characteristics are plotted for the ground state . the chemical potential is then @xmath100 the mott- hubbard gap is expressed through the difference in the chemical potential in the paramagnetic case @xmath101 for @xmath102(the upper part ) and for @xmath103 ( the lower part ) and was discussed earlier @xcite . for @xmath9 the slater split - band picture appears for arbitrary small @xmath0 and @xmath104 increases with increasing @xmath105 , where @xmath106 . in the limit @xmath107 the gap is composed of the slater and the mott - hubbard parts , and when @xmath108 the former merges gradually with the latter . this can be seen explicitly in fig.2 , [ ptbptb ] ffig2.eps where we have shown the ground state energy @xmath109versus @xmath110 in the strong - correlation limit the energy is determined by the kinetic - exchange contribution @xmath111 @xcite , as shown explicitly . the fitted coefficient @xmath112 to the numerical results can not be attributed to any particular 3d structure , since we have used in numerical calculation a constant value of @xmath113 . the energies of para- and antiferro - magnetic states merge for @xmath9 and in the @xmath114 limit . the inset illustrates another interesting characteristic of the solution , namely the magnetic gap _ is not _ proportional to the magnetization , as one would expect from the hartree - fock ( slater ) solution . in other words , the molecular field @xmath51 is a nonlinear function of @xmath115 since from the condition @xmath116 we obtain the relation @xmath117 @xmath118 this is the reason why the antiferromagnetic gap for almost localized fermions can not be regarded as just the slater gap . also , the af solution disappears at the critical band filling @xmath119 to visualize the difference between the magnetic gap and the magnetization we have plotted in fig.3 both quantities as a function of the interaction strength @xmath120 @xmath35for different band filling parameter @xmath121 [ ptb ] ffig3.eps while for @xmath9 the magnetic moment saturates gradually with growing @xmath122 @xmath104 keeps increasing . the magnetic gap for @xmath9 increases and eventually @xmath123 ; this circumstance indicates again that the magnetic gap merges with the hubbard gap , which can be estimated analytically and is @xmath124 @xmath125 for @xmath126 . the double occupancy probability @xmath127 is shown in fig.4 for different band fillings . [ ptb ] ffig4.eps it decreases continuously with growing @xmath105 , i.e. the charge fluctuations are gradually suppressed , while the magnetic moment grows ( cf . the difference in the behavior of @xmath128and @xmath129is caused by the circumstance that the @xmath79 is of intraatomic nature , whereas @xmath62 is determined from the competition between the magnetic energy @xmath130 ( also of intraatomic nature ) and the renormalized band energy @xmath131 . the inset to fig.4 exemplifies the difference between the diminution of @xmath79 with growing @xmath105 for @xmath9 in two situations : for the paramagnetic ( para ) case @xmath132 for @xmath133 this feature is concurrent with the well - known effective mass divergence at the mott - hubbard localization boundary ( the _ brinkman - rice point _ ) . this divergence _ does not emerge _ in the antiferromagnetic state as @xmath79 approaches zero gradually , in the same manner , as @xmath62 approaches saturation @xmath134 in fig.5 we have displayed the inverse band narrowing factor @xmath135 ( for @xmath136 ) which turns into the effective mass renormalization @xmath137 ( for @xmath138 ) as a function of @xmath139 and for different @xmath140 values . [ ptb ] ffig5.eps again , the inset illustrates the difference with the @xmath9 case . one should note that the enhancement factor in af state is very small compared to that in para state , which is equal to @xmath141 ^{-1}.\,\ $ ] the difference between af and para states diminishes with decreasing @xmath142as in that situation the magnetic moment is reduced rapidly . so , the weak band narrowing in the @xmath9 case can be associated with the circumstance that the fermi level falls in the gap . this is the reason why @xmath143 raises rather steeply around @xmath144 . thus , the appearance of the itinerant antiferromagnetism with an isotropic gap changes the brinkman - rice scenario for the mott - hubbard transition , as it has been underlined before @xcite . also , the physical parameters @xmath145 and @xmath146 are all of the same magnitude . this is easy to envisage when estimating e.g. the band narrowing @xmath147 which is in the af state roughly @xmath148 and is of the order of unity . in fig.6 we have displayed the stability regime @xmath149 of the af phase . [ ptb ] ffig6.eps the full circles has been obtained @xcite in the limit of infinite dimension with the help of quantum monte carlo simulation . one should note the range of the filling @xmath3 of stable af phase is the broadest for @xmath150 i.e. when the molecular field is the strongest ( cf . fig.3 ) . note also that the monte - carlo results did note provide the asymptotic behavior for @xmath1 , as it does not reduce correctly to the hartree - fock limit . for the sake of completeness we display in fig.7 the ground state energy as a function of @xmath105 , for different @xmath151close to the half filling . [ ptb ] ffig7.eps in each case ( for @xmath138 ) the system contains the contribution @xmath152 for @xmath153and the contribution @xmath111 for @xmath154 in other words , the solution interpolates between the hartree - fock and the kinetic exchange limits . moreover , the shift of the curves with diminishing @xmath155 @xmath3 @xmath35 in the @xmath70 limit is due to the presence of the band term proportional to @xmath156 one should note that in the paramagnetic state the direct coulomb interaction contribution @xmath157 competes with band energy @xmath158 in the antiferromagnetic state the local magnetic contribution @xmath159 is of the opposite sign than the coulomb part . however , it changes also the band energy . to specify the role of the staggered field we have plotted in fig.8 the difference @xmath160 as a function of @xmath110 [ ptbptb ] ffig8.eps the magnetic contribution exceeds the coulomb part for @xmath161 ( i.e. away from the hartree - fock limit ) , and is particularly strong ( and comparable to the band energy ) in the regime @xmath162 where the mott - hubbard localization would take place for the paramagnetic case . this circumstance indicates why the metal - insulator transition present at @xmath7 in the paramagnetic phase is wiped out when the itinerant af sets in . simply , the magnetic contribution @xmath163 freezes gradually the spatially separated electrons in the antiferromagnetic phase , taking also advantage of nonvanishing kinetic energy ( the kinetic exchange , @xmath164 , contribution ! ) . in the para phase we have instead a direct competition between band and coulomb energies . to summarize , the mott - localization is achieved gradually at @xmath7 in the af state . in other words , the present approach provides a crossover behavior from slater to mott insulator , as shown e.g. in fig.2 . the same holds true even when we include the intersite exchange @xmath165 which may originate from the quantum gaussian fluctuations . the continuous evolution with growing @xmath166does not preclude the first order transition at nonzero temperature , as has been demonstrated some time ago for the paramagnetic state @xcite and subsequently reconfirmed in the limit @xmath167 @xcite . the correlated state builts in gradually with increasing @xmath105 , as can be seen from the gradual increase of the mass enhancement in @xmath168case , i.e. when the ground state is always metallic . the same holds true for the ground state energy shown in fig.7 . we have addressed the question of crossover from slater ( hartree - fock ) to mott - hubbard ( atomic ) picture in the half - filled band case , as well as have analyzed the behavior of quasiparticle physical properties in af state in the half- and non half - filled - band cases . although our analysis is based on the saddle - point solution within the slave - boson functional - integral approach , the results can serve as a mean - field analysis , since they interpolate between those in the hartree - fock approximation in the limit @xmath169 and those in mean - field approximation for the heisenberg model ( for @xmath170 in the @xmath72 limit . they also represent basis for inclusion of gaussian fluctuations @xcite in a magnetically ordered ( af ) state close to the mott - hubbard localization . however , one should realize that for the half - filled band case the effective mass is only weakly renormalized so the renormalization factor @xmath171 . also , for @xmath172 @xmath173 remains always finite so , perhaps the role of the quantum fluctuations is not as crucial for af state , as it is for the paramagnetic state . in any case , it will be much more involved that in the paramagnetic case @xcite . the full analysis of the mott - hubbard boundary should also include the disordered local - moment phase @xcite , so far not included within the present scheme . the physical meaning of the results obtained within the slave - boson approach ( sba ) can be characterized as follows . it is well known @xcite , @xcite - @xcite , @xcite that the saddle point solution of the approach reproduces the results of the gutzwiller approach ( ga ) , as far as the overall ( ground - state ) behavior is concerned . moreover , sba leads to an improvement of the ga by incorporating both the quasiparticle picture of those systems and the quantum gaussian fluctuations ( not considered in this paper ) . in general , the pseudo - fermion fields @xmath174 representing the quasiparticle states are in one - to - one correspondence to the original fermion fields in the physical fock space . the mapping is quite obvious within the above analysis . specifically , the quasiparticle energy @xmath175 leads to the following density of states @xmath176 where @xmath177is the bare density of states in pm phase . thus the enhancement factor due to the correlations is distinct ( and disappear in the first factor ) from the change of the density of states caused by the appearance of the magnetic gap . by analogy , for the paramagnetic state ( @xmath178 ) , for which @xmath179 can be written as @xmath180 and where the self - energy @xmath181 leads to the mass enhancement @xmath182 we can not single out the factor ( @xmath183 ) in ( [ ro ] ) as the corresponding enhancement also in af state . in the doubly - degenerate - band case ( for @xmath9 ) and under the same - type of approximation scheme , the role of the magnetic gap is played by the gap formed by an alternant orbital ordering in the ferromagnetic state @xcite . the gap @xmath184 in the latter case is reproduced in fig.9 for the magnitude of the intraatomic ( hunds - rule ) exchange @xmath185 . again , the gap approaches asymptotically the mott - hubbard gap , which in this case @xcite is equal to @xmath186 . however , here there is a critical value of @xmath68 , at which the system exhibits antiferromagnetic orbital ordering . in the half - filled case for the doubly degenerate case @xmath187 the gap appears for an arbitrary small @xmath0 @xcite in the af case and for the critical value of @xmath0 for the paramagnetic phase @xcite . one should also note that within the present sb scheme the antiferromagnetic slate - type state ( afs ) evolves for @xmath9 gradually into antiferromagnetic mott ( afi ) state with increasing @xmath68 ratio . for @xmath138 , we observe only antiferromagnetic metallic ( afm ) state . this is in contrast with the results obtained with the composite - operator method @xcite , as well as with an original analysis @xcite where a phase border line between itinerant- and localized - staggered moment - bearing phases is drawn . localized - moment phase for @xmath188 @xmath189 can be obtained only when the polaronic effects due to antiferromagnetic moments surrounding the hole in the mott insulator are included @xcite . obviously , such asf - afi border line at temperature @xmath190 ( for @xmath9 ) is present @xcite and is induced by the difference in entropy of afs and afi states . * acknowledgment * j.s . is grateful to claudine lacroix and m. avignon for discussions during his stay at drfmc - ceng in grenoble . this research was supported by kbn grant no . 2 p03b 050 23 . two of the authors ( ml and js ) acknowledge also the support of the project franco - polonais polonium in the period 1998 - 2000 . 99 see : j. des cloizeaux , j. phys . radium . * 20 * , 606 ( 1959 ) , and d.r . penn , phys . rev.*142 * , 350 ( 1966 ) , - for the stability of antiferromagnetic ( af ) phases in the hartree - fock ( hf ) approximation . the stability of af in the present context ( i.e. in the gutzwiller approximation ) has been obtained in : k.kubo and m. uchinami , prog . . phys . * 54 * , 1289 ( 1975 ) . for a comparative analysis see : e.g. a.m. ole and j. spaek z. phys.b * 44 * , 177 ( 1981 ) ; see also : g. seibold et al . b * 57 * , 6937 ( 1998 ) . the phase diagram recovering both hf and strong correlation limits correctly has been discussed in g. kotliar and a.e . ruckenstein , phys . * 57 * , 1362 ( 1986)-within the slave - boson approach ; and in : w. metzner and d. vollhardt , phys . * 62 * , 324 ( 1989 ) ; p. fazekas , b. menge and e. mler - hartmann , z. phys . * 78 * , 69 ( 1990)-in the limit of infinite dimension @xmath191 . the corresponding discussion in two - spatial dimensions was performed in : e. arrigoni and g.c . strinati , phys . b**44 * * , 7455 ( 1991 ) ; w. ziegler et al . , phys . b * 53 * , 1231 ( 1996 ) . in that respect the af stability should be considered for @xmath192 against the onset of the ferromagnetism , for which the pauli principle plays the same role as @xmath72 for antiferromagnetism . however , in the large-@xmath0 limit , the virtual hopping processes , particularly for @xmath9 , lead to a stable af state ; see e.g. j.spaek , a.m. ole , and k.a . chao , phys . solidi ( b ) * 108 * , 329 ( 1981 ) . cf . also : k. a.chao , j. spaek , and a.m. ole , j. phys . c**10 * * , l271 ( 1977 ) , where the effective kinetic exchange has been derived in the @xmath193 limit for an arbitrary @xmath3 . asymptotic values of the magnetic moment and the molecular - field parameter @xmath194 are discussed in : b. mller et al . , j. phys . : condens . matter * 5 * , 4847 ( 1993 ) . m. jarrell , phys . lett . * 69 * , 168 ( 1992 ) ; m. jarrell and t. pruschke , z.phys . b**90 * * , 187 ( 1993 ) . for the comparison the men - field slave - boson and the quantum monte - carlo approaches see : l. lilly et al . * 65 * , 1379 ( 1990 ) . the gaussian fluctuations in para state are treated in ref . 6 , and in : p. wlfle and t. li , z. phys . * 78 * , 45 ( 1990 ) ; r. raimondi and c. castellani , phys . rev . b**48 * * , 11453 ( 1993 ) . for recent treatment see : r. frsard and t. kopp , nucl . b * 594 * , 769 ( 2001 ) ; a. tandon et al . lett . * 83 * , 2046 ( 1999 ) . these effects are considered mainly in the context of large - u situation for a two dimentional case , cf . e.g. p. wrbel and r. eder , phys . b**58 * * , 15160 ( 1998 ) , and references therein ; cf . also : m. imada et al . , 70 * , 1039 ( 1998 ) . the slater gap @xmath195 and the hubbard gap as the functions of interaction strength . the slater gap merges with the mott - hubbard gap as @xmath196 * fig.2 . ground state energy for af state and @xmath9 versus @xmath197 the lower curve is the fit to the expression @xmath198the inset displays the difference in behavior of magnetic moment @xmath62 and half of the slater gap @xmath199 both plotted as a function of the band filling . the slater gap parameter @xmath200(top panel ) and the magnetic moment @xmath201 ( bottom ) versus @xmath105 and for different values of @xmath121 * fig.4 . the double occupancy @xmath202 vs @xmath203and for the different @xmath3 values.the inset display the difference in behavior for para- and antiferro - magnetic cases for @xmath204 * fig.5 . the effective mass enhancement @xmath205 ( with respectto the band value @xmath206 ) vs @xmath105 and for the @xmath3 values shown . the inset shows a rather weak enhancement close to the mott - hubbard limit . * fig.6 . the stability regime of af solution ; the inset : results ofmonte - carlo calculations in the@xmath207limit @xcite . ground state energy vs @xmath208for different band filling @xmath209 . in each case the energy is @xmath152 for @xmath210 and @xmath211 for @xmath212 * fig.8 . comparison of the magnetic @xmath213and the coulomb @xmath214terms in the ground state energy versus @xmath110 * fig.9 . the half - gap parameter @xmath215 ( in units of the bare bandwidth @xmath216 ) vs @xmath68 for a quarter - filled doubly - degenerate band . the hund s rule exchange is @xmath217 .

we supplement ( and critically overview ) the existing extensive analysis of antiferromagnetic solution for the hubbard model with a detailed discussion of two specific features , namely ( i ) the evolution of the magnetic ( slater ) gap ( here renormalized by the electronic correlations ) into the mott - hubbard or atomic gap , and ( ii ) a rather weak renormalization of the effective mass by the correlations in the half - filled - band case , which contrasts with that for the paramagnetic case . the mass remains strongly enhanced in the non - half - filled - band case . we also stress the difference between magnetic and non - magnetic contributions to the gap . these results are discussed within the slave boson approach in the saddle - point approximation , in which there appears a _ non - linear _ staggered molecular field due to the electronic correlations that leads to the appearance of the _ magnetic gap_. they reproduce correctly the ground - state energy in the limit of strong correlations . a brief comparison with the solution in the limit of infinite dimensions and the corresponding situation in the doubly - degenerate - band case with one electron per atom is also made . pacs nos . 71.10.fd , 75.10.lp , 75.50.ee

You are an expert at summarizing long articles. Proceed to summarize the following text: * quantum fisher information . * we start with a brief introduction of qfi and give a useful form of qfi for a special kind of mixed qudit states , which usually represents the output states of qudit cloning . recall that qfi of parameter @xmath3 encoded in @xmath0-dimensional quantum state @xmath4 is generally defined as^@xcite^ @xmath5 where @xmath6 is the so - called symmetric logarithmic derivative , which is defined by @xmath7 with @xmath8 . by diagonalizing the matrix as @xmath9 , one can rewritten the qfi as^@xcite^ @xmath10 where @xmath11 is the qfi for pure state @xmath12 with the form @xmath13.\ ] ] note that eq . ( [ e2 ] ) suggests the qfi of a non - full - rank state is only determined by the subset of @xmath14 with nonzero eigenvalues . physically , the qfi can be divided into three parts^@xcite^. the first term is just the classical fisher information determined by the probability distribution ; the second term is a weighted average over the qfi for all the nonzero eigenstates ; the last term stemming from the mixture of pure states reduces the qfi and hence the estimation precision below the pure - state case . though the eq . ( [ e2 ] ) is powerful , there is no explicit expression for an arbitrary @xmath0-dimensional mixed state . however , it is worth noting that the output reduced states of uqcm and pqcm all have a form as @xmath15 which is completely characterized by a parameter independent shrinking factor @xmath16 and the dimensionality @xmath0 . here , @xmath17 and @xmath18 are the input state and @xmath0-dimensional identity matrix , respectively . although @xmath19 has such a simple form , an analytical expression of qfi is still difficult to achieve . fortunately , if we restrict our discussions to the special case of input states in the form @xmath20 could be given by ( see methods ) @xmath21 the above equation is a key mathematical tool for our analysis of this paper . although this expression only holds for the combination of eqs . ( [ e3 ] ) and ( [ e4 ] ) , it is powerful since the scaling form of @xmath19 is usually satisfied in quantum cloning machines or in the case of a pure state under white noises . on the other hand , the equatorial states are widely employed in the physical implementations of quantum communication ideas ( such as bb84 protocol^@xcite^ ) as well as in the demonstration of fundamental questions in quantum information processing . after a simple calculation , we find that @xmath22 is a monotonically increasing function of the shrinking factor @xmath16 . this is to be expected because the larger @xmath16 indicates more information the reduced output state @xmath19 contains about the relevant parameter . * distributability . * before moving to the discussion of distribution , a proper measure that quantify the distributability of cloning machines should be well defined . note that one can define the distributability in different ways which depends on what is distributed in the procedure . for example , it can be defined from the perspectives of fidelity which quantifies the total information of the state , and qfi which quantifies the information of particular parameters in the state . it is well known that , unlike the symmetric cloning in which all the copies are the same , the outputs of asymmetric cloning are nonidentical . hence , the optimality of asymmetric cloning can be judged by maximizing the sum of all copies fidelities , as discussed in refs . @xcite . intuitively , when we consider the distribution of quantum states , the distributability of asymmetric cloning can be defined as @xmath23 where @xmath24 is the fidelity between the original and the @xmath25th copy . namely , the larger @xmath26 indicates the better capability of distribution on the quantum states . in a seminal work^@xcite^ , the authors pointed that both fidelity and qfi are highly related to the distinguishability of the states , which is measured by bures distance^@xcite^. therefore , from the perspective of qfi , the measure @xmath27 also qualifies the capability of distributing information of relevant parameter encoded in the input state , where @xmath28 denotes the qfi of parameter @xmath3 in the @xmath25th copy . in the following discussions , we will use the definitions ( [ e06 ] ) and ( [ e07 ] ) to quantify the distributabilities of fidelity and qfi respectively in asymmetric cloning machines . * qfi distribution for 2-dimensional cloning . * as we mentioned above , one is particularly interested in the asymmetric cloning machines which produce two copies with different qualities within the framework of quantum cryptography . two typical asymmetric cloning machines are asymmetric uqcm^@xcite^ , which clones all input states equally well , and asymmetric pqcm^@xcite^ , which works equally well only for equatorial input states with the form of eq . ( [ e4 ] ) . we first discuss the 2-dimensional cloning by obtaining the analytical results , and then generalize it to _ d_-dimensional cloning by the assistance of numerical simulations . * asymmetric 2-dimensional uqcm . * the @xmath29 optimal asymmetric uqcm was independently proposed by niu and griffiths^@xcite^ , buek _ _ et al__^@xcite^ and cerf^@xcite^. though their formalisms are slightly different , the results are exactly the same . for the sake of convenience , we adopt the quantum circuit approach developed by buek . the transformation of asymmetric uqcm can be written in the following form @xmath30 where @xmath31 and @xmath32 . the parameters @xmath33 and @xmath34 are real , and satisfy the normalization condition @xmath35 . with the transformation ( [ e6 ] ) in mind , it is now an easy exercise to verify that the two different copies of the original state @xmath36 are @xmath37 from the point of geometry , the asymmetric uqcm shrinks the original bloch vector by two different shrinking factors ( @xmath38 , @xmath39 ) regardless of its direction . as special cases , we can see that if @xmath40 , then no information has been transferred from the original system , while , if @xmath41 , then all of the information in system a has been transferred to system b. in addition , if @xmath42 ( i.e. , @xmath43 ) , then it reduces to the symmetric uqcm case . obviously , the two shrinking factors @xmath44 and @xmath45 are related , and should satisfy the no - cloning inequality^@xcite^ @xmath46 which is an ellipse in the @xmath47 space . an optimal asymmetric uqcm is characterized by a point @xmath48 which lies on the boundary of this ellipse . according to eq . ( [ e5 ] ) , when @xmath49 , the qfi of parameter @xmath3 is proportional to @xmath50 . immediately , the qfis are @xmath51 here and henceforth we omit the subscript @xmath52 for brevity since we restrict our discussions to the single - parameter scenario . remarkably , a trade - off relation exists for the two qfis : if one qfi is large , correspondingly another qfi will become small . combining eqs . ( [ e9 ] ) and ( [ e10 ] ) , the trade - off relation of qfi is expressed as @xmath53 this trade - off tells us that even we only concern cloning the information of a particular parameter encoded in quantum states , two close - to - perfect copies can not be achieved simultaneously , imposed by quantum mechanics . an intuitive presentation of this trade - off relation is shown in fig . [ fig2]b ( dashed line ) . in the following , we consider the distribution of qfi in the asymmetric uqcm . as we defined in eq . ( [ e07 ] ) . the distributability of qfi is measured by @xmath54 therefore , the larger @xmath55 is , the more qfi of the relevant parameter has been distributed to the two copies . we find that the asymmetric uqcm always performs better than symmetric uqcm in distributing qfi , which means @xmath56 this can be proved by the method of lagrange multiplier . one will find three extreme points : @xmath57 , @xmath58 and @xmath59 . it is easy to verify that @xmath60 is the minimum value . in order to show the difference of distributability between qfi and fidelity , we also write the corresponding fidelities defined as @xmath61 @xmath62 according to eq . ( [ e06 ] ) , we adopt @xmath63 to qualify the capability of asymmetric uqcm in distributing the entire quantum state . fig . [ fig1 ] shows the results of asymmetric 2-dimensional uqcm . it is remarkable that , from the perspective of qfi , the asymmetric uqcm is always works better than symmetric uqcm , which is a sharp contrast to the result of fidelity^@xcite^ , where the symmetric uqcm is always optimal . * asymmetric 2-dimensional pqcm . * in the context of quantum cryptography^@xcite^ , the uqcm studied in the previous subsection might be optimal if the detail setup of qkd protocol is not specified . but it may not be optimal for the quantum states involved in a special qkd protocol . practically , it is possible that we already know a priori information of the input states . thus , a state - dependent cloning machine would perform better than uqcm . the best - known example of state - dependent cloning machine is the so - called pqcm . the symmetric pqcm was firstly proposed by bru _ et al _ for the equatorial qubit state^@xcite^ and then an asymmetric version was demonstrated by niu and griffiths^@xcite^. recently , the asymmetric pqcm has been experimentally realized using nmr^@xcite^ and fiber optics^@xcite^. previous studies suggest that the equatorial qubit state pqcm can be realized by both economic^@xcite^ and non - economic^@xcite^ transformations . one can check that both economic and non - economic methods achieve the same distributability of qfi . in the text we discuss the non - economic case as it can be directly generalized to _ d_-dimensional pqcm . the transformation of asymmetric pqcm can be written in the following form @xmath64 with @xmath65 and @xmath66 . @xmath67 is the normalization condition . @xmath68 is a set of orthogonal normalized ancillary state . for the equatorial qubit - state @xmath36 , the output states have the form as @xmath69 then , the shrinking factors are @xmath70 . using the normalization condition , the shrinking factors can be simplified as @xmath71 as is seen , in the scenario of asymmetric pqcm , there are two free parameters to be optimized . therefore , an optimal asymmetric pqcm is defined as the following : if we fix the quality of one copy , then the other copy is optimal with the highest quality . from the eqs . ( [ e18 ] ) and ( [ e19 ] ) , one can eliminate @xmath34 and obtain the trade - off relation between @xmath44 and @xmath45 @xmath72 assuming @xmath44 is constant , the optimal value of @xmath73 can be found , and the optimal trade - off relation reduces to @xmath74 the corresponding qfis are @xmath75 in the symmetric case , we have @xmath76 . remarkably , according to eq . ( [ e11 ] ) , one can immediately find the inequality @xmath77 the meanings of above equation are twofold . on one hand , it indicates that asymmetric pqcm performs better than asymmetric uqcm in distributing qfi by virtue of the known information . this result is essentially in agreement with that of fidelity . to be clear , we plot the trade - off relations between a s fidelity ( qfi ) and b s fidelity ( qfi ) for both asymmetric uqcm and pqcm in fig . it is evident that the lines of asymmetric pqcm are always above those of asymmetric uqcm , except for the start points and end points . on the other hand , it should be noted that , for the 2-dimensional pqcm , the asymmetric case is as good as the symmetric case on the capability of distributing qfi , while the later always performs better than the former with the measure of fidelity . the reason is that the fidelity @xmath78 as a function of @xmath79 is strictly concave , but the qfi @xmath80 as a function of @xmath81 is convex . this results in the sum of two fidelities and qfis achieving its maximal and minimal value , respectively , in the symmetric case . * qfi distribution for _ d_-dimensional cloning . * until now , we have restricted our discussions to the 2-dimensional cloning . although all quantum information tasks can be performed by using only two - level systems , it has been recently recognized that higher - dimensional quantum states ( i.e. , qudits ) can offer significant advantages for improving the security of quantum cryptographic protocols^@xcite^ , achieving higher information - density coding^@xcite^ and reducing the required resources for quantum computation and simulation^@xcite^. based on these considerations , it would be essential to extend above discussions to _ d_-dimensional cloning . one may think the results will be trivial and analogous conclusions will be obtained as well as the 2-dimensional cloning . however , as we will show below , there are some similarities between them , but more importantly , significant differences will appear with increasing dimensionality @xmath0 . * asymmetric @xmath0-dimensional uqcm . * the optimal asymmetric @xmath0-dimensional uqcm was proposed by cerf^@xcite^ and braunstein _ _ et al__^@xcite^. for a @xmath0-dimensional quantum system , the corresponding asymmetric uqcm can be generalized directly from the transformation ( [ e6 ] ) with @xmath82 and @xmath83 instead defined in higher - dimension , @xmath84 , and @xmath85 respectively . hence , the normalization condition now reads @xmath86 . the output reduced density matrices are written in the form of eq . ( [ e3 ] ) @xmath87 with shrinking factors @xmath38 and @xmath39 . in particular , if @xmath88 , we recover the results of symmetric uqcm . similarly , we can obtain a trade - off relation between two shrinking factors @xmath44 and @xmath45 . @xmath89 which corresponds to a set of ellipses in the space of shrinking factors that their eccentricities vary with dimensionality . it should be noted that in the infinite dimensional case , the corresponding ellipse shrinks to the line @xmath90 . now we turn to the calculation of qfi . assuming the input state is a @xmath0-dimensional equatorial state , then the qfis of output states ( [ e23 ] ) and ( [ e24 ] ) are obtained directly by ( [ e5 ] ) . @xmath91 the tradeoff relation between @xmath92 and @xmath93 can be derived by substituting @xmath94 into ( [ e25 ] ) with @xmath95 , but it is too complicated to present in the text . nevertheless , there is no doubt that one can not gain , at the same time , two copies whose qfis are above values allowed by the trade - off relation . for @xmath96 , respectively from top to down . the red circles denote the corresponding results of symmetric uqcm , and the insertion is magnified plot of @xmath97.,title="fig:",scaledwidth=40.0% ] + for @xmath96 , respectively from top to down . the red circles denote the corresponding results of symmetric uqcm , and the insertion is magnified plot of @xmath97.,title="fig:",scaledwidth=40.0% ] + we are concerned with whether the @xmath0-dimensional asymmetric uqcm still performs better than symmetric uqcm in distributing qfi . against all expectations , the results become subtle with increasing @xmath0 . unlike the 2-dimensional case where asymmetric uqcm is always better than symmetric uqcm , the @xmath0-dimensional asymmetric uqcm may be worse than symmetric uqcm under certain conditions . as shown in fig . [ fig3]a , we find the sum of two fidelities still reaches its largest value at the point of @xmath98 , which means , by the measure of fidelity , the symmetric uqcm will optimally copy the state regardless of the dimension^@xcite^. however , fig . [ fig3]b shows that the distributability of qfi achieves its smallest value at the point of @xmath99 when @xmath100 , while it is interesting to note that the point of @xmath99 becomes a local maximum point when @xmath97 . namely , the asymmetric uqcm is no longer always better than symmetric uqcm with increasing @xmath0 . the reason is that even though qfi is a monotonically function of the shrinking factor , it is not a linear function of it . this sophisticated relation between @xmath55 and @xmath16 reveals above interesting results . naturally , we start wondering when the asymmetric uqcm may become worse than symmetric uqcm . the numerical simulation shows that when @xmath1 , the @xmath44 of global minimal @xmath55 is equal to the symmetric case . while a bifurcation appears at @xmath101 , which means @xmath102 is no longer the global point of minimal @xmath55 , as shown in fig . mathematically , we can understand the bifurcation as follows : @xmath55 as a function of @xmath44 has three extreme points which are physically allowed when @xmath1 , and @xmath99 is the point of global minima . when @xmath2 , it has five extreme points , as seen from the insertion in fig . moreover , @xmath99 is no longer the global minima but a local maximum point . thus , the symmetric uqcm may outperform the asymmetric case . however , it is hard to understand why the critical point is @xmath101 in physical , we conjecture this critical point is related to the hilbert space structure of qudits . we leave this as an open question and the further study is underway . * asymmetric @xmath0-dimensional pqcm . * the generalization of asymmetric pqcm to @xmath0-dimensional is much more difficult , and in particular , it is too complicated to present an analytical trade - off relation between two copies . however , with the aid of numerical simulations , we can confirm two main results about the distribution of qfi in asymmetric @xmath0-dimensional pqcm : ( i ) pqcm gains an advantage over uqcm by utilizing the priori information , and ( ii ) a sudden change of the point of minimum also exists in asymmetric pqcm with increasing @xmath0 . the cloning transformation of asymmetric @xmath0-dimensional pqcm can be introduced^@xcite^ @xmath104 with the normalization condition @xmath105 . given the input state in the form of ( [ e4 ] ) , then , the shrinking factors of the output copies read as @xmath106 where we have use the normalization condition to eliminate the parameter @xmath107 . similar to the 2-dimensional case , here we again need to optimize two free tuning parameters . therefore , in the same way , an optimal asymmetric pqcm is defined by optimizing @xmath45 as large as possible when @xmath44 is fixed , and vice versa . by eliminating the parameter @xmath34 , we can obtain the trade - off relation ( not the optimal one ) @xmath108\nonumber\\ & + \frac{\eta_{a}-(d-2)a^2}{a}\sqrt{\frac{1-\big(\frac{\eta_{a}-(d-2)a^2}{2a}\big)^2}{d-1}-a^2}.\end{aligned}\ ] ] the optimal trade - off relation need to be further optimized by choosing a proper value of @xmath109 to make the largest @xmath45 . unfortunately , there is not a closed analytical form of @xmath109 for any dimensionality @xmath0 . however , by simple numerical simulations , we find that asymmetric pqcm indeed always performs better than asymmetric uqcm in distributing qfi as shown in fig . when the dimensionality _ d _ is large ( e.g. , @xmath110 ) , it should be noted that the advantage of pqcm over uqcm almost disappears . moreover , similar to the case of asymmetric @xmath0-dimensional uqcm , the asymmetric @xmath0-dimensional pqcm is not always better than the symmetric case for any dimensionality @xmath0 . [ fig5]b shows that a bifurcation of the point of global minimum also occurs at @xmath101 . this phenomena stresses that the critical point appearing at @xmath101 is not in any sense accidental . the physical reason behind this is worth further study . and @xmath80 for _ d_-dimensional asymmetric uqcm ( dashed lines ) and pqcm ( solid lines ) with @xmath111 , respectively from top to bottom . the insertion is magnified plot of @xmath112 . note that when @xmath110 and 30 , the two lines overlap greatly . ( b ) @xmath44 of minimal @xmath113 as a function of dimensionality @xmath0 . , title="fig:",scaledwidth=40.0% ] + and @xmath80 for _ d_-dimensional asymmetric uqcm ( dashed lines ) and pqcm ( solid lines ) with @xmath111 , respectively from top to bottom . the insertion is magnified plot of @xmath112 . note that when @xmath110 and 30 , the two lines overlap greatly . ( b ) @xmath44 of minimal @xmath113 as a function of dimensionality @xmath0 . , title="fig:",scaledwidth=40.0% ] + * discussions . * in this paper , we have investigated the distribution of qfi in asymmetric cloning machines which produce two nonidentical copies . in particular , we have elucidated four questions as we mentioned before . here , we summarize our results by replying these questions . ( i ) the answer is yes . it is definite that improving the qfi in one copy results in decreasing the qfi of the other copy , and the trade - off relation can be obtained analytically except for the asymmetric @xmath0-dimensional pqcm . ( ii ) the answer is also yes . thanks to a priori knowledge of the input states , pqcm always performs better than uqcm in distributing qfi . ( iii ) the answer is not so straightforward . it should be divided into two categories : for 2-dimensional cloning , asymmetric cloning always outperforms symmetric cloning on the distribution of qfi ; while for the _ d_-dimensional cloning case , the above conclusion only holds when @xmath1 and becomes invalid when @xmath2 , i.e. , the asymmetric cloning is not always better than symmetric cloning for any dimensionality . ( iv ) the most significant difference between fidelity and qfi is that fidelity is a linear function of the shrinking factor while qfi is nonlinear . this leads to the counterintuitive result that symmetric cloning is always optimal from the perspective of fidelity , but asymmetric cloning usually works better than symmetric cloning on the distribution of qfi , except for some particular situations ( e.g. , when @xmath97 and @xmath114 , see the insertion in fig . [ fig3]b ) . in view of these findings , we note that there are some problems in need of further clarifications . the first important issue is to understand why does the critical point appear at @xmath101 , not other numbers . secondly , we should realize that we have confined our discussion to the distributability of single parameter in asymmetric cloning machines . however , from both theoretical and practical points of view , it seems to be interesting to examine the problem of multi - parameter distribution in asymmetric quantum cloning . intuitively , there would be a trade - off relation of the quantum fisher information matrices between two nonidentical copies . these would be very intriguing topics that need further studies . here , we give the details of the derivation of eq.([e5 ] ) from eqs . ( [ e3 ] ) and ( [ e4 ] ) . to be clear , we can rewrite the ( [ e3 ] ) as @xmath115 note that the eigenvalues of @xmath116 consists of only two categories : @xmath117/d$ ] , and @xmath118 with @xmath119 . obviously , @xmath120 is an eigenstate of @xmath116 . thus the problem is converted to construct a complete orthogonal set ( containing @xmath121 bases ) of the operator @xmath122 which is also orthogonal to @xmath123 at the same time . the procedure can be divided into three steps : ( i ) finding @xmath121 bases of @xmath124 which are orthogonal to @xmath123 ; ( ii ) using the gram - schmidt procedure to orthogonalize them and ( iii ) the normalization . by this time , we have diagonalized the state ( [ e3 ] ) in the bases @xmath129 , and then the qfi can be calculated by eq . ( [ e2 ] ) . note that all the parameters @xmath52 is equally weighted due to the symmetry of @xmath123 , thus the qfi of any parameter is the same . furthermore , we observe that the part of classical fisher information vanishes since the probability distribution is independent of parameters @xmath52 , if measured in this bases . the remaining work is to determine the last two terms in eq . ( [ e2 ] ) . after lots of complicated but straightforward calculations , we obtain the qfi of any parameter @xmath52 @xmath130 the authors are supported by the national natural science foundation of china under grants no . 11247006 , 11025527 and 10935010 , the national 973 program under grants no . 2012cb921602 , and the china postdoctoral science foundation under grant no . 2014m550598 . bennett , c. h. & brassard , g. quantum cryptography : public key distribution and coin tossing . _ in proceedings of ieee international conference on computers , systems and signal processing _ pp.175 - 179 ( 1984 ) . ghiu , i. asymmetric quantum telecloning of _ d_-level systems and broadcasting of entanglement to different locations using the `` many - to - many '' communication protocol . a _ * 67 * , 012323 ( 2003 ) . braunstein , s. l. , buek , v. & hillery , m. quantum - information distributors : quantum network for symmetric and asymmetric cloning in arbitrary dimension and continuous limit . a _ * 63 * , 052313 ( 2001 ) .

an unknown quantum state can not be copied on demand and broadcast freely due to the famous no - cloning theorem . approximate cloning schemes have been proposed to achieve the optimal cloning characterized by the maximal fidelity between the original and its copies . here , from the perspective of quantum fisher information ( qfi ) , we investigate the distribution of qfi in asymmetric cloning machines which produce two nonidentical copies . as one might expect , improving the qfi of one copy results in decreasing the qfi of the other copy , roughly the same as that of fidelity . it is perhaps also unsurprising that asymmetric phase - covariant cloning machine outperforms universal cloning machine in distributing qfi since a priori information of the input state has been utilized . however , interesting results appear when we compare the distributabilities of fidelity ( which quantifies the full information of quantum states ) , and qfi ( which only captures the information of relevant parameters ) in asymmetric cloning machines . in contrast to the results of fidelity , where the distributability of symmetric cloning is always optimal for any @xmath0-dimensional cloning , we find that asymmetric cloning performs always better than symmetric cloning on the distribution of qfi for @xmath1 , but this conclusion becomes invalid when @xmath2 . classical information can be replicated perfectly and broadcast without fundamental limitations . however , information encoded in quantum states is subject to several intrinsic restrictions of quantum mechanics , such as heisenberg s uncertainty relations^@xcite^ and quantum no - cloning theorem^@xcite^. the no - cloning theorem tells us that an unknown quantum state can not be perfectly replicated because of the linearity of the time evolution in quantum physics , which is the essential prerequisite for the absolute security of quantum cryptography^@xcite^. nevertheless , it is still possible to clone a quantum state approximately , or instead , clone it perfectly with certain probability^@xcite^. therefore , various types of quantum cloning machines have been designed for different quantum information tasks , including universal quantum cloning machine ( uqcm)^@xcite^ , state - dependent cloning machines^@xcite^ and phase - covariant quantum cloning machine ( pqcm)^@xcite^. so far , the optimality of the approximate cloning machine is judged generally by whether the obtained fidelity between the cloning output state and the ideal state achieves its optimal bound . although the fidelity may have qualified the complete information of the quantum states , in most scenarios , only the information of certain parameters which are physically encoded in quantum states is our practical concern . for example , the relative phase estimation is an extremely important issue in the field of quantum metrology^@xcite^. thus it is not necessary to gain complete information of the whole quantum states themselves , but rather the relevant parameter information . qfi is a natural candidate to quantify the physical information about the involved parameters^@xcite^. in ref . @xcite , the authors pointed that the qfi of relevant parameter encoded in quantum states also can not be cloned perfectly , while it might be broadcast even in some non - commuting quantum states . furthermore , from the perspective of qfi , song _ et al . _ showed that wootters - zurek cloning performs better than universal cloning for the symmetric cloning cases^@xcite^. in our recent work , the multiple phase estimation problem was investigated in the framework of symmetric quantum cloning machines^@xcite^. on the other hand , we note that quantum cloning machines not only provide a good platform for investigating distribution of quantum information , but also have been proved to be very efficient eavesdropping attacks on the quantum key distribution ( qkd ) protocols^@xcite^. in this context , asymmetric quantum cloning machines would be of particular interest since the eavesdropper can adjust the trade - off between the information gained from a quantum communication channel and the error rate of information transmitted to the authorized receiver . motivated by these considerations , we investigate the problem of distributing qfi in asymmetric quantum cloning machines for any dimension . we focus on the following four questions : ( i ) is it possible to improve the qfi of one copy by decreasing that of the other copy ? if yes , what s the trade - off relation between them ? ( ii ) does asymmetric pqcm always perform better than asymmetric uqcm on the capability of distributing qfi ? ( iii ) does asymmetric cloning always outperform symmetric cloning in distributing qfi for any dimensionality ? ( iv ) what s the difference between fidelity and qfi on the characterization of distributability in asymmetric cloning ? except for the fourth question need to be clarified in detail , we can briefly answer the first two questions in the affirmative but the third in the negative . our results shed an alternative light on quantum cloning and may be exploited for quantum phase estimation .

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Proceed to summarize the following text: time - resolved pump / probe experiments have seen a rebirth in recent years as timescales have been pushed well into the femtosecond ( and sometimes attosecond ) range , and as many different types of experiments can now be performed in an ultrafast pump / probe format . much of the experimental work , ranging from ultrafast optical studies to angle - resolved photoemission ( arpes ) have been centered on examining the behavior of charge density wave materials @xcite . in this paper , we will review work on theoretical descriptions of pump / probe time - resolved photoemission spectroscopy on charge - density - wave ( cdw ) materials . because this is a short review , we will focus entirely on our work , which describes many of the experimental features seen in recent studies . pump / probe photoemission experiments really came to life when it was recognized that one could period quadruple a 1.5 ev light pulse to create 6 ev pulses that also are ultrashort ( typical widths are in the few 10s of femtoseconds ) . the higher - energy pulses are sufficient to photoemit electrons from part of the brillouin zone ( bz ) in materials with small workfunctions . the experimental apparatus for this was then applied to a wealth of different materials , with a large focus on cdw systems like tas@xmath0 , tise@xmath0 and tbte@xmath1 . these materials have complex ground states , and often have a number of different competing phases that can be accessed by changing temperature or pressure . tas@xmath0 was the first material studied @xcite . in the 1 t phase , this material goes through a range of different phase transitions , starting with an incommensurate cdw at high temperature , passing to a nearly commensurate cdw , and then a commensurate one , which has a three - sublattice ordering in the form of planar star - of - davids . this lowest temperature commensurate cdw is predicted to be a metal in density functional theory calculations , but is seen to be an insulator in transport measurements ( and other probes ) . it has long been believed that the insulator arises from a mott transition in the band at the fermi energy @xcite , but recent work has called this interpretation into question @xcite , as it is becoming clearer that the stacking of the orbital ordering along the c - axis also plays an important role , and could even be the origin of the insulating behavior . the time - resolved arpes studies showed that the system has its gap collapse when it is driven and it also saw an interesting subgap resonance as the system relaxed back to equilibrium . an ultrafast core - level x - ray photoemission spectroscopy study measured the charge modulation order parameter of the material @xcite and found that it decreases , and then relaxes back , but never goes all the way to zero , even though the gap has collapsed . more recently , it has been found that the system can relax to a metastable metallic nonequilibrium state @xcite that appears to be similar in many respects to the nearly commensurate cdw but with some clear differences . electron diffraction studies found yet another nonequilibrium metastable phase @xcite . other materials have also been studied . tbte@xmath1 is another interesting material which has been examined with time resolved arpes @xcite . here one can watch the closing and re - opening of the gap directly in momentum space . in addition to seeing this behavior , two new results have emerged first , the photoemission rings at a frequency given by the ordering phonon in the system and second , the spectral gap can be reduced , but not all the way to zero , even if there is a high fluence and there are significant subgap states which have closed the gap . the persistence of this spectral gap feature is something that naturally emerges in nesting - based cdws as we describe below . finally , tise@xmath0 has been thoroughly studied @xcite . this material is believed to be an excitonic insulator , which should respond fast to a pump because the ordering is electronic in nature and does not require phonons . indeed , the response time is seen to be quite fast in this system ( under 50 femtoseconds ) , and a recent theory has been developed to describe the experiments @xcite . it is clear that there is much more work needed to fully understand the behavior of these materials . many - body theory has also seen a significant development in recent years . dynamical mean - field theory @xcite ( dmft ) has emerged as one of the most useful methods for describing electron correlations in three - d materials . its extension to nonequilibrium @xcite , enables a wide range of different many - body problems to be solved . the theory can work both in the normal state and in commensurate ordered phases , and recently , there has been a range of work on the nonequilbrum properties of cdws . this started with the exact solution of a simplified bandstructure model , where a range of different phenomena were studied : ( i ) pump / probe photoemission spectroscopy @xcite ; ( ii ) high - harmonic generation of light @xcite ; ( iii ) response to a large dc current and bloch oscillations @xcite ; and ( iv ) how quantum systems are excited as functions of the pump amplitude and frequency @xcite . since then , this approach has been expanded to examine the behavior of nesting - driven electronic cdws as described by the falicov - kimball model @xcite and also being solved exactly @xcite . here , the bloch oscillations in response to a large dc field , and the time - resolved photoemission spectroscopy were both studied . the cdw phase of the falicov - kimball model is quite interesting , because it displays an additional tricritical quantum - critical point , which is unique in that the order parameter is not suppressed to zero at the critical point instead , the system transforms from an insulator to a metal @xcite . we describe this unique phenomena in detail here . this review is not intended to be an exhaustive review of the subject , instead , its intent is to carefully derive the needed formalism to solve these problems , explain how one controls the numerics to implement the solutions , and then discusses the properties of those solutions . our focus has been entirely on our work , which represents nearly all of the work on these systems within the nonequilibrium dmft approach . finally , we discuss how one can relate the results of the theory to experiment , and determine the origin of some of the features seen in these experiments . most cdw order is complex in real materials and does not simply follow the peierls paradigm of an ab ordered phase @xcite . nevertheless , the simplest case of an ab ordered phase provides a rich playground to examine the generic properties of many cdw systems @xcite . it also is the easiest problem to solve , and hence we focus all of our efforts in this work to understanding the properties of such an ordered system . we envision the lattice @xmath2 having @xmath3 sites and being bipartite , which means it consists of two disjoint sublattices : the a sublattice and the b sublattice , which are each connected by the hopping matrix @xmath4 . the hopping only connects between sites in different sublattices and for standard systems like a simple - cubic lattice , or a square lattice , the ordering wavevector is @xmath5 , in which @xmath6 is the lattice spacing ; we work in units where @xmath7 here . this allows us to express the formalism in two representative ways in real space or in momentum space . due to the ordering , the translational symmetry is reduced in real space , requiring a lattice with a basis , while in momentum space , we have a coupling between momentum @xmath8 and @xmath9 for all momenta in the reduced brillouin zone , which is half the size of the original bz . the ordering wavevector satisfies @xmath10 where @xmath11 is the position vector of the @xmath12th lattice site . note that the bandstructure for a periodic lattice , where @xmath4 depends only on @xmath13 , can be expressed as @xmath14 the hamiltonian will be written in terms of spinless fermionic creation @xmath15 and annihilation @xmath16 operators for conduction electrons at site @xmath12 on the lattice . these operators satisfy the conventional anticommutation relations @xmath17 and can be transformed to momentum space via @xmath18 in the ordered phase , the reduced brillouin zone ( rbz ) is defined via @xmath19 when there is only nearest - neighbor hopping and @xmath20 is restricted to the rbz . every momentum @xmath20 is then coupled to the momentum @xmath21 which lies in the original bz , but outside the rbz . in momentum space , the two fermionic operators are denoted @xmath22 and @xmath23 , whereas , for the sublattice basis , we restrict the momentum summations to the respective sublattices , as follows : @xmath24 where the @xmath25 is required so that these fermionic operators satisfy the conventional anticommutation relations . with this notation , we then find that @xmath26 as we work on the development of the many - body formalism for pump / probe experiments on cdw systems , we will find it convenient to use both of these different bases to represent the fermionic creation and annihilation operators . converting between the two is simple , with the unitary transformation that relates them satisfying @xmath27 we will be working with an all electronic model of the cdw , which can be described by the falicov - kimball model ( or a specific , noninteracting limit of the model described below ) @xcite , which has conduction electrons , denoted with a @xmath28 and localized electrons , denoted by an @xmath29 , which mutually interact . the hamiltonian in an external electric field is @xmath30 where @xmath31 is the localized electron number operator ( which can be treated as a classical ising - like variable ) , @xmath32 is the conduction electron chemical potential , @xmath33 is the @xmath29-electron site energy and @xmath34 is the on - site coulomb repulsion . in the limit @xmath35 , and in the case of half - filling , where the density of conduction electrons and the density of localized electrons is each 0.5 , we find that the equilibrium solution has @xmath36 for @xmath37 and @xmath38 for @xmath39 . the simplest model for a cdw fixes @xmath40 at those exact values for all @xmath41 . for the simplified model , the system always has cdw order , and it can be solved by diagonalizing a bandstructure on a lattice with a basis ( although the dynamics are still complex in nonequilibrium ) . while in the falicov - kimball model , the asymmetry between the two sublattices decreases as the temperature increases until we reach @xmath42 , where the density of both particles becomes uniformly distributed , on average , throughout the entire lattice . in eq . ( [ eq : ham ] ) , the hopping matrix is time - dependent to model an electric field via the peierls substitution @xcite @xmath43 with the spatially uniform , but time - dependent electric field given by the negative of the time derivative of the spatially uniform vector potential : @xmath44 . in eq . ( [ eq : hopping ] ) , the hopping matrix @xmath4 is a constant spatially periodic matrix [ which we take to be nonzero only between nearest neighbors , where it is equal to @xmath45 , with @xmath46 the spatial dimension of the system ; we will work in units with @xmath47 and take the @xmath48 limit ] . in other words , we will be working on the infinite - dimensional hypercubic lattice . we re - express the kinetic - energy operator in terms of the bandstructure in eq . ( [ eq : bandstructure ] ) and the conduction - electron operators in the two different representations ( when in the ordered phase ) : @xmath49{\aunderbrace[l1r]{c_{\mathbf{k}a}^\dagger~c_{\mathbf{k}b}^\dagger } } \left ( \begin{array}{c c } 0 & \epsilon(\mathbf{k}-\mathbf{a}(t))\\ \epsilon(\mathbf{k}-\mathbf{a}(t ) ) & 0 \end{array } \right ) \left ( \begin{array}{c } c_{\mathbf{k}a}^{\phantom\dagger}\\ c_{\mathbf{k}b}^{\phantom\dagger } \end{array } \right ) , \label{eq : ke_ab}\\ & = & \sum_{\mathbf{k}\in \textrm{\footnotesize rbz } } \aoverbrace[l1r]{\aunderbrace[l1r]{c_{\mathbf{k}1}^\dagger~c_{\mathbf{k}2}^\dagger } } \left ( \begin{array}{c c } \epsilon(\mathbf{k}-\mathbf{a}(t))&0\\ 0&-\epsilon(\mathbf{k}-\mathbf{a}(t ) ) \end{array } \right ) \left ( \begin{array}{c } c_{\mathbf{k}1}^{\phantom\dagger}\\ c_{\mathbf{k}2}^{\phantom\dagger } \end{array } \right ) . \label{eq : ke_12}\end{aligned}\ ] ] here , the bandstructure shifted by the vector potential is directed along the diagonal of the hypercubic lattice and thereby satisfies @xmath50 with @xmath51 and the field ( and vector potential ) oriented along the diagonal so that @xmath52 . one can think of the second bandstructure @xmath53 as the projection of the velocity onto the direction of the electric field . note how the @xmath54 matrix that represents the kinetic energy is off - diagonal in the sublattice representation and diagonal in the original momentum representation ( @xmath20 and @xmath21 ) . to a maximum time @xmath55 and back , ending with a spur parallel to the negative imaginary axis of length @xmath56.,scaledwidth=48.0% ] the many - body problem is solved with contour - ordered green s functions which are defined on the kadanoff - baym - keldysh contour depicted in fig . [ fig : keldysh ] @xcite . the contour graphically illustrates the time evolution of the operators in the heisenberg representation , which evolve from the intial time ( @xmath57 ) to time @xmath58 , then from time @xmath58 to @xmath59 and finally from time @xmath59 back to the initial time , followed by an evolution along a segment of length @xmath56 parallel to the negative imaginary axis . the momentum - dependent contour - ordered green s function ( for @xmath20 in the rbz ) is defined by @xmath60 with @xmath61 and @xmath62 being the 1 , 2 or a , b subscripts depending upon the representation and @xmath63 $ ] being the partition function . the symbol @xmath64 is the time - ordering operator , which orders times _ along the contour_. the fermionic creation and annihilation operators are in the heisenberg representation where @xmath65 and the evolution operator satisfies @xmath66 where the time ordering is with respect to ordinary time . substituting the evolution operators into the definition of the green s function then yields @xmath67 for @xmath58 ahead of @xmath59 on the contour ( @xmath68 ) ; we employed the identities @xmath69 and @xmath70 . one can now directly see why the contour runs from the initial time to @xmath58 , back to the initial time , and then along the imaginary axis ( if we think of the thermal factor as an evolution along the imaginary - time axis ) . there are two green s functions that we can extract from the contour - ordered green s functions the retarded green s function ( which holds information about the quantum states ) and the lesser green s function ( which tells us how those states are occupied ) . they are defined via @xmath71 and @xmath72 the self - energy is defined via the equation of motion . to begin , we must first determine the noninteracting green s function . this is found by setting @xmath73 in the hamiltonian . because the subsequent hamiltonian commutes with itself at different times , the green s function can be found simply by determining the equation of motion for the fermionic creation and annihilation operators . this yields ( with the integrals between @xmath59 and @xmath58 _ on the contour _ ) @xmath74}\rme^{-i\int_{t'}^td\bar t [ \epsilon(\mathbf{k}-\mathbf{a}(t))-\mu ] } & 0\\ \fl 0 & { \scriptstyle [ f(-\epsilon(\mathbf{k})-\mu)-\theta_c(t , t')]}\rme^{-i\int_{t'}^td\bar t [ -\epsilon(\mathbf{k}-\mathbf{a}(t))-\mu ] } \end{array } \right ) , \end{aligned}\ ] ] since @xmath75 . we introduced the fermi - dirac distribution @xmath76 $ ] and the contour unit step function @xmath77 which is equal to 1 if @xmath68 and 0 if @xmath78 . this is in the @xmath79 momentum representation , where the kinetic energy is diagonal . the self - energy is diagonal in the @xmath80 sublattice representation , given by diagonal elements @xmath81 and @xmath82 ; it has no momentum dependence because we are solving the problem in dynamical mean - field theory , which has a local self - energy . converting to the @xmath79 representation yields @xmath83 & \frac{1}{2}[\sigma^c_a(t , t')-\sigma^c_b(t , t')]\\ \frac{1}{2}[\sigma^c_a(t , t')-\sigma^c_b(t , t ' ) ] & \frac{1}{2}[\sigma^c_a(t , t')+\sigma^c_b(t , t ' ) ] \end{array } \right ) .\ ] ] dyson s equation then yields the interacting green s function @xmath84^{-1}(t , t ' ) , \label{eq : dyson}\ ] ] which is the @xmath85 matrix element of the inverse of the operator inside the square brackets . the inverse is with respect to both the time indices and the @xmath54 structure imposed by the ordered phase . note that the green s functions and the self - energy are continuous matrix operators . hence , they can not be easily evaluated numerically . to do so , requires one to discretize the contour and approximate the operators by finite matrices , which can then be inverted using standard linear algebra methods . the discretization is then extrapolated to zero to produce the approximation to the continuous matrix operator . details for how to do this can be found in ref . @xcite . we are often interested in local quantities , such as the local green s function , which is found by summing the momentum - dependent green s function over all momentum in the rbz . in many cases , it is more convenient to perform the summation over the entire bz , since each term appears twice in the summation and one does not need to trace over the final @xmath54 structure . the difference is whether one sums over @xmath86 terms that are larger than zero . if restricting to the rbz , then we must weight the @xmath87 terms by 0.5 , otherwise , those boundary terms will be over counted . of course , because the self - energy is independent of momentum , the sum over momentum can be replaced by a two - dimensional integral over the joint density of states for @xmath86 and @xmath53 . in the @xmath48 limit , we find that the joint density of states is a double gaussian given by @xcite @xmath88 if restricting to the rbz , then one needs to reweight the joint density of states , and in some cases perform a trace over the final @xmath54 matrix . but , by convention , we often weight the green s functions so that the many - body density of states on the a sublattice and the b sublattice each have spectral weight of 1 . in that case , one averages over both to get the _ average _ local density of states . the simplified cdw case is determined more easily than the full interacting case . the self - energy simplifies to @xmath89 and @xmath90 . then , the evolution operator becomes block - diagonal for each momentum ( or , equivalently for each @xmath86 , @xmath53 pair ) , which is described by a simple @xmath54 landau - zener - like system . the full evolution operator is found by using the trotter formula for a given discretization @xmath91 and can be analytically found to satisfy @xmath92 \mathbb{i}\\ -i\left [ \epsilon\left ( \mathbf{k}-\mathbf{a}\left ( t+\frac{\delta t}{2}\right ) \right ) \sigma_z+\frac{u}{2}\sigma_x\right ] \frac{\sin\left [ \delta t\sqrt{\epsilon^2\left ( \mathbf{k}-\mathbf{a}\left ( t+\frac{\delta t}{2}\right ) \right ) + \frac{u^2}{4}}\right ] } { \sqrt{\epsilon^2\left ( \mathbf{k}-\mathbf{a}\left ( t+\frac{\delta t}{2}\right ) \right ) + \frac{u^2}{4 } } } , \nonumber\end{aligned}\ ] ] where @xmath93 is the @xmath54 identity matrix , @xmath94 and @xmath95 are the corresponding pauli spin matrices , and we employ a midpoint integration rule for the evaluation of the hamiltonian in the trotter factor . the full evolution operator for this @xmath54 block then becomes @xmath96 and then this is repeated for each momentum point . it turns out that the green s function for the simplified cdw model is determined entirely in terms of this evolution operator . namely , we find that the retarded green s function in the 1,2 momentum representation is determined by just evolution operators between times @xmath58 and @xmath59 ( because the quantum states just depend on the instantaneous value of the hamiltonian ) via @xmath97 if we sum over momentum and convert to the @xmath80 representation , we find that @xmath98 , \ ] ] with the @xmath99 sign for the @xmath100 sublattice and the @xmath101 sign for the @xmath102 sublattice ; these green s functions are normalized so that @xmath103 . note that the sum over momentum is replaced by a double integral over the two band energies weighted by the joint density of states . the lesser green s functions are more complicated , because they depend on all times , not just the times between @xmath58 and @xmath59 . this is reasonable , because how the states are occupied depends on the history of how the occupancy has evolved over time . the final expression depends on the initial occupancies of the electrons . these are given by the following @xmath104 here , we use the following notation @xmath105 . the final result for the lesser green s function in the 1,2 representation is cumbersome and is given by @xmath106,\end{aligned}\ ] ] @xmath107,\end{aligned}\ ] ] @xmath108,\end{aligned}\ ] ] and @xmath109.\end{aligned}\ ] ] the shortened symbol @xmath110 is employed in these equations . while the transformation to the @xmath80 representation is straightforward , the resulting equations are so long , that we do not write them down here . however , it is important to calculate the order parameter of the conduction electrons , for both the falicov - kimball model and the simplified model . it is given by @xmath111}=-\frac{\sum_{\mathbf{k}\in { \rm rbz}}\left [ g^<_{12}(\mathbf{k},t , t)+g^<_{21}(\mathbf{k},t , t)\right ] } { 2\sum_{\mathbf{k}\in { \rm rbz}}\left [ g_{11}^<(\mathbf{k},t , t)+g_{22}^<(\mathbf{k},t , t)\right ] } , \ ] ] which is bounded between 0 and 0.5 in equilibrium ( but can become negative in nonequilibrium ) . similarly , the order parameter of the localized electrons is @xmath112 which is fixed at 0.5 for the simplified model , and reaches 0.5 at @xmath113 for the falicov - kimball model . it is always nonegative , because it is fixed at its equilibrium value , and hence also has no time dependence . while we have provided a complete solution of the simplified model , we have not yet described how one solves the dmft for the cdw state in nonequilibrium . it is solved via an iterative algorithm , but we must work with matrices that have time discretized on the contour . the algorithm starts with a guess for the self - energies on the two sublattices ( in the @xmath80 representation , the self - energy is diagonal in the @xmath114 space ) . the iterative approach is then as follows : ( 1 ) for the given self - energies , compute the local green s function @xmath115 by summing dyson s equation in eq . ( [ eq : dyson ] ) over all momenta ( practically speaking , we use an integration over the joint density of states to do this and we do so in the @xmath80 representation ) ; ( 2 ) extract the effective medium @xmath116 from the local dyson s equation ( which has an additional @xmath54 matrix structure in the @xmath80 representation ) : @xmath117 ( 3 ) construct the ( diagonal ) impurity green s function from the diagonal components of the effective medium via @xmath118(t , t'),\ ] ] with @xmath119 or @xmath102 and @xmath120 with @xmath121 or @xmath102 ; ( 4 ) extract the self - energy for the impurity by solving @xmath122 ( 5 ) and finally setting the new self - energy for the lattice to equal that of the impurity and then iterates steps ( 1)(4 ) until it is converged . one of the important checks is the short time behavior of the green s function . it turns out that by carefully examining the definition of the green s function , one can find the coefficients of the taylor series expansion in relative time . in particular , for the retarded green s function , when @xmath123 ( or @xmath124 ) , we immediately know that @xmath125 , because the anticommutator of two fermionic operators is equal to one ; this holds both in momentum space and in real space . higher derivatives can be evaluated by taking commutators with the hamiltonian . remarkably , the first few derivatives do not depend on the field , so they hold in equilibrium and in nonequilibrium . these results are also called moment sum rules , because that is what they look like when one converts the relative time to a frequency via fourier transformation . since the first few moment sum rules ( or equivalently relative time derivatives ) of the green s function can be found exactly , they become an important tool in testing the accuracy of calculations @xcite . for the simplified cdw model , one can evaluate a taylor series expansion of the time evolution operator and immediately find that the expressions given above satisfy the appropriate sum rules @xcite . for the falicov - kimball model , we calculate the moment sum rules by numerically evaluating the first few derivatives of the retarded green s functions . they serve as important numerical consistency checks of the approach , and are critical to ensure accuracy of the final results . the derivation of these sum rules is straightforward , but tedious . while one can do this both for the momentum - dependent green s functions and the local green s functions , we report them only for the local green s functions here . the standard way to report them is in terms of the many - body density of states for each sublattice , which is defined to be @xmath126 , with @xmath127 . here , the average time is @xmath128 and the relative time is @xmath129 . the moments then satisfy @xmath130 which become @xmath131 @xmath132 and @xmath133 the expectation value @xmath134 measures the average density of the heavy particles on each sublattice ; this expectation value does not change with time . when calculating momentum - dependent quantities , like the angle - resolved pes , one must be careful to work with gauge - invariant quantities to ensure that the object being measured is a true observable . in this work , we will focus on the angle - summed pes , or total pes , which , being a local quantity , is manifestly gauge invariant . hence , we do not discuss gauge - invariance issues further here . in addition , we work with a constant matrix element approximation . for a single band model in the normal state , a constant matrix element simply factors out of the pes expressions . but when the system has multiple bands , so that the green s function is represented by a matrix ( here a @xmath54 matrix structure for the @xmath100 , @xmath102 sublattices ) , then the matrix elements can not be constant in all different bases they are constant in one basis , then they are related by a unitary transformation in another basis . while it is tempting to ignore this fact , and approximate the pes signal by the trace of the matrix green s function ( since it is an invariant ) multiplied by a constant , this only holds in the basis where the green s function matrix is diagonal ; the photoemission spectra may not satisfy positivity in this case . otherwise , the pes ( and especially the angle - resolved pes ) can involve more complex contributions . here we take the simplifying assumption that the pes is given by a constant matrix element multiplied by the sum of the local diagonal contributions in the @xmath100 , @xmath102 representation . the averaging over the two sublattices is required because the probe pulse will uniformly irradiate both sublattices . the result for the photoemission from each sublattice is then a two - time , probe - pulse - envelope - weighted , fourier transform , given by @xcite @xmath135 the symbol @xmath136 is the probe pulse envelope function , which we take to be a gaussian centered about the time @xmath137 : @xmath138 with @xmath139 the effective width of the probe pulse ( broader pulses mean more energy resolution , less time resolution and _ vice versa _ ) . for completeness , we discuss two other observables . one is the current , which determines the time rate of change of the energy via @xmath140 and the other is the filling within each of the bands ( essentially the filling within bands with energy larger than zero or smaller than zero ) . for the current , one finds @xmath141\left [ g^<_{\mathbf{k}11}(t , t)+g^<_{\mathbf{k}22}(t , t)\right ] , \ ] ] where @xmath142 is the particle band velocity and the green s function is in the 1,2 basis . the result for the filling into the different bands has only been derived for the simplified model . we refer to ref . @xcite for those complete formulas . , red ) and a low amplitude pulse ( @xmath143 , green ) . note how the sum rule is accurate within a few percent for the large amplitude case , but is quite poor for the smaller amplitude . [ figure reprinted from @xcite , with permission],scaledwidth=60.0% ] , red ) and a low amplitude pulse ( @xmath143 , green ) . note how the sum rule is again accurate within a few percent for the large amplitude case , but is quite poor for the smaller amplitude . [ figure reprinted from @xcite , with permission],scaledwidth=60.0% ] we end this section with a discussion of how to perform the numerical calculations . in all cases , our goal is to determine the contour - ordered green s function or the retarded and lesser green s functions ( which can be extracted from the contour - ordered one ) . the electric field is chosen to satisfy @xmath144 where @xmath145 is the magnitude of the field at time @xmath146 . for the simplified model we have already shown that the green s function is directly found from the evolution operator , which decouples for each momentum . furthermore , the retarded green s function is determined solely by the relative time , so it typically does not require much computation to evaluate it . the lesser green s function knows about the previous history of the system , so it requires longer runs in time to determine it , since we must start from a time in the distant past before the field is turned on . we have chosen , in this work , to evaluate the evolution operator via the trotter formula . the only subtlety is how small of a time step do we take for each of the trotter factors . this is then adjusted to ensure that the results have converged ( best to extrapolate to @xmath147 and use sum rules to verify the convergence ) . one of the benefits of this approach is that we maintain unitarity explicitly for the evolution operator because each trotter factor is determined analytically , and is manifestly unitary . an alternative way to solve this problem is to employ a conventional differential equation solver . the advantage of the differential equation approach is that they can be made adaptive to help ensure appropriate accuracy , but they often suffer from loss of unitarity for long runs over large time intervals , and hence are often less reliable than the trotter - based methods for these problems . we employed the trotter approach for all results shown here . sublattice for ( a ) equilibrium and ( b ) nonequilibrium at an average time well before the pump pulse ( @xmath148 ) . this is for the simplified model of the cdw with @xmath149 , but also holds for the falicov - kimball model at @xmath113 . ( c ) because the retarded green s function has a long tail in the time domain , the density of states for a large range of average times is affected by the pump pulse . inset , one can see the long tail of the green s function which decays like @xmath150 for the equilibrium green s function , perhaps somewhat faster for the nonequilibrium green s function . [ figure reprinted from @xcite , with permission],scaledwidth=90.0% ] the falicov - kimball model calculations are more demanding , because they require the full nonequilibrium dmft algorithm . we discretize the system , usually with @xmath151 , 0.05 , and 0.033 and then quadratically extrapolate to zero @xmath91 . as a check on the accuracy , we compute the zeroth and the first moment sum rules ( the second moment accuracy is poorer during the initial part of the pulse ) . empirically , we find that the equations converge more accurately when the amplitude of the pump pulse is large . this is illustrated in figs . [ fig : moments1 ] and [ fig : moments2 ] , where we plot the extrapolated moment sum rules for @xmath143 and @xmath152 , and find that the large - amplitude case has acceptable errors , while the other has too large errors to be useful . it is surprising that this result is most accurate for the nonequilibrium regions ; when the system is in equilibrium ( left region of the figures ) , the errors are much larger . for this reason , we work exclusively with @xmath152 in this work for the falicov - kimball model results . we begin our discussion of the behavior of these systems by focusing on the density of states which will allow us to immediately discuss the phase diagram an quantum critical behavior . in both models , one can show that the @xmath113 dos diverges as the inverse square root of the frequency at the band edges , which form the spectral band gap for the cdw . this is illustrated in fig . [ fig : cdw_dos ] ( a ) for the @xmath100 sublattice with the divergence at the upper band edge . when we fourier transform this to relative time , the fourier transform has a long tail which decays like @xmath150 . for an accuracy of 0.1% in the dos , one needs to run the relative time out to @xmath153 or more due to the long decay . this extreme nonlocality in time can cause misconceptions when one is working with the nonequilibrium system . in particular , if we fix the average time and fourier transform with respect to the relative time , then once the relative time is large enough , one of the two times in the @xmath58 and @xmath59 basis will be earlier than the time when the field was applied and one will be later . hence even for large negative average times , the dos at that average time will be affected by the presence of the field , as depicted in fig . [ fig : cdw_dos ] ( b ) and ( c ) . when fourier transformed to frequency the dos has significant oscillations that occur due to the slope discontinuity when the field is turned on ( occurring near relative time of @xmath154 for this case ) . this analysis can only be done for the simplified model , because the falicov - kimball model can not be calculated out to long enough times to see this behavior . this behavior is generic , however , for the transient dos when the green s functions in real time have long tails in equilibrium ( in most cases these singularities , or sharp peaks disappear or are broadened in nonequilibrium , so the steady state dos with a field present often have shorter tails in time , but the long - tails return for the quasiequilibrium states after the pump is turned on for a pump / probe experiment ) . , which corresponds to a strongly correlated metal . the curves correspond to different temperatures . note how the singularity disappears at finite temperature and how the subgap states evolve . inset is the order parameter for the corresponding dos , as indicated by the color . [ figure reprinted from @xcite , with permission],scaledwidth=75.0% ] , which corresponds to the quantum critical cdw . the curves correspond to different temperatures . note how the singularity disappears at finite temperature and how the subgap states evolve . inset is the order parameter for the corresponding dos , as indicated by the color . [ figure reprinted from @xcite , with permission],scaledwidth=75.0% ] , which corresponds to the critical point for the mott insulator . the curves correspond to different temperatures . note how the singularity disappears at finite temperature and how the subgap states evolve . inset is the order parameter for the corresponding dos , as indicated by the color . [ figure reprinted from @xcite , with permission],scaledwidth=75.0% ] we next focus on the behavior of the falicov - kimball model in equilibrium at finite - temperature , because the electron correlations bring on a distinctive behavior which is quite different from the standard bardeen - cooper - schrieffer ( bcs ) paradigm @xcite . in particular , if we define the spectral gap for the cdw to be the distance between the maxima that are the remnants of the divergence at @xmath155 in the @xmath113 dos , then we see that for most cases we will consider , the spectral gap remains fixed at @xmath34 all the way up until @xmath156 ( or quite close to @xmath42 ) . this differs completely from the bcs paradigm , where the spectral gap is tied directly to the order parameter ( in that case the superconducting gap function ) , and the spectral gap shrinks as the order parameter shrinks until @xmath42 is reached where the spectral gap vanishes . here , the phenomena is quite different . we instead have two subgap minibands that form once @xmath157 , and they grow both in weight and in bandwidth until they close the subgap and then fill in the dos from below as the spectral gap features diminish from above and both meet at @xmath156 to produce the normal state dos . figures [ fig : cdw_dos_fk_metal ] , [ fig : cdw_dos_fk_crit ] , and [ fig : cdw_dos_fk_mott ] plot the dos in equilibrium for different temperatures for the metal , the quantum critical cdw and the critical mott insulator phases . the different colors correspond to the different temperatures , with the dashed black line being the normal state . inset are the order parameters for the light ( right ) and heavy ( left ) electrons . note how the heavy electrons always have an order parameter that goes to 0.5 as @xmath35 , but the light electrons order parameter is always less than that , although it increases as @xmath34 increases . but the behavior is even more complex than this . the minibands initially start near the upper and lower band edges , but as @xmath34 increases , they migrate toward the center of the bandgap and they meet when @xmath158 . this is the underlying quantum critical point for the cdw , because at this point , the system has a transition from a metal to an insulator at @xmath35 . the metallic phase opens as a fan for higher @xmath41 as the dos becomes nonzero at the chemical potential and traces out the novel metallic cdw phase within the phase diagram . this phase is most stable for temperatures below but near @xmath42 . as @xmath34 is increased further ( up to @xmath159 ) we then have the mott transition . in this case , the minibands form and grow in weight and broaden as @xmath41 increases , but they never broaden to completely fill the gap . instead the band edges stop at the mott insulator band edges , so the subgap region inside the mott gap never fills with any states . once one is above @xmath42 , the dos becomes temperature independent , and either has nonzero weight at the chemical potential for the metal or has no dos at the chemical potential for the mott insulator . , characterized by the opening of a gap in the single - particle density of states , which is the same on each sublattice . in the cdw ordered phase , where the densities of the particles are different on the two sublattices , there are three phases : ( 1 ) a weakly correlated cdw phase , which is continuously connected to @xmath73 ; ( 2 ) a strongly correlated cdw metal , which is present only at nonzero temperature , and emerges from the quantum critical point of the model at @xmath160 ; and ( 3 ) a strongly correlated cdw insulator , which is continuously connected to the mott insulator within the cdw phase . the quantum critical point is nonstandard , because the order parameter for the cdw order varies continuously through the transition.,scaledwidth=65.0% ] the phase diagram showing all of these phases and their region of stability is plotted in fig . [ fig : cdw_phase ] for the spinless falicov - kimball model on a hypercubic lattice . the five different phases are indicated by the different colors and the boundary lines . note how the quantum critical cdw region is extremely narrow for temperatures below 0.02 @xmath161 . it becomes quite difficult to determine the precise phase boundary in this region because it is hard to calculate the dos accurately there . the phase boundaries are more rigorously determined and the existence of the quantum critical point is explicitly proved on an infinite - coordination bethe lattice @xcite ( as opposed to the hypercubic lattice results shown here ) . nevertheless , we work with the hypercubic lattice here , where the phase boundary is more challenging to explicitly determine . note that this quantum critical point is rather unique . an ordinary quantum critical point will occur at the terminus of a phase diagram where an order parameter is suppressed to zero . at that point , there is a zero - temperature phase transition . in many quantum critical points , the original quantum critical region is hidden by a superconducting dome . the case here is different for a number of reasons . first , the spatial order parameter does not disappear at the quantum critical point , instead , it varies continuously at the critical point . instead , the dos has a change of character as @xmath35 versus @xmath113 , the former being metallic and the latter being insulating . nevertheless , there remains a strange metal fan above the quantum critical point , similar to more conventional quantum critical points . as a function of time for different pulse amplitudes ( zero is indicated by the magenta dashed line ) . the purple dashed line is a zoom in of the black curve for long times . the field is shown at the top in red . [ adapted from @xcite , with permission],scaledwidth=55.0% ] now we turn to describing the conduction electron order parameter and how it varies with the pump pulse . as electrons are driven by the field , they will flow through the material moving from the @xmath100 to the @xmath102 sublattice and being excited up to the higher energy band ( or de - excited to the lower energy band ) . as a result , we anticipate that the conduction - electron order parameter will transiently change as a function of time . this is indicated for the simplified cdw model in fig . [ fig : cdw_non_order ] . the cases with a pulse amplitude of 0.75 and 1 both oscillate and are pushed downwards , but settle into values that are positive for the long - time limit , while , when the amplitude is increased to 1.25 and 1.5 , the order parameter actually changes sign ! for larger amplitudes we anticipate that the order parameter will continue to oscillate and the sign of the long - time limit may be difficult to determine without running through the whole calculation . in fig . [ fig : cdw_fk_order ] , we show a similar figure , but this time for a large amplitude pulse ( @xmath152 ) in the falicov - kimball model with different initial temperatures . the simplified cdw model will be closest to the low - temperature results . initially , the system starts in equilibrium , and so the order parameter should be a constant , but the numerical results show some small dependence on time in this region ( on the order of a few percent ) which is an indication of the accuracy of the data after the extrapolations have been done . inset , we show the total electron filling , which should be precisely 1 throughout the simulation . its variation is another indication of the accuracy of these calculations , which are pushing the state - of - the - art to its limits . the metallic case is in panel ( a ) and it shows rather expected behavior . it starts off fairly flat , is reduced as the field is turned on , with large oscillations , and then settles into a rather flat result in the long - time limit , which is reduced from the original equilibrium value , but remains positive . panel ( b ) of the quantum critical cdw is much more interesting . it shows a nearly symmetric curve , where the order parameter is reduced , but in the end , the final value is quite similar to the initial equilibrium value . we will see that this occurs because it is difficult to excite the quantum critical cdw . finally , panel ( c ) shows the critical mott insulator where in all cases , regardless of the starting point , the order parameter is being suppressed almost to zero . it is surprising that this suppression is more prominent for the mott insulator than it is for the metal , and there is no simple explanation for why this would be so . and different temperatures . the convolution washes out a number of the features in the density of states , which will continue as the system is pumped . panel ( e ) shows the same for the simplified model with @xmath162 and @xmath163 . [ panels ( a - d ) reprinted from @xcite , with permission],title="fig:",scaledwidth=45.0% ] and different temperatures . the convolution washes out a number of the features in the density of states , which will continue as the system is pumped . panel ( e ) shows the same for the simplified model with @xmath162 and @xmath163 . [ panels ( a - d ) reprinted from @xcite , with permission],title="fig:",scaledwidth=45.0% ] the formula for the time - resolved photoemission spectroscopy in eq . ( [ eq : pes ] ) involves a double - time fourier transform weighted by the probe envelope given in eq . ( [ eq : probe ] ) . the form of the photoemission integral is that of a convolution . when the probe width @xmath139 is narrow , one has high time resolution but poor spectral resolution and _ vice versa _ when the probe width is broad . to understand the effect this has on the features of the density of states , we show the convolution of the probe function used for the quantum critical cdw state of the falicov - kimball model , where @xmath164 with the local dos for different temperatures in fig . [ fig : dos_conv ] ( a d ) . one immediately sees that the gap region is smoothed out , and the details of the subgap states are lost . furthermore , as one approaches the normal state , the convolution becomes quite close to the original signal because it does not have any sharp features . when the interaction is decreased to @xmath165 and the probe width is increased to @xmath163 , as shown for the @xmath113 data on the right , we see that the smoothing out of the features is much less and the gap is nearly fully formed . the divergent peaks however remain smoothed out , because they require long tails in time before they are fully developed . it is important to keep in mind the smoothed out features due to the convolution when we look at the time - resolved pes next , because those data will also have the smoothed out behavior . with @xmath166 and averaged over the a and b sublattices , for the simplified cdw model . the electric field profile is shown above the plot . [ reprinted from @xcite , with permission],scaledwidth=49.0% ] the results for the transient time - resolved pes for the simplified model with @xmath167 is shown in fig . [ fig : pes_non ] . the false - color plot has a logarithmic scale for the pes to emphasize the data at small values . one can immediately see that the gap closes ( light blue region near @xmath146 ) when the field is present and then re - opens . a substantial number of electrons are excited by the pulse , and there is a small band narrowing , but the spectral gap remains at about the same value for all times ( perhaps it shrinks by a percent or two ) . since the noninteracting system requires a field to be present for both excitation and de - excitation , the number of electrons excited into the upper band does not change once the pump pulse is over . we will use the same false - color scale for all of the pes shown in this work . with @xmath168 , @xmath165 and averaged over the a and b sublattices , plotted in false color . this system is a strongly correlated metal . the electric field is shown above the plot . the different panels are for the different temperatures in fig . [ fig : cdw_dos_fk_metal ] with ( a ) the lowest and ( d ) the highest temperature . ( right ) waterfall image of vertical cuts through the tr - pes data plotted for different delay times and offset for clarity . inset is the total integrated spectral weight . only the data that conserves the spectral weight is shown . this data corresponds to the case of panel ( a ) to the left . [ reprinted and adapted from @xcite , with permission],title="fig:",scaledwidth=49.0% ] with @xmath168 , @xmath165 and averaged over the a and b sublattices , plotted in false color . this system is a strongly correlated metal . the electric field is shown above the plot . the different panels are for the different temperatures in fig . [ fig : cdw_dos_fk_metal ] with ( a ) the lowest and ( d ) the highest temperature . ( right ) waterfall image of vertical cuts through the tr - pes data plotted for different delay times and offset for clarity . inset is the total integrated spectral weight . only the data that conserves the spectral weight is shown . this data corresponds to the case of panel ( a ) to the left . [ reprinted and adapted from @xcite , with permission],title="fig:",scaledwidth=49.0% ] we now move on to the falicov - kimball model , where we are required to run the calculations with a much larger amplitude of the field , otherwise the extrapolated results have not yet converged . the data given here all correspond to @xmath152 . in fig . [ fig : pes_fk_metal ] , we show a series of false color images on the left and a waterfall plot for the lowest temperature on the right . there are a few features to emphasize here which are different from what we saw in the simplified cdw case . some of these may be arising from the fact that the field amplitude is so large now . first , we see a significant narrowing of the band when the field is on ( notice the large blue region near @xmath146 ) . because the total spectral weight is conserved , this narrowing comes with a sharpening of the peaks in the pes as well , which is a bit harder to see in the false color image , but may be clearer in the waterfall . one can also clearly see that the spectral gap is shrinking , as the lower gap edge is being pushed toward 0 in the waterfall . we inset the total spectral weight as a function of the probe delay , to show that the data we use has conserved spectral weight , while for too long or too short delays , our data becomes poor , and spectral weight is lost . we do not show any data for the red points here . the severe band narrowing is quite surprising , as it is coming from a dressing of the electronic states by the pump pulse . the surprise is that the bandwidth is narrowed almost by a factor of two , which is a significant effect . with @xmath168 , @xmath160 and averaged over the a and b sublattices , plotted in false color . this system is a strongly correlated metal . the electric field is shown above the plot . the different panels are for the different temperatures in fig . [ fig : cdw_dos_fk_crit ] with ( a ) the lowest and ( d ) the highest temperature . ( right ) waterfall image of vertical cuts through the tr - pes data plotted for different delay times and offset for clarity . inset is the total integrated spectral weight . only the data that conserves the spectral weight is shown . this data corresponds to the case of panel ( a ) to the left . [ reprinted and adapted from @xcite , with permission],title="fig:",scaledwidth=49.0% ] with @xmath168 , @xmath160 and averaged over the a and b sublattices , plotted in false color . this system is a strongly correlated metal . the electric field is shown above the plot . the different panels are for the different temperatures in fig . [ fig : cdw_dos_fk_crit ] with ( a ) the lowest and ( d ) the highest temperature . ( right ) waterfall image of vertical cuts through the tr - pes data plotted for different delay times and offset for clarity . inset is the total integrated spectral weight . only the data that conserves the spectral weight is shown . this data corresponds to the case of panel ( a ) to the left . [ reprinted and adapted from @xcite , with permission],title="fig:",scaledwidth=49.0% ] as we move to fig . [ fig : pes_fk_crit ] , we find even more surprising results . here , it is virtually impossible to excite the electrons at all into the upper band . at the lowest temperature , the number excited is almost equal to the number de - excited . we continue to have the same band narrowing and peak sharpening due to field dressing of the states and the reduction of the spectral band gap for the cdw as before . but now , the real hallmark is that the number of electrons excited to the upper band is small after the pump pulse is completed . this compares reasonably to what we saw earlier for the order parameter , where we saw that it did not change significantly for the quantum critical cdw . perhaps this behavior is tied to the existence of the conducting channel at the chemical potential in equilibrium ; if that conducting channel is efficient , it can lead to a channel for de - excitation which competes with the excitation and leads to a small net excitation . with @xmath168 , @xmath169 and averaged over the a and b sublattices , plotted in false color . this system is a strongly correlated metal . the electric field is shown above the plot . the different panels are for the different temperatures in fig . [ fig : cdw_dos_fk_mott ] with ( a ) the lowest and ( d ) the highest temperature . ( right ) waterfall image of vertical cuts through the tr - pes data plotted for different delay times and offset for clarity . inset is the total integrated spectral weight . only the data that conserves the spectral weight is shown . this data corresponds to the case of panel ( a ) to the left . [ reprinted and adapted from @xcite , with permission],title="fig:",scaledwidth=49.0% ] with @xmath168 , @xmath169 and averaged over the a and b sublattices , plotted in false color . this system is a strongly correlated metal . the electric field is shown above the plot . the different panels are for the different temperatures in fig . [ fig : cdw_dos_fk_mott ] with ( a ) the lowest and ( d ) the highest temperature . ( right ) waterfall image of vertical cuts through the tr - pes data plotted for different delay times and offset for clarity . inset is the total integrated spectral weight . only the data that conserves the spectral weight is shown . this data corresponds to the case of panel ( a ) to the left . [ reprinted and adapted from @xcite , with permission],title="fig:",scaledwidth=49.0% ] in fig . [ fig : pes_fk_mott ] , we show the same results for the critical mott insulator . here , we see almost complete excitation in the system the weight in the upper and the lower bands appears to be nearly equal . we continue to see the band narrowing and the spectral gap narrowing , which we saw before , but now , this system has quite strong excitation . this result is surprising , because it excites more than the metal does . perhaps , in this case , the de - excitation pathway suffers from some bottlenecks which make it less efficient . in any case , the other curious feature is that the gap closure due to subgap states is much more modest here , and in fact , it closes with the initial and final fields , but reopens when the field amplitude is the largest ! a truly surprising result . in experiment , the most important features seen are the closing of a gap by filling in of subgap states , its reforming well after the pump , along with an oscillation of the pes that modulates at the same frequency as the phonon that is responsible for the cdw order . in addition , for the tbte@xmath1 experiment , as the fluence is turned up , the system has the spectral gap get reduced , but it never goes all the way to zero . some of these features are captured in these simplified models of a cdw while others are not . what these all electronic models see is a gap collapse by filling of subgap states , and a partial reduction of the spectral gap , but not all the way to zero . they do not see the collapse of the gap for long times after the pump , nor do they see the long - time oscillations , both which are likely associated with the ordering phonon ( which is not part of this model ) . the model calculations also show a decoupling of what is happening with the cdw order parameter , as measured by the modulation of the charge , and the spectral gaps , as measured in a pes experiment . the model calculations also see some new phenomena which is yet to be seen in experiment . this includes the preponderence of de - excitation and the difficulty with exciting a quantum critical cdw . it also includes the large band narrowing and peak sharpening due to field dressing when the pump field is on . what is next ? from the theory standpoint , the next step is to include the coupling to the phonons directly . this should reproduce a number of the additional features seen in experiment , and will hopefully allow for a more complete solution of the problem . from the experimental standpoint , we hope that experiments might be done on materials that have there cdw transition driven by nesting , where these additional subgap features and quantum critical behavior may be present . those systems could prove to be an interesting playground for novel physics . in this review , we covered a wide range of the theory needed to describe strongly correlated cdw materials in nonequilibrium . we developed the theory both for the simplest model of a cdw , which is essentially a bandstructure with a basis , to describing the exact solution of an all electronic cdw described by the falicov - kimball model . the latter having a unique quantum critical point within the cdw phase . we used these formalisms to solve these problems for a wide array of different parameters . technical reasons restricted the falicov - kimball model calculations to large amplitude fields only . this work represents starting point for the theory of nonequilibrium pump / probe experiments in strongly correlated materials . much more work needs to be done in the future to treat different kinds of order , the competition between different ordered phases , and different types of electron correlations . in addition , extensions that include the possibility of incorporating more real materials properties , along the lines of density functional theory plus dynamical mean - field theory , but now in nonequilibrium , will also be important . we hope to see and to participate in these developments in the coming years . this work was supported by the department of energy , office of basic energy sciences , division of materials sciences and engineering under contract nos . de - 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in this review , we develop the formalism employed to describe charge - density - wave insulators in pump / probe experiments using ultra short driving pulses of light . the theory emphasizes exact results in the simplest model for a charge - density wave insulator ( given by a noninteracting systems with two bands and a gap ) and by employing nonequilibrium dynamical mean - field theory to solve the falicov - kimball model in its ordered phase . we show both how to develop the formalism and how the solutions behave . care is taken to describe the details behind these calculations and to show how to verify their accuracy via sum - rule constraints . _ keywords _ : pump / probe experiment , charge - density - wave insulator , time - resolved photoemission spectroscopy , nonequilibrium dynamical mean - field theory

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Proceed to summarize the following text: the soft x - ray telescope ( sxt ; * ? ? ? * ) on board yohkoh @xcite revealed that a soft x - ray emitting plasma ejection , or plasmoid ejection , is sometimes observed in solar flares ( e.g. * ? ? ? * ) . it was also found that the plasmoids show blob - like or loop - like shapes and that the strong acceleration of the plasmoid ejection occurs during the peak time of the hard x - ray emission @xcite . their ejection velocities are typically several hundred km s@xmath0 and the ejected plasma is heated to more than 10 mk before the onset of the ejection @xcite . they often start to rise up gradually a few tens of minutes before the onset of a hard x - ray burst and are then strongly accelerated just before or at the impulsive phase of the flare . a similar kinetic evolution is also seen in the case of coronal mass ejections ( cmes ; * ? ? ? * ; * ? ? ? * ; * ? ? ? similarly , slowly drifting radio structures , observed at the beginning of the eruptive solar flares in the 0.6 - 1.5 ghz frequency range , have been interpreted as the radio signatures of plasmoid ejection @xcite . hudson et al . ( 2001 ) , moreover , identified a rapidly moving hard x - ray source associated with a moving microwave source and an x - ray plasmoid ejection . kundu et al . ( 2001 ) also identified moving soft x - ray ejecta associated with moving decimetric / metric radio sources observed by the nanay radioheliograph . sui et al . ( 2004 ) also found a plasmoid ejection in hard x - ray images with rhessi satellite . in the standard model of solar flares , so - called cshkp model @xcite , a filament / plasmoid ejection is included . however , it does not necessarily stress the importance of the role of plasmoid ejection explicitly . shibata et al . ( 1995 ) and shibata & tanuma ( 2001 ) extended the cshkp model by unifying reconnection and plasmoid ejection and stressed the importance of the plasmoid ejection in a reconnection process . the model is called the `` plasmoid - induced - reconnection '' model . in that model , the plasmoid inhibits reconnection and stores magnetic energy in a current sheet . then , once it is ejected , inflow is induced because of the mass conservation , resulting in the enhancement of reconnection rate and the acceleration of the plasmoid due to the faster reconnection outflow . moreover , reconnection theories predict several plasmoids of various scales are generated . the dynamics of plasmoid formation in the solar flare and their subsequent plasmoid ejection affect the reconnection rate in the nonlinear evolution . therefore , plasmoid ejections are observational evidence of magnetic reconnection of solar flares . since plasmoid ejections have been observed in both long duration events and compact flares @xcite , it is shown that the magnetic reconnection model may be applicable even for the compact flares that do not show the other typical features of the magnetic reconnection . on the basis of the results of magnetohydrodynamic ( mhd ) numerical simulations , kliem et al . ( 2000 ) suggested that each individual burst in the slowly pulsating structure is generated by suprathermal electrons , accelerated in the peak of the electric field in the quasi - periodic and bursting regime of the magnetic field reconnection . this is the so - called `` impulsive bursty '' reconnection @xcite . in that regime , several plasmoids can be formed successively as a result of the tearing and coalescence instabilities @xcite . the repeated formation of magnetic islands can induce magnetic reconnection and their subsequent coalescence @xcite . these processes even have a cascading form : secondary tearing , tertiary tearing , and so on , always on smaller and smaller spatial scales @xcite . furthermore the formed plasmoids can merge into larger plasmoids . tanuma et al . ( 2001 ) also showed that an increase in the velocity of the plasmoid ejection leads to an increase in the reconnection rate , and brta et al . ( 2008 ) analyzed the dynamics of plasmoids formed by the current sheet tearing . the unsteady reconnection mentioned above can release a large amount of energy in a quasi - periodic way . the energy released in the upward direction can be observed as plasmoid ejections or coronal mass ejections , while that in the downward direction as downflows @xcite and impulsive bursts at the footpoints of the coronal loops . bursty energy release in solar flares has been observed as highly time variable hard x - ray bursts and microwave bursts @xcite . benz & aschwanden ( 1992 ) and aschwanden ( 2002 ) argued that these impulsive bursts suggest the existence of highly fragmented particle acceleration regions . this fragmented structure of solar flares indicates that a flare is an ensemble of a vast amount of small - scale energy releases and the fractal / turbulent structure of the current sheet can be expected ( see also * ? ? ? * ) . recently the kinematics of multiple plasmoids have been studied by full - particle simulations and how the particles interact with their surroundings has been explained ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? it is interesting to note that the stochastic acceleration mechanism ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) may be related to particle acceleration in fractal / turbulent current sheet ( see also * ? ? ? * ; * ? ? ? * ; * ? ? ? karlick et al . ( 2004 ) showed a unique series of slowly drifting structures during one flare , from which the authors proposed that it indirectly maps a formation of several plasmoids and their interactions . however in most of the previous studies , only one plasmoid or one drifting structure was reported during the solar flare . in this paper , we present for the first time the direct observations of multiple x - ray emitting plasmoid ejections associated with a single solar flare observed by yohkoh / sxt ( firstly reported by takasaki 2006 ) . in section 2 , we describe the multiple plasmoid ejection events . then we analyzed the data in section 3 by examining in detail the relationship between the multiple plasmoid ejections and the nonthermal hard x - ray emissions using yohkoh data . finally we discuss the dynamic features of magnetic reconnection and the roll of plasmoid ejections in the particle acceleration in a solar flare in section 4 . a series of homologous flare - cme events occurred in noaa active region 9236 from 2000 november 24 to november 26 . these events have been reported by several researchers . nitta & hudson ( 2001 ) found that the cme - flare events of the homologous flares show quite similar characteristics in both their coronal / photospheric magnetic structures and their cme properties . zhang & wang ( 2002 ) compared the homologous flares in detail through the use of multiwavelength observations . wang et al . ( 2002 ) reported that the activities of these flares was driven or triggered by newly emerging magnetic flux , which appeared on the western side of the leading sunspot in this active region . figure 1a-1c show snapshot images of the preceding sunspot in noaa 9236 and an associated two - ribbon flare , which occurred on 2000 november 24 observed in white light , ultraviolet and soft x - ray emission . takasaki et al . ( 2004 ) performed a comparison of the physical parameters between the individual flares and from this they could confirm that the plasmoid - induced - reconnection model is reasonable . they then showed that the interaction between the new emerging magnetic flux loops and the pre - existing magnetic field was essential for producing the homologous flares and plasmoid ejections in the active region . these ejections were followed by a single halo - cme which occurred at 15:30 ut on 2000 november 24 . figure 2 shows a cme image observed with the large angle spectroscopic coronagraph ( lasco ; * ? ? ? * ) that occurred following the flare ( e.g. see * ? ? ? * ; * ? ? ? the core of the cme was observed traveling in the northwest direction . moon et al . ( 2003 ) found a good correlation between cme speed and the goes x - ray peak flux of the associated flares in this series of homologous flare - cme events . impulsive hard x - ray bursts were also observed in this flare with the hard x - ray telescope ( hxt ; * ? ? ? * ) on board yohkoh ( see fig . a pair of hard x - ray sources was located at the footpoints of the coronal arcade . we used the hard x - ray emission data observed with the h - band ( 52.7 - 92.8 kev ) of hxt , whose temporal resolution was 0.5 s. we focus on a goes x2.3 class flare that occurred in noaa 9236 ( n19@xmath1 , w06@xmath1 ) at 14:51 ut on 2000 november 24 . this flare was one of the homologous flares which were described previously . in this flare , we observed seven plasmoid ejections in the soft x - ray images of the flare taken with yohkoh / sxt . we mainly used the partial frame images with half- and quarter - resolution for the analysis . the spatial resolutions are about 5 and 10 , respectively . we used the sandwich ( almg ) filter images which were taken with 20 second cadence . figure 3 and supplement movie 1 show the temporal evolution of the flare , which is made of full- , half- and quarter - resolution images taken with yohkoh / sxt almg filter ( negative images ) . full- , half- and quarter - resolution images are different in their spatial resolutions , field of views and exposure times . full - resolution images are of short exposure time and focus on the brightest region of the active region , such as the two ribbon structure . on the other hand , quarter - resolution images are of longer exposure time so that they are applicable for the detection of large - scale and faint phenomena such as plasmoid ejections . the black vertical line in the middle of figure 3 shows the saturation of the quarter - resolution images . we identified seven major ejections which we named p1-p7 . in figure 4 ( supplement movie 2 ) , figure 5 and figure 6 , we marked each plasmoid ejection with a circle . each plasmoid ejection can be seen more clearly in quarter - resolution images of figure 5 , while half - resolution images of figure 6 are convenient to see plasmoids just after ejections . the fields of view of quarter- and half - resolution images are also shown in figure 3 . figure 7 shows the temporal evolution of soft x - ray emission observed with sxt . a full resolution image is inset on figure 7a . figure 7a-7c shows three of the plasmoid ejections denoted as p1 , p4 and p7 . in figure 7d-7f we overlaid contour images of the soft x - ray emission , which show the time evolution of the plasmoid ejection . the directions of the ejections are indicated by the arrows in the panels . to make clear the traveling of the plasmoids , we also overlaid the contours of the sxr images taken at different times ( e.g. 15:09:19 ut , 15:09:39 ut and 15:09:59 ut for fig . 7d ) on figures 7d , 7e , and 7f . from these contour images , we can roughly outline the position and the size of the bright cores of the plasmoids . we can also measure the velocity @xmath2 by taking the time difference of these contour images . those size and apparent velocity of the plasmoids are listed in table 1 . here we note that these are the `` apparent '' velocities , and the motions in the line of sight are ignored . therefore , the actual velocity of the plasmoid @xmath3 will be greater than the plane - of - the - sky value , where @xmath4 representing the expected deprojection over a reasonable range of angles @xmath5 to the plane of the sky . the trajectory of the plasmoid is not necessarily in the straight lines as shown with the arrows in figure 7 . in order to take into account those non - straight motions , we measured the apparent velocity of each plasmoid by averaging the velocities derived from each time differences . in the following discussions , we consider the apparent velocity as the actual velocity . we can see several plasmoids were ejected in the northwest direction , and one plasmoid ejected in the southwest direction in figure 4 ( supplement movie 2 ) and figure 5 , which is marked as p6 in this paper . each plasmoid has a unique velocity , brightness and size . the first ejection was that of p1 , which was followed by p2-p4 which were successively ejected as a group in the same direction . p1 shows blob - like structure , while p2-p4 seem to be a part of an expanding loop . we are not able to clearly define the structure of p5 and p6 due to the faint emission , although we can surely identify p5 and p6 traveling outward from the active region . p7 is the brightest ejection of blob - like structure . it starts to rise up gradually at 15:12 ut and ejected / accelerated upwards at 15:14 ut . in the lasco cme images we can no longer identify the fine structure corresponding to the individual plasmoid ejections , though some complicated structures can be observed ( see fig.2 ) this is probably because the ejected plasmoids merge into a single cme . the average velocity of the cme listed in the cme event catalog is about 1245 km s@xmath0 . this apparent velocity of the cme is faster than those of the plasmoid ejections p1-p7 summarized in table 1 . this observational fact qualitatively suggests that the merged plasmoids are continuously accelerated as they are ejected into interplanetary space , as shown by cheng et al . ( 2003 ) , although we can hardly identify the one - to - one relation between them . here we focus on the time evolution of the plasmoid ejections . we used a time slice image of the plasmoid ejections as we show in figure 8 . the horizontal axis is the time from 15:06 ut to 15:18 ut , and the vertical axis is the 1d image ( negative images ) using a slit line placed along the direction of the several plasmoid ejections ( p1 , p3 , p4 , p5 and p7 ) . this time slice image is made of half - resolution images . the position of the slit line is shown in figure 6 and 7a . those ejections seen in the time slice image are marked with signs @xmath6 , @xmath7 @xmath8 in figure 8b . we can also identify further additional faint ejections in the time slice image . some ejections travel along the slit lines , while others travel on a path which is slightly different from the slit line . as a result , the visibility of each plasmoid ejection is different . initially p1 , p2 and p3 are slowly accelerated , then strongly accelerated during the initial impulsive phase of the hard x - ray emission ( 15:07:40 - 15:08:40 ut ) followed closely by the ejection of p4 . a group of plasmoids gradually rise up 15:09:40- 15:12:20 ut followed by the faint ejection p5 . p6 is ejected in a different direction ( southwest ) and does not cross the slit line , so it does not appear in figures 8a , b . p7 is the brightest ejection and the most clearly visible in figures 8a , b . the apparent velocities of the plasmoids along the slit can also be derived from the slopes of the fitted lines in figure 8b . we note that these are the apparent velocities measured from the time slice images of figure 8b and different from the velocities measured from contour plot images in figure 7 . this is because the former shows the front velocity of thinner density plasma , while the latter shows the velocity of the thick core part of the plasmoids . in figure 8a and 8b , the plasmoid ejections roughly start to rise at the apparent height of approximately 50 ( @xmath935 mm ) and propagate into the upper atmosphere . this probably means that reconnection occurs at around or just below the height of @xmath935 mm . in this paper , we set the start position of plasmoid ejections as the height ( @xmath935 mm ) and define the start times of ejections as the time when each plasmoid crosses the height ( which is shown with dotted line in fig . it is noted that figure 6 and figure 8a , b are drawn with half - resolution images , but figure 5 is a series of quarter - resolution images . therefore we see more of the earlier phase of plasmoid ejection in figure 6 and 8 than in figure 5 , and so figure 8 is more appropriate to determine the time of plasmoid ejection . we confirmed that the start times are comparable to those defined above . in figure 8c , we show the light curve of the hard x - ray emission obtained with the h - band ( 52.7 - 92.8 kev ) of yohkoh / hxt and the goes soft x - ray light curve . we can distinguish the hard x - ray bursts into three separate periods : the first period , a ( 15:08 - 15:09 ut ) , is the brightest phase of the hard x - ray emission , the second period , b ( 15:09 - 15:11 ut ) , show gradual enhancement and the last period , c ( @xmath915:14 ut ) , is an isolated hard x - ray peak . the plasmoid ejections of p1-p3 seem to be ejected during the peak time of period a. p1 seems to be ejected just before the hard x - ray peaks in figure 8c . this is consistent with the report by ohyama & shibata ( 1997 ) who showed that plasmoids are ejected at or just before the hard x - ray peak . during period b , a group of plasmoids gradually rise up and are followed by the faint ejection of p4 and p5 , while the brightest plasmoid p7 is ejected during period c. the correspondence of the plasmoid ejections and hard x - ray peaks are shown by the arrows in figure 8b . since the bursts in period a are superposed and very complex , it is difficult to identify the exact correspondence between the plasmoid ejections ( p1-p3 ) and the hard x - ray bursts . we made a correlation plot of the times of the hard x - ray bursts and those of the plasmoid ejections ( figure 9a ) to make the relation clearer . both the horizontal and vertical axes show the times ( ut ) . the horizontal ( _ light gray _ ) and the vertical ( _ dark gray _ ) lines illustrate the times of the plasmoid ejections and those of the hard x - ray bursts , respectively . the thickness of the lines shows an estimation of the error . figure 9b shows hard x - ray ( 52.7 - 92.8 kev ) light curves obtained with h - band of yohkoh / hxt . the thin solid line shows the points where the times of the plasmoid ejections correspond to those of the hard x - ray bursts . these results appear to show that several plasmoid ejections coincide with hard x - ray bursts . here we note that , although the intense hxr emissions indicate that strong energy releases occur at those times , it does not necessarily mean that certain amount of plasmoids , that is , notable plasmoids are ejected . we have already noticed that there are many fainter ejections , which also tend to appear at hxr bursts , while some of the hard x - ray intensity fluctuations have no associated ejections in figures 8 and 9 . on the other hand , the hxr burst for p5 and p6 does not show a sharp summit but a gentle hump . this is probably because ejections p5 and p6 are parts of continuous outflows from the active region during this time range . this may also suggest a milder energy release compared with the others , resulting into a gradual enhancement of hard x - ray emission between 15:09:40 ut and 15:11:00 ut . we also studied temperature diagnostics on the soft x - ray emitting plasma using the sxt filter ratio method @xcite . we used the half - resolution images taken with the be and thick al filters of sxt for the analysis of plasmoid ejections . we subtracted the background photon flux of plasmoids . the temperature and emission measure are determined using the two filter data of the be and thick al filters . the size of the plasmoid @xmath10 was measured from the contour plot images in figure 7b , which shows the lower limit of the observable size ( summarized in table 1 ) . we assume that the x - ray emitting plasma , measured by the filter ratio method , fills the plasmoid with a filling factor of 1 . we assumed that the volume of the plasmoid is @xmath11 , such that the line of sight width of a plasmoid is equal to the square - root of the size @xmath10 . since the observation times of the be and thick al filters are not exactly the same , we used two images from the thick al filter which were taken just before and just after the be filter observation . the error shown in table 1 mostly results from combining these two images this error is much greater than the background photon noise ( see * ? ? ? * in more detail ) . as for plasmoid ejections p2 and p3 , there are no be and thick al filter images , because p2 and p3 are out of field of view of half - resolution images as shown in figure 6 . in these cases we assumed the temperatures of p2 and p3 as @xmath12 k. table 1 summarizes the physical parameters , such as temperature , emission measure , density , mass , thermal energy and kinetic energy of each plasmoid , of the seven plasmoid ejections identified in figure 5 . since the size of the plasmoids that we derived from the images is just the lower limit , the mass , thermal and kinetic energies calculated with the size are also the lower limit . in table 1 , each plasmoid shows a typical temperature of 10@xmath13 k , a density of 10@xmath14 @xmath15 and an apparent velocity of 200 - 1400 km s@xmath0 , which is similar to results of previous studies ( i.e. * ? ? ? * ; * ? ? ? the kinetic energy of each plasmoid ejection seems to be comparable to or twice as large as their thermal energy . we also estimate the total flare energy from the full - resolution images ( spatial resolution 2.5 " ) of the same filters at the peak time of goes soft x - ray emission . since the only part of the total flare energy is converted into plasmoid ejections , the total energy of plasmoid ejections is smaller than the total energy of solar flare . we analyzed the goes x2.3 class flare which occurred at 14:51 ut on 2000 november 24 . we found multiple plasmoid ejections from a single flare . furthermore , each plasmoid ejection seems to be associated with a peak in the hard x - ray emission . in figure 8 , the plasmoid ejections seem to occur at the height of approximately 50 " ( @xmath9 35 mm ) . this tells us that reconnection occurs at the height of approximately 35 mm . the horizontal light gray lines in the top panel of figure 9 show that the times of plasmoid ejections , when they reach the height of 35 mm , seems to be well correlated to the peak time of hard x - ray emission , which is consistent with previous studies ( e.g. * ? ? ? * ) . since the peak in hard x - ray emission indicates strong energy release , we have demonstrated that each plasmoid ejection occurs during a period of strong energy release , suggesting a series of impulsive energy releases in a single flare . we also performed a temperature and emission measure analysis and investigated the physical parameters of the plasmoid ejections shown in table 1 . figure 8 and table 1 show that the hard x - ray bursts in period a have large intensities in correlation with the large kinetic energy of plasmoid ejections p1-p4 . conversely , the hard x - ray bursts in period b and c have small intensities in correlation with the kinetic energy of plasmoid ejections p5-p7 . figure 10a shows the relation between the kinetic energy of plasmoid ejection and the intensity of the corresponding hard x - ray peak emission . we can see a rough tendency that the larger kinetic energy of plasmoid ejections is associated with the brighter hard x - ray peak emission , and vice versa . the hard x - ray emission is known to show energy release rate ( e.g. neupert effect ; * ? ? ? * ; * ? ? ? * ) , which leads us to the following equation : @xmath16 where we assumed that the released thermal energy of a solar flare in a short period is comparable to the kinetic energy of the plasmoid ejection . this means that a plasmoid ejection with large hard x - ray emission , and therefore with large energy release , can be accelerated strongly . a similar kinetic evolution is also seen in the case of cmes @xcite . figure 10b shows the relation between the apparent velocity and the intensity of the corresponding hard x - ray peak emissions . there seems to be a correlation between the plasmoid velocities and the hard x - ray emissions , although it is difficult to measure the velocities of the plasmoids precisely due to the faint emission . similar to the above equations , we can derive the following relation between hard x - ray emission and the kinetic energy of a plasmoid ejection : @xmath17 here @xmath18 is the typical magnetic field in a current sheet , @xmath19 is the inflow velocity and @xmath20 is the characteristic length of the inflow region . the inflow velocity @xmath19 is thought to be about 0.01@xmath21 from direct observations @xcite . it has been also known that inflow can be controlled by the plasmoid ejection . as a plasmoid is ejected out of the current sheet , the density in the current sheet decreases , and the inflow is enhanced to conserve the total mass under the condition that incompressibility is approximately satisfied . here , we assume that @xmath19 is proportional to the plasmoid velocity @xmath22 . then , we find the relation @xmath23 . moreover , if we can further assume the @xmath22 is proportional to cme velocity @xmath25 , then @xmath23 @xmath24 @xmath25 . this is consistent with the result of yashiro & gopalswamy ( 2009 ) . a correlation between the energy release rate , which is represented by @xmath23 , and the plasmoid velocity @xmath22 is also successfully reproduced by a magnetohydrodynamic simulation @xcite . in the simulation , they clearly showed that the plasmoid velocity controls the energy release rate ( i.e. reconnection rate ) in the nonlinear evolution . observations of plasmoid ejections have been paid attention to as evidence of magnetic reconnection , though they found only one plasmoid ejection per one solar flare . however , magnetic reconnection theory suggests that the impulsive bursty regime of reconnection or fractal reconnection is associated with a series of plasmoids of various scales . it is known that magnetic reconnection is an effective mechanism for energy release in a solar flare . once the current sheet becomes thin enough for the tearing instability to occur , repetitive formation of magnetic islands and their subsequent coalescence drives the `` impulsive bursty '' regime of reconnection @xcite . furthermore , such reconnection can produce a fractal structure in the current sheet , which is not only theoretically predicted but has also been observed ( i.e. * ? ? ? * ; * ? ? ? the plasmoids generated in this cycle control the energy release by inhibiting magnetic reconnection in the current sheet and/or by inflow driven by the ejection . the plasmoid ejection enhances reconnection and promotes further plasmoid ejections from the current sheet . similar intermittent energy release from a solar flare has been reported as multiple downflows associated with hard x - ray bursts in the impulsive phase @xcite . mckenzie et al . ( 2009 ) estimated the total amount of energy of multiple downflows and showed that it is comparable to the total amount of energy released from the magnetic field , which is consistent with the magnetic reconnection model . tanuma et al . ( 2001 ) showed through numerical simulations that plasmoid ejections are closely coupled with the reconnection process and the greatest energy release occurs when the largest plasmoid is ejected . if the strong energy release corresponds to magnetic reconnection , we may conclude that this is evidence of unsteady magnetic reconnection in a solar flare and that plasmoids have a key role in energy release and particle acceleration . we first acknowledge an anonymous referee for his / her useful comments and suggestions . we also thank a. hillier for his careful reading and correction of this paper . we also thank j. kiyohara for her help in making the movies and t. morimoto for his help in finding the multiple plasmoid ejection events . we made extensive use of yohkoh / sxt and hxt data . this work was supported in part by the grant - in - aid for creative scientific research `` the basic study of space weather prediction '' ( head investigator : k.shibata ) from the ministry of education , culture , sports , science , and technology of japan , and in part by the grand - 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15:09:59 & 6.98 - 7.09 & 45.32 - 45.33 & @xmath303.9 - 4.7 & 2.76 - 2.79 & @xmath302.63 - 2.66 & 654 - 818 & @xmath305.63 - 8.91 & @xmath302.51 - 3.28 & northwest & 1000 + p2 & 15:09:59 - 15:10:59 & ( 7.00 ) & ( 45.34 ) & @xmath304.6 - 5.8 & ( 2.65 ) & @xmath30(3.69 ) & 573 - 654 & ( @xmath306.05 - 7.88 ) & ( @xmath303.69 ) & northwest & 1300 + p3 & 15:10:19 - 15:10:59 & ( 7.00 ) & ( 45.32 ) & @xmath304.7 - 8.3 & ( 2.59 ) & @xmath30(3.60 ) & 982 - 1470 & ( @xmath3017.4 - 38.9 ) & ( @xmath303.60 ) & northwest & 1500 + p4 & 15:10:39 - 15:11:19 & 7.02 - 7.07 & 45.80 - 45.80 & @xmath303.1 - 7.1 & 4.85 & @xmath304.30 & 654 - 828 & @xmath309.19 - 14.7 & @xmath304.50 - 5.05 & northwest & 950 + p5 & 15:13:33 - 15:14:27 & 7.02 - 7.08 & 45.73 - 45.74 & @xmath301.5 - 3.1 & 5.76 - 5.83 & @xmath301.12 - 1.13 & 360 - 844 & @xmath300.72 - 4.03 & @xmath301.17 - 1.36 & northwest & 120 + p6 & 15:13:33 - 15:13:51 & 7.07 - 7.11 & 45.49 - 45.50 & @xmath301.6 - 5.2 & 4.37 - 4.42 & @xmath300.85 - 0.86 & 360 - 720 & @xmath300.55 - 2.22 & @xmath300.99 - 1.11 & southwest & 130 + p7 & 15:16:27 - 15:18:43 & 6.97 - 7.03 & 46.05 - 46.06 & @xmath303.2 - 7.8 & 6.95 - 7.03 & @xmath303.99 - 4.04 & 254 - 409 & @xmath301.29 - 3.38 & @xmath303.72 - 4.33 & northwest & 150 + total & - & - & - & - & - & - & - & 43.5 - 84.7 & 20.2 - 22.4 & & & & + flare & 15:18:33 - 15:18:53 & 6.73 - 7.08 & 48.27 - 48.96 & 9.33 & 137 - 304 & 391 - 865 & - & - & 210 - 1040 & & & & +

the soft x - ray telescope ( sxt ) on board yohkoh revealed that the ejection of x - ray emitting plasmoid is sometimes observed in a solar flare . it was found that the ejected plasmoid is strongly accelerated during a peak in the hard x - ray emission of the flare . in this paper we present an examination of the goes x 2.3 class flare that occurred at 14:51 ut on 2000 november 24 . in the sxt images we found `` multiple '' plasmoid ejections with velocities in the range of 250 - 1500 km s@xmath0 , which showed blob - like or loop - like structures . furthermore , we also found that each plasmoid ejection is associated with an impulsive burst of hard x - ray emission . although some correlation between plasmoid ejection and hard x - ray emission has been discussed previously , our observation shows similar behavior for multiple plasmoid ejection such that each plasmoid ejection occurs during the strong energy release of the solar flare . as a result of temperature - emission measure analysis of such plasmoids , it was revealed that the apparent velocities and kinetic energies of the plasmoid ejections show a correlation with the peak intensities in the hard x - ray emissions .

You are an expert at summarizing long articles. Proceed to summarize the following text: as hetnet provides flexible and efficient topology to boost spectral efficiency , it has recently aroused immense interest in both academia and industry . as illustrated in fig . [ fig : hetnet_2tiers ] , a hetnet consists of a diverse set of regular macro base stations ( bs ) overlaid with low power pico bss . since this overlaid structure may lead to severe interference problem , it is extremely critical to control interference via rrm in hetnet . there has been much research conducted on rrm optimization for traditional cellular networks . in @xcite , the authors considered power and user scheduling in single - carrier cellular networks . in @xcite , the game theoretical approaches are proposed for distributed resource allocation . in @xcite , the authors proposed a dynamic fractional frequency reuse scheme to combat the inter - sector interference under a game - based optimization by each sector . the coordinated multipoint transmission ( comp ) @xcite is another important technique to handle the inter - cell interference . for example , in @xcite , the authors exploited the uplink - downlink duality to do joint optimization of power allocation and beamforming vectors . in @xcite , a wmmse algorithm is proposed to find a stationary point of the weighted sum - rate maximization problem for multi - cell downlink systems . while the above algorithms achieve comparably good performance , they require global channel state information ( csi ) for centralized implementation @xcite or over - the - air iterations and global message passing for distributed implementation @xcite . it is quite controversial whether comp is effective or not in lte systems due to large signaling overhead , signaling latency , inaccurate csit , and the complexity of the algorithm . on the other hand , solutions for traditional cellular networks can not be applied directly to hetnet due to the unique difference in hetnet topology . first , the inter - cell interference in hetnet is more complicated , e.g. , there is _ co - tier interference _ among the pico bss and among the macro bss as well as the _ cross - tier interference _ between the macro and pico bss . furthermore , due to load balancing , some of the mobiles in hetnet may be assigned to a pico bs which is not the strongest bs @xcite and the mobiles in the pico cell may suffer from strong interference from the macro bss . to solve these problems , some eicic techniques , such as the abs control @xcite , have been proposed in lte and lte - a @xcite . in @xcite , the authors analyzed the performance for abs in hetnet under different cell range extension ( re ) biases . however , they focused on numerical analysis for the existing heuristic eicic schemes , which are the baselines of this paper . in @xcite , the authors proposed an algorithm for victim pico user partition and optimal synchronous abs rate selection . however , they used a universal abs rate for the whole network , and as a result , their scheme could not adapt to dynamic network loading for different macro cells . in this paper , we focus on the resource optimization in the downlink of a hetnet without comp . we consider _ dynamic abrb _ for interference control and dynamic user scheduling to exploit _ multi - user diversity_. the abrb is similar to the abs but it is scheduled over both time and frequency domain . unlike @xcite , we do not restrict the abrb rate to be the same for all macro bss and thus a better performance can be achieved . however , this also causes several new technical challenges as elaborated below . * * exponential complexity for dynamic abrb : * optimization of abrb patterns is challenging due to the combinatorial nature and exponentially large solution space . for example , in a hetnet with @xmath0 macro bss , there are @xmath1 different abrb pattern combinations . hence , brute force solutions are highly undesirable . * * complex interactions between dynamic user scheduling and dynamic abrb * : there is complex coupling between the dynamic user scheduling and abrb control . for instance , the abrb pattern will affect the user sets eligible for user scheduling . furthermore , the optimization objective of abrb control depends on user scheduling policy and there is no closed form characterization . * * challenges in rrm architecture : * most existing solutions for resource optimization of hetnet requires global knowledge of csi and centralized implementations . yet , such designs are not scalable for large networks and they are not robust with respect to ( w.r.t . ) latency in backhaul . to address the above challenges , we propose a two timescale control structure where the long term controls , such as dynamic abrb , are adaptive to the large scale fading . on the other hand , the short term control , such as the user scheduling , is adaptive to the local csi within a pico / macro bs . such a multi - timescale structure allows _ hierarchical rrm _ design , where the long term control decisions can be implemented on a rrm server for inter - cell interference coordination . the short - term control decisions can be done locally at each bs with only local csi . such design has the advantages of low signaling overhead , good scalability , and robustness w.r.t . latency of backhaul signaling . while there are previous works on two timescale rrm @xcite , those approaches are heuristic ( i.e. the rrm algorithms are not coming from a single optimization problem ) . our contribution in this paper is a formal study of two timescale rrm algorithms for hetnet based on optimization theory . to overcome the exponential complexity for abrb control , we exploit the sparsity in the _ interference graph _ of the hetnet topology and derive structural properties for the optimal abrb control . based on that , we propose a _ two timescale alternative optimization _ solution for user scheduling and abrb control . the algorithm has low complexity and is asymptotically optimal at high snr . simulations show that the proposed solution has significant performance gain over various baselines . _ notations _ : let @xmath2 denote the indication function such that @xmath3 if the event @xmath4 is true and @xmath5 otherwise . for a set @xmath6 , @xmath7 denotes the cardinality of @xmath6 . consider the downlink of a two - tier hetnet as illustrated in fig . [ fig : hetnet_2tiers ] . there are @xmath0 macro bss , @xmath8 pico bss , and @xmath9 users , sharing @xmath10 ofdm subbands . denote the set of the macro bss as @xmath11 , and denote the set of the pico bss as @xmath12 . a two - tier heterogeneous network with macro and pico base stations , width=332 ] the hetnet topology ( i.e. , the network connectivity and csi of each link ) is represented by a topology graph as defined below . [ hetnet topology graph]define the _ topology graph _ of the hetnet as a bipartite graph @xmath13 , where @xmath14 denotes the set of all macro and pico bs nodes , @xmath15 denotes the set of all user nodes , and @xmath16 is the set of all edges between the bss and users . an edge @xmath17 between bs node @xmath18 and user node @xmath19 represents a wireless link between them . each edge @xmath17 is associated with a csi label @xmath20 , where @xmath21 represents the channel coefficient between bs @xmath22 and user @xmath23 on subband @xmath24 . for each bs node @xmath22 , let @xmath25 denote the set of associated users . for each user node @xmath23 , define @xmath26 as the set of neighbor macro bss and @xmath27 as the set of neighbor pico bss . in the topology graph , @xmath28 means that the path gain between user @xmath23 and bs @xmath22 is sufficiently small compared to the direct link path gain , and thus the interference from bs @xmath22 will have negligible effect on the data rate of user @xmath23 . we have the following assumption on the channel fading process @xmath29 . [ asm : fory]the channel fading coefficient has a two timescale structure given by @xmath30 . the small scale fading process @xmath31 is identically distributed w.r.t . the subframe and subband indices ( @xmath32 ) , and it is i.i.d . user and bs indices ( @xmath33 ) . moreover , for given @xmath34 , @xmath31 is a continuous random variable . the large scale fading process @xmath35 is assumed to be a slow ergodic process ( i.e. , @xmath36 remains constant for several _ _ super - frames _ _ subframes ] . ) according to a general distribution . the two timescale fading model has been adopted in many standard channel models . the large scale fading @xmath36 is usually caused by path loss and shadow fading , which changes much slowly compared to the small scale fading . we consider the following _ biased cell selection _ mechanism to balance the loading between macro and pico bss @xcite . let @xmath37 denote the serving bs of user @xmath23 . let @xmath38 denote the cell selection bias and let @xmath39 denote the transmit power of bs @xmath22 on a single sub - band . let @xmath40 and @xmath41 respectively be the strongest macro bs and pico bs for user @xmath23 . if @xmath42 , user @xmath23 will be associated to pico cell @xmath43 , i.e. , @xmath44 ; otherwise @xmath45 . if a user only has a single edge with its serving bs , it will not receive inter - cell interference from other bss and thus its performance is noise limited ; otherwise , it will suffer from strong inter - cell interference if any of its neighbor bss is transmitting data and thus its performance is interference limited . this insight is useful in the control algorithm design later and it is convenient to formally define the interference and noise limited users . [ interference / noise limited user][def : inuser]if a user @xmath23 has a single edge with its serving bs @xmath37 only , i.e. , @xmath46 , then it is called a _ noise limited user _ ( n - user ) ; otherwise , it is called an _ interference limited user _ ( i - user ) . fig . [ fig : abs_intro ] illustrates an example of the hetnet topology graph . in fig . [ fig : abs_intro](a ) , an arrow from a bs to a user indicates a direct link and the dash circle indicates the coverage area of each bs . an i - user which lies in the coverage area of a macro bs is connected to this macro bs , while a n - user does not have connections with the neighbor macro bss in the topology graph as illustrated in fig . [ fig : abs_intro](b ) . we consider a two timescale hierarchical rrm control structure where the control variables are partitioned into _ long - term _ and _ short - term _ control variables . the long - term control variables are adaptive to the large scale fading @xmath47 and they are implemented at the radio resource management server ( rrms ) . the short - term control variables are adaptive to the instantaneous csi @xmath48 and they are implemented locally at each macro / pico bs . abs is introduced in lte systems @xcite for interference mitigation among control channels in hetnet . it can also be used to control the _ co - tier _ and _ cross - tier interference _ among the data channels . in lte systems , abs is only scheduled over time domain . in this paper , we consider dynamic abrb control for interference coordination . the abrb is similar to abs but it is scheduled over both time and frequency domain . it is a generalization of abs and enables more fine - grained resource allocation . when an abrb is scheduled in a macro bs , a rb with blank payload will be transmitted at a given frequency and time slice and this eliminates the interference from this macro bs to the pico bss and the adjacent macro bss . hence , as illustrated in fig . [ fig : abs_intro ] , scheduling abrb over both time and frequency domain allows us to control both the _ macro - macro _ bs and _ macro - pico _ bs interference . we want to control the abrb dynamically w.r.t . the large scale fading because the optimal abrb pattern depends on the hetnet topology graph . for example , when there are a lot of pico cell i - users , we should allocate more abrbs at the macro bs to support more pico cell i - users . on the other hand , when there are only a few pico cell i - users , we should allocate less abrbs to improve the spatial spectrum efficiency . for any given subframe , define @xmath49 to indicate if abrb is scheduled ( @xmath50 ) for subband @xmath24 at macro bs @xmath22 . let @xmath51^{t}\in\mathcal{a}$ ] be the abrb pattern vector for subband @xmath24 and @xmath52 is the set of all possible abrb patterns macro bss can either schedule an abrb or not for the @xmath53-th rb ( i.e. , subframe @xmath54 and subband @xmath24 ) , there are @xmath1 possible abrb patterns . hence , the size of @xmath52 is @xmath1 . ] . in the proposed dynamic abrb control , each macro bs is allowed to dynamically change the ratio of abrb transmission on each subband and this ratio can be any positive real number . to facilitate implementation , we consider randomized abrb control policy as defined below . [ randomized abrb control policy ] an abrb control policy of the @xmath24-th subband @xmath55 is a mapping from the abrb pattern space @xmath52 to a probability in [ 0,1 ] . at any subframe , the instantaneous abrb pattern vector for subband @xmath24 is stochastically determined according to the probabilities @xmath56 , where @xmath57 denote the probability that the @xmath58 subband is in abrb pattern @xmath59 . to facilitate structural abrb design , we partition the users into two types . [ partitioning of user set][def : user - set - partition ] the mobile user set is partitioned into two subsets @xmath60 , where @xmath61 denotes the set of _ type a users _ and is defined as @xmath62 and @xmath63 denotes the set of _ type b users_. the type a users include all pico cell users and macro cell n - users , while the type b users include all macro cell i - users . for type a users , it will not lose optimality by imposing a _ synchronous abrb structure _ where the transmissions of the abrb at all macro bss are aligned as much as possible . the formal definition of the synchronous abrb structure is given in theorem [ thm : absreduce ] . as will be shown in theorem [ thm : absreduce ] , if there is only type a users , imposing the synchronous abrb structure can dramatically reduce the number of abrb control variables from exponential large ( @xmath1 ) to only @xmath0 and this complex reduction is achieved without loss of optimality . on contrast , the performance of the macro cell i - users is very poor under the synchronous abrb structure because aligning the data transmissions of all macro bss will cause strong inter - cell interference for macro cell i - users . motivated by these observations , we partition the @xmath10 subbands into two groups , namely @xmath64 and @xmath65 , and use different abrb control policies for type a and type b users on these two groups of subbands respectively . the variable @xmath66 controls the fraction of _ type a _ subbands . at each subframe , each bs @xmath22 dynamically selects a user from @xmath25 for each subband @xmath24 based on the knowledge of current abrb pattern @xmath59 and channel realization @xmath67 to exploit _ multi - user diversity_. let @xmath68 be the user scheduling variable ( of user @xmath23 at bs @xmath37 ) of subband @xmath24 and @xmath69 $ ] be the associated vectorized variable . the set of all feasible user scheduling vectors at bs @xmath22 for the @xmath24-th subbands with abrb pattern @xmath59 is given by @xmath70 where @xmath71 is the set of users that can not be scheduled on a type a subband under abrb pattern @xmath59 ; and @xmath72 . the physical meaning of @xmath73 is elaborated below . first , if a macro bs is transmitting abrb , none of its associated users can be scheduled for transmission . moreover , due to large cross - tier interference from macro bss , a pico cell i - user can not be scheduled for transmission if any of its neighbor macro bss @xmath74 is transmitting data subframe ( i.e. , @xmath75 ) . as will be seen in section [ sub : structural - properties - ofpa ] , explicitly imposing this user scheduling constraint for the pico cell i - users is useful for the structural abrb design . the user scheduling policy @xmath76 of the @xmath24-th sub - band is defined below . [ user scheduling policy][def : randomized - user - scheduling ] a user scheduling policy of the @xmath22-th bs and @xmath24-th sub - band @xmath77 is a mapping : @xmath78 , where @xmath79 is the csi space . specifically , under the abrb pattern @xmath59 and csi realization @xmath67 , the user scheduling vector of bs @xmath22 is given by @xmath80 . let @xmath81 denote the overall user scheduling policy on sub - band @xmath24 . assuming perfect csi at the receiver ( csir ) and treating interference as noise , the instantaneous data rate of user @xmath23 is given by : @xmath82 where @xmath83 , @xmath84 ; @xmath85 is the mutual information of user @xmath23 contributed by the @xmath24-th subband ; and @xmath86 is the interference - plus - noise power at user @xmath23 on subband @xmath24 . for a given policy @xmath87 and large scale fading state @xmath88 , the average data rate of user @xmath23 is given by : @xmath89=\sum_{m\in\mathcal{m}\left(k\right)}\overline{\mathcal{i}}_{m , k}\left(q_{m},\pi_{m}\right),\ ] ] where the average mutual information on subband @xmath24 is @xmath90 and @xmath91 $ ] . for conciseness , the abrb pattern @xmath59 for a specific subband @xmath24 is denoted as @xmath92 $ ] when there is no ambiguity . the performance of the hetnet is characterized by a utility function @xmath93 , where @xmath94 $ ] is the average rate vector . we make the following assumptions on @xmath93 . [ asm : ufun]the utility function can be expressed as @xmath95 , where @xmath96 is the weight for user @xmath23 , @xmath97 is assumed to be a concave and increasing function . moreover , for any @xmath98 such that @xmath99 and @xmath100 belongs to the domain of @xmath97 , @xmath101 satisfies @xmath102 where @xmath103 and @xmath104 are some scalar functions of @xmath105 . the above assumption is imposed to facilitate the problem decomposition in section [ sub : problem - decomposition ] . this utility function captures a lot of interesting cases below . * * weighted sum throughput : * the utility function is @xmath106 . * * @xmath107-fair @xcite : * @xmath107-fair can be used to compromise between the fairness to users and the utilization of resources . the utility function is * * @xmath108 * * proportional fair @xcite : * this is a special case of @xmath107-fair when @xmath109 . due to the statistical symmetry of the @xmath10 subbands , there is no loss of optimality to consider symmetric policy @xmath110 , where @xmath111 ( @xmath112 ) and @xmath113 ( @xmath114 ) if @xmath115 ( @xmath65 ) . [ optimality of symmetric policy][lem : optimality - of - symmetric]there exists a symmetric policy @xmath116 such that it is the optimal solution of the following optimization problem : @xmath117 please refer to appendix [ proof - of - lemma symp ] for the proof . moreover , we have @xmath118 and @xmath119 . as a result , the utility function under a symmetric policy @xmath120 can be expressed as : @xmath121 where @xmath122 , @xmath123 , and @xmath124 ( @xmath125 ) can be any type a ( type b ) subband . finally , for a given hetnet topology graph @xmath13 , the two timescale rrm optimization is given by : @xmath126 where ( [ eq : qeccon1a ] ) and ( [ eq : qeccon2a ] ) ensure that @xmath127 and @xmath128 satisfy the definition of probability mass function ( pmf ) . using primal decomposition , problem @xmath129 can be decomposed into the following subproblems . * subproblem a * ( * cross - tier interference control * ) : optimization of abrb @xmath130 and user scheduling @xmath131 . @xmath132 * subproblem b * ( * co - tier interference control * ) : * * optimization of abrb @xmath112 and user scheduling @xmath114 . @xmath133 * subproblem c * ( * subband partitioning * ) : optimization of subband partitioning @xmath134 . @xmath135 note that the solution of @xmath136/@xmath137 is independent of the value of @xmath134 because both @xmath138 and @xmath139 are independent of @xmath134 . after solving @xmath136 and @xmath137 , the optimal @xmath134 can be easily solved by bisection search . on the other hand , the optimization of @xmath131/@xmath114 is a stochastic optimization problem because the @xmath140/@xmath141 involves stochastic expectation over csi realizations and they do not have closed form characterization . furthermore , the number of abrb control variables in @xmath136/@xmath137 is exponential w.r.t . the number of macro bss @xmath0 . we shall tackle these challenges in section [ sec : solution - of - pa ] and [ sub : solution - of - subproblempb ] . in this section , we first derive structural properties of @xmath136 and reformulate @xmath136 into a simpler form with reduced solution space . then , we develop an efficient algorithm for @xmath136 . we require the following assumption to derive the results in this section . [ asm : picoiedge]for any @xmath142 , let @xmath143 denote the set of all pico cell i - users in pico cell @xmath22 . then we have @xmath144 and define @xmath145 as the set of neighbor macro bss of pico cell @xmath22 . the above assumption states that a macro bs will interfere with all the i - users in a pico cell as long as it interferes with any user in the pico cell . this is reasonable since the coverage area of a macro bs is much larger than that of a pico bs . we exploit the interference structure in the hetnet topology @xmath146 to derive the structural properties of @xmath136 . throughout this section , we will use the following example problem to illustrate the intuition behind the main results . [ exm - problema]consider @xmath147 for the hetnet in fig . [ fig : abs_intro ] with @xmath148 macro bss and @xmath149 pico bs . the set of type a users is @xmath150 and the objective function is specified as @xmath151 ( i.e. , we consider sum - rate utility ) . for illustration , we focus on the case when the marginal probability that a macro bs is transmitting abrb is transmitting abrb is @xmath152 . ] is fixed as @xmath153^{t}=[0.7,0.5]^{t}$ ] . define two sets of abrb patterns @xmath154 and @xmath155 for each bs @xmath22 . for macro bs , @xmath154 is the set of abrb patterns under which macro bs @xmath22 is transmitting data . for pico bs , @xmath154 is the set of abrb patterns under which all of its neighbor macro bss is transmitting abrb . using the configuration in example [ exm - problema ] , we have @xmath156,\left[1,1\right]\right\ } $ ] , @xmath157,\left[1,1\right]\right\ } $ ] and @xmath158,\left[1,0\right]\right\ } $ ] ( the formal definition of @xmath154 and @xmath159 for general cases is in appendix [ sub : spageneral ] ) . in observation [ clm : effect - of - abrb - mi ] , we find that the abrb pattern @xmath160 only affects the data rate of a type a user in cell @xmath22 by whether @xmath161 or not . based on that , we find that the policy space for both the abrb control @xmath130 and user scheduling @xmath131 can be significantly reduced in observation [ clm : qareduction ] and [ clm : policy - space - reductionpia ] . while these observations are made for the specific configuration in example [ exm - problema ] , they are also correct for general configurations and are formally stated in lemma [ lem : sysman ] , theorem [ thm : absreduce ] and theorem [ thm : policy - space - reductionpia ] in appendix [ sub : spageneral ] . finally , using the above results , we transform the complicated problem @xmath136 to a much simpler problem with @xmath162 as the optimization variables . [ clm : effect - of - abrb - mi]for given csi @xmath163 and a feasible user scheduling vector @xmath164 , the mutual information @xmath165 of a type a user @xmath23 in cell @xmath22 only depends on whether the abrb pattern @xmath161 or not , i.e. , @xmath166,@xmath167 ( or @xmath168 ) . moreover , we have @xmath169 for any @xmath170 . let us illustrate the above observation using the configuration in example [ exm - problema ] . there are 4 type a users : @xmath150 . for user 1 in macro cell 1 ( n - user ) , it is scheduled for transmission whenever macro bs 1 is not transmitting abrb ( i.e. , @xmath171 ) . hence the mutual information is @xmath172 , which only depends on whether @xmath171 ( i.e. , @xmath173 ) or not . moreover , the mutual information is higher if @xmath173 because @xmath174 . for user 4 in pico cell 3 ( i - user ) , if @xmath175 , its neighbor macro bs 2 is transmitting abrb and the mutual information is @xmath176 ; otherwise ( @xmath177 ) , macro bs 2 is transmitting data and we have @xmath178 and @xmath179 . similar observations can be made for user 2 and 3 . based on observation [ clm : effect - of - abrb - mi ] , the abrb control policy space can be significantly reduced . [ clm : qareduction]consider @xmath147 for the configuration in example [ exm - problema ] . the optimal abrb control of @xmath147 conditioned on a given marginal probability vector @xmath180^{t}$ ] , denoted by @xmath181 , has the _ _ synchronous abrb structure__. ] where the transmissions of abrb at the macro bss are aligned as much as possible . as a result , there are only @xmath182 active abrb patterns @xmath183,[0,1],[1,1]\right\ } $ ] and the corresponding pmf @xmath184 is given by a function of @xmath185 as illustrated in fig . [ fig : abs_reduction ] . illustration of the structure of the optimal @xmath186 conditioned on a given @xmath185,width=332 ] in general , for a hetnet with @xmath0 macro bss , there are only @xmath187 active abrb patterns under synchronous abrb structure , which is significantly smaller than the number of all possible abrb patterns @xmath1 . as a result , the optimization @xmath147 w.r.t @xmath130 ( with @xmath1 variables ) can be reduced to an equivalent optimization w.r.t . @xmath185 ( with @xmath0 variables ) with much lower dimensions . observation [ clm : qareduction ] can be understood as follows . by observation [ clm : effect - of - abrb - mi ] , a higher average mutual information can be achieved for user @xmath188 under the abrb patterns @xmath161 . hence , for given marginal probabilities @xmath180^{t}$ ] , the average mutual information region will be maximized if we can simultaneously maximize @xmath189 for all bss @xmath190 . for macro bss @xmath191 , we have @xmath192 , which is fixed for given @xmath185 . for pico bss @xmath193 , we have @xmath194 , and the equality holds if and only if @xmath130 has the synchronous abrb structure in fig . [ fig : abs_reduction ] . similarly , we can reduce the user scheduling policy space using observation [ clm : effect - of - abrb - mi ] . [ clm : policy - space - reductionpia]consider @xmath147 for the configuration in example [ exm - problema ] . for given csi @xmath163 and abrb pattern @xmath160 , the optimal user scheduling at bs @xmath22 is given by @xmath195 by observation [ clm : effect - of - abrb - mi ] , if @xmath196 solves the maximization problem ( [ eq : argmaxruo ] ) for certain @xmath161 , it solves ( [ eq : argmaxruo ] ) for all @xmath161 . hence , it will not loss optimality by imposing an additional constraint on the user scheduling such that @xmath197 ( or @xmath168 ) . for convenience , let @xmath198 denote the set of all feasible user scheduling policies satisfying the above constraint in observation [ clm : policy - space - reductionpia ] ( the formal definition of @xmath198 is given in theorem [ thm : policy - space - reductionpia ] ) . then for given @xmath185 , @xmath199 and under the synchronous abrb , the corresponding objective function of @xmath136 can be rewritten as @xmath200 , where @xmath201 and @xmath202 is the corresponding average mutual information given in ( [ eq : ibarak ] ) of appendix [ sub : spageneral ] . as a result , the subproblem @xmath147 can be transformed into a simpler problem with @xmath203 solution space . [ equivalent problem transformation of @xmath147][cor : equivalent - problem - pa]let @xmath205 denote the optimal solution of the following joint optimization problem . @xmath206 where @xmath207,\forall j\right\ } $ ] . then @xmath208 , @xmath209 is the optimal solution of problem @xmath147 . furthermore , problem ( [ eq : plare ] ) is a bi - convex problem , i.e. , for fixed @xmath131 , problem ( [ eq : plare ] ) is convex w.r.t . @xmath185 , and for fixed @xmath185 , problem ( [ eq : plare ] ) is also convex w.r.t . @xmath131 . please refer to appendix [ sub : proof - of - corollaypa ] for the proof . by corollary [ cor : equivalent - problem - pa ] , we only need to solve the equivalent problem of @xmath136 in ( [ eq : plare ] ) . since problem ( [ eq : plare ] ) is bi - convex , we propose the following _ two timescale alternating optimization ( ao ) _ algorithm . for notation convenience , time index @xmath54 and @xmath210 are used to denote the subframe index and super - frame index respectively , where a super - frame consists of @xmath211 subframes . _ algorithm ao_a _ ( two timescale ao for @xmath147 ) : * * initialization * * : choose proper initial @xmath212,@xmath213 . set @xmath214 . * * step 1 * * ( short timescale user scheduling optimization ) : for fixed @xmath215 , let @xmath216 , where @xmath217 is given by @xmath218 where @xmath219 is the average data rate of user @xmath23 under @xmath215 and user scheduling policy @xmath220 . for each subframe @xmath221 $ ] , the user scheduling vector of bs @xmath22 is given by @xmath222 , where @xmath163 and @xmath223 are the csi and abrb pattern at the @xmath54-th subframe . * step 2 * ( long timescale abrb optimization ) : * * find the optimal solution @xmath224 of problem ( [ eq : plare ] ) under fixed @xmath220 using e.g. , ellipsoid method . let @xmath225 . * * return to step 1 until @xmath226 * * or the maximum number of iterations is reached . while ( [ eq : plare ] ) is a bi - convex problem and ao algorithm is known to converge to local optimal solutions only , we exploit the hidden convexity of the problem and show below that algorithm ao_a can converge to the global optimal solution under certain conditions . [ global convergence of algorithm ao_a][thm : global - convergence - la]let @xmath227=\mathcal{f}\left(\left[\mathbf{q}_{a}^{(t)},\pi_{a}^{(t-1)}\right]\right)$ ] denote the iterate sequence of algorithm ao_a began at @xmath228 , and denote the set of fixed points of the mapping @xmath229 as @xmath230\right.$ ] @xmath231\right)\right\ } $ ] . for any @xmath232 that is not a fixed point of @xmath229 , assume that @xmath233 , where @xmath234=\mathcal{f}\left(\left[\mathbf{q}_{a},\pi_{a}\right]\right)$ ] . then : 1 . algorithm ao_a converges to a fixed point @xmath235\in\delta$ ] of @xmath229 . any fixed point @xmath235\in\delta$ ] is a globally optimal solution of problem ( [ eq : plare ] ) . please refer to appendix [ sub : proof - of - theoremla ] for the proof . step 1 of algorithm ao_a requires the knowledge of the average data rate @xmath236 under @xmath215 and @xmath220 . we adopt a reasonable approximation on @xmath236 using a moving average data rate @xmath237 $ ] given by @xcite @xmath238 where @xmath239 is the data rate delivered to user @xmath23 at subframe @xmath54 . if we replace @xmath236 in step 1 of algorithm ao_a with the approximation @xmath240 in ( [ eq : rbarappr ] ) , the global convergence result in theorem [ thm : global - convergence - la ] no longer holds . however , it has been shown in @xcite that @xmath240 converges to @xmath236 as @xmath241 . hence , with the approximation @xmath240 , algorithm ao_a is still asymptotically optimal for large super frame length @xmath211 . in step 2 of algorithm ao_a , the average mutual information @xmath242 in the optimization objective contains two intermediate problem parameters @xmath243 and @xmath244 defined under ( [ eq : ibarak ] ) in appendix [ sub : spageneral ] . the calculation of @xmath243 s and @xmath244 s requires the knowledge of the distribution of all the channel coefficients , which is usually difficult to obtain offline . however , these terms can be easily estimated online using the time average of the sampled data rates delivered to user @xmath23 under abrb patterns @xmath245 and @xmath246 respectively . the abrb control @xmath224 is then obtained by solving the long timescale problem in step 2 with say , the ellipsoid method based on these estimates . the number of abrb control variables @xmath247 is exponentially large w.r.t . @xmath0 . to simplify @xmath137 , we introduce an auxiliary variable called the _ abrb profile _ and decompose @xmath137 . we first define the abrb profile . [ abrb profile ] the abrb profile @xmath248 is a subset of abrb patterns for type b subbands . using the notion of abrb profile and primal decomposition , @xmath249 can be approximated by two subproblems : * optimization of @xmath112 and @xmath114 for a given abrb profile @xmath250 . * @xmath251 * optimization of abrb profile @xmath250 . * @xmath252 in @xmath253 , we restrict the size of @xmath250 to be no more than @xmath254 . in appendix [ sub : proof - of - theoremequpb ] , we prove that at the asymptotically optimal solution of @xmath249 as snr becomes high , the number of active abrb patterns is indeed less than or equal to @xmath254 . let @xmath256 denote the @xmath257 abrb pattern in @xmath250 . then the average mutual information @xmath141 can be rewritten as @xmath258 where @xmath259 $ ] with @xmath260 denoting the probability that the abrb pattern @xmath256 is used . hence @xmath261 can be reformulated as @xmath262 similar to ( [ eq : plare ] ) , we propose a two timescale ao to solve for problem ( [ eq : plbequ ] ) w.r.t . @xmath114 and @xmath263 . _ algorithm ao_b _ ( two timescale ao for @xmath261 ) : * * initialization * * : choose proper initial @xmath264,@xmath265 . set @xmath214 . * * step 1 * * ( short timescale user scheduling optimization ) : for fixed @xmath266 , let @xmath267 , where @xmath268 is given by @xmath269 where @xmath270 is the average data rate of user @xmath23 under @xmath266 and user scheduling policy @xmath271 . for each subframe @xmath221 $ ] , the user scheduling vector of bs @xmath22 is given by @xmath272 , where @xmath273 and @xmath274 are the csi and abrb pattern at the @xmath54-th subframe . * step 2 * ( long timescale abrb optimization ) : * * find the optimal solution @xmath275 of problem ( [ eq : plbequ ] ) under fixed @xmath276 s using e.g. , interior point method . let @xmath225 . * * return to step 1 until @xmath277 * * or the maximum number of iterations is reached . using similar proof as that in appendix [ sub : proof - of - theoremla ] , it can be shown that algorithm ao_b converges to the global optimal solution for problem ( [ eq : plbequ ] ) under similar assumption as in theorem [ thm : global - convergence - la ] . the average data rate @xmath236 and the mutual information @xmath276 can be estimated online in a similar way as in algorithm ao_a . problem @xmath253 is a difficult combinatorial problem and the complexity of finding the optimal abrb profile is extremely high . in this section , we illustrate the top level method for finding a good abrb profile . the detailed algorithm to solve @xmath253 is given in appendix [ sub : proof - of - theoremequpb ] . a good abrb profile can be found based on an _ interference graph _ extracted from the topology graph . [ interference graph][def : igraph]for a hetnet topology graph @xmath13 , define an undirected interference graph @xmath278 , where @xmath279 is the vertex set and @xmath280 is edge set with @xmath281 denoting the edge between @xmath282 and @xmath283 . for any @xmath284 , if @xmath285 , @xmath286 , otherwise , @xmath287 , where @xmath288 is the void set . [ fig : ifc_graph1 ] illustrates how to extract the interference graph from the topology graph using an example hetnet . given an interference graph @xmath289 for the hetnet , any two links @xmath290 having an edge ( i.e. , @xmath286 ) should not be scheduled for transmission simultaneously . on the other hand , we should `` turn on '' as many `` non - conflicting '' links as possible to maximize the spatial reuse efficiency . this intuition suggests that the optimal abrb profile is highly related to the _ maximal independent set _ of the interference graph @xmath289 . an example of extracting the interference graph from the hetnet topology graph , width=332 ] [ maximal independent set ( mis)]a subset @xmath291 of @xmath292 is an _ independent set _ of @xmath278 if @xmath293 . maximal independent set _ ( mis ) is an independent set that is not a proper subset of any other independent set . for any mis @xmath291 , define @xmath294 as the _ maximal independent macro bs set _ corresponding to @xmath291 . let @xmath295 denote the set of all miss of @xmath289 . for example , the set of all miss of the interference graph in fig . [ fig : ifc_graph1 ] is @xmath296 . define a set @xmath297 then the top level flow of finding a good abrb profile is summarized in table . [ tab : flowpb ] . in step 2 , we need to find a set of miss @xmath298 . for the example in fig . [ fig : ifc_graph1 ] , @xmath299 has a unique element given by @xmath300 and thus @xmath301 . in step 3 , the mapping from @xmath302 to the abrb profile @xmath303 is @xmath304 , where @xmath305 $ ] with @xmath306 and @xmath307 . for example , for the hetnet in fig . [ fig : ifc_graph1 ] , we have @xmath301 and thus @xmath308 , where @xmath309 $ ] , @xmath310 $ ] and @xmath311 $ ] . since @xmath312 in this case , @xmath303 can be reduced to @xmath313,\left[0,1,0\right]\right\ } $ ] . .[tab : flowpb]top level flow of finding a good abrb profile [ cols= " < " , ] fig . [ fig : conva ] shows the utility @xmath314 in ( [ eq : plare ] ) of the type a users versus the number of super - frames . the utility increases rapidly and then approaches to a steady state after only @xmath315 updates . the figure demonstrates a fast convergence behavior of algorithm ao_a . similar convergence behavior was also observed for algorithm ao_b and the simulation result is not shown here due to limited space . utility of the type a users versus the number of super - frames , width=321 ] we compare the complexity of the baselines and proposed rrm algorithms . the complexity can be evaluated in the following 3 aspects . \1 ) for the short term user scheduling , the proposed scheme and baseline 1 - 3 have the same complexity order of @xmath316 , while the baseline 4 has a complexity of @xmath317 @xcite , where @xmath318 is a proportionality constant that corresponds to some matrix and vector operations with dimension @xmath319 , and @xmath320 is the number of bss in each cooperative cluster . \2 ) for the long term abrb control variables @xmath185 and @xmath263 , as they are updated by solving standard convex optimization problems in step 2 of algorithm ao_a and ao_b respectively , the complexities are polynomial w.r.t . the number of the associated optimization variables . specifically , for control variable @xmath185 , the complexity is polynomial w.r.t . the number of macro bss @xmath0 . for control variable @xmath321 , the complexity is polynomial w.r.t . the size of the abrb profile @xmath303 : @xmath322 . in addition , they are only updated once in each super - frame . \3 ) the abrb profile @xmath303 is computed using algorithm b2 in every several super - frames to adapt to the large scale fading . in step 1 of algorithm b2 , the complexity of solving the convex problem ( [ eq : fixthetapdet ] ) is polynomial w.r.t . @xmath323 . in step 2 of algorithm b2 , if the mwis algorithm in @xcite is used to solve problem ( [ eq : mwisp ] ) , the complexity is @xmath324 , where @xmath325 is the number of edges in the interference graph @xmath278 . we propose a two - timescale hierarchical rrm for hetnet with dynamic abrb . to facilitate structural abrb design for cross - tier and co - tier interference , the @xmath10 subbands are partitioned into type a and type b subbands . consequently , the two timescale rrm problem is decomposed into subproblems @xmath136 and @xmath137 which respectively optimizes the abrb control and user scheduling for the type a and type b subbands . both subproblems involve non - trivial multi - stage optimization with exponential large solution space w.r.t . the number of macro bss @xmath0 . we exploit the sparsity in the hetnet interference graph and derive the structural properties to reduce the solution space . based on that , we propose two timescale ao algorithm to solve @xmath136 and @xmath137 . the overall solution is asymptotically optimal at high snr and has low complexity , low signaling overhead as well as robust w.r.t . latency of backhaul signaling . define the average rate region as @xmath326\in\mathbb{r}_{+}^{k}:\ : x_{k}\le\overline{r}_{k}\left(\lambda\right),\forall k\right\ } .\ ] ] for any utility function that is concave and increasing w.r.t . to the average data rates @xmath327 , the optimal policy of @xmath328 must achieve a pareto boundary point of @xmath329 . hence , we only need to show that any pareto boundary point of @xmath329 can be achieved by a symmetric policy . define the average rate region under fixed @xmath134 as @xmath330\in\mathbb{r}_{+}^{k}:\\ & & x_{k}\le\overline{r}_{k}\left(\left\ { q_{s},\left\ { q_{m}\right\ } , \left\ { \pi_{m}\right\ } \right\ } \right),\forall k\bigg\}.\end{aligned}\ ] ] then we only need to show that any pareto boundary point of @xmath331 can be achieved by a symmetric policy @xmath110 . define the average mutual information region _ _ for subband @xmath115 as : @xmath332_{\forall k\in\mathcal{u}_{a}}\in\mathbb{r}_{+}^{\left|\mathcal{u}_{a}\right|}:\nonumber \\ & & x_{k}\le\overline{\mathcal{i}}_{m , k}\left(q_{m},\pi_{m}\right),\forall k\in\mathcal{u}_{a}\bigg\}.\label{eq : ratereg1g-1}\end{aligned}\ ] ] define the average mutual information region _ _ for subband @xmath333 as : @xmath332_{\forall k\in\mathcal{u}_{b}}\in\mathbb{r}_{+}^{\left|\mathcal{u}_{b}\right|}:\nonumber \\ & & x_{k}\le\overline{\mathcal{i}}_{m , k}\left(q_{m},\pi_{m}\right),\forall k\in\mathcal{u}_{b}\bigg\}.\label{eq : ratereg2g-1}\end{aligned}\ ] ] it can be verified that @xmath334 is a convex region in @xmath335 and @xmath336 is a convex region in @xmath337 . moreover , due to the statistical symmetry of the subbands , we have @xmath338 let @xmath339 and @xmath340 . from the convexity of @xmath341 and ( [ eq : ra]-[eq : rb ] ) , we have @xmath342 hence , for any pareto boundary point @xmath343^{t}$ ] of @xmath331 , @xmath344_{k\in\mathcal{u}_{a}}\in\mathbb{r}_{+}^{\left|\mathcal{u}_{a}\right|}$ ] is a pareto boundary point of @xmath345 and @xmath346_{k\in\mathcal{u}_{b}}\in\mathbb{r}_{+}^{\left|\mathcal{u}_{b}\right|}$ ] is a pareto boundary point of @xmath347 . due to the statistical symmetry of the subbands , there exists an abrb control policy and a user scheduling policy @xmath348 such that @xmath349 can be achieved for all subbands @xmath115 . similarly , there exists an abrb control policy and a user scheduling policy @xmath350 such that @xmath351 can be achieved for all subbands @xmath333 . hence , @xmath352 can be achieved using the symmetric policy @xmath353 . this completes the proof . [ [ structural - properties - of - mathcalp_a - for - general - casessubspageneral ] ] structural properties of @xmath136 for general cases[sub : spageneral ] ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 1 . @xmath167 , we have @xmath358 and @xmath359 @xmath360 . the same is true if we replace @xmath154 with @xmath159 . 2 . for given abrb patterns @xmath170 and user scheduling vector @xmath361 , there exists @xmath362 such that @xmath363 . by definition [ def : inuser ] , there is no inter - cell interference for macro cell n - users . by the definition of @xmath364 , a pico cell i - user @xmath357 can not be scheduled for transmission if any of the neighbor macro bss in @xmath365 is transmitting data subframe ( i.e. , the current abrb pattern @xmath366 ) . on the other hand , if all of the neighbor macro bss is transmitting abrb ( i.e. , the current abrb pattern @xmath161 ) , the interference from the macro bss is negligible . finally , by definition [ def : inuser ] , there is no inter - cell interference for pico cell n - users . then lemma 1 follows straightforwardly from the above analysis and the definition of @xmath367 . [ rem : usefulassumptions]note that in the proof of lemma [ lem : sysman ] , we have used the fact that the inter - cell interference seen at a n - user is negligible . we have also used assumption [ asm : picoiedge ] , which states that the sets of neighbor macro bss of the pico cell i - users belonging to the same pico cell are identical . [ policy space reduction for @xmath130][thm : absreduce]given a marginal probability vector that each macro bs is transmitting abrb @xmath369 , the optimal abrb control policy of @xmath147 conditioned on has the following synchronous abrb structure : by lemma [ lem : sysman ] , for given marginal probabilities @xmath185 , the average mutual information region will be maximized if we maximize @xmath189 for all @xmath382 . for @xmath383 , we have @xmath192 . for @xmath384 , we have @xmath385 , where the equality holds if and only if @xmath130 has the structure in theorem [ thm : absreduce ] . this completes the proof . define the _ achievable mutual information region _ for subband @xmath390 as @xmath391_{\forall k\in\mathcal{u}_{a}}:x_{k}\le\overline{\mathcal{i}}_{m_{a},k}\left(q_{a},\pi_{a}\right)\right\ } , \label{eq : ratereg1g}\ ] ] where @xmath392 . it can be verified that @xmath393 is a convex region in @xmath335 . since the utility function @xmath138 is concave and increasing w.r.t . @xmath394 , the optimal policy @xmath395 must achieve a pareto boundary point of @xmath393 . for given abrb pattern @xmath160 and bs @xmath22 , define a region as @xmath396_{\forall k\in\mathcal{u}_{a}\cap\mathcal{u}_{n}}:\ : x_{k}\le i_{m_{a},k}\left(\pi_{a},\mathbf{a}\right)\right\ } .\ ] ] it can be verified that @xmath397 is a convex region . from lemma [ lem : sysman ] , we have @xmath398 for convenience , define @xmath399 then @xmath400 , we have @xmath401 . from ( [ eq : rma]-[eq : rmabar ] ) and the fact that @xmath402_{\forall k\in\mathcal{u}_{a}}$ ] is a pareto boundary point of @xmath393 , it follows that @xmath403_{\forall k\in\mathcal{u}_{a}\cap\mathcal{u}_{n}}$ ] is a pareto boundary point of @xmath404 and @xmath405_{\forall k\in\mathcal{u}_{a}\cap\mathcal{u}_{n}}$ ] is a pareto boundary point of @xmath406 . hence , there exists user scheduling policy @xmath407 satisfying @xmath408 and @xmath409 for all @xmath410 . then it follows that @xmath402_{\forall k\in\mathcal{u}_{a}}$ ] can be achieved by the control policy @xmath411 . [ [ proof - of - corollary - corequivalent - problem - pa - subproof - of - corollaypa ] ] proof of corollary [ cor : equivalent - problem - pa ] [ sub : proof - of - corollaypa ] ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ the first part of the corollary follows straightforward from theorem [ thm : absreduce ] and [ clm : policy - space - reductionpia ] . we only need to prove that problem ( [ eq : plare ] ) is bi - convex . the average mutual information in ( [ eq : uaobj ] ) can be expressed as @xmath412 where @xmath413 , @xmath414 , @xmath415 is the set of macro cell n - users , @xmath416 is the set of pico cell n - users , and @xmath417 is the set of pico cell i - users . it is easy to verify that @xmath418 is a concave function w.r.t . @xmath185 for fixed @xmath131 . using the vector composition rule for concave function @xcite , the objective in ( [ eq : plare ] ) is also concave w.r.t . @xmath185 and thus problem ( [ eq : plare ] ) is convex w.r.t . @xmath185 for fixed @xmath131 . for fixed @xmath185 , @xmath418 is a linear function of the user scheduling variables @xmath419 . hence problem ( [ eq : plare ] ) is also convex w.r.t . @xmath131 for fixed @xmath185 . [ [ proof - of - theorem - thmglobal - convergence - lasubproof - of - theoremla ] ] proof of theorem [ thm : global - convergence - la][sub : proof - of - theoremla ] ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ it is clear that @xmath420 . by the assumption in theorem [ thm : global - convergence - la ] , we have @xmath421 if @xmath422 is not a fixed point of @xmath229 . combining the above and the fact that @xmath314 is upper bounded , ao_a must converge to a fixed point @xmath235\in\delta$ ] . the rest is to prove that any @xmath235\in\delta$ ] is globally optimal for problem ( [ eq : plare ] ) . note that problem ( [ eq : plare ] ) is equivalent to the problem @xmath423_{\forall k\in\mathcal{u}_{a}}\in\mathcal{r}_{m_{a}}.\label{eq : equplapre}\ ] ] since the objective in ( [ eq : equplapre ] ) is a concave function w.r.t . @xmath424 , and @xmath393 is a convex region , the following lemma holds . [ optimality condition for ( [ eq : plare])][lem : opt - cond - pa]a solution @xmath425 , @xmath209 is optimal for problem ( [ eq : plare ] ) if and only if its average mutual information @xmath426 satisfies @xmath427_{\forall k\in\mathcal{u}_{a}}\in\mathcal{r}_{m_{a}}\ ] ] according to the step 1 of ao_a , for any abrb pattern @xmath428 and csi realization @xmath355 , the user scheduling vector @xmath429 of bs @xmath22 under @xmath430 is the optimal solution of @xmath431 where @xmath432 and @xmath433 is the average mutual information under @xmath434 . combining ( [ eq : optpiatuta ] ) and the fact that @xmath435 is the optimal solution of problem ( [ eq : plare ] ) with fixed @xmath430 , we have @xmath436 . this implies that @xmath434 satisfy the optimality condition in lemma [ lem : opt - cond - pa ] , and thus is the globally optimal solution . * * step 1 * * ( update the coefficients @xmath442 ) : for fixed @xmath443 , obtain the optimal solution @xmath444_{j=1, ... ,\left|\theta^{(i)}\right|}$ ] of the following convex optimization problem @xmath445,\forall j\:\textrm{and}\:\sum_{j=1}^{\left|\theta^{(i)}\right|}\check{q}_{j}=1,\nonumber \end{aligned}\ ] ] where @xmath446_{j=1, ... ,\left|\theta^{(i)}\right|}$ ] , and @xmath447 is the @xmath257 mis in @xmath443 . [ asymptotically optimal abrb profile][thm : asymptotic - equivalence - ofpb]algorithm b2 always converges to an abrb profile @xmath303 with @xmath322 . furthermore , the converged result is asymptotically optimal for high snr . i.e. @xmath456 , where @xmath457 for some positive constants @xmath458 s , and @xmath459 is the optimal objective value of @xmath249 . consider problem @xmath460 which is the same as @xmath249 except that there are two differences : 1 ) the fading channel @xmath461 is replaced by a _ deterministic channel _ with the channel gain between bs @xmath22 and user @xmath23 given by the corresponding large scale fading factor @xmath462 ; 2 ) an additional constraint is added to the user scheduling policy such that any two links @xmath290 having an edge ( i.e. , @xmath286 ) in the interference graph @xmath289 can not be scheduled for transmission simultaneously . it can be shown that the optimal solution of problem @xmath460 is asymptotically optimal for @xmath249 at high snr . moreover , using the fact that the achievable mutual information region in the deterministic channel is a convex polytope with @xmath463 as the set of _ pareto boundary _ vertices , it can be shown that @xmath460 is equivalent to the following problem @xmath464 where @xmath465 is the @xmath257 mis in @xmath448 . to complete the proof of theorem [ thm : asymptotic - equivalence - ofpb ] , we only need to further prove that algorithm b2 converges to the optimal solution of problem ( [ eq : pdet ] ) . using the fact that any point in a @xmath466-dimensional convex polytope can be expressed as a convex combination of no more than @xmath254 vertices , it can be shown that there are at most @xmath254 non - zero elements in @xmath467 in step 1 of algorithm b2 . hence @xmath468 . moreover , it can be verified that @xmath469 if @xmath443 is not optimal for ( [ eq : pdet ] ) . combining the above and the fact that @xmath470 is upper bounded by @xmath471 , algorithm b2 must converge to the optimal solution of ( [ eq : pdet ] ) . this completes the proof . in step 2 of algorithm b2 , problem ( [ eq : mwisp ] ) is equivalent to finding a _ maximum weighted independent set _ ( mwis ) in the interference graph @xmath289 with the weights of the vertex nodes given by @xmath472 . the mwis problem has been well studied in the literature @xcite . although it is in general np hard , there exists low complexity algorithms for finding near - optimal solutions @xcite . although the asymptotic global optimality of algorithm b2 is not guaranteed when step 2 is replaced by a low complexity solution of ( [ eq : mwisp ] ) , we can still prove its monotone convergence . d. gesbert , s. kiani , a. gjendemsj _ et al . _ , `` adaptation , coordination , and distributed resource allocation in interference - limited wireless networks , '' _ proceedings of the ieee _ , vol . 95 , no . 12 , pp . 23932409 , 2007 . a. l. stolyar and h. viswanathan , `` self - organizing dynamic fractional frequency reuse in ofdma systems , '' in _ infocom 2008 . the 27th conference on computer communications . ieee_.1em plus 0.5em minus 0.4emieee , 2008 , pp . 691699 . r. irmer , h. droste , p. marsch , m. grieger , g. fettweis , s. brueck , h. mayer , l. thiele , and v. jungnickel , `` coordinated multipoint : concepts , performance , and field trial results , '' _ ieee communications magazine _ , vol . 49 , no . 2 , pp . 102111 , 2011 . q. shi , m. razaviyayn , z .- q . luo , and c. he , `` an iteratively weighted mmse approach to distributed sum - utility maximization for a mimo interfering broadcast channel , '' _ ieee trans . signal processing _ 59 , no . 9 , pp . 43314340 , sept . 2011 . y. wang and k. i. pedersen , `` performance analysis of enhanced inter - cell interference coordination in lte - advanced heterogeneous networks , '' in _ vehicular technology conference ( vtc spring ) , 2012 ieee 75th_. 1em plus 0.5em minus 0.4emieee , 2012 , pp . j. pang , j. wang , d. wang , g. shen , q. jiang , and j. liu , `` optimized time - domain resource partitioning for enhanced inter - cell interference coordination in heterogeneous networks , '' in _ wireless communications and networking conference ( wcnc ) , 2012 ieee_.1em plus 0.5em minus 0.4emieee , 2012 , pp . 16131617 . m. hong , r .- y . sun , h. baligh , and z .- q . luo , `` joint base station clustering and beamformer design for partial coordinated transmission in heterogenous networks , '' 2012 . [ online ] . available : http://arxiv.org/abs/1203.6390 r. ghaffar and r. knopp , `` fractional frequency reuse and interference suppression for ofdma networks , '' in _ proceedings of the 8th international symposium on modeling and optimization in mobile , ad hoc and wireless networks _ , 2010 , pp . 273277 . o. somekh , o. simeone , y. bar - ness , a. haimovich , and s. shamai , `` cooperative multicell zero - forcing beamforming in cellular downlink channels , '' _ ieee trans . inf . theory _ 55 , no . 7 , pp . 32063219 , 2009 .

interference is a major performance bottleneck in heterogeneous network ( hetnet ) due to its multi - tier topological structure . we propose almost blank resource block ( abrb ) for interference control in hetnet . when an abrb is scheduled in a macro bs , a resource block ( rb ) with blank payload is transmitted and this eliminates the interference from this macro bs to the pico bss . we study a two timescale hierarchical radio resource management ( rrm ) scheme for hetnet with _ dynamic abrb _ control . the long term controls , such as dynamic abrb , are adaptive to the large scale fading at a rrm server for co - tier and cross - tier interference control . the short term control ( user scheduling ) is adaptive to the local channel state information within each bs to exploit the _ multi - user diversity_. the two timescale optimization problem is challenging due to the exponentially large solution space . we exploit the sparsity in the _ interference graph _ of the hetnet topology and derive structural properties for the optimal abrb control . based on that , we propose a _ two timescale alternative optimization _ solution for the user scheduling and abrb control . the solution has low complexity and is asymptotically optimal at high snr . simulations show that the proposed solution has significant gain over various baselines . heterogeneous network , dynamic abrb control , two timescale rrm

You are an expert at summarizing long articles. Proceed to summarize the following text: in comparison with stationary black holes , understanding of dynamical black holes is still far from clear . although there is a lot of potentially interesting subjects such as thermodynamical properties , dynamical stability , or hawking radiation , the absence of the preferred time direction makes them intractable . also , lack of concrete exact solutions in simple systems is one of the main reasons for the relatively slow progress . in the present paper , among others , we focus on asymptotically anti - de sitter ( ads ) dynamical black holes . in the history of gravitation physics , ads black holes had been considered unrealistic and eccentric configurations for a long time . however , they stepped into the limelight by the discovery of the ads / cft duality @xcite . now ads black holes have new significance as a stage to study strongly coupled gauge theories and occupy a central position of research in string theory . the motivation of the study in the present paper is twofold . one comes from the ads / cft duality . while a static ads black hole corresponds to the field theory at the boundary which has finite temperature in equilibrium , a dynamical ads black hole would correspond to some field theory in the non - equilibrium state . actually , an asymptotically ads spacetime has been studied in a dynamical setting as a holographic dual to the bjorken flow @xcite . while the dynamical spacetime in @xcite was constructed perturbatively , exact dynamical ads black holes are desirable to derive more specific results . the second motivation comes from the recently - found dynamical instability of the ads spacetime . although the ads vacuum is known to be stable at the linear level , its nonlinear instability was numerically found with a massless klein - gordon field in arbitrary dimensions @xcite . ( see also @xcite . ) it was both numerically and analytically supported that an ads black hole forms as a result of this instability @xcite . however , there is an argument that static ads black holes are also unstable at the nonlinear level @xcite . ( see also @xcite . ) therefore , the final fate of the instability of the ads vacuum or a static ads black hole is still not clear at present . in this context , not only a static configuration but also a dynamical configuration is the candidate of the final state . an example is an oscillating or time - periodic spacetime @xcite . therefore , an exact dynamical black - hole solution might represent the final state or an intermediate stage during the time evolution and must be useful for further study . in the present paper , we consider spacetimes with spherical , plane , or hyperbolic symmetry in arbitrary dimensions . it is well - known in this system that the no - hair theorem holds for a wide class of scalar fields , which prohibits asymptotically flat black holes with non - trivial configuration of scalar fields @xcite . here one assumes staticity to prove the no - hair theorem . for a massless klein - gordon field , even a stronger result is available , namely the no - hair theorem independent of the asymptotic condition for the spacetime and the value of @xmath0 . ( see appendix a for the proof . ) as a result , all the known solutions with a non - trivial scalar field in this system contain naked singularities both for @xmath1 @xcite and @xmath2 @xcite , and the only possible static black hole is the schwarzschild(-(a)ds ) black hole with a constant scalar field or its topological generalization . therefore , in order to obtain non - trivial black - hole solutions , one has to remove the assumption of staticity . in four dimensions , a set of exact dynamical and inhomogeneous solutions has been obtained by many authors @xcite . in the present paper , we generalize this set of solutions and show that some of the solutions describe a locally ads dynamical black hole . in the case where the klein - gordon field is purely imaginary , namely ghost , an ads dynamical wormhole may be realized . in the following section , we give our spacetime ansatz and present the solutions . in sec . iii , we show that the class - i solution represents an ads dynamical black hole or wormhole . in sec . iv , we discuss the properties of other classes of solutions . concluding remarks are summarized in sec . v. the scalar no - hair theorem for a massless klein - gordon field is shown in appendix a. in appendix b , we present the counterpart of our solution in the case without a cosmological constant . in appendix c , it is shown that the class - i solution with a real scalar field does not represent a wormhole . our basic notation follows @xcite . the convention for the riemann curvature tensor is @xmath3v^\mu = { { \cal r}^\mu } _ { \nu\rho\sigma}v^\nu$ ] and @xmath4 . the minkowski metric is taken to be mostly plus sign , and greek indices run over all spacetime indices . we adopt the units such that only the @xmath5-dimensional gravitational constant @xmath6 is retained . we consider the einstein - klein - gordon-@xmath0 system in arbitrary @xmath7 dimensions . the field equations are @xmath8 and @xmath9 , where @xmath10 is defined by the @xmath5-dimensional gravitational constant @xmath6 as @xmath11 . the energy - momentum tensor for the klein - gordon field is @xmath12 in this paper , we consider an @xmath5-dimensional warped product manifold @xmath13 with the line element @xmath14 where @xmath15 is a lorentzian metric on @xmath16 and @xmath17 is a scalar on @xmath16 . @xmath18 is an @xmath19-dimensional unit space of constant curvature , where @xmath20 denotes its curvature taking the values @xmath21 , @xmath22 , and @xmath23 , and @xmath24 is the metric on @xmath18 . namely the riemann tensor on @xmath18 is given by @xmath25 where the superscript @xmath19 means the geometrical quantity on @xmath18 . the generalized misner - sharp quasi - local mass is a scalar on @xmath16 defined by @xmath26,\end{aligned}\ ] ] where @xmath27 $ ] , @xmath28 is the covariant derivative on @xmath16 and @xmath29 @xcite . @xmath30 denotes the volume of @xmath18 if it is compact and otherwise arbitrary . @xmath31 has the monotonicity and positivity properties for arbitrary ( positive ) @xmath30 and is constant in vacuum @xcite . in the asymptotically flat or ads case , that coefficient is fixed in such a way that it converges to the global mass such as the arnowitt - deser - misner mass @xcite or abbott - deser mass @xcite . in the non - static spacetime , there is no timelike killing vector to define a natural time - slicing . in such a case , the kodama vector @xmath32 defines a preferred time direction in the untrapped region , where @xmath33 and @xmath34 is a volume element of @xmath35 @xcite . the kodama vector is timelike ( spacelike ) in the untrapped ( trapped ) region defined by @xmath36 . on the trapping horizon defined by @xmath37 , the kodama vector becomes null . in the present paper , we consider solutions in the following form : @xmath38 , \label{g - sussman } \\ h(\rho)= & \left\ { \begin{array}{ll } \displaystyle{\sqrt{-{\tilde \lambda}}\sin \rho } & \mbox{[class - i~($\lambda<0$)]},\\ \displaystyle{\sqrt{-{\tilde \lambda}}\rho } & \mbox{[class - ii~($\lambda<0$)]},\\ \displaystyle{\sqrt{-{\tilde \lambda}}\sinh \rho } & \mbox{[class - iii~($\lambda<0$)]},\\ \displaystyle{\sqrt{{\tilde \lambda}}\cosh \rho } & \mbox{[class - iii~($\lambda>0$)]}. \end{array } \right . \end{aligned}\ ] ] the physical domain is the region with @xmath39 . the areal radius is given by @xmath40 , where @xmath41 is chosen such that @xmath17 is non - negative . these classes of solutions have been investigated as solutions with a stiff fluid , which is equivalent to a massless klein - gordon field if the gradient of the scalar field is timelike @xcite . they were first obtained by lake @xcite for @xmath42 and @xmath43 and independently obtained by other authors @xcite . the global structure and physical properties were investigated in @xcite . the solutions with @xmath42 and general @xmath20 were obtained by collins and lang @xcite and also in @xcite . keeping in mind this history , we call these classes of solutions the _ generalized lake solution _ in the present paper . the system reduces to the following master equation for @xmath44 : @xmath45 where a dot denotes the derivative with respect to @xmath46 and the constant @xmath47 is @xmath21 , @xmath22 , and @xmath23 for class - i , -ii , and -iii , respectively . the klein - gordon field is given as @xmath48 the scalar field is homogeneous @xmath49 in our coordinate system . the energy - momentum tensor has the form of @xmath50 , where @xmath51 is the energy density of the scalar field . important physical quantities are given as @xmath52 where a prime denotes the derivative with respect to @xmath53 . @xmath54 and @xmath55 are satisfied in the spherically symmetric class - i solution ( @xmath56 ) . using @xmath57 , the kodama vector is written as @xmath58 the master equation ( [ master ] ) is solved analytically in three and four dimensions for any @xmath20 but only for @xmath59 in higher dimensions , which will be presented later . in order to see the qualitative property of the solution , we introduce a new variable @xmath60 and write the master equation as @xmath61 this equation is integrated by parts to give @xmath62 where @xmath63 is an integration constant . this is a simple one - dimensional potential problem for the variable @xmath64 . using the following expression , @xmath65 we obtain simple expressions of @xmath66 , @xmath67 and @xmath31 : @xmath68 it is clear that the energy density of the scalar field and the quasi - local mass are positive ( negative ) for @xmath69 and then the scalar field is real ( purely imaginary , namely ghost ) . in three dimensions , the scalar field becomes trivial and we have @xmath70 and @xmath71constant . the spacetime is then locally ( a)ds . it is shown that , in the case of @xmath72 , @xmath73 corresponds to ads infinity . ( in contrast , @xmath74 can not be zero for @xmath75 in the class - iii solution . ) indeed , @xmath76 is satisfied . we actually show that the affine parameter @xmath77 blows up at @xmath78 along null geodesics . in the spacetime ( [ g - sussman ] ) , there is a conformal killing vector @xmath79 satisfying the conformal killing equation : @xmath80 along a null geodesic , with its tangent vector @xmath81 , there is a conserved quantity @xmath82 . the following expression @xmath83 is integrated to give @xmath84},\\ \displaystyle{{\tilde \lambda}\rho } & \mbox{[class - ii]},\\ \displaystyle{{\tilde \lambda}\tanh \rho } & \mbox{[class - iii~($\lambda<0$)]},\\ \displaystyle{{\tilde \lambda}/\tanh \rho } & \mbox{[class - iii~($\lambda>0$)]}. \end{array } \right . \end{aligned}\ ] ] therefore , @xmath85 for @xmath72 corresponds to @xmath86 . ads infinity is given by @xmath87 in the class - ii and -iii solutions , and by @xmath88 in the class - i solution , where @xmath89 is an integer . it is seen in eqs . ( [ mu2 ] ) and ( [ m2 ] ) that the spacetime is indeed vacuum at ads infinity ( @xmath85 ) , but @xmath31 blows up there . the quasi - local mass @xmath31 with @xmath43 converges to the abbott - deser mass in the asymptotically ads spacetime @xcite under the henneaux - teitelboim fall - off condition @xcite . ( see @xcite for the higher - dimensional version . ) its contraposition means that if @xmath31 blows up , the fall - off rate is slower than the henneaux - teitelboim condition and the spacetime is asymptotically only locally ads @xcite . there is a static solution @xmath90 of the master equation ( [ master ] ) in the case of @xmath91 in four and higher dimensions : @xmath92 where @xmath93 is constant . the energy density and the quasi - local mass are given by @xmath94 while the metric is static , the scalar field is time - dependent . in the class - iii solution with @xmath43 , the klein - gordon field is real , while it is ghost in the class - i solution with @xmath95 . we do nt present the detailed analysis for this static solution , but the penrose diagram is fig . 1(d ) for the class - i solution with @xmath95 and the solution represents a static ads wormhole . ( see @xcite for the wormhole solution without @xmath0 . ) the penrose diagram for the class - iii solution with @xmath43 is fig . 2(a ) for @xmath72 and fig . 2(f ) for @xmath75 . hereafter we do nt consider the static case . we are interested in the class - i solution because the coordinate system covers the maximally extended spacetime and describes an asymptotically locally ads black hole or wormhole . in four dimensions , @xmath44 is given by @xmath96 where @xmath97 is a constant . the energy density and the quasi - local mass are given by @xmath98 the energy density is positive ( negative ) for @xmath99 . the ads vacuum is realized for @xmath100 with @xmath101 . ( for @xmath59 , @xmath102 is not allowed since it gives @xmath103 . ) the scalar field with positive energy density is given by @xmath104},\\ \displaystyle{\sqrt{\frac{1}{2\kappa_4 ^ 2}}\ln\biggl|\frac{1-\cos 2t}{\sin 2t}\biggl| } & \mbox{[for $ k=0$]}. \end{array } \right . \end{aligned}\ ] ] the scalar field with negative energy density is given as @xmath105 where @xmath106 . in arbitrary dimensions , @xmath44 and @xmath66 for the class - i solution are given in closed forms only for @xmath59 : @xmath107^{2/(n-2)},\\ \pm(\phi-\phi_0)=&\sqrt{\frac{n-3}{(n-2)\kappa_n^2}}\ln\biggl|\frac{1-\cos(n-2)t}{\sin(n-2)t}\biggl|.\end{aligned}\ ] ] the energy density and the quasi - local mass are given by @xmath108 ^ 2},\\ m = & \frac{(n-2)v_{n-2}^{(0)}c_1^{(n-1)/2}}{2\kappa_n^2(\varepsilon h)^{n-3}[\sin(n-2)t]^{(n-3)/(n-2)}}.\end{aligned}\ ] ] in three dimensions ( @xmath109 ) , we obtain @xmath110 and @xmath71constant and the solution represents a baados - teitelboim - zanelli black hole in the non - standard coordinates @xcite . it is not difficult to understand the causal structure of the spacetime ( [ g - sussman ] ) . @xmath111 corresponds to curvature singularity , of which existence depends on the parameters . since the metric on @xmath112 in the solution ( [ g - sussman ] ) is conformally flat , a light ray runs along a 45-degree straight line in the @xmath113-plane . the penrose diagrams for this solution are presented in fig . [ adsbh ] . ( see table [ table : ads ] . ) the spacetime represents a dynamical black hole or wormhole depending on the parameters . portions of the @xmath113 plane covering the maximally extended spacetime of the class - i solution . the corresponding penrose diagrams have the same structures . a zigzag and a thick line correspond to a curvature singularity and ads infinity , respectively . bheh stands for the black - hole event horizon . figures . ( a)(c ) represent a black hole , while fig . ( d ) represents a wormhole . ] .[table : ads ] the penrose diagrams for the class - i solution with positive energy density and @xmath114 . in the case of @xmath115 with negative energy density , the penrose diagram is fig . 1(d ) . [ cols="<,^,^",options="header " , ] the corresponding penrose diagrams are drawn in fig . 2 . in those diagrams , the regular coordinate boundary @xmath116 , which consists of null hypersurfaces in fig . 2 drawn by dashed lines , is extendable . this is obvious by eq . ( [ geodesic ] ) since @xmath116 corresponds to a finite value of the affine parameter @xmath77 . furthermore , @xmath117 and @xmath118 are satisfied for @xmath116 along null geodesics . this fact indicates that a variety of @xmath119 extension is possible beyond this coordinate boundary without introducing any matter field on the junction surface . one possible extension is to attach an exact ( a)ds spacetime @xcite . although @xmath116 is regular and extendable along null geodesics , it is singular along spacelike curves with @xmath120constant , where @xmath67 blows up . therefore , although the regions ( ii ) in fig . 2(e ) and ( j ) are maximally extended , there is no regular cauchy surface . the regions ( ii ) in figs . 2(b ) and 2(e ) and ( iii ) in fig . 2(e ) contain a black - hole event horizon . the penrose diagrams for the class - ii and -iii solutions . figures ( a)(e ) [ ( f)(j ) ] correspond to the case with @xmath121 . figures ( c ) , ( d ) , ( h ) , and ( i ) correspond to the special case of the class - iii(c ) solution with @xmath122 . coordinate boundaries represented by dashed lines are extendable . the @xmath113 plane is divided by singularities and ads infinity into several portions and each of them represents a distinct spacetime . the parameters are required to give @xmath39 for the physical spacetime and the left halves of figs . ( a)(e ) are equivalent to the right halves . the region ( ii ) in fig . ( e ) is maximally extended and contains an event horizon , but there is no regular cauchy surface . ] we have presented a set of exact solutions in the einstein - klein - gordon system with a cosmological constant in arbitrary dimensions . the spacetime has spherical , plane , or hyperbolic symmetry and admits a spatially conformal killing vector . the solution is obtained in a closed form in three and four dimensions for any @xmath20 but only for @xmath59 in higher dimensions . even in the case without the explicit form , it is able to understand the qualitative properties of the solutions by analyzing the equivalent one - dimensional potential problem . in three dimensions , the solution reduces to the locally ( a)ds vacuum . for @xmath72 , the spacetime is asymptotically locally ads . the quasi - local mass blows up at ads infinity while the energy density converges to zero , which infers the slow fall - off to the ads infinity . some of the solutions admit a black - hole event horizon . in the class - i solution , the coordinate system covers the maximally extended spacetime and the solution with a real scalar field describes a dynamical ads black hole . if the scalar field is ghost , the solution represents a dynamical ads wormhole . while the solution with @xmath95 in four dimensions represents the dynamical formation of a black hole , the black hole is eternal in the case of @xmath43 in four dimensions and @xmath59 in @xmath123 dimensions . it is still not clear whether the black hole is eternal or not in the case of @xmath124 in higher dimensions . for the class - ii and -iii solutions , we have analyzed the global structures in four dimensions and in higher dimensions with @xmath59 . the regular coordinate boundary is extendable and the @xmath119 extension beyond it would be possible . there are several cases where the spacetime contains a black - hole event horizon , however , the coordinate system does not cover the maximally extended spacetime or there is no regular cauchy surface in the spacetime . in summary , the class - i solution may be a good model of a dynamical ads black hole for further investigations . thermodynamical properties , dynamical stability , and the hawking radiation are interesting subjects to study , of which results will shed light on dynamical properties of ads black holes . the author thanks julio oliva , takashi torii , kei - ichi maeda , cristin martnez , and especially fabrizio canfora for useful comments and discussions . this work has been funded by the fondecyt grants 1100328 , 1100755 ( hm ) and by the conicyt grant `` southern theoretical physics laboratory '' act-91 . this work was also partly supported by the jsps grant - in - aid for scientific research ( a ) ( 22244030 ) . the centro de estudios cientficos ( cecs ) is funded by the chilean government through the centers of excellence base financing program of conicyt . in this appendix , we present a simple proof of the no - hair theorem for a klein - gordon field ; there is no killing horizon in the spacetime represented by the metric ( [ sol2 ] ) if the spacetime is static and the klein - gordon field is static and inhomogeneous . we note that this result is independent of the value of @xmath0 and the asymptotic condition for the spacetime . we adopt the following coordinates for the static spacetime : @xmath125 we can replace @xmath17 by @xmath126 without loss of generality if @xmath17 is not constant . for @xmath127 , the klein - gordon equation @xmath9 gives @xmath128 where @xmath129 is a constant and a prime denotes here the derivative with respect to @xmath126 . if @xmath129 is zero , @xmath66 is constant . the trace of the einstein equation gives the expression of the ricci scalar @xmath130 : @xmath131 where we used eq . ( [ kg1 ] ) . a killing horizon is defined by @xmath132 with @xmath133 , where @xmath134 is its location satisfying @xmath135 . equation ( [ key ] ) shows @xmath136 for @xmath137 , namely a curvature singularity at @xmath134 , unless @xmath138 . therefore , the killing horizon is allowed only in the case where @xmath66 is constant . in this appendix , we present the counterpart of the generalized lake solution in the case without a cosmological constant . the conformal self - similarity naturally reduces to the homothetic self - similarity in the absence of a characteristic scale introduced by the cosmological constant . first let us derive the master equation for the system . introducing the double - null coordinates @xmath139 on @xmath35 and assuming that @xmath35 is flat and @xmath140 , where @xmath141 , the line element for the homothetic self - similar spacetime is given by @xmath142 the einstein tensor and the energy - momentum tensor for the scalar field are written as @xmath143,\\ g^{u}_{~{\bar z}}=&-\frac{n-2}{4u{\cal s}^2}({{\cal s}'}^2 - 2{\cal s}{\cal s}''),\quad g^{{\bar z}}_{~u}=0,\\ g^{{\bar z}}_{~{\bar z}}=&\frac{(n-2)(n-3)}{4u^2{\cal s}^2}(-2k{\cal s}-2{\cal s}{\cal s}'+{\bar z}{{\cal s}'}^2),\\ g^{i}_{~j}=&\frac{n-3}{4u^2{\cal s}^2}\biggl[(n-6){\bar z}{{\cal s}'}^2 - 2(n-4){\cal s}{\cal s } ' \nonumber \\ & -2k(n-4){\cal s}+4{\bar z}{\cal s}{\cal s}''\biggl]\delta^i_{~j}\end{aligned}\ ] ] and @xmath144 where a prime denotes here the derivative with respect to @xmath145 . then , the einstein equation gives the following master equation for @xmath146 : @xmath147 the solution for this master equation is obtained in a closed form for @xmath42 or @xmath59 . ( the solution for @xmath109 is vacuum and hence locally flat , as shown below . ) in the case of @xmath42 , @xmath148 is given by @xmath149 where @xmath150 and @xmath151 are constants . for @xmath152 , @xmath66 is real and given as @xmath153 for @xmath154 , @xmath66 is ghost and given by @xmath155 in the case of @xmath59 , the solution is obtained as @xmath156 it is noted that @xmath157 gives the minkowski spacetime . in the double - null coordinates , the metric and the generalized misner - sharp mass are written as @xmath158 for @xmath42 and @xmath159 for @xmath59 . this solution with @xmath43 and @xmath42 is the roberts solution @xcite . ( see also @xcite . ) the case with @xmath43 in arbitrary dimensions was first discussed in @xcite . this spacetime admits a homothetic killing vector @xmath160 satisfying @xmath161 . since @xmath35 is flat , @xmath162 with constant @xmath163 or @xmath164 with constant @xmath165 corresponds to null infinity . different from the generalized lake solution , the gradient of the scalar field may be spacelike in some domain of spacetime , where the solution is not equivalent to a solution with a stiff fluid . in this appendix , we show that the class - i solution with positive energy density does not represent a wormhole for @xmath114 . this result restricts the possible penrose diagram for the solution without the explicit form . here we define a wormhole spacetime by the existence of causal curves connecting one infinity to another and prove the non - existence of such curves . because the orbits of timelike curves or non - radial null curves run inside the light cone at a given spacetime point in the @xmath113 plane , to show the non - existence for radial null curves is sufficient . first we show that the areal radius @xmath166 blows up at ads infinity in the class - i solution for @xmath167 , which is characterized by @xmath168 and @xmath169 . it is obvious that @xmath170 holds if @xmath39 there . ads infinity where @xmath171 is satisfied is more subtle but it is also the case , as shown below . in the class - i solution with positive energy density ( @xmath172 ) , the master equation ( [ master - x ] ) shows us the behavior of @xmath173 near @xmath174 ( and hence @xmath171 ) : @xmath175 where @xmath176 is the time when @xmath171 . on the other hand , the metric function @xmath177 behaves as @xmath178 near ads infinity . because @xmath179 is satisfied along a radial null curve , @xmath180 is satisfied along such a light ray going to or coming from ads infinity with @xmath171 . along this curve , the areal radius @xmath17 behaves near ads infinity as @xmath181}}{\sqrt{-{\tilde\lambda}}|t - t_0|^{(n-3)/(n-2)}}.\end{aligned}\ ] ] therefore , the areal radius @xmath17 blows up at ads infinity . now we show that there is no radial null curve connecting two distinct ads infinity in the spacetime of the class - i solution with positive energy density . let us consider the einstein equation @xmath8 in the double null coordinates : @xmath182 the @xmath183 and @xmath184 components of the einstein equation are written as @xmath185 the null energy condition requires @xmath186 and @xmath187 . the generalized lake solution is written in the double - null coordinates by introducing @xmath165 and @xmath163 such that @xmath188 the correspondence between ( [ g - sussman ] ) and ( [ metric - double ] ) is @xmath189 if the solution represents an ads wormhole , there is a radial null ray which travels from one ads infinity ( where @xmath190 ) to the other . obviously there is at least one positive local minimum of @xmath17 along its orbit , which locally defines a wormhole throat @xcite . without loss of generality , this orbit is expressed by @xmath191 ( and @xmath192constant ) , where @xmath193 is a constant and the throat condition is then given by @xmath194 with @xmath195 . then , eq . 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we present several classes of exact solutions in the einstein - klein - gordon system with a cosmological constant . the spacetime has spherical , plane , or hyperbolic symmetry and the higher - dimensional solutions are obtained in a closed form only in the plane symmetric case . among them , the class - i solution represents an asymptotically locally anti - de sitter ( ads ) dynamical black hole or wormhole . in four and higher dimensions , the generalized misner - sharp quasi - local mass blows up at ads infinity , inferring that the spacetime is only locally ads . in three dimensions , the scalar field becomes trivial and the solution reduces to the baados - teitelboim - zanelli black hole .

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Proceed to summarize the following text: nanoplasmonics continues to receive substantial interest from various fields of research including biology @xcite , chemistry @xcite and physics @xcite , with applications ranging from renewable energy technology @xcite to homeland security by the sensitive identification of explosive material @xcite . due to the collective excitations of electrons at the surface of a metallic resonator coupling to electromagnetic fields , the local density of optical states ( ldos ) decreases rapidly for spatial positions away from the surface of the metal resonator ; so - called `` hot spots '' , where the local electric field can be enhanced by orders of magnitude in comparison to a bulk medium , are formed near the surface . the unique properties of localized surface plasmon ( lsp ) resonances , manifest in a strong confinement of electric field far below the diffraction limit and result in an exotic electromagnetic response that underlines the wide application of surface plasmons , especially in nanoscience and nanophotonics ; however , the extreme spatial localization of lsps makes them experimentally challenging for direct detection of the spatial field distribution , e.g. , by employing conventional spectroscopy techniques , since the spatial resolution is constrained by the diffraction limit . in addition , optically dark modes have a vanishing dipole moment , and traditional optical methods fail to excite such modes . due to the strong frequency dispersion and losses in metals , which satisfy the kramers - kronig relations , it is a very challenging problem to model the electromagnetic response of arbitrarily shaped metal resonators . consequently , most optical studies of plasmonic structures rely heavily on brute force numerical simulations . direct imaging of the lsp resonances is important both for revealing the exotic physics underlying these resonances and for verifying and testing numerical models used in theoretical studies . there are presently several different experimental schemes developed to detect the electromagnetic component of lsps , including near - field scanning optical microscopy ( nsom ) @xcite , leakage radiation microscopy @xcite , and two - photon induced photoluminescence @xcite , and cathodoluminescence @xcite . of particular interest , electron energy loss spectroscopy ( eels ) has been shown to be capable of accessing the subwavelength spatial variation of the surface plasmon modes of single metallic nanoresonator experimentally @xcite for almost a decade ; during this time , it has been recognized to be @xcite one of the most powerful approaches , in which high speed electron beams ( typically ranging from several tens to several hundreds of kev ) are injected and transmitted across an optically thin sample , with a spatial resolution of around 1 nm ; eels has been applied to various systems such as split ring resonator ( srr ) @xcite and single nanorod / antennas @xcite . the eels method is also quite versatile , allowing one to probe both optically bright and dark resonances over broadband frequencies , and can be used for detecting both localized and extended excitations @xcite . recently , vortex electron beams have also been experimentally demonstrated @xcite , which may find use for probing the magnetic component of lsps @xcite . modelling eels is an extremely challenging and tedious numerical problem , and there has been different theoretical approaches developed to model the eels of plasmonic resonators , including boundary element method ( bem ) @xcite , discrete dipole approximation ( dda ) @xcite , discontinuous galerkin time - domain ( dgtd ) @xcite and finite - difference time - domain ( fdtd ) methods @xcite . with sufficient care and computational resources , all of these approaches can show good agreement with experimental results , since these are basically full dipole solutions to the classical maxwell equations ; typically , these approaches employ numerous dipole point calculations in a 3d spatial grid to obtain the photon green function , which allows one to obtain the eels profile in space and frequency ; non - local effect of the conductive electrons are typically negligible until spatial positions within a few nm from the metal surface @xcite , so most studies of eels in nanoplasmonics have worked with the classical maxwell equations without any non - local effects ; however , as shown very recently , non - local effects may become important for certain metallic resonators and geometries @xcite . unfortunately , most eels calculation methods to date are computational expensive , hard to employ without parallel computers , limited to certain geometries , and offer little physical insight into the lsp resonance structures . it is thus highly desired to find an approach that is both simpler and more intuitive in terms of explaining the features of the eels maps and frequency profiles , and applicable to arbitrarily shaped resonators . recently , the physical meaning and applications of eels has been explored extensively . for example , hrl _ et al_. @xcite show that eels is an efficient tomography probe of the surface plasmon modes , and they propose to get the 3d green function from the eels ; unlike eels , which is related to the projected full electromagnetic ldos ( eldos ) , losquin _ et al_. @xcite theoretically show that cathodoluminescence is related to the so - called projected radiative eldos ( i.e. , nonradiative coupling effects are not captured ) , which they illustrate with a quasistatic mode expansion technique based on the bem . the quasistatic bem approach uses only geometry - defined modes , which gives a nonretarded modal solution with scale invariance @xcite ; in contrast , the mode expansion technique introduced below uses the rigorously defined qnms which are the true open system retarded eigenmodes , and we show only a few of them ( indeed usually only one ) will be needed around the frequency of interest . in this work , we introduce an accurate and physical intuitive method to model the eels map of the surface plasmon modes based on the quasinormal mode ( qnm ) expansion of the photon green function @xcite . the qnms are the eigenfunction of the source - free maxwell equations with open boundary condition @xcite , and complex eigenfrequencies . two key advantages of our qnm technique are as follows : ( i ) after obtaining the normalized qnm , the calculation of the eels is straightforward and essentially instantaneous in the frequency regime of interest ; ( ii ) our calculation includes the contribution of the lsp in a modal theory , and thus has intuitive and analytical insight . after introducing the basic theory of eels and connecting to the qnms , we present several example structures of interest including a gold srr , a single gold nanorod , and a dimer of gold nanorods as is shown schematically in fig . we also use the same qnm green function to obtain the purcell factor ( and projected ldos ) from a coupled dipole emitter and we show how the spectral profile compares and contrasts with the eels as a function of frequency . nm , @xmath0 nm , @xmath1 nm , @xmath2 nm , @xmath3 nm , @xmath4 nm , @xmath5 nm and @xmath6 nm , and we consider material properties for gold . ] for the structures of interest , we consider a general shaped metallic nanoresonator inside a homogeneous background medium with refractive index @xmath7 . we assume the magnetic response is negligible , with permeability @xmath8 ; the electric response is described by the drude model , with permittivity @xmath9 with parameters similar to gold : @xmath10thz and @xmath11thz . the electric - field green function , * g * , of the system is defined as @xmath12 where @xmath13 is the unit dyadic , and @xmath14 inside the metallic nanoresonator with @xmath15 elsewhere . considering the frequency regime of interest where there is only a single qnm , @xmath16 , which gives the mode profile of the lossy / dissipative mode of the source free maxwell equations with open boundary conditions , with complex eigenfrequency @xmath17 , then the contribution to the transverse green function in the near field of the nanoresonator , around the cavity resonance , is given by @xcite @xmath18 the qnm , @xmath19 is normalized here as @xmath20 with @xmath21 . alternative qnm normalization schemes are discussed in @xcite . when discussing eels , it is useful to also connect to common quantities for use in quantum plasmonics . for example , using the normalized qnm , the corresponding effective mode volume for use in purcell factor calculations is defined as @xmath22 , \label{eq : veee}\end{aligned}\ ] ] at some characteristic position @xmath23 the enhancement of spontaneous emission ( se ) , or the enhancement of the projected ldos , at this position is then obtained from @xmath24}{{\rm im}[\hat{n}_\alpha\cdot{\bf g}_{\rm b}({\bf r}_0,{\bf r}_0;\omega)\cdot\hat{n}_\alpha ] } , \label{eq : pf}\end{aligned}\ ] ] where @xmath25 = \frac{\omega^3n_{\rm b}}{6\pi c^3}{\bf i}$ ] is for a lossless homogeneous background with refractive index @xmath26 , and @xmath27 is a unit vector of the dipole emitter aligned along @xmath28 . using the qnm approach , then @xmath29 is simply obtained by using @xmath30 , and the accuracy of this approach can be checked by performing a full dipole calculation of @xmath31 at this position , which we will show later using accurate fdtd techniques @xcite . we consider a high speed electron beam with initial kinetic energy , @xmath32 , e.g. , 50 kev@xmath33200 kev , which gives an electron speed , @xmath34 , 0.55@xmath35 0.70 @xmath36 with @xmath36 the velocity of light in vacuum ; specifically , @xmath37 with @xmath38 the electron rest mass ; the electron passes through the nanoresonator which is a few tens of nanometers thick . generally , scanning transmission electron microscopes will be employed to obtain the eels map , as a function of frequency , for which the relevant length scale of the spatial path over which the electron beam is traveling is around a few hundred nanometers ; under this situation , the energy loss of the electron is negligible , which means that , to a very good approximation , we can take the velocity of the electron as a constant . as the electron comes near the surface , the electric quasistatic interaction can be described by the image charge @xcite , which is negligible until it comes to around a few nanometers from the surface ; at this scale , the local geometry details can be ignored and the surface can be approximated by a slab , and detailed analysis elsewhere shows that the electric quasistatic contribution is typically negligible @xcite for the eels calculation . thus we assume the electron energy loss is primarily induced by the dominant qnm(s ) . in practice there are also `` bulk losses''@xcite coming from other background modes such as evanescent modes in the metal , but these are regularized depending upon the finite size of the cross section of the electron beam and have little influence on a modal interpretation of the eels map . in fact , in @xcite , in order to investigate the modal response of the lsp resonance of the nanoresonator , they eliminated the bulk contribution by subtracting the solution from a different fdtd simulation with a homogeneous metal calculation , thus eliminating fdtd grid - dependent effects . with our qnm approach , there is no need to subtract off such a term , and , moreover , this contribution can be obtained analytically @xcite , and can also formulated as a local field problem for emitters inside lossy resonators @xcite . numerically , we inject a spatial plane wave modulated with a finite pulse length ( fwhm ) , @xmath39 , with a central frequency around the resonance of the qnm ; then a run - time fourier transform with a time window @xmath40 is employed @xcite to get the qnm numerically ; we also use a non - uniform conformal mesh scheme with a fine mesh of 1 - 2 nm around the metallic nanoresonator . the energy loss is defined by @xmath41 where the electric field induced by the qnm is given by @xmath42d\omega\nonumber\\ = & -2\int_{0}^{\infty}d\omega { \rm im}[e^{-{\rm i}\omega t}\frac{1}{\varepsilon_0\omega}\int{\bf g}^{\rm c}({\bf r}_t,{\bf r}';\omega)\cdot{\bf j}({\bf r}',\omega)d{\bf r}']\end{aligned}\ ] ] with the effective current carried by the moving electron , @xmath43 , and @xmath44 is the absolute value of the charge of the electron . in the calculations below , we will assume the electron moving along the -@xmath45-axis , so @xmath46 with @xmath47 the unit vector along @xmath45 . under these assumption , the eels function , @xmath48 , due to the qnm for electrons injected along @xmath45-axis is simply given by @xcite @xmath49dzdz ' , \label{loss } \end{aligned}\ ] ] where @xmath50 and @xmath51 on a 2d spatial map of the image . as is shown in eq . ( [ loss ] ) , in order to calculate the eels for one particular @xmath52 in the plane , the green function along the electron beam should be calculated at various @xmath45 and @xmath53 ; the number of simulations required should be sufficiently large to model electric dipoles scanning over the trajectory of the electron beam , and this is the reason why thousands of dipole simulation are usually employed @xcite . in stark contrast , with the qnm technique , once the qnms are obtained numerically , the green function can be calculated with eq . ( [ gt ] ) analytically , with a computation that is basically instantaneous . below we present a selection of example metal resonators using the drude model for the material properties of the metal . we also show the purcell factor or enhanced se factor at selected positions as well as the full eels as a 2d image . for our first example of the qnm calculation of eels , we study the gold srr , which is the basic `` artificial atom '' unit cell of the negative index metamaterial , with a rich magnetic response to external electromagnetic fields @xcite . while the 2d metamaterial lattice is usually fabricated on a low index semiconductor , for the present study , we will ignore the effects of the substrate and assume the srr is in free space ( @xmath54 ) , though this is not a model restriction . the srr with thickness , @xmath2 nm , is located in @xmath55-plane as is shown in fig . [ f0](a ) with parameters @xmath56 nm , @xmath0 nm , @xmath1 nm and @xmath3 nm . a full dipole fdtd calculation @xcite shows that the dipole resonance of the qnm is around @xmath57thz with mode profile @xmath58 shown in fig . [ f1](a ) . we use an @xmath59-polarized spatial plane wave with central frequency around 216 thz and pulse width @xmath60 fs , which is injected along the @xmath45-axis ; a running fourier transform with a temporal bandwidth @xcite @xmath61fs is used to obtain the qnm . the corresponding effective mode volume ( which is complex in its generalized form @xcite ) is around @xmath62 at the chosen dipole position @xmath63 nm where @xmath64 ( the dipole is shown by the blue arrow in fig . [ f1](a ) , and note we have set the center of the srr as the origin of the coordinate system ) . in order to first check the accuracy of the qnm calculation , we calculate the enhancement of the projected ldos ( or se enhancement of a dipole emitter ) @xmath65 from eq . ( [ eq : pf ] ) . figure [ f1](b ) shows that the single qnm model calculation ( magenta solid , with eq . ( [ gt ] ) ) agrees very well with the full numerical dipole calculation using fdtd @xcite ( grey dashed ) at position @xmath66 . as is shown above , due to the strong confinement of the lsp , extremely small mode volumes are obtained leading to a strong enhancement of se of an electric dipole ( or single photon emitter ) at this near field spatial position . the broad bandwidth of the se enhancement is also a notable feature of metallic nanoresonators , making it much easier to spectrally couple to artificial atoms . we further remark that the spatial dependent spectral function , @xmath67 $ ] , as is discussed in ref . @xcite , usually has a non - lorentzian lineshape that in general changes as a function of position ; this effect is captured by the qnm technique @xcite through the spatial dependent phase factor of the qnm . due to the collective motion of the free electrons , there is an oscillating electric current circling along the srr ; as a result , a temporarily non - zero magnetic dipole is created , which displays a strong magnetic response to external electromagnetic field around the resonance of the qnm and forms the basis of exciting optical properties of metamaterial such as negative index . the working region of the srr could be controlled by changing the length of the srr which determines the qnm resonance . the 2d eels image , @xmath68 for a high energy , @xmath69100 kev , ( @xmath70 ) electron beam injected along @xmath45-axis is shown in fig . [ f1](c ) , with @xmath71 , which is consistent with the calculations in refs . @xcite using full fdtd , nodal dgtg , and bem , respectively . the eels at the position of the blue circle ( near a maximum field position ) is obtained as shown in fig . [ f1](d ) by the dark green solid line as a function of the frequency ; clearly the eels can be used to effectively explore the qnm response of the plasmonic resonator . the magenta dashed line in fig . [ f1](d ) shows that the se enhancement example almost has the same spectral lineshape as the spatially averaged eels calculation . for our second example , we consider a single nanorod as is shown in fig . [ f0](b ) , with radius @xmath72 nm and length @xmath73 nm . frequently such nanorods are embedded inside liquids , so we assume the nanorod is located inside a homogeneous background medium with @xmath74 . using fdtd simulations , the dipole resonance is found around @xmath75thz @xcite . in order to get the qnm , a @xmath76-polarized spatial plane wave with central frequency 325 thz and @xmath77fs is injected along @xmath59-direction , and the running fourier transform with @xmath78fs is employed @xcite . figure [ f2](a ) shows the mode profile of @xmath79 and the effective mode volume is found around @xmath80 ; the enhancement of the projected ldos at position @xmath81 nm is shown in fig . [ f2](b ) ( see arrow for dipole position ) , and the qnm calculation ( magenta solid ) shows excellent agreement with the full numerical calculation ( grey dashed ) . the corresponding 2d eels for an injected electron beam with energy @xmath82kev is shown in fig . [ f2](c ) at the resonance frequency , @xmath83thz . at the in - plane position @xmath84 nm , around which the maximum of the eels is obtained ( blue circle in fig . [ f2](c ) ) , @xmath85 is shown by the blue solid line in fig . [ f2](d ) . it can be seen that the eels again picks up the correct resonant response of the qnm . as is shown in fig . [ f4](d ) , by the magenta dashed line , the se enhancement now has a different lineshape than the eels calculation ; as discussed earlier , this is caused by the spatially varying nature of spectral lineshape . for our final resonator example , we study a dimer composed of two identical gold nanorods in homogeneous background with @xmath86 as is shown in fig . [ f0](c ) . the eigenfrequency of the dipole mode is found at @xmath87thz @xcite , and the correspondent qnm , @xmath88 is shown in fig . [ f4](a ) . here in order to obtain the qnm mode , a @xmath76-polarized plane wave with central frequency 291 thz and @xmath77fs is injected , and @xmath89fs is used for the running fourier transform @xcite . the effective mode volume at @xmath90 nm is found to be @xmath91 ; the enhancement of the ldos , @xmath92 at @xmath66 , using qnm and full numerical fdtd calculations are shown by the magenta solid and gray dashed in fig . [ f4](b ) , respectively . the 2d eels , @xmath93 is shown in fig . [ f4](c ) at @xmath94thz , for injected electrons with energy @xmath95kev ; @xmath85 at @xmath96 nm ( blue circle in fig . [ f4](a ) ) is shown by the dark green solid line in fig . [ f4](d ) ; once again we see that the se enhancement ( magenta dashed line ) displays a rather different lineshape compared to eels calculation , similar to the case of the single nanorod . the se enhancement is also much larger for the dimer which also has a larger output coupling efficiency @xcite . as is shown in @xcite , due to the inherent loss of the plasmonic system , the normalized qnm is in general complex , @xmath97 , and one obtains a position dependent phase factor @xmath98 , which induces both the reshaping of the spectrum of ldos ( that is proportional to the imaginary part of the green function ) and the variation of the position of peak ; this causes the spectral lineshape between the eels and the enhancement of ldos to differ in general . it is important to stress that for the qnm calculation of eels , one only needs two simulations to get the complex eigenfrequency and spatial distribution of the qnm ( or qnms ) , respectively . the rest of the calculation can be done semi - analytically with eq . ( [ gt ] ) . consequently , the qnm technique for eels is many orders of magnitude faster than full fdtd calculation using electric dipoles , and offers more insight . furthermore , by obtaining the magnetic field component of the qnm , i.e. , @xmath99 , the magnetic green function , @xmath100 could be calculated just as easily as the electric green function ; this can be used to simulate vortex - eels as done in ref . @xcite which used brute force fdtd simulations . the qnm could also be applied to model the electromagnetic force on atoms and nanostructures for optical trapping , which as input , usually requires the maxwell stress tensor and/or the green function @xcite . recently , guillaume _ et al_. @xcite proposed an efficient modal expansion dda method to model eels . they show that by choosing a small number , e.g. , 3 - 10 , of eigenvectors their eigenvector expansion technique could get good results with reasonable accuracy , and the number of usual dda operations are decreased considerably . for certain geometries this approach may outperform a qnm fdtd computation , though it is not clear how general the approach is for various shaped resonators . the philosophy of the qnm approach is to use the source free eigenmode solutions , obtained here using fdtd with the aid of the modal response from a scattered plane wave @xcite . as a result , when there are a few dominant qnms around the frequency of interest , in principle a single fdtd simulation is enough to obtain the modes and green function as long as the overlap between the modes and injected field is sufficient . moreover , we stress that the qnm computation is not limited to the fdtd method , e.g. , it can also be obtained using an efficient dipole excitation technique with comsol @xcite . the typical time needed for our qnm calculations with a fine mesh size as small as 1 nm around the metal resonator , and total simulation volume 1 - 2 micron cubed , takes around several days on a high performance workstation . this is certainly not insignificant , however , having the full green function as a function of position and frequency can then solve numerous problems without any more numerical simulations for the electromagnetic response . apart from an efficient calculation of eels , the qnm green functions that we use above can be immediately adopted to efficiently study quantum light - matter interactions and true regimes of quantum plasmonics with quantized fields . for example , using the quantization scheme of the electromagnetic field in lossy structures @xcite , the interaction between a quantum dipole ( two level atom with frequency @xmath101 and dipole @xmath102 ) and electric field operator in the rotating wave approximation ( assuming no external field ) is given by the interaction hamiltonian @xmath103 $ ] ; where the electric field operator is @xmath104 , with @xmath105 the imaginary part of @xmath106 and @xmath107 is the collective excitation operator of the field and medium ; and @xmath108 is the pauli operator . we stress that the electromagnetic response of the lossy structure is rigorously included by the classical green function ( obtained from eq . ( [ gt ] ) ) . for example , in a born - markov approximation , the quantum dynamics of @xmath109 quantum emitters around a metal nanostructures can be described through a reduced density matrix whose coupling terms can be fully described , including emitter - emitter and emitter - lsp interactions , through the analytical properties of the qnm green function . for example , a recent example study of the quantum dynamics between two plasmon - coupled quantum dots is shown in @xcite . note that such an approach is ultimately more powerful than a standard jaynes - cummings model since it can include non - lorentzian decay processes and nonradiative coupling to the resonator in a self - consistent way , and it can also be used to improve the simpler jaynes - cummings models ( in a regime where they are deemed to be approximately valid ) with a rigorous definition of the various required coupling parameters @xcite . we have introduced an efficient and semi - analytic calculation technique for modelling eels using a qnm expansion technique , and exemplified the approach for several different metallic nanostructures . we first showed that the qnm technique works well for the srr , and demonstrated that the qnm could be used to obtain similar 2d eels maps to those shown in refs . @xcite , but with orders of magnitude improvements in efficiency and deeper physical insight . we then showed qnm calculations for a single gold nanorod and dimer of gold nanorods . we also presented example purcell factor calculations and demonstrated how the spectral profiles may differ to eels . z. guo , j. hwang , b. zhao , j. h. chung , s. g. cho , s .- j . baek , and j. choo , `` ultrasensitive trace analysis for 2,4,6-trinitrotoluene using nano - 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understanding light - matter interactions using localized surface plasmons ( lsps ) is of fundamental interest in classical and quantum plasmonics and has a wide range of applications . in order to understand the spatial properties of lsps , electron energy loss spectroscopy ( eels ) is a common and powerful method of spatially resolving the extreme localized fields that can be obtained with metal resonators . however , modelling eels for general shaped resonators presents a major challenge in computational electrodynamics , requiring the full photon green function as a function of two space points and frequency . here we present an intuitive and computationally simple method for computing eels maps of plasmonic resonators using a quasinormal mode ( qnm ) expansion technique . by separating the contribution of the qnm and the bulk material , we give closed - form analytical formulas for the plasmonic qnm contribution to the eels maps . we exemplify our technique for a split ring resonator , a gold nanorod , and a nanorod dimer structure . the method is accurate , intuitive , and gives orders of magnitude improvements over direct dipole simulations that numerically solve the full 3d maxwell equations . we also show how the same qnm green function can be used to obtain the purcell factor ( and projected local density of optical states ) from quantum dipole emitters or two level atoms , and we demonstrate how the spectral features differ in general to the eels spectrum .

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Proceed to summarize the following text: the intriguing phenomenon , the strong collinearity of cores in emulsion experiments @xcite , closely related to coplanar scattering of secondary particles in the interaction , has been observed long time ago . so far there is no simple satisfactory explanation of these cosmic ray observations in spite of numerous attempts to find it ( see , for instance , @xcite and references therein ) . among them , the jet - like mechanism @xcite looks very attractive and gives the natural explanation of alignment of three spots along the straight line which results from momentum conservation in a simple parton picture of scattering . besides , the strong momentum correlation of particles inside a jet and correlation between jet axes due to singularity of qcd matrix elements allow us to suggest the high degree of alignment for more than 3 spots . this has been already demonstrated for the four cores in @xcite but using a simplified picture of hadronization . with increasing energy of colliding hadrons ( nuclei ) hard and semi - hard jets begin to play an important role due to growth of their production cross sections . thus the jet activity is likely to be a feature of all events above certain threshold collision energy . one of the manifestations of this activity and the strict momentum ordering inside a hard enough jet can be the observed strong collinearity of spots in emulsion experiments . the main purpose of the present paper is just to trace this relation in detail . in sect . 2 we formulate the problem on the whole . section 3 describes the results of numerical simulation made under conditions close to emulsion experiments in the framework of pythia @xcite , and some discussion . a summary can be found in sect . in the pamir experiment the observed events ( @xmath4-hadron families with the alignment ) are produced , mostly , by a proton with energy @xmath5 tev interacting at a height of several hundred metres to several kilometres in the atmosphere above the chamber @xcite . the collision products are observed within a radial distance up to several centimetres in the emulsion where the spot separation is of the order of 1 mm . one can estimate the typical transverse momentum in the events under consideration using the ratio ( see also ( [ position ] ) ) : @xmath6 where @xmath7 is the energy deposition in the spot , @xmath8 is its spacing in the @xmath9-ray film , @xmath10 is the height of interaction . so @xmath11 is of the order of 10 gev for @xmath12 mm , @xmath13 km , @xmath14 tev . the particles with such transverse momenta can be typically initiated by a jet with @xmath15 gev or larger , since the most probable value of a fraction of jet energy carried by leading particles is @xmath16 @xcite . such energetic jets are already enough collimated : their effective angular cone size @xmath17 due to the strict ordering of transverse and longitudinal particle momenta in the leading logarithm approximation of perturbative qcd @xcite . qcd teaches " us also that this effective size decreases with the growth of the jet hardness " ( transverse momentum ) as @xmath18 where @xmath19 is the dimensional qcd parameter . the main conjecture is that the particle distribution from the hard enough jets in the @xmath9-ray film plane can lead to the alignment of the emulsion spots due to the strong collimation of such particles and dynamical correlation between jet axis directions . let us consider the kinematics in detail . for our analysis it is convenient to parametrize 4-momentum of each produced particle @xmath20 under consideration with its transverse momentum @xmath21 ( relative to the collision axis @xmath22 ) , azimuthal angle @xmath23 and rapidity @xmath24 in the center - of - mass system : @xmath25.\ ] ] in this case the transformation from the center - of - mass system to the laboratory one reduces to a simple rapidity shift : @xmath26 , where @xmath27 , @xmath28 are the rapidities of particle @xmath20 and the center - of - mass system correspondingly in the laboratory reference frame . if we neglect the further interactions of particles propagating through the atmosphere ( this gives the maximum estimation of the alignment effect ) , then their position in the transverse @xmath29-plane is easily calculated @xmath30 where @xmath31 and @xmath32 are the radial and longitudinal components of particle velocity respectively ( @xmath33 is the particle energy in the laboratory frame ) . since the size of the observation region is of the order of several centimetres , these radial distances must obey the following restriction : @xmath34 @xmath35 we set @xmath36 mm , @xmath37 mm . the restriction ( [ mini ] ) simply means that spots are not mixed with the central one formed by the particles which fly close to the collision axis ( mainly from the fragmentation region of an incident proton ) . the separation of spots in the @xmath9-ray film gives another restriction on the distance between particles @xmath38 it must be larger than 1 mm : @xmath39 in the opposite case the particles must be combined in one particle - cluster until there remain only particles and/or particle - clusters with the mutual distances larger than @xmath40 , each such particle - cluster being considered as a single particle with coordinates defined in the same way as center - of - mass coordinates of two bodies : @xmath41 then we select @xmath42 clusters / particles which are most energetic and obey the restrictions ( [ mini ] , [ max ] , [ dijres ] ) and calculate the alignment @xmath43 using the conventional definition @xcite : @xmath44 and taking into account the central cluster , i.e. @xmath45 . here @xmath46 is the angle between two vectors ( @xmath47 ) and ( @xmath48 ) ( for the central spot @xmath49 ) . this parameter characterizes the location of @xmath50 points just along the straight line and varies from @xmath51 to @xmath52 . for instance , in the case of the symmetrical and close to most probable random configuration of three points in a plane ( the equilateral triangle ) @xmath53 . the ultimate case of perfect alignment is @xmath54 when all points lie exactly along the straight line , while for an isotropic distribution @xmath55 . the alignment degree @xmath56 is defined as a fraction of events with @xmath57 @xcite with the number of cores not less than @xmath50 . if the hypothesis about the relation of alignment to the prevailing jet character of events at super high energies is valid , then this must manifest itself first of all in nucleon - nucleon collisions . therefore , to be specific we consider a collision of two protons and fix a primary energy in the laboratory system @xmath58 tev , that is equivalent to @xmath59 tev just the energy attainable at lhc ( the rapidity shift being @xmath60 after the transformation from the center - of - mass system to the laboratory one ) . to simulate a collision of two protons with such energies we use the monte carlo generator pythia @xcite , which basically well describes jet events in hadron - hadron interactions and is tuned using the available experimental accelerator data . the results of numerical simulation which follows the consideration in the previous section are presented in fig . 1 ( solid curve ) with the parameters @xmath61 mm , @xmath62 mm , @xmath63 m , which are close to the conditions of emulsion experiments , with the additional restriction on the energy threshold of particle registration in the emulsion : @xmath64 tev . the estimated alignment degree @xmath56 for @xmath50 cores is considerably larger than that for randomly selected chaoticly located spots in the @xmath9-ray film , but is still too small ( by a factor of 34 ) to describe the experimental data @xcite even taking into account their large errors . this can mean that the jet activity is not sufficient at such energies or the jet mechanism can not , in principle , give the large experimentally observable alignment . in order to try to answer this question let us consider the influence of the applied restrictions ( [ mini ] , [ max ] ) ( the laboratory acceptance criterion ) on the spectrum of particles selected to calculate the alignment . for particles with high enough transverse momenta @xmath21 relative to their masses @xmath65 these conditions ( [ mini ] , [ max ] ) reduce , mainly , to the restriction on the available particle rapidities in the center - of - mass system : @xmath66 @xmath67 since in this case @xmath68 for @xmath69 , where @xmath70 due to the kinematical restriction @xcite , @xmath71 a production of harder jets with larger rapidities becomes possible with the growth of @xmath72 . the rapidity region ( [ mini1 ] , [ max1 ] ) just corresponds to the transition from soft to hard qcd physics , where the jet activity _ could _ manifest itself . here one should note that ultrarelativistic particles ( @xmath73 ) are detected in the @xmath9-ray film from the restricted rapidity region ( [ mini1 ] , [ max1 ] ) which excludes such configurations as back - to - back hard jets with rapidities close to zero in the center - of - mass system . but just such configurations with scattering of hard partons at angles close to @xmath74 in the considered hadronic center - of - mass system ( which in this case practically coincides with the partonic center - of - mass system ) can be expected to be responsible for the alignment phenomenon . the point is that leading particles from both these hard jets have quantitatively comparable energies in the laboratory frame together with the `` strong memory '' of scattering plane . meanwhile leading particles from any other back - to - back hard jets with the relatively large modulo rapidities @xmath75 in the partonic center - of - mass system have essentially different laboratory energies due to lorentz boost . and the energy distinction is mainly determined by the value of @xmath76 . in the latter case particles from a forward hard jet produce as a rule the most energetic clusters ( apart from the central one ) in the laboratory frame , even if the particles from a backward ( in the partonic center - of - mass system ) jet hit the detection region . however such most energetic clusters from one jet are less correlated with the primary scattering plane and therefore will not be much aligned as clusters from both hard jets . this argumentation is confirmed by our simulation . in this connection it is necessary to comment on the work @xcite in which the jet hypothesis has been suggested for the first time for the explanation of alignment phenomenon . there the high degree of alignment has been demonstrated for the four cores only , using a simplified picture of fragmentation process . in fact , an axis distribution has been calculated in the partonic center - of - mass system in the first order of perturbative qcd theory at the partonic level , considering three partons in the final state only . then the lorentz transformation has been done in order to find their directions and localizations with respect to the central spot in the laboratory frame , considering each parton - jet as one long - living system ( the fragmentation time is of the order of flight time ) with some effective mass and aggregate group velocity . the velocity has been fixed ( @xmath77 in @xcite ) so as to be able to include the events giving the high degree of alignment and corresponding to the two final - state parton - jets in the backward hemisphere and one in the forward hemisphere in the partonic center - of - mass system . in our variables this means that a mass factor @xmath78 must be very small for a such massive system ( @xmath79 ) so that it hits the detection region even with negative rapidities ( i.e. corresponds to the backward hemisphere in the partonic center - of - mass system ) , if one uses the same boost parameters as for the hadronic center - of - mass system without taking into account the possible additional boost due to the distinction between these partonic and hadronic frames . note that for real particles , e.g. @xmath80-mesons which mainly contribute to the multiplicity , this mass factor becomes significant for very small transverse momenta , @xmath81 gev , only . however , as our investigation shows , falling of appropriately correlated particles into the observation region is still not sufficient to obtain the high degree of alignment because of the energy selection procedure , if the total number of particles is large and they generate many distinctly separated spots . for completeness one should also mention that high transverse momentum jet production has a connection to the double - core configuration of cosmic - ray events as it has been pointed out in @xcite . under certain conditions a hard forward ( in the partonic center - of - mass system ) jet together with a central bunch gives two relatively far separated clusters with large energies . the detailed studies of double - core ( or binocular ) phenomena with estimations of event rates and average lateral spread of the @xmath4-family using a pqcd based monte carlo can be found , for instance , in @xcite . ultrarelativistic particles from the central rapidity region in the hadronic center - of - mass system ( as possible sources of appropriately correlated spots ) can hit the observation region owing to the decrease of @xmath82 _ only _ , i.e. the decrease of the height @xmath10 of primary interaction or the increase of the rapidity @xmath28 of the center - of - mass system due to the growth of energy @xmath72 , as it follows from ( [ r0 ] ) . the energy growth seems preferable , if we intend to be closer to emulsion experiments and increase the jet activity . however this demands the extrapolation of pythia parameters and their special tuning to the experimentally untested energy domain . updating can be done appropriately after the lhc operation starts . moreover at present this generator already uses the extrapolation of experimentally tested cross sections and structure functions to the lhc energy region @xmath59 tev in order to estimate the effects expected at such energies . for illustration we utilize the first `` less dangerous '' alternative decrease the interaction height by a factor of @xmath83 rather than increase the energy @xmath72 by the same factor of @xmath83 at the initial height so that particles from both hard jets ( with back - to - back structure ) , hitting the registration region , come from some rapidity range near @xmath84 including adjoint positive and negative values . in this case the alignment degree becomes strongly dependent on the minimum transverse momentum of hard process , @xmath85 , which is a parameter of pythia . at the height @xmath13 km such dependence was not visible , although we might catch some marginal tendency of the alignment degree to grow with the increase of @xmath85 at that height . however without the restriction on @xmath85 from below ( minimum bias ) the result coincides practically with one obtained earlier ( solid curve in fig . 1 ) that shows some general characteristics of jet structure of events . if @xmath86 tev , particles from these hard jets together with particles flying close to @xmath22-axis ( within the transverse radius @xmath87 mm ) result in the alignment degree ( dashed curve ) comparable with the experimentally observed one @xcite . thus the jet - like mechanism can , in principle , attempt to explain the results of emulsion experiments . for such an explanation it is necessary ( but not sufficient ) that particles from both hard jets ( with rapidities near @xmath84 in the center - of - mass system ) hit the observation region . this is possible at the relatively small height @xmath88 m and @xmath59 tev ; or at the height @xmath63 m , but the considerably higher energy @xmath89 tev ; or at some reasonable and acceptable intermediate combination of @xmath10 , @xmath72 and @xmath90 which meets the following condition : @xmath91 where @xmath92 is the proton mass . @xmath93 is needed in order to have particles with @xmath94 that hit the detection region ( see ( [ max1 ] ) ) . we verified the decisive significance of condition ( [ r00 ] ) to allow the observation of large degree of alignment and its dependence on the process hardness for the smaller energy @xmath95 tev ( where the prediction of pythia is quite adequate ) and the height @xmath96 m ( in accordance with ( [ r00 ] ) ) thereby confirming this peculiar kinematic `` scaling '' . at @xmath97 tev jets carry away about half of the energy of colliding protons in the center - of - mass system due to the relationship in a parton picture @xmath98 , where @xmath99 is a fraction of proton energy carried by each interacting parton ( quark or gluon ) . the striking feature of such configurations in the @xmath9-ray film is approximate equality of energy deposition in the central and the rest most energetic clusters , that can be one of the physical guideline to select the events with very hard jets not only at the generator level ( simulation ) . if we simply apply the additional threshold on the minimum energy of detected clusters needed in the alignment analysis , then we still obtain neither the desirable selection of jet hardness nor the increase of the alignment degree . the small variation of resolution parameter @xmath40 does not provide the desirable effect also . however , introduction of another threshold on the total energy of all @xmath100 selected clusters @xmath101 ( without taking into account the energy deposition in the central cluster around @xmath102 ) , @xmath103 allows us to select the events with hard jets only in a `` natural '' physical way and to reduce the hypothesis to the really active mechanism . figure 2 shows that the alignment degree increases with the growth of @xmath104 ( the restriction on @xmath85 is absent at all ! ) , and it becomes large enough ( dashed curve ) and comparable with the experimentally observed one @xcite above the threshold @xmath105 pev . though one should note that our estimations give still too steep dependence on @xmath50 as one can see in figs . 1b , 2b from comparison of slopes of straight lines with the experimental behaviour . to give the reader a feeling for the various measures of alignment we present in figs . 3 and 4 the spatial distributions of most energetic clusters in the @xmath29-plane for a few generated events along with the corresponding values of @xmath43 . some spots are hardly visible because of their small sizes which are proportional to the cluster energies ( especially in the case @xmath106 ) or because they are outside a square @xmath107 mm@xmath108 mm ( but inside a circle @xmath12 mm ) as it sometimes happens in the case @xmath109 . besides for @xmath106 we can distinctly see three relatively large spots resulted from two hard jets and a central bunch . here one should note that there was slightly other criterion for the selection of families for the analysis in the works of pamir collaboration : the families with the total energy of @xmath4-quanta larger than a certain threshold and at least one hadron present were selected and analyzed . the alignment becomes apparent considerably at @xmath110 pev ( the families being produced , mostly , by a proton with energy @xmath111 pev ) . since the adequate comparison of our estimations with experimental data is impossible without a full simulation of particle propagation through the atmosphere , taking into account the energy distribution of primary cosmic particles , etc . , then in order to demonstrate the possibility of appearance of high alignment degree due to the jet mechanism we restrict ourself to the simpler ( but as concerns physics essentially close to experiment ) criterion of selection over the total energy of all particles . these particles are mostly @xmath80-mesons , the neutrals among them being the main source of the detected @xmath4-quanta . it is natural that the threshold on the total energy of all particles must be larger than the similar threshold on the total energy of @xmath4-quanta at the same collision energy . for comparison we estimate also the alignment degree selecting only the most energetic @xmath4-quanta with their total energy larger than certain threshold @xmath112 ( fig . 5 ) : @xmath113 the result is close to that obtained previously with the threshold imposed on the total energy of all particles . besides for jet events @xmath114 with a high accuracy ( see figs . 1b , 2b , 5b , which present the dependence of alignment degree on the number of considered cores at the different values of hardness parameter ( 1b ) and threshold total energy ( 2b , 5b ) in the logarithmic scale ) . this constant depends on @xmath85 , @xmath104 , @xmath112 , decreasing with their growth , and could in principle be determined by the kernels of the gribov altarelli parisi dokshitzer equations @xcite which describe the process of radiation of quarks and gluons in the initial and final states . and , in fact , this process is implemented in the pythia generator together with the subsequent hadronization of quarks and gluons . if nevertheless particles from the central rapidity region @xmath84 and the jet - like mechanism are insufficient to describe the observed alignment and there is another mechanism of its appearance at the energy @xmath115 tev and the height @xmath116 m ( mostly used in emulsion experiment estimations ) , then in any case some sort of alignment should arise at lhc too in the rapidity region ( [ mini1 ] , [ max1 ] ) . this region must be investigated more carefully on the purpose to study the azimuthal anisotropy of energy flux in accordance with the procedure applied in the emulsion and other experiments , i.e. one should analyze the energy deposition in the cells of @xmath117-space in the rapidity interval ( [ mini1 ] , [ max1 ] ) ( the equivalent threshold minimum particle energy being @xmath118 gev in the center - of - mass system ) . note that the absolute rapidity interval can be shifted : it is necessary only that the difference ( @xmath119 is equal to @xmath120 in accordance with the variation of radial distance by a factor of 15 ( @xmath121 ) due to the relationship @xmath122 ( independently of @xmath82 ) . in other words , since we use particle momenta in the center - of - mass system , then future data should be treated in accordance with the algorithm described earlier in sects . 2 , 3 introducing the corresponding laboratory observables . our analysis shows that for @xmath123-collision at a fixed height of primary interaction above the energy @xmath72 , when the condition ( [ r00 ] ) is fulfilled that is ultrarelativistic particles from the rapidity interval near @xmath84 in the center - of - mass system fall into the observation region inside the radius @xmath90 in the laboratory frame due to the large lorentz factor the alignment of spots arises ( this , in principle , explains the existence of the experimental energy threshold of this effect ) and the alignment degree becomes strongly dependent on the process hardness . if the process hardness is close to maximum for the given energy @xmath72 , the estimated degree of alignment is already comparable with the experimentally observed one . introducing another additional threshold ( the scale of which is determined by the energy of an incident proton ) on the total energy of all @xmath100 selected most energetic clusters ( without taking into account the energy deposition in the central cluster ) allows us to select the events with high hardness in a natural physical way and thereby support the jet - like hypothesis , which later on may be accepted ( or refuted ) in further investigations of , for instance , the energy cluster distribution and their particle composition with regard for interactions in the atmosphere , etc . meanwhile we suggest the more careful investigation of the rapidity region ( [ mini1 ] , [ max1 ] ) at lhc in order to reveal the new still unknown mechanisms of alignment if they exist . for this purpose one should perform the analysis of energy deposition in calorimeters of cms and atlas experiments in accordance with the procedure described in sects . 2 , 3 ( i.e. calculating the appropriate observables in the laboratory frame ) . such investigation can clarify the origin of the alignment , test the alternative hypotheses and give the new restrictions on the values of height and energy . it is pleasure to thank a.i . demianov , s.v . molodtsov , s.a . slavatinsky , l.g . sveshnikova , k.yu . teplov and g.t . zatsepin for discussions . this work is supported by grant n 04 - 02 - 16333 of russian foundation for basic research . pamir collaboration , in proceedings of the 21st international cosmic ray conference , adelaide , australia ( 1989 ) , edited by r.j.protheroe ( university of adelaide , australia ) , 227 ( 1990 ) ; + s.a . slavatinsky , in proceedings of the 5th international symposium on very high energy cosmic ray interactions , lodz , poland ( 1988 ) , edited by m. giler ( university of lodz , lodz , poland ) , 90 ( 1989 ) . kopenkin , a.k . managadze , i.v . rakobolskaya , t.m . roganova , phys . d * 52 * , 2766 ( 1995 ) f. halzen , d.a . morris , phys . d * 42 * , 1435 ( 1990 ) t. sjostrand , comp . phys . com . * 135 * , 238 ( 2001 ) i.v . rakobolskaya et al . the peculiarity of hadron interactions of cosmic rays at super high energies ( msu , moscow , 2000 ) ( in russian ) . dokshitzer , d.i . dyakonov , s.i . troyan , phys . * 58 * , 269 ( 1980 ) v.v . kopenkin , a.k . managadze , i.v . rakobolskaya , t.m . roganova , izv . . fiz . * 58 * , 13 ( 1994 ) d. cline , f. halzen , j. luthe , phys . lett . * 31 * , 491 ( 1973 ) z. cao , l.k . ding , q.q . zhu , y.d . he , phys . lett . * 72 * , 1794 ( 1994 ) z. cao , l.k . ding , q.q . zhu , y.d . he , phys . d * 56 * , 7361 ( 1997 ) v.n . gribov , l.n . lipatov , sov . . phys . * 15 * , 438 ( 1972 ) ; + v.n . gribov , l.n . lipatov , sov . 15 * , 675 ( 1972 ) l.n . lipatov , sov . . phys . * 20 * , 94 ( 1974 ) yu.l . dokshitzer , sov . j. jetp * 46 * , 641 ( 1977 ) g. altarelli , g. parisi , nucl . b * 126 * , 298 ( 1977 )

the hypothesis about the relation between the observed alignment of spots in the x - ray film in cosmic ray emulsion experiments and the features of events in which jets prevail at super high energies is tested . due to strong dynamical correlation between jet axis directions and that between momenta of jet particles ( almost collinearity ) , the evaluated degree of alignment is considerably larger than that for randomly selected chaoticly located spots in the x - ray film . it appears comparable with experimental data provided that the height of primary interaction , the collision energy and the total energy of selected clusters meet certain conditions . the monte carlo generator pythia , which basically well describes jet events in hadron - hadron interactions , was used for the analysis . = -15 mm # 1#23.6pt * jet activity versus alignment * i.p . lokhtin@xmath0 , a.k . managadze@xmath1 , l.i . sarycheva@xmath2 , a.m. snigirev@xmath3 + m.v.lomonosov moscow state university , d.v.skobeltsyn institute of nuclear physics , + 119992 , vorobievy gory , moscow , russia + + @xmath0e - mail : igor@lav01.sinp.msu.ru + @xmath1e - mail : mng@dec1.sinp.msu.ru + @xmath2e - mail : lis@alex.sinp.msu.ru + @xmath3e - mail : snigirev@lav01.sinp.msu.ru +

You are an expert at summarizing long articles. Proceed to summarize the following text: cooling in lattice field theory is a technique for exposing the topological features of field configurations @xcite . recently , using this method , evidence has been presented @xcite that instantons play a dominant role in determining hadron properties in quantum chromodynamics ( qcd ) . this conclusion was based on the calculation of long - distance properties of a variety of hadronic correlation functions . it has been argued , however , that the persistence of long - distance effects ( particularly , confinement @xcite ) is an inevitable consequence of the local nature of the cooling procedure and is not indicative of the underlying dynamics @xcite . to gain some more insight into what cooling is doing we have extended our work @xcite on three - dimensional qed to four - dimensional qcd with su(2 ) color . we first examine the behavior of the static quark - antiquark potential ( calculated from wilson loops ) and of the spin - dependent interaction ( calculated from chromo - magnetic field correlations ) under cooling . as in qed@xmath2 , there is a good indication that the confining behavior of the static potential persists even after a significant amount of cooling although the overall scale of the potential is greatly reduced . on the other hand , the spin - dependent potentials ( which are short - ranged ) disappear rapidly upon cooling . in the spin - spin interaction , a residual effect , which one may associate with instantons , of only a few percent of the orginal potential was observed . next chiral symmetry breaking was calculated using staggered fermions in quenched approximation . the chiral symmetry order parameter @xmath3 was calculated at a few nonzero quark mass values and compared to the result obtained for free fermions . our first observation is that @xmath3 relaxes to its free - field value even after a small amount of cooling . this is an indication that dynamical mass generation in the cooled configurations is small compared to that found in the uncooled vacuum . the extrapolation to zero quark mass is problematic since , in principle , it requires an extrapolation to infinite volume first . however , we can conservatively estimate that in the region of intermediate cooling ( 20 to 50 cooling steps ) the chiral condensate has been reduced by at least an order of magnitude . finally , meson correlators for the spin-0 and spin-1 channels were calculated . the behavior of these time - correlation functions upon cooling is in qualitative agreement with what would be anticipated given the behavior of the potentials and the chiral order parameter . the meson correlators do not relax uniformly . the relaxation rate is most rapid at small times and it also decreases as quark mass decreases . at short time separations the spin-0 and spin-1 meson correlators reflect very clearly the effect of cooling on the spin dependent potential ( degeneracy of pseudoscalar and vector states ) and on chiral symmetry breaking ( parity doubling in the spin-1 channel ) . at larger times at fit to the pseudoscalar correlation functions yields `` mass '' values roughly 2/3 the size ( in lattice units ) of the uncooled masses . section ii contains a description of the calculational methods used in this paper . the results are presented in sec . the usual plaquette action @xmath4 is used . periodic boundary conditions were imposed on the gauge field links in all directions . field configurations were generated using a heatbath monte carlo algorithm . for cooling , links were updated `` vectorially '' following the same checkerboard sequence as was used in the monte carlo algorithm . each link in turn was replaced by a link proportional to the inverse of the sum of the `` staples '' of the plaquettes containing the link being updated so as to minimize the local contribution to the action . some test runs were done with adiabatic cooling @xcite . after 10 to 20 cooling steps the results become qualitatively the same as with full local minimization . the physics of cooled configurations has been interpreted in terms of instantons . we also monitor the topological charge as a function of cooling and , following chu _ @xcite , the simple transcription of @xmath5 to the lattice @xcite is used for the topological charge density @xmath6 where @xmath7 , @xmath8 are plaquettes of gauge field links . the total topological charge is @xmath9 as is well known this simple definition of topological charge does not accurately represent the topological charge of the uncooled lattice configurations @xcite . however , for sufficiently smooth ( cooled ) configurations one can see instantons with the expected continuum action . the creutz ratio @xmath10 where @xmath11 denotes the @xmath12 by @xmath13 rectangular wilson loop , can be used to determine the string tension . for large loops , which obey the area law , @xmath14 gives the string tension . the potential @xmath15 between static quarks was also calculated directly by extrapolating wilson loops to large @xmath13 @xmath16 variance reduction methods @xcite were used in the computation of the wilson loops in the uncooled theory . in addition to the confining central potential , spin - dependent interactions can also be calculated @xcite . these are related to chromo - magnetic field correlations ( see ref . @xcite for a simple derivation ) which are computed by making magnetic field insertions on @xmath17 wilson loops . the spin - spin and tensor interactions are then given by @xmath18 and @xmath19 where @xmath20 and @xmath21 are expectation values of wilson loops with plaquette insertions ( at @xmath22 and @xmath23 ) corresponding to magnetic fields parallel and transverse to the spatial direction of the loop . in practise , the magnetic field insertion @xmath24 that was used was the average over the eight spatial plaquettes whose corners lie on the wilson loop @xmath11 . this corresponds to operator ii of ref . @xcite . in addition to observables constructed purely from gauge field variables we are also interested in how quarks behave in the cooled vacuum . in qcd a basic property of the vacuum is chiral symmetry breaking which can be studied most easily if staggered fermions are used . the action for staggered fermions is @xmath25 + \sum_x \ , m \overline{\chi}(x ) \chi(x ) , \nonumber \\ & \equiv & \overline{\chi } \,\ , { \cal m}(\{u\ } ) \ , \chi , \label{sf}\end{aligned}\ ] ] where @xmath26 are single - component fermion fields , @xmath27 is the staggered fermion phase @xcite , @xmath28 is the mass in lattice units and the @xmath29 are gauge field links . antiperiodic boundary conditions were used for the fermion fields in all directions . the chiral symmetry order parameter is calculated from the inverse of the fermion matrix @xmath30 of eq . ( [ sf ] ) @xmath31 where v is the lattice volume and the angle brackets denote the gauge field configuration average . a random source method @xcite was used to calculate @xmath32 . sixteen gaussian random sources were used for each gauge field configuration . meson correlation functions can be constructed from local bilinears of the single - component @xmath33 fields . we consider two such correlators which after integration over the fermion fields , can be expressed in terms of the inverse of the fermion matrix as @xmath34^\dagger \right\},\ ] ] and @xmath35 \mbox{tr } \left\ { { \cal m}^{-1}(\vec x , t ; 0 ) \left [ { \cal m}^{-1}(\vec x , t ; 0 ) \right]^\dagger \right\}.\ ] ] these two functions describe the propagation of zero - momentum meson states of spin 0 and 1 respectively . as is well known , with local staggered fermion operators mixing between states of different parity can in principle occur . in practise , the spin-0 channel is essentially pure pseudoscalar and describes the pseudo - goldstone boson . the spin-1 correlator is dominated by the vector meson ( at least in the uncooled theory ) with some admixture of axial - vector meson states . most of the calculations were done on a @xmath36 lattice at @xmath37 . this value was chosen as it is well into the scaling region for su(2 ) color . for comparison some calculations were done at @xmath38 on a @xmath36 lattice and at @xmath37 on a @xmath39 lattice . however , not all results will be shown here since they are qualitatively the same in all cases . it is useful to consider first the topological properties of the gauge field configurations . a sample of 300 configurations , separated by 100 heat - bath monte - carlo sweeeps after 4000 sweeps of thermalization , was analyzed . in the uncooled configurations our value for the ( lattice ) topological susceptibility @xmath40 of @xmath41 agrees well with the high statistics result of @xmath42 obtained by campostrini _ the simple transcription of the continuum topological charge operator is not a true topological quantity on the lattice @xcite . it need not take integer values as can be seen in fig . 1 which shows a histogram of the number of configurations versus @xmath43 . however in cooled configurations which are sufficiently smooth , the operator @xmath43 does cluster around integer values . this is shown in fig . 2 constructed from our sample of 300 configurations at 25 , 50 , 75 and 100 cooling steps . it is also useful to examine the behavior of the action under cooling . histograms of the average plaquette are plotted in fig . 3 for different amounts of cooling . 4 shows a scatter plot of @xmath43 versus average plaquette . at 100 cooling steps there is a fairly obvious instanton interpretation . the vertical dashed lines in fig . 3d correspond to values of the total action of @xmath44 for n=1 , 2 , 3 and 4 . after about 100 cooling steps the configurations are dominated by a single classical instanton . 75 cooling steps seems to be in a transition region @xcite . single instanton peaks are seen in the action but there are many configurations which have a more complicated structure . the region of 25 to 50 cooling has been interpreted as being dominated by multi - instanton anti - instanton fluctuations . however one has to be aware that , as can be inferred from fig . 3 and 4 , these configurations are not simply superpositions of isolated ( noninteracting ) classical instantons . the creutz ratio and static potential can be calculated from wilson loops as in eqs . ( 4 ) and ( 5 ) . a @xmath36 lattice is too small to see the true asymptotic confining behavior but we can infer the general trend . the creutz ratio @xmath14 calculated from our sample of 300 uncooled configurations is shown in fig . 5(a ) . on the same graph we also show the creutz ratio after 25 cooling steps . further cooling results in the creutz ratios of fig . the corresponding results for the potentials are shown in fig . the potentials are steadily reduced by cooling but the concave upward curvature is consistent with the idea put forward by teper @xcite that the string tension survives cooling , albeit , at increasingly large distances . spin - dependent potentials are calculated from chromo - magnetic field correlations . the spin - spin and tensor potentials are plotted in fig . 7 and 8 respectively . the uncooled potentials are qualitatively consistent with the @xmath45 operator ii results of michael and rakow @xcite . the spin - dependent potentials are short ranged and decrease rapidly upon cooling . in the region of 25 to 50 cooling steps the residual effect in these potentials is only at the level of a few percent of the uncooled values . of course it has to be remembered that the comparison of cooled and uncooled results is being done here in terms of lattice ( not `` physical '' ) units . it is clear that if one attempted to keep the magnitude of the potentials quantitatively similar in physical units while cooling a very large change in the lattice spacing would be required . spontaneous chiral symmetry breaking is a basic property of qcd . a signature for this phenomenon is the persistence of a nonzero value of the `` quark condensate '' in the limit of zero quark mass . to study chiral symmetry breaking we calculate the expectation value of the local staggered fermion operator @xmath0 . for our @xmath36 calculation , 40 configurations ( separated by 200 monte carlo sweeps ) at @xmath37 and staggered fermion masses @xmath46 and @xmath47 were used . fig . 9 shows @xmath0 versus quark mass ( in lattice units ) . the results without cooling are consistent with those of billoire _ @xcite at the same value of @xmath48 . the quantity @xmath0 was also calculated using 20 configurations on a @xmath39 lattice and quark masses down to @xmath49 . the results are shown in fig . 10 . at the masses that are common to both calculations , the @xmath36 and @xmath39 lattice results are consistent . the results for @xmath0 after 10 cooling steps and for massive free staggered fremions are also plottted in figs . 9 and 10 . after 10 cooling steps the values of @xmath0 are already quite close to the free field results at the same value of @xmath50 . therefore , dynamical mass generation seems to be quite small in the cooled vacuum . in principle the extrapolation of @xmath0 to zero quark mass to extract the genuine chiral condensate requires that the infinite volume limit be taken first . this would allow calculations at arbitrarily small quark masses . in practise we can only do a limited number of calculations so the procedure of billoire _ @xcite is adopted . three values of @xmath0 at the lowest nonzero mass are used to determine the coefficients of the expansion @xmath51 the extrapolated values of @xmath52 from this procedure are plotted in fig . 11 versus cooling up to 50 cooling steps . unfortunately the values from the @xmath36 and @xmath39 lattice calculations do not agree indicating that perhaps the correct mass window is not being used for the extrapolation or that the extrapolation function is not adequate . therefore only a qualitative conclusion is possible , namely , that the chiral condensate ( in lattice units ) is greatly reduced in cooled configurations . of course , one could keep the condensate fixed in physical units but this would require a decrease of the lattice spacing by a factor of about 2.5 to 3 . finally we examine the meson correlation functions . first we consider the qualitative effects of cooling by comparing the meson correlators in the vector and pseudoscalar channels . a sample of results at the lightest quark mass values for both @xmath36 and @xmath39 lattice are presented . 12 shows the pseudoscalar and vector zero - momentum correlators on the @xmath36 lattice for @xmath53 . after only a small amount of cooling the correlation functions for small time separations display essentially the same behavior as would be obtained for free fields . the pronounced oscillation of the vector correlator indicates near degeneracy of even and odd parity spin-1 states ( parity doubling ) . it is clear that , after cooling , pseudoscalar and vector states are also nearly degenerate . in other words , after 20 to 50 cooling steps , quark propagation at @xmath53 , at least in the limited time region available on the @xmath36 lattice , shows very little evidence of spin - dependent forces or chiral symmetry breaking . on the @xmath39 lattice the correlations functions can be explored over a larger time range and at smaller masses . the results for @xmath53 and @xmath54 are plotted in figs . 13 and 14 respectively . at large time separations the correlation functions differ from those obtained for free fields . for example , comparing figs . 12 and 13 one sees at the largest time separation some evidence of a difference between the pseudoscalar and vector channels similarly , if the mass is decreased ( compare figs . 13 and 14 ) the difference between pseudoscalar and vector correlators after cooling is enhanced . although our lattices are too small to do very accurate mass determinations we were able to get reasonable fits to the pseudoscalar correlator on the @xmath39 lattice in a restricted time window with the function @xmath55,\ ] ] where @xmath56 is the size of the lattice . the meson decay constant @xmath57 can be calculated @xcite from the coefficient @xmath58 according to @xmath59 . the results for the mass and decay constant are given in table 1 . the pattern is consistent with the findings of chu _ et al . the masses ( in lattice units ) are reduced to about 2/3 of the uncooled values and do not change very much in the range of intermediate cooling ( 20 to 50 cooling steps ) . the meson decay constant , which reflects the short - distance behavior of the wave function , is reduced to about 1/2 of the uncooled value . how is one to interpret the behavior of the meson correlators under cooling ? one way is that the correlators reflect the true dynamics of the cooled vacuum . at sufficiently large times and sufficiently small masses the correlation functions are not affected by cooling very much so hadron properties ( and the uncooled lattice scale ) survive essentially intact . this is the approach of chu _ et al . an obvious question with this interpretation is how to reconcile it with the finding of direct calculation that the spin - dependent potentials and chiral symmetry breaking essentially disappear ( in lattice units ) after only a small amount of cooling . another interpretation is that the effect identified by teper @xcite in the string tension is also present in the meson correlators . namely , correlations functions do not relax uniformly under cooling . at sufficiently large time separations it is inevitable that the behavior of the original uncooled configurations persists and this does not reflect the dynamics of the cooled vacuum . in this work the effect of cooling on a number of observables was calculated in su(2 ) lattice gauge theory . these include the central and spin - dependent potentials , the chiral symmetry breaking order parameter and meson correlators . even after 100 cooling steps a remnant of the confining static potential is seen at large distance . however interactions on the distance scale of a few lattice spacings are essentially eliminated . this includes the spin - dependent interactions induced by chromo - magnetic field correlations . the quantity @xmath0 was found to approach its free - field value very quickly with cooling . this may indicate that on the lattice , with staggered fermions , chiral symmetry breaking is driven more by local fluctuations in the topological charge density rather than by global topological properties . as cooling smooths out local fluctuations , chiral symmetry breaking decreases rapidly , even though instantons remain . this is in line with the study of hands and teper @xcite who suggest that the instanton - induced zero modes for staggered fermions do not become delocalized as would be required for chiral symmetry breaking . meson correlators up to time separations of several lattice spacings relax quickly to free - field values reflecting the behavior of the potentials . at larger times , differences from free - field behavior persist . a fit to the pseudoscalar correlation function yields `` mass '' values about @xmath1 of the uncooled masses . to this extent our results confirm the calculation of chu _ et al . _ @xcite . on the other hand , we can not conclude that our results provide evidence for the dominant role of instantons . there are properties of qcd that are changed by cooling . the lack of a direct signature for spin - dependent forces and chiral symmetry breaking has to be reconciled with the large - time behavior of the hadron correlators . until this can be done the interpretation of the hadron properties after cooling remains imprecise . r. scalettar , d. scalapino and r. sugar , phys . b * 34 * , 7911 ( 1986 ) ; s. gottlieb , w. liu , d. toussaint , r.l . renken and r.l . sugar , phys . d * 35 * , 3972 ( 1987 ) ; k. bitar , a.d.kennedy , r. horsley , s. meyer and p. rossi , nucl . b313 * , 348 ( 1989 ) .

the effect of cooling on a number of observables is calculated in su(2 ) lattice gauge theory . the static quark - antiquark potential and spin - dependent interactions are studied , and the topological charge is monitored . the chiral symmetry breaking order parameter @xmath0 and meson correlators are calculated using staggered fermions . interactions on the distance scale of a few lattice spacings are found to be essentially eliminated by cooling , including the spin - dependent potentials . @xmath0 and meson correlators up to time separations of several lattice spacings relax very quickly to their free - field values . at larger times , there is evidence of a difference between the pseudoscalar and vector channels . a fit to the pseudoscalar correlation function yields `` mass '' values about @xmath1 ( in lattice units ) of the uncooled masses . these results raise the question of how to reconcile the large - time behavior of the hadron correlators with the fact that the spin - dependent potentials and @xmath0 essentially disappear ( in lattice units ) after only a small amount of cooling .

You are an expert at summarizing long articles. Proceed to summarize the following text: non - gaussian quantum states , endowed with properly enhanced nonclassical properties , may constitute powerful resources for the efficient implementation of quantum information , communication , computation and metrology tasks @xcite . indeed , it has been shown that , at fixed first and second moments , gaussian states _ minimize _ various nonclassical properties @xcite . therefore , many theoretical and experimental efforts have been made towards engineering and controlling highly nonclassical , non - gaussian states of the radiation field ( for a review on quantum state engineering , see e.g. @xcite ) . in particular , several proposals for the generation of non - gaussian states have been presented @xcite , and some successful ground - breaking experimental realizations have been already performed @xcite . concerning continuous - variable ( cv ) quantum teleportation , to date the experimental demonstration of the vaidman - braunstein - kimble ( vbk ) teleportation protocol @xcite has been reported both for input coherent states @xcite , and for squeezed vacuum states @xcite . in particular , ref . @xcite has reported the teleportation of squeezing , and consequently of entanglement , between upper and lower sidebands of the same spatial mode . it is worth to remark that the efficient teleportation of squeezing , as well as of entanglement , is a necessary requirement for the realization of a quantum information network based on multi - step information processing @xcite . in this paper , adopting the vbk protocol , we study in full generality , e.g. including loss mechanisms and non - unity gain regimes , the teleportation of input single - mode coherent squeezed states using as non - gaussian entangled resources a class of non - gaussian entangled quantum states , the class of squeezed bell states @xcite . this class includes , for specific choices of the parameters , non - gaussian photon - added and photon - subtracted squeezed states . in tackling our goal , we use the formalism of the characteristic function introduced in ref . @xcite for an ideal protocol , and extended to the non - ideal instance in ref . @xcite . here , in analogy with the teleportation of coherent states , we first optimize the teleportation fidelity , that is , we look for the maximization of the overlap between the input and the output states . but the presence of squeezing in the unknown input state to be teleported prompts also an alternative procedure , depending on the physical quantities of interest . in fact , if one cares about reproducing in the most faithful way the initial state in phase - space , then the fidelity is the natural quantity that needs to be optimized . on the other hand , one can be interested in preserving as much as possible the squeezing degree at the output of the teleportation process , even at the expense of the condition of maximum similarity between input and output states . in this case , one aims at minimizing the difference between the output and input quadrature averages and the quadrature variances . it is important to observe that this distinction makes sense only if one exploits non - gaussian entangled resources endowed with tunable free parameters , so that enough flexibility is allowed to realize different optimization schemes . indeed , it is straightforward to verify that this is impossible using gaussian entangled resources . we will thus show that exploiting non - gaussian resources one can identify the best strategies for the optimization of different tasks in quantum teleportation , such as state teleportation vs teleportation of squeezing . comparison with the same protocols realized using gaussian resources will confirm the greater effectiveness of non - gaussian states vs gaussian ones as entangled resources in the teleportation of quantum states of continuous variable systems . the paper is organized as follows . in section [ secqtelep ] , we introduce the single - mode input states and the two - mode entangled resources , and we recall the basics of both the ideal and the imperfect vkb quantum teleportation protocols . with respect to the instance of gaussian resources ( twin beam ) , the further free parameters of the non - gaussian resource ( squeezed bell state ) allow one to undertake an optimization procedure to improve the efficiency of the protocols . in section [ sectelepfidelity ] we investigate the optimization procedure based on the maximization of the teleportation fidelity . we then analyze an alternative optimization procedure leading to the minimization of the difference between the quadrature variances of the output and input fields . this analysis is carried out in section [ secoptvar ] . we show that , unlike gaussian resources , in the instance of non - gaussian resources the two procedures lead to different results and , moreover , always allow one to improve on the optimization procedures that can be implemented with gaussian resources . finally , in section [ secconcl ] we draw our conclusions and discuss future outlooks . in this section , we briefly recall the basics of the ideal and imperfect vbk cv teleportation protocols ( for details see ref . the scheme of the ( cv ) teleportation protocol is the following . alice wishes to send to bob , who is at a remote location , a quantum state , drawn from a particular set according to a prior probability distribution . the set of input states and the prior distribution are known to alice and bob , however the specific state to be teleported that is prepared by alice remains unknown . alice and bob share a resource , e.g. a two - mode entangled state . the input state and one of the modes of the resource are available for alice , while the other mode of the resource is sent to bob . alice performs a suitable ( homodyne ) bell measurement , and communicates the result to bob exploiting a classical communication channel . then bob , depending on the result communicated by alice , performs a local unitary ( displacement ) transformation , and retrieves the output teleported state . the non - ideal ( realistic ) teleportation protocol includes mechanisms of loss and inefficiency : the photon losses occurring in the realistic bell measurements , and the noise arising in the propagation of optical fields in noisy channels ( fibers ) when the second mode of the resource is sent to bob . the photon losses occurring in the realistic bell measurements are modeled by placing in front of an ideal detector a fictitious beam splitter with non - unity transmissivity @xmath0 ( and corresponding non - zero reflectivity @xmath1 ) @xcite . the propagation in fiber is modeled by the interaction with a gaussian bath with an effective photon number @xmath2 , yielding a damping process with inverse - time rate @xmath3 @xcite . denoting by @xmath4 the input field mode , and by @xmath5 and @xmath6 , respectively , the first and the second mode of the entangled resource , the decoherence due to imperfect photo - detection in the homodyne measurement performed by alice involves the input field mode @xmath4 , and one mode of the resource , e.g. mode @xmath5 . throughout , we assume a pure entangled resource . indeed , it is simple to verify that considering mixed ( impure ) resources is equivalent to a consider a suitable nonvanishing detection inefficiency @xmath7 @xcite . the degradation due to propagation in fiber affects the other mode of the resource , e.g. mode @xmath6 , which has to reach bob s remote place at the output stage . denoting now by @xmath8 and @xmath9 the projectors corresponding , respectively , to a generic pure input single - mode state and a generic pure two - mode entangled resource , the characteristic function @xmath10 of the single - mode output field @xmath11 can be written as @xcite : @xmath12 \\&=e^{- \gamma_{\tau , r}|\alpha|^{2 } } \chi_{in}\left(g t \ , \alpha \right ) \chi_{res}\left(g t \ , \alpha^{*};e^{-\frac{\tau}{2 } } \ , \alpha\right ) , \end{split } \label{chioutfinale}\ ] ] where @xmath13 is the glauber displacement operator , @xmath14 $ ] is the characteristic function of the input state , @xmath15 $ ] is the characteristic function of the resource , @xmath16 is the gain factor of the protocol @xcite , @xmath17 is the scaled dimensionless time proportional to the fiber propagation length , and the function @xmath18 is defined as : @xmath19 we assume in principle to have some knowledge about the characteristics of the experimental apparatus : the inefficiency @xmath7 ( or @xmath20 ) of the photo - detectors , and the loss parameters @xmath21 and @xmath2 of the noisy communication channel . we consider as input state a single - mode coherent and squeezed ( cs ) state @xmath22 with unknown squeezing parameter @xmath23 and unknown coherent amplitude @xmath24 . we then consider as non - gaussian entangled resource the two - mode squeezed bell ( sb ) state @xmath25 , defined as @xcite : @xmath26 here @xmath27 is , as before , the displacement operator , @xmath28 is the single - mode squeezing operator , @xmath29 is the two - mode squeezing operator @xmath30 , with @xmath31 denoting the annihilation operator for mode @xmath32 @xmath33 , @xmath34 is the two - mode fock state ( of modes 1 and 2 ) with @xmath35 photons in the first mode and @xmath36 photons in the second mode , and @xmath37 and @xmath38 are two intrinsic free parameters of the resource entangled state , in addition to @xmath39 and @xmath40 , which can be exploited for optimization . note that particular choices of the angle @xmath38 in the class of squeezed bell states eq . ( [ squeezbell ] ) allow one to recover different instances of two - mode gaussian and non - gaussian entangled states : for @xmath41 the gaussian twin beam ( twb ) ; for @xmath42 $ ] and @xmath43 the two - mode photon - added squeezed ( pas ) state @xmath44 ; for @xmath45 $ ] and @xmath43 the two - mode photon - subtracted squeezed ( pss ) state @xmath46 . the last two non - gaussian states are defined as : @xmath47 and are already experimentally realizable with current technology @xcite . in the following section we study , in comparison with the instance of two - mode gaussian entangled resources , the performance of the optimized two - mode squeezed bell states when used as entangled resources for the teleportation of input single - mode coherent squeezed states . for completeness , in the same context we make also a comparison with the performance , as entangled resources , of the more specific realizations ( [ photaddsqueez ] ) , ( [ photsubsqueez ] ) . the characteristic functions of states ( [ cohsqueezst ] ) , ( [ squeezbell ] ) , ( [ photaddsqueez ] ) , and ( [ photsubsqueez ] ) are computed and their explicit expressions are given in appendix [ appendixstates ] . .summary of the notation employed throughout this work to describe the different parameters that characterize the input coherent squeezed ( cs ) states [ eq . ( [ cohsqueezst ] ) ] , the shared entangled two - mode squeezed bell ( sb ) resources [ eq . ( [ squeezbell ] ) ] , and the characteristics of non - ideal teleportation setups @xcite . see text for further details on the role of each parameter . [ cols="^,^,^",options="header " , ] for ease of reference , table [ tableparam ] provides a summary of the parameters associated with the input states , the shared resources , and the sources of noise in the teleportation protocol . the commonly used measure to quantify the performance of a quantum teleportation protocol is the fidelity of teleportation @xcite , @xmath48 $ ] , which amounts to the overlap between a pure input state @xmath49 and the ( generally mixed ) teleported state @xmath11 . in the formalism of the characteristic function the fidelity reads @xmath50 where @xmath51 is the characteristic function of the single - mode input state @xmath52 , eq . ( [ cohsqueezst ] ) , and @xmath53 is the characteristic function for the output teleported state , eq . ( [ chioutfinale ] ) . in this section , we will make use of eq . ( [ fidelitychi ] ) to analyze the efficiency of the cv teleportation protocol . in the instance of non - gaussian squeezed bell resources ( [ squeezbell ] ) , at fixed squeezing parameter , the optimization procedure amounts to the maximization of the teleportation fidelity ( [ fidelitychi ] ) over the free parameters of the entangled resource . it can be shown that the optimal choice for the phases @xmath40 and @xmath37 is @xmath54 and @xmath55 . the analytic expression for the fidelity @xmath56 of the non - ideal quantum teleportation of coherent squeezed states using squeezed bell resources reads @xmath57 \right . \nonumber \\ & & + \frac{1}{4}e^{-2(2r+\tau ) } \delta_{2}^{2}\sin^{2}\delta \left[\frac{1}{\lambda_{1}^{2}}\left(3+\frac{12 \omega_{1}^{2}}{\lambda_{1}}+\frac{4\omega_{1}^{4}}{\lambda_{1}^{2}}\right)+ \frac{1}{\lambda_{2}^{2}}\left(3-\frac{12 \omega_{2}^{2}}{\lambda_{2}}+\frac{4\omega_{2}^{4}}{\lambda_{2}^{2}}\right)\right . \\ & & \left . + \frac{2}{\lambda_{1}\lambda_{2}}\left(1+\frac{2\omega_{1}^{2}}{\lambda_{1}}-\frac{2\omega_{2}^{2}}{\lambda_{2 } } -\frac{4\omega_{1}^{2}\omega_{2}^{2}}{\lambda_{1}\lambda_{2}}\right)\right ] \right\ } \ , , \nonumber \label{telepfidelitycohsq}\end{aligned}\ ] ] where , introducing @xmath58 , the quantities @xmath59 , @xmath60 , @xmath61 , @xmath62 , @xmath63 , and @xmath64 are defined by the following relations : @xmath65 for different choices of @xmath38 in eq . ( [ telepfidelitycohsq ] ) , see section [ secqtelep ] , one obtains the teleportation fidelities associated to photon - added and photon - subtracted squeezed resource states . let us observe that the fidelity in eq . ( [ telepfidelitycohsq ] ) depends both on the input coherent amplitude @xmath24 , and on the input single - mode squeezing parameter @xmath66 , while it is independent of the input squeezing phase @xmath67 . once again , it is worth stressing that , in the teleportation paradigm , the input state is unknown and only partial ( probabilistic ) knowledge on the alphabet of input states is admitted . it is thus required , in principle , to assume teleportation protocols independent of the input parameters , as it turns out to be the case for the vbk protocol with gaussian entangled resources and input coherent states . however , in more general cases , one can study the behavior of the so - called one - shot fidelity , that is the teleportation fidelity at specific values of the input parameters . suitable averages of the one - shot fidelity over the set of input states and parameters , according to an assigned prior distribution , will then result in the average quantum teleportation fidelity . the latter quantity can then be confronted with so - called classical fidelity thresholds ( benchmarks ) that correspond to the maximum achievable average fidelity between the input state ( measured by alice in order to achieve an optimal estimation of it ) and the output state ( prepared by bob according to alice s measurement outcomes ) , without the use of any shared entanglement @xcite . while teleportation benchmarks are available for the cases of coherent input states ( with completely unknown @xmath24 ) @xcite , purely squeezed input states ( with @xmath68 and completely unknown @xmath66 ) @xcite , as well as for states with known squeezing degree and unknown displacement and phase @xcite , a benchmark for the case of input states with totally unknown displacement and squeezing has not yet been derived , and stands as a challenging problem in quantum estimation theory . henceforth , assuming _ a priori _ that the input parameters ( displacement and squeezing degree ) are completely random , we adopt then the following approach to optimize the quantum teleportation fidelity . we exploit a non - unity gain strategy to remove at least the @xmath24-dependence in the one - shot fidelity ; then , we study the behavior of the @xmath24-independent one - shot fidelity for specific values of the input squeezing parameter @xmath66 , in order to identify an effective , @xmath66-independent approximation . indeed , fixing the gain @xmath16 at the value @xmath69 @xmath70 in eq . ( [ telepfidelitycohsq ] ) yields the @xmath24-independent fidelity @xmath71 : @xmath72 where the quantities @xmath59 , @xmath60 , @xmath61 , @xmath62 , @xmath63 , and @xmath64 are defined in eq . ( [ relations ] ) . for different choices of @xmath38 ( see section [ secqtelep ] ) , one obtains the teleportation fidelities associated to the use of different gaussian and non - gaussian entangled resources : the twin beam , the photon - added , and the photon - subtracted squeezed states . for such resources no optimization procedure is possible as @xmath38 is a specific function of @xmath39 . instead , the optimization of the fidelity ( [ telepfidelitysqvac ] ) with respect to the free non - gaussian parameter @xmath38 identifies the optimal squeezed bell resource associated to the optimal value : @xmath73\!\end{array}\!\!. \label{deltaoptfid}\ ] ] let us notice that , for @xmath74 ( ideal protocol ) and @xmath75 ( input coherent states ) , eq . ( [ deltaoptfid ] ) reduces to @xcite : @xmath76 \ , . \label{deltaoptfid2}\ ] ] the displacement - independent one - shot fidelity @xmath71 and the optimal angle @xmath77 are still dependent on @xmath66 , the input squeezing . unfortunately , the optimization of the non - gaussian resource based on the choice ( [ deltaoptfid ] ) as optimal angle would be practically unfeasible because the input squeezing is not known . in order to circumvent this problem , we introduce a sub - optimal angle @xmath78 such that @xmath79 where @xmath80 is a fixed effective value of the input squeezing chosen , according to a suitable criterion that will be clarified below , in the range of possible values of the squeezing parameter @xmath66 . at fixed @xmath66 , and as a function of the angle @xmath38 parameterized by @xmath80 , expressed in db , i.e. @xmath81 , see eq . ( [ deltasubopt ] ) , both in the instance of the ideal protocol @xmath82 ( full lines ) , and of a non - ideal protocol , with @xmath83 , @xmath84 , and @xmath85 ( dashed lines ) . the one - shot fidelities are drawn for three different values of the input squeezing : @xmath86 db . the curves are ordered from top to bottom for increasing @xmath66.,width=321 ] in the following we will express the squeezing parameters @xmath39 and @xmath66 in decibels , according to the relation @xcite : @xmath87 the practical rationale for introducing a sub - optimal characterization in the maximization of the output fidelity is based on the observation that the assumption of a completely random degree of input squeezing @xmath66 is clearly unrealistic . it is instead very sensible to consider that the range of possible values of @xmath66 falls in a window @xmath88 $ ] db . indeed , to date , the experimentally reachable values of squeezing fall roughly in such a range with @xmath89 db @xcite . we can then study the behavior of @xmath71 corresponding to the angle @xmath78 as a function of the effective input squeezing parameter @xmath80 , at fixed squeezing parameters of the resource and of the input state , respectively @xmath39 and @xmath66 , and at fixed loss parameters @xmath21 , @xmath2 , and @xmath7 . [ fig1sfidsbar ] shows that @xmath71 is quite insensitive to the value of @xmath80 . assuming the realistic range @xmath90 $ ] db , the choice of a sub - optimal angle such that @xmath91 db ( average value of the interval ) , leads to a decrease of the optimized fidelity , compared to the choice of @xmath77 , of at most @xmath92 in ideal conditions , and even smaller in realistic conditions . in other words , the teleportation fidelity is essentially constant in the considered interval of variability for the angle @xmath38 . therefore , throughout in the following , we fix @xmath93 db in the expression eq . ( [ deltaoptfid ] ) to make it @xmath66-independent . in fig . [ fig1sfid ] , we plot the teleportation fidelity associated to the various considered resources ( gaussian twin beam , optimized two - mode squeezed bell - like state , two - mode squeezed photon - subtracted state ) both for the ideal protocol ( panel i ) and for the non - ideal protocol ( panel ii ) . we see that , at fixed ( finite ) squeezing @xmath39 of the resource , the gaussian twin beam is always outperformed by the optimal non - gaussian squeezed bell resource in the ideal protocol . it is worth to remark that for very high values of the squeezing @xmath39 , the advantage of the non - gaussian resources fades and gaussian twin beams perform in practice equally well for the teleportation of the considered input states . this reflects the well known fact that , using the ideal vbk protocol and an ideal einstein podolsky rosen resource ( corresponding , e.g. , to a twin beam in the limit @xmath94 ) , _ any _ quantum state can be unconditionally teleported with unit fidelity @xcite . all the one - shot fidelities decrease for increasing squeezing @xmath66 of the input and , interestingly , in the non - ideal protocol they achieve a maximum at a finite value @xmath95 of the squeezing @xmath39 of the resource . the optimal squeezed bell resource and the twin beam share the same @xmath96 db and coincide at that point . in fig . [ fig1sfid ] we also plot the one - shot fidelities associated with the two - mode photon - subtracted squeezed states , eq . ( [ photsubsqueez ] ) . the two - mode photon - subtracted squeezed state always outperforms the twin beam in the ideal protocol , and at low and intermediate values of the resource squeezing @xmath39 in the non - ideal case . it is always outperformed by the optimized squeezed bell resource . we note that , on the other hand , the two - mode photon - added squeezed states always exhibit a performance worse than the two - mode photon - subtracted squeezed states and the squeezed bell resources ( the corresponding fidelities are omitted in the plots for clarity ) . in a given range of the squeezing @xmath39 , @xmath46 and @xmath25 exhibit comparable levels in the fidelity of teleportation . in conclusion , properly optimized non - gaussian resources maximize the fidelity of teleportation of squeezed coherent states both in the ideal and imperfect vbk protocols , outperforming the corresponding gaussian resources . in the next section we carry out a similar analysis with the aim of identifying the optimal strategy that maximizes the reproduction at the output of the input squeezing . in this section , we introduce a different approach to the optimization of the teleportation protocol , aimed at retaining and faithfully reproducing at the output the variances and thus the squeezing of the input state . the strategy is to constrain the first and second order moments of the output field to reproduce the ones of the input field , by exploiting the free parameters of the non - gaussian resources . we introduce the mean values @xmath97 $ ] , with @xmath98 @xmath99 , and the variances @xmath100 - { \rm tr } [ z_j \rho_{j } ] ^{2}$ ] , and @xmath101 - 2{\rm tr } [ x_j \rho_{j } ] { \rm tr } [ p_j \rho_{j } ] $ ] ( the cross - quadrature variance , with @xmath102 denoting the symmetrization ) of the quadrature operators @xmath103 , @xmath104 , associated with the single - mode input state @xmath49 and the output state @xmath11 of the teleportation protocol . the explicit expressions for the quantities @xmath105 , @xmath106 , and @xmath107 are reported in the appendix [ appendixquadratures ] . the quantities measuring the deviation of the output from the input are the differences between the output and input first and second quadrature moments : @xmath108 with @xmath109 given by eq . ( [ eqc ] ) . from the above equations , we see that the assumption @xmath110 ( i.e. @xmath69 ) yields @xmath111 and @xmath112 . therefore , for @xmath110 , the input and output fields possess equal average position and momentum ( equal first moments),and equal cross - quadrature variance ; then , the optimization procedure reduces to the minimization of the quantity @xmath113 with respect to the free parameters of the non - gaussian squeezed bell resource , i.e. @xmath114 . moreover , as for the optimization procedure of section [ sectelepfidelity ] , it can be shown that the optimal choice for @xmath40 and @xmath37 is , once again , @xmath54 and @xmath55 . the optimization on the remaining free parameter @xmath38 yields the optimal value @xmath115 : @xmath116 \ , . \label{deltaoptvar}\ ] ] the optimal angle @xmath115 , corresponding to the minimization of the differences @xmath117 and @xmath118 between the output and input quadrature variances , is independent of @xmath7 , at variance with the optimal value @xmath77 , eq . ( [ deltaoptfid ] ) , corresponding to the maximization of the teleportation fidelity . it is also important to note that in this case there are no questions related to a dependence on the input squeezing @xmath66 . for @xmath119 eq . ( [ deltaoptvar ] ) reduces to @xmath120 . such a value is equal to the asymptotic value given by eq . ( [ deltaoptfid2 ] ) for @xmath94 , so that , in this extreme limit the two optimization procedures become equivalent . in the particular cases of photon - added and photon - subtracted resources , no optimization procedure can be carried out , and the parameter @xmath38 is simply a given specific function of @xmath39 ( see section [ secqtelep ] ) . we remark that , having automatically zero difference in the cross - quadrature variance at @xmath110 , finding the angles that minimize @xmath117 and @xmath118 precisely solves the problem of achieving the optimal teleportation of both the first moments and the full covariance matrix of the input state at once . in order to compare the performances of the gaussian and non - gaussian resources , and to emphasize the improvement of the efficiency of teleportation with squeezed bell - like states , we consider first the instance of ideal protocol ( @xmath119 , @xmath84 , @xmath121 ) , and compute , and explicitly report below , the output variances @xmath122 of the teleported state associated with non - gaussian resources ( i.e. optimized squeezed bell - like states @xmath123 , photon - added squeezed states @xmath124 , photon - subtracted squeezed states @xmath125 ) , and with gaussian resources , i.e. twin beams @xmath126 . from eqs . ( [ varxout])([eqc3 ] ) , we get : @xmath127 eq . ( [ dzsb ] ) is derived exploiting the optimal angle ( [ deltaoptvar ] ) , which reduces to eq . ( [ deltaoptfid2 ] ) in the ideal case . independently of the resource , the teleportation process will in general result in an amplification of the input variance . however , the use of non - gaussian optimized resources , compared to the gaussian ones , reduces sensibly the amplification of the variances at the output . looking at eq . ( [ dztwb ] ) , we see that the teleportation with the twin beam resource produces an excess , quantified by the exponential term @xmath128 , of the output variance with respect to the input one . on the other hand , the use of the non - gaussian squeezed bell resource eq . ( [ dzsb ] ) yields a reduction in the excess of the output variance with respect to the input one by a factor @xmath129 . let us now analyze the behaviors of the photon - added squeezed resources and of the photon - subtracted squeezed resources , eqs . ( [ dzpas ] ) and ( [ dzpss ] ) , respectively . we observe that , in analogy with the findings of the previous section , the photon - subtracted squeezed resources exhibit an intermediate behavior in the ideal protocol ; indeed for low values of @xmath39 they perform better than the gaussian twin beam , but worse than the optimized squeezed bell states . the photon - added squeezed resources perform worse than both the twin beam and the other non - gaussian resources . these considerations follow straightforwardly from a quantitative analysis of the terms associated with the excess of the output variance in eqs . ( [ dzpas ] ) and ( [ dzpss ] ) . moreover , again in analogy with the analysis of the optimal fidelity , for low values of @xmath39 , there exists a region in which the performance of photon - subtracted squeezed states and optimized squeezed bell states are comparable . finally , again in analogy with the case of the fidelity optimization , the output variance associated with the gaussian twin beam and with the optimized squeezed bell states coincide at a specific , large value of @xmath39 , at which the two resources become identical . the input variances @xmath130 ( [ varxin ] ) and ( [ varpin ] ) , and the output variances @xmath131 , are plotted in panels i and ii of fig . [ figvar ] for the ideal vkb protocol and in panels iii and iv of fig . [ figvar ] for the non - ideal protocol . in the instance of realistic protocol , for small resource squeezing degree @xmath39 , similar conclusions can be drawn , leading to the same hierarchy among the entangled resources . however , analogously to the behavior of the teleportation fidelity , for high values of @xmath39 the photon - subtracted squeezed resources are very sensitive to decoherence . in fact , such resources perform worse and worse than the gaussian twin beam for @xmath39 greater than a specific finite threshold value . rather than minimizing the differences between output and input quadrature variances , one might be naively tempted to consider minimizing the difference between the ratio of the output variances @xmath132 and the ratio of the input variances @xmath133 . this quantity might appear to be of some interest because it is a good measure of how well squeezing is teleported in all those cases in which the input and output quadrature variances are very different , that is those situations in which the statistical moments are teleported with very low efficiency . however , it is of little use to preserve formally a scale parameter if the noise on the quadrature averages grows out of control . the procedure of minimizing the difference between output and input quadrature statistical moments is the only one that guarantees the simultaneous preservation of the squeezing degree and the reduction of the excess noise on the output averages and statistical moments of the field observables . we have studied the efficiency of the vbk cv quantum teleportation protocol for the transmission of quantum states and averages of observables using optimized non - gaussian entangled resources . we have considered the problem of teleporting gaussian squeezed and coherent states , i.e. input states with two unknown parameters , the coherent amplitude and the squeezing . the non - gaussian resources ( squeezed bell states ) are endowed with free parameters that can be tuned to maximize the teleportation efficiency either of the state or of physical quantities such as squeezing , quadrature averages , and statistical moments . we have discussed two different optimization procedures : the maximization of the teleportation fidelity of the state , and the optimization of the teleportation of average values and variances of the field quadratures . the first procedure maximizes the similarity in phase space between the teleported and the input state , while the second one maximizes the preservation at the output of the displacement and squeezing contents of the input . we have shown that optimized non - gaussian entangled resources such as the squeezed bell states , as well as other more conventional non - gaussian entangled resources , such as the two - mode squeezed photon - subtracted states , outperform , in the realistic intervals of the squeezing parameter @xmath39 of the entangled resource achievable with the current technology , entangled gaussian resources both for the maximization of the teleportation fidelity and for the maximal preservation of the input squeezing and statistical moments . these findings are consistent and go in line with previous results on the improvement of various quantum information protocols replacing gaussian with suitably identified non - gaussian resources @xcite . in the process , we have found that the two optimal values of the resource angle @xmath38 associated with the two optimization procedures are different and identified , respectively , by eqs . ( [ deltaoptfid ] ) and ( [ deltaoptvar ] ) . this inequivalence is connected to the fact that , when using entangled non - gaussian resources with free parameters that are amenable to optimization , the fidelity is closely related to the form of the different input properties that one wishes to teleport , e.g. quasi - probability distribution in the phase space , squeezing , statistical moments of higher order , and so on . different quantities correspond to different optimal teleportation strategies . finally , regarding the vbk protocol , it is worth remarking that the maximization of the teleportation fidelity corresponds to the maximization of the squared modulus of the overlap between the input and the output ( teleported ) state , without taking into account the characteristics of the output with respect to the input state . therefore , part of the non - gaussian character of the entangled resource is unavoidably transferred to the output state . the latter then acquires unavoidably a certain degree of non - gaussianity , even if the presence of pure gaussian inputs . moreover , as verified in the case of non - ideal protocols , the output state is also strongly affected by decoherence . thus , in order to recover the purity and the gaussianity of the teleported state , purification and gaussification protocols should be implemented serially after transmission through the teleportation channel is completed @xcite . if the second ( squeezing preserving ) procedure is instead considered , the possible deformation of the gaussian character is not so relevant , because the shape reproduction is not the main goal , while purification procedures are again needed to correct for the extra noise added during teleportation when finite entanglement and realistic conditions are considered . an important open problem is determining a proper teleportation benchmark for the class of gaussian input states with unknown displacement and squeezing . such a benchmark is expected to be certainly smaller than @xmath134 in terms of teleportation fidelity , the latter being the benchmark for purely coherent input states with completely random displacement in phase space @xcite . our results indicate that optimized non - gaussian entangled resources will allow one to beat the classical benchmark , thus achieving unambiguous quantum state transmission via a truly quantum teleportation , with a smaller amount of nonclassical resources , such as squeezing and entanglement , compared to the case of shared gaussian twin beam resources . in this context , [ fig1sfid ] provides strong and encouraging evidence that suitable uses of non - gaussianity in tailored resources , feasible with current technology @xcite , may lead to a genuine demonstration of cv quantum teleportation of displaced squeezed states in realistic conditions of the experimental apparatus . this would constitute a crucial step forward after the successful recent experimental achievement of the quantum storage of a displaced squeezed thermal state of light into an atomic ensemble memory @xcite . we acknowledge financial support from the european union under the fp7 strep project hip ( hybrid information processing ) , grant agreement no . here we report the characteristic functions for the single - mode input states and for the two - mode entangled resources . the characteristic function for the coherent squeezed states ( [ cohsqueezst ] ) , i.e. @xmath135 reads : @xmath136 the characteristic function for the squeezed bell - like resource ( [ squeezbell ] ) , i.e. @xmath137 reads : @xmath138 \ , , \end{split } \label{charfuncsb}\ ] ] where the complex variables @xmath139 are defined as : @xmath140 it is worth noticing that , for @xmath41 , eq . ( [ charfuncsb ] ) reduces to the well - known gaussian characteristic function of the twin beam . given the characteristic functions for the single - mode the input state and for the two - mode entangled resource , eqs . ( [ chiinput ] ) and ( [ charfuncsb ] ) , respectively , it is straightforward to obtain the characteristic function for the single - mode output state of the teleportation protocol by using eq . ( [ chioutfinale ] ) and replacing @xmath141 with @xmath142 . in this appendix , we report the analytical expressions for the mean values @xmath97 $ ] , with @xmath98 @xmath99 , and the variances @xmath100 - { \rm tr } [ z_j \rho_{j } ] ^{2}$ ] of the quadrature operators @xmath103 , @xmath104 , associated with the single - mode input state @xmath49 and the output state @xmath11 of the teleportation protocol . we also compute the cross - quadrature variance @xmath101 - 2{\rm tr } [ x_j \rho_{j } ] { \rm tr } [ p_j \rho_{j } ] $ ] , associated with the non - diagonal term of the covariance matrix of the density operator , where the subscript @xmath102 denotes the symmetrization . the mean values and the variances associated with the input single - mode coherent squeezed state ( [ cohsqueezst ] ) can be easily computed : @xmath143 and @xmath144 the mean values and the variances associated with the output single - mode teleported state , described by the characteristic function ( [ chioutfinale ] ) read : @xmath145 and @xmath146 with in the instance of gaussian resource @xmath149 , such quantity simplifies to : @xmath150 for suitable choices of @xmath38 in eq . ( [ eqc2 ] ) , see section [ secqtelep ] , one can easily obtain the output variances associated with photon - added and photon - subtracted squeezed states . s. suzuki , h. yonezawa , f. kannari , m. sasaki , and a. furusawa , appl . 89 * , 061116 ( 2006 ) ; h. vahlbruch , m. mehmet , n. lastzka , b. hage , s. chelkowski , a. franzen , s. gossler , k. danzmann , and r. schnabel , phys . lett . * 100 * , 033602 ( 2008 ) . o. glckl , u. l. andersen , r. filip , w. p. bowen , and g. leuchs , phys . lett . * 97 * , 053601 ( 2006 ) ; j. heersink , ch . marquardt , r. dong , r. filip , s. lorenz , g. leuchs , and u. l. andersen , phys . lett . * 96 * , 253601 ( 2006 ) ; a. franzen , b. hage , j. diguglielmo , j. fiurasek , and r. schnabel , phys . lett . * 97 * , 150505 ( 2006 ) ; b. hage , a. samblowski , j. diguglielmo , a. franzen , j. fiurasek , and r. schnabel , nature phys . * 4 * , 915 ( 2008 ) ; r. dong , m. lassen , j. heersink , ch . marquardt , r. filip , g. leuchs , and u. l. andersen , nature phys . * 4 * , 919 ( 2008 ) . k. jensen , w. wasilewski , h. krauter , t. fernholz , b. m. nielsen , a. serafini , m. owari , m. b. plenio , m. m. wolf , and e. s. polzik , e print arxiv:1002.1920 ( 2010 ) , nature phys . ( advance online publication , doi:10.1038/nphys1819 ) .

we study the continuous - variable quantum teleportation of states , statistical moments of observables , and scale parameters such as squeezing . we investigate the problem both in ideal and imperfect vaidman - braunstein - kimble protocol setups . we show how the teleportation fidelity is maximized and the difference between output and input variances is minimized by using suitably optimized entangled resources . specifically , we consider the teleportation of coherent squeezed states , exploiting squeezed bell states as entangled resources . this class of non - gaussian states , introduced in references @xcite , includes photon - added and photon - subtracted squeezed states as special cases . at variance with the case of entangled gaussian resources , the use of entangled non - gaussian squeezed bell resources allows one to choose different optimization procedures that lead to inequivalent results . performing two independent optimization procedures one can either maximize the state teleportation fidelity , or minimize the difference between input and output quadrature variances the two different procedures are compared depending on the degrees of displacement and squeezing of the input states and on the working conditions in ideal and non - ideal setups .

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Proceed to summarize the following text: in the primary visual area ( v1 ) of many mammals , most neurons respond to the stimulation of two eyes unevenly : they are either right or left eye dominated . in some species , right / left eye dominated neurons are segregated and form a system of alternating monocular regions known as the ocular dominance pattern ( odp ) ( wiesel and hubel , 1965 , 1969 ) . in others , odp is not observed ( see horton and hocking , 1996b for a comprehensive list of species ) . odps , when observed , vary significantly between different species and even between different parts of v1 in the same animal . most modeling studies of odp ( erwin et al . , 1995 ; swindale , 1996 ) have addressed its development . they succeeded in generating odps of realistic appearance . however , several _ why _ rather than _ how _ questions remained unanswered . for instance , ( 1 ) why , from functional point of view , do the odps exist ? ( 2 ) why do some mammalian species exhibit odps while others do not ( horton and hocking , 1996b ; livingstone , 1996 ) ? ( 3 ) why do the monocular regions have different appearance ( stripes as opposed to patches ) between different parts of v1 within the same animal ( levay et al . , 1985 ) ? the question of functional significance of odps has been addressed theoretically using the wiring economy principle ( mitchison , 1991 ; chklovskii , 2000 ) . the idea is that evolutionary pressure to keep the brain volume to a minimum requires making the wiring ( axons and dendrites ) as short as possible , while maintaining neuronal functional properties ( cajal , 1995 ; allman and kaas , 1974 ; cowey , 1979 ; cherniak , 1992 ; young , 1992 ; chklovskii et al . , 2001 ; koulakov and chklovskii , 2001 ) . in many cases these functional properties are specified by the rules of establishing connections between neurons , or wiring rules . the problem presented by the wiring economy principle is therefore to find , for given wiring rules , the spatial neuronal layout that minimizes the total connection length . this approach allows to understand many features in cortical maps , such as orientation preference maps ( koulakov and chklovskii , 2001 ) , as evolutionary adaptations , which minimize the total cortical volume . the goal of this study is to find the simplest model , which on one hand is supported by experimental evidence , and on the other encompasses most of od phenomenology . the use of the simple model allows us to explore its parameter space completely and to give answers to the set of questions above . we also evaluate the dependence of the odp period on the parameters of our model and compare it to the odp periodicity observed in macaque monkey . we find that the experimentally observed variation of the period is in agreement with the wiring economy theory . for the purposes of minimizing the cortical wiring we consider only intra - cortical connections since they constitute the majority of gray matter wiring ( levay and gilbert , 1976 ; peters and payne , 1993 ; ahmed et al . , 1994 ) . we therefore disregard the thalamic afferents and other extra - cortical projections . in an attempt to make wiring economy argument more quantitative , we propose a model describing the component of intracortical circuitry sensitive to od . the principal elements of our model are therefore the connection rules between cortical neurons . to assess the sensitivity of the intracortical wiring to od we examine the connections in the cortical layer @xmath1 , where od is most strongly pronounced . such sensitivity has been studied by katz et al . they made three observations regarding the wiring rules : _ i ) _ neurons in the layer @xmath1 near the interface between two od columns arborize more in home rather than in the opposite eye column . therefore neurons establish more connections with the neurons dominated by the same rather than by the opposite eye . _ ii ) _ axons and dendrites of these neurons have a tendency to bend away from the interface between od columns . this implies that not only they avoid penetration to the opposite od column but also they attempt to maintain sufficiently high number of connections in the home column . _ iii ) _ axons or dendrites penetrating through the opposite eye column to the next same eye column are _ never _ observed . this means that retinotopy has little effect on connections in layer @xmath1 . indeed the neurons on the edges of two same eye columns separated by one opposite eye column have on average receptive fields centered next to each other . if connections in @xmath1 were sensitive to the retinotopic coordinates , these two edges should be connected ( mitchison , 1991 ) . however out of 21 cells examined katz et al . ( 1989 ) observed _ none _ producing axons reaching the next same eye domain . the only possibility for such cells to be connected is due to the overlap between dendritic and axonic arbors of two cells separated by more than @xmath2 @xmath3 m . such possibility is small because of the strong repulsion of the connections by the opposite eye column located between two cells ( observation _ i ) _ ) . these three observations lay the basis of our model which we now describe . the elementary unit of our model mimics the columnar organization of the cortex ( mountcastle , 1957 ) and uniformity of odp along the direction normal to the slab . the elementary unit is therefore a microcolumn , which is defined as a box , spanning the cortex perpendicular to its surface , whose other two dimensions are smaller than the characteristic scale of odp ( @xmath4 ) , and yet large enough to include many neurons . a possible choice of dimensions for such a microcolumn is _ thickness of cortex _ @xmath5 mm @xmath6 @xmath7 , in which case it includes about 310 cells in v1 ( rockel et al . , 1980 ) . the microcolumn units are therefore arranged on a square lattice with 30@xmath3 period . although the choice of the elementary unit size may seem arbitrary , the results of our calculation are independent of the choice . the size of the unit is analogous to the integration step , which does not affect the value of an integral significantly if chosen to be small enough . , width=144 ] motivated by the second observation in layer @xmath1 listed above , i.e. that neurons maintain a fixed number of connections in the home od column , we make the following assumption about the connection rules . each microcolumn unit must establish connections with @xmath8 distinct units dominated by the same eye and @xmath9 units dominated by the opposite eye . these connection rules are illustrated in fig . . only the relative values of @xmath8 and @xmath9 ( rather than absolute ) are important because of the arbitrariness in the definition of microcolumn . thus , if a @xmath10 microcolumn receives @xmath11 projections from the same od column and @xmath12 projections from the opposite od column , a @xmath13 microcolumn receives four times less projections respectively , or @xmath14 and @xmath15 . this is because with coarser units each projection is more effective : connecting to one @xmath13 unit implies connecting to four @xmath10 units . both these implementations of the model produce the same od pattern , discretized in `` pixels '' of different size . the important quantity , which is invariant with respect to the change of `` pixel''/microcolumn size , is the ratio between @xmath8 and @xmath9 ( equal to 10 in this example ) . this is the first parameter of our model . the second parameter of our model is the filling fraction of the units ( microcolumns ) dominated by the left eye afferents @xmath16 with respect to the total number of units , averaged over several odp periods . this parameter is the fraction of the left eye dominated units @xmath16 by @xmath17 . for the majority of important cases @xmath18 , however on the periphery of visual field one of the eyes ( ipsilateral ) is underrepresented . therefore , its filling fraction is less than @xmath19 . the third observation above implies that the component of connections sensitive to od is not sensitive to the retinotopy , and both numbers @xmath8 and @xmath9 do not depend on the position of the receptive field of the unit . this may be due to significant scatter of the receptive field of the cells within cortical column on the scales of about 1 mm ( hubel and wiesel , 1974 ) . the position of the receptive field of the microcolumn is therefore vaguely defined and can not affect od sensitive connections significantly . given these wiring rules we look for an optimal layout of the microcolumn units which minimizes the total length of connections . to find the layout minimizing the total wirelength we employ a combination of computational and analytical techniques . to make our choice of methods clear we first comment on the expected properties of the solution . a possible solution of our model is the _ salt and pepper _ layout in which the units dominated by right and left eyes are uniformly intermixed . in this layout the units belonging to different eyes are not segregated , odp is not formed , and the local values of the filling fraction are equal to @xmath19 ( by local value we understand an average over a domain including many units yet small compared to the period of odp ) . it should be contrasted to the case when units dominated by the same eye fill in large domains i.e. form the odp . in the latter case the local values of the filling factor of each eye vary from 0 to 1 . however one can imagine an intermediate situation when the local filling fraction varies from @xmath20 to @xmath21 , where the amplitude of variation @xmath22 . this corresponds to the case of _ weak segregation _ into odp . the weak segregation is found in squirrel monkey where odp has fuzzy appearance and until recently was suspected not to be formed ( horton and hocking , 1996 ) . if @xmath23 , i.e. the local filling fraction varies from @xmath24 to @xmath25 , the odp s have sharp appearance . using the general terminology from binary mixtures ( cite diblock copolimer paper ) we call this regime the _ strong segregation _ limit . the methods useful in the strong segregation limit are not good in the weak segregation case and vice versa . we use the simulated annealing to find the optimum phases for the strong and nearly strong segregation cases . having found the optimum phase in the strong segregation case to assess the period of odp we use the exact enumeration technique , which compares layouts belonging to the same phase with different periods . the treatment of the weak segregation case requires the use of continuous variables and is done employing the perturbation theory . below we describe these methods in more detail . the parameters of metropolis monte - carlo method ( metropolis et al . , 1953 ) are optimized to render most consistent results for multiple restarts . we use square @xmath26 array of units with periodic boundary conditions . the units are either left or right eye dominated . at each step the algorithm attempts to change the dominance of one unit to the opposite . the value of the average filling fraction @xmath27 is enforced by adding the following term to the total connection length : @xmath28 where @xmath29 and @xmath16 are the current values of the total wirelength and average filling fraction . such term in the functional keeps the current value @xmath16 close to the required value @xmath27 . to map out the phase diagram the values of @xmath16 change from @xmath30 to @xmath31 in @xmath32 increments . the values of @xmath8 and @xmath9 satisfy the condition @xmath33 and are changed in unit increments , i.e. have the following values : 12 , 18 ; 13 , 17 ; 14 , 16 ; 15 , 15 ; 16 , 14 ; etc . the phases at the intermediate points are taken from the nearest points , where result is available . the monte - carlo temperature is gradually annealed from @xmath34 to @xmath35 ( @xmath36 is the total number of units ) in 5000 sweeps through the entire system ( @xmath37 steps ) . the resulting layout is then examined and the phases visually identified . _ salt and pepper _ layout is relatively easy to study due to its uniformity , and can be solved exactly ( chklovskii and koulakov , 2000 ) . if a layout does not deviate significantly from _ salt and pepper _ , i.e. the weak segregation case takes place , it can also be solved exactly . this implies that the wire length can be written as an explicit functional of density distribution of the units . such functional was evaluated and optimized with respect to the density variations by chklovskii and koulakov ( 2000 ) . the optimization shows that odps are formed for the values of parameter @xmath38 . however the simulated annealing method can not distinguish weak segregated odp from _ salt and pepper _ for @xmath39 . there are two reasons for the failure of simulated annealing to do so : * simulated annealing is performed at small but finite temperature that destroys weakly segregated odp . * the units can be either completely right or left eye dominated . this implies that od can change only sharply in the described annealing version . this is useful for obtaining the strongly segregated phases , which occupy major part of the parameter space . however , in the weak segregation limit the local od changes smoothly . thus used version of simulated annealing performs poorly at @xmath40 . we therefore replace the simulated annealing results by those from chklovskii and koulakov ( 2000 ) at small values of parameter @xmath41 ( see the phase diagram below ) . to evaluate the period of odp precisely , we first determine the phase ( _ salt and pepper _ , _ stripes _ , or _ patches _ ) for the given set of parameters @xmath42 and @xmath16 , using methods described above . we then take a lattice containing a large number of units , which exceeds sufficiently the lattice used in simulated annealing . this is possible because the method of determining period described below is much less time consuming than simulated annealing . we then arrange the two types of units on the lattice , using odp determined by the simulated annealing , and vary the period of the pattern to find the period producing the minimum of the wire length . below we describe the procedure for both _ stripes _ and _ patches _ in more details . _ i ) stripes _ to find the optimum period for stripes we use an array containing 300 by 300 units . this array includes three periods of the stripes , which run parallel to one of the sides of the region . each period therefore includes 100 units , containing @xmath43 left and @xmath44 right eye units , @xmath45 . by varying @xmath43 we accomplish the change in the filling fraction of the left ( ipsilateral ) eye , according to the formula : @xmath46 . we consider a string of 100 units at the center of the array , which is representative of all the units in the configuration . for each of the central units the computer program establishes connections , according to the connection rules . most of the calculations are done for @xmath47 . we check that results change for different @xmath48 in a predictable fashion ( see below , results , subsection [ sp ] ) . stripes therefore have a fixed period in terms of number of units ( 100 ) . to find the optimal spatial period of the stripes we vary the shape of each elementary cell in the 300 by 300 array . thus , if the rectangular cell dimensions are @xmath49 perpendicular and @xmath50 parallel to the stripes , we vary both @xmath49 and @xmath50 , keeping the area of elementary cell @xmath51 constant . by doing so we do not change the density of units , but vary the spatial od period , according to the formula @xmath52 . for each value of @xmath49 the cells are reconnected according to the connection rules . special care is taken about exclusion of the boundary effects by making sure that none of the units on the edges of the array is connected to . after the optimum period is found the period in terms of number of units is changed from 100 to another value , closer to the value of spatial period , to check for the absence of geometric artifacts , associated with distortions of elementary cells . the change of the spatial period after this procedure is typically absent but in extreme cases does not exceed 3% . _ ii ) patches _ since our results indicate that a triangular crystal of _ patches _ is formed ( see fig . [ phases]j ) , we consider an array in the shape of parallelogram commensurate with the triangular arrangement of _ patches_. the lattice sites in the array , representing units , are also arranged on a triangular lattice . their positions are given by @xmath53 and @xmath54 , where @xmath55 and @xmath56 are integers varying between 1 and @xmath57 . here @xmath58 is the period of odp to be optimized . the centers of _ patches _ are located at points @xmath59 and @xmath60 . each patch includes lattice sites at the distance from a center determined by the filling fraction of the ipsilateral eye : @xmath61 . the units within / outside the patch are left / right eye dominated . the units in the configuration are then represented by the central parallelogram : @xmath62 . for each of the units connections are made according to the connection rules with @xmath47 . the optimum period is obtained by varying parameter @xmath58 . to determine the experimental dependence of the odp period on the filling factor , the image of odp in macaque monkey ( horton and hocking , 1996a ) is converted into a digital format . in this format the image is represented by a set of pixels . a pixel with coordinates @xmath63 and @xmath64 is represented by a number @xmath65 , equal to @xmath24 for the right eye dominated and @xmath25 for the left eye dominated area . for each position in the image we then determine the local value of the average filling fraction of the ipsilateral eye and the value of local od period . both these calculations are similar . to do the calculation at a certain point in the map , given by coordinates @xmath66 , we surround the corresponding pixel by a square , containing @xmath67 pixels ( black square in fig . [ monkey1 ] , @xmath68 mm ) . the dimensions of the square are such that one hand it contains a few odp periods ( about 3 ) , which is needed for averaging , and on the other hand it is small compared to the characteristic dimensions over which the properties of odp change ( @xmath69 1 cm , see fig . [ monkey1 ] ) . to determine the filling fraction we average the scanned image over the square : for position @xmath66 in the map the local value of the average filling fraction is given by @xmath70 , width=297 ] to determine the local value of odp period we perform the fourier transform of the @xmath71 in the square . as a result we obtain a set of numbers @xmath72 , representing the fourier transform amplitudes , defined on a @xmath67 set of wave vectors @xmath73 . the spectral power , represented by @xmath74 , is shown in fig . [ tf ] for one of the points in the pattern , corresponding to _ stripes_. it clearly has a bimodal appearance , indicating the average in the square direction of the stripes . we then determine the average value of the wave vector , using the formula : @xmath75 ) . the spectral power in the scale bar is in arbitrary units . [ tf ] , width=297 ] the value of the mean odp period is then defined as @xmath76 this value for each pixel in the shaded area in fig . [ monkey1 ] is shown in fig . we start by finding optimal layouts for three illustrative examples of wiring rules with small numbers of connections , @xmath8 and @xmath9 . we caution the reader that because of the small numbers of connections phase assignments may seem arbitrary . these examples are chosen to illustrate our main results which will be confirmed both in the lattice model with large @xmath8 and @xmath9 later in this section and in the continuous model ( chklovskii and koulakov , 2001 ) . for the first two examples we set equal numbers of left and right dominated neurons , @xmath18 . in the first example each neuron connects with equal numbers of the same - eye and other - eye neurons , @xmath77 . then the optimal layout is the `` chess board '' of left / right neurons , fig.[lat1]a . this layout is a realization of the _ salt and pepper _ phase , fig.[phases]a , because each neuron has an equal number of left and right neurons among its immediate neighbors . to calculate the length of connections per neuron , @xmath58 , we notice that in this layout all neurons have the same pattern of connections . by considering one of them , fig.[lat1]a , we find that @xmath78 . this layout is optimal because each neuron makes all of its connections with immediate neighbors . and @xmath77 . ( a ) a realization of the _ salt and pepper _ phase gives minimal wire length ( lattice constants per neuron ) . ( b ) a realization of the _ stripe _ phase is suboptimal ( ) . [ lat1 ] , width=297 ] a suboptimal layout for the same wiring rules is illustrated by a realization of the _ stripe _ phase , fig.[lat1]b . in this layout each neuron has the same pattern of connections up to a mirror reflection . by considering one of them , fig.[lat1]b , we find @xmath79 , greater than @xmath80 for the _ salt and pepper _ phase . here each neuron has among its immediate neighbors only three other - eye neurons , while the wiring rules require connecting with four other - eye neurons . a connection to a more distant neighbor is longer making the layout suboptimal . we confirm the optimality of the _ salt and pepper _ phase for @xmath81 for large @xmath8 , @xmath9 both numerically and analytically . and @xmath82 , @xmath83 . ( a ) a realization of the _ salt and pepper _ is suboptimal ( ) . ( b ) a realization of the _ stripe _ phase gives minimal wire length ( ) . [ lat2 ] , width=297 ] in the second example each neuron connects with more same - eye than other - eye neurons : @xmath82 , @xmath83 . then a realization of the _ salt and pepper _ phase , fig.[lat2]a is not optimal anymore . the length of connections per neuron is @xmath84 , while the _ stripe _ phase , fig.[lat2]b gives @xmath80 . the _ salt and pepper _ phase loses in wiring efficiency because there are not enough same - eye neurons among immediate neighbors and connections with more distant neighbors are needed . the _ stripe _ phase , fig.[lat2]b rectifies this inefficiency by having each neuron make connections only with immediate neighbors . thus , clustering of same - eye neurons is advantageous if each neuron connects more with the same - eye than with the other - eye neurons . and @xmath82 , @xmath83 . realizations of the ( a ) _ salt and pepper _ ( ) and ( b ) _ stripes _ ( ) are suboptimal . ( c ) a realization of the _ l - patch _ phase gives minimal wire length ( ) . [ lat3 ] , width=297 ] in the third example we use the same wiring rules ( @xmath82 , @xmath83 ) but take different numbers of left / right neurons , @xmath85 , @xmath86 . the realizations of the _ salt and pepper _ phase is shown in fig.[lat3]a and of the _ stripe _ phase in fig.[lat3]b . in these layouts , different neurons have different patterns of connections . to find the wiring length per neuron we average over different patterns and find for the _ salt and pepper _ phase @xmath87 and for the _ stripe _ phase @xmath88 . a more efficient layout is the _ l - patch _ phase , fig.[lat3]c , where @xmath89 . although we can not prove that the _ l - patch _ phase is optimal , this seems likely . thus , the optimal shape of monocular regions depends on the relative numbers of left / right neurons . after giving some examples of odps with small numbers of connections @xmath8 and @xmath9 we discuss the opposite case of large numbers . as we show below in section [ sp ] , the shape of od columns in this case does not depend on the absolute values of parameters @xmath8 and @xmath9 . it is determined by the ratio @xmath42 and by the relative amount of ipsilateral neurons @xmath16 ( assuming that the left eye is ipsilateral ) . depending on the values of parameters @xmath42 , and @xmath16 , optimal layout belongs to the one of the eight phases shown in fig . [ phases ] , where ipsilateral and contralateral - eye dominated neurons are shown by black and white regions respectively . these phases can be divided into three major classes . the first class is represented by the unsegregated _ salt and pepper _ layout , in which two types of neurons are uniformly intermixed ( figure [ phases]a ) . the second class includes _ stripy _ layouts , shown in figures [ phases]c , e , g , i. the third class consists of _ patchy _ layouts , displayed in figures [ phases]d , f , h , g. . * a * : _ salt and pepper _ ; * b * : _ stripes _ mixed with _ patches _ ; * c * and * d * : weakly segregated _ stripes _ and _ patches _ obtained by the perturbation theory ; * e * and * f * : weakly segregated _ stripes _ and _ patches _ obtained by simulated annealing ; * g * : modulated _ stripes _ ; * h * : elongated _ patches _ ; * i * : sharp _ stripes _ ; * j * : sharp _ patches _ ; [ phases ] , width=192 ] we distinguish several subclasses of _ stripy _ phases . first , it is the sharp _ stripes _ ( figure [ phases]i ) , which consists of alternating lamellar monocular regions . second , it is the weakly segregated _ stripes _ ( figure [ phases]c , e ) . in this odp the variation of density of left / right eye dominated neurons is small . this is an intermediate pattern between the unsegregated _ salt an pepper _ and the sharp _ stripe _ layouts . this phase is therefore fragile and difficult to obtain numerically . in some cases simulated annealing can produce such a phase , figure [ phases]e . in the other cases the weak segregated phase can only be obtained by the perturbation theory , which can carefully account for a weak variation of neuronal density . such case is shown in figure [ phases]c . third , we also obtain _ stripy _ phases that show a tendency to become _ patches _ , by e.g. their longitudinal modulation , such as shown in figure [ phases]g . similar subclasses exist among _ patchy _ layouts . we obtain sharp , weakly segregated ( obtained from simulated annealing or perturbation theory ) , and elongated _ patches _ , which are shown in figures [ phases]j , f , d , and h respectively . finally , there are mixed phases containing both _ stripes _ and _ patches _ , such as in figure [ phases]b . these odp s are shown on the phase diagram ( pd ) in figure [ phd ] . the phase diagram shows the optimum phase ( minimizing the total wire length ) for given values of parameters @xmath42 and @xmath16 . the important feature of the pd is its left - right eye symmetry . it is apparent from the symmetry of figure [ phd ] with respect to the line @xmath90 . this is a consequence of the left - right eye symmetry of our model , implying that the connection rules , defined by numbers @xmath8 and @xmath9 are independent on whether a neuron is left or right - eye dominated . for this reason the phase for @xmath91 can be obtained from the point with the same @xmath42 and the value of the filling fraction equal to @xmath92 . this corresponds to the replacement of the white regions in figure [ phases ] by black and vice versa . a similar correspondence takes place when one compares odp s in left and right hemisphere . and @xmath42 . for the color key see figure [ phases ] . [ phd ] , width=297 ] another important feature of the pd is the existence of the _ salt and pepper _ region around the line @xmath93 . this implies that the difference between @xmath8 and @xmath9 is the driving force of segregation into odp . the larger the difference , the sharper the odp becomes . the area of the pd adjacent to @xmath18 is occupied by _ odps . at small values of the filling fraction the phases are _ patchy_. a transition from _ stripes _ to _ patches _ occurs at @xmath94 almost independently on parameter @xmath42 . an example of such transition for @xmath95 is shown in figure [ transition ] . , width=297 ] the reasons for the transition for _ small _ values of @xmath96 are discussed in koulakov and chklovskii , 1999 . for _ larger_differences , when odp becomes sharp , the transition occurs due to the presence of surface contribution to the wire length . to demonstrate this we present the following argument , which is rigorously valid in the asymptotic limit of large number of connections to the same - eye neurons , i.e. @xmath97 . in this limit connections to the same - eye neurons are the most abundant and therefore most costly , from wire length prospective . hence , we can disregard connections to the opposite - eye neurons in the first approximation . consider a unit near the interface between two od columns ( top unit in fig . [ connections ] ) . the connection arbor of this unit to the same od units , shown by empty circles in fig . [ connections ] , is strongly biased toward the home column , since the unit has to maintain certain number of connections there . this effect has been seen by katz et al . , 1989 , in macaque striate cortex ( see also the discussion in the model section above ) . for units away from the interface the connection arbor within the same od column is close to a circle ( fig . [ connections ] bottom unit ) . thus , circular arbor renders the minimum wirelength in the absence of constraints , such as the interface between od columns . with the interface present the connection arbor to the same eye neurons is therefore strongly deformed with respect to the optimum . hence , the presence of the interface between the od columns implies an increase in the wirelength , and is therefore associated with a surface cost ( similar to malsburg , 1979 ) . this surface cost drives the transition from _ stripes _ to _ patches_. indeed if @xmath98 the patchy phases have much shorter length of the surface compared to _ stripes_. this is because _ patches _ shrink when @xmath99 reducing their surface length , whereas _ stripes _ become narrower , keeping their surface length the same . however , this is not true for @xmath18 where _ stripes _ have a shorter surface for numerical reasons . therefore , when @xmath16 is decreased , the _ stripes _ should unavoidably condense into _ patches _ to minimize the surface cost . this is similar to droplets of water assuming circular shape to minimize the surface energy . , width=297 ] we conclude therefore that two important transitions occur in our model . * the transition from unsegregated _ salt and pepper _ to weakly segregated and then sharp odp is driven by the difference between parameters @xmath8 and @xmath9 characterizing the intra - cortical circuitry . * the transition from _ stripy _ to _ patchy _ odp is driven by the decreasing filling fraction of the ipsilateral eye and occurs at @xmath100 , if left eye is underrepresented . our phase diagram in fig . [ phd ] shows that the transition from _ stripes _ to _ patches _ occurs when @xmath101 for a wide range of @xmath42 . this value will be compared now with the value of @xmath16 at which the transition occurs in the experiment , found from the relative area occupied by left eye dominated neurons . the conclusion that the _ patch _ phase becomes optimal when contralateral eye dominates is , indeed , non - trivial , because there may be a system of alternating wide and narrow monocular stripes instead . we test our conclusion on the data from macaque monkey first ( horton and hocking , 1996a ) . the relative area occupied by the left / right eye depends on the location in v1 . in the parafoveal part of v1 both eyes are represented equally , i.e. @xmath102 . odp has a stripy appearance , in agreement with the phase diagram . away from the foveal region contralateral eye becomes dominant . the odp becomes patchy there ( levay et . al . , 1985 ) , just as expected from the theoretical phase diagram . we verify the location of the transition by using the following algorithm . we find @xmath16 for each point of the pattern by calculating the relative area occupied by the left / right regions in a window centered on that point and a few od periods wide ( dashed lines in fig . [ macaque ] ) . then we draw a contour corresponding to @xmath103 , fig . [ macaque ] . we observe in fig . [ macaque ] that stripes indeed become patchy at the black contour indicating @xmath104 . . shown is a fragment of the macaque odp from horton and hocking ( 1996a ) . areas dominated by the left eye are grey and neurons dominated by the left eye are white . the black contour corresponds to @xmath103 averaged over a window , whose dimensions are shown by the dashed square ( @xmath68 mm ) . the points of transition from _ stripes _ to _ patches _ coincides with the black contour . [ macaque ] , width=297 ] in _ cebus _ monkey the odp has a similar transition ( rosa et al . , 1992 ) . for monkey co6l from rosa et al , 1992 , we determine visually that along the horizontal meridian the transition occurs at the eccentricity of @xmath105 . according to the plot of the relative representations given in rosa et al . , 1992 , @xmath16 changes in the range @xmath106 at these eccentricities . our theoretical conclusion about a transition at @xmath103 falls into this interval . for the upper @xmath107 degree meridian of the same monkey the transition occurs at the eccentricity of @xmath108 degrees or at filling fractions @xmath109 . again , the predicted value belongs to this interval . we conclude that these data are consistent with the results of our model . in cats the odps have a patchy appearance ( anderson et al . , 1988 ; jones et al . , 1991 ) . in this case our theory implies that one of the eyes should dominate . according to some reports ( shatz and stryker , 1978 ; crier et al . , 1998 ) the filling fraction of the contralateral eye in cat v1 is about @xmath31 in young animals ( before p22 ) . this may lead to _ patches _ in cat v1 . the strong contralateral bias disappears in older animals ( crier et al . , 1998 ) . this is consistent with other reports ( anderson et al . , 1988 ) that both eyes are represented almost equally in adult cats . one of the general features of our model is that the period of od pattern becomes larger , when the total number of connections is increased . indeed , the size of the connection arbors grows if both @xmath8 and @xmath9 are increased , given that the density of units ( @xmath110 ) is kept constant . this is because one has to go further to find the necessary number of connections to satisfy the wiring rules . since dimensions of the connection arbors set up a characteristic scale for the od pattern , the period of the latter also increases . this property of our model , which we call _ scalability _ , is discussed in this subsection . let us define scalability in a mathematically rigorous fashion . assume that one has found a minimum wire length configuration for certain set of parameters @xmath8 , @xmath9 , and @xmath16 ( or @xmath111 ) . assume that both @xmath8 and @xmath9 are very large . in this case the dimensions of connection arbors are much larger than the lattice spacing , and one can ignore the fine structure imposed on the connection arbors by the square lattice . this is exactly the limit in which our model has some validity , both because realistic numbers of neuronal connections are large and because we would like to avoid artifacts introduced by the square lattice . an example of connection arbors for a neuron for some set of @xmath8 and @xmath9 are shown in fig . [ scalaba ] ( left ) . this neuron and its connection arbors resemble the top neuron , marked by the star , in fig . [ connections ] . the connection arbors in fig . [ scalaba ] look like continuous circular pieces , due to the large @xmath8 and @xmath9 limit ( square lattice makes the boundaries of connection arbors look like staircases , whose steps are too small to show in the picture for large @xmath8 and @xmath9 ) . imagine now a geometric transformation , in which the dimensions of the connection arbors of all of the neurons , as well as the od pattern itself , are blown up by the same scaling factor @xmath112 . the two - dimensional density of the neurons must be preserved during this transformation . the obtained new od pattern and the new connection arbors are shown schematically in fig . [ scalaba ] ( right ) . since the transformation is applied to the two - dimensional objects , and each of the dimensions was stretched by the factor @xmath113 , each neurons in the new configuration will receive @xmath114 and @xmath115 connections from the same and opposite eye neurons . the newly obtained configuration ( fig . [ scalaba ] right ) will satisfy wiring rules with connection numbers given by @xmath114 and @xmath115 . note that the filling fraction is not changed by this transformation . it is @xmath18 in fig . [ scalaba ] . will this be the minimum wire length configuration for this set of parameters ? to prove that the new configuration minimizes the total wire length for the new set of parameters @xmath114 and @xmath115 we notice that the total wirelength for the new configuration is given by @xmath116 , where @xmath29 is the total wirelength for the original configuration ( fig . [ scalaba ] left ) . this is because each neuron now receives the number of connections increased by @xmath117 , and each connection is stretched by @xmath113 . imagine now that one finds a new configuration , which has the total connection length @xmath118 . let us take this more optimal configuration and shrink it by the factor of @xmath113 . we obtain a configuration , satisfying wiring rules for the set @xmath8 and @xmath9 , whose total wirelength is @xmath119 . but this contradicts to our postulate that the original configuration in fig . [ scalaba ] ( left ) is optimal for the set of parameters @xmath8 and @xmath9 . thus the stretched configuration provides the minimum of the wirelength for the new set of parameters @xmath114 and @xmath115 . this property is important , because once the solution for given @xmath8 and @xmath9 is found , one can obtain other solutions , with the set of parameters @xmath114 and @xmath115 , by stretching the original configuration by the factor of @xmath113 uniformly in all the directions . the important property which remains the same for all these related configurations is the ratio between the numbers of the same and other eye connections , @xmath42 . thus , we conclude that this ratio determines the shapes of the od patterns , which is unchanged during the uniform stretching procedure . what is changed in the uniform stretching is the odp period ? since the period is proportional to the stretching parameter @xmath113 , while the total number of connections is proportional to @xmath117 , we conclude that the period is proportional to the square root of the total number of connections , when the ratio @xmath42 is kept constant . this implies that @xmath120 here @xmath121 , where @xmath122 is the size of the microcolumn unit . parameter @xmath123 has a meaning of the average axonal arbor diameter . the quantity @xmath124 is the _ normalized od period _ , calculated in the units of the average axonal diameter . this quantity is introduced here for easier comparison to the experiment . notice that this quantity does not depend on the total number of connections . the latter dependence is entirely absorbed by the parameter @xmath123 . scalability is valid for the limit of large @xmath8 and @xmath9 , when square lattice effects can be ignored , and our model becomes continuous . does scalability apply to our numerical solution , described in subsection [ exenum ] ? to check this we plot the ratio @xmath125 , obtained using methods described in [ exenum ] , for different values of the total number of connections @xmath126 in fig . [ scalab ] . if eq . ( [ scalab_form ] ) is valid , this ratio should not depend on the total number of connections . as evident from fig . [ scalab ] this property is indeed satisfied . hence , below in this subsection we always present the results for @xmath127 , which are independent on the total number of connections , assuming that the latter dependence can be easily recovered using eq . ( [ scalab_form ] ) . , top neuron . the stretched configuration is shown on the right . [ scalaba ] , width=297 ] we now examine the dependence of normalized period @xmath128 [ see eq . ( [ scalab_form ] ) ] on the parameter @xmath42 , for @xmath90 , when we have to consider the stripe phase , according to subsection [ phdsection ] . the results are shown in fig . these results have been obtained using methods described in subsection [ exenum ] . in general , the normalized period increases with increasing parameter @xmath42 . this increase in the od period can be understood considering the interplay between connections to the same and opposite eye units . indeed , the presence of connections between the same eye units implies affinity between the same od neurons . an increase in the relative number of such connections ( @xmath42 ) strengthens such affinity . the od columns provide a neighborhood rich with the same eye neurons ; even more so , on average , for coarser domains . thus stronger affinity between the same eye neurons ( @xmath42 ) leads to an increase in the size of od domains . this effect is produced by wiring economy principle , since the latter provides a substrate for the affinity of connected neurons . , width=297 ] we now examine fig . [ fs ] in more detail . the relative period diverges in the limit @xmath129 . the divergence can be described by the asymptotic formula @xmath130 shown in fig . [ fs ] by the dotted curve . the divergence can be understood as follows . imagine that the neurons do not have to connect to the neurons of the opposite od , i.e. parameter @xmath131 , @xmath132 , which corresponds to the extreme case @xmath129 . in this case the optimum wire length configuration consists of only two large domains , dominated by left and right eye neurons , occupying a half of v1 each . this is because of the notion of surface contribution , introduced in subsection [ phdsection ] . to minimize this interface contribution the system phase segregates into two large domains thus , in the case @xmath131 odp has maximum possible period , spanning the entire v1 . this explains the tendency of the period diverge in the limit @xmath133 ( @xmath134 ) in fig . [ fs ] . what happens if @xmath133 ? since the neurons now have to connect to the opposite eye neurons , phase segregated configuration ( two large domains spanning the entire v1 ) is no longer optimum . the system introduces more interfaces between od columns to shorten distances between opposite eye neurons . more interfaces implies a reduction in the od period . thus , finite @xmath42 leads to the finite od period . this is reflected by the asymptotic dependence ( [ asymp_p ] ) and the dotted curve in fig . [ fs ] . an interesting phenomenon observed in fig . [ fs ] is the presence of a few discontinuous changes in the od period . one such a change occurs at @xmath135 and consists in an abrupt increase in the od period by a factor of about @xmath136 . another discontinuous transition occurs at @xmath137 and the corresponding increase in the period is by a factor of @xmath138 . note that these transitions are truly discontinuous , i.e. they occur at discrete points . they appear smooth in fig.[fs ] due to a sparse sampling ( the real data points are shown by dots ) . note also that the quantity @xmath123 in eq . ( [ scalab_form ] ) changes negligibly in the interval between @xmath139 and @xmath140 , which implies that both od period @xmath141 and the normalized period @xmath142 change approximately by the same factor . such discontinuous changes in the od period in the weakly segregated regime , i.e. when the odp is not well defined , may be responsible for the coarsening of odp in strabismic squirrel monkeys ( see discussion for more details ) . the dependence of the normalized period @xmath124 on the filling fraction of the left eye @xmath16 is shown in fig . these results have been obtained using methods described in subsection [ exenum ] . four dependencies are shown , for four values of the parameter @xmath42 marked on each curve . the general tendency for the period to grow with increasing parameter @xmath42 , described in the previous subsection , is evident in the figure . , width=297 ] for small values of parameter @xmath42 the period increases when the filling fraction moves away from @xmath90 , as long as one stays within the same phase ( _ stripes _ or _ patches _ ) . this behavior is seen for the two bottom curves in fig . [ fr ] . in the transitional region between _ stripes _ and _ patches _ the od period experiences a discontinuity , marked by the dotted lines . for large values of @xmath42 , the dependence of the period on @xmath16 is opposite : the period decreases , as the filling factor deviates from @xmath19 , as demonstrated by the top curve in fig . [ fr ] . , width=297 ] we now compare this behavior of our model to the observations in the striate cortex of macaque monkey ( horton and hocking , 1996a ) , using fourier transform method described in subsection [ fourier ] . to make this comparison possible the following technical consideration is taken into account . the fourier transform method evaluates the average value of the spatial frequency of the odp @xmath143 . the od period is then calculated by the formula @xmath144 . for the _ stripe _ phase this procedure results in the value , which is close to the period of stripes . for _ patches _ it results in the distance between rows of patches , which is smaller than the period by the factor @xmath145 ( see fig . [ rows ] ) . thus , to make comparison to the experiment possible , the values in fig . [ fr ] corresponding to _ patches _ should be multiplied by the factor @xmath146 . the result of this procedure is shown in fig . [ fp ] by the gray line . ) . the distinction between _ patch _ period and distance between rows should be taken into account for accurate comparison to the experimental observations . [ rows ] , width=240 ] fig . [ fp ] shows that the period observed in the experiment decreases when the filling factor of the ipsilateral eye deviates from @xmath19 . this warrants the use of the top curve in fig . [ fr ] to represent the theoretical result . since , the shape of the theoretical dependence does not change much when @xmath147 , the parameter @xmath42 can not be established from the comparison of the theory to the experiment . to obtain the gray curve in fig . [ fp ] the @xmath148 dependence in fig . [ fr ] was multiplied by the fitting parameter @xmath149 mm . this is the only fitting parameter used . as seen in fig . [ fp ] , our theory describes the experimentally observed dependence quite well . . the latter is the top curve in fig . [ fr ] , with the sector of the data corresponding to _ patches _ corrected by the factor for compatibility with the fourier transform . the only fitting parameter used is @xmath149 mm [ see eq . ( [ scalab_form ] ) ] . [ fp ] , width=297 ] the widths of the ipsilateral and contralateral eye stripes in macaques has also been measured independently by tychsen and burkhalter ( 1997 ) . based on their results one can evaluate the odp period and the filling fraction : @xmath150 here @xmath151 , @xmath152 , and @xmath153 are the filling fraction of the ipsilateral eye , and the ipsilateral / contralateral column widths respectively . the dependence of the period on the filling fraction can therefore be established . this dependence is shown in fig.[tychsen ] . the best parabolic fit to the data in fig.[tychsen ] is given by : @xmath154.\ ] ] the coefficient @xmath155 is estimated using bootstrap ( efron and tibshirani , 1993 ) . the expectation value of the coefficient is therefore below zero , as seen from fig.[tychsen ] . the probability of the coefficient to be greater than zero is @xmath156 as evaluated by bootstrap procedure , which is used since the distribution of coefficients @xmath157 is non - gaussian . this implies that with great degree of certainty one can assume that the period of odp does decrease with the filling fraction deviating from @xmath19 . it should be noted that the value of coefficient @xmath157 can be obscured by the variability of odp period between different individuals , since data in fig.[tychsen ] are assembled from three monkeys ( four v1 s ) . to reduce the impact of inter - individual variability we then normalized the period for each area v1 by the mean value for each individual animal . the value of the coefficient is then @xmath158 , with the probability of positive coefficient @xmath159 . thus the decrease of the period with filling fraction is even more certain , when the inter - individual variability is accounted for . the value of coefficient @xmath157 obtained from the theory is @xmath138 ( fig . [ fr ] , @xmath148 ) . it is consistent with both measurements . , width=297 ] in this work we propose a model which can account for most of experimentally observed features of odps . our model has two principal parameters . the first parameter characterizes the intracortical circuitry . it is the difference between the number of connections to the same and to the opposite od neurons . our results suggest that this difference is the driving force of segregation into odps . we argue therefore that the sensitivity of the intra - cortical connectivity to od provides a reason to formation of od columns ( see below ) . the second parameter is the fraction of neurons dominated by the ipsilateral eye . this parameter determines the shape of monocular regions in odp . in the majority of the primary visual area of macaque and _ cebus _ monkeys this parameter is close to @xmath160 , which implies that both ipsi- and contralateral eyes are equally represented . however , in the proximity of monocular crescent the ipsilateral eye becomes underrepresented . this is because the inputs into the eye are blocked by the nose of the animal . our theory suggests that the decrease in the filling fraction of the ipsilateral eye drives the transition in the odp structure from stripy ( zebra skin like ) to patchy ( similar to leopard skin ) . the transition occurs when the fraction of the ipsilateral eye dominated neurons approaches @xmath0 in both macaque and _ cebus _ monkeys ( see below ) . we also analyze the dependence of od period on the parameters of our model and find satisfactory agreement with experimental data . each neuron in our model establishes certain number of intra - cortical connections with neurons dominated by the same and the opposite eye . as suggested by experimental studies in macaque striate cortex , neurons in layer @xmath1 typically make more connections with neurons of the same od ( katz et al . , 1989 ) . thus , from wiring economy prospective , connections with neurons of the same od are more important than the opposite eye connections . therefore , it is advantageous to form od columns , since they provide environment rich with the same od neurons , which results in shortening connections to the same eye neurons . the wiring economy principle thus provides a natural reason for the existence of od patterns , i.e. answers the first question in the program listed in the introduction . our model suggests that in primates with weakly defined od columns , such as squirrel monkey ( hubel et al . , 1976 ; livingstone , 1996 ; horton and hocking , 1996b ) and owl monkey ( livingstone , 1996 ) , the difference between these two components of intracortical connectivity is not large . such difference may be increased in these animals by experimentally induced strabismus . indeed , strabismus reduces correlated activity between opposite od cortical neurons , therefore reducing their connectivity @xmath9 . reduction in @xmath9 unbalanced by the corresponding reduction in @xmath8 increases the parameter @xmath42 and leads to sharpening of od columns , according to our phase diagram in fig . [ phd ] . such sharpening is indeed observed experimentally ( shatz et al . , 1977 ; lowel , 1994 ; livingstone , 1996 ) . this phenomenon was also predicted theoretically by goodhill ( 1993 ) . the two parameters of intra - cortical circuitry , @xmath8 and @xmath9 , represent in our model the interplay between two classes of processing performed by the visual cortex . the first class includes the processing of the monocular image , for which connections to the same od neurons are necessary . the second class includes various tasks related to stereopsis , which require comparison of two monocular images , relying on the connections between the opposite od neurons . we proposed above that the function of od columns is to shorten the connections between the same eye neurons . the impact of stereoscopic vision should therefore be the opposite : strong stereoscopy should make odp weaker . inversely , weak stereoscopy induces sharp odps . the latter statement is justified by the observations in animals with experimental strabismus . however , one should be careful about this statement , since the relation between functional and anatomical properties may not be direct . the situation in the animals with lateral eye positioning , such as mice , rats , tree shrews , etc . , is different . in such animals the visual pathway is almost completely crossed , i.e. v1 in each hemisphere is strongly dominated by the contralateral eye [ drager , 1974 , 1975 , 1978 ; drager and olsen , 1980 ; gordon and stryker , 1996 ( mouse ) ; hubel , 1977 ( rat ) ; casagrande and harting , 1975 ; mully and fitzpatrick , 1992 ( tree shrew ) ; horton and hocking , 1996b ( other species ) ] . as suggested by antonini et al . this implies that the odp contains only two large monocular columns , each spanning the whole hemisphere . this can be interpreted as an od having very large period , spanning both striate cortices . this picture can be fitted into the framework of our model . indeed , we predict that if the number of connections to the other od neurons ( @xmath9 ) is very small the od has a very large period ( fig.[fs ] ) . thus our model predicts that the number of connections received by each neuron from the neurons of the same od ( @xmath8 ) is much larger than the number of opposite eye connections ( @xmath9 ) in the species with lateral eye positioning . this should include the cross - hemispheric projections . this statement should have functional consequences . since @xmath9 is small , synthesis of images from two eyes is weaker . therefore the animals have to find another strategy to implement stereopsis . hooded rats for example use vertical head movements to estimate distances ( legg and lambert , 1990 ) . our conclusion about small @xmath9 should also apply to the superior colliculus in these animals ( colonnese and constantine - paton , 2001 ) , as well as to the tectum in lower vertebrates ( schmidt and tieman , 1985 ) , in which cases the visual inputs cross over almost completely too . to summarize , our model encompasses most of the phenomena related to the sharpness and observability of odp . it relates the interspecies variability in the odp to the relative amount of binocular interaction occurring in different species . thus , we predict , that in the animals with weakly segregated columns ( squirrel monkey ) @xmath40 ; in the animals with sharp columns ( macaque ) @xmath8 is much larger than @xmath9 [ confirmed by katz et al . , ( 1989 ) ] ; and , finally , in the animals with lateral eye positioning , @xmath9 should have a value , whose contribution to the wirelength can be neglected . another consequence of a decreasing @xmath16 in macaque is a decrease in the odp period ( levay et al . , 1985 ) . in fig . [ fp ] we compare the result of our theory to the data from macaque monkey ( fourier transform applied to data from horton and hocking , 1996a ) . from this comparison we conclude that , according to the wiring economy principle , parameter @xmath161 , or cells establish much more connection with the same od cells , than with the opposite . this result of is consistent with the observations of od sensitive circuitry in the striate cortex of macaque by katz et al . ( 1989 ) . we chose the regions proximal to the horizontal meridian for this comparison . this is based on the assumption that cortical properties , such as @xmath8 and @xmath9 , change little along this meridian . this assumption is in part supported by the fact that od periodicity changes little on the large segment of the meridian occupied by stripes , spanning the region between about 2 and 25 degree eccentricity ( notice very little scatter in fig . [ fp ] around the point @xmath90 ) . the changes in the period begin to occur when @xmath16 deviates from @xmath19 . this is illustrated by fig . other authors notice a decrease in the period when comparing vertical to horizontal meridian . studies based of computer reconstructions report about 2 fold decrease in od periodicity comparing these areas ( levay et . al . , 1985 ) , while others , based of flat - mounts ( horton and hocking , 1996a ) , report a more moderate change . such variation can not be accounted for by a decrease in parameter @xmath16 alone , since @xmath16 is about @xmath19 on both meridians in close proximity to parafoveal region ( < 20 degrees of eccentricity ) . our model suggests two possibilities based on the variation in the intracortical circuitry , described by @xmath8 and @xmath9 . since such differences in the circuitry may also be responsible for variability of the od period between different animals , we discuss this possibility in the next subsection . studies in macaque monkeys ( horton and hocking , 1996a ) reveal large inter - individual variability of the stripe period . the stripe period is @xmath162 along the v1 border , after comparison of 6 animals . two factors may contribute to this phenomenon in the framework of our model . ( 1 ) the basic diameter of axonal and dendritic arbors @xmath123 varies from animal to animal . this could be due to changes in @xmath8 , @xmath9 , or neuronal density . ( 2 ) the ratio between monocular and binocular interactions @xmath42 varies . the former reason is justified by eq . ( [ scalab_form ] ) . the latter can be understood from fig . simply speaking , monocular interactions ( @xmath8 ) favor formation of od columns , making them wider , in an effort to provide same od rich environment for all the neurons . binocular interactions ( @xmath9 ) favor interfaces between columns , since interfaces bring opposite od neurons closer to each other . they therefore decrease od period . when @xmath42 increases the od period increases too ( fig . this may occur when comparing different individuals . similar consideration may apply to the experiments in strabismic animals ( lowel , 1994 ; livingstone , 1996 ) . since strabismus reduces correlations between eyes , its effect in our model is to reduce parameter @xmath9 . hence , the ratio @xmath42 is increased . according to our model ( fig . [ fs ] ) this generally leads to an increase in the relative od period ( ratio of the basic od periodicity to the connection range @xmath123 ) . this result is understood from the interplay between affinity between the same eye neurons ( @xmath8 ) , increasing the period , and the affinity between opposite od neurons ( @xmath9 ) , reducing od period . since the latter is reduced by strabismus , the od period grows . the degree of the period change depends on the decrease in the number of interocular connection , and is difficult to estimate . a curious feature displayed by od period in our model is an abrupt increase at @xmath163 by a factor of about @xmath136 , cf . this implies that close to point @xmath164 the od period may be very sensitive to developmental manipulations . this finding may have correlate in squirrel monkey , for which the observed increase in period is by a factor @xmath165 ( horton and hocking , 1996b ) . these data are obtained from comparison to a single strabismic animal . the following scenario is possible , comparing squirrel monkey to the strabismus experiments in owl monkey ( livingstone , 1996 ) , in which no significant increase in periodicity is observed . parameter @xmath42 in squirrel monkey passes the point @xmath164 due to strabismus , leading to the substantial increase in period . in owl monkey parameter @xmath42 is above @xmath166 in normal animal . strabismus therefore has little effect . this scenario is consistent with sharper od columns in normal owl monkeys ( @xmath167 ) than in normal squirrel monkeys ( @xmath168 ) ( livingstone , 1996 ; horton and hocking , 1996b ) . experimentally induced strabismus in cat leads to an increase in the od period by a factor of @xmath169 ( lowel , 1994 ; goodhill , 1993 ; see however jones et al . , 1996 ) . our model suggests that parameter @xmath167 in cat , and the increase in the period is due to the smooth part of the dependence in fig . [ fs ] , which may not be so substantial as in squirrel monkey . the relevance of wiring economy principle to the neuronal spatial organization can be illustrated by the following thought experiment ( koulakov and chklovskii , 2001 ) . imagine taking a cortical area and scrambling neurons in that area , while keeping all the connections between neurons the same . since the circuit is unchanged , the functional properties of the neurons remain intact . therefore , from the functional point of view , the scrambled region is identical to the original one . in fact , the only difference caused by scrambling is in the length of neuronal connections . therefore , it is hard ( if not impossible ) to justify the existence of systematic cortical maps , such as od pattern , without invoking the cost of making long neuronal connections . although some theories of map formation may not explicitly mention the wiring optimization principle , it is present implicitly , usually in requiring the locality of intra - cortical connections . how important is the constraint imposed by wiring minimization ? in principle one can imagine development of an organism , which has 30% excess of wire with respect to the existing ones . it turns out that the existence of such an organism is close to impossible . indeed , imagine that an external object , such as a blood vessel , is introduced in certain area of the gray matter . in this case some of the neuronal connection have to go around the vessel , therefore increasing in length . if the nerve pulses are to be delivered at the original speed and/or intensity , the elongated axons and dendrites have to be made thicker , to increase the pulse propagation speed and decrease dendritic attenuation . this leads to more obstacles on the way of other neuronal connections and so on . thus , introduction of a new blood vessel leads to an infinite series of axonal and dendritic reconstructions . the same is true about the extra connection volume , resulting from wasteful neuronal positioning . such infinite series of reconstructions can diverge , which implies that the connection volume resulting from more and more reconstructions increases indefinitely . in this case the new blood vessel can never be inserted without sacrificing significantly the brain function . it turns out that mammalian brain has reached the verge of this so called _ wiring catastrophe _ ( chklovskii and stevens , 2001 ) , so that it gets increasingly more difficult to accommodate excess volume in the nerve tissues . the wiring catastrophe occurs when the fraction of axons and dendrites in the cortical volume reaches @xmath170 . electron microscopy studies of cortical slices show that the actual volume occupied by neuronal processes is about @xmath171 ( chklovskii and stevens , 2001 ) . the brain therefore has approached the barrier imposed by wiring catastrophe . further increase of the volume fraction of neuronal processed may deteriorate the brain function . as discussed in the previous subsection , wiring optimization is the only known way to relate neuronal layout ( as manifested in the odp ) to the statistics of neuronal connectivity . models of the odp development that do not explicitly rely on wiring optimization invoke it implicitly , usually requiring the locality of intra - cortical connections . in his pioneering work , mitchison ( 1991 ) studied a question whether odp minimize the wiring volume relative to the _ salt and pepper _ layout . he assumed that the inter- neuronal connectivity is determined both by ocular dominance and retinotopy with all neurons having the same connectivity rules . he found that the answer to this question depends on the detailed assumptions about axonal branching rules . in particular it depends on the value of axonal branching exponent @xmath157 . he has shown that if all axonal segments have the same caliber ( @xmath172 ) , than odp s are indeed advantageous for certain range of ratios of same - eye to opposite - eye connections . he also showed that if @xmath173 than the odp do not save wiring volume relative to _ salt and pepper_. however , existing data seems to suggest that axonal caliber branches with @xmath173 ( deschenes and landry 1980 , adal and barker , 1965 ) . the case of axonal branching with the cross - sectional area conservation corresponds closely to our model because we require a separate connection for each neuron . the reason we find that odp minimize wiring length is because we drop the retinotopy requirement on inter - neuronal connection rules , an assumption supported by the experimental data ( katz ) . although , effectively connections are roughly retinotopic , connection rules may vary from neuron to neuron thus providing some flexibility . the advantage of our approach is its simplicity allowing us to map out a complete phase diagram and make experimentally testable predictions . the full theory of the odp will require a detailed analysis of axonal branching which must rely on better knowledge of axonal branching rules . jones et al . ( 1991 ) proposed an explanation for why odp have either stripy or patchy appearance . they assumed that neurons are already segregated into the odp ( by considering units whose size equals the width of monocular regions ) and found that the difference between stripy and patchy appearances of the odp could be due to the boundary conditions , i.e. different shape of v1 in different species . although the correlation between the shape of v1 and the odp layout is observed , the model of jones et al . does not explain why peripheral representation of macaque v1 has patchy odp or why ocular dominance stripes run perpendicular to the long axis of v1 in some parts of v1 but not in others . moreover , it is the local structure of odp that is likely to determine the shape of v1 and not the other way around . therefore , unlike jones et al . , our work proposes a unified theory of odp including _ salt and pepper _ , stripy and patchy layouts , and is based on local inter - neuronal connectivity rules . another model related to wiring length minimization is the elastic net model studied by goodhill and coworkers ( 1993 ) . the original formulation of the model minimized the cost function which penalized for placing nearby neurons whose activity is not correlated , a choice justified by computational convenience . later the elastic model was generalized by the introduction of a c measure . maximization of c measure effectively corresponds to penalizing for placing correlated neurons far apart . unlike wiring optimization the penalty does not increase beyond a distance called cortical interaction . because of this , elastic net often yields solutions where left and right eye neurons are completely segregated into left and right eye maps . our wiring optimization models can be viewed as a sub - set of models described by c measure . the advantage of our wiring optimization approach is that it has a transparent biological justification for the cost of placing neurons far from each other - the cost of wiring . because of this , wiring optimization is a natural choice for questions related to the anatomy of intra - cortical connections . wiring optimization provides a link between neuronal connectivity and spatial layout . thus , it leaves open the connection between connectivity and computational function . unlike most other models , which assume that neurons should be connected only if they are correlated , wiring optimization makes other assumptions about connectivity possible , for example connecting neurons with anti - correlated firing . our theory relates functional requirements on the neuronal circuits to its structural properties . in particular , our model relates the amounts of the neuronal intraocular and interocular interactions , and the filling fraction of ipsilateral neurons , to the structure of the ocular dominance pattern . we conclude that the interspecies variability in the ocular dominance patterns may be explained by differences in the underlying cortical circuitry . adal mn , barker d ( 1965 ) intramuscular branching of fusimotor fibers . j physiol 177:288 - 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3808 . jones dg , murphy km , van sluyters rc ( 1996 ) spacing of ocular dominance columns is not changed by monocular deprivation or strabismus . invest ophtalmol vis sci 37:1964 . katz lc , gilbert cd , wiesel tn ( 1989 ) local circuits and ocular dominance columns in monkey striate cortex . j neurosci 9:1389 - 1399 . koulakov aa , chklovskii db ( 2001 ) orientation preference patterns in mammalian visual cortex : a wire length minimization approach . neuron 29:519 - 527 . levay s , gilbert cd ( 1976 ) laminar patterns of geniculocortical projection in the cat . brain res 113:1 - 19 . levay s , connolly m , houde j , van essen dc ( 1985 ) the complete pattern of ocular dominance stripes in the striate cortex and visual field of the macaque monkey . j neurosci 5:486 - 501 . livingstone ms ( 1996 ) ocular dominance columns in new world monkeys . j neurosci 16:2086 - 2096 . lowel s ( 1994 ) ocular dominance column development : strabismus changes the spacing of adjacent columns in cat visual cortex . j neurosci 14:7451 - 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155 .

we study a mathematical model for ocular dominance patterns ( odps ) in primary visual cortex . this model is based on the premise that odp is an adaptation to minimize the length of intra - cortical wiring . thus we attempt to understand the existing odps by solving a wire length minimization problem . we divide all the neurons into two classes : left- and right - eye dominated . we find that segregation of neurons into monocular regions reduces wire length if the number of connections to the neurons of the same class ( intraocular ) differs from the number of interocular connections . the shape of the regions depends on the relative fraction of neurons in the two classes . we find that if both classes are almost equally represented , the optimal odp consists of interdigitating stripes . if one class is less numerous than the other , the optimal odp consists of patches of the less abundant class surrounded by the neurons of the other class . we predict that the transition from stripes to patches occurs when the fraction of neurons dominated by the underrepresented eye is about @xmath0 . this prediction agrees with the data in macaque and cebus monkeys . we also study the dependence of the periodicity of odp on the parameters of our model .

You are an expert at summarizing long articles. Proceed to summarize the following text: empirically the literature reports several behaviors for the income and wealth distributions in different countries . a century ago , the italian social economist pareto suggested a power - law @xcite distribution in the high - income range , namely , in terms of cumulative distribution : @xmath0 , with @xmath1 being the pareto index @xcite . on the other hand montroll @xcite suggested a lognormal distribution with power law tail for the usa personal income . more recently , wealth and income distributions in the usa and in the united kingdom have been described by an exponential distribution with power law high - end tails @xcite . whereas , the japanese personal income distribution appears to follow lognormal distributions also with power law tails @xcite . in some recent papers zipfs law has also been proposed @xcite . in this paper we add to the above empirical investigations an analysis of the income distribution in australia ( figure [ f.income ] ) . > from the theoretical side , it has been shown that pure multiplicative stochastic ( msp ) processes can explain the lognormal income distribution but they fail to explain the power law tails @xcite . power law tails can be obtained extending msp processes by including -for instance- additive noise and boundary constraints @xcite . these models explain well the emergence of power law distributions , but they are incomplete , neglecting interactions between agents . hence , msps with interacting agents connected through a network have been developed @xcite . these models retrieve power law tails with exponents @xmath1 which are related to the network properties . in this paper , we show that distributions with power law tails can emerge also from additive stochastic processes with interacting agents . in this case , we show that the network of connections among agents plays a crucial role . indeed , the resulting wealth distribution is shaped directly by the degree distribution of the network . the original purpose of the present work was not to construct any realistic model for the wealth distribution . our aim was simply to demonstrate the possibility to obtain ` fat ' tails also without the use of multiplicative stochastic processes . rather surprising we find out that the results from such an additive process are in good qualitative agreement with the empirical data for the income distribution in australia . [ cols="^,^ " , ] we have shown that a mechanism of wealth exchange with additive gaussian noise can produce distributions with power - law tails when the network which connects the agents is of a scale - free type . although the original purpose of this work was not to produce a realistic model for the wealth evolution , we find a good qualitative agreement between the empirical data and the theoretical prediction . more realistic models will be proposed in future works by introducing also multiplicative stochastic terms and a dynamical evolution in the network connectivity . t. di matteo wishes to thank the research school of social sciences , anu , for providing the abs data . this work was partially founded by the arc discovery project : dp0344004 . we acknowledge the stac supercomputer time grant at apac national facility .

we investigate the wealth evolution in a system of agents that exchange wealth through a disordered network in presence of an additive stochastic gaussian noise . we show that the resulting wealth distribution is shaped by the degree distribution of the underlying network and in particular we verify that scale free networks generate distributions with power - law tails in the high - income region . numerical simulations of wealth exchanges performed on two different kind of networks show the inner relation between the wealth distribution and the network properties and confirm the agreement with a self - consistent solution . we show that empirical data for the income distribution in australia are qualitatively well described by our theoretical predictions . [ 1999/12/01 v1.4c il nuovo cimento ]

You are an expert at summarizing long articles. Proceed to summarize the following text: topological superconductor is a novel phase of matter that has been theoretically predicted but yet to be experimentally verified . among topological materials , topological superconductivity is especially interesting because it is a platform to realize majorana particles - an elusive particle that is its own antiparticle . furthermore , topological superconductors have been proposed as a platform for topological quantum computation . @xcite the robustness of the topological surface states makes this avenue an attractive alternative to traditional methods for realizing quantum computation . @xcite a topological superconductor must have a full superconducting gap in the bulk with odd parity pairing , and the fermi surface must enclose an odd number of time reversal invariant momenta in the brillouin zone , i.e. the fermi surface must contain an odd number of high symmetry points such as @xmath3 , z , x , etc . it also has a topologically protected gapless surface state with majorana fermions . @xcite cu@xmath0bi@xmath1se@xmath2 has been proposed as a leading candidate for topological superconductivity @xcite and has sparked a lot of interest . experiments have shown that by intercalating cu between se layers in known topological insulator bi@xmath1se@xmath2 the compound becomes superconducting at 3.8 k. @xcite cu@xmath0bi@xmath1se@xmath2 has been confirmed to be a bulk superconductor with a full pairing gap by specific heat measurement . @xcite there are some reports of surface andreev bound states through the observation of zero bias conductance peak ( zbcp ) @xcite , but other reports that the zbcp can be removed with gating . @xcite recent works using scanning tunneling spectroscopy also did not observed the zbcp . @xcite arpes measurements have argued against the topological superconducting mechanism in cu@xmath0bi@xmath1se@xmath2 by reporting an even number of time reversal invariant momenta in the brillouin zone . @xcite both arpes and quantum oscillation experiments show a dirac dispersion in cu@xmath0bi@xmath1se@xmath2 - a characteristic feature of topological systems . @xcite the continual interest in cu@xmath0bi@xmath1se@xmath2 and surmounting controversy of its exotic phase motivates this study for a more complete look at quantum oscillations in magnetization . this work is a continuation and expansion on our previous study of the de haas - van alphen effect in cu@xmath4bi@xmath1se@xmath2 @xcite and now includes several samples at a variety of doping levels and complete angular dependence . from mapping out the fermi surface , we reveal a closed ellipsoidal fermi surface that becomes increasingly elongated with increased carrier density . at high carrier concentration , the fermi surface crosses the brillouin zone boundary and becomes open and quasi - cylindrical . amplitude damping analysis reveals a strongly anisotropic effective mass . the slope of the energy - momentum dispersion is unchanged with increased fermi momentum confirming a linear , dirac - like band structure in cu@xmath0bi@xmath1se@xmath2 . the manuscript will first introduce torque magnetometry which we use to resolve quantum oscillations , second it will discuss the results of the angular dependence of the quantum oscillation frequencies , then it will cover the various damping mechanisms of the quantum oscillation amplitude and the parameters extracted from that analysis . single crystals of cu@xmath0bi@xmath1se@xmath2 were grown by melting stoichiometric mixtures of high purity elements bi ( 99.999% ) , cu ( 99.99% ) , and se ( 99.999% ) in a sealed evacuated quartz tube then slowly cooling the mixture from 850@xmath5c down to 620@xmath5c at which point the crystal was quenched in cold water . the doping level was determined according to the mole ratio of the reactants used in the crystal growth , but the nominal doping did not end up corresponding with the measured carrier concentration leaving the precise number under suspicion . therefore , in this study , we look at how parameters evolve with increased carrier concentration rather than the unreliable nominal doping . the samples used in the study were cut out of large boule of crystals . they are generally black , and the typical size is about 5 mm @xmath6 2 mm @xmath6 0.5 mm . quantum oscillations are used to resolve fermi surface geometry and to discover electronic properties of topological materials . oscillations in magnetization , the de haas - van alphen effect ( dhva effect ) , arise from the quantization of the fermi surface into landau levels . to measure quantum oscillations in magnetization , @xmath7 , we employed a highly sensitive torque magnetometry method . torque mangetometry measures the magnetic susceptibility anisotropy of the sample by putting the sample in a tilted magnetic field , @xmath8 , where both @xmath8 and @xmath7 are confined to the x - z plane . the torque is then given by @xmath9 and @xmath10 , where @xmath11 is the tilt angle of the magnetic field @xmath12 away from the crystalline @xmath13 axis and @xmath14 . we glue the sample to the head of a thin film cantilever . both brass cantilevers and kapton cantilevers with a metalized surface were used . the thinner 0.001 inch brass cantilevers with a higher young s modulus and the thicker 0.003 inch katpon thin films with a lower young s modulus offer different spring constants that can provide a balance between strength for heavier samples and sensitivity . the magnetic torque was tracked by measuring the capacitance between the metal surface of the cantilever and a thin gold film underneath . an example of oscillations in the torque data after background subtraction is shown in fig . [ figtorque ] with a schematic of the experimental setup in the upper right corner . oscillations arise from landau level quantization . the frequency of this oscillation is proportional to the cross section of the fermi surface , @xmath15 , by the onsager relation : @xmath16 to further analyze the oscillation torque pattern , a polynomial background is subtracted from the @xmath17 curve to get the oscillatory torque @xmath18 . a fast fourier transform ( fft ) of the oscillatory @xmath18 vs. @xmath19 is given in the lower left inset of fig . [ figtorque ] revealing a single fermi pocket . the dhva effect was observed in all of our cu@xmath0bi@xmath1se@xmath2 crystals . a typical example of our torque data as a function of @xmath20 with the polynomial background subtracted is shown in fig . [ figtorque ] . a single frequency of oscillations in magnetization reveal a single fermi pocket . at 300 mk the oscillation frequency was measured as a function of angle up to 90@xmath5 . we further measured the temperature dependence of the oscillation amplitude at two different angles up to 25 k. a detailed analysis of the temperature and angular dependence of the quantum oscillations revealed the following features : 1 ) resolved quantum oscillations up to 90@xmath5 at low carrier concentration shows a closed ellipsoidal fermi surface , 2 ) the fermi surface gets progressively elongated in the z - direction as carrier concentration increases and becomes open at high carrier density , 3 ) there is a strong effective mass anisotropy , and 4 ) the fermi velocity is unchanged with increased carrier concentration supporting previous reports of a linear , dirac - like dispersion in cu@xmath0bi@xmath1se@xmath2 . figure [ figangtorq ] shows the torque signal from sample 4 . in panel a , oscillations are clearly seen up to 90@xmath5 in raw data indicating a closed fermi surface . panel b shows the fft of the raw signal from panel a. clear angular dependence can be tracked up to 90@xmath5 , where @xmath8 is parallel to the plane . previous studies @xciteof the dhva effect measured quantum oscillations up to 35@xmath5 . the observation of quantum oscillations up to 90@xmath5 is important confirmation of the previous result that the fermi surface is an ellipsoid @xcite . we note the sample with this 3d fermi surface is superconducting , and the meissner effect of sample 4 is shown in fig . [ figsupercon ] and discussed in detail there . figure [ figang ] shows the angular dependence of the oscillation frequency for the various samples . the dashed lines are ellipsoidal fits given by @xmath21+(\frac{k_{f}^{x}}{k_{f}^{z}})^{2}\sin^{2}[\phi])^{-\frac{1}{2}}$ ] where @xmath22 is the frequency of the quantum oscillations at a particular @xmath11 , and the fitted parameters are @xmath23 ( the quantum oscillation frequency at @xmath11 = 0@xmath5 ) and @xmath24 ( a measure of the eccentricity of the fermi surface ) . most of the samples are fit well by a closed , ellipsoidal fermi surface ; however , for the highest carrier concentration sample , a closed fermi surface fitting yields @xmath25 = 4.69 nm@xmath26 , which is longer than the brillouin zone height @xcite of 3.28 nm@xmath26 . thus , it is clear that the fermi surface becomes open at high carrier concentration - which was not seen in previous dhva studies where there was only one sample of lower doping . @xcite the angular dependence of the quantum oscillation frequency provides the size of the fermi pocket . from the onsager relation , the frequency of the quantum oscillation is proportional to the cross - section area given by @xmath27@xmath28@xmath29 , with @xmath30 and @xmath31 two semi - axes of the elliptical fermi surface . thus @xmath23 yields @xmath28 = @xmath32 and the eccentricity gives @xmath25 . for sample 4a , @xmath28 = @xmath32 = 0.95nm@xmath26 and @xmath33 = 2.06 . for a closed fermi pocket , the bulk carrier concentration , @xmath34 , is given by @xmath35 . for the sample with the open fermi surface , we calculated the bulk carrier concentration from finding the volume of the fermi surface . this volume is arrived by integrating the ellipsoidal fit up to the brillouin zone boundary . we assume that the deviation from the ellipsoidal fit around the brillouin zone boundary due to bending is small . in this case the carrier concentration is given by @xmath36 where @xmath37 is the @xmath38 distance . this yields a carrier concentration for sample 5 . the inferred carrier densities @xmath34 are listed with other electronic parameters in table [ density ] . .[density ] summary of results in order of increasing carrier concentration . * the value of @xmath39 for sample 5 is ill - defined since @xmath40 is taller than the brillouin zone . this is the value extracted from the ellipsoidal fit . [ cols="^,^,^,^,^",options="header " , ] [ parameter ] in addition to determining the electronic state , we further measured the the superconducting fraction of cu doped bi@xmath1se@xmath2 . the magnetic susceptibility was measured in an quantum design magnetic property measurement system 2 weeks after the high field torque experiments . the sample with the lowest carrier concentration shows a superconducting transition with a 16% superconducting volume as seen in fig . [ figsupercon ] . two of the higher carrier concentration samples showed no superconducting transition suggesting either that the sample quality deteriorates with time and exposure or the superconducting phase is killed in the over - doped regime . at high carrier concentration , the fermi surface becomes quasi - cylindrical and contains both the @xmath3 and z points . this indicates that a topological superconducting state does not exist at high carrier concentration since the fermi surface must enclose an odd number of time reversal invariant momenta in the brillouin zone for a topological superconductor . however , since superconductivity coexists with the closed fermi surface ( which can only contain the @xmath3 point ) in the low carrier density sample , a topological superconducting state can still exist in the lower carrier density samples of cu@xmath0bi@xmath1se@xmath2 . ( color online ) volume susceptibility measurements of 3 different samples . sample 4 , the sample with lowest carrier concentration , shows a superconducting transition at 3 k and a 16% superconducting volume . samples 3 and 5 do not show any superconducting property most likely due to sample quality degradation over time . , width=3 ] we note that sample 4 , though having the highest level of disorder , is the only sample to show a superconducting transition . the sample from the previous dhva study had a dingle temperature of 23.5 k much like the other samples and it also exhibited superconductivity . @xcite therefore , it is not the case that superconductivity only occurs in the extreme case of highly disordered samples , rather we suggest that sample 4 had undergone the least amount of degradation and just happened to be the sample with the highest disorder . quantum oscillations in magnetization were resolved using highly sensitive torque magnetometry up to 31 t. a single fermi pocket was observed to be increasingly elongated with added carriers . the effective mass has strong anisotropy , and the fermi velocity remains unchanged with increasing fermi momentum suggesting a linear , dirac - like dispersion . the nature of the transition of the fermi surface topology is an interesting question . at higher concentration , the elongated 3d ellipsoidal fermi surface touches the fermi surface in the neighboring brillouin zone , mandating the transition from the 3d fermi surface to a 2d quasi - cylindrical one . such an dramatic change of the fermi surface topology suggests a lifshitz transition as the cu brings it extra carriers . two experimental consequences are essential to confirm the dimensionality change and probe the nature of the transition . first , at higher @xmath34 , the quasi-2d fermi surface shall have two quantum oscillation frequencies , a large one from the belly , and a small one from the neck . the large frequency is what we observed in our dhva measurement , @xcite and confirmed by the sdh measurements . @xcite in contrast , the small neck frequency was not observed either in our dhva measurements , nor in the sdh results . @xcite this point is in particular puzzling , though ref . @xcite reports that the signal arising from the small neck may be too small do to large effective mass effects . therefore , the quantum oscillation measurements at dilution refrigerator temperature range is called to resolve the second oscillation frequency to confirm the 2d to 3d transition . another interesting experiment would be to the enhancement of thermopower near the 3d to 2d transition . a topology change in the electronic state usually leads to a large thermopower , a typical signature of lifshitz transition . @xcite further thermoelectric measurements are essential to confirm this nature . if the dimensionality changes indeed occur and enhance the thermopower greatly , the cu doping might lead to another interesting application of topological materials in thermoelectrics . * acknowledgement * we are grateful to discussion with liang fu , kai sun , and a. kanigel , . the work is supported by the national science foundation under award number eccs-1307744 ( low field torque magnetometry ) , the department of energy under award number de - sc0008110 ( high field torque magnetometry ) , by the start up fund and the mcubed project at the university of michigan ( low field magnetic susceptibility characterization ) , and by the national science foundation under award number dmr-1255607 ( sample growth ) . the high - field experiments were performed at the national high magnetic field laboratory , which is supported by nsf cooperative agreement no . dmr-084173 , by the state of florida , and by the doe . we thank the assistance of tim murphy and ju - hyun park of nhmfl . b. j. lawson acknowledges the support by the national science foundation graduate research fellowship under grant no . f031543 . t. asaba thanks the support from the nakajima foundation .

cu@xmath0bi@xmath1se@xmath2 has drawn much attention as the leading candidate to be the first topological superconductor and the realization of coveted majorana particles in a condensed matter system . however , there has been increasing controversy about the nature of its superconducting phase . this study sheds light on present ambiguity in the normal state electronic state , by providing a complete look at the quantum oscillations in magnetization in cu@xmath0bi@xmath1se@xmath2 at intense high fields up to 31 t . our study focuses on the angular dependence of the quantum oscillation pattern in a low carrier concentration . as magnetic field tilts from along the crystalline c - axis to ab - plane , the change of the oscillation period follows the prediction of the ellipsoidal fermi surface . as the doping level changes , the 3d fermi surface is found to transform into quasi - cylindrical at high carrier density . such a transition is potentially a lifshitz transition of the electronic state in cu@xmath0bi@xmath1se@xmath2 .

You are an expert at summarizing long articles. Proceed to summarize the following text: the shapley supercluster ( ssc ) has for long been known as a large region of a high overdensity of galaxies ( shapley 1930 ) . located in in the direction of hydra - centaurus at @xmath4 it is one of the densest large scale concentrations of matter ( fabian 1991 ) in the universe . if clusters of galaxies are taken as mass tracers ( scaramella et al . 1989 ) , the shapley concentration accounts for at least 10 - 20% of the acceleration of the local group towards the great attractor ( lynden - bell et al . 1988 , scaramella et al . 1991 , drinkwater et al . 1998 ) . optical ( vettolani et al . 1990 , raychaudhury et al . 1991 , bardelli et al . 1994 , quintana et al . 1997 ) and x - ray observations ( raychaudhury et al . 1991 , day et al . 1991 , breen et al . 1994 , bardelli et al . 1996 , ettori et al . 1997 ) reveal that the ssc has a remarkable structure dominated by a central high - density core . centered on a3558 ( shapley 8) this central region of the ssc includes five abell clusters ( a3558 , a3556 , a3559 , a3560 , a3562 ) and two rich groups of galaxies ( sc1327 - 312 and sc1329 - 313 ) . this corresponds to an overdensity of @xmath5 with respect to the average density of aco clusters ( vettolani et al . 1990 ) . a comprehensive study of the mass distribution in the ssc inferred from x - ray observations has been reported by ettori et al . their analysis was based on a mosaic of rosat pscp and einstein observatory ipc data enclosed in a sky area of @xmath6 and centered on a3558 . the individual mass estimates for 14 clusters and two groups of galaxies amount to an estimated total mass of the ssc of @xmath7m@xmath8 . assuming a cdm - like power spectrum , ettori et al . ( 1997 ) find that the core of shapley is an overdensity on the level of @xmath9 ( @xmath10 ) to @xmath11 ( @xmath12 ) . in this letter we report on the x - ray observation of the core of the ssc by rosat combining pointed observations and data from the rosat all - sky survey . the region of our study is centered at @xmath13 ; @xmath14 and covers a field of @xmath0 . it contains the central clusters a3562 , a3558 and a3556 and the two rich groups of galaxies sc1327 - 312 and sc1329 - 313 . this part of the core of the ssc is of particular interest since there are strong indications ( bardelli et al . 1994 ) that it forms a single elongated structure on the scale of @xmath15 mpc at a redshift of @xmath16 . our analysis shows that the core of ssc is traced by an elongated x - ray emission on similar scale . this confirms the existence of a physically connected , filamentary structure in the core of shapley . while rosat pspc pointed data is limited by the sky coverage , rosat all - sky survey ( rass ) ( trmper 1993 , voges et al . 1996 ) data offers a virtually unlimited field of view . on the other hand , rass data is limited by the short exposure time ( i.e. @xmath17 sec vs. typical @xmath18 sec of pspc pointings ) . as only a part of the core of the ssc is covered by pointed observations , a completion of this area by rass data offers a possibility to combine both advantages . fig . [ fig_region ] shows the field of our study . the clusters a3562 , a3558 and a3556 are shown as solid circles the radius of which correspond to 1 abell radius ( @xmath19 mpc ) . the rosat pointed pspc observations covering the region are indicated by line - shaded circles of @xmath20 diameter each . table [ tab_pointings ] lists the center and total observation time for each observation . the rass and pointed pspc data has been processed following the standard procedure for extended x - ray sources . in order to obtain the best signal - to - noise - ratio , only the 0.5 - 2 kev energy band ( the hard rosat band ) is used ( snowden et al . 1994 ) . this reduces both the foreground x - ray emission of the galactic interstellar medium and the detector background . the absorption of the x - ray emission due to neutral hydrogen in the line of sight varies only little . the neutral hydrogen column density ranges form @xmath21 @xmath22 towards a3558 to @xmath23 @xmath22 ( a3562 ) and @xmath24 @xmath22 towards a3556 ( stark et al . we adopted in the following the mean value @xmath25 @xmath22 . we began our analysis by merging the data of the pointed rosat pspc observations . we then binned both the rass and the pointed data into bins of size 36@xmath2636 arcsec@xmath27 each and calculated the corresponding exposure maps using exsas . [ fig_contour ] presents the contour plot of the merged count rates in the rosat hard band ( rosat band b , channels 52 - 201 , @xmath28 0.5 - 2 kev ) obtained from the rass and the pointed data . the contour plot is based on data smoothed with a gaussian of fwmh 12 arcmin . this choice compensates for the different fwmh of the point spread functions ( psf ) of pointed and rass observations as well as for the varying psf within the aperture of single pointings . the contour plot is overlaid on the binned flux . due to the shorter exposure time , the region outside the pointed observations ( i.e. the rass data ) exhibits a much lower signal - to - noise ratio resulting in a patchy appearance . the figure shows an overall picture of the region . neither the background nor point sources have been subtracted . local maxima of the count rate clearly trace the clusters a3562 , a3558 and a3556 as well as the two groups sc1327 - 312 and sc1329 - 313 . note also the region in the vicinity of a3556 . the peak of the count rate centered on a3556 is surrounded by faint x - ray emission with an extension of about @xmath29 in the east - west direction and @xmath30 in the north - south direction . we have checked for x - ray sources in this region . clearly visible is the point source 1rxs j132129.4 - 314054 of the rosat all - sky survey bright source catalogue . its position roughly coincidents with the western edge of the emission around a3556 . there are about five other point - like sources in this field , presumably associated with emission of single galaxies ( c.f . section [ sec_discuss ] ) . in analogy with a more comprehensive study of the x - ray emission of a3558 ( bardelli et al . 1996 ) , we find the point - like sources to contribute @xmath31 to the observed count rate . to check the significance of the elongated emission , we rebinned the central region into 31 bins as shown in fig . [ fig_bins ] . the count rates in the various bins as obtained from the unfiltered data is plotted in fig . [ fig_flux ] . the error bars are the @xmath32 ( poisson ) errors . the background shown as dotted line ( @xmath33 cts sec@xmath3 arcmin@xmath34 ) is estimated from the average of the six bins with lowest flux . its value is in good agreement with the background estimated from the smaller northern and south - western fields ( @xmath35 and @xmath36 cts sec@xmath3 arcmin@xmath34 , respectively ) . since the emission found for the background bins shows some structure , we quote here conservative gaussian errors . while the region used for the first background estimation is covered by pointed observations , the two other fields lie mostly within the area of the rass . because of the poor photon statistics , the data from these two fields has not been included in the subsequent analysis . the peaks of the flux seen in fig . [ fig_bins ] are clearly associated with the clusters and rich groups in the field . from fig . [ fig_flux ] it is also seen that the emission between the clusters a3556 and a3558 lies well above the background . thus the x - ray emission traces the the core of the ssc as a filamentary - like structure with an extension of @xmath37 corresponding to @xmath38 @xmath39 mpc . as a final test we checked for the effect of different bin sizes and applied the same analysis to the rosat a band ( channels 11 - 41 , @xmath28 0.1 - 0.4 kev ) , c band ( channels 52 - 90 @xmath28 0.5 - 0.9 kev ) and d band ( channels 91 - 201 @xmath28 0.9 - 2 kev ) . while no corresponding emission could be found in the a band , the emission observed in the c and d band traces the same structure . therefore we can exclude any contamination by galactic foreground emission miming the elongated emission . table [ tab_lum ] shows background subtracted fluxes and luminosities for various combinations of bins . for the conversion of the count rates in the rosat 0.5 - 2.0 kev band to the flux and luminosity in the 0.1 - 2.4 kev band we used a raymond - smith code with metallicity 0.35 ( solar units ) . where available , the plasma temperature for the different regions has been taken from the literature . the plasma temperature of sc1327 - 312 is estimated to be equal to sc1329 - 313 . for the faint emission between the clusters a3558 and a3556 as well as for the elongated emission in the vicinity of a3556 we assumed a plasma temperature of 1 kev , corresponding roughly to the temperature expected for ( poor ) groups of galaxies or for faint filamentary emission ( c.f . also section [ sec_discuss ] ) . the ( background subtracted ) flux and luminosity obtained for a3558 ( i.e. bins 13 - 17 ) are in good agreement with the corresponding values @xmath40 erg @xmath22 s@xmath3 and @xmath41 @xmath42 erg s@xmath3 ( rescaled to the energy range 0.1 - 2.4 kev ) published by bardelli et al . note however , that the values given in table [ tab_lum ] are of limited relevance since the binning used only crudely reflects the actual spatial structure of the different sources . in analogy , the total flux and luminosity found for the whole structure by summing up the respective values for the bins 3 - 27 has to be considered as a lower limit . the combination of flux and temperature data with a spatial model for the emitting region allows for an estimation of the electron density and gas mass in the respective region . in order to get upper limits , we assume here the emitting gas to be uniformly distributed in a cylinder whose projection corresponds to the binned region ( i.e. the axis of the cylinder is perpendicular to the line of sight ) . the volume of a bin then is @xmath43 @xmath44 mpc@xmath45 . the resulting electron density and gas mass estimations are quoted in table [ tab_lum ] . note that the approximation we applied overestimates the actual values since it assumes uniform distributions of the flux and emitting gas . thus the quoted values have to be considered as upper limits . the value @xmath46 we find for a3558 ( i.e. bins 13 - 17 ) is in rough agreement with @xmath47 within @xmath48 mpc calculated by bardelli et al . extrapolating to the whole structure , we find as an estimation of the total gas mass @xmath49 . an more precise determination of the gas mass as well as the estimation of the total mass should involve a more accurate model of both the gas and the flux distribution . based on rosat data , bardelli et al . ( 1998 ) performed a comprehensive analysis of spectral properties of the emission of the central cluster a3558 ( for a comparison with asca data see markevitch & vikhlinin 1997 ) . because of comparable exposure times of a3558 and the rest of the region covered by pointed pspc observations , a corresponding analysis of the whole elongated emission should be possible . however this is beyond the scope of this letter . as a first step towards a spectral analysis of the region we determine here the hardness ratio ( i.e. x - ray color ) related to the rosat c and d energy band @xmath50 where @xmath51 and @xmath52 denote the flux in the rosat bands c and d , respectively . in fig . [ fig_hardness1 ] the hardness ratio of the background corrected flux is shown as a function of the bin number . the errors are the @xmath32 errors . as it is the case for the flux , the hardness ratio traces the structure of the core of shapely . a clearly higher hardness ratio is found for the emission of clusters and groups than for intermediate , connecting emission . this is evidence for higher gas temperatures associated with density peaks , i.e. a result one would expect . for temperatures between 2 and 10 kev , and a neutral hydrogen column density of @xmath25 @xmath22 the theoretical values for the hardness ratio range form 0.33 to 0.355 . these values are in rough agreement with the hardness ratio found for the clusters and groups in the field . the corresponding hardness ratio of the background ( i.e. the hardness ratio of six bins with lowest flux ) is -0.22 . for a gas temperature of 1 kev , the theoretical value for the hardness ratio is @xmath53 . as is clear form fig . [ fig_hardness1 ] , the hardness ratio between a3556 and a3558 lies bellow this value , indicating a temperature bellow 1 kev . the oscillating behavior of the hardness ratio westwards of a3556 ( i.e. for bins @xmath54 ) is due to the background subtraction procedure . by subtracting the average flux of the six bins with lowest flux , the background corrected net flux in some of these bins may become negative giving rise to observed oscillatory feature . the analysis of the rosat rass and pointed data of the core of the ssc shows clear evidence for x - ray emission related to a filamentary superstructure . optical observations ( bardelli et al . 1994 , 1998 ) reveal a striking similarity of isodensity contour lines of galaxy counts and x - ray contours . this strongly suggests a tight connection between the galaxy distribution and the x - ray emission . it is intriguing that galaxy counts and x - ray emission not only correlate on the scale of the single clusters in the field but also on the scale of the whole filamentary structure , including the region of faint emission between a3558 and a3556 and westwards of a3556 . the angular and redshift distribution of galaxies in this field provides strong evidence for a filamentary superstructure connecting the clusters a3562 , a3558 and a3556 ( bardelli et al . 1994 ) . we thus conclude the elongated x - ray emission of the core of the ssc to trace this filamentary superstructure . in principle the elongated x - ray emission could arise from projection effects . however , there is little doubt that this is nt the case . first of all , redshift measurements indicate the structure to lie perpendicular to the line of sight which excludes projection effects . additional evidence comes from the flat , elongated x - ray emission of a3556 which is not comparable to the x - ray emission of compact , isolated clusters . it is interesting to note that there is additional independent evidence for an elongated gas distribution westwards of a3556 . the radio survey of a3556 ( venturi et al . 1997 ) reveals a wide - angle tailed ( wat ) radio galaxy at a distance of @xmath55 mpc westwards from the center of a3556 . normally , wat galaxies are found at the center of groups or clusters and it is assumed that the tails are bent by gas flows due to a merging processes ( gomez et al . 1997 ) . the original suggestion of venturi et al . ( 1997 ) that the bent morphology of the wat galaxy is due to interaction with surrounding relatively cold gas ( i.e. @xmath56 kev ) is confirmed by the present study . x - ray emission from filamentary large scale structures is predicted by models ( e.g. bond et al . 1996 ) and recent n - body / hydro simulations ( e.g. cen & ostriker 1996 ) of structure formation . it is expected to originate from the hot phase of the intergalactic medium ( igm ) of temperature @xmath57 kev , either found in association with filaments themselves or with single cd galaxies or groups of galaxies tracing the structure . there have been only few claims of detection of x - ray emission on scales extending the scale of clusters of galaxies ( i.e. wang et al . 1997 , soltan et al . 1997 , but see also briel & henry 1995 ) . the fact that the present study found clear evidence for x - ray emission on the scale of @xmath1 mpc is reconciled by the extraordinary dense nature of the core of shapley . while the emission centered on the clusters a3558 , a3562 and a3556 as well as the one centered on the poor groups sc1329 - 313 and sc1327 - 312 is due to hot intercluster medium , the nature of the fainter emission connecting the whole structure is less obvious . to some extent , it remains debatable if the x - ray emission connecting the clusters a3558 and a3556 originates from poor groups of galaxies , from igm distributed between the clusters or from overlapping gas distributions . in general , the x - ray luminosity of groups of galaxies in the 0.1 - 2.4 kev band is typically @xmath58 erg s@xmath3 ( henry et al . 1995 , mulchaey et al . 1996 ) . comparing this value to the x - ray luminosity found in the bins 18 - 20 leads to @xmath59 groups in this region , a number which seems rather high . in addition , at least compact groups should be resolved as individual sources . we thus exclude this possibility . likewise , the background subtracted x - ray surface brightness of the bins 18 - 20 is @xmath60 erg @xmath22 s@xmath3 arcmin@xmath34 ( in the 0.1 - 2.4 kev energy band ) corresponding to about 2.5 times the upper limit for emission from filamentary large scale structure ( briel and henry , 1995 ) . while the extraordinary dense nature of the region could explain this difference , the overall elongated structure may also suggest that the emission seen in the bins 18 - 20 is due to overlapping gas distributions of a3558 and a3556 . in this case , the low hardness ratio of the intermediate region could be due to a temperature decrease in the outskirts of the clusters . yet another possibility is that we are observing the beginning of the merging of a3558 and a3556 . a more definite answer to this question should involve the determination of the temperature distribution . if the low temperature ( i.e. @xmath57 kev ) assumed for the gas in the bins 18 - 20 is confirmed , this would be a strong indication for x - ray emission originating from igm distributed on intrasupercluster scale . the low hardness ratio obtained for the questionable bins as well as the fact that we are dealing with an overall exceptional dense region indicates that this actually may be the case . in summary , our analysis shows that the central part of the shapley supercluster is traced by elongated x - ray emission connecting the three clusters of galaxies a3562 , a3558 , a3556 and the two groups sc1327 , sc1329 . the filamentary - like x - ray emission extends over @xmath37 corresponding to @xmath38 @xmath39 mpc . this is strong evidence for the original claim of bardelli et al . ( 1994 ) that the three clusters a3562 , a3558 and a3556 form a physically connected , single structure . while the emission between a3558 and a3556 seems not to originate from groups of galaxies it remains debatable if it is due to overlapping gas distributions of a3558 and a3556 or if its of intrasupercluster origin . a more definite answer to this question as well as to questions related to the dynamical state of the structure could be provided by the temperature distribution of the region . its determination is thus of great interest . bardelli s. , zucca e. , vettolani g. , zamorani g. , scarmella r. , scaramella r. , collins c. a. , macgillivray t. , 1994 , mnras , 267 , 665 bardelli s. , zucca e. , malizia a. , zamorani g. , scarmella r. , vettolani g. 1996 , a&a , 305 , 435 bardelli , s. , zucca , e. , zamorani , g. , vettolani , g. , & scaramella , r. 1998 , mnras , 296 , 599 bond j. r. , kofman l. , pogosyan d. , 1996 , nature , 380 , 603 breen j. , raychaudhury s. , forman w. , jones c. , 1994 , apj , 424 , 59 briel u. g. , henry j. p. 1995 , a&a , 302 , l9 cen r. , ostriker j. p. , 1996 , apj , 464 , 27o dickey j. m. , lockman f. j. , 1990 , ara&a , 28 , 215 drinkwater , m.j . , parker , q. a. , proust , d. , quintana , h. , slezak , e. 1998 , astro - ph 9807118 day c. s. r. , fabian a. c. , edge a. c. , raychaudhury s. , 1991 , 252 , 394 fabian a. c. , 1991 , mnras , 253 , 29 gomez p. l. , pinkney j. , burns j. o. , wang q. , owen f. n. , & voges w. 1997 , apj , 474 , 580 henry j. p. , et al . 1995 , apj , 449 , 422 lynden - bell d. , faber s. m. , burstein d. , davies r. l. , dressler a. , terlevich r. j. , & wegner g. 1988 , apj , 326 , 19 ettori s. , fabian a. c. , white d. a. 1997 , mnras , 289 , 787 markevitch m. , vikhlinin a. 1997 , apj , 474 , 84 mulchaey j. s. , davis d. s. , mushotzky r. f. , burstein d. 1996 , apj , 456 , 80 postman m. , lauer t. r. , 1995 , apj 440 , 28 quintana h. , melnick j. , proust d. , infante l. , 1997 , a&as , 125 , 247 raychaudhury s. , fabian a. c. , edge a. c. , jones , c. , forman w. , 1991 , mnras , 248 , 101 shapley h. , 1930 , harvard obs . bull . 874 , 9 snowden s. c. , mcgammon d. , burrows d. n. , mendenhall j. a. , 1994 , apj , 424 , 714 scaramella r. , baiesi - pillastrini g. , chincarini g. , vettolani g. , zamorani g. , 1989 , natur , 338 , 562 soltan , a. m. , hasinger , g. , egger , r. , snowden , s. , & trmper , j. , 1997 , a&a , 320 , 705 trmper , j. 1993 , science , 260 , 1769 venturi , t. , bardelli , s. , morganti , r. , & hunstead , r. w. 1997 , mnras , 285 , 898 vettolani g. , chingarini g. , scaramella r. , zamorani g. , 1990 , aj 99 , 1709 voges , w. , et al . 1996 , mpe report 263 , 637 wang q. d. , connolly a. j. , brunner , r. j. 1997 , apj , 487 , l13 white d. a. , jones c. , forman w. 1997 , mnras , 292 , 419 @xmath62 } & \mbox{[cts s$^{-1}$ ] } & \mbox{[kev ] } & \mbox{[erg cm$^{-2}$ s$^{-1}$ ] } & \mbox{[$10^{44}$ erg s$^{-1}$ ] } & \mbox{[cm$^{-3}$ ] } & \mbox{[$10^{14 } m_\odot$]}\\ \hline \noalign{\smallskip } a3562 & [ 3 - 7 ] & 1.73 & 3.8^{\rm c } & 3.6\times10^{-11 } & 3.70 & 1.33 & 2.5\\ sc1329 & [ 8 - 9 ] & 0.43 & 3.0^{\rm c } & 9.0\times10^{-12 } & 0.91 & 1.48 & 0.8\\ sc1327 & [ 11 - 12 ] & 0.80 & 3.0^{\rm d } & 1.7\times10^{-11 } & 1.70 & 2.01 & 1.1\\ a3558 & [ 13 - 17 ] & 3.98 & 3.8^{\rm c } & 8.3\times10^{-11 } & 8.51 & 2.82 & 3.8\\ & [ 18 - 20 ] & 0.15 & 1.0^{\rm d } & 2.4\times10^{-12 } & 0.24 & 0.52 & 0.4\\ a3556 & [ 21 - 23 ] & 0.31 & 2.1^{\rm c } & 6.3\times10^{-12 } & 0.64 & 1.02 & 0.8\\ & [ 24 - 26 ] & 0.13 & 1.0^{\rm d } & 2.2\times10^{-12 } & 0.22 & 0.50 & 0.4\\ & [ 3 - 27 ] & 7.53 & - & 1.6\times10^{-10 } & 15.95 & - & 9.8\\ \hline \end{array}\ ] ]

we report on x - ray observations of the core of the shapley supercluster . combining data from pointed observations of the rosat pspc detector and data from the rosat all - sky survey , the observed region covers an area of @xmath0 . it contains the central clusters a3562 , a3558 and a3556 . we find clear evidence for x - ray emission connecting the three clusters . this confirms the existence of a filamentary , physical structure embedding the three clusters a3562 , a3558 and a3556 . we also find evidence for faint emission westwards of a3556 . in total , the extension of the filamentary x - ray emission of the core of the shapley supercluster amounts up to at least @xmath1 mpc . the total luminosity in the 0.1 - 2.4 kev energy band is @xmath2 erg s@xmath3 .

You are an expert at summarizing long articles. Proceed to summarize the following text: neutron stars represent the most dense form of matter known before black hole formation . the equation of state of such material is not well understood but typical expected size scales of @xmath2 km for a @xmath3 object give @xmath4 similar to that of the last stable orbit around a schwarzschild black hole . the gravitational potential seen by the accreting material is then very similar in neutron stars and black holes , but with the obvious difference that the material may then fall seamlessly and invisibly from the edge of the disc through the event horizon in the black holes ( but see e.g. krolik , hawley & hirose 2005 for an alternative view ) , but must impact onto the solid surface for the neutron stars . this boundary layer emission should give a clear distinction between the two types of systems , yet in practice it can be quite hard to discriminate between the disc accreting , low magnetic field neutron stars and galactic black holes ( gbh ) . both show hard spectra at low mass accretion rates ( e.g. barret & vedrenne 1994 ; barret et al . 2000 ) while at higher mass accretion rates the black holes can show low temperature , optically thick comptonized emission from a corona which can look similar to an optically thick boundary layer in the neutron stars systems ( the black hole ` very high state ' , done & gierliski 2003 , hereafter dg03 ) . nonetheless , dg03 showed that there are clear differences between the black hole and neutron star systems . they used the huge database now available on these objects from the _ rossi x - ray timing explorer _ ( _ rxte _ ) to build a picture of the x - ray spectral evolution as a function of overall mass accretion rate ( as measured by luminosity , parameterized in units of the eddington rate , @xmath5 ) . they used data from gbh and both subclasses of disc accreting , low magnetic field neutron stars namely atolls and z sources ( see e.g. hasinger & van der klis 1989 ) . the atolls and gbh cover the same luminosity range in @xmath5 , from @xmath6 to @xmath7 , so their accretion flows should be directly comparable ( z sources are typically more luminous , and may have residual magnetic fields : hasinger & van der klis 1989 ) . dg03 showed that the gbh as a class were all consistent with the same spectral evolution as a function of mass accretion rate , as were the atolls , but that this behaviour _ differed substantially _ between the two types of system even though they covered the same range in @xmath5 . dg03 interpreted this in the light of the recent advances in accretion flow models , which suggest that at low mass accretion rates the optically thick , cool , inner disc can evaporate into an optically thin , hot flow ( e.g. an advection dominated accretion flow : narayan & yi 1996 or a jet dominated accretion flow : falcke , kording & markoff 2004 ) . these models are currently controversial as they conflict with the occasional detection of extremely broad iron lines in gbh spectra in this state ( miller et al 2002 ; miniutti , fabian & miller 2004 ; miller et al 2006 ) , though these extreme widths could also be an artifact of complex absorption ( done & gierliski 2006 ) . nonetheless , the truncated disc / hot inner flow models are very attractive as they can qualitatively explain a wide range of observed properties in x - ray binaries . this includes the spectral evolution of the gbh , which can be broadly explained by the transition radius between the disc and hot flow progressively decreasing as the mass accretion rate increases , while the atolls are consistent with an identical accretion flow behaviour but with the addition of the expected boundary layer / surface emission ( dg03 ) . these observational differences between gbh and atolls are consistent with the existence of the event horizon , the most fundamental predicted property of a black hole ( see also mcclintock , narayan & rybicki 2004 ; narayan & heyl 2002 ; garcia et al . 2001 ; sunyaev & revnivtsev 2000 ) . however , the sample of atolls used by dg03 was fairly small , with only 4 objects ( 1 of which was an accreting millisecond pulsar ) . this contrasts with their gbh selection , which included all available objects which were not heavily absorbed . here we present a much wider study using all the available atolls in the _ rxte _ database which are not heavily absorbed to investigate their behaviour in more detail . we find that while all the atolls ( and millisecond pulsars ) broadly show the same spectral evolution , there are some subtle differences . some of these can be connected to spin / inclination effects , but there are also some differences in transition behaviour which correlate with long term mass accretion rate . we speculate that these may be linked through the jet being affected by any remnant polar magnetic field , which in turn depends on the accretion rate . we searched the _ rxte _ database for low magnetic field , disc accreting neutron star systems . we do not include z sources ( sco x-1 , gx349 + 2 , gx17 + 2 , cyg x-2 , gx51 , gx340 + 0 e.g. kuulkers et al 1996 , cir x-1 , lmc x-2 , hasinger & van der klis 1989 ) as we were interested in the range @xmath8 to compare with the galactic black holes from dg03 . we also exclude all objects with heavy absorption , defined as @xmath9 @xmath10 , as above this the spectral decomposition becomes more uncertain . given that most systems are in the galactic plane , this removes many objects ( 4u 162449 , gro j174428 , grs 1747312 , sax j1747.02853 ) . the absorption criteria also excludes objects where there is substantial intrinsic absorption . this can be continuous , completely blocking our view of the continuum source so that the only scattered flux is seen ( the accretion disc corona sources ) , or the absorption can be episodic , giving discrete dips in the lightcurve ( the dippers ) . we exclude all accretion disc corona sources ( 2s 0921 - 630 , 4u 1624 - 49 , 4u 1822 - 371 , ms 1603 + 206 ) , and dippers where the absorption events are so frequent that most observations are affected by them ( exo 0748 - 676 ) , but we do include dippers where the absorption is limited to contaminating only a few observations ( which we remove , see next section ) . details of all included sources are shown in table [ tab : sources ] . [ cols="^,^,^,^,^,^,^,^,^,^ " , ] the standard products spectra ( and associated background and response files ) from the proportional counter array ( pca ) , available from the high energy astrophysics science archive research centre ( heasarc ) database were gathered for each source , giving a total of over 3000 spectra . following dg03 we fit these with a model of the disc emission and a comptonized continuum , described by diskbb ( mitsuda et al . 1984 ) and thcomp ( zdziarski et al . 1996 ) in xspec , together with a smeared edge and gaussian line to approximately mimic the reflected spectrum . the absorption was fixed at the galactic value for each object ( see table [ tab : sources ] ) . this model gave a good fit to all the spectra ( @xmath11 ) . following dg03 , the best - fitting model to each spectrum was integrated to form intrinsic soft and hard colours ( i.e. absorption corrected and independent of the instrument response ) , defined as flux ratios in the bandpasses 46.4/34 and 9.716/6.49.7 kev , respectively . the unabsorbed model was also used to estimate the bolometric flux in the 0.011000 kev bandpass , and this converted into a luminosity using the distance estimate given in table [ tab : sources ] . as discussed by dg03 , the colours are fairly robust to changes in the model spectra as long as the source is not heavily absorbed and the advantage of using intrinsic colours ( as opposed to the more widely used instrument colours ) is that many objects can be directly compared on the same plot . the bolometric fluxes are more uncertain , as the model is extrapolated outside of the observed energy bandpass . nonetheless , this model corresponds to expected physical components , and we fix a lower limit to the disc temperature of @xmath12 kev and an upper limit to the comptonizing temperature of @xmath13 kev so the continuum components can not produce arbitrarily large luminosities in the unobserved energy ranges ( dg03 ) . we identify and exclude data contaminated by dips and x - ray bursts by using the standard 1 lightcurves ( 0.125s time resolution ) corresponding to each standard 2 spectra . we used the intrinsic r.m.s . variability as a tracer of these . all lightcurves with r.m.s above 50 percent were inspected for bursts , while the 16-s rebinned lightcurves were checked for dips if their r.m.s . was above 20 percent . observations were also excluded if they were contaminated by galactic ridge emission , seen as a softening of the spectra and large iron line contribution at the lowest luminosities in objects with low galactic latitude : see e.g. wardziski et al . ( 2002 ) for an example of this in the black hole gx339 - 4 . [ fig : landr ] shows the resulting colour - colour and colour - luminosity diagram for all 34 atolls and 6 millisecond pulsars combined together . one interesting thing to note is that several atoll sources approach or even exceed eddington luminosities ( 4u 1735 - 44 ; 4u 1744 - 26 ( gx 3 + 1 ) ; 4u 1758 - 20 ( gx 9 + 1 ) ; xte j1806 - 246 : gladstone 2006 ) , showing that the high luminosity sources are not necessarily z sources ( hasinger & van der klis 1989 ) [ fig : landr](a ) shows that while the atolls show the same overall behaviour as claimed by dg03 , they also show subtle but significant differences . this is most evident in terms of the spectral evolution during the transition between the hard ( island ) and soft ( banana ) spectral states , which forms the middle branch of the total z - shaped track on the colour - colour diagram ( even though these are atolls , not z sources , see gierliski and done 2002 ; muno , remillard & chakrabarty 2002 ) . we have highlighted this by picking out in red those objects where the transition makes an almost vertical track on the colour - colour diagram ( hereafter called verticals ) , while the blue symbols show those where the transition from island to banana starts much further to the right on the colour - colour diagram , but ends at approximately the same place , making a much more diagonal middle branch ( hereafter called diagonals ) . this difference in spectral evolution is also picked out in the colour - luminosity diagram as a different luminosity for the hard - soft transition . while the distances are generally rather uncertain , this correlation between the behaviour on the colour - colour and colour - luminosity diagrams give some confidence in the reality of the luminosity difference . we stress that this is not primarily due to hysteresis , the well known effect in both black holes and neutron stars where the luminosity at which the hard - soft transition occurs varies considerably in the same object ( e.g. nowak 1995 , maccarone & coppi 2003 ) . here , the transition luminosity is varying between different objects . we show this difference in transition behaviour by plotting the colour - colour and colour - luminosity diagrams for each individual source which shows a clear state transition . those making a vertical transition are shown in fig . [ fig : verticals ] , while those making a diagonal transition are shown in fig . [ fig : diagonals ] . not all objects have data covering a transition . some , such as all the millisecond pulsars , are only seen in the hard ( island ) state ( see fig . [ fig : msps ] ) , others e.g. 4u 1758 - 20 are seen only in the soft ( banana ) branch . of the ones which do make transitions , showing both hard and soft spectra , the effects of data windowing mean that not all have observations covering the transition period . nonetheless , the size of the _ rxte _ database mean there are 12 sources which do have enough data to constrain the shape of the transition on the colour - colour diagrams . the remaining sources , where there is insufficient transition data available at present are included in the background points plotted on each image , but not plotted individually here ( they can be seen in gladstone 2006 ) . the majority of sources have well defined transition luminosities ( to within a factor 23 ) , irrespective of whether the transition is from hard to soft i.e. on the rising part of the light curve , or from soft to hard i.e. with a decreasing flux . however , there are two sources where the transitions have a much large scatter in their properties , namely 4u 160852 and 4u 1908 + 005 ( aql x-1 ) . the colour - colour and colour - luminosity diagrams for these are plotted as the bottom two panels in fig . [ fig : verticals ] ) , but it is not clear from the data that these sources should be classed as verticals , so here we examine them in more detail . these two systems are also the only two ( apart from the millisecond pulsars ) which show large scale outbursts . in order to investigate their scatter in transition behaviour we select simple outbursts from the light curve , where there is a clear fast rise followed by a monotonic decay . from this data we build a new , simplified colour - luminosity diagram ( fig . [ fig : hysteresis ] ) . the black triangular points show the island state seen on the rise , which looks very similar to the msps ( fig . [ fig : msps ] ) . we then exclude all of the transition data on the rise / peak of each outburst , and plot the remaining banana branch and transition back to island state on the decay as the blue square points . plainly the transition luminosity is much larger in the rise to outburst than during the decay . this effect of hysteresis is also seen in most galactic black holes ( e.g. maccarone & coppi 2003 ) . equally plainly , the decay shows a clear vertical track , confirming the identification of these as verticals . hence we speculate that the msps are similar objects , and that if the msp outburst ever reached high enough luminosities to make a transition then this transition would display hysteresis and the colour - colour track during the outburst decay would be vertical . none of the diagonals show marked hysteresis , though 4u 170544 and ks 173126 may show a small effect in that their top branch ( island state ) extends to higher luminosities than seen during the transition ( fig . [ fig : diagonals ] ) . a similar small hysteresis effect may also be present in the vertical 4u 091954 . there are subtle differences in the spectral evolution of the atolls on the colour - colour and colour - luminosity diagrams : there is the distinction between the transition track ( verticals and diagonals ) , and then within the diagonals there is a difference in spectral hardness of the banana branch , and within the verticals there are objects which show large scale hysteresis . here we examine possible explanations for these effects . a range of inclinations is expected for these sources , and this could be important for the observed spectrum if the intrinsic emission is not isotropic . some degree of anisotropy is certainly expected from the accretion disc due to its planar geometry , and as long as the boundary layer has a different angular dependence then the overall spectrum will change as a function of inclination . to estimate the effects of inclination we use the spectral model consisting of the disc emission ( diskbb ) and the optically thick boundary layer emission , which we model by thermal comptonization ( comptt ) . dg03 showed that the banana branch could be roughly characterized by a disc varying in temperature in the range 1.01.5 kev together with an equal luminosity boundary layer with @xmath14 = 3 kev and @xmath15 = 5 ( see also revnivtsev & gilfanov ( 2006 ) ) . we use these parameters to model the shape of the banana branch . we first assume that the boundary layer is isotropic , while the disc normalization varies as @xmath16 but with angle averaged luminosity equal to that of the boundary layer . thus the apparent ratio of disc to boundary layer flux will change as the inclination angle varies . in fig . [ fig : inclination](a ) we show how the banana branch moves on the colour - colour diagram for inclinations of @xmath17 , @xmath18 and @xmath19 . the change is mainly only in the soft colour , with a greater proportion of disc emission shifting the start of the banana branch to lower soft colours . plainly this can not account for either of the two subtle spectral effects seen , but seems to have more potential to explain the different positions of the banana branch amongst the diagonals than the difference between verticals and diagonals . examination of some of the individual banana branch spectra of the diagonals showed that the comptonization temperature of the boundary layer is also changing , and in a way which is correlated with @xmath20 . this could indicate that the boundary layer emission is itself anisotropic , perhaps with a temperature gradient so that it is hotter close to the equatorial plane where the disc hits the neutron star . [ fig : inclination](b ) shows the effect of changing the temperature of the boundary layer from 2.5 , 3.0 and 3.5 kev , for inclination angles of @xmath17 , @xmath18 and @xmath19 , respectively . this matches very well with the range of banana branches seen in the diagonals . testing this would require knowledge of the inclination , but very few sources have good enough orbital determinations to constrain this . however , any source exhibiting dips must be at fairly high inclination ( @xmath21 ) , and hence would be expected to have a banana branch starting at fairly large hard and soft colours . this can be seen in the case of 4u 1724307 ( see fig [ fig : diagonals ] ) , a dipping source whose banana branch ranges from 1.651.75 in soft colour and 0.7 to 0.8 in hard colour . there are three other dipping sources , 4u 170430 , 4u 174637 and 4u 191505 ( not included in this work ) , that also support this idea , with each exhibiting large values of both hard and soft colours in the banana state ( for details , see gladstone 2006 ) . burst oscillations and khz qpos have finally given observational constraints on lmxb neutron star spins ( e.g. the review by van der klis 2000 ) . for our sample , the inferred range is from 330 to 619 hz . however , this is unlikely to be the origin of the difference in transition track as the spins seem fairly evenly distributed between the two classes . however , spin could also give a difference in the overall luminosity / temperature of the boundary layer . lower spin gives a higher relative speed between the inner edge of the disc and the surface , leading to a lower @xmath20 ratio and a higher boundary layer temperature , as required for the different banana branch hard colours seen . thus it seems likely that some combination of spin and inclination effects are responsible for the different banana branches seen in the diagonals , but neither is likely to explain the origin of the two different types of transition behaviour . of the subsample of systems with data covering the transition , the publically available asm and/or pca galactic bulge long term light curves show large scale transient outbursts only in 4u 160852 , 4u 1908 + 005 and all the millisecond pulsars ( see table 1 ) . this is very intriguing as 4u 160852 and 4u 1908 + 005 are both verticals . however , examination of the long term lightcurves of the other verticals clearly shows that these objects do not undergo dramatic outbursts , so this can not be the origin of the vertical / diagonal distinction . however , 4u 160852 and 4u 1908 + 005 are also the only objects to show clear large - scale hysteresis . thus it seems possible that hysteresis is linked to large amplitude outbursts . there are no counterexamples i.e. no systems that show major disc outbursts ( which push the accretion rate high enough to make a spectral transition ) which do not show hysteresis . all the millisecond pulsar outbursts remain in the hard state , and all other atoll sources with enough data to deliniate the transition and hence constrain hysteresis do not exhibit major outbursts . this hypothesis can be tested on the black hole lmxb systems . unlike the atolls , none of these sources are persistant ( mcclintock & remillard 2006 ) . this distinction in disc instability behaviour between black holes and neutron stars can be broadly explained in the context of the hydrogen ionization models . neutron stars are lower in mass than black holes , so for the same companion star to overflow its roche lobe requires a smaller binary orbit for a neutron star compared to a black hole . smaller orbital separation means a smaller disc due to tidal truncation . the size of the disc , together with the mass loss rate from the companion star ( determined by its evolutionary state ) determines the temperature of the coolest part of the disc . thus the smaller neutron star systems are less likely to have a disc which can drop below the hydrogen ionization temperature required to trigger the disc instability ( king & ritter 1998 ; king , kolb & szuszkiewicz 1997 ; dubus et al 1999 ) . it is well known that most black hole systems show large scale hysteresis ( e.g. maccarone & coppi 2003 ) . thus they support the idea above that hysteresis is linked to large amplitude outbursts . even more support comes from the one obvious exception , which is cyg x-1 . this does not show hysteresis during its hard / soft transitions , and is of course a persistent ( hmxrb ) source ( e.g. maccarone & coppi 2003 ; dg03 ) . maccarone & coppi ( 2003 ) also suggested that hysteresis could be linked to large scale outbursts , but with only cyg x-1 as an counterexample , the connection to outbursts is not quite so unambiguous . cyg x-1 is also one of the few known galactic black holes with a high mass companion , so there is the possibility that its accretion structure is somewhat different due to lower angular momentum material from a stellar wind . by contrast , with the neutron stars being generally stable to the hydrogen ionization trigger for the disc outbursts , there are many counterexamples , all with low mass companions . we can also rule out the alternative origin for hysteresis discussed by maccarone & coppi ( 2003 ) , namely that it is produced simply by a difference in behaviour in low mass x - ray binaries between a hard to soft transition on the rising part of the light curve , and a soft to hard transition as the flux decreases . most of the sources shown in figs . [ fig : diagonals ] and [ fig : verticals ] make the transition in both directions , and do not show large scale hysteresis ( though there are smaller scale effects , consistent with the difference in environment : meyer & meyer - hoffmeister 2005 ) . instead , hysteresis is seen where the mass accretion rate changes dramatically ( from @xmath22 to @xmath23 ) . we speculate that the accretion flow is able to access non - equilibrium states during the rapid changes in accretion disc structure caused by the disc instability . thus the neutron stars clearly show a one to one correlation between the dramatic large amplitude outbursts triggered by the disc instability and hysteresis , consistent this being the origin of hysteresis . this predicts that the millisecond pulsar outbursts should also show hysteresis if their accretion rate at the outburst peak ever goes high enough to trigger a spectral transition ( in which case we would expect them also be verticals , along with 4u 160852 and 4u 1908 + 005 ) . however , there is no such one to one correlation between outbursts and transition behaviour ( vertical / diagonal ) , so we explore further aspects of the systems below . table [ tab : sources ] shows the binary parameters where these are known . we first explore whether there is any correlation between transition behaviour and binary period ( which is a tracer of companion type , and also of superburst behaviour : cumming 2003 ) . for the verticals , the periods span a large range , from 1 hour for the persistent sources 4u 0614 + 091 and 4u 091954 through to 19 and 288 hours for the transients 4u 160852 and 4u 1908 + 005 . this distinction is as expected from the disc instability model ( wide binary implies a much larger disc , so a cooler outer edge which is more likely to trigger the hydrogen instability ) . if the millisecond pulsars are also verticals then these fill in the period distribution from 40 minutes to 4 hours.there is very little data to compare this to the diagonals , as only 2 have periods ( 11 minutes and 3.9 hours ) , but the broad span seen from the verticals encompasses most of the binary periods seen in atolls , so binary period alone is unlikely to be the origin of the transition track dichotomy . the distinction between millisecond pulsars , which plainly retain a residual magnetic field channelling the flow , and other atolls which do not show pulsations is most probably due to the long timescale mass accretion rate ( cumming , zweibel & bildsten 2001 ) . plausibly , high accretion rates can bury the magnetic field below the neutron star surface , but the very low average mass accretion rate in the millisecond pulsars is insufficient to bury the field ( cumming et al . thus there is a potential physical mechanism for the long term mass accretion rate to change the properties of the accretion flow . we estimate the long - term mass accretion rate , @xmath24 , ( or corresponding average luminosity , @xmath25 ) for all of the systems from the observed x - ray emission . the pca data gives estimates for @xmath26 through spectral fitting , but the light curves are highly incomplete , so the data can not simply be used as an indicator of the long term average mass accretion rate . by contrast , the _ rxte _ all sky monitor ( asm ) gives an almost continuous light curve for every bright x - ray source in its field of view , but its lack of spectral resolution means that going from count rate to @xmath27 is highly uncertain . hence we combine the two approaches . we use the pca light curve to define the average @xmath25 during these observations , then select the simultaneous asm points to find the average asm count rate during the pca observations . the ratio of this to the full asm light curve gives the correction for the incompleteness of the pca observations . this approach works well unless the source becomes very faint , in which case contamination of the asm by other nearby sources and/or galactic ridge emission can be a problem . the only sources for which this is an issue are the transients i.e. the verticals 4u 160852 and 4u 1908 + 005 and all the millisecond pulsars . the outbursts of 4u 160852 and 4u 1908 + 005 are so bright that they dominate the average of the asm lightcurves . however , this is not the case for the millisecond pulsars , so for these we use a different approach . here we use the pca data alone , which have good outburst coverage , assuming that the source intensity is negligible in the periods outside the known outbursts . the results of both the millisecond pulsars and atoll sources are shown in table [ tab : lbol ] . ccc source name & source type & @xmath28 + + igr j00291 + 5934 & msp & 1.4 @xmath29 + xte j0929314 & msp & 1.4 @xmath30 + xte j1751305 & msp & 5.2 @xmath29 + xte j1807294 & msp & 2.7 @xmath30 + xte j1808369 & msp & 6.7 @xmath29 + xte j1814338 & msp & 1.6 @xmath30 + + 4u 0614 + 091 & v & 3.2 @xmath31 + 4u 091954 & v & 3.1 @xmath31 + 4u 160852 & v & 6.6 @xmath31 + 4u 1908 + 005 & v & 3.7 @xmath31 + slx 1735 - 269 & v & 1.6 @xmath31 + + 4u 163653 & d & 4.4 @xmath32 + 4u 170243 & d & 7.8 @xmath31 + 4u 170544 & d & 1.9 @xmath32 + 4u 1724307 & d & 7.3 @xmath31 + 4u 172834 & d & 1.9 @xmath32 + 4u 1820303 & d & 1.3 @xmath33 + ks 173126 & d & 3.4 @xmath32 + + it can be seen that there is a systematic difference between the three classes of sources . the millisecond pulsars have the lowest @xmath28 , then the verticals , then the diagonals . thus it seems possible that this is the origin of the difference in transition properties . the millisecond pulsars low @xmath34 allows the field to diffuse out of the crust , and to be strong enough to collimate the flow and produce pulsations . one possibility might be that while the high @xmath35 of the diagonals suppresses the field entirely , the medium @xmath34 in the verticals allows some field to diffuse out . but the verticals show no trace of pulsation , so the magnetic field can not collimate any significant part of the flow . one way around these pulsation limits is to separate the field and accretion flow . any non - spherical accretion flow will predominantly bury the field in the region of the flow , leaving the field to escape in regions with little accretion . even the hot accretion flow envisaged for the hard island states favours the equatorial plane , with a geometry which is more like a thick disc than a truly spherical flow ( e.g. narayan & yi 1995 ) . this nicely circumvents the pulsation limits , but then there is no interaction between the magnetic field and the flow , and so no physical mechanism to change the behaviour of the transition . however , one aspect of the system that a polar magnetic field could affect is the jet . the most recent numerical simulations of the accretion flow magnetohydrodynamics show a causal link between the jet and accretion flow ( hawley & krolik 2006 ; mckinney 2006 ) , so this might give an indirect link between the magnetic field and accretion flow properties . one way to test this is to look at the radio emission from these systems . while theoretical models of jets are not well developed , we can use the observed gbh behaviour as a template . these show a clear correlation between radio and x - ray luminosity in their low / hard state , showing that @xmath36 is important in determining the power of the jet ( gallo et al . 2006 ) . hence to see whether there are differences in the jet emission between the millisecond pulsars , verticals and diagonals we need to compare the radio emission at the same @xmath36 . hysteresis gives potential problems , so ideally the comparison would be between persistent verticals and diagonals in the hard island state at the same @xmath37 . however , there is very little data to make this comparison , with only 4u 0614 + 091 and 4u 172834 for the persistent verticals and diagonals , respectively , and these do not overlap in @xmath37 ( migliari & fender 2006 ) . thus we speculate that the origin of the difference in transition behaviour between the verticals and diagonals is due to the presence of some magnetic field at the pole in the verticals which affects the accretion flow indirectly through jet formation , by contrast to the diagonals which have higher mass accretion rates , sufficient to bury the field everywhere . the atolls and millisecond pulsars are consistent with showing the same overall spectral evolution with changing mass accretion rate , but there are some subtle differences . the spectral shape of the soft ( banana ) branch shows subtle variations from object to object , most probably due to combination of inclination and spin changing the ratio of the observed disc to boundary layer luminosity , and boundary layer temperature . however , there are also clear differences in behaviour during the hard / soft transition which point to a more fundamental distinction . the data shows two types of sources , those where the transition makes a vertical track on the colour - colour diagram , and occurs at @xmath0 , and those which make a diagonal track on the colour - colour diagram with the transition at @xmath38 . there are hysteresis effects in individual sources which introduce dispersion in the transition luminosity , but these are large _ only _ for the outbursting atolls ( which are both verticals ) . splitting these outbursts into the rapid rise and slow decay phases show that the rapid rise looks like the millisecond pulsars ( so they are probably also verticals ) while the slow decay looks like the persistent verticals . thus it seems likely that large scale hysteresis effects are only seen in sources where the disc structure changes rapidly due to the onset of the hydrogen ionization instability . this is also consistent with the observed black hole behaviour . the association of the millisecond pulsars with verticals suggests that the difference in transition is ultimately linked to the surface magnetic field , and indeed , all the verticals have long term mass accretion rates which are smaller than those of the diagonals , though not as small as those of the millisecond pulsars . thus the verticals could have some small b field which is able to affect the inner accretion flow , but this must be indirect as otherwise these systems also would show pulsations . we speculate that the physical link between the magnetic field ( predominantly polar ) and accretion flow ( predominantly equatorial ) may be due to the changes in the jet , which would be testable with more radio data on these sources . this research has made use of data obtained through the high energy astrophysics science archive research center online service , provided by the nasa / goddard space flight center .

we systematically analyze all the available x - ray spectra of disc accreting neutron stars ( atolls and millisecond pulsars ) from the _ rxte _ database . we show that while these all have similar spectral evolution as a function of mass accretion rate , there are also subtle differences . there are two different types of hard / soft transition , those where the spectrum softens at all energies , leading to a diagonal track on a colour - colour diagram , and those where only the higher energy spectrum softens , giving a vertical track . the luminosity at which the transition occurs is _ correlated _ with this spectral behaviour , with the vertical transition at @xmath0 while the diagonal one is at @xmath1 . superimposed on this is the well known hysteresis effect , but we show that classic , large scale hysteresis occurs only in the outbursting sources , indicating that its origin is in the dramatic rate of change of mass accretion rate during the disc instability . we show that the long term mass accretion rate correlates with the transition behaviour , and speculate that this is due to the magnetic field being able to emerge from the neutron star surface for low average mass accretion rates . while this is not strong enough to collimate the flow except in the millisecond pulsars , its presence may affect the inner accretion flow by changing the properties of the jet . = -0.5 cm [ firstpage ] accretion , accretion discs x - rays : binaries , atoll sources

You are an expert at summarizing long articles. Proceed to summarize the following text: it is well known that the exclusive diffractive higgs production provides a very convenient tool for higgs searches at hadron colliders due to a very clean environment unlike the inclusive production @xcite . a qcd mechanism for the diffractive production of heavy central system has been proposed by kaidalov , khoze , martin and ryskin ( durham group ) for higgs production at the lhc ( see refs . @xcite ) . below we will refer to it as the kkmr approach . in the framework of this approach the amplitude of the exclusive @xmath12 process is considered to be a convolution of the hard subprocess amplitude describing fusion of two off - shell gluons producing a heavy system @xmath13 , and the soft hadronic factors containing information about emission of the relatively soft gluons from the proton lines ( see fig . [ fig : fig1 ] ) . in the framework of the @xmath14-factorisation approach these soft parts are written in terms of so - called off - diagonal unintegrated gluon distributions ( ugdfs ) . the qcd factorisation is rigorously justified in the limit of very large factorisation scale being the transverse mass of the central system @xmath15 . _ the qcd mechanism of diffractive production of the heavy central system @xmath16._,scaledwidth=35.0% ] in order to check the underlying production mechanism it is worth to replace the higgs boson by a lighter ( but still heavy enough to provide the qcd factorisation ) meson which is easier to measure . in this respect the exclusive production of heavy quarkonia is under special interest from both experimental and theoretical point of view @xcite . verifying the kkmr approach against various data on exclusive meson production at high energies is a good test of nonperturbative dynamics of parton distributions encoded in ugdfs . recently , the signal from the diffractive @xmath17 charmonia production in the radiative @xmath7 decay channel has been measured by the cdf collaboration @xcite : @xmath18 nb . assuming the absolute dominance of the spin-0 contribution , this result was published by the cdf collaboration in the form : @xmath19 indeed , in the very forward limit the contributions from @xmath6 vanish due to the @xmath20 selection rule @xcite . this is not true , however , for general kinematics @xcite . in particular , it was shown in ref . @xcite that the axial - vector @xmath10 production , due to a relatively large branching fraction of its radiative decay , may not be negligible and gives a significant contribution to the total signal measured by the cdf collaboration . the same holds also for the tensor @xmath3 meson contribution @xcite . recent durham group investigations @xcite support these predictions . the production of the axial - vector @xmath10 meson is additionally suppressed w.r.t . @xmath21 in the limit of on - shell fusing gluons ( with non - forward protons ) due to the landau - yang theorem @xcite . such an extra suppression may , in principle , lead to the dominance of the @xmath3 contribution over the @xmath10 one in the radiative decay channel @xcite . off - shell effects play a significant role even for the scalar @xmath8 production reducing the total cross section by a factor of 2 5 depending on ugdfs @xcite . the major part of the amplitude comes from rather small gluon transverse momenta @xmath4 . this requires a special attention and including all polarisation states @xmath6 . our present goal is to analyze these issues in more detail in the case of tensor charmonium production at the tevatron , to study its energy dependence and to compare with corresponding contributions from scalar and axial - vector charmonia . the paper is organized as follows . section 2 contains the generalities of the qcd central exclusive production mechanism , two different prescriptions for off - diagonal ugdfs are introduced and discussed . in section 3 we derive the hard subprocess amplitude @xmath22 in the nonrelativistic qcd formalism and consider its properties . section 4 contains numerical results for total and differential cross sections of @xmath17 cep and their correspondence to the last cdf data . in section 5 the summary of main results is given . the general kinematics of the central exclusive production ( cep ) process @xmath12 with @xmath16 being the colour singlet @xmath24 bound state has already been discussed in our previous papers on @xmath8 @xcite and @xmath10 @xcite production . in this section we adopt the same notations and consider the matrix element for exclusive @xmath3 production and its properties in detail . according to the kkmr approach the amplitude of the exclusive double diffractive color singlet production @xmath25 is @xcite @xmath26 where @xmath27 are the momentum transfers along the proton lines , @xmath28 is the momentum of the screening gluon , @xmath29 are the momenta of fusing gluons , and @xmath30 are the off - diagonal ugdfs ( see fig . [ fig : fig1 ] ) . traditional ( asymmetric ) form of the off - diagonal ugdfs is taken in the limit of very small @xmath31 in analogy to collinear off - diagonal gluon distributions ( with factorized @xmath32-dependence ) @xcite , i.e. @xmath33 with a quasiconstant prefactor @xmath34 which accounts for the single @xmath35 skewed effect @xcite and is found to be @xmath36 at the tevatron energy and @xmath37 at the lhc energy ( for lo pdf ) , @xmath38 are the effective gluon transverse momenta , as adopted in ref . @xcite , @xmath39 is the proton vertex factor , which can be parameterized as @xmath40 with @xmath41 @xcite , or by the isoscalar nucleon form factor @xmath42 as we have done in ref . below we shall refer to eq . ( [ asym - off ] ) as kmr ugdf . our results in ref . @xcite showed up a strong sensitivity of the kmrs numerical results @xcite on the definition of the effective gluon transverse momenta @xmath43 and the factorisation scales @xmath44 . this behavior is explained by the fact that for @xmath45 production the great part of the diffractive amplitude ( [ ampl ] ) comes from nonperturbatively small @xmath46 . it means that the total diffractive process is dominated by very soft screening gluon exchanges with no hard scale and extremely small @xmath31 . in principle , the factor @xmath34 in eq . ( [ asym - off ] ) should be a function of @xmath47 and @xmath48 or @xmath49 . in this case the off - diagonal ugdfs do not depend on @xmath47 and @xmath50 ( or @xmath51 ) , and their evolution is reduced to diagonal ugdfs evolution corresponding to one `` effective '' gluon . in general , the factor @xmath34 can depend on ugdf and reflects complicated and still not well known dynamics in the small-@xmath52 region . in order to test this small-@xmath52 dynamics and estimate the theoretical uncertainties related to introducing one `` effective '' gluon transverse momentum instead of two ones in eq . ( [ asym - off ] ) , in refs . @xcite we have used more generalized symmetrical prescription for the off - diagonal ugdfs . actually , it is possible to calculate the off - diagonal ugdfs in terms of their diagonal counterparts as follows , @xmath53 arguments must be omitted . ] @xmath54 where @xmath55 this form of skewed two - gluon ugdfs ( [ skewed - ugdfs ] ) is inspired by the positivity constraints for the collinear generalized parton distributions @xcite , and can be considered as a saturation of the cauchy - schwarz inequality for the density matrix @xcite . it allows us to incorporate the actual dependence of the off - diagonal ugdfs on longitudinal momentum fraction of the soft screening gluon @xmath47 and its transverse momentum @xmath50 in explicitly symmetric way . as will be shown below , these symmetric off - diagonal ugdfs lead to results which are consistent with the tevatron data . however , trying to incorporate the actual dependence of ugdfs on ( small but nevertheless finite ) @xmath47 we may encounter a problem . the kinematics of the double diffractive process @xmath12 does not give any precise expression for @xmath47 in terms of the phase space integration variables . from the qcd mechanism under consideration one can only expect the general inequality @xmath31 and upper bound @xmath56 since the only scale appearing in the left part of the gluon ladder is the transverse momentum of the soft screening gluon @xmath57 . to explore the sensitivity of the final results on the values of @xmath47 , staying in the framework of traditional kkmr approach , one can introduce naively @xmath58 with an auxiliary parameter @xmath59 @xcite . in our earlier papers @xcite we considered the limiting case of maximal @xmath47 ( with @xmath60 ) . however , it is worth to compare the predictions of the underlying qcd mechanism for smaller @xmath59 against the available experimental data in order to estimate typical @xmath47 values . we will analyze this issue in greater detail in the results section . projection of the hard amplitude onto the singlet charmonium bound state @xmath61 is given by an 4-dimentional integral over relative momentum of quark and antiquark @xmath62 @xcite : @xmath63.\nonumber\end{aligned}\ ] ] here the function @xmath64 is the momentum space wave function of the charmonium , the clebsch - gordan coefficient in color space is @xmath65 the trace of @xmath32-matrices is @xmath66 , and the projection operator @xmath67 for a small relative momentum @xmath68 has the form @xmath69 since @xmath70-wave function @xmath71 vanishes at the origin , we may expand the trace in eq . ( [ amplitude - diff ] ) in the taylor series around @xmath72 , and only the linear terms in @xmath73 survive . this yields an expression proportional to @xmath74 with the derivative of the @xmath70-wave radial wave function at the origin @xmath75 whose numerical value can be found in ref . @xcite . the general @xmath70-wave result ( [ amplitude - diff ] ) may be further reduced by employing the clebsch - gordan identity which for the tensor @xmath76 charmonium states reads @xmath77 taking into account standard definitions of the light - cone vectors @xmath78 and momentum decompositions @xmath79 and using the gauge invariance property ( gribov s trick ) one gets the following projection ( for any spin @xmath80 ) @xmath81 since we adopt here the definition of the polarization vectors proportional to gluon transverse momenta @xmath82 , then @xmath83 it shows that gluon transverse momenta are necessary to get a nonzeroth diffractive cross section . summarizing all ingredients above , we get the vertex factor @xmath22 in the following covariant form @xmath84 polarization tensor of @xmath76 satisfies the following relations ( see e.g. ref . @xcite ) @xmath85 one can check that it may be represented in the following general form @xmath86\right)\\ \nonumber & & + \frac12\delta_{1|\lambda|}\big(i[n_2^{\mu}n_3^{\nu}+n_3^{\mu}n_2^{\nu } ] \pm[n_1^{\mu}n_3^{\nu}+n_3^{\mu}n_1^{\nu}]\big ) \\ \nonumber & & -\frac12\delta_{2|\lambda|}\big(i[n_1^{\mu}n_2^{\nu}+n_2^{\mu}n_1^{\nu } ] \pm[n_1^{\mu}n_1^{\nu}-n_2^{\mu}n_2^{\nu}]\big ) \ ; , \end{aligned}\ ] ] where @xmath87 are light - like basis vectors satisfying @xmath88 ( with @xmath89 ) , and @xmath90 are the @xmath3 meson helicities . to our best knowledge , there is no explicit decomposition of the meson polarisation tensor @xmath91 in terms of basis vectors @xmath92 like eq . ( [ poltens ] ) in the literature . in practical calculations below it is convenient to use it in a different representation : @xmath93 + \frac{\sqrt{6}}{4}(2-|\lambda|)(1-|\lambda|)n_3^{\mu}n_3^{\nu}+\\ \nonumber & & + \frac14\lambda(1-|\lambda|)[n_1^{\mu}n_1^{\nu}-n_2^{\mu}n_2^{\nu}]+ \frac14i|\lambda|(1-|\lambda|)[n_1^{\mu}n_2^{\nu}+n_2^{\mu}n_1^{\nu}]+\\ \nonumber & & + \frac12\lambda(2-|\lambda|)[n_1^{\mu}n_3^{\nu}+n_3^{\mu}n_1^{\nu}]+ \frac12i|\lambda|(2-|\lambda|)[n_2^{\mu}n_3^{\nu}+n_3^{\mu}n_2^{\nu } ] \ ; .\end{aligned}\ ] ] similarly to what has been done for @xmath10 production in ref . @xcite , in the c.m.s . frame we choose the basis with collinear @xmath94 and @xmath95 vectors ( so , we have @xmath96 ) as a simplest one @xmath97 note , that we choose @xmath98 to be transverse to the c.m.s beam axis ( see fig . [ fig : cms ] ) , while @xmath99 are turned around by the polar angle @xmath100 $ ] between @xmath101 and the c.m.s . beam axis . in the considered basis @xmath102 we have the following coordinates of the incoming protons @xmath103 the gluon transverse momenta with respect to the c.m.s . beam axis are @xmath104 where @xmath105 are the components of the gluon transverse momenta in the basis with the @xmath106-axis collinear to the c.m.s . beam axis . from definition ( [ protons ] ) it follows that energy of the meson and polar angle @xmath107 are related to covariant scalar products in the considered coordinate system as @xcite @xmath108 furthermore , we also see that from @xmath109 and @xmath110 we have @xmath111 relations ( [ epsi ] ) and ( [ x1x2cos ] ) show that the interchange of proton momenta @xmath112 is equivalent to the interchange of the angle @xmath113 , i.e. @xmath114 and @xmath115 simultaneously . the last permutation also provides the interchange of the longitudinal components of gluons momenta @xmath116 . conservation laws provide us with the following relations between components of gluon transverse momenta and covariant scalar products @xmath117 where @xmath118 is the meson transverse momentum with respect to the @xmath106-axis . the appearance of the factor @xmath119 guarantees the applicability of ( [ rel_comp ] ) for positive and negative @xmath120 . note that under permutations @xmath121 implied by bose statistics the components interchange as @xmath122 and @xmath123 . in our notations the quantity @xmath124 plays a role of the noncollinearity of meson in considered coordinates . a straightforward calculation leads to the following vertex function in these coordinates @xmath125\times{\bf n}_1|\,(1-|\lambda|)\,\mathrm{sign}(\sin\psi)\,\mathrm{sign}(\cos\psi)+\nonumber\\ & & 2\,|[{\bf q}_{1,t}\times{\bf q}_{2,t}]\times{\bf n}_3|\,(2-|\lambda|)\big\}-\big[2q_{1,t}^2q_{2,t}^2+(q_{1,t}^2+q_{2,t}^2)(q_{1,t}q_{2,t})\big ] \big\{3m^2(\cos^2\psi+1)\lambda(1-|\lambda|)+\nonumber\\ & & 6me\sin(2\psi)\,\lambda(2-|\lambda|)\,\mathrm{sign}(\sin\psi)\,\mathrm{sign}(\cos\psi)+ \sqrt{6}\,(m^2 + 2e^2)\sin^2\psi\,(1-|\lambda|)(2-|\lambda|)\big\}\biggr ] \ ; , \nonumber\end{aligned}\ ] ] where @xmath126\times{\bf n}_1|=\sqrt{q_{1,t}^2q_{2,t}^2-(q_{1,t}q_{2,t})^2}\,|\cos\psi|,\\ & & |[{\bf q}_{1,t}\times{\bf q}_{2,t}]\times{\bf n}_3|=\frac{e}{m}\sqrt{q_{1,t}^2q_{2,t}^2-(q_{1,t}q_{2,t})^2}\,|\sin\psi|.\end{aligned}\ ] ] the amplitude ( [ chi2-fin ] ) explicitly obeys the bose symmetry under the interchange of gluon momenta and polarizations due to resulting simultaneous permutations @xmath115 , @xmath114 and @xmath127 . it follows from the conservation laws that @xmath128 let us consider first the limit of the `` coherent '' scattering of protons @xmath129 , so @xmath130 the production vertex ( [ chi2-fin ] ) in this limit has the form @xmath131 \big\{3m^2(\cos^2\psi+1)\lambda(1-|\lambda|)+\nonumber\\ & & 6me\sin(2\psi)\,\lambda(2-|\lambda|)\,\mathrm{sign}(\sin\psi)\,\mathrm{sign}(\cos\psi)+ \sqrt{6}\,(m^2 + 2e^2)\sin^2\psi\,(1-|\lambda|)(2-|\lambda|)\big\}\biggr ] \nonumber\end{aligned}\ ] ] we see that in contrast to the axial - vector case considered in ref . @xcite , the diffractive amplitude of @xmath3 production does not turn to zero in this `` coherent '' limit for @xmath132 . in the forward limit @xmath133 ( which is a particular case of the `` coherent '' one ) the amplitude turns to zero at any meson rapidities @xmath134 . indeed , we have @xmath135 and @xmath136 and the amplitude turns into @xmath137 - 2\,|\lambda|\,q_{0t}^xq_{0t}^y\ , \mathrm{sign}(\cos\psi)|_{\psi\to0,\pi}\big\ } \nonumber \ , .\end{aligned}\ ] ] the imaginary part of this vertex function turns out to be antisymmetric w.r.t . interchanging @xmath138 , whereas its real part is antisymmetric w.r.t . changing the sign of @xmath139 or @xmath140 component , i.e. @xmath141 since in this case @xmath142 in the forward limit , then the double integral in the diffractive amplitude has an antisymmetric integrand and turns to zero in the symmetric limit @xmath143 this explicitly confirms the observation made in refs . @xcite . very recently , when our paper was almost complete , a paper by l. harland - lang , v. khoze , m. ryskin and w. stirling ( hkrs ) @xcite appeared where the hard subprocess amplitudes @xmath144 ( based on formalism by kuhn et al for @xmath145 @xcite ) including the gluon virtualities were listed for different spins including the tensor @xmath3 : @xmath146,\label{hkrs - j0}\\ v_{j=1,\lambda}^{\mathrm{hkrs}}&=&-\frac{2ic}{s}p_{1,\nu}p_{2,\alpha}\varepsilon^{\mu\nu\alpha\beta}\epsilon_{\beta } \big[(q_{2,t})_{\mu}q_{1,t}^2-(q_{1,t})_{\mu}q_{2,t}^2\big],\label{hkrs - j1}\\ v_{j=2,\lambda}^{\mathrm{hkrs}}&=&\frac{\sqrt{2}cm}{s}\,\epsilon^{\mu\alpha } \big[s(q_{1,t})_{\mu}(q_{2,t})_{\alpha}+2(q_{1,t}q_{2,t})p_{1,\mu}p_{2,\alpha}\big ] , \label{hkrs - j2}\end{aligned}\ ] ] where the constant prefactor is @xmath147 the first amplitude @xmath148 ( [ hkrs - j0 ] ) is the same as the expression obtained in ref . @xcite ( up to a factor of 2 coming from different normalisations of the hard part @xmath149 in our case and @xmath150 in ref . @xcite ) , where the major role of the gluon virtualities in the hard subprocess amplitude of quarkonia production was claimed to be crucial . in particular , it was shown that an account of the gluon virtualities reduces the previous kmrs result in ref . @xcite for on - mass - shell gluons @xmath151 by a factor of 2 3 . the second amplitude , @xmath152 , looks different from our previous result , obtained in ref . @xcite . however , one can directly check that the difference between the amplitudes ( [ hkrs - j1 ] ) and ( 2.12 ) in ref . @xcite turns to zero when fixing the coordinates in the c.m.s . frame of reference as in eq . ( [ protons ] ) ( see also fig . [ fig : cms ] ) and the meson polarisation vector @xmath153 with the basis as in eq . ( [ basis ] ) . due to the covariant structure of these amplitudes , the last observation means that they are the same in any frame of reference . the calculations proving this equality are rather involved , and we do not show them explicitly here . very similar situation holds for @xmath3 production amplitudes . namely , the amplitudes ( [ hkrs - j2 ] ) and ( [ vgen - chi2 ] ) turned out to be the same under fixing the coordinates as in the previous section . therefore , under the kinematical relations our results for the hard subprocess amplitudes are in complete agreement with the corresponding hkrs results . let us now turn to the discussion of numerical results . results for the differential cross sections @xmath154 of the diffractive @xmath17 meson production at the tevatron energy @xmath155 gev for different ugdfs are shown in table [ table : dy0 ] . in the last column we show the results for the expected signal in the @xmath156 channel summed over all @xmath45 spin states ( and all polarisation states of @xmath6 mesons ) @xmath157 which can be compared with the corresponding value measured by the cdf collaboration @xcite : @xmath158 nb . in refs . @xcite is was assumed that the nlo corrections factor @xmath159 in the @xmath160 vertex is the same as in the @xmath161 width implying that @xmath162 . in general , such corrections depend on spin of @xmath163 resonance . so , the diffractive cross section for each @xmath164 has to be multiplied by not necessarily the same factor @xmath165 , as shown in eq . ( [ sig ] ) . this can be done , however , only for @xmath166 and @xmath167 states , and the corresponding nlo qcd radiative corrections are well - known @xcite : @xmath168 due to the landau - yang theorem the decay of the axial vector charmonium @xmath169 to on - shell gluons is forbidden , and there are no reliable calculations of the nlo qcd corrections to its coupling with off - shell gluons . in the following we take naively @xmath170 . this leads to an additional uncertainty of the model predictions . as has been claimed in refs . @xcite the absorptive corrections are quite sensitive to the meson spin - parity . this was studied before in the context of scalar and pseudoscalar higgs production in ref . we adopt here the following effective gap survival factors , calculated in ref . @xcite for different spins including eikonal and so - called enhanced contributions : @xmath171 the contribution of the scalar @xmath8 cep , which was initially assumed to be the dominant one @xcite , is reduced by a very small branching ratio of its observable radiative decay @xcite . in turn , the strong suppression of the @xmath10 central production in both the on - mass - shell limit of fusing gluons ( due to landau - yang theorem @xcite ) and the forward scattering limit of outgoing protons ( due to the so - called @xmath20 selection rule @xcite ) may be partially compensated by its much higher branching ratio to the observed @xmath156 final state @xcite . analogously to the axial - vector case , the suppression of the tensor @xmath3 cep is likely to be eliminated by its large decay branching ratio @xcite , and the resulting value of the radiative decay signal is under our special interest . .[table : dy0 ] differential cross section @xmath172 ( in nb ) of the exclusive diffractive production of @xmath17 mesons and their partial and total signal in radiative @xmath156 decay channel @xmath173 at tevatron for different ugdfs , cuts on the transverse momentum of the gluons in the loop ( @xmath57 ) and different values of the auxiliary parameter @xmath59 controlling the characteristic @xmath47 values in the symmetric skewed ugdfs prescription ( [ skewed - ugdfs ] ) ( denoted as `` sqrt '' ) . nlo skewedness factor @xmath174 for the kmr asymmetric prescription ( [ asym - off ] ) ( denoted as `` @xmath175 '' ) , nlo correction factors ( [ knlo ] ) and absorptive correction factors ( [ s2eff ] ) are included . contributions from all polarisations are incorporated . [ cols="^,^,^,^,^,^,^,^,^,^,^ " , ] in the case of the kmr ugdf , we observe quite substantial dependence of the predicted observable signal w.r.t . variations of the infrared cut - off on small transverse momenta of the gluons in the most internal loop ( see fig . [ fig : cutdep ] ) . from table [ table : dy0 ] we see that the shift of @xmath176 from the value @xmath177 used in ref . @xcite down to the minimal perturbative scale of the integrated grv94ho distributions @xmath178 @xcite leads to increase of the cross section by a factor of about 3 , approaching the cdf data . for comparison , decrease of the @xmath179 from @xmath180 down to @xmath178 leads to increase of the cross section by a factor of 6 . since we can not estimate the nonperturbative contribution coming from below @xmath178 , this allows us to conclude that perturbatively motivated kmr ugdf leads to infrared unstable result in the case of relatively light charmonium cep . it is clear that the essential part of the qcd dynamics comes from the nonperturbative region of transverse momenta below the hkrs cut - off @xmath177 @xcite . ks and gbw ugdfs allow to incorporate some unknown physics even below the minimal grv scale @xmath181 , avoiding ambiguities in defining the effective gluon momenta . applying the kmr s asymmetrical off - diagonal ugdf according to eq . ( [ asym - off ] ) ( `` @xmath34 '' prescription ) in the case of the gbw models we get strongly overestimated observable signal at tevatron , which means that in this case it is crucial to take into account the @xmath47-dependence of off - diagonal ugdfs when going deeply into the infrared region of small @xmath182 s . the @xmath47-dependent `` sqrt '' prescription , introduced in eq . ( [ skewed - ugdfs ] ) , leads to observable signal , which is much closer to the experimental data . the `` sqrt '' prescription , introduced in eq . ( [ skewed - ugdfs ] ) , provides an agreement with the data ( within a factor of 2 in overall theoretical uncertainty between different ugdfs ) with the ks ( with rather small @xmath183 ) and gbw models giving the cross section @xmath184 ( see table [ table : dy0 ] ) . this practically means that the smaller @xmath57 comes into the game , the smaller @xmath47 w.r.t . @xmath185 is required to get the data description , providing one more argument about importance of nonperturbative effects in charmonia cep . the relative contributions of different charmonium states in the @xmath156 channel ( including absorption effects ) in the case of , e.g. , ks model are found to be : @xmath186 they are not affected by smaller @xmath47 or nonlinear effects in this model . as the normalization point we took the contribution of the @xmath8 meson cep as was done in ref . @xcite . _ distributions @xmath187 in rapidity of @xmath8 ( left panel ) , @xmath10 ( middle panel ) and @xmath3 ( right panel ) mesons for different ugdfs at the tevatron energy @xmath188 = 1.96 tev . the dash - dotted line corresponds to the ks ugdf @xcite in the symmetrical `` sqrt''-prescription with @xmath189 , solid line kmr ugdf @xcite with @xmath190 , @xmath191 and grv94ho pdf @xcite , and short - dashed line represents result with the gbw ugdf @xcite ( @xmath189 ) . absorption effects are not included here._,scaledwidth=130.0% ] _ distributions @xmath187 in rapidity of @xmath8 ( left panel ) , @xmath10 ( middle panel ) and @xmath3 ( right panel ) mesons for different ugdfs at the tevatron energy @xmath188 = 1.96 tev . the dash - dotted line corresponds to the ks ugdf @xcite in the symmetrical `` sqrt''-prescription with @xmath189 , solid line kmr ugdf @xcite with @xmath190 , @xmath191 and grv94ho pdf @xcite , and short - dashed line represents result with the gbw ugdf @xcite ( @xmath189 ) . absorption effects are not included here._,scaledwidth=130.0% ] _ distributions @xmath187 in rapidity of @xmath8 ( left panel ) , @xmath10 ( middle panel ) and @xmath3 ( right panel ) mesons for different ugdfs at the tevatron energy @xmath188 = 1.96 tev . the dash - dotted line corresponds to the ks ugdf @xcite in the symmetrical `` sqrt''-prescription with @xmath189 , solid line kmr ugdf @xcite with @xmath190 , @xmath191 and grv94ho pdf @xcite , and short - dashed line represents result with the gbw ugdf @xcite ( @xmath189 ) . absorption effects are not included here._,scaledwidth=130.0% ] in table [ table : dy0 ] we also presented results with the linear kutak - stato model based on the unified bfkl - dglap framework and the nonlinear one based on the balitsky - kovchegov equation @xcite . it turned out that incorporation of the nonlinear effects responsible for the gluon recombination in this model reduces the @xmath192 cep cross sections by 30 - 50 % . we see that the nonlinear effects play a crucial role in diffractive quarkonia production effectively decreasing the characteristic values of @xmath47 ( controlled by @xmath59 ) . however , reliable predictions including the nonlinear effects require the exact knowledge of the triple pomeron vertex at nllx accuracy , which is yet unknown . it is also interesting to compare the diffractive production of @xmath45 states at different energies . as an example , in table [ table : edep ] we present the integrated ( over full phase space ) cross sections of @xmath17 production at rhic , tevatron and lhc energies . the results show similar energy behavior of the diffractive cross section for different ugdfs as well as for different @xmath45 states . _ distribution in @xmath27 of @xmath8 ( left panel ) , @xmath10 ( middle panel ) and @xmath3 ( right panel ) for meson cep for different ugdfs . the meaning of curves here is the same as in fig . [ fig : chic012-dy]._,scaledwidth=130.0% ] _ distribution in @xmath27 of @xmath8 ( left panel ) , @xmath10 ( middle panel ) and @xmath3 ( right panel ) for meson cep for different ugdfs . the meaning of curves here is the same as in fig . [ fig : chic012-dy]._,scaledwidth=130.0% ] _ distribution in @xmath27 of @xmath8 ( left panel ) , @xmath10 ( middle panel ) and @xmath3 ( right panel ) for meson cep for different ugdfs . the meaning of curves here is the same as in fig . [ fig : chic012-dy]._,scaledwidth=130.0% ] finally , let us turn to differential distributions . in fig . [ fig : chic012-dy ] we show the differential cross section @xmath193 in rapidity @xmath134 for all @xmath45 states . in this figure and in the following , all helicity contributions for @xmath6 cep are taken into account . here and below we show only bare cep cross sections for gbw , ks and kmr ugdfs . in the last case , we present the results computed with the hkrs cut - off parameter @xmath177 @xcite . we see that the shape of the curves is rather similar , however , they have substantially different maxima . the biggest cross section for the @xmath21 states is obtained with the ks ugdf , whereas for @xmath10 the ks and kmr ugdfs give quite similar cross sections . in fig . [ fig : chic012-dt ] we present corresponding distributions in @xmath194 or @xmath195 ( identical ) , again for different ugdfs . except of normalisation the shapes are rather similar . this is because of the @xmath196 and @xmath197 dependencies of form factors ( describing the off - diagonal effect ) are taken the same for different ugdfs . _ distribution in relative azimuthal angle @xmath198 between outgoing protons for @xmath8 ( left panel ) , @xmath10 ( middle panel ) and @xmath3 ( right panel ) meson cep for different ugdfs . the meaning of curves here is the same as in fig . [ fig : chic012-dy]._,scaledwidth=130.0% ] _ distribution in relative azimuthal angle @xmath198 between outgoing protons for @xmath8 ( left panel ) , @xmath10 ( middle panel ) and @xmath3 ( right panel ) meson cep for different ugdfs . the meaning of curves here is the same as in fig . [ fig : chic012-dy]._,scaledwidth=130.0% ] _ distribution in relative azimuthal angle @xmath198 between outgoing protons for @xmath8 ( left panel ) , @xmath10 ( middle panel ) and @xmath3 ( right panel ) meson cep for different ugdfs . the meaning of curves here is the same as in fig . [ fig : chic012-dy]._,scaledwidth=130.0% ] in fig . [ fig : chic012-dphi ] we show the correlation function @xmath199 in relative azimuthal angle @xmath198 between outgoing protons for different @xmath45 states . the shapes of the distributions are somewhat dependent on ugdfs . it is interesting to note here that the ks and kmr ugdfs lead to very similar angular dependence of @xmath199 for all @xmath45 states . in the case when energy resolution is not enough to separate contributions from different states of @xmath45 ( @xmath8 , @xmath10 , @xmath3 ) , which seems to be the case for tevatron , the distribution in relative azimuthal angle may , at least in principle , be helpful . the fact that the angular distributions are not simple functions ( like @xmath200 , @xmath201 ) of the relative azimuthal angle between outgoing nucleons is due to the loop integral in eq . ( [ ampl ] ) which destroys the dependence one would obtain with single fusion of well defined ( spin , parity ) objects ( mesons or reggeons ) @xcite . our results can be summarized as follows : we have derived the qcd amplitude for central exclusive production of tensor @xmath3 meson . this amplitude vanishes in the forward limit of outgoing protons , as demanded by the @xmath20 selection rule . our numerical results show the importance of non - forward corrections , including all polarisation states of @xmath3 and nonperturbative contributions to the @xmath3 cep . inclusion of all the ingredients leads to a noticeable contribution of the @xmath3 meson in the observable radiative decay channel depending on ugdf . we have observed the importance of the @xmath9 state @xmath10 cep and @xmath202 states for @xmath3 cep at @xmath5 0 , whereas the total cep cross section is dominated by maximal helicity contributions . the main contribution to diffractive charmonium production comes from small gluon transverse momenta @xmath203 leading to quite substantial sensitivity of the corresponding cross section on the infrared cut - off in perturbatively modeled kmr ugdf . alternatively one could use ugdfs like kutak - stato and gbw models , which by construction can be used for any values of the gluon transverse momenta . we have tested the symmetrical prescription for off - diagonal ugdfs , following from positivity constraints and incorporating @xmath52 , @xmath182 dependence of both participating gluons , against the present cdf experimental data . a rather good quantitative agreement with the cdf data on charmonium cep in the radiative decay channel is achieved with the nonlinear kutak - stato ugdf model giving the cross section @xmath204 nb without imposing extra normalisation conditions beyond the qcd framework . such a description is achieved by incorporating very soft screening gluons with @xmath205 . we have also calculated total cross sections of @xmath45 cep at different energies ( rhic , tevatron and lhc ) , as well as differential distributions in three phase space variables @xmath206 . overall theoretical uncertainty of the qcd mechanism under consideration is rather high but hard to estimate due to large unknown nonperturbative contributions coming into the game and not well known higher - order qcd corrections to the hard subprocess @xmath207 ( especially , in the axial - vector case ) . also , absorptive corrections may depend on ugdf used in the calculation , and there is no reliable estimation of such a sensitivity in literature . in the present paper we kept the strategy to study different distinct options and analyze the sensitivity of the final results with respect to the ugdfs choice , prescriptions for skewed ugdfs , nonperturbative cut - off parameter and characteristic @xmath47 variations , etc . then a comparison with experimental data would allow to select the most reliable option . however , we observe a variety of such `` good '' options , namely , description of the data ( with , however , pretty large theoretical uncertainties related , in particular , with unknown nlo corrections ) can be , in principle , achieved for a few ugdfs ( gbw , ks and kmr ugdfs , see table [ table : dy0 ] ) . each of them pick up some essential qcd dynamics . further constraints can , in principle , be settled by experimental measurements of separate @xmath192 contributions , the energy dependence of the cross section and the shapes of differential distributions . useful discussions and helpful correspondence with mike albrow , sergey baranov , rikard enberg , wodek guryn , lucian harland - lang , gunnar ingelman , valery khoze , francesco murgia , mikhail ryskin and wolfgang schfer are gratefully acknowledged . this study was partially supported by the polish grant of mnisw n n202 249235 , the russian foundation for fundamental research , grants no . 07 - 02 - 91557 , 08 - 02 - 00896 , 09 - 02 - 00732 and no . 09 - 02 - 01149 . m. g. ryskin , a. d. martin , v. a. khoze and a. g. shuvaev , j. phys . g * 36 * , 093001 ( 2009 ) [ arxiv:0907.1374 [ hep - ph ] ] ; 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( particle data group ) , phys . b667 * 1 ( 2008 ) and 2009 partial update for the 2010 edition .

we discuss central exclusive production ( cep ) of the tensor @xmath0 meson in proton-(anti)proton collisions at tevatron , rhic and lhc energies . the amplitude for the process is derived within the @xmath1-factorisation approach . differential and total cross sections are calculated for several unintegrated gluon distributions ( ugdfs ) . we compare exclusive production of all charmonium states @xmath2 and @xmath3 . equally good description of the recent tevatron data is achieved both with martin - ryskin phenomenological ugdf and ugdf based on unified bfkl - dglap approach . unlike for higgs production , the main contribution to the diffractive amplitude of heavy quarkonia comes from nonperturbative region of gluon transverse momenta @xmath4 . at @xmath5 0 , depending on ugdf we predict the contribution of @xmath6 to the @xmath7 channel to be comparable or larger than that of the @xmath8 one . this is partially due to a significant contribution from lower polarization states @xmath9 for @xmath10 and @xmath11 for @xmath3 meson . corresponding theoretical uncertainties are discussed .

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Proceed to summarize the following text: the scaling relation between black hole mass and host - galaxy properties , e.g. , the black hole mass@xmath5stellar velocity dispersion relation ( @xmath0 ) , suggests a coevolution of black holes and galaxies ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) , motivating various theoretical and observational studies to constrain the origin of the scaling relations and their cosmic evolution @xcite . along with inactive galaxies , galaxies hosting active galactic nuclei ( agn ) also seem to follow the @xmath0 relation with a similar slope ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) , indicating that the present - day galaxies show a similar scaling relation regardless of black hole activity . in contrast , it has been debated whether present - day narrow - line seyfert 1 galaxies ( nls1s ) deviate from the @xmath0 relation ( e.g. , * ? ? ? * ; * ? ? ? * ) . as a sub - class of agns , nls1s were initially identified by the relatively small width of the broad - component of the balmer lines ( fwhm @xmath2 2000 km s@xmath6 ) and a weak [ o iii]-to - h@xmath4 ratio ( [ o iii]/h@xmath4 @xmath2 3 ; osterbrock & pogge 1985 ) . since nls1s are believed to have small black hole masses and high eddington ratios ( boroson 2002 ) , nls1s are often considered as relatively young agns hosting black holes in a growing phase although the time evolution among various types of agns is highly uncertain . thus , it is interesting to investigate the location of nls1s in the @xmath0 plane in the context of black hole - galaxy coevolution . a number of studies have been devoted to studying the @xmath0 relation of nls1s over the last decade , resulting in a controversy . on the one hand , some studies claimed that nls1 lie below the @xmath0 relation on average with smaller black hole masses at fixed stellar velocity dispersions , compared to the broad - line agns and quiescent galaxies ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) . on the other hand , other studies reported that the nls1s are generally on the @xmath0 relation ( e.g. , * ? ? ? * ; * ? ? ? the fundamental limitation of the aforementioned studies is the fact that stellar velocity dispersions were not directly measured . instead , the width of the narrow [ o iii ] emission line at 5007 was used as a surrogate for stellar velocity dispersion , based on the empirical correlation between [ o iii ] width and stellar velocity dispersion @xcite , although there is a considerably large scatter between them . if the ionized gas in the narrow - line region follows the gravitational potential of the host - galaxy , then [ o iii ] line width can be substituted for stellar velocity dispersion . however for individual objects the uncertainty of this substitution is very large as shown by the direct comparison between [ o iii ] width and the measured stellar velocity dispersion ( e.g. , * ? ? ? * ; * ? ? ? moreover , the [ o iii ] line often suffers from the effect of outflow , manifesting an asymmetric line profile and a strong blue - shifted wing component ( e.g. * ? ? ? * ; * ? ? ? * ) . in this case , the width of the [ o iii ] line will become much broader than stellar velocity dispersion , if the blue wing is not properly corrected for . in fact , @xcite showed that when the blue wing component is removed in measuring the width of the [ o iii ] line , the inferred stellar velocity dispersion from [ o iii ] becomes smaller , hence the nls1 show a consistent @xmath0 relation compared to broad - line agns . the solution to this decade - long debate is to investigate the locus of nls1s in the @xmath0 plane , using _ directly measured _ stellar velocity dispersion . although , measuring stellar velocity dispersion of agn host galaxies is difficult due to the presence of strong agn features , i.e. , power - law continuum , fe ii emission , and broad emission lines , it is possible to measure stellar velocity dispersion if high quality spectra are available as demonstrated in a number of studies ( e.g. , * ? ? ? * ; * ? ? ? ? * ; * ? ? ? * ; * ? ? ? * ) . in this paper , we present the direct stellar velocity dispersion measurements and estimates of black hole masses for a sample of 93 nls1s at z @xmath7 selected from sloan digital sky survey data release 7 ( sdss dr7 ) @xcite , in order to investigate the @xmath0 relation of nls1s . we describe the sample selection and properties in section 2 , and the analysis including mass determination and stellar velocity dispersion measurements in section 3 . section 4 presents the results , followed by discussion in section 5 , and summary and conclusions in section 6 . throughout the paper , we adopt a cosmology of @xmath8 km s@xmath9 mpc@xmath9 , @xmath10 and @xmath11 . nls1s are generally defined with two criteria : ( 1 ) the full - width - at - half - maximum ( fwhm ) of broad component of the balmer lines @xmath2 2000 km s@xmath6 , and ( 2 ) the line flux ratio [ o iii]/h@xmath4 @xmath2 3 @xcite . additional characteristics of nls1s include strong fe ii emission @xcite , high eddington ratio and soft x@xmath5ray emission @xcite . in this study , we selected a sample of nls1s from sdss dr7 @xcite , based on the width of balmer lines and the [ o iii]/h@xmath4 flux ratios . first , we selected nls1 candidates by limiting the width of h@xmath4 to 500@xmath52500 km s@xmath9 , using the specline class in the sdss query tool ( http://casjobs.sdss.org ) . since the line width measurements from the sdss pipeline is not precise , we used a wider width range than the conventional definition for the initial selection , obtaining 4,252 nls1 candidates at z @xmath2 0.1 . second , using this initial sample , we performed a multi - component spectral decomposition analysis for each galaxy , to properly measure the width of the broad component of the balmer lines . in the fitting process , we included multiple components , namely , featureless agn continuum , stellar population model , and fe ii emission component , using an idl - based spectral decomposition code ( see * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) . by subtracting the linear combination of featureless agn continuum , stellar component and fe ii emission , we obtained emission line spectra and fit the broad and narrow emission lines ( see section 3.1 for the detailed fitting process ) . based on the measurements from the fitting process , we finalized a sample of 464 nls1s , that satisfy the aforementioned two criteria by limiting the fwhm of broad h@xmath12 between 800 and 2200 km s@xmath6 and the line flux ration [ o iii]/h@xmath4 less than 3 . among these objects , we measured and collected the stellar velocity dispersion for 93 nls1s . for 63 objects , we were able to directly measure stellar velocity dispersion using the sdss spectra ( see section 3.5 ) while for 30 objects we obtained the stellar velocity dispersion measurements from @xcite . thus , using this sample of 93 nls1s , we investigate the properties of nls1 and the @xmath0 relation . note that the distribution of nls1 properties ( i.e. , [ o iii]/h@xmath4 ratio , fe ii / h@xmath4 ratio , h@xmath12 luminosity and width ) of the final sample of 93 objects is similar to that of the initial sample of 464 objects , suggesting that we may treat the final sample as a random subsample of nls1 galaxy population . figure 1 presents the distributions of redshift and the width of h@xmath12 of the final sample ( top panels ) . to demonstrate the weak [ o iii ] emission and strong fe ii emission of the sample as the characteristic features of nls1s @xcite , we also present the distribution of the flux ratio [ o iii]/h@xmath4 ( r5007 ) and fe ii / h@xmath4 ( r4570 ) ratios in figure 1 ( bottom panels ) . since the h@xmath4 is relatively weak and the decomposition of the broad and narrow components of h@xmath4 is uncertain , we used the total h@xmath4 flux to compare with [ o iii ] and fe ii fluxes . in the case of the [ o iii ] strength ( r5007 ) , all galaxies in our sample show low [ o iii]/h@xmath4 ratio ( @xmath23 ) , with a median 1.05 and a mean 1.14 . the fe ii strength ( r4570 ) , defined by the line flux ratio of fe ii emission integrated over the 4434@xmath54684 region , to h@xmath4 ( e.g. , * ? ? ? * ) , is also high with a mean 1.06 , as similarly found by other nls1 studies ( for dependence on the r4570 index , see 4.1 ) . for example , @xcite used the broad component of h@xmath4 to compare with fe ii and reported the mean r4570 as @xmath130.82 , while @xcite adopted the total flux of h@xmath4 and found the mean r4570 @xmath130.7 . the @xmath1 estimated with the line dispersion of broad component of h@xmath12 ranges over an order of magnitude , i.e. , log @xmath1/@xmath14= 5.84@xmath57.38 with a mean 6.72 , which is comparable to that of the previous nls1 @xmath0 relation studies @xcite . the eddington ratio of our nls1s ranges from 5% to the eddington limit with a mean of 0.2 - 0.3 , depending on the mass estimates . we performed multi - components spectral fitting analysis in two separate spectral ranges : h@xmath4 region ( 4400@xmath55580 ) and h@xmath12 region ( 6500@xmath56800 ) . for the h@xmath4 region , we followed the procedure given by our previous studies ( * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * see also barth et al . 2013 ) . after converting all spectra to the rest frame , we modeled the observed spectra with three components , i.e. , featureless agn continuum , host - galaxy starlight , and fe ii emission blends , by respectively using a single power law continuum , a stellar population model based on the sed templates from @xcite , and an fe ii template from @xcite . the best continuum model was determined in the regions 4430@xmath54600 and 5080@xmath55550 , where fe ii emission dominates . we simultaneously fitted all 3 components , using the nonlinear levenberg - marquardt least - squares fitting routine @xmath15 @xcite in idl . after subtracting the featureless agn continuum and host - galaxy starlight from the raw spectra , emission line fitting for h@xmath4 , [ o iii ] @xmath164959 and [ o iii ] @xmath165007 was carried out for this region . since [ o iii ] @xmath165007 shows often complex profile such as velocity shift of [ o iii ] core and asymmetry @xcite , we decomposed the [ o iii ] line into a narrow core and a broad base . if [ o iii ] has the broad base which tends to show blue - asymmetric ( blue wing ) , the [ o iii ] is fitted with double gaussian components . on the other hand , if the [ o iii ] profile is symmetric or of the s / n is low , the [ o iii ] is fitted with a single gaussian component . then , the best - fit model of the [ o iii ] @xmath165007 line was used to model [ o iii ] @xmath164959 and h@xmath4 narrow component by assuming that these narrow lines have the same widths . the flux ratio of the [ o iii ] @xmath164959 to the [ o iii ] @xmath165007 was assumed to be 1:3 , while the height of the h@xmath4 narrow component was set as free parameter . next , we fitted the h@xmath4 broad component with a single gaussian component since the s / n of h@xmath4 is typically lower than [ o iii ] . for the h@xmath12 region , we did not subtract fe ii emission because fe ii is relatively weak in this spectral range . first , we fit the host - galaxy continuum using two spectral regions 6400@xmath56460 and 6740@xmath56800 for determining the best model , where no other emissions are present . after subtracting the stellar features , we fitted [ s ii ] @xmath166716 and [ s ii ] @xmath166731 , respectively with a single gaussian component . we assumed that the widths of [ s ii ] , [ n ii ] , and the h@xmath12 narrow component are the same , and used the width of the [ s ii ] for fitting [ n ii ] and the narrow h@xmath12 , if the spectral quality is high ( s / n of [ s ii ] @xmath17 ) . for low s / n targets , the width of [ s ii ] is not reliable and we fitted the h@xmath12 narrow component and the [ n ii ] doublet with a single gaussian model , without using the best - fit of the [ s ii ] line . the flux ratio between [ n ii ] @xmath166548 and [ n ii ] @xmath166583 is assumed as 1/3 . for the h@xmath12 broad component , gauss - hermitian series were used to model the h@xmath12 profile as done by @xcite . figure [ fig : fitting ] presents an example of the multicomponent fitting . we estimated the uncertainty of the h@xmath12 luminosity based on the s / n of the line flux . in the case of the line widths , we performed monte carlo simulations by randomizing the flux per pixel using the flux noise . for a set of 100 simulated spectra , we repeated spectral decomposition , measured the line width , and adopted the 1-sigma dispersion of the distribution as the uncertainty of the line widths for each object . the estimated uncertainties are included in table 1 . black hole mass can be determined based on the virial theorem : @xmath18 where @xmath19 is the velocity of the broad - line region ( blr ) gas , @xmath20 is the blr size , and g is the gravitational constant @xcite . generally , either the second moment ( line dispersion ; @xmath21 ) or the fwhm of the h@xmath4 line ( @xmath22 ) is used for the velocity of the blr gas . along with each velocity measurements , a virial factor f is needed for mass determination . the determination of the average virial factor , respectively , for @xmath21 and @xmath22can be found in appendix , where we derived the virial factor by comparing the reverberation - mapped agns and quiescent galaxies in the @xmath0 plane . instead of directly measuring the size of blr by reverberation mapping , which requires a long - term spectroscopic monitoring , an empirical size - luminosity relation ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) has been used for @xmath1 estimates . we used the size@xmath5luminosity relation from @xcite , and derive the @xmath1 estimator as follows , @xmath23 for our nls1s , the width of the h@xmath12 line is better determined that that of the h@xmath4 lines since h@xmath4 often have much lower s / n . thus , we used the measurement of h@xmath12 line width and luminosity for @xmath1 estimation , using the following two relations @xcite : @xmath24 @xmath25 assuming the h@xmath4 and h@xmath12 have the same line profile ( i.e. , fwhm = 2 @xmath26 ) , we also converted @xmath27 to @xmath21 using eq . 3 . to test the validity of eq . 3 for our nls1 , we compared the line width of h@xmath4 and h@xmath12 using a subsample of 41 nls1s , for which the s / n ratio of h@xmath4is larger than 20 so that we could obtain reliable emission line fitting results . we find that the relation between h@xmath12 and h@xmath4 of nls1s is consistent with that of reported by @xcite , with a slight offset @xmath28 from the equation ( 3 ) . for comparing @xmath29 with h@xmath12 luminosity , we used all nls1s in our sample , for which @xmath29 was measured from a power - law component in the multi - component fitting process . as shown in figure 3 , the relation between @xmath29 and h@xmath12 luminosity is close to equation ( 4 ) , with a slight offset @xmath30 . this result suggests that using the conversion equation is acceptable for nls1s and that the multi - component fitting results are reasonable , although a proper comparison is difficult due to the limited dynamical range of the nls1 sample compared to that of @xcite . note that we used a gauss - hermite series for the broad h@xmath12 component , and a single gaussian model for the broad h@xmath4 component ( due to low s / n ratio ) , while @xcite used a multicomponent gaussian models for both h@xmath12 and h@xmath4 . the difference of the fitting model may be partly responsible for the slight systematic offset . we derived a black hole mass estimators by combining aforementioned scaling relations as : . @xmath31 @xmath32 we adopted log f = 0.05@xmath330.12 ( f = 1.12 ) for fwhm - based @xmath1 while we used log f = 0.65@xmath330.12 ( f = 4.47 ) for @xmath26-based @xmath1 ( see appendix for detailed discussion ) . directly measuring stellar velocity dispersions is a key to determine the location of nls1 on the @xmath0 plane . to investigate the systematic uncertainties of the stellar velocity dispersion measurements , we measured @xmath34 in three spectral regions : ( 1 ) mg @xmath35-fe region ( 5000 - 5430 ) , which includes strong absorption lines , i.e. , mg @xmath35 triplet ( 5069 , 5154 , 5160 ) and fe ( 5270 , 5335 ) lines ( hereafter @xmath36 ) ; ( 2 ) mg @xmath35-fe region ( 5000 - 5430 ) excluding the mg @xmath35 triplet ( hereafter @xmath37 ) ; and ( 3 ) ca ii region ( 8400 - 8800 ) , where the ca ii triplet ( 8498 , 8542 , 8662 ) is a strong feature ( hereafter @xmath38 ) . the line strength of the mg @xmath35 triplet is much higher in the composite spectra of massive elliptical galaxies than in the nearby stars , hence , the template mismatch due to the @xmath39element enhancement can potentially cause a systematic bias in measuring @xmath34 , although this effect is not significant for late - type host galaxies @xcite . in the case of the ca ii triplet region , agn contamination ( e.g. , fe ii emission ) is relatively weaker than the mg @xmath35 region , while the residual of sky emission lines is often present and the quality of spectra is generally lower than that of the mg @xmath35 region . thus , as a consistency check , we measured stellar velocity dispersion using three different spectral regions ( see similar investigation by greene et al . we find that three measurements are consistent , showing that the effect of the mg abundance is negligible ( see below ) . we corrected for the sdss spectral resolution by subtracting the instrumental resolution from the measured stellar velocity dispersion in quadrature . instead of using a mean constant resolution @xmath13 70 km s@xmath6 , which is often adopted in the literature , we calculated the mean instrumental resolution in the corresponding fitting ranges for each object , using the spectral resolution fits file provided by sdss dr7 . for example , we used the spectral range 5000 - 5430 to calculate the mean instrumental resolution for the mg @xmath35-fe region , which is @xmath1355 - 56 km s@xmath6 . compared to the instrumental resolution , the stellar lines of the objects that we mesured stellar velocity dispersions are well resolved . after masking out agn narrow emission lines ( e.g. , [ fe vii ] @xmath165160 , [ n i ] @xmath165201 , [ ca v ] @xmath165310 ; * ? ? ? * ) , we measured @xmath34 by using both the penalized pixel - fitting ( ppxf ) method @xcite and a python - based code based on the algorithm by @xcite . we used stellar velocity templates from indo@xmath5us stellar library , which includes various spectral type giant stars with a range of metallicity ( [ fe / h ] = -0.49 @xmath5 0.18 ) @xcite . low order polynomials were used to fit the broad curvature in the spectra after masking out the narrow emission lines and bad spectral regions . after intense tests with various polynomial orders and templates for each target , we adopted the mean of the measurements based on each polynomial order and each spectral range with a different mask - out region , as a final measurement of @xmath34 . in this process , we measured @xmath36 for 62 nls1s , that show strong enough stellar lines . among them , we were able to measure @xmath38 for 34 nls1s , while we could not measure the @xmath34 from the ca ii triplet for the other objects , since the sdss spectral range does not cover the rest - frame ca ii triplet region for targets at z @xmath40 , or the strength of the ca ii triplet is too weak to measure @xmath34 ( see figure 4 ) . as a consistency check , we compared the @xmath36 with @xmath37 and @xmath38 in figure [ fig : svd ] . the @xmath36 is slightly higher by a few percent ( 0.015 dex ) than the @xmath37 , and the rms scatter is 0.06 dex . this result confirms the @xmath36 is consistent with the @xmath37 and indicates the influence of the mg @xmath35 triplet is marginal in measuring the @xmath34 of the host galaxies of nls1s . the comparison between @xmath38 and @xmath36 shows slightly larger scatter ( 0.10 dex ) , but the average offset is still close to zero ( i.e. , 0.014 dex ) , suggesting that the @xmath36 is consistent with the @xmath38 . based on these results without strong bias among the measurements from various spectral regions , we adopted @xmath36 as the final measurements . as a consistency check , we compared our measurements with sdss dr7 values . we found stellar velocity dispersion measurements for 5 objects from sdss dr7 , which are consistent with our measurements within the measurement uncertainties . among 93 nls1s , 30 nls1s were studied previously by @xcite , who measured @xmath34 based on high quality spectra with higher spectral resolution obtained with the keck echellette spectrograph and imager ( esi ) and the magellan echellette ( mage ) . thus , including the @xmath34 measurements of 30 nls1s from @xcite , we have a total of 93 measurements of the @xmath34 . we note that the 3 sdss fiber size is larger than the slit size adopted by @xcite . thus , the sdss spectra represent a larger physical scale of the host galaxies than the keck spectra of @xcite , and may show larger influence of rotational broadening . however , it is difficult to perform a direct comparison between sdss - based and keck - based measurements due to the fact that most of 30 nls1s studied by @xcite have smaller velocity dispersion than the sdss instrumental resolution . we found only one object among 30 nls1s , for which both sdss - based and keck - based stellar velocity dispersion measurements are available and show consistency ( @xmath41 vs. @xmath42 km s@xmath6 ) . for galaxies with a rotating stellar disk , the line - of - sight stellar velocity dispersion can be overestimated due to rotational broadening @xcite , therefore it is important to correct for the rotation effect in measuring @xmath34 . since the rotating disk is common among late - type galaxies and the ratio between rotation velocity and velocity dispersion is typically higher in late - type galaxies than in early - type galaxies , the effect of the rotational broadening is expected to be stronger for late type galaxies , particularly for more inclined galaxies toward the line - of - sight . to investigate the rotation effect on the @xmath0 relation , we classified our nls1s into early and late type galaxies , using the sdss colors and the presence of a disk . for late - type galaxies , we further divided them into two groups : more face - on and more edge - on galaxies based on the inclination of the disk . the inclination angle is determined from the minor - to - major axial ratio of the disk as @xmath43 , where @xmath44 is the inclination angle of the galactic disk to the line of sight ( i.e. , @xmath45 for an edge - on disk ) and @xmath46 is the ratio of the minor to major axes of the disk . we classified our sample with @xmath47 ( i.e. , @xmath48 ) as face - on galaxies , and the others with @xmath49 ( i.e. , @xmath50 ) were classified as edge - on galaxies . as a result , 93 nls1 galaxies were divided into 35 early type galaxies and 58 late type galaxies which were further divided into 48 face - on and 10 edge - on late types . lcccc cccc j010409.16 + 000843.6 & 0.071 & 41.29 @xmath33 0.01 & 701 @xmath33 11 & 1375 @xmath33 28 & 6.57 @xmath33 0.02 & 6.55 @xmath33 0.02 & 66 @xmath33 16 & 15 + j030417.78 + 002827.2 & 0.045 & 41.40 @xmath33 0.01 & 728 @xmath33 11 & 1248 @xmath33 19 & 6.65 @xmath33 0.01 & 6.51 @xmath33 0.01 & 88 @xmath33 8 & 30 + j073106.86 + 392644.5 & 0.048 & 41.06 @xmath33 0.01 & 662 @xmath33 7 & 1185 @xmath33 19 & 6.41 @xmath33 0.01 & 6.31 @xmath33 0.01 & 72 @xmath33 14 & 19 + j073714.28 + 292634.1 & 0.080 & 41.46 @xmath33 0.01 & 966 @xmath33 26 & 1553 @xmath33 36 & 6.93 @xmath33 0.03 & 6.74 @xmath33 0.03 & 102 @xmath33 12 & 19 + j080253.18 + 130559.6 & 0.095 & 42.05 @xmath33 0.01 & 1072 @xmath33 8 & 1903 @xmath33 21 & 7.30 @xmath33 0.01 & 7.19 @xmath33 0.01 & 97 @xmath33 17 & 24 + j080538.22 + 244214.8 & 0.099 & 41.61 @xmath33 0.02 & 784 @xmath33 14 & 1242 @xmath33 94 & 6.81 @xmath33 0.02 & 6.61 @xmath33 0.02 & 102 @xmath33 24 & 11 + j080801.75 + 381935.3 & 0.041 & 40.86 @xmath33 0.01 & 896 @xmath33 21 & 1683 @xmath33 41 & 6.59 @xmath33 0.02 & 6.53 @xmath33 0.02 & 100 @xmath33 12 & 20 + j081718.55 + 520147.7 & 0.039 & 41.06 @xmath33 0.01 & 842 @xmath33 12 & 1486 @xmath33 32 & 6.62 @xmath33 0.01 & 6.51 @xmath33 0.01 & 68 @xmath33 14 & 18 + j082007.81 + 372839.6 & 0.082 & 41.42 @xmath33 0.01 & 1085 @xmath33 61 & 1661 @xmath33 92 & 7.02 @xmath33 0.05 & 6.78 @xmath33 0.05 & 141 @xmath33 17 & 22 + j083202.15 + 461425.7 & 0.046 & 41.42 @xmath33 0.01 & 1026 @xmath33 20 & 1646 @xmath33 42 & 6.97 @xmath33 0.02 & 6.77 @xmath33 0.02 & 128 @xmath33 5 & 38 + j083741.94 + 263344.1 & 0.076 & 41.34 @xmath33 0.01 & 1025 @xmath33 44 & 1767 @xmath33 131 & 6.93 @xmath33 0.04 & 6.80 @xmath33 0.04 & 105 @xmath33 16 & 17 + j083949.65 + 484701.4 & 0.039 & 41.56 @xmath33 0.01 & 904 @xmath33 9 & 1495 @xmath33 14 & 6.92 @xmath33 0.01 & 6.75 @xmath33 0.01 & 112 @xmath33 6 & 42 + j084927.36 + 324852.8 & 0.064 & 41.64 @xmath33 0.01 & 1235 @xmath33 23 & 2045 @xmath33 33 & 7.23 @xmath33 0.02 & 7.07 @xmath33 0.02 & 137 @xmath33 11 & 25 + j085504.16 + 525248.3 & 0.089 & 41.86 @xmath33 0.01 & 889 @xmath33 20 & 1540 @xmath33 32 & 7.04 @xmath33 0.02 & 6.92 @xmath33 0.02 & 103 @xmath33 10 & 20 + j092438.88 + 560746.8 & 0.026 & 41.00 @xmath33 0.01 & 899 @xmath33 18 & 1723 @xmath33 38 & 6.66 @xmath33 0.02 & 6.62 @xmath33 0.02 & 146 @xmath33 5 & 39 + j093638.69 + 132529.6 & 0.090 & 41.41 @xmath33 0.01 & 1025 @xmath33 44 & 1916 @xmath33 209 & 6.96 @xmath33 0.04 & 6.91 @xmath33 0.04 & 102 @xmath33 11 & 17 + j094153.41 + 163621.0 & 0.052 & 41.05 @xmath33 0.01 & 1005 @xmath33 17 & 2078 @xmath33 39 & 6.78 @xmath33 0.02 & 6.81 @xmath33 0.02 & 101 @xmath33 11 & 15 + j095848.67 + 025243.2 & 0.079 & 41.07 @xmath33 0.01 & 1004 @xmath33 30 & 1710 @xmath33 51 & 6.79 @xmath33 0.03 & 6.65 @xmath33 0.03 & 117 @xmath33 11 & 18 + j100854.93 + 373929.9 & 0.054 & 41.97 @xmath33 0.01 & 1010 @xmath33 20 & 1750 @xmath33 52 & 7.21 @xmath33 0.02 & 7.08 @xmath33 0.02 & 105 @xmath33 9 & 38 + j102532.09 + 102503.9 & 0.046 & 41.29 @xmath33 0.01 & 930 @xmath33 9 & 1615 @xmath33 20 & 6.82 @xmath33 0.01 & 6.70 @xmath33 0.01 & 111 @xmath33 8 & 26 + j103103.52 + 462616.8 & 0.093 & 41.86 @xmath33 0.01 & 1029 @xmath33 15 & 1806 @xmath33 30 & 7.17 @xmath33 0.01 & 7.06 @xmath33 0.01 & 169 @xmath33 16 & 21 + j103751.81 + 334850.1 & 0.051 & 40.82 @xmath33 0.01 & 1053 @xmath33 49 & 1832 @xmath33 72 & 6.71 @xmath33 0.04 & 6.59 @xmath33 0.04 & 94 @xmath33 12 & 18 + j104153.59 + 031500.6 & 0.093 & 41.69 @xmath33 0.01 & 1155 @xmath33 20 & 1940 @xmath33 35 & 7.20 @xmath33 0.02 & 7.04 @xmath33 0.02 & 126 @xmath33 21 & 18 + j105600.39 + 165626.2 & 0.085 & 41.48 @xmath33 0.01 & 996 @xmath33 24 & 1821 @xmath33 42 & 6.97 @xmath33 0.02 & 6.89 @xmath33 0.02 & 126 @xmath33 15 & 20 + j110016.03 + 461615.2 & 0.032 & 40.91 @xmath33 0.01 & 835 @xmath33 11 & 1646 @xmath33 20 & 6.55 @xmath33 0.01 & 6.54 @xmath33 0.01 & 68 @xmath33 6 & 24 + j111253.12 + 314807.3 & 0.076 & 41.87 @xmath33 0.01 & 1291 @xmath33 20 & 2049 @xmath33 35 & 7.38 @xmath33 0.02 & 7.18 @xmath33 0.02 & 72 @xmath33 18 & 16 + j111407.35 - 000031.1 & 0.073 & 41.41 @xmath33 0.01 & 954 @xmath33 23 & 1519 @xmath33 33 & 6.90 @xmath33 0.02 & 6.70 @xmath33 0.02 & 125 @xmath33 10 & 25 + j112229.65 + 214815.5 & 0.061 & 41.44 @xmath33 0.01 & 994 @xmath33 16 & 1683 @xmath33 36 & 6.95 @xmath33 0.02 & 6.80 @xmath33 0.02 & 125 @xmath33 7 & 29 + j112229.65 + 214815.5 & 0.100 & 41.71 @xmath33 0.01 & 1318 @xmath33 75 & 2027 @xmath33 70 & 7.32 @xmath33 0.05 & 7.09 @xmath33 0.05 & 176 @xmath33 20 & 21 + j112545.34 + 240823.9 & 0.024 & 40.20 @xmath33 0.01 & 688 @xmath33 16 & 1211 @xmath33 33 & 6.05 @xmath33 0.02 & 5.94 @xmath33 0.02 & 82 @xmath33 8 & 25 + j113101.10 + 134539.6 & 0.092 & 41.83 @xmath33 0.01 & 1087 @xmath33 26 & 1826 @xmath33 37 & 7.21 @xmath33 0.02 & 7.06 @xmath33 0.02 & 171 @xmath33 14 & 26 + j113111.93 + 100231.3 & 0.074 & 41.25 @xmath33 0.02 & 930 @xmath33 33 & 1785 @xmath33 112 & 6.80 @xmath33 0.03 & 6.77 @xmath33 0.03 & 130 @xmath33 18 & 14 + j113913.91 + 335551.1 & 0.033 & 41.53 @xmath33 0.01 & 834 @xmath33 19 & 1394 @xmath33 48 & 6.84 @xmath33 0.02 & 6.68 @xmath33 0.02 & 112 @xmath33 15 & 32 + j115333.22 + 095408.4 & 0.069 & 41.62 @xmath33 0.01 & 983 @xmath33 16 & 1844 @xmath33 35 & 7.02 @xmath33 0.02 & 6.97 @xmath33 0.02 & 130 @xmath33 11 & 25 + j120012.47 + 183542.9 & 0.066 & 40.92 @xmath33 0.01 & 862 @xmath33 47 & 1571 @xmath33 61 & 6.58 @xmath33 0.05 & 6.50 @xmath33 0.05 & 136 @xmath33 13 & 19 + j121157.48 + 055801.1 & 0.068 & 41.74 @xmath33 0.01 & 1012 @xmath33 13 & 1984 @xmath33 36 & 7.10 @xmath33 0.01 & 7.09 @xmath33 0.01 & 119 @xmath33 12 & 22 + j122307.79 + 192337.0 & 0.076 & 41.33 @xmath33 0.01 & 1079 @xmath33 35 & 1832 @xmath33 101 & 6.97 @xmath33 0.03 & 6.83 @xmath33 0.03 & 122 @xmath33 12 & 21 + j123651.17 + 453904.1 & 0.030 & 41.24 @xmath33 0.01 & 863 @xmath33 16 & 1601 @xmath33 47 & 6.73 @xmath33 0.02 & 6.67 @xmath33 0.02 & 97 @xmath33 7 & 29 + j123932.59 + 342221.3 & 0.084 & 41.53 @xmath33 0.01 & 898 @xmath33 56 & 1540 @xmath33 109 & 6.90 @xmath33 0.06 & 6.77 @xmath33 0.06 & 84 @xmath33 7 & 32 + j124319.97 + 025256.1 & 0.087 & 41.69 @xmath33 0.01 & 752 @xmath33 16 & 1276 @xmath33 29 & 6.81 @xmath33 0.02 & 6.67 @xmath33 0.02 & 112 @xmath33 12 & 26 + j130456.96 + 395529.7 & 0.028 & 40.42 @xmath33 0.01 & 915 @xmath33 34 & 1431 @xmath33 124 & 6.40 @xmath33 0.03 & 6.19 @xmath33 0.03 & 92 @xmath33 6 & 23 + j131142.56 + 331612.7 & 0.078 & 41.29 @xmath33 0.01 & 1145 @xmath33 33 & 2086 @xmath33 41 & 7.01 @xmath33 0.03 & 6.93 @xmath33 0.03 & 106 @xmath33 14 & 16 + j131305.81 + 012755.9 & 0.029 & 40.85 @xmath33 0.01 & 868 @xmath33 12 & 1599 @xmath33 28 & 6.56 @xmath33 0.01 & 6.49 @xmath33 0.01 & 108 @xmath33 5 & 36 + j131905.95 + 310852.7 & 0.032 & 40.97 @xmath33 0.01 & 1391 @xmath33 38 & 2063 @xmath33 61 & 7.03 @xmath33 0.03 & 6.77 @xmath33 0.03 & 137 @xmath33 6 & 38 + j134240.09 + 022524.4 & 0.075 & 41.03 @xmath33 0.01 & 956 @xmath33 57 & 1842 @xmath33 60 & 6.73 @xmath33 0.05 & 6.70 @xmath33 0.05 & 105 @xmath33 14 & 16 + j134401.90 + 255628.3 & 0.062 & 41.33 @xmath33 0.01 & 1068 @xmath33 58 & 1651 @xmath33 43 & 6.96 @xmath33 0.05 & 6.74 @xmath33 0.05 & 140 @xmath33 9 & 25 + j140659.58 + 231738.6 & 0.061 & 40.73 @xmath33 0.01 & 965 @xmath33 48 & 1400 @xmath33 87 & 6.59 @xmath33 0.05 & 6.31 @xmath33 0.05 & 97 @xmath33 8 & 26 + j141434.52 + 293428.2 & 0.076 & 41.29 @xmath33 0.01 & 844 @xmath33 29 & 1376 @xmath33 39 & 6.73 @xmath33 0.03 & 6.55 @xmath33 0.03 & 75 @xmath33 15 & 20 + j143658.68 + 164513.6 & 0.072 & 40.93 @xmath33 0.01 & 770 @xmath33 24 & 1418 @xmath33 55 & 6.49 @xmath33 0.03 & 6.42 @xmath33 0.03 & 73 @xmath33 10 & 17 + j143708.46 + 074013.6 & 0.087 & 41.24 @xmath33 0.01 & 1089 @xmath33 45 & 1956 @xmath33 66 & 6.94 @xmath33 0.04 & 6.84 @xmath33 0.04 & 98 @xmath33 13 & 16 + j151356.88 + 481012.1 & 0.079 & 41.63 @xmath33 0.01 & 737 @xmath33 30 & 1270 @xmath33 55 & 6.77 @xmath33 0.04 & 6.64 @xmath33 0.04 & 124 @xmath33 16 & 21 + j152209.56 + 451124.0 & 0.066 & 41.32 @xmath33 0.01 & 900 @xmath33 28 & 1886 @xmath33 251 & 6.80 @xmath33 0.03 & 6.85 @xmath33 0.03 & 128 @xmath33 12 & 18 + j152324.42 + 551855.3 & 0.039 & 41.12 @xmath33 0.01 & 1086 @xmath33 34 & 1717 @xmath33 91 & 6.88 @xmath33 0.03 & 6.67 @xmath33 0.03 & 128 @xmath33 7 & 33 + j152940.58 + 302909.3 & 0.036 & 41.69 @xmath33 0.01 & 1073 @xmath33 22 & 1823 @xmath33 54 & 7.13 @xmath33 0.02 & 6.99 @xmath33 0.02 & 107 @xmath33 5 & 44 + j155640.90 + 121717.9 & 0.036 & 41.05 @xmath33 0.01 & 1131 @xmath33 26 & 2002 @xmath33 35 & 6.88 @xmath33 0.02 & 6.78 @xmath33 0.02 & 149 @xmath33 9 & 30 + j160746.00 + 345048.9 & 0.054 & 41.53 @xmath33 0.01 & 749 @xmath33 7 & 1422 @xmath33 14 & 6.74 @xmath33 0.01 & 6.69 @xmath33 0.01 & 80 @xmath33 10 & 28 + j161527.67 + 403153.6 & 0.084 & 41.35 @xmath33 0.01 & 868 @xmath33 39 & 1608 @xmath33 54 & 6.79 @xmath33 0.04 & 6.72 @xmath33 0.04 & 137 @xmath33 18 & 17 + j161809.36 + 361957.8 & 0.034 & 41.16 @xmath33 0.01 & 578 @xmath33 11 & 896 @xmath33 27 & 6.34 @xmath33 0.02 & 6.11 @xmath33 0.02 & 87 @xmath33 8 & 30 + j161951.31 + 405847.3 & 0.038 & 41.27 @xmath33 0.01 & 1020 @xmath33 15 & 1746 @xmath33 26 & 6.89 @xmath33 0.01 & 6.76 @xmath33 0.01 & 114 @xmath33 10 & 26 + j162930.01 + 420703.2 & 0.072 & 41.37 @xmath33 0.01 & 816 @xmath33 14 & 1440 @xmath33 36 & 6.74 @xmath33 0.02 & 6.63 @xmath33 0.02 & 101 @xmath33 11 & 22 + j163501.46 + 305412.1 & 0.054 & 41.63 @xmath33 0.01 & 854 @xmath33 40 & 1261 @xmath33 145 & 6.90 @xmath33 0.04 & 6.63 @xmath33 0.04 & 130 @xmath33 14 & 23 + j210226.54 + 000702.3 & 0.052 & 40.76 @xmath33 0.01 & 806 @xmath33 45 & 1466 @xmath33 46 & 6.45 @xmath33 0.05 & 6.37 @xmath33 0.05 & 96 @xmath33 14 & 15 + j210533.44 + 002829.3 & 0.054 & 41.21 @xmath33 0.01 & 853 @xmath33 17 & 1429 @xmath33 27 & 6.71 @xmath33 0.02 & 6.55 @xmath33 0.02 & 81 @xmath33 9 & 23 + j010712.03 + 140844.9 & 0.077 & 41.42 @xmath33 0.01 & 597 @xmath33 184 & 998 @xmath33 170 & 6.48 @xmath33 0.28 & 6.32 @xmath33 0.28 & 38 @xmath33 4@xmath51 & 15 + j024912.86 - 081525.7 & 0.030 & 40.21 @xmath33 0.01 & 542 @xmath33 19 & 915 @xmath33 46 & 5.84 @xmath33 0.03 & 5.69 @xmath33 0.03 & 53 @xmath33 3@xmath51 & 18 + j080629.80 + 241955.6 & 0.041 & 40.71 @xmath33 0.01 & 629 @xmath33 19 & 1067 @xmath33 39 & 6.20 @xmath33 0.03 & 6.06 @xmath33 0.03 & 71 @xmath33 5@xmath51 & 20 + j080907.57 + 441641.4 & 0.054 & 40.90 @xmath33 0.01 & 692 @xmath33 27 & 1150 @xmath33 42 & 6.38 @xmath33 0.04 & 6.22 @xmath33 0.04 & 65 @xmath33 3@xmath51 & 21 + j081550.23 + 250640.9 & 0.073 & 40.93 @xmath33 0.02 & 568 @xmath33 61 & 895 @xmath33 90 & 6.21 @xmath33 0.10 & 6.00 @xmath33 0.10 & 65 @xmath33 2@xmath51 & 12 + j082912.68 + 500652.3 & 0.044 & 41.28 @xmath33 0.01 & 597 @xmath33 7 & 1002 @xmath33 16 & 6.42 @xmath33 0.01 & 6.26 @xmath33 0.01 & 60 @xmath33 2@xmath51 & 29 + j094057.19 + 032401.2 & 0.061 & 41.46 @xmath33 0.01 & 738 @xmath33 21 & 1206 @xmath33 45 & 6.69 @xmath33 0.03 & 6.51 @xmath33 0.03 & 82 @xmath33 3@xmath51 & 20 + j094529.36 + 093610.4 & 0.013 & 40.52 @xmath33 0.01 & 907 @xmath33 11 & 1767 @xmath33 27 & 6.44 @xmath33 0.01 & 6.42 @xmath33 0.01 & 76 @xmath33 2@xmath51 & 34 + j095151.82 + 060143.6 & 0.093 & 41.00 @xmath33 0.02 & 742 @xmath33 100 & 1192 @xmath33 139 & 6.48 @xmath33 0.12 & 6.29 @xmath33 0.12 & 76 @xmath33 6@xmath51 & 11 + j101627.33 - 000714.5 & 0.094 & 41.17 @xmath33 0.03 & 648 @xmath33 34 & 1109 @xmath33 90 & 6.44 @xmath33 0.05 & 6.31 @xmath33 0.05 & 55 @xmath33 7@xmath51 & 8 + j102348.44 + 040553.7 & 0.099 & 40.96 @xmath33 0.02 & 812 @xmath33 181 & 869 @xmath33 108 & 6.55 @xmath33 0.20 & 5.99 @xmath33 0.20 & 91 @xmath33 13@xmath51 & 9 + j111031.61 + 022043.2 & 0.079 & 41.37 @xmath33 0.01 & 671 @xmath33 15 & 1100 @xmath33 30 & 6.56 @xmath33 0.02 & 6.39 @xmath33 0.02 & 77 @xmath33 3@xmath51 & 16 + j112526.51 + 022039.0 & 0.049 & 41.00 @xmath33 0.01 & 843 @xmath33 30 & 1305 @xmath33 48 & 6.60 @xmath33 0.03 & 6.37 @xmath33 0.03 & 87 @xmath33 5@xmath51 & 20 + j114339.49 - 024316.3 & 0.094 & 41.32 @xmath33 0.01 & 746 @xmath33 40 & 1192 @xmath33 72 & 6.64 @xmath33 0.05 & 6.44 @xmath33 0.05 & 97 @xmath33 5@xmath51 & 22 + j121518.23 + 014751.1 & 0.071 & 41.28 @xmath33 0.01 & 636 @xmath33 22 & 1036 @xmath33 38 & 6.47 @xmath33 0.03 & 6.29 @xmath33 0.03 & 81 @xmath33 3@xmath51 & 18 + j122342.82 + 581446.2 & 0.015 & 40.42 @xmath33 0.01 & 706 @xmath33 13 & 1049 @xmath33 32 & 6.17 @xmath33 0.02 & 5.91 @xmath33 0.02 & 45 @xmath33 2@xmath51 & 26 + j124035.82 - 002919.4 & 0.081 & 41.76 @xmath33 0.01 & 728 @xmath33 11 & 1133 @xmath33 31 & 6.82 @xmath33 0.02 & 6.60 @xmath33 0.02 & 56 @xmath33 3@xmath51 & 19 + j125055.28 - 015556.7 & 0.081 & 41.51 @xmath33 0.02 & 849 @xmath33 21 & 1428 @xmath33 73 & 6.84 @xmath33 0.02 & 6.69 @xmath33 0.02 & 66 @xmath33 4@xmath51 & 15 + j131926.52 + 105610.9 & 0.064 & 41.55 @xmath33 0.01 & 671 @xmath33 13 & 1040 @xmath33 31 & 6.65 @xmath33 0.02 & 6.42 @xmath33 0.02 & 47 @xmath33 3@xmath51 & 23 + j143450.62 + 033842.5 & 0.028 & 40.27 @xmath33 0.01 & 708 @xmath33 34 & 1289 @xmath33 54 & 6.11 @xmath33 0.04 & 6.03 @xmath33 0.04 & 57 @xmath33 3@xmath51 & 22 + j144052.60 - 023506.2 & 0.045 & 41.18 @xmath33 0.01 & 674 @xmath33 18 & 1087 @xmath33 43 & 6.48 @xmath33 0.03 & 6.29 @xmath33 0.03 & 73 @xmath33 8@xmath51 & 28 + j144705.46 + 003653.2 & 0.096 & 41.14 @xmath33 0.02 & 924 @xmath33 44 & 1495 @xmath33 56 & 6.75 @xmath33 0.04 & 6.56 @xmath33 0.04 & 64 @xmath33 4@xmath51 & 9 + j145045.54 - 014752.9 & 0.099 & 41.62 @xmath33 0.01 & 1086 @xmath33 96 & 1690 @xmath33 250 & 7.11 @xmath33 0.08 & 6.89 @xmath33 0.08 & 138 @xmath33 6@xmath51 & 17 + j155005.95 + 091035.7 & 0.092 & 41.73 @xmath33 0.01 & 572 @xmath33 37 & 988 @xmath33 121 & 6.59 @xmath33 0.06 & 6.46 @xmath33 0.06 & 78 @xmath33 6@xmath51 & 18 + j162636.40 + 350242.1 & 0.034 & 40.63 @xmath33 0.01 & 578 @xmath33 21 & 828 @xmath33 35 & 6.09 @xmath33 0.03 & 5.80 @xmath33 0.03 & 52 @xmath33 1@xmath51 & 24 + j163159.59 + 243740.2 & 0.044 & 41.08 @xmath33 0.01 & 649 @xmath33 10 & 958 @xmath33 20 & 6.40 @xmath33 0.02 & 6.13 @xmath33 0.02 & 66 @xmath33 2@xmath51 & 24 + j172759.14 + 542147.0 & 0.100 & 41.28 @xmath33 0.02 & 668 @xmath33 39 & 1055 @xmath33 80 & 6.52 @xmath33 0.05 & 6.31 @xmath33 0.05 & 67 @xmath33 8@xmath51 & 8 + j205822.14 - 065004.3 & 0.074 & 41.61 @xmath33 0.01 & 655 @xmath33 8 & 1101 @xmath33 19 & 6.65 @xmath33 0.01 & 6.50 @xmath33 0.01 & 58 @xmath33 3@xmath51 & 17 + j221139.16 - 010534.9 & 0.092 & 40.95 @xmath33 0.02 & 604 @xmath33 50 & 1104 @xmath33 64 & 6.28 @xmath33 0.07 & 6.20 @xmath33 0.07 & 68 @xmath33 7@xmath51 & 10 + j230649.77 + 005023.3 & 0.061 & 40.93 @xmath33 0.01 & 851 @xmath33 46 & 1508 @xmath33 48 & 6.58 @xmath33 0.05 & 6.47 @xmath33 0.05 & 65 @xmath33 3@xmath51 & 16 [ tab : nls1_table ] we compare the 93 nls1 with the rm agns and quiescent galaxies in the @xmath0 plane in figure 4 . in the left panel , @xmath1 is determined using the line dispersion of the balmer lines and the virial factor log f = 0.65 , while @xmath1 in the right panel is estimated using the fwhm of the balmer lines and the virial factor log f = 0.05 ( see appendix for the determination of the virial factors ) . in general , nls1s seem to show a consistent @xmath0 relation compared to the rm agns . with respect to the best - fit @xmath0 relation obtained for the joint sample of the rm agns and quiescent galaxies ( solid line ) , the average offset of the nls1s is @xmath52 @xmath1@xmath53 in the left panel , and @xmath52 @xmath1@xmath54 in the right panel , suggesting that nls1s follow the same @xmath0 relation as other local galaxies . when we compare nls1s with the best - fit @xmath0 relation of quiescent galaxies , we obtained almost the same result since the best - fit @xmath0 relation is almost identical between quiescent sample and the joint sample of quiescent and rm agns since the quiescent galaxies are dominant in terms of number and dynamical range ( for details , see * ? ? ? * ) . similarly , when we compare nls1s with the best - fit @xmath0 relation of the rm agns only ( dashed line ) , we obtain a slightly increased offset @xmath52 @xmath1@xmath55 and @xmath56@xmath1@xmath57 , respectively for @xmath26-based @xmath1 and @xmath58-based @xmath1 . the best - fit @xmath0 relation of the rm agns suffers from the effect of the limited mass distribution compared to the quiescent galaxy sample . the truncation of the mass distribution of the rm agns caused a shallower slope of the @xmath0 relation as discussed in detail by woo et al . 2013 . in turn , the offset of the nls1s with respect to this shallow @xmath0 slope becomes slightly negative since the nls1s are mainly located at the low @xmath1 and low stellar velocity dispersion region . considering the small offset and the limited mass distribution , nls1s seem to show a consistent @xmath0 relation compared to the rm agns . among nls1s , there is a large scatter with a clear trend with the host galaxy morphology . compared to the best - fit @xmath0 relation of the joint sample of quiescent galaxies and rm agns , early - type nls1s show a positive offset ( @xmath52 @xmath1@xmath59 and @xmath52 @xmath1@xmath60 , respectively in the left and right panels ) while late - type galaxies present a negative offset ( @xmath52 @xmath1@xmath61 and @xmath52 @xmath1@xmath62 , respectively in the left and right panels in figure 4 ) . the large difference of the offset between early - type and late - type nls1 galaxies may stem from the effect of the rotational broadening in the stellar absorption lines since single aperture spectra have been used for measuring the stellar velocity dispersion . to test this scenario , we further divide the late - type nls1 galaxies into two groups , i.e , edge - on and face - on galaxies ( see section 3.4 for morphology classification ) , and calculated the mean offset . clearly , the edge - on late type galaxies , which are expected to have larger rotational broadening in the line - of - sight stellar velocity dispersion measurements , show the largest negative offset ( @xmath52 @xmath1@xmath63 and @xmath52 @xmath1@xmath64 , respectively in the left and right panels in figure 4 ) , while the face - on galaxies do not show a clear offset ( @xmath52 @xmath1@xmath65 and @xmath52 @xmath1@xmath66 , respectively in the left and right panels in figure 4 ) . thus , we suspect that the large scatter of the nls1s in the @xmath0 plane and the systematic trend of the offset with galaxy morphology and inclination are due to the rotational broadening @xcite . in this section , we investigate whether the offset from the @xmath0 relation is correlated with other agn parameters , i.e. , eddington ratio , r5007 and r4570 . note that the offset is calculated with respect to the best - fit @xmath0 relation of the joint sample of quiescent galaxies and rm agns . first , we compare the offset with eddington ratio in figure [ fig : offset ] ( left ) , finding no significant correlation between the offset and @xmath67 ( see consistent results by * ? ? ? * ) . for this comparison , eddington ratio was determined by dividing the bolometric luminosity by the eddington luminosity , l@xmath68 , using the continuum luminosity at 5100 as a proxy ( @xmath69 = 9@xmath16@xmath29 ) @xcite . we also used the h@xmath12 line luminosity instead of the continuum luminosity at 5100based on equation 4 , and obtained the same results . second , we present the comparison between the offset and r5007 in figure [ fig : offset ] . r5007 does not significantly correlate with the offset of nls1s in general and in all three different morphology groups . in figure [ fig : offset_r4570 ] , we compare the fe ii strength ( r4570 ) with the offset from the @xmath0 relation . there is a weak correlation between them : while the weak fe ii emitters show both positive and negative offsets , the strong fe ii emitters mainly show negative offset . this result implies that nls1s with strong fe ii more significantly deviate from the @xmath0 relation . the correlation is slightly different for different morphology groups . however , the sample size in each morphology group is too small to definitely claim any difference . in addition , we tested whether the offset from the @xmath0 relation is related with the agn fraction ( see figure [ fig : offset_r4570 ] right panels ) , using the agn fraction determined from the monochromatic flux ratio of agn to host galaxy at 5100 . we find a good correlation of the offset with the agn fraction : the offset positively increases with increasing agn fraction . this correlation is also present in each morphology group while early - type galaxies have on average higher agn fraction than late type galaxies . the interpretation of this correlation is not straightforward since a strong selection effect is involved in measuring stellar velocity dispersion . for example , if the agn fraction is higher , then it is more difficult to measure stellar velocity dispersion . hence , only massive early - type galaxies are available at high agn fraction regime , while most late - type galaxies hosting high luminosity agns are missing from the distribution . the contribution of the rotation of stellar disks can bias stellar velocity dispersion measurements due to the rotational broadening of the stellar absorption lines . for example , if a single - aperture spectrum , which is integrated over a large portion of a stellar disk , is used for measuring the second moment of the absorption lines , the rotational effect can significantly increase the velocity dispersion measurements . for early - type galaxies the rotation effect is relatively small since the velocity dispersion is typically higher than rotation velocity . for example , @xcite reported that the stellar velocity dispersion changes by @xmath1310% as a function of the aperture size , based on the spatially resolved measurements of 31 early - type galaxies in the @xmath0 sample . in contrast , we expect the rotation effect can be substantially larger in late - type galaxies than in early - type galaxies due to much higher velocity - to - dispersion ( v/@xmath26 ) ratios . for disk - dominant late - type galaxies , the inclination to the line - of - sight can also play a significant role due to the project effect . based on the n - body smoothed particle hydrodynamic simulations , @xcite reported that bulge stellar velocity dispersion measurements can change by 30% depending on the galaxy inclination . since most of the @xmath34 measurements for agn host galaxies are based single - aperture spectra , the effect of rotation and inclination can play a role in comparing bl agns with nls1s . using a sample of low @xmath1 agn sample , @xcite showed a clear dependency of galaxy inclination on the offset from the @xmath0 relation , i.e. , more inclined galaxies tend to have higher @xmath34 and negatively offset , while more face - on galaxies tend to have lower @xmath34 and positively offset . the observed trend of the offset from the @xmath0 relation with galaxy inclination in our study is similar to @xcite , reflecting the same effect of the rotation and inclination of late - type galaxies . thus , the conclusion that nls1s follow the same @xmath0 relation as bl agns is still limited by the lack of spatially resolved measurements . to better understand the effect of rotation and inclination , spatially resolved measurements are required for nls1s , which is beyond the scope of the current study . two different scenarios have been suggested for the evolution of nls1s . on the one hand , nls1s are considered as the precursors of bl agns , evolving into bl agns . the low @xmath1 and the high eddington ratio of nls1s may imply that nls1s are young phase of agns @xcite . on the other hand , nls1s are viewed as an extension of bl agns at the low - mass scale @xcite . if the high eddington ratio of nls1s represents a relatively short - lived accretion phase , and the eddington ratio before and after the strong accretion phase is relatively low , then the black hole growth in nls1 may be insignificant . we find no significant evidence that nls1 offset from the @xmath0 relation of active and inactive galaxies , suggesting that nls1s and bl agns are similar in terms of the current black hole growth . in the case of the host galaxies of nls1s , there seems no strong difference between the environments of nls1s and bl agns @xcite . also , nls1s are not preferentially hosted by merging galaxies @xcite . thus , the growth of black holes and host galaxies seem to be similar between nls1 and bl agns . based on the estimates of the bolometric luminosity of the nls1s in our sample , we calculated the mass accretion rate in order to investigate the black hole growth time scale . for given the range of bolometric luminosity of 10@xmath70 - 10@xmath71 erg s@xmath6 , we estimate the mass accretion rate as @xmath130.002 - @xmath130.02 @xmath14 year@xmath9 . thus , in order to accrete a million solar mass to a black hole with a constant mass accretion rate of @xmath130.002 - @xmath130.02 @xmath14 year@xmath9 , it would take 10@xmath72 - 10@xmath73 yrs . the mean eddington ratio of the nls1s in our sample is @xmath1310% , for which the e - folding growth time scale is 4@xmath74@xmath75 yrs . thus , unless the life time of the agn activity is comparable to this growth time scale , nls1s are not expected to move up to the larger @xmath1 direction in the @xmath0 plane . nls1s are often considered as more inclined ( pole - on ) systems to the line - of - sight than bl agns , implying that the measured line - of - sight velocity dispersion ( line width ) of broad emission lines is relatively narrow due to the projection effect . if this is the case , then the @xmath1 of nls1s are significantly underestimated and their eddington ratios are accordingly overestimated . however , although there are some evidences that nls1s are close to pole - on systems ( e.g. , * ? ? ? * ; * ? ? ? * ) , the inclination effect can not explain the entire nls1 population ( see discussion by peterson 2011 ) . the implication of the potential inclination effect is that the nls1s in our sample would positively offset toward the high @xmath1 direction , if the black hole masses were were estimated after correcting for the velocity projection effect . in this scenario , it is difficult to understand why nls1s have higher black hole to galaxy mass ratios compared to bl agns and quiescent galaxies . we note that 6 nls1s are included in the sample of the reverberation - mapped agns , which are used for deriving the average virial factor for type 1 agns ( see figure 5 ) . the location of the nls1s in the @xmath0 plane is not different from that of bl agns , implying that the virial factor and inclination angle of the nls1s may not be very different from those of bl agns , although the number of nls1s in the reverberation - mapped agn sample is still small to make a firm conclusion . we investigated the @xmath0 relation of the present - day nls1 , using directly measured stellar velocity dispersions for a sample of 93 nls1s at z@xmath20.1 selected from the sdss . we summarize the main results . @xmath76 compared to the @xmath0 relation derived from the joint sample of the reverberation - mapped agns and inactive galaxies , the nls1s in our sample show no significant offset , suggesting that nls1s are an extension of bl agns at lower mass scale . @xmath76 among nls1s , there is a systematic trend with galaxy inclination , i.e. , more inclined galaxies have larger @xmath34 at fixed @xmath1 , probably due to the contribution of the rotational broadening in the stellar absorption lines . @xmath76 by jointly fitting the @xmath0 relation using the most updated reverberation - mapped agns and quiescent galaxies , we obtained the virial factor log f = 0.65 @xmath33 0.12 ( i.e. , f = 4.47 ) and log f = 0.05 @xmath33 0.12 ( i.e. , f = 1.12 ) , respectively for @xmath1 estimators based on the @xmath21 and @xmath22 . we thank the anonymous referee for valuable comments , which improved the clarity of the manuscript . this work was supported by the national research foundation of korea ( nrf ) grant funded by the korea government ( mest ; no . 2012 - 006087 ) . j.h.w acknowledges the support by the korea astronomy and space science institute ( kasi ) grant funded by the korea government ( mest ) . the virial factor f in equation ( 1 ) is difficult to determine for individual objects due to the unknown geometry and distribution of the blr gas ( c.f . * ; * ? ? ? * ; * ? ? ? * ) . instead , an average @xmath77 has been determined by scaling the reverberation - mapped agns to quiescent galaxies in the @xmath0 plane , assuming that agn and non - agn galaxies follow the same @xmath0 relation @xcite . while most of these calibrations have been performed using the virial product ( @xmath78 r@xmath79 /g ) based on @xmath21 as the velocity proxy of the broad - line gas , a number of black hole mass studies used @xmath22 for estimating single - epoch @xmath1 because of the difficulty of measuring @xmath21 due to the low s / n of available spectra ( e.g. , sdss ) . in this case , @xmath22 is converted to @xmath21 with a constant fwhm/@xmath26 ratio . however , the fwhm/@xmath26 ratio has a wide range since the line profile of the h@xmath4 line is not universal @xcite , hence , a systematic uncertainty is added to the mass estimates . here we provide the f factor for @xmath21-based and @xmath22-based virial products , respectively , by fitting the @xmath0 relation . for the reverberation - mapped agns , we collected and updated the time - lag ( e.g. , * ? ? ? * ) , @xmath22 and @xmath21 as well as stellar velocity dispersion measurements from the literature for a sample of 29 agns , as listed in table a1 ( see a recent compilation by woo et al . 2013 and the addition of grier et al . 2013 and bentz et al . 2014 ) , after excluding two objects , pg 1229 + 204 and pg 1617 + 175 since their stellar velocity dispersion measurements are very uncertain ( see for example figure 3 in grier et al . 2013 ) . the @xmath22 and @xmath21 are measured from the rms spectra of each object except for the 4th entry of mrk 817 ( see table a1 ) . when there are multiple measurements available for given objects , we calculated the mean of the virial products . note that we often found typos of the quoted values of the time lag and the h@xmath4 velocity in the literature . thus , we included the reference of the original measurements . in the case of the quiescent galaxy sample , we used 84 galaxies from the compilation of kormendy & ho ( 2013 ) , after excluding 3 galaxies , ngc 2778 , ngc 3945 , ngc 4382 due to the lack of the lower limit of the black hole mass . note that the choice of the quiescent galaxy sample does not significantly change the results presented for the nls1s since the virial factor is determined based on the best - fit @xmath0 relation and the @xmath1 of the nls1s scales accordingly . a careful comparison of the @xmath0 relation based on various subsamples of the quiescent galaxies will be presented by woo et al . ( in preparation ) . we performed a joint - fit analysis for the combined sample of reverberation - mapped agns and quiescent galaxies in order to determine the slope , intercept , and the virial factor , following the joint - fit method as described in woo et al . 2013 : @xmath80 where @xmath81(@xmath1/@xmath14 ) of quiescent galaxies , @xmath82(@xmath83 ) of reverberation - mapped agns , and @xmath84(@xmath34/ 200 km s@xmath6 ) , while @xmath85 , @xmath86 , and @xmath87 are the measurements uncertainties in @xmath88 , @xmath89 , and @xmath90 , respectively , and @xmath91 is intrinsic scatter , which we change for the reduced @xmath92 to be unity . in figure a1 , we present the best - fit @xmath0 relation for the combined sample . when @xmath21 is used as v in eq . 1 , we obtained the intercept @xmath12 = 8.34 @xmath33 0.05 , the slope @xmath4 = 4.97 @xmath33 0.28 , and log f = 0.65 @xmath33 0.12 . in the case of @xmath22 , we derived @xmath12 = 8.34 @xmath33 0.05 , @xmath4 = 5.04 @xmath33 0.28 , and log f = 0.05 @xmath33 0.12 . the intrinsic scatter of the combined sample is 0.43 @xmath33 0.03 and 0.43 @xmath33 0.03 , respectively for @xmath21-based mass and @xmath22-based mass . the derived f factor and the @xmath0 relation based on the updates of the reverberation and stellar velocity dispersion measurements are consistent with those derived by @xcite . in the case of the @xmath22-based @xmath1 , the best - fit virial factor f = 1.12 is consistent with the value derived by @xcite . for future @xmath1 studies , we recommend to use log f = 0.65 @xmath33 0.12 for the @xmath21-based @xmath1 estimates , and log f = 0.05 @xmath33 0.12 for the @xmath22-based @xmath1 estimates . the derived virial factor is consistent with that determined from the dynamical modeling based on the velocity - resolved measurements of five agns ( pancoast et al . 2014 ) , which are log f = 0.68 @xmath33 0.40 and log f = -0.07 @xmath33 0.40 , respectively for the @xmath21-based and @xmath22-based black hole masses . note that we did not attempt to use a different @xmath0 relation for pseudo - bulge galaxies since the @xmath0 relation of the pseudo - bulge galaxies is not well defined due to the limited dynamical range ( see figure a1 ) . it is not clear whether pseudo - bulge galaxies offset from the @xmath0 relation of classical bulges in figure a1 ( see also bennert et al . more detailed comparison of pseudo - bulge galaxies in the @xmath0 plane will be provided by woo et al . ( in preparation ) based on the new measurements from the spatially - resolved kinematics of 9 pseudo - bulge galaxies . thus , in this study we simply combine classical and pseudo bulges in determining the best fit @xmath0 relation . in figure a1 , we used open symbols for pseudo bulge galaxies following the classification from kormendy & ho 2014 and ho & kim 2014 . as a consistency check , we fit the @xmath0 relation for the agn sample only by minimizing @xmath93 where we used log f = 0.65 for the @xmath21-based @xmath1 estimates , and log f = 0.05 for @xmath22-based @xmath1 . using the @xmath21-based @xmath1 , we obtained the best - fit @xmath12 = 8.16 @xmath33 0.18 , @xmath4 = 3.97 @xmath33 0.56 , and the intrinsic scatter @xmath94 = 0.41 @xmath33 0.05 . in the case of the @xmath22-based @xmath1 , we derived @xmath12 = 8.21 @xmath33 0.18 , @xmath4 = 4.32 @xmath33 0.59 , and @xmath94 = 0.43 @xmath33 0.05 . these slopes are consistent with the best - fit slope of the combined sample within the uncertainties . we note that the slope @xmath12 of the agn @xmath0 relation does not depend on the choice of the virial factor in equation a2 . we emphasize that in our study the @xmath0 relation of the reverberation - mapped agns is derived with a consistent method adopted for the quiescent galaxies ( see park et al . 2012 ) , while other studies of agn @xmath0 relation often utilized somewhat different method , without including an iterative fitting process with intrinsic scatter . compared to grier et al . ( 2013 ) , for example , we obtained a different @xmath0 relation , hence , the virial factor even if we used the compiled values in their table . this discrepancy seems to stem from the treatment of the intrinsic scatter since we obtained the same results as grier et al . ( 2013 ) when we excluded the intrinsic scatter in the fitting process . lcccc ccccc c 3c 120 & & @xmath95 & 1 & 1514 @xmath33 65 & 2539 @xmath33 466 & 1 & @xmath96 & @xmath97 & 162 @xmath33 20 & 13 + 3c 390.3 & & @xmath98 & 2 & 5455 @xmath33 278 & 10872 @xmath33 1670 & 2 & @xmath99 & @xmath100 & 273 @xmath33 16 & 14 + ark 120 & & @xmath101 & 3 & 1959 @xmath33 109 & 5536 @xmath33 297 & 4 & @xmath102 & @xmath103 & & + & & @xmath104 & 3 & 1884 @xmath33 48 & 5284 @xmath33 203 & 4 & @xmath105 & @xmath106 & & + & mean & & & & & & @xmath107 & @xmath108 & 192 @xmath33 8 & 15 + + arp 151 & & @xmath109 & 5 & 1295 @xmath33 37 & 2458 @xmath33 82 & 6 & @xmath110 & @xmath111 & 118 @xmath33 4 & 16 + mrk 50 & & @xmath112 & 7 & 1740 @xmath33 101 & 4039 @xmath33 606@xmath51 & 7 & @xmath113 & @xmath114 & 109 @xmath33 14 & 7 + mrk 79 & & @xmath115 & 3 & 2137 @xmath33 375 & 5086 @xmath33 1436 & 4 & @xmath116 & @xmath117 & & + & & @xmath118 & 3 & 1683 @xmath33 72 & 4219 @xmath33 262 & 4 & @xmath119 & @xmath120 & & + & & @xmath121 & 3 & 1854 @xmath33 72 & 5251 @xmath33 533 & 4 & @xmath122 & @xmath123 & & + & & @xmath124 & 3 & 1883 @xmath33 246 & 2786 @xmath33 390 & 4 & @xmath125 & @xmath126 & & + & mean & & & & & & @xmath127 & @xmath128 & 130 @xmath33 12 & 14 + + mrk 110 & & @xmath129 & 3 & 1196 @xmath33 141 & 1494 @xmath33 802 & 4 & @xmath130 & @xmath131 & & + & & @xmath132 & 3 & 1115 @xmath33 103 & 1381 @xmath33 528 & 4 & @xmath133 & @xmath134 & & + & & @xmath135 & 3 & 755 @xmath33 29 & 1521 @xmath33 59 & 4 & @xmath136 & @xmath137 & & + & mean & & & & & & @xmath138 & @xmath139 & 91 @xmath33 7 & 17 + + mrk 202 & & @xmath140 & 5 & 962 @xmath33 67 & 1794 @xmath33 181 & 6 & @xmath141 & @xmath142 & 78 @xmath33 3 & 16 + mrk 279 & & @xmath143 & 3 & 1420 @xmath33 96 & 3385 @xmath33 349 & 4 & @xmath144 & @xmath145 & 197 @xmath33 12 & 14 + mrk 509 & & @xmath146 & 3 & 1276 @xmath33 28 & 2715 @xmath33 101 & 4 & @xmath147 & @xmath148 & 184 @xmath33 12 & 5 + mrk 590 & & @xmath149 & 3 & 789 @xmath33 74 & 1675 @xmath33 587 & 4 & @xmath150 & @xmath151 & & + & & @xmath152 & 3 & 1935 @xmath33 52 & 2566 @xmath33 106 & 4 & @xmath153 & @xmath154 & & + & & @xmath155 & 3 & 1251 @xmath33 72 & 2115 @xmath33 575 & 4 & @xmath156 & @xmath157 & & + & & @xmath158 & 3 & 1201 @xmath33 130 & 1979 @xmath33 386 & 4 & @xmath159 & @xmath160 & & + & mean & & & & & & @xmath161 & @xmath162 & 189 @xmath33 6 & 14 + + mrk 817 & & @xmath163 & 3 & 1392 @xmath33 78 & 3515 @xmath33 393 & 4 & @xmath164 & @xmath165 & & + & & @xmath166 & 3 & 1971 @xmath33 96 & 4952 @xmath33 537 & 4 & @xmath167 & @xmath168 & & + & & @xmath169 & 3 & 1729 @xmath33 158 & 3752 @xmath33 995 & 4 & @xmath170 & @xmath171 & & + & & @xmath172 & 3 & 2025 @xmath33 5@xmath173 & 5627 @xmath33 30@xmath173 & 8 & @xmath174 & @xmath175 & & + & mean & & & & & & @xmath176 & @xmath177 & 120 @xmath33 15 & 14 + + mrk 1310 & & @xmath178 & 5 & 921 @xmath33 135 & 1823 @xmath33 157 & 6 & @xmath179 & @xmath180 & 84 @xmath33 5 & 16 + ngc 3227 & & @xmath181 & 3 & 2018 @xmath33 174 & 5278 @xmath33 1117 & 4 & @xmath182 & @xmath183 & & + & & @xmath184 & 3 & 1376 @xmath33 44 & 3578 @xmath33 83 & 8 & @xmath185 & @xmath186 & & + & mean & & & & & & @xmath187 & @xmath188 & 133 @xmath33 12 & 18 + + ngc 3516 & & @xmath189 & 3 & 1591 @xmath33 10 & 5175 @xmath33 96 & 8 & @xmath190 & @xmath191 & 181 @xmath33 5 & 14 + ngc 3783 & & @xmath192 & 3 & 1753 @xmath33 141 & 3093 @xmath33 529 & 4 & @xmath193 & @xmath194 & 95 @xmath33 10 & 19 + ngc 4051 & & @xmath195 & 3 & 927 @xmath33 64 & 1034 @xmath33 41 & 8 & @xmath196 & @xmath197 & 89 @xmath33 3 & 14 + ngc 4151 & & @xmath198 & 3 & 2680 @xmath33 64 & 4711 @xmath33 750 & 9 & @xmath199 & @xmath200 & 97 @xmath33 3 & 14 + ngc 4253 & & @xmath201 & 5 & 538 @xmath33 82 & 986 @xmath33 251 & 6 & @xmath202 & @xmath203 & 93 @xmath33 32 & 16 + ngc 4593 & & @xmath204 & 3 & 1561 @xmath33 55 & 4141 @xmath33 416 & 10 & @xmath205 & @xmath206 & 135 @xmath33 6 & 14 + ngc 4748 & & @xmath207 & 5 & 791 @xmath33 80 & 1373 @xmath33 86 & 6 & @xmath208 & @xmath209 & 105 @xmath33 13 & 16 + ngc 5273 & & @xmath210 & 11 & 1544 @xmath33 98 & 4615 @xmath33 330 & 11 & @xmath211 & @xmath212 & 74 @xmath33 4 & 20 + ngc 5548 & & @xmath213 & 5 & 3900 @xmath33 266 & 12539 @xmath33 1927 & 6 & @xmath214 & @xmath215 & 195 @xmath33 13 & 16 + ngc 6814 & & @xmath216 & 5 & 1697 @xmath33 224 & 2945 @xmath33 283 & 6 & @xmath217 & @xmath218 & 95 @xmath33 3 & 16 + ngc 7469 & & @xmath219 & 3 & 1456 @xmath33 207 & 2169 @xmath33 459 & 4 & @xmath220 & @xmath221 & 131 @xmath33 5 & 14 + pg 1411 + 442 & & @xmath222 & 3 & 1607 @xmath33 169 & 2398 @xmath33 353 & 4 & @xmath223 & @xmath224 & 209 @xmath33 30 & 5 + pg 1426 + 015 & & @xmath225 & 3 & 3442 @xmath33 308 & 6323 @xmath33 1295 & 4 & @xmath226 & @xmath227 & 217 @xmath33 15 & 21 + pg 2130 + 099 & & @xmath228 & 12 & 1825 @xmath33 65 & 2097 @xmath33 102 & 1 & @xmath229 & @xmath230 & 163 @xmath33 19 & 5 + sbs 1116 + 583a & & @xmath231 & 5 & 1550 @xmath33 310 & 3202 @xmath33 1127 & 6 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narrow - line seyfert 1 galaxies ( nls1s ) are arguably one of the key agn subclasses in investigating the origin of the black hole mass - stellar velocity dispersion ( @xmath0 ) relation because of their high accretion rate and significantly low @xmath1 . currently , it is under discussion whether present - day nls1s offset from the @xmath0 relation . using the directly measured stellar velocity dispersion of 93 nls1s at z@xmath20.1 , and @xmath1 estimates based on the updated mass estimators , we investigate the @xmath0 relation of nls1s in comparison with broad - line agns . we find no strong evidence that the nls1s deviates from the @xmath0 relation , which is defined by reverberation - mapped type 1 agns and quiescent galaxies . however , there is a clear trend of the offset with the host galaxy morphology , i.e. , more inclined galaxies toward the line - of - sight have higher stellar velocity dispersion , suggesting that the rotational broadening plays a role in measuring stellar velocity dispersion based on the single - aperture spectra from the sloan digital sky survey . in addition , we provide the virial factor @xmath3 ( f = 1.12 ) , for @xmath1 estimators based on the fwhm of h@xmath4 , by jointly fitting the @xmath0 relation using quiescent galaxies and reverberation - mapped agns .

You are an expert at summarizing long articles. Proceed to summarize the following text: suppose we are given a finite set @xmath1 , a finite cyclic group @xmath2 acting on @xmath1 , and a polynomial @xmath3 $ ] with integer coefficients . following reiner , stanton , and white @xcite , we say that the triple @xmath4 exhibits the _ cyclic sieving phenomenon _ ( csp ) if for every integer @xmath5 , we have that @xmath6 where @xmath7 is a root of unity of multiplicitive order @xmath8 and @xmath9 is the fixed point set of the action of the power @xmath10 . the sizes of the fixed point sets determine the cycle structure of the canonical image of @xmath11 in the group of permutations of @xmath1 , @xmath12 . therefore , to find the cycle structure of the image of any bijection @xmath13 , it is enough to determine the order of the action of @xmath11 on @xmath1 and find a polynomial @xmath14 such that @xmath15 exhibits the csp . the cyclic sieving phenomenon has been demonstrated in a variety of contexts . the paper of reiner , stanton , and white @xcite itself includes examples involving noncrossing partitions , triangulations of polygons , and cosets of parabolic subgroups of coxeter groups . an example of the csp with standard young tableaux is due to rhoades @xcite and will discussed further in section [ sec : rhoades ] . now we turn to the csp of interest to this note . let @xmath16 denote the longest element in the type @xmath17 coxeter group . given generating set @xmath18 for @xmath17 , ( @xmath19 being the special " reflection ) , we will write a reduced expression for @xmath20 as a word in the subscripts . for example , @xmath21 can be written as @xmath22 we will abbreviate this product by @xmath23 . it turns out that if we cyclically permute these letters , we always get another reduced expression for @xmath20 . said another way , @xmath24 for @xmath25 . the same is not true for longest elements of other classical types . in type a , we have @xmath26 , and for type d , @xmath27 let @xmath0 denote the set of reduced expressions for @xmath20 in type @xmath17 and let @xmath28 denote the action of placing the first letter of a word at the end . then the orbit in @xmath29 of the word above is : @xmath30 @xmath31 as the length of @xmath20 is @xmath32 , we clearly have @xmath33 , and the size of any orbit divides @xmath32 . for an example of a smaller orbit , notice that the word @xmath34 has cyclic order 3 . for any word @xmath35 , ( e.g. , a reduced expression for @xmath20 ) , a _ descent _ of @xmath36 is defined to be a position @xmath37 in which @xmath38.the _ major index _ of @xmath36 , @xmath39 , is defined as the sum of the descent positions . for example , the word @xmath40 has descents in positions @xmath41 , and @xmath42 , so its major index is @xmath43 . let @xmath44 denote the generating function for this statistic on words in @xmath0 : @xmath45 the following is our main result . [ theorem : main ] the triple @xmath46 exhibits the cyclic sieving phenomenon , where @xmath47 for example , let us consider the case @xmath48 . we have @xmath49 let @xmath50 . then we compute : @xmath51 thus , the @xmath52 reduced expressions for @xmath21 split into two orbits of size three ( the orbits of @xmath53 and @xmath54 ) and four orbits of size nine . we prove theorem [ theorem : main ] by relating it to another instance of the csp , namely rhoades recent ( and deep ) result ( * ? ? ? * thm 3.9 ) for the set @xmath55 of rectangular standard young tableaux with respect to the action of _ promotion _ ( defined in section [ sec : prom ] ) . to make the connection , we rely on a pair of remarkable bijections due to haiman @xcite . the composition of haiman s bijections maps to @xmath0 from the set of square tableaux , @xmath56 . in this note our main goal is to show that haiman s bijections carry the orbit structure of promotion on @xmath56 to the orbit structure of @xmath57 on @xmath0 . we conclude this section by remarking that this approach was first outlined by rhoades ( * ? ? ? * thm 8.1 ) . one purpose of this article is to fill some nontrivial gaps in his argument . a second is to justify the new observation that the polynomial @xmath14 can be expressed as the generating function for the major index on @xmath0 . we thank brendon rhoades for encouraging us to write this note . thanks also to kevin dilks , john stembridge , and alex yong for fruitful discussions on this and related topics , and to sergey fomin for comments on the manuscript . for @xmath58 a partition , let @xmath59 denote the set of standard young tableaux of shape @xmath58 . if @xmath58 is a _ strict partition _ , i.e. , with no equal parts , then let @xmath60 denote the set of standard young tableaux of shifted shape @xmath58 . we now describe the action of _ jeu de taquin promotion _ , first defined by schtzenberger @xcite . we will consider promotion as a permutation of tableaux of a fixed shape ( resp . shifted shape ) , @xmath61 ( resp . @xmath62 ) . given a tableau @xmath63 with @xmath64 , we form @xmath65 with the following algorithm . ( we denote the entry in row @xmath66 , column @xmath67 of a tableau @xmath63 , by @xmath68 . ) 1 . remove the entry 1 in the upper left corner and decrease every other entry by 1 . the empty box is initialized in position @xmath69 . 2 . perform jeu de taquin : 1 . if there is no box to the right of the empty box and no box below the empty box , then go to 3 ) . if there is a box to the right or below the empty box , then swap the empty box with the box containing the smaller entry , i.e. , @xmath70 . set @xmath71 , where @xmath72 are the coordinates of box swapped , and go to 2a ) . 3 . fill the empty box with @xmath73 . here is an example : @xmath74 as a permutation , promotion naturally splits @xmath59 into disjoint orbits . for a general shape @xmath58 there seems to be no obvious pattern to the sizes of the orbits . however , for certain shapes , notably haiman s generalized staircases " more can be said @xcite ( see also edelman and greene ( * ? ? ? * cor . 7.23 ) ) . in particular , rectangles fall into this category , with the following result . [ thm : ordern ] if @xmath75 is a rectangle , then @xmath76 for all @xmath77 . thus for @xmath78 square shapes @xmath58 , @xmath79 and the size of every orbit divides @xmath32 . with @xmath48 , here is an orbit of size 3 : @xmath80 there are @xmath52 standard young tableaux of shape @xmath81 , and there are @xmath52 reduced expressions in the set @xmath29 . stanley first conjectured that @xmath0 and @xmath82 are equinumerous , and proctor suggested that rather than @xmath82 , a more direct correspondence might be given with @xmath83 , that is , with shifted standard tableaux of doubled staircase " shape . ( that the squares and doubled staircases are equinumerous follows easily from hook length formulas . ) haiman answers proctor s conjecture in such a way that the structure of promotion on doubled staircases corresponds precisely to cyclic permutation of words in @xmath0 ( * ? ? ? * theorem 5.12 ) . moreover , in ( * ? ? ? * proposition 8.11 ) , he gives a bijection between standard young tableaux of square shape and those of doubled staircase shape that ( as we will show ) commutes with promotion . as an example , his bijection carries the orbit in to this shifted orbit : @xmath84 both of these orbits of tableaux correspond to the orbit of the reduced word @xmath54 . we first describe the bijection between reduced expressions and shifted standard tableaux of doubled staircase shape . this bijection is described in section 5 of @xcite . let @xmath63 in @xmath83 . notice the largest entry in @xmath63 , ( i.e. , @xmath32 ) , occupies one of the outer corners . let @xmath85 denote the row containing this largest entry , numbering the rows from the bottom up . the _ promotion sequence _ of @xmath63 is defined to be @xmath86 , where @xmath87 . using the example above of @xmath88 we see @xmath89 , @xmath90 , and since @xmath91 , we have @xmath92 haiman s result is the following . [ theorem : phiandccommute ] the map @xmath93 is a bijection @xmath94 . by construction , then , we have @xmath95 i.e. , @xmath96 is an orbit - preserving bijection @xmath97 next , we will describe the bijection @xmath98 between squares and doubled staircases . though not obvious from the definition below , we will demonstrate that @xmath99 commutes with promotion . we assume the reader is familiar with the _ robinson - schensted - knuth insertion _ algorithm ( rsk ) . ( see ( * ? ? ? * section 7.11 ) , for example . ) this is a map between words @xmath36 and pairs of tableaux @xmath100 . we say @xmath101 is the _ insertion tableau _ and @xmath102 is the _ recording tableau_. there is a similar correspondence between words @xmath36 and pairs of shifted tableaux @xmath103 called _ shifted mixed insertion _ due to haiman @xcite . ( see also sagan @xcite and worley @xcite . ) serrano defined a semistandard generalization of shifted mixed insertion in @xcite . throughout this paper we refer to semistandard shifted mixed insertion simply as _ mixed insertion_. details can be found in ( * ? ? ? * section 1.1 ) . [ theorem : bijection ] let @xmath36 be a word . if we view @xmath104 as a skew shifted standard young tableau and apply jeu de taquin to obtain a standard shifted young tableau , the result is @xmath105 ( independent of any choices in applying jeu de taquin ) . for example , if @xmath106 , then @xmath107 @xmath108 performing jeu de taquin we see : @xmath109 haiman s bijection is precisely @xmath110 . that is , given a standard square tableau @xmath102 , we embed it in a shifted shape and apply jeu de taquin to create a standard shifted tableau . that this is indeed a bijection follows from theorem [ theorem : bijection ] , but is originally found in ( * ? ? ? * proposition 8.11 ) . haiman s bijection @xmath99 applies more generally between rectangles and shifted trapezoids " , i.e. , for @xmath111 , we have @xmath112 . all the results presented here extend to this generality , with similar proofs . we restict to squares and doubled staircases for clarity of exposition . we will now fix the tableaux @xmath101 and @xmath113 to ensure that the insertion word @xmath36 has particularly nice properties . we will use the following lemma . [ lem : insert ] fix a word @xmath36 . let @xmath114 be the rsk insertion tableau and let @xmath115 be the mixed insertion tableau . then the set of words that mixed insert into @xmath113 is contained in the set of words that rsk insert into @xmath101 . now we apply lemma [ lem : insert ] to the word @xmath116 if we use rsk insertion , we find @xmath101 is an @xmath78 square tableau with all 1s in row first row , all 2s in the second row , and so on . with such a choice of @xmath101 it is not difficult to show that any other word @xmath117 inserting to @xmath101 has the property that in any initial subword @xmath118 , there are at least as many letters @xmath119 as letters @xmath120 . such words are sometimes called ( reverse ) _ lattice words _ or ( reverse ) _ yamanouchi words_. notice also that any such @xmath117 has @xmath73 copies of each letter @xmath37 , @xmath121 . we call the words inserting to this choice of @xmath101 _ square words_. on the other hand , if we use mixed insertion on @xmath36 , we find @xmath113 as follows ( with @xmath122):@xmath123 in general , on the shifted half " of the tableau we see all 1s in the first row , all 2s in the second row , and so on . in the straight half " we see only prime numbers , with @xmath124 on the first diagonal , @xmath125 on the second diagonal , and so on . lemma [ lem : insert ] tells us that every @xmath117 that mixed inserts to @xmath113 is a square word . but since the sets of recording tableaux for @xmath101 and for @xmath113 are equinumerous , we see that the set of words mixed inserting to @xmath113 is precisely the set of all square words . yamanouchi words give a bijection with square standard young tableaux that circumvents insertion completely . in reading the word from left to right , if @xmath126 , we put letter @xmath37 in the leftmost unoccupied position of row @xmath127 . ( see ( * ? ? ? * proposition 7.10.3(d ) ) . ) we will soon characterize promotion in terms of operators on insertion words . first , some lemmas . for a tableau @xmath63 ( shifted or not ) let @xmath128 denote the result of all but step ( 3 ) of promotion . that is , we delete the smallest entry and perform jeu de taquin , but we do not fill in the empty box . the following lemma says that , in both the shifted and unshifted cases , this can be expressed very simply in terms of our insertion word . the first part of the lemma is a direct application of the theory of jeu de taquin ( see , e.g. , ( * ? ? ? * a1.2 ) ) ; the second part is ( * ? ? ? * lemma 3.9 ) . [ lemma : recording ] for a word @xmath129 , let @xmath130 . then we have @xmath131 and @xmath132 the operator @xmath133 acting on words @xmath134 is defined in the following way . consider the subword of @xmath36 formed only by the letters @xmath120 and @xmath135 . consider every @xmath135 as an opening bracket and every @xmath120 as a closing bracket , and pair them up accordingly . the remaining word is of the form @xmath136 . the operator @xmath133 leaves all of @xmath36 invariant , except for this subword , which it changes to @xmath137 . ( this operator is widely used in the theory of _ crystal graphs_. ) as an example , we calculate @xmath138 for the word @xmath139 . the subword formed from the letters @xmath140 and @xmath141 is@xmath142 which corresponds to the bracket sequence @xmath143 . removing paired brackets , one obtains @xmath144 , corresponding to the subword @xmath145 we change the last @xmath141 to a @xmath140 and keep the rest of the word unchanged , obtaining @xmath146 . the following lemma shows that this operator leaves the recording tableau unchanged . the unshifted case is found in work of lascoux , leclerc , and thibon ( * ? ? ? * theorem 5.5.1 ) ; the shifted case follows from the unshifted case , and the fact that the mixed recording tableau of a word is uniquely determined by its rsk recording tableau ( theorem [ theorem : bijection ] ) . [ lemma : operator ] recording tableaux are invariant under the operators @xmath147 . that is , @xmath148 and @xmath149 let @xmath150 denote the composite operator given by applying first @xmath151 , then @xmath152 and so on . it is clear that if @xmath153 is a square word , then @xmath154 is again a square word . [ theorem : commute ] let @xmath153 be a square word . then , @xmath155 and @xmath156 in other words , haiman s bijection commutes with promotion : @xmath157 by lemma [ lemma : recording ] , we see that @xmath158 is only one box away from @xmath159 . further , repeated application of lemma [ lemma : operator ] shows that @xmath160 the same lemmas apply show @xmath161 is one box away from @xmath162 . all that remains is to check that the box added by inserting 1 into @xmath163 ( resp . @xmath164 ) is in the correct position . but this follows from the observation that @xmath154 is a square word , and square words insert ( resp . mixed insert ) to squares ( resp . doubled staircases ) rhoades @xcite proved an instance of the csp related to the action of promotion on rectangular tableaux . his result is quite deep , employing kahzdan - lusztig cellular representation theory in its proof . recall that for any partition @xmath64 , we have that the standard tableaux of shape @xmath58 are enumerated by the frame - robinson - thrall _ hook length formula _ : @xmath165 where the product is over the boxes @xmath166 in @xmath58 and @xmath167 is the _ hook length _ at the box @xmath166 , i.e. , the number of boxes directly east or south of the box @xmath166 in @xmath58 , counting itself exactly once . to obtain the polynomial used for cyclic sieving , we replace the hook length formula with a natural @xmath168-analogue . first , recall that for any @xmath169 , @xmath170_q : = 1 + q + \cdots + q^{n-1}$ ] and @xmath170_q ! : = [ n]_q [ n-1]_q \cdots [ 1]_q$ ] . [ thm : cs ] let @xmath171 be a rectangular shape and let @xmath172 . let @xmath173 act on @xmath1 via promotion . then , the triple @xmath4 exhibits the cyclic sieving phenomenon , where @xmath174_q!}{\pi_{(i , j ) \in \lambda } [ h_{ij}]_q}\ ] ] is the @xmath168-analogue of the hook length formula . now thanks to theorem [ theorem : commute ] we know that @xmath99 preserves orbits of promotion , and as a consequence we see the csp for doubled staircases . [ cor : sieving1 ] let @xmath175 , and let @xmath176 act on @xmath1 via promotion . then the triple @xmath4 exhibits the cyclic sieving phenomenon , where @xmath177_q!}{[n]_q^n\prod_{i=1}^{n-1 } ( [ i]_q\cdot [ 2n - i]_q)^i}\ ] ] is the @xmath168-analogue of the hook length formula for an @xmath178 square young diagram . because of theorem [ theorem : phiandccommute ] the set @xmath0 also exhibits the csp . [ cor : sieving2 ] let @xmath179 and let @xmath14 as in corollary [ cor : sieving1 ] . let @xmath176 act on @xmath1 by cyclic rotation of words . then the triple @xmath4 exhibits the cyclic sieving phenomenon . corollary [ cor : sieving2 ] is the csp for @xmath0 as stated by rhoades . this is nearly our main result ( theorem [ theorem : main ] ) , but for the definition of @xmath14 . in spirit , if @xmath4 exhibits the csp , the polynomial @xmath14 should be some @xmath168-enumerator for the set @xmath1 . that is , it should be expressible as @xmath180 where @xmath181 is an intrinsically defined statistic for the elements of @xmath1 . indeed , nearly all known instances of the cyclic sieving phenomenon have this property . for example , it is known ( ( * ? ? ? * cor 7.21.5 ) ) that the @xmath168-analogue of the hook - length formula can be expressed as follows : @xmath182 where @xmath183 and for a tableau @xmath63 , @xmath184 is the sum of all @xmath37 such that @xmath37 appears in a row above @xmath185 . thus @xmath14 in theorem [ thm : cs ] can be described in terms a statistic on young tableaux . with this point of view , corollaries [ cor : sieving1 ] and [ cor : sieving2 ] are aesthetically unsatisfying . section [ sec : comb ] is given to showing that @xmath14 can be defined as the generating function for the major index on words in @xmath0 . it would be interesting to find a combinatorial description for @xmath14 in terms of a statistic on @xmath186 as well , though we have no such description at present . as stated in the introduction , we will show that @xmath187 if we specialize to square shapes , we see that @xmath188 and @xmath189 thus it suffices to exhibit a bijection between square tableaux and words in @xmath0 that preserves major index . in fact , the composition @xmath190 has a stronger feature . define the _ cyclic descent set _ of a word @xmath191 to be the set @xmath192 that is , we have descents in the usual way , but also a descent in position 0 if @xmath193 . then @xmath194 . for example with @xmath195 , @xmath196 and @xmath197 . similarly , we follow @xcite in defining the cyclic descent set of a square ( in general , rectangular ) young tableau . for @xmath63 in @xmath56 , define @xmath198 to be the set of all @xmath37 such that @xmath37 appears in a row above @xmath185 , along with 0 if @xmath199 is above @xmath32 in @xmath65 . major index is @xmath200 . we will see that @xmath201 preserves cyclic descent sets , and hence , major index . using our earlier example of @xmath195 , one can check that @xmath202 has @xmath203 , and so @xmath204 . first , we observe that both types of descent sets shift cyclically under their respective actions : @xmath208 and @xmath209 for words under cyclic rotation , this is obvious . for tableaux under promotion , this is a lemma of rhoades ( * ? ? ? * lemma 3.3 ) . because of this cyclic shifting , we see that @xmath210 if and only if @xmath211 . thus , it suffices to show that @xmath212 if and only if @xmath213 . ( actually , it is easier to determine if @xmath199 is a descent . ) let @xmath214 be the shifted doubled staircase tableau corresponding to @xmath36 . we have @xmath215 if and only if @xmath32 is in a higher row in @xmath216 than in @xmath217 . but since @xmath32 occupies the same place in @xmath216 as @xmath199 occupies in @xmath217 , this is to say @xmath199 is above @xmath32 in @xmath217 . on the other hand , @xmath218 if and only if @xmath199 is above @xmath32 in @xmath63 . it is straightforward to check that since @xmath217 is obtained from @xmath63 by jeu de taquin into the upper corner , the relative heights of @xmath32 and @xmath199 ( i.e. , whether @xmath32 is below or not ) are the same in @xmath217 as in @xmath63 . this completes the proof . [ thm : interp ] the @xmath168-analogue of the hook length formula for an @xmath178 square young diagram is , up to a shift , the major index generating function for reduced expressions of the longest element in the hyperoctahedral group : @xmath219_q!}{[n]_q^n\prod_{i=1}^{n-1 } ( [ i]_q\cdot [ 2n - i]_q)^i}.\ ] ] theorem [ thm : interp ] , along with corollary [ cor : sieving2 ] , completes the proof of our main result , theorem [ theorem : main ] . because this result can be stated purely in terms of the set @xmath0 and a natural statistic on this set , it would be interesting to obtain a self - contained proof , i.e. , one that does not appeal to haiman s or rhoades work . why must a result about cyclic rotation of words rely on promotion of young tableaux ?

we show that the set @xmath0 of reduced expressions for the longest element in the hyperoctahedral group exhibits the cyclic sieving phenomenon . more specifically , @xmath0 possesses a natural cyclic action given by moving the first letter of a word to the end , and we show that the orbit structure of this action is encoded by the generating function for the major index on @xmath0 .

You are an expert at summarizing long articles. Proceed to summarize the following text: the most luminous known agn with an ionized absorber is the radio - loud lobe - dominated quasar 3c351 ( l@xmath5 erg s@xmath6 , z=0.371 , fiore et al . , 1993 ) . ionized absorbers are common in low redshift , low luminosity seyfert 1 galaxies ( reynolds , 1997 ) , but rare in higher redshift , higher luminosity quasars . high luminosity agns are very likely physically larger and so may exhibit slower time variability and have different physical conditions . 3c 351 has a very high uv to x - ray ratio ( @xmath7 , tanambaun et al . , 1989 ) and its ir to uv spectrum does not show any evidence of reddening , in contrast to several low redshift seyfert 1 galaxies thought to host dusty / warm absorbers . only two other radio - loud quasars , 3c 212 ( mathur et al . , 1994 ) and 3c 273 ( grandi et al , 1997 ) , have candidate ionized absorbers . the main absorption feature in the rosat pspc x - ray spectrum of 3c 351 is a deep edge at @xmath1 kev ( quasar frame ) , which is likely to be due to ovii - oviii . mathur et al . ( 1994 ) built a simple one - zone model of this absorber that also explained the ovi , civ and ly@xmath8 uv absorption lines seen in a nearly simultaneous hst fos spectrum . such simple models , although elegant , have come under criticism . ionized absorbers in some seyfert galaxies show different variability behavior in different absorption features , some of which are not as predicted for gas in photoionization equilibrium . multiple absorbing zones have been introduced to explain these effects ( e.g. reynolds 1997 ) . in another paper ( nicastro et al . , 1998 ) we instead explore additional physics ( non - equilibrium photonionization models and collisional models ) while retaining a single zone of absorbing gas . 3c 351 is a relatively slowly varying source , compared to the rapidly variable seyfert with warm absorbers . this persistent variability on timescales that span possible ionization and recombination times makes it hard to define an initial equilibrium state . in contrast 3c 351 presents a simplified situation in which to study changes in the ionization state in response to luminosity changes . here we show that in 3c 351 at least the simplest photoionization equilibrium , single zone model continues to be sufficient . 3c351 was observed twice by rosat , on 1991 october and 1993 august . the first observation was reported by fiore et al . the second observation was then proposed in order to search for time variability that could test photoionization models . fortunately a factor 1.7 decrease in the pspc count rate was seen , providing just such a test . here we present the second data set and compare the results with predictions . we considered the two pspc observations of 3c351 taken roughly two years apart , in 1991 october and 1993 august . table 1 gives , for each observation , the observation date , its duration , the net exposure time , the net source counts , the count rate , and the signal to noise ratio . the data reduction and the timing analysis were performed using xselect . we reduced the second pspc observation of 3c351 following the procedures used in fiore et al . ( 1993 ) for the first observation . for the first observation we used the spectrum and light curve obtained by fiore et al . the source intensity was consistent with a constant value during both pspc observations , with no more than 20% change in the @xmath9 day and @xmath10 day spans of the two observations . 3c 351 thus appears more stable than the lower luminosity seyferts with ionized absorber ( reynolds , 1997 ) . since 3c 351 has a flat x - ray spectrum and broad optical emission lines it is consistent with the trend exhibited by the three similar but radio - quiet , pg quasars of the fiore et al . ( 1998 ) sample . those quasars show little or no variability on timescales as short as 10 days ( compared to the narrow line objects of that sample which are rapidly variable and have steep x - ray spectra ) . however the mean count rate dropped by a factor @xmath11 during the 22 months between the observations ( see table 1 ) , again consistently with the fiore et al . ( 1998 ) discovery . hereinafter we shall call these the high and low states , respectively . [ cols="^,^,^,^,^,^,^,^ " , ] @xmath12 in @xmath13 ph s@xmath6 @xmath14 kev@xmath6 ( at 1 kev ) . = 4.5 in = 3.in collisional ionization models can fit the data equally well , as they should at low column densities ( nicastro et al . , 1997 ) . to do so however they require arbitrary changes , by a factor 1.5 in temperature or a factor 2 in n@xmath15 . instead photoionization equilibrium models predict the observed correlation with the ionizing continuum . to the authors this is a strong argument in favor of a photoionization model . the behavior of the main physical properties of the absorber can be seen in a color - color diagram . in fig . 3 we plot the hardness ratios hr = h / m against the softness ratio sr = s / m from the count rates in three bands ( s=0.15 - 0.58 kev , m=0.88 - 1.47 kev , and h=1.69 - 3.40 kev , at z=0.371 ) for theoretical curves ( for log(@xmath16)=21 , 21.5 , 22 and 22.14 ) obtained by folding the equilibrium photoionization models ( for log(u ) in the range -0.3 - 1.5 , and galactic @xmath16 ) with the response matrix of the pspc . = 4.0 in = 4.0 in the position of a point in this diagram readily gives the dominant ion in the gas . the rapid change of sr as u increase from 0.5 to 5.5 ( on the curve corresponding to log(@xmath16)=22.14 ) corresponds to large increases of the transparency of the gas at e@xmath17 kev as h and he becomes rapidly fully ionized . as u further increases from 5.5 to 16.5 the sr color changes more slowly , and inverts its trend at u@xmath18 , with hr now changing more rapidly than sr . the inversion point indicates the switch from an ionization state dominated by ovii to that dominated by oix . in the last part of the curve as u increases , sr and hr decrease until all the ions in the gas are fully stripped and the gas is completely transparent to radiation of any energy . the two data points show the two 3c 351 observations . the best fit u in the high observation is marked on the log(@xmath16)=22.14 curve along with the value obtained by scaling by the intensity ratio between the two observations ( a factor 1.7 ) . the two points are consistent with the position of the observed colors of 3c351 in the two observations , so the predictions of the simple equilibrium photoionization model is consistent with the data . the as predicted change of the ionization parameter in the 3c 351 absorber to a change in the ionizing continuum is strong evidence that photoionization is the dominant ionization mechanism . we also see that the absorber comes to ionization equilibrium within 22 months , and can use this to constrain the physical properties of the absorber . the time @xmath19 measures the time the gas needs to reach equilibrium with the instantaneous ionizing flux ( nicastro et al , 1997 ) . this time depends on the particular ionic species considered . the ionic abundances of oxygen in the absorber in 3c 351 are distributed mainly between only two ionic species : ovii and oviii . in this simple case a useful analytical approximation for @xmath19 is : @xmath20_{t+\delta t}}}\nonumber \normalsize\end{aligned}\ ] ] where _ eq _ indicates the equilibrium quantities . by requiring that @xmath19 for ovii and oviii species is shorter than the @xmath21 s ( quasar frame ) elapsed between the two observations , we can find a lower limit to the electron density of the absorber : @xmath22 @xmath3 . since we know the values of u and @xmath23 , this density limit translates into a limit on the distance of the ionized gas from the central source , r@xmath24 pc . these limits are consistent with the upper limit of @xmath25 @xmath3 , and r@xmath26 pc , found by mathur et al . ( 1994 ) using a lower limit for the distance of the cloud from the central source , based on the absorber being outside the broad emission line region . the relative closeness of these two limits ( factor @xmath0100 ) implies that a variation in shorter times ( @xmath27 month ) , would be likely to show non - equilibrium effects ( nicastro et al , 1997 ) . if a factor 2 flux change in @xmath28 week showed no such effects the simplest photoionization model would have to be abandoned . we have tested ionization models for the ionized absorber in 3c 351 . in particular , we tested a simple one zone photoionization equilibrium model on two pspc spectra of 3c 351 that show a factor @xmath29 decrease in flux . the model correctly predicts the sense and amplitude of the observed change in the ionization state of the absorber , correlated with the ionizing continuum flux . given that photoionization equilibrium applies we can derive a lower limit to the electron density of the absorber : @xmath30 @xmath3 . this is consistent with the upper limit of @xmath31 @xmath3 found by mathur et al . the distance of the ionized gas from the central source is then 0.3 pc @xmath4 r @xmath4 50 pc . the closeness of these two limits creates a strong test of photoionization models : factor 2 variations in 3c 351 on timescales of order a week _ must _ show non - equilibrium effects .

we present two rosat pspc observations of the radio - loud , lobe - dominated quasar 3c 351 , which shows an ` ionized absorber ' in its x - ray spectrum . the factor 1.7 change in flux in the @xmath02 years between the observations allows a test of models for this ionized absorber . the absorption feature at @xmath1 kev ( quasar frame ) is present in both spectra but with a lower optical depth when the source intensity - and hence the ionizing flux at the absorber - is higher , in accordance with a simple , single - zone , equilibrium photoionization model . detailed modeling confirms this agrement quantitatively . the maximum response time of 2 years allows us to limit the gas density : @xmath2 @xmath3 ; and the distance of the ionized gas from the central source r @xmath4 19 pc . this produces a strong test for a photoionized absorber in 3c 351 : a factor 2 flux change in @xmath01 week in this source _ must _ show non - equilibrium effects in the ionized absorber .

You are an expert at summarizing long articles. Proceed to summarize the following text: the first theoretical constructions related with directed transport have been formulated in an early work by feynman @xcite . this opened a field whose relevance and activity has been increasing since then . the motivation is twofold , in the first place several fundamental questions have originated from these ideas and have been only partially answered @xcite . as a consequence , the subject grew into a major new field of statistical physics . on the other hand , the wealth of possible applications has determined the emergence of a very broad area of research in physics . ratchets , generically defined as periodic systems having a dissipative dynamics associated with thermal noise and unbiased perturbations ( driving them out of equilibrium ) , can be used to model a wide range of different phenomena . in order to give just a few examples we can mention molecular motors in biology @xcite , nanodevices ( rectifiers , pumps , particle separators , molecular switches and transistors ) @xcite , and coupled josephson junctions @xcite . on the other hand , there is a great interest in the theoretical description and experimental implementations of cold atoms subjected to time - dependent standing waves of light . they play a central role in many studies of the quantum dynamics of nonlinear systems ( dynamical localization , decoherence , quantum resonance , etc . ) in particular , the so - called optical ratchets , i.e. directed transport of laser cooled atoms , have been successfully implemented in this sort of experiments @xcite . the appearance of a net current can be classically explained by the necessary condition of breaking all spatiotemporal symmetries leading to momentum inversion @xcite . this , along with curie s principle indicates that transport should be present , given that it is not forbidden . as an example of this situation we can mention hamiltonian systems ( with necessarily mixed phase spaces ) where a chaotic layer should be asymmetric @xcite . in the more general dissipative case , chaotic attractors having this property are necessary @xcite . it is usual that the same principle translates almost directly into the quantum domain @xcite , and very similar behaviors arise . but sometimes quantum mechanics introduces new effects @xcite and the results depart from the classical ones . for example , the efficiency of a forced thermal quantum ratchet has been calculated in @xcite . in that work the authors find that the quantum current is higher in comparison with the classical one at the lowest values of the temperature . as this parameter increases the discrepancies diminish and finally they become negligibly small . recently , a quantum chaotic dissipative ratchet has been introduced in @xcite . in this example directed transport appears for particles under the influence of a pulsed asymmetric potential in the presence of a dissipative environment at zero temperature . the asymmetry of the quantum strange attractor is at the origin of the quantum current , in close analogy with what happens at the classical level . indeed , this work provides with the parameters needed for a possible implementation using cold atoms in an optical lattice . for a somewhat similar dynamics , the case of weak coupling and low temperature has been studied in @xcite . in the present work we extend the study of @xcite for a wide range of couplings with the environment and different temperature values . we have verified that there is a strong dependence of the current behavior on the coupling strength . if we compare this with the results found in @xcite , for instance , we could not find a generic enhancement of the quantum current for finite temperatures . instead of that , we could identify a close quantum - to - classical correspondence when considering thermal effects only at the classical level . in fact , we have found that the finite @xmath0 effects on the quantum current are analogous to the influence of the classical thermal fluctuations on the classical transport . in the following we describe the organization of this paper . in section ii we present our model for the system and for the environment , explaining the methods we have used to investigate the current behavior . in section iii we show the results where the roles of @xmath0 , the coupling strength and the temperature are analyzed in detail . finally , in section iv we summarize and point out our conclusions . in this section we describe the approach used to model the system plus the environment . we study the motion of a particle in a periodic kicked asymmetric potential given by @xmath1 \sum_{m=-\infty}^{+ \infty } \delta ( t - m \tau)\ ] ] where @xmath2 is the kicking period , @xmath3 is the strength of the kick , and @xmath4 and @xmath5 are parameters that allow to introduce a spatial asymmetry @xcite . the effects of the environment are taken into account by means of a velocity dependent damping and thermal fluctuations . at the classical level , these ingredients are incorporated in the following map : @xmath6 + \xi \\ \overline{x}= x + \tau \overline{n } \end{array } \right.\ ] ] in these expressions , @xmath7 is the momentum variable conjugated to @xmath8 and @xmath9 is the dissipation parameter , with @xmath10 . the thermal noise @xmath11 is related to @xmath9 , according to @xmath12 , where @xmath13 is the boltzmann constant and @xmath14 is the temperature , making the formulation consistent with the fluctuation - dissipation relationship . by performing the change of variables @xmath15 , @xmath16 , and @xmath17 ( where @xmath18 and @xmath19 ) , we can eliminate the period from the classical expressions , and define the new map @xmath20 + \tilde \xi \\ \overline{x}= x + \overline{p } \end{array } \right . \label{cl_map}\ ] ] in the quantum version of the model the system hamiltonian is given by @xmath21 where the quantization has been performed in such a way @xcite that @xmath22 , @xmath23 and @xmath24 . this amounts to saying that , being @xmath25=i\tau$ ] , there is an effective planck constant given by @xmath26 . then , the classical limit corresponds to @xmath27 , while keeping @xmath28 constant . in order to incorporate dissipation and thermalization to the quantum map we consider the coupling of the system to a bath of non interacting oscillators in thermal equilibrium . the degrees of freedom of the bath are eliminated introducing the usual weak coupling , markov and rotating wave approximations @xcite . this leads to a lindblad equation in action representation @xmath29 \ , + \ , \\ \!\!\ ! & + & \ ! g \ , \sqrt{n^+_{th } ( \omega_n , t ) n^+_{th}(\omega_{n'},t ) } \ , \ { [ \hat{l}_n , \hat{\rho } \hat{l}_{n'}^{\dagger } ] + [ \hat{l}_n \hat{\rho},\hat{l}_{n'}^{\dagger}]\ } \\ \!\!\ ! & + & \ ! g \ , \sqrt{n^-_{th } ( \omega_n , t ) n^-_{th}(\omega_{n'},t ) } \ { [ \hat{l}_n^{\dagger } , \hat{\rho } \hat{l}_{n ' } ] + [ \hat{l}_n^{\dagger } \hat{\rho},\hat{l}_{n'}]\ } \end{array } \label{qdiff}\ ] ] where the frequencies @xmath30 are the energy differences between two neighboring levels of the rotator . the population densities of the bath found in eq . [ qdiff ] are given by @xmath31 the system operators @xmath32 describe transitions towards the ground state of the free rotator . requiring quantum to classical correspondence at short times we fix the coupling constant @xmath33 . for @xmath34 we recover the master equation used in @xcite for the pure dissipative case . for finite temperature , the last term in eq . [ qdiff ] describes the thermal excitation of the rotator through absorption of heat bath energy . finally , eq . [ qdiff ] will be integrated numerically without further approximations . since we are interested in chaotic transport , throughout the following calculations we will use the set of parameters given by @xmath35 , @xmath36 , and @xmath37 . in the hamiltonian limit , this case shows no visible stability islands in phase space . we have first studied some classical aspects of our system at zero temperature , beginning with the bifurcation diagrams in terms of @xmath38 as a function of the parameter @xmath39 $ ] ( see fig . [ fig : bifdiag](_a _ ) ) . it should be mentioned that the chaotic attractors set in very fast for small @xmath9 . but this is not the case for larger values where the transient times can be very long . for this reason , we have calculated these diagrams with the last @xmath40 iterations of the map , after the first @xmath41 have been discarded . we have randomly taken @xmath42 initial conditions inside the unit cell ( @xmath43 , @xmath44 $ ] ) . several regular and chaotic windows alternate . the former are characterized by simple attractors ( stable fixed points of the dissipative map ) and the latter are dominated by chaotic attracting sets . the width in @xmath38 of these sets grows as dissipation weakens ( @xmath45 ) . the main quantity characterizing transport is the current @xmath46 , where @xmath47 stands for the average taken on the initial conditions , and @xmath48 is the moment after the @xmath49th iteration of the map . since bifurcations that change the shape of strange attractors also play a role in determining the values of the asymptotic current we restrict our analysis to the chaotic window approximately located at @xmath50 $ ] indicated by an arrow . the inset of fig . [ fig : bifdiag](a ) displays the asymptotic current @xmath51 as a function of @xmath9 ( circles stand for @xmath34 ) . at this @xmath9 range , 100 kicks are enough to reach the stationary behavior , independently of the initial distributions . as pointed out in @xcite , dissipation induces an asymmetry of the strange attractor which is responsible for the directed transport . on the other hand this dissipation mechanism contracts phase space and makes the higher energies inaccessible for the system . the final value of @xmath52 results from the interplay between both effects . in fact , increasing dissipation does not increase the transport and the largest values of @xmath52 are obtained for the lowest values of dissipation , i.e. @xmath53 ( for example @xmath54 for @xmath55 ) . the minimum current in this window is reached for an intermediate value of dissipation ( @xmath56 ) . we then consider the case of finite temperature . the bifurcation diagram corresponding to @xmath57 is shown in fig . [ fig : bifdiag](b ) . it is clear that the effect of temperature consists of smoothing the finer structure of the chaotic attractors that is present for the smallest values of @xmath9 . even for this extremely low value of @xmath14 the detailed features have almost completely disappeared , with the exception of the black lines corresponding to the highest values of the density distributions . the other very interesting effect is that temperature erases the regular windows allowing for a continuously chaotic behavior . this could be of much relevance in obtaining large ratchet currents without the need for an extremely fine tuned , weak dissipation @xcite ( this will addressed in future studies @xcite ) . as shown in the inset , low temperatures ( diamonds correspond to @xmath57 ) lead to a noticeable enhancement of the asymptotic current @xmath58 ( around @xmath59 for @xmath56 ) . we now turn to compare the classical and quantum behaviors . firstly , we analyze the currents ( which in the quantum case is given by @xmath60 ) . in fig . [ fig : allasympcurrentsvst ] we display @xmath58 as a function of @xmath14 , for three different values of @xmath9 and @xmath61 . at the classical level low temperatures lead to an enhancement of the current for intermediate values of the dissipation ( see fig . [ fig : allasympcurrentsvst ] upper and middle panels corresponding to @xmath9= 0.7 , 0.75 ) . in the case of weak dissipation ( @xmath62 in the lower panel of fig . [ fig : allasympcurrentsvst ] ) , which displays larger values of @xmath58 , the effect of thermal noise is negligible . for higher temperatures , the thermal effects reduce @xmath52 in all cases . this can be interpreted as follows : thermal noise reduces the energy loss caused by dissipation ( with no kicks , the system would attain a boltzmann distribution ) , so higher energies can be reached , in comparison with a pure dissipative process . but since this diffusion also tends to blur the asymmetry of the strange attractor , the two effects compete and transport has a maximum for low values of t , and then decreases . at the quantum level we can clearly see that the previously described thermal enhancement is not generally present , at least for the @xmath61 values we have considered . it is important to note that these values are consistent with experimental implementations using for example , cold sodium atoms in a laser field having a wave length @xmath63 ( for more details see @xcite ) . however , we observe a very slight growth of the current for the @xmath64 case with @xmath62 ( see the blue triangles in the lower panel of fig . [ fig : allasympcurrentsvst])indicating that the temperature dependence of the current is very sensitive to the particular dynamics of the system . for a different example , an enhancement of the quantum transport has been observed @xcite , hence it is difficult to extract universal behaviors . for @xmath65 the quantum current is lower than the classical one for all temperatures , as already pointed out in @xcite , but only for @xmath34 . the same happens in the case of weak damping @xmath62 ( nevertheless , for @xmath64 and @xmath66 both currents coincide ) . the case @xmath67 shows a different behavior . at @xmath34 the quantum currents ( for any of the @xmath61 values we have considered ) are larger than the classical ones , that is , there is an enhancement due to the finite size of @xmath61 . also , the maximal quantum current corresponds to @xmath68 , and this is valid for all the temperatures shown ( for @xmath69 it is still greater than the classical one ) . for larger quantum coarse - graining ( see @xmath70 ) quantum currents decrease . in this sense , it seems that the effect of quantum fluctuations on the quantum directed currents is analogous to the effect of those of thermal origin on the classical ones . small fluctuations of thermal or quantum mechanical origin assist directed transport while large fluctuations ( corresponding to high temperatures or to large values of @xmath61 respectively ) blur the asymmetry of the attractor and thus kill the net current . it is interesting to note that in the case @xmath67 the thermal diffusion associated with the temperature which gives the maximal value of the current ( @xmath71 for @xmath72 ) is of the order of the quantum coarse - graining @xmath61 corresponding to the strongest quantum current . for @xmath65 the classical current attains its maximum value at @xmath69 ( @xmath73 ) , which corresponds to a value of @xmath61 that we were not able to consider in our numerical calculations . the analogy between thermal noise and quantum coarse - graining can also be appreciated when looking at the asymptotic poincare sections and husimi distributions ( displayed in fig . [ fig : phasespace ] for @xmath74 , @xmath75 ) . as expected , at zero temperature the quantum husimi function reproduces well the main patterns of the classical attractor but shows less fine structure ( see the upper panels ) . if a small temperature is introduced the fine details of the classical distribution are smoothed out and both distributions look more alike ( see the lower panels corresponding to @xmath76 ) . on the other hand the quantum distributions at zero and finite temperatures are practically indistinguishable , indicating that the quantum coarse - graining is at least of the order of the thermal one for these values of @xmath14 . we finally study @xmath77 as a function of @xmath49 ( i.e. , the number of iterations of the map ) . results for @xmath67 are shown in fig . [ fig : currentvstg075 ] , where different temperatures and @xmath61 values have been considered . besides the mentioned fact that the asymptotic value is reached very rapidly , we notice that the transient behavior shows a very close quantum - to - classical correspondence . the classical current peak observed at @xmath78 for low temperatures ( @xmath76 ) is also present in the quantum current when @xmath64 and @xmath34 . this peak disappears from the classical current at larger temperatures ( @xmath79 ) , and so does the quantum one at @xmath70 and @xmath34 . so the analogy seems to hold at all times . in this work we have analyzed the behavior of a chaotic dissipative system that shows directed transport under the influence of a thermal bath , both in its classical and quantum versions . we have varied the strength of the coupling with the environment and also the temperature . we have found that the transport enhancement effect due to a finite temperature is highly dependent on the system specific properties . in fact , it depends on the coupling strength of the system with the environment and also on the @xmath0 size . moreover , we could find an analogy between the effects caused by thermal and quantum fluctuations . these results open the possibility for many further studies that include finding ways of obtaining large ratchet currents in experimentally realistic situations in kicked becs and cold atoms experiments . these are one of the best candidates to test our results since even becs show an unavoidable fraction of noncondensed atoms when kicked . if kicks become strong , thermal excitations will be of much relevance rather than a negligible effect . with the sort of calculations presented in this paper the effects of this fraction on the transport properties of the system could be estimated . f. l. moore _ et al . _ , lett . * 75 * , 4598 ( 1995 ) ; j. ringot _ et al . _ , 85 * , 2741 ( 2000 ) ; h. ammann _ lett . * 80 * , 4111 ( 1998 ) ; m. b. darcy _ et al . _ , lett . * 87 * , 074102 ( 2001 ) .

we study a chaotic ratchet system under the influence of a thermal environment . by direct integration of the lindblad equation we are able to analyze its behavior for a wide range of couplings with the environment , and for different finite temperatures . we observe that the enhancement of the classical and quantum currents due to temperature depend strongly on the specific properties of the system . this makes difficult to extract universal behaviors . we have also found that there is an analogy between the effects of the classical thermal noise and those of the finite @xmath0 size . these results open many possibilities for their testing and implementation in kicked becs and cold atoms experiments .

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Proceed to summarize the following text: the kondo model and its periodic extension , the kondo lattice model ( klm ) , which describe spin - exchange interaction between a localized spin or a system of localized spins , respectively , and a band of itinerant electrons , has been subject of intense theoretical studies in the past @xcite . this model has been applied to a variety of different problems in solid - state physics using both a ferromagnetic and antiferromagnetic coupling constant @xmath2 . the model with @xmath0 is the one originally known as _ kondo lattice model _ or simply _ kondo model _ in its non - periodic form with a single impurity spin in the system . it was used by kondo to explain the unusual temperature behavior of the resistivity of magnetic impurities in non - magnetic hosts @xcite . the negative spin - exchange interaction can be derived from the hybridization of a correlated `` atomic '' level with a conduction band , the situation described by the anderson model @xcite . in the limit of a low - lying half - filled atomic level and strong correlations , the anderson model can be mapped onto the kondo model with a negative exchange constant @xcite . the kondo lattice model is still subject to much theoretical work , the main objective is the understanding of the unusual physical behavior found in _ heavy - fermion _ materials @xcite . a model with identical operator structure in the hamiltonian , but with positive exchange constant has been known in the literature for a long time by many different names ( double exchange model , @xmath3-@xmath4 model , @xmath3-@xmath5 model , ) @xcite . for clarity , we will refer to this model in the following as _ ferromagnetic kondo lattice model_. the model with ferromagnetic exchange has to be understood as an effective one . the origins of the exchange with @xmath1 are found in the interband coulomb correlations @xcite . this situation is believed to dominate the physical properties of important systems such as the magnetic semiconductors @xcite ( eux ; x = o , s , se , te ) , the diluted magnetic semiconductors @xcite ( cd@xmath6mn@xmath7te , hg@xmath6fe@xmath7se ) , and the `` local moment '' metals @xcite ( gd , dy , tb ) . to these problems , the ferromagnetic klm was successfully applied @xcite . recently , this variant of the klm has gained a lot of interest with the discovery of the colossal magnetoresistance ( cmr ) materials @xcite . in these materials , typically manganese oxides with perovskite structure ( la@xmath6(ca , sr)@xmath7mno@xmath8 ) , the double - exchange model @xcite has been successfully applied to explain the origin of ferromagnetic order and is expected to be a good starting point to investigate the resistivity anomalies @xcite . this double - exchange model , however , is nothing else than the kondo lattice model with ferromagnetic ( positive ) exchange constant in the strong coupling limit . in the cmr materials , the localized @xmath9-spin of the model represents the more or less localized manganese @xmath10-@xmath11 electrons , whereas the conduction band is formed by the @xmath12 electrons . the interband - exchange interaction is nothing else but the intra - shell hund s rule coupling . since the @xmath10-@xmath12 electrons of the manganese form a relatively narrow band ( theoretical results from band - structure calculations : @xmath13 @xcite and experimental estimates : @xmath14 @xcite ) and hund s coupling is assumed to be large , the model has to be taken in the intermediate to strong coupling regime . there are few estimates about the value of the interaction constant in the literature , e.g. @xmath15 @xcite , but these are challenged as to be too small @xcite . most theoretical papers of the last years concerned with colossal magnetoresistance assume classical spins @xmath16 @xcite . this has been justified by the assumption of @xmath17 @xcite . although it is true that the important energy scale is @xmath18 , there are much more implications of @xmath16 that are not justified in the strong - coupling limit for a @xmath9 system . in several papers , it was stated that `` the @xmath12 electrons are oriented parallel to the @xmath11 spins . '' @xcite or equivalently `` so one only need consider configurations with @xmath12 electrons parallel to core spins . '' @xcite . we will show below using exact results as well as several well - defined approximation methods , that for @xmath9 there is a considerable amount of spin-@xmath19 spectral weight located in the main region of the spin-@xmath20 states even for large interaction strengths . the assumption of a half - metallic state @xcite , made in the two citations above can therefore never be met in the klm with quantum spins and is merely an effect of the ( unphysical ) limit of `` classical '' spins . the recently discussed half - metallic behaviour of the manganites @xcite must have a different origin . however , for the opposite sign of @xmath2 , exactly the assumed effect happens in the strong - coupling limit : the lowest - lying excitations in the conduction band density of states will be purely spin-@xmath19 . this already implies that results for the kondo lattice model with @xmath1 and @xmath0 can not simply be reverted into the respective other case . the change of sign changes the whole physics of the system . for @xmath21 an antiparallel ( `` antiferromagnetic '' ) alignment of the conduction band spin and the localized spin lowers the internal energy . for a sufficient band filling , this tends to a screening of the local moments by conduction electrons , well - known from the kondo effect that refers to a single magnetic impurity in a conduction electron sea . from this , the name `` kondo lattice model '' was originally derived for the @xmath0 case . we will further show that already for comparatively low interaction strengths the spin - exchange interaction alone leads to an opening of a gap in the density of states . this extraordinary correlation effect could give hints to the explanation of a recently discovered pseudogap in the managese oxides @xcite . to prove our claims already laid out so far , we will first review two important non - trivial exactly solvable limiting cases of the kondo lattice model in sec . [ sec : theo ] . the first is the _ zero - bandwidth limit _ ( `` atomic limit '' ) where the bandwidth of the conduction band is set to @xmath22 @xcite . the second exactly solvable limiting case is the so - called _ ferromagnetically saturated semiconductor _ @xcite . this is essentially the zero - temperature limit of the model with vanishing electron density and fully aligned spin system . in this limit , striking correlation effects can be observed and discussed . these limiting cases will already give clear evidence to our propositions made above . in sec . [ sec : dmft ] we will present a new _ dynamical mean - field theory _ ( dmft)-based approach for the klm with @xmath23 spins . to circumstantiate our theory , we will introduce three more approximation schemes which also keep the spin as a quantum variable , not relying on the classical spin limit . the first will be a _ second - order perturbation theory _ ( sopt ) for the klm based on the projector operator formalism @xcite . the _ self - consistent cpa _ ( coherent potential approximation ) is a straightforward extension to the well - known cpa for the klm @xcite . it starts from the zero - bandwidth limit discussed before . the third approximation method is a moment - conserving decoupling procedure for the equation of motion of the single - electron green function . this approximation scheme continuously evolves from the exactly solvable limit of the ferromagnetically saturated semiconductor . it will be called _ moment - conserving decoupling approximation _ ( mcda ) . the comparison of the results obtained by these three methods and our dmft scheme , each of which starts from a different limit allows to evaluate the range of applicability of the new approach , and to select the most trustworthy common features of all methods to gain a reliable picture of the physics of the ferromagnetic klm . the kondo - lattice model ( or @xmath3-@xmath5 model ) traces back the characteristic features of the underlying physical system to an interband exchange coupling of itinerant conduction electrons to ( quasi- ) localized magnetic moments described by the following model hamiltonian @xcite @xmath24 @xmath25 ( @xmath26 ) and @xmath27 are , respectively , creation and annihilation operators of a band electron being specified by the lower indices . the hopping integrals @xmath28 are connected by fourier transformation to the single - electron bloch energies @xmath29 . the interband ( @xmath30 ) exchange with coupling strength @xmath2 is taken as an intra - atomic interaction between the conduction electron spin @xmath31 and the localized magnetic moment represented by the spin operator @xmath32 . for practical reasons it is sometimes convenient to use the second quantization representation of the band electron spin @xmath31 which leads to the following form of the interband - interaction term : @xmath33 here we have used the abbreviations @xmath34 , @xmath35 and @xmath36 . the first term in ( [ eq : hamil - sf_2 ] ) describes an ising - like interaction between the z - components of the spin - operators , while the second term incorporates spin exchange processes between the localized and the itinerant system . the last two terms are an extension to the original model : @xmath37 leads to the correlated kondo lattice model , it introduces correlations between the conduction electrons in form of a hubbard - type interaction . we will include this term in some parts of the discussion below . the second term , @xmath38 represents a direct spin - exchange between localized moments on different lattice sites . although the @xmath30 exchange can lead to an effective rkky interaction between the spins for non - vanishing band occupation , a `` superexchange''could also be included . in case of an empty conduction band , this term becomes essential as source of magnetic order . @xmath39 are the superexchange integrals . we state once more that the original kondo - lattice model ( or @xmath3-@xmath5 model ) is defined by @xmath40 , only . the additional terms in eq . ( [ eq : hamiltonian ] ) are used as soon as physical requirements do not allow to neglect them . if we are mainly interested in the conduction electron properties then the single - electron green function @xmath41 is of primary interest . its equation of motion reads for the correlated klm @xmath42 the two types of interaction terms in ( [ eq : hamil - sf_2 ] ) let appear the `` spinflip function '' @xmath43 and the `` ising function '' @xmath44 , while the `` hubbard - function '' @xmath45 comes into play only when the `` hubbard - interaction '' ( last term in eq . ( [ eq : hamiltonian ] ) ) is switched on . the three `` higher '' green functions on the right - hand side of eq . ( [ eq : eom ] ) prevent a direct solution of the equation of motion . a formal solution for the fourier - transformed single - electron green function , @xmath46 defines the electronic selfenergy @xmath47 by the ansatz @xmath48_-;c_{{{\bi{k}}}\sigma}^{\dagger } \rangle \rangle_e = \sigma_{{{\bi{k}}}\sigma}(e ) g_{{{\bi{k}}}\sigma}(e)$ ] for the general case @xmath47 can not be determined rigorously . before introducing some approaches to the not exactly solvable many - body problem of the klm let us discuss in the next sections two rather illustrative limiting cases which can help to test the unavoidable approximations . let us assume that the arbitrarily filled conduction band is shrinked to an @xmath49-fold degenerate level @xmath50 : @xmath51 . nevertheless , we consider the @xmath5-spin system as collectively ordered for @xmath52 by any direct or indirect exchange interaction . in this case , the hierarchy of equations of motion decouples exactly and can rigorously be solved @xcite . the resulting energies and respective spectral weights are @xmath53 the `` hubbard - u '' in @xmath54 and @xmath55 indicates that these excitations are bound to a double occupancy of the respective lattice site . @xmath56 and @xmath57 appear when our `` test electron '' enters an empty site . if it orients its spin parallel to the local @xmath5-spin then the energy @xmath56 is needed . in case of an antiparallel spin orientation a triplet or a singlet state is formed . the first requires the energy @xmath56 , the second @xmath57 . the latter is therefore two - fold degenerate . the spectral weights are , contrary to the energy levels , strongly dependent on the magnetization state of the @xmath5 system and the band filling . for a complete solution one needs the average occupation number @xmath58 and the mixed correlation functions @xmath59 and @xmath60 . the evaluation can selfconsistently be done by use of the spectral theorem for the green functions @xmath61 , @xmath62 and @xmath63 ( cf . it is interesting to observe that in any case from the four poles only three do appear . for less than half - filled bands and @xmath1 ( @xmath0 ) @xmath64 ( @xmath65 ) vanishes , and for more than half - filled @xmath66 ( @xmath67 ) does . it should be mentioned that the spectral weights @xmath68 do not explicitly depend on the coupling constants @xmath2 and @xmath69 . that means , on the one side , that even for @xmath70 the @xmath3-@xmath5 interaction produces a splitting into four not coinciding quasiparticle levels . on the other hand , there is a striking dependence of @xmath71 and @xmath58 on the sign of the @xmath3-@xmath5 coupling . that transfers to the spectral weights giving them some indirect dependence on @xmath72 . for @xmath73 and @xmath21 the order of the energy levels is different resulting via the spectral theorem in different correlation functions . note that the mentioned dependence on @xmath2 concerns only the sign of @xmath2 . the spectral weights are not influenced by the absolute value @xmath74 . there is another very instructive limiting case that can be treated rigorously . it concerns the situation of a single electron in an otherwise empty conduction band at @xmath75 , when the local moment system is ferromagnetically saturated . in this case the coulomb interaction is meaningless , the `` hubbard - function '' @xmath76 is identical to zero . in the zero - bandwidth limit , discussed in the last section , for the @xmath20-spectrum all spectral weights disappear except for @xmath77 , while for the @xmath19-spectrum the levels @xmath56 and @xmath57 survive with @xmath78 and @xmath79 . for finite bandwidth the mentioned special case is that of a ferromagnetically saturated semiconductor @xcite . in this case , the spin-@xmath20 quasiparticle density of states @xmath80 is only rigidly shifted compared to the `` free '' bloch density of states . @xmath81 consequently , the quasiparticle dispersion is undeformed with respect to the bloch energies , @xmath82 . spectral densities ( quasiparticle bandstructures ) of the ferromagnetically saturated semiconductor for @xmath23 and @xmath2 as indicated along several symmetry directions . the left ( right ) column displays the @xmath0 ( @xmath1 ) case , the top - most picture shows the interaction - free spectral density.,scaledwidth=60.0% ] the @xmath19-spectrum is more complicated because a @xmath19-electron has several possibilities to exchange its spin with the antiparallel @xmath5 spins . therefore , the @xmath83-spinflip function is not at all trivial . however , its equation of motion decouples exactly , producing a closed system of equations which can be solved after fourier transformation for the single - electron green function . the corresponding selfenergy @xmath84 reads : @xmath85 @xmath86 are the spin wave energies following from the heisenberg exchange @xmath38 ( cf . ( [ eq : hamiltonian ] ) ) , @xmath87 . @xmath88 is the fourier transform of the exchange integral @xmath39 . usually the spin wave energies will be smaller by about two orders of magnitude than other typical energies of the system as the exchange constant @xmath2 or the bloch bandwidth @xmath89 . as a general result the spectral density @xmath90 consists of two structures corresponding to special elementary excitation processes of the @xmath19 electron . there is a rather broad structure built up by `` scattering states '' which result from magnon emission by the original @xmath19 electron . thereby the excited electron reverses its spin becoming a @xmath20 electron . such a process is possible only if there are @xmath20 band states within reach for the original @xmath19 electron to land after the spinflip . the scattering states therefore occupy the same energy region as the @xmath20-dos ( [ eq : ro_up_magpol ] ) . there is another possibility for the @xmath19 electron to flip its spin . it can also be done by a repeated emission and reabsorption of a magnon by the conduction electron resulting in a `` dressed '' particle propagating through the lattice accompanied by a virtual cloud of magnons . for not too small positive ( negative ) @xmath2 the energy of this `` dressed '' particle lies above ( below ) the scattering spectrum giving even rise to a bound state , i.e. to a quasiparticle with infinite lifetime which we call the `` magnetic polaron '' . outside the scattering region the polaron peak manifests itself as a @xmath91-function . as soon as the peak dips into the scattering part the polaron gets a finite lifetime after which it decays into a @xmath20 electron plus magnon . figure [ fig : mag_pol_spec ] shows the down - spin quasiparticle bandstructure as derived from the respective spectral density as a density plot . the degree of blackening is a measure of the spectral density magnitude . sharp dark lines refer to pronounced peaks in the spectral density representing quasiparticles with long life - time . for weak coupling @xmath92 scattering processes smear out a little bit the `` free '' dispersion but do not lead to strong deformations . however , already for moderate couplings @xmath93 one recognizes for some @xmath94 vectors the appearance of a sharp polaron dispersion . for @xmath1 ( right column ) the magnetic polaron is stable on the high - energy side of the @xmath20 spectrum , for @xmath0 on the low - energy side . in addition the scattering spectrum is clearly visible taking away a great part of the total spectral weight . in the antiferromagnetic klm the magnetic polaron represents the ground state configuration @xcite . for still rather moderate couplings of @xmath95 the polaron dispersion has split off over the full brillouin zone . the magnetic polaron has an infinite lifetime . it is surprising that even the broad scattering structure is obviously bunched together as if it were a rather stable quasiparticle . it is noteworthy to repeat that the results of fig . [ fig : mag_pol_spec ] are exact and free of any uncontrollable approximation . so we have to expect these quasiparticle effects in real systems , too . this holds also for the quasiparticle density of states ( qdos ) plotted in fig . [ fig : mag_pol_dos ] for several exchange couplings @xmath74 . according to eq . ( [ eq : ro_up_magpol ] ) the @xmath20-qdos is only rigidly shifted to higher ( lower ) energies for @xmath0 ( @xmath1 ) . correlation effects appear exclusively in the @xmath19 spectrum . for @xmath96 a band splitting sets in . one of the subbands occupies the same energy region as @xmath80 being therefore built up by the mentioned scattering states . in the ferromagnetic ( @xmath1 ) klm it is the low energy part of the spectrum containing a considerable amount of @xmath19-spectral weight . this is not a specialty of the @xmath23 case or of weak and moderate couplings exhibited in fig . [ fig : mag_pol_dos ] but holds equivalently , e.g. , for @xmath9 and in the strong coupling region ( @xmath97 , see last row in fig . [ fig : mag_pol_dos ] ) . after the band splitting has set in the weights of the two spin-@xmath19 subbands are close to the zero - bandwidth values @xmath98 for the lower , and @xmath99 for the upper part ( @xmath1 ) , i.e. independent of @xmath2 . the very often used assumption , when the klm is applied to the manganites , that the @xmath12 electron orients its spin parallel to the @xmath11-@xmath9 spin @xcite , appears with respect to the exact results in fig . [ fig : mag_pol_dos ] rather questionable . in the antiferromagnetic ( @xmath0 ) klm the low energy quasiparticle subband consists of stable polaron states which build the high energy part in the ferromagnetic ( @xmath1 ) klm . although the @xmath1-qdos and the @xmath0-qdos appear as mirror - images the implied physics turns out to be rather different . it is not possible to directly apply the methods of the dynamical mean - field theory @xcite to the klm . one exception is the case of the classical - spin limit ( @xmath16 ) @xcite , thus removing the quantum nature of the spins . the effect of quantum mechanics is strongest for @xmath23 , but also for @xmath100 , they dramatically influence the spectrum @xcite . the assumption of classical spins for @xmath9 , as done in refs . @xcite , needs therefore careful analysis of the neglected effects . another possibility to derive a dynamical mean - field theory for the kondo lattice model is the fermionization of the localized spin operators as suggested in @xcite . this approach is , however limited to @xmath23 systems ( see also @xcite ) . the first part of our approach will closely follow ref . @xcite , but as discussed below , will then differ from the cited reference . the localized spins in the hamiltonian ( [ eq : hamiltonian ] ) can be expressed in terms of auxiliary fermion operators @xmath101 ( @xmath102 ) @xcite : @xmath103 here , @xmath104 represents the pauli matrices . this is the same transformation that led to ( [ eq : hamil - sf_2 ] ) where it was applied to the conduction electron spin @xmath31 . the `` fermionization '' according to eq . ( [ eq : fermionization ] ) implies the introduction of the constraint : @xmath105 the inclusion of this constraint via a lagrange formalism corresponds to the addition of a hubbard - like interaction term for the @xmath5-fermions to the hamiltonian with the interaction constant @xmath106 @xcite . the one - particle energy of the @xmath5-fermions is located at @xmath107 . the `` fermionized klm '' takes the form : @xmath108 this hamiltonian describes a system of two different kinds of @xmath23 fermions coupled by a local spin - exchange interaction . the @xmath5-fermions are additionally correlated via the hubbard - type of interaction to prevent double - occupancy . this hamiltonian resembles the periodic anderson model ( pam ) @xcite in some way . in fact the only difference in the operator structure is the `` coupling '' between conduction band and @xmath5 state . in the pam , this is simply a kinetic - energy term ( `` hybridization '' ) , whereas here , the two subsystems are coupled by the spin exchange . essentially , the `` fermionized klm '' is a rudimentary version of the general multi - band hubbard model with ( local ) inter - band interaction . for this model , the dmft is discussed by @xcite . it is straightforward to apply the standard methods of the dmft to map model ( [ eq : klm+u ] ) onto an appropriate impurity model with the corresponding self - consistency condition to determine the parameters of the impurity model . this leads to the following hamiltonian for the impurity model ( single - site kondo model , sskm ) with fermionized @xmath5-spins : @xmath109 with only one `` impurity '' site for the @xmath5-fermions . the spin - exchange interaction acts only at this single site denoted by the site - index @xmath110 . it is advisable to express the conduction band part of hamiltonian ( [ eq : sskm-1 ] ) in local space and single out the operators referring to the impurity site @xmath110 : @xmath111 for better readability , we will denote the construction operators at the site @xmath110 with the symbol @xmath112 ( @xmath113 ) , the on - site energy will be denoted @xmath114 . finally we introduce a unitary transformation which diagonalizes the last term of the second line in eq . ( [ eq : transformek ] ) . the transformed fermion operators will be denoted @xmath115 ( @xmath116 ) , the hopping @xmath117 and finally @xmath118 for @xmath119 . in the context of the dmft , one does not need to know the explicit structure of this unitary transformation since @xmath120 and @xmath121 need not be known explicitly either . the parameters of the single - site kondo model will be determined by the self - consistency condition of the dmft ( see below ) . a direct solution of model ( [ eq : sskm-1 ] ) is , to our knowledge not possible . however , it can be further simplified : the hubbard term originating from the constraint ( [ eq : constraint ] ) can be eliminated by simply reversing the `` fermionization '' procedure which led from hamiltonian ( [ eq : hamiltonian ] ) to ( [ eq : klm+u ] ) . the auxiliary fermion operators get replaced by a local spin-@xmath122 operator at the impurity site , and the constraint ( [ eq : constraint ] ) can be dropped . this leads to the final version of the single - site kondo model : @xmath123 this last step , which could be called `` de - fermionization '' , distinguishes our approach from the one used in ref . @xcite and others . this `` de - fermionization '' ensures the exact fulfillment of the constraint ( [ eq : constraint ] ) , which could for example only be kept on average ( @xmath124 instead of @xmath125 ) in @xcite . the parameters determining the conduction band in the hamiltonian ( [ eq : sskm ] ) , namely @xmath120 and @xmath121 have to be specified according to the dmft self - consistency condition : the local conduction band green function ( cf . ( [ eq : gf ] ) ) should be equivalent to the @xmath4-operator green function of the single - site model , @xmath126 : @xmath127 where the right equation follows from the formal solution of the equation of motion of @xmath128 . so instead of the usual definition for the hybridization function , @xmath129 , it has to be determined so that eq . ( [ eq : seco ] ) holds . one will see below that the knowledge of @xmath130 , which can become spin - dependent through this procedure , is sufficient to solve the single - site kondo model ( [ eq : sskm ] ) . its dispersion @xmath120 and the hybridization parameter @xmath121 need not to be determined explicitly . the dmft , i.e. the mapping of the klm onto the single - site model is , except for the limit of infinite spatial dimensions , an approximation , equivalent to the local approximation . in the exactly solvable case of the ferromagnetic semiconductor ( cf . [ sec : magpol ] ) this is equivalent to neglecting the magnon energies which are assumed to be at least one order of magnitude smaller than the energy scales under consideration here , e.g. bandwidth or @xmath2 . next , we will introduce an approximative method to solve the single - site kondo model defined by hamiltonian ( [ eq : sskm ] ) for an arbitrary hybridization function @xmath130 . in the following section , we derive an equation of motion - based method to solve the single - site kondo model ( [ eq : sskm ] ) . since in this section we deal only with this model , we suppress all subscripts distinguishing between quantities in the lattice and the single - site model . starting point is the equation of motion for the @xmath4-green function : @xmath131 where the higher green functions @xmath132 and @xmath133 , corresponding to the green functions @xmath134 and @xmath135 for the lattice case , are introduced . the `` mixed '' green function @xmath136 can be eliminated by investigating its equation of motion : @xmath137 thereby defining the hybridization function @xmath138 . this yields the final equation of motion : @xmath139 eq . ( [ eq : eomgd ] ) looks , except for the @xmath140 in the prefactor of @xmath128 , like the equation of motion of the zero - bandwidth limit . the hybridization function @xmath140 is due to the @xmath121-term in the hamiltonian ( [ eq : sskm ] ) . this term prohibits an exact solution . in fact , the `` hybridization '' via @xmath121 is nothing else than the inter - site hopping which was neglected in the zero - bandwidth limit . this term will force us to make certain approximations in the determination of the higher green functions on the right - hand side of eq . ( [ eq : eomgd ] ) . we will exemplify this using the @xmath141-function . its equation of motion reads @xmath142 where on the one hand , higher `` impurity - site '' green functions are introduced , but on the other hand also a higher `` mixed '' green function , @xmath143 . in analogy to the one - particle mixed green function , eq . ( [ eq : mixedgd ] ) , we use the following substitution : @xmath144 the justification of this procedure can be found in analogy to the `` self - energy substitution '' known from other approximation methods for the klm ( see e.g. @xcite ) by inspecting the spectral representations of the relevant green functions . this reveals that all of them have the same single - particle poles , they differ only in the respective weights given by matrix elements of the type @xmath145 . here @xmath146 are the energy eigenstates and a represents one of the relevant operators : @xmath112 , @xmath115 , @xmath147 or @xmath148 . from eq . ( [ eq : mixedgd ] ) follows that the hybridization function accounts for the differences of the matrix elements of the first two operators ( @xmath112 and @xmath115 ) . assuming that these differences are almost equal to those of the matrix elements built up by the latter two operators leads to the _ hybridization approximation _ ( [ eq : hybridapprox ] ) . this is the only approximation necessary to decouple the hierarchy of equations of motion . in the case of @xmath23 spins , there are only 6 different `` impurity - site '' green functions , whose equations of motion form a closed set of equations . besides the already introduced green functions @xmath128 , @xmath141 and @xmath149 , these are @xmath150 several expectation values , introduced into the theory via the inhomogeneities of the equations of motion can be expressed in terms of the above - mentioned green functions , a self - consistent solution has to be found . at this point , let us comment on the reliability of this approximation . although only one approximation enters our decoupling procedure ( cf . ( [ eq : hybridapprox ] ) ) , it still has to be seen as an uncontrollable approximation meaning that there is no true small parameter . there are two non - trivial limiting cases where the replacement ( [ eq : hybridapprox ] ) becomes exact : the first is the limit @xmath151 , representing the zero - bandwidth klm of sec . [ sec : atomic ] . but already a small , but finite bandwidth in the klm could lead to any ( unknown ) expression for @xmath140 implying that we can not necessarily assume @xmath121 to be small any more . the second limit is the `` classical spin '' limit where the `` spin variable '' @xmath152 has no operator properties any more . here , the replacement of eq . ( [ eq : hybridapprox ] ) reduces to ( [ eq : mixedgd ] ) . however since we are interested in the general case with finite @xmath152 and bandwidth , we have to confirm the trustworthiness of this approximation by a comparison with other well - tested methods . we are now going to introduce three further approximation methods for the klm . these are known from literature , their strengths and weaknesses have been identified . the three methods differ substantially with respect to the theoretical assumptions made for an approximate solution of the klm . common features following from these procedures and the above - introduced dmft scheme should then give some credit of reliability , in particular when they additionally fit the exact limiting cases discussed in sec . [ sec : theo ] . an application of the usual diagrammatic perturbation theory to the kondo model can not be performed due to the absence of wick s theorem . only for low temperatures ( spin - wave approximation ) @xcite or in the classical - spin limit @xcite , this method is applicable . the projection operator formalism of mori @xcite is better suited for the kondo model . it has been used successfully to describe correlation effects in the hubbard model in the weak coupling regime @xcite . the general formula for the second - order contribution @xmath153 can be found there ( eq . ( 3.12 ) in ref . @xcite ) . to allow for a better comparison with the other approximations in this paper , we further approximate the self - energy taking only @xmath94 averaged occupation numbers into account ( local approximation ) . next , we want to introduce a modification of the well - known `` coherent potential approximation '' ( cpa ) @xcite for the klm . the cpa is a standard many - body approach that starts from a fictitious alloy in analogy to the interacting particle system . starting point may be the zero bandwidth limit of sec . [ sec : atomic ] . we think of a four - component alloy each constituent of which is characterized by one of the energy levels @xmath154 in eq . ( [ eq : atomic_energies ] ) . the spectral weights @xmath68 ( [ eq : atomic_energies ] ) are then to be interpreted as the `` concentrations '' of the alloy components as seen by a propagating @xmath155-electron . the fictitious alloy for a @xmath155-electron is built up by the local moments and by the frozen ( @xmath156 ) electrons . the cpa - selfenergy of the @xmath155-electron is found by the well - known formula @xcite . @xmath157 as a consequence of the single - site aspect of the cpa the resulting selfenergy is wave - vector independent . according to eq . ( [ eq : atomic_energies ] ) the `` concentrations '' @xmath68 depend on a sum of the `` higher '' correlation functions @xmath62 and @xmath63 , which can rigorously be expressed by the single - electron green function . @xmath158 the shortcomings of the cpa - procedure lie on hand . the one is the same as that in the conventional alloy analogy of the hubbard model , namely the assumption of frozen ( @xmath156 ) electrons . this is partially removed by our proposed modification of the standard cpa procedure for the klm , namely the selfconsistent calculation of the higher correlation functions @xmath159 and @xmath160 via eq . ( [ eq : spectheo_delta ] ) as well as the band - occupation @xmath161 via the spectral theorem for the one - electron green function . maybe even more serious in the case of the klm is the blocking of repeated spin exchange with the local moment system . magnon emission or absorption is not involved . so we can not expect that the cpa - treatment correctly reproduces the exact limiting case of sec . [ sec : magpol ] . however , some general information about the quasiparticle bandstructure might be possible , in particular in the strong coupling ( `` split band '' ) regime . by construction the method yields the correct zero - bandwidth limit . a green function method which takes the spin dynamics correctly into account has been proposed in ref . @xcite . for details about this approach , we refer the read to the cited paper , here we summarize the result shortly this decoupling approach yields finally a selfenergy of the following structure : @xmath162 the first term is linear in the coupling @xmath2 and proportional to the @xmath163 magnetization @xmath164 . it just represents the result of a mean - field approximation being correct in the weak - coupling limit . the second term in ( [ eq : sestruc ] ) contains all the spin exchange processes which may happen . it is a complicated functional of the selfenergy itself . so ( [ eq : sestruc ] ) is not at all an analytical solution but an implicit equation for the selfenergy . we do not present here the lengthy expression for @xmath165 referring the reader for further details to ref . it should be mentioned , however , that @xmath165 contains several expectation values which must be fixed to get a self - consistent solution . no problems arise with the mixed correlation functions @xmath59 and @xmath166 . they can rigorously be expressed by the spin flip function @xmath63 and the ising function @xmath62 defined previously . both functions are already involved in the procedure , so that no further approximations are necessary to fix @xmath167 and @xmath168 . for pure local - moment correlations such as @xmath169 , @xmath170 , , however , a special treatment is necessary , e.g. as described in ref . @xcite . in the following section , we discuss the results obtained for the dmft and the three approximation schemes of sec . [ sec : approx ] for the ferromagnetic kondo lattice model with @xmath23 . since we are interested in the reaction of the conduction band due to the magnetic order of the spin system , we have not calculated the latter self - consistently . instead , we have simulated the magnetic order by determining @xmath171 using a brillouin function . temperatures are given in units of @xmath172 . within the cpa , our choice of @xmath171 can lead to unphysical results . namely in the case of @xmath173 , some of the weights ( [ eq : atomic_energies ] ) can become negative . there is an upper bound for @xmath164 @xcite . we therefore had to limit @xmath164 to @xmath174 for some of the cpa calculations . within the dmft calculations , we experienced severe numerical problems which forced us to introduce a further approximation : `` mean - field''-decoupling the @xmath175 and @xmath176 functions simplifies the system of equations of motion : @xmath177 all dmft - results presented below were obtained using the hybridization approximation in combination with this unrestricted - mean - field decoupling of @xmath175 and @xmath176 . as function of energy . the temperature is set to @xmath178 , therefore @xmath179 ( cpa : @xmath180 , see text ) and the electron density @xmath181 , the chemical potential is at @xmath182 ; solid line : spin-@xmath20 , dotted line spin-@xmath19 dos , scaledwidth=70.0% ] in all calculations , the conduction band is described by a tight - binding dos for a simple - cubic lattice structure @xcite of unit width ( @xmath183 ) . the curie temperature is taken as @xmath184 . the quasiparticle densities of states ( dos ) for quarter - filling and different values of @xmath2 are plotted in figs . [ fig : sf_t0 ] and [ fig : sf_tc ] for @xmath178 and @xmath185 , respectively . as indicated , the columns correspond to dmft , mcda , cpa and sopt , respectively ( from left to right ) . we will begin the discussion with the dmft results shown in the left column of fig . [ fig : sf_t0 ] . a small value of @xmath2 leads to a spin - dependent shift of the spin-@xmath20 and @xmath19 dos , as one would obtain by a simple mean - field decoupling , i.e. by replacing @xmath186 by its mean value @xmath164 in hamiltonian ( [ eq : hamiltonian ] ) . with increasing @xmath2 , the dos show some striking correlation effects : first a broadening , later the onset of a splitting of the band can be observed . whereas in general , the correlation effects are stronger for spin @xmath19 ( indicated by a stronger quasiparticle damping ) , the splitting is , in contrast to the other two methods , more pronounced in the spin-@xmath20 dos . however , here the split - of ( upper ) peak has much less spectral weight than the original peak which , except for a band - narrowing still resembles strongly the free conduction band dos . the picture is very similar in the mcda . again , a mean - field like spin - dependent bandshift is observed for small @xmath2 . on increasing @xmath2 , the spin-@xmath19 dos broadens , and a two - peak structure emerges . the spin-@xmath20 dos remains , except for a small tail at its upper edge , unchanged . this behavior can easily be understood by comparing with the special case of the ferromagnetically saturated semiconductor as discussion in sec . [ sec : magpol ] since the mcda develops continously into this special case for @xmath187 and @xmath188 . the spin-@xmath19 dos splits into a scattering part ( low energies ) and the polaron - like part at higher energies above the conduction band . unlike the @xmath189 case , however , there are some , but weak modification in the @xmath20 dos , namely the above mentioned tail at its upper edge . this can be interpreted as a scattering contribution for spin-@xmath20 electrons . the origin of this is the finite number of spin-@xmath19 electrons . , but for @xmath185 , therefore @xmath190.,scaledwidth=70.0% ] also the cpa dos show a remarkably similar picture as the previously discussed theories . again , there is the mean - field shift for small @xmath2 . for larger values of @xmath2 , the splitting of the spin-@xmath19 dos sets in similarly to the other two methods . here , this can be contributed to the two single - occupancy quasiparticle energies @xmath56 and @xmath57 known from the zero - bandwidth limit ( cf . sec . [ sec : atomic ] ) . for @xmath191 a third quasiparticle energy carries non - vanishing weight , namely the @xmath55 peak . this is located between the two other peaks since we switched off the conduction band coulomb interaction @xmath69 ( cf . ( [ eq : hamiltonian ] ) ) . for @xmath192 , the appearance of this third peak in between the two main peaks is vaguely visible . a remarkable difference to the dmft and mcda results is the fact , that the spin-@xmath20 and scattering part of the spin-@xmath19 dos do not cover the same energy range . this shortcoming of the cpa is due to the neglection of spin - dynamics ( magnons ) as mentioned in sec . [ sec : self - consistent - cpa ] ( cf.ref . @xcite ) . the sopt results are , for the plotted values of @xmath2 in the weak- and intermediate coupling regime , not too far away from the other results . for small @xmath2 , where the sopt becomes by definition reliable , the mean - field shift as in the other methods is clearly observable . similar to the other methods , the deformation of the spin-@xmath19 dos is much stronger than that of the spin-@xmath20 dos . however , for larger @xmath2 , the sopt never shows a true band splitting , its range of validity is certainly restricted . for @xmath185 , where spin - symmetry is re - established , the dmft , mcda and cpa give a similar overall picture . the resulting dos shows , already for @xmath193 , a two - peak structure ( dmft and mcda ) , in the cpa a third peak is dimly noticeable . the most remarkable observation is the near - coincidence of the dmft and mcda results for all @xmath2 values . the transition from the ( clearly distinct ) @xmath178 dos to the ( resembling ) @xmath185 dos is continuous for both theories . the sopt , however , has to be seen as a complete failure for a paramagnetic system already for relatively small @xmath194 . this can already be read of the formula for calculating the self - energy , which is more or less trivial . the corresponding results can not be connected to any of the exactly solvable limiting cases ( cf . [ sec : sopt ] ) . a sound - standing interpretation of the observations is not difficult since the mcda can be traced back to the exactly solved limiting case of the ferromagnetically saturated semiconductor ( cf.sec . [ sec : magpol ] ) and the cpa to that of the zero - bandwidth limit discussed in sec . [ sec : atomic ] . the sopt , of course , becomes reliable for small @xmath2 . although all four presented methods are of approximate nature and their results do , at least for intermediate - to - large values of @xmath2 , differ , some common properties emerge : for small @xmath2 , the behavior is genuine mean - field like , a bandshift proportional to @xmath195 is observed . for larger @xmath2 , the onset of a band splitting occurs . at @xmath178 this primarily affects the spin-@xmath19 dos . this fact is understandable by examination of the ferromagnetically saturated semiconductor where the band splitting was discussed in terms of a scattering band and the magnetic polaron . in that case the spin-@xmath20 dos remains , except for a simple shift , unaffected by the interaction simply because spin - flip of spin-@xmath20 electrons is suppressed in this case . now for finite @xmath196 , this is only approximately true . the spin-@xmath19 dos still shows strong correlation effects , but also the spin-@xmath20 dos is affected due to the finite number of spin-@xmath19 electrons in the system . this manifests itself differently in the various methods : in the mcda , a tail is seen at the upper edge of the spin-@xmath20 dos , in the cpa , a shoulder develops , and in the dmft approach , a band splitting is observable . from this we conclude that correlation effects are much more pronounced in the dmft than in the other two theories . at @xmath185 , the dip indicating the onset of the band splitting is still existing for all but the sopt method . in cpa , mcda and the dmft results this splitting is of similar size , which can easily be read off the cpa where it is simply given by the difference of the respective energies from the zero - bandwidth limit ( see eq . ( [ eq : atomic_energies ] ) ) . the two most pronounced peaks correspond to the single - occupancy quasiparticle energies @xmath56 and @xmath57 , the band splitting is therefore approximately @xmath197 . this is in contrast to the results obtained by dynamical mean - field theory for the @xmath16 klm ( klm with classical spins ) . the emerging picture for classical spins is the following @xcite : at @xmath178 , the dos is characterized by a mean - field like `` zeeman '' splitting between the bands of both spin directions . with increasing temperature , at each of the respective spin-@xmath20 or @xmath19 band , spectral weight of the opposite spin direction appears , until finally at @xmath185 , the system becomes paramagnetic . the splitting into two subbands separated by @xmath198 stays constant . comparing our results with the @xmath16 ones , phenomenologically the dos s show completely different characteristics at @xmath178 , especially in the spin-@xmath19 band . for @xmath185 , the dos s look quite similar . however , the physics behind the scenes turn out to be completely different as can be seen , e.g. , by the inconsistent size of the band splitting @xmath199 . whereas in the classical - spin limit , the splitting is always due to a mean - field like `` on - site zeeman splitting '' , in the case of quantum spins , the various elementary excitations as discussed in the case of the ferromagnetically saturated semiconductor , are responsible for the sub - band structure . for the ferromagnetically saturated system , the neglection of spin - flip processes and magnons in the @xmath16 calculation lead to the picture of a half - metal @xcite . this does not apply for any finite value of @xmath152 and is therefore clearly an artifact of the @xmath16 limit . let us shortly remark on the situation with anti - ferromagnetic coupling ( @xmath0 ) @xcite . all four approximation methods presented here do not show any signs of kondo screening . an investigation of the special low - temperature physics of the model with @xmath0 is therefore not possible . however , some remarks about the behavior of the model for @xmath200 can be made : in general the excitation spectra are broader than in the @xmath1 case . the resulting exchange splitting of @xmath201 is also larger than for @xmath1 which can already be seen in the zero - bandwidth limit . this is again in sharp contrast to the @xmath202 results , where the size of the splitting is independent of the sign of @xmath2 . in this paper , we investigated the kondo lattice model ( klm ) focusing on the model with positive ( ferromagnetic ) exchange constant @xmath2 . we discussed two exactly solvable , but nevertheless non - trivial limiting cases as well as four different approximation methods . the results obtained by these methods , maybe except for the perturbation theory , compare generally reasonably well . sometimes even nearly perfect matches occur ( cf . fig . [ fig : sf_tc ] ) . in general , the differences between the methods are smaller for the paramagnetic than the ferromagnetic system . from the comparison of common features of these approximation methods in combination with the exact results , the following picture emerges for the quasiparticle structure of the ferromagnetic kondo lattice model : for small @xmath74 , a mean - field like shift of @xmath20 and @xmath19 dos is visible . already for intermediate coupling strengths , a band splitting of size @xmath203 occurs . the relevant energy scale for the splitting is @xmath204 . in the case of ferromagnetic saturation ( @xmath178 ) , the splitting is more pronounced in the @xmath19 than in the @xmath20 dos . this band - splitting should not be confused with the splitting found in the limit of `` classical spins '' ( @xmath16 ) . there , the splitting is simply due to a mean - field like zeeman - splitting and therefore of the size @xmath198 . contrary to that , our results clearly show that the two emerging subbands can be traced back to the two elementary excitations known from the limit of the ferromagnetically saturated semiconductor ( see above ) . the magnetic saturation of the spin - system suppresses these processes for spin-@xmath20 electrons which explains the stronger footprint of the correlations in the spin-@xmath19 dos . for finite temperatures , the magnetic polaron generally remains a well defined quasiparticle , represented by a rather sharp lorentzian peak in the spectral density . now the spin-@xmath20 electrons can also participate in spin - exchange processes since the localized spins are not fully aligned any more . the spin - symmetric dos at @xmath205 also show the characteristic splitting . our results also confirm the fundamental differences between the @xmath1 and the @xmath0 case of the kondo lattice model : it can be read of both exactly solvable limiting cases that the ground state will be different depending on the sign of @xmath2 . the approximative approaches of secs . [ sec : dmft ] and [ sec : approx ] do not allow a deeper investigation of the model with @xmath0 due to the inability to reproduce the special low - temperature properties of that model ( `` kondo physics '' ) . another important conclusion from our calculations can be drawn : there is always finite spin-@xmath19 spectral weight in the region of the spin-@xmath20 dos . this spectral weight does not disappear in the limit @xmath206 but only in the limit @xmath16 ( `` classical spins '' ) . for the ferromagnetic klm ( @xmath1 ) , this implies that the klm for @xmath9 and large @xmath2 , corresponding to the situation found in manganites , will not be a half - metal for @xmath178 , contrary to the predictions of @xmath16 calculations . financial support by the _ volkswagen - foundation _ within the project _ `` phasendiagramm des kondo - gitter - modells '' _ is gratefully acknowledged . this work also benefitted from the financial support of the _ sonderforschungsbereich sfb 290 _ ( `` metallische dnne schichten : struktur , magnetismus und elektronische eigenschaften '' ) of the deutsche forschungsgemeinschaft . one of us ( d. m. ) wants to thank the _ friedrich - naumann - foundation _ for supporting his work .

a new `` dynamical mean - field theory '' based approach for the _ kondo lattice model _ with quantum spins is introduced . the inspection of exactly solvable limiting cases and several known approximation methods , namely the _ second - order perturbation theory _ , the _ self - consistent cpa _ and finally a _ moment - conserving decoupling _ of the equations of motion help in evaluating the new approach . this comprehensive investigation gives some certainty to our results : whereas our method is somewhat limited in the investigation of the @xmath0-model , the results for @xmath1 reveal important aspects of the physics of the model : the energetically lowest states are not completely spin - polarized . a band splitting , which occurs already for relatively low interaction strengths , can be related to distinct elementary excitations , namely magnon emission ( absorption ) and the formation of magnetic polarons . we demonstrate the properties of the ferromagnetic kondo lattice model in terms of spectral densities and quasiparticle densities of states .

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Proceed to summarize the following text: our bubble chambers are insensitive to minimum ionizing particles , allowing us to exploit a new calibration technique using charged pions as wimp surrogates to produce nuclear recoils by strong elastic scattering . we measure the pion scattering angle using silicon pixel detectors . the nuclear recoil kinetic energy can be calculated by @xmath26 on an event by event basis , where @xmath27 is the beam momentum , @xmath28 the scattering angle , and @xmath29 the nuclear mass of the target . for a cf@xmath0i target , a measured scattering angle corresponds to a different recoil energy depending on which nucleus is involved in the interaction ; in this paper , we will refer to iodine equivalent recoil energy , @xmath30 , as the energy given to an iodine nucleus for a given pion scattering angle . for a @xmath1 pion beam , approximately @xmath31 of the rate of pions scattering into angles corresponding to @xmath30 between 5 and 20 kev is due to elastic scattering on iodine , with smaller contributions from carbon , fluorine , and inelastic scattering @xcite . therefore , the bubble nucleation efficiency for iodine recoils in a bubble chamber with seitz threshold between 5 and 20 kev can be inferred from a measurement of the fraction of pion - scattering events that nucleate bubbles in the chamber as a function of @xmath30 . the measurement was performed in the fermilab test beam facility @xcite using a @xmath1 mainly @xmath32 beam with @xmath33 and an angular spread of @xmath34 mrad . the absolute momentum of the beam is known to @xmath21 . the pions were tracked with a silicon pixel telescope @xcite consisting of 4 upstream and 4 downstream silicon pixel plaquettes , with a spatial coverage of 14 mm x 14 mm . the total length of the telescope was 90 cm . the angular resolution was 0.6 mrad ( @xmath6 ) in the horizontal ( @xmath35 ) direction and 0.7 mrad in the vertical ( @xmath36 ) direction , with roughly equal contributions from multiple coulomb scattering ( mcs ) in the target and the spatial resolution of the telescope . plastic scintillators triggered the pixel telescope on each beam particle . a small bubble chamber was designed for this measurement consisting of a quartz test tube with inner diameter 10 mm and 1-mm - thick wall , filled with 7 @xmath37 of cf@xmath0i . the small size is required to minimize mcs in the short radiation length of cf@xmath0i ( @xmath38 mm ) . the bubble chamber was operated at a pressure of @xmath39 psia and a temperature of @xmath40 c with a nominal seitz threshold of @xmath2 . the iodine equivalent threshold scattering angle is 4.7 mrad . an acoustic transducer was attached to the top of the test tube to record the acoustic emission produced during bubble formation , providing the time of bubble nucleation with @xmath2010 @xmath41s resolution . temperature control was provided by a water bath around the bubble chamber . bubble chamber data were taken between march 14 and march 28 , 2012 , with a beam flux of @xmath201000 particles per 4-second beam spill with one spill per minute . the size of the beam spot was wider than both the bubble chamber and the pixel telescope . the chamber was expanded to the superheated state 22 seconds before the arrival of the beam , allowing time for pressure and temperature transients to dissipate after expansion . the observation of bubbles by a 100-hz video camera system created a bubble chamber trigger , causing the video images and associated data to be recorded and the chamber to be recompressed . after recompression , the chamber was dead for the remainder of the beam spill , allowing us to collect at most one bubble event per minute . we collected about four good single - bubble events per hour , with the primary losses due to premature bubble chamber triggers , bubbles forming outside of the region covered by the telescope planes , multiple bubble events and large - angle scatters outside the acceptance of the downstream plaquettes . the last two categories are predominantly the result of inelastic interactions . figure [ fig : signals ] shows an example scattering event . at the end of the run the cf@xmath0i was removed and a target empty data set was taken . in addition , data were taken in a test run in december 2011 with no target , as well as solid targets of quartz , graphite , teflon or ( c@xmath42f@xmath24)@xmath43 , and crystalline iodine . ( color online ) an example event ( @xmath44 mrad ) , including the relative timing of the telescope trigger and acoustic signal , one camera image of the bubble , and the @xmath36 and @xmath45 positions of the telescope hits . the pion beam is in the @xmath46 direction . the camera image is not to scale but the test tube has inner diameter of 10 mm . beam tubes in the water bath to minimize the material traversed by the pion beam are visible to either side of the bubble chamber . ] the primary analysis output is the bubble nucleation fraction as a function of @xmath30 , given by the ratio @xmath47 where @xmath48 is the observed number of pion tracks creating single bubbles , @xmath49 is the total number of pion tracks , @xmath50 is the number of tracks creating multiple bubbles , and @xmath51 is the fraction of scatters that occur in the active cf@xmath52i volume , determined by a comparison of the number of scatters in the target - full data set to the number in the target - empty data set normalized to the number of pion tracks ( @xmath53 ) : @xmath54 an angular smearing correction is made to @xmath53 to include the mcs from the absent cf@xmath0i by convolution with the standard gaussian approximation for mcs @xcite . each pion track is fitted for an upstream and downstream component , with an associated scattering angle and 3-d point of closest approach of the two components . the upstream and downstream track segments are required to have exactly one hit cluster in at least three of the four pixel planes , good fits to straight lines ( @xmath55 ) , and to meet in space to within @xmath56 mm . to exclude pions that passed through little or no cf@xmath0i , the upstream track is required to pass within @xmath57 mm of the center of the @xmath58-mm - diameter bubble chamber in the @xmath35 direction . the @xmath36 location of the track is limited by the vertical extent of the pixel planes . because the uncertainty on the location of the point of closest approach in the beam direction ( @xmath45 ) depends strongly on the scattering angle , we require the @xmath45 location to be within 3@xmath59 of the bubble chamber , where @xmath59 is the uncertainty on @xmath45 for each individual event . events with more than one track are rejected . as these track cuts are applied without regard to the presence of nucleations in the bubble chamber , their efficiency applies equally to @xmath48 , @xmath49 , @xmath50 , and @xmath53 , and therefore cancels in the final ratio , @xmath60 . the next step is to associate a bubble with a unique track using both time and space correlations . the timing requirement for correlating a track with a bubble is chosen to be @xmath61 @xmath41s . the bubble locations are reconstructed using standard coupp techniques @xcite , and the difference between reconstructed bubble position and point of closest approach of the track components is required to be less than 2.1 mm in the @xmath35 direction and less than 0.9 mm in the @xmath36 direction . the combined event acceptance of these timing and spatial cuts is @xmath62 . after these data selection and quality cuts , @xmath63 good single bubble events remain . the final bubble nucleation fraction , @xmath64 , is shown as the points in fig . [ results ] . ( color online ) the fraction of pion scattering events that produced bubbles as a function of iodine equivalent recoil energy . the solid curves show the simulated contribution from individual recoil species ( from high to low at 20 kev , red for iodine , green for fluorine , and pink for carbon and inelastics ) , with the blue dashed curve showing the sum . the iodine curve shown takes a step function efficiency model for iodine recoils using the best fit threshold of @xmath65 . ] to disentangle the iodine component from carbon , fluorine and inelastic scattering , we perform a full simulation using geant4.9.5 @xcite . the simulation was validated by comparing the simulated scattering angular distributions to data for no target , target empty , target full , and the solid targets . in all cases , in the mcs - dominated small scattering angle region there is good ( few percent ) agreement with no adjustable parameters , suggesting that the telescope geometry is accurately modeled in the simulation . in the larger scattering angle region dominated by strong elastic scattering , the simulation systematically overestimates the observed scattering rate by @xmath66 . as this ratio is measured to be the same for teflon ( @xmath67 ) and iodine ( @xmath68 ) , we assume that the relative contributions of iodine , fluorine and carbon are being accurately described by the mc . a significant systematic uncertainty is introduced by our developing understanding of the carbon and fluorine recoil nucleation efficiency in this low energy regime . ongoing studies with ad hoc neutron sources @xcite will reduce this uncertainty in the future , but here we apply the exponential carbon and fluorine efficiency model from @xcite : we test the hypothesis that the iodine recoil nucleation efficiency follows the nominal seitz model of a step function ( @xmath72 ) with 100% efficiency above the seitz threshold by fitting a step function to the data in the region @xmath73 kev , allowing @xmath18 to float . the fit returns @xmath74 , where the error bars are statistical . this value is 2.1@xmath6 higher than the seitz model threshold @xmath75 , where the systematic error includes absolute energy scale uncertainties of 3% in the beam momentum and 1% in the scattering angle stemming from uncertainty in the @xmath45 positions of the plaquettes . the fit is shown as the dashed blue line in fig . [ results ] . figure [ contour ] shows the inferred iodine nucleation efficiency as a function of @xmath30 with the iodine component isolated by subtracting the simulated contributions from carbon , fluorine and inelastic scatters . the dashed blue curve is the best fit step function with @xmath74 . for comparison , the red region represents a step function at the predicted seitz threshold , where the range represents the 1@xmath6 band including the thermodynamic uncertainty and the scale uncertainties in the absolute energy scale of the experiment . given the energy resolution smearing induced by mcs in this experiment , the preference for a value of @xmath18 higher than the prediction can not be easily distinguished from an exponential model like eq . ( [ eq : expo ] ) for iodine nucleation efficiency with a lower threshold energy and a finite value of @xmath76 . previous studies have shown that the seitz model accurately predicts the threshold at which bubble nucleation begins for heavy radon daughter nuclei in cf@xmath0i @xcite . we therefore perform a second fit applying the exponential model to iodine recoils , taking the seitz threshold calculation as an external input to the analysis to explore the allowed range of @xmath76 . the best fit is shown as the black curve in fig . [ contour ] . the inset shows 2@xmath6 contours for fits to the exponential model with the threshold constrained by our prediction ( shaded region ) and free ( unshaded region ) , along with the best fit points . ( color online ) the data points represent the measured iodine nucleation efficiency as a function of iodine equivalent recoil energy , where the contributions from carbon , fluorine and inelastic scatters have been subtracted . the gradual turn on is predominantly due to the angular resolution of the experiment , as illustrated by both the red region , representing the step function model with the threshold varied within the uncertainty on the seitz theory prediction , @xmath77 , and the dashed blue curve , representing the best fit step function with @xmath78 kev . the black curve shows the best fit exponential model with the threshold constrained by the theory as described in the text . the inset shows 2@xmath6 contours for a fit to the exponential model with the threshold allowed to float ( pink ) or constrained by the theory ( solid cyan ) . the colored dots represent the corresponding curves in the main plot . ] to assess the systematic errors associated with carbon and fluorine recoils , we refit the data with two alternative models for carbon and fluorine efficiency : the flat model from @xcite with energy - independent nucleation efficiency , @xmath79 , above threshold , and a step function with @xmath80 . the latter case represents the worst possible scenario for the response of the bubble chamber to iodine recoils , as the response to carbon and fluorine is maximized . we use the exponential model for iodine recoils , allowing @xmath76 to float and treating @xmath18 as a nuisance parameter constrained by the prediction . the results of these fits are summarized in table [ tab : summary_alpha ] . extended fits over the energy interval @xmath81 kev have a negligible effect on the iodine fit parameters but disfavor the flat c / f efficiency model with @xmath79 . .summary of fits to @xmath76 , including @xmath82 lower limits on @xmath76 . the three different c / f efficiency models described in the text are tested , and in all cases the predicted seitz threshold is treated as a nuisance parameter . by maximizing the subtracted c / f contribution , the step function with @xmath80 represents the worst case for iodine efficiency . [ cols="<,^,^,^",options="header " , ] in conclusion , we have directly measured the efficiency for iodine bubble nucleation in a cf@xmath0i bubble chamber operated with a nominal threshold of @xmath2 . for some models of carbon and fluorine efficiency , the response to iodine recoils is consistent with a step function at the seitz threshold , but in all cases there is a preference for either a softer turn on or a slightly higher threshold . even in the worst case scenario for iodine , however , the response of the chamber to iodine recoils is much closer to the nominal seitz model than it is for carbon and fluorine recoils . this was expected from the considerably larger stopping power of iodine , which facilitates the concentration of energy that leads to critical bubble formation . systematic uncertainties from both the absolute beam momentum calibration and the carbon and fluorine response limit the present measurement . this measurement provides confirmation of the sensitivity of coupp bubble chambers to spin - independent wimp interactions with iodine nuclei , a confirmation that was not attainable using standard neutron source calibrations . the technique of employing hadron elastic scattering as a tool to measure bubble nucleation thresholds is now established , enabling the measurement of bubble nucleation energies on an event by event basis . we have begun studies of the feasibility to repeat this technique with different fluids . the coupp collaboration would like to thank fermi national accelerator laboratory , the department of energy and the national science foundation for their support including grants phy-0856273 , phy-1205987 , phy-0937500 and phy-0919526 . we acknowledge technical assistance from fermilab s accelerator , computing , and particle physics divisions , and from a. behnke at iusb .

we have directly measured the energy threshold and efficiency for bubble nucleation from iodine recoils in a cf@xmath0i bubble chamber in the energy range of interest for a dark matter search . these interactions can not be probed by standard neutron calibration methods , so we develop a new technique by observing the elastic scattering of @xmath1 negative pions . the pions are tracked with a silicon pixel telescope and the reconstructed scattering angle provides a measure of the nuclear recoil kinetic energy . the bubble chamber was operated with a nominal threshold of @xmath2 . interpretation of the results depends on the response to fluorine and carbon recoils , but in general we find agreement with the predictions of the classical bubble nucleation theory . this measurement confirms the applicability of cf@xmath0i as a target for spin - independent dark matter interactions and represents a novel technique for calibration of superheated fluid detectors . recent years have seen a resurgence in the use of superheated liquids and bubble chambers as continuously sensitive nuclear recoil detectors searching for dark matter in the form of weakly interacting massive particles ( wimps)@xcite . at a low degree of superheat , bubble chambers are insensitive to minimum ionizing backgrounds that normally plague wimp searches but retain sensitivity to the nuclear recoils that would be characteristic of wimp scattering . in a superheated liquid the process of radiation - induced bubble nucleation is described by the classical `` hot spike '' model @xcite . for the phase transition to occur , the energy deposited by the particle must create a critically sized bubble , requiring a minimum energy deposition in a volume smaller than the critical bubble . under mildly superheated conditions , the latter requirement renders the bubble chamber insensitive to minimum ionizing particles . the radius of the critical bubble is given by the condition that the bubble be in ( unstable ) equilibrium with the surrounding superheated fluid @xcite . this demands the pressure balance @xmath3 where @xmath4 is the pressure inside the bubble , @xmath5 is the pressure in the liquid , @xmath6 is the bubble surface tension , and @xmath7 is the critical bubble radius . the pressure @xmath4 is fixed by the condition that the chemical potential inside and outside the bubble be equal , giving @xmath8 where @xmath9 is the pressure in a saturated system at the given temperature , and @xmath10 and @xmath11 are the liquid and vapor densities in the saturated system @xcite . in seitz s `` hot spike '' model for bubble nucleation , the entire energy necessary to create the critical bubble must come from the particle interaction that nucleates the bubble . this is in contrast to earlier models that required only the work ( free energy ) to come from the particle interaction , with the remaining bubble - formation energy supplied by heat flowing in from the surrounding superheated fluid @xcite . as the name `` hot spike '' implies , the nucleation site in seitz s model begins as a high - temperature seed , so it can not draw heat from the surrounding fluid . once the decision is made to consider the total bubble creation energy rather than just the free energy , the threshold energy calculation is completely described by gibbs @xcite . this energy is given by @xmath12 here , @xmath13 is the temperature of the system , @xmath14 is the bubble vapor density , and @xmath15 and @xmath16 are the specific enthalpies of the bubble vapor and superheated liquid . the surface tension @xmath6 and temperature derivative are taken along the usual saturation curve . the three terms give , from left to right , the heat necessary to create the bubble surface , the heat needed to vaporize the fluid to make the bubble interior , and a reversible work term done in expanding the bubble to the critical size that must be subtracted to avoid double - counting work present in both of the first two terms . to good approximation @xmath17 may be replaced by the normal heat of vaporization at temperature @xmath13 . the greatest uncertainty in determining the thermodynamic @xmath18 is the relation between the surface tension at a flat liquid - vapor interface and the surface tension for a very small bubble . this relation is described by the `` tolman length '' @xmath19 , which is unknown but is expected to be some fraction of the intermolecular distance @xcite . this translates to an uncertainty on @xmath18 of @xmath20@xmath21 . for the rest of this paper , we refer to the calculated threshold in eq . ( [ eq : seitz ] ) as the seitz threshold . the seitz model assumes the efficiency for bubble nucleation is @xmath22 for all interactions that deposit @xmath23 over a volume small compared to the critical bubble . the length scales for nuclear recoil cascades in the energy region between 5 and 20 kev relevant for a wimp search are similar to the critical radius , so the seitz model may or may not give a good description of bubble nucleation , and direct calibrations of bubble nucleation efficiency are necessary . the working fluid discussed in this paper is iodotrifluoromethane or cf@xmath0i , which contains two highly sensitive wimp target nuclei : fluorine , for spin - dependent interactions , and iodine , for spin - independent interactions . neutrons are typically used to mimic wimps in calibrating the nuclear recoil response of a wimp detector , and neutron sources have been used to measure the nucleation threshold for carbon and fluorine recoils in cf@xmath0i , cf@xmath0br @xcite and c@xmath24f@xmath25 @xcite under various superheat conditions . however , iodine recoils contribute only a small fraction to the total neutron - nucleated bubble rate in cf@xmath0i . therefore , neutron sources are an ineffective calibration tool for iodine recoils in coupp . we have used heavy daughter nuclei produced in alpha decays as a proxy @xcite , but these are high energy recoils of @xmath20100 kev . this paper describes a measurement of bubble nucleation efficiency for iodine recoils near our dark matter search thresholds .

You are an expert at summarizing long articles. Proceed to summarize the following text: recent advances in extragalactic astrophysics show the close link of a disk galaxy dynamical evolution and its chemical and photometric behaviour over the hubble timescale . in spite of remarkable succes of the modern theory of galaxy chemical evolution in explaining the properties of evolving galaxies ( @xcite ) its serious shortcomings concern the multiparameter character and practical neglecting of dynamical effects . the inclusion of simplified dynamic into the chemical network ( @xcite ) and vice versa the inclusion of simplified chemical scheme into the sophisticated 3d hydrodynamical code ( @xcite ) gives very promising results and allows to avoid a formal approach typical to standard theory . in this paper the interplay between a disk galaxy dynamical evolution and its chemical behaviour is studied in a frame of a simplified model which provides a realistic description of the process of galaxy formation and evolution over the cosmological timescale . the evolving galaxy is treated as a system of baryonic fragments embedded into the extended halo composed of dark nonbaryonic and baryonic matter . the halo is modelled as a static structure with dark ( dh ) and diluted baryonic ( bh ) halo components having plummer type density profiles ( @xcite ) : @xmath0 and @xmath1 where @xmath2 @xmath3 the dense baryonic matter ( future galaxy disk and bulge ) of total mass of @xmath4 is assumed to be distributed among @xmath5 particles fragments . the single particle density profile is also assumed to be a plummer one . its mass is taken to be @xmath6 and radius @xmath7 kpc . initially all particles are smoothly placed inside the sphere of radius @xmath8 kpc and are involved into the hubble flow ( @xmath9 km / s / mpc ) and solid body rotation around @xmath10 axis . the initial motion of this system is described as : @xmath11 + h_{0}\cdot { \bf r } + { \bf dv}(x , y , z),\ ] ] where @xmath12 is an angular velocity of the rotating sphere , @xmath13 and @xmath14 the components @xmath15 , @xmath16 , @xmath17 of the random velocity @xmath18 vector are assumed to be initially randomly distributed within an interval @xmath19 km / s . the dynamical evolution of baryonic matter fragments which are subjected to gravitational influences of dm baryonic halo and interfragment interactions is followed by means of effective n body integrator with individual time step . the dynamics of such n body system is described by following equations : @xmath20 the acceleration of @xmath21 - th particle @xmath22 is defined as a sum of three components . @xmath23 where the first term @xmath24 accounts for gravitational interactions between fragments . the second one @xmath25 is defined as an external gravitational acceleration caused by the dm and baryonic halo . the last term @xmath26 corresponds to the viscous decceleration of fragment when passing through the baryonic halo . the gravitational interaction between fragments is defined as the interaction of @xmath27 plummer profile elements : @xmath28 here @xmath29 and @xmath30 . accounting for that the halo dm and baryonic components are also plummer spheres the second term becomes : @xmath31 the form of the last term @xmath26 will be discussed in the next subsection . the characteristic time step @xmath32 in the integration procedure for each particle is defined as : @xmath33 \label{eq : def_dt}\ ] ] where the @xmath34 and the @xmath35 . here the @xmath36 is a numerical parameter equal to @xmath37 that provides a nice momentum and energy conservation over the integration interval of about @xmath38 gyr . for example , in the conservative case ( when viscosity of the system is set equal to @xmath39 ) , the final total error in the energy equation is less than @xmath40 . the viscosity term @xmath26 is artificially introduced into the model so as to match the results of more sophisticated sph approach on dynamical evolution of disk galaxies . the best fitness of results of this simplified approach with sph modelling data ( see @xcite ) is achived when the momentum exchange between the baryonic halo and moving particles is modelled by the following expression : @xmath41 here @xmath42 and @xmath43 are numerical parameters set equal to @xmath44 @xmath45 and @xmath46 kpc . the vector @xmath47 is a particle velocity vector . it is to be noted that single particle is assumed to have a total mass which does nt change with time and is defined as sum @xmath48 . but masses of its gas and star components @xmath49 and @xmath50 are variable values and are defined by the temporal evolutionary status of the given fragment . initially @xmath51 . the results of fitness show that for @xmath10 component of viscosity term @xmath52 . in the galaxy plane where it is necessary to account for baryonic halo and baryonic fragments partial corrotation the dynamical friction is decreased and for @xmath53 component of viscosity term @xmath54 reduced to the value @xmath55 . in the frame of the multifragmented model the definition of local gas density is introduced in the sph manner , e.g. local gas density depends on the total mass of matter contained in the sphere of radius @xmath56 around the @xmath21 th particle . for each @xmath21 th particle the value of its smoothing radius @xmath57 is chosed ( using the quicksort algorithm ) requiring that the volume of such radius compraises @xmath58 nearest particles ( i.e. @xmath59 1 % of total number of particles @xmath27 ) . therefore , the total mass @xmath60 and density of gas @xmath61 inside this sphere are defined as @xmath62 where @xmath63 is defined as @xmath64 @xmath65 @xmath66 a forming disk galaxy is modelled as a system of interacting fragments ( named as particles ) embedded into the extended halo . each particle is composed of gas and stellar components and its total mass is defined as @xmath67 . initially all particles are purelly gaseous and , therefore , initially @xmath68 . to follow a particle star formation ( sf ) activity a special timemark @xmath69 is introduced which initially is set equal to @xmath70 . the particles eligible to star formation events are chosen as particles which still have a sufficient amount of the gas component and their densities exceed some critical value @xmath71 during some fixed time interval @xmath72 ( of order of free fall time ) : @xmath73 here @xmath74 myr , and @xmath75 @xmath45 ( this last value is not crucial and is only limiting one ) . if the particle was subjected to sf activity the parameter @xmath76 is set equal to @xmath77 , , and @xmath78 and @xmath79 are redefined as @xmath80 here @xmath81 is a sf efficiency which is defined as @xmath82 therefore , @xmath83 @xmath84 to match available observational data on star formation efficiency ( see @xcite ) parameters @xmath85 and @xmath86 are set equal to @xmath87 and @xmath88 . the chemical evolution of each separate fragment is treated in the frame of one zone close box model with instantaneous recycling . in the frame of this approach for the returned fraction of gas from evolved stars is taken a standard value @xmath89 . following the instantenuous recycling approximation it is assumed that after each sf and sn explosions the heavy element enriched gas is returned to the system and mixed with old ( heavy element deficient ) gas . after each act of sf and sn explosions the value of heavy element abundances of gas in particle is upgraded according to : @xmath90 the value @xmath91 is used as an average value for all values of @xmath92 ( see @xcite ) . initially @xmath93 in all particles . the proposed simple model provides the self - consistent picture of the process of galaxy formation , its dynamical and chemical evolution is in agreement with the results of more sophisticated approaches ( see @xcite ) . * the rapidly rotating protogalaxy finally formed a three - component system resembling a typical spiral galaxies : a thin disk and spheroidal component made of gas and/or stars and a dark matter halo . + fig.1 and fig.2 show respectively the star formation rate and the galaxy total stellar mass as a function of time . fig.3 shows the cylindrical distribution of stellar ( upper curve ) and gaseous ( lower one ) components of the final model disk galaxy as a function of a distance from the galaxy center in the galactic plane . + the total star formation rate ( sfr ) is a succesion of short bursts which does nt exceed @xmath94/yr . during first @xmath95 gyr of evolution only about @xmath96 of total galaxy mass is transforms into the stars . the sfr gradually decreases , during the further evolution , to the value of about @xmath97/yr typical for our own galaxy . the final total stellar and gas mass of the model galaxy disk are about @xmath98 and @xmath99 . all these data as well as surface densities distributions of stellar and gaseous components ( see fig.4 ) are in nice agreement with present date observational data ( @xcite ) . * the disk component posesses a typical spiral galaxy rotation curve and the distribution of radial and @xmath100 - th velocities of baryonic particles clearly show the presence of the central bulge ( see fig.6 and fig.7 ) . * the metallicities and the global metallicity gradient resemble distributions observed in our own galaxy ( fig.5 ) . the averaged observed value of global metallicity @xmath101 ( see @xcite ) is shown in this fig . as a solid line . * acknowledgements : * peter berczik would like to acknowledge the american astronomical society for financial support of this work under international small research grant . figure 1 : the variation of total star formation rate of forming disk galaxy with time . figure 4 : the surface density radial distributions of stellar and gaseous components ( stellar component is shown by filled stars , gaseous one by astericks ) . theoretical distributions ( not scaled ) of surface density for radial exponential scale lengthes @xmath102 and @xmath103 kpc are shown below by lines .

the evolving galaxy is considered as a system of baryonic fragments embedded into the static dark nonbaryonic ( dh ) and baryonic ( bh ) halo and subjected to gravitational and viscous interactions . although the chemical evolution of each separate fragment is treated in the frame of one zone close box model with instantaneous recycling , its star formation ( sf ) activity is a function of mean local gas density and , therefore , is strongly influenced by other interacting fragments . in spite of its simplicity this model provides a realistic description of the process of galaxy formation and evolution over the hubble timescale .

You are an expert at summarizing long articles. Proceed to summarize the following text: beams of light can carry quantized angular momentum , @xmath0 , that are typically referred to as having a photonic topological charge ( ptc ) of @xmath1 @xcite . circularly polarized light carries a spin angular momentum ( sam ) of @xmath2 and so has ptc of @xmath3 , but beams can also be created that twist about their axis of propagation @xcite causing them to be referred to as _ vortex beams _ or simply _ twisted light _ @xcite . at the macro - scale , their interaction with matter generates torque which can be used to manipulate rings of atoms @xcite and particles @xcite , carry out fabrication @xcite , provide control for spintronics @xcite , and for many other applications as well @xcite . at the atomic scale , the absorption of such photons can change the angular momentum state of bound electrons , only recently confirmed experimentally @xcite . the ptc of twisted light also amounts to an additional degree of freedom for carrying information content , and ptc values in excess of @xmath4 have been experimentally realized @xcite . this information perspective on ptc is particularly relevant to emerging technologies in communications , computing and quantum cryptography @xcite , classical data transfer @xcite , quantum key distribution@xcite , quantum entanglement@xcite , and quantum cloning@xcite . there are a variety of ways in which photonic angular momentum can be generated . linearly polarized light can be transformed to a circular polarization by exploiting material birefringence in the form of a quarter - wave plate giving it sam and a topological charge of @xmath3 . only slightly more complicated to carry out , light beams with orbital angular momentum ( oam)i.e . twisted light can be generated using standard optical elements such as spiral phase plates @xcite , computer generated holograms @xcite , and spatial light modulators @xcite . beams with sam can also be converted to those with oam . staged spiral phase plates @xcite , uniaxial crystals @xcite , liquid crystals @xcite , q - plates@xcite , and even metamaterials @xcite have all been demonstrated to produce such conversions . changing the angular momentum of photons that already have a topological charge is less standard though . such manipulations currently exploit higher - order susceptibilities to up / down convert oam along the same lines as is carried out to alter the wavelength of light . for instance , topological charge doubling via second harmonic generation ( shg ) has been demonstrated @xcite , and ptc addition operations have also been realized using shg@xcite . topological charge can even be transferred from a pump beam to a down - converted idler beam via optical parametric oscillation ( opo ) @xcite . along the same lines , four - wave - mixing ( fwm ) with atomic vapors can result in output beams that have either the sum of the input ptcs or no ptc at all @xcite . in each of these nonlinear processes , charge conversion is essentially instantaneous and involves two vortex beam inputs that must illuminate the target material simultaneously . a variation of fwm , applied to bulk crystals , generates output beams with ptcs that are linear combinations of those of the input beams @xcite . intriguingly , there is a time delay between pump and probe implying that ptc is being temporarily stored in the electronic structure of the bulk crystal , but the nature of this structure was not elucidated . here we consider an alternative means of manipulating ptc that exploits the properties of a special type of molecular exciton . the requisite structures have a discrete rotational symmetry group of either @xmath5 or @xmath6 with respect to the propagation direction of light . such n - arm molecules have been previously identified as having excitonic states that can relax by emitting photons with ptc @xcite . we extend this work by first introducing the notion of an _ excitonic topological charge _ ( etc ) that can be generated with azimuthally polarized light @xcite . the resulting electronic state of the molecule will be referred to as a _ twisted exciton_. since both spin and circularly polarized vortex beams can be decomposed into a linear combination of radial and azimuthal vector vortices@xcite , they can create twisted excitons too . the idea is then generalized to consider the sequential excitation of molecules with pulses of light that carry either sam or oam . it is shown that the sum of ptc and etc is conserved in such processes resulting in a simple algebra in which etc can be raised and lowered using photons . subsequent emission , either spontaneous or stimulated , produces photons with the accumulated topological charge of the excitons . in this way , the excitons play the role of an angular momentum bank in which photonic charge can be deposited and withdrawn in different increments . this is illustrated in figure [ calculator ] . within a linear setting , the fact that the real momentum of absorbed photons is transferred to crystal momentum is a foundational plank of solid - state physics @xcite . an analogous discrete - space application of noether s first theorem has also be used to show that real angular momentum can be transferred to crystal angular momentum @xcite . this provides a conceptual basis for understanding the topological charge algebra that can be carried out with molecules . in the molecular setting , periodicity is azimuthal and lattice points are associated with identical molecular arms . etc quantifies the molecular analog of crystalline angular momentum . we first use a tight binding ( tb ) paradigm to show that topological charge is conserved in the molecular setting . this paradigm is used to demonstrate topological charge algebra using sequential laser pulses . the associated hamiltonian is subsequently replaced with one that does not rely on prescribed transition dipoles and for which electron excitations are treated as many - body events with exchange energies and correlation effects included . twisted excitons are no longer just superpositions of two - level excitations on each molecular arm , but topological charge conservation emerges none - the - less . this time - domain density functional theory ( td - dft ) setting is used to simulate the dynamics of topological charge transfer between photonic and excitonic manifestations , and simple charge additions and subtractions are once again demonstrated . in both tb and td - dft settings , the underlying processes are linear in the sense that only first - order electric dipole interactions are necessary . this is in contrast to the nonlinear optics strategy for converting up / down converting angular momentum using higher order susceptibilities . first consider a molecular hamiltonian in the absence of light - matter coupling . the requisite @xmath5 or @xmath6 symmetry is provided by an n - arm molecule in which arm @xmath7 supports two energy levels : ground state @xmath8 and excited state @xmath9 . the tight - binding ( tb ) hamiltonian is taken to be @xmath10 here @xmath11 is the excited state energy of each arm , @xmath12 is the coupling between nearest arms , and @xmath13 is the creation operator for arm @xmath7 . it is straightforward to show @xcite that the ground state is @xmath14 while the n excited states are @xmath15 with @xmath16 and @xmath17 . the etc , @xmath18 , is an integer ranging from @xmath19 to @xmath20 . the corresponding energies are @xmath21 with @xmath22 for @xmath23 and @xmath24 for @xmath25 . a hollow @xmath26 will be used to distinguish exciton energy from electric field , @xmath27 . now introduce semi - classical light - matter coupling via two hamiltonians : @xmath28 that governs light - mediated interactions between the ground state and each molecular eigenstate , while @xmath29 governs the analogous laser interactions that can cause transitions between eigenstates . the angular momentum of incident electric fields may be manifested as a circular polarization , a vector vortex , or linear polarization with a scalar vortex , but we restrict attention to the first two types . an electric dipole approximation is made , and the discrete rotational symmetry ensures that a rotation of the molecule about its axis by @xmath30 maps one dipole into the next . under these conditions , the details of electric field structure and dipole orientations are irrelevant , and the light - matter interactions are well - captured by the following hamiltonians , which are functions of the topological charge of the incident light , @xmath31 : @xmath32 the _ mod _ function returns its argument modulo @xmath33 and use has been made of the fact that @xmath34 . the scalars , @xmath35 and @xmath36 , represent the inner product of electric transition dipole moments with a time - dependent electric field . the total hamiltonian , @xmath37 , is then applied to the schr " odinger equation with solutions assumed to be of the form @xmath38 this results in a set of @xmath39 coupled ode s that can be solved numerically for a prescribed electric field and initial state . the evolving state can then be projected onto each excitonic eigenstate to determine the population of each topological charge state , @xmath18 , as a function of time : @xmath40 within the tb setting , conservation of energy implies conservation of topological charge in association with light - matter interactions . to see this , suppose that the molecule is initially in eigenstate @xmath41 , where subscript @xmath42 indicates an initial etc of @xmath43 . a beam with ptc of @xmath31 , is incident on the molecule , and the molecule is excited into eigenstate @xmath44 as a result . subscripts @xmath45 and @xmath46 delineate photonic and excitonic manifestations while @xmath47 is the etc of the final state . a necessary condition for this transition to occur is that @xmath48 . assuming that @xmath49 , equations [ tbstates ] and [ h12 ] imply that @xmath50 which can be easily reduced to @xmath51 the following cyclic sum orthogonality property of periodic exponentials is then useful : @xmath52 it is applied to both terms in equation [ twoterms ] to give @xmath53 energy conservation implies that exactly one of the two kronecker delta functions will be nonzero . we therefore have the following statements of topological charge conservation : @xmath54 if the initial exciton energy is lower than that of the final state , crystal angular momentum will be removed by the light . the converse is also true . since the sign of the ptc can be either positive or negative , this allows for a number of ways in which photonic angular momentum quanta can be used to manipulate the level of excitonic angular momentum . it also offers a strategy for withdrawing angular momentum from the molecule in a range of denominations . this is next illustrated in two applications . as a first proof - of - concept , a 7-arm molecule with an etc of @xmath55 is subjected to windowed , continuous wave ( cw ) , azimuthally polarized lasers with angular momenta of @xmath56 and the following scalar waveform : @xmath57 as usual , @xmath42 and @xmath58 are the initial and final topological charges of the exciton with corresponding energies , @xmath59 and @xmath60 , given by equation [ tbenergies ] . all tb simulations used the following parameters in atomic units : @xmath61 and a radial distance to the center of each arm of 0.6 . here @xmath62 is the strength of the transition dipole . proportional changes to these parameters do not affect the results of course . the top plots of figure [ tb_am2_up_down ] show how the eigenstate populations evolve as a function of time , and the associated charge conservations are listed above each plot . the bottom plot is a composite of results from both simulations which shows the phase relationship between two adjacent arms before and after application of the laser pulses . the horizontal axis was obtained by rigidly translating sections of the plots of amplitude versus time so that they appear in the same time interval . then time was mapped into phase through multiplication with the associated frequencies of light . in this form , the phase relation between two arms can be easily measured by comparing the amplitude of one arm ( solid ) with its neighbor ( dashed ) . the initial phase progression should be @xmath63 , from equation [ tbstates ] , and this is confirmed in the black and dashed black curves . the addition of ptc results in a @xmath64 etc and the measured phase between the red and dashed red curves exhibits the expected progression of @xmath65 . likewise , the subtraction of ptc leaves the molecule with @xmath66 and the anticipated phase progression of @xmath67 between arms , shown in blue and dashed blue . a second tb simulation , figure [ tb_addition ] , carries out a sequence of charge addition and subtraction that starts and ends in the ground state ( gs ) . the same 7-arm molecule is subjected to three windowed , cw laser pulses . the first laser @xmath68 has @xmath69 and an energy equal to that of the eigenstate for an @xmath70 . the second laser @xmath71 has the same ptc but with an energy equal to the difference of excitonic charge states one and two . the ptc of the third laser is @xmath72 with an energy equal to that of the eigenstate for which @xmath73 . the topological charge balances associated with each laser are given in the figure to more easily interpret the data plotted . these plots also show how the excitonic energy evolves as a function of time , becoming asymptotic to the appropriate eigenenergies after each laser pulse is applied . photons are absorbed and one @xmath74 photon is emitted . the color legend identifies the populations of each etc state.,scaledwidth=70.0% ] many of the idealizations associated with the tb paradigm can be removed by reconsidering the light - matter dynamics using time - domain density functional theory ( td - dft ) . unlike standard ground state density functional theory ( dft ) , td - dft captures the non - equilibrium response of material to an externally applied , time - varying electric field . such real - time simulations are made possible through runge - gross ( rg ) reformulation of the time - dependent schr " odinger equation @xcite . a methodology was developed so that td - dft can be used to quantify topological charge transfers as detailed in the methods section . td - dft calculations are computationally intense and amount to carrying out a standard dft calculation for a series of very small time steps . once again using atomic units , the requisite time step for the calculations of this study , in particular , is @xmath75 au for simulations covering approximately @xmath76 au in total . to reduce the computational cost in this initial proof - of - concept , a ring of radially aligned @xmath77 molecules was used as an idealized n - arm system . the ring was given a radius of @xmath78 bohr with the h - h bond lengths taken to be @xmath79 bohr . for each simulation , a computational domain was constructed as the sum of spheres of radius @xmath80 bohr around each atom . this domain was discretized with a spacing of @xmath81 bohr . the generalized gradient approximation ( gga ) parametrized by perdew , burke , and ernzerhof ( pbe)@xcite was adopted to account for exchange and correlation , and a troullier - martins pseudopotential was used . since the wavelength of laser field is much larger than the dimension of the n - arm system , a point dipole approximation of light - matter interaction is applied in our td - dft simulations . because the limitations on the type of external field that can be inputted to octopus , an approximation scheme was used to create twisted light . specifically , it was taken to have n piecewise homogeneous components . transition dipoles from the ground state were found to be maximal on each arm and so the associated field components were constructed to be arm - centered . in contrast , the transition dipoles between two excited states were maximal at the midpoints between arms , so the associated external field components were centered between the arms . a 5-arm configuration of @xmath77 dimers was considered all of the td - dft simulations . casida s perturbative td - dft methodology @xcite was first performed to obtain excitation energies . this was necessary in order to design laser pulses with the frequency need to excite a given excitonic state . the energies , dominant determinants , and the corresponding topological charges of the first five excited states are given in table [ casidaexs ] . .dominant determinant , @xmath82 , and topological charge , @xmath18 , of the first five lowest excited states as calculated from casida perturbation within td - dft . here @xmath82 is the spin - adapted singlet so that one electron is excited from @xmath83 occupied ks orbital to the @xmath84 unoccupied ks orbital . energies are in hartrees ( ha ) . [ casidaexs ] [ cols="<,^,^,^,^,>",options="header " , ] the determinants listed in table [ casidaexs ] are those that dominate each excited state , representing approximately @xmath85 of the respective wavefunction . approximating an excited state with only a single determinant makes it possible to find the population of etc states , equation [ population ] , as detailed in the methods section . it was found that the more realistic many - body setting is still able to carry out topological charge algebra . three cases were considered which each involve a sequence of two laser pulses . the following radial vector vortex field , with a gaussian envelope , was utilized : @xmath86 as with the tb analyses , conservation of energy determines whether or not the ptc is added to or subtracted from the molecular assembly . first consider a scenario in which a 0.367 ha laser pulse of @xmath72 is used to create one of the two lowest - energy twisted excitons , @xmath87 , of table [ casidaexs ] . a second laser pulse , with an energy equal to the difference between that of the first and second excitonic states , 0.0220 ha , is subsequently applied as shown in figure [ etcm2tom1 ] . this requires that the photon be absorbed since there is no excitonic state between the ground and first - excited energy levels . the associated statement of charge conservation is thus @xmath88 . in contrast , nothing happens if the @xmath87 state is subjected to the same laser energy but of the opposite sam since this would be violate topological charge conservation . the lower - right panel of figure [ etcm2tom1 ] demonstrates this . as is clear in the figure , the second laser needs to be much stronger than the first because the transition dipole between excited states is in the mid - arm region , where the relevant state densities are small , while the transition dipoles from the ground state are located in the arms where the relevant state densities are much larger . . upper panel shows sequenced laser pulses of 0.367 ha and 0.0220 ha , respectively . the system is initially given a charge of @xmath87 using a laser of the same charge . application of second laser with charge @xmath89 transfers the system to a charge of @xmath90 ( lower - left panel ) . the same laser , but with a charge of @xmath56 , does not cause the state to evolve ( lower - right panel ) . envelop parameters are @xmath91 and @xmath92 in atomic units , respectively . , scaledwidth=90.0% ] a second case shows how etc can be either increased or decreased in response to the application of pulses with identical ptc . the system is first given a charge of @xmath64 and , in both cases , a second laser pulse with a ptc of @xmath56 is then applied . in the evolution shown at lower - right , the energy of the second pulse is equal to the difference between the states @xmath64 and @xmath93 , 0.0158 ha ( table [ casidaexs ] ) . this results in the absorption of a photon because the @xmath93 state is of higher energy with the charge balance equation of @xmath94 . on the other hand , tuning the second pulse to an energy of 0.0257 ha causes the system to transition to the @xmath55 state because this is the energy difference between the @xmath64 and @xmath55 states ( table [ casidaexs ] ) . in this case a photon is emitted because the @xmath55 state is of lower energy , and the charge balance is @xmath95 . the oscillation in the population of the @xmath64 state ( blue curve ) is an artifact associated with the single - determinant approximation in concert with our piecewise homogeneous construction of the incident beam . this field approximation results in a larger contribution of the non - primary determinant , and the artificial oscillation in population is an indicator that there is the relative weighting of these determinants is time dependent . in contrast , the analogous curve associated with circularly polarized light ( figure [ stoc ] ) shows no such oscillation because the non - primary determinants make almost no contribution . ( left ) and @xmath96 ( right ) . upper panels shows two laser sequences . in both processes , the parameters for first laser are @xmath97 . the parameters for second lasers are ( left ) : @xmath98 and ( right ) : @xmath99 in atomic units . , scaledwidth=90.0% ] a third scenario , shown in figure [ etc0tom1orp1 ] , demonstrates how photonic charges of opposite sign can be emitted from the same initial etc state . the system is placed its highest energy state , @xmath93 , after application of an appropriate laser pulse . a second laser with ptc of @xmath56 stimulates photon emission and changes the system to an etc of @xmath90 . the associated statement of charge conservation is @xmath100 . on the other hand , illuminating the system with a ptc of @xmath56 also stimulates photon emission but changes the system to an etc of @xmath101 . the associated statement of charge conservation is @xmath102 . ( left ) and @xmath103 ( right ) . upper panels shows two laser sequences . lower panels show that the system is initially given a @xmath93 charge which is subsequently changed to @xmath90 ( left ) by emission of @xmath89 or to @xmath101 by emission of @xmath56 . the parameters for first laser are @xmath104 and those for the second laser are @xmath105 in atomic units.,scaledwidth=90.0% ] all the cases considered so far have focused on excitonic charge arithmetic . when purposed as a photonic charge converter , though , it is important to be able to transfer the final charge into a photonic state . such an operation , already considered in within the tb setting , is shown to hold in for td - dft as well in figure [ etcemission ] . the system is first given a @xmath87 charge which is then changed to @xmath90 using a second laser . taken together , these first two steps can be viewed as ptc addition : @xmath106 . a third laser then stimulates the emission of photonic charge of @xmath56 leaving the system in its ground state . the overall photonic arithmetic is therefore @xmath107 . . upper panels shows two laser sequences . lower panel shows the system is excited into @xmath87 state by the first laser . subsequent absorption of a @xmath108 photon then transfers the system into @xmath90 state . this etc is converted to ptc with a third laser which brings system back to its ground state after the photon is emitted . the laser parameters are : ( first ) @xmath109 ; ( second ) @xmath110 ; and ( third)@xmath111 in atomic units . , scaledwidth=70.0% ] the excited state excitonic populations are small because very weak and short lasers are applied in all td - dft simulations this is a pragmatic step taken to avoid the population of extraneous eigenstates that result from stronger or longer laser pulses . this is purely a computational issue that results because the vector vortex beams were approximated with five piecewise - homogeneous components a computational work - around to the limitations of the input fields allowed by the octopus td - dft software . the result is unphysical light - matter interactions in the regions between each arm that can be remedied by dividing the region into more than five piecewise - homogeneous components . then stronger and/or longer laser illumination can be applied to increase the population of twisted excitons . as an alternative to applying twisted light , photonic topological charge can be inputted in the form of spin angular momentum i.e . using circularly polarized light . twisted excitons can also be used to convert photons with spin angular momentum to those with angular orbital to momentum . this just amounts to a change of basis for describing the molecular dipoles since circularly polarized light can be mathematically decomposed into a combination of radial and azimuthal vector vortices@xcite : @xmath112 here @xmath113 and @xmath114 are the basis vectors in cartesian and polar representations . in the 5-arm @xmath77 system , only the radial vortex components are absorbed . subsequent photon emission would be in the form of a vector vortex , though , so the molecular assembly serves to convert light from spin to orbital angular momentum ( stoc ) . since multiple photons can be absorbed , pulses of circularly polarized light can be used to generate vector vortices with an arbitrary topological charge . the stoc concept is demonstrated in figure [ stoc ] , where a sequence of @xmath115 and @xmath116 spin laser pulses are applied . the photonic subscript @xmath117 highlights the spin nature of light while @xmath118 subscripts indicate that the photonic charge is in the form of a vector vortex . the first laser pulse causes photon absorption and the following charge conservation relation : @xmath119 . the second laser , of opposite spin , results in photon emission : @xmath120 . . upper panels shows two laser sequences . lower panel shows the system is sequentially provided with two topological charges resulting in @xmath121 . subsequent emission would produce a @xmath122 photon . the laser parameters are : ( first ) @xmath123 ; and ( second ) @xmath124 in atomic units.,scaledwidth=70.0% ] figure [ stoc ] shows that the strength of first laser is the same as those used for the vortex beams of figure [ etcm2tom1 ] , [ etc1to2or0 ] , and [ etc0tom1orp1 ] , but the duration is three times longer . this allows a much larger population of proper excited states without involvement of extraneous eigenstates . as compared with figure [ etc1to2or0 ] , we now have a very smooth population of the @xmath64 state ( blue curve ) . this confirms that the origin of the oscillations in our former td - dft simulations is actually the approximation of vector vortex with five piecewise - homogeneous components . beams of light can transfer their angular momentum quanta to the quasi - angular momentum associated with the electronic structure of molecules with @xmath5 or @xmath6 point group symmetry . strictly speaking , this is a transformation of real angular momentum into a molecular analog to the crystal angular momentum of solid - state physics . topological charge is then a convenient bookkeeping scheme since it is conserved in such light - matter interactions . in concert with conservation of energy , it is possible to design processes in which sequential laser pulses are used to increase or decrease the excitonic topological charge . subsequent emission results in photons with a different orbital angular momentum than the input beams . these molecules can be used to convert either spin polarized or vortex beams into vector vortex beams with a range of angular momentum values . unlike existing approaches , this molecular strategy offers a means of manipulating the angular momentum of light which does not rely on the nonlinear optical properties of a mediating crystal . topological charge conservation can be elucidated using simple tight - binding hamiltonians , a setting in which it is straightforward to demonstrate charge algebra . however , time - domain dft gives consistent results within a many - electron setting with the effects of electron correlation and exchange accounted for . the level of tight - binding analysis employed does not require any details of the molecular structure beyond point group , and the time - domain dft analyses adopted , as a computational expedient , molecular arms composed of hydrogen dimers . there exist a panoply of molecules which exhibit @xmath5 or @xmath6 symmetry , though , and several examples are shown in figure [ molecules ] . or @xmath6 symmetry . _ , scaledwidth=90.0% ] these may radiate outward as chiral spoke , such as triphenylphosphene or hexaphenylbenzene , be in a planar , non - spoke arrangement such as cyclohexane , or even be in a three - dimensional band such as cycloparaphenylene @xcite . another intriguing possibility is to functionalize inert scaffolds that have the requisite point symmetry . in the present work , stimulated emission causes photons to be generated that have the polarization of the stimulating beam . however , it is clear that charge conversions may also culminate in spontaneous emission . the resulting photons will then exhibit a vector vortex in the far field with radial and azimuthal components that depend upon the orientation of the electric dipoles of each arm . real - time simulations are made possible through the runge - gross ( rg ) reformulation of time - dependent schr " odinger equation @xcite : @xmath125(\vec r , t ) + \nu_{xc}[\rho](\vec r , t)\big]\psi_i(\vec r , t ) . \label{rg}\ ] ] here the spin - reduced electronic density , @xmath126 , is expressed in terms of the time - dependent kohn - sham ( tdks ) orbitals , @xmath127 , as @xmath128 these orbitals , in turn , can be represented in the basis of their counterparts at time zero , @xmath129 , so that the time - propagated multi - electron wavefunction is constructed from a linear combination of determinants built from these initial orbitals@xcite : @xmath130 ket @xmath131 is the ground state and @xmath132 is a determinant with the @xmath83 occupied kohn - sham ( ks ) orbital replaced by the @xmath84 unoccupied orbital . the first summation is over all occupied ks orbitals , five occupied ks orbitals in the case of 5-arm @xmath77 system , and the second summation is over all unoccupied ks orbitals . in our case , because the frequency of laser is chosen to only access the first five lowest excited states , the only unoccupied ks orbital is the sixth as shown in table [ casidaexs ] . if it was possible to express equation [ tpstate ] in form of equation [ tbstates ] , the associated excitonic charge could be determined directly . such a simple expansion of @xmath133 in basis of @xmath134 does not exist , though , since it is a many - body wavefunction . this is remedied , albeit in an approximate way , by working only with the dominant determinant for which only lowest unoccupied molecular orbital ( lumo ) is involved in equation [ tpstate ] . this makes it possible to combine the determinants corresponding to each etc subspace @xmath135 with @xmath136 , as in equation [ rearranged ] , allowing the etc of @xmath133 to be obtained via ks orbitals using the method detailed below . in all simulations , twisted excitons are a constructed as a linear combination of corresponding pairs of degenerate excited states , consistent with the tb model . focusing on the 5-arm system , if a laser with @xmath137 is applied , then the resulting excited state can be approximated with a determinant involving only @xmath138 . likewise , the application of @xmath139 results in excited states that can be approximated with determinants involving only @xmath140 and @xmath141 , and @xmath142 yields states that are well - approximated with only @xmath143 and @xmath144 . the time - propagated wavefunction for each etc subspaces @xmath145 , @xmath146 and @xmath147 can therefore be expressed , respectively , as : @xmath148 system . _ the five occupied ks orbitals have been expressed in the basis of arm wavefunctions , @xmath134 , with their coefficients labeled on the corresponding arms . the red ( blue ) isosurfaces indicate positive ( negative ) values of the wavefunctions . the lumo is given in order to show that it is symmetric . homo = highest occupied molecular orbital . , scaledwidth=80.0% ] the only difference among these three equations is that the ground state determinant is modified as follows : @xmath149 replaced by @xmath150 ; @xmath151 and @xmath152 are replaced by @xmath153 ; and @xmath154 , @xmath155 and @xmath156 are replaced by @xmath157 and @xmath154 . the ground state determinant @xmath158 is the @xmath93 state of course . these replacement orbitals must therefore be responsible for the etc of excited states . figure [ ksorbitals ] gives the isosurface and decomposition in the basis of @xmath159 with @xmath160 of all the relevant ks orbitals . as shown in figure [ ksorbitals ] , the lumo is symmetrically distributed across all five arms . therefore @xmath154 will not introduce a phase difference among arms in the right side of equation [ rearranged ] . this implies that @xmath153 and @xmath157 will introduce a phase dependence corresponding to @xmath161 and @xmath162 , respectively . the population of each twisted exciton state is therefore given by : @xmath163 here @xmath164 is the eigenstate associated with an etc of @xmath18 , from equation [ tbstates ] , and the factor of two in each expression accounts for the fact that the electron spin can be either up or down . we are grateful to profs . david l. andrews , guillermo f. quinteiro and mark e. siemens for extended discussions on the generation of twisted light . all calculations were carried out using the high performance computing resources provided by the golden energy computing organization at the colorado school of mines . r. baltz and c. f. klingshirn , in _ ultrafast dynamics of quantum systems : physical processes and spectroscopic techniques _ , edited by b. di bartolo and g. gambarota ( springer us , boston , ma , 2002 ) , pp . 381396 .

the molecular absorption of photons with angular momentum can result in _ twisted excitons _ with a well - defined quasi - angular momentum . although they represent different physical properties , photonic and excitonic quanta can both be described in terms of topological charge , a conserved quantity . multiple absorption events can be used to create a wide range of excitonic topological charges . subsequent emission produces photons that exhibit this same range . the molecule can thus be viewed as a mediator for changing the orbital angular momentum of light . this sidesteps the need to exploit nonlinear light - matter interactions based on higher - order susceptibilities . a tight - binding paradigm is used to establish topological charge conservation and demonstrate how it can be exploited to combine , subtract , and change the nature of the angular momentum of light . the approach is then extended to a time - dependent density functional theory setting where the key results are shown to hold in a many - body , multi - level setting .

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Proceed to summarize the following text: synchronization in chaotic systems is a surprising phenomenon , which recently received a lot of attention , see e.g. @xcite . even though the heuristic theory and the classification of the synchronization phenomena are well studied and reasonably well understood , a mathematically rigorous theory is still lacking . generally speaking , a standard difficulty lies in the fact that the phenomenon involves the dynamics of non - uniformly chaotic systems , typically consisting of different sub - systems , whose long - time behavior depends crucially on the sign of the `` central '' lyapunov exponents , i.e. of those exponents that are zero in the case of zero coupling , and become possibly non - trivial in the presence of interactions among the sub - systems . the mathematical control of such exponents is typically very hard . progress in their computation is a fundamental preliminary step for the construction of the srb measure of chains or lattices of chaotic flows , which may serve as toy models for extensive chaotic systems out - of - equilibrium ( i.e. they may serve as standard models for non - equilibrium steady states in non - equilibrium statistical mechanics ) . in a previous paper @xcite , we introduced a simple model for phase synchronization in a three - dimensional system consisting of the suspension flow of arnold s cat map coupled with a clock . the coupling in @xcite was unidirectional , in the sense that it did not modify the suspension flow , but only the clock motion . notwithstanding its simplicity , the model has a non - trivial behavior : in particular , it exhibits phase locking and in @xcite we constructed the corresponding attractive invariant manifold via a convergent expansion . however , because of unidirectionality , the lyapunov spectrum in @xcite was very simple : the `` longitudinal '' exponents ( i.e. , those corresponding to the motion on the invariant manifold ) coincided with the unperturbed ones , and the central exponent was expressed in the form of a simple integral of the perturbation over the manifold . in this paper , we extend the analysis of @xcite to a simple bidirectional model , for which the lyapunov spectrum is non - trivial , and we show how to compute it in terms of a modified expansion , which takes the form of a decorated tree expansion discussed in detail in the following . the model is defined as follows . take arnold s cat map @xmath0 and denote by @xmath1 and @xmath2 the eigenvalues and eigenvectors , respectively , of @xmath3 : @xmath4 with @xmath5 , so that @xmath6 are normalized . we let the suspension flow of arnold s cat be defined as @xmath7 , with @xmath8 , if @xmath9 @xmath10 . formally , @xmath11 is the solution to the following differential equation instead of , but throughout the paper we only used the fact that at all times @xmath12 the variable @xmath13 jumped abruptly from @xmath14 to @xmath15 , and besides these discontinuities the flow was smooth . therefore , all the results and statements of @xcite are correct , modulo this re - interpretation of the flow equation ( * ? ? ? * ( 2.1 ) ) , where @xmath16 should be replaced by @xmath17 . ] on @xmath18 : x=(t)(s ) x , [ 1.susf]where @xmath19 is the @xmath20-periodic delta function such that @xmath21 for all @xmath22 . the model of interest is obtained by coupling the suspension flow of arnold s cat map with a clock by a regular perturbation , so that on @xmath23 the evolution equation is @xmath24x+\varepsilon f(x , w , t ) , & \\ \dot{w}=1+\varepsilon g(x , w , t ) , \end{cases}\ ] ] where @xmath25 and @xmath26 , @xmath27 are @xmath20-periodic in their arguments . for @xmath28 the motions of @xmath13 and @xmath29 are independent . therefore , the relative phase @xmath29 mod @xmath30 among the two flows is arbitrary . if @xmath31 and if the interaction is dissipative ( in a suitable sense , to be clarified in a moment ) , then the phases of the two sub - systems can lock , so that the limiting motion in the far future takes place on an attractor of dimension smaller than 3 , for all initial data in an open neighborood of the attractor . in @xcite , we explicitly constructed such an attractor in terms of a convergent power series expansion in @xmath32 , for @xmath33 and a special class of dissipative functions @xmath27 . in this paper , we generalize the analysis of @xcite to @xmath34 . our first result concerns the construction of the attractive invariant manifold for @xmath34 . [ prop:1 ] let @xmath35 be the flow on @xmath36 associated with the dynamics , with @xmath26 and @xmath27 analytic in their arguments . set @xmath37 and assume there exists @xmath38 such that @xmath39 and @xmath40 , independently of @xmath41 . then there are constants @xmath42 such that for @xmath43 there exist a homemorphism @xmath44 and a continuous function @xmath45 , both hlder - continuous of exponent @xmath46 , such that the surface @xmath47 is invariant under the poincar map @xmath48 and the dynamics of @xmath48 on @xmath49 is conjugated to that of @xmath50 on @xmath51 , i.e. @xmath52 the proof of this theorem is constructive : it provides an explicit algorithm for computing the generic term of the perturbation series of @xmath53 with respect to @xmath32 , it shows how to estimate it and how to prove convergence of the series . as a by - product , we show that the invariant manifold is holomorphic in @xmath32 in a suitable domain of the complex plane , whose boundary contains the origin . the construction also implies that @xmath54 is an attractor . we denote by @xmath55 its basin of attraction and by @xmath56 an arbitrary open neighborood of @xmath54 contained in @xmath55 such that @xmath57 , with @xmath58 the lesbegue measure on @xmath59 . in addition to the construction of the invariant surface , in this paper we show how to compute the invariant measure on the attractor and the lyapunov spectrum , in terms of convergent expansions . more precisely , let @xmath60 be the lesbegue measure restricted to @xmath56 , i.e. , denoting by @xmath61 the characteristic function of @xmath56 , @xmath62 , for all measurable @xmath63 . the `` natural '' invariant measure on the attractor , @xmath64 , is defined by @xmath65 for all continuous functions @xmath66 and @xmath60-a.e . @xmath67 , where @xmath68 . the limiting measure @xmath64 is supported on @xmath54 and such that @xmath69 . on the attractor , @xmath64-a.e . point defines a dynamical base , i.e. a decomposition of the tangent plane as @xmath70 , such that @xmath71 the constants of motion @xmath72 are the lyapunov exponents , and we suppose them ordered as @xmath73 ; in the following we shall call @xmath74 the _ central _ lyapunov exponent . our second main result is the following . [ prop:2 ] there exists @xmath75 such that the following is true . let @xmath66 be an hlder continuous function on @xmath59 . then @xmath76 is hlder continuous in @xmath32 , for @xmath77 . if @xmath66 is analytic , then @xmath76 is analytic in @xmath32 , for @xmath78 and a suitable @xmath66-dependent constant @xmath79 . moreover , the lyapunov exponents @xmath72 , @xmath80 , are analytic in @xmath32 for @xmath77 . in particular , the central lyapunov exponent is negative : @xmath81 , while @xmath82 . the paper is organized as follows . theorem [ prop:1 ] is proved in section [ sec:2 ] below . the proof follows the same strategy of @xcite : ( 1 ) we first write the equations for the invariant surface and solve them recursively at all orders in @xmath32 ; ( 2 ) then we express the result of the recursion ( which is not simply a power series in @xmath32 ) in terms of tree diagrams ( planar graphs without loops ) ; trees with @xmath83 nodes are proportional to @xmath84 times a _ tree value _ , which is also a function of @xmath32 ; ( 3 ) finally , using the tree representation , we derive an upper bound on the tree values . the fact that the dissipation is small , of order @xmath85 , produces bad factors @xmath86 in the bounds of the tree values , for some @xmath87 depending on the tree therefore , we need to show that for any tree @xmath87 is smaller than a fraction of @xmath83 , if @xmath83 is the number of nodes in the tree . this is proved by exhibiting suitable cancellations , arising from the condition @xmath39 . theorem [ prop:2 ] is proved in section [ sec:3 ] . the proof adapts the tree expansion to the computation of the local lyapunov exponents @xmath88 on the invariant surface , in the spirit of ( * ? ? ? * chapter 10 ) and @xcite . the positive local lyapunov exponent @xmath89 plays the role of the gibbs potential for the invariant measure @xmath90 . therefore , given a convergent expansion for @xmath89 , @xmath90 can be constructed by standard cluster expansion methods , as in ( * ? ? ? * chapter 10 ) . finally , @xmath72 can be expressed as averages of the local exponents over the stationary distribution . in section [ sec:4 ] , we present some numerical evidences for a fractal to non - fractal transition of the invariant manifold , and formulate some conjectures . in this section , we define the equations for the invariant manifold , by introducing a conjugation that maps the dynamics restricted to the attractor onto the unperturbed one . the conjugation is denoted by @xmath91 , with @xmath92 where @xmath93 is the identity in @xmath94 . let @xmath95 and @xmath96 be the initial conditions at time @xmath97 . we will look for a solution to of the form @xmath98 for @xmath99 , with boundary conditions @xmath100 the evolution equation for @xmath29 will be written by `` expanding the vector field at first order in @xmath101 and at zeroth order in @xmath102 '' , i.e. as @xmath103 where @xmath104 and g(,t):=g(s h()+a(,t),w_0+t+u(,t),t)-_0(,t)- _ 1(,t)u(,t).[2.g ] the logic in the rewriting is that the ( linear ) approximate dynamics obtained by neglecting @xmath105 is dissipative , with contraction rate proportional to @xmath32 , thanks to the second condition in : this will allow us to control the full dynamics as a perturbation of the approximate one . the approximation obtained by neglecting @xmath105 is the simplest one displaying dissipation . in principle we could have expanded the dynamics at first order both in @xmath101 and in @xmath106 , but the result would be qualitatively the same . we now set @xmath107 and fix @xmath108 such that @xmath109 ; then we obtain @xmath110 the equation for @xmath13 , if expressed in terms of @xmath111 , gives , after integration , @xmath112 for @xmath113 , these give [ eq:2.7 ] @xmath114 it is useful to introduce an auxiliary parameter @xmath115 , to be eventually set equal to @xmath32 , and rewrite and as [ eq:2.8 ] @xmath116 the idea is to first consider @xmath115 as a parameter independent of @xmath117 , then write the solution in the form of a power series in @xmath32 , with coefficients depending on @xmath60 , and finally show that the ( @xmath115-dependent ) radius of convergence of the series in @xmath32 behaves like @xmath118 , @xmath119 , at small @xmath60 : this implies that we will be able to take @xmath120 without spoiling the summability of the series . summarizing , we will look for a solution of in the form @xmath121 with [ eq:2.11 ] @xmath122 and : @xmath123 , with @xmath11 and @xmath124 as in ; @xmath105 is defined in ; @xmath60 must be eventually set equal to @xmath32 . the solution to - is looked for in the form of a power series expansion in @xmath117 ( at fixed @xmath60 , in the sense explained after ) . therefore , we write [ eq:2.10 ] @xmath125 and insert these expansions into . by the analyticity assumption on @xmath26 and @xmath27 , we may expand ( defining @xmath126 and @xmath127 ) @xmath128 where @xmath129 in the first sum denotes the constraint @xmath130 , and @xmath131 and similarly for @xmath132 ( recall that @xmath133 and @xmath6 are the eigenvalues and eigenvectors of @xmath134 ) ; here and henceforth we are denoting by @xmath135 the standard scalar product in @xmath136 . moreover @xmath137 where @xmath138 and @xmath139 . for future reference , we note since now that the analyticity of @xmath26 and @xmath27 yields , by the cauchy inequality , @xmath140 for some constant @xmath141 , uniformly in @xmath142 ( here @xmath143 ) . define @xmath144 , @xmath145 , and @xmath146 , with @xmath147 ) . setting @xmath148 and plugging into , we find for @xmath149 [ eq:2.12 ] @xmath150 introduce the notation @xmath151 where @xmath152 . here and henceforth , if @xmath153 , the product @xmath154 should be interpreted as 1 , and similarly for the other products in the case that @xmath155 and/or @xmath156 . then , defining @xmath157 , we find , for @xmath158 , [ eq:2.13 ] @xmath159 where @xmath160 . we now want to bound the generic term in the series originating from the recursive equations ; the goal is to show that the @xmath83-th order is bounded proportionally to @xmath161 , with @xmath162 and @xmath163 . we find convenient to represent graphically the coefficients in in terms of rooted trees ( or simply trees , in the following ) as in @xcite . we refer to ( * ? ? ? * section v ) for the definition of trees and notations . with respect to the trees in @xcite in the present case there are four _ types _ of nodes . we use the symbols , , and , calling them nodes of type @xmath164 , @xmath165 , @xmath166 and @xmath167 , respectively : they correspond to contributions to @xmath168 , respectively . the constraint @xmath129 forbids a node of type 0 or 1 to be immediately preceded by exactly one node of these two types . recall that a _ tree _ is a partially ordered set of nodes and lines ; the partial ordering relation is denoted by @xmath169 and each line will be drawn as an arrow pointing from the node it exits to the node it enters . we call @xmath170 the set of nodes and @xmath171 the set of lines of the tree @xmath172 . as in @xcite we denote by @xmath173 the node such that @xmath174 for any node @xmath175 : @xmath173 will be called the _ special node _ and the line exiting @xmath173 will be called the _ root line_. the root line can be imagined to enter a further point , called the root , which , however , is not counted as a node . with each node we associate a label @xmath176 to denote its type and a time variable @xmath177 $ ] . with the nodes of types 2 and 3 we also associate a label @xmath178 . denoting by @xmath179 the path of lines connecting @xmath29 to @xmath180 , with both @xmath181 and @xmath29 included , set @xmath182 where @xmath183 if @xmath184 , @xmath185 if @xmath186 , and @xmath187 otherwise . given a node @xmath181 , we denote by @xmath188 the unique node immediately following it ; moreover , we let @xmath189 be the number of nodes of type @xmath190 immediately preceding it ; if @xmath191 , we also define @xmath192 , @xmath193 , to be the number of nodes @xmath29 of type @xmath190 with @xmath194 immediately preceding it ( i.e. such that @xmath195 ) . finally , we let @xmath196 , with @xmath197 , and @xmath198 . a node @xmath181 is called an _ end - node _ if @xmath199 , while it is called an _ internal node _ if it is not an end - node . the _ node factor _ @xmath200 is defined as @xmath201 where @xmath202 is interpreted as equal to @xmath203 , while the _ node integral _ is @xmath204 with the definitions above , we denote by @xmath205 the set of labelled trees with @xmath83 nodes , @xmath206 , and the constraint that nodes of type @xmath164 or @xmath165 can not be immediately preceded by exactly one node of type @xmath164 or @xmath165 ; if @xmath191 , we also denote by @xmath207 the subset of by @xmath205 with @xmath208 . then , one can prove by induction that [ eq:2.14 ] @xmath209 where @xmath210 with the integrals to be performed by following the tree ordering , i.e. by starting from the end - nodes and by moving towards the root . in figure [ fig : n=00003d1 ] the first order contributions are graphically represented , while the contributions of order @xmath211 are shown in figure [ fig : n=00003d2 ] . for each node @xmath181 , the label @xmath212 is drawn superimposed on the line exiting @xmath181 , for clarity purposes , while the label @xmath213 ( to be summed over ) is not explicitly shown . @c=2cm@r=0.5 cm @<-[r]^- _ ( 1.08)v_0 & * + [ o][f*:black ] & u^(1)()=@<-[r]^- _ ( 1.10)v_0 & * + [ o][f ] + @<-[r]^- _ ( 1.08)v_0 & * + [ f*:black ] & h^(1)_()=@<-[r]^- _ ( 1.12)v_0 & * + [ f ] @c=1.5cm@r=0.5 cm .1 cm ^(2)()=@<-[r]^- _ ( 1.12)v_0 & * + [ o][f*:black]@<-[r]^- _ ( 1.20)v_1 & * + [ f ] & + & @<-[r]^- _ ( 1.20)v_0 & * + [ o][f*:black]@<-[r]^- _ ( 1.20)v_1 & * + [ f*:black ] @c=1.5cm@r=0.5 cm u^(2)()=@<-[r]^- _ ( 1.12)v_0 & * + [ o][f]@<-[r]^- _ ( 1.20)v_1 & * + [ f ] & + & @<-[r]^- _ ( 1.20)v_0 & * + [ o][f]@<-[r]^- _ ( 1.20)v_1 & * + [ f*:black ] @c=1.5cm@r=0.5 cm a^(2)_()=@<-[r]^- _ ( 1.12)v_0 & * + [ f*:black]@<-[r]^- _ ( 1.20)v_1 & * + [ f ] & + & @<-[r]^- _ ( 1.20)v_0 & * + [ f*:black]@<-[r]^- _ ( 1.20)v_1 & * + [ f*:black ] + @<-[r]^- _ ( 1.14)v_0 & * + [ f*:black]@<-[r]^- _ ( 1.20)v_1 & * + [ o][f ] & + & @<-[r]^ - _ ( 1.20)v_0 & * + [ f*:black]@<-[r]^- _ ( 1.20)v_1 & * + [ o][f*:black ] @c=1.5cm@r=0.5 cm h^(2)_()=@<-[r]^- _ ( 1.16)v_0 & * + [ f]@<-[r]^- _ ( 1.20)v_1 & * + [ f ] & + & @<-[r]^- _ ( 1.20)v_0 & * + [ f]@<-[r]^- _ ( 1.20)v_1 & * + [ f*:black ] + @<-[r]^- _ ( 1.14)v_0 & * + [ f]@<-[r]^- _ ( 1.20)v_1 & * + [ o][f ] & + & @<-[r]^ - _ ( 1.20)v_0 & * + [ f]@<-[r]^- _ ( 1.20)v_1 & * + [ o][f*:black ] for @xmath214 a few contributions to @xmath215 are shown in figure [ fig : n=00003d3 ] . all the other contributions are obtained by replacing the nodes preceding the special node @xmath173 by nodes of a different type , with the constraint that if only one line enters @xmath173 then it exits a node of type 2 or 3 ; note that in @xcite the linear trees at the bottom of the figure were not possible . @r=0.5 cm & & * + [ o][f ] & & & & * + [ o][f*:black ] & & & & * + [ o][f*:black ] + @<-[r ] _ ( 0.80)v_0 & * + [ o][f]@<-[ru ] _ ( 1.14)v_1 @<-[rd]^- _ ( 1.16)v_2 & & + & @<-[r ] _ ( 0.80)v_0 & * + [ o][f]@<-[ru ] _ ( 1.14)v_1 @<-[rd ] _ ( 1.16)v_2 & & + & @<-[r ] _ ( 0.80)v_0 & * + [ o][f]@<-[ru ] _ ( 1.14)v_1 @<-[rd ] _ ( 1.16)v_2 & & + + & & * + [ f ] & & & & * + [ o][f ] & & & & * + [ o][f*:black ] & + & & * + [ f ] + @<-[r ] _ ( 0.80)v_0 & * + [ o][f]@<-[ru]^- _ ( 1.14)v_1 @<-[rd]^- _ ( 1.16)v_2 & & + & @<-[r ] _ ( 0.80)v_0 & * + [ o][f]@<-[r]^- _ ( 0.70)v_1 & * + [ f]@<-[r]^- _ ( 1.28)v_2 & * + [ o][f*:black ] & + & @<-[r ] _ ( 0.80)v_0 & * + [ o][f]@<-[r]^ _ ( 0.70)v_1 & * + [ f]@<-[r]^- _ ( 1.28)v_2 & * + [ f ] + & & * + [ f ] given @xmath216 , we let @xmath217 be the family of labelled trees differing from @xmath218 just by the choice of the labels @xmath219 . then , using and proceeding as in ( * ? ? ? * section vi ) , we obtain @xmath220 where @xmath221 is a suitable constant and @xmath222 is the number of internal nodes of type @xmath190 in @xmath218 ; see @xcite for details . we are then left with bounding @xmath223 . [ lem:2.1 ] for all @xmath224 one has @xmath225 . the proof is given in appendix [ app : a ] , where it is also shown that such an upper bound on @xmath226 is optimal , i.e. there are trees that saturate the inequality . combining lemma [ lem:2.1 ] with and recalling that the number of distinct families @xmath217 in @xmath227 is bounded by @xmath228 , for a suitable @xmath229 , we get @xmath230 is a positive constant and @xmath231 $ ] is the integer part . implies that the radius of convergence of the series is bounded by @xmath232 . therefore , we can take @xmath233 . hlder - continuity of @xmath234 and @xmath235 can be proved mutatis mutandis like in @xcite . this completes the proof of theorem [ prop:1 ] . in order to compute the lyapunov exponents , we need to understand how the vectors on the tangent space evolve under the interacting dynamics . to this purpose , we set @xmath236 , rewrite as @xmath237 , with @xmath238 , and write the dynamics on the tangent space as follows : @xmath239 + \e \boldsymbol{\partial f}({\boldsymbol{x}},t)\;,\ ] ] where @xmath240 , @xmath241 is the solution to found in section [ sec:2 ] , @xmath242 are the projections into the unperturbed eigendirections @xmath243 , @xmath133 are the corresponding unperturbed eigenvalues , and @xmath244 is the jacobian matrix of @xmath245 . integration of gives the tangent map @xmath246 . we denote by @xmath247 the solution to with initial condition @xmath248 at @xmath97 ( started at @xmath249 ) . for @xmath250 $ ] we obtain @xmath251 we look for a conjugation @xmath252 , @xmath253 where @xmath93 is the identity in @xmath254 and @xmath255 is a @xmath256 matrix , such that , by setting @xmath257 for a suitable @xmath256 matrix @xmath258 , to be determined , one has @xmath259 , that is @xmath260 of course only the part of involving the tangent dynamics has still to be solved , so we study the conjugation equation @xmath261 the matrix @xmath262 will be taken to be diagonal in the basis @xmath263 , where @xmath264 , and @xmath265 . then in the basis @xmath266 one has @xmath267 while the matrix @xmath268 takes diagonal form , with values @xmath269 along the main diagonal . from now on we shall use this basis , and implicitly assume that the indices @xmath270 , run over the values @xmath271 , unless stated otherwise . by setting @xmath272 , we obtain from @xmath273 note that , given a solution @xmath274 of , then also @xmath275 is a solution with @xmath276 replaced with @xmath277 , where @xmath278 are non - zero functions from @xmath51 to @xmath279 . therefore , with no loss of generality , we can require the diagonal elements of @xmath280 to vanish : hence @xmath280 will be looked for as an off - diagonal matrix . we look for the conjugation in the form - , with @xmath281 equations and give @xmath282 where @xmath283 can be computed iteratively via as & & -.8truecm ds_^(())= +_n1^n_0^d_1 ( s^_1_(()),_1 ) _ 0^_1d_2(s^_2_(()),_2 ) + & & 1.15truecm _ 0^_n-1d_n(s^_n_(()),_n):=(+m(,)).[eq:3.9bis]at first order in @xmath32 , gives ( with @xmath284 ) @xmath285 in particular , setting @xmath286 and recalling that @xmath287 is off - diagonal , we find @xmath288 while , for @xmath289 , we have to solve recursively for @xmath290 , the result being ( if @xmath193 ) : [ eq:3.11 ] @xmath291 in order to compute the higher orders , we insert in the left side of , thus getting @xmath292 where @xmath293 . note that , according to , @xmath294 is expressed as a series of iterated integrals of @xmath295 , where @xmath244 is analytic in its argument . therefore , the power series expansion in @xmath32 of @xmath294 can be obtained ( and its @xmath83-th order coefficient can be bounded ) by using the corresponding expansions for the components of @xmath296 ; here the functions @xmath297 , @xmath298 , @xmath299 and @xmath300 are as in with the coefficients given by and bounded as in . we write @xmath301 by using the very definition of @xmath294 and the bounds , it is straightforward to prove that ^(n ) ( ) : = _ i , j \{+,-,3 } | _ i , j ( ) | c_4^n^-[(2n-1)/3][e3.14]for a suitable @xmath302 . now , if @xmath303 , the diagonal part of gives @xmath304 while the off - diagonal part can be solved in a way similar to , i.e. , if @xmath193 , [ eq:3.16 ] @xmath305 , \label{eq:3.16a } \\ k_{\alpha,3}^{(n)}(\varphi ) & = -\a\l_\a^{-1}\sum_{m\in\mathbb z_\a}\l_\a^{-m } \big [ \mathfrak{m}_{\alpha,3}^{(n)}(s^m\f ) + \mathcal q_{\a,3}^{(n)}(s^m\f ) \bigr ] , \label{eq:3.16b } \\ k_{3,-\alpha}^{(n)}(\varphi ) & = - \a\sum_{m\in\mathbb z_{\a}}\l_{-\a}^{m } \big[\mathfrak{m}_{3,-\alpha}^{(n)}(s^m\f)\l_{-\a } + \mathcal q_{3,-\a}^{(n)}(s^m\f ) \bigr ] , \label{eq:3.16c}\end{aligned}\ ] ] where we have set @xmath306 in the simple case that @xmath307 , @xmath308 , while @xmath309 thus recovering the formula for @xmath310 given in ( * ? ? ? * section vii ) . in figure [ fig : rappresentazione - grafica - di gamma^n ] and [ fig : rappresentazione - grafica - di k^n ] we give a graphical representation of and , respectively . the representation of and is the same as in figure [ fig : rappresentazione - grafica - di k^n ] , simply with the labels @xmath311 replaced by @xmath312 and @xmath313 , respectively . @r=0.5cm@c=1.0 cm & & & & & & & & * + [ o][f**:black ] + _ i^(n)()=@<-[r ] ^-ii ^(1.24)(n ) & * + [ f**:black ] & = @<-[r]^-ii _ ( 0.75)v_0 & * + [ f**:black]@ < [ r]^<<^(0.6)ii ^(1.4)(n ) & * + [ o][f**:black ] & + & @<-[r]^-ii _ ( 0.80)v_0 & * + [ f**:black ] @ < [ ru]^-ij_(1.2)(n_1)@<-[rd]^(0.2)^-ji^(1.2)(n_2 ) & + & & & & & & & & * + [ f**:white ] @r=0.5cm@c=1.0 cm & & & & & & & & * + [ o][f**:black ] + k_i , j^(n)()=@<-[r ] ^-i j ^(1.24)(n ) & * + [ f**:white ] & = @<-[r]^-i j _ ( 0.75)v_0 & * + [ f**:white]@ < [ r]^<<^(0.7)i j ^(1.4)(n ) & * + [ o][f**:black ] & + & @<-[r]^-i j _ ( 0.80)v_0 & * + [ f**:white ] @ < [ ru]^-i j_(1.2)(n_1 ) @<-[rd]^(0.2)^(0.6)j j^(1.2)(n_2 ) & + & & & & & & & & * + [ f**:white ] + & & * + [ f**:white ] + + @<-[r]^-i j _ ( 0.80)v_0 & * + [ f**:white ] @<-[ru]^-i j_(1.2)(n_1 ) @<-[rd]^(0.25)^(0.6)j j^(1.2)(n_2 ) & + & & * + [ f**:black ] to iterate the graphical construction and provide a tree representation for both @xmath262 and @xmath280 , we need a few more definitions . we identify three types of _ principal nodes _ , that we call of type @xmath314 , @xmath315 and @xmath316 , and represent graphically , respectively , by , and . with any such node @xmath181 , we associate a label @xmath317 , to denote its type , and two labels @xmath318 , which will be drawn superimposed to the line exiting @xmath181 ; if @xmath181 is of type @xmath315 , then @xmath319 , while if @xmath181 is of type @xmath314 , then @xmath320 . a node is of type @xmath316 if and only if it is an end - node . furthermore , with each node @xmath181 with @xmath321 , we associate a label @xmath322 , while we set @xmath323 for all nodes @xmath181 with @xmath324 ; with each node @xmath181 with @xmath325 , we associate a label @xmath326 such that either @xmath327 or @xmath328 ( recall that if @xmath181 is a nodes of type @xmath314 then @xmath320 , so that either @xmath329 or @xmath330 are @xmath331 ) , and a label @xmath332 ; if @xmath181 is of type @xmath315 or @xmath316 we define @xmath187 . if @xmath333 denotes the number of lines entering @xmath181 and @xmath334 the number of lines of type @xmath190 entering @xmath181 , we have the constraints @xmath335 and @xmath336 . moreover : @xmath337 ; @xmath338 . if @xmath339 and @xmath340 is the node immediately preceding @xmath181 on has @xmath341 and @xmath342 . if @xmath343 , let @xmath344 be the two nodes immediately preceding @xmath181 ; if @xmath345 , with no loss of generality we assume that @xmath340 is of type @xmath346 ( so that @xmath347 is of type @xmath314 ) ; if @xmath348 , with no loss of generality we assume that @xmath340 is of type @xmath314 ( so that @xmath347 is of type @xmath315 ) ; in both cases we impose the constraints that @xmath349 , @xmath350 and @xmath351 . denoting by @xmath188 the node immediately following @xmath181 , we set @xmath352 , and @xmath353 we are finally ready to define the _ node factors _ associated with the nodes : @xmath354 then , by iterating the graphical representation in figures [ fig : rappresentazione - grafica - di gamma^n ] and [ fig : rappresentazione - grafica - di k^n ] , we end up with trees like that in figure [ fig : esempi - di - alberi 1 e 2 ] for @xmath289 ; note that the end - nodes are all of type @xmath316 . if @xmath286 the only difference is that the special node @xmath355 is of type @xmath315 . with the definitions above , we denote by @xmath356 the set of labelled trees such that @xmath357 , @xmath206 , and the constraints and properties described above . then it is straightforward to prove by induction that @xmath358 where @xmath359 given @xmath360 , we let @xmath217 be the family of labelled trees differing from @xmath218 just by the choice of the labels @xmath219 . then , using , it is easy to see that @xmath361},\ ] ] which immediately implies that @xmath362},\ ] ] for a suitable constant @xmath363 . therefore , the radius of convergence in @xmath32 of the series for @xmath255 and @xmath364 is proportional to @xmath365 , which allows us to fix eventually @xmath366 . the lyapunov exponents @xmath367 are the time average of the quantities @xmath368 . however , if @xmath369 denotes the restriction of @xmath370 on the attractor @xmath49 , the dynamical system @xmath371 is conjugated to an asonov system and hence it is ergodic : therefore time - averaged observables are @xmath41-independent . furthermore , there exists a unique srb measure @xmath372 such that @xmath373 the measure @xmath374 can be computed by reasoning as in ( * ? ? ? * chapter 10 ) . let @xmath375 be a markov partition for @xmath134 on @xmath376 and set @xmath377 . call @xmath378 the symbolic code induced by the markov partition @xmath379 and denote by @xmath380 the symbolic representation of a point @xmath381 , i.e. @xmath382 . then the expansion rate of @xmath369 along the unstable manifold of @xmath383 is @xmath384 , where @xmath385 with @xmath386 denoting the shift map and @xmath387 . if @xmath374 denotes the gibbs distribution for the energy function @xmath388 ( see ( * ? ? ? * chapter 5 ) ) , then the srb distribution @xmath389 for the system @xmath371 is @xmath374 and can be computed accordingly ( see ( * ? ? ? * chapter 6 ) ) . moreover , by construction , @xmath388 is analytic in @xmath32 and hlder - continuous in @xmath380 . therefore , for any hlder - continuous function @xmath390 , the expectation value @xmath391 is hlder - continuous in @xmath32 for @xmath77 . if @xmath66 is analytic in @xmath32 , then there exists a positive constant @xmath392 , depending on @xmath66 , such that @xmath393 is analytic for @xmath394 . in particular the lyapunov exponents are analytic in @xmath32 and , from , one finds @xmath395 this completes the proof of theorem [ prop:2 ] . in this section , we discuss informally some of the consequences of our main theorem , and formulate a conjecture about the transition from fractal to smooth(er ) behavior , which is suggested by our result . from theorem [ prop:1 ] we know that the surface @xmath54 of the attractor is h@xmath396lder continuous , but we do not have any control on its possible differentiability . this means that our attractor may be fractal , and we actually expect this to be the case for @xmath32 positive and small enough . an analytic estimate of the fractal dimension of the attractor in terms of the lyapunov exponents is provided by the _ lyapunov dimension _ @xmath397 , which is defined as follows . consider an ergodic dynamical system admitting an srb measure on its attractor , and let @xmath398 be its lyapunov exponents , counted with their multiplicities . then , @xmath399 where @xmath400 is the largest integer such that @xmath401 . the _ kaplan - yorke conjecture _ @xcite states that @xmath402 coincides with the hausdorff dimension of the attractor ( also known as the _ information dimension _ , see , e.g. , ( * ? ? ? * chapt.5.5.3 ) for a precise definition ) . in this section , we take @xmath397 as a heuristic estimate of the fractal dimension of the attractor , without worrying about the possible validity of the conjecture ( which has been rigorously proven only some special cases , see e.g. ) . specializing the expression of @xmath397 to our context , we find that , for @xmath32 sufficiently small , d_l=2+=3++r_2(),[eq : fract]where @xmath403 is the taylor remainder of order 2 in @xmath32 , which is computable explicitly in terms of the convergent expansion derived in the previous sections . note that @xmath404 , so that @xmath405 is smaller than 3 ( as desired ) and is decreasing in @xmath32 , for @xmath32 small . therefore , combined with the kalpan - yorke conjecture , ( [ eq : fract ] ) suggests that the attractor is fractal for @xmath32 small , and its fractal dimension decreases ( as expected ) by increasing the strength @xmath32 of the dissipative interaction . it is now tempting to extrapolate to larger values of @xmath32 ( possibly beyond the range of validity of theorem [ prop:2 ] ) , up to the point where , possibly , the relative ordering of @xmath74 and @xmath406 changes . in the simple case that @xmath33 ( which is the case considered in @xcite ) , the lyapunov exponents @xmath407 are independent of @xmath32 : @xmath408 . therefore , on the basis of , we conjecture that by increasing @xmath32 the hausdorff dimension of the attractor decreases from @xmath167 to @xmath166 until @xmath32 reaches the critical value @xmath409 , where @xmath410 . formally , this critical point is @xmath411(higher orders ) , the higher orders being computable via the expansion described in the previous sections . for @xmath412 , we expect the attractor to be a smooth manifold of dimension two . the transition is illustrated in fig.[fig1 ] and [ fig2 ] for the simple case that @xmath33 and @xmath413 , in which case the expected critical point is @xmath414 . ( 100,30 ) ( 0,0 ) @xmath33 and @xmath413 . a ) : @xmath415 . b ) : @xmath416.,title="fig:",width=264 ] ( 50,0 ) @xmath33 and @xmath413 . a ) : @xmath415 . b ) : @xmath416.,title="fig:",width=264 ] ( 100,30 ) ( 0,0 ) @xmath33 and @xmath27 as in fig . a ) : @xmath417 . b ) : @xmath418.,title="fig:",width=264 ] ( 50,0 ) @xmath33 and @xmath27 as in fig . a ) : @xmath417 . b ) : @xmath418.,title="fig:",width=264 ] if @xmath419 , on the basis of numerical simulations , the attractor does not seem to display a transition from a fractal set to a smooth manifold . still , for suitable choices of @xmath419 , we expect the attractor to display a `` first order phase transition '' , located at the value of @xmath32 where @xmath420 , to be called again @xmath409 . at @xmath421 , the derivative of the hausdorff dimension of the attractor with respect to @xmath32 is expected to have a jump . a possible scenario is that the attractor is fractal both for @xmath422 and for @xmath423 , but it is `` smoother '' at larger values of @xmath32 , in the sense that its closure may be a regular , smooth , manifold of dimension two . an illustration of this smoothing " mechanism is in fig.[fig3 ] . ( 100,30 ) ( 0,0 ) . a ) : @xmath416 . b ) : @xmath424.,title="fig:",width=264 ] ( 50,0 ) . a ) : @xmath416 . b ) : @xmath424.,title="fig:",width=264 ] it would be interesting to investigate the nature of this transition in a more quantitative way , by comparing a numerical construction of the attractor with the theory proposed here , obtained by extrapolating the convergent expansion described in this paper to intermediate values of @xmath32 . such a comparison goes beyond the purpose of this paper , and we postpone the discussion of this issue to future research . one has @xmath426 for @xmath149 and @xmath427 for @xmath211 . given a tree of order @xmath428 one proceed by induction . assume that @xmath429 for all the trees @xmath430 of order @xmath431 and consider a tree @xmath172 of order @xmath83 . let @xmath355 be the special node of @xmath172 , and call @xmath432 the subtrees entering @xmath173 , with @xmath433 . if @xmath355 is a node of type 1 , then @xmath434 , so that the bound follows for @xmath435 . if @xmath436 , then the node preceding @xmath355 can not be of type 1 . call @xmath437 the subtrees entering @xmath438 , with @xmath439 . then one has @xmath440 for all @xmath441 . finally if @xmath355 is not a node of type 1 , then it has not to be counted and the argument follows by using the inductive bounds for the subtrees entering @xmath355 . moreover the bound in lemma [ lem:2.1 ] is optimal . indeed there are trees @xmath172 of order @xmath83 such that @xmath442 . define recursively the level @xmath443 of a node @xmath181 by setting @xmath444 if @xmath181 is an end - node and @xmath445 if at least one line entering @xmath181 exits a node @xmath29 with level @xmath446 . then consider a tree in which all nodes except the end - nodes are circles ( thais is of type 1 or 2 ) and have two entering lines except those with level @xmath165 which have only one entering line ; see figure [ fig:3.8 ] for an example with @xmath447 ( means we can have any kind of square node ) . for such trees one has @xmath448 , where @xmath449 , @xmath450 is the number of internal nodes and @xmath451 is the number of end - nodes . hence @xmath452 . @r=0.5 cm & & & * + [ o][f]@<-[r ] & * + < 3.0pt>[f**:black]+[f**:white ] + & & * + [ o][f]@<-[ru]@<-[rd ] & & + & & & * + [ o][f]@<-[r ] & * + < 3.0pt>[f**:black]+[f**:white ] + @<-[r ] & * + [ o][f]@<-[ruu]@<-[rdd ] & & & + & & & * + [ o][f]@<-[r ] & * + < 3.0pt>[f**:black]+[f**:white ] + & & * + [ o][f]@<-[ru]@<-[rd ] & & + & & & * + [ o][f]@<-[r ] & * + < 3.0pt>[f**:black]+[f**:white ] 10 adkmz a. arenas , a. daz - guilera , j. kurths , y. moreno , ch . zhou , _ synchronization in complex networks _ phys . rep . * 469 * ( 2008 ) , no . 3 , 93 - 153 . blech i.i . blekhman , _ synchronization in science and technology _ , asme press , new york , 1988 . bkovz s. boccaletti , j. kurths , g. osipov , d.l . valladares , c.s . zhou , _ the synchronization of chaotic systems _ , phys . * 366 * ( 2002 ) , no . 1 - 2 , 1 - 101 . bfg04 f. bonetto , p. falco , a. giuliani , _ analyticity of the srb measure of a lattice of coupled anosov diffeomorphisms of the torus _ , j. math * 45 * ( 2004 ) , no . 8 , 3282 - 3309 . eck - ruelle j .- eckmann , d. ruelle , _ ergodic theory of chaos and strange attractors _ , rev . modern phys . * 57 * ( 1985 ) , no . 3 , 617 - 656 . farmer - ott - yorke j.d . farmer , e. ott , j.a . yorke , _ the dimension of chaotic attractors _ , phys . d * 7 * ( 1983 ) , no . 1 - 3 , 153 - 180 . g. gallavotti , _ foundations of fluid dynamics _ , springer - verlag , berlin heidelberg , 2002 . gbg g. gallavotti , f. bonetto , g. gentile , _ aspects of the ergodic , qualitative and statistical theory of motion _ , springer , berlin , 2004 . ggg g. gallavotti , g. gentile , a. giuliani , _ resonances within chaos _ , chaos * 22 * , 026108 ( 2012 ) , 6 pages . gonzalez j.m . gonzlez - miranda , _ synchronization and control of chaos . an introduction for scientists and engineers _ , imperial college press , london , 2004 . kaplan - yorke l. kaplan , j.a . yorke , _ chaotic behavior of multidimensional difference equations _ , functional differential equations and approximation of fixed points , lecture notes in mathematics 730 , 204 - 227 , eds . peitgen , h .- o . walther , springer , berlin , 1979 . pcjmh l. pecora , th.l . carroll , g.a . johnson , d.j . mar , j.f . heagy , _ fundamentals of synchronization in chaotic systems , concepts , and applications _ , chaos * 7 * ( 1997 ) , no . 4 , 520 - 543 . a. pikovsky , m. rosenblum , j. kurths , _ synchronization . a universal concept in nonlinear sciences _ , cambridge university press , cambridge , 2001 .

we consider a three - dimensional chaotic system consisting of the suspension of arnold s cat map coupled with a clock via a weak dissipative interaction . we show that the coupled system displays a synchronization phenomenon , in the sense that the relative phase between the suspension flow and the clock locks to a special value , thus making the motion fall onto a lower dimensional attractor . more specifically , we construct the attractive invariant manifold , of dimension smaller than three , using a convergent perturbative expansion . moreover , we compute via convergent series the lyapunov exponents , including notably the central one . the result generalizes a previous construction of the attractive invariant manifold in a similar but simpler model . the main novelty of the current construction relies in the computation of the lyapunov spectrum , which consists of non - trivial analytic exponents . some conjectures about a possible smoothening transition of the attractor as the coupling is increased are also discussed . * _ keywords : _ * partially hyperbolic systems ; anosov systems ; synchronization ; phase - locking ; lyapunov exponents ; fractal attractor ; srb measure ; tree expansion ; perturbation theory .

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Proceed to summarize the following text: a great deal about galaxy evolution can be learned by studying their broadband properties . broadband observations give an immediate impression of the spectral energy distribution and thereby information on stellar and dust content . even though integrated magnitudes of galaxies can be used to study global properties of galaxies , even more can be learned from examining the detailed distribution of their light and colors . the star formation history in galaxies seems to be related to their surface density properties ( kennicutt @xcite ; ryder and dopita @xcite ; de jong @xcite ) , and therefore it is imperative to have a statistical knowledge of surface brightness distributions in galaxies to understand galaxy evolution . the image data set presented here was collected to study the surface brightness distribution of spiral galaxies . of especial interest was the question whether disks in spiral galaxies have a preferred central surface brightness value as proposed by freeman ( @xcite ) . the observations were made in such a way that they were suitable to study this central surface brightness effect , but this might make the observations less useful for some other studies due to two limitations . ( 1 ) disk central surface brightnesses are in general determined from one - dimensional ( 1d ) luminosity profiles , constructed by some kind of azimuthal averaging of the light distribution . no effort was made to obtain images with high signal - to - noise per pixel , as large numbers of pixels were to be averaged in the process of creating luminosity profiles . furthermore the depth " of the optical images were matched to the near - ir observations , which are more limited by the high sky background level than by signal - to - noise ratios . a considerable fraction of the images have too low signal - to - noise per pixel to allow detailed morphological studies of non - axisymmetric structures ( ie . bars and spiral arms ) except in the highest surface brightness regions . ( 2 ) the used telescope / camera combinations had a limited field - of - view , especially in the near - ir . often only the major axis was imaged of the larger galaxies , as this was sufficient to measure the radial luminosity distribution of the galaxy . this again limits the usefulness of the images to study non - axisymmetric light distributions in the outer part of galaxies . the structure of this paper is as follows : the selection of the sample is described in section 2 and the observations in section 3 . section 4 explains the different data reduction techniques used . in section 5 i describe the format of the fits images on the cd - rom , in section 6 the format of the luminosity profiles and in section 7 the format of the bulge / disk decomposition files . a more detailed description of the selection , observations and data reduction can be found in paper i. the bulge / disk decomposition methods are explained in more detail in paper ii . the galaxies were selected from the uppsala general catalogue of galaxies ( ugc , nilson @xcite ) . only spiral galaxies in the range s1-dwarfsp were selected , excluding galaxies with classifications as s0-s1 , sb0-sb1 , s3-irr , irr and dwarf irr . ideally one would like to have a volume - limited sample of galaxies for a statistical study of galaxy properties , but this is impossible due to selection effects . to create a sample that is correctable for selection effects , the galaxies were selected to have ugc red diameters of at least 2 . the galaxies have red ugc minor over major axis ratios larger than 0.625 to reduce problems with projection effects and dust extinction . this axis ratio range corresponds to inclinations less than approximately 51 . only galaxies with an absolute galactic latitude @xmath6 were selected , to minimize the effect of galactic extinction and to reduce the number of foreground stars . these selection criteria resulted in a sample of 368 galaxies . the final sample of 86 galaxies observed was selected on the basis of hour angle and declination only , in such a way that we had about equal number of observable galaxies during the whole night in the granted observing time . the total selected areas cover about 12.5% of the sky . all global parameters of the observed galaxies are listed in table [ globpar ] . nearly all _ bvri _ images were obtained with the 1 m jacobus kapteyn telescope ( jkt ) at la palma , equipped with a 385x578 gec ccd camera , in march and september 1991 and april 1992 . the kitt peak _ bvri _ filter set ( rgo / la palma technical notes @xcite ) was used , the pixel size was 0.3 . the ccd camera was used in both its normal imaging mode as well as in its driftscan mode . in driftscan mode , optimal use is made of the way ccds are designed : while the telescope is tracking the object , the ccd camera is shifted under the telescope at the same speed as the image is shifted down the columns of the ccd while it is read out . typical exposure times were 600s in @xmath0 and 400s for the other optical passbands . twilight flatfields were obtained at the beginning or at the end of the night and globular cluster fields with standard stars were observed at regular intervals through the night for calibration . a small number of optical observations were obtained from the la palma archive . the near - ir @xmath4 and @xmath5 passband observations were made at the united kingdom infrared telescope at hawaii with ircam ii containing a 58x62 insb array . during the february 1992 run standard @xmath4 and @xmath5 filters were used , but a @xmath7 filter was used in september 1991 . the pixel size was 1.2 . for accurate sky subtraction and flatfielding sky frames were obtained before and after every two object frames at a position offset a few arcmin from the object . images were taken in a strip along the major axis of the galaxies , spending about twice as much time on the outer part of galaxies than on the central region to increase signal - to - noise in these low surface brightness regions . calibration stars from the list of elias et al . ( @xcite ) were imaged at regular intervals . dark frames with exposure times equal to the object exposure times were also obtained at regular intervals . the full observing log with observing method ( driftscan , mosaic ) , exposure times , photometric quality and seeing estimates can be found in paper i. these values are also store in the fits headers of the images . the normal data reduction procedure for ccd data was followed to create calibrated images from the direct imaging data obtained with the jkt . a bias value was subtracted from the images using the average value in the overscan region . the images were divided by normalized flatfields created by averaging several twilight frames . no dark current was subtracted as this was found to be insignificant for this ccd . in general two observations at the same position of an object were made , which allowed cosmic - ray removal when they were averaged . the data reduction of the driftscans was more elaborate . a driftscan image consists of a ramp up part ( rows that were not exposed for a full chip length before being read out ) , a flat fully - exposed part and a ramp down part ( rows that are read out after the shutter has closed ) . the first rows of the ramp up part showed a gradient in the bias level in the cross - scan direction . therefore , the bias level was determined by fitting the first half of the ramp up part of each column , giving a bias level for each column at the first row . the images were flatfielded by flatlines created averaging normal flatfields in column direction . the ramp up and down parts were corrected for the shorter exposure times , extending the field - of - view beyond the area that was exposed to the sky for a full chip length . careful attention had to be given to flatfielding of the near - ir images , as flux levels 5@xmath8 times below the sky level were measured . we first subtracted the dark current from all near - ir images ( object and sky ) using the average of the two dark frames obtained nearest in time . a normalized flatfield image was created for each galaxy by taking the median of the 4 - 5 sky frames observed around the galaxy . after flatfielding , known hot " and dead " pixels were set to undefined " by a bad pixel mask and remaining dubious pixels were set to undefined by hand . these undefined " pixels were not used in further analysis . the different object frames of a galaxy were mosaiced together to create a full image along the major axis . the spatial offset between frames was determined by a cross - correlation technique or by using the telescope offsets if no structure was available to be used in the cross - correlation technique . the relative spatial offsets between all overlapping frame combinations were determined and a least - square - fit determined the relative offset of all frames with respect to the central frame . zero point ( due to sky fluctuations ) and intensity scaling factors ( only for non - photometric observations ) were determined in a similar fashion . all zero point ( and when necessary intensity ) offsets between overlapping frames ( using the just determined spatial offsets ) were calculated and a least - squares - fit through all relative offsets provided the intensity offset with respect to the central frame . all frames were mosaiced together using these spatial and intensity offsets , taking the average in the overlapping areas . the images were calibrated using the standard star fields observed during each night under different airmasses . the optical standard star fields we used were calibrated to landolt ( @xcite ) stars , and therefore our system response has been transformed to johnson @xmath0 and @xmath1 and kron - cousins @xmath2 and @xmath3 . the near - ir was calibrated to @xmath4 and @xmath5 standard stars of elias et al . ( @xcite ) , using the corrections of wainscoat and cowie ( @xcite ) to transform the @xmath7 passband to the @xmath5 passband . instrumental magnitudes ( -2.5log(number of counts ) ) of the different stars in the calibration fields were measured with daophot ( stetson @xcite ) . all photometric calibration measurements of one observing run were combined to least - square - fit equations of the form : @xmath9 where @xmath10 , @xmath3 , @xmath4 and @xmath5 are the standard star magnitudes , @xmath11 , @xmath12 , @xmath13 and @xmath14 the instrumental magnitudes per second , @xmath15 the airmass of the observation and @xmath16 the unknown transformation coefficients . the results of these fits can be found in tables [ mag0tab ] and [ mag0tabir ] and in the fits headers of the images . non - photometric observations were calibrated with aperture photometry from the literature when available . we first determined magnitudes in synthetic apertures of the size of the literature photometry using the calibration of a photometric night . if our magnitude differed more than the expected error from the literature value , all magnitude parameters were corrected for this difference ( indicated by header item corr in the fits files ) . the optical pixel size was determined to be 0.303@xmath170.004 , using images of globular clusters which contained accurately known star positions . this pixel size agreed to within its uncertainty to the instrumental specification , and therefore a value of 0.30 was adopted . the near - ir pixel size was derived from the scaling factor to align the near - ir images with the optical images ( see next paragraph ) . the near - ir pixel size was 1.20 per pixel . we determined the sky background level on the fully reduced images using the box method . average sky values were measured in small boxes around the galaxies . sky level was set to the median value of these measurements . the uncertainty in the sky value was taken to be half the difference between the maximum and minimum average sky values found in these boxes . this uncertainty will reflect errors due to imperfect flatfielding and mosaicing . we aligned the images in the different passbands using foreground stars in common between the different frames . images obtained during the same observing run were only allowed to shift , between different runs also rotation and scaling was allowed . the near - ir data was regridded to the much smaller pixel scale of the optical images , which means that nothing smaller than the original pixel size ( 1.2 ) should be trusted on these images . a linear interpolation was used for regridding and therefore the new smaller pixels contain values that are representative of the original surface brightness in the pixels of the original size . total flux in the image is not conserved in this process , but the original number of counts in an area can easily be calculated by multiplying the new number of counts in an area with the ratio of the square of the pixel sizes , ( pixelsize@xmath18/pixelsize@xmath19)@xmath20 . all aligned images are stored in fits format on the cd - rom in the directory images/ , with a separate directory for each galaxy . the aligned near - ir images in these directories have been compressed with gzip , but the `` raw '' near - ir images ( ie . before aligning and regridding to the optical images ) are available in uncompressed fits format in the directory irimages/. the fits headers contain all the essential information for analysis . the images are in analog - to - digital - units ( adu ) , which corresponds approximately to the number of detected photons for the optical images and to 50 detected photons in the near - ir images . undefined pixels in the images contain the value -999 . the header items of interest are as follows : naxis1 , naxis2 : : number of pixels in ra and dec respectively ctype1 , ctype2 : : ra - tan , dec - tan axis type and projection system crval1 , crval2 : : should contain the ra and dec value at the reference pixel ( * crpix1 , crpix2 * ) , but as the exact position of the galaxies was often unknown , the stored values have no meaning cdelt1 , cdelt2 : : the pixel size in _ degrees_. the same value is stored in arcseconds in header item * pixsizim * filter : : passband filter ( b , v , r , i , h , k or k@xmath21 ) seeing : : full - width - at - half - maximum ( fwhm ) of seeing estimate in arcsec phot : : photometric quality estimate as in paper i ( 1 : photometric , 2 : 0.0 - 0.2 mag , 3 : 0.2 - 0.5 mag , 4 : 0.5 - 1.0 mag and 5 : @xmath221.0 mag error ) qual : : quick look quality estimate , taking into account ( in order of importance ) flatfield quality , area to measure the sky level , signal - to - noise and seeing . the numbers mean , 1 : excellent , 2 : reasonable , but take into account some of the limitations such as limited sky area , 3 : poor , do not use except in case of an emergency mag0 : : zero point calibration constant for a 1 second exposure ( -@xmath23 in eq . [ caleq ] ) ccol : : color calibration constant , when not used 0 ( @xmath24 ) col : : average color of this galaxy used for calibration cair : : airmass calibration constant ( @xmath25 ) airmass : : airmass during the observation ( @xmath15 ) corr : : correction for non - photometric observation to put this image on literature photometry pixsize : : pixel size in arcsec of original image ( before rebinning ) pixsizim : : pixel size in arcsec of this image ( after rebinning / aligning ) exptime : : exposure time calibration constant ( if several images were averaged , this contains the average exposure time ) skylev : : estimate of the sky background level in adu skyerr : : maximum uncertainty in sky background magoff : : for convenience , this constant gives the calibration to convert pixel adu values into mag arcsec@xmath26 . it is equal to * @xmath27mag0@xmath27ccol@xmath28col@xmath27cair@xmath28airmass @xmath27 corr @xmath29pixsize@xmath20@xmath28exptime * ) . the surface brightness in mag arcsec@xmath26 of a pixel with adu counts in the galaxy is * magoff*@xmath30pixel@xmath31adu@xmath32*skylev*@xmath33 . to use this constant to calculate the magitude in an area , take into acount that flux was not conserved per area in the rebinning / alligning proces . the magnitude in an area with total of adu counts is * magoff@xmath30*area@xmath31adu@xmath32*skylev@xmath34pixsize@xmath35pixsizim@xmath36 * the radial luminosity distribution of each galaxy was determined in each passband and these are also present on the cd - rom . the areas in the @xmath2 passband images affected by foreground stars were masked using a polygon editor . this mask was transfered to the other passbands , thus making certain that the same area was used in all passbands . the center of the galaxy was determined by fitting an ellipse to the central peak in the @xmath2 passband image . next , with this center fixed , ellipses were fit to the isophotes at the 23.5 , 24.0 and 24.5 @xmath2-mag arcsec@xmath26 level . the median values found for the minor / major axis ratio ( @xmath37 ) and position angle ( pa ) in the @xmath2-band were used in all passbands to determine the luminosity profiles . average adu values were determined in concentric elliptical annuli of increasing radius with the already determined center , @xmath37 and pa fixed . for face - on galaxies this method gives a better estimate of the average luminostity at each radius than methods which freely fit ellipses at each isophote , if we assume that the galaxy is not strongly warped . bars , spiral arms and hii regions make isophote fitting methods unreliable for face - on spiral galaxies . the profiles are provided in ascii in the directory profiles/ and the graphs can be found in paper i. the surface brightness profiles are in mag arcsec@xmath26 , the radii in arcsec . undefined values are indicated by a . note that the central regions of ugc7540 were saturated in the @xmath1 , @xmath2 and @xmath3 passband . further header information in these files are incl : : inclination in degrees ( actually cos@xmath38(@xmath37 ) ) used for profile extraction pa : : position angle in degrees used for profile extraction , measured from north to east exptime : : exposure time in seconds of image used magoff : : magnitude calibration constant ( see image catalog ) magsky : : sky surface brightness in mag arcsec@xmath26 magskyerr : : uncertainty in * magsky * magtot : : total apparent magnitude of the galaxy derived from the surface brightness profile ( see paper i ) magerr : : uncertainty in apparent magnitude seeing : : fwhm of seeing estimate in arcsec date : : date of observation photq : : photometry quality estimate ( see image catalog ) a number bulge / disk decomposition methods was applied to the data ( see paper ii for details ) and the results are stored in directory b_dratio/. the results of the 1d profile decompositions with @xmath39 , @xmath40 and exponential bulges can be found in the files bd4qfpar.dat , bd4ffpar.dat and bd4efpar.dat respectively . the results of the 2d decompositions with exponential bulges and disks and with freeman bars can be found in bd4fpar.dat . note that not all observations were photometric and that for non - photometric observations the listed numbers are the lower limits in surface brightness flux . obviously the scale parameters are correct for the non - photometric observations . check the file pht.dat for a listing of the photometric quality of the observations . the description of all the columns in these files can be found in file bd4read.me . this research was supported under grant no . 782 - 373 - 044 from the netherlands foundation for research in astronomy ( astron ) , which receives its funds from the netherlands foundation for scientific research ( nwo ) . this paper is based on observations with the jacobus kapteyn telescope and the isaac newton telescope operated by the royal greenwich observatory at the observatorio del roque de los muchachos of the instituto de astrofsica de canarias with financial support from the pparc ( uk ) and nwo ( nl ) and with the uk infrared telescope at mauna kea operated by the royal observatory edinburgh with financial support of the pparc . .global parameters of the galaxies in the observed sample . the positions and the @xmath41 recession velocities ( cz ) are obtained from the rc3 catalog , @xmath42 is the red ugc major axis diameter , @xmath37 is the red ugc minor over major axis diameter ratio . [ globpar ] [ cols="<,<,>,>,>,>,>,>,^,^,^,^,>",options="header " , ] cccc + passband & zero - point ( @xmath23 ) & color coef . ( @xmath24 ) & extinction coef . ( @xmath25 ) + + @xmath0 & -22.251@xmath170.065 & -0.062@xmath170.011 & 0.251@xmath170.027 + @xmath1 & -22.791@xmath170.032 & -0.013@xmath170.007 & 0.216@xmath170.030 + @xmath2 & -22.883@xmath170.030 & -0.001@xmath170.010 & 0.179@xmath170.020 + @xmath3 & -22.060@xmath170.045 & -0.012@xmath170.015 & 0.058@xmath170.058 + + @xmath0 & -21.757@xmath170.111 & -0.161@xmath170.044 & 0.238@xmath170.065 + @xmath1 & -22.215@xmath170.067 & -0.048@xmath170.024 & 0.135@xmath170.025 + @xmath2 & -22.438@xmath170.073 & -0.016@xmath170.046 & 0.141@xmath170.020 + @xmath3 & -21.709@xmath170.081 & -0.034@xmath170.057 & 0.081@xmath170.082 + + @xmath0 & -21.977@xmath170.122 & -0.161@xmath170.044 & 0.279@xmath170.052 + @xmath1 & -22.322@xmath170.072 & -0.048@xmath170.024 & 0.121@xmath170.030 + @xmath2 & -22.558@xmath170.064 & -0.016@xmath170.046 & 0.126@xmath170.026 + @xmath3 & -21.833@xmath170.068 & -0.034@xmath170.057 & 0.023@xmath170.027 + + @xmath0 & -22.157@xmath170.041 & -0.067@xmath170.013 & 0.294@xmath170.011 + @xmath1 & -22.697@xmath170.019 & -0.033@xmath170.005 & 0.198@xmath170.005 + @xmath2 & -22.768@xmath170.036 & -0.002@xmath170.018 & 0.170@xmath170.010 + @xmath3 & -22.063@xmath170.038 & -0.008@xmath170.027 & 0.118@xmath170.012 + ccc + color & zero point ( @xmath23 ) & extinction coefficient ( @xmath25 ) + + @xmath4 & -20.500@xmath170.200 & + @xmath43 & -20.018@xmath170.040 & 0.087@xmath170.032 + + @xmath4 & -20.704@xmath170.032 & 0.147@xmath170.048 + @xmath5 & -20.497@xmath170.032 & 0.119@xmath170.047 +

fits images in the @xmath0 , @xmath1 , @xmath2 , @xmath3 , @xmath4 and @xmath5 passbands are presented of a sample of 86 face - on spiral galaxies . the galaxies were selected from the ugc to have a diameter of at least 2 and a minor over major axis ratio larger than 0.625 . the selected galaxies have an absolute galactic latitude @xmath6 , to minimize the effect of galactic extinction and foreground stars . nearly all _ bvri _ data were obtained with the 1 m jacobus kapteyn telescope at la palma and the @xmath4 and @xmath5 data were obtained at the 3.8 m uk infra - red telescope at hawaii . the field of view of the telescope / camera combinations were often smaller than the observed galaxies , therefore driftscanning and mosaicing techniques were employed to image at least along the major axis of the galaxies . most images were obtained during photometric nights and calibrated using standard stars . a small fraction of the images was calibrated from literature aperture photometry . the azimuthally averaged radial luminosity profiles derived from these galaxy images ( see de jong and van der kruit @xcite , paper i ) are also made available in machine readable format , as are the results of the bulge / disk decompositions described in de jong ( @xcite , paper ii ) . a detailed statistical analysis of the bulge and disk parameters of this data set can be found in de jong ( @xcite , paper iii ) . the dust and stellar content of the galaxies as derived from the color profiles is described in de jong ( @xcite , paper iv ) . evidence for secular evolution as found in this sample is shown in courteau , de jong and broeils ( @xcite ) . * keywords : * surveys - galaxies : fundamental parameters - galaxies : photometry - galaxies : spiral - galaxies : structure

You are an expert at summarizing long articles. Proceed to summarize the following text: although the number of detected @xmath0-ray bursts has increased tremendously due to the batse instrument on board the compton @xmath0-ray observatory @xcite , unique identifications and quick follow - up observations of the celestial object that harbours the burst site are rare . this issue has been successfully addressed by the bepposax satellite , that monitors a large fraction of the sky ( @xmath2 ) in the intermediate x - ray range @xcite and is able to derive burst localisations with an accuracy of @xmath3 . with such improved localisations , the beppo - sax team has succeeded in detecting x - ray afterglows @xcite , by which valuable insight into the emission mechanism was gained . finding @xmath0-ray burst counterparts and performing spectroscopy at wavelengths other than the @xmath0-ray band during the active phase of a burst as well as providing good localisations for follow - up observations is the basic scientific motivation of the hete-2 mission . this publication is structured as follows : after a brief description of hete-2 s instrumentation in section [ cont : instruments ] , the position reconstruction algorithm is outlined in section [ cont : algorithm ] . the performance of the algorithm on monte carlo generated events as well as on data taken in astrometric observations of the x - ray source _ scorpius x-1 _ is presented in section [ cont : result ] . in section [ cont : psf ] findings on the point spread function are shown . a summary in section [ cont : conclusion ] concludes the paper . the high energy transient explorer ( hete-2 ) is a dedicated mission for the localisation and spectroscopy of @xmath0-ray bursts . details of the instrumentation of hete-2 and the mission are given in @xcite . its scientific payload consists of three experiments : _ fregate _ , a scintillation crystal experiment , that is expected to provide triggers because of its high sensitivity and large sky coverage , _ sxc _ , a coded - mask imager based on a x - ray ccd chip design , which is able to localise @xmath0-ray bursts with very high spatial resolution , and the core experiment _ wxm_. the wide field x - ray monitor _ wxm _ consists of two perpendicularly oriented 1-dimensional coded mask cameras , in which photons are detected by position sensitive proportional counters . the detectors , one pair for each of the two orthogonal systems , are filled with xenon gas at a pressure of @xmath4 with an admixture of @xmath5 carbon dioxide as a quenching gas . each detector contains three carbon wire anodes in which the position of an absorbed photon is inferred by comparing the accumulated charges at both ends of a wire . _ wxm _ achieves a position resolution of @xmath6 ( fwhm ) at @xmath7 . a veto system below the counting wires reduces the background due to charged particles . _ wxm _ is sensitive to photons with energies @xmath8 in the range @xmath9 with an relative energy resolution of @xmath10 at @xmath11 . the mask pattern , identical for the @xmath12- and @xmath13-system , is placed @xmath14 above the detectors and is composed of 103 elements , each @xmath15 in width . on third of the elements are open and randomly distributed . the parameters have been optimised with respect to source localisation accuracy for the expected levels of signal and background photon count rates @xcite . the combined @xmath12- and @xmath13-systems are monitoring a field of view of approximately @xmath16 . figure [ fig : hete ] shows the actual mask pattern of a _ wxm _ camera . further details of the _ wxm _ cameras can be found in @xcite . in order to provide other experiments with precise localisations of @xmath0-ray burst sites within seconds after burst onset , an array of twelve burst alert stations has been installed below hete-2 s flight path . once received , the information is relayed to the grb coordinate network ( gcn ) , from where it may be obtained by interested observers . for the sake of readability , the algorithm is described for determining the angle of incidence @xmath17 in the @xmath12-detector , completely analogous formulae apply to the @xmath13-system . figure [ fig : angles ] provides the definition of all quantities involved . the task of localising a single point source is performed by computing the correlation function @xmath18(x)$ ] of the mask pattern @xmath19 and the recorded intensity distribution @xmath20 ; the asterisk denoting complex conjugation : @xmath21(x)=\int m(\lambda)\cdot f^*(\lambda+x ) d\lambda\mbox{. } \label{eqn : correlation}\ ] ] the correlation function @xmath22(x)$ ] will peak at a value @xmath23 , indicating the distance @xmath20 is shifted with respect to @xmath19 . this yields the angle of incidence @xmath17 by @xmath24 , where @xmath14 is the distance between mask and detector . in @xmath19 , open elements of the mask pattern have been assigned a value of @xmath25 , whereas closed elements correspond to a value of @xmath26 , where @xmath27 is the open fraction of hete-2 s imaging system . this method is known as balanced correlation and removes the correlation background . there exist more sophisticated correlation schemes such as unbiased balanced correlation ( e.g. @xcite , @xcite ) , that do take into account partial shadowing of the mask onto the detector by bursts in the periphery of the field of view and increased coding noise caused by steady strong off - axis sources . for hete-2 s flight algorithm we rely on basic balanced correlation for three reasons : the width of the mask is almost twice as large as the active area of the detector and only bursts at angles @xmath28 off the optical axis are affected by partial shadowing . furthermore , the running average of the mask s transparency is close to constant with deviations being very small , so that the modulation of the pattern is insensitive to angle of incidence . thirdly , in hete-2 s nominal survey mode , the satellite is rotated in such a way that strong x - ray sources are avoided and the assumption of isotropically incidenting background radiation is valid . in reality the correlation is complicated by imperfections of the detector : the photons travel a finite path before interacting and the loci of photons interacting in the detector are measured with finite position resolution ( in the case of _ wxm _ to @xmath6 , subtending an angle of @xmath29 ) . whereas the finite position resolution simply results in a blurring of the image , the penetration of energetic photons effectively induces an additional shift in the image , if the event is happening at nonzero angles of incidence , and causes the angles of incidence to be overestimated . this effect increases in a complex fashion with distance from the center of the field of view and is of the order of @xmath30 at @xmath31 from the optical axis for a photon distribution following a power law spectrum with a spectral index of @xmath32 . in contrast to a time consuming monte carlo simulation , in which one determines the image of the mask pattern under estimated angles of incidence , the algorithm presented in this article aims at modelling all effects involved in the delapidation of the mask pattern , enabling a much faster localisation of the @xmath0-ray burst site . imaging aberrations may be described by applying suitable changes to the mask pattern @xmath19 prior to the calculation of the correlation function . this can be archieved by convolving the mask pattern @xmath19 with integration kernels @xmath33 and @xmath34 , that describe the penetration of highly energetic photons and the detector resolution , respectively , @xmath35(x)=\int m(\lambda)\cdot p(x-\lambda ) d\lambda\mbox{. } \label{eqn : convolution}\ ] ] the convolution kernel @xmath33 that is used for describing the penetration of photons is given by equation [ eqn : penetrate ] : @xmath36\cdot x\right)\mbox { , } \label{eqn : penetrate}\ ] ] where @xmath37 is the heaviside function and _ sgn _ the signum function the convolution yields an altered mask pattern @xmath38(x)$ ] , in which the penetration of photons is incorporated . the length scale @xmath39 of the exponential decay , given by equation [ eqn : ceqn ] , is equal to the spectrally averaged attenuation length @xmath40 of photons inside the detector , corrected by a projection factor : @xmath41 estimates for the angles of incidence @xmath17 and @xmath42 follow from a source localisation with the unmodified mask pattern @xmath19 in a first step . the finite spatial resolution of the position sensitive proportional counter is described by convolution with a gaussian integration kernel as in equation [ eqn : resolution ] : @xmath43 the standard deviation corresponds to a value for full width half maximum of @xmath6 . in contrast to @xmath40 , @xmath44 depends only very weakly on the spectral distribution of the incident photons and is considered to be constant . after convolution with @xmath34 , the mask pattern has been modified to @xmath45(x)$ ] . @xmath46 is the image of the mask pattern under ideal statistics and the source reconstruction with @xmath46 should not display any deviation from the ideal behaviour . the correlation @xmath47(x)$ ] yields a corrected value @xmath48 for the angle of incidence . due to their high numerical complexity it is favourable to perform both the convolutions and the image localisation by correlation in the fourier domain . the nomenclature is such that lower case letters denote functions in real space and the corresponding upper case letters their fourier transforms : @xmath49=\frac{1}{2\pi}\int m(x)\exp(-ikx)dx\mbox{. } \label{eqn : fourierpair}\ ] ] convolutions and correlations reduce by virtue of equation [ eqn : fconvolve ] to mere multiplications in fourier space . the asterisk denotes complex conjugation : @xmath50(x)&=&\mathcal{f}^{-1}\left[p(k)m(k)\right]\mbox{,}\\ \left[f\otimes m''\right](x)&=&\mathcal{f}^{-1}\left[f(k)^*m''(k)\right]\mbox{. } \end{array } \label{eqn : fconvolve}\ ] ] diagram [ dia : cc ] summarises all steps . starting from the fourier - transform of the mask pattern @xmath51 $ ] , both convolutions and the correlation are carried out in fourier - space by determining the product @xmath52 . inverse transformation yields the correlation function @xmath53 $ ] , from which the corrected angle of incidence @xmath48 is derived : @xmath54 in comparision to the _ wfc _ camera on board the bepposax satellite , the penetration effect is noticably more pronounced in the case of hete-2 . evaluating the average attenuation length for a photon at the corner of the field of view projected onto the detector plane determines the angular deviation in the worst case . this value for _ wfc _ is @xmath55 times smaller compared to _ wxm _ , because of the narrower field of view , the higher gas pressure , which results in a shorter average attenuation length and the larger distance between the mask and detector . furthermore , being a genuine 2-dimensional camera , the aberration vector field of _ wfc _ is radial and does not have the complex angular dependence described by equation [ eqn : ceqn ] . in order to investigate to which extend the algorithm is capable of correcting systematic deviations , a set of @xmath56 monte carlo images containing a large number of photons ( @xmath57 ) was generated , so that the statistical scatter in the source reconstruction is small . the simulated burst positions were randomly distributed , the angles of incidence ranging from @xmath58 to @xmath59 . the number of incident photons was corrected with a factor of @xmath60 for the purpose of coping for the decreasing projected area of the detector at increasing angles of incidence . the spectral distribution of photons followed a power law with spectral index of @xmath32 , which is considered to be shallow , but not untypical for @xmath0-ray bursts @xcite . because of the flatness of the spectrum , the aberration effect due to penetrating photons is distinctive and easy to study . localisation of photons by the proportional counter was modelled by employing detector response data taken prior to the launch . figure [ fig : pure ] shows the deviation @xmath61 of the uncorrected reconstructed burst position from the nominal burst position as a function of angle of incidence @xmath62 . the aberration increases with increasing distance @xmath62 from the center of the field of view . each point corresponds to the localisation derived from a monte carlo image . furthermore , one observes a widening of the scatter of @xmath61 with increasing distance from the optical axis . this is due to the fact that the aberration , being proportional to the function @xmath63 to first order , is concave for @xmath64 and convex for @xmath65 . the magnitude of this curvature increases with increasing distance from the origin . in order to disentangle the contributions from statistical scatter in the reconstruction from residual systematics , the dataset has been split into @xmath66 intervals in @xmath67 . for each interval @xmath68 the mean value @xmath69 of @xmath70 as well as the standard deviation @xmath71 has been derived . because the intervals are chosen such that they cover an angular range of just @xmath72 , the scatter @xmath71 can be assumed to be dominated by statistics . an increase in photon number @xmath73 per image would have reduced the statistical scatter , according to @xmath74.from the values @xmath69 we derive the quantity @xmath75 , which is the mean quadratic deviation from zero . further on , from @xmath71 we determine the average statistical scatter @xmath76 . from those two quantities , we define the spot size @xmath77.figure [ fig : mcdata_depend ] illustrates the performance of the applied corrections : the mean quadratic deviation @xmath69 as well as the statistical error @xmath71 is plotted against the angle of incidence @xmath67 . as the flatness indicates , @xmath69 is consistent with zero , i.e. the imaging aberrations have been properly corrected . moreover , @xmath71 has become independent from the angle of incidence @xmath67 , which indicates that the second order influence of @xmath78 on the reconstruction has been properly accounted for . residual deviations are caused by detector nonlinearities , especially in the charge division method by which the position of an absorbed photon is determined . the mean statistical scatter has been determined to @xmath79 and the mean systematic deviation from zero to @xmath80 . the algorithm thus fulfills the design requirements . figure [ fig : mcdata_spot ] shows the result of combining the reconstructions in @xmath12- and @xmath13-direction . the scatter plot is circularly symmetric and the width of the spot corresponds to @xmath81 . in the above analysis , the value for the average attenuation length @xmath40 has been optimised is such a way that @xmath82 assumed a minimal value . the smallest deviations resulted in the choice @xmath83 , which is in agreement with the theoretical value @xmath84 . the theoretical value was derived by determining the flux weighted average of tabulated values for the photon attenuation length in xenon , where the photon spectrum has been altered by the spectral photon absorption probability . data from the x - ray source _ scorpius x-1 _ has been taken for calibration purposes . due to hete-2 s antisolar pointing , any source will drift through the field of view at ecliptic rate , i.e. @xmath85 . the orientation of the satellite was such that the _ wxm _ coordinate system was rotated @xmath86 relative to the celestial @xmath87 frame . _ scorpius x-1 _ was viewed at angles of incidence ranging from @xmath88 to @xmath89 and at @xmath90 fixed . the spacecraft attitude was known to an accuracy surpassing @xmath1 . as pointed out in @xcite , the spectrum of _ scorpius x-1 _ , being a low - mass x - ray binary , can be described by a multicolour blackbody spectrum . due to the integration time of @xmath91 , short term spectral changes like quasi - periodic oscillations average out . long term variabilities either happen on significantly longer time scales compared to the data taking , or they affect energies outside hete-2 s sensitivity interval . determining the average attenuation length by a photon flux weighted integral with tabulated values for the attenuation length yielded @xmath92 . thus , the spectrum of _ scorpius x-1 _ is much softer compared to the previous case . the background subtracted images oconsisted roughly of @xmath57 photons . the analysis [ ana : mc ] is repeated for the _ scorpius x-1 _ dataset . for each pointing @xmath69 and @xmath71 have been derived . figure [ fig : scodata_depend ] illustrates the accuracy of the source localisation algorithm . again , the values @xmath69 with their errors @xmath71 are shown as a function of angle of incidence @xmath67 . the average quadratic deviation from zero has been determined to @xmath93 which is slightly larger than in the previous case , but reflects inaccuracies in the satellite pointing . furthermore , this value is probably underestimated because of the incomplete sampling of the field of view . the statistical error is marginally smaller due to larger photon numbers and has been derived to be @xmath94 . figure [ fig : scodata_spot ] shows the combined localisation in @xmath12- and @xmath13-direction , converted into the celestial coordinate frame . the radius of the spot is @xmath95 . the average attenuation length @xmath40 has been varied to yield the minimum value for @xmath82 . in the case of _ scorpius x-1 _ , we find the optimum value to be @xmath96 , which corresponds well to the theoretically expected value @xmath92 . in order to assess the extend to which angular resolving power is affected by the source localisation algorithm , the point spread function of the imaging system has been derived . for a coded mask camera like _ wxm _ , the point spread function is equal to the autocorrelation @xmath97 of the mask pattern @xmath46 , i.e. @xmath98(x)$ ] . to be exact , @xmath97 describes the isotropised point spread function . anisotropies provoke asymmetries in the central correlation peak of the function @xmath99(x)$ ] . again , the determination of @xmath97 is carried out in fourier - space . diagram [ dia : psf ] illustrates the derivation of @xmath100 $ ] : @xmath101 the width @xmath102 of the autocorrelation function @xmath97 has been determined by fitting a gaussian to the central peak . the average width @xmath103 of the point spread function has been defined as the geometric mean of its extend in @xmath12- and @xmath13-direction ( @xmath104 ) , because the quantity of interest is the size of the spot a source is imaged onto . the value of @xmath97 at the origin corresponds to the correlation strength @xmath105 . accordingly , the average correlation strength @xmath106 has been defined to be equal to the arithmetic mean of @xmath105 and @xmath107 because of its analogy to brightness . the angles of incidence have been restricted to @xmath108 , in order to ensure that the mask pattern is fully imaged onto the detector . the source positions were arranged on a square lattice with mesh size of @xmath109 . the average attenuation length @xmath40 has been set to , corresponding to a power law spectrum with spectral index @xmath32 . the left plot in figure [ fig : psf ] illustrates , that the width @xmath103 of the autocorrelation function increases and how the correlation strength decreases correspondingly at larger distances from the optical axis . at increasingly larger distance from the optical axis , the functions @xmath33 and @xmath110 disperse the correlation peak . because the area underneath the peak has to be conserved , the correlation strength @xmath106 drops accordingly . thus , the angular resolution power has decreased thus by @xmath111 for a burst at @xmath112 compared to a burst on the optical axis . [ cols="^,^ " , ] b.m.s . wishes to thank y. shirasaki for help on the monte carlo generator . the support of the german national merit foundation is greatly appreciated . we thank all members of the hete-2 team , who participated in the construction and operation of the satellite as well as the anonymous referee for valuable comments . 00 w. s. paciesas at al . , preprint astro - ph/9903205 r. jager et al . , preprint astro - ph/9706065 l. piro et al , astron . astrophys . 329 , 906 - 910 ( 1998 ) e. costa et al . , preprint astro - ph/9796965 n. kawai et al . , astron . 138 , 563 - 564 ( 1999 ) j.j.m . int zand et al . , proceedings of imaging in high - energy astronomy , capri ( 1994 )

the high - energy transient explorer ( hete-2 ) , launched in october 2000 , is a satellite experiment dedicated to the study of @xmath0-ray bursts in a very wide energy range from soft x - ray to @xmath0-ray wavelengths . the intermediate x - ray range ( 2 - 30 kev ) is covered by the wide - field x - ray monitor _ wxm _ , a coded aperture imager . in this article , an algorithm for reconstructing the positions of @xmath0-ray bursts is described , which is capable of correcting systematic aberrations to approximately @xmath1 throughout the field of view . functionality and performance of this algorithm have been validated using data from monte carlo simulations as well as from astrometric observations of the x - ray source _ scorpius x-1_. epsf , x - ray astronomy , coded - mask imaging , @xmath0-ray bursts 95.55.ka,98.70.rz,95.75.mn

You are an expert at summarizing long articles. Proceed to summarize the following text: the introduction of reflective optical elements revolutionized the design and imaging capabilities of telescopes @xcite . while the possibility of creating large diameter objectives was one of the primary aims , a significant advantage is that reflective optical elements provide a means of reducing or eliminating the spherical and chromatic aberrations that are inherent in lens based systems . here we explore the opposite extreme of microfabricated optics . microfabricated optical elements are key components for the development and integration of optics into a range of research and commercial areas @xcite . to date , the majority of the work in microphotonics has been in refractive elements , i.e. microlenses . however , in this regime microlenses typically have significant numerical apertures and surface curvatures , which introduce large aberrations . a number of groups have in recent years discussed the design and fabrication of concave micromirrors @xcite . these examinations are largely driven by two purposes ; for optical tweezing , and for integration into atom optics . spherical mirrors have been demonstrated to collect light from single ions @xcite . parabolic mirrors , similar to those described here but with larger length scales , have been also been used as highly efficient collectors of light from single ions @xcite , atoms @xcite , and point sources @xcite . in a similar manner they may be used to tightly focus light onto atomic samples , which has to date been shown with refractive optics @xcite . in this work we consider the use of reflective micro optical components for focusing light . we present the construction and optical characterization of parabolic reflectors with an open aperture of radius 10 @xmath0 m and measured focal lengths that range from 24 @xmath0 m to 36 @xmath0 m . detailed mapping of the focused intensity field is made possible by the development of a previously unreported adaptation of a confocal microscope that allows the illumination of the reflector with collimated light , while still maintaining the highly - desirable large - numerical - aperture confocal collection . using this device we obtain 3d data about the focal plane demonstrating diffraction limited focussing . we also discuss the application of the parabolic mirror for use in atomic physics and tweezing experiments . the details of the fabrication of concave paraboloid structures through ion - beam milling are covered in @xcite . briefly , a focused ion - beam ( gallium ions with typical currents of 50 - 300 pa and accelerating voltages of 30 kv ) is used to precisely sculpt a silicon substrate with the required mirror profile , which is subsequently coated with gold to provide a highly - reflective coating . in focused ion - beam milling , controlling the dose of ions to which an area is exposed allows a region of the surface to be sputtered to a known depth , due to a linear relationship between depth and dose in silicon . the applied dose is a function of the beam current , the dwell time , and number of passes the beam makes over an area . by tracing a number of concentric discs of increasing radius , whilst linearly increasing the dose , a parabolic depression can be milled into the substrate . in principle this would create a stepped contour , however due to edge - effects of the milling process , as well as redistribution of etched material , a larger number of passes creates a smooth contour of the parabolic dish . further details of the construction and characterization can be found in @xcite , where an rms roughness of 4.0 nm was measured by afm over the range of the concave parabolic surface . due to the identical manufacturing process similar values are expected in this work . the propagation of light using microscopic optical elements is a well represented topic in the literature . here we restrict our discussion to the behavior of light fields after wavelength scale apertures and curved surfaces . in their work , goldwin and hinds @xcite derive analytic results for spherical mirrors , which they further compare with numerical integration of maxwell s equations . meanwhile , bandi _ et . _ consider the propagation of light after a wavelength - scale aperture using a fresnel representation of the fields @xcite , which offers the possibility of adding focusing to the formalism . here , however , the reflected field was modeled using the angular spectrum method @xcite , which provides a mapping of an electric field from a particular plane into a secondary plane ; @xmath2 where @xmath3 are the cartesian spatial directions , @xmath4 is the wavevector in the @xmath5-direction , @xmath6 is the electric field , and @xmath7 is the 2d fourier spectrum of the electric field in the plane @xmath8 . the result of this compact equation can be understood by first noting that the spatial spectrum of an electric field , @xmath9 can be translated from a plane @xmath10 to another plane @xmath8 using the helmholtz propagator , @xmath11 where the helmholtz propagator in reciprocal space is @xmath12 , @xcite . we then note that the electric field can be calculated from its spatial spectrum in a plane are by the inverse fourier transform , @xmath13 , @xmath14 these relations , eq . ( 2 ) and eq . ( 3 ) then clearly show the result of eq . ( 1 ) , and can be used to calculate the electric field in an arbitrary plane , given that is is known in one plane . the essential details of calculating the final intensity profile using the angular spectrum method can be clearly seen in fig . [ fig : theory_schematic](c ) ; the initial electric field is fourier transformed , the helmholtz propagator is then applied , before the inverse fourier spectrum is taken . finally , the resulting intensity field is found from the modulus - squared of the electric field . example codes are available on request from the corresponding author . for this work , in the plane of @xmath10 a curvature is numerically added to the phase - fronts within the area of the aperture of the mirror . the curvature is expressed as @xmath15 , where @xmath16 is the total wavevector , @xmath17 , @xmath18 is the radial distance from the center of the mirror , @xmath19 is the mirror aperture radius , and @xmath20 is the focal length . this modification of the optical wavefront represents the position dependent optical path - length difference , due to the spatially varying propagation distance to the metal surface , fig . [ fig : theory_schematic ] . care must be taken to include all modes of the field , including the evanescent ones , in order to return the full field simulation . all simulations were performed in matlab with the focal length , @xmath20 and the position of the surface as free parameters for fitting to the measured data . although transmission confocal imaging provides a powerful means of examining microlenses @xcite , reflection images of focusing elements require post analysis to interpret the data . in traditional reflection confocal microscopy the illuminating laser is focused onto the imaging plane at a point that is confocal with a pinhole in the imaging axis of the microscope . while this allows accurate probing of the surface it precludes imaging the focal plane of our concave mirrors directly . instead one may directly image about the center of curvature of the mirror and then infer the transfer function of the mirror assuming the point spread functions of the other optical elements are known . in this work we devise a scheme that allows interrogation of the focus while illuminating a mirror with a collimated laser beam , fig . [ fig : confocal_schematic ] . this arrangement means that the intensity distribution that would be formed in an optical tweezing experiment @xcite , for example , can be faithfully interrogated . probing the focal plane of the parabolic mirrors requires illumination with collimated light co axial with the mirror axis ( orthogonal to the substrate surface ) . the working distance of the objective lens used ( olympus uplansapo 40x/0.9@xmath21 ) is only 180 @xmath0 m , which severely limits introducing further optical elements before the object . to circumvent these restrictions a non - polarizing beam splitter ( npbs ) was introduced between the objective lens and the scanning column of a commercial confocal microscope ( olympus fv1000 scan head with olympus ix81 inverted microscope ) . this enabled an external probe beam to be introduced through the back aperture of the objective lens using the npbs to combine the beam into the imaging axis , see fig . [ fig : confocal_schematic ] . the probe , a commercial solid - state 589 nm laser ( based on sum - frequency generation using two lasing lines of an nd : yag laser ) , was focused , after fiber - coupling to clean up the spatial mode , at the back - focal plane of the objective lens . the specific probe wavelength was chosen for convenience ( good detector efficiency and high optical power in a single transverse mode ) - the numerical simulations were also conducted for this wavelength to account for wavelength - dependent diffraction effects . the collimation of the beam emitted was optimized by varying the axial position of the coupling lens . the npbs , optical fiber , and coupling lens were mounted in a holder with three axis adjustment , which allowed for overlapping of the optical axes . the emitted beam had a diameter of 1 mm , justifying the assumption of a spatially uniform intensity profile across the micromirror aperture ( 20 @xmath0 m diameter ) . the magnification of the combined optical system was calibrated by measuring the aperture diameter of the mirrors . in order to visually verify the surface quality of the mirrors , as well as to measure the aperture diameter of the mirrors the surface of the mirrors was also probed using conventional confocal microscopy @xcite , without removal of the additional optical elements . no aberrations were observed to have been introduced . m parabolic micromirror . a ) shows data collected from the surface of the mirror substrate ; b ) taken 25 @xmath0 m before the focus ( @xmath22 m in fig [ fig : data_vs_simulation ] ) , c ) taken at the position of the focus , and d ) taken 25 @xmath0 m after the focus ( @xmath23 m in fig [ fig : data_vs_simulation ] ) . the black scale bars in a ) , b ) , d ) indicate a length of 10 @xmath0 m , which is the width of image c ) . the linear colormap for each image is normalized from white to black for minimum to maximum value respectively.,width=321 ] optical measurements were taken for four mirrors with different focal lengths , all with 10 @xmath0 m radii apertures . two dimensional scans were taken , with 77 nm pixel size , of the reflected intensity field in defined planes parallel to the surface of the substrate , fig . [ fig : raw_data ] . a series of such scans , separated by 1 @xmath0 m , then return a high - resolution , 3d map of the intensity field . the raw data were then radially averaged about the optical axis of the beam . the optical axis was independently found by fitting a 2d gaussian to individual scans near the focal plane and then linearly extrapolating the center of mass fits to regions where a simple 2d - gaussian fit was not valid , _ e.g. _ , such as in fig . [ fig : raw_data](b ) . ( note , however , that about the focus , fig . [ fig : raw_data](c ) , a 2d - gaussian gives an excellent fit to the data . ) this step was taken to ensure any residual tilts in the optical system were accounted for . the averaged data were then combined , as shown in fig . [ fig : data_vs_simulation](a ) , to visualize the cylindrically symmetric intensity field . the focal length was extracted from the distance between the surface and the peak of the intensity , as well as by the best fit of the numerical model , fig . [ fig : data_vs_simulation](b ) . these data show excellent qualitative agreement as the detailed structure of the field away from the focus , which is emphasized by the logarithmic scale on the color axis . to further illustrate the agreement between experiment and theory the on - axis intensity profile of the data and the simulation are shown in fig . [ fig : data_vs_simulation](c ) . m mirror , using a log - scale to highlight the detail in the intensity field away from the focus . the data in each figure are each normalized , with the same color - axis for both . the color - axis , [ -4 0 ] , corresponds to a range of four orders of magnitude in intensity . the measured ( solid blue line ) and simulated ( dashed line ) on - axis intensity profiles , c ) , and radial intensity profiles at the focus , d ) , show agreement between experiment and model . also included in d ) , as a red dotted line , is a gaussian fit to the simulated data.,width=453 ] quantitative results from the measurements are in table [ table ] , below . the measured waists given are the minimum e@xmath24 radii returned from gaussian fits about the waist . the waists in the orthogonal directions were found to be equal and to occur in the same place , within measurement error . it can be seen in fig . [ fig : data_vs_simulation](d ) that a gaussian is an excellent fit to both simulation and experimental data near the focus . the diffraction limit was calculated as @xmath25 , where @xmath26 is the numerical aperture , @xmath27 which was derived using the measured focal length @xcite . it is clear that there is good agreement between the measured minimum beam waists and the prediction from diffraction theory . .measured beam waists ( e@xmath24 radius ) and focal lengths for a range of parabolic mirrors with 10 @xmath0 m radius aperture . errors , where quoted , are one standard deviation . [ cols="^,^,^,^,^ " , ] the intensity patterns observed in these data suggest that the these mirrors would be favorable for trapping of small clouds of ultra - cold atoms , or even single atoms @xcite . for a total power of 1 w in the illuminating beam we estimate a peak intensity at the focus of @xmath28 w / m@xmath29 ; with a dipole trapping wavelength of @xmath30 nm this corresponds to a trap depth of 370 @xmath0k for rubidium atoms @xcite . the trap depth would scale at least with the square of the aperture of the mirrors ( the power captured scales as the aperture diameter squared , while the peak intensity increases with the numerical aperture ) meaning that deeper traps could readily be designed . two dimensional arrays of single trapped atoms could be formed from one laser beam illuminating a congruent 2d array of mirrors . a feature of this array is that the illumination of individual mirrors would then provide single atom addressability , but on the length scale of the mirror aperture , which is inherently larger than the single atom focus . m. the blue dotted line corresponds to the position of peak intensity , the spot of arago , after an open aperture of diameter equal to that of the simulated mirror.,width=377 ] although we have shown , in fig . [ fig : data_vs_simulation ] , that the angular spectrum method yields agreement with the measured data in the regime probed here , further simulations show that focusing power of parabolic micro - mirrors is limited at longer focal lengths . in fig . [ fig : focal_vs_simulation ] the position of peak intensity as a function of mirror design focal length shows that diffraction from the aperture edge begins to dominate at focal lengths significantly longer than the aperture radius . in the limit of large focal length the location of the peak intensity tends towards that predicted for diffraction from the edges ; this is shown in fig . [ fig : focal_vs_simulation ] by the dotted line which is found from simulating diffraction from an aperture of radius equal the that of the mirrors . for the shortest focal length measured in this work , @xmath31 m , we derive a numerical aperture of @xmath32 for operation in water . as shown in @xcite by merenda _ , stable 3d trapping is to be expected in this regime . further work will be conducted to demonstrate optical tweezing using these mirrors . construction and diffraction limited focusing from parabolic micromirrors with radii of @xmath33 m ( @xmath34 ) has been demonstrated . simulations performed using the angular spectrum method show clear agreement with data and reproduce detailed features evident in the measured 3d intensity field . the results offer promise for use of these micromirrors for experiments in optical trapping in both biological and atomic - physics experiments . the authors are extremely grateful to alastair sinclair for initial project design and for discussions , to jonathan pritchard for help with simulations , and to john harris for assistance . pfg acknowledges the generous support of the royal society of edinburgh .

we report on the fabrication and diffraction - limited characterization of parabolic focusing micromirrors . sub micron beam waists are measured for mirrors with 10-@xmath0 m radius aperture and measured fixed focal lengths in the range from 24 @xmath0 m to @xmath1 m . optical characterization of the 3d intensity in the near field produced when the device is illuminated with collimated light is performed using a modified confocal microscope . results are compared directly with angular spectrum simulations , yielding strong agreement between experiment and theory , and identifying the competition between diffraction and focusing in the regime probed . 99 h. c. king , _ the history of the telescope _ ( dover , 2003 ) . l. novotny , and b. hecht , _ principles of nano - optics _ ( cambridge university , 2012 ) . w. r. jamroz , r. kruzelecky , and e. i. haddad , _ applied microphotonics _ ( taylor & francis , 2006 ) . r. vlkel , m. eisner , and k.j . weible , `` miniaturized imaging systems , '' microelectron . eng . * 67 * , 461 - 472 ( 2003 ) . m. trupke , e. a. hinds , s. eriksson , e. a. curtis , . moktadir , e. kukharenka , and m. kraft `` microfabricated high - finesse optical cavity with open access and small volume , '' appl . phys . lett . * 87 * , 211106 ( 2005 ) . f. merenda , j rohner , j .- m . fournier , and r .- p . salath `` miniaturized high - na focusing - mirror multiple optical tweezers , '' opt . express * 15 * , 6075 - 6086 ( 2007 ) . f. merenda , m. grossenbacher , s. jeney , l. forr , and r .- p . salath `` three - dimensional force measurements in optical tweezers formed with high - na micromirrors , '' opt . lett . * 34 * , 1063 - 1065 ( 2009 ) .. y. s. ow , m. b. h. breese , and sara azimi `` fabrication of concave silicon micro - mirrors , '' opt . express * 18 * , 14511 - 14518 ( 2010 ) . g. w. biedermann , f. m. benito , k. m. fortier , d. l. stick , t. k. loyd , p. d. d. schwindt , c. y. nakakura , r. l. jarecki jr . , and m. g. blain `` ultrasmooth microfabricated mirrors for quantum information , '' appl . . lett . * 97 * , 181110 ( 2010 ) . g. shu , n. kurz , m. dietrich , and b. blinov , `` efficient fluorescence collection from trapped ions with an integrated spherical mirror , '' phys . rev . a * 81 * , 042321 ( 2010 ) . j. merrill , c. volin , d. landgren , j. amini , k. wright , s. doret , c. pai , h. hayden , t. killian , d. faircloth , k. r. brown , a. w. harter , and r. e. slusher , `` demonstration of integrated microscale optics in surface - electrode ion traps , '' new j. phys . * 13 * , 103005 ( 2011 ) . r , maiwald , a , golla , m , fischer , m , bader , s , heugel , b , chalopin , m. sondermann , and g. leuchs `` collecting more than half the fluorescence photons from a single ion , '' phys . . lett . * 86 * 043431 ( 2012 ) . m. fischer , m. bader , r , maiwald , a , golla , m. sondermann , and g. leuchs , `` efficient saturation of an ion in free space , '' appl . phys . b - lasers o. * 117 * , 797 - 801 ( 2014 ) . a. roy , a.b . shen jing , and m.d barrett , `` the trapping and detection of single atoms using a spherical mirror , '' new j. phys . * 14 * , 093007 ( 2012 ) . r. noek , c. knoernschild , j. migacz , t. kim , p. maunz , t. merrill , h. hayden , c. s. pai , and j. kim , `` multiscale optics for enhanced light collection from a point source , '' opt . lett . * 35 * , 2460 - 2462 ( 2010 ) . m. k. tey , z. chen , s. a. aljunid , b. chng , f. huber , g. maslennikov , and c. kurtsiefer , `` strong interaction between light and a single trapped atom without the need for a cavity , '' nat . phys . * 4 * , 924 - 927 ( 2008 ) . m.t . langridge , d.c . cox , r.p . webb , and v. stolojan , `` the fabrication of aspherical microlenses using focused ion - beam techniques , '' micron * 57 * , 56 - 66 ( 2014 ) . j. goldwin , and e. a. hinds , `` tight focusing of plane waves from micro - fabricated spherical mirrors , '' opt . express * 16 * , 17808 - 17816 ( 2008 ) . t. n. bandi , v. g. minogin , , and s. nic chormaic , `` atom microtraps based on near - field fresnel diffraction , '' phys . rev . a * 78 * , 013410 ( 2008 ) . j. goodman , _ introduction to fourier optics _ ( roberts , 2005 ) . e. gu , h. w. choi , c. liu , c. griffin , j. m. girkin , i. m. watson , m. d. dawson , g. mcconnell , and a. m. gurney , `` reflection - transmission confocal microscopy characterization of single - crystal diamond microlens arrays , '' appl . . lett . * 84 * , 2754 - 2756 ( 2004 ) . a. ashkin , `` forces of a single - beam gradient laser trap on a dielectric sphere in the ray optics regime , '' biophys . j. * 61 * , 569 - 582 ( 1992 ) . n. schlosser , g. reymond , i. protsenko , and p. grangier , `` sub - poissonian loading of single atoms in a microscopic dipole trap , '' nature * 411 * , 1024 - 1027 ( 2001 ) . r. grimm , m. weidemller , and y. b. ovhinnikov , `` optical dipole traps for neutral atoms , '' adv . atom mol . opt . phys . * 42 * , 95 - 170 ( 2000 ) .

You are an expert at summarizing long articles. Proceed to summarize the following text: after carrying out a fourier transform and a multipole decomposition , the radial and time parts of the retarded green function for linear fields on a schwarzschild black hole can be written as @xmath15 where @xmath16 , @xmath17 is the multipole number , @xmath18 , @xmath19 and @xmath20 is the wronskian of the two functions @xmath21 and @xmath22 . these functions are linearly independent solutions of the radial ode @xmath23\right\ } \psi_{\ell}=0 \ ] ] where @xmath24 and @xmath25 . the solutions are uniquely determined when @xmath26 by the boundary conditions : @xmath27 as @xmath28 and @xmath29 as @xmath30 . the behaviour of the radial potential at infinity leads to a branch cut in the radial solution @xmath22 @xcite . the contour of integration in eq.([eq : green ] ) can be deformed in the complex-@xmath3 plane @xcite yielding a contribution from a high - frequency arc , a series over the residues ( the qnms ) and a contribution from the branch cut along the nia : @xmath31 where the bcms are @xmath32 with @xmath33 where @xmath34 $ ] is the discontinuity of @xmath22 across the branch cut . we present here methods for the analytic calculation of the bcms . we calculate @xmath21 using the jaff series , eq.39 @xcite . the coefficients of this series , which we denote by @xmath35 , satisfy a 3-term recurrence relation . we calculate @xmath22 using the series in eq.73 @xcite , which is in terms of the confluent hypergeometric @xmath36-function and the coefficients @xmath35 . this series has seldom been used and one must be aware that , in order for @xmath22 to satisfy the correct boundary condition , we must set @xmath37 , which itself has a branch cut . to find an expression for @xmath38 _ on _ the nia we exploit this series by combining it with the known behavior of the @xmath36-function across its branch cut : @xmath39 where we are taking the principal branch both for @xmath40 and for the @xmath36-function . in order to check the convergence of this series , we require the behaviour for large-@xmath41 of the coefficients @xmath35 . using the _ birkhoff series _ as in app.b @xcite , we find the leading order @xmath42 ( we have calculated up to four orders higher in @xcite ) as @xmath43 . we note that this behaviour corrects leaver s eq.46 @xcite in the power ` @xmath44 ' instead of ` @xmath45 ' . the integral test then shows that the series ( [ eq : leaver - liu series for deltagt ] ) converges for any @xmath46 . although convergent , the usefulness of ( [ eq : leaver - liu series for deltagt ] ) at small-@xmath47 is limited since convergence becomes slower as @xmath47 approaches 0 while , for large-@xmath47 , @xmath38 grows and oscillates for fixed @xmath48 and @xmath49 . therefore we complement our analytic method with asymptotic results for small and large @xmath8 . the small-@xmath50 asymptotics are based on an extension of the mst formalism @xcite . we start with the ansatz @xmath51 imposing eq.([eq : radial ode ] ) yields a 3-term recurrence relation for @xmath52 and requiring convergence as @xmath53 yields an equation for @xmath54 , that may readily be solved perturbatively in @xmath3 from starting values @xmath55 and @xmath56 . likewise for the coefficients @xmath52 , taking @xmath57 we obtain @xmath58 \\ & a_{2}^{\mu}= -\frac{(\ell+1-s)^2(\ell+2-s)^2}{4(\ell+1)(2\ell+1)(2\ell+3)^2}\omega^2+o\left(\omega^3\right ) \ ] ] while @xmath59 and @xmath60 are given by the corresponding terms with @xmath61 . ( apparent possible singularities in these coefficients are removable . ) the @xmath62 term in eq.([eq : f small - nu ] ) corresponds to page s eq.a.9 @xcite . to obtain higher order aymptotics we employ the barnes integral representation of the hypergeometric functions @xcite which involves a contour in the complex @xmath63-plane from @xmath64 to @xmath65 threading between the poles of @xmath66 , @xmath67 and @xmath68 . as @xmath69 double poles arise at the non - negative integers from 0 to @xmath70 , however we may move the contour to the right of all these ambient double poles picking up polynomials in @xmath48 with coefficients readily expanded in powers of @xmath3 , leaving a regular contour which admits immediate expansion in powers of @xmath8 . by the method of mst we can also construct @xmath22 and hence determine @xmath71 and @xmath72 . for compactness , we only give the following small-@xmath8 expressions for the case @xmath10 ( cases @xmath73 and @xmath12 are presented in @xcite ) , @xmath74 \nonumber\\ & -\frac { ( -1)^{\ell } \pi } { 2^{2 \ell-1 } } \left(\frac{(2\ell+1)\ell!}{((2 \ell+1)!!)^2}\right)^2 \nu^{2\ell+3 } \left[\frac{4(15\ell^2 + 15\ell-11)}{(2 \ell-1)(2\ell+1 ) ( 2 \ell+3 ) } \left(\ln ( 2 \nu ) + h_{\ell}-4 h_{2 \ell}+ \gamma_e \right)\right . \\ & \left . -4 \left(-8 h_{\ell}{}^2 + 8 h_{\ell}+3 h_\ell^{(2)}+ 2h_{\infty}^{(2 ) } \right)+ \frac{512\ell^6 + 2016 \ell^5 + 1616 \ell^4 - 1472 \ell^3- 1128 \ell^2 + 722 \ell-59}{(2 \ell-1)^2 ( 2 \ell+1)^2 ( 2 \ell+3)^2}\right ] + o(\nu^{2\ell+3 } ) \nonumber\end{aligned}\ ] ] where @xmath75 is the @xmath17-th harmonic number of order @xmath48 . we note that the @xmath76 term at second - to - leading order originates both in @xmath71 and in @xmath72 . in fact , both functions possess a @xmath76 already at next - to - leading order for small-@xmath47 , but they cancel each other out in @xmath77 . similarly , the coefficient of a potential term in @xmath77 of order @xmath78 is actually zero . let us now investigate the branch cut contribution to the black hole response to an initial perturbation given by the field @xmath79 and its time derivative @xmath80 at @xmath81 : @xmath82 \nonumber\end{aligned}\ ] ] we obtain the asymptotics of the response for late times @xmath83 using eqs.([eq : deltag in terms of deltag ] ) and ( [ eq : f small - nu])([q / w^2 s=0 gral l ] ) . we note the following features . the orders @xmath84 and @xmath85 in the bcms @xmath38 yield tail terms behaving like @xmath86 and @xmath87 , respectively . we have thus generalized leaver s eq.56 @xcite to finite values of @xmath48 . furthermore , eq.56 @xcite is an expression containing the leading orders from @xmath79 and from @xmath80 . however , the next - to - leading order from @xmath80 will be of the same order as the leading - order from @xmath79 . in our approach above we consistently give a series in small-@xmath47 , thus obtaining the correct next - to - leading order term for large-@xmath88 in the power - law tail . importantly , we also obtain the following two orders in the perturbation response : @xmath89 and @xmath90 . we note the interesting @xmath89 behaviour , which is due to the @xmath91 term in eq.([q / w^2 s=0 gral l ] ) . to the best of our knowledge , this is the first time in the literature that any of the above features has been obtained . the logarithmic behaviour is not completely surprising given the calculations in @xcite . however , one may be led to a wrong logarithmic behaviour @xcite if the calculations are not performed in detail . in order to exemplify our results , we give the explicit asymptotic behaviour in the case @xmath10 and @xmath92 and initial data @xmath93 and @xmath94 , with @xmath95 . the perturbation response due to the branch cut at @xmath96 at late times is given by @xmath97t^{-7 } + o\left(t^{-7}\right ) . \nonumber\end{aligned}\ ] ] fig . [ fig : perturbation response ] shows that these asymptotics are in excellent agreement with a numerical solution of the wave equation . to the gaussian described above eq.([eq : pert_asymp ] ) compared to the late - time asymptotics . solid - red : numerical solution ; dashed - black : eq.([eq : pert_asymp ] ) ; lower curves : numerical solution minus the first ( green ) , first 2 ( blue ) and first 4 ( cyan ) terms in eq.([eq : pert_asymp ] ) . ] at large-@xmath47 , we obtain the asymptotics @xmath98},&\!\!\ ! s=0,2\\ \dfrac{-\sqrt{\pi}i\lambda\sin(2\pi{\nu})}{{\nu}^{3/2}},&\!\!\!s=1 \end{cases } \\ & { f_{\ell}}(r ,- i\nu ) \sim \begin{cases } ( -1)^{s/2 } e^{\nu r_*}+\sin(2\pi{\nu})e^{-\nu r _ * } , & s=0,2\\ \dfrac{\sqrt{\pi}\lambda}{2{\nu}^{1/2}\sin(2\pi{\nu})}e^{\nu r_*}+e^{-\nu r_*},&s=1 \end{cases } \nonumber\end{aligned}\ ] ] these asymptotics show a divergence in @xmath99 when @xmath100 . they also lead to a divergence in the perturbation response at fixed @xmath101 and @xmath48 for a non - compact gaussian as initial data . both types of divergences are expected to cancel out with the other contributions to the green function . we have thus provided a complete account of the bcms for all frequencies along the nia ; the behaviour is illustrated in fig.[fig : deltag s = l=2 and s=0,l=1 ] . as a function of @xmath47 for @xmath102 and @xmath103 . ( a ) using eq.([eq : leaver - liu series for deltagt ] ) ; dashed - green : @xmath104 , continuous - blue : @xmath10 , @xmath92 , dot - dashed - orange : @xmath105 . note the interesting behaviour near the algebraically - special frequency @xcite at @xmath106 for @xmath107 . ( b ) @xmath10 , @xmath92 for small @xmath47 ; continuous - blue using eq.([eq : leaver - liu series for deltagt ] ) ; dashed - red using eq.([q / w^2 s=0 gral l ] ) to @xmath108 see @xcite . ( c ) @xmath10 , @xmath92 for large @xmath47 ; continuous - blue using eq.([eq : leaver - liu series for deltagt ] ) ; dashed - red using the asymptotics of eq.([eq : large_nu ] ) . , width=309 ] [ cols="^,^ " , ] we present here an analysis for large-@xmath1 of the electromagnetic qnms . we may find solutions of eq.([eq : radial ode ] ) valid for fixed @xmath109 as expansion in powers of @xmath110 as @xmath111 , @xmath112 , starting with the two independent solutions : @xmath113 and @xmath114 . we may express any higher order solution in terms of the @xmath115-order green function as @xmath116\frac{\psi_i^{({k}-1)}(u)}{\sqrt{\nu } } \right . \nonumber \\ & \left.- ( 2u)^{1/2}\left[4d^2 -2d - \lambda\right]\frac{\psi_i^{({k}-2)}(u)}{{\nu } } \right\}\end{aligned}\ ] ] where @xmath117 . from this expression , it follows that @xmath118 \nonumber \end{aligned}\ ] ] where @xmath119 . in addition , for @xmath120 , @xmath121 and @xmath122 $ ] are both real . it follows that along @xmath123 , up to power law corrections , @xmath124 equating asymptotic expansions at @xmath125 yields @xmath126 , @xmath127 and also serves to determine @xmath128 ( except when @xmath129 for @xmath130 , which do not contribute to the qnm condition ) . by matching the @xmath134 to wkb solutions along @xmath120 and @xmath135 we are able to find large-@xmath8 asymptotics for @xmath22 . also , we may use the exact monodromy condition , @xmath136 , to obtain large-@xmath47 asymptotics for @xmath21 . the asymptotic qnm condition ( @xmath137 ) in the 4th quadrant then becomes @xmath138 it is straightforward to find the qnm frequencies to _ arbitrary _ order in @xmath1 in terms of the @xmath132 by systematically solving eq.([eq : qnm cond ] ) . explicitly , using the values in eq.([eq : alpha values ] ) , we have @xmath139}{96n^{5/2 } } + o\!\left({n^{-3}}\right ) \nonumber\end{aligned}\ ] ] it is remarkable that the terms in the expansion show the behaviour @xmath140 to all orders . in fig . [ fig : qnm numeric closed form ] we compare these asymptotics with the numerical data in @xcite . in @xcite we apply the method used to obtain eq.([eq : qnm s=1 ] ) to the cases @xmath10 and @xmath12 and we obtain the corresponding qnm frequencies up to order @xmath141 and have agreement with @xcite . we are thankful to sam dolan and , particularly , to barry wardell for helpful discussions . thanks luis lehner and the perimeter institute for theoretical physics for hospitality and financial support . m. c. is supported by a ircset - marie curie international mobility fellowship in science , engineering and technology . a.o . acknowledges support from science foundation ireland under grant no 10/rfp / phy2847 . 37ifxundefined [ 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 * * , ( ) link:\doibase 10.1103/physrevlett.74.2414 [ * * , ( ) ] , @noop * * , ( ) link:\doibase 10.1088/0264 - 9381/26/16/163001 [ * * , ( ) ] , @noop * * , ( ) @noop * * , ( ) link:\doibase 10.1103/physrevd.78.044006 [ * * , ( ) ] , link:\doibase 10.1103/physrevd.62.024027 [ * * , ( ) ] , @noop * * , ( ) , @noop * * , ( ) link:\doibase 10.1016/0370 - 2693(95)01148-j [ * * , ( ) ] , link:\doibase 10.1103/physrevlett.81.4293 [ * * , ( ) ] , link:\doibase 10.1103/physrevlett.90.081301 [ * * , ( ) ] , link:\doibase 10.1103/physrevlett.100.141301 [ * * , ( ) ] , link:\doibase 10.1103/physrevd.84.084031 [ * * , ( ) ] , @noop * * , ( ) @noop ( ) , link:\doibase 10.1063/1.1626805 [ * * , ( ) ] , @noop * * , ( ) , link:\doibase 10.1016/j.physletb.2007.04.068 [ * * , ( ) ] link:\doibase 10.1103/physrevd.69.044004 [ * * , ( ) ] , link:\doibase 10.1103/physrevd.62.064009 [ * * , ( ) ] , link:\doibase 10.1103/physrevd.5.2419 [ * * , ( ) ] link:\doibase 10.1103/physrevd.5.2439 [ * * , ( ) ] @noop ( ) , @noop * * , ( ) , link:\doibase 10.1088/0264 - 9381/28/9/094021 [ * * , ( ) ] , link:\doibase 10.1038/nphys1907 [ * * , ( ) ] , @noop @noop * * , ( ) , @noop * * , ( ) link:\doibase 10.1103/physrevd.52.2118 [ * * , ( ) ] , @noop _ _ ( , ) @noop * * , ( ) @noop * * , ( ) , @noop _ _ ( , , ) @noop * * , ( ) , @noop _ _ ( , ) @noop

linear field perturbations of a black hole are described by the green function of the wave equation that they obey . after fourier decomposing the green function , its two natural contributions are given by poles ( quasinormal modes ) and a largely unexplored branch cut in the complex - frequency plane . we present new analytic methods for calculating the branch cut on a schwarzschild black hole for _ arbitrary _ values of the frequency . the branch cut yields a power - law tail decay for late times in the response of a black hole to an initial perturbation . we determine explicitly the first three orders in the power - law and show that the branch cut also yields a new logarithmic behaviour @xmath0 for late times . before the tail sets in , the quasinormal modes dominate the black hole response . for electromagnetic perturbations , the quasinormal mode frequencies approach the branch cut at large overtone index @xmath1 . we determine these frequencies up to @xmath2 and , formally , to _ arbitrary _ order . highly - damped quasinormal modes are of particular interest in that they have been linked to quantum properties of black holes . the retarded green function for linear field perturbations in black hole spacetimes is of central physical importance in classical and quantum gravity . an understanding of the make - up of the green function is obtained by performing a fourier transform , thus yielding an integration just above the real - frequency ( @xmath3 ) axis . in his seminal paper , leaver @xcite deformed this real-@xmath3 integration in the case of schwarzschild spacetime into a contour on the complex-@xmath3 plane . he thus unraveled three contributions making up the green function : ( 1 ) a high - frequency arc , ( 2 ) a series over poles of the green function ( quasinormal modes qnms ) , and ( 3 ) an integral of modes around a branch cut originating at @xmath4 and extending down the negative imaginary axis ( nia ) , which we refer to as branch cut modes ( bcms ) . the three contributions dominate the black hole response to an initial perturbation at different time regimes . the high - frequency arc yields a ` direct ' contribution which is expected to vanish after a certain finite time @xcite . the qnm contribution to the green function dominates the black hole response during ` intermediate ' times and it has been extensively investigated ( e.g. , @xcite for a review ) . at ` late times ' the qnm contribution decays exponentially , with a decay rate given by the overtone number @xmath5 . qnms have also triggered numerous interpretations in different contexts in classical and quantum physics , ranging from astrophysical ` ringdown ' @xcite to hawking radiation @xcite , the ` gauge - gravity duality ' ( @xcite for schwarzschild black holes which are asymptotically anti - de sitter and @xcite for asymptotically flat ones ) , black hole area quantization @xcite and structure of spacetime at the shortest length scales @xcite . the quantum interpretations are given in the highly - damped limit , i.e , for large @xmath1 . the highly - damped qnm frequencies in schwarzschild have been calculated up to next - to - leading order in . despite all the efforts , the leading order of the real part of the frequencies for electromagnetic perturbations has remained elusive ( only in @xcite they find numerical indications that it goes like @xmath6 ) . the contribution from the bcms , on the other hand , remains largely unexplored . the technical difficulties of its analysis mean that most of the studies have been constrained to large radial coordinate as well as small @xmath7 along the nia . an exception is a large-@xmath8 asymptotic analysis of the bcms in @xcite ( and near the algebraically - special frequency in @xcite ) solely for gravitational perturbations . the small-@xmath8 bcms are known to give rise to a power - law tail decay at ` late ' times of an initial perturbation @xcite . in general , however , there is an appreciable time interval between when the qnm contribution becomes negligible and when the power - law tail starts @xcite . the calculation of the bcms for general values of the frequency ( i.e. , not in the asymptotically small nor large regimes ) , to the best of our knowledge has only been attempted in @xcite where the radial functions were calculated off the nia via a numerical integration of the radial ode ( [ eq : radial ode ] ) followed by extrapolation to the nia , and only for the gravitational case . in this letter we present the following new results : a new analytic method for the calculation of the bcms directly _ on _ the nia and valid for _ any _ value of @xmath8 . in particular , this method provides analytic access for the first time to the ` mid'-@xmath8 regime . a consistent expansion up to @xmath9th order for small-@xmath8 of the bcms for arbitrary value of the radial coordinate . we explicitly derive a new logarithmic behaviour @xmath0 at late times . a large-@xmath8 asymptotic analysis of the bcms . it shows a formal divergence , which is expected to be cancelled out by the other contributions to the green function . a new asymptotic analysis for large-@xmath1 of the electromagnetic qnms . the analysis is formally valid up to _ arbitrary _ order in @xmath1 ; we explicitly calculate the corresponding frequencies up to @xmath2 . methods in ( 1)(3 ) provide the first full analytic account of the bcms and they are valid for any spin @xmath10 ( scalar ) , @xmath11 ( electromagnetic ) and @xmath12 ( gravitational ) of the field perturbation . for the qnm calculation we focus on spin-1 as this is the least well understood case . we note that spin-1 perturbations are acquiring increasing importance @xcite , although it is expected that only the lowest overtones of the qnms are astrophysically relevant . we present details in @xcite and @xcite . we take units @xmath13 , where @xmath14 is the mass of the black hole .

You are an expert at summarizing long articles. Proceed to summarize the following text: stochastic processes made their appearance in research in physics long time ago and their theory has played an important role in the description of systems which do not behave in a deterministic manner @xcite . in particular , the study of the dynamics of particles lying inside material media has been the object of high interest . a classical example is the study of the brownian motion @xcite . a large deal of those investigations had a non - relativistic character and the random interactions with the background medium were considered as being dependent of the state of motion of the particle , that is , lacking invariance under the changes of the reference system @xcite . another large class of studies in this field had been directed to show the equivalence with random processes of the solutions of quantum relativistic or non - relativistic equations , like the klein - gordon , dirac and schrodinger ones . @xcite . two basic additional subjects in connection with stochastic processes in quantum theory are : the attempts to derive the collapse of the wave function during measurements from the existence of random perturbations in quantum mechanics ( qm ) @xcite , and the study of the decoherence processes and their role in spontaneous transitions from pure to mixed states @xcite . the main objective of the present work is to investigate some consequences on the motion of a particle determined by the action exerted over it by a medium which random properties are defined in a relativistically invariant form . the basic motivation is simple : it is recognized that the copenhagen interpretation of quantum mechanics ( qm ) , is the most successful and dominant from all the existing ones . however , it is also accepted that its approach to measurements constitute one its more puzzling aspects , which up to now is widely debated in the literature@xcite . let us suppose for a moment , that in opposition to the copenhagen interpretation and in accordance with einstein expectations , the particles in nature are in fact localized at definite points of the space at any moment . then , the only way we can imagine for the quantum mechanical properties of the motion to emerge from a model , is that the action of the vacuum on the particles have a stochastic character . but , the relativistic invariance of the vacuum , leads to expect that the acceleration felt by the particle in its proper frame should be a stationary random variable as a function of the proper time . this circumstance motivates the study started here about the motion of particles inside a random media showing the above mentioned property . for the sake of simplicity the one dimensional motion is considered . it is not attempted to show the equivalence of the dynamics in the medium with the one predicted by the quantum mechanical equations . the purpose in this first step , being redundant , is to study general properties of the motion of one and two particles assuming two main conditions : a ) they have a definite localization in the space at any moment and b ) the forces acting on the particles have the above mentioned random properties which are independent the observer s inertial reference frame . the work will proceed as follows . firstly , the equations of motion of the particles under the action of the medium are formulated . for this purpose the properties which ensure the relativistic invariance of the motion under the action of the medium are stated by specifying the form of the random forces . further , the equations of motion of a single particle are written and solved and a statistical analysis of the random properties is done . a main conclusion following is the existence of a conservation law for a mean drift momentum and kinetic energy of a free particle propagating in the medium . it indicates the validity of a kind of stochastic noether theorem which links the relativist invariance of the stochastic motion with the conservation of the mean 4-momentum of the particle . further , the conservation law is studied for the mean of the addition of two four momenta associated to the scattering of two identical particles , which repel each other through an instantaneous coulomb interaction . it is concluded that the random action of the medium does not modify the usual conservation law , valid for the impact in the absence of external forces . a review of the results and future extensions of the work are presented in a conclusion section . some possibilities to extend the study are advanced . in general terms , our view is that the form of the analysis have the chance of being useful in the search for consistent hidden variables models . the study of these possibilities will be considered elsewhere . in this section we will obtain and solve the newton equation of motion for a particle on which a random force @xmath0 is acting . a one dimensional system will be considered to make the discussion as simple as possible . the force will be defined as a vector in the proper reference frame of the particle and will depend on the proper time @xmath1 that means , in each instant we will consider an inertial reference frame moving relative to the observer s fixed frame ( lab frame ) with the velocity of the particle @xmath2 and having the origin of coordinates coinciding with it . in this system of reference , after a time increment @xmath3 , it is possible to write@xmath4 where @xmath5 is the proper mass of the particle . the particle reaches a small velocity @xmath6 relative to this system and a new velocity respect to the lab frame @xmath7 , given by the equation@xmath8 where @xmath9 is the velocity of light . thus , the variation of speed in the lab frame @xmath10 is@xmath11 from expressions ( [ fp ] ) and ( [ dif_vel ] ) the required differential equation for the motion is obtained : @xmath12 it is useful to state the relation between the strength of the force in the lab system and its proper frame counterpart , which is : @xmath13 however , since the relativistic invariance condition will be imposed on @xmath0 this will be the type of force mostly considered in what follows . integrating the equation [ dif_mov ] in the proper time it follows @xmath14 which determines the velocity the lab frame @xmath15 as a function of the proper time @xmath16 , only through the dependence of @xmath17 of the integral of the random force @xmath18 the explicit form of @xmath19 becomes @xmath20 , \label{velocidad_en_tiempo_propio}\ ] ] where @xmath21 is an arbitrary constant . as it was mentioned in the introduction , the medium under study will be defined in the proper frame as randomly acting over the particle being at rest in it . that is , its action in this reference system will be given by a stochastic process showing no preferential spatial direction and assumed to be produced by an external relativistic system which dynamics is unaffected by the presence of the particle . its is also natural to impose the coincidence of the distribution function of the forces of the medium for a large sampling interval of proper time @xmath22 and the one obtained fixing the proper time @xmath17 produced by an ensemble of a large number of samples of the forces taken during long time intervals @xmath22 . these conditions , can be assured by a random force @xmath0 being stationary , ergodic and symmetrical distributed about the zero value of the force . a numerical realization of a band limited white noise distribution obeying these properties is implemented in reference @xcite and will be employed here . concretely , the expression for the stochastic force given in the proper reference frame as a function of the proper time will be taken in the form @xmath23 where the @xmath24 phases @xmath25 @xmath26 are randomly chosen with a uniform distribution between @xmath27 and @xmath28 . the integer number @xmath24 is the number of frequency components of the numerical band limited white noise distribution . the bandwidth will be chosen as a fixed one and equal to @xmath29 . clearly , the exact randomness for an arbitrarily large time interval only will attained in the infinite limit of @xmath24 . the parameter @xmath30 controls the amplitude of the force values . note that the absence of a zero frequency component is implied by the condition of the process not showing a preferential direction in space . figure [ graficofuerzas1 ] shows the force distribution @xmath31 for a @xmath32 of the ensemble of forces for @xmath33 a @xmath32 here is called the time evolution of the force , for the set of randomly fixed phases @xmath34 at the start . the picture qualitatively shows the stationary character of the random force . figure [ distribucion fuerza ] depicts an interpolation curve of the data for the distribution function @xmath35 corresponding to an ensemble of random _ realizations _ of the force . the white noise frequency spectrum was defined by @xmath36 frequency components @xmath37 @xmath38,@xmath39 . the force amplitude fixed was as @xmath40 @xmath41 @xmath42 the picture corresponds to a large sampling time @xmath22 ( it is sufficient to be at most equal to the period of the smallest frequency of the spectrum ) . notice the even character of the distribution and its rapid decay to zero . of course , this occurs inside the interval defined by @xmath43 @xmath44 . this result is natural due to the fact that the force is normalized and its absolute value can not exceed @xmath45 . in this section the existence of a conservation law for the mean momentum ( to be also called from now on the drift momentum ) will be shown for a particle moving in the before defined random medium . to this purpose let us employ the solution ( [ velocidad_en_tiempo_propio ] ) which was found in the previous section we will combine this expression with the results obtained from the definition of the random force ensemble in equation ( [ fuerza_estocastica ] ) , to explicitly determine @xmath46 the resulting relation links the velocity of the particle in the lab frame with the proper time measured by a clock fixed to it . once having @xmath47 we will comment about its stationary random behavior . the existence of a non vanishing conserved mean value of the velocity @xmath48 will be shown . starting form the method defined above , the velocity distribution function will be determined for the frequency spectrum defined before and few representative values of the arbitrary constant @xmath21 . after that , the position of the particle @xmath49 as a function of the time @xmath50 measured at the lab frame will be evaluated . taking @xmath0 to be given by the white noise force @xmath51 we have : @xmath52 then , integrating with respect to @xmath16 produces @xmath53\text { } d\tau,\nonumber\\ & = \frac{f_{0}}{8\pi}[\sum_{i=1}^{n}\frac{1}{i}\sin(\frac{8\pi i\text { } } { n}\tau+\varphi_{i } ) ] . \label{integral_fuerza}\ ] ] substituting ( [ integral_fuerza ] ) into ( [ velocidad_en_tiempo_propio ] ) , gives@xmath54,\\ & = c\text { } \tanh[\frac{1}{m_{0}c}(\frac{f_{0}}{8\pi}[\sum_{i=1}^{n}\frac { 1}{i}\sin(\frac{8\pi i}{n}\tau+\varphi_{i})]+\hat{c } ) ] . \label{v(tao)}\ ] ] it can be seen that the summation within the argument of the hyperbolic tangent is symmetrically distributed around its zero value . since the @xmath55 is an anti - symmetric function , it follows that when @xmath56 there will be a nonvanishing mean velocity of the particle in the medium . as the mean value of the kinetic energy is also conserved , it can be said that the mean 4-momentum of the particle is conserved . moreover , the relativistic invariance of the system also implies that the mean 4-momentum of the solutions for any two values of @xmath21 should be linked by certain lorentz transformation . thus , the particle trajectories for the various values of @xmath21 are simply a fixed trajectory ( in the stochastic sense ) after being lorentz transformed into certain moving frame . the picture in figure [ graficovel1 ] shows the behavior of @xmath57 for the following values of the parameters @xmath58s@xmath59 , @xmath60 s@xmath59 , @xmath61 . note that for fixed values of @xmath21 ( that means , for certain initial conditions ) the velocity rapidly oscillates around a value being close to the quantity @xmath62 $ ] . it illustrates the mentioned conservation of the mean drift velocity for the particle in spite of the random oscillations of the medium . figure [ graficovel2 ] shows the same dependence for specific values of @xmath21 . let us consider now the numerical evaluation of the distribution function @xmath63 ( for measuring a given value of the velocity @xmath19 ) for few representative values of @xmath21 . this function is found on the basis of the ergodic property of the system , that is , by performing a large number of evaluations of the velocity s expression with time running from zero to a `` sampling '' value @xmath22 , for afterwards compute the number of times , for which @xmath19 takes values in a small neighborhood of a given value . further , after interpolating the results , the distribution functions are obtained after dividing by the fixed size of the mentioned neighborhood . in figure ( [ distribuciones ] ) the distribution @xmath63 is plotted for a few values of @xmath21 . these pictures correspond to the frequency spectrum @xmath64 but with the amplitude fixed by @xmath65 the almost independence of the form of these curves on the size of a large `` sampling '' time @xmath66 employed ( whenever @xmath24 is sufficiently large ) indicates the approximate validity of the ergodic property of the white noise implementation employed . note the rapid decay away from the mean value and the deformation of the symmetry around it , of the distribution @xmath63 for different values of @xmath21 . these distributions allow to calculate the mean velocities in each of the cases . this method will be employed in the next sections for finding the mean values as integrals over the domain of the quantity , of its value times the distribution function . it can be noted that the explicit solution for the velocity obtained in ( [ velocidad_en_tiempo_propio ] ) corresponds to this velocity in terms of the proper time @xmath16 . however , in order to integrate the velocity to find the random particle trajectories as functions of the time in the laboratory frame , it becomes necessary to know the functional relationship @xmath67 for the considered trajectory . then , let us consider now the numerical determination of this relation . the parameters of the random force will be @xmath68 @xmath69s@xmath70 , @xmath71 ( s@xmath70 , @xmath72 using the solution @xmath73 it follows @xmath74 and employing it , the values of @xmath75 were numerically found . afterwards , finding the inverse mapping @xmath76 , this function is depicted in figure [ tiempos ] with @xmath21 as a parameter . finding the composed function for various values of @xmath21 , the observer s time dependence of the particle velocity @xmath77 follows . the results are illustrated in figure [ velocidad de la particula libre ] , for the chosen values of @xmath21 . further , after integrating the calculated velocities with respect to @xmath78 @xmath79(starting at @xmath80 and imposing @xmath81 ) the values of the positions @xmath49 respect to the laboratory reference frame are obtained . the trajectories of the particles are shown in figure [ posicion de la particula ] , for the same set of values of @xmath82 note the randomness of the motion in the case of small absolute values of @xmath21 which is not the case for the larger ones . this property can be understood analyzing the pictures in figure ( [ velocidad de la particula libre ] ) . in the cases in which the velocity constantly changes from positive to negative values and vice versa , the randomness is more evident . thus , the aleatory effect is more visible in one case than in the other , because the curve changes from a monotonous to non monotonous behavior . but , after taking into account the relativistic invariance of the model , as noticed before , it can be seen that the curves for different values of @xmath21 should transform any one into another by a lorentz transformation ( exactly , when the same _ realization _ of the force is employed , or statistically , if another _ realization _ is used ) . therefore , the apparent increased randomness for low @xmath21 is only a visual effect enforced by the change of sign of the velocity in frames in which its mean value is sufficiently low . let us consider in this final section the conservation of the mean total momentum of a closed system of two particles which travel in the medium by also interacting between them . a repulsive interaction between the two identical particles @xmath41 and @xmath83 will introduced . the forces between the particles will be defined in the laboratory frame for studying the impact between them there . they will be chosen as satisfying the law of action and reaction and having the coulomb form . if , for example , the forces have an electromagnetic origin , the retardation effects will be disregarded in order to assure that the field will not deliver momentum to the set of two particles . this approximation is appropriate for low particle velocities in its random motion in the lab frame . the system of differential equations will be written and numerically solved . the results for the position and velocity of both particles @xmath84 , @xmath85 , @xmath86 , @xmath87 , will show how after the impact , any of them deliver to the other its mean drift velocity and its type of randomness . as it was mentioned above , the expression for the repulsive force in the lab frame will have the coulomb form@xmath88 consider now the relativistic newton equations for both particles in their respective proper frames and also the two transformations between the common laboratory time @xmath50 and the two different proper times @xmath89 and @xmath90 these relations may be written as @xmath91 in order to be consistent with the non retarded approximation for the coulomb repulsion , as mentioned above , initial velocities for the particles being relatively small with respect to @xmath9 were taken . this assumption makes the considered approximation to be satisfied if the intensity of the repulsive force is weak so that the velocities after the impact will also be small . employing the white noise parameters @xmath92 , @xmath93 and @xmath94 , we obtained the numerical solutions shown in figure [ choque1 ] for various initial conditions . the conservation law of the total mean momentum of the two particles can be noticed . figure [ choque2 ] shows the time variation of the velocities of the particles . the process of exchange of their drift momenta is clearly illustrated and therefore the corresponding conservation of the total momentum . this outcome follows in spite of the strong oscillations produced by the action of the medium , in opposition to what happens within a standard material media . in these systems the stochastic action normally tends to stop the motion of the particle making the drift velocity to vanish , assumed the absence of accelerating external fields . the calculated values for the mean velocities before and after the shock are @xmath95 which numerically confirms the conservation of the total momentum in the shock of two particles forming a closed system immersed and moving in a relativistic random medium . the one - dimensional equation of motion for a particle moving in a medium having relativistic invariant stochastic properties is formulated and numerically solved . the velocity of the particles in the medium is a function of the proper time only through the integral of the force in the proper reference frame . this relation directly shows the existence of a stochastic conservation law : _ a free particle in the defined random medium conserves its mean momentum and kinetic energy along its motion . _ the problem of two particles moving in the medium is also investigated after considering a momentum conserving coulomb repulsion between them . the evaluated solutions generalize the conservation law : _ the sum of the two drift momenta of two particles moving in the medium without any other external action conserves after a shock . _ an interesting outcome is that for two different shocks with identical initial conditions for both particles when they are far apart , the drift velocities before the impact are not uniquely determined by the initial conditions . these velocities also depend on the array of phases utilized for the _ realization _ of the force . _ _ therefore , it follows that the results of the impact will show a dependence on the specific _ realization _ of the random force . this circumstance implies that the probability distribution associated to an ensemble of particles all situated at the beginning at the same point and with a fixed value of the velocity , will not define only one direction of the trajectory for large times . the probability distributions of such an ensemble , on the other hand , should evolve in space and time in a relativistically invariant manner , as the lorentz invariance of the system indicates . the same conclusion can be traced for any other sort of fixed boundary conditions . therefore , the set of possible initial conditions for the particle ( considering also that they can be placed in the medium at different initial spacial and temporal points for the construction of the ensemble ) should be expected to be equivalent to the set of all possible space time evolving ensembles that can be observed . the above remarks suggest some possible extensions of the present work , which are described below : * the indicated dependence of the ending results of the shocks , not only of the initial conditions , but also of the concrete _ realization _ of the random force , led us to think in extending the results to the 2d and 3d cases . the idea is to study the spatial and temporal behavior of the ensemble of outcomes o a series of shocks `` prepared '' with fixed initial mean velocities for both particles , and to compare the results with the corresponding predictions of the quantum scattering . * another task of interest which is suggested by this study is to investigate the existence of preferential bounded states in the case that the coulomb interaction is considered as attractive . @xmath79a particular simpler situation could be to assume one of the particles as very massive , that is , merely acting as an attracting center . in both cases the study could consist in determining the statistical properties of the stationary trajectories given the initial condition . in conclusion , we would like to underline that it seems not without sense that the realization of the above proposed studies could be of use in the justification or search for hidden variable approaches to quantum mechanics ( qm ) . as it can be seen from the discussion , the resulting models could not show the limitations of the bohm approach ( like the absence of predictions for the `` guided '' motions of the particles for real wave functions , for example ) . moreover , the analysis seems of interest in seeking for a theory not requiring to fix a particular random process to each stationary state , but one in which all the statistical properties can emerge from a general framework . in the imagined outcome , the particle could propagate having a probability for transit or stay into each one of a set of stationary random motions which could be associated ( but now within the general context ) to the particular eigenfunctions of the system . as for the allowance of the necessary generalizations needed to make contact with reality , it can guessed that the possible generalization of the statistical noether properties ( detected in the simple model considered here ) could lead to the conservation of the mean values for the angular momentum and other internal quantities , in analogy with the case of the linear momentum . we think these possibilities deserve further examination that will be published elsewhere . d. bohm , in _ quantum theory , radiation and high energy physics _ d. bates ( academic press , new york , 1962 ) . p. vigier , _ lett . nuovo cimento _ * 24 * , 528 ( 1979 ) ; * 24 * , 265 ( 1979 ) . t. e. nelson , _ phys . rev . _ * 150 * , 1079 ( 1966 ) . d. kershaw , _ phys . rev . _ * 136 * , b1079 ( 1964 ) . t. h. boyer , _ phys . _ * 182 * , 1374 ( 1969 ) .

the dynamics of particles moving in a medium defined by its relativistically invariant stochastic properties is investigated . for this aim , the force exerted on the particles by the medium is defined by a stationary random variable as a function of the proper time of the particles . the equations of motion for a single one - dimensional particle are obtained and numerically solved . a conservation law for the drift momentum of the particle during its random motion is shown . moreover , the conservation of the mean value of the total linear momentum for two particles repelling each other according with the coulomb interaction is also following . therefore , the results indicate the realization of a kind of stochastic noether theorem in the system under study . possible applications to the stochastic representation of quantum mechanics are advanced .

You are an expert at summarizing long articles. Proceed to summarize the following text: let @xmath27 . let @xmath2 be an algebraically closed field and @xmath0 a @xmath1-dimensional symplectic vector space over @xmath2 . the symplectic group @xmath15 acts naturally on @xmath0 from the left hand side , and hence on the @xmath28-tensor space @xmath18 . let @xmath29 be the brauer algebra over @xmath2 with generators @xmath30 @xmath31 and parameter @xmath32 ( see [ dfn1 ] for their definitions ) . there is a right action of @xmath17 on @xmath9 which commutes with the left action of @xmath15 . let @xmath33 be the following natural @xmath2-algebra homomorphisms : @xmath34 for any positive integer @xmath35 , a composition of @xmath35 is a sequence of non - negative integers @xmath36 with @xmath37 . we use @xmath38 to denote the largest integer @xmath39 such that @xmath40 . a composition @xmath36 of @xmath35 is said to be a partition if @xmath41 ; in that case , we write @xmath42 . the following results are referred as brauer schur weyl duality . [ thm11 ] ( @xcite , @xcite , @xcite ) [ ddh1 ] 1 ) the natural left action of @xmath15 on @xmath18 commutes with the right action of @xmath17 ; \2 ) both @xmath43 and @xmath44 are surjective ; \3 ) if @xmath45 then @xmath43 is injective , and hence an isomorphism ; \4 ) if @xmath46 , then there is a decomposition of @xmath18 as a direct sum of irreducible @xmath47-@xmath17-bimodules : @xmath48}\bigoplus_{\substack{{\lambda}\vdash n-2f\\ \ell({\lambda})\leq m}}\delta({{\lambda}})\otimes d^{(f,{\lambda})},\ ] ] where @xmath49 ( respectively , @xmath50 ) denotes the irreducible @xmath15-module ( respectively , the irreducible @xmath17-module ) corresponding to @xmath24 ( respectively , corresponding to @xmath51 ) . there is a variant of the above brauer schur weyl duality as we shall describe . let @xmath52 be the two - sided ideal of @xmath17 generated by @xmath53 . we set @xmath54 [ w1n ] we call @xmath55 the subspace of harmonic tensors or traceless tensors . note that our definition of harmonic tensors looks slightly different from that given in @xcite and @xcite . the two definitions are reconciled in corollary [ 2dfn ] . if @xmath46 , then we shall write @xmath56 instead of @xmath57 in order to emphasize the base field . we have that @xmath58 . the right action of @xmath17 on @xmath18 gives rise to a right action of @xmath59 on @xmath55 . ( ( * ? ? ? * ( 10.2.7 ) ) , @xcite ) [ gw1 ] the natural left action of @xmath15 on @xmath55 commutes with the right action of @xmath59 . if @xmath60 , then there is a decomposition of @xmath61 as a direct sum of irreducible @xmath47-@xmath62-bimodules : @xmath63 where @xmath64 ( respectively , @xmath65 ) denotes the irreducible @xmath47-module ( respectively , the irreducible @xmath62-module ) corresponding to @xmath24 . as before , we have two natural @xmath2-algebra homomorphisms : @xmath66 in @xcite , de concini and strickland proved that @xmath67 is independent of the field @xmath2 and @xmath68 is always surjective . furthermore , they showed that if @xmath45 , then @xmath68 is an isomorphism . their proof makes use of the previous results in @xcite and @xcite on multilinear invariants of a variety and symplectic standard tableaux which eventually relies on some algebro - geometric arguments . in @xcite , using the theory of rational representations of symplectic group , maliakas proved that @xmath69 has a good filtration whenever @xmath45 and he claimed that it is true for arbitrary @xmath70 . the starting point of this paper is , on one hand , to generalize the above duality to the case of partially harmonic tensors of arbitrary valence @xmath8 , and on the other hand , to provide a self - contained and purely representation - theorietic approach which makes it possible to work also in the quantized case ) which prevent us from generalizing the main results of this paper to the quantized case . the main difficulty lies in that there are various choices of tangle which can specialize to the same brauer diagram and it also becomes much harder to describe the action of a tangle on tensor space in a simple combinatorial way ( as in the brauer algebra case ) . ] . we are mostly interested in the non - semisimple case . to describe our main results , we need some more notations and definitions . for each integer @xmath6 $ ] , let @xmath3 be the two - sided ideal of @xmath17 generated by @xmath5 . by convention , @xmath71 and @xmath72 + 1)}=\{0\}$ ] . set @xmath73 following ( * ? ? ? * ( 10.3.1 ) ) , we call @xmath74 the space of _ partially harmonic tensors of valence @xmath8_. classically , these spaces play an important role in the study of the @xmath75-module structure on the @xmath28-tensor spaces @xmath76 ( cf . one of our motivation for studying them is to try to use them to construct some natural ( integral ) quotient of the symplectic schur algebras . by ( * ( 10.3.14 ) ) , it is easy to see that if @xmath46 then @xmath77 as a @xmath78-linear space . in particular , @xmath79 as a @xmath78-linear space . note that the space @xmath0 can be defined over arbitrary field ( and even over @xmath80 ) . so we can consider the spaces @xmath81 , @xmath82 and @xmath83 over an arbitrary field @xmath2 . all the results we obtain in this paper are valid over these more general ground fields . however , we shall always assume that @xmath2 is algebraically closed whenever notions and results from algebraic groups theory ( e.g. , good filtration , weyl filtration ) are needed . the transition between an arbitrary field and its algebraic closure usually follows from some standard arguments in commutative algebras . note that @xmath83 is a @xmath15-@xmath84-bimodule . we use @xmath85 to denote the following natural homomorphism : @xmath86 let @xmath87 denote the value of the usual kronecker delta . for each integer @xmath88 , we set @xmath89 . we fix an ordered basis @xmath90 of @xmath0 such that @xmath91 we use @xmath92 to denote the free @xmath80-submodule of @xmath93 generated by @xmath94 . for any commutative @xmath80-algebra @xmath95 , we set @xmath96 . the brauer algebra @xmath17 can also be defined over @xmath95 and we denote it by @xmath97 . to simplify notations , the two - sided ideal of @xmath97 generated by @xmath5 will be still denoted by @xmath3 . the main results in this paper are the following theorems and corollaries . [ mainthm0 ] for each integer @xmath98 $ ] , \1 ) @xmath99 is a pure @xmath80-submodule of @xmath100 , equivalently , @xmath101 is a free @xmath80-module ; \2 ) both @xmath102 and @xmath83 are stable under base change , i.e. , for any commutative @xmath80-algebra @xmath95 , the canonical maps @xmath103 are isomorphisms . in particular , the character formulae of the left @xmath15-modules @xmath104 are both independent of the field @xmath2 . [ mainthm1 ] let @xmath2 be an algebraically closed field . for each integer @xmath8 with @xmath98 $ ] , both @xmath83 and @xmath102 have a good filtration as @xmath15-modules . [ maincor1 ] let @xmath2 be an algebraically closed field . let @xmath6 $ ] . then the dimension of @xmath82 is independent of @xmath2 , and there is a @xmath15-@xmath105-bimodule isomorphism : @xmath106 in particular , the dimension of @xmath74 is independent of @xmath2 too . [ maincor15 ] let @xmath2 be an algebraically closed field . for each @xmath6 $ ] , the @xmath15-module @xmath82 always has a good filtration and the @xmath15-module @xmath74 always has a weyl filtration . [ mainthm2 ] let @xmath2 be an algebraically closed field . for each integer @xmath8 with @xmath98 $ ] , \1 ) the dimension of the endomorphism algebra @xmath13 is independent of @xmath2 ; \2 ) @xmath107 . the proof of the above results are _ completely self - contained _ and use purely representation - theorietic knowledge . as a consequence of these theorems and corollaries , we recover and extend the previously mentioned results of de concini and strickland @xcite and the result of maliakas @xcite . in the way of our proof , we also obtain the following result , which seems of independent interest . [ mainthm3 ] 1 ) as a @xmath15-@xmath17-bimodule , @xmath18 has a filtration such that each successive quotient is isomorphic to some @xmath108 with @xmath20 , @xmath21 and @xmath22 $ ] , where @xmath23 is the co - weyl module associated to @xmath24 and @xmath25 is a maximal vector of weight @xmath24 ( see [ zdfn ] for its definition ) ; \2 ) for any partition @xmath24 of @xmath109 with @xmath22 $ ] and @xmath21 and any commutative @xmath80-algebra @xmath95 , the canonical map @xmath110 is always an isomorphism . in particular , the dimension of @xmath26 is independent of @xmath2 . in fact , if @xmath46 then @xmath111 is a simple right @xmath112-module . therefore , for any field @xmath2 , the dimension of @xmath26 is always equal to the number of @xmath113-permissible up - down tableaux of shape @xmath114 and length @xmath28 ( cf . * theorem 1.1 , theorem 1.2 ) and @xcite ) , where @xmath114 denotes the conjugate of @xmath24 . the paper is organized as follows . in section 2 we recall some basic knowledge about brauer algebras and their actions on @xmath28-tensor spaces . in particular , we show that our definition [ w1n ] of harmonic tensors coincides with that given in @xcite and @xcite . in section 3 , we give the proof of theorem [ mainthm0 ] , [ mainthm1 ] and corollary [ maincor1 ] . the main idea is to show that @xmath102 can be identified with the image of @xmath18 under a truncation functor @xmath115 associated with a saturated set @xmath116 of dominant weights . the proof makes use of the main result obtained in @xcite , some results on weyl filtration ( resp . , good filtration ) and a key lemma [ hu2lem ] . as a consequence , we prove the first part of theorem [ mainthm3 ] . section 4 is devoted to the proof of lemma [ hu2lem ] . the proof relies on lusztig s theory of canonical bases and based modules . as a result , we get the second part of theorem [ mainthm3 ] . in section 5 we prove theorem [ mainthm2 ] , which gives one side of the brauer - schur - weyl duality between @xmath117 and @xmath118 on @xmath119 . we conjecture that the other side of this duality is also true . let @xmath120 . the brauer algebra @xmath17 with parameter @xmath121 and size @xmath28 was first introduced by richard brauer ( see @xcite ) when he studied how the @xmath28-tensor space @xmath122 decomposes into irreducible modules over @xmath123 . in his language , @xmath17 was defined as the @xmath78-linear space with a basis being the set @xmath124 of all the brauer @xmath28-diagrams . by definition , a brauer @xmath28-diagram is a diagram with specific @xmath125 vertices arranged in two rows of @xmath28 each , the top and bottom rows , and exactly @xmath28 edges such that every vertex is joined to another vertex ( distinct from itself ) by exactly one edge . we label the vertices in each row of a brauer @xmath28-diagram by the integers @xmath126 from left to right . the multiplication of two brauer @xmath28-diagrams is defined as follows . we compose two diagrams @xmath127 by identifying the bottom row of @xmath128 with the top row of @xmath129 ( such that the vertex @xmath130 in the bottom row of @xmath128 is identified with the vertex @xmath130 in the top row of @xmath129 for each @xmath131 ) . the result is a graph , with a certain number , @xmath132 , of interior loops . after removing the interior loops and the identified vertices , retaining the edges and remaining vertices , we obtain a new brauer @xmath28-diagram @xmath133 , the composite diagram . then we define @xmath134 . in general , the multiplication of two elements in @xmath17 is given by the linear extension of a product defined on diagrams . for example , let @xmath135 be the following brauer @xmath136-diagram . let @xmath137 be the following brauer @xmath136-diagram . then @xmath138 is equal to [ 0.55 ] alternatively , one can define the brauer algebra using generators and relations . [ dfn1 ] let @xmath2 be a field . the brauer algebra @xmath17 over @xmath2 is a unital associative @xmath2-algebra with canonical generators @xmath139 and relations ( see @xcite ) : @xmath140 replacing @xmath2 by any commutative @xmath80-algebra @xmath95 , we can define the brauer algebra @xmath97 over @xmath95 in a similar way . the algebra @xmath97 is a free @xmath95-module with rank @xmath141 , and the canonical map @xmath142 is an isomorphism for any commutative @xmath80-algebra @xmath95 . the two definitions of the brauer algebra @xmath17 can be identified as follows . let @xmath130 be an integer with @xmath143 . the generator @xmath144 corresponds to the brauer @xmath28-diagram with edges connecting vertices @xmath130 ( respectively , @xmath145 ) on the top row with @xmath145 ( respectively , @xmath130 ) on the bottom row , and all other edges are vertical , connecting vertices @xmath35 on the top and bottom rows for all @xmath146 . the generator @xmath147 corresponds to the brauer @xmath28-diagram with horizontal edges connecting vertices @xmath148 on the top and bottom rows , and all other edges are vertical , connecting vertices @xmath35 on the top and bottom rows for all @xmath146 . note that the subalgebra of @xmath149 generated by @xmath150 is isomorphic to the group algebra of the symmetric group @xmath151 over @xmath80 . the brauer algebra was studied in a number of literatures , see @xcite , @xcite , @xcite , @xcite , @xcite , @xcite , @xcite . we are mainly interested in their actions on @xmath28-tensor space @xmath18 and related schur weyl dualities involving symplectic groups . from now on let @xmath2 be an algebraically closed field and @xmath0 a @xmath1-dimensional @xmath2-vector space equipped with a non - degenerate skew - symmetric bilinear form @xmath152 . let @xmath15 be the corresponding symplectic group . recall that for each integer @xmath88 , @xmath89 . let @xmath153 be a @xmath2-basis of @xmath0 such that @xmath91 for each integer @xmath88 , we define @xmath154 then @xmath153 and @xmath155 are dual bases for @xmath0 in the sense that @xmath156 for any @xmath157 . there is a right action of @xmath17 on @xmath18 which is defined on generators by @xmath158 assume @xmath159 . for any pair of integers @xmath160 we define the @xmath161-contraction operator @xmath162 by @xmath163 where @xmath164 , @xmath165 mean that we omit the tensor factors @xmath166 and @xmath167 in the tensor product , and we define the @xmath161-expansion operator @xmath168 by @xmath169 ( @xcite ) for each pair of integers @xmath160 , both @xmath170 and @xmath171 are @xmath15-module homomorphisms . for each pair of integers @xmath160 , we use @xmath172 to denote the unique brauer @xmath28-diagram which satisfies the following two conditions : 1 . for any integer @xmath173 , the vertex labeled by @xmath174 in the top row is connected with the vertex labeled by @xmath174 in the bottom row ; 2 . the vertex labeled by @xmath175 in either the top row or the bottom row is connected with the vertex labeled by @xmath39 in the same row . in particular , @xmath176 for any integer @xmath177 . [ contra ] for any pair of integers @xmath160 , we have @xmath178 . in particular , @xmath179 by definition , for any integers @xmath180 , @xmath181 it follows that for any @xmath182 , @xmath183 . on the other hand , it is clear that @xmath171 maps the basis of simple @xmath184-tensors to a set of @xmath2-linearly independent elements in @xmath18 . hence it is an injective map from @xmath185 to @xmath18 . it follows that @xmath178 . in particular , @xmath186 [ br ] let @xmath187 . the following three statements are equivalent : 1 . @xmath188 contains exactly two horizontal edges ( one edge in each of the top and the bottom rows in the diagram ) ; 2 . @xmath189 , for some @xmath190 and two integers @xmath191 ; 3 . @xmath192 for some @xmath193 and two integers @xmath194 . by @xcite , @xmath188 contains exactly two horizontal edges if and only if @xmath195 for some @xmath196 , @xmath197 , where @xmath198 is the set of distinguished right coset representatives of @xmath199 in @xmath200 . it follows that @xmath201 since both @xmath202 and @xmath203 are of the form @xmath172 for two distinct integers @xmath160 , the lemma follows at once . for each integer @xmath204 $ ] , we denote by @xmath3 the two - sided ideal of @xmath17 generated by @xmath5 . note that @xmath3 is spanned by all the brauer @xmath28-diagrams which contain at least @xmath205 horizontal edges . recall the definition of @xmath55 given in definition [ w1n ] . [ 2dfn ] with the notations as above , we have that @xmath206 this follows directly from lemma [ contra ] and [ br ] . note that the above corollary ensures that our definition [ w1n ] of harmonic tensors coincides with that given in @xcite and @xcite . @xmath207 " be the @xmath2-algebra anti - automorphism ( of order two ) of @xmath17 which is defined on generators by @xmath208 for any right @xmath17-module @xmath209 , the dual space @xmath210 is naturally endowed with a right @xmath17-module structure via the anti - involution @xmath207 " . that is , @xmath211 . for each integer @xmath212 , we set @xmath213 for any @xmath214 , we write @xmath215 . the bilinear form @xmath216 on @xmath0 naturally induces a non - degenerate bilinear form on @xmath18 such that @xmath217 by definition and an easy check , we see that for any @xmath218 , @xmath219 in other words , the bilinear form @xmath220 induces a @xmath15-@xmath17-bimodule isomorphism @xmath221 by definition and ( [ asso ] ) , we deduce that @xmath222 restricts to an inclusion @xmath223 in the next section , we shall show that @xmath224 is actually an isomorphism , see corollary [ comdim ] . the purpose of this section is to give a proof of theorem [ mainthm0 ] , [ mainthm1 ] and corollary [ maincor1 ] . one of the key steps is the use of lemma [ hu2lem ] , whose proof will be given in the next section . let @xmath225 . we identify @xmath0 with @xmath226 and @xmath15 with @xmath227 by using the ordered basis @xmath228 . then @xmath229 where @xmath230 denotes the matrix transpose of @xmath231 , @xmath232 and for each @xmath233 , @xmath234 denotes the matrix unit which is @xmath235 on the @xmath236th position and @xmath237 elsewhere . recall that @xmath227 is a connected semisimple linear algebraic group over @xmath238 . let @xmath239 . let @xmath240 be the subgroup consisting of all diagonal matrices in @xmath227 . that is , @xmath241 then @xmath240 is a maximal torus of @xmath227 . let @xmath242 be the weight lattice of @xmath227 . an element in @xmath243 will also be called a @xmath240-weight . let @xmath244 be the set of dominant @xmath240-weights of @xmath227 . let @xmath245 be the maximal torus consisting of all the diagonal matrices in @xmath246 . for each @xmath88 , let @xmath247 be the function which sends a matrix in @xmath245 to its @xmath130th element in the diagonal . then @xmath240 is a subtorus of @xmath245 . let @xmath248 , which is the set of simple roots of @xmath249 . let @xmath250 be the corresponding positive borel subgroup . we refer the reader to ( * ? ? ? * , page 144145 ) for the explicit description of @xmath250 . for each @xmath251 , we denote by @xmath252 and @xmath253 the simple module , co - weyl module and weyl module for @xmath227 associated to @xmath24 respectively . we identify the weight @xmath254 with @xmath255 . thus a weight @xmath256 is dominant if and only if @xmath257 is a partition . for any @xmath258 , we define @xmath259 let @xmath8 be an integer with @xmath6 $ ] . let @xmath264 be the set of dominant @xmath240-weights of @xmath227 appearing in @xmath265 . by @xcite , @xmath266$. } \end{matrix}\biggr\}.\ ] ] it is well - known that @xmath116 is saturated in the sense of ( * ? ? ? * part ii , a.2 ) . let @xmath267 denote the category of finite dimensional @xmath227-modules @xmath209 such that all composition factors of @xmath209 have the form @xmath268 with @xmath269 . following ( * ? ? ? * part ii , chapter a ) , we define a functor @xmath115 from the category of finite dimensional @xmath227-modules to the category @xmath267 as follows : @xmath270 let @xmath209 be a finite dimensional @xmath227-module . recall that ( ( * ? ? ? * part ii , 4.16 ) ) an ascending chain @xmath271 of @xmath227-submodules is called a good filtration of @xmath209 if each @xmath272 is isomorphic to some @xmath273 with @xmath274 . we use @xmath275 to denote the number of factors in the filtration isomorphic to @xmath273 . similarly , an ascending chain @xmath276 of @xmath227-submodules is called a weyl filtration of @xmath209 if each @xmath277 is isomorphic to some @xmath278 with @xmath279 . we use @xmath280 to denote the number of factors in the filtration isomorphic to @xmath278 . note that the left action of @xmath227 on @xmath18 commutes with the right action of @xmath17 . by definition , @xmath291 is a sum of some submodules of the form @xmath292 where @xmath293 . applying the result 1 ) that we have proved , we get that ( as a @xmath227-module ) @xmath294 therefore , it follows from definition that @xmath295 . we want to show that @xmath296 . before proving this equality , we need some preparation . for simplicity , for any two finite dimensional @xmath227-modules @xmath297 , we use @xmath298 to denote the sum of all the image subspaces @xmath299 , where @xmath300 runs over all the @xmath227-module homomorphisms from @xmath209 to @xmath301 . it is clearly a @xmath227-submodule of @xmath301 . we use @xmath302 to denote the following embedding : @xmath303 recall that there is a natural right action of the brauer algebra @xmath315 on the @xmath316-tensor space @xmath317 . we use @xmath318 to denote the natural @xmath2-algebra homomorphism from @xmath319 to @xmath320 . by ( * theorem 1.4 ) , we know that @xmath318 is surjective . therefore , @xmath320 is spanned by the elements @xmath321 , where @xmath135 runs over the brauer @xmath316-diagrams in @xmath322 . let @xmath323 . we regard @xmath321 as an element in @xmath324 using the isomorphism @xmath325 . by the definition of @xmath318 , it is easy to see that @xmath326 for some @xmath327 . now using the isomorphism @xmath328 one can check the image of @xmath321 . it follows that @xmath329 is spanned by all @xmath305 , where @xmath306 . it remains to prove that @xmath330 . it suffices to show that the image of @xmath331 is contained in @xmath102 for each @xmath323 . note that @xmath332 . in particular , we can write @xmath333 where @xmath334 are @xmath335 independent summation indices . the positions of each pair of @xmath336 in the above sum are uniquely determined by @xmath337 and hence by @xmath135 . for example , if then @xmath338 each independent summation index @xmath339 was attached with a pair of integers @xmath340 such that @xmath341 appear in the position @xmath342 respectively . using ( * ? ? ? * lemma 4.2 ) and the definition of @xmath328 , it suffices to show that @xmath343 we fix a bijection between the horizontal edges in the top row of @xmath135 and the horizontal edges in the bottom row of @xmath135 . we consider a vertex @xmath231 in the top row of @xmath135 which is labeled by an integer @xmath188 with @xmath344 . there are only three possibilities : 1 . @xmath231 is connected with a vertex @xmath250 in the top row of @xmath135 which is labeled by an integer @xmath345 with @xmath346 , then the horizontal edge @xmath347 must correspond to a horizontal edge @xmath348 in the bottom row of @xmath135 . applying ( * ? ? ? * lemma 4.2 ) and the definition of @xmath325 , we see that the edges @xmath349 determine two integers @xmath350 , such that @xmath351 ; 2 . @xmath231 is connected with a vertex @xmath250 in the top row of @xmath135 which is labeled by an integer @xmath345 with @xmath352 , then the horizontal edge @xmath347 must correspond to a horizontal edge @xmath348 in the bottom row of @xmath135 . applying ( * ? ? ? * lemma 4.2 ) and the definition of @xmath325 , we see that the edges @xmath348 determines an integer @xmath353 , such that @xmath354 ; 3 . @xmath231 is connected with a vertex @xmath250 in the bottom row of @xmath135 , then applying ( * ? ? ? * lemma 4.2 ) and the definition of @xmath325 , we see that the vertical edge @xmath347 determines an integer @xmath353 , such that @xmath355 . [ bp2 ] let @xmath209 be a @xmath227-module and @xmath372 . suppose @xmath373 is a surjective @xmath227-module homomorphism such that @xmath374 for some highest weight vector @xmath375 of weight @xmath24 , then @xmath325 is an isomorphism . in particular , @xmath376 . suppose that @xmath387 . then there exist some integers @xmath388 such that @xmath389 comparing the coefficients of @xmath247 for each @xmath390 and adding all of them together , we get that @xmath391 which is a contradiction to the assumption that @xmath392 . as a @xmath227-module , @xmath393 is a tilting module . by ( * ? ? ? * proposition e.7 ) , we know that @xmath18 is a tilting module too . in particular , @xmath18 has both a weyl filtration and a good filtration . we fix an integer @xmath8 with @xmath6 $ ] . in view of lemma [ bp ] and lemma [ compare ] , we can find a good filtration of @xmath18 : @xmath394 such that 1 . @xmath395 for @xmath396 , where @xmath397 , @xmath398 for some integers @xmath399 $ ] and @xmath400 ; 2 . @xmath401 for any @xmath402 , and @xmath403 only if @xmath404 ; 3 . there exists an integer @xmath405 , such that @xmath406 if and only if @xmath407 . \1 ) if @xmath409 is a ( non - zero ) maximal vector of weight @xmath410 in @xmath18 , then @xmath411 is also a ( non - zero ) maximal vector of weight @xmath410 in @xmath412 . moreover , every maximal vector of weight @xmath410 in @xmath412 arises in this way ; \2 ) since @xmath418 has a good filtration with each section being isomorphic to some @xmath419 with @xmath420 , it follows from lemma [ bp ] that @xmath421 therefore , the natural embedding @xmath422 becomes an equality , from which we deduce that @xmath423 . \3 ) this follows from the following simple fact : if @xmath209 is a finite dimensional @xmath227-module with a good filtration and @xmath372 is minimal such that @xmath23 appears in a good filtration of @xmath209 , then @xmath424 is a direct sum of copies of @xmath23 and @xmath425 has a good filtration without factor @xmath23 . in particular , @xmath426 is a right module for @xmath427 and @xmath425 has an @xmath427-module structure . let @xmath35 be an integer and @xmath24 a composition of @xmath35 . the conjugate of @xmath24 is the partition @xmath428 where @xmath429 . the young diagram of @xmath24 is the set @xmath430:=\bigl\{(a , b)\bigm|1\leq b\leq{\lambda}_a\bigr\}.\ ] ] assume that @xmath24 is a partition of @xmath28 . a @xmath24-tableau is a bijective map @xmath431\rightarrow\{1,2,\dots , n\}$ ] . a standard @xmath24-tableau is a @xmath24-tableau in which the entries increase along each row and down each column . let @xmath432 ( resp . , @xmath433 ) be the standard @xmath24-tableau such that the numbers @xmath126 appear in order along the rows ( resp . , along the columns ) . let @xmath434 be the young subgroup of @xmath200 corresponding to @xmath24 , which is the subgroup fixing the sets @xmath435 . let @xmath436 be the set of distinguished right coset representatives of @xmath434 in @xmath200 . let @xmath437 such that @xmath438 . then @xmath439 . let @xmath440 . recall the definition of @xmath290 in definition [ alpha ] . recall that @xmath357 is the unipotent radical of the positive borel subgroup @xmath250 . for each @xmath227-module @xmath209 , we use @xmath443 to denote the subspace of maximal vectors in @xmath209 of weight @xmath24 . the following lemma plays a key role in this section . the proof will be given in the next section . [ hu2lem ] let @xmath441 be an integer with @xmath22 $ ] and @xmath24 be a partition of @xmath109 with @xmath21 . then @xmath444 . in particular , @xmath25 is a non - zero maximal vector of weight @xmath24 in @xmath18 . moreover , the dimension of @xmath445 ( and hence of @xmath26 ) is independent of @xmath2 ; by construction , @xmath448 . we fix a decomposition : @xmath449 where @xmath450 for each @xmath451 . let @xmath452 be a fixed non - zero maximal vector of weight @xmath410 in @xmath413 . for each @xmath451 , let @xmath453 be a fixed non - zero maximal vector of weight @xmath410 in @xmath454 and let @xmath455 be the unique isomorphism which sends @xmath452 to @xmath453 . then it is easy to see ( by definition and comparing dimensions ) that the following map @xmath456 is a left @xmath417-module isomorphism . we claim that it is also a right @xmath17-module homomorphism . it suffices to show that @xmath457 for each @xmath451 , @xmath416 and @xmath458 . we write @xmath459 , where @xmath460 for each @xmath39 . then by definition , @xmath461 . by the commuting action between @xmath417 and @xmath17 , it is easy to check that @xmath462 is a left @xmath417-homomorphism from @xmath413 to @xmath272 . by direct verification , @xmath463 , hence @xmath464 for any @xmath465 . since @xmath466 and @xmath467 , it follows that any non - zero map in @xmath468 must be injective . we deduce that @xmath464 for any @xmath458 , as required . this proves that @xmath469 is a right @xmath17-module homomorphism as well . by lemma [ keylem1 ] , it is enough to show that @xmath472 since @xmath265 has a weyl filtration such that each section is isomorphic to some @xmath473 with @xmath269 and @xmath474 has a good filtration such that each section is isomorphic to some @xmath475 with @xmath476 , it follows that @xmath477 hence @xmath478 which implies that @xmath479 as required . first , by lemma [ jakey ] 2 ) , we have that @xmath484 note that the character @xmath485 of @xmath23 is given by the weyl character formula . in particular , @xmath486 is independent of @xmath2 . note also that the character @xmath487 is equal to @xmath488 which is also independent of @xmath2 . so the filtration multiplicities @xmath489 is nothing but the coefficient of the weyl character @xmath490 in the expansion of @xmath488 into a linear combination of weyl characters . in particular , @xmath491,\ ] ] which is independent of @xmath2 . since the set @xmath116 is also independent of @xmath2 , we conclude that @xmath492 is independent of @xmath2 . moreover , by the construction of @xmath493 , it is clear that @xmath494 let @xmath353 be an integer with @xmath507 . by the properties of the filtration ( [ filtra1 ] ) and the filtration ( [ filtra2 ] ) we have constructed and lemma [ bp ] 4 ) , it is clear that @xmath508 . it follows that the natural embedding @xmath509 becomes an equality . by lemma [ jakey2 ] , we see that the following map @xmath510 is an embedding . it follows that @xmath511 for each @xmath507 . in particular , @xmath512 induces a natural map from @xmath513 to @xmath272 . recall that for each @xmath408 , @xmath514 . in particular , @xmath515 for each @xmath407 . for each @xmath407 , we define @xmath516 . then @xmath517 is a maximal vector of weight @xmath410 in @xmath265 and @xmath518 ( cf . lemma [ hu2lem ] and proposition [ mainprop2 ] ) . it is clear that @xmath519 . let @xmath520 be a @xmath227-submodule of @xmath513 such that @xmath521 . applying lemma [ bp2 ] , we deduce that @xmath522 . therefore , combining with our induction hypothesis , we can find a @xmath227-submodule @xmath523 of @xmath18 such that @xmath524 by the proof of proposition [ mainprop2 ] , we can deduce that @xmath525 . as a consequence , we get that @xmath526 since @xmath527 , it follows that @xmath528 , as required . with theorem [ keystep3 ] in hand , theorem [ mainthm0 ] and [ mainthm1 ] follow almost trivially from some standard arguments in commutative algebras . for the reader s convenience , we include the details below . by theorem [ keystep3 ] and lemma [ jakey ] , we know that @xmath102 has a good filtration . note that @xmath18 has a good filtration too . applying ( * ? ? ? * part ii , 4.17 ) , we deduce that both @xmath119 and @xmath530 have a good filtration . this proves theorem [ mainthm1 ] and the first statement of corollary [ maincor15 ] . it is clear that @xmath99 is a free @xmath80-module of finite rank @xmath531 . since @xmath532 , it follows that @xmath533 . by theorem [ keystep3 ] , we know that @xmath482 is independent of @xmath2 and hence @xmath534 . thus the canonical surjection @xmath535 must be an isomorphism . now let @xmath536 be an arbitrary field and @xmath537 be the algebraic closure of @xmath536 . recall that @xmath538 , @xmath539 . it is clear that the canonical map @xmath540 is an isomorphism . it follows that @xmath541 . in particular , the first statement of corollary [ maincor1 ] also follows . for any commutative @xmath80-algebra @xmath536 which is a field , the canonical surjection @xmath542 must be an isomorphism . for any commutative @xmath80-algebra @xmath95 , let @xmath536 be a field such that @xmath543 for some maximal ideal @xmath70 of @xmath95 . we have the following commutative diagram of maps : @xmath544 where @xmath545 are all canonical maps . since @xmath546 is an isomorphism , it follows that the canonical surjection @xmath547 must be injective and hence an isomorphism . this proves the first isomorphism in part 2 ) of theorem [ mainthm0 ] . using the same argument as before , we know that @xmath548 note that @xmath549 . in order to show that @xmath550 is a free @xmath80-module of rank @xmath551 , we consider the following commutative diagram of maps : @xmath552^{\quad\iota\otimes\operatorname{id } } \ar[d]^{\wr } & v_{{\mathbb z}}^{\otimes n}\otimes_{{\mathbb z}}f \ar@{>>}[r]^{\pi\otimes\operatorname{id}\quad\quad\,}\ar[d]^{\wr } & v_{{\mathbb z}}^{\otimes n}/v_{{\mathbb z}}^{\otimes n}{\mathfrak{b}}_n^{(f)}\otimes_{{\mathbb z}}f \ar@{>>}[d]^{\theta}\\ v_{f}^{\otimes n}{\mathfrak{b}}_n^{(f ) } \ar@{^{(}->}[r ] & v_{f}^{\otimes n } \ar@{>>}[r ] & v_{f}^{\otimes n}/v_{f}^{\otimes n}{\mathfrak{b}}_n^{(f)},}\ ] ] where @xmath553 denotes the natural injection @xmath554 , @xmath555 denotes the natural projection @xmath556 . by diagram chasing , it is easy to see that the natural surjection @xmath557 is an injection and hence an isomorphism . in particular , @xmath558 we claim that @xmath559 must be a free @xmath80-module . in fact , suppose this is not the case , then @xmath559 must contain a non - zero @xmath560-torsion element for some prime number @xmath560 . it follows that @xmath561 which is a contradiction . this proves our claim . as a consequence , we conclude that @xmath99 is a pure @xmath80-submodule of @xmath100 . now using a commutative diagram similar to ( [ cd1 ] ) ( replacing @xmath99 with @xmath101 ) , we can argue as before that for any commutative @xmath80-algebra @xmath95 , the canonical map @xmath562 is always an isomorphism . this completes the proof of theorem [ mainthm0 ] . it remains to prove the second statement of corollary [ maincor1 ] . let @xmath2 be an algebraically closed field . recall the bimodule isomorphism @xmath222 introduced in the paragraph below ( [ asso ] ) . @xmath222 induces a map @xmath563 from @xmath74 to @xmath564 . the second equality in ( [ asso ] ) and the definition of @xmath74 imply that the image of @xmath563 is contained in @xmath565 . therefore , @xmath563 is a bimodule homomorphism from @xmath81 to @xmath566 . we claim that @xmath563 is injective . let @xmath567 such that @xmath568 . now for any @xmath569 , we can write ( by applying corollary [ maincor25 ] ) @xmath570 where @xmath571 . using the assumption that @xmath568 and applying the two equalities in ( [ asso ] ) we get that @xmath572 since the bilinear form @xmath573 on @xmath18 is non - degenerate , it follows that @xmath574 . hence @xmath563 is injective . this proves our claim . to emphasize the base field , we use the notations @xmath575 , @xmath576 . note that we have already known that the dimension of @xmath291 is independent of the field @xmath2 . the space @xmath74 can be identified as the solution space of a homogeneous system of linear equations whose coefficients are all defined over @xmath80 . in particular , @xmath577 by ( * ? ? ? * ( 10.3.14 ) ) , we know that @xmath578 by the injectivity of @xmath563 , we know that @xmath579 it follows that @xmath580 . therefore , @xmath563 must be an isomorphism . this completes the proof of corollary [ maincor1 ] . note that the space @xmath55 can be identified as the solution space of a homogeneous system of linear equations whose coefficients are all defined over @xmath80 . this implies that @xmath582 . now applying corollary [ maincor1 ] and ( [ embed ] ) , we prove the corollary . suppose that @xmath954 . by the proof of lemma [ jakey2 ] , @xmath102 is a sum of some submodules @xmath955 such that @xmath956 for each @xmath130 . it follows that @xmath957 since @xmath265 is a tilting module , @xmath265 has a weyl filtration . for each @xmath958 , @xmath959 only if @xmath960 for some integer @xmath961 $ ] . by lemma [ bp ] , we get that @xmath965\\ 0\leq s < f}}\bigl(v^{\otimes n-2f}:\delta({\lambda})\bigr ) \bigl(v^{\otimes n}/v^{\otimes n}{\mathfrak{b}}_n^{(f)}:\nabla({\lambda})\bigr)\\ & = 0 , \end{aligned}\ ] ] which is a contradiction . we have the following exact sequence of maps : @xmath968 since @xmath18 has a weyl filtration , and by theorem [ mainthm2 ] , @xmath102 has a good filtration , it follows that @xmath969 . this implies that the canonical map @xmath966 is surjective . this proves 1 ) . the above exact sequence implies that @xmath970 since @xmath18 has a weyl filtration as well as a good filtration , and @xmath102 has a good filtration , and both the character formula of @xmath18 and of @xmath291 are independent of @xmath2 , it follows that @xmath971 which is independent of @xmath2 . note that @xmath972 so 2 ) also follows . * proof of theorem [ mainthm2 ] : * let @xmath973 denote the natural @xmath2-algebra homomorphism : @xmath974 . then @xmath975 in view of lemma [ lm42 ] , it suffices to show that @xmath973 is surjective . we consider the following commutative diagram of maps : @xmath976^{\varphi_k}\ar[d]_{\varphi''_k } & \operatorname{end}_{kg}\bigl(v^{\otimes n}\bigr ) \ar@{>>}[d]^{\theta_1}\\ \operatorname{end}_{kg}\bigl(v^{\otimes n}/v^{\otimes n}{\mathfrak{b}}_n^{(f)}\bigr ) \ar[r]_{\iota_1\,\,\,\quad}^{\sim\,\,\quad } & \operatorname{hom}_{kg}\bigl(v^{\otimes n},v^{\otimes n}/v^{\otimes n}{\mathfrak{b}}_n^{(f)}\bigr),}\ ] ] by @xcite , the top horizontal map is surjective . by lemma [ lm42 ] , @xmath977 is surjective . by lemma [ hom0 ] , @xmath978 is an isomorphism . it follows that @xmath973 must be surjective , as required . let @xmath981 be the symplectic schur algebra . by theorem [ thm11 ] , there is a natural surjection @xmath982 . the action of @xmath983 on @xmath83 factors through an action of @xmath984 . it follows that @xmath985 let @xmath986 be the algebraic closure of @xmath2 . by @xcite and @xcite , @xmath987 . it follows that @xmath988 in particular , @xmath989 hence @xmath979 is independent of the infinite field @xmath2 . note that @xmath990 . it follows that @xmath991 from which the proposition follows immediately . let @xmath8 be an integer with @xmath204 $ ] and @xmath2 an infinite field . proposition [ mainprop ] proves one side of the brauer - schur - weyl duality between @xmath117 and @xmath118 on @xmath119 . we conjecture that the other side of the brauer - schur - weyl duality is also true . that is , the dimension of the endomorphism algebra @xmath998 is independent of @xmath2 and the natural @xmath2-algebra homomorphism @xmath999 is also surjective . the purpose of this section is to give a proof of lemma [ hu2lem ] as well as theorem [ mainthm3 ] 2 ) . the key step in our proof is to show that ( for any @xmath583 ) @xmath584 and @xmath585 has a weyl filtration as a @xmath417-module . the proof makes use of lusztig s theory of canonical basis and based modules . to this end , we have to work in a quantized setting . let @xmath586 be an indeterminate over @xmath80 . let @xmath587 $ ] be the ring of laurent polynomials in @xmath586 . let @xmath588 be the specialized birman - murakami - wenzl algebra ( @xcite , @xcite ) . by definition , it has generators @xmath589 which satisfy the following relations : 1 . @xmath590 , for @xmath591 , 2 . @xmath592 , for @xmath591 , 3 . @xmath593 , for @xmath594 , 4 . @xmath595 , for @xmath596 , 5 . @xmath597 , for @xmath594 , 6 . @xmath598,for @xmath594 , 7 . @xmath599 , for @xmath591 . 8 . @xmath600,for @xmath594 . let @xmath601 be the quantized enveloping algebra of @xmath602 over @xmath603 . let @xmath604 be the chevalley generators of @xmath605 . then it is a hopf algebra with coproduct @xmath606 , counit @xmath607 and antipode @xmath608 defined on generators by @xmath609 where @xmath610 we regard @xmath80 as an @xmath611-algebra by specializing @xmath586 to @xmath235 . for each @xmath611-module @xmath301 and @xmath612 , we set @xmath613 and @xmath614 . we call @xmath615 the specialization of @xmath188 at @xmath616 . let @xmath617 be lusztig s @xmath611-form in @xmath618 . let @xmath619 be the free @xmath611-module spanned by @xmath153 . it is well - known that there are natural commuting actions ( cf . * section 2,3 ) ) of @xmath617 and @xmath588 on @xmath620 . if we specialize @xmath586 to @xmath235 , then each @xmath621 ( resp . , @xmath622 ) specializes to @xmath623 ( resp . , to @xmath147 ) , and @xmath624 will become the brauer algebra @xmath4 in this paper . moreover , the action of @xmath624 on @xmath620 becomes the action of @xmath4 on @xmath100 in this paper . we recall that the representation @xmath625 of @xmath588 on @xmath620 is defined on generators as follows : @xmath626 for all @xmath627 , where @xmath628 and each @xmath234 is the matrix unit ( i.e. , @xmath629 for each @xmath212 ) . let @xmath630 be the iwahori hecke algebra associated to the symmetric group @xmath200 , defined over @xmath611 and with parameter @xmath586 . by definition , @xmath630 has generators @xmath631 which satisfy the following relations : @xmath632 let @xmath633 . for each composition @xmath24 of @xmath28 , we use @xmath634 to denote the hecke algebra over @xmath603 associated to the young subgroup @xmath434 . for each @xmath635 , we use @xmath636 to denote the minimal integer @xmath35 such that @xmath637 ; in that case , we call @xmath638 a reduced expression of @xmath639 . we define @xmath640 if @xmath641 with @xmath642 . it is well - known that this is well defined , i.e. , independent of the choice of the reduced expression . let @xmath643 be the free @xmath611-submodule of @xmath619 generated by @xmath644 . we recall that the representation @xmath645 of @xmath630 on @xmath646 is defined on generators as follows : @xmath647 where @xmath648 let @xmath24 be a partition with @xmath21 . recall the definition of @xmath649 in definition [ zdfn ] . we have that @xmath650 for each @xmath651 . let @xmath668 . let @xmath669 be the quantized enveloping algebra of the general linear lie algebra @xmath670 over @xmath603 . let @xmath671 be lusztig s @xmath611-form in @xmath672 . there is a natural representation of @xmath671 on @xmath643 and hence on @xmath646 ( cf . @xcite ) . by ( * ( 27.3 ) ) , the @xmath673-module @xmath674 is a based module . there is a bar involution @xmath675 which is defined on @xmath674 . for any integers @xmath676 , there is a unique element @xmath677 , such that 1 . @xmath678 , and 2 . @xmath679 is equal to @xmath680 plus a linear combination of elements @xmath681 with @xmath682 and with coefficients in @xmath683 $ ] , where @xmath684 is a partial order root system . ] defined in ( * ? ? ? * ( 27.3.1 ) ) . let @xmath685 . then @xmath686 is the set of lusztig s canonical bases of @xmath674 . in particular , @xmath686 is an @xmath611-basis of @xmath646 . similarly , @xmath687 is a based module as a @xmath601-module . let @xmath250 be the set of canonical bases of @xmath209 . then @xmath250 is actually an @xmath611-basis of @xmath688 . recall that @xmath689 is the set of dominant @xmath240-weights appeared in @xmath209 ( cf . ( [ pif ] ) ) . by ( * ( 27.2.1 ) ) , there is a partition @xmath690.\ ] ] for each @xmath691 , we set @xmath692:=\bigsqcup_{{\lambda}\leq\mu\in\pi_0}b[\mu],\quad b[>\!{\lambda}]:=\bigsqcup_{{\lambda}<\mu\in\pi_0}b[\mu].\ ] ] let @xmath693_{{\mathscr a } } , m[>\!{\lambda}]_{{\mathscr a}}$ ] be the @xmath611-submodule of @xmath694 generated by the canonical basis elements in @xmath695 , b[>\!{\lambda}]$ ] respectively . by @xcite , both @xmath693_{{\mathscr a}}$ ] and @xmath696_{{\mathscr a}}$ ] are @xmath697-stable and we have that @xmath698_{{\mathscr a}}/m[>\!{\lambda}]_{{\mathscr a}}\cong \bigl(\delta({\lambda})_{{\mathscr a}}\bigr)^{\oplus n_{{\lambda}}},\ ] ] for some @xmath699 . moreover , the canonical image of each @xmath700 $ ] in @xmath693_{{\mathscr a}}/m[>\!{\lambda}]_{{\mathscr a}}$ ] is mapped to a canonical basis element of some direct summand @xmath701 in the right - hand side of ( [ filtra3 ] ) . recall that a weight of @xmath669 is identified as an element @xmath702 by setting @xmath703 for @xmath390 ; in this case , @xmath383 is said to be dominant if @xmath704 . for any two weights @xmath705 of @xmath669 , we write @xmath706 if @xmath707 for some @xmath708 . let @xmath709 be the set of dominant weights of @xmath669 appeared in @xmath710 . then @xmath711 is nothing but the set of partitions of @xmath28 with no more than @xmath70 parts . for each @xmath712 , let @xmath713^{hi}$ ] be as defined in @xcite . using @xmath711 and the partial order @xmath714 , one can define the subsets @xmath715 , \hat{b}[>\!{\lambda}]$ ] and the @xmath611-submodule @xmath716_{{\mathscr a } } , \hat{m}[>\!{\lambda}]_{{\mathscr a}}$ ] in a similar way as before . we set @xmath717^{hi},\quad { b}^{hi}:=\bigsqcup_{{\lambda}\in { \pi}_{0}}{b}[{\lambda}]^{hi}.\ ] ] let @xmath718 be a partition of @xmath28 satisfying @xmath719 and @xmath720 . we define @xmath721 which is ( by definition ) a canonical basis element in @xmath686 . for each integer @xmath723 , let @xmath724 denote the kashiwara operators as defined in @xcite . since we are following the notations and definitions of coproduct in lusztig s book , the actions of kashiwara operators on the tensor product of crystal bases are given by the following formulae ( cf . @xcite ) @xmath725 where @xmath726 , @xmath727 are the crystal bases of the @xmath669-modules @xmath728 respectively , and @xmath729 let @xmath723 and @xmath730 . recall that @xmath731 to prove the lemma , it suffices to show that for each @xmath390 , @xmath732 but this follows from a direct verification ( using the above tensor product rule ) and an inductive argument . we write @xmath741 , where @xmath742 . by ( * theorem 27.3.2 ) , there is a bar involution @xmath743 on @xmath666 , and for each @xmath180 , there is a unique canonical basis element @xmath744 satisfying : we claim that @xmath751 for any @xmath752 . we use induction on @xmath28 . suppose that our claim is true when @xmath28 is replaced by @xmath753 . let @xmath754 , @xmath755 be the bar involutions defined on @xmath756 , @xmath757 respectively . let @xmath222 be the quasi-@xmath758-matrix of @xmath601 introduced in @xcite . we shall mainly follow the notations in @xcite without any further explanation , except that we use @xmath586 to denote the quantum parameter instead of @xmath409 in lusztig s book . we define @xmath759 in a similar way , one can define the subalgebras @xmath760 , @xmath761 of @xmath669 using the chevalley generators of @xmath669 . it is well - known that both @xmath762 and @xmath760 can be presented by their chevalley generators and quantum serre relations ( @xcite ) . using the pbw bases ( @xcite ) for @xmath762 and @xmath760 , it is easy to see that @xmath760 can be identified with the subalgebra of @xmath763 generated by @xmath764 . a similar statement holds for @xmath761 and @xmath765 . let @xmath766 be defined as in @xcite and @xmath767 be a @xmath603-basis of @xmath766 such that @xmath768 is a basis of @xmath769 for any @xmath770 . let @xmath771 be the basis of @xmath766 dual to @xmath767 under the bilinear form @xmath772 introduced in @xcite . then @xmath773 , where @xmath774 applying ( * ? ? ? * ( d ) ) , we see that @xmath771 is homogeneous too . moreover , @xmath775 and @xmath776 are bases of @xmath761 and @xmath760 respectively . recall that @xmath777 is a part of the canonical bases of the @xmath601-module @xmath778 . we refer the reader to ( * ? ? ? * section 6 ) for the description of the whole canonical bases of @xmath778 and the action of @xmath601 on @xmath778 . the important facts that we need here are @xmath779 and @xmath780 for any @xmath781 . for any @xmath782 and any @xmath783 , we see that @xmath784 must appear in every monomial occurring in @xmath785 . in particular , this implies that @xmath786 since @xmath787 . now we have that @xmath790 on the other hand , by construction , we have that @xmath791v_{j_1}\otimes\cdots\otimes v_{j_n}.\ ] ] suppose that @xmath792 . then we can find an @xmath793 , such that @xmath794v_{j_1}\diamond_c\cdots\diamond_c v_{j_n},\ ] ] where @xmath795 $ ] . applying @xmath743 to both sides of the above equality , we get a contradiction to ( [ bar1 ] ) . this proves that @xmath796 and the first statement of the lemma follows . the second statement follows from the definition of @xmath797^{hi}$ ] ( see @xcite ) , the facts that @xmath798 for any @xmath390 and the action of the kashiwara operators @xmath799 are compatible with the natural embedding @xmath800 . finally , note that for any @xmath801 $ ] , @xmath801^{hi}$ ] if and only if the canonical image of @xmath802 in @xmath803_{{\mathscr a}}/m[>\!\mu]_{{\mathscr a}}$ ] is mapped to a highest weight vector of some direct summand @xmath701 in the right - hand side of ( [ filtra3 ] ) . it follows that @xmath804_{{\mathscr a}}=\sum_{\substack{b\in b[\nu]^{hi}\\ \mu<\nu\in\pi_0}}\mathbb{u}_{{\mathscr a}}(\mathfrak{sp}_{2m})b,\quad \hat{m}[>\!\mu]_{{\mathscr a}}=\sum_{\substack{b\in\hat{b}[\nu]^{hi}\\ \mu<\nu\in\hat\pi_0}}\mathbb{u}_{{\mathscr a}}(\mathfrak{gl}_{m})b.\ ] ] now the third statement follows from 2 ) and the fact for any @xmath805 , @xmath806 and every chevalley generator of @xmath618 acts on @xmath409 in the same way as the corresponding chevalley generator of @xmath807 . now we are in position to prove the key lemma in this section . the most difficult part is the second statement of the following lemma . the main idea of its proof is to show that there exists a canonical basis element @xmath808^{hi}$ ] of the @xmath601-module @xmath666 such that @xmath809 always appears with coefficient @xmath235 in the linear expansion of @xmath810 into the specialization at @xmath616 of the canonical basis elements of @xmath666 . the element @xmath802 will be identified with a canonical basis element in the @xmath669-module @xmath710 and hence eventually identified with a parabolic kazhdan - lusztig basis of certain permutation module over a type @xmath231 hecke algebra . it is well - known that @xmath812 ( cf . in particular , @xmath813 . note that @xmath814 . thus the statement 1 ) follows from lemma [ lmhx ] . since @xmath815 , it follows easily that @xmath816 . it is also well - known ( cf . @xcite ) that @xmath817 as a right @xmath59-module . in particular , @xmath818 which is independent of the field @xmath2 . this proves the statement 3 ) . it remains to prove the statement 2 ) . we divide the proof into three steps : _ step 1 . _ we claim that there exists a canonical basis element @xmath819^{hi}$ ] , such that @xmath820^{hi}}{c}'_{b'}b'\!\!\downarrow_{q=1}\!\!\!\pmod{\hat{m}[>\!{\lambda}]_{{\mathbb z}}},\ ] ] where @xmath821 for each @xmath822 . for each composition @xmath383 of @xmath28 , let @xmath823 be the corresponding weight subspace of @xmath646 ( with respect to the action of @xmath671 ) . it is well known that there is a right @xmath630-module isomorphism : @xmath824 where @xmath825 is the one dimensional representation of @xmath826 which is defined on generators by @xmath827 for each @xmath828 . let @xmath300 be the anti - linear involution on @xmath829 which is defined on generators by @xmath830 for each @xmath143 . by the main result of @xcite , @xmath300 naturally induces an anti - linear involution on @xmath831 so that one can define the parabolic kazhdan - lusztig bases @xmath832 of @xmath831 . recall that @xmath831 also has a standard basis @xmath833 . for each @xmath834 , we have that @xmath835\hat{y}_{{\lambda}}\hat{t}_{d'}},\ ] ] where @xmath836 " is the usual bruhat order defined on the symmetric group @xmath200 . we identify @xmath837 with @xmath838 via the isomorphism @xmath839 . it follows from ( * ? ? ? * theorem 2.5 ) that the bases @xmath832 coincide with the canonical bases of @xmath710 which are of weight @xmath24 . here one should understand the notations @xmath840 in this paper as the notations @xmath841 in @xcite . let @xmath842 such that @xmath843 . for each @xmath834 , @xmath844 . note that if @xmath845 , then @xmath846 . we want to use the isomorphism @xmath839 and ( [ can1 ] ) to express @xmath810 into a linear combination of the specializations at @xmath616 of some canonical basis elements . to this end , we first express @xmath810 into a linear combination of standard basis elements . let @xmath847 be the longest element in the young subgroup @xmath848 of @xmath200 . note that @xmath849 is row - standard , which implies that @xmath850 . since @xmath851 , we have that @xmath852 . note also that @xmath853 . it follows that @xmath854 where @xmath855 for each @xmath856 and @xmath857 . since @xmath858 for any @xmath859 , we deduce that @xmath860 only if @xmath861 . that is , @xmath862 recall that in this paper @xmath200 acts on @xmath18 by sign permutation action . thus , @xmath863 , @xmath864 . note that @xmath865 , @xmath866 for each @xmath834 . it follows from ( [ can1 ] ) , ( [ can2 ] ) and the isomorphism @xmath839 that @xmath867 where @xmath868 . we set @xmath869 and write @xmath718 , where @xmath870 and @xmath871 . then it is easy to check that @xmath872 which implies that @xmath873 by lemma [ lmkey41 ] . hence @xmath874^{hi}$ ] as @xmath802 is a weight vector of weight @xmath24 . note that @xmath875 only if @xmath822 is a weight vector of weight @xmath24 . on the other hand , if @xmath876 $ ] and @xmath822 is a weight vector of weight @xmath24 then we must have that @xmath24 is a weight of the weyl module of @xmath669 associated to @xmath383 , which implies that @xmath877 . therefore , we can rewrite ( [ expan1 ] ) as @xmath820^{hi}}{c}'_{b'}b'\!\!\downarrow_{q=1}\!\!\!\pmod{\hat{m}[>\!{\lambda}]_{{\mathbb z}}},\ ] ] where @xmath878^{hi}$ ] , @xmath821 for each @xmath822 . this proves our claim . recall the equality ( [ filt6 ] ) we obtained in step 2 . let @xmath700^{hi}$ ] be the canonical basis element we obtained in step 2 . using lemma 3.7 3 ) , we can get a sequence @xmath886 of dominant weights in @xmath689 and construct a weyl filtration of @xmath694 as follows : @xmath887 such that 1 . for each integer @xmath888 , @xmath889 is spanned by the canonical basis elements it contains ; 2 . for each integer @xmath888 , there is a @xmath697-module isomorphism : @xmath890 , such that if @xmath891 then the canonical image of @xmath822 in @xmath892 is mapped either to @xmath237 or to a canonical basis element of @xmath893 ; 3 . @xmath894 only if @xmath404 ; 4 . @xmath895^{hi}\setminus\{b\}\subseteq m^{{\mathscr a}}_{(i_0 - 1)}$ ] , where @xmath896 is the unique integer such that @xmath897 . in particular , @xmath898 as @xmath808^{hi}$ ] . thus we can write @xmath899 , where @xmath900 , such that for each @xmath888 , the elements in the subset @xmath901 form an @xmath611-basis of @xmath889 . moreover , @xmath902 specializing at @xmath616 , we get a @xmath903-submodules filtration of @xmath100 . we define a new @xmath903-submodules filtration of @xmath100 as follows : @xmath904 where @xmath905 we claim that ( [ filtra5 ] ) is a weyl filtration of @xmath906 . in fact , to prove this claim , it suffices to show that for each @xmath907 , @xmath908 is isomorphic to some weyl module . let @xmath536 be any field which is an @xmath80-algebra . let @xmath909 . replacing @xmath80 by @xmath536 in ( [ filtra5 ] ) , we get a @xmath910-submodules filtration of @xmath911 as follows : @xmath912 recall that @xmath810 is a maximal vector of weight @xmath24 . using ( [ filt6 ] ) , ( [ generator ] ) , together with the third and the fourth properties of the filtration ( [ filtra4 ] ) , we see that @xmath913 by the second property of the filtration ( [ filtra4 ] ) , we know that @xmath914 it follows that each @xmath915 must be a homomorphic image of @xmath916 , where @xmath917 noting that @xmath918 and comparing the dimensions , we deduce that the natural surjection from @xmath916 onto @xmath915 must be an isomorphism . in particular , @xmath919 since @xmath810 is a maximal vector of weight @xmath24 , we have a natural surjection from @xmath920 onto @xmath921 , which induces a surjection from @xmath922 onto @xmath923 . on the other hand , we also have a natural surjection from @xmath923 onto @xmath924 . it follows again by comparing dimensions that the natural surjection from @xmath922 onto @xmath923 is an isomorphism for any field @xmath536 . hence the natural surjection from @xmath920 onto @xmath921 must be an isomorphism as well . by a similar argument , we can show that the natural surjection from @xmath925 onto @xmath908 must be an isomorphism as well for each @xmath130 . as a result , the @xmath881-module @xmath926 has a weyl filtration . this proves our claim . finally , by taking @xmath927 and noting that @xmath928 ( cf . @xcite ) whenever @xmath2 is an algebraically closed field , we deduce that @xmath584 , and @xmath929 has a weyl filtration . this completes the proof of the statement 2 ) . [ cor45 ] let @xmath441 be an integer with @xmath22 $ ] and @xmath24 a partition of @xmath109 satisfying @xmath21 . then there exists an embedding @xmath930 such that @xmath931 has a weyl filtration . in particular , @xmath932 in the remaining part of this section , we fix an integer @xmath933 $ ] and a partition @xmath24 of @xmath109 with @xmath934 . for simplicity , we shall write @xmath25 instead of @xmath935 . by lemma [ lmhx ] and specializing @xmath586 to @xmath235 , we get that @xmath25 is a non - zero maximal vector of weight @xmath24 with respect to the action of @xmath227 on @xmath18 . that is , @xmath936 as a consequence , @xmath937 . on the other hand , we have @xmath938 since @xmath18 has a good filtration , it follows from lemma [ bp ] that the dimensions of @xmath939 and hence of @xmath445 are independent of @xmath2 . therefore , to complete the proof of lemma [ hu2lem ] as well as the second part of theorem [ mainthm3 ] , it suffices to prove that @xmath940 in view of the above discussion and lemma [ lmk40 ] , to prove ( [ equaz1 ] ) , it suffices to show that @xmath943 . using corollary [ cor45 ] , we have an embedding @xmath930 such that @xmath931 has a weyl filtration . therefore , we have the following commutative diagram of homomorphisms : @xmath944 & \operatorname{hom}_{kg}\bigl(v^{\otimes n-2 g } , v^{\otimes n}{\mathfrak{b}}_n^{(g)}\bigr ) \ar[r]^{\,\quad\sim}\ar[d]_{\beta } & \operatorname{hom}_{kg}\bigl(v^{\otimes n-2 g } , v^{\otimes n}\bigr ) \ar@{>>}[d]\\ 0 \ar[r ] & \operatorname{hom}_{kg}\bigl(\delta({\lambda } ) , v^{\otimes n}{\mathfrak{b}}_n^{(g)}\bigr ) \ar@{^{(}->}[r ] & \operatorname{hom}_{kg}\bigl(\delta({\lambda } ) , v^{\otimes n}\bigr),}\ ] ] where by lemma [ keylem1 ] the top horizontal map is an isomorphism and the fact that @xmath945 forces that the right vertical map is a surjection . since the bottom horizontal map is an injection , it follows that the left vertical map @xmath946 must be a surjection and the bottom horizontal map must be an isomorphism . now applying lemma [ keylem1 ] , we get that @xmath947 is spanned by @xmath948 for @xmath306 , where @xmath949 is defined in ( [ tauf ] ) . therefore , we can deduce that the subspace of maximal vectors of weight @xmath24 in @xmath9 is spanned by all @xmath950 , where @xmath306 . hence @xmath951 . this completes the proof of lemma [ hu2lem ] as well as the second part of theorem [ mainthm3 ] .

let @xmath0 be a @xmath1-dimensional symplectic vector space over an algebraically closed field @xmath2 . let @xmath3 be the two - sided ideal of the brauer algebra @xmath4 over @xmath2 generated by @xmath5 , where @xmath6 $ ] . let @xmath7 be the subspace of partially harmonic tensors of valence @xmath8 in @xmath9 . in this paper , we prove that @xmath10 and @xmath11 are both independent of @xmath2 , and the natural homomorphism from @xmath12 to @xmath13 is always surjective . we show that @xmath7 has a weyl filtration and is isomorphic to the dual of @xmath14 as a @xmath15-@xmath16-bimodule . we obtain a @xmath15-@xmath17-bimodules filtration of @xmath18 such that each successive quotient is isomorphic to some @xmath19 with @xmath20 , @xmath21 and @xmath22 $ ] , where @xmath23 is the co - weyl module associated to @xmath24 and @xmath25 is an explicitly constructed maximal vector of weight @xmath24 . as a byproduct , we show that each right @xmath17-module @xmath26 is integrally defined and stable under base change .

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Proceed to summarize the following text: the lep experiments at the resonance of @xmath1-boson have tested the standard model ( sm ) at quantum level , measuring the @xmath1-decay into fermion pairs with an accuracy of one part in ten thousands . the good agreement of the lep data with the sm predictions have severely constrained the behavior of new physics at the @xmath1-pole . taking these achievements into account one can imagine that the physics of @xmath1-boson will again play the central role in the frontier of particle physics if the next generation @xmath1 factory comes true with the generated @xmath1 events several orders of magnitude higher than that of the lep . this factory can be realized in the gigaz option of the international linear collider ( ilc)@xcite . the ilc is a proposed electron - positron collider with tunable energy ranging from @xmath12 to @xmath13 and polarized beams in its first phase , and the gigaz option corresponds to its operation on top of the resonance of @xmath1 boson by adding a bypass to its main beam line . given the high luminosity , @xmath14 , and the cross section at the resonance of @xmath1 boson , @xmath15 , about @xmath16 @xmath1 events can be generated in an operational year of @xmath17 of gigaz , which implies that the expected sensitivity to the branching ratio of @xmath1-decay can be improved from @xmath18 at the lep to @xmath19 at the gigaz@xcite . in light of this , the @xmath1-boson properties , especially its exotic or rare decays which are widely believed to be sensitive to new physics , should be investigated comprehensively to evaluate their potential in probing new physics . among the rare @xmath1-decays , the flavor changing ( fc ) processes were most extensively studied to explore the flavor texture in new physics @xcite , and it was found that , although these processes are severely suppressed in the sm , their branching ratios in new physics models can be greatly enhanced to @xmath19 for lepton flavor violation decays @xcite and @xmath20 for quark flavor violation decays @xcite . besides the fc processes , the @xmath1-decay into light higgs boson(s ) is another type of rare process that was widely studied , e.g. the decay @xmath21 ( @xmath22 ) with the particle @xmath0 denoting a light higgs boson was studied in @xcite , the decay @xmath23 was studied in the two higgs doublet model ( 2hdm)@xcite and the minimal supersymmetric standard model ( mssm)@xcite , and the decay @xmath4 was studied in a model independent way @xcite , in 2hdm@xcite and also in mssm@xcite . these studies indicate that , in contrast with the kinematic forbidden of these decays in the sm , the rates of these decays can be as large as @xmath18 in new physics models , which lie within the expected sensitivity of the gigaz . in this work , we extend the previous studies of these decays to some new models and investigate these decays altogether . we are motivated by some recent studies on the singlet extension of the mssm , such as the next - to - minimal supersymmetric standard model ( nmssm ) @xcite and the nearly minimal supersymmetric standard model ( nmssm ) @xcite , where a light cp - odd higgs boson @xmath0 with singlet - dominant component may naturally arise from the spontaneous breaking of some approximate global symmetry like @xmath24 or peccei - quuin symmetry @xcite . these non - minimal supersymmetric models can not only avoid the @xmath25-problem , but also alleviate the little hierarchy by having such a light higgs boson @xmath0 @xcite . we are also motivated by that , with the latest experiments , the properties of the light higgs boson are more stringently constrained than before . so it is worth updating the previous studies . so far there is no model - independent lower bound on the lightest higgs boson mass . in the sm , it must be heavier than @xmath26 gev , obtained from the null observation of the higgs boson at lep experiments . however , due to the more complex structure of the higgs sector in the extensions of the sm , this lower bound can be significantly relaxed according to recent studies , e.g. , for the cp - odd higgs boson @xmath0 we have @xmath27 gev in the nmssm @xcite , @xmath28 gev in the nmssm @xcite , and @xmath29 gev in the lepton - specific 2hdm ( l2hdm ) @xcite . with such a light cp - odd higgs boson , the z - decay into one or more @xmath0 is open up . noting that the decay @xmath30 is forbidden due to bose symmetry , we in this work study the rare @xmath1-decays @xmath6 ( @xmath22 ) , @xmath31 and @xmath4 in a comparative way for four models , namely the type - ii 2hdm@xcite , the l2hdm @xcite , the nmssm and the nmssm . in our study , we examine carefully the constraints on the light @xmath0 from many latest experimental results . this work is organized as follows . in sec . ii we briefly describe the four new physics models . in sec . iii we present the calculations of the rare @xmath1-decays . in sec . iv we list the constraints on the four new physics models . in sec . v we show the numerical results for the branching ratios of the rare @xmath1-decays in various models . finally , the conclusion is given in sec . as the most economical way , the sm utilizes one higgs doublet to break the electroweak symmetry . as a result , the sm predicts only one physical higgs boson with its properties totally determined by two free parameters . in new physics models , the higgs sector is usually extended by adding higgs doublets and/or singlets , and consequently , more physical higgs bosons are predicted along with more free parameters involved in . the general 2hdm contains two @xmath32 doublet higgs fields @xmath33 and @xmath34 , and with the assumption of cp - conserving , its scalar potential can be parameterized as@xcite : @xmath35,\end{aligned}\ ] ] where @xmath36 ( @xmath37 ) are free dimensionless parameters , and @xmath38 ( @xmath39 ) are the parameters with mass dimension . after the electroweak symmetry breaking , the spectrum of this higgs sector includes three massless goldstone modes , which become the longitudinal modes of @xmath40 and @xmath1 bosons , and five massive physical states : two cp - even higgs bosons @xmath41 and @xmath42 , one neutral cp - odd higgs particle @xmath0 and a pair of charged higgs bosons @xmath43 . noting the constraint @xmath44 with @xmath45 and @xmath46 denoting the vacuum expectation values ( vev ) of @xmath33 and @xmath34 respectively , we choose @xmath47 as the input parameters with @xmath48 , and @xmath49 being the mixing angle that diagonalizes the mass matrix of the cp - even higgs fields . the difference between the type - ii 2hdm and the l2hdm comes from the yukawa coupling of the higgs bosons to quark / lepton . in the type - ii 2hdm , one higgs doublet @xmath34 generates the masses of up - type quarks and the other doublet @xmath33 generates the masses of down - type quarks and charged leptons ; while in the l2hdm one higgs doublet @xmath33 couples only to leptons and the other doublet @xmath34 couples only to quarks . so the yukawa interactions of @xmath0 to fermions in these two models are given by @xcite @xmath50 with @xmath51 denoting generation index . obviously , in the type - ii 2hdm the @xmath52 coupling and the @xmath53 coupling can be simultaneously enhanced by @xmath54 , while in the l2hdm only the @xmath53 coupling is enhanced by @xmath55 . the structures of the nmssm and the nmssm are described by their superpotentials and corresponding soft - breaking terms , which are given by @xcite @xmath56 where @xmath57 is the superpotential of the mssm without the @xmath25 term , @xmath58 and @xmath59 are higgs doublet and singlet superfields with @xmath60 and @xmath61 being their scalar component respectively , @xmath62 , @xmath63 , @xmath64 , @xmath65 , @xmath66 and @xmath67 are soft breaking parameters , and @xmath68 and @xmath69 are coefficients of the higgs self interactions . with the superpotentials and the soft - breaking terms , one can get the higgs potentials of the nmssm and the nmssm respectively . like the 2hdm , the higgs bosons with same cp property will mix and the mass eigenstates are obtained by diagonalizing the corresponding mass matrices : @xmath70 where the fields on the right hands of the equations are component fields of @xmath71 , @xmath72 and @xmath61 defined by @xmath73 @xmath74 and @xmath75 are respectively the cp - even and cp - odd neutral higgs bosons , @xmath76 and @xmath77 are goldstone bosons eaten by @xmath1 and @xmath78 , and @xmath79 is the charged higgs boson . so both the nmssm and nmssm predict three cp - even higgs bosons , two cp - odd higgs bosons and one pair of charged higgs bosons . in general , the lighter cp - odd higgs @xmath0 in these model is the mixture of the singlet field @xmath80 and the doublet field combination , @xmath81 , i.e. @xmath82 and its couplings to down - type quarks are then proportional to @xmath83 . so for singlet dominated @xmath0 , @xmath84 is small and the couplings are suppressed . as a comparison , the interactions of @xmath0 with the squarks are given by@xcite @xmath85 i.e. the interaction does not vanish when @xmath86 approaches zero . just like the 2hdm where we use the vevs of the higgs fields as fundamental parameters , we choose @xmath68 , @xmath69 , @xmath87 , @xmath88 , @xmath66 and @xmath89 as input parameters for the nmssm@xcite and @xmath68 , @xmath54 , @xmath88 , @xmath65 , @xmath90 and @xmath91 as input parameters for the nmssm@xcite . about the nmssm and the nmssm , three points should be noted . the first is for the two models , there is no explicit @xmath92term , and the effective @xmath25 parameter ( @xmath93 ) is generated when the scalar component of @xmath59 develops a vev . the second is , the nmssm is actually same as the nmssm with @xmath94@xcite , because the tadpole terms @xmath95 and its soft breaking term @xmath96 in the nmssm do not induce any interactions , except for the tree - level higgs boson masses and the minimization conditions . and the last is despite of the similarities , the nmssm has its own peculiarity , which comes from its neutralino sector . in the basis @xmath97 , its neutralino mass matrix is given by @xcite @xmath98 where @xmath99 and @xmath100 are @xmath101 and @xmath102 gaugino masses respectively , @xmath103 , @xmath104 , @xmath105 and @xmath106 . after diagonalizing this matrix one can get the mass eigenstate of the lightest neutralino @xmath107 with mass taking the following form @xcite @xmath108 this expression implies that @xmath107 must be lighter than about @xmath109 gev for @xmath110 ( from lower bound on chargnio mass ) and @xmath111 ( perturbativity bound ) . like the other supersymmetric models , @xmath107 as the lightest sparticle acts as the dark matter in the universe , but due to its singlino - dominated nature , it is difficult to annihilate sufficiently to get the correct density in the current universe . so the relic density of @xmath107 plays a crucial way in selecting the model parameters . for example , as shown in @xcite , for @xmath112 , there is no way to get the correct relic density , and for the other cases , @xmath107 mainly annihilates by exchanging @xmath1 boson for @xmath113 , or by exchanging a light cp - odd higgs boson @xmath0 with mass satisfying the relation @xmath114 for @xmath115 . for the annihilation , @xmath54 and @xmath25 are required to be less than 10 and @xmath116 respectively because through eq.([mass - exp ] ) a large @xmath87 or @xmath25 will suppress @xmath117 to make the annihilation more difficult . the properties of the lightest cp - odd higgs boson @xmath0 , such as its mass and couplings , are also limited tightly since @xmath0 plays an important role in @xmath107 annihilation . the phenomenology of the nmssm is also rather special , and this was discussed in detail in @xcite . in the type - ii 2hdm , l2hdm , nmssm and nmssm , the rare @xmath1-decays @xmath118 ( @xmath22 ) , @xmath3 and @xmath4 may proceed by the feynman diagrams shown in fig.[fig1 ] , fig.[fig2 ] and fig.[fig3 ] respectively . for these diagrams , the intermediate state @xmath119 represents all possible cp - even higgs bosons in the corresponding model , i.e. @xmath41 and @xmath42 in type - ii 2hdm and l2hdm and @xmath41 , @xmath42 and @xmath120 in nmssm and nmssm . in order to take into account the possible resonance effects of @xmath119 in fig.[fig1](c ) for @xmath2 and fig.[fig3 ] ( a ) for @xmath11 , we have calculated all the decay modes of @xmath119 and properly included the width effect in its propagator . as to the decay @xmath121 , two points should be noted . one is , unlike the decays @xmath6 and @xmath11 , this process proceeds only through loops mediated by quarks / leptons in the type - ii 2hdm and l2hdm , and additionally by sparticles in the nmssm and nmssm . so in most cases its rate should be much smaller than the other two . the other is due to cp - invariance , loops mediated by squarks / sleptons give no contribution to the decay@xcite . in actual calculation , this is reflected by the fact that the coupling coefficient of @xmath122 differs from that of @xmath123 by a minus sign ( see eq.([asqsq ] ) ) , and as a result , the squark - mediated contributions to @xmath121 are completely canceled out . with regard to the rare decay @xmath11 , we have more explanations . in the lowest order , this decay proceeds by the diagram shown in fig.[fig3 ] ( a ) , and hence one may think that , as a rough estimate , it is enough to only consider the contributions from fig.[fig3](a ) . however , we note that in some cases of the type - ii 2hdm and l2hdm , due to the cancelation of the contributions from different @xmath119 in fig.[fig3 ] ( a ) and also due to the potentially largeness of @xmath124 couplings ( i.e. larger than the electroweak scale @xmath125 ) , the radiative correction from the higgs - mediated loops may dominate over the tree level contribution even when the tree level prediction of the rate , @xmath126 , exceeds @xmath20 . on the other hand , we find the contribution from quark / lepton - mediated loops can be safely neglected if @xmath127 in the type - ii 2hdm and the l2hdm . in the nmssm and the nmssm , besides the corrections from the higgs- and quark / lepton - mediated loops , loops involving sparticles such as squarks , charginos and neutralinos can also contribute to the decay . we numerically checked that the contributions from squarks and charginos can be safely neglected if @xmath127 . we also calculated part of potentially large neutralino correction ( note that there are totally about @xmath128 diagrams for such correction ! ) and found they can be neglected too . since considering all the radiative corrections will make our numerical calculation rather slow , we only include the most important correction , namely that from higgs - mediated loops , in presenting our results for the four models . one can intuitively understand the relative smallness of the sparticle contribution to @xmath11 as follows . first consider the squark contribution which is induced by the @xmath129 interaction ( @xmath130 denotes the squark in chirality state ) and the @xmath131 interaction through box diagrams . because the @xmath132 interaction conserves the chirality of the squarks while the @xmath133 interaction violates the chirality , to get non - zero contribution to @xmath11 from the squark loops , at least four chiral flippings are needed , with three of them provided by @xmath131 interaction and the rest provided by the left - right squark mixing . this means that , if one calculates the amplitude in the chirality basis with the mass insertion method , the amplitude is suppressed by the mixing factor @xmath134 with @xmath135 being the off diagonal element in squark mass matrix . next consider the chargino / neutralino contributions . since for a light @xmath0 , its doublet component , parameterized by @xmath84 in eq.([mixing ] ) , is usually small , the couplings of @xmath0 with the sparticles will never be tremendously large@xcite . so the chargino / neutralino contributions are not important too . in our calculation of the decays , we work in the mass eigenstates of sparticles instead of in the chirality basis . for the type - ii 2hdm and the l2hdm , we consider the following constraints @xcite : * theoretical constraints on @xmath136 from perturbativity , unitarity and requirements that the scalar potential is finit at large field values and contains no flat directions @xcite , which imply that @xmath137 * the constraints from the lep search for neutral higgs bosons . we compute the signals from the higgs - strahlung production @xmath138 ( @xmath139 ) with @xmath140 @xcite and from the associated production @xmath141 with @xmath142 @xcite , and compare them with the corresponding lep data which have been inputted into our code . we also consider the constraints from @xmath138 by looking for a peak of @xmath143 recoil mass distribution of @xmath1-boson @xcite and the constraint of @xmath144 mev when @xmath145 @xcite . + these constraints limit the quantities such as @xmath146 \times br ( h_i \to \bar{b } b ) $ ] on the @xmath147 plane with the the subscript @xmath148 denoting the coupling coefficient of the @xmath149 interaction . they also impose a model - dependent lower bound on @xmath150 , e.g. , @xmath151 for the type - ii 2hdm ( from our scan results ) , @xmath152 for the l2hdm@xcite , and @xmath153 for the nmssm @xcite . these bounds are significantly lower than that of the sm , i.e. @xmath154 , partially because in new physics models , unconventional decay modes of @xmath155 such as @xmath156 are open up . as to the nmssm , another specific reason for allowing a significantly lighter cp - even higgs boson is that the boson may be singlet - dominated in this model . + with regard to the lightest cp - odd higgs boson @xmath0 , we checked that there is no lower bound on its mass so long as the @xmath157 interaction is weak or @xmath155 is sufficiently heavy . * the constraints from the lep search for a light higgs boson via the yukawa process @xmath158 with @xmath22 and @xmath61 denoting a scalar @xcite . these constraints can limit the @xmath159 coupling versus @xmath160 in new physics models . * the constraints from the cleo - iii limit on @xmath161 and the latest babar limits on @xmath162 . these constraints will put very tight constraints on the @xmath163 coupling for @xmath164 . in our analysis , we use the results of fig.8 in the second paper of @xcite to excluded the unfavored points . * the constraints from @xmath165 couplings . since the higgs sector can give sizable higher order corrections to @xmath165 couplings , we calculate them to one loop level and require the corrected @xmath165 couplings to lie within the @xmath166 range of their fitted value . the sm predictions for the couplings at @xmath1-pole are given by @xmath167 and @xmath168 @xcite , and the fitted values are given by @xmath169 and @xmath170 , respectively@xcite . we adopt the formula in @xcite to the 2hdm in our calculation . * the constraints from @xmath171 leptonic decay . we require the new physics correction to the branching ratio @xmath172 to be in the range of @xmath173 @xcite . we use the formula in @xcite in our calculation . + about the constraints ( 5 ) and ( 6 ) , two points should be noted . one is all higgs bosons are involved in the constraints by entering the self energy of @xmath171 lepton , the @xmath174 vertex correction or the @xmath175 vertex correction , and also the box diagrams for @xmath176@xcite . since the yukawa couplings of the higgs bosons to @xmath171 lepton get enhanced by @xmath54 and so do the corrections , @xmath54 must be upper bounded for given spectrum of the higgs sector . generally speaking , the lighter @xmath0 is , the more tightly @xmath54 is limited@xcite . the other point is in the type - ii 2hdm , @xmath177 , b - physics observables as well as @xmath178 decays discussed above can constraint the model in a tighter way than the constraints ( 5 ) and ( 6 ) since the yukawa couplings of @xmath171 lepton and @xmath179 quark are simultaneously enhanced by @xmath54 . but for the l2hdm , because only the yukawa couplings of @xmath171 lepton get enhanced ( see eq.[yukawa ] ) , the constraints ( 5 ) and ( 6 ) are more important in limiting @xmath54 . * indirect constraints from the precision electroweak observables such as @xmath180 , @xmath181 and @xmath182 , or their combinations @xmath183 @xcite . we require @xmath184 to be compatible with the lep / sld data at @xmath185 confidence level@xcite . we also require new physics prediction of @xmath186 is within the @xmath187 range of its experimental value . the latest results for @xmath188 are @xmath189 ( measured value ) and @xmath190 ( sm prediction ) for @xmath191 gev @xcite . in our code , we adopt the formula for these observables presented in @xcite to the type - ii 2hdm and the l2hdm respectively . + in calculating @xmath180 , @xmath181 and @xmath182 , we note that these observables get dominant contributions from the self energies of the gauge bosons @xmath1 , @xmath192 and @xmath193 . since there is no @xmath194 coupling or @xmath195 coupling , @xmath0 must be associated with the other higgs bosons to contribute to the self energies . so by the uv convergence of these quantities , one can infer that , for the case of a light @xmath0 and @xmath196 , these quantities depend on the spectrum of the higgs sector in a way like @xmath197 at leading order , which implies that a light @xmath0 can still survive the constraints from the precision electroweak observables given the splitting between @xmath150 and @xmath198 is moderate@xcite . * the constraints from b physics observables such as the branching ratios for @xmath199 , @xmath200 and @xmath201 , and the mass differences @xmath202 and @xmath203 . we require their theoretical predications to agree with the corresponding experimental values at @xmath187 level . + in the type - ii 2hdm and the l2hdm , only the charged higgs boson contributes to these observables by loops , so one can expect that @xmath198 versus @xmath54 is to be limited . combined analysis of the limits in the type - ii 2hdm has been done by the ckmfitter group , and the lower bound of @xmath204 as a function of @xmath87 was given in fig.11 of @xcite . this analysis indicates that @xmath198 must be heavier than @xmath205 at @xmath185 c.l . regardless the value of @xmath54 . in this work , we use the results of fig.11 in @xcite to exclude the unfavored points . as for the l2hdm , b physics actually can not put any constraints@xcite because in this model the couplings of the charged higgs boson to quarks are proportional to @xmath206 and in the case of large @xmath54 which we are interested in , they are suppressed . in our analysis of the l2hdm , we impose the lep bound on @xmath198 , i.e. @xmath207@xcite . * the constraints from the muon anomalous magnetic moment @xmath208 . now both the theoretical prediction and the experimental measured value of @xmath208 have reached a remarkable precision , but a significant deviation still exists : @xmath209 @xcite . in the 2hdm , @xmath208 gets additional contributions from the one - loop diagrams induced by the higgs bosons and also from the two - loop barr - zee diagrams mediated by @xmath0 and @xmath155@xcite . if the higgs bosons are much heavier than @xmath25 lepton mass , the contributions from the barr - zee diagrams are more important , and to efficiently alleviate the discrepancy of @xmath208 , one needs a light @xmath0 along with its enhanced couplings to @xmath25 lepton and also to heavy fermions such as bottom quark and @xmath171 lepton to push up the effects of the barr - zee diagram@xcite . the cp - even higgs bosons are usually preferred to be heavy since their contributions to @xmath208 are negative . + in the type - ii 2hdm , because @xmath54 is tightly constrained by the process @xmath210 at the lep@xcite and the @xmath178 decay@xcite , the barr - zee diagram contribution is insufficient to enhance @xmath208 to @xmath187 range around its measured value@xcite . so in our analysis , we require the type - ii 2hdm to explain @xmath208 at @xmath211 level . while for the l2hdm , @xmath54 is less constrained compared with the type - ii 2hdm , and the barr - zee diagram involving the @xmath171-loop is capable to push up greatly the theoretical prediction of @xmath208@xcite . therefore , we require the l2hdm to explain the discrepancy at @xmath187 level . + unlike the other constraints discussed above , the @xmath208 constraint will put a two - sided bound on @xmath54 since on the one hand , it needs a large @xmath54 to enhance the barr - zee contribution , but on the other hand , too large @xmath54 will result in an unacceptable large @xmath208 . * since this paper concentrates on a light @xmath0 , the decay @xmath212 is open up with a possible large decay width . we require the width of any higgs boson to be smaller than its mass to avoid a too fat higgs boson@xcite . we checked that for the scenario characterized by @xmath213 , the coefficient of @xmath214 interaction is usually larger than the electroweak scale @xmath125 , and consequently a large decay width is resulted . for the nmssm and nmssm , the above constraints become more complicated because in these models , not only more higgs bosons are involved in , but also sparticles enter the constraints . so it is not easy to understand some of the constraints intuitively . take the process @xmath199 as an example . in the supersymmetric models , besides the charged higgs contribution , chargino loops , gluino loops as well as neutralino loops also contribute to the process@xcite , and depending on the susy parameters , any of these contributions may become dominated over or be canceled by other contributions . as a result , although the charged higgs affects the process in the same way as that in the type - ii 2hdm , charged higgs as light as @xmath215 is still allowed even for @xmath216@xcite . since among the constraints , @xmath208 is rather peculiar in that it needs new physics to explain the discrepancy between @xmath217 and @xmath218 , we discuss more about its dependence on susy parameters . in the nmssm and the nmssm , @xmath208 receives contributions from higgs loops and neutralino / chargino loops . for the higgs contribution , it is quite similar to that of the type - ii 2hdm except that more higgs bosons are involved in@xcite . for the neutralino / chargino contribution , in the light bino limit ( i.e. @xmath219 ) , it can be approximated by@xcite @xmath220 for @xmath221 with @xmath222 being smuon mass . so combining the two contributions together , one can learn that a light @xmath0 along with large @xmath54 and/or light smuon with moderate @xmath87 are favored to dilute the discrepancy . because more parameters are involved in the constraints on the supersymmetric models , we consider following additional constraints to further limit their parameters : * direct bounds on sparticle masses from the lep1 , the lep2 and the tevatron experiments @xcite . * the lep1 bound on invisible z decay @xmath223 ; the lep2 bound on neutralino production @xmath224 and @xmath225@xcite . * dark matter constraints from the wmap relic density 0.0975 @xmath226 0.1213 @xcite . note that among the above constraints , the constraint ( 2 ) on higgs sector and the constraint ( c ) on neutralino sector are very important . this is because in the supersymmetric models , the sm - like higgs is upper bounded by about @xmath227 at tree level and by about @xmath228 at loop level , and that the relic density restricts the lsp annihilation cross section in a certain narrow range . in our analysis of the nmssm , we calculate the constraints ( 3 ) and ( 5 - 7 ) by ourselves and utilize the code nmssmtools @xcite to implement the rest constraints . we also extend nmssmtools to the nmssm to implement the constraints . for the extension , the most difficult thing we faced is how to adapt the code micromegas@xcite to the nmssm case . we solve this problem by noting the following facts : * as we mentioned before , the nmssm is actually same as the nmssm with the trilinear singlet term setting to zero . so we can utilize the model file of the nmssm as the input of the micromegas and set @xmath229 . * since in the nmssm , the lsp is too light to annihilate into higgs pairs , there is no need to reconstruct the effective higgs potential to calculate precisely the annihilation channel @xmath230 with @xmath61 denoting any of higgs bosons@xcite . we thank the authors of the nmssmtools for helpful discussion on this issue when we finish such extension@xcite . with the above constraints , we perform four independent random scans over the parameter space of the type - ii 2hdm , the l2hdm , the nmssm and the nmssm respectively . we vary the parameters in following ranges : @xmath231 for the type - ii 2hdm , @xmath232 for the l2hdm , @xmath233 for the nmssm , and @xmath234 for the nmssm . in performing the scans , we note that for the nmssm and the nmssm , some constraints also rely on the gaugino masses and the soft breaking parameters in the squark sector and the slepton sector . since these parameters affect little on the properties of @xmath0 , we fix them to reduce the number of free parameters in our scan . for the squark sector , we adopt the @xmath235 scenario which assumes that the soft mass parameters for the third generation squarks are degenerate : @xmath236 800 gev , and that the trilinear couplings of the third generation squarks are also degenerate , @xmath237 with @xmath238 . for the slepton sector , we assume all the soft - breaking masses and trilinear parameters to be 100 gev . this setting is necessary for the nmssm since this model is difficult to explain the muon anomalous moment at @xmath239 level for heavy sleptons@xcite . finally , we assume the grand unification relation @xmath240 for the gaugino masses with @xmath241 being fine structure constants of the different gauge group . with large number of random points in the scans , we finally get about @xmath242 , @xmath243 , @xmath244 and @xmath242 samples for the type - ii 2hdm , the l2hdm , the nmssm and the nmssm respectively which survive the constraints and satisfy @xmath245 . analyzing the properties of the @xmath0 indicates that for most of the surviving points in the nmssm and the nmssm , its dominant component is the singlet field ( numerically speaking , @xmath246 ) so that its couplings to the sm fermions are suppressed@xcite . our analysis also indicates that the main decay products of @xmath0 are @xmath247 for the l2hdm@xcite , @xmath248 ( dominant ) and @xmath247 ( subdominant ) for the type - ii 2hdm , the nmssm and the nmssm , and in some rare cases , neutralino pairs in the nmssm@xcite . in fig.[fig4 ] , we project the surviving samples on the @xmath249 plane . this figure shows that the allowed range of @xmath54 is from @xmath250 to @xmath251 in the type - ii 2hdm , and from @xmath252 to @xmath253 in the l2hdm . just as we introduced before , the lower bounds of @xmath254 come from the fact that we require the models to explain the muon anomalous moment , while the upper bound is due to we have imposed the constraint from the lep process @xmath255 , which have limited the upper reach of the @xmath256 coupling for light @xmath61 @xcite(for the dependence of @xmath256 coupling on @xmath54 , see sec . this figure also indicates that for the nmssm and the nmssm , @xmath54 is upper bounded by @xmath257 . for the nmssm , this is because large @xmath87 can suppress the dark matter mass to make its annihilation difficult ( see @xcite and also sec . ii ) , but for the nmssm , this is because we choose a light slepton mass so that large @xmath54 can enhance @xmath208 too significantly to be experimentally unacceptable . we checked that for the slepton mass as heavy as @xmath258 , @xmath259 is still allowed for the nmssm . in fig.[fig5 ] and fig.[fig6 ] , we show the branching ratios of @xmath260 and @xmath261 respectively . fig.[fig5 ] indicates , among the four models , the type - ii 2hdm predicts the largest ratio for @xmath260 with its value varying from @xmath262 to @xmath263 . the underlying reason is in the type - ii 2hdm , the @xmath264 coupling is enhanced by @xmath54 ( see fig.[fig4 ] ) , while in the other three model , the coupling is suppressed either by @xmath265 or by the singlet component of the @xmath0 . fig.[fig6 ] shows that the l2hdm predicts the largest rate for @xmath266 with its value reaching @xmath5 in optimum case , and for the other three models , the ratio of @xmath261 is at least about one order smaller than that of @xmath267 . this feature can be easily understood from the @xmath268 coupling introduced in sect . we emphasize that , if the nature prefers a light @xmath0 , @xmath260 and/or @xmath269 in the type - ii 2hdm and the l2hdm will be observable at the gigaz . then by the rates of the two decays , one can determine whether the type - ii 2hdm or the l2hdm is the right theory . on the other hand , if both decays are observed with small rates or fail to be observed , the singlet extensions of the mssm are favored . in fig.[fig7 ] , we show the rate of @xmath3 as the function of @xmath270 . this figure indicates that the branching ratio of @xmath121 can reach @xmath271 , @xmath272 , @xmath273 and @xmath274 for the optimal cases of the type - ii 2hdm , the l2hdm , the nmssm and the nmssm respectively , which implies that the decay @xmath121 will never be observable at the gigaz if the studied model is chosen by nature . the reason for the smallness is , as we pointed out before , that the decay @xmath121 proceeds only at loop level . comparing the optimum cases of the type - ii 2hdm , the nmssm and the nmssm shown in fig.5 - 7 , one may find that the relation @xmath275 holds for any of the decays . this is because the decays are all induced by the yukawa couplings with similar structure for the models . in the supersymmetric models , the large singlet component of the light @xmath0 is to suppress the yukawa couplings , and the @xmath0 in the nmssm has more singlet component than that in the nmssm . next we consider the decay @xmath11 , which , unlike the above decays , depends on the higgs self interactions . in fig.[fig8 ] we plot its rate as a function of @xmath270 and this figure indicates that the @xmath276 may be the largest among the ratios of the exotic @xmath1 decays , reaching @xmath277 in the optimum cases of the type - ii 2hdm , the l2hdm and the nmssm . the underlying reason is , in some cases , the intermediate state @xmath119 in fig.[fig3 ] ( a ) may be on - shell . in fact , we find this is one of the main differences between the nmssm and the nmssm , that is , in the nmssm , @xmath119 in fig.[fig3 ] ( a ) may be on - shell ( corresponds to the points with large @xmath278 ) while in the nmssm , this seems impossible . so we conclude that the decay @xmath11 may serve as an alternative channel to test new physics models , especially it may be used to distinguish the nmssm from the nmssm if the supersymmetry is found at the lhc and the @xmath11 is observed at the gigaz with large rate . before we end our discussion , we note that in the nmssm , the higgs boson @xmath0 may be lighter than @xmath279 without conflicting with low energy data from @xmath178 decays and the other observables ( see fig.[fig4]-[fig8 ] ) . in this case , @xmath0 is axion - like as pointed out in @xcite . we checked that , among the rare @xmath1 decays discussed in this paper , the largest branching ratio comes from @xmath280 which can reach @xmath281 . since in this case , the decay product of @xmath0 is highly collinear muon pair , detecting the decay @xmath280 may need some knowledge about detectors , which is beyond our discussion . in this paper , we studied the rare @xmath1-decays @xmath2 ( @xmath7 ) , @xmath282 and @xmath4 in the type - 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various new physics models predict a light cp - odd higgs boson ( labeled as @xmath0 ) and open up new decay modes for @xmath1-boson , such as @xmath2 , @xmath3 and @xmath4 , which could be explored at the gigaz option of the ilc . in this work we investigate these rare decays in several new physics models , namely the type - ii two higgs doublet model ( type - ii 2hdm ) , the lepton - specific two higgs doublet model ( l2hdm ) , the nearly minimal supersymetric standard model ( nmssm ) and the next - to - minimal supersymmetric standard model ( nmssm ) . we find that in the parameter space allowed by current experiments , the branching ratios can reach @xmath5 for @xmath6 ( @xmath7 ) , @xmath8 for @xmath3 and @xmath9 for @xmath4 , which implies that the decays @xmath10 and @xmath11 may be accessible at the gigaz option . moreover , since different models predict different patterns of the branching ratios , the measurement of these rare decays at the gigaz may be utilized to distinguish the models .

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Proceed to summarize the following text: entanglement is one of the most peculiar features of quantum mechanics and it also plays the role of a fundamental resource in many applications of quantum information @xcite . on the other hand , entangled systems unavoidably interact with their environments causing decoherence and a loss of entanglement . since entanglement is by definition a nonlocal resource , one expects that any attempt to restore it must involve the use of nonlocal operations . we consider physical situations where two subsystems , for example two qubits , are prepared in an entangled state and subsequently decoupled @xcite . due to the interaction with their local environment , entanglement dynamics may exhibit a non - monotonous behaviour , with the occurrence of revivals alternating to dark periods @xcite . in some cases , this phenomenon is due to the fact that entanglement is transferred to quantum environments , and then back - transferred to the system @xcite . in other cases , the environment can be modelled as a classical system @xcite and no entanglement between the system and the environment is established at any time . in the latter cases , the occurrence of entanglement revivals may appear paradoxical , since the effect of the noise is analogous to a local operation on a subsystem . a first interpretation of this phenomenon has been given in terms of correlations present in a classical - quantum state of environments and qubits @xcite . in a recent work @xcite , we have proposed to solve the apparent paradox by introducing the concept of _ hidden _ entanglement ( he ) , which measures the amount of entanglement that may be recovered without the help of any nonlocal operation . the definition of he is based on the quantum trajectory description of the system dynamics , that allows to point out the presence of entanglement in the system even if the density operator formalism does not reveal it : this entanglement is thus not accessible ( hidden ) due to the lack of classical information @xcite . relevant examples of situations where the environment can be modelled as a classical system may be found in solid - state implementations of qubits . for example , in superconducting nanocircuits one of the most relevant sources of decoherence @xcite are fluctuating background charges localized in the insulating materials surrounding superconducting islands @xcite . each impurity produces a bistable fluctuation of the island polarization . the collective effect of an ensemble of these random telegraph processes , with a proper distribution of switching rates , gives rise to @xmath0-noise @xcite routinely observed nanodevices @xcite . in this paper we exploit the concept of hidden entanglement to explain the occurrence of entanglement revivals in a simple system . in particular , we consider two noninteracting qubits , one of them affected by a random telegraph noise at pure dephasing @xcite . the paper is organized as follows . in section [ sec : model ] , we introduce the hamiltonian model . in section [ sec : entanglement - dynamics ] we discuss the entanglement dynamics , showing that the revivals of entanglement are due to the presence of hidden entanglement . in section [ conclusions ] we summarize obtained results and present some final comments . we consider two noninteracting qubits @xmath1 and @xmath2 , initially prepared in a pure maximally entangled state @xmath3 , evolving according to the hamiltonian ( @xmath4 ) @xmath5 where @xmath6 , @xmath7 and @xmath8 represents a random telegraph ( rt ) process @xmath9 @xcite acting on qubit @xmath1 . the rt process induces a random switching of qubit @xmath1 frequency between @xmath10 and @xmath11 , with an overall switching rate @xmath12 ( without loss of generality , we assume @xmath13 ) . we consider a symmetric rt process where the transition rates between the two states are equal , that is @xmath14 . our first aim is to find the system density matrix at any time @xmath15 . the two qubits independently evolve under the hamiltonian ( [ eq : hamiltonian1 ] ) : qubit @xmath2 freely evolves whereas qubit @xmath1 displays a pure dephasing dynamics due to the effect of the stochastic process @xmath16 . the dynamics of single qubit subject to rt noise at pure dephasing has been solved in @xcite . a possible way to obtain @xmath15 is to solve a _ stochastic _ schrdinger equation which gives the following formal expression for @xmath15 @xmath17 p[\xi(t)]\,\rho_\xi(t ) \ , , \label{eq : evolved_qubit_densityoperator}\ ] ] where @xmath18 with @xmath19 , and the probability of the realization @xmath16 can be written as @xmath20=\lim_{m\to \infty } \eta_{m+1}(\xi_m , t_m ; \ldots;\xi_1,t_1;\xi_0,t_0 ) , \label{eq : probability_evolved_qubit_state}\ ] ] where @xmath21 is a ( @xmath22 ) joint probability for the sampled @xmath16 at regular intervals @xmath23 , @xmath24 , @xmath25 ( @xmath26 ) @xcite . since the qubits evolve independently , the above procedure leads to a simple form depending on the single qubit coherences . in the computational basis @xmath27 , where @xmath28 , with @xmath29 and @xmath30 , and assuming an initial bell state @xmath31 , we obtain @xmath32 where the coherence decay factor @xmath33 reads @xcite @xmath34 , \label{eq : coherence - decayfactor}\ ] ] with @xmath35 , @xmath36 and @xmath37 . in the following we shall exploit @xmath15 given by eq.([eq : densityoperator ] ) to analyze the two - qubit entanglement dynamics . to quantify the degree of entanglement of the system state @xmath15 we use the entanglement of formation @xmath38 @xcite that can be readily calculated by the formula @xcite @xmath39 where @xmath40 is the concurrence and @xmath41 . for the state @xmath15 of eq . ( [ eq : densityoperator ] ) we obtain @xmath42 , where @xmath33 is given in eq . ( [ eq : coherence - decayfactor ] ) . it is worth to notice that the evolved state @xmath15 belongs to the hilbert space spanned by the bell states @xmath43 , therefore the entanglement of formation equals the entanglement cost @xcite . in the strong coupling regime , @xmath44 @xcite entanglement revivals occur during the system dynamics @xcite . to understand the nature of this phenomenon we initally consider the limiting case of an extraordinarily slow rt process , @xmath45 ( @xmath46 ) . this regime physically describes situations where the stochastic process is slow enough to be considered static during the system time evolution lasting @xmath47 , i. e. we assume @xmath48 @xcite . the evolution expressed by eq.([eq : densityoperator ] ) describes an average resulting from the collection of several time evolutions each lasting @xmath47 . the average includes the possibility that the rt process takes any of the two values @xmath49 or @xmath50 at time @xmath51 with equal probability . by a straightforward calculation we find that the concurrence in this case is given by @xmath52 . under these conditions we do not find any entanglement decay , rather the concurrence is equal to one at times @xmath53 and it vanishes at times @xmath54 , where @xmath55 is a non - negative integer . ( color online ) entanglement of formation @xmath56 as a function of the dimensionless time @xmath57 for the @xmath58 . from top to bottom , the solid curves correspond to @xmath59 and @xmath60 , the dashed curves represent @xmath61 for the same values of @xmath62.,scaledwidth=65.0% ] the entanglement revivals ( see the top solid curve in fig . [ fig : rt ] ) are not due to periodic entanglement death and rebirth by nonlocal operations . indeed , the hamiltonian evolution described by eq . ( [ eq : hamiltonian1 ] ) only includes local operations . since local operations can not increase entanglement @xcite , its increase during the intervals @xmath63\tilde{t}_n,{t}_n]$ ] must be attributed to the manifestation of quantum correlations that were already present at times @xmath64 , but were _ hidden _ , in the sense that the density operator formalism does not capture them . these correlations are evident in the quantum trajectory description of the system dynamics @xcite . the system evolution in fact results from averaging on only two possible quantum trajectories . the first trajectory corresponds to @xmath65 and the bloch vector of qubit @xmath1 rotates around its @xmath66-axis with frequency @xmath10 . the second trajectory orresponds to @xmath67 and the bloch vector of qubit @xmath1 rotates around the @xmath66-axis with a different frequency @xmath11 . the bloch vector of qubit @xmath2 instead rotates in both cases around its @xmath66-axis with frequency @xmath68 . thus , during the first trajectory the system state evolves , up to an irrelevant global phase factor , as @xmath69 while during the second trajectory the system evolves , apart from an irrelevant global phase factor , as @xmath70 the two quantum trajectories only differ by the fact that the basis states @xmath71 and @xmath72 they acquire the additional relative phase @xmath73 in the quantum superpositions of eqs . ( [ eq : state-1 ] ) and ( [ eq : state-2 ] ) . since the two quantum trajectory occur with equal probability , the system s state is described by the quantum ensemble @xmath74 where @xmath75 . the entanglement associated to the quantum ensemble @xmath76 can be suitably identified by its average entanglement given by @xcite @xmath77 since both states @xmath78 and @xmath79 are maximally entangled at any time ( @xmath80 is the _ entropy of entanglement _ @xcite ) . the _ hidden entanglement _ @xcite of the ensemble @xmath76 is defined as the difference between the average entanglement of eq . ( [ eq : average_entanglement ] ) and the entanglement of the corresponding density operator @xmath81 : @xmath82 the meaning of hidden entanglement is simple : it measures the entanglement that can not be exploited as a resource due to the lack of classical knowledge about which state in the ensemble @xmath76 we are dealing with . once this classical information is provided , the entanglement can be recovered . coming back to the interpretation of entanglement revivals , the ensemble description ( average entanglement @xmath83 ) tells us that at times @xmath64 the system is always in a maximally entangled state ( @xmath84 or @xmath85 ) but the lack of classical knowledge about which of the two states in the ensemble @xmath76 we are dealing with prevents us from distilling any entanglement : in fact , entanglement is _ hidden _ being @xmath86 and @xmath87 . at times @xmath88 this lack of knowledge is irrelevant since the random relative phase becomes meaningless at @xmath89 : all the initial entanglement is recovered , @xmath90 and @xmath91 . we now investigate the case when the rt process evolves during the systen evolution time , i. e. we consider the regime @xmath92 . this situation is illustrated in fig . [ fig : rt ] where we observe that the amplitude of revivals decreases as @xmath12 increases ( g decreases ) . aslo in this case there is hidden entanglement . the possible quantum trajectories the system undergoes are now infinite . the system state is described by the quantum ensemble @xmath93,{|{\varphi_\xi(t)}\rangle } \}$ ] and the average entanglement of @xmath76 is calculated by solving the path - integral @xmath94 p[\xi(t)]\,e\big({|{\varphi_\xi(t)}\rangle } \big).\ ] ] once again , we obtain @xmath83 since during each trajectory the state remains in a maximally entangled state at any time . on the other hand , the entanglement of formation assumes lower values with respect to the static noise case , @xmath45 . in particular , the amplitude of revivals does not reach anymore the initial maximum value . this is due to the fact that , in general , the action of the rt process during the time evolution makes the coherences of @xmath95 ( in the basis @xmath96 ) no longer in phase at times @xmath88 . in the time interval @xmath630,{t}_n]$ ] one or more transitions can occur between the two rt states , such that we can have a random extra phase at the times @xmath88 given by @xmath97 where @xmath55 is a non - negative integer . this unknown phase difference is responsible for the decay of the absolute values of coherences @xmath98 in the evolved two - qubit state @xmath99 of eq . ( [ eq : densityoperator ] ) : if we knew the phase difference @xmath100 for each state @xmath101 , we would be able to restore the coherence absolute value to 1 , and therefore recover all the initial entanglement , by simply applying the unitary local operation @xmath102 . for completeness we point out , that the relative maxima of the entanglement of formation occur at @xmath103 , as one can derive from equation ( [ eq : coherence - decayfactor ] ) . notice that the amount of the decay of the amplitude of entanglement revivals is monotonously related to the normalized autocorrelation function of the symmetric rt process @xmath104 @xcite . indeed , as we have already mentioned , the reduction of the amplitude revivals is related to the transitions of the rt occurred in @xmath105 $ ] , whose mean number is @xmath106 . from a quantitative point of view , for @xmath107 the coherences decay factor eq . ( [ eq : coherence - decayfactor ] ) can be approximated as @xmath108 $ ] , so that @xmath109 and @xmath110 , with @xmath111 defined in eq . ( [ eq : entanglement_formation ] ) . this clearly shows that the decay of the entanglement amplitude revivals is due to the decrease of the rt correlations , or in other words , to the memory loss of the stochastic process @xmath16 itself . in this paper we have exploited the concept of hidden entanglement to interpret the occurrence of entanglement revivals in a particular system where back - action from the environment is absent . namely , we have considered a system composed of two noninteracting qubits where one qubit is subject to random telegraph noise at pure dephasing . during the dynamics , entanglement vanishes and revives always `` remaining '' inside the system , as it is signalled by the average entanglement , @xmath112 at any time . at certain times @xmath113 this entanglement is completely hidden , in the sense that the entanglement of formation @xmath114 while the hidden entanglement @xmath115 . for this reason , the two - qubit entanglement can be simply recovered at subsequent times without the help of any nonlocal operation : in the considered case in fact the hamiltonian only involves local operations . finally we remark that the concept of hidden entanglement can be of practical relevance in solid - state devices , where dominant noise sources tipically have large amplitude components at low frequencies . in these systems entanglement revivals may also be induced by applying local pulses to the qubits @xcite . 50 nielsen m a and chuang i l 2000 _ quantum computation and quantum information _ ( cambridge university press , cambridge ) . lpez c e , romero g , lastra f , solano e and retamal j c 2008 _ phys . * 101 * 080503 ; lpez c e , romero g and retamal j c 2010 _ phys . a _ * 81 * 062114 ; bai y - k , ye m - y and wang z d 2009 _ phys . a _ * 80 * 044301 .

entanglement dynamics of two noninteracting qubits , locally affected by random telegraph noise at pure dephasing , exhibits revivals . these revivals are not due to the action of any nonlocal operation , thus their occurrence may appear paradoxical since entanglement is by definition a nonlocal resource . we show that a simple explanation of this phenomenon may be provided by using the ( recently introduced ) concept of _ hidden _ entanglement , which signals the presence of entanglement that may be recovered with the only help of local operations .

You are an expert at summarizing long articles. Proceed to summarize the following text: the relation between the casimir - polder ( cp ) interaction of a single atom with a body @xcite and its van der waals ( vdw ) interaction with the atoms comprising the body @xcite was first discussed in detail by renne @xcite , with special emphasis on an atom interacting with a dielectric half space . approaching the problem from the microscopic side and modelling all atoms by identical harmonic oscillators , he showed that the sum of all many - atom vdw interactions between the single atom and the body atoms corresponds to the result for the cp potential derived earlier on the basis of a macroscopic approach @xcite . milonni and lerner generalised this result to bodies of arbitrary shapes @xcite . using the ewald - oseen extinction theorem which follows from the clausius - mosotti relation , they demonstrated that the cp potential of an atom in the presence of a nonabsorbing homogeneous dielectric body can be obtained by summing over an infinite series of many - atom vdw potentials . in this paper , we approach the problem from the macroscopic side and under more general conditions . to that end , based on macroscopic qed in linear , causal media ( sec . [ sec2 ] ) , we first consider the cp interaction of an atom with a macroscopic body as well as the vdw interaction of two atoms ( sec . [ sec3 ] ) , and then make use of the born expansion and the clausius - mosotti relation to establish the microscopic origin of the cp potential ( sec . [ sec4 ] ) , followed by a summary ( sec . [ sec5 ] ) . according to the multipolar scheme , the hamiltonian for a system of neutral atoms and/or molecules briefly referred to as atoms in the following interacting with the electromagnetic field in the presence of dispersing and absorbing bodies is given by @xcite @xmath0 where the hamiltonian @xmath1 governs the ( unperturbed ) internal dynamics of atom @xmath2 , @xmath3 is the hamiltonian of the system composed of the electromagnetic field and the magnetodielectric medium including dissipative interactions , with @xmath4 and @xmath5 being the dynamical variables of the system , which satisfy bosonic commutation relations , and the atom - field interaction in electric dipole approximation is given by @xmath6 ( @xmath7 : atomic electric dipole moment , @xmath8 : centre - of - mass position ) . in eq . ( [ eq6 ] ) , @xmath9 is the medium - assisted electric field expressed in terms of the dynamical variables , with the quantities @xmath10 being given in terms of the classical green tensor @xmath11 , @xmath12 \bm{g}(\mathbf{r},\mathbf{r}',\omega ) = \bm{\delta}(\mathbf{r}-\mathbf{r}').\ ] ] the ( macroscopic ) permittivity @xmath13 and permittivity @xmath14 satisfy the kramers - kronig relations and the conditions @xmath15 @xmath16 @xmath17 and @xmath18 @xmath16 @xmath17 imposed for absorbing media . note that the green tensor has the useful properties @xcite @xmath19 dispersion forces can be derived from the associated potentials @xmath20 , which are commonly identified as the position - dependent part of the ground - state energy shift @xmath21 induced by the atom - field coupling . for a single atom @xmath2 the ground state of the system is given by @xmath22 @xmath23 @xmath24 [ @xmath25 , atomic ground state ; @xmath26 @xmath23 @xmath17 ] , and the leading , second - order energy shift reads @xmath28 note that the primed sum includes ( principal - value ) integrals over the continuous degrees of freedom . inspection of eq . ( [ eq6 ] ) reveals that only intermediate states of the type @xmath29 @xmath23 @xmath30 contribute . substituting the respective matrix elements @xmath31_i\ ] ] [ @xmath32 @xmath23 @xmath33 and energy denominators @xmath34 @xmath23 @xmath35 [ @xmath36 @xmath23 @xmath37 into eq . ( [ eq15 ] ) , using eq . ( [ eq13 ] ) , and separating the green tensor into its bulk and scattering parts according to @xmath38 one arrives , after some calculation , at [ @xmath39 @xmath40 @xmath41 @xcite @xmath42 where , for simplicity , an isotropic atomic ground - state polarizability is considered , @xmath43 the potential ( [ eq19 ] ) implies the cp force ( @xmath44 @xmath45 @xmath46 ) @xmath47 on a ground - state atom @xmath2 due to the presence of an arbitrary arrangement of dispersing and absorbing bodies [ which are accounted for by @xmath48 . to illustrate the use of eq . ( [ eq19 ] ) , consider an atom placed at distance @xmath49 from the centre of a small , homogeneous , magnetodielectric sphere of radius @xmath50 @xmath51 @xmath52 . substituting the respective green tensor @xcite into eq . ( [ eq19 ] ) and retaining only the leading - order term in @xmath53 ( cf.ref . @xcite ) , one obtains @xmath54,\\ \label{eq23 } g_{ee}(x ) = & \ ; 2e^{-2x}(3 + 6x+5x^2 + 2x^3+x^4),\\ \label{eq24 } g_{em}(x ) = & \ ; 2e^{-2x}(1 + 2x+x^2),\end{aligned}\ ] ] with the electric and magnetic polarizabilities of the sphere being given by @xcite @xmath55 in the nonretarded limit , where the atom - sphere separation is much larger than the characteristic transition wavelengths of the atom and the sphere medium , one may approximate @xmath56 @xmath57 @xmath58 , @xmath59 @xmath57 @xmath60 , so eq . ( [ eq22 ] ) reduces to @xmath61 and @xmath62 for electric and magnetic spheres , respectively , while in the opposite retarded limit the approximations @xmath63 @xmath57 @xmath64 , @xmath65 @xmath57 @xmath66 , @xmath67 @xmath57 @xmath68 lead to @xmath69 } { 64\pi^3\varepsilon_0 ^ 2r_\mathrm{a}^7}\,.\end{aligned}\ ] ] to calculate the vdw interaction of two atoms @xmath2 and @xmath70 in the presence of dispersing and absorbing bodies , we start from the ground state @xmath22 @xmath23 @xmath71 and consider those contributions to the energy shift that depend on the positions of both atoms . the leading contributions are hence contained in the fourth - order perturbative shift @xmath72 where @xmath73 @xmath23 @xmath74 @xmath75 @xmath76 . a typical set of possible intermediate states is given by @xmath77 upon using eq . ( [ eq16 ] ) as well as @xmath78_{i ' } \delta_{\lambda\lambda^{\prime\prime}}\delta_{ii^{\prime\prime } } \delta(\mathbf{r } -\mathbf{r}^{\prime\prime } ) \delta(\omega-\omega^{\prime\prime})\nonumber\\ & \hspace{4ex}\quad\,+\bigl[\mathbf{d}_a^{k0}\!\cdot\ ! \bm{g}_{\lambda^{\prime\prime}}(\mathbf{r}_a , \mathbf{r}^{\prime\prime},\omega^{\prime\prime } ) \bigr]_{i^{\prime\prime } } \delta_{\lambda\lambda'}\delta_{ii ' } \delta(\mathbf{r } -\mathbf{r } ' ) \delta(\omega-\omega')\bigr\}\end{aligned}\ ] ] and recalling eq . ( [ eq13 ] ) , substitution of the intermediate states ( [ eq31 ] ) into eq . ( [ eq30 ] ) leads , after some calculation , to @xmath79,\end{aligned}\ ] ] where we have assumed real dipole - matrix elements . under this assumption , the various two - atom contributions @xmath80 to the energy shift @xmath81 only differ by the denominators in the square brackets of eq . ( [ eq33 ] ) , and hence after a lengthy calculation one may show that [ @xmath81 @xmath40 @xmath82 @xcite @xmath83.\ ] ] from the two - atom potential ( [ eq37 ] ) one can calculate the vdw force on atom @xmath2(@xmath70 ) due to atom @xmath70(@xmath2 ) in the presence of arbitrary dispersing and absorbing magnetodielectric bodies according to @xmath84 as a simple example , consider two atoms embedded in bulk magnetodielectric material . substitution of the respective green tensor @xcite into eq . ( [ eq37 ] ) leads to @xmath85}{\varepsilon^2(iu)}\,,\end{aligned}\ ] ] @xmath86 @xmath45 @xmath87 , @xmath88 @xmath23 @xmath89 , recall eq . ( [ eq23 ] ) , which reduces to @xmath90 for nonretarded and retarded interatomic separations , respectively . equations ( [ eq39])([eq41 ] ) show that the presence of a medium leads to a reduction of the potential w.r.t . its well - known free - space value @xcite , while comparison of eqs . ( [ eq22 ] ) and ( [ eq39 ] ) reveals that in free space the dispersion interaction of an atom with a small dielectric sphere has the same form as that of two atoms . we now turn to the question how the cp interaction of a single atom with dielectric bodies can be related to its many - atom vdw interactions with the atoms comprising the bodies . for simplicity , we will speak of a single dielectric body in the following . we assume the dielectric body to be given by @xmath91 , and we allow for the presence of an arbitrary magnetodielectric background of additional bodies characterised by @xmath92 and @xmath14 , such that @xmath93 the green tensor corresponding to this scenario can formally be written as a born series @xcite @xmath94\nonumber\\ & \hspace{4ex}\times\ , \overline{\bm{g}}(\mathbf{r},\mathbf{s}_1,\omega)\!\cdot\ ! \overline{\bm{g}}(\mathbf{s}_1,\mathbf{s}_2,\omega)\!\cdots\ ! \overline{\bm{g}}(\mathbf{s}_k,\mathbf{r}',\omega),\end{aligned}\ ] ] where @xmath95 is the green tensor corresponding to the magnetodielectric background . substituting eq . ( [ eq43 ] ) into eq . ( [ eq19 ] ) , the cp potential can be written as @xmath96 where @xmath97 is the cp potential due to the magnetodielectric background , which is not of further interest here , and @xmath98 \nonumber\\ & & \times\mathrm{tr}\bigl [ \overline{\bm{g}}(\mathbf{r}_a,\mathbf{s}_1,iu)\!\cdot\ ! ! \overline{\bm{g}}(\mathbf{s}_k,\mathbf{r}_a , iu)\bigr]\end{aligned}\ ] ] is the contribution to the potential that is of @xmath99th order in @xmath100 . assuming the dielectric body described by @xmath100 to be comprised of atoms of polarizabilities @xmath101 and number densities @xmath102 , the gap between the macroscopic and microscopic descriptions can be bridged by means of the clausius - mosotti law @xmath103 note that since @xmath100 is the fourier transform of a linear response function , it must satisfy the condition @xmath104 @xmath16 @xmath105 @xmath16 @xmath17 for @xmath106 @xmath16 @xmath17 , which implies that the inequality @xmath107 must hold . substituting eq . ( [ eq48 ] ) into eq . ( [ eq47 ] ) and splitting off the singular part of the green tensor according to @xmath108 one obtains @xmath109 where @xmath110\\ \times\ , v_{ab_1\ldots b_l } ( \mathbf{r}_a,\mathbf{s}_1,\ldots,\mathbf{s}_l , iu)\end{gathered}\ ] ] with @xmath111\end{gathered}\ ] ] denotes the sum of all @xmath112-atom terms that are of order @xmath99 in @xmath100 , and each power of the factor @xmath113 is due to the integration of one term containing @xmath114 . summing eq . ( [ eq51 ] ) over @xmath99 , one may rearrange the double sum as follows : @xmath115v_{ab_1\ldots b_l } ( \mathbf{r}_a,\mathbf{s}_1,\ldots,\mathbf{s}_l),\end{gathered}\ ] ] where we have performed the geometric sums @xmath116 by means of eq . ( [ eq54 ] ) . note that the convergence of these sums requires @xmath117 @xmath118 @xmath119 , which by means of eq . ( [ eq49 ] ) is equivalent to @xmath120 finally , we symmetrize the many - atom terms by introducing the symmetrization operator @xmath121 where @xmath122 denotes the permutation group of the numbers @xmath123 . from the cyclic property of the trace together with eq . ( [ eq12 ] ) it follows that @xmath124 = \mathrm{tr}\bigl [ \overline{\bm{h } } ( \mathbf{r}_{\pi(1)},\mathbf{r}_{\pi(2)},\omega)\cdots \overline{\bm{h } } ( \mathbf{r}_{\pi(j)},\mathbf{r}_{\pi(1)},\omega)\bigr],\ ] ] if @xmath125 is either a cyclic permutation or the reverse of a cyclic permutation . thus the sum on the r.h.s . of eq . ( [ eq59 ] ) contains classes of @xmath126 terms that give the same result . by forming a set @xmath127 @xmath128 @xmath129 containing exactly one representative of each class , we can remove this redundancy , leading to @xmath130 \nonumber\\ & \qquad=\sum_{\pi\in \overline{p}(j)}\!\ ! \mathrm{tr}\bigl [ \overline{\bm{h } } ( \mathbf{r}_{\pi(1)},\mathbf{r}_{\pi(2)},\omega)\cdots \overline{\bm{h } } ( \mathbf{r}_{\pi(j)},\mathbf{r}_{\pi(1)},\omega)\bigr].\end{aligned}\ ] ] introducing the factor @xmath131 in eq . ( [ eq56 ] ) and summing over all @xmath132 possible ways of renaming the variables @xmath133 and @xmath134 , the representative of each class in eq . ( [ eq61 ] ) is generated exactly twice ( only once for @xmath135 @xmath23 @xmath136 @xmath23 @xmath137 ) , so that the cp potential of atom @xmath2 due to the dielectric body @xmath100 can be written as @xmath138 u_{ab_1\ldots b_l } ( \mathbf{r}_a,\mathbf{s}_1,\ldots,\mathbf{s}_l),\ ] ] where @xmath139\end{gathered}\ ] ] is nothing but the @xmath135-atom vdw potential on an arbitrary magnetodielectric background @xmath140 , @xmath14 . we have hence proved that the cp interaction of an atom with a macroscopic dielectric body which is described within the framework of macroscopic qed is the result of all possible microscopic many - atom vdw interactions between the atom under consideration and the atoms forming the body , provided that the susceptibility is of clausius - mosotti type ( [ eq48 ] ) and the convergence condition ( [ eq58 ] ) holds generalising the result in ref . conversely , our proof shows that when eqs . ( [ eq62 ] ) and ( [ eq63 ] ) hold and the convergence condition ( [ eq58 ] ) is satisfied , then the electric susceptibility must have the clausius - mosotti form . in addition , the proof has delivered the general many - atom vdw potentials ( [ eq63 ] ) on an arbitrary magnetodielectric background . for @xmath135 @xmath23 @xmath137 , eq . ( [ eq63 ] ) agrees with the two - atom potential ( [ eq37 ] ) derived in sec . [ sec3.2 ] , while for higher @xmath135 , it presents a generalisation of the free - space vdw potentials derived in ref . @xcite . the applicability of the microscopic expansion ( [ eq62 ] ) depends crucially on the validity of the convergence condition ( [ eq58 ] ) . recalling eq . ( [ eq20 ] ) and estimating @xmath141 [ @xmath142 , electron charge ; @xmath143 , electron mass ; @xmath144 @xmath23 @xmath145 ; @xmath146 @xmath23 @xmath147 ; @xmath148 @xmath23 @xmath149 ; @xmath150 , volume accessible per atom within the body ; @xmath151 @xmath23 @xmath152 , , species - dependent factor ] , eq . ( [ eq58 ] ) can be reformulated as @xmath150 @xmath153 @xmath154 , stating simply that the atoms must be well - separated within the body . if this is not the case , the microscopic expansion ( [ eq62 ] ) does not converge , while the more general macroscopic expression ( [ eq19 ] ) for the cp potential remains valid . we have demonstrated that on the basis of macroscopic qed in linear , causal media , leading - order perturbation theory can be employed to derive general expressions for both the single - atom cp potential and the two - atom vdw potential in the presence of an arbitrary arrangement of magnetodielectric bodies . moreover , starting from this very general , geometry - independent basis , we have used the born expansion of the green tensor together with the clausius - mosotti law to prove that the cp interaction of a single atom with inhomogeneous , dispersing and absorbing dielectric bodies in the presence of an arbitrary magnetodielectric background can be written as a sum of many - atom vdw potentials . the proof demonstrates the equivalence of the microscopic and macroscopic descriptions provided that the microscopic picture is applicable , while at the same time delivering explicit expressions for the general many - atom vdw potentials in the presence of magnetodielectric media . this work was supported by the deutsche forschungsgemeinschaft . would like to thank the ministry of science , research , and technology of iran . h.t.d . would like to thank the alexander von humboldt stiftung and the national program for basic research of vietnam .

we establish a general relation between dispersion forces . first , based on qed in causal media , leading - order perturbation theory is used to express both the single - atom casimir - polder and the two - atom van der waals potentials in terms of the atomic polarizabilities and the green tensor for the body - assisted electromagnetic field . endowed with this geometry - independent framework , we then employ the born expansion of the green tensor together with the clausius - mosotti relation to prove that the macroscopic casimir - polder potential of an atom in the presence of dielectric bodies is due to an infinite sum of its microscopic many - atom van der waals interactions with the atoms comprising the bodies . this theorem holds for inhomogeneous , dispersing , and absorbing bodies of arbitrary shapes and arbitrary atomic composition on an arbitrary background of additional magnetodielectric bodies .

You are an expert at summarizing long articles. Proceed to summarize the following text: one would not expect that dwarf elliptical galaxies ( des ) in dense environments contain a significant interstellar medium ( ism ) . several arguments support this statement . supernova - explosions are able to transfer enough energy to the ism to heat it above the escape velocity in the least massive dwarfs @xcite . alternatively , the frequent high - speed interactions with giant cluster - members to which a small late - type disk galaxy is subjected can transform it into a gasless spheroidal de - like object . this `` galaxy harassment '' process @xcite induces a dramatic morphological evolution on a time - span of about 3 gyr . moreover , hydrodynamical simulations of dwarf galaxies moving through the hot , thin intergalactic medium in clusters @xcite or groups @xcite show that ram - pressure stripping can completely remove the ism of a dwarf galaxy less massive than @xmath10 within a few 100 myrs . a quite different point of view on the origin of des comes to the same conclusion . if des are related to other dwarf galaxies such as blue compact dwarfs ( bcds ) or dwarf irregular galaxies ( dirrs ) , the `` fading model '' conjectures that star - forming dwarf galaxies will fade and reach an end - state similar to present - day des after they have used up their gas supply and star - formation has ended @xcite . interactions may have sped up the gas - depletion process @xcite , explaining both the abundance of des and the paucity of bcds / dirrs in high - density environments . for all these reasons , des in dense environments were generally thought to be virtually gas - depleted systems . however , evidence is building up that at least some des have retained part of their gas . in their multi - wavelength study of the local group dwarf galaxies , young & lo presented vla hi observations of ngc147 , ngc185 , and ngc205 @xcite . these were the first observations that painted a detailed picture of the complex , multi - phase interstellar medium ( ism ) of the most nearby representatives of the class of the des @xcite . while ngc147 was not detected with a 3 @xmath0 mass upper limit of @xmath11 for an 8 km s@xmath2 velocity width , ngc205 was found to contain @xmath12 of neutral hydrogen and the total hi mass of ngc185 was estimated at @xmath13 . the neutral ism of both detected galaxies turned out to be very clumpy , making a meaningful determination of their velocity fields rather difficult . still , the stars and hi gas in ngc205 seem to have different rotation velocities while in ngc185 , neither the hi or the stars show significant rotation @xcite . single - dish observations of @xmath14co emission provide evidence that the molecular and atomic gas are kinematically linked . ngc205 was not detected on h@xmath15[nii ] narrow - band images while ngc185 contains an extended emission region , about 50 pc across @xcite . more recently , hi surveys of the virgo cluster de population ( see conselice et al . ( 2003 ) and references therein ) have shown that roughly 15% of the des contain a neutral ism . the detected hi masses range between 0.03 and @xmath16 . processes that remove gas , such as galaxy interactions and ram - pressure stripping @xcite , act most vigorously near the cluster center . accordingly , the gas - rich dwarf galaxies in the virgo cluster tend to have positions towards the outskirts of the cluster , suggesting that they are recent acquisitions of the cluster or are moving on orbits that avoid the cluster center . in a spectroscopic survey of the fornax cluster , drinkwater et al . ( 2001 ) discovered h@xmath6 emission in about 25% of the des . again , most of these galaxies lie towards the outskirts of the cluster , while des near the center of the cluster are generally devoid of ionized gas . in this paper , we present new hi 21 cm line observations of two des , obtained with the australia telescope compact array ( atca ) . with optical systemic velocities @xmath17 km / s ( fcc032 ) and @xmath18 km / s ( fcc336 ) @xcite , these des are bona fide members of the fornax cluster , located in the sparsely populated outskirts of the cluster ( see fig . [ cat ] ) . in section [ hi ] , we present our hi observations , followed by a discussion of our results in section [ disc ] . we summarize our conclusions in section [ conc ] . we have used the australia telescope compact array on 20 and 23 december 2004 to observe two des in the fornax cluster . we preferred interferometry observations above single - dish observations to avoid confusion with other galaxies that can be located within the large beam , which is a common nuisance in crowded environments such as the fornax cluster . the observations were made during night time to avoid solar rfi . we used the atca in the 1.5d configuration , with baselines ranging from 107 m to 4439 m. to be able to detect hi emission in three independent channels and since both sources had an estimated velocity width of about 50 km s@xmath2 , we selected a correlator setup that yielded 512 channels of width 15.6 khz . to increase the signal - to - noise ratio the data were on - line hanning smoothed which resulted in a velocity resolution of @xmath19 km s@xmath2 . at the start of each observation we observed the source 1934 - 638 as primary calibrator for 15 minutes . the source 0332 - 403 was observed every 40 minutes for 5 minutes as a secondary calibrator . the total integration time ( including calibration ) for each galaxy was 12h . the usual data reduction steps ( phase , amplitude and bandpass calibration ) were performed with the miriad package @xcite , the standard atca data analysis program . we subtracted the continuum by performing a first order fit to the visibilities over the line - free channels which were not affected by the edge effects of the band ( selected in advance by eye ) . the data cubes were created by using natural weighting and were subsequently smoothed with a gaussian beam of 1@xmath20 ( which corresponds to the optical spatial radius of our sources ) and off - line hanning smoothed to increase the signal - to - noise . the final data cubes had a spectral resolution of @xmath21 km s@xmath2 . due to the faintness of these objects , we did not attempt a deconvolution of our images . fcc032 was selected as a suitable target because it was known to contain a sizable ionised ism @xcite . this galaxy harbors a large ionised gas complex , about 850 pc across , containing star - formation regions and bubble - shaped gaseous filaments , probably supernova remnants . this offers strong evidence for recent or ongoing star formation in this de . our final data cube had a synthesised beam of @xmath22 due to the smoothing , resulting in a noise of 4 mjy / beam . these data cubes were inspected by eye for emission by plotting them over an optical image . the data were clipped at 1.5@xmath0=6 mjy / beam . weak emission near the optical center of the galaxy was found in 4 adjacent channels located around the optical systemic velocity . to derive a final spectrum ( containing all detected hi ) , we summed the flux within a @xmath23 box centered on the central radio position of fcc032 which was derived from the total hi intensity map ( see fig.[total032 ] ) . this resulted in a detection of the galaxy near the optical systemic velocity of 1318 km s@xmath2 , spread over 4 channels ( see fig . [ spec032 ] ) . we fitted a gaussian to the global hi profile and adopted the method of verheijen & sancisi ( 2001 ) to measure the systemic velocity and line widths at the 20% and 50% levels ( corrected for broadening and random motions ) . we applied the instrumental correction given by the formulae @xmath24,\ ] ] @xmath25,\ ] ] with @xmath26 the velocity resolution or @xmath21 km s@xmath2 in our case . this correction is based on approximating the edges of the global profile , which are mostly due to turbulent motion , with a gaussian of dispersion @xmath27 km s@xmath2 . the results show a maximum flux of 24 mjy / beam at a velocity of 1318 km s@xmath2 , which we adopt as the radio systemic velocity of fcc032 and which is in excellent agreement with the optically derived systemic velocity @xcite . for the widths at 20% and 50% we found respectively 52 km s@xmath2 and 34 km s@xmath2 . by summing the flux within the 4 channels that contain the 21 cm emission of fcc032 , we created a total hi intensity map ( see fig . [ total032 ] ) . the noise at a certain position in the total intensity map can be calculated by means of the formula @xcite : @xmath28 where @xmath29 stands for the number of channels that have been added at that position and with @xmath30 the noise in the hanning smoothed channel maps ( 4 mjy / beam or 53.2 mjy / beam km s@xmath2 ) . not all channels contribute to the flux at a given position in the total intensity map so @xmath29 is not the same everywhere . typically , @xmath31 in this case . this yields an average spatial rms error @xmath32 jy / beam km s@xmath2 . we rebinned the channels plotted in fig . [ spec032 ] by taking together 4 adjacent channels , such that all the 21 cm flux of fcc032 ends up in one single bin , 53.2 km s@xmath2 wide , and then calculated the rms noise from the other bins . this way , we find a total velocity integrated hi flux density of 0.66@xmath10.22 jy km s@xmath2 and calculated a total estimated hi mass of 5.0@xmath11.7@xmath7 , based on the @xmath33 velocity width by means of the formula : @xmath34 with @xmath35mpc the distance to the fornax cluster and @xmath36 the total flux density in units of jy km s@xmath2 . the hi intensity - weighted velocity field of fcc032 is presented in fig . [ map032 ] . a velocity gradient can be observed , suggesting rotation . fcc336 was selected according to its position in the outskirts of the fornax cluster and thus , in accordance with the ram - pressure stripping theory , could be expected to harbor a relatively large amount of neutral hydrogen . smoothing was again applied to the data cube to increase the signal - to - noise ratio , resulting in a synthesised beam of @xmath37 and a noise of 4 mjy / beam . in a similar way as for fcc032 , we found weak emission in three channels near the optical systemic velocity . we summed the flux within a @xmath23 box centered on the central radio position of the galaxy , which again was derived by the total hi intensity map , ( see fig . [ total336 ] ) in order to create a global hi profile of the galaxy . we found a strong intensity peak of 34.4 mjy / beam at a velocity of 2004 km s@xmath2 ( see fig . [ spec336 ] ) , which is in fair agreement with the optical systemic velocity of 1956 @xmath1 67 km s@xmath2 @xcite . a total hi intensity map shows that the emission , allowing for the coarse resolution of the hi map , coincides spatially with the optical image of the galaxy ( see fig . [ total336 ] ) . we fit a gaussian to the global hi profile and find a velocity width of 33 km s@xmath2 and 22 km s@xmath2 at respectively the 20% and 50% levels . we measure a total hi flux density of 0.37@xmath10.10 jy km s@xmath2 , which , by means of formula ( [ massa ] ) , corresponds to an estimated total hi mass of 2.8@xmath10.7 @xmath3 . here , we use the summed rms within the @xmath23 box and the @xmath33 velocity width . due to our velocity resolution of 13.3 km s@xmath2 and the very small velocity width of fcc336 , we did not attempt constructing an hi intensity - weighted velocity field . in fig . [ plotje ] , we plot the b - band luminosities of fcc032 and fcc336 versus their hi masses , along with the virgo des compiled by conselice et al . ( 2003 ) , the local group des ngc185 and ngc205 @xcite , the local group dwarf spheroidals ( dsphs ) and dwarf irregulars ( dirrs ) , taken from mateo ( 1998 ) , field dwarf irregulars and spheroidal galaxies , taken from roberts et al . ( 2004 ) , and virgo cluster blue compact dwarfs ( bcds ) and late - type dwarf galaxies ( sd - sm - im ) , taken from gavazzi et al . ( 2005 ) and sabatini et al . we included only those galaxies in this diagram that were actually detected at 21 cm . many des , however , are too gas - poor to have been detected in hi at the distances of the virgo and fornax clusters . hi studies in these clusters typically have mass - limits of the order of @xmath38 . only inside the local group have des and dsphs with a gas content as low as @xmath39 been detected . it should therefore be kept in mind that many des reside in the upper left part of this figure that have so far evaded detection . clearly , bcds and dirrs seem to trace a sequence , defined roughly by the relation @xmath40 . all these galaxies reside either in the virgo cluster or in the local group , with secure distance estimates @xcite . this makes us confident that this sequence is not a spurious result of the distance - dependence of both @xmath41 and @xmath42 . the local group dsphs and the local group des ngc185 and ngc205 ( both are satellites of m31 ) deviate from this sequence by being gas deficient . non - detections at 21 cm were not plotted in this figure . hence , many undetected gas - poor dwarf galaxies ( like ngc147 , with a 3@xmath0 upper limit of @xmath11 for @xmath43 ( young & lo , 1997 ) , or the 20 virgo des that were not detected by conselice et al . ( 2003 ) , with mass upper limits of @xmath44 ) are expected to occupy the left part of the diagram . therefore , the @xmath45 vs. @xmath46 sequence of gas - rich dwarf galaxies in fig . [ plotje ] is best seen as a boundary , enclosing the most hi - rich galaxies while many gas - poor dwarfs ( like des and dsphs ) lie significantly to the left of this sequence . in order to interpret this diagram , we overplotted the observed data points with theoretical predictions for the @xmath41 vs. @xmath42 relation , based on the analytical models of pagel & tautvaiien ( 1998 ) for the chemical evolution of the large and small magellanic clouds . in the formalism of these models , galaxies are formed by the infall of pristine gas . stars are born at a rate proportional to the gas mass and supernova explosions eject gas at a rate proportional to the star - formation rate ( sfr ) . the build - up of the elemental abundances is calculated using the delayed - recycling approximation in order to include the contribution of snia . we used the b - band mass - to - light ratios of single - age @xmath47 , single - metallicity @xmath48 stellar populations ( or ssps ) , denoted by @xmath49 , presented by vazdekis et al . ( 1996 ) , in order to calculate the present - day @xmath50 ratio as @xmath51 with @xmath52 the assumed age of the galaxies , @xmath53 the gas mass at time @xmath47 , and @xmath54 the star - formation efficiency ( or the inverse time - scale for star formation ) . the time - dependence of @xmath55 and @xmath48 is taken from pagel & tautvaiien ( 1998 ) ( their equations ( 6 ) , ( 8) , ( 11 ) , and ( 14 ) ) . the rightmost curve in fig . [ plotje ] , labeled with `` lmc '' , corresponds to parameter values fine - tuned to reproduce the elemental abundances observed in the lmc ( star - formation efficiency @xmath56 gyr@xmath2 and outflow parameter @xmath57 ) ; the middle curve , labeled with `` smc '' , corresponds to the smc ( star - formation efficiency @xmath58 gyr@xmath2 and outflow parameter @xmath59 ) . cleary , these simple models nicely reproduce the observed locus of the gas - rich dwarf late types , bcds , and des . the green curve in fig . [ plotje ] traces the @xmath41 vs. @xmath42 relation of late - type galaxies predicted by semi - analytical models ( sams ) of galaxy formation via hierarchical merging in a @xmath60cdm universe @xcite . sams make use of a monte - carlo technique to construct the hierarchical merger tree that leads up to the formation of a galaxy of a given mass . they moreover contain prescriptions for star - formation , energy feedback from supernova explosions , gas cooling , tidal stripping , dust extinction , and the dynamical response to starburst - induced gas ejection . despite the inevitable oversimplifications in the description of immensely complex processes such as star formation , they are able to account pretty well for many observed properties of galaxies . the green curve in fig . [ plotje ] indicates the amount of cold gas present in simulated galaxies that were classified as late - types ( see also fig . 4 in nagashima & yoshii , 2004 ) . thus , it seems that the locus of gas - rich sd - sm - im galaxies and bcds in fig . [ plotje ] can be reproduced quite satisfactorily by chemical evolution models of isolated galaxies in which slow star formation does not exhaust all available gas within a hubble time . using the same formalism , one can produce more gas - poor systems by raising the star - formation efficiency @xmath54 . for instance , interactions may have sped up the gas - depletion process @xcite , explaining both the abundance of gas - poor des and the paucity of gas - rich bcds / dirrs in high - density environments . e.g. , the leftmost curve in fig . [ plotje ] , labeled with `` enhanced sfr '' , corresponds to a model with a star - formation efficiency that is a factor of 3 higher than in the lmc model ( star - formation efficiency @xmath61 gyr@xmath2 and outflow parameter @xmath57 ) . this way , one can reproduce the locus of gas - poor dwarf galaxies such as the local group dwarf dsphs . all models presented in fig . [ plotje ] have luminosity - weighted mean metallicities in the range [ fe / h@xmath62=-0.65 $ ] to @xmath63 . the models with a high sfr consume their gas reservoir in a strong starburst at an early epoch , when the gas was not yet enriched with metals , and hence , even though they form stars more efficiently , do not have significantly higher mean metallicities than the low sfr models ( although they do contain a sprinkling of recently formed metalrich stars which are absent in low sfr models ) . as dicussed in grebel et al . ( 2003 ) , dsphs indeed have more metalrich red giants than dirrs and show evidence for a more vigorous early enrichment than dirrs . raising the gas - ejection efficiency can also enhance the @xmath64 ratio although , at the same time , it significantly lowers the mean metallicity by effectively terminating further star - formation after the first star - forming event ( e.g. making the gas - ejection by supernovae 30 times as efficient as in the magellanic clouds leads to @xmath65 and [ fe / h@xmath62 \approx -1.6 $ ] ) . however , the star - formation histories of the local group dsphs seem rather continuous ( with the exception of the carina dsph ) and show no evidence for major starbursts which would be able to expell significant amounts of gas @xcite . on the other hand , as argued by ferrara & tolstoy ( 2000 ) , maclow & ferrara ( 1999 ) , and de young & heckman ( 1994 ) , a centralised star - burst event in a round galaxy is much more efficient at transfering energy to the ism ( and hence at expelling gas out of a galaxy ) than a similar star - burst in a disk galaxy since , in the latter case , the hot supernova - driven gas can break through the disky ism along the minor axis . hence , this suggests that rotationally flattened galaxies , such as dwarf late - types , indeed have lower gas - ejection efficiencies than des and dsphs . the metallicity - flattening relation observed in des @xcite , with more flattened des tending to be less metal - rich , seems to indicate that in flattened des , individual supernova - explosions or star - formation sites are better able to eject the hot , enriched gas via a chimney perpendicular to the disk , without appreciably affecting the surrounding ism , than in round des . this leads to flattening as a second parameter in controlling a de s metallicity besides total mass . hydrodynamical simulations of dwarf galaxies moving through the hot , rarefied intracluster medium show that ram - pressure stripping can completely remove the ism of a low - mass dwarf galaxy @xcite . interactions and ram - pressure stripping are most efficient at removing gas from galaxies near the cluster center . indeed , the gas - rich dwarf galaxies in the virgo cluster tend to have positions towards the outskirts of the cluster ( e.g. conselice et al . ( 2003 ) ) , suggesting that they are recent acquisitions of the cluster or are moving on orbits that avoid the cluster center . in a spectroscopic survey of the fornax cluster , drinkwater et al . ( 2001 ) discovered h@xmath6 emission in about 25% of the des . again , most of these galaxies lie towards the cluster periphery , while des near the center of the cluster are generally devoid of ionized gas . likewise , both hi - rich des presented in this paper are located in the sparsely populated outskirts of the fornax cluster ( see fig . [ cat ] ) . based on our hi 21 cm observations of des in the fornax cluster and on hi 21 cm observations of des , bcds , and late - type dwarf galaxies in the virgo cluster @xcite , and the local group dwarfs , we conclude that the gas - content of the most gas - rich dwarf galaxies is consistent with a continuous , slow star - formation history . after one hubble time , these galaxies still have a large gas reservoir left and roughly trace a sequence defined by the relation @xmath66 . however , the majority of the dwarf spheroidals and dwarf ellipticals contain significantly less gas than predicted by this relation . external gas - removal mechanisms such as a star - formation rate enhanced by gravitational interactions or ram - pressure stripping can account very well for the existence of these gas - poor systems . such external mechanisms will act most vigorously in high - density environments , offering a natural explanation for the trend for hi mass to increase with distance from the nearest massive galaxy @xcite , and the fact that gas - rich des are observed predominantly in the outskirts of clusters . we wish to thank c. de breuck for his kind help and advice and the anonymous referee for the very helpful suggestions . dm and bp wish to thank the bijzonder onderzoeksfonds ( bof ) of ghent university for financial support . this paper is based on data obtained with the australia telescope compact array ( atca ) . the australia telescope is funded by the commonwealth of australia for operation as a national facility managed by csiro . 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 . 99 barazza , f. d. & binggeli , b. , 2002 , a&a , 394 , l15 conselice , c. , oneil , k. , gallagher , j. s. , wyse , r. f. g. , 2003 , apj , 591 , 167 de young , d. s. & heckman , t. m. , 1994 , apj , 431 , 598 de rijcke , s. , zeilinger , w. w. , dejonghe , h. , hau , g. k. t. , 2003 , mnras , 339 , 225 drinkwater , m. j. , gregg , m. d. , holman , b. a. & brown , m. j. i. , 2001 , mnras , 326 , 1076 ferguson , h. c. & binggeli , b. , 1994 , a&arv , 6 , 67 ferrara , a. & tolstoy , e. , 2000 , mnras , 313 , 291 gavazzi , g. , boselli , a. , van driel , w. , oneil , k. , 2005 , a&a , 429 , 439 grebel , e. k. , 2000 , in `` star formation from the small to the large scale '' , eslab symposium . f. favata , a. kaas , and a. wilson , p. 87 grebel , e. k. , gallagher iii , j. s. , harbeck , d. , 2003 , apj , 125 , 1926 maclow , m. & ferrara , a. , 1999 , apj , 513 , 142 marcolini , a. , brighenti , f. , dercole , a. , 2003 , mnras , 345 , 1329 marlowe , a. t. , meurer , g. r. , heckman , t. m. , 1999 , apj , 522 , 183 mateo , m. l. , 1998 , ara&a , 36 , 435 michielsen , d. , de rijcke , s. , zeilinger , w. w. , prugniel , p. , dejonghe , h. , roberts , s. , 2004 , mnras , 353 , 1293 michielsen , d. , de rijcke , s. , dejonghe , h. , 2004 , ans , 325 , 122 moore b. , katz n. , lake g. , dressler a. , oemler a. , jr . , 1996 , nature , 379 , 613 mori m. & burkert a. , 2000 , apj , 538 , 559 mori , m. , yoshii , y. , tsujimoto , t. , nomoto , k. , 1997 , apj , 479 , l21 nagashima , m. & yoshii , y. , 2004 , apj , 610 , 23 pagel , b. e. j. , tautvaiien , g. , 1998 , mnras , 299 , 535 roberts , s. , davies , j. , sabatini , s. , van driel , w. , oneil , k. , baes , m. , linder , s. , smith , r. , evans , r. , 2004 , mnras , 352 , 478 roediger , e. & hensler , g. , accepted by a&a , astro - ph/0412518 sabatini , s. , davies , j. , van driel , w. , baes , m. , roberts , s. , smith , r. , linder , s. , oneil , k. , 2005 , mnras , 357 , 819 sault , r.j . , teuben , p.j . , wright , m.c.h . , 1995 , in shaw r. , payne h.e . , hayes j.j.e . , eds , asp conf . ser . vol . 77 , astronomical data analysis software and systems iv . astron . soc , san fransisco , p.433 simien , f. & prugniel , ph . , 2002 , a&a , 384 , 371 tully , r.b . & fouque , p. , 1985 , apjs , 58 , 67 van zee , l. , skillman , e. d. , haynes , m. p. , 2004 , aj , 128 , 121 vazdekis , a. , casuso , e. , peletier , r. f. , beckman , j. e. , 1996 , apjs , 106 , 307 verheijen , m.a.w . & sancisi , r. , 2001 , a&a , 370 , 765 vollmer , b. , 2003 , a&a , 398 , 525 young , l. m. & lo , k. y. , 1997 , apj , 476 , 127 young , l. m. & lo , k. y. , 1996 , apj , 464 , l59

we present hi 21 cm line observations , obtained with the australia telescope compact array , of two dwarf elliptical galaxies ( des ) in the fornax cluster : fcc032 and fcc336 . the optical positions and velocities of these galaxies place them well within the fornax cluster . fcc032 was detected at the 3@xmath0 significance level with a total hi flux density of 0.66@xmath10.22 jy km s@xmath2 or an hi mass of 5.0@xmath11.7@xmath3 . based on our deep h@xmath4 $ ] narrow - band images , obtained with fors2 mounted on the vlt , this de was already known to contain 600 - 1800 @xmath5 of ionised hydrogen ( depending on the relative strengths of the h@xmath6 and [ nii ] emission lines ) . hence , this is the first study of the complex , multi - phase interstellar medium of a de outside the local group . fcc336 was detected at the same significance level : 0.37@xmath10.10 jy km s@xmath2 or a total hi mass of 2.8@xmath10.7@xmath7 . using a compilation of hi data of dwarf galaxies , we find that the observed high hi - mass boundary of the distribution of dirrs , bcds , and des in a @xmath8 versus @xmath9 diagram is in good agreement with a simple chemical evolution model with continuous star formation . the existence of many gas - poor des ( undetected at 21 cm ) suggest that the environment ( or more particularly , a galaxy s orbit within a cluster ) also plays a crucial role in determining the amount of gas in present - day des . e.g. , fcc032 and fcc336 are located in the sparsely populated outskirts of the fornax cluster . this is in agreement with hi surveys of des in the virgo cluster and an h@xmath6 survey of the fornax cluster , which also tend to place gas - rich dwarf galaxies in the cluster periphery . = = = = = = = = # 1 # 1 # 1 # 1 @mathgroup@group @mathgroup@normal@groupeurmn @mathgroup@bold@groupeurbn @mathgroup@group @mathgroup@normal@groupmsamn @mathgroup@bold@groupmsamn = `` 019 = ' ' 016 = `` 040 = ' ' 336 = " 33e = = = = = = = = # 1 # 1 # 1 # 1 = = = = = = = = [ firstpage ] galaxies : individual : fcc032 , fcc336 galaxies : dwarf galaxies : ism radio lines : galaxies

You are an expert at summarizing long articles. Proceed to summarize the following text: the rate of mixing , which has not yet been observed , would provide an important window on new physics . the contribution of box diagrams is very small due to gim suppression , and the mixing rate is believed to be dominated by long - distance processes . these processes are themselves suppressed by @xmath7-flavor symmetry . the mixing rate is usually expressed in terms of the parameters @xmath8 and @xmath9 , where @xmath10 and @xmath11 are the mass- and width - differences of the physical states , and @xmath12 is the average decay width . numerous predictions for @xmath13 and @xmath14 exist @xcite ; a recent calculation gives @xmath15 @xcite . non - standard - model ( sm ) processes are expected to enhance @xmath13 relative to @xmath14 , and measuring @xmath16 would be a strong indication of new physics @xcite . in this paper we report a measurement of the parameter @xmath17 which is equal to @xmath14 in the limit where cp is conserved . if cp is violated , the widths of @xmath18 and @xmath19 decays into the cp eigenstate @xmath20 differ . these widths can be expressed as @xcite @xmath21 \\ \hat{\gamma}(\overline{d}{}^{\,0 } \ra k^+ k^- ) & = & \gamma \left[\,1 + r_m^{-1 } ( y \cos{\phi } + x \sin{\phi})\right]\,,\end{aligned}\ ] ] where @xmath22 and @xmath23 are cp - violating parameters defined in @xcite . cp violation in mixing results in @xmath24 , and cp violation due to interference between mixed and unmixed decay amplitudes results in @xmath25 . we investigate whether cp violation is present by measuring the quantity @xmath26 the sm predicts @xmath27 @xcite ; a large value would indicate physics beyond the sm . our measurement of @xmath28 is obtained by measuring the lifetime difference between @xmath2 and @xmath3 decays . ( throughout this paper , charge - conjugate modes are included unless noted otherwise . ) our measurement of is obtained by measuring the lifetime difference between @xmath3 and @xmath29 decays . the analysis is based on 158 fb@xmath30 of data , and all results are preliminary . to identify the flavor of the @xmath18 and also reduce background , we require that all @xmath18 s originate from @xmath1 decays . the belle detector is a large - solid - angle magnetic spectrometer that consists of a three - layer silicon vertex detector ( svd ) , a 50-layer central drift chamber ( cdc ) , an array of aerogel threshold erenkov counters ( acc ) , a barrel - like arrangement of time - of - flight scintillation counters ( tof ) , and an electromagnetic calorimeter comprised of csi(tl ) crystals ( ecl ) located inside a super - conducting solenoid coil that provides a 1.5 t magnetic field . an iron flux - return located outside of the coil is instrumented to detect @xmath31 mesons and to identify muons ( klm ) . the detector is described in detail elsewhere @xcite . we select well - measured charged tracks that have impact parameters less than 1.0 cm in the radial direction and less than 2.0 cm along the beam direction ( @xmath32 ) with respect to the interaction point ( ip ) . the @xmath18 daughter tracks are required to have at least two hits in each of the @xmath33-@xmath23 and @xmath32-measuring svd layers . this requirement is not applied to the slow pion from @xmath34 decays . for charged particle identification , information from the cdc , tof , and acc subsystems is combined to form an overall hadron likelihood : @xmath35 where @xmath36 stands for @xmath37 or @xmath38 . charged particles are identified as pions or kaons using the likelihood ratios : @xmath39 kaon candidates are required to have @xmath40 . this requirement is 88% efficient and has a pion misidentification probability of 8% . pion candidates are required to satisfy @xmath41 . both continuum and @xmath5 data are used in this analysis . each event is required to contain at least one @xmath42 meson decaying to @xmath43 . to remove @xmath42 s resulting from @xmath44 meson decays , the @xmath42 is required to have a momentum in the @xmath6 center - of - mass frame ( cms ) greater than 2.5 @xmath45 . @xmath18 meson candidates are reconstructed from @xmath2 and @xmath3 decays . all @xmath18 candidates must satisfy @xmath46 , where @xmath47 and @xmath48 are the mean and standard deviation of a gaussian fit to the @xmath18 mass peak in data . these parameters are calculated independently for each @xmath18 decay mode in order to help keep the @xmath49 and @xmath20 vertex resolution functions unbiased by the @xmath18 mass requirement . from the fits , @xmath48 is found to be 5.1 @xmath50 for @xmath2 and 4.5 @xmath50 for @xmath51 ( see table [ tab - datamass ] ) . the momentum of the slow @xmath52 from @xmath34 decays is obtained by refitting using the beam position constraint , i.e. , the track must project back to the ip region . the invariant mass of the @xmath42 candidates after slow @xmath52 refitting is required to satisfy @xmath53 . the proper time between the production and decay of the @xmath18 candidates is determined by projecting the momentum @xmath55 and the flight length @xmath56 onto the @xmath57 plane transverse to the beam axis : @xmath58 the momentum @xmath55 is the vector sum of the @xmath38 and @xmath37 momenta , and @xmath56 is the displacement vector between the production and decay vertices . information from the @xmath32 projections is not included due to the large uncertainty in the @xmath32 component of the ip . to improve the spatial resolution , the vertex fit of @xmath59 decays is done in three steps . first , the decay point is fit using only information from the @xmath18 daughter tracks . then , the @xmath18 production vertex is found by fitting the reconstructed @xmath18 momentum vector with the ip region , whose profile is obtained from a separate study of the full data set . finally , the @xmath18 decay vertex is corrected using information from the @xmath18 production vertex fit . no mass constraint is applied to the @xmath18 candidates during the vertex fit to avoid mode - dependent bias that could result from a correlation between the invariant mass and a global momentum correction procedure . the error in the position of the @xmath18 decay vertex obtained from the fit is required to be less than 150 @xmath60 m in each of the @xmath13 and @xmath14 directions . [ sigxy ] illustrates the high efficiency of this cut . this procedure produces typical @xmath18 decay vertex resolutions of @xmath61 m in both @xmath13 and @xmath14 . ( left plot ) and @xmath62 ( right plot ) returned by the @xmath18 vertex fit after the ip position correction . the horizontal scale is in cm . vertical lines indicate the position of the fit quality cut : @xmath63m.,title="fig:",scaledwidth=48.0% ] ( left plot ) and @xmath62 ( right plot ) returned by the @xmath18 vertex fit after the ip position correction . the horizontal scale is in cm . vertical lines indicate the position of the fit quality cut : @xmath63m.,title="fig:",scaledwidth=48.0% ] candidates ( left plot ) and @xmath3 candidates ( right plot ) . the horizontal scale is in gev@xmath64 . the bin size is 0.5 mev@xmath64 and the solid curves show the fit results.,title="fig:",scaledwidth=48.0% ] candidates ( left plot ) and @xmath3 candidates ( right plot ) . the horizontal scale is in gev@xmath64 . the bin size is 0.5 mev@xmath64 and the solid curves show the fit results.,title="fig:",scaledwidth=48.0% ] the invariant mass spectra of the selected @xmath2 and @xmath3 candidates are shown in fig . [ fig - datamass ] . the distributions are fit with a sum of two gaussians representing the signal and a first - order polynomial representing the background . the results of the fit are summarized in table [ tab - datamass ] . @c@ @c@ @c@ @c@ c@ mode & # events & main gaussian @xmath65 & main gaussian fraction & signal purity + & & ( mev/@xmath66 ) & ( % ) & ( % ) + @xmath49 & 448000 & @xmath67 & 78.4 & 99.1 + @xmath20 & 36480 & @xmath68 & 74.0 & 97.6 + ( left plot ) and @xmath3 ( right plot ) obtained from mc simulation . the horizontal scale is in fs . the bin size of the data points is 40 fs and the solid curves show the results of the fit . ( the fit is performed using a bin size of 10 fs . ) , title="fig:",scaledwidth=48.0% ] ( left plot ) and @xmath3 ( right plot ) obtained from mc simulation . the horizontal scale is in fs . the bin size of the data points is 40 fs and the solid curves show the results of the fit . ( the fit is performed using a bin size of 10 fs . ) , title="fig:",scaledwidth=48.0% ] the proper time distribution is represented by an exponential function with time constant @xmath69 convolved with the detector vertex resolution . the properties of the detector resolution function are studied using monte carlo ( mc ) simulation of the signal decay modes . approximately @xmath70 @xmath2 decays and @xmath71 @xmath3 decays were simulated , corresponding to two times the data sample and three times the data sample , respectively . [ fig - res ] shows the difference between the generated and reconstructed decay times of the @xmath18 mesons . the distributions are fit with a sum of five gaussians constrained to a common central value . the results of the fit are summarized in table [ tab - res ] ; the confidence level ( cl ) is 22.2% . due to similar kinematics , the @xmath20 decay mode is fit with the same resolution function as that used for @xmath49 . however , to account for small differences between @xmath20 and @xmath49 , the width of each gaussian component in the @xmath20 fit is multiplied by a common scale factor @xmath72 . fitting first the @xmath49 distribution and then the @xmath20 distribution , we obtain @xmath73 . the cl for the @xmath20 fit is 71.3% . the common central values of the gaussians are found to be @xmath74 fs for @xmath49 and @xmath75 fs for @xmath20 . in the lifetime difference fits for both mc and data , the relative fractions of the gaussian components of the resolution function are fixed to the values determined in the above study . the central value of the @xmath20 resolution function is fixed to that of the @xmath49 resolution function , which is consistent with the results above . the possible bias due to this assumption is included in the systematic error . @c@ @c@ @c@ fit parameter & fraction ( % ) & value ( fs ) + @xmath76 & 26.1 & @xmath77 + @xmath78 & 50.4 & @xmath79 + @xmath80 & 19.8 & @xmath81 + @xmath82 & 3.1 & @xmath83 + @xmath84 & 0.6 & @xmath85 + the @xmath18 lifetime difference is obtained from a simultaneous , binned , maximum likelihood fit to the measured proper lifetime distributions of @xmath3 and @xmath2 decays . the consistency and quality of the fitting procedure is tested with mc events . the lifetime distributions of @xmath86 decays were fit with an exponential function convolved with a resolution function of the form described above . the cl of the fit is 61% . [ fig - mcfit ] and table [ tab - mcfit ] show the results of the fit and a comparison to the parameters obtained from the resolution function study . the parameters are seen to agree to within the quoted errors . the resolution function scale factor @xmath72 determined from the fit is @xmath87 , also in agreement with the result from the resolution function study . the @xmath18 lifetime obtained is @xmath88 fs , which is consistent with the generated value of 412.6 fs . the lifetime difference obtained is @xmath89 fs , which is consistent with the generated value of exactly 0 fs . ( left plot ) and @xmath3 ( right plot ) performed on mc - generated events . the horizontal scale is in fs . the bin size of the data points is 40 fs and the solid curves show the results of the fit . ( the fit was done using a bin size of 10 fs . ) , title="fig:",scaledwidth=48.0% ] ( left plot ) and @xmath3 ( right plot ) performed on mc - generated events . the horizontal scale is in fs . the bin size of the data points is 40 fs and the solid curves show the results of the fit . ( the fit was done using a bin size of 10 fs . ) , title="fig:",scaledwidth=48.0% ] @c@ @c@ @c@ fit parameter & resolution function ( fs ) & lifetime fit ( fs ) + @xmath76 & @xmath77 & @xmath90 + @xmath78 & @xmath79 & @xmath91 + @xmath80 & @xmath81 & @xmath92 + @xmath82 & @xmath83 & @xmath93 + @xmath84 & @xmath85 & @xmath94 + @xmath95 & @xmath96 & @xmath97 + @xmath72 & @xmath98 & @xmath87 + to fit the lifetime difference in data , one must include small contributions from background processes . to include these , the proper time distribution of background is obtained from events in the @xmath18 mass sidebands . the sidebands chosen begin @xmath99 mev@xmath64 from @xmath100 and cover the same total range as the signal region ( @xmath101 ) . ( left plot ) and @xmath3 ( right plot ) . the horizontal scale is in fs . the bin size is 40 fs and the solid curves show the results of the fit.,title="fig:",scaledwidth=48.0% ] ( left plot ) and @xmath3 ( right plot ) . the horizontal scale is in fs . the bin size is 40 fs and the solid curves show the results of the fit.,title="fig:",scaledwidth=48.0% ] the background proper time distributions are parameterized as the sum of an exponential convolved with a gaussian and a breit - wigner function . the resulting spectra are shown in fig . [ fig - bkg ] . the systematic uncertainties due to the background fitting procedure are evaluated by varying the fitting parameters . since the background contribution is small compared to the signal , the effect of the background magnitude and shape uncertainty is small ( approximately @xmath102 in @xmath28 ) . it was checked with mc simulation that the small fraction of signal events remaining in the mass sidebands does not affect the @xmath28 measurement above the quoted systematic error on background . the decay time function obtained for background is included ( with normalization fixed ) in the lifetime fit to the signal region . ( left plot ) and @xmath3 ( right plot ) . the horizontal scale is in fs . the bin size is 40 fs and the solid curves show the results of the fit . the dashed curves show the background component included in the fit.,title="fig:",scaledwidth=48.0% ] ( left plot ) and @xmath3 ( right plot ) . the horizontal scale is in fs . the bin size is 40 fs and the solid curves show the results of the fit . the dashed curves show the background component included in the fit.,title="fig:",scaledwidth=48.0% ] @c@ @c@ @c@ fit parameter & resolution function ( fs ) & lifetime fit ( fs ) + @xmath76 & @xmath77 & @xmath103 + @xmath78 & @xmath79 & @xmath104 + @xmath80 & @xmath81 & @xmath105 + @xmath82 & @xmath83 & @xmath106 + @xmath84 & @xmath85 & @xmath107 + @xmath95 & @xmath96 & @xmath108 + @xmath72 & @xmath98 & @xmath109 + fig . [ fig - datafit ] and table [ tab - datafit ] show the results of the lifetime difference fit to the data . the cl of the fit is 94% . the fitted @xmath2 lifetime is @xmath110 fs , and the fitted lifetime difference between @xmath2 and @xmath3 is @xmath111 fs . together these values give @xmath112% . the parameters obtained by the fit for the resolution function ( see table [ tab - datafit ] ) differ from those obtained from the mc study by more than the quoted errors . this is nominally due to vertex detector misalignment , ip mismeasurement , material unaccounted for in the mc simulation , and momentum dependence . these effects are included when evaluating the overall systematic error on @xmath113 ( table [ tab - ycp - syst ] ) . @l@ @c@ source & value ( % ) + @xmath114 assumption & 0.15 + @xmath72 difference on data and mc & 0.20 + bin size & 0.10 + @xmath18 mass window & 0.20 + background & 0.10 + selection requirements & 0.15 + total & 0.38 + the cp - violating parameter in @xmath3 decays is obtained from fits to the data using the charge of the slow pion from @xmath115 or @xmath116 to distinguish @xmath18 decays from @xmath19 decays . the @xmath18 and @xmath19 subsamples are fit separately using the resolution function parameters obtained from the full statistics fit . background is treated as described in the previous section . the same procedure is performed for @xmath2 decays and also for mc events in order to estimate the systematic uncertainty of the fit . all results are listed in table [ tab - cpv ] . the systematic uncertainty of the measurement is taken to be @xmath117 . @c@ @c@ @c@ @c@ @xmath18 decay mode & data / mc & @xmath118 ( fs ) & ( % ) + @xmath119 & mc & @xmath120 & @xmath121 + @xmath122 & mc & @xmath123 & @xmath124 + @xmath119 & data & @xmath125 & @xmath126 + @xmath122 & data & @xmath127 & @xmath128 + in conclusion , we report a preliminary measurement of the mixing parameter @xmath28 and the cp - violating parameter obtained by measuring lifetime differences in @xmath129 and @xmath130 decays . the flavor of the @xmath18 or @xmath19 is identified via @xmath131 decays . the results obtained are @xmath132 and @xmath133 . we wish to thank the kekb accelerator group for the excellent operation of the kekb accelerator . we acknowledge support from the ministry of education , culture , sports , science , and technology of japan and the japan society for the promotion of science ; the australian research council and the australian department of education , science and training ; the national science foundation of china under contract no . 10175071 ; the department of science and technology of india ; the bk21 program of the ministry of education of korea and the chep src program of the korea science and engineering foundation ; the polish state committee for scientific research under contract no . 2p03b 01324 ; the ministry of science and technology of the russian federation ; the ministry of education , science and sport of the republic of slovenia ; the national science council and the ministry of education of taiwan ; and the u.s . department of energy .

we report a preliminary measurement of the mixing parameter @xmath0 and the cp - violating parameter using the decay @xmath1 followed by @xmath2 and @xmath3 . the results are obtained from a @xmath4 data sample collected near the @xmath5 resonance with the belle detector at the kekb asymmetric energy @xmath6 collider .

You are an expert at summarizing long articles. Proceed to summarize the following text: in 1983 thouless @xcite proposed a simple pumping mechanism to produce , even in the absence of an external bias , a quantized electron current through a quantum conductor by an appropriate time - dependent variation of the system parameters . experimental realizations of quantum pumps using quantum dots ( qds ) were already reported in the early 90 s @xcite . more recently , due to the technological advances in nano - lithography and control , such experiments have risen to a much higher sophistication level , making it possible to pump electron @xcite and spin @xcite currents through open nanoscale conductors , as well as through single and double qds @xcite . early theoretical investigations where devoted to the adiabatic pumping regime within the single - particle approximation @xcite . this is well justified for experiments with open qds , where interaction effects are believed to be weak @xcite and the typical pumping parameters are slow with respect the characteristic transport time - scales , such as the electron dwell time @xmath0 . this time - scale separation enormously simplifies the analysis of the two - time evolution of the system . within the adiabatic regime , inelastic and dissipation @xcite effects of currents generated by quantum pumps were analyzed . furthermore , issues like counting statistics @xcite , memory effects @xcite , and generalizations of charge pumping to adiabatic quantum spin pumps were also proposed and studied @xcite . non - adiabatic pumping has been theoretically investigated within the single - particle picture , either by using keldysh non - equilibrium green s functions ( negf ) with an optimal parametrization of the carrier operators inspired by bosonization studies @xcite , or by a flouquet analysis of the @xmath1-matrix obtained from the scattering approach @xcite . while the first approach renders complicated integro - differential equations for the green s functions associated to the transport , the second one gives a set of coupled equations for the flouquet operator . it is worth to stress that , in both cases the single - particle picture is crucial to make the solution possible and it is well established that both methods are equivalent @xcite . several works have provided a quite satisfactory description of quantum pumping for weakly interacting systems . in contrast , the picture is not so clear for situations where interaction effects are important . different approximation schemes have been proposed to deal with pumping in the presence of interactions and to address charging effects , which are not accounted for in a mean - field approximation . typically , two limiting regimes have been studied , namely , the one of small pumping frequencies @xmath2 , such that @xmath3 ( adiabatic limit ) @xcite and the one of very high frequencies , @xmath4 ( sudden or diabatic limit ) @xcite . nonadiabatic pumping is mainly studied as a side effect of photon - assisted tunneling @xcite , where @xmath4 . unfortunately , it is quite cumbersome to calculate corrections to these limit cases . for instance , the analysis of higher - order corrections to the adiabatic approximation for the current gives neither simple nor insightful expressions @xcite . in addition to the theoretical interest , a comprehensive approach bridging the limits of @xmath4 and @xmath5 has also a strong experimental motivation : most current experimental realizations of quantum pumping deal with qds in the coulomb blockade regime and @xmath6 . this regime was recently approached ( from below ) by means of a diagrammatic real - time transport theory with a summation to all orders in @xmath2 @xcite . however , the derivation implied the weak tunnel coupling limit , whereas experiments @xcite typically rely on tunnel coupling variations which include both weak and strong coupling . to address the above mentioned issues and to account for the different time scales involved it is natural to use a propagation method in the _ time domain _ @xcite . in this work we express the current operator in terms of density matrices in the heisenberg representation . we obtain the pumped current by truncating the resulting equations - of - motion for the many - body problem . the time - dependence is treated exactly by means of an auxiliary - mode expansion @xcite . this approach provides a quite amenable path to circumvent the usual difficulties of dealing with two - time green s functions @xcite . moreover , it has been successfully applied to systems coupled to bosonic reservoirs @xcite and to the description of time - dependent electron - transport using generalized quantum master equations for the reduced density matrix @xcite . since the auxiliary - mode expansion is well controlled @xcite , the accuracy of our method is determined solely by the level of approximation used to treat the many - body problem . the formalism we put forward is illustrated by the study of the charge pumped through a qd in the coulomb - blockade regime by varying its resonance energy and couplings to the leads . the external drive is parametrized by a single pulse , whose duration and amplitude can be arbitrarily varied . by doing so , the formalism is capable to reproduce all known results of the adiabatic limit and to explore transient effects beyond this simple limit . the paper is organized as follows . in sec . [ sec : model ] we present the resonant - level model , as well the theoretical framework employed in our analysis . in sec . [ sec : prop ] we introduce the general propagation scheme , suitable to calculate the pumping current at the adiabatic regime and beyond it . next , in sec . [ sec : app ] , we discuss few applications of the method . finally , in sec . [ sec : conclusion ] we present our conclusions . the standard model to address electron transport through qds is the anderson interacting single - resonance model coupled to two reservoirs , one acting as a source and the other as a drain . despite its simplicity , the model provides a good description for coulomb - blockade qds and for qds at the kondo regime , where the electrons are strongly correlated . in this paper we address the coulomb - blockade regime , for qds whose typical line width @xmath7 is much smaller than the qd mean level spacing @xmath8 , justifying the use of the anderson single - resonance model . in addition , in the coulomb blockade regime @xmath7 is much smaller than the resonance charging energy @xmath9 . the total hamiltonian is given by the usual threefold decomposition into a quantum dot hamiltonian @xmath10 , a hamiltonian @xmath11 representing the leads , and a coupling term @xmath12 , namely [ eq : hamiltonian ] @xmath13 the qd is modeled by a single level of energy @xmath14 , which can be occupied by spin - up and spin - down electrons , which interact through a contact interaction of strength @xmath9 . the qd hamiltonian reads @xmath15 where @xmath16 , @xmath17 and @xmath18 stand for electron number , creation and annihilation operator for the respective spin state @xmath19 in the dot . the two reservoirs , labeled as @xmath20 ( left ) and @xmath21 ( right ) , are populated by non - interacting electrons , whose hamiltonian reads @xmath22 where @xmath23 and @xmath24 stand for the electron creation and annihilation operators for the @xmath25-reservoir state @xmath26 , respectively . the reservoir single - particle energies have the general form @xmath27 with the @xmath28 accounting for a time - dependent bias . the stationary current due to a time - dependent bias was already addressed several years ago @xcite . for pumping , we take @xmath29 , as usual . finally , the coupling hamiltonian is given by @xmath30 with @xmath31 denoting the coupling matrix element between the qd and the reservoir @xmath25 . we are interested in the electronic current from reservoir @xmath25 to the qd state @xmath32 , which can be obtained from current operator @xmath33\,.\end{aligned}\ ] ] here and in the following we use units where the elementary charge @xmath34 and the reduced planck constant @xmath35 , unless otherwise indicated . to calculate @xmath36 we use the following equations of motion , which are obtained from the hamiltonian [ eqs . ] by means of the heisenberg equation , [ eq : eomtheisenberg ] @xmath37\;.\label{eq : eomtheisenberg3}\end{aligned}\ ] ] analogous equations hold for @xmath38 and @xmath39 . in the spirit of the scheme introduced by caroli and co - workers @xcite , we assume an initially uncorrelated density operator of the combined system , _ i.e. _ , we set @xmath40 for @xmath41 . further , we apply the so - called wide - band limit @xcite , where the square of the tunneling element @xmath42 is inversely proportional to the density of states @xmath43 at energy @xmath44 . by means of the lead green function @xcite @xmath45\ ] ] we can define the decay rate [ eq : gammawbl ] @xmath46 which becomes local in time in the wide - band limit , namely @xmath47 in the following we replace the sum in eq . by the expression involving the @xmath8-function in eq . . the equation of motion for the reservoir operators @xmath48 , eq . , is now readily integrated , yielding [ eq : eomtresop ] @xmath49 where we have used the lead green functions , eq . , and introduced @xmath50 equations are used to rewrite eq . as @xmath51 { \hat{c}}_{s}(t ) \notag\\ & + \sum\limits_{\alpha k } t^*_{\alpha k}(t ) { \hat{b}}_{\alpha k s}(t)\ , . \label{eq : ceom}\end{aligned}\ ] ] here the wide - band limit , eq . , is employed to obtain the decay term , proportional to @xmath52 . similarly , we can rewrite eq . as @xmath53 - { \mathrm{i}}\gamma(t ) { \hat{n}}_{\bar{s}}(t)\;.\notag\end{aligned}\ ] ] here again the time integral of @xmath54 is reduced to a decay width due to the wide - band limit . the expression for the time - dependent current is given by the expectation value of the current operator @xmath55 defined in eq . . as will become clear later on , it is useful to write this expectation value as @xmath56 with the _ current matrices of the first order _ @xmath57 these current matrices are an essential ingredient of our propagation scheme , which is based on finding equations of motion for @xmath58 . such equations have been derived starting from a negf formalism for non - interacting electrons @xcite . exactly as for the operator equations above we can use @xmath59 from eq . and employ the wide - band limit for the current matrices defined in eq . . this leads to the following decomposition [ eq : negfdefpi1 ] @xmath60 having derived all relevant equations of motion for the operators we can specify the respective equations for the two contributions @xmath61 and @xmath62 . the term @xmath61 is the simplest and is basically given by the equation of motion for @xmath63 , cf . the corresponding equation for the occupation @xmath64 reads @xmath65 the above relation can be viewed as the charge conservation equation for the qd . the rate by which the charge in the qd changes is equal to the total electronic currents . the first term at the _ r.h.s . _ of the equation can be interpreted as the current flowing into the qd , whereas the second term gives the current flowing out . since we do not consider a spin - dependent driving or spin - polarized initial states it is @xmath66 . this relation is not explicitly used in the derivation , but is employed as a consistency check throughout the analysis . the evaluation of @xmath62 requires the solutions for both , the lead operator @xmath48 and the dot operator @xmath18 . using those , we write @xmath67 here we have introduced the abbreviation @xmath68\ ] ] and used that @xmath69 with @xmath70 the fermi function describing the equilibrium occupation of lead @xmath25 . the last term in eq . uses the _ auxiliary current matrices of the second order _ @xmath71 which will be subject to further approximations in the following . before we turn to the approximations , we would like to briefly discuss the physical meaning of @xmath72 . the equation of motion for the two - electron density matrix @xmath73 reads @xmath74 which follows from eq . . the two - electron density matrix may be interpreted as the occupation of one quantum - dot level under the condition that the other one is occupied . the rate of change of this conditional occupation is consequently given by tunneling into and out of the respective dot state under the same condition . the latter process is described by the first term on the _ r.h.s . _ of eq . . the former process is governed by the auxiliary current matrices @xmath72 , which can be rewritten in the suggestive form @xmath75 consequently , the current matrices @xmath72 describe the _ conditional current _ from reservoir @xmath25 into the quantum - dot level with spin @xmath32 . the simplest approximation to @xmath72 consists in using the following factorization @xmath76 inserting this expression into eq . , results in the following equation of motion @xmath77 { \pi''}_{\alpha k s}(t ) \notag\\ & + t^*_{\alpha k}(t ) f_{\alpha k}.\end{aligned}\ ] ] this result is equivalent to the hartree - fock approximation applied to the anderson model standard two - electron green function @xcite . as any mean field approach , it does not lead to a double resonance green function , which is required to properly account for charging effects . hence , as it is well known , a good description of the coulomb - blockade regime requires going beyond this level of truncation in the equations of motion . instead of factorizing @xmath72 directly , we proceed by deriving its equation of motion . by means of eqs . we get @xmath78 { \phi}_{\alpha k s}(t ) + \sum\limits_{\alpha',k ' } t^*_{\alpha ' k'}(t ) { \left\langle { \hat{b}}^\dagger_{\alpha k s}(t ) { \hat{b}}_{\alpha ' k ' s}(t ) { \hat{n}}_{\bar{s}}(t ) \right\rangle}\notag\\ & + \sum\limits_{\alpha',k ' } \left [ t_{\alpha ' k'}(t ) { \left\langle { \hat{b}}^\dagger_{\alpha k s}(t ) { \hat{c}}_{s}(t ) { \hat{b}}^\dagger_{\alpha ' k ' \bar{s}}(t ) { \hat{c}}_{\bar{s}}(t ) \right\rangle } - t^*_{\alpha ' k'}(t ) { \left\langle { \hat{b}}^\dagger_{\alpha k s}(t ) { \hat{c}}_{s}(t ) c^\dagger_{\bar{s}}(t ) { \hat{b}}_{\alpha ' k ' \bar{s}}(t ) \right\rangle } \right ] . \label{eq : phieom}\end{aligned}\ ] ] note that the term proportional to @xmath9 has only four operators in the expectation values because of @xmath79 . the approximation consists in neglecting matrix elements involving opposite spins , which renders the following factorizations @xmath80 this approximation for the density matrices is equivalent to the truncation scheme employed in the negf approach used for the study of coulomb blockade regime ( high - temperature limit of the anderson model ) @xcite . as a result of the factorization , we obtain the following compact equation of motion for the approximated second - order current matrices @xmath81 { \tilde{\phi}}_{\alpha k s}(t ) \notag\\ & + t^*_{\alpha k}(t)\ , f_{\alpha k}\ , n_{\bar{s}}(t)\;. \label{eq : pieomhubbard}\end{aligned}\ ] ] the equations of motion for @xmath82 [ eq . ] , @xmath83 [ eq . with @xmath72 replaced by @xmath84 and for @xmath85 [ eq . ] form a closed set of equations , which can be solved by means of an auxiliary - mode expansion discussed below . the general idea of the auxiliary - mode expansion consists in making use of a contour integration and the residue theorem to perform the energy integration , for instance , in eq . . to this end the fermi function is expanded in a sum over simple poles ( or auxiliary modes ) and the respective integrals are given as finite sums , cf . appendix [ sec : appexp ] . the transition to auxiliary modes ( denoted by the index @xmath86 ) is facilitated by the following set of rules [ eq : rules ] @xmath87 which are derived in appendix [ sec : appexp ] . the first rule replaces the reservoir energy @xmath44 by the ( complex ) pole @xmath88 of the expansion , cf . the second rule replaces the fermi function by the respective weight , which is the same for all auxiliary modes . finally , the third rule provides the actual expansion for the current matrices . applying these rules , the current matrices become @xmath89 the equation of motion for the auxiliary matrix @xmath90 is obtained from eq . . one arrives at @xmath91 { \pi''}_{\alpha s p } ( t ) \nonumber\\ & & + \frac{1}{\beta}t_{\alpha}(t ) + u \:{\phi}_{\alpha s p } ( t ) \ , . \label{eq : auxeomtpi}\end{aligned}\ ] ] the equations of motion for the auxiliary matrices @xmath92 are quite similar to those of eq . , namely , @xmath93 { \tilde{\phi}}_{\alpha s p } ( t ) \nonumber\\ & + \frac{1}{\beta}t_{\alpha}(t)\ , n_{\bar{s}}(t)\;. \label{eq : auxeomtphi}\end{aligned}\ ] ] the solution of the above equations still requires a complete description of the population dynamics given by @xmath82 . the latter can be directly obtained from eq . in terms of the current matrices @xmath94 this concludes the derivation of the auxiliary mode propagation scheme . the set of equations to , with initial conditions @xmath95 , @xmath96 , and @xmath97 , can be solved numerically using standard algorithms . before the desired time dependence of the parameters @xmath98 and @xmath99 sets in , the system has to be propagated until a steady state is reached . in this way , transient effects arising from the choice of the initial state are avoided . for convenience we derive in appendix [ sec : appstat ] the expressions for the stationary occupations , which may also be used as initial values for @xmath100 . in this section we present two applications of the formalism developed above . as shown below , one of the interesting features of non - adiabatic pumping is an increasing delay in the current response to the external drive with growing driving speed . hence , in distinction to the adiabatic limit , the current caused by a train of pulses can show interesting transient effects , whenever the pulse period is shorter than the system response time . to better understand non - adiabatic driving effects , we focus our analysis on single pulses and vary the speed by which their shape is changed . it is worth stressing that our propagation method does not possess restrictions on the time dependence of the system driving parameters . in other words , the external time - dependent drive can be just a single pulse or a train of pulses , it can also be either fast or slow as compared with the system internal time scales . let us begin by discussing the current generated by a single gaussian voltage pulse changing the resonance energy as @xmath101\;.\end{aligned}\ ] ] here @xmath102 sets the pumping time - scale . we take @xmath103 to be time - independent and equal for both leads , @xmath104 . since thereby @xmath105 , we will consider only @xmath106 in the following . figure [ fig : single]a shows the time dependence of the resonance energy according to eq . . the two bottom panels show the instantaneous current @xmath106 as a function of time for both the non - interacting ( @xmath107 ) and the interacting ( @xmath108 ) case . in the limit of large @xmath102 , we use as a check for our results an analytical expression for the pumped current @xmath106 , obtained for @xmath107 within the adiabatic approximation @xcite . for different pulse lengths @xmath102 and @xmath107 and @xmath109 . parameters used : @xmath110 , @xmath111 , @xmath112 , @xmath113 and @xmath114 ( number of auxiliary modes ) . dots denote the adiabatic limit @xcite . ] here , due to the l / r symmetry , there is no net charge flowing through the qd . at any given time both leads pump the same amount of charge in or out . in the driving scheme defined by eq . , the qd is initially nearly empty . at @xmath115 , the resonance energy favors an almost full occupation . for very slow pumping , large @xmath116 , the current @xmath106 depends only on the resonance energy @xmath14 : as the resonance dives into the fermi sea , the qd is loaded with charge and the process is reversed as @xmath14 starts increasing . this is no longer true when the drive is faster and @xmath116 decreases : now one observes a retardation effect , namely , the @xmath106 depends not only on the resonance position , but also on driving speed . for fast driving one needs to integrate @xmath106 over times much longer than @xmath102 to observe a vanishing net charge per pulse . the pumped currents @xmath117 characterize the time - dependent electron response to the external drive . however , in most applications one is only interested in the charge pumped per cycle @xmath118 or per pulse @xmath119 . in the latter case , @xmath119 is given as time integral over the current which we write in a symmetric way @xmath120\;.\ ] ] one of the beautiful lessons learned form the investigation of adiabatic pumping , establishes a proportionality relation between @xmath119 and the area swapped by the time - dependent driving forces in parameter space @xcite . in other words , the total charge flowing through a qd per cycle ( or per pulse ) in a _ single - parameter adiabatic pump _ vanishes . due to the constraints of single - parameter pumps , in most applications at least two parameters are used @xcite . on the other hand , by using a single - gate modulation one can realize a constrained two - parameter pump @xcite , which implies that the time - dependence of the parameters is ultimately coupled due to the modulation of only a single gate voltage . in the following we will investigate the implications of this scenario for non - adiabatic pumping . ( upper row ) and the decay widths @xmath121 ( lower row , blue / full line ) and @xmath122 ( lower row , red / broken line ) for three different cases : a ) @xmath123 , b ) @xmath124 , and c ) @xmath125 . the dotted lines indicate the chemical potential in the reservoirs and the shaded area shows the times when the resonance energy is below the chemical potential . ] specifically , let us consider voltage pulses of the form @xmath126 \;.\ ] ] here @xmath102 measures the characteristic pulse time , whereas @xmath8 governs the time the pulse sets in . the numerical factor @xmath127 ensures that @xmath102 is the full width at half maximum of the pulse , which simplifies the following discussion . by tuning the delay one can conveniently switch between a single parameter ( @xmath128 ) and a two parameter setup ( @xmath129 ) . further , the time - dependence of the resonance energy and the coupling strengths ( decay widths ) are chosen as @xmath130\;,\\ \gamma_{\rm l}(t ) & = \frac{\gamma_0}{2 } \left [ 1 + s(t , \delta_{\rm l } ) \right]\;.\end{aligned}\ ] ] this choice takes into account that the coupling strengths depend exponentially on the gate voltage @xcite . the constraint is imposed by setting @xmath131 and the specific value of @xmath132 . for this driving parameterization , the resonance and the decay widths are @xmath133 and @xmath134 , respectively , for both asymptotic limits of @xmath135 . in the following , the parameters are taken as @xmath136 , @xmath137 , @xmath138 , and interaction energy either @xmath107 or @xmath109 . in fig . [ fig : appscheme3 ] the time dependence of @xmath139 and @xmath140 is illustrated for three cases @xmath124 and @xmath141 . as mentioned above , in each case the coupling to the right reservoir @xmath122 follows the time dependence of the resonance energy . when the latter attains its minimal value at @xmath115 , which brings the energy well below the chemical potential of the reservoirs , the coupling to the right reservoir is _ minimal_. on the other hand , the behavior of the coupling to the left reservoir can be influenced by the value of @xmath132 . for @xmath123 the _ maximum _ of @xmath121 comes before @xmath115 , while for @xmath142 it is attained after @xmath115 . in the case @xmath124 the coupling to the left reservoir is maximal simultaneously with @xmath122 being minimal at @xmath115 . in the following the response to these drivings will be investigated . vs pulse - shift @xmath132 in the long - pulse limit ( upper panel ) and at @xmath143 ( lower panel ) . the dashed lines indicate half of the non - interacting result . ] knowing the time dependence shown in fig . [ fig : appscheme3 ] one can readily predict the behavior of @xmath119 in the adiabatic limit . in this case , electron flow occurs when the resonance energy matches the chemical potential of the reservoirs . in our pulse scheme , @xmath14 equals the chemical potential at @xmath144 and @xmath145 corresponding to the onset of charging and de - charging of the qd , respectively . further , the direction of the net current is determined by the difference of the couplings to the reservoirs at these very times . for example , for @xmath123 one finds @xmath146 while charging and @xmath147 while de - charging . consequently , the net current is directed from left to right and @xmath119 is expected to be _ positive_. for @xmath142 the situation is opposite and @xmath119 should be _ finally , for @xmath124 the couplings are equal at both instants of time and the net current is vanishing . these expectations are confirmed by our results for the adiabatic regime , @xmath148 , and different values of @xmath149 , which are shown in fig . [ fig : appmonoqpphi]a . as already mentioned , one observes @xmath150 for @xmath124 ( monoparametric pumping ) . as @xmath151 begins to increase , @xmath152 increases as well . in this scenario , when the resonance energy matches the chemical potential , electrons load the dot from the left ( or right ) and later they are unloaded to the right ( or left , depending on the sign on @xmath8 ) . for larger values of @xmath151 , the left reservoir participates less in the loading or unloading of the qd and the charge per pulse vanishes accordingly . for interaction strengths @xmath153 the double occupation of the qd is suppressed and consequently , in the adiabatic regime , @xmath119 is half the value of @xmath119 for the non - interacting case . the numerical results indicate that within the hubbard i approximation , @xmath108 does not introduce new time scales to the problem for @xmath148 , and its major effect is to correct the spin degeneracy factor in the equations for the @xmath107 case . none of the aforementioned features are observed in the non - adiabatic pumping regime . figure [ fig : appmonoqpphi]b shows , for example , that , for short pulses there is no simple relation between @xmath119 for @xmath107 and for @xmath108 . moreover , compared to the adiabatic regime the charge per pulse can be substantially larger in this regime . unfortunately , the behavior of @xmath119 in this regime is not as easily predicted in general , since the evolution of the parameters @xmath154 , @xmath121 , @xmath155 after the onset of loading and unloading has to be taken into account . this is because in the non - adiabatic regime the qd charging and de - charging is delayed with respect to the external system changes , as it was shown in sec . [ sec : comparison ] . taking , for example , the case @xmath124 one finds from fig . [ fig : appscheme3]b , that @xmath146 while the resonance energy is below the chemical potential and charging occurs . during the de - charging , when @xmath156 , one finds @xmath157 . consequently , the current is expected to flow mainly from left to right , which leads to a positive charge per cycle . this is confirmed by the results shown in fig . [ fig : appmonoqpphi]b . the quantitative behavior depends on the precise magnitude of the delay , which is determined by the pulse length . however , from the analysis presented above and for sufficiently short pulses one concludes that @xmath119 has to be positive independent of @xmath132 . the interesting implications of this result will be discussed at the end of this section . finally , in fig . [ fig : appmonoqp ] we summarize and corroborate the discussion of the non - adiabatic pumping . it shows the charge pumped due the pulse as a function of pulse length @xmath116 in the non - interacting ( @xmath107 ) and the coulomb blockade regime ( @xmath158 ) . in the latter case @xmath159 for all pulse lengths . as discussed above , the amount of pumped charge @xmath119 depends very strongly on the value of @xmath116 . in the limit of large pulse lengths , @xmath119 approaches the respective adiabatic value , while for @xmath160 the charge per pulse vanishes . moreover , one finds that @xmath119 is indeed positive for small pulse lengths . this has the intriguing consequence that the charge per pulse can change its sign sweeping from short to long pulses . this is shown in fig . [ fig : appmonoqp]b for @xmath125 , where @xmath119 is negative in the adiabatic regime . a more general and quantitative analysis of this effect is certainly desirable , but beyond the scope of this article . it may lead , however , to interesting new applications . it is also worth to mention , that by changing the pumping parameters it is possible to optimize the charge pumped per pulse and in particular to find situations where @xmath161 , which may be very interesting for metrology purposes @xcite . we presented a new method to analyze non - adiabatic charge pumping through single - level quantum dots that takes into account coulomb interactions . the method is based on calculating the time evolution for single - electron density matrices . the many - body aspects of the problem are approximated by truncating the equations of motion one order beyond mean field . the novelty is the way the time evolution is treated : by means of an auxiliary - mode expansion , we obtain a propagation scheme that allows for dealing with arbitrary driving parameters , fast and slow . the method presented in this paper can be applied to a wide range of coupling parameters @xmath162 , provided one avoids the kondo regime . hence , we are not restricted to the weak coupling limit where @xmath119 , the charge pumped per pulse , is rather small . the presented results for single - pulses are also valid for pulse trains , provided the time between the pulses is sufficiently long . one can expect to find qualitatively new and interesting effects by decreasing the time lag . the propagation scheme allows , in principle , for studying transient effects . in addition , by propagating over a periodic sequence of pulses it constitutes a complementary approach to the more familiar periodic driving . in this regard , our propagation scheme has the potential to be a valuable tool and provide deeper insights into non - adiabatic quantum pumps . this work is supported in part by cnpq ( brazil ) . here we motivate the rules given in sect . [ sec : prop ] . to begin with we introduce correlation functions , which can be approximated by finite sums . then we write the current matrices in terms of these finite sums . as we will show later , in the present case we have to consider the following reservoir correlation function @xmath163 \right\ } \;,\label{eq : corrfuncint}\end{aligned}\ ] ] where the line - width function @xmath164 is defined as usual @xcite @xmath165 in the second line we have used the wide - band limit . in order to perform the energy integration in eq . we expand the fermi function @xmath166 as a finite sum over simple poles @xmath167 with @xmath168 and @xmath169 . instead of using the matsubara expansion @xcite , with poles @xmath170 , we use a partial fraction decomposition of the fermi function @xcite , which converges much faster than the standard matsubara expansion . in this case the poles @xmath171 are given by the eigenvalues @xmath172 of a @xmath173 matrix @xcite . the poles are arranged such that all poles @xmath174 ( @xmath175 ) are in the upper ( lower ) complex plane . as in the matsubara expansion all poles have the same weight . employing the expansion given by eq . , one can evaluate the energy integrals by contour integration in the upper or lower complex plane depending on the sign of @xmath176 . thereby , the integral in eq . becomes a ( finite ) sum of the residues . for @xmath177 one gets [ eq : corrfuncexp ] @xmath178 with the auxiliary modes for reservoir @xmath25 given by @xmath179+x_p/\beta.\ ] ] here , @xmath180 is the chemical potential and @xmath181 is due to the time - dependent single - particle energies @xmath182 of the reservoir hamiltonian [ eq.([eq : reshamilop ] ) ] . the set of equations and can be formally solved . in order to write down these solutions we define the following functions # 1^#1 @xmath183 } \;,\\ g^u_s ( t , t ' ) & \equiv \exp { - { \mathrm{i}}\int^t_{t ' } dt '' \left[\varepsilon_s(t '' ) + u - { \mathrm{i}}\frac{\gamma(t'')}{2}\right ] } \;.\end{aligned}\ ] ] with these definitions the formal solution of eq . reads @xmath184\,,\end{aligned}\ ] ] where we have assumed @xmath185 , corresponding to our choice of an initially uncorrelated density matrix ( see sec . [ sec : setup ] ) . an analogous equation holds for @xmath85 , again with @xmath186 . we can combine these two expressions to get for the second part of the current matrix @xmath187 where we have used the definition of the correlation function @xmath188 given by eq . . finally , by means of the expansion of the correlation functions we obtain an expansion of the current matrices @xmath189 which resembles the last rule of eqs using the explicit expression for @xmath90 and taking the time derivative one can easily verify the first two rules given by eqs . . similarly , one also obtains an expression for @xmath190 , which reads @xmath191 the time derivative of this expression is given by eq . . if neither the couplings @xmath42 ( and thus @xmath7 ) nor the levels @xmath192 or @xmath44 depend on time the level occupations @xmath193 and the currents @xmath194 converge to stationary values . these values can be obtained by setting all time derivatives in the respective equations of motion to zero . in order to simplify the notation we characterize the stationary values by omitting the time argument . within the hartree - fock approximation [ sec . [ sec : hfapp ] ] we get from eq . @xmath196 plugging this into eq . , changing the @xmath197 summation into an integral over @xmath198 and using the definition , we get for the wide - band limit [ eq . ] @xmath199 equation is a non - linear equation for @xmath193 and has to be solved numerically . we obtain the stationary conditional current @xmath200 for the hubbard i approximation [ sec . [ sec : hiapp ] ] from eq . as @xmath201 this expression can be used for the stationary @xmath62 in eq . @xmath202 } \end{aligned}\ ] ] we use eq . and the definition and finally get for the occupation the following integral @xmath203\\ a'(\varepsilon ) & \equiv \frac{1}{(\varepsilon{-}\varepsilon_{s})^{2}+\left(\frac{\gamma}{2}\right)^{2 } } \\ a''(\varepsilon ) & \equiv a'(\varepsilon ) \frac{u\left[4(\varepsilon{-}\varepsilon_{s})-u\right ] } { ( \varepsilon{-}\varepsilon_{s}{-}u)^{2}+\left(\frac{3\gamma}{2}\right)^{2 } } \,.\end{aligned}\ ] ] this time the equation is linear in @xmath204 and can be solved explicitly . in the limits @xmath205 and @xmath206 it is @xmath207 and @xmath208 , respectively . the former limit corresponds to non - interacting electrons and eq . gives the correct expression for the occupation @xcite . the latter case describes the situation with very strong interactions .

we study non - adiabatic charge pumping through single - level quantum dots taking into account coulomb interactions . we show how a truncated set of equations of motion can be propagated in time by means of an auxiliary - mode expansion . this formalism is capable of treating the time - dependent electronic transport for arbitrary driving parameters . we verify that the proposed method describes very precisely the well - known limit of adiabatic pumping through quantum dots without coulomb interactions . as an example we discuss pumping driven by short voltage pulses for various interaction strengths . such finite pulses are particular suited to investigate transient non - adiabatic effects , which may be also important for periodic drivings , where they are much more difficult to reveal .

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Proceed to summarize the following text: metal oxides demonstrate a variety of physical and chemical properties , sometimes in intriguing combinations . apart from being ubiquitous in nature , metal - oxide surfaces and interfaces find diverse technological applications and are being explored for potential use in future electronic devices . in particular , surfaces of magnetoelectric antiferromagnets such as possess an equilibrium surface magnetization , @xcite making them suitable for use as active layers in electrically - switchable magnetic nanostructures . @xcite the ( 0001 ) surface has been a subject of many experimental @xcite and theoretical @xcite studies , but its surface remains poorly understood . low - energy electron diffraction ( leed ) experiments for a thin ( 0001 ) film grown on a cr ( 110 ) single crystal revealed an unusual reentrant structural phase transition,@xcite in which the surface structure changes from @xmath5 to @xmath6 and back to @xmath5 under cooling from room temperature to 150 k and then further down to 100 k. the origin of these phase transitions is not understood . while the high - temperature transition may , as suggested by the leed data , @xcite be a conventional order - disorder transition , the second one is unusual in that a more symmetric phase appears at lower temperatures . the situation is further complicated by the fact that , as shown by takano _ @xcite , both phase transitions disappear for thicker films grown in a similar way . this suggests that the epitaxial strain has an important effect on the surface energetics . oxidation of the cr @xmath7 surface was investigated by leed and auger spectroscopy , @xcite and it was found that under growth conditions similar to those of ref . the thin ( 0001 ) film is subject to a tensile epitaxial strain of about 1.5% . in this paper we study the structure of the ( 0001 ) surface using first - principles electronic structure calculations and monte carlo simulations . our results suggest that the dynamics of the system is driven by the occupation of two competing surface cr sites . the system can be mapped to an ising model on a two - dimensional hexagonal lattice in external field . the first phase transition is clearly identified as a conventional ordering transition , and the theoretical transition temperature is found to be in good agreement with experiment . our calculations further reveal a strong effect of tensile epitaxial strain , which parametrically drives the system towards configurational frustration , particularly in combination with antiferromagnetic ordering . an explanation of the second phase transition is offered based on these results . the paper is organized as follows . in section [ method ] we describe the computational methods . section [ surf - struc ] presents the results on the configurational and magnetic energetics of the ( 0001 ) surface , including the identification of the competing surface cr sites , the construction of the configurational hamiltonian , the analysis of magnetic interactions , and the evaluation of the ground - state phase diagram . section [ thermo ] deals with configurational thermodynamics of the surface , and section [ sec - exper ] discusses the relation of the results to experiments . the electronic structure of the ( 0001 ) surface is presented in section [ el - str ] , and its magnetic properties in section [ surf - mag ] . section [ summary ] draws the conclusions . electronic structure calculations were performed using the projector - augmented wave method @xcite implemented in the vasp code . @xcite for the cr @xmath8 shell we employed the rotationally - invariant lsda@xmath9 method @xcite with @xmath10 ev and @xmath11 ev . this method was preferred over gga@xmath9 adopted in ref . due to its better description of the structural , electronic , and magnetic properties of bulk .@xcite different surface superstructures were modeled using supercells representing symmetric slabs with eight atomic layers of o and 16 atomic layers of cr stacked along the ( 0001 ) direction . the periodically repeating slab is separated from its image by 1.5 nm of vacuum . we considered @xmath5 , @xmath12 , @xmath13 , and @xmath6 surface supercells ( where @xmath5 corresponds to the hexagonal unit cell of bulk ) . the lateral dimensions of the unstrained supercell were fixed to the calculated equilibrium bulk values;@xcite for the strained case these values were used as a reference . apart from these constraints , the ionic positions were relaxed until the hellmann - feynman forces were converged to less than @xmath14 ev@xmath15 . the plane - wave energy cutoff was fixed to @xmath16 ev and the brillouin zone integration was performed using @xmath17-centered monkhorst - pack grids . @xcite for relaxation we used gaussian smearing of @xmath18 ev and a @xmath19-point mesh equivalent to or denser than @xmath20 for the @xmath5 surface supercell . we checked the convergence with respect to the number of @xmath19-points , the energy cutoff for the plane wave expansion , the size of the vacuum region , and the thickness of the slab . these tests indicate that the total energies are generally converged to within 1 mev . density of states ( dos ) calculations were performed using gaussian smearing of @xmath21 ev and a @xmath19-point mesh equivalent to or denser than @xmath22 for the @xmath5 supercell . the energy barriers for the thermally - activated jumping of cr ions between the two competing surface sites were calculated using the nudged elastic band method . @xcite seven images were inserted between the two energy minima , and in each image the ions were relaxed so that forces perpendicular to the reaction path were smaller than 0.05 ev / . crystalizes in the corundum structure with the @xmath23 space group . it can be viewed as a stacking of buckled honeycomb cr double layers along the ( 0001 ) direction with quasi - hexagonal closed - packed o layers in between , see fig.[slab ] . the ( 0001 ) surface is polar , and simple electrostatic arguments suggest that non - stoichiometric terminations by an o layer or by a cr double layer should lead to divergent electrostatic potential in the bulk . on the other hand , the surface can terminate in the middle of the buckled cr layer so that only half of the cr ions from this layer remain on the surface . although still polar , this termination is stoichiometric , and the electrostatic potential in the bulk is not divergent . it can therefore be expected that this termination is energetically favorable . indeed , surface termination by a single cr layer was consistent with leed @xcite and scanning tunneling microscope @xcite measurements of the ( 0001 ) surface in ultrahigh vacuum . further , first principles calculations by rohrbach _ @xcite based on the gga@xmath9 method have shown that this termination has the lowest surface energy ( compared to all others considered ) over the entire range of oxygen chemical potential where is stable . note that earlier results based on the gga method , which leads to grossly incorrect electronic and magnetic properties,@xcite were quite different.@xcite in this work we only consider the ( 0001 ) surface terminated by a single layer of cr . the location of the cr ions within the single cr terminating layer has been debated . within the double cr layer there are three possible octahedral sites , two of them being occupied in the bulk . they give rise to three nonequivalent surface sites ( a , c and d , see fig . [ slab ] ) that surface cr ions can occupy . occupation of site a corresponds to the continuation of the bulk structure . further , as pointed out by gloege _ et . al . _ , @xcite the surface cr ion can jump below the oxygen subsurface layer and occupy the empty octahedral site within the underlying cr double layer . this interstitial site is directly underneath the surface site a , and we denote it by b ( see fig . [ slab ] ) . surface terminations . gray and red spheres represent cr and o atoms , respectively.,scaledwidth=45.0% ] in order to identify the energetically preferable sites , we therefore considered four @xmath5 surface models corresponding to the exclusive occupation of sites a , b , c , or d , respectively . in all cases a significant inward relaxation was observed , as expected for a nominally polar surface . the relaxation data for models a and b are included in appendix [ app - a ] . we define the surface energy as @xmath24 here @xmath25 is the ground state energy of the slab for the given surface model with magnetic structure corresponding to bulk , @xmath26 is the ground state energy per unit cell of bulk , @xmath27 and @xmath28 are the numbers of atoms in the slab and in the bulk unit cell , and @xmath29 is the number of surface cr atoms on one side of the slab . the surface energies for the four @xmath5 surface terminations are given in table [ tab1 ] . the surface energy is the lowest when site a is occupied . occupation of sites c and d leads to much higher surface energies , and we therefore do not consider their occupation in the subsequent analysis . on the other hand , the surface energy of model b is only slightly higher than that of model a. thus , sites a and b can both be partially occupied , which can lead to non - trivial ordered terminations and phase transitions ; these issues are addressed in the following subsections . .surface energies of @xmath5 surface models with different surface sites occupied . [ cols="^,^,^,^,^",options="header " , ] [ tab4 ] the upper panel of fig . [ relax ] shows the vertical coordinate of surface cr ions occupying site a for different surface models as a function of @xmath30 defined after eq . ( [ model2 ] ) . this coordinate is referenced with respect to that of the cr ions occupying sites b averaged over different surface models . ( the subsurface o layer was used as an anchor for measuring the @xmath31 coordinate . ) for surface supercells with two inequivalent a - site cr ions their vertical coordinates were similar , and we used their average . one can see that the calculated data agree very well with eq . ( [ model2 ] ) for both unstrained and strained surfaces . of a cr ion at site a for different surface models as a function of @xmath30 defined after eq . ( [ model2 ] ) . circles ( squares ) correspond to the unstrained ( strained ) surface . solid lines are linear fits to the data . lower panel : @xmath32 as a function of @xmath33 ( see eq . ( [ ehat ] ) ) for different surface models . red ( light ) symbols correspond to the ground state , and blue ( dark ) symbols to the paramagnetic state . circles ( squares ) : data for unstrained ( strained ) surface . solid lines are fits to a quadratic function with zero constant term , see eq . ( [ ehat ] ) . from bottom to top , the curves are shifted upward by 0 , 0.01 , 0.01 and 0.03 ev , respectively.,scaledwidth=45.0% ] here we demonstrate the quality of the fit of _ ab initio _ energies to the configurational hamiltonian and explain the importance of the three - body term in eq . ( [ model3 ] ) . note that for all surface models for which _ ab initio _ energies were calculated , the a sites are equivalent ( ignoring the directionality of the bonds on the actual surface , see section [ config ] ) . in this case , the surface energy from eq . ( [ model3 ] ) can be rewritten as @xmath34 where @xmath33 is the value of @xmath30 for the a sites . setting @xmath35 , where @xmath36 and @xmath37 are the surface energies for models a ( @xmath5 ) and b ( @xmath5 ) , we can define @xmath38 in the lower panel of fig . [ relax ] we plotted @xmath32 as a function of r using the _ ab initio _ energies for all considered surface models . we included the data for both strained and unstrained surfaces using both ground state and paramagnetic surface energies . the resulting plots are very well fitted by the quadratic function with a zero constant term , demonstrating the high fidelity of the fit . the value of the parameter @xmath39 extracted from the fit ranges from @xmath40 to @xmath41 , which , as expected , is a small number . however , the relative importance of the three - body term compared to the two - body term is @xmath42 . since for the considered surface models @xmath33 varies between @xmath43 and @xmath44 , the relative importance of the three - body term is substantial and reaches 50% . in the diagrammatic cluster - expansion language one can say that although @xmath39 is small , the number of corresponding diagrams is large . h. jnsson , g. mills , and k. m. jacobsen , in nudged elastic band method for finding minimum energy paths of transitions in classical and quantum dynamics in condensed phase simulations , edited by b. j. berne , g. ciccotti , and d. f. coker ( world scientific , singapore , 1998 ) . the shaded areas in fig . [ temppd ] denote regions where the ground - state structure was not included in mfa . typically these structures have complicated orderings and should undergo first - order transitions to simpler orderings like a@xmath1b@xmath0 or ab at fairly low temperatures . therefore , with exception of the small rectangle on the left for @xmath45 , the high - temperature behavior in the shaded areas is correct within mfa .

the surface of a cr@xmath0o@xmath1 ( 0001 ) film epitaxially grown on cr undergoes an unusual reentrant sequence of structural phase transitions ( @xmath2 ) . in order to understand the underlying microscopic mechanisms , the structural and magnetic properties of the cr@xmath0o@xmath1 ( 0001 ) surface are here studied using first - principles electronic structure calculations . two competing surface cr sites are identified . the energetics of the surface is described by a configurational hamiltonian with parameters determined using total energy calculations for several surface supercells . effects of epitaxial strain and magnetic ordering on configurational interaction are also included . the thermodynamics of the system is studied using monte carlo simulations . at zero strain the surface undergoes a @xmath3 ordering phase transition at @xmath4 . tensile epitaxial strain together with antiferromagnetic ordering drive the system toward strong configurational frustration , suggesting the mechanism for the disordering phase transition at lower temperatures .

You are an expert at summarizing long articles. Proceed to summarize the following text: at energies below the electroweak scale the weak interactions are described by local four - fermi operators multiplied by effective coupling constants , the wilson coefficients . the formal framework to achieve this is the operator product expansion ( ope ) which allows one to separate the calculation of a physical amplitude into two distinct parts : the short distance ( perturbative ) calculation of the wilson coefficients and the long distance ( generally non - perturbative ) calculation of the hadronic matrix elements of the operators @xmath11 . we calculate on the lattice @xmath0 and @xmath1 . this allows us to calculate the low energy constants in chiral perturbation @xcite which , after incorporating the non - perturbative renormalization factors are then translated into @xmath12 matrix elements . the cp - pacs collaboration has also presented a very similar calculation at this meeting @xcite . we have used the wilson gauge action , quenched , at @xmath7 on a @xmath13 lattice which corresponds to an inverse lattice spacing @xmath14 . the domain wall fermion height @xmath8 and fifth dimension @xmath15 give a residual symmetry breaking @xmath16 @xcite ; 400 configurations separated by 10000 heat - bath sweeps were used in this analysis . @xmath0 matrix elements were calculated in the @xmath17 limit for 5 light quark masses @xmath18 . since the @xmath1 matrix elements vanish in the @xmath19 limit these matrix elements were calculated with non - degenerate quark propagators for 10 mass combinations subject to the constraint @xmath20 . we have also calculated the so called eye diagrams with an active charm quark for @xmath21 ( the physical charm quark is around 0.5 ) . however , the analysis for charm - in is still in progress ; in this presentation we concentrate on the case with 3-active flavors wherein charm is integrated out assuming it is very heavy . the calculation took about 4 months on 800 gflops ( peak ) . quark propagators were calculated using the conjugate gradient method with a stopping residual of @xmath22 with periodic and anti - periodic boundary conditions which amounts to doubling the lattice size in time direction . the two wall source propagators at @xmath23 and @xmath24 were fixed to coulomb gauge . for eye diagrams we employed random wall sources spread over time slices @xmath25 with 2 hits per configuration . dividing the three - point correlation functions by the wall - wall pseudoscalar - pseudoscalar correlation function yields the desired matrix elements up to a factor of @xmath26 which is determined from a covariant fit to the wall - point two - point function in the range @xmath27 for each mass . since our results unambiguously show that re@xmath4 and re@xmath5 come essentially from the current - current operators ( recall these have the largest wilson coefficients ) we will concentrate on these operators from now on . quenched chiral perturbation theory predicts @xmath28 % \end{equation } % \end{small}\ ] ] we find a quenched chiral logarithm coefficient @xmath29 which has a negligible contribution in our matrix element calculation . unlike the quenched chiral logarithms , the conventional logarithms coming from quenched chiral perturbation theory induce large corrections to the @xmath3 @xmath0 matrix element as can be seen in figure [ fig : o2_ktopi_3_2 ] . we fit these amplitudes to @xcite @xmath30 + b_2^{(27,1 ) } m_m^4 % \end{equation}\ ] ] where @xmath31 , @xmath32 , @xmath33 . the conventional chiral logarithm @xmath34 is almost linear over the mass range we have used so the fitting routine can not distinguish this term from the linear term if we leave the coefficient of the logarithm as a free parameter . since the large coefficient -6 of the logarithm makes the contribution of this term comparable to the contribution of the linear term omitting this term would change @xmath35 by almost a factor of two . the quenched chiral log contribution is very small . = -0.3 in -0.2 in @xmath2 @xmath0 matrix elements mix with @xmath36 with a power divergent coefficient @xmath37 . we define a subtracted matrix element @xmath38 by @xmath39 where @xmath40 is obtained from a linear fit to @xmath41 . for an explanation of this subtraction of the power divergence we refer the reader to @xcite . the quenched chiral perturbation theory corrections to @xmath42 our data is consistent with a linear fit @xmath43 with the slope @xmath44 determining the low energy constants @xmath45 and the intercept @xmath46 arising from residual chiral symmetry breaking . -0.1 in = -0.3 in -0.2 in we use chiral perturbation theory to compute the lattice @xmath49 matrix elements . using non - perturbative z factors we obtain the continuum matrix elements which are then multiplied by wilson coefficients to yield the physical amplitudes . we present an extrapolation to the kaon mass scale to lowest order in chiral perturbation theory and a second extrapolation which includes one loop logarithmic effects . we multiply the pseudoscalar masses by @xmath50 so that for @xmath51 the chiral perturbation theory extrapolation is increasingly accurate but we need the extrapolation at @xmath52 , the physical point . in figure [ fig : re_a0_re_a2 ] we present re@xmath47 and re@xmath48 as a function of the parameter @xmath50 . the chiral logarithm correction for re@xmath47 is large ( about @xmath53 ) . in addition one expects a large correction ( not included here ) coming from the tree level @xmath54 terms necessary to cancel the dependence on the chiral perturbation theory scale @xmath55 . -0.1 in = -0.3 in -0.2 in if the z factors and the wilson coefficients were calculated to all orders in perturbation theory the physical amplitudes that we calculate would not depend on the scale @xmath56 where the transition between the lattice and the continuum operators is made . to a good approximation this is what we find , even though at @xmath57 one expects non - perturbative effects in the z factors and at @xmath58 the discretization errors may be large . in figure [ fig : delta_i_1_2_mu_dep ] we present the ratio re@xmath47/re@xmath48 , the so called @xmath59 rule which shows a large enhancement in the @xmath60 channel in accord with experiment ( note , the chiral logarithm corrections largely cancel in the ratio so a large enhancement is seen for both extrapolation choices ) . the residual scale dependence in the physical amplitudes is slight ( see figure [ fig : delta_i_1_2_mu_dep ] ) . = -0.3 in -0.2 in in conclusion , re@xmath47 , re@xmath48 and especially the ratio re@xmath47/re@xmath48 were found reasonably close to the experimental values . we see this as an important success of the lattice method . however there were a number of major approximations in our calculation , the hardest to quantify is the use of quenched qcd . also the chiral logarithms in quenched @xmath0 , @xmath61 are not known and we have included only the logarithmic portion of the next - to - leading - order , 1-loop corrections in @xmath12 extrapolations . 6 c. bernard , _ et . * d32 * ( 1985 ) 2343 . j. noaki , _ et . hep - lat/0108013 , j. noaki , these proceedings . t. blum , _ et . al . _ , hep - lat/0007038 m. golterman and e. pallante , jhep * 08 * , 023 ( 2000 ) , hep - lat/0006029 . t. blum , _ et . al . _ ( rbc ) , hep - lat/0110075 , r. mawhinney , these proceedings . j. bijnens , phys . lett . b * 152 * ( 1985 ) 226 .

we have used domain wall fermions to calculate @xmath0 and @xmath1 matrix elements which can be used to study the @xmath2 rule for k decays in the standard model . nonlinearities in the @xmath3 matrix elements due to chiral logarithms are explored and the subtractions needed for the @xmath2 matrix elements are discussed . using renormalization factors calculated using non - perturbative renormalization then yields values for real @xmath4 and @xmath5 . we present the details of our quenched @xmath6 , @xmath7 , @xmath8 simulation , where a previous calculation showed that the finite @xmath9 chiral symmetry breaking effects are small ( @xmath10 ) .

You are an expert at summarizing long articles. Proceed to summarize the following text: interference is a major performance - limiting factor in modern wireless communication systems . many interference mitigation strategies have been proposed to improve network spectral efficiency . by allowing partial or full cooperation among interfering base stations , interference can effectively be managed and spectral efficiency can be improved . joint beamforming @xcite and network mimo ( or multicell processing ) @xcite among base stations have been shown to be effective interference mitigation techniques . however , if cooperation among transmitters is not allowed , orthogonal multiple access has been a traditional solution to interference . in a @xmath0-user single - input single - output ( siso ) interference channel ( ic ) , for example , each user can achieve @xmath2 degrees of freedom ( dof ) by time division multiple access . in recent years , interference alignment ( ia ) techniques have received much attention @xcite . the basic concept of ia is to align the interfering signals in a small dimensional subspace . in a @xmath0-user siso ic , @xmath3 dof have been shown to be achievable using ia @xcite . although ia provides a substantial asymptotic capacity gain in interference channels , there are many practical challenges for implementation . ia requires global channel state information at the transmitter ( csit ) , and imperfect channel knowledge severely degrades the gain of ia . in some channel configurations , symbols should be extended in the time / frequency domain to align interfering signals . the high computational complexity is also considered as a major challenge . to ameliorate these difficulties , many ia algorithms have been proposed such as iterative ia @xcite and a subspace ia @xcite . for interference suppression , multiuser diversity can also be exploited by opportunistic user selection for minimizing interference . the interference reduction by multiuser diversity can be enjoyed without heavy burden on global channel knowledge because user selection in general requires only a small amount of feedback @xcite . in this context , opportunistic interference alignment ( oia ) has been recently proposed in @xcite and has attracted much attention . in a 3-transmitter @xmath4 mimo interfering broadcast channel ( ibc ) , the authors of @xcite proved that @xmath5 ( where @xmath6 $ ] ) dof per transmitter is achievable when the number of users scales as @xmath7 . in @xcite and @xcite , a @xmath0-transmitter @xmath8 simo ibc and a @xmath0-transmitter @xmath9 simo ibc have been studied , respectively . for simo interfering multiple access channel ( imac ) constituted by @xmath0-cell uplink channels with @xmath10 transmit antennas and single antenna users , the authors of @xcite showed that @xmath11 dof are achievable when the number of users scales as @xmath12 . in these schemes , user dimensions are used to align the interfering signals ; each transmitter opportunistically selects a user whose interfering signals are most aligned among the users associated with the transmitter . contrary to the conventional opportunistic user selection techniques @xcite , the oia scheme exploits the multiuser dimensions for interference alignment . in this paper , we investigate the optimal role of multiuser diversity for the target dof in the ibc with @xmath0-transmitters and generalize the results of @xcite@xcite . for the @xmath0-transmitter simo ibc , each transmitter selects and serves a single user in its user group consisting of @xmath13 users . once after @xmath0-transmitters select their serving users , a @xmath0-user ic is constructed . each user has @xmath14 antennas less than or equal to the number of interferers , i.e. , @xmath15 . thus , without help of multiuser diversity , interference at each user can not be perfectly removed so that the achievable rate of each transmitter goes to zero as signal - to - noise ratio ( snr ) increases . consequently , the achievable dof per transmitter becomes zero . however , non - zero dof per transmitter is achievable by exploiting multiuser diversity as the number of users increases . since opportunistic user selection can focus on either enhancing the desired signal or decreasing interference , non - zero dof can be obtained by properly enhancing the desired signal strength and reducing interference via user selection . that is , the non - zero dof @xmath16 comprises a dof gain term @xmath17 from the desired signal and a dof loss term @xmath18 caused by interference such that @xmath19 , and the target dof @xmath16 can be obtained by a proper combination of @xmath20 and @xmath21 . however , many questions remain unsolved ; what is the feasible and optimal combination of @xmath22 for the target dof @xmath23 and what is the sufficient number of users for the target dof achieving strategy . we answer these fundamental questions and analytically investigate how the multiuser dimensions can be optimally exploited for the target dof in the ibc . specifically , from geometric interpretation of interfering channels , we define an interference alignment measure that indicates how well interference signals are aligned at each user . using the interference alignment measure , we first consider the @xmath0- transmitters simo ibc and show that the dof gain term @xmath20 can be achieved if the number of users scales in terms of transmit power @xmath24 as @xmath25 and the dof loss term can be reduced to @xmath21 if the number of users scales as @xmath26 . from these results , we find the optimal strategy of exploiting multiuser diversity for the target dof @xmath16 in terms of the required number of users ; the optimal target dof achieving strategies @xmath27 are @xmath28 and @xmath29 for the target dof @xmath30 $ ] and @xmath31 , respectively . we also investigate how the optimal target dof achieving strategy @xmath27 can be realized by practical user selection schemes . then , we extend our results to the @xmath0-transmitter mimo ibc where each transmitter has @xmath32 multiple antennas and serves multiple users with @xmath14 receive antenna each . our generalized key findings are summarized as follows : * for the target dof @xmath33 $ ] , @xmath34 is the optimal target dof achieving strategy that minimizes the required number of users . that is , the multiuser dimensions should be exploited to make the dof loss @xmath35 . the sufficient number of users for this strategy scales like @xmath36 . * for the target dof @xmath37 , @xmath38 is the optimal target dof achieving strategy which minimizes the required number of users . that is , the multiuser dimensions should be exploited to make the dof loss term zero as well as to make the dof gain term @xmath16 . the sufficient number of users for this strategy scales like @xmath39 . the rest of this paper is organized as follows . in section ii , we describe the system model . in section iii , a geometric interpretation of interfering channels is provided , and the interference alignment measure is defined . section iv derives the optimal strategies of achieving the target dof in terms of the required number of users . in section v , we show how various practical user selection schemes exploit multiuser diversity for the target dof and discuss their optimality to achieve the target dof . the system model is extended for the mimo ibc in section vi . numerical results are shown in section vii , and we conclude our paper in section viii . _ notations _ throughout the paper , we use boldface to denote vectors and matrices . the notations @xmath40 , @xmath41 , and @xmath42 denote the conjugate transpose , the @xmath43th largest eigenvalue , and the eigenvector of matrix @xmath44 corresponding to the @xmath43th largest eigenvalue . for convenience , the smallest eigenvalue , the largest eigenvalue , and the eigenvectors corresponding eigenvectors of @xmath44 are denoted as @xmath45 , @xmath46 , @xmath47 , and @xmath48 , respectively . also , @xmath49 , @xmath50 , and @xmath51 indicate the @xmath52 identity matrix , the @xmath53-dimensional complex space , and the set of @xmath54 complex matrices , respectively . our system model is depicted in fig . [ fig : system_model ] . the system corresponds to the interfering broadcast channel ( ibc ) of which capacity is unknown . there are @xmath0 transmitters with @xmath32 transmit antennas each , and each transmitter has its own user group consisting of @xmath13 users with @xmath14 antennas each . first , each transmitter is assumed to have a single antenna , i.e. , @xmath55 , and serves a single user selected in its user group so that @xmath0-transmitter simo ic is opportunistically constituted . the system model with multiple transmit antennas ( i.e. , @xmath56 ) becomes statistically identical with the single transmit antenna model if each transmitter uses a random precoding vector . in section [ sec : extension ] , we extend our system model to the @xmath0-transmitter mimo ibc where each transmitter with multiple antennas serves the multiple users through orthonormal random beams . in this paper , we focus on the cases that the number of receive antennas is smaller than the number of transmitters , i.e. , @xmath57 . otherwise ( i.e. , if @xmath58 ) , each user can suppress all interfering signals through zero - forcing like schemes so that dof one is trivially guaranteed at each transmitter . we also assume that collaboration or information sharing among the transmitters is not allowed . since the user selection at each transmitter is independent of the other transmitters , we only consider the achievable rate of the first transmitter without loss of generality . note that the average achievable rate per transmitter will be same if the configurations of the transmitters are identical . at the first transmitter , the received signal at the @xmath53th user denoted by @xmath59 is given by @xmath60 where @xmath61 is the vector channel from the @xmath62th transmitter to the @xmath53th user whose elements are independent and identically distributed ( i.i.d . ) circularly symmetric complex gaussian random variables with zero means and unit variance . also , @xmath63 is the transmitted signal using random gaussian codebook from the @xmath62th transmitter such that @xmath64 , where @xmath24 is the power budget at each transmitter . also , @xmath65 is a circularly symmetric complex gaussian noise with zero mean and an identity covariance matrix , i.e. , @xmath66 . assuming perfect channel estimation at each receiver , the channel state information @xmath67 is available at the @xmath53th user . the received signal is postprocessed at each user using multiple receive antennas . let @xmath68 be the postprocessing vector of the @xmath53th user such that @xmath69 . then , the received signal after postprocessing becomes @xmath70 to aid user selection at the transmitter , each user feeds one scalar value back to the transmitter . various user selection criteria and corresponding feedback information will be discussed in the following sections . since no information is shared among the transmitters , each transmitter independently selects a single user based on the collected information . let @xmath71 be the index of the selected user at the first transmitter . then , the average achievable rate of the first transmitter is given by @xmath72 we decompose @xmath73 into two terms @xmath74 and @xmath75 such that @xmath76 , which are given , respectively , by @xmath77 then , the achievable dof of the first transmitter becomes @xmath78 we call @xmath79 and @xmath80 as _ dof gain term _ and _ dof loss term _ , respectively . the achievable rate of each transmitter depends on the number of users because multiuser dimensions are exploited for a rate increase . when there are fixed number of users , the achievable rate of each transmitter will be saturated in the high snr region due to interferences because the number of receive antennas at each user is smaller than the number of total transmitters . consequently , the first transmitter can not obtain any dof , i.e. , @xmath81 in this case , both the dof gain term and the dof loss term become one , i.e. , @xmath82 on the other hand , when the transmit power is fixed , the achievable rate of the selected user can increase to infinity as the number of users increases , i.e. , @xmath83 then , how much dof can be achieved when both the number of users and the transmit power increase ? obviously , non - zero dof can be obtained by exploiting multiuser dimensions , and the achievable dof @xmath84 \label{eqn : opportunistic_dof}\end{aligned}\]]will depend on the increasing speeds of @xmath13 and @xmath24 . in this case , dof @xmath85 at the first transmitter comprises the dof gain term @xmath86 and the dof loss term @xmath87 such that @xmath88 , i.e. , @xmath89 we call @xmath90 as a _ target dof achieving strategy _ if @xmath91 for the target dof @xmath16 . since each strategy requires different user scaling , we need to find the optimal dof achieving strategy that exploits multiuser diversity most efficiently , i.e. , which requires the minimum user scaling . for the _ target dof _ per transmitter @xmath92 , we find the optimal target dof achieving strategy @xmath27 satisfying @xmath93 and derive the required user scaling . note that the definition of dof in this paper is extended from the conventional definition of dof in order to properly capture multiuser diversity gain in terms of achievable rate . achievable dof defined in depends on increasing speeds of @xmath13 and @xmath24 ; and can have non - zero values even larger than one if the number of users properly scales with the transmit power . from the definitions of the rate gain term and the rate loss term given in and , respectively , the strategies which achieve the target dof @xmath16 are given by @xmath94 the following lemma shows that we do not need to consider all of the candidate strategies in but take into account only a subset of to find the optimal target dof achieving strategy . [ lemma : dof_terms_range ] for any non - negative target dof , the optimal dof achieving strategy is in the set @xmath95,~d_1-d_2=d \big\}. \label{eqn : dof_term_range}\end{aligned}\ ] ] at each channel realization , the achievable dof has the form of @xmath96 where @xmath97 and @xmath98 are its own signal power and the interfering signal power at the selected user , respectively . since the function is an increasing function of @xmath99 and a decreasing function of @xmath100 , for an increase of , the multiuser dimension should be used for increasing @xmath99 , for decreasing @xmath100 , or mixture of them . this fact results in . lemma 1 provides a basic guideline of using the multiuser dimension ; multiuser diversity should not be used for either decreasing dof gain term or increasing dof loss term . since the optimal target dof achieving strategy is obtained in the reduced set of candidate strategies , we consider the dof gain term larger than one and dof loss term smaller than one , i.e. , @xmath101 and @xmath102 $ ] , in the latter parts of this paper . in our system model , each user suffers from @xmath1 interfering channels which is larger than or equal to the number of receive antennas , i.e. , @xmath103 . since the interfering channels are isotropic and independent of each other , they span @xmath14-dimensional space . thus , the whole signal space at the receiver is corrupted by interfering signals , and hence the dof loss term becomes one if no effort is made to align interfering signals . on the other hand , the dof loss can be reduced by aligning interfering signals in smaller dimensional subspace . for example , if the interfering signals are perfectly aligned in @xmath104-dimensional subspace , they can be nullified by postprocessing so that we can make the dof loss zero . the transmitter can exploit the multiuser dimensions to align interfering signals by simply selecting a user whose interfering channels are most aligned . thus , each user needs to measure how much the interfering channels are aligned in @xmath104-dimensional subspace at the receiver . we call this measure as the _ interference alignment measure_. in this section , we geometrically interpret the interfering channels and define the interference alignment measure at each user . the interference alignment measure will be used for computing the reducible dof loss via multiuser diversity in section iv . let @xmath105 be the surface of the @xmath14-dimensional unit hypersphere centered at the origin , i.e. , @xmath106 for an arbitrary unit vector @xmath107 and an arbitrary non - negative real number @xmath108 , we can divide @xmath105 into two parts , @xmath109 and @xmath110 , given by @xmath111 when @xmath112 , two parts @xmath113 and @xmath114 are represented in fig . [ fig : sphere_s1s2 ] . let @xmath115 be the surface area of @xmath116 for @xmath117 . the surface area of an @xmath14-dimensional complex unit hypersphere is given by @xmath118 , and it was shown that @xcite @xmath119 which is invariant with @xmath120 . therefore , we obtain @xmath121 from the relationship @xmath122 . from this fact , we obtain the following lemma . [ lemma : p_n_lambda ] let @xmath123 be independent and isotropic unit vectors in @xmath124 . for an arbitrary unit vector @xmath125 and @xmath126 $ ] , the probability that @xmath127 contains @xmath128 becomes @xmath129 = \left(1-(1-\lambda)^{n_r-1}\right)^m , \label{eqn : prob_m_lambda}\end{aligned}\ ] ] which is invariant with @xmath120 . from the ratio of @xmath130 and @xmath131 , we obtain @xmath132 = \frac{a(s_2(\mathbf{c } , \lambda))}{a(s_0 ) } = 1-(1-\lambda)^{n_r-1 } , \quad \forall i.\end{aligned}\ ] ] since @xmath123 are independent of each other , it is satisfied that @xmath133 & = \pr[\mathbf{g}_1 \in s_2 ( \mathbf{c } , \lambda)]^m , { \nonumber}\end{aligned}\ ] ] which is given in . in this subsection , we define the interference alignment measure at each user . the dof loss is determined by how much the interfering channels are closely aligned in @xmath104-dimensional subspace . only if interfering channels are perfectly aligned in @xmath104-dimensional subspace , we can have zero dof loss . the interference alignment measure is used for computing the dof loss at each user . let @xmath134 be the @xmath135 normalized interfering channels at a user and @xmath136 be the interference alignment measure among them . consider the following optimization problem : @xmath137.{\nonumber}\end{aligned}\ ] ] from the definition of @xmath138 given in , this problem is equivalent to @xmath139,{\nonumber}\end{aligned}\ ] ] which can be solved by linear programming @xcite . let @xmath140 be the solution of the above problem . then , @xmath141 has the smallest surface area among all @xmath142 containing @xmath134 . using @xmath143 , we can divide an @xmath14-dimensional space into two subspaces which are the one - dimensional subspace spanned by @xmath143 and the @xmath104-dimensional complementary subspace denoted by @xmath144 . if there exists @xmath143 such that @xmath145 , it is satisfied that @xmath146 and @xmath147 , and hence @xmath148 becomes zero . in this case , we can say that the interfering channels are perfectly aligned in @xmath104-dimensional subspace in @xmath124 . note that @xmath149 is an @xmath104-dimensional subspace orthogonal to @xmath150 , and @xmath151 is the @xmath14-dimensional complex hypersphere , @xmath105 . when @xmath148 is the smaller , the vectors are the more aligned in the @xmath104-dimensional subspace , @xmath144 . thus , we will use @xmath152 as an _ interference alignment measure _ to quantify how much the interfering channels are closely aligned in an @xmath104-dimensional subspace , i.e. , @xmath153).\label{eqn : lambda*}\end{aligned}\ ] ] in other words , we use the mini - max distance of the interfering channels from an @xmath104-dimensional subspace . in fig . [ fig : iam_concept ] , the interference alignment measure is geometrically represented . the more the interfering channels are aligned , the smaller the interference alignment measure becomes . since the interference alignment measure is obtained from the optimization problem , the exact distribution is difficult to find . instead , we obtain the lower bound for the cumulative distribution function ( cdf ) of the interference alignment measure in the following lemma . [ lemma : prob_bar_d_m ] when @xmath154 , the probability that the interference alignment measure @xmath136 is smaller than @xmath155 $ ] is lower bounded on @xmath156 } \ge \left(1-(1-\lambda)^{n_r-1}\right)^{k - n_r}. \label{eqn : am1_prob}\end{aligned}\ ] ] we consider two events : @xmath157 where @xmath158 is the @xmath14-dimensional unit vector such that @xmath159 . by the definition of the interference alignment measure given in , @xmath160 is true whenever @xmath161 is true , equivalently , @xmath162 } \ge { \mathrm{pr}\left[\mathrm{(e2)}\right]}$ ] . the probability of @xmath161 is obtained by @xmath163 } & = { \mathrm{pr}\left[s_2(\bar{\mathbf{c } } , \lambda ) \supset \{\mathbf{g}_1 , \ldots , \mathbf{g}_{k-1}\}\right]}{\nonumber\\}&\stackrel{(a)}{= } { \mathrm{pr}\left[s_2(\bar{\mathbf{c } } , \lambda ) \supset \{\mathbf{g}_{n_r } , \ldots , \mathbf{g}_{k-1}\}\right]}{\nonumber\\}&\stackrel{(b)}{= } \left(1-(1-\lambda)^{n_r-1}\right)^{k - n_r},\end{aligned}\ ] ] where the equality @xmath164 is from the definition of @xmath158 such that @xmath165 . also , the equality @xmath166 holds from lemma [ lemma : p_n_lambda ] and from the fact that @xmath158 is independent of @xmath167 . thus , we obtain @xmath168 } \ge \left(1-(1-\lambda)^{n_r-1}\right)^{k - n_r}.\end{aligned}\ ] ] the remaining question is how much we can reduce the interference alignment measure via user selection . in the first user group , the @xmath53th user has @xmath1 interfering channels , @xmath169 . the interference alignment measure at the @xmath53th user can be written by @xmath170 where @xmath171 is the normalized interfering channel , i.e. , @xmath172 . thus , the achievable smallest interference alignment measure via user selection is given by @xmath173 obviously , the smallest interference alignment measure will decrease as the number of users increases . in the following lemma , we find the relationship between and the number of total users ( i.e. , @xmath13 ) . [ lemma : lambda_n1 ] when there are @xmath13 users , the expectation of the smallest interference alignment measure is upper bounded on @xmath174 < n^{-\frac{1}{k - n_r}}.\label{eqn : am1_bound}\end{aligned}\ ] ] the complementary cdf of is bounded on @xmath175 } & \qquad={\mathrm{pr}\left[\mathfrak{q } \big ( \tilde{\mathbf{h } } _ { n,2 } , \ldots , \tilde{\mathbf{h}}_{n , k}\big ) \ge \lambda \textrm{~for all~ } n\right]}{\nonumber\\}&\qquad=\prod_{n=1}^n{\mathrm{pr}\left [ \mathfrak{q } \big ( \tilde{\mathbf{h } } _ { n,2 } , \ldots , \tilde{\mathbf{h}}_{n , k}\big ) \ge \lambda\right]}{\nonumber\\}&\qquad=\left(1-{\mathrm{pr}\left [ \mathfrak{q } \big ( \tilde{\mathbf{h } } _ { n,2 } , \ldots , \tilde{\mathbf{h}}_{n , k}\big ) \le \lambda\right]}\right)^n{\nonumber\\}&\qquad\stackrel{(a ) } { < } \big[1- ( 1-(1 - \lambda)^{n_r-1})^{k - n_r}\big]^n , \label{eqn : prob_bound}\end{aligned}\ ] ] where @xmath176 $ ] , and the inequality @xmath164 holds from lemma [ lemma : prob_bar_d_m ] . using this bound , we obtain as @xmath177 & = \int_0 ^ 1 { \mathrm{pr}\left[\min_n \mathfrak{q } \big ( \tilde{\mathbf{h } } _ { n,2 } , \ldots , \tilde{\mathbf{h}}_{n , k}\big)\ge \lambda\right ] } d \lambda { \nonumber\\}&\le \int_0 ^ 1 \big[1- ( 1 - ( 1- \lambda ) ^{n_r-1})^{k - n_r}\big]^n d \lambda{\nonumber\\}&\stackrel{(a)}{\le } \int_0 ^ 1 \big[1- ( 1 - ( 1- \lambda))^{k - n_r}\big]^n d \lambda{\nonumber\\}&\stackrel{(b)}{= } \frac{1}{k - n_r}\beta\left(\frac{1}{k - n_r } , n+1\right){\nonumber\\}&\stackrel{(c)}{= } \frac{\gamma\left(1+\frac{1}{k - n_r}\right)\gamma(n+1 ) } { \gamma\left(n+ 1 + \frac{1}{k - n_r}\right ) } { \nonumber\\}&\stackrel{(d ) } { < } n^{-\frac{1}{k - n_r}},{\nonumber}\end{aligned}\ ] ] where the inequality @xmath164 is due to @xmath178 for @xmath108 , and the equality @xmath166 holds from the representation of beta function @xcite @xmath179 the equality @xmath180 comes from the definition of the beta function @xmath181 and the property of the gamma function @xmath182 . in the right - hand - side of the equality @xmath180 , it holds @xmath183 because @xmath184 for @xmath185 . also , it is satisfied that @xmath186 where @xmath187 is from the gautschi s inequality @xcite given by @xmath188 with @xmath189 and @xmath190 . thus , the inequality @xmath191 holds . in this section , we derive the optimal strategies of exploiting multiuser diversity for the target dof @xmath16 . we first decompose the target dof @xmath16 into the dof gain term @xmath20 and the dof loss term @xmath21 such that @xmath192 , and find the required user scalings for @xmath20 and @xmath21 , respectively . then , the optimal target dof achieving strategy is derived by determining the optimal combination @xmath193 which requires the minimum user scaling for the target dof @xmath16 . in this subsection , we find the required user scaling to reduce the dof loss . via user selection , the rate loss term given in can be minimized by @xmath194 . \label{eqn : rloss_min}\end{aligned}\ ] ] this value is upper bounded on @xmath194 & \stackrel{(a)}{= } \mathbb{e}_{\vert\mathbf{h}\vert , \tilde{\mathbf{h } } } \left[\min_{n,\mathbf{v}_n } ~\log_2\left(1 + p \sum_{k=2}^{k } \vert\mathbf{h}_{n , k}\vert^2\vert \mathbf{v}_{n}^\dagger \tilde{\mathbf{h}}_{n , k } \vert^2\right ) \right]{\nonumber\\}&\stackrel{(b)}{\le } \mathbb{e}_{\tilde{\mathbf{h } } } \left[\min_{n,\mathbf{v}_n } ~\mathbb{e}_{\vert\mathbf{h}\vert}\log_2\left(1 + p \sum_{k=2}^{k } \vert\mathbf{h}_{n , k}\vert^2\vert \mathbf{v}_{n}^\dagger \tilde{\mathbf{h}}_{n , k } \vert^2\right ) \right]{\nonumber\\}&\stackrel{(c)}{\le } \mathbb{e}_{\tilde{\mathbf{h } } } \left[\min_{n,\mathbf{v}_n } ~\log_2\left(1 + n_rp \sum_{k=2}^{k } \vert \mathbf{v}_{n}^\dagger \tilde{\mathbf{h}}_{n , k } \vert^2\right ) \right]{\nonumber\\}&\stackrel{(d)}{\le } \mathbb{e}_{\tilde{\mathbf{h } } } \left[\min_{n } ~\log_2\left(1 + n_rp ( k-1)\mathfrak{q}(\tilde{\mathbf{h}}_{n,2 } , \ldots , \tilde{\mathbf{h}}_{n , k } ) \right)\right]{\nonumber\\}&\stackrel{(e)}{\le } ~\log_2\left(1 + n_rp ( k-1 ) \mathbb{e}_{\tilde{\mathbf{h}}}\left [ \min_{n } \mathfrak{q}(\tilde{\mathbf{h}}_{n,2 } , \ldots , \tilde{\mathbf{h}}_{n , k } ) \right]\right){\nonumber\\}&\stackrel{(f)}{\le } ~\log_2\left(1 + n_rp ( k-1 ) n^{-\frac{1}{k - n_r}}\right ) , \label{eqn : rloss_bound}\end{aligned}\ ] ] where the equality @xmath164 is obtained by decomposing the channel vector into direction and magnitude independent of each other such that @xmath195 . the inequality @xmath166 holds because the minimum of the average is larger than the average of the minimum . the inequality @xmath180 is from the jensen s inequality and @xmath196 . also , the inequality @xmath191 holds from the fact that @xmath197 & \le \min_{\mathbf{v}_n } \left[(k-1 ) \max_{2\le k\le k } \vert \mathbf{v}_n^\dagger \tilde{\mathbf{h}}_{n , k}\vert^2 \right]{\nonumber\\}&= ( k-1 ) \mathfrak{q } ( \tilde{\mathbf{h}}_{n,2 } , \ldots , \tilde{\mathbf{h}}_{n , k}),\end{aligned}\ ] ] where @xmath198 is the interference alignment measure at the user @xmath53 given in . the inequality @xmath187 is from the jensen s inequality , and the inequality @xmath199 holds from lemma [ lemma : lambda_n1 ] . we obtain the following theorem . [ theorem : n_scaling_dof_loss ] we can obtain the dof loss term @xmath200 $ ] when the number of users in each group scales as @xmath201 to obtain the dof loss term @xmath21 , it is enough to make satisfying @xmath202 which is achieved if @xmath203 . a tighter upper bound of the rate loss term than could exist , but the derived upper bound in enables us to compare the increasing speeds of the transmit power and the required number of users , which is the crucial factor of dof calculation . the scaling law of the required number of users obtained from , which is derived in theorem [ theorem : n_scaling_dof_loss ] , is enough to find the optimal target dof achieving strategy as shown in section [ sec : dof_achieving_strategy ] . we also find the required user scaling to increase the dof gain term . from the definition of the rate gain term given in , the maximum rate gain term obtained by user selection is @xmath204 . \label{eqn : rgain_max}\end{aligned}\ ] ] this value is lower bounded on @xmath205 & \ge\mathbb{e}\left [ \max_{n , \mathbf{v}_n } \log_2\left(1 + p \vert \mathbf{v}_{n}^\dagger { \mathbf{h}_{n,1 } } \vert^2\right)\right]{\nonumber\\}&= \mathbb{e}\left [ \max_{n } \log_2\left(1 + p \vert { \mathbf{h}_{n,1 } } \vert^2\right)\right],\label{eqn : rgain_lower_bound}\end{aligned}\ ] ] and upper bounded on @xmath205 & \le \mathbb{e}\left [ \max_{n } \log_2\left(1 + p \sum_{k=1}^{k } \vert { \mathbf{h}_{n , k } } \vert^2\right)\right]{\nonumber\\}&\le \mathbb{e}\left [ \max_{n } \log_2\left(1 + pk \max_k \vert { \mathbf{h}_{n , k } } \vert^2\right)\right].\label{eqn : rgain_upper_bound}\end{aligned}\ ] ] since all @xmath206 are i.i.d . @xmath207 random variables , for sufficiently large @xmath13 , the bounds and acts like @xcite @xmath208 & \sim \log_2(1+p\log n ) { \nonumber\\}\mathbb{e}\left [ \max_{n , k } \log_2\left(1 + pk \vert { \mathbf{h}_{n , k } } \vert^2\right)\right ] & \sim \log_2(1+pk\log ( kn)).{\nonumber}\end{aligned}\ ] ] thus , when both @xmath13 and @xmath24 are large enough , act like @xmath209 , i.e , @xmath210 \sim \log_2(p\log n ) . \label{eqn : rgain_asymptotic}\end{aligned}\ ] ] therefore , we establish the following theorem . [ theorem : n_scaling_dof_gain ] the dof gain term @xmath211 is achievable when the number of users in each group scales as @xmath212 we use . by setting @xmath213 we obtain the required user scaling for the dof gain term @xmath20 given by @xmath25 . in theorem [ theorem : n_scaling_dof_loss ] and theorem [ theorem : n_scaling_dof_gain ] , we found the required user scalings for the dof loss term @xmath21 and the dof gain term @xmath20 , respectively . in this subsection , we find the optimal target dof achieving strategy which requires the minimum user scaling . we start with the following theorem . [ theorem : scaling1 ] for the target dof up to one , the whole multiuser dimensions should be devoted to minimizing the dof loss caused by interfering signals . the optimal dof achieving strategy for the target dof @xmath214 $ ] is @xmath215 , and the corresponding sufficient user scaling is @xmath216 in theorem [ theorem : n_scaling_dof_loss ] , we have shown that the target dof @xmath217 $ ] is achievable by reducing the dof loss term with the user scaling @xmath218 . on the other hand , this user scaling can not increase the dof gain term . substituting @xmath218 into the dof gain term @xmath20 becomes @xmath219 which is the same as when there is a fixed number of users as described in . that is , any other combinations @xmath220 which achieve the target dof @xmath16 requires larger user scaling than @xmath218 , where @xmath221 since @xmath222 . therefore , the optimal target dof achieving strategy is given by @xmath215 , and the sufficient user scaling is @xmath218 . now , we derive the target dof achieving strategy when the target dof @xmath16 is greater than one . to find the optimal dof achieving strategy , we firstly find the sufficient user scaling for an arbitrary strategy @xmath90 achieving dof @xmath223 . then , we show that the optimal target dof achieving strategy for the target dof @xmath224 is @xmath225 . [ lemma : general_scaling_d>1 ] for the target dof @xmath226 , the sufficient user scaling for an arbitrary strategy @xmath90 achieving dof @xmath227 is given by @xmath228 where @xmath222 and @xmath102 $ ] . as a target dof achieving scheme , we consider a two - stage user selection scheme ; the first stage is to increase the dof gain term , and the second stage is to decrease the dof loss term . the considered two stage user selection strategy is illustrated in fig . [ fig : user_selection_2stage ] . we randomly divide total @xmath13 users into @xmath229 subgroups having @xmath230 users each such that @xmath231 . then , the user selection in each stage is performed as follows . * stage 1 : in each subgroup , a single user having the largest channel gain is selected among @xmath230 users . as a result , we have @xmath229 selected users after stage 1 . * stage 2 : among the @xmath229 users , the transmitter selects a single user to minimize the dof loss term . in stage 1 , the dof gain term @xmath20 is obtained at each selected user when @xmath232 as stated in theorem [ theorem : n_scaling_dof_gain ] . in stage 2 , we can make the dof loss term @xmath21 when @xmath233 as shown in theorem [ theorem : n_scaling_dof_loss ] . thus , the target dof @xmath226 with the strategy @xmath90 such that @xmath192 can be obtained by the user scaling @xmath234 , which is given in . from lemma [ lemma : general_scaling_d>1 ] , we obtain the optimal dof achieving strategy for the target dof @xmath235 in following theorem . [ theorem : scaling2 ] the optimal target dof achieving strategy for @xmath236 is to increase the dof gain term to @xmath16 and to perfectly eliminate the dof loss , i.e. , @xmath237 . consequently , the sufficient user scaling for target dof @xmath226 becomes @xmath238 the proof is similar to that of theorem [ theorem : scaling1 ] . from lemma [ lemma : general_scaling_d>1 ] , we can obtain the target dof @xmath226 by the strategy @xmath239 with the sufficient user scaling given in . however , this scaling can not increase the dof gain term larger than @xmath16 even when the user scaling is only used to increase the dof gain term . substituting into , we still have @xmath240 this implicates that the user scaling given in is sufficient for the strategy @xmath239 but not enough for other strategies @xmath241 as well as @xmath242 which requires the higher user scaling than that of @xmath241 , where @xmath243 $ ] . therefore , the optimal strategy for the target dof @xmath226 becomes @xmath244 . in fig . [ fig : dof_achieving_strategy ] , the optimal dof achieving strategy @xmath27 is plotted according to the target dof @xmath23 . in this section , we discuss how the optimal target dof achieving strategy can be realized by practical user selection schemes . the practical schemes considered in this section require no cooperation and no information exchange among the transmitters . for practical scenarios , we assume that each user has knowledge of channel state information ( csi ) of the direct channel and the covariance matrix of the received signal without explicit knowledge of csi of the interfering channels . that is , the @xmath53th user knows csi of its own desired channel @xmath245 and the covariance matrix of the received signal @xmath246 = \mathbf{i}_{n_r } + p\sum_{k=1}^k \mathbf{h}_{n , k } \mathbf{h}_{n , k } ^\dagger$ ] . from these values , the user @xmath53 easily obtains the interference covariance matrix denoted by @xmath247 such as @xmath248 - p\mathbf{h}_{n,1}\mathbf{h}_{n,1}^\dagger -\mathbf{i}_{n_r}.\end{aligned}\ ] ] therefore , the achievable rate at the first transmitter given in can be rewritten by @xmath249 to increase @xmath73 , various user selection schemes can be considered , but we focus on several popular techniques in the following subsections to maximize the postprocessed snr @xmath250 , to minimize the postprocessed inr @xmath251 , and to maximize the postprocessed sinr @xmath252 . in the max - snr user selection scheme , each user maximizes the postprocessed snr , and the transmitter selects the user having the maximum postprocessed snr . consequently , the postprocessed snr at the selected user becomes @xmath253 \stackrel{(a)}{= } \max_n p\vert { \mathbf{h}_{n,1}}\vert^2,\end{aligned}\ ] ] where the equality @xmath164 holds when the @xmath53th user adopts the postprocessing vector @xmath254 . thus , the selected user denoted by @xmath255 becomes @xmath256 and the desired channel gain at each user ( @xmath257 for the user @xmath53 ) should be informed to the transmitter . in the min - inr user selection scheme , each user minimizes the postprocessed inr , and the transmitter selects the user having the minimum postprocessed inr . thus , the postprocessed inr at the selected user becomes @xmath263 \stackrel{(a)}{= } \min_n \left[\lambda _ { \min } \left(\mathbf{r}_{n}\right)\right],\end{aligned}\ ] ] where the equality @xmath164 is obtained by the postprocessing vector of the @xmath53th user @xmath264 the required feedback information from the @xmath53th user is @xmath265 , and index of the selected user denoted by @xmath266 becomes @xmath267.\end{aligned}\ ] ] note that this scheme minimizes the rate loss term defined in . using the min - inr scheme , the transmitter can decrease the dof loss term while the dof gain term remains to be one . therefore , the min - inr scheme realizes the optimal target dof achieving strategy @xmath268 for the target dof @xmath30 $ ] . the required number of users by the min - inr scheme for the target dof @xmath30 $ ] scales like @xmath269 which is the required user scaling of the optimal target dof achieving strategy when the target dof is @xmath270 $ ] as shown in theorem [ theorem : scaling1 ] . the max - sinr user selection scheme is known to maximize the achievable rate at the transmitter although it requires additional complexity for postprocessing at the receivers . the achievable rate by the max - sinr scheme denoted by @xmath271 becomes @xmath272 . \label{eqn : r_sinr}\end{aligned}\ ] ] at each channel realization , the postprocessed sinr at the selected user is given by @xmath273 . \label{eqn : max_sinr}\end{aligned}\ ] ] to maximize the postprocessed sinr , the @xmath53th user adopts the postprocessing vector given by @xmath274 which is identical with the mmse - irc in @xcite . the corresponding postprocessed sinr at user @xmath53 becomes @xmath275 @xcite , and hence the selected user at the transmitter denoted by @xmath276 is given by @xmath277 [ lemma : scaling1 ] to obtain the target dof @xmath30 $ ] , the required user scaling of the max - sinr scheme is exactly the same as that of the min - inr scheme . from the fact that @xmath278 we obtain @xmath279 where the equality @xmath164 is because @xmath280 as shown in the proof of theorem [ theorem : scaling1 ] . therefore , the required user scaling for @xmath281 is exactly the same as the required user scaling for @xmath282 , equivalently , for @xmath283 . lemma [ lemma : scaling1 ] indicates that the max - sinr scheme realizes the optimal dof achieving strategy @xmath284 for the target dof @xmath30 $ ] . [ lemma : scaling2 ] the max - sinr scheme realizes the dof achieving strategy @xmath285 whenever the target dof @xmath16 is greater than 1 . since the max - sinr scheme is the optimal user selection scheme , it achieves dof @xmath226 with the user scaling @xmath286 as stated in theorem [ theorem : scaling2 ] . from the definition of , we obtain @xmath287 , and hence we have @xmath288 . as shown in , the sufficient user scaling for the max - sinr scheme to obtain the target dof @xmath16 can not increase the dof gain term larger than @xmath16 even if the whole user scaling is only devoted to increasing the dof gain term . this implicates that when we obtain the target dof @xmath226 by the max - sinr scheme with the user scaling @xmath286 , we obtain @xmath289 and have the dof gain @xmath16 at most ( i.e. , @xmath290 ) . therefore , the max - sinr scheme can only have @xmath291 if @xmath292 . for the target dof @xmath226 , the two - stage user selection scheme described in the proof of lemma [ lemma : general_scaling_d>1 ] can be adopted . more specifically , the transmitter selects the users by the max - snr scheme in the first stage . then , in the second stage , the transmitter selects a single user by the min - inr scheme or the max - sinr scheme . as shown in the proof of lemma [ lemma : general_scaling_d>1 ] , the two - stage user selection scheme can realize the optimal target dof achieving strategy for the target dof @xmath226 . in this section , we extend our system to interfering mimo bc cases . more specifically , each transmitter with @xmath32 antennas sends @xmath32 independent streams over @xmath32 orthonormal random beams using equal power allocation . similar to the user selection procedure in @xcite , each transmitter broadcasts @xmath32 orthonormal random beams , and each user feeds @xmath32 scalar values corresponding to all beams back to the transmitter . the feedback information corresponding to each stream such as snr , inr , and sinr can be easily found in a similar way to the simo case . a single user is selected for each beam , but the same user can be selected for different beams . however , it rarely occurs that multiple streams are transmitted for a single user as the number of users increases . when multiple streams are transmitted for a single user , the user is assumed to decode each stream treating the other streams as interferences . we denote the orthonormal random beams by @xmath293 which satisfies that @xmath294 and @xmath295 for all @xmath296 . we start from the following remark . let @xmath297 be the channel matrix whose elements are i.i.d . gaussian random variables . then , for an arbitrary unitary matrix @xmath298 ( i.e. , @xmath299 ) , the distributions of @xmath300 and @xmath301 are identical . since the @xmath14 columns of @xmath300 are independent and isotropic random vectors in @xmath124 , so are the @xmath14 columns of @xmath302 . owing to remark 1 , the @xmath0-transmitter mimo interfering bc is statistically identical with the @xmath303-transmitter simo interfering bc . let @xmath304 be the channel matrix from the @xmath62th transmitter to the @xmath53th user in the first user group . since the random beams satisfy that @xmath305^\dagger [ \mathbf{u}_1 , \ldots , \mathbf{u}_{n_t } ] = [ \mathbf{u}_1 , \ldots , \mathbf{u}_{n_t } ] [ \mathbf{u}_1 , \ldots , \mathbf{u}_{n_t}]^\dagger = \mathbf{i}_{n_t}$ ] , the user @xmath53 in the first user group has @xmath303 independent and isotropic channel vectors @xmath306 ~~ i\in[n_t],{\nonumber}\end{aligned}\ ] ] formed by the random beams and channel matrices from all transmitters . if the @xmath53th user in the first group is served by the @xmath43th random beam , the user has desired channel @xmath307 and the @xmath308 interfering channels , which correspond to @xmath309 inter - stream interfering channels @xmath310 and @xmath311 inter - transmitter interfering channels @xmath312 } \{\mathbf{h}_{n , k}\mathbf{u}_j\}_{k=2}^k.{\nonumber}\end{aligned}\ ] ] consequently , each random beam can be regarded as a single antenna transmitter with the transmit power @xmath313 . this fact leads to the following theorems . [ theorem : mimo_bc ] in a mimo bc where a transmitter with @xmath32 antennas supports @xmath32 users among @xmath13 users with @xmath314 antennas each , the optimal dof achieving strategy for the target dof @xmath315)$ ] is @xmath316 and requires the number of users to scale as @xmath317 . for the target dof @xmath318 , the optimal dof achieving strategy is @xmath239 and requires the number of users to scale as @xmath319 . the dof gain term @xmath320 is obtained when each stream achieves dof gain term @xmath321 . thus , with the same procedure given in section [ sec : dof_gain_scaling ] , we can easily show that the dof gain term @xmath320 is obtained when @xmath322 . on the other hand , the dof loss term @xmath323 is obtained when each stream achieves dof loss term per stream @xmath324 . as stated earlier , using @xmath32 orthonormal random beams at the transmitter each of which independently supports a single user , the mimo bc can be translated into an @xmath32-transmitter simo ic where each transmitter supports one of @xmath13 users with @xmath14 antennas . in this case , dof loss term @xmath323 , i.e. , dof loss @xmath325 per stream , is obtained when @xmath326 . therefore , we can conclude that the optimal dof achieving strategy for the target dof @xmath315)$ ] is @xmath327 and requires the number of users to scale as @xmath328 . also , for the target dof @xmath318 , the optimal dof achieving strategy is @xmath239 and requires the number of users to scale as @xmath329 . [ theorem : mimo_ibc ] consider a @xmath0-transmitter interfering mimo bc where the @xmath62th transmitter with @xmath330 antennas supports @xmath330 users among @xmath331 users with @xmath332 antennas each . at the @xmath62th transmitter , the optimal dof achieving strategy for the target dof @xmath333)$ ] is @xmath334 and requires the number of users to scale as @xmath335 . for the target dof @xmath37 , the optimal dof achieving strategy becomes @xmath239 and requires the number of users to scale as @xmath336 . the proof is similar to that of theorem [ theorem : mimo_bc ] . the @xmath62th transmitter obtains dof gain term @xmath337 when each stream obtains dof gain term @xmath338 , and the required user scaling is exactly given by @xmath339 . on the other hand , the @xmath62th transmitter obtains @xmath340 when the dof loss term per stream becomes @xmath341 . using @xmath330 orthonormal random beams at each transmitter each of which independently supports a single user , the interfering mimo bc can be translated into an @xmath342-transmitter simo ic where each transmitter supports a single user among @xmath331 users with @xmath343 antennas . thus , the @xmath62th transmitter obtains the dof loss term @xmath340 when @xmath344 . therefore , we can conclude that the optimal dof achieving strategy of the @xmath62th transmitter for the target dof @xmath333)$ ] is @xmath334 and obtained when the number of users scales as @xmath345 . also , for the target dof @xmath346 , the optimal dof achieving strategy is @xmath239 and requires the number of users to scale as @xmath336 . in this section , we first compare achievable rates of the practical user selection schemes for given number of users . then , we check if the target dof can be achievable with increasing number of users by showing achievable rates per transmitter for the practical user selection schemes . we have also considered two time division multiple access ( tdma ) schemes . in the first tdma scheme ( tdma1 ) , a single transmitter operates at each time so that @xmath2 dof is achieved at each transmitter . in the second tdma scheme ( tdma2 ) , only @xmath14 of @xmath0 transmitters operate at each time so that @xmath347 dof is achieved at each transmitter . [ fig : oia_k4_a10_n10 ] shows the achievable rates of each transmitter for various user selection schemes in ibc when there are 4 transmitters and each transmitter has 10 users with three receive antennas each . it is confirmed that the achievable rates are saturated in the high snr region and the achievable dof per transmitter becomes zero for the fixed number of users . now , we show that the target dof can be achievable if the number of users properly scales . in fig . [ fig : oia_k4_a10_kn ] , the number of users scales as @xmath348 , i.e. , @xmath349 for the target dof one . specifically , two user scaling @xmath350 and @xmath351 are considered , and other configurations except the number of users are the same as those in fig . [ fig : oia_k4_a10_n10 ] . fig . [ fig : oia_k4_a10_kn ] verifies that the min - inr and the max - sinr schemes achieve dof one per transmitter as predicted in theorem [ theorem : scaling1 ] and lemma [ lemma : scaling1 ] . in fig . [ fig : oia_k4_a10_n_powerp ] , we consider two different user scaling @xmath352 and @xmath353 from those in fig . [ fig : oia_k4_a10_kn ] . according to theorem [ theorem : scaling1 ] and lemma [ lemma : scaling1 ] , the achievable dof at each transmitter by either the max - sinr scheme or the min - inr scheme is @xmath16 when the number of users scales as @xmath354 . as predicted , fig . [ fig : oia_k4_a10_n_powerp ] shows that the achieved dof per transmitter is 0.5 and 1 when @xmath352 and @xmath353 , respectively , by either the min - inr scheme or the max - sinr scheme . we first studied the optimal way of exploiting multiuser diversity in the @xmath0-transmitter simo ibc where each transmitter with a single antenna selects a user and the number of transmitters is larger than the number of receive antennas at each user . we proved that the multiuser dimensions should be used first for decreasing the dof loss caused by interfering signals ; the whole multiuser dimensions should be exploited to reduce the dof loss term to @xmath355 for the target dof @xmath356 $ ] , while the multiuser dimensions should be devoted to making the dof loss zero and then to increasing the dof gain term to @xmath16 for the target dof @xmath357 . we also derived the sufficient user scaling for the target dof . the dof per transmitter @xmath33 $ ] is obtained when the number of users scales as @xmath358 , and the dof per transmitter @xmath357 is achieved when the number of users scales as @xmath359 . also , we extended the results to the @xmath0-transmitter mimo ibc where each transmitter having the multiple antennas supports the multiple users . k. gomadam , v. r. cadambe , s. a. jafar , `` approaching the capacity of wireless networks through distributed interference alignment , '' in _ proc . of ieee global telecommunications conference _ , pp . 16 , dec . 2008 . y. huang and b. rao , random beamforming with heterogeneous users and selective feedback : individual sum rate and individual scaling laws " , _ ieee trans . wireless commun . 5 , pp . 20802090 , may 2013 . j. h. lee and w. choi , opportunistic interference aligned user selection in multi - user mimo interference channels , " in _ proc . of ieee global telecommunications conference _ , miami , fl , usa , dec . j. h. lee and w. choi , interference alignment by opportunistic user selection in 3-user mimo interference channels , " in _ proc . of ieee international conference on communications _ , kyoto , japan , june , 2011 . j. h. lee and w. choi , on the achievable dof and user scaling law of opportunistic interference alignment in 3-transmitter mimo interference channels , " _ ieee trans . wireless commun . 12 , no . 6 , pp . 2743 - 2753 , june 2013 . j. h. lee and w. choi , opportunistic interference alignment by receiver selection in a k - user 1 x 3 simo interference channel , " in _ proc . of ieee global telecommunications conference _ , houston , tx , usa , dec . 2011 . t. kim , d. j. love , and b. clerckx , instantaneous degrees of freedom of downlink interference channels with multiuser diversity , " in _ proc . of asilomar conference on signals , systems , and computers _ , pp.12931297 , nov . b. c. jung , d. park , and w .- y . shin , opportunistic interference mitigation achieves optimal degrees - of - freedom in wireless multi - cell uplink networks , " _ ieee trans . 60 , no . 7 , july 2012 . j. joung and a. h. sayed , user selection methods for multiuser two - way relay communications using space division multiple access , " _ ieee trans . wireless commun . _ , vol . 9 , no . 7 , pp 21302136 , july 2010 . y. ohwatari , n. miki , t. asai , t. abe , and h. taoka , performance of advanced receiver employing interference rejection combining to suppress inter - cell interference in lte - advanced downlink " , in _ proc . ieee vehicular technology conference ( vtc - fall ) _ , sep . y. tokgoz , b. d. rao , m. wengler , and b. judson , performance analysis of optimum combining in antenna array systems with multiple interferers in flat rayleigh fading , " _ ieee trans . 10471050 , july 2004 .

this paper investigates how multiuser dimensions can effectively be exploited for target degrees of freedom ( dof ) in interfering broadcast channels ( ibc ) consisting of @xmath0-transmitters and their user groups . first , each transmitter is assumed to have a single antenna and serve a singe user in its user group where each user has receive antennas less than @xmath0 . in this case , a @xmath0-transmitter single - input multiple - output ( simo ) interference channel ( ic ) is constituted after user selection . without help of multiuser diversity , @xmath1 interfering signals can not be perfectly removed at each user since the number of receive antennas is smaller than or equal to the number of interferers . only with proper user selection , non - zero dof per transmitter is achievable as the number of users increases . through geometric interpretation of interfering channels , we show that the multiuser dimensions have to be used first for reducing the dof loss caused by the interfering signals , and then have to be used for increasing the dof gain from its own signal . the sufficient number of users for the target dof is derived . we also discuss how the optimal strategy of exploiting multiuser diversity can be realized by practical user selection schemes . finally , the single transmit antenna case is extended to the multiple - input multiple - output ( mimo ) ibc where each transmitter with multiple antennas serves multiple users . multiuser diversity , degrees of freedom , interference alignment measure , interfering broadcast channel

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Proceed to summarize the following text: the quantum dynamics of the radical - pair mechanism @xcite , underlying the avian magnetic compass @xcite and spin transport in photosynthetic reaction centers @xcite , have recently attracted the attention of the quantum physics community @xcite , since it was shown @xcite that radical - pairs offer an ideal system to study quantum coherence effects and explore quantum information processing in a complex biochemical setting . radical - pair reactions consist of a coherent spin motion in a multi - spin system embedded in a biomolecule , interrupted by an electron transfer that results in the spin - dependent charge recombination of the radical - ion - pair and the termination of the reaction . it is known that the coherent spin motion as well as the measurable reaction yields in radical - pair reactions are also influenced by the external magnetic field through the unpaired electrons zeeman interaction . hence radical - pair reactions are no different than other quantum systems used to measure a classical parameter , as for example are the well - developed atomic magnetometers @xcite using e.g. alkali vapors @xcite or nitrogen vacancy centers @xcite . central in these studies have been the fundamental measurement precision limits set by the quantum dynamics of the system under consideration . we here establish a venue for studying quantum metrology in a biological context @xcite . we introduce the full machinery of quantum parameter estimation @xcite in order ( i ) to establish the _ exact _ value of @xmath0 , the fundamental magnetic sensitivity of the reaction , and ( ii ) design an optimal molecular system approaching this fundamental limit . to this end we consider the quantum fisher information obtained from the radical - pair reaction and the resulting cramr - rao bound . we then treat the intra - molecule hyperfine couplings as free design parameters , and obtain their optimum value by maximizing the quantum fisher information . this leads to the fundamental limit @xmath0 , which we explicitly derive for any radical - pair . knowing the absolute quantum limit on @xmath0 , we address the well - known measurement of reaction yields and show it is sub - optimal . we then modify a recently proposed method of reaction control @xcite , introducing a quantum circuit analysis of the controlled reaction , and reducing @xmath0 by a factor of 3 compared to @xcite . the outline of the paper is the following . in sec . ii we briefly introduce the dynamics of radical - pair reactions , and in sec . iii the basic tools of quantum metrology , in particular the analytic form of the parameter - generator , a useful tool recently introduced @xcite . in sec . iv , the eigenvalues of this operator are then used to find the maximum quantum fisher information and the resulting bounds on @xmath0 for radical - pair reactions . in sec . v we discuss a common observable in radical - pair reactions , the reaction yield , in the context of magnetic sensitivity . we demonstrate that the resulting maximum possible sensitivity is an order of magnitude smaller than the absolute quantum limit . in sec . vi we present the optimum measurement scheme that can realize the optimum quantum limit on @xmath0 . since this scheme does not appear to be chemically realistic , a natural question is whether some sort of quantum reaction control can improve the magnetic sensitivity of reaction yields . this is indeed the case as shown in sec . vii , where we take advantage of the spin - exchange interaction naturally occurring in radical - pairs , and known from quantum metrology work to simulate a controlled - not gate . the spin - exchange interaction effects a state - preparation and readout before and after the actual magnetometric state evolution , respectively , reminding of ramsey interferometry . together with the reaction control method of @xcite , which is a factor of 6 away from the absolute quantum limit , our measurement scheme is shown to approach this limit within a factor of 2 . radical - pairs ( rps ) are the cornerstone system of spin chemistry , the field of physical chemistry and photochemistry dealing with the effect of electron and nuclear spins on chemical reactions . the radical - pair mechanism was introduced by closs and closs @xcite and by kaptein and oosterhoff @xcite as a reaction intermediate explaining anomalously large epr and nmr signals observed in organic molecule reactions in the 1960 s . the quantum degrees of freedom of radical - ion pairs are formed by a multi - spin system embedded in a biomolecule . the multi - spin system is comprised of the two unpaired electrons of the two radical - ions and a usually large number of nuclei . their coherent spin motion is driven by intramolecule magnetic interactions , as for example hyperfine couplings between each radical s magnetic nuclei and the respective unpaired electron . the magnetic field effects resulting from such interactions in this spin - dependent biochemical reaction have been extensively explored theoretically and experimentally @xcite . in particular , a charge transfer following the photoexcitation of a donor - acceptor dyad da leads to the radical - pair ( also called charge - separated state ) @xmath1 , where the two dots represent the two unpaired electron spins of the two radicals . the initial spin state of the two unpaired electrons of the radical - pair is usually a singlet , denoted by @xmath2 . now , both d and a contain a number of magnetic nuclei which hyperfine - couple to the respective electron . neither singlet - state nor triplet - state rps are eigenstates of the magnetic hamiltonian , @xmath3 , hence the initial formation of @xmath2 is followed by singlet - triplet ( s - t ) mixing , i.e. a coherent oscillation of the spin state of the electrons , designated by @xmath4 . concomitantly , nuclear spins also precess , and hence the total electron / nuclear spin system undergoes a coherent spin motion driven by @xmath3 . as will be detailed later , the subscript @xmath5 in @xmath3 is a reminder that the hamiltonian depends parametrically on the magnetic field @xmath5 to be estimated . this coherent spin motion has a finite lifetime . charge recombination , i.e. charge transfer from a back to d , terminates the reaction and leads to the formation of the neutral reaction products , conserving during the process the electronic angular momentum . that is , there are two kinds of neutral products , singlet ( the original da molecules ) and triplet , @xmath6da . the percentage of the initial radical - pair population ending up in the singlet ( triplet ) neutral product defines the singlet ( triplet ) reaction yield . singlet and triplet recombination takes place at the rate @xmath7 and @xmath8 , respectively . both rates are in principle known parameters of the specific molecular system under consideration , as of course are the hyperfine couplings entering @xmath3 . the whole process is schematically depicted in fig.[schematic]a . , which is coherently converted to the triplet radical - pair , @xmath9 , due to intramolecule magnetic interactions embodied in the spin hamiltonian @xmath3 . simultaneously , spin - selective charge recombination leads to singlet ( da ) and triplet neutral products ( @xmath6da ) . ( b ) quantum metrology aspect of the radical - pair reaction , where an initial spin state is transformed into a final spin state , which depends on the magnetic field @xmath5 through the spin hamiltonian . the measurement of the final state conveys information about @xmath5 . ] the schematic of fig.[schematic]a should not be taken too literally , as it suggests that only pure singlet ( triplet ) radical - pairs can recombine to the singlet ( triplet ) neutral reaction products . this is not the case , like it is not the case for a two - level atom that only an excited - state atom can decay to the ground state . as in atoms having ground - excited state coherence , radical - pairs can be in coherent superpositions of singlet and triplet states , continuously evolving by @xmath3 . a major aspect of our previous work has been to understand the physics of this coherence , its fundamental dissipation properties , and the role thereof in establishing the fundamental master equation , @xmath10 , accounting for the radical - pair reaction s quantum dynamics . this issue is still hotly debated @xcite . this work , however , is decoupled from this debate , as we consider only the hamiltonian contribution to the quantum metrology aspect of the reaction . the non - trivial quantum dynamics previously alluded to mainly appear in the case of unequal recombination rates , @xmath11 . here we consider the simple exponential model , where @xmath12 and we further neglect the presence of s - t decoherence @xcite , which is unavoidable even in the case @xmath13 . the rationale of this approach is that it allows a step - wise understanding of the quantum metrology aspect of radical - pair reactions , starting from the most evident features stemming from the coherent spin motion , and progressing to more complex properties of the system , which is an open and leaky quantum system . earlier works @xcite attempted to explore some metrological aspects of these reactions , however using the traditional ( called haberkorn s ) master equation and considering phenomenological sources of decoherence . our understanding is ( i ) that haberkorn s master equation scrambles the quantum dynamics of the system and is a phenomenological description valid only in the regime of strong spin relaxation , and ( ii ) consideration of decoherence and its role in this new kind of biochemical quantum metrology is a non - trivial task , intertwined with the understanding of the fundamental master equation . therefore we opt to first establish the fundamental limits to the magnetic sensitivity of radical - pair reactions in the simple case of equal recombination rates and a purely coherent spin motion . thus , the only effect of the finite radical - pair s lifetime @xmath14 relevant to this work is that the radical - pair population decays exponentially as @xmath15 . hence the sensitivity limit @xmath0 @xcite will be shown to be directly dependent on @xmath16 . this is not unexpected , since measurement time is a central resource in quantum metrology . in other words , we here explore the equivalence of fig.[schematic]a with fig.[schematic]b , which describes the usual scheme of quantum estimating a classical parameter like a magnetic field . an initial state @xmath17 evolves under the unitary action of the hamiltonian @xmath3 into @xmath18 . measuring the final state conveys information about @xmath5 . in radical - pair reactions , an initial spin state ( comprising the spin of the two electrons and the present nuclei ) evolves under @xmath3 , and the recombination process effects the measurement in the singlet / triplet basis , i.e. the measurement stage is naturally inbuilt into the radical - pair mechanism . we here treat radical - pair reactions as scalar magnetometers , and conclusively address the questions : ( a ) what is the fundamental quantum limit , @xmath0 , to the precision of estimating @xmath5 ? ( b ) is this limit realized when the physical observable carrying the information on @xmath5 is the reaction yield ? ( c ) if not , can we control the reaction in order to better approach the fundamental limit @xmath0 ? if we consider a radical - pair with @xmath19 nuclear spins in the donor and @xmath20 nuclear spins in the acceptor , the hyperfine hamiltonian in the presence of an external magnetic field @xmath21 , where @xmath5 is to be estimated , is _ b =- b(s_dz+s_az)+_j=1^n_d_d_j_j+_k=1^n_a_a_k_k.[ham ] here we denote by @xmath22 ( @xmath23 ) the electron spin of the donor ( acceptor ) radical , @xmath24 ( @xmath25 ) is the @xmath26-th ( @xmath27-th ) nuclear spin of the donor ( acceptor ) radical , and @xmath28 ( @xmath29 ) the hyperfine tensor coupling the @xmath26-th ( @xmath27-th ) nuclear spin of the donor ( acceptor ) radical to the donor s ( acceptor s ) electron . the gyromagnetic ratio of the electrons setting the frequency scale , @xmath30 , has been set to @xmath31 in eq . and we will keep this convention from now on . we also set @xmath32 , thus the hyperfine couplings and the magnetic field @xmath5 have units of frequency , while the spin operators are dimensionless . in the following , in order to get actual magnetic field values , one should divide the derived expressions for @xmath5 with @xmath33 . before embarking on our analysis , we will first lay out the tools of quantum parameter estimation and apply them to two pedagogical cases , a single electron zeeman interaction and a two - electron zeeman interaction . these considerations will form a baseline for comparing fundamental sensitivity limits in radical - pairs . the formalism developed by brun and coworkers @xcite is ideally suited to treat the estimation of @xmath5 , which is not a multiplicative parameter of the radical - pair hamiltonian . specifically , the authors in @xcite consider the time - evolution operator , @xmath34 , which obviously depends on the parameter @xmath5 we wish to estimate . if the initial state of the system is @xmath35 , then the time - evolved state will be @xmath36 , where the subscript in @xmath37 reminds us that the time evolved state also depends on @xmath5 . the generator of @xmath5 translations is then shown to be @xmath38 . the utility of this generator is that it directly leads to the _ maximum _ quantum fisher information f_b^max=[_max(h_b)-_min(h_b)]^2 , where @xmath39 ( @xmath40 ) is the maximum ( minimum ) eigenvalue of @xmath41 . the authors in @xcite then derive two general results . first , knowing the @xmath42 different eigenvalues , @xmath43 , and corresponding eigenvectors , @xmath44 of @xmath3 , where @xmath45 , with @xmath46 being the degeneracy of eigenvalue @xmath43 , one can obtain @xmath47 from @xmath48 by @xmath49 we denote the projector to the @xmath27-th eigenspace of @xmath3 . secondly , the maximum fisher information is obtained for the initial state = 1(+e^i),[opt_init ] where @xmath50 and @xmath51 are the eigenkets of @xmath41 corresponding to its maximum and minimum eigenvalues , respectively . finally , the uncertainty @xmath0 in estimating @xmath5 is limited by the cramr - rao bound @xcite b,[dbfmax ] where @xmath52 is the number of independent repetitions ( number of radical - pairs in our case ) of the measurement . before proceeding with radical - pairs , we analyze two intuitive and simple examples . consider first a single electron in a magnetic field @xmath53 , the hamiltonian being @xmath54 . the eigenvectors and eigenvalues of @xmath55 are @xmath56 and @xmath57 , respectively . since the eigenvectors are @xmath5-independent , the second term in eq . is zero , while the first term leads to @xmath58 . the maximum and minimum eigenvalues of @xmath41 are @xmath59 and @xmath60 , respectively , hence @xmath61 . thus , we recover the well - known time - scaling limit , namely the magnetic sensitivity resulting from measuring the electron s larmor frequency during a time interval @xmath62 is limited by @xmath63 . repeating this measurement @xmath52 times , the so - called shot - noise limited sensitivity will be @xmath64 . we will now demonstrate the particle - number scaling limit by considering a 4-dimensional hamiltonian consisting of the zeeman interaction of two electrons , @xmath65 . now the eigenvalues are @xmath66 with corresponding eigenstate @xmath67 , @xmath68 with corresponding eigenstate @xmath69 , and @xmath70 , which is doubly degenerate , with corresponding eigenstates @xmath71 and @xmath72 . again , the eigenstates are @xmath5-independent , hence @xmath73 . the eigenvalues of @xmath41 are @xmath74 , hence now it is @xmath75 . the magnetic sensitivity is now @xmath76 . this is @xmath77 times better than repeating the one - electron measurement two times . generalizing to an @xmath78-electron system , where @xmath79 , we find @xmath80 . this is @xmath81 times better than repeating the one - electron measurement @xmath78 times . this enhancement is due to a possible multi - partite entanglement in the @xmath78-electron state . that is , a notable feature of @xmath82 is that it automatically takes into account of such a possibility in the system s state preparation . it should be noted that the scaling with the particle number should not be confused with the scaling with the number @xmath52 of the experiment s repetition . since the experimental realizations are independent , the scaling with @xmath52 is the ordinary statistical scaling @xmath83 . that is , repeating the 2-electron measurement @xmath52 times , the so - called heisenberg limited magnetic sensitivity will be @xmath84 . similarly , in the @xmath78-electron system , the heisenberg - limited magnetic sensitivity obtained by averaging @xmath52 independent measurements will be @xmath85 . as shown previously with the simple scenario of free electrons in a magnetic field , the sensitivity @xmath0 depends on the measurement time @xmath62 . in radical - pair reactions there is a natural time scale limiting the magnetic sensitivity , the radical - pair s lifetime . this is determined by the recombination rates @xmath7 and @xmath8 . for the reasons outlined in sec . ii.a , we here consider the so - called exponential model , where @xmath12 . when @xmath86 , the quantum dynamics of the radical - pair reaction simplify considerably . this is because , neglecting singlet - triplet decoherence which we understand is inherent in the system ( even when @xmath13 ) , in the exponential model radical - pairs can be considered to evolve unitarily by the magnetic hamiltonian @xmath3 , while their population decays exponentially at the rate @xmath27 . equivalently , at the single - molecule level , each radical - pair evolves unitarily until the random instant in time when it recombines . this time follows the exponential distribution @xmath87 , where @xmath14 . since the quantum fisher information is time - dependent , we have to take into account the fact that from each radical - pair in the ensemble we can extract a different fisher information , depending on the time it recombined . if @xmath88 is the radical - pair population at time @xmath62 and @xmath89 is the initial population , then in each time interval @xmath90 there will be @xmath91 radical - pairs contributing to @xmath82 . if the variance of the magnetic field estimate resulting from one molecule is @xmath92 , then the @xmath91 molecules contribute independently to the measurement during @xmath90 . the inverse uncertainties of @xmath5 add in quadrature , hence the inverse variance stemming from those @xmath91 molecules will be @xmath93 . since @xmath94 , the magnetic sensitivity for the whole reaction is b=1.[delta_b_f ] clearly , we are not concerned with the absolute value of @xmath0 as determined by how many molecules participate in the experiment . we rather focus on optimizing @xmath82 , which depends on the state preparation and measurement scheme . thus , in the following we will take @xmath95 . for those cases where @xmath96 , where @xmath97 is some constant , it follows that @xmath98 . we will first derive exact analytic results for @xmath82 and @xmath0 for a radical - pair with one nuclear spin-1/2 contained in e.g. the donor . the hyperfine hamiltonian is _ b =- b(s_dz+s_az)+a_xs_dxi_x+a_ys_dyi_y+a_zs_dzi_z.as it formally turns out , the maximum quantum fisher information does not depend on @xmath99 . intuitively , this is because the term @xmath100 just produces a shift in the magnetic field `` seen '' by the electron spin @xmath101 along the z - axis and hence does not `` produce '' any information on @xmath5 . we therefore have to distinguish two cases : @xmath102 and @xmath103 . after dealing with the single - nuclear - spin radical - pair , we generalize to multiple nuclear spins . for the spheroidal hyperfine coupling it b =- b(s_dz+s_az)+as_dxi_x+as_dyi_y+as_dzi_z . a special case , occurring when @xmath104 , is the commonly encountered isotropic hyperfine coupling . the eight eigenvalues of @xmath3 are given in appendix a , along with the eigenvalues of @xmath41 calculated from eq . . it is found that the maximum and minimum eigenvalues of @xmath41 are @xmath62 and @xmath105 , respectively . hence in this case , the maximum quantum fisher information is @xmath75 , leading to the quantum limit b_f=1.[deltab_f ] : the minimum uncertainty , @xmath0 , for determining a magnetic field @xmath5 by using a radical - ion - pair reaction , the single nuclear spin of the radical - pair having a spheroidal hyperfine coupling , is given by eq . . it is worthwhile noting that the maximum fisher information , @xmath106 , is the same with the case of two free electrons studied in section iii.b . one would perhaps expect that having three particles in the system ( two electrons and one nucleus ) , the optimal sensitivity should gain ( according to the heisenberg scaling ) a factor of 3 compared to the single - electron case , or a factor of 3/2 compared to the two - electron case . the reason behind the absence of such enhancement is that the nuclear spin does not strongly couple to the magnetic field . hence it does not provide any independent information on the magnetic field , but only serves to drive the time evolution of the radical - pair s electronic spin state . the lack of enhancement by the nuclear spin is not because we omitted the nuclear zeeman interaction in the hamiltonian . indeed , if we include the nuclear zeeman term in @xmath3 , we find that @xmath107 , where @xmath108 is the nuclear gyromagnetic ratio ( scaled to @xmath33 ) . thus the correction to @xmath109 is on the order of @xmath110 and hence negligible . however , if it were @xmath111 , then we would get the expected factor of 3 in sensitivity gain compared to the single - electron case of sec . iii.a . in other words , the information about the magnetic field essentially stems from the strength of the field s coupling to the spins . we will now consider the general hyperfine coupling , where @xmath103 , _ b =- b(s_dz+s_az)+a_xs_dxi_x+a_ys_dyi_y+as_dzi_z . again , we can find analytic expressions for the eigenvalues of @xmath41 , which are given in appendix b. there it is shown that @xmath112 and @xmath113 , hence the resulting maximum quantum fisher information is bound by @xmath106 , which we found previously for the spheroidal hyperfine coupling . * we thus arrive at our second general result * : for a radical - pair with a single nuclear spin-1/2 , the spheroidal hyperfine coupling ( the isotropic being a special case ) leads to the smallest uncertainty , @xmath0 , for determining a magnetic field @xmath5 along the spheroid s symmetry axis . this uncertainty depends only on the radical - pair s lifetime @xmath16 , and is given by @xmath114 . as a numerical estimate , for @xmath115 and @xmath116 radical - pairs we obtain @xmath117 . radical - pairs contain many ( sometimes tens ) of nuclear spins . based on the above , we can readily generalize and state * the third general result of this work * : for any radical - pair with a spin - independent lifetime ( i.e. @xmath118 ) , the maximum magnetic sensitivity ( minimum @xmath0 ) that can be obtained with _ any measurement method and any initial state _ is @xmath119 . this follows from the same physical argument used in sec . iv.a , namely that the uncertainty @xmath0 is determined just by the two electron spins . the nuclear spins do not couple to the external magnetic field , i.e. they are spectators just driving the spin state evolution . a formal proof of this general result follows . for any operator @xmath120 depending parametrically on @xmath5 , it is @xcite @xmath121 . we take @xmath122 , calculate the above derivative with @xmath5 and multiply with @xmath123 in order to find @xmath124 . taking the operator norm we get @xmath125 , hence indeed the maximum ( minimum ) eigenvalue of @xmath41 is smaller ( larger ) or equal than @xmath62 ( @xmath105 ) . typically , when studying the magnetic sensitivity of radical - pair reactions , one considers the singlet reaction yield , which quantifies the percentage of the reactants ( number of radical - pairs starting out in the electronic singlet state at @xmath126 ) ending up in the singlet neutral product state . to define the singlet reaction yield , @xmath127 , we first need to introduce two basic operators , the singlet and triplet projectors , @xmath128 and @xmath129 , respectively . for a radical - pair with a single nuclear spin they are written as @xmath130 and @xmath131 . they leave the nuclear spin state untouched and project out of a general state @xmath132 the electronic singlet or triplet component . the expectation value of @xmath128 in the state @xmath132 is thus @xmath133 , hence the singlet reaction yield is written as @xmath134 , where @xmath135 . it is obviously irrelevant whether one chooses to measure the singlet or the triplet reaction yield , since it is always @xmath136 , where @xmath137 . now , since the magnetic field enters @xmath138 through the hamiltonian , the reaction yield is a function of @xmath5 . in particular , in order to find the magnetic sensitivity @xmath0 , we first need to distinguish two cases . ( a ) if one can measure the instantaneous singlet yield given by @xmath139 , one can estimate @xmath5 just from those radical - pairs that recombined into the singlet channel during @xmath90 through the relation @xmath140 , where @xmath141 is the variance of @xmath128 at time @xmath62 , and @xmath142 is the magnetic sensitivity of the instantaneous yield . all such estimates can then be statistically combined ( inverse uncertainties add in quadrature ) to yield the total uncertainty b=^-1/2,[dbin ] where in the expression for the variance of @xmath128 appearing in the denominator of the integrand in eq . we took into account that @xmath143 , since @xmath128 is a projector . for this measurement scheme to be realistic , the time resolution of the measurement of the instantaneous yield must be much better than @xmath144 . if this is not the case , we are led to case ( b ) integration over the whole reaction , i.e. measurement of the total yield @xmath127 . then the magnetic sensitivity @xmath0 is given by @xmath145 , where @xmath146 is the precision with which @xmath127 is measured . this is calculated as follows . in each time step @xmath90 , the instantaneous yield , proportional to @xmath147 , is a random variable following a binomial distribution with probability @xmath147 . thus , the total yield follows the sum of binomials having different probabilities , which is the poisson binomial distribution . its variance is @xmath148 , hence b=^1/2.[dbav ] it is expected that the magnetic sensitivity of case ( b ) is smaller than case ( a ) , or equivalently @xmath149 , since in case ( a ) we have access to much more information along the reaction than the integrated yield relevant to case ( b ) . nevertheless , we here opt to provide exact expressions for @xmath0 in the integrated case , as we think that this is most relevant for physiological conditions . for completeness , we then report the corresponding sensitivities for case ( a ) . we first consider an isotropic hyperfine hamiltonian , @xmath150 . we calculate @xmath0 for initial state ( i ) @xmath151 , ( ii ) @xmath152 , and ( iii ) an equal mixture of ( i ) and ( ii ) , which is usually taken to describe the initial state of radical - pair reactions , as it accounts for thermal equilibrium ( practically zero ) nuclear spin polarization . we denote the respective uncertainties by @xmath153 , @xmath154 and @xmath155 . the analytic expressions for these uncertainties follow from the analytic expressions for the reaction yields @xmath156 , @xmath157 and @xmath158 and their derivatives with respect to @xmath5 entering the denominator of eq . , as well as from the analytic expressions for the respective nominators . the resulting formulas are too cumbersome to list here . in appendix c we provide for reference the exact expressions for the reaction yields . we here use the obtained analytic expressions for the uncertainties @xmath0 to display their inverses as a function of the hamiltonian parameters @xmath5 and @xmath159 in the contour plots of fig.[db](a)-(c ) for the cases ( i)-(iii ) , respectively . we first note that the minimum of @xmath154 ( see fig.[db]b ) is smaller by about 30% than the minimum of @xmath153 ( see fig.[db]a ) , and both minima appear at a finite ( and different in each case ) value of the hyperfine coupling @xmath159 and at a different field @xmath5 . this is due to the different singlet - triplet mixing frequencies caused by the nuclear spin in the @xmath160 or in the @xmath161 state . in the @xmath161 state the nuclear magnetic field opposes @xmath5 and hence reduces the mixing frequency , thus its @xmath5-dependence becomes relatively more significant . in case ( iii ) , shown in fig.[db]c , the minimum of @xmath155 is achieved for @xmath162 . although the sensitivity @xmath163 is linear in the density matrix , the magnetic sensitivity @xmath0 depends on the absolute value of @xmath163 , hence @xmath155 is not trivially related to @xmath153 and @xmath154 . for example , at low @xmath5 and @xmath159 where @xmath153 and @xmath154 are close to their minimum , the respective derivatives @xmath163 are opposite in sign , and this is why @xmath155 is large in this region . in any case , taking the limit of large @xmath159 we find the exact expression b_s^iso(b)^1/2.the minimum occurs at @xmath164 and takes the value b_s^iso=5.14.[dbs ] : for a radical - pair with one isotropically coupled nuclear spin , the maximum possible magnetic sensitivity obtained by measuring the time - integrated reaction yield is 15 times lower , @xmath165 , than the highest possible sensitivity allowed by quantum physics and given by eq . . this means that there is ample room for improvement . by changing to an anisotropic hyperfine interaction we can already get about a factor of 2 improvement in @xmath0 . that is , we repeat the calculation for @xmath0 taking @xmath166 . we find that @xmath0 is minimized either for @xmath167 and @xmath168 or for @xmath169 and @xmath170 . for both cases the minimum is the same for both initial states ( i ) @xmath151 and ( ii ) @xmath152 , and hence the same for ( iii ) the mixed singlet initial state . this is expected , since both pure initial states are symmetric with respect to the hamiltonian anisotropy . we thus denote the uncertainty common to all three initial states ( i)-(iii ) by @xmath171 . as in the isotropic case , the resulting expressions are long . in appendix c we provide for reference the reaction yield . as shown in fig.[db]d , @xmath172 increases with increasing @xmath159 . like before , we take the limit @xmath162 and find b_s^aniso(b)^1/2.the minimum occurs at @xmath173 , and takes the value b_s^aniso=2.27 , which is still a factor of 6.4 away from @xmath174 . * to summarize our fifth main result * : the measurement of the integrated reaction yield can at best provide 6.4 times worse magnetic sensitivity than the absolute quantum limit , and this is achieved for the maximally anisotropic hyperfine interaction . the reason the anisotropic coupling outperforms the isotropic in the reaction yield magnetic sensitivity will be given in sec . vi.a after we introduce the optimal measurement strategy . furthermore , we stress that for a given magnetic field @xmath5 to be estimated , the optimum reaction - yield sensitivity @xmath0 is obtained for a particular lifetime of the radical - pair on the order of @xmath175 . the reason will be given in sec . vii.c . for completeness , we produce in figs.[db]e , f the results of eq . , i.e. the case when our measurement time resolution is enough to monitor the instantaneous yield along the reaction . in both cases studied , isotropic and anisotropic , this kind of measurement yields about a factor of 2 improvement in magnetic sensitivity . specifically , we find @xmath176 and @xmath177 , obtained for @xmath162 at @xmath178 . moreover , both minimums become broader , i.e. there is a larger range of @xmath5 values close to the optimal @xmath0 . the usual measurement scheme of radical - pair reactions , namely the singlet initial state and the measurement of the singlet reaction yield , is enforced by the very nature of these reactions . as shown in the previous section , this measurement is sub - optimal . towards a possible improvement in magnetic sensitivity , we first need to point to the optimal initial state and the optimal measurement operator . according to the general result of eq . , the optimal initial state for a single - nuclear spin radical - pair is the greenberger - horne - zeilinger state @xmath179 . clearly , @xmath17 belongs to the triplet manifold , and exhibits maximum tri - partite entanglement . it is expected that by measuring the electronic spin precession of this state in the magnetic field one would obtain the optimum sensitivity . indeed , for the isotropic hyperfine hamiltonian , which we know already is optimal ( see section iv.a ) , the time - evolved state ( taking @xmath180 ) is @xmath181 . we choose @xcite as measurement operator @xmath182 . we will now analyze the two scenarios mentioned in section v , that of a time - resolved measurement and that of an integrated measurement . in the former case we get @xmath183 , hence @xmath184 . now during @xmath90 there will be @xmath185 molecules contributing to this measurement of @xmath186 . the resulting inverse variance in @xmath5 is @xmath187 , where @xmath188 is the variance of @xmath186 . thus we find @xmath189 , which is exactly equal to the maximum quantum fisher information , leading to @xmath190 . it thus follows that with this measurement strategy one achieves the limit @xmath174 at any @xmath5 . in contrast , an integrated measurement is not as capable . now the integrated @xmath186-``yield '' is @xmath191 , and its magnetic sensitivity is @xmath192 . the square error in @xmath193 will be the integrated variance of @xmath186 , weighted by the exponential population decay , i.e. @xmath194^{1/2}=[8b^2/(16b^2+k^2)]^{1/2}$ ] . finally , the magnetic sensitivity will be @xmath195^{1/2}$ ] . it is seen that @xmath196 , with the equality sign valid only for @xmath197 . that is , in the integrated measurement with the optimal initial state and optimal measurement operator we achieve the optimal sensitivity only at @xmath197 . the optimum magnetic sensitivity follows from the optimal measurement strategy outlined before , choosing as initial state a maximally entangled state of the triplet electronic manifold , and measuring its spin coherence while it is evolving , always within the triplet manifold . clearly , this is far from how radical - pairs evolve in reality . this leads to a natural question that we will affirmatively address in the following section , i.e. can we control the reaction in a chemically and physically realistic way in order to approach the optimum magnetic sensitivity ? before addressing the previous question , we will explain the fact that the maximally anisotropic hyperfine interaction gives a factor of 2 improvement in @xmath0 , as was demonstrated in sec . this can be seen to result from the overlap of the state evolved by the magnetic hamiltonian @xmath3 , which is @xmath198 , with the optimal state @xmath199 previously defined . for the isotropic hamiltonian the overlap is zero , while for the anisotropic hamiltonian it is @xmath200 . finally , it might sound as contradicting that on the one hand we obtain the maximum fisher information for the isotropic case , while the maximum _ reaction - yield _ magnetic sensitivity for the anisotropic case . the latter finding does not contradict the former , as the _ reaction - yield _ sensitivity limit is well below the quantum limit defined by the fisher information . in section v we have rigorously proved that the singlet reaction yield with a maximally anisotropic hyperfine interaction can at most provide a magnetic sensitivity 6.4 times worse than the absolute quantum limit . a natural question is , how can one do better ? in particular , given the discussion of the previous section , how can one do better in a chemically realistic way ? towards addressing this question we will ( i ) take advantage of a very promising approach of optically switching the conformation of the radical - pair , recently proposed in @xcite , and ( ii ) include a realistic exchange interaction in the hamiltonian , which changes ( a ) the initial spin state before the radical - pair commences its magnetometric state evolution , and ( b ) the effective measurement basis before it recombines . in summary , given the maximally anisotropic coupling that resulted from the optimization of section v.c , the reaction control proposed in @xcite , and the modified initial state and measurement basis we introduce in the following , we will show that the obtained sensitivity @xmath0 is just a factor 2 away from the quantum limit @xmath174 of eq . . moreover , compared to the approach of @xcite , we reduce @xmath0 by a factor of 3 . , where @xmath201 , for a mixed singlet initial state @xmath202 , and ( a ) an isotropic and ( c ) a maximally anisotropic hyperfine interaction . in ( b ) and ( d ) we depict the corresponding probability per unit time , @xmath203 , for the molecular switches to close by the reaction control laser pulses , which are tuned to coincide with the positive swings of @xmath204 . in the middle of trace ( b ) there are a number of pulses missing , since there the corresponding @xmath204 is on average zero and hence will not contribute to the singlet yield magnetic sensitivity . for all plots it was @xmath205 and @xmath206.,width=6 ] we briefly reiterate the method of @xcite , since the added advantage we introduce by the exchange hamiltonian is based on the same method of optically pulsing the conformation of the radical - pair . in particular , the authors in @xcite suggest binding the donor and acceptor parts of the radical - pair to the two ends of a molecular switch , the conformation of which can be laser controlled . schematically , this is shown in fig.[rc ] . the rationale behind this idea is the following . as shown previously , the magnetic sensitivity depends on @xmath207 , where @xmath208 is the so - called singlet fidelity of the radical - pair state at time @xmath62 , and @xmath159 , @xmath5 the hyperfine coupling and the magnetic field . for reference , the functions @xmath204 are given in appendix d for the hamiltonians considered in this work . in fig.[pm]a we plot an example of @xmath209 , which is seen to be symmetric about zero . thus , when integrated with the exponential population decay , @xmath210 , and the lifetime @xmath144 is long enough to contain many positive and negative swings of @xmath211 , magnetic sensitivity is suppressed . the idea of @xcite is to pulse the conformation of the molecular switch by an external laser . when the switch is open , the radical - pair evolves unitarily by the magnetic hamiltonian , which for later use we call @xmath212 . for example , this would be either @xmath213 for the isotropic or @xmath214 for the anisotropic case . when the switch is closed , the authors in @xcite argue , the short distance between d and a will turn on the exchange interaction @xcite , @xmath215 . for large exchange coupling @xmath216 , pertinent to the small d - a separation at the closed switch position , the singlet and triplet energy levels separate by @xmath216 and singlet - triplet mixing is suppressed , so only recombination can take place . if the reaction control laser in turned on at those instances ( fig.[pm]b ) where @xmath211 is positive ( and does not have fast oscillations , as in the middle part of fig.[pm]a ) , then the reaction yield magnetic sensitivity will be enhanced , as demonstrated in @xcite . taking advantage of the @xmath217 modulation that can be externally controlled by the reaction control laser pulses , we now analyze our measurement scheme approaching the absolute quantum limit @xmath174 . as shown in fig.[scheme ] , we first prepare the radical - pair state in the electron singlet state . the nuclear spin is usually in an equal mixture of the states @xmath160 and @xmath161 . towards better exhibiting the connection of this biochemical reaction with quantum metrology , we take the quantum circuit perspective and depict the electron singlet state as produced from @xmath218 by a hadamard gate followed by a controlled - not gate . in radical - pairs , this state preparation is naturally realized by the electron transfer producing the charge separated state , since the precursor neutral molecule is already in the singlet state . at @xmath126 all molecular switches are in the `` closed '' conformation and the radical - pairs in the state @xmath219 , which describes a singlet state for the electrons and a mixed state for the nuclear spin . now , while the authors in @xcite open the switch at this time , using a laser pulse strong enough to open all molecular switches , we wait for a time @xmath220 and act on the initial state with the hamiltonian @xmath221 . while the authors in @xcite consider an exchange coupling @xmath216 too large to allow any s - t mixing , we take @xmath216 to be a finite optimization parameter . we thus take @xmath222 . the duration @xmath220 of the action of @xmath221 is a free parameter , however constrained by @xmath223 , so that the radical - pairs do nt have enough time to recombine through the singlet channel . essentially , the action of @xmath221 for a time @xmath220 prepares the initial state of the radical - pair in a state other than @xmath35 . at time @xmath224 a strong reaction control laser pulse opens all molecular switches , and the two radicals are now far apart , so that @xmath225 is a good approximation , given the exponential dependence of @xmath216 on inter - radical distance @xcite . from @xmath224 until @xmath226 the hamiltonian @xmath212 effects the singlet - triplet conversion forming the main magnetometric state evolution . at time @xmath226 a weak reaction control laser pulse closes some of the switches . the pulse energy is chosen so that the rate of closing is equal to the radical - recombination rate @xmath27 . this pulse is the first pulse shown in the pulse sequence of fig.[pm]d . now in our model , the hamiltonian @xmath221 will act again until the radical - pairs of those switches that closed recombine . in the model of @xcite , the radical - pairs just recombine at some time after the switches close without any state evolution taking place before recombination . the radical - pairs of those switches that did not close in step 3 continue to evolve under @xmath212 . step 3 is then repeated with the next weak reaction control pulse , and so on . thus , the pulse repetition period is @xmath227 , which is the envelope period of the function @xmath209 shown in fig.[pm]c , while the pulse width is @xmath228 , so that only the positive swings of @xmath209 contribute to the yield s magnetic sensitivity . hence for any given radical - pair , the time @xmath229 during which @xmath212 is acting is some odd multiple of @xmath228 , plus the time within the pulse , at which this radical - pair recombines . the magnetic sensitivity resulting from the quantum circuit of fig.[scheme ] is shown in fig.[result ] in two equivalent ways . in fig.[result]a we plot the yield sensitivity @xmath230 , in order to directly compare with the result of @xcite . in fig.[result]b we plot the absolute value of @xmath0 , normalized to the optimum quantum limit @xmath114 . it is evident that ( a ) the choice of the maximally anisotropic hamiltonian and ( b ) the inclusion of the action of the exchange interaction in @xmath221 leads to an enhancement by a factor of 3 compared to @xcite , and puts the scheme of fig.[scheme ] a factor of 2 away from the absolute quantum limit . the factor of 3 is equally attributed to ( a ) and ( b ) . the physical interpretation of the enhancement of the magnetic sensitivity by the exchange interaction is the initial phase difference between the singlet and triplet states resulting from the initial action of @xmath221 . due to this phase difference , the action of @xmath212 fully transforms the @xmath231 into a @xmath232 state , thus sensitively affecting the results of the recombination measurement . without this phase , i.e. setting @xmath233 , the singlet and triplet states both have significant populations at the end of the circuit and dilute the magnetic sensitivity of the recombination products . our quantum circuit scheme of fig.[scheme ] reminds of ramsey spectroscopy , where an initial @xmath234 pulse produces an atomic hyperfine coherence , which evolves under the clock transition hyperfine hamiltonian , and is refocused by the final @xmath234 pulse . to further clarify the workings of this quantum reaction control the following remarks are in order . \(1 ) the pulse sequence of the reaction control laser shown in fig.[pm]d is synchronized with the positive swings of @xmath204 shown in fig.[pm]c . this necessitates some prior ( and approximate ) knowledge of the magnetic field , a feature common with the reaction control scheme of @xcite . \(2 ) the time interval @xmath220 during which @xmath221 acts before the molecular switch opens is taken @xmath235 , so that radical - pair recombination is negligible ( it actually increases the obtained @xmath0 by 5% ) . after the switch closes , @xmath221 acts for time @xmath236 before the radical - pairs recombine . this time is taken to follow the exponential distribution with parameter @xmath27 , i.e. @xmath221 acts for a time as long as the radical - pair takes to recombine on average , i.e. @xmath237 . \(3 ) the inclusion of @xmath238 , which includes the exchange interaction was motivated by ( i ) other works @xcite , where a controlled - not gate is shown to be a crucial element in metrology , and ( ii ) the fact that the controlled - not gate is naturally realized by the exchange interaction , as analyzed in @xcite . we let the exchange coupling @xmath216 be a free optimization parameter . the minimum @xmath0 was found for @xmath239 . for a typical hyperfine coupling @xmath159 of several gauss , the resulting value of @xmath216 is also on the order of several gauss . now , @xmath240 , where @xmath241 is the donor - acceptor distance , and typical values @xcite of @xmath242 and @xmath243 are @xmath244 and @xmath245 , respectively . for @xmath216 to be on the order of several gauss , the distance @xmath241 in the closed position of the switch must be around 1.8 nm . this is quite larger than the d - a distance of 0.5 nm in the closed position of azobenzene @xcite , proposed in @xcite as a molecular switch . so for the reaction control studied here azobenzene is not an ideal candidate . furthermore , in fig.[result]c we plot the minimum value of the obtained sensitivity , @xmath246 ( i.e. the minimum of the red solid trace of fig.[result]b ) as a function of the exchange coupling @xmath216 . however , as the exchange coupling depends on inter - radical separation , which is modulated by molecular vibrations , in reality we have to average the trace of fig.[result]c . indeed , evaluating @xmath240 around @xmath247 , and taking a variation of @xmath241 by 0.05 nm , which is typical for studies on the relaxation effect of @xmath216-modulation due to molecular vibrations @xcite , leads to a factor of 2 change in @xmath216 , similar to the @xmath216-range of fig.[result]c . we thus obtain a final @xmath248 , i.e. 10% higher than the value for a constant ( and optimum ) @xmath216 . \(4 ) further , there are two points that might cause a misunderstanding . we first note that although the reaction control pulse sequence introduces a timing in the measurement of reaction yields , the measurement is not of the instantaneous type described in sec . v.a , since we still measure an integrated yield , as does the scheme in @xcite . secondly , the reader might argue that we use an exchange interaction , which was absent in the optimization presented in secs . iv and v. however , the exchange interaction is used in @xmath221 , which is just a state - preparation process , changing the initial singlet state and the final measurement basis . we thus engineer an initial state which is more optimal than @xmath35 , and the actual magnetometry takes place during the action of @xmath212 , which does not include any spin exchange . , and ( b ) error @xmath0 in the estimation of the magnetic field , normalized by the absolute quantum limit @xmath114 . the black dashed line reproduces the result of the reaction control scheme of @xcite , while the red solid line is the result of this work . our reaction control scheme approaches @xmath114 within a factor of 2 . ( c ) the minimum of the red solid line in ( b ) is plotted as a function of the exchange coupling @xmath216 . for @xmath249 , we obtain @xmath250 . but nearby values of @xmath216 are induced by molecular vibrations , hence averaging trace ( c ) leads to the realistic uncertainty 2.2 times away from @xmath174.,width=8 ] finally , we elaborate on a subtle point regarding practical implementation . in section v , evaluating the optimum sensitivity of the reaction yield in case of the maximally anisotropic coupling , we found @xmath171 to be 6.4 times away from the absolute quantum limit @xmath114 . this optimum , however , is realized for a specific value of @xmath5 , e.g. @xmath251 for the anisotropic case , and a hyperfine coupling @xmath162 . in other words , if one wants to realize the limit @xmath171 at e.g. earth s field , one needs to find a radical - pair having a lifetime @xmath252 . we can now explain this earlier finding : because at that lifetime the reaction is almost complete during one ( positive or negative ) swing of the sensitivity function @xmath204 , and further swings do not suppress sensitivity . now , it is evident by looking at fig.[result]b that the optimum sensitivity @xmath171 we obtain just by using the optimal rp lifetime ( i.e. without any reaction control ) is the same as the one achieved by the authors of @xcite using the reaction control , but taking @xmath253 , which is far from the magnetic field value at which @xmath171 is optimized . this leads to the following statement summarizing our findings . one can realize the optimum uncertainty @xmath0 at a desired magnetic field @xmath5 if it is possible to engineer a radical - pair with the specified lifetime and an anisotropic hyperfine coupling approaching the maximal anisotropy . for example , the lifetime engineering could result from molecular bridges @xcite interleaving the donor and acceptor . on the other hand , if such experimental control of the radical - pair s lifetime is not possible , then the reaction control scheme of @xcite and its modification presented here offer a generally useful alternative . in this work we introduced the tools of quantum metrology to put formal and fundamental limits to the magnetic sensitivity of radical - pair reactions , a class of spin - dependent biochemical reactions central in the field of spin chemistry and relevant to the avian compass mechanism . knowing what is the fundamental limit is crucial for understanding how successful a particular measurement scheme is , and for motivating the search for new measurement schemes if there is room for improvement . this has been shown to be the case with the reaction yield measurement , which we have shown to be sub - optimal by almost an order of magnitude . we then took advantage of a recently proposed reaction control scheme , modified the scheme by inclusion of the exchange interaction along the lines of a quantum circuit and ramsey interferometry , and demonstrated a close approach to the absolute quantum limit . regarding future work , we point to two venues of research naturally following from here . a recurring discussion @xcite in the quantum dynamics of radical - pair reactions , in particular in relevance to the avian compass , is whether electron spin entanglement is a resource . in other words , whether the initial singlet electron state , which is maximally entangled , and its subsequent evolution , more or less maintaining the initial entanglement , enhances whatever biological performance radical - pair reactions have . regarding the radical - pair magnetometer we have considered in this work , the answer is clear : considering a radical - pair with a spin - independent lifetime ( @xmath118 ) , and neglecting the intrinsic singlet - triplet decoherence mechanism we introduced @xcite , electron spin entanglement obviously helps _ in principle_. indeed , based on the discussion of sec . iii.a and sec . iii.b , for a system consisting of just two _ uncorrelated _ electron spins the optimum magnetic sensitivity is @xmath254 . allowing quantum correlations one can in principle obtain a @xmath77 improvement , i.e. @xmath255 . our reaction control scheme of fig.[scheme ] leads to @xmath256 , but this does not imply that entanglement `` does not help '' . in other words , it is not straightforward to arrive at a definitive statement with such comparisons . on the one hand it is inconceivable how to experiment with two free electron spins in a chemical environment . radical - pairs offer such a possibility . similarly , there is no immediate way to controllably `` switch - off '' entanglement within the radical - pair reactions . put differently , even though the achieved sensitivity @xmath256 happens to be worse than the two - uncorrelated - spins case , further analysis is required to demonstrate whether or not ( or what part of ) @xmath256 is attributed to entanglement . moreover , according to our understanding @xcite , singlet - triplet decoherence is an unavoidable feature of the radical - pair mechanism itself , and in the case of equal recombination rates ( @xmath86 ) leads to a master equation for @xmath257 that reads @xmath258-k({{\rm q}_{\rm s}}\rho+\rho{{\rm q}_{\rm s}}-2{{\rm q}_{\rm s}}\rho{{\rm q}_{\rm s}})-k\rho$ ] . in other words , in this work we omitted the second term of the previous equation , firstly because its validity is not generally accepted and we wish to decouple this work from the relevant debate , secondly because omitting it considerably simplifies the calculations , and thirdly we obtain the sought after fundamental limits in the idealized and intuitive physical context of unitary evolution . nevertheless , the role of decoherence in the magnetic sensitivity @xmath0 ought to be addressed in detail , as it is known that the advantage due to entangled states might deteriorate @xcite . hence it remains an unsettled issue if entanglement is a resource for this kind of biochemical magnetometers . a natural extension of this work is to study the fundamental limit @xmath259 in estimating the angle of the magnetic field with respect to a molecular frame of reference . this is directly relevant to the avian compass function of radical - pair reactions , and the relevant study will be undertaken elsewhere . for the spheroidal hyperfine interaction @xmath260 considered in sec . iv.a , the eigenvalues of @xmath3 are @xmath261 ( doubly degenerate ) , @xmath262 , @xmath263 and @xmath264 . taking care of the degeneracy in the calculation of @xmath41 , the eigenvalues of @xmath41 are found to be 0 ( doubly degenerate ) , @xmath265 , @xmath266^{1/2}/2(a^2+b^2)$ ] , and @xmath267^{1/2}/2(a^2+b^2)$ ] . by inspection it is seen that @xmath268 , but due to the cosine term it is not immediately obvious how @xmath269 compares to @xmath270 . we can prove that for all times @xmath271 . indeed , take @xmath272 and @xmath273 and subtract from both the common term @xmath59 . we need to show that @xmath274^{1/2}/2(a^2+b^2)$ ] is less than @xmath59 , or their ratio smaller than 1 . the maximum value of the term involving the cosine occurs at @xmath275 , where @xmath276 . then the maximum value of the ratio is @xmath277 for all @xmath42 . thus the maximum and minimum eigenvalues of @xmath41 are @xmath62 and @xmath105 , respectively . for the ellipsoidal hyperfine coupling discussed in sec . iv.b , the eigenvalues of @xmath41 are found to be @xmath278 , @xmath279^{1/2}/[(a_x+a_y)^2 + 4b^2]$ ] , @xmath280^{1/2}/[(a_x+a_y)^2 + 4b^2]$ ] , and @xmath281^{1/2}/[(a_x - a_y)^2 + 4b^2]$ ] . now it is less straightforward to find the maximum ( and similarly the minimum ) eigenvalue , as for some times @xmath272 is the maximum , while at other times it is @xmath273 . however , we can prove as in appendix a that _ at any time _ the maximum eigenvalue is smaller or equal than @xmath62 , and similarly the minimum eigenvalue is larger or equal than @xmath105 . hence the ellipsoidal case can not exceed the spheroidal @xmath82 . for the hamiltonian @xmath282 , we calculate the singlet reaction yields @xmath283 and @xmath284 corresponding to the initial states @xmath151 and @xmath152 , respectively . for the hamiltonian @xmath285 , and for all three initial states considered before we find a common singlet reaction yield @xmath286 . the results are the + ( - ) sign in the third term of corresponds to @xmath151 ( @xmath152 ) . taking the average @xmath287 , we reproduce the result of @xcite . the sensitivities @xmath288 can be readily evaluated , but are too long expressions to list here . the magnetic field sensitivity of the singlet fidelity @xmath289 , where @xmath201 , is given ( after setting @xmath290 ) by the expressions , and for the isotropic hamiltonian @xmath150 , and initial states ( a ) @xmath151 , ( b ) @xmath152 and ( c ) an equal mixture of ( a ) and ( b ) , respectively . for the maximally anisotropic hamiltonian @xmath285 , all three initial states produce the same expression for @xmath291 , given ( after setting @xmath292 ) by . @xmath293\big[\alpha\sin({{(a+b)t}\over 2})+b\sin({{\alpha t}\over 2})\big]\label{e1}\\ g_{t}(a , b)&={a^2\over { 4\alpha^4}}\big[\alpha t\cos({{\alpha t}\over 2})-2\sin({{\alpha t}\over 2})\big]\big[\alpha\sin({{(a - b)t}\over 2})-b\sin({{\alpha t}\over 2})\big]\label{e2}\\ g_{t}(a , b)&=-{a^2\over { 4\alpha^4}}\big[\alpha t\cos({{\alpha t}\over 2})-2\sin({{\alpha t}\over 2})\big]\big[\alpha\cos({{at}\over 2})\sin({{bt}\over 2})+b\sin({{\alpha t}\over 2})\big]\label{e3}\\ g_{t}(a , b)&=-{a^2\over \beta^4}\sin({{bt}\over 2})\big[\beta t\cos({{\beta t}\over 4})-4\sin({{\beta t}\over 4})\big]\big[\beta\cos({{bt}\over 2})\cos({{\beta t}\over 4})+2b\sin({{bt}\over 2})\sin({{\beta t}\over 4})\big]\label{e4}\end{aligned}\ ] ] e. daviso et al . , _ the electronic structure of the primary electron donor of reaction centers of purple bacteria at atomic resolution as observed by photo - cidnp @xmath294c nmr _ , proc . usa * 106 * , 22281 ( 2009 ) . i. f. c@xmath295spedes - camacho and j. matysik , _ spin in photosynthetic electron transport _ , in _ the biophysics of photosynthesis _ goldbeck j , van der est a ( eds . ) , springer science + business media , new york ( 2014 ) . xu , j. zou , j .- g . li and b. shao , _ estimating the hyperfine coupling parameters of the avian compass by comprehensively considering the available experimental results _ , phys . e * 88 * , 032703 ( 2013 ) . m. kritsotakis and i. k. kominis , _ retrodictive derivation of the radical - ion - pair master equation and monte carlo simulation with single - molecule quantum trajectories _ , phys . e * 90 * , 042719 ( 2014 ) . i. k. kominis , _ reply to the comment on `` quantum trajectory tests of radical - pair quantum dynamics in cidnp measurements of photosynthetic reaction centers '' by g. jeschke _ , lett . * 648 * , 204 ( 2016 ) .

radical - ion pairs and their reactions have triggered the study of quantum effects in biological systems . this is because they exhibit a number of effects best understood within quantum information science , and at the same time are central in understanding the avian magnetic compass and the spin transport dynamics in photosynthetic reaction centers . here we address radical - pair reactions from the perspective of quantum metrology . since the coherent spin motion of radical - pairs is effected by an external magnetic field , these spin - dependent reactions essentially realize a biochemical magnetometer . using the quantum fisher information , we find the fundamental quantum limits to the magnetic sensitivity of radical - pair magnetometers . we then explore how well the usual measurement scheme considered in radical - pair reactions , the measurement of reaction yields , approaches the fundamental limits . in doing so , we find the optimal hyperfine interaction hamiltonian that leads to the best magnetic sensitivity as obtained from reaction yields . this is still an order of magnitude smaller than the absolute quantum limit . finally , we demonstrate that with a realistic quantum reaction control reminding of ramsey interferometry , here presented as a quantum circuit involving the spin - exchange interaction and a recently proposed molecular switch , we can approach the fundamental quantum limit within a factor of 2 . this work opens the application of well - advanced quantum metrology methods to biological systems .

You are an expert at summarizing long articles. Proceed to summarize the following text: a central set in a euclidean space has symmetry with respect to reflection through a point , called its center . a closed embedded smooth hypersurface of @xmath3 is called an ovaloid if all its principal curvatures , with respect to the outer unit normal , are positive everywhere . an ovaloid of dimension one is also called an oval . the compact transverse cross - sections of a cylinder over a central ovaloid in @xmath3 , @xmath4 , with hyperplanes are central ovaloids . a similar result holds also for quadrics , which are the level sets of quadratic polynomials in @xmath3 , @xmath4 . their compact transverse cross - sections with hyperplanes are ellipsoids , which are central ovaloids . following solomon who showed that these two kinds of examples provide the only complete smooth hypersurfaces in @xmath5 , whose ovaloid cross - sections are central , we show that the same conclusion also holds in @xmath3 , @xmath1 . roughly , we will say that a smooth hypersurface @xmath6 , @xmath1 , has the _ central ovaloid property _ , or _ cop _ , if * @xmath2 intersects at least one hyperplane transversally along an ovaloid , and * every such ovaloid is central . given this set - theoretic definition of _ cop _ , our main result can be stated as follows : _ a complete , connected smooth hypersurface @xmath7 in @xmath3 , @xmath1 , with _ cop _ is either a cylinder over a central ovaloid , or a quadric . _ more precisely , the _ central ovaloid property _ and our main theorem can be stated in terms of mappings as follows : [ ds1.1:ccut ] if @xmath7 is a smooth manifold , @xmath8 , @xmath1 , is an immersion , and @xmath9 is a hyperplane transversal to @xmath10 , a cross - cut of @xmath10 relative to @xmath11 is a compact connected component @xmath12 . [ ds1.1:cop ] let @xmath7 be a smooth manifold , and @xmath13 , be an immersion , @xmath1 , then @xmath10 is said to have the _ central ovaloid property _ , or , _ cop _ , if * there exists at least one cross - cut whose image is an ovaloid , and * every such ovaloid is central . given this precise terminology , the main theorem can be stated as follows : let @xmath7 be a smooth , connected manifold , and @xmath13 , @xmath1 , be a proper , complete immersion with _ cop _ , then @xmath14 is either a cylinder over a central ovaloid or a quadric . historically , the first theorem of this sort was proved by w. blaschke . @xcite suppose every plane transverse , and nearly tangent to , a smooth convex surface @xmath15 intersects @xmath16 along a central loop . then @xmath16 is a quadric . b. solomon , in 2009 , removed the convexity assumption and first considered hypersurfaces of revolution as follows : @xcite let @xmath6 , @xmath4 , be a hypersurface of revolution . if @xmath2 intersects every hyperplane nearly perpendicular to its axis of rotation in a central set , then @xmath2 is a quadric . using the rotationally symmetric case , solomon then proved the @xmath17 analogue of our main theorem in 2012 . @xcite let @xmath18 have the _ central oval property _ , i.e. , @xmath2 intersects some plane along an oval , and every such oval is central . then a complete , connected , smooth surface @xmath2 with _ central oval property _ must either be a cylinder over a central oval or a quadric . our main result is the generalization of solomon s result @xcite to all dimensions @xmath4 . the proof is not inductive , and although it is , at certain times , a direct generalization of solomon s method , some essentially new difficulties arose . moreover , the technical tools required to carry on the argument are more sophisticated and the calculations are more elaborate . as in solomon s paper , we obtain our main result by first working the local version . if an immersed hypersurface @xmath2 intersects some hyperplane along an ovaloid as our definition of _ cop _ requires , then some neighborhood , in @xmath2 , of that ovaloid embeds into @xmath3 as a tube with _ cop_. the proof that these tubes are either cylindrical or quadric constitute the local version of our work . unless noted otherwise , @xmath19 denotes the open interval @xmath20 . suppose @xmath21 , @xmath4 , is an embedding of the form @xmath22 where @xmath23 and @xmath24 are smooth , and for each fixed @xmath25 , the map @xmath26 parameterizes an ovaloid with center of mass at the origin . a transversely convex tube is any embedded annulus that , after an affine isomorphism , can be parameterized this way . a transversely convex tube in _ standard position _ is the image of an embedding @xmath27 of the form . if we discard the central curve @xmath28 of a transversely convex tube @xmath29 in _ standard position _ , we get the rectification @xmath29 , denoted @xmath30 , the image of @xmath31 we say that @xmath30 splits when we can separate variables : @xmath32 for some functions @xmath33 and @xmath34 parameterizing a fixed ovaloid . as in solomon s paper we get a splitting lemma : * proposition * [ ps5.2:split ] ( splitting lemma)*. * _ if a transversely convex tube @xmath29 in _ standard position _ has _ cop _ , then its rectification @xmath35 splits . _ to obtain this splitting lemma we derive a pair of partial differential equations satisfied by the function @xmath36 which , for each @xmath25 , yields the support function @xmath37 of the ovaloid @xmath38 . using the second pde we cook up a strictly elliptic operator with bounded coefficients and use harnack s inequality to get our splitting lemma . after we are able to split the function @xmath39 we use the first pde to obtain the following local version of our main result for the rectified tube : * proposition * [ ps5.2:classofrcttube]*. * _ suppose @xmath29 is a transversely convex tube in _ standard position _ with _ cop_. then its rectification @xmath30 is either _ 1 . a cylinder over a central ovaloid , or 2 affinely isomorphic to a hypersurface of revolution . * proposition * [ ps5.3:axslemm ] ( axis lemma)*. * combining axis lemma and @xcite we can conclude that a transversely convex tube with _ cop _ is either cylindrical or quadric . this local version of our main result can be stated as * proposition * [ ps5.4:lclvrs ] ( local version)*. * _ a transversely convex tube with _ cop _ is either a cylinder over a central ovaloid or a quadric . _ the final steps of the proof of our main theorem are as follows : given any immersion with _ cop _ and a cross - cut , lemma [ ls3.3:exttubnbd ] ( tubular neighborhood ) shows the existence of a neighborhood of this cross - cut foliated by diffeomorphic copies . then lemma [ ls3.3:exsttrncvxtube ] ( existence of transversely convex tube ) shows that this neighborhood is embedded onto a transversely convex tube with _ cop _ about the image of the cross - cut , when that image is an ovaloid . by proposition [ ps5.4:lclvrs ] ( local version ) , the tube is either cylindrical or quadric . but the boundaries of such a tube , in either case , are again images of cross - cuts and therefore , using lemma [ ls3.3:exttrncvxtube ] ( extension of transversely convex tube ) we can push the boundaries of the annular region in @xmath2 and the tubular region in @xmath3 a little further . since @xmath2 is connected and complete , the extension process stops only when the annular region fills up @xmath2 and the image is either a complete cylinder or a quadric . we would like to thank our advisor bruce solomon for giving us this project , and providing guidance . we would also like to express our gratitude to the indiana university mathematics department for creating an academically challenging and supportive environment , and to the graduate school for the financial support that made our studies possible . in this section , we provide the necessary basic definitions and auxiliary lemmas from convex geometry . all definitions and lemmas come from the book of schneider @xcite . this section ends with a computation of the support function of an ellipsoid . in this subsection , we provide the definition of convex set , convex body , and strictly convex body . a set @xmath40 is convex if for every @xmath41 it contains the segment @xmath42 = \big\{\ , ( 1-\lambda ) x + \lambda y \colon 0 \leq \lambda \leq 1 \,\big\}.\ ] ] a set @xmath43 is called a convex body if @xmath44 , convex , and compact . by @xmath45 we denote the set of all convex bodies in @xmath3 . @xmath40 , the affine hull of @xmath46 , @xmath47 , is the smallest affine subspace of @xmath48 containing @xmath46 . the relative boundary , @xmath49 , is the boundary of @xmath46 relative to its affine hull . a convex body @xmath50 is called a strictly convex body if @xmath51 does not contain any line segment . in this subsection , we introduce the support function and list its important properties that will be relevant to us in later parts . in particular , we remark how the gradient of a support function determines the boundary points of a convex set . we also give a rigorous definition of the centrix and we end this subsection with a calculation of the support function of an ellipsoid . let @xmath43 , @xmath52 , be a convex body . the support function @xmath53 of @xmath54 is defined on @xmath3 by @xmath55 [ ls2.2:spfnpr ] let @xmath50 , then the support function @xmath56 of @xmath54 has the following properties : 1 . @xmath56 is sublinear . [ ls2.2:spfnpr2 ] 2 . @xmath57 and @xmath58 for all @xmath59 . [ ls2.2:spfnpr3 ] 3 . given @xmath60 , @xmath61 . [ ls2.2:spfnpr5 ] 4 . @xmath62 for all @xmath63 and for all @xmath64 , @xmath65 . [ ls2.2:spfnpr6 ] the item of lemma [ ls2.2:spfnpr ] shows that a support function is sublinear . however , the converse is also true . [ ls2.2:subvcxbd ] @xcite if @xmath66 is a sublinear function , then there exists a unique convex body @xmath67 with support function @xmath68 . let @xmath50 be a convex body and @xmath63 , then @xmath69 is called the support hyperplane of @xmath54 with outer normal @xmath64 , @xmath70 is called the supporting halfspace of @xmath54 with outer normal @xmath64 , and @xmath71 is called the support set of @xmath54 with outer normal @xmath64 . for @xmath72 , @xmath73 is the signed distance of the support hyperplane @xmath74 to the origin . note that the support set of convex body does not need to be a singleton . in particular , the convex body @xmath75 in has a support set , which is a line segment . however , if the support set is a singleton then we have an important analytical consequence about the support function . [ ls2.2:diffsptfct ] @xcite let @xmath50 and @xmath76 . the support function @xmath56 is differentiable at @xmath64 if and only if @xmath77 . in this case @xmath78 [ ds2.2:centrix ] let @xmath50 and @xmath56 be differentiable in @xmath79 , then the centrix of @xmath54 is defined as @xmath80 note that the function @xmath81 and hence the centrix of @xmath54 are positively homogeneous of degree zero . we end this subsection with a computation of the support function of an ellipsoid . this support function will play an important rule in the proof of the main theorem . [ es2.5:sptfnctellipsoid ] [ support function of ellipsoid ] given @xmath82 positive numbers @xmath83 @xmath84 , the function @xmath85 defined as @xmath86^{1/2}\ ] ] is sublinear and differentiable in @xmath79 . by lemma [ ls2.2:subvcxbd ] there exists a unique convex body @xmath50 , with @xmath87 and @xmath88 we want to show that the set consisting of the gradients of @xmath56 equals the complete ellipsoid @xmath89 let @xmath90 , then the gradient of @xmath56 at @xmath91 equals @xmath92^{2}}{\lambda_{i}^{2 } } & = \frac{1 } { h_{k}^{2}(x ) } \sum_{i=1}^{n } \lambda_{i}^{2 } x_{i}^{2 } = 1 . \end{aligned}\ ] ] on the other hand let @xmath93 satisfy @xmath94 so we can conclude that @xmath95 and since @xmath54 is convex the equality must hold , and @xmath54 is the convex body with boundary an origin centered ellipsoid . from now on @xmath87 will also be called the support function of the origin centered ellipsoid @xmath96 given in . given any ellipsoid @xmath97 in @xmath3 there exist @xmath60 and @xmath65 so that @xmath98 , hence the support function @xmath99 of @xmath100 can be calculated using the properties and of the support function listed in lemma [ ls2.2:spfnpr ] in the following manner : @xmath101 for every @xmath63 . here , we provide the necessary basic definitions and auxiliary lemmas from multilinear algebra and differential geometry . the references for this material are the books of federer @xcite , hirsch @xcite . we end this section by sketching proofs of the existence and extension of a transversely convex tube . in this subsection , we introduce a principal geometric object of interest , the ovaloid in @xmath3 . using hadamard s theorem @xcite , in the case @xmath4 , we note that every ovaloid has a support parameterization . writing the laplacian in polar coordinates and using lemma [ ls2.2:diffsptfct ] we give an analytical expression for the support parameterization . lastly , we recall that ovaloids are preserved under affine isomorphisms of @xmath3 , find a transversely convex tube about the image of a cross - cut when that image is an ovaloid , and determine sufficient conditions for being able to extend it . an ovaloid @xmath102 , @xmath103 is a closed embedded smooth hypersurface with all principal curvatures positive everywhere with respect to the outer unit normal . in particular , an ovaloid of dimension one , @xmath104 , also called an oval , is an embedded smooth loop in @xmath105 such that , given the outer unit normal vector field @xmath106 and the unit tangent vector field @xmath107 along @xmath108 @xmath109 with @xmath110 , called the geodesic curvature , is positive everywhere . it is a simple exercise of differential and convex geometry to show that that the gauss map of an oval is a diffemorphism and that an oval bounds a strictly convex body . the analogous statement holds in higher dimensions by a theorem due to hadamard @xcite , with an alternate proof in @xcite . hadamard s theorem implies that any ovaloid @xmath102 , @xmath4 is the boundary of a strictly convex body and that its gauss map is a diffeomorphism . using this remark we can make the following definition . the support parameterization of the ovaloid @xmath102 , @xmath103 , that bounds a strictly convex body @xmath54 is defined as the inverse of the gauss map @xmath111 since @xmath112 is bijective and for each @xmath72 , the outer unit normal vector @xmath64 to the hyperplane @xmath74 at the points of @xmath113 is also normal to the boundary @xmath114 , we can conclude that @xmath115 must be a singleton . now using this observation and lemma [ ls2.2:diffsptfct ] we can write @xmath116 so by the sublinearity property of the support function , introduced in item of lemma , and the representation of @xmath117 in polar coordinates we can conclude that the support parameterization is also equal to @xmath118 for all @xmath72 . using the the @xmath119-operator defined in @xcite we get , up to a sign , a unique coordinate representation for the unit normal field defined along a smooth orientable hypersurface . [ rs3.2:outrunitnorfld ] if @xmath6 is a hypersurface and @xmath120 is a local parameterization of @xmath2 then up to a sign the unique unit normal field along @xmath121 is defined as @xmath122 when @xmath6 is a closed embedded hypersurface we let the mapping in define the unit normal field pointing into the unbounded component of @xmath123 . during the course of our proof of the main theorem we want to use the fact that the hypotheses and the conclusion are invariant under affine isomorphisms of @xmath3 . to achieve this , we use the fact that ovaloids are preserved under affine isomorphisms of @xmath3 . [ ls3.2:glnprsvovld ] if @xmath103 , @xmath124 is an affine isomorphism of @xmath3 , and @xmath102 is an ovaloid , then @xmath125 is again an ovaloid . normal curvatures are preserved under isometries and hence ovaloids are preserved under orthogonal maps and translations . since each invertible linear map can be written as the composition of a symmetric linear map and an orthogonal map , it suffices to show that ovaloids are preserved under invertible symmetric maps . the last claim can be easily shown by writing the ovaloid and its image locally as a graph . in this subsection , our main goal is to construct a transversely convex tube @xmath29 about the image of a cross - cut when that image is an ovaloid . in order to achieve this goal , we first need lemma [ ls3.3:exttubnbd ] ( tubular neighborhood ) , which provides a neighborhood for a cross - cut foliated by diffeomorphic copies . since the cross - sections of @xmath29 that are close enough to the image of the cross - cut must also be ovaloids we can use lemma [ ls3.3:exsttrncvxtube ] ( existence of transversely convex tube ) to get the required tube . another important goal is to give sufficient condition for extensibility of a transversely convex tube . in particular , we point out in lemma [ ls3.3:exttrncvxtube ] ( extension of transversely convex tube ) that whenever the boundary of the tube is the image of a cross - cut then one can extend the tube a little further . given @xmath76 , define the linear function @xmath126 by @xmath127 . for any given @xmath76 and @xmath128 , the level hyperplane @xmath129 is defined by @xmath130 [ ls3.3:exttubnbd ] suppose @xmath2 is a smooth manifold , and @xmath131 is a compact connected embedded hypersurface of @xmath2 . assume that for some @xmath76 and @xmath128 , @xmath10 is transversal to @xmath129 along @xmath131 , then there exist @xmath132 and an embedding @xmath133 \to m$ ] with the following properties : 1 . @xmath134 for all @xmath135 , 2 . @xmath136 for all @xmath135 , @xmath137 , 3 . @xmath138 for all @xmath135 , @xmath137 . [ rs3.3:ccutfoliate ] according to lemma [ ls3.3:exttubnbd ] ( tubular neighborhood ) , @xmath131 is a cross - cut of @xmath10 relative to @xmath139 and the neighborhood @xmath140 \,\bigr)$ ] of @xmath131 in @xmath2 is foliated by cross - cuts @xmath141 diffeomorphic to @xmath131 , each a level set of @xmath142 . since the gradient @xmath143 does not vanish on the compact submanifold @xmath131 we can define the vectorfield @xmath144 in an open neighborhood of @xmath131 . using a standard flow box argument for the vectorfield @xmath27 we then get the required embedding @xmath145 . [ ls3.3:exsttrncvxtube ] suppose @xmath1 , the map @xmath8 is an immersion , @xmath146 is a hyperplane , and @xmath147 is a compact connected embedded hypersurface of @xmath2 . if @xmath148 is an ovaloid and the function @xmath149 has no critical point on @xmath131 then @xmath10 maps some neighborhood of @xmath131 in @xmath2 onto a transversely convex tube . we can compose the immersion @xmath10 with the embedding @xmath145 , obtained in lemma [ ls3.3:exsttrncvxtube ] , to get an embedding because @xmath10 embeds @xmath131 onto a simply connected ovaloid and the set of embeddings is open in the strong topology @xcite . since the principal curvatures are continuous there exists an open neighborhood of @xmath148 on the image of @xmath150 so that each horizontal cross section is again an ovaloid . now suppose @xmath10 , @xmath131 are as above and @xmath151 is the transversely convex tube whose existence is guaranteed by lemma [ ls3.3:exsttrncvxtube ] . note that if @xmath10 has _ cop _ so does @xmath151 . let @xmath152 \bigr)$ ] be the neighborhood of @xmath131 that maps under @xmath10 onto @xmath151 as described in lemma [ ls3.3:exttubnbd ] and lemma [ ls3.3:exsttrncvxtube ] . [ ls3.3:exttrncvxtube ] let @xmath8 be an immersion with _ cop _ , and @xmath153 a cross - cut of @xmath10 relative to @xmath154 , then given a transversely convex tube @xmath151 , with _ cop _ , about @xmath148 , where @xmath142 has no critical point in @xmath155 , there exists @xmath156 so that @xmath157 is a transversely convex tube , with _ cop _ , about @xmath148 , where @xmath142 has no critical point in a larger neighborhood @xmath158 of @xmath131 . since each boundary of the transversely convex tube @xmath151 is the image of a cross - cut we can use lemma [ ls3.3:exsttrncvxtube ] to construct a transversely convex tube about each boundary . then it is an easy exercise to glue these two transversely convex tubes about each boundary to the tube @xmath151 to get the required extension . finally , the _ cop _ of @xmath10 implies that the extended tube has also _ here , we give the proof of the main theorem . we first show that the conclusion holds locally and then use standard differential topology arguments to get the global result . the two papers that are cited , @xcite and @xcite , were written by solomon . unless stated otherwise , we assume that @xmath4 throughout this section . in this subsection , we classify the support function of an origin centered ellipsoid as a function whose square is the solution of some differential equation . we define the centrix of an ovaloid and characterize central symmetry in terms of the centrix . [ ls5.1:orgcntellipsoid ] let @xmath159 , @xmath4 , be a non vanishing function on @xmath160 , then @xmath161(u ) = 0\ ] ] for each @xmath162 if and only if @xmath163 or @xmath164 is the support function of an origin centered ellipsoid . @xmath165 : define the linear differential operator @xmath166 as @xmath167 , then given @xmath168 the equality @xmath169 implies @xmath170 , for some constant @xmath171 . the function @xmath172 can be written as @xmath173 , where @xmath174 , @xmath175 is the particular solution given by @xmath176 @xmath177 . the homogeneous solution @xmath178 , @xmath179 satisfies @xmath180 and using the fact that the spherical harmonics are the eigenspaces of the laplace - beltrami operator , @xmath181 @xcite , we can conclude that @xmath182 is a spherical harmonic of degree @xmath183 in @xmath184 . therefore , @xmath182 must be of the form @xmath185 , @xmath186 for some trace - free symmetric matrix @xmath46 . then the function @xmath172 equals @xmath187 is a positive definite symmetric matrix . the set @xmath188 is therefore an origin centered ellipsoid . there exists @xmath189 so that @xmath190 where @xmath191 for each @xmath192 and the set @xmath96 equals @xmath193 px = 1\bigr\}\\ & = \bigl\ { x \in \mathbb{r}^{n-1 } \colon ( px)^{\operatorname{\mathrm{t } } } \operatorname{\mathrm{diag}}\bigl[\ , \lambda_{1}^{-2 } , \dotsc , \lambda_{n-1 } ^{-2 } \,\bigr ] px = 1\bigr\}\\ & = p^{-1 } \bigl\ { x \in \mathbb{r}^{n-1 } \colon x^{\operatorname{\mathrm{t } } } \operatorname{\mathrm{diag}}\bigl[\ , \lambda_{1}^{-2 } , \dotsc , \lambda_{n-1 } ^{-2 } \,\bigr ] x = 1\bigr\}\\ & = p^{-1 } \left\ { x \in \mathbb{r}^{n-1 } \colon \sum_{i=1}^{n-1 } \frac{x_{i}^{2}}{\lambda_{i}^{2 } } = 1 \right \}\\ pe & = \left\ { x \in \mathbb{r}^{n-1 } \colon \sum_{i=1}^{n-1 } \frac{x_{i}^{2}}{\lambda_{i}^{2 } } = 1 \right\}. \end{aligned}\ ] ] the support function of @xmath194 is calculated in example [ es2.5:sptfnctellipsoid ] as @xmath195 and its square satisfies for all @xmath196 @xmath197^{2 } = \sum_{i=1}^{n-1 } \lambda_{i}^{2 } x_{i}^{2 } & = ( x^{\operatorname{\mathrm{t } } } p)b(p^{-1}x ) = ( p^{-1 } x)^{\operatorname{\mathrm{t}}}b(p^{-1}x)\\ \rightarrow \bigl [ ( h_{pe } \circ p)(x ) \bigr]^{2 } & = x^{\operatorname{\mathrm{t}}}bx = h^{2}(x ) . \end{aligned}\ ] ] @xmath198 so we can conclude that @xmath199^{2}$ ] , @xmath200 is the support function of an origin centered ellipsoid and @xmath201 or @xmath202 , depending on whether the nonvanishing function @xmath163 is positive or negative . @xmath203 : for any given @xmath83 , @xmath192 , let @xmath204 x = 1 \bigr\}\ ] ] be an origin centered ellipsoid with principal axes lying on the coordinate axes and @xmath200 the support function of @xmath96 . we need to show that @xmath205 on @xmath160 for some constant @xmath171 . since @xmath206 is positively homogeneous of degree @xmath183 , using the expression of the laplacian in polar coordinates we can compute for any @xmath162 @xmath207 @xmath208 holds for all @xmath162 . thus we can conclude that @xmath209 \equiv 0 \quad \text{on $ \mathbb{s}^{n-2}$.}\ ] ] a general ellipsoid @xmath97 centered at the origin is of the form @xmath210 , where @xmath211 . according to the item of lemma [ ls2.2:spfnpr ] , the support function @xmath99 satisfies @xmath212 for each @xmath162 , by applying the representation of @xmath213 in general coordinates to the particular case @xmath214 we get @xmath215 and hence the spherical laplacian of @xmath216 equals @xmath217 - \left . \frac{\partial^{2}}{\partial r^{2 } } \right\rvert_{r=1 } h_{e}^{2}\bigl[rp(u)\bigr]\\ & = \delta_{\mathbb{r}^{n-1 } } h_{e}^{2}(pu ) - 2(n-1 ) h_{e}^{2 } ( pu ) = \delta_{\mathbb{s}^{n-2 } } h_{e}^{2 } ( pu ) \end{aligned}\ ] ] therefore , for each @xmath162 we can compute @xmath218 so we can conclude that @xmath219 \equiv 0 $ ] on @xmath160 . the centrix of an ovaloid @xmath102 is defined as the centrix of the strictly convex body @xmath54 that it bounds . namely , following definition [ ds2.2:centrix ] , if @xmath163 is the support function of @xmath54 , and hence of @xmath108 by definition , then the centrix of @xmath108 is defined as @xmath220 where @xmath221 is the support parameterization of @xmath108 , @xmath222 , and the second the equality follows because @xmath223 is positively homogeneous of degree zero . [ ls5.1:oddsptparam ] the centrix @xmath225 , when restricted to @xmath226 , coincides with @xmath227 . the centrix is constant if and only if @xmath108 has central symmetry . in that case , @xmath228 support parameterizes the origin centered ovaloid @xmath229 , where @xmath230 . the first claim follows directly from the definition of @xmath28 and @xmath231 . to prove the second claim , assume first that @xmath232 is constant and consider the map @xmath233 which is the reflection through @xmath234 . given @xmath72 @xmath235 = 2 c_{0 } - \gamma(u ) = 2 \frac{\gamma(u ) + \gamma(-u)}{2 } - \gamma(u ) = \gamma(-u).\ ] ] since @xmath72 is arbitrary , we can conclude that @xmath236 and @xmath237 has central symmetry with center @xmath234 . on the other hand , if @xmath108 is central with center @xmath234 , then for every @xmath72 @xmath238}{2 } = \frac{\gamma(u ) + \gamma(-u)}{2 } = c(u)\ ] ] end hence the centrix @xmath28 is constant . in order to show the last claim assume that @xmath108 has the center of symmetry @xmath234 and @xmath239 according to the item of lemma [ ls2.2:spfnpr ] , for every @xmath72 , the support parameterization of @xmath240 satisfies @xmath241 and hence @xmath242 so we can conclude that the odd part @xmath243 of @xmath244 support parameterizes the origin centered ovaloid @xmath229 . our goal is to reparameterize the tube @xmath29 with the use of the support map . using this new parameterization we will construct the height function @xmath245 that lets us parameterize nearly horizontal ovaloid cross - sections as graphs over horizontal ovaloid cross - sections . given any @xmath246 , @xmath247 , the @xmath248-tilted hyperplane is given by @xmath249 where @xmath250 is called the tilt direction , @xmath251 is the @xmath252-intercept , and @xmath248 is the slope of this hyperplane along @xmath250 direction a transversely convex tube @xmath29 in _ standard position _ can be reparameterized as follows : @xmath253 where @xmath254 is a smooth map such that for every @xmath25 , the map @xmath255 support parameterizes the ovaloid @xmath256 , where @xmath257 is the ovaloid with center of mass at @xmath258 . the smooth map @xmath112 is called the support map of @xmath29 . in particular , the construction of the map @xmath112 is carried out as follows : @xmath262 = \bar{\nu } \bigl(\ , c(z ) + \alpha(u , z ) , z \,\bigr)\\ & = \left ( \parbox{4.5cm}{\centering{outer unit normal in $ \mathbb{r}^{n-1}$ to the ovaloid $ \mathcal{o}(z ) - ( c(z ) , z)$ at $ ( \alpha(u , z ) , 0)$ } } , \quad z \right)\\ & = \left ( \frac{\star \bigl[\ , \bigwedge \nolimits_{n-2 } \partial_{1 } \alpha(u , z ) \,\bigr ] \bigl ( \frac{\partial}{\partial u_{1 } } \wedge \cdots \wedge \frac{\partial}{\partial u_{n-2 } } \bigr)}{\bigl\lvert \star \bigl[\ , \bigwedge \nolimits_{n-2 } \partial_{1 } \alpha(u , z ) \,\bigr ] \bigl ( \frac{\partial}{\partial u_{1 } } \wedge \cdots \wedge \frac{\partial}{\partial u_{n-2 } } \bigr ) \bigr\rvert } , \quad z \right ) \end{aligned}\ ] ] the map @xmath263 \bigl ( \frac{\partial } { \partial u_{1 } } \wedge \cdots \wedge \frac{\partial}{\partial u_{n-2 } } \bigr)}{\bigl\lvert \star \bigl[\ , \bigwedge \nolimits_{n-2 } \partial_{1 } \alpha(u , z ) \,\bigr ] \bigl ( \frac{\partial}{\partial u_{1 } } \wedge \cdots \wedge \frac{\partial}{\partial u_{n-2 } } \bigr ) \bigr\rvert}\ ] ] is smooth because @xmath119 is an @xmath264-linear map and @xmath265 is smooth . the map @xmath266 is bijective because for every @xmath25 the ovaloid @xmath267 has a globally defined unit normal field , and the differential has a maximal rank because for every @xmath25 @xmath268 \bigl ( \frac{\partial}{\partial u_{1 } } \wedge \cdots \wedge \frac{\partial}{\partial u_{n-2 } } \bigr)}{\bigl\lvert \star \bigl[\ , \bigwedge \nolimits_{n-2 } \partial_{1 } \alpha(u , z ) \,\bigr ] \bigl ( \frac{\partial}{\partial u_{1 } } \wedge \cdots \wedge \frac{\partial}{\partial u_{n-2 } } \bigr ) \bigr\rvert}\ ] ] is a diffemorphism , which follows from a simple observation in the case @xmath17 ( oval ) , and from hadamard s theorem @xcite in the case @xmath1 . using the inverse function theorem we can conclude that @xmath269 is a diffeomorphism and we can define the reparameterization @xmath270 of @xmath29 as follows : @xmath271 @xmath272 = x(u , z ) = \bigl ( c(z ) + \alpha(u , z ) , z \bigr)$ ] where @xmath273 \bigl ( \frac{\partial}{\partial u_{1 } } \wedge \cdots \wedge \frac{\partial}{\partial u_{n-2 } } \bigr)}{\bigl\lvert \star \bigl[\ , \bigwedge \nolimits_{n-2 } \partial_{1 } \alpha(u , z ) \,\bigr ] \bigl ( \frac{\partial}{\partial u_{1 } } \wedge \cdots \wedge \frac{\partial}{\partial u_{n-2 } } \bigr ) \bigr\rvert}\ ] ] is the outer unit normal in @xmath274 to the ovaloid @xmath275 at @xmath276 . or equivalently , @xmath277 is the outer unit normal to the ovaloid @xmath256 at the point @xmath278.therefore , if we define the map @xmath279 as @xmath280 we can conclude that @xmath281 support parameterizes the ovaloid @xmath282 for each @xmath25 and @xmath283 for each @xmath284 . because of the transverse convexity of @xmath29 and the proof of lemma [ ls3.3:exsttrncvxtube ] , given any @xmath285 , there exists @xmath289 so that for every @xmath247 , @xmath290 , @xmath291 is again an ovaloid . with this remark , our objective now is to show that for any given @xmath285 there exist @xmath292 and a height function @xmath293 so that for every @xmath294 , @xmath295 lets us parameterize the ovaloid @xmath291 by the map @xmath296 we define the smooth map @xmath297 as @xmath298 consider @xmath290 , and observe that @xmath299 \iff y(u , z ) \in \overline{\mathcal{o}}_{\tau}(z_{0 } , \epsilon)\\ & \iff \bigl(\ , \gamma(u , z ) , z \,\bigr ) \in \mathcal{t } \cap p_{\tau , z_{0}}(\epsilon)\\ & \iff \bigl(\ , \gamma(u , z ) , z \,\bigr ) \in p_{\tau , z_{0}}(\epsilon)\\ & \iff z = z_{0 } + \epsilon \bigl [ \tau \cdot \gamma(u , z ) \bigr]\\ & \iff g(\epsilon , u , z ) = 0 . \end{aligned}\ ] ] fix a local parameterization @xmath302 of @xmath160 and let @xmath303 , then the map @xmath304 satisfies @xmath305 therefore , the implicit function theorem applied to @xmath304 yields the existence of an open neighborhood of @xmath306 , where @xmath307 is an open interval about @xmath308 , @xmath309 is an open subset of @xmath310 , and a unique smooth function @xmath311 such that @xmath312 & = 0 \quad \text{for all } \epsilon \in v_{1 } \text { and } v \in v_{2}\\ \rightarrow g \circ \phi \bigl [ \epsilon , v , g(\epsilon , v ) \bigr ] & = g ( \epsilon , v ) - z_{0 } - \epsilon \tau \cdot \gamma \bigl ( \eta(v ) , g(\epsilon , v ) \bigr ) = 0\\ g \circ \phi \bigl [ 0 , v , g(0 , v ) \bigr ] & = g(0 , v ) - z_{0 } = 0\\ g(0 , v ) & = z_{0 } \quad \text{for all $ v \in v_{2}$.}\end{aligned}\ ] ] so there exists a unique smooth function @xmath313 defined as @xmath314 since @xmath160 can be covered by finitely many overlapping open sets @xmath315 , @xmath316 , by taking the intersection @xmath317 , using the uniqueness of the smooth function @xmath318 and by letting @xmath319 we can construct a smooth function @xmath320 that satisfies @xmath321 using the observation in we get for every @xmath294 , @xmath322 & \iff g(\epsilon , u , z ) = 0\\ & \iff z = z(\epsilon , u)\end{aligned}\ ] ] and hence the ovaloid @xmath291 can be parameterized as @xmath323 therefore , the projected ovaloid @xmath324 , for each @xmath247 and @xmath294 , can be parameterized as @xmath325 using the equation we can compute the derivative for each @xmath294 and @xmath162 as @xmath326 [ ps5.2:sptreparam ] given @xmath285 and @xmath247 there exist @xmath332 and a differentiable @xmath333-parameter family of diffeomorphisms @xmath334 such that given the map @xmath335 support parameterizes @xmath324 for each @xmath294 . the initial map @xmath336 is the identity map on @xmath160 , with the initial @xmath248-derivative given by @xmath337 \tau^{\operatorname{\mathrm{t}}}(u)\\ & = \left ( u \cdot \frac{\partial \gamma}{\partial z}(u , z_{0 } ) \right ) \tau^{\operatorname{\mathrm{t}}}(u ) . \end{aligned}\ ] ] let @xmath338 be the interval obtained in definition [ es5.2:impfctdom ] of the height function @xmath245 , then for each @xmath339 the map @xmath340 parameterizes @xmath324 , and @xmath341 is smooth . define the smooth map @xmath342 and for every @xmath294 the map @xmath343 which , according to remark [ rs3.2:outrunitnorfld ] , is the outer unit normal field along the closed embedded hypersurface @xmath344 . the map @xmath345 is a diffeomorphism because @xmath346 parameterizes the ovaloid @xmath324 and the outer unit normal field @xmath347 , in , is a diffeomorphism as a result of a simple observation in the case @xmath17 ( oval ) , and from hadamard s theorem @xcite in the case @xmath1 . since @xmath348 is bijective and its differential has full rank , by the inverse function theorem , @xmath348 is a diffeomorphism with the smooth inverse @xmath349 where @xmath350 , @xmath294 . since @xmath351 is smooth , @xmath352 is a @xmath333-parameter family of diffeomorphisms . for each @xmath353 , since @xmath354 parameterizes @xmath355 , the map @xmath356 support parameterizes @xmath324 because the unit vector @xmath162 is the outer unit normal to the ovaloid @xmath357 at @xmath358 fix @xmath162 , @xmath364 , and choose a smooth curve @xmath365 so that @xmath366 and @xmath367 . then for every @xmath339 @xmath368 = \left . \frac{\partial}{\partial t } \right\rvert_{t=0 } u \cdot ( \gamma_{\epsilon } \circ \theta_{\epsilon } ) \bigr [ c(t ) \bigr ] . \notag\\ \intertext{taking the $ \epsilon$-derivative at $ \epsilon = 0 $ and using the smoothness of $ \gamma$ we obtain } 0 & = \left . \frac{\partial}{\partial \epsilon } \right\rvert_{\epsilon=0 } \left . \frac{\partial}{\partial t } \right\rvert_{t=0 } \bigl\ { u \cdot \gamma \bigl(\ , \theta_{\epsilon}(c(t ) ) , z\bigl ( \epsilon , \theta_{\epsilon}(c(t ) ) \bigr ) \,\bigr ) \bigr \ } \notag\\ & = \left . \frac{\partial}{\partial t } \right\rvert_{t=0 } \left\ { u \cdot \left . \frac{\partial}{\partial \epsilon } \right \rvert_{\epsilon=0 } \gamma \bigl(\ , \theta_{\epsilon}(c(t ) ) , z\bigl ( \epsilon , \theta_{\epsilon}(c(t ) ) \bigr ) \ , \bigr)\right\}. \label{es5.2:drvcompsptfnc1 } \end{aligned}\ ] ] @xmath369 is a smooth curve on @xmath370 whose @xmath248-derivative at @xmath359 equals @xmath371 since @xmath372 , @xmath373 , using the equation we can compute @xmath374 and hence the tangent vector equals @xmath375 the computation of the derivative in can be continued as @xmath376 & = ( \partial_{1}\gamma ) \bigl ( c(t ) , z_{0 } \bigr ) \left ( \left . \frac{\partial}{\partial \epsilon } \right\rvert_{\epsilon=0 } \theta_{\epsilon } \bigl [ c(t ) \bigr ] \right)\\ & \phantom{(\partial_{1}\gamma ) \bigl [ c(t ) , z_{0 } \bigr ] + } + ( \partial_{2 } \gamma ) \bigl ( c(t ) , z_{0 } \bigr ) \bigl(\ , \tau \cdot \gamma \bigl ( c(t ) , z_{0 } \bigr ) \,\bigr ) \end{aligned}\ ] ] @xmath377 \right\}\\ & \phantom{\left . \frac{\partial}{\partial t } \right\rvert_{t=0 } \left\ { u \cdot ( \partial_{1}\gamma ) \bigl ( c(t ) \right\ } } + \left . \frac{\partial}{\partial t } \right\rvert_{t=0 } \bigl\ { u \cdot ( \partial_{2 } \gamma ) \bigl ( c(t ) , z_{0 } \bigr ) \bigl ( \tau \cdot \gamma \bigl ( c(t ) , z_{0 } \bigr ) \bigr ) \bigr\}\\ & = u \cdot \left . \frac{\partial}{\partial t } \right\rvert_{t=0 } \left\ { \operatorname{\mathrm{d}\!}\gamma \bigl ( c(t ) , z_{0 } \bigr ) \left ( \left . \frac{\partial}{\partial \epsilon } \right\rvert_{\epsilon=0 } \bigl ( \theta_{\epsilon } \bigl [ c(t ) \bigr ] , z_{0 } \bigr ) \right ) \right\}\\ & \phantom{\left . \frac{\partial}{\partial t } \right\rvert_{t=0 } \left\ { u \cdot ( \partial_{1}\gamma ) \bigl ( c(t ) \right\ } } + u \cdot \left . \frac{\partial}{\partial t } \right\rvert_{t=0 } \left\ { \left ( \frac{\partial}{\partial z } \right \rvert_{z = z_{0 } } \gamma \bigl ( c(t ) , z \bigr ) \right ) \bigl ( \tau \cdot \gamma \bigl ( c(t ) , z_{0 } \bigr ) \bigr)\\ & = u \cdot \left . \frac{\partial}{\partial t } \right\rvert_{t=0 } \left . \frac{\partial}{\partial \epsilon } \right \rvert_{\epsilon=0 } \gamma \bigl(\ , \theta_{\epsilon } \bigl ( c(t ) \bigr ) , z_{0 } \,\bigr)\\ & \phantom{\left . \frac{\partial}{\partial t } \right\rvert_{t=0 } \left\ { u \cdot ( \partial_{1}\gamma ) \bigl ( c(t ) \right\ } } + u \cdot \left\ { \left ( \left . \frac{\partial}{\partial t } \right\rvert_{t=0 } \left . \frac{\partial}{\partial z } \right\rvert_{z = z_{0 } } \gamma \bigl ( c(t ) , z \bigr ) \right ) \bigl(\tau \cdot \gamma ( u , z_{0 } ) \bigr ) \right\}\\ & \phantom{\left . \frac{\partial}{\partial t } \right\rvert_{t=0 } \left\ { u \cdot ( \partial_{1}\gamma ) \bigl ( c(t ) \right\ } } + u \cdot \left\ { \left ( \left . \frac{\partial}{\partial z } \right\rvert_{z = z_{0 } } \gamma(u , z ) \right ) \left ( \tau \cdot \left . \frac{\partial}{\partial t } \right\rvert_{t=0 } \gamma \bigl ( c(t ) , z_{0 } \bigr ) \right ) \right\ } \end{aligned}\ ] ] @xmath378 = \operatorname{\mathrm{d}\!}\gamma_{0}(u)v , \quad \text{we continue as}\ ] ] @xmath379 \bigl ( \tau \cdot \operatorname{\mathrm{d}\!}\gamma_{0}(u)v \bigr ) . \end{aligned}\ ] ] for each @xmath380 , @xmath281 support parameterizes @xmath381 , and @xmath382 \in \operatorname{\mathrm{t}}_{\gamma(u , z ) } \mathcal{o } ( z ) = \operatorname{\mathrm{t}}_{u } \mathbb{s}^{n-2}\\ & \rightarrow u \cdot \left . \frac{\partial}{\partial t } \right\rvert_{t=0 } \gamma \bigl ( c(t ) , z\bigr ) = 0 \end{aligned}\ ] ] therefore , we can continue the computation as @xmath383 \bigl ( \tau \cdot \operatorname{\mathrm{d}\!}\gamma_{0}(u)v \bigr)\\ & = u \cdot \left . \frac{\partial}{\partial \epsilon } \right\rvert_{\epsilon=0 } \operatorname{\mathrm{d}\!}\gamma_{0 } \bigl [ \theta_{\epsilon } ( u ) \bigr ] \bigl ( \operatorname{\mathrm{d}\!}\theta_{\epsilon}(u)v \bigr ) + \left [ u \cdot \frac{\partial \gamma}{\partial z } ( u , z_{0 } ) \right ] \bigl ( \tau \cdot \operatorname{\mathrm{d}\!}\gamma_{0}(u)v \bigr ) . \end{aligned}\ ] ] in summary , we have the equality @xmath384 \bigl ( \operatorname{\mathrm{d}\!}\theta_{\epsilon}(u)v \bigr ) + \left [ u \cdot \frac{\partial \gamma}{\partial z } ( u , z_{0 } ) \right ] \bigl ( \tau \cdot \operatorname{\mathrm{d}\!}\gamma_{0}(u)v \bigr ) . \end{aligned}\ ] ] note that @xmath385 support parameterizes @xmath386 and so for every @xmath387 @xmath388 \bigl ( \operatorname{\mathrm{d}\!}\theta_{\epsilon}(u)v \bigr ) = 0.\ ] ] when we take the @xmath248-derivative of the equation at @xmath359 we get @xmath389 \bigl ( \operatorname{\mathrm{d}\!}\theta_{\epsilon}(u)v \bigr ) \bigr\}\\ & = \left [ \left . \frac{\partial}{\partial \epsilon } \right\rvert_{\epsilon=0 } \theta_{\epsilon}(u ) \right ] \cdot \operatorname{\mathrm{d}\!}\gamma_{0}(u)v + u \cdot \left . \frac{\partial}{\partial \epsilon } \right\rvert_{\epsilon=0 } \operatorname{\mathrm{d}\!}\gamma_{0 } \bigl [ \theta_{\epsilon}(u ) \bigr ] \bigl ( \operatorname{\mathrm{d}\!}\theta_{\epsilon}(u)v \bigr ) \end{aligned}\ ] ] @xmath390 \bigl(\operatorname{\mathrm{d}\!}\theta_{\epsilon}(u)v \bigr ) = - \left [ \left . \frac{\partial}{\partial \epsilon } \right \rvert_{\epsilon=0 } \theta_{\epsilon}(u ) \right ] \cdot \operatorname{\mathrm{d}\!}\gamma_{0}(u)v.\ ] ] using the equality in the equation , we get @xmath391 \cdot \operatorname{\mathrm{d}\!}\gamma_{0}(u)v + \left [ u \cdot \frac{\partial \gamma}{\partial z}(u , z_{0 } ) \right ] \bigl ( \tau \cdot \operatorname{\mathrm{d}\!}\gamma_{0}(u)v \bigr)\\ & = \left [ \left ( u \cdot \frac{\partial \gamma}{\partial z}(u , z_{0 } ) \right ) \tau - \left . \frac{\partial}{\partial \epsilon } \right\rvert_{\epsilon=0 } \theta_{\epsilon}(u ) \right ] \cdot \operatorname{\mathrm{d}\!}\gamma_{0}(u)v\\ \intertext{since $ \operatorname{\mathrm{d}\!}\gamma_{0}(u)v \in \operatorname{\mathrm{t}}_{\gamma(u , z_{0 } ) } \mathcal{o}(z_{0 } ) = \operatorname{\mathrm{t}}_{u}\mathbb{s } ^{n-2}$ we can replace $ \tau$ with $ \tau^{\operatorname{\mathrm{t}}}(u)$ } & = \left [ \left ( u \cdot \frac{\partial \gamma}{\partial z}(u , z_{0 } ) \right ) \tau^{\operatorname{\mathrm{t}}}(u ) - \left . \frac{\partial } { \partial \epsilon } \right\rvert_{\epsilon=0 } \theta_{\epsilon}(u ) \right ] \cdot \operatorname{\mathrm{d}\!}\gamma_{0}(u)v \end{aligned}\ ] ] holds for every @xmath392 . since @xmath393 is an isomorphism we can conclude that @xmath394 holds for every @xmath162 . our goal is to get an analytic formulation of central symmetry . since the constancy of the centrix is equivalent to central symmetry , an appropriate derivative of the centrix , which we call the symmetry obstruction , will provide the needed formulation . given any tilt direction @xmath247 , @xmath285 , and @xmath332 as in proposition [ ps5.2:sptreparam ] , define the @xmath333-parameter family of smooth maps @xmath395 + \gamma \bigl [ \theta_{\epsilon}(-u ) \bigr]}{2 } \end{aligned}\ ] ] namely , for each @xmath294 , @xmath396 is the centrix of the projected ovaloid @xmath397 . let @xmath398 denote the odd part of @xmath281 for each @xmath399 and define the maps @xmath400 & \bar{\theta}_{\epsilon } \colon \mathbb{s}^{n-2 } \to \mathbb{s}^{n-2 } & & \\ & \bar{\theta}_{\epsilon } = ( \theta_{\epsilon } \circ \rho)(u ) . & & \end{aligned}\ ] ] then we can compute for each @xmath162 @xmath401 @xmath402 which can be equivalently written as @xmath403 supose the horizontal cross - section @xmath404 of a transversely convex tube @xmath29 is central about @xmath405 for each @xmath399 . then for any tilt direction @xmath247 and @xmath285 , we have @xmath406\\ & + \bigl [ \tau \cdot c(z_{0 } ) \bigr ] c'(z_{0 } ) + \bigl [ \tau \cdot \gamma^{-}(u , z_{0})\bigr ] \frac{\partial \gamma^{-}}{\partial z } ( u , z_{0 } ) . \end{aligned}\ ] ] using the notation in the centrix can be written as @xmath407 and the @xmath248-derivative at @xmath359 of the first summand equals @xmath408 \right ) . \label{es5.2:dervcompcnt1 } \end{aligned}\ ] ] since @xmath409 , it follows that @xmath410 and using the equation the derivative in simplifies to @xmath411 + \frac{\partial \gamma}{\partial z}(u , z_{0 } ) \bigl [ \tau \cdot \gamma(u , z_{0 } ) \bigr]\ ] ] for each @xmath25 , @xmath281 support parameterizes the projected ovaloid @xmath404 with center of symmetry at @xmath412 , @xmath413 , and according to lemma [ ls5.1:oddsptparam ] , @xmath414 support parameterizes the ovaloid @xmath415 . using the previous equality and the equation we can conclude that @xmath416 for each @xmath162 and @xmath25 . using these observations we can compute the @xmath248-derivative of @xmath417 at @xmath359 as follows @xmath418 + \frac{\partial \gamma}{\partial z}(u , z_{0 } ) \bigl [ \tau \cdot \gamma(u , z_{0 } ) \bigr ] \right\}\\ & \quad+ \frac{1}{2 } \left\{(\partial_{1 } \gamma ) ( -u , z_{0 } ) \left [ \left . \frac{\partial}{\partial \epsilon } \right \rvert_{\epsilon=0 } \bar{\theta}_{\epsilon}(u ) \right ] + \frac{\partial \gamma}{\partial z}(-u , z_{0 } ) \bigl [ \tau \cdot \gamma(-u , z_{0 } ) \bigr ] \right\}\\ & = \frac{1}{2 } \left\ { ( \partial_{1 } \gamma^{- } ) ( u , z_{0 } ) \left [ \left . \frac{\partial}{\partial \epsilon } \right \rvert_{\epsilon=0 } \theta_{\epsilon}(u ) \right ] \right\}\\ & \quad + \frac{1}{2 } \left\ { \left ( c'(z_{0 } ) + \frac{\partial \gamma^{-}}{\partial z } ( u , z_{0 } ) \right ) \bigl ( \tau \cdot \bigl [ c(z_{0 } ) + \gamma^{-}(u , z_{0 } ) \bigr ] \bigr ) \right\}\\ & \qquad + \frac{1}{2 } \left\ { ( \partial_{1 } \gamma ) ( u , z_{0 } ) \left [ \left . \frac{\partial}{\partial \epsilon } \right \rvert_{\epsilon=0 } \bar{\theta}_{\epsilon}(u ) \right ] \right\}\\ & \quad \qquad + \frac{1}{2 } \left\ { \left ( c'(z_{0 } ) - \frac{\partial \gamma^{-}}{\partial z } ( u , z_{0 } ) \right ) \bigl ( \tau \cdot \bigl [ c(z_{0 } ) - \gamma^{-}(u , z_{0 } ) \bigr ] \bigr ) \right\ } \end{aligned}\ ] ] @xmath419\right\ } + \bigl [ \tau \cdot c(z_{0 } ) \bigr ] c'(z_{0})\\ & \qquad + \bigl [ \tau \cdot \gamma^{-}(u , z_{0 } ) \bigr ] \frac{\partial \gamma^{-}}{\partial z}(u , z_{0 } ) \end{aligned}\ ] ] the proposition [ ps5.2:sptreparam ] gives the @xmath248-derivatives of @xmath327 and @xmath420 as @xmath421 \tau^{\operatorname{\mathrm{t}}}(u)\ ] ] @xmath422 \tau^{\operatorname{\mathrm{t}}}(-u)\\ & = u \cdot \left [ \frac{\partial \gamma^{-}}{\partial z}(u , z_{0 } ) - c'(z_{0 } ) \right ] \tau^{\operatorname{\mathrm{t}}}(u ) \end{aligned}\ ] ] plugging the derivatives and into the equation we get @xmath423\\ & + \bigl [ \tau \cdot c(z_{0 } ) \bigr ] c'(z_{0 } ) + \bigl [ \tau \cdot \gamma^{-}(u , z_{0})\bigr ] \frac{\partial \gamma^{-}}{\partial z } ( u , z_{0 } ) . \qedhere \end{aligned}\ ] ] [ the symmetry obstruction ] given any tilt direction @xmath247 and @xmath285 , @xmath324 is central if and only if @xmath424 is constant , or equivalently , @xmath425 for every @xmath162 and @xmath364 , where the vector field @xmath426 along @xmath427 is extended radially constant in an open neighborhood of @xmath160 in @xmath184 to a vector field in an open subset of @xmath184 . when @xmath324 has central symmetry for all sufficiently small @xmath428 , which _ cop _ requires , we have @xmath429_{u}\ ] ] for every @xmath162 and @xmath364 . therefore , the derivative in forms an obstruction to central symmetry and hence to _ cop_. the spherical divergence of the vectorfield @xmath430 equals @xmath431(u ) & = \sum_{j=1}^{n-2 } e_{j}(u ) \cdot \nabla_{e_{j}(u)}^{\mathbb{s}^{n-2 } } \left [ \left . \frac{\partial } { \partial \epsilon } \right\rvert_{\epsilon=0 } c_{\epsilon}(\cdot , z_{0 } ) \right]_{u}\\ & = \sum_{j=1}^{n-2 } e_{j}(u ) \cdot \overline{\nabla}_{e_{j}(u ) } \left [ \left . \frac{\partial}{\partial \epsilon } \right\rvert_{\epsilon=0 } c_{\epsilon}(\cdot , z_{0 } ) \right]_{u}\end{aligned}\ ] ] where @xmath432 is the local orthonormal field constructed in definition [ ds3.2:orthnmfield ] . since the equality above holds for all @xmath433 we can define a function for each tilt direction @xmath247 as @xmath434(u).\end{aligned}\ ] ] the central ovaloid property of the tube @xmath29 implies @xmath435 for every @xmath162 , @xmath25 , and @xmath247 . for each @xmath436 , @xmath414 support parameterizes the horizontal cross - section @xmath437 of the rectified tube @xmath35 . therefore , there exists a smooth function @xmath438 which , for each @xmath436 , yields the support function of the ovaloid @xmath415 . we call @xmath163 the transverse support function of @xmath35 . [ ps5.2:pde ] on a transversely convex tube @xmath35 with central ovaloid property , the transverse support function @xmath163 of @xmath439 satisfies the two partial differential equations : @xmath440 \ , \bigr\rvert_{u}\\ & \ + \frac{\partial}{\partial z } \bigl[\ , \delta_{\mathbb{s}^{n-2 } } h(u , z ) + ( n-2 ) h(u , z ) \,\bigr ] \nabla_{\mathbb{s}^{n-2 } } h(\cdot , z ) \ , \bigr\rvert_{u}\\ & \ + 2 \left\ { h(u , z ) \nabla_{\mathbb{s}^{n-2 } } \frac{\partial h}{\partial z}(\cdot , z ) \ , \bigr\rvert_{u } + \nabla_{\textstyle \nabla_{\mathbb{s}^{n-2 } } \left . \frac{\partial h}{\partial z } ( \cdot , z ) \,\right\rvert_{u } } ^{\mathbb{s}^{n-2 } } \nabla_{\mathbb{s}^{n-2 } } h ( \cdot , z ) \ , \bigr\rvert_{u } \right\ } = 0 \end{aligned}\ ] ] @xmath441\\ & - \bigl [ \delta_{\mathbb{s } ^{n-2 } } h(u , z ) + ( n-2 ) h(u , z ) \bigr ] \frac{\partial h}{\partial z}(u , z ) = 0 \end{aligned}\ ] ] our aim is to show that for an given @xmath162 , @xmath443 , and @xmath247 @xmath444 \bigr\rvert_{u } \right.\\ & \phantom{= f_{\tau}(u , z_{0 } ) = \tau^{\operatorname{\mathrm{t}}}(u ) \cdot \ } + \frac{\partial}{\partial z } \left [ \delta h(u , z_{0 } ) + ( n-2)h(u , z_{0 } ) \right ] \nabla h(\cdot , z_{0 } ) \ , \bigr\rvert_{u } \notag\\ & \phantom{= f_{\tau}(u , z_{0 } ) = \tau^{\operatorname{\mathrm{t}}}(u ) \cdot \ } + 2 \left . \left ( h(u , z_{0 } ) \nabla \frac{\partial h}{\partial z}(\cdot , z_{0 } ) \ , \bigr\rvert_{u } + \nabla_{\textstyle \nabla \left . \frac{\partial h}{\partial z } ( \cdot , z_{0 } ) \ , \right \rvert_{u } } \nabla h ( \cdot , z_{0 } ) \ , \bigr\rvert_{u } \right ) \right\ } \notag\\ & \phantom{= f_{\tau}(u , z_{0 } ) } + ( \tau \cdot u ) \left\ { h(u , z_{0 } ) \frac{\partial}{\partial z } \bigl [ \delta h(u , z_{0 } ) + ( n-2 ) h(u , z_{0 } ) \bigr ] \right.\\ & \phantom{= f_{\tau}(u , z_{0 } ) = \tau^{\operatorname{\mathrm{t}}}(u ) \quad}\left . - \bigl [ \delta h(u , z_{0 } ) + ( n-2 ) h(u , z_{0 } ) \bigr ] \frac{\partial h}{\partial z}(u , z_{0 } ) \right\}\notag \end{aligned}\ ] ] since @xmath247 is arbitrary and the expressions in curly brackets do not depend on @xmath250 we can conclude that each one of the two expressions must vanish and thus we get the two partial differential equations and . throughout the proof assume that each @xmath455 , @xmath456 , @xmath457 , @xmath452 and @xmath106 is extended radially constant in a neighborhood of @xmath160 in @xmath184 to a vectorfield in an open subset of @xmath184 . using the equation the function @xmath458 can be expanded as @xmath459(u)\\ & = \operatorname{\mathrm{div}}\left [ v \mapsto \bigl ( \tau \cdot \gamma^{-}(v , z_{0 } ) \bigr ) \frac{\partial \gamma^{-}}{\partial z}(v , z_{0 } ) \right](u ) + \operatorname{\mathrm{div}}\bigl [ v \mapsto \bigl ( \tau \cdot c(z_{0 } ) \bigr ) c'(z_{0 } ) \bigr](u)\\ & \quad + \operatorname{\mathrm{div}}\left [ v \mapsto \left ( \frac{\partial \gamma^{-}}{\partial z}(v , z_{0 } ) \cdot v \right ) ( \partial_{1 } \gamma^{-})(v , z_{0 } ) \bigl [ \tau^{\operatorname{\mathrm{t}}}(v ) \bigr ] \right](u ) . \end{aligned}\ ] ] clearly the second summand vanishes and @xmath460 is equal to @xmath461(u)}_{{\ensuremath{\text{\textcircled{\raisebox{-0.2ex}{\textbf{1}}}}}}}\\ & \quad + \underbrace{\operatorname{\mathrm{div}}\left [ v \mapsto \left ( \frac{\partial \gamma^{-}}{\partial z } ( v , z_{0 } ) \cdot v \right ) ( \partial_{1 } \gamma^{-})(v , z_{0 } ) \bigl [ \tau^{\operatorname{\mathrm{t}}}(v ) \bigr ] \right](u)}_{{\ensuremath{\text{\textcircled{\raisebox{-0.3ex}{\textbf{2 } } } } } } } \end{aligned}\ ] ] the computation of the part @xmath462 of the equation is carried out as follows : @xmath463(u)\\ & = \sum_{j=1}^{n-2 } e_{j}(u ) \cdot \overline{\nabla}_{e_{j}(u ) } \left [ v \mapsto \bigl ( \tau \cdot \gamma^{- } ( v , z_{0})\bigr ) \frac{\partial \gamma^{-}}{\partial z}(v , z_{0 } ) \right]_{u}\\ & = \underbrace{\sum_{j=1}^{n-2 } e_{j}(u ) \cdot \left\ { \underbrace{e_{j}\bigl [ v \mapsto \tau \cdot \gamma^{-}(v , z_{0})\bigr](u)}_{{\ensuremath{\text{\textcircled{\raisebox{-0.2ex}{\textbf{1 } } } } } } } \frac{\partial \gamma^{-}}{\partial z}(u , z_{0 } ) \right\}}_{{\ensuremath{\text{\textcircled{\raisebox{-0.3ex}{\textbf{2}}}}}}}\\ & \quad + \underbrace{\sum_{j=1}^{n-2}\bigl [ \tau \cdot \gamma^{-}(u , z_{0 } ) \bigr ] e_{j}(u ) \cdot \underbrace{\overline{\nabla}_{e_{j}(u ) } \left [ v \mapsto \frac{\partial \gamma^{-}}{\partial z}(v , z_{0 } ) \right]_{u}}_{{\ensuremath{\text{\textcircled{\raisebox{-0.2ex}{\textbf{3}}}}}}}}_{{\ensuremath{\text{\textcircled{\raisebox{-0.2ex}{\textbf{4 } } } } } } } \end{aligned}\ ] ] the part @xmath462 of the equation can be expanded as @xmath464(u ) = \tau \cdot \overline{\nabla}_{e_{j}(u ) } \bigl [ v \mapsto \gamma^{-}(v , z_{0 } ) \bigr ] \ , \bigr\rvert_{u}\\ & = \tau \cdot \overline{\nabla}_{e_{j}(u ) } \bigl [ v \mapsto \nabla h(\cdot , z_{0 } ) \ , \bigr\rvert_{v } + h(v , z_{0})v \bigr ] \ , \bigr\rvert_{u}\\ & = \tau \cdot \overline{\nabla}_{e_{j}(u ) } \nabla h(\cdot , z_{0 } ) \ , \bigr\rvert_{u } + \tau \cdot \overline{\nabla}_{e_{j}(u ) } \bigl[v \mapsto h(v , z_{0})v \bigr ] \ , \bigr\rvert_{u}\\ & = \tau^{\operatorname{\mathrm{t}}}(u ) \cdot \nabla_{e_{j}(u ) } \nabla h(\cdot , z_{0 } ) \ , \bigr\rvert_{u } + \bigl[(\tau \cdot u)u \bigr ] \cdot \overline{\nabla}_{e_{j}(u ) } \nabla h(\cdot , z_{0 } ) \ , \bigr\rvert_{u}\\ & \qquad + \tau \cdot \left\ { e_{j}\bigr[h(\cdot , z_{0})\bigr](u)u + h(u , z_{0 } ) \overline{\nabla}_{e_{j}(u)}n \ , \bigr\rvert_{u } \right\ } \end{aligned}\ ] ] recalling the fact that @xmath465 for each @xmath162 we obtain @xmath466(u)}_{{\ensuremath{\text{\textcircled{\raisebox{-0.2ex}{\textbf{1 } } } } } } } & = \tau^{\operatorname{\mathrm{t}}}(u ) \cdot \underbrace{\nabla_{e_{j}(u ) } \nabla h(\cdot , z_{0 } ) \ , \bigr\rvert_{u}}_{{\ensuremath{\text{\textcircled{\raisebox{-0.3ex}{\textbf{2}}}}}}}\\ & + ( \tau \cdot u ) \underbrace{u \cdot \overline{\nabla}_{e_{j}(u ) } \nabla h(\cdot , z_{0 } ) \ , \bigr\rvert_{u } } _ { { \ensuremath{\text{\textcircled{\raisebox{-0.2ex}{\textbf{3}}}}}}}\\ & + \tau \cdot \bigl\ { e_{j}\bigl[h(\cdot , z_{0})\bigr](u)u + h(u , z_{0})e_{j}(u)\bigr\}. \end{aligned}\ ] ] the part @xmath467 of the equation can be written as @xmath468 e_{k } \right]_{u}\\ & = \sum_{k=1}^{n-2 } \nabla_{e_{j}(u ) } \bigl ( e_{k } \bigl [ h(\cdot , z_{0 } ) \bigr ] e_{k } \bigr ) \bigr\rvert_{u}\\ & = \sum_{k=1}^{n-2 } e_{j}\bigl ( e_{k } \bigl [ h(\cdot , z_{0 } ) \bigr ] \bigr)(u ) e_{k}(u ) + e_{k } \bigl [ h(\cdot , z_{0 } ) \bigr](u ) \underbrace{\nabla_{e_{j}(u)}e_{k } \ , \bigr\rvert_{u}}_{=0}\\ & = \sum_{k=1}^{n-2 } e_{j}\bigl ( e_{k } \bigl [ h(\cdot , z_{0 } ) \bigr ] \bigr)(u ) e_{k}(u ) . \end{aligned}\ ] ] the part @xmath469 of the equation can be simplified as @xmath470 \ , \big\rvert_{u } - \overline{\nabla}_{e_{j}(u ) } n \ , \bigr\rvert_{u } \cdot \nabla h(\cdot , z_{0 } ) \ , \bigr\rvert_{u}\\ & = - e_{j}(u ) \cdot \nabla h(\cdot , z_{0 } ) \ , \bigr\rvert_{u}. \end{aligned}\ ] ] since @xmath471 the part @xmath462 of the equation can be written as @xmath472(u)\\ & = \tau^{\operatorname{\mathrm{t}}}(u ) \cdot \left\ { \sum_{k=1}^{n-2 } e_{j } \bigl ( e_{k } \bigl [ h(\cdot , z_{0 } ) \bigr ] \bigr)(u ) e_{k}(u ) + h(u , z_{0 } ) e_{j}(u ) \right\}\\ & \phantom{\tau^{\operatorname{\mathrm{t}}}(u ) \cdot } + ( \tau \cdot u ) \bigl\ { e_{j } \bigl [ h(\cdot , z_{0 } ) \bigr](u ) - e_{j}(u ) \cdot \nabla h(\cdot , z_{0 } ) \ , \bigr\rvert_{u}\bigr\}. \end{aligned}\ ] ] in order to compute the part @xmath467 of the equation we first observe that @xmath473 \notag\\ & = e_{j}(u ) \cdot \left [ \nabla \frac{\partial h}{\partial z}(\cdot , z_{0 } ) \ , \bigr\rvert_{u } + \frac{\partial h } { \partial z}(u , z_{0})u \right ] = e_{j}(u ) \cdot \nabla \frac{\partial h}{\partial z}(\cdot , z_{0 } ) \ , \bigr\rvert_{u } \end{aligned}\ ] ] and then by using the equation we can write @xmath474(u ) \left [ e_{j}(u ) \cdot \nabla \frac{\partial h}{\partial z}(\cdot , z_{0 } ) \ , \bigr\rvert_{u } \right]\\ & = \sum_{j=1}^{n-2 } \left\ { \tau^{\operatorname{\mathrm{t}}}(u ) \cdot \left ( \sum_{k=1}^{n-2 } e_{j } \bigl ( e_{k } \bigl [ h(\cdot , z_{0 } ) \bigr ] \bigr)(u ) e_{k}(u ) + h(u , z_{0 } ) e_{j}(u ) \right ) \right.\\\ & \hspace{1 cm } + ( \tau \cdot u ) \bigl ( e_{j } \bigl [ h(\cdot , z_{0 } ) \bigr](u ) - e_{j}(u ) \cdot \nabla h(\cdot , z_{0 } ) \ , \bigr\rvert_{u}\bigr ) \biggr\ } \left [ e_{j}(u ) \cdot \nabla \frac{\partial h}{\partial z}(\cdot , z_{0 } ) \ , \bigr\rvert_{u } \right ] \end{aligned}\ ] ] @xmath475 e_{j } \bigl ( e_{k } \bigl [ h(\cdot , z_{0 } ) \bigr ] \bigr)(u ) \right ) e_{k}(u)\\ & + \tau^{\operatorname{\mathrm{t}}}(u ) \cdot \sum_{j=1}^{n-2 } \left [ e_{j}(u ) \cdot \nabla \frac{\partial h}{\partial z}(\cdot , z_{0 } ) \ , \bigr\rvert_{u } \right ] h(u , z_{0 } ) e_{j}(u)\\ & + ( \tau \cdot u ) \left ( \sum_{j=1}^{n-2 } \left [ e_{j}(u ) \cdot \nabla \frac{\partial h}{\partial z}(\cdot , z_{0 } ) \ , \bigr \rvert_{u } \right ] e_{j } \bigl [ h(\cdot , z_{0 } ) \bigr](u ) - \nabla h(\cdot , z_{0 } ) \ , \bigr\rvert_{u } \cdot \nabla \frac{\partial h}{\partial z}(\cdot , z_{0 } ) \ , \bigr\rvert_{u } \right ) \end{aligned}\ ] ] where the last dot product is obtained as @xmath476 e_{j}(u ) \cdot \nabla h(\cdot , z_{0 } ) \ , \bigr\rvert_{u}\\ & = \nabla h(\cdot , z_{0 } ) \ , \bigr\rvert_{u } \cdot \sum_{j=1}^{n-2 } \left [ e_{j}(u ) \cdot \nabla \frac{\partial h } { \partial z}(\cdot , z_{0 } ) \ , \bigr\rvert_{u } \right ] e_{j}(u)\\ & = \nabla h(\cdot , z_{0 } ) \ , \bigr\rvert_{u } \cdot \nabla \frac{\partial h}{\partial z } ( \cdot , z_{0 } ) \ , \bigr\rvert_{u } \end{aligned}\ ] ] so we can conclude that the part @xmath467 of the equation equals @xmath477(u ) e_{j}(u ) \cdot \frac{\partial \gamma^{-}}{\partial z}(u , z_{0 } ) \\ & = \tau^{\operatorname{\mathrm{t}}}(u ) \cdot \biggl\ { \sum_{k=1}^{n-2 } \biggl ( \sum_{j=1}^{n-2 } \biggl [ e_{j}(u ) \cdot \nabla \frac{\partial h}{\partial z}(\cdot , z_{0 } ) \ , \bigr\rvert_{u } \biggr ] e_{j } \bigl ( e_{k } \bigl [ h(\cdot , z_{0 } ) \bigr ] \bigr)(u ) \biggr ) e_{k}(u ) \\ & \hspace{2.5 cm } + h(u , z_{0 } ) \nabla \frac{\partial h}{\partial z}(\cdot , z_{0 } ) \ , \bigr\rvert_{u } \biggr\ } \end{aligned}\ ] ] note that @xmath478 terms vanishes and is obtained because @xmath479 e_{j}\bigl [ h(\cdot , z_{0 } ) \bigr](u ) = \nabla h(\cdot , z_{0 } ) \ , \bigr\rvert_{u } \cdot \nabla \frac{\partial h}{\partial z } ( \cdot , z_{0 } ) \ , \bigr\rvert_{u}.\ ] ] in oder to compute the part @xmath480 of the equation , which is @xmath481 \sum_{j=1}^{n-2 } e_{j}(u ) \cdot \overline{\nabla}_{e_{j}(u)}\left [ v \mapsto \frac{\partial \gamma^{-}}{\partial z}(v , z_{0 } ) \right]_{u}\ ] ] we first expand the part @xmath469 of the same equation as @xmath482_{u } = \overline{\nabla}_{e_{j}(u ) } \left [ v \mapsto \nabla \frac{\partial h}{\partial z}(v , z_{0 } ) + \frac{\partial h}{\partial z}(v , z_{0})v \right]_{u}\\ & = \overline{\nabla}_{e_{j}(u ) } \left [ v \mapsto \nabla \frac{\partial h}{\partial z}(v , z_{0 } ) \right]_{u } + \overline{\nabla}_{e_{j}(u ) } \left [ v \mapsto \frac{\partial h}{\partial z}(v , z_{0 } ) n_{v } \right]_{u}\\ & = \overline{\nabla}_{e_{j}(u ) } \left [ \nabla \frac{\partial h}{\partial z}(\cdot , z_{0 } ) \right]_{u } + e_{j } \left [ \frac{\partial h}{\partial z}(\cdot , z_{0 } ) \right ] ( u ) \ , u + \frac{\partial h}{\partial z } ( u , z_{0 } ) \overline{\nabla}_{e_{j}(u ) } n \ , \bigr\rvert_{u } \end{aligned}\ ] ] and when we sum over @xmath483 the dot product with @xmath484 we get @xmath485_{u}\\ & = \sum_{j-1}^{n-2 } e_{j}(u ) \cdot \overline{\nabla}_{e_{j}(u ) } \left [ \nabla \frac{\partial h}{\partial z } ( \cdot , z_{0 } ) \right]_{u } + \sum_{j=1}^{n-2 } \frac{\partial h}{\partial z}(u , z_{0})\\ & = \operatorname{\mathrm{div}}\left [ \nabla \frac{\partial h}{\partial z}(\cdot , z_{0 } ) \right](u ) + ( n - 2 ) \frac{\partial h}{\partial z}(u , z_{0 } ) \\ & = \delta \frac{\partial h}{\partial z}(\cdot , z_{0 } ) \ , \bigr\rvert_{u } + ( n - 2 ) \frac{\partial h}{\partial z}(u , z_{0 } ) = \left . \frac{\partial}{\partial z } \right\rvert_{z = z_{0 } } \bigl [ \delta h(\cdot , z ) \ , \bigr\rvert_{u } + ( n - 2)h(u , z ) \bigr ] . \end{aligned}\ ] ] meanwhile , @xmath486\\ & = \tau^{\operatorname{\mathrm{t}}}(u ) \cdot \nabla h(\cdot , z_{0 } ) \ , \bigr\rvert_{u } + ( \tau \cdot u ) h(u , z_{0 } ) . \end{aligned}\ ] ] therefore , the part @xmath480 of the equation equals @xmath487 \sum_{j=1}^{n-2 } e_{j}(u ) \cdot \overline{\nabla}_{e_{j}(u ) } \left [ v \mapsto \frac{\partial \gamma^{-}}{\partial z}(v , z_{0 } ) \right]_{u}\\ & = \tau^{\operatorname{\mathrm{t}}}(u ) \cdot \left\ { \left . \frac{\partial}{\partial z } \right\rvert_{z = z_{0 } } \bigl [ \delta h(\cdot , z ) \ , \bigr\rvert_{u } + ( n - 2)h(u , z ) \bigr ] \nabla h ( \cdot , z_{0 } ) \ , \bigr\rvert_{u}\right\}\\ & \qquad + ( \tau \cdot u ) h(u , z_{0 } ) \left . \frac{\partial}{\partial z } \right\rvert_{z = z_{0 } } \bigl [ \delta h(\cdot , z ) \ , \bigr\rvert_{u } + ( n - 2)h(u , z ) \bigr ] \end{aligned}\ ] ] then adding the equations and we can conclude that the part @xmath462 of the equation equals @xmath488(u)\\ & = \tau^{\operatorname{\mathrm{t}}}(u ) \cdot \biggl\ { \left . \frac{\partial}{\partial z } \right\rvert_{z = z_{0 } } \bigl [ \delta h(\cdot , z ) \ , \bigr\rvert_{u } + ( n - 2 ) h(u , z ) \bigr ] \nabla h(\cdot , z_{0 } ) \ , \bigr\rvert_{u}\\ & \hspace{0.8cm}+ h(u , z_{0 } ) \nabla \frac{\partial h}{\partial z } ( \cdot , z_{0 } ) \ , \bigr\rvert_{u}\\ & \hspace{0.8cm}+ \sum_{k=1}^{n-2 } \left(\sum_{j=1}^{n-2 } \left [ e_{j}(u ) \cdot \nabla \frac{\partial h } { \partial z } ( \cdot , z_{0 } ) \ , \bigr\rvert_{u } \right ] e_{j}\bigl ( e_{k}\bigl [ h ( \cdot , z_{0 } ) \bigr ] \bigr)(u)\right ) e_{k}(u ) \biggr\}\\ & + ( \tau \cdot u ) \biggl\ { h(u , z_{0 } ) \left . \frac{\partial}{\partial z } \right\rvert_{z = z_{0 } } \bigl [ \delta h(\cdot , z ) \ , \bigr\rvert_{u } + ( n - 2 ) h(u , z ) \bigr ] \biggr\ } \end{aligned}\ ] ] now we want to compute the part @xmath467 of the equation @xmath489 \right](u).\ ] ] note that @xmath490 is a vector field on @xmath160 and @xmath491 support parameterizes the ovaloid @xmath492 , where for every @xmath493 @xmath494 thus we can conclude that @xmath495 $ ] is a vector field on @xmath496 . @xmath497 & = \overline{\nabla}_{\tau^{\operatorname{\mathrm{t}}}(v ) } \gamma^{- } ( \cdot , z_{0 } ) \ , \bigr\rvert_{v } = \overline{\nabla}_{\tau^{\operatorname{\mathrm{t}}}(v ) } \bigl [ \nabla h(\cdot , z_{0 } ) + h(\cdot , z_{0 } ) n \bigr ] \ , \bigr\rvert_{v}\\ & = \overline{\nabla}_{\tau^{\operatorname{\mathrm{t}}}(v ) } \nabla h(\cdot , z_{0 } ) \ , \bigr\rvert_{v } + \overline{\nabla}_{\tau^{\operatorname{\mathrm{t}}}(v ) } h(\cdot , z_{0})n \ , \bigr\rvert_{v}\\ & = \overline{\nabla}_{\tau^{\operatorname{\mathrm{t}}}(v ) } \nabla h(\cdot , z_{0 } ) \ , \bigr\rvert_{v } + \bigl [ \tau^{\operatorname{\mathrm{t}}}h(\cdot , z_{0 } ) \bigr](v ) v + h(v , z_{0 } ) \overline{\nabla}_{\tau^{\operatorname{\mathrm{t}}}(v ) } n \ , \bigr\rvert_{v}\\ & = \overline{\nabla}_{\tau^{\operatorname{\mathrm{t}}}(v ) } \nabla h(\cdot , z_{0 } ) \ , \bigr\rvert_{v } + \bigl [ \tau^{\operatorname{\mathrm{t}}}h(\cdot , z_{0 } ) \bigr](v ) v + h(v , z_{0 } ) \tau^{\operatorname{\mathrm{t}}}(v ) . \end{aligned}\ ] ] since @xmath498 \in \operatorname{\mathrm{t}}_{v } \mathbb{s}^{n-2}$ ] , @xmath499 decomposes as @xmath500(v)v\ ] ] and @xmath501 & = \bigl ( \nabla_{\tau^{\operatorname{\mathrm{t}}}(v ) } \nabla h(\cdot , z_{0 } ) \ , \bigr\rvert_{v } - \bigl [ \tau^{\operatorname{\mathrm{t}}}h(\cdot , z_{0 } ) \bigr](v)v \bigr)\\ & \quad + \bigl [ \tau^{\operatorname{\mathrm{t}}}h(\cdot , z_{0 } ) \bigr](v ) v + h(v , z_{0 } ) \tau^{\operatorname{\mathrm{t}}}(v)\\ & = \nabla_{\tau^{\operatorname{\mathrm{t}}}(v ) } \nabla h(\cdot , z_{0 } ) \ , \bigr\rvert_{v } + h(v , z_{0 } ) \tau^{\operatorname{\mathrm{t}}}(v ) . \end{aligned}\ ] ] therefore , the divergence in can be expanded as @xmath502_{u}\\ & = \underbrace{\sum_{j=1}^{n-2 } e_{j}(u ) \cdot \biggl\ { e_{j } \left [ v \mapsto \frac{\partial \gamma^{-}}{\partial z } ( v , z_{0 } ) \cdot v \right](u ) \bigl ( \nabla_{\tau^{\operatorname{\mathrm{t}}}(u ) } \nabla h(\cdot , z_{0 } ) \ , \bigr\rvert_{u}+ h(u , z_{0 } ) \tau^{\operatorname{\mathrm{t}}}(u ) \bigr ) \biggr\}}_{\textstyle { \ensuremath{\text{\textcircled{\raisebox{-0.2ex}{\textbf{6}}}}}}}\\ & + \underbrace{\sum_{j=1}^{n-2 } e_{j}(u ) \cdot \biggl\ { \left ( \frac{\partial \gamma^{-}}{\partial z}(u , z_{0 } ) \cdot u \right ) \underbrace{\nabla_{e_{j}(u ) } \nabla_{\tau^{\operatorname{\mathrm{t } } } } \nabla h(\cdot , z_{0 } ) \ , \bigr \rvert_{u}}_{\textstyle { \ensuremath{\text{\textcircled{\raisebox{-0.2ex}{\textbf{1 } } } } } } } \biggr\}}_{\textstyle { \ensuremath{\text{\textcircled{\raisebox{-0.3ex}{\textbf{2}}}}}}+ { \ensuremath{\text{\textcircled{\raisebox{-0.2ex}{\textbf{3}}}}}}+ { \ensuremath{\text{\textcircled{\raisebox{-0.2ex}{\textbf{4}}}}}}+ { \ensuremath{\text{\textcircled{\raisebox{-0.2ex}{\textbf{5}}}}}}}\\ & + \underbrace{\sum_{j=1}^{n-2 } e_{j}(u ) \cdot \biggl\ { \left ( \frac{\partial \gamma^{-}}{\partial z}(u , z_{0 } ) \cdot u\right ) \underbrace{\overline{\nabla}_{e_{j}(u ) } \bigl [ v \mapsto h(v , z_{0 } ) \tau^{\operatorname{\mathrm{t}}}(v ) \bigr ] \ , \bigr\rvert_{u}}_{\textstyle { \ensuremath{\text{\textcircled{\raisebox{-0.2ex}{\textbf{7 } } } } } } } \biggr\}}_{\textstyle { \ensuremath{\text{\textcircled{\raisebox{-0.2ex}{\textbf{8 } } } } } } } \end{aligned}\ ] ] using the fact @xmath503 and gauss equations we get @xmath504 } z & = - ( s_{n}x \cdot z ) s_{n}y + ( s_{n}y \cdot z ) s_{n}x\\ & = - ( x \cdot z)y + ( y \cdot z)x \end{aligned}\ ] ] for all @xmath27 , @xmath270 , @xmath505 . so if we let @xmath506 , @xmath507 , and @xmath508 then the part @xmath462 of the equation can be written as @xmath509}_{\textstyle { \ensuremath{\text{\textcircled{\raisebox{-0.2ex}{\textbf{1 } } } } } } } } \nabla h(\cdot , z_{0})}_{\textstyle { \ensuremath{\text{\textcircled{\raisebox{-0.3ex}{\textbf{2}}}}}}}\\ & \underbrace{- \bigl [ s_{n}e_{j } \cdot \nabla h(\cdot , z_{0 } ) \bigr ] s_{n}\tau^{\operatorname{\mathrm{t}}}}_{\textstyle { \ensuremath{\text{\textcircled{\raisebox{-0.2ex}{\textbf{4 } } } } } } } { } + \underbrace{\bigl[s_{n}\tau^{\operatorname{\mathrm{t } } } \cdot \nabla h(\cdot , z_{0})\bigr ] s_{n}e_{j}}_{\textstyle { \ensuremath{\text{\textcircled{\raisebox{-0.2ex}{\textbf{5 } } } } } } } \end{split}\ ] ] the parts @xmath467 , @xmath469 , @xmath480 , and @xmath510 of the equation yield the corresponding parts of the equation . we will first compute the parts of the equation . the part @xmath462 of the equation can be calculated as @xmath511(u ) = \nabla_{e_{j}(u ) } \tau^{\operatorname{\mathrm{t } } } \ , \bigr\rvert_{u } - \nabla_{\tau^{\operatorname{\mathrm{t}}}(u ) } e_{j } \ , \bigr \rvert_{u } = \overline{\nabla}_{e_{j}(u ) } \tau^{\operatorname{\mathrm{t } } } \ , \bigr\rvert_{u } - \overline{\nabla}_{\tau^{\operatorname{\mathrm{t}}}(u ) } e_{j } \bigr\rvert_{u}\\ & = \overline{\nabla}_{e_{j}(u ) } \bigl [ v \mapsto \tau - ( \tau \cdot v)v \bigr ] \ , \bigr\rvert_{u } - \bigl [ \underbrace{\nabla_{\tau^{\operatorname{\mathrm{t}}}(u ) } e_{j } \ , \bigr\rvert_{u}}_{= 0 } { } + \bigl ( \overline{\nabla}_{\tau^{\operatorname{\mathrm{t } } } ( u ) } e_{j } \ , \bigr\rvert_{u } \cdot u\bigr)u \bigr ] \end{aligned}\ ] ] @xmath512u + ( \tau \cdot u ) \overline{\nabla}_{e_{j}(u ) } n \ , \bigr\rvert_{u } \bigr)\\ & \hspace{5.5 cm } - \bigl ( \underbrace{\overline{\nabla}_{\tau^{\operatorname{\mathrm{t}}}(u ) } e_{j } \cdot n \ , \bigr\rvert_{u}}_{=0 } { } - e_{j}(u ) \cdot \overline{\nabla}_{\tau^{\operatorname{\mathrm{t}}}(u ) } n \ , \bigr\rvert_{u } \bigr)u\\ & = - \bigl ( \bigl [ \tau \cdot e_{j}(u ) \bigr ] u + ( \tau \cdot u ) e_{j}(u ) \bigr ) + \bigl [ e_{j}(u ) \cdot \tau^{\operatorname{\mathrm{t}}}(u ) \bigr]u = - ( \tau \cdot u ) e_{j}(u ) . \end{aligned}\ ] ] the parts @xmath480 and @xmath510 of the equation can be written as @xmath513 \cdot \nabla h(\cdot , z_{0 } ) \ , \bigr\rvert_{u } \bigr ) s_{n } \bigl [ \tau^{\operatorname{\mathrm{t}}}(u ) \bigr ] & = - \bigl ( e_{j}(u ) \cdot \nabla h(\cdot , z_{0 } ) \ , \bigr\rvert_{u } \bigr ) \tau^{\operatorname{\mathrm{t}}}(u)\\ \big ( s_{n } \bigl [ \tau^{\operatorname{\mathrm{t}}}(u ) \bigr ] \cdot \nabla h(\cdot , z_{0 } ) \ , \bigr\rvert_{u } \bigr ) s_{n } \bigl [ e_{j}(u ) \bigr ] & = \bigl ( \tau^{\operatorname{\mathrm{t}}}(u ) \cdot \nabla h(\cdot , z_{0 } ) \ , \bigr\rvert_{u } \bigr ) e_{j}(u ) . \end{aligned}\ ] ] using the part @xmath462 , the part @xmath467 of the equation can be written as @xmath514 } \nabla h(\cdot , z_{0 } ) \ , \bigr\rvert_{u } & = - ( \tau \cdot u ) \nabla_{e_{j}(u ) } \nabla h(\cdot , z_{0 } ) \ , \bigr\rvert_{u}\\ & = - ( \tau \cdot u ) \nabla_{e_{j}(u ) } \left ( \sum_{k=1}^{n-2 } e_{k } \bigl [ h(\cdot , z_{0 } ) \bigr ] e_{k } \right)_{u}. \end{aligned}\ ] ] recall that @xmath515 is the support parameterization of the ovaloid @xmath516 given by @xmath517 and hence the part @xmath467 of the equation equals @xmath518(u ) } \nabla h(\cdot , z_{0 } ) \ , \bigr\rvert_{u } \right ] \notag\\ & = \frac{\partial h}{\partial z}(u , z_{0 } ) \sum_{j=1}^{n-2 } e_{j}(u ) \cdot \nabla_{-(\tau \cdot u)e_{j}(u ) } \nabla h(\cdot , z_{0 } ) \ , \bigr\rvert_{u } \notag\\ & = - ( \tau \cdot u ) \frac{\partial h}{\partial z}(u , z_{0 } ) \sum_{j=1}^{n-2 } e_{j}(u ) \cdot \nabla_{e_{j}(u ) } \nabla h(\cdot , z_{0 } ) \ , \bigr\rvert_{u } \notag\\ & = ( \tau \cdot u ) \left [ - \delta h(\cdot , z_{0 } ) \ , \bigr\rvert_{u } \frac{\partial h}{\partial z}(u , z_{0})\right ] . \label{es5.2:pde2;1p2 } \end{aligned}\ ] ] using the equation we can easily compute @xmath519(u ) & = e_{j } \left [ \frac{\partial h}{\partial z}(\cdot , z_{0 } ) \right](u ) . \end{aligned}\ ] ] the part @xmath520 of the equation is equal to @xmath521 \ , \bigr\rvert_{u}\\ & = e_{j } \bigl [ h(\cdot , z_{0 } ) \bigr](u ) \tau^{\operatorname{\mathrm{t}}}(u ) + h(u , z_{0 } ) \overline{\nabla}_{e_{j}(u ) } \bigl[v \mapsto \tau - ( \tau \cdot v)v \bigr ] \ , \bigr\rvert_{u}\notag\\ & = e_{j } \bigl [ h(\cdot , z_{0 } ) \bigr](u ) \tau^{\operatorname{\mathrm{t}}}(u ) + h(u , z_{0 } ) \bigl [ - \overline{\nabla}_{e_{j}(u ) } ( \tau \cdot n)n \ , \bigr\rvert_{u } \bigr ] \notag\\ & = e_{j } \bigl [ h(\cdot , z_{0 } ) \bigr](u ) \tau^{\operatorname{\mathrm{t}}}(u ) - h(u , z_{0 } ) \bigl [ \bigl ( \tau \cdot \overline{\nabla}_{e_{j } ( u)}n \ , \bigr\rvert_{u } \bigr)u + ( \tau \cdot u ) \overline{\nabla}_{e_{j}(u ) } n \ , \bigr\rvert_{u } \bigr ] \notag\\ & = e_{j } \bigl [ h(\cdot , z_{0 } ) \bigr](u ) \tau^{\operatorname{\mathrm{t}}}(u ) - h(u , z_{0 } ) \bigl [ \tau \cdot e_{j}(u ) \bigr]u - h(u , z_{0 } ) ( \tau \cdot u ) e_{j}(u ) . \end{aligned}\ ] ] using the part @xmath520 , part @xmath522 of the equation can be computed as @xmath523 \ , \bigr\rvert_{u } \right\ } \notag\\ & = \frac{\partial h}{\partial z}(u , z_{0 } ) \left\ { \sum_{j=1}^{n-2 } e_{j } \bigl [ h(\cdot , z_{0 } ) \bigr](u ) \bigl ( \tau^{\operatorname{\mathrm{t}}}(u ) \cdot e_{j}(u ) \bigr ) - ( n - 2)(\tau \cdot u ) h(u , z_{0 } ) \right\ } \notag\\ & = \tau^{\operatorname{\mathrm{t}}}(u ) \cdot \left\ { \sum_{j=1}^{n-2 } e_{j } \bigl [ h(\cdot , z_{0 } ) \bigr](u ) \frac{\partial h}{\partial z}(u , z_{0 } ) e_{j}(u ) \right\ } \notag\\ & \hspace{6 cm } + ( \tau \cdot u ) \left [ - ( n - 2 ) h(u , z_{0 } ) \frac{\partial h}{\partial z}(u , z_{0 } ) \right ] \notag\\ & = \tau^{\operatorname{\mathrm{t}}}(u ) \left\ { \frac{\partial h}{\partial z}(u , z_{0 } ) \nabla h(\cdot , z_{0 } ) \ , \bigr\rvert_{u } \right\ } + ( \tau \cdot u ) \left [ - ( n - 2 ) h(u , z_{0 } ) \frac{\partial h}{\partial z}(u , z_{0 } ) \right ] . \label{es5.2:pde2;1p3 } \end{aligned}\ ] ] the part @xmath469 of the equation equals @xmath524 \ , \bigr\rvert_{u } \notag\\ & = \frac{\partial h}{\partial z}(u , z_{0 } ) \nabla_{\tau^{\operatorname{\mathrm{t}}}(u ) } \left ( \sum_{j=1}^{n-2 } e_{j } \cdot \overline{\nabla}_{e_{j } } \nabla h(\cdot , z_{0 } ) \right)_{u } \notag\\ & = \frac{\partial h}{\partial z}(u , z_{0 } ) \nabla_{\tau^{\operatorname{\mathrm{t}}}(u ) } \delta h(\cdot , z_{0 } ) \ , \bigr\rvert_{u } \notag\\ & = \frac{\partial h}{\partial z}(u , z_{0 } ) \nabla_{\tau^{\operatorname{\mathrm{t}}}(u ) } \bigr [ v \mapsto \delta h(v , z_{0 } ) \bigr ] \ , \bigr \rvert_{u } \notag\\ & = \tau^{\operatorname{\mathrm{t}}}(u ) \cdot \left\ { \frac{\partial h}{\partial z}(u , z_{0 } ) \nabla \bigl [ \delta h(\cdot , z_{0 } ) \bigr ] \ , \bigr \rvert_{u } \right\}. \label{es5.2:pde2;1p3 } \end{aligned}\ ] ] the part @xmath480 of the equation can be written as @xmath525 \tau^{\operatorname{\mathrm{t}}}(u ) \right\ } \notag\\ & = - \frac{\partial h}{\partial z}(u , z_{0 } ) \sum_{j=1}^{n-2 } \bigl [ e_{j}(u ) \cdot \nabla h(\cdot , z_{0 } ) \ , \bigr \rvert_{u } \bigr ] e_{j}(u ) \cdot \tau^{\operatorname{\mathrm{t}}}(u ) \notag\\ & = \tau^{\operatorname{\mathrm{t}}}(u ) \cdot \left\ { - \frac{\partial h}{\partial z}(u , z_{0 } ) \nabla h(\cdot , z_{0 } ) \ , \bigr\rvert_{u } \right\}. \label{es5.2:pde2;1p4 } \end{aligned}\ ] ] the part @xmath510 of the equation is equal to @xmath526 e_{j}(u)\right\ } \notag\\ & = \frac{\partial h}{\partial z}(u , z_{0 } ) \bigl [ \tau^{\operatorname{\mathrm{t}}}(u ) \cdot \nabla h(\cdot , z_{0 } ) \ , \bigr\rvert_{u } \bigr ] \sum_{j=1}^{n-2 } e_{j}(u ) \cdot e_{j}(u ) \notag\\ & = \tau^{\operatorname{\mathrm{t}}}(u ) \cdot \left\{(n - 2 ) \frac{\partial h}{\partial z}(u , z_{0 } ) \nabla h(\cdot , z_{0 } ) \ , \bigr\rvert_{u } \right\}. \label{es5.2:pde2;1p5 } \end{aligned}\ ] ] by summing the equations , , , and we get @xmath527 \ , \bigr\rvert_{u } - \frac{\partial h}{\partial z}(u , z_{0 } ) \nabla h(\cdot , z_{0 } ) \ , \bigr\rvert_{u}\\ & \phantom{\tau^{\operatorname{\mathrm{t}}}(u ) \cdot \bigl\ { \frac{\partial h}{\partial z}(u , z_{0 } ) \nabla \bigl [ \delta h(\cdot , z_{0 } ) \bigr ] \ , \bigr\rvert_{u } - { } } + ( n -2 ) \frac{\partial h}{\partial z}(u , z_{0 } ) \nabla h(\cdot , z_{0 } ) \ , \bigr \rvert_{u } \bigr\}\\ & \qquad + ( \tau \cdot u ) \left\ { - \frac{\partial h}{\partial z}(u , z_{0 } ) \delta h(\cdot , z_{0 } ) \ , \bigr\rvert_{u } \right\ } \end{aligned}\ ] ] the part @xmath528 of the equation can be computed as @xmath529(u ) \bigl ( \nabla_{\tau^{\operatorname{\mathrm{t}}}(u ) } \nabla h(\cdot , z_{0 } ) \ , \bigr\rvert_{u}+ h(u , z_{0 } ) \tau^{\operatorname{\mathrm{t}}}(u ) \bigr ) \biggr\ } \notag\\ & = \sum_{j=1}^{n-2 } e_{j}(u ) \cdot \left\ { e_{j}\left [ \frac{\partial h}{\partial z}(\cdot , z_{0 } ) \right](u ) \ \nabla_{\tau^{\operatorname{\mathrm{t}}}(u ) } \nabla h(\cdot , z_{0 } ) \ , \bigr\rvert_{u}\right\}\notag\\ & \hspace{4 cm } + \sum_{j=1}^{n-2 } e_{j}(u ) \cdot \left\ { e_{j } \left [ \frac{\partial h}{\partial z}(\cdot , z_{0 } ) \right](u ) \ h(u , z_{0 } ) \tau^{\operatorname{\mathrm{t}}}(u ) \right\}\notag \end{aligned}\ ] ] @xmath530(u ) \ \bigl\ { e_{j}(u ) \cdot \nabla_{\tau^{\operatorname{\mathrm{t}}}(u ) } \nabla h(\cdot , z_{0 } ) \ , \bigr\rvert_{u } \bigr\}\notag\\ & \hspace{4 cm } + \tau^{\operatorname{\mathrm{t}}}(u ) \cdot \sum_{j=1}^{n-2 } h(u , z_{0 } ) e_{j } \left [ \frac{\partial h}{\partial z}(\cdot , z_{0 } ) \right](u ) \ e_{j}(u ) \notag\\ & = \sum_{j=1}^{n-2 } e_{j } \left [ \frac{\partial h}{\partial z}(\cdot , z_{0 } ) \right](u ) \ \nabla_{\tau^{\operatorname{\mathrm{t}}}(u ) } \bigl [ e_{j } \cdot \nabla h(\cdot , z_{0 } ) \bigr ] \ , \bigr\rvert_{u } + \tau^{\operatorname{\mathrm{t}}}(u ) \cdot h(u , z_{0 } ) \nabla \frac{\partial h}{\partial z}(\cdot , z_{0 } ) \ , \bigr\rvert_{u } \notag\\ & = \sum_{j=1}^{n-2 } e_{j } \left [ \frac{\partial h}{\partial z}(\cdot , z_{0 } ) \right](u ) \ \bigl\ { \tau^{\operatorname{\mathrm{t}}}(u ) \cdot \nabla \bigl [ e_{j } \cdot \nabla h(\cdot , z_{0 } ) \bigr ] \ , \bigr\rvert_{u } \bigr\ } \notag\\ & \hspace{4 cm } + \tau^{\operatorname{\mathrm{t}}}(u ) \cdot h(u , z_{0 } ) \nabla \frac{\partial h}{\partial z}(\cdot , z_{0 } ) \ , \bigr\rvert_{u } \notag\\ & = \tau^{\operatorname{\mathrm{t}}}(u ) \cdot \left\ { \sum_{j=1}^{n-2 } e_{j } \left [ \frac{\partial h}{\partial z}(\cdot , z_{0 } ) \right](u ) \ \nabla \bigl [ e_{j } \cdot \nabla h(\cdot , z_{0 } ) \bigr ] \ , \bigr\rvert_{u } + h(u , z_{0 } ) \nabla \frac{\partial h}{\partial z } ( \cdot , z_{0 } ) \ , \bigr\rvert_{u } \right\}. \label{es5.2:pde2;1p1 } \end{aligned}\ ] ] by summing the equations , and we find that the divergence at @xmath162 @xmath531 \right ] ( u)\ ] ] equals @xmath532 \ , \bigr\rvert_{u } + h(u , z_{0 } ) \nabla \frac{\partial h}{\partial z}(\cdot , z_{0 } ) \ , \bigr \rvert_{u}\\ & \phantom{\tau^{\operatorname{\mathrm{t}}}(u ) \cdot \biggl\ { \frac{\partial h}{\partial z}(u , z_{0 } ) \nabla \bigl [ \delta h(\cdot , z_{0})}+ \sum_{j=1}^{n-2 } e_{j } \left [ \frac{\partial h}{\partial z}(\cdot , z_{0 } ) \right](u ) \ \bigl [ e_{j } \cdot \nabla h(\cdot , z_{0 } ) \bigr ] \ , \bigr\rvert_{u } \biggr\}\\ & \quad + ( \tau \cdot u ) \left\ { - \bigl [ \delta h(\cdot , z_{0 } ) \ , \bigr\rvert_{u } + ( n -2)h(u , z_{0 } ) \bigr ] \frac{\partial h}{\partial z}(u , z_{0})\right\ } . \end{aligned}\ ] ] finally the value of the function @xmath458 at @xmath533 can be found by adding the equations and @xmath534(u)\\ & = \operatorname{\mathrm{div}}\left [ v \mapsto \bigl ( \tau \cdot \gamma^{-}(v , z_{0 } ) \bigr ) \frac{\partial \gamma^{-}}{\partial z}(v , z_{0 } ) \right](u)\\ & \hspace{4 cm } + \operatorname{\mathrm{div}}\left [ v \mapsto \left ( \frac{\partial \gamma^{-}}{\partial z}(v , z_{0 } ) \cdot v \right ) ( \partial_{1}\gamma^{-})(v , z_{0 } ) \bigl [ \tau^{\operatorname{\mathrm{t}}}(u ) \bigr ] \right ] \end{aligned}\ ] ] @xmath535 \nabla h(\cdot , z_{0 } ) \ , \bigr\rvert_{u}\\ & \hspace{1.8 cm } + 2 h(u , z_{0 } ) \nabla \frac{\partial h}{\partial z}(u , z_{0 } ) \ , \bigr\rvert_{u } + \underbrace{\sum_{j=1}^{n-2 } e_{j } \left [ \frac{\partial h}{\partial z}(\cdot , z_{0 } ) \right](u ) \ \nabla \bigl [ e_{j } \cdot \nabla h(\cdot , z_{0 } ) \bigr](u)}_{\textstyle { \ensuremath{\text{\textcircled{\raisebox{-0.2ex}{\textbf{1 } } } } } } } \\ & \hspace{3.8 cm } + \underbrace{\sum_{k=1}^{n-2 } \left ( \sum_{j=1}^{n-2 } e_{j } \left [ \frac{\partial h } { \partial z}(\cdot , z_{0})\right](u ) \ e_{j } \bigl [ e_{k}h(\cdot , z_{0 } ) \bigr](u ) \right)e_{k}(u)}_{\textstyle { \ensuremath{\text{\textcircled{\raisebox{-0.3ex}{\textbf{2}}}}}}}\\ & \hspace{3.8 cm } + \frac{\partial h}{\partial z}(u , z_{0 } ) \nabla \bigl [ \delta h(\cdot , z_{0 } ) + ( n -2 ) h(\cdot , z_{0 } ) \bigr ] \ , \bigr\rvert_{u } \biggr\ } \end{aligned}\ ] ] @xmath536\\ - \bigl [ \delta h(\cdot , z_{0 } ) \ , \bigr\rvert_{u } + ( n - 2 ) h(u , z_{0 } ) \bigr ] \frac{\partial h}{\partial z}(u , z_{0 } ) \biggr\ } \end{gathered}\ ] ] @xmath537(u ) \ \bigl [ e_{j } \cdot \nabla h(\cdot , z_{0 } ) \bigr](u)\\ & = \sum_{j=1}^{n-2 } e_{j } \left [ \frac{\partial h}{\partial z}(\cdot , z_{0 } ) \right](u ) \sum_{k=1}^{n-2 } e_{k } \bigl [ e_{j } \cdot \nabla h(\cdot , z_{0 } ) \bigr](u ) e_{k}(u)\\ & = \sum_{j , k=1}^{n-2 } e_{j } \left [ \frac{\partial h}{\partial z}(\cdot , z_{0 } ) \right](u ) \bigl\ { \underbrace{\nabla_{e_{k}(u ) } e_{j } \ , \bigr\rvert_{u}}_{=0 } \cdot \nabla h(\cdot , z_{0})\\ & \hspace{5 cm } + e_{j}(u ) \cdot \nabla_{e_{k}(u ) } \nabla h(\cdot , z_{0 } ) \ , \bigr\rvert_{u}\bigr\ } e_{k}(u)\\ & = \sum_{j , k=1}^{n-2 } e_{j } \left [ \frac{\partial h}{\partial z}(\cdot , z_{0 } ) \right](u ) \left\ { e_{j}(u ) \cdot \sum_{l=1}^{n-2 } \nabla_{e_{k}(u ) } \bigl ( e_{l } \bigl [ h(\cdot , z_{0 } ) \bigr ] e_{l } \bigr)_{u } \right\ } e_{k}(u)\\ & = \sum_{j , k , l=1}^{n-2 } e_{j } \left [ \frac{\partial h}{\partial z}(\cdot , z_{0})\right](u ) \biggl\ { e_{j}(u ) \cdot \bigl [ e_{k } \bigl [ e_{l } ( h ( \cdot , z_{0 } ) ) \bigr](u ) \ e_{l}(u)\\ & \hspace{5 cm } + e_{l } \bigl[h(\cdot , z_{0})\bigr](u ) \underbrace{\nabla_{e_{k}(u)}e_{l } \ , \bigr\rvert_{u } } _ { = 0}\bigr ] \biggr\ } e_{k}(u)\\ & = \sum_{j , k=1}^{n-2 } e_{j } \left [ \frac{\partial h}{\partial z}(\cdot , z_{0 } ) \right](u ) \left\ { \sum_{l=1}^{n-2 } e_{j}(u ) \cdot \bigl [ e_{k } \bigl [ e_{l}(h ( \cdot , z_{0 } ) ) \bigr](u ) \bigr ] e_{l}(u)\right\ } e_{k}(u ) \end{aligned}\ ] ] @xmath538(u ) \ e_{k } \bigl [ e_{j}(h(\cdot , z_{0 } ) ) \bigr](u ) \ , e_{k}(u)\\ & \mspace{-58mu } = \sum_{k=1}^{n-2 } \left\ { \sum_{j=1}^{n-2 } e_{j } \left [ \frac{\partial h}{\partial z } ( \cdot , z_{0 } ) \right](u ) \ e_{k } \bigl [ e_{j}(h(\cdot , z_{0 } ) ) \bigr](u ) \right\ } e_{k}(u)\\ & \mspace{-58mu } = \sum_{k=1}^{n-2 } \left\ { \sum_{j=1}^{n-2 } e_{j } \left [ \frac{\partial h}{\partial z } ( \cdot , z_{0 } ) \right](u ) \ e_{j } \bigl [ e_{k}(h(\cdot , z_{0 } ) ) \bigr](u ) \right\ } e_{k}(u ) . \end{aligned}\ ] ] the change of order of differentiation at the last step is permissible because for every smooth function @xmath539 , @xmath162 and for every @xmath540 , @xmath541 @xmath542 f \bigr)(u ) & = \bigl [ \nabla_{e_{j}(u ) } e_{k } \ , \bigr\rvert_{u } - \nabla_{e_{k}(u ) } e_{j } \ , \bigr\rvert_{u } \bigr ] f = 0 \\ e_{j}(e_{k}f)(u ) & - e_{k}(e_{j}f)(u ) = 0 \end{aligned}\ ] ] so we can conclude the parts @xmath462 and @xmath467 of the equation are equal and now we want to show that they are also equal to @xmath543 @xmath544 \nabla_{e_{j}(u ) } \bigl [ \nabla h(\cdot , z_{0 } ) \bigr ] \ , \bigr\rvert_{u}\\ & = \sum_{j=1}^{n-2}e_{j } \left [ \frac{\partial h}{\partial z}(\cdot , z_{0 } ) \right ] ( u ) \sum_{k=1}^{n-2 } \nabla_{e_{j}(u ) } \bigl ( v \mapsto e_{k } \bigl [ h(\cdot , z_{0 } ) \bigr](v ) \ , e_{k}(v ) \bigr)_{u } \end{aligned}\ ] ] @xmath545(u ) \bigl\ { e_{j } \bigl [ e_{k } \bigl [ h(\cdot , z_{0 } ) \bigr ] \big](u ) \ , e_{k}(u ) + e_{k } \bigl [ h(\cdot , z_{0 } ) \bigr](u ) \underbrace{\nabla_{e_{j } ( u ) } e_{k } \ , \bigr\rvert_{u}}_{=0}\bigr\}\\ & = \sum_{j , k=1}^{n-2 } e_{j } \left [ \frac{\partial h}{\partial z}(\cdot , z_{0 } ) \right](u ) \ e_{j } \bigl [ e_{k } \bigl [ h(\cdot , z_{0 } ) \bigr ] \big](u ) \ , e_{k}(u)\\ & = \sum_{k=1}^{n-2 } \left\ { \sum_{j=1}^{n-2 } e_{j } \left [ \frac{\partial h}{\partial z}(\cdot , z_{0 } ) \right ] ( u ) \ e_{j } \bigl [ e_{k } \bigl [ h(\cdot , z_{0 } ) \bigr ] \bigr ] ( u ) \right\ } e_{k}(u ) . \end{aligned}\ ] ] @xmath546 \ , \bigr\rvert_{u}\\ & \hspace{3.3cm}+ \left . \frac{\partial}{\partial z } \right\rvert_{z = z_{0 } } \bigl [ \delta h(\cdot , z_{0 } ) + ( n -2)h(u , z ) \bigr ] \nabla h(\cdot , z_{0 } ) \ , \bigr\rvert_{u}\\ & \hspace{3.3cm}+2 \left ( h(u , z_{0 } ) \nabla \frac{\partial h}{\partial z}(\cdot , z_{0 } ) \ , \bigr\rvert_{u } + \nabla_{\textstyle \nabla \left . \frac{\partial h}{\partial z}(\cdot , z_{0 } ) \ , \right\rvert_{u } } \nabla h ( \cdot , z_{0 } ) \ , \bigr\rvert_{u}\right ) \bigg\}\\ & \phantom{f_{\tau}(u , z_{0 } ) = } + ( \tau \cdot u ) \ , \biggl\{h(u , z_{0 } ) \left . \frac{\partial}{\partial z } \right \rvert_{z = z_{0 } } \bigl [ \delta h(\cdot , z_{0 } ) \ , \bigr\rvert_{u } + ( n -2 ) h(u , z ) \bigr]\\ & \phantom{f_{\tau}(u , z_{0 } ) = } \hspace{3.8 cm } - \bigl [ \delta h(\cdot , z_{0 } ) \ , \bigr\rvert_{u } + ( n - 2 ) h(u , z_{0 } ) \bigr ] \frac{\partial h}{\partial z}(u , z_{0 } ) \biggr\}. \end{aligned}\ ] ] as we mentioned in the beginning the expressions in the curly brackets do not depend on @xmath247 , @xmath547 for all @xmath247 and hence these expressions must individually vanish for each @xmath162 and @xmath285 . let @xmath548 be the transverse support function of the rectified tube @xmath549 and @xmath550 be the support map of @xmath551 given by @xmath552 if we can show that @xmath553 is the product of @xmath554 and the support function @xmath555 of a fixed ovaloid , then @xmath556 + \bigl [ r(z)h_{0}(u ) \bigr]u\\ & = r(z ) \bigl [ \nabla_{\mathbb{s}^{n-2 } } h_{0}(u ) + h_{0}(u)u \bigr ] = r(z ) \gamma^{\circ}(u ) \end{aligned}\ ] ] where @xmath557 is the support parameterization of a fixed ovaloid . according to proposition [ ps5.2:pde ] , the transverse support function @xmath163 satisfies the partial differential equation , which is equivalent to @xmath558 at height @xmath559 , using @xcite , we have @xmath560 for every @xmath162 , where @xmath561 is the @xmath562 principal curvature of the ovaloid @xmath563 , @xmath564 . therefore , the equation is equivalent to @xmath565 for every @xmath162 and @xmath436 . and the condition is equivalent to @xmath566 define the operator @xmath567 , then the transverse support function h satisfies @xmath568 let @xmath569 be the support function of the ovaloid @xmath570 . our aim is to show that @xmath571 for some positive function @xmath554 of height @xmath245 . by construction @xmath572 for each @xmath162 and assume that there exists @xmath573 satisfying @xmath574 and @xmath575 for each @xmath162 . since @xmath576 is a strictly positive function , @xmath576 and @xmath577 are continuous on a compact set , we can fix @xmath578 so that @xmath579 on @xmath160 . define the positive constants @xmath580 and @xmath581 as follows : @xmath582 the subset @xmath583 defined as @xmath584 is nonempty because there exists @xmath162 with @xmath585 the subset @xmath586 is also closed because @xmath587 is a continuous function . now we want to show that @xmath586 is also open and then conclude that @xmath588 . let @xmath589 , and consider a local coordinate system @xmath590 of @xmath160 about @xmath64 @xmath591 consider the function @xmath592 , which is smooth and satisfies @xmath593 because @xmath594 for every @xmath493 . the operator @xmath595 , under the local coordinate system @xmath590 , is strictly elliptic in the domain @xmath596 for some @xmath597 . therefore , the operator @xmath166 and the smooth function @xmath598 in @xmath596 satisfy the hypotheses of harnack s inequality @xcite and hence exists a positive constant @xmath599 depending only on @xmath600 and @xmath166 so that for some closed ball @xmath601 @xmath602 and the last equality holds because @xmath603 and @xmath604 . so we can conclude that @xmath605 } = 0.\ ] ] for the open ball @xmath606 of radius @xmath607 and centered at the origin , @xmath608 $ ] is an open neighborhood of @xmath589 in which @xmath587 vanishes identically and hence @xmath609 \subseteq u.\ ] ] so we can conclude that @xmath610 on @xmath160 . we show that when @xmath29 is a transversely convex tube in _ standard position _ , and @xmath35 is not a cylinder , then there exists a linear isomorphism that fixes the @xmath252-axis while making each horizontal cross - section @xmath615 of @xmath35 simultaneously spherical . according to lemma [ ps5.2:split ] ( splitting lemma ) , the transverse support function @xmath163 can be written as @xmath616 for each @xmath436 and @xmath162 . the partial differential equation can therefore be written as @xmath617 ( u ) + ( n -2 ) \nabla h_{0}(u)\bigr\}\\ & \quad + r(z)r'(z ) \bigl\ { \bigl [ \delta h_{0}(u ) \bigr ] \nabla h_{0}(u ) + ( n -2 ) h_{0}(u ) \nabla h_{0}(u ) \bigr \}\\ & \quad + 2 \left\ { r(z)r'(z ) h_{0}(u ) \nabla h_{0 } ( u)+ \nabla_{\textstyle \nabla r'(z ) h_{0}(u ) } \nabla r(z)h_{0}(u)\right\}\\ & = r(z)r'(z ) \bigl\ { h_{0}(u ) \nabla \bigl [ \delta h_{0}\bigr](u ) + 2(n -1)h_{0}(u ) \nabla h_{0}(u)\\ & \qquad \phantom{r(z)r'(z ) \bigl\ { } + \bigl [ \delta h_{0}(u ) \bigr ] \nabla h_{0}(u ) + 2 \nabla_{\textstyle \nabla h_{0}(u ) } \nabla h_{0}(u ) \bigr\}. \end{aligned}\ ] ] if @xmath35 is not cylindrical , then there exists @xmath443 , @xmath618 and since @xmath619 we must have @xmath620 \nabla h_{0}(u ) + h_{0}(u ) \nabla \bigl [ \delta h_{0 } \bigr](u ) + 2(n-1 ) h_{0 } ( u ) \nabla h_{0}(u ) \\ & \phantom{{}= \bigl [ \delta h_{0}(u ) \bigr ] \nabla h_{0}(u ) + h_{0}(u ) \nabla \bigl [ \delta h_{0 } \bigr](u ) } + \underbrace{2 \nabla_{\textstyle \nabla h_{0}(u ) } \nabla h_{0}(u)}_{\textstyle { \ensuremath{\text{\textcircled{\raisebox{-0.2ex}{\textbf{1 } } } } } } } \end{aligned}\ ] ] for each @xmath162 . let s consider the smooth function @xmath621 on @xmath160 and calculate its spherical gradient @xmath622(u ) = \nabla \bigl [ \delta h_{0}^{2 } \bigr](u ) + 2 ( n -1 ) \nabla h_{0}^{2}(u)\\ & = \nabla \bigl [ 2 h_{0 } \delta h_{0 } + 2 \lvert \nabla h_{0 } \rvert^{2 } \bigr](u ) + 4(n - 1 ) h_{0}(u ) \nabla h_{0}(u ) \\ & = 2 \bigl [ \delta h_{0}(u ) \bigr ] \nabla h_{0}(u ) + 2 h_{0}(u ) \nabla \bigl [ \delta h_{0}\bigr](u ) + 4 ( n -1)h_{0}(u ) \nabla h_{0}(u)\\ & \phantom{{}= 2 \bigl [ \delta h_{0}(u ) \bigr ] \nabla h_{0}(u ) + 2 h_{0}(u ) \nabla \bigl [ \delta h_{0}\bigr](u)}+ \underbrace{2 \nabla \bigl [ \nabla h_{0 } \cdot \nabla h_{0 } \bigr]}_{\textstyle { \ensuremath{\text{\textcircled{\raisebox{-0.2ex}{\textbf{1}}}}}}}. \end{aligned}\ ] ] the part @xmath462 of the equation can be expanded using the @xmath623 as follows : @xmath624(u ) \notag\\ & = 2 \sum_{j , k=1}^{n-2 } \bigl [ ( e_{k}h_{0})(u ) \ e_{k } ( e_{j}h_{0})(u ) \bigr ] e_{j}(u ) \notag\\ \phantom{2 \nabla_{\textstyle \nabla h_{0}(u ) } \nabla h_{0}(u ) } & = 2 \sum_{j=1}^{n-2 } \left [ \sum_{k=1}^{n-2 } ( e_{k}h_{0 } ) ( u ) \ e_{k}(e_{j } h_{0})(u ) \right ] e_{j}(u ) \notag\\ & = 2 \sum_{j=1}^{n-2 } \left [ \sum_{k=1}^{n-2 } ( e_{k}h_{0 } ) ( u ) \ e_{j}(e_{k } h_{0})(u ) \right ] e_{j}(u ) \end{aligned}\ ] ] where the change of the order of differentiation at the last equality is possible because of the observation . again by using the @xmath625 , the part @xmath462 of the equation can expanded as follows : @xmath626(u ) = 2 \sum_{j=1}^{n-2 } e_{j } \bigl [ \nabla h_{0 } \cdot \nabla h_{0 } \bigr](u ) e_{j}(u)\\ & = 2 \sum_{j=1}^{n-2 } 2 \bigl [ \nabla_{e_{j}(u ) } \nabla h_{0}(u ) \cdot \nabla h_{0}(u ) \bigr ] e_{j}(u ) \\ & = 4 \sum_{j=1}^{n-2 } \left\ { \nabla_{e_{j}(u ) } \left [ \sum_{k=1}^{n-2 } ( e_{k}h_{0 } ) \ , e_{k}\right](u ) \cdot \nabla h_{0}(u ) \right\ } e_{j}(u)\\ \end{aligned}\ ] ] @xmath627 \cdot \nabla h_{0}(u ) \bigr\ } e_{j}(u ) \notag\\ & = 4 \sum_{j , k=1}^{n-2 } e_{j}(e_{k}h_{0})(u ) \left\ { e_{k}(u ) \cdot \sum_{l=1}^{n-2 } ( e_{l}h_{0})(u ) \ , e_{l } ( u ) \right\ } e_{j}(u ) \notag\\ & = 4 \sum_{j=1}^{n-2 } \left [ \sum_{k=1}^{n-2 } ( e_{k}h_{0})(u ) \ e_{j}(e_{k}h_{0})(u ) \right ] e_{j}(u ) . \end{aligned}\ ] ] therefore we can conclude that @xmath628(u ) = 2 \nabla_{\textstyle \nabla h_{0}(u ) } \nabla h_{0}(u)\ ] ] and hence @xmath629(u ) & = \bigl [ \delta h_{0}(u ) \bigr ] \nabla h_{0}(u ) + h_{0}(u ) \nabla \bigl [ \delta h_{0 } \bigr](u)\\ & \qquad + 2(n-1 ) h_{0}(u ) \nabla h_{0}(u ) + 2 \nabla_{\textstyle \nabla h_{0}(u ) } \nabla h_{0}(u)\\ & = 0 . \end{aligned}\ ] ] for each @xmath162 . using lemma [ ls5.1:orgcntellipsoid ] we can conclude that the positive function @xmath576 on @xmath160 is the support function of an origin centered ellipsoid . since the support function @xmath163 satisfies @xmath630 , the item of lemma [ ls2.2:spfnpr ] implies that every horizontal cross section @xmath631 of @xmath35 is a homothetic copy of @xmath632 . therefore , there exists an affine isomorphism of @xmath633 mapping @xmath632 onto the unit sphere @xmath634 . extend this affine isomorphism to @xmath3 by fixing the last coordinate and hence @xmath35 is affinely isomorphic to a hypersurface of revolution in @xmath635 . [ linearity criterion ] [ ls5.3:affcrv ] a curve @xmath636 defined on an open interval is affine if and only if it is locally odd , i.e. , for each @xmath637 there exists @xmath638 so that @xmath639 for each @xmath640 . without loss of generality we can assume that @xmath29 lies in _ standard position _ because a general transversely convex tube , by definition , is affinely isomorphic to the one in _ standard position _ and the central curve is preserved under affine isomorphisms . using proposition [ ps5.2:classofrcttube ] and the invariance of the central curve under affine isomorphisms we can further assume that @xmath35 is either a cylinder or a hypersurface of revolution with @xmath641 as its axis and show that the _ cop _ assumption forces the central curve of @xmath29 to be a straight line . * cylindrical case : * assume that @xmath35 is a cylinder over a central ovaloid @xmath570 at height @xmath308 . each horizontal cross - section @xmath404 at height @xmath642 translates to @xmath570 . let @xmath112 be the support parameterization of @xmath570 , then a parameterization of @xmath29 can be obtained as @xmath643 fix @xmath644 , @xmath162 , consider the @xmath64-diameter of @xmath645 , which is the line segment joining the unique two points at which @xmath645 has normal line @xmath646 . since @xmath112 support parameterizes @xmath570 , the endpoints of the @xmath64-diamater are @xmath647 and @xmath648 . the tangent hyperplanes of @xmath29 at @xmath647 and @xmath648 are @xmath649 , \ , \partial_{2}x(u , b)(1 ) \bigr\}\\ & = u^{\perp } \oplus \mathbb{r}(c'(b ) , 1)\\ & = ( -u)^{\perp } \oplus \mathbb{r}(c'(b ) , 1)\\ & = \mathrm{span } \bigl\ { \partial_{1}x(-u , b ) \bigl [ ( -u)^{\perp } \bigr ] , \ , \partial_{2}x(-u , b)(1 ) \bigr\}\\ & = \operatorname{\mathrm{t}}_{x(-u , b ) } \mathcal{t}. \end{aligned}\ ] ] fix the @xmath64-diameter of @xmath645 as axis and tilt the horizontal hyperplane @xmath650 in a direction perpendicular to the axis with some small slope @xmath651 to get a new hyperplane @xmath652 . for sufficiently smal @xmath651 , the intersection @xmath653 will remain a central ovaloid because @xmath29 has _ cop_. since @xmath652 contains the @xmath64-diameter of @xmath645 , the endpoints @xmath647 and @xmath648 of that diameter remain on @xmath654 independently on @xmath651 . since the tangent hyperplanes to @xmath655 at these points are parallel and their intersections with @xmath652 form codimension one planes tangent to @xmath656 at @xmath647 and @xmath648 , those planes are also parallel . we can conclude that the @xmath64-diameter of @xmath645 remains a diameter of @xmath656 independently on @xmath651 , and hence @xmath657 remains the center of symmetry of @xmath656 for each @xmath162 and for all sufficiently small @xmath651 . the center of @xmath656 remains fixed as we vary @xmath651 . every point sufficiently close to @xmath645 on @xmath29 belongs to @xmath656 for some @xmath658 and small enough @xmath651 , so that by _ cop _ , its reflection through @xmath657 also lies on @xmath29 . it follows that an entire neighborhood @xmath586 of @xmath645 in @xmath29 has reflection symmetry through @xmath657 . in this neighborhood , pick two points @xmath659 , @xmath660 , which are the endpoints of a diameter of horizontal cross - section and denote the center of symmetry @xmath657 as @xmath661 . we can define the reflection through @xmath661 as @xmath662 which maps @xmath659 and @xmath660 respectively to @xmath663 and @xmath664 , which belong to a horizontal cross - section in the neighborhood @xmath586 . since the differentials satisfy @xmath665 and @xmath666 and @xmath667 we can conclude that @xmath668 . this implies that the segment @xmath669 $ ] is a diameter and the midpoint of @xmath670 $ ] is mapped to the midpoint of @xmath669 $ ] . since this holds in the neighborhood @xmath586 of @xmath645 we can choose @xmath638 so that @xmath671\\ & = c(b ) - c(b - t ) \end{aligned}\ ] ] for each @xmath672 . since @xmath644 is arbitrary then by using lemma [ ls5.3:affcrv ] ( linearity criterion ) we can conclude that @xmath28 is affine . * horizontally spherical case : * each horizontal plane @xmath650 cuts the tube @xmath29 in an @xmath264- sphere centered at @xmath657 , @xmath644 . let @xmath673 denote the square of the radius of this sphere and @xmath674 for each @xmath644 . fix @xmath683 , @xmath684 , let @xmath685 denote the orthogonal complemet of @xmath250 in @xmath686 with basis @xmath687 , and @xmath688 . since @xmath29 has _ cop _ , and lies in _ standard position _ we can find a small slope @xmath689 , and an @xmath690-intercept @xmath691 such that the plane @xmath692 given in the new coordinates by @xmath693 intersects @xmath29 in a central ovaloid @xmath108 with center having @xmath694 coordinate @xmath680 . from now on we will identify the curve @xmath695 , the plane @xmath692 , the tube @xmath29 , and the ovaloid @xmath108 with their coordinate representations under the basis @xmath696 because such an identification constitutes an orthogonal isomorphism . namely , we make the following identifications : for the curve @xmath23 given by @xmath697 we let @xmath698 , where @xmath699 . the plane @xmath692 becomes @xmath700 the tube is replaced with @xmath701^{2 } + \sum_{i=1}^{n-2 } \bigl [ y_{i } - \bar{c}_{i+1}(z ) \bigr]^{2 } = f(z ) \right\}\ ] ] and lastly the ovaloid @xmath702 is replaced with @xmath703^{2 } + \sum_{i=1}^{n-2 } \bigl [ y_{i } - \bar{c}_{i+1}(z ) \bigr]^{2 } = f(z ) \right\ } \label{es5.3:ovaloidyz - crd}\\ & = \bigl\ { ( y , z ) \in \mathbb{r}^{n-2 } \times \mathbb{r } \colon \lvert z \rvert < 1 , \ , \lvert y - c^{\perp}(z ) \rvert^{2 } = \lambda(z ) \bigr\ } \notag \end{aligned}\ ] ] let @xmath707 , @xmath708 , @xmath709 , @xmath710 , @xmath711 , then the cross - section @xmath712 of the ovaloid is an @xmath713-sphere centered at @xmath714 with radius @xmath715 . when @xmath716 and @xmath717 , choose two antipodal points @xmath718 pick the point @xmath719 and let @xmath244 be the support parameterization of @xmath108 , then there exists a unique @xmath720 with @xmath721 . since @xmath108 has the center of symmetry whose @xmath694-coordinate is @xmath680 , @xmath722 must also lie on @xmath723 . the ovaloid @xmath108 has the same tangent hyperplane @xmath724 at the two endpoints of the diameter connecting @xmath719 to @xmath725 . this hyperplane , when intersected with @xmath726 , becomes the tangent plane to @xmath727 at @xmath719 and @xmath728 . therefore , we can conclude that @xmath729 , and the chord joining @xmath719 and @xmath730 passes through the center of @xmath108 which has coordinates @xmath731 in the @xmath732 coordinate system . by using the expression of the ovaloid @xmath108 in the @xmath732 coordinate system we get @xmath736^{2 } + \bigl\lvert ( c^{\perp}(\beta ) + y ) - c^{\perp}(\beta + t ) \bigr\rvert^{2 } \\ & = \left [ \frac{\bar{\beta } + t}{m } - \mu_{\beta}^{+}\bar{c}_{1}(t ) - \mu_{\beta}^{-}\bar{c}_{1}(t ) \right]^{2 } + \bigl\lvert c^{\perp}(\beta ) + y - \mu_{\beta}^{+}c^{\perp}(t ) - \mu_{\beta}^{-}c^{\perp}(t ) \bigr \rvert^{2}. \end{aligned}\ ] ] @xmath737^{2 } + \bigl\lvert ( c^{\perp}(\beta ) - y ) - c^{\perp } ( \beta - t ) \bigr\rvert^{2}\\ & = \left [ \frac{\bar{\beta } - t}{m } - \mu_{\beta}^{+}\bar{c}_{1}(t ) + \mu_{\beta}^{-}\bar{c}_{1}(t ) \right]^{2 } + \bigl\lvert c^{\perp}(\beta ) - y - \mu_{\beta}^{+}c^{\perp}(t ) + \mu_{\beta}^{-}c^{\perp}(t ) \bigr \rvert^{2}. \end{aligned}\ ] ] when we subtract the equation from the equation we get @xmath738^{2}\!+\ ! \bigl\lvert ( c^{\perp}(\beta ) - \mu_{\beta}^{+}c^{\perp}(t ) ) + ( y - \mu_{\beta}^{- } c^{\perp}(t ) ) \bigr\rvert^{2}\\ & - \left [ \left ( \frac{\bar{\beta}}{m } - \mu_{\beta}^{+}\bar{c}_{1}(t ) \right ) - \left ( \frac{t}{m } - \mu_{\beta } ^{- } \bar{c}_{1}(t ) \right ) \right]^{2}\ ! - \ ! \bigl\lvert ( c^{\perp}(\beta ) - \mu_{\beta}^{+}c^{\perp}(t ) ) - ( y - \mu_{\beta}^{-}c^{\perp}(t ) ) \bigr\rvert^{2}\\ & = 4 \left ( \frac{\bar{\beta}}{m } - \mu_{\beta}^{+}\bar{c}_{1}(t ) \right)\left ( \frac{t}{m } - \mu_{\beta}^{- } \bar{c}_{1}(t ) \right ) + 4 \bigl(c^{\perp}(\beta ) - \mu_{\beta}^{+}c^{\perp}(t)\bigr ) \cdot \bigl(y - \mu_{\beta } ^{-}c^{\perp}(t)\bigr ) \end{aligned}\ ] ] recall that @xmath739^{2 } > 0 \end{aligned}\ ] ] for sufficiently small @xmath740 because the ovaloid @xmath108 can not contain its center of symmetry and hence @xmath741 . so we can conclude that @xmath712 is a nondegenerate sphere @xmath742 \times \{\beta + t\}\ ] ] for small enough @xmath740 . pick any two distinct points @xmath743 , @xmath744 , @xmath745 , \beta + t \bigr)\\ \bar{y } ' = ( y ' , \beta + t ) & = ( c^{\perp}(\beta + t ) + \xi ' , \beta + t ) \in \mathcal{o}\\ & = \bigl ( c^{\perp}(\beta ) + \bigl [ c^{\perp}(\beta + t ) - c^{\perp}(\beta ) + \xi ' \bigr ] , \beta + t \bigr ) \end{aligned}\ ] ] for two distinct points @xmath746 , @xmath747 . the central symmetry condition of the ovaloid @xmath108 and the equations , allow us to write the following two equalities @xmath748 - \mu_{\beta}^{-}c^{\perp}(t)\bigr ) \end{aligned}\ ] ] using the coordinates of @xmath743 and @xmath749 - \mu_{\beta}^{-}c^{\perp}(t)\bigr ) \end{aligned}\ ] ] using the coordinates of @xmath750 . subtracting the equation from the equation we get @xmath751 since the equation holds for every @xmath746 , @xmath747 we can conclude that @xmath752 \end{aligned}\ ] ] holds for small enough @xmath753 depending on @xmath754 and hence using lemma [ ls5.3:affcrv ] ( linearity criterion ) we can conclude that @xmath755 is affine . the curve @xmath23 has coordinates @xmath756 with respect the the orthonormal basis @xmath757 and since @xmath755 is affine it must be of the form @xmath758 therefore , the curve @xmath28 must be of the form @xmath759 with respect to the basis @xmath760 . using @xmath761 as the tilt direction and the argument above we get @xmath762 with respect to the basis @xmath760 and hence we can conclude that the curve @xmath763 is affine . let @xmath29 be a transversely convex tube with _ cop_. by definition @xmath29 is affinely isomorphic to a transversely convex tube @xmath764 in _ standard position_. as central ovaloids are mapped to central ovaloids by affine isomorphisms , @xmath764 also has _ cop_. the central curve of @xmath765 is affine by lemma [ ps5.3:axslemm ] ( axis lemma ) and hence @xmath764 is affinely isomorphic to a rectified transversely convex tube @xmath766 with _ cop _ in _ standard position_. using proposition [ ps5.2:classofrcttube ] we can conclude that @xmath766 is either a cylinder over a central ovaloid or affinely isomorphic to a hypersurface of revolution with _ cop_. as @xmath29 is affinely isomorphic to @xmath767 we can conclude that @xmath29 is either a cylinder over a central ovaloid or affinely isomorphic to a hypersurface of revolution with _ cop_. in the case when @xmath29 is affinely isomorphic to a hypersurface of revolution with _ cop _ we can use @xcite to conclude that @xmath29 is affinely isomorphic to a quadric and hence it is itself a quadric . for every affine isomorphism @xmath768 , @xmath769 is again a proper , complete , immersion with cop if @xmath10 has these properties and @xmath770 is again a cylinder or a quadric depending on whether @xmath14 is a cylinder or a quadric . according to the _ cop _ definition [ ds1.1:cop ] , there exists a cross - cut @xmath12 so that @xmath771 is an ovaloid . by of the affine invariance of the problem we can assume that @xmath772 and use lemma [ ls3.3:exsttrncvxtube ] ( existence of transversely convex tube ) to construct a transversely convex tube with _ cop _ about @xmath771 with the following properties : * case * @xmath778 in this case , @xmath774 has no critical point in @xmath779 , @xmath780 is an embedding and @xmath10 embeds @xmath781 into @xmath782 as a transversely convex tube @xmath783 with _ cop_. therefore , using proposition [ ps5.4:lclvrs ] ( local version ) , we can conclude that @xmath783 is either a cylinder over a central ovaloid , or a quadric , and because of the affine invariance of the problem we can further assume that @xmath641 is the axis of @xmath783 . assume first that @xmath783 is a cylinder . our aim is to show that this is not possible by producing a contradiction to the choice of maximal and minimal heights . since @xmath780 is an embedding and @xmath784 = \dim m$ ] , using the inverse function theorem we can conclude that it is a diffeomorphism onto its image , which is an open subset of @xmath2 . define the set @xmath785 as @xmath786 we eventually want to show that @xmath787 is a submanifold . but we first argue that the set @xmath787 exists , is compact and belongs to the boundary of @xmath788 . the limit @xmath789 exists : fix a cauchy sequence @xmath790 with @xmath791 , @xmath651 , then there exist @xmath792 , @xmath793 smooth curve connecting @xmath794 to @xmath795 so that for each @xmath449 and @xmath796 @xmath797 then the distance between the points @xmath798 and @xmath799 has a bound @xmath800 \leq \mathrm{length } \ , \beta_{ij } = \mathrm{length } \ , \alpha_{ij } < \epsilon\ ] ] whenever @xmath449 , @xmath796 because @xmath2 has the pullback metric . therefore , @xmath801 is a cauchy sequence and the limit @xmath789 exists because @xmath802 is a complete metric space , which follows because @xmath10 is a complete immersion . @xmath803 \cap f^{-1 } \bigl [ \mathcal{o}(b ) \bigr]$ ] and it is compact . take @xmath804 , then @xmath805 and hence @xmath806 $ ] , @xmath807 , @xmath808 = b$ ] . this implies that @xmath809 and we get one inclusion . on the other hand , if we take @xmath810 , @xmath811 with @xmath812 , @xmath813 , @xmath814 , then @xmath815 is a cauchy sequence , @xmath816 and @xmath817 . therefore , the equality holds and the set @xmath787 is compact because @xmath10 is a proper map . moreover the definition of @xmath787 implies that @xmath818 . a similar argument shows that @xmath819 \cap f^{-1 } \bigl [ \mathcal{o}(a ) \bigr]\ ] ] is compact and we can conclude that @xmath820 \setminus \psi_{a , b}(\gamma \times ( a , b))\\ & = \gamma_{a } \cup \gamma_{b}. \end{aligned}\ ] ] now we want to show that @xmath787 is a submanifold of @xmath2 of dimension @xmath821 and conclude that @xmath787 is a cross - cut of @xmath10 relative to @xmath822 . this conclusion , together with lemma [ ls3.3:exttrncvxtube ] ( extension of transversely convex tube ) will contradict the maximality of @xmath777 . because of the affine invariance of the problem we can assume that @xmath823 , @xmath824 and denote the map @xmath825 as @xmath145 . note that because @xmath826 is a cylinder , for every @xmath827 @xmath828 hence @xmath829 . therefore , we can conclude that @xmath830 does not vanish on the compact set @xmath831 and we can define the vectorfield @xmath832 in an open neighborhood @xmath833 of @xmath834 . using the existence and uniqueness of flow box for @xmath270 we obtain a triple @xmath835 , where 1 . @xmath836 is an open neighborhood of @xmath834 in @xmath2 + 2 . @xmath837 is smooth + 3 . for each @xmath838 , @xmath839 is the integral curve of @xmath270 and @xmath840 + 4 . for each @xmath841 , @xmath842 is a diffeomorphism onto its image . for each @xmath838 the curve @xmath843 satisfies @xmath844 y(c_{q}(t ) ) = 1\ ] ] for each @xmath841 . therefore , for each @xmath838 the height function @xmath845 has no local maximum or minimum in @xmath846 . choose @xmath171 so that @xmath847 , @xmath848 , and @xmath849 . define the submanifold @xmath850 of @xmath2 , which is diffeomorphic to @xmath851 as @xmath852^{-1 } \mathcal{o}(c).\ ] ] its @xmath853-distance to @xmath834 satisfies @xmath854 = \lvert c \rvert < \operatorname{\mathrm{dist}}_{m}(\gamma_{0 } , m \setminus v_{1})\ ] ] and hence we can conclude that @xmath855 . similarly for every @xmath856 the submanifold @xmath857 satisfies @xmath858 . @xmath864^{-1 } \mathcal{o } ( c + s)\\ & \iff \phi_{s}(x ) \in \psi(\gamma \times ( a , 0 ) ) \text { and } ( f \circ \phi_{s})(x ) \in \mathcal{o}(c + s)\\ & \iff \phi_{s}(x ) \in \psi ( \gamma \times ( a , 0 ) ) . \end{aligned}\ ] ] the last equivalence is true because @xmath865 implies that its @xmath10 image @xmath866 and @xmath867 yields @xmath868 . by definition @xmath869 \bigr)$ ] , and if this image is not a subset of @xmath870 then there must exist @xmath871 so that @xmath872 , which implies the existence of a local extremum in @xmath873 . this contradicts the observation in and hence we can conclude that @xmath874 . assume that there exists @xmath875 , then there is an open neighborhood @xmath876 of @xmath660 with @xmath877 > 0 $ ] . choose @xmath878 so that for each @xmath879 @xmath880 > 0 $ ] . then for each @xmath881 @xmath882 \cap \phi_{-c+s}(\gamma_{c } ) & = \emptyset\\ \bigl[\ , w \cap \psi ( \gamma \times ( a , 0 ) \,\bigr ] \cap \gamma_{s } & = \emptyset \end{aligned}\ ] ] and @xmath883 \cap f(\gamma_{s } ) = \emptyset$ ] because @xmath884 is a bijection onto its image . therefore , we must have for each @xmath881 @xmath885 \cap \mathcal{o}(s ) = \emptyset\ ] ] which is not possible . this implies that @xmath886 . let @xmath863 be arbitrary and consider the limit @xmath887 for some sequence @xmath888 in @xmath112 and hence we have @xmath889 \cap f^{-1 } \bigl [ \mathcal{o}(0 ) \bigr ] = \gamma_{0}.\ ] ] so we get @xmath890 and since @xmath891 is a diffeomorphism and @xmath850 is an @xmath821 dimensional compact connected submanifold of @xmath2 we can conclude that @xmath834 is also an @xmath821 dimensional compact connected submanifold of @xmath2 . therefore , using lemma [ ls3.3:exttubnbd ] ( tubular neighborhood ) we can conclude that @xmath834 is actually a cross - cut and lemma [ ls3.3:exsttrncvxtube ] ( existence of transversely convex tube ) shows that there exists a transversely convex tube about @xmath892 . finally , lemma [ ls3.3:exttrncvxtube ] ( extension of transversely convex tube ) implies that tube about @xmath893 extends @xmath894 , contradicting the maximality of @xmath895 . it follows that the transversely convex tube @xmath896 is a quadric hypersurface of revolution about the @xmath897-axis . therefore there exist a vertical dilation @xmath898 and a vertical translation @xmath899 such that the affine isomorphism @xmath900 satisfies @xmath901 for some @xmath171 . since @xmath783 intersects some horizontal hyperplane along a compact set so does @xmath902 . after we remove the cylindrical case we can conclude that @xmath902 is one of the following quadrics : therefore , @xmath783 is affinely isomorphic to one of the quadrics above . discarding the cone , on all these hypersurfaces , horizontal cross - sections are spheres and hence the maximality and minimality of @xmath777 and @xmath46 , respectively , must be dictated by the condition alone . this implies that @xmath774 must have critical points on both boundaries of the open set @xmath788 . namely , there exist @xmath903 , @xmath904 so that @xmath905 but among the quadrics listed above , @xmath906 has multiple critical points only on the sphere , where it attains both a maximum and a minimum . so we can conclude that @xmath907 $ ] equals the complete ellipsoid @xmath908 . since @xmath2 is connected we must have @xmath909 . * case * @xmath910 in this case @xmath911 is an embedding and since @xmath912 the map @xmath145 must be a diffeomorphism onto its image , which is both open and closed in @xmath2 . since @xmath2 is connected @xmath913 . @xmath914 & = \bigcup_{r > 0 } f \bigl [ \psi(\gamma \times ( -r , r ) ) \bigr]\\ & = \bigcup_{r > 0 } f(u_{-r , r } ) \end{aligned}\ ] ] where @xmath915 for each @xmath916 . for each @xmath916 , @xmath917 is a transversely convex tube with _ cop_. using proposition [ ps5.4:lclvrs ] ( local version ) we can conclude that @xmath917 is either a cylinder over a cental ovaloid with axis @xmath918 or a quadric hypersurface of revolution about @xmath641 . therefore , @xmath917 must either be a cylinder over a central ovaloid or a tube hyperboloid for each @xmath916 . assume that there exists @xmath919 and @xmath920 is a cylinder over a central ovaloid , then for every @xmath921 @xmath922 and hence @xmath917 is a cylinder over a central ovaloid . so we can conclude that @xmath923 is a cylinder over a central ovaloid . similarly , if there exists @xmath919 so that @xmath920 is a tube hyperboloid , then @xmath14 is a again a tube hyperboloid . * case * @xmath924 since the reflection about @xmath925 is an affine isomorphism , these two cases are equivalent . so we assume @xmath926 . for each @xmath927 @xmath928 \text { is an open subset of $ m$ and}\\ & f(u_{a , b } ) \text { is a transversely convex tube with \emph{cop}. } \end{aligned}\ ] ] @xmath929 can not be a cylinder over a central ovaloid because we can get , as in * case 1 * , a contradiction to the minimality of @xmath46 . so we can conclude that for each @xmath927 , @xmath929 is either a spherical paraboloid or a convex hyperboloid . assume that there exists @xmath930 so that @xmath931 is a spherical paraboloid , then for each @xmath932 , @xmath929 is again a spherical paraboloid and @xmath933 is a complete paraboloid . since @xmath2 is connected , @xmath934 and @xmath14 is a complete paraboloid . similarly , if there exists @xmath930 so that @xmath931 is a convex hyperboloid , then @xmath935 is a convex hyperboloid . 99 wilhelm blaschke , _ affine differentialgeometrie . differentialgeometrie der kreis - und kugelgruppen_. gesammelte werke , band 4 , edited by burau , s.s . chern , k. leichtwei , h.r . mller , l.a . santal , simon and k.strubecker , thales - verlag , essen , 1985 .

a hypersurface @xmath0 , @xmath1 , has the central ovaloid property , or _ cop _ , if * @xmath2 meets some hyperplane transversally along an ovaloid , and * every such ovaloid on @xmath2 has central symmetry . generalizing work of b. solomon to higher dimensions , we show that a complete , connected , smooth hypersurface with _ cop _ must either be a cylinder over a central ovaloid , or else quadric .

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Proceed to summarize the following text: the requirement of higher integration scales in electronic circuits , the onset of nanosensory applications in biomedicine , but also the fascinating capabilities of modern experimental setup with its enormous potential in polymer and surface research recently led to an increasing interest at the hybrid interface of organic and inorganic matter @xcite . this also includes numerous detailed studies , e.g. , of polymer film wetting phenomena @xcite , pattern recognition @xcite , protein ligand binding and docking @xcite , charged adsorbed polymers @xcite as well as deposition and growth of polymers at surfaces @xcite . in most theoretical and computational studies the polymer is anchored at the substrate with one of its ends which reduces the entropic freedom of the polymer . these surface - grafted polymers @xcite are , e.g. , of particular interest in studies of shape transformations @xcite , e.g. , as reaction to external fields @xcite . however , in many recent experiments of organic inorganic interfaces the setup is different @xcite and is more adequately described by a polymer moving in a cavity with one adsorbing surface @xcite . the main difference of such nongrafted polymers considered in this work is of entropic kind : in the desorbed phase the polymer can move freely within the cavity , and the polymer can fold into conformations , where the ends have no contact with the surface . this paper is organized as follows . in sect . [ secmod ] we describe the details of a minimalistic model for the hybrid system . the main result , the solubility - temperature pseudo - phase diagram , is presented and discussed in sect . [ sechyb ] . the interpretation is consolidated by exemplified studies of fluctuations and correlations of relevant thermodynamic quantities such as numbers of contacts between monomers and monomer - substrate contacts as well as the gyration tensor , in the different phases . the contact numbers turn out to be adequate system parameters for the description of the macrostate of the system , and therefore the free energy in dependence of these contact numbers is subject of a detailed study in sect . [ secfree ] . this quantity is also useful for classifying the conformational transitions between the phases which are also discussed there . eventually , we conclude in sect . [ secsum ] with a summary of the main results . we employ a minimalistic simple - cubic ( sc ) lattice model @xcite which allows a systematic analysis of the conformational phases experienced by a nongrafted polymer in a cavity with one adhesive surface . an example for the cavity model is shown in fig . [ figmod ] . the polymer can move between the two infinitely extended parallel planar walls , separated by a distance @xmath0 expressed in lattice units . the substrate is short - range attractive to the monomers of the polymer chain , while the influence of the other wall is purely steric . denoting the number of nearest - neighbor , but nonadjacent monomer - monomer contacts by @xmath1 and the number of nearest - neighbor monomer - substrate contacts by @xmath2 , the energy of the hybrid system can be expressed in the simplest model as @xmath3 where @xmath4 and @xmath5 are the respective contact energy scales , which are left open in the following . for simplicity , we perform a simple rescaling and set @xmath6 and @xmath7 . here we have introduced the overall energy scale @xmath8 and the dimensionless reciprocal solubility @xmath9 that controls the quality of the implicit solvent surrounding the polymer ( the larger the @xmath9 , the worse the solvent ) . since contacts with the substrate usually entail a reduction of monomer - monomer contacts , there are two competing forces ( rated against each other by the energy scales ) affecting the formation of intrinsic and surface contacts . in this paper we mainly focus on the conformational transitions the polymer experiences under different environmental conditions . concretely , we are interested in the dependence of energetic and structural quantities on temperature @xmath10 and reciprocal solubility @xmath9 in equilibrium . the probability ( per unit area ) for a conformation with @xmath2 surface and @xmath1 monomer - monomer contacts at temperature @xmath10 and reciprocal solubility @xmath9 is given by @xmath11 where @xmath12 is the contact density and @xmath13 the partition sum . in this decomposition , @xmath14 stands for the density of unbound conformations , whereas @xmath15 is the density of surface and intrinsic contacts of all conformations bound to the substrate . obviously , the number of the conformations without contact to the attractive substrate , @xmath14 , depends on the distance @xmath0 between the cavity walls . for a sufficiently large distance @xmath0 from the substrate the influence of the neutral surface on the unbound polymer is small . for @xmath16 , however , @xmath14 formally diverges . therefore , the non - adhesive , impenetrable steric wall is necessary for regularization . we studied polymers with up to 200 monomers by applying the contact - density chain - growth algorithm which is an improved variant of the recently developed multicanonical chain - growth sampling method @xcite . all these methods set up on a variant of the pruned - enriched variant @xcite of rosenbluth sampling @xcite . the main advantage of the improved method is that it directly samples the contact density @xmath17 , which is very useful for problems , where the model provides different energy scales . this generalizes the ordinary multicanonical version @xcite which samples the density of states , i.e. , the number of states for given energy . here we can set the two independent energy scales @xmath5 and @xmath4 or their ratio @xmath9 , respectively , _ after _ the simulation . this allows to introduce the reciprocal solubility @xmath9 as a second environmental parameter in addition to the temperature @xmath10 . the partition sum of the system as a function of these two parameters is simply @xmath18 and the statistical average of any function @xmath19 is given by the formula @xmath20 which is very convenient since it only requires to estimate the contact density @xmath17 in the simulation . denoting contact correlation matrix elements as @xmath21 with @xmath22 , the specific heat can be written as @xmath23 all quantities depending only on the contact numbers @xmath1 and @xmath2 can therefore simply be calculated from the estimate of the contact density @xmath17 provided by our simulation method . although the two contact parameters are sufficient to describe the macrostate of the system and their fluctuations characterize the main pseudo - phase transition lines , it is often useful to introduce also nonenergetic quantities such as the end - to - end distance and the gyration tensor for gaining more detailed structural information of the polymer . for our specific problem at hand it is particularly useful to study the structural anisotropy of the adsorbed polymer in the different phases . to this end , we define the general gyration tensor for a polymer chain of @xmath24 beads with the components @xmath25 where @xmath26 , @xmath27 , is the @xmath28th cartesian coordinate of the @xmath29th monomer and @xmath30 is the center of mass with respect to the @xmath28th coordinate . anisotropy in the polymer fluctuations is connected with the system s geometry and therefore it will be sufficient to study the components of the gyration tensor parallel ( @xmath31 , @xmath32 components ) and perpendicular ( in @xmath33 direction ) to the planar walls , @xmath34\ ] ] and @xmath35 the gyration radius is then simply the trace of the gyration tensor , @xmath36 . the calculation of statistical averages for quantities @xmath37 that are not necessarily functions of the contact numbers @xmath2 and @xmath1 can not be performed via eq . ( [ eqav ] ) . in this case only the more general relation @xmath38 holds , where the sum runs over all polymer conformations @xmath39 . introducing the accumulated density @xmath40 , where @xmath41 is the kronecker symbol , the expectation value can be expressed , however , in a form similar to eq . ( [ eqav ] ) : @xmath42 the quantity @xmath43 can easily be measured in simulations with the contact density chain - growth algorithm . in the following we use natural units , i.e. , we set @xmath44 . for our exemplified study of the hybrid system in equilibrium we chose a polymer with 179 monomers . since this is a prime number , the polymer is unable to form perfect cuboid conformations on the sc lattice , as it is , e.g. , the case for a 100-mer . @xcite there we found two low - temperature subphases dominated by the same @xmath45 cuboid . in one subphase it had 20 surface contacts , while in the other the cuboid was simply rotated , entailing 25 surface contacts this is a typical example , where the exact number of monomers in the linear chain is directly connected with the occurrence of such specific pseudo - phases which are not , of course , phases in the traditional view . nonetheless , the enormous progress in high - resolution experimental structure analyses and in the technological equipment for precise polymer deposition , as well as the natural finite length of classes of polymers ( e.g. , peptides and proteins ) , explain the growing interest in pseudo - phases and the conformational transitions between them . here we mainly focus on the expected thermodynamic phase transitions @xcite and low - temperature higher - order layering pseudo - phase transitions @xcite . the following results were obtained from contact - density chain - growth simulations of the 179-mer in a cavity with @xmath46 ( see fig . [ figmod ] ) , choosing uniformly distributed starting points at random . in eight independent runs @xmath47 polymer conformations were generated in total . the resulting contact density @xmath17 and accumulated densities like @xmath43 are independent of external parameters such as temperature @xmath10 and reciprocal solubility @xmath9 . concrete values of statistical quantities for specific parameter settings are obtained by simple reweighting as in eqs . ( [ eqprob ] ) and ( [ eqavb ] ) . = 8.8 cm discontinuities or divergences of energetic and nonenergetic fluctuations as functions of external parameters reveal typically dramatic cooperative transitions in the collective , macroscopic behavior of the system s microscopic degrees of freedom in the thermodynamic limit . these transitions separate then thermodynamically stable phases and the transitions can uniquely be identified by certain values of the external parameters , e.g. , the transition temperature . usually , all fluctuations collapse at the same parameter sets . but , this `` traditional view '' is only true in the thermodynamic limit . finite - size systems usually exhibit a zoo of crossover- or pseudo - transitions , most of which disappearing in the thermodynamic limit . in special cases , e.g. , proteins , where the specific amino acid sequence is of finite length , no phase transitions in the strict sense happen at all . still , peaks in curves of fluctuating quantities _ can be _ signatures for `` cooperative activity '' , but this is not necessarily indicated by all fluctuations considered , and if , then typically at different parameter values . @xcite nonetheless , in protein science , pseudo - transitions such as conformational transitions are important in the understanding of secondary structure formation and the tertiary hydrophobic - core collapse . for polymers , mainly the @xmath48 collapse transition , which is probably of second order , is of particular interest . @xcite this is a real thermodynamic phase transition . nonetheless , at least for finite systems , an additional first - order - like glassy or crystallization transition at lower temperatures is also conjectured for polymers . @xcite all these peculiarities of finite polymer systems are also relevant for the adsorption problem we consider here . in fig . [ figpd ] we have plotted the projection of the specific heat profile onto the solubility - temperature plane as obtained from our simulation of the 179-mer in a cavity with @xmath46 . the color code reflects the value of the specific heat and the brighter the shading , the larger the value of @xmath49 . black and white lines emphasize the ridges of the profile . since we consider the specific heat as appropriate to identify pseudo - phases , these ridges mark the pseudo - phase boundaries . as expected , the pseudo - phase diagram is divided into two main parts , the phases of adsorption and desorption . the two desorbed pseudo - phases dc ( desorbed - compact conformations ) and de ( desorbed - expanded structures ) are separated by the collapse transition line which corresponds to the @xmath48 transition of the infinite - length polymer which is allowed to extend into the three spatial dimensions @xcite . the region of the adsorbed pseudo - phases is much more complex , and little is known about its details , since it is relevant at lower temperatures , where conventional monte carlo methods with pivot - like updates usually tend to fail . the presence of general phases of adsorbed - expanded ( ae ) @xcite and adsorbed - compact ( ac1 , ac2 ) conformations was postulated in adsorption studies of grafted polymers and the existence of an additional phase of surface - attached globules ( ag ) @xcite was assumed @xcite . in a recent study @xcite , it was argued that the layering transition between ac1 and ac2 is a thermodynamic phase transition . although the polymer in our study is still relatively small , we can clearly identify pseudo - phases in fig . [ figpd ] which can be assigned these labels , too . those regions are separated by the black lines indicating the transitions between them . we expect that these are transitions in the thermodynamic meaning ; only the precise location of the transition lines will still change with increasing length of the polymer . thus , this picture confirms the previously assumed phases and it provides evidence that the ag phase is indeed there . furthermore , we have also highlighted by white lines transitions between pseudo - phases which will probably not survive in the thermodynamic limit . this concerns , e.g. , the higher - order layering transitions among the compact pseudo - phases ac2a@xmath50-d . in the following sections we will analyse the properties of the pseudo - phases in more detail . the contact numbers @xmath2 and @xmath1 can be considered as system parameters appropriately describing the state of the system and are therefore useful to identify the pseudo - phases . peaks and dips in the external - parameter dependence of self - correlations @xmath51 , @xmath52 and cross - correlations @xmath53 indicate activity in the contact - number fluctuations and , analysing the expectation values @xmath54 and @xmath55 in these active regions of the external parameters @xmath10 and @xmath9 , allow for an interpretation of the respective conformational transitions between the pseudo - phases . = 8.8 cm in fig . [ figc179 ] , we have plotted for the 179-mer these quantities and , for comparison , the specific heat as functions of the temperature @xmath10 at a fixed solvent parameter @xmath56 . this example is quite illustrative as the system experiences several conformational transitions when increasing the temperature starting from @xmath57 ( see fig . [ figpd ] ) . at temperatures very close to @xmath57 ( pseudo - phase ac1 ) all 179 monomers have contact to the substrate and 153 monomer - monomer contacts are formed . this is the most compact contact set being possible for _ topologically two - dimensional _ , film - like conformations . it should be noted , however , that approximately @xmath58 conformations ( self - avoiding walks ) belong to this contact set . @xcite this high degeneracy is an artefact of the minimalistic lattice polymer model used . it is remarkable that the conformations with the highest number of total contacts @xmath59 are film - like compact ( @xmath60 ) . all other conformations we found possess less contacts , even the most compact contact set that dominates the five - layer pseudo - phase ac2a@xmath61 ( @xmath62 , @xmath63 , i.e. , @xmath64 ) . the reason is that for low temperatures , those macrostates are formed which are energetically favored . entropy is not yet relevant for the @xmath56 example @xmath54 drops to 149 only up to @xmath65 . increasing the temperature further , the situation dramatically changes , as can be seen in fig . [ figc179 ] . in a highly cooperative process , the average number of intrinsic contacts @xmath55 significantly increases ( to @xmath66 ) at the expense of surface contacts ( @xmath54 drops to approximately 104 ) . consequently , the strong fluctuations @xmath67 signalize a conformational transition , and the anticorrelation indicated by @xmath53 confirms that surface contacts turn into intrinsic contacts , which indirectly leads to the conclusion that the film - like structure is given up in favor of layered , spatially three - dimensional conformations . the system has entered subphase age which is the part of the phase ag , where two - layer conformations dominate . the subphase transition near @xmath68 from the two - layer ( age ) to the bulky regime of ag is due to the ongoing , rather unstructured expansion of the polymer into the @xmath33 direction by forming so - called surface - attached globules @xcite . this is accompanied by a further reduction of surface contacts , while the number of intrinsic contacts changes weakly . approaching @xmath69 , the situation is just vice versa . intrinsic contacts dissolve and the system experiences a conformational phase transition from globular conformations in ag to random strands in ae . crossing this transition line , the system enters the good - solvent regime . eventually , close to @xmath70 , the polymer unbinds off the substrate . a clear signal is observed in the fluctuations of @xmath2 , i.e. , the number of average surface contacts rapidly decreases . the expanded polymer is `` free '' and the influence of _ both _ walls is effectively steric . this phase ( ae ) is closely related to the typical random - coil phase of entirely free and dissolved polymers in good solvent . this example shows that a study of the contact number fluctuations is indeed sufficient to qualitatively identify and describe the conformational transitions between the pseudo - phases of the hybrid system . for this reason , @xmath2 and @xmath1 are adequate system parameters playing a similar role as order parameters in thermodynamic phase transitions . one of the most interesting structural quantities in studies of polymer phase transitions is the gyration tensor ( [ eqgyr ] ) . for our hybrid system we expect that the respective components parallel ( [ eqgyrpara ] ) and perpendicular ( [ eqgyrperp ] ) to the substrate will behave differently when the polymer passes pseudo - transition lines . in order to prove this anisotropy explicitly , we have plotted in fig . [ figg179 ] the expectation values , @xmath71 , and the fluctuations of these two components , @xmath72 , again for the polymer in solvent with @xmath56 . for interpreting the peaks of the fluctuations , we have also included once more the specific heat curve for comparison . the immediate observation is that the temperatures , where one or both gyration tensor components exhibit peaks , almost perfectly coincide with those of the specific heat . this is a strong confirmation for the phase diagram in fig . [ figpd ] which is based on the specific heat . obviously , even for the rather short polymer with 179 monomers , we encounter the onset of fluctuation collapse near the ( pseudo-)phase transitions . this is very promising for future quantitative finite - size scaling analyses . at very low temperatures , i.e. , in pseudo - phase ac1 , we have argued in the previous section that the polymer - conformation is the most compact single - layer film . this is confirmed by the behavior of @xmath73 and @xmath74 , the latter being zero in this phase . a simple argument that the structure is indeed maximally compact is as follows . it is well known that the most compact shape in the two - dimensional continuous space is the circle . for @xmath29 monomers residing in it , @xmath75 , where @xmath76 is the ( dimensionless ) radius of this circle . the usual squared gyration radius is @xmath77 and therefore @xmath78 for @xmath79 . indeed , this is close to the value @xmath80 of the ground - state conformation we identified in phase ac1 . note that the most compact shape in the simple lattice polymer model we used in our study is a square and not a discretized circle . @xcite = 8.8 cm near @xmath65 , the strong layering transition from ac1 to age is accompanied by an immediate decrease of @xmath73 , while @xmath74 rapidly increases from zero to about @xmath81 which is exactly the gyration radius ( perpendicular to the layers ) of a two - layer system , where both layers cover approximately the same area . note that the single layers are still compact , but not maximally . applying the same approximation as in eq . ( [ eqcirc ] ) , the planar gyration radius for each of the two layers is now ( with @xmath82 ) @xmath83 , while we measured in this phase ( age ) @xmath84 . this separates the subphase age from the other two - layer pseudo - phase ac2d in fig . [ figpd ] , where the dominating conformation has perfect two - layer ( lattice ) structure with @xmath85 ( this is the same 2% difference between continuous and lattice calculation for perfect shapes as above ) . we will discuss the conformational peculiarities in the following in more detail . the subphase transition from age to ag near @xmath68 is accompanied by a further decrease of @xmath73 whereas @xmath74 increases , i.e. , the height of the surface - adsorbed globule increases at the expense of the width . this tendency is stopped when approaching the transition ( @xmath69 ) from the globular regime ag to the phase of expanded , but still adsorbed conformations . while @xmath74 remains widely constant ( the fluctuation does not signalize any transition ) , the polymer strongly extends in the directions parallel to the substrate , as indicated by the peak of @xmath86 . after unbinding from the substrate , parallel and perpendicular gyration radii behave widely isotropically ( @xmath87 ) as the influence of the isotropy - disturbing walls is weak in this regime . it was shown in sect . [ ssecfl ] that the contact numbers @xmath2 and @xmath1 are unique system parameters for the pseudo - phase identification of the hybrid system . we define the restricted partition sum for a macrostate with @xmath2 surface contacts and @xmath1 monomer - monomer contacts by @xmath88 such that @xmath89 . assuming as usual that the dominating macrostate is given by the minimum of the free energy as a function of appropriate system parameters , it is useful to define the specific contact free energy as a function of the contact numbers @xmath2 and @xmath1 , @xmath90 identifying @xmath91 as a `` micro - contact '' entropy . for given external parameters @xmath10 and @xmath9 , this relation can be used to determine the minimum of the contact free energy and therefore allows the identification of the dominant macrostate with respect to the contact numbers . in turn , this quantity allows for an alternative representation of the pseudo - phase diagram , complementary to the one shown in fig . [ figpd ] in that it is related to the contact numbers @xmath2 and @xmath1 . this is done by determining for ( in principle ) all values of the external parameters @xmath10 and @xmath9 the minima of the contact free energy ( [ eqrfreeb ] ) . then the pair of values @xmath2 and @xmath1 of the minimum contact free - energy state are marked in an @xmath2-@xmath1 phase diagram . this is shown in fig . [ figmap179 ] , where all free - energy minima of the 179-mer for the parameter set @xmath92 $ ] and @xmath93 $ ] are included and , based on the arguments of the previous section , differently shaded according to the pseudo - phase they belong to . the nice thing of this representation is that it allows the differentiation of continuous and discontinuous pseudo - phase transitions . = 8.8 cm the first important observation is that the diagram is divided into two separate regions , the pseudo - phases of desorbed conformations ( dc and de ) and the remaining different phases of adsorption . the `` space '' in between is blank , i.e. , none of these ( possible ) conformations was found to be a free - energy minimum conformation . this shows that transitions between the adsorbed and desorbed pseudo - phases are always first - order - like . it should be noted , that the regime of contact pairs @xmath94 lying _ above _ the shown compact phases is forbidden , i.e. , conformations with such contact numbers do not exist on the sc lattice . the second remarkable result is that the pseudo - phases dc , de , ae , and ag are `` bulky '' , while all ac subphases are highly localized in the plot of the free - energy minima . comparing with fig . [ figpd ] , the conclusion is that conformations in the ac phases are energetically favored ( more explicitly , for @xmath95 in ac1 and @xmath96 in the ac2 subphases ) , while the behavior in the other pseudo - phases is entropy - dominated : the number of conformations with similar contact numbers in the globular or expanded regime is higher than the rather exceptional conformations in the compact phases , i.e. , for sufficiently small @xmath97 ratios the entropic effect overcompensates the energetic contribution to the free energy . .[tabconf ] representative minimum free - energy examples of conformations in the different pseudo - phases of a 179-mer in a cavity . the substrate is shaded in lightgray . [ cols="^,^,^,^",options="header " , ] the subphases ac2a@xmath50-d are strongly localized , thorn - like `` peninsulas '' standing out from the ag regime . the discrete number and their separation leads to the conclusion that they have related structures . indeed , as can be seen in table [ tabconf ] , where we have listed representative conformations for all pseudo - phases , the few conformations dominating these subphases exhibit compact layered structures . the most compact three - dimensional conformation with 263 monomer - monomer contacts and 36 surface contacts is favored in subphase ac2a@xmath61 and possesses five layers . starting from this subphase and increasing the temperature , two things may happen . a rather small change is accompanied with the transition to ac2a@xmath98 , where the number of intrinsic contacts is reduced but the global five - layer structure remains . on the other hand , passing the transition line towards ac2b , the monomers prefer to arrange in compact four - layer conformations . advancing towards ac2d , the typical conformations reduce layer by layer in order to increase the number of surface contacts . in ac2d there are still two layers lying almost perfectly on top of each other . this is similar in subphase age , where also two - layer but less compact conformations dominate . in pseudo - phase ac1 only the film - like surface layer remains . the reason for the differentiation of the phases ac1 and ac2 of layered conformations is that the transition from single- to double - layer conformations is expected to be a real phase transition , while the transitions between the higher - layer ac2 subphases are assumed to disappear in the thermodynamic limit . @xcite = 8.8 cm as can be seen in fig . [ figpd ] , a transition between ac1 and the phase of adsorbed , expanded conformations , ae , is possible . since these two phases are connected in fig . [ figmap179 ] , we expect that the transition in between is second - order - like . indeed , this transition is strongly related with the _ two - dimensional _ @xmath48 transition since , close to the transition line , all monomers form a planar ( surface-)layer . similarly , there is also a second - order - like transition line @xmath99 between ag and ae which separates the regions of poor ( ag : @xmath100 ) and good ( ae : @xmath101 ) solvent . also , the transition between the desorbed compact ( dc ) and expanded ( de ) conformations is second - order - like : this transition is strongly related with the well - known @xmath48 transition in three dimensions . @xcite eventually , the transitions from the layer - phases ac2a@xmath98 , ac2b , ac2c , and age to the globular pseudo - phase ag as well as transitions between pseudo - phases dominated by the same layer type ( i.e. , between the two - layer subphases ac2d and age , and between the five - layer subphases ac2a@xmath61 and ac2a@xmath98 ) are expected to be continuous . on the other hand , the transitions among the energetically caused compact low - temperature pseudo - phases are rather first - order - like , due to their noticeable localization in the map of free - energy minima ( fig . [ figmap179 ] ) . the possible transitions ( see fig . [ figpd ] ) are ac2a@xmath50ac2b , ac2b ac2c , and ac2c ac2d , respectively . even more interesting , however , are the transitions from the single - layer pseudo - phase ac1 to the double - layer subphases ac2d and age . in the previous sections we already discussed this transition for the special choice @xmath56 , where near @xmath65 the fluctuations of the contact numbers and the components of the gyration tensor exhibit a strong activity . we have included into fig . [ figmap179 ] the `` path '' of macrostates the system passes by increasing the temperature from @xmath57 to @xmath102 . at @xmath57 the system is in a film - like , single - layer state . near @xmath65 it indeed suddenly rearranges into two layers and enters subphase age in a single step . in fig . [ figp179](a ) we have plotted the probability distribution @xmath103 for @xmath56 and @xmath104 and it can clearly be seen that two distinguished macrostates coexist . @xcite increasing the temperature further , the system undergoes the continuous transitions from age via ag until it unfolds when entering pseudo - phase ae . the system is still in contact with the substrate . close to a temperature @xmath105 , however , the unbinding of the polymer off the substrate happens ( from ae to de ) . comparing figs . [ figmap179 ] and [ figp179](b ) , where the probability distribution at @xmath106 is shown , we see also a clear indication for a discontinuous transition . note that we consider here the transition state , where the two minima of the free energy coincide @xcite ( see also the black dashed line in fig . [ figpd ] ) and not the point , where the width of the distribution , i.e. , the specific heat , is maximal . since the system is finite , the transition temperature ( @xmath70 ) , as signalled by the fluctuations studied in the previous sections , deviates slightly from the transition - state temperature reported here . in this paper , we have studied in detail the solubility - temperature ( pseudo-)phase diagram of a polymer in a cavity with an attractive substrate . we identified the thermodynamic phases of adsorbed compact and expanded ( ac , ae ) and desorbed ( dc @xcite , de ) conformations as well as the previously not yet clearly confirmed phase of adsorbed globules ( ag ) . although the polymer in our study possessed only @xmath107 monomers , these phases are expected to be stable also in the thermodynamic limit @xmath108 . other noticeable phase transitions in the compact - globular adsorbed regime ( ac1-ac2d , ac1-age ) are the energetic layering transitions from film - like surface - layer to double - layer conformations which are also believed to survive the thermodynamic limit . @xcite in addition , further subphases of higher - order layers were found in low - temperature regions and bad solvent ( ac2a@xmath50 , ac2b , and ac2c ) . the most compact three - dimensional conformation found is cube - like and forms five layers ( in subphase ac2a@xmath61 ) . the ( pseudo-)phase diagram is based on the specific - heat profile as a function of temperature and reciprocal solubility . although this profile allows for the identification of phases and their boundaries it does tell little about the conformational transitions between the phases . for this purpose we considered expectation values and fluctuations for the numbers of monomer - surface contacts , @xmath2 , and intrinsic monomer - monomer contacts , @xmath1 , separately . these contact numbers turned out to be sufficient to describe the macrostate of the system and therefore are useful to describe the conformations dominating the different phases . this view was completed by an exemplified study of the anisotropic behavior of the gyration tensor components of the polymer parallel and perpendicular to the substrate . another central aspect was the classification of the conformational transitions between the ( pseudo-)phases . based on the contact numbers @xmath2 and @xmath1 , we defined an appropriate free energy and studied the distribution of the minima in the @xmath2-@xmath1 space . from this kind of free - energy landscape , we found strong indications that the binding - unbinding transitions between the adsorbed and desorbed phases are first - order - like . this was also observed for the layering transitions . on the other hand , the transitions across the line separating good and poor solvent , i.e. , between the compact ( or globular ) and the expanded conformations , are rather second - order - like . this is in coincidence with the known behavior of free polymers at the @xmath48 collapse transition in two and three dimensions . since the experimental equipment and the technological capabilities have nowadays reached an enormous standard of high single - molecular resolution , we expect that it should be possible to verify experimentally not only the existence of the described thermodynamic phases , but also the pseudo - phases being only relevant for finite polymers and specific to their lengths . this work is partially supported by the dfg ( german science foundation ) grant under contract no . ja 483/24 - 1 . some simulations were performed on the supercomputer jump of the john von neumann institute for computing ( nic ) , forschungszentrum jlich , under grant no . 199 s. brown , nature biotechnol . * 15 * , 269 ( 1997 ) . s. r. whaley , d. s. english , e. l. hu , p. f. barbara , a. m. belcher , nature * 405 * , 665 ( 2000 ) . k. goede , p. busch , and m. grundmann , nano lett . * 4 * , 2115 ( 2004 ) . r. l. willett , k. w. baldwin , k. w. west , and l. n. pfeiffer , proc . acad.sci . 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[ ] .

in a contact density chain - growth study we investigate the solubility - temperature pseudo - phase diagram of a lattice polymer in a cavity with an attractive surface . in addition to the main phases of adsorbed and desorbed conformations we find numerous subphases of collapsed and expanded structures .

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Proceed to summarize the following text: the detection of compact signals ( sources ) embedded in a background is a recurrent problem in many fields of astronomy . some common examples are the separation of individual stars in a crowded optical image , the identification of local features ( lines ) in noisy one - dimensional spectra or the detection of faint extragalactic objects in microwave frequencies . the detection , identification and removal of the extragalactic point sources ( eps ) is fundamental for the study of the cosmic microwave background radiation ( cmb ) data ( franceschini et al . 1989 , toffolatti et al . 1998 , de zotti et al . 1999 ) . in particular , the contribution of eps is expected to be very relevant at the lowest and highest frequency channels of the future esa planck mission ( mandolesi et al . 1998 , puget et al . 1998 ) . the heterogeneous nature of the eps that appear in cmb maps as well as their unknown spatial distribution make difficult to separate them from the other physical components ( cmb , galactic dust , synchrotron , etc ) by means of statistical component separation methods . techniques based on the use of linear filters , however , are well - suited for the task of detecting compact spikes on a background . several techniques based on different linear filters have been proposed in the literature : the mexican hat wavelet ( mhw , cayn et al . 2000 , vielva et al . 2001a , b , 2003 ) , the classic _ matched _ filter ( mf , tegmark and oliveira - costa 1998 ) , the adaptive top hat filter ( chiang et al 2002 ) and the scale - adaptive filter ( saf , sanz et al . 2001 ) . a certain deal of controversy has appeared about which one , if any , of the previous filters is _ optimal _ for the detection of point sources in cmb data . in order to answer that question it is necessary to consider first a more fundamental issue , the concept of _ detection _ itself . the detection process can be posed as follows : given an observation , the problem is to _ decide _ whether or not a certain signal was present at the input of the receiver . the decision is not obvious since the observation is corrupted by a random process that we call ` noise ' or ` background ' . formally , the _ decision _ is performed by choosing between two complementary hypotheses : that the observed data is originated by the background alone ( _ null hypothesis _ ) , and the hypothesis that the observation corresponds to a combination of the background and the signal . to decide , the detector should use all the available information in terms of the probabilities of both hypotheses given the data . the _ decision device _ separates the space @xmath1 of all possible observations in two disjoint subspaces , @xmath2 and @xmath3 , so that if an observation @xmath4 the null hypothesis is accepted , and if @xmath5 the null hypothesis is rejected , that is , a source is ` detected ' ( @xmath2 is called the region of acceptance ) . hence , we will call any generic decision device of this type a _ detector_. the simplest example of detector , and one that has been extensively used in astronomy , is _ thresholding _ : if the intensity of the field is above a given value ( e.g. 5@xmath6 ) , a detection of the signal is accepted , on the contrary one assumes that only background is present . thresholding has a number of advantages , among them the facts that it is straightforward and that it has a precise meaning in the case of gaussian backgrounds in the sense of controlling the probability of spurious detections . however , it does not use all the available information contained in the data to perform decisions . for example , the inclusion of spatial information ( such as the curvature ) could help to distinguish the sources from fluctuations in the background with similar scale but a different shape . a general detector that can use more information than simple thresholding is given by the neyman - pearson ( np ) decision rule : @xmath7 where @xmath8 is called the likelihood ratio , @xmath9 is the probability density function ( _ pdf _ ) associated to the null hypothesis ( i.e. there is no source ) whereas @xmath10 is the _ pdf _ corresponding to the alternative hypothesis ( i.e. there is a source ) . @xmath11 are a set of variables which are measured from the data . @xmath12 is an arbitrary constant , which defines the region of acceptance @xmath2 , and must be fixed using some criterion . for instance , one can adopt a scheme for object detection based on maxima . the procedure would consist on considering the intensity maxima of the image as candidates for compact sources and apply to each of them the np rule to decide whether they are true or spurious . for a 1d image , the ratio of probabilities would then correspond to the probability of having a maximum with a given intensity and curvature ( which are the variables @xmath11 in this case ) in the presence of background plus signal over the probability of having a maximum when only background is present . if this ratio is larger than a given value @xmath12 , the candidate is accepted as a detection , if not , it is rejected . unfortunately , in many cases the sources are very faint and this makes very difficult to detect them . in order to improve the performance of the detector , a prior processing of the image could be useful . here is where _ filtering _ enters in scene . the role of filtering is to transform the data in such a way that a detector can perform better than before filtering . once the detector is fixed , it is interesting to compare the performance of different filters , which has been rarely considered in the literature . in a recent work , barreiro et al . ( 2003 ) introduce a novel technique for the detection of sources based on the study of the number density of maxima for the case of a gaussian background in the presence or absence of a source . in order to define the region of acceptance the neyman - pearson decision rule is used with _ pdf _ s associated to the previous number densities and using the information of both the intensities @xmath13 and the curvatures @xmath14 of the peaks in a data set . in addition , @xmath12 is fixed by maximising the _ significance _ , which is the weighted difference between the probabilities of having and not having a source . in that work the performances of several filters ( saf , mf and mhw ) is compared in terms of their _ reliability _ , defined as the ratio between the number density of true detections over the number density of spurious detections . they find that , on the basis of this quantity , the choice of the optimal filter depends on the statistical properties of the background . however , the criterion chosen to fix @xmath12 based on the significance does not necessarily leads to the optimal reliability . therefore , if we are considering the reliability as the main criterion to compare filters , a different criterion for @xmath12 , based on number densities must be used . in a posterior article , vio et al . ( 2004 ) , following the previous work , adopt the same neyman - pearson decision rule , based on the _ pdf _ s of maxima of the background and background plus source , to define the region of acceptance . however , they propose to find @xmath12 by fixing the number density of spurious detections and compare the performance of the filters based on the number density of true detections . in this case , the mf outperforms the other two filters . note that in these last two works different criteria have been used to fix @xmath12 , thus leading to different results . in the present work , our goal will also be to find an optimal filter that gives a maximum number density of detections fixing a certain number density of spurious sources . in order to define the detector , we will use a decision rule based on the neyman - pearson test . we will consider some standard filters ( mf , saf and mh ) introduced in the literature as well as a new filter that we call the biparametric scale adaptive filter ( bsaf ) . in all the filters appears in a natural way the scale of the source . we will modify such a scale introducing an extra parameter . in fact , it has been shown by lpez - caniego et al . ( 2004 ) that the standard matched filter can be improved under certain conditions by filtering at a different scale from that of the source . the performance of the bsaf will be compared with the other filters . the overview of this paper is as follows . in section 2 , we introduce two useful quantities : number of maxima in a gaussian background in the absence and presence of a local source . in section 3 , we introduce the detection problem and define the region of acceptance . in section 4 , we introduce an estimator of the amplitude of the source that is proven to be unbiased and maximum efficient . in section 5 and 6 , we obtain different analytical and numerical results regarding weak point sources and scale - free background spectra and compare the performance of the new filter with others used in the literature . in section 7 , we describe the simulations performed to test some theoretical aspects and give the main results and finally , in section 8 , we summarize the conclusions and applications of this paper . appendix a is a sketch to obtain a sufficient linear detector whereas we obtain the linear unbiased and maximum efficient estimator in appendix b. let us assume a 1d background ( e. g. one - dimensional scan on the celestial sphere or time ordered data set ) represented by a gaussian random field @xmath15 with average value @xmath16 and power spectrum @xmath17 : @xmath18 , where @xmath19 is the fourier transform of @xmath15 and @xmath20 is the 1d dirac distribution . the distribution of maxima was studied by rice ( 1954 ) in a pioneering article . the expected number density of maxima per intervals @xmath21 , @xmath22 and @xmath23 is given by @xmath24 being @xmath25 the expected total number density of maxima ( i.e. number of maxima per unit interval @xmath26 ) @xmath27 @xmath28 where @xmath29 and @xmath30 represent the normalized field and curvature , respectively . @xmath31 is the moment of order @xmath32 associated to the field . @xmath33 are the coherence scale of the field and maxima , respectively . as an example , figure [ fig : fig1 ] shows the values of the ratio @xmath34 for the case @xmath35 ( a typical value for the backgrounds we are considering ) . in this case , the expected density of maxima has a peak around @xmath36 and @xmath37 , that is , most of the peaks appear at a relatively low threshold and curvature , and the density of peaks decreases quickly for extreme values of @xmath38 and @xmath14 . if the original field is linear - filtered with a circularly - symmetric filter @xmath39 , dependent on @xmath40 parameters ( @xmath41 defines a scaling whereas @xmath42 defines a translation ) @xmath43 we define the filtered field as @xmath44 then , the moment of order @xmath32 of the linearly - filtered field is @xmath45 being @xmath46 the power spectrum of the unfiltered field and @xmath47 the fourier transform of the circularly - symmetric linear filter . now , let us consider a position @xmath48 in the image where a gaussian source ( i.e. profile given by @xmath49 , where @xmath41 is the beam width ) is superimposed to the previous background . then , the expected number density of maxima per intervals @xmath21 , @xmath22 and @xmath50 , given a source of amplitude @xmath51 in such spatial interval , is given by ( barreiro et al . 2003 ) @xmath52 @xmath53 where @xmath29 and @xmath30 , @xmath54 is the normalized amplitude of the source and @xmath55 is the normalized curvature of the filtered source . the last expression can be obtained as @xmath56 note that due to the statistical homogeneity and isotropy of the background , the previous equations are independent of the position of the source . we consider that the filter is normalized such that the amplitude of the source is the same after linear filtering : @xmath57 . we want to choose between different filters based on _ detection_. to make such a decision , we will focus on the following two fundamental quantities : a ) the number of spurious sources which emerge after the filtering and detection processes and b ) the number of real sources detected . as we will see in this section , these quantities are properties of the gaussian field and source that can be calculated from equations ( [ nbackground ] ) and ( [ nsource ] ) . as we will see , the previous properties are not only related to the snr gained in the filtering process but depend on the filtered momenta up to 4th - order ( in the 1d case ) , i.e. the amplification and the normalized curvature . let us consider a local peak in the 1d data set characterised by the normalized amplitude and curvature @xmath58 . let @xmath59 : n.d.f . @xmath60 represents the _ null _ hypothesis , i.e. the local number density of background maxima , and @xmath61 : n.d.f . @xmath62 represents the _ alternative _ hypothesis , i.e. the local number density of maxima when there is a compact source : @xmath63 in the previous equation , we have introduced a priori information about the probability distribution of the sources : we get the number density of source detections weighting with the a priori probability @xmath64 . to construct our detector , we will assume a neyman - pearson ( np ) decision rule using number densities instead of probabilities : @xmath65 where @xmath12 is a constant . the previous equation defines a region in @xmath66 , the so - called region of acceptance @xmath2 . therefore , the decision rule is expressed such that if the values of @xmath66 of the candidate maximum is inside @xmath2 ( i.e. @xmath67 ) we decide that the signal is present . on the contrary , if @xmath68 we decide that the signal is absent . ) is equivalent to the one defined by the usual neyman - pearson test in terms of probabilities @xmath69 where @xmath70 , @xmath71 are the _ pdf _ s associated to the number densities given by equations ( [ nbackground ] ) and ( [ eq : number_b+s ] ) and , in order to compare different filters , the constant @xmath72 must be found by fixing the number density of spurious sources in the region of acceptance instead of the _ false alarm _ probability . ] it can be proved that the previous region of acceptance @xmath2 is equivalent to the sufficient linear detector ( see appendix a ) @xmath73 where @xmath74 is a constant and @xmath75 is given by @xmath76 we remark that the assumed criterion for detection leads to a _ linear _ detector @xmath75 ( i.e. linear dependence on the threshold @xmath38 and curvature @xmath14 ) . moreover , this detector is independent of the _ pdf _ of the source amplitudes . using this detector , the expected number density of spurious sources and of true detections are given by @xmath77 @xmath78 we remark that in order to get the true number of real source detections such a number must be multiplied by the probability to have a source in a pixel in the data set . note that for a fixed number density of spurious sources @xmath79 , the np detector leads to the maximum number density of true detections @xmath80 . taking into account equations ( [ eq : r _ * ] ) to ( [ eq : ndet ] ) , one can find @xmath79 and @xmath80 for a gaussian background . after a straightforward calculation , the number density of spurious sources found using the np rule is given by : @xmath81 , \label{eq : nb*}\end{aligned}\ ] ] @xmath82 similarly , the number density of detections is obtained as : @xmath83e^{- \frac{(1 - \rho^2)\varphi^2}{2{(1 - \rho y_s)}^2 } } , \label{eq : nb}\end{aligned}\ ] ] where @xmath84 @xmath85 @xmath86 the signal has an unknown parameter , the amplitude @xmath51 , that has to be estimated from the data @xmath87 . we shall assume that the most probable value of the distribution @xmath88 gives an estimation of the amplitude of the source ( criterion for amplitude estimation ) . the result @xmath89 is given by the equation @xmath90 where the function @xmath75 is given by equation ( [ eq : phi ] ) . one can prove that the previous expression corresponds to a linear estimator that is unbiased and maximum efficient ( minimum variance ) , i.e. @xmath91 where @xmath92 denotes average value over realizations ( see appendix b ) . we will consider as an application the detection of compact sources characterised by a gaussian profile @xmath93 , and fourier transform @xmath94 , though the extension to other profiles will be considered in the future . such a profile is physically and astronomically interesting because it represents the convolution of a point source ( dirac @xmath95 distribution ) with a gaussian beam . the source profile above includes a `` natural scale '' @xmath41 that characterises the source . this is a fundamental scale that will appear in all the filters we will consider here . by construction , the standard mf and saf operate on this scale , as well as the canonical mhw at the scale of the source . however , it has been shown that changing the scale at which the mhw and the mf filter the image can improve its performance in terms of detection ( vielva et al . 2001a , lpez - caniego et al . following this idea , we will introduce another degree of freedom in all the filters that allows us to change their scale in a continuous way ( similarly to the scaling of a continuous wavelet ) . this degree of freedom is obtained by multiplying the scale @xmath41 by a new parameter @xmath96 . we will show that with this new parameter the improvement in the results is significant . the idea of a scale - adaptive filter ( or optimal pseudo - filter ) has been recently introduced by the authors ( sanz et al . 2001 ) . by introducing a circularly - symmetric filter , @xmath97 , we are going to express the conditions in order to obtain a scale - adaptive filter for the detection of the source @xmath98 at the origin taking into account the fact that the source is characterised by a single scale @xmath99 . the following conditions are assumed : @xmath100 , i.e. @xmath101 is an _ unbiased _ estimator of the amplitude of the source ; @xmath102 the variance of @xmath103 has a minimum at the scale @xmath99 , i.e. it is an _ efficient _ estimator ; @xmath104 @xmath105 has a maximum with respect to the scale at @xmath106 . then , the filter satisfying these conditions is given by ( sanz et al . 2001 ) @xmath107 , \nonumber\end{aligned}\ ] ] @xmath108 assuming a scale - free power spectrum , @xmath109 , and a gaussian profile for the source , the previous set of equations lead to the filter @xmath110 , \ \ q\alpha r , \nonumber\end{aligned}\ ] ] @xmath111 @xmath112 where we have modified the scale as @xmath113 . in this case the filter parameters @xmath114 and the curvature of the source @xmath115 are given by @xmath116 @xmath117 @xmath118 @xmath119 figure [ fig : fig3 ] shows the saf for two values of the spectral index . if one removes condition ( 3 ) defining the saf in the previous subsection , it is not difficult to find another type of filter after minimization of the variance ( condition ( 2 ) ) with the constraint ( 1 ) @xmath120 this will be called _ matched _ filter as is usual in the literature . note that in general the matched and adaptive filters are different . for the case of a gaussian profile for the source and a scale - free power spectrum given by @xmath121 , the previous formula leads to the following modified matched filter @xmath122 @xmath123 where @xmath124 and @xmath125 is given by equation ( [ eq : saf_m ] ) and we have included the scale parameter @xmath126 . figure [ fig : fig3 ] shows the mf for the case @xmath127 ( standard mf ) and values of the spectral index @xmath128 . we remark that for @xmath129 the scale - adaptive filter and the matched filter coincide , and for @xmath130 ( not shown in the figure ) , the matched filter and the mexican hat wavelet are equal . for the mf the parameters @xmath131 and the curvature of the source @xmath115 are given by @xmath132 we remark that the linear detector @xmath133 is reduced to @xmath134 for the standard matched filter ( @xmath135 ) . i.e. curvature does not affect the region of acceptance for such a filter . the mh is defined to be proportional to the laplacian of the gaussian function : @xmath136 thus , in fourier space @xmath137 in this case the filter parameters @xmath138 and the curvature of the source @xmath115 are given by @xmath139 the generalization of this type of wavelet for two dimensions has been extensively used for point source detection in 2d images ( cayn et al . 2000 , vielva et al . 2001 , 2003 ) . as for the previous filter , the mh is modified by including the scale parameter @xmath126 in the form @xmath140 @xmath141 for the mh the parameters @xmath142 and the curvature of the source @xmath115 are given by @xmath143 figure [ fig : fig3 ] shows the mh for different values of the spectral index . if one removes condition ( 3 ) defining the saf in subsection 5.1.1 and introduces the condition that @xmath144 has a spatial maximum in the filtered image at @xmath145 , i.e. @xmath146 @xmath147 , it is not difficult to find another type of filter @xmath148 where @xmath149 is an arbitrary constant that can be related to the curvature of the maximum . we remark that the constraint @xmath150 is automatically satisfied for any circularly - symmetric filter if the source profile has a maximum at the origin . for the case of a scale - free power spectrum , the filter is given by the parametrized equation @xmath151 where we have modified the scale as @xmath113 . hereinafter , we will call this new filter containing two arbitrary parameters , @xmath152 and @xmath149 , the biparametric scale - adaptive filter ( bsaf ) . a calculation of the different moments leads to @xmath153 where m and t are defined in equation ( [ eq : saf_m ] ) and @xmath154 and @xmath155 are given by @xmath156\gamma ( m),\ ] ] @xmath157 \delta^m \gamma ( m).\ ] ] note that the bsaf contains all the other considered filters as particular cases : the mf is recovered for @xmath158 , when @xmath159 the bsaf defaults to the saf and , finally , the mh wavelet is obtained in the two cases : @xmath160 , @xmath161 and @xmath162 , @xmath163 . we will test two different _ pdf _ @xmath164 : a uniform distribution in the interval @xmath165 and a scale - free distribution with a lower and upper cut - off @xmath166 . in particular , we will especially focus on values for the cut - off s that lead to distributions dominated by weak sources . it is in this regime where sophisticated detection methods are needed , since bright sources can be easily detected with simple techniques . in this case , @xmath167.\ ] ] this allows us to obtain @xmath168,\ ] ] @xmath169 in general , we will consider a cut - off in the amplitude of the sources such that @xmath170 after filtering with he standard mf . note that this correspond to different thresholds for the rest of the filters . in this case , @xmath171,\ \ \beta \neq 1,\ ] ] where the normalization constant n and @xmath172 are @xmath173 @xmath174}.\\\ ] ] in general , we will consider @xmath175 and @xmath176 , @xmath177 after filtering with the standard mf and the corresponding thresholds for the other filters . for a fixed number density of spurious sources @xmath178 , we want to find the optimal filter that produces the maximum number density of true detections @xmath80 for different spectral indices ( @xmath179 ) , values of @xmath41 and point source distributions . in order to do this , we first obtain implicitly the value of @xmath74 from equation ( [ eq : nb * ] ) ( for a fixed value of @xmath178 ) and then substitute it in equation ( [ eq : nb ] ) to calculate @xmath80 . we consider two different distributions of sources to test the robustness of the method : a uniform distribution and a scale - free distribution . given that bright point sources are relatively easy to detect , we mainly concentrate on the more interesting case of weak sources . in any case , we also mention some results for distributions containing bright sources . we remark that the bsaf has an additional degree of freedom , the parameter @xmath149 , as it appears in equation ( [ eq : eqnndf ] ) . note that the bsaf and the saf are not the same filter . the parameter @xmath149 in the bsaf can take any positive or negative value , while the coefficient @xmath180 , for the saf , is a known function of @xmath179 . by construction , the bsaf always outperforms the mf and saf or , in the worst case , defaults to the best of them . as a first case , we consider a uniform distribution of sources with amplitudes in the interval @xmath181\sigma_0 $ ] , where @xmath182 is the zero - order moment of the linearly - filtered map with the standard mf . therefore , the threshold @xmath38 in the image filtered with this filter is in the interval @xmath183 $ ] . thus , the corresponding upper limit for @xmath38 in the original ( unfiltered ) map is below 2 , what means that we are considering the detection of weak sources . as a reference example , in figure [ fig : numdet_alfa_unif_g0 ] , we plot @xmath80 , the number density of detections , as a function of @xmath126 for the case @xmath184 , @xmath185 and @xmath186 , where @xmath41 is given in pixel units . for completeness , the theoretical values of @xmath80 are given , in this figure , for values of @xmath126 down to zero ( note that @xmath80 @xmath187 @xmath178 when @xmath188 ) . however , from a practical point of view , we do not expect the theoretical results to reproduce the values obtained for a pixelized image when filtering at small scales ( since the effect of the pixel is not taken into account ) . therefore , hereinafter , we will only consider those results obtained when filtering at scales larger ( or of the order ) of the pixel size , which corresponds to @xmath189 . taking into account this constraint , the best results are obtained for @xmath190 for the bsaf , that clearly outperforms the standard mf ( i.e. , @xmath127 ) with an improvement of the @xmath0 in @xmath80 . if we compare with the mf at @xmath191 , the improvement is of @xmath192 . in figure [ fig : numdet_alfa_unif_g05 ] , we give the same results for the case @xmath193 . in this case , the bsaf at @xmath191 improves again significantly the standard mf , with an increase in the number density of detections of @xmath194 . as @xmath179 increases , the improvement of the bsaf with respect to the standard mf decreases . in fact , for values of @xmath195 they produce very similar results . as an example , we give the number of detections achieved for each filter for the case @xmath196 , @xmath197 and @xmath186 in fig . [ fig : unif_g1p5 ] . it can be seen that the maximum number of detections is approximately found for the standard mf . however , we would like to point out that the saf and mh wavelet at the optimal scale give approximately the same number of detections as the standard mf . these results show the importance of filtering at scales @xmath113 instead of the usual scale of the source @xmath41 . this can also be seen in fig . [ fig : unif_gamma ] , that summarizes how the relative performance of the considered filters with respect to the standard mf changes with the spectral index @xmath179 ( again for @xmath197 and @xmath186 ) . for each filter , the results are given for the optimal scale ( and parameter @xmath149 in the case of bsaf ) . the improvement of the bsaf with respect to the mf ranges from @xmath198 ( for white noise ) to zero ( for the largest values of @xmath199 . we would also like to point out that the mh at the optimal scale performs similarly to the standard mf . in addition , the mh has an analytical expression which makes it very robust and easy to implement . therefore , it can be a useful filter in some practical cases . we have also explored how the previous results change when varying @xmath178 and @xmath41 . in particular , we have considered vaules of @xmath178 in the interval 0.01 - 0.05 , @xmath200 and @xmath186 and values of @xmath201 . the results are summarized in table [ tab : tabla1_n ] for the bsaf and the standard mf ( we present only those cases where the bsaf improves at least a few per cent the standard mf ) . the values of @xmath126 and @xmath149 for the bsaf are found as the ones that maximise @xmath80 in each case . . number density of detections @xmath80 for the standard mf ( @xmath127 ) and the bsaf with optimal values of c and @xmath126 . rd means relative difference of number densities in percentage : @xmath202.[tab : tabla1_n ] [ cols="^,^,^,^,^,^,^,^",options="header " , ] we have also explored how the results depend on the values of @xmath41 and @xmath79 . in table [ tab : tabla2_n ] , we show the number density of detections for the bsaf and for the standard mf ( @xmath127 ) for @xmath200 and @xmath186 , with @xmath178 ranging from 0.01 to 0.05 , and for values of @xmath203 ( we only include the results for those cases where the relative difference between the bsaf and standard mf is at least a few per cent ) . we also give the optimal values of @xmath149 and @xmath126 where the bsaf performs better ( taking into account the constraint @xmath204 ) . as for the previous case of the uniform distribution , the relative performance of the bsaf improves when increasing @xmath41 and @xmath79 . it is also interesting to consider other values of the parameter @xmath205 . for instance @xmath206 $ ] has an intrinsic interest for astronomy , because they describe the distribution of compact sources in the sky at microwave wavelengths . we have also explored the performance of the filters for these values ( for the reference case @xmath184 , @xmath197 , @xmath186 ) and the improvement of the bsaf ( with optimal values of @xmath191 and @xmath207 versus the standard mf is still significant and of the order @xmath208 . to test the effect of the presence of bright sources on our results , we have also considered a scale - free distribution with @xmath209 \sigma_0 $ ] ( i.e. , a mixture of weak and bright sources ) for @xmath175 . we find , for the reference case ( @xmath184 , @xmath197 , @xmath186 ) , that the bsaf improves the standard mf around a @xmath210 , with optimal parameters @xmath191 and @xmath211 . we would like to point out that for for a given set of @xmath179 , @xmath41 and @xmath178 , this distribution of weak and bright sources leads to very similar optimal parameters for the bsaf as the scale - free distribution of weak sources . in addition , we have also tested the performance of the filters for a scale - free distribution of bright sources with @xmath212 \sigma_0 $ ] , for the same case as before ( @xmath184 , @xmath197 , @xmath186 ) . we explore the parameter space of @xmath213 , looking for the best filter regarding detection . we find that , for this distribution , the optimal parameters for the bsaf are @xmath158 and @xmath127 , that is , the bsaf defaults to the standard mf . the filters considered here depend on a number of parameters ( @xmath126 in the case of saf , mf and mh and @xmath126 and @xmath149 in the case of bsaf ) that must be determined in order to get the maximum number of detections for a fixed number of spurious detections . while for a given filter the region of acceptance is explicitly independent of the source distribution , the methodology presented here for the estimation of the optimal filter parameters depends on some assumed parameters of the source distribution ( namely @xmath205 , @xmath214 and @xmath215 ) and the noise power spectrum ( @xmath179 ) . a full study of the robustness of the method for all the filters is out of the scope of this work . however , we have considered some interesting cases as tests of the robustness of the method . in order to ascertain to what extent the uncertainties in the @xmath205 parameter of the source distribution affects the determination of the optimal filter parameters , we repeated our calculations using wrong assumptions on its value . an interesting case corresponds to assume that the source distribution is uniform when , in reality , it is scale - free and vice versa . in order to do this , we first construct the bsaf using the optimal @xmath216 values that were obtained for the uniform distribution . we calculate then the number density of sources @xmath80 obtained from a map that contains sources that follow a scale - free distribution with @xmath175 . we find that the differences in the number of detections when using the wrong filter with respect to using the optimal one are very small ( lower than @xmath217 ) . the same happens if the filter is constructed assuming an underlying scale - free distribution and applied to a map with sources uniformly distributed . this is not surprising , since tables [ tab : tabla1_n ] and [ tab : tabla2_n ] show that both uniform and scale - free distributions lead to similar values of the optimal @xmath126 and @xmath149 parameters . another source of uncertainty that appears in any real case is the value of the limiting cut - offs of the source distribution , @xmath214 and @xmath215 . we have seen in the previous subsections that for a given set of values @xmath79 , @xmath179 and @xmath41 , different cut - offs for the same distribution lead to similar optimal @xmath126 and @xmath149 parameters . for instance , in our reference example ( @xmath184 , @xmath197 , @xmath186 ) and a uniform distribution of sources with @xmath218 and @xmath219 the optimal filter parameters are @xmath191 and @xmath220 , whereas if the upper cut - off value is @xmath221 , the parameters take the values @xmath191 and @xmath222 . then , the shape of the optimal filter is only weakly dependent on the value of the cut - offs . this suggests that the methodology presented here is robust against uncertainties in the prior knowledge of the cut - offs of the distribution . in order to test this idea we proceeded in an analogous way to the case of the @xmath205 parameter explained before : we apply wrong filters ( that is , filters whose parameters have been determined assuming wrong values of the cutoffs ) to test cases with real distributions of sources . we tested several interesting cases : for uniform distributions , we studied the case when the sources are assumed to be weak but in reality some of them are bright ( in our example , @xmath215 is assumed to be 2 but in reality its true value is @xmath221 ) and the opposite situation . for the scale - free distribution , we studied the effect of mistaking the lower cut - off value ( assuming @xmath223 instead of its true value @xmath224 and vice versa ) . for all the cases , we plotted the curves @xmath225 versus @xmath126 . we observe that using a wrong filter changes the number of detections of all the filters , but the qualitative behaviour of the @xmath225 @xmath126 curves does not change . the relative behaviour of the filters is basically the same , and therefore the conclusions we obtained in the previous sections are still valid . thus , we conclude that the uncertainty in the knowledge of the source distribution is , in general , not a critical issue in the cases we have considered . a more delicate issue is the one related to the assumption of the @xmath179 parameter . if one assumes a value for @xmath179 that is very different from the true one , the shape of the filters changes dramatically ( except in the case of the mh whose shape is independent of @xmath179 ) with respect to the optimal ones and this may lead to wrong results . however , note that there are very well established techniques to estimate the power spectrum . albeit in this academic case we consider power law - type backgrounds , it is straightforward to apply the method to any kind of power spectrum that can be present in the data . as an example , we consider a case where the background corresponds to a true value @xmath226 whereas the filters have been constructed with a wrong @xmath227 , that is , a @xmath228 error in the determination of @xmath179 . the resulting @xmath229 curves are given in figure [ fig : gamma_falsa ] . the behaviour of the bsaf , the mf and the saf is qualitatively similar to the case where the noise power spectrum is perfectly known , but the performance of the three filters is poorer . the mh curve is identical to the ideal case since the shape of the filter does not depend on @xmath179 . the bsaf still outperforms all the other filters , although the improvement in the number of detections with respect to the mf slightly decreases . in order to see how our theoretical framework works in a practical example , we run a large set of simulations and study the performance of the np detector after filtering them with the different filters considered in the previous sections . we choose as a practical example the interesting case of a gaussian background characterised by a white noise power spectrum ( @xmath179=0 ) and sources whose intensity distribution is uniform . we will focus on the detection of weak sources . for the sake of simplicity , we give the results only for the bsaf and the mf since the other two filters ( saf and mh ) perform worse in the considered case . the different simulations are performed as follows . the images contain a number @xmath230 pixels , which is sufficiently large so that the addition of a single source does not modify significantly the dispersion of the images . the background is generated as a random field with dispersion ( before filtering ) @xmath231 ( in arbitrary units ) . the sources that we have considered for this example have a characteristic scale @xmath186 pixels . since we are interested in the detection of weak sources , we add point sources with a uniform amplitude distribution in the interval @xmath232 $ ] in the same arbitrary units of the background . the images filtered with the standard mf ( @xmath127 ) for this scale ( @xmath186 ) have dispersion @xmath233 . thus , the sources are distributed in the interval @xmath234 $ ] , where @xmath235 is the normalized amplitude of the sources with respect to the dispersion of the field filtered with the standard mf . for every maximum in a given image , it is possible to apply an empirical np criterion to decide whether the maximum corresponds or not to a source . the quantities in equations ( [ eq : r _ * ] ) and ( [ eq : phi ] ) can be obtained from simulations in the following way : the momenta @xmath182 , @xmath236 and @xmath237 ( and , therefore , the quantities @xmath238 , @xmath115 needed to know the value of the linear detector @xmath75 ) can be straightforwardly calculated from the image . for every maximum in the image , it is possible as well to measure directly its amplitude @xmath51 and curvature @xmath14 . the normalized curvature is easily obtained by fourier transforming the image , multiplying by @xmath239 and going back to real space . this gives the value of @xmath240 at each point and @xmath14 is obtained dividing by @xmath237 . the critical value @xmath74 that defines the acceptance region using equation ( [ eq : r _ * ] ) can be obtained as well directly from the simulations . for each considered filter , it is in principle possible to calculate @xmath74 semi - empirically , inverting equation ( [ eq : nb * ] ) ( with the empirically obtained values of @xmath238 and @xmath115 ) just as we did in the previous sections , and hence to proceed with the np decision rule . instead , since we are dealing with simulations , we will follow a fully empirical approach . the argument goes as follows . we fix the number density of spurious detections , i.e. , the number of maxima of the background that are misidentified as sources " by our detection criterion . then we simulate a set of images containing only background and filter them with the filter under study . we focus on the background maxima and try to determine the value of @xmath74 that makes the np rule to produce the specified number of spurious detections . for example let us consider that we perform @xmath241 noise realisations and focus on what happens in a certain pixel ( we choose the central pixel of the simulation in order to avoid border effects ) . for every realisation , we check if there is a maximum at this position or not . if a maximum is present , the value of @xmath75 is calculated . all the values of @xmath75 obtained in that way are sorted into descending numbers ( large to small ) . the value of @xmath74 is then given by the @xmath75 corresponding to the r - th element @xmath242 of the sorted list ( that is , @xmath74 is the value of @xmath75 so that there are @xmath243 background maxima with @xmath244 ) . for this example , we considered @xmath185 and therefore @xmath245 . once the value of @xmath74 has been empirically determined we add a source with a gaussian profile of dispersion @xmath246 pixels at the central position ( pixel @xmath247 ) of the unfiltered background image and then we filter it with the considered filter . we proceed to apply the detector to any maxima located at the pixel we are considering . finally , @xmath80 , for each of the filters , will be the number of sources with an estimated @xmath244 divided by the total number of realizations @xmath248 . we have performed numerical simulations for our reference case ( @xmath185 , @xmath186 and @xmath184 ) , assuming a uniform distribution with @xmath170 . we have compared the performance of the filters and the empirical np detector in the simulated images with the theoretical predictions as a function of the parameter @xmath126 . for each @xmath126 value and each filter , we have done five sets of simulations of the background , each one with a number of realizations large enough to have 5000 of them containing a maximum in the central pixel . ] . we used these simulations to obtain @xmath74 as explained before . in the figure [ fig : numdet_sims_todo ] we present the results from the simulations for this case and the comparison with the theoretical calculations . the lines in this plot show the theoretical results for each filter , the triangles the result from the simulations for the bsaf and the squares the results for the mf . we concentrate on the bsaf , which corresponds to the dot - dash line . as we mentioned in previous sections , the bsaf significantly improves the standard mf for @xmath184 . the simulations follow the theoretical results well . in the region where @xmath190 , there is a small deviation from theory which we believe is related to the fact that we are close to the scale of the pixel , but still , significantly close to the expected theoretical value . we can estimate the amplitude of a source using the unbiased and maximum efficient estimator from equation ( [ eq : estim_amplitude ] ) and then compare it with the amplitude that we have randomly generated . in figure [ fig : estim_ampli1 ] , we plot the real amplitude versus the estimated one for the bsaf ( top panels ) and the mf ( bottom panels ) . two amplitude regimes have been explored . in the left panels , we have simulated a uniform distribution of weak sources , @xmath181\sigma_0 $ ] . the parameters used for these simulations are @xmath185 , @xmath184 and @xmath186 . the optimal filter parameters have been chosen at each case . the points and the error bars are calculated as the average and the dispersion of the detected sources that fall in each of the amplitude bins from a total of @xmath249 10000 detected sources . we find a similar positive bias in the determination of the amplitude for the bsaf ( @xmath250 , @xmath220 ) and mf ( @xmath250 ) . however , the error bars corresponding to the bsaf are slightly smaller than those of the mf . in the right panels , we give the results for a uniform distribution of sources with @xmath251\sigma_{0}$ ] . the simulation parameters are the same as before ( @xmath185 , @xmath184 and @xmath186 ) . as before , the points and the error bars are calculated as the average and the dispersion of the detected sources that fall in each of the amplitude bins from a total of @xmath249 20000 detected sources . we find that the bsaf ( @xmath191 , @xmath222 ) is unbiased for bright sources whereas the estimation of the amplitude in the case of the standard mf ( @xmath127 ) shows some bias even for bright sources . therefore , the bsaf with @xmath191 outperforms the standard mf in both the detection and estimation for this distribution . the fact that sources with small amplitudes are significantly affected by a positive bias can be explained taking into account that these sources are more easily detected if they lie over a positive contribution of the background . this contributes systematically to the overestimation of the amplitude . we would like to point out that this estimator produces appreciably better results than a naive estimation using directly the measured values at the maxima . nowadays , the detection of compact sources on a background is a relevant problem in many fields of science . a number of detection techniques use linear filters and thresholding - based detectors . our approach to the problem of detector design is different . we use a neyman - pearson rule that takes into account _ a priori _ information of the distribution of sources and the number density of maxima to define the region of acceptance . in our case , we take advantage not only of the amplification but also of the spatial information : the curvature of the background is different from that of the sources , and we use this to improve our detection rule . the background is modelled by a homogeneous and isotropic gaussian random field , characterized by a scale - free power spectrum @xmath252 , @xmath253 . we design a new filter that we call bsaf in such a way that the use of our improved detection rule based on amplification and curvature on the filtered field will increase the number of detections for a fixed number of spurious sources . we generalize the functional form of this filter , as well as other standard filters , and introduce another degree of freedom , @xmath126 , that allows us to filter at any scale , including that of the source @xmath41 . we have shown the benefits of filtering at scales smaller than @xmath41 , which significantly improves the number of detections . as an example , we have considered two different distributions of sources . a uniform distribution in the interval @xmath234 $ ] and a scale - free power law distribution in the interval @xmath254 $ ] ( where the threshold @xmath38 corresponds to the field filtered with the standard mf ) , i.e. we are considering weak sources . the bsaf has proven to be significantly better than the standard mf , the saf and mh wavelet in certain cases . in particular , we have considered a reference case with parameters @xmath184 , @xmath197 and @xmath186 , where the improvement in the number of detections of the bsaf at @xmath191 with respect to the standard mf is @xmath255 . we have also tested the performance of the filters for a mixture of weak , intermediate and bright sources . for a uniform distribution with @xmath256 $ ] and for a scale - free distribution with @xmath257 $ ] , the bsaf also improves the mf . however , for a scale - free distribution with @xmath258 $ ] , i.e. , dominated by bright sources , we find that the optimal bsaf defaults to the standard mf , which gives the maximum number of detections in this case . we find that the bsaf gives in any case the best performance among the considered filters . indeed , the saf and the mf are particular cases of the bsaf and the strategy we follow , i.e. maximization of the detections , guarantees that the parameters of the bsaf will default to the best possible of these filters in each case . in addition , we also find that the bsaf performs at least as well as the mh in all the considered cases . therefore , the number density of detections obtained with the bsaf will be at least equal to the best of the other three filters , and in certain cases superior . however , in some other cases , the gain is small and it is justified to use an analitically simpler filter . our results suggest that for power law spectra , from the practical point of view , one could use the bsaf when @xmath259 since , in this range , clearly improves the number of detections with respect to the other filters . however , for @xmath260 the usage of the mh is justified due to its robustness ( since it has an analytical form ) and it gives approximately the same number of detections obtained either with the bsaf or mf . for all the studied cases of source distributions ( except for the one dominated by bright sources ) and fixing the values of @xmath179 , @xmath79 and @xmath41 , we find that the optimal parameters of the bsaf are only weakly dependent on the distribution of the sources . we have done some simple tests in order to study the robustness of the method when the knowledge about the source _ pdf _ or the background spectral index is not perfect . we find that the values of the optimal filter parameters vary slightly when we assume that the source distribution is uniform when , in reality , it is scale - free and vice versa . the uncertainties in the cut - off values of the source _ pdf _ affect the number of detections , but in a similar way for all the filters , and therefore the relative behaviour of the filters do not change . errors in the estimation of the spectral index @xmath179 reduces the efectiveness of the bsaf , but it still outperforms the other filters . all of this indicates that our detection scheme is robust against uncertainties in the knowledge of the distribution of the sources and spectral index . to test the validity of our results in a practical example , we have tested our ideas with simulations for the uniform distribution ( using our reference case @xmath197 , @xmath186 , @xmath184 ) and find that the results follow approximately the expected theoretical values regarding source estimation , we propose a linear estimator which is unbiased and of maximum efficiency , that we have also tested with simulations . the ideas presented in this paper can be generalized : application to other profiles ( e.g. multiquadrics , exponential ) and non - gaussian backgrounds is physically and astronomically interesting . the extension to include several images ( multi - frequency ) is relevant . the generalization to two - dimensional data sets ( flat maps and the sphere ) and nd images is also very interesting . finally the application of our method to other fields is without any doubt . we are currently doing research in some of these topics . the authors thank enrique martnez - gonzlez and patricio vielva for useful discussions . mlc thanks the ministerio de ciencia y tecnologa ( mcyt ) for a predoctoral fpi fellowship . rbb thanks the mcyt and the universidad de cantabria for a ramn y cajal contract . dh acknowledges support from the european community s human potential programme under contract hprn - ct-2000 - 00124 , cmbnet . we acknowledge partial support from the spanish mcyt project esp2002 - 04141-c03 - 01 and from the eu research training network ` cosmic microwave background in europe for theory and data analysis ' . de zotti , g. , toffolatti , l. , argeso , f. , davies , r.d . , mazzotta , p. , partridge , r.b . , smoot g.f . & vittorio , n. , 1999 , 3k cosmology , proceedings of the ec - tmr conference held in rome , italy , october , 1998 . woodbury , n.y . : american institute of physics , vol . 476 , 204 . the ratio @xmath261 can be explicitly written as @xmath262 and taking into account the np criterion for detection , we find @xmath263 where @xmath12 is a constant . by differentiating the previous equation with respect to @xmath75 @xmath264 therefore , @xmath265 is equivalent to @xmath244 , where @xmath74 is a constant , i.e. @xmath133 given by equation ( [ eq : phi ] ) is a sufficient linear detector . let us assume a linear estimator combination of the normalized amplitude @xmath38 and normalized curvature @xmath14 with the constraint @xmath266 if the estimator is unbiased , i.e. @xmath267 , taking into account that @xmath268 and @xmath269 , we obtain the constraint @xmath270 on the other hand , the variance is given by @xmath271 where we have taken into account that @xmath272 . by minimizing the previous expression with the constraint ( [ eq : constraint_appc ] ) , one obtains @xmath273 therefore , one obtains : @xmath274 @xmath275

this paper considers the problem of compact source detection on a gaussian background . we make a one - dimensional treatment ( though a generalization to two or more dimensions is possible ) . two relevant aspects of this problem are considered : the design of the detector and the filtering of the data . our detection scheme is based on local maxima and it takes into account not only the amplitude but also the curvature of the maxima . a neyman - pearson test is used to define the region of acceptance , that is given by a sufficient linear detector that is independent on the amplitude distribution of the sources . we study how detection can be enhanced by means of linear filters with a scaling parameter and compare some of them that have been proposed in the literature ( the mexican hat wavelet , the matched and the scale - adaptive filters ) . we introduce a new filter , that depends on two free parameters ( biparametric scale - adaptive filter ) . the value of these two parameters can be determined , given the a priori _ pdf _ of the amplitudes of the sources , such that the filter optimizes the performance of the detector in the sense that it gives the maximum number of real detections once fixed the number density of spurious sources . the new filter includes as particular cases the standard matched filter and the scale - adaptive filter . then , by construction , the biparametric scale adaptive filter outperforms these filters . the combination of a detection scheme that includes information on the curvature and a flexible filter that incorporates two free parameters ( one of them a scaling ) improves significantly the number of detections in some interesting cases . in particular , for the case of weak sources embedded in white noise the improvement with respect to the standard matched filter is of the order of @xmath0 . finally , an estimation of the amplitude of the source ( most probable value ) is introduced and it is proven that such an estimator is unbiased and it has maximum efficiency . we perform numerical simulations to test these theoretical ideas in a practical example and conclude that the results of the simulations agree with the analytical ones . methods : analytical - methods : data analysis - techniques : image processing

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Proceed to summarize the following text: the central regions of clusters of galaxies are very interesting and enigmatic places . the intracluster medium ( icm ) in the central region is often so dense that its radiative cooling time is significantly shorter than the hubble time . therefore , the icm will cool down unless it is heated significantly . unbalanced radiative cooling would cause a `` cooling flow '' in the cluster center ( see fabian 1994 for a review ) . x - ray imaging analyses by _ rosat _ and _ exosat _ indicated that mass deposition rates , @xmath1 , were more than @xmath2 for many clusters ( edge , stewart , & fabian 1992 ; allen & fabian 1997 ; peres et al . however , the fate of the resulting cold gas , or whether icm in `` cooling flow '' clusters really cools down to low temperatures , is still unclear . the cd galaxies in cooling flows have an excess blue star component , which suggests that the cooled icm forms new stars ( mcnamara & oconnell 1989 ) . however , the implied star formation rates are much lower than the cooling flow mass deposition ones ( mcnamara & oconnell 1989 ) . furthermore , recent _ xmm - newton _ high resolution spectroscopic observations have imposed a strong constraint on the amount of gas cooling down to low temperatures ( peterson et al . 2001 ; kaastra et al . 2001 ; tamura et al . 2001 ) , which is consistent with the fact that the mass deposition rates determined spectroscopically by _ asca _ tended to be lower than those determined by _ exosat _ and _ rosat _ ( see makishima et al . 2001 for a review ) . the cd galaxies in the cooling flow clusters usually host strong radio sources , with a central radio core and/or radio lobes . observations utilizing _ chandra _ s excellent spatial resolution have revealed interactions between these radio lobes and icm , which may affect the dynamical and thermal history of icm in the cluster center . in hydra a ( mcnamara et al . 2000 ; david et al . 2001 ) , perseus ( fabian et al . 2000 ) , and abell 2052 ( blanton et al . 2001 ) , radio lobes coincide with x - ray low surface brightness regions , which suggests that the radio plasma expands to displace the icm . on the other hand , x - ray cavities without radio emission are also found at larger distances in abell 2597 ( mcnamara et al . they are believed to be old radio lobes which can not emit ( at least at high radio frequencies ) because the high energy relativistic electrons have already suffered radiative losses . these radio quiet cavities also indicate that cd galaxies provided the icm with high energy electrons and magnetic fields intermittently . abell 3112 is a cooling flow cluster at a redshift of @xmath3 . there is a powerful radio source , pks 0316 - 444 , in the cluster center . previous x - ray imaging analyses with _ exosat _ ( edge , stewart , & fabian 1992 ) and _ rosat _ ( allen & fabian 1997 ; peres et al . 1998 ) indicated that abell 3112 had a strong cooling flow , with a mass deposition rate of @xmath4 @xmath5 yr@xmath6 . the temperature profile at a larger scale than the cooling radius was studied by markevitch et al . ( 1998 ) and by irwin , bregman , & evrard ( 1999 ) with _ asca _ data . markevitch et al . ( 1998 ) found that the temperature decreased outward from 6 kev to 3 kev with _ asca _ data , although they explicitly assumed a cooling flow component in the central region . on the other hand , irwin et al . ( 1999 ) argued that the data were consistent with an isothermal distribution . an increase in the iron abundance toward the center was reported by finoguenov , david , & ponman ( 2000 ) . however , there was no detailed study of the temperature and abundance structure within the cooling radius ( @xmath7 kpc ) because of the limited spatial resolution of _ asca_. furthermore , any interaction region between the radio source and the icm would be too small to have been resolved with _ rosat _ or _ asca_. in this paper , we present _ chandra _ observations of central region of abell 3112 . we assume @xmath8 km s@xmath6 mpc@xmath6 , @xmath9 , and @xmath10 . at a redshift @xmath3 , 1 corresponds to 1.94 kpc . the errors correspond to the 90% confidence level throughout the paper . abell 3112 was observed twice with _ chandra _ on 2001 may 24 for 7,257 seconds and on 2001 september 15 for 17,496 seconds . each observation was pointed so that the cluster center would fall near the aimpoint of the acis - s3 detector . however , the roll angles were different , so the outer parts of the field of view differ for the two observations . the two observations were merged , using the positions of bright point sources to register the two images . here , we analyze data from the s3 chip only . the data from the s1 chip was used to search for the background flares . a few short periods with background flares were found and removed using the lc_clean software provided by maxim markevitch , leaving a total exposure of 21,723 seconds . only events with asca grades of 0 , 2 , 3 , 4 , and 6 were included . standard bad pixels and columns were removed . exposure maps and background data were generated for each observation separately , and then merged appropriately . blank sky background data were taken from the compilation by maxim markevitch . the data are possibly affected by the low energy qe degradation of acis . we tried to correct this using the corrarfprogram . however , the correction did not appear to work well for this data ; specifically , corrarf seemed to over - correct the data . as a result , we obtained absorption column densities which were much lower than the galactic value in our spectral fits . we expect that the absorbing column should be at least as large as the galactic value . therefore , we will show the results without the correction for the low energy qe degradation . fortunately , this correction significantly affects only the very low energy ( @xmath11 1 kev ) part of the _ chandra _ band . as a result , neither temperature nor abundance determinations were seriously affected by this correction . on the other hand , mass deposition rates of cooling flow components increased by a factor of about two when we fitted the spectra after the correction . the observations were made with very large array at a center frequency of 1320 mhz on 1996 october 18 . the array was in the ` a ' configuration , which provided an angular resolution of 6.9 @xmath12 1.5 arcsec . a total of 1.2 hours were obtained on source using 32 channels across a 6.25 mhz band . both right and left circular polarizations were observed . phase and bandpass calibration was obtained by short ( 1 min ) observations of the strong ( 4.9 jy ) calibrator j0440 - 4333 taken every @xmath1320 minutes . atomic hydrogen absorption was searched for towards the compact core over a velocity range from 22,10023,440 km s@xmath6 at a resolution of 44 km s@xmath6 , but not detected down to a 3@xmath14 limit on the maximum optical depth of 0.012 . the x - ray image was adaptively smoothed using the ciao csmooth program with a minimum signal - to - noise of 3 per smoothing beam . an identically - smoothed background was subtracted , and the image was divided by an identically - smoothed exposure map . the exposure map was corrected for vignetting , using the typical energy of the observed photons to take account of the energy dependence of the vignetting . the resulting image is shown in figure [ fig : corimage ] . as in other typical cooling flow clusters , x - ray emission is distributed symmetrically and is fairly concentrated in the center . the image is smooth and quite symmetric on large scales . the cluster emission is elliptical , with a major axis along the direction from nne to ssw . on large scales , there appears to be no substructure except for some point sources . we used a wavelet detection algorithm ( wavedetect in ciao ) to detect individual point sources . sources were visually confirmed on the x - ray image , and a few low - level detections were removed . using this method , we found sixteen point sources . one of these corresponds to the radio source pks 0316 - 444 , which is located at the center of the central cd galaxy eso 248- g 006 . another corresponds to a galaxy in the cluster ( lcrs b031619.9 - 442441 ) . the regularity of the cluster x - ray emission on large scales does nt continue into the very central region . figure [ fig : center1 ] shows the x - ray contour map of the @xmath15 region around the central point source . in the very center , an elongated structure from the central point source towards the se is clearly seen . we can see another filament - like component from @xmath16 south of the central point source towards the sw . to clarify structures which deviate from symmetric cluster emission , we fitted the entire cluster image with a concentric elliptical isophotal model , and then subtracted the best - fit model from the image to get the residual component . in the fitting procedure , we fixed the center of the isophotes to the center determined from the larger scale isophotes about which the image was symmetric , rather than the central point source . the ellipticity and position angle for each isophote were allowed to vary . it is most likely that relaxed cluster potential structure is not spheroidal but triaxial ( jing & suto 2001 ) . in this case , the icm hydrostatic density distribution also will be triaxial ( lee & suto 2003 ) . therefore , the ellipticities and the position angles of x - ray isophotes can vary with radius unless our line - of - sight coincides with one of the three symmetry axes . in figure [ fig : center2 ] , the greyscale shows the residual produced by subtracting our best - fit elliptical isophotal model from the adaptively smoothed image of the central region of the cluster . the sharp elliptical edge near the outside is an artifact because the elliptical model extended only to this isophote . dark areas are positive residuals ( excess emission ) , while light areas are negative residuals . the dark and light regions at the center are due to the central point source , which is not exactly at the center of the x - ray isophotes at large radii . there are regions of excess emission to the south of the central source . the contours show the 1.32 ghz vla radio image . there is a very bright radio core , which is unresolved in the radio image ; the elliptical contours for the core show the beam of the radio observation . the very bright radio core is coincident with the central x - ray source . there are two diffuse radio regions to the se and sw of the radio core , which may be connected to the core . the ns extent of these regions is presumably exaggerated due to the elongated observing beam . roughly speaking , the excess x - ray regions appear to surround two lobe - like radio components . however , part of the sw radio lobe is coincident with a region of x - ray excess emission . unfortunately , the low resolution of the radio data , bright radio core , and elongated radio beam make it difficult to decide definitely whether the excess is in the radio lobes or partly surrounding it . in order to examine the temperature and abundance structure quantitatively , we determined the x - ray spectra in annuli . we fitted the data between 0.5 and 10.0 kev with a photo - absorbed single temperature mekal model ( kaastra 1992 ; liedahl , osterheld , & goldstein 1995 ) after masking the point sources . we tried to fit the data fixing the absorbing column density to the galactic value ( @xmath17 @xmath18 ; dickey & lockman 1990 ) or allowing it to vary freely . the fitting results depend somewhat on whether we fixed the absorption to the galactic value or not . figure [ fig : radtemabund ] shows the radial temperature and abundance profiles . the solid crosses are the values obtained when we allowed the absorption to vary , while the dashed crosses are the values when we fixed the absorption to the galactic value . the dashed crosses are shifted slightly to the left in order to be more easily seen . the results with fixed or varying absorption are similar . at @xmath19 , both the temperature and abundance are nearly constant at @xmath20 kev and @xmath21 , respectively . the temperature decreases from @xmath22 kev at 70 to 3.5 kev in the central 20 . the abundance increases from @xmath23 at 70 to 1.3 @xmath24 in the central 20 . the temperature values obtained with variable absorption , along with their associated errors , are fitted to a power - law profile given by @xmath25 . for @xmath26 , @xmath27 kev and @xmath28 . for @xmath29 , @xmath30 kev and @xmath31 . we will use these fitted temperature profiles later when we analyze conduction timescale and integrated gravitational mass . to examine the non - radial temperature structure , we made a two - dimensional temperature map of the central @xmath32 region . we divided the region into 64 ( @xmath33 ) square regions , each of which is @xmath34 . then , we fitted the data of each region with the photo - absorbed mekal model where absorption is treated as a free parameter . figure [ fig : tmap ] shows a two - dimensional temperature map of the central @xmath32 region overlaid with x - ray surface brightness contours . black and white represent lower ( @xmath35 kev ) and higher ( @xmath22 kev ) temperatures , respectively . there is no significant azimuthal structure in the temperature map while temperature decrease toward the center is again clearly seen . we also made a two - dimensional abundance map in the same way . however , abundances in the individual fits were too poorly constrained to provide a useful map . we tried to fit the annular spectra with a model for a multiphase cooling flow to constrain the contribution from a cooling flow component and a mass deposition rate in each annulus , although the data within each annulus are almost consistent with a single temperature mekal model . as a cooling flow spectral model , we used the mkcflow model based on the mekal model . we added a mekal model representing the emission from the icm outside the cooling flow . both mkcflow and mekal were assumed to be subject to the same absorption column , which we fixed to the galactic value or let vary freely . we fixed the metallicity and initial gas temperature of the mkcflow model to the same values of metallicity and temperature as the mekal model , respectively . the low temperature in the mkcflow model is fixed to the lowest value ( 0.001 kev ) in xspec . the results are shown in table [ tab : cflow1 ] and [ tab : cflow2 ] . compared with the fitting results without cooling flows , the results are not improved dramatically . figure [ fig : radmdot ] shows a radial profile of the mass deposition rate derived from the fitting . the circles and solid error bars are the values obtained from fits with freely varying absorption . the squares and dashed error bars are the values from the fits with fixed galactic absorption . when absorption is set as a free parameter , forth bin and the outermost three bins are consistent with a model without a cooling flow component . only the inner three bins and the fifth bin have the cooling flow component , although their errors are fairly large and comparable to themselves . the resultant total mass deposition rate is @xmath36 yr@xmath6 , which is significantly smaller than the rates derived from the previous image analyses of _ exosat _ and _ rosat _ data , which are @xmath37 yr@xmath6 ( edge , stewart , & fabian 1992 ; allen & fabian 1997 ; peres , et al . the cooling flow component contributes less than @xmath38% of the total x - ray emission . when we adopt the galactic absorption value , the mass deposition rate becomes even lower and consistent with a model without a cooling flow component in all bins . even though multiphase gas is not found within small regions of the cluster , we do see an overall temperature gradient , with gas in the center that is cooler by a factor of @xmath39 compared with gas in the outer regions of the cluster . therefore , if we fit a spectrum extracted from the whole cluster , a significant mass deposition rate and a low temperature cutoff might be expected . we fit the data within @xmath40 with the mekal plus mkcflow model where the low temperature in mkcflow is allowed to vary . the results are shown in table [ tab : totspec ] . the upper and lower rows show the results with the absorption column allowed to vary and fixed to the galactic value , respectively . in contrast to the spectral analyses in annuli , the mass deposition rate is comparable with the former values based on imaging analysis . however , the low temperature in the mkcflow model is not very low ( @xmath41 kev ) , which means that the observations are consistent with significant cooling , but only over a limited temperature range . this is consistent with what has been observed with many cooling flow clusters using chandra and xmm - newton ( e.g. , peterson et al . please note that the goodness of the fit is marginal ( the reduced chi - squared is about 1.5 ) . this might indicate that the emission measure distribution with the temperature is not that of the standard cooling flow model even in the temperature range between @xmath42 and @xmath43 . to quantify the radial structure of the icm , we made a radial profile of the x - ray surface brightness in the 0.3 - 10.0 kev band ( figure [ fig : radsfpden]a ) . the bins for x - ray surface brightness were chosen to be the same as those used below to determine the radial variation in the x - ray spectrum ( [ sec : spectrum ] ) . the surface brightness values were deprojected to determine the x - ray emissivity and gas density ( figure [ fig : radsfpden]b ) , assuming the emissivity is constant in spherical shells . when we convert the emissivity into the gas density , we use a mekal code and assume that the temperature is constant in spherical shells and equal to that derived from the spectral fitting of projected data . some fluctuations are seen in the density and pressure profiles near the outer boundary ( figure [ fig : radsfpden]b and [ fig : radsfpden]c ) , which are artifacts of the deprojection since we assume zero emissivity outside the outer boundary . however , many simulations of our deprojection method show that this affects only a few outermost points , and has no significant effect anywhere near the center . the observed gas density and pressure profiles are quite smooth outside of the region of the radio source . based on the smooth radial profiles and lack of substructure in the x - ray image outside of the radio source region , we conclude that this cluster is dynamically relaxed and that the icm is generally in hydrostatic equilibrium . the gravitational mass distribution was determined from the equation of hydrostatic equilibrium , @xmath44 we use the fitted power - law temperature profiles when @xmath45 is calculated . to reduce the noise in the density gradient term , we determine the gradient at @xmath46 by differencing the densities at @xmath47 and @xmath48 . uncertainties in the density gradient at @xmath46 are also calculated from the density uncertainties at @xmath47 and @xmath48 . the density gradients and their uncertainties are assumed to be equal in the innermost three points and the outermost three points . the gas mass profile is also determined from the icm density profile assuming spherical symmetry . the gas mass is then given by @xmath49 the integrated gravitational mass and gas mass are shown in figure [ fig : radmrmg ] ( a ) . a radial profile of the gas mass fraction is shown in figure [ fig : radmrmg ] ( b ) . the gas mass fraction increases from @xmath50 at @xmath51 to @xmath52 at @xmath53 . this trend is similar to those in other well - relaxed clusters ( david , jones , & forman 1995 ; ettori , & fabian 1999 ; allen , schmidt , & fabian 2002 ) . our gas mass fraction is lower than that of mohr , mathiesen , & evrard ( 1999 ) at much larger radii ( @xmath54 1 mpc ) , but the gas fractions of clusters generally increase with radius , and this appears to apply to abell 3112 . using the same beta - model fits as in mohr et al . , we find that the gas fraction at @xmath53 is predicted to be @xmath1312% , which agrees well with our values . because the temperature is higher than @xmath35 kev over the whole cluster , the main emission mechanism is thermal bremsstrahlung . therefore , the isobaric cooling time is ( sarazin 1986 ) @xmath55 radiative cooling would make a radial temperature gradient in the central region of the icm through its density dependence , as is observed . [ fig : radtemabund ] ) . on the other hand , thermal conduction would have the effect of reducing the temperature gradient . the conduction timescale is generally expressed as ( see also sarazin 1986 ) @xmath56 where @xmath57 is electron number density , @xmath58 is the scale length of the temperature gradient , and @xmath59 is the boltzmann constant . if we consider only the coulomb scattering process , the thermal conductivity for hydrogen plasma is ( spitzer 1962 ) @xmath60 where @xmath61 , the coulomb logarithm , is @xmath62.\ ] ] figure [ fig : radtctcond ] shows radial profiles of @xmath63 and @xmath64 . when we calculated @xmath64 , we used the fitted power - law temperature profile to reduce the noise in the temperature gradient . also , @xmath64 is evaluated only for @xmath65 because an isothermal distribution is consistent with the temperature data in the outer regions . the cooling time of the innermost bin is @xmath66 yr . if we define a cooling radius as where @xmath63 is @xmath67 yr , it becomes @xmath68 , or @xmath69 kpc . both results are consistent with the former _ rosat _ ( allen & fabian 1997 ; peres et al . 1998 ) and _ exosat _ results ( edge , stewart , & fabian 1992 ) . at the innermost three bins , @xmath63 and @xmath64 are comparable with each other . outside of this region , the conduction timescale is clearly shorter than the cooling timescale . if conduction proceeded at the spitzer rate , it would erase the temperature gradient . since the observations show a significant gradient in this region ( fig . [ fig : radtemabund]a ) , thermal conduction must be significantly suppressed by , e.g. , tangled magnetic fields and/or plasma instabilities . the cooling rate of gas with solar abundance is only @xmath13 10% higher than that of hydrogen - helium mixture gas at 3.5 kev ( see figure 9 - 9 of binney & tremaine 1987 ) . therefore , actual cooling timescales in the central regions are probably @xmath13 10% shorter at the most , which does not change the situation here dramatically . although our estimation here is rough , it is still useful for an estimation of an order of magnitude . calculations taking account of time evolution will be helpful to make more precise models . we detected an x - ray point source in the cluster center . the position is coincident with the optical core of the cd galaxy and the radio core . the position of the x - ray point source is ( epoch j2000 ) @xmath70 @xmath71 with an uncertainty of 0.04 arcsec in each coordinate . the spectrum of the point source is shown in figure [ fig : cpssp ] . we fitted the data with a mekal model plus a power - law model . in addition to the absorption column density common for both components , we added intrinsic absorption for the power - law component only . as a result , the applied model is @xmath72.\ ] ] the abundance of the mekal component was fixed to the value derived from the region surrounding the point source . the absorbing column @xmath73 was also fixed either to the value derived from the surrounding region or the galactic value . the fitting results are shown in table [ tab : centps ] . the spectral model consists of a 1.26 kev thermal plasma component and a power - law component with a photon index of @xmath74 . the power - law component has an extra absorbing column of @xmath75 @xmath18 . this is consistent with the central source being a strongly absorbed agn . we found that the central radio source associated with the central cd galaxy in abell 3112 is interacting with the surrounding icm ( fig . [ fig : center1 ] and [ fig : center2 ] ) . the radio image shows a central core and two diffuse lobes to the se and sw of the core . the central x - ray image has an asymmetric structure , and the excess x - ray emission over an elliptical isophotal model roughly appears to surround two radio lobes . this configuration is naturally explained if the radio lobes have swept up the icm which was where the radio lobes are now . similar scenarios have been proposed in other cooling flow clusters with radio sources ( e.g. , mcnamara et al . 2000 ; fabian et al . 2000 ; blanton et al . in addition , there is excess x - ray emission coincident with part of the sw radio lobe . it is possible that the excess x - ray emission is due to the cool gas that was originally near the very center ; this cool gas might have been entrained by hot buoyant bubbles . this scenario was originally proposed to explain the x - ray morphology of the central region of m87 ( churazov et al . indeed , the filamentary structures in the x - ray image of the central region of abell 3112 ( figure [ fig : center1 ] ) are similar to those in m87 ( feigelson et al . 1987 ; bhringer et al . 1995 ) . this model may also apply to abell 133 ( fujita et al . 2002 ) . in order to determine which model applies to abell 3112 , temperature and abundance measurements of the excess x - ray components would be useful because the excess components are expected to have higher abundances and lower entropy in the entrainment scenario . unfortunately , the present data have too few counts in these regions to allow a detailed spectral analysis . a deeper radio image would also be useful , as would lower frequency observations to search for steep spectrum emission . note that the effect of the central radio source is limited to a region very close ( @xmath76 ) to the center . this region is much smaller than the cooling flow region itself ( @xmath77 ) . outside the interacting region , there is no evidence of the interaction such as x - ray cavities , instead the x - ray image is quite symmetric and smooth . therefore , the central radio source does not directly affect the structure of the cooling flow outside of the very central region , at least at present . note that we only have high frequency radio data . low frequency radio observations might give another picture . in the case of m87 in the virgo cluster , for instance , 327 mhz radio data show an morphology which fills out a large portion of the central region of the cluster ( owen , eilek , & kassim 2000 ) , although similar structures can not be seen in higher frequency data . radiative cooling produces a temperature decrease towards the center of the cluster . on the other hand , thermal conduction would reduce this temperature gradient . if we adopt the standard spitzer conductivity , conduction would be expected to be very effective and to have eliminated the central temperature gradient ( figure [ fig : radtctcond ] ) . however , the observations show that the temperature does decrease significantly at the center of the cluster . this implies that thermal conduction is significantly suppressed below the spitzer value . although we did not find a large amount of locally cooling multiphase gas , the existence of a large - scale temperature gradient indicates that the icm is cooling in a global scale . the mass deposition rate of abell 3112 determined from our spectroscopy of the total spectrum is comparable to that determined previously based on _ rosat _ and _ exosat _ image analyses ( edge , stewart , & fabian 1992 ; allen & fabian 1997 ; peres et al . however , the temperature range of the cooled gas is limited to @xmath0 kev . in addition , emission measure distribution about temperature is probably different than that expected from the standard cooling flow model . this suggests that some other heating mechanism may affect the cooling gas . one probable solution is heat conduction which is reduced from the spitzer value by some physical mechanism but still is energetically important ( e.g. , hattori & umetsu 2000 ; malyshkin & kulsrud 2001 ) . the relativistic electrons from the central radio source might also supply the required heating ( e.g. , fabian et al . 2000 ; mcnamara et al . 2000 ; david et al . 2001 ; blanton et al . 2001 ) . however , it is unlikely that the radio - emitting electrons are heating all of the cooling gas in abell 3112 at present , because the radio lobes are so small that they do not directly affect the cooling flow on large scales . in abell 3112 , there is no evidence for ghost bubbles which might indicate a history of past radio source interactions ( mcnamara et al . another possible form of radio source heating would be high energy protons from the central agn . since the minimum energy density in relativistic electrons is too small to support radio bubbles against the surrounding hot gas ( e.g. , blanton et al . 2001 ) , most of the energy in the radio source may be in the form of relativistic protons . the accelerated protons could diffuse and heat the icm through coulomb interactions ( rephaeli & silk 1995 ; inoue & sasaki 2001 ) . although they do not emit any observable radio radiation , protons would produce several hundred mev @xmath78-rays through the decay of neutral pions produced in collisions with thermal protons . future @xmath78-ray observations will test this hypothesis . we would like to thank h. bhringer , j. c. kempner , and y. fujita for very useful comments . support for this work was provided by the national aeronautics and space administration through _ chandra _ award number go1 - 2133x issued by the _ chandra _ x - ray observatory center , which is operated by the smithsonian astrophysical observatory for and on behalf of nasa under contract nas8 - 39073 . m. t. was supported in part by a grant - in - aid from the ministry of education , science , sports , and culture of japan ( 13440061 ) . support for e. l. b. was provided by nasa through the _ chandra _ fellowship program , grant award number pf1 - 20017 , under nasa contract number nas8 - 39073 . allen , s. w. , & fabian , a. c. 1997 , mnras , 286 , 583 allen , s. w. , schmidt , r. w. , & fabian , a. c. 2002 , mnras , 334 , l11 binney , j. , & tremaine , s. 1987 , galactic dynamics ( princeton : princeton univ . press ) blanton , e. l. , sarazin , c. l. , mcnamara , b. r. , & wise , m. w. 2001 , apj , 558 , l15 bhringer , h. , nulsen , p. e. j. , braun , r. , & fabian , a. c. 1995 , mnras , 274 , l67 churazov , e. , brggen , m. , kaiser , c. r. , bhringer , h. , & forman , w. 2001 , apj , 554 , 261 david , l. p. , jones , c. , & forman , w. 1995 , apj , 445 , 578 dickey , j. m. , & lockman , f. j. 1990 , ara&a , 28 , 215 edge , a. c. , stewart , g. c. , & fabian , a. c. 1992 , mnras , 258 , 177 ettori , s. , & fabian , a. c. 1999 , 305 , 834 fabian , a. c. 1994 , ar&aa , 32 , 277 fabian , a. c. , et al . 2000 , mnras , 318 , 65 feigelson , e. d. , wood , p. a. d. , schreier , e . j. , harris , d. e. , & reid , m. j. 1987 apj , 312 , 101 finoguenov , a. , david , l. p. , & ponman , t. j. 2000 , apj , 544 , 188 fujita , y. , sarazin , c. l. , kempner , j. c. , rudnick , l. , slee , o. b. , roy , a. l. , andernach , h. , & ehle , m. 2002 , apj , 575 , 764 hattori , m. , & umetsu , k. 2000 , apj , 533 , 84 inoue , s. , & sasaki , s. 2001 , apj , 562 , 618 irwin , j. a. , bregman , j. n. , & evrard , a. e. apj , 1999 , 519 , 518 jing , y. p. , & suto , y. 2002 , apj , 574 , 538 kaastra , j. s. 1992 , an x - ray spectral code for optically thin plasmas ( internal sron - leiden report , updated version 2.0 ) kaastra , j. s. , et al . , 2001 , a&a 365 , l99 lee , j. & suto , y. 2003 , apj , 585 , 151 liedahl , d. a. , osterheld , a. l. , & goldstein , w. h. 1995 , apj , 522 , 82 makishima , k. , et al . , 2001 , pasj , 53 , 401 malyshkin , l. , & kulsrud , r. 2001 , apj , 549 , 402 markevitch , m. , forman , w. r. , sarazin , c. l. , & vikhlinin , a. 1998 , apj , 503 , 77 mcnamara , b. r. & oconnell , r. , w. 1989 , aj , 98 , 2018 mcnamara , b. r. , et al . 2000 , apj , 534 , l135 mcnamara , b. r. , et al . 2001 , apj , 562 , l149 mohr , j. j. , mathiesen , b. , & evrard , a. e. 1999 , apj , 517 , 627 peres , c. b. , fabian , a. c. , edge , a. c. , allen , s. w. , johnstone , r. m. , & white , d. a. 1998 , mnras , 298 , 416 owen , f. n. , eilek , j. a. , & kassim , n. e. 2000 , apj , 543 , 611 peterson , j. r. , et al . 2001 , a&a , 365 , l104 peterson , j. r. , et al . 2003 , apj submitted , astro - ph/0210662 rephaeli , y. & silk , j. 1995 , apj , 442 , 91 sarazin , c. s. , 1986 , rev . phys . , 58 , 1 spitzer , l. , jr . 1962 , physics of fully ionized gases ( new york : wiley ) tamura , t. , et al . 2001 , a&a , 365 , l87 cccccc + @xmath79 & @xmath80 & @xmath81 & @xmath73 & @xmath1 & @xmath82 + ( arcsec ) & ( kev ) & ( @xmath83 ) & ( @xmath84 @xmath18 ) & ( @xmath5 yr@xmath6 ) & + @xmath85 & @xmath86 & @xmath87 & @xmath88 & @xmath89 & 284.5/200 + @xmath90 & @xmath91 & @xmath92 & @xmath93 & @xmath94 & 285.3/233 + @xmath95 & @xmath96 & @xmath97 & @xmath98 & @xmath99 & 226.2/216 + @xmath100 & @xmath101 & @xmath102 & @xmath103 & @xmath104 & 234.3/205 + @xmath105 & @xmath106 & @xmath107 & @xmath108 & @xmath109 & 222.9/195 + @xmath110 & @xmath111 & @xmath112 & @xmath113 & @xmath114 & 216.8/188 + @xmath115 & @xmath116 & @xmath117 & @xmath118 & @xmath119 & 202.5/181 + @xmath120 & @xmath121 & @xmath122 & @xmath123 & @xmath124 & 204.5/175 + cccccc + @xmath79 & @xmath80 & @xmath81 & @xmath73 & @xmath1 & @xmath82 + ( arcsec ) & ( kev ) & ( @xmath83 ) & ( @xmath84 @xmath18 ) & ( @xmath5 yr@xmath6 ) & + @xmath85 & @xmath125 & @xmath126 & ( @xmath127 ) & @xmath128 & 315.6/201 + @xmath90 & @xmath129 & @xmath130 & ( @xmath127 ) & @xmath131 & 302.5/234 + @xmath95 & @xmath132 & @xmath133 & ( @xmath127 ) & @xmath134 & 237.1/217 + @xmath100 & @xmath135 & @xmath136 & ( @xmath127 ) & @xmath137 & 236.9/206 + @xmath105 & @xmath138 & @xmath139 & ( @xmath127 ) & @xmath140 & 230.1/196 + @xmath110 & @xmath141 & @xmath142 & ( @xmath127 ) & @xmath143 & 217.7/189 + @xmath115 & @xmath144 & @xmath145 & ( @xmath127 ) & @xmath146 & 202.5/182 + @xmath120 & @xmath147 & @xmath148 & ( @xmath127 ) & @xmath149 & 205.85/176 + cccccc + @xmath80 & @xmath150 & @xmath81 & @xmath73 & @xmath1 & @xmath82 + ( kev ) & ( kev ) & ( @xmath83 ) & ( @xmath84 @xmath18 ) & ( @xmath5 yr@xmath6 ) & + @xmath151 & @xmath152 & @xmath153 & @xmath154 & @xmath155 & @xmath156 + @xmath157 & @xmath158 & @xmath159 & @xmath160 & @xmath161 & @xmath162 + cccccc + @xmath163 & @xmath81 & @xmath73 & @xmath164 & @xmath165 & @xmath82 + ( kev ) & ( @xmath83 ) & ( @xmath84 @xmath18 ) & & ( @xmath84 @xmath18 ) & + @xmath166 & @xmath167 & @xmath168 & @xmath169 & @xmath170 & 12.9/21 + @xmath171 & @xmath172 & @xmath160 & @xmath173 & @xmath174 & 12.9/21 + note . the abundance and @xmath73 of the mekal component were derived from the fit of the region surrounding the point source . upper and lower rows show the results when the @xmath73 is allowed to vary or fixed to the galactic value , respectively .

we present the results of a _ chandra _ observation of the central region of abell 3112 . this cluster has a powerful radio source in the center and was believed to have a strong cooling flow . the x - ray image shows that the intracluster medium ( icm ) is distributed smoothly on large scales , but has significant deviations from a simple concentric elliptical isophotal model near the center . regions of excess emission appear to surround two lobe - like radio - emitting regions . this structure probably indicates that hot x - ray gas and radio lobes are interacting . from an analysis of the x - ray spectra in annuli , we found clear evidence for a temperature decrease and abundance increase toward the center . the x - ray spectrum of the central region is consistent with a single - temperature thermal plasma model . the contribution of x - ray emission from a multiphase cooling flow component with gas cooling to very low temperatures locally is limited to less than 10% of the total emission . however , the whole cluster spectrum indicates that the icm is cooling significantly as a whole , but in only a limited temperature range ( @xmath0 kev ) . inside the cooling radius , the conduction timescales based on the spitzer conductivity are shorter than the cooling timescales . we detect an x - ray point source in the cluster center which is coincident with the optical nucleus of the central cd galaxy and the core of the associated radio source . the x - ray spectrum of the central point source can be fit by a 1.3 kev thermal plasma and a power - law component whose photon index is 1.9 . the thermal component is probably plasma associated with the cd galaxy . we attribute the power - law component to the central agn .

You are an expert at summarizing long articles. Proceed to summarize the following text: the periodically forced sir model @xmath2 @xmath3 @xmath4 and variants of it , are extensively used to model seasonally recurrent diseases @xcite . here @xmath5 are the fractions of the population which are susceptible , infective , and recovered , @xmath6 denotes the birth and death rate , @xmath7 the recovery rate , and @xmath8 , which we assume is a positive continuous @xmath0-periodic function , is the seasonally - dependent transmission rate ( so that @xmath0 is the yearly period ) . when simulating this model numerically ( see section [ numerical ] ) , it is observed that : \(i ) if @xmath9 , where @xmath10 @xmath11 then all solutions tend to the disease free equilibrium @xmath12 this fact can be rigorously proved , see @xcite . \(ii ) if @xmath13 , then depending on the values of the parameters , one observes convergence to @xmath0-periodic orbits , or to @xmath14-periodic orbits with @xmath15 ( subharmonics ) , or chaotic behavior . a fundamental question , that is addressed here , is the _ existence _ of a @xmath0-periodic solution of the system . we demand , of course that the components @xmath16 of this solution will be positive . obviously when @xmath9 , a positive periodic solution can not exist , because such a solution would not converge to the disease - free equilibrium . we will prove , however , that [ main ] whenever @xmath1 , there exists at least one @xmath0-periodic solution @xmath17 of ( [ s])-([r ] ) , all of whose components are positive . thus , when @xmath0-periodic behavior is not observed in simulations , this is not due to the fact that such a solution does not exist , but rather to the fact that all @xmath0-period solutions are unstable . despite the fact that the existence of a @xmath0-periodic solution of the @xmath0-periodically forced sir model is a fundamental issue , the only paper in the literature of which we are aware to have dealt with this question is the recent paper of jdar , villanueva and arenas @xcite . they treated a more general system then we do here ( including loss of immunity and allowing other coefficients besides @xmath8 to be @xmath0-periodic ) . restricting their existence result to the case of the sir model ( [ s])-([r ] ) , they proved , using mawhin s continuation theorem , that a @xmath0-periodic solution exists whenever the condition @xmath18 holds . note that the condition ( [ jodar ] ) implies that @xmath19 , that is @xmath1 , but it is a stronger condition . theorem [ main ] uses only the condition @xmath1 , so that together with the fact noted above , that when @xmath9 a @xmath0-periodic solution does _ not _ exist , we have that @xmath1 is a _ necessary and sufficient _ condition for the existence of a @xmath0-periodic solution with positive components . our technique for proving theorem [ main ] relies on nonlinear functional analysis , for which we refer to the textbooks @xcite . reformulating the problem as one of solving an equation in an infinite dimensional space of periodic functions , we define a homotopy between the periodically forced problem and the autonomous problem in which @xmath8 is replaced by the mean @xmath20 . the autonomous problem has an endemic equilibrium , which is a trivial periodic solution . we then employ leray - schauder degree theory to continue this solution along the homotopy . the challenge here lies in the fact that there we always have a trivial periodic solution , given by the disease - free equilibrium , which lies on the boundary of the relevant domain @xmath21 in the functional space , which requires us to construct a smaller domain @xmath22 excluding the trivial solution , and to show that the conditions for applying the leray - schauder theory hold for the domain @xmath23 . we note that our proof of theorem [ main ] is easily extended to give the same result for the sirs model , which includes loss of immunity @xmath24 @xmath25 @xmath26 we present the proof for the sir model ( @xmath27 ) in order to avoid notational clutter . in section [ proof ] we prove theorem [ main ] . in section [ numerical ] we discuss how to obtain the @xmath0-periodic solution numerically , which can not be done by direct numerical simulation when it is unstable , and present some results obtained by using the galerkin method . finally , in section [ discussion ] we mention some other works providing rigorous mathematical results on forced sir models , beyond numerical simulation . since @xmath16 are fractions of the population we have @xmath28 for all @xmath29 - note that by adding the equations ( [ s])-([r ] ) we have @xmath30 . since @xmath31 does not appear in ( [ s]),([i ] ) , the equation ( [ r ] ) can be ignored , and it suffices to proved the existence of a periodic solution of ( [ s]),([i ] ) satisfying @xmath32 where the third condition is equivalent to @xmath33 . we decompose @xmath8 as @xmath34 setting , for @xmath35 $ ] , @xmath36 we consider the system @xmath37 @xmath38 which is homotopy between an unforced system with @xmath39 and our system ( [ s]),([i ] ) , which corresponds to @xmath40 . for @xmath41 , ( [ s1]),([i1 ] ) has exactly two periodic solutions , which are constant , given by @xmath42 @xmath43 we note that @xmath44 ( the disease - free equilibrium ) is in fact a ( trivial ) periodic solution of ( [ s1])-([i1 ] ) for _ all _ @xmath45 . our aim is to continue the solution @xmath46 with respect to @xmath45 in order to prove the existence of a periodic solution for @xmath40 . to this end , we now reformulate the problem in a functional - analytic setting , which will enable us to employ degree theory . we rewrite ( [ s1]),([i1 ] ) as @xmath47 @xmath48 let @xmath49 be the banach spaces @xmath50 @xmath51 define the linear operator @xmath52 by @xmath53 and the nonlinear operator @xmath54 @xmath55 then the periodic problem for ( [ s2])-([i2 ] ) can be rewritten as @xmath56 it is easy to check that @xmath57 is invertible , that is the equations @xmath58 and @xmath59 have unique @xmath60 @xmath0-periodic solutions @xmath61 for any @xmath62 , and the mapping @xmath63 given by @xmath64 is bounded . we can thus rewrite ( [ re ] ) as @xmath65 where @xmath66 is given by @xmath67 since @xmath63 is bounded , and since , by the arzela - ascoli theorem , @xmath68 is compactly embedded in @xmath69 , we can consider @xmath70 as a compact operator from @xmath69 to itself , and since @xmath54 is continuous , @xmath71 is compact as an operator from @xmath69 to itself . we therefore consider ( [ re1 ] ) in the space @xmath69 , and we note that any solution in @xmath69 will in fact be in @xmath68 , hence a classical solution of ( [ s2]),([i2 ] ) . since @xmath72 is a compact perturbation of the identity on @xmath69 , leray - schauder theory is applicable . since we want our solution to satisfy ( [ pos ] ) , we want to solve ( [ re1 ] ) in the subset @xmath73 given by @xmath74 note that for @xmath41 the solution @xmath46 given by ( [ i0 ] ) lies in @xmath21 . our aim is to continue this solution in @xmath45 up to @xmath40 . we recall that the leray - schauder degree theory ( see e.g. @xcite ) implies that , given a bounded open set @xmath75 , the existence of a solution @xmath76 of ( [ re1 ] ) for all @xmath35 $ ] will be assured if the following conditions hold : \(i ) @xmath77 , \(ii ) @xmath78 , \(iii ) @xmath79 for all @xmath80 , @xmath35 $ ] . the most obvious choice for @xmath23 would be @xmath81 . however , this will not do , since @xmath44 ( given by ( [ i00 ] ) ) satisfies @xmath82 and @xmath83 , so that ( iii ) does not hold . to satisfy ( iii ) we will need to choose @xmath23 so as to exclude @xmath44 from its boundary . we take @xmath23 to be the open subset of @xmath21 given by @xmath84 where @xmath85 is fixed . note that @xmath86 . we will show below that @xmath23 satisfies ( i)-(iii ) if @xmath87 is chosen so that @xmath88 . we first show that @xmath44 is the _ only _ solution of ( [ re1 ] ) on @xmath89 . [ only ] if @xmath90 is a solution of ( [ re1 ] ) for some @xmath35 $ ] , then @xmath91 , as given by ( [ i00 ] ) . assume that @xmath92 is a solution of ( [ s2]),([i2 ] ) . note that @xmath92 , if an only if @xmath93 and at least one of the following conditions holds : \(i ) there exists @xmath94 so that @xmath95 . \(ii ) there exists @xmath94 so that @xmath96 . \(iii ) there exists @xmath94 so that @xmath97 . we now consider each of these three cases : \(1 ) assume ( i ) holds . let @xmath98 be the solution of @xmath99 and let @xmath100 . then @xmath101 is a solution of the initial - value problem ( [ s2]),([i2 ] ) with initial condition @xmath102 by uniqueness of the solution for the initial - value problem , we conclude that @xmath103 , @xmath104 . thus @xmath105 satisfies @xmath106 , and since the only periodic solution of this equation is @xmath107 , we conclude that @xmath108 , as we wanted to prove . \(2 ) assume now that ( ii ) holds . then from ( [ s2 ] ) we get @xmath109 . but this implies that @xmath110 for @xmath111 sufficiently close to @xmath112 , which contradicts ( [ cl ] ) . thus this case is impossible . \(3 ) assume now that ( iii ) holds . moreover since we have already proven the result in the case that ( i ) holds , we may assume that @xmath113 for all @xmath29 . adding ( [ s1 ] ) and ( [ i1 ] ) we get @xmath114 therefore we conclude that @xmath115 for @xmath111 sufficiently close to @xmath112 , contradicting ( [ cl ] ) . therefore this case is impossible . we can now show that @xmath23 , defined by ( [ u ] ) , satisfies ( iii ) . assume @xmath1 . if @xmath116 then , for any @xmath35 $ ] there are no solutions @xmath76 of ( [ re1 ] ) with @xmath117 . suppose @xmath117 . then either @xmath92 or @xmath118 and @xmath119 in the first case , lemma [ only ] and the fact that @xmath120 imply that @xmath76 is not a solution of ( [ re1 ] ) . we therefore assume that @xmath118 and ( [ bc ] ) holds , which implies that @xmath121 assume by way of contradiction that @xmath76 solves ( [ re1 ] ) , or equivalently @xmath76 solves ( [ s2]),([i2 ] ) . using the assumption @xmath118 , we have that @xmath122 is everywhere positive , so we can divide ( [ i2 ] ) by @xmath122 , and integrate over @xmath123 $ ] , to obtain @xmath124 but from ( [ bc1 ] ) we get @xmath125 by the assumption @xmath126 we have @xmath127 , so that ( [ vv ] ) implies @xmath128 contradicting ( [ ee ] ) . to apply the leray - schauder degree it remains to verify that ( i ) and ( ii ) hold . since @xmath129 , the condition @xmath130 implies @xmath77 , so ( i ) holds . to prove ( ii ) , it suffices to show that the frchet derivative @xmath131 is invertible . since @xmath132 is a compact perturbation of the identity so that @xmath131 is fredholm , it suffices to prove that the kernel of @xmath131 is trivial . indeed , let us assume that @xmath133 , and prove that @xmath134 . we have @xmath135 , or , equivalently , @xmath136 note that @xmath137 so that ( [ ker ] ) is equivalent to @xmath138 the characteristic polynomial of the above matrix is @xmath139 noting that @xmath140 and that , for @xmath141 , @xmath142 , we see that the matrix has no imaginary or @xmath143 eigenvalues , so that ( [ sys ] ) has no periodic solutions except @xmath144 , and the claim is proved . we have thus proven that ( i)-(iii ) hold , which completes the proof of theorem [ main ] . as we have noted in the introduction , the period solution whose existence was proved whenever @xmath1 is observable in numerical simulation of ( [ s])-([r ] ) only for those parameter regimes for which it is stable . therefore , if we are interested in examining the shape and amplitude of the @xmath0-periodic solution for values of the parameters for which the system displays subharmonic or chaotic behavior , we need a different computational approach . we now describe a simple approach that we successfully implemented , which allows us to observe the @xmath0-periodic solution for arbitrary parameters . we used the galerkin method , expanding the periodic solution @xmath145 in a fourier series . we used the maple system , whose symbolic capabilities make the implementation particularly easy . we take @xmath146 , @xmath147 and search for approximate periodic solutions of ( [ s]),([i ] ) the form @xmath148,\nonumber\\ \tilde{i}(t)&=&a_i^{0}+\sum_{n=1}^n [ a_i^{n}\cos(nt)+b_i^{n}\sin(nt)].\end{aligned}\ ] ] plugging ( [ fo ] ) into ( [ s])-([i ] ) , and then taking the fourier coefficients of both sides of the equations with respect to @xmath149 we get @xmath150 algebraic equations in @xmath150 variables , which we numerically solve for @xmath151 , obtaining the approximate @xmath0-periodic solution ( [ fo ] ) . we have found that the numerical iteration for solving the algebraic equations , using maple s fsolve command , works well , when we start with the initial conditions for the iteration given by the endemic equilibrium of the autonomous case , that is @xmath152 ( see ( [ i0 ] ) ) and @xmath153 for @xmath154 . we check that the functions @xmath155 indeed approximate a periodic solution of ( [ s]),([i ] ) by observing that the highest fourier coefficients @xmath156 are very small , and by plugging @xmath155 into ( [ s]),([i ] ) and checking that the residual is small . we note that theoretical justification of the galerkin method for approximating periodic solutions can be found , e.g. , in @xcite . we now present some examples of results obtained by the method described above . with the period @xmath157 of the forcing representing one year , we took take @xmath7 corresponding to a @xmath158-week infectious period , @xmath159 , @xmath6 corresponding to @xmath160 population growth rate per year , giving @xmath161 . these parameters are approximately those estimated for measles . we consider different values of the strength of seasonality @xmath45 ( see ( [ for ] ) ) . in figure 1 we plot , for different value of @xmath45 , the periodic solution found by the galerkin method ( with @xmath162 ) , together with a solution of ( [ s])-([i ] ) obtained by direct simulation , starting the plot at @xmath163 to ensure that transients have decayed . [ fig1 ] -periodic forcing ( @xmath146 ) , obtained by direct simulation , and the @xmath0-periodic solution obtained by the galerkin method ( dashed line ) , for varying strength of seasonality @xmath45 . top row , from left to right : @xmath164 , bottom row : @xmath165 . other parameters : @xmath166 , @xmath159 , @xmath167.,title="fig:",width=170,height=170]-periodic forcing ( @xmath146 ) , obtained by direct simulation , and the @xmath0-periodic solution obtained by the galerkin method ( dashed line ) , for varying strength of seasonality @xmath45 . top row , from left to right : @xmath164 , bottom row : @xmath165 . other parameters : @xmath166 , @xmath159 , @xmath167.,title="fig:",width=170,height=170]-periodic forcing ( @xmath146 ) , obtained by direct simulation , and the @xmath0-periodic solution obtained by the galerkin method ( dashed line ) , for varying strength of seasonality @xmath45 . top row , from left to right : @xmath164 , bottom row : @xmath165 . other parameters : @xmath166 , @xmath159 , @xmath167.,title="fig:",width=170,height=170 ] + -periodic forcing ( @xmath146 ) , obtained by direct simulation , and the @xmath0-periodic solution obtained by the galerkin method ( dashed line ) , for varying strength of seasonality @xmath45 . top row , from left to right : @xmath164 , bottom row : @xmath165 . other parameters : @xmath166 , @xmath159 , @xmath167.,title="fig:",width=170,height=170]-periodic forcing ( @xmath146 ) , obtained by direct simulation , and the @xmath0-periodic solution obtained by the galerkin method ( dashed line ) , for varying strength of seasonality @xmath45 . top row , from left to right : @xmath164 , bottom row : @xmath165 . other parameters : @xmath166 , @xmath159 , @xmath167.,title="fig:",width=170,height=170]-periodic forcing ( @xmath146 ) , obtained by direct simulation , and the @xmath0-periodic solution obtained by the galerkin method ( dashed line ) , for varying strength of seasonality @xmath45 . top row , from left to right : @xmath164 , bottom row : @xmath165 . other parameters : @xmath166 , @xmath159 , @xmath167.,title="fig:",width=170,height=170 ] [ fig2 ] -periodic solution obtained by the galerkin method , for varying strength of seasonality @xmath168.,title="fig:",width=377,height=188 ] when @xmath169 , the system behavior is @xmath157-periodic , so the solution of the simulated system coincides with the @xmath157-periodic solution found by the galerkin method . at @xmath170 , the @xmath157-periodic solution has lost stability , and we see bifurcation to a subharmonic of order 2 ( period @xmath171 ) , which is still quite close to the @xmath157-periodic solution , with larger and smaller epidemics alternating . at @xmath172 the @xmath171-periodic subharmonic solution is already quite different , with a large epidemic every two years . at @xmath173 we observe that the system has a subharmonic of order 4 ( period @xmath174 ) , while at @xmath175 we observe chaotic behavior . the @xmath157-periodic solution ( which is unstable except for the case @xmath169 ) increases in amplitude and becomes less sinusoidal as @xmath45 increases ( note the differences in scales in the different plots ) . in figure 2 we plot the @xmath157-periodic solutions for all values of @xmath45 , for a better view . the forced sir model is a beautiful example of a simple nonlinear dynamical system which displays complicated behaviors which are difficult to understand in intuitive terms . moreover , these complicated behaviors are relevant to explaining the epidemiology of infectious diseases in humans , as studies comparing the behavior of the sir and variants of it to surveillance data have shown @xcite . we have proven the fundamental result that a @xmath0-periodic solution exists for the @xmath0-periodically forced sir model whenever @xmath1 . as we have stressed , this does not mean that the dynamics of the model is periodic , since the periodic solution whose existence is proved need not be stable , although one can use standard perturbation theory to prove that the @xmath0-periodic solution _ is _ stable provided the seasonality parameter @xmath45 in ( [ deco ] ) is sufficiently small . numerical simulations show that complex dynamics - subharmonic and chaotic behavior - is very common in the forced sir model . it is interesting to ask to what extent the complex dynamics of the forced sir model can be rigorously understood , beyond numerical simulations . while we do not expect to be able to precisely characterize the dynamics of the model for different parameter values , it is of great interest even to be able to rigourously prove that complicated dynamics occurs for at least _ some _ parameter values . in this context we mention the work of h.l . smith @xcite , who proved that the forced sir model can have multiple stable subharmonic oscillations in certain parameter ranges . chaotic behavior has been rigorously established by glendinning & perry @xcite for a variant of the forced sir model , in which the dependence of the incidence term on @xmath122 is nonlinear . for the standard sir model ( [ s])-([r ] ) , we are not aware of a proof of chaotic behavior . classifying and explaining the dynamical patterns observed in simulations of the forced sir model is still very challenging , so that , like other well - known ` simple ' models such as the forced pendulum equation , the forced sir model can serve as a stimulus and as a benchmark problem for new developments in nonlinear analysis .

we prove that the seasonally - forced sir model with a @xmath0-periodic forcing has a periodic solution with period @xmath0 whenever the basic reproductive number @xmath1 . the proof uses the leray - schauder degree theory . we also describe some numerical results in which we compute the @xmath0-periodic solution , where in order to obtain the @xmath0-periodic solution when the behavior of the system is subharmonic or chaotic , we use a galerkin scheme . * msc * : 34v25 , 37j45 , 92d30 .

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Proceed to summarize the following text: title of program : siliconiap computer hardware and operating system : any shared memory computer running under unix or linux programming language : fortran90 with openmp compiler directives memory requirements : roughly 150 words per atom no . of bits in a word : 64 no . of processors used : tested on up to 4 processors has the code been vectorized or parallelized : parallelized withe openmp no . of bytes in distributed program , including test data , etc : 50 000 distribution format : compressed tar file keywords : silicon , interatomic potential , force field , molecular dynamics nature of physical problem : condensed matter physics method of solution : interatomic potential restrictions on the complexity of the problem : none typical running time : 30 @xmath0sec per step and per atom on a compaq dec alpha unusual features of the program : none due to its technological importance , silicon is one of the most studied materials . for small system sizes ab - initio density functional calculations @xcite are the preferred approach . unfortunately this kind of calculation becomes unfeasible for larger systems required to study problems such as interfaces or extended defects . for this type of calculations one resorts to force fields which are several orders of magnitude faster . recent progress in the development of force fields has demonstrated that they can be a reliable tool for such studies . a highly accurate silicon force field has been developed by lenosky and coworkers @xcite . its transferability has been demonstrated by extensive tests containing both bulk and cluster systems @xcite . its accuracy is in part due to the fact that second nearest neighbor interactions are included . this makes it unfortunately somewhat slower than force fields containing only nearest neighbor interactions . in the following a highly optimized parallel implementation of this force field will be presented that allows large scale calculations with this force field . the parallelization is achieved by using openmp , an emerging industry standard for medium size shared memory parallel computers . molecular dynamics calculations @xcite have also been parallelized on distributed memory supercomputers @xcite . this approach is considerably more complex than the one presented here . since few researches have access to massively parallel supercomputers and are willing to overcome the complexities of doing molecular dynamics on such machines , medium scale parallelization @xcite of molecular dynamics has an important place in practice . user friendliness was one of the major design goals in the development of this routine . using fortran90 made it possible to hide all the complexities in an object oriented fashion from the user . the calling sequence is just .... call lenosky(nat , alat , rxyz , fxyz , ener , coord , ener_var , coord_var , count ) .... on input the user has to specify the number of atoms , @xmath1 , the vector @xmath2 containing the 3 lattice constant of the orthorhombic periodic volume and the atomic positions @xmath3 . the program then returns the total energy , @xmath4 , the forces , @xmath5 , the average coordination number , the variation of the energy per atom and of the coordination number as well as an counter that is increased in each call . in particular the user has not to supply any verlet list . since the calculation of the forces is typically much more expensive than the update of the atomic positions in molecular dynamics or geometry optimizations , we expect that the subroutine will be called in most cases from within a serial program . in case the user is on a shared memory machine the subroutine will then nevertheless be executed in parallel if the program is compiled with the appropriate openmp options . in addition the subroutine can of course also be used on a serial machine . in this case all the parallelization directives are considered by the compiler to be comments . the verlet list gives all the atoms that are contained within the potential cutoff distance @xmath6 of any given atom . typically the verlet list consists of two integer arrays . the first array , called @xmath7 in this work , points to the first / last neighbor position in the second array @xmath8 that contains the numbering of the atoms that are neighbors . a straightforward implementation for a non - periodic system containing @xmath1 atoms is shown below . in this simple case the search through all atoms is sequential with respect to their numbering and it is redundant to give both the starting positions @xmath9 and the ending position @xmath10 , since @xmath11 . but in the more complicated linear scaling algorithm to be presented below , both will be needed . .... indc=0 do 10 iat=1,nat c starting position lsta(1,iat)=indc+1 do 20 jat=1,nat if ( jat.ne.iat ) then xrel1= rxyz(1,jat)-rxyz(1,iat ) xrel2= rxyz(2,jat)-rxyz(2,iat ) xrel3= rxyz(3,jat)-rxyz(3,iat ) rr2=xrel1**2 + xrel2**2 + xrel3**2 if ( rr2 .le . cut**2 ) then indc = indc+1 c nearest neighbor numbers lstb(indc)=jat endif endif 20 continue c ending position lsta(2,iat)=indc 10 continue .... this straightforward implementations has a quadratic scaling with respect to the numbers of atoms . due to this scaling the calculation of the verlet list starts to dominate the linear scaling calculation of the energies and forces for system sizes of more than 10 000 atoms . it is therefore good practice to calculate the verlet list with a modified algorithm that has linear scaling @xcite as well . to do this one first subdivides the system into boxes that have a side length that is equal to or larger than @xmath6 and then finds all the atoms that are contained in each box . the cpu time for this first step is less than 1 percent of the entire verlet list calculation . hence this part was not parallelized . it could significantly affect the parallel performance according to amdahls law @xcite only if more than 50 processors are used . the largest smp machines at our disposal had however only 4 processors . to implement periodic boundary conditions all atoms within a distance @xmath6 of the boundary of the periodic volume are replicated on the opposite part as shown in figure [ period ] . this part is equally well less than 1 percent of the cpu time for the verlet list for a 8000 atom system . being a surface term it becomes even smaller for larger systems . consequently it was nt parallelized either . ( -1.5,0.5 ) after these two preparing steps one has to search only among all the atoms in the reference cell containing the atom for which one wants to find its neighbors as well as all the atoms in the cells neighboring this reference cell ( 26 cells in 3 dimensions ) . this implies that starting and ending positions @xmath7 for the atoms @xmath12 to @xmath1 are not calculated in a sequential way , necessitating , as mentioned before , separate starting and ending positions in the array @xmath7 . the corresponding parallel code is shown below . the indices @xmath13 refer to the cells , @xmath14 contains the number of atoms in cell @xmath13 and @xmath15 their numbering . the array @xmath16 saves the relative positions and distances that will again be needed in the loop calculating the forces and energies . each thread has its own starting position @xmath17 in the shared memory space @xmath8 and these starting positions are uniformly distributed . this approach allows the different threads to work independently . the resulting speedup is much higher than the one that one would obtain by calculating in the parallel version an array @xmath8 that is identical to the one from the serial version . if there are on the average more neighbors than expected ( 24 by default ) the allocated space becomes too small . in this case the array @xmath8 is deallocated and a new larger version is allocated . this check for sufficient memory requires some minimal amount of coordination among the processors and is implemented by a critical section . if a reallocation is necessary , a message is written into an file to alert the user of the inefficiency due to the need of a second calculation of the verlet list . .... allocate(lsta(2,nat ) ) nnbrx=24 2345 nnbrx=3*nnbrx/2 allocate(lstb(nnbrx*nat),rel(5,nnbrx*nat ) ) indlstx=0 ! $ omp parallel & ! $ omp private(iat , cut2,iam , ii , indlst , l1,l2,l3,myspace , npr ) & ! $ omp shared ( indlstx , nat , nn , nnbrx , ncx , ll1,ll2,ll3,icell , lsta , lstb , lay , & ! $ omp rel , rxyz , cut , myspaceout ) npr=1 ! $ npr = omp_get_num_threads ( ) iam=0 ! $ iam = omp_get_thread_num ( ) cut2=cut**2 myspace=(nat*nnbrx)/npr if ( iam.eq.0 ) myspaceout = myspace ! verlet list , relative positions indlst=0 do 6000,l3=0,ll3 - 1 do 6000,l2=0,ll2 - 1 do 6000,l1=0,ll1 - 1 do 6600,ii=1,icell(0,l1,l2,l3 ) iat = icell(ii , l1,l2,l3 ) if ( ( ( iat-1)*npr)/nat .eq . iam ) then lsta(1,iat)=iam*myspace+indlst+1 call sublstiat(iat , nn , ncx , ll1,ll2,ll3,l1,l2,l3,myspace , & rxyz , icell , lstb(iam*myspace+1),lay , rel(1,iam*myspace+1),cut2,indlst ) lsta(2,iat)=iam*myspace+indlst endif 6600 continue 6000 continue ! $ omp critical indlstx = max(indlstx , indlst ) ! $ omp end critical ! $ omp end parallel if ( indlstx.gt.myspaceout ) then write(10 , * ) count,'nnbrx too small ' , nnbrx deallocate(lstb , rel ) goto 2345 endif subroutine sublstiat(iat , nn , ncx , ll1,ll2,ll3,l1,l2,l3,myspace , & rxyz , icell , lstb , lay , rel , cut2,indlst ) implicit real*8 ( a - h , o - z ) dimension rxyz(3,nn),lay(nn),icell(0:ncx,-1:ll1,-1:ll2,-1:ll3 ) , & lstb(0:myspace-1),rel(5,0:myspace-1 ) do 6363,k3=l3 - 1,l3 + 1 do 6363,k2=l2 - 1,l2 + 1 do 6363,k1=l1 - 1,l1 + 1 do 6363,jj=1,icell(0,k1,k2,k3 ) jat = icell(jj , k1,k2,k3 ) if ( jat.eq.iat ) goto 6363 xrel= rxyz(1,iat)-rxyz(1,jat ) yrel= rxyz(2,iat)-rxyz(2,jat ) zrel= rxyz(3,iat)-rxyz(3,jat ) rr2=xrel**2 + yrel**2 + zrel**2 if ( rr2 .le . cut2 ) then indlst = min(indlst , myspace-1 ) lstb(indlst)=lay(jat ) tt = sqrt(rr2 ) tti=1.d0/tt rel(1,indlst)=xrel*tti rel(2,indlst)=yrel*tti rel(3,indlst)=zrel*tti rel(4,indlst)=tt rel(5,indlst)=tti indlst= indlst+1 endif 6363 continue return end .... the computationally most important part taking some 80 percent of the cpu time is the calculation of the energies and forces . the energy expression for the lenosky force field is given by @xmath18 all the functions ( @xmath19 , @xmath20 , @xmath21 , @xmath22 , @xmath23 ) in this energy expression are given by cubic splines . the subroutine for evaluating the cubic spline is listed below . the case that the argument is outside the cubic spline interval @xmath24 $ ] is rare and unimportant for performance considerations . the important cubic spline case is characterized by many dependencies . in the case of such dependencies the latency of the functional unit pipeline comes into play and reduces the attainable speed @xcite . a latency of some 20 cycles comes from the first two statements ( tt=(x - tmin)*hi ; klo = tt ) alone , requiring arithmetic operations and a floating point to integer conversion . for this reason the calculation of tt was taken out of the ( most likely occurring ) else block to overlap its evaluation with the evaluation of the if clauses . to further speed up the evaluation of the splines the structure of the energy expression [ energy ] was exploited . in the computationally most important loop over @xmath25 and @xmath26 two splines ( @xmath27 and @xmath28 ) have to be evaluated . inlining by hand the subroutine splint for both evaluations and calculating alternatingly one step of the first spline evaluation and one step of the second spline evaluation introduces two independent streams . this reduces the effect of latencies and boosts speed . compilers are not able to do these complex type of optimizations . the best performance after these optimizations was obtained with low level compiler optimization flags ( -o3 -qarch = pwr3 -qtune - pwr3 on ibm power3 , -o2 on the compaq dec alpha , -o2 -xw on intel pentium4 ) .... subroutine splint(ya , y2a , tmin , tmax , hsixth , h2sixth , hi , n , x , y , yp ) implicit real*8 ( a - h , o - z ) dimension y2a(0:n-1),ya(0:n-1 ) ! interpolate if the argument is outside the cubic spline interval [ tmin , tmax ] tt=(x - tmin)*hi if ( x.lt.tmin ) then yp = hi*(ya(1)-ya(0 ) ) - & ( y2a(1)+2.d0*y2a(0 ) ) * hsixth y = ya(0 ) + ( x - tmin)*yp else if ( x.gt.tmax ) then yp = hi*(ya(n-1)-ya(n-2 ) ) + & ( 2.d0*y2a(n-1)+y2a(n-2 ) ) * hsixth y = ya(n-1 ) + ( x - tmax)*yp ! otherwise evaluate cubic spline else klo = tt khi = klo+1 ya_klo = ya(klo ) y2a_klo = y2a(klo ) b = tt - klo a=1.d0-b ya_khi = ya(khi ) y2a_khi = y2a(khi ) b2=b*b y = a*ya_klo yp = ya_khi - ya_klo a2=a*a cof1=a2 - 1.d0 cof2=b2 - 1.d0 y = y+b*ya_khi yp = hi*yp cof3=3.d0*b2 cof4=3.d0*a2 cof1=a*cof1 cof2=b*cof2 cof3=cof3 - 1.d0 cof4=cof4 - 1.d0 yt1=cof1*y2a_klo yt2=cof2*y2a_khi ypt1=cof3*y2a_khi ypt2=cof4*y2a_klo y = y + ( yt1+yt2)*h2sixth yp = yp + ( ypt1 - ypt2 ) * hsixth endif return end .... the final single processor performance for the entire subroutine is 460 mflops on a compaq dec alpha at 833 mhz , 300 mflops on a ibm power3 at 350 mhz and 550 mflops on a pentium 4 . in order to obtain a high parallel speedup in this central part of the subroutine the threads are completely decoupled . this was done by introducing private copies for each thread to accumulate the energies @xmath29 and forces @xmath30 . the global energy and force are summed up in an additional loop at the end of the parallel region in a critical section . .... ! $ omp parallel & ! $ omp private(iam , npr , iat , iat1,iat2,lot , istop , tcoord , tcoord2 , & ! $ omp tener , tener2,txyz , f2ij , f3ij , f3ik , npjx , npjkx ) & ! $ omp shared ( nat , nnbrx , lsta , lstb , rel , ener , ener2,fxyz , coord , coord2,istopg ) npr=1 ! $ npr = omp_get_num_threads ( ) iam=0 ! $ iam = omp_get_thread_num ( ) npjx=300 ; npjkx=3000 istopg=0 if ( npr.ne.1 ) then ! parallel case ! create temporary private scalars for reduction sum on energies and ! temporary private array for reduction sum on forces ! $ omp critical allocate(txyz(3,nat),f2ij(3,npjx),f3ij(3,npjkx),f3ik(3,npjkx ) ) ! $ omp end critical if ( iam.eq.0 ) then ener=0.d0 ener2=0.d0 coord=0.d0 coord2=0.d0 do 121,iat=1,nat fxyz(1,iat)=0.d0 fxyz(2,iat)=0.d0 121 fxyz(3,iat)=0.d0 endif lot = nat / npr+.999999999999d0 iat1=iam*lot+1 iat2=min((iam+1)*lot , nat ) call subfeniat(iat1,iat2,nat , lsta , lstb , rel , tener , tener2 , & tcoord , tcoord2,nnbrx , txyz , f2ij , npjx , f3ij , npjkx , f3ik , istop ) ! $ omp critical ener = ener+tener ener2=ener2+tener2 coord = coord+tcoord coord2=coord2+tcoord2 istopg = istopg+istop do 8093,iat=1,nat fxyz(1,iat)=fxyz(1,iat)+txyz(1,iat ) fxyz(2,iat)=fxyz(2,iat)+txyz(2,iat ) fxyz(3,iat)=fxyz(3,iat)+txyz(3,iat ) 8093 continue ! $ omp end critical deallocate(txyz , f2ij , f3ij , f3ik ) else ! serial case iat1=1 iat2=nat allocate(f2ij(3,npjx),f3ij(3,npjkx),f3ik(3,npjkx ) ) call subfeniat(iat1,iat2,nat , lsta , lstb , rel , ener , ener2 , & coord , coord2,nnbrx , fxyz , f2ij , npjx , f3ij , npjkx , f3ik , istop ) deallocate(f2ij , f3ij , f3ik ) endif ! $ omp end parallel if ( istopg.gt.0 ) stop ' dimension error ( see warning above ) ' ener_var = ener2/nat-(ener / nat)**2 coord = coord / nat coord_var = coord2/nat - coord**2 deallocate(rxyz , icell , lay , lsta , lstb , rel ) end subroutine subfeniat(iat1,iat2,nat , lsta , lstb , rel , tener , tener2 , & tcoord , tcoord2,nnbrx , txyz , f2ij , npjx , f3ij , npjkx , f3ik , istop ) implicit real*8 ( a - h , o - z ) dimension lsta(2,nat),lstb(nnbrx*nat),rel(5,nnbrx*nat),txyz(3,nat ) dimension f2ij(3,npjx),f3ij(3,npjkx),f3ik(3,npjkx ) initialize data ........ ! create temporary private scalars for reduction sum on energies and tener=0.d0 tener2=0.d0 tcoord=0.d0 tcoord2=0.d0 istop=0 do 121,iat=1,nat txyz(1,iat)=0.d0 txyz(2,iat)=0.d0 121 txyz(3,iat)=0.d0 ! calculation of forces , energy do 1000,iat = iat1,iat2 dens2=0.d0 dens3=0.d0 jcnt=0 jkcnt=0 coord_iat=0.d0 ener_iat=0.d0 do 2000,jbr = lsta(1,iat),lsta(2,iat ) jat = lstb(jbr ) jcnt = jcnt+1 if ( jcnt.gt.npjx ) then write(6 , * ) ' warning : enlarge npjx ' istop=1 endif fxij = rel(1,jbr ) fyij = rel(2,jbr ) fzij = rel(3,jbr ) rij = rel(4,jbr ) sij = rel(5,jbr ) ! coordination number calculated with soft cutoff between first and ! second nearest neighbor if ( rij.le.2.36d0 ) then coord_iat = coord_iat+1.d0 else if ( rij.ge.3.83d0 ) then else x=(rij-2.36d0)*(1.d0/(3.83d0 - 2.36d0 ) ) coord_iat = coord_iat+(2*x+1.d0)*(x-1.d0)**2 endif ! pairpotential term call splint(cof_phi , dof_phi , tmin_phi , tmax_phi , & hsixth_phi , h2sixth_phi , hi_phi,10,rij , e_phi , ep_phi ) ener_iat = ener_iat+(e_phi*.5d0 ) txyz(1,iat)=txyz(1,iat)-fxij*(ep_phi*.5d0 ) txyz(2,iat)=txyz(2,iat)-fyij*(ep_phi*.5d0 ) txyz(3,iat)=txyz(3,iat)-fzij*(ep_phi*.5d0 ) txyz(1,jat)=txyz(1,jat)+fxij*(ep_phi*.5d0 ) txyz(2,jat)=txyz(2,jat)+fyij*(ep_phi*.5d0 ) txyz(3,jat)=txyz(3,jat)+fzij*(ep_phi*.5d0 ) ! 2 body embedding term call splint(cof_rho , dof_rho , tmin_rho , tmax_rho , & hsixth_rho , h2sixth_rho , hi_rho,11,rij , rho , rhop ) dens2=dens2+rho f2ij(1,jcnt)=fxij*rhop f2ij(2,jcnt)=fyij*rhop f2ij(3,jcnt)=fzij*rhop ! 3 body embedding term call splint(cof_fff , dof_fff , tmin_fff , tmax_fff , & hsixth_fff , h2sixth_fff , hi_fff,10,rij , fij , fijp ) do 3000,kbr = lsta(1,iat),lsta(2,iat ) kat = lstb(kbr ) if ( kat.lt.jat ) then jkcnt = jkcnt+1 if ( jkcnt.gt.npjkx ) then write(6 , * ) ' warning : enlarge npjkx ' istop=1 endif ! begin optimized version rik = rel(4,kbr ) if ( rik.gt.tmax_fff ) then fikp=0.d0 ; fik=0.d0 gjik=0.d0 ; gjikp=0.d0 ; sik=0.d0 costheta=0.d0 ; fxik=0.d0 ; fyik=0.d0 ; fzik=0.d0 else if ( rik.lt.tmin_fff ) then fxik = rel(1,kbr ) fyik = rel(2,kbr ) fzik = rel(3,kbr ) costheta = fxij*fxik+fyij*fyik+fzij*fzik sik = rel(5,kbr ) fikp = hi_fff*(cof_fff(1)-cof_fff(0 ) ) - & ( dof_fff(1)+2.d0*dof_fff(0 ) ) * hsixth_fff fik = cof_fff(0 ) + ( rik - tmin_fff)*fikp tt_ggg=(costheta - tmin_ggg)*hi_ggg if ( costheta.gt.tmax_ggg ) then gjikp = hi_ggg*(cof_ggg(8 - 1)-cof_ggg(8 - 2 ) ) + & ( 2.d0*dof_ggg(8 - 1)+dof_ggg(8 - 2 ) ) * hsixth_ggg gjik = cof_ggg(8 - 1 ) + ( costheta - tmax_ggg)*gjikp else klo_ggg = tt_ggg khi_ggg = klo_ggg+1 cof_ggg_klo = cof_ggg(klo_ggg ) dof_ggg_klo = dof_ggg(klo_ggg ) b_ggg = tt_ggg - klo_ggg a_ggg=1.d0-b_ggg cof_ggg_khi = cof_ggg(khi_ggg ) dof_ggg_khi = dof_ggg(khi_ggg ) b2_ggg = b_ggg*b_ggg gjik = a_ggg*cof_ggg_klo gjikp = cof_ggg_khi - cof_ggg_klo a2_ggg = a_ggg*a_ggg cof1_ggg = a2_ggg-1.d0 cof2_ggg = b2_ggg-1.d0 gjik = gjik+b_ggg*cof_ggg_khi gjikp = hi_ggg*gjikp cof3_ggg=3.d0*b2_ggg cof4_ggg=3.d0*a2_ggg cof1_ggg = a_ggg*cof1_ggg cof2_ggg = b_ggg*cof2_ggg cof3_ggg = cof3_ggg-1.d0 cof4_ggg = cof4_ggg-1.d0 yt1_ggg = cof1_ggg*dof_ggg_klo yt2_ggg = cof2_ggg*dof_ggg_khi ypt1_ggg = cof3_ggg*dof_ggg_khi ypt2_ggg = cof4_ggg*dof_ggg_klo gjik = gjik + ( yt1_ggg+yt2_ggg)*h2sixth_ggg gjikp = gjikp + ( ypt1_ggg - ypt2_ggg ) * hsixth_ggg endif else fxik = rel(1,kbr ) tt_fff = rik - tmin_fff costheta = fxij*fxik fyik = rel(2,kbr ) tt_fff = tt_fff*hi_fff costheta = costheta+fyij*fyik fzik = rel(3,kbr ) klo_fff = tt_fff costheta = costheta+fzij*fzik sik = rel(5,kbr ) tt_ggg=(costheta - tmin_ggg)*hi_ggg if ( costheta.gt.tmax_ggg ) then gjikp = hi_ggg*(cof_ggg(8 - 1)-cof_ggg(8 - 2 ) ) + & ( 2.d0*dof_ggg(8 - 1)+dof_ggg(8 - 2 ) ) * hsixth_ggg gjik = cof_ggg(8 - 1 ) + ( costheta - tmax_ggg)*gjikp khi_fff = klo_fff+1 cof_fff_klo = cof_fff(klo_fff ) dof_fff_klo = dof_fff(klo_fff ) b_fff = tt_fff - klo_fff a_fff=1.d0-b_fff cof_fff_khi = cof_fff(khi_fff ) dof_fff_khi = dof_fff(khi_fff ) b2_fff = b_fff*b_fff fik = a_fff*cof_fff_klo fikp = cof_fff_khi - cof_fff_klo a2_fff = a_fff*a_fff cof1_fff = a2_fff-1.d0 cof2_fff = b2_fff-1.d0 fik = fik+b_fff*cof_fff_khi fikp = hi_fff*fikp cof3_fff=3.d0*b2_fff cof4_fff=3.d0*a2_fff cof1_fff = a_fff*cof1_fff cof2_fff = b_fff*cof2_fff cof3_fff = cof3_fff-1.d0 cof4_fff = cof4_fff-1.d0 yt1_fff = cof1_fff*dof_fff_klo yt2_fff = cof2_fff*dof_fff_khi ypt1_fff = cof3_fff*dof_fff_khi ypt2_fff = cof4_fff*dof_fff_klo fik = fik + ( yt1_fff+yt2_fff)*h2sixth_fff fikp = fikp + ( ypt1_fff - ypt2_fff ) * hsixth_fff else klo_ggg = tt_ggg khi_ggg = klo_ggg+1 khi_fff = klo_fff+1 cof_ggg_klo = cof_ggg(klo_ggg ) cof_fff_klo = cof_fff(klo_fff ) dof_ggg_klo = dof_ggg(klo_ggg ) dof_fff_klo = dof_fff(klo_fff ) b_ggg = tt_ggg - klo_ggg b_fff = tt_fff - klo_fff a_ggg=1.d0-b_ggg a_fff=1.d0-b_fff cof_ggg_khi = cof_ggg(khi_ggg ) cof_fff_khi = cof_fff(khi_fff ) dof_ggg_khi = dof_ggg(khi_ggg ) dof_fff_khi = dof_fff(khi_fff ) b2_ggg = b_ggg*b_ggg b2_fff = b_fff*b_fff gjik = a_ggg*cof_ggg_klo fik = a_fff*cof_fff_klo gjikp = cof_ggg_khi - cof_ggg_klo fikp = cof_fff_khi - cof_fff_klo a2_ggg = a_ggg*a_ggg a2_fff = a_fff*a_fff cof1_ggg = a2_ggg-1.d0 cof1_fff = a2_fff-1.d0 cof2_ggg = b2_ggg-1.d0 cof2_fff = b2_fff-1.d0 gjik = gjik+b_ggg*cof_ggg_khi fik = fik+b_fff*cof_fff_khi gjikp = hi_ggg*gjikp fikp = hi_fff*fikp cof3_ggg=3.d0*b2_ggg cof3_fff=3.d0*b2_fff cof4_ggg=3.d0*a2_ggg cof4_fff=3.d0*a2_fff cof1_ggg = a_ggg*cof1_ggg cof1_fff = a_fff*cof1_fff cof2_ggg = b_ggg*cof2_ggg cof2_fff = b_fff*cof2_fff cof3_ggg = cof3_ggg-1.d0 cof3_fff = cof3_fff-1.d0 cof4_ggg = cof4_ggg-1.d0 cof4_fff = cof4_fff-1.d0 yt1_ggg = cof1_ggg*dof_ggg_klo yt1_fff = cof1_fff*dof_fff_klo yt2_ggg = cof2_ggg*dof_ggg_khi yt2_fff = cof2_fff*dof_fff_khi ypt1_ggg = cof3_ggg*dof_ggg_khi ypt1_fff = cof3_fff*dof_fff_khi ypt2_ggg = cof4_ggg*dof_ggg_klo ypt2_fff = cof4_fff*dof_fff_klo gjik = gjik + ( yt1_ggg+yt2_ggg)*h2sixth_ggg fik = fik + ( yt1_fff+yt2_fff)*h2sixth_fff gjikp = gjikp + ( ypt1_ggg - ypt2_ggg ) * hsixth_ggg fikp = fikp + ( ypt1_fff - ypt2_fff ) * hsixth_fff endif endif ! end optimized version tt = fij*fik dens3=dens3+tt*gjik t1=fijp*fik*gjik t2=sij*(tt*gjikp ) f3ij(1,jkcnt)=fxij*t1 + ( fxik - fxij*costheta)*t2 f3ij(2,jkcnt)=fyij*t1 + ( fyik - fyij*costheta)*t2 f3ij(3,jkcnt)=fzij*t1 + ( fzik - fzij*costheta)*t2 t3=fikp*fij*gjik t4=sik*(tt*gjikp ) f3ik(1,jkcnt)=fxik*t3 + ( fxij - fxik*costheta)*t4 f3ik(2,jkcnt)=fyik*t3 + ( fyij - fyik*costheta)*t4 f3ik(3,jkcnt)=fzik*t3 + ( fzij - fzik*costheta)*t4 endif 3000 continue 2000 continue dens = dens2+dens3 call splint(cof_uuu , dof_uuu , tmin_uuu , tmax_uuu , & hsixth_uuu , h2sixth_uuu , hi_uuu,8,dens , e_uuu , ep_uuu ) ener_iat = ener_iat+e_uuu ! only now ep_uu is known and the forces can be calculated , lets loop again jcnt=0 jkcnt=0 do 2200,jbr = lsta(1,iat),lsta(2,iat ) jat = lstb(jbr ) jcnt = jcnt+1 txyz(1,iat)=txyz(1,iat)-ep_uuu*f2ij(1,jcnt ) txyz(2,iat)=txyz(2,iat)-ep_uuu*f2ij(2,jcnt ) txyz(3,iat)=txyz(3,iat)-ep_uuu*f2ij(3,jcnt ) txyz(1,jat)=txyz(1,jat)+ep_uuu*f2ij(1,jcnt ) txyz(2,jat)=txyz(2,jat)+ep_uuu*f2ij(2,jcnt ) txyz(3,jat)=txyz(3,jat)+ep_uuu*f2ij(3,jcnt ) ! 3 body embedding term do 3300,kbr = lsta(1,iat),lsta(2,iat ) kat = lstb(kbr ) if ( kat.lt.jat ) then jkcnt = jkcnt+1 txyz(1,iat)=txyz(1,iat)-ep_uuu*(f3ij(1,jkcnt)+f3ik(1,jkcnt ) ) txyz(2,iat)=txyz(2,iat)-ep_uuu*(f3ij(2,jkcnt)+f3ik(2,jkcnt ) ) txyz(3,iat)=txyz(3,iat)-ep_uuu*(f3ij(3,jkcnt)+f3ik(3,jkcnt ) ) txyz(1,jat)=txyz(1,jat)+ep_uuu*f3ij(1,jkcnt ) txyz(2,jat)=txyz(2,jat)+ep_uuu*f3ij(2,jkcnt ) txyz(3,jat)=txyz(3,jat)+ep_uuu*f3ij(3,jkcnt ) txyz(1,kat)=txyz(1,kat)+ep_uuu*f3ik(1,jkcnt ) txyz(2,kat)=txyz(2,kat)+ep_uuu*f3ik(2,jkcnt ) txyz(3,kat)=txyz(3,kat)+ep_uuu*f3ik(3,jkcnt ) endif 3300 continue 2200 continue tener = tener+ener_iat tener2=tener2+ener_iat**2 tcoord = tcoord+coord_iat tcoord2=tcoord2+coord_iat**2 1000 continue return end .... in addition to the energy and the forces the program still returns the coordination number as well as the variance of the energy per atom and the coordination number . the coordination number is calculated using a soft cutoff between the first and second nearest neighbor distance . these extra calculations are very cheap and not visible as an increase in the cpu time table [ speedup_dec ] shows the final overall speedups obtained by the program . the results were obtained for an 8000 atom system , but the cpu time per call and atom is nearly independent of system size . . timings in @xmath0sec for a combined evaluation of the forces and the energy per particle as well as the corresponding speedups ( in parentheses ) on an ibm sp3 based on a 375 mhz power3 processor , on a compaq sc 232 based on a 833mhz ev67 processor and on an intel pentium4 biprocessor at 2 ghz [ speedup_dec ] [ cols="<,^,^,^",options="header " , ] obtaining such high speedups was not straightforward . only the compaq fortran90 compiler was able to use in the original version of the program the openmp parallel do directive to obtain a good speedup . both the ibm compiler and the intel compiler failed . in order to get the performances of table [ speedup_dec ] , it was necessary to encapsulate the workload of the different threads into the subroutines sublstias and subfen , which amounts to doing the parallelization quasi by hand . using allocatable arrays in connection with openmp turned also out to be tricky . because of these problems , the parallelization was much more painful that one might expect for a shared memory model . the results show that simulations for very large silicon systems are feasible on relatively cheap serial or parallel computers accessible to a large number of researches .

the force field by lenosky and coworkers is the latest force field for silicon which is one of the most studied materials . it has turned out to be highly accurate in a large range of test cases . the optimization and parallelization of this force field using openmp and fortan90 is described here . the optimized program allows us to handle a very large number of silicon atoms in large scale simulations . since all the parallelization is hidden in a single subroutine that returns the total energies and forces , this subroutine can be called from within a serial program in an user friendly way . -0.75 in

You are an expert at summarizing long articles. Proceed to summarize the following text: observations of accretion powered pulsars began with the discovery of periodic x - ray pulsations from cen x-3 by _ uhuru _ ( giacconi et al . 1971 ; schreier et al . . qualitative understanding of accretion powered pulsars was achieved in the 1970s ( pringle & rees 1972 ; davidson & ostriker 1973 ; lamb et al . 1973 ) . ghosh and lamb presented an accretion disk theory to address the accretion powered pulsar observations in the 1970s in terms of a fastness parameter , material and magnetic torques in the case of a stable prograde accretion disk ( ghosh&lamb , 1979a , b ) . in the absence of a stable accretion disk , numerical simulations were used to probe the nature of accretion ( anzer et al . 1987 ; taam&fryxell 1988a,1988b,1989 ; blondin et al . 1990 ) . observations of pulse frequency changes in accretion powered pulsars are direct signs of torques exerted on the pulsar . these torques can originate either outside or inside the star ( lamb et al . 1978 ; baykal & gelman 1993 ) . internal torques depend on the coupling between interior components , in particular the core superfluid , and the solid outer crust ( baykal et al . external torques depend on the magnetic field strength of the neutron star and on the type of accretion flow to the neutron star . if the neutron star accretes mass from an accretion disk , torques are produced either by the angular momentum transfer of the plasma to the magnetic field in the magnetospheric radius via interaction of the inner boundary of the disk and the magnetic field lines ( causing material torques ) or by the interaction of the disk and the magnetic field ( causing magnetic torques ) ( ghosh & lamb 1979a , b ) . if the accretion results from roche lobe overflow of the companion , a persistent prograde keplerian accretion disk forms and the disk creates material and magnetic torques causing the neutron star to spin - up or spin - down . for such a configuration , material torques can only give spin - up contribution to the net torque , while magnetic torques may give either spin - up or spin - down contribution . if the companion does not fill its roche lobe , then the neutron star may still accrete mass from its companion s wind . from the hydrodynamic simulations , it is seen that the stellar wind is disrupted in the vicinity of a compact x - ray source ( the neutron star for our case ) which causes plasma to lose its homogeneity . the interaction of the incident flow with the shock fronts around the neutron star can produce retrograde and prograde temporary accretion disks ( anzer et al . 1987 , taam & fryxell 1988a , 1988b , 1989 ; blondin et al . 1990 ) . the relations between x - ray luminosity , torque and specific angular momentum may lead to important clues about the accretion process . if the neutron star accretes mass from a stable prograde accretion disk , we expect a positive correlation between x - ray flux and torque ( ghosh&lamb 1979a , b ) . for the case of continuous changes in accretion geometry , we can expect a correlation between specific angular momentum and torque which may be the sign of significant torque changes while the luminosity does not vary significantly ( taam & fryxell 1988a , 1988b , 1989 ; blondin et al . 1990 ) . in this paper , we use batse ( burst and transient source experiment ) 20 - 60 kev band x - ray flux and pulse frequency time series of three high mass systems ( vela x-1 , gx 301 - 2 , and oao 1657 - 415 ) . this database is a part of the flux and pulse frequency database for accretion powered pulsars which was discussed before by bildsten et al . ( 1997 ) . using these time series , we investigate the correlations of torque , x - ray luminosity and specific angular momentum . detailed studies on torque and x - ray luminosity using the batse x - ray flux and pulse frequency data were presented before for gx 1 + 4 ( chakrabarty 1996,1997 ) and oao 1657 - 415 ( baykal 1997 ) . baykal(1997 ) also discussed correlations of specific angular momentum with torque and x - ray luminosity for oao 1657 - 415 . gx 1 + 4 , which was continuously spinning - up in the 1970 s , later exhibited a continuous spin - down trend with an anticorrelation of torque and x - ray luminosity , i.e the spin - down rate is increased with increasing x - ray luminosity ( chakrabarty 1996,1997 ) . this spin - down episode was interpreted as evidence for a retrograde keplerian accretion disk ( nelson et al . 1997 ) which may originate from the slow wind of a red giant ( murray et al . 1998 ) . other explanations for this spin - down episode were the radially advective sub - keplerian disk ( yi et al . 1997 ) and warped disk ( van kerkwijk et al . 1998 ) models . x - ray luminosity , torque and specific angular momentum correlations for oao 1657 - 415 were studied earlier ( baykal , 1997 ) . that work employed a flux and pulse frequency data string covering a @xmath030% shorter time interval compared to the content of the oao 1657 - 415 data studied in the present paper . in that paper , correlations of pulse frequency derivative ( proportional to torque exerted on the neutron star ) , pulse frequency derivative over flux ( proportional to specific angular momentum of the accreted plasma ) and flux ( proportional to luminosity ) were discussed . it was found that the most natural explanation of the observed x - ray flux and pulse frequency derivative fluctuations is the formation of temporary accretion disks in the case of stellar wind accretion . the present paper extends the analysis on oao 1657 - 415 to cover a larger data string . we also present the results of a similar analysis in two other pulsars , vela x-1 and gx 301 - 2 . in the next section , database is introduced , and pulse frequency , pulse frequency derivative and flux time series are presented . a discussion of the results and conclusions are given in section 3 . batse is made up of eight detector modules located at the corners of cgro ( compton gamma ray observatory ) . these detectors have enabled continuous all sky monitoring for both pulsed and unpulsed sources above 20 kev since 1991 . batse daily monitors the pulse frequency and x - ray flux of three low mass binaries , five high mass binaries and seven previously known transients . it has also discovered new transients ( bildsten et al . 1997 ) . this paper is based on orbitally corrected batse 20 - 60 kev band x - ray flux and pulse frequency time series of vela x-1 , gx 301 - 2 and oao 1657 - 415 which are obtained from the ftp site `` ftp.cossc.gsfc.nasa.gov '' . in this paper , we assume that time variation of the 20 - 60 kev band flux represents the time variation of the bolometric x - ray flux . it should be noted that the flux time series , reported in fig . 1 , 4 , and 7 might not be representative of the time variation of the bolometric x - ray flux . pulse frequency and flux time series are binned by considering that measurement errors dominate in short time lags or high frequencies in the pulse frequency derivative power spectrum ( bildsten et al . 1997 ) . we choose our bin sizes to the extent that the measurement errors do not dominate on pulse frequency derivatives . bin sizes are 45 days , 30 days and 16 days for vela x-1 , gx 301 - 2 and oao 1657 - 415 respectively . pulse frequency derivatives are found by averaging left and right derivatives of pulse frequency values so that each pulse frequency derivative corresponds to a single flux and time value . we present x - ray flux , pulse frequency and pulse frequency derivative time series in fig . 1,4 and 7 . in fig . 2 , 5 and 8 , we present the plot of pulse frequency derivative and flux values corresponding to the same time value . since pulse frequency derivative and x - ray flux are directly proportional to torque and x - ray luminosity , these figures show the relation between torque and x - ray luminosity . pulse frequency derivative over flux is directly proportional to the specific angular momentum of the plasma ( @xmath1 where l is the specific angular momentum , @xmath2 is the frequency derivative and @xmath3 is the mass accretion rate ) . fig . 3 , 6 , 9 which are plots of pulse frequency derivative over flux and pulse frequency derivative show the relation between specific angular momentum and torque . pulse frequency derivative over flux time series are created by dividing each pulse frequency derivative value with the flux value corresponding to the same time . 283s pulsations from vela x-1 were discovered by _ sas-3 in 1975 ( mcclintock et al . 1976 ) . it is the brightest persistent accretion powered pulsar in the 20 - 60 kev energy band ( bildsten et al . . optical companion of vela x-1 is the b0.5 ib supergiant hd77581 ( vidal et al . this system is an eclipsing binary with eccentricity of @xmath4 and period of 8.96days ( rappaport et al . _ x - ray flux , pulse frequency and pulse frequency derivative values of vela x-1 cover the interval between 48371 mjd and 50580 mjd ( fig.1 ) . in fig.2 , pulse frequency derivative and corresponding x - ray flux data were plotted . no correlation between frequency derivative and flux is found . from fig.3 , it is seen that there is a correlation between pulse frequency derivative over flux and frequency derivative . 700s pulsations from gx 301 - 2 ( 4u 1223 - 62 ) were discovered by ariel 5 in 1975 ( white et al . gx 301 - 2 was , on average , neither spinning up nor spinning down between 1975 and 1985 . after 1985 , a spin - up episode began ( nagase 1989 ) , reaching the current pulsar spin period of @xmath5 . gx 301 - 2 , being in a 41.5 day eccentric orbit ( @xmath6 ) , is an accreting pulsar with the supergiant companion wray 977 ( sato et al . this source exhibited two rapid spin - up episodes at @xmath7 mjd and @xmath8 mjd . these two spin - up episodes suggest the existence of a long - lived ( @xmath9days ) accretion disk ( koh et al . 1997 ) . x - ray flux , pulse frequency and pulse frequency derivative values of gx 301 - 2 cover the interval between 48371 mjd and 50577 mjd ( fig.4 ) . fig.5 is the plot of pulse frequency derivative and corresponding x - ray flux values . no correlation between pulse frequency derivative and flux is found . from fig.6 , we see that there is a correlation between pulse frequency derivative over flux and pulse frequency derivative . in fig . 5 and 6 , there exist two points with pulse frequency derivative values greater than @xmath10 hz / day . these points correspond to two rapid spin - up episodes which was interpreted as a sign of transient prograde accretion disk ( koh et al . 1997 ) . 1657 - 415 was first detected by the copernicus satellite ( poldan et al . 1978 ) . 38.22s pulsations from oao 1657 - 415 were found in 1978 from heao 1 observations ( white et al . the optical companion of oao 1657 - 415 is probably a ob type star . its binary orbit period with an eccentricity of @xmath11 was found to be 10.4 days from the eclipse due to its companion from timing observations of this source with the batse observations ( chakrabarty 1993 ) . detailed studies on x - ray flux and pulse frequency derivative changes were performed earlier on a less extensive batse 20 - 60kev x - ray flux and pulse frequency data string ( baykal 1997 ) . in the current data string , x - ray flux and pulse frequency and pulse frequency derivative values of oao 1657 - 415 cover the interval between 48372 mjd and 50302 mjd ( fig.7 ) . 8 presents pulse frequency derivative and corresponding x - ray flux time series . no correlation between pulse frequency derivative and flux is found . there is a correlation between pulse frequency derivative over flux and pulse frequency derivative as seen from fig . the torque on the neutron star can be expressed in terms of the specific angular momentum ( @xmath12 ) added to the neutron star by the accreted plasma and the mass accretion rate ( @xmath3 ) : this equation is a general expression which is valid for both accretion from a keplerian disk and accretion from the stellar wind . in the case of accretion from a keplerian disk ( ghosh & lamb 1979b ) we have , where @xmath17 is the fastness parameter and @xmath18 , the dimensionless torque , represents the ratio of the total ( magnetic plus material ) torque to the material torque . @xmath19 is the radius of the inner edge of the disk , which can be written as @xmath20 where @xmath21 is the magnetic moment of the neutron star . dimensionless torque can approximately be written as where @xmath23 is a critical value for the fastness parameter which defines the boundary between spin - up and net spin - down phases . dimensionless torque may be positive or negative depending on the fastness parameter @xmath24 which is defined as where @xmath26 is the neutron star s spin angular velocity and @xmath27 is the angular velocity corresponding to the keplerian velocity at the magnetospheric radius . for a slowly rotating neutron star for which @xmath28 ( ghosh&lamb 1979b ; wang 1995 ; li&wang 1996,1999 ) , we expect a spin - up torque and for a very fast rotating neutron star ( @xmath29 ) , we expect a spin - down torque . we also expect to see positive correlation between torque and mass accretion rate if the disk is prograde . for a disk formed from roche lobe overflow of the companion we expect the plasma to carry positive specific angular momentum , so a prograde disk should be formed . the total torque exerted on the neutron star is proportional to the material torque for a given fastness parameter . since the specific angular momentum weakly depends on mass accretion rate ( @xmath30 ) , the net torque becomes proportional to the mass accretion rate ( @xmath3 ) . the bolometric x - ray luminosity is also correlated with the mass accretion rate ( @xmath31 where r is the radius of the neutron star ) . thus , we expect a correlation between torque and x - ray luminosity for the sources accreting from prograde accretion disks ( torques are positive ) . for the similar reasons , an anticorrelation between torque and x - ray luminosity is expected from a retrograde accretion disk ( torques are negative ) . however , for the pulsars we have considered , we see no correlation of pulse frequency derivative and flux(fig . 2,5,8 ) . moreover , there are several transitions from spin - up and spin - down . these results suggest that we do not have stable accretion disks for these sources . there is a model which does not exclude the possibility of a stable keplerian accretion disk(anzer&brner 1995 ) . this model is proposed to explain the torque reversals in vela x-1 suggesting the existence of a stable accretion disk lying just outside the magnetosphere mass of which exhibits small variations . it is found that change of the disk s mass in a random way can produce variations of torque in vela x-1 . however , the authors have not identified a specific physical process which is responsible for such random mass fluctuations . where i is the moment of inertia of the neutron star , l is the specific angular momentum of the accreting matter , @xmath34 , @xmath35 , and @xmath36 are the variations of spin frequency derivative , mass accretion rate ( proportional to x - ray flux ) and specific angular momentum ( proportional to pulse frequency derivative over flux ) respectively . when the variations of torques for the three pulsars are concerned , we have considerable changes and transitions from negative values to positive values and vice versa . for the case of an accretion from the wind , numerical simulations show that the changes in the sign of specific angular momentum is possible ( anzer et al . 1987 ; taam & fryxell 1988a,1988b,1989 ; blondin et al . 1991 , murray et al . so , we can observe transitions from spin - up to spin - down or vice versa even if there is not a significant change in mass accretion rate . for wind accreting sources , continuos change in accretion geometry rules out the existence of a stable accretion disk . thus , it is unlikely to see a correlation between torque and x - ray luminosity which is the case for our sources as well . for such sources , a correlation between specific angular momentum and torque shows that there are considerable changes in torque while there are not very considerable changes in x - ray luminosity . this indicates changes in accretion geometry . the correlation between specific angular momentum and torque exists for all of the three sources ( fig . 3 , 6 and 9 ) . there are recent developments which explain the torque reversals in accretion powered pulsars . negative torques may come from a retrograde keplerian accretion disk ( nelson et al . 1997 ) which may , for instance , originate from a red giant ( murray et al . 1998 ) . these spin - down torques may be the result of an advection dominated sub - keplerian disk for which the fastness parameter should be higher than that of a corresponding keplerian disk causing a net spin - down ( yi et al . 1997 ) or the warping of the disk so that the inner disk is tilted by more than 90 degrees ( van kerkwijk et al . 1998 ) . torque and x - ray luminosity correlation is expected from these models which is not found for our sources . moreover , in these models timescales for torque reversals are either not certain or of the order of years . so , these ideas about torque reversals are not supported by the behaviour of the three pulsars we have considered . our considerations about correlations between pulse frequency derivative , flux and specific angular momentum give insights about the physics of the plasma flow in the vicinity of the neutron stars accreting from the winds of their companions . we had similar conclusions for all three sources . we found that it is unlikely for these sources to have stable prograde keplerian accretion disks since they show both spin - up and spin - down episodes and they do not show correlation between torque and luminosity . correlation between specific angular momentum and torque for these sources may indicate the continuous change in accretion geometry . this shows the possibility of temporary prograde and retrograde accretion disk formation . it would be also possible to have stronger idea about the accretion geometry if we had measurements of the pulse frequency derivative with a time resolution of the order of hours , which is a typical time scale of accretion geometry changes for these systems ( taam & fryxell 1988a , 1988b , 1989 ; blondin et al . for such a case , we would allow a better comparison of the changes in flux and specific angular momentum and it would be interesting to detect both high flux and low specific angular momentum points corresponding to the radial flow cases and to see low flux and high specific angular momentum points corresponding to the accretion from prograde ( high positive specific angular momentum ) or retrograde ( high negative specific angular momentum ) accretion disks . our flux and pulse frequency time series are found using the x - ray flux and pulse frequency between 20 - 60 kev . observatories with higher time resolution and capable of detecting photons from a wider energy band will be useful to observe the torque , x - ray luminosity and specific angular momentum changes for these pulsars . an encouraging example is given by rxte ( rossi x - ray timing explorer ) observations of the accretion powered pulsar 4u 1907 + 09 . dipping activity in the x - ray intensity was found which was interpreted as a consequence of inhomogeneity of the wind from the companion(int zand et al . 1997 ) , and more recently , it was shown from the rxte observations that 4u 1907 + 09 exhibited transient @xmath37s qpo oscillations during a flare which was superposed on long term spin - down rate . this was interpreted as a sign of transient retrograde accretion disk ( int zand et al . 1998a , b ) . rxte observations of vela x-1 , gx 301 - 2 , and oao 1657 - 415 may provide more understanding on the nature of accretion flow through tests on flux and frequency time series at higher resolution . we acknowledge dr.ali alpar and dr.len balman for critical reading of the manuscript . it is a pleasure to thank dr.matthew scott and dr.bob wilson for their help to our questions about the database . we thank the compton gamma ray observatory team at heasarc for the archival data . anzer , u. et al . 1987 , a&a , 176 , 235 anzer , u. , brner , g. 1995 , a&a , 299 , 62 baykal , a. et al . 1991 , a&a , 252 , 664 baykal , a. 1997 , a&a , 319,515 baykal , a. , gelman , h. 1993 , a&a , 267 , 119 bildsten , l. et al . 1997 , apjs , 113,367 blondin , j.m . et al . 1990 , apj , 356 , 591 chakrabarty , d. 1993 , apj , 403 , l33 chakrabarty , d. 1996 , ph.d thesis submitted to california institute of technology chakrabarty , d. 1997 , apj , 481 , l101 davidson , k. , ostriker , j.p . 1973 , apj , 179 , 585 ghosh , p. , lamb , f.k . 1979a , apj , 232 , 259 ghosh , p. , lamb , f.k . 1979b , apj , 234 , 296 giacconi , r. et al . 1971 , apj , 167 , l67 int zand , j.j.m . , strohmayer , t.e . , baykal , a. 1997 , apj , 479 , l47 int zand , j.j.m . , baykal , a. , strohmayer , t.e . 1998a , apj , 496 , 386 int zand , j.j.m . , strohmayer , t.e . , baykal , a. 1998b , nuclear physics b ( proc.suppl . ) 69/1 - 3 , 224 koh , d.t . 1997 , apj , 479 , 913 lamb , f.k . 1973 , apj , 184,271 lamb , f.k . 1978 , apj , 224 , 969 li , x .- d . , wang , z .- r . 1996 , a&a , 307l , 5 li , x .- d . , wang , z .- r . 1999 , astro - ph 9901083 mcclintock , j.e . 1976 , apj , 206 , l99 murray , j.r . 1998 , astro - ph 9810118 nagase , f. 1989 , pasj , 41 , 1 nelson , r.w . 1997 , apj , 488 , l117 polidan , r.s . 1978 , nature , 275 , 296 pringle , j.e . , rees m.j . , 1972 , a&a , 21 , 1 rappaport , s. et al . 1976 , apj , 206 , l103 sato , n. et al . 1986 apj , 304 , 241 schreier , e. et al . 1972 , apj , 172 , l79 taam , r.e . , fryxell , b.a . , 1988a , apj , 327 , l73 taam , r.e . , fryxell , b.a . , 1988b , apj , 335 , 862 taam , r.e . , fryxell , b.a . , 1989 , apj , 339 , 297 van kerkwijk , m.h . 1998 , astro - ph 9802162 vidal , n.v . 1973 , apj , 265 , 1036 yi , i. et al . 1997 , apj , 481 , l51 wang , y .- m . 1995 , apj , 449 , l153 white , n.e . 1976 , apj , 209 , l119 white , n.e . 1979 , apj , 233 , l121

using archival batse ( burst and transient source experiment ) 20 - 60 kev band x - ray flux and pulse frequency time series , we look for correlations between torque , luminosity and specific angular momentum for three high mass x - ray binary pulsars vela x-1 , gx 301 - 2 and oao 1657 - 415 . our results show that there is no correlation between pulse frequency derivative and flux which may be an indication of the absence of stable prograde accretion disk . from the strong correlation of specific angular momentum and torque , we conclude that the accretion geometry changes continuously as suggested by the hydrodynamic simulations(blondin et al . 1990 ) .

You are an expert at summarizing long articles. Proceed to summarize the following text: the present eps database consists of a rather heterogeneous sample of planets , showing great variety in all the measured quantities . statistical analysis provides a necessary means to find correlations among various physical parameters involved in planetary formation and evolution . nevertheless , we caution that statistical analysis may not be able to disclose important relationships due to the complex and -mostly unknown- interplay of the involved parameters . in order to overcome such problems , we performed a global statistical analysis of the eps , using a novel approach . the underlying idea is to find groups of _ similar _ eps and to distinguish different ep groups on the basis of their _ diversity_. in this work , the concept of similarity and diversity among eps is quantified by means of a distance measure in the multifold space of physical parameters . this goal has been achieved with the aid of principal component analysis and hierarchical cluster analysis . in the present analysis we followed the procedure used by @xcite , and updated in @xcite . the database used in this paper is updated to july 9@xmath0 , 2008 . we restricted our analysis to those eps having measurements for five _ input variables _ , that is : planetary mass ( @xmath1 ) , semimajor axis ( @xmath2 ) , eccentricity ( @xmath3 ) , stellar mass ( @xmath4 ) and metallicity ( [ fe / h ] ) . of 308 eps ( including jupiter ) , 252 were finally selected for the analysis . the purpose of our analysis is to identify planets which are similar with respect to the 5-fold space of the input variables . as a result , 6 robust ep clusters have been identified . before focusing on hps , we briefly outline the general nature of the clusters . + cluster c1 is characterized by sub- to jovian - like @xmath1 , @xmath5 au , and super - solar [ fe / h ] . cluster c2 has sub - jovian @xmath1 , @xmath5 au , sub - solar @xmath4 and sub - solar [ fe / h ] . both clusters c1 and c2 also have low mean eccentricity . cluster c3 is the least populated cluster ( 14 eps ) and probably has no strong significance except the fact that it contains many peculiar eps that for different reasons have been rejected by the other clusters . they are mostly jovian mass planets , having a remarkably super - solar [ fe / h ] . cluster c4 has mostly jovian mass planets , a relatively large @xmath2 , orbiting solar mass stars with a widespread [ fe / h ] distribution characterized by sub - solar values ( jupiter belongs to this cluster ) . cluster c5 has super - jovian mass , @xmath6 au , super - solar @xmath4 and sub - solar [ fe / h ] . the same holds for cluster c6 , except for its super - solar [ fe / h ] and its higher average eccentricity . all the input variables have an important role in defining the clusters , in particular @xmath2 and [ fe / h ] . beyond the general traits of the solutions , which will be described in more detail elsewhere , we focus in this paper on the hps defined as those having a period less than 12 days ( see later for further comments on this selection ) . according to the adopted definition , 69 hps are present in the database . the main results of the taxonomy is that hps have been split into two main groups , belonging to clusters c1 and c2 . in addition , there are a few outliers placed in other clusters . one of the most interesting features about the hps , is the shape of their mass distribution ( see fig . [ mp_a_hist ] ) . it has two peaks . these peaks are thought to be real , even if the shape of the present distribution is severely affected by observational biases that make the discovery of low mass hps difficult . in the literature , the bodies belonging to the lower mass peak are referred to as hot neptunes ( hns ) while those belonging to the higher mass peak as hot jupiters ( hjs ) . theoretical models also predict a double peaked distribution @xcite , where hns are expected to be much more abundant than hjs . + it is interesting that our taxonomy splits the majority of hps into two different clusters ( except for a few outliers ) . most of the hps belonging to the peak at @xmath7 are placed in cluster c1 ( fig . [ mp_a_hist ] ) . hps of the cluster c2 have a broad and flatter mass distribution , and it contains most of the hns . concerning the semimajor axis distribution , similar considerations hold . hps of cluster c1 are strongly clustered at @xmath8 ( @xmath9 au ) ; while those of cluster c2 have a flatter @xmath2 distribution ( fig . [ mp_a_hist ] ) . the two groups of hps also have clearly distinctive traits in terms of properties of stellar metallicity and -to a lesser extent- to stellar mass . these variables are important because they account for the environment where the planets formed . figure [ fe_ms_hist ] shows a remarkable result : hps of cluster c1 have super - solar [ fe / h ] , while those of c2 have a sub - solar [ fe / h ] . notice that this result is also valid for the whole of c1 and c2 , and not only for their hps . a similar , but less pronounced , result also holds for the stellar mass : hps of c1 have mostly super - solar @xmath4 ; while those of c2 have mostly sub - solar @xmath4 ( fig . [ fe_ms_hist ] ) . + the hps of cluster c1 and c2 also have other distinctive traits . we found some significant intracluster correlations which hold for one cluster but not for the other . these are the correlations @xmath10 , @xmath11[fe / h ] , @xmath12 ( see fig . [ ae ] ) . the semimajor axis of the hps of cluster c1 strongly correlate with @xmath3 , while this correlation is absent for cluster c2 . on the other hand , hps of cluster c2 exhibit a strong correlation of @xmath11[fe / h ] and @xmath12 , while cluster c1 does not ( fig . these two plots in turn clearly show that hns of cluster c1 are very different from the few belonging to cluster c2 . the latter have remarkable sub - solar [ fe / h ] and sub - solar @xmath4 . + another important point is why some hps have been placed in clusters other than c1 and c2 . they are : hd73256b , hd68988b , hat - p-7b and hd118203b ( c3 ) ; xo-4b , hat - p-6b and hd162020b ( c4 ) ; wasp-14b , corot - exo-3b and xo-3b ( c5 ) ; hat - p-2b ( c6 ) . it is not clear if they are real outliers or if they are misplaced by the clustering algorithm . however , a close look at their properties shows that the main characteristic of these hps is that they have high @xmath1 and high @xmath4 . among the outliers , the most massive ( @xmath13 ) hps are present , namely hat - p-2b , wasp-14b , corot - exo-3b , xo-3b and hd162020b ( see fig . [ mp_a_hist ] , left panel ) . notice that the massive hd41004bb is placed instead in c2 , due to the very low stellar mass . in the present database , the combination of high @xmath1 and high @xmath4 is rather unusual , and this suggests that these hps may be real outliers . of course , if more hps having such characteristics are discovered , it is possible that they may be grouped into a third class of hps . finally , it is also important to note that previous results are robust to changes in the selection of hp periods , at least in the range from 10 d to 30 d. our present theoretical understanding of the formation of hps is based on two models : planetary migration within a protoplanetary disk @xcite and planet - planet scattering followed by tidal circularization @xcite . planetary migration has been shown to be an efficient mechanism to produce hps . migration can occur when planetary embryos are still embedded within the disk ( type i ) , or when they are large enough to open a gap in the disk ( type ii ) . according to the present state of the art , the migration seems the best candidate for hp formation . on the other hand , early works on planet - planet scattering showed that the probability for the scattering model to produce hps was very low . however , @xcite found that the kozai mechanism enhances the probability significantly . therefore , it is possible that the scattering contributes to the formation of hot planets as well as type ii migration , although the latter may be a main channel . + type ii migration is more efficient for moderate mass planets since the planets have to be massive enough to clear a gap in the disk , but not too massive to efficiently exchange angular momentum and energy with the disk itself . previous simulations have shown that the formation of hps via gravitational scattering among planets and the subsequent kozai mechanism combined with tidal dissipation is more likely for dynamically active systems of multiple planets , typically containing three or more gas giants ( e.g. * ? ? ? the formation of many giant planets is preferred in high dust surface density disks @xcite . dust surface density scales as @xmath14}m_d$ ] , where @xmath15 is the total disk mass . the latter scales , according to theoretical models , as @xmath16 , where @xmath17 @xcite . therefore high stellar masses and high - metallicity disks favor the onset of a scattering phase . it must also be noted that smaller planets tend to be scattered inward during the scattering phase and that the tidal circularization is more efficient for planets with small mass and large physical radius . the detailed final orbital configuration of the hps may vary according to several parameters . however , there are a number of general features that can be outlined . + in the case of type ii migration , the hps final position is close to the location of the disk s inner edge . this is placed near the corotation radius ( namely the distance where the keplerian period matchs the star spin period ) , where disk material accretes onto the stellar poles following the magnetic field lines . the spin period for young stars having @xmath18 may vary considerably , from 1 d up to 20 d or more @xcite . therefore , taking also into account a wide variety of disk parameters , the final location of hps formed by type ii migration tends to be spread , in the range from 0.01 to 0.1 au . concerning the eccentricity , at the end of the migration phase low @xmath3 values are expected . recent simulations @xcite suggest that , when hps reach the stellar magnetosphere cavities , they may further evolve to smaller @xmath2 . for moderate - to - high @xmath1 , the @xmath3 also increases . on the other hand , the location of hps formed by scattering is determined by the location where the tidal force is effective . the tidal strength depends on several parameters ( planetary radius , planetary mass , etc ) but the final location tends to be in the range 0.03 - 0.08 au , for typical parameter values . the eccentricity of the inner planet is excited to values close to unity by close scattering and via the subsequent kozai mechanism . the resultant small pericenter distance enables the planet s eccentricity and semi - major axis to be decreased by the tidal dissipation and moderate eccentricities can remain in some cases @xcite . we show this in fig . [ scatter_ae ] . the outcome of the simulations vary according to tide efficiency . for the simulations shown here , we followed the tidal evolution of test planets , according to @xcite , for a time span of @xmath19 yr . a large number of planetary radii , planetary masses and stellar masses have been considered , choosen at random within the following intervals : @xmath20 , @xmath21 , @xmath22 . the initial eccentricity is randomly chosen from the distribution obtained by @xcite . figure [ scatter_ae ] ( left panel ) shows that at the end of the scattering and tidal evolution phases the eccentricity and semimajor axes are correlated ( compared to the observed distribution in fig . [ ae ] ) , while the right panel shows that the resulting semimajor axis distribution is peaked . the value of the peak depends on the strength of the tide , and for the values used here it is peaked at @xmath23 au , as observed in cluster c1 ( compare with the observed distribution in fig . [ mp_a_hist ] ) . the two main processes of hp formation produce different orbital distributions . on the basis of our taxonomic analysis , we identify two types of hps , which may retain the footprints of these two different formation processes . in this respect , hps of cluster c2 and c1 may have been formed by migration and scattering , respectively . this scenario is supported by a number of facts . first of all the @xmath2 distribution and the strong @xmath10 relationship for cluster c1 , which do not hold for cluster c2 . moreover , also the @xmath1 distribution of the two clusters support this conclusion : higher mass bodies are located in cluster c1 , indicating more massive protoplanetary disks , which in turn is confirmed by high [ fe / h ] and high @xmath4 . in this case , the detected hps are expected to be the least massive for each system , since in a multibody scattering the least massive planets are more effectively pushed inward . the orbits of the most massive planets are , however , only slightly affected by the scattering phase , therefore they tend to stay close to their formation regions and therefore on relatively large @xmath2 . although hjs originated via the scattering process are expected to be accompanied by outer companions , the latter would be beyond the detectable limit by present surveys . we find no significant difference between c1 and c2 about the multiplicity of planets , but we caution that this result is affected by low number statistics . some jupiter - like mass hps are also present in cluster c2 , but in this case they may be the most massive bodies formed in these systems , given also the moderate - to - low @xmath4 and low [ fe / h ] . another interesting point is that most of the hns belong to cluster c2 . + therefore , if we extrapolate directly the percentage of hps belonging to c1 and c2 into the efficiency of formation of the two processes , we end up with 50% of hps formed via scattering and 30% via migration . the remaining 20% are outliers and may be formed either way . these numbers , however , have to be taken with caution , since some degree of mixing between the two clusters is expected . on the other hand , from a theoretical point of view ( see discussion in previous sections ) type ii is expected to be more efficient in producing close - in eps . this fact is not in contradiction with our findings since it is possible that planets migrating inward by type ii may stop before becoming hps . this would be the case , for instance , if the gas in the disk dissipates before the planet reaches the magnetospheric cavity . in this respect , it is interesting that many giant eps exist in the range 0.1 - 3 au and that their semimajor axis distribution is well explained by the type ii migration model @xcite . + an alternative scenario is that the two groups of hps were formed by the same process , and the cluster analysis splits them on the basis of their diversity in the input variables . in this case , the most distinctive variable would be the metallicity . formation in low metallicity , moderate - to - low @xmath4 , environments would have produced the hps of cluster c2 . on the other hand , high [ fe / h ] and moderate - to - high @xmath4 would have produced the hps of cluster c1 . the combinations of these diversities would have also produced the observed differences in the @xmath1 and @xmath2 distribution of the two groups . although this is a possibility , we think the peculiar traits of cluster c1 and c2 clearly shown the influences of the two formation mechanisms .

the extrasolar planets ( eps ) so far detected are very different to the planets in our own solar system . many of them have jupiter - like masses and close - in orbits ( the so - called hot planets , hps ) , with orbital periods of only a few days . in this paper , we present a new statistical analysis of the observed eps , focusing on the origin of the hps . among the several hp formation mechanisms proposed so far , the two main formation mechanisms are type ii migration and scattering . in both cases , planets form beyond the so - called snow - line of the protoplanetary disk and then migrate inward due to angular momentum and energy exchange with either the protoplanetary disk or with companion planets . although theoretical studies produce a range of observed features , no firm correspondence between the observed eps and models has yet been established . in our analysis , by means of principal component analysis and hierarchical cluster analysis , we find convincing indications for the existence of two types of hps , whose parameters reflect physical mechanisms of type ii migration and scattering . [ firstpage ] planets and satellites : formation planetary systems : formation planetary systems : protoplanetary discs .

You are an expert at summarizing long articles. Proceed to summarize the following text: understanding brain function requires knowing connections between neurons . however , experimental studies of inter - neuronal connectivity are difficult and the connectivity data is scarce . at the same time neuroanatomists possess much data on cellular morphology and have powerful techniques to image neuronal shapes . this suggests using morphological data to infer inter - neuronal connections . such inference must rely on rules which relate shapes of neurons to their connectivity . the purpose of this paper is to derive such a rule for a frequently encountered feature in the brain organization : a topographic projection . two layers of neurons are said to form a topographic projection if adjacent neurons of the input layer connect to adjacent neurons of the output layer , figure [ fig : cd ] . as a result , output neurons form an orderly map of the input layer . i characterize inter - neuronal connectivity for a topographic projection by divergence and convergence factors defined as follows , fig . [ fig : cd ] . _ divergence _ , @xmath0 , of the projection is the number of output neurons which receive connections from an input neuron . _ convergence _ , @xmath1 , of the projection is the number of input neurons which connect with an output neuron . i assume that these numbers are the same for each neuron in a given layer . furthermore , each neuron makes the required connections with the nearest neurons of the other layer . in most cases , this completely specifies the wiring diagram . a typical topographic wiring diagram shown in fig . [ fig : cd ] misses an important biological detail . in real brains , connections between cell bodies are implemented by neuronal processes : axons carrying nerve pulses away from the cell bodies and dendrites carrying signals towards cell bodies.@xcite therefore each connection is interrupted by a synapse which separates an axon of one neuron from a dendrite of another . both axons and dendrites branch away from cell bodies forming arbors . in general , a topographic projection with given divergence and convergence may be implemented by axonal and dendritic arbors of different sizes , which depend on the locations of synapses . for example , consider a wiring diagram with @xmath2 and @xmath3 , figure [ fig : menorah]a . narrow axonal arbors may synapse onto wide dendritic arbors , figure [ fig : menorah]b or wide axonal arbors may synapse onto narrow dendritic arbors , figure [ fig : menorah]c . i call these arrangements type i and type ii , correspondingly . the question is : which arbor sizes are preferred ? i propose a rule which specifies the sizes of axonal arbors of input neurons and dendritic arbors of output neurons in a topographic projection : _ high divergence / convergence ratio favors wide axonal and narrow dendritic arbors while low divergence / convergence ratio favors narrow axonal arbors and wide dendritic arbors . _ alternatively , this rule may be formulated in terms of neuronal densities in the two layers : _ sparser layer has wider arbors . _ in the above example , divergence / convergence ( and neuronal density ) ratio is 1/6 and , according to the rule , type i arrangement , figure [ fig : menorah]b , is preferred . in this paper i derive a quantitative version of this rule from the principle of wiring economy which can be summarized as follows . @xcite,@xcite,@xcite,@xcite,@xcite space constraints require keeping the brain volume to a minimum . because wiring ( axons and dendrites ) takes up a significant fraction of the volume , evolution has probably designed axonal and dendritic arbors in a way that minimizes their total volume . therefore we may understand the existing arbor sizes as a result of wiring optimization . to obtain the rule i formulate and solve a wiring optimization problem . the goal is to find the sizes of axons and dendrites which minimize the total volume of wiring in a topographic wiring diagram for fixed locations of neurons . i specify the wiring diagram with divergence and convergence factors . throughout most of the paper i assume that the cross - sectional area of dendrites and axons are constant and equal . therefore , the problem reduces to the wiring _ length _ minimization . my results are trivially extended to the case of unequal fiber diameters as shown below . purves and coworkers@xcite,@xcite,@xcite have previously reported empirical observations which may be relevant to the present theory . they found a correlation between convergence and complexity of dendritic arbors in sympathetic ganglia . conclusive comparison of this data with the theory requires establishing topographic ( or some other ) wiring diagram and measuring axonal arbor sizes in this system . in the next section i consider a one - dimensional version of the problem . in this case , wirelength is minimized by wide dendritic and no axonal arbors ( type i ) for divergence less than convergence and by no dendritic and wide axonal arbors ( type ii ) in the opposite case . in section [ sec:2d ] , i consider a two - dimensional version of the problem . if both convergence and divergence are much greater than one , the optimal ratio of dendritic and axonal arbors equals to the square root of convergence / divergence ratio . i test the rule on the available anatomical data in section [ sec : data ] . for several projections between retinal and cerebellar neurons , arbor sizes agree with the rule . in section [ sec : other ] i consider other factors which may affect arbor sizes . consider two parallel rows of evenly - spaced neurons , figure [ fig : cd ] , with a topographic wiring diagram characterized by divergence , @xmath0 , and convergence , @xmath1 . the goal is to find axonal and denritic arbor sizes which minimize the combined length of axons and dendrites . i compare different arbor arrangements by calculating wirelength per unit length of the rows , @xmath4 . i assume that input / output rows are close to each other and include in the calculation only those parts of the wiring which are parallel to the neuronal rows . i start by considering a special case where each input neuron connects with only one output neuron ( @xmath2 ) , figure [ fig : menorah]a . there are two limiting arrangements satisfying the wiring diagram : type i has wide dendritic arbors and no axonal arbors , figure [ fig : menorah]b ; type ii has wide axonal arbors and no dendritic arbors , figure [ fig : menorah]c . intuitively , the former arrangement has smaller wirelength : short axons synapsing onto a common bus - like dendrite is better than long axons from each input neuron synapsing onto a short dendrite . to confirm this i calculate wirelength in the two extreme arrangements for @xmath2 , see methods . @xmath5 @xmath6 these results show that for @xmath2 and @xmath7 the two arrangements have the same wirelength . for @xmath2 and @xmath8 the arrangement with wide dendritic arbors and no axonal arbors ( type i ) has smaller wirelength than the arrangement with wide axonal arbors and no dendritic arbors ( type ii ) . i can readily apply this result to another special case , @xmath9 , by invoking the symmetry of the problem in respect to the direction of the signal propagation . i can interchange words `` axons '' and `` dendrites '' and variables @xmath0 and @xmath1 in the derivation and use the above argument . for @xmath9 and @xmath10 the two extreme arrangements have the same wirelength , while for @xmath11 the arrangement with wide axonal arbors ( type ii ) has shorter wiring than the arrangement with wide dendritic arbors ( type i ) . next , i consider the case when both convergence and divergence are greater than one ( @xmath12 ) . for the two extreme arrangements i get ( see methods ) : @xmath13 @xmath14 comparison of the two expressions reveals the following . if divergence is less than convergence then the optimal arrangement has wide dendritic and no axonal arbors ( type i ) . if divergence is greater than convergence then the optimal arrangement has wide axonal and no dendritic arbors ( type ii ) . if convergence and divergence are equal both arrangements have the same wirelength . i can restate this result by using the identity between the divergence / convergence ratio and the neuronal density ratio ( see methods ) : in the optimal arrangement the sparser layer has wide arbors , while the denser layer has none . so far i compared extreme arrangements with wide arbors in one row and none in the other . what about intermediate arrangements , with both axonal and dendritic arbors of non - zero width ? to address this question i consider the limit of large divergence and convergence factors ( @xmath15 . i find wirelength as a function of the axonal arbor size @xmath16 ( see methods ) : @xmath17 because @xmath18 i find the following . if @xmath19 then the minimal wirelength is achieved when @xmath20 , arrangement with wide dendritic and no axonal arbors ( type i ) . if @xmath21 then the minimal wirelength is achieved when @xmath22 , arrangement with wide axonal and no dendritic arbors ( type ii ) . if @xmath23 then all possible axonal arbor widths give the same wirelength . this proves that for @xmath24 extreme arrangements minimize wirelength . in cases of small @xmath1 and @xmath0 i checked intermediate solutions one by one . in many cases intermediate arrangements have the same wirelength as the extreme solution . however , only for a few `` degenerate '' @xmath25 pairs there are equally good intermediate arrangements with the reverse ratio of average axonal and dendritic arbor sizes relative to the extreme solution . my results are conveniently summarized on the phase diagram , figure [ fig : phase ] , which shows optimal arrangements for various pairs of divergence and convergence factors . i mark the `` degenerate '' @xmath25 pairs by diamonds on the phase diagram , figure [ fig : phase ] . what if axons and dendrites have different crossectional areas ? the principle of wiring economy requires that wire volume rather than wire length should be minimized . i can modify the arguments of this section by including the cross - sectional areas of the processes . i find for @xmath26 that if divergence / convergence ratio is less than the ratio of axonal and dendritic cross - sections then the optimal arrangement has wide dendritic and no axonal arbors ( type i ) . in the opposite case i find wide axonal and no dendritic arbors ( type ii ) . consider two parallel layers of neurons with densities @xmath27 and @xmath28 . the topographic wiring diagram has divergence and convergence factors , @xmath0 and @xmath1 , requiring each input neuron to connect with @xmath0 nearest output neurons and each output neuron with @xmath1 nearest input neurons . again , the problem is to find the arrangement of arbors which minimizes the total length of axons and dendrites . for different arrangements i compare the wirelength per unit area , @xmath4 . i assume that the two layers are close to each other and include only those parts of the wiring which are parallel to the layers . i start with a special case where each input neuron connects with only one output neuron ( @xmath2 ) . consider an example with @xmath29 and neurons arranged on a square grid in each layer , figure [ fig : por]a . two extreme arrangements satisfy the wiring diagram : type i has wide dendritic arbors and no axonal arbors , figure [ fig : por]b ; type ii has wide axonal arbors and no dendritic arbors , figure [ fig : por]c . i take the branching angles equal to @xmath30 , an optimal value for constant crossectional area.@xcite assuming `` point '' neurons the ratio of wirelength for type i and type ii arrangements : @xmath31 thus , the type i arrangement with wide dendritic arbors has shorter wire length . this conclusion holds for other convergence values much greater than one , provided @xmath2 . however , there are other arrangements with non - zero axonal arbors that give the same wire length . one of them is shown in figure [ fig : por]d . degenerate arrangements have axonal arbor width @xmath32 where the upper bound is given by the approximate inter - neuronal distance . this means that the optimal arbor size ratio for @xmath2 @xmath33 by using the symmetry in respect to the direction of signal propagation i adapt this result for the @xmath9 case . for @xmath34 , arrangements with wide axonal arbors and narrow dendritic arbors ( @xmath35 ) have minimal wirelength . these arrangements have arbor size ratio @xmath36 next , i consider the case when both divergence and convergence are greater than one . due to complexity of the problem i study the limit of large divergence and convergence ( @xmath26 ) . i find analytically the optimal layout which minimizes the total length of axons and dendrites . unlike the one - dimensional projection , optimal sizes of both axons and dendrites turn out to be non - zero . notice that two neurons may form a synapse only if the axonal arbor of the input neuron overlaps with the dendritic arbor of the output neuron in a two - dimensional projection , figure [ fig : ar4 ] . thus the goal is to design optimal dendritic and axonal arbors so that each dendritic arbor intersects @xmath1 axonal arbors and each axonal arbor intersects @xmath0 dendritic arbors . to be specific , i consider a wiring diagram with convergence exceeding divergence , @xmath37 ( the argument can be readily adapted for the opposite case ) . i make an assumption , to be verified later , that dendritic arbor diameter @xmath38 is greater than axonal one , @xmath16 . in this regime each output neuron s dendritic arbor forms a sparse mesh covering the area from which signals are collected , figure [ fig : ar4 ] . each axonal arbor in that area must intersect the dendritic arbor mesh to satisfy the wiring diagram . this requires setting mesh size equal to the axonal arbor diameter . by using this requirement i express the total length of axonal and dendritic arbors as a function of only the axonal arbor size , @xmath16 . then i find the axonal arbor size which minimizes the total wirelength . details of the calculation are given in methods . here , i give an intuitive argument for why in the optimal layout both axonal and dendritic size are non - zero . consider two extreme layouts . in the first one , dendritic arbors have zero width , type ii . in this arrangement axons have to reach out to every output neuron . for large convergence , @xmath39 , this is a redundant arrangement because of the many parallel axonal wires whose signals are eventually merged . in the second layout , axonal arbors are absent and dendrites have to reach out to every input neuron . again , because each input neuron connects to many output neurons ( large divergence , @xmath40 ) many dendrites run in parallel inefficiently carrying the same signal . a non - zero axonal arbor rectifies this inefficiency by carrying signals to several dendrites along one wire . i find that the optimal ratio of dendritic and axonal arbor diameters equals to the square root of the convergence / divergence ratio , or , alternatively , to the square root of the neuronal density ratio : @xmath41 since i considered the case with @xmath37 this result also justifies the assumption about axonal arbors being smaller than dendritic ones . for arbitrary axonal and dendritic cross - sectional areas , @xmath42 and @xmath43 , expressions of this section are modified . the wiring economy principle requires minimizing the total volume occupied by axons and dendrites resulting in the following relation for the optimal arrangement : @xmath44 notice that in the optimal arrangement the total axonal volume of input neurons is equal to the total dendritic volume of the output neurons . this theory predicts a relationship between the con-/divergence ratio and the sizes of axonal and dendritic arbors . i test these predictions on several cases of topographic projection in two dimensions . the predictions depend on whether divergence and convergence are both greater than one or not . therefore , i consider the two regimes separately . first , i focus on topographic projections of retinal neurons whose divergence factor is equal or close to one . because retinal neurons use mostly graded potentials the difference between axons and dendrites is small and i assume that their cross - sectional areas are equal . the theory predicts that the ratio of dendritic and axonal arbor sizes must be greater than the square root of the input / output neuronal density ratio , @xmath45 ( eq.[sql ] ) . i represent the data on the plot of the relative arbor diameter , @xmath46 , vs. the square root of the relative densities , @xmath47 , ( figure [ fig : data ] ) . because neurons located in the same layer may belong to different classes , each having different arbor size and connectivity , i plot data from different classes separately . all the data points lie above the @xmath48 line in agreement with the prediction . second , i apply the theory to cerebellar neurons whose divergence and convergence are both greater than one . i consider a projection from granule cell axons ( parallel fibers ) onto purkinje cells . ratio of granule cells to purkinje cells is 3300,@xcite , indicating a high convergence / divergence ratio . this predicts a ratio of dendritic and axonal arbor sizes of 58 . this is qualitatively in agreement with wide dendritic arbors of purkinje cells and no axonal arbors on parallel fibers . quantitative comparison is complicated because the projection is not strictly two - dimensional : purkinje dendrites stacked next to each other add up to a significant third dimension . naively , given that the dendritic arbor size is about 400@xmath49 m eq.[res ] predicts axonal arbor of about 7@xmath49 m . this is close to the distance between two adjacent purkinje cell arbors of about 9@xmath49 m . because the length of parallel fibers is greater than 7@xmath49 m absence of axonal arbors comes as no surprise . in general , application of the rule requires some care because it was derived for a simplified model . i considered a topographic projection only between a single pair of layers . however , neurons often make connections to different layers . in particular , dendritic arbors of the output layer may be determined by connections other than to the input layer . for example , consider the topographic projection from thalamus to the primary visual cortex . one may think that because the density of magnocellular thalamic afferents is smaller than neurons in layer 4c@xmath50 ( @xmath51 compared with @xmath52)@xcite then the axonal arbors should be wider than the dendritic ones . although this is true ( @xmath53@xcite compared with @xmath54@xcite ) the majority of inputs to layer 4c@xmath50 are intra - cortical.@xcite therefore , the dendritic arbor size may be determined by these other projections . one may argue that dendrites and axons have functions other than linking cell bodies to synapses and , therefore , the size of the arbors may be dictated by other considerations . although i can not rule out this possibility , the _ primary _ function of axons and dendrites is to connect cell bodies to synapses in order to conduct nerve pulses between them . indeed , if neurons were not connected more sophisticated effects such as non - linear interactions between different dendritic inputs could not take place . hence the most basic parameters of axonal and dendritic arbors such as their size should follow from considerations of connectivity . another possibility is that the size of dendritic arbors is dictated by the surface area needed to arrange all the synapses . this argument does not specify the arbor size , however : a compact dendrite of elaborate shape can have the same surface area as a wide dendritic arbor . finally , agreement of the predictions with the existing anatomical data suggests that the rule is based on correct principles . further extensive testing of the rule is desirable . violation of the rule in some system would suggest the presence of other overriding considerations in the design of that system , which is also interesting . in conclusion , i propose a rule relating connectivity of neurons to their morphology based on the wiring economy principle . this rule may be used to infer connections between neurons from the sizes of their axonal and dendritic arbors . i frequently use the following identity@xcite relating convergence / divergence ratio and neuronal densities ratio : @xmath55 to prove it , i calculate the number of connections ( or synapses , if connections are monosynaptic ) per unit length in two ways . the number of connections ( or synapses ) is the number of input neurons , @xmath27 , times divergence , @xmath0 . at the same time , the number of connections ( or synapses ) is the number of output neurons , @xmath28 , times convergence , @xmath1 . since the answer should not depend on the argument , @xmath56 and eq.[cnd ] follows trivially . first , consider the case of @xmath2 . in type i arrangement ( figure [ fig : menorah]b ) , the size of a dendritic arbor , @xmath38 , is the inter - neuronal spacing @xmath57 times the number of inter - neuronal intervals covered by the arbor , @xmath58 : @xmath59 the number of dendritic arbors per unit length is equal to the density of output neurons @xmath28 . the combined dendritic arbor length per unit length is @xmath60 . since the axonal arbors do not contribute , the total wire length per unit length : @xmath61 by using eq.[cnd ] and recalling that @xmath2 i get eq.[li ] . in type ii arrangement ( figure [ fig : menorah]c ) , the wire length is equal to the sum of the lengths of axons converging on each output neuron multiplied by the neuronal density in the output layer @xmath28 : @xmath62 using eq.[cnd ] i express the result in terms of convergence alone @xmath63 and get eq.[lii ] . now consider the case of @xmath12 . by using eq.[cnd ] i find from eq.[lnn ] that @xmath64 this is eq.[dd ] of the main text . by using the symmetry in respect to the direction of signal propagation i find eq.[cc ] of the main text . next , i consider an arrangement with arbitrary sizes of axonal , @xmath16 , and dendritic , @xmath38 , arbors , figure [ fig : ar1 ] , in the limit of @xmath26 . to satisfy the wiring diagram each input neuron must connect with @xmath0 output neurons and each output neuron must connect with @xmath1 input neurons . this places a constraint on the sum of axonal and dendritic arbor widths : @xmath65 therefore , axonal arbor width can take values @xmath18 . the total wirelength per unit length is : @xmath66 using eq.[cnd],[ss ] i get eq.[cll ] of the main text . i consider the case of @xmath67 , fig.[fig : ar4 ] . the following calculation is valid to the leading order in @xmath0 and @xmath1 . i omit numerical factors of order one which depend on the precise geometry of axonal and dendritic arbors . the total length of a dendritic arbor , @xmath68 , is equal to the number of periods in the mesh @xmath69 times the mesh size , @xmath16 : @xmath70 the size of the dendritic arbor , @xmath38 follows from expressing convergence as the product of the area covered by the dendritic arbor times the density of input neurons @xmath71 : @xmath72 substituting this into eq.[sa ] i find : @xmath73 the length of an axonal arbor is approximately given by its size : @xmath74 then the total wirelength per unit area is : @xmath75 in order to find the optimal axonal arbor size @xmath16 , i differentiate wirelength in respect to @xmath16 and set the derivative to zero . @xmath76 solution of this equation gives the optimal size of an axonal arbor , @xmath16 : @xmath77 by using eq.[la ] i get the size of the dendritic arbor , @xmath78 the last two equations combined give eq.[res ] of the main text . i have benefited from helpful discussions with e.m . callaway , e.j . chichilnisky , h.j . karten , c.f . stevens and t.j . i am grateful to a.a . koulakov for making several valuable suggestions . i thank g.d . brown for suggesting that the size of axonal and dendritic arbors may be related to con-/divergence . this research was supported by the sloan foundation .

i consider a topographic projection between two neuronal layers with different densities of neurons . given the number of output neurons connected to each input neuron ( divergence ) and the number of input neurons synapsing on each output neuron ( convergence ) i determine the widths of axonal and dendritic arbors which minimize the total volume of axons and dendrites . analytical results for one - dimensional and two - dimensional projections can be summarized qualitatively in the following rule : neurons of the sparser layer should have arbors wider than those of the denser layer . this agrees with the anatomical data from retinal and cerebellar neurons whose morphology and connectivity are known . the rule may be used to infer connectivity of neurons from their morphology . 2

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Proceed to summarize the following text: over the last several decades algorithmic advances coupled with the development of high speed supercomputers , have made _ ab initio _ quantum mechanical simulations possible for a wide range of physical systems . these methods have been used to provide a theoretical framework for interpreting experimental results and even to accurately predict the material properties before experimental data were available . however , the calculations are currently restricted to systems containing a few hundred atoms.@xcite this limitation is set by the available computer power , and the scaling of the computational work with the number of atoms . one of the most successful of the recent techniques is the car - parrinello method@xcite in which the electronic orbitals are expanded in plane - wave basis functions and the resulting hamiltonian is iteratively diagonalized . the practical and efficient extension of _ ab initio _ quantum methods to larger and more difficult systems may be accomplished by the refinement and improvement of traditional methods or by the development of new techniques . although highly successful , traditional plane - wave methods encounter considerable difficulties when they are applied to physical systems with large length scales , or containing first - row or transition - metal atoms . these difficulties may be partially eliminated by the use of preconditioned conjugate - gradient techniques,@xcite optimized pseudopotentials,@xcite augmented - wave methods,@xcite or plane waves in adaptive coordinates.@xcite however , these methods are still constrained by the use of a plane - wave basis set , and the necessity of performing fast fourier transforms ( fft ) between the real and reciprocal spaces . while fft s may be implemented in a highly efficient manner on traditional vector supercomputers , the current trend in supercomputer design is massively parallel architectures . it is difficult to implement efficient fft algorithms on such machines , due to the required long range communications . real - space methods are inherently local and therefore do not lead to a large communication overhead . the scaling of several critical parts of large calculations is improved from @xmath1 in a plane - wave representation to @xmath2 , where @xmath3 is the number of atoms . furthermore , preconditioning and convergence acceleration are most effectively carried out in real space ( q.v . section iii ) . a real - space formulation is also required for efficient implementations of @xmath4 electronic structure methods , in which the computational work required scales linearly with the number of atoms . these methods impose a localization constraint on the electronic orbitals@xcite or the electron charge density,@xcite which eliminates the @xmath5 orthogonalization step . orbital - based real - space approaches , _ e.g. _ atom - centered or floating gaussians , are very well established . recently , however , there has been substantial interest in developing real - space orbital - independent methods , which permit systematic studies of convergence in the spirit of plane - wave methods . these methods include finite elements,@xcite grids , and wavelets.@xcite the finite - element method was applied by white _ @xcite to one - electron systems . they used both conjugate - gradient and multigrid acceleration@xcite to find the ground - state wavefunction . two of the present authors @xcite used a basis with a high density of grid points in the regions where the ions are located , and a lower density of points in the vacuum regions , in conjunction with multigrid acceleration , to calculate the electronic properties of atomic and diatomic systems . the core electrons were explicitly included and nearly singular pseudopotentials were used . the non - uniform grid led to order of magnitude savings in the basis size and total computational effort . the multigrid iterations , which provide automatic preconditioning on all length scales , reduced the number of iterations needed to converge the electronic wavefunctions by an order of magnitude in these multi - length - scale systems . et al._@xcite used a similar method to solve for the ground state of h@xmath6 . wavelet bases@xcite were used to solve the lda equations for atoms and the o@xmath7 molecule . et al._@xcite have used high - order finite - difference methods and soft nonlocal pseudopotentials on uniform grids to calculate the electronic structure , geometry , and short - time dynamics of small si clusters and of an isolated so@xmath7 molecule . et al._@xcite have used uniform grids and a smeared nuclear potential to examine the energetics and structures of atoms and small molecules . they employed multigrid iteration techniques to improve the convergence rates of the kohn - sham functional . real - space grids in curvilinear coordinates have been used by gygi and galli@xcite to compute the properties of atoms and co@xmath7 , and zumbach _ et al._@xcite tested it on o@xmath7 . et al._@xcite used a uniform grid approach with pseudopotentials and a conjugate gradients scheme to calculate the electronic structure of p@xmath7 , and to study a positron trapped by a cd vacancy in cdte . in a previous communication@xcite the present authors have outlined a multigrid - based approach suitable for large scale calculations , together with a number of test applications . these included calculations for a vacancy in a 64-atom diamond supercell , an _ isolated _ @xmath8 molecule using non - periodic boundary conditions , a highly elongated diamond supercell , and a 32-atom supercell of gan that included the ga @xmath9 electrons in valence . uniformly spaced grids were used ; in this case an effective `` cut - off '' may be defined , which is equal to that of the plane - wave calculation that uses the same real - space grid for the fft s . whenever feasible , the corresponding calculations were also carried out using plane - wave techniques , and the two sets of results were in excellent agreement with each other . this paper provides a comprehensive description of the real - space multigrid method , and reports extensions to non - uniform grids , non - cubic grids , and to molecular dynamics simulations with highly accurate forces . several computational issues absent from plane - wave and orbital - based methods arise when using a real - space grid approach . in the plane - wave basis the action of the kinetic energy operator on the basis functions can be computed exactly , and the wavefunctions , potentials , and the electron charge density can be trivially expanded in the basis . every basis - set integral , except those involving the lda exchange - correlation functional , can be computed exactly . the computational errors in the calculations are mainly due to the truncation of the basis . in a real - space grid implementation , the kohn - sham equations must be discretized explicitly , which presents important trade - offs between accuracy and computational efficiency . furthermore , the quantum - mechanical operators are known only at a discrete set of grid points , which can introduce a spurious systematic dependence of the kohn - sham eigenvalues , the total energy , and the ionic forces on the relative position of the atoms and the grid . we have developed a set of techniques that overcome these difficulties and have been used to compute accurate static and dynamical properties of large physical systems , while taking advantage of the rapid convergence rates afforded by multigrid methods . this paper is organized as follows : in section ii a method for the accurate and efficient real - space discretization of the kohn - sham equations for cubic , orthorhombic , and hexagonal symmetries is described . section iii focuses on the multigrid algorithms , which greatly accelerate convergence of the electronic wavefunctions and of the hartree potential . tests of the convergence acceleration are described in section iv . the calculation of ionic forces that are sufficiently accurate for large step molecular dynamics requires special methods , which are described in section v. section vi discusses performance issues for massively parallel supercomputers , and describes a highly scalable and efficient implementation on the cray - t3d , which has been tested on up to 512 processors . the summary in section vii is followed by several technical appendices . electronic structure calculations that use a real - space mesh to represent the wavefunctions , charge density , and ionic pseudopotentials must address a new set of technical difficulties when compared with plane - wave methods . in a plane wave representation the form of the kinetic energy operator is obvious . in contrast , the representation of the kinetic energy operator on a real - space grid is approximated by some type of finite differencing , the accuracy of which must be carefully tested . below , we describe real - space discretizations for uniform cubic , orthorhombic , and hexagonal grids , as well as nonuniform scaled cubic grids that increase the resolution locally . for uniform cubic grids we also describe and test the extension to periodic systems with arbitrary sampling of the brillioun zone . previously,@xcite we described a real - space approach that uses uniform cubic grids with @xmath10 point k - space sampling . in direct comparisons with plane - wave calculations , we found nearly perfect agreement between the two methods for several test systems . we now provide further details of our method . in real - space the wavefunctions , the electron charge density , and the potentials are directly represented on a uniform three - dimensional real - space grid of @xmath11 points with linear spacing @xmath12 . the physical coordinates of each point are @xmath13 the ions are described by norm - conserving pseudopotentials@xcite in the kleinman - bylander nonlocal form.@xcite these potentials are interpolated onto the grid from their radial representation . exchange and correlation effects are treated using the local density approximation ( lda ) of density functional theory , in which the total electronic energy of a system of electrons and ions may be expressed as @xmath14 - \mu_{xc}[\rho({\bf r } ) ] \ } \nonumber\\ & & -{1 \over 2 } \int \ , d{\bf r } \ , \rho({\bf r } ) \ , v_{hartree}({\bf r } ) + e_{ion - ion}. \label{lda - energy}\end{aligned}\ ] ] the minimization of this functional requires the solution of the kohn - sham equations @xmath15 = -{1 \over 2}\nabla^2 \psi_n + v_{eff } \psi_n = \epsilon_n \psi_n , \label{ksequations}\ ] ] subject to the orthonormality constraint on the eigenfunctions @xmath16 . the accurate discretization of these equations on the grid structure described by eq . ( [ grid1 ] ) requires appropriate numerical representations of the integral and differential operators . all integrations are performed using the three - dimensional trapezoidal rule : @xmath17 we have found that for high accuracy it is essential that the integrand @xmath18 be band - limited in the sense that its fourier transform should have minimal magnitude in the frequency range @xmath19 . this is explicit in a plane - wave calculation since the basis is cut - off at a specific plane - wave energy . the discrete real - space grid also provides a kinetic - energy cut - off of approximately @xmath20 . unlike the plane - wave basis , high - frequency components above this cut - off can nonetheless manifest themselves on the grid . this high - frequency behavior , which can introduce unphysical variation in the total energy or the electron charge density , is perhaps best seen when the ions , and hence their pseudopotentials , shift relative to the grid points.@xcite if the pseudopotentials contain significant high - frequency components near or above @xmath21 then , as the ions shift , the high frequency components are aliased to lower frequency components in an unpredictable manner . this effect can be decreased by _ explicitly _ eliminating the high - frequency components in the pseudopotentials by fourier filtering . in the context of plane - wave calculations , king - smith _ et al._@xcite recognized that the real - space integration of the nonlocal pseudopotentials could differ significantly from the exact result computed in momentum space , unless the potentials were modified so that fourier components near @xmath21 were removed . fourier filtering of the pseudopotentials is thus required in real - space calculations for accurate results . it is , of course , possible to use unfiltered potentials on real - space grids provided the grid spacing is sufficiently small , but our experience shows that the total energy and the electron charge density are often sufficiently well converged for significantly larger grid spacings provided that explicit pseudopotential filtering is used . we use a somewhat different fourier filtering method than that proposed by king - smith _ et al . _ , but it produces essentially the same effect ( see appendix b ) . the differential operator in the kohn - sham equations is approximated using a _ generalized _ eigenvalue form : @xmath22 = & & \nonumber\\ { 1 \over 2 } { \bf a}_{mehr}[\psi_n ] + { \bf b}_{mehr}[v_{eff } \psi_n ] = & & \epsilon_n { \bf b}_{mehr}[\psi_n ] , \label{compact - implicit}\end{aligned}\ ] ] where @xmath23 and @xmath24 are the components of the _ mehrstellen _ discretization,@xcite which is based on hermite s generalization of taylor s theorem . it uses a weighted sum of the wavefunction and potential values to improve the accuracy of the discretization of the _ entire _ differential equation , not just the kinetic energy operator . in contrast to the central finite - differencing method , this discretization uses more _ local _ information ( next - nearest neighbors , for example ) . the definition of the fourth - order _ mehrstellen _ operator used in the present work is specified by the weights listed in table [ mehrstellen - ortho ] , which pertain to both cubic and orthorhombic grids ( see below ) . a more detailed analysis of the _ mehrstellen _ operator and eq . ( [ compact - implicit ] ) is given in appendix a. this representation of the kohn - sham hamiltonian is short - ranged in real space in the sense that the operator can be applied to any orbital in @xmath25 operations . specifically , the application of the @xmath23 operator at a point involves a sum over 19 orbital values while the application of the @xmath24 operator requires a sum over 7 points . the local potential multiplies the orbital pointwise , and the short - ranged nonlocal projectors require one integration over a fixed volume around each ion and a pointwise multiplication . this sparseness permits the use of iterative diagonalization techniques , and the short - ranged representation of the hamiltonian leads to an efficient implementation on massively parallel computers . the discussion up to this point has been restricted to uniform cubic grids , but the extension to a general orthorhombic grid is straightforward . there are now three separate grid spacings @xmath26 , @xmath27 , and @xmath28 with the coordinates of each grid point given by : @xmath29 the orthorhombic _ mehrstellen _ operator described in table [ mehrstellen - ortho ] is used to discretize the kohn - sham equations and numerical integration is performed according to @xmath30 . \label{integration1}\ ] ] in the preceding section the wavefunctions were assumed to be real , with the brillioun - zone sampling restricted to the @xmath10 point . when these restrictions are lifted the kohn - sham equations become @xmath31 where the @xmath32 are now the periodic parts of the bloch functions . they are complex - valued , which presents no additional difficulties in discretization . the nonlocal projectors in @xmath33 have been multiplied by the phase factor @xmath34 . the gradient term @xmath35 is computed using a central finite difference expression , which in one dimension has the form : @xmath36 where @xmath37 , @xmath38 , @xmath39 , and @xmath40 . for a cubic grid structure the three dimensional generalization of this is the sum of the individual expressions for each coordinate axis . denoting this finite - difference operator by @xmath41 , the discretization of the kohn - sham equation becomes @xmath42 & & = { 1 \over 2 } { \bf a}_{mehr}[\psi_{nk } ] + \nonumber\\ & & { \bf b}_{mehr}[i{\bf k } \cdot \tilde { \bf \nabla } \psi_{nk } + { 1 \over 2 } { \bf k}^2\psi_{nk } + v_{eff}({\bf k } ) \psi_{nk } ] \nonumber\\ & & = \epsilon_n { \bf b}_{mehr}[\psi_{nk } ] , \label{complexcompact - implicit}\end{aligned}\ ] ] where @xmath23 and @xmath24 are again the components of the _ mehrstellen _ operator . the accuracy of the discretization was tested by calculating the lattice constant and bulk modulus for an 8-atom si supercell . a 26-ry equivalent cut - off was used with k - space sampling restricted to the baldereschi point.@xcite the calculated lattice constant was 5.38 with a bulk modulus of 0.922 mbar . these are in good agreement with the corresponding plane - wave calculation with the cut - off of 26 ry : 5.39 and 0.960 mbar , respectively.@xcite bulk aluminum was selected as an additional test case . a 4-atom cell was used with a 23 ry equivalent cut - off and k - space sampling of 35 special points in the irreducible part of the brillouin zone . since al has partially occupied orbitals , the bands near the fermi level were occupied using a fermi - dirac broadening function of width 0.1 ev . the convergence of the total energy with respect to energy cut - off and the number of k - points was tested by increasing the cut - off from 23 to 49 ry , which produced a change of only 10 mev / atom , and by increasing the number of k - points to 56 , which changed the total energy by 3 mev / atom . the calculated lattice constant and bulk modulus of 4.02 and 0.734 mbar are in excellent agreement with the experimental values of 4.02 and 0.722 mbar , and with previous theoretical results of lam and cohen@xcite of 4.01 and 0.715 mbar . unlike plane - wave methods , where different symmetry groups can be handled easily , an _ efficient _ real - space implementation for periodic systems with non - orthogonal lattice translation vectors requires considerable modifications to the orthorhombic - symmetry implementation . the hexagonal grid describing the unit cell or supercell is generated by @xmath43 where the @xmath44 are the hexagonal bravais - lattice vectors.@xcite the @xmath45 ratio can be chosen arbitrarily by varying the two independent grid spacings , @xmath46 and @xmath28 , and the number of grid points , @xmath47 and @xmath48 . however , one should use @xmath49 in order to maintain similar resolution both in the _ xy _ plane and along the _ z_-axis . because the indexing of this hexagonal grid is isomorphic to the cubic one , the computer representation of potentials and wavefunctions does not change . the most important difference is in the discretization of the kohn - sham equations . we have implemented a mixed sixth - order kinetic energy operator . this discretization is described in appendix c , as well as the modifications required in the multigrid restriction and interpolation procedures ( q.v . section iii ) . the above implementation has been tested on a 32-atom supercell of aln in the wurtzite phase . the accuracy of the results was confirmed by comparison with plane - wave car - parrinello calculations on the same supercell . generalized norm - conserving@xcite al and n pseudopotentials were used for both calculations with k - space sampling restricted to the @xmath10 point . the cohesive energy from the real - space calculation was 11.5 ev per aln unit , which compares well with the value of 11.6 ev obtained in the car - parrinello calculations . the eigenvalue degeneracies were identical in both calculations and the maximum difference in any eigenvalue was 0.04 ev . the extension of the real - space grid representation to other bravais lattices proceeds in a similar manner , the only requirement being the existence of an accurate finite difference discretization . in real - space calculations it is possible to add resolution locally . this is especially valuable for systems with a wide range of length scales such as surface or cluster calculations . a high density of grid points can be used near the ions , with a low density in the vacuum regions . other possible applications are simulations of impurities in bulk materials , where the impurity ions may require higher resolutions to be accurately represented . by using locally enhanced regions the required resolution may be added only where needed , thereby greatly reducing the total number of points required . local enhancement of the grid resolution may be achieved by adding small high - resolution grids@xcite onto a uniform global grid , or by using a coordinate transform to warp the grid structure . our focus here will be on the second approach , which was first proposed by gygi@xcite for plane - wave basis sets and recently extended to real space by several workers.@xcite in the real - space approach , a continuous coordinate transform is applied to a uniform grid . in general the transformation is non - separable , but we prefer a separable coordinate transformation in order to avoid mixed derivatives in the kinetic - energy operator . as a test of the utility and accuracy of this scaled - grid approach , we examined an interstitial oxygen impurity in si@xcite for two grid layouts : a dense uniform grid with a 76-ry cut - off and a scaled grid with a cut - off that varied from 18 ry to 76 ry . the scaling - transformation that maps the fictitious computational grid @xmath50 to the physical warped grid @xmath51 is @xmath52 where @xmath53 is the @xmath50-coordinate of the oxygen atom , @xmath54 is the length of the supercell in the @xmath50-direction , and @xmath55 is an adjustable parameter between zero and one that controls the degree of scaling . the @xmath56 and @xmath57 coordinates are scaled analogously . the scaled grid required four times fewer points than the uniform grid to achieve the same convergence of the total energy . since the coordinate transforms are continuous functions , the integration weights and the coefficients of the discretized kinetic - energy operator may be generated from the uniform grid values using the metric tensor of the transform in the manner outlined by gygi.@xcite with these modifications , the calculations proceed as for the uniform grids , but a sixth - order central finite - difference operator is used to discretize the first and second derivatives because we have not constructed a _ operator for the scaled grid . in each case ( uniform and scaled grids ) , a 64-atom supercell was used and the silicon atoms were fully relaxed . the oxygen atom was held fixed in order to avoid pulay corrections@xcite to the ionic forces , which would have been required if it and the scaled grid were free to move . the uniform and scaled - grid calculations are in very good agreement : the maximum difference in kohn - sham eigenvalues was 40 mev and the maximum difference in ionic coordinates was 0.03 . to efficiently solve eq . ( [ compact - implicit ] ) , we have used multigrid - iteration techniques that accelerate convergence by employing a sequence of grids of varying resolutions . the solution is obtained on a grid fine enough to accurately represent the pseudopotentials and the electronic wavefunctions . if the solution error is expanded in a fourier series , it may be shown that iterations on any given grid level will quickly reduce the components of the error with wavelengths comparable to the grid spacing but are ineffective in reducing the components with wavelengths large relative to the grid spacing.@xcite the solution is to treat the lower frequency components on a sequence of auxiliary grids with progressively larger grid spacings , where the remaining errors appear as high frequency components . this procedure provides excellent preconditioning for all length scales present in a system and leads to very rapid convergence rates . the operation count to converge one wavefunction with a fixed potential is @xmath25 , compared to @xmath58 for fft - based approaches.@xcite there is no one multigrid algorithm but rather a collection of algorithms that share certain common features . in order to describe the implementation used in this work , we start with a description of a multigrid solver for poisson s equation . this will then be used as a building block for the more sophisticated algorithms actually employed . a standard numerical problem that illustrates the multigrid algorithm is the poisson equation , @xmath59 , defined on a rectangular cell of dimensions @xmath60 with periodic boundary conditions . a standard method of solving for @xmath61 is to discretize the equation on a uniform three dimensional grid with spacing @xmath12 and @xmath11 total points . the differential operator @xmath62 is represented by some form of finite differencing . this produces a set of linear algebraic equations @xmath63 where @xmath64 and @xmath65 are the discretized forms of @xmath66 and @xmath67 , respectively , and @xmath68 is the finite - difference representation of @xmath62 . if the system is small , direct matrix methods are an acceptable means of solving the equations ; however , for large systems the work required scales as @xmath69 , which is prohibitive . an alternative approach is to use an _ iterative _ relaxation scheme such as the jacobi method.@xcite in this technique , the solution is iteratively improved . first , define @xmath70 as an approximate solution of eq . ( [ mg1 ] ) , and the residual @xmath71 , a measure of the solution error , as @xmath72 an improved @xmath70 is generated using @xmath73 where @xmath74 is a pseudo time step and k is the inverse of the diagonal component of @xmath68 . this approach will always converge to a solution for some value of @xmath74 if @xmath68 is diagonally dominant.@xcite however , the number of iterations required to reduce the magnitude of the residual to a specified accuracy is proportional to @xmath75 , so that the algorithmic cost to converge , @xmath76 , is too great.@xcite while more sophisticated relaxation methods such as gauss - seidel , successive overrelaxation ( sor ) , or the alternating direction implicit method ( adi)@xcite have improved convergence rates , they still scale as @xmath77 with @xmath78 , and are too slow for the grid sizes required in electronic structure calculations . the slow convergence of the jacobi method can be qualitatively understood by noting that because @xmath68 is a short - ranged operator , the updated approximate solution , eq . ( [ res2 ] ) , is a linear combination of nearby values . if the error in the current estimate of @xmath70 is decomposed into fourier components , it can be shown that one jacobi iteration considerably reduces the high frequency components of the error , but many jacobi iterations are needed to affect the longest wavelength components of the error . the overall convergence rate is then limited by that of the lowest frequency components . as the problem becomes larger , the lowest frequency representable on the grid becomes smaller and the convergence rate decreases . the essence of the multigrid approach is the observation that the individual frequency components of the error are best reduced on a grid where the resolution is of the same order of magnitude as the wavelength of the error component . this approach will maintain a high convergence rate for all frequency components of the error even when the problem size ( and the grid ) becomes very large . we first describe a multigrid algorithm to solve poisson s equation that uses two grids ; a fine grid of spacing @xmath79 and a coarser auxiliary one of spacing @xmath80 . in this work , we use a coarse - to - fine grid ratio @xmath81 of 2 , but other ratios are possible . the solution is generated as follows : the high - frequency components of the solution error with wavelength @xmath82 are reduced by one or two jacobi iterations . the residual @xmath83 , which should be devoid of high - frequency variation , is computed and transferred to the coarse grid by restriction ( see below ) . next , poisson s equation on the coarse grid with the residual as a source term is solved by using the same iteration procedure as in eq . ( [ res2 ] ) , but with an initial estimate of zero . the jacobi iteration on this level removes error components with wavelength @xmath84 . finally , the coarse grid solution is interpolated to the fine grid and added to the fine grid solution . this process is referred to as a coarse - grid - correction scheme ( cgc ) . a few applications of the cgc cycle are generally sufficient to solve poisson s equation to machine precision even for extremely large systems . an obvious question is how the solution is obtained on the coarse grid . if the total number of grid points in the coarse grid is small , a direct matrix method will be sufficient . if this number is so large that a direct method is impractical , then a second , coarser grid level is introduced and the two - grid algorithm is repeated in a recursive manner . when multiple grid levels are used , the pattern of cycling through the grids also needs to be considered . we use a simple progression from the finest to the coarsest grid level and then back to the finest level , which is referred to as a v - cycle . more complicated cycling schemes exist , but we have found that the v - cycle works as well in electronic structure calculations as the more sophisticated approaches . another consequence of the multigrid approach is the reduction in the size of the grid ( and consequently in the work required ) on each level . for a uniform grid in three dimensions a doubling of @xmath12 with each level leads to a factor of eight reduction in @xmath11 , so that the addition of extra coarse grid levels is computationally inexpensive . the simplest choice for the restriction operator is to copy every other point in the fine grid directly to the coarse grid . this so - called straight injection , while easy to implement , does not always yield good convergence rates . a better choice is a weighted restriction , in which each coarse - grid value is the average of the 27 fine - grid values surrounding it . in our work , the weight assigned to each fine - grid point is proportional to the volume it occupies at a given coarse - grid point . a good choice for interpolating from the coarse grid to the fine grid is the adjoint of the restriction operator , which in this case becomes simple tri - linear interpolation.@xcite the final accuracy of the solution is determined by the finite - difference representation of @xmath62 on the finest grid level . it is neither necessary nor desirable to use the same representation of @xmath62 on all grid levels ; i.e. , @xmath85 may differ from @xmath86 in the form of the discretization as well as in the grid spacing . the technique of changing discretization on different levels is referred to as deferred defect correction ( ddc ) or double discretization.@xcite it is especially valuable for problems where the operator used on the finest grid level is numerically unstable on the coarser grids , or when it is inconvenient to apply . the accuracy of the fine - grid solution does not depend on the choice of this coarse - grid operator , which effects only the convergence rate . in solving poisson s equation , we use the _ mehrstellen _ operator on the finest grid level for high accuracy . however , it is unsuitable for convergence acceleration on the coarser grids because of stability problems . thus , on the coarser grids a 7-point central finite - difference operator is used , which provides excellent stability and rapid high - frequency attenuation . the extension of multigrid concepts to the solution of the kohn - sham equations introduces several complications . first , the equations are non - linear since the eigenvalues and the orbitals must be computed simultaneously . second , when the kleinman - bylander form of the nonlocal pseudopotentials is used , the equations become a set of integro - differential eigenvalue equations . finally , the hamiltonian depends upon the density , and must be solved self - consistently . brandt _ et al._@xcite have given a multigrid algorithm for the standard eigenvalue problem . below we describe an alternative means of linearizing the eigenvalue equations . the multigrid technique recommended in the literature for non - linear integro - differential equations is the full approximation storage ( fas ) method.@xcite in fas the _ entire _ problem is discretized and solved on _ all _ grid levels . in contrast , the cgc method outlined above generates the full solution only on the finest level . while the theoretical performance of fas on this problem is superior , its implementation is significantly more complex . in addition , it is difficult to obtain an accurate representation of the nonlocal pseudopotentials on the coarser grid levels . furthermore , the kohn - sham equations need not be converged to maximum accuracy at every iteration , because the electron charge density ( and therefore the hamiltonian ) change after each multigrid step . for these reasons , a modification of the double discretization approach described above was used . the discretized operator on the finest grid is the _ mehrstellen _ approximation of the kohn - sham hamiltonian , while a 7-point central finite - difference representation of @xmath62 alone is used on the coarse grids . effectively , the coarse - grid equation for the wavefunction residual becomes poisson s equation , where the source term is the residual generated by the _ mehrstellen _ operator on the fine grid , eq.([mehrres ] ) . our multigrid procedure begins with the selection of some initial wavefunctions and electron charge density . we postpone discussion of initialization techniques until later and assume that an adequate start has been generated . the following steps are then performed for each individual wavefunction : first , an estimate of the eigenvalue is calculated from the rayleigh quotient of the generalized eigenvalue equation , eq.([compact - implicit ] ) : @xmath87\rangle \over \langle\psi_n| { \bf b}_{mehr } [ \psi_n]\rangle } . \label{rd1a}\ ] ] in the case of complex orbitals , the eigenvalue is calculated using the rayleigh quotient of the real ( or imaginary ) part of the generalized eigenvalue equation , eq . ( [ complexcompact - implicit ] ) : @xmath88| re[{\bf h}_{bloch } [ \psi_{nk}]]\rangle \over \langle re[\psi_{nk}]| re[{\bf b}_{mehr } [ \psi_{nk}]]\rangle } . \label{rd1b}\ ] ] next , several jacobi iterations are applied to the orbital on the finest grid using eqs . ( [ res1 ] ) and ( [ res2 ] ) , where the residual is computed as @xmath89 - { \bf h}_{mehr}[\psi_n ] . \label{mehrres}\ ] ] the fictitious time step @xmath90 used in the jacobi iteration is typically chosen between @xmath91 and @xmath92 a.u . in the case of complex orbitals , the real and imaginary components of the orbital are updated separately , using the appropriate generalization of eq . ( [ mehrres ] ) . next , the residual is restricted to the first coarse grid . a ddc coarse - grid cycle begins using the 7-point central finite - difference representation of @xmath62 instead of @xmath93 . several auxiliary coarse grids can be used . when the coarse - grid correction is interpolated onto the finest grid , only a fraction @xmath94 of it is added to the orbital , for reasons of stability . a value of @xmath95 has been found to work for almost all systems . ( larger values may produce much higher convergence rates on some systems while being unstable for others , so some experimentation is necessary . ) before transferring the residual to the coarse grid , it is essential that enough jacobi iterations be performed to eliminate the high frequency components from the residual . since the residual is used as the right hand side of a cgc correction cycle , any high frequency components will eventually be transferred to a coarser grid where they can not be represented correctly , greatly reducing the effectiveness of the multigrid cycle . in some cases they may even make the process numerically unstable . in the above approach , the difficulties of discretizing the nonlocal pseudopotentials on the coarse grid levels are avoided because the potential term is computed on the finest grid and frozen thereafter . the steps outlined above in the ddc apply only to a single wavefunction . the full solution process also requires the application of the orthonormality constraints and an update of the electron charge density . the full solution process ( one scf step ) consists of the following cycle . first , the ddc is applied to all of the wavefunctions . next , the orthonormality constraints are applied using the gram - schmidt procedure : @xmath96 the new electron charge density is generated by linear mixing : @xmath97 where @xmath98 is the occupation of the @xmath99 state and @xmath100 is a mixing parameter , generally set to a value between 0.5 and 0.9 . next , the hartree potential is recomputed for the new charge density using a _ ddc cycle , and a new exchange - correlation potential is generated . finally , a subspace diagonalization may be performed at this point . this need only be done occasionally ( every 10 - 20 scf steps is generally adequate ) in order to unmix eigenstates that may be close in energy . because the _ mehrstellen _ hamiltonian leads to a non - hermitian generalized eigenvalue equation ( see appendix a ) , subspace diagonalization requires a brief discussion : we look for a unitary transformation of the current wavefunctions that better represents the eigenvectors of the hamiltonian , and are led to the following eigenvalue equation for the subspace : @xmath101 where @xmath102\rangle \\ b^{sub}_{m , n } = & & \langle\psi_m| { \bf b}_{mehr } [ \psi_n]\rangle,\end{aligned}\ ] ] and @xmath103 is the matrix of coefficients of the unitary transformation for the @xmath104 state . because @xmath105 is invertible ( see appendix a ) , the subspace equations are equivalent to @xmath106 where @xmath107 . the matrix @xmath108 is not hermitian except when the subspace is a subset of the space of eigenvectors . thus , we do _ not _ diagonalize @xmath108 because its eigenvectors are not necessarily orthogonal , which would spoil the orthogonality of the electronic orbitals . instead , we discard the anti - hermitian part of @xmath108 , which is smaller than the hermitian part of @xmath108 by @xmath109 , and diagonalize the hermitian part . this approximation works well in practice , and is exact at convergence . the hermitian approximation does not affect the final accuracy of the solution because the multigrid - assisted jacobi iterations ultimately converge the orbitals . nonetheless , accurate subspace rotations are essential for good convergence : compare the convergence rates in figs.[siliconconvergence]-[diamondconvergence ] . as a test , we compared subspace diagonalizations with the _ mehrstellen _ and the hermitian sixth - order discretizations , and found that the convergence is significantly improved with the former . the cycle described above is repeated until the electronic system converges to the desired tolerance , which may be monitored by computing the rms value of the residual vector for each wavefunction ( see eq . ( [ mehrres ] ) ) . when this reaches a value of @xmath110 a.u . for all wavefunctions in the occupied subspace , the convergence is sufficient for the computation of forces that are accurate enough for large step molecular dynamics with excellent energy conservation . as was mentioned previously , the convergence rates depend on the choice of the initial wavefunctions and electron charge density . a poor choice can lead to slow convergence rates or in some cases the system will not converge at all . apart from random initial wavefunctions or an approximate solution that is generated using an lcao basis set , one can also use a double - grid scheme . in the latter method the initial solutions are generated on a grid with a spacing twice as large as that used for the final grid . the computational work on this coarse grid is eight times smaller than what is needed on the fine grid . the approximate coarse - grids wavefunctions are then interpolated to the fine grid and used as the initial guess . this process can reduce the number of scf cycles needed on the finest grid level by a factor of two to three , thereby achieving significant savings in the computational effort . the theoretical convergence rates of multigrid methods may , in principle , be calculated exactly for certain types of problems . for well - behaved partial differential equations such as poisson s equation discretized on @xmath11 points , @xmath25 total operations are required to obtain a solution accurate to the the grid - truncation error . this compares well with fft based methods which require @xmath111 $ ] operations . for the kohn - sham equations , an exact theoretical bound on multigrid convergence rates is difficult to obtain due to self - consistency effects , and to the best of our knowledge this analysis does not yet exist . we have therefore elected to study convergence properties in an empirical fashion by performing tests on physical systems typical of the problems normally examined with density functional theory . in previous work@xcite the present authors examined convergence rates for 8-atom supercells of perfect diamond as a function of the effective kinetic - energy cut - off determined by the grid resolution , for a 32-atom supercell of gan that included the ga @xmath9 electrons in valence , and for a highly elongated 96-atom diamond supercell . it was found that multigrid convergence rates were largely independent of energy cut - off and cell geometries . while promising , these results were obtained for perfect crystal configurations of semiconductor compounds , which are generally fairly easy to converge . in this article we present the results of a more systematic study that includes disordered systems . the first system selected was a 64-atom supercell of bulk silicon . the ions were represented by a generalized norm - conserving pseudopotential@xcite and the grid spacing used corresponded to an energy cut - off of 12 ry . the ionic positions were generated by a molecular dynamics simulation at a temperature of 1000 k. because the work required to converge to the ground state depends on the quality of the initial wavefunctions and charge density , we used random initial wavefunctions and a constant initial electron charge density to minimize any possible bias from the choice of a starting configuration . a small number ( 10% of the total ) of conduction band states was included in the calculations . fig.[siliconconvergence ] shows the convergence rate defined as the @xmath112 , plotted as a function of iteration number , where each iteration represents a single scf step . results are shown for calculations performed with and without multigrid acceleration , where the latter used a steepest - descents algorithm . in addition , the two calculations were repeated with , and without subspace diagonalizations of the orbitals . for the calculations that included subspace diagonalizations , the procedure was applied every 8 scf steps , which led to small discontinuities in the smooth evolution of the total energy . the results show that maximum convergence rates are obtained when multigrid iterations are combined with subspace diagonalization . the slowest convergence occurs for steepest descents with no subspace diagonalization . for the two runs where subspace diagonalizations were performed , the multigrid run converged at roughly 2.5 times the rate of the steepest descents approach . while these results are encouraging , bulk silicon is a relatively straight - forward test and is well handled by standard plane - wave methods . as an example of a more difficult system we have considered a 64-atom diamond supercell with a substitutional nitrogen impurity . standard pseudopotentials@xcite were used for both c and n. the strong n @xmath113 potential required an energy cut - off of 63 ry . the presence of a localized nitrogen donor level together with the 63 ry cut - off makes the system more difficult to converge . random initial wavefunctions were used , and fig . [ diamondconvergence ] shows the observed convergence rates . the convergence rates for the two runs that use subspace diagonalization are a factor of 4 better for multigrid than for steepest descents . this relative improvement is considerably greater than that observed for the silicon cell and is the consequence of the automatic preconditioning provided by multigrid techniques for all of the length and energy scales present in the problem . the multigrid convergence rates are largely independent of the grid spacing , which roughly corresponds to the kinetic energy cut - off in plane - wave approaches . this is not true of the steepest descents algorithm , where the maximum stable time step that may be used decreases as the energy cut - off increases . when comparing the convergence rates of the multigrid and steepest descents approaches , the computational workload involved in each technique must also be considered . a particular advantage of multigrid methods , when compared to other convergence acceleration schemes , is their low computational cost . this is due to the factor of 8 reduction in the number of grid points on each successive multigrid level . the computational time per scf step in the silicon and diamond runs described above increased by less than 10% when multigrid was used instead of steepest descents . for bigger systems , where the costs of orthogonalizing the orbitals and applying the nonlocal pseudopotentials begin to dominate the total computational time , the extra work needed for the multigrid accelerations becomes negligible . in terms of computational time , the 64-atom si supercell described above required 1.6 seconds per scf step on 64 processors of a cray - t3d . efficient structure optimizations and the calculation of dynamical quantities such as phonon frequencies and thermodynamic properties require accurate ionic forces . in plane - wave methods the ionic forces are computed by applying the hellmann - feynman theorem.@xcite since the derivative of the pseudopotentials may be expressed exactly within the plane - wave basis , the accuracy of the ionic forces is limited only by machine precision and the degree of convergence to the born - oppenheimer surface . for the grid - based approach the accuracy of hellmann - feynman forces is limited by the numerical error in computing the integrals of the derivatives of the pseudopotentials . this error decreases with grid spacing . the differentiation of the radial potentials and projectors must be performed with care to include the effects of the fourier filtering . alternatively , a derivative - free implementation of the hellmann - feynman forces can be used , which we term virtual displacements . in this scheme , the ionic pseudopotentials are numerically differentiated directly on the real - space grid . the ions are moved through a set of small displacements , while the electron charge density and the wavefunctions are held fixed . the potential energy is calculated for each displacement and finite - differenced to form the derivative . the forces computed by the two methods agree well . most of the forces we have calculated to date have been computed using virtual displacements . a stringent test of the accuracy of the ionic forces is a constant - energy molecular dynamics simulation . over the course of the simulation any systematic errors in the forces will manifest themselves as poor energy conservation . a distinction has to be made between small random errors that appear as bounded oscillations in the total energy and errors that increase in magnitude with simulation time . the small random errors are expected in the real - space approach because the energy of an ion varies by a small amount as its position changes relative to the grid points.@xcite this is of no particular concern as long as the magnitude of the variation is small and oscillatory in nature . of greater concern are errors that are unbounded . these could arise from errors in the forces , errors in integrating the equations of motion of the ions , and lack of self - consistency due to inadequate convergence of the electronic wavefunctions . the first source of error was minimized by fourier filtering of the ionic pseudopotentials . the second is generally not a problem unless the ionic time step is too large . for small time steps even a simple integrator such as the verlet algorithm is sufficient , and larger time steps may be handled by using higher order integrators , such as the beeman - verlet method.@xcite the last source of error is the most significant because hellmann - feynman forces are only accurate to first order in the error of the wavefunctions . a high degree of self - consistency is thus necessary to obtain good energy conservation . a 64-atom silicon supercell was selected to test energy conservation on a typical system . the ions were given random initial displacements from the perfect crystal configuration , and several velocity rescaling steps were performed in order to attain an average ionic temperature of 1100 k. a constant - energy molecular - dynamics simulation over 1 ps was then carried out , using 80 a.u . time steps and third - order beeman - verlet@xcite integration of the ionic equations of motion . the potential , kinetic , and total energy of the system vs. simulation time are plotted in fig.[simd ] . we observed good energy conservation : the maximum variation in the total energy was 1.75 mev , which corresponds to 27 @xmath0ev per atom . the performance of a given algorithm when solving complicated problems depends not only on the theoretical efficiency , which may be quite high , but also on how adaptable the algorithm is to modern computer architectures . one example are certain classical molecular dynamics algorithms , which perform only slightly better on vector supercomputers than on low cost engineering workstations , even though the supercomputer s theoretical peak performance may be an order of magnitude larger . a particular strength of the car - parrinello method has been its efficient implementation on vector supercomputers , such as the cray - ymp . however , vector performance , while improving steadily , is unlikely to increase by several orders of magnitude per decade as has occurred in the past . at the same time , the development of powerful , low cost microprocessors and memory , has led to massively parallel architectures consisting of a large number of microprocessors , linked by a high speed communication network . although efficient implementations of plane - wave - based methods on massively parallel architectures exist , the fft - based algorithms do not scale well with the number of processors because the fft is a global operation . below , we will describe a massively parallel implementation of the multigrid method . although some of the code - optimization issues are architecture - specific , most are generic and thus applicable to any massively parallel computation . the target machine is the cray - t3d , which uses up to 2048 dec - alpha microprocessors , each with a peak performance of 150 mflops . each processor has 8 kb direct - mapped data and instruction caches and 8 mw of local memory . the processors are linked together in a three - dimensional torus arrangement for data communication . three issues have to be addressed in order to write an efficient code for this type of machine : minimizing communication costs between processors , balancing the work - load on each processor , and code optimization on the individual processors . the majority of the data storage in the multigrid method consists of the wavefunction values on the real - space grid . we will consider the case where the points are distributed on a uniform three - dimensional rectangular grid . if @xmath114 is the total number of wavefunctions , then @xmath115 total storage is required . the simplest possible decomposition of data is to store complete wavefunctions on each processing element ( pe ) , where each pe stores @xmath116 orbitals . while conceptually simple , this approach will perform poorly for large systems with many wavefunctions , because orthogonalizing wavefunctions residing on different pe s requires sending large amounts of data between processors . an alternative approach , and the one adopted by us , is to use real - space data decomposition . in this method , each pe is mapped to a specific region of space . the electron charge density , hartree potential , and each wavefunction are distributed by regions over the processors . with this approach interprocessor communication is restricted to two areas : the computation of integrals on the real space grid ( see eq.([integration ] ) ) , and the application of the finite - differencing operators . for integration , the ideal optimization strategy is to organize the calculation so that as many integrals as possible are computed at once . this can be understood by considering the time required for interprocessor communication , which consists of a latency period and a transfer phase . the latency period is significant and is the same whether 1 or 1000 words of data is transferred . our integration procedure is as follows : first , calculate the intra - processor contributions to the integral ( i.e. , integrate over the subdomains ) ; second , store as many of these local integrals as possible ; finally , transfer them between processors in blocks and complete the integration by summing the local integrals . it was straightforward to implement the above procedure in most cases , but the orthogonalization step required significant modifications . in a standard implementation of the gram - schmidt orthogonalization algorithm , wavefunction overlaps and updates are computed sequentially , and the integrals can not be computed in parallel . to reduce the number of data transfers , the following implementation of gram - schmidt orthogonalization was adopted . first , the overlap matrix @xmath117 is computed as above : the local parts of the overlap integrals are computed and stored on each processor ; the integration is then completed by transferring them in blocks to the other processors . second , the cholesky factorization@xcite of the overlap matrix is computed : @xmath118 . the cholesky factor , @xmath119 , is relevant because its components are the overlaps between the new orthogonal wavefunctions and the original ones.@xcite finally , the diagonal components of the cholesky factor are used to normalize the wavefunctions , and the off - diagonal ones are used to complete the orthogonalization : @xmath120 for simplicity , the cholesky factorization of @xmath121 is currently performed on each processor . the computational time to factorize scales as @xmath122 , but has not yet become a bottleneck . however , for very large systems ( greater than 800 orbitals ) , a parallelized cholesky factorization will save significant computer time and memory . the second area where interprocessor communication is required is the finite differencing of the wavefunctions and hartree potential , since finite differencing is nonlocal . however , in the _ mehrstellen _ discretization , the nonlocality is restricted to points within one grid unit in each cartesian direction . interprocessor communication is thus always limited to nearest neighbor pe s regardless of the size of the system . this low communication cost is a particular advantage of a _ mehrstellen _ type approach as opposed to a central finite - difference approach , which requires a higher degree of nonlocality to achieve the same level of accuracy . load balancing and the efficient use of all pe s is a major concern for any parallel algorithm . with the method described above , the load balancing is essentially perfect for all parts of the calculation except for the application of the nonlocal pseudopotentials . these are applied to the wavefunctions in localized volumes around each ion . if the distribution of ions in space is nonuniform , then some of the pe s will be idle for a fraction of each scf step . however , actual calculations on many systems have shown that the application of the nonlocal potentials typically requires less than 10% of the total computational time on any pe , so that processor utilization will always exceed 90% . the efficiency of the massively parallel implementation described here is illustrated in fig . [ speedup ] , which shows the speedup in execution time per step for a given problem as the number of pe s is increased . the graph indicates a superlinear relationship , which is an artifact due to single processor cache effects . there are two competing factors here . the first is the increased communication cost as the number of processors increases , which tends to reduce the speedup . the second is the reduction in the amount of data stored on each processor and a consequent increase in the number of cache hits . as was discussed earlier , the communication costs are relatively small with the data model being used ; since the cache is relatively small , cache hit effects outweigh these . an apparent superlinear speedup is observed . we have described the development of a multigrid - based method that uses a real - space grid as a basis . the multigrid techniques provide preconditioning and convergence acceleration at all length scales , and therefore lead to particularly efficient algorithms . a specific implementation of multigrid methodology in the context of density functional theory was described and illustrated with several applications . the salient points of our implementation include : ( i ) the development of new compact discretization schemes in real space for systems with cubic , orthorhombic , and hexagonal symmetry , and ( ii ) the development of new multilevel algorithms for the iterative solution of kohn - sham and poisson equations . the accuracy of the discretizations was tested by direct comparison with plane - wave calculations when possible and were found to be in excellent agreement in all cases . these algorithms are very suitable for use on massively parallel computers and in @xmath4 methods . we described an implementation on the cray - t3d massively parallel computer that led to a linear speedup in the calculations with the number of processors . the above methodology was tested on a large number of systems . a prior communication described tests on c@xmath123 molecule , and diamond and gan supercells . the present article examined convergence properties in detail for a supercell of disordered si and the n impurity in diamond . the multigrid techniques increased the convergence rates by factors of 2 and 4 , respectively , when compared to the steepest descents algorithm . an extension to non - uniform grids that uses a separable coordinate transform to change grid resolution locally , e.g. , at the surface or near an impurity , was developed and tested on the o interstitial in si . this extension results in only minor changes in methodology and coding , while the reduction in basis set size and thus in computational effort can be significant . a complex version of multigrid code , capable of an arbitrary sampling of the brillouin zone , was also was developed and tested on bulk al . large time - step molecular dynamics simulations require very accurate forces , which can potentially lead to difficulties in real space methods as the atoms move relative to the grid points . we have described a set of techniques based on fourier filtering of pseudopotentials that eliminate these difficulties for grid spacings of sizes similar to those used in plane - wave calculations . a 1 ps test simulation of bulk si at 1100 k conserved the total energy to within 27 @xmath0ev per atom , and illustrated the high quality of these forces . further applications of this methodology are in progress , including a simulation of surface melting of si,@xcite structural properties of large biomolecules that contain over 400 atoms,@xcite and electronic and structural properties of in@xmath124ga@xmath125n quantum wells.@xcite the multigrid methodology is also very suitable for @xmath4 implementations , and tests results for a 216-atom cell of bulk si were described recently.@xcite the authors wish to thank c. j. brabec for designing some of the parallel multigrid routines ; m. buongiorno nardelli for his help with the complex - wavefunction and brillioun zone sampling routines ; and m. g. wensell for supplying his molecular dynamics routines . this work was supported by onr grant number n00014 - 91-j-1516 , nsf grant number dmr-9408437 , and onr grant number n00014 - 96 - 1 - 0161 . the calculations were performed at the pittsburgh supercomputer center and the north carolina supercomputing center . the _ mehrstellen _ discretization differs from central finite - differencing in two important respects : first , higher accuracy in the discretization is achieved by using more local information , but this accuracy is fully realized only at convergence ; and second , the discretized kohn - sham eigenvalue equation eq . ( [ compact - implicit ] ) is non - hermitian because the operator @xmath24 does not commute with the potential operator . in this appendix , we examine the accuracy of the _ mehrstellen _ discretization , and prove that the non - hermitian nature of @xmath126 does not change the nature of the wavefunctions : they remain orthogonal . for simplicity , we analyze only the _ mehrstellen _ discretization of the orthorhombic lattice . the fourth - order _ mehrstellen _ discretization ( see table [ mehrstellen - ortho ] ) samples the hamilton and the wavefunction at 19 points @xmath127 = & & a f({\bf x } ) + \sum_{n=1}^3 b_n f({\bf x } \pm h_n \hat { \bf x}_n ) \nonumber\\ & & + \sum_{n < m } c_{n , m } f({\bf x } \pm h_n \hat { \bf x}_n \pm h_m \hat { \bf x}_m ) \\ { \bf b}_{mehr}[f({\bf x } ) ] = & & a ' f({\bf x } ) + \sum_{n=1}^3 b'_n f({\bf x } \pm h_n \hat { \bf x}_n ) . \label{appa - cubic}\end{aligned}\ ] ] the accuracy of the _ mehrstellen _ discretization is one order higher than the corresponding central finite - differencing one , but this accuracy is achieved only at convergence.@xcite the small @xmath79 expansions of the @xmath23 and @xmath24 demonstrate this principle : @xmath128 note that by construction , @xmath129 to @xmath130 . thus , the _ mehrstellen _ discretization of the kohn - sham equations is equivalent to @xmath131 - { \bf b}_{mehr } [ \epsilon_n \psi_n ] = \nonumber\\ & & { \bf b}_{mehr}[{\bf h}_{ks}\psi_n -\epsilon_n\psi_n ] + o(h^4 ) . \label{appa-2}\end{aligned}\ ] ] the @xmath132 terms , implicit in the right - hand side , vanish at convergence , when @xmath133 . a similar analysis applies to the discretization of the poisson equation : @xmath134 - { \bf b}_{mehr}[4\pi\rho ] = \nonumber\\ & & { \bf b}_{mehr}[-\nabla^2 v_h - 4\pi \rho ] + o(h^4 ) . \label{appa-3}\end{aligned}\ ] ] unlike a plane - wave or central finite - differencing representation of the kohn - sham equations , the _ mehrstellen _ discretization eq.([compact - implicit ] ) leads to a _ non_-hermitian , _ generalized _ eigenvalue equation . nonetheless , we prove that the right eigenvectors of the discretized operator , _ i.e. _ the electronic orbitals , are orthogonal because they are also eigenvectors of a _ hermitian _ hamiltonian . the generalized eigenvalue equation can be recast into a more familiar form by multiplication by @xmath135 ( the invertibility of @xmath24 is discussed below ) : @xmath136 where @xmath137 is a non - compact discretization of @xmath62 of the same order as @xmath23 . the solutions of this equation , the @xmath138 and @xmath139 , are the solutions of the original equation . because @xmath140 and @xmath24 are finite - differencing operators with constant coefficients , they are translationally invariant and thus commute . they are also hermitian . thus , @xmath141 is hermitian , and the wavefunctions of eq.([compact - implicit ] ) are orthogonal . ( [ compact - implicit ] ) is the preferred discretization for computation , and the equivalent eq . ( [ newmehr ] ) is of formal interest only because the operators @xmath135 and hence @xmath137 are long - ranged and therefore computationally expensive to apply . finally , we consider the invertibility of the @xmath24 operator . we show that under reasonable conditions @xmath24 has no _ zero _ eigenvalues ( in fact , it is a positive definite operator ) by arguing that its null space is empty . it is straightforward to show that the null space of @xmath24 is comprised only of plane waves of maximum kinetic energy ; namely , @xmath142 ( or @xmath143 , etc . ) ; see eq . ( [ bofg ] ) below . thus , the null space of @xmath144 is empty whenever these plane waves can not be represented on the real - space mesh . this condition can be realized in two ways : choice of grid size , or explicit projection . for periodic boundary conditions , when one or more of the linear dimensions @xmath145 , @xmath146 , or @xmath48 is odd , the maximum g - vector along that dimension is @xmath147 . second , if the grid discretization can not be chosen to meet the formal invertibility condition , the pseudo inverse@xcite of @xmath24 exists and can be used ; that is , the few vectors in the null space of @xmath24 are projected out from the wavefunctions . on physical grounds any orbital of such rapid variation should be excluded from the calculation because it is marginally representable on the mesh . the pseudo inverse of @xmath24 is @xmath148 where the discrete fourier transform of @xmath24 is @xmath149 the pseudopotentials are short - ranged : the coulomb tail of the local potential is explicitly canceled and added to the madelung summation of the electrostatic energy , and by construction the nonlocal projectors have no coulomb tail . the nonlocal projectors and short - ranged local pseudopotentials are fourier filtered only once , when the appropriate potentials and grid spacing are selected . the filtering procedure attenuates the high - frequency components while maintaining the localization of the projectors and potentials . the unfiltered potentials or projectors are defined on a real - space radial grid , and are transformed to momentum space in order to filter the high - frequency components : @xmath150 where the cut - off function @xmath151 smoothly attenuates the radial fourier transform beyond @xmath152 . the cut - off wave vector is determined by the grid spacing : @xmath153 . the cut - off function is unity for @xmath154 and equals @xmath155 for @xmath152 . the parameters @xmath100 and @xmath156 depend on the atomic species and are carefully adjusted to achieve the best results . after the momentum - space filtering , the back - transformed potentials and projectors will extend beyond the original core radius . for computational efficiency , it is important that the nonlocal pseudopotentials be short - ranged . accordingly , a second filtering in real space is applied to reduce the large - radius oscillations beyond an empirically - determined radius @xmath157 . the second filtering function is unity below the cut - off radius and equals @xmath158 above it . example values for a carbon generalized norm - conserving pseudopotential with @xmath159 and @xmath113 nonlocalities are @xmath160 and @xmath161 , @xmath162 bohr , and @xmath163 . since the filtering procedure modifies the pseudopotentials , it is necessary to determine whether the modified potentials affect the system s physical properties . because the degree of filtering is set by the real - space grid spacing @xmath12 , the effect is similar to performing an under - converged plane - wave calculation . the last effects are well understood and can be measured quantitatively by progressively increasing the plane - wave cut - off . in particular , the main results of plane - wave calculations remain valid , even if they are significantly underconverged . this is due to the uniform convergence properties of plane waves , which form a translationally invariant basis set . similarly , the convergence effects may be monitored for a real - space calculation by decreasing the grid spacing . in our tests we found that the total energy of the system converges to an asymptotic value in a manner similar to that observed with plane waves . the hexagonal grid structure described in eq . ( [ hexgrid ] ) is a simple hexagonal lattice . because the @xmath57 axis is orthogonal to the @xmath164 plane , the @xmath62 operator may be written in separable form @xmath165 along the @xmath57 direction a sixth - order central finite - difference operator was selected @xmath166 where @xmath167 and the @xmath168 are given in table [ hexoperatorcoefficients ] . for the @xmath164 plane the the lattice translational vectors are not orthogonal , and a central finite - difference expression is not applicable . instead a composite form was selected @xmath169 + o(h_{xy}^6),\end{aligned}\ ] ] where @xmath170 and the @xmath171 are given in table [ hexoperatorcoefficients ] . in the multigrid solution process these sixth - order operators are only used on the finest grid level to compute the kinetic energy and the residual on coarser grid levels , a second - order operator is used to represent @xmath62 ; viz . , @xmath172 and @xmath173 + o(h_{xy}^2),\end{aligned}\ ] ] where the discretization weights are listed in table [ hexoperatorcoefficients ] . the multigrid restriction operator uses a volume weighting scheme with the weights adjusted for the hexagonal grid , and similarly , the hexagonal generalization of tri - linear interpolation is used to transfer the coarse - grid correction to the fine grid . the wavefunctions and hartree potential are generated using multigrid iterations in exactly the same manner as was described in section iii except for the modifications described here . see , for example , i. stich , m. c. payne , r. d. king - smith , and j .- s . lin , , 1351 ( 1992 ) ; k. d. brommer , m. needels , b. e. larson , and j. d. joannopoulos , _ ibid . _ * 68 * , 1355 ( 1992 ) ; p. bogusawski , q .- m . zhang , z. zhang , and j. bernholc , _ ibid . _ * 72 * , 3694 ( 1994 ) . r. car and m. parrinello , , 2471 ( 1985 ) . m. p. teter , m. c. payne , and d. c. allan , , 12255 ( 1989 ) . t. a. arias , m. c. payne , and j. d. joannopoulos , , 1077 ( 1992 ) ; , 1538 ( 1992 ) . d. vanderbilt , , 7892 ( 1990 ) . a. m. rappe , k. m. rabe , e. kaxiras , and j. d. joannopoulos , , 1227 ( 1990 ) . lin , a. qteish , m. c. payne , v. heine , , 4174 ( 1993 ) . g. li and s. rabii ( 1992 ) , unpublished . p. e. blchl , , 5414 ( 1990 ) . f. gygi , europhys . * 19 * , 6617 ( 1992 ) ; f. gygi , , 11692 ( 1993 ) . d. r. hamann , , 7337 ( 1995 ) ; _ ibid . _ * 51 * , 9508 ( 1995 ) . w. yang , , 1438 ( 1991 ) ; g. galli and m. parrinello , _ ibid . _ * 69 * , 3547 ( 1992 ) ; f. mauri , g. galli , and r. car , , 9973 ( 1993 ) ; p. ordejn , d. a. drabold , m. p. grumbach , and r. m. martin , _ ibid . _ * 48 * , 14646 ( 1993 ) ; j. kim , f. mauri , and g. galli , , 1640 ( 1995 ) . s. baroni and p. giannozzi , europhys . lett . * 17 * , 547 ( 1992 ) ; x .- p . li , r. nunes , and d. vanderbilt , , 10891 ( 1993 ) ; m. daw , _ ibid . _ * 47 * , 10895 ( 1993 ) ; w. hierse and e. stechel , _ ibid . _ * 50 * , 17811 ( 1994 ) ; s. goedecker and l. colombo , , 122 ( 1994 ) ; e. hernandez and m. j. gillan , , 10157 ( 1995 ) ; e. hernandez , c. m. goringe , and m. j. gillan , _ ibid _ * 53 * , 7147 ( 1996 ) . s. r. white , j. w. wilkins , and m. p. teter , , 5819 ( 1989 ) . j. bernholc , j .- y . yi , and d. j. sullivan , faraday disc . soc . * 92 * , 217 ( 1991 ) . j. r. chelikowsky , n. troullier , and y. saad , , 1240 ( 1994 ) ; j. r. chelikowsky , n. troullier , k. wu , and y. saad , , 11355 ( 1994 ) ; x. jing , n. troullier , d. dean , n. binggeli , j. r. chelikowsky , k. wu , and y. saad , _ ibid . _ * 50 * , 12234 ( 1994 ) ; j. r. chelikowsky , x. jing , k. wu , and y. saad , preprint ( 1995 ) . e. l. briggs , d. j. sullivan , and j. bernholc , , r5471 ( 1995 ) . k. a. iyer , m. p. merrick , and t. l. beck , j. chem . phys . * 103 * , 227 ( 1995 ) ; t. l. beck , k. a. iyer , and m. p. merrick , proc . sixth international conference on density functional theory , paris ( 1995 ) . e. j. bylaska , s. r. kohn , s. b. baden , a. edelman , r. kawai , m. elizabeth , g. ong , and j. h. weare , presented at the sixth siam conference on parallel processing for scientific computing , san francisco ( 1995 ) . f. gygi and g. galli , , r2229 ( 1995 ) . g. zumbach , n. a. modine , and e. kaxiras , preprint ( 1995 ) . k. cho , t. a. arias , j. d. joannopoulos , and p. k. lam , , 1808 ( 1993 ) . s. wei and m. y. chou , preprint ( 1995 ) . a. brandt , math . comp . * 31 * , 333 ( 1977 ) ; gmd studien , * 85 * , 1 ( 1984 ) . a. p. seitssonen , m. j. puska , and r. m. nieminen , , 14057 ( 1995 ) . d. r. hamann , , 2980 ( 1989 ) . g. b. bachelet , d. r. hamann , and m. schluter , , 4199 ( 1982 ) . d. r. hamann , m. schluter , and c. chang , , 1494 ( 1979 ) . l. kleinman and d. m. bylander , , 1425 ( 1982 ) . r. d. king - smith , m. c. payne , and j .- s . lin , , 13063 ( 1991 ) . l. collatz , _ the numerical treatment of differential equations _ , 3rd ed . 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( a12 ) and ( a2 ) for the definitions of @xmath174 , @xmath175 , and @xmath176 . the cubic - grid operator corresponds to @xmath177 . [ cols="<,^,^,^ " , ]

we describe a set of techniques for performing large scale _ ab initio _ calculations using multigrid accelerations and a real - space grid as a basis . the multigrid methods provide effective convergence acceleration and preconditioning on all length scales , thereby permitting efficient calculations for ill - conditioned systems with long length scales or high energy cut - offs . we discuss specific implementations of multigrid and real - space algorithms for electronic structure calculations , including an efficient multigrid - accelerated solver for kohn - sham equations , compact yet accurate discretization schemes for the kohn - sham and poisson equations , optimized pseudopotentials for real - space calculations , efficacious computation of ionic forces , and a complex - wavefunction implementation for arbitrary sampling of the brillioun zone . a particular strength of a real - space multigrid approach is its ready adaptability to massively parallel computer architectures , and we present an implementation for the cray - t3d with essentially linear scaling of the execution time with the number of processors . the method has been applied to a variety of periodic and non - periodic systems , including disordered si , a n impurity in diamond , aln in the wurtzite structure , and bulk al . the high accuracy of the atomic forces allows for large step molecular dynamics ; e.g. , in a 1 ps simulation of si at 1100 k with an ionic step of 80 a.u . , the total energy was conserved within 27 @xmath0ev per atom .

You are an expert at summarizing long articles. Proceed to summarize the following text: the transverse degree of freedom of an optical field is the fundamental aspect of light that contains spatial information . utilization of this information is the basic resource in traditional imaging systems and in applications such as microscopy , lithography , holography or metrology . in addition , use of the transverse modes of light has recently been demonstrated to be an important resource in optical communication @xcite , high - dimensional entanglement studies @xcite , and quantum key distribution @xcite . having control of the spatial coherence properties of a light beam provides an additional degree of control compared to using fully coherent light only , and has been shown to be advantageous for a number of applications . beams of decreased coherence allow access to spatial frequencies that are twice those available in a purely coherent system @xcite . greater spatial frequencies can enable improvements in imaging based systems and has been shown to be particularly useful in lithography @xcite . partial coherence also allows for the suppression of unwanted coherent effects by decreasing the coherence , such as suppression of speckle @xcite which enables lower noise and opens the door to novel imaging modalities @xcite . it has also been suggested that partial coherence can improve the deleterious effects of optical propagation through random or turbulent media @xcite . in addition , the coherent property of optical beams can be used for novel beam shaping @xcite as well as a method for control over soliton formation due to modulation instabilities in the study of nonlinear beam dynamics @xcite . the ability to generate arbitrary optical beams could also be used as a tool in basic research , such as in optical propagation @xcite or testing of novel methods in quantum state tomography dealing with the transverse wavefunction of light that has seen a great deal of interest recently @xcite . traditional methods used to generate partially coherent beams of light often rely on imprinting a changing pattern of random phase or speckle onto a coherent beam , such as with a spatial light modulator ( slm ) @xcite or rotating diffuser @xcite . it has even been demonstrated that slms allow the statistics of the speckle patterns to be varied across the beam to give spatially varying coherence properties @xcite . however none of these methods have been shown to allow for complete arbitrary control over the spatial coherence of an optical beam . in this paper we demonstrate how to generate any arbitrary quasi - monochromatic partially coherent field that can be specified by a cross - spectral density function @xmath1 , i.e. for fields fully specified by their two point spatial correlations . this is done by first computing the coherent mode decomposition of @xmath1 , which is an incoherent mixture of orthogonal coherent modes . for each of these coherent modes a computer generated hologram ( cgh ) is computed for a digital micromirror device ( dmd ) that acts as a binary amplitude spatial light modulator with rapid modulation speeds . the dmd then switches between each coherent mode on timescales slower than the coherence time of the source laser , but long relative to the detection time of the ccd . this creates an incoherent averaging that physically reproduces the coherent mode decomposition . section [ sec : coherentmodedecomposition ] details computation of the coherent mode decomposition , section [ sec : binarygratings ] describes the algorithm to compute binary amplitude cghs for the generation of coherent modes and section [ sec : experiment ] details the experimental demonstration of this technique . the transverse wavefront of a deterministic and coherent scalar beam is described by a complex field , @xmath2 . for a stochastic beam , @xmath2 is a random variable and it becomes necessary to represent the field in a more sophisticated way . the standard way of doing this is with the cross - spectral density function . at a single frequency the cross - spectral density function is defined as @xmath3 and represents the average intensity ( @xmath4 ) , as well as the correlations ( up to second order ) of such a partially coherent field @xcite . @xmath0 can be decomposed into an incoherent sum of orthogonal spatial modes @xmath5 , written as @xmath6 where @xmath7 are real and nonnegative , and @xmath8 is the relative weight of the field in mode @xmath5 @xcite . the modes @xmath5 can be computed as the eigenfunctions with corresponding eigenvalues @xmath7 from the fredholm integral equation @xmath9 this representation is often referred to as a coherent mode decomposition of @xmath0 . mathematically eq . [ eq : decomposition ] is a sum over an infinite number of modes , but in practice @xmath10 is bounded by the maximum spatial frequency content of @xmath1 , i.e. there is some maximum @xmath11 such that for @xmath12 , @xmath13 will be negligibly small . for example , gaussian schell - model beams are a common example of a partially coherent beam . such a beam is defined by having a gaussian intensity @xmath14 , as well as a gaussian degree of coherence @xmath15 , which gives a cross - spectral density function @xmath16 a coherent mode decomposition of such a gaussian schell - model beam shows that the number of coherent modes necessary to describe eq . [ eqn : gaussianschell ] is given by the number of independent coherent regions within the beam which is quantified by @xmath17 @xcite . physically eq . [ eq : decomposition ] can be realized if one can create a beam that alternates between the coherent modes @xmath5 in time with relative frequency weighted by @xmath13 . for measurement to yield the intended field , the switching time @xmath18 must be much faster than any detector integration time @xmath19 in order to create the intended averaging over the inputs . in addition , for the mixture to be an incoherent mixture , the various modes must not have any correlations in time . thus the switching time must be slower than the coherence time @xmath20 of the source . together these form the condition @xmath21 if eq . [ eqn : coherencetimes ] is met , then one has a physically realized implementation of @xmath1 . in order to generate an arbitrary partially coherent field @xmath1 , one only needs to find a way to create the coherent fields @xmath5 in rapid succession . such rapid mode generation was recently demonstrated by using dmds to create quickly addressable binary amplitude modulated cghs @xcite , though this comes at the cost of having a maximum efficiency around 10% . dmds are devices that provide both the speed and resolution desired for rapid generation and switching of coherent fields @xcite . a dmd consists of a 2-dimensional array of mirrors that can be in one of two positions , which can be used to act as an on or off state at each pixel . each pixel can be individually addressed and changed very rapidly , at frame rates exceeding @xmath22 . ( 1,0.35 ) ( 0,0 ) ( 0.16937818,0.07624671)(0,0)[lb ] ( 0.15969989,0.33046017)(0,0)[b ] ( 0.24935869,0.28885149)(0,0)[lb ] ( 0.07004105,0.28885149)(0,0)[rb ] ( 0.50968643,0.07624671)(0,0)[lb ] ( 0.58924682,0.28885149)(0,0)[lb ] ( 0.8499949,0.07624705)(0,0)[lb ] ( 0.92955516,0.28885149)(0,0)[lb ] the fact that dmds have 2 settings , allows us to make a binary grating . any periodic structure acts as a diffraction grating . a transverse shift in this diffraction grating will induce a phase shift or detour phase in the diffracted orders , even if the grating is an amplitude only structure . in addition , the form of each period will determine the scattering efficiency into the diffracted order . taken together , modulating the grating position and each periodic form locally within the hologram allows one to control both the amplitude and phase , and thus create any field , @xmath23 in the diffracted order . a well known method of encoding binary holograms is to create a periodic array of binary fringes or rectangular ` pulses . ' a one dimensional representation of this is shown in fig . [ fig : binarygrating ] . a shift in the location of these pulses will change the overall phases into the diffracted orders , while changing the widths or duty cycles of the pulses will change the diffracted efficiency . these two methods are known as pulse position and pulse width modulation respectively @xcite , and such a modulation represents a generalization of the moir technique @xcite . mathematically , a periodic binary grating can be written as a fourier series @xmath24 where @xmath25 is the grating wavevector . the grating consists of rectangular pulses of width @xmath26 spaced at a period of @xmath27 and @xmath28 $ ] is the relative location of the array within each period . looking only at the first diffraction order @xmath29 the field is given by @xmath30 where @xmath31 is the input field , which we ll assume to be a constant plane wave . in addition all optics after the dmd are aligned along the axis of the first diffraction order , allowing us to ignore any phase tilt from @xmath31 as well as the @xmath32 tilt from the grating in our description of @xmath33 . we can allow @xmath34 and @xmath35 to become functions of position and the previous results still hold so long as @xmath36 and @xmath37 vary much slower than the grating period @xmath27 . then any complex field @xmath38 can be generated by allowing @xmath39 where the phase is @xmath40,$ ] which is defined symmetrically around 0 to avoid encoding errors in the presence of a varying amplitude @xcite . this full procedure can be represented in the following fashion . first one chooses the field @xmath41 that one wishes to create . then @xmath36 and @xmath37 are computed from eq . [ eqn : pulsepostionandwidth ] and a periodic sinusoidal function is computed to give @xmath42 to convert this into a binary hologram , this function is thresholded by @xmath43 to create a binary pulse train with local pulse width @xmath44 . this can be written in the compact form @xmath45 , \label{eqn : gratingmodulation}\ ] ] where @xmath46 is the heaviside step function defined as @xmath47 a schematic of the experimental setup is shown in fig . [ fig : setup ] . a hene laser is spatially filtered using a 4f system to provide an initial coherent plane wave incident on the dmd . the various coherent modes , @xmath48 , are created in rapid succession with a spatially modulated binary diffraction grating on the dmd that gives the desired field in the first diffraction order . a second 4f system and pinhole are used to filter out all other diffraction orders and the resultant beam is imaged onto a ccd camera . . a fully spatially coherent plane wave is prepared by spatial filtering of a hene laser . this collimated beam is reflected off a cgh generated by the dmd and the desired diffracted order is filtered by a 4f system and imaged onto a ccd.,scaledwidth=47.0% ] the dmd is a type of micro - electronic mechanical system , commonly known as a mems , that can function as an amplitude only slm @xcite . the device consists of a two dimensional pixelated array of micromirrors each mounted on an individually addressed mems that can be in one of two positions . in order to use the device as a slm , the device is aligned such that the light is reflected and collected by the optics after the dmd if the micromirrors are in the on position , but scattered out of the system if the mirrors are in the off position . the device used in the experiment was a texas instrument dlp3000 . this device has a display resolution of @xmath49 pixels , a micromirror size of @xmath50 , and the pixels can be switched at rates up to 4khz which is much faster than a typical phase based slm @xcite . the ccd operates at 60hz , thus the detector integration time is @xmath51 . the dlp3000 dmd used in this experiment has a switching rate of 4khz , thus @xmath52 , which fulfills the first inequality in eq . [ eqn : coherencetimes ] . the bandwidth of the hene is 1.5ghz which gives @xmath53 which meets the second part of the inequality in eq . [ eqn : coherencetimes ] . as a demonstration of the ability to generate a single coherent state the field @xmath54 was generated . this represents a coherent superposition of two plane wave states , which form a sinusoidal interference pattern as shown in fig [ fig : coherentdata ] . for this experiment the mode was generated using a grating with wavevector @xmath55{px}}(\mathbf{\hat{x } } + \mathbf{\hat{y}}),\ ] ] which represents a period of @xmath56{pixels } \approx \unit[275]{\mu m}$ ] oriented at @xmath57 . this value of @xmath58 was chosen to be large enough to allow enough separation in the fourier plane to allow for filtering of the 1st diffracted order with an iris . in addition a nonzero value was chosen for both the @xmath59 and @xmath60 components of @xmath58 in order to minimize the noise by ensuring that the diffracted order did not overlap with any specular reflection due to the dmd s imperfect pixel fill - fraction . the underlying grating can be seen in the left image in fig . [ fig : coherentdata ] which have the appearance of the small diagonally oriented slivers . the plane wave transverse wavenumber was chosen to be @xmath61{px } } \approx \frac{2\pi}{\unit[780]{\mu m}}.\ ] ] @xmath62 and thus is slowly varying enough to allow us to use the procedure in section [ sec : binarygratings ] to construct the cgh to create this state . since we are perfectly interfering 2 plane waves , the intensity varies as @xmath63 . therefore @xmath64 , while @xmath65 . next we created a superposition of the plane waves @xmath66 and @xmath67 as before , but this time the degree of coherence between the two beams was spatially varied , creating a partially coherent mix of modes . the coherent modes used to represent this is given by @xmath68 where the relative probability weightings are given as @xmath69 , and where @xmath70 is related to the fringe visibility @xmath71 by @xmath72 the intensity for this beam is @xmath73 which is the sum of an incoherent and a coherent term which can be continuously tuned from fully coherent ( @xmath74 ) to incoherent ( @xmath75 ) . the visibility function chosen for the experiment is given by @xmath76 where @xmath77{px } } \approx \frac{2\pi}{\unit[580]{\mu m}}.\ ] ] since @xmath78 was chosen to be real , eq . [ eqn : visibility ] also represents our spectral degree of coherence at @xmath79 . the cghs necessary to create the modes @xmath80 and @xmath81 ( eq . [ eqn : partiallycoherentmodes ] ) for this spatially varying fringe visibility are shown in the top row of fig . [ fig : partiallycoherent ] . the cgh parameters are @xmath82 where @xmath83 is the maximum value of @xmath84 and @xmath85 along the 1d slice of @xmath86 for @xmath87 , i.e. along the @xmath59-axis ( solid blue line ) . also shown as the black dotted line is the theoretical envelope of the maximum and minimum intensities based on the intended visibility function @xmath88.,scaledwidth=47.0% ] in order to compare the intended visibility given by eq . [ eqn : visibility ] with the image shown in fig . [ fig : partiallycoherent ] , a one dimensional slice of the intensity is plotted in fig . [ fig : fringevisibility ] . this slice is a radial slice @xmath86 along the @xmath59 axis ( i.e. at an orientation of @xmath87 ) , and is plotted over an entire period of @xmath89 of the visibility . in addition the theoretical envelope of the visibility equal to @xmath90 is plotted for comparison . as can be seen in both the original coherent and partially coherent cases , the intended and measured patterns are in excellent agreement with one another . in this paper we have demonstrated a novel method of generating arbitrary fields of light . any partially coherent field that is described by the cross - spectral density function @xmath1 can be generated by computing the coherent mode decomposition into an incoherent sum of coherent modes . this incoherent mix of modes was physically realized by rapid generation of spatial holograms on a dmd and was temporally averaged in detection . we acknowledge mayukh lahiri , joe vornehm and alex radunsky for helpful discussions . our work was supported by the defense advanced research projects agency ( darpa ) inpho program and osml also acknowledges support from the conacyt . j. wang , j .- y . yang , i. m. fazal , n. ahmed , y. yan , h. huang , y. ren , y. yue , s. dolinar , m. tur , and a. e. wilner , `` terabit free - space data transmission employing orbital angular momentum multiplexing , '' nature photonics * 6 * , 488496 ( 2012 ) . b. rodenburg , m. p. j. lavery , m. malik , m. n. osullivan , m. mirhosseini , d. j. robertson , m. j. padgett , and r. w. boyd , `` influence of atmospheric turbulence on states of light carrying orbital angular momentum , '' * 37 * , 3735 ( 2012 ) . a. c. dada , j. leach , g. s. buller , m. j. padgett , and e. andersson , `` experimental high - dimensional two - photon entanglement and violations of generalized bell inequalities , '' nature physics * 7 * , 677680 ( 2011 ) . r. w. boyd , a. jha , m. malik , c. osullivan , b. rodenburg , and d. j. gauthier , `` quantum key distribution in a high - dimensional state space : exploiting the transverse degree of freedom of the photon , '' proceedings of spie * 7948 * , 79480l6 ( 2011 ) . m. malik , m. n. osullivan , b. rodenburg , m. mirhosseini , j. leach , m. p. j. lavery , m. j. padgett , and r. w. boyd , `` influence of atmospheric turbulence on optical communications using orbital angular momentum for encoding , '' * 20 * , 13195 ( 2012 ) . z. chen , s. m. sears , h. martin , d. n. christodoulides , and m. segev , `` clustering of solitons in weakly correlated wavefronts , '' proceedings of the national academy of sciences * 99 * , 52235227 ( 2002 ) . c. rickenstorff , e. flores , m. olvera - santamara , and a. ostrovsky , `` modulation of coherence and polarization using nematic 90 degree - twist liquid - crystal spatial light modulators , '' revista mexicana de fsica * 58 * , 270273 ( 2012 ) . p. zhang , z. zhang , j. prakash , s. huang , d. hernandez , m. salazar , d. n. christodoulides , and z. chen , `` trapping and transporting aerosols with a single optical bottle beam generated by moir techniques . '' optics letters * 36 * , 14913 ( 2011 ) .

we describe an experimental technique to generate a quasi - monochromatic field with any arbitrary spatial coherence properties that can be described by the cross - spectral density function , @xmath0 . this is done by using a dynamic binary amplitude grating generated by a digital micromirror device ( dmd ) to rapidly alternate between a set of coherent fields , creating an incoherent mix of modes that represent the coherent mode decomposition of the desired @xmath0 . this method was then demonstrated experimentally by interfering two plane waves and then spatially varying the coherence between them . it is then shown that this creates an interference pattern between the two beams whose fringe visibility varies spatially in an arbitrary and prescribed way .

You are an expert at summarizing long articles. Proceed to summarize the following text: early investigators studied the spatial distribution of galaxies because they hoped to learn about the structure of the universe on the largest scales . their influential work was superseded , in the end , by its competition . problems began with the demonstration that galaxies contained only a small fraction of the matter in the universe . galaxy formation remained too poorly understood to quell doubts about how faithfully galaxies traced underlying distribution of dark matter . other observations improved gravitational lensing , peculiar velocities , intergalactic absorption lines , and so on and seemed easier to relate to matter fluctuations . computers became fast enough to predict the evolution of the large - scale matter distribution from the the initial conditions that microwave - background missions were measuring with increasing precision . as it became clear that the simulations and observations agreed remarkably well , most researchers concluded that the large - scale structure of the universe could be understood completely as the product of gravitational instability amplifying small inflationary perturbations . galaxies , once believed to be the primary constituent of the universe , came to be seen as small test particles swept into ever larger structures by converging dark matter flows . the spatial distribution of galaxies remains interesting because it can teach us about galaxy formation . galaxy formation must be closely related to larger process of gravitational structure formation , since the formation of a galaxy begins with gas streaming into a massive potential well and ends with stars drifting in the cosmic flow . lessons from 20 years of numerical investigations into structure formation should therefore carry over to the analysis of galaxy clustering . one example is the known positive correlation between clustering strength and mass for virialized dark matter halos . since galaxies reside in dark matter halos , their clustering strength provides an indication of the mass of the halos that contain them . the resulting mass estimate depends on the assumptions that the microwave background and other observations have given us reliable estimates of cosmological parameters and of the initial matter power - spectrum , that numerical simulations can correctly trace the evolution of the matter distribution , at least for moderate densities , and that no process can significantly separate baryons from dark matter on mpc scales assumptions that are at least as plausible as those behind competing techniques for mass estimation . another example is the evolution in the clustering strength of a population of galaxies once it has formed . this is driven solely by gravity and is easy to predict from numerical simulations . comparing the clustering of ( say ) galaxies in the local universe and those at high redshift can therefore suggest or rule out possible links between them . few other methods are as useful for unifying into evolutionary sequences the galaxy populations we observe at different look - back times . an excellent review of the history of these techniques has been written by giavalisco ( 2002 ; pp 620 - 624 ) . this paper has two aims . the first is to present measurements of the clustering of uv - selected star - forming galaxies in a redshift range @xmath2 that is only partially explored . section [ sec : data ] describes the way we obtained our data , [ sec : methods ] describes and justifies the techniques we used to estimate the spatial clustering in our galaxy samples , and [ sec : results ] presents our estimates of the galaxy correlation function at redshifts @xmath30 , @xmath31 , and @xmath32 . the survey analyzed here is several times larger than its predecessors ; the surveyed area of @xmath3 square degrees is roughly 700 times larger than the area analyzed by arnouts et al . ( 2002 ) and 4 times larger than the areas analyzed by giavalisco & dickinson ( 2001 ) and ouchi et al . the second aim is to discuss what our measurements imply about the galaxies and their descendants . in [ sec : correspondence ] we show that the galaxies correlation functions are indistinguishable from those of virialized dark matter halos with mass @xmath33 . in [ sec : evolution ] we show that the galaxies , dragged by gravity for billions of years , caught in the press of structure formation , would by redshift @xmath28 have a correlation function that is indistinguishable from that of the elliptical galaxies that surround us . our results are summarized in [ sec : summary ] . the data we analyzed were drawn from our ongoing surveys of high - redshift star - forming galaxies . a brief description of the surveys follows ; see steidel et al . ( 2003 , 2004 ) for further details . deep , multi - hour @xmath7 images of 21 fields scattered around the sky were obtained with various 4m - class telescopes ( table [ tab : fields ] ) . tens of thousands of objects were visible in these images . for the analysis of this paper we ignored all but the subset ( @xmath34 20% ) with ab magnitude @xmath35 and ab colors satisfying the `` lbg '' selection criteria of steidel et al . ( 2003 ) , @xmath36 the `` bx '' selection criteria of adelberger et al . ( 2004 ) , @xmath37 or the `` bm '' selection criteria of adelberger et al . ( 2004 ) , @xmath38 in this range of @xmath14 magnitudes , the colors are characteristic of galaxies at @xmath39 . ( the restriction to @xmath40 helps eliminate most interlopers ; see adelberger et al . 2004 and steidel et al . 2004 . ) by obtaining spectra of more than @xmath8 galaxies with these colors , steidel et al . ( 2003 , 2004 ) established that the mean redshift and @xmath41 range of galaxies in the three samples is @xmath42 ( these values exclude galaxies with @xmath43 or @xmath44 as well as the handful of low - redshift `` interloper '' galaxies with @xmath45 . ) redshift histograms are shown in figure [ fig : zhistos ] . we will use the term `` photometric candidates '' to describe the objects with @xmath46 whose colors satisfy one set of the selection criteria presented above , and the term `` spectroscopic sample '' to describe the subset of photometric candidates that had a spectroscopic redshift measured by steidel et al . ( 2003 ) or steidel et al . although the spectroscopic sample is sizeable , it contains only a small fraction ( @xmath47% ) of the photometric candidates ( see figure [ fig : specfrac ] and table [ tab : fields ] ) . lrrrrrrr 3c324 & @xmath48 & 11/49 & 0/166 & 0/126 & 0.0035 & 0.0034 & 0.0042 + b20902 & @xmath49 & 31/65 & 1/207 & 0/189 & 0.0036 & 0.0035 & 0.0043 + cdfa & @xmath50 & 34/99 & 0/336 & 0/280 & 0.0029 & 0.0028 & 0.0036 + cdfb & @xmath51 & 20/120 & 0/316 & 0/273 & 0.0028 & 0.0028 & 0.0035 + dsf2237a & @xmath51 & 39/100 & 1/367 & 0/328 & 0.0028 & 0.0028 & 0.0035 + dsf2237b & @xmath52 & 44/161 & 1/516 & 0/309 & 0.0028 & 0.0028 & 0.0035 + hdf & @xmath53 & 54/187 & 128/735 & 37/587 & 0.0022 & 0.0022 & 0.0028 + q0201 & @xmath54 & 18/90 & 4/339 & 0/285 & 0.0029 & 0.0029 & 0.0036 + q0256 & @xmath55 & 45/126 & 1/346 & 0/243 & 0.0030 & 0.0029 & 0.0037 + q0302 & @xmath56 & 46/824 & 0/1778 & 0/749 & 0.0018 & 0.0018 & 0.0023 + q0933 & @xmath57 & 63/192 & 0/435 & 0/273 & 0.0028 & 0.0028 & 0.0035 + q1307 & @xmath58 & 16/483 & 47/1352 & 9/936 & 0.0017 & 0.0017 & 0.0023 + q1422 & @xmath59 & 96/253 & 1/728 & 0/491 & 0.0024 & 0.0024 & 0.0030 + q1623 & @xmath60 & 6/462 & 189/1220 & 2/847 & 0.0016 & 0.0016 & 0.0022 + q1700 & @xmath61 & 15/406 & 62/1456 & 1/948 & 0.0018 & 0.0018 & 0.0024 + q2233 & @xmath62 & 44/76 & 1/267 & 0/181 & 0.0028 & 0.0028 & 0.0035 + q2343 & @xmath63 & 10/385 & 148/938 & 8/541 & 0.0016 & 0.0016 & 0.0022 + q2346 & @xmath64 & 1/362 & 34/1142 & 1/754 & 0.0016 & 0.0017 & 0.0022 + ssa22a & @xmath65 & 42/151 & 10/360 & 1/253 & 0.0029 & 0.0028 & 0.0036 + ssa22b & @xmath65 & 29/73 & 5/281 & 1/308 & 0.0029 & 0.0028 & 0.0036 + westphal & @xmath66 & 172/270 & 43/724 & 20/632 & 0.0018 & 0.0018 & 0.0024 + total & 2907 & 836/4934 & 676/14009 & 80/9533 & 0.0021 & 0.0021 & 0.0029 + [ tab : fields ] all redshifts were measured with the low - resolution imaging spectrograph ( lris ; oke et al . 1995 ) on the keck telescopes . the number of redshifts in each field was determined by the number of clear nights that were allocated . photometric candidates were selected for spectroscopy more - or - less at random , but in one way the selection was far from random : spectroscopic objects in each field were constrained to fit together in a non - interfering way on one of a small number of multislit masks . this introduced artificial angular clustering to the spectroscopic samples , particularly in fields where our image s size significantly exceeded the spectrograph s @xmath67 field - of - view . in some cases the artificial angular clustering was increased by our desire to obtain particularly dense spectroscopic sampling in some parts of an image , e.g. , near a background qso . table [ tab : fields ] lists the number of bm , bx , and lbg photometric candidates and spectroscopic galaxies in each field . the field - to - field variations in the number of photometric candidates per square arcminute were caused primarily by differences in exposure times , seeing , sky brightness , telescope plus instrumental throughput , and so on , from one run to the next . recall that we used many different telescopes and cameras during our imaging survey . the expected variations in intrinsic surface density ( also shown in the table , and calculated for galaxies with bias @xmath68 as described in [ sec : angularmethods ] below ) are significantly smaller . we estimated the angular correlation functions from the lists of photometric candidates . inferring a comoving correlation length @xmath69 from the measured angular clustering required an estimate of the objects redshift distribution . for this we took the measured redshift distributions of the spectroscopic samples . since the spectroscopic samples are large several hundred for the bx and lbg criteria , nearly 100 for bm random fluctuations are unlikely to have given redshift distributions to them that are significantly different from those of the parent photometric samples . we were able to measure a redshift for only @xmath70% of the objects we observed spectroscopically , however , and it is therefore possible that various systematics ( e.g. , difficulties measuring spectroscopic redshifts for galaxies in certain redshift ranges ) could have caused the spectroscopic and photometric samples to have somewhat different redshift distributions . repeated observations of a subset of the initial spectroscopic failures show that these objects have the same redshift distribution as the initial successes , implying that any systematics are not severe . to help us interpret our observations , we refered at times the gif - lcdm numerical simulation of structure formation in a cosmology with @xmath71 , @xmath72 , @xmath73 , @xmath74 , and @xmath75 . this gravity - only simulation contained @xmath76 particles with mass @xmath77 in a periodic cube of comoving side - length @xmath78 mpc , used a softening length of @xmath79 comoving kpc , and was released publicly , along with its halo catalogs , by frenk et al . astro - ph/0007362 . further details can be found in jenkins et al . ( 1998 ) and kauffmann et al . ( 1999 ) . since the gif - lcdm cosmology is consistent with the wilkinson microwave - anisotropy probe results ( spergel et al . 2003 ) , and since modeling the gravitational growth of perturbations on large ( @xmath34 mpc ) scales is not numerically challenging , we will assume that the growth of structure found in this simulation closely mirrors the growth of structure in the actual universe . for our purposes the most interesting aspect of the simulation is the spatial distribution of virialized `` halos '' , or overdensities with @xmath80 , since these deep potential wells are the sites where galaxies can form from cooling gas . the public halo catalogs were created , by the gif team , by running a halo - finding algorithm at various time steps in the simulation . we will say that all the halos identified by the algorithm at time - step @xmath81 ( say ) had @xmath81 as their time of identification . correlation functions for halos at the time of identification were calculated directly from the public halo catalogs . in subsequent time - steps these halos were progressively displaced by gravity . some grew ; others were destroyed as they were subsumed into larger structures . any galaxies within the halos would be likely to survive intact , however , and it is interesting to trace the expected evolution in their correlation function over time . to do this , we assumed that the galaxies in a halo would be displaced by gravity by the same amount , and in the same direction , as the halo s most bound particle . if @xmath82 denotes the set of particles that were the most bound particle in a halo at @xmath83 , we assumed that the correlation function at time @xmath84 of the galaxies that lay within halos identified at time @xmath83 would be roughly equal to the correlation function of particles @xmath82 at time @xmath84 . at spatial separations that are large compared to the typical halo radius , the expected evolution of the galaxy correlation function is insensitive to the details of this procedure . these are the only spatial separations we will consider . our focus on large spatial scales also justifies our ignoring the possibility of galaxy mergers . although mergers can strongly affect the correlation function on small scales , on large scales the effect is more subtle . it can be understood as follows . at time @xmath84 , the galaxies identified at time @xmath83 will be found in halos with a range of masses , and their large - scale correlation function will be a weighted average of the correlation functions of the halos . since the weighting depends on typical number of descendant galaxies in halos of each mass , and since halos with different masses have different correlation functions , the large - scale correlation function of descendant galaxies will be altered by mergers if the merger frequency depends on halo mass . in practice , however , the difference in correlation functions between the more massive and less massive halos that host the descendants is not enormous , and as a result the merger frequency would have to be an implausibly strong function of halo mass to alter the descendant correlation functions on large scales in a significant way . two approaches will be used to estimate the clustering strength . the first approach , which is standard , relies almost exclusively on the angular positions of the galaxies . the second relies almost exclusively on the galaxies measured redshifts . these approaches exploit different aspects of our data and are subject to different systematics . the level of agreement between them provides an important test of our conclusions robustness . this section describes and justifies the two approaches . readers interested primarily in our scientific results may wish to skip ahead to [ sec : results ] . the observed clustering of galaxies on the plane of the sky is related to the galaxies three - dimensional correlation function in a straightforward way . let @xmath85 denote a galaxy s redshift and @xmath86 denote its angular position . @xmath86 is written in bold face because two numbers ( e.g. , right ascension and declination ) are required to specify the galaxy s angular position . if @xmath87 is the probability that a galaxy at known position @xmath88 has a neighbor at position @xmath89 , then elementary identities show that the probability that a galaxy at angular position @xmath90 will have a neighbor at angular position @xmath91 is @xmath92 observations indicate that the reduced correlation function is well approximated by an isotropic power law , @xmath93,\ ] ] where @xmath69 and @xmath94 parametrize the shape of the power law and @xmath95 is the distance between @xmath89 and @xmath88 . this implies , in the circumstances of interest to us , that the reduced angular correlation function will also be a power law , @xmath96 with @xmath97 . if the angular separation @xmath98 between the galaxies is small , @xmath99 , and if the comoving correlation length @xmath69 does not change significantly from the front to the back of the survey , then @xmath100 ^ 2 \label{eq : alimber}\end{aligned}\ ] ] ( see , e.g. , totsuji & kihara 1969 ) where @xmath101 is the survey selection function , @xmath102 is the beta function in the convention of press et al . ( 1992 ) , @xmath103 is the change in comoving distance with redshift , @xmath104 is the change in comoving distance with angle , and @xmath105 is the angular diameter distance . this follows from the relationship @xmath106^{\gamma/2 } = r_0^\gamma r^{1-\gamma } b(1/2,(\gamma-1)/2)$ ] . our first approach to estimating the three dimensional clustering strength will be to measure the parameters @xmath107 and @xmath108 of the reduced angular correlation function @xmath109 , then infer values for @xmath69 and @xmath94 using the relationships above . our estimates of @xmath110 in different angular bins will be based on the landy - szalay ( 1993 ) estimator @xmath111 where @xmath112 is the observed number of unique galaxy pairs with separation @xmath113 , @xmath114 is the number of unique pairs with separation in the same range between the observed galaxy catalog and a galaxy catalog with random angular positions , and @xmath115 is the number of unique pairs in the random catalog with separations in the same range . in practice we reduce the noise in the random pair counts by creating random catalogs with many times more objects than the data catalogs ( @xmath116 ) , calculating @xmath117 and @xmath118 , then multiplying @xmath117 and @xmath118 by @xmath119 and @xmath120 $ ] , respectively . unless fluctuations on the size of our typical field - of - view are negligible , the number of detected galaxies in any field will be somewhat higher or lower than in a fair sample of the universe , and the number of galaxies in the field s ideal random catalog would therefore be lower or higher than the observed number . as a result the values @xmath117 and @xmath118 that we calculate with our approach will be incorrect to some degree . in a single field this can make the clustering appear stronger or weaker than it truly is , but when many fields are averaged it tends to make the observed clustering appear artificially weak . this can be shown as follows . assume that the observed mean density in a field differs from the global average by the unknown factor @xmath121 , i.e. , @xmath122 , and let @xmath117 and @xmath118 be the pair counts calculated from scaling the random catalogs to the observed density . one might guess that the estimator of equation [ eq : landyszalay ] ought to have @xmath117 and @xmath118 replaced by values corrected to the true mean density , i.e. , by @xmath123 and @xmath124 , and indeed hamilton ( 1993 , 3 ) has shown that the estimator @xmath125 is equal to the true angular correlation function on average , @xmath126 this equation does not help us directly , since we do not know @xmath127 and can not calculate @xmath128 , but it does show that the estimator @xmath129 must be biased : @xmath130 where @xmath131 and the approximation assumes the weak clustering ( @xmath132 ) limit . it is therefore customary to estimate @xmath110 by adding a constant @xmath133 to the calculated values @xmath134 . the constant @xmath135 depends on the unknown values of @xmath127 in the observed field or fields and can not be calculated exactly . if @xmath136 , so that field - to - field fluctuations are in the linear regime and have a nearly gaussian distribution , and if our data are drawn from @xmath137 independent fields with measured pair counts @xmath138 , @xmath139 , and @xmath140 , then the value of @xmath135 appropriate to a given angular bin in our data set will have a variance of @xmath141 \\ & \simeq & \frac{1}{rr_{\rm tot}^2}\sum_{i=1}^n 2\sigma_i^4 dd_i^2 \label{eq : var_i}\end{aligned}\ ] ] around its expectation value @xmath142 here @xmath143 is the sum over all fields of the random pair counts in the chosen angular bin . in practice @xmath144 and @xmath145 depend very weakly on which bin is chosen . below we will take @xmath144 at @xmath146 as our best guess at the correction @xmath135 . when it matters we will discuss the effect of the uncertainty in @xmath135 . we use two approaches to estimate the size of the uncertainty @xmath147 in the mean galaxy density of the @xmath148th field . since @xmath149 it can be estimated numerically as @xmath150 if @xmath151 is known ( infante 1994 ; roche & eales 1999 ) . unfortunately the iterative approach suggested by equations [ eq : actualestimator ] and [ eq : rocheeales ] can be unstable , at least when the correlation function slope is allowed to vary : a large value of @xmath134 will imply a large correction @xmath152 , which implies an even larger @xmath134 and even larger correction , and so on . the instability is undoubtedly worse for large images , where the estimate of the integral constraint correction for one iteration is completely dominated by the assumed correction from the previous . we were unable to use equation [ eq : rocheeales ] as anything other than a consistency check . a more robust estimate of @xmath152 follows from theoretical considerations . since matter fluctuations will still be in the linear regime on the large scales of our observations , the relative variance of mass from one surveyed volume to the next can be estimated from the linear cold - dark matter power - spectrum @xmath153 ( bardeen et al . 1986 ; we adopt the parameters @xmath154 , @xmath75 , @xmath155 ) with parseval s relationship @xmath156 where @xmath157 is the fourier transform of a survey volume . the shape of the observed volume in any one of our fields is reasonably approximated in the radial direction by a gaussian with comoving width ( rms ) @xmath158 and in the transverse directions by a rectangle with comoving dimensions @xmath159 . in this case @xmath160 \frac{\sin(k_xl_x/2)}{k_xl_x/2}\frac{\sin(k_yl_y/2)}{k_yl_y/2}.\ ] ] the implied value of @xmath161 for each sample in each field is shown in table [ tab : fields ] ; the values assume the powerspectrum normalization required for the r.m.s . fluctuation as a function of redshift in spheres of comoving radius @xmath162 mpc to obey @xmath163 where @xmath164 and @xmath165 is the linear growth factor to redshift @xmath85 . the desired corrections @xmath152 are then given by @xmath166 where @xmath167 , the galaxy bias , is calculated from the ratio of galaxy to matter fluctuations in spheres of comoving radius @xmath162 mpc : @xmath168 here the galaxy variance @xmath169 ( peebles 1980 eq . 59.3 ) can be derived from the fit to the galaxy correlation function . this approach also requires an iterative solution , since the correction @xmath152 to @xmath110 depends on @xmath110 , but the advantage is that the assumed size of large - scale fluctuations is anchored in reality by our requirement that the slope of the correlation function match other observations on very large scales . the discussion so far assumes that we will know the precise shape of the selection function @xmath101 . in fact this is not true , and uncertainty in the true shape of our selection function is a source of error in the derived values of @xmath69 . means that errors in @xmath101 alter the inferred value of @xmath94 as well . we will neglect this small effect . ] a larger width for the selection function means that projection effects are stronger , and therefore implies a larger value of @xmath69 for given angular clustering ( see equation [ eq : alimber ] ) . if the selection function is a gaussian with mean @xmath170 and standard deviation @xmath171 , and if the weak redshift variations of @xmath172 and @xmath173 can be ignored , then the constant @xmath107 in equation [ eq : alimber ] is proportional to @xmath174 , and the implied value of @xmath69 is proportional to @xmath175 . measuring @xmath137 redshifts drawn from this selection function determines @xmath170 to a precision @xmath176 and @xmath177 to a relative precision @xmath178 . excluding interlopers with @xmath179 , we have measured roughly 800 , 700 , and 80 redshifts for galaxies in the lbg , bx , and bm samples , and the selection function width is @xmath180 for each . the relative uncertainty in @xmath171 is therefore approximately @xmath181% for the lbg and bx samples and @xmath182% for the bm sample , which implies @xmath183% uncertainty in @xmath69 for the lbg and bx samples and @xmath184% uncertainty in the bm sample . variations of the selection function from one field to the next ( owing , for example , to differences in the depth of the data or to systematic errors in our photometric zero points ) are another source of concern , especially at the redshifts @xmath185 where galaxies colors are insensitive to redshift and small color errors mimic large redshift differences . suppose for simplicity that all fields have the same number of photometric candidates , let the rms width of the selection function in the @xmath148th field be written @xmath186 , where @xmath187 is the mean width among all fields , and let the mean redshift of the selection function be written @xmath188 where @xmath170 is the mean redshift among all fields . then the rms width of the total selection function @xmath189^{1/2 } = s[1+{\rm var}(\delta)+{\rm var}(\epsilon)]^{1/2},\ ] ] exceeds the value @xmath187 that should be used in determining @xmath107 . since we will ( by necessity ) use the total selection function in estimating @xmath69 , our estimates will be biased high . systematic errors in our zero points are unlikely to be larger than @xmath190 , and variations in photometric depth will at most change our characteristic color uncertainties from @xmath191 to @xmath192 magnitudes . the measured variations in galaxy redshift with @xmath7 color ( see , e.g. , adelberger et al . 2004 ) imply ( a ) that zero point errors with @xmath190 will shift the mean redshifts of galaxies that satisfy the lbg , bx , and bm selection criteria by @xmath193 , @xmath194 , and @xmath195 , respectively , and ( b ) that increasing the photometric uncertainty from @xmath196 to @xmath197 will increase the widths of the lbg , bx , and bm selection functions by @xmath198 , @xmath199 , and @xmath199% . the upper limits on @xmath200 and @xmath201 are therefore @xmath202 ( @xmath203 ) and @xmath204 ( @xmath205 ) , respectively , for the lbg ( bx , bm ) sample . the required reduction in @xmath69 is negligible for the lbg sample but could be as large as @xmath206% for the other two . we will account for uncertainties in the selection function by decreasing the best - guess value of @xmath69 for the bm and bx samples by 3.5% and increasing the uncertainty in quadrature by @xmath207 . as figure [ fig : zhistos ] shows , some fraction of the objects in the bx and bm samples will be low redshift interlopers . we correct for the resulting dilution in the clustering strength by using the full selection function , starting at @xmath28 , in our estimate of @xmath69 from equation [ eq : alimber ] . this is the optimal correction only if the interlopers have the same comoving correlation length @xmath208 mpc ( see below ) as the galaxies in the primary samples . this should be nearly true , since budavri et al . ( 2003 ) estimate @xmath209 mpc for the blue star - forming galaxy population at @xmath210 from which our interlopers are drawn . in any case , since the correction itself is small eliminating the tail with @xmath179 from @xmath101 alters the inferred values of @xmath69 for the bm and bx samples by only @xmath198%errors in it should not have an appreciable effect on our estimates of @xmath69 . we face three significant obstacles in trying to estimate the clustering strength from the spectroscopic catalogs . \(1 ) the objects in a given field that were selected for spectroscopy were not distributed randomly across the field , but were instead constrained to lie on one of a small number of multislit masks . since only a small fraction of the galaxies were observed spectroscopically in the typical field , the finite size of the masks coupled with the need to avoid spectroscopic conflicts produced significant artificial clustering in the angular positions of sources in the spectroscopic catalog . the effect was worsened in some fields by our decision to obtain particularly dense spectroscopy near background qsos . ( 2 ) because galaxies @xmath7 colors change slowly with redshift near @xmath185 , the expected redshift distribution @xmath101 of our bm and bx color - selected samples depends sensitively on the quality of the photometry . the larger color errors from noisy photometry will lead to a broader @xmath101 , while relatively small systematic shifts in the photometric zero points can significantly alter the mean of @xmath101 . adelberger et al . ( 2004 ) and [ sec : selfn_uncertainties ] of this paper discuss this point in more detail , but the upshot is that we can not estimate the selection function @xmath101 with great precision for the bx and bm samples . ( 3 ) peculiar velocities and redshift uncertainties render imprecise our estimate of each galaxy s position in the @xmath85 direction . this limits the accuracy of our estimate of the distance from one galaxy to its neighbors , complicating our efforts to measure the correlation function on small spatial scales . effects ( 1)(3 ) are usually compensated with the aid of detailed simulations . although this approach should work in principle , in practice it is hard for outsiders to evaluate whether the simulations were flawed . the remainder of this section describes the alternate approach that we adopt . it is based on analyzing observable quantities that are not affected by systematics ( 1)(3 ) . the spurious angular clustering signal can be eliminated if we take the angular positions of spectroscopic galaxies as given and estimate the clustering strength solely from their redshifts . let @xmath211 be the comoving distance to redshift @xmath85 , and let @xmath212 be the transverse comoving separation implied by the angular separation @xmath213 between a galaxy and some reference position , e.g. , the center of the observed field . according to elementary probability identities , if we know that one galaxy has position @xmath214 , then the probability that a second galaxy at transverse position @xmath215 has radial position @xmath216 is @xmath217 and the expected distribution of radial separations @xmath218 for galaxies with transverse separation @xmath219 is @xmath220\times\nonumber\\ & & \quad \int_0^{\infty } dz_2\,\frac{p(z_2)p(z_2+z_{12})}{1+p(z_2)r_0^\gamma r_{12}^{1-\gamma}\beta(\gamma ) } \label{eq : pzgivenr}\end{aligned}\ ] ] where we have used results from the previous section and adopted the shorthand @xmath221 for the beta - function given above , @xmath222 . ( equations [ eq : pzgivenrrz ] through [ eq : pzgivenr ] assume that the quantity @xmath223 is constant with redshift , an approximation that is valid for the small separations @xmath224 comoving mpc between the galaxy pairs we will use in this analysis . ) equation [ eq : pzgivenr ] shows that the observable quantity @xmath225 is sensitive to the clustering strength but independent of angular variations in the spectroscopic sampling density . unfortunately the correlation function @xmath226 can be estimated from @xmath225 with equation [ eq : pzgivenr ] only if we have a reasonably accurate estimate of the selection function shape @xmath227 . this can be seen more clearly by taylor - expanding the integral in equation [ eq : pzgivenr ] around @xmath228 and approximating the selection function as a gaussian with standard deviation @xmath171 that is centered many standard deviations from @xmath28 . one finds @xmath229\times\nonumber\\ & & \quad\quad\bigl[\frac{a_0}{\sigma_{\rm sel } } + \frac{a_2}{2\sigma_{\rm sel}^3}z_{12}^2+\frac{a_4}{24\sigma_{\rm sel}^5}z_{12}^4\bigr ] \label{eq : pzgivenrapprox}\end{aligned}\ ] ] where @xmath230 @xmath231 , @xmath232 , @xmath233 , @xmath234 and @xmath235 . the coefficients @xmath236 all have similar sizes since the integrals are dominated by contributions from @xmath237 where the integrands are of the same order . in the angular clustering case above , inaccuracies in the adopted width @xmath171 of the selection function affected the inferred amplitude of the correlation function but not its shape . here they affect both . moreover errors in @xmath171 are multiplied not by @xmath226 , but by @xmath238 , which implies that they can easily dominate the true clustering signal when @xmath239 . equation [ eq : pzgivenrapprox ] shows that one must be careful estimating the strength of redshift clustering when the shape of the selection function is poorly known . our solution exploits the fact that @xmath240 for our survey , which implies @xmath241 for all separations where @xmath226 is large enough to measure . as long as @xmath241 , the terms proportional to @xmath242 and @xmath243 can be neglected in equation [ eq : pzgivenrapprox ] , and @xmath225 will be very nearly equal to @xmath244 $ ] , with @xmath245 a function that does not depend on @xmath246 . the function @xmath245 does depend on the unknown selection function , but it can be eliminated by taking ratios of pair counts in a manner we discuss below . ratios of @xmath225 at fixed @xmath247 and different @xmath246 will therefore be the basis of our estimate of the clustering strength in the spectroscopic sample ; they are nearly immune to systematics from the irregular spectroscopic sampling and from the unknown selection function shape . the final complication is the significant uncertainty @xmath248 in each object s radial position @xmath211 from peculiar velocities and redshift uncertainties . this uncertainty can be treated in various ways . we will follow the standard approach and estimate the value of the correlation function only within bins whose radial size @xmath249 is large compared to @xmath248 . we are now ready to present the estimator that we adopt . letting @xmath250 denote the observed number of galaxy pairs with transverse separation @xmath251 and redshift separation @xmath252 , the discussion of the preceding paragraphs shows that the expected total number of pairs with radial separation @xmath252 ( and any transverse separation ) is @xmath253 \label{eq : expntot}\ ] ] where @xmath254 as long as @xmath255 is large enough that @xmath256 the ratio of pair counts @xmath257 will have expectation value @xmath258 regardless of angular selection effects , of uncertainties in the selection function , is independent of uncertainties in the selection function width . the expectation value @xmath259 _ will _ be affected by errors in the mean redshift of the selection function if these errors are large enough to significantly alter the mapping of redshifts and angles to distances . ] of peculiar velocities , and of redshift measurement errors , provided @xmath260 , @xmath261 , @xmath262 , @xmath263 , the selection function does not have strong features on scales smaller than @xmath171 , and @xmath223 varies slowly with @xmath85 . here @xmath264 is given by equation [ eq : expntot ] and @xmath265.\nonumber\ ] ] the second approximate equality in equation [ eq : exp_kb ] exploits the fact that @xmath266 is a very weak function of @xmath267 in realistic situations . we estimate the correlation function from the spectroscopic sample by finding the parameters required to match the observed ratio @xmath268 . in principle @xmath268 could be calculated separately for pairs in different bins of transverse separation @xmath267 , producing an estimate of the function @xmath269 and allowing one to estimate both @xmath69 and @xmath94 from the data . in practice a much larger sample is needed to fit for both @xmath69 and @xmath94 , so we hold @xmath94 fixed and estimate @xmath69 only . fortunately , as we will see , the best fit value of @xmath69 hardly changes as @xmath94 is varied across the range allowed by the galaxies angular clustering . the dependence of this estimator on the clustering strength is easy to understand intuitively . if the galaxies were unclustered ( @xmath270 ) , we would observe the same number of pairs at every separation and @xmath268 would be equal , on average , to the ratio of the bin sizes @xmath271 . correlation functions that peak near @xmath272 will produce more pairs in bins at smaller separations , driving @xmath268 away from @xmath273 . the difference between @xmath268 and @xmath273 is sensitive to the strength of the clustering , and therefore can be used to estimate it . adelberger ( 2005 ) uses monte carlo simulations to analyze the behavior of @xmath268 in more detail . figure [ fig : lsraw ] shows the raw ( integral - constraint correction @xmath274 ) values of the landy - szalay estimator @xmath134 ( equation [ eq : landyszalay ] ) as a function of angular separation for galaxies in the three samples . we limited these data , and our subsequent fits , to angular separations @xmath275 , since at larger scales the weak angular - clustering signal could be swamped by various low - level systematics . the uncertainty @xmath147 in each bin was taken to be the larger of @xmath276 ( peebles 1980 , 48 ) and the observed standard deviation of the mean of @xmath277 among the different fields in the survey . typically the two were comparable . numerical @xmath278 minimization produced the power - law fits shown with dashed lines . the correlation function parameters implied by the lbg fit , @xmath279 comoving mpc , @xmath280 , agree well with the estimates of giavalisco & dickinson ( 2001 ) which also assumed @xmath274 . it is clear , however , that these parameters can not be correct . substituting them into equations [ eq : s2_from_r0 ] , [ eq : bias_from_sigma ] , and [ eq : ic_cdm ] shows that a significant correction @xmath135 should have been applied to account for fluctuations on scales larger than the field - of - view . ( porciani & giavalisco 2002 reached a similar conclusion , and derived a result for lbgs that agrees well with the integral - constraint - corrected result we present below . ) figure [ fig : wthet_converg ] shows how our best - fit estimates of @xmath69 and @xmath94 change as the correction @xmath135 is applied . in our first iteration , described above , we assumed @xmath274 and calculated the correlation function @xmath281 . for the second iteration we assumed the value of @xmath135 implied by @xmath282 ( equations [ eq : exp_i ] , [ eq : s2_from_r0 ] , [ eq : bias_from_sigma ] , and [ eq : ic_cdm ] ) and estimated @xmath283 . for the third iteration we calculated @xmath135 from @xmath283 . the process continued in this way until convergence . it settled on the same final parameters if we initially assumed a value for @xmath135 that was too large . as figure [ fig : wthet_converg ] shows , the applied integral constraint corrections were comparable for each of the three samples . this is because the increase in @xmath135 implied by the longer correlation lengths at lower redshifts happened to be cancelled by a decrease in @xmath135 that resulted from the lower - redshift samples greater comoving depths . to check the plausibility of our adopted values for @xmath135 , we inserted into equation [ eq : rocheeales ] the best power - law fits to @xmath110 from each sample s final iteration . the equation returned @xmath284 , @xmath285 , and @xmath286 as the empirical estimates of @xmath135 for bm , bx , and lbg samples . these values differ somewhat from the ones we adopt ( figure [ fig : wthet_converg ] ) , because the empirical and cdm approaches ( see [ sec : integralconstraint ] ) make different assumptions about the behavior of @xmath110 on the scales @xmath287 where we can not measure it , but they are consistent within their large @xmath288 uncertainties and small changes ( @xmath289 ) to the best - fit values of @xmath94 would make them agree perfectly . readers may also be reassured to recall that our estimates of @xmath69 and @xmath94 agree well with those of porciani & giavalisco ( 2002 ) , who corrected for the integral constraint in a completely different way . we estimated the random uncertainty in @xmath69 and @xmath94 in two ways . first , we analyzed many alternate realizations of our @xmath110 measurements that were generated under the assumption that the uncertainties were uncorrelated . to create a single alternate realization , we added to each measured value @xmath151 a gaussian random deviate with standard deviation equal to its uncertainty @xmath147 . after creating numerous alternate realizations , we calculated and tabulated the values of @xmath69 and @xmath94 implied by each . our @xmath288 confidence interval on @xmath69 was defined as the range that contained 68.3% of the measured values of @xmath69 among the alternate data sets . the @xmath94 confidence interval is defined in the same way . we found @xmath290 mpc , @xmath291 ( lbg ) , @xmath292 mpc , @xmath293 ( bx ) , and @xmath294 mpc , @xmath295 ( bm ) . these numbers assume uncorrelated error bars and neglect the uncertainty in our selection functions . the uncertainty in @xmath135 is also neglected , since each alternate realization had the same integral constraint correction . our second approach was to extract random subcatalogs from our full galaxy catalog , estimate @xmath69 and @xmath94 for each with the iterative solution for @xmath110 described above , measure how the r.m.s . dispersion in best - fit parameter values depended on the number of sources in the subcatalog , and extrapolate to the full catalog size . in fact we created our random subcatalogs in pairs , with both subcatalogs in a pair containing a random fraction @xmath296 of the sources in the full catalog and no sources in common between them , and estimated the uncertainty in @xmath69 at a given value of @xmath172 as @xmath297 times the r.m.s . difference in @xmath69 among pair members . this prevented us from underestimating the random uncertainty in @xmath69 as @xmath298 , when random subcatalogs could otherwise contain nearly the same galaxies . with this approach we estimate @xmath299 mpc , @xmath300 ( lbg ) , @xmath301 mpc , @xmath302 ( bx ) , and @xmath303 mpc , @xmath304 ( bm ) . these numbers neglect the uncertainty in our selection function and do not fully account for the uncertainty in @xmath135 . they do not assume uncorrelated error bars , however , and we will therefore assume that they are more accurate than the numbers from the preceding paragraph . the uncertainty in @xmath135 is not negligible . according to equation [ eq : var_i ] the @xmath288 uncertainty in @xmath135 is @xmath305@xmath306% as large as @xmath135 itself for our 21 fields . as @xmath135 varies over its @xmath288 allowed range , the best - fit parameters @xmath69 and @xmath94 change by roughly as much as the uncertainties quoted above . adding these changes in quadrature to the random uncertainties above , and making the minor corrections for the selection function uncertainties discussed in [ sec : angularmethods ] , we arrive at the following estimates : @xmath307 mpc , @xmath308 ( lbg ) , @xmath309 mpc , @xmath310 ( bx ) , and @xmath311 mpc , @xmath312 ( bm ) . other investigators ( e.g. , giavalisco & dickinson 2001 ; foucaud et al . 2003 ) have claimed that at redshift @xmath313 bright galaxies cluster more strongly than faint galaxies . our data support this conclusion . figure [ fig : clust_seg ] shows that in the bx and lbg samples the correlation lengths of galaxies with @xmath314 ( @xmath315 mpc ) exceed those of galaxies with @xmath316 ( @xmath317 mpc ) by a significant amount . if we split the bx and lbg samples into two halves at random , rather than by apparent magnitude , the difference in correlation lengths between the two halves is this large only about @xmath318% ( bx ) to @xmath319% ( lbg ) of the time . the situation is less clear for the bm sample at @xmath30 , where the uncertainties are larger owing to the poor determination of @xmath101 from the small number of measured redshifts , but the data do not seem to suggest stronger clustering for brighter galaxies . on the one hand it makes sense that uv - brightness should become less associated with strong clustering as redshift decreases , since uv - bright galaxies are known to be weakly clustered at @xmath28 and @xmath29 . on the other , the overall clustering of the bm sample is still quite strong , stronger than one expects for typical collapsed objects at @xmath30 ( see below ) , and so it seems that the numerous objects too faint to satisfy our selection criteria must be less clustered than the bright objects in our sample . we will wait for additional spectroscopic observations of bm galaxies before commenting further . the dependence of clustering strength on luminosity can produce a false impression of a change in @xmath69 with redshift , since lower redshift samples will tend to reach fainter absolute luminosities . truncating the samples at a fixed absolute luminosity does not seem a good solution to us , however , since the bright end of the uv luminosity function rises rapidly towards higher redshifts ( e.g. , adelberger & steidel 2000 ) and one would therefore be comparing rare objects at lower redshifts to common objects at higher redshifts . a better approach is to compare galaxy samples of roughly the same comoving number density . since selection with a constant apparent magnitude limit @xmath320 happens to produce similar comoving number densities for the three samples ( see equations [ eq : nlbg ] and [ eq : nbxbm ] and the related discussion ) , we will continue to use the constant apparent magnitude limits of equation [ eq : maglimits ] for our samples in the remainder of the paper . readers should be aware that the reported value of each sample s correlation length is somewhat arbitrary for this reason . it reflects the characteristics of the sample as defined here , not of the general galaxy population at high redshift . for our estimator of the redshift clustering strength we took @xmath321 , the ratio of the number of galaxy pairs with comoving radial separation @xmath322 to those with comoving radial separation @xmath323 . since @xmath79 mpc is significantly larger than the uncertainty in each galaxy s radial position ( @xmath324 comoving mpc ) , and since @xmath325 mpc is significantly smaller than the selection functions widths ( @xmath326 comoving mpc ) , the expected value of @xmath321 should be given by equation [ eq : exp_k ] . we limited our analysis to pairs with transverse separations @xmath327 , equivalent to @xmath328 comoving mpc at @xmath329 , to reduce the sensitivity of our results to any deviations of the correlation function from a @xmath330 power - law on large scales . only the bx and lbg spectroscopic samples were large enough to allow meaningful measurements of @xmath321 . for @xmath330 , the right - hand side of equation [ eq : exp_k ] is equal to the observed ratio @xmath321 when @xmath331 ( lbg ) or @xmath332 ( bx ) comoving mpc . the values change by roughly @xmath333% as @xmath94 is varied from 1.45 to 1.65 . when analyzed with this technique , mock galaxy catalogs from the gif simulation ( [ sec : simulated_data ] ) with sizes similar to our observed catalogs show a @xmath41 dispersion in @xmath69 around the true mean of @xmath334 ( lbg ) and @xmath335 ( bx ) comoving mpc , so we adopt @xmath336 and @xmath337 mpc as the best fit values to @xmath69 for our spectroscopic catalogs . the results do not change significantly if we eliminate pairs with @xmath338 from the analysis , showing that we have measured genuine large - scale clustering and not merely the clumping of objects within individual halos . figure [ fig : kclust ] presents the result in a more graphical way . we divided our lists of galaxy pairs into bins according to transverse separation @xmath267 , then calculated @xmath321 separately for each bin . points with error bars show the values we found . the solid lines show the values predicted by the @xmath330 correlation function described in the preceding paragraph . the plot shows that the derived correlation function parameters provide a reasonable fit to the clustering of the galaxies in the spectroscopic sample . we presented two independent estimates of the correlation function for each of our galaxy samples . the estimates were consistent with each other , but the first , based on the galaxies angular clustering , had somewhat smaller uncertainties . this resulted from the larger size of the photometric sample , and was accentuated by the serious systematics in the spectroscopic sample that made us throw much of our data away . we will adopt the angular clustering results for the remainder of the paper . on small scales , smaller than roughly the typical radius @xmath339 of a virialized halo , the spatial clustering of galaxies is difficult to predict or interpret . it depends on the ease with which nearby galaxies merge with each other , on the ability of a galaxy to maintain its star - formation rate as it orbits within a larger potential well , on the possible impact of a galaxy s feed - back on its surroundings , and so on . on larger scales these baryonic complications have little effect and the correlation function of galaxies should be virtually identical to the correlation function of the halos that host them . to see that this is true , consider the galaxy correlation function in a simplified situation where every galaxy is associated with a single halo and the probability that a galaxy lies a distance @xmath340 from its halo s center is @xmath341 . in this case the galaxy distribution will be a poisson realization of the continuous density field @xmath342 that is created when the discrete halo distribution is convolved by @xmath172 , and the galaxy correlation function @xmath343 will be equal to the correlation function of @xmath344 ( see , e.g. , peebles 1980 , eq . 33.6 ) . since the halo powerspectrum is @xmath345 ( peebles 1980 , eq . 41.5 ) , where @xmath346 is the halo correlation function and @xmath137 is the halo number density , since the powerspectrum of @xmath344 is equal to @xmath347 , where @xmath348 is the fourier transform of @xmath172 , and since the powerspectrum and correlation function are fourier - transform pairs , the galaxy and halo correlation functions will be related through @xmath349 where @xmath350 denotes convolution and @xmath351 . now @xmath352 when @xmath340 is greater than some maximum separation @xmath353 ; galaxies can not be located arbitrarily far from the center of their halo . the first term will therefore be zero for @xmath354 . the second term will be almost exactly equal to @xmath355 at the same large separations , because plausible correlation functions do not not change significantly from @xmath340 to @xmath356 when @xmath357 . this shows that @xmath358 for large @xmath340 . although it was derived for a simplistic model , the result is more general . as long as galaxies are associated in some way with dark matter halos , and as long as there is some maximum separation @xmath353 between each galaxy and the center of its halo , galaxies will have the same correlation function as the halos that host them for @xmath359 . we will accordingly focus our attention solely on separations @xmath360 comoving mpc that are many times larger than the typical virial radius . readers who are skeptical of the claimed similarity between galaxy and halo correlation functions at @xmath359 may wish to consider figure [ fig : xi_halo_0z3 ] , which compares observed galaxy correlation functions at redshifts @xmath361 , @xmath362 , @xmath363 to the halo correlation functions in the gif - lcdm simulation outputs at the same redshifts ( see [ sec : simulated_data ] ) . the agreement is excellent . the implied association of galaxies with massive potential wells is hardly surprising . the interesting result is the characteristic mass of the virialized halos that contain the galaxies . this can be estimated since more massive halos cluster more strongly . figure [ fig : xi_halo ] compares the correlation functions at @xmath364 for galaxies in the bm , bx , and lbg samples to the correlation functions of virialized dark matter halos above various mass thresholds in the gif - lcdm simulation ( see [ sec : simulated_data ] ) . the agreement is best if galaxies in the bm , bx , and lbg samples are associated with halos of mass @xmath365 , @xmath366 , and @xmath367 , respectively . uncertainties in our measured correlation functions lead to uncertainties in the estimated characteristic masses . the size of these uncertainties can be gauged most cleanly by comparing our measured angular correlation function to the angular correlation functions of halos above different mass thresholds . we calculated halo angular correlation functions numerically from equation [ eq : limber0 ] after substituting in the observed redshift distributions for the different samples ( fig . [ fig : zhistos ] ) and the gif - lcdm halo correlation functions at the mean redshift of each sample . typical results are shown in figure [ fig : wthet_halo ] . the halo angular correlation functions fall significantly below the data on small scales , since ( by definition ) a halo can not have another halo as a neighbor within the radius @xmath339 . as argued above , however , it is the larger scales @xmath360 comoving mpc ( i.e. , @xmath368 ) that are relevant for comparing to galaxies , and here the agreement is good . the uncertainty in the galaxies implied mass scale is dominated by the uncertainty in the integral constraint correction @xmath135 that was applied to the data , since this moves all points up or down together . rough @xmath288 limits on the threshold masses of the hosting halos are @xmath369 readers unfamiliar with the idea of threshold masses may wish to see footnote 6 in [ sec : summary ] , below , for further discussion of how to interpret them . one implication of figure [ fig : wthet_halo ] is that some halos are occupied by more than one of the galaxies in our samples ; the data on small scales are strongly inconsistent with the correlation functions that assumed one object per halo . ( this was first pointed out by wechsler et al . ( 2001 ) and later denied by porciani & giavalisco ( 2002 ) . our analysis agrees far better with that of wechsler et al . ) the fraction of lbgs that reside in the same halo as another lbg can be calculated as follows . if there were never more than one lbg per halo , the expected number of lbgs within @xmath370 of a randomly chosen lbg would be @xmath371 where @xmath372 is the surface density of lbgs , @xmath373 , and @xmath374 is halo angular correlation function that fits our data best for @xmath375 . the actual number of lbgs within @xmath370 , @xmath376 , is given by the same equation with the halo correlation function @xmath374 replaced by the galaxy correlation function @xmath377 . the expected number of additional lbgs in a halo that is known to contain one lbg , @xmath378 , is equal to the difference between @xmath376 and @xmath379 . numerically integrating the angular correlation functions for observed galaxies and simulated halos , and multiplying by the lbg surface density from table [ tab : fields ] , we estimate @xmath380 . the numbers for the other samples , estimated with a similar approach , are @xmath381 and @xmath382 . _ some _ galaxies must share the same halo to explain the data , but the required number is small . note that we are referring solely to galaxies that satisfy our color and magnitude selection criteria ; we obviously can not say anything about the spatial distribution of galaxies that are not in our samples . having established a rough characteristic mass for the halos that contain our galaxies , we can compare the galaxy and halo number densities and estimate what fraction of the most massive halos at redshifts @xmath383 do not contain a galaxy that is detectable with our techniques . according to adelberger & steidel ( 2000 ) the comoving number density of lbgs brighter than @xmath384 is @xmath385 for @xmath71 , @xmath72 . combining the bm and bx completeness coefficients in table 3 of adelberger et al . ( 2004 ) with the surface densities in table [ tab : fields ] of this paper , we estimate @xmath386 where the assigned uncertainties of 50% are approximate guesses that will be refined later with monte carlo simulations . number densities for the population of halos that contain the galaxies can be estimated from the gif - lcdm simulation given the range of halo masses ( equation [ eq : mass_scales ] ) that are compatible with the galaxies clustering strength . figure [ fig : nhalo ] shows that the number density of galaxies in the bm , bx , and lbg samples is comparable to the number density of halos that can host them . as we will discuss in [ sec : summary ] , below , this implies that the duty cycle of star - formation in the galaxies must be of order unity and shows that our surveys can not be severely incomplete . similar arguments have been made by adelberger et al . ( 1998 ) , giavalisco & dickinson ( 2000 ) , martini & weinberg ( 2001 ) , and others . the spatial distribution of a population of galaxies evolves in an easily predictable way as it responds to the gravitational pull of dark matter . we used a simple approach to estimate this evolution from the lcdm - gif simulation . after connecting the observed galaxies to halos with a range of masses ( equation [ eq : mass_scales ] ) , we measured the evolution in the clustering of those halos in the simulation and assumed that the galaxies clustering would evolve in the same way . [ sec : simulated_data ] . figure [ fig : r0_vs_z ] shows the implied change in correlation length @xmath69 with time for the galaxies in our samples . by @xmath185 galaxies in the lbg sample will have a correlation length similar to measured correlation lengths of galaxies in the bm and bx samples . by @xmath27 their correlation length will be similar to the correlation length of early - type ( i.e. , `` absorption line '' ) galaxies in the sample of coil et al . by @xmath210 their correlation length will be equal to the observed correlation length of ellipticals ( budavri et al . 2003 ) . the evolution of @xmath69 for galaxies in the bm and bx samples is similar . figures [ fig : z1gals ] and [ fig : z0gals ] present a more detailed view of the possible relationships between the descendants of galaxies in our samples and various galaxy populations at lower redshift . figure [ fig : z1gals ] shows that at @xmath29 the descendants clustering strength will significantly exceed that of average galaxies in optical magnitude - limited surveys . since these surveys are dominated by star - forming ( `` emission - line '' ) galaxies , we can conclude that the typical descendant is no longer forming stars by @xmath27 . emission - line sample . ( our estimate of the number density in the emission - line sample , shown in figure [ fig : nhalo ] , is from a. coil , private communication . ) ] a similar point was made by adelberger ( 2000 ) and coil et al . the stronger clustering of redder and brighter sub - populations at @xmath27 is more compatible with the descendants expected clustering , but the match is best for the sub - population with early - type spectra . this is especially true for descendants of the brightest galaxies in the bx and lbg samples . although the observed number density of early - type galaxies at redshift @xmath27 , roughly @xmath387 ( chen et al . 2003 ) , is consistent with the idea that most had a bm / bx / lbg galaxy as a progenitor , we can not rule out the idea that some had multiple merged bm / bx / lbg progenitors and others had none . figure [ fig : z0gals ] is an analogous plot for redshift @xmath210 . correlation lengths for various populations in the sloan digital sky survey ( sdss ) were taken from budavri et al . number densities were calculated assuming a surveyed volume of @xmath388 mpc@xmath389 with an uncertainty of @xmath390% ( t. budavri 2004 , private communication ) . the expected @xmath361 clustering strength of typical lbg descendants ( darker shaded box ) agrees best with the clustering of galaxies with early - type seds in the budavri et al . ( 2003 ) sample . the galaxies in our high - redshift samples are roughly as numerous as these early - type galaxies , though the possibility of merging prevents us from estimating the number density of their bm / bx / lbg descendants at @xmath210 . the descendants of brighter lbgs will have a correlation length closer to that of bright ellipticals , though there are probably not enough descendants to account for the entire bright elliptical population . the first part of the paper was concerned with measuring the spatial clustering of large samples of star - forming galaxies at redshifts @xmath30 , @xmath5 , and @xmath6 . we fit a three - dimensional correlation of the form @xmath391 to the galaxies angular clustering with standard techniques and to the galaxies redshift clustering with a new estimator . the new estimator , @xmath13 , is insensitive to many of the possible systematic biases in our spectroscopic surveys . we reached consistent conclusions about the correlation function with the two approaches , but adopted the angular results since their random uncertainties were somewhat smaller . as given in [ sec : angularresults ] , the best - fit correlation - function parameters from the angular clustering are @xmath392 where bm , bx , and lbg are the names we have given the @xmath7 color - selection criteria that we used to find galaxies at @xmath30 , @xmath5 , and @xmath6 ( [ sec : observeddata ] ) . the quoted @xmath288 errors include random uncertainties , uncertainties in the integral constraint corrections , and uncertainties in the shapes of the selection functions . since the value of @xmath69 depends on the apparent - magnitude limit of the samples , at least in the two higher - redshift bins ( figure [ fig : clust_seg ] ) , the reported values of @xmath69 are somewhat arbitrary . we chose to limit each sample to a fixed range of apparent magnitudes , @xmath393 , on the grounds that this resulted in a similar comoving density in each redshift bin . different magnitude limits would have resulted in different correlation lengths . readers should be aware that the numbers we give are appropriate to the samples as we have defined them , not to the general galaxy population at high redshifts . the second part of the paper was based on the proposition that wmap ( spergel et al . 2003 ) and other experiments have given us reliable measurements of the cosmological parameters and of the shape of the dark matter power - spectrum . this implies that we know what sorts of virialized dark - matter halos existed at different epochs in the past and how their spatial distribution evolved over time . since galaxies reside within dark - matter halos , they will have the same correlation function as the halos on large scales ( equation [ eq : xigxih ] ) . the galaxies clustering should therefore tell us what sort of halo they reside within . we found a good match ( figures [ fig : xi_halo ] and [ fig : wthet_halo ] ) between the correlation functions of the galaxies and of halos with threshold masses ranging from @xmath394 ( lbg ) to @xmath395 ( bm ) . , we mean the subset of halos with mass @xmath396 . the median mass of this subset is @xmath397 in the gif - lcdm simulation at @xmath398 . halo subsets can be defined with schemes more elaborate than our simple mass threshold ( e.g. , kauffmann et al . 1999 , bullock , wechsler , & somerville 2002 ) , but the differences between the possible schemes are too small to affect our analysis when the subsets they produce are constrained to have the same clustering on large scales . ] equation [ eq : mass_scales ] gives rough @xmath288 limits on the halos total masses . similar masses for lyman - break galaxies have been derived with the same approach by jing & suto ( 1998 ) , adelberger et al . ( 1998 ) , giavalisco & dickinson ( 2000 ) , porciani & giavalisco ( 2002 ) , and others . although the estimated masses were derived solely from the galaxy clustering , they seem reasonable on other grounds . they can not be much higher . the number density of halos would be lower than the number density of lbgs , for example , if the halo mass were greater than @xmath399 . such large halo masses would be possible only if significantly more than one lbg resided in the typical halo , and that is something that our observations rule out ( [ sec : correspondence ] ) . nor can they be much lower . the halos that contain lbgs would not have enough baryons to form the median lbg stellar mass of roughly @xmath400 ( shapley et al . 2001 times lower than the value shapley et al . calculated for an imf with a salpeter slope between @xmath401 and @xmath402 . their assumed salpeter imf is probably unrealistic since the imf in the solar neighborhood is known to flatten near @xmath403 and eventually turn over at lower masses . see leitherer ( 1998 ) or renzini ( 2004 ) for further discussion . ] ) unless their total mass were greater than about @xmath404 . we should mention in passing that our best - fit halo masses seem to imply that only a small fraction of the baryons in the halos are associated with the observed galaxies . for example , the best - fit mass threshold of @xmath394 for lbgs corresponds to a median total mass of @xmath405 and median baryonic mass of @xmath406 ( for @xmath407 , spergel et al . 2003 ) , roughly ten times larger than the observed stellar masses of lbgs . since the @xmath408 supernovae that explode during the assembly of the typical lbg s stellar mass will eject roughly @xmath409 of metals ( e.g. , woosley & weaver 1995 ) , enough to enrich at most @xmath410 of gas to lbgs typical metallicities of @xmath411 ( pettini et al . 2002 ) , their observed interstellar gas can not contain a large fraction of the remaining baryons . these baryons need not be associated with other objects in the halo , however . they may be locked in dim stars that formed in previous episodes of star - formation ( e.g. , papovich , dickinson , & ferguson 2001 ) , or may have been heated by various processes to undetectably high temperatures . the latter is presumably the case for nearby galaxies , whose ratios of mass in stars and gas to total mass are usually also smaller than the wmap value @xmath407 . the milky way , for example , has a total mass of @xmath412 ( zaritsky 1999 ; wilkinson & evans 1999 ) and a mass in gas and stars of only @xmath413 ( k. freeman 2004 , private communication ) , yet few would assert that its missing baryons belong to another galaxy in its halo . after establishing plausible total masses for the halos associated with the galaxies , we considered some of the implications . our arguments were not new ( see , e.g. , moustakas & somerville 2002 , martini & weinberg 2001 , adelberger et al . they seemed worth revisiting only because our knowledge of the cosmogony , of the local universe , and of high - redshift galaxies has improved so much in the last few years . we began by estimating the completeness of our surveys from a comparison of the galaxies number densities to the number densities of halos with similar clustering strength ( figure [ fig : nhalo ] ) . similar number densities would imply that almost all of the most massive halos contained a galaxy that satisfied our selection criteria ; a much lower galaxy number density would imply that most of the galaxies in massive halos are missed by our survey . defining @xmath273 as the ratio of galaxy to halo number density , we found rough @xmath288 limits of @xmath414 , @xmath415 , and @xmath416 . these limits were derived from the clustering at radii @xmath360 comoving mpc . the clustering on smaller scales , sensitive to the possible presence of more than one galaxy in a halo , implies that the upper limits on @xmath417 and @xmath418 should be revised downwards to @xmath419 . the data appear consistent with the claim of franx et al . ( 2003 ) that our selection criteria find roughly half of the most massive galaxies at @xmath185 . a completeness of order @xmath306% seems plausible to us for other reasons as well . shapley et al . ( 2001 ) estimate a lifetime for the typical lbg of @xmath420 yr , for example , which implies that the typical lbg will be bright enough for us to detect for only about half of the time that elapsed between the survey selection limits of @xmath421 and @xmath422 . we considered next the way the clustering of the galaxies would evolve ( figure [ fig : r0_vs_z ] ) . analysis of the gif - lcdm simulation suggested that the correlation length of lbg descendants would be similar by @xmath31 to the correlation length of galaxies in the bx sample and by @xmath30 to the correlation length of galaxies in the bm sample . the spatial clustering is therefore consistent with the idea that we are seeing the same population at all three redshifts , though the selection criteria s @xmath423% incompleteness leaves room for the populations to be distinct and the difference in stellar masses between the lbg ( @xmath400 ) and bx ( @xmath424 ; steidel et al . 2005 , in preparation ) samples may not be consistent with continuous star - formation at observed rates through the elapsed time . turning our attention to lower redshifts , we found that at @xmath27 our descendants clustering would most closely match the observed clustering of galaxies that are red and bright and have early - type spectra ( figure [ fig : z1gals ] ) . by @xmath210 the estimated clustering of the descendants suggested elliptical galaxies as the most likely counterparts ( figure [ fig : z0gals ] ) . the correspondence is especially hard to dispute for the descendants of the brightest and most strongly clustered galaxies in the high - redshift samples . one conclusion seems difficult to escape : the descendants of the galaxies in our samples must have significantly larger stellar masses than their high - redshift forebears . only @xmath425% of the total stellar mass in the local universe is found in galaxies with stellar masses smaller than @xmath426 ( kauffmann et al . 2003 ) , similar to the values in our high - redshift samples , and these faint galaxies are too weakly clustered to have descended from the galaxies we find at @xmath2 . only elliptical galaxies have a spatial distribution consistent with our expectations for the descendants , and the characteristic stellar mass of ellipticals is @xmath404 ( padmanabhan et al . the increase in stellar mass from @xmath185 to @xmath26 could have been produced by ongoing star formation or by mergers . our results at redshift @xmath27 may favor the latter , but in any case our findings are strongly inconsistent with traditional notions of monolithic collapse . it would be unfair to close without mentioning one population of high - redshift galaxies that we have ignored completely . these are the bright ( @xmath427 ) near - infrared - selected galaxies . their reported star - formation rates ( @xmath428/yr ) , correlation lengths ( @xmath429 mpc , daddi et al . 2004 ) and stellar masses ( @xmath430 ; van dokkum et al . 2004 ) are extraordinary , far larger than the corresponding values for typical galaxies in our samples . see , e.g. , shapley et al . 2004 and adelberger et al . 2005 ] galaxies with similarly extreme properties are not a negligible component of the high redshift universe . the shape of the @xmath431 m background implies that up to a third ( cowie , barger , & kneib 2002 ) of all stars could have formed in objects with star - formation rates greater than @xmath432 ; halos at @xmath313 with the large masses @xmath433 implied by @xmath434@xmath435 mpc contain in total almost @xmath199% as many baryons as the more numerous and smaller halos with @xmath436 that contain lbgs ; objects with stellar masses @xmath437 contain nearly @xmath318% of all stars in the local universe ( kauffmann et al . 2003 ) and @xmath199% of the stars in local elliptical galaxies ( padmanabhan et al . no treatment will be entirely complete if it neglects galaxies similar to those found in near - ir surveys . the galaxies we studied are neither the most massive , nor the most rapidly star - forming , nor the most clustered galaxies in the high redshift universe , but it is precisely this that makes them plausible progenitors for the early - type galaxies that surround us today . we are grateful to the virgo consortium for its public release of the gif simulation data and to g. kauffmann for bringing the data to our attention . a. coil and t. budavri responded helpfully to our questions about their data . m. giavalisco , the referee , gave us an insightful report . kla , aes , and nar were supported by fellowships from the carnegie institute of washington , the miller foundation , and the national science foundation . dke and ccs were supported by grant ast 03 - 07263 from the national science foundation . adelberger , k.l . , steidel , c.c . , giavalisco , m. , dickinson , m. , pettini , m. & kellogg , m. 1998 , apj , 505 , 18 adelberger , k.l . 2000 , in `` clustering at high redshift , '' asp conf series 200 , eds . mazure , le fvre , le brun , p13 adelberger , k.l . & steidel , c.c . 2000 , apj , 544 , 218 adelberger , k.l . , steidel , c.c . , shapley , a.e . , hunt , m.p . , erb , d.k . , reddy , n.a . , & pettini , m. 2004 , apj , 607 , 226 adelberger , k.l . , erb , d.k . , et al . 2005 , apj , submitted arnouts , s. , moscardini , l. , vanzella , e. , colombi , s. , cristiani , s. , fontana , a. , giallongo , e. , matarrese , s. , & saracco , p. 2002 , mnras , 329 , 355 baldry , i.k . & glazebrook , k. 2003 , apj , 593 , 258 budavri , t. et al . 2003 , apj , 595 , 59 bullock , j.s . , wechsler , r.h . , & somerville , r.s . 2002 , mnras , 329 , 246 chen , h .- w . , marzke , r.o . , mccarthy , p.j . , martini , p. , carlberg , r.g . , persson , s.e . , bunker , a. , bridge , c.r . , & abraham , r.g . , 2003 , apj , 586 , 745 coil , a.l . et al . , 2004 , apj , in press ( astro - 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we analyzed the spatial distribution of @xmath0 photometrically selected galaxies with magnitude @xmath1 and redshift @xmath2 in 21 fields with a total area of @xmath3 square degrees . the galaxies were divided into three subsamples , with mean redshifts @xmath4 , @xmath5 , @xmath6 , according to the @xmath7 selection criteria of adelberger et al . ( 2004 ) and steidel et al . ( 2003 ) . combining the galaxies measured angular clustering with redshift distributions inferred from @xmath8 spectroscopic redshifts , we find comoving correlation lengths at the three redshifts of @xmath9 , @xmath10 , and @xmath11 mpc , respectively , and infer a roughly constant correlation function slope of @xmath12 . we derive similar numbers from the @xmath8 object spectroscopic sample itself with a new statistic , @xmath13 , that is insensitive to many possible systematics . galaxies that are bright in @xmath14 ( @xmath15@xmath16 ) cluster more strongly than fainter galaxies at @xmath17 and @xmath18 but not , apparently , at @xmath19 . comparison to a numerical simulation that is consistent with recent wmap observations suggests that galaxies in our samples are associated with dark matter halos of mass @xmath20@xmath21 ( @xmath17 ) , @xmath22@xmath23 ( @xmath18 ) , @xmath24@xmath25 ( @xmath19 ) , and that a small fraction of the halos contain more than one galaxy that satisfies our selection criteria . adding recent observations of galaxy clustering at @xmath26 and @xmath27 to the simulation results , we conclude that the typical object in our samples will evolve into an elliptical galaxy by redshift @xmath28 and will already have an early - type spectrum by redshift @xmath29 . we comment briefly on the implied relationship between galaxies in our survey and those selected with other techniques .

You are an expert at summarizing long articles. Proceed to summarize the following text: the way in which a galaxy has assembled its stars is reflected in the distribution of stellar orbits . for example , collisionless @xmath0-body collapse simulations predict a predominance of radial orbits in the final remnant @xcite . in contrast , collisionless galaxy merger simulations predict a variety of orbital compositions , depending on progenitor properties @xcite , the merging geometry @xcite , the progenitor mass ratios @xcite and the presence of dissipational components @xcite . stars in galaxies are approximately collisionless and the orbital structure once a galaxy has approached a quasi - steady - state is conserved for a long time . to a certain extent then , the assembly mechanism of early - type galaxies can be constrained from their present - day orbital structure . a global characteristic of the distribution of stellar orbits is its anisotropy . traditionally , anisotropies of elliptical galaxies have been inferred from the ( @xmath1 ) diagram . in particular , the rotation of bright ellipticals has been shown to be insufficient to account for their flattening @xcite . however , whether fainter , fast - rotating ellipticals are flattened by rotation is less easy to determine from the ( @xmath1 ) diagram , because isotropic as well as anisotropic systems can rotate . in fact , fully general axisymmetric dynamical models recently have revealed an anisotropic orbital structure in even the flattest , fast rotating objects @xcite . one goal of this paper is to investigate numerically the connection between anisotropy , rotation and flattening in spheroidal stellar systems . in addition , we present global anisotropies for a sample of coma early - type galaxies . these anisotropies are derived by analysing long - slit stellar absorption line kinematics with axisymmetric orbit models . our dynamical models include dark matter halos . previous anisotropy determinations for larger samples of ellipticals ( including dark matter ) were restricted to round and non - rotating systems , assuming spherical symmetry @xcite . spherical models do not account for galaxy flattening . in the simplest case , a flattened system is axially symmetric . early axisymmetric models , however , did not cover all possible anisotropies ( and orbital structures , respectively ; e.g. @xcite ) . fully general , orbit - based axisymmetric dynamical models have so far only been applied to the inner regions of ellipticals and the orbital analysis was made under the assumption that mass follows light ( e.g. @xcite and @xcite ) . by the mass - anisotropy degeneracy , the neglect of dark matter could translate to a systematic bias in the corresponding orbital structure ( e.g. @xcite ) . comparison of anisotropies derived with and without dark matter will allow one to quantify such a possible bias . we also discuss anisotropies derived from modelling mock observations of synthetic @xmath0-body merger remnants . one motivation to do so is that dynamical models of axisymmetric systems may not be unique . for example , the deprojection of an axisymmetric galaxy is intrinsically degenerate @xcite . uncertainties in the intrinsic shape thereby propagate into uncertainties on the derived masses and anisotropies ( e.g. @xcite ) . moreover , the reconstruction of an axisymmetric orbital system is suspected to be further degenerate with the recovered mass ( e.g. the discussion in @xcite ) . the case for a generic degeneracy , beyond the effects of noise and incompleteness of the data , is still uncertain ( e.g. @xcite ) . numerical studies of a few idealised axisymmetric toy models indicate degeneracies to be moderate when modelling realistically noisy data sets ( @xcite , @xcite ) . since we know the true structure of our @xmath0-body modelling targets , we can extend on these studies and further investigate potential systematics in the models over a broader sample of test cases . another motivation to model @xmath0-body merger remnants is to probe whether ellipticals have formed by merging . this requires a comparison of the orbital structure in real ellipticals with predictions of @xmath0-body simulations ( e.g. @xcite ) . however , because of the symmetry assumptions in models of real galaxies , it is not straight forward to compare _ intrinsic _ properties of @xmath0-body simulations with _ models _ of real galaxies . to avoid the related systematics , we here compare models of real galaxies with _ similar models _ of synthetic @xmath0-body merger simulations and both are indicative for true differences between real galaxies and merger predictions . the galaxy and @xmath0-body merger samples and the modelling technique are briefly outlined in sec . [ sec : data ] . toy models of various flattening and anisotropy are discussed in sec . [ sec : theory ] . the anisotropies of real galaxies are presented in sec . [ sec : galaxies ] and compared with models of @xmath0-body merger remnants in sec . [ sec : merger ] . implications for the formation process of early - type galaxies are discussed in sec . [ sec : discussion ] and we summarise our results in sec . [ sec : summary ] . the influence of regularisation and the inclusion of dark matter halos on reconstructed galaxy anisotropies is discussed in app . [ sec : app ] . in app . [ app : entropy ] we briefly discuss the connection between anisotropy and the shape of the circular velocity curve in maximum entropy models . we assume that the coma cluster is at a distance of 100 mpc . .summary of coma galaxy anisotropies . ( 1 - 2 ) : galaxy i d ( gmp numbers from @xcite ) ; ( 3 ) : intrinsic ellipticity @xmath2 ; ( 4 - 6 ) : anisotropy parameters @xmath3 , @xmath4 and @xmath5 ( cf . equations [ eq : delta]-[eq : gamma ] ) of the best - fit dynamical model ; ( 7 ) : @xmath6 , i.e. @xmath7 normalised by the approximate value @xmath8 of an ( edge - on ) isotropic rotator with the same flattening . note that @xmath6 is an observable , i.e. it combines _ observed _ ellipticities @xmath9 ( from column 10 of tab . 1 of @xcite ) and observed velocities @xmath10 and @xmath11 , without reference to any dynamical model . [ cols="<,^,^,^,^,^,^,^ " , ] a complete description of a stellar system is given by its distribution function @xmath12 ( df ; the density in 6-dimensional phase - space ) . in a steady - state system the df @xmath12 depends on the phase - space coordinates only through the integrals of motion @xcite . axisymmetric potentials , which are considered here , admit the two classical integrals of motion energy ( @xmath13 ) and z - component of the angular momentum ( @xmath14 ) . in addition , many orbits in astrophysically relevant potentials are characterised by another , non - classical , so - called third integral ( @xmath15 ; @xcite ) . since integrals of motion label orbits and vice - versa , a steady - state system can be viewed as a superposition of orbits , each with constant phase - space density . let @xmath16 denote the phase - space density along orbit @xmath17 , then the total amount of light @xmath18 on the orbit equals @xmath19 ( @xmath20 is the orbit s phase - space volume ) . the df or the weights of a suitable orbit superposition model determine the spatial density @xmath21 and intrinsic velocity dispersions @xmath22 via @xmath23 and @xmath24 @xmath25 in the following we will only consider @xmath26 , where @xmath27 is the short axis of the density distribution , @xmath28 is the azimuth around this axis , @xmath29 is a fixed cartesian coordinate parallel to the equatorial plane and @xmath30 is a cylindrical radius . let @xmath31 denote the total ( unordered ) kinetic energy in coordinate direction @xmath17 , then the global anisotropy of an axisymmetric stellar system can be quantified , for example , by the ratios @xmath32 @xmath33 and @xmath34 @xcite . in axisymmetric systems , the three anisotropy parameters are related via @xmath35 non - rotating , isotropic spherical systems as well as classical isotropic rotators obey @xmath36 . the df of real galaxies is not known , but has to be reconstructed from photometric and kinematic observations . in the next two subsections we will describe the two samples of real and simulated galaxies discussed in this paper and will briefly outline our modelling method . our sample of observed galaxies ( coma in the following ) consists of 19 coma early - types from @xcite . it comprises 2 central cd galaxies , 10 ordinary giant ellipticals and 7 s0 or intermediate galaxies with luminosities @xmath37 ( a single fainter object with @xmath38 is also included in the sample ) . the galaxies are drawn from the luminosity limited sample of @xcite and are distributed all over the cluster . high - resolution radial profiles of surface brightness , ellipticity and isophotal shape parameters @xmath39 and @xmath40 ( up to @xmath41 in some cases ; cf . @xcite for a definition of the isophotal shape parameters ) derived from a combination of hst and ground - based imaging were used to calculate the deprojected 3d luminosity distribution for several inclinations . the photometric data are complemented by long - slit stellar absorption line kinematics along @xmath42 position - angles per galaxy . the kinematic data consists of radial profiles of mean velocity , velocity dispersion and higher - order moments of the line - of - sight velocity distribution and reach out to @xmath43 . details about the photometric and kinematic data can be found in @xcite , @xcite , @xcite , @xcite and @xcite . these data were modelled with our implementation of schwarzschild s ( 1979 ) orbit superposition technique for axisymmetric potentials @xcite . for each galaxy , we probed for a variety of mass models , composed of a stellar mass density ( from the deprojected light profile ) and a parametric dark halo profile . the parameter space for the mass models spans the inclination , the stellar mass - to - light ratio and the dark halo parameters . in each trial potential the best - fit orbit model is calculated by maximising @xmath44 where @xmath45 quantifies deviations between observed and modelled kinematics - body merger remnants . ] . the function @xmath46 is used to smooth the orbit models . in the absence of any other constraints the maximisation of s yields orbital weights @xmath47 @xcite , such that the yet not specified @xmath48 can be regarded as weight - factors for the @xmath18 . when modelling real galaxies or mock observations of @xmath0-body merger remnants , we assume that there is no preferred region is phase - space and each orbit is given an a priori - weight equal to its phase - space volume : @xmath49 . then , @xmath50 equals the boltzmann entropy , which drives models towards a constant density in phase - space . the ( binned ) deprojected luminosity density is used as a boundary condition to solve equation ( [ orbitmaxs ] ) and the regularisation parameter @xmath51 in equation ( [ orbitmaxs ] ) has been calibrated by means of monte - carlo simulations @xcite . the final , best - fit orbit model is obtained from a @xmath45-analysis . we have applied the same modelling code to mock observations of synthetic @xmath0-body merger simulations . in brief , we have modelled six merger remnants , each projected along its three principal axes ( models of projections along the long , intermediate and short axis of the merger remnants will be shortly referred to as x , y and z - models later on ) . the six merger remnants are taken from the sample of collisionless disk+bulge+halo mergers of @xcite . they have mass ratios between 1:1 and 4:1 and sample the entire distribution of intrinsic shapes and orbital make - ups , including extreme cases . an orbital analysis of the @xmath0-body systems is given in @xcite . we have simulated typical coma observations for each projection : the merger remnants were placed at a distance of 100 mpc and photometric and kinematic profiles with similar resolution and spatial coverage as in the coma sample have been extracted . for a detailed discussion of the models the reader is referred to @xcite . oblate stellar systems can owe their shapes to a variety of different orbital configurations . classically , one has often distinguished between two proto - typical cases : flattening by rotation and flattening by anisotropy . thereby , flattening by rotation is used to term an otherwise round and isotropic system which appears flattened and rotating by extra - light on near - equatorial , high angular - momentum orbits ( populated with the same sense of rotation ) . flattening by anisotropy refers to systems with a depression of stars with high velocities perpendicular to the equatorial plane ( @xmath52 ) . however , in fact there are infinitely many orbit superpositions that account for a given galaxy shape . some of these are discussed in @xcite . different orbital structures can be distinguished by their different anisotropies . in the following we will numerically construct ( self - consistent ) toy models that are designed to ( 1 ) reproduce a given , flattened , density distribution exactly , but ( 2 ) have different intrinsic anisotropies . the models are orbit - based and similar to those described in sec . [ sec : data ] . however , here we only require the models to reproduce a given density ( @xmath53 in equation [ orbitmaxs ] ) . various expressions for the factors @xmath48 in equation ( [ sgen ] ) will be used to impose different anisotropy structures ( see below ) . for our simple toy models we assume a stellar density @xmath54 @xcite with @xmath55 equations ( [ eq : dehnen ] ) and ( [ eq : flattening ] ) describe systems with constant flattening @xmath56 . they approximate the light profiles of elliptical galaxies reasonably well . [ [ flattening - and - maximum - entropy . ] ] flattening and maximum entropy . let @xmath57 denote the df that maximises the entropy of equation ( [ maxs ] ) subject to the density constraints . the squares in the top panel of fig . [ aniso_prof ] illustrate the connection between anisotropy and flattening for @xmath57 : the three panels show the anisotropy parameters from equations ( [ eq : delta]-[eq : gamma ] ) as a function of the intrinsic ellipticity @xmath58 ( cf . equation [ eq : flattening ] ) . while @xmath3 and @xmath4 increase with flattening , @xmath5 is roughly constant . in maximum - entropy models the flattening thus arises from a suppression of energy in @xmath27-direction , while the balance between the energies in @xmath30 and @xmath28 is roughly conserved . in this sense , the maximum entropy models @xmath57 resemble the classical case of flattening by anisotropy . the only difference is that @xmath59 ( cf . [ app : entropy ] for a discussion of @xmath5 ) . note that we calculated the toy models with the same library setup as used for the coma galaxy models . [ [ flattening - by - a - classical - ffel_z . ] ] flattening by a classical @xmath60 . a classical two - integral df @xmath61 , which only depends on @xmath13 and @xmath14 , can be approximated via equations ( [ orbitmaxs],[sgen ] ) with @xmath62 @xmath63 and @xmath64 ( @xmath0 is the total number of orbits ) . equation ( [ newv ] ) derives from the constraint that for @xmath65 , the phase - space density of any orbit @xmath17 with energy @xmath66 and angular momentum @xmath67 has to equal the mean phase - space density of all orbits @xmath68 with the same @xmath69 and @xmath70 , i.e. @xmath71 this case is included in fig . [ aniso_prof ] by the circles . that the @xmath48 from equation ( [ newv ] ) indeed yield @xmath72 is demonstrated by @xmath73 . the flattening of the corresponding systems comes from an excess energy in @xmath28-direction with respect to the isotropic case ( @xmath74 ; orbits with high angular momentum are strongly populated ) . the relationship between @xmath3 and @xmath2 is similar as in maximum entropy models . note that dfs @xmath72 develop noticeable phase - space density peaks on orbits with high angular momentum @xcite . it is likely this property that lowers their entropy as compared to the @xmath57 models . flattening by anisotropy mainly involves shell orbits which approach closely the intrinsic minor - axis . their phase - space volumes are much larger than those of equatorial near - circular orbits with high angular momentum . even a small change in the phase - space density along shell orbits can reduce the amount of light near the minor - axis considerably and , thus , result in a significant flattening . the larger fraction of phase - space involved in this type of flattening , compared with a strong overpopulation of the relatively small region in phase - space occupied by near - circular orbits ( as in cases where @xmath75 ) explains why objects which are flattened by anisotropy have the higher entropy . [ [ flattening - with - radial - anisotropy . ] ] flattening with radial anisotropy . model dfs @xmath76 obtained with @xmath77 ^ 4 \times v(i)\ ] ] are biased towards orbits with a large difference @xmath78 between apocentre and pericentre radius ( radially extended orbits ) . such models are radially anisotropic ( @xmath79 ; cf . triangles in fig . [ aniso_prof ] ) . the relationship between @xmath3 and @xmath2 is again similar as in the previous models . the latter is no surprise , as for self - consistent ellipsoids with constant flattening , @xmath80 can be calculated from the tensor virial theorem @xcite : @xmath81 where @xmath82 and @xmath83 the solid line in the upper - left panel of fig . [ aniso_prof ] shows relation ( [ eq : deltavir ] ) . our numerically constructed orbit models follow this line well . note that , if @xmath0 dfs @xmath84 project each to the same spatial density , then any convex linear combination @xmath85 with @xmath86 will do so . the properties of @xmath87 will be intermediate between those of the individual @xmath84 . the just discussed toy models ( and any linear combination of them ) are non - rotating , because in our choices for @xmath48 we havent distinguished between prograde and retrograde orbits . a large variety of rotation patterns can be constructed from any df @xmath12 as follows : each orbit in an axisymmetric potential comes in two flavours , one prograde ( with positive @xmath88 ) and one retrograde ( @xmath89 ) . both share the same spatial shape but differ only in the sign of the velocity component around the axis of symmetry . thus , the spatial density will only depend on the sum @xmath90 of light on corresponding prograde and retrograde orbits . the amount of rotation , instead , will depend on the difference between the population of the prograde and retrograde orbits , respectively . this can be quantified , for example , by the fraction @xmath91 of light on the prograde of each orbit pair : @xmath92 ( @xmath93 ) . for simplicity , let s assume from now on that @xmath91 is the same for all orbits . then , any @xmath94 with @xmath95 $ ] ( to remain positive definite ) will give rise to the same density profile as @xmath12 ( @xmath96 ) , but with different degrees of internal rotation . for example , in case of @xmath97 prograde and retrograde orbits are populated equally and there will be no rotation in the corresponding system . with @xmath98 ( @xmath99 ) only prograde ( retrograde ) orbits are populated ( maximum rotation ) . the bottom panel of fig . [ aniso_prof ] shows anisotropies for the toy models of sec . [ subsec : toy ] with @xmath100 . while @xmath4 is independent of the amount of rotation , @xmath3 decreases and @xmath5 increases with increasing rotation . the latter reflects that in our toy models the total energy in @xmath28-direction is constant . any increase of the ordered motion is thus to the expense of a smaller @xmath101 . [ eps_vsig ] illustrates where the toy models appear in the @xmath102 diagram . the figure shows the three cases @xmath103 ( no rotation ) , @xmath104 ( intermediate rotation ) and @xmath105 ( maximum rotation ) . on the y - axis , the ratio @xmath106 of the maximum rotation velocity ( @xmath10 , along the projected major - axis ) and the central velocity dispersion ( @xmath107 , averaged inside @xmath108 ) is shown . all models are edge - on . the highest rotation rates at a given flattening are obtained for @xmath61 , because of its strongly populated high angular momentum orbits ( @xmath109 ) . however , @xmath57 models , which are not flattened by an excess of light on high angular momentum orbits ( relative to the isotropic case ) but instead by a suppression of orbits with large @xmath27-velocities ( @xmath110 ) can reach @xmath111 as well . the dashed lines in fig . [ eps_vsig ] approximate classical isotropic rotators by @xmath112 @xcite . up to @xmath113 , @xmath57 models can appear in the same region as classical isotropic rotators , although they are not flattened by rotation in the classical sense ( e.g. @xmath114 ) . radially anisotropic models are dominated by orbits with low angular momentum and have generally low rotation rates . a complete picture of an axisymmetric galaxy s flattening mechanism requires knowledge of the amount of rotation ( e.g. @xmath115 ) and at least one anisotropy parameter ( e.g. @xmath4 , @xmath5 or @xmath3 or the parameter @xmath51 in the notation of @xcite ) . alternatively , two anisotropy parameters also specify the global orbital structure . in any case , the full information about the anisotropy and the flattening mechanism can not be provided by the ( @xmath1 ) diagram alone . for example , four among the five intrinsically most flattened coma early - types are very close to the isotropic rotator line in fig . [ eps_vsig ] . however , they are shaped by a combination of @xmath116 and @xmath117 ( cf . tab . [ tab : aniso ] ) . the presence of dark matter around a galaxy affects the shape of the stellar orbits . some of the models just discussed may not exist , if an additional dark matter halo reshapes the potential significantly . to check this , we have recalculated all our toy models in a potential , where a spherical , logarithmic dark halo has been added to the stellar potential . the parameters of the halo ( its core radius and its asymptotic circular velocity ) have been set according to the dark matter scaling relations in coma early - types @xcite . the derived anisotropies in the new potential differ in no case by more than @xmath118 from the original ones ( but see the discussion in app . [ app : entropy ] ) . especially , the relationship between @xmath4 and @xmath2 , that arises from the maximisation of the orbital entropy also appears in potentials with a realistic dark matter halo . this does not necessarily imply that the neglect of dark matter in models of real galaxies has no effect on the derived anisotropies , because it may enforce a redistribution of the orbits ( cf . next sec . [ sec : galaxies ] ) . fig . [ aniso_eps ] shows the connection between anisotropy and flattening in real galaxies . the intrinsic flattening of coma galaxies is expressed in terms of @xmath119 @xcite . here , @xmath30 is the radius along the projected major - axis and @xmath120 and @xmath121 are the surface - brightness profile and ellipticity profile in the edge - on projection . for an axisymmetric system ( with flattening q ) @xmath122 . lines in fig . [ aniso_eps ] trace three different toy models @xmath123 ( cf . [ sec : theory ] ; the three models are designed to rotate by using @xmath124 in equation [ flammu ] ) . dfs @xmath72 are inconsistent with the global orbital structure of most galaxies ( because @xmath125 in observed galaxies ) . most galaxies have orbital properties between those of @xmath57 and @xmath61 ( with some rotation ) . [ aniso_eps ] also includes anisotropies and flattenings of 24 early - types from @xcite . these galaxies are a subsample of the 48 es / s0s of the sauron survey @xcite , which uniformly covers the plane of observed flattening @xmath9 and @xmath126 ( for @xmath127 ) . the galaxies of @xcite are drawn from this survey according to various requirements , among them consistency with axial symmetry ( according to 2d kinematical maps ) . the galaxies of @xcite ( shortly sauron in the following ) are on average fainter than the coma galaxies . although the samples do not match exactly , the anisotropies of coma and sauron galaxies are found in the same range . however , the coma sample contains relatively more anisotropic but nearly round galaxies on the one hand and more highly flattened but isotropic galaxies ( @xmath128 ) on the other . as a result , the trend for @xmath3 and @xmath4 to increase with @xmath2 which is seen in the sauron sample is not obvious when considering the complete coma sample ( even not if the two coma galaxies with the most uncertain anisotropies are ignored the two central galaxies gmp2921 and gmp3329 ) . the relation between @xmath4 and @xmath2 is weaker in the coma galaxies in part due to a few round but anisotropic galaxies for example gmp1750 and gmp5568 with @xmath129 and @xmath130 . both galaxies show weak minor - axis rotation @xcite and could be slightly triaxial systems . in addition to differences among nearly round galaxies , anisotropies in coma and sauron galaxies also slightly differ at high @xmath2 . the latter is most clearly seen in @xmath3 versus @xmath2 : two highly flattened coma galaxies ( gmp1990 and gmp2440 , @xmath131 ) have @xmath128 . one of these galaxies is likely close to edge - on , because of its high observed ellipticity ( @xmath132 , cf . the radial profile in @xcite ) and its significant isophotal shape distortions . we expect the model of gmp1990 to be well constrained , because of the near edge - on inclination ( minimal uncertainties in the deprojection ) and its far - extending multi - slit kinematic data . for the other galaxy ( gmp2440 ) @xcite quote only a modest observed ellipticity @xmath133 at @xmath134 and the intrinsic flattening comes mostly from the low inclination of the model . note that this galaxy is far above the isotropic rotator line in the right panel of fig . [ eps_vsig ] ( gmp2440 is the only non edge - on galaxy above the isotropic rotator line ) . a maximum - entropy like df is ruled out for this galaxy , because even the maximally rotating version of the @xmath57 model would not allow for the high observed rotation rate . thus , even if we would have underestimated the inclination of this system , fig . [ eps_vsig ] shows that its orbital structure must be significantly deviant from maximum - entropy models . all in all then , modelling uncertainties are unlikely to explain the outstanding anisotropies of gmp1990 and gmp2440 . in fact , a comparison with fig . 3 in @xcite reveals that the sauron sample does not include galaxies like gmp1990 and gmp2440 , because ( 1 ) for only one object the observed ellipticity is significantly larger than @xmath135 ( ngc4550 ) and ( 2 ) even the fastest rotators in the sauron sample are closer to the isotropic rotator line than gmp2440 . in addition to differences in the sample selection also the modelling methods differ in the details . @xcite use similar orbit - based dynamical models as we do here , but sauron anisotropies are calculated inside a fixed aperture with a radius of @xmath136 . a fixed aperture encloses different fractions of the stellar mass in different galaxies , depending on system size and distance . for the coma galaxies we give anisotropies inside @xmath134 . in some galaxies local anisotropies vary significantly with radius @xcite , such that the radius of comparison is crucial . in addition , sauron models are based on the assumption that mass follows light . as it has been stated already in the introduction , the assumption of a constant mass - to - light ratio can result in artificially large @xmath28-energies ( @xcite , @xcite ) or low @xmath5 , respectively . regarding fig . [ aniso_eps ] , sauron galaxies do not have systematically lower @xmath5 than coma objects . for the only two exceptions ( ngc4473 and ngc4550 ) @xcite report evidence for counter - rotating , disk - like components that likely cause their large @xmath28-energies . the small effect that the neglect of dark matter has on the anisotropies likely reflects the fact that we only consider anisotropies averaged inside @xmath137 , where the assumption that mass follows light is most closely fulfilled ( e.g. @xcite , @xcite ) . for the coma galaxies a quantitative comparison of models with and without dark matter is made in app . [ sec : app ] . the spatial coverage with kinematic data in the inner regions is sparse in the coma galaxies ( long - slit data ) compared to the sauron objects ( 2d kinematical maps ) . in regions of phase - space that are not well constrained by the observed kinematics , the dynamical models are mainly driven by regularisation . thus , because the spatial coverage is lower in the coma galaxies , their anisotropies could be biased towards the adopted regularisation scheme . specifically , coma galaxy models are regularised towards maximum entropy @xcite . however , the middle panel in the top row of fig . [ aniso_eps ] does not show any bias of the coma models towards the maximum entropy relation . in fact , sauron galaxies are on average closer to this relation than coma galaxies . this indicates that regularisation is not the main driver for the coma galaxy models . also , in app . [ sec : app ] we give an explicit comparison of coma galaxy models with standard and with weak regularisation . we do not find significant differences . both the intrinsic ellipticity and the anisotropy depend on the inclination of the models . for the coma galaxies , we probe three different inclinations and use the one that fits best @xcite , while inclinations for sauron galaxies are derived from two - integral jeans models @xcite . the inclination is best constrained for highly flattened galaxies , because these have to be close to edge - on . for three of the coma galaxies ( gmp0756 , gmp1176 and gmp1990 ) large ellipticities together with significantly discy / boxy isophotes indeed indicate close to edge - on inclinations ( for example , gmp1176 exhibits @xmath138 ; @xcite ) . in contrast , two among the five galaxies with @xmath139 owe their flattening in part from the relatively low inclination of the best - fit model ( gmp0282 , gmp2440 ; cf . tab . [ tab : aniso ] ) . these galaxies provide the smallest and largest anisotropies , respectively , at high @xmath2 ( cf . middle panel in the top row of fig . [ aniso_eps ] ) . this suggests that the method to determine the inclination for the coma galaxies does not result in a specific bias of the derived anisotropies . we conclude that slight differences between the sauron and the coma anisotropies are mostly due to the different sample selections , while differences in the modelling methodology ( including differences in the data coverage ) seem to be negligible the lower panels of fig . [ aniso_eps ] display the models of @xmath0-body merger remnants ( cf . [ sec : data ] ) . in terms of @xmath3 versus @xmath2 and @xmath4 versus @xmath2 , these models do not differ strongly from models of real galaxies ( see also @xcite ) . however , while @xmath140 in models of merger remnants , @xmath5 is often negative in models of real galaxies . is this discrepancy in @xmath5 indicative for the merger remnants having a different orbital structure than real galaxies , or does it merely reflect systematics caused by the symmetry assumptions in our models ? reconstructed and true intrinsic anisotropies in axisymmetric systems , @xmath141 in the merger remnants . for the intrinsic @xmath3 of the merger remnants we use the average @xmath142 instead of @xmath143 in equation ( [ eq : delta ] ) . ] and flattenings of the merger remnants are compared in fig . [ aniso_recon ] . the one merger remnant closest to oblate axial symmetry ( oblate ) , is reconstructed with high accuracy from the x and y - projections ( edge - on ) . this is plausible , because for this remnant the assumption of axial symmetry is a good approximation . furthermore , in the edge - on case the deprojection becomes unique and the intrinsic degeneracies in the dynamics are likely smallest . however , the general trend in the axisymmetric models is to underestimate both , the flattening and the anisotropy of the merger remnants . x and y - projections allow a better reconstruction of shape and anisotropy than z - projections . it has already been discussed in @xcite that the assumption of axial symmetry enforces an inclination mismatch in the z - models : while the triaxial remnants appear flattened in the @xmath144 projection ( face - on ) , axisymmetric systems are necessarily round when seen face - on . then , because the models are forced towards a wrong viewing - angle , ( 1 ) the intrinsic flattening is underestimated and ( 2 ) x , y and z - axes of models and remnants do no longer correspond to each other . for example , a z - model s @xmath5 measures a different energy ratio as the remnant s @xmath5 @xcite . had we compared the z - models with the apparent shape of the remnant in z - projection and with the energy ratios along axes of models and remnants that correspond to each other , then the differences would have been much smaller ( for example @xmath145 ) . real galaxies are seen at random viewing angles . starting from our models of principal projections it is difficult to predict directly the analogous distributions of @xmath80 or @xmath146 for the realistic case of random projections . however , because the projections along principal axes yield extreme kinematical and photometrical properties of the merger remnants @xcite , it can be expected that dynamical models of projections along intermediate viewing - angles will have properties intermediate between those of the models from principal projections . we have verified this for two out of the six merger remnants ( elong and oblate ) by modelling additional 11 projections ( at intermediate viewing - angles ) . assuming that this result can be generalised to other remnants as well , then figs . [ aniso_eps ] and [ aniso_recon ] suggest the following : if real galaxies would resemble the modelled merger remnants , then one would see approximately the same relationships @xmath80 and @xmath146 as in the coma and sauron galaxies . however , it is clear from fig . [ aniso_recon ] that @xmath140 , for a sample of randomly projected objects like our modelled @xmath0-body merger remnants . thus , in this respect , models of many real galaxies differ from our comparison sample of synthetic @xmath0-body merger remnants : models of merger remnants are always radially anisotropic ( @xmath147 ) , while models of real galaxies are characterised by @xmath148 . this fact is further illustrated in fig . [ aniso_aniso ] , which shows correlations among the anisotropy parameters . even though shape and anisotropy can not be recovered simultaneously ( in some cases ) , the anisotropy correlations in the models of the merger remnants and in the merger remnants itself are very similar to each other . again , the main difference between real galaxies and merger models is the offset between their @xmath5 distributions . besides the fact that merger remnants have on average positive @xmath149 , while real galaxies have @xmath150 ( on average ) , fig . [ aniso_aniso ] shows that the distribution of anisotropies in the merger remnants is tighter than in real galaxies . this may reflect the similarity in the initial conditions of the @xmath0-body simulations ( most noteworthy the similarity in the progenitors and the fact that we only consider collisionless mergers ) . the anisotropy parameters defined in sec . [ sec : theory ] are only global measures of the orbital structure . a full understanding of the formation process of early - type galaxies can only be provided by spatially resolved anisotropy profiles . for example , equatorial near circular orbits obey , in the epicycle approximation , the local relation @xmath151 where @xmath152 is the circular velocity @xcite . in a typical galaxy potential the circular velocity curve is flat ( @xmath153 ) and equation ( [ eq : epicyc ] ) predicts @xmath154 . since the epicycle approximation holds for perturbed rotating disks , we do not expect the majority of early - types in our sample to be well described by equation ( [ eq : epicyc ] ) . however , it might be relevant for the most flattened , rotating and discy objects in our sample . instead , at least some of these ( for example gmp1176 and gmp3958 ) have negative @xmath5 ( i.e. @xmath155 ) . this does not rule out a disk heating scenario for these galaxies , however , because locally we find @xmath156 near the equatorial plane in these galaxies ( around @xmath157 ; cf . the radial anisotropy profiles in @xcite ) . in case of the collisionless @xmath0-body merger simulations , already the averaged anisotropy parameters reveal significant differences to the models of real galaxies . which physical processes are responsible for this discrepancy ? the orbital structure of the models of merger remnants is largely driven by a population of central box orbits in the @xmath0-body systems @xcite . they cause the centres of the merger remnants to become triaxial / prolate and are , for example , largely responsible for the wrong viewing angle of the z - models . dissipation during a merger can have a significant effect on the shape and the projected properties of the final remnant @xcite . already 10 percent of gas are sufficient to suppress central box orbits and to produce an approximately axisymmetric remnant in binary mergers @xcite ( but this result is based on simulations without star formation ) . multiple , simultaneous minor mergers likewise produce remnants less triaxial than collisionless binary merger remnants @xcite , but the corresponding kinematics have not yet been studied in detail . successive minor merging does not necessarily lead to different final remnants , at least if the cumulative merged mass becomes similar to the most massive progenitor @xcite . again , detailed predictions for the orbital make - up and the shapes of the line - of - sight velocity distributions have not yet been worked out . note that the central dark matter densities in coma ellipticals are larger than in present - day spirals @xcite . even if ellipticals have formed by some variant of merging , present - day spiral galaxies are unlikely the progenitors for the bulk of giant ellipticals ( see also @xcite ) . @xcite pointed out that @xmath0-body systems , which have assembled hierarchically in their cosmological simulations @xcite , or by binary mergers with star - formation and black - hole feedback are consistent with the trend between @xmath3 and @xmath2 in observed galaxies . we have discussed the relationship between anisotropy and flattening in toy models , in models of real galaxies , in merger remnants and in models of merger remnants . models of observed galaxies generally exhibit @xmath158 and @xmath150 . we do not find strong correlations of the anisotropy parameters @xmath3 , @xmath4 and @xmath5 with intrinsic ellipticity @xmath2 . in toy models with maximum entropy for a given density distribution we find @xmath4 to increase with @xmath2 , while @xmath117 . observed galaxies appear close to these maximum - entropy relations , but exhibit a large degree of individuality . rotation appears in anisotropic ( @xmath158 ) as well as isotropic systems ( @xmath73 ) , suggesting that the flattening of the galaxies largely arises from a suppression of stars with large energies perpendicular to the equatorial plane . this is similar to the classical notion of flattening by anisotropy and rules out dfs @xmath72 for most early - type galaxies . the global similarity between models of observed galaxies and our maximum - entropy toy models suggests that early - type galaxies are largely relaxed stellar systems . however , there are differences in the details that probably contain valuable information about the assembly mechanism of the galaxies and will be addressed in a future paper . numerical simulations indicate that both strongly radially anisotropic ( @xmath159 ) and strongly tangentially anisotropic systems ( @xmath160 ) can become unstable ( e.g. @xcite ) . maximum entropy models have intermediate anisotropies and are likely stable . thus , the anisotropies of observed galaxies may not only be understood as being the most likely ones ( in the sense of yielding the maximum entropy at a given flattening ) but could also reflect stability constraints . so far we lack detailed studies exploring the stability of axisymmetric systems with dark matter halos and various intrinsic anisotropies . since our ( three integral ) toy models can be easily transformed to @xmath0-body systems ( cf . @xcite ) they provide a suitable tool to setup both artificially anisotropic as well as realistic and observationally motivated initial conditions . in models of real galaxies the unordered kinetic energy in the azimuthal direction , @xmath161 , can exceed the radial energy @xmath162 by up to 40 percent . this separates real galaxy models from similar models of collisionless @xmath0-body binary disk mergers , which are instead characterised by radial anisotropy ( @xmath147 ) . because we have applied the same modelling machinery to both , the real galaxies as well as the synthetic @xmath0-body merger remnants , our findings indicate a true difference between their intrinsic properties . especially , we have shown that if real galaxies would resemble our merger remnants , then corresponding dynamical models of real data would be radially anisotropic , irrespective of the systematics introduced by the assumption of axial symmetry . the radial anisotropy of the merger remnants is related to a population of central box orbits . because dissipation during a merger can efficiently suppress box orbits , our results suggest that dissipation played an important role during the formation of intermediate mass to massive early - type galaxies . in this paper we focussed on the comparison of real galaxies with collisionless binary disk merger simulations . a similar analysis , but for gaseous mergers with star formation and/or for galaxies formed in cosmological simulations could give more insight into the actual formation paths of elliptical galaxies this work was supported by dfg sonderforschungsbereich 375 `` astro - teilchenphysik '' and dfg priority program 1177 . emc receives support from grant cpda068415/06 by padua university . 99 barnes j. e. , 1992 , apj , 393 , 484 barnes j. e. , hernquist l. , 1996 , apj , 471 , 115 bender r. , mllenhoff c. , 1987 , a&a , 177 , 71 binney j. , 1978 , mnras , 183 , 501 binney j. , tremaine s. , 1987 , galactic dynamics ( princeton : princeton university press ) binney j. , 2005 , mnras , 363 , 937 bournaud f. , jog c. g. , combes f. , 2007 , a&a , 476 , 1179 burkert a. , naab t. , 2005 , mnras , 363 , 597 burkert a. , naab t. , johansson p. h. , jesseit r. , 2008 , apj , 685 , 897 cappellari m. et al . , 2006 , mnras , 366 , 112 cappellari m. et al . , 2007 , mnras , 379 , 418 carollo c. m. , de zeeuw p. t. , van der marel r. p. , danziger i. j. , qian e. e. , 1995 , apjl , 441 , 25 contopoulos g. , 1963 , aj , 68 , 1 corsini e. m. , wegner g. , saglia r. p. , thomas j. , bender r. , thomas d. , 2008 , apjs , 175 , 462 cox t. j. , jonsson p. , primack j. r. , somerville r. s. , 2006 , mnras , 373 , 1013 dehnen w. , 1993 , mnras , 265 , 250 dehnen w. , gerhard o. e. , 1993 , mnras , 261 , 311 dehnen w. , gerhard o. e. , 1994 , mnras , 268 , 1019 dubinski j. , 1998 , apj , 502 , 141 forestell a. , gebhardt k. , 2008 , astro - 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carlo simulations of isotropic rotators @xcite . to check how much the choice of @xmath51 affects our results , we ( 1 ) determined the best - fit dynamical model at @xmath164 and ( 2 ) recalculated the anisotropies of all galaxies from these weakly regularised models ( at @xmath165 , the minimum @xmath45 is usually reached ) . in the top row of fig . [ aniso_aniso : test ] we show both models with standard and with weak regularisation for comparison . as can be seen , lowering the regularisation has almost no effect on the derived anisotropies . especially , there are still at least four galaxies with significantly negative @xmath166 . in the bottom row of fig . [ aniso_aniso : test ] we make a similar comparison for models with and without dark matter : squares indicate the anisotropies of our best - fit models with a constant mass - to - light ratio ( no dark matter halo ) . as expected , when assuming that mass - follows - light , @xmath5 become smaller ( the amount of @xmath28-energy is increased to compensate for the missing dark mass ) . from the bottom - right panel one would expect that the average @xmath5 becomes negative when the radial increase of the mass - to - light ratio ( caused by a dark halo ) is neglected . this is not the case in the sauron sample , however , although @xcite assumed that the mass - to - light ratio is constant with radius in their models . that neglecting dark matter has a stronger effect in coma galaxies is likely related to the fact that our kinematical data reach out into the region where dark matter becomes noticeable ( @xmath167 ) , which is probably not the case in many sauron galaxies ( where the data extend only out to @xmath168 ) ( top ) and @xmath169 ( middle ) for maximum - entropy models ( @xmath57 ; flattening @xmath170 ) with dark matter halos . left - hand side / squares : halo tuned to result in an approximately flat circular velocity curve ; right - hand side / triangles : halo leading to an increasing circular velocity in the outer parts of the model . for comparison , the case without halo is shown on both sides ( circles ) . circular velocity curves of the models ( scaled to the maximum circular velocity @xmath171 without halo ) are shown in the bottom panels.,width=317 ] the maximum - entropy toy models @xmath57 discussed in sec . [ subsec : toy ] resemble the classical flattening by anisotropy , except that they are only approximately isotropic in @xmath30 and @xmath172 . to investigate where this anisotropy comes from , we have constructed maximum - entropy toy models in potentials that include a dark matter halo . the halo density distribution is assumed to follow @xmath173 @xcite . to mimick realistic halos ( cf . @xcite ) we choose a flat central density core ( @xmath174 ) and we set the flattening @xmath56 of the halo equal to the flattening of the luminous component of the toy model ( cf . equation [ eq : flattening ] ) . we investigated three mass models : ( 1 ) no halo , ( 2 ) a mass model that has a roughly constant circular velocity curve and ( 3 ) a mass model with a rising @xmath175 in the outer parts of the model . the corresponding circular velocity curves for @xmath170 are shown in the bottom panels of fig . [ aniso_vcirc ] . the upper panels of fig . [ aniso_vcirc ] display the radial profiles of the local anisotropies @xmath177 and @xmath178 along the equatorial plane ( averaged within @xmath179 , where @xmath180 is the latitude ) . as one can see , the anisotropy in the meridional plane ( @xmath181 ) does not depend on the shape of the gravitational potential . thus , the relation between @xmath4 and @xmath2 is largely independent from the gravitational potential and closely related to the entropy maximisation . however , beyond @xmath134 , where dark matter starts to influence the shape of the circular velocity curve , @xmath169 is different in the three different potentials . [ aniso_vcirc2 ] shows that the local value of @xmath169 along the equatorial plane is directly connected to the logarithmic slope @xmath182 of the circular velocity curve . in general then , because @xmath5 from equation ( [ eq : gamma ] ) is the spatial average of @xmath169 ( and the local anisotropies along other position angles in the meridional plane ) , its exact value is not set uniquely by the entropy maximisation but also depends on the shape of the circular velocity curve . in practice , however , deviations with respect to the model without halo become noticeable only beyond @xmath134 , such that even the spatially averaged @xmath5 of the toy models does not depend strongly on whether a halo is included or not . note that the relation revealed by fig . [ aniso_vcirc2 ] is different from the epicycle relation ( [ eq : epicyc ] ) . this is expected , because the azimuthal velocity dispersion @xmath101 in the toy models largely results from the fact that they do not rotate . instead , the dispersion predicted by the epicycle approximation arises from perturbations on circular orbits in a rotating disk .

we use oblate axisymmetric dynamical models including dark halos to determine the orbital structure of intermediate mass to massive coma early - type galaxies . we find a large variety of orbital compositions . averaged over all sample galaxies the unordered stellar kinetic energy in the azimuthal and the radial direction are of the same order , but they can differ by up to 40 percent in individual systems . in contrast , both for rotating and non - rotating galaxies the vertical kinetic energy is on average smaller than in the other two directions . this implies that even most of the rotating ellipticals are flattened by an anisotropy in the stellar velocity dispersions . using three - integral axisymmetric toy models we show that flattening by stellar anisotropy maximises the entropy for a given density distribution . collisionless disk merger remnants are radially anisotropic . the apparent lack of strong radial anisotropy in observed early - type galaxies implies that they may not have formed from mergers of disks unless the influence of dissipational processes was significant . [ firstpage ] galaxies : kinematics and dynamics galaxies : elliptical and lenticular , cd galaxies : formation