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Tel Aviv University 27

[sec:introduction]introduction the leptonic decays of a charged pseudoscalar meson @xmath7 are processes of the type @xmath8, where @xmath9, @xmath10, or @xmath11. because no strong interactions are present in the leptonic final state @xmath12, such decays provide a clean way to probe the complex, strong interactions that bind the quark and antiquark within the initial - state meson. in these decays, strong interaction effects can be parametrized by a single quantity, @xmath13, the pseudoscalar meson decay constant. the leptonic decay rate can be measured by experiment, and the decay constant can be determined by the equation (ignoring radiative corrections) @xmath14 where @xmath15 is the fermi coupling constant, @xmath16 is the cabibbo - kobayashi - maskawa (ckm) matrix @xcite element, @xmath17 is the mass of the meson, and @xmath18 is the mass of the charged lepton. the quantity @xmath13 describes the amplitude for the @xmath19 and @xmath20-quarks within the @xmath21 to have zero separation, a condition necessary for them to annihilate into the virtual @xmath22 boson that produces the @xmath12 pair. the experimental determination of decay constants is one of the most important tests of calculations involving nonperturbative qcd. such calculations have been performed using various models @xcite or using lattice qcd (lqcd). the latter is now generally considered to be the most reliable way to calculate the quantity. knowledge of decay constants is important for describing several key processes, such as @xmath23 mixing, which depends on @xmath24, a quantity that is also predicted by lqcd calculations. experimental determination @xcite of @xmath24 with the leptonic decay of a @xmath25 meson is, however, very limited as the rate is highly suppressed due to the smallness of the magnitude of the relevant ckm matrix element @xmath26. the charm mesons, @xmath27 and @xmath28, are better instruments to study the leptonic decays of heavy mesons since these decays are either less ckm suppressed or favored, _ i.e. _, @xmath29 and @xmath30 are much larger than @xmath31. thus, the decay constants @xmath32 and @xmath33 determined from charm meson decays can be used to test and validate the necessary lqcd calculations applicable to the @xmath34-meson sector. among the leptonic decays in the charm - quark sector, @xmath35 decays are more accessible since they are ckm favored. furthermore, the large mass of the @xmath11 lepton removes the helicity suppression that is present in the decays to lighter leptons. the existence of multiple neutrinos in the final state, however, makes measurement of this decay challenging. physics beyond the standard model (sm) might also affect leptonic decays of charmed mesons. depending on the non - sm features, the ratio of @xmath36 could be affected @xcite, as could the ratio @xcite @xmath37. any of the individual widths might be increased or decreased. there is an indication of a discrepancy between the experimental determinations @xcite of @xmath33 and the most recent precision lqcd calculation @xcite. this disagreement is particularly puzzling since the cleo - c determination @xcite of @xmath32 agrees well with the lqcd calculation @xcite of that quantity. some @xcite conjecture that this discrepancy may be explained by a charged higgs boson or a leptoquark. in this article, we report an improved measurement of the absolute branching fraction of the leptonic decay @xmath0 (charge - conjugate modes are implied), with @xmath1, from which we determine the decay constant @xmath33. [sec:detector]data and the cleo- detector we use a data sample of @xmath38 events provided by the cornell electron storage ring (cesr) and collected by the cleo - c detector at the center - of - mass (cm) energy @xmath39 mev, near @xmath3 peak production @xcite. the data sample consists of an integrated luminosity of @xmath40 @xmath41 containing @xmath42 @xmath3 pairs. we have previously reported @xcite measurements of @xmath43 and @xmath0 with a subsample of these data. a companion article @xcite reports measurements of @xmath33 from @xmath43 and @xmath0, with @xmath44, using essentially the same data sample as the one used in this measurement. the cleo - c detector @xcite is a general - purpose solenoidal detector with four concentric components utilized in this measurement : a small - radius six - layer stereo wire drift chamber, a 47-layer main drift chamber, a ring - imaging cherenkov (rich) detector, and an electromagnetic calorimeter consisting of 7800 csi(tl) crystals. the two drift chambers operate in a @xmath45 t magnetic field and provide charged particle tracking in a solid angle of @xmath46% of @xmath47. the chambers achieve a momentum resolution of @xmath48% at @xmath49 gev/@xmath50. the main drift chamber also provides specific - ionization (@xmath51) measurements that discriminate between charged pions and kaons. the rich detector covers approximately @xmath52% of @xmath47 and provides additional separation of pions and kaons at high momentum. the photon energy resolution of the calorimeter is @xmath53% at @xmath54 gev and @xmath55% at @xmath56 mev. electron identification is based on a likelihood variable that combines the information from the rich detector, @xmath51, and the ratio of electromagnetic shower energy to track momentum (@xmath57). we use a geant - based @xcite monte carlo (mc) simulation program to study efficiency of signal - event selection and background processes. physics events are generated by evtgen @xcite, tuned with much improved knowledge of charm decays @xcite, and final - state radiation (fsr) is modeled by the photos @xcite program. the modeling of initial - state radiation (isr) is based on cross sections for @xmath3 production at lower energies obtained from the cleo - c energy scan @xcite near the cm energy where we collect the sample. [sec:analysys]analysis method the presence of two @xmath58 mesons in a @xmath3 event allows us to define a single - tag (st) sample in which a @xmath58 is reconstructed in a hadronic decay mode and a further double - tagged (dt) subsample in which an additional @xmath59 is required as a signature of @xmath60 decay, the @xmath59 being the daughter of the @xmath60. the @xmath61 reconstructed in the st sample can be either primary or secondary from @xmath62 (or @xmath63). the st yield can be expressed as @xmath64 where @xmath65 is the produced number of @xmath3 pairs, @xmath66 is the branching fraction of hadronic modes used in the st sample, and @xmath67 is the st efficiency. the @xmath68 counts the candidates, not events, and the factor of 2 comes from the sum of @xmath28 and @xmath61 tags. our double - tag (dt) sample is formed from events with only a single charged track, identified as an @xmath69, in addition to a st. the yield can be expressed as @xmath70 where @xmath71 is the leptonic decay branching fraction, including the subbranching fraction of @xmath1 decay, @xmath72 is the efficiency of finding the st and the leptonic decay in the same event. from the st and dt yields we can obtain an absolute branching fraction of the leptonic decay @xmath71, without needing to know the integrated luminosity or the produced number of @xmath3 pairs, @xmath73 where @xmath74 (@xmath75) is the effective signal efficiency. because of the large solid angle acceptance with high segmentation of the cleo - c detector and the low multiplicity of the events with which we are concerned, @xmath76, where @xmath77 is the leptonic decay efficiency. hence, the ratio @xmath78 is insensitive to most systematic effects associated with the st, and the signal branching fraction @xmath71 obtained using this procedure is nearly independent of the efficiency of the tagging mode. to minimize systematic uncertainties, we tag using three two - body hadronic decay modes with only charged particles in the final state. the three st modes and @xmath79 are shorthand labels for @xmath80 events within mass windows (described below) of the @xmath81 peak in @xmath82 and the @xmath83 peak in @xmath84, respectively. no attempt is made to separate these resonance components in the @xmath85 dalitz plot.] are @xmath86, @xmath79, and @xmath87. using these tag modes also helps to reduce the tag bias which would be caused by the correlation between the tag side and the signal side reconstruction if tag modes with high multiplicity and large background were used. the effect of the tag bias @xmath88 can be expressed in terms of the signal efficiency @xmath74 defined by @xmath89 where @xmath90 is the st efficiency when the recoiling system is the signal leptonic decay with single @xmath59 in the other side of the tag. as the general st efficiency @xmath67, when the recoiling system is any possible @xmath91 decays, will be lower than the @xmath90, sizable tag bias could be introduced if the multiplicity of the tag mode were high, or the tag mode were to include neutral particles in the final state. as shown in sec. [sec : results], this effect is negligible in our chosen clean tag modes. the @xmath92 decay is reconstructed by combining oppositely charged tracks that originate from a common vertex and that have an invariant mass within @xmath93 mev of the nominal mass @xcite. we require the resonance decay to satisfy the following mass windows around the nominal masses @xcite : @xmath94 (@xmath95 mev) and @xmath96 (@xmath97 mev). we require the momenta of charged particles to be @xmath56 mev or greater to suppress the slow pion background from @xmath98 decays (through @xmath99). we identify a st by using the invariant mass of the tag @xmath100 and recoil mass against the tag @xmath101. the recoil mass is defined as @xmath102 where @xmath103 is the net four - momentum of the @xmath4 beam, taking the finite beam crossing angle into account ; @xmath104 is the four - momentum of the tag, with @xmath105 computed from @xmath106 and the nominal mass @xcite of the @xmath91 meson. we require the recoil mass to be within @xmath107 mev of the @xmath108 mass @xcite. this loose window allows both primary and secondary @xmath91 tags to be selected. to estimate the backgrounds in our st and dt yields from the wrong tag combinations (incorrect combinations that, by chance, lie within the @xmath109 signal region), we use the tag invariant mass sidebands. we define the signal region as @xmath110 mev @xmath111 mev, and the sideband regions as @xmath112 mev @xmath113 mev or @xmath114 mev @xmath115 mev, where @xmath116 is the difference between the tag mass and the nominal mass. we fit the st @xmath109 distributions to the sum of double - gaussian signal function plus second - degree chebyshev polynomial background function to get the tag mass sideband scaling factor. the invariant mass distributions of tag candidates for each tag mode are shown in fig. [fig : dm] and the st yield and @xmath109 sideband scaling factor are summarized in table [table : data - single]. we find @xmath117 summed over the three tag modes..[table : data - single] summary of single - tag (st) yields, where @xmath118 is the yield in the st mass signal region, @xmath119 is the yield in the sideband region, @xmath120 is the sideband scaling factor, and @xmath68 is the scaled sideband - subtracted yield. [cols="<,>,>,>,>",options="header ",] we considered six semileptonic decays, @xmath121 @xmath122, @xmath123, @xmath124, @xmath125, @xmath126, and @xmath127, as the major sources of background in the @xmath128 signal region. the second dominates the nonpeaking background, and the fourth (with @xmath129) dominates the peaking background. uncertainty in the signal yield due to nonpeaking background (@xmath130) is assessed by varying the semileptonic decay branching fractions by the precision with which they are known @xcite. imperfect knowledge of @xmath131 gives rise to a systematic uncertainty in our estimate of the amount of peaking background in the signal region, which has an effect on our branching fraction measurement of @xmath132. we study differences in efficiency, data vs mc events, due to the extra energy requirement, extra track veto, and @xmath133 requirement, by using samples from data and mc events, in which _ both _ the @xmath134 and @xmath2 satisfy our tag requirements, i.e., `` double - tag '' events. we then apply each of the above - mentioned requirements and compare loss in efficiency of data vs mc events. in this way we obtain a correction of @xmath135 for the extra energy requirement and systematic uncertainties on each of the three requirements of @xmath136 (all equal, by chance). the non-@xmath69 background in the signal @xmath69 candidate sample is negligible (@xmath137) due to the low probability (@xmath138 per track) that hadrons (@xmath139 or @xmath140) are misidentified as @xmath69 @xcite. uncertainty in these backgrounds produces a @xmath141 uncertainty in the measurement of @xmath142. the secondary @xmath69 backgrounds from charge symmetric processes, such as @xmath143 dalitz decay (@xmath144) and @xmath145 conversion (@xmath146), are assessed by measuring the wrong - sign signal electron in events with @xmath147. the uncertainty in the measurement from this source is estimated to be @xmath148. other possible sources of systematic uncertainty include @xmath68 (@xmath137), tag bias (@xmath149), tracking efficiency (@xmath148), @xmath59 identification efficiency (@xmath150), and fsr (@xmath150). combining all contributions in quadrature, the total systematic uncertainty in the branching fraction measurement is estimated to be @xmath151. [sec:conclusion]summary in summary, using the sample of @xmath152 tagged @xmath28 decays with the cleo - c detector we obtain the absolute branching fraction of the leptonic decay @xmath153 through @xmath154 @xmath155 where the first uncertainty is statistical and the second is systematic. this result supersedes our previous measurement @xcite of the same branching fraction, which used a subsample of data used in this work. the decay constant @xmath33 can be computed using eq. ([eq : f]) with known values @xcite @xmath156 gev@xmath157, @xmath158 mev, @xmath159 mev, and @xmath160 s. we assume @xmath161 and use the value @xmath162 given in ref. we obtain @xmath163 combining with our other determination @xcite of @xmath164 mev with @xmath43 and @xmath0 (@xmath165) decays, we obtain @xmath166 this result is derived from absolute branching fractions only and is the most precise determination of the @xmath91 leptonic decay constant to date. our combined result is larger than the recent lqcd calculation @xmath167 mev @xcite by @xmath168 standard deviations. the difference between data and lqcd for @xmath33 could be due to physics beyond the sm @xcite, unlikely statistical fluctuations in the experimental measurements or the lqcd calculation, or systematic uncertainties that are not understood in the lqcd calculation or the experimental measurements. combining with our other determination @xcite of @xmath169, via @xmath44, we obtain @xmath170 using this with our measurement @xcite of @xmath171, we obtain the branching fraction ratio @xmath172 this is consistent with @xmath173, the value predicted by the sm with lepton universality, as given in eq. ([eq : f]) with known masses @xcite. we gratefully acknowledge the effort of the cesr staff in providing us with excellent luminosity and running conditions. d. cronin - hennessy and a. ryd thank the a.p. sloan foundation. this work was supported by the national science foundation, the u.s. department of energy, the natural sciences and engineering research council of canada, and the u.k. science and technology facilities council. c. amsler _ et al. _ (particle data group), phys. b * 667 *, 1 (2008). k. ikado _ et al. _ (belle collaboration), phys. lett. * 97 *, 251802 (2006). b. aubert _ et al. _ (babar collaboration), phys. rev. d * 77 *, 011107 (2008). a. g. akeroyd and c. h. chen, phys. d * 75 *, 075004 (2007) ; a. g. akeroyd, prog. phys. * 111 *, 295 (2004). j. l. hewett, arxiv : hep - ph/9505246. w. s. hou, phys. d * 48 *, 2342 (1993). e. follana, c. t. h. davies, g. p. lepage, and j. shigemitsu (hpqcd collaboration), phys. lett. * 100 *, 062002 (2008). b. i. eisenstein _ et al. _ (cleo collaboration), phys. rev. d * 78 *, 052003 (2008). b. a. dobrescu and a. s. kronfeld, phys. * 100 *, 241802 (2008). d. cronin - hennessy _ et al. _ (cleo collaboration), arxiv:0801.3418. m. artuso _ et al. _ (cleo collaboration), phys. lett. * 99 *, 071802 (2007). k. m. ecklund _ et al. _ (cleo collaboration), phys. rev. lett. * 100 *, 161801 (2008). j. p. alexander _ et al. _ (cleo collaboration), phys. rev. d * 79 *, 052001 (2009). y. kubota _ et al. _ (cleo collaboration), nucl. instrum. a * 320 *, 66 (1992). d. peterson _ et al. _, instrum. methods phys., sec. a * 478 *, 142 (2002). m. artuso _ et al. _, nucl. instrum. methods phys. a * 502 *, 91 (2003). s. dobbs _ et al. _ (cleo collaboration), phys. rev. d * 76 *, 112001 (2007). j. p. alexander _ et al. _ (cleo collaboration), phys. rev. lett. * 100 *, 161804 (2008). e. barberio and z. was, comput.. commun. * 79 *, 291 (1994).

we have studied the leptonic decay @xmath0, via the decay channel @xmath1, using a sample of tagged @xmath2 decays collected near the @xmath3 peak production energy in @xmath4 collisions with the cleo - c detector. we obtain @xmath5 and determine the decay constant @xmath6 mev, where the first uncertainties are statistical and the second are systematic.

0901.1147

Introduction additive models @xcite provide an important family of models for semiparametric regression or classification. some reasons for the success of additive models are their increased flexibility when compared to linear or generalized linear models and their increased interpretability when compared to fully nonparametric models. it is well - known that good estimators in additive models are in general less prone to the curse of high dimensionality than good estimators in fully nonparametric models. many examples of such estimators belong to the large class of regularized kernel based methods over a reproducing kernel hilbert space @xmath0, see e.g. @xcite. in the last years many interesting results on learning rates of regularized kernel based models for additive models have been published when the focus is on sparsity and when the classical least squares loss function is used, see e.g. @xcite, @xcite, @xcite, @xcite, @xcite, @xcite and the references therein. of course, the least squares loss function is differentiable and has many nice mathematical properties, but it is only locally lipschitz continuous and therefore regularized kernel based methods based on this loss function typically suffer on bad statistical robustness properties, even if the kernel is bounded. this is in sharp contrast to kernel methods based on a lipschitz continuous loss function and on a bounded loss function, where results on upper bounds for the maxbias bias and on a bounded influence function are known, see e.g. @xcite for the general case and @xcite for additive models. therefore, we will here consider the case of regularized kernel based methods based on a general convex and lipschitz continuous loss function, on a general kernel, and on the classical regularizing term @xmath1 for some @xmath2 which is a smoothness penalty but not a sparsity penalty, see e.g. @xcite. such regularized kernel based methods are now often called support vector machines (svms), although the notation was historically used for such methods based on the special hinge loss function and for special kernels only, we refer to @xcite. in this paper we address the open question, whether an svm with an additive kernel can provide a substantially better learning rate in high dimensions than an svm with a general kernel, say a classical gaussian rbf kernel, if the assumption of an additive model is satisfied. our leading example covers learning rates for quantile regression based on the lipschitz continuous but non - differentiable pinball loss function, which is also called check function in the literature, see e.g. @xcite and @xcite for parametric quantile regression and @xcite, @xcite, and @xcite for kernel based quantile regression. we will not address the question how to check whether the assumption of an additive model is satisfied because this would be a topic of a paper of its own. of course, a practical approach might be to fit both models and compare their risks evaluated for test data. for the same reason we will also not cover sparsity. consistency of support vector machines generated by additive kernels for additive models was considered in @xcite. in this paper we establish learning rates for these algorithms. let us recall the framework with a complete separable metric space @xmath3 as the input space and a closed subset @xmath4 of @xmath5 as the output space. a borel probability measure @xmath6 on @xmath7 is used to model the learning problem and an independent and identically distributed sample @xmath8 is drawn according to @xmath6 for learning. a loss function @xmath9 is used to measure the quality of a prediction function @xmath10 by the local error @xmath11. _ throughout the paper we assume that @xmath12 is measurable, @xmath13, convex with respect to the third variable, and uniformly lipschitz continuous satisfying @xmath14 with a finite constant @xmath15. _ support vector machines (svms) considered here are kernel - based regularization schemes in a reproducing kernel hilbert space (rkhs) @xmath0 generated by a mercer kernel @xmath16. with a shifted loss function @xmath17 introduced for dealing even with heavy - tailed distributions as @xmath18, they take the form @xmath19 where for a general borel measure @xmath20 on @xmath21, the function @xmath22 is defined by @xmath23 where @xmath24 is a regularization parameter. the idea to shift a loss function has a long history, see e.g. @xcite in the context of m - estimators. it was shown in @xcite that @xmath22 is also a minimizer of the following optimization problem involving the original loss function @xmath12 if a minimizer exists : @xmath25 the additive model we consider consists of the _ input space decomposition _ @xmath26 with each @xmath27 a complete separable metric space and a _ hypothesis space _ @xmath28 where @xmath29 is a set of functions @xmath30 each of which is also identified as a map @xmath31 from @xmath3 to @xmath5. hence the functions from @xmath32 take the additive form @xmath33. we mention, that there is strictly speaking a notational problem here, because in the previous formula each quantity @xmath34 is an element of the set @xmath35 which is a subset of the full input space @xmath36, @xmath37, whereas in the definition of sample @xmath8 each quantity @xmath38 is an element of the full input space @xmath36, where @xmath39. because these notations will only be used in different places and because we do not expect any misunderstandings, we think this notation is easier and more intuitive than specifying these quantities with different symbols. the additive kernel @xmath40 is defined in terms of mercer kernels @xmath41 on @xmath27 as @xmath42 it generates an rkhs @xmath0 which can be written in terms of the rkhs @xmath43 generated by @xmath41 on @xmath27 corresponding to the form ([additive]) as @xmath44 with norm given by @xmath45 the norm of @xmath46 satisfies @xmath47 to illustrate advantages of additive models, we provide two examples of comparing additive with product kernels. the first example deals with gaussian rbf kernels. all proofs will be given in section [proofsection]. [gaussadd] let @xmath48, @xmath49 $] and @xmath50 ^ 2.$] let @xmath51 and @xmath52.\]] the additive kernel @xmath53 is given by @xmath54 furthermore, the product kernel @xmath55 is the standard gaussian kernel given by @xmath56 define a gaussian function @xmath57 on @xmath58 ^ 2 $] depending only on one variable by @xmath59 then @xmath60 but @xmath61 where @xmath62 denotes the rkhs generated by the standard gaussian rbf kernel @xmath63. the second example is about sobolev kernels. [sobolvadd] let @xmath64, @xmath65 $] and @xmath58^s.$] let @xmath66 : = \bigl\{u\in l_2([0,1]) ; d^\alpha u \in l_2([0,1]) \mbox{~for~all~}|\alpha|\le 1\bigr\}\]] be the sobolev space consisting of all square integrable univariate functions whose derivative is also square integrable. it is an rkhs with a mercer kernel @xmath67 defined on @xmath68 ^ 2 $]. if we take all the mercer kernels @xmath69 to be @xmath67, then @xmath70 $] for each @xmath71. the additive kernel @xmath72 is also a mercer kernel and defines an rkhs @xmath73\right\}.\]] however, the multivariate sobolev space @xmath74^s)$], consisting of all square integrable functions whose partial derivatives are all square integrable, contains discontinuous functions and is not an rkhs. denote the marginal distribution of @xmath6 on @xmath27 as @xmath75. under the assumption that @xmath76 for each @xmath71 and that @xmath43 is dense in @xmath29 in the @xmath77-metric, it was proved in @xcite that @xmath78 in probability as long as @xmath79 satisfies @xmath80 and @xmath81. the rest of the paper has the following structure. section [ratessection] contains our main results on learning rates for svms based on additive kernels. learning rates for quantile regression are treated as important special cases. section [comparisonsection] contains a comparison of our results with other learning rates published recently. section [proofsection] contains all the proofs and some results which can be interesting in their own. Main results on learning rates in this paper we provide some learning rates for the support vector machines generated by additive kernels for additive models which helps improve the quantitative understanding presented in @xcite. the rates are about asymptotic behaviors of the excess risk @xmath82 and take the form @xmath83 with @xmath84. they will be stated under three kinds of conditions involving the hypothesis space @xmath0, the measure @xmath6, the loss @xmath12, and the choice of the regularization parameter @xmath85. the first condition is about the approximation ability of the hypothesis space @xmath0. since the output function @xmath19 is from the hypothesis space, the learning rates of the learning algorithm depend on the approximation ability of the hypothesis space @xmath0 with respect to the optimal risk @xmath86 measured by the following approximation error. [defapprox] the approximation error of the triple @xmath87 is defined as @xmath88 to estimate the approximation error, we make an assumption about the minimizer of the risk @xmath89 for each @xmath90, define the integral operator @xmath91 associated with the kernel @xmath41 by @xmath92 we mention that @xmath93 is a compact and positive operator on @xmath94. hence we can find its normalized eigenpairs @xmath95 such that @xmath96 is an orthonormal basis of @xmath94 and @xmath97 as @xmath98. fix @xmath99. then we can define the @xmath100-th power @xmath101 of @xmath93 by @xmath102 this is a positive and bounded operator and its range is well - defined. the assumption @xmath103 means @xmath104 lies in this range. [assumption1] we assume @xmath105 and @xmath106 where for some @xmath107 and each @xmath108, @xmath109 is a function of the form @xmath110 with some @xmath111. the case @xmath112 of assumption [assumption1] means each @xmath113 lies in the rkhs @xmath43. a standard condition in the literature (e.g., @xcite) for achieving decays of the form @xmath114 for the approximation error ([approxerrordef]) is @xmath115 with some @xmath116. here the operator @xmath117 is defined by @xmath118 in general, this can not be written in an additive form. however, the hypothesis space ([additive]) takes an additive form @xmath119. so it is natural for us to impose an additive expression @xmath120 for the target function @xmath121 with the component functions @xmath113 satisfying the power condition @xmath110. the above natural assumption leads to a technical difficulty in estimating the approximation error : the function @xmath113 has no direct connection to the marginal distribution @xmath122 projected onto @xmath27, hence existing methods in the literature (e.g., @xcite) can not be applied directly. note that on the product space @xmath123, there is no natural probability measure projected from @xmath6, and the risk on @xmath124 is not defined. our idea to overcome the difficulty is to introduce an intermediate function @xmath125. it may not minimize a risk (which is not even defined). however, it approximates the component function @xmath113 well. when we add up such functions @xmath126, we get a good approximation of the target function @xmath121, and thereby a good estimate of the approximation error. this is the first novelty of the paper. [approxerrorthm] under assumption [assumption1], we have @xmath127 where @xmath128 is the constant given by @xmath129 the second condition for our learning rates is about the capacity of the hypothesis space measured by @xmath130-empirical covering numbers. let @xmath131 be a set of functions on @xmath21 and @xmath132 for every @xmath133 the * covering number of @xmath131 * with respect to the empirical metric @xmath134, given by @xmath135 is defined as @xmath136 and the * @xmath130-empirical covering number * of @xmath137 is defined as @xmath138 [assumption2] we assume @xmath139 and that for some @xmath140, @xmath141 and every @xmath142, the @xmath130-empirical covering number of the unit ball of @xmath43 satisfies @xmath143 the second novelty of this paper is to observe that the additive nature of the hypothesis space yields the following nice bound with a dimension - independent power exponent for the covering numbers of the balls of the hypothesis space @xmath0, to be proved in section [samplesection]. [capacitythm] under assumption [assumption2], for any @xmath144 and @xmath145, we have @xmath146 the bound for the covering numbers stated in theorem [capacitythm] is special : the power @xmath147 is independent of the number @xmath148 of the components in the additive model. it is well - known @xcite in the literature of function spaces that the covering numbers of balls of the sobolev space @xmath149 on the cube @xmath150^s$] of the euclidean space @xmath151 with regularity index @xmath152 has the following asymptotic behavior with @xmath153 : @xmath154 here the power @xmath155 depends linearly on the dimension @xmath148. similar dimension - dependent bounds for the covering numbers of the rkhss associated with gaussian rbf - kernels can be found in @xcite. the special bound in theorem [capacitythm] demonstrates an advantage of the additive model in terms of capacity of the additive hypothesis space. the third condition for our learning rates is about the noise level in the measure @xmath6 with respect to the hypothesis space. before stating the general condition, we consider a special case for quantile regression, to illustrate our general results. let @xmath156 be a quantile parameter. the quantile regression function @xmath157 is defined by its value @xmath158 to be a @xmath159-quantile of @xmath160, i.e., a value @xmath161 satisfying @xmath162 the regularization scheme for quantile regression considered here takes the form ([algor]) with the loss function @xmath12 given by the pinball loss as @xmath163 a noise condition on @xmath6 for quantile regression is defined in @xcite as follows. to this end, let @xmath164 be a probability measure on @xmath165 and @xmath166. then a real number @xmath167 is called @xmath159-quantile of @xmath164, if and only if @xmath167 belongs to the set @xmath168\bigr) \ge \tau \mbox{~~and~~ } q\bigl([t, \infty)\bigr) \ge 1-\tau\bigr\}\,.\]] it is well - known that @xmath169 is a compact interval. [noisecond] let @xmath166. 1. a probability measure @xmath164 on @xmath165 is said to have a * @xmath159-quantile of type @xmath170 *, if there exist a @xmath159-quantile @xmath171 and a constant @xmath172 such that, for all @xmath173 $], we have @xmath174 2. let @xmath175 $]. we say that a probability measure @xmath20 on @xmath176 has a * @xmath159-quantile of @xmath177-average type @xmath170 * if the conditional probability measure @xmath178 has @xmath179-almost surely a @xmath159-quantile of type @xmath170 and the function @xmath180 where @xmath181 is the constant defined in part (1), satisfies @xmath182. one can show that a distribution @xmath164 having a @xmath159-quantile of type @xmath170 has a unique @xmath159-quantile @xmath183. moreover, if @xmath164 has a lebesgue density @xmath184 then @xmath164 has a @xmath159-quantile of type @xmath170 if @xmath184 is bounded away from zero on @xmath185 $] since we can use @xmath186\}$] in ([tauquantileoftype2formula]). this assumption is general enough to cover many distributions used in parametric statistics such as gaussian, student s @xmath187, and logistic distributions (with @xmath188), gamma and log - normal distributions (with @xmath189), and uniform and beta distributions (with @xmath190 $]). the following theorem, to be proved in section [proofsection], gives a learning rate for the regularization scheme ([algor]) in the special case of quantile regression. [quantilethm] suppose that @xmath191 almost surely for some constant @xmath192, and that each kernel @xmath41 is @xmath193 with @xmath194 for some @xmath195. if assumption [assumption1] holds with @xmath112 and @xmath6 has a @xmath159-quantile of @xmath177-average type @xmath170 for some @xmath196 $], then by taking @xmath197, for any @xmath198 and @xmath199, with confidence at least @xmath200 we have @xmath201 where @xmath202 is a constant independent of @xmath203 and @xmath204 and @xmath205 please note that the exponent @xmath206 given by ([quantilerates2]) for the learning rate in ([quantilerates]) is independent of the quantile level @xmath159, of the number @xmath148 of additive components in @xmath207, and of the dimensions @xmath208 and @xmath209 further note that @xmath210, if @xmath211, and @xmath212 if @xmath213. because @xmath214 can be arbitrarily close to @xmath215, the learning rate, which is independent of the dimension @xmath216 and given by theorem [quantilethm], is close to @xmath217 for large values of @xmath177 and is close to @xmath218 or better, if @xmath211. to state our general learning rates, we need an assumption on a _ variance - expectation bound _ which is similar to definition [noisecond] in the special case of quantile regression. [assumption3] we assume that there exist an exponent @xmath219 $] and a positive constant @xmath220 such that @xmath221 assumption [assumption3] always holds true for @xmath222. if the triple @xmath223 satisfies some conditions, the exponent @xmath224 can be larger. for example, when @xmath12 is the pinball loss ([pinloss]) and @xmath6 has a @xmath159-quantile of @xmath177-average type @xmath225 for some @xmath196 $] and @xmath226 as defined in @xcite, then @xmath227. [mainratesthm] suppose that @xmath228 is bounded by a constant @xmath229 almost surely. under assumptions [assumption1] to [assumption3], if we take @xmath198 and @xmath230 for some @xmath231, then for any @xmath232, with confidence at least @xmath200 we have @xmath233 where @xmath234 is given by @xmath235 and @xmath202 is constant independent of @xmath203 or @xmath204 (to be given explicitly in the proof). Comparison of learning rates we now add some theoretical and numerical comparisons on the goodness of our learning rates with those from the literature. as already mentioned in the introduction, some reasons for the popularity of additive models are flexibility, increased interpretability, and (often) a reduced proneness of the curse of high dimensions. hence it is important to check, whether the learning rate given in theorem [mainratesthm] under the assumption of an additive model favourably compares to (essentially) optimal learning rates without this assumption. in other words, we need to demonstrate that the main goal of this paper is achieved by theorem [quantilethm] and theorem [mainratesthm], i.e. that an svm based on an additive kernel can provide a substantially better learning rate in high dimensions than an svm with a general kernel, say a classical gaussian rbf kernel, provided the assumption of an additive model is satisfied. our learning rate in theorem [quantilethm] is new and optimal in the literature of svm for quantile regression. most learning rates in the literature of svm for quantile regression are given for projected output functions @xmath236, while it is well known that projections improve learning rates @xcite. here the projection operator @xmath237 is defined for any measurable function @xmath10 by @xmath238 sometimes this is called clipping. such results are given in @xcite. for example, under the assumptions that @xmath6 has a @xmath159-quantile of @xmath177-average type @xmath170, the approximation error condition ([approxerrorb]) is satisfied for some @xmath239, and that for some constants @xmath240, the sequence of eigenvalues @xmath241 of the integral operator @xmath117 satisfies @xmath242 for every @xmath243, it was shown in @xcite that with confidence at least @xmath200, @xmath244 where @xmath245 here the parameter @xmath246 measures the capacity of the rkhs @xmath247 and it plays a similar role as half of the parameter @xmath147 in assumption 2. for a @xmath193 kernel and @xmath112, one can choose @xmath246 and @xmath147 to be arbitrarily small and the above power index @xmath248 can be taken as @xmath249. the learning rate in theorem [quantilethm] may be improved by relaxing assumption 1 to a sobolev smoothness condition for @xmath121 and a regularity condition for the marginal distribution @xmath250. for example, one may use a gaussian kernel @xmath251 depending on the sample size @xmath203 and @xcite achieve the approximation error condition ([approxerrorb]) for some @xmath252. this is done for quantile regression in @xcite. since we are mainly interested in additive models, we shall not discuss such an extension. [gaussmore] let @xmath48, @xmath49 $] and @xmath50 ^ 2.$] let @xmath51 and the additive kernel @xmath72 be given by ([gaussaddform]) with @xmath253 in example [gaussadd] as @xmath52.\]] if the function @xmath121 is given by ([gaussfcn]), @xmath191 almost surely for some constant @xmath192, and @xmath6 has a @xmath159-quantile of @xmath177-average type @xmath170 for some @xmath196 $], then by taking @xmath197, for any @xmath145 and @xmath199, ([quantilerates]) holds with confidence at least @xmath200. it is unknown whether the above learning rate can be derived by existing approaches in the literature (e.g. @xcite) even after projection. note that the kernel in the above example is independent of the sample size. it would be interesting to see whether there exists some @xmath99 such that the function @xmath57 defined by ([gaussfcn]) lies in the range of the operator @xmath254. the existence of such a positive index would lead to the approximation error condition ([approxerrorb]), see @xcite. let us now add some numerical comparisons on the goodness of our learning rates given by theorem [mainratesthm] with those given by @xcite. their corollary 4.12 gives (essentially) minmax optimal learning rates for (clipped) svms in the context of nonparametric quantile regression using one gaussian rbf kernel on the whole input space under appropriate smoothness assumptions of the target function. let us consider the case that the distribution @xmath6 has a @xmath159-quantile of @xmath177-average type @xmath170, where @xmath255, and assume that both corollary 4.12 in @xcite and our theorem [mainratesthm] are applicable. i.e., we assume in particular that @xmath6 is a probability measure on @xmath256 $] and that the marginal distribution @xmath257 has a lebesgue density @xmath258 for some @xmath259. furthermore, suppose that the optimal decision function @xmath260 has (to make theorem [mainratesthm] applicable with @xmath261 $]) the additive structure @xmath207 with each @xmath104 as stated in assumption [assumption1], where @xmath262 and @xmath263, with minimal risk @xmath86 and additionally fulfills (to make corollary 4.12 in @xcite applicable) @xmath264 where @xmath265 $] and @xmath266 denotes a besov space with smoothness parameter @xmath267. the intuitive meaning of @xmath248 is, that increasing values of @xmath248 correspond to increased smoothness. we refer to (*??? * and p. 44) for details on besov spaces. it is well - known that the besov space @xmath268 contains the sobolev space @xmath269 for @xmath270, @xmath271, and @xmath272, and that @xmath273. we mention that if all @xmath41 are suitably chosen wendland kernels, their reproducing kernel hilbert spaces @xmath43 are sobolev spaces, see (*??? * thm. 10.35, p. 160). furthermore, we use the same sequence of regularizing parameters as in (*??? 4.9, cor. 4.12), i.e., @xmath274 where @xmath275, @xmath276, @xmath277 $], and @xmath278 is some user - defined positive constant independent of @xmath279. for reasons of simplicity, let us fix @xmath280. then (*??? 4.12) gives learning rates for the risk of svms for @xmath159-quantile regression, if a single gaussian rbf - kernel on @xmath281 is used for @xmath159-quantile functions of @xmath177-average type @xmath170 with @xmath255, which are of order @xmath282 hence the learning rate in theorem [quantilethm] is better than the one in (*??? 4.12) in this situation, if @xmath283 provided the assumption of the additive model is valid. table [table1] lists the values of @xmath284 from ([explicitratescz2]) for some finite values of the dimension @xmath216, where @xmath285. all of these values of @xmath284 are positive with the exceptions if @xmath286 or @xmath287. this is in contrast to the corresponding exponent in the learning rate by (*?? * cor. 4.12), because @xmath288 table [table2] and figures [figure1] to [figure2] give additional information on the limit @xmath289. of course, higher values of the exponent indicates faster rates of convergence. it is obvious, that an svm based on an additive kernel has a significantly faster rate of convergence in higher dimensions @xmath216 compared to svm based on a single gaussian rbf kernel defined on the whole input space, of course under the assumption that the additive model is valid. the figures seem to indicate that our learning rate from theorem [mainratesthm] is probably not optimal for small dimensions. however, the main focus of the present paper is on high dimensions..[table1] the table lists the limits of the exponents @xmath290 from (*??? * cor. 4.12) and @xmath291 from theorem [mainratesthm], respectively, if the regularizing parameter @xmath292 is chosen in an optimal manner for the nonparametric setup, i.e. @xmath293, with @xmath294 for @xmath295 and @xmath296. recall that @xmath297 $]. [cols= " >, >, >, > ",]

additive models play an important role in semiparametric statistics. this paper gives learning rates for regularized kernel based methods for additive models. these learning rates compare favourably in particular in high dimensions to recent results on optimal learning rates for purely nonparametric regularized kernel based quantile regression using the gaussian radial basis function kernel, provided the assumption of an additive model is valid. additionally, a concrete example is presented to show that a gaussian function depending only on one variable lies in a reproducing kernel hilbert space generated by an additive gaussian kernel, but does not belong to the reproducing kernel hilbert space generated by the multivariate gaussian kernel of the same variance. * key words and phrases. * additive model, kernel, quantile regression, semiparametric, rate of convergence, support vector machine.

1405.3379

Introduction studies of laser beams propagating through turbulent atmospheres are important for many applications such as remote sensing, tracking, and long - distance optical communications. howerver, fully coherent laser beams are very sensitive to fluctuations of the atmospheric refractive index. the initially coherent laser beam acquires some properties of gaussian statistics in course of its propagation through the turbulence. as a result, the noise / signal ratio approaches unity for long - distance propagation. (see, for example, refs.@xcite-@xcite). this unfavourable effect limits the performance of communication channels. to mitigate this negative effect the use of partially (spatially) coherent beams was proposed. the coherent laser beam can be transformed into a partially coherent beam by means of a phase diffuser placed near the exit aperture. this diffuser introduces an additional phase (randomly varying in space and time) to the wave front of the outgoing radiation. statistical characteristics of the random phase determine the initial transverse coherence length of the beam. it is shown in refs. @xcite,@xcite that a considerable decrease in the noise / signal ratio can occur under following conditions : (i) the ratio of the initial transverse coherence length, @xmath0, to the beam radius, @xmath1, should be essentially smaller than unity ; and (ii) the characteristic time of phase variations, @xmath2, should be much smaller than the integration time, @xmath3, of the detector. however, only limiting cases @xmath4 and @xmath5 have been considered in the literature. (see, for example, refs. @xcite,@xcite and ref. @xcite, respectively). it is evident that the inequality @xmath6 can be easily satisfied by choosing a detector with very long integration time. at the same time, this kind of the detector can not distinguish different signals within the interval @xmath3. this means that the resolution of the receiving system might become too low for the case of large @xmath3. on the other hand, there is a technical restriction on phase diffusers : up to now their characteristic times, @xmath2, are not smaller than @xmath7. besides that, in some specific cases (see, for example, ref. @xcite), the spectral broadening of laser radiation due to the phase diffuser (@xmath8) may become unacceptably high. the factors mentioned above impose serious restrictions on the physical characteristics of phase diffusers which could be potentially useful for suppressing the intensity fluctuations. an adequate choice of diffusers may be facilitated if we know in detail the effect of finite - time phase variation, introduced by them, on the photon statistics. in this case, it is possible to control the performance of communication systems. in what follows, we will obtain theoretically the dependence of scintillation index on @xmath9 without any restrictions on the value of this ratio this is the main purpose of our paper. further analysis is based on the formalism developed in ref. @xcite and modified here to understand the case of finite - time dynamics of the phase diffuser. The method of photon distribution function in the problem of scintillations. the detectors of the absorbed type do not sense the instantaneous intensity of electromagnetic waves @xmath10. they sense the intensity averaged over some finite interval @xmath3 i.e. @xmath11 usually, the averaging time @xmath3 (the integration time of the detector) is much smaller than the characteristic time of the turbulence variation, @xmath12, (@xmath13). therefore, the average value of the intensity can be obtained by further averaging of eq. [one] over many measurements corresponding various realizations of the refractive - index configurations. the scintillation index determining the mean - square fluctuations of the intensity is defined by @xmath14\bigg /\big < \bar{i}\big > ^2= \frac{\big < : \bar i(t) ^2:\big>}{\big<\bar i \big>^2}-1,\]] where the symbol @xmath15 indicates the normal ordering of the creation and annihilation operators which determine the intensity, @xmath10. (see more details in refs. @xcite,@xcite). the brackets @xmath16 indicate quantum - mechanical and atmospheric averagings. the intensity @xmath17 depends not only on @xmath18, but also on the spatial variable @xmath19. therefore, the detected intensity is the intensity @xmath20 averaged not only over @xmath18 as in eq. [one], but also over the detector aperture. for simplicity, we will restrict ourselves to calculations of the intensity correlations for coinciding spatial points that correspond to `` small '' detector aperture. this simplification is quite reasonable for a long - distance propagation path of the beam. in the case of quasimonochromatic light, we can choose @xmath20 in the form @xmath21 where @xmath22 and @xmath23 are the creation and annihilation operators of photons with momentum @xmath24. they are given in the heisenberg representation. @xmath25 is the volume of the system. it follows from eqs. [two],[three] that @xmath26 can be obtained if one knows the average @xmath27 it is a complex problem to obtain this value for arbitrary turbulence strengths and propagation distances. nevertheless, the following qualitative reasoning can help to do this in the case of strong turbulence. we have mentioned that the laser light acquires the properties of gaussian statistics in the course of its propagation through the turbulent atmosphere. as a result, in the limit of infinitely long propagation path, @xmath28, only diagonal " terms, i.e. terms with (i) @xmath29 or (ii) @xmath30, @xmath31 contribute to the right part of eq. [four]. for large but still finite @xmath28, there exist small ranges of @xmath32 in case (i) and @xmath33, @xmath34 in case (ii) contributing into the sum in eq. the presence of the mentioned regions is due to the two possible ways of correlating of four different waves (see ref. @xcite) which enter the right hand side of eq. [four]. as explained in ref. @xcite, the characteristic sizes of regions (i) and (ii) depend on the atmospheric broadening of beam radii as @xmath35, thus decreasing with increasing @xmath28. in the case of long - distance propagation, @xmath36 is much smaller than the component of photon wave - vectors perpendicular to the @xmath28 axis. the last quantity grows with @xmath28 as @xmath37. (see ref. @xcite). for this reason, the overlapping of regions (i) and (ii) can be neglected. in this case eq. [four] can be rewritten in the convenient form : @xmath38 @xmath39 where the value @xmath40, confining summation over @xmath41, is chosen to be greater than @xmath42 but much smaller than the characteristic transverse wave vector of the photons ; this is consistent with the above explanations. the two terms in the right - hand side correspond to the two regions of four - wave correlations. the quantity @xmath43 entering the right side of eq. [five] is the operator of photon density in phase space (the photon distribution function in @xmath44 space). it was used in refs. @xcite,@xcite and @xcite for the description of photon propagation in turbulent atmospheres. by analogy, we can define the two - time distribution function @xmath45 then eq. [five] can be rewritten in terms of the distribution functions as @xmath46 let us represent @xmath47 in the form @xmath48. we assume that @xmath49, as explained in the text after eq.[one]. in this case the hamiltonian of photons in a turbulent atmosphere can be considered to be independent of time. as a result, both functions defined by eqs. [six] and [seven] satisfy the same kinetic equation, i.e. @xmath50 @xmath51 where @xmath52 is the photon velocity, @xmath53 is a random force, caused by the turbulence. this force is equal to @xmath54, where @xmath55 is the frequency of laser radiation. @xmath56 is the refractive index of the atmosphere. the general solution of the equation for @xmath48 can be written in the form @xmath57 where @xmath58 @xmath59 the functions @xmath60 and @xmath61 obey the equations of motion @xmath62 with the boundary conditions @xmath63. the instant @xmath64 is equal to @xmath65, where @xmath66 is the speed of light. @xmath64 is the time of the exit of photons from the source. this choice of @xmath64 makes it possible to neglect the influence of the turbulence on the initial values of operators @xmath67 (their dependence on time is as in vacuum). the term for @xmath68 can be obtained from eq. [twelve] by putting @xmath69. substituting both distribution functions into eq. [eight], we obtain @xmath70 @xmath71 @xmath72:\big>,\]] where @xmath73 and @xmath74 are solutions of eqs. [twelve] with the initial conditions @xmath63 and @xmath75, respectively. A phase diffuser with finite correlation time the operators on the right side of eq. [thirteen] are related through matching conditions with the amplitudes of the exiting laser radiation (see ref. @xcite) by the relation @xmath76 where @xmath77 is the operator of the laser field which is assumed to be a single - mode field and the subscript (@xmath78) means perpendicular to the @xmath28-axis component. the function @xmath79 describes the profile of the laser mode, which is assumed to be gaussian - type function [@xmath80. @xmath1 desribes the initial radius of the beam. to account for the effect of the phase diffuser, a factor @xmath81 or @xmath82 should be inserted into the integrand of eq. [fourteen]. the quantity @xmath83 is the random phase introduced by the phase diffuser. a similar consideration is applicable to each of four photon operators entering both terms in square brackets of eq. [thirteen]. it can be easily seen that the factor @xmath84},\]] describing the effect of phase screen on the beam, enters implicitly the integrand of eq. [thirteen] (the indices @xmath78 are omitted here for the sake of brevity). there are integrations over variables @xmath85 as shown in eq. [fourteen]. furthermore, the brackets @xmath16, which indicate averaging over different realizations of the atmosperic inhomogeneities, also indicate averaging over different states of the phase diffuser. as long as both types of averaging do not correlate, the factor ([fifteen]) entering eq. [thirteen] must be averaged over different instants, @xmath64. to begin with, let us consider the simplest case of two phase correlations @xmath86}\big >.\]] it is evident that in the case @xmath87, as shown schematically in fig. 1, the factor ([sixteen]) is sizable if only points @xmath19 and @xmath88 are close to one another. two curves correspond to different instants @xmath18 and @xmath89.] therefore, the term given by eq. [sixteen] can be replaced by @xmath90 where @xmath91 is considered to be a gaussian random variable with the mean - square values given by @xmath92 ^ 2\rangle = \langle [\frac { \partial \varphi ({ \bf r},t_0)}{\partial y}]^2\rangle = 2\lambda _ c^{-2}$], where @xmath93 is the correlation length of phase fluctuations. (see fig.1). as we see, in this case the effect of phase fluctuations can be described by the schell model @xcite-@xcite,@xcite-@xcite. a somewhat more complex situation is for the average value of @xmath94 given by eq. [fifteen]. there is an effective phase correlation not only in the case of coincident times, but also for differing times. for @xmath95, two different sets of coordinates contribute considerably to phase correlations. this can be described mathematically as @xmath96}\big > \approx \big < e^{i[\varphi ({ \bf r},t_0)-\varphi ({ \bf r^\prime},t_0)]}\big > \times\]] @xmath97}\big > + \big < e^{i[\varphi ({ \bf r},t_0)-\varphi ({ \bf r^\prime _ 1},t_0+\tau)] } \big > \big < e^{i[\varphi ({ \bf r_1},t_0+\tau) -\varphi ({ \bf r^\prime }, t_0)]}\big >.\]] repeating the arguments leading to eq. [seventeen], we represent the difference in the last term @xmath98 as @xmath99 then, considering the random functions @xmath100 and @xmath101 as independent gaussian variables, we obtain a simple expression for @xmath102. it is given by @xmath103}+ e^{-\lambda _ c^{-2}[({\bf r - r^\prime}_1)^2+({\bf r^\prime -r_1})^2]-2\nu^2\tau ^2},\]] where @xmath104 ^ 2\rangle = 2\nu^2 $]. as we see, the effect of the phase screen can be described by two parameters, @xmath93 and @xmath105, which characterize the spatial and temporal coherence of the laser beam. in the limiting case, @xmath106, the second term in eq. [twenty] vanishes and the problem is reduced to the case considered in refs. @xcite,@xcite. in the opposite case, @xmath107, both terms in eq. [twenty] are important. this is shown in ref. @xcite. in what follows, we will see that these two limiting cases have physical interpretations where where @xmath108 (slow detector) and @xmath109 (fast detector), respectively. there is a specific realization of the diffuser in which a random phase distribution moves across the beam. (this situation can be modeled by a rotating transparent disk with large diameter and varying thickness.) the phase depends here on the only variable @xmath110, i.e. @xmath111 where @xmath112 is the velocity of the drift. then we have @xmath113}+e^{-\lambda _ c^{-2}[({\bf r - r^\prime_1+v}\tau)^2+({\bf r^\prime -r_1+v}\tau)^2]}.\]] comparing eqs. [twenty] and [twtw], we see that the quantity, @xmath114, stands for the characteristic parameter describing the efficiency of the phase diffuser. the criterion of slow " detector requires @xmath115. qualitatively, the two scenarios of phase variations, given by eqs. [twenty] and [twtw], affect in a similar way the intensity fluctuations. in what follows, we consider the first of them as the simplest one. (this is because the spatial and temporal variables in @xmath102, given by eq. [twenty], are separable.) _ vs _ propagation distance @xmath28 in the case of `` slow '' detector : @xmath116. the parameter @xmath117 indicates different initial coherence length. in the absence of phase diffuser @xmath118 (solid line). @xmath119 is the conventional parameter describing a strength of the atmospheric turbulence.] substituting the expressions for operators given by eq. [fourteen] with account for the phase factors @xmath120 and averaging over time as shown in eq. [one], we obtain @xmath121 @xmath122\bigg >, \]] where the notation @xmath16 after sums indicates averaging over different realizations of the atmospheric refractive index. the parameter @xmath123 describes the initial coherence length modified by the phase diffuser. other notations are defined by following relations @xmath124 @xmath125 @xmath126 further calculations follow the scheme described in ref @xcite. 2 illustrates the effect of the phase diffuser on scintillations in the limit of a slow " detector (@xmath127). we can see a considerable decrease in @xmath128 caused by the phase diffuser. at the same time, the effect of the phase screen on @xmath128 becomes weaker for finite values of @xmath129. moreover, comparing the two upper curves in fig. 3, we see the opposite effect : slow phase variations (@xmath130) result in increased scintillations. there is a simple explanation for this phenomenon : the noise generated by the turbulence is complemented by the noise arising from the random phase screen. the integration time of the detector, @xmath3, is not sufficiently large for averaging phase variations generated by the diffuser. the function, @xmath131, has a very simple form in the two limits : (i)@xmath132, when @xmath133 ; and (ii) @xmath134, when @xmath109. then, in case (i) and for small values of the initial coherence [@xmath135, the asymptotic term for the scintillation index (@xmath136) is given by @xmath137 the right - hand side of eq. [twfo] differs from analogous one in ref. @xcite by the value @xmath138 that is much less than unity but, nevertheless, can be comparable or even greater than @xmath139. in case (ii), the asymptotic value of @xmath26 is close to unity, coinciding with the results of refs. @xcite and @xcite. this agrees with well known behavior of the scintillation index to approach unity for any source distribution, provided the response time of the recording instrument is short compared with the source coherence time. (see, for example, survey @xcite). a similar tendency can be seen in both figs. 3 and 4 : the curves with the smallest @xmath129, used for numerical calculations (@xmath130), are close to the curves without diffuser " in spite of the small initial coherence length [@xmath140. it can also be seen that all curves approach their asymptotic values very slowly. describing diffuser dynamics. the solid curve is calculated for @xmath118 (without diffuser). other curves are for @xmath141.] .] Discussion it follows from our analysis that the scintillation index is very sensitive to the diffuser parameters, @xmath0 and @xmath142, for long propagation paths. on the other hand, the characteristics of the irradience such as beam radius, @xmath143, and angle - of - arrival spread, @xmath144, do not depend on the presence of the phase diffuser for large values of @xmath28. to see this, the following analysis is useful. the beam radius expressed in terms of the distribution function is given by @xmath145 straightforward calculations using eq. [ten] with @xmath69 (see ref. @xcite) result in the following explicit form : @xmath146 where @xmath147 and @xmath148 is the inner radius of turbulent eddies, which in our previous calculations was assumed to be equal @xmath149 m. as we see, the third term does not depend on the diffuser parameters and it dominates when @xmath150. a similar situation holds for the angle - of - arrival spread, @xmath144. (this physical quantity is of great importance for the performance of communication systems based on frequency encoded information @xcite.) it is defined by the distribution function as @xmath151 simple calculations, which are very similar to those while obtaining @xmath152, result in @xmath153 ^ 2=\frac 2{r_1 ^ 2q_0 ^ 2}+12tz-\frac { 4z^2}{q_0 ^ 4r^2}(r_1^{-2}+3tq_0 ^ 2z)^2.\]] for long propagation paths,. [twei] reduces to @xmath154, which like @xmath152 does not depend on the diffuser parameters. as we see, for large distances @xmath28, the quantities @xmath152 and @xmath144 do not depend on @xmath93 and @xmath105. this contrasts with the case of the scintillation index. so pronounced differences can be explained by differences in the physical nature of these characteristics. it follows from eq. [two] that the functional, @xmath26, is quadratic in the distribution function, @xmath155. hence, four - wave correlations determine the value of scintillation index. the main effect of a phase diffuser on @xmath26 is to destroy correlations between waves exited at different times. (see more explanations in ref. this is achieved at sufficiently small parameters @xmath93 and @xmath156. in contrast, @xmath152 and @xmath144 depend on two wave - correlations, both waves being given at the same instant. therefore, the values of @xmath152 and @xmath144 do not depend on the rate of phase variations [@xmath105 does not enter the factor ([seventeen]) describing the effect of phase diffuser]. moreover, these quantities become independent of @xmath93 at long propagation paths because light scattering on atmospheric inhomogeneities prevails in this case. the plots in figs. 3 anf 4 show that the finite - time effect is quite sizable even for very slow " detectors (@xmath157). our paper makes it possible to estimate the actual utility of phase diffusers in several physical regimes. Conclusion we have analyzed the effects of a diffuser on scintillations for the case of large - amplitude phase fluctuations. this specific case is very convenient for theoretial analysis because only two parameters are required to describe the effects of the diffuser. phase fluctuations may occur independently in space as well as in time. also, our formalism can be applied for the physical situation in which a spatially random phase distribution drifts across the beam. [twtw].) our results show the importance of both parameters, @xmath93 and @xmath142, on the ability of a phase diffuser to suppress scintillations. Acknowledgment this work was carried out under the auspices of the national nuclear security administration of the u.s. department of energy at los alamos national laboratory under contract no. de - ac52 - 06na25396. we thank onr for supporting this research.

the effect of a random phase diffuser on fluctuations of laser light (scintillations) is studied. not only spatial but also temporal phase variations introduced by the phase diffuser are analyzed. the explicit dependence of the scintillation index on finite - time phase variations is obtained for long propagation paths. it is shown that for large amplitudes of phase fluctuations, a finite - time effect decreases the ability of phase diffuser to suppress the scintillations.

0903.5449

Introduction the transport properties of nonlinear non - equilibrium dynamical systems are far from well - understood@xcite. consider in particular so - called ratchet systems which are asymmetric periodic potentials where an ensemble of particles experience directed transport@xcite. the origins of the interest in this lie in considerations about extracting useful work from unbiased noisy fluctuations as seems to happen in biological systems@xcite. recently attention has been focused on the behavior of deterministic chaotic ratchets@xcite as well as hamiltonian ratchets@xcite. chaotic systems are defined as those which are sensitively dependent on initial conditions. whether chaotic or not, the behavior of nonlinear systems including the transition from regular to chaotic behavior is in general sensitively dependent on the parameters of the system. that is, the phase - space structure is usually relatively complicated, consisting of stability islands embedded in chaotic seas, for examples, or of simultaneously co - existing attractors. this can change significantly as parameters change. for example, stability islands can merge into each other, or break apart, and the chaotic sea itself may get pinched off or otherwise changed, or attractors can change symmetry or bifurcate. this means that the transport properties can change dramatically as well. a few years ago, mateos@xcite considered a specific ratchet model with a periodically forced underdamped particle. he looked at an ensemble of particles, specifically the velocity for the particles, averaged over time and the entire ensemble. he showed that this quantity, which is an intuitively reasonable definition of ` the current ', could be either positive or negative depending on the amplitude @xmath0 of the periodic forcing for the system. at the same time, there exist ranges in @xmath0 where the trajectory of an individual particle displays chaotic dynamics. mateos conjectured a connection between these two phenomena, specifically that the reversal of current direction was correlated with a bifurcation from chaotic to periodic behavior in the trajectory dynamics. even though it is unlikely that such a result would be universally valid across all chaotic deterministic ratchets, it would still be extremely useful to have general heuristic rules such as this. these organizing principles would allow some handle on characterizing the many different kinds of behavior that are possible in such systems. a later investigation@xcite of the mateos conjecture by barbi and salerno, however, argued that it was not a valid rule even in the specific system considered by mateos. they presented results showing that it was possible to have current reversals in the absence of bifurcations from periodic to chaotic behavior. they proposed an alternative origin for the current reversal, suggesting it was related to the different stability properties of the rotating periodic orbits of the system. these latter results seem fundamentally sensible. however, this paper based its arguments about currents on the behavior of a _ single _ particle as opposed to an ensemble. this implicitly assumes that the dynamics of the system are ergodic. this is not true in general for chaotic systems of the type being considered. in particular, there can be extreme dependence of the result on the statistics of the ensemble being considered. this has been pointed out in earlier studies @xcite which laid out a detailed methodology for understanding transport properties in such a mixed regular and chaotic system. depending on specific parameter value, the particular system under consideration has multiple coexisting periodic or chaotic attractors or a mixture of both. it is hence appropriate to understand how a probability ensemble might behave in such a system. the details of the dependence on the ensemble are particularly relevant to the issue of the possible experimental validation of these results, since experiments are always conducted, by virtue of finite - precision, over finite time and finite ensembles. it is therefore interesting to probe the results of barbi and salerno with regard to the details of the ensemble used, and more formally, to see how ergodicity alters our considerations about the current, as we do in this paper. we report here on studies on the properties of the current in a chaotic deterministic ratchet, specifically the same system as considered by mateos@xcite and barbi and salerno@xcite. we consider the impact of different kinds of ensembles of particles on the current and show that the current depends significantly on the details of the initial ensemble. we also show that it is important to discard transients in quantifying the current. this is one of the central messages of this paper : broad heuristics are rare in chaotic systems, and hence it is critical to understand the ensemble - dependence in any study of the transport properties of chaotic ratchets. having established this, we then proceed to discuss the connection between the bifurcation diagram for individual particles and the behavior of the current. we find that while we disagree with many of the details of barbi and salerno s results, the broader conclusion still holds. that is, it is indeed possible to have current reversals in the absence of bifurcations from chaos to periodic behavior as well as bifurcations without any accompanying current reversals. the result of our investigation is therefore that the transport properties of a chaotic ratchet are not as simple as the initial conjecture. however, we do find evidence for a generalized version of mateos s conjecture. that is, in general, bifurcations for trajectory dynamics as a function of system parameter seem to be associated with abrupt changes in the current. depending on the specific value of the current, these abrupt changes may lead the net current to reverse direction, but not necessarily so. we start below with a preparatory discussion necessary to understand the details of the connection between bifurcations and current reversal, where we discuss the potential and phase - space for single trajectories for this system, where we also define a bifurcation diagram for this system. in the next section, we discuss the subtleties of establishing a connection between the behavior of individual trajectories and of ensembles. after this, we are able to compare details of specific trajectory bifurcation curves with current curves, and thus justify our broader statements above, after which we conclude. Regularity and chaos in single-particle ratchet dynamics the goal of these studies is to understand the behavior of general chaotic ratchets. the approach taken here is that to discover heuristic rules we must consider specific systems in great detail before generalizing. we choose the same @xmath1-dimensional ratchet considered previously by mateos@xcite, as well as barbi and salerno@xcite. we consider an ensemble of particles moving in an asymmetric periodic potential, driven by a periodic time - dependent external force, where the force has a zero time - average. there is no noise in the system, so it is completely deterministic, although there is damping. the equations of motion for an individual trajectory for such a system are given in dimensionless variables by @xmath2 where the periodic asymmetric potential can be written in the form @xmath3 + \frac{1}{4 } \sin [4\pi (x -x_0)] \bigg].\]] in this equation @xmath4 have been introduced for convenience such that one potential minimum exists at the origin with @xmath5 and the term @xmath6. (a) classical phase space for the unperturbed system. for @xmath7, @xmath8, two chaotic attractors emerge with @xmath9 (b) @xmath10 (c) and a period four attractor consisting of the four centers of the circles with @xmath11.,title="fig:",width=302] the phase - space of the undamped undriven ratchet the system corresponding to the unperturbed potential @xmath12 looks like a series of asymmetric pendula. that is, individual trajectories have one of following possible time - asymptotic behaviors : (i) inside the potential wells, trajectories and all their properties oscillate, leading to zero net transport. outside the wells, the trajectories either (ii) librate to the right or (iii) to the left, with corresponding net transport depending upon initial conditions. there are also (iv) trajectories on the separatrices between the oscillating and librating orbits, moving between unstable fixed points in infinite time, as well as the unstable and stable fixed points themselves, all of which constitute a set of negligible measure. when damping is introduced via the @xmath13-dependent term in eq. [eq : dyn], it makes the stable fixed points the only attractors for the system. when the driving is turned on, the phase - space becomes chaotic with the usual phenomena of intertwining separatrices and resulting homoclinic tangles. the dynamics of individual trajectories in such a system are now very complicated in general and depend sensitively on the choice of parameters and initial conditions. we show snapshots of the development of this kind of chaos in the set of poincar sections fig. ([figure1]b, c) together with a period - four orbit represented by the center of the circles. a broad characterization of the dynamics of the problem as a function of a parameter (@xmath14 or @xmath15) emerges in a bifurcation diagram. this can be constructed in several different and essentially equivalent ways. the relatively standard form that we use proceeds as follows : first choose the bifurcation parameter (let us say @xmath0) and correspondingly choose fixed values of @xmath16, and start with a given value for @xmath17. now iterate an initial condition, recording the value of the particle s position @xmath18 at times @xmath19 from its integrated trajectory (sometimes we record @xmath20. this is done stroboscopically at discrete times @xmath21 where @xmath22 and @xmath23 is an integer @xmath24 with @xmath25 the maximum number of observations made. of these, discard observations at times less than some cut - off time @xmath26 and plot the remaining points against @xmath27. it must be noted that discarding transient behavior is critical to get results which are independent of initial condition, and we shall emphasize this further below in the context of the net transport or current. if the system has a fixed - point attractor then all of the data lie at one particular location @xmath28. a periodic orbit with period @xmath29 (that is, with period commensurate with the driving) shows up with @xmath30 points occupying only @xmath31 different locations of @xmath32 for @xmath27. all other orbits, including periodic orbits of incommensurate period result in a simply - connected or multiply - connected dense set of points. for the next value @xmath33, the last computed value of @xmath34 at @xmath35 are used as initial conditions, and previously, results are stored after cutoff and so on until @xmath36. that is, the bifurcation diagram is generated by sweeping the relevant parameter, in this case @xmath0, from @xmath27 through some maximum value @xmath37. this procedure is intended to catch all coexisting attractors of the system with the specified parameter range. note that several initial conditions are effectively used troughout the process, and a bifurcation diagram is not the behavior of a single trajectory. we have made several plots, as a test, with different initial conditions and the diagrams obtained are identical. we show several examples of this kind of bifurcation diagram below, where they are being compared with the corresponding behavior of the current. having broadly understood the wide range of behavior for individual trajectories in this system, we now turn in the next section to a discussion of the non - equilibrium properties of a statistical ensemble of these trajectories, specifically the current for an ensemble. Ensemble currents the current @xmath38 for an ensemble in the system is defined in an intuitive manner by mateos@xcite as the time - average of the average velocity over an ensemble of initial conditions. that is, an average over several initial conditions is performed at a given observation time @xmath39 to yield the average velocity over the particles @xmath40 this average velocity is then further time - averaged ; given the discrete time @xmath39 for observation this leads to a second sum @xmath41 where @xmath25 is the number of time - observations made. for this to be a relevant quantity to compare with bifurcation diagrams, @xmath38 should be independent of the quantities @xmath42 but still strongly dependent on @xmath43. a further parameter dependence that is being suppressed in the definition above is the shape and location of the ensemble being used. that is, the transport properties of an ensemble in a chaotic system depend in general on the part of the phase - space being sampled. it is therefore important to consider many different initial conditions to generate a current. the first straightforward result we show in fig. ([figure2]) is that in the case of chaotic trajectories, a single trajectory easily displays behavior very different from that of many trajectories. however, it turns out that in the regular regime, it is possible to use a single trajectory to get essentially the same result as obtained from many trajectories. further consider the bifurcation diagram in fig. ([figure3]) where we superimpose the different curves resulting from varying the number of points in the initial ensemble. first, the curve is significantly smoother as a function of @xmath0 for larger @xmath44. even more relevant is the fact that the single trajectory data (@xmath45) may show current reversals that do not exist in the large @xmath44 data. current @xmath38 versus the number of trajectories @xmath44 for @xmath7 ; dashed lines correspond to a regular motion with @xmath46 while solid lines correspond to a chaotic motion with @xmath47. note that a single trajectory is sufficient for a regular motion while the convergence in the chaotic case is only obtained if the @xmath44 exceeds a certain threshold, @xmath48.,title="fig:",width=302] current @xmath38 versus @xmath0 for different set of trajectories @xmath44 ; @xmath45 (circles), @xmath49 (square) and @xmath50 (dashed lines). note that a single trajectory suffices in the regular regime where all the curves match. in the chaotic regime, as @xmath44 increases, the curves converge towards the dashed one.,title="fig:",width=302] also, note that single - trajectory current values are typically significantly greater than ensemble averages. this arises from the fact that an arbitrarily chosen ensemble has particles with idiosyncratic behaviors which often average out. as our result, with these ensembles we see typical @xmath51 for example, while barbi and salerno report currents about @xmath52 times greater. however, it is not true that only a few trajectories dominate the dynamics completely, else there would not be a saturation of the current as a function of @xmath44. all this is clear in fig. ([figure3]). we note that the * net * drift of an ensemble can be a lot closer to @xmath53 than the behavior of an individual trajectory. it should also be clear that there is a dependence of the current on the location of the initial ensemble, this being particularly true for small @xmath44, of course. the location is defined by its centroid @xmath54. for @xmath45, it is trivially true that the initial location matters to the asymptotic value of the time - averaged velocity, given that this is a non - ergodic and chaotic system. further, considering a gaussian ensemble, say, the width of the ensemble also affects the details of the current, and can show, for instance, illusory current reversal, as seen in figs. ([current - bifur1],[current - bifur2]) for example. notice also that in fig. ([current - bifur1]), at @xmath55 and @xmath56, the deviations between the different ensembles is particularly pronounced. these points are close to bifurcation points where some sort of symmetry breaking is clearly occuring, which underlines our emphasis on the relevance of specifying ensemble characteristics in the neighborhood of unstable behavior. however, why these specific bifurcations should stand out among all the bifurcations in the parameter range shown is not entirely clear. to understand how to incorporate this knowledge into calculations of the current, therefore, consider the fact that if we look at the classical phase space for the hamiltonian or underdamped @xmath57 motion, we see the typical structure of stable islands embedded in a chaotic sea which have quite complicated behavior@xcite. in such a situation, the dynamics always depends on the location of the initial conditions. however, we are not in the hamiltonian situation when the damping is turned on in this case, the phase - space consists in general of attractors. that is, if transient behavior is discarded, the current is less likely to depend significantly on the location of the initial conditions or on the spread of the initial conditions. in particular, in the chaotic regime of a non - hamiltonian system, the initial ensemble needs to be chosen larger than a certain threshold to ensure convergence. however, in the regular regime, it is not important to take a large ensemble and a single trajectory can suffice, as long as we take care to discard the transients. that is to say, in the computation of currents, the definition of the current needs to be modified to : @xmath58 where @xmath59 is some empirically obtained cut - off such that we get a converged current (for instance, in our calculations, we obtained converged results with @xmath60). when this modified form is used, the convergence (ensemble - independence) is more rapid as a function of @xmath61 and the width of the intial conditions. armed with this background, we are now finally in a position to compare bifurcation diagrams with the current, as we do in the next section. The relationship between bifurcation diagrams and ensemble currents our results are presented in the set of figures fig. ([figure5]) fig. ([rev - nobifur]), in each of which we plot both the ensemble current and the bifurcation diagram as a function of the parameter @xmath0. the main point of these numerical results can be distilled into a series of heuristic statements which we state below ; these are labelled with roman numerals. for @xmath7 and @xmath8, we plot current (upper) with @xmath62 and bifurcation diagram (lower) versus @xmath0. note that there is a * single * current reversal while there are many bifurcations visible in the same parameter range.,title="fig:",width=302] consider fig. ([figure5]), which shows the parameter range @xmath63 chosen relatively arbitrarily. in this figure, we see several period - doubling bifurcations leading to order - chaos transitions, such as for example in the approximate ranges @xmath64. however, there is only one instance of current - reversal, at @xmath65. note, however, that the current is not without structure it changes fairly dramatically as a function of parameter. this point is made even more clearly in fig. ([figure6]) where the current remains consistently below @xmath53, and hence there are in fact, no current reversals at all. note again, however, that the current has considerable structure, even while remaining negative. for @xmath66 and @xmath8, plotted are current (upper) and bifurcation diagram (lower) versus @xmath0 with @xmath62. notice the current stays consistently below @xmath53.,title="fig:",width=302] current and bifurcations versus @xmath0. in (a) and (b) we show ensemble dependence, specifically in (a) the black curve is for an ensemble of trajectories starting centered at the stable fixed point @xmath67 with a root - mean - square gaussian width of @xmath68, and the brown curve for trajectories starting from the unstable fixed point @xmath69 and of width @xmath68. in (b), all ensembles are centered at the stable fixed point, the black line for an ensemble of width @xmath68, brown a width of @xmath70 and maroon with width @xmath71. (c) is the comparison of the current @xmath38 without transients (black) and with transients (brown) along with the single - trajectory results in blue (after barbi and salerno). the initial conditions for the ensembles are centered at @xmath67 with a mean root square gaussian of width @xmath68. (d) is the corresponding bifurcation diagram.,title="fig:",width=302] it is possible to find several examples of this at different parameters, leading to the negative conclusion, therefore, that * (i) not all bifurcations lead to current reversal*. however, we are searching for positive correlations, and at this point we have not precluded the more restricted statement that all current reversals are associated with bifurcations, which is in fact mateos conjecture. we therefore now move onto comparing our results against the specific details of barbi and salerno s treatment of this conjecture. in particular, we look at their figs. (2,3a,3b), where they scan the parameter region @xmath72. the distinction between their results and ours is that we are using _ ensembles _ of particles, and are investigating the convergence of these results as a function of number of particles @xmath44, the width of the ensemble in phase - space, as well as transience parameters @xmath73. our data with larger @xmath44 yields different results in general, as we show in the recomputed versions of these figures, presented here in figs. ([current - bifur1],[current - bifur2]). specifically, (a) the single - trajectory results are, not surprisingly, cleaner and can be more easily interpreted as part of transitions in the behavior of the stability properties of the periodic orbits. the ensemble results on the other hand, even when converged, show statistical roughness. (b) the ensemble results are consistent with barbi and salerno in general, although disagreeing in several details. for instance, (c) the bifurcation at @xmath74 has a much gentler impact on the ensemble current, which has been growing for a while, while the single - trajectory result changes abruptly. note, (d) the very interesting fact that the single - trajectory current completely misses the bifurcation - associated spike at @xmath75. further, (e) the barbi and salerno discussion of the behavior of the current in the range @xmath76 is seen to be flawed our results are consistent with theirs, however, the current changes are seen to be consistent with bifurcations despite their statements to the contrary. on the other hand (f), the ensemble current shows a case [in fig. ([current - bifur2]), at @xmath77 of current reversal that does not seem to be associated with bifurcations. in this spike, the current abruptly drops below @xmath53 and then rises above it again. the single trajectory current completely ignores this particular effect, as can be seen. the bifurcation diagram indicates that in this case the important transitions happen either before or after the spike. all of this adds up to two statements : the first is a reiteration of the fact that there is significant information in the ensemble current that can not be obtained from the single - trajectory current. the second is that the heuristic that arises from this is again a negative conclusion, that * (ii) not all current reversals are associated with bifurcations. * where does this leave us in the search for ` positive'results, that is, useful heuristics? one possible way of retaining the mateos conjecture is to weaken it, i.e. make it into the statement that * (iii) _ most _ current reversals are associated with bifurcations. * same as fig. ([current - bifur1]) except for the range of @xmath0 considered.,title="fig:",width=302] for @xmath78 and @xmath8, plotted are current (upper) and bifurcation diagram (lower) versus @xmath0 with @xmath62. note in particular in this figure that eyeball tests can be misleading. we see reversals without bifurcations in (a) whereas the zoomed version (c) shows that there are windows of periodic and chaotic regimes. this is further evidence that jumps in the current correspond in general to bifurcation.,title="fig:",width=302] for @xmath7 and @xmath79, current (upper) and bifurcation diagram (lower) versus @xmath0.,title="fig:",width=302] however, a * different * rule of thumb, previously not proposed, emerges from our studies. this generalizes mateos conjecture to say that * (iv) bifurcations correspond to sudden current changes (spikes or jumps)*. note that this means these changes in current are not necessarily reversals of direction. if this current jump or spike goes through zero, this coincides with a current reversal, making the mateos conjecture a special case. the physical basis of this argument is the fact that ensembles of particles in chaotic systems _ can _ have net directed transport but the details of this behavior depends relatively sensitively on the system parameters. this parameter dependence is greatly exaggerated at the bifurcation point, when the dynamics of the underlying single - particle system undergoes a transition a period - doubling transition, for example, or one from chaos to regular behavior. scanning the relevant figures, we see that this is a very useful rule of thumb. for example, it completely captures the behaviour of fig. ([figure6]) which can not be understood as either an example of the mateos conjecture, or even a failure thereof. as such, this rule significantly enhances our ability to characterize changes in the behavior of the current as a function of parameter. a further example of where this modified conjecture helps us is in looking at a seeming negation of the mateos conjecture, that is, an example where we seem to see current - reversal without bifurcation, visible in fig. ([hidden - bifur]). the current - reversals in that scan of parameter space seem to happen inside the chaotic regime and seemingly independent of bifurcation. however, this turns out to be a ` hidden'bifurcation when we zoom in on the chaotic regime, we see hidden periodic windows. this is therefore consistent with our statement that sudden current changes are associated with bifurcations. each of the transitions from periodic behavior to chaos and back provides opportunities for the current to spike. however, in not all such cases can these hidden bifurcations be found. we can see an example of this in fig. ([rev - nobifur]). the current is seen to move smoothly across @xmath80 with seemingly no corresponding bifurcations, even when we do a careful zoom on the data, as in fig. ([hidden - bifur]). however, arguably, although subjective, this change is close to the bifurcation point. this result, that there are situations where the heuristics simply do not seem to apply, are part of the open questions associated with this problem, of course. we note, however, that we have seen that these broad arguments hold when we vary other parameters as well (figures not shown here). in conclusion, in this paper we have taken the approach that it is useful to find general rules of thumb (even if not universally valid) to understand the complicated behavior of non - equilibrium nonlinear statistical mechanical systems. in the case of chaotic deterministic ratchets, we have shown that it is important to factor out issues of size, location, spread, and transience in computing the ` current'due to an ensemble before we search for such rules, and that the dependence on ensemble characteristics is most critical near certain bifurcation points. we have then argued that the following heuristic characteristics hold : bifurcations in single - trajectory behavior often corresponds to sudden spikes or jumps in the current for an ensemble in the same system. current reversals are a special case of this. however, not all spikes or jumps correspond to a bifurcation, nor vice versa. the open question is clearly to figure out if the reason for when these rules are violated or are valid can be made more concrete. Acknowledgements a.k. gratefully acknowledges t. barsch and kamal p. singh for stimulating discussions, the reimar lst grant and financial support from the alexander von humboldt foundation in bonn. a.k.p. is grateful to carleton college for the ` sit, wallin, and class of 1949'sabbatical fellowships, and to the mpipks for hosting him for a sabbatical visit, which led to this collaboration. useful discussions with j.- m. rost on preliminary results are also acknowledged. p. hnggi and bartussek, in nonlinear physics of complex systems, lecture notes in physics vol. 476, edited by j. parisi, s.c. mueller, and w. zimmermann (springer verlag, berlin, 1996), pp.294 - 308 ; r.d. asturmian, science * 276 *, 917 (1997) ; f. jlicher, a. ajdari, and j. prost, rev. mod. phys. * 69 *, 1269 (1997) ; c. dring, nuovo cimento d*17 *, 685 (1995) s. flach, o. yevtushenko, and y. zolotaryuk, phys. rev. lett. * 84 *, 2358 (2000) ; o. yevtushenko, s. flach, y. zolotaryuk, and a. a. ovchinnikov, europhys. lett. * 54 *, 141 (2001) ; s. denisov et al. e * 66 *, 041104 (2002)

in 84, 258 (2000), mateos conjectured that current reversal in a classical deterministic ratchet is associated with bifurcations from chaotic to periodic regimes. this is based on the comparison of the current and the bifurcation diagram as a function of a given parameter for a periodic asymmetric potential. barbi and salerno, in 62, 1988 (2000), have further investigated this claim and argue that, contrary to mateos claim, current reversals can occur also in the absence of bifurcations. barbi and salerno s studies are based on the dynamics of one particle rather than the statistical mechanics of an ensemble of particles moving in the chaotic system. the behavior of ensembles can be quite different, depending upon their characteristics, which leaves their results open to question. in this paper we present results from studies showing how the current depends on the details of the ensemble used to generate it, as well as conditions for convergent behavior (that is, independent of the details of the ensemble). we are then able to present the converged current as a function of parameters, in the same system as mateos as well as barbi and salerno. we show evidence for current reversal without bifurcation, as well as bifurcation without current reversal. we conjecture that it is appropriate to correlate abrupt changes in the current with bifurcation, rather than current reversals, and show numerical evidence for our claims.

nlin0608019

Introduction the segmentation process as a whole can be thought of as consisting of two tasks : recognition and delineation. recognition is to determine roughly `` where '' the object is and to distinguish it from other object - like entities. although delineation is the final step for defining the spatial extent of the object region / boundary in the image, an efficient recognition strategy is a key for successful delineation. in this study, a novel, general method is introduced for object recognition to assist in segmentation (delineation) tasks. it exploits the pose relationship that can be encoded, via the concept of ball scale (b - scale) @xcite, between the binary training objects and their associated images. as an alternative to the manual methods based on initial placement of the models by an expert @xcite in the literature, model based methods can be employed for recognition. for example, in @xcite, the position of an organ model (such as liver) is estimated by its histogram. in @xcite, generalized hough transform is succesfully extended to incorporate variability of shape for 2d segmentation problem. atlas based methods are also used to define initial position for a shape model. in @xcite, affine registration is performed to align the data into an atlas to determine the initial position for a shape model of the knee cartilage. similarly, a popular particle filtering algorithm is used to detect the starting pose of models for both single and multi - object cases @xcite. however, due to the large search space and numerous local minimas in most of these studies, conducting a global search on the entire image is not a feasible approach. in this paper, we investigate an approach of automatically recognizing objects in 3d images without performing elaborate searches or optimization. Methods the proposed method consists of the following key ideas and components : * 1. model building : * after aligning image data from all @xmath0 subjects in the training set into a common coordinate system via 7-parameter affine registration, the live - wire algorithm @xcite is used to segment @xmath1 different objects from @xmath0 subjects. segmented objects are used for the automatic extraction of landmarks in a slice - by - slice manner @xcite. from the landmark information for all objects, a model assembly @xmath2 is constructed. b - scale encoding : * the b - scale value at every voxel in an image helps to understand `` objectness '' of a given image without doing explicit segmentation. for each voxel, the radius of the largest ball of homogeneous intensity is weighted by the intensity value of that particular voxel in order to incorporate appearance (texture) information into the object information (called intensity weighted b - scale : @xmath3) so that a model of the correlations between shape and texture can be built. a simple and proper way of thresholding the b - scale image yields a few largest balls remaining in the image. these are used for the construction of the relationship between the segmented training objects and the corresponding images. the resulting images have a strong relationship with the actual delineated objects. relationship between @xmath2 and @xmath3 : * a principal component @xmath4 system is built via pca for the segmented objects in each image, and their mean @xmath5 system, denoted @xmath6, is found over all training images. @xmath6 has an origin and three @xmath5 axes. similarly the mean @xmath5 system, denoted @xmath7, for intensity weighted b - scale images @xmath8 is found. finally the transformation @xmath9 that maps @xmath7 to @xmath6 is found. given an image @xmath10 to be segmented, the main idea here is to use @xmath9 to facilitate a quick placement of @xmath2 in @xmath10 with a proper pose as indicated in step 4 below. * hierarchical recognition : * for a given image @xmath10, @xmath3 is obtained and its @xmath5 system, denoted @xmath11 is computed subsequently. assuming the relationship of @xmath11 to @xmath6 to be the same as of @xmath7 to @xmath6, and assuming that @xmath6 offers the proper pose of @xmath2 in the training images, we use transformation @xmath9 and @xmath11 to determine the pose of @xmath2 in @xmath10. this level of recognition is called coarse recognition. further refinement of the recognition can be done using the skin boundary object in the image with the requirement that a major portion of @xmath2 should lie inside the body region delimited by the skin boundary. moreover, a little search inside the skin boundary can be done for the fine tuning, however, since offered coarse recognition method gives high recognition rates, there is no need to do any elaborate searches. we will focus on the fine tuning of coarse recognition for future study. the finest level of recognition requires the actual delineation algorithm itself, which is a hybrid method in our case and called gc - asm (synergistic integration of graph - cut and active shape model). this delineation algorithm is presented in a companion paper submitted to this symposium @xcite. Model building a convenient way of achieving incorporation of prior information automatically in computing systems is to create and use a flexible _ model _ to encode information such as the expected _ size _, _ shape _, _ appearance _, and _ position _ of objects in an image @xcite. among such information, _ shape _ and _ appearance _ are two complementary but closely related attributes of biological structures in images, and hence they are often used to create statistical models. in particular, shape has been used both in high and low level image analysis tasks extensively, and it has been demonstrated that shape models (such as active shape models (asms)) can be quite powerful in compensating for misleading information due to noise, poor resolution, clutter, and occlusion in the images @xcite. therefore, we use asm @xcite to estimate population statistics from a set of examples (training set). in order to guarantee 3d point correspondences required by asm, we build our statistical shape models combining semi - automatic methods : (1) manually selected anatomically correspondent slices by an expert, and (2) semi - automatic way of specifying key points on the shapes starting from the same anatomical locations. once step (1) is accomplished, the remaining problem turns into a problem of establishing point correspondence in 2d shapes, which is easily solved. it is extremely significant to choose correct correspondences so that a good representation of the modelled object results. although landmark correspondence is usually established manually by experts, it is time - consuming, prone to errors, and restricted to only 2d objects @xcite. because of these limitations, a semi - automatic landmark tagging method, _ equal space landmarking _, is used to establish correspondence between landmarks of each sample shape in our experiments. although this method is proposed for 2d objects, and equally spacing a fixed number of points for 3d objects is much more difficult, we use equal space landmarking technique in pseudo-3d manner where 3d object is annotated slice by slice. let @xmath12 be a single shape and assume that its finite dimensional representation after the landmarking consisting of @xmath13 landmark points with positions @xmath14, where @xmath15 are cartesian coordinates of the @xmath16 point on the shape @xmath17. equal space landmark tagging for points @xmath18 for @xmath19 on shape boundaries (contours) starts by selecting an initial point on each shape sample in training set and equally space a fixed number of points on each boundary automatically @xcite. selecting the starting point has been done manually by annotating the same anatomical point for each shape in the training set. figure [img : landmarking_abd] shows annotated landmarks for five different objects (skin, liver, right kidney, left kidney, spleen) in a ct slice of the abdominal region. note that different number of landmarks are used for different objects considering their size. [cols="^ ",] Conclusion \(1) the b - scale image of a given image captures object morphometric information without requiring explicit segmentation. b - scales constitute fundamental units of an image in terms of largest homogeneous balls situated at every voxel in the image. the b - scale concept has been previously used in object delineation, filtering and registration. our results suggest that their ability to capture object geography in conjunction with shape models may be useful in quick and simple yet accurate object recognition strategies. (2) the presented method is general and does not depend on exploiting the peculiar characteristics of the application situation. (3) the specificity of recognition increases dramatically as the number of objects in the model increases. (4) we emphasize that both modeling and testing procedures are carried out on the ct data sets that are part of the clinical pet / ct data routinely acquired in our hospital. the ct data set are thus of relatively poor (spatial and contrast) resolution compared to other ct - alone studies with or without contrast. we expect better performance if higher resolution ct data are employed in modeling or testing. Acknowledgement this paper is published in spie medical imaging conference - 2010. falcao, a.x., udupa, j.k., samarasekera, s., sharma, s., hirsch, b.e., and lotufo, r.a., 1998 user - steered image segmentation paradigms : live wire and live lane. graph. models image process. 60 (4), pp. 233260. kokkinos, i., maragos, p., 2009. synergy between object recognition and image segmentation using the expectation - maximization algorithm. ieee transactions on pattern analysis and machine intelligience, vol.31 (8), pp.14861501. brejl, m., sonka, m., 2000. object localization and border detection criteria design in edge - based image segmentation : automated learning from examples. ieee transactions on medical imaging, vol.19 (10), pp.973985. fripp, j., crozier, s., warfield, s.k., ourselin, s., 2005. automatic initialisation of 3d deformable models for cartilage segmentation. in proceedings of digital image computing : techniques and applications, pp.

this paper investigates, using prior shape models and the concept of ball scale (b - scale), ways of automatically recognizing objects in 3d images without performing elaborate searches or optimization. that is, the goal is to place the model in a single shot close to the right pose (position, orientation, and scale) in a given image so that the model boundaries fall in the close vicinity of object boundaries in the image. this is achieved via the following set of key ideas : (a) a semi - automatic way of constructing a multi - object shape model assembly. (b) a novel strategy of encoding, via b - scale, the pose relationship between objects in the training images and their intensity patterns captured in b - scale images. (c) a hierarchical mechanism of positioning the model, in a one - shot way, in a given image from a knowledge of the learnt pose relationship and the b - scale image of the given image to be segmented. the evaluation results on a set of 20 routine clinical abdominal female and male ct data sets indicate the following : (1) incorporating a large number of objects improves the recognition accuracy dramatically. (2) the recognition algorithm can be thought as a hierarchical framework such that quick replacement of the model assembly is defined as coarse recognition and delineation itself is known as finest recognition. (3) scale yields useful information about the relationship between the model assembly and any given image such that the recognition results in a placement of the model close to the actual pose without doing any elaborate searches or optimization. (4) effective object recognition can make delineation most accurate.

1002.1288

Supplementary information: + numerical simulations the simulations we discuss here allowed us to obtain spectra of the shg response. we employed comsol multiphysics software (www.comsol.com) in order to perform simulations of the linear optical response as described in ref. @xcite. a unit cell containing a single sic pillar attached to the sic substrate was constructed, with floquet boundary conditions along the @xmath30-axis and periodic boundary conditions along the @xmath31-axis, perpendicular to the plane of incidence. a _ p_-polarized plane wave at frequency @xmath32 was launched towards the sic structure at an incident angle of 25@xmath33. originating from the _ p_-polarised light source, the electric field @xmath34 inside the sic pillar and substrate was recorded. then, this electric field was translated into the nonlinear polarization @xmath16, according to eq. (1) in the main manuscript. there, the following non - zero components of the nonlinear susceptibility were accounted for : @xmath14, @xmath17 and @xmath18. as a next step, @xmath16 in the pillar and the substrate was regarded as the source of scattered electric field @xmath35 inside the unit cell. the shg output was then obtained by integrating the power flow through a _ xy_-plane set above the substrate. sweeping the fundamental frequency and taking the dispersion of both linear and nonlinear sic susceptibilities from refs.@xcite, we calculated several shg spectra for samples with various periodicity of the pillars. we note that in these calculations, the input power density at fundamental frequency was kept constant, and the shg output power was normalized to the area of the _ xy_-plane @xmath36. the results of the simulations presented in fig. [analysis] in the main manuscript demonstrate a single pronounced peak corresponding to the excitation of the sphp monopole mode. barnes, w. l. ; dereux, a. ; ebbesen, t. w. surface plasmon subwavelength optics. _ nature (london) _ * 2003 *, _ 424 _, 824830 maier, s. a. _ plasmonics : fundamentals and applications _ ; springer : new york, 2007 schuller, j. a. ; barnard, e. s. ; cai, w. ; jun, y. c. ; white, j. s. ; brongersma, m. l. plasmonics for extreme light concentration and manipulation. mater. _ * 2010 *, _ 9 _, 193204 boriskina, s. v. ; ghasemi, h. ; chen, g. plasmonic materials for energy : from physics to applications. _ mater. today _ * 2013 *, _ 16 _, 375386 brongersma, m. l. ; halas, n. j. ; nordlander, p. plasmon - induced hot carrier science and technology. _ * 2015 *, _ 10 _, 2534 kauranen, m. ; zayats, a. v. nonlinear plasmonics. photon. _ * 2012 *, _ 6 _, 737748 hentschel, m. ; utikal, t. ; metzger, b. ; giessen, h. ; lippitz, m. in _ progress in nonlinear nano - optics _ ; sabake, s., lienau, c., grunwald, r., eds. ; springer international publishing : switzerland, 2015 ; chapter 9, pp 153181 butet, j. ; brevet, p.- f. ; martin, o. j. f. optical second harmonic generation in plasmonic nanostructures : from fundamental principles to advanced applications. _ acs nano _ * 2015 *, _ 9 _, 1054510562 vahala, k. j. optical microcavities. _ nature (london) _ * 2003 *, _ 424 _, 839846 barnes, w. l. surface plasmon - polariton length scales : a route to sub - wavelength optics. _ j. opt. a : pure appl _ * 2006 *, _ 8 _, s87s93 greffet, j.- j. ; carminati, r. ; joulain, k. ; mulet, j.- p. ; mainguy, s. ; chen, y. coherent emission of light by thermal sources. _ nature (london) _ * 2002 *, _ 416 _, 6164 hillenbrand, r. ; taubner, t. ; keilmann, f. phonon - enhanced light - matter interaction at the nanometre scale. _ nature (london) _ * 2002 *, _ 418 _, 159162, b. ; korobkin, d. ; fietz, c. ; carole, d. ; ferro, g. ; shvets, g. critically coupled surface phonon - polariton excitation in silicon carbide. expr. _ * 2007 *, _ 34 _, 26672669 caldwell, j. d. ; glembocki, o. j. ; francescato, y. ; sharac, n. ; giannini, v. ; bezares, f. j. ; long, j. p. ; owrutsky, j. c. ; vurgaftman, i. ; tischler, j. g. ; wheeler, v. d. ; bassim, n. d. ; shirey, l. m. ; kasica, r. ; maier, s. a. low - loss, extreme subdiffraction photon confinement via silicon carbide localized surface phonon polariton resonators. _ nano lett. _ * 2013 *, _ 13 _, 36903697 chen, y. ; francescato, y. ; caldwell, j. d. ; giannini, v. ; ma, t. w. w. ; glembocki, o. j. ; bezares, f. j. ; taubner, t. ; kasica, r. ; hong, m. ; maier, s. a. spectral tuning of localized surface phonon polariton resonators for low - loss mid - ir applications. _ acs photonics _ * 2014 *, _ 1 _, 718724 caldwell, j. d. ; kretinin, a. v. ; chen, y. ; giannini, v. ; fogler, m. m. ; francescato, y. ; ellis, c. t. ; tischler, j. g. ; woods, c. r. ; giles, a. j. ; hong, m. ; watanabe, k. ; taniguchi, t. ; maier, s. a. ; novoselov, k. s. sub - diffractional volume - confined polaritons in the natural hyperbolic material hexagonal boron nitride. commun. _ * 2014 *, _ 5 _, 5221 dai, s. et al. tunable phonon polaritons in atomically thin van der waals crystals of boron nitride. _ science _ * 2014 *, _ 343 _, 11251129 caldwell, j. d. ; vurgaftman, i. ; tischler, j. g. probing hyperbolic polaritons. _ nat. photon. _ * 2015 *, _ 9 _, 638640 li, p. ; lewin, m. ; kretinin, a. v. ; caldwell, j. d. ; novoselov, k. s. ; taniguchi, t. ; watanabe, k. ; gaussmann, f. ; taubner, t. hyperbolic phonon - polaritons in boron nitride for near - field optical imaging and focusing. commun. _ * 2015 *, _ 6 _, 7507 dai, s. ; ma, q. ; andersen, t. ; mcleod, a. s. ; fei, z. ; liu, m. k. ; wagner, m. ; watanabe, k. ; taniguchi, t. ; thiemens, m. ; keilmann, f. ; jarillo - herrero, p. ; fogler, m. m. ; basov, d. n. subdiffractional focusing and guiding of polaritonic rays in a natural hyperbolic material. commun. _ * 2015 *, _ 6 _, 6963 caldwell, j. d. ; lindsay, l. ; giannini, v. ; vurgaftman, i. ; reinecke, t. l. ; maier, s. a. ; glembocki, o. j. low - loss, infrared and terahertz nanophotonics using surface phonon polaritons. _ nanophotonics _ * 2014 *, _ 4 _, 4468 ferguson, b. ; zhang, x.- c. materials for terahertz science and technology. photon. _ * 2002 *, _ 1 _, 2633 ulbricht, r. ; hendry, e. ; shan, j. ; heinz, t. f. ; bonn, m. carrier dynamics in semiconductors studied with time - resolved terahertz spectroscopy. _ _ * 2011 *, _ 83 _, 543586 kampfrath, t. ; tanaka, k. ; nelson, k. a. resonant and nonresonant control over matter and light by intense terahertz transients. photon. _ * 2013 *, _ 7 _, 680690 schllkopf, w. ; gewinner, s. ; junkes, h. ; paarmann, a. ; von helden, g. ; bluem, h. ; todd, a. m. m. the new ir and thz fel facility at the fritz haber institute in berlin. spie _ * 2015 *, _ 9512 _, 95121l paarmann, a. ; razdolski, i. ; melnikov, a. ; gewinner, s. ; schllkopf, w. ; wolf, m. second harmonic generation spectroscopy in the reststrahl band of sic using an infrared free - electron laser. lett. _ * 2015 *, _ 107 _, 081101, a. ; razdolski, i. ; gewinner, s. ; schoellkopf, w. ; wolf, m. effects of crystal anisotropy on optical phonon resonances in the mid - infrared second harmonic response of sic. * 2016 *, arxiv:1607.05207 [condmat.mtrlsci] carpetti, a. ; walsh, g. f. ; minissale, s. ; trevino, j. ; forestiere, c. ; miano, g. ; negro, l. d. multipolar second harmonic generation from planar arrays of au nanoparticles. _ opt. expr. _ * 2012 *, _ 20 _, 1579715806 canfield, b. k. ; husu, h. ; laukkanen, j. ; bai, b. ; kuittinen, m. ; turunen, j. ; kauranen, m. local field asymmetry drives second - harmonic generation in noncentrosymmetric nanodimers. _ nano lett. _ * 2007 *, _ 7 _, 12511255 raether, h. _ surface plasmons on smooth and rough surfaces and on gratings _ ; springer - verlag : berlin, 1988 agarwal, g. s. ; jha, s. s. surface - enhanced second - harmonic generation at a metallic grating. _ b _ * 1982 *, _ 26 _, 482 coutaz, j. experimental study of second - harmonic generation from silver gratings of various groove depths. _ j. opt b _ * 1987 *, _ 4 _, 105106 quail, j. c. ; simon, h. j. second - harmonic generation from a silver grating with surface plasmons. b _ * 1988 *, _ 5 _, 325329 razdolski, i. ; parchenko, s. ; stupakiewicz, a. ; semin, s. ; stognij, a. ; maziewski, a. ; kirilyuk, a. ; rasing, t. second - harmonic generation from a magnetic buried interface enhanced by an interplay of surface plasma resonances. _ acs phot. _ * 2015 *, _ 2 _, 2026 gubbin, c. r. ; martini, f. ; politi, a. ; maier, s. a. ; de liberato, s. strong and coherent coupling between localized and propagating phonon polaritons. _ * 2016 *, _ 116 _, 246402 shan, j. ; dadap, j. ; stiopkin, i. ; reider, g. ; heinz, t. experimental study of optical second - harmonic scattering from spherical nanoparticles. b _ * 2006 *, _ 73 _, 023819 kujala, s. ; canfield, b. k. ; kauranen, m. ; svirko, y. ; turunen, j. phonon - enhanced light - matter interaction at the nanometre scale. _ phys. _ * 2007 *, _ 98 _, 167403 bachelier, g. ; butet, j. ; russier - antoine, i. ; jonin, c. ; benichou, e. ; brevet, p.- f. origin of optical second - harmonic generation in spherical gold nanoparticles : local surface and nonlocal bulk contributions. b _ * 2010 *, _ 82 _, 235403 butet, j. ; bachelier, g. ; russier - antoine, i. ; jonin, c. ; benichou, e. ; brevet, p.- f. interference between selected dipoles and octupoles in the optical second - harmonic generation from spherical gold nanoparticles. _ phys. lett. _ * 2010 *, _ 105 _, 077401 razdolski, i. ; gheorghe, d. g. ; melander, e. ; hjrvarsson, b. ; patoka, p. ; kimel, a. v. ; kirilyuk, a. ; papaioannou, e. t. ; rasing, t. nonlocal nonlinear magneto - optical response of a magnetoplasmonic crystal. b _ * 2013 *, _ 88 _, 075436 kruk, s. ; weismann, m. ; bykov, a. y. ; mamonov, e. a. ; kolmychek, i. a. ; murzina, t. ; panoiu, n. c. ; neshev, d. n. ; kivshar, y. s. enhanced magnetic second - harmonic generation from resonant metasurfaces. _ acs photonics _ * 2015 *, _ 2 _, 10071012 hille, a. ; moeferdt, m. ; wolff, c. ; matyssek, c. ; rodriguez - oliveros, r. ; prohm, c. ; niegemann, j. ; grafstrom, s. ; eng, l. m. ; ; busch, k. second harmonic generation from metal nano - particle resonators : numerical analysis on the basis of the hydrodynamic drude model. _ j. phys. c _ * 2016 *, _ 120 _, 11631169 bluet, j. ; chourou, k. ; anikin, m. ; madar, r. weak phonon modes observation using infrared reflectivity for 4h, 6h and 15r polytypes. eng. _ * 1999 *, _ b6162 _, 212216 patrick, l. infrared absorption in sic polytypes. _ phys. rev. _ * 1967 *, _ 167 _, 809813 feldman, d. w. ; parker, jr., j. h. ; choyke, w. j. ; patrick, l. phonon dispersion curves by raman scattering in sic, polytypes 3c, 4h, 6h, 15r, and 21r. _ rev. _ * 1968 *, _ 173 _, 787793 engelbrecht, f. ; helbig, r. effect of crystal anisotropy on the infrared reflectivity of 6h sic. b _ * 1993 *, _ 48 _, 1569815707 metzger, b. ; gui, l. ; fuchs, j. ; floess, d. ; hentschel, m. ; giessen, h. strong enhancement of second harmonic emission by plasmonic resonances at the second harmonic wavelength. _ nano lett. _ * 2015 *, _ 15 _, 39173922 husu, h. ; canfield, b. k. ; laukkanen, j. ; bai, b. ; kuittinen, m. ; turunen, j. ; kauranen, m. local - field effects in the nonlinear optical response of metamaterials. _ metamaterials _ * 2008 *, _ 2 _, 155168 palomba, s. ; novotny, l. nonlinear excitation of surface plasmon polaritons by four - wave mixing. lett. _ * 2008 *, _ 101 _, 056802 carpetti, a. ; forestiere, c. ; negro, l. d. ; miano, g. full - wave analytical solution of second - harmonic generation in metal nanospheres. _ plasmonics _ * 2014 *, _ 9 _, 151166 razdolski, i. ; makarov, d. ; schmidt, o. g. ; kirilyuk, a. ; rasing, t. ; temnov, v. v. nonlinear surface magnetoplasmonics in kretschmann multilayers. _ acs photonics _ * 2016 *, _ 3 _, 179183 dai, s. et al. graphene on hexagonal boron nitride as a tunable hyperbolic metamaterial. nano. _ * 2015 *, _ 10 _, 682687 caldwell, j. d. ; vurgaftman, i. ; tischler, j. g. ; glembocki, o. j. ; owrutsky, j. c. ; reinecke, t. l. atomic - scale photonic hybrids for mid - infrared and terahertz nanophotonics. nano. _ * 2015 *, _ 11 _, 915

we report on strong enhancement of mid - infrared second harmonic generation (shg) from sic nanopillars due to the resonant excitation of localized surface phonon - polaritons within the reststrahlen band. the magnitude of the shg peak at the monopole mode experiences a strong dependence on the resonant frequency beyond that described by the field localization degree and the dispersion of linear and nonlinear - optical sic properties. comparing the results for the identical nanostructures made of 4h and 6h sic polytypes , we demonstrate the interplay of localized surface phonon polaritons with zone - folded weak phonon modes of the anisotropic crystal. tuning the monopole mode in and out of the region where the zone - folded phonon is excited in 6h - sic, we observe a prominent increase of the already monopole - enhanced shg output when the two modes are coupled. envisioning this interplay as one of the showcase features of mid - infrared nonlinear nanophononics, we discuss its prospects for the effective engineering of nonlinear - optical materials with desired properties in the infrared spectral range. light localization in sub - wavelength volumes is a core of modern nanophotonics. conventional methods of achieving strong confinement of the electromagnetic fields extensively utilize unique properties of surface plasmons. a remarkable variety of objects and materials supporting these excitations ensures the key role of plasmonics in a broad range of applications @xcite. apart from unparalleled sensitivity of plasmonic structures to the optical properties of the environment, strong light localization facilitates nonlinear - optical effects @xcite. owing to the spectral tunability of the localized plasmon resonances and their sizeable nonlinearity, metallic nanostructures of different shapes and sizes have earned their place in nonlinear photonics. despite obvious advantages of plasmon - based nanophotonics, metallic nanoobjects exhibit significant optical losses, which lower the quality factor of the localized surface plasmon modes. fast plasmon damping (typically on the order of 10 fs) due to ohmic losses @xcite thus inhibits nonlinear - optical conversion. an alternative, promising metal - free approach has been suggested, utilizing polar dielectrics such as sic @xcite or bn @xcite for high - quality light confinement in the mid - infrared (ir). in these materials, the subdiffractional confinement of electromagnetic radiation relies upon surface phonon polaritons (sphp) in the reststrahlen band : the electric polarization is created due to coherent oscillations of the ions instead of the electron or hole densities. due to the significantly longer scattering times associated with optical phonons as compared to surface plasmons, the lifetimes of sphps tend to be on the order of picoseconds, much longer than their plasmonic counterparts @xcite. in addition, due to energies associated with optical phonons, sphps with typical frequencies within the mid - ir (@xmath0) to the thz domain hold high promise for spectroscopic and nanophotonic applications @xcite. in this letter , we undertake a first step towards the largely unexplored domain of mid - ir nonlinear nanophononics. we study the nonlinear - optical response of localized sphps using nanostructures made of different polytypes of sic. using free electron laser (fel) radiation in the mid - ir spectral range @xcite , we probe second harmonic generation from rectangular arrays of sub - diffractional, cylindrical sic nanopillars. the shg yield in the reststrahlen band of sic demonstrates prominent enhancement at the wavelengths associated with the excitation of the sphp eigenmodes of the pillars. depending on both the size and the spatial periodicity of the pillars, the shg - probed eigenmode exhibits a spectral shift accompanied with strong variations of the shg intensity. analyzing different sic polytypes , we demonstrate the interplay of the localized sphps with the zone - folded optical phonon modes. we further conclude that strong coupling of the two modes allows for a significant additional modulation of the sphp - enhanced shg output. the schematic of our experimental approach is outlined in fig. [general],a. we employed a non - collinear shg configuration discussed elsewhere @xcite to perform spectroscopic shg measurements on square arrays of 1 @xmath1m - tall 4h - sic and 6h - sic pillars with the main axis of the arrays in the _ xz _ plane of incidence. the fundamental radiation incident at 28 and 62 degrees with respect to the normal to the sample surface was focused onto the sample with a peak fluence on the order of 10 mj/@xmath2. both 4h and 6h - sic samples were c - cut so that the _ c_-axis of the crystals was parallel to the surface normal. typical shg and linear reflectivity spectra collected using the fel radiation for the two incident polarisations are presented in fig. [general],b - e. there, the respective spectra of the bare substrate are shown for comparison. for _ p_-polarised fundamental radiation (fig. [general],d), the shg response features two pronounced peaks located at the zone - center frequencies of transverse and longitudinal optical phonons in sic @xcite, around 797 @xmath3 and 965 @xmath3, respectively. the corresponding shg spectrum from the nanopillars demonstrate a much stronger shg signal at around 900 @xmath3. due to the absence of this peak in the shg spectrum when the fundamental radiation is _ s_-polarised, we attribute this shg feature to the excitation of the monopole sphp mode in the nanopillars @xcite. in general, the outgoing shg field @xmath4 is related to the incident electromagnetic fields @xmath5 via the so - called local field factors @xmath6 : @xmath7 where @xmath8 is the nonlinear polarisation and @xmath9 is the nonlinear susceptibility tensor. the excitation of the sphp monopole mode leads to strong localization of the _ z_-projection of the fundamental electric field @xmath10 (normal to the surface plane) and thus a resonant enhancement of @xmath11. the latter results in a pronounced increase of the shg output when the fundamental radiation is _ p_-polarised. however, the sphp dipole modes observed in the range of 920960 @xmath3 rely on the resonant enhancement of the in - plane electric fields described by the local field factors @xmath12 and thus can be excited with both _ p_- and _ s_-polarised fundamental radiation. the total shg response is given by a vector sum of the terms on the right hand side of eq. (1) originating from various tensor components of the nonlinear susceptibility @xmath13. as the strength of @xmath14 is the largest in this spectral range @xcite, the sphp monopole mode, with an enhancement of @xmath11, naturally results in a higher shg output than the dipole modes. the results of the systematic studies of the shg response of various arrays of nanopillars are summarized in fig. [shifts] for the 4h - sic (a, c) and 6h - sic (b, d) samples. the panels (a - b) illustrate the evolution of the shg spectra upon varying pillar diameter @xmath15. it is seen that upon decreasing @xmath15, the sphp monopole - driven shg peak exhibits a clear redshift. remarkably, while the sphp monopole mode shifts in the range of about 890 - 910 @xmath3, the shg enhancement factor associated with the excitation of the sphp monopole mode varies strongly with @xmath15. the panels (c - d) offer a zoom - in into the evolution of the sphp monopole mode - driven shg for a large variety of nanopillar arrays, indicating that the variations of the shg enhancement are observed for both 4h and 6h samples. the dependences of the sphp monopole - driven shg output on the spectral position of the sphp monopole mode for the two sic polytypes are shown in fig. [analysis],a with open symbols. = 1200, 1100 and 1000 nm, respectively. , scaledwidth=47.0%] the strong dispersion in the sic reststrahlen band suggests that the observed spectral dependence of the shg enhancement could be captured in numerical simulations. we calculated shg response using both linear and nonlinear sic dispersion@xcite, the results of the simulation of the linear optical response@xcite, and nonlinear polarization @xmath16 from eq. (1) spatially integrated over the sic volume, with the following non - zero components of the nonlinear susceptibility : @xmath14, @xmath17 and @xmath18. the resultant shg spectra simulated with comsol multiphysics software (www.comsol.com) are shown in fig. [analysis],b with full symbols. it is seen that the steep experimental dependence can not be quantitatively described within the simple model used in the calculations, which yields only a moderate increase of the shg output when the sphp monopole mode frequency is decreased. we note the work of carpetti _ et al_. @xcite where the authors thoroughly examined plasmon - induced shg enhancement from arrays of au nanoparticles as a function of the inter - particle distance @xmath19. in the regime of @xmath20 (as it is in our case), the dependence of the shg output on @xmath19 was explained in terms of the changing filling factor (and thus the associated number of active nanoemitters). for very small inter - particle gaps (@xmath21), a modulation of the shg output has been attributed to the modification of the electromagnetic field localization in the gaps @xcite. further, a large mismatch between the spatial period of the nanopillars @xmath22 and the resonant light wavelength @xmath23 rules out the excitation of propagating surface polaritons @xcite which are known to enhance the shg output @xcite. since all these effects are included in our simulations, we conclude that the origin of the observed shg enhancement is unrelated to the periodicity of the array. an alternative scenario for the observed trend could invoke modifications of the non - local shg contribution enhanced by a regular array structure. the importance of the non - local shg was already demonstrated in a number of subwavelength - separated plasmonic nanoobjects @xcite. the amplitude and phase of this non - local shg source depends on the electric field distribution, i.e. on both the pillar size @xmath15 and the periodicity @xmath24. as such, the interference conditions between the shg sources vary for different samples, thus resulting in a strong modification of the shg output. further, the apparent difference in the shape of the resonant shg output for the 4h and 6h samples (see fig. [shifts]) is related to the anisotropic nature of sic. the hexagonal sic polytypes are known to exhibit zone - folded weak modes in the reststrahlen region @xcite originating from particular stacking of the atomic layers along the _ c_-axis of the crystal @xcite. these weakly ir - active zone - folded modes can be visualised in the reflectivity measurements at oblique incidence @xcite. although zone - folded modes exist in both 4h and 6h polytypes, different stacking of the sic atomic layers is responsible for them having different frequencies, as illustrated in fig. [analysis],c - d. the additional periodicity in the crystal results in folding of the large brillouin zone thus modifying the phonon dispersion and making the excitation of phonons with non - zero wavevectors (@xmath25) possible. it is seen in fig. [analysis],d that the zone - folded mode in the 6h - sic polytype with the reduced wavevector @xmath26 @xmath27 can be excited in the range of @xmath28 @xmath3 which is close to the typical monopole sphp resonant frequency of the sic nanopillars discussed above. in particular, the interaction of the sphp and zone - folded mode which shifted the apparent spectral positions of the monopole sphp eigenmode in the linear response @xcite, is seen responsible for the complex structure of the resonant shg output in our experiments (fig. [shifts],d). the weak ir activity of the zone - folded modes is related to the large negative dielectric permittivity of sic in the reststrahlen band. as such, the out - of - plane component of the electromagnetic field @xmath10 remains small which inhibits the coupling of incident light to the zone - folded phonon mode. however, the excitation of the sphp monopole mode in the nanopillars drives a strong increase of @xmath10, which facilitates the sphp monopole interaction with the zone - folded phonon (fig. [analysis],c). figure [analysis],a illustrates the effect of the zone - folded mode on the observed shg output. the open red squares (4h) and blue circles (6h) depict the shg intensity obtained at the sphp monopole (resonant) frequencies, and the dashed lines illustrate a clear correlation between the maximum shg output and the spectral position of the sphp monopole peak. similar trends are observed for the 4h and 6h - sic samples as long as the sphp monopole and the zone - folded mode are well - separated. when these two resonances start to overlap, an additional enhancement of the shg output produced at the sphp monopole resonance in 6h - sic samples is observed. moreover, the dependence of the shg signal at the frequency of the zone - folded mode @xmath29 @xmath3 (green triangles) exhibits a much faster increase when the two resonances are brought together (dotted line), indicating an efficient interplay between the sphp monopole and the zone - folded mode. we note that the interaction of the localized sphp eigenmodes and the intrinsic excitations of the medium is a unique fingerprint of mid - ir nanophononics. indeed , surface plasmon excitations in metals rely on the free electron gas, which is essentially isotropic. as such, the shg output of plasmonic nanostructures is (i) largely determined by the metal of choice, usually au, (ii) exhibits only weak spectral dependence @xcite and (iii) is limited by robust phase relations in the likely case of multiple shg sources @xcite. on the contrary, the flexibility of the sphps is provided by the coupling of the surface phonon polariton excitations to the intrinsic bulk phonon modes. the latter can be engineered by designing artificial metamaterials based on hybrid multilayer structures @xcite, thus allowing for an effective control of their optical properties. to summarize, we have observed shg output enhancement associated with the excitation of the sphp eigenmodes in an array of nanopillars grown from sic of different polytypes. the strongest shg output is associated with the excitation of the monopole sphp mode characterized by strong localization of the normal to surface projection of the electric field @xmath10. the spectral positions of the shg peaks shift according to the geometric parameters of the nanophononic structures. we found a strong dependence of the magnitude of the shg enhancement on the resonant frequency. this experimentally observed dependence can not be quantitatively described by simply taking into account the sub - diffractional field localization and the dispersion of the linear and nonlinear optical properties of sic. further, we discuss the interplay of the sphps and the intrinsic crystalline anisotropy for an efficient nonlinear - optical conversion. this mechanism is supported by the shg spectral measurements on the 6h - sic nanopillars where excitation of the sphp monopole mode interacting with the weak zone - folded phonon resulted in an additional enhancement of the shg output. the presence of intrinsic resonances strongly alters the phase relations in their vicinity, providing a natural way for optimizing the shg response in a relatively narrow spectral range. our findings demonstrate high potential of mid - ir nonlinear nanophononics as a novel and promising platform for nonlinear optics and illustrate the rich opportunities it provides for efficient control over nonlinear - optical response. the authors thank g. kichin and a. kirilyuk (radboud university nijmegen) for stimulating discussions. s.a.m. acknowledges the office of naval research, the royal society, and the lee - lucas chair in physics. a.j.g. acknowledges financial support from the nrc / nrl postdoctoral fellowship program. funding for j.d.c. was provided by the office of naval research through the naval research laboratory s nanoscience institute.

1607.05158

Introduction the so - called `` nucleon spin crisis '' raised by the european muon collaboration (emc) measurement in 1988 is one of the most outstanding findings in the field of hadron physics @xcite,@xcite. the renaissance of the physics of high energy deep inelastic scatterings is greatly indebted to this epoch - making finding. probably, one of the most outstanding progresses achieved recently in this field of physics is the discovery and the subsequent research of completely new observables called generalized parton distribution functions (gpd). it has been revealed that the gpds, which can be measured through the so - called deeply - virtual compton scatterings (dvcs) or the deeply - virtual meson productions (dvmp), contain surprisingly richer information than the standard parton distribution functions @xcite@xcite. roughly speaking, the gpds are generalization of ordinary parton distributions and the elastic form factors of the nucleon. the gpds in the most general form are functions of three kinematical variables : the average longitudinal momentum fraction @xmath1 of the struck parton in the initial and final states, a skewdness parameter @xmath3 which measures the difference between two momentum fractions, and the four - momentum - transfer square @xmath4 of the initial and final nucleon. in the forward limit @xmath5, some of the gpds reduce to the usual quark, antiquark and gluon distributions. on the other hand, taking the @xmath0-th moment of the gpds with respect to the variable @xmath1, one obtains the generalizations of the electromagnetic form factors of the nucleon, which are called the generalized form factors of the nucleon. the complex nature of the gpds, i.e. the fact that they are functions of three variable, makes it quite difficult to grasp their full characteristics both experimentally and theoretically. from the theoretical viewpoint, it may be practical to begin studies with the two limiting cases. the one is the forward limit of zero momentum transfer. we have mentioned that, in this limit, some of the gpds reduce to the ordinary parton distribution function depending on one variable @xmath1. however, it turns out that, even in this limit, there appear some completely new distribution functions, which can not be accessed by the ordinary inclusive deep - inelastic scattering measurements. very interestingly, it was shown by ji that one of such distributions contains valuable information on the total angular momentum carried by the quark fields in the nucleon @xcite@xcite. this information, combined with the known knowledge on the longitudinal quark polarization, makes it possible to determine the quark orbital angular momentum contribution to the total nucleon spin purely experimentally. another relatively - easy - to - handle quantities are the generalized form factors of the nucleon @xcite,@xcite, which are given as the non - forward nucleon matrix elements of the spin-@xmath0, twist - two quark and gluon operators. since these latter quantities are given as the nucleon matrix elements of local operators, they can be objects of lattice qcd simulations. (it should be compared with parton distributions. the direct calculation of parton distributions is beyond the scope of lattice qcd simulations, since it needs to treat the nucleon matrix elements of quark bilinears, which are _ nonlocal in time_.) in fact, two groups, the lhpc collaboration and the qcdsf collaboration independently investigated the generalized form factors of the nucleon, and gave several interesting predictions, which can in principle be tested by the measurement of gpds in the near future @xcite@xcite. although interesting, there is no _ a priori _ reason to believe that the predictions of these lattice simulations are realistic enough. the reason is mainly that the above mentioned lattice simulation were carried out in the heavy pion regime around @xmath6 with neglect of the so - called disconnected diagrams. our real world is rather close to the chiral limit with vanishing pion mass, and we know that, in this limit, the goldstone pion plays very important roles in some intrinsic properties of the nucleon. the lattice simulation carried out in the heavy pion region is in danger of missing some important role of chiral dynamics. on the other hand, the chiral quark soliton model (cqsm) is an effective model of baryons, which maximally incorporates the chiral symmetry of qcd and its spontaneous breakdown @xcite,@xcite. (see @xcite@xcite for early reviews.) it was already applied to the physics of ordinary parton distribution functions with remarkable success @xcite@xcite. for instance, an indispensable role of pion - like quark - antiquark correlation was shown to be essential to understand the famous nmc measurement, which revealed the dominance of the @xmath7-quark over the @xmath8-quark inside the proton @xcite,@xcite,@xcite. then, it would be interesting to see what predictions the cqsm would give for the quantities mentioned above. now, the main purpose of the present study is to study the generalized form factors of the nucleon within the framework of the cqsm and compare its predictions with those of the lattice qcd simulations. of our particular interest here is to see the change of final theoretical predictions against the variation of the pion mass. such an analysis is expected to give some hints for judging the reliability of the lattice qcd predictions at the present level for the observables in question. the plan of the paper is as follows. in sect.ii, we shall briefly explain how to introduce the nonzero pion mass into the scheme of the cqsm with pauli - villars regularization. in sect.iii, we derive the theoretical expressions for the generalized form factors of the nucleon. sect.iv is devoted to the discussion of the results of the numerical calculations. some concluding remarks are then given in sect.v. Model lagrangian with pion mass term we start with the basic effective lagrangian of the chiral quark soliton model in the chiral limit given as @xmath9 with @xmath10 which describes the effective quark fields, with a dynamically generated mass @xmath11, strongly interacting with pions @xcite,@xcite. since one of the main purposes of the present study is to see how the relevant observables depend on pion mass, we add to @xmath12 an explicit chiral symmetry breaking term @xmath13 given by @xcite @xmath14. \label{eq : lsb}\]] here the trace in ([eq : lsb]) is to be taken with respect to flavor indices. the total model lagrangian is therefore given by @xmath15 naturally, one could have taken an alternative choice in which the explicit chiral - symmetry - breaking effect is introduced in the form of current quark mass term as @xmath16. we did not do so, because we do not know any consistent regularization of such effective lagrangian with finite current quark mass within the framework of the pauli - villars subtraction scheme, as explained in appendix of @xcite. the effective action corresponding to the above lagrangian is given as @xmath17 = s_f [u] + s_m [u], \label{eq : energsol}\]] with @xmath18 = - \,i \,n_c \,\mbox{sp } \, \ln (i \not\!\partial - m u^{\gamma_5 }), \]] and @xmath19 = \int \frac{1}{4 } \,f_{\pi}^2 \,m_{\pi}^2 \, \mbox{tr}_f \, [u (x) + u^{\dagger } (x) - 2] \,d^4 x.\]] here @xmath20 with @xmath21 and @xmath22 representing the trace of the dirac gamma matrices and the flavors (isospins), respectively. the fermion (quark) part of the above action contains ultra - violet divergences. to remove these divergences, we must introduce physical cutoffs. for the purpose of regularization, here we use the pauli - villars subtraction scheme. as explained in @xcite, we must eliminate not only the logarithmic divergence contained in @xmath23 $] but also the quadratic and logarithmic divergence contained in the equation of motion shown below. to get rid of all these troublesome divergence, we need at least two subtraction terms. the regularized action is thus defined as @xmath24 = s_f^{reg } [u] + s_m [u], \]] where @xmath25 = s_f [u] - \sum_{i = 1}^2 \,c_i \,s_f^{\lambda _ i } [u].\]] here @xmath26 is obtained from @xmath27 $] with @xmath11 replaced by the pauli - villars regulator mass @xmath28. these parameters are fixed as follows. first, the quadratic and logarithmic divergence contained in the equation of motion (or in the expression of the vacuum quark condensate) can, respectively, removed if the subtraction constants satisfy the following two conditions : @xmath29 (we recall that the condition which removes the logarithmic divergence in @xmath23 $] just coincides with the 1st of the above conditions.) by solving the above equations for @xmath30 and @xmath31, we obtain @xmath32 which constrains the values of @xmath30 and @xmath31, once @xmath33 and @xmath34 are given. for determining @xmath33 and @xmath34, we use two conditions @xmath35 and @xmath36 which amounts to reproducing the correct normalization of the pion kinetic term in the effective meson lagrangian and also the empirical value of the vacuum quark condensate. to derive soliton equation of motion, we must first write down a regularized expression for the static soliton energy. under the hedgehog ansatz @xmath37 for the background pion fields, it is obtained in the form : @xmath38 = e_f^{reg } [f (r)] + e_m [f (r)], \]] where the meson (pion) part is given by @xmath39 = - \,f_{\pi}^2 \,m_{\pi}^2 \int d^3 x \,[\cos f (r) - 1], \]] while the fermion (quark) part is given as @xmath40 = e_{val } + e_{v p}^{reg }, \label{eq : estatic}\]] with @xmath41 here @xmath42 are the quark single - particle energies, given as the eigenvalues of the static dirac hamiltonian in the background pion fields : @xmath43 with @xmath44, \label{eq : dirach}\]] while @xmath45 denote energy eigenvalues of the vacuum hamiltonian given by ([eq : dirach]) with @xmath46 (or @xmath47). the particular state @xmath48, which is a discrete bound - state orbital coming from the upper dirac continuum under the influence of the hedgehog mean field, is called the valence level. the symbol @xmath49 in ([eq : regenerg]) denotes the summation over all the negative energy eigenstates of @xmath50, i.e. the negative energy dirac continuum. the soliton equation of motion is obtained from the stationary condition of @xmath51 $] with respect to the variation of the profile function @xmath52 : @xmath53 \nonumber \\ & = & 4 \pi r^2 \,\left\ { - \,m \,[s (r) \sin f (r) - p (r) \cos f (r)] + f_{\pi}^2 m_{\pi}^2 \sin f (r) \right\ }, \end{aligned}\]] which gives @xmath54 here @xmath55 and @xmath56 are regularized scalar and pseudoscalar quark densities given as @xmath57 with @xmath58 while @xmath59 and @xmath60 are the corresponding densities evaluated with the regulator mass @xmath28 instead of the dynamical quark mass @xmath11. we also note that @xmath61 and @xmath62. as usual, a self - consistent soliton solution is obtained as follows with use of kahana and ripka s discretized momentum basis @xcite,@xcite. first by assuming an appropriate (though arbitrary) soliton profile @xmath52, the eigenvalue problem of the dirac hamiltonian is solved. using the resultant eigenfunctions and their associated eigenenergies, one can calculate the regularized scalar and pseudoscalar densities @xmath55 and @xmath56. with use of these @xmath63 and @xmath64, eq.([eq : profile]) can then be used to obtain a new soliton profile @xmath52. the whole procedure above is repeated with this new profile @xmath52 until the self - consistency is attained. Generalized form factors in the cqsm since the generalized form factors of the nucleon are given as moments of generalized parton distributions (gpds), it is convenient to start with the theoretical expressions of the unpolarized gpds @xmath65 and @xmath66 within the cqsm. following the notation in @xcite,@xcite, we introduce the quantities @xmath67 and @xmath68 here, the isoscalar and isovector combinations respectively correspond to the sum and the difference of the quark flavors @xmath69 and @xmath70. the relation between these quantities and the generalized parton distribution functions @xmath71 and @xmath72 are obtained most conveniently in the so - called breit frame. they are given by @xmath73 where @xmath74 these two independent combinations of @xmath65 and @xmath66 can be extracted through the spin projection of @xmath75 as @xmath76 where tr " denotes the trace over spin indices, while @xmath77. now, within the cqsm, it is possible to evaluate the right - hand side (rhs) of ([eq : hetrace]) and ([eq : emtrace]). since the answers are already given in several previous papers @xcite@xcite, we do not repeat the derivation. here we comment only on the following general structure of the theoretical expressions for relevant observables in the cqsm. the leading contribution just corresponds to the mean field prediction, which is independent of the collective rotational velocity @xmath78 of the hedgehog soliton. the next - to - leading order term takes account of the linear response of the internal quark motion to the rotational motion as an external perturbation, and consequently it is proportional to @xmath78. it is known that the leading - order term contributes to the isoscalar combination of @xmath79 and to the isovector combination of @xmath80, while the isoscalar part of @xmath79 and the isovector part of @xmath80 survived only at the next - to - leading order of @xmath78 (or of @xmath81). the leading - order gpds are then given as @xmath82 here the symbol @xmath83 denotes the summation over the occupied (the valence plus negative - energy dirac sea) quark orbitals in the hedgehog mean field. on the other hand, the theoretical expressions for the isovector part of @xmath84 and the isoscalar part of @xmath85, which survive at the next - to - leading order, are a little more complicated. they are given as double sums over the single quark orbitals as @xmath86 \, e^{i x m_n z^0 } \nonumber \\ & \, & \hspace{30 mm } \times \ \langle n | \tau^a | m \rangle \langle m | \,\tau^a \,(1 + \gamma^0 \gamma^3) \, e^{-i (z^0/2) \hat{p}_3 } \, e^{i { \mbox{\boldmath$\delta$ } } \cdot { \mbox{\boldmath$x$ } } } \,e^{-i (z^0/2) \hat{p}_3 } | n \rangle. \label{eq : gpdheiv}\end{aligned}\]] and @xmath87 e^{i x m_n z^0 } \nonumber \\ & \, & \hspace{25 mm } \times \ \langle n | \tau^b | m \rangle \langle m | \,(1 + \gamma^0 \gamma^3) \,e^{-i (z^0/2) \hat{p}_3 } \, \frac{\varepsilon^{3 a b } \delta^a}{{\mbox{\boldmath$\delta$}}_{\perp}^2 } \, e^{i { \mbox{\boldmath$\delta$ } } \cdot { \mbox{\boldmath$x$ } } } \,e^{-i (z^0/2) \hat{p}_3 } | n \rangle . \label{eq : gpdemiv}\end{aligned}\]] these four expressions for the unpolarized gpds, i.e. ([eq : gpdheis]) @xmath88 ([eq : gpdemiv]), are the basic starting equations for our present study of the generalized form factors of the nucleon within the cqsm. there are infinite tower of generalized form factors, which are defined as the @xmath0-th moments of gpds. in the present study, we confine ourselves to the 1st and the 2nd moments, which respectively corresponds to the standard electromagnetic form factors of the nucleon and the so - called gravitational form factors. we are especially interested in the second one, since they are believed to contain valuable information on the spin contents of the nucleon through ji s angular momentum sum rule @xcite,@xcite. for each isospin channel, the 1st and the 2nd moments of @xmath89 define the sachs - electric and gravito - electric form factors as @xmath90 and @xmath91 on the other hand, the 1st and the 2nd moments of @xmath92 respectively define the sachs - magnetic and gravito - magnetic form factors as @xmath93 and @xmath94 in the following, we shall explain how we can calculate the generalized form factors based on the theoretical expressions of corresponding gpds, by taking @xmath95 and @xmath96 as examples. setting @xmath97 and integrating over @xmath98 in ([eq : gpdheis]), we obtain @xmath99 putting this expression into ([eq : ge10def]), we have @xmath100 it is easy to see that, using the generalized spherical symmetry of the hedgehog configuration, the term containing the factor @xmath101 identically vanishes, so that @xmath102 is reduced to a simple form as follows : @xmath103 aside from the factor @xmath104, this is nothing but the known expression for the isoscalar sachs - electric form factor of the nucleon in the cqsm @xcite. a less trivial example is @xmath96, which is defined as the 2nd moment of @xmath105. inserting ([eq : heis]) into ([eq : ge10def]) and carrying out the integration over @xmath1, we obtain @xmath106 using the partial - wave expansion of @xmath107, this can be written as @xmath108 this can further be divided into four pieces as @xmath109 where @xmath110 with @xmath111 to proceed further, we first notice that, by using the generalized spherical symmetry, @xmath112 survives only when @xmath113, i.e. @xmath114 which leads to the result : @xmath115 to evaluate @xmath116, we first note that @xmath117^{(\lambda) }.\]] here, the generalized spherical symmetry dictates that @xmath118 must be zero, so that the rhs of the above equation is effectively reduced to @xmath119^{(0) }.\]] this then gives @xmath120^{(0) } \, \phi_n ({ \mbox{\boldmath$x$ } }).\end{aligned}\]] owing to the identity @xmath121 we therefore find that @xmath122 next we investigate the third term @xmath123. using @xmath124^{(\lambda) } \ \sim \ - \,\frac{1}{\sqrt{3 } } \,\delta_{l,1 } \,\delta_{m,0 } \, [y_1 (\hat{x }) \times \hat{{\mbox{\boldmath$p$ } } }] ^{(0) }, \end{aligned}\]] we obtain @xmath125^{(0) } \, \phi_n ({ \mbox{\boldmath$x$ } }).\end{aligned}\]] this term vanishes by the same reason as @xmath116 does. the last term @xmath126 is a little more complicated. we first notice that @xmath127^{(\lambda)}_0 \nonumber \\ & \sim & \sum_{\lambda } \, \langle 1 0 1 0 | \lambda 0 \rangle \, \langle l m \lambda 0 | 00 \rangle \, [y_l (\hat{x }) \times [{ \mbox{\boldmath$\alpha$ } } \times \hat{{\mbox{\boldmath$p$ } } }] ^{(\lambda) }] ^{(0) } \nonumber \\ & = & \delta_{m, 0 } \,\langle 1 0 1 0 | l 0 \rangle \, \langle l 0 l 0 | 0 0 \rangle \, [y_l (\hat{x }) \times [{ \mbox{\boldmath$ \alpha$ } } \times \hat{{\mbox{\boldmath$p$ } } }] ^{(l) }] ^{(0) }, \end{aligned}\]] which dictates that @xmath128 must be 0 or 2. inserting the above expression into ([eq : m4]), and using the explicit values of clebsch - gordan coefficients, @xmath126 becomes @xmath129^{(0) } \, \phi_n ({ \mbox{\boldmath$x$ } }) \nonumber \\ & + & \frac{\sqrt{4 \pi}}{\sqrt{6 } } \cdot \frac{n_c}{m_n } \int d^3 x \,\sum_{n \leq 0 } \,\phi_n^{\dagger } ({ \mbox{\boldmath$x$ } }) \, j_2 (\delta_{\perp } x) \, [y_2 (\hat{x }) \times [{ \mbox{\boldmath$\alpha$ } } \times \hat{{\mbox{\boldmath$p$ } } }] ^{(2) }] ^{(0) } \, \phi_n ({ \mbox{\boldmath$x$ } }).\end{aligned}\]] using the identities @xmath130^{(0) } & = & -\frac{1}{\sqrt{3 } } { \mbox{\boldmath$\alpha$ } } \cdot \hat{{\mbox{\boldmath$p$ } } }, \\ \, [y_2 (\hat{x }) \times [{ \mbox{\boldmath$\alpha$ } } \times { \mbox{\boldmath$p$ } }] ^{(2) }] ^{(0) } & = & [[y_2 (\hat{x }) \times \hat{{\mbox{\boldmath$p$ } } }] ^{(1) } \times { \mbox{\boldmath$\alpha$ } }] ^{(0) }, \end{aligned}\]] @xmath126 can also be written as @xmath131^{(1) } \times { \mbox{\boldmath$\alpha$ } }] ^{(0) } \, \phi_n ({ \mbox{\boldmath$x$ } }).\end{aligned}\]] collecting the answers for @xmath132 and @xmath126, we finally obtain @xmath133^{(1) } \times { \mbox{\boldmath$\alpha$ } }] ^{(0) } \,\phi_n ({ \mbox{\boldmath$x$ } }).\end{aligned}\]] up to now, we have obtained the theoretical expressions for the isoscalar combination of the generalized form factors @xmath95 and @xmath96. for notational convenience, we summarize these results in a little more compact forms as follows : @xmath134 and @xmath135^{(1) } \times { \mbox{\boldmath$\alpha$ } }] ^{(0) } \, | n \rangle \right\ }. \label{eq : inthe20is}\end{aligned}\]] as pointed out before, @xmath95 is @xmath104 times the isoscalar combination of standard sachs - electric form factor of the nucleon. analogously, we may call @xmath96 the gravitoelectric form factor of the nucleon (its quark part), since it is related to the nonforward nucleon matrix elements of the quark part of the qcd energy momentum tensor. the other generalized form factors can be obtained in a similar way. the isovector part of the generalized electric form factors survive only at the next - to - leading order of @xmath78. they are given as @xmath136 and @xmath137^{(1) } \times { \mbox{\boldmath$\alpha$ } }] ^{(0) } \,{\mbox{\boldmath$\tau$ } } \,|| n \rangle \right\ }.\end{aligned}\]] the isoscalar combination of the generalized magnetic form factors also survive only at the next - to - leading order of @xmath78, so that they are given as double sums over the single - quark orbitals in the hedgehog mean field as @xmath138 and @xmath139 we recall that @xmath140 just coincides with the known expression of the isoscalar sachs - magnetic form factor of the nucleon in the cqsm @xcite. on the other hand, @xmath141 is sometimes called the gravitomagnetic form factor of the nucleon (its isoscalar part), which we can evaluate within the qcsm based on the above theoretical expression. finally, the leading - order contribution to the isovector part of the generalized magnetic form factors are given as @xmath142 and @xmath143 especially interesting to us are the values of the generalized form factors in the forward limit @xmath5. the consideration of this limit is also useful for verifying consistency of our theoretical analyses, since it leads to fundamental sum rules discussed below. we first consider the forward limit of @xmath95. from ([eq : inthe10is]), we find that @xmath144 subtracting the corresponding vacuum contribution, this reduces to @xmath145. if we remember the relation @xmath146 the forward limit of ([eq : inthe10is]) just leads to the sum rule : @xmath147 which denotes that the sum of the @xmath69-quark and @xmath70-quark numbers in the proton is three. next we turn to the forward limit of @xmath96, which gives @xmath148 it is easy to see that, after regularization and vacuum subtraction, the first term of the rhs of the above equation reduces to the fermion (quark) part of the soliton energy, i.e. @xmath149 in ([eq : estatic]). it was proved in @xcite that, in the cqsm with vanishing pion mass, the following identity holds : @xmath150 in the case of finite pion mass, which we are handling, this identity does not hold. instead, we can prove (see appendix) that @xmath151 that is, the second term in the parenthesis of rhs of eq.([eq : momsum]) just coincides with the pion part of the soliton energy (or mass). since the sum of the quark and pion part give the total soliton mass @xmath152, we then find that @xmath153 in consideration of eq.([eq : inthe20is]), this relation can also be expressed as @xmath154 which means that the total momentum fraction carried by quark fields (the @xmath69- and @xmath70-quarks) is just unity. this is an expected result, since the cqsm contains quark fields only (note that the pion is not an independent field of quarks), so that the total nucleon momentum should be saturated by the quark fields alone. taking the forward limit of @xmath155, we are again led to a trivial sum rule, constrained by the conservation low. in fact, we have @xmath156 thereby leading to @xmath157 which denotes that the difference of the @xmath69-quark and the @xmath70-quark numbers in the proton is just unity. on the other hand, the forward limit of @xmath158 leads to the first nontrivial sum rule as @xmath159 since this quantity, which represents the difference of momentum fraction carried by the @xmath69-quark and the @xmath70-quark in the proton, is not constrained by any conservation law, its actual value can be estimated only numerically. next we turn to the discussion of the forward limit of the generalized magnetic form factors. first, the forward limit of @xmath140 gives @xmath160 which reproduces the known expression of the isoscalar magnetic moment of the nucleon in the cqsm @xcite. on the other hand, the forward limit of @xmath141 gives @xmath161 it was shown in @xcite that the rhs of the above equation is just unity, i.e. @xmath162 in consideration of ([eq : gm2nd]), this identity can be recast into a little different form as @xmath163 \,d x.\end{aligned}\]] assuming the familiar angular momentum sum rule due to ji @xmath164 \,d x = j^{u + d }, \]] the above identity claims that @xmath165 which means that the nucleon spin is saturated by the quark fields alone. this is again a reasonable result, because the cqsm is an effective quark model which contains no explicit gluon fields. the derived identity ([eq : gm20ist0]) has still another interpretation. remembering the fact that @xmath141 consists of two parts as @xmath166 eq.([eq : gm20ist0]) dictates that @xmath167 since it also holds that (the momentum sum rule) @xmath168 it immediately follows that @xmath169 which is interpreted as showing the absence of the _ net quark contribution to the anomalous gravitomagnetic moment of the nucleon_. finally, we investigate the forward limit of the isovector combination of the generalized magnetic form factors. from eq. ([eq : mgivform]), we get @xmath170 which reproduces the known expression of the isovector magnetic form factor of the nucleon in the cqsm. on the other hand, letting @xmath5 in ([eq : mg2ivform]), we have @xmath171 as shown in @xcite, this sum rule can be recast into the form : @xmath172 where @xmath173 consists of two parts as @xmath174 here, the first part is given as a proton matrix element of the _ free field expression _ for the isovector total angular momentum operator of quark fields as @xmath175 with @xmath176 \, \psi ({ \mbox{\boldmath$x$ } }) \,d^3 x \nonumber \\ & = & \hat{l}_f^{(i = 1) } + \frac{1}{2 } \,\hat{\sigma}^{(i = 1) }.\end{aligned}\]] on the other hand, the second term is given as @xmath177 | n \rangle.\]] Numerical results and discussions the model in the chiral limit contains two parameters, the weak pion decay constant @xmath178 and the dynamical quark mass @xmath11. as usual, @xmath178 is fixed to its physical value, i.e. @xmath179. for the mass parameter @xmath11, there is some argument based on the instanton liquid picture of the qcd vacuum that it is not extremely far from @xmath180 @xcite. the previous phenomenological analysis of various static baryon observables based on this model prefer a slightly larger value of @xmath11 between @xmath180 and @xmath181 @xcite@xcite. in the present analysis, we use the value @xmath182. with this value of @xmath182, we prepare self - consistent soliton solutions for seven values of @xmath183, i.e. @xmath184 and @xmath185, in order to see the pion mass dependence of the generalized form factors etc. favorable physical predictions of the model will be obtained by using the value of @xmath182 and @xmath186, since this set gives a self - consistent soliton solution close to the phenomenologically successful one obtained with @xmath187 and @xmath188 in the single - subtraction pauli - villars regularization scheme @xcite@xcite. we first show in fig.[fig : profile] the soliton profile functions @xmath52 obtained with several values of @xmath183, i.e. @xmath189, and @xmath185. one sees that the spatial size of the soliton profile becomes more and more compact as the pion mass increases. and @xmath190, and @xmath185.,width=340,height=264] we are now ready to show the theoretical predictions of the cqsm for the generalized form factors. since the corresponding lattice predictions are given for the generalized form factors @xmath191 and @xmath192, which are the generalization of the standard dirac and pauli form factors, we first write down the relations between these form factors and the generalized sachs - type factors, which we have calculated in the cqsm. they are given by @xmath193 \,/\, (1 + \tau), \\ a_{20}^{u + d } (t) & = & \left[\, g_{e, 20}^{(i = 0) } (t) + \tau \, g_{m, 20}^{(i = 0) } (t) \right] \,/\, (1 + \tau), \\ a_{10}^{u - d } (t) & = & \left[\, g_{e, 10}^{(i = 1) } (t) + \tau \, g_{m, 10}^{(i = 1) } (t) \right] \,/\, (1 + \tau), \\ a_{20}^{u - d } (t) & = & \left[\, g_{e, 20}^{(i = 1) } (t) + \tau \, g_{m, 20}^{(i = 1) } (t) \right] \,/\, (1 + \tau), \end{aligned}\]] and @xmath194 \,/\, (1 + \tau), \\ b_{20}^{u + d } (t) & = & \left[\, g_{m, 10}^{(i = 0) } (t) - g_{e, 20}^{(i = 0) } (t) \right] \,/\, (1 + \tau), \\ b_{10}^{u - d } (t) & = & \left[\, g_{m, 10}^{(i = 1) } (t) - g_{e, 10}^{(i = 1) } (t) \right] \,/\, (1 + \tau), \\ b_{20}^{u - d } (t) & = & \left[\, g_{m, 20}^{(i = 1) } (t) - g_{e, 20}^{(i = 1) } (t) \right] \,/\, (1 + \tau), \end{aligned}\]] where @xmath195. we recall that @xmath196 and @xmath197 are nothing but the standard dirac and pauli form factors of the nucleon : @xmath198 since the lattice simulations by the lhpc and qcdsf collaborations were carried out in the heavy pion region around @xmath199 and since the simulation in the small pion mass region is hard to perform, we think it interesting to investigate the pion mass dependence of the generalized form factors within the framework of the cqsm. for simplicity, we shall show the pion mass dependence of the generalized form factors at the zero momentum transfer only. we think it enough for our purpose because the generalized form factors at the zero momentum transfer contain the most important information for clarifying the underlying spin structure of the nucleon. at zero momentum transfer, the relations between the generalized dirac and pauli form factors and the generalized sachs - type form factors are simplified to become @xmath200 and @xmath201 and @xmath202 as functions of @xmath183 (the filled diamonds), together with the corresponding lattice predictions. here, the open triangles correspond to the predictions of the lhpc group @xcite, while the open squares to those of the qcdsf collaboration @xcite.,width=604,height=264] fig.[fig : a1020is] shows the predictions of the cqsm for @xmath203 and @xmath202 as functions of @xmath183, together with the corresponding lattice predictions. as for @xmath203, the cqsm predictions and the lattice qcd predictions are both independent of @xmath183 and consistent with the constraint of the quark number sum rule : @xmath204 with high numerical precision. turning to @xmath202, one finds a sizable difference between the predictions of the cqsm and of the lattice qcd. the lattice qcd predicts that @xmath205 which means that only about @xmath206 of the total nucleon momentum is carried by the quark fields, while the rest is borne by the gluon fields. on the other hand, the cqsm predictions for the same quantity is @xmath207 which means that the quark fields saturates the total nucleon momentum. this may certainly be a limitation of an effective quark model, which contains no explicit gluon fields. note, however, that the total quark momentum fraction @xmath202 is a scale dependent quantity. the lattice result corresponds to the energy scale of @xmath208 @xcite, while the cqsm prediction should be taken as that of the model energy scale around @xmath209 @xcite. we shall later make more meaningful comparison by taking care of the scale dependencies of relevant observables. and @xmath210 as functions of @xmath183, together with the corresponding lattice predictions @xcite,@xcite. the meaning of the symbols are the same as in fig.[fig : a1020is].,width=604,height=264] next, in fig.[fig : a1020iv], we show the isovector combination of the generalized form factors @xmath211 and @xmath210. the meaning of the symbols are the same as in fig.[fig : a1020is]. as for @xmath211, both the cqsm and the lattice simulation reproduce the quark number sum rule @xmath212 with good prediction. turning to @xmath210, one observes that the prediction of the cqsm shows somewhat peculiar dependence on the pion mass. starting from a fairly small value in the chiral limit (@xmath213), it first increases as @xmath183 increases, but as @xmath183 further increases it begins to decrease, thereby showing a tendency to match the lattice prediction in the heavy pion region. very interestingly, letting put aside the absolute value, a similar @xmath183 dependence is also observed in the chiral extrapolation of the lattice prediction for the momentum fraction @xmath214 shown in fig.25 of @xcite. physically, the quantity @xmath210 has a meaning of the difference of the momentum fractions carried by the @xmath69-quark and the @xmath70-quark. the empirical value for it is @xmath215 @xcite. one sees that the prediction of the cqsm in the chiral limit is not far from this empirical information, although more serious comparison must take account of the scale dependence of @xmath216. and @xmath217 as functions of @xmath183, together with the corresponding lattice predictions @xcite,@xcite. the meaning of the symbols are the same as in fig.[fig : a1020is].,width=604,height=264] next, shown in fig.[fig : b1020is] are the cqsm predictions for @xmath218 and @xmath217. the former quantity is related to the isoscalar combination of the nucleon anomalous magnetic moment as @xmath219. (we recall that its empirical value is @xmath220.) we find that this quantity is very sensitive to the variation of the pion mass. it appears that the cqsm prediction @xmath221 corresponding to chiral limit underestimates the observation significantly. however, the difference is exaggerated too much in this comparison. in fact, if we carry out a comparison in the total isoscalar magnetic moment of the nucleon @xmath222, the cqsm in the chiral limit gives @xmath223 in comparison with the observed value @xmath224. to our knowledge, no theoretical predictions are given for this quantity by either of the lhpc or qcdsf collaborations. the right panel of fig.4 shows the predictions for @xmath217, which is sometimes called the isoscalar part of the nucleon anomalous gravitomagnetic moment, or alternatively the _ net quark contribution to the nucleon anomalous gravitomagnetic moment_. as already pointed out, the prediction of the cqsm for this quantity is exactly zero : i.e. @xmath169 the explicit numerical calculation also confirms it. it should be recognized that the above result @xmath225 obtained in the cqsm is just a necessary consequence of the _ momentum sum rule _ and the _ total nucleon spin sum rule _, both of which are saturated by the quark field only in the cqsm as @xmath226 and @xmath227 \ = \ \langle j \rangle^{u + d } \ = \ \frac{1}{2 }.\]] in real qcd, the gluon also contributes to these sum rules, thereby leading to more general identities : @xmath228 + [a_{20}^g (0) + b_{20}^g (0)] = 1, \end{aligned}\]] which constrains that only the sum of @xmath217 and @xmath229 is forced to vanish as @xmath230 (while we neglect here the contributions of other quarks than the @xmath69- and @xmath70-quarks, it loses no generality in our discussion below. in fact, to include them, we have only to replace the combination @xmath231 by @xmath232.) the above nontrivial identity claims that the net contributions of quark and gluon fields to the anomalous gravitomagnetic moment of the nucleon must be zero. an interesting question is whether the quark and gluon contribution to the anomalous gravitomagnetic moment vanishes separately or they are both large with opposite sign. a perturbative analysis based on a very simple toy model indicates the latter possibility @xcite. on the other hand, a nonperturbative analysis within the framework of the lattice qcd indicates that the net quark contribution to the anomalous gravitomagnetic moment is small or nearly zero, @xmath233 @xcite,@xcite. (to be more precise, we sees that the prediction of the lhpc collaboration for @xmath217 is slightly negative @xcite, while that of the qcdsf group is slightly positive @xcite.) this strongly indicates a surprising possibility that the quark and gluon contribution to the anomalous gravitomagnetic moment of the nucleon may separately vanish. worthy of special mention here is an interesting argument given by teryaev some years ago, claiming that the vanishing net quark contributions to the anomalous gravitomagnetic moment of the nucleon, violated in perturbation theory, is expected to be restored in full nonperturbative qcd due to the confinement @xcite,@xcite,@xcite. very interestingly, once it actually happens, it leads to a surprisingly simple result, i.e. the proportionality of the quark momentum and angular momentum fraction @xmath234 as advocated by teryaev @xcite,@xcite,@xcite. a far reaching physical consequence resulting from this observation was extensively discussed in our recent report @xcite. (see also the discussion at the end of this section.) and @xmath235 as functions of @xmath183, together with the corresponding lattice predictions @xcite,@xcite. the meaning of the symbols are the same as in fig.[fig : a1020is].,width=604,height=264] next, we show in fig.[fig : b1020iv] the predictions for the isovector case, i.e. @xmath236 and @xmath235. we recall first that the quantity @xmath236 represents the isovector combination of the nucleon anomalous magnetic moment @xmath237, the empirical value of which is known to be @xmath238. one find that this quantity is extremely sensitive to the variation of the pion mass especially near @xmath239. this is only natural if one remembers the important role of the pion cloud in the isovector magnetic moment of the nucleon. (one may notice that the prediction of the cqsm for @xmath240 underestimates a little its empirical value even in the chiral limit. we recall, however, that, within the framework of the cqsm, there is an important @xmath241 correction or the 1st order rotational correction to some kind of isovector quantities like the isovector magnetic moment of the nucleon in question or the axial - vector coupling constant of the nucleon @xcite@xcite. this next - to - leading correction in @xmath241 should also be taken into account in more advanced investigations.) shown in the right panel of fig.[fig : b1020iv] is the theoretical predictions for @xmath235, the half of which can be interpreted as the difference of the total angular momentum carried by the @xmath69-quark and the @xmath70-quark fields according to ji s angular momentum sum rule @xcite. the cqsm predicts fairly small value for this quantity, in contrast to the lattice predictions of sizable magnitude. it seems that the pion mass dependence rescues this discrepancy only partially. here we argue that, the reason why the cqsm (in the chiral limit) gives rather small prediction for this quantity is intimately connected with the characteristic @xmath1 dependence of the quantity @xmath242, the forward limit of the isovector unpolarized spin - flip gpd of the nucleon. to show it, we first recall that, within the theoretical frame work of the cqsm, @xmath236 as well as @xmath243 are calculated as difference of @xmath244 and @xmath245 and of @xmath246 and @xmath247, respectively, as @xmath248 although the quantities of the rhs can be calculated directly without recourse to any distribution functions, they can also be evaluated as @xmath1-weighted integrals of the corresponding gpds as @xmath249 the distribution function @xmath242 has already been calculated within the cqsm in our recent paper @xcite. as shown there, the dirac sea contribution to this quantity has a sizably large peak around @xmath250. since this significant peak due to the deformed dirac - sea quarks is approximately symmetric with respect to the reflection @xmath251, it hardly contributes to the second moment @xmath252, whereas it gives a sizable contribution to the first moment @xmath253. the predicted significant peak of @xmath242 around @xmath250 can physically be interpreted as the effects of pion cloud. it can be convinced in several ways. first, we investigate how this behavior of @xmath242 changes as the pion mass is varied. obtained with @xmath182 and @xmath239.,width=340,height=264] dependence of @xmath254.,width=604,height=264] shown in fig.[fig : eiv_pi0] and in fig.[fig : eiv_pi24] are the cqsm predictions for @xmath242 with several values of @xmath183. i.e. @xmath255, and @xmath256. one clearly sees that the height of the peak around @xmath250, due to the deformed dirac - sea quarks, decreases rapidly as @xmath183 increases. this supports our interpretation of this peak as the effects of pion clouds. on the other hand, one also observes that the magnitude of the valence quark contribution, peaked around @xmath257, gradually increases as @xmath183 becomes large. this behavior of @xmath242 turns out to cause a somewhat unexpected @xmath183 dependence of @xmath253 and @xmath252. as a function of @xmath183, the dirac sea contribution to @xmath253 decreases fast, whereas the valence quark contribution to it increases slowly, so that the total @xmath253 becomes a decreasing function of @xmath183. on the other hand, owing to the approximate odd - function nature of the dirac sea contribution to @xmath258 with respect to @xmath1, it hardly contributes to @xmath259 independent of the pion mass, while the valence quark contribution to @xmath258 is an increasing function of @xmath183, thereby leading to the result that the net @xmath259 is a increasing function of @xmath183. obtained with @xmath182 and @xmath239.,width=340,height=264] dependence of @xmath260.,width=604,height=264] we can give still another support to the above - mentioned interpretation of the contribution of the dirac - sea quarks. to see it, we first recall that the theoretical unpolarized distribution function @xmath261 appearing in the decomposition @xmath262 also has a sizable peak around @xmath250 due to the deformed dirac - sea quarks. as shown in fig.[fig : fiv_pi0] and in fig.[fig : fiv_pi24], this peak is again a rapidly decreasing function of @xmath183, supporting our interpretation of it as the effects of pion clouds. here, we can say more. we point out that this small-@xmath1 behavior of @xmath261 is just what is required by the famous nmc measurement @xcite. to confirm it, first remember that the distribution @xmath263 in the negative @xmath1 region should actually be interpreted as the distribution of antiquarks. to be explicit, it holds that @xmath264 the large and positive value of @xmath261 in the negative @xmath1 region close to @xmath250 means that @xmath265 is negative, i.e. the dominance of the @xmath7-quark over the @xmath8-quark inside the proton, which has been established by the nmc measurement @xcite. evolved to @xmath266 and @xmath267 in comparison with the hermes and nutev data at the corresponding energy scales @xcite,@xcite.,width=340,height=283] shown in fig.[fig : dbdu] are the predictions of the cqsm for @xmath268 evolved to the high energy scales corresponding to the experimental observation @xcite. (the theoretical predictions here were obtained with @xmath182 and @xmath269.) the model reproduces well the observed behavior of @xmath268, although the magnitude of the flavor asymmetry in smaller @xmath1 region seems to be slightly overestimated. it is a widely accepted fact that this flavor asymmetry of the sea quark distribution in the proton can physically be understood as the effects of pion cloud at least qualitatively @xcite@xcite. this then supports our interpretation of the effects of the deformed dirac - sea quarks in @xmath242 and @xmath261 as the effects of pion clouds. .the @xmath270 dependencies of @xmath271, @xmath272, and @xmath273 in the cqsm with @xmath182. [tabivb20] [cols="^,^,^,^",options="header ",] Concluding remarks in this paper, we have investigated the generalized form factors of the nucleon, which will be extracted through near - future measurements of the generalized parton distribution functions, within the framework of the cqsm. a particular emphasis is put on the pion mass dependence as well as the scale dependence of the model predictions, which we compare with the corresponding predictions of the lattice qcd by the lhpc and the qcdsf collaborations carried out in the heavy pion regime around @xmath199. the generalized form factors contain the ordinary electromagnetic form factors of the nucleon such as the dirac and pauli form factors of the proton and the neutron. we have shown that the cqsm with good chiral symmetry reproduces well the general behaviors of the observed electromagnetic form factors, while the lattice simulations by the above two groups have a tendency to underestimate the electromagnetic sizes of the nucleon. undoubtedly, this can not be unrelated to the fact that the above two lattice simulations were performed with unrealistically heavy pion mass. we have also tried to figure out the underlying spin contents of the nucleon through the analysis of the gravitoelectric and gravitomagnetic form factors of the nucleon, by taking care of the pion mass despondencies as well as of the scale dependencies of the relevant quantities. after taking account of the scale dependencies by means of the qcd evolution equations at the nlo in the @xmath274 scheme, the cqsm predicts, at @xmath275, that @xmath276, and @xmath277, which means that the quark orbital angular momentum carries sizable amount of total nucleon spin even at such a relatively high energy scale. it contradicts the conclusion of the lhpc and qcdsf collaborations indicating that the total orbital angular momentum of quarks is very small or consistent with zero. it should be recognized, however, that the prediction of the cqsm for the total quark angular momentum is not extremely far from the corresponding lattice prediction @xmath278 at the same renormalization scale. the cause of discrepancy can therefore be traced back to the lhpc and qcdsf lattice qcd predictions for the quark spin fraction @xmath279 around 0.6, which contradicts not only the prediction of the cqsm but also the emc observation. as was shown in our recent paper @xcite, @xmath279 is such a quantity that is extremely sensitive to the variation of the pion mass, especially in the region close to the chiral limit. more serious lattice qcd studies on the @xmath270-dependence of @xmath279 is highly desirable. worthy of special mention is the fact that, once we accept a theoretical postulate @xmath280, i.e, the absence of the net quark contribution to the anomalous gravitomagnetic moment of the nucleon, which is supported by both of the lhpc and qcdsf lattice simulations, we are necessarily led to a surprisingly simple relations, @xmath281 and @xmath282, i.e. the proportionality of the linear and angular momentum fractions carried by the quarks and the gluons. using these relations, together with the existing empirical information for the unpolarized and the longitudinally polarized pdfs, we can give _ model - independent predictions _ for the quark and gluon contents of the nucleon spin. for instance, with combined use of the mrst2004 fit @xcite and the dns2005 fit @xcite, we obtain @xmath283, @xmath284, and @xmath285 at @xmath275. since @xmath286 (as well as @xmath287) is a rapidly decreasing function of the energy scale, while the scale dependence of @xmath279 is very weak, we must conclude that the former is even more dominant over the latter at the scale below @xmath288 where any low energy models are supposed to hold. the situation is a little more complicated in the flavor - nonsinglet (or isovector) channel, because @xmath289, and also because the cqsm and the lattice qcd give fairly different predictions for @xmath271. as compared with the lattice prediction for @xmath271 around @xmath290, the predictions of the cqsm turns out to be around @xmath291. we have argued that the relatively small value of @xmath246 obtained in the cqsm is intimately connected with the small @xmath1 enhancement of the generalized parton distribution @xmath292, which is dominated by the clouds of pionic @xmath293 excitation around @xmath294. (we recall that the 2nd moment of @xmath292 gives @xmath246.) unfortunately, such a @xmath1-dependent distribution as @xmath292 can not be accessed within the framework of lattice qcd. still, the predicted small @xmath1 behavior of @xmath295 as well as of @xmath296 indicates again the importance of chiral dynamics in the nucleon structure function physics, which has not been fully accounted for in the lattice qcd simulation at the present level. this work is supported in part by a grant - in - aid for scientific research for ministry of education, culture, sports, science and technology, japan (no. c-16540253) Proof of the momentum sum rule here we closely follow the proof of the momentum sum rule given in @xcite, by taking into account a necessary modification in the case of @xmath297. the starting point is the following expression for the soliton mass (or the static soliton energy) : @xmath298 \ - \ (h \rightarrow h_0) \ + \ e_m, \]] with @xmath299 \ = \ - \,f_\pi^2 \,m_\pi^2 \,\int \, [ \cos f(r) - 1] \,d^3 x. \label{eq : emeson}\]] the soliton mass must be stationary with respect to an arbitrary variation of the chiral field @xmath300 or equivalently the soliton profile @xmath52, which lead to a saddle point equation : @xmath301 \ + \ \delta e_m \ = \ 0.\]] here we consider a particular (dilatational) variation of chiral field @xmath302 for infinitesimal @xmath3, we have @xmath303 so that @xmath304 \ = \ \xi \, ([x^k \partial_k, h] - i \gamma^0 \gamma^k \partial_k).\end{aligned}\]] noting the identity @xmath305) \ = \ \mbox{sp } ([h, \theta (e_0 - h + i \varepsilon)] x^k \partial_k) \ = \ 0, \end{aligned}\]] we therefore obtain a key identity @xmath306 = - \,\delta e_m. \label{eq : keyid}\]] now, by using ([eq : emeson]) together with the relations, @xmath307 we get @xmath308 here, taking account of the boundary condition @xmath309 we can manipulate as @xmath310 we thus find an important relation : @xmath311 putting this relation into ([eq : keyid]), we have @xmath312 or @xmath313 if we evaluate the trace sum above by using the eigenstates of the static dirac hamiltonian @xmath50 as a complete set of basis, ([eq : traceap]) can also be written as @xmath314 which is the relation quoted in ([eq : alfp]). we point out that our result has a correct chiral limit, since @xmath315 as @xmath316 and therefore @xmath317 in conformity with the proof given in ref.@xcite.

with a special intention of clarifying the underlying spin contents of the nucleon, we investigate the generalized form factors of the nucleon, which are defined as the @xmath0-th @xmath1-moments of the generalized parton distribution functions, within the framework of the chiral quark soliton model. a particular emphasis is put on the pion mass dependence of final predictions, which we shall compare with the predictions of lattice qcd simulations carried out in the so - called heavy pion region around @xmath2. we find that some observables are very sensitive to the variation of the pion mass. it will be argued that the negligible importance of the quark orbital angular momentum indicated by the lhpc and qcdsf lattice collaborations might be true in the unrealistic heavy pion world, but it is not necessarily the case in our real world close to the chiral limit.

hep-ph0605279

Introduction with significant research efforts being directed to the development of neurocomputers based on the functionalities of the brain, a seismic shift is expected in the domain of computing based on the traditional von - neumann model. the @xmath0 @xcite, @xmath1 @xcite and the ibm @xmath2 @xcite are instances of recent flagship neuromorphic projects that aim to develop brain - inspired computing platforms suitable for recognition (image, video, speech), classification and mining problems. while boolean computation is based on the sequential fetch, decode and execute cycles, such neuromorphic computing architectures are massively parallel and event - driven and are potentially appealing for pattern recognition tasks and cortical brain simulations to that end, researchers have proposed various nanoelectronic devices where the underlying device physics offer a mapping to the neuronal and synaptic operations performed in the brain. the main motivation behind the usage of such non - von neumann post - cmos technologies as neural and synaptic devices stems from the fact that the significant mismatch between the cmos transistors and the underlying neuroscience mechanisms result in significant area and energy overhead for a corresponding hardware implementation. a very popular instance is the simulation of a cat s brain on ibm s blue gene supercomputer where the power consumption was reported to be of the order of a few @xmath3 @xcite. while the power required to simulate the human brain will rise significantly as we proceed along the hierarchy in the animal kingdom, actual power consumption in the mammalian brain is just a few tens of watts. in a neuromorphic computing platform, synapses form the pathways between neurons and their strength modulate the magnitude of the signal transmitted between the neurons. the exact mechanisms that underlie the `` learning '' or `` plasticity '' of such synaptic connections are still under debate. meanwhile, researchers have attempted to mimic several plasticity measurements observed in biological synapses in nanoelectronic devices like phase change memories @xcite, @xmath4 memristors @xcite and spintronic devices @xcite, etc. however, majority of the research have focused on non - volatile plasticity changes of the synapse in response to the spiking patterns of the neurons it connects corresponding to long - term plasticity @xcite and the volatility of human memory has been largely ignored. as a matter of fact, neuroscience studies performed in @xcite have demonstrated that synapses exhibit an inherent learning ability where they undergo volatile plasticity changes and ultimately undergo long - term plasticity conditionally based on the frequency of the incoming action potentials. such volatile or meta - stable synaptic plasticity mechanisms can lead to neuromorphic architectures where the synaptic memory can adapt itself to a changing environment since sections of the memory that have been not receiving frequent stimulus can be now erased and utilized to memorize more frequent information. hence, it is necessary to include such volatile memory transition functionalities in a neuromorphic chip in order to leverage from the computational power that such meta - stable synaptic plasticity mechanisms has to offer. [drawing1] (a) demonstrates the biological process involved in such volatile synaptic plasticity changes. during the transmission of each action potential from the pre - neuron to the post - neuron through the synapse, an influx of ionic species like @xmath5 and @xmath6 causes the release of neurotransmitters from the pre- to the post - neuron. this results in temporary strengthening of the synaptic strength. however, in absence of the action potential, the ionic species concentration settles down to its equilibrium value and the synapse strength diminishes. this phenomenon is termed as short - term plasticity (stp) @xcite. however, if the action potentials occur frequently, the concentration of the ions do not get enough time to settle down to the equilibrium concentration and this buildup of concentration eventually results in long - term strengthening of the synaptic junction. this phenomenon is termed as long - term potentiation (ltp). while stp is a meta - stable state and lasts for a very small time duration, ltp is a stable synaptic state which can last for hours, days or even years @xcite. a similar discussion is valid for the case where there is a long - term reduction in synaptic strength with frequent stimulus and then the phenomenon is referred to as long - term depression (ltd). such stp and ltp mechanisms have been often correlated to the short - term memory (stm) and long - term memory (ltm) models proposed by atkinson and shiffrin @xcite (fig. [drawing1](b)). this psychological model partitions the human memory into an stm and an ltm. on the arrival of an input stimulus, information is first stored in the stm. however, upon frequent rehearsal, information gets transferred to the ltm. while the `` forgetting '' phenomena occurs at a fast rate in the stm, information can be stored for a much longer duration in the ltm. in order to mimic such volatile synaptic plasticity mechanisms, a nanoelectronic device is required that is able to undergo meta - stable resistance transitions depending on the frequency of the input and also transition to a long - term stable resistance state on frequent stimulations. hence a competition between synaptic memory reinforcement or strengthening and memory loss is a crucial requirement for such nanoelectronic synapses. in the next section, we will describe the mapping of the magnetization dynamics of a nanomagnet to such volatile synaptic plasticity mechanisms observed in the brain. Formalism let us first describe the device structure and principle of operation of an mtj @xcite as shown in fig. [drawing2](a). the device consists of two ferromagnetic layers separated by a tunneling oxide barrier (tb). the magnetization of one of the layers is magnetically `` pinned '' and hence it is termed as the `` pinned '' layer (pl). the magnetization of the other layer, denoted as the `` free layer '' (fl), can be manipulated by an incoming spin current @xmath7. the mtj structure exhibits two extreme stable conductive states the low conductive `` anti - parallel '' orientation (ap), where pl and fl magnetizations are oppositely directed and the high conductive `` parallel '' orientation (p), where the magnetization of the two layers are in the same direction. let us consider that the initial state of the mtj synapse is in the low conductive ap state. considering the input stimulus (current) to flow from terminal t2 to terminal t1, electrons will flow from terminal t1 to t2 and get spin - polarized by the pl of the mtj. subsequently, these spin - polarized electrons will try to orient the fl of the mtj `` parallel '' to the pl. it is worth noting here that the spin - polarization of incoming electrons in the mtj is analogous to the release of neurotransmitters in a biological synapse. the stp and ltp mechanisms exhibited in the mtj due to the spin - polarization of the incoming electrons can be explained by the energy profile of the fl of the mtj. let the angle between the fl magnetization, @xmath8, and the pl magnetization, @xmath9, be denoted by @xmath10. the fl energy as a function of @xmath10 has been shown in fig. [drawing2](a) where the two energy minima points (@xmath11 and @xmath12) are separated by the energy barrier, @xmath13. during the transition from the ap state to the p state, the fl has to transition from @xmath12 to @xmath11. upon the receipt of an input stimulus, the fl magnetization proceeds `` uphill '' along the energy profile (from initial point 1 to point 2 in fig. [drawing2](a)). however, since point 2 is a meta - stable state, it starts going `` downhill '' to point 1, once the stimulus is removed. if the input stimulus is not frequent enough, the fl will try to stabilize back to the ap state after each stimulus. however, if the stimulus is frequent, the fl will not get sufficient time to reach point 1 and ultimately will be able to overcome the energy barrier (point 3 in fig. [drawing2](a)). it is worth noting here, that on crossing the energy barrier at @xmath14, it becomes progressively difficult for the mtj to exhibit stp and switch back to the initial ap state. this is in agreement with the psychological model of human memory where it becomes progressively difficult for the memory to `` forget '' information during transition from stm to ltm. hence, once it has crossed the energy barrier, it starts transitioning from the stp to the ltp state (point 4 in fig. [drawing2](a)). the stability of the mtj in the ltp state is dictated by the magnitude of the energy barrier. the lifetime of the ltp state is exponentially related to the energy barrier @xcite. for instance, for an energy barrier of @xmath15 used in this work, the ltp lifetime is @xmath16 hours while the lifetime can be extended to around @xmath17 years by engineering a barrier height of @xmath18. the lifetime can be varied by varying the energy barrier, or equivalently, volume of the mtj. the stp - ltp behavior of the mtj can be also explained from the magnetization dynamics of the fl described by landau - lifshitz - gilbert (llg) equation with additional term to account for the spin momentum torque according to slonczewski @xcite, @xmath19 where, @xmath20 is the unit vector of fl magnetization, @xmath21 is the gyromagnetic ratio for electron, @xmath22 is gilberts damping ratio, @xmath23 is the effective magnetic field including the shape anisotropy field for elliptic disks calculated using @xcite, @xmath24 is the number of spins in free layer of volume @xmath25 (@xmath26 is saturation magnetization and @xmath27 is bohr magneton), and @xmath28 is the spin current generated by the input stimulus @xmath29 (@xmath30 is the spin - polarization efficiency of the pl). thermal noise is included by an additional thermal field @xcite, @xmath31, where @xmath32 is a gaussian distribution with zero mean and unit standard deviation, @xmath33 is boltzmann constant, @xmath34 is the temperature and @xmath35 is the simulation time step. equation [llg] can be reformulated by simple algebraic manipulations as, @xmath36 hence, in the presence of an input stimulus the magnetization of the fl starts changing due to integration of the input. however, in the absence of the input, it starts leaking back due to the first two terms in the rhs of the above equation. it is worth noting here that, like traditional semiconductor memories, magnitude and duration of the input stimulus will definitely have an impact on the stp - ltp transition of the synapse. however, frequency of the input is a critical factor in this scenario. even though the total flux through the device is same, the synapse will conditionally change its state if the frequency of the input is high. we verified that this functionality is exhibited in mtjs by performing llg simulations (including thermal noise). the conductance of the mtj as a function of @xmath10 can be described by, @xmath37 where, @xmath38 (@xmath39) is the mtj conductance in the p (ap) orientation respectively. as shown in fig. [drawing2](b), the mtj conductance undergoes meta - stable transitions (stp) and is not able to undergo ltp when the time interval of the input pulses is large (@xmath40). however, on frequent stimulations with time interval as @xmath41, the device undergoes ltp transition incrementally. [drawing2](b) and (c) illustrates the competition between memory reinforcement and memory decay in an mtj structure that is crucial to implement stp and ltp in the synapse. Results and discussions we demonstrate simulation results to verify the stp and ltp mechanisms in an mtj synapse depending on the time interval between stimulations. the device simulation parameters were obtained from experimental measurements @xcite and have been shown in table i. [table] table i. device simulation parameters [cols="^,^ ",] + the mtj was subjected to 10 stimulations, each stimulation being a current pulse of magnitude @xmath42 and @xmath43 in duration. as shown in fig. [drawing3], the probability of ltp transition and average device conductance at the end of each stimulation increases with decrease in the time interval between the stimulations. the dependence on stimulation time interval can be further characterized by measurements corresponding to paired - pulse facilitation (ppf : synaptic plasticity increase when a second stimulus follows a previous similar stimulus) and post - tetanic potentiation (ptp : progressive synaptic plasticity increment when a large number of such stimuli are received successively) @xcite. [drawing4] depicts such ppf (after 2nd stimulus) and ptp (after 10th stimulus) measurements for the mtj synapse with variation in the stimulation interval. the measurements closely resemble measurements performed in frog neuromuscular junctions @xcite where ppf measurements revealed that there was a small synaptic conductivity increase when the stimulation rate was frequent enough while ptp measurements indicated ltp transition on frequent stimulations with a fast decay in synaptic conductivity on decrement in the stimulation rate. hence, stimulation rate indeed plays a critical role in the mtj synapse to determine the probability of ltp transition. the psychological model of stm and ltm utilizing such mtj synapses was further explored in a @xmath44 memory array. the array was stimulated by a binary image of the purdue university logo where a set of 5 pulses (each of magnitude @xmath45 and @xmath46 in duration) was applied for each on pixel. the snapshots of the conductance values of the memory array after each stimulus have been shown for two different stimulation intervals of @xmath47 and @xmath48 respectively. while the memory array attempts to remember the displayed image right after stimulation, it fails to transition to ltm for the case @xmath49 and the information is eventually lost @xmath50 after stimulation. however, information gets transferred to ltm progressively for @xmath51. it is worth noting here, that the same amount of flux is transmitted through the mtj in both cases. the simulation not only provides a visual depiction of the temporal evolution of a large array of mtj conductances as a function of stimulus but also provides inspiration for the realization of adaptive neuromorphic systems exploiting the concepts of stm and ltm. readers interested in the practical implementation of such arrays of spintronic devices are referred to ref. Conclusions the contributions of this work over state - of - the - art approaches may be summarized as follows. this is the first theoretical demonstration of stp and ltp mechanisms in an mtj synapse. we demonstrated the mapping of neurotransmitter release in a biological synapse to the spin polarization of electrons in an mtj and performed extensive simulations to illustrate the impact of stimulus frequency on the ltp probability in such an mtj structure. there have been recent proposals of other emerging devices that can exhibit such stp - ltp mechanisms like @xmath52 synapses @xcite and @xmath53 memristors @xcite. however, it is worth noting here, that input stimulus magnitudes are usually in the range of volts (1.3v in @xcite and 80mv in @xcite) and stimulus durations are of the order of a few msecs (1ms in @xcite and 0.5s in @xcite). in contrast, similar mechanisms can be exhibited in mtj synapses at much lower energy consumption (by stimulus magnitudes of a few hundred @xmath54 and duration of a few @xmath55). we believe that this work will stimulate proof - of - concept experiments to realize such mtj synapses that can potentially pave the way for future ultra - low power intelligent neuromorphic systems capable of adaptive learning. Acknowledgements the work was supported in part by, center for spintronic materials, interfaces, and novel architectures (c - spin), a marco and darpa sponsored starnet center, by the semiconductor research corporation, the national science foundation, intel corporation and by the national security science and engineering faculty fellowship. j. schemmel, j. fieres, and k. meier, in _ neural networks, 2008. ijcnn 2008.(ieee world congress on computational intelligence). ieee international joint conference on_.1em plus 0.5em minus 0.4emieee, 2008, pp. 431438. b. l. jackson, b. rajendran, g. s. corrado, m. breitwisch, g. w. burr, r. cheek, k. gopalakrishnan, s. raoux, c. t. rettner, a. padilla _ et al. _, `` nanoscale electronic synapses using phase change devices, '' _ acm journal on emerging technologies in computing systems (jetc) _, vol. 9, no. 2, p. 12, 2013. m. n. baibich, j. m. broto, a. fert, f. n. van dau, f. petroff, p. etienne, g. creuzet, a. friederich, and j. chazelas, `` giant magnetoresistance of (001) fe/(001) cr magnetic superlattices, '' _ physical review letters _, 61, no. 21, p. 2472, 1988. g. binasch, p. grnberg, f. saurenbach, and w. zinn, `` enhanced magnetoresistance in layered magnetic structures with antiferromagnetic interlayer exchange, '' _ physical review b _, vol. 39, no. 7, p. 4828, 1989. w. scholz, t. schrefl, and j. fidler, `` micromagnetic simulation of thermally activated switching in fine particles, '' _ journal of magnetism and magnetic materials _, vol. 233, no. 3, pp. 296304, 2001. pai, l. liu, y. li, h. tseng, d. ralph, and r. buhrman, `` spin transfer torque devices utilizing the giant spin hall effect of tungsten, '' _ applied physics letters _, vol. 101, no. 12, p. 122404, 2012. h. noguchi, k. ikegami, k. kushida, k. abe, s. itai, s. takaya, n. shimomura, j. ito, a. kawasumi, h. hara _ et al. _, in _ solid - state circuits conference-(isscc), 2015 ieee international_.1em plus 0.5em minus 0.4emieee, 2015, pp. t. ohno, t. hasegawa, t. tsuruoka, k. terabe, j. k. gimzewski, and m. aono, `` short - term plasticity and long - term potentiation mimicked in single inorganic synapses, '' _ nature materials _, vol. 10, no. 8, pp. 591595, 2011. r. yang, k. terabe, y. yao, t. tsuruoka, t. hasegawa, j. k. gimzewski, and m. aono, `` synaptic plasticity and memory functions achieved in a wo3- x - based nanoionics device by using the principle of atomic switch operation, '' _ nanotechnology _, vol. 24, no. 38, p. 384003

synaptic memory is considered to be the main element responsible for learning and cognition in humans. although traditionally non - volatile long - term plasticity changes have been implemented in nanoelectronic synapses for neuromorphic applications, recent studies in neuroscience have revealed that biological synapses undergo meta - stable volatile strengthening followed by a long - term strengthening provided that the frequency of the input stimulus is sufficiently high. such `` memory strengthening '' and `` memory decay '' functionalities can potentially lead to adaptive neuromorphic architectures. in this paper, we demonstrate the close resemblance of the magnetization dynamics of a magnetic tunnel junction (mtj) to short - term plasticity and long - term potentiation observed in biological synapses. we illustrate that, in addition to the magnitude and duration of the input stimulus, frequency of the stimulus plays a critical role in determining long - term potentiation of the mtj. such mtj synaptic memory arrays can be utilized to create compact, ultra - fast and low power intelligent neural systems.

1511.00051

Introduction let @xmath1. let @xmath2\longrightarrow [0,\pi_{p, q}/2]$] be the integral @xmath3 where @xmath4. the @xmath0-_sine functions _, @xmath5, $] are defined to be the inverses of @xmath6, @xmath7\]] extended to @xmath8 by the rules @xmath9 which make them periodic, continuous, odd with respect to 0 and even with respect to @xmath10. these are natural generalisations of the sine function, indeed @xmath11 and they are known to share a number of remarkable properties with their classical counterpart @xcite. among these properties lies the fundamental question of completeness and linear independence of the family @xmath12 where @xmath13. this question has received some attention recently @xcite, with a particular emphasis on the case @xmath14. in the latter instance, @xmath15 is the set of eigenfunctions of the generalised eigenvalue problem for the one - dimensional @xmath16-laplacian subject to dirichlet boundary conditions @xcite, which is known to be of relevance in the theory of slow / fast diffusion processes, @xcite. see also the related papers @xcite. set @xmath17, so that @xmath18 is a schauder basis of the banach space @xmath19 for all @xmath20. the family @xmath21 is also a schauder basis of @xmath22 if and only if the corresponding _ change of coordinates map _, @xmath23, extends to a linear homeomorphism of @xmath22. the fourier coefficients of @xmath24 associated to @xmath25 obey the relation @xmath26 for @xmath27, let @xmath28 (note that @xmath29 for @xmath30) and let @xmath31 be the linear isometry such that @xmath32. then @xmath33 so that the change of coordinates takes the form @xmath34 notions of `` nearness '' between bases of banach spaces are known to play a fundamental role in classical mathematical analysis, @xcite, @xcite or @xcite. unfortunately, the expansion strongly suggests that @xmath21 is not globally `` near '' @xmath18, e.g. in the krein - lyusternik or the paley - wiener sense, @xcite. therefore classical arguments, such as those involving the paley - wiener stability theorem, are unlikely to be directly applicable in the present context. in fact, more rudimentary methods can be invoked in order to examine the invertibility of the change of coordinates map. from it follows that @xmath35 in @xcite it was claimed that the left side of held true for all @xmath36 where @xmath37 was determined to lie in the segment @xmath38. hence @xmath21 would be a schauder basis, whenever @xmath39. further developments in this respect were recently reported by bushell and edmunds @xcite. these authors cleverly fixed a gap originally published in (*??? * lemma 5) and observed that, as the left side of ceases to hold true whenever @xmath40 the argument will break for @xmath14 near @xmath41. therefore, the basisness question for @xmath21 should be tackled by different means in the regime @xmath42. more recently @xcite, edmunds, gurka and lang, employed in order to show invertibility of @xmath43 for general pairs @xmath44, as long as @xmath45 since is guaranteed whenever @xmath46 this allows @xmath47 for @xmath48. however, note that a direct substitution of @xmath14 in, only leads to the sub - optimal condition @xmath49. in section [linearind] below we show that the family @xmath21 is @xmath50-_linearly independent _ for all @xmath1, see theorem [likernelandspan]. in section [ribap] we establish conditions ensuring that @xmath43 is a homeomorphism of @xmath51 in a neighbourhood of the region in the @xmath44-plane where @xmath52 see theorem [inprovement] and also corollary [beyonda]. for this purpose, in section [criteria] we find two further criteria which generalise in the hilbert space setting, see corollaries [main_1] and [main_2]. in this case, the _ riesz constant _, @xmath53 characterises how @xmath21 deviates from being an orthonormal basis. these new statements yield upper bounds for @xmath54, which improve upon those obtained from the right side of, even when the latter is applicable. the formulation of the alternatives to presented below relies crucially on work developed in section [toep_s]. from lemma [multareshifts] we compute explicitly the wold decomposition of the isometries @xmath31 : they turn out to be shifts of infinite multiplicity. hence we can extract from the expansion suitable components which are toeplitz operators of scalar type acting on appropriate hardy spaces. as the theory becomes quite technical for the case @xmath55 and all the estimates analogous to those reported below would involve a dependence on the parameter @xmath56, we have chosen to restrict our attention with regards to these improvements only to the already interesting hilbert space setting. section [casep = q] is concerned with particular details of the case of equal indices @xmath14, and it involves results on both the general case @xmath20 and the specific case @xmath57. rather curiously, we have found another gap which renders incomplete the proof of invertibility of @xmath43 for @xmath58 originally published in @xcite. see remark [rem_gap]. moreover, the application of (*??? * theorem 4.5) only gets to a _ basisness threshold _ of @xmath59, where @xmath60 is defined by the identity @xmath61 see also (*??? * remark 2.1). in theorem [fixingbbcdg] we show that @xmath21 is indeed a schauder basis of @xmath22 for @xmath62 where @xmath63, see (*??? * problem 1). as @xmath64, basisness is now guaranteed for all @xmath65. see figure [impro_fig_p = q]. in section [nume] we report on our current knowledge of the different thresholds for invertibility of the change of coordinates map, both in the case of equal indices and otherwise. based on the new criteria found in section [criteria], we formulate a general test of invertibility for @xmath43 which is amenable to analytical and numerical investigation. this test involves finding sharp bounds on the first few coefficients @xmath66. see proposition [beyond2]. for the case of equal indices, this test indicates that @xmath21 is a riesz basis of @xmath51 for @xmath67 where @xmath68. all the numerical quantities reported in this paper are accurate up to the last digit shown, which is rounded to the nearest integer. in the appendix we have included fully reproducible computer codes which can be employed to verify the calculations reported. Linear independence a family @xmath69 in a banach space is called @xmath50-linearly independent @xcite, if @xmath70 [likernelandspan] for all @xmath1, the family @xmath21 is @xmath50-linearly independent in @xmath22. moreover, if the linear extension of the map @xmath23 is a bounded operator @xmath71, then @xmath72 for the first assertion we show that @xmath73. let @xmath74 be such that @xmath75 where the series is convergent in the norm of @xmath22. then @xmath76 hence @xmath77 we show that all @xmath78 by means of a double induction argument. suppose that @xmath79. we prove that all @xmath80. indeed, clearly @xmath81 from with @xmath82. now assume inductively that @xmath29 for all @xmath83. from for @xmath84 we get @xmath85 then @xmath80 for all @xmath86. as this would contradict the fact that @xmath87, necessarily @xmath88. suppose now inductively that @xmath89 and @xmath90. we prove that again all @xmath80. firstly, @xmath81 from with @xmath91, because @xmath92 secondly, assume by induction that @xmath29 for all @xmath93. from for @xmath94 we get @xmath95 the latter equality is a consequence of the fact that, for @xmath96 with @xmath97 and @xmath98, either @xmath99 (indices for the @xmath100) or @xmath101 (indices for the @xmath102). hence @xmath80 for all @xmath86. as this would again contradict the fact that @xmath87, necessarily all @xmath103 so that @xmath104. the second assertion is shown as follows. assume that @xmath105. if @xmath106, then @xmath107 for all @xmath108, so @xmath109 which in turns means that @xmath110 for all @xmath111. on the other hand, if the latter holds true for @xmath112, then @xmath113 for all @xmath114, so @xmath115, as required. therefore, @xmath21 is a riesz basis of @xmath51 if and only if @xmath105 and @xmath116. a simple example illustrates how a family of dilated periodic functions can break its property of being a riesz basis. [ex1] let @xmath117 $]. take @xmath118 by virtue of lemma [toep] below, @xmath119 is a riesz basis of @xmath51 if and only if @xmath120. for @xmath121 we have an orthonormal set. however it is not complete, as it clearly misses the infinite - dimensional subspace @xmath122. The different components of the change of coordinates map the fundamental decomposition of @xmath43 given in allows us to extract suitable components formed by toeplitz operators of scalar type, @xcite. in order to identify these components, we begin by determining the wold decomposition of the isometries @xmath31, @xcite. see remark [diri]. [multareshifts] for all @xmath123, @xmath124 is a shift of infinite multiplicity. define @xmath125 then @xmath126 for @xmath127, @xmath128, and @xmath129 one - to - one and onto for all @xmath130. therefore indeed @xmath31 is a shift of multiplicity @xmath131. let @xmath132. the hardy spaces of functions in @xmath133 with values in the banach space @xmath134 are denoted below by @xmath135. let @xmath136 be a holomorphic function on @xmath137 and fix @xmath138. let @xmath139 let the corresponding toeplitz operator (*??? * (5 - 1)) @xmath140 let @xmath141 by virtue of lemma [multareshifts] (see (*??? * and 5.2)), there exists an invertible isometry @xmath142 such that @xmath143. below we write @xmath144 [generic_toep] @xmath145 in is invertible if and only if @xmath146. moreover @xmath147 observe that @xmath148 is scalar analytic in the sense of @xcite. since @xmath149 is holomorphic in @xmath137, then @xmath150 and @xmath151 (*??? * theorem a(iii)). if @xmath152, then @xmath153 is also holomorphic in @xmath137. the scalar toeplitz operator @xmath154 is invertible if and only if @xmath146. moreover, @xcite, @xmath155 the matrix of @xmath148 has the block representation @xcite @xmath156 the matrix associated to @xmath154 has exactly the same scalar form, replacing @xmath157 by @xmath158. then, @xmath148 is invertible if and only if @xmath154 is invertible, and @xmath159 hence @xmath160 [pert] let @xmath161 for @xmath145 as in. if @xmath162, then @xmath43 is invertible. moreover @xmath163 since @xmath145 is invertible, write @xmath164. if additionally @xmath165, then @xmath166 [diri] it is possible to characterise the change of coordinates @xmath43 in terms of dirichlet series, and recover some of the results here and below directly from this characterisation. see for example the insightful paper @xcite and the complete list of references provided in the addendum @xcite. however, the full technology of dirichlet series is not needed in the present context. a further development in this direction will be reported elsewhere. Invertibility and bounds on the riesz constant a proof of can be achieved by applying corollary [pert] assuming that @xmath167 our next goal is to formulate concrete sufficient condition for the invertibility of @xmath43 and corresponding bounds on @xmath54, which improve upon whenever @xmath57. for this purpose we apply corollary [pert] assuming that @xmath145 has now the three - term expansion @xmath168 let @xmath169 let @xmath170 see figure [region]. optimal region of invertibility in lemma [toep]. in this picture the horizontal axis is @xmath171 and the vertical axis is @xmath172.,height=377] [toep] let @xmath57. let @xmath173. the operator @xmath174 is invertible if and only if @xmath175. moreover @xmath176 let @xmath177 be associated with @xmath145 as in section [toep_s]. the first assertion is a consequence of the following observation. if @xmath178, then @xmath179 has roots @xmath180 conjugate with each other and @xmath181 if and only if @xmath182. otherwise @xmath179 has two real roots. if @xmath183 and @xmath184, then the smallest in modulus root of @xmath179 would lie in @xmath137 if and only if @xmath185. if @xmath186 and @xmath187, then the root of @xmath179 that is smallest in modulus would lie in @xmath137 if and only if @xmath188. for the second assertion, let @xmath175 and @xmath189. by virtue of the maximum principle on @xmath179 and @xmath190, @xmath191 since @xmath192 then @xmath193 if and only if @xmath194. for @xmath195, we get @xmath196 and @xmath197. for @xmath198, we get @xmath199 with the condition @xmath200. by virtue of theorem [generic_toep], we obtain the claimed statement. since @xmath201 for all @xmath202, then @xmath203. below we substitute @xmath204 and @xmath205, then apply lemma [toep] appropriately in order to determine the invertibility of @xmath43 whenever pairs @xmath44 lie in different regions of the @xmath44-plane. for this purpose we establish the following hierarchy between @xmath206 and @xmath207 for @xmath208, whenever the latter are non - negative. [ajint] for @xmath209 or @xmath210, we have @xmath211. firstly observe that @xmath212 is continuous, it increases for all @xmath213 and it vanishes at @xmath214. let @xmath209. set @xmath215 { \mathrm{d}}x \qquad \text{and } \\ i_1=\int_{\frac14}^{\frac12 } \sin_{p, q}(\pi_{p, q}x) [\sin(\pi x)-\sin(3\pi x)] { \mathrm{d}}x. \end{gathered}\]] since @xmath216 then @xmath217 and @xmath218. as @xmath219 is odd with respect to @xmath220 and @xmath221 is increasing in the segment @xmath222, then also @xmath223. hence @xmath224 ensuring the first statement of the lemma. let @xmath210. a straightforward calculation shows that @xmath225 if and only if, either @xmath226 or @xmath227. thus, @xmath228 has exactly five zeros in the segment @xmath229 $] located at : @xmath230 set @xmath231 and @xmath232 { \mathrm{d}}x.\]] then @xmath233 for @xmath234 and @xmath235 for @xmath236. since @xmath237 for all @xmath238, then @xmath239 hence @xmath240 the next two corollaries are consequences of corollary [pert] and lemma [toep], and are among the main results of this paper. [main_1] @xmath241 let @xmath161 where @xmath242 the top on left side of and the fact that @xmath203 imply @xmath243 thus, the bottom on the left side of yields @xmath244 so indeed @xmath43 is invertible. the estimate on the riesz constant is deduced from the triangle inequality. since @xmath203, supersedes, only when the pair @xmath44 is such that @xmath245. from this corollary we see below that the change of coordinates is invertible in a neighbourhood of the threshold set by the condition. see proposition [beyond2] and figures [impro_fig_p = q] and [th10ab]. [main_2] @xmath246 the proof is similar to that of corollary [main_1]. we see below that corollary [main_1] is slightly more useful than corollary [main_2] in the context of the dilated @xmath0-sine functions. however the latter is needed in the proof of the main theorem [inprovement]. it is of course natural to ask what consequences can be derived from the other statement in lemma [toep]. for @xmath247 we have @xmath248. hence the same argument as in the proofs of corollaries [main_1] and [main_2] would reduce to, and in this case there is no improvement. Riesz basis properties beyond the applicability of our first goal in this section is to establish that the change of coordinates map associated to the family @xmath21 is invertible beyond the region of applicability of. we begin by recalling a calculation which was performed in the proof of (*??? * proposition 4.1) and which will be invoked several times below. let @xmath249 be the inverse function of @xmath250. then @xmath251 indeed, integrating by parts twice and changing the variable of integration to @xmath252 yields @xmath253'\sin(j\pi x) { \mathrm{d}}x \\ & = -\frac{2 \sqrt{2 } \pi_{p, q}}{j^2\pi^2 } \int_0 ^ 1 \sin \left (\frac{j \pi}{\pi_{p, q } } a(t)\right) { \mathrm{d}}t . \end{aligned}\]] [inprovement] let @xmath57. suppose that the pair @xmath254 is such that the following two conditions are satisfied 1. [improa] @xmath255 2. [improc] @xmath256. then there exists a neighbourhood @xmath257, such that the change of coordinates @xmath43 is invertible for all @xmath258. from the dominated convergence theorem, it follows that each @xmath259 is a continuous function of the parameters @xmath16 and @xmath260. therefore, by virtue of and a further application of the dominated convergence theorem, also @xmath261 is continuous in the parameters @xmath16 and @xmath260. here @xmath262 can be any fixed set of indices, but below in this proof we only need to consider @xmath263 for the first possibility and @xmath264 for the second possibility. write @xmath265. the hypothesis implies @xmath266, because @xmath267 therefore @xmath268 for a suitable neighbourhood @xmath269. two possibilities are now in place. note that @xmath271 is an immediate consequence of [improa] and [improc]. by continuity of all quantities involved, there exists a neighbourhood @xmath272 such that the left hand side and hence the right hand side of hold true for all @xmath273. substitute @xmath275 and @xmath276. if @xmath277, then @xmath278 indeed, the conditions on @xmath171 and @xmath172 give @xmath279 as @xmath280, @xmath281 thus @xmath282 which is. hence @xmath283 thus, once again by continuity of all quantities involved, there exists a neighbourhood @xmath284 such that the left hand side and hence the right hand side of hold true for all @xmath285. the conclusion follows by defining either @xmath286 or @xmath287. we now examine other further consequences of the corollaries [main_1] and [main_2]. [impro_implicit] any of the following conditions ensure the invertibility of the change of coordinates map @xmath288. 1. [aimplicit] (@xmath20) : @xmath289 2. [bimplicit] (@xmath57) : @xmath290, @xmath245, @xmath291 and @xmath292 3. [cimplicit] (@xmath57) : @xmath290, @xmath245, @xmath293 and @xmath294 from, it follows that @xmath295 hence the condition [aimplicit] implies that the hypothesis is satisfied. by virtue of lemma [ajint], it is guaranteed that @xmath296 in the settings of [bimplicit] or [cimplicit]. from, it also follows that @xmath297 combining each one of these assertions with and, respectively, immediately leads to the claimed statement. we recover (*??? * corollary 4.3) from the part [aimplicit] of this theorem by observing that for all @xmath1, @xmath298 in fact, for @xmath299, the better estimate @xmath300 ensures invertibility of @xmath43 for all @xmath20 whenever @xmath301 see figures [th10ab] and [th10c]. The case of equal indices we now consider in closer detail the particular case @xmath302. our analysis requires setting various sharp upper and lower bounds on the coefficients @xmath303 for @xmath304. this is our first goal. employed to show bound [a3l] in lemma [ajpositive]. for reference we also show @xmath305, @xmath306, @xmath307 and @xmath308. [fig : interp], width=340] [ajpositive] 1. [a3l] @xmath309 for all @xmath310 2. [a5l] @xmath311 for all @xmath312 3. [a7l] @xmath313 for all @xmath312 4. [a9l] @xmath314 for all @xmath315 all the stated bounds are determined by integrating a suitable approximation of @xmath316. each one requires a different set of quadrature points, but the general structure of the arguments in all cases is similar. without further mention, below we repeatedly use the fact that in terms of hypergeometric functions, @xmath317.\]] let @xmath318 for @xmath319 let @xmath320 see figure [fig : interp]. since @xmath321 and @xmath322 is an increasing function of @xmath323, then @xmath324 according to (*??? * corollary 4.4), @xmath316 increases as @xmath16 decreases for any fixed @xmath325. let @xmath16 be as in the hypothesis. then @xmath326 and similarly @xmath327 by virtue of (*??? * lemma 3) the function @xmath322 is strictly concave for @xmath323. then, in fact, @xmath328 let @xmath329 since @xmath330 for @xmath331 and @xmath332, @xmath333 note that @xmath334 set @xmath335 then @xmath336 and so @xmath337 also @xmath338 so @xmath339 let @xmath16 be as in the hypothesis. then, similarly to the previous case [a3l], @xmath340 set @xmath341 by strict concavity and, @xmath342 let @xmath343 then @xmath344 as claimed. let @xmath16 be as in the hypothesis. set @xmath345 then @xmath346 hence @xmath347 put @xmath348 then, @xmath349 let @xmath350 since @xmath351 is negative for @xmath352 and positive for @xmath353, then @xmath354. hence @xmath355 note that @xmath356 let @xmath16 be as in the hypothesis. set @xmath357 then @xmath358 hence @xmath347 put @xmath359 then, @xmath360 let @xmath361 then @xmath362. hence @xmath363 the next statement is a direct consequence of combining [a3l] and [a9l] from this lemma with theorem [inprovement]. [beyonda] set @xmath57 and suppose that @xmath364 is such that @xmath365 there exists @xmath366 such that @xmath43 is invertible for all @xmath367. see figure [impro_fig_p = q]. [rem_gap] in @xcite it was claimed that the hypothesis of held true whenever @xmath36 for a suitable @xmath368. the argument supporting this claim @xcite was separated into two cases : @xmath369 and @xmath370. with our definition by a factor of @xmath371. note that the ground eigenfunction of the @xmath16-laplacian equation in @xcite is denoted by @xmath372 and it equals @xmath373 as defined above. a key observation here is the @xmath16-pythagorean identity @xmath374.] of the fourier coefficients, in the latter case it was claimed that @xmath375 was bounded above by @xmath376 as it turns, there is a missing power 2 in the term @xmath377 for this claim to be true. this corresponds to taking second derivatives of @xmath378 and it can be seen by applying the cauchy - schwartz inequality in. the missing factor is crucial in the argument and renders the proof of (*??? * theorem 1) incomplete in the latter case. in the paper @xcite published a few years later, it was claimed that the hypothesis of held true for @xmath379 where @xmath60 is defined by. it was then claimed that an approximated solution of was near @xmath380. an accurate numerical approximation of, based on analytical bounds on @xmath381, give the correct digits @xmath382. therefore neither the results of @xcite nor those of @xcite include a complete proof of invertibility of the change of coordinates in a neighbourhood of @xmath383. accurate numerical estimation of @xmath381 show that the identity is valid as long as @xmath384, which improves slightly upon the value @xmath60 from @xcite. however, as remarked in @xcite, the upper bound @xmath385 ensuring and hence the validity of theorem [impro_implicit]-[aimplicit], is too crude for small values of @xmath16. note for example that the correct regime is @xmath386 whereas @xmath387 as @xmath388 (see appendix [ap1]). therefore, in order to determine invertibility of @xmath43 in the vicinity of @xmath389, it is necessary to find sharper bounds for the first few terms @xmath375, and employ directly. this is the purpose of the next lemma. see figure [impro_fig_p = q]. [ajbounds] let @xmath312. then 1. [a1l] @xmath390 2. [a3u] @xmath391 3. [a5u] @xmath392 4. [a7u] @xmath393 we proceed in a similar way as in the proof of lemma [ajpositive]. let @xmath16 be as in the hypothesis. set @xmath394 then @xmath395 and so @xmath347 let @xmath396 then, @xmath397 hence @xmath398 set @xmath399 then @xmath400 let @xmath401 then, @xmath402 and hence @xmath403 set @xmath404 and let @xmath405 then, @xmath406, so @xmath407 set @xmath408 and @xmath409 then, @xmath406 and @xmath410, so @xmath411 the following result fixes the proof of the claim made in (*??? * claim 2) and improves the threshold of invertibility determined in (*??? * theorem 4.5). [fixingbbcdg] there exists @xmath412, such that @xmath413\pi^2}{2\sqrt{2}\left (\frac{\pi^2}{8}-1- \frac19-\frac{1}{25}-\frac{1}{49 } \right) } \qquad \forall p\in\left(p_3, \frac{6}{5}\right).\]] the family @xmath21 is a schauder basis of @xmath414 for all @xmath415 and @xmath20. both sides of are continuous functions of the parameter @xmath416. the right hand side is bounded. the left side is decreasing as @xmath16 increases and @xmath387 as @xmath388. by virtue of lemma [ajbounds], @xmath417 hence the first statement is ensured as a consequence of the intermediate value theorem. from, it follows that @xmath418 for all @xmath419. lemma [ajpositive] guarantees positivity of @xmath207 for @xmath420. then, by re - arranging this inequality, the second statement becomes a direct consequence of. a sharp numerical approximation of the solution of the equation with equality in gives @xmath421. see figure [impro_fig_p = q]. The thresholds for invertibility and the regions of improvement if sharp bounds on the first few fourier coefficients @xmath259 are at hand, the approach employed above for the proof of theorem [fixingbbcdg] can also be combined with the criteria or. a natural question is whether this would lead to a positive answer to the question of invertibility for @xmath43, whenever @xmath422 in the case of, we see below that this is indeed the case. the key statement is summarised as follows. [beyond2] let @xmath57 and @xmath423. suppose that 1. @xmath290, @xmath245 and @xmath424 for all other @xmath425. 2. @xmath426. if @xmath427 then @xmath43 is invertible. assume that the hypotheses are satisfied. the combination of and gives @xmath428 then @xmath429 and so the conclusion follows from. we now discuss the connection between the different statements established in the previous sections with those of the papers @xcite, @xcite and @xcite. for this purpose we consider various accurate approximations of @xmath207 and @xmath430. these approximations are based on the next explicit formulae : @xmath431 and @xmath432 here @xmath433 is the incomplete beta function, @xmath434 is the beta function and @xmath435 is the gamma function. moreover, by considering exactly the steps described in @xcite for the proof of (*??? * (4.15)), it follows that @xmath436{\mathrm{d}}x \\ & = \frac{\sqrt{2}}{\pi}\int_0 ^ 1 \log\left[\cot \left (\frac{\pi}{4 } \ { \mathcal{i}}\! \!\left(\frac1q,\frac{p-1}{p};x^q\right) \right)\right] { \mathrm{d}}x. \end{aligned}\]] . the positions of @xmath37, @xmath437 and the value of @xmath438 are set only for illustration purposes, as we are only certain that @xmath439. black indicates relevance to the general case @xmath20 while red indicates relevance for the case @xmath57. [impro_fig_p = q],height=241] let us begin with the case of equal indices. see figure [impro_fig_p = q]. as mentioned in the introduction, @xmath440 for @xmath41. the condition @xmath441 is fulfilled for all @xmath442 where @xmath443. the fourier coefficients @xmath444 for all @xmath445 whenever @xmath446. remarkably we need to get to @xmath447, for a numerical verification of the conditions of proposition [beyond2] allowing @xmath448. indeed we remark the following. 1. for @xmath449, the condition hold true only for @xmath450 where @xmath451. 2. for @xmath447 the condition does hold true for @xmath452 where @xmath68. this indicates that that the threshold for invertibility of @xmath43 in the hilbert space setting for @xmath14 is at least @xmath453. now we examine the general case. the graphs shown in figures [th10ab] and [th10c] correspond to regions in the @xmath44-plane near @xmath454. curves on figure [th10ab] that are in red are relevant only to the hilbert space setting @xmath57. black curves pertain to @xmath20. figure [th10ab]-_(a) _ and a blowup shown in figure [th10ab]-_(b) _, have two solid (black) lines. one that shows the limit of applicability of theorem [impro_implicit]-[aimplicit] and one that shows the limit of applicability of the result of @xcite. the dashed line indicates where occurs. to the left of that curve is not applicable. there are two filled regions of different colours in _ (a) _, which indicate where @xmath455 and where @xmath456 for @xmath208. proposition [beyond2] is not applicable in the union of these regions. we also show the lines where @xmath457 and @xmath458. the latter forms part of the boundary of this union. the solid red line corresponding to the limit of applicability of theorem [impro_implicit]-[bimplicit] is also included in figure [th10ab]-_(a)(d)_. to the right of that line, in the white area, we know that @xmath43 is invertible for @xmath57. the blowup in figure [th10ab]-_(b) _ clearly shows the gap between theorem [impro_implicit]-[aimplicit] and theorem [impro_implicit]-[bimplicit] in this @xmath57 setting. certainly @xmath459 is a point of intersection for all curves where @xmath29 for @xmath123. these curves are shown in figure [th10ab]-_(c) _ also for @xmath460 and @xmath461. in this figure, we also include the boundary of the region where @xmath455 and the region where @xmath456 now for @xmath462. note that the curves for @xmath463 and @xmath458 form part of the boundary of the latter. comparing _ (a) _ and _ (c) _, the new line that cuts the @xmath16 axis at @xmath464 corresponds to the limit of where proposition [beyond2] for @xmath465 is applicable (for @xmath16 to the right of this line). the gap between the two red lines (case @xmath57) indicates that proposition [beyond2] can significantly improve the threshold for basisness with respect to a direct application of theorem [impro_implicit]-[bimplicit]. as we increase @xmath466, the boundary of the corresponding region moves to the left, see the blowups in figure [th10ab]-_(d) _ and _ (e)_. the two further curves in red located very close to the vertical axis, correspond to the precise value of the parameter @xmath466 where proposition [beyond2] allows a proof of invertibility for the change of coordinates which includes the break made by. for @xmath467 the region does not include the dashed black line, for @xmath447 it does include this line. the region shown in blue indicates a possible place where corollary [main_1] may still apply, but further investigation in this respect is needed. figure [th10c] concerns the statement of theorem [impro_implicit]-[cimplicit]. the small wedge shown in green is the only place where the former is applicable. as it turns, it appears that the conditions of corollary [main_2] prevent it to be useful for determining invertibility of @xmath43 in a neighbourhood of @xmath454. however in the region shown in green, the upper bound on the riesz constant consequence of is sharper than that obtained from. The shape of @xmath468 as @xmath469 part of the difficulties for a proof of basisness for the family @xmath21 in the regime @xmath470 has to do the fact that the fourier coefficients of @xmath471 approach those of the function @xmath472. in this appendix we show that, indeed @xmath473 note that @xmath474 let @xmath475 be the (unique) value, such that @xmath476 then @xmath477 let @xmath478 be the line passing through the points @xmath479 and @xmath480. there exists a unique value @xmath481 such that @xmath482 this value is unique because of monotonicity of both sides of this equality, and it exists by bisection. as all the functions involved are continuous in @xmath16, then also @xmath483 is continuous in the parameter @xmath16. moreover, @xmath484 indeed, by clearing the equation defining @xmath483, we get @xmath485 the right hand side, and thus the left hand side, approach 0 as @xmath388. then, one (and hence both) of the two terms multiplying on the left should approach 0. let @xmath486 be the polygon which has as vertices (ordered clockwise) @xmath487 as @xmath488 @xmath489\times\{0\ }) \cup (\{1\}\times [0,1])$] in hausdorff distance. then the area of @xmath486 approaches 0 as @xmath388. moreover, @xmath486 covers the graph of @xmath490 for @xmath491. thus @xmath492 hence, there is a point @xmath493 on the graph of @xmath494 such that @xmath495 and @xmath496 the proof of is completed from the fact that, as @xmath494 is concave (because its inverse function is convex), the piecewise linear interpolant of @xmath494 for the family of nodes @xmath497 has a graph below that of @xmath494. Basic computer codes the following computer codes written in the open source languages octave and python can be used to verify any of the numerical estimations presented in this paper. function for computing @xmath207 with 10-digits precision..... # -- function file : [a, err, np]=apq(k, p, q) # a is the kth fourier coefficient of the p, q sine function # err is the residual # np number of quadrature points # function [a, err, np]=apq(k, p, q) if mod(k,2)==0, disp('error : k should be odd') ; return ; end [i, err, np]=quadcc(@(y) cos(k*pi*betainc(y.^q,1/q,(p-1)/p)/2),0,1,1e-10) ; a = i*2*sqrt(2)/k / pi ;.... function for computing @xmath498 with 10-digits precision..... # -- function file : [s, err, np]=apqsum(k, p, q) # s is the sum of the fourier coefficient of the p, q sine function # err is the residual # np number of quadrature points # function [s, err, np]=apqsum(p, q) [i, err, np]=quadcc(@(y) log(cot(pi*betainc(y.^q,1/q,(p-1)/p)/4)),0,1,1e-10) ; s = i*sqrt(2)/pi ;.... Python function for computing @xmath207 with variable precision..... def a(k, p, q) : " " " computes the kth fourier coefficient of the p, q sine function. returns coefficient and residual. > > > from sympy.mpmath import * > > > mp.dps = 25 ; mp.pretty = true > > > a(1,mpf(12)/11,mpf(12)/11) > > > (0.8877665848468607372062737, 1.0e-59) " " " if isint(fraction(k,2)) : apq=0 ; e=0 ; return apq, e f= lambda x : cos(k*pi*betainc(1/q,(p-1)/p,0,x**q, regularized = true)/2) ; (i, e)=quad(f,[0,1],error = true, maxdegree=10) ; apq = i*2*sqrt(2)/k / pi ; return apq, e.... function for computing @xmath498 with variable precision..... def suma(p, q) : " " " computes the sum of the fourier coefficient of the p, q sine function. returns sum and residual. > > > from sympy.mpmath import * > > > mp.dps = 25 ; mp.pretty = true > > > suma(mpf(12)/11,mpf(12)/11) > > > (1.48634943002852603038783, 1.0e-56) " " " f= lambda x : log(cot(pi*betainc(1/q,(p-1)/p,0,x**q, regularized = true)/4)) ; (i, e)=quad(f,[0,1],error = true, maxdegree=10) ; sumapq = i*sqrt(2)/pi ; return sumapq, e.... Acknowledgements the authors wish to express their gratitude to paul binding who suggested this problem a few years back. they are also kindly grateful with stefania marcantognini for her insightful comments during the preparation of this manuscript. we acknowledge support by the british engineering and physical sciences research council (ep / i00761x/1), the research support fund of the edinburgh mathematical society and the instituto venezolano de investigaciones cientficas. -plane where theorem [impro_implicit]-[aimplicit] and [bimplicit], as well as proposition [beyond2] (with different values of @xmath466) apply. in all graphs @xmath16 corresponds to the horizontal axis and @xmath260 to the vertical axis and the dotted line shows @xmath14. [th10ab],title="fig:",height=226] -plane where theorem [impro_implicit]-[aimplicit] and [bimplicit], as well as proposition [beyond2] (with different values of @xmath466) apply. in all graphs @xmath16 corresponds to the horizontal axis and @xmath260 to the vertical axis and the dotted line shows @xmath14. [th10ab],title="fig:",height=226] -plane where theorem [impro_implicit]-[aimplicit] and [bimplicit], as well as proposition [beyond2] (with different values of @xmath466) apply. in all graphs @xmath16 corresponds to the horizontal axis and @xmath260 to the vertical axis and the dotted line shows @xmath14. [th10ab],title="fig:",height=226] -plane where theorem [impro_implicit]-[aimplicit] and [bimplicit], as well as proposition [beyond2] (with different values of @xmath466) apply. in all graphs @xmath16 corresponds to the horizontal axis and @xmath260 to the vertical axis and the dotted line shows @xmath14. [th10ab],title="fig:",height=226] -plane where theorem [impro_implicit]-[aimplicit] and [bimplicit], as well as proposition [beyond2] (with different values of @xmath466) apply. in all graphs @xmath16 corresponds to the horizontal axis and @xmath260 to the vertical axis and the dotted line shows @xmath14. [th10ab],title="fig:",height=226] -plane where theorem [impro_implicit]-[aimplicit] and [bimplicit], as well as proposition [beyond2] (with different values of @xmath466) apply. in all graphs @xmath16 corresponds to the horizontal axis and @xmath260 to the vertical axis and the dotted line shows @xmath14. [th10ab],title="fig:",height=226] -plane where theorem [impro_implicit]-[cimplicit] applies. even when we know @xmath43 is invertible in this region as a consequence of theorem [impro_implicit]-[aimplicit], the upper bound on the riesz constant provided by improves upon that provided by (case @xmath57). in this graph @xmath16 corresponds to the horizontal axis and @xmath260 to the vertical axis and the dotted line shows @xmath14. [th10c],title="fig:",height=226] -plane where theorem [impro_implicit]-[cimplicit] applies. even when we know @xmath43 is invertible in this region as a consequence of theorem [impro_implicit]-[aimplicit], the upper bound on the riesz constant provided by improves upon that provided by (case @xmath57). in this graph @xmath16 corresponds to the horizontal axis and @xmath260 to the vertical axis and the dotted line shows @xmath14. [th10c],title="fig:",height=226]

we improve the currently known thresholds for basisness of the family of periodically dilated @xmath0-sine functions. our findings rely on a beurling decomposition of the corresponding change of coordinates in terms of shift operators of infinite multiplicity. we also determine refined bounds on the riesz constant associated to this family. these results seal mathematical gaps in the existing literature on the subject.

1405.7337

Introduction one surprising result that has come out of the more than 200 extrasolar planet discoveries to date is the wide range of eccentricities observed. unlike our own solar system, many of the extrasolar planets which are not tidally locked to their host stars have moderate eccentricities (@xmath1), and 15 planets have high eccentricities (@xmath0). these observations have spawned several theories as to the origin of highly eccentric extrasolar planets. one such method, planet - planet scattering, occurs when multiple jovian planets form several astronomical units (au) from the host star and then interact, leaving one in an eccentric orbit and often ejecting the other @xcite. this method has been proposed to explain the architecture of the @xmath2 and planetary system @xcite, which contains a hot jupiter as well as two jovian planets in moderately eccentric orbits. @xcite suggested a merger scenario in which inner protoplanets perturb each other and merge to form a single massive, eccentric planet with @xmath3 and @xmath4 au. interactions with stellar companions are another possible way to boost a planet s eccentricity. of the 15 stars hosting a planet with @xmath0, six are also known to possess stellar - mass companions in wide binary orbits : hd 3651 @xcite, hd 20782 @xcite, hd 80606, hd 89744 @xcite, 16 cyg b, and hd 222582 @xcite. if the inclination angle between the planetary orbit and a stellar companion is large, the kozai mechanism @xcite can induce large - amplitude oscillations in the eccentricity of the planet (e.g. malmberg et al. these oscillations can be damped by general relativistic effects and by interaction with other planets, and hence are most effective in systems with a single planet in an orbit @xmath51 au from the host star @xcite. the kozai mechanism has been suggested to explain the high eccentricity of 16 cyg bb @xcite and hd 80606b @xcite. @xcite found the inclination of 16 cyg b orbiting the system barycenter to lie between 100 and 160 degrees, where 90 degrees is an edge - on orientation. however, it is the difference in inclination between the orbital planes of the planetary and stellar companion that is critical in determining the importance of the kozai mechanism, and the inclination of the planet s orbit is generally not known for non - transiting systems. of the 192 known planetary systems, 23 (12%) are multi - planet systems. recent discoveries of additional objects in systems known to host at least one planet @xcite suggest that multiple - planet systems are common. of particular interest are systems which host a jovian planet and a low - mass `` hot neptune, '' e.g. 55 cnc (= hd 75732), gj 876, @xmath6 arae (= hd 160691), gl 777a (= hd 190360). motivated by the discoveries of hot neptunes in known planetary systems, we have undertaken an intensive survey of selected single - planet systems to search for additional low - mass companions. three of the planetary systems discussed in this paper (hd 3651, hd 80606, hd 89744) are part of this campaign. the excellent radial - velocity precision of the high resolution spectrograph on the hobby - eberly telescope (het), combined with queue - scheduling, allow us to time the observations in such a way as to minimize phase gaps in the orbit of the known planet, and also to act quickly on potential new planet candidates. the use of the het in this manner is discussed further in @xcite with regard to the discovery of hd 37605b. in this work, we aim to combine observational limits on additional planets in known planetary systems with dynamical constraints obtained by n - body simulations. the observations address the question : what additional planets are (or are not) in these systems? the dynamical simulations can answer the question : where are additional planets possible? section 2 describes the observations and the test particle simulations for six highly eccentric planetary systems : hd 3651, hd 37605, hd 45350, hd 80606, hd 89744, and 16 cyg b. we have chosen these systems based on two criteria : (1) each hosts a planet with @xmath0, and (2) each has been observed by the planet search programs at mcdonald observatory. in 3, we present and discuss the results of the updated orbital fits, dynamical simulations, and detection limit computations. Observations and data analysis five of the six stars considered in this work have been observed with the mcdonald observatory 9.2 m hobby - eberly telescope (het) using its high resolution spectrograph (hrs) @xcite. a full description of the het planet search program is given in @xcite. for 16 cyg b, observations from mcdonald observatory were obtained only with the 2.7 m harlan j. smith (hjs) telescope ; the long - term planet search program on this telescope is described in @xcite. all available published data on these systems were combined with our data from mcdonald observatory in the orbit fitting procedures. to place constraints on the architecture of planetary systems, we would like to know where additional objects can remain in stable orbits in the presence of the known planet(s). we performed test particle simulations using swifthal / swift.html.] @xcite to investigate the dynamical possibility of additional low - mass planets in each of the six systems considered here. low - mass planets can be treated as test particles since the exchange of angular momentum with jovian planets is small. we chose the regularized mixed - variable symplectic integrator (rmvs3) version of swift for its ability to handle close approaches between massless, non - interacting test particles and planets. particles are removed if they are (1) closer than 1 hill radius to the planet, (2) closer than 0.05 au to the star, or (3) farther than 10 au from the star. since the purpose of these simulations is to determine the regions in which additional planets could remain in stable orbits, we set this outer boundary because the current repository of radial - velocity data can not detect objects at such distances. the test particle simulations were set up following the methods used in @xcite, with the exception that only initially circular orbits are considered in this work. for each planetary system, test particles were placed in initially circular orbits spaced every 0.002 au in the region between 0.05 - 2.0 au. we have chosen to focus on this region because the duration of our high - precision het data is currently only 2 - 4 years for the objects in this study. the test particles were coplanar with the existing planet, which had the effect of confining the simulation to two dimensions. input physical parameters for the known planet in each system were obtained from our keplerian orbit fits described in 3.1, and from recent literature for 16 cyg b @xcite and hd 45350 @xcite. the planetary masses were taken to be their minimum values (sin @xmath7). the systems were integrated for @xmath8 yr, following @xcite and allowing completion of the computations in a reasonable time. we observed that nearly all of the test - particle removals occurred within the first @xmath9 yr ; after this time, the simulations had essentially stabilized to their final configurations. Results and discussion we present updated keplerian orbital solutions for hd 3651b, hd 37605b, hd 80606b, and hd 89744b in table 1. a summary of the data used in our analysis is given in table 2, and the het radial velocities are given in tables 3 - 6. the velocity uncertainties given for the het data represent internal errors only, and do not include any external sources of error such as stellar `` jitter. '' the parameters for the remaining two planets, hd 45350b and 16 cyg bb, are taken from @xcite and @xcite, respectively. radial velocity measurements from the het are given for hd 45350 in @xcite, and velocities for 16 cyg b from the hjs telescope are given in @xcite. as in @xcite, all available published data were combined with those from mcdonald, and the known planet in each system was fit with a keplerian orbit using gaussfit @xcite, allowing the velocity offset between each data set to be a free parameter. examination of the residuals to our keplerian orbit fits revealed no evidence for additional objects in any of the six systems in this study. the saturn - mass (m sin @xmath10) planet hd 3651b was discovered by @xcite using observations from lick and keck. we fit these data, which were updated in @xcite, in combination with observations from the hjs and het at mcdonald observatory. the het data, which consist of multiple exposures per visit, were binned using the inverse - variance weighted mean value of the velocities in each visit. the standard error of the mean was added in quadrature to the weighted rms about the mean velocity to generate the error bar of each binned point (n=29). the rms about the combined fit for each dataset is : lick & keck6.6, het9.4, hjs12.2. the fitted orbital parameters for hd 3651b are of comparable precision to those reported in @xcite, and agree within 2@xmath11. the recent discovery of a t dwarf companion to hd 3651 @xcite prompts an interesting exercise : can the radial - velocity trend due to this object be detected in the residuals after removing the planet? we detect a slope of @xmath12 yr@xmath13, indicating that we are indeed able to discern a trend which is possibly due to the binary companion. however, the reduced @xmath14 of the orbital solution is not significantly improved by the inclusion of a linear trend (@xmath15=0.18). the parameters given in table 1 were obtained from the fit which did not include a trend. we present 23 new het observations for hd 37605 obtained since its announcement by @xcite. the data now span a total of 1065 days. the best fit is obtained by including an acceleration of @xmath16 yr@xmath13, indicating a distant orbiting body. such a finding would lend support to the hypothesis that very eccentric single - planet systems originate by interactions within a wide binary system. the shortest period that this outer companion could have and still remain consistent with the observed acceleration and its uncertainty over the timespan of the observations is about 40 yr, assuming a circular orbit. this object would then have a minimum mass in the brown dwarf range. the planet orbiting hd 80606, first announced by @xcite, is the most eccentric extrasolar planet known, with @xmath17 (table 1). we have fit the coralie data in combination with the keck data given in @xcite and 23 observations from het. the extreme velocity variations induced by this planet greatly increase the sensitivity of orbit fits to the weighting of individual measurements. since the uncertainties of the het velocities given in tables 3 - 6 represent internal errors only, we experimented with adding 1 - 7 of radial - velocity `` jitter '' in quadrature before fitting the data for hd 80606. for all of these jitter values, the fitted parameters remained the same within their uncertainties. table 1 gives the parameters derived from a fit which added 3.5 of jitter @xcite to the het data. the rms about the combined fit is : coralie18.7, het7.5, keck5.6. @xcite noted that the eccentricity @xmath18 and the argument of periastron @xmath19 had to be held fixed in their fit to the keck data alone. however, the large number of measurements included in this work allowed gaussfit to converge with all parameters free. for hd 89744b, we combine data from the het with 6 measurements from the hjs telescope and lick data from @xcite. the het data were binned in the same manner as for hd 3651, resulting in n=33 independent visits. the rms about the combined fit for each dataset is : lick17.1, het10.7, hjs9.5. as with hd 3651b, our derived parameters agree with those of @xcite within 2@xmath11. the scatter about our fit remains large, most likely due to the star s early spectral type (f7v), which hinders precision radial - velocity measurements due to the smaller number of spectral lines. for example, the f7v star hd 221287 was recently found to host a planet @xcite ; despite the superb instrumental precision of the harps spectrograph, that orbital solution has a residual rms of 8.5. the results of the dynamical simulations are shown in figures 1 - 3. the survival time of the test particles is plotted against their initial semimajor axis. as shown in figure 1, the short - period planets hd 3651 and hd 37605 sweep clean the region inside of about 0.5 au. in both of these systems, however, a small number of test particles remained in low - eccentricity orbits near the known planet s apastron distance, near the 1:2 mean - motion resonance (mmr). in the hd 3651 system, particles remained stable beyond about 0.6 au, which is not surprising given the low mass of the planet. for hd 37605, two distinct strips of stability are seen in fig. 1, corresponding to the 1:2 and 1:3 mmrs. the eccentricity of the test particles in the region of the 1:2 mmr oscillated between 0.00 and 0.06. particles in 1:3 mmr oscillated in eccentricity with a larger range, up to @xmath20, which is expected due to secular forcing. as with hd 3651, the region beyond about 0.8 au was essentially unaffected by the planet. figure 2 shows the results for the hd 45350 and hd 80606 systems. the long period (963.6 days) and relatively large mass (m sin @xmath21=1.8) of hd 45350b restricted stable orbits to the innermost 0.2 au. these test particles oscillated in eccentricity up to @xmath22. the 4 planet orbiting hd 80606 removed all test particles to a distance of about 1.5 au, and only beyond 1.75 au did test particles remain in stable orbits for the duration of the simulation (@xmath8 yr). a region of instability is evident at 1.9 au due to the 8:1 mmr. figure 3 shows that hd 89744b eliminated all test particles except for a narrow region near the 8:3 resonance. for the 16 cyg b system, only particles inside of about 0.3 au remained stable, leaving open the possibility of short - period planets. the surviving particles oscillated in eccentricity up to @xmath23, but these simulations treat the star as a point mass, and hence tidal damping of the eccentricity is not included. our results are consistent with those of @xcite, who investigated dynamical stability in extrasolar planetary systems and found that no test particles survived in the habitable zones of the hd 80606, hd 89744, and 16 cyg b systems. three of these systems (hd 3651, hd 80606, hd 89744) were monitored intensely with the het as part of a larger effort to search for low - mass, short period planets. no evidence was found for any such objects in these or any of the six systems in this work. we then asked what limits can be set on additional planets using the high - precision het data we have obtained. the procedure for determining companion limits was identical to the method described in @xcite, except that in this work, the best - fit keplerian orbit for the known planet (see 3.1) was removed before performing the limits computations. in this way, we determined the radial - velocity amplitude @xmath24 for which 99% of planets would have been detected in the residuals. the eccentricity of the injected test signals was chosen to be the mean eccentricity of the surviving particles from the simulations described in 3.2. only the regions in which test particles survived were considered in these limits computations. the results of these computations were highly varied, reflecting the differing observing strategies employed for these six objects. in particular, hd 3651, hd 80606, and hd 89744 were monitored intensely with the het as part of a search for short - period objects, whereas hd 37605 and hd 45350 were only observed sporadically after the known planet orbits were defined and published @xcite, and 16 cyg b has only been observed with the hjs telescope at a frequency of at most once per month. the companion limits are shown in figures 4 - 6 ; planets with masses above the solid line can be ruled out by the data with 99% confidence. not surprisingly, the tightest limits were obtained for hd 3651 (figure 4), which had a total of 195 measurements, including 29 independent het visits. for periods less than about 1 year, we can exclude planets with m sin @xmath21 2 neptune masses. similar results were obtained for 16 cyg b (n=161), where the limits approach a neptune mass (figure 6). since the detection limits generally improve with the addition of more data and with higher - quality data, we can define a quantity to measure the goodness of the limits. a simple choice would be @xmath25, where @xmath26 is the total number of observations, and @xmath27 is the mean uncertainty of the radial - velocity measurements. the values of @xmath26 and @xmath27 are given in table 2. in the hd 45350 system, the results of the dynamical simulations complement those of the detection limit determinations. very tight limits are obtained in close orbits (@xmath280.2 au). in this region, test particles were stable (fig. 2) and our observations can exclude planets with m sin _ i _ between about 1 and 4 neptune masses. similar results were obtained for the 16 cyg b system, in which test particles remained stable inward of @xmath29 0.3 au. in that region, planets of 1 - 3 neptune masses can be excluded by our limits determinations (fig. 6). in most of the limits determinations, there are multiple `` blind spots '' evident where the periodogram method failed to significantly recover the injected signals. typically this occurs at certain trial periods for which the phase coverage of the observational data is poor, and often at the 1-month and 1-year windows. for none of hd 37605 (fig. 4), hd 80606 (fig. 5), or hd 89744 (fig. 6) could additional companions be ruled out below about 0.7, and for most orbital periods tested, the limits were substantially worse. one possible explanation for this result is that the sampling of the observations was poorly distributed in phase for many of the injected test signals, making significant recovery by the periodogram method difficult. this is evidenced by the `` jagged '' regions in the plots. also, the intrinsic scatter for those three systems was too large to permit tight limits determination. this is certainly reasonable for the f7 star hd 89744. the three systems with the best limits (hd 3651, hd 45350, and 16 cyg b) also had the lowest rms scatter about their orbital solutions (mean=@xmath30 ; table 1). in contrast, the mean rms for the remaining three systems was @xmath31. additional factors such as a paucity of data (hd 37605) and short time baselines (hd 80606, hd 89744) made the determination of useful companion limits challenging for some of the planetary systems in this study. Summary we have shown that for a sample of six highly eccentric extrasolar planetary systems, there is no evidence for additional planets. test particle simulations show that there are regions detectable by current surveys (i.e. for @xmath32 au) where additional objects can exist. for hd 3651 and hd 37605, we find that protected resonances are also present. combining these simulations with detection limits computed using new high - precision het data combined with all available published data is particularly effective for the hd 3651 and hd 45350 systems. additional short - period planets can be ruled out down to a few neptune masses in the dynamically stable regions in these systems. this material is based upon work supported by the national aeronautics and space administration under grant nos. nng04g141 g and nng05g107 g issued through the terrestrial planet finder foundation science program. we are grateful to the het tac for their generous allocation of telescope time for this project. we also would like to thank barbara mcarthur for her assistance with gaussfit software. we thank the referee, greg laughlin, for his careful review of this manuscript. this research has made use of nasa s astrophysics data system (ads), and the simbad database, operated at cds, strasbourg, france. the hobby - eberly telescope (het) is a joint project of the university of texas at austin, the pennsylvania state university, stanford university, ludwig - maximilians - universit " at mnchen, and georg - august - universit " at g " ottingen the het is named in honor of its principal benefactors, william p. hobby and robert e. eberly. lllllllll hd 3651 b & 62.197@xmath330.012 & 53932.2@xmath330.4 & 0.630@xmath330.046 & 250.7@xmath336.3 & 15.6@xmath331.1 & 0.20@xmath330.01 & 0.280@xmath330.006 & 7.1 + hd 37605 b & 55.027@xmath330.009 & 52992.8@xmath330.1 & 0.677@xmath330.009 & 218.4@xmath331.7 & 201.5@xmath333.9 & 2.39@xmath330.12 & 0.263@xmath330.006 & 13.0 + hd 45350 b & 963.6@xmath333.4 & 51825.3@xmath337.1 & 0.778@xmath330.009 & 343.4@xmath332.3 & 58.0@xmath331.7 & 1.79@xmath330.14 & 1.92@xmath330.07 & 9.1 + hd 80606 b & 111.428@xmath330.002 & 53421.928@xmath330.004 & 0.933@xmath330.001 & 300.4@xmath330.3 & 470.2@xmath332.5 & 4.10@xmath330.12 & 0.460@xmath330.007 & 13.5 + hd 89744 b & 256.78@xmath330.05 & 53816.1@xmath330.3 & 0.689@xmath330.006 & 194.1@xmath330.6 & 263.2@xmath333.9 & 7.92@xmath330.23 & 0.91@xmath330.01 & 14.4 + 16 cyg b b & 799.5@xmath330.6 & 50539.3@xmath331.6 & 0.689@xmath330.011 & 83.4@xmath332.1 & 51.2@xmath331.1 & 1.68@xmath330.07 & 1.68@xmath330.03 & 10.6 + lllll hd 3651 & 163 & 3.4 & & @xcite + hd 3651 & 3 & 6.1 & & hjs + hd 3651 & 29 & 2.1 & & het + hd 3651 (total) & 195 & 3.2 & 7083 & + hd 37605 (total) & 43 & 2.9 & 1065 & het + hd 45350 & 38 & 2.8 & & @xcite + hd 45350 & 28 & 4.2 & & het + hd 45350 & 47 & 8.9 & & hjs + hd 45350 (total) & 113 & 5.7 & 2265 & + hd 80606 & 61 & 13.7 & & @xcite + hd 80606 & 46 & 5.1 & & @xcite + hd 80606 & 23 & 2.5 & & het + hd 80606 (total) & 130 & 8.7 & 2893 & + hd 89744 & 50 & 11.2 & & @xcite + hd 89744 & 33 & 3.2 & & het + hd 89744 & 6 & 9.4 & & hjs + hd 89744 (total) & 89 & 8.1 & 2687 & + 16 cyg b & 95 & 6.3 & & @xcite + 16 cyg b & 29 & 19.7 & & hjs phase ii + 16 cyg b & 37 & 7.4 & & hjs phase iii + 16 cyg b (total) & 161 & 9.0 & 6950 & + lrr [tbl-3] 53581.87326 & -19.1 & 2.9 + 53581.87586 & -19.4 & 2.7 + 53581.87846 & -20.7 & 2.7 + 53600.79669 & -11.5 & 2.4 + 53600.79860 & -15.5 & 3.0 + 53600.80050 & -22.8 & 2.9 + 53604.79166 & -15.8 & 1.9 + 53604.79356 & -18.8 & 2.1 + 53604.79548 & -21.3 & 2.1 + 53606.78169 & -19.3 & 1.8 + 53606.78360 & -14.8 & 2.1 + 53606.78551 & -24.0 & 1.8 + 53608.77236 & -18.8 & 1.9 + 53608.77426 & -18.0 & 1.9 + 53608.77617 & -18.8 & 1.8 + 53615.96280 & -28.0 & 2.6 + 53615.96471 & -31.9 & 2.4 + 53615.96662 & -37.8 & 2.5 + 53628.74050 & -6.8 & 2.2 + 53628.74240 & -14.5 & 2.4 + 53628.74431 & -5.5 & 2.2 + 53669.61012 & -18.2 & 2.1 + 53669.61203 & -19.2 & 2.2 + 53669.61394 & -17.7 & 2.4 + 53678.78954 & -10.6 & 2.4 + 53678.79141 & -8.6 & 2.3 + 53678.79332 & -2.3 & 2.1 + 53682.78423 & -15.4 & 2.2 + 53682.78609 & -15.0 & 2.3 + 53682.78801 & -11.9 & 2.3 + 53687.77684 & 11.3 & 2.2 + 53687.77875 & 8.7 & 2.2 + 53687.78066 & 15.9 & 2.2 + 53691.75967 & 9.6 & 2.2 + 53691.76158 & 20.3 & 2.1 + 53691.76349 & 15.9 & 2.0 + 53696.75837 & 16.1 & 1.8 + 53696.76028 & 18.6 & 1.8 + 53696.76220 & 20.0 & 2.0 + 53694.75275 & 18.0 & 1.9 + 53694.75466 & 15.1 & 2.0 + 53694.75656 & 17.8 & 2.0 + 53955.83401 & -0.5 & 1.9 + 53955.83593 & -1.2 & 2.0 + 53955.83785 & 1.3 & 1.9 + 53956.82850 & 0.4 & 2.0 + 53956.83046 & -1.0 & 2.0 + 53956.83236 & -5.4 & 2.2 + 53957.82201 & -2.1 & 2.0 + 53957.82392 & -1.3 & 2.0 + 53957.82583 & -3.6 & 2.0 + 53973.80721 & 9.8 & 7.3 + 53973.81020 & 3.5 & 2.3 + 53973.81200 & -3.5 & 2.0 + 53976.78393 & -10.4 & 2.4 + 53976.78586 & -5.4 & 2.1 + 53976.78778 & -6.7 & 2.3 + 53978.97197 & -3.8 & 2.6 + 53985.95886 & -9.0 & 2.3 + 53985.96079 & 4.3 & 3.3 + 53987.95335 & -8.3 & 2.2 + 53987.95527 & -8.0 & 2.2 + 53987.95719 & -12.0 & 2.3 + 53989.73817 & -13.2 & 2.2 + 53989.74009 & -13.2 & 2.1 + 53989.74203 & -18.6 & 2.1 + 54003.70719 & 2.0 & 2.2 + 54003.70915 & 4.7 & 2.4 + 54005.68297 & 7.0 & 2.5 + 54005.68488 & 11.1 & 2.0 + 54005.68690 & 10.2 & 2.1 + 54056.77919 & -7.5 & 2.2 + 54056.78110 & -11.5 & 2.1 + 54056.78302 & -9.6 & 2.3 + 54062.55119 & 20.1 & 1.8 + 54062.55312 & 21.9 & 2.0 + 54062.55505 & 20.9 & 2.0 + 54064.54710 & 12.8 & 2.0 + 54064.54902 & 16.7 & 2.1 + 54064.55094 & 16.6 & 2.1 + 54130.55316 & 19.1 & 2.4 + 54130.55508 & 16.9 & 2.5 + 54130.55701 & 17.6 & 2.5 + lrr [tbl-4] 53002.67151 & 487.6 & 3.8 + 53003.68525 & 495.5 & 3.0 + 53006.66205 & 496.2 & 3.0 + 53008.66407 & 501.3 & 2.9 + 53010.80477 & 499.8 & 2.9 + 53013.79399 & 482.1 & 2.6 + 53042.72797 & 269.7 & 2.8 + 53061.66756 & 489.0 & 2.6 + 53065.64684 & 479.0 & 2.8 + 53071.64383 & 463.8 & 2.6 + 53073.63819 & 460.4 & 2.6 + 53082.62372 & 422.8 & 2.5 + 53083.59536 & 422.2 & 2.8 + 53088.59378 & 418.6 & 4.0 + 53089.59576 & 379.1 & 2.2 + 53092.59799 & 343.7 & 2.5 + 53094.58658 & 323.2 & 2.4 + 53095.58642 & 302.1 & 2.4 + 53096.58744 & 302.1 & 3.2 + 53098.57625 & 193.8 & 2.7 + 53264.95137 & 164.9 & 3.0 + 53265.94744 & 112.9 & 3.0 + 53266.94598 & 113.2 & 3.7 + 53266.95948 & 74.6 & 3.6 + 53266.97396 & 119.2 & 8.0 + 53283.92241 & 471.6 & 2.7 + 53318.81927 & 213.3 & 3.0 + 53335.92181 & 496.9 & 2.6 + 53338.90602 & 493.9 & 2.6 + 53377.81941 & 109.1 & 2.7 + 53378.81189 & 214.6 & 2.7 + 53379.80225 & 338.3 & 2.6 + 53381.64429 & 436.1 & 2.7 + 53384.64654 & 482.9 & 2.8 + 53724.85584 & 468.2 & 2.6 + 53731.69723 & 435.4 & 2.7 + 53738.67472 & 404.3 & 2.6 + 53743.81020 & 400.5 & 2.6 + 53748.64724 & 348.4 & 2.7 + 54039.85015 & 272.5 & 3.1 + 54054.96457 & 437.4 & 2.7 + 54055.95279 & 422.0 & 2.9 + 54067.76282 & 376.4 & 2.6 + lrr [tbl-5] 53346.88103 & -20.8 & 3.0 + 53358.02089 & -49.5 & 2.7 + 53359.82400 & -60.4 & 3.0 + 53361.02985 & -64.7 & 2.5 + 53365.03079 & -77.4 & 2.4 + 53373.98282 & -88.4 & 3.0 + 53377.80112 & -105.5 & 2.4 + 53379.75230 & -109.3 & 2.7 + 53389.74170 & -115.3 & 2.5 + 53391.74400 & -129.4 & 2.4 + 53395.72763 & -146.4 & 2.3 + 53399.72518 & -158.4 & 2.5 + 53401.72497 & -174.7 & 2.7 + 53414.67819 & -219.8 & 3.0 + 53421.85529 & 261.0 & 2.2 + 53423.86650 & 322.1 & 2.0 + 53424.85231 & 245.9 & 2.1 + 53432.87120 & 87.5 & 1.9 + 53433.60628 & 70.0 & 2.1 + 53446.79322 & 4.5 & 1.9 + 54161.85400 & -109.5 & 2.8 + 54166.83797 & -119.3 & 2.4 + 54186.76189 & -184.2 & 2.3 + lrr [tbl-6] 53709.89685 & -184.5 & 2.3 + 53723.85188 & -238.6 & 2.2 + 53723.85367 & -238.2 & 2.5 + 53723.85546 & -227.7 & 2.3 + 53727.84394 & -238.9 & 2.5 + 53727.84573 & -244.9 & 2.4 + 53727.84752 & -242.9 & 2.6 + 53736.81887 & -257.6 & 2.5 + 53736.82100 & -248.2 & 2.9 + 53736.82315 & -253.4 & 2.4 + 53738.03261 & -246.7 & 2.8 + 53738.03441 & -243.3 & 2.4 + 53738.03620 & -236.0 & 2.5 + 53738.80860 & -240.5 & 2.6 + 53738.81040 & -258.9 & 2.4 + 53738.81219 & -249.3 & 2.5 + 53734.81795 & -242.8 & 2.6 + 53734.81973 & -243.9 & 2.8 + 53734.82152 & -248.5 & 2.4 + 53742.79119 & -252.0 & 2.8 + 53742.79299 & -257.2 & 2.8 + 53742.79479 & -239.7 & 2.8 + 53751.78199 & -257.4 & 2.9 + 53751.78378 & -263.1 & 2.5 + 53751.78558 & -268.0 & 2.3 + 53753.78155 & -273.1 & 2.5 + 53753.78381 & -278.7 & 2.5 + 53753.78607 & -266.4 & 2.4 + 53755.76038 & -286.6 & 2.3 + 53755.76218 & -266.5 & 2.6 + 53755.76397 & -274.9 & 2.7 + 53746.81506 & -257.1 & 1.9 + 53746.81778 & -250.9 & 2.1 + 53746.82051 & -245.2 & 2.3 + 53757.77002 & -277.6 & 2.4 + 53757.77181 & -280.3 & 2.4 + 53757.77360 & -288.7 & 2.2 + 53797.64609 & -439.8 & 3.1 + 53797.64834 & -462.6 & 2.8 + 53797.65059 & -452.5 & 2.9 + 53809.62428 & -658.6 & 2.4 + 53809.62700 & -658.8 & 2.5 + 53809.62972 & -659.2 & 2.3 + 53837.76359 & -304.3 & 3.0 + 53837.76670 & -324.0 & 2.9 + 53837.78731 & -308.6 & 2.7 + 53837.79077 & -285.2 & 2.6 + 53866.69987 & -215.9 & 1.7 + 53866.70329 & -228.3 & 1.7 + 53866.70670 & -220.4 & 1.8 + 53868.68349 & -251.6 & 3.8 + 53868.68562 & -208.6 & 2.9 + 53868.68777 & -247.4 & 9.7 + 53875.66956 & -215.7 & 1.6 + 53883.65565 & -213.8 & 1.8 + 53883.65837 & -209.2 & 1.7 + 53883.66109 & -200.4 & 1.7 + 53890.63776 & -203.4 & 1.7 + 53890.63954 & -202.6 & 1.9 + 53890.64134 & -203.2 & 1.9 + 53893.62959 & -193.8 & 2.0 + 53893.63139 & -189.3 & 1.9 + 53893.63318 & -189.7 & 1.8 + 54047.94811 & -375.2 & 4.8 + 54047.94991 & -353.2 & 4.5 + 54047.95172 & -362.6 & 4.4 + 54050.96248 & -415.0 & 2.6 + 54050.96453 & -423.0 & 2.5 + 54050.96657 & -420.1 & 2.4 + 54052.96488 & -426.8 & 2.3 + 54052.96762 & -437.1 & 2.5 + 54052.97035 & -447.6 & 2.5 + 54056.94606 & -468.0 & 3.0 + 54056.94786 & -466.4 & 2.6 + 54056.94964 & -479.4 & 2.8 + 54063.92981 & -599.1 & 2.1 + 54063.93166 & -594.8 & 2.3 + 54063.93348 & -592.3 & 2.4 + 54073.91213 & -685.8 & 2.8 + 54073.91476 & -688.7 & 2.9 + 54073.91739 & -704.4 & 2.7 + 54122.01039 & -220.8 & 2.5 + 54122.01243 & -219.1 & 2.6 + 54122.01447 & -218.4 & 2.8 + 54129.74214 & -215.7 & 2.6 + 54129.74491 & -224.4 & 3.0 + 54129.74768 & -223.7 & 3.1 + 54160.65850 & -189.5 & 3.2 + 54160.66031 & -181.8 & 2.7 + 54160.66212 & -204.8 & 3.2 + 54163.66458 & -213.9 & 3.1 + 54163.66643 & -200.8 & 2.9 + 54163.66828 & -208.0 & 3.2 + 54165.88148 & -208.5 & 2.7 +

long time coverage and high radial velocity precision have allowed for the discovery of additional objects in known planetary systems. many of the extrasolar planets detected have highly eccentric orbits, which raises the question of how likely those systems are to host additional planets. we investigate six systems which contain a very eccentric (@xmath0) planet : hd 3651, hd 37605, hd 45350, hd 80606, hd 89744, and 16 cyg b. we present updated radial - velocity observations and orbital solutions, search for additional planets, and perform test particle simulations to find regions of dynamical stability. the dynamical simulations show that short - period planets could exist in the hd 45350 and 16 cyg b systems, and we use the observational data to set tight detection limits, which rule out additional planets down to a few neptune masses in the hd 3651, hd 45350, and 16 cyg b systems.

0706.1962

Introduction studies of charm decays are pursued for several different reasons. first of all, there is the possibility of directly observing new physics beyond the standard model (sm), since the effects of cp violation due to sm processes is highly suppressed allowing new physics contributions to be more easily seen than in @xmath8 decays where the sm processes typically have large effects @xcite. @xmath1 mixing also is interesting because it could come from either sm or new physics (np) processes, and could teach us interesting lessons. another important reason for detailed charm studies is that most @xmath8 s, @xmath999%, decay into charm, so knowledge about charm decays is particularly useful for @xmath8 decay studies. especially interesting are absolute branching ratios, resonant substructures in multi - body decays, phases on dalitz plots, etc.. other heavier objects such as top quarks decay into @xmath8 quarks and higgs particles may decay with large rates to @xmath10, again making charm studies important. furthermore, charm can teach us a great deal about strong interactions, especially decay constants and final state interactions. Experimental techniques charm has been studied at @xmath11 colliders at threshold, first by the mark iii collaboration and more recently by bes and cleo - c, at higher @xmath11 energies, and at fixed target and hadron collider experiments @xcite. the detection techniques are rather different at threshold than in other experiments. the @xmath12 resonance decays into @xmath13 ; the world average cross - section is 3.72@xmath140.09 nb for @xmath15 production and 2.82@xmath140.09 nb for @xmath16 production @xcite. @xmath5 production is studied at 4170 mev, where the cross - section for @xmath17+@xmath18 is @xmath91 nb @xcite. the underlying light quark continuum " background is about 14 nb. the relatively large cross - sections, relatively large branching ratios and sufficient luminosities, allow experiments to fully reconstruct one @xmath19 as a tag. " since the charge and flavor of the tag is then uniquely determined, the rest of the event can be examined for characteristics of the other known " particle. to measure absolute branching ratios, for example at the @xmath12, the rest of the event is fully reconstructed, as well as the tag. at the @xmath12 @xmath19 meson final states are reconstructed by first evaluating the difference in the energy, @xmath20, of the decay products with the beam energy. candidates with @xmath20 consistent with zero are selected and then the @xmath19 beam - constrained mass is evaluated, @xmath21 where @xmath22 runs over all the final state particles. examples of single and double reconstruction are presented in fig. [cleo - double](a) that shows the @xmath23 distribution for a @xmath24 or @xmath25 final states. these single tags " show a large signal and a very small background. [cleo - double](b) shows a double " tag sample where both @xmath3 and @xmath26 candidates in the same event are reconstructed. distributions for candidates from either @xmath24 or @xmath25 modes. (b) the @xmath27 distribution for candidates for candidates from @xmath24 and @xmath25 modes. the solid curves are a fits to the signals plus the backgrounds, that are indicated by the dashed shapes. the signals are asymmetric due to radiation of the electron beams.,title="fig:",width=302] distributions for candidates from either @xmath24 or @xmath25 modes. (b) the @xmath27 distribution for candidates for candidates from @xmath24 and @xmath25 modes. the solid curves are a fits to the signals plus the backgrounds, that are indicated by the dashed shapes. the signals are asymmetric due to radiation of the electron beams.,title="fig:",width=294] other experiments make use of the both the approximately picosecond lifetimes of charm to identify detached vertices, and the decay @xmath28, which also serves as a flavor tag in the case of @xmath29 transitions. Absolute charm meson branching ratios and other hadronic decays in charm meson decays, usually a single branching ratio sets the scale for determinations of most other rates, that are measured relative to it. for @xmath2 and @xmath3 these modes are @xmath30 and @xmath31, respectively. cleo - c, on the other hand uses a different technique where the branching ratios of several modes are determined simultaneously and all absolutely. consider an ensemble of modes @xmath22, that are both singly reconstructed and also doubly reconstructed, where all combinations of modes may be used. i denote the number of observed single tag charmed particles as @xmath32, anti - charmed particles as @xmath33, and double tags as @xmath34. they are related to the number of @xmath13 events (either charged or neutral) through their branching ratios @xmath35 as @xmath36 where @xmath37 and @xmath38 are the reconstruction efficiencies in single and double tag events for each mode. (in practice the differences in each mode between single and double tag events are small, and @xmath39.) solving these equations we find @xmath40 cleo - c has recently updated their absolute branching ratio measurements using a 281 pb@xmath41 data sample, an approximately 5 times larger data sample than used by them for their previous publication @xcite. the new preliminary results are shown in table [tab : dbr] @xcite. (in this table when two errors follow a number, the first error is statistical and the second systematic ; this will be true for all results quoted in this paper unless specifically indicated.) the absolute branching fractions for charm mesons have been measured with unprecedented accuracy. combining the pdg values with the preliminary cleo - c results for @xmath2 and @xmath3 decays, and using the cleo - c results for @xmath5, i find @xmath42 cleo - c does not quote a branching ratio for @xmath43 mode because of interferences on the dalitz plot. the @xmath44 or the @xmath45 modes should be used for normalization. since most of the @xmath46 decay modes have been measured as ratios to the @xmath47 mode, i extract an effective branching ratio @xmath48 these rates can be used for many purposes. for example, adding up the number of charm quarks produced in each @xmath49 meson decay at the @xmath50 resonance by summing the @xmath2, @xmath3, @xmath5, charmed baryon and twice the charmonium yields gives a rate of 1.09@xmath140.04, where the largest error comes from the @xmath2 yield. there is no definitive evidence for @xmath52 mixing. the best limits yet are @xmath53% and @xmath54 both at 95% c. l. the limit on @xmath55 of about 8% is just beginning to probe an interesting range. there are two hints that mixing may be soon found. belle finds consistency with no mixing at 3.9% c. l. in wrong - sign @xmath30 decays and babar finds consistency with no mixing at 4.5% c. l. in wrong sign @xmath56 decays, thus making further searches more interesting. there have not been any observations of cp or t violation. this work was supported by the national science foundation under grant # 0553004. i thank m. artuso, d. asner, r. faccini, s. malvezzi, n. menaa, p. onysi, r. sia and s. stroiney for interesting discussions and providing data and plots used in this review. m. artuso, charm decays within the standard model and beyond, " in _ proc. of the xxii int. symp. on lepton & photon interactions at high energies _, ed. r. brenner, c. p. de los heros, and j. rathsman, world scientific, singapore (2006) [hep - ex/0510052. s. malvezzi, @xmath19-meson dalitz fit from focus, " prepared for 7th conference on intersections between particle and nuclear physics (cipanp 2000), quebec city, quebec, canada, 22 - 28 may 2000. published in aip conf. proc. * 549 *, 569 (2002). the @xmath57 modes include @xmath58, @xmath59, and @xmath60, that sum to 4.3@xmath140.44 times @xmath61. the @xmath62 modes include the analgous modes to those for the @xmath57 and, inaddition, feeddown from the @xmath57 modes ; they sum to 9.93@xmath140.95 times @xmath61. a. petrov, charm physics : theoretical review, " invited talk at flavor physics and cp violation (fpcp 2003), paris, france, 3 - 6 jun 2003 ; published in econf c030603 : mec05 (2003) http://www.slac.stanford.edu / econf / c030603/. r. godang _ _ (cleo), phys.rev. lett. * 84 *, 5038 (2000) [hep - ex/0001060]. j. m. link _ et al. _ (focus), phys. b * 618 *, 23 (2005) hep - ex/0412034]. k. abe _ et al. _ (belle), phys. lett. * 94 *, 071801 (2005) hep - ex/0408125]. b. aubert _ et al. _ (babar), phys. rev. lett. * 91 *, 171801 (2003) [hep - ex/0304007]. e. m. aitala _ et al. _ (e791), phys. 83 *, 32 (1999) [hep - ex/9903012]. j. link _ et al. _ (focus), phys. lett. b * 485 *, 62 (2000). (cleo), phys. d * 65 *, 092001 (2002). et al. _ (belle), phys. lett. * 88 *, 162001 (2002). _, meaurement of the @xmath52 lifetime difference using @xmath63 decays, " submitted to lepton - photon conference [belle - conf-347] (2003). c. cawlfield _ et al. _ (cleo), phys. d * 71 *, 077101 (2005) [hep - ex/0502012]. b. aubert _ et al. _ (babar), phys. rev. d * 70 *, 091102 (2004) [hep - ex/0408066]. u. bitenc _ et al. _ (belle), phys. rev. * d72 *, 071101 (2005) [hep - ex/0507020]. d. m. asner _ et al. _ (cleo), phys. rev. d * 72 *, 012001 (2005) [hep - ex/0503045]. as usual it takes two interfering amplitudes to generate an asymmetry, so no direct sm cp asymmetries can arise in pure cabibbo allowed or doubly - cabibbo suppressed decays. see ref. @xcite p. 258. asymmetries in singly - cabibbo suppressed decays are very small, on the order of @xmath64 in the sm. x. c. tian _ et al. _ (belle), phys. lett. * 95 *, 231801 (2005) [hep - ex/0507071]. d. cronin - hennessy _ et al. _ (cleo), phys. d * 72 *, 031102 (2005) [hep - ex/0503052]. s. kopp _ et al. _ (cleo), phys. rev. d * 63 *, 092001 (2001) [hep - ex/0011065].

i discuss new results on absolute branching ratios of charm mesons into specific exclusive final states, cabibbo suppressed decay rates, inclusive decays to @xmath0 mesons, limits on @xmath1 mixing, cp violation and t violation. preliminary results from cleo - c now dominate the world average absolute branching fractions. for the most important normalization modes involving @xmath2 and @xmath3, the averages are @xmath4 for the @xmath5 cleo - c measures @xmath6. using this rate, i derive an effective branching ratio @xmath7, that is appropriate for use in extracting other branching fractions that have often been measured relative to this mode. this number is compared with other determinations.

hep-ph0605134

Introduction formation and bose - einstein condensation (bec) of molecules @xcite have recently been achieved based on ultracold atoms with magnetically - tuned feshbach resonances @xcite. in these experiments, feshbach coupling is induced by tuning a foreign molecular state near the scattering continuum, which allows for an efficient transfer of colliding atoms into molecules. this method works for virtually all alkali atoms, and can create ultracold molecules from various sources including bose condensates @xcite, degenerate fermi gases @xcite, or normal thermal gases @xcite. feshbach molecules have special and unique properties. they typically populate only one weakly - bound quantum state, and the bound state can strongly couple to the scattering continuum via feshbach resonance. we may ask the following question : should feshbach molecules rather be considered as molecules in a specific rovibrational state or as pairs of scattering atoms near the continuum? this distinction is particularly crucial in the studies of the bec to bcs (bardeen - cooper - schrieffer state) crossover in degenerate fermi gases, which call for a clarification of the quantum nature of the feshbach molecules @xcite. molecular states near feshbach resonances have been recently investigated based on sophisticated and complete two - body or many - body theory @xcite and multi - channel scattering calculations @xcite. all works suggest that the feshbach molecule is generally a coherent mixture of the foreign molecule in the closed channel and long - range atom pair in the open scattering channel. near resonances with large resonance widths, the molecules can be well approximated as pairs in the open channel. for narrow resonances, as suggested by numerical calculation @xcite, the closed channel dominates and a short - range molecule picture is appropriate. in this paper, we use a simple two - channel model to describe two interacting atoms near a feshbach resonance (sec. ii). to account for the finite interaction range of real atoms, we introduce a spherical box potential, which allows us to analytically calculate the molecular bound state in different regimes and their threshold behavior (sec. iii and sec. iv). finally, we apply our model to feshbach molecules in recent fermi gas experiments and to characterize the associated feshbach resonances (sec. Model we model the interaction of two identical, ultracold atoms with mass @xmath0 based on an open channel @xmath1 that supports the scattering continuum and a closed channel @xmath2 that supports the foreign bound state. the wave function of the atoms is generally expressed as @xmath3, where @xmath4 and @xmath5 are the amplitudes in the open and closed channels, respectively, and @xmath6 is the inter - atomic separation. we assume the interaction @xmath7 is described by a spherical box potential with an interaction range of @xmath8, see fig. (1). for @xmath9, the potential energy of the open channel is 0 and the closed channel @xmath10. for @xmath11, the open (closed) channel has an attractive potential of @xmath12 @xmath13, and a coupling term @xmath14 between the channels. the wave function satisfies the schrdinger equation : @xmath15 . a bound state with energy @xmath16 relative to the scattering continuum is supported by the closed channel.,width=211] the solution of the above equation for zero scattering energy @xmath17 can be expressed as : @xmath18 where the scattering length @xmath19 and @xmath20 are constants, @xmath21 are the eigen wave numbers " for @xmath11 associated with the eigen states @xmath22. based on the boundary conditions @xmath23 and @xmath24, we get @xmath25 the latter equation shows how in general, each channel contributes to the scattering length. in cold atom systems, feshbach resonances are, in most cases, induced by hyperfine interactions or spin - spin interactions. both interactions are many orders of magnitude weaker than the relevant short range exchange potential. it is an excellent approximation to assume @xmath26 and @xmath27. hence, we have @xmath28, @xmath29 and @xmath30. in this limit, the closed channel contribution is significant only when the foreign state is close to the continuum, in which case the last term in eq. (7) diverges. given the energy of the closed channel state as @xmath31 and @xmath32, the boundary condition @xmath33 allows us to expand the last term in eq. (7) as @xmath34. here @xmath35 characterizes the feshbach coupling strength. to the same order of expansion, the middle term in eq. (7) is a constant across the resonance and can be identified as @xmath36, where @xmath37 is the background scattering length. equation (7) reduces to @xmath38 Scattering length and the molecular eigen state experimentally, the relative energy between the continuum and the bare state can be adjusted linearly by a magnetic field @xmath39-induced zeeman shift @xmath40, where @xmath41 and @xmath42 is the magnetic moment of the open(closed) channel. replacing @xmath16 by @xmath43, we can rewrite eq. (8) in terms of the magnetic field as @xmath44 where the resonance width @xmath45 and the resonance position @xmath46 are given by @xmath47 several interesting features are shown here. first of all, we find the resonance width is proportional to both the feshbach coupling @xmath48 and the background scattering properties @xmath49. the latter dependence is due to the fact that the scattering amplitude at short range is proportional to the scattering length. a larger short range scattering amplitude leads to a stronger coupling to the closed channel. secondly and importantly, the resonance position is offset by exactly @xmath45 relative to the crossing of the bare state and the continuum, @xmath50, see eq. (11). for a positive scattering length @xmath51, this shift is negative @xmath52. this feature leads to the renormalization " of the feshbach resonance location discussed in ref. @xcite. to understand the origin of the resonance shift, we should return to eq. the divergence of the scattering length occurs when the open channel contribution (middle term) is exactly canceled by the closed channel one (last term). for systems with large background scattering lengths @xmath53 and strong feshbach couplings @xmath48, this cancelation can occur even when the bare state is far away from the continuum. a large resonance shift then results. now we turn to the binding energy of the molecules. assuming a bound eigen state @xmath54 exists near the continuum at @xmath55, where @xmath56 is the binding energy, we can determine @xmath57 by following essentially the same calculation as eq. (1)-(7). the equivalence of eq. (7) gives @xmath58 where @xmath59. assuming @xmath28 and the bound states in both channels are close to the continuum, namely, @xmath53 and @xmath60, we can expand the two terms on the right side of eq. (12) to leading order as @xmath36 and @xmath61, respectively. equation (12) then reduces to @xmath62 this result shows the evolution of the eigen state near the resonance. similar result is obtained in ref. @xcite based on a contact potential. we can immediately see that in the absence of the feshbach coupling @xmath63, the solutions of eq. (13) are @xmath64 and @xmath65 (for @xmath51), which exactly correspond to the bare bound states in the closed channel and the open channel (for @xmath51), respectively. in the presence of the feshbach coupling @xmath66, eq. (13) suggests an avoided level crossing "- like energy structure, see fig. (2), which also illustrates the resonance position shifts. the level crossing, however, is not hyperbolic as it is in a two - level system. in particular, at small binding energies, the bound state energy approaches the continuum quadratically, see fig. (2) inset. far below the continuum, the bound state approaches the bare state in the closed channel. , solid lines). we assume (a) @xmath67 and @xmath68 and (b) @xmath69 and @xmath68, where @xmath70 can be any relevant length scale. arrows mark the offset resonance positions. insets show the threshold behavior of the bound state.,width=268] to better quantify the role of the open and closed channel, we can write the wave function of the eigen state as @xmath71 and, with our approximations, @xmath72 satisfies @xmath73 and the mixing angle @xmath74 is defined below. we show in eq. (15) that the eigen state generally occupies both the closed channel and open channel. we can introduce a mixing amplitude @xmath75 as the amplitude in the closed channel @xmath76 the mixing fraction @xmath77 can be evaluated by a direct integration of the closed channel wave function. alternatively, noticing that the mixing also leads to a dependence of the eigen state on the bare state, we can also derive @xmath77 from the dependence of @xmath57 on @xmath16, or from the magnetic moment of the feshbach molecule @xmath78. all methods lead to the same result @xmath79 Threshold regime in open channel despite the seemingly complex equations shown in previous sections, the feshbach molecules are simple and universal near the scattering continuum. expanding eq. (13) with small @xmath57 and using eq. (9)-(11), we find the binding energy of the feshbach molecules has a simple dependence on the scattering length and increases quadratically in magnetic field near the resonance, namely, @xmath80 where @xmath81. equation (20) shows identical dependence on scattering length and interaction range as of single channel molecules in the threshold regime @xcite. furthermore, taking the limit @xmath82 in eq. (14) and (19), we find the molecular wave function here is purely in the open channel. its spatial extent is determined by quantum uncertainty, @xmath83, and can be much larger than the interaction range @xmath84. in this limit, the feshbach molecules are identical to long - range atom pairs in a single open channel. by expanding eq. (19) at small @xmath85 and using eq. (13), we find the closed channel fraction can be expressed as @xmath86 where @xmath87. from eq. (22), we see that @xmath88 provides the leading order estimation of the closed channel admixture. when @xmath89 or @xmath90 (this condition applies when @xmath91), the feshbach molecule is purely in the open channel. as expected, the threshold regime is wider for resonances with larger @xmath48 and @xmath92. we can further determine the open channel - dominated " regime by setting @xmath93 in eq. (19). for resonances with small @xmath94, this condition corresponds to @xmath95, which, in terms of magnetic field, maps to only a small fraction of @xmath96 near the resonance @xmath46. for resonances with large @xmath97, the open channel dominates when @xmath98, which covers the full resonance width when @xmath99, and covers the entire upper branch of the bound state when @xmath100. based on the range of the single channel regime, we suggest the broad(narrow) resonances be defined as those with @xmath97 (@xmath101). within the width of the feshbach resonance, the molecules associated with a broad (narrow) resonance are better described as long range pairs in the open channel (short range molecules in the closed channel). we note that this definition is purely based on two - body physics. Feshbach molecules in @xmath102li and @xmath103k .parameters of the @xmath102li and @xmath103k feshbach resonances. interaction range @xmath8 is derived from ref. @xcite, see text. feshbach coupling @xmath48 is derived from eq. @xmath104 is bohr radius and @xmath105 is bohr magneton. [cols="^,^,^,^,^,^,^,^",options="header ",] [table1] (dotted lines) and mixing fractions @xmath77 (solid lines) of the molecules near the @xmath102li and @xmath103k feshbach resonances. the curves are calculated from eq. (13), eq. (19) and the parameters in table 1. binding energies from multi - channel calculation @xcite (dashed lines), from jila group measurement @xcite (open square) and the mixing fractions measurement from rice group @xcite (open circles) are shown for comparison. the shaded areas indicate the typical bec - bcs crossover regimes, @xmath106.,width=268] finally, we apply our model to the @xmath102li@xmath107 and @xmath103k@xmath107 feshbach molecules created in recent bec - bcs experiments @xcite. these molecules are stable near the resonance and both the molecular binding energies and the scattering lengths have been well measured and studied @xcite. to model the interaction of atoms, we adopt @xmath46, @xmath45, @xmath108 and @xmath109 from recent measurements and numerical calculations. to account for the finite range of the atomic interaction, which at low temperatures is determined by the van der waals potential of @xmath110, we choose the interaction range @xmath8 in our model to be the mean scattering length @xmath111 defined in ref. this choice ensures the same behavior of the scattering length in the threshold regime @xcite. all parameters are given in table i. in fig. (3), we show the calculated binding energy @xmath57 and the mixing fraction @xmath77 of the feshbach molecules for the two @xmath102li resonances and one @xmath103k resonance. the results agree very well with the multi - channel calculation @xcite and the measurements on molecular binding energy @xcite, magnetic moment @xcite and mixing fraction @xcite. both the li resonance at 834 g and the k resonance are broad with @xmath112 and @xmath113, respectively. the open channel - dominated regimes of @xmath114 210 mhz for the 834 g li resonance is also larger than the fermi energy of @xmath115 20 khz in the experiments.(here the fermi wave number is @xmath116.) for the k resonance, the full upper branch of the molecular state is open channel dominated with mixing fractions less than @xmath117. therefore, we conclude the open channel description of these feshbach molecules in the crossover regime to be a good approximation. for the narrower li resonance at @xmath118543 g, we obtain @xmath119 and @xmath120 31 hz @xmath121 20 khz. this indicates an extremely narrow open channel regime of less than 50 @xmath109 g near the resonance, where the gas parameter is still over @xmath122. crossover experiments based on these feshbach molecules can not be described by open channel atom pairs and may lead to qualitatively different physics. we attribute the large difference between the two li resonances to their different couplings @xmath48 and very different background scattering length @xmath37, see table i. in the above discussions, we note that fermi energy @xmath123 is an external parameter which depends on the density of the sample. whether the molecules in the crossover regime can be described by single channel strongly depends on the density. the @xmath124 parameter, however, provides a better and independent measure to classify feshbach resonances. we find that the two feshbach resonances in @xmath102li are the two extremes of broad and narrow resonances with @xmath112 and @xmath125. in summary, the two - channel model provides a simple picture to understand the molecular state near the feshbach resonances. the analytic results of the molecular binding energy and mixing fraction on @xmath102li and @xmath103k agree with the measurements and other sophisticated calculations very well. based on the threshold behavior of the bound state, we suggest a dimensionless parameter to assess the broadness " of the feshbach resonance. Acknowledgements we thank p.s. julienne and n. nygaard for stimulating discussions and r. grimm s lithium and cesium groups in innsbruck for the support during our visit. the author is partially supported by the lise - meitner program of the austrian science fund (fwf). s. jochim, m. bartenstein, a. altmeyer, g. hendl, s. riedl, c. chin, j. hecker denschlag and r. grimm, science * 302 *, 2101 (2003) ; m. greiner, c.a. regal, d.s. jin, nature * 426 *, 537 (2003) ; m. zwierlein, c.a. stan, c.h. schunck, s.m.f. raupach, s. gupta, z. hadzibabic, and w. ketterle, phys. 91 *, 250401 (2003). j. herbig, t. kraemer, m. mark, t. weber, c. chin, h.- c. ngerl, and r. grimm, science * 301 *, 1510 (2003) ; s. drr, t. volz, a. marte, and g. rempe phys. rev. * 92 *, 020406 (2004) ; k. xu, t. mukaiyama, j.r. abo - shaeer, j.k. chin, d. miller, and w. ketterle, phys. rev. lett. * 91 *, 210402 (2003). b. marcelis, e.g.m. van kempen, b.j. verhaar, and s.j.j.m.f. kokkelmans, phys. a * 70 *, 012701 (2004) ; s.j.j.m.f. kokkelmans, j.n. milstein, m.l. chiofalo, r. walser, and m.j. holland, phys. a * 65 *, 053617 (2002).

we present a two - channel model to describe the quantum state of two atoms with finite - range interaction near a feshbach resonance. this model provides a simple picture to analytically derive the wave function and the binding energy of the molecular bound state. the results agree excellently with the measurements and multichannel calculations. for small binding energies, the system enters a threshold regime in which the feshbach molecules are identical to long range atom pairs in single channel. according to their threshold behavior , we find feshbach resonances can be classified into two types.

cond-mat0506313

Introduction for fixed integers @xmath0 and @xmath1, we consider the admissible sequences of @xmath2 lattice paths in a colored @xmath3 square given in @xcite. each admissible sequence of paths can be associated with a partition @xmath10 of @xmath4. in section [paths], we show that the number of self - conjugate admissible sequences of paths associated with @xmath10 is equal to the number of standard young tableaux of shape @xmath10, and thus can be calculated using the hook length formula. we extend this result to include the non - self - conjugate admissible sequences of paths and show that the number of all such admissible sequences of paths is equal to the sum of squares of the number of standard young tableaux of partitions of @xmath4 with height less than or equal to @xmath11. using the rsk correspondence in @xcite, it is shown in (@xcite, corollary 7.23.12) that the sum of squares of the number of standard young tableaux of partitions of @xmath4 with height less than or equal to @xmath11 is equal to the number of @xmath6-avoiding permutations of @xmath7. in section [multiplicities], we apply our results to the representation theory of the affine kac - moody algebra @xmath8. let @xmath12, @xmath13 and @xmath14 denote the simple roots, simple coroots, and fundamental weights respectively. note that @xmath15. for @xmath16, set @xmath17 and @xmath18. as shown in @xcite, @xmath19 are maximal dominant weights of the irreducible @xmath8-module @xmath9. we show that the multiplicity of the weight @xmath19 in @xmath9 is the number of @xmath6-avoiding permutations of @xmath7, which proves conjecture 4.13 in @xcite. Lattice paths for fixed integers @xmath0 and @xmath1, consider the @xmath3 square containing @xmath20 unit boxes in the fourth quadrant so that the top left corner of the square is at the origin. we assign color @xmath21 to a box if its upper left corner has coordinates @xmath22. this gives the following @xmath3 colored square @xmath23 : a lattice path @xmath25 on @xmath23 is a path joining the lower left corner @xmath26 to the upper right corner @xmath27 moving unit lengths up or right. for two lattice paths @xmath28 on @xmath23 we say that @xmath29 if the boxes above @xmath30 are also above @xmath25. now, we draw @xmath2 lattice paths, @xmath31 on @xmath23 such that @xmath32. for integers @xmath33, where @xmath34, @xmath35, we define @xmath36 to be the number of @xmath37-colored boxes between @xmath38 and @xmath39. we define @xmath40 to be the number of @xmath37-colored boxes below @xmath41 and @xmath42 to be the number of @xmath37-colored boxes above @xmath43. denote by @xmath49 the set of all admissible sequences of @xmath2 paths. notice that there are @xmath4 0-colored boxes in @xmath23 and hence for any admissible sequence of paths, @xmath50. in addition, it follows from definition [pathsdef](2) that @xmath51 for any admissible sequence of paths. thus, we can and do associate an admissible sequence of paths @xmath44 on @xmath23 with a partition @xmath52 of @xmath4. in this case, we say that this admissible sequence of paths is of type @xmath10 and often draw @xmath10 as a young diagram. figure [adseq](a) is an element of @xmath53, where @xmath54 and @xmath55 are shown in figures [adseq](b), [adseq](c), and [adseq](d), respectively. notice that this admissible sequence of paths is of type @xmath56.

for @xmath0 and @xmath1, we consider certain admissible sequences of @xmath2 lattice paths in a colored @xmath3 square. we show that the number of such admissible sequences of lattice paths is given by the sum of squares of the number of standard young tableaux of partitions of @xmath4 with height @xmath5, which is also the number of @xmath6-avoiding permutations of @xmath7. finally, we apply this result to the representation theory of the affine lie algebra @xmath8 and show that this quantity gives the multiplicity of certain maximal dominant weights in the irreducible module @xmath9.

1508.06930

The relativistic stern-gerlach force the time varying stern - gerlach, sg, interaction of a relativistic fermion with an e.m. wave has been proposed to separate beams of particles with opposite spin states corresponding to different energies@xcite. we will show how spin polarized particle will exchange energy with the electromagnetic field of an rf resonator. let us denote with @xmath1 the coordinates of a particle in the laboratory, and with @xmath2 the coordinates in the particle rest frame, prf. in the latter the sg force that represents the action of an inhomogeneous magnetic field on a particle endowed with a magnetic moment @xmath3 is f_sg = (^ * b) = x (^ * b) + y (^ * b) + z (^ * b) [fsg] with = ge2 m s [mu]. here @xmath4 is the elementary charge with @xmath5 for protons and positrons, @xmath6, and @xmath7 for antiprotons and electrons, @xmath8, making @xmath9 and @xmath10 either parallel or antiparallel to each other, respectively. @xmath11 is the rest mass of the particle, @xmath12 the gyromagnetic ratio and @xmath13 the anomaly defined as a = g-22 = \ { ll 1.793 (g=5.586) & p,|p + + 1.160 10 ^ -3 & e^ .. [anomalies] notice that in eq.([fsg]) we have defined the magnetic moment as @xmath14 in the rest frame, rather than as @xmath15. in the rest frame the quantum vector @xmath10, or spin, has modulus @xmath16 and its component parallel to the magnetic field lines can only take the following values s_m=(-s, -s+1,....,s-1, s), [s_main] where @xmath17 is the reduced planck s constant. combining eqs.([mu]) and ([s_main]) we obtain for the magnetic moment in the prf = ||=g|e|4 m = \ { l 1.41 10 ^ -26 jt^-1 + + 9.28 10 ^ -24 jt^-1 .. [mumod] for a particle traveling along the axis @xmath18, the lorentz transformations of the differential operators and of the force yield \ { lll x = x & y = y & z = (z + c t) + + f_= 1f _ & f_= f _ & (f_z = f_z) [dz].. [ftransf]the force ([fsg]) is boosted to the laboratory system as f_sg = 1 x (^ * b) + 1 y (^ * b) + z (^ * b). [f - sg] because of the lorentz transformation of the fields@xcite @xmath19 and @xmath20 \ { l e = (e + cb) - ^2 + 1(e) + + b = (b - ce) - ^2 + 1(b) .. [br] the energy in the rest frame @xmath21 becomes (^ * b) = ^*_x (b_x + ce_y) + ^*_y (b_y - ce_x) + ^*_zb_z. [magen] combining eqs.([magen]) and ([f - sg]), by virtue of eq.([dz]), after some algebra we can finally obtain the sg force components in the laboratory frame : \ { l f_x = ^*_x(b_xx + c e_yx) + ^*_y(b_yx - c e_xx) + 1^*_zb_zx + + f_y = ^*_x(b_xy + c e_yy) + ^*_y(b_yy - c e_xy) + 1^*_zb_zy + + f_z = ^*_xc_zx + ^*_yc_zy + ^*_zc_zz, . [fsgz] with \ { l c_zx = ^2 + + c_zy = ^2 + + c_zz = (b_zz + c b_zt) .. [czz] these results can also be obtained from the quantum relativistic theory of the spin-@xmath0 charged particle@xcite. let us introduce the dirac hamiltonian h = e+ c(p - ea) + _ 0 mc^2 [ham1] having made use of the dirac s matrices = (), _ 0 = (), = _ 0 = (), [mat - d] where @xmath22 is a vector whose components are the pauli s matrices _ x = (), _ y = (), _ z = (), [mat - p] @xmath23 is the @xmath24 identity matrix, @xmath25 the null matrix and having chosen the @xmath26-axis parallel to the main magnetic field. a standard derivation leads to the non relativistic expression of the hamiltonian exhibiting the sg interaction with the `` normal '' magnetic moment = e+ (p - ea)^2 - (b) [hamp] which coincides with the pauli equation and is valid in the prf. to complete the derivation we must add the contribution from the anomalous magnetic moment to the sg energy term in the previous equation, with a factor @xmath27, yielding - b = - ^ * b ^ * = g. [sgenergy] in order to obtain the @xmath28-component of the sg force in the laboratory frame along the direction of motion of the particle, we must boost the whole pauli term of eq.([hamp]) by using the unitary operator @xmath29 in the hilbert space@xcite, which expresses the lorentz transformation u^-1 u = g(_0b) [hilb] that can be written in terms of the equivalent transformation in the @xmath30 spinor space s = = + () [str] with =, u = = =, u = (=). [defs] from eqs.([hilb]) and ([str]), due to the algebraic structure of the @xmath31 and @xmath32 matrices, we obtain in the laboratory frame the three components of the sg force \ { lll s^-1 (_ 0 _ x) s & = & _ 0 _ x + s^-1 (_ 0 _ y) s & = & _ 0 _ y + s^-1 (_ 0 _ z) s & = & (_ 0 _ z) + i_0_5 ., [evxyz] with _ 5 = _ x_y_z_0 = i (). [g5] from eqs.([evxyz]) we can deduce the expectation values of the sg force in the laboratory system with a defined spin -along the @xmath26-axis in our case- via the expectation values of the pauli matrices and of the pauli interaction term of the proper force f_z = _ 0_y^2^*. [fz4] in our case only the second of eqs.([evxyz]) gives a non vanishing result, while both the first and third produce a null contribution to the force, because of the orthogonality of the two spin states @xmath33 and the properties of the @xmath32 matrices. The radio-frequency system let us consider the standing waves built up inside a rectangular radio - frequency resonator, tuned to a generic te mode@xcite. resonator dimensions are : width @xmath13, height @xmath34 and length @xmath35, as shown in fig.[fig : box]. on the cavity axis, which coincides with the beam axis, the electric and magnetic fields are@xcite @xmath36 \\ \\ b_x & = & - { b_0\over k_c^2 } \left({m\pi\over a}\right) \left({p\pi\over d}\right) \sin\left({m\pi x\over a}\right) \cos\left({n\pi y\over b}\right) \cos\left({p\pi z\over d}\right) \cos\,\omega t \\ \\ b_y & = & - { b_0\over k_c^2 } \left({n\pi\over b}\right) \left({p\pi\over d}\right) \cos\left({m\pi x\over a}\right) \sin\left({n\pi y\over b}\right) \cos\left({p\pi z\over d}\right) \cos\,\omega t \\ \\ b_z & = & b_0 \cos\left({m\pi x\over a}\right) \cos\left({n\pi y\over b}\right) \sin\left({p\pi z\over d}\right) \cos\,\omega t \end{array}\]] where @xmath37 is the rf peak magnetic field, @xmath11, @xmath38 and @xmath39 are integer mode indeces, and k_c =. [kc] the angular frequency of the e.m. wave from the rf generator is = _ = 2c _ = c. [omegen] in contrast with an open waveguide, in a bounded cavity we can define a phase velocity @xmath40 and a cavity wavelength @xmath41, as typical of any e.m. in a refractive media, according to the relations = _ = dp. [rphv] and _ = _ ph_. [cav - wl] it is also = _ c = _ = notice that @xmath42 can take any value, even larger than one, since it is freely dependent on the cavity geometrical parameters. moreover, combining eqs.([omegen]) and ([rphv]) we obtain d = p _ _ = p _ [cav - length] which describes the connection between the cavity length @xmath35 and the wavelengths, as shown in fig.[fig : edge]. for simplicity, let s choose the transverse electric mode @xmath43, so eqs.([omegen]) and ([rphv]) reduce respectively to = _ = c = _ c _ = [ombetap] or, setting the mode index @xmath44, = _ = c = _ c _ =, [ombeta1] which are the quantities pertaining to the preferred @xmath45 mode whose non zero field components on the cavity axis are \ { l b_y(z, t) = - b_0 () t + + e_x(z, t) = -b_0 () t .. [byex] it is important to emphasize that in all the field components met so far there is a clear separation between spatial and temporal contributions, as typical of standing waves. besides, the boundary conditions of the electric and magnetic fields of the e.m. dictate the shape of the spatial component which, in turn, oscillates in time with the frequency @xmath46. then, at the cavity entrance and exit the field components ([byex]) become on axis \ { l b_y(0,t) = -b_0 t + + e_x(0,t) = 0 .. [edge - in] and \ { l b_y(d, t) = -b_0 t = b_0 t + + e_x(d, t) = -t = 0 .. [edge - out] where @xmath47 is a generic time. the null values of @xmath48 at the cavity ends confirm a typical pattern of the transverse electric mode. Stern-gerlach interaction with the cavity field from eq.([fz4]), after some algebra, we obtain tha a charged fermion which crosses a radio - frequency resonator, tuned on the te@xmath49 mode, acquires (or loses) an energy amount when interacts with the field component in the `` body '' of the cavity shown in fig. [fig : edge]@xcite (u) _ = _ 0^df_z dz = _ 0^d^*c_zydz = ^2 ^ 2 b_0^ * (1 +) [du - x] still assuming that the spin is not precessing. however, since the cavity can not be completely enclosed but must have apertures at both ends to allow the particle bean to pass through and consequently will have fringe fields, in order to calculate the full sg interaction it is necessary to deal with the interaction with these fields. this is discussed right below. in order to fulfill the boundary conditions ([edge - in]) and ([edge - out]), a cavity tuned in its @xmath45 mode must be exactly filled by either an even or an odd number of cavity dependent half wave - lengths, eq.([cav - wl]), as illustrated in figs. [fig : edge] and [fig : edges]. consider now a bunch of particles crossing the cavity in synchronism with the rf field. this requires that the bunch centre of mass that enters the cavity at the instant @xmath50 and would leave the cavity at @xmath51, at magnetic field values, respectively b_y(0,0) = -b_0 b_y(d, _) = b_0 [in - out] the field values at both ends fade rapidly to zero over a small distance @xmath52 just outside the cavity (see figures.) we may consider these fringe fields as small - valued functions in the @xmath53-plane, since the time @xmath54 necessary for a particle to proceed through this distances can be very small in comparison with @xmath55, depending of course by the size of the beam channel, or \ { llllll _ & = & -b_0g(z) & & g(-)=0, & g(0)=1 + + _ & = & b_0h(z) & & h(d)=1, & h(d+)=0 .. [edge - flds] under these conditions, a relativistic fermion with its spin directed along the @xmath26-axis and traversing the cavity will experience a sg force parallel to the @xmath28-axis (direction of motion), see eq.([fsgz]) f_z = ^ * c_zy [fz] where @xmath56 is given by the second of the set of eqs.([czz]). for the moment we assume that the spin will conserve its orientation during traversal the electric field @xmath48 and its derivatives in this equation are almost constantly zero, because of the boundary conditions on the walls of the cavity and at the extreme points @xmath57 and @xmath58. furthermore, the function@xmath59 is almost zero along the fringe segments because of its proportionality to @xmath60, with @xmath47 equal to the @xmath54 mentioned before. consequently we have c_zy ^2, [redczy] and for the entire fringe field \ { lll _ & = & -b_0^*^2 () + + _ & = & b_0^*^2 () .. [edge - forces] making use of eqs. ([redczy]) and ([edge - forces]), the energy increments @xmath61_{\m in}$] and @xmath61_{\m out}$] related to the fringe fields are easily evaluated since the integrals @xmath62 and @xmath63 only depend upon the extreme points ([edge - flds]) and do not depend on the curve that connects them. in fact @xmath64 becomes an exact differential. then we obtain for the energy exchange at both edges (u) _ = (u) _ = - b_0^*^2. [cav - en] the total energy exchange at the edges is therefore (u) _ = (u) _ + (u) _ = - 2b_0^*^2. [ff - en] by adding the fringe contributions ([ff - en]) to the cavity body crossing contribution ([du - x]) seen before, obtain (u) _ = (u) _ + (u)_x = -^2 b_0^ * f (_,) [du - tot] with f (_,) =. [beta - beta] for ultra relativistic particles (@xmath65) eq. ([du - tot]) reduces to (u) _ -^2 b_0^ * (1- _). [du - ur1] this last result deserves a few comments. in fact, if we set _ = 2 d = _ = _ [b2] the total energy contribution ([du - ur1]) vanishes, implying a full cancellation of the effect. on the other hand if we set _ = _ [b3] the total energy contribution ([du - ur1]) becomes (u) _ -2 ^ 2 b_0^ * [du - ur2] as deduced from eq.([cav - length]). in table i we gather values calculated from eq.([beta - beta]) for non - relativistic and ultra - relativistic particles for, either @xmath66 or @xmath67 at two proton energies. each @xmath42 is accompanied by the corresponding ratio cavity - length over cavity - height. * table i : * @xmath68 [cols="^,^,^ ",] furthermore, if we consider two contiguous cavities, there will be a gradient between the positive @xmath69 at the end of the first cavity and a negative @xmath69 at the beginning of the second cavity, as shown in fig. [fig : cavx]. in this case we may consider the magnetic field at the interface as linearly dependent on @xmath28, that is _ [gradx] reiterating what done before, obtain rcl _ & = & -2b_0 + + f_z & = & -2b_0^*^2 + + (u) _ = (u) _ & = & _ -^f_z dz = -2b_0^ * ^2 = -2b_0^*^2 which means that, for @xmath70 cavities, we shall have as final result for ultra relativistic particles (u) _ = n(u) _ - (n-1)(u) _ - (u) _ = \ { ll 0 & _ ph = 2 + + b_0^*^2 & _ ph = 3 .. [full - end] conversely, if @xmath42 is even, particles with their spin pointing always in the same direction can not exchange energy with the standing wave of a te resonator. a spin rotator@xcite can align the particle magnetic moments either parallel or anti - parallel to the directions of the magnetic field gradients, thus allowing the desired energy interaction. this situation would be similar to what happens in a multi - stage tandem van de graaff, where the ions are repetitively accelerated by the same electrostatic field, becoming alternatively negative, via an addition of electrons, or positive, via electron stripping. unfortunately, the field integral (@xmath71 tm, for @xmath72) for attaining a spin rotation is so large that this solution is unpractical. instead, the example of @xmath42 equal to an odd number seems much more suitable since does not require cumbersome magnets, but only longer cavities (compare eqs. ([b2]) and ([b3])). in fact, the magnetic moments are (de)accelerated by the field tails at the cavity ends, while do nt change their energy when crossing the cavities. this situation resembles the wideroe linac where the charged particles are accelerated by the electric fields between two contiguous drift tubes, but do nt change their energy while crossing the tubes themselves. Concluding remarks on the basis of the previous estimates, we feel ready to propose the time varying sg interaction as a method for attaining a spin state separation of an unpolarized beam of, say (anti)protons, since the energy of particles with opposite spin orientations will differ and beams in the two states can be separated. in a first stage of the study of a sensible practical design, we intend to proceed with numerical simulations. as a first step, we intend to verify the correctness of eqs.([du - tot]) and ([beta - beta]) setting once @xmath66 and then @xmath67, in a cavity where the field line pattern can be realistically controlled. beyond the verification of the present theory, there is also the aim of studying the effects generated by the spin precession inside the cavity, that we did not yet address in this note. next, we shall consider a spin splitter scheme based on the lattice of an existing or planned (anti)proton ring endowed with an array of splitting cavities. the principal aim of the latter implementations is to check the mixing effect@xcite@xcite of the longitudinal phase - plane filamentation, i.e. the actual foe which could frustrate the entire spin splitting process. Acknowledgments first, we want to thank waldo mackay, who has participated on so many discussions on the whole idea but who was regrettably prevented by numerous commitments from participate to the editing of the present note. we thank renzo parodi for his help for us to better understand the subtleties of the standing waves building up. thanks are also due to chris tschalaer for fruitful discussions on the role of the fringe fields. m. conte, m. ferro, g. gemme, w.w. mackay, r. parodi, m. pusterla : the stern - gerlach interaction between a traveling particle and a time varying magnetic field, infn / tc-00/03, 22 marzo 2000. (http : xxx.lanl.gov / listphysics/0003, preprint 0003069) p. cameron, m. conte, a. luccio, w.w. mackay, m. palazzi and m. pusterla : the relativistic stern - gerlach interaction and quantum mechanics implications, proceedings of the spin2002 symposium, 9 - 14 september 2002, brookhaven, eds. makdisi, a.u. luccio and w.w. mackay, aip conference proceedings 675 (2003) p. 786. j.d.jackson, _ classical elecrodynamics _, john wiley & sons inc., new york 1975 r.p. feynman, _ quantum electrodynamics _ benjamin inc., new york 1961. s. ramo, j.r. whinnery and t. van duzer, _ fields and waves in communication electronics _, john wiley and & sons, new york, 1965. m.conte,a.u.luccio,w.w.mackay and m.pusterla _ stern gerlach force on a precessing magnetic moment _ proc. pac07, albuquerque, nm (2007), p.3729 m. conte, w.w. mackay and r. parodi : an overview of the longitudinal stern - gerlach effect, bnl-52541, uc-414, november 17 1997. m. palazzi : ph.d thesis, genoa university, june 6 2003.

the general expression of the stern - gerlach force is deduced for a relativistic charged spin-@xmath0 particle which travels inside a time varying magnetic field. this result was obtained either by means of two lorentz boosts or starting from dirac s equation. then, the utilization of this interaction for attaining the spin states separation is reconsidered in a new example using a new radio - frequency arrangement.

0907.2161

Introduction excitons in semiconductors have been the subject of many experimental and theoretical investigations of bose condensation. low - energy exciton - exciton interactions are characterized by the exciton - exciton scattering length, @xmath5, which determines the thermodynamics of a low density gas and is crucial for modeling the thermalization time of a dilute exciton gas. despite its importance, the exciton - exciton scattering length is an elusive quantity, being difficult to measure experimentally or to estimate theoretically. as is well - known in atomic physics, scattering lengths can be extremely sensitive to the details of the interactions between particles. in particular, the existence of a weakly bound or nearly bound state causes the scatter length to become quite large. therefore, _ a priori _ one should suspect that exciton - exciton scattering may be a very material dependent property of semiconductors. reliable theoretical predictions of exciton - exciton scatting lengths require both a very accurate hamiltonian for the semiconductor, and an accurate solution to the (four - particle) scattering problem. in this paper we provide an essentially exact solution to exciton - exciton scattering for a commonly used single - band effective mass hamiltonian. this solution allows us to study three important questions : (1) how sensitive is the scattering length to the mass ratio @xmath6, (2) how does the scattering length depend on spin states (singlet or triplet) of the scattering excitons, and (3) to what degree can inter - exciton exchange of electrons or holes cause excitons to scatter into different spin states? this calculations also serve as a benchmark for the single band limit of more complicated scattering hamiltonians. one experimental method for measuring the exciton scattering cross section is to look at line width broadening of the recombination spectra in a gas of excitons. collisions between excitons increase the line width, causing the line width to depend on the exciton - exciton scattering rate, @xmath7, where @xmath8 is the density and @xmath9 is a typical exciton velocity. extracting cross sections from a line width requires that (1) the density and velocity distribution are known, and (2) elastic scattering is the fastest process. as discussed below, @xmath10 is a good material for comparison to the model studied in this work. et al._@xcite have performed such experiments on @xmath10 and have found a line width broadening that suggests an upper bound of @xmath11 on the scattering length. although our simulations do not exactly model @xmath10, we will compare our results to this value. Theoretical background theoretical approaches to this problem start with the effective mass approximation, in which the system under consideration consists of two electrons, labeled @xmath12 and @xmath13, and two holes, labeled @xmath14 and @xmath15. the hamiltonian is @xmath16 where @xmath17. the hamiltonian has symmetry under exchange of electrons and exchange of holes, so eigenstates may be denoted by two exchange quantum numbers. the s - wave states are symmetric under exchange of excitons ; a condition which is satisfied by states @xmath18 and @xmath19, where the @xmath20 signs refer to (anti)symmetry under exchange of electrons and holes, respectively. although this hamiltonian is a well - accepted model for exciton - exciton scattering, we should point out a few of its deficiencies. for small excitons, such as those in @xmath10, that have radii not much larger than the lattice spacing, non - parabolic terms in the kinetic energy and other corrections to the potential energy may be necessary. for many semiconductors, such as si and ge, the valence band is a mixture of three bands and can not be described by a single parabolic band. in the case of @xmath10, the valence band is the parabolic spin - orbit split off band, and there are fewer complications. interband exchange (virtual electron - hole recombination) is an important effect that has been neglected, and could be modeled by an additional spin - dependent potential term. this hamiltonian also describes a family of scattering processes for other particles, including hydrogen - hydrogen, positronium - positronium, and muonium - muonium scattering. the equal mass case is at an extreme (positronium scattering), where the born - oppenheimer approximation is the least applicable. there have been several theoretical estimates of exciton - exciton scattering for bulk systems@xcite and quantum wells@xcite as well as calculations on biexciton - biexciton scattering,@xcite only the bulk, elastic scattering calculations@xcite are directly comparable to the results of this paper, but the techniques presented here could be generalized to the other scattering problems. also, the results presented here provide a benchmark for evaluating the approximations used in other theoretical treatments, and could lend insight into the reliability of the approximations in more complicated situations. one standard theoretical approach is diagrammatic perturbation theory, as presented in the work of keldysh and kolsov@xcite and haug and hanamura.@xcite they estimate the exciton - exciton scattering matrix as arising from a single term, @xmath21, where @xmath22 represents a state of two noninteracting excitons with momentum @xmath23 and @xmath24 and @xmath25 is the inter - exciton coulomb interaction. this method gives an estimate of @xmath26 (independent of the mass ratio), where @xmath27 is the exciton radius, but it is an uncontrolled approximation which may have limited validity in the low energy limit. one serious drawback of the method is that it does not include effects of the biexciton in the scattering. as we show later, biexciton vibrational states cause strong dependence of the scattering length on the mass ratio @xmath6, which is not captured by the low order perturbation theory. a second common approach was developed by elkomoss and munchy,@xcite and uses an effective exciton - exciton potential defined by @xmath28, where @xmath29 is the wavefunction for two free excitons a distance @xmath30 apart. the effective potential @xmath31 arises from the hartree term and is used in a two - particle central - field calculation. while an exciton - exciton scattering pseudo - potential would be a very useful tool, this approximate form has some serious drawbacks. among its deficiencies are a lack of correlation, no van der waals attraction, a failure to reproduce biexciton states, and a vanishing interaction potential for @xmath32. the cross sections calculated by this method are small and lack qualitative agreement with the results of the present work. some insight into exciton - exciton scattering can be gained by considering the bound states, biexcitons. since the number of bound states, @xmath33, enters in the phase shift at zero energy, @xmath34, it is necessary that a good computation method for low energy scattering be able accurately calculate biexciton binding energies.@xcite for the mass ratios considered (and far beyond, including deuterium) the biexcitons can not bind in the @xmath19 states, so biexcitons in rotational @xmath35 states always have @xmath18 wavefunctions. detailed theoretical descriptions of biexcitons can be found in ref. the equal mass case was shown to have a bound biexciton by hyllerass and ore using a variational argument,@xcite and a better variational estimate of the binding energy was given by brinkman, rice and bell,@xcite who found @xmath36, where @xmath37 is the exciton binding energy. however, because of the importance of correlation energy, the latter variational treatment was missing _ half _ of the biexciton binding energy, as shown by diffusion monte carlo (dmc) calculations,@xcite which find @xmath38. dmc is a quantum monte carlo (qmc) method that uses a random walk to project out the ground state wavefunction from a variational wavefunction, in order to stochastically sample the exact ground state energy. the success of dmc for calculating biexciton energies has been a motivation for its use in the present scattering calculations. Method: quantum monte carlo calculation of scattering the @xmath30-matrix approach to scattering is to examine the standing waves of the system. as shown by carlson, pandharipande, and wiringa@xcite and alhassid and koonin,@xcite by fixing nodes in the standing waves the scattering problem may be cast as a stationary state problem suitable for qmc methods. for an elastic scattering process, we label the distance between the products by @xmath30, and the reduced mass of the products by @xmath39. in exciton - exciton scattering there is a subtlety in the definition of r due to inter - exciton exchange, which we will address below in our discussion of the exciton - exciton scattering wavefunctions. nonetheless, for large separation @xmath30, the relative motion of the products is free - particle like, so the many - body wavefunction depends on @xmath30 as @xmath40, \label{asymptoticform}\]] where @xmath41 is the relative angular momentum, @xmath42 is the scattering momentum, and @xmath43 is the phase shift. if we constrain the wavefunction to have a node at a large exciton separation @xmath44, we find a discrete energy spectrum @xmath45, which may be computed by ground state or excited state methods, such as dmc. each choice of @xmath44 gives a spectrum of states @xmath46, with energies @xmath47 that determine values of @xmath43, @xmath48 where @xmath49. the scattering matrix elements are determined by the phase shifts, @xmath50 - 1.\]] carlson has also proposed fixing the logarithmic derivative of the wavefunction at the boundary instead of setting wavefunction to zero, @xcite @xmath51 were @xmath52 is the normal to the boundary surface, at a fixed radius @xmath53, and @xmath54 parameterizes the boundary condition. this formulation has the advantage of separating the choice of simulation size @xmath53 (subject to @xmath53 lying in the asymptotic region), from the sampling of energy, which is handled by varying @xmath54, and is particularly well suited for finding the scattering length. the application to vmc calculations is straight - forward, but preserving the boundary condition in dmc calculations requires a method of images. @xcite the results presented here do not use the logarithmic - derivative boundary condition. the use of excited states @xmath55 is necessary when there is a bound state and, more generally, when the scattering state being studied has its first node before the asymptotic region is reached. we use a qmc method to calculate excited states developed by ceperley and bernu,@xcite to adapt vmc and dmc methods for a hilbert space of several low energy wavefunctions. a set of @xmath56 trial wavefunctions is chosen, @xmath57. the generalized eigenvalue equation to be solved is,@xcite @xmath58d_{k\beta}(t)=0, \label{eq : eigen}\]] where @xmath59 is the @xmath60 eigenvector with eigenvalue @xmath61 and the matrices @xmath62 and @xmath63 are the overlap and hamiltonian matrices in our trial basis, given by @xmath64 the parameter @xmath65 is the projection time. the eigenvalues @xmath61 are energy eigenvalues @xmath66 within the hilbert space spanned by the projected trial functions @xmath67, and approach the exact energy eigenvalues in the limit of large @xmath65. the matrices @xmath62 and @xmath63 are sampled with random walks, using a guiding function @xmath68 which must be positive everywhere. the guiding function must have significant overlap with all basis functions, and should be optimized to decrease the variance of the sampled matrices. at each step @xmath69 in the random walk, the coordinates @xmath70 of the particles are updated using @xmath71 where @xmath72 is a normally distributed random variate with zero mean and unit variance and @xmath73 is the time step. in the limit of small @xmath73, eq. ([scat_step_eq]) describes a process for sampling @xmath74. the matrix element of @xmath75 is estimated by integrating the local energy of the guiding function, @xmath76, along the random walk, @xmath77}.\]] the estimators for matrices @xmath62 and @xmath63 are @xmath78 where @xmath79 and @xmath80 are the local energies of the trial basis states. , at large exciton - exciton separation. the asymptotic form of the @xmath81-wave scattering states are symmetric (@xmath18) or antisymmetric (@xmath19) combinations of these configurations. this symmetric / antisymmetric form is used for the trial wavefunctions @xmath82 and @xmath83, using @xmath84 and @xmath85 from eq. ([trial_ex_scat_psi_eq]).] we now discuss the form for the exciton scattering functions @xmath86 and @xmath87. as mentioned before, inter - exciton exchange of particles complicates the definition of exciton - exciton separation. there are two configurations for well - separated excitons, as shown in fig. [configfig]. configuration i has the electrons and hole paired as @xmath88, @xmath89 ; and configuration ii as @xmath90, @xmath91. we choose wavefunctions @xmath92 and @xmath93 to represent these states, @xmath94 where @xmath95, and parameters in the function @xmath96 are variational. these wavefunctions represent two excitons in a relative s - wave state. since these are not eigenstates of the exchange operator for electrons @xmath97 or holes, @xmath98, we take linear combinations of the two for our trial wavefunctions, @xmath99 and @xmath100. for large separation of excitons, the exponential factors prohibit both configurations from simultaneously contributing to the wavefunction. thus, a node can be approximated in the scattering wavefunction by simply requiring that @xmath101 be zero for all @xmath102. the error introduced by this approximation is of order @xmath103, and is another limit on the use of small values for @xmath44. since we only do calculations for low energy scattering, large @xmath44, the lack of a well - defined exciton - exciton separation distance for short distances does not matter. of the dmc states relative to the vmc states as a function of dmc projection time @xmath65, for basis states @xmath104. the eigenvalue equation is given by eqs. ([eq : eigen])([eq : hmat]), where the hamiltonian and overlap matrices have been sampled using eqs. ([eq : overlapest]) and ([eq : hmatest]).] this method for calculating scattering properties is very sensitive to the energy spectra @xmath105. to get accurate energies, we do not try to construct and optimize elaborate variational wavefunctions, but rather use dmc to project the energy from trial wavefunctions of the form given in eq. ([trial_ex_scat_psi_eq]). the coefficients @xmath106, @xmath107, @xmath108, and @xmath109 are chosen to obey the cusp conditions on the wavefunction for small particle separations. the s - wave envelope functions @xmath101 are taken as solutions to an empirical exciton - exciton scattering potentials, @xmath110 where @xmath111 and @xmath112 have been self - consistently fit to approximate the energy spectrum of the four particle scattering states. we take the guiding function @xmath68 to have the same form as the @xmath82 wavefunctions with @xmath113^{1/2}$]. the parameters @xmath114 are chosen to bias sampling towards the collision : @xmath115 for @xmath18 states and @xmath116 for @xmath19 states to check for convergence of the energies in dmc, we plot the energy difference, @xmath117, as a function of projection time in fig. [evstfig]. we see convergence after a projection time of 3 @xmath118. , for (a) symmetric @xmath18 states, and (b) antisymmetric @xmath19 states, with @xmath119. the lowest energy curve in (a) is the biexciton with binding energy @xmath120. these functions @xmath121 determine the phase shifts @xmath122 and @xmath123, by the relationship in eq. ([kfrome]).] we thus find two energy spectra for each value of @xmath44, as shown in figs. [evsrnfig](a) and [evsrnfig](b). the spectra for the symmetric states @xmath18, show a clearly bound biexciton state, as seen in fig. [evsrnfig](a). the antisymmetric states @xmath19, as shown in fig. [evsrnfig](b), have no bound state. the binding energy of the biexciton is @xmath124, in agreement with other ground state calculations, and is insensitive to the position of the node @xmath125 because it is localized. in contrast, the delocalized scattering states are quite sensitive to @xmath125, and their dependence on @xmath125 is a measure of the elastic scattering matrix elements. for the two @xmath35-wave scattering states, for @xmath119, calculate using eq. ([kfrome]) and the data from fig. [evsrnfig].].coefficients for polynomomial fit to the low energy part of the phase shift functions for the case @xmath119. [cols="^,^,^,^,^",options="header ",] [scattab] the collision of two excitons with well defined initial and final spin states can be determined by decomposing the scattering events into the two channels, @xmath126 and @xmath127. using the change of basis matrix (table [spbasistab]), we find @xmath128, where the coefficients @xmath129 and @xmath130 for all non - zero s - wave scattering processes are given in table [scattab]. the s - wave scattering cross sections are given by @xmath131, where there is a factor of two enhancement due to the identical particle statistics. the spin - dependent cross sections take the form, @xmath132, \label{eq : cs}\end{aligned}\]] where @xmath133, @xmath134, and @xmath135 depend on the initial and final spin states and are tabulated in table [scattab]. the scattering lengths @xmath5 are given by @xmath136, where the derivatives of the phase shifts are determined from the linear coefficients in table [phasecoeftab]. ), for exciton - exciton scattering with @xmath119, for the processes : (a) singlet - singlet @xmath137 singlet - singlet, (b) triplet - triplet @xmath137 triplet - triplet for total spin @xmath138 (dashed line) and @xmath139 (solid line), (c) triplet - singlet @xmath137 triplet - singlet with @xmath140, and (d) triplet - triplet @xmath137 singlet - singlet and singlet - singlet @xmath137 triplet - triplet, both with @xmath140.] Results the calculated spin - dependent scattering lengths for the case @xmath119 are presented in the last column of table [scattab]. these are the low - energy limit the cross - sections shown in fig. [cs1fig]. in fig. [cs1fig] we have plotted the s - wave scattering cross sections versus scattering momentum for the case @xmath119 and all non - zero spin configurations. fig. [cs1fig](a) shows scattering of two singlet - excitons. scattering of two triplet - excitons is shown in fig. [cs1fig](b), where the solid line represents the spin aligned @xmath141 state, and the dashed line represents the @xmath142 state. the @xmath142 state scatters particularly strong because it has a large contribution from the @xmath143 channel, which is enhanced by the weakly bound biexciton. triplet - excitons in a relative @xmath144 state are spatially antisymmetric and thus have no s - wave scattering. we show s - wave scattering of triplet - excitons from singlet - excitons in fig. [cs1fig](c). this state has two distinguishable excitons, and can scatter by both s - wave and p - wave processes. as can be seen in table [scattab], the only contribution to the cross section is from the weaker @xmath145 channel. the coefficient for s - wave scattering is particularly small because only half the scattering process is symmetric (s - wave), and there is an additional factor of one - half which cancels the identical particle factor. there is also an triplet- to singlet - exciton conversion cross section given in fig. [cs1fig](d). although this is an inelastic process in experimental situations, it conserves energy according to our model hamiltonian because we do not have an explicit interband spin coupling. the conversion of two triplet - excitons to two singlet - excitons can be understood as an inter - exciton exchange of a pair of electrons (or holes). since the spins of the individual excitons do not correspond to symmetries of the hamiltonian, they need not remain constant during scattering. this conversion process is a physical consequence of the two inequivalent scattering channels @xmath143 and @xmath145. this effect has been reported in experimental@xcite and theoretical@xcite work on exciton scattering in quantum wells. and @xmath146 as a function of the electron - hole mass ratio. the divergence in @xmath147 near @xmath148 is due to the appearance of second bound biexciton state. solid lines are a guide to the eye.] the dependence of the cross sections on mass may be numerically studied by our methods. in fig. [avmfig] we show our calculated @xmath147 and @xmath146 as a function of mass ratio, @xmath149. we find that the scattering length is remarkably insensitive to the mass ratio for @xmath150 (corresponding to a wide range of semiconductors), but then diverges near @xmath151. this feature is lost in previously published theoretical treatments of exciton - exciton scattering. the divergence in @xmath147 near @xmath148 is due to the acquisition of a biexciton vibrational state. for @xmath119 the biexciton has no bound excited states, while a @xmath152 molecule has 15 bound vibrational states. our calculations have shown the first of these appears near @xmath148, with dramatic effects on the scattering length. the @xmath146 curve is relatively featureless because there are are no bound antisymmetric states in this range. we interpret the upward drift of @xmath146 for larger mass ratios as a systematic error due to difficulties in projecting states in excitonic systems with very different electron and hole masses. the heavy particle determines the projection time @xmath153 while the light particle determines the diffusion time step @xmath73. the difficulty in handling large mass ratios make the method (as presented here) complementary to calculations that use the born - oppenheimer approximation. it is important to realize that similar relationships must exist between the scattering length and other material parameters, such as the luttinger - kohn parameters describing realistic hole states, external strain, and spin - orbit coupling, to name a few. theoretical studies of such effects will need similar high - accuracy scattering calculations, but applied to more accurate hamiltonians, and are an area for future research. Conclusion to summarize, we have shown that there are several significant elastic scattering processes for excitons, and have given numerically exact values for a widely used theoretical model. we find strong triplet - triplet and singlet - singlet scattering, with weaker triplet - singlet scattering and triplet - triplet to singlet - singlet conversion processes. scattering is relatively insensitive to the mass ratio for @xmath154, but becomes very sensitive and actually diverges near @xmath155. dmc has been found be a good tool for this four - particle excited state calculation, since the detection of weakly bound states requires very accurate evaluation of the correlation energy. this computational approach should be extended in many ways. the extension to higher angular momentum states would give important corrections at higher scattering energies. application to biexciton - biexciton scattering are possible, but would be a bit more difficult because the scattering wavefunction would then have to describe eight interacting particles. most importantly, the method should be adapted to better hamiltonians so that the sensitivity of the scattering length to material properties for a wide range of materials can be studied. this approach should be quite useful for quantum well problems, which have similarities to the bulk problem study here, but have many more experimental parameters that can affect exciton - exciton interaction. we would like to thank k. ohara and j. carlson for useful discussions. this work was supported by nsf grant no. dmr 98 - 02373, computer resources at ncsa, and the department of physics at the university of illinois urbana - champaign. for example, the location of the scattering length divergence at @xmath156, reported later in this article, is extremely sensitive to an accurate treatment of correlation energy, since it relies on the appearance of a second, weakly bound biexciton state at that mass ratio.

we calculate cross sections for low energy elastic exciton - exciton scattering within the effective mass approximation. unlike previous theoretical approaches, we give a complete, non - perturbative treatment of the four - particle scattering problem. diffusion monte carlo is used to calculate the essentially exact energies of scattering states, from which phase shifts are determined. for the case of equal - mass electrons and holes, which is equivalent to positronium - positronium scattering, we find @xmath0 for scattering of singlet - excitons and @xmath1 for triplet - excitons, where @xmath2 is the excitonic radius. the spin dependence of the cross sections arises from the spatial exchange symmetry of the scattering wavefunctions. a significant triplet - triplet to singlet - singlet scattering process is found, which is similar to reported effects in recent experiments and theory for excitons in quantum wells. we also show that the scattering length can change sign and diverge for some values of the mass ratio @xmath3/@xmath4, an effect not seen in previous perturbative treatments.

cond-mat9907309

Introduction a magnitude limited complete census of variable stars in nearby dwarf galaxies allows important contributions to the star formation history of these systems. measurements of some variable stars can supply improved distance determinations for the host galaxies, others will provide important constraints for the population analysis. different classes of variables can further improve the understanding of the star formation history of these system, functioning as tracers of star formation during different epochs. we expect the data set of our long term monitoring program to be especially well suited to study the contents of red long - period variables and to re - investigate the paucity of cepheids with @xmath1 days as reported by sandage & carlson (1985). Observations and data reduction we selected a sample of six local group dwarf irregular galaxies which are visible with the 0.8 m telescope of our institute at mt. the names and additional data from the literature compilation by mateo (1998) are shown in table 1..names, variable star counts, absolute @xmath2-band brightness in mag, and current distance estimation in kpc for the dwarf galaxies observed in our project. the data are taken from the literature compilation by mateo (1995). for leo a the data are from the work of dolphin et. al (2002) and from this work. [cols="<,<,^,^,^,^,^ ",] @xmath3 this work the observations so far were carried out in @xmath4 and @xmath2-band, sparsely sampling a three year period starting with test observations in 1999. this part of the data set should be sensitive for long period variable stars with periods up to @xmath5 days. additional observations in @xmath4, @xmath2 and @xmath6-band were obtained during 3 observing campaigns at the 1.23 m telescope on calar alto densely sampling three two week long periods. these observations should provide a ground for a search for variable stars with shorter periods ranging from @xmath7 days up to @xmath8 days. the acquired data were bias subtracted, flat - fielded and cosmic ray rejected. then, the images from one night were astrometrically aligned to a common reference frame and combined with individual weights proportional to their @xmath9. for each epoch, consisting of all the stacked images of a single night, a difference image against a common deep reference frame was created using an implementation (gssl & riffeser, 2002, 2003) of the alard algorithm (alard & lupton, 1998). finally, these difference images were convolved with a stellar psf. to extract lightcurves from the reduced data, first all pixels deviating significantly (@xmath10) from the reference image in a minimum number of epochs @xmath11 were flagged, utilizing the complete per - pixel error propagation of our data reduction pipeline. then, using these coordinates as input, values and associated errors are read from the difference images and the lightcurve data are assembled. to search for periodic signals in the extracted difference fluxes, a lomb (1976) algorithm using the interpretation from scargle (1982) is applied. the photometric calibration was conducted using the hst data published by schulte - ladbeck et al. Preliminary results for the galaxies leo a, and ugca 92, we have a very good monitoring and a large fraction of the data passed already the pipeline. the leo a data set serves as test case : a total of 26 variable star candidates were detected. among them, we identified 16 secure long period variables (typical average values @xmath12, and @xmath13 period [days] @xmath14), and we have 8 further candidates for lpvs. in addition we were able to identify two good candidates for @xmath0 cephei stars with best fitting periods of 6.4 and 1.69 days. the later candidate was previously described by dolphin et al. (2002) as c2-v58 with a period of 1.4 days. the dolphin et al. period solution fails in deriving a reliable lightcurve with our data, yet, applying our period value to their data set yields reasonable results. the phase convolved lightcurves for the two @xmath0 cephei variables are shown in figure 1. Comparison with published work the color magnitude diagram shown in the left panel of figure 2 is based upon the hst data published by tolstoy et al. (1996) and schulte - ladbeck et al. flagged by bigger symbols are those variables from our sample that lie inside the hst field of view, two @xmath0 cephei variables in the instability strip (crosses) and the candidates for long term variability (triangles) in the regime of the red giants. tolstoy et al. (1996) based on ground - based data found a distance modulus for leo a of 24.2 and a resulting distance of 690 kpc (see also schulte - ladbeck et al.). this result got further support by the search for short periodic variables with the wiyn telescope within 3 consecutive days in dec. 2000 (dolphin et al. our data complement this dataset for longer periods. the right hand panel of figure 2 shows the period - luminosity (pl) relation of the smc shifted to the distance determined by tolstoy et al. the short period variables measured by dolphin coincide with the shown pl relation. the overplotted values for the two cepheids from our survey (crosses) support this relation also in the regime of longer periods. Summary we presented preliminary results for our survey for variable stars in a sample of irregular local group dwarf galaxies. for the leo a dwarf galaxy, the best analysed case so far, we already identified a total of 26 candidates for variability, 16 of these as long period variables and 2 @xmath0 cephei stars. we compared the later with the period - luminosity relation and the short period variables discussed by dolphin et al. we found, that our cepheids fully support their findings and the resulting distance estimate for leo a. this result is further in good agreement with the trgb distance (tolstoy et al., schulte - ladbeck et al.). the location of the lpvs in the color - magnitude diagram indicate that most of them are early asymptotic giant branch stars. while a complete census of these intermediate age stars is missing for most of the local group members, a proper statistic of their appearance can guide the reconstruction of the star formation history at the age of several gyr by - passing the age metalicity degeneracy inherent to color magnitude diagram studies. we like to thank drs. i. drozdovsky, c. maraston, r.e. schulte - ladbeck, and e. tolstoy for helpful discussion. we acknowledge the support of the calar alto and wendelstein staff. j. fliri and a. riffeser carried out some of our observations. the project is supported by the deutsche forschungsgemeinschaft grant ho 1812/3 - 1 and ho 1812/3 - 2. alard, c. & lupton, r. h.,, 503, 325 dolphin, a. e. et al. 2002,, 123, 3154 gssl c. a. & riffeser a. 2002,, 381, 1095 gssl, c. a. & riffeser, a. 2003, asp conf. 295, 229 lomb n. r. 1976,, 39, 447 mateo m. l. 1998,, 36, 435 sandage, a. & carlson, g. 1985,, 90, 1464 scargle j. d. 1982,, 263, 835 schulte - ladbeck r. et al. 2002,, 124, 896 tolstoy e. et al. 1996,, 116, 1244

dwarf galaxies in the local group provide a unique astrophysical laboratory. despite their proximity some of these systems still lack a reliable distance determination as well as studies of their stellar content and star formation history. we present first results of our survey of variable stars in a sample of six local group dwarf irregular galaxies. taking the leo a dwarf galaxy as an example we describe observational strategies and data reduction. we discuss the lightcurves of two newly found @xmath0 cephei stars and place them into the context of a previously derived p - l relation. finally we discuss the lpv content of leo a.

astro-ph0310704

Introduction biological aggregations such as fish schools, bird flocks, bacterial colonies, and insect swarms @xcite have characteristic morphologies governed by the group members interactions with each other and with their environment. the _ endogenous _ interactions, _ i.e. _, those between individuals, often involve organisms reacting to each other in an attractive or repulsive manner @xcite when they sense each other either directly by sound, sight, smell or touch, or indirectly via chemicals, vibrations, or other signals. a typical modeling strategy is to treat each individual as a moving particle whose velocity is influenced by social (interparticle) attractive and repulsive forces @xcite. in contrast, the _ exogenous _ forces describe an individual s reaction to the environment, for instance a response to gravity, wind, a chemical source, a light source, a food source, or a predator. the superposition of endogenous and exogenous forces can lead to characteristic swarm shapes ; these equilibrium solutions are the subject of our present study. more specifically, our motivation is rooted in our previous modeling study of the swarming desert locust _ schistocerca gregaria _ @xcite. in some parameter regimes of our model (presented momentarily), locusts self - organize into swarms with a peculiar morphology, namely a bubble - like shape containing a dense group of locusts on the ground and a flying group of locusts overhead ; see figure [fig : locust](bc). the two are separated by an unoccupied gap. with wind, the swarm migrates with a rolling motion. locusts at the front of the swarm fly downwards and land on the ground. locusts on the ground, when overtaken by the flying swarm, take off and rejoin the flying group ; see figure [fig : locust](cd). the presence of an unoccupied gap and the rolling motion are found in real locust swarms @xcite. as we will show throughout this paper, features of swarms such as dense concentrations and disconnected components (that is, the presence of gaps) arise as properties of equilibria in a general model of swarming. the model of @xcite is [eq : locusts] @xmath1 which describes @xmath2 interacting locusts with positions @xmath3. the direction of locust swarm migration is strongly correlated with the direction of the wind @xcite and has little macroscopic motion in the transverse direction, so the model is two - dimensional, _ i.e. _, @xmath4 where the @xmath5 coordinate is aligned with the main current of the wind and @xmath6 is a vertical coordinate. as the velocity of each insect is simply a function of position, the model neglects inertial forces. this so - called kinematic assumption is common in swarming models, and we discuss it further in section [sec : discretemodel]. the first term on the right - hand side of ([eq : locusts]) describes endogenous forces ; @xmath7 measures the force that locust @xmath8 exerts on locust @xmath9. the first term of @xmath7 describes attraction, which operates with strength @xmath10 over a length scale @xmath11 and is necessary for aggregation. the second term is repulsive, and operates more strongly and over a shorter length scale in order to prevent collisions. time and space are scaled so that the repulsive strength and length scale are unity. the second term on the right - hand side of ([eq : locusts]) describes gravity, acting downwards with strength @xmath12. the last term describes advection of locusts in the direction of the wind with speed @xmath13. furthermore, the model assumes a flat impenetrable ground. since locusts rest and feed while grounded, their motion in that state is negligible compared to their motion in the air. thus we add to ([eq : locusts]) the stipulation that grounded locusts whose vertical velocity is computed to be negative under ([eq : locusts]) remain stationary. as mentioned above, for some parameters, ([eq : locusts]) forms a bubble - like shape. this can occur even in the absence of wind, that is, when @xmath14 ; see figure [fig : locust](b). the bubble is crucial, for it allows the swarm to roll in the presence of wind. as discussed in @xcite, states which lack a bubble in the absence of wind do not migrate in the presence of wind. conditions for bubble formation, even in the equilibrium state arising in the windless model, have not been determined ; we will investigate this problem. some swarming models adopt a discrete approach as in our locust example above because of the ready connection to biological observations. a further advantage is that simulation of discrete systems is straightforward, requiring only the integration of ordinary differential equations. however, since biological swarms contain many individuals, the resulting high - dimensional systems of differential equations can be difficult or impossible to analyze. furthermore, for especially large systems, computation, though straightforward, may become a bottleneck. continuum models are more amenable to analysis. one well - studied continuum model is that of @xcite, a partial integrodifferential equation model for a swarm population density @xmath15 in one spatial dimension : @xmath16 the density @xmath17 obeys a conservation equation, and @xmath18 is the velocity field, which is determined via convolution with the antisymmetric pairwise endogenous force @xmath19, the one - dimensional analog of a social force like the one in ([eq : locusts]). the general model ([eq : introeq]) displays at least three solution types as identified in @xcite. populations may concentrate to a point, reach a finite steady state, or spread. in @xcite, we identified conditions on the social interaction force @xmath19 for each behavior to occur. these conditions map out a `` phase diagram '' dividing parameter space into regions associated with each behavior. similar phase diagrams arise in a dynamic particle model @xcite and its continuum analog @xcite. models that break the antisymmetry of @xmath19 (creating an asymmetric response of organisms to each other) display more complicated phenomena, including traveling swarms @xcite. many studies have sought conditions under which the population concentrates to a point mass. in a one - dimensional domain, collapse occurs when the force @xmath19 is finite and attractive at short distances @xcite. the analogous condition in higher dimensions also leads to collapse @xcite. one may also consider the case when the velocity includes an additional term describing an exogenous force, @xmath20 in this case, equilibrium solutions consisting of sums of point - masses can be linearly and nonlinearly stable, even for social forces @xmath19 that are repulsive at short distances @xcite. these results naturally lead to the question of whether a solution can be continued past the time at which a mass concentrates. early work on a particular generalization of ([eq : introeq]) suggests the answer is yes @xcite. for ([eq : introeq]) itself in arbitrary dimension, there is an existence theory beyond the time of concentration @xcite. some of the concentration solutions mentioned above are equilibrium solutions. however, there may be classical equilibria as well. for most purely attractive @xmath19, the only classical steady states are constant in space, as shown via a variational formulation of the steady state problem @xcite. however, these solutions are non - biological, as they contain infinite mass. there do exist attractive - repulsive @xmath19 which give rise to compactly - supported classical steady states of finite mass. for instance, in simulations of ([eq : introeq]), we found classical steady state solutions consisting of compactly supported swarms with jump discontinuities at the edges of the support @xcite. in our current work, we will find equilibria that contain both classical and nonclassical components. many of the results reviewed above were obtained by exploiting the underlying gradient flow structure of ([eq : introeq2]). there exists an energy functional @xmath21 = \frac{1}{2 } \int_\mathbb{r } \int_\mathbb{r } \rho(x) \rho(y) q(x - y)\,dx\,dy + \int_\mathbb{r } f(x)\rho(x)\,dx,\]] which is minimized under the dynamics. this energy can be interpreted as the continuum analog of the summed pairwise energy of the corresponding discrete (particle) model @xcite. we will also exploit this energy to find equilibrium solutions and study their stability. in this paper, we focus on equilibria of swarms and ask the following questions : * what sorts of density distributions do swarming systems make? are they classical or nonclassical? * how are the final density distributions reached affected by endogenous interactions, exogenous forces, boundaries, and the interplay of these? * how well can discrete and continuum swarming systems approximate each other? to answer these questions, we formulate a general mathematical framework for discrete, interacting swarm members in one spatial dimension, also subject to exogenous forces. we then derive an analogous continuum model and use variational methods to seek minimizers of its energy. this process involves solution of a fredholm integral equation for the density. for some choices of endogenous forces, we are able to find exact solutions. perhaps surprisingly, they are not always classical. in particular, they can involve @xmath0-function concentrations of mass at the domain boundary. the rest of this paper is organized as follows. in section [sec : formulation], we create the mathematical framework for our study, and derive conditions for a particular density distribution to be an equilibrium solution, and to be stable to various classes of perturbations. in sections [sec : repulsive] and [sec : morse], we demonstrate different types of swarm equilibria via examples. in section [sec : repulsive], we focus on purely repulsive endogenous interactions. we consider a bounded domain with no exogenous forces, a half - line subject to gravitational forces, and an unbounded domain subject to a quadratic exogenous potential, modeling attraction to a light, chemical, or nutrient source. for all three situations, we find exact solutions for swarm equilibria. for the first two examples, these equilibria consist of a density distribution that is classical in the interior of the domain, but contains @xmath0-functions at the boundaries. for the third example, the equilibrium is compactly supported with the density dropping discontinuously to zero at the edge of the support. for all three examples, we compare analytical solutions from the continuum framework to equilibria obtained from numerical simulation of the underlying discrete system. the two agree closely even for small numbers of discrete swarm members. section [sec : morse] is similar to section [sec : repulsive], but we now consider the more complicated case of endogenous interactions that are repulsive on short length scales and attractive over longer ones ; such forces are typical for swarming biological organisms. in section [sec : locust - ground], we revisit locust swarms, focusing on their bubble - like morphology as described above, and on the significance of dimensionality. in a one - dimensional model corresponding to a vertical slice of a wide locust swarm under the influence of social interactions and gravity, energy minimizers can reproduce concentrations of locusts on the ground and a group of locusts above the ground, but there can not be a separation between the two groups. however, a quasi - two - dimensional model accounting for the influence of the swarm s horizontal extent does, in contrast, have minimizers which qualitatively correspond to the biological bubble - like swarms. Mathematical formulation consider @xmath2 identical interacting particles (swarm members) in one spatial dimension with positions @xmath22. assume that motion is governed by newton s law, so that acceleration is proportional to the sum of the drag and motive forces. we will focus on the case where the acceleration is negligible and the drag force is proportional to the velocity. this assumption is appropriate when drag forces dominate momentum, commonly known in fluid dynamics as the low reynolds number or stokes flow regime. in the swarming literature, the resulting models, which are first - order in time, are known as _ kinematic models have been used in numerous studies of swarming and collective behavior, including @xcite). we now introduce a general model with both endogenous and exogenous forces, as with the locust model ([eq : locusts]). the endogenous forces act between individuals and might include social attraction and repulsion ; see @xcite for a discussion. for simplicity, we assume that the endogenous forces act in an additive, pairwise manner. we also assume that the forces are symmetric, that is, the force induced by particle @xmath9 on particle @xmath8 is the opposite of that induced by particle @xmath8 on particle @xmath9. exogenous forces might include gravity, wind, and taxis towards light or nutrients. the governing equations take the form [eq : discretesystem] @xmath23 eventually we will examine the governing equations for a continuum limit of the discrete problem. to this end, we have introduced a _ social mass _ @xmath24 which scales the strength of the endogenous forces so as to remain bounded for @xmath25. @xmath26 is the total social mass of the ensemble. ([eq : vee]) defines the velocity rule ; @xmath27 is the endogenous velocity one particle induces on another, and @xmath28 is the exogenous velocity. from our assumption of symmetry of endogenous forces, @xmath19 is odd and in most realistic situations is discontinuous at the origin. each force, @xmath28 and @xmath19, can be written as the gradient of a potential under the relatively minor assumption of integrability. as pointed out in @xcite, most of the specific models for mutual interaction forces proposed in the literature satisfy this requirement. many exogenous forces including gravity and common forms of chemotaxis do so as well. under this assumption, we rewrite ([eq : discretesystem]) as a gradient flow, @xmath29 where the potential @xmath30 is [eq : discrete_gradient] @xmath31 the double sum describes the endogenous forces and the single sum describes the exogenous forces. also, @xmath32 is the mutual interaction potential, which is even, and @xmath33 is the exogenous potential. the flow described by ([eq : discretegradient1]) will evolve towards minimizers of the energy @xmath30. up to now, we have defined the problem on @xmath34. in order to confine the problem to a particular domain @xmath35, one may use the artifice of letting the exogenous potential @xmath33 tend to infinity on the complement of @xmath35. while this discrete model is convenient from a modeling and simulation standpoint, it is difficult to analyze. presently, we will derive a continuum analog of ([eq : discretesystem]). this continuum model will allow us to derive equilibrium solutions and determine their stability via the calculus of variations and integral equation methods. to derive a continuum model, we begin by describing our evolving ensemble of discrete particles with a density function @xmath15 equal to a sum of @xmath0-functions. (for brevity, we suppress the @xmath36 dependence of @xmath17 in the following discussion.) our approach here is similar to @xcite. these @xmath0-functions have strength @xmath24 and are located at the positions of the particles : @xmath37 the total mass is @xmath38 where @xmath39 is the domain of the problem. using ([eq : deltafuncs]), we write the discrete velocity @xmath40 in terms of a continuum velocity @xmath41. that is, we require @xmath42 where @xmath43 by conservation of mass, the density obeys @xmath44 with no mass flux at the boundary. we now introduce an energy functional @xmath45 $] which is analogous to the discrete potential @xmath30 in ([eq : generic_potential]) : @xmath46 = \frac{1}{2 } \int_{{\omega}}\int_{{\omega}}\rho(x) \rho(y) q(x - y)\,dx\,dy + \int_{{\omega}}f(x)\rho(x)\,dx.\]] this expression follows from the discrete potential, @xmath21 = m { { \cal{w}}}(x_1,\ldots, x_n),\]] remembering the @xmath0-function definition of the density ([eq : deltafuncs]). the rate of energy dissipation is @xmath47}{dt } = - \int_\omega \rho(x) \left\{v(x)\right\}^2\,dx,\]] where we assume that there is no mass flux at the boundary of @xmath35. a consequence of this boundary condition is that under some conditions, mass may concentrate at the boundary of the domain, and we will later see this manifest. since energy is dissipated, we conclude that stable equilibria correspond to minimizers of @xmath48. imagine now that the energy ([eq : continuum_energy]) is defined for _ any _ density distribution @xmath17, not just ensembles of @xmath0-functions. we will find minimizers for the continuous energy ([eq : continuum_energy]) and show that they approximate solutions to the discrete problem in the limit of large @xmath2. to establish a correspondence between the two frameworks, consider a continuous distribution @xmath49 with total mass @xmath26. define the cumulative density function @xmath50 where the dummy coordinate @xmath51 is taken to the left of the support of @xmath49. we seek a discrete approximation of @xmath2 @xmath0-functions, @xmath52 the associated cumulative density function @xmath53 is @xmath54 & x = x_i,\quad i = 1,\ldots, n \\ i m & x_i < x < x_{i+1 }, \quad i = 1,\ldots, n-1\\ m & x > x_n, \end{cases}\]] where we have used the convention that integrating up to a @xmath0-function yields half the mass of integrating through it. to establish our correspondence, we require that @xmath55, which in turn determines the particle positions @xmath22. as @xmath25 for fixed @xmath26, this step function @xmath53 converges uniformly to @xmath56. the correspondence goes in the opposite direction as well. we can begin with an ensemble of @xmath0-functions @xmath57 placed at the positions of discrete swarm members, as shown in figure [fig : delta_schematic](a). we can find the corresponding cumulative density @xmath53 via ([eq : psid]) and interpolate to construct the continuum cumulative density @xmath56. the functions @xmath53 and @xmath56 are shown as the dotted red step function and the blue curve, respectively, in figure [fig : delta_schematic](b). we may then differentiate to find an approximation @xmath49, as shown in figure [fig : delta_schematic](c). we use this correspondence in sections [sec : repulsive] through [sec : locust - ground] to compare analytical results for the continuum system ([eq : pde]) with numerical simulations of the discrete system ([eq : discretesystem]). we close this subsection by reiterating why we have made a correspondence between the discrete and continuum systems. we use the continuous framework to find equilibrium solutions analytically via variational and integral equation methods. the correspondence above allows a direct comparison to numerical simulation of the discrete system. we use a variational calculation to determine conditions for a density distribution to be a minimizer of @xmath48. our starting point is the energy functional ([eq : continuum_energy]) subject to the mass constraint ([eq : mass_constraint]). let @xmath58 here @xmath59 is an equilibrium solution of mass @xmath26 and @xmath60 is a small perturbation of zero mass, so we have @xmath61 inspired by biological observations of swarms, we focus on equilibria with finite extent and take the support of @xmath59 to be a finite subset of the domain @xmath35. we refer to the support of @xmath59 as @xmath62. this assumption, combined with the fact that the density is nonnegative, restricts the perturbation @xmath63 to be nonnegative on @xmath64, the complement of @xmath62. we write @xmath65 = w[{{\bar \rho } }] + \epsilon w_1[{{\bar \rho}},{{\tilde \rho } }] + \epsilon^2 w_2[{{\tilde \rho}},{{\tilde \rho}}],\]] where @xmath66 and @xmath67 are the first and second variations respectively. this expression is exact because @xmath48 is quadratic in @xmath17 (see eq. [eq : continuum_energy]). we analyze these variations to determine necessary and sufficient conditions for a candidate solution @xmath59 to be a minimizer of @xmath48. our strategy is to consider two classes of perturbations @xmath63. first, we consider perturbations whose support lies in @xmath62. in order for @xmath59 to be extremal, @xmath66 must vanish. for it to be a minimizer, @xmath67 must be positive. since ([eq : expand_w]) is exact, @xmath68 guarantees that @xmath59 is both a local and global minimum with respect to this first class of perturbations. the second class of perturbations we consider (of which the first class is a subset) consists of perturbations on the entire domain @xmath39. as mentioned above, these perturbations must be nonnegative in @xmath64 in order to maintain positivity of @xmath17. a necessary condition for @xmath59 to be a local minimizer is that @xmath69 for this class of perturbations. if in addition @xmath67 remains positive for this larger class of perturbations, then @xmath59 is a global minimizer as well. we derive the first variation @xmath66 by substituting ([eq : expand_rho]) into ([eq : expand_w]) and expanding to first order in @xmath70, which yields @xmath71 = \int_\omega { { \tilde \rho}}\left [\int_\omega q(x - y) { { \bar \rho}}(y)\,dy+f(x) \right] \,dx.\]] consider the first class of perturbations, whose support lies in @xmath62. as the perturbation @xmath63 is arbitrary and of zero mass, for the first variation @xmath66 to vanish, it must be true that @xmath72 the same result can be found by a lagrange multiplier argument including the constant mass as a constraint. the multiplier @xmath73 has a physical interpretation ; it is the energy per unit mass an additional test mass would feel due to the exogenous potential and the interaction with @xmath59. thus far, we have shown that a necessary condition for @xmath59 to be an equilibrium solution is that it satisfies the fredholm integral equation of the first kind for the nonnegative density @xmath59, @xmath74 = \lambda - f(x), \quad { \cal i}[{{\bar \rho}}(x)] \equiv \int_{{\omega_{{{\bar \rho}}}}}q(x - y) { { \bar \rho}}(y)\,dy, \label{eq : fie}\]] as well as the mass constraint ([eq : mass1]). in order for @xmath59 to be a minimizer with respect to the first class of perturbations, the second variation must be positive. substituting ([eq : expand_rho]) into ([eq : expand_w]) yields @xmath75 = \int_{{\omega_{{{\bar \rho}}}}}\int_{{{\omega_{{{\bar \rho } } } } } } q(x - y) { { \tilde \rho}}(x) { { \tilde \rho}}(y)\,dx\,dy.\]] the sign of @xmath67 can be assessed in a number of ways. we first derive a sufficient condition on the fourier transform of @xmath32 for @xmath67 to be nonnegative for the first class of perturbations. define the fourier transform @xmath76 then we have @xmath77 & = & \int_{{\omega_{{{\bar \rho}}}}}\int_{{{\omega_{{{\bar \rho } } } } } } q(x - y) { { \tilde \rho}}(x) { { \tilde \rho}}(y)\,dx\,dy,\\ & = & \int_{-\infty}^{\infty } \int_{-\infty}^{\infty } q(x - y) { { \tilde \rho}}(x) { { \tilde \rho}}(y)\,dx\,dy, \\ & = & \int_{-\infty}^{\infty } { { \tilde \rho}}(x) \left [q(x) * { { \tilde \rho}}(x) \right]\,dx, \\ & = & \frac{1}{2\pi } \int_{-\infty}^{\infty } | \widehat{{{\tilde \rho}}}(k)|^2 \widehat{q}(k)~dk.\end{aligned}\]] we have used the fact that @xmath63 is compactly supported to extend the range of integration to infinity. we have also used the convolution theorem, parseval s theorem, and the fact that @xmath63 is real. we see, then, that @xmath78 is a sufficient condition for @xmath68 (assuming a nontrivial perturbation). as shown in @xcite, this condition is actually equivalent to that for the linear stability of a constant density state in the absence of exogenous forces. a necessary and sufficient condition for @xmath68 for the first class of perturbations comes from considering the spectrum of @xmath79 in @xmath62. note that ([eq : second_variation]) may be written as @xmath80 = \int_{{\omega_{{{\bar \rho}}}}}{\cal i}[{{\tilde \rho}}]{{\tilde \rho}}(x)\, dx \equiv \langle { { \tilde \rho } }, { \cal i}[{{\tilde \rho } }] \rangle,\]] where the angle brackets denote the usual @xmath81 inner product on @xmath62. if the eigenvalues of the integral operator @xmath79 are positive, then @xmath68 (again assuming a nontrivial perturbation). we now turn to the second class of perturbations, which have support in @xmath39 and which are positive in @xmath64. to analyze these perturbations, we extend the definition of the constant @xmath73 to a function @xmath82 that is defined over all of @xmath35. we set @xmath83 trivially from ([eq : lagrange]), @xmath84 for @xmath85. we now rewrite the first variation as @xmath86 = \int_{{\omega}}{{\tilde \rho}}(x) \lambda(x)\,dx,\]] directly from ([eq : first_variation]). remembering that @xmath87 in @xmath64 and that @xmath63 has zero mass in @xmath39, we see that a necessary and sufficient condition for @xmath69 is @xmath88 in @xmath64, that is @xmath89 physically, this guarantees that a parcel of mass transported from @xmath62 to its complement increases the total energy. if we wish for @xmath59 to be a minimizer with respect to the second class of perturbations, it suffices for @xmath68 (for example, by having @xmath78). however, this condition is not necessary. the necessary condition is that @xmath90 + w_2[{{\tilde \rho}},{{\tilde \rho } }] > 0 $] for nontrivial perturbations, which follows from ([eq : expand_w]) being exact. to summarize, we have obtained the following results : * equilibrium solutions @xmath59 satisfy the fredholm integral equation ([eq : fie]) and the mass constraint ([eq : mass1]). * the solution @xmath59 is a local and global minimizer with respect to the first class of perturbations (those with support in @xmath62) if @xmath67 in ([eq : second_variation]) is positive. * the solution @xmath59 is a local minimizer with respect to the second (more general zero - mass) class of perturbations if @xmath59 satisfies ([eq : stable]). if in addition @xmath67 is positive for these perturbations, then @xmath59 is a global minimizer as well. in practice, we solve the integral equation ([eq : fie]) to find candidate solutions. then, we compute @xmath82 to determine whether @xmath59 is a local minimizer. finally, when possible, we show the positivity of @xmath67 to guarantee that @xmath59 is a global minimizer. as the continuum limit replaces individual particles with a density, we need to make sure the continuum problem inherits a physical interpretation for the underlying problem. if we think about perturbing an equilibrium configuration, we note that mass can not `` tunnel '' between disjoint components of the solution. as such we define the concept of a multi - component swarm equilibrium. suppose the swarm s support can be divided into a set of @xmath24 disjoint, closed, connected components @xmath91, that is @xmath92 we define a swarm equilibrium as a configuration in which each individual swarm component is in equilibrium, @xmath93 we can still define @xmath82 in @xmath39 @xmath94 + f(x) = \int_{{\omega_{{{\bar \rho}}}}}q(x - y) { { \bar \rho}}(y)~dy + f(x),\]] but now @xmath95 in @xmath96. we can now define a swarm minimizer. we say a swarm equilibrium is a swarm minimizer if @xmath97 for some neighborhood of each component @xmath96 of the swarm. in practice this means that the swarm is an energy minimizer for infinitesimal redistributions of mass in the neighborhood of each component. this might also be called a lagrangian minimizer in the sense that the equilibrium is a minimizer with respect to infinitesimal lagrangian deformations of the distributions. it is crucial to note that even if a solution @xmath59 is a global minimizer, other multi - component swarm minimizers may still exist. these solutions are local minimizers and consequently a global minimizer may not be a global attractor under the dynamics of ([eq : pde]). Examples with a repulsive social force in this section we discuss the minimization problem formulated in section [sec : formulation]. it is helpful for expository purposes to make a concrete choice for the interaction potential @xmath32. as previously mentioned, in many physical, chemical, and biological applications, the pairwise potential @xmath32 is symmetric. additionally, repulsion dominates at short distances (to prevent collisions) and the interaction strength approaches zero for very long distances. a common choice for @xmath32 is the morse potential with parameters chosen to describe long - range attraction and short - range repulsion @xcite. for the remainder of this section, we consider a simpler example where @xmath32 is the laplace distribution @xmath98 which represents repulsion with strength decaying exponentially in space. when there is no exogenous potential, @xmath99, and when the domain is infinite, _ e.g. _, @xmath100, the swarm will spread without bound. the solutions asymptotically approach the barenblatt solution to the porous medium equation as shown in @xcite. however, when the domain @xmath39 is bounded or when there is a well in the exogenous potential, bounded swarms are observed both analytically and numerically, as we will show. figure [fig : repulsion_schematic] shows solutions @xmath101 for three cases : a bounded domain with no exogenous potential, a gravitational potential on a semi - infinite domain, and a quadratic potential well on an infinite domain. in each case, a bounded swarm solution is observed but the solutions are not necessarily continuous and can even contain @xmath0-function concentrations at the boundaries. we discuss these three example cases in detail later in this section. first, we will formulate the minimization problem for the case of the laplace potential. we will attempt to solve the problem classically ; when the solution has compact support contained within the domain we find solutions that are continuous within the support and may have jump discontinuities at the boundary of the support. however, when the boundary of the support coincides with the boundary of the domain, the classical solution may break down and it is necessary to include a distributional component in the solution. we also formulate explicit conditions for the solutions to be global minimizers. we then apply these results to the three examples mentioned above. recall that for @xmath59 to be a steady solution, it must satisfy the integral equation ([eq : fie]) subject to the mass constraint ([eq : mass1]). for @xmath101 to be a local minimizer, it must also satisfy ([eq : stable]), @xmath102 finally, recall that for a solution @xmath59 to be a global minimizer, the second variation ([eq : second_variation]) must be positive. we saw that if @xmath78, this is guaranteed. for ([eq : laplace]), @xmath103 and so for the remainder of this section, we are able to ignore the issue of @xmath67. any local minimizer that we find will be a global minimizer. additionally, for the remainder of this section, we restrict our attention to cases where the support of the solution @xmath62 is a single interval in @xmath39 ; in other words, the minimizing solution has a connected support. the reason that we are able to make this restriction follows from the notion of swarm minimization, discussed above. in fact, we can show that there are no multi - component swarm minimizers for the laplace potential as long as the exogenous potential @xmath104 is convex, that is, @xmath105 on @xmath39. to see this, assume we have a swarm minimizer with a at least two disjoint components. consider @xmath82 in the gap between two components so that @xmath106. we differentiate @xmath82 twice to obtain @xmath107 note that @xmath108 as @xmath109 in @xmath64. @xmath105 by assumption. consequently, @xmath110 in @xmath64 and so @xmath82 is convex upwards in the gap. also, @xmath111 at the endpoints of the gap. we conclude from the convexity that @xmath82 must be less than @xmath112 near one of the endpoints. this violates the condition of swarm minimization from the previous section, and hence the solution is not a swarm minimizer. since swarm minimization is a necessary condition for global minimization, we now, as discussed, restrict attention to single - component solutions. for concreteness, assume the support of the solution @xmath59 is @xmath113 $]. we transform the integral equation ([eq : fie]) for @xmath59 into a differential equation by noting that @xmath114 is the green s function of the differential operator @xmath115, so that @xmath116, where @xmath117 is the dirac @xmath0-function. applying @xmath118 to both sides of ([eq : fie]) yields the dirac @xmath0-function under the integral. integration and the sifting property of the @xmath0 function lead to @xmath119 = \frac{\lambda}{2}-\frac{1}{2}\left [f- f''(x) \right] \quad { \rm in } \ { { \omega_{{{\bar \rho}}}}}. \label{eq : fie - sol}\]] equation ([eq : fie - sol]) is a necessary condition on a solution @xmath120 but not sufficient. in fact, to verify the candidate solution ([eq : fie - sol]), we must substitute back into the governing integral equation ([eq : fie]). we find @xmath121 & \equiv & \int_\omega e^{-|x - y| } \left \ { \frac{\lambda}{2}-\frac{1}{2}\left [f(y)- f''(y) \right] \right \ } dy, \\ & = & \lambda - f(x) + \frac{e^{\alpha - x}}{2}[f(\alpha)-f'(\alpha)-\lambda] \\ & & \mbox { } + \frac{e^{x-\beta}}{2}[f(\beta)+f'(\beta)-\lambda], \nonumber\end{aligned}\]] where the terms other than @xmath122 must vanish for ([eq : fie - sol]) to be a solution. these error terms are spanned by @xmath123 which is the null space of @xmath124. the constraint that these terms must vanish leads to the conditions @xmath125 for a given @xmath104, both these conditions will only be satisfied for particular choices of @xmath126 and @xmath127. the allowed @xmath126 and @xmath127 then specify @xmath73 which in turn determines the mass @xmath26 through ([eq : mass1]). as we discuss below, the solution derived here is sometimes, but not always, a true minimizer of the energy @xmath48. what we have presented above is the classical view of the solution of ([eq : fie]). the difficulty with this calculation is that we expect on physical grounds that for every mass @xmath26 and interval @xmath128 there should be a minimizer. it is easy to see that if @xmath33 and @xmath32 are bounded from below, the energy @xmath48 is bounded from below also. let @xmath129 then directly from ([eq : continuum_energy]), @xmath130 since @xmath48 is bounded from below, solutions to the minimization problem exist in the space of measure valued functions @xcite. sometimes these solutions are not classical ; when there is finite attraction at small distances, mass can concentrate in a @xmath0-function @xcite. for the more biologically relevant case of repulsion at short scales, in free space and in the absence of an external potential, solutions are classical for all time @xcite. we now show that (under sufficient hypotheses) a @xmath0-function can not occur in the interior of @xmath62. though we have restricted ourselves (above) to considering the laplace potential ([eq : laplace]), this is true more generally. in fact, suppose that at a point @xmath51 in the interior of @xmath39, * the exogenous potential @xmath104 is twice - differentiable in a neighborhood of @xmath51, * @xmath114 is twice - differentiable for @xmath131, * @xmath114 has nonzero repulsion at short distances, that is, @xmath132 then a minimizer does not contain a @xmath0-function at @xmath133. to see this, we will compare the energy of a density containing a @xmath0-function at @xmath133 to one where the @xmath0-function has been replaced by a narrow, unit mass top - hat distribution @xmath134, @xmath135 and show that the energy is reduced by @xmath136. consider a candidate distribution with a @xmath0-function of mass @xmath24 at @xmath51, @xmath137. define the change of energy @xmath138 - w[{{\bar \rho}}_0 + m \delta(x - x_0)].\]] by direct calculation using ([eq : continuum_energy]), @xmath139 \,dx\,dy \nonumber \\ & & \mbox{}+ m \int_{{\omega}}f(x) \left[\delta_\varepsilon(x - x_0) - \delta (x - x_0) \right]\,dx. \nonumber\end{aligned}\]] note that near @xmath140, @xmath141 and @xmath142. expanding to @xmath143, only the first term of ([eq : nodeltas]) persists, yielding @xmath144 for @xmath145, corresponding to nonzero repulsion at short distances, @xmath146, indicating that the energy is reduced by replacing the @xmath0-function with a narrow top - hat. we conclude that @xmath0-functions may not occur in the interior of @xmath39 under the assumed conditions. however, the above reasoning breaks down for @xmath0-functions on the domain boundary, where they may not be replaced by a narrow, symmetric top - hat distribution. based on the result above, we introduce a candidate solution @xmath147 where @xmath148 and @xmath149 are to be determined. the classical solution @xmath150 is supplemented with @xmath0-functions at the boundary of @xmath62. we will show, in agreement with the calculation above, that @xmath148 and @xmath149 must vanish unless the boundary of the support, @xmath62, coincides with the boundary of the domain, @xmath39. we verify the new candidate solution ([eq : fie - sol2]) by substituting back into the governing equation ([eq : fie]). we find @xmath151 & \equiv & \int_\omega e^{-|x - y| } \left \ { { \bar \rho(y) } + a \delta(y-\alpha) + b\delta(y-\beta) \right \}\,dy, \\ & = & \lambda - f(x) + \frac{e^{\alpha - x}}{2}[2a+f(\alpha)-f'(\alpha)-\lambda] \\ & & \mbox{}+\frac{e^{x-\beta}}{2}[2b+f(\beta)+f'(\beta)-\lambda], \nonumber\end{aligned}\]] where the terms other than @xmath122 must vanish for ([eq : fie - sol2]) to be a solution. this constraint leads to the following conditions on @xmath148 and @xmath149, the coefficients of the nonclassical parts of the solution : @xmath152 because @xmath73 is undetermined as of yet, we can find a solution for any mass @xmath26. substituting the solution @xmath101 into the mass constraint ([eq : mass1]) yields @xmath153,\\ & = & \lambda \left (1 + \frac{(\beta -\alpha)}{2 } \right) -\frac{1}{2 } \left [f(\alpha)+f(\beta)+ \int_{{\omega_{{{\bar \rho}}}}}f(x)\,dx \right].\end{aligned}\]] solving for @xmath73 in terms of @xmath26 yields @xmath154 we ve shown that for any @xmath113 $] and any mass @xmath26 we can find a solution to ([eq : fie]) with @xmath101 smooth in the interior and with a concentration at the endpoints. however, we havent yet addressed the issue of @xmath101 being non - negative, nor have we considered whether it is a minimizer. we next consider whether the extremal solution @xmath59 is a minimizer, which involves the study of ([eq : stable2]). we present a differential operator method that allows us to compute @xmath82 and deduce sufficient conditions for @xmath59 to be a minimizer. we start by factoring the differential operator @xmath155 where @xmath156. applying these operators to the interaction potential @xmath32, we see that @xmath157 substituting @xmath59 in ([eq : fie - sol2]) into our definition of @xmath82 in ([eq : stable2]) yields @xmath158 now consider applying @xmath159 to ([eq : one]) at a point @xmath5 in @xmath62. we see that @xmath160 = { \mathcal{d}^- } [\lambda],\]] where we ve used the fact that @xmath161 in @xmath62. if we let @xmath162 and let @xmath6 decrease to zero, the integral term vanishes and @xmath163 solving for @xmath148 yields the first half of ([eq : ab]). a similar argument near @xmath164 yields the value of @xmath149. assuming @xmath126 does not coincide with an endpoint of @xmath39, we now consider the region @xmath165, which is to the left of the support @xmath62. again, applying @xmath159 to ([eq : fie]) simplifies the equation ; we can check that both the integral term and the contribution from the @xmath0-functions are annihilated by this operator, from which we deduce that @xmath166 = 0 \qquad \rightarrow \qquad f(x)- \lambda(x) = ce^x,\]] where @xmath167 is an unknown constant. a quick check shows that if @xmath104 is continuous, then @xmath82 is continuous at the endpoints of @xmath62 so that @xmath168. this in turn determines @xmath167, yielding @xmath169e^{x-\alpha } \qquad { \rm for } \quad x \leq \alpha.\]] a similar argument near @xmath164 yields @xmath170e^{\beta -x } \qquad { \rm for } \quad x \geq \beta.\]] as discussed in section [sec : minimizers], for @xmath101 to be a minimizer we wish for @xmath88 for @xmath171 and @xmath172. a little algebra shows that this is equivalent to [eq : mincon] @xmath173 if @xmath126 and @xmath127 are both strictly inside @xmath39, then ([eq : mincon]) constitutes sufficient conditions for the extremal solution @xmath59 to be a global minimizer (recalling that @xmath68). we may also derive a necessary condition at the endpoints of the support from ([eq : mincon]). as @xmath5 increases to @xmath126, we may apply lhpital s rule and this equation becomes equivalent to the condition @xmath174, as expected. a similar calculation letting @xmath5 decrease to @xmath127 implies that @xmath175. however, since @xmath59 is a density, we are looking for positive solutions. hence, either @xmath176 or @xmath126 coincides with the left endpoint of @xmath39. similarly, either @xmath177 or @xmath127 coincides with the right edge of @xmath39. this is consistent with the result ([eq : nodeltas2]) which showed that @xmath0-functions can not occur in the interior of @xmath39. in summary, we come to two conclusions : * a globally minimizing solution @xmath59 contains a @xmath0-function only if a boundary of the support of the solution coincides with a boundary of the domain. * a globally minimizing solution @xmath59 must satisfy ([eq : mincon]). we now consider three concrete examples for @xmath39 and @xmath104. we model a one - dimensional biological swarm with repulsive social interactions described by the laplace potential. we begin with the simplest possible case, namely no exogenous potential, @xmath178 and a finite domain which for convenience we take to be the symmetric interval @xmath179 $]. as @xmath180 we know from ([eq : singlecomponent]) that the minimizing solution has a connected support, _ i.e. _, it is a single component. we will see that the minimizing solution has an equipartition of mass between @xmath0-functions at the boundaries of the domain and a constant solution in the interior, as shown schematically in figure [fig : repulsion_schematic](a). we now proceed with calculating the solution. from ([eq : fie - sol]), we find that @xmath181 the nonclassical solution is @xmath182 eq. ([eq : mform]) gives @xmath73 as @xmath183 eq. ([eq : ab]) specifies @xmath148 and @xmath149 as @xmath184 since @xmath185, it follows that @xmath186 and @xmath66 vanishes according to ([eq : w1lambda]). therefore, the solution is a global minimizer. the solution @xmath101 is shown schematically in figure [fig : repulsion_schematic](a). figure [fig : repulsion_numerics](a) compares analytical and numerical results for an example case where we take the total mass to be @xmath187 and the finite domain to be @xmath188 $] with @xmath189. cross - hatched boxes indicate the boundary of the domain. the solid line is the classical solution @xmath59. dots correspond to the numerically - obtained equilibrium of the discrete system ([eq : discretesystem]) with @xmath190 swarm members. the density at each lagrangian grid point is estimated using the correspondence discussed in section ([sec : contmodel]) and pictured in figure [fig : delta_schematic]. each `` lollipop '' at the domain boundary corresponds to a @xmath0-function of mass @xmath191 in the analytical solution, and simultaneously to a superposition of @xmath192 swarm members in the numerical simulation. hence, we see excellent agreement between the continuum minimizer and the numerical equilibrium even for this relatively small number @xmath190 of lagrangian points. we now consider repulsive social interactions and an exogenous gravitational potential. the spatial coordinate @xmath193 describes the elevation above ground. consequently, @xmath39 is the semi - infinite interval @xmath194. then @xmath195 with @xmath196, shown in figure [fig : repulsion_schematic](f). as @xmath180 we know from ([eq : singlecomponent]) that the minimizing solution has a connected support, _ i.e. _, it is a single component. moreover, translating this component downward decreases the exogenous energy while leaving the endogenous energy unchanged. thus, the support of the solution must be @xmath197 $], potentially with @xmath198. in fact, we will see that there are two possible solution types depending on @xmath12. for strong enough values of @xmath12, @xmath127 indeed equals zero, and the mass accumulates on the ground @xmath140, as shown schematically in figure [fig : repulsion_schematic](d). for weaker @xmath12, the mass is partitioned between a @xmath0-function on the ground and a classical solution for @xmath131, as shown in figure [fig : repulsion_schematic](e). we now proceed with calculating the solution. from ([eq : fie - sol]), we find that @xmath199 the nonclassical solution is @xmath200 eq. ([eq : mform]) gives @xmath73 as @xmath201 eq. ([eq : ab]) specifies @xmath148 and @xmath149 as @xmath202 for @xmath59 to be a global minimizer, it must be true that @xmath177 as shown in section [sec : funcmin]. solving, we find that @xmath203. for @xmath204, we see that @xmath205 in which case the minimizing solution is @xmath206 it follows directly that @xmath207. figure [fig : repulsion_schematic](e) shows a schematic drawing of this solution. we still must consider the condition ([eq : minconb]). to see it is satisfied, first note that @xmath208. then ([eq : minconb]) becomes @xmath209 we note that by lhpital s rule that equality is obtained as @xmath210. to show the inequality holds for @xmath211 let @xmath212 and @xmath213 (so @xmath214). the inequality becomes @xmath215 we can interpret the left - hand side as the slope of a chord connecting the points @xmath216 and @xmath217. consequently, if the function @xmath218 is concave upward, the slope of the chord will be increasing as @xmath219 increases away from @xmath220, and the inequality will hold. recalling that @xmath221, we compute @xmath222^{\prime \prime } = g / s$] which is positive, and hence the solution is globally stable. for @xmath223 our previous calculation naively implies @xmath224. since @xmath127 can not be negative, the minimizer in this case is a @xmath0-function at the origin, namely @xmath225, shown in figure [fig : repulsion_schematic](d). in this case, @xmath226 from ([eq : gravitylambda]) and @xmath227 from ([eq : lambdaright]). it follows that @xmath228 the first inequality follows from a taylor expansion. the second follows from our assumption @xmath229. since @xmath88 the solution is a global minimizer. in summary, there are two cases. when @xmath223, the globally stable minimizer is a @xmath0-function at the origin. when @xmath230 there is a globally stable minimizer consisting of a @xmath0-function at the origin which is the left - hand endpoint of a compactly - supported classical swarm. the two cases are shown schematically in figures [fig : repulsion_schematic](de). figure [fig : repulsion_numerics](b) compares analytical and numerical results for the latter (@xmath230) case with @xmath187 and @xmath231. we use @xmath190 swarm members for the numerical simulation. the numerical (dots) and analytical (line) @xmath232 agree, as does the nonclassical part of the solution, pictured as the `` lollipop '' which represents a superposition of @xmath233 swarm members in the numerical simulation having total mass @xmath234, and simultaneously a @xmath0-function of mass @xmath235 in the analytical solution. we now consider the infinite domain @xmath100 with a quadratic exogenous potential well, pictured in figure [fig : repulsion_schematic](c). this choice of a quadratic well is representative of a generic potential minimum, as might occur due to a chemoattractant, food source, or light source. thus @xmath236 where @xmath237 controls the strength of the potential. as @xmath238 we know from ([eq : singlecomponent]) that the minimizing solution is a single component. we take the support of the solution to be @xmath113 $]. we will find that the compactly supported density is classical and has an inverted parabolic shape, shown in figure [fig : repulsion_schematic](b). our calculation proceeds as follows. we know from section [sec : funcmin] that because @xmath126 and @xmath127 are assumed to be finite, @xmath239 and so the solution is classical. from ([eq : fie - sol]), we find that @xmath240 and, from ([eq : ab]), @xmath241 eliminating @xmath73 from these equations and recalling that @xmath242, it follows that @xmath243. hence, the solution is symmetric around the center of the potential. for convenience, we now define @xmath244. ([eq : mform]) gives @xmath73 in terms of the mass @xmath26 as @xmath245 however, from the second half of ([eq : abquad]) we know that @xmath246 equating these two expressions for @xmath73 yields that @xmath247 note that @xmath248 increases monotonically from @xmath249 with increasing @xmath26. finally, writing the solution @xmath59 in terms of @xmath26, we have @xmath250 & |x| \leq \left(\frac{3m}{2\gamma } + 1\right)^{1/3 } - 1 \\ 0 & |x| > \left(\frac{3m}{2\gamma } + 1\right)^{1/3 } - 1. \end{cases}\]] it follows directly that @xmath207, and that there is a discontinuity at the edge of the support. we now must show that the candidate @xmath59 is a global minimizer, which is done by demonstrating ([eq : mincon]). by the same argumentation used for the right endpoint in section [sec : grav], it suffices to show that @xmath218 is concave up for @xmath251 which follows directly from algebra. hence, the solution is a global minimizer. the solution @xmath59 is shown schematically in figure [fig : repulsion_schematic](b). figure [fig : repulsion_numerics](b) compares analytical and numerical results for the case with @xmath187 and @xmath252, with @xmath253 swarm members used for the numerical solution. the numerical (dots) and analytical (line) @xmath59 agree closely. Examples with a morse-type social force in many physical, chemical, and biological applications, the pairwise potential @xmath32 is isotropic, with a repulsion dominating at short distance and the interaction strength approaching zero for very long distances. a common choice for @xmath32 is the morse potential @xmath254 with appropriately chosen parameters. here, @xmath5 is the distance between particles, @xmath255 is the characteristic length scale of attraction, and @xmath256 is the characteristic velocity induced by attraction. we have scaled the characteristic repulsive strength and length scale to be unity. in this section we are concerned with the solution of ([eq : mass1]) and ([eq : fie]) with @xmath32 given by ([eq : morse]). the morse potential has been studied extensively and has become a canonical model for attractive - repulsive interactions @xcite. a key characteristic of the potential ([eq : morse]) is whether or not the parameters @xmath257 are chosen to be in the _ h - stable _ or the _ catastrophic _ regime ; see @xcite for a review. consider ([eq : discretesystem]) with @xmath178. if the parameters @xmath257 are chosen in the h - stable regime @xmath258, then as the number of particles @xmath2 increases, the density distribution of particles approaches a constant, as does the energy per particle. stated differently, the particles form a crystalline lattice where the nearest - neighbor distance is approximately equal for all particles (excluding edge effects). as more individuals are added to the group, the inter - organism spacing is preserved and the group grows to cover a larger spatial region. if the parameters are chosen outside of the h - stable regime, _ i.e. _, @xmath259, the system is catastrophic. in this case, the energy per particle is unbounded as @xmath260, and particles pack together more and more closely as n increases. our work in @xcite classifies the asymptotic behaviors of the continuum problem ([eq : pde]) with morse - type interactions in the absence of an external potential, @xmath99, and on the real line. in the h - stable regime, the continuum model displays spreading density profiles, while in the catastrophic regime, it forms compactly - supported steady states. the distinction between catastrophic and h - stable is related to the fourier transform of the potential @xmath114. note that ([eq : morse]) has fourier transform [eq : qhatmorse] @xmath261}{(1+{k}^{2})(1+(kl)^2)}.\end{aligned}\]] the condition for h - stability is equivalent to @xmath262. in this case, @xmath78 for all @xmath263, and from section [sec : minimizers], this is sufficient for @xmath68, or equivalently, linear stability of constant density states. in the catastrophic case, @xmath264 intuitively, in the catastrophic case, the constant density state is unstable to long wave perturbations. the system is attracted to states of finite extent shorter than the length scale of the instability. in the h - stable case, the constant density state is stable to perturbations, and initial density profiles spread evenly to become flat. we will now study minimizers for the case of morse - type interactions, and will see qualitatively different solutions for the catastrophic and h - stable cases. we follow the procedure used in section [sec : repulsive], namely we first look for a classical solution on the interior of @xmath265 $] and then allow for @xmath0-functions at the boundaries. once again, we will see that minimizers contain @xmath0-functions only when the boundary of the support, @xmath62, coincides with the boundary of the domain, @xmath39. for convenience, define the differential operators @xmath266 and @xmath267, and apply @xmath268 to ([eq : fie]) to obtain @xmath269, \quad x \in { { \omega_{{{\bar \rho } } } } }, \label{eq : morselocal}\]] where @xmath270 thus, we guess @xmath271 the full solution to the problem is obtained by substituting ([eq : morseansatz]) into ([eq : mass1]) and ([eq : fie]) which yields @xmath272 where @xmath84 in @xmath62. we begin by considering the amplitudes @xmath148 and @xmath149 of the distributional component of the solution. we factor the differential operators @xmath273 where @xmath274 and @xmath275 where @xmath276. note that @xmath277 h(x - y),\]] where @xmath248 is the heaviside function. now we apply @xmath278 to ([eq : morseintegraleq]) at a point @xmath5 in @xmath62, which yields @xmath279 { { { { { \bar \rho}}_*}}}(y)\,dy & & \\ \mbox{}+a { \mathcal{p}^-}{\mathcal{q}^-}q(x-\alpha) = { \mathcal{p}^-}{\mathcal{q}^-}\ { \lambda -f(x)\}. \nonumber\end{aligned}\]] taking the limit @xmath280 yields @xmath281 where we have used the fact that @xmath282. a similar calculation using the operators @xmath283 and focusing near @xmath164 yields that @xmath284 eqs. ([eq : morseboundary1]) and ([eq : morseboundary2]) relate the amplitudes of the @xmath0-functions at the boundaries to the value of the classical solution @xmath232 there. further solution of the problem requires @xmath104 to be specified. in the case where @xmath285, solving ([eq : morselocal]) for @xmath286 and solving ([eq : morseboundary1]) and ([eq : morseboundary2]) for @xmath148 and @xmath149 yields an equilibrium solution. one must check that the solution is non - negative and then consider the solutions stability to determine if it is a local or global minimizer. in the case where @xmath62 is contained in the interior of @xmath39, we know that @xmath287 as discussed in section [sec : absence]. we consider this case below. suppose @xmath62 is contained in the interior of @xmath39. then @xmath287. following section [sec : funcmin], we try to determine when @xmath88 in @xmath35 and when @xmath68, which constitute necessary and sufficient conditions for @xmath59 to be a global minimizer. we apply @xmath278 to ([eq : morseintegraleq]) at a point @xmath165. the integral term and the terms arising from the @xmath0 functions vanish. the equation is simply @xmath288. we write the solution as @xmath289 the two constants @xmath290 are determined as follows. from ([eq : morseintegraleq]), @xmath82 is a continuous function, and thus @xmath291 we derive a jump condition on the derivative to get another equation for @xmath290. we differentiate ([eq : morseintegraleq]) and determine that @xmath292 is continuous. however, since @xmath84 for @xmath85, @xmath293. substituting this result into the derivative of ([eq : lambdamorse]) and letting @xmath5 increase to @xmath294, we find @xmath295 the solution to ([eq : lambdacontinuity]) and ([eq : lambdaprime]) is [eq : kvals] @xmath296 now that @xmath82 is known near @xmath297 we can compute when @xmath88, at least near the left side of @xmath62. taylor expanding @xmath82 around @xmath297, we find @xmath298 the quadratic term in ([eq : lambdatayl]) has coefficient @xmath299 where the second line comes from substituting ([eq : morseboundary1]) with @xmath176 and noting that the classical part of the solution @xmath300 must be nonnegative since it is a density. furthermore, since we expect @xmath301 (this can be shown a posteriori), we have that the quadratic term in ([eq : lambdatayl]) is positive. a similar analysis holds near the boundary @xmath164. therefore, @xmath302 for @xmath5 in a neighborhood outside of @xmath62. stated differently, the solution ([eq : morseansatz]) is a swarm minimizer, that is, it is stable with respect to infinitesimal redistributions of mass. the domain @xmath62 is determined through the relations ([eq : morseboundary1]) and ([eq : morseboundary2]), which, when @xmath239, become [eq : bcs] @xmath303 in the following subsections, we will consider the solution of the continuum system ([eq : fie]) and ([eq : mass1]) with no external potential, @xmath99. we consider two cases for the morse interaction potential ([eq : morse]) : first, the catastrophic case on @xmath100, for which the above calculation applies, and second, for the h - stable case on a finite domain, in which case @xmath304 and there are @xmath0-concentrations at the boundary. exact solutions for cases with an exogenous potential, @xmath305 can be straightforwardly derived, though the algebra is even more cumbersome and the results unenlightening. in this case, @xmath99 in ([eq : fie]) and @xmath306 in ([eq : morse]) so that @xmath307 in ([eq : morselocal]). the solution to ([eq : morselocal]) is @xmath308 where @xmath309 in the absence of an external potential, the solution is translationally invariant. consequently, we may choose the support to be an interval @xmath310 $] which is symmetric around the origin. hence, by symmetry, @xmath311. while the solution ([eq : freespacerhobar]) satisfies the ordinary differential equation ([eq : morselocal]), substituting into the integral equation ([eq : fie]) and the mass constraint ([eq : mass1]) will determine the constants @xmath167, @xmath73 and @xmath248. the integral operator produces modes spanned by @xmath312. from these modes follow two homogeneous equations for @xmath167 and @xmath73 which simplify to @xmath313c - \frac{1}{{{g}}l^2 -1}\lambda & = & 0, \label{eq : catastrophiceig1 } \\ \frac{2}{1+\mu^2 l^2}\left [\cos (\mu h) - \mu l \sin (\mu h) \right]c - \frac{1}{{{g}}l^2 -1}\lambda & = & 0.\end{aligned}\]] for these equations to have a nontrivial solution for @xmath167 and @xmath73 the determinant of the coefficient matrix must vanish, which yields a condition specifying @xmath248, @xmath314 the mass constraint ([eq : mass1]) yields @xmath315 solving ([eq : catastrophiceig1]) and ([eq : catastrophicmass]) for @xmath167 and @xmath73 yields the full solution for the coefficients in ([eq : freespacerhobar]) and the half - width @xmath248 of the solution, @xmath316 } \left\ { \frac{{{g}}l-1}{\sqrt{(1-{{g}})({{g}}l^2 - 1) } } \right\ }, \\ c & = & \frac{m}{2(h+l+1) } \frac{\sqrt{{{g}}}(l^2 - 1)}{l(1-{{g } }) }, \\ \lambda & = & \frac{m(1-{{g}}l^2)}{h + l+1}.\end{aligned}\]] as we have shown in section [sec : morsenonclass], this solution is a swarm minimizer. in fact, the solution is also a local minimizer. to see this, note that from ([eq : lambdamorse]) and ([eq : kvals]) that @xmath317 & |x| \geq h. \end{cases}\]] since @xmath318, we see that @xmath319 for @xmath320, ensuring that the solution is a local minimizer. while we suspect that @xmath232 is a global minimizer, this is not immediately apparent because @xmath321 in ([eq : qhatmorse]) has mixed sign in this catastrophic case, and hence @xmath67 is of indeterminate sign. to establish that @xmath232 is a global minimizer one might study the quantity @xmath322 but we leave this analysis as an open problem. figure [fig : morse_numerics](a) compares analytical and numerical results for an example case with total mass @xmath187 and interaction potential parameters @xmath323 and @xmath324. the solid line is the compactly supported analytical solution @xmath59. dots correspond to the numerically - obtained equilibrium of the discrete system ([eq : discretesystem]) with @xmath190 swarm members. as described in @xcite, the asymptotic behavior of ([eq : pde]) for the h - stable case is a spreading self - similar solution that approaches the well - known barenblatt solution of the porous medium equation. hence, there is no equilibrium solution for the h - stable case on an unbounded domain (one can verify this by considering the analogous problem to that of the previous section and showing explicitly that there is no solution). here, we assume a bounded domain @xmath188 $]. as before, @xmath99 in ([eq : fie]) but now @xmath258 in ([eq : morse]) so that @xmath325 in ([eq : morselocal]). the classical solution to ([eq : morselocal]) is @xmath326 where @xmath327 we will again invoke symmetry to assume @xmath311. the minimizer will be the classical solution together with @xmath0-functions on the boundary, @xmath328 again by symmetry, @xmath329. consequently, the solution can be written as @xmath330.\]] substituting into the integral equation ([eq : fie]) and the mass constraint ([eq : mass1]) will determine the constants @xmath167, @xmath73 and @xmath148. the integral operator produces modes spanned by @xmath312. this produces two homogeneous, linear equations for @xmath167, @xmath148 and @xmath73. the mass constraint ([eq : mass1]) produces an inhomogeneous one, namely an equation linear in @xmath167, @xmath148, and @xmath73 for the mass. we have the three dimensional linear system @xmath331 the solution is [eq : hstablesoln] @xmath332}{2\phi }, \\ c & = & -\frac{{{\tilde \mu}}m (1-{{\tilde \mu}}^2) (1-{{\tilde \mu}}^2 l^2)}{\phi }, \\ \lambda & = & \frac{m { { \tilde \mu}}(1-{{g}}l^2)\left [(1+{{\tilde \mu}})(1+{{\tilde \mu}}l) + (1-{{\tilde \mu}})(1-{{\tilde \mu}}l) \right]}{\phi},\end{aligned}\]] where for convenience we have defined @xmath333 for this h - stable case, @xmath78 which ensures that @xmath68 for nontrivial perturbations. this guarantees that the solution above is a global minimizer. in the limit of large domain size @xmath334, the analytical solution simplifies substantially. to leading order, the expressions ([eq : hstablesoln]) become @xmath335 note that @xmath336 is exponentially small except in a boundary layer near each edge of @xmath39, and therefore the solution is nearly constant in the interior of @xmath39. figure [fig : morse_numerics](b) compares analytical and numerical results for an example case with a relatively small value of @xmath334. we take total mass @xmath187 and set the domain half - width to be @xmath189. the interaction potential parameters @xmath323 and @xmath324. the solid line is the classical solution @xmath232. dots correspond to the numerically - obtained equilibrium of the discrete system ([eq : discretesystem]) with @xmath190 swarm members. each `` lollipop '' at the domain boundary corresponds to a @xmath0-function of mass @xmath191 in the analytical solution, and simultaneously to a superposition of @xmath192 swarm members in the numerical simulation. Modeling a locust swarm: examples with a gravitational potential we now return to the locust swarm model of @xcite, discussed also in section [sec : intro]. recall that locust swarms are observed to have a concentration of individuals on the ground, a gap or `` bubble '' where the density of individuals is near zero, and a sharply delineated swarm of flying individuals. this behavior is reproduced in the model ([eq : locusts]) ; see figure [fig : locust](b). in fact, figure [fig : locust](c) shows that the bubble is present even when the wind in the model is turned off, and only endogenous interactions and gravity are present. to better understand the structure of the swarm, we consider the analogous continuum problem. to further simplify the model, we note that the vertical structure of the swarm appears to depend only weakly on the horizontal direction, and thus we will construct a _ quasi - two - dimensional _ model in which the horizontal structure is assumed uniform. in particular, we will make a comparison between a one - dimensional and a quasi - two - dimensional model. both models take the form of the energy minimization problem ([eq : fie]) on a semi - infinite domain, with an exogenous potential @xmath337 describing gravity. the models differ in the choice of the endogenous potential @xmath32, which is chosen to describe either one - dimensional or quasi - two - dimensional repulsion. the one - dimensional model is precisely that which we considered in section [sec : grav]. there we saw that minimizers of the one - dimensional model can reproduce the concentrations of locusts on the ground and a group of individuals above the ground, but there can not be a separation between the grounded and airborne groups. we will show below that for the quasi - two - dimensional model, this is not the case, and indeed, some minimizers have a gap between the two groups. as mentioned, the one - dimensional and quasi - two - dimensional models incorporate only endogenous repulsion. however, the behavior we describe herein does not change for the more biologically realistic situation when attraction is present. we consider the repulsion - only case in order to seek the minimal mechanism responsible for the appearance of the gap. we consider a swarm in two dimensions, with spatial coordinate @xmath338. we will eventually confine the vertical coordinate @xmath339 to be nonnegative, since it describes the elevation above the ground at @xmath340. we assume the swarm to be uniform in the horizontal direction @xmath341, so that @xmath342. we construct a quasi - two - dimensional interaction potential, @xmath343 letting @xmath344 and @xmath345, this yields @xmath346 it is straightforward to show that the two - dimensional energy per unit horizontal length is given by @xmath347 = \frac{1}{2 } \int_{{\omega}}\int_{{\omega}}\rho(x_1) \rho(y_1) q_{2d}(x_1-y_1)\,dx_1\,dy_1 + \int_{{\omega}}f(x_1)\rho(x_1)\,dx_1,\]] where the exogenous force is @xmath348 and the domain @xmath39 is the half - line @xmath349. this is exactly analogous to the one - dimensional problem ([eq : continuum_energy]), but with particles interacting according to the quasi - two - dimensional endogenous potential. similarly, the corresponding dynamical equations are simply ([eq : cont_velocity]) and ([eq : pde]) but with endogenous force @xmath350. for the laplace potential ([eq : laplace]), the quasi - two - dimensional potential is @xmath351 this integral can be manipulated for ease of calculation, @xmath352 where ([eq : q2db]) comes from symmetry, ([eq : q2dc]) comes from letting @xmath353, ([eq : q2dd]) comes from letting @xmath354, and ([eq : q2de]) comes from the trigonometric substitution @xmath355. from an asymptotic expansion of ([eq : q2dd]), we find that for small @xmath356, @xmath357 whereas for large @xmath356, @xmath358.\]] in our numerical study, it is important to have an efficient method of computing values of @xmath359. in practice, we use ([eq : smallz]) for small @xmath356, ([eq : largez]) for large @xmath356, and for intermediate values of @xmath356 we interpolate from a lookup table pre - computed using ([eq : q2de]). the potential @xmath360 is shown in figure [fig : q2d]. note that @xmath360 is horizontal at @xmath361, and monotonically decreasing in @xmath356. the negative of the slope @xmath362 reaches a maximum of @xmath363 the quantity @xmath364 plays a key role in our analysis of minimizers below. the fourier transform of @xmath360 can be evaluated exactly using the integral definition ([eq : q2dc]) and interchanging the order of integration of @xmath6 and @xmath365 to obtain @xmath366 which we note is positive, so local minimizers are global minimizers per the discussion in section [sec : minimizers]. we model a quasi - two - dimensional biological swarm with repulsive social interactions of laplace type and subject to an exogenous gravitational potential, @xmath221. the spatial coordinate @xmath193 describes the elevation above ground. consequently, @xmath39 is the semi - infinite interval @xmath194. from section [sec : grav], recall that for the one - dimensional model, @xmath367 is a minimizer for some @xmath26, corresponding to all swarm members pinned by gravity to the ground. we consider this same solution as a candidate minimizer for the quasi - two - dimensional problem. in this case, @xmath59 above is actually a minimizer for any mass @xmath26. to see this, we can compute @xmath82, @xmath368 since @xmath369, @xmath370 increases away from the origin and hence @xmath59 is at least a swarm minimizer. in fact, if @xmath371, @xmath101 is a global minimizer because @xmath372 which guarantees that @xmath82 is strictly increasing for @xmath373 as shown in figure [fig : lambda](a). because it is strictly increasing, @xmath374 for @xmath131. given this fact, and additionally, since @xmath68 as previously shown, @xmath59 is a global minimizer. this means that if an infinitesimal amount of mass is added anywhere in the system, it will descend to the origin. consequently, we believe this solution is the global attractor (though we have not proven this). note that while the condition @xmath375 is sufficient for @xmath59 to be a global minimizer, it is not necessary. as alluded above, it is not necessary that @xmath376 be strictly increasing, only that @xmath377 for @xmath131. this is the case for for @xmath378, where @xmath379. figure [fig : lambda](b) shows a case when @xmath380. although @xmath377 for @xmath131, @xmath82 has a local minimum. in this situation, although the solution with the mass concentrated at the origin is a global minimizer, it is _ not _ a global attractor. we will see that a small amount of mass added near the local minimum of @xmath82 will create a swarm minimizer, which is dynamically stable to perturbations. figure [fig : lambda](c) shows the critical case when @xmath381. in this case the local minimum of @xmath82 at @xmath382 satisfies @xmath383 and @xmath384. figure [fig : lambda](d) shows the case when @xmath385 and now @xmath386 in the neighborhood of the minimum. in this case the solution with the mass concentrated at the origin is only a swarm minimizer ; the energy of the system can be reduced by transporting some of the mass at the origin to the neighborhood of the local minimum. when @xmath387 it is possible to construct a continuum of swarm minimizers. we have conducted a range of simulations for varying @xmath26 and have measured two basic properties of the solutions. we set @xmath388 and use @xmath389 in all simulations of the discrete system. initially, all the swarm members are high above the ground and we evolve the simulation to equilibrium. figure [fig : quasi2dnumerics](a) measures the mass on the ground as a percentage of the total swarm mass. the horizontal blue line indicates (schematically) that for @xmath390, the equilibrium consists of all mass concentrated at the origin ; as discussed above, this state is the global minimizer and (we believe) the global attractor. as mass is increased through @xmath391, the equilibrium is a swarm minimizer consisting of a classical swarm in the air separated from the origin, and some mass concentrated on the ground. as @xmath26 increases, the proportion of mass located on the ground decreases monotonically. figure [fig : quasi2dnumerics](b) visualizes the support of the airborne swarm, which exists only for @xmath392 ; the lower and upper data represent the coordinates of the bottom and top of the swarm, respectively. as mass is increased, the span of the swarm increases monotonically. as established above, when @xmath392, swarm minimizers exist with two components. in fact, there is a continuum of swarm minimizers with different proportions of mass in the air and on the ground. which minimizer is obtained in simulation depends on initial conditions. figure [fig : lambda2] shows two such minimizers for @xmath388 and @xmath393, and the associated values of @xmath82 (each obtained from a different initial condition). recalling that for a swarm minimizer, each connected component of the swarm, @xmath73 is constant, we define @xmath394 for the grounded component and @xmath395 for the airborne component. in figure [fig : lambda2](ab), @xmath396 of the mass is contained in the grounded component. in this case, @xmath397 indicating that the total energy could be reduced by transporting swarm members from the air to the ground. in contrast, in figure [fig : lambda2](cd), @xmath398 of the mass is contained in the grounded component. in this case, @xmath399 indicating that the total energy could be reduced by transporting swarm members from the ground to the air. note that by continuity, we believe a state exists where @xmath400, which would correspond to a global minimizer. however, this state is clearly not a global attractor and hence will not necessarily be achieved in simulation. we ve demonstrated that for @xmath401 one can construct a continuum of swarm minimizers with a gap between grounded and airborne components, and that for @xmath402 these solutions can have a lower energy than the state with the density concentrated solely on the ground. by contrast with the one - dimensional system of section [sec : grav] in which no gap is observed, these gap states appear to be the generic configuration for sufficiently large mass in the quasi - two - dimensional system. we conclude that dimensionality is crucial element for the formation of the bubble - like shape of real locust swarms. Conclusions in this paper we deeveloped a framework for studying equilibrium solutions for swarming problems. we related the discrete swarming problem to an associated continuum model. this continuum model has an energy formulation which enables analysis equilibrium solutions and their stability. we derived conditions for an equilibrium solution to be a local minimizer, a global minimizer, and/or a swarm minimizer, that is, stable to infinitesimal lagrangian deformations of the mass. we found many examples of compactly supported equilibrium solutions, which may be discontinuous at the boundary of the support. in addition, when a boundary of the support coincides with the domain boundary, a minimizer may contain a @xmath0-concentration there. for the case of exogenous repulsion modeled by the laplace potential, we computed three example equilibria. on a bounded domain, the minimizer is a constant density profile with @xmath0-functions at each end. on a half - line with an exogenous gravitational potential, the minimizer is a compactly supported linear density profile with a @xmath0-function at the origin. in free space with an exogenous quadratic potential, the minimizer is a compactly supported inverted parabola with jump discontinuities at the endpoints. each of the aforementioned solutions is also a global minimizer. to extend the results above, we also found analytical solutions for exogenous attractive - repulsive forces, modeled with the morse potential. in the case that the social force was in the catastrophic statistical mechanical regime, we found a compactly supported solution whose support is independent of the total population mass. this means that within the modeling assumptions, swarms become denser with increasing mass. for the case of an h - stable social force, there is no equilibrium solution on an infinite domain. on a finite domain, mass is partitioned between a classical solution in the interior and @xmath0-concentrations on the boundary. we recall that for the locust model of @xcite (see figure [fig : locust]) a concentration of locusts occurs on the ground, with a seemingly classical component above, separated by a gap. none of the one - dimensional solutions (for the laplace and morse potentials) discussed above contain a gap, that is, multiple swarm components that are spatially disconnected, suggesting that this configuration is intrinsically two - dimensional. to study this configuration, we computed a quasi - two - dimensional potential corresponding to a horizontally uniform swarm. we demonstrated numerically that for a wide range of parameters, there exists a continuous family of swarm minimizers that consist of a concentration on the ground and a disconnected, classical component in the air, reminiscent of our earlier numerical studies of a discrete locust swarm model. we believe that the analytical solutions we found provide a sampling of the rich tapestry of equilibrium solutions that manifest in the general model we have considered, and in nature. we hope that these solutions will inspire further analysis and guide future modeling efforts. Acknowledgments cmt acknowledges support from the nsf through grants dms-0740484 and dms-1009633. ajb gratefully acknowledges the support from the nsf through grants dms-0807347 and dms-0730630, and the hospitality of robert kohn and the courant institute of mathematical sciences. we both wish to thank the institute for mathematics and its applications where portions of this work were completed. , _ optimal transportation, dissipative pde s and functional inequalities _, in optimal transportation and applications (martina franca, 2001), vol. 1813 of lecture notes in math., springer, berlin, 2003, pp.

we study equilibrium configurations of swarming biological organisms subject to exogenous and pairwise endogenous forces. beginning with a discrete dynamical model , we derive a variational description of the corresponding continuum population density. equilibrium solutions are extrema of an energy functional, and satisfy a fredholm integral equation. we find conditions for the extrema to be local minimizers, global minimizers, and minimizers with respect to infinitesimal lagrangian displacements of mass. in one spatial dimension, for a variety of exogenous forces, endogenous forces, and domain configurations, we find exact analytical expressions for the equilibria. these agree closely with numerical simulations of the underlying discrete model.the exact solutions provide a sampling of the wide variety of equilibrium configurations possible within our general swarm modeling framework. the equilibria typically are compactly supported and may contain @xmath0-concentrations or jump discontinuities at the edge of the support. we apply our methods to a model of locust swarms, which are observed in nature to consist of a concentrated population on the ground separated from an airborne group. our model can reproduce this configuration ; quasi - two - dimensionality of the model plays a critical role. swarm, equilibrium, aggregation, integrodifferential equation, variational model, energy, minimizer, locust

1008.0881

Introduction the study of supernovae (sne) has greatly advanced in the last few years. intensive and highly automated monitoring of nearby galaxies (e.g., li et al. 1996 ; treffers et al. 1997 ; filippenko et al. 2001 ; dimai 2001 ; qiu & hu 2001), wide - field, moderately deep surveys (e.g., reiss et al. 1998 ; gal - yam & maoz 1999, 2002 ; hardin et al. 2000 ; schaefer 2000), and cosmology - oriented, deep, high - redshift sn search projects (perlmutter et al. 1997 ; schmidt et al. 1998) now combine to yield hundreds of new sn discoveries each year. ambitious programs that are currently planned or underway [e.g., the nearby supernova factory aldering et al. 2001 ; the supernova / acceleration probe (snap) perlmutter et al. 2000 ; automated sn detections in sloan digital sky survey (sdss) data vanden berk et al. 2001 ; miknaitis et al. 2001b ; see also @xmath9 promise to increase these numbers by at least an order of magnitude. sne are heterogeneous events, empirically classified into many subtypes, with the main classification criteria based on spectral properties. briefly, sne of type ii show hydrogen lines in their spectra while sne of type i do not. each of these types is further divided into subtypes, the commonly used ones including ia, ib, and ic, as well as ii - p, ii - l, iin, and iib. see filippenko (1997) for a thorough review and @xmath10 for more details. it is widely accepted that sne ia are produced from the thermonuclear disruption of a white dwarf at or near the chandrasekhar limit, while all other types of sne (ib, ic, and ii) result from the core collapse of massive stars. while understanding sne, their properties, and their underlying physics is of great interest, sne are also useful tools in the study of various other important problems. sne ia are excellent distance indicators, and their hubble diagram has been used to determine the local value of the hubble constant (e.g., parodi et al. 2000, and references therein). the extension of the hubble diagram to higher redshifts (@xmath11) probes the geometry and matter - energy content of the universe (e.g., goobar & perlmutter 1995). two independent groups using large samples of high-@xmath12 sne ia presented a strong case for a current acceleration of the universe (riess et al. 1998 ; perlmutter et al. 1999 ; see filippenko 2001 for a summary), consistent with a nonzero cosmological constant, @xmath13. subsequent work (riess et al. 2001) based on a single sn ia at @xmath14 possibly shows the transition from matter - dominated deceleration to @xmath13-dominated acceleration at @xmath15. sne ii - p can also be used as primary distance estimators through the expanding photosphere method (epm ; kirshner & kwan 1974 ; schmidt, kirshner, & eastman 1992 ; schmidt et al. 1994), as was most recently demonstrated by hamuy et al. (2001) and leonard et al. (2002a, b). leonard et al. (2002a ; see also hflich et al. 2001) suggest that distances good to @xmath16% (@xmath17) may be possible for sne ii - p by simply measuring the mean plateau visual magnitude, obviating the need for a complete epm analysis unless a more accurate distance is desired. hamuy & pinto (2002) refine this technique, showing that a measurement of the plateau magnitude and the ejecta expansion velocity potentially yields a considerably smaller uncertainty in the derived distance. sn rates as a function of redshift probe the star - formation history of the universe, the physical mechanisms leading to sne ia, and the cosmological parameters (jorgensen et al. 1997 ; sadat et al. 1998 ; ruiz - lapuente & canal 1998 ; madau, della valle, & panagia 1998 ; yungelson & livio 2000). sn rates are also important for understanding the chemical enrichment and energetics of the interstellar medium (e.g., matteucci & greggio 1986) and the intracluster medium (e.g., brighenti & mathews 1998, 2001 ; lowenstein 2000 ; gal - yam, maoz, & sharon 2002). once discovered, the study of a particular sn, and its use as a tool for any of the applications above, is almost always based on spectroscopic verification and classification. the information extracted from sn spectra usually includes (but is not limited to) the sn type, redshift, and age (relative to the dates of explosion or peak brightness). spectroscopic followup may not always be possible or practical. sne, especially at high redshift, may be too faint for spectroscopy, even with the largest, 10-m - class telescopes currently available. spectroscopy is also not practical if large numbers (hundreds or thousands) of sne are detected within a relatively short time, as is expected to happen in the case of the sdss southern strip (miknaitis et al. 2001b ; see also @xmath18). finally, spectroscopy is impossible for sne discovered in archival data (gal - yam & maoz 2000 ; riess et al. 2001 ; gal - yam et al. 2002), which have long faded by the time they are found. the discovery of sne in archival data is expected to become frequent as high - quality astronomical databases become larger and more accessible, especially with the development of projects such as astrovirtel (http://www.stecf.org/astrovirtel) and the national virtual observatory (brunner, djorgovski, & szalay 2001). the goal of the present work is to facilitate the scientific exploitation of sne for which no spectroscopic observations exist. the obvious alternative for spectroscopy is multi - color broadband photometry. the potential utility of such an approach is demonstrated, in principle, by the use of the `` photometric redshift '' method to infer the redshift and type of galaxies and quasars that are too faint or too numerous to observe spectroscopically (e.g., weymann et al. 1999 ; richards et al. 2001). applying a similar method to sne is not straightforward since, unlike galaxies, the spectra of sne vary strongly with type, redshift, and time. while photometric approaches to the study of faint sne have been discussed before (dahln & fransson 1999 ; sullivan et al. 2000 ; riess et al. 2001), no general treatment has been presented thus far. in a recent paper, dahln & goobar (2002, hereafter dg2002) tackle the issue of sn classification for the case of deep, cosmology - oriented, high-@xmath12 sn searches. their treatment relies heavily on the particular observational setup used by these programs i.e., the comparison of images obtained @xmath194 weeks apart, designed to detect sne before, or at, peak brightness. as these authors show, this setup is strongly biased toward the discovery of sne ia. the main theme of dg2002 is the selection of the high-@xmath12 sne ia from an observed sample, using photometry of the host galaxies and the sne themselves. for the particular observational setup they consider, dg2002 also wish to minimize the contamination of the sample by non - ia sne. to do this, they use the models of dahln & fransson (1999) and demonstrate their ability to reject most non - ia sne based on their colors and lower brightness. however, the models used for the population of non - ia sne are based on many parameters and assumptions, such as the ultraviolet (uv) spectra, peak brightness, light curves, and relative fractions of non - ia sne, that are not precisely known even in the local universe, let alone at high redshift. while dg2002 seek only to identify high-@xmath12 sne ia detected by a particular observational setup, we present general methods that apply to all the major sn subtypes and can be used for a wide range of sn surveys, including searches that are similar to those described by dg2002. classification of sne using broadband colors is such a complex problem that, as we show below, a full solution (i.e., deriving sn type, redshift, and age from a few broadband colors) is probably impossible. however we will argue here that the problem may be simplified. type, redshift, and age need not all be determined simultaneously. most sne are associated with host galaxies that are readily detectable. one can therefore infer the redshift of the sn from that of the host, derived either from spectroscopy (that may be obtained long after the sn has faded) or using a photometric redshift. in fact, for sne that are detected in well - studied parts of the sky, the redshift information often already exists. gal - yam et al. (2002) recently demonstrated this with sne they discovered in archival _ hubble space telescope (hst) _ images of well - studied galaxy - cluster fields. five of the six apparent sne they found (and all of those clearly associated with a host) had redshift information in the literature. for the majority of sne, the redshift can thus be obtained from observations of the host, and treated as a known parameter. in the absence of a spectrum, the sn age may sometimes be revealed by measuring the photometric light curve of the event. admittedly, such followup may still pose a problem for faint or very numerous events, and is generally not possible for archival sne (see riess et al. 2001 for an exception). however, for many of the scientific uses of sne (e.g., the derivation of sn rates), the age of each event is immaterial. determination of the type of a sn is crucial for most of the applications discussed above. the sn type can be securely determined only from observations of the active sn itself. while some information can be gleaned from the host (e.g., elliptical galaxies are known to contain only sne ia), this information is limited and such reasoning may not hold at high redshifts. we therefore concentrate our efforts on using multi - color broadband photometry to classify sne. Method since we want to constrain the type of a sn with an arbitrary (but known) redshift, it is not simple to utilize broadband photometry ; k - corrections for sne of all types, and at all ages, are currently unknown. instead, we have compiled a large spectral database of nearby, well - observed sne. these spectra are used to calculate synthetic broadband colors for various sn types through a given filter set at a given redshift. the temporal coverage of our compilation allows us to draw paths in color space which show the time evolution of each sn type. inspecting the resulting diagrams, one can then look for regions which are either populated by a single type of sn, or that are avoided by various sn types. the observed photometric colors of a candidate sn with a known redshift can then be plotted on the relevant diagrams, and constraints on its type and age may be drawn. as we show in @xmath20, the type of a sn can sometimes be uniquely determined. when this is not the case, the type may still be deduced by supplementing the color information with other available data on the sn, such as constraints on its brightness, and information (even if very limited) on its variability (e.g., whether its flux is rising or declining). our sn classification method is based on colors, determined by the spectral energy distribution (sed) of each event. by definition, such a method can differentiate only between sn subtypes with unique spectral characteristics. hypothetically, two sne with very different photometric properties (e.g., light curves, peak magnitudes), but having similar seds at all ages, would not be distinguished by our method. the sn classification scheme we adopt needs, therefore, to rely only on spectral properties. fortunately, the common classification of sne (see filippenko 1997 for a review) is mostly based on their spectra, with one notable exception which we discuss below. it is customary to divide the sn population into two main subgroups : type ii sne, which show prominent hydrogen lines in their spectra, and type i sne, which do not (minkowski 1941). type i sne are further divided into type ia whose spectra are characterized by a deep absorption trough around @xmath21 , attributed to blueshifted si ii @xmath226347, 6371 lines, type ib that show prominent he i lines, and type ic that lack both si ii and he i (e.g., matheson et al. 2001, and references therein). type ii sne are divided (e.g., barbon, ciatti, & rosino 1979 ; doggett & branch 1985) into two subclasses according to their light - curve shape : sne ii - p show a pronounced plateau during the first @xmath23 d of their evolution, and sne ii - l decline in a linear fashion in a magnitude vs. time plot, similar to sne i. further studies (e.g., filippenko 1997) show that sne ii - p have characteristic spectra, with the h@xmath24 line showing a p - cygni profile. on the other hand, sne ii - l have not been well characterized spectroscopically, and the best spectroscopically studied events (sne 1979c and 1980k) are considered to be photometrically peculiar (`` overluminous '' ; miller & branch 1990) relative to other sne ii - l. during the 1990s, the focus in studies of sne ii shifted from photometry to spectroscopy. two new subtypes have emerged. sne iin show narrow h@xmath24 emission (e.g., schlegel 1990 ; filippenko 1997, and references therein), and sne iib are transition objects that first appear as relatively normal sne ii, but evolve to resemble sne ib (filippenko 1988 ; filippenko, matheson, & ho 1993 ; matheson et al. 2001, and references therein). for our study, only sn subtypes with well - defined spectral properties are meaningful. we therefore consider henceforth only sne ii - p, iin, and iib as subtypes of the sn ii population. sne iin and iib are probably a large fraction of the sn population that was previously classified as type ii - l based only on their light - curve shape (approximately half of the sn ii population, e.g., cappellaro et al. reviewing the latest spectral databases available to us, we estimate that most of the sne ii discovered by current surveys belong to the spectroscopically defined subtypes ii - p, iin, and iib (see below). sne are a veritable zoo, with many peculiar and even unique events, from luminous hypernovae (e.g., sn 1998bw galama et al. 1998 ; patat et al. 2001 ; sn 1997cy germany et al. 2000 ; turatto et al. 2000 ; sn 1997ef iwamoto et al. 2000 ; matheson et al. 2001) to faint sn 1987a - like events (e.g., arnett et al. 1989, and references therein), and sn `` impostors '' (e.g., filippenko et al. 1995a ; van dyk et al. 2000, and references therein). currently, we limit our discussion to sn types that are not extremely rare, and are well - defined and well - characterized spectroscopically. assuming that the sne reported in the iau circulars are representative of the sn population that is discovered by current programs, we estimate that, collectively, the well - defined subtypes we consider (ia, ib, ic, ii - p, iin, and iib) constitute at least 83%, and probably more, of the entire population. for instance, we consider all the sne that were discovered during the year 2000 and whose type was either reported in iau circulars, or alternatively could be constrained by data available to us. among 110 such events, there were 54 type ia, 5 type ib, 5 type ic, 8 type iin, one type iib, two peculiar sne ia, one peculiar sn ib, and 34 events which were reported just as `` type ii. '' using available spectra or data reported in the iau circulars, we learn that of the latter 34 events, 18 show a p - cygni profile in h@xmath24, typical of sne ii - p, 13 events may or may not belong to the ii - p, iin, or iib subtypes, and 3 events are definitely not ii - p, iin, or iib events. hence 91 events (83%) belong to our spectroscopically defined sn sample (types ia, ib, ic, ii - p, iin, and iib), 13 events (12%) may or may not be of these types, and just 6 events (5%) are known to be of types not included in our database. it is therefore likely that analysis based on our database will be relevant to the large majority of sne discovered by current programs. this is probably also true for future programs that will use similar search methods. we are aware that a large population of intrinsically faint sne (e.g., sn 1987a - like) may be underrepresented in current sn statistics. analysis of future surveys that will be sensitive to such events may require their inclusion in the spectral database. the core of our classification algorithm is the spectral database : a compilation of optical spectra of nearby sne. most of the these spectra were obtained with the shane 3-m reflector at lick observatory. the typical resolution is better than @xmath25 (full width at half maximum), and the majority of the data used cover the range 320010000 . some spectra with narrower coverage (approximately 40008000 ) are also used. all spectra were either obtained at the parallactic angle (filippenko 1982), or calibrated by simultaneous photometry, to ensure that spectral shape distortions due to wavelength - dependent atmospheric refraction are not significant. telluric lines were removed, generally through division by the spectrum of a featureless star. according to the galactic coordinates of the observed sn, the spectra were corrected for galactic reddening, using the maps of schlegel, finkbeiner, & davis (1998) and the extinction curve of cardelli, clayton, & mathis (1989). the working edition of the database was constructed in the following manner. for each of the well - defined sn subtypes mentioned above, we have selected a prototype. the main criterion was the availability of spectra with wide wavelength coverage and multiple epochs. in addition, a few complementary spectra of other events were included in order to verify consistency and fill gaps in the temporal evolution, when needed. table 1 lists the sne whose spectra constitute our database, with the prototypical event for each subtype listed first. we have specifically avoided events that were known to either suffer considerable extinction, or to be otherwise peculiar. for sne ia, peculiar (overluminous, sn 1991t - like, or underluminous, sn 1991bg - like) events are rather well characterized, and may be quite common, at least in low - redshift environments (li et al. our main database includes only spectra of normal sne ia, but we examine the effects of peculiar sne ia in @xmath26 below. all spectra were de - redshifted according to the host - galaxy redshift taken from the ned database. after redshifting the spectra again to a chosen redshift (see below), the telluric features were re - applied before deriving synthetic colors. using our spectral database, we calculate the synthetic colors of sne at a chosen redshift through the johnson - cousins @xmath4 (johnson 1965 ; cousins 1976 ; see moro & munari 2000 for details), bessell @xmath27 (bessell & brett 1988), and sdss @xmath6 (fukugita et al. 1996 ; stoughton et al. 2002) filter systems. if a filter s bandpass is not fully covered by a sn spectrum, the flux in the missing spectral region is extrapolated linearly using the median value of the spectrum. each calculated magnitude that includes such an extrapolation is assigned an error equal to the amount of flux in the extrapolated part of the spectrum, thus providing a conservative error estimate. the resulting synthetic photometry is then displayed on color - color diagrams. note that in all the plots we present, the ordinate and abscissa are composed of colors consisting of adjacent bands (e.g., @xmath28 vs. @xmath29, @xmath29 vs. @xmath1 etc.). since most of the spectra in our database have a wide wavelength range, error bars will hence generally appear only on one axis. in the following section we present such diagrams, discuss their use in the classification of sne discovered by various programs, and derive a few general rules. note that the choice of colors that gives the best sn type differentiation, apart from the obvious dependence on the filter system used, also depends on redshift. for optimal results, for each sn (having a known redshift and some observed colors) one needs to search for color - color diagrams that give the maximum information content. obvious space limitations allow us to present below just a few representative examples of such plots. similar plots for arbitrary values of @xmath12 may be obtained from our website (http://wise-obs.tau.ac.il/$\sim$dovip/typing). Results in figure 1, we compare our data on sne ia to template colors of normal sne ia compiled by leibundgut (1988). one can see that our calculated colors follow the expected path in color space, with a small scatter (@xmath30 mag), consistent with the previously measured dispersion in sne ia colors (e.g., tripp 1998 ; phillips et al. 1999, and references therein). since our spectral database includes spectra of sn 1994d, which was somewhat bluer at early times than average sne ia (richmond et al. 1995), our treatment covers an even wider variety in the colors of early sne ia. from this plot, we can estimate the expected width of color - color paths for normal sne ia to be @xmath300.3 mag. note that sne ia follow a complex curve in color space, with a distinct turn - around point @xmath16 d after peak brightness. the underlying spectral evolution is nontrivial. like most sne, the continuum slope of early sne ia is very blue, and becomes redder as the sn grows older. this process is dominant during the first weeks of sn ia evolution, but is later countered by the emergence of strong emission lines in the blue part of the spectrum (restframe 40005500 ). the line - dominated spectrum results in unique colors for late - time (@xmath31 d) sne ia (e.g., figure 3, below). in figure 2 our results for sne ii - p are plotted against the observed photometry of the recent, well - observed sn 1999em. it can be clearly seen that the results agree perfectly up to around 70 d past maximum brightness, where a bifurcation appears in our calculated color path. our spectral data for sne ii - p is based on three different events (sne 1992h, 1999em, and 2001x). the three points that have considerably lower @xmath28 values (at 110, 127, and 165 d) have all been calculated from the spectra of sn 2001x, which up to that age was consistent with the other sne ii - p used. this demonstrates the well - known variety of late - type spectra of sne ii (filippenko 1997). in subsequent plots, we retain both branches of `` old '' sne ii - p (with the one derived from spectra of sn 2001x marked, when significantly different, in a darker shade), and treat the color space bounded by them as a possible location for such sne. contrary to sne ia, the color path of sne ii - p is quite smooth. the underlying spectral evolution is driven mostly by the change in the continuum slope, from very blue at early ages, growing redder with time (e.g., filippenko 1997). late - time (@xmath31 d) spectra become increasingly dominated by strong and broad emission lines, especially h@xmath24, that give sne ii - p their distinct late - time colors (see, e.g., fig. 3, below). in figure 3 (top panel), we show the color paths of all types of sne at zero redshift in @xmath28 vs. @xmath29. one sees that many sne before maximum light are blue, with @xmath32 mag. while most young sne ii have @xmath33 mag, young sne ia have bluer @xmath29 colors (@xmath34 mag) but redder @xmath28 colors, around @xmath35 mag. it is also evident that old (@xmath36 d) sne ia have unique colors (@xmath37 mag, @xmath38 mag), as do some of the sne ii - p (ages 19100 d) which have the reddest colors of all sne. the arrow shows the reddening effect corresponding to @xmath39 mag of extinction, assuming the galactic reddening curve of cardelli et al. one can see that the unique colors of young (@xmath40 d) and very old (@xmath31 d) sne ia can not be masked even by significant reddening in their host. furthermore, we note that high extinction values are rare for sne ia. for example, only 3 among 49 sne ia (@xmath41) studied by the cfa and caln / tololo programs, and presented in riess et al. (1999), have @xmath42 mag. conversely, because reddening works only in one direction, some candidates with appropriate observed colors can be uniquely determined to be sne ia. however, the center of the diagram is populated by sne of all types. for example, a zero - redshift sn, of unknown type, with colors of @xmath43 mag, @xmath44 mag, would be impossible to classify based on these colors only. but the middle panel of figure 3 shows that if one examines an additional color, such as @xmath1, this degeneracy can be partly lifted. for the same @xmath45 mag, there is a spread in @xmath1 that can shed some light on the sn type. a value of @xmath46 mag would favor a type ia classification, while a positive value would suggest one of the core - collapse types. addition of an @xmath2 measurement (fig. 3, bottom panel) would narrow the options even further, since in this color, at this redshift, different types have distinct values, from the bluest sne ia to the reddest types (ib, ic, and ii - p). looking at the reddening vectors in the middle and bottom panels of figure 3, we again note that only heavy extinction (@xmath47 mag) can move sne ia into color regions populated by core - collapse sne. before we proceed, we briefly comment on the effects of host - galaxy contamination. since our approach assumes the redshift of a given sn is a known parameter, and this is commonly available through the observation of the host galaxy, some data on the galaxy should exist. it should therefore generally be possible to subtract the galaxy s light from the sn photometry. to study the effect of non - subtraction of host - galaxy light, or imperfect subtraction, we parameterize the amount of the contaminating galactic flux, at a given epoch, by the ratio @xmath48 where @xmath49 and @xmath50 are the mean flux of the host galaxy and sn spectra, respectively, calculated over the wavelength range of the spectrum of the sn. we consider contamination as negligible, when it is not likely to make one type of sn look like another. first, we find that for all types of sne and hosts, when @xmath51 the effect of contamination is indeed negligible. for young sne ia, the worst - case scenario would be the contamination by a red elliptical host that could make them look like redder core - collapse sne. when observed in restframe blue colors (e.g., @xmath52), this effect becomes significant only when @xmath53, but if red bands (e.g., @xmath54) are used, contaminated sne ia attain colors similar to those of the core - collapse population already at @xmath55. the opposite effect (i.e., of bluer host galaxies on red core - collapse sne) is much weaker, so contamination by spiral hosts is unlikely to mix core - collapse sne with young sne ia. only heavy contamination (@xmath56) by a very blue, star - forming host galaxy can make a red sn ii - p appear similar to a bluer sn ia on a @xmath52 color - color diagram. in redder bands (e.g., @xmath54), even this extreme case causes negligible shifts in color. we conclude that if the host galaxy light has been subtracted, at least roughly, from the measured sn photometry, residual contamination has no significant implication. a large contribution from an underlying host (e.g., with a flux similar to that of the sn itself) that has not been removed may mask blue sne ia as core - collapse events, but is unlikely to make red core - collapse events appear as sne ia. in the discussion below, we therefore neglect contamination of the sn photometry by light from the underlying host galaxy. with increasing redshift, the available spectral information shifts to longer - wavelength filters. the top panel of figure 4 demonstrates how well sne ia can be differentiated from other types at @xmath57 on a @xmath1 vs. @xmath2 diagram. on the @xmath2 axis the only sne with negative values are sne ia. they are also the only type with @xmath1 color greater than 0.6 mag while @xmath2 is smaller than 0.3 mag. during most of the temporal evolution of sne ia, their distinction from other types is practically unaffected by host galaxy dust reddening, as can be seen from the vector plotted (the reddening is calculated in the sn rest frame). the middle panel of figure 4 shows a similar diagram computed at @xmath58. classification is obviously more difficult. still, sne ii - p (including `` sn 2001x - like, '' see @xmath59) at ages above 100 d have @xmath2 values higher than sne of other types (at all ages). close to peak flux, sne ia have @xmath1 colors of a few tenths of a magnitude bluer than other types, and again, this distinction is not sensitive to reddening. even where all types seem to mingle, at @xmath60 mag and @xmath61 mag, sne ia and ii - p have @xmath1 values that are 0.2 mag higher than other types. should one of these types be rendered unlikely by other data, such as a sn that is too bright to be a sn ii - p, the classification can become unique or nearly so. for example, sne ii - p can be unambiguously identified by their light curves ; no other sn type has a roughly constant @xmath54 brightness over such a long period. in principle, two @xmath62-band (restframe) observations separated by, say, 30 d can unambiguously identify a sn ii - p. at redshifts higher than 0.6, the @xmath62 band samples the restframe uv, not covered by our spectral database. thus, only one color (@xmath2) remains in the johnson - cousins system. the sdss filter system has 3 filters (@xmath63, @xmath64, and @xmath12) in the approximate range covered by the @xmath65 and @xmath66 filters, but extending farther into the red by @xmath67 . using this system, we can extend our analysis to higher redshifts. this is demonstrated in the bottom panel of figure 4 for sne at @xmath7. again, late - time sne ii - p are significantly redder than other types (@xmath68 mag and @xmath69 mag), while early - time sne ia are significantly bluer (@xmath70 mag and @xmath71 mag), and remain so even in the presence of significant reddening by dust. at later times (over two months), sne ia escape from the vicinity of other types and reach @xmath72 values above 1.2 mag. sne iin at this redshift appear to be isolated from other types. as we move even farther up the redshift scale the use of infrared (ir) filters is required. between @xmath73 and @xmath74, classification would require a combination of optical and ir photometry. figure 5 (top panel) shows that at @xmath75, sne ia have distinct blue colors (@xmath76 mag and @xmath77 mag) at almost all ages, sne ii - p develop very red @xmath78 colors (@xmath79 mag) two months after peak, while sne ib and ic are the only ones that have @xmath80 colors that are redder than @xmath81 mag. as before, these distinctions are hardly affected by extinction. between @xmath82 and 2.5 our spectra shift into the region covered by the bessell @xmath27 filters. in the bottom panel of figure 5 (@xmath83), one can see that sne ia have generally lower @xmath80 colors (@xmath84 mag), except between ages of one to two months, shared perhaps by very early type iin, ii - p, and ic events. sne ib and ic are the only objects to have both @xmath80 and @xmath85 above unity. the use of synthetic colors, calculated from spectra of local sne, to classify high-@xmath12 events, assumes that sn spectra do not evolve strongly with redshift. the similarity, to a first approximation, between the spectra of distant sne ia and their local counterparts has been demonstrated (riess et al. 1998 ; coil et al. 2000). however, for all other types of sne, high - quality spectra of distant (@xmath86) events are largely unavailable. since we use synthetic broadband colors, only significant evolution in either the spectral slope or strong emission or absorption features will influence our classification method. nevertheless, this is a caveat that must be addressed once spectra of distant core - collapse sne become available. we now examine the colors of peculiar sne ia and their implication for our classification method. following the same approach used earlier, we have compiled spectra of the most extreme varieties of peculiar sne ia : overluminous (sn 1991t - like ; e.g., filippenko et al. 1992a) and underluminous (sn 1991bg - like ; e.g., filippenko et al. 1992b) objects, as well as spectra of sn 2000cx, a unique sn ia that may be considered a subtype of its own (li et al. table 2 provides observational details for these objects. inspecting figure 6 (top panel), which shows the @xmath28 vs. @xmath29 colors of the various sn ia subtypes at zero redshift, we see that, although the paths of peculiar sne ia in color space are different from those of `` normal '' sne ia, they populate the same general region. on the other hand, in the middle panel of figure 6, showing @xmath1 vs. @xmath2 colors at zero redshift, one can see that underluminous sne ia have significantly larger @xmath2 values (by 0.20.6 mag) than those of all other sne ia, and are therefore liable to mix with the redder core - collapse population. this is illustrated in the bottom panel of the same figure, where indeed, underluminous sne ia at early ages (prior to @xmath87 d) can not be distinguished from core - collapse sne. later in their evolution, these events acquire unique colors (@xmath88 mag, @xmath89 mag) that are different from all other types of sne. as sn 1991t - like and sn 2000cx - like events are almost always bluer than `` normal '' sne ia, they are generally easier to distinguish from core - collapse sne, and pose no special problem in our analysis. sne ia of the underluminous, redder, sn 1991bg - like variety are hard to distinguish from core - collapse sne at early ages, and may be lost in color - based classification. this problem is quite limited at low redshift, where underluminous sne ia are rare (@xmath90% of the sn ia population ; li et al. 2001a), and may be totally negligible at higher redshifts where, empirically, sn 1991bg - like events have not been found at all (li et al. 2001a). high - redshift sn search programs that focus on finding sne ia for use as distance indicators encounter the problem of sample contamination by other sn types, especially a luminous variety of sne ic (riess et al. 1998 ; clocchiatti et al. 2000). the problem is two - fold. first, unwanted non - ia sne take up precious telescope resources for followup spectroscopy. second, when only low signal - to - noise ratio spectra of faint, high-@xmath12 events are available, luminous sne ic may masquerade as sne ia. the latter problem is acute at high @xmath12, since at these redshifts the sn ia hallmark (the si ii @xmath91 trough) is shifted out of the optical range, and sne ic spectra lack the telltale h or he lines that distinguish sne ii and sne ib, respectively. spectra with high signal - to - noise ratios can provide definitive classifications (coil et al. 2000), but these are time - consuming to obtain. > from figure 7, one sees that color information may help, at least partially, to alleviate this problem. we confirm and quantify the well known `` rule of thumb '' (e.g., riess et al. 2001) : sne ic are red compared to sne ia at a similar redshift. one can see that, for sne at ages that are relevant for high-@xmath12 search programs, designed to discover sne near peak brightness, sne ia are typically 0.5 mag bluer in @xmath72 than sne ic, well above the typical effects of reddening. while it is true that rising (pre - maximum) sne ic have colors similar to those of older (@xmath92 weeks past maximum) sne ia, even the most basic variability information (e.g., whether the object is rising or declining) breaks this degeneracy. figure 7 also illustrates the fact that most of the information at this redshift can be obtained from the @xmath72 color alone, and the inclusion of @xmath12-band information does not appear to be cost - effective. sn 2001fg was the third sn reported from sdss data. this event was discovered on 15 october 2001 ut by vanden berk et al. (2001) with @xmath93, @xmath94, and @xmath95 mag. inspecting the color - color diagram calculated for the appropriate redshift (@xmath96, see below), one can see (fig. 8) that the type and approximate age of the sn candidate can be deduced. the diagram clearly indicates this is a sn ia, around one month old. followup spectra by filippenko & chornock (2001) using the keck ii 10-m telescope reveal that the object is indeed a sn ia, at @xmath96. the spectrum is similar to those of sne ia about two months past maximum brightness. this age, at the time of the spectral observation (18 november ut), implies an age around one month past maximum brightness at discovery, confirming our diagnosis. a fully `` blind '' application would have, of course, required an independent photometric or spectroscopic redshift for the sn host galaxy. Future prospects and conclusions the method presented here can become significantly more powerful when more observational data become available. faint, high-@xmath12 sne are usually observed at optical wavelengths (i.e., @xmath62, @xmath65, or @xmath66 bands). in order to use these bands for classification, knowledge of the restframe uv spectrum is required. currently, uv spectra are available for only a handful of objects, while methods like those discussed here require multi - epoch coverage of several objects for each sn subtype. further characteristics of the various sn types, most notably the absolute peak magnitudes and their scatter for every sn type, the typical light curves, and the relative rates of the different types, would also be useful. with such data in hand, one could use the limiting magnitude of a survey to calculate, for each sn type, at what epochs it is bright enough to be observed, thus removing some degeneracy in the classification. a further enhancement could be introduced to deal with the regions in color space that are populated by several different sn types. using the typical light curves and known peak magnitudes for all types, and the relative rates, one could calculate the amount of time a sn of a certain type spends within a region and deduce the likelihood that an observed event belongs to a particular type. as already mentioned in @xmath97, work along these lines has been presented by dg2002, who replace missing observational data with rough estimates or models. sne are generally discovered by their variability. however, the tools developed in this paper may enable one to detect sne by their colors alone. indeed, such methods have recently been discussed for the detection of another kind of transient phenomenon gamma - ray burst afterglows (rhoads 2001). multi - color wide - field surveys (e.g., becker et al. 2001 ; jannuzi et al. 2001 ; sdss york et al. 2000) that are either planned or already underway may be used to discover large numbers of sne using similar methods. to illustrate this point, we plot in figure 9 the locations of sne of all types at @xmath98, together with a polygon within which fall the vast majority of stars, asteroids, cataclysmic variables, cepheids, and quasars, as discussed by krisciunas, margon, & szkody (1998). except for late - time sne ic, all sne have lower @xmath99 values and higher @xmath72 values than these objects. similar results are obtained for the sn population at @xmath100 00.3, the redshifts of sne probed by the sdss. this demonstrates the feasibility of color selection of sne, a method which may emerge as an invaluable resource when applied to upcoming, large data sets. an exciting application of the methods presented in this paper would be the study of sne discovered in the sdss southern strip. during the months of september to november, when the primary target of the sdss, the north galactic pole, is not visible, the sdss telescope observes parts of the sky near the south galactic pole. the so called `` south equatorial strip '' is a patch of sky that will be imaged repeatedly during the 5-year planned operations of the survey. according to the sdss website (http://www.sdss.org/documents/5yearbaseline.pdf), this 270 square degree area will be imaged @xmath101 times during the 5-year period, or about 4 times each fall, on average. assuming that this sampling enables the discovery of all sne that occur during the @xmath19 months this strip is imaged each year, for five years, the total surveyed area is @xmath102 deg@xmath103-months, or 337.5 deg@xmath103-years. pain et al. (1996) have measured a sn ia rate of 34 per year per square degree in the magnitude range @xmath104, and the limiting magnitude of the sdss images in this band is given as @xmath105 mag. we would then expect some 2300 sne ia each year or a total of over 11,000 sne ia in this strip. the numbers of sne of all types (not just ia) will undoubtedly be even larger. miknaitis et al. (2001b) report that a systematic search for sne in these data is underway, and the first sne have already been reported (miknaitis et al. 2001a ; rest, miceli, & covarrubias 2001 ; miknaitis & krisciunas 2001). this huge expected dataset, which may contain an order of magnitude more events than all previously known sne, all with five - band photometry and possibly some variability information, offers an unprecedented resource for sn studies. as an example, sn rates, based on thousands of events, can be computed as a function of sn type, host - galaxy morphology, and redshift. however, it is clear that obtaining spectra for thousands of faint sne each fall is not feasible. photometric classification could therefore be a valuable option. the sdss supplies five - band information, along with ready - made tools for the derivation of photometric redshifts for all host galaxies, and spectral identification for part of the brighter ones. this makes our approach of using the sn photometry for classification, assuming that the redshift is known, even more appropriate. since no followup is required, such an analysis need not be done in real time in principle, it may even be carried out long after the project is complete, and all the data are publicly available. in summary, we have presented a method for the classification of sne, using multi - color broadband photometry. we have shown that the type of a sn may be uniquely determined in some cases. even when this is not the case, constraints on the type may still be drawn. in particular, sne ia may be distinguished from core - collapse sne during long periods in their evolution, for most of the redshifts and filter combinations studied. finally, we have demonstrated the application of our method to a recently discovered sn from the sdss, and we have shown how this technique may become a valuable tool for the analysis of the large sn samples expected to emerge from this and other programs that are already active or will begin soon. this work proves that spectroscopic followup is not always a prerequisite for sne to be a valuable scientific resource, and lays the foundations for the exploitation of large sn samples for which such followup may not be possible. we thank the many people in a.v.f.s group at the university of california, berkeley (especially a. j. barth, r. chornock, l. c. ho, and j. c. shields) who, over the years, helped obtain and calibrate the spectra that constitute much of the dataset used in this study. we are grateful to a. fassia, m. hamuy, and y. qiu for supplying us with digital copies of their spectra of sn 1998s, 1999ee, and 1996cb, respectively. b. leibundgut generously provided help with sn light curves, and we thank d. branch, e. o. ofek, g. richards, and o. shemmer for useful suggestions. we acknowledge the assistance of the staffs of various observatories (especially lick) where the data were taken. 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. d. m. acknowledges support by the israel science foundation the jack adler foundation for space research, grant 63/01 - 1. a.v.f. is grateful for the financial support of nsf grants ast9417213 and ast9987438, as well as of the guggenheim foundation. aldering, g., supernova cosmology project collaboration, & nearby supernova factory collaboration. 2001, baas, 199.67.01 arnett, w. d., bahcall, j. n., kirshner, r. p., & woosley, s. e. 1989, araa, 27, 629 barbon, r., ciatti, f., & rosino, l. 1979, a&a, 72, 287 becker, a., et al. 2001, baas, 199.1101.14 ; see also http://dls.bell-labs.com/team.html bessell, m. s., & brett, j. m. 1988,, 100, 1134 brighenti, f., & mathews, w. g. 1998, apj, 515, 542 brighenti, f., & mathews, w. g. 2001, apj, 553, 103 brunner, r. j., djorgovski, s. g., & szalay, a. s. 2001, eds., virtual observatories of the future (san francisco : asp, conf. ser. 225) cappellaro, e., et al. 1997, a&a, 322, 431 cardelli, j. a., clayton, g. c., & mathis, j. s. 1989, apj, 345, 245 clocchiatti, a., et al. 2000,, 529, 661 coil, a. l., et al. 2000, apj, 544, l111 cousins, a. w. j. 1976 ; see moro & munari 2000 for details dahln, t., & fransson, c. 1999, a&a, 350, 349 dahln, t., & goobar, a. 2002, pasp, 114, 284 (dg2002) dimai, a. 2001, iau circ. 7719 doggett, j. b., & branch, d. 1985, aj, 90, 2303 fassia, a., et al. 2001, mnras, 325, 907 filippenko, a. v. 1982, pasp, 94, 715 filippenko, a. v. 1988, aj, 96, 1941 filippenko, a. v. 1997, ara&a, 35, 309 filippenko, a. v. 2001, pasp, 113, 1441 filippenko, a. v., barth, a. j., bower, g. c., ho, l. c., stringfellow, g. c., goodrich, r. w., & porter, a. c. 1995a, aj, 110, 2261 [erratum : 112, 806 (1996)] filippenko, a. v., & chornock, r. 2001, iau circ. 7754 filippenko, a. v., li, w. d., treffers, r. r., & modjaz, m. 2001, in small - telescope astronomy on global scales, ed. w. p. chen, c. lemme, & b. paczyski (san francisco : asp, conf. 246), 121 filippenko, a. v., matheson, t., & ho, l. c. 1993, apj, 415, l103 filippenko, a. v., et al. 1992a, apj, 384, l15 filippenko, a. v., et al. 1992b, aj, 104, 1543 filippenko, a. v., et al. 1995b, apj, 450, l11 fukugita, m., ichikawa, t., gunn, j. e., doi, m., shimasaku, k., & schneider, d. p. 1996,, 111, 1748 galama, t. j., et al. 1998, nature, 395, 670 gal - yam, a., & maoz, d. 1999, in `` cosmic explosions, '' ed. s. s. holt & w. w. zhang (new york : aip), p. 107 gal - yam, a., & maoz, d. 2000, iau circ. 7405 gal - yam, a., & maoz, d. 2002, in preparation gal - yam, a., maoz, d., & sharon, k. 2002, mnras, in press, astro - ph/0109089 gal - yam, a., & shemmer, o. 2001, iau circ. 7602 ; see http://wise-obs.tau.ac.il/$\sim$avishay/local.html germany, l. m., reiss, d. j., sadler, e. m., schmidt, b. p., & stubbs, c. w. 2000,, 533, 320 goobar, a., & perlmutter, s. 1995, apj, 450, 14 hamuy, m., & pinto, p. a. 2002, apjl, 566, l63 hamuy, m., et al. 2001, apj, 558, 615 hamuy, m., et al. 2002, aj, in press, astro - ph/0203491 hardin, d., et al. 2000, iau circ. 7406 hflich, p., straniero, o., limongi, m., dominguez, i., & chieffi, a. 2001, in rev. mexicana astron. 10, the seventh texas - mexico conference on astrophysics : flows, blows, and glows, ed. w. lee & s. torres - peimbert (mexico, df : inst. astron., unam), 157 iwamoto, k., et al. 2000, apj, 534, 660 jannuzi, b., et al. 2001 ; see http://www.noao.edu/noao/noaodeep johnson, h. l. 1965 ; see moro & munari 2000 for details jorgensen, h. e., lipunov, v. m., panchenko, i. e., postnov, k. a., & prokhorov, m. e. 1997,, 486, 110 kirshner, r. p., & kwan, j. 1974, apj, 193, 27 krisciunas, k., margon, b., & szkody, p. 1998, pasp, 110, 1342 leibundgut, b. 1988, ph.d. thesis, university of basel leonard, d. c., filippenko, a. v., barth, a. j., & matheson, t. 2000, apj, 536, 239 leonard, d. c., et al. 2002a, pasp, 114, 35 leonard, d. c., et al. 2002b, aj, submitted li, w., filippenko, a. v., treffers, r. r., riess, a. g., hu, j., & qiu, y. 2001a,, 546, 734 li, w., et al. 1996, iau circ. 6379 li, w., et al. 2001b, pasp, 113, 1178 loewenstein, m. 2000, apj, 532, 17 madau, p., della valle, m., & panagia, n. 1998, mnras, 297, l17 matheson, t., filippenko, a. v., li, w., leonard, d. c., & shields, j. c. 2001, aj, 121, 1648 matteucci, f., & greggio, l. 1986,, 154, 279 miknaitis, g., & krisciunas, k. 2001, iau circ. 7778 miknaitis, g., miceli, a., stubbs, c., covarrubias, r., & lawton, b. 2001a, iau circ. 7731 miknaitis, g., rest, a., stubbs, c., stoughton, c., & sdss collaboration. 2001b, baas, 199.84.04 miller, d. l., & branch, d. 1990, aj, 100, 530 minkowski, r. 1941, pasp, 53, 224 moro, d., & munari, u. 2000, a&as, 147, 361 pain, r., et al. 1996,, 473, 356 parodi, b. r., saha, a., sandage, a., & tammann, g. a. 2000, apj, 540, 634 patat, f., et al. 2001, apj, 555, 900 perlmutter, s., et al. 1997, apj, 483, 565 perlmutter, s., et al. 1999, apj, 517, 565 perlmutter, s., & snap collaboration. 2000, baas, 197.61.01 phillips, m. m., lira, p., suntzeff, n. b., schommer, r. a., hamuy, m., & maza, j. 1999,, 118, 1766 qiu, y., & hu, j. y. 2001, iau circ. 7753 qiu, y., li, w., qiao, q., & hu, j. 1999, aj, 117, 736 reiss, d., germany, l. m., schmidt, b. p., & stubbs, c. w. 1998, aj, 115, 26 rest, a., miceli, a., & covarrubias, r. 2001, iau circ. 7740 rhoads, j. e. 2001, apj, 557, 943 richards, g. t., et al. 2001,, 122, 1151 richmond, m. w. et al. 1995,, 109, 2121. riess, a. g., et al. 1998, aj, 116, 1009 riess, a. g., et al. 1999, aj, 117, 707 riess, a. g., et al. 2001, apj, 560, 49 ruiz - lapuente, p., & canal, r. 1998, apj, 497, l57 sadat, r., blanchard, a., guiderdoni, b., & silk, j. 1998, a&a, 331, l69 schaefer, b. e. 2000, iau circ. 7387 schlegel, d. j., finkbeiner, d. p., & davis, m. 1998, apj, 500, 525 schlegel, e. m. 1990, mnras, 244, 269 schmidt, b. p., kirshner, r. p., & eastman, r. g. 1992, apj, 395, 366 schmidt, b. p., et al. 1994,, 432, 42 schmidt, b. p., et al. 1998, apj, 507, 46 stoughton, c., et al. 2002,, 123, 485 sullivan, m., ellis, r., nugent, p., smail, i., & madau, p. 2000,, 319, 549 treffers, r. r., peng, c. y., filippenko, a. v., & richmond, m. w. 1997, iau circ. 6627 tripp, r. 1998,, 331, 815 turatto, m., et al. 2000, apj, 534, 57 vanden berk, d. e., et al. 2001, baas, 199.84.05 van dyk, s. d., peng, c. y., king, j. y., filippenko, a. v., treffers, r. r., li, w., & richmond, m. w. 2000, pasp, 112, 1532 weymann, r. j., storrie - lombardi, l. j., sawicki, m., & brunner, r. 1999, eds., photometric redshifts and high - redshift galaxies (san francisco : asp) york, d. g., et al. 2000, aj, 120, 1579 yungelson, l., & livio, m. 2000, apj, 528, 108 clccc ia & 1994d & 22 & 0.0015 & 1,9 + & 1987l & 2 & 0.0074 & 1 + & 1995d & 4 & 0.0066 & 9 + & 1999dk & 5 & 0.0150 & 9 + & 1999ee & 12 & 0.0114 & 10 + ib & 1984l & 12 & 0.0051 & 1,9 + & 1991ar & 1 & 0.0152 & 2 + & 1998dt & 2 & 0.0150 & 2 + & 1999di & 1 & 0.0164 & 2 + & 1999dn & 3 & 0.0093 & 2 + ic & 1994i & 14 & 0.0015 & 1,3 + & 1990u & 8 & 0.0079 & 2 + & 1990b & 4 & 0.0075 & 2 + ii - p & 1999em & 27 & 0.0024 & 4 + & 1992h & 13 & 0.0060 & 1,9 + & 2001x & 12 & 0.0049 & 6,9 + iin & 1998s & 13 & 0.0030 & 5,7 + & 1994y & 1 & 0.0080 & 1 + & 1994ak & 1 & 0.0085 & 1 + iib & 1993j & 12 & 0 & 1 + & 1996cb & 3 & 0.0024 & 8 + total & & 172 & & + clccc 1991t - like & 1991 t & 12 & 0.0058 & 1,5 + & 1998es & 7 & 0.0106 & 5 + 1991bg - like & 1991bg & 5 & 0.0035 & 2,5 + & 1998bp & 2 & 0.0104 & 5 + & 1998de & 3 & 0.0166 & 5 + & 1999da & 4 & 0.0127 & 5 + 2000cx - like & 2000cx & 23 & 0.0079 & 3,4 + total & & 56 & & +

large numbers of supernovae (sne) have been discovered in recent years, and many more will be found in the near future. once discovered, further study of a sn and its possible use as an astronomical tool (e.g., as a distance estimator) require knowledge of the sn type. current classification methods rely almost solely on the analysis of sn spectra to determine their type. however, spectroscopy may not be possible or practical when sne are faint, numerous, or discovered in archival studies. we present a classification method for sne based on the comparison of their observed colors with synthetic ones, calculated from a large database of multi - epoch optical spectra of nearby events. we discuss the capabilities and limitations of this method. for example, type ia sne at redshifts @xmath0 can be distinguished from most other sn types during the first few weeks of their evolution, based on @xmath1 vs. @xmath2 colors. type ii - p sne have distinct (very red) colors at late (@xmath3 d) stages. broadband photometry through standard johnson - cousins @xmath4 filters can be useful to classify sne out to @xmath5. the use of sloan digital sky survey (sdss) @xmath6 filters allows the extension of our classification method to even higher redshifts (@xmath7), and the use of infrared bands, to @xmath8. we demonstrate the application of this method to a recently discovered sn from the sdss. finally, we outline the observational data required to further improve the sensitivity of the method, and discuss prospects for its use on future sn samples. community access to the tools developed is provided by a dedicated website.

astro-ph0202198

Introduction inflation generically predicts a primordial spectrum of density perturbations which is almost precisely gaussian @xcite. in recent years the small non - gaussian component @xcite has emerged as an important observable @xcite, and will be measured with good precision by the _ planck surveyor _ satellite @xcite. in the near future, as observational data become more plentiful, it will be important to understand the non - gaussian signal expected in a wide variety of models, and to anticipate what conclusions can be drawn about early - universe physics from a prospective detection of primordial non - gaussianity. in this paper, we present a novel method for calculating the primordial non - gaussianity produced by super - horizon evolution in two - field models of inflation. our method is based on the real - space distribution of inflationary field values on a flat hypersurface, which can be thought of as a probability density function whose evolution is determined by a form of the collisionless boltzmann equation. using a cumulant representation @xcite to expand our density function around an exact gaussian, we derive ordinary differential equations which evolve the moments of this distribution. further, we show how these moments are related to observable quantities, such as the dimensionless bispectrum measured by @xmath1 @xcite. we present numerical results which show that this method gives good agreement with other techniques. it is not necessary to make any assumptions about the inflationary model beyond requiring a canonical kinetic term and applying the slow - roll approximation. while there are already numerous methods for computing the super - horizon contribution to @xmath1, including the widely used @xmath0 formalism, we believe the one reported here has a number of advantages. first, this new technique is ideally suited to unraveling the various contributions to @xmath1. this is because we follow the moments of the inflaton distribution directly, which makes it straightforward to identify large contributions to the skewness or other moments. the evolution equation for each moment is simple and possesses clearly identifiable source terms, which are related to the properties of the inflationary flow on field space. this offers a clear separation between two key sources of primordial non - gaussianity. one of these is the intrinsic non - linearity associated with evolution of the probability density function between successive flat hypersurfaces ; the other is a gauge transformation from field fluctuations to the curvature peturbation, @xmath2. it would be difficult or impossible to observe this split within the context of other calculational schemes, such as the conventional @xmath0 formalism. a second advantage of our method is connected with the computational cost of numerical implementation. analytic formulas for @xmath1 are known in certain cases, mostly in the context of the @xmath0 framework, but only for very specific choices of the potential @xcite or hubble rate @xcite. these formulas become increasingly cumbersome as the number of fields increases, or if one studies higher moments @xcite. in the future, it seems clear that studies of complex models with many fields will increasingly rely on numerical methods. the numerical @xmath0 framework requires the solution to a number of ordinary differential equations which scales exponentially with the number of fields. since some models include hundreds of fields, this may present a significant obstacle @xcite. moreover, the @xmath0 formalism depends crucially on a numerical integration algorithm with low noise properties, since finite differences must be extracted between perturbatively different initial conditions after @xmath3 e - folds of evolution. thus, the background equations must be solved to great accuracy, since any small error has considerable scope to propagate. in this paper we ultimately solve our equations numerically to determine the evolution of moments in specific models. our method requires the solution to a number of differential equations which scales at most polynomially (or in certain cases perhaps even linearly) with the number of fields. it does not rely on extracting finite differences, and therefore is much less susceptible to numerical noise. these advantages may be shared with other schemes, such as the numerical method recently employed by lehners & renaux - petel @xcite. a third advantage, to which we hope to return in a future publication, is that our formalism yields explicit evolution equations with source terms. from an analysis of these source terms, we hope that it will be possible to identify those physical features of specific models which lead to the production of large non - gaussianities. this paper is organized as follows. in [sec : computing_fnl], we show how the non - gaussian parameter @xmath1 can be computed in our framework. the calculation remains in real space throughout (as opposed to fourier space), which modifies the relationship between @xmath1 and the multi - point functions of the inflaton field. our expression for @xmath1 shows a clean separation between different contributions to non - gaussianity, especially between the intrinsic nonlinearity of the field evolution and the gauge transformation between comoving and flat hypersurfaces. in [sec : transport], we introduce our model for the distribution of inflaton field values, which is a moment expansion " around a purely gaussian distribution. we derive the equations which govern the evolution of the moments of this distribution in the one- and two - field cases. in [sec : numerics], we present a comparison of our new technique and those already in the literature. we compute @xmath1 numerically in several two - field models, and find excellent agreement between techniques. we conclude in [s : conclusions]. throughout this paper, we use units in which @xmath4, and the reduced planck mass @xmath5 is set to unity. Frameworks for computing @xmath1 in this section, we introduce our new method for computing the non - gaussianity parameter @xmath1. this method requires three main ingredients : a formula for the curvature perturbation, @xmath2, in terms of the field values on a spatially flat hypersurface ; expressions for the derivatives of the number of e - foldings, @xmath6, as a function of field values at horizon exit ; and a prescription for evolving the field distribution from horizon exit to the time when we require the statistical properties of @xmath2. the first two ingredients are given in eqs. and , found at the end of [ss : sep_universe] and [sec : derivative - n] respectively. the final ingredient is discussed in [sec : transport]. once it became clear that non - linearities of the microwave background anisotropies could be detected by the wmap and _ planck _ survey satellites @xcite, many authors studied higher - order correlations of the curvature perturbation. in early work, direct calculations of a correlation function were matched to the known limit of local non - gaussianity @xcite. this method works well if isocurvature modes are absent, so that the curvature perturbation is constant after horizon exit. in the more realistic situation that isocurvature modes cause evolution on superhorizon scales, all correlation functions become time dependent. various formalisms have been employed to describe this evolution. lyth & rodrguez @xcite extended the @xmath0 method @xcite beyond linear order. this method is simple and well - suited to analytical calculation. rigopoulos, shellard and van tent @xcite worked with a gradient expansion, rewriting the field equations in langevin form. the noise term was used as a proxy for setting initial conditions at horizon crossing. a similar ` exact'gradient formalism was written down by langlois & vernizzi @xcite. in its perturbative form, this approach has been used by lehners & renaux - petel to obtain numerical results @xcite. another numerical scheme has been introduced by huston & malik @xcite. what properties do we require of a successful prediction? consider a typical observer, drawn at random from an ensemble of realizations of inflation. in any of the formalisms discussed above, we aim to estimate the statistical properties of the curvature perturbation which would be measured by such an observer. some realizations may yield statistical properties which are quite different from the ensemble average, but these large excursions are uninteresting unless anthropic arguments are in play. next we introduce a collection of comparably - sized spacetime volumes whose mutual scatter is destined to dominate the microwave background anisotropy on a given scale. neglecting spatial gradients, each spacetime volume will follow a trajectory in field space which is slightly displaced from its neighbors. the scatter between trajectories is determined by initial conditions set at horizon exit, which are determined by promoting the vacuum fluctuation to a classical perturbation. a correct prediction is a function of the trajectories followed by every volume in the collection, taken as a whole. one never makes a prediction for a single trajectory. each spacetime volume follows a trajectory, which we label with its position @xmath7 at some fixed time, to be made precise below. throughout this paper, superscript ` @xmath8'denotes evaluation on a spatially flat hypersurface. consider the evolution of some quantity of interest, @xmath9, which is a function of trajectory. if we know the distribution @xmath10 we can study statistical properties of @xmath9 such as the @xmath11 moment @xmath12, @xmath13^m, \]] where we have introduced the ensemble average of @xmath9, @xmath14 in eqs. , @xmath7 stands for a collection of any number of fields. it is the @xmath12 which are observable quantities.. defines what we will call the exact separate universe picture. it is often convenient to expand @xmath15 as a power series in the field values centered on a fiducial trajectory, labelled ` fid,'@xmath16 when eq. is used to evaluate the @xmath12, we refer to the ` perturbative'separate universe picture. if all terms in the power series are retained, these two versions of the calculation are formally equivalent. in unfavorable cases, however, convergence may occur slowly or not at all. this possibility was discussed in refs. although our calculation is formally perturbative, it is not directly equivalent to eq. . we briefly discuss the relation of our calculation to conventional perturbation theory in [s : conclusions]. by definition, the curvature perturbation @xmath2 measures local fluctuations in expansion history (expressed in e - folds @xmath6), calculated on a comoving hypersurface. in many models, the curvature perturbation is synthesized by superhorizon physics, which reprocesses a set of gaussian fluctuations generated at horizon exit. in a single - field model, only one gaussian fluctuation can be present, which we label @xmath17. neglecting spatial gradients, the total curvature perturbation must then be a function of @xmath17 alone. for @xmath18, this can be well - approximated by @xmath19 where @xmath1 is independent of spatial position. defines the so - called `` local '' form of non - gaussianity. it applies only when quantum interference effects can be neglected, making @xmath2 a well - defined object rather than a superposition of operators @xcite. if this condition is satisfied, spatial correlations of @xmath2 may be extracted and it follows that @xmath1 can be estimated using the rule @xmath20 where we have recalled that @xmath2 is nearly gaussian, or equivalently that @xmath21. with @xmath1 spatially independent, eq. strictly applies only in single - field inflation. in this case one can accurately determine @xmath1 by applying eq. to a single trajectory with fixed initial conditions, as in the method of lehners & renaux - petel @xcite. where more than one field is present, @xmath1 may vary in space because it depends on the isocurvature modes. in this case one must determine @xmath1 statistically on a bundle of adjacent trajectories which sample the local distribution of isocurvature modes. is then indispensible. following maldacena @xcite, and later lyth & rodrguez @xcite, we adopt eq. as our definition of @xmath1, whatever its origin. in real space, the coefficient @xmath22 in eq. depends on the convention @xmath23. more generally, this follows from the definition of @xmath12, eq. . in fourier space, either prescription is automatically enforced after dropping disconnected contributions, again leading to eq. . to proceed, we require estimates of the correlation functions @xmath24 and @xmath25. we first describe the conventional approach, in which ` @xmath8'denotes a flat hypersurface at a fixed initial time. the quantity @xmath26 denotes the number of e - foldings between this initial slice and a final comoving hypersurface, where @xmath27 indexes the species of light scalar fields. the local variation in expansion can be written in the fiducial picture as @xmath28 where @xmath29. subject to the condition that the relevant scales are all outside the horizon, we are free to choose the initial time set by the hypersurface ` @xmath8'at our convenience. in the conventional approach, ` @xmath8'is taken to lie a few e - folds after our collection of spacetime volumes passes outside the causal horizon @xcite. this choice has many virtues. first, we need to know statistical properties of the field fluctuations @xmath30 only around the time of horizon crossing, where they can be computed without the appearance of large logarithms @xcite. second, as a consequence of the slow - roll approximation, the @xmath31 are uncorrelated at this time, leading to algebraic simplifications. finally, the @xmath0 formula subsumes a gauge transformation from the field variables @xmath30 to the observational variable @xmath2. using eqs. , and, one finds that @xmath1 can be written to a good approximation @xcite @xmath32 where @xmath33 and for simplicity we have dropped the ` @xmath8'which indicates time of evaluation. a similar definition applies for @xmath34.. is accurate up to small intrinsic non - gaussianities present in the field fluctuations at horizon exit. as a means of predicting @xmath1 it is pleasingly compact, and straightforward to evaluate in many models. unfortunately, it also obscures the physics which determines @xmath35. for this reason it is hard to infer, from eq. alone, those classes of models in which @xmath35 is always large or small. is dynamically allowed. see, for example, refs. @xcite.] our strategy is quite different. we choose ` @xmath8'to lie around the time when we require the statistical properties of @xmath2. the role of the @xmath0 formula, eq. , is then to encode _ only _ the gauge transformation between the @xmath36 and @xmath2. in [sec : derivative - n] below, we show how the appropriate gauge transformation is computed using the @xmath0 formula. in the present section we restrict our attention to determining a formula for @xmath1 under the assumption that the distribution of field values on ` @xmath8'is known. in [sec : transport], we will supply the required prescription to evolve the distribution of field values between horizon exit and ` @xmath8 '. although the @xmath31 are independent random variables at horizon exit, correlations can be induced by subsequent evolution. one must therefore allow for off - diagonal terms in the two - point function. remembering that we are working with a collection of spacetime volumes in real space, smoothed on some characteristic scale, we write @xmath37 @xmath38 does not vary in space, but it may be a function of the scale which characterizes our ensemble of spacetime volumes. in all but the simplest models it will vary in time. it is also necessary to account for intrinsic non - linearities among the @xmath31, which are small at horizon crossing but may grow. we define @xmath39 likewise, @xmath40 should be regarded as a function of time and scale. the permutation symmetries of an expectation value such as guarantee that, for example, @xmath41. when written explicitly, we place the indices of symbols such as @xmath42 in numerical order. neglecting a small (@xmath43) intrinsic four - point correlation, it follows that @xmath44 now we specialize to a two - field model, parametrized by fields @xmath45 and @xmath46. using eqs. , and , it follows that the two - point function of @xmath2 satisfies @xmath47 the three - point function can be written @xmath48 where we have identified two separate contributions, labelled ` 1'and ` 2 '. the ` 1'term includes all contributions involving _ intrinsic _ non - linearities, those which arise from non - gaussian correlations among the field fluctuations, @xmath49 the ` 2'term encodes non - linearities arising directly from the gauge transformation to @xmath2 @xmath50 after use of eq. , can be used to extract the non - linearity parameter @xmath1. this decomposes likewise into two contributions @xmath51, which we shall discuss in more detail in [sec : numerics]. to compute @xmath1 in concrete models, we require expressions for the derivatives @xmath52 and @xmath34. for generic initial and final times, these are difficult to obtain. lyth & rodrguez @xcite used direct integration, which is effective for quadratic potentials and constant slow - roll parameters. vernizzi & wands @xcite obtained expressions in a two - field model with an arbitrary sum - separable potential by introducing gaussian normal coordinates on the space of trajectories. their approach was generalized to many fields by battefeld & easther @xcite. product - separable potentials can be accommodated using the same technique @xcite. an alternative technique has been proposed by yokoyama _ _ @xcite. a considerable simplification occurs in the present case, because we only require the derivative evaluated between flat and comoving hypersurfaces which coincide in the unperturbed universe. for any species @xmath27, and to leading order in the slow - roll approximation, the number of e - folds @xmath6 between the flat hypersurface ` @xmath8'and a comoving hypersurface ` @xmath53'satisfies @xmath54 where @xmath55 and @xmath56 are the field values evaluated on ` @xmath8'and ` @xmath53,'respectively. under an infinitesimal shift of @xmath57, we deduce that @xmath52 obeys @xmath58 note that this applies for an arbitrary @xmath59, which need not factorize into a sum or product of potentials for the individual species @xmath27. in principle a contribution from variation of the integrand is present, which spoils a nave attempt to generalize the method of refs. @xcite to an arbitrary potential. this contribution vanishes in virtue of our supposition that ` @xmath8'and ` @xmath53'are infinitesimally separated. to compute @xmath60 it is helpful to introduce a quantity @xmath61, which in the sum - separable case coincides with the conserved quantity of vernizzi & wands @xcite. for our specific choice of a two - field model, this takes the form @xmath62 where the integrals are evaluated on a single spatial hypersurface. in an @xmath63-field model, one would obtain @xmath64 conserved quantities which label the isocurvature fields. the construction of these quantities is discussed in refs. @xcite. for sum - separable potentials one can show using the equations of motion that @xmath61 is conserved under time evolution to leading order in slow - roll. it is not conserved for general potentials, but the variation can be neglected for infinitesimally separated hypersurfaces. under a change of trajectory, @xmath61 varies according to the rules @xmath65 and @xmath66 the comoving hypersurface ` @xmath53'is defined by @xmath67 we are assuming that the slow - roll approximation applies, so that the kinetic energy may be neglected in comparison with the potential @xmath59. therefore on ` @xmath53'we have @xmath68 combining eqs. , and we obtain expressions for @xmath60, namely @xmath69 where we have defined @xmath70 eqs. can alternatively be derived without use of @xmath61 by comparing eq. with the formulas of ref. @xcite, which were derived using conventional perturbation theory. applying , we obtain @xmath71 to proceed, we require the second derivatives of @xmath6. these can be obtained directly from , after use of eqs.. we find @xmath72^\star \cos^2 \theta + 2 \left (\frac{v}{v_{,1 } } \right)^{\star 2 } \cos^2 \theta \\ \hspace{-6 mm } \mbox { } \times \left [ \frac{v_{,11}}{v } \sin^2 \theta - \frac{v_{,1 } v_{,12}}{v v_{,2 } } \sin^4 \theta - \left ( \frac{v_{,11}}{v } - \frac{v_{,22}}{v } + \frac{v_{,2 } v_{,12}}{v v_{,1 } } \right) \cos^2 \theta \sin^2 \theta \right]^c. \end{aligned}\]] an analogous expression for @xmath73 can be obtained after the simultaneous exchange @xmath74. the mixed derivative satisfies @xmath75^c \\ \hspace{-6 mm } \mbox { } + \cos^2 \theta \left ( \frac{v_{,2}}{v_{,1 } } - \frac{v v_{,12}}{v_{,1}^2 } \right)^c. \end{aligned}\]] now that the calculation is complete, we can drop the superscripts ` @xmath8'and ` @xmath53,'since any background quantity is the same on either hypersurface. once this is done it can be verified that (despite appearances) eq. is invariant under the exchange @xmath76. Transport equations in this section we return to the problem of evolution between horizon exit and the time of observation, and supply the prescription which connects the distribution of field values at these two times. we begin by discussing the single - field system, which lacks the technical complexity of the two - field case, yet still exhibits certain interesting features which recur there. among these features are the subtle difference between motion of the statistical mean and the background field value, and the hierarchy of moment evolution equations. moreover, the structure of the moment mixing equations is similar to that which obtains in the two - field case. for this reason, the one - field scenario provides an instructive example of the techniques we wish to employ. recall that we work in real space with a collection of comparably sized spacetime volumes, each with a slightly different expansion history, and the scatter in these histories determines the microwave background anisotropy on a given angular scale. within each volume the smoothed background field @xmath77 takes a uniform value described by a density function @xmath78, where in this section we are dropping the superscript ` @xmath8'denoting evaluation of spatially flat hypersurfaces. our ultimate goal is to calculate the reduced bispectrum, @xmath1, which describes the third moment of @xmath78. in the language of probability this is the skewness, which we denote @xmath42. a gaussian distribution has skewness zero, and inflation usually predicts that the skew is small. for this reason, rather than seek a distribution with non - zero third moment, as proposed in ref. @xcite, we will introduce higher moments as perturbative corrections to the gaussian. such a procedure is known as a _ cumulant expansion_. the construction of cumulant expansions is a classical problem in probability theory. we seek a distribution with centroid @xmath79, variance @xmath80, and skew @xmath42, with all higher moments determined by @xmath81 and @xmath42 alone. a distribution with suitable properties is @xmath82, \]] where @xmath83\]] is a pure gaussian and @xmath84 denotes the @xmath85 hermite polynomial, for which there are multiple normalization conventions. we choose to normalize so that @xmath86 which implies that the leading term of @xmath87 is @xmath88. this is sometimes called the `` probabilist s convention. '' we define expectation values @xmath89 by the usual rule, @xmath90 the probability density function in eq. has the properties, and do not depend on the approximation that @xmath42 is small. however, for large @xmath42 the density function may become negative for some values of @xmath77. it then ceases to be a probability density in the strict sense. this does not present a problem in practice, since we are interested in distributions which are approximately gaussian, and for which @xmath42 will typically be small. moreover, our principal use of eq. is as a formal tool to extract evolution equations for each moment. for this reason we will not worry whether @xmath78 defines an honest probability density function in the strict mathematical sense.] @xmath91 the moments @xmath79, @xmath81, and @xmath42 may be time - dependent, so evolution of the probability density in time can be accommodated by finding evolution equations for these quantities. the density function given in eq. is well - known and has been applied in many situations. it is a solution to the problem of approximating a nearly - gaussian distribution whose moments are known. ([e : p1d]) is in fact the first two terms of the _ gram charlier ` a'series _, also sometimes called the _ hermite series_. in recent years it has found multiple applications to cosmology, of which our method is closest to that of taylor & watts @xcite. other applications are discussed in refs. @xcite. for a review of the ` a'series and related nearly - gaussian probability distributions from an astrophysical perspective, see @xcite. in this paper, we will refer to eq. and its natural generalization to higher moments as the `` moment expansion. '' in the slow - roll approximation, the field in each spacetime volume obeys a simple equation of motion @xmath92 where @xmath6 records the number of e - foldings of expansion. we refer to @xmath93 as the velocity field. expanding @xmath94 about the instantaneous centroid @xmath79 gives @xmath95 where @xmath96 the value of @xmath79 evolves with time, so each expansion coefficient is time - dependent. hence, we do not assume that the velocity field is _ globally _ well - described by a quadratic taylor expansion, but merely that it is well - described as such in the neighborhood of the instantaneous centroid. we expand the velocity field to second order, although in principle this expansion could be carried to arbitrary order. it remains to specify how the probability density evolves in time. conservation of probability leads to the transport equation @xmath97 eq. can also be understood as the limit of a chapman kolmogorov process as the size of each hop goes to zero. it is well known for example, from the study of starobinsky s diffusion equation which forms the basis of the stochastic approach to inflation @xcite that the choice of time variable in this equation is significant, with different choices corresponding to the selection of a temporal gauge. we have chosen to use the e - folding time, @xmath6, which means that we are evolving the distribution on hypersurfaces of uniform expansion. these are the spatially flat hypersurfaces whose field perturbations enter the @xmath0 formulas described in [sec : computing_fnl]. in principle, eq. can be solved directly. in practice it is simpler to extract equations for the moments of @xmath98, giving evolution equations for @xmath79, @xmath81 and @xmath42. to achieve this, one need only resolve eq. into a hermite series of the form @xmath99 the hermite polynomials are linearly independent, and application of the orthogonality condition shows that the @xmath100 must all vanish. this leads to a hierarchy of equations @xmath101, which we refer to as the moment hierarchy. at the top of the hierarchy, the equation @xmath102 is empty and expresses conservation of probability. the first non - trivial equation requires @xmath103 and yields an evolution equation for the centroid @xmath79, @xmath104 the first term on the right - hand side drives the centroid along the velocity field, as one would anticipate based on the background equation of motion, eq. . however, the second term shows that the centroid is also influenced as the wings of the probability distribution probe the nearby velocity field. this influence is not captured by the background equation of motion. if we are in a situation with @xmath105, then the wings of the density function will be moving faster than the center. hence, the velocity of the centroid will be larger than one might expect by restricting attention to @xmath79. accordingly, the mean fluctuation value is not following a solution to the background equations of motion. evolution equations for the variance @xmath80 and skew @xmath42 are obtained after enforcing @xmath106, yielding @xmath107 in both equations, the first term on the right - hand sides describes how @xmath81 and @xmath42 scale as the density function expands or contracts in response to the velocity field. these terms force @xmath80 and @xmath42 to scale in proportion to the velocity field. specifically, if we temporarily drop the second terms in each equation above, one finds that @xmath108 and @xmath109. this precisely matches our expectation for the scaling of these quantities. hence, these terms account for the jacobians associated with infinitesimal transformations induced by the flow @xmath93. for applications to inflationary non - gaussianity, the second terms in and are more relevant. these terms describe how each moment is sourced by higher moments and the interaction of the density function with the velocity field. in the example above, if we are in a situation where @xmath105, the tails of the density function are moving faster than the core. this means that one tail is shrinking and the other is extending, skewing the probability density. the opposite occurs when @xmath110. these effects are measured by the second term in . hence, by expanding our pdf to the third moment, and our velocity field to quadratic order, we are able to construct a set of evolution equations which include the leading - order source terms for each moment. there is little conceptually new as we move from one field to two. the new features are mostly technical in nature. our primary challenge is a generalization of the moment expansion to two fields, allowing for the possibility of correlation between the fields. with this done, we can write down evolution equations whose structure is very similar to those found in the single - field case. the two - field system is described by a two - dimensional velocity field @xmath111, defined by @xmath112 where again we are using the number of e - folds @xmath6 as the time variable. the index @xmath27 takes values in @xmath113. while we think it is likely that our equations generalize to any number of fields, we have only explicitly constructed them for a two - field system. as will become clear below, certain steps in this construction apply only for two fields, and hence we make no claims at present concerning examples with three or more fields. the two - dimensional transport equation is @xmath114 = 0.\]] here and in the following we have returned to our convention that repeated species indices are summed. as in the single - field case, we construct a probability distribution which is nearly gaussian, but has a small non - zero skewness. that gives @xmath115 where @xmath116 is a pure gaussian distribution, defined by @xmath117.\]] in this equation, @xmath118 defines the center of the distribution and @xmath119 describes the covariance between the fields. we adopt a conventional parametrization in terms of variances @xmath120 and a correlation coefficient @xmath121, @xmath122 the matrix @xmath81 defines two - point correlations of the fields, @xmath123 all skewnesses are encoded in @xmath124. before defining this explicitly, it is helpful to pause and notice a complication inherent in eqs. which was not present in the single - field case. to extract a hierarchy of moment evolution equations from the transport equation, eq., we made the expansion given in and argued that orthogonality of the hermite polynomials implied the hierarchy @xmath101. however, hermite polynomials of the form @xmath125 $] are _ not _ orthogonal under the gaussian measure of eq. . following an expansion analogous to eq. the moment hierarchy would comprise linear combinations of the coefficients. the problem is essentially an algebraic question of gram schmidt orthogonalization. to avoid this problem it is convenient to diagonalize the covariance matrix @xmath119, introducing new variables @xmath126 and @xmath127 for which eq. factorizes into the product of two measures under which the polynomials @xmath128 and @xmath129 are separately orthogonal. the necessary redefinitions are @xmath130\]] and @xmath131.\]] a simple expression for @xmath116 can be given in terms of @xmath126 and @xmath127, @xmath132 we now define the non - gaussian factor, which encodes the skewnesses, to be @xmath133 in these variables we find @xmath134, but @xmath135. in addition, we have @xmath136 in order for eq. to be useful, it is necessary to express the skewnesses associated with the physical variables @xmath137 in terms of @xmath126 and @xmath127. by definition, these satisfy @xmath138 after substituting for the definition of these quantities inside the expectation values in eq. we arrive at the relations @xmath139 the moments @xmath118, @xmath38 and @xmath40 are time - dependent, but for clarity we will usually suppress this in our notation. next we must extract the moment hierarchy, which governs evolution of @xmath118, @xmath140, @xmath121 and @xmath40. we expand the velocity field in a neighborhood of the instantaneous centroid @xmath118 according to @xmath141 where we have defined @xmath142 as in the single - field case, these coefficients are functions of time and vary with the motion of the centroid. the expansion can be pursued to higher order if desired. our construction of @xmath126 and @xmath127 implies that the two - field transport equation can be arranged as a double gauss hermite expansion, @xmath143 = p_g \sum_{m, n \ge 0 } c_{mn } h_m(x) h_n(y) = 0.\]] because the hermite polynomials are orthogonal in the measure defined by @xmath116, we deduce the moment hierarchy @xmath144 we define the rank " @xmath145 of each coefficient @xmath146 by @xmath147. we terminated the velocity field expansion at quadratic order, and our probability distribution included only the first three moments. it follows that only @xmath146 with rank five or less are nonzero. if we followed the velocity field to higher order, or included higher terms in the moment expansion, we would obtain non - trivial higher - rank coefficients. inclusion of additional coefficients requires no qualitative modification of our analysis and can be incorporated in the scheme we describe below. a useful feature of the expansion in eq. is that the rank-@xmath145 coefficients give evolution equations for the order-@xmath145 moments. written explicitly in components, the expressions that result from are quite cumbersome. however, when written as field - space covariant expressions they can be expressed in a surprisingly compact form. : : the rank-0 coefficient @xmath148 is identically zero. this expresses the fact that the total probability is conserved as the distribution evolves. : : the rank-1 coefficients @xmath149 and @xmath150 give evolution equations for the centroid @xmath118. these equations can be written in the form @xmath151 we remind the reader that here and below, terms like @xmath152, @xmath153 and @xmath154 represent the velocity field and its derivatives evaluated at the centroid @xmath118. the first term in expresses the non - anomalous motion of the centroid, which coincides with the background velocity field of eq. . the second term describes how the wings of the probability distribution sample the velocity field at nearby points. narrow probability distributions have small components of @xmath119 and hence are only sensitive to the local value of @xmath155. broad probability distributions have large components of @xmath119 and are therefore more sensitive to the velocity field far from the centroid. : : the rank-2 coefficients @xmath156, @xmath157 and @xmath158 give evolution equations for the variances @xmath120 and the correlation @xmath121. these can conveniently be packaged as evolution equations for the matrix @xmath119 @xmath159 this equation describes the stretching and rotation of @xmath119 as it is transported by the velocity field. it includes a sensitivity to the wings of the probability distribution, in a manner analogous to the similar term appearing in . hence the skew @xmath40 acts as a source for the correlation matrix. : : the rank-3 coefficients @xmath160, @xmath161, @xmath162 and @xmath163 describe evolution of the moments @xmath40. these are @xmath164 the first term describes how the moments flow into each other as the velocity field rotates and shears the @xmath165 coordinate frame relative to the @xmath137 coordinate frame. the second term describes sourcing of non - gaussianity from inhomogeneities in the velocity field and the overall spread of the probability distribution. some higher - rank coefficients in our case, those of ranks four and five are also nonzero, but do not give any new evolution equations. these coefficients measure the error " introduced by truncating the moment expansion. if we had included higher cumulants, these higher - rank coefficients would have given evolution equations for the higher moments of the probability distribution. in general, all moments of the density function will mix so it is always necessary to terminate our expansion at a predetermined order both in cumulants and powers of the field fluctuation. the order we have chosen is sufficient to generate evolution equations containing both the leading - order behavior of the moments namely, the first terms in eqs. , and and the leading corrections, given by the latter terms in these equations. Numerical results at this point we put our new method into practice. we study two models for which the non - gaussian signal is already known, using the standard @xmath0 formula. for each case we employ our method and compare it with results obtained using @xmath0. to ensure a fair comparison, we solve numerically in both cases. our new method employs the slow - roll approximation, as described above. therefore, when using the @xmath0 approach we produce results both with and without slow - roll simplifications. first consider double quadratic inflation, which was studied by rigopoulos, shellard & van tent @xcite and later by vernizzi & wands @xcite. the potential is @xmath166 we use the initial conditions chosen in ref. @xcite, where @xmath167, and the fiducial trajectory has coordinates @xmath168 and @xmath169. we plot the evolution of @xmath1 in fig. [fig1], which also shows the prediction of the standard @xmath0 formula (with and without employing slow roll simplifications). we implement the @xmath0 algorithm using a finite difference method to calculate the derivatives of @xmath6. a similar technique was used in ref. this model yields a very modest non - gaussian signal, below unity even at its peak. if inflation ends away from the spike then @xmath1 is practically negligible. shows that the method of moment transport allows us to separate contributions to @xmath1 from the intrinsic non - gaussianity of the field fluctuations, and non - linearities of the gauge transformation to @xmath2. as explained in [ss : sep_universe], we denote the former @xmath170 and the latter @xmath171, and plot them separately in fig. [fig2]. inspection of this figure clearly shows that @xmath1 is determined by a cancellation between two much larger components. its final shape and magnitude are exquisitely sensitive to their relative phase. initially, the magnitudes of @xmath170 and @xmath171 grow, but their sum remains small. the peak in fig. [fig1] arises from the peak of @xmath171, which is incompletely cancelled by @xmath170. it is remarkable that @xmath170 initially evolves in exact opposition to the gauge transformation, to which it is not obviously connected. in the double quadratic model, @xmath1 is always small. however, it has recently been shown by byrnes _ _ that a large non - gaussian signal can be generated even when slow - roll is a good approximation @xcite. the conditions for this to occur are incompletely understood, but apparently require a specific choice of potential and strong tuning of initial conditions. in figs. [fig3][fig4] we show the evolution of @xmath1 in a model with the potential @xmath172 which corresponds to example a of ref. * 5) when we choose @xmath173 and initial conditions @xmath174, @xmath175. it is clear that the agreement is exact. in this model, @xmath1 is overwhelmingly dominated by the contribution from the second - order gauge transformation, @xmath171, as shown in fig. [fig4]. this conclusion applies equally to the other large-@xmath1 examples discussed in refs. @xcite, although we make no claim that this is a general phenomenon. in conclusion, figs. [fig1] and [fig3] show excellent agreement between our new method and the outcome of the numerical @xmath0 formula. these figures also compare the moment transport method and @xmath0 without the slow - roll approximation. we conclude that the slow - roll estimate remains broadly accurate throughout the entire evolution. Discussion non - linearities are now routinely extracted from all - sky observations of the microwave background anisotropy. our purpose in this paper has been to propose a new technique with which to predict the observable signal. present data already give interesting constraints on the skewness parameter @xmath1, and over the next several years we expect that the _ planck _ survey satellite will make these constraints very stringent. it is even possible that higher - order moments, such as the kurtosis parameter @xmath176 @xcite will become better constrained @xcite. to meet the need of the observational community for comparison with theory, reliable estimates of these non - linear quantities will be necessary for various models of early - universe physics. a survey of the literature suggests that the ` conventional'@xmath0 method, originally introduced by lyth & rodrguez, remains the method of choice for analytical study of non - gaussianity. in comparison, our proposed moment transport method exhibits several clear differences. first, the conventional method functions best when we base the @xmath0 expansion on a flat hypersurface immediately after horizon exit. in our method, we make the opposite choice and move the flat hypersurface as close as possible to the time of observation. after this, the role of the @xmath0 formula is to provide no more than the non - linear gauge transformation between field fluctuations and the curvature perturbation. we substitute the method of moment transport to evolve the distribution of field fluctuations between horizon exit and observation. second, in integrating the transport equation one uses an expansion of the velocity field such as the one given in eqs.. this expansion is refreshed at each step of integration, so the result is related to conventional perturbative calculations in a very similar way to renormalization - group improved perturbation theory @xcite. in this interpretation, derivatives of @xmath111 play the role of couplings. at a given order, @xmath177, in the moment hierarchy, the equations for lower - order moments function as renormalization group equations for the couplings at level-@xmath177, resumming potentially large terms before they spoil perturbation theory. this property is shared with any formalism such as @xmath0 which is non - perturbative in time evolution, but may be an advantage in comparison with perturbative methods. we also note that although @xmath0 is non - perturbative as a point of principle, practical implementations are frequently perturbative. for example, the method of vernizzi & wands @xcite and battefeld & easther @xcite depends on the existence of quantities which are conserved only to leading order in @xmath178, and can lose accuracy after @xmath179 e - foldings. numerical calculations confirm that our method gives results in excellent agreement with existing techniques. as a by - product of our analysis, we note that the large non - gaussianities which have recently been observed in sum- and product - separable potentials @xcite are dominated by non - linearities from the second - order part of the gauge transformation from @xmath180 to @xmath2. the contribution from intrinsic non - linearities of the field fluctuations, measured by the skewnesses @xmath40, is negligible. in such cases one can obtain a useful formula for @xmath1 by approximating the field distribution as an exact gaussian. the non - gaussianity produced in such cases arises from a distortion of comoving hypersurfaces with respect to adjacent spatially flat hypersurfaces. our new method joins many well - established techniques for estimating non - gaussian properties of the curvature perturbation. in our experience, these techniques give comparable estimates of @xmath1, but they do not exactly agree. each method invokes different assumptions, such as the neglect of gradients or the degree to which time dependence can be accommodated. the mutual scatter between different methods can be attributed to the theory error inherent in any estimate of @xmath1. the comparison presented in [sec : numerics] shows that while all of these methods slightly disagree, the moment transport method gives good agreement with other established methods. dm is supported by the cambridge centre for theoretical cosmology (ctc). ds is funded by stfc. dw acknowledges support from the ctc. we would like to thank chris byrnes, jim lidsey and karim malik for helpful conversations. References 10 t. falk, r. rangarajan, and m. srednicki, _ the angular dependence of the three - point correlation function of the cosmic microwave background radiation as predicted by inflationary cosmologies _, _ astrophys. j. _ * 403 * (1993) l1, [http://xxx.lanl.gov/abs/astro-ph/9208001 [arxiv : astro - ph/9208001]]. a. gangui, f. lucchin, s. matarrese, and s. mollerach, _ the three - point correlation function of the cosmic microwave background in inflationary models _, _ astrophys. j. _ * 430 * (1994) 447457, [http://xxx.lanl.gov/abs/astro-ph/9312033 [arxiv : astro - ph/9312033]]. t. pyne and s. m. carroll, _ higher - order gravitational perturbations of the cosmic microwave background _ * d53 * (1996) 29202929, [http://xxx.lanl.gov/abs/astro-ph/9510041 [arxiv : astro - ph/9510041]]. v. acquaviva, n. bartolo, s. matarrese, and a. riotto, _ second - order cosmological perturbations from inflation _, _ nucl. phys. _ * b667 * (2003) 119148, [http://xxx.lanl.gov/abs/astro-ph/0209156 [arxiv : astro - ph/0209156]]. e. komatsu and d. n. spergel, _ acoustic signatures in the primary microwave background bispectrum _, _ phys. * d63 * (2001) 063002, [http://xxx.lanl.gov/abs/astro-ph/0005036 [arxiv : astro - ph/0005036]]. f. r. bouchet and r. juszkiewicz, _ perturbation theory confronts observations : implications for the ` initial'conditions and @xmath181 _, http://xxx.lanl.gov/abs/astro-ph/9312007 [arxiv : astro - ph/9312007]. p. fosalba, e. gaztanaga, and e. elizalde, _ gravitational evolution of the large - scale density distribution : the edgeworth & gamma expansions _, http://xxx.lanl.gov/abs/astro-ph/9910308 [arxiv : astro - ph/9910308]. m. sasaki and e. d. stewart, _ a general analytic formula for the spectral index of the density perturbations produced during inflation _, _ prog. * 95 * (1996) 7178, [http://xxx.lanl.gov/abs/astro-ph/9507001 [arxiv : astro - ph/9507001]]. g. i. rigopoulos, e. p. s. shellard, and b. j. w. van tent, _ non - linear perturbations in multiple - field inflation _ rev. _ * d73 * (2006) 083521, [http://xxx.lanl.gov/abs/astro-ph/0504508 [arxiv : astro - ph/0504508]]. h. r. s. cogollo, y. rodrguez, and c. a. valenzuela - toledo, _ on the issue of the @xmath2 series convergence and loop corrections in the generation of observable primordial non - gaussianity in slow - roll inflation. part i : the bispectrum _, _ jcap _ * 0808 * (2008) 029, [http://xxx.lanl.gov/abs/0806.1546[arxiv:0806.1546]]. y. rodrguez and c. a. valenzuela - toledo, _ on the issue of the @xmath2 series convergence and loop corrections in the generation of observable primordial non - gaussianity in slow - roll inflation. part ii : the trispectrum _, http://xxx.lanl.gov/abs/0811.4092 [arxiv:0811.4092]. choi, l. m. h. hall, and c. van de bruck, _ spectral running and non - gaussianity from slow - roll inflation in generalised two - field models _, _ jcap _ * 0702 * (2007) 029, [http://xxx.lanl.gov/abs/astro-ph/0701247 [arxiv : astro - ph/0701247]]. c. gordon, d. wands, b. a. bassett, and r. maartens, _ adiabatic and entropy perturbations from inflation _, _ phys. _ * d63 * (2001) 023506, [http://xxx.lanl.gov/abs/astro-ph/0009131 [arxiv : astro - ph/0009131]]. s. matarrese, l. verde, and r. jimenez, _ the abundance of high - redshift objects as a probe of non- gaussian initial conditions _, _ astrophys. j. _ * 541 * (2000) 10, [http://xxx.lanl.gov/abs/astro-ph/0001366 [arxiv : astro - ph/0001366]]. l. amendola, _ the dependence of cosmological parameters estimated from the microwave background on non - gaussianity _, _ astrophys. j. _ * 569 * (2002) 595599, [http://xxx.lanl.gov/abs/astro-ph/0107527 [arxiv : astro - ph/0107527]]. m. loverde, a. miller, s. shandera, and l. verde, _ effects of scale - dependent non - gaussianity on cosmological structures _, _ jcap _ * 0804 * (2008) 014, [http://xxx.lanl.gov/abs/0711.4126 [arxiv:0711.4126]]. d. seery and j. c. hidalgo, _ non - gaussian corrections to the probability distribution of the curvature perturbation from inflation _, _ jcap _ * 0607 * (2006) 008, [http://xxx.lanl.gov/abs/astro-ph/0604579 [arxiv : astro - ph/0604579]]. s. blinnikov and r. moessner, _ expansions for nearly gaussian distributions _, _ astron. * 130 * (1998) 193205, [http://xxx.lanl.gov/abs/astro-ph/9711239 [arxiv : astro - ph/9711239]]. g. i. rigopoulos, e. p. s. shellard, and b. j. w. van tent, _ quantitative bispectra from multifield inflation _ rev. _ * d76 * (2007) 083512, [http://xxx.lanl.gov/abs/astro-ph/0511041 [arxiv : astro - ph/0511041]]. m. sasaki, j. valiviita, and d. wands, _ non - gaussianity of the primordial perturbation in the curvaton model _, _ phys. rev. _ * d74 * (2006) 103003, [http://xxx.lanl.gov/abs/astro-ph/0607627 [arxiv : astro - ph/0607627]].

we present a novel method for calculating the primordial non - gaussianity produced by super - horizon evolution during inflation. our method evolves the distribution of coarse - grained inflationary field values using a transport equation. we present simple evolution equations for the moments of this distribution, such as the variance and skewness. this method possesses some advantages over existing techniques. among them , it cleanly separates multiple sources of primordial non - gaussianity, and is computationally efficient when compared with popular alternatives, such as the @xmath0 framework. we adduce numerical calculations demonstrating that our new method offers good agreement with those already in the literature. we focus on two fields and the @xmath1 parameter, but we expect our method will generalize to multiple scalar fields and to moments of arbitrarily high order. we present our expressions in a field - space covariant form which we postulate to be valid for any number of fields. * keywords * : inflation, cosmological perturbation theory, physics of the early universe, quantum field theory in curved spacetime.

0909.2256

Introduction it is well known that the classical magnetoresistance (mr) in metals or semiconductors with a closed free electron fermi surface increases quadratically with increasing magnetic field @xmath2 for @xmath3 and saturates when @xmath4. here @xmath5 is the zero - magnetic - field mobility. hence, the extraordinarily high and linear mr (lmr), which breaks this familiar rule, has been gaining much attention as soon as its discovery. in the past decade, this unexpected lmr has been reported in silver chalcogenide,@xcite indium antimonide,@xcite silicon,@xcite mnas - gaas composite material,@xcite and graphene.@xcite kapitza s linear law@xcite indicates that the metal shows a magnetoresistance linear in perpendicular magnetic field when it has an open fermi surface and a mean free path longer than the electronic larmor radius. recently, another two models, irrespective of the open fermi surface, have been constructed to provide possible mechanisms for the lmr phenomenon. abrikosov suggested a quantum - limit origin of lmr for the homogenous system with a gapless linear energy spectrum.@xcite his model requires that landau levels are well formed and the carrier concentration is small that all electrons occupy only the lowest landau band. alternatively, parish and littlewood developed a classical model without involving linear spectrum.@xcite ignoring the concrete microscopic mechanism, they attributed this unusual mr to the mobility fluctuations in a strongly inhomogenous system. topological insulators@xcite (tis) are novel materials with a full energy gap in bulk, while there are gapless surface states. due to its unique band structure with only one helical dirac cone and linear energy dispersion,@xcite the surface states of the ti bi@xmath0se@xmath1 become an excellent platform for the study of quantum - limit lmr. the recent experiment in this flat surface system, however, reported that a large positive mr, which becomes very linear above a characteristic field of @xmath6@xmath7@xmath8 t, was observed even in an opposite situation where the carrier sheet density is high that electrons occupy more than one landau levels.@xcite moreover, they found that raising temperature to room temperature almost has no influence on the observed lmr. it is striking that this observation is in conflict with abrikosov s model and also with the classical parish - littlewood model. so far a reliable theoretical scheme capable of explaining this novel experiment has still been lacking. in this paper, we generalize the balance - equation approach@xcite to a system modeling the surface states of a three - dimensional ti to investigate the two - dimensional magnetotransport in it. we find that a positive, nonsaturating and dominantly linear magnetoresistance can appear within quite wide magnetic - field range in the ti surface state having a positive and finite effective g - factor. this linear magnetoresistance shows up in the system of high carrier concentration and low mobility when electrons are in extended states and spread over many smeared landau levels, and persists up to room temperature, providing a possible mechanism for the recently observed linear magnetoresistance in topological insulator bi@xmath0se@xmath1 nanoribbons.@xcite Balance-equation formulation for magnetoresistivity we consider the surface state of a bi@xmath0se@xmath1-type large bulk gap ti in the @xmath9-@xmath10 plane under the influence of a uniform magnetic field @xmath11 applied along the @xmath12 direction.@xcite following the experimental observation,@xcite we assume that the fermi energy locates in the gap of the bulk band and above the dirac point, i.e. the surface carriers are electrons. further, the separations of the fermi energy from the bottom of bulk band and dirac point are much larger than the highest temperature (@xmath13) considered in this work. hence, the contribution from the bulk band to the magnetotransport is negligible. these electrons, scattered by randomly distributed impurities and by phonons, are driven by a uniform in - plane electric field @xmath14 in the topological surface. the hamiltonian of this many - electron and phonon system consists of an electron part @xmath15, a phonon part @xmath16, and electron - impurity and electron - phonon interactions @xmath17 and @xmath18 : @xmath19 here, the electron hamiltonian is taken in the form @xmath20, \]] in which @xmath21, @xmath22, @xmath23 and @xmath24, stand, respectively, for the canonical momentum, coordinate, momentum and spin operators of the @xmath25th electron having charge @xmath26, @xmath27 is the vector potential of the perpendicular magnetic field @xmath28 in the landau gauge, @xmath29 is the fermi velocity, @xmath30 is the effective g - factor of the surface electron, and @xmath31 is the bohr magneton with @xmath32 the free electron mass. the sum index @xmath25 in eq.([helectron]) goes over all electrons of total number @xmath33 in the surface state of unit area. in the frame work of balance equation approach,@xcite the two - dimensional center - of - mass (c.m.) momentum and coordinate @xmath34 and @xmath35, and the relative - electron momenta and coordinates @xmath36 and @xmath37 are introduced to write the hamiltonian @xmath15 into the sum of a single - particle c.m. part @xmath38 and a many - particle relative - electron part @xmath39 : @xmath40, with @xmath41.\end{aligned}\]] in this, @xmath42 is the canonical momentum of the center - of - mass and @xmath43 is the canonical momentum for the @xmath25th relative electron. here we have also introduced c.m. spin operators @xmath44 and @xmath45. the commutation relations between the c.m. spin operators @xmath46 and @xmath47 and the spin operators @xmath48, @xmath49 and @xmath50 of the @xmath25th electron are of order of @xmath51 : @xmath52= n^{-1}2\,{\rm i}\,\varepsilon_{\beta_1\beta_2\beta_3}\sigma_j^{\beta_3}$] with @xmath53. therefore, for a macroscopic large @xmath33 system, the c.m. part @xmath38 actually commutes with the relative - electron part @xmath54 in the hamiltonian, i.e. the c.m. motion and the relative motion of electrons are truly separated from each other. the couplings between the two emerge only through the electron impurity and electron phonon interactions. furthermore, the electric field @xmath55 shows up only in @xmath38. and, in view of @xmath56={\rm i}\delta_{\alpha \beta}(\delta_{ij}-1/n)\simeq { \rm i}\delta_{\alpha\beta}\delta_{ij}$], i.e. the relative - electron momenta and coordinates can be treated as canonical conjugate variables, the relative - motion part @xmath54 is just the hamiltonian of @xmath33 electrons in the surface state of ti in the magnetic field without the presence of the electric field. in terms of the c.m. coordinate @xmath57 and the relative electron density operator @xmath58, the electron impurity and electron phonon interactions can be written as@xcite @xmath59 here @xmath60 and @xmath61 are respectively the impurity potential (an impurity at randomly distributed position @xmath62) and electron phonon coupling matrix element in the plane - wave representation, and @xmath63 with @xmath64 and @xmath65 being the creation and annihilation operators for a phonon of wavevector @xmath66 in branch @xmath67 having frequency @xmath68. velocity (operator) @xmath69 is the time variation of its coordinate : @xmath70= v_{\rm f}(\sigma_{\rm c}^y\, \hat{i}-\sigma_{\rm c}^x\, \hat{j})$]. to derive a force - balance equation for steady state transport we consider the heisenberg equation for the rate of change of the c.m. canonical momentum @xmath71 : @xmath72= - n e({\bm v}\times { \bm b})- n e{\bm e}+{\bm { f}}_{\rm i}+{\bm { f}}_{\rm p},\]] in which the frictional forces @xmath73 and @xmath74 share the same expressions as given in ref.. the statistical average of the operator equation can be determined to linear order in the electron impurity and electron phonon interactions @xmath17 and @xmath18 with the initial density matrix @xmath75 at temperature @xmath76 when the in - plane electric field @xmath77 is not strong. for steady - transport states we have @xmath78, leading to a force - balance equation of the form @xmath79 here @xmath80, the statistically averaged velocity of the moving center - of - mass, is identified as the average rate of change of its position, i.e. the drift velocity of the electron system driven by the electric field @xmath77, and @xmath81 and @xmath82 are frictional forces experienced by the center - of - mass due to impurity and phonon scatterings : @xmath83,\label{fp}\end{aligned}\]] in which @xmath84 is the bose distribution function, @xmath85, and @xmath86 stands for the imaginary part of the fourier spectrum of the relative - electron density correlation function defined by @xmath87\big\rangle_{0},\]] where @xmath88 and @xmath89 denotes the statistical averaging over the initial density matrix @xmath90.@xcite the force - balance equation describes the steady - state two - dimensional magnetotransport in the surface state of a ti. note that the frictional forces @xmath81 and @xmath82 are in the opposite direction of the drift velocity @xmath91 and their magnitudes are functions of @xmath92 only. with the drift velocity @xmath93 in the @xmath9 direction, the force - balance equation eq. yields a transverse resistivity @xmath94, and a longitudinal resistivity @xmath95. the linear one is in the form @xmath96 Density correlation function in the landau representation for calculating the electron density correlation function @xmath97 we proceed in the landau representation.@xcite the landau levels of the single - particle hamiltonian @xmath98 of the relative - electron system in the absence of electric field are composed of a positive `` @xmath99 '' and a negative `` @xmath100 '' branch@xcite @xmath101 with @xmath102 and @xmath103, and a zero (@xmath104) level @xmath105 the corresponding landau wave functions are @xmath106 and @xmath107 for @xmath108 ; and @xmath109 for @xmath104. here @xmath110 is the wavevector of the system along @xmath9 direction ; @xmath111 with @xmath112 ; and @xmath113 is the harmonic oscillator eigenfunction with @xmath114 being the hermite polynomial, @xmath115, and @xmath116. each landau level contains @xmath117 electron states for system of unit surface area. the positive branch @xmath118 and the @xmath104 level @xmath119 of the above energy spectra are indeed quite close to those of the surface states in the bulk gap of bi@xmath0se@xmath1-family materials derived from microscopic band calculation.@xcite the landau levels are broadened due to impurity, phonon and electron - electron scatterings. we model the imaginary part of the retarded green s function, or the density - of - states, of the broadened landau level @xmath120 (written for `` +'' -branch and @xmath104 levels), using a gaussian - type form:@xcite @xmath121,\]] with a half - width @xmath122 of the form:@xcite @xmath123^{1/2}$]. here @xmath124 is the single - particle lifetime and @xmath125 is the cyclotron frequency of linear - energy - dispersion system with @xmath126 being the zero - temperature fermi level. using a semi - empirical parameter @xmath127 to relate @xmath124 with the transport scattering time @xmath128, and expressing @xmath129 with the zero - field mobility @xmath5 at finite temperature,@xcite we can write the landau - level broadening as @xmath130^{1/2}.\]] in the present study we consider the case of @xmath120-doping, i.e. the fermi level is high enough above the energy zero of the dirac cone in the range of `` +'' -branch levels and the states of `` @xmath100''-branch levels are completely filled, that they are irrelevant to electron transport. special attention has to be paid to the @xmath104 level, since, depending on the direction of exchange potential the effective g - factor of a ti surface state, @xmath30, can be positive, zero or negative.@xcite the sign and magnitude of the effective g - factor determines how many states of the zero level should be included in or excluded from the available states for electron occupation in the case of @xmath120-doping at a magnetic field. (i) if @xmath131, the @xmath104 level center is exactly at @xmath132 and the system is electron - hole symmetric. the total number of negative energy states (including the states of the lower half of the @xmath104 level and states of the @xmath100"-branch levels) and that of positive energy states (including the states of the upper half of the @xmath104 level and states of the @xmath99"-branch levels) do not change when changing magnetic field. therefore, the lower - half negative energy states of this level are always filled and the upper - half positive - energy states of it are available for the occupation of particles which are counted as electrons participating in transport in the case of @xmath120-doping. (ii) for a finite positive @xmath133, the @xmath104 level @xmath134 moves downward to negative energy and its distance to the nearest @xmath100"-branch level is @xmath135 closer than to the nearest + " -branch level at finite magnetic field strength @xmath2. this is equivalent to the opening of an increasingly enlarged (with increasing @xmath2) energy gap between the + " -branch states and the states of the zero - level and the @xmath100"-branch levels. the opening of a sufficient energy gap implies that with increasing magnetic field the states in the + " -branch levels would no longer shrink into the zero - level, and thus the @xmath104 level should be completely excluded from the conduction band, i.e. only particles occupying the + " -branch states are counted as electrons participating in transport in the case of @xmath120-doping, when the magnetic field @xmath2 gets larger than a certain value (depending on the magnitude of @xmath30). (iii) for a finite negative @xmath136, the @xmath104 level @xmath134 moves upward to positive energy and an increasingly enlarged energy gap will be opened between the states of the zero - level and the + " -branch and the states of @xmath100"-branch levels, and particles occupying the @xmath104 level and + " -branch states are electrons participating in transport when the magnetic field @xmath2 gets larger than a certain value. as a result, the experimentally accessible sheet density @xmath33 of electrons participating in transport is related to the fermi energy @xmath137 by the following equation valid at finite @xmath30 for the magnetic field @xmath2 larger than a certain value : @xmath138 in which @xmath139 + 1\}^{-1}$] is the fermi distribution function at temperature @xmath76 and the summation index @xmath120 goes over @xmath140 for @xmath133, or @xmath141 for @xmath136. in the case of @xmath131, @xmath142\]] valid for arbitrary magnetic field, in which @xmath143. the imaginary part of relative - electron density correlation function in the presence of a magnetic field, @xmath86, can be expressed in the landau representation as@xcite @xmath144 in which the transform factor @xmath145 ^ 2,\end{aligned}\]] with @xmath146, @xmath147, @xmath148, and @xmath149 being associated laguerre polynomials. the landau - representation correlation function @xmath150 in eq.([piqw]) can be constructed with the imaginary part of the retarded green s function @xmath151, or the density - of - states, of the @xmath120th landau level as@xcite @xmath152\nonumber\\ & \hspace{1.2cm}\times{\rm im}g_n(\epsilon+\omega){\rm im}g_{n'}(\epsilon).\end{aligned}\]] the summation indices @xmath120 and @xmath153 in eq.([piqw]) are taken over @xmath140 for @xmath133, or @xmath154 for @xmath136. in the case of @xmath131, eq.([piqw]) still works and the summation indices @xmath120 and @xmath153 go over @xmath154 but with @xmath155 replaced by @xmath156 in eq.([p2nn]). Numerical results and discussions numerical calculations are performed for the magnetoresistivity @xmath157 of surface state in a uniform ti bi@xmath0se@xmath1. at zero temperature the elastic scattering contributing to the resistivity is modeled by a coulomb potential due to charged impurities:@xcite @xmath158 with @xmath159 being the impurity density, which is determined by the zero - magnetic - field mobility @xmath5. at temperatures higher than @xmath160,@xcite phonon scatterings play increasingly important role and the dominant inelastic contribution comes from optical phonons. for this polar material, the scattering by optical phonons via the deformation potential can be neglected. hence, we take account of inelastic scattering from optical phonons via frhlich coupling : @xmath161. in the numerical calculation we use the following parameters:@xcite fermi velocity @xmath162, static dielectric constant @xmath163, optical dielectric constant @xmath164, and phonon energy @xmath165. the broadening parameter is taken to be @xmath166. as a function of the magnetic field @xmath2 having different effective g - factors : @xmath167 and @xmath168 for a ti surface system with electron sheet density @xmath169 in the cases of zero - magnetic - field mobility @xmath170 (a) and @xmath171 (b). several integer - number positions of filling factor @xmath172 are marked in (b).,scaledwidth=40.0%] fig.[diffg] shows the calculated magnetoresistivity @xmath157 versus the magnetic field strength @xmath2 for a ti surface system with electron sheet density @xmath169 but having different effective g - factors : @xmath167 and @xmath168 for two values of zero - magnetic - field mobility @xmath170 and @xmath171, representing different degree of landau - level broadening. in the case without zeeman splitting (@xmath131) the resistivity @xmath157 exhibits almost no change with changing magnetic field up to 10 t, except the shubnikov - de haas (sdh) oscillation showing up in the case of @xmath171. this kind of magnetoresistance behavior was indeed seen experimentally in the electron - hole symmetrical massless system of single - layer graphene.@xcite in the case of a positive g - factor, @xmath173, the magnetoresistivity increases linearly with increasing magnetic field ; while for a negative g - factor, @xmath174, the magnetoresistivity decreases linearly with increasing magnetic field. is shown as a function of the magnetic field @xmath2 for different values of zero - magnetic - field mobility : (a) @xmath175, (b) @xmath176, (c) @xmath177, (d) @xmath178, (e) @xmath179, and (f) @xmath180. the inset of (a) illustrates the same for a larger magnetic - field range @xmath181. the filling factor @xmath182 is plotted versus the magnetic field in (f) ; and several integer - number positions of @xmath182 are also marked in (d) and (e). here the surface electron density @xmath169 and the lattice temperature @xmath183.,scaledwidth=47.0%] in the following we will give more detailed examination on the linearly increasing magnetoresistance in the positive @xmath30 case. fig.[rhob] shows the calculated resistivity @xmath157 versus the magnetic field strength @xmath2 at lattice temperature @xmath183 for system of carrier sheet density @xmath169 and @xmath173, having different zero - field mobility @xmath184 and @xmath180. all resistivity curves for mobility @xmath185 exhibit clear linearity in the magnetic - field range and appear no tendency of saturation at the highest field shown in the figure. especially, for the case @xmath170, the linear behavior extends even up to the magnetic field of @xmath186, as illustrated in the inset of fig.[rhob](a). this feature contradicts the classical mr which saturates at sufficiently large magnetic field @xmath187. note that here we only present the calculated @xmath157 for magnetic field @xmath2 larger than @xmath188 t, for which a sufficient energy gap @xmath135 is assumed to open that with further increase of the magnetic field the states in the `` +'' -branch levels no longer shrink into the zero level and thus it should be excluded from the conduction band. this is of course not true for very weak magnetic field. when @xmath189 the energy gap @xmath190, the situation becomes similar to the case of @xmath131 : the whole upper half of the zero - level states are available to electron occupation and we should have a flat resistivity @xmath157 when changing magnetic field. with increasing @xmath2 the portion of the zero - level states available to conduction electrons decreases until the magnetic field reaches @xmath191. as a result the resistivity @xmath157 should exhibit a crossover from a flat changing at small @xmath2 to positively linear increasing at @xmath192. this is just the behavior observed in the ti bi@xmath0se@xmath1.@xcite note that in the case of @xmath170, the broadened landau - level widths are always larger than the neighboring level interval : @xmath193, which requires @xmath194 ^ 2 $], even for the lowest landau level @xmath195, i.e. the whole landau - level spectrum is smeared. with increasing the zero - field mobility the magnitude of resistivity @xmath157 decreases, and when the broadened landau - level width becomes smaller than the neighboring level interval, @xmath196, a weak sdh oscillation begin to occur around the linearly - dependent average value of @xmath157 at higher portion of the magnetic field range, as seen in fig.[rhob](c), (d) and (e) for @xmath197 and @xmath198. on the other hand, in the case of large mobility, e.g. @xmath199, where the broadened landau - level widths @xmath200 are much smaller than the neighboring level interval even for level index @xmath120 as large as @xmath201, the magnetoresistivity shows pronounced sdh oscillation and the linear - dependent behavior disappears, before the appearance of quantum hall effect,@xcite as shown in fig.[rhob](f). abrikosov s model for the lmr requires the applied magnetic field large enough to reach the quantum limit at which all the carriers are within the lowest landau level,@xcite while it is obvious that more than one landau levels are occupied in the experimental samples in the field range in which the linear and non - saturating magnetoresistivity was observed.@xcite for the given electron surface density @xmath202, the number of occupied landau levels, or the filling factor @xmath172, at different magnetic fields is shown in fig.[rhob](f), as well as in the fig.[rhob](d) and (e), where the integer - number positions of @xmath203, i.e. filling up to entire @xmath182 landau levels, coincide with the minima of the density - of - states or the dips of sdh oscillation. this is in contrast with @xmath131 case, where the integer number of @xmath203, which implies a filling up to the center position of the @xmath182th landau levels, locates at a peak of sdh oscillation, as shown in fig.[diffg]b. the observed sdh oscillations in the bi@xmath0se@xmath1 nanoribbon exhibiting nonsaturating surface lmr in the experiment@xcite favor the former case : a finite positive effective @xmath133. is plotted as a function of the surface electron density @xmath33 at magnetic field @xmath204 : (a) at different values of zero - field mobility @xmath5, and (b) at different values of zero - field conductivity @xmath205.,scaledwidth=40.0%] at various lattice temperatures. here the zero - magnetic - field mobility at zero temperature is @xmath206.,scaledwidth=35.0%] next, we examine the density - dependence of the linear magnetoresistivity. to compare with abrikosov s quantum magnetoresistance which suggests a @xmath207 behavior,@xcite we show the calculated @xmath208 for above lmr versus the carrier sheet density @xmath33 in fig.[rhon] at fixed magnetic field @xmath209 t. the mobility is taken respectively to be @xmath210 and @xmath211m@xmath212/vs to make the resistivity in the lmr regime. a clearly linear dependence of @xmath213 on the surface density @xmath33 is seen in all cases, indicating that this non - saturating linear resistivity is almost inversely proportional to the carrier density. in the figure we also show @xmath208 versus @xmath33 under the condition of different given conductivity @xmath214 and @xmath215. in this case the half - width @xmath216 is independent of surface density. the linear dependence still holds, indicating that this linear behavior is not sensitive to the modest @xmath33-dependence of landau level broadening @xmath216 as long as the system is in the overlapped landau level regime. from the above discussion, it is obvious that lmr shows up in the system having overlapped landau levels and the separation of landau levels makes the mr departure from the linear increase. at high temperature, the thermal energy would smear the level separation and phonon scatterings further broaden landau levels. hence, it is believed that this lmr will be robust against raising temperature. this is indeed the case as seen in fig.[rhot], where we plot the calculated magnetoresistivity @xmath157 for the above system with zero - temperature linear mobility @xmath217m@xmath212/vs versus the magnetic field at different lattice temperatures. we can see that raising temperature to room temperature has little effect on the linearity of mr. due to the decreased mobility at higher temperature from phonon scattering, the weak sdh oscillation on the linear background tends to vanish. these features are in good agreement with the experimental report.@xcite Summary in summary, we have studied the two - dimensional magnetotransport in the flat surface of a three - dimensional ti, which arises from the surface states with a wavevector - linear energy dispersion and a finite, positive zeeman splitting within the bulk energy gap. when the level broadening is comparable to or larger than the landau - level separation and the conduction electrons spread over many landau levels, a positive, dominantly linear and non - saturating magnetoresistance appears within a quite wide range of magnetic field and persists up to room temperature. this remarkable lmr provides a possible mechanism for the recently observed linear magnetoresistance in topological insulator bi@xmath0se@xmath1 nanoribbons.@xcite in contrast to quantum hall effect which appears in the case of well formed landau levels and to abrikosov s quantum magnetotransport,@xcite which is limited to the extreme quantum limit that all electrons coalesce into the lowest landau level, the discussed lmr is a phenomena of pure classical two - dimensional magnetotransport in a system having linear - energy - dispersion, appearing in the regime of overlapped landau levels, irrespective of its showing up in relatively high magnetic field range. furthermore, the present scheme deals with spatially uniform case without invoking the mobility fluctuation in a strongly inhomogeneous system, which is required in the classical parish and littlewood model to produce a lmr.@xcite the appearance of this significant positive - increasing linear magnetoresistance depends on the existence of a positive and sizable effective g - factor. if the zeeman energy splitting is quite small the resistivity @xmath157 would exhibit little change with changing magnetic field. in the case of a negative and sizable effective g - factor the magnetoresistivity would decrease linearly with increasing magnetic field. therefore, the behavior of the longitudinal resistivity versus magnetic field may provide a useful way for judging the direction and the size of the effective zeeman energy splitting in ti surface states. Acknowledgments this work was supported by the national science foundation of china (grant no. 11104002), the national basic research program of china (grant no. 2012cb927403) and by the program for science&technology innovation talents in universities of henan province (grant no. 2012hastit029).

a positive, non - saturating and dominantly linear magnetoresistance is demonstrated to occur in the surface state of a topological insulator having a wavevector - linear energy dispersion together with a finite positive zeeman energy splitting. this linear magnetoresistance shows up within quite wide magnetic - field range in a spatially homogenous system of high carrier density and low mobility in which the conduction electrons are in extended states and spread over many smeared landau levels, and is robust against increasing temperature, in agreement with recent experimental findings in bi@xmath0se@xmath1 nanoribbons.

1208.5351

Introduction 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. The new physics models 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. Calculations 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. Constraints on the new physics models 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. Numerical results and discussions 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. Conclusion in this paper, we studied the rare @xmath1-decays @xmath2 (@xmath7), @xmath282 and @xmath4 in the type - ii 2hdm, lepton - specific 2hdm, nmssm and nmssm, which predict a light cp - odd higgs boson @xmath0. in the parameter space allowed by current experiments, the branching ratio can be as large as @xmath5 for @xmath118, @xmath8 for @xmath3 and @xmath9 for @xmath4, which implies that the decays @xmath2 and @xmath283 may be accessible at the gigaz option. since different models predict different size of branching ratios, these decays can be used to distinguish different model through the measurement of these rare decays. Acknowledgment this work was supported in part by hastit under grant no. 2009hastit004, by the national natural science foundation of china (nnsfc) under grant nos. 10821504, 10725526, 10635030, 10775039, 11075045 and by the project of knowledge innovation program (pkip) of chinese academy of sciences under grant no.. for some reviews, see, e.g., m. a. perez, g. tavares - velasco and j. j. toscano, int. j. mod. a * 19 *, 159 (2004) ; j. m. yang, arxiv:1006.2594. j. i. illana, m. masip, 67, 035004 (2003) ; j. cao, z. xiong, j. m. yang, 32, 245 (2004). d. atwood _ et al_., 66, 093005 (2002). j. kalinowski, and s. pokorski, 219, 116 (1989) ; a. djouadi, p. m. zerwas and j. zunft, 259, 175 (1991) ; a. djouadi, j. kalinowski, and p. m. zerwas, z. phys. c * 54 *, 255 (1992). m. krawczyk, _ et al. _, 19, 463 (2001) ; 8, 495 (1999). j. f. gunion, g. gamberini and s. f. novaes, 38, 3481 (1988) ; thomas j. weiler and tzu - chiang yuan, 318, 337 (1989) ; a. djouadi, _ et al. _, 1, 163 (1998)[hep - ph/9701342]. d. chang and w. y. keung, phys. lett. * 77 *, 3732 (1996). e. keith and e. ma, 57, 2017 (1998) ; m. a. perez, g. tavares - velasco and j. j. toscano, int. j. mod.phys. a * 19 *, 159 (2004). f. larios, g. tavares - velasco and c. p. yuan, 64, 055004 (2001) ; 66, 075006 (2002). a. djouadi, _ et al. _, 10, 27 (1999) [hep - ph/9903229]. for a detailed introduction of the nmssm, see f. franke and h. fraas, int. j. mod. a * 12 * (1997) 479 ; for a recent review of the nmssm, see for example, u. ellwanger, c. hugonie, and a. m. teixeira, arxiv : 0910.1785. see, e.g., j. r. ellis, j. f. gunion, h. e. haber, l. roszkowski and f. zwirner, phys. rev. d * 39 * (1989) 844 ; m. drees, int. j. mod. phys. a * 4 * (1989) 3635 ; u. ellwanger, m. rausch de traubenberg and c. a. savoy, phys. b * 315 * (1993) 331 ; nucl. b * 492 * (1997) 21 ; d.j. miller, r. nevzorov, p.m. zerwas, 681, 3 (2004). c. panagiotakopoulos, k. tamvakis, 446, 224 (1999) ; 469, 145 (1999) ; c. panagiotakopoulos, a. pilaftsis, 63, 055003 (2001) ; a. dedes, _ et al. _, 63, 055009 (2001) ; a. menon, _ et al. _, 70, 035005 (2004) ; v. barger, _ et al. _, 630, 85 (2005). c. balazs, _ et al. _, 0706, 066 (2007). b. a. dobrescu, k. t. matchev, 0009, 031 (2000) ; a. arhrib, k. cheung, t. j. hou, k. w. song, hep - ph/0611211 ; 0703, 073 (2007) ; x. g. he, j. tandean, and g. valencia, 98, 081802 (2007) ; 0806, 002 (2008) ; f. domingo _ et al_., 0901, 061 (2009) ; gudrun hiller, 70, 034018 (2004) ; r. dermisek, and john f. gunion, 75, 075019 (2007) ; 79, 055014 (2009) ; 81, 055001 (2010) ; r. dermisek, john f. gunion, and b. mcelrath, 76, 051105 (2007) ; z. heng, _ et al_., 77, 095012 (2008) ; a. belyaev _ et al_., 81, 075021 (2010) ; d. das and u. ellwanger, arxiv:1007.1151 [hep - ph]. s. andreas, o. lebedev, s. ramos - sanchez and a. ringwald, arxiv:1005.3978 [hep - ph]. j. f. gunion, jhep * 0908 *, 032 (2009) ; r. dermisek and j. f. gunion, phys. rev. d * 81 *, 075003 (2010). r. dermisek and j. f. gunion, phys. lett. * 95 *, 041801 (2005) ; phys. d * 73 *, 111701 (2006). j. cao, h. e. logan, j. m. yang, 79, 091701 (2009). j. cao, p. wan, l. wu, j. m. yang, 80, 071701 (2009). j. f. gunion and h. e. haber, 67, 075019 (2003). r. m. barnett, _ et al. _, phys. b * 136 *, 191 (1984) ; r. m. barnett, g. senjanovic and d. wyler, phys. d * 30 *, 1529 (1984) ; y. grossman, nucl. b * 426 *, 355 (1994). h. s. goh, l. j. hall and p. kumar, jhep * 0905 *, 097 (2009) ; a. g. akeroyd and w. j. stirling, nucl. b * 447 *, 3 (1995) ; a. g. akeroyd, phys. b * 377 *, 95 (1996) ; h. e. logan and d. maclennan, phys. rev. d * 79 *, 115022 (2009) ; m. aoki, _ et al. _, arxiv:0902.4665 [hep - ph]. v. barger, p. langacker, h. s. lee and g. shaughnessy, phys. d * 73 *, 115010 (2006). s. hesselbach, _ et. _, arxiv:0810.0511v2 [hep - ph]. de vivie and p. janot [aleph collaboration], pa13 - 027 contribution to the international conference on high energy physics, warsaw, poland, 2531 july 1996 ; j. kurowska, o. grajek and p. zalewski [delphi collaboration], cern - open-99 - 385. [aleph collaboration and delphi collaboration and l3 collaboration], phys. rept. * 427 *, 257 (2006). j. cao and j. m. yang, jhep * 0812 *, 006 (2008). m. krawczyk and d. temes, eur. j. c * 44 *, 435 (2005). g. altarelli and r. barbieri, 253, 161 (1991) ; m. e. peskin, t. takeuchi, 46, 381 (1992). c. amsler, _ et al. _, (particle data group), 667, 1 (2008). o. deschamps, s. descotes - genon, s. monteil, v. niess, s. tjampens and v. tisserand, arxiv:0907.5135 [hep - ph]. s. su and b. thomas, phys. d * 79 *, 095014 (2009). g. abbiendi, _ et al. _, eur. phys. j. c * 32 *, 453 (2004). m. davier, _ et al. _, 66, 1 (2010). k. cheung, _ et al. _, phys. d * 64 *, 111301 (2001). k. cheung and o. c. w. kong, phys. d * 68 *, 053003 (2003). t. besmer, c. greub, t.hurth, 609, 359 (2001) ; f. borzumati, _ et al. _, 62, 075005(2000). j. cao, k. i. hikasa, w. wang, j. m. yang and l. x. yu, phys. d * 82 *, 051701 (2010) [arxiv:1006.4811 [hep - ph]]. j. f. gunion, _ et. d * 73 *, 015011 (2006). martin and j. d. wells, phys. d * 64 *, 035003 (2001). j. abdallah _ et al. _, eur. j. c * 31 *, 421 (2004) ; g. abbiendi _ et al. _, eur. j. c * 35 *, 1 (2004). j. dunkley _ et al. _ [wmap collaboration], astrophys. j. suppl. * 180 *, 306 (2009) [arxiv:0803.0586 [astro - ph]]. u. ellwanger _ et al. _, 02, 066 (2005). g. belanger, f. boudjema, a. pukhov and a. semenov, comput. commun. * 174 *, 577 (2006) ; comput. phys. commun. * 176 *, 367 (2007). g. belanger, f. boudjema, c. hugonie, a. pukhov and a. semenov, jcap * 0509 *, 001 (2005).

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.

1007.1918

Introduction invariants are a popular concept in object recognition and image retrieval @xcite. they aim to provide descriptions that remain constant under certain geometric or radiometric transformations of the scene, thereby reducing the search space. they can be classified into global invariants, typically based either on a set of key points or on moments, and local invariants, typically based on derivatives of the image function which is assumed to be continuous and differentiable. the geometric transformations of interest often include translation, rotation, and scaling, summarily referred to as _ similarity _ transformations. in a previous paper @xcite, building on work done by schmid and mohr @xcite, we have proposed differential invariants for those similarity transformations, plus _ linear _ brightness change. here, we are looking at a _ non - linear _ brightness change known as _ gamma correction_. gamma correction is a non - linear quantization of the brightness measurements performed by many cameras during the image formation process. the idea is to achieve better _ perceptual _ results by maintaining an approximately constant ratio between adjacent brightness levels, placing the quantization levels apart by the _ just noticeable difference_. incidentally, this non - linear quantization also precompensates for the non - linear mapping from voltage to brightness in electronic display devices @xcite. gamma correction can be expressed by the equation @xmath0 where @xmath1 is the input intensity, @xmath2 is the output intensity, and @xmath3 is a normalization factor which is determined by the value of @xmath4. for output devices, the ntsc standard specifies @xmath5. for input devices like cameras, the parameter value is just inversed, resulting in a typical value of @xmath6. the camera we used, the sony 3 ccd color camera dxc 950, exhibited @xmath7. for the kodak megaplus xrc camera] [fig : gammacorr] shows the intensity mapping of 8-bit data for different values of @xmath4. it turns out that an invariant under gamma correction can be designed from first and second order derivatives. additional invariance under scaling requires third order derivatives. derivatives are by nature translationally invariant. rotational invariance in 2-d is achieved by using rotationally symmetric operators. The invariants the key idea for the design of the proposed invariants is to form suitable ratios of the derivatives of the image function such that the parameters describing the transformation of interest will cancel out. this idea has been used in @xcite to achieve invariance under linear brightness changes, and it can be adjusted to the context of gamma correction by at least conceptually considering the _ logarithm _ of the image function. for simplicity, we begin with 1-d image functions. let @xmath8 be the image function, i.e. the original signal, assumed to be continuous and differentiable, and @xmath9 the corresponding gamma corrected function. note that @xmath8 is a special case of @xmath10 where @xmath11. taking the logarithm yields @xmath12 with the derivatives @xmath13, and @xmath14. we can now define the invariant @xmath15 under gamma correction to be @xmath16{0mm}{13 mm } & = $ \frac{\gamma\, \frac{f'(x)}{f(x) } } { \gamma\, \frac{f(x)\,f''(x) - f'(x)^2}{f(x)^2}}$ \\ & = $ \frac{f(x)\,f'(x)}{f(x)\,f''(x) - f'(x)^2}$ \end{tabular}\]] the factor @xmath3 has been eliminated by taking derivatives, and @xmath4 has canceled out. furthermore, @xmath15 turns out to be completely specified in terms of the _ original _ image function and its derivatives, i.e. the logarithm actually does nt have to be computed. the notation @xmath17 indicates that the invariant depends on the underlying image function @xmath8 and location @xmath18 the invariance holds under gamma correction, not under spatial changes of the image function. a shortcoming of @xmath15 is that it is undefined where the denominator is zero. therefore, we modify @xmath15 to be continuous everywhere : @xmath19{0mm}{8 mm } { \normalsize $ \theta_{m12\gamma}=$ } & $ \frac{f\,f'}{f\,f '' - { f'}^2}$ & { \normalsize if $ |f\,f'| < |f\,f '' - { f'}^2|$ } \\ & $ \frac{f\,f '' - { f'}^2}{f\,f'}$ & { \normalsize else } \\ \end{tabular}\]] where, for notational convenience, we have dropped the variable @xmath18. the modification entails. note that the modification is just a heuristic to deal with poles. if all derivatives are zero because the image function is constant, then differentials are certainly not the best way to represent the function. if scaling is a transformation that has to be considered, then another parameter @xmath20 describing the change of size has to be introduced. that is, scaling is modeled here as variable substitution @xcite : the scaled version of @xmath8 is @xmath21. so we are looking at the function @xmath22 where the derivatives with respect to @xmath18 are @xmath23, @xmath24, and @xmath25. now the invariant @xmath26 is obtained by defining a suitable ratio of the derivatives such that both @xmath4 and @xmath20 cancel out : @xmath27{0mm}{10 mm } & = $ \frac{g^2 g'\,g''' - 3\,g\,{g'}^2 g '' + 2\,{g'}^4 } { g^2 { g''}^2 - 2\,g\,{g'}^2 g '' + \ { g'}^4}$ \end{tabular}\]] analogously to eq. ([eq : thm12 g]), we can define a modified invariant @xmath28{0mm}{8 mm } { \normalsize $ \theta_{m123\gamma}=$ } & $ \frac{g^2 g'\,g''' - 3\,g\,{g'}^2 g '' + 2\,{g'}^4 } { g^2 { g''}^2 - 2\,g\,{g'}^2 g '' + \ { g'}^4}$ & { \normalsize if cond2 } \\ & $ \frac{g^2 { g''}^2 - 2\,g\,{g'}^2 g '' + \ { g'}^4 } { g^2 g'\,g''' - 3\,g\,{g'}^2 g '' + 2\,{g'}^4}$ & { \normalsize else } \\ \end{tabular}\]] where condition cond1 is @xmath29 @xmath30, and condition cond2 is @xmath31 @xmath32 @xmath33. again, this modification entails. it is a straightforward albeit cumbersome exercise to verify the invariants from eqs. ([eq : th12 g]) and ([eq : th123 g]) with an analytical, differentiable function. as an arbitrary example, we choose @xmath34 the first three derivatives are @xmath35, @xmath36, and @xmath37. then, according to eq. ([eq : th12 g]), @xmath38. if we now replace @xmath8 with a gamma corrected version, say @xmath39, the first derivative becomes @xmath40, the second derivative is @xmath41, and the third is @xmath42. if we plug these derivatives into eq. ([eq : th12 g]), we obtain an expression for @xmath43 which is identical to the one for @xmath17 above. the algebraically inclined reader is encouraged to verify the invariant @xmath44 for the same function. [fig : analyex] shows the example function and its gamma corrected counterpart, together with their derivatives and the two modified invariants. as expected, the graphs of the invariants are the same on the right as on the left. note that the invariants define a many - to - one mapping. that is, the mapping is not information preserving, and it is not possible to reconstruct the original image from its invariant representation. if @xmath45 or @xmath46 are to be computed on images, then eqs. ([eq : th12 g]) to ([eq : thm123 g]) have to be generalized to two dimensions. this is to be done in a rotationally invariant way in order to achieve invariance under similarity transformations. the standard way is to use rotationally symmetric operators. for the first derivative, we have the well known _ gradient magnitude _, defined as @xmath47 where @xmath48 is the 2-d image function, and @xmath49, @xmath50 are partial derivatives along the x - axis and the y - axis. for the second order derivative, we can use the linear _ laplacian _ @xmath51 horn @xcite also presents an alternative second order derivative operator, the _ quadratic variation _ @xmath52 since the qv is not a linear operator and more expensive to compute, we use the laplacian for our implementation. for the third order derivative, we can define, in close analogy with the quadratic variation, a _ cubic variation _ as @xmath53 the invariants from eqs. ([eq : th12 g]) to ([eq : thm123 g]) remain valid in 2-d if we replace @xmath54 with @xmath55, @xmath56 with @xmath57, and @xmath58 with @xmath59. this can be verified by going through the same argument as for the functions. recall that the critical observation in eq. ([eq : th12 g]) was that @xmath4 cancels out, which is the case when all derivatives return a factor @xmath4. but such is also the case with the rotationally symmetric operators mentioned above. for example, if we apply the gradient magnitude operator to @xmath60, i.e. to the logarithm of a gamma corrected image function, we obtain @xmath61 returning a factor @xmath4, and analogously for @xmath62, qv, and cv. a similar argument holds for eq. ([eq : th123 g]) where we have to show, in addition, that the first derivative returns a factor @xmath20, the second derivative returns a factor @xmath63, and the third derivative returns a factor @xmath64, which is the case for our 2-d operators. while the derivatives of continuous, differentiable functions are uniquely defined, there are many ways to implement derivatives for _ sampled _ functions. we follow schmid and mohr @xcite, ter haar romeny @xcite, and many other researchers in employing the derivatives of the gaussian function as filters to compute the derivatives of a sampled image function via convolution. this way, derivation is combined with smoothing. the 2-d zero mean gaussian is defined as @xmath65 the partial derivatives up to third order are @xmath66, @xmath67, @xmath68, @xmath69, @xmath70, @xmath71, @xmath72, @xmath73, @xmath74. they are shown in fig. [fig : gausskernels]. we used the parameter setting @xmath75 and kernel size @xmath76 with these kernels, eq. ([eq : th12 g]), for example, is implemented as @xmath77 at each pixel @xmath78, where @xmath79 denotes convolution. Experimental data and results we evaluate the invariant @xmath45 from eq. ([eq : thm12 g]) in two different ways. first, we measure how much the invariant computed on an image without gamma correction is different from the invariant computed on the same image but with gamma correction. theoretical, this difference should be zero, but in practice, it is not. second, we compare template matching accuracy on intensity images, again without and with gamma correction, to the accuracy achievable if instead the invariant representation is used. we also examine whether the results can be improved by prefiltering. a straightforward error measure is the _ absolute error _, @xmath80 where `` 0gc '' refers to the image without gamma correction, and gc stands for either `` sgc '' if the gamma correction is done synthetically via eq. ([eq : gammacorr]), or for `` cgc '' if the gamma correction is done via the camera hardware. like the invariant itself, the absolute error is computed at each pixel location @xmath81 of the image, except for the image boundaries where the derivatives and therefore the invariants can not be computed reliably. [fig : imas] shows an example image. the sgc image has been computed from the 0gc image, with @xmath82. note that the gamma correction is done _ after _ the quantization of the 0gc image, since we do nt have access to the 0gc image before quantization. [fig : accuinv] shows the invariant representations of the image data from fig. [fig : imas] and the corresponding absolute errors. since, we have. the dark points in fig. [fig : accuinv], (c) and (e), indicate areas of large errors. we observe two error sources : * the invariant can not be computed robustly in homogeneous regions. this is hardly surprising, given that it is based on differentials which are by definition only sensitive to spatial changes of the signal. * there are outliers even in the sgc invariant representation, at points of very high contrast edges. they are a byproduct of the inherent smoothing when the derivatives are computed with differentials of the gaussian. note that the latter put a ceiling on the maximum gradient magnitude that is computable on 8-bit images. in addition to computing the absolute error, we can also compute the relative error, in percent, as @xmath83 then we can define the set @xmath84 of _ reliable points _, relative to some error threshold @xmath85, as @xmath86 and @xmath87, the percentage of reliable points, as @xmath88 where @xmath89 is the number of valid, i.e. non - boundary, pixels in the image. [fig : reliapts] shows, in the first row, the reliable points for three different values of the threshold @xmath85. the second row shows the sets of reliable points for the same thresholds if we gently prefilter the 0gc and cgc images. the corresponding data for the ten test images from fig. [fig : imadb] is summarized in table [tab : reliaperc]. derivatives are known to be sensitive to noise. noise can be reduced by smoothing the original data before the invariants are computed. on the other hand, derivatives should be computed as locally as possible. with these conflicting goals to be considered, we experiment with gentle prefiltering, using a gaussian filter of size @xmath90=1.0. the size of the gaussian to compute the invariant @xmath45 is set to @xmath91=1.0. note that @xmath90 and @xmath91 can _ not _ be combined into just one gaussian because of the non - linearity of the invariant. with respect to the set of reliable points, we observe that after prefiltering, roughly half the points, on average, have a relative error of less than 20%. gentle prefiltering consistently reduces both absolute and relative errors, but strong prefiltering does not. template matching is a frequently employed technique in computer vision. here, we will examine how gamma correction affects the spatial accuracy of template matching, and whether that accuracy can be improved by using the invariant @xmath45. an overview of the testbed scenario is given in fig. [fig : templloca]. a small template of size @xmath92, representing the search pattern, is taken from a 0gc intensity image, i.e. without gamma correction. this query template is then correlated with the corresponding cgc intensity image, i.e. the same scene but with gamma correction switched on. if the correlation maximum occurs at exactly the location where the 0gc query template has been cut out, we call this a _ correct maximum correlation position _, or cmcp. the correlation function @xmath93 employed here is based on a normalized mean squared difference @xmath94 @xcite : @xmath95 where @xmath1 is an image, @xmath96 is a template positioned at @xmath78, @xmath97 is the mean of the subimage of @xmath1 at @xmath78 of the same size as @xmath96, @xmath98 is the mean of the template, and @xmath99. the template location problem then is to perform this correlation for the whole image and to determine whether the position of the correlation maximum occurs precisely at @xmath78. [fig : matchtempl] demonstrates the template location problem, on the left for an intensity image, and on the right for its invariant representation. the black box marks the position of the original template at (40,15), and the white box marks the position of the matched template, which is incorrectly located at (50,64) in the intensity image. on the right, the matched template (white) has overwritten the original template (black) at the same, correctly identified position. [fig : correlexmpl] visualizes the correlation function over the whole image. the white areas are regions of high correlation. the example from figs. [fig : matchtempl] and [fig : correlexmpl] deals with only _ one _ arbitrarily selected template. in order to systematically analyze the template location problem, we repeat the correlation process for all possible template locations. then we can define the _ correlation accuracy _ ca as the percentage of correctly located templates, @xmath100 where @xmath101 is the size of the template, @xmath102 is the set of correct maximum correlation positions, and @xmath89, again, is the number of valid pixels. we compute the correlation accuracy both for unfiltered images and for gently prefiltered images, with @xmath103. [fig : corrcorrelpts] shows the binary correlation accuracy matrices for our example image. the cmcp set is shown in white, its complement and the boundaries in black. we observe a higher correlation accuracy for the invariant representation, which is improved by the prefiltering. we have computed the correlation accuracy for all the images given in fig. [fig : imadb]. the results are shown in table [tab : ca] and visualized in fig. [fig : correlaccuras]. we observe the following : * the correlation accuracy ca is higher on the invariant representation than on the intensity images. * the correlation accuracy is higher on the invariant representation with gentle prefiltering, @xmath103, than without prefiltering. we also observed a decrease in correlation accuracy if we increase the prefiltering well beyond @xmath103. by contrast, prefiltering seems to be always detrimental to the intensity images ca. * the correlation accuracy shows a wide variation, roughly in the range 30%@xmath10490% for the unfiltered intensity images and 50%@xmath104100% for prefiltered invariant representations. similarly, the gain in correlation accuracy ranges from close to zero up to 45%. for our test images, it turns out that the invariant representation is always superior, but that does nt necessarily have to be the case. * the medians and means of the cas over all test images confirm the gain in correlation accuracy for the invariant representation. * the larger the template size, the higher the correlation accuracy, independent of the representation. a larger template size means more structure, and more discriminatory power. Conclusion we have proposed novel invariants that combine invariance under gamma correction with invariance under geometric transformations. in a general sense, the invariants can be seen as trading off derivatives for a power law parameter, which makes them interesting for applications beyond image processing. the error analysis of our implementation on real images has shown that, for sampled data, the invariants can not be computed robustly everywhere. nevertheless, the template matching application scenario has demonstrated that a performance gain is achievable by using the proposed invariant. Acknowledgements bob woodham suggested to the author to look into invariance under gamma correction. his meticulous comments on this work were much appreciated. jochen lang helped with the acquisition of image data through the acme facility @xcite. d. forsyth, j. mundy, a. zisserman, c. coelho, c. rothwell, `` invariant descriptors for 3-d object recognition and pose '', _ ieee transactions on pattern analysis and machine intelligence _, vol.13, no.10, pp.971 - 991, oct.1991. d. pai, j. lang, j. lloyd, r. woodham, `` acme, a telerobotic active measurement facility '', sixth international symposium on experimental robotics, sydney, 1999. see also : http://www.cs.ubc.ca/nest/lci/acme/

_ this paper presents invariants under gamma correction and similarity transformations. the invariants are local features based on differentials which are implemented using derivatives of the gaussian. the use of the proposed invariant representation is shown to yield improved correlation results in a template matching scenario. _

cs0003079

Introduction in 1957 g.a. askaryan pointed out that ionisation and cavitation along a track of an ionising particle through a liquid leads to hydrodynamic radiation @xcite. in the 1960s, 1970s and 1980s, theoretical and experimental studies have been performed on the hydrodynamic radiation of beams and particles traversing dense media @xcite. the interest in characterising the properties of the acoustic radiation was, among other reasons, lead by the idea that the effect can be utilised to detect ultra - high energy (@xmath1) cosmic, i.e. astrophysical neutrinos, in dense media like water, ice and salt. in the 1970s this idea was discussed within the dumand optical neutrino detector project @xcite and has been studied in connection with cherenkov neutrino detector projects since. the detection of such neutrinos is considerably more challenging than the search for high - energy neutrinos (@xmath2) as currently pursued by under - ice and under - water cherenkov neutrino telescopes @xcite. due to the low expected fluxes, detector sizes exceeding 100km@xmath3 are needed @xcite. however, the properties of the acoustic method allow for sparsely instrumented arrays with @xmath4100 sensors / km@xmath3. to study the feasibility of a detection method based on acoustic signals it is necessary to understand the properties of the sound generation by comparing measurements and simulations based on theoretical models. according to the so - called thermo - acoustic model @xcite, the energy deposition of particles traversing liquids leads to a local heating of the medium which can be regarded as instantaneous with respect to the hydrodynamic time scales. due to the temperature change the medium expands or contracts according to its bulk volume expansion coefficient @xmath5. the accelerated motion of the heated medium generates an ultrasonic pulse whose temporal signature is bipolar and which propagates in the volume. coherent superposition of the elementary sound waves, produced over the cylindrical volume of the energy deposition, leads to a propagation within a flat disk - like volume in the direction perpendicular to the axis of the particle shower. in this study, the hydrodynamic signal generation by two types of beams, interacting with a water target, was investigated : pulsed protons and a pulsed laser, mimicking the formation of a hadronic cascade from a neutrino interaction under laboratory conditions. with respect to the aforementioned experimental studies of the thermo - acoustic model, the work presented here can make use of previously unavailable advanced tools such as geant4 @xcite for the simulation of proton - induced hadronic showers in water. good agreement was found in the comparison of the measured signal properties with the simulation results, providing confidence to apply similar simulation methods in the context of acoustic detection of ultra - high energy neutrinos. a puzzling feature observed in previous studies a non - vanishing signal amplitude at a temperature of 4@xmath6c, where for water at its highest density no thermo - acoustic signal should be present was investigated in detail. such a residual signal was also observed for the proton beam experiment described in this article, but not for the laser beam, indicating that the formation of this signal is related to the charge or the mass of the protons. Derivation of the model in the following, the thermo - acoustic model @xcite is derived from basic assumptions, using a hydrodynamic approach. basis is the momentum conservation, i.e., the euler equation @xmath7 for mass density @xmath8, velocity vector field of the medium @xmath9 and momentum - density tensor @xmath10 including the pressure @xmath11 @xcite. equation ([eq_euler]) can be derived from momentum conservation. in the derivation, energy dissipation resulting from processes such as internal friction or heat transfer are neglected. motions described by the euler equation hence are adiabatic. taking the three partial derivatives of eq. ([eq_euler]) with respect to @xmath12 and using for the density the continuity equation @xmath13 a non - linear wave equation can be derived : @xmath14 to solve this equation, the problem is approached in two separated spatial regions : firstly, a region @xmath15 (_ ` beam'_), where the energy is deposited in the beam interactions with the fluid and thus the wave excited in a non - equilibrium process ; and secondly, a hydrodynamic (_ ` acoustic'_) region @xmath16, where the acoustic wave propagates through the medium and where linear hydrodynamics in local equilibrium can be assumed. this splitting can be reflected by the momentum density tensor, rewriting it as @xmath17 in local equilibrium the changes in mass density are given by @xmath18 with the bulk volume expansion coefficient @xmath19, the energy deposition @xmath20, the adiabatic speed of sound @xmath21 in the medium and the specific heat @xmath22. in the acoustic regime, where @xmath23, the momentum density tensor can be expressed as @xmath24 (using eqs.([eq_densitytensor]) and ([eq_density_change])), where we assume an adiabatic density change with pressure. the non - linear kinetic term @xmath25 entering @xmath26 according to eq. ([eq_densitytensor]) can be neglected for small deviations @xmath27 from the static density @xmath28 and small pressure differences @xmath29 from the static pressure @xmath30 @xcite. in the region b, where non - equilibrium deposition occur, one may make the _ ansatz _ @xmath31 with the direction @xmath32 of the beam which breaks the isotropy of the energy - momentum tensor and describes with the parameter @xmath33 in an effective way the momentum transfer on the fluid. although in non - equilibirum we apply eq. ([eq_density_change]) with the energy deposition density @xmath34 of the beam. then, with the additional energy - momentum tensor due to the beam @xmath35 the wave equation ([eq_wave_base]) reads @xmath36 the general solution for the wave equation can be written using a green function approach as @xmath37 \nonumber \end{aligned}\]] with the components of the unit vector @xmath38 and the retarded time @xmath39. for the last conversion, partial integration and the total derivative @xmath40 have been used repeatedly. note that @xmath41 for @xmath42, so that the integration is carried out over the volume of the energy deposition region @xmath15. assuming an energy deposition without momentum transfer to the medium, the kinetic term in the _ ansatz _ ([eq : ansatz]) can be neglected (@xmath43) yielding @xmath44 for a thermo - acoustic wave generated solely by heating of the medium. the signal amplitude @xmath45 can be shown to be proportional to the dimensionless quantity @xmath46 when solving eq. ([eq_pressure]) for the case of an instantaneous energy deposition. equation ([eq_pressure]) is equivalent to the results obtained from the approaches presented in @xcite. the derivation pursued above, however, uses a different approach starting with the euler equation and an anisotropic energy - momentum tensor, yielding a more general expression in eq. ([eq : pressure - approx]). only if assuming an isotropic energy deposition one arrives at the expression for the pressure deviation @xmath45 given in eq. ([eq_pressure]). note that the validation of the last assumption @xmath43 being a good approximation would require a detailed knowledge of the momentum transfer from the beam to the medium. taking it into account would result in an additional dipol term @xmath47 in eq. ([eq_pressure]) which may become the dominant contribution to wave generation if @xmath48 close to 4.0@xmath49c. however, for @xmath43 the pressure field resulting from a beam interaction in a medium is determined by the spatial and temporal distribution of the energy deposition density @xmath50 alone. the amplitude of the resulting acoustic wave is governed by the thermodynamic properties @xmath21, @xmath51 and @xmath5, the latter three depending primarily on the temperature of the medium. a controlled variation of these parameters in the conducted laboratory experiments and a study of the resulting pressure signals therefore allows for a precise test of the thermo - acoustic model. simulations based on the thermo - acoustic model, as performed to interpret the results of the experiments described in the next section, will be discussed in sec. [sec_simulation]. note that the energy deposition density @xmath50 and its temporal evolution for the proton and laser beam interactions discussed in this paper are quite different from those expected for the interaction of ultra - high energy neutrinos. however, if a simulation starting from basic principles allows for a good reproduction of the experimental results, it is reasonable to assume that these simulation methods are transferable to neutrino interactions, as they are governed by the same underlying physical processes. Experimental setup and beam characteristics the experiments presented in this paper were performed with a pulsed infrared nd : yag laser facility (@xmath52) located at the erlangen centre for astroparticle physics (ecap) of the university of erlangen, and the pulsed @xmath53 proton beam of the `` gustaf werner cyclotron '' at the `` theodor svedberg laboratory '' in uppsala, sweden. the beam properties allow for a compact experimental setup. in both cases, the beams were dumped into a dedicated @xmath54 water tank, where the acoustic field was measured with several position - adjustable acoustic sensors (see fig. [fig_test_setup]). the sensors (also called hydrophones) could be positioned within the tank with absolute uncertainties below 1 cm. the temperature of the water could be varied between @xmath55 and @xmath56 with a precision of @xmath57. the temperature was brought to a particular value by first cooling the water with ice ; subsequently the whole water volume was heated to the desired temperature in a controlled, gradual procedure. once the water temperature had been established, at least 10min remained for measurements until the water volume heated up by @xmath57 through heat transfer from the environment. this time span was sufficient for all measurements conducted at water temperatures below the ambient temperature. the explored range of spill energies for the proton beam was from @xmath58 to @xmath59, the beam diameter was approximately @xmath60 and the spill time @xmath61. for @xmath62 protons, the energy deposition in the water along the beam axis (@xmath63-axis, beam entry into the water at @xmath64) is relatively uniform up to @xmath65 ending in the prominent bragg - peak at @xmath66 (see fig. [fig_energy_deposition_z]). to adjust the spill energy, the number of protons per bunch was varied. the total charge of a bunch was calibrated with two independent methods (faraday cups and scintillation counters), leading to an uncertainty on the order of 15%, with some higher values for low spill energies. to obtain the beam intensity and profile for the proton interactions in the water tank, the distance of about 1.2 m that the beam was travelling from the exit of the beam pipe through air and its entering into the water tank were included in the geant4 simulation. for the laser experiment, the pulse energy was adjusted between @xmath67 and @xmath68 and calibrated using a commercial power meter. the beam had a diameter of approximately @xmath69 mm and the pulse length was fixed at @xmath70. for the infrared light used, the laser energy density deposited along the beam axis has an exponential decrease with an absorption length of @xmath71 (see fig. [fig_energy_deposition_z]). for both beam types the lateral energy deposition profile was gaussian (the aforementioned beam diameters are the @xmath72 s of the profiles). the two experiments allow to explore different spatial and temporal distributions of the energy deposition as well as two different mechanisms of energy transfer into the medium. for both beams, energy is deposited via excitation, in addition the medium is ionised in the case of the proton beam. , width=294] for the signal recording, sensors based on the piezo - electric effect @xcite were used. a full characterisation of these sensors had been performed prior to the experiments. they are linear in amplitude response, the frequency response is flat starting from a few khz up to the main resonance at @xmath73 with a sensitivity of @xmath74 (@xmath40.02v / pa). the main resonance is more sensitive by @xmath45@xmath75 and sensitivity drops rapidly at higher frequencies ; at 90khz, the sensitivity has dropped by 20db. the absolute uncertainty in the determination of the sensitivity is at a level of 2db in the frequency range of interest. above 90khz the uncertainty exceeds 5db. to calculate the response of the sensors to an external pressure pulse a parametrised fit of an equivalent circuit model as described in @xcite was used. the sensitivity dependence on temperature was measured and the relative decrease was found to be less than @xmath76 (or about 0.13db) per @xmath77. for every set of fixed experimental parameters (temperature, energy, sensor position, etc.) the signals of 1000 beam pulses were recorded with a digital oscilloscope at a sampling rate in excess of @xmath78. this rate is sufficient for the signals with spectral components up to 100khz, where the sensitivity of the sensors is negligibly small. these individual pulses were averaged to reduce background and environmental noise in the analysis, thereby obtaining a very high statistical precision. Basic features of the measured signals figure [fig_signals] shows typical signals measured in the proton and the laser experiment, respectively, using the same sensors and experimental setup. the general shapes of the two signals differ : a typical signal for the proton beam shows a bipolar signature, the one for the laser deviates from such a generic form. the laser signal has high frequency components up to several mhz due to the high energy deposition density at the point of beam entry and the almost instantaneous energy deposition compared to the @xmath79s pulse of the proton beam ; therefore the resonance of the sensor is excited causing a ringing in the measured signal. the spatial distribution of the energy density @xmath50 deposited by the laser leads to the two separate signals : the first originates in the beam area at the same @xmath63-region as the sensor placement (_ ` direct signal'_), the second from the beam entry, a point of discontinuity where most of the energy is deposed (_ ` beam entry signal'_). the signal of the proton beam is deteriorated with respect to an ideal bipolar signal. three main contributions to this distortion can be discerned : the recorded signal starts before the expected onset of the acoustic signal (55.2@xmath79s for the given position, see fig. [fig_signals], given by the sonic path length) ; reflections of the acoustic wave on the beam entry window overlay the original wave starting in the first rare - faction peak ; and finally there are frequency components of the signal exciting a resonant response of the sensor, slightly changing the signal shape and causing ringing. the first point was studied and found to be consistent with an electric charge effect in the sensors caused by the proton beam. its starting time was always coincident with the beam pulse entry into the water, even for sensor distances of up to 1 m, hinting at an electromagnetic origin of the distorting signal. the shape of this non - acoustic signal is consistent with the integrated time - profile of the beam pulse with a subsequent exponential decay. this deformation of the main signal is considered a systematic uncertainty on the signal properties and treated as such in the analysis. for the most part, its shape was fitted and subtracted from the signal. in order to minimise the impact of the signal deformation caused by the described effects on the analysis of the recorded signals, robust characteristics were used : the peak - to - peak amplitude and the signal length from maximum to minimum of the signal. for the laser experiment these features were extracted for the direct signals only. = 10 cm and @xmath63=20 cm and are shown within the same time interval. the dash - dotted line in the upper graph indicates the charge effect described in the text. [fig_signals], width=264] Simulation of thermo-acoustic signals for an in - depth validation of the thermo - acoustic model, comparisons of the signal properties with simulation results based on the model are essential. to this end, a simulation of the expected signals was developed. it is based on the thermo - acoustic model using a numeric solution of eq. ([eq_pressure]). the input parameters to the simulation were either measured at the experiments, i.e. medium temperature and beam profiles, or simulated, i.e. the energy deposition of the protons (using geant4). the thermodynamic parameters bulk volume expansion coefficient, heat capacity and speed of sound were derived from the measured water temperature using standard parametrisations. tap water quality was assumed. a series of simulations was conducted, where the input parameters were varied individually in the range given by the experiment, including uncertainties. especially the spatial and temporal beam profile have a substantial impact on amplitude, duration and shape of the signal. simulated signals and the respective sensor response corresponding to the measured signals of fig. [fig_signals] are shown in fig.[fig_signals_sim]. to minimise systematic effects from the setup caused e.g. by reflections on the surfaces, the sensor response was convoluted onto the simulated signals, rather than deconvoluted from the measured ones. thus in the analysis voltage rather than pressure signals are compared. . the corresponding dashed - lined signals mark the simulated signals after convolution with the sensor response. for better comparability, the signal maxima were normalised to 1. [fig_signals_sim], width=264] the shapes of the simulated pulses are altered by the sensor response, especially the high - frequency components above the resonant frequency of the sensor. in the case of the laser pulse, mostly the resonance of the sensor is excited, leading to a strong ringing. for the proton beam, the primarily bipolar shape is again prominent, whereas the laser pulse is segmented into the two parts described above. the direct pulse of the laser experiment exhibits a bipolar shape as well, albeit less symmetric than for the proton beam. figure [fig_systematics] exemplifies the dependency of the signal amplitude on the input parameters of the simulation : water temperature, pulse energy, beam profile in @xmath80 and @xmath81, pulse length and the position of the sensor. as nominal positions of the sensor @xmath82 m, @xmath83 m, and @xmath84 m were used. all parameters were varied by @xmath85 around the value of the best agreement with measurement, i.e. the values used for the simulations of the signals shown in fig. [fig_signals_sim]. some of the characteristics of thermo - acoustic sound generation are observable. as discussed in sec. [sec_model], the dependence on temperature enters through the factor @xmath46 (where the dependence on the speed of sound is negligible) and is roughly linear in the range investigated for this study. the dependence on energy is strictly linear. the dependency on the beam pulse parameters is diverse. it is governed by the integral in eq.([eq_pressure]) and therefore depends on both the spatial and the temporal beam profiles and the interaction of the particles with the water. for a given point in space and time, the elementary waves produced in the volume of energy deposition may interfere constructively or destructively depending on the beam properties. accordingly, the length of the laser pulse has no influence on the amplitude, as with 9ns it is much shorter than the transit time of the acoustic signal through the energy deposition area. for the several ten @xmath79s long proton spill, the spill time is comparable to the transit time. thus the acoustic signal shows a strong dependence on the spill time. the dependence on the radial coordinate @xmath86 w.r.t. the beam axis @xmath86 follows roughly the expected 1/@xmath87 and @xmath88 fall - off of a cylindric source in the near and the far field, respectively. the @xmath81-position was varied between @xmath892 and 2 cm, as the signals were recorded within the @xmath90-plane. the resulting change in amplitude is below 1%. the @xmath63-dependency for the laser experiment follows the exponential fall - off expected from the light absorption. the one for the proton experiment shows the only non - strictly monotonic behaviour due to the form of the energy deposition with the prominent bragg peak. using these dependencies the systematic uncertainties of the model were obtained using the experimental uncertainties of the various parameters. the main uncertainty for the proton beam is given by the temporal profile of the pulse, which was simplified to a gaussian profile for this chapter (however not for the rest of this work). for the laser beam experiment this parameter influences the signal amplitude only on a one percent level. the second main influence on the amplitude is the sensor position along the beam axis (@xmath63-direction). table [tab_systematics] gives the parameters and their uncertainties (@xmath91) used for the simulation of the signals. the resulting systematic uncertainties in the amplitude (@xmath92) are given as well. the combined uncertainties are @xmath93 for the proton signal and @xmath94 for the laser signal, respectively. .beam parameters used for the simulated signals in fig. [fig_signals_sim] with their associated experimental uncertainties (@xmath91) and resulting uncertainty in signal amplitude (@xmath92). the pulse length of the laser is set to a value much higher than in the experiment, to save calculation time. the resulting uncertainty in the signal amplitude is below @xmath95 [cols="<,^,^,^,^ ",] Analysis results in the following, the results obtained in the comparison of experiment and simulation are presented. figure [fig_proton_sim_meas] shows a comparison between simulated and measured signals for the proton beam experiment at different sensor positions. for better visibility only the main part of the signals (first bipolar part) is plotted. the input parameters of the simulation were varied within the experimental uncertainties until the best agreement with the measured signal in amplitude and duration was obtained for the reference point at @xmath80=0.40 m and @xmath63=0.11 m. for this optimisation, a simple procedure of adjusting the parameters manually and scanning the resulting agreement visually was found to be sufficient. the other signals were simulated with the same parameter set, only the sensor positions were changed. the signal shapes differ for different @xmath63-positions due to the geometry of the energy deposition profile described in sec. [sec_setup] with cylindric form in the @xmath96-plane and almost flat energy density in @xmath63-direction up to the bragg peak at @xmath63=0.22 m. due to this geometry an almost cylindrical wave is excited in the medium, with coherent emission perpendicular to the beam axis. in the region @xmath97 m the signals are of bipolar shape. along the beam axis (@xmath98 m, @xmath99 m) the main part of the observed signal originates from the bragg peak as a nearly spheric source and no clear bipolar shape evolves. the agreement between simulation and measurement is good for all positions. not only amplitude and duration match (see also the following sections) but also the signal shape is reproduced to a very high degree. the small discrepancies, primarily in the rare - faction part of the bipolar pulse, have contributions stemming from a non - ideal sensor calibration and reflections on the tank surfaces. however, a significant part of the discrepancies may lie in the beam pulse modelling or even the thermo - acoustic model itself. the prominent feature of the energy deposition of the laser beam is at the beam entry into the medium. overall, the geometry of this deposition is mainly a cylindric one with rotational symmetry around the @xmath63-axis, as for the proton beam, with coherent emission perpendicular to the beam axis (direct signal). the signal from the discontinuity at the beam entry is emitted almost as from a point source (beam entry signal). in contrast to the proton beam, the shapes of signals at different positions along the @xmath63-axis do not vary much, only the relative timing between the two signal components varies. therefore, fig. [fig_laser_sim_meas] shows only signals for a @xmath63-position in the middle of the water tank. here, both signal parts are well described by the simulation. the beam entry part of each signal is less well reproduced in the simulation due to its high frequency components where the sensor calibration is less well understood. + m for different sensor positions along @xmath80 (reference at @xmath100 m). for more details [fig_laser_sim_meas], width=325] to compare the characteristics of the signals in simulation and measurement in the following studies, their amplitudes and the point in time of the recording of the signal maximum were further studied. for the laser signal, the direct signal part is considered only. equation ([eq_pressure]) yields as velocity of propagation for the thermo - acoustic signal the speed of sound in the medium, here water. to verify the hydrodynamic origin of the measured signals the variation of the arrival times for different sensor positions perpendicular to the beam axis were analysed. figure [fig_time_vs_distance] shows the measured data and a linear fit for each beam type. the data is compatible with an acoustic sound propagation in water, as the fits yield a speed of sound compatible with pure water at the temperature used. for the proton beam @xmath101 (@xmath102) was obtained for a water temperature of @xmath103c, where literature @xcite gives for pure water at normal pressure @xmath104 which is in complete agreement. for the laser measurements the water temperature of @xmath105c and thus the speed of sound were significantly higher, the observed @xmath106 (@xmath107) is again in perfect agreement with the theoretical value of @xmath108. the offset between proton and laser beam data in fig. [fig_time_vs_distance] is due to a differing delay time in between trigger time and arrival time of the different beams in the water and is irrelevant for the calculation of the speed of sound. position). the straight lines represent linear fits to the data points yielding @xmath109 and @xmath110 for the proton and laser beam, respectively. [fig_time_vs_distance], width=325] the durations of the complete, unclipped signals vary with sensor positions mainly due to reflections, which were not simulated and can therefore not be compared. the comparison of simulated and measured signals for different sensor positions within the water tank, excluding the parts of the signals dominated by reflections, was shown in fig. [fig_proton_sim_meas]. the good agreement between model and measurement also manifests itself in the development of the signal amplitude with distance of sensors from the beam axis, shown in fig. [fig_amp_vs_x_protons]. to minimise systematic effects from reflections, only the amplitude of the leading maximum is analysed. though there are sizeable deviations, the overall shape of the curve is reproduced. the behaviour is again different for the two @xmath63-positions. the development at @xmath111 m follows the one expected from a cylindric source with a @xmath112 behaviour in the near field up to @xmath4@xmath113 m and a @xmath88 behaviour in the far field beyond that distance. at the smallest measured distances, the simulated behaviour deviates from the measured one. presumably this is due to simplifications made in the derivation of the model in sec. [sec_model]. at @xmath114 m the amplitude falls off more uniformly, this is again a combination of the point - like emission characteristic of the bragg peak interfering with the cylindric emission at @xmath115 m. ) for the proton beam experiment. points mark the measured amplitudes for two different @xmath63-positions and the lines the respective simulation results. [fig_amp_vs_x_protons], width=325] the behaviour for the laser experiment is not as well reproduced by the simulation (see fig. [fig_amp_vs_x_laser]). this is again attributed to the high - frequent signal components of the laser, where minor uncertainties in the simulation may result in big changes of the signal amplitude. especially the overlap of the direct with the beam entry window signal distort the signal shape. at distances exceeding 0.5 m the two signal parts can not be distinguished. ) for the laser beam experiment. points mark the measured amplitudes at @xmath116 m and the line the simulation result. [fig_amp_vs_x_laser], width=325] assuming otherwise unchanged settings, the energy deposition density @xmath50 scales linearly with total deposited energy. thus the spill energy can be written as a pre - factor in eq. ([eq_pressure]) effecting the pressure and thus signal amplitude linearly. as shown in fig. [fig_amplitude_vs_energy] this behaviour was observed in the experiments yielding a zero - crossing of the pulse energy at @xmath117mpa for the proton beam and @xmath118mpa for the laser beam. both values are consistent with zero. the slope of the line depends on the energy deposition and the sensor positioning along the beam axis and can therefore not be compared for the two beams. as expected from the model, the signal duration and signal shape showed no significant dependence on energy. m. there is a ten percent systematic uncertainty in the absolute determination of the pulse energy. the lines represent linear fits to the data points yielding a zero - crossing of the amplitude compatible with no energy in a pulse. the insert shows the data for the proton beam.[fig_amplitude_vs_energy], width=325] the main feature of the thermo - acoustic model is its dependence on the temperature of the medium. figure [fig_temp_laser] shows the temperature dependence of the signal peak - to - peak amplitude for the laser beam, where a positive (negative) sign denotes a leading positive (negative) peak of the signal. the two data sets shown in the figure were recorded by two sensors simultaneously, which were positioned at @xmath119 perpendicular to the beam axis and at @xmath120 and @xmath121 along the beam axis, respectively. in the case of the proton beam setup, which will be discussed below, these hydrophone positions correspond roughly to the @xmath63-position of the bragg - peak and a @xmath63-position half way between the bragg - peak and the beam entry into the water, respectively. for comparability, the same positions and the same sensors were chosen for the laser and proton beam experiments. at @xmath122. the insert shows a blow - up of the region around @xmath123c where the sign of the amplitude changes. [fig_temp_laser],width=325] the laser beam signal shown in fig. [fig_temp_laser] changes its polarity around @xmath124, as expected from the thermo - acoustic model. the theoretical expectation for the signal amplitude, which is proportional to @xmath125 and vanishes at @xmath126 for the given temperature and pressure, is fitted to the experimental data. in the fit, an overall scaling factor and a shift in temperature (for the experimental uncertainty in the temperature measurement) were left free as fit parameters. the fit yielded a zero - crossing of the amplitude at @xmath127, where the error is dominated by the systematic uncertainty in the temperature setting. c, fitted with the model expectation as described in the text. a systematic deviation from the model expectation as the amplitude changes its sign is clearly visible. the amplitudes were normalised to @xmath128 at @xmath122. [temp_amp_protons_uncorr], width=325] . to allow for an easy comparison of the signal shapes, a point in time at the onset of the acoustic signals was chosen as zero time and for all signal amplitudes the corresponding offset was added or subtracted to yield a zero amplitude at that time. [temp_comp_signals], width=325] analysing the proton data in the same fashion resulted in a fit that deviated from the model expectation, and a zero - crossing significantly different from @xmath129 at @xmath130, see fig. [temp_amp_protons_uncorr]. the data strongly indicate the presence of a systematic effect near the zero - crossing of the signal amplitude. to understand this effect, the signal shapes near the temperature of @xmath129 were investigated (fig. [temp_comp_signals]). a non - vanishing signal is clearly observable at @xmath129 and the signal inverts its polarity between @xmath129 and @xmath131. in view of the results from the laser beam measurements and the obvious systematic nature of the deviation from the model visible in fig. [temp_amp_protons_uncorr], we subtracted the residual signal at @xmath129, which has an amplitude of @xmath132 of the @xmath122 signal, from all signals. thus a non - temperature dependent effect in addition to the thermo - acoustic signal was assumed. the resulting amplitudes shown in fig. [fig_temp_proton] are well described by the model prediction. c was subtracted at every temperature. the amplitudes were afterwards normalised to @xmath128 at @xmath122. the insert shows a blow - up of the region around @xmath123c where the sign of the amplitude changes. [fig_temp_proton], width=325] the production mechanism of the underlying signal at @xmath133, which was only observed in the proton experiment, could not be unambiguously determined with the performed measurements. from the model point of view, the main simplification for the derivation of eq. ([eq_pressure]) was to neglect all non - isotropic terms and momentum transfer to the medium in the momentum density tensor @xmath134 by setting @xmath43 in eq. ([eq : ansatz]). as discussed in sec. [sec_model], dipole radiation could contribute significantly near the disappearance of the volume expansion coefficient for the case @xmath135. also other non - thermo - acoustic signal production mechanism have been discussed in the literature which could give rise to an almost temperature independent signal, see e.g.@xcite. the obvious difference to the laser experiment are the charges involved both from the protons themselves and the ionisation of the water which could lead to an interaction with the polar water molecules. another difference are the massive protons compared to massless photons. residual signals at @xmath133 were found in previous works as well @xcite, as will be discussed in more detail in sec. [subsec : comparison]. for clarification further experiments are needed either with ionising neutral particles (e.g. synchrotron radiation) or with charged particles (e.g. protons, @xmath136-particles) with more sensors positioned around the bragg - peak. with such experiments it might be possible to distinguish between the effect of ionisation in the water and of net charge introduced by charged particles. with the analysis that has been described above, the signal production according to the thermo - acoustic model could be unambiguously confirmed. while the model has been confirmed in previous experiments, the simulations presented in this work constitute a new level of precision. the most puzzling feature, a residual signal at 4@xmath6c that was also observed in previous experiments, was investigated with high precision by scanning the relevant temperature region in steps of @xmath137c. the observed shift of the zero - crossing of the amplitude towards values higher than 4@xmath6c, caused by a leading rarefaction non - thermal residual signal at 4@xmath6c, is in qualitative agreement with @xcite. in @xcite, a residual signal at @xmath123c was also reported. since in that work the zero - crossing of the amplitude is observed at @xmath138c, i.e.a higher value than the expected @xmath123c, it can be assumed that the corresponding residual signal has a leading rarefaction. in @xcite, a residual signal is found at @xmath139c, however with a leading compression rather than rarefaction. the authors conclude that this may lead to a signal disappearance point _ below _ the expected value, in contrast to @xcite and the work presented in this article. for the measurements with a laser beam reported in @xcite, a residual signal was also observed at @xmath123c, albeit with a leading compression and a subsequent reduction of the temperature of the zero - crossing of the signal amplitude to about @xmath140c. this observation is in contrast to the laser experiment presented in this article. in conclusion, the works of all authors discussed here indicate a non - thermal residual signal for proton beams, albeit with varying results concerning the size of the effect and the shape of the underlying non - thermal signal. the results for the laser beam reported in @xcite differ from those described in the article at hands. it should be pointed out, however, that in @xcite and @xcite results are reported by the same authors for proton and for laser beams, respectively. a comparison of these two publications shows that a different behaviour near the temperature of @xmath123c was observed for the two types of beams. hence, to the best of our knowledge, there are currently no results in contradiction with the notion of different non - thermal effects in the interaction of proton vs. laser beams with water. the available data does not allow for a more detailed analysis of the correlation between experimental conditions and the temperature of the zero - crossing of the signal amplitude. Applications in astroparticle physics efforts to detect neutrinos at ultra - high energies are at the frontier of research in the field of astroparticle physics. neutrinos are the only viable messengers at ultra - high energies beyond the local universe, i.e. distances well beyond several tens of megaparsecs. if successful, the investigation of these elusive particles will not only enhance the understanding of their own nature, but also provide important complementary information on the astrophysical phenomena and the environments that accelerate particles to such extreme energies. for acoustic particle detection, not only the technical aspects such as optimal design and detector layout are subject of research. but also the underlying physics processes the formation of hadronic cascades resulting from neutrino interactions in dense media have never been observed directly in detector experiments at these energies. producing ever more reliable extrapolation of reaction properties to ultra - high energies is an ongoing effort. with advancements in the simulations of cascades forming in water and improvements of detector simulation tools, the discrepancy between cascade parameters from independent simulations decreased : recent studies differ only slightly @xcite. at the same time it is necessary to gain a solid understanding of the sound signals generated from the energy depositions by particle cascades. for this purpose, laboratory measurements are required. this work, together with others @xcite has established the validity of the thermo - acoustic model with uncertainties at the @xmath141 level. the input to a model as discussed in this article is an energy deposition in water as it is also produced by a cascade that evolves from a neutrino interaction. in comparison with the uncertainty in the thermo - acoustic model, uncertainties due to the simulations of hadronic cascades and cascade - to - cascade variations are large @xcite, dominating the challenge to detect and identify sound signals resulting from neutrino interactions. it can hence be concluded that the current level of precision in modelling sound signals in the context of the thermo - acoustic model is fully sufficient for the understanding of acoustic neutrino signatures. the latter is necessary to improve the selection efficiency and background rejection for neutrino detection algorithms in potential future acoustic neutrino detectors. several experiments have been conducted @xcite to understand the acoustic background at the sites of potential future large - scale acoustic neutrino telescopes in sea water, fresh water and ice. the combination of simulation efforts, laboratory measurements and studies with in - situ test arrays will allow for a conclusion of the feasibility of acoustic neutrino detection. Conclusions we have demonstrated that the sound generation mechanism of intense pulsed beams is well described by the thermo - acoustic model. in almost all aspects investigated, the signal properties are consistent with the model. the biggest uncertainties of the experiments are on the 10@xmath0 level. one discrepancy is the non - vanishing signal at 4@xmath49c for the proton beam experiment, which can be described with an additional non - temperature dependent signal with a @xmath132 contribution to the amplitude at 15@xmath49c. the model allows for calculations of the characteristics of sound pulses generated in the interaction of high energy particles in water with the input of the energy deposition of the resulting cascade. a possible application of this technique would be the detection of neutrinos with energies @xmath142 Acknowledgements this work was supported by the german ministry for education and research (bmbf) by grants 05cn2we1/2, 05cn5we1/7 and 05a08we1. parts of the measurements were performed at the `` theodor svedberg laboratory '' in uppsala, sweden. the authors wish to thank all involved personnel and especially the acoustics groups of desy zeuthen and of uppsala university for their support.

the generation of hydrodynamic radiation in interactions of pulsed proton and laser beams with matter is explored. the beams were directed into a water target and the resulting acoustic signals were recorded with pressure sensitive sensors. measurements were performed with varying pulse energies, sensor positions, beam diameters and temperatures. the obtained data are matched by simulation results based on the thermo - acoustic model with uncertainties at a level of 10@xmath0. the results imply that the primary mechanism for sound generation by the energy deposition of particles propagating in water is the local heating of the medium. the heating results in a fast expansion or contraction and a pressure pulse of bipolar shape is emitted into the surrounding medium. an interesting, widely discussed application of this effect could be the detection of ultra - high energetic cosmic neutrinos in future large - scale acoustic neutrino detectors. for this application a validation of the sound generation mechanism to high accuracy, as achieved with the experiments discussed in this article, is of high importance. cosmic neutrinos, acoustic neutrino detection, thermo - acoustic model, ultra - high energy cosmic rays, beam interaction

1501.01494

Introduction quantum - mechanical fluctuations during an early epoch of inflation provide a plausible mechanism to generate the energy - density perturbations responsible for observed cosmological structure. while it has been known for quite some time that inflation is consistent with open spatial hypersurfaces (gott 1982 ; guth & weinberg 1983), attention was initially focussed on models in which there are a very large number of @xmath17-foldings during inflation, resulting in almost exactly flat spatial hypersurfaces for the observable part of the present universe (guth 1981 ; also see kazanas 1980 ; sato 1981a, b). this was, perhaps, inevitable because of strong theoretical prejudice towards flat spatial hypersurfaces and their resulting simplicity. however, to get a very large number of @xmath17-foldings during inflation it seems necessary that the inflation model have a small dimensionless parameter (j. r. gott, private communication 1994 ; banks et al. 1995), which would require an explanation. attempts to reconcile these favoured " flat spatial hypersurfaces with observational measures of a low value for the clustered - mass density parameter @xmath1 have concentrated on models in which one postulates the presence of a cosmological constant @xmath18 (peebles 1984). in the simplest flat-@xmath18 model one assumes a scale - invariant (harrison 1970 ; peebles & yu 1970 ; zeldovich 1972) primordial power spectrum for gaussian adiabatic energy - density perturbations. such a spectrum is generated by quantum - mechanical fluctuations during an early epoch of inflation in a spatially - flat model, provided that the inflaton potential is reasonably flat (fischler, ratra, & susskind 1985, and references therein). it has been demonstrated that these models are indeed consistent with current observational constraints (e.g., stompor, grski, & banday 1995 ; ostriker & steinhardt 1995 ; ratra & sugiyama 1995 ; liddle et al. 1996b ; ganga, ratra, & sugiyama 1996b, hereafter grs). an alternative, more popular of late, is to accept that the spatial hypersurfaces are not flat. in this case, the radius of curvature for the open spatial sections introduces a new length scale (in addition to the hubble length), which requires a generalization of the usual flat - space scale - invariant spectrum (ratra & peebles 1994, hereafter rp94). such a spectrum is generated by quantum - mechanical fluctuations during an epoch of inflation in an open - bubble model (rp94 ; ratra & peebles 1995, hereafter rp95 ; bucher et al. 1995, hereafter bgt ; lyth & woszczyna 1995 ; yamamoto et al. 1995, hereafter yst), provided that the inflaton potential inside the bubble is reasonably flat. such gaussian adiabatic open - bubble inflation models have also been shown to be consistent with current observational constraints (rp94 ; kamionkowski et al. 1994 ; grski et al. 1995, hereafter grsb ; liddle et al. 1996a, hereafter llrv ; ratra et al. 1995 ; grs). inflation theory by itself is unable to predict the normalization amplitude for the energy - density perturbations. currently, the least controversial and most robust method for the normalization of a cosmological model is to fix the amplitude of the model - predicted large - scale cmb spatial anisotropy by comparing it to the observed cmb anisotropy discovered by the @xmath0-dmr experiment (smoot et al. 1992). previously, specific open cold dark matter (cdm) models have been examined in light of the @xmath0-dmr two - year results (bennett et al. grsb investigated the cmb anisotropy angular spectra predicted by the open - bubble inflation model (rp94), and compared large - scale structure predictions of this dmr - normalized model to observational data. cayn et al. (1996) performed a related analysis for the open model with a flat - space scale - invariant spectrum (wilson 1983, hereafter w83), and yamamoto & bunn (1996, hereafter yb) examined the effect of additional sources of quantum fluctuations (bgt ; yst) in the open - bubble inflation model. in this paper, we study the observational predictions for a number of open cdm models. in particular, we employ the power spectrum estimation technique devised by grski (1994) for incomplete sky coverage to normalize the open models using the @xmath0-dmr four - year data (bennett 1996). in @xmath19 we provide an overview of open - bubble inflation cosmogonies. in @xmath20 we detail the various dmr data sets used in the analyses here, discuss the various open models we consider, and present the dmr estimate of the cmb rms quadrupole anisotropy amplitude @xmath21 as a function of @xmath1 for these open models. in @xmath22 we detail the computation of several cosmographic and large - scale structure statistics for the dmr - normalized open models. these statistics are confronted by various current observational constraints in @xmath23. our results are summarized in @xmath24. Open-bubble inflation models the simplest open inflation model is that in which a single open - inflation bubble nucleates in a (possibly) spatially - flat, inflating spacetime (gott 1982 ; guth & weinberg 1983). in this model, the first epoch of inflation smooths away any preexisting spatial inhomogeneities, while simultaneously generating quantum - mechanical zero - point fluctuations. then, in a tunnelling event, an open - inflation bubble nucleates, and for a small enough nucleation probability the observable universe lies inside a single open - inflation bubble. fluctuations of relevance to the late - time universe can be generated via three different quantum mechanical mechanisms : (1) they can be generated in the first epoch of inflation ; (2) they can be generated during the tunnelling event (thus resulting in a slightly inhomogeneous initial hypersurface inside the bubble, or a slightly non - spherical bubble) ; and (3) they can be generated inside the bubble. the tunneling amplitude is largest for the most symmetrical solution (and deviations from symmetry lead to an exponential suppression), so it has usually been assumed that the nucleation process (mechanism [2]) does not lead to the generation of significant inhomogeneities. quantum - mechanical fluctuations generated during evolution inside the bubble (rp95) are significant. assuming that the energy - density difference between the two epochs of inflation is negligible (and so the bubble wall is not significant), one may estimate the contribution to the perturbation spectrum after bubble nucleation from quantum - mechanical fluctuations during the first epoch of inflation (bgt ; yst). as discussed by bucher & turok (1995, hereafter bt) (also see yst ; yb), the observable predictions of these simple open - bubble inflation models are almost completely insensitive to the details of the first epoch of inflation, for the observationally - viable range of @xmath1. this is because the fluctuations generated during this epoch affect only the smallest wavenumber part of the energy - density perturbation power spectrum, which can not contribute significantly to observable quantities because of the spatial curvature length cutoff " in an open universe (e.g., w83 ; kamionkowski & spergel 1994 ; rp95). inclusion of such fluctuations in the calculations alter the predictions for the present value of the rms linear mass fluctuations averaged over an @xmath25 mpc sphere, @xmath26 $], by @xmath27 (which is comparable to our computational accuracy). besides the open - bubble inflation model spectra, a variety of alternatives have also been considered. predictions for the usual flat - space scale - invariant spectrum in an open model have been examined (w83 ; abbott & schaefer 1986 ; gouda, sugiyama, & sasaki 1991 ; sugiyama & gouda 1992 ; kamionkowski & spergel 1994 ; sugiyama & silk 1994 ; cayn et al. the possibility that the standard formulation of quantum mechanics is incorrect in an open universe, and that allowance must be made for non - square - integrable basis functions has been investigated (lyth & woszczyna 1995), and other spectra have also been considered (e.g., w83 ; abbott & schaefer 1986 ; kamionkowski & spergel 1994). these spectra, being inconsistent with either standard quantum mechanics or the length scale set by spatial curvature, are of historical interest. more recently, the open - bubble inflation scenario has been further elaborated on. yst have considered a very specific model for the nucleation of the open bubble in a spatially - flat de sitter spacetime, and demonstrated a possible additional contribution from a non - square - integrable basis function which depends on the form of the potential, and on the assumed form of the quantum state prior to bubble nucleation. however, since the non - square - integrable basis function contributes only on the very largest scales, the spatial curvature cutoff " in an open universe makes almost all of the model predictions insensitive to this basis function, for the observationally - viable range of @xmath1 (yst ; yb). for example, at @xmath28 its effect is to change @xmath26 $] by @xmath29. an additional possible effect determined for the specific model of an open - inflation bubble nucleating in a spatially - flat de sitter spacetime is that fluctuations of the bubble wall behave like a non - square - integrable basis function (hamazaki et al. 1996 ; garriga 1996 ; garca - bellido 1996 ; yamamoto, sasaki, & tanaka 1996). while there are models in which these bubble - wall fluctuations are completely insignificant (garriga 1996 ; yamamoto et al. 1996), there is as yet no computation that accounts for both the bubble - wall fluctuations as well as those generated during the evolution inside the bubble (which are always present), so it is not yet known if bubble - wall fluctuations can give rise to an observationally significant effect. finally, again in this very specific model, the effects of a finite bubble size at nucleation seem to alter the zero bubble size predictions only by a very small amount (yamamoto et al. 1996 ; cohn 1996). while there is no guarantee that there is a spatially - flat de sitter spacetime prior to bubble nucleation, these computations do illustrate the important point that the spatial curvature length cutoff " in an open universe (e.g., rp95) does seem to ensure that what happens prior to bubble nucleation does not significantly affect the observable predictions for observationally - viable single - field open - bubble inflation models. it is indeed reassuring that accounting only for the quantum mechanical fluctuations generated during the evolution inside the bubble (rp94) seems to be essentially all that is required to make observational predictions for the single - field open - bubble inflation models. that is, the observational predictions of the open - bubble inflation scenario seem to be as robust as those for the spatially - flat inflation scenario. Cmb anisotropy normalization procedure in this paper, we utilize the dmr four - year 53 and 90 ghz sky maps in both galactic and ecliptic coordinates. we thus quantify explicitly the expected small shifts in the inferred normalization amplitudes due to the small differences between the galactic- and ecliptic - coordinate maps. the maps are coadded using inverse - noise - variance weights derived in each coordinate system. the least sensitive 31 ghz maps have been omitted from the analysis, since their contribution is minimal under such a weighting scheme. the dominant source of emission in the dmr maps is due to the galactic plane. we are unable to model this contribution to the sky temperature to sufficient accuracy to enable its subtraction, thus we excise all pixels where the galactic - plane signal dominates the cmb. the geometry of the cut has been determined by using the dirbe 140 @xmath30 m map as a tracer of the strongest emission, as described completely in banday (1996a). all pixels with galactic latitude @xmath31 20@xmath32@xmath33 are removed, together with regions towards scorpius - ophiucus and taurus - orion. there are 3881 surviving pixels in galactic coordinates and 3890 in ecliptic. this extended (four - year data) galactic plane cut has provided the biggest impact on the analysis of the dmr data (see grski et al. 1996, hereafter g96). the extent to which residual high - latitude galactic emission can modify our results has been quantified in two ways. since the spatial morphology of galactic synchrotron, free - free and dust emission seems to be well described by a steeply falling power spectrum (@xmath34 kogut 1996a, g96), the cosmological signal is predominantly compromised on the largest angular scales. as a simple test of galactic contamination, we perform all computations both including and excluding the observed sky quadrupole. a more detailed approach (g96) notes that a large fraction of the galactic signal can be accounted for by using the dirbe 140 @xmath30 m sky map (reach 1995) as a template for free - free and dust emission, and the 408 mhz all - sky radio survey (haslam 1981) to describe synchrotron emission. a correlation analysis yields coupling coefficients for the two templates at each of the dmr frequencies. we have repeated our model analysis after correcting the coadded sky maps by the galactic templates scaled by the coefficients derived in g96. in particular, we adopt those values derived under the assumption that the cmb anisotropy is well - described by an @xmath35 = 1 power law model with normalization amplitude @xmath21 @xmath36 18 @xmath30k and coupling coefficient amplitudes. in fact, we have investigated this for a sub - sample of the models considered here in which we varied @xmath1 but fixed @xmath2 and @xmath10. no statistically significant changes were found in the derived values of either @xmath21 or the coupling coefficients.]. one might make criticisms of either technique : excluding information from an analysis, in this case the quadrupole components, can obviously weaken any conclusions simply because statistical uncertainties will grow ; at the same time, it is not clear whether the galactic corrections applied are completely adequate. we believe that, given these uncertainties, our analysis is the most complete and conservative one that is possible. the power spectrum analysis technique developed by grski (1994) is implemented. orthogonal basis functions for the fourier decomposition of the sky maps are constructed which specifically include both pixelization effects and the galactic cut. (these are linear combinations of the usual spherical harmonics with multipole @xmath37.) the functions are coordinate system dependent. a likelihood analysis is then performed as described in grski (1994). we consider four open model energy - density perturbation power spectra : (1) the open - bubble inflation model spectrum, accounting only for fluctuations that are generated during the evolution inside the bubble (rp94) ; (2) the open - bubble inflation model spectrum, now also accounting for the fluctuations generated in the first epoch of inflation (bgt ; yst) ; (3) the open - bubble inflation model spectrum, now also accounting for both the usual fluctuations generated in the first epoch of inflation and a contribution from a non - square - integrable basis function (yst) ; and, (4) an open model with a flat - space scale - invariant spectrum (w83). in all cases we have ignored the possibility of tilt or primordial gravity waves, since it is unlikely that they can have a significant effect in viable open models. with the eigenvalue of the spatial scalar laplacian being @xmath38, where @xmath39 is the radial coordinate spatial wavenumber, the gauge - invariant fractional energy - density perturbation power spectrum of type (1) above is @xmath40 where @xmath41 is the transfer function and @xmath42 is the normalization amplitude) generalize the primordial part of the spectrum of eq. (1) by multiplying it with @xmath43. as yet, only the specific @xmath44 generalized spectrum (i.e., eq. [1]) is known to be a prediction of an open - bubble inflation model and therefore consistent with the presence of spatial curvature. it is premature to draw conclusions about open cosmogony on the basis of the @xmath45 version of the spectrum considered by bw.]. in the simplest example, perturbations generated in the first epoch of inflation introduce an additional multiplicative factor, @xmath46, on the right hand side of eq. (1). for a discussion of the effects of the non - square - integrable basis function see yst and yb. the energy - density power spectrum of type (4) above is @xmath47 and in this case one can also consider, e.g., @xmath48 (w83), but because of the spatial curvature cutoff " in an open model the predictions are essentially indistinguishable. at small @xmath49 the asymptotic expressions are @xmath50 (type 1), @xmath51 (type 2), and @xmath52 (type 4). conventionally, the cmb fractional temperature perturbation, @xmath53, is expressed as a function of angular position, @xmath54, on the sky via the spherical harmonic decomposition, @xmath55 the cmb spatial anisotropy in a gaussian model can then be characterized by the angular perturbation spectrum @xmath56, defined in terms of the ensemble average, @xmath57 the @xmath56 s used here were computed using two independent boltzmann transfer codes developed by ns (e.g., sugiyama 1995) and rs (e.g., stompor 1994). some illustrative comparisons are shown in fig. we emphasize that the excellent agreement between the @xmath56 s computed using the two codes is mostly a reflection of the currently achievable numerical accuracy. currently, the major likely additional, unaccounted for, source of uncertainty is that due to the uncertainty in the modelling of various physical effects. the computations here assume a standard recombination thermal history, and ignore the possibility of early reionization. the simplest open models (with the least possible number of free parameters) have yet to be ruled out by observational data (grsb ; ratra et al. 1995 ; grs ; this paper), so there is insufficient motivation to expand the model - parameter space by including the effect of early reionization, tilt or gravity waves values determined from the dmr data here (assuming no early reionization) are unlikely to be very significantly affected by early reionization. however, since structure forms earlier in an open model, other effects of early reionization might be more significant in an open model. while it is possible to heuristically account for such effects, an accurate quantitative estimate must await a better understanding of structure formation.]. for the @xmath58 of types (1), (2), and (4) above, we have evaluated the cmb anisotropy angular spectra for a range of @xmath1 spanning the interval between 0.1 and 1.0, for a variety of values of @xmath2 (the hubble parameter @xmath59) and the baryonic - mass density parameter @xmath10. the values of @xmath2 were selected to cover the lower part of the range of ages consistent with current requirements (@xmath60 10.5 gyr, 12 gyr, or 13.5 gyr, with @xmath2 as a function of @xmath1 computed accordingly ; see, for example, jimenez et al. 1996 ; chaboyer et al. the values of @xmath10 were chosen to be consistent with current standard nucleosynthesis requirements (@xmath61 0.0055, 0.0125, or 0.0205 ; e.g., copi, schramm, & turner 1995 ; sarkar 1996). to render the problem tractable, @xmath56 s were determined for the central values of @xmath62 and @xmath63, and for the two combinations of these parameters which most perturb the @xmath56 s from those computed at the central values (i.e., for the smallest @xmath62 we used the smallest @xmath63, and for the largest @xmath62 we used the largest @xmath63). specific parameter values are given in columns (1) and (2) of tables 16, and representative anisotropy spectra can be seen in figs. 2 and 3. we therefore improve on our earlier analysis of the dmr two - year data (grsb) by considering a suitably broader range in the (@xmath10, @xmath2) parameter space. the cmb anisotropy spectra for @xmath58 of type (3) above were computed for a range of @xmath1 spanning the interval between 0.1 and 0.9, for @xmath64 and @xmath65. specific parameter values are given in columns (1) and (2) of table 7, and these spectra are shown in fig. 4. in fig. 5 we compare the various spectra considered here. the differences in the low-@xmath66 shapes of the @xmath56 s in the various models (figs. 25) are a consequence of three effects : (1) the shape of the energy - density perturbation power spectrum at low wavenumber ; (2) the exponential suppression at the spatial curvature scale in an open model ; and (3) the interplay between the usual " (fiducial cdm) sachs - wolfe term and the integrated " sachs - wolfe (hereafter sw) term in the expression for the cmb spatial anisotropy. the relative importance of these effects is determined by the value of @xmath1, and leads to the non - monotonic behaviour of the large - scale @xmath56 s as a function of @xmath1 seen in figs. more precisely, the contributions to the cmb anisotropy angular spectrum from the usual " and integrated " sw terms have a different @xmath66-dependence as well as a relative amplitude that is both @xmath1 and @xmath58 dependent. on very large angular scales (small @xmath66 s), the dominant contribution to the usual " sw term comes from a higher redshift (when the length scales are smaller) than does the dominant contribution to the integrated " sw term (hu & sugiyama 1994, 1995). as a result, in an open model on very large angular scales, the usual " sw term is cut off more sharply by the spatial curvature length scale than is the integrated " sw term (hu & sugiyama 1994), i.e., on very large angular scales in an open model the usual " sw term has a larger (positive) effective index @xmath35 than the integrated " sw term. on slightly smaller angular scales the integrated " sw term is damped (i.e., it has a negative effective index @xmath35) while the usual " sw term plateaus (hu & sugiyama 1994). as a consequence, going from the largest to slightly smaller angular scales, the usual " term rises steeply and then flattens, while the integrated " term rises less steeply and then drops (i.e., it has a peak). the change in shape, as a function of @xmath66, of these two terms is both @xmath1 and @xmath58 dependent. these are the two dominant effects at @xmath67 ; at higher @xmath66 other effects come into play. more specifically, for @xmath68 the curvature length scale cutoff and the precise large - scale form of the @xmath58 considered here are relatively unimportant the cmb anisotropy angular spectrum is quite similar to that for @xmath69, and the dominant contribution is the usual " sw term. for a @xmath58 that does not diverge at low wavenumber, as with the flat - space scale - invariant spectrum in an open model, for @xmath70 the exponential cutoff " at the spatial curvature length dominates, and the lowest-@xmath66 @xmath56 s are suppressed (figs. 3 and 5). for this @xmath58, as @xmath1 is reduced, the usual " term continues to be important on the largest angular scales down to @xmath28. as @xmath1 is reduced below @xmath71 the integrated " term starts to dominate on the largest angular scales, and as @xmath1 is further reduced the integrated " term also starts to dominate on smaller angular scales. from fig. 3(a) one will notice that the integrated " sw term peak " first makes an appearance at @xmath72 the central line in the plot at @xmath73 and that as @xmath1 is further reduced (in descending order along the curves shown) the integrated " term peak " moves to smaller angular scales. the @xmath74 case is where the integrated " term peaks at @xmath75, and the damping of this term on smaller angular scales (@xmath76) is compensated for by the steep rise of the usual " sw term the two terms are of roughly equal magnitude at @xmath77 and these effects result in the almost exactly scale - invariant spectrum at @xmath9 (this case is more scale - invariant than fiducial cdm). a discussion of some of these features of the cmb anisotropy angular spectrum in the flat - space scale - invariant spectrum open model is given in cayn et al. (1996). open - bubble inflation models have a @xmath58 that diverges at low wavenumber (rp95 ; note that no physical quantity diverges), and this increases the low-@xmath66 @xmath56 s (figs. 2 and 5) relative to those of the flat - space scale - invariant spectrum open model (figs. 3 and 5). the @xmath56 s for low @xmath1 models increase more than the higher @xmath1 ones, since, for a fixed wavenumber - dependence of @xmath58, the divergence is more prominent at lower @xmath1 (rp94). the non - square - integrable basis function (yst) contributes even more power on large angular scales, and so, at low-@xmath66, the @xmath56 s of fig. 4 are slightly larger than those of fig. 2 (also see fig. 5). again, spectra at lower values of @xmath1 are more significantly influenced. as is clear from figs. 2 and 5, in an open - bubble inflation model, quantum - mechanical zero - point fluctuations generated in the first epoch of inflation scarcely affect the @xmath56 s, although at the very lowest values of @xmath1 the very lowest order @xmath56 coefficients are slightly modified. the effect is concentrated in this region of the parameter space since the fluctuations in the first inflation epoch only contribute to, and increase, the lowest wavenumber part of @xmath58. in simple open - bubble inflation models, the precise value of this small effect is dependent on the model assumed for the first epoch of inflation (bt). since the dmr data is most sensitive to multipole moments with @xmath78 810, one expects the effect at @xmath78 23 to be almost completely negligible (bt ; also see yst ; yb). figs. 35 show that both the flat - space scale - invariant spectrum open model, and the contribution from the non - square - integrable mode, do lead to significantly different @xmath56 s (compared to those of fig. the results of the dmr likelihood analyses are summarized in figs. 621 and tables 17 and 13. two representative sets of likelihood functions @xmath79 are shown in figs. 6 and 7. figure 6 shows those derived from the ecliptic - frame sky maps, ignoring the correction for faint high - latitude foreground galactic emission, and excluding the quadrupole moment from the analysis. figure 7 shows the likelihood functions derived from the galactic - frame sky maps, accounting for the faint high - latitude foreground galactic emission correction, and including the quadrupole moment in the analysis. together, these two data sets span the maximum range of normalizations inferred from our analysis (the former providing the highest, and the latter the lowest @xmath21). tables 17 give the @xmath21 central values and 1-@xmath80 and 2-@xmath80 ranges for spectra of type (1), (3), and (4) above, computed from the appropriate posterior probability density distribution function assuming a uniform prior. each line in tables 17 lists these values at a given @xmath1 for the 8 possible combinations of : (1) galactic- or ecliptic - coordinate map ; (2) faint high - latitude galactic foreground emission correction accounted for or ignored ; and, (3) quadrupole included (@xmath81) or excluded (@xmath82) value of varying cosmological parameters like @xmath10. since they do not quote derived @xmath21 values for this model we are not able to compare to their results.]. the corresponding ridge lines of maximum likelihood @xmath21 value as a function of @xmath1 are shown in figs. 810 for some of the cosmological - parameter values considered here. although we have computed these values for spectra of type (2) above (i.e., those accounting for perturbations generated in the first epoch of inflation) we record only a subset of them in column (4) of table 13. these should be compared to columns (2) and (6) of table 13, which show the maximal 2-@xmath80 @xmath21 range for spectra of types (1) and (3). while the differences in @xmath21 between spectra (1) and (2) [cols. (2) and (4) of table 13] are not totally insignificant, more importantly the differences between the @xmath26 $] values for the three spectra [cols. (3), (5), and (7) of table 13] are observationally insignificant. the entries in tables 16 illustrate the shift in the inferred normalization amplitudes due to changes in @xmath2 and @xmath10. these shifts are larger for models with a larger @xmath1, since these models have cmb anisotropy spectra that rise somewhat more rapidly towards large @xmath66, so in these cases the dmr data is sensitive to somewhat smaller angular scales where the effects of varying @xmath2 and @xmath63 are more prominent. figure 11 shows the effects that varying @xmath62 and @xmath63 have on some of the ridge lines of maximum likelihood @xmath21 as a function of @xmath1, and fig. 13 illustrates the effects on some of the conditional (fixed @xmath1 slice) likelihood densities for @xmath83 on the whole, for the cmb anisotropy spectra considered here, shifts in @xmath2 and @xmath84 have only a small effect on the inferred normalization amplitude. the normalization amplitude is somewhat more sensitive to the differences between the galactic- and ecliptic - coordinate sky maps, to the foreground high - latitude galactic emission treatment, and to the inclusion or exclusion of the @xmath85 moment. for the purpose of normalizing models, we choose for our 2-@xmath80 c.l. bounds values from the likelihood fits that span the maximal range in the @xmath21 normalizations. specifically, for the lower 2-@xmath80 bound we adopt the value determined from the analysis of the galactic - coordinate maps accounting for the high - latitude galactic emission correction and including the @xmath85 moment in the analysis, and for the upper 2-@xmath80 value that determined from the analysis of the ecliptic - coordinate maps ignoring the galactic emission correction and excluding the @xmath85 moment from the analysis. these values are recorded in columns (5) and (8) of tables 912, and columns (2), (4), and (6) of table 13) were used in the likelihood analyses of the various model spectra, and different interpolation methods were used in the determination of the @xmath21 values, there are small (but insignificant) differences in the quoted @xmath21 values for some identical models in these tables.]. figure 12 compares the ridge lines of maximum likelihood @xmath21 value, as a function of @xmath1, for the four different cmb anisotropy angular spectra considered here, and fig. 14 compares some of the conditional (fixed @xmath1 slice) likelihood densities for @xmath21 for these four cmb anisotropy angular spectra. approximate fitting formulae may be derived to describe the above two extreme 2-@xmath80 limits. for the open - bubble inflation model (rp94 ; bgt ; yst), not including a contribution from a non - square - integrable basis function, we have @xmath86, \eqno(5)\]] which is good to better than @xmath87 for all values of @xmath1 (and to better than @xmath88 over the observationally - viable range of @xmath89). for those models including a contribution from the non - square - integrable basis function (yst), we have @xmath90, \eqno(6)\]] mostly good to better than @xmath88. the flat - space scale - invariant spectrum open model fitting formula is @xmath91, \eqno(7)\]] generally good to better than @xmath92, except near @xmath93 and @xmath94 where the deviations are larger. further details about these fitting formulae may be found in stompor (1996). the approximate fitting formulae (5)(7) provide a convenient, portable normalization of the open models. it is important, however, to note that they have been derived using the @xmath21 values determined for a given @xmath2 and @xmath10, and hence do not account for the additional uncertainty (which could be as large as @xmath88) due to allowed variations in these parameters. we emphasize that in our analysis here we make use of the actual @xmath21 values derived from the likelihood analyses, not these fitting formulae. figures 15 and 16 show projected likelihood densities for @xmath1, for some of the models and dmr data sets considered here. note that the general features of the projected likelihood densities for the open - bubble inflation model only accounting for the fluctuations generated during the evolution inside the bubble (spectrum [1] above), are consistent with those derived from the dmr two - year data (grsb, fig. 3). however, since we only compute down to @xmath95 here, only the rise to the prominent peak at very low @xmath1 (grsb) is seen. bw show in the middle left - hand panel of their fig. 11 (presumably) the projected likelihood density for @xmath1 for the same open - bubble inflation model, the general features of which are consistent with those derived here. figures 1721 show marginal likelihood densities for @xmath1, for some of the models and dmr data sets considered here. for the open - bubble inflation model accounting only for the fluctuations generated during the evolution inside the bubble (rp94), the dmr two - year data galactic - frame (quadrupole moment excluded and included) marginal likelihoods are shown in fig. 3 of grsb, and are in general concord with those shown in fig. 17 here (although, again, only the rise to the prominent low-@xmath1 peak is seen here). note that now, especially for the quadrupole excluded case, the peaks and troughs are more prominent (although still not greatly statistically significant). furthermore, comparing the solid line of fig. 17(b) here to the heavy dotted line of fig. 3 of grsb, one notices that the intermediate @xmath1 peak is now at @xmath96, instead of at @xmath97 for the dmr two - year data. (since bw chose not to compute for the case when the quadrupole moment is excluded from the analysis, they presumably did not notice the peak at @xmath98 in the marginalized likelihood density for the open - bubble inflation model see fig. 17.) for the open - bubble inflation model now also accounting for both the fluctuations generated in the first spatially - flat epoch of inflation (bgt ; yst), and those from the non - square - integrable basis function (yst), the dmr two - year data ecliptic - frame quadrupole - included marginal likelihood (shown as the solid line in fig. 3 of yb) is in general agreement with the dot - dashed line of fig. however, yb did not compute for the case where the quadrupole moment was excluded from the analysis and so did not find the peak at @xmath99 in fig. 19. given the shapes of the marginal likelihoods in figs. 1721, it is not at all clear if it is meaningful to derive limits on @xmath1 without making use of other (prior) information. as an example, it is not at all clear what to use for the integration range in @xmath1. focussing on fig. 21(a) (which is similar to the other quadrupole excluded cases), the only conclusion seems to be that @xmath9 is the value most consistent with the dmr data (at least amongst those models with @xmath14 some of the models have another peak at @xmath100, grsb). however, when the quadrupole moment is included in the analysis (as in fig. 21b), the open - bubble inflation model peaks are at @xmath12 (at least in the range @xmath14, grsb), while the flat - space scale - invariant spectrum open model peak is at @xmath11. at the 95% c.l. no value of @xmath1 over the range considered, 0.11, is excluded. (the yb and bw claims of a lower limit on @xmath1 from the dmr data alone are, at the very least, premature.) Computation of large-scale structure statistics the @xmath58 (e.g., eqs. [1] and [2]) were determined from a numerical integration of the linear perturbation theory equations of motion. as before, the computations were performed with two independent numerical codes. for some of the model - parameter values considered here the results of the two computations were compared and found to be in excellent agreement. illustrative examples of the comparisons are shown in fig. again, we emphasize that the excellent agreement is mostly a reflection of the currently available numerical accuracy, and the most likely additional, unaccounted for, source of uncertainty is that due to the uncertainty in the modelling of various physical effects. table 8 list the @xmath58 normalization amplitudes @xmath42 (e.g., eqs. [1] and [2]) when @xmath101k. examples of the power spectra normalized to @xmath21 derived from the mean of the dmr four - year data analysis extreme upper and lower 2-@xmath80 limits discussed above are shown in figs. one will notice, from fig. 23(e), the good agreement between the open - bubble inflation spectra. when normalized to the two extreme 2-@xmath80 @xmath21 limits (e.g., cols. [5] and [8] of table 10), the @xmath58 normalization factor (eq. [1] and table 8) for the open - bubble inflation model (rp94 ; bgt ; yst), may be summarized by, for the lower 2-@xmath80 limit, @xmath102, \eqno(8)\]] and for the upper 2-@xmath80 limit, @xmath103. \eqno(9)\]] these fits are good to @xmath104 for @xmath14. note however that they are derived using the @xmath21 values determined for given @xmath62 and @xmath63 and hence do not account for the additional uncertainty introduced by allowed variations in these parameters (which could affect the power spectrum normalization amplitude by as much as @xmath105). from fig. 23(e), and given the uncertainties, we see that the fitting formulae of eqs. (8) and (9) provide an adequate summary for all the open - bubble inflation model spectra. the extreme @xmath106-@xmath80 @xmath58 normalization factor (eq. [2] and table 8) for the flat - space scale - invariant spectrum open model (w83) may be summarized by, for the lower 2-@xmath80 limit, @xmath107, \eqno(10)\]] and for the upper 2-@xmath80 limit, @xmath108. \eqno(11)\]] these fits are good to better than @xmath88 for @xmath109 ; again, they are derived from @xmath21 values determined at given @xmath62 and @xmath63. given the uncertainties involved in the normalization procedure (born of both statistical and other arguments) it is not yet possible to quote a unique dmr normalization amplitude (g96). as a central " value for the @xmath58 normalization factor, we currently advocate the mean of eqs. (8) and (9) or eqs. (10) and (11) as required. we emphasize, however, that it is incorrect to draw conclusions about model viability based solely on this central " value. in conjunction with numerically determined transfer functions, the fits of eqs. (8)(11) allow for a determination of @xmath26 $], accurate to a few percent. here the mean square linear mass fluctuation averaged over a sphere of coordinate radius @xmath110 is @xmath111 ^ 2\right\rangle & = & { 2 \over \pi^2 \left[{\rm sinh } (\bar\chi)\, { \rm cosh}(\bar\chi) - \bar\chi\right]^2 } \nonumber \\ { \ } & { \ } & \times \int^\infty_0 { dk \over (1 + k^2)^2 } \left[{\rm cosh}(\bar\chi)\, { \rm sin}(k\bar\chi) - k\, { \rm sinh}(\bar\chi) \, { \rm cos}(k\bar\chi) \right]^2 p(k), \end{aligned}\]] which, on small scales, reduces to the usual flat - space expression @xmath112 \int^\infty_0 dk\, k^2 p(k) \left[{\rm sin}(k\bar\chi) - k\bar\chi \, { \rm cos}(k\bar\chi) \right]^2/(k\bar\chi)^6 $]. if instead use is made of the bardeen et al. (1986, hereafter bbks) analytic fit to the transfer function using the parameterization of eq. (13) below (sugiyama 1995) and numerically determined values for @xmath42, the resultant @xmath113 $] values are accurate to better than @xmath87 (except for large baryon - fraction, @xmath114, models where the error could be as large as @xmath115). use of the analytic fits of eqs. (8)(11) for @xmath42 (instead of the numerically determined values) slightly increases the error, while use of the bbks transfer function fit parameterized by an earlier version of eq. (13) below, @xmath116 $], results in @xmath26 $] values that could be off by as much as @xmath117. nevertheless, as has been demonstrated by llrv, the approximate analytic fit to the transfer function greatly simplifies the computation and allows for rapid demarcation of the favoured part of cosmological - parameter space. numerical values for some cosmographic and large - scale structure statistics for the models considered here are recorded in tables 915. we emphasize that when comparing to observational data we make use of numerically - determined large - scale structure predictions, not those derived using an approximate analytic fitting formula. tables 912 give the predictions for the open - bubble inflation model accounting only for the perturbations generated during the evolution inside the bubble (rp94), and for the flat - space scale - invariant spectrum open model (w83). each of these tables corresponds to a different pair of @xmath118 values. the first two columns in these tables record @xmath1 and @xmath2, and the third column is the cosmological baryonic - matter fraction @xmath119. the fourth column gives the value of the matter power spectrum scaling parameter (sugiyama 1995), @xmath120 which is used to parameterize approximate analytic fits to the power spectra derived from numerical integration of the perturbation equations. the quantities listed in columns (1)(4) of these tables are sensitive only to the global parameters of the cosmological model. columns (5) and (8) of tables 912 give the dmr data 2-@xmath80 range of @xmath21 that is used to normalize the perturbations in the models considered here. the numerical values in table 12 are for @xmath121 gyr, @xmath122. we did not analyze the dmr data using @xmath56 s for these models, and in this case the perturbations are normalized to the @xmath21 values from the @xmath123 gyr, @xmath124 analyses. (as discussed above, shifts in @xmath2 and @xmath63 do not greatly alter the inferred normalization amplitude.) columns (6) and (9) of tables 912 give the 2-@xmath80 range of @xmath125 $]. these were determined using the @xmath58 derived from numerical integration of the perturbation equations. for about two dozen cases, these rms mass fluctuations determined using the two independent numerical integration codes were compared and found to be in excellent agreement. (at fixed @xmath21, they differ by @xmath126 depending on model - parameter values, with the typical difference being @xmath127. we again emphasize that this is mostly a reflection of currently achievable numerical accuracy.). to usually better than @xmath128 accuracy, for @xmath129, the 2-@xmath80 @xmath130 $] entries of columns (6) and (9) of tables 912 may be summarized by the fitting formulae listed in table 14. these fitting formulae are more accurate than expressions for @xmath26 $] derived at the same cosmological - parameter values using an analytic approximation to the transfer function and the normalization of eqs. (8)(11). for open models, as discussed below, it proves most convenient to characterize the peculiar velocity perturbation by the parameter @xmath131 where @xmath132 is the linear bias factor for @xmath133 galaxies (e.g., peacock & dodds 1994). the 2-@xmath80 range of @xmath134 are listed in columns (7) and (10) of tables 912. table 13 compares the @xmath113 $] values for spectra of types (1)(3) above. clearly, there is no significant observational difference between the predictions for the different spectra. in what follows, for the open - bubble inflation model we concentrate on the type (1) spectrum above. again, the ranges in tables 914 are those determined from the maximal 2-@xmath80 @xmath21 range. table 15 lists central dmr - normalized " values for @xmath130 $], defined as the mean of the maximal @xmath1352-@xmath80 entries of tables 912. (the mean of the @xmath1352-@xmath80 fitting formulae of table 14 may be used to interpolate between the entries of table 15.) we again emphasize that it is incorrect to draw conclusions about model viability based solely on these central " values for the purpose of constraining model - parameter values by, e.g., comparing numerical simulation results to observational data one must make use of computations at a few different values of the normalization selected to span the @xmath1352-@xmath80 ranges of tables 912. Current observational constraints on dmr-normalized models the dmr likelihoods do not meaningfully exclude any part of the (@xmath1, @xmath2, @xmath63) parameter space for the models considered here. in this section we combine current observational constraints on global cosmological parameters with the dmr - normalized model predictions to place constraints on the range of allowed model - parameter values. it is important to bear in mind that some measures of observational cosmology remain uncertain thus our analysis here must be viewed as tentative and subject to revision as the observational situation approaches equilibrium. to constrain our model - parameter values we have employed the most robust of the current observational constraints. tables 912 list some observational predictions for the models considered here, and the boldface entries are those that are inconsistent with current observational data at the 2-@xmath80 significance level. for each cosmographic or large - scale parameter, we have generally chosen to use constraints from a single set of observations or from a single analysis. we generally use the most recent analyses since we assume that they incorporate a better understanding of the uncertainties, especially those due to systematics. the specific constraints we use are summarized below, where we compare them to those derived from other analyses. the model predictions depend on the age of the universe @xmath62. to reconcile the models with the high measured values of the hubble parameter @xmath2, we have chosen to focus on @xmath60 10.5, 12, and 13.5 gyr, which are near the lower end of the ages now under discussion. for instance, jimenez et al. (1996) find that the oldest globular clusters have ages @xmath136 gyr (also see salaris, deglinnocenti, & weiss 1996 ; renzini et al. 1996), and that it is very unlikely that the oldest clusters are younger than 9.7 gyr. the value of @xmath1 is another input parameter for our computations. as summarized by peebles (1993, @xmath137), on scales @xmath138 mpc a variety of different observational measurements indicate that @xmath1 is low. for instance, virial analyses of x - ray cluster data indicates @xmath139, with a 2-@xmath80 range : @xmath140 (carlberg et al. 1996 we have added their 1-@xmath80 statistical and systematic uncertainties in quadrature and doubled to get the 2-@xmath80 uncertainty). in a cdm model in which structure forms at a relatively high redshift (as is observed), these local estimates of @xmath1 do constrain the global value of @xmath1 (since, in this case, it is inconceivable that the pressureless cdm is much more homogeneously distributed than is the observed baryonic mass). we hence adopt a 2-@xmath80 upper limit of @xmath141 to constrain the cdm models we consider here. (this large upper limit allows for the possibility that the models might be moderately biased.) the boldface entries in column (1) of tables 912 indicates those @xmath1 values inconsistent with this constraint. column (2) of tables 912 gives the value of the hubble parameter @xmath2 that corresponds to the chosen values of @xmath1 and @xmath62. current observational data favours a larger @xmath2 (e.g., kennicutt, freedman, & mould 1995 ; baum et al. 1995 ; van den bergh 1995 ; sandage et al. 1996 ; ruiz - lapuente 1996 ; riess, press, & kirshner 1996 ; but also see schaefer 1996 ; branch et al. for the purpose of our analysis here we adopt the @xmath142 value @xmath143 (1-@xmath80 uncertainty, tanvir et al. 1995) ; doubling the uncertainty, the 2-@xmath80 range is @xmath144. the bold face entries in column (2) of tables 912 indicates those model - parameter values which predict an @xmath2 inconsistent with this range. comparison of the standard nucleosynthesis theoretical predictions for the primordial light element abundances to what is determined by extrapolation of the observed abundances to primordial values leads to constraints on @xmath63. it has usually been argued that @xmath145he and @xmath146li allow for the most straightforward extrapolation from the locally observed abundances to the primordial values (e.g., dar 1995 ; fields & olive 1996 ; fields et al. 1996, hereafter fkot). the observed @xmath145he and @xmath146li abundances then suggest @xmath147, and a conservative assessment of the uncertainties indicate a 2-@xmath80 range : @xmath148 (fkot ; also see copi et al. 1995 ; sarkar 1996). observational constraints on the primordial deuterium (d) abundance should, in principle, allow for a tightening of the allowed @xmath63 range. there are now a number of different estimates of the primordial d abundance, and since the field is still in its infancy it is, perhaps, not surprising that the different estimates are somewhat discrepant. songaila et al. (1994), carswell et al. (1994), and rugers & hogan (1996a, b) use observations of three high - redshift absorption clouds to argue for a high primordial d abundance and so a low @xmath63. tytler, fan, & burles (1996) and burles & tytler (1996) study two absorption clouds and argue for a low primordial d abundance and so a high @xmath63. carswell et al. (1996) and wampler et al. (1996) examine other absorption clouds, but are not able to strongly constrain @xmath63. while the error bars on @xmath63 determined from these d abundance observations are somewhat asymmetric, to use these results to qualitatively pick the @xmath63 values we wish to examine we assume that the errors are gaussian (and where needed add all uncertainties in quadrature to get the 2-@xmath80 uncertainties). the large d abundance observations suggest @xmath149 with a 2-@xmath80 range : @xmath150 (rugers & hogan 1996a). when these large d abundances are combined with the observed @xmath145he and @xmath146li abundances, they indicate @xmath151, with a 2-@xmath80 range : @xmath152 (fkot). the large d abundances are consistent with the standard interpretation of the @xmath145he and @xmath146li abundances, and with the standard model of particle physics (with three massless neutrino species) ; they do, however, seem to require a modification in galactic chemical evolution models to be consistent with local determinations of the d and @xmath153he abundances (e.g., fkot ; cardall & fuller 1996). the low d abundance observations favour @xmath154 with a 2-@xmath80 range : @xmath155 (burles & tytler 1996). the low d abundance observations seem to be more easily accommodated in modifications of the standard model of particle physics, i.e., they are difficult to reconcile with exactly three massless neutrino species ; alternatively they might indicate a gross, as yet unaccounted for, uncertainty in the observed @xmath145he abundance (burles & tytler 1996 ; cardall & fuller 1996). the low d abundance is approximately consistent with locally - observed d abundances, but probably requires some modification in the usual galactic chemical evolution model for @xmath146li (burles & tytler 1996 ; cardall & fuller 1996). to accommodate the range of @xmath63 now under discussion, we compute model predictions for @xmath124 (table 9), 0.007 (table 12), 0.0125 (table 10), and 0.0205 (table 11). we shall find that this uncertainty in @xmath63 precludes determination of robust constraints on model - parameter values. fortunately, recent improvements in observational capabilities should eventually lead to a tightening of the constraints on @xmath63, and so allow for tighter constraints on the other cosmological parameters. column (3) of tables 912 give the cosmological baryonic - mass fraction for the models we consider here. the cluster baryonic - mass fraction is the sum of the cluster galactic - mass and gas - mass fractions. assuming that the white et al. (1993) 1-@xmath80 uncertainties on the cluster total, galactic, and gas masses are gaussian and adding them in quadrature, we find for the 2-@xmath80 range of the cluster baryonic - mass fraction : @xmath156 elbaz, arnaud, & bhringer (1995), white & fabian (1995), david, jones, & forman (1995), markevitch et al. (1996), and buote & canizares (1996) find similar (or larger) gas - mass fractions. note that elbaz et al. (1995) and white & fabian (1995) find that the gas - mass error bars are somewhat asymmetric ; this non - gaussianity is ignored here. assuming that the cluster baryonic - mass fraction is an unbiased estimate of the cosmological baryonic - mass fraction, we may use eq. (15) to constrain the cosmological parameters. the boldface entries in column (3) of tables 9 - 12 indicates those model - parameter values which predict a cosmological baryonic - mass fraction inconsistent with the range of eq. (15). viana & liddle (1996, hereafter vl) have reanalyzed the combined galaxy @xmath58 data of peacock & dodds (1994), ignoring some of the smaller scale data where nonlinear effects might be somewhat larger than previously suspected. using an analytic approximation to the @xmath58, they estimate that the scaling parameter (eq. [13]) in the exponent of eq. (13), so the numerical values of their constraint on @xmath157 should be reduced slightly. we ignore this small effect here.] @xmath158, with a 2-@xmath80 range, @xmath159 this estimate is consistent with earlier ones than eq. (16) this is one reason why llrv favour a higher @xmath1 for the open - bubble inflation model than do grsb.]. it might be of interest to determine whether the wiggles in @xmath58 due to the pressure in the photon - baryon fluid, see figs. 23, can significantly affect the determination of @xmath157, especially in large @xmath119 models. (these wiggles are not well described by the analytic approximation to @xmath58.) the boldface entries in column (4) of tables 912 indicates those model - parameter values which predict a scaling parameter value inconsistent with the range of eq. (16). to determine the value of the linear bias parameter @xmath160, @xmath161 where @xmath162 is the rms fractional perturbation in galaxy number, we adopt the apm value (maddox, efstathiou, & sutherland 1996) of @xmath163 = 0.96 $], with 2-@xmath80 range : @xmath164 where we have added the uncertainty due to the assumed cosmological model and due to the assumed evolution in quadrature with the statistical 1-@xmath80 uncertainty (maddox et al. 1996, eq. [43]), and doubled to get the 2-@xmath80 uncertainty. the range of eq. (18) is consistent with that determined from eqs. (7.33) and (7.73) of peebles (1993). the local abundance of rich clusters, as a function of their x - ray temperature, provides a tight constraint on @xmath113 $]. eke, cole, & frenk (1996, hereafter ecf) (and s. cole, private communication 1996) find for the open model at 2-@xmath80 : @xmath165 where we have assumed that the ecf uncertainties are gaussian, and that in general it depends weakly on the value of @xmath157 (and so on the value of @xmath2 and @xmath10) see fig. 13 of ecf. in our preliminary analysis here we ignore this mild dependence on @xmath2 and @xmath10. also note that the constraint of eq. (19) is approximately that required for consistency with the observed cluster correlation function.]. the constraints of eq. (19) are consistent with, but more restrictive than, those derived by vl = 0.60 $] for fiducial cdm, which is at the @xmath1662-@xmath80 limit of eq. (as discussed in ecf, this is because vl normalize to the cluster temperature function at 7 kev, where there is a rise in the temperature function.) this is one reason why llrv favour a higher value of @xmath1 for the open - bubble inflation model than did grsb.]. this is because ecf use observational data over a larger range in x - ray temperature to constrain @xmath167, and also use n - body computations at @xmath168 0.3 and 1 to calibrate the press - schechter model (which is used in their determination of the constraints). furthermore, ecf also make use of hydrodynamical simulations of a handful of individual clusters in the fiducial cdm model (@xmath69) to calibrate the relation between the gas temperature and the cluster mass, and then use this calibrated relation for the computations at all values of @xmath1. the initial conditions for all the simulations were set using the analytical approximation to @xmath58, so again it might be of interest to see whether the wiggles in the numerically integrated @xmath58 could significantly affect the determination of the constraints of eq. kitayama & suto (1996) use x - ray cluster data, and a method that allows for the fact that clusters need not have formed at the redshift at which they are observed, to directly constrain the value of @xmath1 for cdm cosmogonies normalized by the dmr two - year data. their conclusions are in resonable accord with what would be found by using eq. (19) (derived assuming that observed clusters are at their redshifts of formation). however, kitayama & suto (1996) note that evolution from the redshift of formation to the redshift of observation can affect the conclusions, so a more careful comparison of these two results is warranted. the boldface entries in columns (6) and (9) of tables 912 indicate those model - parameter values whose predictions are inconsistent with the constraints of eq. (19) (1-@xmath80) uncertainty of eq. (19), approximate analyses based on using the analytic bbks approximation to the transfer function should make use of the more accurate parameterization of eq. (13) (rather than that with @xmath169 in the exponent), as this gives @xmath26 $] to better than @xmath87 in the observationally viable part of parameter space (provided use is made of the numerically determined values of @xmath42).]. from large - scale peculiar velocity observational data zaroubi et al. (1996) estimate @xmath26 = (0.85 \pm 0.2)\omega_0{}^{-0.6}$] (2-@xmath80). it might be significant that the large - scale peculiar velocity observational data constraint is somewhat discordant with (higher than) the cluster temperature function constraint. since @xmath170 is less sensitive to smaller length scales (compared to @xmath26 $]), observational constraints on @xmath170 are more reliably contrasted with the linear theory predictions. however, since @xmath170 is sensitive to larger length scales, the observational constraints on @xmath170 are significantly less restrictive than the @xmath171 (1-@xmath80) constraints of eq. (19), and so we do not record the predicted values of @xmath170 here. observational constraints on the mass power spectrum determined from large - scale peculiar velocity observations provide another constraint on the mass fluctuations. kolatt & dekel (1995) find at the 1-@xmath80 level @xmath172 where the 1-@xmath80 uncertainty also accounts for sample variance (t. kolatt, private communication 1996). since the uncertainties associated with the constraint of eq. (19) are more restrictive than those associated with the constraint of eq. (20), we do not tabulate predictions for this quantity here. however, comparison may be made to the predicted linear theory mass power spectra of figs. 23, bearing in mind the @xmath173 (2-@xmath80) uncertainty of eq. (20) (the uncertainty is approximately gaussian, t. kolatt, private communication 1996),-@xmath80, significance level, eq. (20) provides a strong upper limit on @xmath174, especially at larger @xmath1 because of the @xmath1 dependence.] and the uncertainty in the dmr normalization (not shown in figs. 23). columns (7) and (10) of tables 912 give the dmr - normalized model predictions for @xmath134 (eq. [14]). cole, fisher, & weinberg (1995) measure the anisotropy of the redshift space power spectrum of the @xmath133 1.2 jy survey and conclude @xmath175 with a 2-@xmath80 c.l. range : @xmath176 where we have doubled the error bars of eq. (5.1) of cole et al. (1995) to get the 2-@xmath80 range. cole et al. (1995, table 1) compare the estimate of eq. (21) to other estimates of @xmath134, and at 2-@xmath80 all estimates of @xmath134 are consistent. it should be noted that the model predictions of @xmath134 (eq. [14]) in tables 912 assume that for @xmath133 galaxies @xmath163 = 1/1.3 $] holds exactly, i.e., they ignore the uncertainty in the rms fractional perturbation in @xmath133 galaxy number, which is presumably of the order of that in eq. (18). as the constraints from the deduced @xmath134 values, eq. (21), are not yet as restrictive as those from other large - scale structure measures, we do not pursue this issue in our analysis here. the boldface entries in columns (7) and (10) of tables 912 indicate those model - parameter values whose predictions are inconsistent with the constraints of eq. (21). the boldface entries in tables 912 summarize the current constraints imposed by the observational data discussed in the previous section on the model - parameter values for the open - bubble inflation model (spectra of type [1] above), and for the flat - space scale - invariant spectrum open model (type [4] above). the current observational constraints on the models are not dissimilar, but this is mostly a reflection of the uncertainty on the constraints themselves since the model predictions are fairly different. in the following discussion of the preferred part of model - parameter space we focus on the open - bubble inflation model (rp94). note from table 13 that the large - scale structure predictions of the open - bubble inflation model do not depend on perturbations generated in the first epoch of inflation (bgt ; yst), and also do not depend significantly on the contribution from the non - square - integrable basis function (yst). table 9 corresponds to the part of parameter space with maximized " small - scale power in matter fluctuations. this is accomplished by picking a low @xmath123 gyr (and so large @xmath2), and by picking a low @xmath124 (this is the lower 2-@xmath80 limit from standard nucleosynthesis and the observed @xmath145he, @xmath146li, and high d abundances, fkot). the tightest constraints on the model - parameter values come from the matter power spectrum observational data constraints on the shape parameter @xmath157 (table 9, col. [4]), and from the cluster x - ray temperature function observational data constraints on @xmath26 $] (col. note that for @xmath177 the predicted upper 2-@xmath80 value of @xmath113 = 0.69 $], while ecf conclude that at 2-@xmath80 the observational data requires that this be at least 0.74, so an @xmath177 case fails this test. the constraints on @xmath134 (col. [7]) are not as restrictive as those on @xmath113 $]. for these values of @xmath62 and @xmath178 the cosmological baryonic - mass fraction at @xmath177 is predicted to be 0.033 (col. [3]), while at 2-@xmath80 white et al. (1993) require that this be at least 0.039 (at @xmath179), so again this @xmath177 model just fails this test. given the observational uncertainties, it might be possible to make minor adjustments to model - parameter values so that an @xmath180 model with @xmath181 gyr and @xmath182 is just consistent with the observational data. however, it is clear that current observational data do not favour an open model with @xmath183 the observed cluster @xmath184 $] favours a larger @xmath1 while the observed cluster baryonic - mass fraction favours a smaller @xmath1, and so are in conflict. table 10 gives the predictions for the @xmath121 gyr, @xmath185 models. this value of @xmath63 is consistent with the 2-@xmath80 range determined from standard nucleosynthesis and the observed @xmath145he and @xmath146li abundances : @xmath148 (fkot, also see copi et al. 1995 ; sarkar 1996). it is, however, somewhat difficult to reconcile @xmath8 with the 2-@xmath80 range derived from the observed @xmath145he, @xmath146li, and current high d abundances @xmath152 (fkot), or with that from the current observed low d abundances @xmath186 (burles & tytler 1996). in any case, the observed d abundances are still under discussion, and must be viewed as preliminary. in this case, open - bubble inflation models with @xmath187 are consistent with the observational constraints. the current central observational data values for @xmath157 and @xmath134 favour @xmath74, while that for the cluster baryonic - mass fraction prefers @xmath188, and that for @xmath130 $] favours @xmath189, so in this case the agreement between predictions and observational data is fairly impressive (although the tanvir et al. 1995 central @xmath2 value favours @xmath190). note that in this case models with @xmath191 are quite inconsistent with the data. table 11 gives the predictions for @xmath192 gyr, @xmath193 models. this baryonic - mass density value is consistent with that determined from the current observed low d abundances, but is difficult to reconcile with the current standard nucleosynthesis interpretation of the observed @xmath145he and @xmath146li abundances (cardall & fuller 1996). the larger value of @xmath63 (and smaller value of @xmath2) has now lowered small - scale power in mass fluctuations somewhat significantly, opening up the allowed @xmath1 range to larger values. models with @xmath194 are consistent with the observational data, although the higher @xmath1 part of the range is starting to conflict with what is determined from the small - scale dynamical estimates, and the models do require a somewhat low @xmath2 (but not yet inconsistently so at the 2-@xmath80 significance level while the tanvir et al. 1995 central @xmath2 value requires @xmath100, at 2-@xmath80 the @xmath2 constraint only requires @xmath195). the central observational values for @xmath157, the cluster baryonic - mass fraction, @xmath26 $], and @xmath134 favour @xmath97, so the agreement with observational data is fairly impressive, and could even be improved by reducing @xmath62 a little to raise @xmath2. table 12 gives the predictions for another part of model - parameter space. here we show @xmath122 models (at @xmath121 gyr), consistent with the central value of @xmath63 determined from standard nucleosynthesis using the observed @xmath145he, @xmath146li, and high d abundances (fkot). the larger value of @xmath63 (compared to table 9) eases the cluster baryonic - mass fraction constraint, which now requires only @xmath196. the increase in @xmath63 also decreases the mass fluctuation amplitude, making it more difficult to argue for @xmath177 ; however, models with @xmath197 seem to be consistent with the observational constraints when @xmath4 and @xmath198 gyr. it is interesting that in this case the central observational data values we consider for @xmath157, for @xmath26 $], and for @xmath134 prefer @xmath9 ; however, that for the cluster baryonic - mass fraction (as well as that for @xmath2) favours @xmath190 (although at 2-@xmath80 the cluster baryonic - mass fraction constraint only requires @xmath196). hence, while @xmath199 open - bubble inflation models with @xmath200 and @xmath198 gyr are quite consistent with the observational constraints, in this case the agreement between predictions and observations is not spectacular. note that in this case models with @xmath201 are quite inconsistent with the observational data. in summary, open - bubble inflation models based on the cdm picture (rp94 ; bgt ; yst) are reasonably consistent with current observational data provided @xmath202. the flat - space scale - invariant spectrum open model (w83) is also reasonably compatible with current observational constraints for a similar range of @xmath1. the uncertainty in current estimates of @xmath63 is one of the major reasons why such a large range in @xmath1 is consistent with current observational constraints. our previous analysis of the dmr two - year data led us to conclude that only those open - bubble inflation models near the lower end of the above range (@xmath203) were consistent with the majority of observations (grsb). the increase in the allowed range to higher @xmath1 values @xmath204 can be ascribed to a number of small effects. specifically, these are : (1) the slight downward shift in the central value of the dmr four - year normalization relative to the two - year one (g96) ; (2) use of the full 2-@xmath80 range of normalizations allowed by the dmr data analysis (instead of the 1-@xmath80 range allowed by the galactic - frame quadrupole - excluded dmr two - year data set used previously) ; (3) use of the 2-@xmath80 range of the small - scale dynamical estimates of @xmath1 instead of the 1-@xmath80 range used in our earlier analysis ; (4) we consider a range of @xmath63 values here (in grsb we focussed on @xmath8) ; and (5) we consider a range of @xmath62 values here (in grsb we concentrated on @xmath121 gyr). we emphasize, however, that the part of parameter space with @xmath205 is only favoured if @xmath63 is large (@xmath206), @xmath2 is low @xmath207), and the small - scale dynamical estimates of @xmath1 turn out to be biased somewhat low. the observational results we have used to constrain model - parameter values in the previous sections are the most robust currently available. in addition, there are several other observational results which we do not consider to be as robust, and any conclusions drawn from these should be treated with due caution. in this section we summarize several of the more tentative constraints from more recent observations. in our analysis of the dmr two - year data normalized models, we compared model predictions for the rms value of the smoothed peculiar velocity field to results from the analysis of observational data (bertschinger et al. we do not do so again here since, given the uncertainties, the conclusions drawn in grsb are not significantly modified. in particular, comparison of the appropriate quantities implies that we can treat the old 1-@xmath80 upper limits essentially as 2-@xmath80 upper limits for the four - year analysis. in grsb we used @xmath134 determined by nusser & davis (1994), @xmath208 (2-@xmath80), to constrain the allowed range of models to @xmath209. here we use the cole et al. (1995) estimate, @xmath210 (2-@xmath80), which, for the models of table 10, requires @xmath211. this value is just slightly below the lower limit (@xmath212) derived from the bertschinger et al. (1990) results in grsb. we hence conclude that the large - scale flow results of bertschinger et al. (1990) indicates a lower 2-@xmath80 limit on @xmath1 that is about @xmath213 higher than that suggested by the redshift - space distortion analysis of cole et al. (1995).$].] we however strongly emphasize that the central value of the large - scale flow results of bertschinger et al. (1990) does favour a significantly larger value of @xmath1 than the rest of the data we have considered here. furthermore, as discussed in detail in grsb, there is some uncertainty in how to properly interpret large - scale velocity data in the open models, particularly given the large sample variance associated with the measurement of a single bulk velocity (bond 1996, also see llrv). a more careful analysis, as well as more observational data, is undoubtedly needed before it will be possible to robustly conclude that the large - scale velocity data does indeed force one to consider significantly larger values of @xmath1 than is favoured by the rest of the observational constraints (and hence rules out the models considered here). it might be significant that on comparing the mass power spectrum deduced from a refined set of peculiar velocity observations to the galaxy power spectrum determined from the apm survey, kolatt & dekel (1995) estimate that for the optically - selected apm galaxies @xmath214 with a 2-@xmath80 range, @xmath215 (note that it has been argued that systematic uncertainties preclude a believable determination of @xmath134 from a comparison of the observed large - scale peculiar velocity field to the @xmath133 1.2 jy galaxy distribution, davis, nusser, & willick 1996.) this range is consistent with other estimates now under discussion. the stromlo - apm comparison of loveday et al. (1996) indicates @xmath216, with a 2-@xmath80 upper limit of 0.75, while baugh (1996) concludes that @xmath217 (2-@xmath80), and ratcliffe et al. (1996) argue for @xmath218. using the apm range for @xmath163 $], (18), the kolatt & dekel (1995) estimate of @xmath219, eq. (22), may be converted to an estimate of @xmath167, and at 2-@xmath80, @xmath220 it is interesting that at @xmath69 the lower part of this range is consistent with that determined from the cluster x - ray temperature function data, eq. (19), although at lower @xmath1 eq. (23) indicates a larger value then does eq. (19) because of the steeper rise to low @xmath1. zaroubi et al. (1996) have constrained model - parameter values by comparing large - scale flow observations to that predicted in the dmr two - year data normalized open - bubble inflation model. they conclude that the open - bubble inflation model provides a good description of the large - scale flow observations if, at 2-@xmath80, @xmath221 from table 12 we see that an open - bubble inflation model with @xmath222 and @xmath223 provides a good fit to all the observational data considered in @xmath224. for @xmath223 zaroubi et al. (1996) conclude that at 2-@xmath80 @xmath225 (eq. [24]), just above our value of @xmath222. since the zaroubi et al. (1996) analysis does not account for the uncertainty in the dmr normalization (t. kolatt, private communication 1996), it is still unclear if the constraints from the large - scale flow observations are in conflict with those determined from the other data considered here (and so rule out the open - bubble inflation model). it might also be significant that on somewhat smaller length scales there is support for a smaller value of @xmath1 from large - scale velocity field data (shaya, peebles, & tully 1995). the cluster peculiar velocity function provides an alternate mechanism for probing the peculiar velocity field (e.g., croft & efstathiou 1994 ; moscardini et al. 1995 ; bahcall & oh 1996). bahcall & oh (1996) conclude that current observational data is well - described by an @xmath177 flat-@xmath18 model with @xmath226 and @xmath227 = 0.67 $]. this normalization is somewhat smaller than that indicated by the dmr data (e.g., ratra & sugiyama 1995). while bahcall & oh (1996) did not compare the cluster peculiar velocity function data to the predictions of the open - bubble inflation model, approximate estimates indicate that this data is consistent with the open - bubble inflation model predictions for the range of @xmath1 favoured by the other data we consider in @xmath228 see the @xmath26 $] values for the allowed models in tables 912. bahcall & oh (1996) also note that it is difficult, if not impossible, to reconcile the cluster peculiar velocity observations with what is predicted in high density models like fiducial cdm and mdm. at fixed @xmath113 $], low - density cosmogonies form structure earlier than high density ones. thus observations of structure at high redshift may be used to constrain the matter density. as benchmarks, we note that scaling from the results of the numerical simulations of cen & ostriker (1993), in a open model with @xmath229 = 0.8 $] galaxy formation peaks at a redshift @xmath230 when @xmath231 and at @xmath232 when @xmath72. thus the open - bubble inflation model is not in conflict with observational indications that the giant elliptical luminosity function at @xmath233 is similar to that at the present (e.g., lilly et al. 1995 ; glazebrook et al. 1995 ; i m et al. 1996), nor is it in conflict with observational evidence for massive galactic disks at @xmath233 (vogt et al. these models can also accommodate observational evidence of massive star - forming galaxies at @xmath234 (cowie, hu, & songaila 1995), as well as the significant peak at @xmath235 in the number of galaxies as a function of (photometric) redshift found in the hubble deep field (gwyn & hartwick 1996), and it is not inconceivable that objects like the @xmath236 protogalaxy " candidate (yee et al. 1996 ; ellingson et al. 1996) can be produced in these models. it is, however, at present unclear whether the open - bubble inflation model can accommodate a substantial population of massive star - forming galaxies at @xmath237 (steidel et al. 1996 ; giavalisco, steidel, & macchetto 1996), and if there are many more examples of massive damped lyman@xmath238 systems like the one at @xmath239 (e.g., lu et al. 1996 ; wampler et al. 1996 ; fontana et al. 1996), then, depending on the masses, these might be a serious problem for the open - bubble inflation model. on the other hand, the recent discovery of galaxy groups at @xmath240 (e.g., francis et al. 1996 ; pascarelle et al. 1996) probably do not pose a serious threat for the open - bubble inflation model, while massive clusters at @xmath241 (e.g., luppino & gioia 1995 ; pell et al. 1996) can easily be accommodated in the model. it should be noted that in adiabatic @xmath69 models normalized to fit the present small - scale observations, e.g., fiducial cdm (with a normalization inconsistent with that from the dmr), or mdm, or tilted cdm (without a cosmological constant), it is quite difficult, if not impossible, to accommodate the above observational indications of early structure formation (e.g., ma & bertschinger 1994 ; ostriker & cen 1996). with the recent improvements in observational capabilities, neoclassical cosmological tests hold great promise for constraining the world model. it might be significant that current constraints from these tests are consistent with that region of the open - bubble inflation model parameter space that is favoured by the large - scale structure constraints. these tests include the @xmath142 elliptical galaxy number counts test (driver et al. 1996), an early application of the apparent magnitude - redshift test using type ia supernovae (perlmutter et al. 1996), as well as analyses of the rate of gravitational lensing of quasars by foreground galaxies (e.g., torres & waga 1996 ; kochanek 1996). it should be noted that these tests are also consistent with @xmath69 models, and plausibly with a time - variable cosmological constant " dominated spatially - flat model (e.g., ratra & quillen 1992 ; torres & waga 1996), but they do put pressure on the flat-@xmath18 cdm model. smaller - scale cmb spatial anisotropy measurements will eventually significantly constrain the allowed range of model - parameter values. fig. 24 compares the 1-@xmath80 range of cmb spatial anisotropy predictions for a few representative open - bubble inflation (as well as flat - space scale - invariant spectrum open) models to available cmb spatial anisotropy observational data. from a preliminary comparison of the predictions of dmr two - year data normalized open - bubble inflation models to available cmb anisotropy observational data, ratra et al. (1995) concluded that the range of parameter space for the open - bubble inflation model that was favoured by the other observational data was also consistent with the small - scale cmb anisotropy data. this result was quantified by grs, who also considered open - bubble inflation models normalized to the @xmath1351-@xmath80 values of the dmr two - year data (and hence considered open - bubble inflation models normalized at close to the dmr four - year data value, see figs. 5 and 6 of grs). grs discovered that (given the uncertainties associated with the smaller - scale measurements) the 1-@xmath80 uncertainty in the value of the dmr normalization precludes determination of robust constraints on model - parameter values, although the range of model - parameter space for the open - bubble inflation model favoured by the analysis here was found to be consistent with the smaller - scale cmb anisotropy observations, and @xmath93 open - bubble inflation models were not favoured by the smaller - scale cmb anisotropy observational data (grs, figs. 5 and 6). is favoured, but even at 1-@xmath80 @xmath242 is allowed this broad range is consistent with the conclusion of grs that it is not yet possible to meaningfully constrain cosmological - parameter values from the cmb anisotropy data alone. note also that hancock et al. (1996b) do not consider the effects of the systematic shifts between the various dmr data sets, and also exclude a number of data points, e.g., the four msam points and the max3 mup point (which is consistent with the recent max5 mup result, lim et al. 1996), which do not disfavour a lower value of @xmath1 for the open - bubble inflation model (ratra et al. 1995 ; grs).] a detailed analysis of the ucsb south pole 1994 cmb anisotropy data (gundersen et al. 1995) by ganga et al. (1996a) reaches a similar conclusion : at 1-@xmath80 (assuming a gaussian marginal probability distribution) the data favours open - bubble inflation models with @xmath243, while at 2-@xmath80 the ucsb south pole 1994 data is consistent with the predictions of the open - bubble, flat-@xmath18, and fiducial cdm inflation models. Discussion and conclusion we have compared the dmr 53 and 90 ghz sky maps to a variety of open model cmb anisotropy angular spectra in order to infer the normalization of these open cosmogonical models. our analysis explicitly quantifies the small shifts in the inferred normalization amplitudes due to : (1) the small differences between the galactic- and ecliptic - coordinate sky maps ; (2) the inclusion or exclusion of the @xmath85 moment in the analysis ; and, (3) the faint high - latitude galactic emission treatment. we have defined a maximal 2-@xmath80 uncertainty range based on the extremal solutions of the normalization fits, and a maximal 1-@xmath80 uncertainty range may be defined in a similar manner. for this maximal 1-@xmath80 @xmath21 range the fractional 1-@xmath80 uncertainty, at fixed @xmath10 and @xmath2 (but depending on the assumed cmb anisotropy angular spectrum and model - parameter values), ranges between @xmath244 and @xmath245 (statistical and systematic) uncertainty of bw (footnote 4, also see bunn, liddle, & white 1996), @xmath246, is smaller than the dmr four - year data 1-@xmath80 uncertainty estimated in, e.g., g96, wright et al. (1996), and here. this is because we explicitly estimate the effect of all known systematic uncertainties for each assumed cmb anisotropy angular spectrum, and account for them, in the most conservative manner possible, as small shifts. (in particular : we do not just account for the small systematic difference between the galactic- and ecliptic - frame maps ; we do not assume that any of the small systematic differences lead to model - independent systematic shifts in the inferred @xmath21 values ; and we do not add the systematic shifts in quadrature with the statistical uncertainty.) since our accounting of the uncertainties is the most conservative possible, our conclusions about model - viability are the most robust possible.]. (compare this to the @xmath247, 1-@xmath80, uncertainty of eq. [19].) since part of this uncertainty is due to the small systematic shifts, the maximal 2-@xmath80 fractional uncertainty is smaller than twice the maximal 1-@xmath80 fractional uncertainty. for the largest possible 2-@xmath80 @xmath21 range defined above, the fractional uncertainty varies between @xmath248 and @xmath249. note that this accounts for intrinsic noise, cosmic variance, and effects (1)(3) above. other systematic effects, e.g., the calibration uncertainty (kogut et al. 1996b), or the beamwidth uncertainty (wright et al. 1994), are much smaller than the effects we have accounted for here. it has also been shown that there is negligible non - cmb contribution to the dmr data sets from known extragalactic astrophysical foregrounds (banday et al. 1996b). by analyzing the dmr maps using cmb anisotropy spectra at fixed @xmath1 but different @xmath2 and @xmath10, we have also explicitly quantified the small shifts in the inferred normalization amplitude due to shifts in @xmath2 and @xmath10. although these shifts do depend on the value of @xmath1 and the assumed model power spectrum, given the other uncertainties, it is reasonable to ignore these small shifts when normalizing the models considered in this work. we have analyzed the open - bubble inflation model, accounting only for the fluctuations generated during the evolution inside the bubble (rp94), including the effects of the fluctuations generated in the first epoch of spatially - flat inflation (bgt ; yst), and finally accounting for the contribution from a non - square - integrable basis function (yst). for observationally viable open - bubble models, the observable predictions do not depend significantly on the latter two sources of anisotropy. the observable predictions of the open - bubble inflation scenario seem to be robust it seems that only those fluctuations generated during the evolution inside the bubble need to be accounted for. as discussed in the introduction, a variety of more specific realizations of the open - bubble inflation scenario have recently come under scrutiny. these are based on specific assumptions about the vacuum state prior to open - bubble nucleation. in these specific realizations of the open - bubble inflation scenario there are a number of additional mechanisms for stress - energy perturbation generation (in addition to those in the models considered here), including those that come from fluctuations in the bubble wall, as well as effects associated with the nucleation of a nonzero size bubble. while current analyses suggest that such effects also do not add a significant amount to the fluctuations generated during the evolution inside the bubble, it is important to continue to pursue such investigations both to more carefully examine the robustness of the open - bubble inflation scenario predictions, as well as to try to find a reasonable particle physics based realization of the open - bubble inflation scenario. as has been previously noted for other cmb anisotropy angular spectra (g96), the various different dmr data sets lead to slightly different @xmath21 normalization amplitudes, but well within the statistical uncertainty. this total range is slightly reduced if one considers results from analyses either ignoring or including the quadrupole moment. the dmr data alone can not be used to constrain @xmath1 over range @xmath14 in a statistically meaningful fashion for the open models considered here. it is, however, reasonable to conclude that when the quadrupole moment is excluded from the analysis, the @xmath9 model cmb anisotropy spectral shape is most consistent with the dmr data, while the quadrupole - included analysis favours @xmath12 (for the open - bubble inflation model in the range @xmath250). current cosmographic observations, in conjunction with current large - scale structure observations compared to the predictions of the dmr - normalized open - bubble inflation model derived here, favour @xmath202. the large allowed range is partially a consequence of the current uncertainty in @xmath10. this range is consistent with the value weakly favoured (@xmath9) by a quadrupole - excluded analysis of the dmr data alone. it might also be significant that mild bias is indicated both by the need to reconcile these larger values of @xmath1 with what is determined from small - scale dynamical estimates, as well as to reconcile the smaller dmr - normalized @xmath251 $] values (for this favoured range of @xmath1) with the larger observed galaxy number fluctuations (e.g., eq. [18]). in common with the low - density flat-@xmath18 cdm model, we have established that in the low - density open - bubble cdm model one may adjust the value of @xmath1 to accommodate a large fraction of present observational constraints. for a broad class of these models, with adiabatic gaussian initial energy - density perturbations, this focuses attention on values of @xmath1 that are larger than the range of values for @xmath10 inferred from the observed light - element abundances in conjunction with standard nucleosynthesis theory. whether this additional cdm is nonbaryonic, or is simply baryonic material that does not take part in standard nucleosynthesis, remains a major outstanding puzzle for these models. we acknowledge the efforts of those contributing to the @xmath0-dmr. @xmath0 is supported by the office of space sciences of nasa headquarters. we also acknowledge the advice and assistance of c. baugh, s. cole, j. garriga, t. kolatt, c. park, l. piccirillo, g. rocha, g. tucker, d. weinberg, and k. yamamoto. rs is supported in part by a pparc grant and kbn grant 2p30401607. 1.fractional differences, @xmath253, between the cmb spatial anisotropy multipole coefficients @xmath56 computed using the two boltzmann transfer codes (and normalized to agree at @xmath254). heavy type is for the open - bubble inflation model spectrum accounting only for perturbations that are generated during the evolution inside the bubble (type [1] spectra above), and light type is for the open - bubble inflation model spectrum now also accounting for perturbations generated in the first epoch of inflation (type [2] spectra). solid lines are for @xmath255 and dashed lines are for @xmath256. these are for @xmath64 and @xmath65. note that @xmath257. 2.(a) cmb anisotropy multipole coefficients for the open - bubble inflation model, accounting only for fluctuations generated during the evolution inside the bubble (rp94, solid lines), and also accounting for fluctuations generated in the first epoch of inflation (bgt ; yst, dotted lines these overlap the solid lines, except at the lowest @xmath1 and smallest @xmath66), for @xmath258 0.1, 0.2, 0.25, 0.3, 0.35, 0.4, 0.45, 0.5, 0.6, 0.8, and 1.0, in ascending order. these are for @xmath121 gyr and @xmath8. the coefficients are normalized relative to the @xmath259 amplitude, and different values of @xmath1 are offset from each other to aid visualization. in (b) are the set of cmb anisotropy spectra for the open - bubble inflation model, accounting only for fluctuations generated during the evolution inside the bubble (rp94), with @xmath255 and @xmath256 for the three different pairs of values (@xmath62, @xmath63) : (@xmath260 gyr, @xmath261), (@xmath262 gyr, @xmath263), and (@xmath264 gyr, @xmath265). spectra in the two sets are normalized to have the same @xmath259, and @xmath63 increases in ascending order on the right axis. 3.cmb spatial anisotropy multipole coefficients for the flat - space scale - invariant spectrum open model (w83). conventions and parameter values are as in the caption of fig. 2 (although only one set of spectra are shown in fig. 3a). fig. 4.cmb spatial anisotropy multipole coefficients for the open - bubble inflation spectrum, also accounting for both fluctuations generated in the first epoch of inflation and that corresponding to a non - square - integrable basis function (yst, solid lines), and ignoring both these fluctuations (rp94, dotted lines). they are, in ascending order, for @xmath258 0.1 to 0.9 in steps of 0.1, with @xmath64 and @xmath65, normalized relative to the @xmath259 amplitude, and different values of @xmath1 are offset from each other to aid visualization. 5.cmb spatial anisotropy multipole coefficients, as a function of @xmath66, for the various spectra considered in this paper, at @xmath255 and @xmath266 (vertically offset). light solid and heavy solid lines show the open - bubble inflation cases accounting for (type [2] spectra above) and ignoring (type [1] spectra, at @xmath256 these completely overlap the type [2] spectra) fluctuations generated in the first epoch of inflation. dashed lines show the open - bubble inflation models, now also accounting for the contribution from the non - square - integrable basis function (type [3] spectra). dotted lines show the flat - space scale - invariant spectrum open model spectra (type [4] spectra). all spectra are for @xmath64 and @xmath65. 6.likelihood functions @xmath79 (arbitrarily normalized to unity at the highest peak at @xmath74) derived from a simultaneous analysis of the dmr 53 and 90 ghz ecliptic - frame data, ignoring the correction for faint high - latitude foreground galactic emission, and excluding the quadrupole moment from the analysis. these are for the @xmath64, @xmath65 models. panel (a) is for the flat - space scale - invariant spectrum open model (w83), (b) is for the open - bubble inflation model accounting only for perturbations generated during the evolution inside the bubble (rp94), and (c) is for the open - bubble inflation model now also accounting for both the fluctuations generated in the first epoch of inflation and those corresponding to a non - square - integrable basis function (yst). 7.likelihood functions @xmath79 (arbitrarily normalized to unity at the highest peak near either @xmath93 or @xmath267), derived from a simultaneous analysis of the dmr 53 and 90 ghz galactic - frame data, accounting for the faint high - latitude foreground galactic emission correction, and including the quadrupole moment in the analysis. conventions and parameter values are as for fig. 6. fig. 8.ridge lines of the maximum likelihood @xmath21 value as a function of @xmath1, for the open - bubble inflation model accounting only for fluctuations generated during the evolution inside the bubble (type [1] spectra), for the eight different dmr data sets considered here, and for @xmath121 gyr, @xmath8. heavy lines correspond to the case when the quadrupole moment is excluded from the analysis, while light lines account for the quadrupole moment. these are for the ecliptic - frame sky maps, accounting for (dashed lines) and ignoring (solid lines) the faint high - latitude foreground galactic emission correction, and for the galactic - frame maps, accounting for (dot - dashed lines) and ignoring (dotted lines) this galactic emission correction. the general features of this figure are consistent with that derived from the dmr two - year data (grsb, fig. 2). fig. 9.ridge lines of the maximum likelihood @xmath21 value as a function of @xmath1, for the flat - space scale - invariant spectrum open model (type [4] spectra), for the eight different dmr data sets, and for @xmath121 gyr, @xmath8. heavy lines correspond to the ecliptic - frame analyses, while light lines are from the galactic - frame analyses. these are for the cases ignoring the faint high - latitude foreground galactic - emission correction, and either including (dotted lines) or excluding (solid lines) the quadrupole moment ; and accounting for this galactic emission correction, and either including (dot - dashed lines) or excluding (dashed lines) the quadrupole moment. the general features of this figure are roughly consistent with that derived from the dmr two - year data (cayn et al. 1996, fig. 3). fig. 10.ridge lines of the maximum likelihood @xmath21 value as a function of @xmath1, for the open - bubble inflation model now also accounting for both the fluctuations generated in the first epoch of inflation (bgt ; yst) and those from a non - square - integrable basis function (yst), for the eight different dmr data sets considered here, and for @xmath64, @xmath65. heavy lines correspond to the cases where the faint high - latitude foreground galactic emission correction is ignored, while light lines account for this galactic emission correction. these are from the ecliptic frame analyses, accounting for (dotted lines) or ignoring (solid lines) the quadrupole moment ; and from the galactic - frame analyses, accounting for (dot - dashed lines) or ignoring (dashed lines) the quadrupole moment. the general features of this figure are consistent with that derived from the dmr two - year data (yb, fig. 2). fig. 11.ridge lines of the maximum likelihood @xmath21 value as a function of @xmath1, for the two extreme dmr data sets, and two different cmb anisotropy angular spectra, showing the effects of varying @xmath62 and @xmath63. heavy lines are for @xmath192 gyr and @xmath268, while light lines are for @xmath123 gyr and @xmath124. two of the four pairs of lines are for the open - bubble inflation model accounting only for fluctuations generated during the evolution inside the bubble (type [1] spectra), either from the ecliptic - frame analysis without the faint high - latitude foreground galactic emission correction and ignoring the quadrupole moment in the analysis (solid lines), or from the galactic - frame analysis accounting for this galactic emission correction and including the quadrupole moment in the analysis (dotted lines). the other two of the four pairs of lines are for the flat - space scale - invariant spectrum open model (type [4] spectra), either from the ecliptic - frame analysis without the faint high - latitude foreground galactic emission correction and ignoring the quadrupole moment in the analysis (dashed lines), or from the galactic - frame analysis accounting for this galactic emission correction and including the quadrupole moment in the analysis (dot - dashed lines). given the other uncertainties, the effects of varying @xmath62 and @xmath63 are fairly negligible. fig. 12.ridge lines of the maximum likelihood @xmath21 value as a function of @xmath1, for the two extreme dmr data sets, for the four cmb anisotropy angular spectra models considered here, and for @xmath64, @xmath65. heavy lines are from the ecliptic - frame sky maps ignoring the faint high - latitude foreground galactic emission correction and excluding the quadrupole moment from the analysis, while light lines are from the galactic - frame sky maps accounting for this galactic emission correction and including the quadrupole moment in the analysis. solid, dotted, and dashed lines show the open - bubble inflation cases, accounting only for the fluctuations generated during the evolution inside the bubble (type [1] spectra, solid lines), also accounting for the fluctuations generated in the first epoch of inflation (type [2] spectra, dotted lines these overlap the solid lines except for @xmath269 and @xmath12), and finally also accounting for the fluctuations corresponding to the non - square - integrable basis function (type [3] spectra, dashed lines). dot - dashed lines correspond to the flat - space scale - invariant spectrum open model (type [4] spectra). 13.conditional likelihood densities for @xmath21, derived from @xmath79 (which are normalized to be unity at the peak, for each dmr data set, cmb anisotropy angular spectrum, and set of model - parameter values). panel (a) is for the open - bubble inflation model accounting only for fluctuations generated during the evolution inside the bubble (type [1] spectra), while panel (b) is for the flat - space scale - invariant spectrum open model (type [4] spectra). the heavy lines are for @xmath255, while the light lines are for @xmath256. two of the four pairs of lines in each panel correspond to the results from the analysis of the galactic - frame maps accounting for the faint high - latitude foreground galactic emission correction and with the quadrupole moment included in the analysis, either for @xmath123 gyr and @xmath124 (dot - dashed lines), or for @xmath192 gyr and @xmath268 (dashed lines). the other two pairs of lines in each panel correspond to the results from the analysis of the ecliptic - frame maps ignoring this galactic emission correction and with the quadrupole moment excluded from the analysis, either for @xmath123 gyr and @xmath124 (dotted lines), or for @xmath192 gyr and @xmath268 (solid lines). given the other uncertainties, the effects of varying @xmath62 and @xmath63 are fairly negligible. 14.conditional likelihood densities for @xmath21 normalized as in the caption for fig. 13. panel (a) is from the analysis of the ecliptic - frame maps ignoring the faint high - latitude foreground galactic emission correction and excluding the quadrupole moment from the analysis, while panel (b) is from the analysis of the galactic - frame maps accounting for this galactic emission correction and including the quadrupole moment in the analysis. these are for @xmath64 and @xmath65. the heavy lines are for @xmath255 and the light lines are for @xmath256. there are eight lines (four pairs) in each panel, although in each panel two pairs almost identically overlap. solid, dotted, and dashed lines show the open - bubble inflation cases, accounting only for the fluctuations generated during the evolution inside the bubble (type [1] spectra, solid lines), also accounting for the fluctuations generated in the first epoch of inflation (type [2] spectra, dotted lines these almost identically overlap the solid lines), and finally also accounting for the fluctuations corresponding to the non - square - integrable basis function (type [3] spectra, dashed lines). dot - dashed lines correspond to the flat - space scale - invariant spectrum open model (type [4] spectra). 15.projected likelihood densities for @xmath1 derived from @xmath79 (normalized as in the caption of fig. panel (a) is for the open - bubble inflation model accounting only for the fluctuations generated during the evolution inside the bubble (type [1] spectra), and panel (b) is for the flat - space scale - invariant spectrum open model (type [4] spectra). two of the curves in each panel correspond to the results from the analysis of the galactic - frame maps accounting for the faint high - latitude foreground galactic emission correction and with the quadrupole moment included in the analysis, for @xmath123 gyr and @xmath124 (dot - dashed lines) and for @xmath192 gyr and @xmath268 (dashed lines). the other two curves in each panel are from the analysis of the ecliptic - frame maps ignoring the galactic emission correction and excluding the quadrupole moment from the analysis, for @xmath123 gyr and @xmath124 (dotted lines) and for @xmath192 gyr and @xmath268 (solid lines). 16.projected likelihood densities for @xmath1 derived from @xmath79 (normalized as in the caption of fig. panel (a) is from the analysis of the ecliptic - frame sky maps ignoring the faint high - latitude foreground galactic emission correction and excluding the quadrupole moment from the analysis. panel (b) is from the analysis of the galactic - frame sky maps accounting for this galactic emission correction and including the quadrupole moment in the analysis. there are four curves in each panel, although in each panel two of them almost overlap. solid, dotted, and dashed lines show the open - bubble inflation cases, accounting only for the fluctuations generated during the evolution inside the bubble (type [1] spectra, solid lines), also accounting for the fluctuations generated in the first epoch of spatially - flat inflation (type [2] spectra, dotted lines these almost exactly overlap the solid lines), and finally also accounting for the fluctuations corresponding to the non - square - integrable basis function (type [3] spectra, dashed lines). dot - dashed lines correspond to the flat - space scale - invariant spectrum open model (type [4] spectra). these are for @xmath64 and @xmath270. 17.marginal likelihood densities [@xmath271 for @xmath1, normalized to unity at the peak, for the open - bubble inflation model accounting only for fluctuations generated during the evolution inside the bubble (rp94), for the eight different dmr data sets, and for @xmath121 gyr, @xmath8. panel (a) is from the ecliptic - frame analyses, and panel (b) is from the galactic - frame analyses. two of the four lines in each panel are from the analysis without the faint high - latitude foreground galactic emission correction, either accounting for (dot - dashed lines) or ignoring (solid lines) the quadrupole moment. the other two lines in each panel are from the analysis with this galactic emission correction, either accounting for (dotted lines) or ignoring (dashed lines) the quadrupole moment. 19.marginal likelihood densities for @xmath1, for the open - bubble inflation model now also accounting for both the fluctuations generated in the first spatially - flat epoch of inflation and those that correspond to the non - square - integrable basis function (yst), computed for @xmath64 and @xmath65. conventions are as in the caption of fig. 20.marginal likelihood densities for @xmath1 (normalized as in the caption of fig. panel (a) is for the open - bubble inflation model accounting only for the fluctuations generated during the evolution inside the bubble (rp94), while panel (b) is for the flat - space scale - invariant spectrum open model (w83). two of the lines in each panel are the results from the analysis of the galactic - frame data sets accounting for the faint high - latitude foreground galactic emission correction and with the quadrupole moment included in the analysis, for @xmath123 gyr and @xmath124 (dot - dashed lines), and for @xmath192 gyr and @xmath268 (dashed lines). the other two lines in each panel are the results from the analysis of the ecliptic - frame data sets ignoring this galactic emission correction and with the quadrupole moment excluded from the analysis, for @xmath272 gyr and @xmath124 (dotted lines), and for @xmath273 gyr and @xmath268 (solid lines). 21.marginal likelihood densities for @xmath1 (normalized as in the caption of fig. 17), computed for @xmath64 and @xmath65. panel (a) is from the analysis of the ecliptic - frame sky maps ignoring the faint high - latitude foreground galactic emission correction and excluding the quadrupole moment from the analysis. panel (b) is from the analysis of the galactic - frame sky maps accounting for this galactic emission correction and including the quadrupole moment in the analysis. there are four lines in each panel, although in each panel two of the lines almost overlap. solid, dotted, and dashed curves are the open - bubble inflation cases, accounting only for the fluctuations generated during the evolution inside the bubble (rp94, solid lines), also accounting for the fluctuations generated in the first epoch of spatially - flat inflation (bgt ; yst, dotted lines these almost identically overlap the solid lines), and finally also accounting for the fluctuations corresponding to the non - square - integrable basis function (yst, dashed lines). dot - dashed curves correspond to the flat - space scale - invariant spectrum open model (w83). 22.fractional differences, @xmath274, as a function of wavenumber @xmath49, between the energy - density perturbation power spectra @xmath58 computed using the two independent numerical integration codes (and normalized to give the same @xmath21). the heavy curves are for the open - bubble inflation model spectrum accounting only for fluctuations that are generated during the evolution inside the bubble (type [1] spectra), and the light curves are for the open - bubble inflation model spectrum now also accounting for fluctuations generated in the first epoch of inflation (type [2] spectra). these are for @xmath255 (solid lines) and @xmath256 (dashed lines), with @xmath64 and @xmath65. 23.fractional energy - density perturbation power spectra @xmath58 as a function of wavenumber @xmath49. these are normalized to the mean of the extreme upper and lower 2-@xmath80 @xmath21 values (as discussed in 3.3). panels (a)(d) correspond to the four different sets of (@xmath62, @xmath84) of tables 912, and each panel shows power spectra for three different models at six values of @xmath1. solid lines show the open - bubble inflation model @xmath58 accounting only for fluctuations generated during the evolution inside the bubble (rp95) ; dotted lines are for the open - bubble inflation model now also accounting for fluctuations generated in the first epoch of inflation (bgt ; yst) ; and, dashed lines are for the flat - space scale - invariant spectrum open model (w83). starting near the center of the lower horizontal axis, and moving counterclockwise, the spectra shown correspond to @xmath258 0.1, 0.2, 0.3, 0.45, 0.6, and 1. note that at @xmath69 all three model spectra are identical and so overlap ; also note that at a given @xmath1 the open - bubble inflation model @xmath58 accounting for the fluctuations generated in the first epoch of inflation (bgt ; yst, dotted lines) essentially overlap those where this source of fluctuations is ignored (rp95, solid lines). panel (a) corresponds to @xmath123 gyr and @xmath124, (b) to @xmath121 gyr and @xmath275, (c) to @xmath192 gyr and @xmath276, and (d) to @xmath121 gyr and @xmath122 (normalized using the results of the dmr analysis of the @xmath123 gyr, @xmath277 models). panel (e) shows the three @xmath64, @xmath65 open - bubble inflation spectra of table 13 at five different values of @xmath1. the spectra are for the open - bubble inflation model accounting only for fluctuations generated during the evolution inside the bubble (rp95, solid lines), also accounting for fluctuations generated in the first epoch of inflation (bgt ; yst, dotted lines), and also accounting for the contribution from the non - square - integrable basis function (yst, dashed lines). starting near the center of the lower horizontal axis and moving counterclockwise, the models correspond to @xmath258 0.1, 0.2, 0.3, 0.5, and 0.9. note that at a given @xmath1 the three spectra essentially overlap, especially for observationally - viable values of @xmath212. the solid triangles represent the redshift - space da costa et al. (1994) ssrs2 + cfa2 (@xmath278 mpc depth) optical galaxies data (and were very kindly provided to us by c. park). the solid squares represent the [@xmath279 weighting] redshift - space results of the tadros & efstathiou (1995) analysis of the @xmath133 qdot and 1.2 jy infrared galaxy data. the hollow pentagons represent the real - space results of the baugh & efstathiou (1993) analysis of the apm optical galaxy data (and were very kindly provided to us by c. baugh). it should be noted that the plotted model mass (not galaxy) power spectra do not account for any bias of galaxies with respect to mass. they also do not account for nonlinear or redshift - space - distortion (when relevant) corrections nor for the survey window functions. it should also be noted that the observational data error bars are determined under the assumption of a specific cosmological model and a specific evolution scenario, i.e., they do not necessarily account for these additional sources of uncertainty (e.g., gaztaaga 1995). we emphasize that, because of the different assumptions, the different observed galaxy power spectra shown on the plots are defined somewhat differently and so can not be directly quantitatively compared to each other. 24.cmb anisotropy bandtemperature predictions and observational results, as a function of multipole @xmath66, to @xmath280. the four pairs of wavy curves (in different linestyles) demarcating the boundaries of the four partially overlapping wavy hatched regions (hatched with straight lines in different linestyles) in panel (a) are dmr - normalized open - bubble inflation model (rp94) predictions for what would be seen by a series of ideal, kronecker - delta window - function, experiments (see ratra et al. 1995 for details). panel (b) shows dmr - normalized cmb anisotropy spectra with the same cosmological parameters for the flat - space scale - invariant spectrum open model (w83). the model - parameter values are : @xmath177, @xmath281, @xmath282, @xmath283 gyr (dot - dashed lines) ; @xmath72, @xmath284, @xmath8, @xmath285 gyr (solid lines) ; @xmath256, @xmath286, @xmath287, @xmath288 gyr (dashed lines) ; and, @xmath69, @xmath289, @xmath8, @xmath290 gyr (dotted lines) for more details on these models see ratra et al. (1995). for each pair of model - prediction demarcation curves, the lower one is normalized to the lower 1-@xmath80 @xmath21 value determined from the analysis of the galactic - coordinate maps accounting for the high - latitude galactic emission correction and including the @xmath291 moment in the analysis, and the upper one is normalized to the upper 1-@xmath80 @xmath21 value determined from the analysis of the ecliptic - coordinate maps ignoring the galactic emission correction and excluding the @xmath85 moment from the analysis. amongst the open - bubble inflation models of panel (a), the @xmath72 model is close to what is favoured by the analysis of table 10, and the @xmath256 model is close to that preferred from the analysis of table 11. the @xmath177 model is on the edge of the allowed region from the analysis of table 12, and the @xmath69 fiducial cdm model is incompatible with cosmographic and large - scale structure observations. a large fraction of the smaller - scale observational data in these plots are tabulated in ratra et al. (1995) and ratra & sugiyama (1995). note that, as discussed in these papers, some of the data points are from reanalyses of the observational data. there are 69 detections and 22 2-@xmath80 upper limits shown. since most of the smaller - scale data points are derived assuming a flat bandpower cmb anisotropy angular spectrum, which is more accurate for narrower (in @xmath66) window functions, we have shown the observational results from the narrowest windows available. the data shown are from the dmr galactic frame maps ignoring the galactic emission correction (grski 1996, open octagons with @xmath292) ; from firs (ganga et al. 1994, as analyzed by bond 1995, solid pentagon) ; tenerife (hancock et al. 1996a, open five - point star) ; bartol (piccirillo et al. 1996, solid diamond, note that atmospheric contamination may be an issue) ; sk93, individual - chop sk94 ka and q, and individual - chop sk95 cap and ring (netterfield et al. 1996, open squares) ; sp94 ka and q (gundersen et al. 1995, the points plotted here are from the flat bandpower analysis of ganga et al. 1996a, solid circles) ; bam 2-beam (tucker et al. 1996, at @xmath293 with @xmath294 spanning 16 to 92, and accounting for the @xmath295 calibration uncertainty, open circle) ; python - g, -l, and -s (e.g., platt et al. 1996, open six - point stars) ; argo (e.g., masi et al. 1996, both the hercules and aries+taurus scans are shown note that the aries+taurus scan has a larger calibration uncertainty of @xmath296, solid squares) ; max3, individual - channel max4, and max5 (e.g., tanaka et al. 1996, including the max5 mup 2-@xmath80 upper limit @xmath297k at @xmath298, lim et al. 1996, open hexagons) ; msam92 and msam94 (e.g., inman et al. 1996, open diamonds) ; wdh13 and wdi, ii (e.g., griffin et al. 1996, open pentagons) ; and cat (scott et al. 1996 cat1 at @xmath299 with @xmath294 spanning 351 to 471, and cat2 at @xmath300 with @xmath294 spanning 565 to 710, both accounting for calibration uncertainty of @xmath301, solid hexagons). detections have vertical 1-@xmath80 error bars. solid inverted triangles inserted inside the appropriate symbols correspond to nondetections, and are placed at the upper 2-@xmath80 limits. vertical error bars are not shown for non - detections. as discussed in ratra et al. (1995), all @xmath302 (vertical) error bars also account for the calibration uncertainty (but in an approximate manner, except for the sp94 ka and q results from ganga et al. 1996a see ganga et al. 1996a for a discussion of this issue). the observational data points are placed at the @xmath66-value at which the corresponding window function is most sensitive (this ignores the fact that the sensitivity of the experiment is also dependent on the assumed form of the sky - anisotropy signal, and so gives a somewhat misleading impression of the multipoles to which the experiment is sensitive see ganga et al. 1996a for a discussion of this issue). excluding the dmr points at @xmath303, the horizontal lines on the observational data points represent the @xmath66-space width of the corresponding window function (again ignoring the form of the sky - anisotropy signal). note that from an analysis of a large fraction of the data (corresponding to detections of cmb anisotropy) shown in these figures, grs (figs. 5 and 6) conclude that all the models shown in panel (a), including the fiducial cdm one, are consistent with the cmb anisotropy data.

cut - sky orthogonal mode analyses of the @xmath0-dmr 53 and 90 ghz sky maps are used to determine the normalization of a variety of open cosmogonical models based on the cold dark matter scenario. to constrain the allowed cosmological - parameter range for these open cosmogonies, the predictions of the dmr - normalized models are compared to various observational measures of cosmography and large - scale structure, viz. : the age of the universe ; small - scale dynamical estimates of the clustered - mass density parameter @xmath1 ; constraints on the hubble parameter @xmath2, the x - ray cluster baryonic - mass fraction @xmath3, and the matter power spectrum shape parameter ; estimates of the mass perturbation amplitude ; and constraints on the large - scale peculiar velocity field. the open - bubble inflation model (ratra & peebles 1994 ; bucher, goldhaber, & turok 1995 ; yamamoto, sasaki, & tanaka 1995) is consistent with current determinations of the 95% confidence level (c.l.) range of these observational constraints. more specifically, for a range of @xmath2, the model is reasonably consistent with recent high - redshift estimates of the deuterium abundance which suggest @xmath4, provided @xmath5 ; recent high - redshift estimates of the deuterium abundance which suggest @xmath6 favour @xmath7, while the old nucleosynthesis value @xmath8 requires @xmath9. small shifts in the inferred @xmath0-dmr normalization amplitudes due to : (1) the small differences between the galactic- and ecliptic - coordinate sky maps, (2) the inclusion or exclusion of the quadrupole moment in the analysis, (3) the faint high - latitude galactic emission treatment, and, (4) the dependence of the theoretical cosmic microwave background anisotropy angular spectral shape on the value of @xmath2 and @xmath10, are explicitly quantified. the dmr data alone do not possess sufficient discriminative power to prefer any values for @xmath1, @xmath2, or @xmath10 at the 95% c.l. for the models considered. at a lower c.l., and when the quadrupole moment is included in the analysis, the dmr data are most consistent with either @xmath11 or @xmath12 (depending on the model considered). however, when the quadrupole moment is excluded from the analysis, the dmr data are most consistent with @xmath13 in all open models considered (with @xmath14), including the open - bubble inflation model. earlier claims (yamamoto & bunn 1996 ; bunn & white 1996) that the dmr data require a 95% c.l. lower bound on @xmath1 (@xmath15) are not supported by our (complete) analysis of the four - year data : the dmr data alone can not be used to meaningfully constrain @xmath1. # 10= 0.00em0 - 0 0.03em0 - 0 0.00em.04em0 - 0 0.03em.04em0 - 00 # 1;#2;#3;#4;#5 # 1 # 2, # 3, # 4, # 5 # 1;#2;#3 # 1 # 2, # 3 # 1;#2;#3 # 1 # 2, # 3 # 1@xmath16 mit - ctp-2548, kuns 1399 0.5 cm august 1996

astro-ph9608054

Introduction the epoch of reionization signified the appearance of the first stars and galaxies within the first billion years after the big bang, and the transformation of the intergalactic medium (igm) from opaque to transparent. despite recent progress, however, it is not yet fully understood. it is now well established that reionization is completed by @xmath14 thanks to observations of the ly@xmath15 forest (e.g. @xcite), and that the universe was substantially ionized around redshift @xmath4 when its age was less than 600 myr, based on the electron scattering optical depth measured by planck @xcite. however, there is still substantial uncertainty regarding the sources of reionization. can galaxies form with sufficient efficiency at such early times to provide enough reionizing photons (e.g. @xcite), or is the process possibly driven by other classes of objects such as agn @xcite? observationally, recent progress in near - ir detector technology has dramatically advanced our ability to search for galaxies during this epoch. following the installation of the wide field camera 3 (wfc3) on the _ hubble space telescope _ (_ hst _), a continuously growing sample of galaxy candidates at @xmath16 is accumulating thanks to a variety of surveys. these range from small - area ultradeep observations such as the hubble ultra - deep field (hudf, @xcite), to shallower, larger - area searches for @xmath17 galaxies either in legacy fields such as the cosmic assembly near - infrared deep extragalactic legacy survey (candels ; @xcite), or taking advantage of random - pointing opportunities like in the brightest of reionizing galaxies (borg) survey (go 11700, 12572, 13767 ; pi trenti). overall, a sample approaching 1000 galaxy candidates at @xmath18 is known today @xcite, and we are beginning to identify the first galaxy candidates from the first 500 million years (@xmath19 ; @xcite). these observations provide solid constraints on the galaxy luminosity function (lf) out to @xmath4, which appears to be overall well described by a @xcite form, @xmath20, as at lower redshift @xcite. however, other studies suggest that bright galaxy formation might not be suppressed as strongly at @xmath16, and either a single power law @xcite or a double power law @xcite fit to the bright end of the lf has been explored. this change in the shape of the bright end is in turn connected theoretically to the physics of star formation in the most overdense and early forming environments where the brightest and rarest galaxies are expected to live @xcite. a departure from a schechter form could indicate a lower efficiency of feedback processes at early times, which in turn would imply an increase in the production of ionizing photons by galaxies. additionally, at @xmath21, the observed number density of bright galaxies is affected by magnification bias @xcite, and this bias can cause the lf to take on a power - law shape at the bright end. currently, the samples at @xmath22 are still too small to draw any conclusion on which scenario is realized, since only a handful of @xmath23 candidates are known. in addition to constraining the shape of the lf, the brightest high-@xmath24 candidates identified by _ hst _ observations are also ideal targets for follow - up observations to infer stellar population properties such as ages and stellar masses @xcite, ionization state of the igm @xcite, and spectroscopic redshift. for the latter, confirmation of photometric candidates relies typically on detection of a lyman break in the galaxy continuum, (e.g., @xcite) and/or of emission lines, primarily lyman-@xmath15 (e.g. @xcite) or other uv lines such as ciii] or civ @xcite. spectroscopic follow - up for sources at @xmath25 is extremely challenging, with only limits on line emission resulting from most observations. yet, the brightest targets show significant promise of detection based on the latest series of follow - ups which led to spectroscopic confirmation out to @xmath26 @xcite, with several other ly@xmath15 detections at @xmath25 @xcite. with the goal of complementing the discovery of the rarest and most luminous sources in the epoch of reionization from legacy fields such as candels, the brightest of reionizing galaxies survey (borg, see @xcite) has been carrying out pure - parallel, random pointing observations with wfc3 since 2010. borg identified a large sample (@xmath27) of @xmath4 @xmath28-band dropouts with @xmath17 (@xcite ; see also @xcite). this represents a catalog of galaxies that is not affected by large scale structure bias (sample or `` cosmic '' variance ; see @xcite), which is especially severe for rare sources sitting in massive dark matter halos (@xmath29), as inferred from clustering measurements at @xmath18 @xcite. follow - up spectroscopy of the borg dropouts with keck and vlt has provided evidence for an increase of the igm neutrality at @xmath4 compared to @xmath30 @xcite. currently, a new campaign of observations is ongoing, with a revised filter - set optimized for the new frontier of redshift detection at @xmath23 (borg[z9 - 10] ; go 13767, pi trenti). initial results from @xmath31 of the dataset (@xmath32 arcmin@xmath5) led to the identification of two candidates at @xmath3 @xcite with @xmath33, which are similar in luminosity to the spectroscopically confirmed @xmath26 source reported by @xcite, but significantly brighter than the six @xmath34-dropouts with @xmath35 identified in the goods / candels fields from a comparable area @xcite. these recent developments indicate that it might be possible for a small number of ultra - bright sources (@xmath36) to be present as early as 500 myr after the big bang. thus, they prompted us to systematically analyze the borg archival data from observations in the previous cycles, which cover @xmath37 arcmin@xmath5, to search for bright @xmath3 candidates and constrain their number density. this paper presents the results of this search, and is organized as follows : section [section2] briefly introduces the borg dataset. section [section3] discusses our selection criteria for @xmath3 sources (@xmath38-band dropouts), with results presented in section [section4]. in section [lfsection], we determine the galaxy uv luminosity function at @xmath39, and compare with previous determinations. section [sec : conclusions] summarizes and concludes. throughout the paper we use the @xcite cosmology : @xmath40, @xmath41 and @xmath42. all magnitudes are quoted in the ab system @xcite. The borg survey we use data acquired as part of the brightest of reionizing galaxies (borg[z8]) survey, which consists of core borg pointings (go 11700, 12572, 12905), augmented by other pure parallel archival data (go 11702, pi yan, @xcite) and cos gto coordinated parallel observations. for an in - depth description of the survey, we refer the reader to @xcite. here, we use the 2014 (dr3) public release of the data, which consists of 71 independent pointings covering a total area of @xmath43350 arcmin@xmath5. all fields were imaged using the wfc3/ir filters f098 m, f125w and f160w, and in the optical @xmath44 band, using either the wfc3 f606w or f600lp filter. we refer to the wfc3 f098 m, f125w and f160w images as the @xmath45, @xmath38 and @xmath46 images, and to the f606w and f600lp images as @xmath47 and @xmath48, respectively. exposure times in each filter vary on a field - by - field basis, and 5@xmath49 limiting magnitudes for point sources and aperture @xmath50 are between @xmath51, with a typical value of @xmath52 @xcite. we note that since the dataset originates from parallel observations when the primary instrument is a spectrograph (cos or stis), there is no dithering of the exposures. to compensate for the lack of dithering, the borg data reduction pipeline has been augmented with a customized laplacian edge filtering algorithm developed by @xcite. overall, the lack of dithering has a minimal impact (@xmath53) on the image and photometric quality, as it has been established through comparison between primary (dithered) versus pure - parallel observations of the same field @xcite. since the borg[z8] survey was designed to have @xmath38 as primary detection band, some fields have only a single short exposure in the @xmath46-band. to ensure a consistently high image quality, here we include in the analysis only those fields with total exposure time @xmath54 in @xmath46. this resulted in the exclusion of 9 fields out of 71, so that the area included in our study is @xmath55 arcmin@xmath5. the distribution of the exposure time in @xmath46 for the fields in borg[z8] is shown in figure [fieldhist]. the borg[z8] public data release consists of reduced and aligned science images produced with ` multidrizzle ` @xcite with a pixel scale of @xmath56, as well as associated weight maps (see @xcite). following our standard analysis pipeline to search for dropouts in the data @xcite, we create rms maps from the weight maps, and normalize them to account for correlated noise induced by ` multidrizzle ` (see @xcite). in short, this is done for each field and filter by measuring the noise in the image at random positions not associated with detected sources (i.e. the `` sky '' noise), and comparing the measurement with the value inferred from the rms map, which can then be corrected by a multiplicative factor to match the measurement. our rescaling factors are on average @xmath57 for the ir filters and @xmath58 for @xmath44 (see also @xcite). in addition, photometric zero - points are corrected to account for galactic reddening along each line of sight, according to @xcite. using ` sextractor ` @xcite in dual - image mode, we construct catalogues of sources in each field, using the @xmath46-band image for detection. colors and signal to noise ratios are defined based on isophotal fluxes / magnitudes (flux_iso), while we adopt mag_auto for the total magnitude of each source. Selection of @xmath38-dropouts to select @xmath19 galaxy candidates, we use the dropout technique @xcite. at high @xmath24, neutral hydrogen in the igm almost completely absorbs uv photons, leading to a break at the galaxy rest wavelength of ly@xmath15 (1216 ). for galaxies between @xmath59, this implies a drop in the @xmath38-filter, and non - detection in the @xmath44 and @xmath45 bands. our focus on @xmath38-dropouts implies that our sample of candidates are essentially detected only in @xmath46. therefore, to minimize the risk of introducing spurious sources, we require a clear detection in @xmath46, with s / n@xmath60. we also impose a strong @xmath61 break, trading sample completeness for higher purity, and require a color cut : @xmath62, which is more conservative than the typical @xmath63 applied to legacy fields (e.g. @xcite) since we do not have the availability of multi - observatory data to constrain the continuum of candidates at longer wavelengths and help control contamination. overall, we impose the following criteria for selection as @xmath38-dropouts : @xmath64 when computing the @xmath65 color, if the @xmath38-band flux has s / n @xmath66 we use the @xmath67 limit instead. finally, to reduce the risk of contamination from detector defects surviving the data reduction pipeline, we further impose a stellarity cut through the ` sextractor ` class_star parameter. we require class_star @xmath68, where 1 corresponds to a point source, and 0 to a diffuse light profile. we then visually inspected the dropouts that meet these criteria to reject any remaining detector artifacts and diffraction spikes. all the sources that meet all criteria and pass the visual inspection are listed in table [table1], and discussed below. Results lccccccccccc borg_0240 - 1857_25 & 40.1195 & -18.9726 & 26.24 @xmath69 0.18 & @xmath70 & 8.1 & @xmath71 & -0.1 & 1.0 & @xmath72 & 0.71 & @xmath8 + borg_0240 - 1857_129 & 40.1289 & -18.9678 & 24.74 @xmath69 0.07 & @xmath73 & 14.5 & 2.5 & 0.6 & 0.9 & @xmath74 & 0.02 & @xmath75 + borg_0240 - 1857_369 & 40.1274 & -18.9612 & 25.22 @xmath69 0.11 & @xmath76 & 9.6 & 2.2 & -1.7 & 0.2 & @xmath77 & 0.00 & @xmath78 + borg_0456 - 2203_1091 & 73.9774 & -22.0372 & 26.09 @xmath69 0.13 & @xmath79 & 8.1 & -1.3 & -0.4 & 0.1 & @xmath80 & 0.51 & @xmath81 + borg_1153 + 0056_514 & 178.1972 & 0.9270 & 26.31 @xmath69 0.24 & @xmath82 & 8.0 & 0.02 & -0.1 & -0.6 & @xmath83 & 0.01 & @xmath84 + borg_1459 + 7146_785 & 224.7239 & 71.7814 & 25.82 @xmath69 0.14 & 1.57 & 12.8 & 3.7 & -1.1 & 1.3 & @xmath85 & 0.91 & @xmath86 + + + + + + we performed a search for @xmath38-dropouts over @xmath55 arcmin@xmath5 of archival borg data. we find six sources that satisfy the @xmath38-dropout selection with s / n@xmath87. the candidates are detected over a range of magnitudes, with four candidates between @xmath88, and two brighter candidates at @xmath89 and @xmath90. at @xmath91, this corresponds to @xmath92 to @xmath93. three candidates are detected only in @xmath46, while the remaining three are detected in both @xmath38 and @xmath46. the photometry of the candidates is reported in table [table1], and postage stamps of @xmath44, @xmath45, @xmath38, and @xmath46 are shown in figure [fig1]. we derive photometric redshifts for these six candidates using the photo-@xmath24 code bpz @xcite, assuming a flat prior on redshift, motivated by the uncertainty in the density of sources at intermediate redshifts with colors similar to those of @xmath22 galaxies. for the single band (@xmath46) detections, the photometric redshift distribution is flat over the range @xmath94. for the two - band (@xmath38 and @xmath46) detections, the photometric redshifts are sharply peaked around @xmath9. the photometric probability distributions are included in figure [fig1] alongside the images of the candidates. a comparison of the apparent @xmath46 magnitude against the photometric redshift of our candidates against @xmath21 candidates from other _ hst_/wfc3 surveys is shown in figure [hvszplot]. while two of our candidates are particularly bright in @xmath46, they are consistent with previously - discovered candidates at @xmath39 by @xcite. we also determine the size of the candidates, starting from the observed half - light (effective) radius as determined by ` sextractor `, which is translated into an intrinsic source size taking into account the effects of the point - spread function (psf) broadening and surface brightness limits following @xcite. the empirical relation has been constructed by inserting and recovering artificial sources with known input size and magnitude into borg images. source size is very helpful to help discriminate between high- and low-@xmath24 sources, since direct measurements by @xcite on candels galaxies show that @xmath95 sources are more compact than @xmath96 contaminants with similar colors. this empirical separation might be related to an approximate scaling of galaxy sizes as @xmath97 with @xmath98 @xcite, although a recent study by @xcite highlights that the intrinsic sizes likely evolve less strongly with redshift (@xmath99) compared to observed sizes. we discuss the contamination of our sample further in section [contamination]. this candidate is the brightest in the sample, with magnitude @xmath100. it is robustly detected in @xmath46 at @xmath101, and marginally detected in @xmath38 at @xmath102, even though it is close to the edge of the chip. the source has a very red @xmath103 color, with @xmath104. it also shows extended structure, and has @xmath105. its photometric redshift solution is sharply peaked at @xmath106, with a broad higher - redshift wing. this candidate, in the same field as the previous one is detected with magnitude @xmath107, making it the second - brightest source in the sample. it is detected with s / n @xmath108 in @xmath46, and again marginally detected with s / n @xmath109 in @xmath38. it is the most extended source in the sample, with @xmath110. its photometric redshift, like borg_0240 - 1857_129, is peaked at @xmath111, with a broad higher - redshift wing. field borg_0240 - 1857 includes a third bright candidate with @xmath112, detected at s / n @xmath113. this source is not detected in the other bands (@xmath38, @xmath45 or @xmath44@xmath114). unlike the two brighter candidates, this object is more compact, with a measured half - light radius @xmath115. this is smaller than the psf of the image (@xmath116), indicating that it could be a point - source - like contaminant such as a cool dwarf star, although the stellarity of this source is @xmath117, which is lower than the value expected for a point source (e.g. @xcite uses class_star @xmath118 and @xcite class_star @xmath119 to exclude stars). this candidate is close to a foreground galaxy with @xmath120, with a centre - to - centre projected separation of @xmath121. while this foreground galaxy has an uncertain photometric redshift solution, it is likely to be at @xmath122, based on its compact size. using the framework developed by @xcite and @xcite, we estimate the gravitational lensing of this source. magnification pdfs are obtained by estimating velocity dispersions from @xmath46 magnitudes, using the empirical redshift - dependent faber - jackson relations given in @xcite and @xcite. velocity dispersion is the best tracer of the strength of a strong gravitational lens @xcite. the einstein radii of the foreground objects are modelled as singular isothermal spheres (e.g. @xcite) which depend on the velocity dispersion and the angular diameter distance to the source, and between the lens and source (where we use the best photo-@xmath24 values). assuming that the foreground source is at @xmath123 (which maximizes lensing magnification), we infer a magnification @xmath124. this object has a magnitude @xmath125 (@xmath126), and is detected in the @xmath46 only, with an extended but compact structure (effective radius @xmath127). the source is located relatively close (@xmath128 separation) to a hot pixel, which appears in the @xmath45 and @xmath38 images. the @xmath46-band image is unaffected since it was acquired in a later orbit than the images in bluer bands. we carefully examined the individual flt files and conclude that since the separation between the source center and the hot pixel is larger than twice @xmath129, and there is no sign of a hot pixel in the @xmath46-band, the identification of the candidate as a @xmath38-dropout is robust. this candidate is detected with a magnitude @xmath130, and has s / n @xmath131. it is not detected in @xmath38, @xmath45 or @xmath44. it has an effective radius of @xmath132. this candidate is close to a foreground object (@xmath133 centre - to - centre projected separation). the foreground object has an apparent magnitude @xmath134, and is at an indeterminate photometric redshift. we use the same modelling framework as for borg_0240 - 1857_25 to estimate the lensing magnification of this source. assuming that the source is at @xmath123, we find a maximum @xmath135. analysis of the flt images of this field highlighted the presence of a bad pixel, correctly identified and masked by the data reduction pipeline, at the outer edge of the segmentation map of the dropout candidate in one of the two @xmath46 frames. to determine the impact on the final photometry, we measured the source flux in the flt frames separately, finding that the candidate is detected with s / n @xmath136 in the unaffected image and also s / n @xmath137 in the image affected by the bad pixel. hence, we are confident that the source is real and that the photometry from the final drizzled image is robust. the sixth and final candidate is confidently detected at s / n@xmath138 in @xmath46 (@xmath120), and also in the @xmath38 with s / n = 3.7. its photometric redshift is sharply peaked at @xmath139, with a secondary solution at @xmath140. this candidate is also very compact, with measured half - light radius @xmath141, and the highest stellarity of the sample (class_star = 0.91). combining compactness with high stellarity from a high s / n source, a stellar nature (cool dwarf) for this source is relatively likely, as we discuss in section [contamination]. Number density and luminosity function of @xmath142 galaxies to translate the results on the search of possible candidates at @xmath3 from the archival borg[z8] data into a number density / luminosity function determination, we need to assess both the impact of contamination in our sample, and the effective volume probed by the data. there are multiple classes of lower-@xmath24 sources that may have similar @xmath103 colors to @xmath19 lyman - break galaxies (lbgs), such as galactic stars, intermediate - redshift passive galaxies, and strong line emitters. cool, red stars in the milky way may be possible contaminants of our sample, although typical colors lack a strong @xmath103 drop. at low signal - to - noise ratio, the separation of point - like galactic stars from resolved galaxies using the ` sextractor ` class_star parameter is not fully reliable. we use class_star @xmath143 in our selection of @xmath38-dropouts in section [section3] to reject artifacts remaining from the reduction process, but this is not a strict enough criterion to reject all stars from our sample. in this case, five of our candidates identified as @xmath38-dropouts have class_star @xmath144, with only borg_1459 + 7146_785 having class_star @xmath145 (a value considered by @xcite as indicative of a stellar nature). therefore we conclude that this source is most likely a stellar contaminant with unusual colors. emission - line galaxies are another source of contamination for @xmath19 galaxy samples. for example, galaxies at @xmath146 with strong [oii] emission may appear bright in @xmath46-band while the galaxy continuum is too faint to be detected in the other bands @xcite. @xcite find that, at @xmath147, the average density of extreme line emitters that enter the photometric selection is @xmath148 per arcmin@xmath149, by creating mock catalogs of extreme emission line galaxies with varying @xmath38 magnitude and spectral slope @xmath150. extrapolating this result to @xmath39, we expect to find @xmath151 potential contaminants of this type over our survey area. this value is in line with previous spectroscopic observations of @xmath147 borg candidates by @xcite using the mosfire spectrograph on the keck telescope, and by @xcite using xshooter on the very large telescope. these studies found no emission lines in the spectroscopy of @xmath147 candidates, and are able to rule out emission lines from a low-@xmath24 extreme emission line contaminant to @xmath152, assuming that all of the @xmath46 flux is due to a strong emission line. @xcite also find that, with a 3hr exposure, only a small part of the spectrum (@xmath153) is so affected by atmospheric transmission and absorption by oh lines that a strong emission line would not be detected to @xmath13. the last, and probably most severe class of contaminants, is that of passive and dusty galaxies that thus show a strong balmer break and a very faint uv continuum. under these conditions, @xmath154 sources may mimic properties of lbgs and thus enter into our selection. observations with _ spitzer_/irac at @xmath155 and @xmath156 m can efficiently distinguish between high- and low - redshift sources. in fact, dusty @xmath123 galaxies will appear 1 - 2 magnitudes brighter in [3.6] and [4.5] than in @xmath46, while @xmath19 lbgs will have a relatively flat spectrum. without spitzer data, we rely on the size of the sources as proxy for the @xmath157 $] color, considering sources with @xmath158 as likely contaminants. @xcite find that, while the mean size of candidates in the @xmath39 sample from @xcite is @xmath72, low - redshift, irac - red interlopers have a mean size of @xmath159, but can be as small as @xmath160, and there are no high-@xmath24 candidates with sizes greater than @xmath161 (figure 4, @xcite). hence, we take @xmath162 as a threshold. the two brightest sources in our sample are so extended to fall into such classification. the sources considered in @xcite are fainter than the @xmath39 candidates in our sample, and so it is conceivable that the larger sizes of borg_0240 - 1857_129 and borg_0240 - 1857_369 are due to their higher luminosities. using the size - luminosity relation derived in @xcite, we infer that the size of a @xmath163 galaxy at @xmath164 (the brightest in our sample) would be @xmath165, below our threshold of @xmath162. this size can not be used to completely reject extreme emission - line galaxies, which are likely to be more compact. for example, @xcite find that their sample of 52 extreme emission line galaxies in clash have fwhm @xmath166, similar to our @xmath167 criterion for @xmath39 candidates. in addition to finding the redshift probability distribution of our six candidates using bpz, we also fit sed templates described in @xcite. from this, we find an average probability @xmath168 of @xmath169. we conclude that three out of the six candidates, borg_1459 + 7146_785, borg_0240 - 1857_129, and borg_0240 - 1857_369, are likely to be contaminants. for the remaining three, contamination is still a possibility, and hence we make a conservative assumption of @xmath170 contamination (two out of three sources at @xmath171), based on the estimate from the borg[z9 - 10] survey @xcite using the average probability @xmath168 for the candidates in their sample. of the six possible candidates identified in the full borg[z8] survey, three of them, including the two brightest, are found in the same field (borg_0240 - 1857). the exposure time for this pointing is similar to the median of the survey (@xmath172 s in @xmath46), and thus such a catch is highly unlikely under the assumptions of a uniform distribution of candidates in the 62 fields analyzed here. based on theoretical expectations, the presence of clustering can be used to verify the identity of bright high-@xmath24 sources @xcite, under the broad assumption that uv luminosity is correlated with dark - matter halo mass (e.g., @xcite). overdensities have also been identified at @xmath4 in lbg samples @xcite. however, one alternative possibility, more consistent with the relatively large size of the two bright @xmath34-dropouts, would be the presence of an overdensity of passive and dusty satellite galaxies within an intermediate redshift group. in either case, a further exploration of this configuration is very interesting since it can either identify an exciting overdensity of unexpectedly bright sources at @xmath11, or shed light on the properties of intermediate redshift galaxies with extreme @xmath173 colors. we perform source recovery simulations to determine the efficiency and completeness of our selection, following @xcite. to do this, we insert and recover artificial galaxies with a srsic luminosity profile in the images. half of the sources follow a de vaucouleurs profile (srsic index @xmath174), and the other half follow an exponential profile (srsic index @xmath175), spanning a range of magnitudes (@xmath176), redshifts (@xmath177) and sizes (logarithmic distribution with mean 0175 at @xmath178, scaling as @xmath179). the spectra of the sources are modeled as power law @xmath180 with @xmath181 (gaussian distribution) with a sharp cut - off at rest - frame @xmath182. the intrinsic profiles of the artificial sources are convolved with the wfc3 psf for each corresponding filter, before being inserted into the borg science images at random positions. sources are then identified with, and the statistics of the recovery rate is quantified. this is through the definition of @xmath183 which is the completeness of the source detection, that is the probability of recovering an artificial source of magnitude @xmath184 in the image, and of @xmath185, which is the probability of identifying an artificial source of magnitude @xmath184 and redshift @xmath24 within the dropout sample, assuming that the source is detected. one example of the selection function for the dropout search in field borg_0440 - 5244 is shown in the bottom panel of figure [figvolume], while the upper panel of the same figure shows the overall effective volume probed by our search over all borg archival fields as a function of the apparent @xmath46 magnitude. + recovered from simulations, as a function of the apparent @xmath46 magnitude. bottom panel : an example of the selection function @xmath185 for field borg_0440 - 5244. the selection function is derived from simulations, by inserting and recovering artificial sources. [figvolume],title="fig:",scaledwidth=46.0%] from the discussion in section [contamination], we consider the two brightest sources to be likely contaminants because of their large half - light radii, and we exclude the point - like source borg_1459 + 7146_785 as well. for the surviving three candidates we assume a contamination rate of @xmath170, e.g. we expect two sources to be at @xmath3. after taking into account the effective volume probed by our selection, our estimates for the bright - end of the luminosity function at @xmath3 is reported in table [table3], and shown in figure [figlf]. interestingly, we infer a higher number density of bright sources than previous determinations by @xcite around @xmath186, although the uncertainty is very large because of the small number of candidates. for brighter sources (@xmath187), our upper limits on @xmath3 density are similar to those obtained in legacy fields, and strengthen the evidence for suppression of the abundance of galaxies at the bright - end of the luminosity function. when compared to the initial results from the ongoing borg[z9 - 10] survey @xcite, assuming that our two brightest sources are low - redshift contaminants, we do not find evidence of ultra - bright (@xmath188) galaxies despite analyzing data covering more than twice the area. if follow - up observations of our brightest sources indicate that they are likely at high redshift, we would instead determine that the lf is higher at the bright end than the upper limit from @xcite, and is instead consistent with the determination by @xcite at @xmath189. overall, our lf determination is higher, but still consistent at @xmath190 with the theoretical model of @xcite, shown as grey shaded area in figure [figlf]. previous studies did not attempt unconstrained fits to the @xmath3 lf, likely because of the small number of candidates. to evaluate the status of the situation with our additional datapoints, we attempt to derive schechter parameters for a maximum likelihood fit to the stepwise lf data, carried out assuming a poisson distribution for the number of galaxies expected in each bin (see @xcite). due to the non - detection at @xmath191 by @xcite, the lf is suppressed at the faint end. this leads to a likelihood landscape that is very flat over a wide region of the parameter space, and hence, we are unable to sufficiently constrain the schechter parameters. our fit attempt thus highlights that the dataset is still too small for tight quantitative constraints, but future growth in the number of candidates identified will allow rapid improvements. finally, we note that our conclusions rest on the assumption that the two brightest candidates we identified in field borg_0240 - 1857 are contaminants. if we were to include them in the analysis as @xmath3 sources, we would infer that the lf would favor a power law at the bright end, rather than a schechter form. evidence for a single or double power - law form at high redshift has been seen in determinations of the lf at @xmath192 (@xcite, also earlier considered by @xcite), and potentially at @xmath11 by @xcite, and may be naturally interpreted as a decrease in mass quenching from processes such as agn feedback at high redshift @xcite. magnification bias, however, can also produce this effect on an otherwise schechter - like lf. thus, the astrophysical interpretation of our search ultimately rests on follow - up observations to establish the nature of the candidates borg_0240 - 1857_129 and borg_0240 - 1857_369. in any case, it is very interesting to note that the number of potential candidate @xmath34-dropouts that we identified is small (just six in over 60 fields), making further observations time - efficient, especially because half of the sources are clustered in a single pointing. cc @xmath193 & @xmath194 + @xmath195 & @xmath196 + @xmath197 & @xmath198 Conclusions in this paper we presented a search for @xmath19 candidates in archival data of the 2010 - 2014 brightest of reionizing galaxies (borg[z8]) survey, a pure - parallel optical and near - infrared survey using _ hst_/wfc3. while the survey was optimized to identify @xmath4 sources as @xmath45-dropouts, we searched over the deepest 293 arcmin@xmath5 of the survey for @xmath38-dropout sources with @xmath199, motivated by recent identification of very bright sources with @xmath200. our key results are : * we identify six @xmath39 galaxy candidates, detected in @xmath46 at s / n @xmath6 and satisfying a conservative @xmath65 color selection with non - detection in bluer bands of the survey. the candidate s magnitudes vary from @xmath201 to @xmath202. analysis of the surface brightness profile leads to the tentative identification of three contaminants, with the two brightest sources likely being intermediate redshift passive galaxies due to their size, and one faint source a galactic cool dwarf star because of the compact size and high stellarity. * of the six candidates, three are in the same field, borg_0240 - 1857, including the two brightest of the sample. such strong clustering would be naturally explained if the sources were @xmath3 (see @xcite), despite contrary indication from @xmath129, but an alternative explanation of sub - halo clustering at intermediate redshift would also be viable. * based on our best estimate of the lf, we infer a higher galaxy number density for sources at @xmath186 compared to the observations of bouwens et al. (2015a, b) and with the theoretical model of mason et al. (2015b). however, our measurement is still consistent at the @xmath13 level with these studies. * irrespective of the nature of the two brightest sources in the sample, the selection criteria that we adopted yield a small number of candidates, very manageable for follow - up observations. this is quite remarkable, since the borg[z8] survey was not designed with @xmath11 in mind, and the number of contaminants could have been much larger given the absence of a second detection band and the lack of a near - uv color to help remove passive and dusty intermediate redshift galaxies. * targeted follow - up observations can efficiently clarify the nature of the candidates we identified, help to further constrain the bright - end of the lf and characterize the properties of the yet unstudied population of compact intermediate redshift passive galaxies that mimic the colors of @xmath203 sources. the efficiency of targeted follow - ups and the overall potential to complement searches for @xmath3 sources traditionally carried out in legacy fields are demonstrated by the very recent award of _ spitzer _ irac time to our team to investigate the nature of the sources discussed here (pid # 12058, pi bouwens). with these observations, it will be possible to clarify the behavior of the bright end of the lf at @xmath3, as well as to confirm ideal targets for further spectroscopic follow - up investigations without having to wait for _ james webb space telescope_. sb, mt, and jw thank the vatican observatory, where part of this work was carried out, for kind hospitality. this work was partially supported by grants _ hst_/go 13767, 12905, and 12572. this research was conducted by the australian research council centre of excellence for all - sky astrophysics (caastro), through project number ce110001020. alvarez, m. a., finlator, k., trenti, m. 2012,, 759, l38 atek, h., siana, b., scarlata, c. et al. 2011,, 743, 121 barone - nugent, r. l., trenti, m., wyithe, j. s. b. et al. 2014,, 793, 17 barone - nugent, r. l., wyithe, j. s. b., trenti, m. et al. 2015a,, 450, 1224 barone - nugent, r. l., sonnenfeld, a., wyithe, j. s. b. et al. 2015b,, 453, 3068 bentez, n., 2000,, 536, 571 bertin, e. & arnouts, s. 1996,, 117, 393 - 404 bradley, l. d., trenti, m., oesch, p. a. et al. 2012,, 760, 108 bouwens, r. j., illingworth, g. d., blakeslee, j. p., broadhurst, t. j. & franx, m. 2004,, 611, l1 bouwens, r. j., illingworth, g. d., blakesless, j. p. & franx, m. 2004,, 653, 53 bouwens, r. j., illingworth, g. d., franx, m. & ford, h. 2007,, 670, 928 bouwens, r. j., illingworth, g. d., labb, i. et al. 2011a,, 469, 504 bouwens. r. j., illingworth, g. d., oesch, p. a. et al. 2011b,, 737, 90 bouwens, r. j., bradley, l., zitrin, a. et al. 2014,, 795, 126 bouwens, r. j., illingworth, g. d., oesch, p. a. et al. 2015a,, 803, 34 bouwens, r. j., oesch, p. a., labb, i. et al. 2015b, arxiv:1506.01035v2 bowler, r. a. a., dunlop, j. s., mclure, r. j. et al. 2014,, 440, 2810 calvi, v., trenti, m., stiavelli, m. et al. 2015, arxiv:1512.05363 caruana, j., bunker, a. j., wilkins, s. m. et al. 2012,, 427, 3055 casertano, s., de mello, d., dickinson, m. et al. 2000,, 120, 2747 coe, d., zitrin, a., carrasco, m. et al. 2013,, 762, 32 curtis - lake, e., mclure, r. j., dunlop, j. s. et al. 2016,, 457, 440 ellis, r. s., mclure, r. j., dunlop, j. s. et al. 2013,, 763, l7 fall., s. m. & efstathiou, g. 1980,, 193, 189 fialkov, a. & loeb, a. 2015,, 806, 256 finkelstein, s. l., papovich, c., dickinson, m. et al. 2013,, 502, 524 finkelstein, s. l., ryan, r. e., jr, papovich, c. et al. 2015,, 810, 71 giallongo, e., menci, n., castellano, m., fontana, a., grazian, a. & pentericci, l. 2012,, 755, 124 grazian, a., fontana, a., santini, p. et al. 2015,, 575, a96 grogin, n. a., kocevski, d. d., faber, s. m. et al. 2011,, 197, 35 holwerda, b. w., bouwens, r., oesch, p., et al. 2015,, 808, 6 huang, x., zheng, w., wang, j. et al. 2015,, 801, 12 illingworth, g. d., magee, d., oesch, p. a. et al. 2013,, 209, 6 infante, l., zheng, w., laporte, n. et al. 2015,, 815, 18 ishigaki, m., kawamata, r., ouchi, m., oguri, m., shimasaku, k., & ono, y. 2015,, 799, 12 ishigaki, m., ouchi, m. & harikane, y., 2015, arxiv150901751 koekemoer, a. m., fruchter, a. s., hook, r. n. & hack, w. 2003, in the 2002 hst calibration workshop : hubble after the installation of the acs and the nicmos cooling system, ed. s. arribas, a. koekemoer, & b. whitmore (baltimore, md : stsci), 337 koekemoer, a. m., faber, s. m., ferguson, h. c. et al. 2011,, 197, 36 labb, i., gonzlez, v., bouwens, r. j. et al. 2010,, 716, l103 labb, i., oesch, p. a., illingworth, g. d. et al. 2015, arxiv:1507.08313 madau, p., haardt, f., & rees, m. j. 1999,, 514, 648 madau, p. & haardt, f. 2015,, 813, l8 malhotra, a., rhoads, j. e., pirzkal, n. et al. 2005,, 626, 666 mason, c. a., treu, t., schmidt, k. b. et al. 2015a,, 805, 79 mason, c. a., trenti, m. & treu, t. 2015b,, 813, 21 mcleod, d. j., mclure, r. j., dunlop, j. s., robertson, b. e., ellis, r. s., & targett, t. a. 2015,, 450, 3032 mclure, r. j., dunlop, j. s., bowler, r. a. a. et al. 2013,, 432, 2696 muoz, j. a. & loeb, a. 2008,, 385, 2175 muoz, j. a. & loeb, a. 2011,, 729, 99 oesch, p. a., stiavelli, m., carollo, c. m. et al. 2007,, 671,1212 oesch, p. a., carollo, c. m., stiavelli, m. et al. 2009,, 690, 1350 oesch, p. a., bouwens, r. j., carollo, c. m. et al. 2010,, 709, l21 oesch, p. a., bouwens, r. j., illingworth, g. d. et al. 2012,, 759, 135 oesch, p. a., bouwens, r. j., illingworth, g. d. et al. 2013,, 773, 75 oesch, p. a., bouwens, r. j., illingworth, g. d. et al. 2014,, 786, 108 oesch, p. a., van dokkum, p. g., illingworth, g. d. et al. 2015,, 804, l30 oke, j. b. & gunn, j. e. 1983,, 266, 713 pentericci, l., fontana, a., vanzella, e. et al. 2011,, 743, 132 pentericci, l., vanzella, e., fontana, a. et al. 2014,, 793, 113 planck collaboration, ade, p. a. r., aghanim, n. et al. 2015, arxiv:1502.01589 postman, m., coe, d., bnitez, n. et al. 2012,, 199, 25 roberts - borsani, g. w., bouwens, r. j., oesch, p. a. et al. 2015, arxiv:1506.00854 schechter, p. 1976,, 203, 297 schenker, m. a., stark, d. p., ellis, r. s. et al. 2012,, 744, 179 schlafly, e. f., finkbeiner, d. p. 2011,, 737, 103 schmidt, k. b., treu, t., trenti, m. et al. 2014,, 786, 57 schneider, p., kochanek, c. s. & wambsganss, j. 2006, gravitational lensing : strong, weak and micro (berlin : springer) stark, d. p., ellis, r. s., bunker, a. et al. 2009,, 697, 1493 stark, d. p., ellis, r. s., chiu, k., ouchi, m. & bunker, a. 2010,, 408, 1628 stark, d. p., richard, j., charlot, s. et al. 2015,, 450, 1846 stark, d. p., walth, g., charlot, s. et al. 2015,, 454, 1393 steidel, c. c., giavalisco, m., pettini, m., dickinson, m., & adelberger, k. l. 1996,, 462, l17 tacchella, s., trenti, m. & carollo, c. m., 768, l37 tilvi, v., papovich, c., finkelstein, s. l. et al. 2014,, 794, 5 trenti, m. & stiavelli, m. 2008,, 676, 767 trenti, m., stiavelli, m., bouwens, r. j. et al. 2010,, 714, l202 trenti, m., bradley, l. d., stiavelli, m. et al. 2011,, 727, l39 trenti, m., bradley, l. d., stiavelli, m. et al. 2012,, 746, 55 treu, t. 2010,, 48, 87 treu, t., trenti, m., stiavelli, m., auger, m. w. & bradley, l. d. 2012,, 747, 27 treu, t., schmidt, k. b., trenti, m., bradley, l. d., & stiavelli, m. 2013,, 775, l29 turner, e. l., ostriker, j. p. van dokkum, p. g. 2001,, 113, 1420 vanzella, e., fontana, a., pentericci, l. et al. 2014,, 569, a78 willott, c. j., delorme, p., omont, a. et al. 2007,, 134, 2435 wyithe, j. s. b., yan, h., windhorst, r. & mao, s. 2011,, 469, 181 yan, h., yan, l., zamojski, m. a. et al. 2011,, 728, l22 zheng, w., postman, m., zitrin, a. et al. 2012,, 489, 406 zheng, w., shu, x., moustakas, j. et al. 2014,, 795, 93 zitrin, a., zheng, w., broadhurst, t. et al. 2014,, 793, l12 zitrin, a., labb, i., belli, s. et al. 2015, arxiv:1507.02679

the wide field camera 3 (wfc3) on the _ hubble space telescope _ (_ hst _) enabled the search for the first galaxies observed at @xmath0 (@xmath1 myr after the big bang). to continue quantifying the number density of the most luminous galaxies (@xmath2) at the earliest epoch observable with _ hst _, we search for @xmath3 galaxies (f125w - dropouts) in archival data from the brightest of reionizing galaxies (borg[z8]) survey, originally designed for detection of @xmath4 galaxies (f098m - dropouts). by focusing on the deepest 293 arcmin@xmath5 of the data along 62 independent lines of sight, we identify six @xmath3 candidates satisfying the color selection criteria, detected at s / n @xmath6 in f160w with @xmath7 to @xmath8 if at @xmath9. three of the six sources, including the two brightest, are in a single wfc3 pointing (@xmath10 arcmin@xmath5), suggestive of significant clustering, which is expected from bright galaxies at @xmath11. however, the two brightest galaxies are too extended to be likely at @xmath11, and one additional source is unresolved and possibly a brown dwarf. the remaining three candidates have @xmath12, and given the area and completeness of our search, our best estimate is a number density of sources that is marginally higher but consistent at @xmath13 with searches in legacy fields. our study highlights that @xmath11 searches can yield a small number of candidates, making tailored follow - ups of _ hst _ pure - parallel observations viable and effective.

1606.05955

Introduction in quantum information processing, information is stored and processed with a quantum system. a quantum system is always in contact with its surrounding environment, which leads to decoherence in the quantum system. decoherence must be suppressed for quantum information stored in qubits to be intact. there are several proposals to fight against decoherence. quantum error correction, abriviated as qec hereafter, is one of the most promising candidate to suppress environmental noise, which leads to decoherence @xcite. by adding extra ancillary qubits, in analogy with classical error correction, it is possible to encode a data qubit to an @xmath4-qubit codeword in such a way that an error which acted in the error quantum channel is identified by measuring another set of ancillary qubits added for error syndrome readout. then the correct codeword is recovered from a codeword suffering from a possible error by applying a recovery operation, whose explicit form is determined by the error syndrome readout. in contrast with the conventional scheme outlined in the previous paragraph, there is a scheme in which neither syndrome readouts nor syndrome readout ancilla qubits are required @xcite. in particular, in @xcite, a general efficient scheme was proposed. a data qubit is encoded with encoding ancilla qubits by the same encoding circuit as the conventional one, after which a noisy channel is applied on the codeword. subsequently, the inverse of the encoding circuit is applied on a codeword, which possibly suffers from an error. the resulting state is a tensor product of the data qubit state with a possible error and the ancilla qubit state. it is possible to correct erroneous data qubit state by applying correction gates with the ancilla qubits as control qubits and the data qubit as a target qubit. this paper presents two examples of error correcting codes falling in the second category. the noisy quantum channel is assumed to be fully correlated @xcite, which means all the qubits constituting the codeword are subject to the same error operators. in most physical realizations of a quantum computer, the system size is typically on the order of a few micrometers or less, while the environmental noise, such as electromagnetic wave, has a wavelength on the order of a few millimeters or centimeters. then it is natural to assume all the qubits in the register suffer from the same error operator. to demonstrate the advantage of the second category, we restrict ourselves within the noise operators @xmath7 in the following, where @xmath3 is the number of constituent qubits in the codeword. we show that there exists an @xmath4-qubit encoding which accommodates an @xmath5-qubit data state if @xmath4 is odd and an @xmath6-qubit date state if @xmath4 is even. although the channel is somewhat artificial as an error channel, we may apply our error correction scheme in the following situation. suppose alice wants to send qubits to bob. their qubit bases differ by unitary operations @xmath8 or @xmath9. even when they do not know which basis the other party employs, the can correctly send qubits by adding one extra qubits (when @xmath4 is odd) or two extra qubits (when @xmath4 is even). we state the theorems and prove them in the next section. the last section is devoted to summary and discussions. Main theorems in the following, @xmath10 denotes the @xmath11th component of the pauli matrices and we take the basis vectors @xmath12 so that @xmath13 is diagonalized. we introduce operators @xmath14 and @xmath15 acting on the @xmath4-qubit space @xmath16, where @xmath3 as mentioned before. let @xmath17 be @xmath18 complex matrices, and let @xmath19. denote by @xmath20 the (joint) rank-@xmath21 numerical range of @xmath22, which is the collection of @xmath23 such that @xmath24 for some @xmath18 rank-@xmath21 orthogonal projection @xmath25 @xcite. a quantum channel of the form @xmath26 has a @xmath21-dimensional quantum error correcting code (qecc) if and only if @xmath27. to prove this statement, we need to recall the knill - laflamme correctability condition, which asserts that given a quantum channel @xmath28 with error operators @xmath29, @xmath30 is a qecc of @xmath31 if and only if @xmath32, where @xmath33 is the projection operator with the range space @xmath30 @xcite. it should be clear that @xmath34 if and only if there is a qecc with dimension @xmath21. now it follows from @xmath35 and the relations @xmath36 when @xmath4 is even and @xmath37 when @xmath4 is odd that the channel ([eq : corrch]) has a @xmath21-dimensional qecc if and only if @xmath38 by noting that @xmath39 irrespective of rank @xmath25, we find @xmath40 if and only if @xmath41. [thm1] suppose @xmath3 is odd. then @xmath42. our proof is constructive. for @xmath43, denote @xmath44. let @xmath45 then @xmath46, where @xmath47 is the number of @xmath48-combinations from @xmath4 elements. since @xmath49 we have @xmath50 let @xmath25 be the orthogonal projection onto @xmath30. then the above observation shows that @xmath51 and @xmath52. therefore, @xmath53, which shows that @xmath54 and hence @xmath30 is shown to be a @xmath55-dimensional qecc. _ now let us turn to the even @xmath4 case. we first state a lemma which is necessary to prove the theorem. [lem1] let @xmath56 be a normal matrix. then the rank-@xmath21 numerical range of @xmath57 is the intersection of the convex hulls of any @xmath58 eigenvalues of @xmath57. the proof of the lemma is found in @xcite. [thm2] suppose @xmath3 is even. then @xmath59 but @xmath60. _ proof let @xmath61. by theorem [thm1], @xmath62. consider @xmath63 observe that the projection @xmath25 onto @xmath64 satisfies @xmath65 and @xmath52 and hence @xmath66, which proves @xmath59. _ since @xmath67 is a commuting family, @xmath8 and @xmath9 can be diagonalized simultaneously. we may assume that @xmath68 since @xmath69, we have @xmath70 let us show that @xmath71. we first note the identity @xmath72 for hermitian @xmath73. let us replace @xmath74 by @xmath75 and @xmath76 by @xmath77 to obtain @xmath78. since @xmath75 and @xmath77 commute, @xmath79 is normal and lemma [lem1] is applicable. from eqs. ([eq1]) and ([eq2]), we find @xmath79 has eigenvalues @xmath80 and each eigenvalue is @xmath81-fold degenerate. by taking @xmath82 and @xmath83 in lemma [lem1], we find the rank-@xmath55 numerical range of @xmath79 is the intersection of the convex hulls of any @xmath84 eigenvalues. since each eigenvalue has multiplicity @xmath81, each convex hull involves at least three eigenvalues. by inspecting four eigenvalues plotted in the complex plane, we easily find the intersection of all the convex hulls is a single point @xmath85, which proves @xmath71. similarly, we prove @xmath86. from these equalities we obtain @xmath87 suppose @xmath42. let @xmath25 be a rank-@xmath55 projection such that @xmath88. let @xmath89\]] where each @xmath90 has size @xmath91. from @xmath92 and @xmath93, we have four independent equations @xmath94 let @xmath95 be the singular value decomposition of @xmath96, where @xmath97 is a nonnegative diagonal matrix and @xmath98. then the above equations are solved as @xmath99 by collecting these results, we find the projection operator is decomposed as @xmath100 \[\begin{array}{cc}d&d\\ d&d\end{array}\]\[\begin{array}{cc}u^\dagger&0\\ 0&v^\dagger\end{array}\]\,.\]] since rank @xmath101 and @xmath102, it follows from @xmath103 that @xmath104. let @xmath105 then both @xmath57 and @xmath106 are non - singular. on the other hand, the assumption @xmath107 implies @xmath108 and hence @xmath109, which is a contradiction. therefore, @xmath60. in the following, we give an explicit construction of qecc for @xmath31 in eq. ([eq : corrch]) with odd @xmath4. the technique is based on theorem [thm1] and the results in @xcite. let @xmath110 be the @xmath111 matrix with columns in the set @xmath112 define the @xmath113 matrix @xmath114 $]. in our qec, an @xmath5-qubit state @xmath115 is encoded with one ancilla qubit @xmath116 as @xmath117. then a noisy quantum channel @xmath31 is applied on the encoded state and subsequently the recovery operation @xmath118 is applied so that the decoded state automatically appears in the output with no syndrome measurements. our qec is concisely summarized as @xmath119 where @xmath120. choosing an encoding amounts to assigning each of @xmath81 column vectors in @xmath110 a basis vector of the whole hilbert space without repetition. therefore there are large degrees of freedom in the choice of encoding. in the following examples, we have chosen encoding whose quantum circuit can be implemented with the least number of cnot gates. since our decoding circuit is the inverse of the encoding circuit, it is also implemented with the least number of cnot gates. when @xmath121, the unitary operation @xmath122 can be chosen as @xmath123 when @xmath124, @xmath122 can be chosen as @xmath125 figure [nodd] shows quantum circuits of the matrix @xmath122 for @xmath126 and @xmath127. [nodd] -qubit state @xmath115 with a single ancilla qubit initially in the state @xmath128. (a) is for @xmath126 while (b) is for @xmath127. the quantum channel in the box represents a quantum operation with fully correlated noise given in eq. the output ancilla state is @xmath129 for error operators @xmath130 and @xmath131 (@xmath132 and @xmath133) for @xmath126 and @xmath129 for @xmath134 and @xmath135 (@xmath136 and @xmath137) for @xmath127.,width=529] it follows from eq. ([qecc]) that the recovery circuit is the inverse of the encoding circuit. it seems, at first sight, that the implementations given in fig. 1 contradict with eq. ([qecc]) since the controlled not gate in the end of the recovery circuit is missing in the encoding circuit. note, however, that the top qubit is set to @xmath138 initially and the controlled not gate is safely omitted without affecting encoding. we construct a decoherence - free encoding when @xmath4 is even as follows. the codeword in this case is immune to the noise operators, which is an analogue of noiseless subspace / subsystem introduced in @xcite. let @xmath139 then evidently a vector @xmath140 is separately invariant under the action of @xmath141 and @xmath9. there are @xmath142 orthogonal vectors of such form, e.g. we have four vectors, @xmath143 for @xmath144. thus we find a decoherence - free encoding for @xmath145 qubits by projecting onto this invariant subspace spanned by these basis. it should be noted that the projection operator @xmath25 to the subspace @xmath146 spanned by the four vectors in eq. ([nss]) satisfies rank @xmath147 and @xmath148, which shows @xmath149. it is easy to generalize this result to cases with arbitrary @xmath150. figure [neven] (a) and (b) depict quantum circuits for (a) @xmath144 and (b) @xmath151, respectively. [neven] -qubit state @xmath115 with two ancilla qubit initially in the state @xmath152. (a) is for @xmath144 while (b) is for @xmath151. the quantum channel in the box represents a quantum operation with fully correlated noise given in eq. the output ancilla state is always @xmath153, irrespective of error operators acted in the channel.,width=529] Summary and discussions we have shown that there is a quantum error correction which suppresses fully correlated errors of the form @xmath154, in which @xmath4 qubits are required to encode (i) @xmath155 data qubit states when @xmath4 is odd and (ii) @xmath156 data qubit states when @xmath4 is even. we have proved these statements by using operator theoretical technique. neither syndrome measurements nor ancilla qubits for syndrome measurement are required in our scheme, which makes physical implementation of our scheme highly practical. examples with @xmath126 and @xmath127 are analyzed in detail and explicit quantum circuits implementing our qec with the least number of cnot gate were obtained. since the error operators are closed under matrix multiplication, errors can be corrected even when they act on the codeword many times. a somewhat similar qec has been reported in @xcite. they analyzed a partially correlated noise, where the error operators acts on a fixed number of the codeword qubits simultaneously. they have shown that the quantum packing bound was violated by taking advantage of degeneracy of the codes. justification of such a noise physically, however, seems to be rather difficult. they have also shown that correlated noise acting on an arbitrary number @xmath4 of qubits can encode @xmath157 data qubits. in contrast, we have analyzed a fully correlated noise, which shows the highest degeneracy, and have shown that @xmath158 data qubits can be encoded with an @xmath4-qubit codeword when @xmath4 is odd. clearly, our qec suppressing fully correlated errors is optimal as it is clear that one can not encode @xmath4 qubits as data qubits for odd @xmath4 and we have shown that one can not encode @xmath155 qubits for even @xmath4. Acknowledgement ckl was supported by a usa nsf grant, a hk rgc grant, the 2011 fulbright fellowship, and the 2011 shanxi 100 talent program. he is an honorary professor of university of hong kong, taiyuan university of technology, and shanghai university. mn and ht were supported by `` open research center '' project for private universities : matching fund subsidy from mext (ministry of education, culture, sports, science and technology). ytp was supported by a usa nsf grant. nss was supported by a hk rgc grant.

we investigate an efficient quantum error correction of a fully correlated noise. suppose the noise is characterized by a quantum channel whose error operators take fully correlated forms given by @xmath0, @xmath1 and @xmath2, where @xmath3 is the number of qubits encoding the codeword. it is proved that (i) @xmath4 qubits codeword encodes @xmath5 data qubits when @xmath4 is odd and (ii) @xmath4 qubits codeword implements an error - free encoding, which encode @xmath6 data qubits when @xmath4 is even. quantum circuits implementing these schemes are constructed. quantum error correction, higher rank numerical range, recovery operator, mixed unitary channel

1104.4750

Introduction the current observations, such as sneia (supernovae type ia), cmb (cosmic microwave background) and large scale structure, converge on the fact that a spatially homogeneous and gravitationally repulsive energy component, referred as dark energy, accounts for about @xmath1 % of the energy density of universe. some heuristic models that roughly describe the observable consequences of dark energy were proposed in recent years, a number of them stemming from a certain physics @xcite and the others being purely phenomenological @xcite. dark energy can even behave as a phantom and effectively violate the weak energy condition@xcite. in various cosmological models, fundamental quantities are either geometrical (if they are constructed from a spacetime geometry directly) or physical (if they depend upon physical fields). physical quantities are certainly model - dependent, while geometrical quantites are more universal. about thirty years ago, the bouncing cosmological model with torsion was suggested in ref.@xcite, but the torsion was imagined as playing role only at high densities in the early universe. goenner et al. made a general survey of the torsion cosmology @xcite, in which the equations for all the pgt (poincar gauge theory of gravity) cases were discussed although they only solved in detail a few particular cases. recently some authors have begun to study torsion as a possible reason of the accelerating universe @xcite. nester and collaborators @xcite consider an accounting for the accelerated universe in term of a riemann - cartan geometry : dynamic scalar torsion. they explore the possibility that the dynamic pgt connection, reflecting the nature of dynamic pgt torsion, provides the accelerating force. with the usual assumptions of homogeneity and isotropy in cosmology and specific cases of the suitable parameters and initial conditions, they find that torsion field could play a role of dark energy. one of the motivation was to avoid singularities in the initial investigations of torsion cosmology @xcite. however, it soon was found that non - linear torsion effects were more likely to produce stronger singularities @xcite. the non - linear effects turn out to play a key role for the outstanding present day mystery : the accelerated universe. in the various pgt, the connection dynamics decomposed into six modes with certain spin and parity : @xmath2, @xmath3, @xmath4. some investigations showed that @xmath4 may well be the only acceptable dynamic pgt torsion modes @xcite. the pseudoscalar mode @xmath5 is naturally driven by the intrinsic spin of elementary fermions, therefore it naturally interacts with such sources. consequently, it is generally thought that axial torsion must be small and have small effects at the late time of cosmological evolution. this is a major reason why one does not focus on this mode at the late time. on the other hand, the scalar mode @xmath6 does not interact in any direct obvious fashion with any known type of matter @xcite, therefore one can imagine it as having significant magnitude and yet not being conspicuously noticed. furthermore, there is a critical non - zero value for the affine scalar curvature since @xmath6 mode can interact indirectly through the non - linear equations. the homogeneity and isotropy of cosmology have received strong confirmation from modern observations, which greatly restrict the possible types of non - vanishing fields. under the assumption of homogeneity and isotropy, @xmath6 mode has only a time component and it can be specified as the gradient of a time - dependent function. therefore, the cosmological models with the scalar mode offer a situation where dynamic torsion may lead to observable effect at late time. we emphasize again that one does not focus on the early universe, where one could indeed expect large effects (though their signature would have to be separated from other large effects), and substitutionally asks about traces of torsion effects at the late time of cosmological evolution @xcite. obviously, the fine - tuning problem is one of the most important issues for the torsion cosmology @xcite. and a good model should limit the fine - tuning as much as possible. the dynamical attractor of the cosmological system has been employed to make the later time behaviors of the model insensitive to the initial condition of the field and thus alleviates the fine - tuning problem @xcite. furthermore, nester et al @xcite have shown that the hubble parameter and @xmath7 have an oscillatory form for the scalar torsion cosmology. the traditional geometrical parameters, i.e., the hubble parameter @xmath8 and the deceleration parameter @xmath9, are two elegant choices to describe the expansion state of universe but they can not distinguish various accelerating mechanism uniquely, because a quite number of models may just correspond to the same current values of @xmath10 and @xmath11. however, sahni, saini, starobinsky and alam @xcite have introduced the statefinder pair @xmath12 : @xmath13, @xmath14. it is obviously a natural next step beyond @xmath10 and @xmath11. fortunately, as is shown in the literatures @xcite, the statefinder parameters which are also geometrical diagnostics, are able to differentiate a series of cosmological models successfully. using the discussion of statefinder parameters in the scalar torsion cosmology, we explain easily why the present field equations modify the expansion of the universe only at late time. if the evolving trajectory of statefinder have a decelerating phase (@xmath15) at early time, then we can understand why the expansion of the universe until @xmath16 remains unchanged in the scalar torsion models. in this paper, we apply the statefinder diagnostics to the torsion cosmology. we find that there are some characteristics of statefinder parameters for the torsion cosmology that can be distinguished from the other cosmological models. the statefinder diagnostics show that the universe naturally has an accelerating expansion at low redshifts (late time) and a decelerating expansion at high redshifts (early time). therefore, scalar torsion cosmology can avoid some of the problems which occur in other models. especially, the effect of torsion can make the expansion rate oscillate when torsion parameter @xmath17 or @xmath18. whether the universe has properties which are easier to explain within the scalar torsion context is a remarkable possibility demanding further exploration. the oscillatory feature of hubble parameter had earlier been reported for the braneworld cosmology @xcite and the quasi - steady state cosmology @xcite. we show that statefinder diagnostic has a direct bearing on the critical points of the dynamical system. one of the most interesting characteristic of the trajectories is that there are loop and curves with the shape of tadpole in the case of the torsion parameter @xmath19. in this case, we fit the scalar torsion model to current type ia supernova data and find it is consistent with the observations. furthermore, we analyze preliminarily the relevance for realistic observation of the found statefinder parameters. The equations of motion pgt @xcite has been regarded as an interesting alternative to general relativity because of its gauge structure and geometric properties. pgt based on a riemann - cartan geometry, allows for dynamic torsion in addition to curvature. the affine connection of the riemann - cartan geometry is @xmath20 where @xmath21 is the levi - civita connection and @xmath22 is the torsion tensor. meantime, the ricci curvature and scalar curvature can be written as @xmath23 where @xmath24 and @xmath25 are the riemannian ricci curvature and scalar curvature, respectively, and @xmath26 is the covariant derivative with the levi - civita connection (for a detailed discussion see ref.. theoretical analysis of pgt led us to consider tendentiously dynamic `` scalar mode ''. in this case, the restricted expression of the torsion can be written as @xcite @xmath27\kappa},\label{trestrct}\]] where the vector @xmath28 is the trace of the torsion. then, the ricci curvature and scalar curvature can be expressed as @xmath29 the gravitational lagrangian density for the scalar mode is @xmath30 where @xmath31 and @xmath0 is a torsion parameter. consider that the parameter @xmath32 is associated with quadratic scalar curvature term @xmath33, so that @xmath32 should be positive @xcite. therefore, the field equations of the scalar mode are @xmath34 where @xmath35 is the source energy - momentum tensor and @xmath36 is the contribution of the scalar torsion mode to the effective total energy - momentum tensor : @xmath37 since current observations favor a flat universe, we will work in the spatially flat robertson - walker metric @xmath38 $], where @xmath39 is the scalar factor. this engenders the riemannian ricci curvature and scalar curvature : @xmath40 where @xmath39 is the scalar factor, and @xmath10 is the hubble parameter. the torsion @xmath41 should also be only time dependent, therefore one can let @xmath42 (@xmath43 is the torsion field) and the spatial parts vanish. the corresponding equations of motion in the matter - dominated era are as follows @xmath44 where @xmath45 and the energy density of matter component @xmath46 one can scale the variables and the parameters as @xmath47 where @xmath48 is the present value of hubble parameter and @xmath49 is the planck length. under the transform ([scale]), eqs. ([dth])-([dtr]) remain unchanged. after transform, new variables @xmath50, @xmath10, @xmath51 and @xmath52, and new parameters @xmath53, @xmath0, @xmath54 and @xmath32 are all dimensionless. obviously, the newtonian limit requires @xmath55. for the case of scalar torsion mode, the effective energy - momentum tensor can be represented as @xmath56\,,\label{torpre}\end{aligned}\]] and the off - diagonal terms vanish. the effective energy density @xmath57 which is deduced from general relativity. @xmath58 is an effective pressure, and the effective equation of state is @xmath59 which is induced by the dynamic torsion., the temporal component of the torsion @xmath51, the affine scalar curvature @xmath52 and the deceleration parameter @xmath11 as functions of time. we have chosen the parameters @xmath60, @xmath61 and the initial values @xmath62, @xmath63, @xmath64.,title="fig:",width=264], the temporal component of the torsion @xmath51, the affine scalar curvature @xmath52 and the deceleration parameter @xmath11 as functions of time. we have chosen the parameters @xmath60, @xmath61 and the initial values @xmath62, @xmath63, @xmath64.,title="fig:",width=268] in the case of @xmath65, nester et al showed that the scalar mode can contribute an oscillating aspect to the expansion rate of the universe @xcite. this oscillatory nature can be illustrated in fig. [hqevol] where we have chosen @xmath60, @xmath61, @xmath62, @xmath63, @xmath64 and set the current time @xmath66. according to scaling ([scale]), the present value of the hubble parameter is unity. obviously, @xmath10 is damped - oscillating at late time and @xmath67 is negative today, which means the expansion of the universe is currently accelerating. the value of @xmath11 turns from positive to negative when the time is around @xmath68, which is the epoch the universe began to accelerate. however, the above result is dependent on the choice of initial data and the values of the parameters. then, the scalar torsion cosmology is unsuited to solving the fine - tuning problem in the case of @xmath65. in the following sections, we ll investigate the statefinder and give the dynamics analysis for all ranges of the parameter @xmath0. Statefinder diagnostic for the spatially flat @xmath69cdm model the statefinder parameters correspond to a fixed point @xmath70 while @xmath71 for the standard cold dark matter model (scdm) containing no radiation. since the torsion cosmology have used the dynamic scalar torsion (a geometry quantity in the riemann - cartan spacetime), the torsion accelerating mechanism is bound to exhibit an essential distinction in contrast with various dark energy models. therefore, its statefinder diagnostic is sure to reveal differential feature. let us now study the torsion cosmological model in detail. using eqs. ([dth])-([fieldrho]), we have the deceleration parameter @xmath72(432a_{1}h^{2})^{-1}\,,\label{torsionq } \end{aligned}\]] and the statefinder parameters @xmath73(108a_{1}bh^{3})^{-1}\,,\nonumber\\ \label{torsionr } \end{aligned}\]] and @xmath74}{3bh\left[(6\mu + br)(36h^{2}-24h\phi + 4\phi^{2}-3r)-54\mu r\right]}\,. \label{torsions } \end{aligned}\]] in the following we will discuss the statefinder for four differential ranges of the torsion parameter @xmath0, respectively. firstly, we consider the time evolution of the statefinder pairs @xmath12 and @xmath75 in the case of @xmath76. in fig. [planecase4], we plot evolution trajectories in the @xmath77 and @xmath78 planes, where we have chosen @xmath79 and @xmath80. we see easily that cosmic expansion alternates between deceleration and acceleration in the evolving trajectories of @xmath77 plane, and the amplitude becomes larger and larger as increase of time. the trajectories in the @xmath78 plane is quite complicated, so we mark its sequence by the ordinal number. every odd number curve evolves from finite to infinite, but even number curve evolves from infinite to finite. these are quasi - periodic behaviors which corresponds to the numerical solution of ref. noticeably, the trajectories will never pass @xmath69cdm point @xmath70. secondly, we discuss the time evolution of the statefinder pairs @xmath12 and @xmath75 for the case of @xmath81. we plot evolving trajectories in fig. [planecase3], where we have chosen @xmath82 and @xmath83. we see clearly that the cosmic acceleration is guaranteed by the dynamic scalar torsion in the evolving trajectories of @xmath77 plane, and the curves will converge into @xmath69cdm point. the evolving trajectories go through a climbing - up stage first, then get into a rolling - down stage in the @xmath78 plane. lastly, trajectories tend to @xmath69cdm point @xmath70. furthermore, the only one forms a loop that starts from @xmath70 then evolves back to @xmath70, and others show in the shape of tadpole. and @xmath78 planes for the case of torsion parameter @xmath84, where we choose the parameters @xmath85 and @xmath86. the arrows show the direction of the time evolution.,title="fig:",width=264] and @xmath78 planes for the case of torsion parameter @xmath84, where we choose the parameters @xmath85 and @xmath86. the arrows show the direction of the time evolution.,title="fig:",width=264] and @xmath78 planes for the case of torsion parameter @xmath87, where we choose the parameters @xmath88 and @xmath89. the arrows show the direction of the time evolution.,title="fig:",width=264] and @xmath78 planes for the case of torsion parameter @xmath87, where we choose the parameters @xmath88 and @xmath89. the arrows show the direction of the time evolution.,title="fig:",width=264] and @xmath90 planes for the case of @xmath91, where we choose @xmath92 and @xmath93. the arrows show the direction of the time evolution.,title="fig:",width=264] and @xmath90 planes for the case of @xmath91, where we choose @xmath92 and @xmath93. the arrows show the direction of the time evolution.,title="fig:",width=257] and @xmath78 planes for the case of torsion parameter @xmath94, where we choose the parameters @xmath95 and @xmath86. the arrows show the direction of the time evolution.,title="fig:",width=264] and @xmath78 planes for the case of torsion parameter @xmath94, where we choose the parameters @xmath95 and @xmath86. the arrows show the direction of the time evolution.,title="fig:",width=257] thirdly, we discuss the time evolution of the trajectories for the case of @xmath96. we plot evolving trajectories in fig. [planecase2], where we have chosen @xmath97 and @xmath98. obviously, the cosmic acceleration can happen since deceleration parameter is negative. @xmath99, @xmath100 and @xmath101 become larger and larger first, then less and less as the cosmic time increase. finally, we consider the time evolution of the statefinder pairs @xmath12 and @xmath75 in the case of @xmath18. in fig. [planecase1], we plot evolving trajectories in the @xmath77 and @xmath78 planes, where we have chosen @xmath102 and @xmath80. we find easily that the evolving trajectories analogous to the case of @xmath103 except trajectories pass the @xmath69cdm point. to sum up, it is very interesting to see that the scalar torsion naturally provide the accelerating force in the universe for any torsion parameter @xmath0. however, it is dependent on torsion parameters that there is a decelerating (@xmath15) expansion before an accelerating (@xmath104) expansion. the statefinder diagnostics show that the universe naturally has an accelerating expansion at low redshifts (late time) and a decelerating expansion at high redshifts (early time) for the cases of @xmath105 and @xmath18. obviously, scalar torsion cosmology can avoid some of the problems which occur in other models. if we refuse the possibility of non - positivity of the kinetic energy, we will employ normal assumption, i. e., @xmath17. in this case, the effect of torsion can make the expansion rate oscillate. with suitable adjustments of the torsion parameters, it is possible to change the quasi - period of the expansion rate as well as its amplitudes. it is worth noting that the true values of the statefinder parameters of the universe should be determined in model - independent way. in principle, @xmath106 can be extracted from some future astronomical observations, especially the snap - type experiment. why there are new features for the statefinder diagnostic of torsion cosmology? why the torsion parameter @xmath0 is divided into differential ranges by the statefinder answer is very simple. in fact, the statefinder diagnostic has a direct bearing on the attractor of cosmological dynamics. therefore, we will discuss the dynamic analysis in next section. Dynamics analysis eqs. ([dth])-([dtr]) is an autonomous system, so we can use the qualitative method of ordinary differential equations. critical points are always exact constant solutions in the context of autonomous dynamical systems. these points are often the extreme points of the orbits and therefore describe the asymptotic behavior. if the solutions interpolate between critical points they can be divided into a heteroclinic orbit and a homoclinic orbit (a closed loop). the heteroclinic orbit connects two different critical points and homoclinic orbit is an orbit connecting a critical point to itself. in the dynamical analysis of cosmology, the heteroclinic orbit is more interesting @xcite. if the numerical calculation is associated with the critical points, then we will find all kinds of heteroclinic orbits. according to equations ([dth])-([dtr]), we can obtain the critical points and study the stability of these points. substituting linear perturbations @xmath107, @xmath108 and @xmath109 near the critical points into three independent equations, to the first orders in the perturbations, gives the evolution of the linear perturbations, from which we could yield three eigenvalues. stability requires the real part of all eigenvalues to be negative. there are five critical points @xmath110 of the system as follows @xmath111 where @xmath112, @xmath113 and @xmath114. the corresponding eigenvalues of the critical points (i)-(v) are @xmath115 using eq. ([criticalpoints]), we find that there is only a critical point @xmath116 in the case of @xmath76. from eq. ([eigenvalues]), the corresponding eigenvalue is @xmath117, so @xmath118 is an asymptotically stable focus. if we consider the linearized equations, then eqs. ([dth])-([dtr]) are reduced to @xmath119 + the linearized system ([dthphirlinear]) has an exact periodic solution @xmath120 where @xmath121, @xmath122, @xmath123 and @xmath124, @xmath125 and @xmath126 are initial values. obviously, @xmath127 is a critical line of center for the linearized eqs. ([hrphips]). in other words, there are only exact periodic solutions for the linearized system, but there are quasi - periodic solutions near the focus for the coupled nonlinear equations. this property of quasi - periodic also appears in the statefinder diagnostic with the case of @xmath76. @l*15@l critical points & property & @xmath128 & stability + (` i `) & saddle & @xmath129 & unstable + (` ii`)&positive attractor&-1&stable + (` iii `) & negative attractor&-1&unstable + (` iv `) & saddle&-1&unstable + (` v `) & saddle&-1&unstable + @l*15@l critical points & property & @xmath128 & stability + (` i `) & focus & @xmath130 & stable + (` ii`)&saddle&-1&unstable + (` iii `) & saddle&-1&unstable + (` iv `) & saddle&-1&unstable + (` v `) & saddle&-1&unstable + in the case of @xmath131, the critical point (ii) is a late time de sitter attractor and (iii) is a negative attractor. the properties of the critical points are shown in table [cripointsa1l0]. the de sitter attractor indicates that torsion cosmology is an elegant scheme and the scalar torsion mode is an interesting geometric quantity for physics. in the dynamical analysis of cosmology, the heteroclinic orbit is more interesting. using numerical calculation, we plot the heteroclinic orbit connects the critical point case (iii) to case (ii) in fig. [heteroclinicorbit]. this heteroclinic orbit is just corresponding to the loop in fig. [planecase3], which is from @xmath69cdm point to @xmath69cdm point. furthermore, the trajectories with the shape of tadpole correspond to saddles. with @xmath132. the heteroclinic orbit connects the critical points case (iii) to case (ii). we take @xmath133.,width=321] with @xmath94. we take @xmath134 and the initial value @xmath135. @xmath136 is an asymptotically stable focus point.,width=321] in the case of @xmath96, there is only an unstable saddle @xmath116 where the effective equation of state @xmath128 tends to @xmath129. therefore, the trajectories in fig. [planecase2] show that @xmath99,@xmath100 and @xmath101 become larger and larger, then less and less as time increases. in the case of @xmath18, the properties of the critical points are shown in table [cripointsa1l-1]. the trajectories correspond to the stable focus (see fig. 6) and unstable saddles with @xmath137. therefore, the trajectories pass through the @xmath69cdm point. Fit the torsion parameters to supernovae date in ref.@xcite, the authors have compared the numerical values of the torsion model with the observational data, in which they fixed the initial values @xmath48, @xmath138 and @xmath139, and torsion parameters @xmath0 and @xmath32. in this section, we fixed the initial value, then fit the torsion parameters to current type ia supernovae data. the scalar torsion cosmology predict a specific form of the hubble parameter @xmath140 as a function of redshifts @xmath141 in terms of two parameters @xmath0 and @xmath32 when we chose initial values. using the relation between @xmath142 and the comoving distance @xmath143 (where @xmath141 is the redshift of light emission) @xmath144 and the light ray geodesic equation in a flat universe @xmath145 where @xmath146 is the scale factor. in general, the approach towards determining the expansion history @xmath140 is to assume an arbitrary ansatz for @xmath140 which is not necessarily physically motivated but is specially designed to give a good fit to the data for @xmath147. given a particular cosmological model for @xmath148 where @xmath149 are model parameters, the maximum likelihood technique can be used to determine the best fit values of parameters as well as the goodness of the fit of the model to the data. the technique can be summarized as follows : the observational data consist of @xmath150 apparent magnitudes @xmath151 and redshifts @xmath152 with their corresponding errors @xmath153 and @xmath154. these errors are assumed to be gaussian and uncorrelated. each apparent magnitude @xmath155 is related to the corresponding luminosity distance @xmath156 by @xmath157 + 25,\]] where @xmath158 is the absolute magnitude. for the distant sneia, one can directly observe their apparent magnitude @xmath159 and redshift @xmath141, because the absolute magnitude @xmath158 of them is assumed to be constant, i.e., the supernovae are standard candles. obviously, the luminosity distance @xmath147 is the ` meeting point'between the observed apparent magnitude @xmath160 and the theoretical prediction @xmath140. usually, one define distance modulus @xmath161 and express it in terms of the dimensionless ` hubble - constant free'luminosity distance @xmath162 defined by@xmath163 as @xmath164 where the zero offset @xmath165 depends on @xmath48 (or @xmath166) as @xmath167 the theoretically predicted value @xmath168 in the context of a given model @xmath169 can be described by @xcite @xmath170 therefore, the best fit values for the torsion parameters (@xmath171) of the model are found by minimizing the quantity @xmath172 ^ 2}{\sigma_i^2}.\]] since the nuisance parameter @xmath165 is model - independent, its value from a specific good fit can be used as consistency test of the data @xcite and one can choose _ a priori _ value of it (equivalently, the value of dimensionless hubble parameter @xmath166) or marginalize over it thus obtaining @xmath173 where @xmath174 ^ 2}{\sigma_i^2},\]] @xmath175}{\sigma_i^2},\]] and @xmath176 in the latter approach, instead of minimizing @xmath177, one can minimize @xmath178 which is independent of @xmath165. the eqs. ([gradr]-[tort]) can be solved explicitly by a series in the form @xmath179,\]] where @xmath180 and @xmath181 using the general relation between hubble parameter @xmath182 and the redshift @xmath141, @xmath141 can be written as a function of @xmath50 @xmath183}-1,\]] however, the convergence radius of the series ([e1]) is @xmath184, so we can use the expansion directly in the case of the redshift being @xmath185. by the numerical calculation, we find that @xmath186 corresponds to @xmath187 for the valuses of parameters @xmath139 and @xmath138 in the fig. [192clcontours3]. for the case of @xmath188, we should use a direct analytic continuation. weierstrass @xcite had built the whole theory of analytic functions from the concept of power series. given a point @xmath189 (@xmath190), the function @xmath182 has a taylor expansion @xmath191.\]] where the coefficients @xmath192 is still expressed as eq. ([e2]) and @xmath193 can be decided by eq. ([e1]). the new series defines an analytic function @xmath194 which is said to be obtained from @xmath195 by direct analytic continuation. this process can be repeated any number of times. in the general case we have to consider a succession of power series @xmath195, @xmath194,...,@xmath196, each of which is a direct analytic continuation of the preceding one. by using this method we have the evolution of hubble parameter @xmath182. furthermore, we have the function @xmath140 from eq. ([e3]). in fact, we need only to consider the case of @xmath197 for the essence supernovae data. we now apply the above described maximum likelihood method using the essence supernovae data which is one of the reliable published data set consisting of 192 sneia (@xmath198). beside the 162 data points given in table 9 of ref. @xcite, which contains 60 essence sneia, 57 snls sneia and 45 nearby sneia, we add 30 sneia detected at @xmath199 by the hubble space telescope @xcite as in ref.@xcite. in table [differentvalues], we show the best fit of torsion parameters at different initial values of @xmath139 and @xmath138. @l*15@l @xmath139 & @xmath138 & @xmath0 & @xmath32 + 0.25&0.35 & -0.10&1.44 + 0.20&0.34 & -0.08&1.80 + 0.15&0.34 & -0.06&2.40 + 0.10&0.33 & -0.04&3.60 + in fig. [192clcontours3], contours with 68.3%, 95.4% and 99.7% confidence level are plotted, in which we take a marginalization over the model - independent parameter @xmath165. the best fit as showed in the figure corresponds to @xmath200 and @xmath201, and the minimum value of @xmath202. for @xmath203, one can get @xmath204 and the best fit @xmath205. therefore, it s easy to know that @xmath206 is consistent at the @xmath207 level with the best fits of scalar torsion cosmology. in fig. [zu], we show a comparison of the essence supernovae data along with the theoretically predicted curves in the context of scalar torsion and @xmath206. we can see that the scalar torsion model(@xmath208, @xmath209, @xmath210, @xmath211) gives a close curve behavior to the one from @xmath206 (@xmath212). clearly, the allowed ranges of the parameters @xmath0 and @xmath32 favor the case of @xmath213 if we chose @xmath214 and @xmath215. with the essence supernovae data via the relation between the redshift @xmath141 and the distance modulus @xmath54. the scalar torsion model(@xmath208, @xmath209, @xmath210, @xmath211) gives a close curve behavior to the one from @xmath206 (@xmath212).,title="fig:",width=264] with the essence supernovae data via the relation between the redshift @xmath141 and the distance modulus @xmath54. the scalar torsion model(@xmath208, @xmath209, @xmath210, @xmath211) gives a close curve behavior to the one from @xmath206 (@xmath212).,title="fig:",width=272] and @xmath32 using the essence sneia dataset. here we have assumed @xmath216, @xmath214 and @xmath215.,width=321] Conclusion and discussion we have studied the statefinder diagnostic to the torsion cosmology, in which an accounting for the accelerated universe is considered in term of a riemann - cartan geometry : dynamic scalar torsion. we have shown that statefinder diagnostic has a direct bearing on the critical points. the statefinder diagnostic divides the torsion parameter @xmath0 into four ranges, which is in keeping with the requirement of dynamical analysis. therefore, the statefinder diagnostic can be used to an exceedingly general category of models including several for which the notion of equation of state is not directly applicable. the statefinder diagnostic has the advantage over the dynamical analysis at the simplicity, but the latter can provide more information. the most interesting characteristic of the trajectories is that there is a loop in the case of @xmath217. this behavior corresponds to the heteroclinic orbit connecting the negative attractor and de sitter attractor. the trajectories with the shape of tadpole show that they pass through the @xmath69cdm fixed point along the time evolution, then the statefinder pairs are going along with a loop and they will pass through the @xmath69cdm fixed point again in the future. it is worth noting that there exists closed loop in the ref. @xcite, but there is no closed loop which contains the @xmath69cdm fixed point. these behaviors indicate that torsion cosmology is an elegant scheme and the scalar torsion mode is an interesting geometric quantity for physics. furthermore, the quasi - periodic feature of trajectories in the cases of @xmath76 or @xmath18 shows that the numerical solutions in ref. @xcite are not periodic, but are quasi - periodic near the focus for the coupled nonlinear equations. we fixed only the initial values, then fitted the torsion parameters to current sneia dataset. we find that the scalar torsion naturally explain the accelerating expansion of the universe for any torsion parameter @xmath0. however, it is dependent @xmath0 and @xmath32 that there is a decelerating expansion before an accelerating expansion. the statefinder diagnostics show that the universe naturally have an accelerating expansion at late time and a decelerating expansion at early time for the case of @xmath218 and @xmath18. if we refuse the possibility of non - positivity of the kinetic energy, we have to employ normal assumption (@xmath17). under this assumption, the effect of torsion can make the expansion rate oscillate. furthermore, with suitable adjustments of the torsion parameters and initial value, it is possible to change the quasi - period of expansion rate as well as its amplitudes. in order to have a quantitative understanding of the scalar torsion cosmology, the matter density @xmath219, the effective mass density @xmath220, and the quantity @xmath221 are important. this scenario bears a strong resemblance to the braneworld cosmology in a very different context by sahni, shtanov and viznyuk @xcite. the @xmath222 parameters in the torsion cosmology and in the @xmath69cdm cosmology can nevertheless be quite different. therefore, at high redshift, the torsion cosmology asymptotically expands like a matter - dominated universe with the value of @xmath222 inferred from the observations of the local matter density. at low redshift, the torsion model behaves like @xmath69cdm but with a renormalized value of @xmath223. the difference between @xmath222 and @xmath223 is dependent on the present value of statefinder parameters @xmath224. a more detailed estimate, however, lies beyond the scope of the present paper, and we will study it in a future work. finally, @xmath225 and @xmath226 should be extracted from some future astronomical observations in principle, especially the snap - type experiments. Acknowledgments this work is supported by national science foundation of china grant no. 10847153 and no. 10671128 References 10 sahni v and starobinsky a 2000 _ int. _ d*9 * 373 [astro - ph/9904398] + peebles p j e and ratra b 2003 _ rev. phys. _ * 75 * 559 [astro - ph/0207347] + padmanabhan t 2003 _ phys. rept. _ * 380 * 235 [hep - th/0212290] + copeland e j, sami m and tsujikawa s 2006 _ int. j. mod. _ d*15 * 1753 [hep - th/0603057] + sahni v and starobinsky a 2006 _ int. j. mod _ d*15 * 2105 [astro - ph/0610026] endo m and fukui t 1977 _ gen. * 8 * 833 + overduin j m and cooperstock f i 1998 _ phys. _ d*58 * 043506 [astro - ph/9805260] + hao j g and li x z 2005 _ phys. _ b*606 * 7 [astro - ph/0404154] + liu d j and li x z 2005 _ phys. _ b*611 * 8 [astro - ph/0501596] + beesham a 1993 _ phys. rev _ d*48 * 3539 caldwell r r 2002 _ phys. lett. _ b*545 * 23 [astro - ph/9908168] + sahni v and shtanov yu 2003 _ jcap _ * 11 * 014 [astro - ph/0202346] + hao j g and li x z 2003 _ phys. _ d*67 * 107303 [gr - qc/0302100] + liu d j and li x z 2003 _ phys. _ d*68 * 067301 [hep - th/0307239] + li x z and hao j g 2004 _ phys. _ d*69 * 107303 [hep - th/0303093] + hao j g and li x z 2003 _ phys. _ d*68 * 083514 [hep - th/0306033] kerlick g d 1976 _ ann. phys. _ * 99 * 127 goenner h and mller - hoissen f 1984 _ class. grav. _ * 1 * 651 capozziello s, carloni s and troisi a 2003 _ recent res. astrophys. _ * 1 * 625 [astro - ph/0303041] + boehmer c g and burnett j 2008 _ phys. _ d*78 * 104001 [arxiv:0809.0469 [gr - qc]] + boehmer c g 2005 _ acta phys. _ b**36 * * 2841 [gr - qc/0506033] + mielke e w and romero e s 2006 _ phys. _ d*73 * 043521 + minkevich a v, garkin a s and kudin v i 2007 _ class. * 24 * 5835 [arxiv:0706.1157 [gr - qc] shie k f, nester j m and yo h j 2008 _ phys. _ d*78 * 023522 [arxiv:0805.3834 [gr - qc]] + yo h j and nester j m 2007 _ mod. _ a*22 * 2057 [astro - ph/0612738] + li x z, sun c b and xi p 2009 _ phys. _ d*79 * 027301 [arxiv:0903.3088 [gr - qc]] kopczyski w 1972 _ phys. _ a*39 * 219 nester j m and isenberg j a 1977 _ phys. _ d*15 * 2078 hecht r d, nester j m and zhytnikov v v 1996 _ phys. _ a*222 * 37 + yo h j and nester j m 1999 _ int _ d*8 * 459 [gr - qc/9902032] shapiro i l 2002 _ phys. rep. _ * 357 * 113 [hep - th/0103093] hao j g and li x z 2004 _ phys. _ d*70 * 043529 [astro - ph/0309746] sahni v, saini t d, starobinsky a a and alam u 2003 _ jetp lett. _ * 77 * 201 [astro - ph/0201498] alam u, sahni v and saini t d and starobinsky a a 2003 _ mon. not. soc. _ * 344 * 1057 [astro - ph/0303009] zhang x 2007 _ jcap _ * 0703 * 007 [gr - qc/0611084] + chang b, liu h, xu l, zhang c and ping y 2007 _ jcap _ * 0701 * 016 [astro - ph/0612616] + liu d j and liu w z 2008 _ phys. _ d*77 * 027301 [arxiv:0711.4854 [astro - ph]] tretyakov p, toporensky a, shtanov y and sahni v 2006 _ class. * 23 * 3259 [gr - qc/0510104] hoyel f, burbidge g and narlikar j v 1993 _ astrophys. j. _ * 410 * 437 [astro - ph/9412045] hehl f w, mccrea j d, mielke e w and neeman y 1995 _ phys. rept. _ * 258 * 1 [gr - qc/9402012] li x z, zhao y b and sun c b 2005 _ class _ * 22 * 3759 [astro - ph/0508019] starobinsky a a 1998 _ jetp lett. _ * 68 * 757 [astro - ph/9810431] + liu d j, sun c b and li x z 2006 _ phys. _ b*634 * 442 [astro - ph/0512355] choudury t r and padmanabhan t 2005 _ astron. _ * 429 * 807 [astro - ph/0311622] ahlfors v 1979 _ complex analysis _ (new york : mcgraw hill) wood - vasey w m et al. 2007 _ astrophys. j _ * 666 * 694 [astro - ph/0701041] riess a g et al. 2007 _ astrophys. j _ * 659 * 98 [astro - ph/0611572] davis t m et al. 2007 _ astrophys. j _ * 666 * 716 [astro - ph/0701510] sahni v, shtanov y and viznyuk a 2005 _ j. cosmol. * 0512 * 005 [astro - ph/0505004]

we apply the statefinder diagnostic to the torsion cosmology, in which an accounting for the accelerated universe is considered in term of a riemann - cartan geometry : dynamic scalar torsion. we find that there are some typical characteristic of the evolution of statefinder parameters for the torsion cosmology that can be distinguished from the other cosmological models. furthermore, we also show that statefinder diagnostic has a direct bearing on the critical points. the statefinder diagnostic divides the torsion parameter @xmath0 into differential ranges, which is in keeping with the requirement of dynamical analysis. in addition, we fit the scalar torsion model to essence supernovae data and give the best fit values of the model parameters.

0903.4724

Introduction it is believed that solar magnetic fields are created in the tachocline, the interface between the convection zone and the radiative interior. due to buoyancy instabilities, they move upward and emerge into the solar atmosphere in the form of @xmath0-shaped flux tubes @xcite. the largest emerging active regions produce sunspots with magnetic fluxes in excess of @xmath4 mx and lifetimes of several weeks to months. smaller active regions consist of pores and contain an order of magnitude less flux, persisting over days to weeks. the smallest emerging regions detected to date are the so - called ephemeral regions. they have fluxes between @xmath5 and @xmath6 mx and lifetimes in the range from hours to days (e. g. *??? * ; *??? * ; *??? *). outside of active regions, the quiet sun has proved to be full of magnetic fields with strengths roughly in equipartition with the photospheric convective flows @xcite. an important question is the origin of these fields. @xcite suggested that horizontal internetwork fields represent concentrated loops of flux carried to the surface by the upflows of granular convection or by magnetic buoyancy. in a recent paper, @xcite indirectly traced the emergence of magnetic flux and reconstructed, for the first time, the three dimensional topology of the magnetic field vector in quiet regions of the solar photosphere. it was found that at least 20% of the magnetic flux in the quiet sun is connected by low - lying magnetic loops. later, @xcite studied time series of spectropolarimetric observations taken with the solar optical telescope aboard _ hinode_. these authors followed the time evolution of one magnetic loop in the internetwork, showing that they appear on spatial scales smaller than 2@xmath7. @xcite and @xcite demonstrated that the emergence of magnetic flux on granular scales brings large amounts of horizontal fields to the photosphere both in plage regions and in the quiet sun. another form of flux emergence has been reported by @xcite. it involves the appearance and subsequent disappearance of what seem to be _ vertical _ fields at the center of granular cells. the observations strongly suggest that a significant fraction of the magnetic flux in the quiet sun might be the result of the emergence of small - scale magnetic loops. but, where do the loops come from? are they created by the global solar dynamo, by a local dynamo, or by recycling of flux from decaying active regions? is the emergence process a local phenomenon confined to the photosphere or does the magnetic flux reach higher atmospheric layers? the answers to these questions bear important consequences for our understanding of the magnetic and thermal structure of the solar atmosphere. for example, @xcite claim that the magnetic energy stored in the quiet photosphere is sufficient to balance the radiative losses of the chromosphere. quiet sun magnetic fields are excellent candidates to solve the chromospheric and coronal heating problem, but a mechanism capable of transferring their energy to the upper layers has not been identified yet. from a theoretical point of view, it is not clear whether the fields of the quiet solar photosphere can rise to the chromosphere. @xcite have presented mhd simulations in which the magnetic field emerges into the photosphere in the form of small - scale @xmath0-loops. they reach the chromosphere and get reconnected with the local expanding vertical magnetic fields, heating the plasma and generating high frequency mhd waves that propagate into the corona. however, the magnetoconvection simulations of @xcite show @xmath0-loops that disintegrate as they rise through the solar atmosphere. these discrepancies emphasize the need for observational studies aimed at determining whether magnetic fields emerging into the quiet photosphere are able to reach higher atmospheric layers. here we use multi - wavelength observations taken by _ hinode _ and the dutch open telescope to address this question. we also characterize the physical properties of small - scale magnetic loops in the quiet sun, providing estimates of their magnetic fluxes, emergence rates, lifetimes, sizes, and velocities. Observations and data reduction the data analyzed in this paper consist of time series of polarimetric and imaging observations of quiet sun regions at disk center. they were acquired in seven different days (25 - 29 september, 1 and 6 october 2007) using the instruments of the solar optical telescope aboard hinode @xcite and the dutch open telescope (dot ; *??? *) at observatorio de el roque de los muchachos (la palma, spain). the observations belong to the _ hinode _ operation plan 14, entitled `` hinode / canary islands campaign ''. the _ hinode _ spectro - polarimeter (sp ; *??? *) recorded the full stokes vector of the pair of lines at 630 nm in a narrow field of view (fov) of @xmath8. this region was scanned with a cadence of 28 s during 2 - 6 hours per day (table [tabla_obs]). the exposure time per slit position was set to 1.6 s to track very rapid events. however, this mode of operation also led to a noise level of @xmath9 in units of the continuum intensity @xmath10. with a pixel size of @xmath11 along the slit and @xmath12 perpendicular to it, the sp measurements have a spatial resolution of about @xmath13. the hinode narrowband filter imager (nfi ; tsuneta et al. 2008) acquired stokes i and v filtergrams in the wings of the chromospheric mg i b 517.3 nm line, @xmath14 pm away from its center. the nfi was operated in shutterless mode to reach an effective exposure time of 9.6 s per wavelength and polarization state, covering a fov of @xmath15. the original filtergrams had a pixel size of 0.08@xmath7, but we rebined them to the sp pixel size in order to further reduce the noise. the _ hinode _ broadband filter imager (bfi ; *??? *) acquired simultaneous images of the photosphere in the cn bandhead at 388.3 nm (filter width of 0.52 nm) and the chromosphere in the caiih line at 396.85 nm (filter width of 0.22 nm). the exposure times were 0.1 s and 0.3 s, respectively. the bfi covered a region of @xmath16 with a pixel size of @xmath17. both the nfi and the bfi took images with a cadence of 30 s. the area scanned by the sp represents a small part of the total fov of the nfi and the bfi. therefore, we have cospatial and cotemporal observations of the quiet sun tracing different heights in the atmosphere. the dot observed photospheric and chromospheric layers by means of a tunable lyot filter that scanned the intensity profile of the h@xmath18 line at five wavelength positions (@xmath19, @xmath20, and 0 ). the passband of the filter was 0.25 . speckle bursts of 100 frames were taken at each wavelength position every 30 s. following the standard reduction procedure at the dot, the individual filtergrams were reconstructed using a speckle masking technique (see *??? * for details). the reconstructed images cover a fov of @xmath21 and have a spatial resolution of about 0.2. the dot and _ hinode _ carried out simultaneous observations, but there is little overlap between them because of bad weather conditions. the sp data have been corrected for dark current, flat - field, and instrumental cross - talk using the sp_prep.pro routine included in the solarsoft package. the algorithm applied to the hinode filtergrams (fg_prep.pro) removed dark current, hot pixels, and cosmic rays. the spectropolarimetric maps and the various filtergrams have been aligned with pixel accuracy using the granulation, g - band bright points, and network elements as a reference. in fig. [calcio] we show time - averaged h filtergrams and b magnetograms for the seven days of observation. the h images have been trimmed to the size of the b fov. the rectangles represent the areas scanned by the sp. note the absence of strong brigthenings in the h maps, as expected for very quiet regions largely devoid of network elements. + ccccc date & 1st period & 2nd period & observed & detected + & (ut) & (ut) & time & loops + 25@xmath22 sep & 13:00 - 15:59 & & 3.0 h & 18 + 26@xmath22 sep & 08:15 - 14:14 & & 6.0 h & 16 + 27@xmath22 sep & 06:16 - 09:59 & 11:25 - 13:59 & 6.3 h & 7 + 28@xmath22 sep & 07:00 - 09:59 & 11:20 - 13:59 & 5.7 h & 11 + 29@xmath22 sep & 06:51 - 09:44 & & 2.9 h & 5 + 1@xmath23 oct & 08:21 - 10:09 & & 1.8 h & 3 + 6@xmath22 oct & 08:01 - 10:18 & & 2.3 h & 9 + Data analysis loop - like magnetic structures leave clear signatures in spectropolarimetric maps : linear polarization flanked by two circular polarization signals of opposite polarity. one of the main goals of this paper is to trace the possible ascent of small - scale magnetic loops through the solar atmosphere. to this end, we use photospheric and chromospheric observables. the sp data make it possible to investigate the topology of the field in the photosphere. the information is complemented by the cn filtergrams, where bright points associated with magnetic fields are easily visible. we define the total circular polarization as the integral of the unsigned stokes @xmath24 spectrum. the integration is carried out in the wavelength range from @xmath25 pm to @xmath26 pm. the total linear polarization is computed as the integral of the stokes @xmath27 profile of 603.25 nm, using the same initial and final wavelengths. we visually inspect the polarization maps to search for weak linear signals between two patches of circular polarization with opposite polarity. since in general the linear polarization signals are very small in the quiet sun, we ascribe those cases to loop - like structures only after corroborating that the linear polarization is produced by symmetric stokes @xmath28 and @xmath29 profiles. figure [primer_ejemplo] shows the first stages of the emergence of a small - scale magnetic loop as seen in the 630.25 nm line. the different panels represent continuum intensity (top), total linear polarization (middle), and total circular polarization (bottom). red contours indicate enhanced linear polarization. black and white contours mark the location of strong negative and positive circular signals. at @xmath30 s, a patch of linear polarization shows up at the center of the image. it corresponds to the horizontal part of a magnetic structure emerging into the photosphere. between @xmath31 and 60 s, two circular polarization patches of opposite polarity appear next to it, at the edge of a granular cell. the fact that the linear signal is detected earlier than the circular polarization indicates that the magnetic structure has the shape of an @xmath0-loop : the apex creates linear polarization and the vertical fields of the footpoints give rise to circular polarization @xcite. the linear polarization disappears below the noise at @xmath32 s while the footpoints separate with time. this sequence of events is consistent with a loop that emerges and travels up in the atmosphere. the distance between the footpoints keeps increasing until they reach the edges of the area scanned by the sp. the subsequent evolution of this loop will be studied in [description]. the stokes @xmath33 and @xmath24 filtergrams acquired in the red and blue wings of the b line give information about the upper photosphere / temperature minimum region @xcite. we have used them to construct longitudinal magnetograms (@xmath34) and dopplergrams (@xmath35) as @xmath36 where the subscripts r and b represent the measurements at @xmath37 and @xmath38 pm from line center, respectively. to first order, the magnetograms computed in this way are not affected by mass motions. the quantities @xmath34 and @xmath35 have been transformed into magnetic flux densities and line - of - sight velocities according to @xmath39 with @xmath40 in mx cm@xmath1 and @xmath42 in km s@xmath3. equation [dopplergram] is valid in the range @xmath43 km s@xmath3. these expresions have been obtained through calibration of the b line shape in the fourier transform spectrometer atlas of the quiet sun @xcite and give only rough estimates of the magnetic flux density and velocity at the height of formation of the b measurements. since stokes q and u were not recorded, the mg i magnetograms can only be used to detect relatively vertical fields such as those expected at the footpoints of magnetic loops. by definition, fields pointing towards the observer and upflows will both be positive. note that our sign convention for the velocity differs from that commonly used in astrophysics. in the chromosphere we do not have polarimetric information. however, magnetic fields can be detected through brightness enhancements in the h filtergrams. the passband of the _ hinode _ h filter includes a significant photospheric contribution, but it has a long tail that extends well into the chromosphere @xcite. finally, information on the upper chromosphere is provided by the h@xmath18 measurements taken at the dot. we have used them to construct dopplergrams at different heights in the chromosphere. we determine the magnetic flux density from the stokes @xmath24 profiles of the 630 nm lines using the weak field approximation (e.g., *??? *) @xmath44 where @xmath45 is the longitudinal flux density, @xmath46 the filling factor, @xmath47 the field strength, @xmath48 the inclination of the field with respect to the vertical, @xmath49 a proportionality constant that depends on the central wavelength @xmath50 and the effective land factor @xmath51 of the transition, and @xmath33 represents the intensity profile. the units of @xmath40 are mx cm@xmath1 when @xmath50 is expressed in . the longitudinal flux density is obtained from a least - squares minimization of the form @xmath52}{\partial \phi}=0, \frac{\partial}{\partial \phi}\left[\sum_i \left (v_i+\phi c\frac{\partial i}{\partial \lambda}_i\right)^2\right]=0, \label{weakfield}\]] which uses all the wavelength samples across the profile (index @xmath53) and is therefore more accurate that determinations based on single magnetogram measurements. this calculation is repeated for each pixel and each spectral line separately. the final result is @xmath54 to estimate the uncertainty caused by photon noise we simulated a data set containing only gaussian noise with a standard deviation of @xmath55 @xmath10. the analysis of this data set using eq. [eq2] leads to a gaussian - shaped histogram for the magnetic flux density which is centered at 0 and has a standard deviation of @xmath56 mx cm@xmath1. in figure [hist_ratio_flux] we check the assumption made on the zeeman regime. the plot shows an histogram of the ratio between the magnetic flux densities derived from 630.15 and 630.25 nm. only the stokes v spectra at the footpoints of the loops having amplitudes above 5 times the noise level have been considered. as can be seen, the histogram is narrow and peaks at 1. the figure also shows a scatter plot of the magnetic flux densities obtained with the two lines. the fact that most of the points are located near the diagonal implies that in the majority of cases both spectral lines measure the same magnetic flux. this strongly supports the idea that the fields are weak. Emergence of small-scale magnetic loops in the quiet sun during 28 hours of observations we have detected the appearance of 69 loop - like structures in the @xmath57 region scanned with the hinode spectropolarimeter. table [tabla] summarizes their properties. the observed loops followed very similar evolution patterns. in this section we describe specific cases to illustrate the emergence process, distinguishing between loops that rise to the chromosphere and loops that remain low - lying. one of the clearest examples of a loop that reached the chromosphere was loop ml09, observed on september 25, 2007 at 14:39 ut. its evolution is summarized in figs. [ej1_crom] and [ej1_crom2]. each row represents a time step, for a total of 1020 s. from left to right we show cn filtergrams, photospheric los velocities derived from the stokes @xmath24 zero - crossing shifts of 630.25 nm, total circular polarization maps in 632.05 nm, magnetic flux densities computed from b 517.3 nm, los velocities at the height of formation of the b line, and h line - core filtergrams. positive velocities indicate upflows. note that the time steps are not evenly spaced ; rather, we have adjusted them to better describe the various phases of the process. the red contours mark regions of large linear polarization signals in 630.25 nm. when the footpoints of the loop are visible in the photosphere, we plot contours of 630.25 nm circular polarization in black and white for the negative and positive signals, respectively. when the footpoints are only visible in the b magnetogram we plot them in blue and turquoise. the first row of fig. [ej1_crom] shows the emergence of linear polarization above a granule (cf. the white arrow in the cn filtergram). it is caused by the horizontal part of the loop reaching the photosphere. the footpoints are not yet visible but will appear 30 s later, very close to the patch of linear polarization. at @xmath58 the loop is completely formed. it emerges in a granular region, perhaps because the upward granular motions help the field lines to rise from below the solar surface @xcite. the los velocity maps show photospheric upflows at the position of the footpoints, confirming that the loop is rising. for the moment, however, the magnetic field remains in the lower photosphere : we do not observe circular polarization signals in the b magnetograms or brightenings in caiih that could be associated with the loop. interestingly, the strong photospheric network element with negative polarity located towards the bottom of the scan, at a height of about one fourth of the displayed fov, is well observed both in the b magnetograms and the ca filtergrams, and will remain so during most of the time sequence. this nicely illustrates the capabilities of our observations : magnetic structures that are visible in the polarization maps but not in b or h are intrinsically lower in the solar atmosphere. between @xmath59 s and @xmath60 s the linear polarization signals disappear below the noise. the positive footpoint has drifted to an intergranular lane and is concentrated, whereas the negative footpoint continues to be rooted in the granule and is more diffuse. the distance between them increases steadily. we still see upward motions in the photospheric velocity maps. since the loop is moving to higher layers, it is reasonable to conclude that the linear polarization disappears because the apex of the loop leaves the formation region of the 630 nm lines. however, no traces of the loop are detected yet in b or h. at @xmath61 s, weak circular polarization signals cospatial with the photospheric footpoints are observed in the b magnetograms for the first time. this indicates that the loop has reached the upper photospheric / lower chromospheric layers where the central part of the b line forms. interestingly, the b dopplergram exhibits downflows of about @xmath62 km s@xmath3 at the position of the positive footpoint. the downflows could represent plasma moving along the legs of the loop as the whole structure reaches high atmospheric layers. these motions may be essential for the loop to get rid of part of its mass before it can emerge into a less dense medium. at @xmath63 s, the loop is nearly out of the region scanned by the sp. the signals in the b magnetogram are much more intense and correspond to footpoints rooted in intergranular lanes. from now on the distance between the footpoints will increase, but at a slower rate than when they were crossing granular structures. this inflection point can be seen in fig. [dist], where we plot the footpoint separation as a function of time. the distance is computed only when the two footpoints are visible, both in the maps (squares) and in the b magnetograms (triangles). in the first 500 s of the loop evolution, the distance between the footpoints increases linearly at a rate of 5.9 km s@xmath3. therefore, the mean velocity of the footpoints is 2.95 km s@xmath3, a value compatible with the motion of the granular plasma. the linear increase of the separation with time is a common feature of the loops and indicates that they do not undergo a free random walk (otherwise the distance would increase as the square root of time). towards the end of the loop evolution the separation rate slows down, coinciding with the arrival of the footpoints to intergranular lanes. summarizing, the loop emerges in a granule and the horizontal granular motions drive the magnetic field lines to the closest intergranular space, where strong downdrafts capture and stabilize them. when this happens, the separation between the footpoints is about 4000 km. at @xmath64 s, the footpoints are clearly visible in the b magnetogram and exhibit downflows in the b dopplergram. the whole structure is rising because the footpoints continue to separate. however, no brightenings are detected in the h filtergrams. we mention in passing that a new loop appear in the fov at this time, very close to the site of emergence of the structure we are describing in detail. they show a linear polarization signal in between opposite polarities, which makes it easy to identify. at @xmath65 s, the loop has reached the chromosphere since we observe two h brightenings associated with the footpoints (in fig. [ej1_crom2], the contours have been substituted by arrows for clarity). the legs of the loop still show downflows in the b dopplergrams and, for the first time, bright points are observed in the cn images at the position of the footpoints. the last panels of fig. [ej1_crom2] displays the beginning of the loop decay. the positive footpoint is very weak, although it still shows downflows at the height of formation of the b measurements. it will disappear below the noise level, together with the downdrafts, at the end of the sequence. the negative footpoint is approaching a negative polarity patch with whom it will eventually mix. the negative footpoint shows downflows and is associated with a bright point in h. these features will survive the disappearance of the footpoint thanks to the interaction with the network element. in this section we present a typical example of a loop which do not show chromospheric signatures and thus remain low - lying. figure [ej1_nocrom] shows all the data available for this loop (ml23), arranged as in fig. [ej1_crom]. in the first frame, a patch of linear polarization is observed to emerge at the border of a granule (see the white arrow). the footpoints can already be detected in the intergranular lane, but they are very weak. at this time the photospheric velocity map exhibits a patch of upflows at the position of the loop, confirming its rise through the solar atmosphere. at @xmath66 s the loop reaches its largest extent (520 km) while the upflows start to weaken. the linear polarization and the upward plasma motions are almost gone by @xmath67 s. in the next frame, at @xmath68 s, the loop is no longer seen. the evolution of the loop is so rapid that it appears and disappears almost at the same place. interestingly, the footpoints never approach each other. this rules out submergence below the solar surface as the cause of the loop disappearance. all the loops that stay in the photosphere show very similar behaviors. in general, the evolve very quickly, disappearing not far from the region where they emerged. none of these loops exhibit downflows in the b line or brightenings in the cn or ca filtergrams. Effects in the upper chromosphere in the previous section we have seen that small - scale magnetic loops in the quiet sun may rise through the atmosphere and reach the layers where the central part of the b 517.3 nm line is formed. some of them also produce h brightness enhancements. in this context, the question naturally arises as to the maximum height that these structures can attain. are they able to reach the upper chromosphere or even the corona? here we use the h@xmath18 observations of the dot to provide a partial answer to this question. unfortunately, there is little overlap between the _ hinode _ and dot measurements because of bad seeing conditions. nevertheless, for one of the loops detected by _ there is simultaneous coverage from the dot. we use these data to attempt to observe the rise of the loop to the upper chromosphere. the analysis is not complete and must be refined with euv and x - ray observations tailored to the detection of such magnetic structures in the hot corona. the loop observed simultaneously by _ hinode _ and the dot (ml20) appeared on september 26, 2007, at 09:06 ut. in fig. [ej2_crom_todos] we show its evolution in the photosphere / temperature minimum region using the 630.25 nm circular polarization maps and the b magnetograms (first and second columns, respectively). the figure also displays h@xmath18 line core filtergrams, as well as h@xmath18 dopplergrams at @xmath20 and @xmath19 from line center. if the loop reaches the layers where h@xmath18 is formed, it should first appear in the fifth column, then in the fourth, and finally in the third. the loop emerged as a small patch of linear polarization at the border of a granule (@xmath69 s, not shown). its subsequent evolution is similar to that of the loop considered in fig. [ej1_crom]. the footpoints of the loop are detected in the photosphere for the first time at @xmath70 s. the whole structure shows upflows in the stokes v zero - crossing maps, indicating its ascent. unfortunately, the negative footpoint is close to the border of the fov scanned by the sp and soon disappears from the photospheric maps. the loop becomes visible in the b magnetograms at @xmath71 s. at this time there is clear signal in the negative footpoint and weaker polarization in the positive leg. at @xmath65 s both footpoints show stronger magnetogram signals but the distance between them has not increased. the ascent of the loop to the chromosphere is associated with downflows in the b dopplergram and brightenings in the caiih line - core images. the and b signals start to fade at @xmath72 s until the loop disappears simultaneously from the low and the upper photospheric layers. the h@xmath18 line core images and the dopplergrams do not show any particular feature that could be associated with the appearence of an arch filament system in the chromosphere. it is important to remark that the maximum separation between the footpoints of this loop was only 760 km. it may well be that a larger separation is required for the apex of the loop to reach the upper chromosphere. in fact, excessive magnetic tension might prevent the field lines from rising. keeping in mind these considerations, we do not discard that magnetic loops with larger separations may be seen in future h@xmath18 observations. Physical properties of the emerging loops in this section we characterize the physical properties of the small - scale magnetic loops observed with _ weak examples and loops appearing close to strong network elements or in crowded areas are omitted from the analysis to maintain the quality of the results. this leaves us with 33 loops, which represents 48% of the total sample. table [tabla] lists the basic parameters of the loops, including lifetimes (@xmath73), maximum distances between footpoints (@xmath74), speeds at which the footpoints separate initially (@xmath75), total magnetic fluxes (@xmath76) and maximum flux densities (@xmath77) in the photosphere, estimations of maximum magnetic flux densities at the height of formation of the b measurements (@xmath78), and an estimation of the largest downflows detected in the b dopplergrams (@xmath79). the lifetime is the time elapsed between the appearance and disappearance of the polarization signals. when two numbers are given, the first indicates the time passed until one of the footpoints interacts with a neighboring magnetic element. the second is the time of disappearance proper ; if it is accompanied by an asterisk, then the polarization signatures of the loop were still visible at the end of the observations. to compute the total magnetic flux we define the footpoints as those regions where the flux density is larger than @xmath80 at the position of the loop. the value of @xmath81 reported in table [tabla] is the maximum flux detected in one of the footpoints during the loop evolution, and the error indicates the uncertainty in @xmath81 caused by photon noise. the last four columns of table [tabla] give the time intervals between the appearance of the loops in the photosphere and their detection in the b magnetograms, the b dopplergrams, the h line core images, and the cn filtergrams. we consider that a loop is present in any of these maps when at least one of the footpoints shows up clearly. the two numbers in each column correspond to the positive footpoint (left) and the negative one (right). cccccccccccccc name & date & t@xmath82 & @xmath83 t & d@xmath84 & @xmath85 & @xmath86 & @xmath87 & @xmath88 & @xmath89 & @xmath83t@xmath90 & @xmath83t@xmath91 & @xmath83t@xmath92 & @xmath83t@xmath93 + & (09/07) & (ut) & (s) & (km) & (km / s) & (mx) & (mx/@xmath94) & (mx/@xmath94) & (km / s) & (s) & (s) & (s) & (s) + ml01 & 25 & 13:36:15 & 240 & 800 & 3.9 & 4.6@xmath9510@xmath9610@xmath97 & @xmath98 & & & & & & + ml02 & 25 & 13:35:45 & 150/1020 & 1220 & 4.0 & 1.5@xmath9510@xmath9910@xmath97 & @xmath100 & & & & & & + ml03 & 25 & 13:37:45 & 330 & 560 & 0.9 & 7.6@xmath9510@xmath9610@xmath97 & @xmath101 & & & & & & + ml04 & 25 & 13:46:15 & 60/480 & 690 & 1.7 & 8.2@xmath9510@xmath9610@xmath97 & @xmath102 & & & & & & + ml05 & 25 & 13:45:15 & 240 & 790 & 1.1 & 1.1@xmath9510@xmath10310@xmath97 & @xmath104 & & & & & & + ml06 & 25 & 13:36:15 & 180 & 910 & 6.0 & 3.6@xmath9510@xmath10510@xmath106 & @xmath107 & & & & & & + ml07 & 25 & 13:54:15 & 90 & 490 & 1.4 & 3.5@xmath9510@xmath10810@xmath106 & @xmath109 & & & & & & + ml08 & 25 & 13:41:45 & 630 & 800 & 6.2 & 3.9@xmath9510@xmath9610@xmath97 & @xmath110 & & & & & & + ml09 & 25 & 14:39:30 & 960/1200 & 4000 & 3.9 & 1.3@xmath9510@xmath10310@xmath97 & @xmath111 & 15.0@xmath112 & -1.1 & 270/540 & 270/750 & 780/780 & 780/1230 + ml10 & 25 & 14:42:00 & 120 & 560 & 1.7 & 1.5@xmath9510@xmath11310@xmath106 & @xmath114 & & & & & & + ml11 & 25 & 14:51:00 & 300/2550 * & 1670 & 2.5 & 1.2@xmath9510@xmath10310@xmath97 & @xmath115 & 10.4@xmath116 & -0.58 & 180/450 & 180/450 & 420/660 & /660 + ml12 & 25 & 14:51:00 & 300/2550 * & 2040 & 0.7 & 1.3@xmath9510@xmath10310@xmath97 & @xmath117 & 8.3@xmath1180.8 & & 690/750 & & & + ml13 & 25 & 15:04:00 & 660/1230 & 1440 & 2.1 & 7.8@xmath9510@xmath9610@xmath97 & @xmath119 & 28.6@xmath1180.9 & -0.70 & 390/390 & 690/ & 750/ & 690/ + ml14 & 25 & 15:06:00 & 180 & 480 & 3.7 & 6.1@xmath9510@xmath9610@xmath97 & @xmath120 & & & & & & + ml15 & 25 & 15:14:00 & 360/1590 * & 3170 & 0.5 & 2.0@xmath9510@xmath9910@xmath97 & @xmath121 & 5.4@xmath1180.5 & -0.91 & 330/330 & 570/420 & 750/420 & 690/570 + ml16 & 25 & 15:24:30 & 510 & 790 & 0.9 & 1.3@xmath9510@xmath9910@xmath97 & @xmath122 & 7.3@xmath1180.5 & -0.59 & 240/240 & 270/270 & & + ml17 & 25 & 14:50:30 & 90 & 620 & 1.0 & 5.5@xmath9510@xmath9610@xmath97 & @xmath123 & & & & & & + ml18 & 26 & 08:32:00 & 510/1110 * & 990 & 1.1 & 1.1@xmath9510@xmath10310@xmath97 & @xmath124 & & & & & & + ml19 & 26 & 08:23:30 & 660 & 1670 & 3.1 & 6.2@xmath9510@xmath9610@xmath97 & @xmath125 & 10.0@xmath1180.9 & -0.33 & /180 & /180 & /210 & /180 + ml20 & 26 & 09:06:00 & 1050 & 1450 & 2.2 & 1.5@xmath9510@xmath11310@xmath106 & @xmath126 & 5.0@xmath1180.4 & -0.38 & 390/360 & 510/390 & 570/420 & 600/390 + ml21 & 26 & 09:47:30 & 240 & 700 & 2.8 & 3.8@xmath9510@xmath10510@xmath106 & @xmath127 & & & & & & + ml22 & 26 & 11:24:00 & 1380 * & 1570 & 0.9 & 2.4@xmath9510@xmath9910@xmath97 & @xmath128 & 11.8@xmath1181.1 & -0.48 & 360/450 & 600/750 & 630/750 & + ml23 & 26 & 11:48:00 & 150 & 520 & 3.6 & 3.3@xmath9510@xmath10510@xmath106 & @xmath129 & & & & & & + ml24 & 26 & 12:16:00 & 120/390 & 830 & 3.4 & 1.9@xmath9510@xmath13010@xmath97 & @xmath131 & & & & & & + ml25 & 26 & 12:28:00 & 1230 & 2840 & 0.1 & 2.0@xmath9510@xmath13210@xmath106 & @xmath133 & 7.50@xmath1180.8 & -0.56 & 210/420 & 540/630 & 690/690 & + ml26 & 27 & 09:25:00 & 1110 & 760 & 0.9 & 7.0@xmath9510@xmath9610@xmath97 & @xmath134 & 26.7@xmath1180.8 & -0.42 & 270/270 & 540/ & & + ml27 & 27 & 11:08:00 & 420 & 960 & 4.5 & 8.6@xmath9510@xmath9610@xmath97 & @xmath135 & & & & & & + ml28 & 27 & 12:20:30 & 90 & 670 & 0.9 & 3.0@xmath9510@xmath10510@xmath106 & @xmath136 & & & & & & + ml29 & 27 & 12:26:30 & 1110 & 2290 & 4.0 & 1.7@xmath9510@xmath9910@xmath97 & @xmath137 & 14.5@xmath1180.8 & -0.49 & 150/150 & 450/390 & /450 & + ml30 & 28 & 11:28:30 & 300/750 & 1300 & 0.9 & 7.8@xmath9510@xmath9610@xmath97 & @xmath138 & & & & & & + ml31 & 28 & 11:38:00 & 180 & 1560 & 1.5 & 7.8@xmath9510@xmath9610@xmath97 & @xmath139 & & & & & & + ml32 & 28 & 12:32:30 & 720/720 & 710 & 1.5 & 1.9@xmath9510@xmath10310@xmath97 & @xmath140 & 19.0@xmath1180.5 & & 240/240 & & 360/ & 330/ + ml33 & 29 & 08:39:00 & 240/630 & 860 & 2.4 & 2.1@xmath9510@xmath10310@xmath97 & @xmath128 & & & & & & + * mean * & & & * 741 * & * 1234 * & * 2.2 * & * 9.13@xmath9510@xmath141 * & * 26.1 * & * 13.0 * & * -0.60 * & * 295 * & * 406 * & * 513 * & * 514 * + + as can be seen in table [tabla], there is a wide range of loop parameters. the lifetimes vary from some 2 min up to 40 min, although most of the loops disappear in less than 10 minutes. the maximum separation between the footpoints is a strong function of the lifetime and ranges from @xmath142 km to 4000 km. many loops reach horizontal dimensions comparable to, or larger than, those of granules. therefore, they must be viewed as coherent structures capable of withstanding the conditions of the granular environment for a relatively long time. the initial velocity of separation between the footpoints does not seem to have any relationship with the other parameters listed in the table. values of 0.1 to 6 km s@xmath3 are typical. as already mentioned, the separation speed tends to decrease when the footpoints reach the intergranular space, likely because horizontal motions there are not as vigorous as in the interior of granular cells. the longitudinal magnetic fluxes measured in the footpoints range from @xmath143 to @xmath144 mx at the level of formation of the 630 nm lines mx, considerably larger than the rest of values. this might be an artifact caused by the difficult separation of ml24 and the strong network element with which it interacts.]. therefore, the loops have smaller fluxes than ephemeral regions and should be placed at the lower end of the flux distribution observed in emerging active regions. the magnetic flux density in the footpoints is typically 2040 mx cm@xmath3. to infer the magnetic field strength from the magnetic flux density values we need to know the filling factor of the field lines that build the loop structure and their inclination. the footpoints should be relatively vertical because of geometrical reasons. assuming that the fields occupy most of the resolution element, i.e., that the magnetic filling factor is close to unity the field strength of the loops can be estimated to be of order 10 - 100 g. only if the filling factor is much smaller than unity would the field strength increase to kg values, but we consider this possibility unlikely in view of fig. [hist_ratio_flux]. an important result is that 23% of the loops are detected in the b magnetograms after their appearance in the photosphere (16 cases out of 69). it takes an average of 5 minutes for the loops to move from the photosphere to the height at which the b measurements form, although faster and slower ascents have been observed too. all the loops detected in the b magnetograms develop downflows at the same heights. in addition, 15% of the loops are seen as bright points in h line - core filtergrams. this means that an important fraction of the magnetic flux that emerges into the photosphere reaches the chromosphere. as they travel upward, the loops are observed in the magnetograms, the b magnetograms, the b dopplergrams, the h line - core images, and the cn filtergrams (in this order). by contrast, 77% of the loops never make it to the chromosphere. we have been unable to identify any parameter determining whether a given loop will rise or not. this includes the total magnetic flux and the magnetic flux density. however, low - lying loops tend to have lifetimes shorter than 500 s and separations smaller than 500 km. thus it may simply be that they do not last long enough to reach high atmospheric layers. further work is clearly needed to explain why a substantial fraction of the loops remain low - lying. also, the relation between these structures and the transient horizontal fields described by @xcite should be investigated, given their similar lifetimes and magnetic topologies. Sites of emergence, evolution, and tilt angles the observations described above demonstrate that magnetic fields do emerge into the quiet solar atmosphere in the form of small - scale loops, confirming the results of @xcite and @xcite. the loops are detected as a patch of linear polarization flanked by two circular polarization signals of opposite polarity. in nearly all the cases the linear polarization appears before or at the same time than the stokes @xmath24 signals, as can be expected from @xmath0-shaped loops rising through the atmosphere. only in two cases out of 69 have we detected linear polarization after the loop had already disappeared. in those cases, the footpoints were approaching each other. this behavior is compatible with a loop that emerges and then submerges in the photosphere, or with a `` magnetic bubble '', i.e., a circle of magnetic field lines. the long duration of our time series has permitted us to discover the existence of emergence centers in which several loops appear one after the other. for example, there is a @xmath146 region of the solar surface where we have detected 9 events in a time interval of 1 h. the complete set of observations covering this period is provided as an mpeg animation in the electronic edition of the astrophysical journal. some of the loops even appear at the very same position. the example shown in fig. [ej1_crom] belongs to this area. the existence of emergence centers may have important consequences for the origin of the loops. these regions act as subsurface reservoirs of magnetic flux that is transferred intermittently to the photosphere by an as yet unkwnown mechanism. the loops generally emerge in granules or at their edges, although there are exceptions of loops appearing in dark areas. as the loops emerge the footpoints separate and the linear polarization fades away. in most cases, the footpoints do not describe rectilinear trajectories. figure [trayectorias] shows the paths followed by the two polarities of the loops observed on september 25, 2007 (loops ml09 to ml17 in the table 2). the red curves correspond to the example discussed in fig.[ej1_crom]. in this case the footpoints described quite a rectilinear path, similarly to emerging active regions and ephemeral regions. however, the majority of loops show more complicated trajectories. the reason is that, in general, they emerge in granules and drift toward the closest intergranular lane. when the footpoints reach the intergranular space they stay there and are passively advected by the flow. this creates complicated trajectories. the important point, however, is that the magnetic field is sufficiently weak as to be pushed and moved around by the granular flow, but _ without being destroyed in the process_. the loops remain coherent during all their lifetime, as if the granular flow did not exist. during their evolution, the loops interact with other magnetic flux concentrations that cross their paths. if the loops stay long in the photosphere, the footpoints cancel with elements of opposite polarity or are absorbed by patches of the same polarity. loops that experience a fast evolution have more probabilities of avoiding other magnetic elements and often disappear without undergoing any interaction. in seeking the origin of the loops it is of interest to determine the magnetic orientation of their footpoints. if the loops are caused by the global solar dynamo, one may expect a regular ordering of the footpoints at the moment of emergence. this is what happens in active regions, where the signs of the leader and follower polarities are governed by hale s rules (see, e.g., *??? all the loops considered here appeared in the northern hemisphere during solar cycle 23. in that cycle, leader polarities were positive in the northern hemisphere and negative in the southern hemisphere. figure [tilt] shows an histogram of the orientation of our small - scale emerging loops. the tilt angle is defined to be the angle between the solar equator and the line joining the positive footpoint with the negative one, measured from the west. the angles compatible with the orientation of sunspot polarities during solar cycle 23 in the northern hemisphere are those between 90 and 270@xmath147. even if the statistical sample is not very large, the loops seem to have nearly random orientations. thus, we conclude that they do not obey hale s polarity rules, much in the same way as the shortest - lived ephemeral regions @xcite. Discussion and conclusions recent observations of hanle - sensitive lines @xcite and zeeman - sensitive lines @xcite suggest that a significant fraction of the quiet sun is occupied by magnetic fields. apparently, these fields are weak and isotropically distributed in inclination @xcite. one way to shed light on their nature is to study how they emerge in the surface and what their contribution is to the energy budget of the solar atmosphere. on granular scales, magnetic flux appears in the solar photosphere as transient horizontal fields @xcite and small - scale magnetic loops @xcite. we have studied the latter in detail using seeing - free observations made by _ hinode_. in 28 hours of _ hinode _ data we have detected 69 small - scale loops emerging in a quiet sun region of size @xmath148 at disk center. the occurrence rate is thus 0.02 events hr@xmath3 arcsec@xmath1. the loops show clear spectropolarimetric signatures with a central region of linear polarization and two patches of circular polarization of opposite polarity. the longitudinal flux observed in each footpoint ranges from @xmath143 to @xmath144 mx, with an average of @xmath149 mx. this means that the loops represent the smallest emerging flux regions detected to date (ephemeral regions have fluxes above 10@xmath150 mx ; zwaan 1987). the rate at which magnetic flux is carried to the quiet photosphere by the loops can be estimated to be @xmath151 mx s@xmath3 arcsec@xmath1, or @xmath152 mx over the solar surface per day. this is about half the value derived by lites et al. (1996) for horizontal internetwork fields, but still enormous (see @xcite for a comparison with the flux emergence rates in active and ephemeral regions). in the photosphere, the linear polarization associated with the top of the loop disappears soon, while the circular signals tracing the loop legs are observed to separate with time. this behavior is consistent with field lines moving upward through the solar atmosphere. also the upflows observed in the stokes v zero - crossing velocities at the position of the footpoints confirm the ascent of the loops. 23% of the loops are detected in b magnetograms that sample the upper photosphere or the temperature minimum region (say, 400 km above the continuum forming layer). there is a time delay of about 5 minutes between the first detection in the photosphere and the appearance in the b magnetograms, implying an ascent speed of the order of 1 km @xmath3. some of the loops continue to travel upward and become visible in h line - core filtergrams as small brightness enhancements. thus, a fraction of the loops are able to reach the low chromosphere, carrying magnetic flux with them. the rise of small - scale magnetic loops may provide an efficient mechanism to transfer substantial amounts of energy from the photosphere to the chromosphere. this would support claims by @xcite and @xcite that the tangled fields of the quiet sun store sufficient energy to heat the chromosphere. a related question is whether the small - scale loops rise up to the transition region or even the corona. the observations required to answer this question are quite challenging due to the different spatial resolutions attainable with present day optical, euv, and x - ray instruments, but should be pursued. about 77% of the loops that appear in the solar surface never rise to the chromosphere. these loops have the shortest lifetimes and show the smallest footpoint separations ; other than that, they do not differ from those reaching higher layers. usually, they disappear close to their emergence sites. the fields associated with these loops might represent the tangled quiet sun fields deduced from hanle measurements @xcite, but a definite conclusion can not be made without studying the compatibility of hanle and zeeman measurements. what is the origin of the small - scale magnetic loops? one possibility is that they are created by the solar dynamo at the bottom of the convection zone, as part of a larger toroidal flux tube. @xcite presented three - dimensional mhd simulations of the last stages of the emergence of one such tube. they placed a horizontal tube at the top of the convection zone, just beneath the photosphere. when the initial magnetic flux is smaller than @xmath15310@xmath150 mx, the tube is not sufficiently buoyant to rise coherently against the convective flows and fragments. at the surface, the process of flux emergence occurs on very small spatial scales (typically 1000 - 2000 km) and short time scales (5 min). these properties are compatible with our observations. thus, the small - scale loops we have detected may simply be the result of weak flux tubes distorted by the granulation as they emerge from the convection zone into the photosphere. the fragmentation of the tubes might explain why there are emergence centers where loops appear recurrently one after the other. a preliminary analysis of the footpoint orientations suggests that the loops do not show a tendency to be aligned according to hale s rules. this can be regarded as a considerable difficulty against the idea that the origin of the loops is the solar dynamo. however, it may also be a natural consequence of the interaction of the tube s fragments with the near - surface granular convection if it removes all the information carried originally by the tube. another possibility is that the magnetic loops represent flux recycled from decaying active regions. in a sense, the mhd simulations of abbett (2007), @xcite, and @xcite model such a process, because all of them assume an initial magnetic field in the computational box which could be provided by decaying active regions. in the simulations, the field evolve and interact with the granular flows. this interaction creates a significant amount of horizontal fields, even if the initial field is purely vertical. moreover, the simulations show the emergence of magnetic loops on granular scales. the loops are less coherent than classical flux tubes and do not connect to deeply rooted field lines. in this scenario, the magnetic fields of the quiet sun, and thus the emergence events we have described, would be the consequence of local processes acting on the remnants of decaying active regions. yet another possibility is that the loops represent submerged horizontal magnetic fields carried to the surface by the upward motions of granules or by magnetic buoyancy, as modeled by @xcite. even in that case, the origin of such submerged fields would be unknown. nowadays, we do not have enough observational constraints to distinguish between a surface dynamo or a `` exploding '' magnetic flux tube emerging from the solar interior. determining the nature of the magnetic loops observed in internetwork regions is important for a better understanding of the magnetism of the quiet sun and its role in the heating of the solar atmosphere. future efforts should concentrate on the solution of these problems. in addition to high - resolution photospheric observations, polarization measurements in the chromosphere are recquired to track the evolution of the loops with height. these data can now be provided by two - dimensional spectrometers like ibis, crisp, or imax. we thank andrs asensio ramos, pascal dmoulin and rafael manso sainz for very helpful discussions, and vronique bommier for carefully reading the manuscript. we are grateful to all the observers who participated in the _ hinode _ operation plan 14, both at isas / jaxa and at the ground - based telescopes. special thanks are due to suguru kamio (naoj) for coordinating the campaign and to peter stterlin (utrecht university) for making the observations at the dutch open telescope and reducing them. hinode is a japanese mission developed and launched by isas / jaxa, with naoj as a domestic partner, and nasa and stfc (uk) as international partners. it is operated by these agencies in cooperation with esa and nsc (norway). part of this work was carried out while one of us (mjmg) was a visiting scientist at the instituto de astrofsica de andaluca. we acknowledge financial support from the spanish micinn through projects esp2006 - 13030-c06 - 02, pci2006-a7 - 0624, and aya2007 - 63881, and from junta de andaluca through project p07-tep-2687. , d., bellot rubio, l. r., del toro iniesta, j. c., tsuneta, s., lites, b. w., ichimoto, k., katsukawa, y., nagata, s., shimizu, t., shine, r. a., suematsu, y., tarbell, t. d., & title, a. m. 2007, apj, 670, 61

we investigate the emergence of magnetic flux in the quiet sun at very small spatial scales, focusing on the magnetic connection between the photosphere and chromosphere. the observational data consist of spectropolarimetric measurements and filtergrams taken with the hinode satellite and the dutch open telescope. we find that a significant fraction of the magnetic flux present in internetwork regions appears in the form of @xmath0-shaped loops. the emergence rate is 0.02 loops per hour and arcsec@xmath1, which brings @xmath2 mx s@xmath3 arcsec@xmath1 of new flux to the solar surface. initially, the loops are observed as small patches of linear polarization above a granular cell. shortly afterwards, two footpoints of opposite polarity become visible in circular polarization within or at the edges of the granule and start to move toward the adjacent intergranular space. the orientation of the footpoints does not seem to obey hale s polarity rules. the loops are continuously buffeted by convective motions, but they always retain a high degree of coherence. interestingly, 23% of the loops that emerge in the photosphere reach the chromosphere (16 cases out of 69). they are first detected in 630 nm magnetograms and 5 minutes later in b 517.3 nm magnetograms. after about 8 minutes, some of them are also observed in h line - core images, where the footpoints produce small brightness enhancements.

0905.2691

Introduction developments are currently underway to promote the sensitivity of ligo and to improve its prospect for detecting gravitational waves emitted by compact object binaries @xcite. of particular interest are the detection of gravitational waves released during the inspiral and merger of binary black hole (bbh) systems. detection rates for bbh events are expected to be within 0.41000 per year with advanced ligo @xcite. it is important that rigorous detection algorithms be in place in order to maximize the number of detections of gravitational wave signals. the detection pipeline currently employed by ligo involves a matched - filtering process whereby signals are compared to a pre - constructed template bank of gravitational waveforms. the templates are chosen to cover some interesting region of mass - spin parameter space and are placed throughout it in such a way that guarantees some minimal match between any arbitrary point in parameter space and its closest neighbouring template. unfortunately, the template placement strategy generally requires many thousands of templates (e.g. @xcite) evaluated at arbitrary mass and spin ; something that can not be achieved using the current set of numerical relativity (nr) waveforms. to circumvent this issue, ligo exploits the use of analytical waveform families like phenomenological models @xcite or effective - one - body models @xcite. we shall focus here on the phenomenological b (phenomb) waveforms developed by @xcite. this waveform family describes bbh systems with varying masses and aligned - spin magnitudes (i.e. non - precessing binaries). the family was constructed by fitting a parameterized model to existing nr waveforms in order to generate a full inspiral - merger - ringdown (imr) description as a function of mass and spin. the obvious appeal of the phenomb family is that it allows for the inexpensive construction of gravitational waveforms at arbitrary points in parameter space and can thus be used to create arbitrarily dense template banks. to optimize computational efficiency of the detection process it is desirable to reduce the number of templates under consideration. a variety of reduced bases techniques have been developed, either through singular - value decomposition (svd) @xcite, or via a greedy algorithm @xcite. svd is an algebraic manipulation that transforms template waveforms into an orthonormal basis with a prescription that simultaneously filters out any redundancies existing within the original bank. as a result, the number of templates required for matched - filtering can be significantly reduced. in addition, it has been shown in @xcite that, upon projecting template waveforms onto the orthonormal basis produced by the svd, interpolating the projection coefficients provides accurate approximations of other imr waveforms not included in the original template bank. in this paper, we continue to explore the use of the interpolation of projection coefficients. we take a novel approach that utilizes both the analytic phenomb waveform family @xcite and nr hybrid waveforms @xcite. we apply svd to a template bank constructed from an analytical waveform family to construct an orthonormal basis spanning the waveforms, then project the nr waveforms onto this basis and interpolate the projection coefficients to allow arbitrary waveforms to be constructed, thereby obtaining a new waveform approximant. we show here that this approach improves upon the accuracy of the original analytical waveform family. the original waveform family shows mismatches with the nr waveforms as high as @xmath0 when no extremization over physical parameters is applied (i.e., a measure of the faithfulness " of the waveform approximant), and mismatches of @xmath1 when maximized over total mass (i.e., a measure of the effectualness " of the waveform approximant). with our svd accuracy booster, we are able to construct a new waveform family (given numerically) with mismatches @xmath2 even without extremization over physical parameters. this paper is organized as follows. we begin in section [sec : mbias] where we provide definitions to important terminology used in our paper. we then compare our nr hybrid waveforms to the phenomb family and show that a mass - bias exists between the two. in section [sec : method] we present our svd accuracy booster applied to the case study of equal - mass, zero - spin binaries. in section [sec:2d] we investigate the feasibility of extending this approach to include unequal - mass binaries. we finish with concluding remarks in section [sec : discussion]. Gravitational waveforms a gravitational waveform is described through a complex function, @xmath3, where real and imaginary parts store the sine and cosine components of the wave. the specific form of @xmath4 depends on the parameters of the system, in our case the total mass @xmath5 and the mass - ratio @xmath6. while @xmath4 is a continuous function of time, we discretize by sampling @xmath7, where the sampling times @xmath8 have uniform spacing @xmath9. we shall also whiten any gravitational waveform @xmath4. this processes is carried out in frequency space via @xmath10 where @xmath11 is the ligo noise curve and @xmath12 is the fourier transform of @xmath4. the whitened time - domain waveform, @xmath13, is obtained by taking the inverse fourier transform of. in the remainder of the paper, we shall always refer to whitened waveforms, dropping the subscript `` w ''. for our purposes it suffices to take @xmath11 to be the initial ligo noise curve. using the advanced ligo noise curve would only serve to needlessly complicate our approach by making waveforms longer in the low frequency domain. as a measure of the level of agreement between two waveforms, @xmath4 and @xmath14, we will use their match, or overlap, @xmath15 @xcite. we define @xmath16 where @xmath17 is the standard complex inner product and the norm @xmath18. we always consider the overlap maximized over time- and phase - shifts between the two waveforms. the time - maximization is indicated in, and the phase - maximization is an automatic consequence of the modulus. note that @xmath19. for discrete sampling at points @xmath20 we have that @xmath21 where @xmath22 is the complex conjugate of @xmath14. without whitening, would need to be evaluated in the frequency domain with a weighting factor @xmath23. the primary advantage of is its compatibility with formal results for the svd, which will allow us to make more precise statements below. when maximizing over time - shifts @xmath24, we ordinarily consider discrete time - shifts in integer multiples of @xmath25, as this avoids interpolation. after the overlap has been maximized, it is useful to speak in terms of the mismatch, @xmath26, defined simply as @xmath27 we use this quantity throughout the paper to measure the level of disagreement between waveforms. we use numerical waveforms computed with the spectral einstein code spec @xcite. primarily, we use the 15-orbit equal - mass (mass - ratio @xmath28), zero - spin (effective spin @xmath29) waveform described in @xcite. in section [sec:2d], we also use unequal mass waveforms computed by @xcite. the waveforms are hybridized with a taylort3 post - newtonian (pn) waveform as described in @xcite at matching frequencies @xmath30 and @xmath31 for mass - ratios @xmath32 and @xmath33, respectively. taylort4 at 3.5pn order is known to match nr simulations exceedingly well for equal - mass, zero - spin bbh systems @xcite (see also fig. 9 of @xcite). for @xmath34, a taylort3 hybrid is very similar to a taylort4 hybrid, cf.figure 12 of @xcite. the mismatch between taylort3 and taylort4 hybrids is below @xmath35 at @xmath36, dropping to below @xmath37 for @xmath38, and @xmath39 for @xmath40. these mismatches are significantly smaller than mismatches arising in the study presented here, so we conclude that our results are not influenced by the accuracy of the utilized @xmath28 pn - nr hybrid waveform. for higher mass - ratios, the pn - nr hybrids have a larger error due to the post - newtonian waveform @xcite. the error - bound on the hybrids increases with mass - ratio, however, is mitigated in our study here, because we use the @xmath41 hybrids only for total mass of @xmath42, where less of the post - newtonian waveform is in band. because nr simulations are not available for arbitrary mass ratios, we will primarily concentrate our investigation to the equal - mass and zero - spin nr hybrid waveforms described above. the full imr waveform can be generated at any point along the @xmath43 line through a simple rescaling of amplitude and phase with total mass @xmath44 of the system. despite such a simple rescaling, the @xmath43 line lies orthogonal to lines of constant chirp mass @xcite, therefore tracing a steep gradient in terms of waveform overlap, and encompassing a large degree of waveform structure. since our procedure for constructing an orthonormal basis begins with phenomb waveforms, let us now investigate how well these waveforms model the nr waveforms to be interpolated. for this purpose, we adopt the notation @xmath45 and @xmath46 to represent nr and phenomb waveforms of total mass @xmath44, respectively. we quantify the faithfulness of the phenomb family by computing the mismatch @xmath47 $] as a function of mass. the result of this calculation for @xmath48 is shown as the dashed curve in the top panel of figure [figure : bias]. the mismatch starts off rather high with @xmath49 at @xmath50 and then slowly decreases as the mass is increased, until eventually flattening to @xmath51 at high mass. the mismatch between nr and phenomb waveforms can be reduced by optimizing over a mass - bias. this is accomplished by searching for the mass @xmath52 for which the mismatch @xmath53 $] is a minimum. the result of this process is shown by the solid line in the top panel of figure [figure : bias]. allowing for a mass bias significantly reduces the mismatch for @xmath54. the mass @xmath55 that minimizes mismatch is generally smaller than the mass @xmath44 of our nr `` signal '' waveform, @xmath56 over almost all of the mass range considered. apparently, phenomb waveforms are systematically underestimating the mass of the `` true '' nr waveforms, at least along the portion of parameter space considered here. the solid line in the bottom panel of figure [figure : bias] plots the relative mass - bias, @xmath57. at @xmath50 this value is @xmath58, and it rises to just above @xmath59 for @xmath60. and @xmath29. a more comprehensive minimization over mass, mass ratio, and effective spin might change this result.] it is useful to consider how this mass bias compares to the potential parameter estimation accuracy in an early detection. for a signal with a matched - filter signal - to - noise ratio (snr) of 8 characteristic of early detection scenarios template / waveform mismatches will influence parameter estimation when the mismatch is @xmath61 @xcite. placing a horizontal cut on the top panel of figure [figure : bias] at @xmath62, we see that for @xmath63 phenomb waveform errors have no observational consequence ; for @xmath64 a phenomb waveform with the wrong mass will be the best match for the signal. for @xmath65 the missmatch between equal - mass phenomb waveforms and nr (when optimizing over mass) grows to @xmath66. optimization over mass - ratio will reduce this mismatch, but we have not investigated to what degree. Interpolated waveform family we aim to construct an orthonormal basis via the svd of a bank of phenomb template waveforms, and then interpolate the coefficients of nr waveforms projected onto this basis to generate a waveform family with improved nr faithfulness. the first step is to construct a template bank of phenomb waveforms, with attention restricted to equal - mass, zero - spin binaries. an advantage of focusing on the @xmath43 line is that template bank construction can be simplified by systematically arranging templates in ascending order by total mass. with this arrangement we define a template bank to consist of @xmath67 phenomb waveforms, labelled @xmath68 (@xmath69), with @xmath70 and with adjacent templates satisfying the relation : @xmath71 where @xmath72 is the desired overlap between templates and @xmath73 is some accepted tolerance in this value. the template bank is initiated by choosing a lower mass bound @xmath74 and assigning @xmath75. successive templates are found by sequentially moving toward higher mass in order to find waveforms satisfying until some maximum mass @xmath76 is reached. throughout each trial, overlap between waveforms is maximized continuously over phase shifts and discretely over time shifts. for template bank construction we choose to refine the optimization over time by considering shifts in integer multiples of @xmath77. we henceforth refer to our fiducial template bank which employs the parameters @xmath78, @xmath79, @xmath80, and @xmath81. the lower mass bound was chosen in order to obtain a reasonably sized template bank containing @xmath82 waveforms ; pushing downward to @xmath50 results in more than doubling the number of templates. template waveforms each have a duration of @xmath83 and are uniformly sampled at @xmath84 (a sample frequency of @xmath85). @xmath86 of memory is required to store this template bank using double - precision waveforms. the next step is to transform the template waveforms into an orthonormal basis. following the presentation in @xcite, this is achieved by arranging the templates into the rows of a matrix @xmath87 and factoring through svd to obtain @xmath88 where @xmath89 and @xmath90 are orthogonal matrices and @xmath91 is a diagonal matrix whose non - zero elements along the main diagonal are referred to as singular values. the svd for @xmath87 is uniquely defined as long as the singular values are arranged in descending order along the main diagonal of @xmath91. the end result of is to convert the @xmath67 complex - valued templates into @xmath92 real - valued orthonormal basis waveforms. the @xmath93 basis waveform, @xmath94, is stored in the @xmath93 row of @xmath89, and associated with this mode is the singular value, @xmath95, taken from the @xmath93 element along the main diagonal of @xmath91. one of the appeals of svd is that the singular values rank the basis waveforms with respect to their ability to represent the original templates. this can be exploited in order to construct a reduced basis that spans the space of template waveforms to some tolerated mismatch. for instance, suppose we choose to reduce the basis by considering only the first @xmath96 basis modes while discarding the rest. template waveforms can be represented in this reduced basis by expanding them as the sum @xmath97 where @xmath98 are the complex - valued projection coefficients, @xmath99 the prime in is used to stress that the reduced basis is generally unable to fully represent the original template. we are guaranteed from to completely represent the template.] it was shown in @xcite that the mismatch expected from reducing the basis in this way is @xmath100 given @xmath91, can be inverted to determine the number of basis waveforms, @xmath101, required to represent the original templates for some expected mismatch @xmath102. provides a useful estimate to the mismatch in represeting templates from a reduced svd basis. in order to investigate its accuracy, however, we should compute the mismatch explicitly for each template waveform. using the orthonormality condition @xmath103, it is easy to show from that the mismatch between the template and its projection can be expressed in terms of the projection coefficients : @xmath104 this quantity is minimized continuously over phase and discretely over time shifts in integer multiples of @xmath25. choosing @xmath105, predicts that @xmath106 of the @xmath107 basis waveforms from our fiducial template bank are required to represent the templates to the desired accuracy. in figure [figure : rec] we compare the expected mismatch of @xmath108 to the actual mismatches computed from for each phenomb waveform in the template bank. the open squares in this plot show that the actual template mismatch has a significant amount of scatter about @xmath102, but averaged over a whole remains well bounded to the expected result. the phenomb template waveforms can thus be represented to a high degree from a reasonably reduced svd basis. we are of course more interested in determining how well nr waveforms can be represented by the same reduced basis of phenomb waveforms. since nr and phenomb waveforms are not equivalent, can not be used to estimate the mismatch obtained when projecting nr waveforms onto the reduced basis. we must therefore compute their representation mismatch explicitly. a general waveform, @xmath109, can be represented by the reduced basis in analogy to by expressing it as the sum : @xmath110 where @xmath111. as before, the represented waveform @xmath112 will in general be neither normalized nor equivalent to the original waveform @xmath109. the mismatch between them is @xmath113 where we remind the reader that we always minimize over continuous phase shifts and discrete time shifts of the two waveforms. in figure [figure : rec] we use open circles to plot the representation mismatch of nr waveforms evaluated at the same set of masses @xmath114 from which the phenomb template bank was constructed. we see that nr waveforms can be represented in the reduced basis with a mismatch less than @xmath35 over most of the template bank boundary. this is about a factor of five improvement in what can be achieved by using phenomb waveforms optimized over mass. since nr waveforms were not originally included in the template bank, and because a mass - bias exists between the phenomb waveforms which were included, we can expect that the template locations have no special meaning to nr waveforms. this is evident from the thin dashed line which traces the nr representation mismatch for masses evaluated between the discrete templates. this line varies smoothly across the considered mass range and exhibits no special features at the template locations. this is in contrast to the thin solid line which traces phenomb representation mismatch evaluated between templates. in this case, mismatch rises as we move away from one template and subsequently falls back down as the next template is approached. the representation tolerance @xmath115 of the svd is a free parameter, which so far, we have constrained to be @xmath116. when this tolerance is varied, we observe the following trends : (i) phenomb representation mismatch generally follows @xmath117 ; (ii) nr representation mismatch follows @xmath117 at first and then _ saturates _ to a minimum as the representation tolerance is continually reduced. these trends are observed in figure [figure : trun] where we plot nr and phenomb representation mismatch averaged over the mass boundary of the template bank evaluated both at and between templates. the saturation in nr representation mismatch occurs when the reduced basis captures all of the nr waveform structure contained within the phenomb basis. reducing the basis further hits a point of diminishing returns as the increased computational cost associated with a larger basis outweighs the benefit of marginally improving nr match. we now wish to examine the possibility of using the reduced svd basis of phenomb template waveforms to construct a new waveform family with improved nr representation. the new waveform family would be given by a numerical interpolation of the projection coefficients of nr waveforms expanded onto the reduced basis. here we test this using the fiducial template bank and reduced basis described above. the approach is to sample nr projection coefficients, @xmath118, at some set of locations, @xmath119, and then perform an interpolation to obtain the continuous function @xmath120 that can be evaluated for arbitrary @xmath121. the accuracy of the interpolation scheme is maximized by finding the space for which @xmath118 are smooth functions of @xmath121. it is reasonable to suppose that the projection coefficients will vary on a similar scale over which the waveforms themselves vary. hence, a suitable space to sample along is the space of constant waveform overlap. we define this to be the space @xmath122 $] for which the physical template masses are mapped according to : @xmath123 moving a distance @xmath124 in this space is thus equivalent to moving a distance equal to the overlap between adjacent templates. in this space, we find the real and imaginary components, @xmath125 and @xmath126, of the complex projection coefficients to be oscillatory functions that can roughly be described by a single frequency. this behaviour is plotted for the basis modes @xmath127, 50, and 123 in figure [figure : interpcoeff]. another trend observed in this plot is that the projection coefficients become increasingly complex (i.e. show less structure) for higher - order modes. this is a direct result of the increasing complexity of higher - order basis waveforms themselves. we find that the low - order waveforms are smoothest while the high - order modes feature many of the irregularities associated with the multiple frequency components and merger features of the templates. though they are more complex, higher - order modes have smaller singular values and are therefore less important in representing waveforms. this is evident from the steady decline in amplitude of the projection coefficients when moving down the different panels of figure [figure : interpcoeff]. we shall use chebyshev polynomials to interpolate the projection coefficients. these are a set of orthogonal functions where the @xmath128 chebyshev polynomial is defined as @xmath129. \label{eq : chebyn } \end{aligned}\]] the orthogonality of chebyshev polynomials can be exploited to perform an @xmath130 order chebyshev interpolation by sampling @xmath131 at the @xmath132 so - called collocation points given by the gauss - lobatto chebyshev nodes @xcite @xmath133 for @xmath134. in general, the interpolation will not be exact and some residual, @xmath135, will be introduced : @xmath136 here @xmath137 is the actual coefficient of @xmath45 projected onto the basis waveform @xmath94, @xmath138 and @xmath139 is the coefficient obtained after interpolation. the new waveform family is expressed numerically as a function of mass through the relation @xmath140{\bf u}_k,\]] where the subscript intp " reminds the reader that this is computed from an interpolation over @xmath98. an interpolated waveform of total mass @xmath44 can be compared to the original nr waveform (which we consider to be the true " signal), where the latter is expressed as @xmath141 with @xmath142 denoting the component of @xmath45 that is orthogonal to the svd basis (i.e. orthogonal to all phenomb waveforms in the template bank). @xmath45 differs from @xmath143 by an amount @xmath144 to compute the impact of the various approximations influencing, we calculate the overlap between the interpolated waveform, and the exact waveform, @xmath145 $]. to begin this calculation, it is useful to consider the square of the overlap, @xmath146 where we have dropped the explicit mass - dependence and subscripts for convenience. using @xmath147 and taylor - expanding the right - hand - side of to second order in @xmath148, we find @xmath149 to second order in @xmath148, the mismatch is therefore @xmath150 we note that the right - hand - side of can be written as @xmath151, where @xmath152 is the part of @xmath148 orthogonal to @xmath109, @xmath153 however, for simplicity, we proceed by dropping the last term in : @xmath154 using, this gives @xmath155 \le \\ \frac{1}{2}\sum_{k=1}^{n'} \left|r_k(m)\right|^2 + \frac{1}{2}\sum_{k = n'+1}^{2n}\left|\mu_k(m)\right|^2 + \frac{1}{2}|{\bf h}_\perp|^2 \end{gathered}\]] we thus see three contributions to the total mismatch : (i) the interpolation error, @xmath156 ; (ii) the truncation error from the discarded waveforms of the reduced basis, @xmath157 ; (iii) the failure of the svd basis to represent the nr waveform, @xmath158. the sum of the last two terms, which together make up the representation error, is traced by the dashed line in figure [figure : rec]. the goal for our new waveform family is to have an interpolation error that is negligible compared to the representation error. to remove the mass - dependence of interpolation error in, we introduce the maximum interpolation error of each mode, @xmath159 this allows the bound @xmath160 to place an upper limit on the error introduced by interpolation. figure [figure : rk] plots @xmath161 as a function of mode - number @xmath162 as well as the cumulative sum @xmath163. the data pertains to an interpolation performed using @xmath164 chebyshev polynomials on the reduced svd basis containing the frist @xmath106 of @xmath107 waveforms. in this case, we find the interpolation error to be largely dominated by the lowest - order modes and also partially by the highest - order modes. interpolated coefficients for various modes are plotted in figure [figure : interpcoeff] and help to explain the features seen in figure [figure : rk]. in the first place, interpolation becomes increasingly more difficult for higher - order modes due to their increasing complexity. this problem is mitigated by the fact that high - order modes are less important for representing waveforms, as evidenced by the diminishing amplitude of projection coefficients. although low - order modes are much smoother and thus easier to interpolate, their amplitudes are considerably larger meaning that interpolation errors are amplified with respect to high - order modes. summarizes the three components adding to the final mismatch of our interpolated waveform family. their total contribution can be computed directly from the interpolated coefficients in a manner similar to : @xmath165 = 1 - \sqrt{\sum_{k=1}^{n^\prime } \mu_k^\prime(m) \mu_k^{\prime^*}(m)}. \label{eq : interpolationmismatch3}\]] in the case of perfect interpolation for which @xmath166, and reduce to and respectively, and the total mismatch is simply the representation error of the reduced basis. in figure [figure : interperror] open circles show the total mismatch between our interpolated waveform family and the true nr waveforms for various masses. also plotted is the nr representation error without interpolation and the mismatch between nr and phenomb waveforms minimized over mass. we see that interpolation introduces only small additional mismatch to the interpolated waveform family, and remains well below the optimized nr - phenomb mismatch. this demonstrates the efficacy of using svd coupled to nr waveforms to generate a _ faithful _ waveform family with improved accuracy over the _ phenomb family that was originally used to create templates. this represents a general scheme for improving phenomenological models and presents an interesting new opportunity to enhance the matched - filtering process employed by ligo. Higher dimensions so far, we have focused on the total mass axis of parameter space. as already discussed, this served as a convenient model - problem, because the @xmath28 nr waveform can be rescaled to any total mass, so that we are able to compare against the `` correct '' answer. the natural extension of this work is to expand into higher dimensions where nr waveforms are available only at certain, discrete mass - ratios @xmath167. in this section we consider expanding our approach of interpolating nr projection coefficients from a two - dimensional template bank containing unequal - mass waveforms. we compute a template bank of phenomb waveforms covering mass - ratios @xmath167 from 1 to 6 and total masses @xmath168. this mass range is chosen to facilitate comparison with previous work done by @xcite. for the two - dimensional case the construction of a template bank is no longer as straightforward as before due to the additional degree of freedom associated with varying @xmath167. one method that has been advanced for this purpose is to place templates hexagonally on the waveform manifold @xcite. using this procedure we find @xmath169 templates are required to satisfy a minimal match of 0.97. following the waveform preparation of @xcite, templates are placed in the rows of a matrix @xmath87 with real and imaginary components filled in alternating fashion where the whitened waveforms are arranged in such a way that their peak amplitudes are aligned. the waveforms are sampled for a total duration of @xmath170 with uniform spacing @xmath9 so that @xmath171 of memory is required to store the contents of @xmath87 if double precision is desired. application of transforms the 16 complex - valued waveforms into 32 real - valued orthonormal basis waveforms. the aim is to sample the coefficients of nr waveforms projected onto the svd basis of phenomb waveforms using mass - ratios for which nr data exists, and then interpolate amongst these to construct a numerical waveform family that can be evaluated for arbitrary parameters. this provides a method for evaluating full imr waveforms for mass - ratios that have presently not been simulated. to summarize, we take some nr waveform, @xmath172, or total mass @xmath44 and mass - ratio @xmath167, and project it onto the basis waveform @xmath94 in order to obtain @xmath173 next we apply some two - dimensional interpolation scheme on to construct continuous functions @xmath174 that can be evaluated for arbitrary values of @xmath44 and @xmath167 bounded by the regions of the template bank. the interpolated waveform family is given numerically by the form : @xmath175 as before, the interpolation process works best if we can develop a scheme for which the projection coefficients are smoothly varying functions of @xmath44 and @xmath167. following the procedure described in @xcite, the complex phase of the first mode is subtracted from all modes : @xmath176}\mu_k(m, q). \label{eq : musmooth}\]] to motivate why might be useful, let us consider modifying the phenomb waveform family with a parameter - dependent complex phase @xmath177 : @xmath178 when constructing a template bank, or when using a template bank, such a complex phase @xmath177 is irrelevant, because the waveforms are always optimized over a phase - shift. however, @xmath177 will appear in the projection coefficients,, @xmath179 therefore, if one had chosen a function @xmath177 with fine - scale structure, this structure would be inherited by the projection coefficients @xmath180. for traditional uses of waveform families the overall complex phase @xmath177 is irrelevant, and therefore, little attention may have been paid to how it varies with parameters @xmath181. the transformation removes the ambiguity inherent in @xmath177 by choosing it such that @xmath182. this choice ties the complex phase to the physical variations of the @xmath183 coefficient, and does therefore eliminate all unphysical phase - variations on finer scales. in the leftmost panels of figure [figure : coeff2d] we plot the real part of the smoothed coefficients @xmath184 for phenomb waveforms projected onto the basis modes @xmath185 and @xmath186. the middle panels show the same thing except using the nr waveforms evaluated at the set of mass - ratios @xmath167 = \{1, 2, 3, 4, 6 } for which we have simulated waveforms. obviously, the refinement along the @xmath167 axis is much finer for the phenomb waveforms since they can be evaluated for arbitrary mass - ratio, whereas we are limited to sampling at only 5 discrete mass - ratios for nr waveforms. for comparison purposes, the rightmost panels of figure [figure : coeff2d] show the phenomb projection coefficients coarsened to the same set of mass - ratios for which the nr waveforms are restricted to. we find the same general behaviour as before that low - order modes display the smoothest structure, while high - order modes exhibit increasing complexity. a plausible interpolation scheme would be to sample @xmath187 for nr waveforms of varying mass for constant mass ratio (i.e. as we have done previously) and then stitch these together across the @xmath167 axis. since the projection coefficients in figure [figure : coeff2d] show sinusoidal structure they must be sampled with at least the nyquist frequency along both axes. however, looking at the middle and rightmost panels it appears as though this is not yet possible given the present set of limited nr waveforms. at best the 5 available mass - ratios are just able to sample at the nyquist frequency along the @xmath167 axis for high - order modes. in order to achieve a reasonable interpolation from these projection coefficients the current nr data thus needs to be appended with more mass - ratios. based on the left panels of figure [figure : coeff2d] a suitable choice would be to double the current number of mass - ratios to include @xmath167 = \{1.5, 2.5, 3.5, 4.5, 5, 5.5}. hence, though it is not yet practical to generate an interpolated waveform family using the svd boosting scheme applied to nr waveforms, the possibility remains open as more nr waveforms are generated. Discussion we have shown that svd can be used to improve the representation of nr waveforms from a phenomb template bank. a reasonably reduced svd basis was able to reduce mismatch by a factor of five compared to phenomb waveforms optimized over mass. there was also no mass - bias associated with the svd basis and therefore no optimization over physical parameters required. this occurs because svd unifies a range of waveform structure over an extended region of parameter space so that any biases become blended into its basis. svd therefore represents a generalized scheme through which phenomenological waveform families can be de - biased and enhanced for use as matched - filter templates. we were able to calibrate an svd basis of phenomb templates against nr waveforms in order to construct a new waveform family with improved accuracy. this was accomplished by interpolating the coefficients of nr waveforms projected onto the phenomb basis. only marginal error was introduced by the interpolation scheme and the new waveform family provided a more faithful representation of the `` true '' nr signal compared to the original phenomb model. this was shown explicitly for the case of equal - mass, zero - spin binaries. we proceeded to investigate the possibility of extending this approach to phenomb template banks containing unequal - mass waveforms. at present, however, this method is not yet feasible since the current number of mass - ratios covered by nr simulations are unable to sample the projection coefficients with the nyquist frequency. this method will improve as more nr waveforms are simulated and should be sufficient if the current sampling rate of mass - ratios were to double. we thank ilana macdonald for preparing the hybrid waveforms used in this study. kc, jde and hpp gratefully acknowledge the support of the national science and engineering research council of canada, the canada research chairs program, the canadian institute for advanced research, and industry canada and the province of ontario through the ministry of economic development and innovation. dk gratefully acknowledges the support of the max planck society.

matched - filtering for the identification of compact object mergers in gravitational - wave antenna data involves the comparison of the data stream to a bank of template gravitational waveforms. typically the template bank is constructed from phenomenological waveform models since these can be evaluated for an arbitrary choice of physical parameters. recently it has been proposed that singular value decomposition (svd) can be used to reduce the number of templates required for detection. as we show here, another benefit of svd is its removal of biases from the phenomenological templates along with a corresponding improvement in their ability to represent waveform signals obtained from numerical relativity (nr) simulations. using these ideas, we present a method that calibrates a reduced svd basis of phenomenological waveforms against nr waveforms in order to construct a new waveform approximant with improved accuracy and faithfulness compared to the original phenomenological model. the new waveform family is given numerically through the interpolation of the projection coefficients of nr waveforms expanded onto the reduced basis and provides a generalized scheme for enhancing phenomenological models.

1211.7095

Introduction it has become a generally agreed concept that the euv dimming observed during the onset of a _ coronal mass ejection (cme) _ manifests the coronal mass loss of the cme, and thus we basically expect a one - to - one correlation between the detections of cmes and euv dimmings, unless there exist special circumstances. for instance, the cme could originate behind the limb, in which case the euv dimming is obscured, or the cme could start in the upper corona, where there is little euv emission because of the gravitational stratification. the latter case would imply very low masses compared with a cme that originates at the base of the corona, i.e., @xmath0 at two thermal scale heights. however, there exists a case with an average cme mass that did not leave any footprints behind in euv (robbrecht et al. a statistical study on the simultaneous detection of euv dimmings and cmes has recently been performed by bewsher et al. this study based on soho / cds and lasco data confirms a 55% association rate of dimming events with cmes, and vice versa a 84% association rate of cmes with dimming events. some of the non - associated events may be subject to occultation, cme detection sensitivity, or incomplete temperature coverage in euv and soft x - rays. perhaps the cme - dimming association rate will reach 100% once the stereo spacecraft arrive at a separation of @xmath1 and cover all equatorial latitudes of the sun. a number of studies have been carried out by using the detection of coronal dimming to identify cme source regions, focusing on transient coronal holes caused by filament eruptions (rust 1983 ; watanabe et al. 1992), euv dimming at cme onsets (harrison 1997 ; aschwanden et al. 1999), soft x - ray dimming after cmes (sterling & hudson 1997), soft x - ray dimming after a prominence eruption (gopalswamy & hanaoka 1998), simultaneous dimming in soft x - rays and euv during cme launches (zarro et al. 1999 ; harrison & lyons 2000 ; harrison et al. 2003), determinations of cme masses from euv dimming from spectroscopic data (harrison & lyons 2000 ; harrison et al. 2003) or from euv imaging data (zhukov and auchere 2004 ; aschwanden et al. all these studies support the conclusion that dimmings in the corona (either detected in euv, soft x - rays, or both) are unmistakable signatures of cme launches, and thus can be used vice versa to identify the mutual phenomena. in this study here we attempt for the first time to model the kinematics of a cme and the resulting euv dimming quantitatively, which provides us unique physical parameters of the cme source region and on the cme kinematics in the initial acceleration phase. Model assumptions the dynamics of a cme can often be characterized by a rapid expansion of a magnetically unstable coronal volume that expands from the lower corona upward into the heliosphere. different shapes have been used to approximately describe the 3d geometry of a cme, such as a spherical bubble, an ice - cone, a crescent, or a helical flux rope, which expand in a self - similar fashion and approximately maintain the aspect ratio in vertical and horizontal directions during the initial phase of the expansion. here we develop a four - dimensional (4d=3d+t) model that describes the 3d evolution of the cme geometry in time (t) in terms of 4d electron density distributions @xmath2 that allow us also to predict and forward - fit a corresponding euv intensity image data cube @xmath3 in an observed wavelength. for the sake of simplicity we start in our model here with the simplest case, assuming : (1) spherical 3d geometry for the cme front and cavity ; (2) self - similar expansion in time ; (3) density compression in the cme front and adiabatic volume expansion in the cme cavity ; (4) mass conservation for the sum of the cme front, cavity, and external coronal volume ; (5) hydrostatic (gravitational stratification) or super - hydrostatic density scale heights ; (6) line - tying condition for the magnetic field at the cme base ; and (7) a magnetic driving force that is constant during the time interval of the initial expansion phase. this scenario is consistent with the traditional characterization of a typical cme morphology in three parts, including a cme front (leading edge), a cavity, and a filament (although we do not model the filament part). the expanding cme bubble sweeps up the coronal plasma that appears as a bright rim at the observed `` cme front '' or leading edge. the interior of the cme bubble exhibits a rapid decrease in electron density due to the adiabatic expansion, which renders the inside of the cme bubble darker in euv and appears as the observed `` cme cavity ''. the assumption of adiabatic expansion implies no mass and energy exchange across the outer cme boundary, and thus is consistent with the assumption of a low plasma @xmath4-parameter in the corona with perfect magnetic confinement, while the cme bubble will become leaking in the outer corona and heliosphere, where the plasma @xmath4-parameter exceeds unity (not included in our model here). Analytical model a spherical 3d geometry can be characterized by one single free parameter, the radius @xmath5 of the sphere. the self - similar expansion maintains the spherical shape, so the boundary of the cme bubble can still be parameterized by a single time - dependent radius @xmath6. the time - dependence of the cme expansion is controlled by magnetic forces, e.g., by a lorentz force or hoop force. for sake of simplicity we assume a constant force during the initial phase of the cme expansion, which corresponds to a constant acceleration @xmath7 and requires three free parameters (@xmath8) to characterize the radial cme expansion, @xmath9 where @xmath10 is the initial radius at starting time @xmath11, @xmath12 is the initial velocity and @xmath7 is the acceleration of the radial expansion. for the motion of the cme centroid at height @xmath13 we assume a similar quadratic parameterization, @xmath14 where @xmath15 is the initial height at starting time @xmath11, @xmath16 is the initial velocity and @xmath17 is the acceleration of the vertical motion. this parameterization is consistent with a fit to a theoretical mhd simulation of a breakout cme (lynch et al. 2004) as well as with kinematic fits to observed cmes (byrne et al. 2009). , the outer radius of the cme sphere is @xmath6, and the inner radius of the cme front is @xmath18. these parameters increase quadratically with time. the circular footpoint area of the cme with radius @xmath19 stays invariant during the self - similar expansion in order to satisfy the line - tying condition of the coronal magnetic field at the footpoints.,width=313] further we constrain the cme geometry with a cylindrical footpoint area of radius @xmath19, which connects from the solar surface to the lowest part of the spherical cme bubble. in order to ensure magnetic line - tying at the footpoints, the cme bubble should always be located above the cyclidrical footpoint base, which requires that the initial height satisfies @xmath20 and the acceleration constants are @xmath21. we visualize the model geometry in fig. 1. the time - invariant cme footprint area allows us a simple mass estimate of the cme from the cylindrical volume integrated over a vertical scale height, since the spherical cme bubble will eventually move to large heights with no additional mass gain (at time @xmath22, see right - hand panel in fig.1). assuming adiabatic expansion inside the cme cavity, the electron density in the confined plasma decreases reciprocally to the expanding volume, i.e., @xmath23 so it drops with the third power as a function of time from the initial value @xmath24 (of the average density inside the cme). for the mass distribution inside the cme we distinguish between a compression region at the outer envelope, containing the cme front, and a rarefaction region in the inside, which is also called cme cavity. we define an average width @xmath25 of the cme front that is assumed to be approximately constant during the self - similar expansion of the cme. while the radius @xmath6 demarcates the outer radius of the cme front, we denote the inner radius of the cme front or the radius of the cavity with @xmath18, @xmath26 which has an initial value of @xmath27. the total volume @xmath28 of the cme is composed of a spherical volume with radius @xmath6 and the cylindrical volume beneath the cme with a vertical height of @xmath29, @xmath30 \, \]] which has an initial volume value of @xmath28, @xmath31 \.\]] the volume of the cme front @xmath32 is essentially the difference between the outer and inner sphere (neglecting the cylindrical base at the footpoint) @xmath33 while the volume @xmath34 of the cavity is, @xmath35 we have now to define the time - dependent densities in the cme, for both the cme front, which sweeps up plasma during its expansion, as well as for the cme cavity, which rarifies due to the adiabatic expansion. the total mass @xmath36 of the plasma that is swept up from the external corona in a cme corresponds to the total cme volume @xmath37 minus the initial volume of the cme cavity, @xmath38 \\ & = m_p < n_e > { 4 \over 3 } \pi [r(t)^3 - r_0 ^ 3] \end{array } \, \]] where @xmath39 is the mass of the hydrogen atom and @xmath40 is the electron density in the external corona averaged over the cme volume. the same mass has to be contained inside the volume @xmath41 of the cme front, @xmath42 \end{array } \, \]] thus, mass conservation yields a ratio of the average electron density @xmath43 in the cme front and the average external density @xmath40 of @xmath44 this density ratio amounts to unity at the initial time, i.e., @xmath45 and monotonically increases with time. the maximum value of the density jump in mhd shocks derived from the _ rankine - hugoniot relations _ (e.g., priest 1982) is theoretically @xmath46. fast cmes are expected to be supersonic and will have a higher compression factor at the cme front than slower cmes. thus we keep the maximum compression factor @xmath47 as a free parameter, keeping in mind that physically meaningful solutions should be in the range of @xmath48. since we prescribe both the width @xmath25 of the cme front as well as a maximum density compression ratio @xmath47 we obtain a definition of the critical value @xmath49 for the cavity radius @xmath18 when the prescribed maximum density compression @xmath47 is reached (using eq. 11), @xmath50 which yields the critical radius @xmath49, @xmath51^{1/3 } \.\]] therefore, only plasma outside this critical radius @xmath49 can be compressed in the cme front, while the plasma inside this critical radius dilutes by adiabatic expansion and forms the cavity, yielding an average density ratio @xmath52 inside the cavity (according to eq. 3), @xmath53}^3 & { \rm for}\ \rho(t) \ge r(t) \\ { \left [r_0 / r(t) \right]}^3 & { \rm for}\ \rho(t) < r(t) \end{array } \right. \.\]] our numerical model of a spherical cme expansion has a total of 14 free parameters : 3 positional parameters (the heliographic coordinates (@xmath54) and height @xmath55 of the initial cme centroid position), 5 kinematic parameters (starting time @xmath11, velocities @xmath56, accelerations @xmath57), 2 geometric parameters (initial radius @xmath19 and width @xmath25 of the cme front), and 4 physical parameters (coronal base density @xmath24, maximum density compression factor @xmath47 in the cme front, the mean coronal temperature @xmath58 at the observed wavelength filter), and a vertical density scale height factor (or super - hydrostaticity factor) @xmath59 that expresses the ratio of the effective density scale height to the hydrostatic scale height at temperature @xmath58). the temperature @xmath58 defines the hydrostatic scale height @xmath60 of the corona in the observed temperature range, which enters the definition of the effective density scale height @xmath61 (e.g., eq. [3.1.16] in aschwanden 2004), @xmath62}\.\]] thus, assuming an exponentially stratified atmosphere (eq. 15), a density compression factor @xmath63 in the cme front (eq. 12), and adiabatic expansion inside the cme cavity (eq. 14), we have the following time - dependent 3d density model : @xmath64 where @xmath65 is the distance of an arbitrary location with 3d coordinates @xmath66 to the instantaneous center position @xmath67 $] of the cme, @xmath68 ^ 2+[y - y_0(t)]^2+[z - z_0(t)]^2 } \, \]] which is located at height @xmath13 vertically above the heliographic position (@xmath54). Forward-fitting of model to observations one cme event observed with stereo that appears as a spherically expanding shell, and thus is most suitable for fitting with our analytical model, is the 2008-mar-25, 18:30 ut, event. this cme occurred near the east limb for spacecraft stereo / ahead, and was observed on the frontside of the solar disk with spacecraft stereo / behind. some preliminary analysis of this event regarding cme expansion and euv dimming can be found in aschwanden et al. (2009a), the cme mass was determined in white light with stereo / cor-2 (colaninno and vourlidas 2009) and with stereo / euvi (aschwanden et al. 2009b), and the 3d geometry was modeled with forward - fitting of various geometric models to the white - light observations (thernisien et al. 2009 ; temmer et al. 2009 ; maloney et al. 2009 ; mierla et al. 2009a, b). while most previous studies model the white - light emission of this cme, typically a few solar radii away from the sun, our model applies directly to the cme source region in the lower corona, as observed in euv. we follow the method outlined in aschwanden et al. (2009a). our model also quantifies the geometry and kinematics of the cme, as well as the euv dimming associated with the launch of the cme. figures 2 and 3 show sequences of 16 (partial) euv images, simultaneously observed with stereo / a and b with a cadence of 75 s during the time interval of 18:36 - 18:56 ut on 2008-mar-25. in order to determine the positional parameters of the cme as a function of time we trace the outer envelope of the cme bubble (by visual clicking of 3 points) in each image and each spacecraft and fit a circle through the 3 points in each image. the selected points for fitting the position of the cme bubble were generally chosen in the brightest features of the lateral cme flanks, but could not always been traced unambiguously in cases with multiple flank features. in those cases we traced the features that were closest to a continuously expanding solution. the radii and y - positions of the circular fits are fully constrained from the stereo / a images, so that only the x - positions of the centroid of the spherical shell need to be fitted in the epipolar stereo / b images. we note that the fits of the cme bubble roughly agree with the envelopes of the difference flux in the stereo / b images initially (up to 18:48 ut), while there is a discrepancy later on. apparently the cme has a more complex geometry than our spherical bubble model, which needs to be investigated further. this procedure yields the cme centroid positions @xmath69 $] and @xmath70 $] for the time sequence @xmath71. the images in fig. 2 and 3 are displayed as a baseline difference (by subtracting a pre - cme image at 18:36 ut) to enhance the contrast of the cme edge. the circular fits to the cme outer boundaries are overlayed in fig. 2 and 3. both images have been coaligned and rotated into epipolar coordinates (inhester et al. 2006), so that the y - coordinates of a corresponding feature are identical in the spacecraft a and b images, while the x - coordinates differ according to the spacecraft separation angle @xmath72, which amounts to @xmath73 at the time of the cme event. the epipolar coordinates measured from both spacecraft are then related to the heliographic longitude @xmath74, latitude @xmath75, and distance @xmath76 from sun center as follows, @xmath77 which can directly be solved to obtain the spherical (epipolar) coordinates @xmath78, @xmath79 ^ 2 } \\ b = \arcsin { (y_a / r_c) } \end{array}\]] therefore, using stereoscopic triangulation, we can directly determine the spherical coordinates @xmath80, @xmath81 for all 16 time frames, as well as obtain the cme curvature radii @xmath82 from the circular fits to the cmes. we plot the so obtained observables @xmath83, @xmath84, @xmath6, and @xmath85 in fig. 3 and determine our model parameters @xmath74 and @xmath75 from the averages. we obtain a longitude of @xmath86 (for spacecraft stereo / a), @xmath87 (for spacecraft stereo / b), and a latitude @xmath88. thus, the cme source region is clearly occulted for stereo / a. these epipolar coordinates can be rotated into an ecliptic coordinate system by the tilt angle @xmath89 of the spacecraft a / b plane. viewed from earth, the longitude is approximately @xmath90. thus, the cme source region is @xmath91 behind the east limb when seen from earth. this explains why the euv dimming is seen uncontaminted from post - flare loops, which are seen by stereo / b but hidden for stereo / a. (top panel) and latitude @xmath84 (second panel) of the cme centroid, the radius @xmath6 of the cme sphere (third panel), and cme centroid height @xmath13 (bottom). the average values are @xmath92 and @xmath93 for the heliographic position. the cme radius @xmath6 and height @xmath13 are fitted with quadratic functions (dashed curves), yielding the constants @xmath94 hrs (18:38 ut), @xmath95 mm, @xmath96 mm, and accelerations @xmath97 km s@xmath98 and @xmath99 km s@xmath98 (see eqs. 1 - 2).,width=313] we plot also the observables @xmath13 and @xmath6 in fig. 4 and determine the model parameters @xmath55, @xmath100, @xmath17, @xmath7 by fitting the quadratic functions @xmath6 (eq. 1) and @xmath13 (eq. 2), for which we obtain the starting time @xmath94 hrs (18:38 ut), the initial cme radius @xmath95 mm, the initial height @xmath96 mm, and the accelerations @xmath97 km s@xmath98 for the cme radius expansion, and @xmath99 km s@xmath98 for the height motion. the initial velocity is found to be negligible (@xmath101 and @xmath102). we estimate the accuracy of the acceleration values to be of order @xmath0, based on the uncertainty of defining the leading edge of the cme. thus, we determined 9 out of the 14 free parameters of our model sofar. note that the acceleration measured here refers to the very origin of the cme in low coronal heights of @xmath103 solar radii observed in euvi data. the acceleration is expected to be initially high and to rapidly decline further out, when the driving magnetic forces decrease at large altitudes. this explains why our values for the acceleration in low coronal heights are significantly higher than measured further out in the heliosphere, typically in the order of tens of m s@xmath104 in height ranges of 5 - 22 solar radii, as measured with soho / lasco. soho / lasco reported even a slightly negative acceleration at altitutes of 5 - 22 solar radii. the driving magnetic forces that accelerate a cme are clearly confined to much lower altitudes. we model the 3d geometry of the cme bubble with the time - dependent radius @xmath6 and the width @xmath25 of the cme compression region. in fig. 5 we show cross - sectional euv brightness profiles across the cme in horizontal direction (parallel to the solar surface) and in vertical direction for the euvi / a 171 observations (indicated with dotted lines in fig. these baseline - subtracted profiles clearly show a progressive dimming with a propagating bright rim at the cme boundary, which corresponds to the density compression region at the lateral expansion fronts of the cme. the bright rims are clearly visible in the images during 18:46@xmath10518:56 ut shown in fig. 2. the average width of the observed bright rims is @xmath106 mm, a value we adopt in our model. finally we are left with the four physical parameters @xmath107, and @xmath47. since we show here only data obtained with the 171 filter, the mean temperature is constrained by the peak temperature of the instrumental euvi response function, which is at @xmath108 mk. this constrains the thermal scale height to @xmath109 km. the remaining free parameters @xmath59, @xmath24, and @xmath47 need to be determined from best - fit solutions by forward - fitting of simulated euv brightness images (or profiles, as shown in fig. 5) to observed euv brightness images (or profiles). the euv emission measure in each pixel position @xmath110 can be computed by line - of - sight integration along the @xmath111-axis in our 3d density cube @xmath112 per pixel area @xmath113 for each time @xmath114, @xmath115 from which the intensity @xmath116 in the model image in units of dn s@xmath104 can be directly obtained by multiplying with the instrumental response function @xmath117 of the 171 filter, @xmath118 where @xmath119 dn s@xmath104 @xmath120 mk@xmath104 and the fwhm of the 171 filter is @xmath121 mk. in fig. 6 we show best - fit solutions of horizontal and vertical brightness profiles. the absolute flux level is proportional to the coronal base density squared, which we obtain by minimizing the mean flux difference between simulated and observed flux profiles. we obtain a best - fit value of @xmath122 @xmath123. the super - hydrostaticity factor is most sensitive to the vertical flux profile (fig. 5, right - hand side panels), for which we find a best - fit value of @xmath124. thus, the average density scale height in the cme region is slightly super - hydrostatic, as expected for dynamic processes. these values are typical for quiet - sun corona and active region conditions (see figs. 6 and 10 in aschwanden and acton 2001). the last free parameter, the maximum density compression factor @xmath47, affects mostly the brightness of the cme rims. fitting the brightness excess at the cme rims at those times where bright rims are visible are consistent with a value of @xmath125. ll parameter & best - fit value + spacecraft separation angle & @xmath73 + spacecraft plane angle to ecliptic & @xmath126 + heliographic longitude a & @xmath127 + heliographic longitude b & @xmath128 + heliographic longitude earth & @xmath129 + heliographic latitude a, b & @xmath130 + start time of acceleration & @xmath11=2008-mar-25 18:38 ut + start time of modeling & @xmath131=2008-mar-25 18:36 ut + end time of modeling & @xmath131=2008-mar-25 18:56 ut + initial height of cme center & @xmath95 mm + initial radius of cme & @xmath96 mm + width of cme front & @xmath106 mm + acceleration of vertical motion & @xmath99 m s@xmath98 + acceleration of radial expansion & @xmath97 m s@xmath98 + inital vertical velocity & @xmath102 m s@xmath104 + inital expansion velocity & @xmath132 m s@xmath104 + maximum density compression & @xmath133 + corona / cme base density & @xmath122 @xmath123 + super - hydrostaticity factor & @xmath124 + mean temperature (171 filter) & @xmath108 mk + temperature width (171 filter) & @xmath134 mk + after we constrained all 14 free parameters (listed in table 1) of our analytical 4d model by fitting some observables, such as measured coordinates (fig. 4) and cross - sectional horizontal and vertical brightness profiles (fig. 5), we are now in the position to inter - compare the numerically simulated images with the observed images, as shown in fig. 6 for 5 selected times, for both the stereo / a and b spacecraft. the comparison exhibits a good match for the extent of the dimming region the the bright lateral rims, both extending over about 1.5 thermal scale heights above the solar surface. the base - difference images of euvi / a reveal a fairly symmetric cme (as the model is by design), surrounded by spherical bright rims at the northern and southern cme boundaries (as the model is able to reproduce it). the model, however, is less in agreement with the observed euvi / b images. the extent of the euv dimming region matches relatively exactly, although the observed dimming region is somewhat cluttered with bright postflare loops that appear in the aftermath of the cme, which are mostly hidden in the euvi / a observations. the biggest discrepancy between the model and the euvi / b observations is the location of the brightest rim of the cme boundary. the combination of projection effects and gravitational stratification predict a brighter rim on the west side, where we look through a longer and denser column depth tangentially to the cme bubble, which is not apparent in the observations of euvi / b. instead, there is more bright emission just above the eastern limb that can not be reproduced by the model. apparently there exists stronger density compression on the eastern side of the cme bubble than the model predicts. another inconsistency is the bright loop seen in euvi / b at 18:51 ut, which does not match the surface of the modeled cme sphere as constrained by euvi / a. apparently, there are substantial deviations from a spherically symmetric cme bubble model that are visible in euvi / b but not in euvi / a. perhaps a flux rope model could better fit the observations than a spherical shell model. these discrepancies between the observations and our simple first - cut model provide specific constraints for a more complex model (with more free parameters) that includes inhomogeneities in the density distribution of the cme. our model allows us, in principle, to estimate the cme mass by integrating the density @xmath2 over the entire cme sphere, which is of course growing with time, but expected to converge to a maximum value once the cme expands far out into the heliosphere. a simple lower limit can analytically be obtained by integrating the density in the cylindrical volume above the footpoint area, @xmath135 from our best - fit values @xmath95 mm, @xmath136, @xmath137 @xmath123 and the thermal scale height of @xmath138 mm, we obtain a lower limit of @xmath139 g. however, this cme appears to expand in a cone - like fashion in the lowest density scale height, so the total volume and mass is likely to be about a factor of @xmath140 higher. moreover, the mass detected in 171 amounts only to about a third of the total cme mass (aschwanden et al. 2009b), so a more realistic estimate of the total cme mass is about a factor 6 higher than our lower limit, i.e., @xmath141 g, which brings it into the ballpark of previous cme mass determinations of this particular event, i.e., @xmath142 g from stereo / cor-2 white - light observations (colaninno and vourlidas 2009), or @xmath143 g from stereo / euvi observations (aschwanden et al. Conclusions we developed an analytical 4d model that simulates the cme expansion and euv dimming in form of a time sequence of euv images that can directly be fitted to stereoscopic observations of stereo / euvi. the dynamic evolution of the cme is characterized by a self - similar adiabatic expansion of a spherical cme shell, containing a bright front with density compression and a cavity with density rarefaction, satisfying mass conservation of the cme and ambient corona. we forward - fitted this model to stereo / euvi observations of the cme event of 2008-mar-25 and obtained the following results : 1. the model is able to track the true 3d motion and expansion of the cme by stereoscopic triangulation and yields an acceleration of @xmath144 m s@xmath98 for both the vertical centroid motion and radial expansion during the first half hour after cme launch. fitting the euv dimming region of the model to the data mostly constrains the coronal base density in the cme region (@xmath145 @xmath123) and the density scale height, which was found to be super - hydrostatic by a factor of @xmath136. the average cme expansion speed during the first 10 minutes is approximately @xmath146 km s@xmath104, similar to the propagation speeds measured for eit waves in the initial phase (i.e., @xmath147 km s@xmath104, veronig et al. 2008 ; @xmath148 km s@xmath104, long et al. 2008 ; @xmath149 km s@xmath104, patsourakos et al. 2009), and thus the cme expansion speed seems to be closely related to the associated propagation kinematics of eit waves. the derived base density, scale height, and footpoint area constrain the cme mass, but accurate estimates require a more complete temperature coverage with other euv filters (e.g., see aschwanden et al. 2009b). 5. the width and density compression factor of the cme front are also constrained by our model, but accurate values require a perfectly homogeneous cme shell. 6. while the spherical shell geometry reproduces the euvi / a observations well, significant deviations are noted in the euvi / b observations, indicating substantial inhomogeneities in the cme shell, possibly requiring a hybrid model of bubble and flux rope geometries. the most important conclusion of this modeling study is that euv dimming can be understood in a quantitative manner and that it provides a direct measurement of the coronal mass loss of a cme. this exercise has shown that the spherical shell geometry can reproduce a number of observed features of an evolving cme, which constrains the physical and kinematic parameters of the initial phase of the cme launch, but reveals also significant deviations that require a modification of the idealized homogeneous spherical shell model. the method of analytical 4d models with forward - fitting to stereoscopic euv images appears to be a promising tool to investigate quantitatively the kinematics of cmes. combining with simultaneous magnetic or mhd modeling may further constrain the physical parameters and ultimately provide the capability to discriminate between different theoretical cme models. full 4d modeling of the initial cme expansion may also provide a self - consistent treatment of eit waves and cme expansion (e.g., chen et al. 2005), for which we find similar kinematic parameters. this work is supported by the nasa stereo under nrl contract n00173 - 02-c-2035. the stereo/ secchi data used here are produced by an international consortium of the naval research laboratory (usa), lockheed martin solar and astrophysics lab (usa), nasa goddard space flight center (usa), rutherford appleton laboratory (uk), university of birmingham (uk), max - planck - institut fr sonnensystemforschung (germany), centre spatiale de lige (belgium), institut doptique thorique et applique (france), institute dastrophysique spatiale (france). the usa institutions were funded by nasa ; the uk institutions by the science & technology facility council (which used to be the particle physics and astronomy research council, pparc) ; the german institutions by deutsches zentrum fr luft- und raumfahrt e.v. (dlr) ; the belgian institutions by belgian science policy office ; the french institutions by centre national detudes spatiales (cnes), and the centre national de la recherche scientifique (cnrs). the nrl effort was also supported by the usaf space test program and the office of naval research. References # 1 [a]schwanden, m.j. 2004, _ physics of the solar corona. an introduction _, praxis : chichester / uk and springer : berlin. [a]schwanden, m.j., fletcher, l., schrijver, c., & alexander, d. 1999, apj 520, 880. [a]schwanden, m.j. and acton, l.w. 2001, apj 550, 475. [a]schwanden, m.j., nitta, n.v., wuelser, j.p., and lemen 2009a, 256, 3. [a]schwanden, m.j., nitta, n.v., wuelser, j.p., lemen, j.r., sandman, a., vourlidas, a., and colaninno, r. 2009b, apj (subm). [b]ewsher, d., harrison, r.a., & brown, d.s. 2008, a&a 478, 897. [b]yrne, j.p., gallagher, p.t., mcateer, r.t.j., and young, c.a. 2009, a&a 495, 325. [c]olaninno, r.c. and vourlidas, a. 2009, apj (in press) [g]opalswamy, n. & hanaoka, y. 1998, apj 498, l179. [h]arrison, r.a. 1997, proc. 31st eslab symp., correlated phenomena at the sun, in the heliosphere and in geospace, esa sp-415, 121. [h]arrison, r.a. & lyons, m. 2000, a&a 358, 1097. [h]arrison, r.a., bryans, p., simnett, g.m., & lyons, m. 2003, a&a 400, 1071. [i]nhester, b. 2006, _ stereoscopy basics for the stereo mission _, eprint arxiv : astro - ph/0612649. [l]ong, d.m., gallagher, p.t., mcateer, r.t.j., and bloomfield, d.s. 2008, apj 680, l81. [l]ynch, b.j., antiochos, s.k., macneice, p.j., zurbuchen, t.h., and fisk, l.a. 2004, apj 617, 589. [m]aloney, s.a., gallagher, p.t., and mcateer, r.t.j. 2009, sp 256, 149. [m]ierla, m., inhester, b., marque, c., rodriguez, l., gissot, s., zhukov, a.n., berghmans, d., and davila j. 2009a, sp (subm). [m]ierla, m. et al. 2009b, ann.geophys. (in preparation). [p]atsourakos, s. and vourlidas, a. 2009, apj 700, l182. [p]riest, e.r. 1982, _ solar magnetohydrodynamics _, geophysics and astrophysics monographs, vol. 21, d. reidel publishing company, dordrecht, netherlands. [r]obbrecht, e., patsourakos, s., and vourlidas, a. 2009, apj (in press). [r]ust, d.m. 1983, ssrv 34, 21. [s]terling, a.c. & hudson, h.s. 1997, apj 491, l55. [t]emmer, m., preiss, s., and veronig, a.m. 2009, sp 256, 183. [t]hernisien, a., vourlidas, a., and howard, r.a. 2009, sp (in press). [v]eronig, a.m., temmer, m., and vrsnak, b. 2008, apj 681, l113. [w]atanabe, t., kozuka, y., ohyama, m., kojima, m., yamaguchi, k., watari, s.i., tsuneta, s., jeselyn, j.a., harvey, k.l., acton, l.w., & klimchuk j.a. 1992, pasj 44, l199. [z]arro, d.m., sterling, a.c., thompson, b.j., hudson, h.s., & nitta, n. 1999, apj 520, l139. [z]hukov, a.n. and auchere, f. 2004, 427, 705.

this is the first attempt to model the kinematics of a cme launch and the resulting euv dimming quantitatively with a self - consistent model. our 4d - model assumes self - similar expansion of a spherical cme geometry that consists of a cme front with density compression and a cavity with density rarefaction, satisfying mass conservation of the total cme and swept - up corona. the model contains 14 free parameters and is fitted to the 2008 march 25 cme event observed with stereo / a and b. our model is able to reproduce the observed cme expansion and related euv dimming during the initial phase from 18:30 ut to 19:00 ut. the cme kinematics can be characterized by a constant acceleration (i.e., a constant magnetic driving force). while the observations of euvi / a are consistent with a spherical bubble geometry, we detect significant asymmetries and density inhomogeneities with euvi / b. this new forward - modeling method demonstrates how the observed euv dimming can be used to model physical parameters of the cme source region, the cme geometry, and cme kinematics.

0908.1913

Introduction supersymmetry (susy) is one of the most attractive extensions of the standard model. this symmetry solves the naturalness problem and predicts gauge coupling unification at the gut scale @xmath1. it also predicts the existence of superpartner of the standard model (sm) particles. from the naturalness argument, their masses should be below tev range, hence these particles will be discovered at tevatron or large hadron collider (lhc). mechanisms of susy breaking and its mediation to the minimal supersymmetric standard model (mssm) sector are one of the most important problems in the susy phenomenology. in many models, this dynamics is related to high energy physics far above the electroweak(ew) scale, e.g., gut scale or planck scale. once the mechanism is specified, mass spectrum and flavor structure of susy particle at the ew scale can be determined by a small number of parameters. hence it may be possible to confirm or exclude the mechanism by direct search or flavor - changing - neutral - current (fcnc) experiments in near future. if susy breaking is mediated by gravity, the structure of susy breaking masses of scalars are determined by khler potential. in the present paper, we focus on the no - scale type khler potential, in which the hidden sector and the observable sector are separated as follows : @xmath2 where @xmath3 and @xmath4 are hidden sector fields and observable sector fields, respectively. this type of khler potential is originally investigated in ref. @xcite with @xmath5 and @xmath6. characteristic features of the khler potential eq.([eq : noscalekahler]) is that all scalar masses and trilinear scalar couplings (a - terms) vanish as the cosmological constant vanishes@xcite. the only source of susy breaking is gaugino masses. hence this scenario is highly predictive, various phenomenological consequences are obtained with a few parameters. the separation in eq.([eq : noscalekahler]) implies that couplings of the hidden sector and the observable sector is flavor blind, and contributions of susy particles to fcnc are suppressed. therefore this khler potential is also interesting from the viewpoint of the susy flavor problem. the no - scale structure of the khler potential is obtained in various models. it has been shown that in some classes of string theory, for example weakly coupled @xmath7 heterotic string theory, khler potential becomes the no - scale type@xcite. if the hidden sector and the observable sector are separated in the superspace density in the supergravity lagrangian, the khler potential is indeed given as in eq. ([eq : noscalekahler]). in the two cases, the gaugino masses can be induced if the hidden sector fields couple to the gauge multiplets via the gauge kinetic function. recently it has been pointed out that the form eq.([eq : noscalekahler]) is realized naturally in a five - dimensional setting with two branes, namely, sequestered sector scenario@xcite. in this scenario, the hidden sector fields live on one brane and the visible sector fields live on the other. it has been shown that the form of the khler potential of the effective theory obtained by dimensional reduction is indeed eq.([eq : noscalekahler])@xcite. if the sm gauge fields dwell in the bulk, gaugino mediate the susy breaking on the hidden sector brane to the visible sector brane and the no - scale boundary condition is given at the compactification scale of the fifth dimension (gaugino mediation @xcite). in the no - scale scenario, degrees of freedom of susy particle mass spectrum is limited because only non - zero soft susy breaking masses are gaugino masses and higgs mixing mass @xmath8 at the energy scale where the boundary condition is given. hence phenomenological aspects of this scenario have been investigated in the literature, mainly focusing on the mass spectrum. direct search bounds and the cosmological constraint (i.e., a charged particle can not be the lsp if the r - parity is conserved) were considered and allowed region in the parameter space was identified. for the boundary condition, the following three cases were considered. first, universal gaugino masses are given at the gut scale. in this case, the cosmological constraint is severe and only the region @xmath9 and @xmath10 is allowed since stau tends to be light@xcite. the second case is that universal gaugino masses are given above the gut scale. and the third case is that non - universal gaugino masses are given at the gut scale. in this case wino, higgsino or sneutrino can be the lsp. in the latter two cases, it is shown that the cosmological constraint is not severer than the first case. in the present paper, current limits from the lightest higgs mass @xmath11 and the branching ratio for @xmath0 are also used to constrain the no - scale scenario. combining these constraints, we will show that almost all the parameter region is excluded when universal gaugino masses are given at the gut scale. however, when the boundary condition is given above the gut scale, relatively large parameter region is allowed. we also consider the case that the non - universal gaugino masses are given at the gut scale. we will show that these constraints are important when the higgsino - like neutralino is the lsp. this paper is organized as follows. in section [sec : noscalsebc], we review some phenomenological aspects of the no - scale models, especially indications of the direct search bounds and the cosmological bound. in section [sec : higgsbsgamma], we further constrain these models from the higgs mass bound and @xmath12 result. indications of these bounds for the tevatron are also discussed. our conclusions are given in section [sec : conclusions]. Models with no-scale boundary condition in this section, we briefly review phenomenological aspects of susy models with no - scale boundary condition, mainly indications of the cosmological bound and direct search limit at lep 2. we consider the following three cases. * universal gaugino masses are given at the gut scale. hereafter we call this case the minimal scenario. * universal gaugino masses are given above the gut scale @xmath13. throughout this paper, we take the minimal su(5) to be the theory above the gut scale as a typical example. * non - universal gaugino masses are given at the gut scale. once one of the above boundary conditions is given, mass spectrum of susy particles at the ew scale and their contributions to fcnc can be calculated. in this paper we solve the one - loop level rges to obtain the soft susy breaking mass parameters at the ew scale. the higgsino mass parameter @xmath14 is determined by potential minimum condition at the one - loop level. first, we discuss the minimal scenario. in this case, the following boundary condition is given at the gut scale, @xmath15 where @xmath16 is the common scalar mass and @xmath17 is universal trilinear scalar coupling. with this boundary condition, bino and right - handed sleptons are lighter than other susy particles. their masses are approximately, @xmath18 from eq.([eq : minimalmneumtau]) we see that the charged right - handed slepton is the lsp if the d - term @xmath19 is negligible, i.e., @xmath20. hence this parameter region is excluded by the cosmological consideration. on the other hand, lep 2 experiments yields the upper bound on the cross section for smuon pair production, @xmath21 for @xmath22 and @xmath23@xcite, so the parameter region @xmath24 is excluded in fig. [fig : limitmimposmu] and [fig : limitmimnegmu], allowed region of the parameter space are shown in the @xmath25 plane. the regions above the dash - dotted line and the left side of the dash - dot - dotted line are excluded by cosmological bound and lep 2 bound on smuon pair production, respectively. therefore the minimal scenario is constrained severely. next we see the case that the universal gaugino masses are given above the gut scale. in the minimal su(5) case, the right - handed slepton belongs to 10-plet, so the large group factor makes slepton masses heavier. for example, when @xmath26 the bino mass and the right - handed slepton mass at the weak scale are approximately given, @xmath27 hence the cosmological constraint is not severe because the stau mass is large enough and neutralino is the lsp in the large parameter region @xcite. in the fig.[fig : limitmbc1e17posmu] and [fig : limitmbc1e17negmu], the same figures as in the fig.[fig : limitmimposmu] and [fig : limitmimnegmu] are shown. unlike in the minimal case, the stau search bound at lep @xcite is also plotted because mass difference between @xmath28 and @xmath29 is larger than in the minimal case and it can be stronger than the smuon search. from these figures we see that the @xmath29 lsp is avoided unless @xmath30 is larger than about 20. the charged stau lsp can also be avoided if gaugino masses at the gut scale are non - universal@xcite, i.e., the following boundary condition is given, @xmath31 this boundary condition can be given naturally within the gut framework @xcite. in this case, not only bino - like neutralino, but also wino - like, higgsino - like neutralino or sneutrino can be the lsp. for @xmath32 and @xmath33, the lsp is wino - like neutralino. for example, when @xmath34 and @xmath35, then wino mass and charged slepton mass are (notice that in this case the left - handed sleptons are lighter than right - handed sleptons) ; @xmath36 the higgsino is the lsp if @xmath37. for example, when @xmath38 and @xmath39, then the higgsino mass and the right - handed slepton mass are @xmath40 in the two cases given above, neutral wino or higgsino is the lsp. in fact from fig.[fig : limit12r432r2posmu] - [fig : limit12r232r0.5negmu] we find that neural particle is the lsp in large parameter region, thus it is cosmologically viable. Higgs mass and @xmath41 constraint on no-scale scenario in the previous section we take into account only lep 2 bound and the cosmological constraint. we find that the minimal scenario is severely constrained, but the other two scenarios are not. in this section we also include the current higgs mass bound and @xmath41 constraint. as we will see, combining the above four constraint, not only the minimal case but also the other two scenarios can be constrained more severely. we also discuss the possibility whether this scenario can be seen at the tevatron run 2 or not. before we show the numerical results, some remarks on our calculation of the higgs mass and @xmath12 are in order. it is well known that radiative correction is important when the lightest higgs mass is evaluated @xcite. in the present paper, the lightest higgs mass is evaluated by means of the one - loop level effective potential@xcite. this potential is evaluated at the renormalization point of the geometrical mean of the two stop mass eigenvalues @xmath42. we compared our result with a two - loop result by using _ feynhiggs_@xcite, and checked that the difference between these two results is smaller than 5 gev as long as @xmath30 is bigger than 5. when @xmath30 is close to 2, the difference can be 7 gev. however, as we will see later, higgs mass bound plays an important rule around @xmath43. and the two - loop effects always make the higgs mass lighter than that obtained at the one - loop level. so our conclusion is conservative and is not significantly changed by the two - loop effect. we exclude the parameter region where the lightest higgs mass is lighter than the current 95% c.l. limit from lep 2 experiments, @xmath44 @xcite. in the present paper, @xmath12 is calculated including leading order (lo) qcd corrections@xcite, and compare it to the current cleo measurement. in the mssm, chargino contribution can be comparable to the sm and charged higgs contributions. they interfere constructively (destructively) each other when @xmath45 (@xmath46). the difference between the lo and the next - to - leading order (nlo) result can be sizable only when cancellation among different contributions at the lo is spoiled by the nlo contributions. as we will see, however, the @xmath41 constraint is severe when the interference is constructive. in the case of destructive interference where the deviation from the nlo result may be large, this constraint is not so important. hence we expect that our conclusion is not changed significantly by the inclusion of the nlo corrections. for the experimental value, we use 95% c.l. limit from cleo, @xmath47 @xcite. first we show the numerical results for the minimal case. the case for @xmath48 is shown in fig.[fig : limitmimposmu]. in this case, for small @xmath30 region, the stop mass is not so large that radiative correction factor @xmath49 which raises the higgs mass is small. (for example, @xmath50 gev and @xmath51 gev for @xmath52 gev and @xmath53). hence the higgs mass limit constrains this scenario severely. in fig.[fig : limitmimposmu], the higgs mass bound and @xmath12 constraints in the @xmath54 plane are shown. the regions below the solid line and above the dashed line are excluded by the higgs mass and @xmath12 bound, respectively. the indication of @xmath55 reported by lep 2@xcite is also shown in this figure. from the figure we find that the higgs mass bound almost excludes the region where the stau lsp is avoided. note that, as we discussed earlier, the bound we put on the higgs mass may be conservative, because the two loop correction may further reduce the higgs mass. the same figure but for @xmath56 is shown in fig.[fig : limitmimnegmu]. now @xmath12 also constrains parameter region strongly since chargino contribution to @xmath0 interferes with sm and charged higgs ones constructively. the region above the dashed line is excluded by @xmath12 constraint. we find that only one of the two constraints is enough to exclude all the region where cosmological bound and the smuon mass bound are avoided. hence if r - parity is conserved, i.e., the cosmological bound is relevant, this scenario with @xmath56 is excluded. next we show the numerical results in the case that the cutoff scale is larger than the gut scale. as a typical example, we choose the minimal su(5) as the theory above the gut scale. in fig.[fig : limitmbc1e17posmu] and [fig : limitmbc1e17negmu], results are shown for positive and negative @xmath14, respectively. in both figures, we take @xmath58 gev. for @xmath48 case, large parameter region is allowed and susy scale @xmath59 can be as small as about 180 gev, which indicates the lsp mass @xmath60 gev. for @xmath56, as in the minimal case, @xmath12 constraint is severer, and @xmath59 must be larger than around 280 gev. we also considered other values of the boundary scale @xmath61 from @xmath62 to @xmath63, and checked that the behavior of the contour plot does not change so much. according to ref.@xcite, tevatron run 2 experiment can explore up to @xmath64 gev for integrated luminosity @xmath65. hence if @xmath66 and @xmath67, susy particles can be discovered at the experiment. in this range, trilepton from chargino - neutralino associated production @xmath68, @xmath69, @xmath70 is one of clean signals for susy search. notice that now two body decay @xmath71 opens. so same flavor, opposite sign dilepton from @xmath72 decay may be useful. the two body decay allows us to observe the peak edge of invariant mass of two leptons at the @xmath73. it is expressed in terms of the neutralino masses and the slepton mass as, @xmath74 in table [tab : mllmax], the dependence of @xmath73 on @xmath61 is shown. here we fix @xmath75. notice that as @xmath61 changes, the right - handed mass changes sizably while the neutralino masses do not. hence we can obtain the mass relation among them and also cutoff scale @xmath61, which corresponds to the compactification scale in the sequestered sector scenario, by measuring @xmath73. on the other hand, since only @xmath76 gev is allowed for @xmath45, the tevatron run 2 can not survey this scenario, and we have to wait lhc experiment. next, we turn to the case that gaugino masses are non - universal at the gut scale. we explore the following three cases, wino - like neutralino lsp, higgsino - like neutralino lsp and the tau sneutrino lsp. we will see that in the wino - like neutralino lsp and tau sneutrino lsp cases, constraint is not so severe even if we combine higgs mass bound and @xmath12 data, but in the higgsino - like lsp case where stops are as light as sleptons and charginos, the predicted higgs mass tends to be small, and thus the higgs mass bound becomes important. first, we discuss the wino - lsp case. the results for @xmath34, @xmath35 are shown in fig.[fig : limit12r432r2posmu] and fig.[fig : limit12r432r2negmu], for @xmath46 and @xmath45, respectively. in this case, we obtain a relatively large higgs mass since @xmath77 is large and so are the masses of stops. hence, for @xmath46, @xmath78 can be as small as 100 gev at @xmath43, where the mass of the lsp @xmath79 is about 90 gev. for @xmath45, though @xmath12 constraint is slightly severer than in the @xmath46 case, @xmath80 is allowed, which corresponds to @xmath81. hence the wino - lsp with mass around 100 gev is allowed. examples of the mass spectrum in this case are listed as point a (@xmath46) and point b (@xmath45) in table [tab : spectrum]. at the both points, @xmath59 is chosen to be near the smallest value such that all constraints are avoided. in general, however, masses of @xmath28 and @xmath82 are highly degenerate when wino is the lsp. in fact, from table [tab : spectrum], we see that the mass difference is less than 1 gev. therefore a lepton from @xmath70 is very soft and trilepton signal search is not useful because acceptance cut usually requires the smallest transverse momentum of the three leptons @xmath83 to be larger than 5 gev@xcite. recently collider phenomenology in such cases are studied in ref. it is shown that certain range of @xmath84 and @xmath85, susy signals which are different from those in the minimal case can be detected. the high degeneracy requires to include radiative corrections to calculate @xmath85 @xcite, which is beyond of this work. it deserves detail study to estimate the mass difference in the scenario. since the constraint for the sneutrino lsp case in the @xmath86 plane is similar to those in the wino - lsp case, we show the result for @xmath46 only in fig.[fig : limit12r2.532r1.5posmu]. in the figure, we take @xmath87 and @xmath88. notice that the decomposition of the lsp depends on @xmath30 and the sneutrino is the lsp for @xmath89. an example of the mass spectrum is listed as the point c in table [tab : spectrum]. in this case, trilepton signal comes from @xmath90, @xmath91, @xmath92. since @xmath93, @xmath94 of a lepton from @xmath82 decay is small and this signal may be hard to be detected. we may need unusual trigger to explore this scenario. next, we turn to the higgsino lsp case. higgsino lsp scenario is realized when @xmath77 is smaller than half of @xmath78, which indicates that colored particles are lighter than in the universal gaugino mass case. hence the one - loop correction to the higgs potential which enhances the higgs mass is small and the higgs mass constraint is important. the same figures as fig.[fig : limitmimposmu] and [fig : limitmimnegmu] are shown in fig.[fig : limit12r232r0.5posmu] (@xmath46) and fig.[fig : limit12r232r0.5negmu] (@xmath45) for @xmath95 and @xmath39. in order to satisfy the higgs mass bound, @xmath78 must be larger than around 300 gev. combining the bound with @xmath12, constraint becomes severer, especially for @xmath45 case where @xmath96 is required. example of mass spectrum in this scenario is listed as point d and e in table [tab : spectrum]. again we choose almost the smallest value of @xmath78 where all constraints are avoided. we see that the lsp mass must be at least @xmath97 for @xmath46 and @xmath98 for @xmath45. hence this scenario can not be explored at the tevatron run 2. Conclusions the no - scale type boundary conditions are obtained in various types of susy models. this scenario is attractive because it is highly predictive and can be a solution to the susy flavor problem. in this paper we investigated the indication of the current higgs mass and @xmath41 constraint on susy models with the boundary condition. first we considered the minimal case where the universal gaugino mass are given at the gut scale. this scenario has been already constrained by direct search at lep and the cosmological bound severely, under the assumption of the exact r - parity. we showed that the higgs mass bound and @xmath41 constraint are also taken into account, then almost all the parameter region is excluded, leaving very narrow allowed region for @xmath48. next we considered the case that the boundary condition is given above the gut scale. since the cosmological constraint is not severe, wide region of the parameter space is allowed. in the @xmath46 case, tevatron have a chance to observe susy signatures like trilepton events. the scale @xmath61 may be explored by measuring the peak edge of invariant mass of two leptons at the @xmath73. however for the @xmath45 case, since @xmath99 is required, we have to wait lhc. finally we considered the case where non - universal gaugino masses are given at the gut scale. we see that the higgs mass bound is strong in the higgsino lsp case because stop masses are as light as sleptons and charginos. the mass of the higgsino - like neutralino must be larger than about 245 gev and 370 gev for @xmath46 and @xmath45, respectively. in the wino lsp and sneutrino lsp case, the mass of the lsp can be as small as 150 gev. however, the mass difference between the lsp and parent particles produced at the collider is much smaller than in the minimal case, unusual acceptance cut may be required. Acknowledgment the author would like to thank m. yamaguchi for suggesting the subject, fruitful discussion and careful reading of the manuscript. he also thanks t. moroi and m. m. nojiri for useful discussion. j. ellis, c. kounnas and d.v. nanopoulos, nucl. b247 (1984) 373 ; j. ellis, a.b. lahanas, d.v. nanopoulos and k. tamvakis, phys. b134 (1984) 439 ; a.b. lahanas and d.v. nanopoulos, phys. (1987) 1. e. witten, phys. b155 (1985) 151. t. banks and m. dine, nucl. b505 (1997) 445 ; h.p. nilles, m. olechowski and m. yamaguchi, phys. b415 (1997) 24 ; nucl. phys. b530 (1998) 43 ; z. lalak and s. thomas, nucl. phys. b515 (1998) 55 ; a. lukas, b.a. ovrut and d. waldram, nucl. phys. b532 (1998) 43 ; phys. d57 (1998) 7529 ; k. choi, h.b. kim and c. muoz, phys. d57 (1998) 7521. y. okada, m. yamaguchi and t. yanagida, prog. 85 (1991) 1, phys. b262 (1991) 54 ; j. ellis, g. ridolfi and f. zwirner, phys. b257 (1991) 83 ; h. e. haber and r. hempfling, phys. 66 (1991) 1815. aleph collabolation, phys. b495 (2000) 1 ; l3 collabolation, phys. b495 (2000) 18 ; delphi collabolation, phys. b499 (2001) 23 ; opal collabolation, phys. b499 (2001) 38 ; p. igo - kemenes, talk at the lepc open session on nov. 3rd, http://lephiggs.web.cern.ch/lephiggs/talks/index.html. plane for the minimal case with @xmath48. in the region above the dash - dotted line, the stau is the lsp. the left side of the dash - dot - dotted line is excluded by the upper bound on smuon pair production cross section at lep. the current higgs mass bound excludes the region below the solid line. in the region above the dashed line, @xmath12 is smaller than the lower limit obtained by the cleo. the shaded region is allowed. the @xmath100 curve is also shown as the dotted line. noewsb means that radiative breaking does not occur.,width=340,height=264] but @xmath56. the region above the dashed line is excluded since @xmath12 is larger than the upper limit obtained by the cleo. the other lines are the same as in fig.[fig : limitmimposmu]. notice that all region is excluded.,width=340,height=264] plane for universal gaugino masses, @xmath26 and @xmath48. the left sides of the dash - dot - dotted and dash - dot - dot - dotted line are excluded by the upper bound on the smuon and stau pair production cross section, respectively. the other lines are the same as in fig.[fig : limitmimposmu], width=340,height=264] plane for the higgsino lsp case, @xmath95, @xmath39, @xmath101 and @xmath48. in the region between the two dashed line, @xmath12 is smaller than the lower limit of the cleo result., width=340,height=264]

no - scale structure of the khler potential is obtained in many types of supersymmetric models. in this paper, phenomenological aspects of these models are investigated with special attention to the current higgs mass bound at lep and @xmath0 result at the cleo. when the boundary condition is given at the gut scale and gaugino masses are universal at this scale, very narrow parameter region is allowed only for positive higgsino mass region if r - parity is conserved. the negative higgsino mass case is entirely excluded. on the other hand, relatively large parameter region is allowed when the boundary condition is given above the gut scale, and tevatron can discover susy signals for the positive higgsino mass case. the no - scale models with wino, higgsino or sneutrino lsp are also considered. we show that the higgs mass constraint is important for the higgsino lsp case, which requires the lsp mass to be larger than about 245 gev. 0.0 mm 0.0 mm 159.2 mm -16.0 mm 240.0 mm

hep-ph0102030

Introduction one of the fundamental problems in the classification of complex surfaces is to find a new family of complex surfaces of general type with @xmath0. in this paper we construct new simply connected _ numerical campedelli surfaces _ with an involution, i.e. simply connected minimal complex surfaces of general type with @xmath0 and @xmath1, that have an automorphism of order @xmath5. there has been a growing interest for complex surfaces of general type with @xmath0 having an involution ; cf. j. keum - y. lee @xcite, calabri - ciliberto - mendes lopes @xcite, calabri - mendes lopes - pardini @xcite, y. lee - y. shin @xcite, rito @xcite. a classification of _ numerical godeaux surfaces _ (i.e. minimal complex surfaces of general type with @xmath0 and @xmath2) with an involution is given in calabri - ciliberto - mendes lopes @xcite. it is known that the quotient surface of a numerical godeaux surface by its involution is either rational or birational to an enriques surface, and the bicanonical map of the numerical godeaux surface factors through the quotient map. however, the situation is more involved in the case of numerical campedelli surfaces, because the bicanonical map may not factor through the quotient map ; cf. calabri - mendes lopes - pardini @xcite. in particular it can happen that the quotient is of general type. more precisely, let @xmath6 be a numerical campedelli surface with an involution @xmath7. if @xmath7 has only fixed points and no fixed divisors, then the minimal resolution @xmath8 of the quotient @xmath9 is a numerical godeaux surface and @xmath7 has only four fixed points ; cf. barlow @xcite. conversely, if @xmath8 is of general type, then @xmath7 has only four fixed points and no fixed divisors ; calabri - mendes lopes - pardini @xcite. there are some examples of numerical campedelli surfaces @xmath6 with an involution @xmath7 having only four fixed points. barlow @xcite constructed examples with @xmath10. barlow @xcite also constructed examples with @xmath11 whose minimal resolution of the quotient by the involution is the first example of a _ simply connected _ numerical godeaux surface. also all catanese s surfaces @xcite have such an involution and @xmath12. recently calabri, mendes lopes, and pardini @xcite constructed a numerical campedelli surface with torsion @xmath13 and two involutions. frapporti @xcite showed that there exists an involution having only four fixed points on the numerical campedelli surface with @xmath14 constructed first in bauer - catanese - grunewald - pignatelli @xcite. it is known that the orders of the algebraic fundamental groups of numerical campedelli surfaces are at most @xmath15 and the dihedral groups @xmath16 and @xmath17 can not be realized. recently, the existence question for numerical campedelli surfaces with @xmath18 was settled by the construction of examples with @xmath19 ; frapporti @xcite and h. park - j. shin @xcite. hence it would be an interesting problem to construct numerical campedelli surfaces having an involution with @xmath20 for each given group @xmath21 with @xmath22. especially we are concerned with the simply connected case because the fundamental groups of all the known examples with an involution have large order : @xmath23. furthermore the first example of _ simply connected _ numerical campedelli surfaces is very recent (y. lee - j. park ), but we have no information about the existence of an involution in their example. the main theorem of this paper is : there are simply connected minimal complex surfaces @xmath6 of general type with @xmath24 and @xmath25 which have an involution @xmath7 such that the minimal resolution @xmath8 of the quotient @xmath9 is a simply connected minimal complex surface of general type with @xmath26 and @xmath27. we also show that the minimal resolution @xmath8 of the quotient @xmath9 has a local deformation space of dimension @xmath28 corresponding to deformations @xmath29 of @xmath8 such that its general fiber @xmath30 is the minimal resolution of a quotient @xmath31 of a numerical campedelli surface @xmath32 by an involution @xmath33 ; theorem [theorem : invariant - part]. in addition, we show that the resolution @xmath8 should be always simply connected if the double cover @xmath6 is already simply connected ; proposition [proposition : simply - connected=>simply - connected]. conversely barlow @xcite showed that if the resolution @xmath8 is a simply connected numerical godeaux surface then the possible order of the algebraic fundamental group of the double cover @xmath6 is @xmath34, @xmath35, @xmath36, @xmath37, or @xmath15. as far as we know, the example in barlow @xcite was the only one whose quotient is simply connected. it has @xmath38 as mentioned earlier. here we find an example with @xmath39. hence it would be an intriguing problem in this context to construct an example with @xmath40. in order to construct the examples, we combine a double covering and a @xmath3-gorenstein smoothing method developed in y. lee - j. park . first we build singular surfaces by blowing up points and then contracting curves over a specific rational elliptic surface. these singular surfaces differ by contracting certain @xmath41-curves. if we contract all of the @xmath41-curves, we obtain a stable surface @xmath42 in the sense of kollr shepherd - barron @xcite, and we prove that the space of @xmath3-gorenstein deformations of @xmath42 is smooth and @xmath43 dimensional ; proposition [propsotion : stable - godeaux]. a (@xmath3-gorenstein) smoothing of @xmath42 in this space produces simply connected numerical godeaux surfaces. in particular, the smoothing of @xmath42 gives the existence of a two dimensional family of simply connected numerical godeaux surfaces with six @xmath41-curves ; corollary [corollary : six]. we also prove that a four dimensional family in this space produces simply connected numerical godeaux surfaces with a @xmath5-divisible divisor consisting of four disjoint @xmath41-curves ; theorem [theorem : q - smoothing - of - y] and theorem [theorem : invariant - part]. these numerical godeaux surfaces are used to construct the numerical campedelli surfaces with an involution. the desired numerical campedelli surfaces are obtained by taking double coverings of the numerical godeaux surfaces branched along the four disjoint @xmath41-curves ; theorem [theorem : campedelli]. on the other hand we can also obtain the campedelli family explicitly from a singular stable surface @xmath44. it comes from blowing up points and contracting curves over a certain rational elliptic surface ; proposition [proposition : campexplicit]. the @xmath3-gorenstein space of deformations of @xmath44 is smooth and @xmath45 dimensional ; proposition [proposition : stable - campedelli]. in both godeaux and campedelli cases we compute @xmath46 to show no local - to - global obstruction to deform them ; theorem [theorem : h2(ty)=0] and theorem [theorem : h2(tx)=0]. this involves a new technique (theorem [theorem : burns - wahl]) which generalizes a result of burns - wahl @xcite describing the space of first order deformations of a singular complex surface with only rational double points. a cyclic quotient singularity (germ at @xmath47 of) @xmath48, where @xmath49 with @xmath50 a @xmath51-th primitive root of @xmath34, @xmath52, and @xmath53, is denoted by @xmath54. @xmath55 means @xmath56. a @xmath57-curve (or @xmath41-curve) in a smooth surface is an embedded @xmath58 with self - intersection @xmath59 (respectively, @xmath60). throughout this paper we use the same letter to denote a curve and its proper transform under a birational map. a singularity of class @xmath61 is a quotient singularity which admits a @xmath3-gorenstein one parameter smoothing. they are either rational double points or @xmath62 with @xmath63 and @xmath64 ; see kollr shepherd - barron @xcite. for a normal variety @xmath6 its tangent sheaf @xmath65 is @xmath66. the dimension of @xmath67 is @xmath68. the authors would like to thank professor yongnam lee for helpful discussion during the work, careful reading of the draft version, and many valuable comments. the authors also wish to thank professor jenia tevelev for indicating a mistake in an earlier version of this paper, and the referee especially for the remark on the proof of proposition 3.7 which makes it simpler. heesang park was supported by basic science research program through the national research foundation of korea (nrf) grant funded by the korean government (2011 - 0012111). dongsoo shin was supported by basic science research program through the national research foundation of korea (nrf) grant funded by the korean government (2010 - 0002678). giancarlo urza was supported by a fondecyt inicio grant funded by the chilean government (11110047). Numerical godeaux surfaces with a @xmath5-divisible divisor in this section we construct a family of simply connected numerical godeaux surfaces having a @xmath5-divisible divisor consisting of four disjoint @xmath41-curves by smoothing a singular surface @xmath69 ; theorem [theorem : q - smoothing - of - y]. this is the key to construct numerical campedelli surfaces with an involution. in addition, we describe the explicit stable model of the singular surface @xmath69. in fact, we construct a rational normal projective surface @xmath42 with four singularities @xmath70, @xmath70, @xmath71, @xmath72 and @xmath73 ample. hence @xmath42 is a stable surface (cf. kollr - shepherd - barron @xcite, hacking @xcite). we will prove that the versal @xmath3-gorenstein deformation space @xmath74 (cf. hacking @xcite) is smooth and @xmath43 dimensional, and that the @xmath3-gorenstein smoothings of @xmath42 are simply connected numerical godeaux surfaces. in particular, this shows that there are simply connected numerical godeaux surfaces whose canonical model has precisely two @xmath70 singularities ; corollary [corollary : godeaux]. furthermore a four dimensional family in @xmath74 produces the above simply connected numerical godeaux surfaces with a @xmath5-divisible divisor consisting of four disjoint @xmath41-curves ; theorem [theorem : q - smoothing - of - y] and theorem [theorem : invariant - part]. we start with a rational elliptic surface @xmath75 with an @xmath76-singular fiber, an @xmath77-singular fiber, and two nodal singular fibers. in fact we will use the same rational elliptic surface @xmath75 in the papers h. park - j. shin @xcite. however, we need to sketch the construction of @xmath75 to show the relevant curves that will be used to build the singular surfaces @xmath69 and @xmath42. let @xmath78, @xmath79, @xmath80, and @xmath81 be lines in @xmath82 and let @xmath83 be a smooth conic in @xmath82 given by the following equations. they intersect as in figure [figure : pencil]. @xmath84 we consider the pencil of cubics @xmath85 \in \mathbb{cp}^1 \}\]] generated by the two cubic curves @xmath86 and @xmath87. this pencil has four base points @xmath88, @xmath89, @xmath90, @xmath91, and four singular members corresponding to @xmath92=[1:0], [0:1], [2:3\sqrt{3 }], [2:-3\sqrt{3}]$]. the latter two singular members are nodal curves, denoted by @xmath93 and @xmath94 respectively. they have nodes at @xmath95 $] and @xmath96 $], respectively. in order to obtain a rational elliptic surface @xmath75 from the pencil, we resolve all base points (including infinitely near base - points) of the pencil by blowing - up @xmath15 times as follows. we first blow up at the points @xmath88, @xmath89, @xmath90, @xmath91. let @xmath97, @xmath98, @xmath99, @xmath100 be the exceptional divisors over @xmath88, @xmath89, @xmath90, @xmath91, respectively. we blow up again at the three points @xmath101, @xmath102, @xmath103. let @xmath104, @xmath105, @xmath106 be the exceptional divisors over the intersection points, respectively. we finally blow up at each intersection points @xmath107 and @xmath108. let @xmath109 and @xmath110 be the exceptional divisors over the blown - up points. we then get a rational elliptic surface @xmath111 over @xmath58 ; see figure [figure : e]. the four exceptional curves @xmath100, @xmath106, @xmath109, @xmath110 are sections of the elliptic fibration @xmath75, which correspond to the four base points @xmath91, @xmath90, @xmath88, @xmath89, respectively. the elliptic fibration @xmath75 has one @xmath76-singular fiber @xmath112 containing all @xmath113 (@xmath114) : @xmath115, @xmath116, and @xmath117 ; cf. figure [figure : e]. we will use frequently the sum @xmath118, which will be shown to be @xmath5-divisible. the surface @xmath75 has also one @xmath77-singular fiber consisting of @xmath81 and @xmath83, and it has two more nodal singular fibers @xmath93 and @xmath94. ] there is a special bisection on @xmath75. let @xmath119 be the line in @xmath82 passing through the point @xmath120 $] and the two nodes @xmath95 $] and @xmath96 $] of @xmath93 and @xmath94. since @xmath119 meets every member in the pencil at three points but it passes through only one base point, the proper transform of @xmath119 is a bisection of the elliptic fibration @xmath121 ; cf. figure [figure : e]. note that @xmath122 in @xmath75. let @xmath123 be the class of the pull - back of a line in @xmath82. we denote again by @xmath124 the class of the pull - back of the exceptional divisor @xmath125. we have the following linear equivalences of divisors in @xmath75 : @xmath126 let @xmath127. note that the divisor @xmath118 is @xmath5-divisible because of the relation @xmath128 in the construction of @xmath130, we use only one section @xmath131. we first blow up at the two nodes of the nodal singular fibers @xmath93 and @xmath94 so that we obtain a blown - up rational elliptic surface @xmath132 ; figure [figure : w]. let @xmath133 and @xmath134 be the exceptional curves over the nodes of @xmath93 and @xmath94, respectively. we further blow up at each three marked points @xmath135 in figure [figure : w], and we blow up twice at the marked point @xmath136 (that is, we first blow - up @xmath136 and then again on the intersection point of the section and the exceptional curve ; see figure [figure : z]). we then get @xmath137 as in figure [figure : z]. there exist two linear chains of the @xmath58 in @xmath130 whose dual graphs are given by : @xmath138 where @xmath139 consists of @xmath81, @xmath8, @xmath93, @xmath133, and @xmath140 contains @xmath94, @xmath134, @xmath119. ] ] we construct rational singular surfaces which produce under @xmath3-gorenstein smoothings simply connected surfaces of general type with @xmath0 and @xmath2. we first contract the two chains @xmath139 and @xmath140 of @xmath58 s from the surface @xmath130 so that we have a normal projective surface @xmath69 with two singularities @xmath141, @xmath142 of class @xmath61 : @xmath143. denote the contraction morphism by @xmath144. let @xmath145 be the surface obtained by contracting the four @xmath41-curves @xmath146 in @xmath69. we denote the contraction morphism by @xmath147. then @xmath145 is also a normal projective surface with singularities @xmath141, @xmath142 from @xmath69, and four @xmath148 s (ordinary double points), denoted by @xmath149. we finally contract @xmath139, @xmath140, @xmath150 and @xmath151 in @xmath130 to obtain @xmath42. it has the singularities @xmath152, and two @xmath153 s. let @xmath154 be the contraction. in section [section : obstruction] we will prove that the obstruction spaces to local - to - global deformations of the singular surfaces @xmath69, @xmath145 and @xmath42 vanish. that is : [theorem : h2(ty)=0] @xmath155, @xmath156, and @xmath157. the singular surface @xmath42 is the stable model of the singular surfaces @xmath69 and @xmath145 : [propsotion : stable - godeaux] the surface @xmath42 has @xmath158, @xmath159, and @xmath73 is ample. the space @xmath74 is smooth and @xmath43 dimensional. a @xmath3-gorenstein smoothing of @xmath42 is a simply connected canonical surface of general type with @xmath0 and @xmath2. for a surface @xmath42 with only singularities of type @xmath61 we have @xmath160 where @xmath161 is the number of exceptional curves over @xmath88 and @xmath162 is the milnor number of @xmath88. in our case, @xmath163. we have @xmath164 because of the rationality of the singularities. we now compute @xmath165 in a @xmath3-numerically effective way. let @xmath166 be the general fiber of the elliptic fibration in @xmath130. let @xmath167, @xmath168, @xmath169 be the exceptional curves over @xmath170, @xmath171, @xmath172 respectively. let @xmath173, @xmath174 be the exceptional curves over @xmath175 with @xmath176. then, @xmath177. we also have @xmath178. writing @xmath179 in @xmath180 and adding the discrepancies from @xmath141 and @xmath142, we obtain @xmath181 we now intersect @xmath165 with all the curves in its support, which are not contracted by @xmath182, to check that @xmath73 is nef. moreover, if @xmath183 for a curve @xmath184 not contracted by @xmath182, then @xmath184 is a component of a fiber in the elliptic fibration which does not intersect any curve in the support. this is because @xmath93, @xmath133, @xmath167, and @xmath168 belong to the support, and they are the components of a fiber. one easily checks that @xmath184 does not exist, proving that @xmath73 is ample. therefore any @xmath3-gorenstein smoothing of @xmath42 over a (small) disk will produce canonical surfaces ; cf. kollr - mori @xcite. to compute the fundamental group of a @xmath3-gorenstein smoothing we use the recipe in y. lee - j. we follow the argument as in y. lee - j. consider the normal circles around @xmath119 and @xmath133. we can compare them through the transversal sphere @xmath167. since the orders of the circles are @xmath185 and @xmath186, which are coprime, we obtain that both end up being trivial. the smoothness of @xmath74 follows from theorem [theorem : h2(ty)=0] and hacking @xcite. to compute the dimension, we observe that if @xmath187 is a @xmath3-gorenstein smoothing of @xmath188 and @xmath189 is the dual of @xmath190, then @xmath189 restricts to @xmath191 as @xmath192 (tangent bundle of @xmath191) when @xmath193, and @xmath194 with cokernel supported at the singular points of @xmath195 ; cf. wahl @xcite. then the flatness of @xmath189 and semicontinuity in cohomology plus the fact that @xmath196 gives @xmath197 for any @xmath198. but then, since @xmath191 is of general type, hirzebruch - riemann - roch theorem says @xmath199 this proves the claim. [corollary : godeaux] there is a two dimensional family of simply connected canonical numerical godeaux surfaces with two @xmath70 singularities. [corollary : six] we consider the sequence @xmath200 at the end of hacking @xcite. we just proved that @xmath201 is @xmath43 dimensional, and we know that @xmath202 is @xmath43 dimensional, since each @xmath70 gives @xmath35 dimensions and each @xmath203 gives @xmath34 dimension. therefore @xmath204. to produce the claimed family we need to smooth up at the same time @xmath141 and @xmath142. a simply connected numerical godeaux surfaces with a @xmath5-divisible divisor consisting of four disjoint @xmath41-curves is obtained from a @xmath3-gorenstein smoothing of the singular surface @xmath69 : [theorem : q - smoothing - of - y] a. there is a @xmath3-gorenstein smoothing @xmath205 over a disk @xmath206 with central fiber @xmath207 and an effective divisor @xmath208 such that the restriction to a fiber @xmath209 over @xmath210 @xmath211 is @xmath5-divisible in @xmath209 consisting of four disjoint @xmath41-curves and @xmath212. b. there is a @xmath3-gorenstein deformation @xmath213 of @xmath145 with central fiber @xmath214 such that a fiber @xmath215 over @xmath216 has four ordinary double points as its only singularities and the minimal resolution of @xmath215 is the corresponding fiber @xmath209 of @xmath217. we apply a similar method in y. lee - j. park @xcite. since any local deformations of the singularities of @xmath145 can be globalized by theorem [theorem : h2(ty)=0], there are @xmath3-gorenstein deformations of @xmath145 over a disk @xmath206 which keep all four ordinary double points and smooth up @xmath141 and @xmath142. let @xmath218 be such deformation, with @xmath145 as its central fiber, and @xmath219 (@xmath216) a normal projective surface with four @xmath148s as its only singularities. we resolve simultaneously these four singularities in each fiber @xmath219. we then get a family @xmath217 that is a @xmath3-gorenstein smoothing of the central fiber @xmath69, which shows that a @xmath3-gorenstein smoothing of @xmath145 can be lifted to a @xmath3-gorenstein smoothing of the pair @xmath220, i.e. the @xmath5-divisible divisor @xmath221 on @xmath69 is extended to an effective divisor @xmath208. we finally show that the effective divisor @xmath222 is 2-divisible in @xmath209 for @xmath216. according to manetti (*??? * lemma 2), the natural restriction map @xmath223 is injective for every @xmath210 and bijective for @xmath224. here we are using that @xmath225. since the divisor @xmath221 is nonsingular, @xmath226 is also nonsingular. since @xmath227 in, it follows that @xmath228, where @xmath229 is extended to a line bundle @xmath230 and @xmath231 is the corresponding restriction. Numerical campedelli surfaces with an involution the main purpose of this section is to construct simply connected numerical campedelli surfaces with an involution. along the way, we will introduce a rational normal projective surface @xmath44 with @xmath45 singularities (two @xmath148, two @xmath232, and two @xmath72) and @xmath233 ample. a certain four dimensional @xmath3-gorenstein deformation of @xmath44 will produce numerical campedelli surfaces with an involution. recall that the rational surface @xmath130 has a @xmath5-divisible divisor @xmath118 ; cf. . let @xmath234 be the double cover of @xmath130 branched along the divisor @xmath221, where the double cover is given by the data @xmath227, @xmath235. we denote the double covering by @xmath236. the surface @xmath234 has two @xmath139 s and two @xmath140 s. on the other hand the surface @xmath234 can be obtained from a certain rational elliptic surface by blowing - ups, as we now explain. the morphism @xmath237 blows down to a double cover @xmath238 branched along @xmath221. the ramification divisor @xmath239 consists of four disjoint @xmath57-curves @xmath240. we blow down them from @xmath241 to obtain a surface @xmath242 ; cf. figure [figure : e]. in figure [figure : e] the pull - back of the @xmath243 are the @xmath244, of the curve @xmath81 is @xmath245, of the section @xmath8 is @xmath246, and of the double section @xmath119 is @xmath247. each @xmath77 in @xmath242 is the pull - back of each @xmath248 in @xmath75. ] note that the surface @xmath242 has an elliptic fibration structure with two @xmath249-singular fibers and two @xmath77-singular fibers. in fact, the surface @xmath242 can be obtained from the pencil of cubics in @xmath82 @xmath250 \in \mathbb{cp}^1 \}\]] where the @xmath249-singular fibers come from @xmath251 and @xmath252, and the @xmath77-singular fibers come from @xmath253 and @xmath254. the two double sections @xmath255 and @xmath256 are defined by the lines @xmath257 and @xmath258. in summary : [proposition : campexplicit] the surface @xmath241 is the blow - up at four nodes of one @xmath249-singular fiber of the rational elliptic fibration @xmath259. hence the surface @xmath234 can be obtained from @xmath241 by blowing - up in the obvious way. let @xmath260 be the double cover of the singular surface @xmath69 branched along the divisor @xmath221. note that the surface @xmath260 is a normal projective surface with four singularities of class @xmath61 whose resolution graphs consist of two @xmath139 s and two @xmath140 s. the ramification divisor in @xmath260 consists of the four disjoint @xmath57-curves @xmath261. let @xmath262 be the double covering. on the other hand the surface @xmath260 can be obtained from the rational surface @xmath234 by contracting the two @xmath139 s and two @xmath140 s. let @xmath263 be the contraction morphism. let @xmath6 be the surface obtained by blowing down the four @xmath57-curves @xmath240 from @xmath260. we denote the blowing - down morphism by @xmath264. then there is a double covering @xmath265 branched along the four ordinary double points @xmath149. finally, let @xmath44 be the contraction of the @xmath41-curves @xmath266 and @xmath267 in @xmath6. let @xmath268 be the contraction. we then get a double covering @xmath269. to sum up, we have the following commutative diagram : @xmath270 \ar[d]^{\psi'} & v \ar[l] \ar[d]^{\psi } \ar[r]^{\widetilde{\beta } } & \widetilde{x } \ar[d]^{\widetilde{\phi } } \ar[r]^{\beta } & x \ar[d]^{\phi } \ar[r] & x'\ar[d]\\ & e(1) & \ar[l] z \ar[r]^{\widetilde{\alpha } } & \widetilde{y } \ar[r]^{\alpha } & y \ar[r] & y'}\]] we will show in section [section : obstruction] the obstruction spaces to local - to - global deformations of the singular surfaces @xmath260, @xmath6 and @xmath44 vanish : [theorem : h2(tx)=0] @xmath271, @xmath272, and @xmath273. the singular surface @xmath44 is the stable model of @xmath260 and @xmath6 : [proposition : stable - campedelli] the surface @xmath44 has @xmath274, @xmath275, and @xmath233 ample. the space @xmath276 is smooth and @xmath45 dimensional. a @xmath3-gorenstein smoothing of @xmath44 is a simply connected canonical surface of general type with @xmath0 and @xmath1. the proof goes as the one for @xmath42 in proposition [propsotion : stable - godeaux], using the explicit model we have for @xmath234 by blowing - up @xmath242 in proposition [proposition : campexplicit]. one can check that an intersection computation as in proposition [propsotion : stable - godeaux] verifies ampleness for @xmath233. the proof of the next main result follows easily from theorem [theorem : q - smoothing - of - y]. [theorem : campedelli] there exist @xmath3-gorenstein smoothings @xmath277 of @xmath260 and @xmath278 of @xmath6 that are compatible with the @xmath3-gorenstein deformations of @xmath279 of @xmath69 and @xmath218 of @xmath145 in theorem [theorem : q - smoothing - of - y], respectively ; that is, the double coverings @xmath280 and @xmath265 extend to the double coverings @xmath281 and @xmath282 between the fibers of the @xmath3-gorenstein deformations. by theorem [theorem : h2(tx)=0], the obstruction @xmath283 to local - to - global deformations of the singular surface @xmath6 vanishes. the point of the above theorem is that there is a @xmath3-gorenstein smoothing of the cover @xmath6 that is compatible with the @xmath3-gorenstein deformation of the base @xmath145. [corollary : campedelli - with - involution] a general fiber @xmath32 of the @xmath3-gorenstein smoothing @xmath278 of @xmath6 is a simply connected numerical campedelli surface with an involution @xmath33 such that the minimal resolution of the quotient @xmath284 is a simply connected numerical godeaux surface. let @xmath6 be a minimal complex surface of general type with @xmath0 and @xmath1. suppose that the group @xmath4 acts on @xmath6 with just @xmath28 fixed points. let @xmath285 be the quotient and let @xmath286 be the minimal resolution of @xmath145. barlow (*??? * proposition 1.3) proved that if @xmath287 then @xmath288. conversely : [proposition : simply - connected=>simply - connected] if @xmath6 is simply connected, then so is @xmath8. let @xmath289 be the quotient map. then @xmath290 is a double covering which is branched along the four ordinary double points of @xmath145. let @xmath291 be the complement of the four branch points (i.e., the four @xmath148-singularities) of @xmath145 and let @xmath292, that is, @xmath293 is the complement of the four fixed points of the involution @xmath7. then we get an tale double covering @xmath294. since @xmath295, we have @xmath296. note that the boundary @xmath297 of an arbitrary small neighborhood @xmath298 of one of the four nodes of @xmath145 is a lens space @xmath299. let @xmath300 $] be a generator of @xmath301 represented by a loop @xmath302 contained in @xmath297. since the lifting of @xmath302 by the covering @xmath294 is not a closed path and @xmath303, @xmath304 is generated by @xmath300 $]. then it follows by van kampen theorem that @xmath305 is trivial. hence @xmath306 is trivial because @xmath8 is obtained from @xmath145 by resolving only four @xmath148-singularities. The obstruction spaces to local-to-global deformations in this section we prove theorem [theorem : h2(ty)=0] which says that the obstruction spaces to local - to - global deformations of the singular surfaces @xmath69, @xmath145, and @xmath42 vanish. that is, we will prove that @xmath307. at the end, we also prove the analogues, theorem [theorem : h2(tx)=0], for @xmath260, @xmath6, and @xmath44. at first the vanishing of the obstruction spaces of a singular surface can be proved by the vanishing of the second cohomologies of a certain logarithmic tangent sheaf on the minimal resolution of the singular surface : [proposition : lee - park - log] if @xmath308 be the minimal resolution of a normal projective surface @xmath8 with only quotient singularities, and @xmath309 is the reduced exceptional divisor of the resolution @xmath310, then @xmath311. [proposition : flenner - zaidenberg] let @xmath61 be a nonsingular surface and let @xmath309 be a simple normal crossing divisor in @xmath61. let @xmath312 be the blow - up of @xmath61 at a point @xmath88 of @xmath309. let @xmath313. then @xmath314. we can add or remove disjoint @xmath57-curves. [add - delete] let @xmath61 be a nonsingular surface and let @xmath309 be a simple normal crossing divisor in @xmath61. let @xmath315 be a @xmath57-curve in @xmath61 such that @xmath316 is again simple normal crossing. then @xmath317. we can also add or remove disjoint exceptional divisors of rational double points. the following theorem may give a new general way to prove unobstructedness for deformations of surfaces. [theorem : burns - wahl] let @xmath8 be a normal projective surface with only rational double points @xmath318 as singularities. let @xmath319 be the minimal resolution of @xmath8 with exceptional reduced divisor @xmath320. let @xmath321 be a simple normal crossing divisor such that @xmath322. then @xmath323. let @xmath324 and @xmath325 be the prime decompositions of @xmath119 and @xmath321. we have three short exact sequences : @xmath326 we then have the following commutative diagram of cohomologies : @xmath327 & 0 \ar[d] & & \\ 0 \ar[r] & h^1(\sheaf{t_{\widetilde{s}}}(-\log(c+m))) \ar[r] \ar[d] & h^1(\sheaf{t_{\widetilde{s}}}(-\log{c })) \ar[r]^{\phi } \ar[d]^{\xi } & \oplus h^1(\sheaf{n_{m_i/{\widetilde{s } } } }) \ar@{=}[d] & \\ 0 \ar[r] & h^1(\sheaf{t_{\widetilde{s}}}(-\log{m })) \ar[r] \ar[d] & h^1(\sheaf{t_{\widetilde{s } } }) \ar[r]^{\psi } \ar[d]^{\zeta } & \oplus h^1(\sheaf{n_{m_i/{\widetilde{s } } } }) \ar[r] & 0\\ & \oplus h^1(\sheaf{n_{c_i/{\widetilde{s } } } }) \ar@{=}[r] & \oplus h^1(\sheaf{n_{c_i/{\widetilde{s } } } }) & & } } \]] here all horizontal and vertical sequences are exact. especially the second row is a short exact sequence, which we explain now briefly : it is shown in burns - wahl @xcite (see also wahl @xcite) that the composition @xmath328 is an isomorphism because the @xmath329 s are _ rational double points _ ; hence, one has a direct sum decomposition @xmath330 and an isomorphism @xmath331. therefore the second row is exact. in order to prove the assertion, it is enough to show that @xmath332 is surjective. let @xmath333. since @xmath334 is surjective, we have @xmath335 for some @xmath336. by we have @xmath337 for some @xmath338 and @xmath339 such that @xmath182 is mapped to @xmath340 under the composition @xmath341. since @xmath182 is supported on @xmath119 and @xmath342, its image @xmath343 under @xmath344 vanishes. therefore @xmath345, and so @xmath346 for some @xmath347 ; hence, @xmath348, which shows that @xmath349 is surjective. [proposition : e - v - sequence] let @xmath350 be a simple normal crossing divisor on a smooth surface @xmath61. then one has the following exact sequences : a. @xmath351. b. @xmath352. we first claim that @xmath353 by duality, we have to show that @xmath354 since @xmath355, it follows by proposition [proposition : e - v - sequence] that @xmath356 hence it suffices to show that @xmath357. on the other hand, we obtain a long exact sequence from proposition [proposition : e - v - sequence] : @xmath358 since @xmath359, it is enough to show that the connecting homomorphism @xmath360 is injective. note the map @xmath360 is the first chern class map. but @xmath93 and @xmath133 are linearly independent in the picard group of @xmath361 ; hence, the map @xmath360 is injective. therefore the claim follows. let @xmath362. by theorem [theorem : burns - wahl] we have @xmath363 we use propositions [add - delete] and [proposition : flenner - zaidenberg] to obtain @xmath364 where @xmath365. in this way, it follows by the above claim that @xmath366 then, by proposition [proposition : lee - park - log], we have @xmath367. notice we can modify @xmath368 to obtain vanishing for @xmath369 and @xmath370 as well. we now prove that @xmath371. we will use our explicit model of @xmath44 in proposition [proposition : campexplicit]. the proof goes along the same lines as the proof of the above theorem [theorem : h2(ty)=0]. we may only need to mention that we start with the elliptic fibration @xmath242, and @xmath5 @xmath77 fibers (instead of @xmath5 @xmath248 s). The invariant part of the deformation space the involution of a general fiber @xmath32 induced by the double covering @xmath282 extends to a @xmath4-action on the deformation space of @xmath32. we will count the dimension of the subspace of @xmath372 which is fixed by the @xmath4-action ; theorem [theorem : invariant - part]. let @xmath373 be the minimal resolution and let @xmath374 be the blowing - up at the four ramification points. we then have the following commutative diagram where the vertical morphisms are double covers : @xmath375^{\beta_t } \ar[d]_{\widetilde{\phi}_t } & x_t \ar[d]^{\phi_t } \\ \widetilde{y}_t \ar[r]^{\alpha_t } & y_t } \]] recall that the branch divisor @xmath222 of the double covering @xmath376 consists of four disjoint @xmath41-curves @xmath377 and the corresponding ramification divisor @xmath378 consists of four disjoint @xmath57-curves @xmath379. as before the involution of @xmath380 induced by the double covering @xmath281 extends to a @xmath4-action on the deformation space of @xmath380. [lemma : invariant] @xmath381. by pardini (* lemma 4.2), the invariant part of @xmath382 under the @xmath4-action is @xmath383. therefore we have @xmath384 we know that @xmath385, where @xmath386 defines the double cover @xmath387. therefore @xmath388 by theorem [proposition : stable - campedelli]. then we have @xmath389. hence, by proposition [proposition : e - v - sequence], there is a short exact sequence @xmath390 since @xmath222 consists of four disjoint @xmath41-curves, we have @xmath391. by proposition [propsotion : stable - godeaux], we know that @xmath392. therefore @xmath393. [theorem : invariant - part] the subspace of the deformation space of @xmath32 invariant under the @xmath4-action is four dimensional. we apply a similar strategy in werner @xcite. we have the exact sequence @xmath394 since each @xmath395 is a @xmath57-curve, we have @xmath396. on the other hand, it follows by catanese (*??? * lemma 9.22) that @xmath397 where @xmath398 is the ideal sheaf of the four points in @xmath32 obtained by contacting the exceptional divisors @xmath399. let @xmath400 be the set of these four points. from the ideal sequence, we have @xmath401 therefore the invariant parts of each space satisfies : @xmath402 according to werner @xcite, we have @xmath403. therefore it follows by and lemma [lemma : invariant] that @xmath404 Another example we briefly describe another rational surface @xmath130 which makes it possible to construct simply connected numerical campedelli surfaces with an involution as before. the associated godeaux surfaces come from a rational surface @xmath42 with @xmath73 ample having three @xmath148-singularities, one @xmath70-singularity, and only one singularity of class @xmath61. the elliptic fibration @xmath75 is the one in section [section : godeaux]. in the construction of @xmath130, we will use the sections @xmath100, @xmath106, @xmath109 among the four sections of @xmath75. we denote the sections @xmath100, @xmath106, @xmath109 by @xmath406, @xmath407, @xmath408, respectively. we first blow up at the two nodes of the nodal singular fibers @xmath93 and @xmath94 so that we obtain a blown - up rational elliptic surface @xmath409 ; figure [figure : w-2]. let @xmath133 and @xmath134 be the exceptional curves over the nodes of @xmath93 and @xmath94, respectively. we further blow up at each two marked points @xmath135 and blow up four times at the marked point @xmath136 in figure [figure : w-2]. we then get a rational surface @xmath410 ; figure [figure : z-2]. there exists one linear chain of @xmath58s in @xmath130 whose dual graph is @xmath411 notice that the @xmath57-curve @xmath408 is contracted in the way down, which fixes the configuration so that we obtain one singular point of class @xmath61 whose resolution graph is given by @xmath412 the divisor @xmath413 is the @xmath5-divisible one as before. the @xmath414 and the @xmath41-curves @xmath415, @xmath416, @xmath151, and @xmath119 are contracted to obtain a singular surface @xmath42. one can use again theorem [theorem : burns - wahl] to show that the space @xmath74 is smooth (of dimension @xmath43).

we construct a simply connected minimal complex surface of general type with @xmath0 and @xmath1 which has an involution such that the minimal resolution of the quotient by the involution is a simply connected minimal complex surface of general type with @xmath0 and @xmath2. in order to construct the example , we combine a double covering and @xmath3-gorenstein deformation. especially, we develop a method for proving unobstructedness for deformations of a singular surface by generalizing a result of burns and wahl which characterizes the space of first order deformations of a singular surface with only rational double points. we describe the stable model in the sense of kollr and shepherd - barron of the singular surfaces used for constructing the example. we count the dimension of the invariant part of the deformation space of the example under the induced @xmath4-action.

1108.0797

Introduction there are reasons to believe that cosmic rays (crs) around the ankle at @xmath0 gev are dominated by extragalactic protons @xcite. scattering processes in the cosmic microwave background (cmb) limit the propagation of ultra high energy (uhe) charged particles in our universe. a continuation of a power - like cr spectrum above the greisen - zatsepin - kuzmin (gzk) cutoff @xcite at about @xmath1 gev is only consistent with the proton dominance if the sources lie within the proton attenuation length of about 50 mpc. very few astrophysical accelerators can generate crs with energies above the gzk cutoff (see e.g. @xcite for a review) and so far none of the candidate sources have been confirmed in our local environment. it has been speculated that decaying superheavy particles, possibly some new form of dark matter or remnants of topological defects, could be a source of uhe crs, but also these proposals are not fully consistent with the cr spectrum at lower energies @xcite. the observation of gzk excesses has led to speculations about a different origin of uhe crs. berezinsky and zatsepin @xcite proposed that _ cosmogenic _ neutrinos produced in the decay of the gzk photopions could explain these events assuming a strong neutrino nucleon interaction. we have followed this idea in ref. @xcite and investigated the statistical goodness of scenarios with strongly interacting neutrinos from optically thin sources using cr data from agasa @xcite and hires @xcite (see fig. [cr]) and limits from horizontal events at agasa @xcite and contained events at rice @xcite. -branes, and string excitations (see ref. @xcite).] Strongly interacting neutrinos from optically thin sources the flux of uhe extragalactic protons from distant sources is redshifted and also subject to @xmath2 pair production and photopion - production in the cmb which can be taken into account by means of propagation functions. the resonantly produced photopions provide a _ guaranteed _ source of cosmogenic uhe neutrinos observed at earth. in astrophysical accelerators inelastic scattering of the beam protons off the ambient photon gas in the source will also produce photopions which provide an additional source of uhe neutrinos. the corresponding spectrum will in general depend on the details of the source such as the densities of the target photons and the ambient gas @xcite. we have used the flux of crs from _ optically thin _ sources using the luminosities given in ref. @xcite in the goodness - of - fit test. for a reasonable and consistent contribution of extragalactic neutrinos in vertical crs one has to assume a strong and rapid enhancement of the neutrino nucleon interaction. the realization of such a behavior has been proposed in scenarios beyond the (perturbative) sm (see ref. @xcite). for convenience, we have approximated the strong neutrino nucleon cross section in our analysis by a @xmath3-behavior shown in fig. [fig], parameterized by the energy scale and width of the transition, and the amplification compared to the standard model predictions. our analysis showed that uhe crs measured at agasa and hires can be interpreted to the 90% cl as a composition of extragalactic protons and strongly interacting neutrinos from optically thin sources in agreement with experimental results from horizontal events at agasa and contained events at rice (see fig. [fig]). the pierre auger observatory combines the experimental techniques of agasa and hires as a hybrid detector. with a better energy resolution, much higher statistics and also stronger bounds on horizontal showers it will certainly help to clarify our picture of uhe crs in the future. Acknowledgements the author would like to thank the organizers of the erice school on nuclear physics 2005 _ `` neutrinos in cosmology, in astro, particle and nuclear physic '' _ for the inspiring workshop and vihkos (_ `` virtuelles institut fr hochenergiestrahlungen aus dem kosmos '' _) for support. m. ahlers, a. ringwald, and h. tu, _ astropart. (to appear), preprint astro - ph/0506698. v. berezinsky, a. z. gazizov and s. i. grigorieva, preprint hep - ph/0204357 ; v. berezinsky, a. z. gazizov and s. i. grigorieva,. m. ahlers _ et al. _,. k. greisen, ; g. t. zatsepin and v. a. kuzmin,. d. f. torres and l. a. anchordoqui,. d. v. semikoz and g. sigl,. v. s. beresinsky and g. t. zatsepin,. m. takeda _ et al. _ [agasa],. d. j. bird _ et al. _ [hires], ; r. u. abbasi _ et al. _ [hires], ; r. u. abbasi _ et al. _ [hires],. s. yoshida _ _ [agasa],.

the origin and chemical composition of ultra high energy cosmic rays is still an open question in astroparticle physics. the observed large - scale isotropy and also direct composition measurements can be interpreted as an extragalactic proton dominance above the _ ankle _ at about @xmath0 gev. photopion production of extragalactic protons in the cosmic microwave background predicts a cutoff at about @xmath1 gev in conflict with excesses reported by some experiments. in this report we will outline a recent statistical analysis @xcite of cosmic ray data using strongly interacting neutrinos as primaries for these excesses.

astro-ph0511483

Introduction high q cavities such as whispering gallery mode (wgm) cavities have recently demonstrated quality factors (@xmath0) as high as @xmath3 and have shown the potential to reach even higher q values @xcite. however, there are difficulties in measurement of the linewidth and q of such high q cavities. while in theory, the q factor could be as high as @xmath4 and is limited only by rayleigh scattering @xcite, in practice, it is limited by other losses in the cavity. they include absorption and scattering losses due to impurities in the cavity material, and light - induced losses due to nonlinear processes. due to the extremely small mode volume and high q - factor of the cavity, the cavity build - up intensity is extremely high, even in the case of an input with small power (as small as several mw). such a high resonator intensity leads to very efficient nonlinear processes inside wgm cavities, such as raman scattering, second harmonic generation, and four - wave mixing @xcite. whereas this is beneficial in many applications, it causes additional losses in the cavity and thus makes the q factor measurement unreliable (at least, making it power - dependent) @xcite. squeezed states of vacuum or light have been used in many applications such as improvement in interferometric @xcite and absorption @xcite measurements, for quantum teleportation @xcite and quantum cryptography @xcite, and for quantum imaging @xcite. however, to the best of our knowledge, no experiment for measurements of cavity parameters by use of squeezing has yet been reported. in this paper we propose and demonstrate an alternative method of measuring q factors by use of a squeezed vacuum field which is equivalent to a field with correlated quantum sidebands @xcite. this technique is advantageous over traditional optical methods in that it utilizes the injection of squeezed vacuum into a test cavity not to excite any nonlinear processes in the cavity. when the input field is detuned from the cavity resonance frequency, it transmits only the upper or lower quantum sidebands within the cavity linewidth while reflecting the counterparts (associated upper or lower sidebands) and all the other sidebands. the linewidth of the cavity can then be measured by observing the destruction of the correlation between the upper and lower quantum sidebands with respect to the carrier frequency. we show that the linewidth and q factor of a test cavity using the method agrees with those measured by traditional optical methods. this paper is organized as follows : in sec. [sect : theory1], we describe the theoretical framework for the measurement method. in sec. [sect : theory2], we explain the validity of the use of squeezed vacuum as a probe for non - invasive measurements and compare the technique to using a classical state. in sec. [sect : experiment], we demonstrate the method using a test cavity with known cavity parameters and compare the parameter values obtained by the new method and the traditional optical methods. the conclusions of the paper are summarized in sec. [sect : conclusions]. Theory consider a squeezed vacuum field with carrier and sideband frequencies, @xmath5 and @xmath6 respectively. as shown in fig. [cavity], when the upper sideband of the squeezed vacuum field @xmath7 is injected into an optical cavity with resonance frequency @xmath8 and mirror reflectivities @xmath9, and @xmath10, the reflected field @xmath11 and its adjoint @xmath12 are given in terms of @xmath13 and its adjoint @xmath14 by @xmath15 where @xmath16 is the frequency - dependent cavity reflection coefficient and @xmath17 is the vacuum noise coupling coefficient associated with transmission and intra - cavity losses. when the cavity is not perfectly mode - matched, the reflected field contains the cavity - coupled reflection @xmath18 @xcite and the promptly reflected field @xmath19 that does not couple to the cavity due to mode mismatch such that @xmath20 where @xmath18 and @xmath19 are spatially orthogonal and @xmath21}}{1-\sqrt{r_1 r_2 r_3}e^{-i\left[\phi_c(\omega_d) \pm \phi_s(\omega)\right] } }, \\ r_m & = & \sqrt{r_1}.\end{aligned}\]] here, @xmath22 is the detuning from the cavity resonance given by @xmath23 and we have assumed that the resonance frequency of @xmath19 is far from that of @xmath18 such that the reflection coefficient @xmath24 can be treated as a frequency - independent constant at frequencies around the resonance frequency of @xmath19. the vacuum noise coupling coefficients are then given by @xmath25 the cavity mirror reflectivity and transmission of each mirror satisfies @xmath26 where l@xmath27 is the loss of each mirror. the intra - cavity losses can be absorbed into @xmath10. , r@xmath28, and r@xmath29, respectively. @xmath30 is the upper sideband of an injected field at frequency @xmath31, @xmath32 is the cavity - filtered reflection at the frequency, @xmath33 is the transmission at the frequency, and @xmath34 is the vacuum field that couples in due to losses in the cavity at the frequency. @xmath8 is the cavity resonance frequency. the carrier field at frequency @xmath5 transmits through the cavity when @xmath35.] since the carrier is detuned from the cavity resonance frequency, the reflection acquires extra frequency - dependent phase shifts at the detuned carrier frequency and the sideband frequencies, respectively given by @xmath36 where @xmath37 and @xmath38 are the round - trip length and free spectral range of the cavity, and @xmath33 is the speed of light in vacuum. for simplicity, we transform into the rotating frame of the carrier frequency @xmath5 in the frequency domain, such that eqs. and become @xmath39 where @xmath40 and @xmath41 satisfy the commutation relations @xmath42 = 2\pi\delta(\omega-\omega^{'}),\]] and all others vanish (similarly for @xmath43, @xmath44, @xmath45, @xmath46, @xmath47, and @xmath48). in the two - photon representation @xcite, the amplitude and phase quadratures of @xmath18 are defined by @xmath49,\end{aligned}\]] respectively (similarly for @xmath19, @xmath32, @xmath50, and @xmath51). a little algebra yields the amplitude and phase quadrature fields of the reflected light in compact matrix form, @xmath52 where we use the two - photon matrix representation @xmath53 for the operator @xmath18 (similarly for @xmath19, @xmath32, @xmath50, and @xmath51), @xmath54 is a matrix representing propagation through the cavity, and @xmath55 @xmath56 comprises an overall phase shift @xmath57, rotation by angle @xmath58, and attenuation by factor @xmath59. here we have defined @xmath60, \\ a_{\pm } & \equiv & \frac{1}{2}\left[|r_c(\omega)|\pm l_{\pm } & \equiv & \frac{1}{2}\left[l_c(\omega_d+\omega) \pm l_c(\omega_d-\omega)\right].\end{aligned}\]] in the case of no carrier detuning (@xmath61), @xmath62, and @xmath58 and a@xmath63 vanish, giving neither quadrature angle rotation nor asymmetrical amplitude attenuation. in the case of cavity detunings (@xmath64), nonzero @xmath58 gives quadrature angle rotation. from eq., when we perform homodyne detection of the reflected field with a local oscillator (lo) field, the measured amplitude and phase quadrature variances of the field, defined by @xmath65 and @xmath66 (similarly for @xmath67, @xmath68, @xmath69, and @xmath70), are found in terms of the mode - matched input amplitude and phase quadrature variances @xmath67 and @xmath68 to be @xmath71 \left(\begin{array}{cc } 1\\1 \end{array}\right) \nonumber\\ & & \hspace{-0.5 cm } + \eta_m (1-r_m^2) \left(\begin{array}{cc } 1 \\ 1 \end{array}\right) + \eta_l \left(\begin{array}{cc } 1 \\ 1 \end{array}\right),\end{aligned}\]] where @xmath72 and @xmath73 are the composite efficiencies of detection associated with the cavity - coupled and cavity - mismatched modes respectively, @xmath74 is the coupling of detection losses, and @xmath75. the detection efficiency is a product of the quantum efficiency of the photodiodes and the mode - overlap efficiency with the lo mode. eq. can be rewritten in terms of the quadrature variances of the incident field @xmath76 since the cavity - coupled reflection @xmath77 and the mode - mismatch reflection @xmath78 originate from the same incident field @xmath76, such that @xmath79 and therefore, @xmath80 \left(\begin{array}{cc } v_1^{a } \\ v_2^{a } \end{array}\right) \nonumber\\ & & \hspace{-0.4cm}+ \left[1-\eta_c\left(a_+^2 + a_-^2\right) - \eta_mr_m^2\right] \left(\begin{array}{cc } 1\\1 \end{array}\right).\end{aligned}\]] note that if the input field is in a vacuum or coherent state such that @xmath81, then @xmath82, as expected, and no cavity information is contained in the output state @xmath32. if the carrier frequency is detuned downward from the cavity resonance frequency, the cavity transmits only the upper sidebands within the cavity linewidth and replaces them by vacuum at those frequencies while reflecting the associated lower sidebands and all the other sidebands. hence, the cavity - coupled reflected field is composed of the uncorrelated sidebands within the linewidth and the reflected correlated sidebands outside of it. the consequence is the destruction of the correlation within the linewidth between the upper and lower quantum sidebands. this is analogous to the destruction of the correlation between electro - optically modulated coherent sidebands in pairs, in which the beat between the carrier and the upper or lower sideband can be measured only when either sideband is absorbed into the cavity, reflecting the carrier and other sideband. the beat could not be observed if all the fields were reflected. similar measurements could be done with the transmission of the squeezed vacuum field through the cavity. however, the signal - to - noise ratio would not be as good as in the reflection method because the background of the transmission signal is shot noise. it is convenient to define the test cavity linewidth @xmath83, the quality factor @xmath0, and the finesse @xmath84, as @xmath85 \nonumber\\ & \simeq & \frac{1-\sqrt{r_1r_2r_3}}{\pi(r_1 r_2 r_3)^{1/4}}\,\omega_{\rm fsr},\end{aligned}\]] @xmath86 and @xmath87 respectively. the approximations made in eqs. and are valid for high q cavities. @xmath88, @xmath89, @xmath22, and @xmath38 will be treated as free fitting parameters. we also assume the input mirror is lossless such that @xmath90. since we are interested in having as little light (at the carrier frequency) as possible in the test cavity, it is instructive to calculate the average photon number in the field we use. the average photon number in squeezed light with squeeze factor @xmath91 and squeeze angle @xmath92 is given by @xcite @xmath93 where @xmath94 is the coherent amplitude of the light. as the number of coherent photons becomes zero (@xmath95), resulting in squeezed vacuum, eq. becomes @xmath96 this is the average photon number in squeezed vacuum generated by squeezing. note that if the field is unsqueezed (@xmath97), @xmath98. for a squeeze factor of 1.5 corresponding to the squeezed or anti - squeezed level of @xmath99 db which is the current experimental limit @xcite, @xmath100. therefore, it is fair to say that the optical influence of ideal squeezed vacuum on cavities is negligible. and @xmath101 are shown as solid and dashed curves, respectively, for different input states. (a) and (b) show the (impure) input state with @xmath102 db and @xmath103 db, _ in the absence of the cavity_. (c) and (d) show the cavity - coupled response to the squeezed and anti - squeezed vacuum injection, respectively. (e) and (f) show the cavity - coupled response to injection of a classically noisy state with @xmath104 db, @xmath105 db. comparing (e) and (c), we note that squeezing improves the signal contrast, but the classical noise and the anti - squeezed quadrature behave almost identically [cf. (d) and (f)].] similarly, it is instructive to compare this technique to using a classical state. for simplicity, assuming that the quadrature variance in both quadratures is frequency - independent, we consider the case in which the lower sideband is fully transmitted through an impedance - matched cavity and the upper sideband is fully reflected at the input mirror such that @xmath106 and @xmath107 at @xmath108, respectively, which gives @xmath109 from eq. thus, the amplitude and phase quadrature variances of the reflected field are found to be @xmath110 in the absence of coherent light, the signal contrast can be defined as the quadrature variance at detuning frequency @xmath22 compared to the cavity - uncoupled quadrature variance at off - resonance frequencies (@xmath111), in which case @xmath112 and @xmath113, and the signal contrasts at the two orthogonal quadratures are respectively given by @xmath114 in the limiting case of @xmath115 and @xmath116, we obtain @xmath117 we see that @xmath118 has about the same limiting level as in the classical case, while @xmath119 grows if @xmath120 gets smaller. classically, @xmath121 (the shot noise limit), but using squeezed vacuum we can obtain @xmath122, or improved signal contrast for a measurement in the squeezed quadrature. this is illustrated in fig. [cl_vs_sq], where we compare the signal contrast for measurement of the cavity linewidth using a classical field with the signal contrast for squeezed field injection. the cavity - coupled responses of the classical and anti - squeezed quadrature variances behave almost identically in the case of the impedance - matched cavity, whereas squeezing improves the signal contrast of the measurement. it is important to note that even in the absence of technical noise, quadrature variance measurements are intrinsically contaminated by quantum noise itself. the standard deviation of the quadrature variances is given by @xcite @xmath123 thus, the noise of the measurement is proportional to the measured value itself, and many averages can be performed to achieve smaller uncertainty levels. this is different from the classical case where the parameters of a cavity are measured by measuring the transmission of a probe optical field incident on the cavity as a function of cavity detuning. in this case, the measurements are fundamentally limited by shot noise : the number of measured photons (n) has uncertainty proportional to @xmath124. therefore, the signal - to - noise ratio grows as the number of the transmitted photons increases. Experiment mhz and @xmath125 mhz respectively. the squeezed vacuum generator is composed of an optical parametric oscillator (opo) and a second harmonic generator (shg) that pumps the opo. the cavity length is locked to the laser frequency by the pdh - locking servo and pzt (pzt2). the homodyne angle is locked by the noise - locking servo and pzt (pzt1).] the experiment is schematically shown in fig. [setup]. the nd : yag laser (lightwave model 126) gives an output of cw 700 mw at 1064 nm, which is injected into the squeezed vacuum generator (squeezer). the squeezer is composed of a second harmonic generator (shg) and an optical parametric oscillator (opo), both using 5@xmath126 mgo : linbo@xmath29 nonlinear crystals placed within optical cavities (hemilith for the shg and monolith for the opo) in the type i phase - matching configuration. the shg pumped by the nd : yag laser generates 250 mw at 532 nm, which then pumps the opo below threshold with a vacuum seed. the resultant field generated by the opo is a squeezed vacuum field with a squeezing bandwidth of 66.2 mhz defined by the opo cavity linewidth. a sub - carrier field, frequency - shifted by an acousto - optic modulator (aom) to a frequency that is coincident with the cavity tem@xmath127 mode, is injected into the other end of the opo cavity. the cavity is thus locked to the tem@xmath127 mode, offset by 220 mhz from the carrier frequency, using the pound - drever - hall (pdh) locking technique @xcite. the frequency - shift is necessary to ensure that no cavity transmitted light at the fundamental frequency is injected into the opo cavity since it acts as a seed and degrades broadband squeezing due to the imperfect isolation of the faraday isolator @xcite. this is especially important for high q cavities with linewidths as narrow as khz because low - frequency squeezing is difficult to achieve. the squeezed vacuum is injected into a triangular test cavity with the fsr of 713 mhz and fwhm of @xmath2 khz, both measured by traditional methods using light. the frequency shift, of the subcarrier is @xmath128 mhz so that the carrier frequency is detuned from the tem@xmath129 mode by @xmath130 mhz. as a result of this frequency shift, only the upper sidebands are within the cavity linewidth, destroying the correlation between the upper and lower sidebands and, therefore, destroying the squeezing or anti - squeezing. this cavity - coupled squeezed vacuum reflection is measured by balanced homodyne detection, where the field to be measured interferes with a local oscillator (lo) field and is detected by two (nearly) identical photodetectors. the difference of the two photodetector signals is sent to an hp4195a spectrum analyzer (sa) to measure the noise variance of the squeezed or anti - squeezed quadrature. the results are shown in fig. [linewidth]. the experimental data are exponentially averaged 100 times. the resolution bandwidth of the spectrum analyzer is 100 khz. since the squeezed vacuum does not carry any coherent amplitude, the noise - locking technique @xcite is employed to lock the homodyne angle to either the squeezed or anti - squeezed quadrature at 2 mhz. khz.] before fitting the experimental data points, the homodyne efficiencies @xmath131 and @xmath132, and the quantum efficiency of the photo - detectors @xmath133 need to be taken into account. the sum of the homodyne efficiencies and the quantum efficiency were independently measured to be @xmath134 and @xmath135 respectively. the sum of the efficiencies @xmath136 in eq. is given by @xmath137. we ignore @xmath132 since the cavity mode - matching efficiency is @xmath138 and hence @xmath139, which yields @xmath140. moreover, we have assumed that the input mirror m@xmath141 is lossless. this assumption is valid since it is a single - pass loss and does not influence the linewidth of the cavity. we then fit eq. to the measured data points with free parameters @xmath88,@xmath142, and @xmath22 ; both the data and the fits are shown in fig. [linewidth]. the resulting fitting values are @xmath143, and @xmath144 mhz. therefore, the fwhm linewidth of the cavity is found to be @xmath145 khz, which agrees with the classically measured linewidth of the cavity within the uncertainty (@xmath2 khz). we note that @xmath38 can be determined from the fit, but here we have used the optically measured value to estimate the linewidth. this is valid because any loss in the cavity does not change the fsr. Conclusion we have proposed and experimentally demonstrated a method for non - invasive measurements of optical cavity parameters by use of squeezed vacuum. the technique has the advantage over traditional optical methods that the injection of a squeezed vacuum field as a probe for cavity parameters does not excite any nonlinear processes in cavities, and is, therefore, useful for ultrahigh q cavities such as whispering gallery mode (wgm) cavities. we have shown that when a squeezed vacuum field is injected into a detuned cavity, the linewidth and @xmath0 factor of a test cavity can be determined by measuring the destruction of upper and lower quantum sidebands with respect to the carrier frequency. the linewidth of a test cavity is measured to be @xmath146 khz, which agrees with the classically measured linewidth of the cavity within the uncertainty (@xmath147 khz). we have also show that the use of squeezed fields leads to better signal contrast, as expected. Acknowledgments we would like to thank our colleagues at the ligo laboratory, especially thomas corbitt and christopher wipf, and stan whitcomb for his valuable comments on the manuscript. we gratefully acknowledge support from national science foundation grant nos. phy-0107417 and phy-0457264.

we propose and experimentally demonstrate a method for non - invasive measurements of cavity parameters by injection of squeezed vacuum into an optical cavity. the principle behind this technique is the destruction of the correlation between upper and lower quantum sidebands with respect to the carrier frequency when the squeezed field is incident on the cavity. this method is especially useful for ultrahigh @xmath0 cavities, such as whispering gallery mode (wgm) cavities, in which absorption and scattering by light - induced nonlinear processes inhibit precise measurements of the cavity parameters. we show that the linewidth of a test cavity is measured to be @xmath1 khz, which agrees with the classically measured linewidth of the cavity within the uncertainty (@xmath2 khz).

quant-ph0605207

Introduction the origin - destination (od) matrix is important in transportation analysis. the matrix contains information on the number of travellers that commute or the amount of freight shipped between different zones of a region. the od matrix is difficult and often costly to obtain by direct measurements / interviews or surveys, but by using incomplete traffic counts and other available information one may obtain a reasonable estimate. a particular application of the od matrix estimation is in the area of public transport. in order to improve their service, the responsible managers are looking for on - going evaluation of the passenger flow and the reasons that would influence this flow. this is typically the case for the city rail, sydney bus and sydney ferry organisations, which handle the public transport in the region around the city of sydney, australia. cityrail and co are handling a large number of stations (wharfs, bus stops) for trains (buses and ferries) across the state. they carry thousands of passengers every day, and periodically optimise the time - table schedule to best meet the changing demand. + + an ideal optimization of the schedule would consider the resources in trains, drivers, stations and passengers. while the primary informations (trains, drivers, stations) are known to cityrail and co, the number of passenger on each train between each station can not be deduced easily given their current passenger flow data collection processes. + + various approaches to estimating the od matrix using traffic counts have been developed and tested @xcite using traffic counts, or road traffic flows @xcite, @xcite. most of the papers in the literature solve this problem by postulating a general model for the trip distribution, for example a gravity type model @xcite, which aims at introducing a prior knowledge on the traffic flows and assigning a cost to each journey. then the inference is produced to estimate the parameters of this model. all these papers _ are not passengers oriented_. + most of the work relating to od matrix estimation are based on passengers observations assuming the knowledge of where the people get in and out of the public transport. lo et al @xcite developed a framework centred on the passenger choice, which they called the random link choice, and model this to obtain a maximum likelihood estimator. nandi et al @xcite applied a strategy centred on a fixed cost per person per kilometre assumption on the air - route network of india and provide some comparisons with the real data. + when the information is not available (for example we have no data on when passengers get off the bus), kostakos @xcite offers to use a wireless detection of the passengers trips, and lundgren and peterson s model @xcite is based on a target od - matrix previously defined. however, none of the cited work considered using survey data. indeed, if no complete information is available about the passengers destinations, the simplest solution is to use an appropriate survey to estimate destination information. furthermore, what characteristics of the survey are required for the estimation to be accurate? bierliaire and toint @xcite introduces a structure - based estimation of the origin - destination matrix based on parking surveys. in their article, they used the parking surveys to infer an a priori estimate of the od matrix, and they used this prior in coordination with the partial observations of the network flows to derive a generalized least square estimator of the od matrix. despite its novelty, this article assume that the behaviour of car - user and public transport users are the same, at least regarding their respective od matrix. given that the public transport network topology is often different from the road network topology, one may doubt the accuracy of this assumption. moreover, they just use the partial structure extracted from the surveys. + the purpose of this paper is then to develop an estimation procedure for the origin - destination matrix based on the ticket records available for the transport network and/or on previous surveys. unlike the article from bierliaire @xcite, we use survey data collection from public transport users, and estimate the approximate whole matrix structure through the estimation of its eigenvectors. we propose a robust version of the estimator to avoid biases induced by the survey. we also construct a regression estimation procedure that accounts for the influence of exogenous variable such as the weather conditions or the time of the year. + we first briefly present the passenger model, and then move on to outlining the observations model. in section [sec : om], we explain how the measurements are obtained, and what measurements error should be expected. in section [sec : mam], we explain the assumptions we make on the measurements, and how this affects our estimation procedure. we present in section [sec : est] the maximum likelihood (ml) estimation procedure, by providing a system of equation to be solved, for deriving estimators. we improve on this ml estimation to make it robust to survey biases in section [sec : rob]. finally, we present a simulation example and an application to a real world case in section [sec : app]. we finally comment on the results and outline some future research opportunities. The passenger model let @xmath0 be the matrix of passengers number between the stations in the rail network over time period @xmath1 so that @xmath2 is the number of passengers who depart from station @xmath3 and arrive at station @xmath4 at time period @xmath1. given that there is an obvious time dependency here, denoted by @xmath1 the period in which the commuting occur (for example a day). the purpose of this work is to provide an estimation of @xmath0 given the observations specified in section [sec : om]. The observations model the observations provided about the passengers are very different, and only considering them all allow a direct estimation of @xmath0. we list in the subsections [om - casual],[om - deparr] and [om - regular] the different kind of observations. a casual commuter is defined as a single or return journey that is not repeated regularly (e.g. daily). typically, people going to a once - in - a - year event will buy their ticket for that trajectory and will probably return on the same day. accordingly for single and day return tickets, we have complete information under the assumption that they take the next train after purchasing their ticket and that they take the shortest route. let @xmath5 be that matrix of measurements. each journey between major stations, the passenger has to validate his ticket through the machines at the entrance of the station, and do it again at the exit. between minor stations we assume they take the next train to arrive at the station they purchased their ticket at and assume they take the trio planners recommended route for that time. two scenarios are considered. in the first one, (called @xmath6), every station in the network have these machines. in the second case (called @xmath7) only major stations have these machines. in any case, let call @xmath8 the vector corresponding to the departures at the stations, and @xmath9 the vector of arrivals. fortunately we can have regular passengers with specific departure and destination, and this matrix will be denoted @xmath10, where the rows stand for the departure stations and the columns for the arrival stations. this matrix is observed, and assumed distributed according to a poisson probability function with mean @xmath11. + the main part of the information, however, remains unknown. indeed most of the passengers will probably have a zone ticket for a period of time, from 1 week to 1 year. the nature of these tickets make the station of departure and arrival unknown, and is the main challenge of this paper. let call @xmath12 the matrix of zone passengers numbers. + to make a proper statistical inference, we need two assumptions ; * the traveller will act independently of the validity duration of his ticket ; * the regular traveller commits to a return journey on each working day. the observations linked to this model are two - folds. for major stations, we have the total number of passengers that crossed the boom gates, in and out. for stations without boom gates, the observations have to estimated using a survey. we also have access to the total number of people with a valid zone ticket at time @xmath1 (e.g. the day of the analysis), denoted @xmath13, @xmath14 in the end, the total number of regular passenger at the time period @xmath1 will be denoted @xmath15, and we have, @xmath16 Model assumptions and modelling with these very different observations, we need a good fitting model based on reasonable assumptions. sections [mam - gm], [mam - cm], [mam - da] and [mam - rm] presents these assumptions for each parameter in our model. recall that @xmath17 is a matrix of count, the main assumption on that matrix is that the number of passenger is the sum of the casual passengers (@xmath18) plus the regular passengers (@xmath15) plus a matrix stating the unusual big events such as major sporting events, or large concerts (called @xmath19), @xmath20 the casual commuter journey could be assumed to be poisson distributed i.e. @xmath21 is supposed to be drawn with a poisson distribution which parameter belong to the matrix @xmath22.] where @xmath22 is the matrix of means for the counts. + however, the variance of the counts are not expected to be equal to their mean and so the poisson counts assumption may be unrealistic. therefore, we decided to use a negative binomial regression model for @xmath23, which can be over - dispersed in order to better describe the distribution of the counts. we specify that @xmath22 is distributed according to a gamma distribution, @xmath24. for a purpose of simplicity, let @xmath23 be distributed as negative binomial with parameters @xmath25 and @xmath26 (we will denote @xmath27). according to the definition of the measurements, the following relationships hold : @xmath28 where @xmath8 and @xmath9 are the vectors of the total number of departures and arrivals at each station during time period @xmath1. for the same reasons as described for the casual matrix, we will use a negative binomial distribution to model the uncertainty around the regular traveller s information. however, unlike the casual commuter, we do not suspect an over - dispersion but an under - dispersion, so that, @xmath29 and @xmath30. let @xmath31 and @xmath32 be the expectation of @xmath33 and @xmath34. Naive maximum likelihood (ml) estimation when the model is well defined, the estimation procedure is computationally straight forward, e.g., between major stations where we have complete information of arrivals and departures. meaning that the maximum likelihood estimation method accuracy, practically depends on the efficient solving of the optimization problem. in this section, the stationary model parameters are estimated from the data. since the process is unlikely to be stationary, we present a second option (section [est : reg]), a multivariate spatio - temporal model that we expect to fit the data better. the estimation procedure will be carried out in well - defined steps. if we ignore the time dependence, the successive observations can be considered independent, identical random counts from negative binomial or poisson distribution. this means that simple maximum likelihood estimation should work well, especially for large sample sizes. we observe @xmath18 for several realizations. given no space - time dependencies we assume that @xmath5 is independently distributed as @xmath35. the likelihood is then, @xmath36\end{aligned}\]] where @xmath37 stands for one element of the matrix @xmath5. we thus can estimate the parameters through, @xmath38 despite the absence of closed form solution to this problem, the optimization algorithms can quickly lead to a global maximum. unfortunately we do nt have complete information for those with weekly, monthly, quarterly or anuual tickets (long - term tickets). we have information of the times they enter and departs at major stations but we do nt have complete information for the long term tickets either to or from minor stations. our assumption here is that only a proportion @xmath39 of the @xmath32 people will travel on day @xmath1, where @xmath40. @xmath39 is an additional parameter that reflects the passengers habit. it does exist because when performing the estimation, one may find a bigger estimation of travellers than what is observed. some of the difference is due to the randomness of @xmath41, but it might also be explained by the fact that travellers with prepaid long term tickets will not necessary travel each of the working day of the week. + however, we may provide the same estimation for the @xmath31 parameters as we did in the previous section, that is, @xmath42 where @xmath43 stands for the same likelihood function as above. + this leads us to the final estimation, the contribution this paper makes to the literature. the aim is to estimate the matrix @xmath12 with the available departure and arrival data. the first step is to estimate the general shape of the @xmath10 matrix. the problem is to achieve this in a simple way given that @xmath12 is to be estimated with @xmath44 parameters, and only @xmath45 equations. the following paragraph presents an elegant solution to this problem. + recall @xmath32 as the expectation of @xmath46. it is assumed symmetric, we can diagonalize it, so that, @xmath47 where @xmath48 is a projection matrix of eigenvectors of @xmath32 and @xmath49 is a @xmath44 diagonal matrix, with terms equal to the respective eigenvalues. therefore, if the structure of @xmath32 is known (i.e. the eigenvectors are known) and constant, then we have reduced the problem to solving a system of @xmath45 unknown parameters with @xmath45 equations. [eq : odeq] and the previous estimations, we have the following system, @xmath50 where @xmath51 and @xmath52 are obtained by simple subtraction. the probability density function of the observations @xmath53 can then be written, @xmath54 \quad p \big (y_{di}^t \vert r_z, p_{rz } \big) & \sim & \mathcal{nb } (\sum_j r^{ij}_z, p_{rz})\end{aligned}\]] where @xmath55 and @xmath56. according to this equation, we then have @xmath57 likelihood equations (@xmath58 $]) but @xmath59 parameters to estimate (@xmath32 being symmetric). to perform the estimation, we then need to reduce the number of parameters. according to eq. [eq : diag], we have, @xmath60 then, if we knew @xmath48, the maximum likelihood would be tractable and provide an estimation of @xmath32 (with @xmath61 being the @xmath3th eigenvalue). the @xmath57 likelihoods would look like, @xmath62 \quad l \big (y_{di}^t \vert r_z, p_{rz } \big) & = & \prod_t p \big (y_{di}^t \vert r_z, p_{rz } \big)\end{aligned}\]] where @xmath63. the maximum likelihood estimating equations are then, @xmath64 moreover, this complex equation can be simplified by assuming that the observations are independent conditionally on knowing the parameters (e.g. the errors are independents), and then, @xmath65 the dimension of the parameter space is reduced according to the knowledge of @xmath48. to estimate @xmath48, remember that @xmath31 is symmetric, we have, @xmath66 then, we make the assumption that all the regular passengers behave identically over time, that is, @xmath48 is not a function of time. then we have, @xmath67 then @xmath48 can be estimated from @xmath68, and introduced into equation [eq : density] to obtain eq. [eq : mle]. finally, we can use @xmath69 and @xmath70 to estimate @xmath71. finding the solution of equation [eq : mle] is a classic optimization problem. the simple likelihood shape insures the existence of a solution, and it can be found by any standard optimization function. let this solution be @xmath72. the problem with this estimator is that we can not guarantee that it will fulfil the underlying constraint of the density parameters. indeed, some values in the matrix will be negative, and there will be some element in the diagonal that wo nt be null. therefore, some constraints have to be added to the ml estimating equations. these are : + * constraint 1 * _ all the elements in the matrix @xmath73 are greater than or equal to zero, or equivalently, @xmath74 _ + * constraint 2 * _ all the diagonal elements in the matrix @xmath71 must be zero, or equivalently, @xmath75 _ + * constraint 3 * _ the last set of parameters to be estimated is the probability matrix @xmath76. therefore, all the elements should belong to the interval @xmath77 $], @xmath78 \end{aligned}\]] _ most of the optimization algorithms that deal with the constraint require an initialization which belong to the constrained space. one could be tempted to address as a starting point the mean value of the observations, according to the one - dimensional (@xmath79) result. however, it is very unlikely that this initial point will satisfy the constraints. therefore, the best choice so far seems to be the diagonal elements of the matrix @xmath80, given that they naturally fill * constraint 1 * and * constraint 2*. + + the complete optimization program therefore becomes, @xmath81 with the initial value @xmath82. this optimization program can be replaced by an explicit expression of the estimator, subject to some constraints stated in [ann : a]. the main constraint is the poisson distribution assumption, so that we have, + * proposition 1 : * _ assume that @xmath83, then @xmath84^{-1 } \bar{y}\end{aligned}\]] where @xmath85 is the matrix of estimated eigenvalues of @xmath32. _ + if now we consider a gaussian likelihood instead of poisson, the following maximum likelihood estimator is found, + * proposition 2 : * _ assume that @xmath86, then @xmath87 where @xmath88 is the matrix of estimated eigenvalues of @xmath32. _ + the proofs of propositions 1 and 2 are presented in [ann : a]. we can also derive the follwing theorem, that ensures us of the quality of the estimation, + * theorem 1 * + assume that @xmath89{\ a.s.\ } p$] (see anderson @xcite,@xcite). then we have, @xmath90{\ \mathcal{p } \ } \lambda\end{aligned}\]] the proof is presented in [ann : b]. if we want to deal with a more realistic modelling, it seems obvious that we have to consider spatial, temporal and multivariate influences on the number of passengers, and then on the parameters of our modelling. + let @xmath91 be a set of exogenous features, and @xmath92 their corresponding (unknown) influence on the number of passengers. therefore, the regression model can be written, + @xmath54 \quad p \big (y_{di}^t \vert r_z, p_{rz }, (x_i)_i \big) & \sim & \mathcal{nb } \big (\sum_{k } \lambda_k p_{ik } \sum_j p_{jk } + \sum_l x_l \beta_l, p_{rz}\big)\end{aligned}\]] where the parameters to be estimated are @xmath93. the likelihood is expressed as in eq. [eq : like], and we have, @xmath94 then, if we consider the projection matrix to be constant over time, then we have the following proposition (proved in [ann : c]), + * proposition 3 : * _ if we apply the same methodology as before we obtain the following estimator, @xmath95^{-1 } y { } ^t x (x { } ^t x)^{-1}\end{aligned}\]] where @xmath96 is the matrix of the exogenous variables. _ + a more complex solution would be to perform the same technique as before, except that @xmath48 will no longer be constant. therefore, the varying @xmath97 estimated from @xmath31 will help in the recovery of the matrix @xmath32. the reason why we can not perform a direct estimation of the @xmath98 s through this technique is because @xmath32 can be diagonalize, but we do nt know how the @xmath98 influence the @xmath99. then a space - time model for the regular zone passenger is required. a further sensitivity analysis has to be run in order to figure out if the results are significantly affected by the increased uncertainty. Ad hoc approach the quality of the estimator will strongly depend on the survey sub - matrix @xmath31. in section [sec : est], we assume that the similarity between the matrix @xmath31 and @xmath32 relies on the projection matrix @xmath48 and @xmath100. moreover, we assume @xmath101 (eq. [eq : ass1]), and this assumption is the key to derive the propositions @xmath102. + it is however difficult to design and implement a survey that provides accurate information. therefore, eq. [eq : ass1] is no longer valid. to overcome this, we propose to consider an ad hoc estimation of the o - d matrix. + despite the survey having some unknown biases, it provides useful information that we need. let @xmath88 be the estimation based on the eigenvalues of @xmath32, relying on the assumption @xmath101. while inaccurate, this estimation looks like the perfect prior information. we define, @xmath103 now, given that the survey may be biased, we need to emphasize the influence of the observations @xmath104, by building an observation - based matrix @xmath105, such that, @xmath106 where @xmath107 and @xmath108. @xmath109 is symmetric, and roughly correspond to an equal partition of the passengers in the different stations. let @xmath110. now, we need to integrate the information provided by @xmath109. then we define the final ad hoc estimator @xmath111, @xmath112 we define the robustness of our estimation as its ability to overcome biased survey results. to analyse the performance of the estimators, we considered that the parameter matrix of the survey was biased according to the following two equations, @xmath113 where @xmath114 $] stands for the scale of the survey, @xmath115 $] stands for the varying noise level, and @xmath116 means the poisson distribution, and, @xmath117 the difference between the two equations relies in the bias structure. in eq. [eq : noise], integer values are randomly added regardless of the real value of the parameter. it is the kind of errors we expect to find in badly designed surveys or respondents responding randomly. in eq. [eq : prop.noise], we consider that the bias keeps the same structure as the @xmath32 matrix. this is more an equivalent of a measurement error usually described in the literature with centred gaussian distributions. therefore, we expect the estimators to provide better estimates when the survey s parameters are driven by eq. [eq : prop.noise]. + figure [fig : sensnoise] shows the ml and ad hoc estimation results for the two kind of bias, with different number of observations and different strength of bias. as expected, the second type of error (eq. [eq : prop.noise]) is more easily overcome by the estimators. the important conclusion we can make from this result is that the ad hoc estimation performs better than the naive ml estimation, no matter what kind of noise we add. the ad - hoc estimation is plotted in green. the upper line stands for a bias designed according to eq. [eq : noise], and the bottom line for eq. [eq : prop.noise]. the mse is calculated among @xmath118 simulations., title="fig:",width=264,height=226] ci. the ad - hoc estimation is plotted in green. the upper line stands for a bias designed according to eq. [eq : noise], and the bottom line for eq. [eq : prop.noise]. the mse is calculated among @xmath118 simulations., title="fig:",width=264,height=226] + ci. the ad - hoc estimation is plotted in green. the upper line stands for a bias designed according to eq. [eq : noise], and the bottom line for eq. [eq : prop.noise]. the mse is calculated among @xmath118 simulations., title="fig:",width=264,height=226] ci. the ad - hoc estimation is plotted in green. the upper line stands for a bias designed according to eq. [eq : noise], and the bottom line for eq. [eq : prop.noise]. the mse is calculated among @xmath118 simulations., title="fig:",width=264,height=226] finally, and also very important, the ad hoc estimator seems more consistent and its performances are less affected by the number of available observations. for instance, we can observe that the log(mse) profile is identical in fig. [fig : sensnoise] for @xmath119 observations and for @xmath120. the importance of that being to allow some reliable time - dependency analysis (on a monthly basis, it would mean @xmath121 observations), which would be more complicated with the ml estimation. Application let m be a @xmath122 matrix, representing departure and arrival stations (origin and destination). we assume that any value of m is a random number, generated according to the matrix of parameters @xmath123, and the probability @xmath124. the matrix @xmath123 has the following values, @xmath125\]] let @xmath126 be an observed sub - sample of m, i.e. only a proportion @xmath127 of m is represented in @xmath126. we assume that for every value of m, @xmath127 will be the same, only perturbed by some low - level additive noise. a realization of @xmath126 is the following matrix, @xmath128\]] the value of @xmath127 in this example is roughly equal to @xmath129. the estimation of @xmath130 is performed with the observations of this matrix. then, @xmath130 being symmetric, we diagonalize it, and using the optimization program described in section [sec : oi] we are able to provide the following estimated @xmath131 matrix (for @xmath132, @xmath133, @xmath134 and @xmath135 observations respectively), @xmath136 & \left [\begin{array}{ccccc } 0 & 68 & 24 & 42 & 54 \\ 68 & 0 & 18 & 36 & 124 \\ 24 & 18 & 0 & 126 & 12 \\ 42 & 36 & 126 & 0 & 18 \\ 54 & 124 & 12 & 18 & 0 \end{array } \right] \\ & \\ \left [\begin{array}{ccccc } 0 & 78 & 12 & 60 & 42 \\ 78 & 0 & 24 & 48 & 78 \\ 12 & 24 & 0 & 102 & 12 \\ 60 & 48 & 102 & 0 & 24 \\ 42 & 78 & 12 & 24 & 0 \end{array } \right] & \left [\begin{array}{ccccc } 0 & 72 & 12 & 54 & 48 \\ 72 & 0 & 24 & 42 & 78 \\ 12 & 24 & 0 & 96 & 12 \\ 54 & 42 & 96 & 0 & 24 \\ 48 & 78 & 12 & 24 & 0 \end{array } \right] \end{array}\]] table [tble : mse_bin] provides the mean square error of the estimator for different number of observations and different value of dispersion in the case of a negative binomial modelling. we also display in table [tble : norm_test] the p - values for the cramer - von mises normality test of @xmath137 estimation procedures for each eigenvalue..[tble : mse_bin] negative binomial modelling : values of the mean square error of the parameter estimation with different number of observations and different distribution assumption over @xmath137 replication of the simulation, and the according variance (under brackets). [cols="^,^,^,^,^",options="header ",] we consider as an example the passengers commuting from the sydney region ferry, according to five surveys organized in @xmath138, @xmath139, and @xmath140 by the bureau of transport statistics of new south wales. the overall survey could have been analysed, but in order to be understandable, we will focus our analysis on the eastern suburbs route, composed with @xmath141 different wharfs and @xmath142 different days. [fig : lpw20xx] represents the links between the wharfs, meaning that you can have a direct access by taking only one ferry. + as we can see from the plots, all the wharfs seem to be able to be reached starting from any other wharf, except for darling point, watsons bay and garden island. + starting from this, the objective is the estimation of the od matrix, which is almost like the ferry links matrix presented in the figures except that instead of logical values, it will be filled by counts of passengers. however, this survey does not provide enough data to perform this estimation. in particular, it does nt give any observation on the ticket sales at the wharfs. then, to provide an estimation, we use a different survey from the same institute, which aimed at understanding the preferred mean of transport for people going to work. from this database, we only kept those who declared taking the ferry. + before we could provide an analysis of these data, we had to consider that the origins and destination of the people in the second survey did nt correspond to the wharfs of the first survey. therefore, we decided to attribute to every passenger the starting point of their journey as the closest wharf to their home, and a destination point the closest wharf to their office. the distance have been calculated according to each wharf and location longitude and latitude values. this being done, we can provide a new origin - destination matrix. it is plotted in fig. [fig : jw2006]. + this od matrix is then supposed to have the same structure as the regular od of the barrier counts. therefore, we will apply the methodology presented in section [sec : cpe] to reconstruct the 2010, 2011 and 2012 od matrix according to the barrier counts. the eigenvalues and eigenvectors are then calculated, and the reconstruction of the ferry passengers are presented in figs. [fig : est1] and [fig : est2]. Conclusion and further work we presented in this paper a new estimation technique for the od matrix. we use the information available from surveys to infer the correct projection matrix and reduce significantly the size of the parameter space. by using a maximum likelihood approach, we compute the estimating equations and provide an ad hoc estimator of the od matrix. we demonstrated its robustness in section [sec : rob]. + we also demonstrated that a regression analysis could be performed on this kind of data, and showed that this estimation procedure is also consistent. to the best of our knowledge, this is the first time that a such multivariate approach is used to estimate the od matrix. this approach will improve the prediction ability of passengers journeys. + we finally applied our techniques to simulated data and real - case scenario in sydney ferry transport using the data from the bureau of transport statistics. + the estimation of the od matrix is a first step for the analysis of the passengers flow over the transport network. then, beyond this estimation point, we may cite : 1. [monit] monitoring the passengers count ; 2. [forecast] forecasting of the passengers count (1 week in advance for example) ; 3. [predict] predicting the passengers flow in case of spatio - temporal topological change in the network in order to address the monitoring task ([monit]), several strong assumptions have to be made, which will require ground verifications before being tested. among them, we can cite the time between ticket validation and getting into the train or the regularity at which people take their train if they are regular passengers. moreover, a real - time access to the data is necessary. while difficult, this seems achievable. + the forecasting of passenger counts ([forecast]) can be done without additional information (if sufficient temporal information has been provided in a first place), even if more observation would probably mean smaller variances. these forecasts could be helpful for efficient scheduling the trains (for example), but further study have to be done in order to understand the influence of complex variables such as the temperature or the humidity. + real - time prediction of passenger flows ([predict]) is more difficult, but is theoretically achievable. what we denote a spatio - temporal topological change is a change in the timetable, or in the public transport route. _ to make it clear, we will explore a simple density example, where the poisson distribution will be used instead of the negative binomial. therefore, the pdf can be expressed as, @xmath143 where @xmath144 is the parameter we are interested in. then, if we make the assumption that the number of passengers in every station are independent, each @xmath145 is distributed according to a poisson distribution, with the parameter @xmath146, which can be re - written according to eq. [eq : diag2], @xmath147 where the @xmath148 are the eigenvalues, and the @xmath149 the element of the matrix p. if we denote @xmath150, the transformed density can then be written, @xmath151 + therefore, the log - likelihood can be expressed as follow, @xmath152 the maximum likelihood estimation is then equivalent to solve the following system, @xmath153 still under the constraint * c1 * and * c2*. * c3 * is excluded because this set of parameter does nt exist in the poisson modelling. if @xmath154, @xmath155 and * c2 * does nt stand. then estimated value corresponds to the classical one dimensional poisson unbiased mean estimator @xmath156. + the system of equations [eq : sys_pois] seems at first a quite complicated one. nevertheless, it can be simplify so as to become, @xmath157 where @xmath158 contains the unknown parameters. then, if we denote @xmath159, then, @xmath160 where @xmath161 and @xmath162. then, we can keep simplifying the expression, @xmath163 finally, the same reasoning leads to the following estimator, @xmath164 where @xmath165. this estimator will probably not be the best estimator given that it relies on the inversion of @xmath166, but has the advantage to be asymptotically unbiased, with variance decreasing to zero. _ _ let @xmath167 be a probability density function. if @xmath168 denotes the pdf of @xmath169, and @xmath170 the pdf of @xmath100, we can write, @xmath171 let consider the estimator presented in eq. [eq : linsol_lamfg], and make the assumption that we are in a large value case, meaning @xmath172. then, @xmath173 where, @xmath174 \tilde{s}_d^{-1 } \end{array } \right.\]] and @xmath100 is an estimation of @xmath48 according to the first observations. + + _ to prove the convergence in probability, we need to demonstrate that, @xmath175 where @xmath176. starting with the left hand side, we have, @xmath177 and we know that @xmath178{\ a.s.\ } p s_d \lambda $]. then, @xmath179 and we have, @xmath180 where @xmath181 stands for the probability density function of @xmath182. + the first integral decreases towards @xmath183 as @xmath57 grows to infinity according to eq. [eq : as]. the argument for the second integral is the following. according to the assumption of strong convergence of @xmath100, @xmath181 converge towards the dirac function @xmath184 as @xmath185 goes to infinity. @xmath186 being strictly positive, this ends the proof. @xmath187. [[calculation - in - case - of - poisson - regression - and - log - link - function]] * calculation in case of poisson regression (and log link function) * + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + _ the beginning of the reasoning is similar to the previous one. then, if we assume that exogenous variables have impacts on the number of passengers, we can write, @xmath188 where @xmath98 are symmetric matrices reflecting the intercept (@xmath189) for baseline commuter flows and the variable influences (@xmath190) for changes in commuter flows from known daily influences. moreover, we assume that the same diagonalization (meaning with the same eigenvectors) can be applied, which lead us to, @xmath191 therefore, @xmath192 will be distributed according to a poisson distribution with the following parameter, @xmath193 where the parameters to be estimated are @xmath194, which means we have to estimate @xmath195 parameters. + the probability of one observation can then be written, @xmath196 which gives the following log - likelihood, @xmath197 therefore, to obtain the final system of equation, we need to calculate the derivatives of the log - likelihood with respect to each parameter @xmath198. _

the estimation of the number of passengers with the identical journey is a common problem for public transport authorities. this problem is also known as the origin - destination estimation (od) problem and it has been widely studied for the past thirty years. however, the theory is missing when the observations are not limited to the passenger counts but also includes station surveys. + our aim is to provide a solid framework for the estimation of an od matrix when only a portion of the journey counts are observable. + our method consists of a statistical estimation technique for od matrix when we have the sum - of - row counts and survey - based observations. our technique differs from the previous studies in that it does not need a prior od matrix which can be hard to obtain. instead, we model the passengers behaviour through the survey data, and use the diagonalization of the partial od matrix to reduce the space parameter and derive a consistent global od matrix estimator. we demonstrate the robustness of our estimator and apply it to several examples showcasing the proposed models and approach. we highlight how other sources of data can be incorporated in the model such as explanatory variables, e.g. rainfall, indicator variables for major events, etc, and inference made in a principled, non - heuristic way. constraint maximum likelihood estimation, eigenvectors, counts estimation

1305.7010

Introduction in solid - core photonic crystal fibers (pcf) the air - silica microstructured cladding (see fig. [fig1]) gives rise to a variety of novel phenomena @xcite including large - mode area (lma) endlessly - single mode operation @xcite. though pcfs typically have optical properties very different from that of standard fibers they of course share some of the overall properties such as the susceptibility of the attenuation to macro - bending. macrobending - induced attenuation in pcfs has been addressed both experimentally as well as theoretically / numerically in a number of papers @xcite. however, predicting bending - loss is no simple task and typically involves a full numerical solution of maxwell s equations as well as use of a phenomenological free parameter, _ e.g. _ an effective core radius. in this paper we revisit the problem and show how macro - bending loss measurements on high - quality pcfs can be predicted with high accuracy using easy - to - evaluate empirical relations. Predicting macro-bending loss predictions of macro - bending induced attenuation in photonic crystal fibers have been made using various approaches including antenna - theory for bent standard fibers @xcite, coupling - length criteria @xcite, and phenomenological models within the tilted - index representation @xcite. here, we also apply the antenna - theory of sakai and kimura @xcite, but contrary to refs. @xcite we make a full transformation of standard - fiber parameters such as @xmath1, @xmath2, and @xmath0 @xcite to fiber parameters appropriate to high - index contrast pcfs with a triangular arrangement of air holes. in the large - mode area limit we get (see appendix) @xmath3 for the power - decay, @xmath4, along the fiber. for a conversion to a db - scale @xmath5 should be multiplied by @xmath6. in eq. ([alpha_lma]), @xmath7 is the bending radius, @xmath8 is the effective area @xcite, @xmath9 is the index of silica, and @xmath10 is the recently introduced effective v - parameter of a pcf @xcite. the strength of our formulation is that it contains no free parameters (such as an arbitrary core radius) and furthermore empirical expressions, depending only on @xmath11 and @xmath12, have been given recently for both @xmath8 and @xmath13 @xcite. from the function @xmath14 we may derive the parametric dependence of the critical bending radius @xmath15. the function increases dramatically when the argument is less than unity and thus we may define a critical bending radius from @xmath16 where @xmath17. typically the pcf is operated close to cut - off where @xmath18 @xcite so that the argument may be written as @xmath19 this dependence was first reported and experimentally confirmed by birks _ et al. _ @xcite and recently a pre - factor of order unity was also found experimentally in ref. Experimental results we have fabricated three lma fibers by the stack - and - pull method and characterized them using the conventional cut - back technique. all three fibers have a triangular air - hole array and a solid core formed by a single missing air - hole in the center of the structure, see fig. [fig1]. for the lma-20 macro - bending loss has been measured for bending radii of r=8 cm and r=16 cm and the results are shown in fig. the predictions of eq. ([alpha_lma]) are also included. it is emphasized that the predictions are based on the empirical relations for @xmath8 and @xmath13 provided in refs. @xcite and @xcite respectively and therefore do not require any numerical calculations. similar results are shown in figs. [fig3] and [fig4] for the lma-25 and lma-35 fibers, respectively. Discussion and conclusion the pcf, in theory, exhibits both a short and long - wavelength bend - edge. however, the results presented here only indicate a short - wavelength bend - edge. the reason for this is that the long - wavelength bend - edge occurs for @xmath20 @xcite. for typical lma - pcfs it is therefor located in the non - transparent wavelength regime of silica. in conclusion we have demonstrated that macro - bending loss measurements on high - quality pcfs can be predicted with good accuracy using easy - to - evaluate empirical relations with only @xmath21 and @xmath22 as input parameters. since macro - bending attenuation for many purposes and applications is the limiting factor we believe that the present results will be useful in practical designs of optical systems employing photonic crystal fibers. Appendix the starting point is the bending - loss formula for a gaussian mode in a standard - fiber @xcite @xmath23 where @xmath8 is the effective area, @xmath24 is the core radius, @xmath7 is the bending radius, and the standard - fiber parameters are given by @xcite @xmath25 substituting these parameters into eq. ([alpha1]) we get @xmath26 in the relevant limit where @xmath27. here, @xmath28 and @xmath29 in eqs. ([alpha_lma]) and ([v_pcf]) have been introduced. for large - mode area fibers we make a further simplification for the isolated propagation constant ; using that @xmath30 we arrive at eq. ([alpha_lma]). Acknowledgments m. d. nielsen acknowledges financial support by the danish academy of technical sciences.

we report on an easy - to - evaluate expression for the prediction of the bend - loss for a large mode area photonic crystal fiber (pcf) with a triangular air - hole lattice. the expression is based on a recently proposed formulation of the v - parameter for a pcf and contains no free parameters. the validity of the expression is verified experimentally for varying fiber parameters as well as bend radius. the typical deviation between the position of the measured and the predicted bend loss edge is within measurement uncertainty. 10 url # 1`#1`urlprefix[2][]#2 j. c. knight, `` photonic crystal fibres, '' nature * 424 *, 847851 (2003). t. a. birks, j. c. knight, and p. s. j. russell, `` endlessly single mode photonic crystal fibre, '' opt. lett. * 22 *, 961963 (1997). t. srensen, j. broeng, a. bjarklev, e. knudsen, and s. e. b. libori, `` macro - bending loss properties of photonic crystal fibre, '' electron. lett. * 37 *, 287289 (2001). t. srensen, j. broeng, a. bjarklev, t. p. hansen, e. knudsen, s. e. b. libori, h. r. simonsen, and j. r. jensen, `` spectral macro - bending loss considerations for photonic crystal fibres, '' iee proc.- opt. * 149 *, 206 (2002). 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 *, 163167 (2003). j. c. baggett, t. m. monro, k. furusawa, v. finazzi, and d. j. richardson, `` understanding bending losses in holey optical fibers, '' opt. commun. * 227 *, 317335 (2003). j. sakai and t. kimura, `` bending loss of propagation modes in arbitrary - index profile optical fibers, '' appl. opt. * 17 *, 14991506 (1978). j. sakai, `` simplified bending loss formula for single - mode optical fibers, '' appl. opt. * 18 *, 951952 (1979). a. w. snyder and j. d. love, _ optical waveguide theory _ (chapman & hall, new york, 1983). n. a. mortensen, `` effective area of photonic crystal fibers, '' opt. express * 10 *, 341348 (2002). http://www.opticsexpress.org/abstract.cfm?uri=opex-10-7-341. 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 *, 18791881 (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 *, 23092311 (2003). m. d. nielsen and n. a. mortensen, `` photonic crystal fiber design based on the @xmath0parameter, '' opt. express * 11 *, 27622768 (2003). http://www.opticsexpress.org / abstract.cfm?uri = opex-11 - 21 - 2762% [http://www.opticsexpress.org / abstract.cfm?uri = opex-11 - 21 - 2762%].

physics0404057

Introduction to enhance chemical reactivity of cu surfaces with nitrogen oxides (no@xmath0) is an important issue for development of new catalytic materials effective in the no@xmath0 reduction process @xcite. the dissociative adsorption of no@xmath0, for example, was found to be less expected on cu, compared with highly reactive rh, ir, ru, co, and ni surfaces, although dissociative adsorption was reported at finite temperatures in experiments @xcite. in order to provide active surfaces for no dissociation, cu thin films and low index surfaces were considered on one hand @xcite. in several electronic structure calculations based on the density functional theory (dft), on the other hand, dissociative adsorption of no was found to be possible but energetically un - favored compared with molecular adsorption @xcite. we note that the simulations were often performed with respect to reactions on stable bulk surfaces. although the theoretical data suggested less reactivity of cu bulk surfaces for no@xmath0 reduction, there could be remarkable reactivity on some surface - like atomic structures of cu. when we considered wider classes of nano - scale structures other than defined surfaces of bulk cu crystals, one could find another clue. in this line of approach, indeed, many theoretical investigations with computer simulations had been done intending to explore efficiency of _ e.g. _ step - like structures of various metals @xcite. to explore possible no dissociation, we consider ultra thin cu structures. in this study, we focus on a cu atomic layer, that is the triangular lattice of cu. we adopted structural optimization simulations based on electronic structure calculations to find a stable cu triangular lattice (cu - tl). on this thin structure, we adsorbed an no molecule and performed an optimization simulation. after finding molecular adsorbed structures, we searched possible dissociative adsorption on the cu structures. to find a possible reaction path and to conclude a reduction process, we performed simulations for reaction path estimation. in the discussion of this paper, by comparing the obtained adsorption energies with each other, we will discuss a possible no@xmath0 reduction mechanism by using cu nano - structures. Methods we adopted the electronic structure calculation based on the density functional theory@xcite to estimate the electronic state, and to obtain inter - atomic forces. in this simulation, the kohn - sham wavefunctions were expanded in the plane - waves and the electron charge density was given both on a real space mesh and on the fourier mesh. an approximation for the exchange - correlation energy functional by perdew, burke, and ernzerhof@xcite in the scheme of the generalized gradient approximation was adopted. the ultra - soft pseudo - potential@xcite was utilized to describe the valence electron state. all of our simulations were done using the simulation package, the quantum espresso @xcite. the calculation conditions are summarized as follows. the energy cut - off for the wave function expansion was 30 [ry], while the cut - off for the charge density was 240 [ry]. the brillouin zone integration was done using a @xmath1 mesh of 8@xmath28@xmath21 for the largest super cell adopted. these values were refined, if the computation facility allowed much accurate calculations. the convergence criterion for the force constant was that the simulation ended, when the absolute value of the total force vector became less than 1@xmath210@xmath3 [ry / a.u.]. Atomic layer of cu to explore possible high reactivity of cu nano - structures, we considered atomic - layer structures. an important structure for our discussion is the cu triangular lattice (cu - tl). in this section, we show data for structural and electronic properties of cu - tl. we obtained an optimized lattice structure using a cu atomic layer in a primitive super cell. major calculation conditions were the same as those given in section [methods]. the @xmath1-point mesh was 24@xmath224@xmath21 in this simulation. the cell was given in a hexagonal structure. the vacuum layer had thickness of 15 . in this simulation, the value of the lattice constant was optimized. the bond length was found to be 2.43 . [fig : cu - tl - ene - a]) this value is rather small compared to the bond length 2.55 of the bulk fcc cu. the reason for shrink in the bond length is mainly to reduce the total band energy. the total energy of tl was energetically higher than the bulk cu by 1.2 ev per a cu atom. of the triangular lattice of cu. the value of @xmath4 in rydberg is given as a function of the lattice constant @xmath5[]., height=302] starting from some initial conditions, we found appearance of cu - tl in optimized structures. as another evidence to show the local stability of cu - tl, we considered an atomic two - layer structure (atls). this structure was obtained by cutting the bulk fcc cu crystal and was placed in a simulation super - cell. the layer structure was perpendicular to the (100) direction of bulk cu and thus was contained in an orthorhombic unit cell. energy difference between atls and cu - tl was 3.11 ev per a cu atom. an optimization calculation of the structure concluded local stability. but, atls was not kept against global reconstruction which was happened when an no molecule was adsorbed on it. furthermore, we found a strongly reacted structure starting from an no molecule adsorbed on atls. it means that using atls as an initial structure, naively speaking, we realized simulated annealing in our simulation. compared with this un - stable structure, cu - tl was found to be stable. once the molecule was adsorbed on atls, reconstruction of atls happened and formation of cu - tl was observed in our simulations. conversely, we can say that cu - tl is stable against distortion making corrugation toward atls. even when one observed local stability of an atomic structure in simulation, however, a final evidence of the structure would be requested to be given using real experiments. realization of an atomically thin layer, _ i.e. _ cu - tl, will need development of a fabrication method. recently, formation of an atomic layer of pb on the si(111) surface was reported @xcite. in this superconducting pb system, positions of pb atoms are affected by the atomic structure of the substrate and inter - atomic distance between pb atoms is not determined independently from the substrate. the most remarkable example of natural realization of the atomic layer is graphene @xcite. this unique flexible structure of carbon is possible to be supported in air according to the strong c - c sp@xmath6 bonding. peeling a graphene sheet and pasting it on a silicon - di - oxide surface, graphene is obtained efficiently from graphite. in case of cu, we may expect formation of an atomic layer on a suitable inert substrate. we might be able to keep the atomic layer as a film pasted on a support with a nano - meter - scale hole. then, mechanical properties of the atomic layer would be paid attention like a graphene sheet @xcite. but for our consideration of no adsorption, a local structure of cu is important. so, we assume an atomic - scale local structure in a part of nano - scale cu. the density of states (dos) of cu - tl is shown in fig. [fig : cu111 - 1-tetra]. the major peaks are characterized similarly to the bulk copper. looking dos from the low energy region, we see that the 4s band starts from -6.38 ev and spreads over above the fermi energy. sharp peaks of 3d levels are seen from -4.04 ev but the 3d bands end below the fermi level. thus, parfectly filled 3d band with the @xmath7 configuration is kept and the structure behaves as an @xmath8 metal. these characteristic features are seen in the electronic band structure of cu - tl, too. [fig : cu-2d - band]) along the @xmath9-m line, or the k-@xmath9 line, we see hybridization of the 4s band and a 3d band. comparison with the cu (111) surface allows us to evaluate similarity and difference between cu - tl and the bulk cu surface. the density of states in the 4s band leveled almost around 0.2 owing to two - dimensional nature. dos in the 3d bands peaked well above 10 for cu - tl in the unit of states / ev per a unit cell, while the value is from 4 to 6, except for a singularity at the top of the 3d bands, for the cu (111) surface. the shorter bond length of cu atoms, the height in dos should be lower in a fixed lattice structure. thus, the higher dos peak for cu - tl than for the bulk cu suggests that two - dimensional nature of cu - tl affects dos. since dos at the fermi level is almost the same for both cu - tl and the cu (111) surface, the chemical reactivity of no is expected to be similar, if the structure is kept undeformed. however, we should note that the top of cu 3d bands is much closer to the fermi energy for cu - tl than the cu (111) surface. this tendency suggests higher reactivity of cu - tl against no. when the stable adsorption site is the on - top site, similarity in characteristic energy like the adsorption energy would be expected. however, if a bridge site or a hollow site became stable for no on cu - tl, we could have difference even in the chemical reactivity from that on the cu (111) surface. this is because much easy deformation of the cu network structure is expected for cu - tl and the bond formation between no and cu - tl will create distortion. in the next section, we discuss occurrence of strong reactions between no and cu - tl. Adsorption of no we consider adsorption of no on cu - tl and an atomic step like structure (ass) created on an cu atomic layer. the second structure was found in optimization simulations of no adsorption on atls. observing results of no - adsorbed structures starting from atls, we identified a stable substrate structure in a super cell as cu - tl with ass in our simulations. therefore, we regarded cu - tl and ass as typical atomic - scale layer structures of cu. several characteristic adsorption sites for no were found on these structures. molecular adsorbed structures were obtained by structural optimization. starting from an initial structure with a no molecule a little separated from a substrate, cu - tl, ass or atls, each adsorbed structure was determined. by a series of simulations, we found the next general rules for molecular adsorption. on cu - tl, adsorption on a hollow site is energetically most favorable. on ass, a bridge site on the cu array is energetically most favorable among sites including an on - top site, a hollow site in the back surface, and a bridge site in the back surface. thus, we treat these locally stable structures only in the following discussion. while, structures corresponding to dissociative adsorption were given by locating n and o atoms a little separated on the substrate and by optimizing the whole structure. we have found two locally stable dissociative adsorbed structures on cu - tl and on ass. the structures are depicted in fig. [fig : ss - sideview]. as typical structures, we consider these structures only. we define the adsorption energy by the next formula. @xmath10 here, @xmath11 is the molecular adsorption energy, while @xmath12 is the dissociative adsorption energy. the values of @xmath13 and @xmath14 are the total energy of a cu slab with no and that of another slab with a n atom and an o atom adsorbed on the cu slab, @xmath15 is the total energy of a cu slab without no, and @xmath16 is the total energy of the no molecule contained in a super cell with the same size as the other calculations. molecular dissociation energy is defined as, @xmath17 adsorbed structures found in our simulations are itemized in the next list. the adsorption energy is also shown in each parenthesis for convenience. molecular adsorption on cu - tl : : in adsorption of a no molecule on a surface of cu - tl, a hollow site (-0.83 ev) is selected. see the center figure of fig. [fig : ss - sideview] (a). molecular adsorption on ass : : in adsorption of a no molecule on an atomic step - like structure, a bridge site (-1.32 ev) is selected. see the center figure of fig. [fig : ss - sideview] (b). dissociative adsorption : : dissociative adsorption of no is found on tl (-1.92 ev) and on ass (-1.69 ev). see the right figures of fig. [fig : ss - sideview] (a) and (b). now, dissociative adsorption structures are discussed. we have two typical dissociative adsorption structures on cu - tl and on ass. in the first structure, the nitrogen atom locates at a center of five surrounding cu atoms. (see the right figures of fig. [fig : ss - sideview] (a).) this structure may be regarded as a nitrogen interstitial impurity in a cu lattice. arrangement of cu is largely distorted from pure tl, so that the n atom is embedded in cu layer. the oxygen atom locates at a hollow site and it is embedded in cu layer. the local structure of these impurity sites are ncu@xmath18 and ocu@xmath19. here, the ocu@xmath19 structure is planer. a reason for appearance of the high coordination numbers for n and o is that cu valence is not largely modified and that 3d@xmath20 configuration is almost kept. 4s electrons are in extended states so that the local n@xmath3 and o@xmath21 are efficiently screened by neighboring five and four copper atoms. on ss, we have another dissociative adsorbed structure for no. the nitrogen atom is again embedded in the cu structure. (see the right figure of fig. [fig : ss - sideview] (b).) in this structure, the oxygen atom is at a bridge site and keeps two - fold coordination, while the nitrogen atom has four - fold coordination. now we compare the obtained values of adsorption energy with those in the literature. the molecular adsorption energy is higher for ass than those on cu - tl. except for a case of a bond - center site on the back surface of ass, molecular adsorption favors the bridge site of the step - like structure. this general tendency is natural in comparison to the other examples known in the literature. the most notable feature of our results is the finding of the large dissociative adsorption energy. we conclude that the dissociative adsorption may happen, when the nitrogen atom can go to an interstitial site of a cu structure. the large value of @xmath12 (-1.92 ev on tl and -1.69 ev on ass) is actually possible, since these structures possess high coordination of cu around the nitrogen atom. on the clean cu (111) surface, no favors the molecular adsorption (with the adsorption energy of -1.22 ev estimated in ref. @xcite) against the dissociation of no, where the dissociative adsorption energy is estimated to be -0.79 ev in ref. @xcite. the qualitative difference between data for known bulk surfaces and our result should be attributed on movement of cu atoms in the reconstruction process. in our simulation, positions of cu atoms are rather easily modified because the atomic structure of cu is just a single layer. even in the optimization simulation, we can reach the nitrogen insertion in the cu structures. from the present result, we conjecture the following picture. if a cu structure allows large configurational distortion owing to the chemical reaction with no, the nitrogen atom can move into the cu structure and form the local ncu@xmath22 configuration. (@xmath23 or 5.) owing to the energy reduction coming from the large formation energy of the local ncu@xmath22 structure, we can expect even the dissociative adsorption of no on cu. in the real cu nano - structures, there can happen large distortion of cu configuration owing to finite temperatures and possible local strain. therefore, our simulation, which is prepared using the atomic layer of cu, might have derived a hidden possible path of the no dissociative adsorption on cu structures. to estimate a reaction path on the no reduction on cu - tl, we estimated the local structure and the energy of a transition state using the nudged - elastic - band method. the initial configuration was the molecular adsorbed structure on cu - tl and the final configuration was the dissociative adsorbed structure obtained in [ads - structure]. the dissociation reaction was determined by obtaining the transition state with an energy @xmath24 for the no reduction process. the activation energy for dissociation of no on the atomic layer is estimated using the next definition. @xmath25 an upper bound of the transition potential barrier is estimated to be 1.4 ev. in the initial state, the no molecule adsorbed on cu - tl with the nitrogen atom binding to the cu surface. in the transition state, the oxygen atom had local bond connections with surrounding cu atoms to reduce the total energy. to form this distorted structure in the transition state, the whole cu atomic configuration were optimized, creating drastic change in cu - tl. Summary and conclusion utilizing the dft - gga simulations, we have shown that no dissociative adsorption may happen on an atomic cu layer, which is the triangular lattice of cu atoms (cu - tl). the reactivity of cu - tl against molecular adsorption of no was found to be similar to the cu(111) surface. some stable sites for no were found to give molecular adsorption. however, our optimization simulation revealed that there was a co - adsorbed structure of n and o atoms, which was energetically stabler by 1.08ev than the molecular adsorbed cu - tl structure. a reaction path estimation showed existence of a path with an energy barrier of 1.4ev. thus, we may conclude molecular dissociation of no on the cu atomic layer. the large dissociation energy appears owing to formation of local n - cu or o - cu bondings and creation of local n - cu@xmath26 and o - cu@xmath22 structures. we further considered an atomic step - like structure (ass) of cu, which was an atomic - scale wrinkle in the cu - tl structure. the absolute value of the molecular adsorption energy on the step was larger than the values found for cu - bulk surfaces or cu - tl. our simulation revealed that there existed a dissociative adsorbed structure in which a nitrogen impurity site embedded in a cu structure was created. the estimated dissociation energy of no became -0.37 ev on ass. flexibility against modification of this cu atomic structure in the nano - meter scale is decisive both to stabilize dissociative adsorption of no and to reduce the energy barrier on the no - reduction path. catalytic activity of cu to reduce no should appear on the atomically flexible cu networks. therefore, in order to realize cu - based no@xmath0 reduction catalysts, it is important to create atomic structures of cu, _ i.e. _ atomic layers, atomic scale clusters, and atomic scale networks, which allow conformational change. this work was supported by the elements science and technology project and also by grant - in - aid for scientific research in priority area (no. 19051016) and a grants - in - aid for scientific research (no. 22360049). the computation is partly done using the computer facility of issp, univ. of tokyo, and ri2 t, kyushu university. References 10 satterfied c n 1996 _ heterogeneous catalysis in industrial practice _, 2nd ed. (krieger publishing company, malabar) brown w a and king d a 2000 _ j. phys. chem. b _ * 104 * 2578 balkenende a r, gijzeman o l j and geus j w 1989 _ appl. surf. * 37 * 189 godowski p j, onsgaard j, gagor a, kondys m and li z s 2005 _ chem. lett. _ * 406 * 441 matllb m h and roberts m w 1977 _ j. chem. soc., faraday trans. _ * 73 * 1393 jonson d w, matloob m h and roberts m w 1979 _ j. chem., faraday trans. _ * 75 * 2143 so s k, franchy r and ho w 1991 _ j. chem. phys. _ * 95 * 1385 damus p, suhren m, chabal y j, hirschmugl c j and williams g p 1997 _ surf. sci. _ * 371 * 200 van daelen m a, li y s, newsam j m and van santen r a 1996 _ j. phys. chem. _ * 100 * 2279 gojdo@xmath27 m, hafner j and eichler a 2006 _ j. phys. : condens. matter _ * 18 * 41 gonzlez s, sousa c and illas f 2006 _ j. catal. _ * 239 * 431 yen m - y and ho j - j 2010 _ chem. * 373 * 300 loffreda d, simon d and sauter p 2003 _ j. catal. _ * 213 * 211 hammer b 1998 _ faraday discuss. _ * 110 * 323 hammer b 1999 _ phys. lett. _ * 83 * 3681 hammer b 2000 _ surf. sci. _ * 459 * 323 hammer b 2001 _ j. catal. _ * 199 * 171 liu z p, jenkins s j and king d a 2003 _ j. am. soc. _ * 125 * 14660 ge q and neurock m 2004 _ j. am. soc. _ * 126 * 1551 backus e h g, eichler a, grecea m l, kleyn a w and bonn m 2004 _ j. chem. phys. _ * 121 * 7946 rempel j. greeley j. hansen l.b. nielsen o.h. nrskov j.k. and mavrikakis m. 2009 _ j. phys. c _ * 113 * 20623 hohenberg p and kohn w 1964 _ phys. rev. _ * 136 * b864 kohn w and sham l j 1965 _ phys. rev. _ * 140 * a1133 giannozzi p, baroni s, bonini n, calandra m, car r, cavazzoni c, ceresoli d, chiarotti g l, cococcioni m, dabo i, dal corso a, de gironcoli s, fabris s, fratesi g, gebauer r, gerstmann u, gougoussis c, kokalj a, lazzeri m, martin - samos l, marzari n, mauri f, mazzarello r, paolini s, pasquarello a, paulatto l, sbraccia c, scandolo s, sclauzero g, seitsonen a p, smogunov a, umari p and wentzcovitch r m 2009 _ j. phys. : condens. matter _ * 21 * 395502 perdew j p, burke k and ernzerhof m 1996 _ phys. lett. _ * 77 * 3865 perdew j p, burke k and ernzerhof m 1997 _ phys. lett. _ * 78 * 1396 vanderbilt d 1990 _ phys. b _ * 41 * 7892 zhang t, cheng p, li w - j, sun, y - j, wang g, zhu x - g, he k, wang l, ma x, chen x, wang y, liu y, lin h - q, jia j - f, and xue q - k 2010 _ nature physics _ * 6 * 104 novoselov k s and geim a k 2004 _ science _ * 306 * 666 novoselov k s and geim a k 2005 _ nature _ * 438 * 197 mohr m, papagelis k, maultzsch j, and thomsen c 2009 _ phys. rev. b _ * 80 * 205410

to investigate chemical reactivity of cu atomic - scale structures, we performed simulations based on the generalized gradient approximation in the density functional theory. an atomic layer of cu forming a triangular lattice (tl) was found to give a stable structure. the nitrogen monoxide molecule (no) was adsorbed on some atomic sites of tl or on an atomic step structure (ass) of cu. the molecular adsorption energy on tl was -0.83 ev. our data suggested that dissociative adsorption of no with a dissociation energy of -1.08 ev was possible with an energy barrier of order 1.4 ev. in this optimized structure, the nitrogen and oxygen atoms were embedded in the cu layer. on the step, no adsorbed at a bridge site and the formation energy of cu-(no)-cu local bond connections was estimated to be around -1.32 ev. molecular dissociation of no with a dissociation energy of -0.37 ev was also possible around ass.

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Introduction a multistable system is one that possesses a large number of coexisting attractors for a fixed set of parameters. there is ample evidence for such phenomena in the natural sciences, with examples coming from neurosciences and neural dynamics @xcite - @xcite, optics @xcite @xcite, chemistry @xcite @xcite @xcite, condensed matter @xcite and geophysics @xcite. multistability also seems to be an essential complexity - generating mechanism in a large class of agent - based models @xcite. in view of this, it is important to identify the dynamical mechanisms leading to multistability and, in particular, to construct simple models where this phenomenon might be under control. the first mathematical result in this direction was obtained by newhouse @xcite @xcite @xcite who proved that, near a homoclinic tangency, a class of diffeomorphisms in a two - dimensional manifold has infinitely many attracting periodic orbits (sinks), a result that was later extended to higher dimensions @xcite. it has also been proved @xcite that, in addition to infinitely many sinks, infinitely many strange attractors exist near the homoclinic tangencies. the stability of the phenomena under small random perturbations has been studied @xcite @xcite. a second dynamical mechanism leading to multistability is the addition of small dissipative perturbations to conservative systems. conservative systems have a large number of coexisting invariant sets, namely periodic orbits, invariant tori and cantori. by adding a small amount of dissipation to a conservative system one finds that some of the invariant sets become attractors. not all invariant sets of the conservative system will survive when the dissipation is added. however, for sufficiently small dissipation, many attractors (mainly periodic orbits) have been observed in typical systems @xcite @xcite @xcite. the problem of migration between attractors and their stability in multiple - attractor systems has also been studied by other authors @xcite @xcite. most of results are based on numerical evidence. however, using the techniques of deformation stability @xcite @xcite @xcite @xcite some rigorous mathematical results @xcite may be obtained. finally, it has been found recently @xcite that, for parameter values near the feigenbaum period - doubling accumulation point, quadratic maps coupled by convex coupling may have a large number of stable periodic orbits. this is one of the phenomena we study in detail in this paper. the emphasis on quadratic maps near the feigenbaum accumulation point has a motivation close to the idea of control of chaos @xcite @xcite. the typical situation in control of chaos, is that of a strange attractor with an infinite number of embedded periodic orbits, all of them unstable. these orbits are then stabilized by several methods. if, instead of a large number of unstable periodic orbits, one has, for example, a large number of sinks, the controlling situation would seem more promising and robust, because the control need not be so accurate. it would suffice to keep the system inside the desired basin of attraction. at the period - doubling accumulation point the feigenbaum attractor, because of the properties of the flip bifurcations, coexists with an infinite set of unstable periodic orbits. by coupling, as we will show, an arbitrarily large number of orbits may become stable. the existence of a large number of stable periodic orbits for just two coupled quadratic maps, provides a simple model where multistability is well under control, in the sense that not only the nature of the phenomenon is completely understood as one may also compute the range of parameters that provides any desired number of stable orbits. this should be contrasted, for example, with concrete models for the newhouse phenomenon @xcite. rather than merely focusing on multistability, we also study the phenomenology of two coupled quadratic maps, in particular the bifurcations of periodic orbits and the regime of synchronization.. the stabilization of orbits in the coupled system is similar to that obtained in higher dimensional coupled map lattices @xcite with the exception that, due to the restricted dimension of the phase space, the types of bifurcations are different in our system. the results concerning the multistability phenomenon at @xmath0 also considerably extend, and also correct, some imprecise statements in @xcite. Coupled quadratic maps coupled map lattices (cml) are discrete dynamical systems generated by the composition of a local nonlinearity and a coupling. the phase space of the cml considered in this letter is the square @xmath1^{2}$] and the dynamics is generated by the map @xmath2 defined as follows. given a point @xmath3, its image by @xmath2, denoted @xmath4 is given by @xmath5 where @xmath6, @xmath7 and @xmath8. the map @xmath9 maps @xmath1 $] into itself. therefore, the convex combination in ([def]) ensures that @xmath10^{2})\subset [-1,1]^{2}$] and the dynamics is well - defined. we denote the orbit issued from the initial condition @xmath11 by the sequence @xmath12, that is to say, @xmath13 and @xmath14 for all @xmath15. for the sake of simplicity, we will often employ the variables @xmath16 and @xmath17. the previous notation of orbits also applies to these variables for which relation ([def]) becomes @xmath18 where @xmath19. finally, note that the dynamics commutes with the symmetry @xmath20 or @xmath21 in the original variables. Synchronization if @xmath22, then @xmath23 and @xmath24. in this case, the orbit is said to be synchronized (from @xmath25 on). more generally, an orbit is said to synchronize if @xmath26 and if all orbits synchronize, then we say to have synchronization of the map. synchronization is the simplest dynamical regime exhibited by two - dimensional cml. to determine a sufficient condition for synchronization in our system, we note that for any orbit, one has @xmath27 for all @xmath15. it follows from ([def2]) that the condition @xmath28 ensures an exponential decay of @xmath29, and hence synchronization.. since @xmath30, the condition @xmath28 is equivalent to the following ones (see figure 1). @xmath31 from now on, we assume that @xmath32. the condition @xmath33 is not necessary. indeed, if for instance @xmath34, then @xmath9 has an attracting fixed point in @xmath1 $], and one can prove that synchronization occurs. when @xmath35 is sufficiently small, this happens even though @xmath36. Non-synchronized period-2 orbits starting from synchronization and modifying the parameters, non - synchronized (periodic) orbits appear from bifurcations of synchronized (periodic) ones. to understand this phenomenon, as well as the bifurcations of subsequent orbits, we now study analytically the periodic orbits of period 1 and 2. let @xmath37 be the synchronized fixed point denoted in @xmath38-variables. it exists for any values of the parameters and @xmath2 has no other fixed point in @xmath1^{2}$]. in @xmath38-variables, the jacobian of @xmath39 at this fixed point is diagonal. one eigenvalue is @xmath40 and the corresponding eigendirection is the diagonal @xmath41. the other eigenvalue is @xmath42 and the corresponding direction, orthogonal to the diagonal is referred as the anti - diagonal. the condition @xmath43 then determines a period - doubling bifurcation, which is of co - dimension 1 if @xmath44. this is the well - known period - doubling bifurcation of @xmath9 which creates a synchronized period-2 orbit of @xmath2. moreover, one checks that the derivative @xmath45 is negative for any @xmath46. hence the conditions @xmath44 and @xmath47, i.e. @xmath48 determine another co - dimension 1 period - doubling bifurcation of the synchronized fixed point. indeed the conditions of the corresponding bifurcation theorem (see e.g. @xcite) are satisfied when the curve @xmath49 is crossed upward. the period-2 orbit created at this bifurcation is non - synchronized and symmetric. to show this, denote by @xmath50 and @xmath51 its components. since the multiplier @xmath42 is negative and the bifurcating direction is the anti - diagonal, we have @xmath52 (sufficiently close to the bifurcation). because of the @xmath53 symmetry, the map @xmath2 also has a period-2 orbit with components @xmath54 and @xmath55. consequently if @xmath56, the system would have two periodic orbits created by a co - dimension 1 bifurcation. this is impossible by the unicity in the bifurcation theorem. therefore, sufficiently close to the bifurcation, we have @xmath57 and @xmath58 which is the desired conclusion. by continuity in the parameters of @xmath2, sufficiently close to the bifurcation, this symmetric orbit is stable with respect to perturbations in one direction (the anti - diagonal direction at the bifurcation) and since @xmath59, it is unstable in the direction orthogonal to the latter. the bifurcations will now be computed. the orbit with @xmath60 and @xmath58 exists for any @xmath61 and is the unique (up to time translations) period-2 non - synchronized symmetric orbit of @xmath2 in @xmath1^{2}$]. computing the corresponding jacobian, one obtains the equation for the multipliers @xmath62 \lambda + \left (1 - 2\varepsilon \right) ^{2}(s^{2}-d^{2})^{2}=0\]] where @xmath63 direct calculations show that, if @xmath64, the multipliers, say @xmath65 and @xmath66, have zero imaginary part iff @xmath67. under this condition, we have @xmath68 if @xmath69 and @xmath70 iff @xmath71. (the inequality @xmath72 indeed holds if @xmath73 and @xmath74, see figure 1.) consequently, by increasing @xmath46, the symmetric orbit suffers an inverse pitchfork bifurcation at @xmath75. this bifurcation is generic for a symmetric orbit in a system with symmetry @xcite and the conditions of the bifurcation theorem hold when the curve @xmath76 is crossed upward. this bifurcation creates two non - symmetric period-2 orbits (one orbit and its symmetric). we have checked that these orbits exist for any @xmath77 and @xmath78. for @xmath79, their components are combinations of a fixed point of @xmath9 and the components of a period-2 orbit. when the imaginary part of @xmath80 and @xmath81 is not zero, we have @xmath82. (once again, if @xmath73 and @xmath83, the inequality @xmath84 is satisfied, see figure 1.) the symmetric orbit is thus stable in the interval @xmath85. if @xmath44 and the curve @xmath86 is crossed upward, this orbit suffers a hopf bifurcation creating a locally stable invariant circle. a numerical calculation shows that the latter is destroyed when @xmath46 is sufficiently large or when @xmath87 is sufficiently small. obviously, if @xmath79, it does not exist and the bifurcation at @xmath88 which is a period - doubling bifurcation of @xmath9 creates a period-4 orbit. note that invariant circles in two - dimensional cml resulting from the destabilization of a symmetric orbit and their normal form had already been reported in @xcite. in that work, the system is also defined by ([def]), but the local map is @xmath89 and @xmath35 may be larger than @xmath90. figure 2 shows an example of the phenomenology described above. numerically, it is more convenient to follow the orbits from @xmath79 to increasing values of the coupling. in this picture, as well as in the following ones the map parameter is @xmath0 (the accumulation point of the period - doubling cascade). in figure 2, from @xmath79 (the circle) to @xmath91 (the point labelled 1), the symmetric orbit is unstable. the figure also shows the invariant circle for @xmath92. between the points 1 and 2, the symmetric orbit is stable. at point 2, the pitchfork occurs, the symmetric orbit becomes unstable and the non - symmetric orbits are created. finally, the point 3 (@xmath93) corresponds to the collapse on the synchronized fixed point. The phase opposition orbits the previous phenomenology is not restricted to small periods but extend to any power of 2. in particular, the synchronized period-@xmath94 orbit may destabilize to create a symmetric (non - synchronized) orbit of twice the period. given @xmath95, let @xmath96 be the components of the period-@xmath94 orbit of @xmath9. the points @xmath97 are the components of the synchronized period-@xmath94 orbit of @xmath39. by the chain rule and since each jacobian at @xmath98 is diagonal, the corresponding multiplier along the anti - diagonal direction is @xmath99 the condition that this multiplier equals @xmath100 determines, if @xmath44, a co - dimension 1 period - doubling bifurcation. applying the reasoning of the previous section to each component @xmath98, we conclude that this bifurcation creates an orbit with the property @xmath101 and @xmath102 for all @xmath15, which is called a phase - opposition period-@xmath103 orbit. since @xmath104, this bifurcation occurs only if the bifurcation along the diagonal direction has occurred (the local period - doubling bifurcation of @xmath96). in other words, the phase opposition period-@xmath94 orbit exists only if the synchronized period-@xmath94 orbit does. moreover it follows from figure 3 that, at least for @xmath0, the phase opposition period-@xmath103 orbit exists only if the phase opposition period-@xmath94 orbit does. this is confirmed analytically for the period@xmath105 orbit whose existence condition is the instability of the synchronized period@xmath106 orbit in the anti - diagonal direction. one obtains @xmath107 and @xmath108 if @xmath109 and @xmath110 and @xmath111 if @xmath112 (see figure 1). furthermore, a numerical calculation at @xmath0, reported in figure 3, shows that the succession of bifurcations of a phase opposition orbit does not depend on the period. on this picture, we have plotted the values of @xmath35 for the hopf bifurcation, the pitchfork bifurcation and the period - doubling bifurcation creating the orbit, versus the power of the period. for each period, the phenomenology is identical to that described in the previous section, with an adequate change of scale in @xmath87. in addition, the picture shows that several phase opposition orbits may be stable for @xmath44 fixed. this stabilization is an effect of the coupling that will be discussed below. finally, since the phase opposition orbits are the first orbits to appear when the parameters are varied from synchronization and since the first such orbit that is created is of period 2, it follows that a necessary and sufficient condition for synchronization is @xmath113, the condition for the existence of the latter. The non-symmetric orbits we now analyze the existence and the stability of other period-@xmath94 orbits for @xmath0. our interest for this value of @xmath46 is that the scaling properties of @xmath9 are reflected on scaling laws for the periods and values of @xmath35 at which the bifurcations occur (see figure 3 and 8). we only consider the orbits which for @xmath79 have the same period on projection to both axis @xmath114 and @xmath115. these orbits are followed numerically when @xmath35 increases and are referred using the phase shift of their components at @xmath79. for @xmath0, the map @xmath9 has a period-@xmath94 orbit for each @xmath116, whose components for @xmath117 up to 5 are shown in figure 5. in this picture, the numbers reflect the order in which the components are visited and the tree structure represents the origin of each component in the bifurcation cascade. an important notion is the dyadic distance @xmath118 between the components of an orbit. @xmath118 is the number of steps one has to go back in the bifurcation tree to meet a common component. the dyadic distance is used to characterize the families of periodic orbits that we are considering. for instance, the coordinates of each component of a synchronized orbit are at distance 0, those of a phase - opposition orbit are at distance 1. accordingly, when we speak of distance@xmath119 orbit we refer to the dyadic distance of the coordinates of its components. for any @xmath120, there are @xmath121 different orbits with distance @xmath118 which have coordinates out of phase by @xmath122 steps, with @xmath123. the distance of a period-@xmath124 orbit is at most @xmath117 (@xmath125). for @xmath79, the only symmetric orbits are those at distance 0 and 1. this property is preserved for @xmath44 as shows figure 6 for @xmath126. the succession of bifurcations of orbits with distance @xmath127 should then differ from those with distance 1. the differences are seen in figure 6 which shows the evolution of the eigenvalues. for @xmath79, the orbit is unstable. when @xmath35 increases, it suffers two collisions with orbits of twice the period when the eigenvalues cross @xmath100 and then becomes stable. (when decreasing @xmath35, these collisions would be period - doubling bifurcations.) if @xmath35 increases further, the orbit collides with an unstable one of the same period in a saddle - node bifurcation when the larger eigenvalue reaches 1. for larger values of @xmath87, the orbit does not exist. the unstable orbit with which it collides is the one that at @xmath79 has period @xmath94 in one projection and @xmath128 in the other. for higher dyadic distances, the overall variation of the eigenvalues is similar to the @xmath129 case. figure 7 shows a typical example of these phenomena for the case @xmath129. between @xmath79 and the point labelled 1 in the figure, the orbit is unstable. the point 1 corresponds to the smaller eigenvalue crossing -1 (see figure 6). therefore, between the point 1 and 2, the orbit is stable. it disappear at the point 2 when it collides with an unstable orbit of the same period. Multistability we have seen that the coupling stabilizes the orbits with distance larger than 0 at @xmath0. there are indeed two mechanisms responsible for this stabilization.. the determinant of the jacobian of a period-@xmath94 orbit is @xmath130 the term @xmath131 coming from the coupling decreases when @xmath117 increases. however, there is yet a second stabilizing mechanism. denote by @xmath132 the remaining factor in the determinant @xmath133 without coupling, @xmath134 is simply the square of the multiplier of @xmath9 for the periodic orbit. from the properties of the feigenbaum - cvitanovic functional equation it follows @xcite that this factor converges to a fixed value around @xmath135 when @xmath117 increases. the coupling however, changes the position of the orbit components in such a way that this factor also decreases. it is the combined action of this decrease with the contraction of the coupling that brings the eigenvalues into the interior of the unit circle and stabilizes the orbits. for small @xmath35 there is a simple geometrical interpretation for the variation of @xmath136. the reason why in the one dimensional map the product @xmath137 remains constant, when @xmath117 grows, is because each time the period doubles, the doubling in the number of factors greater than one is compensated by the fact that the component of the orbit closest to zero approaches zero a little more. for the unstable orbits along the period - doubling chain, the orbit components closest to zero alternate on each side of the origin. the contracting effect of the convex coupling tends to bring the orbits back in the period - doubling hierarchy. therefore, because the component closest to zero has to move across the origin for the orbit to approach the one with half the period, this implies that the product of the coordinates is going to decrease. the greater the dyadic distance between the orbit projections on the axis, the greater will be the perturbation that the original (one - dimensional) orbits suffer. therefore one expects the contracting effect in @xmath136 to increase with the dyadic distance. this effect is quite apparent on figure 8 which shows the stabilizing and destabilizing lines for orbits with distance from 1 to 4. the shift downwards of the stable regions for successively larger dyadic distances implies that the smaller @xmath35 is, the larger the number of distinct stable orbits that are obtained. an accurate numerical estimate of the number of distinct orbits is obtained by computing the derivative @xmath138 at @xmath79 for each @xmath117 and dyadic distance @xmath139. actually this derivative provides an accurate estimate of @xmath140 itself, because this one varies almost linearly with @xmath35 for most of the stable range of the orbits. on figure 9, the scaling properties, when @xmath117 grows, of this derivative are shown. from these results one computes @xmath141 with @xmath142 notice that in figure 9 there is more than one data point for each pair @xmath143 which correspond to non - equivalent orbits with the same dyadic distance. * the value of the smallest @xmath35 parameter that stabilizes an orbit of dyadic distance @xmath118 equal to the power @xmath117 @xmath144 * the value of the largest @xmath35 parameter for which a @xmath145 orbit is stable @xmath146 from this, one obtains the result that at least @xmath147 distinct stable orbits are obtained if @xmath148 @xmath147 is only a lower bound on the number of distinct stable periodic orbits, because here we have studied only orbits with the same period under projection in the two axis. in conclusion : _ for sufficiently small _ @xmath35 _ _ an arbitrarily large number of distinct stable periodic orbits is obtained__. however, for any fixed @xmath35, it is an arbitrarily large number that is obtained, not an infinite number. most orbits either synchronize (and are then unstable) or disappear as @xmath35 grows. as a result, a reasoning based on the implicit function theorem, as used in @xcite is misleading. given a sequence of orbits of different periods, even if they remain as orbits for a small perturbation, that does not mean that their (smallest) periods remain distinct.

the phenomenology of a system of two coupled quadratic maps is studied both analytically and numerically. conditions for synchronization are given and the bifurcations of periodic orbits from this regime are identified. in addition, we show that an arbitrarily large number of distinct stable periodic orbits may be obtained when the maps parameter is at the feigenbaum period - doubling accumulation point. an estimate is given for the coupling strength needed to obtain any given number of stable orbits.

nlin0005053

Introduction a fair number of astronomers and astronomy students have a physical challenge. it is our responsibility to learn the basics of accessibility to be able to help our library patrons to gain access to things that they need for their studies and work. astronomy is often seen as a very visual science. after all, its origins lie in looking at the skies. hence, it is a common belief that you need to use your sight to be able to study astronomy. this is strictly not true. in reality, we have been using assistive technologies telescopes, sensors, computers for a long time now to gain access to data that the human eye does not see unaided. visual information is coming to us as large streams of bytes. the modern astronomer is hardly bound by physical limitations. one can produce solid research sitting comfortably in front of one s personal computer. there are many examples of physically challenged individuals who have made successful careers in science. those who have seen the movie _ contact _ based on carl sagan s novel are familiar with the blind astronomer who is listening to radio signals instead of watching them on the screen. his character is based on a real scientist, dr. d. kent cullers. there are other success stories in fact, too many to enumerate here. What does accessibility look like? but, you ask, is nt the sheer amount of information a major hindrance to those who can not browse it easily? yes, it is to some degree. electronic textual materials provide both a possibility and a challenge for those with low vision. in theory, it is possible for almost anyone to access online information, but in practice, this requires know - how and proper tools. plenty of assistive technologies exist to overcome hindrances. the daisy standard for digital talking books has been an important tool for making electronic texts easy to browse. not all hindrances are in the visual domain. imagine an elderly astronomer who has the full use of his or her intelligence, but whose hands are shaking, and who might have some difficulty with pointing a mouse when navigating a webpage and filling out search forms. it is a challenging task for librarians and information specialists to make our services and search forms accessible to people with a diversity of abilities so that they can do the research necessary for building careers as active contributors in their chosen fields of research. but what does accessibility look like? there is a pervasive myth that it looks boring. this is strictly not true. accessible design should be functional enough, not just pretty. with proper html code and other techniques, we can make the text compliant with technological aids. if the html coding is poor, a document may be impossible to open with such aids or it could be impossible to navigate the text. the author of this paper was involved with an university - wide accessibility project that was undertaken by the university of helsinki in 20052006, with a follow up in 20082009. it was recognized that accessibility must cover not only our physical surroundings, but also the online environment as well. in spring 2009, we noticed that the new national online system for applying for university education was not accessible to blind students. the system was provided by the finnish ministry of education, and we challenged them to fix it. to our big surprise, they did, working in collaboration with us and the finnish federation of the visually impaired. figure 1 shows a page from the application system. it looks exactly the same both before and after accessibility changes were made. differences can be seen on the coding level, but otherwise one can not tell the old version from the new one by visual inspection alone. the change has resulted in a major functional improvement. the old version could not even be opened with assistive technology, and blind students could not use it. now they can. Standards and how to apply them accessibility needs some muscle to drive it. it is not just about good people doing good deeds it is also about ensuring that everyone has access to things that matter to them. we need guidelines and standards, preferably with legislation to back them up. in the united states, section 508 of the rehabilitation act regulates purchases made with federal funding. it is about `` access to and use of information and data that is comparable to that provided to others. '' a market for accessible products helps big publishers to take accessibility into account. when a publisher has a large enough number of customers who need to buy accessible products, they will be motivated to sell accessible products. we also need strong standards. the world wide consortium has updated its web content accessibility guidelines (wcag) version 2 dates back to 2008. this new version of wcag is meant to be a practical tool, evidenced by its three levels of accessibility : * a : minimum * aa : medium * aaa : as accessible as possible you will find a good wcag2 checklist online. the ideal thing to do would be to make your website as accessible as possible, but in practice you need to read the guidelines and identify the accessibility level best suited to serving your users. let s look at a concrete example by applying an a - level guideline to an existing search form. the guideline states : `` form inputs have associated text labels or, if labels can not be used, a descriptive title attribute. '' let s look at a part of an ads search form with its original coding. this piece of code is from the section which requires an object for selection. 0.2 in _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ ` < input name = obj_req value = yes type = checkbox > require object for selection ` _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 0.2 in let s add some more coding (in boldface). rather than just a checkbox, we now have a _ text label_. 0.2 in _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ ` < input id = obj_req name = obj_req value = yes type = checkbox > < label for = obj_req > require object for selection</label > ` _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 0.2 in figure 2 shows what has changed. the text label in question has been highlighted. it is no longer necessary to hit the small checkbox it is enough if you just click the associated text. this makes the box much easier to check. Structure is the key you can do clever things with html. there are however many other formats to consider : pdf, flash, and office products, to name just a few. no matter what the material at hand, it needs structure above all else. otherwise, a blind person who tries to read a text has to read everything from beginning to end and is not able to navigate to a chapter or a footnote. even pdf which used to be an accessibility nightmare can now boast of a structure to make it more accessible it s called tagged pdf. as a general guideline, no matter what kind of document you are writing, you will need to stick to structure. do you use subtitles that are bold and in a different font? please, use proper titles instead and use styles to control the fonts and such. let s take a peek at an html page that has structure. there are tools to make the structure visible. the box in figure 3 has been done with a wave toolbar. this example is taken from _ planetary and space science_. a good amount of structure has been revealed. the html structure of _ earth, moon and planets _, shows next to nothing. its only structure is a references header, `` h2 references. '' there is no subtitle structure at all that you can jump to. most publishers make their electronic materials available in pdf format. usually, those files are without any structure. figure 4 shows the acrobat reader results of an accessibility quick check there is no structure. Big publishers and small what is the current situation with different astronomy publishers and journals? table 1 shows accessibility elements for a selection of publishers based on inspection of a few papers published in 2009 by university of helsinki astronomers. we asked some questions about the basic properties of each paper. is there html fulltext? does it have structure? and does the pdf have structure? if not, are there at least pdf bookmarks? you can see that these results leave a lot to hope for. the only consistently good results are from _ planetary and space science _, which is published by elsevier. unfortunately, however, not all elsevier products are equally accessible. llcccc title & publisher & html & html & pdf & + & & fulltext & structure & structure & bookmarks + astronomy & astrophysics & edp sciences & yes & ok & no & yes + astrophysical journal & iop & yes & none & no & yes + monthly notices r.a.s. & wiley & yes & none & no & no + astron. nachrichten & wiley & no & & no & no + planetary space sci. & elsevier & yes & ok & yes & yes + earth, moon & planets & springer & yes & none & no & yes + elsevier was the winner of this brief check. it has been making some efforts to increase accessibility of its products, which sets a good example for other big publishers. @xcite have inspected the overall accessibility compliance and practices of major database vendors, elsevier included. even if major publishers are making some progress, it is not enough. there are also smaller publishers, and beyond that there are institutes and libraries producing their own online materials or making their own search forms. many of them are unaware of current accessibility standards. standards can seem difficult to apply. but really, they are easy to follow if we make the guidelines clear enough so that everyone can understand and use them. remember that new technologies are taken into use all the time. we will be constantly facing new challenges to make them accessible, but they will also bring new possibilities with them. Dont forget copyright law and license agreements there is one last thing that you need to be aware of do nt forget about copyright. it is not a given fact that a library can freely distribute electronic material to a patron who could then read it on a personal computer or some other device. the copyright laws in different countries vary surprisingly on this point. moreover, even when the right to access is written into a law, thus making special exceptions to copyright for disabled persons, a license agreement between a library and a publisher might take this right away for particular electronic materials or products. a publisher or a consortium will not allow you to do things that are not specifically stated in the signed agreement. please always remember to check the accessibility options in agreements you sign. to give an example, the current finnish national electronic library (finelib) consortium agreement with elsevier specifies that `` coursepacks in nonelectronic, non - print perceptible form (e.g. braille) may be offered for [the] visually impaired. '' this is not, however, how visually impaired users would like to use the materials. this is a standard clause that should be modified to meet real needs. unfortunately, when the consortium was formed, this clause did not receive the proper attention it should have. An afterthought practically everyone who lives long enough has to face physical challenges at some point. an astronomer who is able - bodied today could have accessibility issues tomorrow. we can not expect that she or he is willing to give up practicing science. in her essay _ the blind astronomer _ @xcite, the new zealand astronomer tracy farr eloquently describes the changes brought by the gradual loss of her vision. with a different approach to looking at the research data, she can continue to access the universe : _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ i am freeing myself from the fixedness of the seen. with my mind open to the universe, i hear the heavens ebb and flow as music. it is the incomprehensibly wonderful revelation of music first heard after only ever having seen black spots and lines on a white page. as my ears open and my eyes close, i hear the planets dance. _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _

making online resources more accessible to physically challenged library users is a topic deserving informed attention from astronomy librarians. recommendations like wcag 2.0 standards and section 508, in the united states, have proven valuable, and some vendors are already making their products compliant with them. but what about the wide variety of databases and other resources produced by astronomy information professionals themselves ? few, if any, of these are currently compliant with accessibility standards. here we discuss some solutions to these accessibility challenges.

1006.1803

Introduction among the great variety of the works devoted to random motions at finite speed in the euclidean spaces @xmath8 (see @xcite, @xcite, @xcite, @xcite for the markovian case and @xcite, @xcite for different non - markovian cases), the markov random flight in the three - dimensional euclidean space @xmath1 is, undoubtedly, the most difficult and hard to study. while in the low even - dimensional spaces @xmath9 and @xmath10 one managed to obtain the distributions of the motions in an explicit form (see @xcite, @xcite and @xcite, respectively), in the important three - dimensional case only a few results are known. the absolutely continuous part of the transition density of the symmetric markov random flight with unit speed in the euclidean space @xmath1 was presented in (*??? * formulas (1.3) and (4.21) therein). it has an extremely complicated form of an integral with variable limits whose integrand involves inverse hyperbolic tangent function. this formula has so complicated form that can not even be evaluated by means of standard computer environments. moreover, the lack of the speed parameter in this formula impoverishes somewhat the model because it does not allow to study the limiting behaviour of the motion under various scaling conditions (under kac s condition, for example). the presence of both parameters (i.e. the speed and the intensity of switchings) in any process of markov random flight makes it, undoubtedly, the most adequate and realistic model for describing the finite - velocity diffusion in the euclidean spaces. these parameters can not be considered as independent because they are connected with each other through the time (namely, the speed is the distance passed _ per unit of time _ and the intensity is the mean number of switchings _ per unit of time _). another question concerning the density presented in @xcite is the infinite discontinuity at the origin @xmath11. while the infinite discontinuity of the transition density on the border of the diffusion area is a quite natural property in some euclidean spaces of low dimensions (see @xcite for the euclidean plane @xmath12 and (*??? * the second term of formulas (1.3) and (4.21)), @xcite, (*??? * formula (3.12)) in the space @xmath1), the discontinuity at the origin looks somewhat strange and hard to explain. the difficulty of analysing the three - dimensional markov random flight and, on the other hand, the great theoretical and applied importance of the problem of describing the finite - velocity diffusion in the space @xmath1 suggest to look for other methods of studying this model. that is why various asymptotic theorems yielding a good approximation would be a fairly desirable aim of the research. such asymptotic results could be obtained by using the characteristic functions technique. in the case of the three - dimensional symmetric markov random flight some important results for its characteristic functions were obtained. in particular, the closed - form expression for the laplace transform of the characteristic function was obtained by different methods in (*??? * formulas (1.6) and (5.8)) (for unit speed) and in (*??? * formula (45)), @xcite (for arbitrary speed). a general relation for the conditional characteristic functions of the three - dimensional symmetric markov random flight conditioned by the number of changes of direction, was given in (*??? * formula (3.8)). the key point in these formulas is the possibility of evaluating the inverse laplace transforms of the powers of the inverse tangent functions in the complex right half - plane. this is the basic idea of deriving the series representations of the conditional characteristic functions corresponding to two and three changes of direction given in section 3. based on these representations, an asymptotic formula, as time @xmath5, for the unconditional characteristic function is obtained in section 4 and the error in this formula has the order @xmath6. the inverse fourier transformation of the unconditional characteristic function yields an asymptotic formula for the transition density of the process which is presented in section 5. this formula shows that the density is discontinuous on the border, but it is continuous at the origin @xmath11, as it must be. the unexpected and interesting peculiarity is that the conditional density corresponding to two changes of direction contains a term having an infinite discontinuity on the border of the diffusion area. from this fact it follows that such conditional density is discontinuous itself on the border and this differs the 3d - model from its 2d - counterpart where only the conditional density of the single change of direction has an infinite discontinuity on the border. the error in the obtained asymptotic formula has the order @xmath6. in section 6 we estimate the accuracy of the asymptotic formula and show that it gives a good approximation on small time intervals whose lengths depend on the intensity of switchings. finally, in appendices we prove a series of auxiliary lemmas that have been used in our analysis. Description of the process and structure of distribution consider the stochastic motion of a particle that, at the initial time instant @xmath13, starts from the origin @xmath14 of the euclidean space @xmath1 and moves with some constant speed @xmath15 (note that @xmath15 is treated as the constant norm of the velocity). the initial direction is a random three - dimensional vector with uniform distribution on the unit sphere @xmath16 the motion is controlled by a homogeneous poisson process @xmath17 of rate @xmath3 as follows. at each poissonian instant, the particle instantaneously takes on a new random direction distributed uniformly on @xmath18 independently of its previous motion and keeps moving with the same speed @xmath15 until the next poisson event occurs, then it takes on a new random direction again and so on. let @xmath19 be the particle s position at time @xmath20 which is referred to as the three - dimensional symmetric markov random flight. at arbitrary time instant @xmath20 the particle, with probability 1, is located in the closed three - dimensional ball of radius @xmath21 centred at the origin @xmath22 : @xmath23 consider the probability distribution function @xmath24 of the process @xmath4, where @xmath25 is the infinitesimal element in the space @xmath1. for arbitrary fixed @xmath20, the distribution @xmath26 consists of two components. the singular component corresponds to the case when no poisson events occur on the time interval @xmath27 and it is concentrated on the sphere @xmath28 in this case, at time instant @xmath29, the particle is located on the sphere @xmath30 and the probability of this event is @xmath31 if at least one poisson event occurs on the time interval @xmath32, then the particle is located strictly inside the ball @xmath33 and the probability of this event is @xmath34 the part of the distribution @xmath26 corresponding to this case is concentrated in the interior @xmath35 of the ball @xmath33 and forms its absolutely continuous component. let @xmath36 be the density of distribution @xmath37. it has the form @xmath38 where @xmath39 is the density (in the sense of generalized functions) of the singular component of @xmath26 concentrated on the sphere @xmath30 and @xmath40 is the density of the absolutely continuous component of @xmath26 concentrated in @xmath41. the singular part of density ([struc2]) is given by the formula : @xmath42 where @xmath43 is the dirac delta - function. the absolutely continuous part of density ([struc2]) has the form : @xmath44 where @xmath45 is some positive function absolutely continuous in @xmath41 and @xmath46 is the heaviside unit - step function given by @xmath47 asymptotic behaviour of the transition density ([struc2]) on small time intervals is the main subject of this research. since its singular part is explicitly given by ([denss]), then our efforts are mostly concentrated on deriving the respective asymptotic formulas for the absolutely continuous component ([densac]) of the density. our main tool is the characteristic functions technique because, as it was mentioned above, some closed - form expressions for the characteristic functions (both conditional and unconditional ones) of the three - dimensional symmetric markov random flight @xmath4 are known. Conditional characteristic functions in this section we obtain the series representations of the conditional characteristic functions corresponding to two and three changes of direction. these formulas are the basis for our further analysis leading to asymptotic relations for the unconditional characteristic function and the transition density of the three - dimensional symmetric markov random flight @xmath4 on small time intervals. the main result of this section is given by the following theorem. * theorem 1. * _ the conditional characteristic functions @xmath48 and @xmath49 corresponding to two and three changes of direction are given, respectively, by the formulas : _ @xmath50 @xmath51 @xmath52 _ where @xmath53 is bessel function, @xmath54 is the generalized hypergeometric function given by _ ([hypergeom54]) _ (see below) and the coefficients @xmath55 are given by the formula _ @xmath56 0.2 cm _ proof. _ it was proved in (*??? * formula (3.8)) that, for arbitrary @xmath20, the characteristic function @xmath57 (that is, fourier transform @xmath58 with respect to spatial variable @xmath59) of the conditional density @xmath60 of the three - dimensional markov random flight @xmath4 corresponding to @xmath61 changes of directions is given by the formula @xmath62(\boldsymbol\alpha) = \frac{n!}{t^n } (c\vert\boldsymbol\alpha\vert)^{-(n+1) } \mathcal l_s^{-1 } \left [\left (\text{arctg } \frac{c\vert\boldsymbol\alpha\vert}{s } \right)^{n+1 } \right](t), \]] @xmath63 where @xmath64 is the inverse laplace transformation with respect to complex variable @xmath65 and @xmath66 is the right half - plane of the complex plane @xmath67. in particular, in the case of two changes of directions @xmath68, formula ([eq1]) yields : @xmath69(\boldsymbol\alpha) = \frac{2!}{t^2 } (c\vert\boldsymbol\alpha\vert)^{-3 } \mathcal l_s^{-1 } \left [\left (\text{arctg } \frac{c\vert\boldsymbol\alpha\vert}{s } \right)^3 \right](t), \qquad \boldsymbol\alpha\in\bbb r^3, \quad s\in\bbb c^+.\]] applying lemma b3 of the appendix b to the power of inverse tangent function in ([eq2]), we obtain : @xmath70(t) \\ & = \frac{2}{\sqrt{\pi } \ ; t^2 } \ ; \sum_{k=0}^{\infty } \frac{\gamma\left (k+\frac{1}{2 } \right)}{k! \ ; (2k+1) } \ ; (c\vert\boldsymbol\alpha\vert)^{2k } \\ & \qquad \times \ ; _ 5f_4\left (1,1,1,-k,- k-\frac{1}{2 } ; \ ; -k+\frac{1}{2 }, -k+\frac{1}{2 }, \frac{3}{2 }, 2 ; \ ; 1 \right) \mathcal l_s^{-1 } \biggl [\frac{1}{\left (s^2 + (c\vert\boldsymbol\alpha\vert)^2 \right)^{k+3/2 } } \biggr](t). \endaligned\]] note that evaluating the inverse laplace transformation of each term of the series separately is justified because it converges uniformly in @xmath65 everywhere in @xmath71 and the complex functions @xmath72 are holomorphic and do not have any singular points in this half - plane. moreover, each of these functions contains the inversion complex variable @xmath73 in a negative power and behaves like @xmath74, as @xmath75, and, therefore, all these complex functions rapidly tend to zero at infinity. according to (* table 8.4 - 1, formula 57), we have @xmath76(t) = \frac{\sqrt{\pi}}{\gamma\left (k+\frac{3}{2 } \right) } \left (\frac{t}{2c\vert\boldsymbol\alpha\vert } \right)^{k+1 } j_{k+1}(ct\vert\boldsymbol\alpha\vert).\]] substituting this into ([eq3]), after some simple calculations we obtain ([char2]). for @xmath77, formula ([eq1]) yields : @xmath78(\boldsymbol\alpha) = \frac{3!}{t^3 } (c\vert\boldsymbol\alpha\vert)^{-4 } \mathcal l_s^{-1 } \left [\left (\text{arctg } \frac{c\vert\boldsymbol\alpha\vert}{s } \right)^4 \right](t), \qquad \boldsymbol\alpha\in\bbb r^3, \quad s\in\bbb c^+.\]] applying lemma b4 of the appendix b to the power of inverse tangent function in ([eq4]) and taking into account that @xmath79(t) = \frac{\sqrt{\pi}}{(k+1)! } \left (\frac{t}{2c\vert\boldsymbol\alpha\vert } \right)^{k+3/2 } j_{k+3/2}(ct\vert\boldsymbol\alpha\vert), \]] we obtain : @xmath80(t) \\ & = 3\pi^{3/2 } \ ; \sum_{k=0}^{\infty } \frac{\gamma_k \ ; (ct\vert\boldsymbol\alpha\vert)^{k-3/2}}{2^{k+3/2 } \ ; (k+1)! } \ ; j_{k+3/2}(ct\vert\boldsymbol\alpha\vert), \endaligned\]] where the coefficients @xmath55 are given by ([coef1]). the theorem is proved. @xmath81 _ _ the series in formulas ([char2]) and ([char3]) are convergent for any fixed @xmath20, however this convergence is not uniform in @xmath82. therefore, we can not invert each term of these series separately. moreover, one can see that the inverse fourier transform of each term does not exist for @xmath83. thus, while there exist the inverse fourier transforms of the whole series ([char2]) and ([char3]), it is impossible to invert their terms separately and, therefore, we can not obtain closed - form expressions for the respective conditional densities. these formulas can, nevertheless, be used for obtaining the important asymptotic relations and this is the main subject of the next sections. Asymptotic formula for characteristic function using the results of the previous section, we can now present an asymptotic relation on small time intervals for the characteristic function @xmath84 of the three - dimensional symmetric markov random flight, where @xmath85 are the conditional characteristic functions corresponding to @xmath86 changes of direction. this result is given by the following theorem. * theorem 2. * _ for the characterictic function @xmath87 of the three - dimensional markov random flight @xmath4 the following asymptotic formula holds : _ @xmath88 \\ & \qquad\qquad\qquad + \frac{\lambda^2 t}{c\vert\boldsymbol\alpha\vert } j_1(ct\vert\boldsymbol\alpha\vert) + \frac{\lambda^3 \ ; \sqrt{\pi } \ ; t^{3/2}}{(2 c\vert\boldsymbol\alpha\vert)^{3/2 } } \ ; j_{3/2}(ct\vert\boldsymbol\alpha\vert)\biggr\ } + o(t^3), \endaligned\]] @xmath52 _ where _ @xmath89 _ and _ @xmath90 _ are the incomplete integral sine and cosine, respectively, given by the formulas : _ @xmath91 0.2 cm _ proof. _ we have : @xmath92.\]] since all the conditional characteristic functions are uniformly bounded in both variables, that is, @xmath93 then @xmath94 and, therefore, @xmath95.\]] in view of ([char2]), we have : @xmath96. \endaligned\]] from the asymptotic formula @xmath97 we get @xmath98 and, therefore, @xmath99 thus, we obtain the following asymptotic relation : @xmath100 similarly, according to ([char3]), we have : @xmath101. \endaligned\]] in view of ([asbes]), we have @xmath102 and, therefore, @xmath103 thus, taking into account that @xmath104 (see ([coef1])), we arrive at the formula : @xmath105 since (see (*??? * formula (3.11))) @xmath106\]] and @xmath107 (that is, characteristic function of the uniform distribution on the surface of the three - dimensional sphere of radius @xmath21), then by substituting these formulas, as well as ([eq8]) and ([eq9]) into ([eq7]), we finally obtain asymptotic relation ([eq6]). the theorem is completely proved. Asymptotic relation for the transition density asymptotic formula ([eq6]) for the unconditional characteristic function enables us to obtain the respective asymptotic relation for the transition density of the process @xmath4. this result is given by the following theorem. * theorem 3. * _ for the transition density @xmath108 of the three - dimensional markov random flight @xmath4 the following asymptotic relation holds : _ @xmath109 \theta(ct-\vert\bold x\vert) + o(t^3), \endaligned\]] @xmath110 0.2 cm _ proof. _ applying the inverse fourier transformation @xmath111 to both sides of ([eq6]), we have : @xmath112(\bold x) \\ & \qquad\quad + \mathcal f_{\boldsymbol\alpha}^{-1 } \biggl [\frac{\lambda}{c^2 t \vert\boldsymbol\alpha\vert^2 } \biggl (\sin{(ct\vert\boldsymbol\alpha\vert) } \text{si}(2ct\vert\boldsymbol\alpha\vert) + \cos{(ct\vert\boldsymbol\alpha\vert) } \text{ci}(2ct\vert\boldsymbol\alpha\vert) \biggr) \biggr](\bold x) \\ & \qquad\quad + \mathcal f_{\boldsymbol\alpha}^{-1 } \biggl [\frac{\lambda^2 t}{c\vert\boldsymbol\alpha\vert } j_1(ct\vert\boldsymbol\alpha\vert) \biggr](\bold x) \\ & \qquad\quad + \mathcal f_{\boldsymbol\alpha}^{-1 } \biggl [\frac{\lambda^3 t}{2(c\vert\boldsymbol\alpha\vert)^2 } \left (\frac{\sin{(ct\vert\boldsymbol\alpha\vert)}}{ct\vert\boldsymbol\alpha\vert } - \cos{(ct\vert\boldsymbol\alpha\vert) } \right) \biggr](\bold x) \biggr\ } + o(t^3). \endaligned\]] note that here we have used the fact that, due to the continuity of the inverse fourier transformation, the asymptotic formula @xmath113(\bold x) = o(t^3)$] holds. let us evaluate separately the inverse fourier transforms on the right - hand side of ([dens2]). the first one is well known (see @xcite) : @xmath114(\bold x) = \frac{1}{4\pi (ct)^2 } \ ; \delta(c^2t^2-\vert\bold x\vert^2)\]] that is the uniform density concentrated on the surface of the sphere @xmath115 of radius @xmath21 centred at the origin @xmath11. the second fourier transform on the right - hand side of ([dens2]) is also well known (see (*??? *, the theorem) or (*??? * formulas (3.11) and (3.12))) : @xmath116(\bold x) \\ & \hskip 4 cm = \frac{\lambda}{4\pi c^2 t \vert\bold x\vert } \ln\left (\frac{ct+\vert\bold x\vert}{ct-\vert\bold x\vert } \right) \ ; \theta(ct-\vert\bold x\vert). \endaligned\]] applying the hankel inversion formula, we have for the third fourier transform on the right - hand side of ([dens2]) : @xmath117(\bold x) = \frac{\lambda^2 t}{c } \ ; (2\pi)^{-3/2 } \vert\bold x\vert^{-1/2 } \int_0^{\infty } j_{1/2}(\vert\bold x\vert \xi) \ ; \xi^{3/2 } \ ; \xi^{-1 } j_1(ct\xi) \ ; d\xi.\]] taking into account that @xmath118 and applying (*??? * formula 2.12.15(2)), we have : @xmath119(\bold x) & = \frac{\lambda^2 t}{2\pi^2 c \vert\bold x\vert } \int_0^{\infty } \sin{(\vert\bold x\vert \xi) } \ ; j_1(ct\xi) \ ; d\xi \\ & = \frac{\lambda^2 t}{2\pi^2 c \vert\bold x\vert } \ ; (c^2t^2-\vert\bold x\vert^2)^{-1/2 } \ ; \left (\frac{\vert\bold x\vert}{ct } \right) \ ; \theta(ct-\vert\bold x\vert)\\ & = \frac{\lambda^2}{2\pi^2 c^2 \ ; \sqrt{c^2t^2-\vert\bold x\vert^2 } } \ ; \theta(ct-\vert\bold x\vert). \endaligned\]] this is a fairly unexpected result showing that the conditional density @xmath120 corresponding to two changes of direction has an infinite discontinuity on the border of the three - dimensional ball @xmath33. this property is similar to that of the conditional density @xmath121 corresponding to the single change of direction (for the respective joint density see ([dens4])). applying the hankel inversion formula and taking into account ([bessin]), we have for the fourth term on the right - hand side of ([dens2]) : @xmath122(\bold x) \\ & = \frac{\lambda^3 \ ; \sqrt{\pi } \ ; t^{3/2}}{(2 c)^{3/2 } } \ ; (2\pi)^{-3/2 } \vert\bold x\vert^{-1/2 } \int_0^{\infty } j_{1/2}(\vert\bold x\vert \xi) \ ; \xi^{3/2 } \ ; \xi^{-3/2 } j_{3/2}(ct\xi) \ ; d\xi \\ & = \frac{\lambda^3 \ ; \sqrt{2 } \ ; t^{3/2}}{8c^{3/2 } \ ; \pi\sqrt{\pi } \ ; \vert\bold x\vert } \int_0^{\infty } \xi^{-1/2 } \ ; \sin{(\vert\bold x\vert \xi) } \ ; j_{3/2}(ct\xi) \ ; d\xi. \endaligned\]] using (*??? * formula 6.699(1)), we obtain : @xmath123(\bold x) \\ & = \frac{\lambda^3 \ ; \sqrt{2 } \ ; t^{3/2}}{8c^{3/2 } \ ; \pi\sqrt{\pi } \ ; \vert\bold x\vert } \ ; \frac{2^{-1/2 } \ ; \sqrt{\pi } \ ; \vert\bold x\vert \ ; (ct)^{-3/2}}{\gamma(1) } \ ; \theta(ct-\vert\bold x\vert) \\ & = \frac{\lambda^3}{8\pi c^3 } \ ; \theta(ct-\vert\bold x\vert). \endaligned\]] substituting now ([dens3]), ([dens4]), ([dens5]) and ([dens6]) into ([dens2]) we arrive at ([dens1]). the theorem is proved. @xmath81 ) at instant @xmath124 @xmath125 (for @xmath126) on the interval @xmath127_,width=377,height=302] -1 cm the shape of the absolutely continuous part of density ([dens1]) at time instant @xmath124 (for @xmath128) on the interval @xmath127 is plotted in fig. the error in these calculations does not exceed 0.001. we see that the density increases slowly as the distance @xmath129 from the origin @xmath11 grows, while near the border this growth becomes explosive. from this fact it follows that, for small time @xmath29, the greater part of the density is concentrated outside the neighbourhood of the origin @xmath11 and this feature of the three - dimensional markov random flight is quite similar to that of its two - dimensional counterpart. the infinite discontinuity of the density on the border @xmath130 is also similar to the analogous property of the two - dimensional markov random flight (see, for comparison, (*??? * formula (20) and figure 2 therein)). note that density ([dens1]) is continuous at the origin, as it must be. _ remark 2. _ using ([dens1]), we can derive an asymptotic formula, as @xmath5, for the probability of being in a subball @xmath131 of some radius @xmath132 centred at the origin @xmath11. applying (*??? * formula 4.642) and (*??? * formula 1.513(1)), we have : @xmath133 this series can be expressed through the special lerch @xmath134-function. applying again (*??? * formula 4.642), we get : @xmath135 where we have used the easily checked equality : @xmath136 then, by integrating the absolutely continuous part of ([dens1]) over the ball @xmath137 and taking into account ([dens7]) and ([dens8]). we have (for arbitrary @xmath132) : @xmath138 \\ & = e^{-\lambda t } \biggl [\frac{\lambda}{4\pi c^2 t } \ ; 8\pi r ct \sum_{k=1}^{\infty } \frac{1}{4k^2 - 1 } \ ; \left (\frac{r^2}{c^2t^2 } \right)^k \\ & \qquad\qquad + \frac{\lambda^2}{2\pi^2 c^2 } \biggl (2\pi (ct)^2 \arcsin\left (\frac{r}{ct } \right) - 2\pi r \sqrt{c^2t^2-r^2 } \biggr) + \frac{\lambda^3}{8\pi c^3 } \ ; \frac{4}{3 } \pi r^3 \biggr], \endaligned\]] and after some simple computations we finally arrive at the following asymptotic formula (for @xmath132) : @xmath139, \qquad t\to 0. \endaligned\]] Estimate of the accuracy the error in asymptotic formula ([dens1]) has the order @xmath6. this means that, for small @xmath29, this formula yields a fairly good accuracy. to estimate it, let us integrate the function in square brackets of ([dens1]) over the ball @xmath33. for the first term in square brackets of ([dens1]) we have : @xmath140 because the second integrand is the conditional density corresponding to the single change of direction (see (*??? * the theorem) or (*??? * formula (3.12))) and, therefore, the second integral is equal to 1. applying (*??? * formula 4.642), we have for the second term in square brackets of ([dens1]) : @xmath141 for the third term in square brackets of ([dens1]) we get : @xmath142 hence, in view of ([est1]), ([est2]) and ([est3]), the integral of the absolutely continuous part in asymptotic formula ([dens1]) is : @xmath143 dx_1 dx_2 dx_3 \\ & = e^{-\lambda t } \left (\lambda t + \frac{\lambda^2t^2}{2 } + \frac{\lambda^3 t^3}{6 } \right). \endaligned\]] note that ([est4]) can also be obtained by passing to the limit, as @xmath144, in asymptotic formula ([dens9]). on the other hand, according to ([struc1]) and ([densac]), the integral of the absolutely continuous part of the transition density of the three - dimensional markov random flight @xmath4 is @xmath145 the difference between the approximating function @xmath146 and the exact function @xmath147 given by ([est4]) and ([est5]) enables us to estimate the value of the probability generated by all the terms of the density aggregated in the term @xmath6 of asymptotic relation ([dens1]). the shapes of functions @xmath147 and @xmath146 on the time interval @xmath148 for the values of the intensity of switchings @xmath149 are presented in figures 2 and 3. + + + + we see that, for @xmath150, the function @xmath146 yields a very good coincidence with function @xmath147 on the subinterval @xmath151 (fig. 2 (left)), while for @xmath152 (fig. 2 (right)) such coincidence is good only on the subinterval @xmath153. the same phenomenon is also clearly seen in figure 3 where, for @xmath154, the function @xmath146 yields a very good coincidence with function @xmath147 on the subinterval @xmath155 (fig. 3 (left)), while for @xmath156 such good coincidence takes place only on the subinterval @xmath157 (fig. 3 (right)). thus, we can conclude that the greater is the intensity of switchings @xmath7, the shorter is the subinterval of coincidence. this fact can easily be explained. really, the greater is the intensity of switchings @xmath7, the shorter is the time interval, on which no more than three changes of directions can occur with big probability. this means that, for increasing @xmath7, the asymptotic formula ([dens1]) yields a good accuracy on more and more small time intervals. however, for arbitrary fixed @xmath7, there exists some @xmath158 such that formula ([dens1]) yields a good accuracy on the time interval @xmath159 and the error of this approximation does not exceed @xmath160. this is the essence of the asymptotic formula ([dens1]). * appendices * in the following appendices we establish some lemmas that have been used in the proofs of the above theorems. note that some of them are of a separate mathematical interest because no similar results can be found in the mathematical handbooks. Auxiliary lemma * lemma a1. * _ for arbitrary integer @xmath161 and for arbitrary real @xmath162, the following formula holds : _ @xmath163 @xmath164 0.2 cm _ proof. _ using the well - known relations for pochhammer symbol @xmath165 and the formula for euler gamma - function @xmath166 we can easily check that the sum on the left - hand side of ([appa1]) is @xmath167 where @xmath168 is the generalized hypergeometric function. according to (* item 7.4.4, page 539, formula 88) @xmath169 substituting this into ([appa3]), we obtain ([appa1]). the lemma is proved. Powers of the inverse tangent function in this appendix we derive series representations for some powers of the inverse tangent function that have been used in the proofs of the above theorems. moreover, these results are of a more general mathematical interest because, to the best of the author s knowledge, there are no series representations, similar to ([appb2]), ([appb4]) and ([appb6]) (see below), in mathematical handbooks, including @xcite, @xcite, @xcite. @xmath184 substituting these coefficients into ([appb3]) we obtain ([appb2]). the uniform convergence of the series in formula ([appb2]) can be established similarly to that of lemma b1. this completes the proof of the lemma. @xmath81 * lemma b3. * _ for arbitrary @xmath185, the following series representation holds : _ @xmath186 _ where _ @xmath187 _ is the generalized hypergeometric function. the series in _ ([appb4]) _ is convergent uniformly in @xmath172. _ 0.2 cm _ proof. _ from ([appb1]) and ([appb2]) it follows that @xmath188 where the coefficients @xmath55 are given by @xmath189 applying ([apa3]), ([appa2]) and the formula @xmath190 after some simple computations, we arrive at the relation @xmath191 substituting these coefficients into ([appb5]) we obtain ([appb4]). the lemma is proved. @xmath81 * lemma b4. * _ for arbitrary @xmath177, the following series representation holds : _ @xmath192 _ where the coefficients @xmath55 are given by the formula _ @xmath193 _ the series in _ ([appb6]) _ is convergent uniformly in @xmath172. _ _ proof. _ according to lemma b2, we have : @xmath194 where the coefficients @xmath195 are : @xmath196\\ & = \frac{2}{k+2 } \sum_{l=0}^k \frac{l! \ ; (k - l)!}{(l+1) \ ; \gamma\left (l+\frac{3}{2 } \right) \ ; \gamma\left (k - l+\frac{3}{2 } \right) }. \endaligned\]] substituting this into ([appb7]), we get the statement of the lemma. @xmath81

we consider the markov random flight @xmath0 in the three - dimensional euclidean space @xmath1 with constant finite speed @xmath2 and the uniform choice of the initial and each new direction at random time instants that form a homogeneous poisson flow of rate @xmath3. series representations for the conditional characteristic functions of @xmath4 corresponding to two and three changes of direction, are obtained. based on these results, an asymptotic formula, as @xmath5, for the unconditional characteristic function of @xmath4 is derived. by inverting it , we obtain an asymptotic relation for the transition density of the process. we show that the error in this formula has the order @xmath6 and, therefore, it gives a good approximation on small time intervals whose lengths depend on @xmath7. estimate of the accuracy of the approximation is analysed. * asymptotic relation for the transition density + of the three - dimensional markov random flight + on small time intervals * alexander d. kolesnik + institute of mathematics and computer science + academy street 5, kishinev 2028, moldova + e - mail : kolesnik@math.md 0.2 cm 0.1 cm _ keywords : _ markov random flight, persistent random walk, conditional density, fourier transform, characteristic function, asymptotic relation, transition density, small time intervals 0.2 cm _ ams 2010 subject classification : _ 60k35, 60k99, 60j60, 60j65, 82c41, 82c70

1604.08362

Introduction multiscale dynamics is present in many phenomena, e.g., turbulence @xcite, finance @xcite, geosciences @xcite, etc, to quote a few. it has been found in many multiscale dynamics systems that the self - similarity is broken, in which the concept of multiscaling or multifractal is relevant @xcite. this is characterized conventionally by using the structure - functions (sfs), i.e., @xmath10, in which @xmath11 is an increment with separation scale @xmath2. note that for the self - similarity process, e.g., fractional brownian motion (fbm), the measured @xmath0 is linear with @xmath12. while for the multifractal process, e.g., turbulent velocity, it is usually convex with @xmath12. other methods are available to extract the scaling exponent. for example, wavelet based methodologies, (e.g., wavelet leaders, wavelet transform modulus maxima @xcite), hilbert - based method @xcite, or the scaling analysis of probability density function of velocity increments @xcite, to name a few. each method has its owner advantages and shortcomings. for example, the classical sfs is found to mix information of the large- (resp. known as infrared effect) and small - scale (resp. known as ultraviolet effect) structures @xcite. the corresponding scaling exponent @xmath0 is thus biased when a large energetic structure is present @xcite. previously the influence of the large - scale structure has been considered extensively by several authors @xcite. for example, praskvosky et al., @xcite found strong correlations between the large scales and the velocity sfs at all length scales. sreenivasan & stolovitzky @xcite observed that the inertial range of the sfs conditioned on the large scale velocity show a strong dependence. huang et al., @xcite showed analytically that the influence of the large - scale structure could be as large as two decades down to the small scales. blum et al., @xcite studied experimentally the nonuniversal large - scale structure by considering both conditional eulerian and lagrangian sfs. they found that both sfs depend on the strength of large - scale structures at all scales. in their study, the large - scale structure velocity is defined as two - point average, i.e., @xmath13/2 $], in which @xmath14 is the vertical velocity in their experiment apparatus. note that they conditioned sfs on different intensity of @xmath15. later, blum et al., @xcite investigated systematically the large - scale structure conditioned sfs for various turbulent flows. they confirmed that in different turbulent flows the conditioned sfs depends strongly on large - scale structures at all scales. in this paper, a detrended structure - function (dsf) method is proposed to extract scaling exponents @xmath0. this is accomplished by removing a @xmath1st - order polynomial within a window size @xmath2 before calculating the velocity increment. this procedure is designated as detrending analysis (da). by doing so, scales larger than @xmath2, i.e., @xmath3, are expected to be removed or constrained. hence, the da acts as a high - pass filter in physical domain. meanwhile, the intermittency is still retained. a velocity increment @xmath16 is then defined within the window size @xmath2. a @xmath12th - order moment of @xmath16 is introduced as @xmath12th - order dsf. the dsf is first validated by using a synthesized fractional brownian motion (fbm) and a lognormal process with an intermittent parameter @xmath17 respectively for mono - fractal and multifractal processes. it is found that dsfs provide comparable scaling exponents @xmath0 and singularity spectra @xmath4 with the ones provided by the original sfs. when applying to a turbulent velocity with a reynolds number @xmath18, the @xmath19rd - order dsf shows a clear inertial range @xmath7, which is consistent with the one predicted by the fourier power spectrum @xmath20, e.g., @xmath8. moreover, a compensated height of the @xmath19rd - order dsf is @xmath21. this value is consistent with the famous kolmogorov four - fifth law. the directly measured scaling exponents @xmath0 (resp. singularity spectrum @xmath4) agree very well with the lognormal model with an intermittent parameter @xmath9. due to the large - scale effect, known as infrared effect, the sfs are biased. note that the scaling exponents are extracted directly without resorting to the extended - self - similarity (ess) technique. the method is general and could be applied to different types of data, in which the multiscale and multifractal concepts are relevant. Detrending analysis and detrended structure-function we start here with a scaling process @xmath22, which has a power - law fourier spectrum, i.e., @xmath23 in which @xmath24 is the scaling exponent of @xmath25. the parseval s theorem states the following relation, i.e., @xmath26 in which @xmath27 is ensemble average, @xmath25 is the fourier power spectrum of @xmath22 @xcite. we first divide the given @xmath22 into @xmath28 segments with a length @xmath2 each. a @xmath29th - order detrending of the @xmath30th segment is defined as, i.e., @xmath31 in which @xmath32 is a @xmath29th - order polynomial fitting of the @xmath33. we consider below only for the first - order detrending, i.e., @xmath34. to obtain a detrended signal, i.e., @xmath35 $], a linear trend is removed within a window size @xmath2. ideally, scales larger than @xmath2, i.e., @xmath36 are removed or constrained from the original data @xmath22. this implies that the da procedure is a high - pass filter in the physical domain. the kinetic energy of @xmath37 is related directly with its fourier power spectrum, i.e., @xmath38 in which @xmath39 and @xmath40 is the fourier power spectrum of @xmath37. this illustrates again that the da procedure acts a high - pass filter, in which the lower fourier modes @xmath41 (resp. @xmath36) are expected to be removed or constrained. for a scaling process, i.e., @xmath42, it leads a power - law behavior, i.e., @xmath43 the physical meaning of @xmath44 is quite clear. it represents a cumulative energy over the fourier wavenumber band @xmath45 $] (resp. scale range @xmath46 $]). we emphasize here again that the da acts as a high - pass filter in physical domain and the intermittency nature of @xmath22 is still retained. the above mentioned detrending analysis can remove / constrain the large - scale influence, known as infrared effect. this could be utilized to redefine the sf to remove / constrain the large - scale structure effect as following. after the da procedure,, the velocity increment can be defined within a window size @xmath2 as, i.e., @xmath47 in which @xmath30 represents for the @xmath30th segment. we will show in the next subsection why we define an increment with a half width of the window size. a @xmath12th - order dsf is then defined as, i.e., @xmath48 for a scaling process, we expect a power - law behavior, i.e., @xmath49 in which the scaling exponent @xmath0 is comparable with the one provided by the original sfs. to access negative orders of @xmath12 (resp. the right part of the singularity spectrum @xmath4, see definition below), the dsfs can be redefined as, i.e., @xmath50 in which @xmath51 is local average for the @xmath30th segment. a power - law behavior is expected, i.e., @xmath52. it is found experimentally that when @xmath53, eqs. ([eq : dsf]) and ([eq : rdsf]) provide the same scaling exponents @xmath0. in the following we do not discriminate these two definitions for dsfs. for different methods : structure - function @xmath54 (dashed line), first - order detrending analysis @xmath55 (thin solid line), and the detrended structure - function @xmath56 (thick solid line). the detrended scale @xmath2 is demonstrated by a vertical solid line with @xmath39. ideally, scales larger than @xmath2, i.e., @xmath57 (resp. @xmath41) are expected to be removed after the detrending process.] to understand better the filter property of the detrending procedure and dsfs, we introduce here a weight function @xmath58, i.e., @xmath59 in which @xmath25 is the fourier power spectrum of @xmath22, and @xmath60 is a second - order moment, which could be one of @xmath44 or @xmath61, or @xmath62, respectively. the weight function @xmath58 characterizes the contribution of the fourier component to the corresponding second - order moment. note that an integral constant is neglected in the eq. ([eq : weight]). for the second - order sfs, one has the following weight function @xcite, i.e., @xmath63 for a scaling process, one usually has a fast decaying fourier spectrum, i.e. @xmath42 with @xmath64. hence, the contribution from small - scale (resp. high wavenumber fourier mode) is decreasing. the sfs might be more influenced by the large - scale part for large values of @xmath24 @xcite. for the detrended data, the corresponding weight function is ideally to be as the following, i.e., @xmath65 the dsfs (resp. the combination of the da and sf) have a weight function, i.e., @xmath66 comparing with the original sfs, the dsfs defined here can remove / constrain the large - scale effect. figure [fig : weight] shows the corresponding @xmath58 for the sf, detrending analysis, and dsf, respectively. the detrended scale @xmath2 is illustrated by a vertical line, i.e., @xmath39. we note here that with the definition of eq. ([eq : dvi]), @xmath61 provides a better compatible interpretation with the fourier power spectrum @xmath25 since we have @xmath67. this is the main reason why we define the velocity increment with the half size of the window width @xmath2. we provide some comments on eq. ([eq : weight]). the above argument is exactly valid for linear and stationary processes. in reality, the data are always nonlinear and nonstationary for some reasons, see more discussion in ref. @xcite. therefore, eq. ([eq : weight]) holds approximately for real data. another comment has to be emphasized here for the detrending procedure. several approaches might be applied to remove the trend @xcite. however, the trend might be linear or nonlinear. therefore, different detrending approaches might provide different performances. in the present study, we only consider the @xmath1st - order polynomial detrending procedure, which is efficient for many types of data. for fractional brownian motion with @xmath68 on the range @xmath69. the inset shows the singularity spectrum @xmath70 on the range @xmath71. the errorbar is the standard deviation estimated from 100 realizations. ideally, one should have @xmath72 and @xmath73. both methods provide the same @xmath74 and @xmath4 and statistical error.] for the lognormal process with an intermittent parameter @xmath17. the errorbar is the standard deviation from the 100 realizations. the theoretical singularity curve is illustrated by a solid line. both estimators provide the same singularity spectra @xmath4 and statistical error.] Numerical validation we first consider here the fractional brownian motion as a typical mono - scaling process. fbm is a gaussian self - similar process with a normal distribution increment, which is characterized by @xmath75, namely hurst number @xmath76 @xcite. a wood - chan algorithm is used to synthesize the fbm with a hurst number @xmath68. we perform 100 realizations with a data length @xmath77 points each. power - law behavior is observed on a large - range of scales for @xmath78. the corresponding singularity spectrum is, i.e., @xmath79 ideally, one should have a single point of singularity spectrum with @xmath72 and @xmath73. however, in practice, the measured singularity spectrum @xmath4 is always lying in a narrow band. figure [fig : fbm] shows the measured singularity spectrum @xmath4 for sfs (@xmath80) and dsfs (@xmath81) for @xmath78, in which the inset shows the singularity spectra @xmath4 estimated on the range @xmath82. visually, both estimators provide the same @xmath4 and the same statistical error, which is defined as the standard deviation from different realizations. we now consider a multifractal random walk with a lognormal statistics @xcite. a multiplicative discrete cascade process with a lognormal statistics is performed to simulate a multifractal measure @xmath83. the larger scale corresponds to a unique cell of size @xmath84, where @xmath85 is the largest scale considered and @xmath86 is a dimensional scale ratio. in practice for a discrete model, this ratio is often taken as @xmath87 @xcite. the next scale involved corresponds to @xmath88 cells, each of size @xmath89. this is iterated and at step @xmath90 (@xmath91) @xmath92 cells are retrieved. finally, at each point the multifractal measure @xmath83 is as the product of @xmath12 cascade random variables, i.e., @xmath93 where @xmath94 is the random variable corresponding to position @xmath95 and level @xmath28 in the cascade @xcite. following the multifractal random walk idea @xcite, a nonstationary multifractal time series can be synthesized as, i.e., @xmath96 where @xmath97 is brownian motion. taking a lognormal statistic for @xmath98, the scaling exponent @xmath0 for the sfs, i.e., @xmath99, is written as, @xmath100 where @xmath101 is the intermittency parameter (@xmath102) characterizing the lognormal multifractal cascade @xcite. synthetic multifractal time series are generated following eq. ([eq : multitime]). an intermittent parameter @xmath17 is chosen for @xmath103 levels each, corresponding to a data length @xmath104 points each. a total of 100 realizations are performed. the statistical error is then measured as the standard deviation from these realizations. figure [fig : wfbm] shows the corresponding measured singularity spectra @xmath4, in which the theoretical value is illustrated by a solid line. graphically, the theoretical singularity spectra @xmath4 are recovered by both estimators. statistical error are again found to be the same for both estimators. we would like to provide some comments on the performance of these two estimators. for the synthesized processes, they have the same performance since there is no intrinsic structure in these synthesized data. but for the real data, as we mentioned above, they possess nonstationary and nonlinear structures @xcite. therefore, as shown in below, they might have different performance. Application to turbulent velocity and @xmath105 from experimental homogeneous and nearly isotropic turbulent flow. they are respectively 3rd - order sfs with (@xmath80) and without (@xmath81) absolute value, and 3rd - order dsfs with (@xmath106) and without (@xmath107) absolute value. the horizontal solid line indicates the kolmogorovs four - fifth law. an observed plateau for @xmath108 indicates an inertial range on the range @xmath7, corresponding to a wavenumber range @xmath8. roughly speaking, a plateau for @xmath109 indicates an inertial range on the range @xmath110. the height of the inertial range are respectively @xmath111 (@xmath81), @xmath112 (@xmath80), @xmath113 (@xmath107) and @xmath21 (@xmath106), in which the statistical error is the standard deviation obtained from the inertial range. note that the inertial range are @xmath110 for the sfs and @xmath7 for the dsfs. the corresponding scaling exponents @xmath114 are @xmath115, @xmath116, @xmath117 and @xmath118. the statistical error is the 95% fitting confidence on the inertial range.] . the errorbar is the standard deviation from 120 realizations. the inset shows the corresponding scaling exponents @xmath0. for comparison, the lognormal model with an intermittent parameter @xmath9 is illustrated by a solid line.] we consider here a velocity database obtained from a high reynolds number wind tunnel experiment in the johns - hopkins university with reynolds number @xmath18. an probe array with four x - type hot wire anemometry is used to record the velocity with a sampling wavenumber of @xmath119khz at streamwise direction @xmath120, in which @xmath121 is the size of the active grid. these probes are placed in the middle height and along the center line of the wind tunnel to record the turbulent velocity simultaneously for a duration of 30 second. the measurement is then repeated for 30 times. finally, we have @xmath122 data points (number of measurements @xmath123 number of probes @xmath123 duration time @xmath123 sampling wavenumber). therefore, there are 120 realizations (number of measurements @xmath123 number of probes). the fourier power spectrum @xmath20 of the longitudinal velocity reveals a nearly two decades inertial range on the wavenumber range @xmath8 with a scaling exponent @xmath124, see ref. this corresponds to time scales @xmath7. here @xmath125 is the kolmogorov scale. note that we convert our results into spatial space by applying the taylors frozen hypothesis @xcite. more detail about this database can be found in ref.@xcite. to determine the inertial range in real space, we plot the measured compensated 3rd - order moments in fig.[fig : third] for the sfs (@xmath126 with (@xmath80) and without (@xmath81) absolute value), dsfs (@xmath105 with (@xmath106) and without (@xmath107) absolute value), respectively. a horizontal solid line indicates the kolmogorovs four - fifth law. a plateau is observed for @xmath105 on the range @xmath7, which agrees very well with the inertial range predicted by @xmath20, i.e., on the range @xmath8. the corresponding height and scaling exponent are @xmath21 with absolute value (resp. @xmath113 without absolute value) and @xmath127 (resp. @xmath128), respectively. the statistical error is the standard deviation obtained from the range @xmath7. note that the kolmogorovs four - fifth law indicates a linear relation @xmath129. it is interesting to note that, despite of the sign, we have @xmath130 on nearly two - decade scales. for comparison, the 3rd - order sfs are also shown. roughly speaking, a plateau is observed on the range @xmath110. this inertial range is shorter than the one predicted by the fourier analysis or dsfs, which is now understood as the large - scale influence. the corresponding height and scaling exponent are @xmath111 without absolute value (resp. @xmath112 with absolute value) and @xmath115 (resp. @xmath131). therefore, the dsfs provide a better indicator of the inertial range since it removes / constrains the large - scale influence. we therefore estimate the scaling exponents for the @xmath132 on the range @xmath7 for @xmath69 directly without resorting to the extended self - similarity technique @xcite. for the sfs, we calculate the scaling exponents @xmath0 on the range @xmath110 for @xmath82 directly. figure [fig : singularity] shows the measured singularity spectra @xmath4 for @xmath78, in which the errorbar is a standard deviation from 120 realizations. the inset shows the corresponding scaling exponents @xmath0. for comparison, the lognormal model @xmath133 with an intermittent parameter @xmath9 is shown as a solid line. visually, the dsfs curve fully recovers the lognormal curve not only on the left part (resp. @xmath134) but also on the right part (resp. @xmath135). due to the large - scale contamination, the sfs underestimates the scaling exponents @xmath0 when @xmath134 @xcite. this leads an overestimation of the left part of singularity spectrum @xmath4 (see @xmath80 in fig.[fig : singularity]). however, if one resorts the ess algorithm when measuring the sf scaling exponent @xmath0, the corresponding singularity spectrum @xmath4 is then horizontal shifted to the theoretical curve. this has been interpreted as that the ess technique suppresses the finite reynolds number effect. we show here that if one removes / constrains the effect of large - scale motions, one can retrieve the scaling exponent @xmath0 (resp. singularity spectrum @xmath4) without resorting the ess technique. or in other words, the finite reynolds number effect manifests at large - scale motions, which is usually anisotropic too. Conclusion in this paper, we introduce a detrended structure - function analysis to remove / constrain the influence of large - scale motions, known as the infrared effect. in the first step of our proposal, the @xmath1st - order polynomial trend is removed within a window size @xmath2. by doing so, the scales larger than @xmath2, i.e., @xmath136, are expected to be removed / constrained. in the second step, a velocity increment is defined with a half of the window size. the dsf proposal is validated by the synthesized fractional brownian motion for the mono - fractal process and a lognormal random walk for the multifractal process. the numerical test shows that both sfs and dsfs estimators provide a comparable performance for synthesized processes without intrinsic structures. when applying to the turbulent velocity obtained from a high reynolds number wind tunnel experiment, the 3rd - order dsfs show a clearly inertial range on the range @xmath7 with a linear relation @xmath137. the inertial range provided by dsfs is consistent with the one predicted by the fourier power spectrum. note that, despite of the sign, the kolmogorovs four - fifth law is retrieved for the 3rd - order dsfs. the corresponding 3rd - order sfs are biased by the large - scale structures, known as the infrared effect. it shows a shorter inertial range and underestimate the 3rd - order scaling exponent @xmath114. the scaling exponents @xmath0 are then estimated directly without resorting to the ess technique. the corresponding singularity spectrum @xmath4 provided by the dsfs fully recovers the lognormal model with an intermittent parameter @xmath9 on the range @xmath69. however, the classical sfs overestimate the left part singularity spectrum @xmath4 (resp. underestimate the corresponding scaling exponents @xmath0) on the range @xmath138. this has been interpreted as finite reynolds number effect and can be corrected by using the ess technique. here, to our knowledge, we show for the first time that if one removes / constrains the influence of the large - scale structures, one can recover the lognormal model without resorting to the ess technique. the method we proposed here is general and applicable to other complex dynamical systems, in which the multiscale statistics are relevant. it should be also applied systematically to more turbulent velocity databases with different reynolds numbers to see whether the finite reynolds number effect manifests on large - scale motions as well as we show for high reynolds number turbulent flows. Acknowledgements this work is sponsored by the national natural science foundation of china under grant (nos. 11072139, 11032007,11161160554, 11272196, 11202122 and 11332006), pu jiang project of shanghai (no. 12pj1403500), innovative program of shanghai municipal education commission (no. 11zz87) and the shanghai program for innovative research team in universities. y.h. thanks prof. schmitt for useful comments and suggestions. we thank prof. meneveau for sharing his experimental velocity database, which is available for download at c. meneveau s web page : http://www.me.jhu.edu/meneveau/datasets.html. we thank the two anonymous referees for their useful comments and suggestions. f. schmitt, y. huang, z. lu, y. liu, and n. fernandez, _ analysis of velocity fluctuations and their intermittency properties in the surf zone using empirical mode decomposition _, j. mar. 77 (2009), pp. 473481. j. muzy, e. bacry, and a. arneodo, _ multifractal formalism for fractal signals : the structure - function approach versus the wavelet - transform modulus - maxima method _, e 47 (1993), pp. 875884. y. huang, f.g. schmitt, j.p. hermand, y. gagne, z. lu, and y. liu, _ arbitrary - order hilbert spectral analysis for time series possessing scaling statistics : comparison study with detrended fluctuation analysis and wavelet leaders _, phys. e 84 (2011), p. 016208. y. huang, f. schmitt, q. zhou, x. qiu, x. shang, z. lu, and y. liu, _ scaling of maximum probability density functions of velocity and temperature increments in turbulent systems _, phys. fluids 23 (2011), p. 125101. praskovsky, e.b. gledzer, m.y. karyakin, and y. zhou, _ the sweeping decorrelation hypothesis and energy - inertial scale interaction in high reynolds number flows _, j. fluid mech. 248 (1993), p. 493. blum, g.p. bewley, e. bodenschatz, m. gibert, a. gylfason, l. mydlarski, g.a. voth, h. xu, and p. yeung, _ signatures of non - universal large scales in conditional structure functions from various turbulent flows _, new j. phys. 13 (2011), p. 113020. n. huang, z. shen, s. long, m. wu, h. shih, q. zheng, n. yen, c. tung, and h. liu, _ the empirical mode decomposition and the hilbert spectrum for nonlinear and non - stationary time series analysis london, ser. a 454 (1998), pp. 903995.

the classical structure - function (sf) method in fully developed turbulence or for scaling processes in general is influenced by large - scale energetic structures, known as infrared effect. therefore, the extracted scaling exponents @xmath0 might be biased due to this effect. in this paper, a detrended structure - function (dsf) method is proposed to extract scaling exponents by constraining the influence of large - scale structures. this is accomplished by removing a @xmath1st - order polynomial fitting within a window size @xmath2 before calculating the velocity increment. by doing so, the scales larger than @xmath2, i.e., @xmath3, are expected to be removed or constrained. the detrending process is equivalent to be a high - pass filter in physical domain. meanwhile the intermittency nature is retained. we first validate the dsf method by using a synthesized fractional brownian motion for mono - fractal processes and a lognormal process for multifractal random walk processes. the numerical results show comparable scaling exponents @xmath0 and singularity spectra @xmath4 for the original sfs and dsfs. when applying the dsf to a turbulent velocity obtained from a high reynolds number wind tunnel experiment with @xmath5, the 3rd - order dsf demonstrates a clear inertial range with @xmath6 on the range @xmath7, corresponding to a wavenumber range @xmath8. this inertial range is consistent with the one predicted by the fourier power spectrum. the directly measured scaling exponents @xmath0 (resp. singularity spectrum @xmath4) agree very well with a lognormal model with an intermittent parameter @xmath9. due to large - scale effects, the results provided by the sfs are biased. the method proposed here is general and can be applied to different dynamics systems in which the concepts of multiscale and multifractal are relevant. fully developed turbulence ; intermittency ; detrended structure - function

1402.0371

Introduction the transformation, upon charge doping, of an antiferromagnetic (af) mott insulator into a superconducting (sc) metal and the role of af correlations in the appearance of superconductivity have challenged researchers since the discovery of high-@xmath12 superconductivity in cuprates. is the af order an indispensable component or a competitor for the high-@xmath12 phenomenon? in a prototype high-@xmath12 cuprate la@xmath6sr@xmath7cuo@xmath8, the long - range af order is destroyed by doped holes way before the superconductivity sets in @xcite, which has led to a general belief that the spin frustration is a prerequisite for metallic conduction and superconductivity. the destructive impact of static spin order on superconductivity was further supported by the observation of sc suppression at a peculiar 1/8 doping in la@xmath6ba@xmath7cuo@xmath8 @xcite. on the other hand, spin excitations are often suggested to provide glue for sc pairing, implying the ultimate importance of af correlations, be they static or dynamic. besides, the incompatibility of static af order and sc may be not necessarily a general feature of cuprates. in @xmath1ba@xmath2cu@xmath3o@xmath4 (@xmath1 is a rare - earth element), for instance, the long - range af order survives up to much higher doping levels than in la@xmath6sr@xmath7cuo@xmath8 @xcite, though the possibility of its coexistence with superconductivity still remains to be clarified. in strongly anisotropic high-@xmath12 cuprates, the @xmath0-axis charge transport appears to be remarkably sensitive to the spin ordering in cuo@xmath2 planes. in @xmath1ba@xmath2cu@xmath3o@xmath4 crystals, for example, the @xmath0-axis resistivity @xmath13 exhibits a steep increase at the nel temperature @xmath14 @xcite. even relatively weak modifications of the spin structure such as spin - flop or metamagnetic transitions result in surprisingly large changes by up to an order of magnitude in the @xmath0-axis resistivity of both hole - doped la@xmath6sr@xmath7cuo@xmath8 @xcite and electron - doped pr@xmath15la@xmath16ce@xmath17cuo@xmath8 @xcite and nd@xmath6ce@xmath17cuo@xmath8 crystals @xcite. this sensitivity of the interplane charge transport in cuprates to the spin order can be, and actually is, employed for tracing the evolution of the spin state with doping, temperature, or magnetic fields @xcite. while electrical resistivity measurements have proved to be a very convenient tool for mapping the magnetic phase diagrams in cuprates, their usage has an obvious limitation ; namely, they fail as the superconductivity sets in. because of this limitation, previous resistivity studies of @xmath1ba@xmath2cu@xmath3o@xmath4 crystals @xcite could not clarify whether the long - range af order vanishes by the onset of superconductivity, or extends further, intervening the sc region. it sounds tempting to employ strong magnetic fields to suppress the superconductivity and to use the @xmath0-axis resistivity technique of detecting the spin order in otherwise inaccessible regions of the phase diagram. in the present paper, we use this approach to study the very region of the af - sc transformation in luba@xmath2cu@xmath3o@xmath4 and yba@xmath2cu@xmath3o@xmath4 single crystals. Experiment @xmath1ba@xmath2cu@xmath3o@xmath4 single crystals with nonmagnetic rare - earth elements @xmath1@xmath5lu and y were grown by the flux method and their oxygen stoichiometry was tuned to the required level by high - temperature annealing with subsequent quenching @xcite. in order to ensure that no oxygen - enriched layer was formed at the crystal surface during the quenching process, one of the crystals was dissolved in acid in several steps ; resistivity measurements detected no considerable change in the sc transition upon the crystal s surface destruction. the @xmath0-axis resistivity @xmath13 was measured using the ac four - probe technique. to provide a homogeneous current flow along the @xmath0-axis, two current contacts were painted to almost completely cover the opposing @xmath18-faces of the crystal, while two voltage contacts were placed in small windows reserved in the current ones @xcite. the magnetoresistance (mr) was measured by sweeping temperature at fixed magnetic fields up to 16.5 t applied along the @xmath0 axis of the crystals. Results and discussion a representative @xmath13 curve obtained for a luba@xmath2cu@xmath3o@xmath4 single crystal with a doping level slightly lower than required for the onset of superconductivity is shown in fig. 1. in general, the @xmath0-axis resistivity in @xmath1ba@xmath2cu@xmath3o@xmath4 crystals of non - sc composition exhibits two peculiar features upon cooling below room temperature, both of which can be seen in fig. 1. the first one is a pronounced crossover at @xmath19 (@xmath20@xmath21@xmath22k for the particular composition in fig. 1), indicating a change with decreasing temperature of the dominating conductivity mechanism from some kind of thermally activated hopping to a coherent transport @xcite. it is worth noting that a similar coherent - incoherent crossover was observed in other layered oxides as well @xcite. the second feature is a sharp growth of the resistivity associated with the long - range af ordering @xcite. if the crystals were less homogeneous, the low - temperature resistivity upturn would be easy to confuse with a usual disorder - induced charge localization. however, this sharp resistivity anomaly with a characteristic negative peak in the derivative (inset in fig. 1) is definitely related to the spin ordering at the nel temperature @xmath14 : it has been traced from the parent compositions @xmath1ba@xmath2cu@xmath3o@xmath23 with well - known @xmath14 to avoid any doubt in its origin. , of a luba@xmath2cu@xmath3o@xmath24 single crystal. the sharp growth of the resistivity upon cooling below @xmath25@xmath26k is caused by the af ordering. inset : anomaly in the normalized derivative @xmath27 associated with the nel transition.,width=287] in carefully prepared crystals, the af transitions remain sharp for all compositions, including crystals with very low @xmath28@xmath29@xmath30k, that is @xmath3120 times lower than original @xmath32@xmath25@xmath33k in parent @xmath1ba@xmath2cu@xmath3o@xmath23. it is important to emphasize that the transitions at @xmath28@xmath29@xmath30k remain virtually as sharp as in undoped parent crystals even though spin freezing into a spin - glass state is usually expected for such low temperatures and high hole concentrations @xcite. moreover, the impact of the af ordering on @xmath13 does not weaken with decreasing nel temperature ; as can be seen in fig. 1, the resistivity of the crystal with @xmath28@xmath29@xmath34k increases by more than 50% upon cooling below @xmath14, while in crystals with @xmath28@xmath35@xmath36k the corresponding @xmath37 growth does not exceed 15 - 20% @xcite. what do these observations tell about the impact of doped holes on copper spins? apparently, the sharp af transitions are hard to reconcile with strongly frustrated spin states and disordered spin textures in cuo@xmath2 planes that are usually expected to emerge in cuprates with doping @xcite. besides, a strong frustration - induced reduction of the staggered magnetization @xmath38 required to account for the decrease in @xmath14 would necessarily diminish the impact of the interplane spin ordering on @xmath37, which again disagrees with observations. on the other hand, if the role of mobile doped holes is not to introduce a uniform spin frustration, but simply to break the long - range af order into two - dimensional domains in cuo@xmath2 planes, the observed behavior is easier to understand. in this case, it is the af domains in cuo@xmath2 planes that become the elementary magnetic units and the long - range af state should develop through ordering of their phases, which can occur rather abruptly. correspondingly, the @xmath14 evolution with doping should be governed by the decreasing af domain size, rather than @xmath38. in turn, the ordering of af domains whose local staggered magnetization does nor change appreciably with hole doping leaves room for large changes in the @xmath0-axis resistivity even at low @xmath14. the @xmath14 and @xmath12 values determined from the @xmath0-axis resistivity of @xmath1ba@xmath2cu@xmath3o@xmath4 crystals have been used to establish the doping - temperature phase diagram in fig. 2. a peculiarity of @xmath1ba@xmath2cu@xmath3o@xmath4 crystals is that their doping level is determined both by the oxygen content and by the degree of its ordering @xcite. for characterizing the doping level, we use therefore the in - plane conductivity @xmath39 instead of the oxygen content ; the former is a good measure of the hole density given that the hole mobility stays almost constant in the doping region under discussion @xcite. ba@xmath2cu@xmath3o@xmath4 (@xmath1 = lu, tm, y) crystals near the af - sc transformation. the af and sc transition temperatures, @xmath14 and @xmath12, are presented as a function of the in - plane conductivity @xmath39 which is a good measure of the hole density in the shown doping region @xcite. the nel temperature was determined either at the position of the jump (middle point) in the derivative @xmath40 (open circles), or at the position of the negative peak in @xmath40 (solid circles).,width=326] as can be seen in fig. 2, the long - range af order in @xmath1ba@xmath2cu@xmath3o@xmath4 appears to be much more stable than in la@xmath6sr@xmath7cuo@xmath8 where it vanishes well in advance before the onset of superconductivity. in @xmath1ba@xmath2cu@xmath3o@xmath4, the af phase boundary flattens upon approaching the sc compositions and hits the sc region, crossing the @xmath12 line at @xmath25@xmath41k. the observed overlap of the sc and af regions is very close to the area where @xmath42sr studies of yba@xmath2cu@xmath3o@xmath4 ceramics revealed the coexistence of superconductivity with spontaneous static magnetism @xcite. given that the @xmath0-axis resistivity studied here is sensitive to the _ interlayer _ spin ordering, the static magnetism detected by @xmath42sr @xcite should in fact be related to the three - dimensional af order. the superconductivity in @xmath1ba@xmath2cu@xmath3o@xmath4 thus develops directly from the af - ordered state without any intervening paramagnetic or spin - glass region. cu@xmath3o@xmath4 single crystal for two oxygen concentrations near the af - sc transformation. for the superconducting composition @xmath43, the data were taken at several magnetic fields from zero up to 16.5 t applied along the @xmath0 axis. the sharp upturn in the resistivity associated with the nel transition shows up as the superconductivity is suppressed with the magnetic field.,width=326] according to the established phase diagram (fig. 2), an increase of the hole density in cuo@xmath2 planes by @xmath441% per cu (from @xmath215% to @xmath256%, assuming the onset of superconductivity at @xmath255% doping) turns an af @xmath1ba@xmath2cu@xmath3o@xmath4 crystal without any sign of superconductivity into a bulk superconductor with @xmath9@xmath44@xmath45k. what happens with the af order upon entering the sc region, does it vanish abruptly? zero - field @xmath13 curves measured on the same luba@xmath2cu@xmath3o@xmath4 crystal for two hole doping levels (that differ by @xmath250.5 - 0.6% per cu) indeed demonstrate a switch from an af state with @xmath13 sharply growing below @xmath14 to a sc state (fig. however, when the superconductivity is suppressed with the magnetic field @xmath46@xmath47@xmath48, the steep increase in @xmath37 associated with the af ordering is recovered back (fig. 3). moreover, the recovered nel temperature is merely several kelvin lower than for the non - sc composition (upper curve in fig. 3) and the resistivity increase is not reduced appreciably either. as long as the sc in luba@xmath2cu@xmath3o@xmath4 and yba@xmath2cu@xmath3o@xmath4 crystals is weak enough to be killed by the 16.5-t field, the unveiled @xmath13 curves keep demonstrating the anomalous growth below 15 - 20@xmath49k associated with the nel transition. this behavior indicates that, at least when superconductivity is suppressed with magnetic fields, the af order extends to considerably higher doping levels than the sc onset. consequently, at zero magnetic field the af and sc orders either coexist with each other in a certain range of doping, or the af order is frustrated in the sc state but revives as the superconductivity is destroyed with the magnetic field. a switching between the af and sc orders was indeed suggested based on early @xmath42sr studies @xcite, yet no further proofs were collected. the close location and even overlapping of the af and sc orders on the phase diagram raise another question of whether the af and sc orders reside in nanoscopically separated phases in cuo@xmath2 planes or coexist on the unit - cell scale, which calls for local microscopic tools to be clarified. j. m. tranquada, a. h. moudden, a. i. goldman, p. zolliker, d. e. cox, g. shirane, s. k. sinha, d. vaknin, d. c. johnston, m. s. alvarez, a. j. jacobson, j. t. lewandowski, and j. m. newsam, phys. rev. b * 38 *, 2477 (1988).

the remarkable sensitivity of the @xmath0-axis resistivity and magnetoresistance in cuprates to the spin ordering is used to clarify the doping - induced transformation from an antiferromagnetic (af) insulator to a superconducting (sc) metal in @xmath1ba@xmath2cu@xmath3o@xmath4 (@xmath1@xmath5lu, y) single crystals. the established phase diagram demonstrates that the af and sc regions apparently overlap : the superconductivity in @xmath1ba@xmath2cu@xmath3o@xmath4, in contrast to la@xmath6sr@xmath7cuo@xmath8, sets in before the long - range af order is completely destroyed by hole doping. magnetoresistance measurements of superconducting crystals with low @xmath9@xmath10@xmath11k give a clear view of the magnetic - field induced superconductivity suppression and recovery of the long - range af state. what still remains to be understood is whether the af order actually persists in the sc state or just revives when the superconductivity is suppressed, and, in the former case, whether the antiferromagnetism and superconductivity reside in nanoscopically separated phases or coexist on an atomic scale. keywords : phase diagram, antiferromagnetism, magnetoresistance, c - axis conductivity.

0810.4598

Introduction polluted white dwarfs (typed with a suffix `` z '') provide an opportunity to investigate the ultimate fate of planetary systems. although planets have not yet been detected around white dwarfs, the evidence for the presence of planetary debris around these objects lies in their polluted atmospheres. approximately one quarter of white dwarfs show the presence of elements heavier than helium in their atmospheres @xcite and approximately one fifth of these have a mid - infrared (ir) excess that is consistent with a circumstellar, debris disc @xcite. more recently using the cosmic origins spectrograph on the _ hubble space telescope _ @xcite have shown that about half of da white dwarfs with effective temperatures ranging from 17000 k to 27000 k have polluted atmospheres. several detailed studies of polluted white dwarfs have uncovered large variations in the composition of the accreted material. based on a study of ultraviolet (uv) spectra of a sample of white dwarfs (19000 < @xmath0 k), @xcite showed that the abundance diversity in the accreted material is similar to that observed among solar system meteorites, although the effect of selective radiative radiation pressure on accretion rate calculations was neglected. @xcite demonstrated that selective radiation pressure on trace elements, silicon for instance, shapes observed abundance patterns in hot white dwarfs (@xmath1 k). after including this effect in their calculations, @xcite concluded that at least 27% of their white dwarf sample, which includes the @xcite sample, would be currently accreting, while in 29% of these objects, usually among the warmest in their sample, the effect of radiative levitation dominates the abundance pattern. the inclusion of this effect also leads to a reduction in the estimated accretion flow in some objects with @xmath2 k (e.g., wd0431 + 126). an analysis of uv and optical spectra of two additional white dwarfs by @xcite show the accreting source to be of a rocky nature where the abundance of refractory elements is enhanced compared to volatile elements. also, @xcite showed that the cool, hydrogen - rich and magnetic white dwarf nltt 43806 (typed dazh) is enriched in aluminium but poor in iron which suggests that the accreting material is similar to the earth lithosphere. oxygen has been detected in several white dwarfs (e.g., galex j1931 + 0117 ; *??? *), and, in some of these objects, the amount of oxygen with respect to the other heavier elements detected suggests that the accreted material contains significant amount of water. for example, in gd 61 @xcite found that the accreted material contains oxygen in excess of the amount expected to be carried by metal oxides, suggesting that the parent material contained water. a similar finding, but with a higher fraction of water, was found in the case of sdss j124231.07@xmath3522626.6 @xcite. the material accreted at the surface of a white dwarf is subjected to diffusion processes : trace elements are quickly mixed in the convective envelope of cool white dwarfs, and diffuse - out below the convection zone in a period of time much shorter than evolutionary timescales @xcite. recent estimates @xcite of diffusion timescales show that relics of an accretion event remain visible in the photosphere of a cool (6000 k) hydrogen - rich white dwarf for nearly @xmath4 years and much longer (several @xmath5 years) for cool helium - rich white dwarfs. however, the observed abundance would follow details of the accretion history, and the presence of heavy elements is likely transitory when compared to the cooling age of old white dwarfs (@xmath6 years). we present a spectroscopic and photometric analysis of an hitherto unknown cool, polluted white dwarf (nltt 19868) from the revised nltt catalogue of @xcite. we provide details of the new observations in section 2 : we obtained new low- and high - dispersion spectra as well as new and archival photometric measurements allowing to build a spectral energy distribution (sed). in section 3, we analyse our new data and derive atmospheric parameters : temperature, surface gravity, and composition. next, in section 4, we attempt to reconstruct recent accretion history onto this object. then, we draw a comparison with the sample of cool white dwarfs highlighting the peculiar photospheric composition of the cool white dwarf nltt 19868, and, finally, we summarize our results. Observations we present detailed spectroscopic and photometric observations of the newly identified white dwarf nltt 19868. we first observed nltt 19868 with the eso faint object spectrograph and camera (efosc2) attached to the new technology telescope (ntt) at la silla observatory on ut 2009 march 3. using grism number 11 (300 lines / mm) with the slit - width set to 1 arcsec, we obtained a resolution of @xmath7 . the two consecutive spectra of 1800 s each revealed a cool daz white dwarf with strong h&k lines. we followed up on our initial observations with four sets of echelle spectra of using the x - shooter spectrograph @xcite attached to the ut3 at paranal observatory on ut 2014 may 1, 29 and june 1. the slit - width was set to 0.5, 0.9 and 0.6 arcsec for the uvb, vis and nir arms, respectively. this setup provided a resolving power of 9900, 7450 and 7780 for the uvb, vis and nir arms, respectively. the exposure times for the uvb and vis arms were 2940 and 3000 s, respectively, and for the nir arm we obtained five exposures of 600 s each. we used the acquisition images from the efosc2 and x - shooter observations to obtain estimates of @xmath8 and @xmath9 magnitudes of nltt 19868, respectively. first, we measured the instrumental magnitudes of nltt 19868 and of a brighter comparison star (ra[j2000]=08h 36 m 03.44s, dec[j2000]=@xmath101005525) with published photometry (@xmath11 mag, @xmath12 mag, @xmath13 mag, and @xmath14 mag) from the aavso photometric all - sky survey (apass). apass is an all - sky survey conducted in five filters (johnson @xmath15 and @xmath9, and sloan @xmath16, @xmath17 and @xmath18) with a magnitude range from approximately 10 up to 17. we converted the sloan @xmath17 magnitude to the johnson @xmath8 magnitude using the transformation equations of lupton (2005):[multiblock footnote omitted] @xmath19 or @xmath20 we calculated @xmath8 using both equations and used the average of the two measurements (@xmath21 mag). finally, using the difference between the instrumental magnitudes of nltt 19868 and the comparison star, we calculated @xmath22 mag and @xmath23 mag for nltt 19868. note that the uncertainties for @xmath9 and @xmath8 are statistical only and neglect any possible systematic effects. we obtained ir photometric measurements from the two micron all sky survey (2mass ; *??? *) and _ _ w__ide - field infrared survey explorer (_ wise _ ; *??? the measurements which are all on the vega system are listed in table [tbl_phot]. the @xmath24 and @xmath25 measurements are not listed because only upper limits were available for this object. we have examined the _ wise _ images in combination with our x - shooter @xmath9 acquisition images. these images show that there is a nearby star 5.2 arcsec away at a position angle p.a. = 98@xmath26. the crowded star does not share the white dwarf proper motion and, consequently, is not physically related. using the proper motion of nltt 19868 the distance between the nearby star and nltt 19868 would have been 5.2 arcsec at p.a. = 107@xmath26 at the time the _ wise _ images were obtained (2010). since the point spread function (psf) of @xmath27 and @xmath28 are 6.1 and 6.4 arcsec, respectively, the _ wise _ photometric measurements listed in table [tbl_phot] should combine both nltt 19868 and the nearby object. the amount of contamination is unknown since the spectral type of the nearby star is unknown, although its sed shows it to be a cool object (@xmath29k). we have examined the 2mass images which show that nltt 19868 and the nearby object are clearly separated, and we conclude that the 2mass photometric measurements of nltt 19868 are not contaminated by the nearby star. the 2mass catalog did not contain any flags that would suggest problems with the @xmath30 photometry..photometry and astrometry [cols="<,^,^",options="header ",] Discussion and summary abundance studies of cool, hence old white dwarfs allow to determine the frequency of planetary debris at an advanced cooling age, i.e., long after the parent star left the main - sequence. also, because of the longer diffusion timescales predicted in cool white dwarfs, diffusion effects may become more apparent, particularly following a discrete accretion event. in this case, a spread of diffusion timescales within a group of elements would lead to time - dependent alteration to abundance ratios allowing for a critical examination of the physical conditions at the base of the convection zone (see *??? *). the number of cool daz white dwarfs remains low compared to the number of their hot daz counterparts, or to the helium - rich dz white dwarfs. concentrating our efforts on the hydrogen - rich sample may help establish whether their environment is similar to the more common dzs. in this context, our analysis of the cool daz white dwarf nltt 19868 and similar objects is timely. nltt 19868 lies among the coolest known daz white dwarfs. only three other daz white dwarfs have temperatures below 5500 k : g174 - 14 (@xmath31 k, @xcite), nltt 10480 (@xmath32 k, @xcite) and g77 - 50 (@xmath33 k, @xcite). the effective temperature adopted for g77 - 50 is the weighted average of the two measurements from @xcite and @xcite. fig. [fig_ratio] (top panel) plots the calcium abundance of all known cool daz white dwarfs with temperatures lower than 7000 k. the effective temperature for seven of these objects were updated with the results of @xcite. the calcium abundance varies by several orders of magnitudes, consistent with other studies (e.g., *??? * ; *??? this range of abundances is possibly a result of large variations in the accreted mass and of a possible time lapse since the last accretion event resulting in a diffusion - induced decline in the observed abundance. [fig_ratio] (bottom panel) also shows the fe / ca abundance ratio for stars with a measurable iron abundance. even though the sample is small, a large dispersion (@xmath34) in the fe / ca abundance ratio is observed. within the small sample of objects depicted in fig. [fig_ratio], @xmath35 averages @xmath36, which is lower than the bulk - earth abundance ratio of @xmath37 @xcite but is still consistent given the @xmath38 of the sample. we compared our cool sample to the larger sample of 50 polluted white dwarfs presented by @xcite which includes white dwarfs with @xmath39 k to @xmath40 k. our cool sample has a slightly smaller average but a larger dispersion than what is observed in the @xcite sample which has an average of @xmath41 and a @xmath42. the likely reason for the difference in the dispersion is that in the hotter sample the dispersion is smaller than in the cooler sample and hence bringing down the dispersion in the @xcite sample. hence, the extrema at @xmath43 (nltt 888 ; *??? *) and @xmath44 (nltt 19868) are notable. the abundance ratio observed in the nltt 888 is only second to that of the cool dz white dwarf sdss j1043 + 3516 (@xmath45, *??? *) while fe / ca in nltt 19868 is slightly lower than in the cool daz nltt 43806 (@xmath46, *??? the observed abundance ratio @xmath47 in nltt 19868 is also the lowest known among known polluted white dwarfs, underlining the low relative abundance of fe with respect to the other detected elements. this abundance ratio is slightly lower than that of nltt 43806 (@xmath48=-0.2, *???. finally, the abundance ratio @xmath49 is among the lowest in polluted white dwarf atmospheres, in fact it is the second lowest after the heavily polluted dbaz gd 362 (@xmath50, *??? *). a comparison with the population of the 60 polluted white dwarfs showing mg @xcite, for which the average is @xmath51 with a dispersion of 0.4, clearly places nltt 19868 at the magnesium - deficient end of the distribution. diffusion at the bottom of the convection zone alters the observed abundance pattern over a time period much shorter than cooling timescales. elements with short diffusion timescales relative to other elements are depleted faster and observed abundance ratios must vary over time. in a regime of steady state accretion, the hypothetical z1/z2 abundance ratio is simply given by @xmath52 while in the declining phase, i.e., after mass accretion ceased, the time (@xmath53) dependent abundance ratio is given by @xmath54 where @xmath55 are the diffusion timescales at the bottom of the convection zone which is assumed to be homogeneously mixed. unfortunately, diffusion timescales are uncertain. few calculations are available for objects with effective temperatures below 6000 k. the abundance pattern in the accreted material @xmath56 is also uncertain and may, for example, correspond to solar (see *??? *), bulk, core, or mantle earth @xcite. other types of material, based on an analogy with solar system bodies such as those of meteorites or asteroids can also be considered. a study of the fe / ca abundance ratio in the extreme cases of nltt 888 (@xmath57 k) and nltt 19868 (@xmath58 k) and of applicable scenarios supports intrinsic abundance variations in the accreted material with likely alterations brought upon by diffusion effects. the observed fe / ca ratio is 58 in nltt 888 and 0.59 in nltt 19868. in both cases, steady - state accretion regime does not significantly alter the observed ratio relative to the parent material ratio. first, we examine the case of nltt 888. interpolating the tables of @xcite between 5000 and 6000 k, we find @xmath59 and the abundance ratio in both media are nearly identical. using timescales from @xcite, @xmath60, leads to the same conclusion. therefore, the estimated ratio in the accreted material fe / ca @xmath61 largely exceeds that of bulk- (@xmath6213.4) or mantle - earth (@xmath621.8) suggesting that the accreted material in nltt 888 consists of 66% core material and 34% mantle using mass fractions converted to numbers fractions for these various media from @xcite and @xcite. considering that the fe / ca abundance ratio is likely to decrease over a diffusion time scale (@xmath63 years) in the eventuality that accretion is turned off, the inferred fraction of core material found in the parent body of the accreted material of nltt 888 must be considered a lower limit. the predominance of core iron material implies the likely presence of the, yet undetected, core elements such as nickel and sulfur @xcite. a significant amount of silicon (6% by mass, *? *) is also predicted to be present in the core, although if both core and mantle are to be accreted, most of the silicon would come from the mantle. among the larger population of polluted white dwarfs, including higher temperature stars, such as in the sample of @xcite iron - enrichment is also observed, for example, when compared to silicon, both pg 0843@xmath3516 and pg 1015@xmath3161 accrete material where the fe / si ratio is comparable to the core earth. however, a comparison of the fe / ca ratio for stars in their sample, galex 1931@xmath30117 is the one with the highest fe / ca ratio @xcite and is only slightly lower than that of nltt 888. scenarios for nltt 19868 suggest that, on the contrary, core - type material is largely absent and that the accreted material is most likely composed of mantle - type material. assuming steady - state accretion, the low iron content in the atmosphere implies a similar deficiency in the parent material, i.e., significantly lower than the earth s mantle composition. such a deficiency of iron in the parent material, i.e., below that of the earth s mantle, may not be necessary. the observed fe / ca abundance ratio is well reproduced if the accretion of earth s mantle like material turned off @xmath64 years ago, allowing to further reduce the fe / ca abundance ratio in the convection zone to the observed level. note that diffusion timescales tabulated by @xcite are systematically longer than those tabulated by @xcite by up to a factor of three. the difference arises because the convection zone reaches deeper in envelope models used by @xcite where diffusion would operate in a denser medium. the depth of the convection zone at low temperatures (@xmath65 k) is not affected by the various treatments of the mixing length theory but, instead by different treatments of the equation - of - state and of the conductive opacity @xcite. interpolating the tables of @xcite and @xcite between 5000 and 6000 k, we find @xmath66 and 0.96, respectively. assuming mantle composition in the source, the original fe / ca ratio of 1.8 would be reduced to the observed ratio of 0.59 in @xmath67 years following @xcite or a much longer time of @xmath68 years following @xcite. a longer elapsed time implied by the calculations of @xcite requires an unrealistically large accretion event (@xmath69). using the shorter elapsed time implied by the calculations of @xcite results we can calculate an initial iron abundance of fe / h @xmath70 using : @xmath71 where the elapsed time since the accretion event is @xmath72 years and the diffusion timescale is @xmath73 years. assuming that the mass of the convection zone is @xmath74 and @xmath75 @xcite, then @xmath76 g, and the total mass of iron accreted onto the white dwarf is @xmath77 g, or, assuming an iron mass fraction of 6% in the mantle, the original accretion event would have weighed @xmath78 g. this mass corresponds to less than one thousandth of the mass of the earth (@xmath79). the same diffusion scenario, but employing earth bulk material characterized by a higher fe / ca abundance ratio than in the mantle, would require a larger settling time scale (elapsed time @xmath80 years) to match the low, present - day fe / ca abundance ratio. assuming an accretion event as large as the convection zone itself, even calcium would have disappeared below the detection limit (@xmath81) after @xmath80 years. in summary, we have identified a new cool, polluted white dwarf showing strong lines of calcium among weaker lines of magnesium, aluminium, and iron. our model atmosphere analysis revealed the lowest iron to calcium abundance ratio of any cool polluted white dwarf. applying heavy element diffusion models, we found that the accretion event involving another peculiar daz white dwarf, nltt 888, and that involving nltt 19868 are clearly distinguishable. the material accreted into the surface of nltt 888 is composed mainly of the iron - rich planetary core material, while the material accreted onto the surface of nltt 19868 is more akin to earth mantle material. in the case of nltt 19868, the accretion event most likely occurred several diffusion timescales ago. although these scenarios appear reliable, details of the calculations rest upon uncertain diffusion timescale calculations. Acknowledgements a.k. and s.v. acknowledge support from the grant agency of the czech republic (13 - 14581s and 15 - 15943s) and ministry of education, youth and sports (lg14013). this work was also supported by the project rvo:67985815 in the czech republic. this publication makes use of data products from the wide - field infrared survey explorer, which is a joint project of the university of california, los angeles, and the jet propulsion laboratory / california institute of technology, funded by the national aeronautics and space administration. this publication makes use of data products from the two micron all sky survey, which is a joint project of the university of massachusetts and the infrared processing and analysis center / california institute of technology, funded by the national aeronautics and space administration and the national science foundation. 99 allgre c., manhs g., lewin ., 2001, e&psl, 185, 49 asplund m., grevesse n., sauval a. j., scott p., 2009, ara&a, 47, 481 benvenuto o. g., althaus l. g., 1999, mnras, 303, 30 bergfors c., farihi j., dufour p., rocchetto m., 2014, mnras, 444, 2147 chayer p., 2014, mnras, 437, l95 debes j. h., sigurdsson s., hansen b., 2007, aj, 134, 1662 del peloso e. f., da silva l., porto de mello g. f., arany - prado l. i. 2005, a&a, 440, 1153 farihi j., zuckerman b., becklin e. e., 2008, apj, 674, 431 farihi j., jura m., zuckerman b., 2009, apj, 694, 805 farihi j., dufour p., napiwotzki r., koester d., 2011, mnras, 413, 2559 farihi j., gnsicke b. t., koester d., 2013, science, 342, 218 gnsicke b. t., koester d., farihi j., girven j., parsons s. g., breedt e., 2012, mnras, 424, 333 giammichele n., bergeron p., dufour p., 2012, apjs, 199, 29 hogg d. w., blanton m. r., roweis s. t., johnston k. v., 2005, apj, 629, 268 johnson d. r. h., soderblom d. r., 1987, aj, 93, 864 jura m., xu s., 2013, aj, 145, 30 kawka a., vennes s., 2006, apj, 643, 402 kawka a., vennes s., 2011, a&a, 532, a7 kawka a., vennes s., 2012, a&a, 538, a13 kawka a., vennes s., 2014, mnras, 439, l90 koester d., 2009, a&a, 498, 517 koester d., gnsicke b. t., farihi j., 2014, a&a, 566, a34 koester d., girven j., gnsicke b. t., dufour p., 2011, a&a, 530, a114 koester d., rollenhagen k., napiwotzki r., voss b., homeier d., reimers d., 2005, a&a, 432, 1025 koester d., wilken d., 2006, a&a, 453, 1051 liu w. m., chaboyer b., 2000, apj, 544, 818 mcdonough w.f., 2001, in teisseyre r., majewski e., eds, earthquake thermodynamics and phase transformation in the earth s interior. academic press, san diego, p. 5 paquette c., pelletier c., fontaine g., michaud g., 1986, apjs, 61, 197 raddi r., gnsicke b. t., koester d., farihi j., hermes j. j., scaringi s., breedt e., girven j., 2015, mnras, 450, 2083 romero a. d., campos f., kepler s. o., 2015, mnras, 450, 3708 salim s., gould a., 2003, apj, 582, 1011 skrutskie m. f., et al., 2006, aj, 131, 1163 soubiran c., bienaym o., siebert a., 2003, a&a, 398, 141 tassoul m., fontaine g., winget d. e., 1990, apjs, 72, 335 vennes s., kawka a., nmeth p., 2010, mnras, 404, l40 vernet j., et al., 2011, a&a, 536, a105 wright e. l., et al., 2010, aj, 140, 1868 xu s., jura m., 2012, apj, 745, 88 xu s., jura m., koester d., klein b., zuckerman b., 2014, apj, 783, 79 zuckerman b., koester d., melis c., hansen b. m., jura m., 2007, apj, 671, 872 zuckerman b., koester d., reid i. n., hnsch m., 2003, apj, 596, 477 zuckerman b., et al., 2011, apj, 739, 101 zuckerman b., melis c., klein b., koester d., jura m., 2010, apj 722, 725

we present an analysis of intermediate - dispersion spectra and photometric data of the newly identified cool, polluted white dwarf nltt 19868. the spectra obtained with x - shooter on the very large telescope (vlt)-melipal show strong lines of calcium, and several lines of magnesium, aluminium and iron. we use these spectra and the optical - to - near infrared spectral energy distribution to constrain the atmospheric parameters of nltt 19868. our analysis shows that nltt 19868 is iron poor with respect to aluminium and calcium. a comparison with other cool , polluted white dwarfs shows that the fe to ca abundance ratio (fe / ca) varies by up to approximately two orders of magnitudes over a narrow temperature range with nltt 19868 at one extremum in the fe / ca ratio and, in contrast, nltt 888 at the other extremum. the sample shows evidence of extreme diversity in the composition of the accreted material : in the case of nltt 888, the inferred composition of the accreted matter is akin to iron - rich planetary core composition, while in the case of nltt 19868 it is close to mantle or bulk - earth composition depleted by subsequent chemical separation at the bottom of the convection zone. [firstpage] diffusion stars : abundances stars : atmospheres stars : individual (nltt 888, nltt 19868) white dwarfs

1602.05000

Acknowledgments massimo capone is thanked for useful discussions. the sezione infn di pavia is acknowledged for granting the computing time necessary to perform dft calculations. this work was supported by miur - prin2012 project no. 2012x3yfz2. 99 n. f. mott, proc. phys london, sect. a * 62 *, 416 (1949). t. wu, h.mayaffre, s. krmer, m. horvati, c. berthier, w.n. hardy, r. liang, d.a. bonn, and m.- h. julien, nature (london) * 477 *, 191 (2011). tranquada, b.j. sternlieb, j.d. axe, y. nakamura and s. uchida, nature * 375 *, 561(1995). g. ghiringhelli, m. le tacon, m. minola, s. blanco - canosa, c. mazzoli, n.b. brookes, g.m. de luca, a. frano, d.g. hawthorn, f. he, t. loew, m. moretti sala, d.c. peets, m. salluzzo, e. schierle, r. sutarto, g.a. sawatzky, e. weschke, b. keimer and l. braicovich, science * 337 *, 821 (2012). m. hcker, n.b. christensen, a.t. holmes, e. blackburn, e.m. forgan, r. liang, d.a. bonn, w.n. hardy, o. gutowski, m. v. zimmermann, s.m. hayden and j. chang, phys. b * 90 *, 054514 (2014). y. kamihara, t. watanabe, m. hirano, and h. hosono, j. am. soc. * 130 *, 3296 (2008). johnston, adv. 59 *, 803 (2010). i. i. mazin, d. j. singh, m. d. johannes, and m. h. du, phys. * 101 *, 057003 (2008) ; k. kuroki, s. onari, r. arita, h. usui, y. tanaka, h. kontani, and h. aoki, phys.rev.lett. * 101 *, 087004 (2008). jasek, k. komedera, a. blachowski, k. ruebenbauer, z. bukowski, j.g. storey and j. karpinski, j. alloys comp. 609, 150 (2014) p. s. wang, p. zhou, j. dai, j. zhang, x. x. ding, h. lin, h. h. wen, b. normand, r. yu, and w. yu, phys. b * 93 *, 085129 (2016). f. f. tafti, a. ouellet, a. juneau - fecteau, s. faucher, m. lapointe - major, n. doiron - leyraud, a. f. wang, x.- luo, x. h. chen, and l. taillefer, phys. b * 91 *, 054511 (2015). f. eilers, k. grube, d. a. zocco, t. wolf, m. merz, p. schweiss, r. heid, r. eder, r. yu, j.- x. zhu, q. si, t. shibauchi, and h. v. lhneysen, phys. 116 *, 237003 (2016) y. p. wu, d. zhao, a. f. wang, n. z. wang, z. j. xiang, x. g. luo, t. wu, and x. h. chen, phys.rev.lett. 116, 147001 (2016) ; see also the corresponding supplemental material at http://journals.aps.org/prl/supplemental/ + 10.1103/physrevlett.116.147001 a. k. pramanik, m. abdel - hafiez, s. aswartham, a. u. b. wolter, s. wurmehl, v. kataev, and b. bchner, phys. b * 84 *, 064525 (2011). g. li, w. z. hu, j. dong, z. li, p. zheng, g. f. chen, j. l. luo, and n. l. wang, phys. * 101 *, 107004 (2008) ; m. yi, d. h. lu, j. g. analytis, j.- h. chu, s.- k. mo, r.- h. he, r. g. moore, x. j. zhou, g. f. chen, j. l. luo, n. l. wang, z. hussain, d. j. singh, i. r. fisher, and z.- x. shen, phys. b * 80 *, 024515 (2009). l. de medici, s.r. hassan, m. capone and x. dai, phys.rev.lett. * 102 *, 126401 (2009) l. de medici, g. giovannetti and m. capone, phys.rev.lett. * 112 *, 177001 (2014) e. gull, m. ferrero, o. parcollet, a. georges, and a. j. millis, phys. b * 82 *, 155101 (2010). v. j. emery and s.a. kivelson, physica c * 209 *, 597 (1993) ; u. low, v. j. emery, k. fabricius, and s.a. kivelson, phys. lett. * 72 *, 1918 (1994). c. castellani, c. di castro and m. grilli, phys. lett. * 75 *, 4650 (1995). e. dagotto, a. moreo, a. nicholson, q. luo, s. liang and x. zhang, front. phys. * 6 *, 379 (2011) a. abragam, in _ principles of nuclear magnetism _, oxford university press (1983). h. alloul, t. ohno and p. mendels, phys. lett. * 63 *, 1700 (1989). h. ding, t. yokoya, j.c. campuzano, t. takahashi, m. randeira, m.r. norman, t. mochiku, h. kadowaki and j. giapintzakis, nature * 382 *, 51 (1996). b. batlogg, h. y. hwang, h. takagi, r.j. cava, h.l. kao and j. kwo, physica c * 235 - 240 *, 130 (1994). z. bukowski, s. weyeneth, r. puzniak, j. karpinski and b. batlogg, physica c * 470 *, s328 (2010). z. shermadini, h. luetkens, a. maisuradze, r. khasanov, z. bukowski, h.- h. klauss, and a. amato, phys. b * 86 *, 174516 (2012). for details on the sample preparation and characterization, dft calculations as well as on nqr and nmr measurements see the supplemental material [url] which includes refs. [29 - 36]. elk code, version 3.3.17, http://elk.sourceforge.net j. p. perdew, a. ruzsinszky, g. i. csonka, o. a. vydrov, g e. scuseria, l. a. constantin, x. zhou, and k. burke, phys. lett. * 100 *, 136406 (2008) h. j. monkhorst and j. d. pack, phys. b * 13 *, 5188 (1976). m. methfessel and a. paxton, phys. b * 40 *, 3616 (1989). lehmann - horn, r.yong, d.g.miljak and t.j.bastow, solid state nucl. mag. * 71 *, 87 (2015) r. e. walstedt and s.- w. cheong, phys. b * 51 *, 3163 (1995). l. bossoni, p. carretta, w. p. halperin, s. oh, a. reyes, p. kuhns, and p. c. canfield, phys b * 88 *, 100503 (2013). maclaughlin, j.d. williamson and j. butterworth, phys. b * 4 *, 60 (1971). s. backes, h.o. jeschke, and r. valenti, phys. b * 92 *, 195128 (2015). g. lang, h.- j. grafe, d. paar, f. hammerath, k. manthey, g. behr, j. werner, and b. bchner, phys. rev.lett. * 104 *, 097001 (2010). p. butaud, p. sgransan, c. berthier, j. dumas, and c. schlenker, phys. rev. lett. * 55 *, 253 (1985). j. h. ross, z. wang, and c.p. slichter phys. lett. * 56 *, 663 (1986) c. berthier and p. sgransan, in _ low - dimensional conductors and superconductors _, eds. d.jrome and l.g.caron, (plenum pub., 1987), p.455 k.i. kugel and d.i. khomskii, sov. jetp * 37 *, 725 (1973) f. hardy, a.e. bhmer, l. de medici, m. capone, g. giovannetti, r. eder, l. wang, m.he, t. wolf, p. schweiss, r. heid, a. herbig, p. adelmann, r. a. fisher, and c. meingast, arxiv:1605.05485 w.e. blumberg, phys. * 119 *, 79 (1960) a. ishigaki and t. t. moriya, j. phys. 65 *, 3402 (1996) ; ibid. * 67 *, 3924 (1998). f. hammerath, p. bonf, s. sanna, g. prando, r. de renzi, y. kobayashi, m. sato, and p. carretta, phys. rev. b * 89 *, 134503 (2014). s. caprara, c. di castro, g. seibold and m. grilli, arxiv:1604.07852v1 l. de medici, arxiv:1609.01303v1 I. sample synthesis and characterization a polycrystalline sample of rbfe@xmath3as@xmath3 was synthesized in two steps.@xcite first, rbas and fe@xmath3as precursors were prepared from stoichiometric amounts of rubidium, arsenic and iron. the components were mixed and heated in evacuated and sealed silica tubes at 350 c and at 800 c, respectively. then, the obtained rbas and fe@xmath3as were mixed together in a molar ratio 1:1, pressed into pellets and placed in an alumina crucible and sealed in an evacuated silica ampoule. the sample was annealed at 650 c for three days, ground and annealed for another three days at the same temperature. it should be emphasized that the annealing temperature and time are crucial parameters. the annealing at higher temperature or extended annealing time causes decomposition of the compound. the phase purity was checked by x - ray powder diffraction. the diffraction lines (shown in fig.[xrd]) can be indexed with a tetragonal thcr@xmath3si@xmath3 type unit cell with lattice parameters @xmath58 and @xmath59 , in good agreement with those reported in ref.. the ac - susceptibility measurements were performed on heating with an ac field of 10 oe at 1111 hz. the real component (fig.[chiac]) reveals the onset of diamagnetism and of bulk superconductivity below @xmath60 k.@xcite Ii. dft calculations first - principles dft calculations of the electronic structure were performed using the full - potential linearized augmented plane - wave method as implemented in the elk package @xcite. for the exchange - correlation functional we used the generalized gradient approximation of perdew, burke, and ernzerhof @xcite. the atomic positions used in the calculation are those obtained from room temperature x - ray diffraction. in order to calculate the electric field gradient (efg) tensor components @xmath61 we solved the poisson equation for the charge distribution to determine the electrostatic potential @xmath62 and derived @xmath61 from @xmath63 where @xmath64 runs over the nuclei at @xmath65. since the efg tensor is extremely sensitive to the charge distribution a well converged basis set is needed to grant the convergence with respect to the efg tensor components. we used muffin tin radii of 2.6 @xmath66 for rb and 2.4 @xmath66 for fe and as, with @xmath67, where @xmath68 is the smallest muffin tin (mt) radius inside the mt spheres and @xmath69 the magnitude of the reciprocal space vectors. we choose 9 for the cut off of the angular momentum quantum number in the lattice harmonics expansion inside the mts. reciprocal space was sampled with the monkhorst - pack @xcite scheme on a @xmath70 grid. a smearing of 270 mev was used within the methfessel - paxton scheme @xcite and convergence of the efg components with respect to all these parameters has been carefully checked. once the efg tensor components are known the nqr frequency at @xmath0as and @xmath1rb can be calculated from eq. 1 in the main article. the obtained values, (@xmath71)@xmath72 14.12 mhz and (@xmath73)@xmath72 6.7 mhz are in good agreement with the experimental values (@xmath71)@xmath7414.6 mhz and (@xmath73)@xmath74 6.2 mhz and the discrepancy represents an estimate of the accuracy of the dft calculation which is known to not properly account for the electronic correlations. Iii. nqr and nmr spectra @xmath0as nqr and @xmath1rb nmr spectra were derived by recording the integral of the echo signal after a @xmath75 pulse sequence as a function of the irradiation frequency. at a few temperatures the @xmath0as nqr spectra was also obtained by merging the fourier transforms of half of the echo recorded at different frequencies but no relevant additional features appeared in the spectra. we point out that any tiny amount of spurious phases as feas and fe@xmath3as (not detected in x - ray diffraction) will not affect the @xmath0as nqr spectra since these materials are magnetically ordered and the internal field shifts the resonance frequency to much higher values. also in feas@xmath3 the @xmath0as nqr line is in a completely different frequency range.@xcite the narrow @xmath13rb nqr spectra were obtained from the fourier transform of half of the echo signal obtained after the same echo pulse sequence. @xmath1rb nmr powder spectrum for the central @xmath76 transition is displayed in fig.[rbnmr]. the spectrum is fully compatible with the @xmath1rb quadrupole frequency determined from the nqr spectra. Iv. spin-spin relaxation rate @xmath40 @xmath0as spin - spin relaxation rate @xmath40 was derived in nqr by recording the decay of the echo amplitude @xmath77 after a @xmath75 pulse sequence. the decay could be fit in general with @xmath78 with @xmath79. a value of @xmath80 lower than 2 and the slight temperature dependence of @xmath81 (fig.[figt2]) should be associated with redfield contribution to the relaxation @xmath82. then one can write @xmath83, with @xmath40 the spin - spin relaxation rate. in case of an anysotropic spin - lattice relaxation rate, et al_. @xcite calculated a general expression for @xmath84. in case of a nuclear spin @xmath16, with the @xmath85 axes of the efg along the @xmath86 axes one should have : @xmath87 where the symbols @xmath88 and @xmath89 refers to the external field orientation with respect to the crystallographic @xmath86 axis. in particular, @xmath90 corresponds to @xmath0as nqr @xmath14. @xmath91 was determined by assuming an anisotropy in @xmath14 equal to the one found in electron - doped bafe@xmath3as@xmath3 @xcite. once the data have been corrected by redfield contribution one finds that @xmath0as @xmath40 is temperature independent (fig.[figt2]), with @xmath92. the deviation of @xmath93 from 2, as it is expected in the case of nuclear dipolar interaction in a dense system @xcite is likely a consequence of the partial irradiation of the nqr spectrum. taking into account of @xmath94 corrections we have measured the @xmath95-dependence of @xmath0as nqr spectrum amplitude below 50 k and did not observe any significant change. V. nuclear spin-lattice relaxation rate @xmath14 @xmath1rb nmr @xmath14 was measured in a @xmath96 tesla magnetic field by irradiating just the high frequency shoulder of the powder spectrum of the central line shown in fig.[rbnmr], corresponding to grains with the @xmath86-axes perpendicular to @xmath97. the recovery of @xmath1rb nmr central line magnetization after a saturation recovery pulse sequence was fit according to @xmath98 the recovery is shown in fig.[t1rbnmr] and one observes, similarly to what one finds in @xmath0as nqr, two components appearing at low temperature. the long component were measured just in nmr since in nqr the very long @xmath1rb relaxations and the much lower signal intensity make the measurements quite demanding. the fast component, the only one present at @xmath99 k, was measured in nqr irradiating either @xmath1rb @xmath100 transition or @xmath2rb @xmath101. the recovery of nuclear magnetization was fit according to the recovery laws expected for a magnetic relaxation mechanism @xcite @xmath102 for @xmath1rb and for @xmath2rb. the ratio between the @xmath14 of the two nuclei for the fast relaxing component was measured at a few selected temperatures between 4 and 25 k and @xmath104 (fig.[t1rbnqr]), in good agreement with the ratio between the square of the gyromagnetic ratios of the two nuclei @xmath43, confirming the adequacy of the recovery laws we have used to estimate @xmath14 and the fact that the relaxation is driven by electron spin fluctuations. notice that if the relaxation was driven by efg fluctuations @xmath14 should scale with the square of the nuclear electric quadrupole moment and one should have @xmath105 a value about 50 times smaller than the experimental one. we have measured the frequency dependence of @xmath0as @xmath14 across the nqr spectrum by decreasing the intensity of the radiofrequency field so that we irradiated a width of about 30 khz in the spectrum. we found that the recovery laws did not change significantly across the spectrum (fig.[t1vsfreq]). namely, @xmath31, @xmath32, @xmath27 and @xmath28 show a negligible frequency dependence. it should be mentioned that @xmath32 and @xmath31 appear to slightly depend on the thermal history (i.e. on the cooling rate), an aspect that will be the subject of future studies. finally, we have checked that @xmath32, the amplitude of the slow relaxing component, is zero above 20 k by recording the recovery up to more than @xmath106 @xmath39 (fig.[t1vsfreq]).

@xmath0as, @xmath1rb and @xmath2rb nuclear quadrupole resonance (nqr) and @xmath1rb nuclear magnetic resonance (nmr) measurements in rbfe@xmath3as@xmath3 iron - based superconductor are presented. we observe a marked broadening of @xmath0as nqr spectrum below @xmath4 k which is associated with the onset of a charge order in the feas planes. below @xmath5 we observe a power - law decrease in @xmath0as nuclear spin - lattice relaxation rate down to @xmath6 k. below @xmath7 the nuclei start to probe different dynamics owing to the different local electronic configurations induced by the charge order. a fraction of the nuclei probes spin dynamics associated with electrons approaching a localization while another fraction probes activated dynamics possibly associated with a pseudogap. these different trends are discussed in the light of an orbital selective behaviour expected for the electronic correlations. the parent compounds of high temperature superconducting cuprates are emblematic examples of mott - hubbard insulators at half band filling,@xcite where the large electron coulomb repulsion @xmath8 overcomes the hopping integral @xmath9 and induces both charge localization and an antiferromagnetic (af) coupling among the spins. electronic correlations remain sizeable even when the cuprates become superconducting and give rise to a rich phase diagram at low hole doping levels characterized by the onset of a charge density wave (cdw) which progressively fades away as the doping increases @xcite and eventually, in the overdoped regime, a fermi liquid scenario is restored. the comprehension of the role of electronic correlations in iron - based superconductors (ibs)@xcite is more subtle. at variance with the cuprates ibs are characterized by similar nearest neighbour and next - nearest neighbour hopping integrals, the parent compounds of the most studied families of ibs (e.g. bafe@xmath3as@xmath3 and lafeaso)@xcite are not characterized by half - filled bands and, moreover, in ibs the fermi level typically crosses five bands associated with the different fe 3@xmath10 orbitals, leading to a rich phenomenology in the normal as well as in the superconducting state.@xcite moreover, even if signs have been reported @xcite, the evidence for a charge order in the phase diagram of ibs still remains elusive. nominally, half band filling can be approached in bafe@xmath3as@xmath3 ibs by replacing ba with an alkali atom a = k, rb or cs, resulting in 5.5 electrons per fe atom.@xcite transport measurements show that afe@xmath3as@xmath3 compounds are metals@xcite with sizeable electronic correlations and it has been recently pointed out that their behaviour shares many similarities with that of heavy fermion compounds.@xcite indeed, the effective mass progressively increases as one moves from bafe@xmath3as@xmath3 to afe@xmath3as@xmath3,@xcite even if clear discrepancies in the values derived by the different techniques are found depending on their sensitivity to the electrons from a single band or from all the five bands.@xcite de medici et al. @xcite pointed out that if electronic correlations are sizeable, namely @xmath11 is of the order of the unity, the local atomic physics starts to be relevant and hund coupling may promote the single electron occupancy of fe @xmath10 orbitals (i.e. half band - filling) and decouple the interband charge correlations. accordingly the mott transition becomes orbital selective@xcite so that while the electrons of a given band localize the electrons of other bands remain delocalized, leading to a metallic behaviour and eventually to superconductivity. this orbital selective behaviour should give rise to markedly @xmath12-dependent response functions @xcite and to a sort of @xmath12-space phase separation of metallic and insulating - like domains. the point is, what happens in the real space? will one probe the sum of the insulating and metallic response functions or should one detect a real space phase separation@xcite also in afe@xmath3as@xmath3 ibs @xcite, with different local susceptibilities? more interestingly, if electronic correlations become significant in afe@xmath3as@xmath3 one could envisage the onset of a charge order@xcite as in the cuprates.@xcite nuclear quadrupole resonance (nqr) and nuclear magnetic resonance (nmr) are quite powerful tools which allow to probe the local response function and charge distribution. moreover, in nqr experiments @xcite the magnetic field, which often acts as a relevant perturbation, is zero. here we show, by combining @xmath0as and @xmath13rb nqr and @xmath1rb nmr measurements, that in rbfe@xmath3as@xmath3 a charge order develops in the normal state below @xmath4 k, possibly leading to a differentiation in real space of fe atoms with different orbital configurations. below @xmath5, @xmath0as and @xmath1rb nuclear spin - lattice relaxation rates (@xmath14) show a power law behaviour, as it is expected for a strongly correlated electron system and in good agreement with @xmath0as nmr results reported by wu et al.@xcite. however, at @xmath6 k we observe that a fraction of @xmath0as (or @xmath1rb) nuclei probes spin dynamics characteristic of a system approaching localization while others probe dynamics possibly associated with a metallic phase with a pseudogap.@xcite upon further decreasing the temperature the volume fraction of the heavy electron phase vanishes while the one of the metallic phase, which eventually becomes superconducting below @xmath15 k, grows. thus, we present a neat evidence for a charge order in rbfe@xmath3as@xmath3 akin to underdoped cuprates. the charge order favours a phase separation into metallic and nearly insulating regions, which could result from the theoretically predicted orbital selective behaviour.@xcite nqr and nmr measurements were performed on a rbfe@xmath3as@xmath3 polycrystalline sample with a mass of about 400 mg, sealed in a quartz tube under a 0.2 bar ar atmosphere in order to prevent deterioration. the superconducting transition temperature derived from ac susceptibility measurements turned out @xmath15 k, in good agreement with previous findings @xcite. further details on the sample preparation and characterization are given in the supplemental material.@xcite first of all we shall discuss the appearance of a charge order in the feas planes of rbfe@xmath3as@xmath3, as detected by @xmath0as nqr spectra. for a nuclear spin @xmath16, as it is the case of @xmath0as and @xmath1rb, the nqr spectrum is characterized by a single line at a frequency@xcite @xmath17 with @xmath18 the nuclear quadrupole moment, @xmath19 the main component of the electric field gradient (efg) tensor and @xmath20 its asymmetry @xmath21. hence the nqr spectrum probes the efg at the nuclei generated by the surrounding charge distribution. above 140 k, @xmath0as nqr spectrum (fig.[figspec]) is centered around 14.6 mhz, with a linewidth of about 170 khz, while @xmath1rb nqr spectrum is centered around 6.2 mhz with a width of about 20 khz. the relatively narrow nqr spectra confirms the good quality of our sample. we performed density functional theory (dft) calculations using elk code in the generalized gradient approximation@xcite in order to derive _ ab initio _ the electric field gradient and nqr frequency. for @xmath0as and @xmath1rb we obtained @xmath22 mhz and @xmath23 mhz, respectively, in reasonable agreement with the experimental values in spite of the significant electronic correlations.@xcite this shows that dft is still able to provide a fair description of the system as far as it remains a normal metal. upon cooling the sample below @xmath4 k significant changes are detected in @xmath0as nqr spectra (fig. [figspec]). the spectrum is observed to progressively broaden with decreasing temperature and below 50 k one clearly observes that the spectrum is actually formed by two humps nearly symmetrically shifted with respect to the center (fig.[figspec]a). the presence of two peaks in the @xmath0as nqr spectra has already been detected in different families of ibs and associated with a nanoscopic phase separation in regions characterized by different electron doping levels.@xcite however, at variance with what we observe here, the two peaks observed in other ibs do not show the same intensity @xcite and the spectra show little temperature dependence, namely the nanoscopic phase separation is likely pinned. under both high magnetic field and high pressure an asymmetric splitting of @xmath0as nmr spectrum was detected also in kfe@xmath3as@xmath3 which, however, is absent in zero field (nqr).@xcite here we observe the emergence of an nqr spectrum which recalls the one expected for an incommensurate cdw,@xcite which causes a periodic modulation of the efg at the nuclei and gives rise to two symmetrically shifted peaks in the spectrum. the efg modulation could involve also the onset of an orbital order@xcite or a structural distortion, possibly coupled to the charge order. although it is not straightforward from our data to discriminate among these scenarios, it is clear that we detect a symmetry breaking below @xmath5 to a low temperature phase characterized by a spatial modulation of the efg, namely by a charge order. @xmath1rb nqr spectrum does not show a significant broadening upon decreasing the temperature but is characterized by a @xmath24 which, at @xmath25 k, shows a temperature dependence similar to that of the @xmath0as nqr spectra full width at half maximum (fwhm), proportional to the charge order parameter (fig.[figspec]c). below @xmath7 @xmath1rb @xmath24 flattens and deviates from @xmath0as nqr fwhm. the fact that the nqr spectrum of the out of plane @xmath1rb nuclei is less sensitive than @xmath0as one to the charge order is an indication that the order develops in the feas planes and that the modulation of the efg at @xmath0as nuclei should occur over a few lattice steps, otherwise one should expect a splitting also of the narrow @xmath1rb nqr spectrum. it is interesting to notice that at a temperature of the order of @xmath7 an abrupt change in the uniaxial thermal expansion occurs,@xcite evidencing also a change in the lattice properties. now we discuss the temperature dependence of the low - energy dynamics probed by @xmath0as and @xmath1rb @xmath14. the nuclear spin - lattice relaxation rate was determined from the recovery of the nuclear magnetization after exciting the nuclear spins with a saturation recovery pulse sequence. the recovery of @xmath0as magnetization in nqr is shown in fig.[figrec]a. one notices that a single exponential recovery describes very well the recovery of the nuclear magnetization at @xmath26 k, as it can be expected for a homogeneous system where all nuclei probe the same dynamics . however, below @xmath6 k one observes the appearance of a second component characterized by much longer relaxation times. namely, a part of the nuclei probes dynamics causing a fast relaxation (@xmath27) and a part of the nuclei a slow relaxation (@xmath28). accordingly, the recovery was fit to @xmath29 \,\,\ ,, \]] with @xmath30 the nuclear magnetization at thermal equilibrium, @xmath31 and @xmath32 the fraction of fast relaxing and slow relaxing nuclei, respectively, @xmath33 a factor accounting for a non perfect saturation by the radiofrequency pulses and @xmath34 a stretching exponent characterizing the slowly relaxing component. as the temperature is lowered one observes a progressive increase of @xmath32 with respect to @xmath31 and at the lowest temperature (@xmath35 k), about 80% of the nuclei are characterized by the slow relaxation (fig.[figrec]c). it is important to notice that in rbfe@xmath3as@xmath3 wu et al.@xcite (in nmr, not in nqr) did not observe a clear separation of the recovery in two components as we do here but they did observe deviations from a single exponential recovery below 20 k which, however, were fitted with a stretched exponential, likely yielding an average @xmath14 value between @xmath28 and @xmath27. remarkably also @xmath1rb nmr @xmath14 clearly shows two components below 25 k and just one above.@xcite @xmath0as @xmath14 was measured both on the high frequency and on the low - frequency shoulder of the nqr spectrum and it was found to be the same (fig.[figt1]a) over a broad temperature range. moreover, at @xmath36 k we carefully checked the frequency dependence of @xmath37, @xmath38, @xmath31 and @xmath32 and found that neither the two relaxation rates nor their amplitude vary across the spectrum (fig.[figrec]b, see also ref.). this means that nuclei resonating at different frequencies probe the same dynamics which implies that the charge modulation induced by the charge order has a nanoscopic periodicity.@xcite one could argue that the two components are actually present at all temperatures but that they arise only at low temperature once nuclear spin diffusion@xcite is no longer able to establish a common spin temperature (i.e. a common @xmath39) among the nuclei resonating at different frequencies. however, we remark that since the nuclear spin - spin relaxation rate (@xmath40) is constant @xcite and the width of the nqr spectrum is nearly constant below 40 k (fig.[figspec]c) the poor efficiency of nuclear spin diffusion should not vary, at least for @xmath41 k. hence, the appearance of different relaxation rates below @xmath7 should arise from a phase separation causing a slight change in the average electronic charge distribution causing little effect on the nqr spectra (see fig. [figspec]) but a marked differentiation in the low - energy excitations @xcite, which starts to be significant at low temperature once the effect of electronic correlations is relevant. one has to clarify if the relaxation mechanism is magnetic, driven by electron spin fluctuations, or quadrupolar, driven by efg fluctuations, typically induced by cdw amplitude and phase modes.@xcite in order to clarify this point we measured the ratio between @xmath1rb and @xmath2rb @xmath14 (fast component) at a few selected temperatures below 25 k. the ratio @xmath42, in good agreement with the ratio between the square of the gyromagnetic ratios of the two nuclei @xmath43, showing that the relaxation is driven by the correlated spin fluctuations and not by charge fluctuations associated with cdw excitations. since @xmath0as shows a temperature dependence of the relaxation analogous to the one of @xmath1rb (fig.[figt1]a) we argue that also @xmath0as @xmath14 is driven by spin fluctuations. thus we can write that @xmath44 the form factor giving the hyperfine coupling with the collective spin excitations at wave - vector @xmath45, and @xmath46 the imaginary part of the dynamic susceptibility at the resonance frequency @xmath47. now we turn to the temperature dependence of @xmath14 above @xmath6 k and of @xmath28 and @xmath27 below that temperature. above @xmath7 @xmath14 increases with a power law @xmath48, with @xmath49 for @xmath0as, and flattens around @xmath4 k (fig.[figt1]a), in very good agreement with the results reported by wu et al.@xcite from @xmath0as nmr. notice that @xmath5 corresponds to the temperature below which we start to observe a significant broadening of @xmath0as nqr spectrum. hence, the power law behaviour of @xmath14 seems to arise from the onset of the charge order. below @xmath6 k @xmath27 deviates from the power law behaviour and progressively flattens on decreasing temperature (fig.[figt1]a). the same behaviour is detected for @xmath1rb nmr @xmath14, although the flattening starts at a higher temperature, suggesting that @xmath7 might be field dependent. on the other hand, @xmath28 gets progressively longer as the temperature is lowered and follows an activated trend with an energy barrier @xmath50 k. the behaviour of @xmath27 is characteristic of a system approaching a qcp where localization occurs. in fact, from moriya self - consistent renormalization (scr) approach for a quasi-2d system with af correlations, one should have @xmath51,@xcite with @xmath52 the static susceptibility at the af wave - vector. in the proximity of the qcp @xmath53, leading to a weak logarithmic divergence of @xmath54 for @xmath55, while at higher temperature @xmath52 should show a curie - weiss behaviour, yielding a nearly flat @xmath14, as we do observe in rbfe@xmath3as@xmath3 (fig.[figt1]a). the corresponding behaviour of @xmath56 is reported in fig.[figt1]b. on the other hand, @xmath57, corresponding to the relaxation rate of the majority phase at low temperature, shows the opposite trend (fig.[figt1]b), decreasing upon cooling. being the system metallic at low temperature, the deviation of @xmath57 from the constant korringa - like behaviour @xcite expected for a metal should possibly be associated with the opening of a pseudogap, similarly to what one observes in the underdoped regime of the cuprates,@xcite and in agreement with theoretical predictions for hole - doped ibs.@xcite in conclusion, our results show that, akin to cuprates, a charge order develops also in the normal state of ibs when the electronic correlations are sizeable. accordingly, the presence of a charge order appears to be a common feature in the phase diagram of cuprate and iron - based superconductors and could play a key role in determining the superconducting state properties.@xcite moreover, we observe a local electronic separation in two phases characterized by different excitations which could possibly be explained in terms of the orbital selective behaviour@xcite predicted for ibs. finally we remark that the occurrence of an electronic phase separation is theoretically supported by a recent study of the electron fluid compressibility.

1608.04532

Introduction compact symmetric objects (csos) are a family of extragalactic radio sources comprising roughly 3% of flux - limited samples selected at high frequencies (taylor, readhead, & pearson 1996 ; peck & taylor 2001). their defining characteristic is the presence of high luminosity radio components on both sides of a central engine on sub - kiloparsec scales with little or no extended emission present. csos typically exhibit terminal hotspots which move apart at subluminal speeds (owsianik & conway 1998 ; gugliucci et al. jet components en route between the core and the hot spots where they terminate, appear to move faster, and superluminal speeds in the jets are seen in a few sources (taylor, readhead & pearson 2000). the jets can be similar or much brighter than the counterjets. csos on the whole exhibit weak radio variability, have low radio polarization and low core luminosities. their hosts are in general elliptical galaxies (readhead et al. 1996) though there are a few notable exceptions identified with seyferts and quasars. the general picture that has emerged is that csos are young radio galaxies, with ages between 100 and 10000 years. since they are selected on the basis of relatively unbeamed hot - spots and lobes, their orientation on the sky is random. orientation may still affect the presence of detectable linear polarization (gugliucci et al. 2007), or absorption from atomic gas associated with the hypothesized gas and dust torus that surrounds the central engine and is thought to be oriented perpendicular to the jet axis (peck et al. 2000 ; pihlstrm et al. 2003). however, as doppler boosting effects are less dominant, we have the added benefit of studying the emission from both sides of the nucleus. csos provide a unique opportunity to determine the hubble constant, as a direct measurement of the distance to an object can be obtained by observing angular motion in it, if the intrinsic linear velocity can be ascertained independently. basically, one has the time derivative of a standard ruler, with the additional constraint that no intrinsic motion can be faster than light. from five epochs of observations of the cso 1946 + 708 at 5 ghz taylor & vermeulen (1997) placed a lower limit of h@xmath4 @xmath2 37 km s@xmath3 mpc@xmath3. we explore what can be learned from continuing this analysis using more comprehensive observations. we also use this symmetric system, for which the geometry of the jets can be precisely determined, to probe the details of jet propagation. kinematic ages for the radio emission in csos can be obtained by measuring the separation speed between the hot spots over time and comparing this to the overall spatial separation (owsianik & conway 1998 ; gugliucci et al. 2005). at the moment, the cso age distribution seems to be disproportionally stacked towards younger sources. however, there are a number of selection effects that need to be addressed before the meager data collected can be properly analyzed. larger sources are over - resolved by vlbi observations so that measuring the expansion becomes difficult ; more slowly expanding sources require longer time baselines to obtain age estimates. in this work we present observations of the cso 1946 + 708 spanning 16 years. it is of considerable interest to pursue this line of research to address the evolution of radio galaxies. we assume @xmath1 = 71 km s@xmath3 mpc@xmath3 and the standard cosmology so that 1= 1.835 kpc and an angular velocity of 1 mas / y = 5.98 c at the redshift (0.1008) of the radio galaxy 1946 + 708 (peck & taylor 2001). Observations and data reduction the observations were carried out at 8.4 ghz and 15.4 ghz over the course of eight epochs, taken on 1995 march 22, 1996 july 07, 1998 february 06, 1999 july 11, 2001 may 17, and at 8.4 ghz only on 2003 jan 24/2003 feb 10, 2006 feb 18, and 2008 mar 9 (see table 1). observations in 1995, 1998, and 2001 were observed using all ten elements of the vlba of the nrao, and observations from 2003 on were observed using a global array including all ten elements of the vlba, and five elements from the european vlbi network (evn) including the 100 m telescope at effelsberg, the westerbork phased array, onsala, medicina, and in 2003 only, the telescope at noto. due to problems at the st. croix and north liberty stations, epochs in 1996 and 1999 were performed using nine vlba antennas each. problems in 2003 prevented effelsberg from participating. both right and left circular polarizations were recorded for the first 5 epochs, while the last three where observed only in right circular polarization. all epochs employed 2 bit sampling across if bandwidths of 8 mhz. the vlba correlator produced 16 frequency channels across each if during every 2 s integration. we also include in our analysis the 5 ghz observations acquired between 1992 and 1995 by taylor & vermeulen (1997), some of which are contemporaneous with the 8.4 and 15 ghz observations reported here. parallactic angle effects resulting from the altitude - azimuth antenna mounts were removed using the aips task clcor. amplitude calibration for each antenna was derived from measurements of antenna gain and system temperatures during each run. delays between the stations clocks were determined using the aips task fring (schwab & cotton 1983). calibration was applied by splitting the multi - source data set immediately prior to preliminary editing, imaging, deconvolution and self - calibration in difmap (shepherd, pearson, & taylor 1994, 1995). multiple iterations of phase self - calibration and imaging were applied to each source before any attempt at amplitude self - calibration was made. the preliminary models developed in difmap were subsequently applied in aips to make phase corrections, to determine the leakage terms between the rcp and lcp feeds and to correct for residual phase differences between polarizations. final imaging and self - calibration were performed in difmap. model fitting of gaussian components to the self - calibrated visibility data was also performed in difmap. the shapes of the components were fixed after fitting to the 1995 epoch ; in the rest of the epochs each component was allowed only to move and to vary in flux density in order to fit the independently self - calibrated visibility data. each frequency band (5, 8.4 and 15.4 ghz) was modelfit independently in order to allow for differences in component shapes between frequencies. for our last three epochs from 2003 to 2008, all global 8.4 ghz observations, an additional pair of components had to be fit on either side of the core (see 4.1). uncertainties in the sizes and positions for components were derived from signal - to - noise ratios and component sizes (fomalont 1999). this assumes that parameters are not covariant, which should be the case for the strong, well - separated components that we present in this analysis of 1946 + 708. Results while one of the first objects to be confirmed as a bona - fide compact symmetric object, 1946 + 708 is in a few ways unusual for this class. the source is remarkably `` rotation '' symmetric (see figures 1 and 2). narrow, well collimated jets emerge from a compact, flat spectrum core and bend gradually before terminating in well defined hot spots. faint `` lobes '' of emission are evident at 90 angles to the jets. distinct components in the jet are seen to move out from the core to the hot spots on both the northern (jet) side and the southern (counter - jet) side. our most uniform, sensitive, and highest resolution sequence of images is that at 8.4 ghz. the 8.4 ghz images cover the period from march 1995 to march 2008, and have noise levels ranging from 163 @xmath5jy / beam down to 38 @xmath5jy / beam (table 1). no polarized flux has been detected from 1946 + 708 at either 8.4 or 15.4 ghz in any epoch. typical 2@xmath6 limits on the linearly polarized flux density are @xmath7100 @xmath5jy at 8.4 ghz and @xmath7170 @xmath5jy at 15.4 ghz. Discussion this source is quite steady in the production of strong and distinct jet components. in addition to the previously studied pairs n2/s2 and n5/s5, we report here on the discovery of a new set of components which we name n6/s6 (fig. 2). like the previous sets, these new components appear to have been launched simultaneously as a pair, and appear to be moving away from the core in opposite directions. the angular separation between the new pair (n6 and s6) observed in our 2008 epoch is 1.288 mas. the apparent proper motion between n6 and s6 and the core is difficult to determine owing to possible blending of these components with the core, but the total separation can be compared with the 2006 epoch in which it was 1.045 mas. the total apparent expansion speed between the pair over the past two years is thus 0.238 mas / y or 0.72 c. this puts the ejection age for the new pair at 1997.3. indeed evidence of the new component set can be seen in the 2003 epoch, and to a lesser extent in 2001. the sum of the core flux density and the flux density of the new components n6 and s6 during this period (fig. 3) was rising steadily, and only seems to be declining later on, as the new components move well away from the core. in fig. 4 we plot the positions of the 4 strongest jet components (n2, n5, s5 and s2) derived from modelfitting elliptical gaussians to the visibility data at 8.4 ghz. the well - tracked jet components close to the core, n5 and s5, appear to move out with the highest apparent velocity (table 2 and fig. 5 ; and see also the moviegbtaylor/]), with n5 being the fastest at an apparent velocity of 1.088 c, and s5 being about one third that in very nearly the opposite direction. note that these are apparent velocities projected in the plane of the sky, as opposed to intrinsic, three - dimensional velocities. the more distant pair of components (n2/s2) appear to be moving at roughly half the apparent velocity of the inner (n5/s5) pair. from this fact one might be tempted to assert that the jet starts out fast and decelerates with time. if this was the case, and the deceleration is uniform, then we would expect to see the n5/s5 pair slowing down, but this is not observed. even after 12 years of monitoring and a distance covered corresponding to 10% of the length of the jet, there is no compelling evidence for slowing of any of the components. the @xmath8 values indicate reasonably good fits to a straight line for all components (the @xmath8 values may be systematically better in declination than in right ascension owing to the north - south orientation of most of the components). if current velocities remain unchanged then the n5/s5 pair will catch up with the n2/s2 pair in @xmath075 years, close to the time that they enter the hot spots. alternatively, it could be that there is a sudden deceleration of the jet components at a distance from the core between 3.5 and 9 mas (on the northern side). this gap is defined by the minimum distance of 9 mas for n2 in 1995 and the maximum distance for n5 observed in 2008. this is in the region where the jet bends the most (@xmath9) towards the northern hot spot, so a deceleration region should not be ruled out prematurely. considering the newest pair of components, n6/s6, their total apparent separation speed is 0.72c, more like that of the n2/s2 pair (0.63 c), than the n5/s5 pair (1.4 c). observations over the next @xmath05 years should settle the issue of whether components emerge with their own, intrinsic set of velocities, or if they partake in some fixed pattern of acceleration and deceleration. if there is a fixed, repeating pattern, then it might be possible to relate it to a helical jet model. in a helical model, the intrinsic velocity could be constant while the orientation changes in time to produce the observed variations in the projected apparent velocities. such a model naturally reproduces the rotational symmetry of 1946 + 708 and has been found to explain observations over many years of the galactic jet in ss433 (roberts et al. 2008, hjellming & johnston 1981). the path of the inner jet components of 1946 + 708 (n5 and s5) both appear fairly straight to within the errors in the measurements. there is a suggestion of a bend in the trajectory of n5 after the 2006 epoch (epoch 9 in fig. 4), but this relies almost entirely on the position at the 2008 epoch. for the more distant northern component (n2) there are kinks in the apparent motion near epochs 1994, 2001 and 2006 (epochs 2, 7 and 9 in fig. 4). there also appears to be an offset in the 15 ghz positions (see fig. this suggests a flatter spectrum on the eastern side of this component. a similar offset is suggested by the multi - frequency trajectory plot of s2 based on the 5 and 8 ghz observations, but since the motions are smaller and the component is weaker it is difficult to be sure if the spectral gradient is real. a kink is observed in component s2 near the 2003 epoch (epoch 8 in fig. 4). in well studied core - jet sources, such as 3c345 (zensus et al. 1995, unwin et al. 1997), jet components are found to travel along curved trajectories and to change their speeds. this acceleration has been taken as justification for magnetically driven jets, as opposed to purely hydrodynamical structures (vlakhis & konigl 2004). by analogy with these core - jet sources, we draw a similar conclusion that the jets of 1946 + 708 are likely to be magnetically driven. note, however, that intrinsic velocities of jet components in core - jet agns are typically in the range @xmath10, considerably larger than we observe in the cso 1946 + 708 of 1.3 @xmath11 2.3 (see below). even before the core of 1946 + 708 was identified based on 15 ghz observations, the first two epochs at 5 ghz suggested its location by exhibiting bi - directional motions away from the center of the source. given these motions and the symmetry of the source (figs. 1 and 2), it is reasonable to assert that n5 and s5 were ejected at the same time, and likewise n2 and s2. indeed, we are fortunate enough to witness the ejection of n6 and s6 in @xmath01997 as discussed above. sensitive observations at 8.4 ghz (see fig. 2), reveal that there is continuous emission from the core out to both hot spots. the jet is probably not made up of discrete blobs that can be well described by the elliptical gaussian components that we identify and modelfit. rather the jet appears to be a continuous flow, with features of enhanced emission (shocks?) that propagate down the jet. under the assumption that components are ejected in pairs, and that all differences are due to doppler boosting and light travel time effects, we can use the observed apparent velocity ratios and differences to solve for the intrinsic component velocity and orientation. for simultaneously ejected components moving in opposite directions at an angle @xmath12 to the line of sight at an intrinsic velocity @xmath13, it follows directly from the light travel time difference that the ratio of apparent projected distances from the origin (@xmath14 for the approaching side, @xmath15 for the receding side) as well as the ratio of apparent motions (approaching : @xmath16, receding : @xmath17) is given at any time by (taylor & vermeulen 1997) : @xmath18 our other important constraint on the two parameters @xmath13 and @xmath12 can be obtained from the separation rate @xmath19, is not subject to the uncertainty in the reference point. from geometry and the conversion of angular to linear velocity we have : @xmath20 where @xmath21 is the projected separation velocity, @xmath22 is the angular size distance to the source, and @xmath23 is the redshift. note that eq. 2 has a distance dependence, while eq. 1 does not, so we can solve the system jointly for the distance and hence the hubble constant. for the n5/s5 pair we find an apparent velocity ratio @xmath24 of 3.50 @xmath25 0.44. this leads to @xmath26 = 0.56 @xmath25 0.04. at the same time, the apparent separation speed of the n5/s5 pair is 1.40 @xmath25 0.01 c. the two relations above are shown graphically in fig. 6 (bottom panel). assuming a standard cosmology and h@xmath4 = 71 km s@xmath3 mpc@xmath3 we find a common solution for an intrinsic velocity of 0.88 @xmath25 0.03 c, at an inclination of 50 @xmath25 5. a similar analysis for the n2/s2 pair (top panel of fig. 6), yields in intrinsic velocity of 0.57 @xmath25 0.11 c and with an inclination of 36 @xmath25 10. the intrinsic velocity therefore changes from 0.9 c for n5/s5 to 0.6 c for n2/s2 (a change of 2.7 @xmath6). there is no single intrinsic velocity that can fit the observations for both pairs, and can be used to directly measure the hubble constant. we can, however, place a lower limit on the hubble constant based on the fact that the intrinsic velocity must be less than the speed of light. we find that h@xmath4 @xmath2 57 km s@xmath3 mpc@xmath3 in order to achieve a valid solution for the n5/s5 pair, and the weaker constraint that h@xmath4 @xmath2 28 km s@xmath3 mpc@xmath3 in order to achieve a valid solution for the n2/s2 pair. since the velocities of n2/s2 differ from those of n5/s5 anyway, we see no reason why the latter should be particularly close to c, so for the rest of this discussion we adopt h@xmath4 = 71 km s@xmath3 mpc@xmath3. the solutions discussed above using equations 1 and 2 are based on assuming that the components in a pair, as observed, are oppositely directed and equally fast. we derive angles near 45 ; the line of sight depth difference between the components in a pair would therefore be an appreciable fraction of their total distance along the connecting line through the core, and roughly equal to the distance projected onto the plane of the sky. we derive component speeds that are an appreciable fraction of the speed of light. in combination, the speed and angle solutions imply that the receding component, due to a significant light travel time difference, should be observed at a rather younger age (25%-75%) than its approaching counterpart, and at a correspondingly smaller projected distance from the core. however, the observed arm length ratios (@xmath27) are lower by a factor of about 2 compared to the apparent velocity ratios (1.33 vs. 2.76 for n2/s2 ; and 1.80 vs.3.50 for n5/s5). this requires that currently the light travel time differences between the approaching and receding components are less significant with respect to the ages of the components than our simple speed and angle solutions suggest. the arm length ratios depend on the entire history of motion of the components, and the greater degree of symmetry in arm lengths seems to imply that for a significant part of their lifetime the components were slower and/or moving more along the plane of the sky than they are now. a fully self - consistent solution therefore will incorporate a model for the true core location and for this evolution in velocity, since, inevitably, it implies that the receding component is seen at a younger age and therefore with a different velocity than its counterpart. this leads to an extension of equations 1 and 2, into a set where the arm length ratio is directly incorporated ; helical jet models may provide a framework for this extension. however, it is beyond the scope of this paper. to properly verify such a model in the case of 1946 + 708 will require observations spanning several decades. nevertheless, we believe the velocities we have derived, while not fully self - consistent, are indicative of the fact that the jets of this source contain features that move with speeds that are an appreciable fraction of the speed of light, and certainly much faster than the advance speeds of the hot spots. furthermore, the differences in angle (by at least 15 degrees) as well as in speed (by at least 50%) between the n2/s2 and n5/s5 pairs are good indications of the kinds of changes in component velocity that are evidently occurring in and along the jets. both the discrepant arm length ratios and the observed bending in projection of the overall jets on the plane of the sky provide interesting constraints. the time evolution of the flux densities of the n2/s2 and n5/s5 pair is shown in fig. the n2 component has shown a steady rise by @xmath020% over 13 years of monitoring at 8.4 ghz. during that same time s2 has only slightly decreased. meanwhile both n5 and s5 have been declining though that appears to have leveled off starting in 2006. the amount of the decline was 75% for n5 and 57% for s5. we are currently in the unbalanced situation where s5 appears to be brighter than n5, which would at first sight seem contrary to the expectations from doppler boosting. however, we have to keep in mind that the light from s5 is delayed compared to the light that we see from n5 (by 27 years for the 2008 epoch according to the geometry derived above) so that the history of variability must be taken into account when attempting to interpret flux density ratios between component pairs. a more detailed analysis of the flux density ratios, taking into account the time variability, should eventually be possible. even so, it may not be possible to explain the very significant changes in flux density shown in fig. 7. given the stability of the component velocities (see fig. 5). local circumstances (e.g., variations in magnetic field strength or particle populations), and possible interactions of the jet components with their environments, may influence the synchrotron emissivity of the jet components. the curvature in the jets of 1946 + 708 (fig. 1) could be explained by either (1) ballistic motion from a precessing nucleus ; or (2) helical confinement of the components. in the first case we would expect components to travel in straight lines at constant apparent velocities. in the latter case we would expect that components might travel along helical trajectories at apparent velocities that changed in time. although there is no strong evidence yet for deviations from constant apparent velocities, there do appear to be some wiggles or kinks in the jets, and in particular if one considers also the earliest 5 ghz measurement (c1 in fig. 4), then there appears to have been a change in direction of several degrees. this favors the second explanation of a helically confined jet. likewise, the presence of continuous emission along the jet, well confined, even if traveling at different intrinsic velocities (n5 vs n2), supports the notional model of an intrinsically helical jet. from the modelfit analysis it is difficult to ascertain the expansion of the source. certainly it is clear that the motion is much slower than the jet components. unfortunately the southern hot spot is fairly faint and diffuse, so is correspondingly less amenable to modelfitting than the compact jet components. the northern hot spot does have a bright and compact feature, and one can measure with considerable accuracy its position relative to the core. the observed motion is @xmath70.01 mas / y, at the level that one has to be concerned about possible motions of the core component due to the ejection of the n6/s6 pair. since the modelfit results are referenced to the core component, any apparent proper motion of the core translates into the addition of a systematic apparent velocity to all components. an alternative approach is to compute a difference map between two images, well separated in time. apparent motions then show up as alternating positive and negative structures. 8 shows such an image constructed by differencing the images from the 1996 and 2008 epochs - both of which are of very high quality and low noise, and were convolved with the same restoring beam. lighter regions in fig. 8 indicate where the source was brighter in 2008. in fig. 8 one can see a relatively clear signature of motion away from the core of the southern hot spot, which appears as a positive (dark) structure (closer to the core) and a negative (whiter) structure (further from the core) in the difference image. the more rapidly moving jet components show similar structures. from measuring the advance of the edge of the southern hot spot we estimate this apparent motion to be 0.008 @xmath25 0.002 mas / y. since the apparent motion measured is relative to the northern hot spot, this is the overall apparent expansion rate of the source, and the apparent motion of the southern hot spot would presumably be half this value, or 0.004 mas / y, corresponding to a projected intrinsic velocity of 0.024 c (7000 km / s). based on the measurement of the overall expansion rate, we can derive a kinematic age for 1946 + 708 assuming a constant expansion rate. the velocity measurement derived above yields a kinematic age of 4000 @xmath25 1000 years. this is on the long end of cso ages measured to date (gugliucci et al. 2005), which have been measured between 100 and 3000 years, though the statistics are admittedly still poor. furthermore, there is a selection effect that slow expansions take longer to measure. the ratio of 15:1 for the northern to southern hot spot flux densities (derived from modelfitting), is difficult to explain in terms of doppler boosting given the very low apparent velocities measured for the hot spots. a more likely explanation is that the interstellar medium may be enhanced on the northern side, consistent with the observation of greater hi opacity to the north (peck & taylor 2001). the arm length ratio between northern and southern hot spots is close to parity, 0.94:1, with the northern hot spot being a little closer to the core. in the case of doppler boosting we would expect the northern hot spot to be further away from the core, thus we favor a denser medium on the northern side to be responsible for the difference in flux densities and arm length ratios between the two hot spots. the low observed polarization (less than 0.4% for the core, less than @xmath00.3% for the jet components, and less than 0.1% for the northern hot spot at 8.4 ghz in 1996), can be explained by faraday depolarization due to ionized gas and a magnetic field tangled on scales smaller than the angular resolution of the observations. this situation could naturally arise due to magnetic fields and free electrons associated with the accretion disk. detection of polarization has only been found in a few csos to date (gugliucci et al. those few incidents of detected polarization occur in the approaching jets of csos which are more asymmetric and core - dominated than typical csos, probably indicating a smaller angle to the line - of - sight, and therefore a more shallow faraday screen. Conclusions after a detailed, multi - frequency, multi - epoch study of the compact symmetric object 1946 + 708, we find the kinematic age of the extant radio emission to be @xmath04000 years. on timescales of @xmath010 years, outbursts occur producing symmetric components that emerge from the core and travel at intrinsic speeds between 0.6 and 0.9 c towards the hot spots. some of the individual jet components are observed to move along slightly bent, or kinked paths, but no components have yet been observed to change their speed. the jet components in general appear to be well confined, and to lie within the overall `` s - symmetric '' shape of the jets. we suggest that the jets are helically confined, rather than ballistic in nature. no linear polarization from the jets or hot - spots is detected down to quite low levels (0.1 to 0.4%), consistent with observations of csos in general. future observations over the next decade should allow for a detailed analysis of the newly ejected component pair n6/s6. together with continued observations of n2/s2 and n5/s5 it should be possible to measure accelerations in the jet components, and to tell if each component pair has an intrinsic velocity that is established upon ejection. a more sophisticated analysis, taking into account the flux history of the jet components, could provide a more stringent test of doppler boosting in 1946 + 708 and thereby ascertain the extent to which interactions with the environment are important. we thank h. smith for help with the data reduction of the 1998 and 1999 epochs. taylor gratefully acknowledges the university of bordeaux for hosting a visit during which much of this work was undertaken. this work has benefited from research funding from the european community s sixth framework programme under radionet r113ct 2003 5058187. the national radio astronomy observatory is operated by associated universities, inc., under cooperative agreement with the national science foundation. the european vlbi network is a joint facility of european, chinese, south african and other radio astronomy institutes funded by their national research councils. gugliucci, n.e., taylor, g.b., peck, a.b., & giroletti, m. 2005 apj, 622, 136 gugliucci, n.e., taylor, g.b., peck, a.b., & giroletti, m. 2007 apj, 661, 78 hjellming, r. m., & johnston, k. j. 1981,, 246, l141 fomalont, e. b. 1999, synthesis imaging in radio astronomy ii, 180, 301 owsianik, i. & conway, j. e. 1998, a&a, 337, 69 peck, a. b., taylor, g. b., fassnacht, c. .d., readhead, a. c. s., & vermeulen, r. c. 2000, apj, 534, 104 peck, a.b., & taylor, g.b. 2001, apjl, 554, l147 pihlstrm, y. m., conway, j. e., & vermeulen, r. c. 2003, a&a, 404, 871 readhead, a. c. s., taylor, g. b., xu, w., pearson, t. j., wilkinson, p. n., & polatidis, a. g. 1996, apj, 460, 612 roberts, d. h., wardle, j. f. c., lipnick, s. l., selesnick, p. l., & slutsky, s. 2008,, 676, 584 schwab, f. r., & cotton, w. d. 1983, aj, 88, 688 shepherd, m. c., pearson, t. j., & taylor, g. b. 1994, baas, 26, 987 shepherd, m. c., pearson, t. j., & taylor, g. b. 1995, baas, 27, 903 taylor, g. b., vermeulen, r. c., pearson, t. j., readhead, a. c. s., henstock, d. r., browne, i. w. a., & wilkinson, p. n. 1994, apjs, 95, 345 taylor, g. b., readhead, a. c. s., & pearson, t. j. 1996, apj, 463, 95 taylor, g. b., & vermeulen, r. c. 1997, apjl, 485, l9 taylor, g. b., marr, j. m., readhead, a. c. s., & pearson, t. j. 2000, apj, 541, 112 unwin, s. c. wehrle, a. e., lobanov, a. p., zensus, j. a., madejski, g. m., aller, m. f., & aller, h. d. 1997, apj, 480, 596 vlahakis, n. & konigl, a. 2004, apj, 605, 656 zensus, j. a., cohen, m. h., & unwin, s. c. 1995, apj, 443, 35 -12pt @xmath28 2003 feb 2 is the mean epoch for observations taken with the vlba alone on january 24, 2003 and with a global array consisting of the vlba, westerbork phased array, onsala, medicina and noto on 2003 february 10. the observations were combined for calibration and imaging. lccccccc core & reference component &... &... &... &... &... &... + n2 & 0.076 @xmath25 0.0008 & 0.90 & @xmath110.014 @xmath25 0.0002 & 3.0 & 0.077 @xmath25 0.0008 & 0.461 @xmath25 0.005 & 10.4 + s2 & @xmath110.026 @xmath25 0.0008 & 0.80 & 0.011 @xmath25 0.0004 & 3.0 & 0.028 @xmath25 0.0009 & 0.167 @xmath25 0.005 & @xmath11157.1 + n5 & 0.126 @xmath25 0.0013 & 1.4 & @xmath110.132 @xmath25 0.0014 & 2.3 & 0.182 @xmath25 0.0019 & 1.088 @xmath25 0.011 & 46.3 + s5 & @xmath110.031 @xmath25 0.0008 & 0.53 & 0.042 @xmath25 0.0007 & 2.1 & 0.052 @xmath25 0.001 & 0.311 @xmath25 0.006 & @xmath11126.4 + -5pt

we report on a multi - frequency, multi - epoch campaign of very long baseline interferometry observations of the radio galaxy 1946 + 708 using the vlba and a global vlbi array. from these high - resolution observations we deduce the kinematic age of the radio source to be @xmath04000 years, comparable with the ages of other compact symmetric objects (csos). ejections of pairs of jet components appears to take place on time scales of 10 years and these components in the jet travel outward at intrinsic velocities between 0.6 and 0.9 c. from the constraint that jet components can not have intrinsic velocities faster than light, we derive @xmath1 @xmath2 57 km s@xmath3 mpc@xmath3 from the fastest pair of components launched from the core. we provide strong evidence for the ejection of a new pair of components in @xmath01997. from the trajectories of the jet components we deduce that the jet is most likely to be helically confined, rather than purely ballistic in nature.

0904.1879

Introduction the statistical mechanics of pure systems most prominently the topic of phase transitions and their associated surface phenomena has been a subject of fairly intensive research in recent years. several physical principles for pure systems (the gibbs phase rule, wulff construction, etc.) have been put on a mathematically rigorous footing and, if necessary, supplemented with appropriate conditions ensuring their validity. the corresponding phenomena in systems with several mixed components, particularly solutions, have long been well - understood on the level of theoretical physics. however, they have not received much mathematically rigorous attention and in particular have not been derived rigorously starting from a local interaction. a natural task is to use the ideas from statistical mechanics of pure systems to develop a higher level of control for phase transitions in solutions. this is especially desirable in light of the important role that basic physics of these systems plays in sciences, both general (chemistry, biology, oceanography) and applied (metallurgy, etc.). see e.g. @xcite for more discussion. among the perhaps most interesting aspects of phase transitions in mixed systems is a dramatic _ phase separation _ in solutions upon freezing (or boiling). a well - known example from `` real world '' is the formation of brine pockets in frozen sea water. here, two important physical phenomena are observed : 1. migration of nearly all the salt into whatever portion of ice / water mixture remains liquid. clear evidence of _ facetting _ at the water - ice boundaries. quantitative analysis also reveals the following fact : 1. salted water freezes at temperatures lower than the freezing point of pure water. this is the phenomenon of _ freezing point depression_. phenomenon (1) is what `` drives '' the physics of sea ice and is thus largely responsible for the variety of physical effects that have been observed, see e.g. @xcite. notwithstanding, (13) are not special to the salt - water system ; they are shared by a large class of the so called _ non - volatile _ solutions. a discussion concerning the general aspects of freezing / boiling of solutions often referred to as _ colligative _ properties can be found in @xcite. of course, on a heuristic level, the above phenomena are far from mysterious. indeed, (1) follows from the observation that, macroscopically, the liquid phase provides a more hospitable environment for salt than the solid phase. then (3) results by noting that the migration of salt increases the entropic cost of freezing so the energy - entropy balance forces the transition point to a lower temperature. finally, concerning observation (2) we note that, due to the crystalline nature of ice, the ice - water surface tension will be anisotropic. therefore, to describe the shape of brine pockets, a wulff construction has to be involved with the caveat that here the crystalline phase is on the outside. in summary, what is underlying these phenomena is a phase separation accompanied by the emergence of a crystal shape. in the context of pure systems, such topics have been well understood at the level of theoretical physics for quite some time @xcite and, recently (as measured on the above time scale), also at the level of rigorous theorems in two @xcite and higher @xcite dimensions. the purpose of this and a subsequent paper is to study the qualitative nature of phenomena (13) using the formalism of equilibrium statistical mechanics. unfortunately, a microscopically realistic model of salted water / ice system is far beyond reach of rigorous methods. (in fact, even in pure water, the phenomenon of freezing is so complex that crystalization in realistic models has only recently and only marginally been exhibited in computer simulations @xcite.) thus we will resort to a simplified version in which salt and both phases of water are represented by discrete random variables residing at sites of a regular lattice. for these models we show that phase separation dominates a non - trivial _ region _ of chemical potentials in the phase diagram a situation quite unlike the pure system where phase separation can occur only at a single value (namely, the transition value) of the chemical potential. the boundary lines of the phase - separation region can be explicitly characterized and shown to agree with the approximate solutions of the corresponding problem in the physical - chemistry literature. the above constitutes the subject of the present paper. in a subsequent paper @xcite we will demonstrate that, for infinitesimal salt concentrations scaling appropriately with the size of the system, phase separation may still occur dramatically in the sense that a non - trivial fraction of the system suddenly melts (freezes) to form a pocket (crystal). in these circumstances the amount of salt needed is proportional to the _ boundary _ of the system which shows that the onset of freezing - point depression is actually a surface phenomenon. on a qualitative level, most of the aforementioned conclusions should apply to general non - volatile solutions under the conditions when the solvent freezes (or boils). notwithstanding, throughout this and the subsequent paper we will adopt the _ language _ of salted water and refer to the solid phase of the solvent as ice, to the liquid phase as liquid - water, and to the solute as salt. our model will be defined on the @xmath0-dimensional hypercubic lattice @xmath1. we will take the (formal) nearest - neighbor hamiltonian of the following form : @xmath2 here @xmath3 is the inverse temperature (henceforth incorporated into the hamitonian), @xmath4 and @xmath5 are sites in @xmath1 and @xmath6 denotes a neighboring pair of sites. the quantities @xmath7, @xmath8 and @xmath9 are the ice (water), liquid (water) and salt variables, which will take values in @xmath10 with the additional constraint @xmath11 valid at each site @xmath4. we will say that @xmath12 indicates the _ presence of ice _ at @xmath4 and, similarly, @xmath8 the _ presence of liquid _ at @xmath4. since a single water molecule can not physically be in an ice state, it is natural to interpret the phrase @xmath12 as referring to the collective behavior of many particles in the vicinity of @xmath4 which are enacting an ice - like state, though we do not formally incorporate such a viewpoint into our model. the various terms in are essentially self - explanatory : an interaction between neighboring ice points, similarly for neighboring liquid points (we may assume these to be attractive), an energy penalty @xmath13 for a simultaneous presence of salt and ice at one point, and, finally, fugacity terms for salt and liquid. for simplicity (and tractability), there is no direct salt - salt interaction, except for the exclusion rule of at most one salt `` particle '' at each site. additional terms which could have been included are superfluous due to the constraint. we will assume throughout that @xmath14, so that the salt - ice interaction expresses the negative affinity of salt to the ice state of water. this term is entirely and not subtly responsible for the general phenomenon of freezing point depression. we remark that by suitably renaming the variables, the hamiltonian in would just as well describe a system with boiling point elevation. as we said, the variables @xmath7 and @xmath8 indicate the presence of ice and liquid water at site @xmath4, respectively. the assumption @xmath15 guarantees that _ something _ has to be present at @xmath4 (the concentration of water in water is unity) ; what is perhaps unrealistic is the restriction of @xmath7 and @xmath8 to only the extreme values, namely @xmath16. suffice it to say that the authors are confident (e.g., on the basis of @xcite) that virtually all the results in this note can be extended to the cases of continuous variables. however, we will not make any such mathematical claims ; much of this paper will rely heavily on preexisting technology which, strictly speaking, has only been made to work for the discrete case. a similar discussion applies, of course, to the salt variables. but here our restriction to @xmath17 is mostly to ease the exposition ; virtually all of our results directly extend to the cases when @xmath9 takes arbitrary (positive) real values according to some _ a priori _ distribution. it is not difficult to see that the `` ice - liquid sector '' of the general hamiltonian reduces to a ferromagnetic ising spin system. on a formal level, this is achieved by passing to the ising variables @xmath18, which in light of the constraint gives @xmath19 by substituting these into, we arrive at the interaction hamiltonian : @xmath20 where the new parameters @xmath21 and @xmath22 are given by @xmath23 we remark that the third sum in is still written in terms of `` ice '' indicators so that @xmath24 will have a well defined meaning even if @xmath25, which corresponds to prohibiting salt entirely at ice - occupied sites. (notwithstanding, the bulk of this paper is restricted to finite @xmath13.) using an appropriate restriction to finite volumes, the above hamitonian allows us to define the corresponding gibbs measures. we postpone any relevant technicalities to section [sec2.1]. the hamiltonian as written foretells the possibility of fluctuations in the salt concentration. however, this is _ not _ the situation which is of physical interest. indeed, in an open system it is clear that the salt concentration will, eventually, adjust itself until the system exhibits a pure phase. on the level of the description provided by it is noted that, as grand canonical variables, the salt particles can be explicitly integrated, the result being the ising model at coupling constant @xmath21 and external field @xmath26, where @xmath27 in this context, phase coexistence is confined to the region @xmath28, i.e., a simple curve in the @xmath29-plane. unfortunately, as is well known @xcite, not much insight on the subject of _ phase separation _ is to be gained by studying the ising magnet in an external field. indeed, under (for example) minus boundary conditions, once @xmath22 exceeds a particular value, a droplet will form which all but subsumes the allowed volume. the transitional value of @xmath22 scales inversely with the linear size of the system ; the exact constants and the subsequent behavior of the droplet depend on the details of the boundary conditions. the described `` failure '' of the grand canonical description indicates that the correct ensemble in this case is the one with a fixed amount of salt per unit volume. (the technical definition uses conditioning from the grand canonical measure ; see section [sec2.1].) this ensemble is physically more relevant because, at the moment of freezing, the salt typically does not have enough `` mobility '' to be gradually released from the system. it is noted that, once the total amount of salt is fixed, the chemical potential @xmath30 drops out of the problem the relevant parameter is now the salt concentration. as will be seen in section [sec2], in our ising - based model of the solvent - solute system, fixing the salt concentration generically leads to _ sharp _ phase separation in the ising configuration. moreover, this happens for an _ interval _ of values of the magnetic field @xmath22. indeed, the interplay between the salt concentration and the actual external field will demand a particular value of the magnetization, even under conditions which will force a droplet (or ice crystal, depending on the boundary condition) into the system. [rem0] we finish by noting that, while the parameter @xmath22 is formally unrelated to temperature, it does to a limited extent play the role of temperature in that it reflects the _ a priori _ amount of preference of the system for water _ vs _ ice. thus the natural phase diagram to study is in the @xmath31-plane. the reasoning which led to formula allows for an immediate heuristic explanation of our principal results. the key simplification which again boils down to the absence of salt - salt interaction is that for any ising configuration, the amalgamated contribution of salt, i.e., the gibbs weight summed over salt configurations, depends only on the overall magnetization and not on the details of how the magnetization gets distributed about the system. in systems of linear scale @xmath32, let @xmath33 denote the canonical partition function for the ising magnet with constrained overall magnetization @xmath34. the total partition function @xmath35 at fixed salt concentration @xmath36 can then be written as @xmath37 where @xmath38 denotes the sum of the salt part of the boltzmann weight which only depends on the ising spins via the total magnetization @xmath34over all salt configurations with concentration @xmath36. as usual, the physical values of the magnetization are those bringing the dominant contribution to the sum in. let us recapitulate the standard arguments by first considering the case @xmath39 (which implies @xmath40), i.e., the usual ising system at external field @xmath22. here we recall that @xmath41 can approximately be written as @xmath42},\]] where @xmath43 is a suitably chosen constant and @xmath44 is a (normalized) canonical free energy. the principal fact about @xmath44 is that it vanishes for @xmath45 in the interval @xmath46 $], where @xmath47 denotes the spontaneous magnetization of the ising model at coupling @xmath21, while it is strictly positive and strictly convex for @xmath45 with @xmath48. the presence of the `` flat piece '' on the graph of @xmath44 is directly responsible for the existence of the phase transition in the ising model : for @xmath49 the dominant contribution to the grand canonical partition function comes from @xmath50 while for @xmath51 the dominant values of the overall magnetization are @xmath52. thus, once @xmath53which happens for @xmath54 with @xmath55 whenever @xmath56a phase transition occurs at @xmath57. the presence of salt variables drastically changes the entire picture. indeed, as we will see in theorem [thm1], the salt partition function @xmath38 will exhibit a nontrivial exponential behavior which is characterized by a _ strictly convex _ free energy. the resulting exponential growth rate of @xmath58 for @xmath59 is thus no longer a function with a flat piece instead, for each @xmath22 there is a _ unique _ value of @xmath45 that optimizes the corresponding free energy. notwithstanding (again, due to the absence of salt - salt interactions) once that @xmath45 has been selected, the spin configurations are the typical ising configurations with overall magnetizations @xmath60. in particular, whenever @xmath35 is dominated by values of @xmath60 for an @xmath61, a _ macroscopic droplet _ develops in the system. thus, due to the one - to - one correspondence between @xmath22 and the optimal value of @xmath45, phase separation occurs for an _ interval _ of values of @xmath22 at any positive concentration ; see fig. [fig1]. we finish with an outline of the remainder of this paper and some discussion of the companion paper @xcite. in section [sec2] we define precisely the model of interest and state our main results concerning the asymptotic behavior of the corresponding measure on spin and salt configurations with fixed concentration of salt. along with the results comes a description of the phase diagram and a discussion of freezing - point depression, phase separation, etc., see section [sec2.3]. our main results are proved in section [sec3]. in @xcite we investigate the asymptotic of infinitesimal salt concentrations. interestingly, we find that, in order to induce phase separation, the concentration has to scale at least as the inverse linear size of the system. Rigorous results with the (formal) hamiltonian in mind, we can now start on developing the _ mathematical _ layout of the problem. to define the model, we will need to restrict attention to finite subsets of the lattice. we will mostly focus on rectangular boxes @xmath62 of @xmath63 sites centered at the origin. our convention for the boundary, @xmath64, of the set @xmath65 will be the collection of sites outside @xmath66 with a neighbor inside @xmath66. for each @xmath67, we have the water and salt variables, @xmath68 and @xmath17. on the boundary, we will consider fixed configurations @xmath69 ; most of the time we will be discussing the cases @xmath70 or @xmath71, referred to as plus and minus boundary conditions. since there is no salt - salt interaction, we may as well set @xmath72 for all @xmath73. we will start by defining the interaction hamiltonian. let @xmath65 be a finite set. for a spin configuration @xmath69 and the pair @xmath74 of spin and salt configurations, we let @xmath75 here, as before, @xmath6 denotes a nearest - neighbor pair on @xmath1 and the parameters @xmath21, @xmath22 and @xmath13 are as discussed above. (in light of the discussion from section [sec1.3] the last term in has been omitted.) the probability distribution of the pair @xmath74 takes the usual gibbs - boltzmann form : @xmath76 where the normalization constant, @xmath77, is the partition function. the distributions in @xmath78 with the plus and minus boundary conditions will be denoted by @xmath79 and @xmath80, respectively. for reasons discussed before we will be interested in the problems with a fixed salt concentration @xmath81 $]. in finite volume, we take this to mean that the total amount of salt, @xmath82 is fixed. to simplify future discussions, we will adopt the convention that `` concentration @xmath36 '' means that @xmath83, i.e., @xmath84. we may then define the finite volume gibbs probability measure with salt concentration @xmath36 and plus (or minus) boundary conditions denoted by @xmath85 (or @xmath86). in light of, these are given by the formulas @xmath87 both measures @xmath88 depend on the parameters @xmath21 and @xmath13 in the hamiltonian. however, we will always regard these as fixed and suppress them from the notation whenever possible. in order to describe our first set of results, we will need to bring to bear a few standard facts about the ising model. for each spin configuration @xmath89 let us define the overall magnetization in @xmath78 by the formula @xmath90 let @xmath91 denote the magnetization of the ising model with coupling constant @xmath21 and external field @xmath92. as is well known, cf the proof of theorem [thm3.1], @xmath93 continuously (and strictly) increases from the value of the spontaneous magnetization @xmath94 to one as @xmath22 sweeps through @xmath95. in particular, for each @xmath96, there exists a unique @xmath97 such that @xmath98. next we will use the above quantities to define the function @xmath99, which represents the canonical free energy of the ising model in . as it turns out see theorem [thm3.1] in section [sec3]we simply have @xmath100 as already mentioned, if @xmath101, where @xmath102 is the critical coupling constant of the ising model, then @xmath103 and thus @xmath104 for @xmath105 $]. (since @xmath106 only for @xmath56, the resulting `` flat piece '' on the graph of @xmath107 appears only in dimensions @xmath56.) from the perspective of the large - deviation theory, cf @xcite, @xmath107 is the large - deviation rate function for the magnetization in the (unconstrained) ising model ; see again theorem [thm3.1]. let @xmath108 denote the entropy function of the bernoulli distribution with parameter @xmath109. (we will set @xmath110 whenever @xmath111 $].) for each @xmath112, each @xmath81 $] and each @xmath113 $], let @xmath114 as we will show in section [sec3], this quantity represents the entropy of configurations with fixed salt concentration @xmath36, fixed overall magnetization @xmath45 and fixed fraction @xmath115 of the salt residing `` on the plus spins '' (and fraction @xmath116 `` on the minus spins ''). having defined all relevant quantities, we are ready to state our results. we begin with a large - deviation principle for the magnetization in the measures @xmath88 : [thm1] let @xmath117 and @xmath14 be fixed. for each @xmath118, each @xmath119 and each @xmath112, we have @xmath120 here @xmath121 is given by @xmath122}{\mathscr{g}}_{h, c}(m,\theta),\]] where @xmath123 the function @xmath121 is finite and strictly convex on @xmath124 with @xmath125. furthermore, the unique minimizer @xmath126 of @xmath121 is continuous in both @xmath36 and @xmath22 and strictly increasing in @xmath22. on the basis of the above large - deviation result, we can now characterize the typical configurations of the measures @xmath88. consider the ising model with coupling constant @xmath21 and zero external field and let @xmath127 be the corresponding gibbs measure in volume @xmath78 and @xmath128-boundary condition. our main result in this section is then as follows : [thm2] let @xmath117 and @xmath14 be fixed. let @xmath118 and @xmath119, and define two sequences of probability measures @xmath129 on @xmath130 $] by the formula @xmath131\bigr)=p_l^{\pm, c, h}(m_l\le ml^d), \qquad m\in[-1,1].\]] the measures @xmath129 allow us to write the spin marginal of the measure @xmath88 as a convex combination of the ising measures with fixed magnetization ; i.e., for any set @xmath132 of configurations @xmath133, we have @xmath134 moreover, if @xmath126 denotes the unique minimizer of the function @xmath121 from, then the following properties are true : 1. given the spin configuration on a finite set @xmath65, the salt variables on @xmath66 are asymptotically independent. explicitly, for each finite set @xmath65 and any two configurations @xmath135 and @xmath136, @xmath137 where the numbers @xmath138 $] are uniquely determined by the equations @xmath139 2. the measure @xmath129 converges weakly to a point mass at @xmath126, @xmath140 in particular, the ising - spin marginal of the measure @xmath88 is asymptotically supported on the usual ising spin configurations with the overall magnetization @xmath141, where @xmath45 minimizes @xmath121. the fact that conditioning @xmath88 on a fixed value of magnetization produces the ising measure under same conditioning which is the content of is directly related to the absence of salt - salt interaction. the principal conclusions of the previous theorem are thus parts (1) and (2), which state that the presence of a particular amount of salt _ forces _ the ising sector to choose a particular value of magnetization density. the underlying variational principle provides insight into the physical mechanism of phase separation upon freezing of solutions. (we refer the reader back to section [sec1.4] for the physical basis of these considerations.) we will proceed by discussing the consequences of these results for the phase diagram of the model and, in particular, the phenomenon of freezing point depression. theorems [thm1] and [thm2] are proved in section [sec3.2]. [fig1] . the horizontal axis marks the concentration of the salt in the system, the vertical line represents the external field acting on the ising spins see formula . for positive concentrations @xmath142, the system stays in the liquid - water phase throughout a non - trivial range of negative values of @xmath22a manifestation of the freezing - point depression. for @xmath143 in the shaded region, a non - trivial fraction of the system is frozen into ice. once @xmath143 is on the left of the shaded region, the entire system is in the ice state. for moderate values of @xmath13, the type of convexity of the transition lines may change from concave to convex near @xmath144 ; see the companion paper @xcite., width=336] . the horizontal axis marks the concentration of the salt in the system, the vertical line represents the external field acting on the ising spins see formula . for positive concentrations @xmath142, the system stays in the liquid - water phase throughout a non - trivial range of negative values of @xmath22a manifestation of the freezing - point depression. for @xmath143 in the shaded region, a non - trivial fraction of the system is frozen into ice. once @xmath143 is on the left of the shaded region, the entire system is in the ice state. for moderate values of @xmath13, the type of convexity of the transition lines may change from concave to convex near @xmath144 ; see the companion paper @xcite., width=336] the representation along with the asymptotic allow us to characterize the distribution @xmath88 in terms of the canonical ensemble of the ising ferromagnet. indeed, these formulas imply that the distribution of ising spins induced by @xmath88 is very much like that in the measure @xmath127 conditioned on the event that the overall magnetization @xmath145 is near the value @xmath146. as a consequence, the asymptotic statements (e.g., the wulff construction) that have been (or will be) established for the spin configurations in the ising model with fixed magnetization will automatically hold for the spin - marginal of the @xmath88 as well. a particular question of interest in this paper is phase separation. recall that @xmath47 denotes the spontaneous magnetization of the ising model at coupling @xmath21. then we can anticipate the following conclusions about typical configurations in measure @xmath88 : 1. if @xmath147, then the entire system (with plus boundary condition) will look like the plus state of the ising model whose external field is adjusted so that the overall magnetization on the scale @xmath148 is roughly @xmath146. 2. if @xmath149, then the system (with minus boundary condition) will look like the ising minus state with similarly adjusted external field. if @xmath150, then, necessarily, the system exhibits phase separation in the sense that typical configurations feature a large droplet of one phase inside the other. the volume fraction taken by the droplet is such that the overall magnetization is near @xmath151. the outer phase of the droplet agrees with the boundary condition. the cases (1 - 2) with opposite boundary conditions that is, the minus boundary conditions in (1) and the plus boundary conditions in (2)are still as stated ; the difference is that now there has to be a large contour near the boundary flipping to the `` correct '' boundary condition. [rem1] we have no doubt that the aforementioned conclusions (1 - 3) hold for all @xmath56 and all @xmath101 (with a proper definition of the _ droplet _ in part (3), of course). however, the depth of conclusion (3) depends on the level of understanding wulff construction, which is at present rather different in dimensions @xmath152 and @xmath153. specifically, while in @xmath152 the results of @xcite allow us to claim that for all @xmath101 and all magnetizations @xmath61, the system will exhibit a unique large contour with appropriate properties, in @xmath153 this statement is known to hold @xcite only in `` @xmath154-sense '' and only for @xmath61 which are near the endpoints. (moreover, not all values of @xmath101 are, in principle, permitted ; cf @xcite for a recent improvement of these restrictions.) we refer to @xcite for an overview of the situation. notwithstanding the technical difficulties of wulff construction, the above allows us to characterize the phase diagram of the model at hand. as indicated in fig. [fig1], the @xmath155 and @xmath156 quadrant splits into three distinct parts : the _ liquid - water _ region, the _ ice _ region and the _ phase separation _ region, which correspond to the situations in (1 - 3), respectively. the boundary lines of the phase - separation region are found by setting @xmath157 which in light of strict monotonicity of @xmath158 allows us to calculate @xmath22 as a function of @xmath36. the solutions of can be obtained on the basis of the following observation : [prop2b] let @xmath105 $] and @xmath81 $] and define the quantities @xmath159 by . let @xmath22 be the solution to @xmath160. then we have : @xmath161 in particular, there exist two continuous and decreasing functions @xmath162 $] with @xmath163 for all @xmath142, such that @xmath164 is equivalent to @xmath165 for all @xmath142. proposition [prop2b] is proved at the very end of section [sec3.2]. here is an informal interpretation of this result : the quantities @xmath166 represent the _ mole fractions _ of salt in liquid - water and ice, respectively. in mathematical terms, @xmath167 is the probability of having a salt particle on a given plus spin, and @xmath168 is the corresponding quantity for minus spins, see . formula quantifies the shift of the chemical potential of the solvent (which is given by @xmath169 in this case) due to the presence of the solute. this is a manifestation of _ freezing point depression _, see also remark [rem0]. in the asymptotic when @xmath170 we have @xmath171 this relation, derived in standard chemistry and physics books under the auspicies of the `` usual approximations, '' is an essential ingredient in the classical analyses of colligative properties of solutions @xcite. here the derivation is a direct consequence of a microscopic (albeit simplistic) model which further offers the possibility of systematically calculating higher - order corrections. Proofs the proofs of our main results are, more or less, straightforward exercises in large - deviation analysis of gibbs distributions. we first state and prove a couple of technical lemmas ; the actual proofs come in section [sec3.2]. the starting point of the proof of theorem [thm1] (and, consequently, theorem [thm2]) is the following large - deviation principle for the ising model at zero external field : [thm3.1] consider the ising model with coupling constant @xmath172 and zero external field. let @xmath127 be the corresponding (grand canonical) gibbs measure in volume @xmath78 and @xmath128-boundary conditions. then for all @xmath173 $], @xmath174 where @xmath145 is as in and @xmath175 is as defined in. proof the claim is considered standard, see e.g. (*??? * section ii.1), and follows by a straightforward application of the thermodynamic relations between the free energy, magnetization and external field. for completeness (and reader s convenience) we will provide a proof. consider the function @xmath176, where @xmath177 is the expectation with respect to @xmath127, and let @xmath178. the limit exists by subadditivity arguments and is independent of the boundary condition. the function @xmath179 is convex on @xmath180, real analytic (by the lee - yang theorem @xcite) on @xmath181, and hence it is strictly convex on @xmath180. by the @xmath182 symmetry there is a cusp at @xmath57 whenever @xmath183. it follows that for each @xmath184 there is a unique @xmath185 such that @xmath186, with @xmath187 increasing continuously from @xmath188 to @xmath189 as @xmath45 increases from @xmath190 to @xmath191. the plus - minus symmetry shows that a similar statement holds for the magnetizations in @xmath192 $]. let @xmath193 denote the legendre transform of @xmath194, i.e., @xmath195 $]. by the above properties of @xmath179 we infer that @xmath196 when @xmath197 and @xmath198, while @xmath199 for @xmath105 $]. applying the grtner - ellis theorem (see (*??? * theorem v.6) or (*??? * theorem 2.3.6)), we then have with @xmath200 for all @xmath201$]which is the set of so called exposed points of @xmath193. since @xmath202 and the derivative of @xmath203 is @xmath187, this @xmath175 is given by the integral in. to prove when @xmath105 $], we must note that the left - hand side of is nonpositive and concave in @xmath45. (this follows by partitioning @xmath78 into two parts with their own private magnetizations and disregarding the interaction through the boundary.) since @xmath44 tends to zero as @xmath45 tends to @xmath204 we thus have that for @xmath105 $] as well. the `` first '' part of the grtner - ellis theorem (*??? * theorem v.6) actually guarantees the following _ large - deviation principle _ : @xmath205 for any closed set @xmath206 while @xmath207}\phi^\star(m)\]] for any open set @xmath208. (here @xmath209 for @xmath173 $] and @xmath210 otherwise.) the above proof follows by specializing to @xmath211-neighborhoods of a given @xmath45 and letting @xmath212. the @xmath105 $] cases i.e, the non - exposed points have to be dealt with separately. the above is the core of our proof of theorem [thm1]. the next step will be to bring the quantities @xmath36 and @xmath22 into play. this, as we shall see, is easily done if we condition on the total magnetization. (the cost of this conditioning will be estimated by.) indeed, as a result of the absence of salt - salt interaction, the conditional measure can be rather precisely characterized. let us recall the definition of the quantity @xmath213 from which represents the total amount of salt in the system. for any spin configuration @xmath89 and any salt configuration @xmath214, let us introduce the quantity @xmath215 representing the total amount of salt `` on the plus spins. '' then we have : [lemma3.2] for any fixed spin configuration @xmath216, all salt configurations @xmath217 with the same @xmath213 and @xmath218 have the same probability in the conditional measure @xmath219. moreover, for any @xmath220 with @xmath221 and for any @xmath173 $], @xmath222 where the normalization constant is given by @xmath223 here @xmath177 is the expectation with respect to @xmath127. proof the fact that all salt configurations with given @xmath213 and @xmath218 have the same probability in @xmath219 is a consequence of the observation that the salt - dependent part of the hamiltonian depends only on @xmath218. the relations ([3.3][3.4]) follow by a straightforward rewrite of the overall boltzmann weight. the characterization of the conditional measure @xmath224 from lemma [lemma3.2] allows us to explicitly evaluate the configurational entropy carried by the salt. specifically, given a spin configuration @xmath89 and numbers @xmath225, let @xmath226 the salt entropy is then the rate of exponential growth of the size of @xmath227 which can be related to the quantity @xmath228 from as follows : [lemma3.3] for each @xmath229 and each @xmath230 there exists a number @xmath231 such that the following is true for any @xmath225, any @xmath112 that obey @xmath232, @xmath233 and any @xmath234 : if @xmath89 is a spin configuration with @xmath235, then @xmath236 proof we want to distribute @xmath237 salt particles over @xmath238 positions, such that exactly @xmath239 of them land on @xmath240 plus sites and @xmath241 on @xmath242 minus sites. this can be done in latexmath:[\[\label{3.8a } number of ways. now all quantities scale proportionally to @xmath238 which, applying stirling s formula, shows that the first term is within, say, @xmath244 multiples of @xmath245 once @xmath234, with @xmath246 depending only on @xmath247. a similar argument holds also for the second term with @xmath115 replaced by @xmath116 and @xmath45 by @xmath248. combining these expressions we get that @xmath249 is within @xmath250 multiples of @xmath251 once @xmath32 is sufficiently large. for the proof of theorem [thm2], we will also need an estimate on how many salt configurations in @xmath227 take given values in a finite subset @xmath252. to that extent, for each @xmath253 and each @xmath135 we will define the quantity @xmath254 as a moment s thought reveals, @xmath255 can be interpreted as the probability that @xmath256 occurs in (essentially) any homogeneous product measure on @xmath214 conditioned to have @xmath257 and @xmath258. it is therefore not surprising that, for spin configurations @xmath259 with given magnetization, @xmath260 will tend to a product measure on @xmath261. a precise characterization of this limit is as follows : [lemma3.4] for each @xmath262, each @xmath263 and each @xmath230 there exists @xmath231 such that the following holds for all @xmath234, all @xmath252 with @xmath264, all @xmath45 with @xmath232 and all @xmath265 $] for which @xmath266 satisfy @xmath267 $] : if @xmath89 is a spin configuration such that @xmath235 and @xmath135 is a salt configuration in @xmath66, then @xmath268 proof we will expand on the argument from lemma [lemma3.3]. indeed, from we have an expression for the denominator in. as to the numerator, introducing the quantities @xmath269 and the shorthand @xmath270 the same reasoning as we used to prove allows us to write the object @xmath271 as @xmath272, where the various parameters are as follows : the quantities @xmath273 represent the total number of pluses and minuses in the system, respectively, @xmath274 are the numbers of salt particles on pluses and minuses, and, finally, @xmath275 are the corresponding quantities for the volume @xmath66, respectively. since and the restrictions on @xmath232 and @xmath265 $] imply that @xmath276, @xmath277, @xmath278, @xmath279, @xmath280 and @xmath281 all scale proportionally to @xmath238, uniformly in @xmath259 and @xmath282, while @xmath283 and @xmath284 are bounded by @xmath285which by our assumption is less than @xmath286we are in a regime where it makes sense to seek an asymptotic form of quantity @xmath287. using the bounds @xmath288 which are valid for all integers @xmath289 and @xmath290 with @xmath291, we easily find that @xmath292 since @xmath293 and @xmath294 as @xmath295, while @xmath283, @xmath296, @xmath284 and @xmath297 stay bounded, the desired claim follows by taking @xmath32 sufficiently large. the reader may have noticed that, in most of our previous arguments, @xmath115 and @xmath45 were restricted to be away from the boundary values. to control the situation near the boundary values, we have to prove the following claim : [lemma3.5] for each @xmath298 and each @xmath299, let @xmath300 be the event @xmath301 then for each @xmath118 and each @xmath119 there exists an @xmath262 such that @xmath302 proof we will split the complement of @xmath300 into four events and prove the corresponding estimate for each of them. we begin with the event @xmath303. the main tool will be stochastic domination by a product measure. consider the usual partial order on spin configurations defined by putting @xmath304 whenever @xmath305 for all @xmath4. let @xmath306 be the conditional probability that @xmath307 occurs given a spin configuration @xmath308 in @xmath309 and a salt configuration @xmath310 in @xmath78, optimized over all @xmath308, @xmath310 and also @xmath311 and the system size. since @xmath312 reduces to (the exponential of) the local interaction between @xmath313 and its ultimate neighborhood, we have @xmath314. using standard arguments it now follows that the spin marginal of @xmath88 stochastically dominates the product measure @xmath315 defined by @xmath316 for all @xmath4. in particular, we have @xmath317 let @xmath318. then @xmath319namely, the expectation of @xmath313 with respect to @xmath315exceeds the negative of @xmath320 and so cramr s theorem (see (*??? * theorem i.4) or (*??? * theorem 2.1.24)) implies that the probability on the right - hand side decays to zero exponentially in @xmath238, i.e., @xmath321 the opposite side of the interval of magnetizations, namely, the event @xmath322, is handled analogously (with @xmath323 now focusing on @xmath324 instead of @xmath325). the remaining two events, marking when @xmath218 is either less than @xmath211 or larger than @xmath320 times the total number of plus spins, are handled using a similar argument combined with standard convexity estimates. let us consider the event @xmath326which contains the event @xmath327and let us emphasize the dependence of the underlying probability distribution on @xmath13 by writing @xmath88 as @xmath328. let @xmath329 denote the expectation with respect to @xmath328 and note that @xmath330. we begin by using the chernoff bound to get @xmath331 a routine application of jensen s inequality gives us @xmath332 it thus suffices to prove that there exists a @xmath333 such that @xmath334 is positive. (indeed, we take @xmath211 to be strictly less than this number and set @xmath335 to observe that the right - hand side decays exponentially in @xmath238.) to show this we write @xmath336 as the sum of @xmath337 over all @xmath311. looking back at, we then have @xmath338, where @xmath323 is now evaluated for @xmath339, and so @xmath340 thus, once @xmath341, the probability @xmath342 decays exponentially in @xmath238. as to the complementary event, @xmath343, we note that this is contained in @xmath344, where @xmath345 counts the number of plus spins with no salt on it. since we still have @xmath346, the proof boils down to the same argument as before. on the basis of the above observations, the proofs of our main theorems are easily concluded. however, instead of theorem [thm1] we will prove a slightly stronger result of which the large - deviation part of theorem [thm1] is an easy corollary. [thm3.5] let @xmath117 and @xmath347 be fixed. for each @xmath348, each @xmath119 and each @xmath112, let @xmath349 be the set of all @xmath350 for which @xmath351 and @xmath352 hold. then @xmath353 \end{subarray } } { \mathscr{g}}_{h, c}(m',\theta'),\]] where @xmath354 is as in. proof since the size of the set @xmath227 depends only on the overall magnetization, let @xmath355 denote this size for the configurations @xmath259 with @xmath356. first we note that, by lemma [lemma3.2], @xmath357 where @xmath358 here @xmath359 is the normalization constant from which in the present formulation can also be interpreted as the sum of @xmath360 over the relevant (discrete) values of @xmath45 and @xmath115. let @xmath361 denote the sum of @xmath362 over all @xmath363 and @xmath364 for which @xmath365 and @xmath366 are integers and @xmath367 and @xmath368. (this is exactly the set of magnetizations and spin - salt overlaps contributing to the set @xmath369.) applying to extract the exponential behavior of the last probability in, and using to do the same for the quantity @xmath355, we get @xmath370 where @xmath247 is as in. as a consequence of the above estimate we have @xmath371 for any @xmath112 and any @xmath372. next we will attend to the denominator in. pick @xmath373 and consider the set latexmath:[\[{\mathcal m}_\delta=\bigl\{(m,\theta)\colon write @xmath359 as a sum of two terms, @xmath375, with @xmath376 obtained by summing @xmath377 over the admissible @xmath378 and @xmath379 collecting the remaining terms. by lemma [lemma3.5] we know that @xmath380 decays exponentially in @xmath238 and so the decisive contribution to @xmath359 comes from @xmath376. assuming that @xmath381, let us cover @xmath382 by finite number of sets of the form @xmath383\times[\theta_\ell'-\epsilon,\theta_\ell'+\epsilon]$], where @xmath384 and @xmath385 are such that @xmath386 and @xmath387 are integers. then @xmath376 can be bounded as in @xmath388 moreover, the right - hand side is bounded by the left - hand side times a polynomial in @xmath32. taking logarithms, dividing by @xmath238, taking the limit @xmath295, refining the cover and applying the continuity of @xmath389 allows us to conclude that @xmath390}{\mathscr{g}}_{h, c}(m,\theta).\]] combining these observations, is proved. proof of theorem [thm1] the conclusion follows from by similar arguments that prove. the only remaining thing to prove is the strict convexity of @xmath121 and continuity and monotonicity of its minimizer. first we note that @xmath391 is strictly convex on the set of @xmath115 where it is finite, which is a simple consequence of the strict convexity of @xmath392. hence, for each @xmath45, there is a unique @xmath393 which minimizes @xmath391. our next goal is to show that, for @xmath394, the solution @xmath393 will satisfy the inequality @xmath395 (a heuristic reason for this is that @xmath396 corresponds to the situation when the salt is distributed independently of the underlying spins. this is the dominating strategy for @xmath397 ; once @xmath14 it is clear that the fraction of salt on plus spins _ must _ increase.) a formal proof runs as follows : we first note that @xmath398 solves for @xmath115 from the equation @xmath399 where @xmath228 is as in. but @xmath400 is strictly concave and its derivative vanishes at @xmath401. therefore, for @xmath394 the solution @xmath393 of must obey. let @xmath402 be the set of @xmath403 for which holds and note that @xmath402 is convex. a standard second - derivative calculation now shows that @xmath354 is strictly convex on @xmath402. (here we actually differentiate the function @xmath404which is twice differentiable on the set where it is finite and then use the known convexity of @xmath44. the strict convexity is violated on the line @xmath401 where @xmath389 has a flat piece for @xmath105 $].) now, since @xmath405 minimizes @xmath354 for a given @xmath45, the strict convexity of @xmath354 on @xmath402 implies that for any @xmath406, @xmath407 hence, @xmath121 is also strictly convex. the fact that @xmath408 diverges as @xmath409 is a consequence of the corresponding property of the function @xmath107 and the fact that the rest of @xmath410 is convex in @xmath45. as a consequence of strict convexity and the abovementioned `` steepness '' at the boundary of the interval @xmath124, the function @xmath121 has a unique minimizer for each @xmath119 and @xmath142, as long as the quantities from satisfy @xmath411. the minimizer is automatically continuous in @xmath22 and is manifestly non - decreasing. furthermore, the continuity of @xmath412 in @xmath36 allows us to conclude that @xmath405 is also continuous in @xmath36. what is left of the claims is the _ strict _ monotonicity of @xmath45 as a function of @xmath22. writing @xmath413 as @xmath414 and noting that @xmath415 is continuously differentiable on @xmath124, the minimizing @xmath45 satisfies @xmath416 but @xmath417 is also strictly convex and so @xmath418 is strictly increasing. it follows that @xmath45 has to be strictly increasing with @xmath22. theorem [thm3.1] has the following simple consequence that is worth highlighting : [cor3.6] for given @xmath119 and @xmath118, let @xmath419 be the minimizer of @xmath354. then for all @xmath262, @xmath420 proof on the basis of and the fact that @xmath354 has a unique minimizer, a covering argument same as used to prove implies that the probability on the left - hand side decays to zero exponentially in @xmath238. before we proceed to the proof of our second main theorem, let us make an observation concerning the values of @xmath421 at the minimizing @xmath45 and @xmath115 : [lemma3.7] let @xmath119 and @xmath118 be fixed and let @xmath419 be the minimizer of @xmath354. define the quantities @xmath159 by and @xmath422 by. then @xmath423 moreover, @xmath166 are then related to @xmath22 via whenever @xmath105 $]. proof first let us ascertain that @xmath166 are well defined from equations. we begin by noting that the set of possible values of @xmath424 is the unit square @xmath425 ^ 2 $]. as is easily shown, the first equation in corresponds to an increasing curve in @xmath425 ^ 2 $] connecting the corners @xmath426 and @xmath427. on the other hand, the second equation in is a straight line with negative slope which by the fact that @xmath428 intersects both the top and the right side of the square. it follows that these curves intersect at a single point the unique solution of. next we will derive equations that @xmath421 have to satisfy. let @xmath419 be the unique minimizer of @xmath354. the partial derivative with respect to @xmath115 yields @xmath429 and from the very definition of @xmath421 we have @xmath430 noting that @xmath431, we now see that @xmath421 satisfies the same equations as @xmath166 and so, by the above uniqueness argument, must hold. to prove relation, let us also consider the derivative of @xmath354 with respect to @xmath45. for solutions in @xmath46 $] we can disregard the @xmath175 part of the function (because its vanishes along with its derivative throughout this interval), so we have @xmath432 a straightforward calculation then yields. now we are ready to prove our second main result : proof of theorem [thm2] the crucial technical step for the present proof has already been established in lemma [lemma3.2]. in order to plug into the latter result, let us note that the sum of @xmath433 over all salt configurations @xmath214 with @xmath221 is a number depending only on the total magnetization @xmath434. lemma [lemma3.2] then implies @xmath435 where @xmath436 is a positive number depending on @xmath45, the parameters @xmath36, @xmath22, @xmath21 and the boundary condition @xmath128 but not on the event @xmath132. noting that @xmath129 is simply the distribution of the random variables @xmath437 in measure @xmath88, this proves . in order to prove the assertion, we let @xmath438, pick @xmath252 and fix @xmath439. since lemma [lemma3.2] guarantees that, given @xmath440, all salt configurations with fixed @xmath218 and concentration @xmath36 have the same probability in @xmath219, we have @xmath441 where @xmath442 is defined in. pick @xmath230 and assume, as in lemma [lemma3.4], that @xmath443 $], @xmath444 $] and @xmath445 for some @xmath45 with @xmath232. then the aforementioned lemma tells us that @xmath446 is within @xmath211 of the probability that @xmath447 occurs in the product measure where the probability of @xmath448 is @xmath449 if @xmath450 and @xmath451 if @xmath452. let @xmath419 be the unique minimizer of @xmath354. taking expectation of over @xmath453 with @xmath454 fixed, using corollary [cor3.6] to discard the events @xmath455 or @xmath456 and invoking the continuity of @xmath421 in @xmath45 and @xmath115, we find out that @xmath457 indeed converges to @xmath458 with @xmath421 evaluated at the minimizing @xmath419. but for this choice lemma [lemma3.7] guarantees that @xmath459, which finally proves ([2.7][2.8]). the last item to be proved is proposition [prop2b] establishing the basic features of the phase diagram of the model under consideration : proof of proposition [prop2b] from lemma [lemma3.7] we already know that the set of points @xmath160 for @xmath105 $] is given by the equation. by the fact that @xmath460 is strictly increasing in @xmath22 and that @xmath461 as @xmath462 we thus know that defines a line in the @xmath143-plane. specializing to @xmath463 gives us two curves parametrized by functions @xmath464 such that at @xmath143 satisfying @xmath165 the system magnetization @xmath460 is strictly between @xmath465 and @xmath190, i.e., @xmath143 is in the phase separation region. it remains to show that the above functions @xmath466 are strictly monotone and negative for @xmath142. we will invoke the expression which applies because on the above curves we have @xmath467 $]. let us introduce new variables @xmath468 and, writing @xmath22 in in terms of @xmath469, let us differentiate with respect to @xmath36. (we will denote the corresponding derivatives by superscript prime.) since gives us that @xmath470, we easily derive @xmath471 thus, @xmath472 and @xmath473 have opposite signs ; i.e., we want to prove that @xmath474. but that is immediate : by the second equation in we conclude that at least one of @xmath475 must be strictly positive, and by @xmath470 we find that both @xmath476. it follows that @xmath466 are strictly decreasing, and since @xmath477, they are also negative once @xmath142. Acknowledgments the research of k.s.a. was supported by the nsf under the grants dms-0103790 and dms-0405915. the research of m.b. and l.c. was supported by the nsf grant dms-0306167. p. curie, _ sur la formation des cristaux et sur les constantes capillaires de leurs diffrentes faces _, bull.. mineral. * 8 * (1885) 145 ; reprinted in _ uvres de pierre curie _, gauthier - villars, paris, 1908, pp. 153157.

using the formalism of rigorous statistical mechanics, we study the phenomena of phase separation and freezing - point depression upon freezing of solutions. specifically, we devise an ising - based model of a solvent - solute system and show that, in the ensemble with a fixed amount of solute, a macroscopic phase separation occurs in an interval of values of the chemical potential of the solvent. the boundaries of the phase separation domain in the phase diagram are characterized and shown to asymptotically agree with the formulas used in heuristic analyses of freezing point depression. the limit of infinitesimal concentrations is described in a subsequent paper. = 1

math-ph0407034

Introduction the variability of ex lup was discovered by miss e. janssen in 1944 while examining spectral plates at the harvard observatory (mclaughlin 1946). herbig (1950) first pointed out the similarity of ex lupi s spectral characteristics and t tauri stars with strong emission lines of h, caii, feii, and hei. in one of the spectrograms he obtained in 1949/1950 the h and caii lines clearly show an inverse p cygni profile. herbig (1977a) assigned the spectral type of m0:ev using the 5850 - 6700 range. photographic and visual light - curves covering a century of observations revealed the irregular photometric behaviour of the star (mclaughlin 1946, bateson et al. outbursts of up to 5 magnitudes may occur, but the star normally shows only small amplitude irregular variations. the most prominent events last about one year. the typical recurrence time scale of outbursts is of the order of a decade. + up to now there are only a few other stars known with comparable outburst characteristics (herbig 1989). this small group of very active t tauri stars has been called exors or sometimes sub - fuors. both names point to an affinity to the so called fu orionis stars (fuors). fuors are another group of young low mass stars with even stronger outbursts lasting for decades. unlike exors, during an outburst fuor spectra turn from t tauri characteristics to that of much earlier f or g supergiants lacking strong line emission (herbig 1977b). fuors have high mass accretion rates (@xmath0,hartmann 1991) and strong winds (e.g. calvet et al. 1993) and they may be the source that drive herbig - haro flows (reipurth 1989). + exors are little studied, but potentially of great interest because they may represent an intermediate level of activity between ordinary active t tauri stars and full blown fu orionis eruptions. in order to cast further light on this interpretation, we have followed some exors spectroscopically and photometrically during 1993 and 1994. Observations the star ex lup has been at a low level of activity during the 1980 s. in the early 1990 s this situation changed and the star became active (jones et al. 1993, hughes et al. amateur observations (variable star section of the royal astronomical society of new zealand, unpublished) indicated a strong brightening in february / march 1994. patten (1994) reports some follow - up photometric and low resolution spectroscopic observations of the same outburst. + in this paper we present part of our optical observations of ex lup taken at eso, la silla. we concentrate on data obtained during the outburst in march 1994 and include some spectroscopic observations carried out in august 1994 when the star only exhibited post - outburst low level activity. a complete presentation of our data will appear in a future paper. Photometric results differential ccd photometry has been carried out at the 0.9m - dutch and the 1.54m - danish telescopes. this photometry was later calibrated with respect to standard stars including extinction and colour corrections. all reductions have been made with the apphot package in iraf. typical errors (1@xmath1) in the differential photometry are @xmath2b=0.005, @xmath2v=0.004, @xmath2r=0.004 whereas the absolute magnitude scale itself is accurate to about 0.01 in all three colours. + the resulting lightcurves in b, v, and r are presented in fig. 1. the maximum occurred between february 25 and march 4 (herbig, priv. the fading tail of the eruption can be described as an exponential decline with small fluctuations superimposed. variability of more than 0.1mag is present on timescales of less than one hour (e.g. march 6.3, see also patten 1994). figure 2 displays the colour change in b - v during the decline. the star clearly becomes redder when fading. for comparison we have included some points close to minimum light taken from the literature. the outburst amplitude was about @xmath2v=2.0mag and @xmath2b=2.6mag . Spectroscopic results spectroscopic observations in the blue spectral range were carried out during the first few nights in march 1994 on the eso-1.52 m telescope using the boller & chivens spectrograph at 1.2 resolution. after the decline of ex lup we obtained post - outburst spectra in the same wavelength region at resolutions of 1.5 and 12 at the 3.5m - ntt with emmi in august 1994. all spectra have been reduced with the ctioslit package in iraf. observations of spectrophotometric standards and nightly extinction curves allowed for a flux calibration. + in fig.3 we present two spectra of ex lup : one close to the outburst maximum and the other at low activity almost half a year after the eruption. some of the emission lines of h, caii, feii, hei, and heii are indicated. under the assumption that the total light can be decomposed into an underlying t tauri star photosphere, a continuum source, and superimposed emission lines, we now discuss the different spectral components and their variability. a powerful method to determine the continuum excess emission is to determine the veiling by comparison with spectra of stars of the same spectral type and luminosity class but lacking any disk signature (hartigan et al. 1989, 1991). the accuracy of the veiling determination decreases rapidly when the emission component exceeds the photospheric luminosity. in the case of ex lup during its eruption we therefore did not intend to derive the veiling and the true excess emission spectrum by comparison with spectral type standards, but we could examine the spectral variability caused by the outburst. + no photospheric absorption features are seen during the outburst (upper spectrum in fig.3) but they appear in the post - outburst spectrum. thus the major source of variability presumably is a featureless continuum. therefore, a difference spectrum between outburst and post - outburst spectra should be a good measure of the continuum emission spectrum. in fig. 4 we plot two difference spectra at low resolution. the first shows the difference between an outburst (march 3) and a post - outburst (august 16) spectrum, while the second shows the difference between two post - outburst (august 18 and 16) spectra which displays normal low - level variability. the continuum emission spectrum displaying the normal low - level activity is bluer than the continuum emission present during outburst. the most intriguing features in the spectra of ex lup are strong emission lines. the balmer series can be seen up to h15 especially during times of minimum activity. equivalent widths and fluxes of individual lines are given in table 1. essentially all strong emission lines have increasing fluxes as the star brightens. however due to the steep rise of the continuum the equivalent widths decrease, which is also evident in the data from patten (1994) at h@xmath3, h@xmath4, and h@xmath5 during the maximum. obviously the caii lines have a larger flux amplification during the outburst than the balmer lines. there is some indication that line fluxes of metals do not increase while the star goes into outburst (cai, feii, srii). + lrrrr & & & & + identification & w@xmath6 & w@xmath7 & flux@xmath8 & flux@xmath9 + & & & @xmath10 & @xmath10 + & & & & + & & & & + h11 3771 & -1.0 & -4.2 & 16 & 6 + h10 3798 & -1.5 & -7.0 & 25 & 9 + h 9 3835 & -1.2 & -12.1 & 20 & 16 + h 8 3889 & -2.7 & -13.0 & 44 & 18 + sii 3906 & -0.2 & -0.8 & 3 & 1 + caii 3934 & -7.7 & -12.0 & 123 & 15 + .comparison of selected emission lines at different levels of activity. equivalent widths and line fluxes during the outburst measured on march 3 (@xmath8) and in the post - outburst spectrum on august 16 (@xmath9) [cols= " < ",] @xmath11 & -0.4 & -1.7 & 8 & 3 + heii 4686 & -0.3 & -0.9 & 6 & 2 + h@xmath4 4861 & -9.4 & -16.8 & 196 & 30 + feii 4924 & & -1.5 & & 2 + the presence of inverse p cygni profiles in the strongest emission lines during outburst, as first noted by herbig (1950), is here corroborated. at balmer lines higher than h9 the equivalent width of the redshifted absorption dip is even larger than the width of the emission component. comparing the sequence of spectra between march 3 and 6 we can see a substantial fading of the absorption. the mean velocity displacement of the absorption measured in these spectra is @xmath12 km / s. this absorption component is still visible in our spectrum taken on august 18 (fig.5a). we also plot the difference between the two spectra from august 18 and august 16 to enhance the visibility of the absorption dip and to remove possible contamination due to photospheric lines. the displacement of the absorption dip measured in the post - outburst difference spectrum corresponds to a velocity of @xmath13 km / s. photospheric features of the underlying t tauri star can be seen only in the post - outburst spectra. figure 6 shows the region around cai 4227, which is the strongest stellar absorption line, in two post - outburst spectra. the difference of these two spectra no longer exhibits the absorption line, and the change of total flux by about 40% is therefore due to continuum emission rather than photospheric variability. + the photospheric lines of the t tauri star are veiled, even at minimum brightness. the superimposed emission line spectrum additionally fills in many absorption lines. the measurement of the veiling is therefore rather difficult. we find a good fit to the observed strength of absorption lines by introducing a flat continuum emission equal to the photospheric continuum of the underlying star (veiling r=1, comparison with hd 202560, spectral type m0v) at 4200 when the brightness of ex lup is v=13.0. Discussion and conclusions the outburst of ex lup can be understood in terms of a mass accretion event causing increased continuum emission in a hot region close to the surface of the star where the infalling matter finally releases its kinetic energy. the total light of the photosphere and the hot region becomes dominated by the latter and therefore it is much bluer during the outburst. furthermore all photospheric lines are heavily veiled (assuming that r=1 at minimum light then the veiling during outburst would be r@xmath1420). the different slope of the continuum emission in the outburst compared to the post - outburst (see fig.4) indicates that the hot region is _ cooler _ during outburst (assuming no change in extinction due to circumstellar matter). this interpretation then implies a dramatic expansion of the hot region in order to account for the observed rise in luminosity during the outburst. + the inverse p cygni profiles of many emission lines prove the infall motion of accreted material. the velocity derived from the redward displacement of the absorption component of these lines are of the order of 300 km / s and therefore much higher than those assumed in the classical boundary layer model for t tauri stars (lynden - bell & pringle, 1974). however, these high infall velocities may result from magnetospherically mediated disk accretion (camenzind 1990, knigl 1991, hartmann et al. high resolution studies of classical t tauri stars have revealed a large fraction of stars exhibiting inverse p cygni structures (e.g. appenzeller 1977, edwards et al., 1994). the usual low level variability might be caused by geometrical effects during the rotation of the star. the more dramatic outbursts could be attributed to episodic changes in the magnetosphere, resulting in more extended infall flows of circumstellar material onto the star. + _ acknowledgements : _ we thank g.herbig for alerting us to the outburst of ex lup in early march 1994. also we are grateful to the following observers for kindly providing part of their observing time : t.abbott, j.f.claeskens, d.de winter, c.flynn, h.jerjen, a.manchado, f.patat, n.robichon, p.stein. tl & wb were supported by student fellowships of the european southern observatory. wb acknowledges support by the deutsche forschungsgemeinschaft (dfg) under grant yo 5/16 - 1. appenzeller i., 1977, _ a&a _ * 61 *, 21 bastian u., & mundt r., 1979, _ a&as _ * 36 *, 57 bateson f.m., mcintosh r., & brunt d., 1990, _ publ. of var.star section of the roy.astron.soc. of new zealand _ no.16, 49 calvet n., hartmann l., & kenyon s.j., 1993, _ apj _ * 402 *, 623 camenzind m., 1990, _ rev. astron. _ * 3 *, 234 edwards s., hartigan p., ghandour l., & andrulis c., 1994, _ aj _ * 108 *, 1056 hartigan p., hartmann l., kenyon s., & hewett r., 1989, _ apjs _ * 70 *, 899 hartigan p., kenyon s.j., hartmann l. et al., 1991, _ apj _ * 382 *, 617 hartmann l., 1991, in _ the physics of star formation and early stellar evolution _, ed. c.j. lada & n.d. kylafis, kluwer academic publishers, p.623 hartmann l., hewett r., & calvet n., 1994, _ apj _ * 426 *, 669 herbig g.h., 1950, _ pasp _ * 62 *, 211 herbig g.h., 1977a, _ apj _ * 214 *, 747 herbig g.h., 1977b, _ apj _ * 217 *, 693 herbig g.h., 1989, in _ low mass star formation and pre - main sequence objects _, b. reipurth, eso conference and workshop proceedings no.33, p.233 herbig g.h., gilmore a.c., & suntzeff n., 1992, _ ibvs _ no.3808 hughes j., hartigan p., krautter j., kelemen j., 1994, _ aj _ * 108 *, 1071 jones a.f., albrecht w.b., gilmore a.c., & kilmartin p.m., 1993, _ iauc _ no.5791 knigl a., 1991, _ apj _ * 370 *, l39 lynden - bell d., & pringle j.e., 1974, _ mnras _ * 168 *, 603 mclaughlin d.b., 1946, _ aj _ * 52 *, 109 patten b.m., 1994, _ no.4049 reipurth b., 1989, _ nature _ * 340 *, 42

we have observed an outburst of the t tauri star ex lup in march 1994. we present both photometric (bvr) and spectroscopic (low and medium resolution) observations carried out during the decline after outburst. the star appears much bluer during outburst due to an increased emission of a hot continuum. this is accompanied by a strong increase of the veiling of photospheric lines. we observe inverse p cygni profiles of many emission lines over a large brightness range of ex lup. we briefly discuss these features towards the model of magnetospherically supported accretion of disk material. 2.5 cm # 1to -1.5pt#1

astro-ph9506066

Introduction the liquid - gas (lg) phase transition in nuclear matter remains illusive and a hot research topic despite of the great efforts devoted to understanding its nature and experimental manifestations by the nuclear physics community over many years@xcite. for a recent review, see, e.g., refs.@xcite. most of the previous studies have focused on the lg phase transition in symmetric nuclear matter. while in an asymmetric nuclear matter, the lg phase transition is expected to display some distinctly new features because of the isospin degree of freedom and the associated interactions and additional conservation laws@xcite. this expectation together with the need to understand better properties of asymmetric nuclear matter relevant for both nuclear physics and astrophysics have stimulated a lot of new work recentlyliko97,ma99,wang00,su00,lee01,li01,natowitz02,li02,chomaz03,sil04,lizx04,chomaz06,li07. moreover, the study on the lg phase transition in asymmetric nuclear matter has received recently a strong boost from the impressive progress in developing more advanced radioactive beams that can be used to create transiently in terrestrial laboratories large volumes of highly asymmetric nuclear matter. though significant progress has been made recently in studying properties of isospin asymmetric nuclear matter and the lg phase transition in it, there are still many challenging questions to be answered. among the main difficulties are our poor understanding about the isovector nuclear interaction and the density dependence of the nuclear symmetry energy @xcite. fortunately, recent analyses of the isospin diffusion data in heavy - ion reactions have allowed us to put a stringent constraint on the symmetry energy of neutron - rich matter at sub - normal densities betty04,chen05,lichen05. it is therefore interesting to investigate how the constrained symmetry energy may allow us to better understand the lg phase transition in asymmetric nuclear matter. moreover, both the isovector (i.e., the nuclear symmetry potential) and isoscalar parts of the single nucleon potential should be momentum dependent. however, effects of the momentum - dependent interactions on the lg phase transition in asymmetric nuclear matter were not thoroughly investigated previously. we report here our recent progress in investigating effects of the isospin and momentum dependent interactions on the lg phase transition in hot neutron - rich nuclear matter within a self - consistent thermal model using three different interactions@xcite. the first one is the isospin and momentum dependent mdi interaction constrained by the isospin diffusion data in heavy - ion collisions. the second one is a momentum - independent interaction (mid) which leads to a fully momentum independent single nucleon potential, and the third one is an isoscalar momentum - dependent interaction (emdyi) in which the isoscalar part of the single nucleon potential is momentum dependent but the isovector part of the single nucleon potential is momentum independent. we note that the mdi interaction is realistic, while the other two are only used as references in order to explore effects of the isospin and momentum dependence of the nuclear interaction. Theoretical models in the isospin and momentum - dependent mdi interaction, the potential energy density @xmath0 of a thermally equilibrated asymmetric nuclear matter at total density @xmath1, temperature @xmath2 and isospin asymmetry @xmath3 is expressed as follows @xcite, @xmath4 in the mean field approximation, eq. ([mdiv]) leads to the following single particle potential for a nucleon with momentum @xmath5 and isospin @xmath6 in the thermally equilibrated asymmetric nuclear matter das03,chen05 @xmath7 in the above the isospin @xmath8 is @xmath9 for neutrons and @xmath10 for protons, and @xmath11 is the phase space distribution function at coordinate @xmath12 and momentum @xmath5. the detailed values of the parameters @xmath13 and @xmath14 can be found in ref. @xcite and have been assumed to be temperature independent here. the isospin and momentum - dependent mdi interaction gives the binding energy per nucleon of @xmath15 mev, incompressibility @xmath16 mev and the symmetry energy of @xmath17 mev for cold symmetric nuclear matter at saturation density @xmath18 @xmath19 @xcite. the different @xmath20 values in the mdi interaction are introduced to vary the density dependence of the nuclear symmetry energy while keeping other properties of the nuclear equation of state fixed @xcite. we note that the mdi interaction has been extensively used in the transport model for studying isospin effects in intermediate - energy heavy - ion collisions induced by neutron - rich nuclei li04b, chen04,chen05,lichen05,li05pion, li06,yong061,yong062,yong07. in particular, the isospin diffusion data from nscl / msu have constrained the value of @xmath20 to be between @xmath21 and @xmath22 for nuclear matter densities less than about @xmath23 @xcite. we will thus in the present work consider the two values of @xmath24 and @xmath25 with @xmath24 giving a softer symmetry energy while @xmath25 giving a stiffer symmetry energy. the potential part of the symmetry energy @xmath26 at zero temperature can be parameterized by @xcite @xmath27 (% \frac{\rho } { \rho _ { 0}})^{g(x) }, \label{epotsym}\]]where the values of @xmath28 and @xmath29 for different @xmath20 can be found in ref. @xcite. in the momentum - independent mid interaction, the potential energy density @xmath30 of a thermally equilibrated asymmetric nuclear matter at total density @xmath1 and isospin asymmetry @xmath3 can be written as @xmath31the parameters @xmath32, @xmath33 and @xmath34 are determined by the incompressibility @xmath35 of cold symmetric nuclear matter at saturation density @xmath36 as in ref. @xcite and @xmath35 is again set to be @xmath37 mev as in the mdi interaction. and @xmath26 is just same as eq. ([epotsym]). so the mid interaction reproduces very well the eos of isospin - asymmetric nuclear matter with the mdi interaction at zero temperature for both @xmath24 and @xmath25. the single nucleon potential in the mid interaction can be directly obtained as @xmath38with @xmath39 { \tau } { \delta } \notag \\ & + & (18.6-f(x))(g(x)-1)(\frac{\rho } { \rho _ { 0}})^{g(x)}{\delta } ^{2}. \label{uasy}\end{aligned}\]] the momentum - dependent part in the mdi interaction is also isospin dependent while the mid interaction is fully momentum independent. in order to see the effect of the momentum dependence of the isovector part of the single nucleon potential (nuclear symmetry potential), we can construct an isoscalar momentum - dependent interaction, called extended mdyi (emdyi) interaction since it has the same functional form as the well - known mdyi interaction for symmetric nuclear matter @xcite. in the emdyi interaction, the potential energy density @xmath40 of a thermally equilibrated asymmetric nuclear matter at total density @xmath1, temperature @xmath2 and isospin asymmetry @xmath3 is written as @xmath41here @xmath42 is the phase space distribution function of _ symmetric nuclear matter _ at total density @xmath1 and temperature @xmath2. again @xmath26 has the same expression as eq. ([epotsym]). we set @xmath43 and @xmath44, and @xmath45, @xmath46 and @xmath14 have the same values as in the mdi interaction, so that the emdyi interaction also gives the same eos of asymmetric nuclear matter as the mdi interaction at zero temperature for both @xmath24 and @xmath25. the single nucleon potential in the emdyi interaction can be obtained as@xmath47where @xmath48and @xmath49 is the same as eq. ([uasy]) which implies that the symmetry potential is identical for the emdyi and mid interactions. therefore, in the emdyi interaction, the isoscalar part of the single nucleon potential is momentum dependent but the nuclear symmetry potential is not. at zero temperature, @xmath50 @xmath51 and all the integrals in above expressions can be calculated analytically @xcite, while at a finite temperature @xmath2, the phase space distribution function becomes the fermi distribution @xmath52where @xmath53 is the chemical potential of proton or neutron and can be determined from@xmath54 in the above, @xmath55 is the proton or neutron mass and @xmath56 is the proton or neutron single nucleon potential in different interactions. from a self - consistency iteration scheme @xcite, the chemical potential @xmath53 and the distribution function @xmath57 can be determined numerically. from the chemical potential @xmath53 and the distribution function @xmath58, the energy per nucleon @xmath59 can be obtained as @xmath60. \label{e}\]]furthermore, we can obtain the entropy per nucleon @xmath61 as @xmath62dp \label{s}\]]with the occupation probability@xmath63finally, the pressure @xmath64 can be calculated from the thermodynamic relation @xmath65 \rho+\sum_{\tau } \mu _ { \tau } \rho _ { \tau }. \label{p}\end{aligned}\]] Lg phase transition with the above theoretical models, we can now study the lg phase transition in hot asymmetric nuclear matter. the phase coexistence is governed by the gibbs conditions and for the asymmetric nuclear matter two - phase coexistence equations are @xmath66where @xmath67 and @xmath68 stand for liquid phase and gas phase, respectively. the chemical stability condition is given by @xmath69the gibbs conditions ([coexistencemu]) and ([coexistencep]) for phase equilibrium require equal pressures and chemical potentials for two phases with different concentrations and asymmetries. for a fixed pressure, the two solutions thus form the edges of a rectangle in the proton and neutron chemical potential isobars as a function of isospin asymmetry @xmath3 and can be found by means of the geometrical construction method @xcite. at @xmath70 mev for the mdi and mid interactions(left) and the emdyi interaction(right) with @xmath24 and @xmath25. the geometrical construction used to obtain the isospin asymmetries and chemical potentials in the two coexisting phases is also shown. taken from ref. @xcite.] at @xmath70 mev for the mdi and mid interactions(left) and the emdyi interaction(right) with @xmath24 and @xmath25. the geometrical construction used to obtain the isospin asymmetries and chemical potentials in the two coexisting phases is also shown. taken from ref. @xcite.] we calculate the chemical potential isobars at @xmath70 mev, which is a typical temperature of lg phase transition. the solid curves shown in the left panel of fig. [mudelta] are the proton and neutron chemical potential isobars as a function of the isospin asymmetry @xmath71 at a fixed temperature @xmath70 mev and pressure @xmath72 mev@xmath73@xmath74 by using the mdi and mid interactions with @xmath24 and @xmath25. the resulting rectangles from the geometrical construction are also shown by dotted lines in the left panel of fig. [mudelta]. when the pressure increases and approaches the critical pressure @xmath75, an inflection point will appear for proton and neutron chemical potential isobars. above the critical pressure, the chemical potential of neutrons (protons) increases (decreases) monotonically with @xmath3 and the chemical instability disappears. in the left panel of fig. [mudelta], we also show the chemical potential isobar at the critical pressure by the dashed curves. at the critical pressure, the rectangle is degenerated to a line vertical to the @xmath3 axis as shown by dash - dotted lines. the values of the critical pressure are @xmath76, @xmath77, @xmath78 and @xmath79 mev@xmath73@xmath80 for the mdi interaction with @xmath24, mid interaction with @xmath24, mdi interaction with @xmath25 and mid interaction with @xmath25, respectively. shown in the right panel of fig. [mudelta] is the chemical potential isobar as a function of the isospin asymmetry @xmath3 at @xmath70 mev by using the emdyi interaction with @xmath24 and @xmath25. compared with the results from the mdi and mid interactions, the main difference is that the left (and right) extrema of @xmath81 and @xmath82 do not correspond to the same @xmath3 but they do for the mdi and mid interactions as shown in the left panel. the chemical potential of neutrons increases more rapidly with pressure than that of protons in this temperature. at lower pressures, for example, @xmath72 mev/@xmath80 as shown in panel (a), the rectangle can be accurately constructed and thus the gibbs conditions ([coexistencemu]) and ([coexistencep]) have two solutions. due to the asynchronous variation of @xmath81 and @xmath82 with pressure, we will get a limiting pressure @xmath83 above which no rectangle can be constructed and the coexistence equations ([coexistencemu]) and ([coexistencep]) have no solution. panel (b) shows the case at the limiting pressure with @xmath84 and @xmath85 mev/@xmath80 for @xmath24 and @xmath25, respectively. with increasing pressure, in panel (c) @xmath81 passes through an inflection point while @xmath82 still has a chemically unstable region, and in panel (d) @xmath82 passes through an inflection point while @xmath81 increases monotonically with @xmath3. for each interaction, the two different values of @xmath3 correspond to two different phases with different densities and the lower density phase (with larger @xmath3 value) defines a gas phase while the higher density phase (with smaller @xmath71 value) defines a liquid phase. collecting all such pairs of @xmath86 and @xmath87 thus forms the binodal surface. mev in the mdi and mid interactions with @xmath24 and @xmath25. the critical point (cp), the points of equal concentration (ec) and maximal asymmetry (ma) are also indicated. (b) the section of binodal surface at @xmath88 mev in the emdyi interaction with @xmath24 and @xmath25. lp represents the limiting pressure. taken from ref. @xcite.] in fig. [pdelta] (a), we show the section of the binodal surface at @xmath88 mev for the mdi and mid interactions with @xmath24 and @xmath25. on the left side of the binodal surface there only exists a liquid phase and on the right side only a gas phase exists. in the region of filet mignon is the coexistence phase of liquid phase and gas phase. interestingly, we can see from fig. [pdelta] (a) that the stiffer symmetry energy (@xmath25) significantly lowers the critical point (cp) and makes the maximal asymmetry (ma) a little smaller. meanwhile, the momentum dependence in the interaction (mdi) lifts the cp in a larger amount, while it seems to have no effects on the ma point. in addition, just as expected, the value of @xmath20 does not affect the equal concentration (ec) point while the momentum dependence lifts it slightly (by about @xmath89 mev/@xmath80). these features clearly indicate that the critical pressure and the area of phase - coexistence region in hot asymmetric nuclear matter is very sensitive to the stiffness of the symmetry energy with a softer symmetry energy giving a higher critical pressure and a larger area of phase - coexistence region. meanwhile, the critical pressure and the area of phase - coexistence region are also sensitive to the momentum dependence. the mdi interaction has a larger area of phase coexistence region and a larger value of the critical pressure, compared to the result of mid interaction in the temperature of @xmath70 mev. [pdelta] (b) displays the section of the binodal surface at @xmath70 mev by using the emdyi interaction with @xmath24 and @xmath25. we can see that the curve is cut off at the limiting pressure with @xmath84 and @xmath90 mev/@xmath80 for @xmath24 and @xmath25, respectively. we can also see that the limiting pressure and the area of phase - coexistence region are still sensitive to the stiffness of the symmetry energy with a softer symmetry energy (@xmath24) giving a higher limit pressure and a larger area of phase - coexistence region in this temperature. comparing the results of the mdi and mid interactions shown in fig. [pdelta] (a), we can see that for pressures lower than the limiting pressure, the binodal surface from the emdyi interaction is similar to that from the mdi interaction. this feature implies that the momentum dependence of the symmetry potential has little influence on the lg phase transition in hot asymmetric nuclear matter while the momentum dependence of the isoscalar single nucleon potential significantly changes the area of phase - coexistence region for pressures lower than the limiting pressure. for pressures above the limiting pressure, the momentum dependence of both the isoscalar and isovector single nucleon potentials becomes important. Summary in summary, we have studied the liquid - gas phase transition in hot neutron - rich nuclear matter within a self - consistent thermal model using three different nuclear effective interactions, namely, the isospin and momentum dependent mdi interaction constrained by the isospin diffusion data in heavy - ion collisions, the momentum - independent mid interaction, and the isoscalar momentum - dependent emdyi interaction. at zero temperature, the above three interactions give the same eos for asymmetric nuclear matter. by analyzing liquid - gas phase transition in hot neutron - rich nuclear matter with the above three interactions, we find that the boundary of the phase - coexistence region is very sensitive to the density dependence of the nuclear symmetry energy. a softer symmetry energy leads to a higher critical pressure and a larger area of the phase - coexistence region. in addition, the area of phase - coexistence region are also seen to be sensitive to the isospin and momentum dependence of the nuclear interaction. for the isoscalar momentum - dependent emdyi interaction, a limiting pressure above which the liquid - gas phase transition can not take place has been found. Acknowledgements this work was supported in part by the national natural science foundation of china under grant nos. 10334020, 10575071, and 10675082, moe of china under project ncet-05 - 0392, shanghai rising - star program under grant no. 06qa14024, the srf for rocs, sem of china, the china major state basic research development program under contract no. 2007cb815004, the us national science foundation under grant no. phy-0652548 and the research corporation under award no.

the liquid - gas phase transition in hot neutron - rich nuclear matter is investigated within a self - consistent thermal model using different interactions with or without isospin and/or momentum dependence. the boundary of the phase - coexistence region is shown to be sensitive to the density dependence of the nuclear symmetry energy as well as the isospin and momentum dependence of the nuclear interaction.

0711.1717

Introduction cp violation, initially observed @xcite only in the @xmath7@xmath8 system, is one feature of the standard model (sm) that still defies clear theoretical understanding. the ckm picture, which describes _ all _ the _ observed _ cp violation in terms of a single phase in the quark - mixing matrix, has been vindicated by the recent measurements of @xmath9@xmath10 mixing at belle and babar @xcite. cp violation is in fact one of the necessary ingredients for generating the observed excess of baryons over antibaryons in the universe @xcite. the amount of cp violation present in the quark sector is, however, too small to generate a baryon asymmetry of the observed level of @xmath11 @xcite. new sources of cp violation _ beyond _ the sm are therefore a necessity @xcite. supersymmetry (susy) is arguably the most attractive extension of the sm, as it solves, for instance, the problem of the instability of the electroweak symmetry - breaking scale against radiative corrections. already the minimal supersymmetric standard model (mssm) @xcite provides possible new sources of cp violation through additional cp - violating phases, which can not be rotated away by simple field redefinitions. a large number of these phases, particularly those involving sparticles of the first and to a large extent of the second generation, are severely constrained by measurements of the electric dipole moments (edms) of the electron, muon, neutron as well as @xmath12hg and @xmath13tl. however, these constraints are model - dependent. it has been demonstrated @xcite that cancellations among different diagrams allow certain combinations of these phases to be large in a general mssm. furthermore, if the sfermions of the first two generations are sufficiently heavy, above the 1 tev range, the edm constraints on the phase of the higgsino mass parameter @xmath14, in general constrained to @xmath15, get weaker ; the sfermions of the third generation can still be light. non - vanishing phases of @xmath16 and/or the trilinear scalar couplings @xmath17 can induce explicit cp violation in the higgs sector via loop corrections @xcite. though these phases generate edms independently of the first two generations of sfermions, the edms are suppressed by the mass scale of the two heavy higgses @xcite. for a thorough discussion of the edms see @xcite and references therein. the above mentioned phases can also have a significant influence on the higgs production rates in the gluon fusion mode at the tevatron and the lhc @xcite. mssm cp phases can hence change the higgs phenomenology at colliders quite substantially. all this makes the mssm with cp - violating phases a very attractive proposition. it has therefore been the subject of many recent investigations, studying the implications of these phases on neutralino / chargino production and decay @xcite, on the third generation of sfermions @xcite, as well as the neutral @xcite and charged @xcite higgs sector. various cp - even and cp - odd (t - odd) observables, which can give information on these phases, have been identified. it is interesting to note that cp - even observables such as masses, branching ratios, cross sections, etc., often afford more precise probes thanks to the larger magnitude of the effects. for direct evidence of cp violation, however, cp - odd / t - odd observables as discussed e.g. in @xcite have to be measured. the latest study of the @xmath18 sector in @xcite demonstrates that it may be possible to determine the real and imaginary parts of @xmath19 to a precision of 23% from a fit of the mssm lagrange parameters to masses, cross sections and branching ratios at a future @xmath20 linear collider (lc). this requires that both the @xmath21, @xmath22 mass eigenstates can be produced at the lc and the branching ratios measured with high precision. in the @xmath23 sector @xcite the precision on @xmath24 is worse, around 1020% for low @xmath25 and about 37% for large @xmath25. in this paper, we show that the longitudinal polarization of fermions produced in sfermion decays, i.e. @xmath26 and @xmath27 with @xmath28 a third generation (s)quark or (s)lepton, can also be used as a probe of cp phases. the fermion polarization can give complementary information to the decay branching ratios and will in particular be useful if the branching ratios can not be measured with high enough precision or if one decay channel dominates. the average polarization of fermions produced in sfermion decays carries information on the @xmath29@xmath30 mixing as well as on the gaugino higgsino mixing @xcite. the polarizations that can be measured are those of top and tau ; both can be inferred from the decay lepton distributions. it is its large mass that causes the @xmath31 to decay before hadronization and thus the decay products can carry information about its polarization. for taus, also the energy distribution of the decay pions can be used. the polarization of the decay fermions has been used for studies of mssm parameter determination in the cp - conserving case in @xcite. for the cp - violating case, the phase dependence of the longitudinal fermion polarization has been mentioned in @xcite. we extend these studies by discussing in detail the sensitivity of the fermion polarization to cp - violating phases in the mssm. the paper is organized as follows : in section 2, we summarize our notation for the description of the sfermion, neutralino and chargino systems in the mssm with cp violation. in section 3, we discuss fermion polarization in sfermion decays to neutralinos, @xmath32 with @xmath33. we present numerical results on the polarization as a function of different mssm parameters and discuss the sensitivity to cp - violating phases in the sfermion and neutralino sectors. in section 4 we perform an analogous analysis for @xmath34 decays. in section 5 we summarize the results and present our conclusions. Notation and conventions ignoring intergenerational mixing, the sfermion mass matrices can be written as a series of @xmath35 matrices, each of which describes sfermions of a specific flavour : @xmath36 with @xmath37 { m_{\sf_{r}}}^2 & = & m^2_{\ti r } + e_f\,m_z^2 \cos 2\b\,\sin^2\t_w + m_f^2, \\[1 mm] a_f & = & a_f - \mu^*\, \ { \cot\b, \tan\b \ } = |a_f|\, e^{i\phsf } \,, \label{eq : aq}\end{aligned}\]] for @xmath38up, down@xmath39-type sfermions ; @xmath40, @xmath41 and @xmath42 are the mass, electric charge and the third component of the weak isospin of the partner fermion, respectively ; @xmath43, @xmath44 and @xmath45 are soft susy - breaking parameters for each family, and @xmath16 is the higgsino mass parameter ; @xmath45 and @xmath16 can have complex phases : @xmath46 and @xmath47. + according to eq., @xmath48 is diagonalized by a unitary rotation matrix @xmath49. the weak eigenstates @xmath50 and @xmath51 are thus related to their mass eigenstates @xmath52 and @xmath53 by @xmath54 with @xmath55 and @xmath56 the sfermion mixing angle and phase. since the off - diagonal element of @xmath48 is proportional to @xmath40, this mixing is mostly relevant to the third generation, @xmath57, on which we concentrate in the following. the mass eigenvalues are given by @xmath58 by convention, we choose @xmath52 to be the lighter mass eigenstate, @xmath59. notice also that @xmath60. for the mixing angle @xmath55 we choose @xmath61 which places @xmath55 in the 2nd quadrant of the unit circle. the @xmath62@xmath63 mixing is large if @xmath64, with @xmath65 if @xmath66 and @xmath67 if @xmath68. moreover, we see that the phase dependence of @xmath69 and @xmath49 is determined by @xmath70. this dependence is strongest if @xmath71. this issue will be discussed in more detail in the numerical analyses of sections 3 and 4. in the basis @xmath72 the neutralino mass matrix is : @xmath73 the gaugino mass parameters @xmath74 and the higgsino mass parameter @xmath16 can in principle all be complex. the phase of @xmath75 can be rotated away, which leaves us with two phases in this sector : @xmath76, the phase of @xmath77, and @xmath78, the phase of @xmath16. + the matrix of eq. is diagonalized by the unitary mixing matrix @xmath79 : @xmath80 where @xmath81, @xmath82, are the (non - negative) masses of the physical neutralino states. we choose the ordering @xmath83. a concise discussion of the neutralino sector with complex phases can be found in @xcite. the chargino mass matrix is : @xmath84 it is diagonalized by the two unitary matrices @xmath85 and @xmath86, @xmath87 where @xmath88 are the masses of the physical chargino states with @xmath89. Fermion polarization in @xmath90 decays the sfermion interaction with neutralinos is (@xmath91 ; @xmath82) @xmath92 where @xmath93 the @xmath94 and @xmath95 couplings are @xmath96 for stops and sbottoms, and @xmath97 for staus, with the yukawa couplings @xmath98 given by @xmath99 the gaugino interaction conserves the helicity of the sfermion while the higgsino interaction flips it. in the limit @xmath100, the average polarization of the fermion coming from the @xmath101 decay can therefore be calculated as @xcite @xmath102 using eqs. , and as well as @xmath103, we obtain, for the @xmath104 decay (omitting the overall factor @xmath105 and dropping the sfermion and neutralino indices for simplicity) : @xmath106 \,.\end{aligned}\]] we see that the phase dependence of @xmath107 is the largest for maximal sfermion mixing (@xmath108) and if the neutralino has both sizeable gaugino and higgsino components. it is, moreover, enhanced if the yukawa coupling @xmath98 is large. furthermore, @xmath107 is sensitive to cp violation even if just one phase, in either the neutralino or the sfermion sector, is non - zero. in particular, if only @xmath45 and thus only the sfermion mixing matrix has a non - zero phase, the phase - dependent term becomes @xmath109 if, on the other hand, only @xmath76 is non - zero we get @xmath110\,\sin 2\t \,.\]] the polarization @xmath111, eq., depends only on couplings but not on masses. for the numerical analysis we therefore use @xmath77, @xmath75, @xmath16, @xmath112, @xmath55 and @xmath113 as input parameters, assuming @xmath114 to satisfy edm constraints more easily : assuming cancellations for the 1-loop contributions and the cp - odd higgs mass parameter @xmath115 gev, 1-loop and 2-loop contributions to the electron edm (eedm), as well as their sum, stay below the experimental limit. we use the formulae of @xcite for the 2-loop contributions. in order not to vary too many parameters, we use, moreover, the gut relation @xmath116 and choose @xmath117 and @xmath118 (large but not maximal mixing) throughout this section. the free parameters in our analysis are thus @xmath75, @xmath119, and the phases @xmath76, @xmath113. before we present the numerical results, a comment is in order : cp violation in the neutralino sector is determined by the phases of @xmath77 and @xmath16, while @xmath113 originates from relative phases of @xmath45 and @xmath16. for stops the mixing is dominated by @xmath19, while for sbottoms and staus it is dominated by @xmath120 ; quite generally we have @xmath121 unless @xmath122, and @xmath123 unless @xmath124. more precisely, @xmath125 for @xmath126, any @xmath113 can be reached by an appropriate choice of @xmath127, independent of @xmath128. for @xmath129, however, @xmath113 is restricted by @xmath128. in the special case of @xmath130 and @xmath131, @xmath132. in the stop sector this is not a problem since @xmath133 can in general be easily achieved. for sbottoms and staus, choosing @xmath134 freely leads, however, to quite large @xmath135, which may in some cases create problems with charge- or colour - breaking minima. (148,66) (0,0) (30,0) (104,0) (-2,37) (72,37) (-2,64) (72,64) (16,14) (90,14) (148,72) (0,0) (30,0) (104,0) (-2,37) (72,37) (-2,64) (72,64) (16,14) (110,14) shows the average tau polarization in @xmath136 decays as functions of @xmath75 and @xmath119 for @xmath117, @xmath137 and various choices of @xmath76 and @xmath138. the lower limits of @xmath75 and @xmath119 are given by the lep2 constraint of @xmath139 gev @xcite, which automatically takes care of all other lep constraints on the gaugino higgsino sector. as can be seen, @xmath140 is quite sensitive to cp phases for @xmath141, that is if the @xmath142 has a sizeable higgsino component. analogously, shows the average top polarization in @xmath143 decays. we observe again a strong dependence on the cp phases if the neutralino has a sizeable higgsino component. unlike the case of @xmath140, for @xmath144 the dependence is still significant when @xmath145. we also note that some phase combinations lead to very similar polarizations, e.g. @xmath146 and @xmath147. at a future @xmath20 linear collider (lc), one expects to be able to measure the tau polarization to about 35% and the top polarization to about 10% @xcite. we see from figs. [fig : ptau_m2mu] and [fig : ptop_m2mu] that the effects of cp - violating phases may well be visible in @xmath144 and/or @xmath140, provided @xmath16 is not too large. we next choose specific values of @xmath75 and @xmath119 to discuss the phase dependences in more detail. a shows @xmath140 as a function of @xmath76, for @xmath148 gev, @xmath149 gev and @xmath150, @xmath151, @xmath152 and @xmath153. since for fixed @xmath75 and @xmath119 the @xmath142 mass changes with @xmath76, we show in addition in b @xmath140 as a function of @xmath154 for various values of @xmath76, with @xmath149 gev and @xmath75 adjusted such that @xmath155 gev. @xmath140 varies over a large range depending on @xmath76 and @xmath154 ; if the neutralino mass parameters, @xmath25 and @xmath156 are known, @xmath140 can hence be used as a sensitive probe of these phases (although additional information will be necessary to resolve ambiguities and actually determine the various phases). at a lc, the parameters of the neutralino / chargino sector and also sfermion masses and mixing angles can be determined very precisely, exploiting tunable beam energy and beam polarization @xcite. the actual precision depends of course on the specific scenario. to illustrate the influence of uncertainties in the knowledge of the model parameters, we take the case of @xmath148 gev, @xmath149 gev and vanishing phases as reference point and assume that the following precisions can be achieved : @xmath157, @xmath158, @xmath159, and @xmath160. varying the parameters within this range around the reference point leads to @xmath161 at @xmath150, which is indicated as an error bar in b. (the 35% uncertainty in the measurement of @xmath162 is comparatively negligible). we conclude that in our particular scenario, if no phase has been observed in the neutralino / chargino sector, a measurement of @xmath140 would be sensitive to @xmath163. if @xmath25 can be measured to @xmath164, this improves to @xmath165 and @xmath166. according to, a measurement of a non - zero @xmath167 implies a lower limit on @xmath168 ; in our example where @xmath131, @xmath169 gev (1 tev) for @xmath170 (@xmath171). increasing the precision in @xmath172, @xmath172 and @xmath173 from 0.5% to 0.1% barely improves these limits. (148,72) (0,0) (36,-1) (109,-1) (-2,37) (72,37) (-2,64) (72,64) (148,72) (0,0) (36,-1) (108,-1) (-2,37) (72,37) (-2,64) (72,64) we perform a similar analysis for @xmath144, using @xmath174 gev and @xmath175 gev as reference point. the results are shown in a, b in analogy to a, b. again a high sensitivity to both @xmath76 and @xmath176 is observed. for the case of vanishing phases, we get @xmath177. a variation of the parameters around the reference point as above (with @xmath178) leads to a parametric uncertainty of @xmath179. adding the experimental resolution @xmath180 in quadrature gives @xmath181 at @xmath182, indicated as an error bar in b. we see that in this scenario @xmath144 would be sensitive to @xmath183. if @xmath184 can be measured to @xmath185 this improves to @xmath186 (@xmath187) and @xmath188 ; if @xmath77, @xmath75, @xmath119 can be measured to 0.1% and @xmath25 to @xmath189, @xmath190 becomes negligible with respect to the experimental resolution of @xmath191. since @xmath192, a measurement of @xmath144 can be used to derive information on @xmath19. in particular, if both mass eigenstates are known, @xmath19 is given by @xmath193 an analogous relation with @xmath194 holds for @xmath195, although the precision on @xmath24 is in general much worse than on @xmath19. in this context note that @xmath111 can also be useful to resolve the sign ambiguity in the @xmath196 determination from cross section measurements @xcite in the cp - conserving case. this corresponds to distinguishing the cases @xmath197 and @xmath198. even though we have presented results of our analysis for @xmath199, chosen in order to satisfy the edm constraints without having to appeal to cancellations, we have also investigated the case of a non - zero @xmath128. we found that a non - zero @xmath128 shifts the curves in [fig : ptop_phases] but does not cause a qualitative change of the results. (148,72) (0,0) (38,-1) (112,-1) (-2,37) (72,37) (-2,64) (72,64) last but not least we note that giving up the gut relation between @xmath200 and @xmath75 changes the picture completely, as the pattern of gaugino higgsino mixing is strongly affected @xcite. this is illustrated in, where we plot @xmath140 and @xmath144 as functions of @xmath201 for @xmath202 gev, @xmath203 gev and the other parameters as in and [fig : ptop_m2mu]. a detailed study of the implications of non - universal gaugino masses will be presented elsewhere. + _ to sum up, _ both @xmath144 and @xmath140 can vary over a large range depending on @xmath76 and @xmath204 (and also @xmath128, though we did not discuss this case explicitly) and may thus be used as sensitive probes of these phases. to this aim, however, the neutralino mass parameters, @xmath25 and the sfermion mixing angles need to be known. given the complexity of the problem, a combined fit of all available data seems to be the most convenient method for the extraction of the mssm parameters. Fermion polarization in @xmath205 decays the sfermion interaction with charginos is (@xmath206) @xmath207 where @xmath208 (@xmath209) stands for up - type (s)quark and (s)neutrinos, and @xmath210 (@xmath211) stands for down - type (s)quark and charged (s)leptons. the couplings @xmath212 and @xmath213 are @xmath214 for stops and sbottoms and @xmath215 for staus and sneutrinos. analogous to the decay into a neutralino, eq., the average polarization of the fermion coming from the @xmath216 decay is given by @xmath217 since only top and tau polarizations are measurable, we only discuss @xmath218 and @xmath219 decays. the latter case is especially simple because @xmath220 depends only on the parameters of the chargino sector : @xmath221 a measurement of @xmath220 may hence be useful to supplement the chargino parameter determination. however, the dependence of @xmath222 on @xmath199 turns out to be very small, the effects being in general well below 1% (i.e. @xmath223). only for the decay into the heavier chargino, the effect of a non - zero phase is severely restricted by the non - observation of the eedm.] may be sizeable. as an example, shows the differences in @xmath224 between @xmath131 and @xmath225 in the @xmath226 plane for @xmath227 gev. @xmath228 can go up to @xmath229. however, it requires quite heavy sneutrinos for this decay to be kinematically allowed. moreover, the measurement of @xmath224 will be diluted by @xmath230 decays. + (70,74) (0,0) (27,-3) (-2,71) (7,60) (36,60) (22,50) (24,38) (28,26.5) (34,18) let us now turn to top polarization in @xmath218 decays. for @xmath231 decays, we have @xmath232 \cos\varphi + { { \cal i}m\, } [u_{j1}^ { } u_{j2}^ { * }] \sin\varphi) \,. \label{eq : sbot1}\end{aligned}\]] for @xmath233 decays, the corresponding expression is given by the rhs of with @xmath234, @xmath235 interchanged, and a change in sign of the term @xmath236. we see that the phase dependence of @xmath237 is proportional to @xmath238 and the amount of gaugino higgsino mixing of the charginos ; it will therefore be largest for @xmath239, @xmath240 and large @xmath112. again, there is a non - zero effect even if there is just one phase in either the sbottom or chargino sector. note, however, that the only cp phase in the chargino sector is @xmath128, which also enters the sfermion mass matrices. complex @xmath85 and @xmath86 hence imply @xmath241. more precisely, @xmath242 for medium and large @xmath112, and thus @xmath243 unless @xmath244 ; see eq. and the related discussion. for the sake of a general discussion of the phase dependence of @xmath237 (and since @xmath245 is still a free parameter), we nevertheless use @xmath199 and @xmath246 as independent input parameters. shows the average top polarization in @xmath247 decays as a function of @xmath119 for @xmath248 gev, @xmath249 and 30, and various combinations of @xmath199 and @xmath246. here we have fixed @xmath250, since from renormalization - group running one expects @xmath251. as in the previous section, we find large effects from cp - violating phases if the @xmath252 has a sizeable higgsino component ; as expected, these effects are enhanced for large @xmath25. the results stay the same if both @xmath199 and @xmath246 change their signs. moreover, @xmath253. if @xmath199 and @xmath246 have the same sign, the difference in @xmath237 from the case of vanishing phases is larger than if they have opposite signs. in particular, we find @xmath254 over large regions of the parameter space. with an experimental resolution of the top polarization of about 10% this implies that in many cases @xmath255 can not be distinguished from @xmath256 by measurement of @xmath237. furthermore, the value of @xmath237 is quite sensitive to the running @xmath257 quark mass, which enters the bottom yukawa coupling of eq. and is subject to possibly large susy loop corrections. for the lines in we have used @xmath258 gev the grey bands show the range of @xmath237 when @xmath259 is varied between 2.5 and 4.5 gev. as can be seen, the uncertainty in @xmath259 more precisely in @xmath260 tends to wash out small effects of cp - violating phases, specially in the case of large @xmath25. (148,72) (0,0) (32,-1) (104,-1) (-2,37) (72,37) (-2,64) (72,64) (30,67) (102,67) (148,72) (5,3.3) (81,2) (33,-1) (112,-1) (-4,37) (73,37) (-4,63) (73,63) (14,68) (92,68) in order to see what information can be extracted from a @xmath237 measurement, we pick two values of @xmath119 from a ; namely @xmath261 gev and @xmath262 gev, and show the phase dependences at these points in. a shows @xmath237 as a function of @xmath246, for @xmath261 gev, @xmath249, @xmath263, and various values of @xmath128. @xmath75 is chosen such that @xmath264 gev (i.e. @xmath174 gev for @xmath131). the range obtained by varying @xmath259 within 2.54.5 gev is shown as grey bands for two of the curves, for @xmath131 and @xmath265. we estimate the effect of an imperfect knowledge of the model parameters in the same way as in the previous section. for @xmath266 gev, @xmath267 gev, @xmath268, @xmath269 and @xmath270, we get @xmath271 at @xmath272. varying in addition @xmath2734.5 gev gives @xmath274. adding a 10% measurement error on @xmath237 in quadrature, we end up with @xmath275 (@xmath276) without (with) the @xmath259 effect. these are shown as error bars in a. we see that the case of @xmath277 can not be distinguished from @xmath278 in this scenario. however, @xmath237 turns out to be quite a sensitive probe of @xmath279, i.e. the deviation from the ` natural'alignment @xmath277. in the example of a, @xmath280 @xmath281 can be resolved if @xmath260 is (not) known precisely, quite independently of @xmath199. observing such a @xmath282 also implies a bound on @xmath283 of @xmath284 @xmath285 gev. if the precision on @xmath75 and @xmath119 is 0.1% and @xmath164, we get @xmath286 at @xmath272, so that the error is dominated by the experimental uncertainty. however, the resultant improvement in the sensitivity is limited to @xmath287 and @xmath288 gev. b shows @xmath237 as a function of @xmath246, for @xmath262 gev, @xmath289 gev, and the other parameters as above (@xmath174 gev at @xmath131). the effect of an uncertainty in @xmath260 is again shown as grey bands for @xmath131 and @xmath265. estimating the parametric uncertainty in the same way as above, but with @xmath290 gev, we get @xmath291 at @xmath272. varying in addition @xmath2734.5 gev gives @xmath292. adding a 10% measurement error on @xmath237 in quadrature, we end up with @xmath293 (@xmath294) without (with) the effect of @xmath259, shown as error bars in b. in a three - dimensional plot, @xmath237 has a bell - like shape in the @xmath128@xmath246 plane, with contours of constant @xmath237 being ellipses in this plane. if @xmath199 is not known, a measurement of @xmath237 may therefore be useful to put limits on @xmath128 and @xmath246, but not on @xmath282, which restricts @xmath283. in our case study, we have assumed that @xmath295 is known. in this case, a measurement of @xmath296, for instance, would restrict @xmath297 at @xmath298, while a measurement of @xmath299 would disfavour @xmath300 as well as @xmath301. the latter would also allow a constraint on @xmath282. as mentioned above, a lower limit on @xmath282 implies a lower limit on @xmath283. an upper limit on @xmath302 can be used to set an upper limit on @xmath283 as a function of @xmath303 : @xmath304. note, however, that this becomes unbounded for @xmath305. we have also investigated the case of large @xmath25 (@xmath306). it reveals a @xmath246 dependence similar to that of a, with almost no dependence on @xmath199 and the @xmath307 dependence accordingly more pronounced. we encounter, however, a large parametric uncertainty, which practically washes out the sensitivity to @xmath302. + _ to sum up, _ tau polarization in @xmath219 decays depends only little on @xmath199. @xmath220 is hence not a promising quantity to study cp phases, but may be useful for (consistency) tests of the gaugino higgsino mixing. top polarization in @xmath218 decays, on the other hand, can be useful to probe @xmath199, @xmath246 and/or @xmath308 in some regions of the parameter space. the measurement of @xmath237, revealing phases or being consistent with vanishing phases, may also constrain @xmath283. Conclusions we have discussed the influence of cp - violating phases on the fermion polarization in sfermion decays to neutralinos or charginos, @xmath309 and @xmath310 (@xmath206 ; @xmath82 ; @xmath311). this polarization is considered as a useful tool for the mssm parameter determination @xcite. in decays into charginos, the polarization depends on the phase of @xmath16. since this dependence is weak in the case of @xmath219, @xmath220 does not provide a promising probe of cp phases (on the other hand, exactly this feature can make @xmath220 useful for consistency tests of gaugino higgsino mixing). in @xmath312 decays, the dependence on @xmath199 can be rather large ; in addition, also the phase of the sbottom - mixing matrix plays a role. if @xmath313, @xmath314. we found that this case can be difficult to distinguish from the cp - conserving case by measuring @xmath315. if, however, a deviation from @xmath316 is observed, these phases can be constrained and also limits on @xmath245 can be derived. the decays @xmath317 and @xmath318 provide a more effective probe of cp violation because an additional phase, the phase of @xmath77, contributes. we found that cp phases can have a significant effect on the top and tau polarizations, especially if the involved neutralino has a sizeable higgsino component. if the parameters of the neutralino sector can be measured precisely, e.g. in @xmath20 annihilation with polarized beams, @xmath319 and @xmath320 can be useful for the determination of cp phases. in particular, since @xmath192 unless @xmath119 is very large, a measurement of @xmath144 can give information on @xmath19. in this respect it is important to note that (for fixed masses) the sfermion production cross sections do not depend on cp phases. in the sfermion sector, these can be manifest in branching ratios as discussed in @xcite, polarization of the decay fermions as discussed in this paper, and cp - odd asymmetries. branching ratios are in general rather difficult to measure with high precision. the information that can be gained from branching ratios is also limited if one decay channel dominates, e.g. @xmath136 in case of a light stau. this makes the polarization of the decay fermions a very interesting possibility to explore cp phases. last but not least we note that the computations in this paper, leading to effects of a few percent, have been performed at tree level. the influence of radiative corrections @xcite can be of comparable size and will therefore have to be taken into account for precision analyses. a measurement of the cp phases in the sfermion/@xmath321 sector will also complement cp studies of the higgs sector @xcite, since in the mssm higgs - sector cp violation is generated through quantum corrections @xcite. last but not least we emphazise that, since the effects can be large, the possibility of cp violation should be taken into account in precision susy parameter analyses, especially in a general analysis project as envisaged in @xcite. Acknowledgements t.g. acknowledges the financial support of the institut fr hochenergiephysik in vienna and the cern theory division. r.g. wishes to thank the cern theory division and lapth for hospitality and financial support, where this work was completed. thanks the centre for high energy physics in bangalore for hospitality and financial support. rg wishes to acknowledge the partial support of the department of science and technology, india, under project number sp / s2/k-01/2000-ii. Standard model constants the sm constants used in the numerical analysis are : @xmath322 a. d. sakharov, pisma zh. teor. fiz. * 5 * (1967) 32, jetp lett. 6, 24 (1967). for a recent summary, see a. d. dolgov, hep - ph/0211260. c. l. bennett _ et al. _, astrophys. j. suppl. * 148 *, 1 (2003) [astro - ph/0302207]. for a review, see e.g. m. dine and a. kusenko, rev. phys. * 76 *, 1 (2004) [hep - ph/0303065]. see, e.g.h. e. haber and g. l. kane, phys. rept. * 117 * (1985) 75. t. ibrahim and p. nath, phys. b * 418 *, 98 (1998) [hep - ph/9707409] ; phys. rev. d * 57 *, 478 (1998), erratum - ibid. d * 58 *, 019901 (1998) [hep - ph/9708456] ; phys. d * 58 *, 111301 (1998), erratum - ibid. d * 60 *, 099902 (1999) [hep - ph/9807501] ; phys. d * 61 *, 093004 (2000) [hep - ph/9910553]. m. brhlik, g. j. good and g. l. kane, phys. d * 59 *, 115004 (1999) [hep - ph/9810457]. a. bartl, t. gajdosik, w. porod, p. stockinger and h. stremnitzer, phys. d * 60 *, 073003 (1999) [hep - ph/9903402]. t. falk and k. a. olive, phys. b * 439 *, 71 (1998) [hep - ph/9806236]. t. falk, k. a. olive, m. pospelov and r. roiban, nucl. b * 560 *, 3 (1999) [hep - ph/9904393]. a. pilaftsis, phys. d * 58 * (1998) 096010 [hep - ph/9803297] ; phys. b * 435 * (1998) 88 [hep - ph/9805373] ; a. pilaftsis and c. e. wagner, nucl. b * 553 *, 3 (1999) [hep - ph/9902371]. d. a. demir, phys. d * 60 * (1999) 055006 [hep - ph/9901389]. s. y. choi, m. drees and j. s. lee, phys. b * 481 *, 57 (2000) [hep - ph/0002287]. m. carena, j. r. ellis, a. pilaftsis and c. e. m. wagner, nucl. b * 586 *, 92 (2000) [hep - ph/0003180]. d. chang, w. y. keung and a. pilaftsis, phys. lett. * 82 *, 900 (1999) [erratum - ibid. * 83 *, 3972 (1999)] [arxiv : hep - ph/9811202]. a. pilaftsis, phys. b * 471 *, 174 (1999) [arxiv : hep - ph/9909485]. d. a. demir, o. lebedev, k. a. olive, m. pospelov and a. ritz, nucl. b * 680 * (2004) 339 [arxiv : hep - ph/0311314]. a. dedes and s. moretti, phys. * 84 * (2000) 22, [hep - ph/9908516] ; nucl. b * 576 * (2000) 29, [hep - ph/990941]. s. y. choi, k. hagiwara and j. s. lee, phys. b * 529 * (2002) 212 [hep - ph/0110138]. s. y. choi, a. djouadi, h. s. song and p. m. zerwas, eur. j. c * 8 *, 669 (1999) [hep - ph/9812236]. s. y. choi, m. guchait, h. s. song and w. y. song, phys. b * 483 *, 168 (2000) [hep - ph/9904276]. s. y. choi, m. guchait, h. s. song and w. y. song, hep - ph/0007276. s. y. choi, j. kalinowski, g. moortgat - pick and p. m. zerwas, eur. j. c * 22 *, 563 (2001), addendum - ibid. c * 23 *, 769 (2002) [hep - ph/0108117]. v. d. barger, t. falk, t. han, j. jiang, t. li and t. plehn, phys. d * 64 *, 056007 (2001) [hep - ph/0101106]. a. bartl, h. fraas, o. kittel and w. majerotto, phys. d * 69 * (2004) 035007 [hep - ph/0308141]. a. bartl, t. kernreiter and o. kittel, phys. b * 578 *, 341 (2004) [hep - ph/0309340]. s. y. choi, m. drees, b. gaissmaier and j. song, phys. d * 69 *, 035008 (2004) [hep - ph/0310284]. a. bartl, h. fraas, o. kittel and w. majerotto, hep - ph/0402016. s. y. choi, m. drees and b. gaissmaier, hep - ph/0403054. a. bartl, t. kernreiter and w. porod, phys. b * 538 *, 59 (2002) [hep - ph/0202198]. a. bartl, k. hidaka, t. kernreiter and w. porod, phys. b * 538 * (2002) 137 [hep - ph/0204071]. a. bartl, k. hidaka, t. kernreiter and w. porod, phys. d * 66 * (2002) 115009 [hep - ph/0207186]. a. bartl, h. fraas, t. kernreiter and o. kittel, eur. j. c * 33 * (2004) 433 [hep - ph/0306304]. a. bartl, s. hesselbach, k. hidaka, t. kernreiter and w. porod, phys. b * 573 * (2003) 153 [hep - ph/0307317]. a. bartl, s. hesselbach, k. hidaka, t. kernreiter and w. porod, hep - ph/0311338. s. y. choi, k. hagiwara and j. s. lee, phys. rev. d * 64 * (2001) 032004 [hep - ph/0103294] ; a. arhrib, d. k. ghosh and o. c. w. kong, phys. b * 537 * (2002) 217 [hep - ph/0112039]. s. y. choi, m. drees, j. s. lee and j. song, eur. j. c * 25 *, 307 (2002) [hep - ph/0204200]. m. carena, j. r. ellis, s. mrenna, a. pilaftsis and c. e. m. wagner, nucl. b * 659 * (2003) 145 [hep - ph/0211467]. f. borzumati, j. s. lee and w. y. song, hep - ph/0401024. e. christova, h. eberl, w. majerotto and s. kraml, nucl. b * 639 * (2002) 263, erratum - ibid. b * 647 * (2002) 359 [hep - ph/0205227] ; e. christova, h. eberl, w. majerotto and s. kraml, jhep * 0212 * (2002) 021 [hep - ph/0211063]. m. m. nojiri, phys. d * 51 * (1995) 6281 [hep - ph/9412374]. m. m. nojiri, k. fujii and t. tsukamoto, phys. d * 54 *, 6756 (1996) [hep - ph/9606370]. e. boos, h. u. martyn, g. moortgat - pick, m. sachwitz, a. sherstnev and p. m. zerwas, eur. j. c * 30 * (2003) 395 [hep - ph/0303110]. lepsusywg, aleph, delphi, l3 and opal experiments, note lepsusywg/01 - 03, http://lepsusy.web.cern.ch/lepsusy j. a. aguilar - saavedra _ et al. _ [ecfa / desy lc physics working group collaboration], _ tesla technical design report part iii : physics at an @xmath20 linear collider _, hep - ph/0106315. a. bartl, h. eberl, s. kraml, w. majerotto, w. porod and a. sopczak, z. phys. c * 76 * (1997) 549 [hep - ph/9701336] ; a. bartl, h. eberl, s. kraml, w. majerotto and w. porod, eur. j. directc * 2 * (2000) 6 [hep - ph/0002115]. see, for instance, g. belanger, f. boudjema, a. cottrant, r. m. godbole and a. semenov, phys. b * 519 * (2001) 93 [hep - ph/0106275]. for a review see : w. majerotto, arxiv : hep - ph/0209137, and references therein.

the longitudinal polarization of fermions (tops and taus) produced in sfermion decays to neutralinos or charginos can be a useful tool for the determination of susy parameters. we discuss this fermion polarization in the context of the mssm with complex parameters. we show that the dependence on cp - violating phases can be large and that the fermion polarization may hence be used as a sensitive probe of cp phases in the mssm. cern - ph - th/2004 - 086 + hephy - pub 790/04 + iisc - chep/7/04 + fi2004 - 15 + hep - ph/0405167 * fermion polarization in sfermion decays as + a probe of cp phases in the mssm * + thomas gajdosik@xmath0, rohini m. godbole@xmath1, sabine kraml@xmath2 + _ @xmath3 institute of physics, vilnius lt-2600, lithuania + @xmath4 centre for high energy physics, indian institute of science, bangalore 560012, india + @xmath5 inst. f. hochenergiephysik, sterr. akademie d. wissenschaften, 1050 vienna, austria + @xmath6 department of physics, cern, theory division, 1211 geneva 23, switzerland + _

hep-ph0405167

Introduction the study of the phase transition of frustrated spin systems on two - dimensional (2d) lattices is a central problem in modern condensed mater physics. a competition of exchange interaction can lead to frustration, where spatial arrangement of magnetic ions in a crystal for which a simultaneous antiparallel ordering of all interacting spin is impossible. in particular, one of the frustrated 2d models most discussed is the quantum spin-@xmath0 heisenberg antiferromagnet on a square lattice with competing nearest - neighbor (nn) and next - nearest - neighbor (nnn) antiferromagnetic exchange interactions (known as @xmath12 model) @xcite. the criticality of this @xmath12 heisenberg model on a square lattice are relatively well known at @xmath13. there are two magnetically long - range ordered phases at small and at large values of @xmath6 separated by an intermediate quantum paramagnetic phase without magnetic long - range order in the region between @xmath14 and @xmath15, where the properties of these disordered phase are still under intensive debate. for @xmath16 @xmath17, the system possesses antiferromagnetic (af) long - range order with wave vector @xmath18, with a staggered magnetization smaller than the saturated value (quantum fluctuations), which vanished continuously when @xmath19. for @xmath20 we have two degenerate collinear states which are the helical states with pitch vectors @xmath21 and @xmath22. these two collinear states are characterized by a parallel spin orientation of nearest neighbors in vertical (or horizontal) direction and an antiparallel spin orientation of nearest neighbors in horizontal (or vertical) direction, and therefore exhibit nel order within the initial sublattice a and b. at @xmath23, the magnetization jumps from a nonzero to a zero value. the phase transition from nel to the quantum paramagnetic state is second order, whereas the transition from the collinear to the quantum paramagnetic state is first order@xcite. et al_.@xcite have shown that the intermediate quantum paramagnetic is a (_ singlet _) plaquette crystal, and the ground and first excited states are separated by a finite gap. the interest to study the two - dimensional @xmath12 heisenberg antiferromagnet have been greatly stimulated by its experimental realization in vanadium phosphates compounds@xcite, such as li@xmath24vosio@xmath25, li@xmath24vogeo@xmath25, and vomoo@xmath25, which might be described by this frustrated model in the case of @xmath26 (@xmath27). these isostructural compounds are characterized by a layered structure containing v@xmath28 (@xmath29) ions. the structure of v@xmath28 layer suggest that the superexchange is similar. in these compounds a second order phase transition to a long - range ordered magnetic phase has been observed. nmr spin - lattice relaxation measurements@xcite below @xmath30 shows that the order is collinear. due to the two - fold degeneracy of the ground - state for @xmath31 it is not possible to say _ a priori _ which will be the magnetic wave vector (i.e., @xmath21 and @xmath22) below @xmath30. on the other hand, such a scenario can change by considering spin - lattice coupling which will lift the degeneracy of the ground - state and will lower its energy@xcite. then, any structural distortion should inevitably reduce this competing interactions and thus reduces the frustration. in the case of this frustrated magnetic materials, the competing interactions are inequivalent but their topology and magnitudes can be tuned so that the strong quantum fluctuations destroy the long - range ordering. experimentally the ground state phase diagram of frustrated compounds, described by the @xmath12 model, can be explored continuously from high to the low @xmath6 regime by applying high pressures (p), which modify the bonding lengths and angles. recent results from x - ray diffraction measurements@xcite on the li@xmath24vosio@xmath25 compound has shown that the ratio @xmath8 decreases by about @xmath32 when the pressure increases from @xmath33 to @xmath34gpa. a generalization of the @xmath12 heisenberg antiferromagnetic model on a square lattice was introduced by nersesyan and tsvelik@xcite and studied by other groups@xcite, the so - called @xmath35 model. in the @xmath36 model is considered inequivalence nn couplings @xmath1 and @xmath37 in the two orthogonal spatial lattice dimensions with all the nnn bonds across the diagonals to have the same strength @xmath3. study of extensive band structure calculations@xcite for the vanadium phosphates abvo(po@xmath38)@xmath24 (ab = pb@xmath24, srzn, bazn, and bacd) have indicated four inequivalent exchange couplings : @xmath1 and @xmath2 between nn and @xmath3 and @xmath39 between nnn. for example, in srznvo(po@xmath25)@xmath24 was estimated @xmath40 and @xmath41 causing a distortion of the spin lattice. this spatial anisotropy tends to narrow the critical region and destroys it completely at a certain value of the interchain parameter @xmath7. on the other hand, by using the continuum limit of the @xmath36 spin-@xmath0 model starykh and balents@xcite have shown that this transition splits into two, with the presence of an intermediate quantum paramagnetic (columnar dimer) phase for @xmath42. bishop, _ _ et al__@xcite, by using coupled cluster treatment found the surprising and novel result that there exists a quantum triple point (* qtp *) with coordinates at (@xmath43), below which there is a second - order phase transition between the * af * and * caf * phases while above this * qtp * are these two ordered phases separated by the intermediate magnetically disordered phase (vbs or rvb). the order parameters of both the * af * and * caf * phases vanish continuously both below and above the * qtp *, which is typical of second - order phase transition. there is some evidence that the transition between the * caf * and intermediate phases is of first - order. using exact diagonalization@xcite with small lattice of @xmath44 (@xmath45) size, the intermediate * qp * phase for all interval of @xmath46 $] has been obtained for the pure spin-@xmath0 @xmath12 model on a square lattice. these results are in accordance with results obtained by starykh and balentes@xcite, that predicted not the * qtp * in the ground - state phase diagram recently observed by bishop, _ _ et al.__@xcite. the ground state (gs) properties of the two - dimensional frustrated heisenberg antiferromagnet have been investigated by various methods. the exact diagonalization starts from singlet states on pairs of sites, which cover the whole 2d lattice. however, the manifold of these states which can be constructed is nonorthogonal and overcomplete. this numerical methods are limited to small clusters @xmath47 due to storage problems. the computation on the largest cluster @xmath45 has been performed by schulz and co - workes@xcite @xmath48 years ago. in spite of the great improvements achieved during this time, it is not possible so far to repeat this calculation for the next interesting cluster @xmath49. this is only possible with other technique, as the quantum monte carlo simulation. due to the progress in computer hardware and the increased efficiency in programing, very recently@xcite the gs of the quantum spin-1/2 @xmath12 model have been calculated by the lanczos algoritm for a square lattice with @xmath50 sites. the theoretical treatment of the frustrated quantum models is far from being trivial. many of the standard many - body methods, such as quantum monte carlo techniques, may fail or become computationally infeasible to implement if frustration is present due to the minus - sign problem. hence, there is considerable interest in any method that can deal with frustrated spin systems. this considerable qualitative difference in the ground state phase diagram in the @xmath51 plane of the quantum spin-@xmath0 @xmath52 model further motivates us to study this issue by alternative methods. using a variational approximation, in which plaquettes of four spins are treated exactly, oliveira@xcite has studied the ground state phase diagram of the pure @xmath12 heisenberg antiferromagnet on a square lattice, where the quantitative results are in good accordance with a more sophisticated method (exact diagonalization). in this work, we generalize this variational method to treat the anisotropic square lattice (@xmath52 model). the rest of this paper is organized as follows : in sec. ii, the model is presented and a brief discussion of results. in sec. iii, the method is applied for the case of one plaquette with four spins interacting with other plaquette type mean field approximation. main results will be presented in sec. iv, as well as some discussions. finally, in sec. v we will give a brief summary. Model the critical behavior of the quantum spin-@xmath0 @xmath12 heisenberg model has been studied for many years, but very little has been done in the anisotropic square lattice case, which is described by following hamiltonian : @xmath53 where @xmath54 is the spin-@xmath0 pauli spin operators, the index @xmath55 labels the @xmath56 (row) and @xmath57 (column) components of the lattice sites. the first sum runs over all nn and the second sum runs over all nnn pairs. we denote the hamiltonian (1) by @xmath35 model, with strength @xmath1 along the row direction, @xmath37 along the column direction, @xmath58 along the diagonals, and we assume all couplings to be positive with @xmath59. the classical (@xmath60) model (1) has only two ordered ground - states : * af * (or nel) for @xmath61 and columnar stripe (* caf *) for @xmath62 separated by a first - order line at @xmath63. quantum fluctuations play a significant role in the magnetic phase diagram of the system at zero temperature. we will investigate the role of quantum fluctuations on the stability of the nel and collinear phases. in the @xmath29 case (quantum limit), the line splits into two phase transitions, where the ordered states (* af * and * caf *) are separated by an intermediate quantum paramagnetic (* qp *) phase, both on a square lattice. exact diagonalization@xcite has estimated a critical line at @xmath64, for the transition between the * caf * and * qp * states, and at @xmath65 between the * af * and * qp * states. the phase diagram in the @xmath51 plane obtained is in accordance with starykh and balents@xcite. however, the existence of * qtp * (_ quantum triple point _) that was predicted by bishop, _ et al_.@xcite, is not present in their obtained phase diagram. moreover, they found only presence of second - order phase transitions in the phase diagram. this contradictory qualitative results (existence or not of * qtp *) is the primary motivation behind this present work. on the other hand, a critical endpoint (* ce *) is a point in the phase diagram where a critical line meets and is truncated by a first - order line. this * ce * appear in the phase diagram of many physical systems such as binary fluid mixtures, superfluids, binary alloys, liquid crystals, certain ferromagnets, etc, and have been known for over a century@xcite. despite the * ce * long history, new singularities at the * ce * were predicted. fisher and upton@xcite argued that a new singularity in the curvature of the first - order phase transition line should arise at a * ce*. this prediction was confirmed by fisher and barbosas@xcite phenomenological studies for an exactly solvable spherical model. in conclusion of the analysis of the multicritical behavior observed in the ground - state phase diagram in the @xmath66 plane for the @xmath52 model, we have the presence of a _ quantum critical endpoint _ (* qce *) and not * qtp * as mentioned other works@xcite therefore, the objective of this work is to obtain the * qce * using the variational method, that was developed previously by oliveira@xcite in the pure limit (@xmath67) case. Method we first express the fluctuations around the classical ground state (* af * and * caf * phases), where consider a trial vector state @xmath68 for the ground state as a product of plaquette state @xmath69. we denote the plaquettes by @xmath70 label, that is composed of four spins, where it do not overlap (_ mean field _) on the square lattice as illustrated in figure 1. each plaquette state is given by and @xmath71 spin operators that are considered in eq. (2).] @xmath72 where \{@xmath73, @xmath74 } is the vector basis with @xmath75, \{@xmath76 } are real variational parameters obeying the normalization condition @xmath77. with this choice of vector states, the mean value of the spin operator in each site of the plaquette is given by @xmath78 @xmath79, where the components in the @xmath56 and @xmath57 directions are null. using the trial vector state defined in the eq. (2), we obtain the magnetizations at each site that are given by @xmath80@xmath81@xmath82 and@xmath83 where we have used the same set of parameters (_ canonical transformation _) of ref.@xcite, i.e., @xmath84, and @xmath85, which obeys the normalization condition @xmath86. the ground state energy per spin and unit of @xmath1, @xmath87, is given by @xmath88 with @xmath89 and@xmath90 , \tag{9}%\end{aligned}\]] where @xmath91 is the mean value of a given observable @xmath92 calculated in the vector state of the @xmath93 plaquette as illustrated in fig. the variational energy can be evaluated using the properties of the spin-@xmath0 pauli operator components, i.e., @xmath94,@xmath95 and @xmath96, that is expressed for@xmath97 + \nonumber\\ & 2\left ( 1-\lambda\right) \left ( y^{2}v^{2}-z^{2}\omega^{2}\right) + 2x\left ( y\lambda+z\right) + \alpha\left [ \dfrac{1}{2}-\left ( y - z\right) ^{2}-v^{2}-\omega^{2}+6x^{2}u^{2}-6y^{2}v^{2}-6z^{2}\omega^{2}\right] . \tag{10}%\end{aligned}\]] to obtain the minimum energy with a boundary condition given by normalization @xmath98, we use the lagrange multiplier method which correspond the minimization of the functional@xmath99 the stationary solutions (@xmath100) are obtained by solving the set of nonlinear equations @xmath101{c}% -\left ( \lambda+1\right) x-4\left ( \lambda+1\right) xu^{2}+2\left (y\lambda+z\right) + 12\alpha xu^{2}=2\eta x\\ \left ( 1-\lambda\right) \left ( y+4yv^{2}\right) + 2x\lambda-2\alpha\left (y - z\right) -12\alpha yv^{2}=2\eta y\\ -\left ( 1-\lambda\right) z-4\left ( 1-\lambda\right) z\omega^{2}% + 2x+2\alpha\left (y - z\right) -12\alpha z\omega^{2}=2\eta z\\ -\left ( \lambda+1\right) u=2\eta u\\ \left ( 1-\lambda\right) v-2\alpha v=2\eta v\\ -\left ( 1-\lambda\right) \omega-2\alpha\omega=2\eta\omega \end{array } \right. , \tag{12}%\]] where @xmath102 is the lagrange multiplier. Results the variational parameters @xmath103, and @xmath102 are determined simultaneously solving the system of equations (12) combined with the normalization condition @xmath98 for each phase. in the quantum paramagnetic (* qp *) phase we have @xmath104. we note that in the isotropic limit (@xmath67), our results reduce the same expression obtained by oliveira@xcite. in this disordered phase, the ground state vector @xmath105 is an eigenvector of @xmath106, where @xmath107 is the total spin of the @xmath70th plaquette of four spins, with zero eigenvalue (singlet state). in the * af * ordered phase we have the boundary condition @xmath108, and in the * caf * phase @xmath109. the order parameters @xmath110 and @xmath111 are numerically obtained as a function of frustration parameter @xmath8 for a given value of spatial anisotropy @xmath7. we observe that the order parameter @xmath112 goes smoothly to zero when the frustration parameter (@xmath8) increases from zero to @xmath113 with @xmath114 characterizing a second - order phase transition. a simple fitting of the form @xmath115 in the vicinity of the second - order transition gives the same classical value for the critical exponent @xmath116. on the other hand, for @xmath114 and @xmath117 the staggered magnetization @xmath118 increases monotonically with the frustration parameter @xmath8 in the * caf * phase, with a discontinuity of @xmath118 at @xmath119, which is a first - order phase transition. for @xmath120, the * qp * intermediate phase between the two ordered states (* af * and * caf *) disappears, and a direct transition between the magnetically ordered * af * and * caf * located at the crossing point @xmath11 correspondent to the classical value. the ground state (@xmath13) phase diagram in the @xmath66 plane is displayed in fig. the solid line indicate the critical points and the dashed lines represent first - order frontiers. we observe three different phases, namely : * af * (antiferromagnetic), * caf * (collinear antiferromagnetic) and * qp * (quantum paramagnetic). the * af * and * qp * phases are separated by a second - order transition line @xmath121, while the * qp * and * caf * phases are separated by a first - order transition line @xmath122. the presence of the interchain parameter @xmath7 has the general effect of suppressing the * qp * phase. the * qp * region decreases gradually with the decrease of the @xmath7 parameter, and it disappears completely at the _ quantum _ _ critical endpoint _ * * qce**@xmath123(@xmath124) where the boundaries between these phases emerge. below this * qce *, i.e., for @xmath120, there is a direct first - order phase transition between the * af * and * caf * phases, with a transition point @xmath125 (classical value). in order to illustrate the nature of the phase transition, we also show, in inset fig. 2, the behavior of the staggered magnetization (order parameter) as a function of the frustration parameter (@xmath8) for @xmath126 and @xmath127 @xmath128. from curves such as those shown in fig. 2 we see that for @xmath129 there exists an intermediate region between the critical point @xmath130 at which @xmath131 for the * af * phase, characterizing a second - order transition, and the point @xmath132 at which the @xmath133 order parameter presents a discontinuity for the * caf * phase, characterizing a first - order transition. for @xmath134, the order parameter of the * af * phase decreases monotonically with increase of the frustration parameter from @xmath135, for @xmath136, to zero for @xmath137 (@xmath138). in the * caf * phase @xmath133 decreases from @xmath139 for @xmath140 to @xmath141 for @xmath142, characterizing a direct first - order transition between the magnetically ordered * af * and * caf * phases located at the crossing point. we note that the definition of the order parameter @xmath143 (@xmath144) difer of @xmath0 factor when compared with calculations which use other methods (i.e., @xmath145). therefore, in the limit of the not frustrated (@xmath136) square lattice (@xmath67) antiferromagnetic, solving the equations (12) and applying the corrections factor we found @xmath146 which is consistent with the numerical results obtained by various methods such as series expansion, quantum monte carlo simulation, and others@xcite, and can also be compared with experimental results for the k@xmath24nif@xmath25, k@xmath24mnf@xmath25, and rb@xmath147mnf@xmath25 compounds@xcite. plane for the quantum spin-@xmath0 @xmath35 model on a square lattice, where @xmath6 and @xmath148. the dashed and solid lines corresponds the first- and second - order transitions lines, respectively. the black point represents the quantum critical endpoint (* qce *). the notations indicated by * af *, * caf * and * qp * corresponds the antiferromagnetic, collinear antiferromagnetic and quantum paramagnetic phases, respectively. the dotted line correspond the classical solution @xmath63.] Conclusion in summary, we have studied the effects of quantum fluctuations due to spatial (@xmath7) and frustration (@xmath8) parameter in the quantum spin-@xmath0 @xmath149 heisenberg model. using a variational method we calculated the sublattice magnetization for the * af * and * caf * phases. for values of @xmath150 the frustration contributes significantly to the existence of a disordered intermediate state (* qp *) between the two * af * and * caf * ordered phases, while for @xmath151, we have a direct first - order transition between the * af * and * caf * phases. we have observed, by analyzing the order parameters of the * af * and * caf * phases, that the phase transitions are of second and first - order between the * af - qp * and * caf - qp *, respectively. the obtained phase diagram can be compared with recent results which used effective - field theory@xcite and coupled - cluster method@xcite, showing the same qualitative results predicting a paramagnetic region for small interlayer parameter (i.e., @xmath152), and for @xmath120 this * qp * phase disappears by presenting a direct first - order transition between the * af * and * caf * phases. on the other hand, recent calculations of second order spin wave theory@xcite have indicated that the intermediate * qp * phase exists for all @xmath42 in accordance with results of exact diagonalization@xcite. we speculate that by using a more sophisticated method, for example, quantum monte carlo simulations@xcite and density matrix renormalization group (dmrg) method@xcite, this disordered region should disappear for certain values of @xmath120. r. f. bishop, p. h. y. li, r. darradi, j. schulenburg, and j. richter, _ phys. b _ * 78 *, 054412 (2008) ; r. f. bishop, p. h. y. li, r. darradi, j. schulenburg, j. richter, and c. e. campbell, _ j. phys. : condens. matter _ * 20 *, 415213 (2008). see also, r. darradi, j. richter, j. schulenburg, r. f. bishop, and p. h. y. li, _ j. phys. : conference series _ * 145 *, 012049 (2009). p. carretta, r. melzi, n. papinutto, and p. millet, _ phys. * 88 *, 047601 (2002) ; p. carretta, n. papinutto, c. b. azzoni, m. c. mozzati, e. pavarini, s. gonthier, and o. millet, _ phys. b_. * 66 *, 094420 (2002).

the phase transition of the quantum spin-@xmath0 frustrated heisenberg antiferroferromagnet on an anisotropic square lattice is studied by using a variational treatment. the model is described by the heisenberg hamiltonian with two antiferromagnetic interactions : nearest - neighbor (nn) with different coupling strengths @xmath1 and @xmath2 along x and y directions competing with a next - nearest - neighbor coupling @xmath3 (nnn). the ground state phase diagram in the (@xmath4) space, where @xmath5 and @xmath6, is obtained. depending on the values of @xmath7 and @xmath8 , we obtain three different states : antiferromagnetic (* af *), collinear antiferromagnetic (* caf *) and quantum paramagnetic (* qp *). for an intermediate region @xmath9 we observe a * qp * state between the ordered * af * and * caf * phases, which disappears for @xmath7 above some critical value @xmath10. the boundaries between these ordered phases merge at the _ quantum critical endpoint _ (* qce *). below this * qce * there is again a direct first - order transition between the * af * and * caf * phases, with a behavior approximately described by the classical line @xmath11. * pacs numbers * : 75.10.jm, 05.30.-d, 75.40.-s, 75.40.cx

1205.5710

Parameter adjustment in adaptive control and recursive parameter estimation one often needs to adjust recursively an estimate @xmath0 of a vector @xmath1, which comprises @xmath2 constant but unknown parameters, using measurements of a quantity @xmath3 here @xmath4 is a vector of known data, often called the regressor, and @xmath5 is a measurement error signal. the goal of tuning is to keep both the estimation error @xmath6 and the parameter error @xmath7 as small as possible. there are several popular methods for dealing with the problem above, for instance least - squares. maybe the most straightforward involve minimizing the prediction error via gradient - type algorithms of the form : @xmath8 where @xmath9 is a constant, symmetric, positive - definite gain matrix. let us define @xmath10 and analyze differential equations and, which under the assumption that @xmath11 is identically zero read : @xmath12 the nonnegative function @xmath13 has time derivative @xmath14 hence @xmath15 inspection of the equation above reveals that @xmath16 is limited in time, thus @xmath17, and also that the error @xmath18 (norms are taken on the interval @xmath19 where all signals are defined). these are the main properties an algorithm needs in order to be considered a suitable candidate for the role of a tuner in an adaptive control system. often @xmath20 or something similar is also a desirable property. to obtain the latter, normalized algorithms can be used ; however, the relative merits of normalized versus unnormalized tuners are still somewhat controversial. another alternative is to use a time - varying @xmath9, as is done in least - squares tuning. in [sec : acceleration] we present a tuner that sets the second derivative of @xmath0, and in [sec : covariance] the effects of a white noise @xmath5 on the performance of the two algorithms are compared. then we show some simulations and make concluding remarks. The accelerating algorithm classical tuners are such that the _ velocity _ of adaptation (the first derivative of the parameters) is set proportional to the regressor and to the prediction error @xmath21. we propose to set the _ acceleration _ of the parameters : @xmath22 notice that the the formula above is implementable (using @xmath23 integrators) if measurement error is absent, because the unknown @xmath24 appears only in scalar product with @xmath25. choose another function of lyapunovian inspiration : @xmath26 taking derivatives along the trajectories of gives @xmath27 integrating @xmath28 we obtain @xmath29 which leads immediately to the desired properties : @xmath30 the slow variation property @xmath31 follows without the need for normalization, and now we obtain @xmath32 instead of @xmath33 as before. we might regard @xmath34 as a modified error, which can be used in the stability analysis of a detectable or `` tunable '' adaptive system via an output - injection argument ; see @xcite. a generalization of is @xmath35 with @xmath36 and @xmath37 constant, symmetric, positive - definite @xmath38 matrices such that @xmath39 and @xmath40. the properties of tuner, which can be obtained using the positive - definite function @xmath41 in the same manner as before, are @xmath42 Covariance analysis we now consider the effects on the expected value and covariance of @xmath43 of the presence of a measurement error. the assumptions are that @xmath11 is a white noise with zero average and covariance @xmath44 and that @xmath45 are given, deterministic data. for comparison purposes, first consider what happens when the conventional tuner is applied to in the presence of measurement error @xmath5 : @xmath46 the solution to the equation above can be written in terms of @xmath47 s state transition matrix @xmath48 as follows @xmath49 hence @xmath50 because @xmath51 by assumption. here the notation @xmath52, denoting the expectation with respect to the random variable @xmath5, is used to emphasize that the stochastic properties of @xmath25 are not under consideration. the conclusion is that @xmath43 will converge to zero in average as fast as @xmath53 does. the well - known persistency of excitation conditions on @xmath54 are sufficient for the latter to happen. to study the second moment of the parameter error, write @xmath55 the covariance of @xmath43 can be written as the sum of four terms. the first is deterministic. the second term @xmath56 because @xmath11 has zero mean, and the third term is likewise zero. the fourth term @xmath57 where fubini s theorem and the fact @xmath58 were used. performing the integration and adding the first and fourth terms results in @xmath59 this equation can be given the following interpretation : for small @xmath60, when @xmath53 is close to the identity, the covariance of @xmath43 remains close to @xmath61, the outer product of the error in the initial guess of the parameters with itself. as @xmath62, which will happen if @xmath54 is persistently exciting, @xmath63 tends to @xmath64. this points to a compromise between higher convergence speeds and lower steady - state parameter error, which require respectively larger and smaller values of the gain @xmath9. algorithms that try for the best of both worlds parameter convergence in the mean - square sense often utilize time - varying, decreasing gains ; an example is the least - squares algorithm. we shall now attempt a similar analysis for the acceleration tuner applied to, which results in the differential equation @xmath65 let @xmath66 where @xmath67, @xmath68, each @xmath69 is a function of @xmath70 unless otherwise noted, and the dot signifies derivative with respect to the first argument. if @xmath71, @xmath72 following the same reasoning used for the velocity tuner, one concludes that @xmath73 and that @xmath74 however the properties of the acceleration and velocity tuners are not yet directly comparable because the right - hand side of does not lend itself to immediate integration. to obtain comparable results, we employ the ungainly but easily verifiable formula, @xmath75 ''' '' valid for arbitrary scalars @xmath76 and @xmath77, and make the [[simplifying - assumption]] simplifying assumption : + + + + + + + + + + + + + + + + + + + + + + + + for @xmath78, and 3, @xmath79, where @xmath80 are scalars and @xmath81 is the @xmath82 identity matrix. premultiplying by @xmath83 $], postmultiplying by @xmath83^\top$], integrating from 0 to @xmath60, and using the simplifying assumption gives formula. @xmath84 ''' '' taking @xmath85 in, @xmath86 results positive - semidefinite, therefore @xmath87 the combination of and shows that @xmath88 can be increased without affecting @xmath24 s steady - state covariance. on the other hand, to decrease the covariance we need to increase @xmath89, which roughly speaking means increasing damping in. since @xmath88 and @xmath89 can be increased without affecting the stability properties shown in [sec : acceleration], a better transient @xmath90 steady - state performance compromise might be achievable with the acceleration tuner than with the velocity tuner, at least in the case when @xmath91, @xmath92, and @xmath37 are `` scalars. '' notice that @xmath93 by construction. [[approximate - analysis]] approximate analysis : + + + + + + + + + + + + + + + + + + + + + + the derivation of inequality does not involve any approximations, and therefore provides an upper bound on @xmath94, valid independently of @xmath54. a less conservative estimate of the integral in can be obtained by replacing @xmath95 by its average value @xmath96 in the definition of @xmath86 in. this approximation seems reasonable because @xmath86 appears inside an integral, but calls for more extensive simulation studies. to obtain a useful inequality, we require @xmath97 ; namely, using the schur complement @xmath98 or, using the simplifying assumption and substituting @xmath95 by its approximation @xmath96 @xmath99 suppose further that @xmath100. looking for the least conservative estimate, we pick @xmath101, the least value of @xmath76 that keeps @xmath97. thus @xmath102 with @xmath103 \bar{m}_1 \left[\begin{smallmatrix}{\phi}^\top_{11}(t,0) \\ { \phi}^\top_{12}(t,0) \end{smallmatrix}\right]}{4m_1 ^ 2 m_2m_3r(1+\mu_2) -r}.$] taking @xmath104 we repeat the previous, exact result. for large positive values of @xmath77 the first term of the right - hand side of tends to @xmath105, which indicates that the steady - state covariance of the parameter error decreases when the signal @xmath25 increases in magnitude, and that it can be made smaller via appropriate choices of the gains @xmath88 and @xmath106. the situation for the accelerating tuner is hence much more favorable than for the conventional one. Simulations the simulations in this section compare the behavior of the accelerating tuner with those of the gradient tuner and of a normalized gradient one. all simulations were done in open - loop, with the regressor a two - dimensional signal, and without measurement noise. figure [fig : step] shows the values of @xmath107 and @xmath108 respectively when @xmath25 is a two - dimensional step signal. in figure [fig : sin] the regressor is a sinusoid, in figure [fig : sia] an exponentially increasing sinusoid, and in figure [fig : prb] a pseudorandom signal generated using matlab. no effort was made to optimize the choice of gain matrices (@xmath91, @xmath92, and @xmath37 were all chosen equal to the identity), and the effect of measurement noise was not considered. the performance of the accelerating tuner is comparable, and sometimes superior, to that of the other tuners. = 2.5 in = 2.5 in = 2.5 in = 2.5 in = 2.5 in = 2.5 in = 2.5 in = 2.5 in Concluding remarks other ideas related to the present one are replacing the integrator in with a positive - real transfer function @xcite, and using high - order tuning (@xcite). high - order tuning generates as outputs @xmath0 as well as its derivatives up to a given order (in this sense we might consider the present algorithm a second - order tuner), but unlike the accelerating tuner requires derivatives of @xmath25 up to that same order. we expect that accelerating tuners will find application in adaptive control of nonlinear systems and maybe in dealing with the topological incompatibility known as the `` loss of stabilizability problem '' in the adaptive control literature. the stochastic analysis in [sec : covariance] indicates that the performance and convergence properties of the accelerating tuner, together with its moderate computational complexity, may indeed make it a desirable tool for adaptive filtering applications. it seems that a better transient @xmath90 steady - state performance compromise is achievable with the accelerating tuner than with the velocity tuner. to verify this conjecture, a study of convergence properties of the accelerating tuner and their relation with the persistence of excitation conditions is in order, as well as more extensive simulations in the presence of measurement noise.

we propose a tuner, suitable for adaptive control and (in its discrete - time version) adaptive filtering applications, that sets the second derivative of the parameter estimates rather than the first derivative as is done in the overwhelming majority of the literature. comparative stability and performance analyses are presented. * key words : * adaptive control ; parameter estimation ; adaptive filtering ; covariance analysis.

math9903022

Introduction the entanglement entropy, as a tool to detect and classify quantum phase transitions, has been playing an important role in the last fifteen years (see @xcite and references therein). in one dimension, where most of the critical quantum chains are conformal invariant, the entanglement entropy provides a powerful tool to detect, as well to calculate, the central charge @xmath11 of the underlying cft. for example, for quantum chains, the ground - state entanglement entropy of a subsystem formed by contiguous @xmath7 sites of an infinite system, with respect to the complementary subsystem has the leading behavior @xmath12 if the system is critical or @xmath13, when the system is noncritical with correlation length @xmath14 @xcite. although there are plenty of proposals to measure this quantity in the lab @xcite the actual experiments were out of reach so far. strictly speaking the central charge of quantum spin chains has never been measured experimentally. recently other quantities, that are also dependent of the central charge has been proposed @xcite. among these proposals interesting measures that, from the numerical point of view, are also efficient in detecting the phase transitions as well as the universality class of critical behavior, are the shannon and rnyi mutual informations @xcite (see also the related works @xcite). the rnyi mutual information (the exact definition will be given in the next section) has a parameter @xmath5 that recovers the shannon mutual information at the value @xmath15. the results derived in @xcite indicate that the shannon and rnyi mutual informations of the ground state of quantum spin chains, when expressed in some special local basis, similarly as happens with the shannon and rnyi entanglement entropy, show a logarithmic behavior with the subsystem s size whose coefficient depends on the central charge. recently additional new results concerning the shannon and rnyi mutual information in quantum systems were obtained, see @xcite. there are also studies of the mutual information in classical two dimensional spin systems @xcite. it is worth mentioning that the shannon and rnyi mutual informations studied in the above papers, as will be defined in the next section, are basis dependent quantities. it is important to distinguish them from the more known basis independent quantity, namely, the von neumann mutual information. for recent developments on the calculation of the von neumann mutual information in thermal equilibrium and non - equilibrium systems see @xcite. most of the results regarding the shannon and the rnyi mutual information, except for the case of harmonic chains, are based on numerical analysis, especially for systems with central charge not equal to one. one of the main problems in a possible analytical derivation comes from the presence of a discontinuity at @xmath15 of the rnyi mutual information. this discontinuity prevents the use of the replica trick, which is normally a necessary step for the analytical derivation of the shannon mutual information. in this paper we will consider, for many different quantum chains, another version of the mutual information, which is also parametrized by a parameter @xmath5 that reduces at @xmath15 to the shannon mutual information. the motivation for our calculations is two fold. firstly this definition is more appropriate from the point of view of a measure of shared information among parts of a system, since it has the expected properties. this will be discussed in the appendix. secondly, this quantity does not show any discontinuity at @xmath15, so it might be a good starting point for the analytical calculation of the shannon mutual information with some sort of analytical continuation of the parameter @xmath5. from now on we will call this new quantity generalized mutual information. having the above motivations in mind we firstly calculated numerically (using exact diagonalization) the generalized mutual information for several critical quantum spin chains. we considered models with @xmath1 symmetries like the @xmath16-state potts modes for @xmath17 and @xmath18, the z(4) ashkin - teller model and the @xmath1 parafermionic models with @xmath19. we then calculated the generalized mutual information for quantum critical harmonic chains (discrete version of klein - gordon field theory) and also for quantum spin chains with @xmath4 symmetry like the xxz and the spin-1 fateev - zamolodchikov quantum chains. the structure of the paper is as follows : in the next section we will present the essential definitions of the shannon and rnyi mutual informations as well as generalized mutual information. in section three we will present the numerical results of the generalized mutual information for many different critical quantum spin chains. finally in the last section we present our conclusions. the generalized mutual informations: definitions consider the normalized ground state eigenfunction of a quantum spin chain hamiltonian @xmath20, expressed in a particular local basis @xmath21, where @xmath22 are the eigenvalues of some local operators defined on the lattice sites. the rnyi entropy is defined as @xmath23 where @xmath24 is the probability of finding the system in the particular configuration given by @xmath25. the limit @xmath26 gives us the shannon entropy @xmath27. since we are considering only local basis it is always possible to decompose the configurations as a combination of the configurations inside and outside of the subregions as @xmath28. one can define the marginal probabilities as @xmath29 and @xmath30. in a previous paper @xcite we studied the naive definition of the rnyi mutual information : @xmath31 from now on instead of using @xmath32 we will use just @xmath33. the known results of the rnyi mutual informations of quantum critical chains are obtained by using the definition ([renyi2]). for special basis, usually the ones where part of the hamiltonian is diagonal (see @xcite), the definition ([renyi2]) for the rnyi mutual information gives us a logarithmic behavior with the subsystem size, for arbitrary values of @xmath5. however, as observed numerically for several quantum chains (see @xcite), it shows a discontinuity at @xmath15, that forbids the use of large-@xmath5 analysis to obtain the most interesting case where @xmath15, namely the standard shannon mutual information. although the definition ([renyi2]) has its own uses it is not the one which normally has been considered in information sciences. for example @xmath34 for @xmath35 is not necessarily a positive function, a property that we naturally expect to be hold for the mutual informations. in this paper we consider a definition that is common in information sciences @xcite. the generalized mutual information with the desired properties, as a measure of shared information (see appendix), is defined as @xcite : @xmath36 where @xmath37 and @xmath38, as before, are the probabilities that the subsystems are independently in the configurations @xmath39 and @xmath40 that forms the configuration @xmath25 that occurs with probability @xmath33. hereafter @xmath6 will represent the size of the whole system and @xmath7 and @xmath8 the sizes of the subsystems. with this new notation one can write @xmath41 as @xmath42. this definition of the generalized mutual information comes from the natural extension of the relative entropy to the rnyi case and measures the distance of the full distribution from the product of two independent distributions. in the limit @xmath43 one easily recovers the shannon mutual information @xmath44, where @xmath27 is the standard shannon entropy. one of the important properties of @xmath0, that is not shared by @xmath34, is its nondecreasing behavior as a function of @xmath5 (see appendix). our calculations for a set of distinct quantum spin chains will be done numerically, since up to our knowledge an analytical method to consider these quantum chains is still missing. The generalized mutual information in quantum chains in this section we will numerically calculate the ground - state generalized mutual information of two series of critical quantum spin chains with slightly different structure. in the first part we will calculate the generalized mutual information for systems with discrete symmetries such as the @xmath16-state potts models with @xmath17 and @xmath18, the ashkin - teller model and the parafermionic @xmath1-quantum spin chain @xcite for the values of @xmath45 and @xmath46. in the second part we will calculate the generalized mutual information for systems with @xmath4 symmetry such as the klein - gordon field theory, the xxz model and the fateev - zamolodchikov model with different values of their anisotropy parameters. in this subsection we will study the generalized mutual information of the ground state of different critical spin chains with @xmath1 discrete symmetries. the results we present were obtained by expressing the ground - state wavefunction in two specific basis where the systems show some universal properties. [htb] [fig1] of the @xmath47 sites periodic ising quantum chain, as a function of @xmath48. the ground - state wavefunction is in the basis where the matrices @xmath49 are diagonal (@xmath50 basis).,title="fig:",scaledwidth=35.0%] our results show that the q - state potts model and the ashkin - teller model share a similar behavior. for this reason we discuss them together. the critical @xmath16-state potts model in a periodic lattice is defined by the hamiltonian @xcite @xmath51 where @xmath49 and @xmath52 are @xmath53 matrices satisfying the following @xmath1 algebra : @xmath54=[s_i, s_j]=[s_i, r_j]=0 $] for @xmath55 and @xmath56 and @xmath57. the model has its critical behavior governed by a cft with central charge @xmath58 where @xmath59. the @xmath60 potts chain is just the standard ising quantum chain. [htb] [fig2] of the @xmath47 sites periodic ising quantum chain, as a function of @xmath48. the ground - state wavefunction is in the basis where the matrices @xmath52 are diagonal (@xmath61 basis).,title="fig:",scaledwidth=35.0%] the ashkin - teller model has a @xmath62 symmetry and a hamiltonian given by : @xmath63 where @xmath49 and @xmath52 are the same matrices introduced in the @xmath64 potts model. the model is critical and conformal invariant for @xmath65 with the central charge @xmath66. it is worth mentioning that at @xmath67 we recover the @xmath64 potts model and at @xmath68 the model is equivalent to two decoupled ising models. in the paper @xcite we already showed that the shannon and rnyi mutual informations, as defined in ([renyi2]), are basis dependent. in other words one can get quite distinct different finite - size scaling behaviors by considering different basis. surprisingly in some particular basis, that we called conformal basis, the results shows some universality. for example, the results for the @xmath16-state potts model and for the ashkin - teller model in the basis where the matrices @xmath52 or the matrices @xmath49 are diagonal are the same, and follow the asymptotic behavior @xmath69 with @xmath70 we should mention that in @xcite, based on numerical results, it was claimed that for @xmath15 the coefficient @xmath71 might not be exactly equal to the central charge. as it was discussed in @xcite it is quite likely that @xmath34 is not a continuous function around @xmath15 and so any attempt to do the replica trick using this definition of rnyi mutual information will be useless. this makes the analytical calculation a challenge. this is an additional reason to examine the behavior of @xmath0, besides being the correct extension, from the point of view of a measure of shared information. having this in mind we calculated the @xmath0 for @xmath17 and @xmath64 potts chains and for the ashkin - teller model in the @xmath61 and the @xmath50 basis. we found that in some regimes of variation of the parameter @xmath5 one can fit the data nicely to @xmath72 being @xmath10 a monotonically nondecreasing function of @xmath5, consistent with what we expect for the mutual information, since it is a good measure of shared information (see the appendix). [htb] [fig3] of the coefficient of the logarithm in the equation ([renyi mi potts]) and the central charge @xmath11 for the @xmath16-state potts model with @xmath17 and 4, and for the ashkin - teller model (a - t) with different anisotropies @xmath73. the ashkin - teller model at the isotropic point (@xmath74) is equivalent to the @xmath18-state potts model. the ground - state wavefunctions are in the basis where the @xmath49 matrices are diagonal. the lattice sizes of the models are shown and the coefficients @xmath10 were estimated by using the subsystem sizes @xmath75$].,title="fig:",scaledwidth=35.0%] here we summarize the results for the @xmath16-state potts and ashkin - teller quantum chains : 1. the results in general depend on the basis we choose to express the ground - state wavefunction. 2. the generalized mutual information follows ([renyi mi potts]) in the @xmath50 and @xmath61 basis but with different coefficients for different basis. to illustrate the logarithmic behavior we show in fig. 1 and fig. 2 the mutual information @xmath0 for the ising model (@xmath60) with @xmath47 sites and ground - state eigenfunctions in the @xmath50 and @xmath61 basis, respectively. we see, from these figures, that for subsystem sizes @xmath76 we have the logarithmic behavior given by ([renyi mi potts]) up to @xmath77 in the @xmath50-basis and @xmath78 in the @xmath61-basis. as we can see our results does not exclude the existence of some relevant @xmath7-dependent terms in ([renyi mi potts]) for large values of @xmath5. + [htb] [fig4] are diagonal. the lattice sizes of the models are shown in the figure, as well as the subsystems sizes @xmath7 used to estimate @xmath10.,title="fig:",scaledwidth=35.0%] 3. the coefficient of the logarithm @xmath10 in ([renyi mi potts]) is a continuous monotonically non - decreasing function of @xmath5 and it follows the following formula in the @xmath50 basis : @xmath79 where @xmath11 is the central charge and @xmath80 seems to be a continuous universal function independent of the model, as we can see in fig. 3. in the case of the ashkin - teller model the results start to deviate around @xmath81 from the ones obtained for the potts models. as we can see in fig. 3, the deviation point is dependent on the anisotropy parameter @xmath73 of the model. 4. in the case of the @xmath61 basis, as one can see in fig. 4, equation ([coefficient]) is still valid for values of @xmath5 up to @xmath824. however the function @xmath80 is distinct from the one obtained in the @xmath50 basis. as shown in fig. 4, up to @xmath83 the form of the function @xmath80 seems to be also independent of the model. this figure also shows that the ashkin - teller model has stronger deviations in this basis, as compared with the results obtained in the @xmath50 basis. in order to better see the difference of the coefficients @xmath10 in the @xmath50 and @xmath61 basis, we present in fig. 5 the data of figs. 3 and 4 for the @xmath17 and 4 state potts models. + [htb] [fig5] of figs. 3 and 4 for the @xmath17 and 4-state potts model are shown in the same figure, for comparison.,title="fig:",scaledwidth=35.0%] 5. the coefficient of the logarithm in the @xmath50 basis always goes to zero as @xmath84, differently from the @xmath61 basis where it approaches to a non - trivial number. this simply means that probably in the continuum limit all the probabilities in the @xmath50 basis are positive but in the @xmath61 basis some of them are zero. for the definition of the @xmath85 case see the appendix. our numerical results indicate that @xmath10 is a continuous function of @xmath5 around @xmath15. this means that @xmath0 should be a continuous function with respect to @xmath5 and so it is a better candidate to be used in techniques exploring the analytical continuation of the value @xmath5, as happens for example in the replica trick. however, the appropriate technique that may be used is still unclear to us. it is important to mention that the results obtained for the ratio @xmath86 in this section (fig. 3) and in the subsequent ones (figs. 8, 9, 11 and 13) are based on the linear fit with the @xmath87 $] dependence. these fittings were done by choosing a set of subsystems sizes. in all the presented figures we only depict results where a small variation of the number of subsystem sizes gives us estimated values of @xmath10 that differs a few percent. as an example we consider the fittings obtained from the data of figs. 1 and 2 for the ising model with @xmath47 sites and ground - state eigenfunction in the @xmath50 and @xmath61 basis, respectively. this is shown in fig. as we can see, while for the @xmath50 basis the fitting is reasonable up to @xmath88 in the @xmath61 basis we do not have reliable results for @xmath89. [htb] [fig6] obtained from the data of figs. 1 and 2 for the ising quantum chain with @xmath47 sites and eigenfunction expressed in @xmath50 and @xmath61 basis.,title="fig:",scaledwidth=35.0%] [htb] [fig7] of the @xmath90 sites periodic @xmath91-parafermionic quantum chain, as a function of @xmath48. the ground - state wavefunction is in the basis where the @xmath49 matrices are diagonal (@xmath50 basis).,title="fig:",scaledwidth=35.0%] in this subsection we consider the generalized mutual information for some critical spin chains with discrete @xmath1 symmetry and central charge bigger than one. the quantum chains we consider are the parafermionic @xmath1-quantum spin chains @xcite with hamiltonian given by @xcite @xmath92 where again @xmath49 and @xmath52 are the @xmath53 matrices that appeared in ([potts hamiltonian]). this model is critical and conformal invariant with a central charge @xmath93. for the case where @xmath60 and @xmath94 we recover the ising and 3-state potts model, and for the case where @xmath64 we obtain the ashkin - teller model with the anisotropy value @xmath95. we have considered the models with @xmath45 and @xmath46 and the ground - state wavefunctions expressed in the @xmath50 or @xmath61 basis. the results for the several values of @xmath16 are shown in figs. 7, 8 and 9. to illustrate the logarithmic dependence with the subsystem size @xmath7 we show in fig. 7 @xmath42, as a function of @xmath96 for the @xmath91 parafermionic quantum chain with @xmath90 sites, with the ground - state wavefunction expressed in the @xmath50 basis. in figs. 8 and 9 we show the ratio @xmath86 of the logarithmic coefficient of ([renyi mi potts]) with the central charge @xmath11 for the @xmath1-parafermionic models with ground - state wavefunction in the @xmath50 and @xmath61 basis, respectively. the maximum lattice sizes we used for the @xmath1-parafermionic models are @xmath97 and @xmath98 for @xmath45 and 8, respectively. the results we obtained are very similar to the ones we already discussed in the previous case of the @xmath16-state potts models. all the five properties that we discussed in that subsection are equally valid also for the @xmath1-parafermionic models. by comparing the results of figs. 8 and 9 with figs. 3 and 4 we observe that the function @xmath80 in ([coefficient]) are quite similar for the two set of models, at least for values of @xmath5 up to @xmath99. probably the matching of these curves is not perfect due to the small system sizes we consider, specially for @xmath100. [htb] [fig8] of the coefficient of the logarithm in equation ([renyi mi potts]) and the central charge @xmath11 for the @xmath1-parafermionic models with @xmath45 and 8. the ground - states are in the basis where the @xmath49 matrices are diagonal. the lattice sizes of the models are shown in the figure and the coefficients @xmath10 were estimated by using the subsystem sizes @xmath75$].,title="fig:",scaledwidth=35.0%] in this section we consider the generalized mutual information of critical chains having a continuous @xmath4 symmetry. we studied a set of coupled harmonic oscillators which gives a discrete version of klein - gordon field theory as well as the spin-1/2 xxz and the spin-1 fateev - zamolodchikov quantum chains. the last two models are interesting since, like the ashkin - teller model, they have an anisotropy that gives us a critical line of continuously varying critical exponents but with a fixed central charge. [htb] [fig9] are diagonal. the lattice sizes of the models, as well as the subsystems sizes @xmath7 used to estimate @xmath10 are shown.,title="fig:",scaledwidth=35.0%] in this subsection we will first consider the generalized mutual information of the ground state of a system of generic coupled harmonic oscillators. then at the very end we will confine ourselves to the simple case where we have only the nonzero couplings at the next - nearest sites, that in the continuum limit gives us the klein - gordon field theory. consider the hamiltonian of @xmath6-coupled harmonic oscillators, with coordinates @xmath101 and conjugated momenta @xmath102 : @xmath103 the ground state of the above hamiltonian has the following form @xmath104 for the general hamiltonian ([harmonicosc]), one can calculate the two point correlators @xmath105 and @xmath106 using the @xmath107 matrix defined in ([harmonicosc]). the squared root of this matrix, as well as its inverse, can be split up up into coordinates of the subsystems @xmath108 (size @xmath7) and @xmath109 (size @xmath8), i. e., @xmath110 here we chose the couplings so that we always keep the equalities @xmath111 and @xmath112. the spectra of the matrix @xmath113, can be used to calculate the rnyi entanglement entropy (see @xcite and references therein) as @xmath114. \nonumber\end{aligned}\]] in this formulation we only need the correlators inside the region @xmath108. note that the above quantity is basis independent and is considered as an usual measure of the quantum entanglement. here we need to introduce this quantity just for later use. to calculate the generalized mutual information for a system of coupled harmonic oscillators one first needs to fix the basis. here we work in the position coordinate basis, however all the results are valid also in the momentum basis. one should notice that the same is not true if one works in a generic basis obtained through canonical transformations from the position or momentum basis. in order to calculate @xmath0 first we find @xmath115 and @xmath116 as @xmath117 where @xmath118 and @xmath119. since @xmath120 takes continuum values one needs to consider the integral version of the equation ([renyi new]) as follows : @xmath121 where @xmath122. plugging eqs. ([groundswave]), ([subsystem probabilities1]) ([subsystem probabilities2]) in the equation ([new renyi continuum]) and performing the gaussian integral one can derive the generalized mutual information @xmath123 the following determinant formulas @xmath124 allow us to write @xmath125 where @xmath126 there is an important remark that we should mention : in principle eq. ([renyi mi hoc]) makes sense only if @xmath127 is a symmetric positive definite matrix. if we start with a symmetric positive definite matrix @xmath128 this is already warrantied for @xmath129 but for @xmath130 one needs to check its validity. this will be an important point when we study the short - range coupled harmonic oscillators. finally one can write @xmath131 where @xmath132 is the only @xmath5 dependent part. we notice here that by changing @xmath5 to @xmath133 we just change the sign of the second term, i. e., @xmath134. [htb] [fig10]), @xmath132, as a function of @xmath135 for periodic quantum harmonic chain with @xmath136 sites.,title="fig:",scaledwidth=35.0%] when @xmath26 the second term vanishes and we recover the result of @xcite @xmath137 for massless klein - gordon theory the above result in one dimension gives, as a consequence the well known result for the rnyi entanglement entropy @xcite, @xmath138 where the dots are the subleading terms. our numerical analyses indicate that for short - range quantum harmonic oscillators the matrix @xmath127 is symmetric positive definite up to just @xmath139 is not exactly equal to @xmath140, however, by increasing the lattice size it approaches the value @xmath140. we conjecture that @xmath141 is exact in the thermodynamic limit.]. the numerical results show that for the values @xmath142 the equation ([renyi mi potts]) is a very good approximation, as we can see for example in fig. the coefficient @xmath10 of the logarithmic term in ([renyi mi potts]) is obtained from the fitting of the model with @xmath136 sites is shown in fig. 11 and in the range @xmath143 surprisingly it follows the simple formula : @xmath144 this is the red line in fig. 11. at @xmath85 we expect zero mutual information for our system, this means that based on the symmetry @xmath145 the coefficient for @xmath83 should be @xmath146. finally one can conclude that for integer values of @xmath147 the coefficient of the logarithm is @xmath148 [htb] [fig11] in the equation ([renyi mi potts]). the lattice size @xmath136 and the coefficients @xmath10 were estimated by using the subsystem sizes @xmath75 $]. the red line is given by eq. ([c for ho]).,title="fig:",scaledwidth=35.0%] the hamiltonian of the xxz chain is defined as @xmath149 where @xmath150, @xmath151 and @xmath152 are spin-@xmath153 pauli matrices and @xmath73 is an anisotropy. the model is critical and conformal invariant for @xmath154 with a constant central charge @xmath66, giving us a good example to test the universality of our results with respect to the change of the anisotropy. the long - distance critical fluctuations are ruled by a cft with central charge @xmath66 described by a compactified boson whose action is given by @xmath155 where the compactification radius depends upon the values of @xmath73, namely : @xmath156 as it is shown in fig. 12, in the @xmath152 basis, the generalized mutual information @xmath42 shows the logarithmic behavior given in ([renyi mi potts]) only for @xmath157. this can be simply understood based on what we observed for the chain of harmonic oscillators. one can look to the klein - gordon field theory as a non - compactified version of the action ([compactified boson]). since we showed that in that case the generalized mutual information is not defined beyond @xmath83 we expect the same behavior also in the compactified version. note that in our numerical calculations one can actually derive spurious big numbers for the generalized mutual information even for @xmath158, but we expect all of them go to infinity in the thermodynamic limit. this behavior seems to be independent of the anisotropy parameter @xmath73. the coefficient of the logarithm in ([renyi mi potts]) for @xmath157 is again given by ([coefficient]), as we can see in fig. 13, with a function @xmath80 which fits to the results of the harmonic chain perfectly. we also considered the results in the case where the ground state wavefunction is expressed in the @xmath150 basis and, except around @xmath15, the equation ([renyi mi potts]) is not a good approximation. [htb] [fig12] of the periodic xxz quantum chain with anisotropy @xmath159, as a function of @xmath160. the ground - state wavefunction is in the basis where the @xmath161 matrices are diagonal (@xmath152 basis). the results are for lattice sizes @xmath47 and @xmath162 and give an idea of the finite - size corrections.,title="fig:",scaledwidth=35.0%] the second @xmath4-symmetric model we considered is the spin-1 fateev - zamolodchikov quantum chain whose hamiltonian is given by @xcite @xmath163 where @xmath164 are spin-1 @xmath165 matrices, @xmath166 and @xmath167. the model is antiferromagnetic for @xmath168 and ferromagnetic for @xmath169. it has a line of critical points (@xmath170) with a quite distinct behavior in the antiferromagnetic (@xmath168) and ferromagnetic (@xmath171) cases. the antiferromagnetic version of the model is governed by a cft with central charge @xmath172 @xcite while the ferromagnetic one is ruled by a @xmath66 cft @xcite. we calculated @xmath42 in both critical regimes where @xmath66 and @xmath172, and for different values of the anisotropy. we found a very similar pattern as that of the xxz quantum chain, as can be seen in fig. 13 the equation ([renyi mi potts]) is valid for values of @xmath157 and the coefficient of the logarithm follows ([coefficient]) with a function @xmath80 which is quite similar to the one we found for the quantum harmonic oscillators and the xxz chain. this shows an interesting universal pattern for critical chains with continuous @xmath4 symmetry. [htb] [fig13] of the coefficient of the logarithm in equation ([renyi mi potts]) with the central charge @xmath11 for the xxz and for the spin-1 fateev - zamolodchikov quantum chains (f - z). the xxz (fateev - zamolodchikov) ground - state wavefunction are in the @xmath152 (@xmath173) basis. the results for the xxz are for the anisotropies @xmath174 and in the case of the fateev - zamolodchikov model their are for the couplings @xmath175. the lattice sizes of the models are shown and the coefficients @xmath10 were estimated by using the subsystem sizes @xmath176.,title="fig:",scaledwidth=35.0%] Conclusions in this paper we calculated the generalized mutual information @xmath42, as defined in ([renyi new]), for quantum chains describing the dynamics of quantum systems with continuous or discrete degrees of freedom. most of our analysis was purely numerical due to the absence, at the moment, of suitable analytical methods to treat this problem. we considered several integrable quantum spin chains. these quantum chains either have a @xmath1 symmetry (like the @xmath16-state potts model with @xmath17 and 4, the ashkin - teller model, and the @xmath1-parafermionic model with @xmath45 and 8) or a @xmath4 symmetry (xxz quantum chain and the spin-1 fateev - zamolodchikov model). we also considered the discrete version of the klein - gordon field theory given by a set of coupled harmonic oscillators. in this case we have a continuum hilbert space. we observed that by expressing the ground - state wavefunctions in general basis the obtained results are distinct. however, similarly as happens for the quantity @xmath34 given in ([renyi2]) (see @xcite), our results on some special basis reveal some general features. these basis are the ones where the @xmath50 or @xmath61 operators are diagonal, for the models with @xmath1 symmetry or the ones where @xmath152 or @xmath173 are diagonal for the models with @xmath4 symmetry. in a continuum field theory description of these quantum chains these basis are expected to be associated to the boundaries that do not destroy the conformal invariance of the bulk underlying euclidean conformal field theory, and for this reason we call them conformal basis @xcite. our results indicate that in these special basis the mutual information @xmath0 has the same kind of leading behavior with the subsystem size @xmath7 as we have in the rnyi entanglement entropy, namely @xmath177, with a function @xmath178, with @xmath179. differently from the rnyi entanglement entropy where the equivalent function @xmath80 is universal (for any model and any basis) in the case of @xmath0 our results indicate that the function @xmath80 depends on the special basis chosen to express the ground - state eigenfunction of the particular model. for the set of @xmath1-symmetric models we considered the function @xmath80, for @xmath180, although different for the @xmath50 and @xmath61 basis are similar as the ones of the @xmath16-state potts chain (@xmath2) and the parafermionic @xmath1 quantum chains (@xmath3). in the case of the ashkin - teller model our results indicate that @xmath80, for @xmath158, also depends on the anisotropy @xmath73 of the model. on the other hand the models with continuum symmetry showed a similar behavior only for @xmath157. for @xmath158 we have strong evidences that most probably the generalized mutual information is not defined. it is quite interesting that in these cases one can understand most of the results by just studying simple short - range coupled harmonic oscillators. in order to conclude we should mention that an analytical approach for the shannon entropy or the shannon mutual information (@xmath181 or @xmath182 in ([renyi2]) and ([renyi new])) is a theoretical challenge. the analytical methods to treat this kind of problem normally use some sort of analytical continuation, in the parameter @xmath5, like the usual replica trick. the results we present showing the continuity of @xmath0 around @xmath15, differently from what happens with @xmath34, indicate that @xmath0 is probably more appropriate for an analytical treatment. _ acknowledgments _ this work was supported in part by fapesp and cnpq (brazilian agencies). we thank j. a. hoyos, r. pereira and v. pasquier for useful discussions. Appendix: the relative entropy and the rnyi divergence in this appendix we review the definitions of the relative entropy and its generalization : the rnyi divergence. the relative entropy is defined as the expectation of the difference between the logarithm of the two distribution of probabilities @xmath183 and @xmath184, from the point of view of the distribution @xmath183, i. e., @xmath185 it can be considered as a measure of the difference between the two distributions @xmath183 and @xmath184. although it is not a symmetric quantity it helps us to define the mutual information of the subsets @xmath186 ans @xmath187 of the system as follows @xmath188 in words, the mutual information between two parts of a system is just the relative entropy between the distribution probability for the whole system and the product of the probability distributions of the different parts. it tells how much the different parts are correlated. the natural generalization of the relative entropy is the rnyi divergence and can be defined (see @xcite for example), as @xmath189 it has the following properties : for @xmath190 we have @xmath191 and if @xmath192 then we have @xmath193. the especial case @xmath26 gives the usual relative entropy. we also define the @xmath85 case by : @xmath194 it is worth mentioning that using the above definition @xmath195 is not zero except when for all @xmath196 s for which @xmath197 also @xmath198 holds. comparing ([renyi divergence]) with ([mutual information from relative entropy]) and ([relative entropy]) the natural definition of the generalized mutual information is @xmath199 the above definition is different from @xmath200, as given by ([renyi2]), and has been frequently used in different areas of information science. 99 l. amico, r. fazio, a. osterloh, and v. vedral, rev. phys. * 80 *, 517 (2008) ; k. modi, a. brodutch, h. cable, t. paterek, and v. vedral, rev. * 84 *, 1655 (2012) ; u. schollwck, rev. phys. * 77 *, 259 (2005) p. calabrese, j. cardy, j. phys. a 42:504005 (2009) j. cardy, phys. lett. * 106 *, 150404 (2011). d. a. abanin and e. demler, phys.. lett. * 109 *, 020504 (2012). a. j. daley, h. pichler, j. schachenmayer, and p. zoller, phys. rev. lett. * 109*,020505 (2012). f. c. alcaraz, m. a. rajabpour, phys. * 111 *, 017201 (2013) j - m stphan, j. stat. (2014) p05010 j - m stphan, phys. b * 90 *, 045424 (2014) f. c. alcaraz, m. a. rajabpour, phys. b * 90 *, 075132 (2014) m. m. wolf, f. verstraete, m. b. hastings, j. i. cirac, phys. lett. * 100 *, 070502 (2008). j - m stphan, s. furukawa, g. misguich, and v. pasquier, phys. rev. b, * 80 *, 184421 (2009). j - m stphan, g. misguich, and v. pasquier, phys. rev. b, * 82 *, 125455 (2010) ; m. oshikawa [arxiv:1007.3739] j - m stphan, g. misguich, and v. pasquier, phys. rev. b * 84 *, 195128 (2011). j. um, h. park and h. hinrichsen, j. stat (2012) p10026 h. w. lau and p. grassberger, phys. e * 87 *, 022128 (2013). d. j. luitz, f. alet, n. laflorencie, phys. rev. lett. * 112 *, 057203 (2014) d. j. luitz, f. alet, n. laflorencie, phys. b * 89 *, 165106 (2014) d. j. luitz, n. laflorencie, f. alet, j. stat (2014) p08007 d. j. luitz, x. plat, n. laflorencie, f. alet, phys. b * 90 *, 125105 (2014) j. wilms, m. troyer, f. verstraete, j. stat. (2011) p10011 and j. wilms, j. vidal, f. verstraete, s. dusuel, j. stat. (2012) p01023 j. iaconis, s. inglis, a. b. kallin, r. g. melko, phys. rev. b * 87 *, 195134 (2013) j - m stphan, s. inglis, p. fendley, r. g. melko, phys. rev. lett. * 112 *, 127204 (2014) a. rahmani, g - w chern, phys. b * 88 *, 054426 (2013) o. cohen, v. rittenberg, t. sadhu, [arxiv:1409.5520] h. bernigau, m. j. kastoryano and j. eisert, j. stat. (2015) p02008 v. eisler, z. zimboras, phys. rev. a * 89 *, 032321 (2014) jos. c. principe, _ information theoretic learning _, springer, berlin (2010) f. y. wu, rev. phys, * 54 *, 235 (1982). v. a. fateev and a. b. zamolodchikov, phys. lett.a * 92 *, 37 (1982) f.c. alcaraz and a. lima santos, nucl. b, * 275 *, 436 (1986) f. c. alcaraz, j. phys. a, * 20 *, 2511 (1987) m. cramer, j. eisert, m. b. plenio, j. dreissig, phys. rev. a * 73 *, 012309 (2006) h. casini and m. huerta, j. phys. a, * 42 *, 504007 (2009) a. b. zamolodchikov and v. a. fateev, yad. fiz. * 32 *, 581 (1980) [sov.. phys. * 32 *, 298 (1980)].

we study the generalized mutual information @xmath0 of the ground state of different critical quantum chains. the generalized mutual information definition that we use is based on the well established concept of the rnyi divergence. we calculate this quantity numerically for several distinct quantum chains having either discrete @xmath1 symmetries (q - state potts model with @xmath2 and @xmath1 parafermionic models with @xmath3 and also ashkin - teller model with different anisotropies) or the @xmath4 continuous symmetries (klein - gordon field theory, xxz and spin-1 fateev - zamolodchikov quantum chains with different anisotropies). for the spin chains these calculations were done by expressing the ground - state wavefunctions in two special basis. our results indicate some general behavior for particular ranges of values of the parameter @xmath5 that defines @xmath0. for a system, with total size @xmath6 and subsystem sizes @xmath7 and @xmath8, the @xmath0 has a logarithmic leading behavior given by @xmath9 where the coefficient @xmath10 is linearly dependent on the central charge @xmath11 of the underlying conformal field theory (cft) describing the system s critical properties.

1501.02852

Introduction it has been known for a long time that owing to planar property and mutual focussing, colliding plane waves (cpw) result in spacelike singularities [1]. these singularities are somewhat weakened when the waves are endowed with a relative cross polarization prior to the collision. a solution given by chandrasekhar and xanthopoulos (cx) [2], however constitutes an example contrasting this category, namely, it possesses a cauchy horizon (ch) instead of a spacelike singularity. naturally, this solution initiated a literature devoted entirely on the quest of stability of horizons formed hitherto. ch formed in spacetimes of cpw was shown by yurtsever to be unstable against plane - symmetric perturbations [3]. a linear perturbation analysis by cx reveals also an analogues result [4]. any such perturbation applied to a cpw spacetime will turn the ch into an essential singularity. a second factor that proved effective in weakening the strength of a singularity in cpw is the electromagnetic (em) field itself. in other words, the degree of divergence in the curvature scalars of colliding pure gravitational waves turn out to be stronger than the case when em field is coupled to gravity. in particular, collision of pure em waves must have a special significance as far as singularity formation is concerned. such an interesting solution was given by bell and szekeres (bs) which describes the collision of two linearly polarized step em waves [5]. the singularity (in fact a ch) formed in the interaction region of the bs solution was shown to be removable by a coordinate transformation. on the null boundaries, however it possesses esential curvature singularities which can not be removed by any means. since cross polarization and em field both play roles in the nature of resulting singularity it is worthwhile to purse these features together. this invokes a cross polarized version of the bs (cpbs) solution which was found long time ago [6,7]. this metric had the nice feature that the weyl scalars are all regular in the interaction region. cross polarization, however, does not remove the singularities formed on the null boundaries. in this paper we choose cpbs solution as a test ground, instead of bs, with various added test fields to justify the validity of a ch stability conjecture proposed previously by helliwell and konkowski (hk) [8,9]. unlike the tedious perturbation analysis of both cx and yurtsever the conjecture seems to be much economical in reaching a direct conclusion about the stability of a ch. this is our main motivation for considering the problem anew, for the case of untested solutions in cpw. in this paper we look at the spacetimes : a) single plane wave with added colliding test fields and b) colliding plane waves having non - singular interaction regions with test field added, fig.1 illustrates these cases. the terminology of singularities should be follwed from the classification presented by ellis and schmidt [10]. singularities in maximal four dimensional spacetimes can be divided into three basic types : quasiregular (qr), scalar curvature (sc) and non - scalar curvature (nsc). the ch stability conjecture due to hk is defined as follows. for all maximally extended spacetimes with ch, the backreaction due to a field (whose test stress - energy tensor is @xmath0) will affect the horizon in one of the following manners. @xmath1, @xmath2 and any null dust density @xmath3 are finite, and if the stress energy tensor @xmath4 in all parallel propagated orthonormal (ppon) frames is finite, then the ch remains non - singular. b)if @xmath1, @xmath2 and any null dust density @xmath3 are finite, but @xmath4 diverges in some ppon frames, then an nsc singularity will be formed at the ch. c)if @xmath1, @xmath2 and any null dust density @xmath3 diverges, then an sc singularity will be formed at the ch. expressed otherwise, the conjecture suggests to put a test field into the background geometry and study the reaction it will experience. if certain scalars diverge then in an exact back - reaction solution the field will respond with an infinite strength to the geometry (i.e action versus reaction). such an infinite back - reaction will render a ch unstable and convert it into a scalar singularity. + the paper is organized as follows. in section ii, we review the cpbs solution and the correct nature of the singularity structure is presented in appendix a. section iii, deals with geodesics and test em and scalar field analyses. in section iv, we present an exact back reaction calculation for the collision of cross polarized em field coupled with scalar field. the derivation of weyl and maxwell scalars are given in appendix b. the insertion of test null dusts to the background cpbs spacetime and its exact back reaction solution is studied in section v. appendix c is devoted for the properties of this solution. the paper is concluded with a discussion in section vi. The cross-polarized bs (cpbs) metric the metric that describes collision of em waves with the cross polarization was found to be [7] + @xmath5 in this representation of the metric our notations are + @xmath6 in which @xmath7 is a constant measuring the second polarization, @xmath8 are constant of energy and @xmath9 stand for the usual null coordinates. it can be seen easily that for @xmath10 the metric reduces to bs. unlike the bs metric, however, this is conformally non - flat for @xmath11, where the conformal curvature is generated by the cross polarization. as a matter of fact this solution is a minimal extension of the bs metric. a completely different generalization of the bs solution with second polarization was given by cx [11]. their solution, however, employs an ehlers transformation and involves two essential parameters which is therefore different from ours. both solutions form ch in the interaction region. our result drown out in this paper, namely, that the horizon is unstable against added sources can also be shown to apply to the cx metric as well. as it was shown before the interaction region @xmath12 of the above metric is of type - d without scalar curvature singularities. we wish to check now the possible singularities of metric (1). the single component of the weyl scalar suffices to serve our purpose. we find that the real part of the weyl scalar @xmath13 is given by @xmath14\end{aligned}\]] where we have used the abbreviations @xmath15 as @xmath16 we obtain @xmath17 which reduces to the singularity form of the bs spacetime given by @xmath18. we see that the same singularity remains unaffected by the introduction of the cross polarization. a similar calculation for @xmath19 gives the symmetrical singular hypersurface sitting on @xmath20. now in order to explore the true nature of the singularity we concentrate our account on the incoming region ii @xmath21. the metric in this region is expressed in the form @xmath22\]] where @xmath23 we observe that for @xmath24, @xmath25 is a bounded positive definite function which suggests that no additional singularities arise except the one occuring already in the bs case, namely at @xmath26. to justify this we have calculated all riemann components in local and ppon frames (see appendix a). it is found that all riemann tensor components vanish as @xmath27. in the ppon frame, however, they are all finite and according to the classification scheme of ellis and schmidt such a singularity is called a quasiregular (qr) singularity. this is said to be the mildest type among all types of singularities. to check whether the qr is stable or not we consider generic test fields added to such a background geometry and study the effects. this we will do in the follwing sections. Geodesics behaviour, test em and test scalar fields we are interested in the stability of qr singularities that are developed at @xmath28 in region ii and @xmath29 in region iii. to investigate their stability we will express geodesics and behaviour of test em and scalar fields by calculating stress - energy tensor in local and ppon frames. + our discussion on geodesics will be restricted in region ii only. we shall consider the geodesics that originate at the wave front and move toward the quasiregular singularity. solution of geodesics equations in region ii can be obtained by geodesics lagrangian method and using @xmath30 as a parameter. the results are @xmath31}{a}\tan(au) + \frac{3p_{x_{0}}\left[5q^{2}+2 - 2\sqrt{1+q^{2}}\right]u}{4 } \nonumber \\ & & - \frac{p_{x_{0}}\left[\sqrt{1+q^{2}}-1\right]^{2}}{8a } \sin(2au) - \frac{2p_{y_{0}}q}{a\cos(au) } \nonumber \\ y - y_{0}&=&-\frac{2qp_{x_{0}}}{a\cos(au)}-\frac{2p_{y_{0}}\tan(au)}{a } \nonumber \\ v - v_{0}&=&\frac{\tan(au)}{a}\left[p^{2}_{x_0}(1 + 2q^2) + p^{2}_{y_{0}}\right] + u\left[\frac{\epsilon}{4}(1 + 3\sqrt{1+q^{2}})\right. \nonumber \\ & & \left.-\frac{3p^2_{x_0}}{8}(5q^{2}+2 - 2\sqrt{1+q^{2}})\right] + \frac{2p_{x_{0}}p_{y_{0}}q}{a\cos(au) } \nonumber \\ & & + \frac { (\sqrt{1+q^{2}}-1) \sin(2au)}{8a}\left[\frac{p^{2}_{x_{0}}}{2 } ( \sqrt{1+q^{2}}-1)-\epsilon\right]\end{aligned}\]] where @xmath32 for null and @xmath33 for time like geodesics and @xmath34 and @xmath35 are constants of integration. it can be checked easily that for @xmath10 our geodesics agree with those of the region ii of the bs metric [8]. it is clear to see that if either @xmath36 or @xmath35 is nonzero then @xmath37 becomes positive for @xmath38, and particles can pass from region ii to the region iv. geodesics that remain in region ii are @xmath39 where @xmath40. the effect of cross polarization is that more geodesics remains in region ii relative to the parallel polarization case. on physical grounds this result could be anticipated because cross polarization behaves like rotation which creates a pushing out effect in the non - inertial frames. to test the stability of quasiregular singularity, let us consider a test em field whose vector potential is choosen appropriately as in [9] to be @xmath41 with arbitrary functions @xmath42 and @xmath43. the only nonzero energy - momentum for this test em field is @xmath44\]] in which a prime denotes derivative with respect to @xmath37. both of scalars @xmath45 and @xmath2 vanish, predicting that qr singularities are not transformed into a scs. in the ppon frame. @xmath46 we find that @xmath47 are given in terms of @xmath48 by @xmath49 for@xmath50 and @xmath51, otherwise. the divergence of this quantity predicts the occurence of nscs and therefore qr singularity must be unstable. + the stability conjecture therefore correctly finds that these qr singularities are unstable. however, the same stability conjecture does not find correctly the nature of the singularity. as we have discussed in section ii, the interior of the interaction region has no scs. the only scs is on the null boundaries. clarke and hayward have analysed these singular points for a collinear bs spacetime and found that the singularity nature of surfaces @xmath52 and @xmath53 are qr. this observation can also be used in the cross polarized version of bs spacetime, because the order of diverging terms in @xmath54 and @xmath13 are the same. the qr singularity structure formed in the incoming region of bs problem remains unchanged in the case of cross polarized version of the same problem. let us now consider the stability of these qr singularities by imposing a test scalar field in region ii which is the one of the incoming region bounded by the qr singularity. the massless scalar field equation is given by @xmath55 where we consider @xmath56 independent scalar waves so that a particular solution to this equation is obtained as in the ref () @xmath57 where @xmath58 and @xmath59 are arbitrary functions. the stress energy tensor is given by @xmath60 the corresponding non - zero stress - energy tensors for the test scalar wave is obtained by taking @xmath61 as, @xmath62f(v)f'(v) } { 8\pi f^{2 } } \nonumber \\ t_{xy}&=&t_{yx}=\frac{aq\sin(au) \tan(au) f(v)f'(v)}{8 \pi f^{2}}\end{aligned}\]] it is observed that each component diverges as the qr singularity @xmath63 is approached. + next we consider the stress energy tensor in a ppon frame. such frame vectors are given in equation (11). the stress - energy tensor is @xmath64 the nonzero components are ; @xmath65 \nonumber \\ t_{01}&=&t_{10}=\left(\frac{\sec^{2}(au)}{16 \pi}\right)\left[\frac{a^{2 } \tan^{2}(au)f^2}{f^{2}}-f'^{2}(v)\right] \nonumber \\ t_{22}&=&t_{33}=\left(\frac{a\sec^{2}(au)\tan(au)f(v)f'(v)}{8\pi f^{3}}\right)\left[f^{2 } + 2q^{2}\sin^{2}(au)\right] \nonumber \\ t_{32}&=&t_{23}=\frac{aq\sec^{3}(au)\sin^{2}(au)f'(v)f(v)\sqrt{f^{2 } + q^{2}\sin^{2}(au)}}{4\pi f^{3}}\end{aligned}\]] these components also diverge as the singularity @xmath66 is approached. by the conjecture, this indicates that the qr singularity will be transformed into a curvature singularity. finally we calculate the scalar @xmath67. @xmath68 \right\}\end{aligned}\]] which also diverges as @xmath69. from these analyses we conclude that the curvature singularity formed will be an scs. + hence, the conjecture predicts that the qr singularities of cross polarized version of bs spacetime are unstable. it is predicted that the qr singularities will be converted to scalar curvature singularities if generic waves are added. the similar results have also been obtained by hk for the bs spacetime. hk was unable to compare the validity of the conjecture by an exact back - reaction solution. in the next section we present an explicit example that represents cross - polarized em field coupled with scalar field. Testing the conjecture for a class of einstein-maxwell-scalar (ems) solutions. in the former sections, we applied hk stability conjecture to test the stability of qr singularities in the incoming region of cpbs spacetime. according to the conjecture these mild singularities are unstable. in order to see the validity of the conjecture we introduce this new solution. + let the metric be ; @xmath70\]] the new solution is obtained from the electrovacuum solution. the ems solution is generated in the following manner. the lagrangian density of the system is defined by @xmath71\end{aligned}\]] which correctly generate all ems field equations from a variational principle. here @xmath72 is the scalar field and we define the em potential one - form (with coupling constant @xmath73) by @xmath74 where @xmath75 and @xmath76 are the components in the killing directions. variation with respect to @xmath77 and @xmath72 yields the following ems equations. @xmath78 \\ 2b_{uv}&=&-v_{v}b_{u}-v_{u}b_{v}-tanhw\left(w_{v}b_{u}+w_{u}b_{v}\right) \nonumber \\ & & -e^{v}\left[2b_{uv}tanhw + w_{u}b_{v}+w_{v}b_{u}\right]\end{aligned}\]] where @xmath79 and @xmath80 are the newmann - penrose spinors for em plane waves given as follows @xmath81 \nonumber \\ \phi_{0}&= & \frac{e^{u/2}}{\sqrt{2}}\left[e^{-v/2}\left(isinh\frac{w}{2 } + cosh\frac{w}{2}\right)a_{v}\right. \nonumber \\ & & \left.+e^{v/2}\left(icosh\frac{w}{2}+sinh\frac{w}{2}\right)b_{v}\right] \end{aligned}\]] the remaining two equations which do not follow from the variations, namely @xmath82 are automatically satisfied by virtue of integrability equations. + the metric function @xmath83 can be shifted now in accordance with + @xmath84 where @xmath85 and @xmath86 satisfy the electrovacuum em equations. integrability condition for the equation (31) imposes the constraint, @xmath87=0\]] from which we can generate a large class of ems solution. as an example, for any @xmath88 satisfying the massless scalar field equation the corresponding @xmath89 is obtained from @xmath90 the only effect of coupling a scalar field to the cross polarized em wave is to alter the metric into the form, @xmath91 here @xmath92 and @xmath93 represents the solution of electrovacuum equations and @xmath89 is the function that derives from the presence of the scalar field. + it can be easily seen that for @xmath10 our solution represents pure em bs solution coupled with scalar field. it constitutes therefore an exact back reaction solution to the test scalar field solution in the bs spacetime considered by hk (). it is clear to see that the weyl scalars are nonzero and scs is forming on the surface @xmath94. this is in aggrement with the requirement of stability conjecture introduced by hk. for @xmath24 the obtained solution forms the exact back reaction solution of the test scalar field solution in the cpbs spacetime. in appendix b, we present the weyl and maxwell scalars explicitly. + from the explicit solutions we note that, the coulomb component @xmath95 remains regular but @xmath13 and @xmath54 are singular when @xmath96 or @xmath97. this indicates that the singularity structure of the new solution is a typical scs. this result is in complete agreement with the stability conjecture. Oppositely moving null dusts in cpbs spacetime * a) * let us assume first null test dusts moving in the cpbs background. for simplicity we consider two different cases the @xmath98 and @xmath99 projections of the spacetime. we have in the first case @xmath100 where we have used the coordinates @xmath101 according to @xmath102 the energy - momentum tensor for two oppositely moving null dusts can be chosen as @xmath103 where @xmath104 and @xmath105 are the finite energy densities of the dusts. the null propagation directions @xmath106 and @xmath107 are @xmath108 with @xmath109 @xmath110 we observe from (1) that @xmath111 which is finite for @xmath112. the components of energy - momentum tensors in ppon frames are proportional to the expression (38). this proportionality makes all the components of energy - momentum tensors finite. in the limit as @xmath113 which reduces our line element to the bs this expression diverges on the horizon @xmath114. trace of the energy - momentum is obviously zero therefore we must extract information from the scalar @xmath115. one obtains, @xmath116 the projection on @xmath99, however is not as promising as the @xmath98 case. consider the reduced metric @xmath117 a similar analysis with the null vectors @xmath118 with @xmath119 @xmath120 yields the scalar @xmath121 from the metric (1) we see that @xmath122 which diverges on the horizon @xmath123. the scalar @xmath124 constructed from the test dusts therefore diverges. the ppon components of the energy momentum tensors are also calculated and it is found that all of the components are proportional to the expression (42). this indicates that the components of energy - momentum tensor diverges as the singularity is approached. when we consider the hk stability conjecture an exact back reaction solution must give a singular solution. we present now an exact back reaction solution of two colliding null shells in the interaction region of the cpbs spacetime. + * b) * the metric given by @xmath125 where @xmath126 with @xmath127 positive constants was considered by wang [12] to represent collision of two null shells (or impulsive dusts). the interaction region is transformable to the de sitter cosmological spacetime. in other words the tail of two crossing null shells is energetically equivalent to the de sitter space. it can be shown also that the choice of the conformal factor @xmath128, with @xmath129 positive constants becomes isomorphic to the anti - de sitter space. + the combined metric of cpbs and colliding shells can be represented by @xmath130 this amounts to the substitutions @xmath131 where @xmath132 correspond to the metric functions of the cpbs solution. under this substitutions the scale invariant weyl scalars remain invariant (or at most multiplied by a conformal factor) because @xmath133 is the combination that arise in those scalars. the scalar curvature, however, which was zero in the case of cpbs now arises as nonzero and becomes divergent as we approach the horizon. appendix c gives the scalar quantities @xmath134 and @xmath135. thus the exact back reaction solution is a singular one as predicted by the conjecture. it is further seen that choosing @xmath136, which removes one of the shells leaves us with a single shell propagating in the @xmath37- direction. from the scalars we see that even a single shell gives rise to a divergent back reaction by the spacetime. the horizon, in effect, is unstable and transforms into a singularity in the presence of two colliding, or even a single propagating null shell. let us note as an alternative interpretation that the metric (43) may be considered as a colliding em waves in a de sitter background. collision of em waves creates an unstable horizon in the de sitter space which is otherwise regular for @xmath137 and @xmath138. Discussion in this paper we have analysed the stability of qr singularities in the cpbs spacetime. three types of test fields are used to probe the stability. first we have applied test em field to the background cpbs spacetime. from the analyses we observed that the qr singularity in the incoming region becomes unstable according to the conjecture, and it is transformed into nsc singularity. this is the prediction of the conjecture. however, the exact back - reaction solution shoes that beside the true singularities on the null boundaries @xmath139 and @xmath140. there are quasiregular singularities in the incoming regions. the interior of interaction region is singularity free and the hypersurface @xmath141 is a killing - cauchy horizon. as it was pointed out by hk in the case of colliding em step waves, conjecture fails to predict the correct nature of the singularity in the interaction region. we have also discovered the same behaviour for the cross polarized version of colliding em step waves. the addition of cross polarization does not alter the existing property. + secondly we have applied test scalar field to the background cpbs spacetime. the effect of scalar field on the qr singularity is stronger than the effect of em test field case. we have obtained that the qr singularity is unstable and transformes into a scs. to check the validity of the conjecture, we have constructed a new solution constituting an exact back reaction solution to the test scalar field in the cpbs spacetime. the solution represents the collision of cross polarized em field coupled to a scalar field. an examination of weyl and maxwell scalars shows that @xmath142 and @xmath143 diverge as the singularity is approached and unlike the test em field case the conjecture predicts the nature of singularity formed correctly. + finally, we have introduced test null dusts into the interaction region of cpbs spacetime. the conjecture predicts that the ch are unstable and transforms into scs. this result is compared with the exact back - reaction solution and observed that the conjecture works. Appendix a: + riemann components for region ii to determine the type of singularity in the incoming region of cpbs spacetime, the riemann tensor in local and in ppon frame must be evaluated. non - zero riemann tensors in local coordinates are found as follows. @xmath144 \nonumber \\ -r_{uyuy}&= & e^{-u - v } \left [\phi_{22 } coshw + (i m \psi_4) sinhw - re \psi_4 \right] \nonumber \\ r_{uxuy}&= & e^{-u}\left [\phi_{22 } sinhw + (im\psi_4)coshw\right]\end{aligned}\]] where @xmath145 \nonumber \\ i m \psi_4&=&-\frac{i}{2}\left(w_{uu } -u_uw_u + m_uw_u -v^2_u coshw sinhw \right) \nonumber \\ \phi_{22}&=&\frac{1}{4}\left(2u_{uu}-u^2_u - w^2_u - v^2_u cosh^2w \right)\end{aligned}\]] note that in region ii the weyl scalar @xmath146, therefore only @xmath143 exists. it is clear that @xmath147 in the limit @xmath27, so that all of the components vanish @xmath148 to find the riemann tensor in a ppon frame, we define the following ppon frame vectors ; @xmath149 in this frame the non - zero components of the riemann tensors are : @xmath150 in the limit of @xmath27, we have the following results @xmath151 which are all finite. this indicates that the apparent singularity in region ii (one of the incoming region) is a quasiregular singularity. Appendix b: + properties of the ems solution in order to calculate the weyl and maxwell scalars we make use of the cx line element @xmath152 where @xmath153 @xmath154 is given in equation (4) and we have chosen @xmath155, such that the new coordinates @xmath156 are related to @xmath157 by @xmath158 the weyl and maxwell scalars are found as @xmath159 \\ & & \nonumber \\ \psi_0&=&-e^{\gamma - i\lambda } \left [3r + \frac{1}{4f\sigma \sin \theta \sin \psi } \left(\sigma \sin(\psi -\theta)-\sigma_{\theta } \sin \psi \sin\theta \right. \nonumber \\ & & \nonumber \\ & & \left.\left.+ i\sin \alpha \sin^2 \theta \sin \psi \right) \left(\gamma_{\psi}+\gamma_{\theta}\right)\right] \\ & & \nonumber \\ 2\phi_{00}&= & e^{\gamma}\left[\frac{\cos \alpha}{\sigma^2}-\frac { \sin(\psi+\theta)(\gamma_{\theta}+\gamma_{\psi})}{2f\sin \psi \sin \theta } \right] \\ & & \nonumber \\ 2\phi_{22}&= & e^{\gamma}\left[\frac{\cos \alpha}{\sigma^2}-\frac{\sin(\theta- \psi)(\gamma_{\theta}-\gamma_{\psi})}{2f\sin \psi \sin \theta } \right] \\ & & \nonumber \\ -2\phi_{02}&=&e^{\gamma + i\lambda}\frac{\cos \alpha}{\sigma^2}\end{aligned}\]] where @xmath160 \\ & & \\ e^{i\lambda}&=&\frac{\sin \theta + \sigma \sin \psi + i \sin \psi \sin \theta \cos \psi } { \sin \theta + \sigma \sin \psi - i \sin \psi \sin \theta \cos \psi}\end{aligned}\]] Appendix: c + the weyl and maxwell scalars the non - zero weyl and maxwell scalars for the collision of null shells in the background of cpbs spacetime are found as follows. @xmath161 \theta(u) \theta(v) \\ & & \nonumber \\ 4\phi e^{-m}\lambda&= & \left [(a\beta + \alpha b)\tan(au+bv)+(a \beta -\alpha b)\tan(au - bv)\right. \nonumber \\ & & \nonumber \\ & & \left.+\frac{4\alpha \beta}{\phi } \right] \theta(u) \theta(v) \\ & & \nonumber \\ \phi_{22}&=&(\phi_{22})_{cpbs } + \left(\frac{\alpha e^m } { \phi}\right)\left [ \delta(u) \right.\nonumber \\ & & \nonumber \\ & & \left. - \theta(u)\left(a \pi + \frac{u}{(1-u^2)(1-v^2)}\right) \right]\\ & & \nonumber \\ \phi_{00}&=&(\phi_{00})_{cpbs } + \left(\frac{\beta e^m } { \phi}\right)\left [ \delta(v) \right. \nonumber \\ & & \nonumber \\ & & \left.+ \theta(v)\left(b \pi - \frac{v}{(1-u^2)(1-v^2)}\right)\right] \\ & & \nonumber \\ \phi_{02}&= & (\phi_{02})_{cpbs}+\left (\frac{e^m}{4fy\phi}\right) \left[\frac{1}{f}\left(\alpha q \theta(u) + \beta p \theta(v)\right) \right. \nonumber \\ & & \nonumber \\ & & \left. + iq\left(\alpha l \theta(u) + \beta k \theta(v)\right)\right]\end{aligned}\]] where @xmath162 \\ & & \\ p&=&a\left[2q^2\sin(au+bv)\cos(au - bv)+f^2\left(\tan(au - bv)-\tan(au+bv)\right) \right. \\ & & \\ & & \left. + 2f\cos(au - bv)\sin(au - bv)\left(\sqrt{1+q^2}-1\right)\right] \\ & & \\ y&=&\left(1+\frac{q^2}{f^2}\tan(au+bv)\sin(au+bv)\cos(au - bv)\right)^{1/2}\\ & & \\ k&=&\frac{a}{\sqrt{\cos(au+bv)\cos(au - bv)}}\left[\frac{\cos(au - bv) } { \cos(au+bv)}+\sin2au \right. \\ & & \\ & & \left. -\frac{2\left(\sqrt{1+q^2}-1\right)\sin(au+bv)\cos(au - bv) \tan(au - bv)}{f}\right] \\ & & \\ l&=&\frac{b}{\sqrt{\cos(au+bv)\cos(au - bv)}}\left[\frac{\cos(au - bv) } { \cos(au+bv)}+\sin2bv \right. \\ & & \\ & & \left. + \frac{2\left(\sqrt{1+q^2}-1\right)\sin(au+bv)\cos(au - bv) \tan(au - bv)}{f}\right] \\ & & \\ \pi&=&\frac{\left(\sqrt{1+q^2}-1\right)\sin(2au-2bv)}{\sqrt{1+q^2}+1 + \left(\sqrt{1+q^2}-1\right)\sin^2(au - bv)}\end{aligned}\]] 99 j.b. griffiths, colliding plane waves in general relativity, oxford, clarendon press (1991). s. chandrasekhar and b.c. xanthopoulos, proc. london * a 408 *, 175 (1986). u. yurtsever, phys. rev. * d 36 *, 1662 (1987). s. chandrasekhar and b.c. xanthopoulos, proc. london * a 415 *, 329 (1988). p. bell and p. szekeres, gen. * 5 *, 275 (1974). m. halilsoy, phys. rev. * d,37 *, 2121, (1988). m. halilsoy, j. math. phys. * 31 *, 2694, (1990). d. a. konkowski. and t. m. helliwell, phys. rev. * d 43 *, 609 (1991). d. a. konkowski. and t. m. helliwell, class. quantum grav. * 16 *, 2709 (1999). g. f. r. ellis and b. g. schmidt, gen. * 8 *, 915 (1977). s. chandrasekhar and b.c. xanthopoulos, proc. london * a 410 *, 311 (1987). a. z. wang, j. math. phys. * 33 *, 1065, (1992). fig.1(a) : single plane waves with added colliding test fields indicated by arrows. ch exists on the surface @xmath163. + (b) : colliding plane wave spacetime with ch s in the incoming regions at @xmath164 and @xmath165. test fields are added to test the stability of ch existing in region iv.

the quasiregular singularities (horizons) that form in the collision of cross polarized electromagnetic waves are, as in the linear polarized case, unstable. the validity of the helliwell - konkowski stability conjecture is tested for a number of exact backreaction cases. in the test electromagnetic case the conjecture fails to predict the correct nature of the singularity while in the scalar field and in the null dust cases the aggrement is justified.

gr-qc0006038

Introduction the construction of background field formalism for n=2 super - yang - mills theory (sym) in projective hyperspace (@xmath0) @xcite is an open problem. such a formalism is desirable for any (non-)supersymmetric theory as it simplifies (loop) calculations and even intermediate steps respect gauge covariance. a major obstacle in solving this problem for the n=2 case seems to be the lack of knowledge relating the gauge connections to the tropical hyperfield @xmath1, which describes the sym multiplet for all practical purposes @xcite. we note that the closely related @xcite n=2 harmonic superspace (@xmath2) @xcite does nt encounter this issue as the hyperfield, @xmath3 describing the sym multiplet is itself a connection, @xmath4. in fact, background field formalism in harmonic superspace has quite a straightforward construction @xcite. although the construction has some subtleties, it has been refined in a series of papers along with relevant calculations @xcite. in this paper, we solve the problem of constructing the background field formalism in projective superspace without the need for knowing the connections explicitly in terms of @xmath1. this is possible by choosing the background fields to be in a ` real'representation (@xmath5) and the quantum fields to be in the ` analytic'representation (@xmath6). this is reminiscent of the quantum - chiral but background - real representation used in n=1 superspace @xcite. what this does is make the effective action independent of @xmath7 and dependent on background fields (like @xmath8) with ` dimension'greater than @xmath9 (since the lowest one is a spinor). non - existence of @xmath9-dimension background fields (like @xmath7) is a crucial requirement for the non - renormalization theorems to hold as discussed in @xcite. this directly leads to a proof of finiteness beyond 1-loop. (a different approach for proof of finiteness has been discussed in @xcite.) the coupling of quantum fields to background fields comes through the former s projective constraint alone, which simplifies the vertex structure a lot. the calculations are also simplified at 1-hoop as most @xmath10-integrals turn out to be trivial since the background fields have trivial @xmath10-dependence. this means that the @xmath10-integration effectively vanishes from the effective action and as expected from the supergraph rules, only one @xmath11-integration survives at the end of the calculations. we also work in fermi - feynman gauge so there are no ir issues to worry about while evaluating the super - feynman graphs. another important aspect is the ghost structure of the theory in this background gauge. apart from the expected faddeev - popov (fermionic @xmath12) and nielsen - kallosh (bosonic @xmath13) ghosts, we require two more extra ghosts, namely real bosonic @xmath14 and complex fermionic @xmath15. this is in contrast to n=1 sym but very similar to the harmonic treatment of n=2 theory. heuristically, we can even see that such a field content would give a vanishing @xmath16-function for n=4. moreover, we will see that the loop contributions of @xmath1 and extra ghosts have spurious divergences arising due to multiple @xmath17 s. these are very similar to the ` coinciding harmonic'singularities in the @xmath2 case, which manifest themselves at 1-loop level via the subtleties regarding regularization of similar looking determinants. however, in @xmath0 case, we do not encounter such striking similarities. only the divergences turn out to be similar, leading to a cancellation between the vector hyperfield s contribution and that of the extra ghosts. the finite pieces in the effective action are contributed by these extra ghosts only. Construction this section is mostly built on the ordinary projective superspace construction of sym detailed in @xcite. we review it briefly below for the sake of continuity. we also use the 6d notation to simplify some useful identities involving background covariant derivatives and moreover, the results carry over to n=1 6d sym in a trivial manner with this notation. the projective hyperspace comprises of usual spacetime coordinates (@xmath18), four fermionic ones (@xmath11) and a complex coordinate on cp@xmath19 (@xmath10). the full n=2 superspace requires four more fermionic coordinates (@xmath20) in addition to these projective ones. the super - covariant derivatives corresponding to these extra @xmath20 s define a projective hyperfield (@xmath21) via the constraint @xmath22. the algebra of the covariant derivatives will be given below but we note here that in the ` real'representation (called ` reflective'in @xcite and the one we use extensively in this paper) the @xmath23 s are @xmath10-dependent. their anti - commutation relation at different @xmath10 s is all that we need here : @xmath24 the scalar hypermultiplet is described by an ` arctic'hyperfield (@xmath25) that contains only non - negative powers of @xmath10 and the vector hypermultiplet by a ` tropical'@xmath1, which contains all powers of @xmath10. to construct the relevant actions, the integration over this internal coordinate is defined to be the usual contour integration, with the contour being a circle around the origin (for our purposes in this paper). so, the projective measure simply reads : @xmath26 (with the usual factor of @xmath27 being suppressed). now, we are ready to delve into the details of the background field formalism. the gauge covariant derivatives, @xmath28, describing n=2 sym satisfy the following (anti-) commutation relations (written in 6d notation) : @xmath29=-_{}w_a^{}\,,\\ \{_{a},w_b^{}\}={{\cal d}}_{ab}_{}^{}-\tfrac{{\dot{\iota}}}{2}c_{ab}f_{}^{}\,,\\ [_{},^{}]=f_^{[}_^{]}\,,\\ [_{},_y]=_{}\,,\quad [_{},_y]=0\,, \label{dyd}\end{gathered}\]] where the su(2) index @xmath30, @xmath31 and @xmath32 are the field strengths, and @xmath33 are the triplet of auxiliary scalars. the 4d scalar chiral field strength, @xmath34 is related to the spinor field strength via appropriate spinor derivatives. we solve the commutation relation for @xmath35 by writing @xmath36, where @xmath37 is an unconstrained complex hyperfield. we can do a background splitting of @xmath37 (similar to n=1 superspace) such that @xmath38 with @xmath39 being the background covariant derivative. we can now choose ` real'representation for the background derivatives independently such that @xmath40. this simplifies the @xmath10-dependence of the connections : @xmath41 since these connections have simple @xmath10-dependence, the @xmath10-integrals in the effective action can be trivially done. moreover, the quantum part of the full covariant derivatives then can be chosen to be in ` analytic'representation, @xmath42, @xmath43 and @xmath44. the projective (analytic) constraint on hyperfields ` lifts'to @xmath45 so we can now define a background projective hyperfield @xmath46 as @xmath47 such that @xmath48. then, the scalar hypermultiplet s action reads : @xmath49 the vector hyperfield @xmath1 s action looks the same as in the ordinary case ; the difference being that the @xmath1 appearing below is only the quantum piece and is background projective : @xmath50 we know from @xcite that this action should give an expression for @xmath7 and hence the ` analytic'representation for quantum hyperfields is a consistent choice. the background dependence of @xmath1 comes through the projective constraint and the background covariant derivatives only. the following identities will be useful in showing that and deriving other results in the following sections : @xmath51_^4\,,\\ _{1}^4_{2}^4=\left[y_{12}{{\cal d}}_{}+\tfrac{1}{2}y_{12}^2{\check{}}+\tfrac{1}{2}y_{12}^3{\left(}_{,}^{}_{,}+w_{}^ _{,}+2{{\cal d}}_{}{\right)}+y_{12}^4_{2}^4\right]_{2}^4\,,\end{gathered}\]] where @xmath52 is the gauge - covariant dalembertian and @xmath53. as the quantum connections do not appear explicitly in the calculations, we will drop the usage of curly fonts to denote the background fields (as has been done above) and also the subscript ` @xmath54'on @xmath55 from now on. the quantization procedure in the background gauge proceeds similar to the ordinary case. the ordinary derivatives are now background - covariant derivatives so @xmath56 gets replaced by @xmath57 (or @xmath58) everywhere. moreover, we need extra ghosts for the theory to be consistent in this formalism as we elaborate further in the following subsections. the scalar hypermultiplet is background projective but the structure of its action is still the same as in the ordinary case. that means the kinetic operator appearing in the equations of motion is @xmath59, @xmath42, @xmath60 still holds. so the derivation of the propagator performed in @xcite goes through after employing these changes : @xmath61 and @xmath62 : @xmath63 the gauge - fixing for the vector hypermultiplet leading to faddeev - popov (fp) ghosts is still similar to the ordinary case and we just quote the results with suitable modifications : @xmath64v_2\,;\label{sgfinv}\\ { { \cal s}}_{fp}&=-{\text{tr}}\int dx\,d^4\theta\,dy\,\left[\bar{b}\,c+\bar{c}\,b+(y\,b+\bar{b})\frac{v}{2}\left(c+\frac{\bar{c}}{y}\right)+... \right].\label{fpaction}\end{aligned}\]] the propagators for the fp ghosts are similar to the scalar hypermultiplet and will be written down later. we will always work in fermi - feynman gauge (@xmath65) but let us derive the propagator for @xmath1 with arbitrary @xmath66 as this technique will be useful later. we first combine the terms quadratic in @xmath1 from the above equation and the vector hypermultiplet action to get @xmath67_{1}^4 v_2\nonumber\\ = & -\frac{{\text{tr}}}{2g^2}\int dx\,d^4\,dy_1\,dy_2\,v_1\frac{1}{y_{12}^2}\left[1+\frac{1}{}{\left(}-1+\frac{y_1+y_2}{2}(y_{12}){\right)}\right]y_{12}^2\left(\frac{1}{2}{\check{}}+\cdots\right) v_2\nonumber\\ = & -\frac{{\text{tr}}}{2g^2}\int dx\,d^4\,dy_1\,dy_2\,v_1\left[1+\frac{-1+y_1(y_{12})}{}\right]\left(\frac{1}{2}{\check{}}+\cdots\right) v_2\,.\end{aligned}\]] then, we add a generic real source @xmath68 to the quadratic gauge - fixed vector action : @xmath69\frac{1}{2y_{12}^2}v_2-dx\,d^8\,dy_2\,j_2v_2\right\}\nonumber\\ = & -\frac{{\text{tr}}}{g^2}\left\{\int dx\,d^4\,dy_{1,2}\,v_1\left[1+\frac{-1+y_1(y_{12})}{}\right]\frac{_{1}^4}{2y_{12}^2}v_2-dx\,d^4\,dy_2\,{{\cal j}}_2v_2\right\}.\label{svj}\end{aligned}\]] here, @xmath70 is now defined to be (background) projective. now, equation of motion for @xmath1 reads @xmath71\frac{_{2}^4}{y_{12}^2}={{\cal j}}_2\,,\label{eomf}\]] which we can solve to write @xmath1 in terms of @xmath70. this amounts to inverting the kinetic operator for @xmath1 as we will see. assuming the following ansatz for @xmath1 : @xmath72 and demanding it satisfy ([eomf]), we are led to @xmath73 because @xmath74\left[1+\frac{-1+y_1(y_{12})}{}\right]=(y_{02})\,.\]] plugging ([eomf]) and ([fjk]) in the action ([svj]), we get @xmath75 which leads to the required propagator, first derived (for the ordinary case) in @xcite @xmath76 this expression simplifies @xcite for @xmath65 to @xmath77 as does the quadratic part of the vector action @xmath78 in background field gauge, the gauge fixing function leads to additional ghosts apart from the fp ghosts, which contribute to the 1-loop calculations. to see that, consider the effective action @xmath79 defined by the following functional : @xmath80 where @xmath81 is found by the normalization condition @xmath82. it gives @xmath83 so ([expg]) simplifies to @xmath84 we can rewrite the last factor as @xmath85 where @xmath86 are unconstrained hyperfields. proceeding similar to the harmonic case @xcite, we redefine @xmath87 and introduce nielsen - kallosh (nk) ghost @xmath13 to account for the resulting jacobian. this means the 1-loop contribution for n=2 sym coupled to matter simplifies to : @xmath88 for n=4, the scalar hypermultiplet is in adjoint representation and its contribution will cancel the joint fp and nk ghosts contributions. the remaining two terms have spurious divergences due to multiple @xmath17 s but their joint contribution has to be finite, which will turn out to be the case as we develop this section further. to incorporate the effect of @xmath86 fields directly in the path integral, we choose to introduce a real scalar @xmath14 and a complex fermion @xmath15 as follows : @xmath89 where @xmath90 so the background field requires 3 fermionic ghosts @xmath91 and 2 bosonic ghosts @xmath92 and the full quantum action for n=2 sym coupled to matter reads : @xmath93+s_{fp}(v, b, c)+s_{nk}(v, e)+s_{xr}(v, x, r)+s_{}(v,).\]] the fp and nk ghosts are background projective hyperfields. the actions for these ghosts look the same as those in the case of non - background gauge. the action for fp ghosts is given in equation ([fpaction]) and that for nk ghost is similar to the scalar hypermultiplet s action. that means their propagators are straightforward generalizations and read @xmath94 now, we focus on the new ingredient of the background field formalism : the extra ghosts. in the same vein as the vector hypermultiplet, we can simplify the actions of these ghosts. let us just concentrate on the scalar ghost action as the fermionic ghost can be treated similarly : @xmath95x_2\\ = & \,-\frac{{\text{tr}}}{4}dx d^4dy_{1,2}\,x_1\left[{\left(}\frac{y_1}{y_{21}}+\frac{y_2}{y_{12}}{\right)}\frac{1}{y_{12}^2}{\widehat{}}\right]x_2\\ = & \,-\frac{{\text{tr}}}{4}dx d^4dy_{1,2}\,x_1\left[\frac{-1+y_1{\left(}y_{12}{\right)}}{y_{12}^2}{\widehat{}}\right]x_2\,.\end{aligned}\]] the @xmath14 propagator can then be derived in a similar way as the vector propagator with arbitrary @xmath66. lets add a source term to the action for x ghost : @xmath96x_2+{\text{tr}}dx\,d^8\,dy_2\,j_2x_2\nonumber\\ = & -\frac{{\text{tr}}}{4}dx\,d^4\,dy_{1,2}\,x_1{\left(}\frac{-1+y_1(y_{12})}{y_{12}^2}{\right)}{\widehat{}}x_2+{\text{tr}}dx\,d^4\,dy_2\,{{\cal j}}_2x_2\,.\end{aligned}\]] the equation of motion for @xmath14 now reads @xmath97 adopting an ansatz for @xmath14 (similar to what was done for @xmath1 before), @xmath98\frac{1}{\frac{1}{2}{\widehat{}}^2}2{{\cal j}}_0\,,\]] we find that @xmath99 and @xmath100 satisfy ([xeom]). collecting all the results, the action reduces to @xmath101 which leads to the required propagator @xmath102 the propagator for the fermionic @xmath15 ghost has a similar expression. Calculations given this new construction of the background field formalism for sym, we can now employ it to calculate contributions to the effective action coming from different hypermultiplets. the general rules for constructing diagrams in the background field formalism are similar to the ordinary case discussed in @xcite. however, as expected in this formalism, the quantum propagators form the internal lines of the loops and the external lines correspond to the background fields. the @xmath58 and @xmath57 operators in the propagators need to be expanded around @xmath103 (the connection - independent part of @xmath56), which will generate the vertices with the vector connection and background fields. for the extra ghosts, we can further simplify the nave rules by noticing that the vertices have @xmath104-factor and the propagator will generate such a factor in the numerator due to the presence of @xmath105. thus, we can remove them from the very start and work with the revised propagator and vertex for the purpose of calculating diagrams. let us now collect all the relevant feynman rules below. @xmath106{\left(}{\widehat{}}-_0{\right)}\end{aligned}\]] [[scalar]] scalar + + + + + + the one - loop contribution from the scalar hypermultiplet to the effective action can not be written in a fully gauge covariant form with a projective measure. thus, the diagrammatic calculation required to get this contribution (which includes the uv - divergent piece too) is not accessible via the formalism constructed here. we note that such an issue appears in the n=1 background formalism too when the scalar multiplets in complex representation are considered. the calculations can not be performed covariantly and explicit gauge fields appear in addition to the connections. [[vector]] vector + + + + + + the contribution to one - loop n - point diagrams from vector hypermultiplet running in the loop would be given by the following : @xmath107 where the numerical subscript on @xmath108 denotes the external momenta dependence. as usual, to kill the extra @xmath109-function, at least four @xmath110 should be available from the vertices and so @xmath111. the first non - vanishing contribution is from the 4-point diagram : @xmath112 too many @xmath17 s lead to spurious @xmath113 singularity, similar to ` coinciding harmonic'singularities in @xmath2. these will cancel when we take into account the @xmath114 ghosts. [[extra - ghosts]] extra ghosts + + + + + + + + + + + + their combined contribution to one - loop n - point diagrams reads : @xmath115\nonumber\\ & { \left(}w^(1)_{,}+... {\right)}... \,_{n}^4^8(_{n1 }) \frac{(y_{nb,1b})}{y_{nb}}\frac{1}{k_n^2}\left[{\left(}-1+y_{na } (y_{na, nb}){\right)}\right]{\left(}w^(n) _{,}+... {\right)}\nonumber\\ \sim&-d^4kd^4_ndy_{1a,...,1b}_{1b}^4^8(_{n1})\frac{{\left(}-1+y_{1a}(y_{1a,2b}){\right)}}{y_{1a}}\frac{1}{k_1 ^ 2}\nonumber\\ & { \left(}w^(1) _{,}+... {\right)}... \,\frac{{\left(}-1+y_{na}(y_{nb,1b}){\right)}}{y_{1b}}\frac{1}{k_n^2}{\left(}w^(n) _{,}+... {\right)}.\label{fxn}\end{aligned}\]] again, the first non - vanishing contribution is from @xmath116 that has the same @xmath117 singularity structure as the vector in ([4pdiv]) leading to a cancellation, in addition to the following finite part : @xmath118 the last line follows because only @xmath10-independent pieces of @xmath119 s can survive the @xmath10-integrals. till here, we have treated @xmath119 s as fields depending on individual external momenta and eq. ([4pfinite]) is the complete 4-point effective action. assuming them to be momentum independent, we can further simplify this expression in case of the u(1) gauge group and perform the integral over loop - momentum to get @xmath120 where we used the reduction to 4d for @xmath121. using this and the fact that @xmath122 is related to @xmath123, we get the same non - holomorphic 4-point contribution (with the full superspace measure @xmath124) to n=4 sym action rather directly when compared to the calculation done in @xcite (for similar calculations in @xmath2 see, for example, @xcite). [[loops]] 2-loops + + + + + + + we can also see that there are no uv divergences at two - loops. the proof is similar to that given in the ordinary case, _ i.e. _, absence of sufficient @xmath125 s. only 3 diagrams shown in fig. [v2hbg] are supposed to contribute at 2-loops. all of them will vanish due to the @xmath126-algebra unless we get at least 4 @xmath110 s from the expansion of the propagators. this, as we have seen before, brings in 4 more @xmath56 s making these 2-loop diagrams convergent. furthermore, we note that the arguments of @xcite apply in our case since there is no background connection @xmath7, there can not be any divergences at 2 or more loops from just power counting. this situation is different than @xmath2 where such ` 0-dimensional'connections are present and arguments similar to the one given above involving number of @xmath35 s have to be used and at higher loops they can be quite involved @xcite. Conclusion we have formulated the background field formalism for n=2, 4d projective superspace. the crucial ingredient was to recognize that different representations for background and quantum pieces of the hypermultiplets are required. choosing real representation for the background fields allowed non - renormalization theorems to be applicable here as the lowest - dimensional fields available were spinors. the usual choice of analytic representation for the quantum fields allowed us to make a simple extension of the existing ` ordinary'super - feynman rules to the background covariant rules. moreover, there are extra ghosts required (apart from fp and nk ghosts) to evaluate the full sym effective action. these extra ghosts also appear in the harmonic case but in projective case, they cancel the spurious ` harmonic'divergences coming from vector hypermultiplet in a straightforward manner and the resultant finite pieces are as expected for n=4. the uv divergent parts come only from the usual (fp and nk) ghosts and scalar hypermultiplet. however, their contribution can not be directly calculated in the formalism developed here for reasons mentioned in section [examples]. we also gave a diagrammatic 2-loops argument for finiteness of n=2 sym coupled with matter. this is easily supplanted by the power counting argument of @xcite in general, which directly leads to a proof for finiteness beyond 1-loop. for n=1 background formalism, there exist improved rules as showcased in @xcite and our hope is that such techniques could be applied to what we have developed in this paper. that would lead to a further simplification of the higher - loop calculations while also allowing explicit inclusion of the scalar hypermultiplet s 1-loop contribution. Acknowledgements this research work is supported in part by nsf grant no. phy-0969739. [99] u. lindstrm and m. roek, _ commun. math. phys. _ http://inspirehep.net/search?ln=en&ln=en&p=find+j+"commun.math.phys.,115,21"&of=hb&action_search=search&sf=&so=d&rm=&rg=25&sc=0[*115 * (1988) 21] ; _ commun. http://inspirehep.net/search?ln=en&ln=en&p=find+j+"commun.math.phys.,128,191"&of=hb&action_search=search&sf=&so=d&rm=&rg=25&sc=0[*128 * (1990) 191]. w. siegel, 2010, http://arxiv.org/abs/1005.2317[arxiv:1005.2317] ; + d. jain and w. siegel, _ phys. * d83 * (2011) 105024 [http://arxiv.org/abs/1012.3758[arxiv:1012.3758]] ; + d. jain and w. siegel, _ phys. rev. _ * d86 * (2012) 065036 [http://arxiv.org/abs/1106.4601[arxiv:1106.4601]]. d. jain and w. siegel, _ phys. _ * d86 * (2012) 125017 [http://arxiv.org/abs/1203.2929[arxiv:1203.2929]]. s. m. kuzenko, _ int. * a14 * (1999) 1737 [http://arxiv.org/abs/hep-th/9806147[arxiv:hep-th/9806147]]. d. jain and w. siegel, _ phys. * d80 * (2009) 045024 [http://arxiv.org/abs/0903.3588[arxiv:0903.3588]]. a. galperin, e. ivanov, s. kalitzin, v. ogievetsky and e. sokatchev, _ class. http://inspirehep.net/search?ln=en&ln=en&p=find+j+"class.quant.grav.,1,469"&of=hb&action_search=search&sf=&so=d&rm=&rg=25&sc=0[*1 * (1984) 469] ; + e. ivanov, a. galperin, v. ogievetsky and e. sokatchev, _ class. http://inspirehep.net/search?ln=en&ln=en&p=find+j+"class.quant.grav.,2,601"&of=hb&action_search=search&sf=&so=d&rm=&rg=25&sc=0[*2 * (1985) 601] ; _ class. http://inspirehep.net/search?ln=en&ln=en&p=find+j+"class.quant.grav.,2,617"&of=hb&action_search=search&sf=&so=d&rm=&rg=25&sc=0[*2 * (1985) 617] ; + a.s. galperin, e.a. ivanov, v.i. ogievetsky, and e.s. sokatchev, _ harmonic superspace _ (cambridge univ. press, 2001). i. l. buchbinder, e. i. buchbinder, s. m. kuzenko and b. a. ovrut, _ phys. lett. _ * b417 * (1998) 61 [http://arxiv.org/abs/hep-th/9704214[arxiv:hep-th/9704214]] ; + i. l. buchbinder and s. m. kuzenko, _ mod. phys. * a13 * (1998) 1623 [http://arxiv.org/abs/hep-th/9804168[arxiv:hep-th/9804168]]. i. l. buchbinder, e. i. buchbinder, e. a. ivanov, s. m. kuzenko and b. a. ovrut, _ phys. lett. _ * b412 * (1997) 309 [http://arxiv.org/abs/hep-th/9703147[arxiv:hep-th/9703147]] ; + i. l. buchbinder, s. m. kuzenko and b. a. ovrut, _ phys. lett. _ * b433 * (1998) 335 [http://arxiv.org/abs/hep-th/9710142[arxiv:hep-th/9710142]]. e. i. buchbinder, i. l. buchbinder and s. m. kuzenko, _ phys. _ * b446 * (1999) 216 [http://arxiv.org/abs/hep-th/9810239[arxiv:hep-th/9810239]] ; + s. m. kuzenko and i. n. mcarthur, _ phys. * b506 * (2001) 140 [http://arxiv.org/abs/hep-th/0101127[arxiv:hep-th/0101127]] ; + s. m. kuzenko and i. n. mcarthur, _ phys. lett. _ * b513 * (2001) 213 [http://arxiv.org/abs/hep-th/0105121[arxiv:hep-th/0105121]]. i. l. buchbinder, e. a. ivanov and a. y. petrov, _ nucl. phys. _ * b653 * (2003) 64 [http://arxiv.org/abs/hep-th/0210241[arxiv:hep-th/0210241]]. i. l. buchbinder and a. yu. petrov, _ phys. _ * b482 * (2000) 429 [http://arxiv.org/abs/hep-th/0003265[arxiv:hep-th/0003265]]. m. t. grisaru, w. siegel and m. roek, _ nucl. _ * b159 * (1979) 429. m. t. grisaru and w. siegel, _ nucl. phys. _ * b201 * (1982) 292. p. s. howe, k. s. stelle, p. c. west, _ phys. lett. _ * b124 * (1983) 55. f. gonzalez - rey, 1997, http://arxiv.org/abs/hep-th/9712128[arxiv:hep-th/9712128]. f. gonzalez - rey and m. roek, _ phys. _ * b434 * (1998) 303 [http://arxiv.org/abs/hep-th/9804010[arxiv:hep-th/9804010]]. m. t. grisaru and d. zanon, _ phys. lett. _ * b142 * (1984) 359 ; + m. t. grisaru and d. zanon, _ nucl. phys. _ * b252 * (1985) 578. t. r. morris, _ phys. _ * b164 * (1985) 315.

we introduce a new background field method for n=2 superspace. (we treat projective hyperspace, but similar remarks apply for the harmonic case.) in analogy to n=1, background gauge fields are in the real representation, so the lowest - dimension potentials are spinor and the usual non - renormalization theorems are manifest. another consequence is that the r - coordinates disappear from the effective action.

1302.3277

Introduction although the sunspot number varies periodically with time with an average period of 11 year, the individual cycle period (length) and also the strength (amplitude) vary in a random way. it is observed that the stronger cycles have shorter periods and vice versa. this leads to an important feature of solar cycle known as waldmeier effect. it says that there is an anti - correlation between the rise time and the peak sunspot number. we call this as we1. now instead of rise time if we consider the rise rate then we get very tight positive correlation between the rise rate and the peak sunspot number. we call this as we2. another important aspect of solar activity are the grand minima. these are the periods of strongly reduced activity. a best example of these is the during during 16451715. it was not an artifact of few observations, but a real phenomenon (hoyt & schatten 1996). from the study of the cosmogenic isotope @xmath0c data in tree rings, usoskin et al. (2007) reported that there are @xmath1 grand minimum during last @xmath2 years. Methodology and results we want to model these irregularities of solar cycle using flux transport dynamo model (choudhuri et al. 1995 ; dikpati & charbonneau 1999 ; chatterjee et al. 2004). in this model, the turbulent diffusivity is an important ingredient which is not properly constrained. therefore several groups use different value of diffusivity and this leads to two kinds of flux transport dynamo model high diffusivity model and low diffusivity model. in the earlier model, the value of diffusivity usually used is @xmath3 @xmath4 s@xmath5 (see also jiang et al. 2007 and yeates et al. 2008 for details), whereas in the latter model, it is @xmath6 @xmath4 s@xmath5. we mention that the mixing length theory gives the value of diffusivity as @xmath7 @xmath4 s@xmath5. another important flux transport agent in this model is the meridional circulation. only since 1990 s we have some observational data of meridional circulation near the surface and therefore we do not know whether the varied largely with solar cycle in past or not. however if the flux transport dynamo is the correct dynamo for the solar cycle, then one can consider the solar cycle period variation as the variation for the because the cycle period is strongly determined by the strength of the meridional circulation in this model. now the periods of the solar cycle indeed had much variation in past, then we can easily say that the had significant variation with the solar cycle. therefore the main sources of randomness in the flux transport dynamo model are the stochastic fluctuations in process of generating poloidal field and the stochastic fluctuations in the meridional circulation. in this paper we explore the effects of fluctuations of the latter. we model last @xmath8 cycles by fitting the periods with variable meridional circulation in a high diffusivity model based on chatterjee et al. (2004) model. the solid line in fig. [fit23](a) shows the variation of the amplitude of @xmath9 used to model the periods of the cycles. note that we did not try to match the periods of each cycles accurately which is bit difficult. we change @xmath9 between two cycles and not during a cycle. in addition, we do not change @xmath9 if the period difference between two successive cycles is less than @xmath10 of the average period. (in m s@xmath5) with time (in yr). the solid line is the variation of @xmath9 used to match the theoretical periods with the observed periods. (b) variation of theoretical sunspot number (dashed line) and observed sunspot number (solid line) with time. (c) scatter diagram showing peak theoretical sunspot number and peak observed sunspot number. the linear correlation coefficients and the corresponding significance levels are given on the plot.,scaledwidth=100.0%] in fig. [fit23](b), we show the theoretical sunspot series (eruptions) by dashed line along with the observed sunspot series by solid line. the theoretical sunspot series has been multiplied by a factor to match the observed value. it is very interesting to see that most of the amplitudes of the theoretical sunspot cycle have been matched with the observed sunspot cycle. therefore, we have found a significant correlation between these two (see fig. [fit23](c)). this study suggests that a major part of the fluctuations of the amplitude of the solar cycle may come from the fluctuations of the meridional circulation. this is a very important result of this analysis. now we explain the physics of this result based on yeates et al. toroidal field in the flux transport model, is generated by the stretching of the poloidal field in the tachocline. the production of this toroidal field is more if the poloidal field remains in the tachocline for longer time and vice versa. however, the poloidal field diffuses during its transport through the convection zone. as a result, if the diffusivity is very high, then much of the poloidal field diffuses away and very less amount of it reaches the tachocline to induct toroidal field. therefore, when we decrease @xmath9 in high diffusivity model to match the period of a longer cycle, the poloidal field gets more time to diffuse during its transport through the convection zone. this ultimately leads to a lesser generation of toroidal field and hence the cycle becomes weaker. on the other hand, when we increase the value of @xmath9 to match the period of a shorter cycle, the poloidal field does not get much time to diffuse in the convection zone. hence it produces stronger toroidal field and the cycle becomes stronger. consequently, we get weaker amplitudes for longer periods and vice versa. however, this is not the case in low diffusivity model because in this model the diffusive decay of the fields are not much important. as a result, the slower meridional circulation means that the poloidal field remains in the tachocline for longer time and therefore it produces more toroidal field, giving rise to a strong cycle. therefore, we do not get a correct correlation between the amplitudes of theoretical sunspot number and that of observed sunspot number when repeat the same analysis in low diffusivity model based on dikpati & charbonneau (1999) model. we study the using flux transport dynamo model. we have seen that the stochastic fluctuations in the process and the stochastic fluctuations in the are the two main sources of irregularities in this model. therefore, to study we first introduce suitable stochastic fluctuations in the poloidal field source term of process. we see that this study can not reproduce we1 (fig. [pol](a)). however it reproduces we2 (fig. [pol](b)). finally we introduce stochastic fluctuations in both the poloidal field source term and the meridional circulation. we see that both we1 and we2 are remarkably reproduced in this case (see fig. [both]). we repeat the same study in low diffusivity model based on dikpati & charbonneau (1999) model. however in this case we are failed to reproduce we1, only we2 is reproduced. the details of this work can be found in karak & choudhuri (2011). we have realized that the is important in modeling many aspects of solar cycle. therefore we check whether a large decrease of the leads to a maunder - like grand minimum. to answer this question, we decrease @xmath9 to a very low value in both the hemispheres. we have done this in the decaying phase of the last sunspot cycle before maunder minimum. we keep @xmath9 at low value for around 1 yr and then we again increase it to the usual value but at different rates in two hemispheres. in northern hemisphere, @xmath9 is increased at slightly lower rate than southern hemisphere. (in m s@xmath5) in northern and southern hemispheres with time. (b) the butterfly diagram. (c) the dashed and dotted lines show the sunspot numbers in southern and northern hemispheres, whereas the solid line is the total sunspot number. (d) variation of energy density of toroidal field at latitude 15@xmath11 at the bottom of the convection zone.,scaledwidth=100.0%] in fig. [mm], we show the theoretical results covering the maunder minimum episode. fig. [mm](a), shows the maximum amplitude of meridional circulation @xmath9 varied over this period in two hemispheres. in fig. [mm](b), we show the butterfly diagram of sunspot numbers, whereas in fig. [mm](c), we show the variation of total sunspot number along with the individual sunspot numbers in two hemispheres (see the caption). in order to facilitate comparison with observational data, we have taken the beginning of the year to be 1635. note that our theoretical results reproduce the sudden initiation and the gradual recovery, the north - south asymmetry of sunspot number observed in the last phase of maunder minimum and the cyclic oscillation of solar cycle found in cosmogenic isotope data. we also mention that if we reduce the poloidal field to a very low value at the beginning of the maunder minimum then also we can reproduce maunder - like grand minimum (choudhuri & karak 2009). however in both the cases, either we need to reduce the or the poloidal field at the beginning of the maunder minimum. however if we reduce the poloidal field little bit, then one can reproduce maunder - like grand minimum at a moderate value of meridional circulation. the details of this study can be found in karak (2010). we have shown that with a suitable stochastic fluctuations in the meridional circulation, we are able to reproduce many important irregular features of solar cycle including waldmeier effect and maunder like grand minimum. however we are failed to reproduce these results in low diffusivity model. therefore this study along with some earlier studies (chatterjee, nandy & choudhuri 2004 ; chatterjee & choudhuri 2006 ; goel & choudhuri 2009 ; jiang, chatterjee & choudhuri 2007 ; karak 2010 ; karak & choudhuri 2011 ; karak & choudhuri 2012) supports the high diffusivity model for solar cycle. chatterjee, p., nandy, d., & choudhuri, a. r. 2004, a&a, 427, 1019 choudhuri, a. r., chatterjee, p., & jiang, j., 2007, phys., 98, 1103 choudhuri, a. r., & karak, b. b. 2009, raa 9, 953 choudhuri, a. r., schssler, m., & dikpati, m. 1995, a&a, 303, l29 dikpati, m., & charbonneau, p. 1999, apj, 518, 508 jiang, j., chatterjee, p., & choudhuri, a. r. 2007, mnras, 381, 1527 hoyt, d. v., & schatten, k. h., 1996, sol. phys., 165, 181 karak, b. b. 2010, apj, 724, 1021 karak, b. b., & choudhuri, a. r. 2011, mnras, 410, 1503 karak, b. b., & choudhuri, a. r. 2012, sol., 278:137 usoskin, i. g., solanki, s. k., & kovaltsov, g. a. 2007, a&a, 471, 301 yeates, a. r., nandy, d., & mackay, d. h. 2008, apj, 673, 544

the sunspot number varies roughly periodically with time. however the individual cycle durations and the amplitudes are found to vary in an irregular manner. it is observed that the stronger cycles are having shorter rise times and vice versa. this leads to an important effect know as the waldmeier effect. another important feature of the solar cycle irregularity are the grand minima during which the activity level is strongly reduced. we explore whether these solar cycle irregularities can be studied with the help of the flux transport dynamo model of the solar cycle. we show that with a suitable stochastic fluctuations in a regular dynamo model, we are able to reproduce many irregular features of the solar cycle including the waldmeier effect and the grand minimum. however, we get all these results only if the value of the turbulent diffusivity in the convection zone is reasonably high. [firstpage]

1205.0614

The jhu turbulence database in @xcite a database containing a solution of the 3d incompressible navier - stokes (ns) equations is presented. the equations were solved numerically with a standard pseudo - spectral simulation in a periodic domain, using a real space grid of @xmath0 grid points. a large - scale body force drives a turbulent flow with a taylor microscale based reynolds number @xmath1. out of this solution, @xmath2 snapshots were stored, spread out evenly over a large eddy turnover time. more on the simulation and on accessing the data can be found at http://turbulence.pha.jhu.edu. in practical terms, we have easy access to the turbulent velocity field and pressure at every point in space and time. Vortices within vortices one usual way of visualising a turbulent velocity field is to plot vorticity isosurfaces see for instance the plots from @xcite. the resulting pictures are usually very `` crowded '', in the sense that there are many intertwined thin vortex tubes, generating an extremely complex structure. in fact, the picture of the entire dataset from @xcite looks extremely noisy and it is arguably not very informative about the turbulent dynamics. in this work, we follow a different approach. first of all, we use the alternate quantity @xmath3 first introduced in @xcite. secondly, the tool being used has the option of displaying data only inside clearly defined domains of 3d space. we can exploit this facility to investigate the multiscale character of the turbulent cascade. because vorticity is dominated by the smallest available scales in the velocity, we can visualize vorticity at scale @xmath4 by the curl of the velocity box - filtered at scale @xmath4. we follow a simple procedure : * we filter the velocity field, using a box filter of size @xmath5, and we generate semitransparent surfaces delimitating the domains @xmath6 where @xmath7 ; * we filter the velocity field, using a box filter of size @xmath8, and we generate surfaces delimitating the domains @xmath9 where @xmath10, but only if these domains are contained in one of the domains from @xmath6 ; and this procedure can be used iteratively with several scales (we use at most 3 scales, since the images become too complex for more levels). additionally, we wish sometimes to keep track of the relative orientation of the vorticity vectors at the different scales. for this purpose we employ a special coloring scheme for the @xmath11 isosurfaces : for each point of the surface, we compute the cosine of the angle @xmath12 between the @xmath13 filtered vorticity and the @xmath5 filtered vorticity : @xmath14 the surface is green for @xmath15, yellow for @xmath16 and red for @xmath17, following a continuous gradient between these three for intermediate values. Observations the opening montage of vortex tubes is very similar to the traditional visualisation : a writhing mess of vortices. upon coarse - graining, additional structure is revealed. the large - scale vorticity, which appears as transparent gray, is also arranged in tubes. as a next step, we remove all the fine - scale vorticity outside the large - scale tubes. the color scheme for the small - scale vorticity is that described earlier, with green representing alignment with the large - scale vorticity and red representing anti - alignment. clearly, most of the small - scale vorticity is aligned with the vorticity of the large - scale tube that contains it. we then remove the fine - grained vorticity and pan out to see that the coarse - grained vortex tubes are also intricately tangled and intertwined. introducing a yet larger scale, we repeat the previous operations. the relative orientation properties of the vorticity at these two scales is similar to that observed earlier. next we visualize the vortex structures at all three scales simultaneously, one inside the other. it is clear that the small vortex tubes are transported by the larger tubes that contain them. however, this is not just a passive advection. the small - scale vortices are as well being distorted by the large - scale motions. to focus on this more clearly, we now render just the two smallest scales. one can observe the small - scale vortex tubes being both stretched and twisted by the large - scale motions. the stretching of small vortex tubes by large ones was suggested by orszag and borue @xcite as being the basic mechanism of the turbulent energy cascade. as the small - scale tubes are stretched out, they are `` spun up '' and gain kinetic energy. here, this phenomenon is clearly revealed. the twisting of small - scale vortices by large - scale screw motions has likewise been associated to helicity cascade @xcite. the video thus allows us to view the turbulent cascade in progress. next we consider the corresponding view with three levels of vorticity simultaneously. since the ratio of scales is here 1:15:49 we are observing less than two decades of the turbulent cascade. one must imagine the complexity of a very extended inertial range with many scales of motion. not all of the turbulent dynamics is tube within tube. in our last scene we visualize in the right half domain all the small - scale vortices, and in the left domain only the small - scale vortices inside the larger scale ones. in the right half, the viewer can observe stretching of the small - scale vortex structures taking place externally to the large - scale tubes. the spin - up of these vortices must contribute likewise to the turbulent energy cascade. 6ifxundefined [1] ifx#1 ifnum [1] # 1firstoftwo secondoftwo ifx [1] # 1firstoftwo secondoftwo `` `` # 1''''@noop [0]secondoftwosanitize@url [0] + 12$12 & 12#1212_12%12@startlink[1]@endlink[0]@bib@innerbibempty link:\doibase 10.1080/14685240802376389 [* * (), 10.1080/14685240802376389] @noop * * (), in http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=1592886[__] () p. @noop _ _, () link:\doibase 10.1017/s0022112097008306 [* *, ()] http://journals.cambridge.org / production / action / cjogetfulltext?fulltextid=4% 00523 [* *, ()]

the jhu turbulence database @xcite can be used with a state of the art visualisation tool @xcite to generate high quality link : anc / dfdsubmissionquarterres.mpg[fluid dynamics videos]. in this work we investigate the classical idea that smaller structures in turbulent flows, while engaged in their own internal dynamics, are advected by the larger structures. they are not advected undistorted, however. we see instead that the small scale structures are sheared and twisted by the larger scales. this illuminates the basic mechanisms of the turbulent cascade.

1210.3325

Introduction in recent years electron transfer (et) between molecular adsorbates and semiconductor nanomaterials and surfaces has been subject of much research @xcite. the injection of an electron into the conduction band is a prototype reaction for a lot of electrochemical and photoelectrochemical interfacial processes such as photography, solar energy conversion, quantum dot devices, etc. interfacial et between discrete molecular levels and a conducting surface is the simplest of all surface reactions : it involves only the exchange of an electron, and so no bonds are broken @xcite. the ultrafast nature of the charge injection from adsorbed molecules to the conduction band of semiconductor surfaces was shown in recent experiments @xcite. the theoretical description of such experiments demands an adequate treatment of the et dynamics to be able to describe short time - scale phenomena such as coherences. this can be done within the reduced density matrix (rdm) description used in the present contribution. recently @xcite the electron injection from a chromophore to a semiconductor conduction band was described using the time - dependent schrdinger equation, thus neglecting relaxation processes. the neglect of relaxation processes was motivated by the experimental finding that injected electrons relax only within 150 fs in the perylene - tio@xmath0 system. here we include relaxation to be able to treat a larger class of experiments where, for example, the adsorbed molecule is surrounded by a liquid environment, and longer times. Theory in the rdm theory the full system is divided into a relevant system part and a heat bath. therefore the total hamiltonian consists of three terms the system part @xmath1, the bath part @xmath2, and the system - bath interaction @xmath3 : @xmath4 the rdm @xmath5 is obtained from the density matrix of the full system by tracing out the degrees of freedom of the environment. this reduction together with a second - order perturbative treatment of @xmath3 and the markov approximation leads to the redfield equation @xcite : @xmath6 + { \mathcal r } \rho = { \mathcal l } \rho. \label{eq : redfield}\]] in this equation @xmath7 denotes the redfield tensor. if one assumes bilinear system - bath coupling with system part @xmath8 and bath part @xmath9 @xmath10 one can take advantage of the following decomposition @xcite : @xmath11 + [\lambda\rho, k]+ [k,\rho\lambda^{\dagger }] . \label{eq : pf - form}\]] the @xmath12 operator can be written in the form @xmath13 where @xmath14 is the operator @xmath8 in the interaction representation. the system bath interaction is taken to be linear in the reaction coordinate as well as in the bath coordinates. neither the rotating wave nor the secular approximation have been invoked. the so - called diabatic damping approximation which has numerical advantages @xcite is not used because it could lead to wrong results in the present system studied @xcite. in the following we direct our attention to et between an excited molecular state and a conduction band. the hamiltonian modeling this system consists of the ground and one excited state of the molecule and a quasi - continuum describing the conduction band together with one vibrational coordinate @xmath15 here @xmath16 can be equal to @xmath17 for the ground state, @xmath18 for the excited state, and @xmath19 for the quasi - continuum. as in ref. @xcite we choose the frequency of the vibrational mode to be @xmath20. the coupling between the excited state and the continuum states is assumed to be constant : @xmath21. a box - shaped uniform density of states is used. instead of modeling the excitation from the ground state explicitly we assume a @xmath22-pulse. the excited state potential energy surface is shifted 0.1 along the reaction coordinate with respect to the ground state potential energy surface. this results in an initial vibrational wave packet on the excited state with significant population in the lowest 4 - 5 vibrational states. the shift between the excited state energy surface and the continuum parabola is 0.2 . the thermal bath is characterized by its spectral density @xmath23. because all system oscillators have the same frequency the coupling to the bath can be given by one parameter @xmath24 in the diabatic damping approximation. denoting the effective mass of the harmonic oscillator by @xmath25 the strength of the damping is chosen as @xmath26. to be able to study the effects of dissipation we do not model the quasi - continuum with such a large number of electronic states as in ref. @xcite. in that work a band of width 2 ev was described using an energy difference of 2.5 mev leading to 801 electronic surfaces. these calculations are already demanding using wave packet propagation but almost impossible using direct density matrix propagation. for doing such a large system one would have to use the monte carlo wave function scheme @xcite. we use a much simpler model and describe only that part of the conduction band which really takes part in the injection process. the total width of the conduction band may be significantly larger. in the following, a band of width 0.75 ev is treated with 31 electronic surfaces. in each of these electronic states five vibrational states are taken into account. we are aware that this is only a minimal model but hope that it catches the effects of dissipation on the electron injection process. Results here we look at two different populations arising in the process of electron injection. the time - dependent population of the electronic states in the conduction band is calculated as the sum over the vibrational levels of each electronic surface @xmath27. as a second quantity we look at the time - dependent population of the vibrational levels of the excited molecular state @xmath28. these two probability distributions give some hints on the effect of dissipation. figure 1 shows the electronic population for the quasi - continuum, i.e. the probability distribution of the injected electron, versus the energy of the conduction band. as described above, the four lowest vibrational states are populated significantly at @xmath29. the structure arising in the upper panel of fig. 1 was already explained by ramakrishna et al. it can be estimated using the golden rule. the electronic probabilities in the quasi - continuum are given as @xmath30 where @xmath31 is the initial vibronic distribution in the excited state and @xmath32 and @xmath33 are the vibronic parts of the wave packet in the excited and quasi - continuum states, respectively. the energy @xmath34 denotes the middle of the band. turning on dissipation two effects can be seen. first, the vibrational populations in the excited state of the molecule no longer only decay into the quasi - continuum states but also relax within the excited state (see fig. 2). second, the vibrational populations also relax within the quasi - continuum states. the recurrences back into the excited state become much smaller. only those parts of the wave packet which are still high enough in energy can go back to the molecule. in summary, we extended the work by ramakrishna, willig, and may @xcite by including relaxation processes into the description of electron injection into the conduction band of a semiconductor. this will, at least, become important for modeling electron injection in the presence of a fluid surrounding the attached molecule.

electron injection from an adsorbed molecule to the substrate (heterogeneous electron transfer) is studied. one reaction coordinate is used to model this process. the surface phonons and/or the electron - hole pairs together with the internal degrees of freedom of the adsorbed molecule as well as possibly a liquid surrounding the molecule provide a dissipative environment, which may lead to dephasing, relaxation, and sometimes excitation of the relevant system. in the process studied the adsorbed molecule is excited by a light pulse. this is followed by an electron transfer from the excited donor state to the quasi - continuum of the substrate. it is assumed that the substrate is a semiconductor. the effects of dissipation on electron injection are investigated. electron transfer, density matrix theory, molecules at surfaces

physics0109012

Introduction the experimental data used in this paper were collected by the forward looking radar of the us army research laboratory @xcite. that radar was built for detection and possible identification of shallow explosive - like targets. since targets are three dimensional objects, one needs to measure a three dimensional information about each target. however, the radar measures only one time dependent curve for each target, see figure 5. therefore, one can hope to reconstruct only a very limited information about each target. so, we reconstruct only an estimate of the dielectric constant of each target. for each target, our estimate likely provides a sort of an average of values of its spatially distributed dielectric constant. but even this information can be potentially very useful for engineers. indeed, currently the radar community is relying only on the energy information of radar images, see, e.g. @xcite. estimates of dielectric constants of targets, if taken alone, can not improve the current false alarm rate. however, these estimates can be potentially used as an additional piece of information. being combined with the currently used energy information, this piece of the information might result in the future in new classification algorithms, which might improve the current false alarm rate. an inverse medium scattering problem (imsp) is often also called a coefficient inverse problem (cip). imsps / cips are both ill - posed and highly nonlinear. therefore, an important question to address in a numerical treatment of such a problem is : _ how to reach a sufficiently small neighborhood of the exact coefficient without any advanced knowledge of this neighborhood? _ the size of this neighborhood should depend only on the level of noise in the data and on approximation errors. we call a numerical method, which has a rigorous guarantee of achieving this goal, _ globally convergent method _ (gcm). in this paper we develop analytically a new globally convergent method for a 1-d inverse medium scattering problem (imsp) with the data generated by multiple frequencies. in addition to the analytical study, we test this method numerically using both computationally simulated and the above mentioned experimental data. first, we derive a nonlinear integro - differential equation in which the unknown coefficient is not present. _ element _ of this paper is the method of the solution of this equation. this method is based on the construction of a weighted least squares cost functional. the key point of this functional is the presence of the carleman weight function (cwf) in it. this is the function, which is involved in the carleman estimate for the underlying differential operator. we prove that, given a closed ball of an arbitrary radius @xmath1 with the center at @xmath2 in an appropriate hilbert space, one can choose the parameter @xmath3 of the cwf in such a way that this functional becomes strictly convex on that ball. the existence of the unique minimizer on that closed ball as well as convergence of minimizers to the exact solution when the level of noise in the data tends to zero are proven. in addition, it is proven that the gradient projection method reaches a sufficiently small neighborhood of the exact coefficient if its starting point is an arbitrary point of that ball. the size of that neighborhood is proportional to the level of noise in the data. therefore, since restrictions on @xmath4 are not imposed in our method, then this is a _ globally convergent _ numerical method. we note that in the conventional case of a non convex cost functional a gradient - like method converges to the exact solution only if its starting point is located in a sufficiently small neighborhood of this solution : this is due to the phenomenon of multiple local minima and ravines of such functionals. unlike previously developed globally convergent numerical methods of the first type for cips (see this section below), the convergence analysis for the technique of the current paper does not impose a smallness condition on the interval @xmath5 of the variations of the wave numbers @xmath6. the majority of currently known numerical methods of solutions of nonlinear ill - posed problems use the nonlinear optimization. in other words, a least squares cost functional is minimized in each problem, see, e.g. chavent, engl, gonch1,gonch2. however, the major problem with these functionals is that they are usually non convex. figure 1 of the paper scales presents a numerical example of multiple local minima and ravines of non - convex least squares cost functionals for some cips. hence, convergence of the optimization process of such a functional to the exact solution can be guaranteed only if a good approximation for that solution is known in advance. however, such an approximation is rarely available in applications. this prompts the development of globally convergent numerical methods for cips, see, e.g. @xcite. the first author with coauthors has proposed two types of gcm for cips with single measurement data. the gcm of the first type is reasonable to call the tail functions method ``. this development has started from the work @xcite and has been continued since then, see, e.g. @xcite and references cited therein. in this case, on each step of an iterative process one solves the dirichlet boundary value problem for a certain linear elliptic pde, which depends on that iterative step. the solution of this pde allows one to update the unknown coefficient first and then to update a certain function, which is called the tail function ''. the convergence theorems for this method impose a smallness condition on the interval of the variation of either the parameter @xmath7 of the laplace transform of the solution of a hyperbolic equation or of the wave number @xmath8 in the helmholtz equation. recall that the method of this paper does not impose the latter assumption. in this paper we present a new version of the gcm of the second type. in any version of the gcm of the second type a weighted cost functional with a cwf in it is constructed. the same properties of the global strict convexity and the global convergence of the gradient projection method hold as the ones indicated above. the gcm of the second type was initiated in klib95,klib97,kt with a recently renewed interest in @xcite. the idea of any version of the gcm of the second type has direct roots in the method of @xcite, which is based on carleman estimates and which was originally designed in @xcite only for proofs of uniqueness theorems for cips, also see the recent survey in @xcite. another version of the gcm with a cwf in it was recently developed in bau1 for a cip for the hyperbolic equation @xmath9 where @xmath10 is the unknown coefficient. this gcm was tested numerically in @xcite. in bau1,bau2 non - vanishing conditions are imposed : it is assumed that either @xmath11 or @xmath12 or @xmath13 in the entire domain of interest. similar assumptions are imposed in @xcite for the gcm of the second type. on the other hand, we consider in the current paper, so as in @xcite, the fundamental solution of the corresponding pde. the differences between the fundamental solutions of those pdes and solutions satisfying non - vanishing conditions cause quite significant differences between klib95,klib97,kt, ktsiap and @xcite of corresponding versions of the gcm of the second type. recently, the idea of the gcm of the second type was extended to the case of ill - posed cauchy problems for quasilinear pdes, see the theory in klquasi and some extensions and numerical examples in bakklkosh, klkosh. cips of wave propagation are a part of a bigger subfield, inverse scattering problems (isps). isps attract a significant attention of the scientific community. in this regard we refer to some direct methods which successfully reconstruct positions, sizes and shapes of scatterers without iterations @xcite. we also refer to @xcite for some other isps in the frequency domain. in addition, we cite some other numerical methods for isps considered in @xcite. as to the cips with multiple measurement, i.e. the dirichlet - to - neumann map data, we mention recent works @xcite and references cited therein, where reconstruction procedures are developed, which do not require a priori knowledge of a small neighborhood of the exact coefficient. in section 2 we state our inverse problem. in section 3 we construct that weighted cost functional. in section 4 we prove the main property of this functional : its global strict convexity. in section 5 we prove the global convergence of the gradient projection method of the minimization of this functional. although this paper is mostly an analytical one (sections 3 - 5), we complement the theory with computations. in section 6 we test our method on computationally simulated data. in section 7 we test it on experimental data. concluding remarks are in section 8. Problem statement let the function @xmath14 be the spatially distributed dielectric constant of the medium. we assume that@xmath15@xmath16fix the source position @xmath17 for brevity, we do not indicate below dependence of our functions on @xmath18 consider the 1-d helmholtz equation for the function @xmath19,@xmath20@xmath21let @xmath22 be the solution of the problem ([2.4]), ([2.6]) for the case @xmath23 then@xmath24our interest is in the following inverse problem : * inverse medium scattering problem (imsp)*. _ let _ @xmath25\subset \left (0,\infty \right) $] _ _ be an interval of wavenumbers _ _ @xmath26__. reconstruct the function _ _ @xmath27 _ _ assuming that the following function _ _ @xmath28 _ _ is known _ _ @xmath29. \label{2.8}\]] denote@xmath30it follows from ([2.8]), ([2.100]) and @xcite that @xmath31, \label{2.101}\]]@xmath32. \label{2.160}\]] in this subsection we briefly outline some results of @xcite, which we use below in this paper. existence and uniqueness of the solution @xmath33 for each @xmath8 was established in @xcite. also, it was proven in @xcite that @xmath34, \forall k>0. \label{2.9}\]]in particular, @xmath35.$] in addition, uniqueness of our imsp was proven in klibloc. also, the following asymptotic behavior of the function @xmath36 takes place : @xmath37 \left (1+% \widehat{u}\left (x, k\right) \right), k\rightarrow \infty, \forall x\in % \left [0,1\right], \label{2.10}\]]@xmath38 given ([2.9]) and ([2.10]) we now can uniquely define the function @xmath39 as in @xcite. the difficulty here is in defining @xmath40 since this number is usually defined up to the addition of @xmath41 where @xmath42 is an integer. for sufficiently large values of @xmath26 we define the function @xmath39 using ([2.60]), ([2.100]), ([2.10]) and ([2.1000]) as @xmath43where @xmath44hence, for sufficiently large @xmath26, @xmath45which eliminates the above mentioned ambiguity. suppose that the number @xmath46 is so large that ([2.12]) is true for @xmath47 then @xmath48 is defined as in ([2.11]). as to not large values of @xmath26, we define the function ([2.11])@xmath49 as @xmath50by ([2.9]) @xmath51, \forall \xi > 0.$] differentiating both sides of ([2.13]) with respect to @xmath26, we obtain @xmath52multiplying both sides of ([2.14]) by @xmath53, we obtain @xmath54 hence, there exists a function @xmath55 independent on @xmath26 such that @xmath56setting in ([2.15]) @xmath57 and using the fact that by (2.13) @xmath58, we obtain @xmath59. \label{2.150}\]]hence, ([2.13]) and ([2.15]) imply that @xmath60 is defined as @xmath61 The weighted cost functional in this section we construct the above mentioned weighted cost functional with the cwf in it. * lemma 3.1 * (carleman estimate). _ for any complex valued function _ @xmath62 _ _ with _ _ @xmath63 _ _ and for any parameter _ _ @xmath64 _ _ the following carleman estimate holds _ _ @xmath65, \label{3.00}\]]__where the constant _ _ @xmath66 _ is _ _ independent of _ @xmath67__and _ _ @xmath68 * proof*. in the case when the integral with @xmath69 is absent in the right hand side of ([3.00]) this lemma was proved in klibloc. to incorporate this integral, we note that @xmath70. \label{3.02}\]]let @xmath71 then ([3.02]) implies ([3.00]) where @xmath72 is replaced with @xmath73 @xmath74 for @xmath75,k\in \lbrack \underline{k},\overline{k}]$] consider the function @xmath76 and its @xmath77derivative @xmath78, where @xmath79 hence,@xmath80consider the function @xmath81, which we call the tail function ", and this function is unknown,@xmath82 let @xmath83 note that since @xmath84 for @xmath85 then equation ([2.4]) and the first condition ([2.6]) imply that @xmath86 for @xmath87 hence, ([2.60]) and ([2.100]) imply that @xmath88 for @xmath87 it follows from ([2.4]), ([2.60]), ([2.100])([2.160]), ([2.15]) and ([2.150]) that @xmath89@xmath90using ([2.15]), ([2.150]), ([3.0]) and ([3.3]), we obtain @xmath91differentiate ([3.5]) with respect to @xmath26 and use ([3.0])-([3.4]). we obtain @xmath92@xmath93@xmath94where@xmath95, k\in % \left [\underline{k},\overline{k}\right]. \label{3.70}\]] we have obtained an integro - differential equation ([3.6]) for the function @xmath96 with the overdetermined boundary conditions ([3.7]). the tail function @xmath97 is also unknown. first, we will approximate the tail function @xmath98. next, we will solve the problem ([3.6]), ([3.7]) for the function @xmath96. to solve this problem, we will construct the above mentioned weighted cost functional with the cwf @xmath99 in it, see ([3.00]). this construction, combined with corresponding analytical results, is the _ central _ part of our paper. thus, even though the problem ([3.6])-([3.70]) is the same as the problem (65), (66) in @xcite, the numerical method of the solution of the problem ([3.6])-([3.70]) is _ radically _ different from the one in @xcite. now, suppose that we have obtained approximations for both functions @xmath100 and @xmath78. then we obtain the unknown coefficient @xmath101 via backwards calculations. first, we calculate the approximation for the function @xmath102 via (3.1) and ([3.2]). next, we calculate the function @xmath103 via ([3.5]). we have learned from our numerical experience that the best value of @xmath26 to use in ([3.5]) for the latter calculation is @xmath104 the approximation for the tail function is done here the same way as the approximation for the so - called first tail function " in section 4.2 of @xcite. however, while tail functions are updated in @xcite, we are not doing such updates here. it follows from ([2.100])-([2.110]) and ([3.0])-([3.2]) that there exists a function @xmath105 $] such that @xmath106hence, assuming that the number @xmath107 is sufficiently large, we drop terms @xmath108 and @xmath109 in ([3.8]). next, we set@xmath110set @xmath111 in ([3.6]) and ([3.7]). next, substitute (3.9) in ([3.6]) and ([3.7]) at @xmath57. we obtain @xmath112 recall that functions @xmath113 and @xmath114 are linked via ([2.160]). thus, @xmath115where functions @xmath113 and @xmath116 are defined in ([2.101]) and (2.160) respectively. it seems to be at the first glance that one can find the function @xmath98 as, for example cauchy problem for ode ([3.10]) with data @xmath117 and @xmath118 however, it was noticed in remark 5.1 of @xcite that this approach, being applied to a similar problem, does not lead to good results. we have the same observation in our numerical studies. this is likely to the approximate nature of ([3.9]). thus, just like in @xcite, we solve the problem ([3.10]), ([3.11]) by the quasi - reversibility method (qrm). the boundary condition @xmath119 provides a better stability property. so, we minimize the following functional @xmath120 on the set @xmath121, where @xmath122@xmath123where @xmath124 is the regularization parameter. the existence and uniqueness of the solution of this minimization problem as well as convergence of minimizers @xmath125 in the @xmath126norm to the exact solution @xmath127 of the problem ([3.11]), (3.12) with the exact data @xmath128 as @xmath129 were proved in @xcite. we note that in the regularization theory one always assumes existence of an ideal exact solution with noiseless data @xcite. recall that by the embedding theorem @xmath130 $] and @xmath131 } \leq c\left\vert f\right\vert _ { h^{2}\left (0,1\right) }, \forall f\in h^{2}\left (0,1\right), \label{3.130}\]] where @xmath66 is a generic constant@xmath132 theorem 3.1 is a reformulation of theorem 4.2 of @xcite. * theorem 3.1. * _ let the function _ @xmath133 _ _ satisfying conditions ([2.1])-([2.2]) be the exact solution of our imsp with the noiseless data _ _ $] _ _, where _ _ @xmath135 _ _ and _ _ @xmath136 _ _ is the solution of the forward problem ([2.4]), ([2.6]). let the exact tail function _ _ @xmath137 _ _ and the function _ _ @xmath138__have the form ([3.9]) with _ _ @xmath139 _ _ assume that for _ _ $] _ _ _ _ @xmath141__where _ _ @xmath142 _ _ is a sufficiently small number, which characterizes the level of the error in the boundary data. let in ([3.12]) _ _ @xmath143 _ _ let the function _ _ @xmath144 _ _ be the minimizer of the functional ([3.12]) on the set of functions _ _ @xmath121 _ _ defined in ([3.13]). then there exists a constant _ _ @xmath145__depending only on _ _ @xmath107 _ _ and _ _ @xmath146 _ _ such that _ _ @xmath147 } \leq c\left\vert v_{\alpha \left (\delta \right) } \left (x\right) -v^{\ast } \left (x,% \overline{k}\right) \right\vert _ { h^{2}\left (0,1\right) } \leq c_{1}\delta. \label{3.15}\]] * remark 3.1*. we have also tried to consider two terms in the asymptotic expansion for @xmath98 in ([3.8]) : the second one with @xmath148 this resulted in a nonlinear system of two equations. we have solved it by via minimizing an analog of the functional of section 3.3. however, the quality of resulting images deteriorated as compared with the above function @xmath149 in addition, we have tried to iterate with respect to the tail function @xmath98. however, the quality of resulting images has also deteriorated. consider the function @xmath78 satisfying ([3.6])-(3.70). in sections 5.2 and 5.3 we use lemma 2.1 and theorem 2.1 of bakklkosh. to apply theorems, we need to have zero boundary conditions at @xmath150 hence, we introduce the function @xmath151@xmath152denote @xmath153also, replace in ([3.6]) @xmath98 with @xmath154 then ([3.6]), ([3.7]) and ([3.16]) and ([3.170]) imply that@xmath155@xmath156@xmath157 introduce the hilbert space @xmath158 of pairs of real valued functions @xmath159 @xmath160 as@xmath161 ^{1/2}<\infty% \end{array}% \right\ }. \label{3.19}\]]here and below @xmath162 based on ([3.17]) and ([3.18]), we define our weighted cost functional as@xmath163let @xmath1 be an arbitrary number. let @xmath164 be the closure in the norm of the space @xmath158 of the open set @xmath165 of functions @xmath166 defined as @xmath167 * minimization problem*. _ minimize the functional _ @xmath168 _ _ on the set _ _ @xmath169 * remark 3.1*. the analytical part of this paper below is dedicated to this minimization problem. since we deal with complex valued functions, we consider below @xmath170 as the functional with respect to the 2-d vector of real valued functions @xmath171 thus, even though we the consider complex conjugations below, this is done only for the convenience of writing. below @xmath172 $] is the scalar product in @xmath158. even though we use in ([3.16]) and ([3.17]) the functions @xmath173 @xmath174 it is always clear from the context below what do we actually mean in each particular case : the first component of @xmath175 of the vector function @xmath166 or the above functions @xmath176 The global strict convexity of @xmath177 theorem 4.1 is the main analytical result of this paper. * theorem 4.1*. _ assume that conditions of theorem 3.1 are satisfied. then the functional _ @xmath170 _ _ has the frecht derivative _ _ @xmath178__for all _ _ @xmath179 _ also, there exists a sufficiently large number _ @xmath180 }, r\right) > 1 $] _ depending only on listed parameters and a generic constant _ @xmath66 _ _, such that for all _ _ @xmath181__the functional _ _ @xmath170 _ _ is strictly convex on _ _ @xmath182 _ _ i.e. for all _ _ @xmath183 _ _ _ _ @xmath184 * proof. * everywhere below in this paper @xmath185 }, r\right) > 0 $] denotes different constants depending only on listed parameters. since conditions of theorem 3.1 are satisfied, then by ([3.15])@xmath186 } \leq \left\vert v^{\ast } \right\vert _ { c^{1}\left [0,1\right] } + c_{1}\delta \leq c_{2}. \label{3.220}\]]let @xmath187 where @xmath188 then ([3.130]), ([3.19]) and ([3.21]) imply that @xmath189 } ^{2}dk\leq c_{2}. \label{3.23}\]]using ([3.23]), we obtain@xmath190@xmath191 } ^{2}dk\leq c_{2}.\]] we use the formula @xmath192where @xmath193 is the complex conjugate of @xmath194. denote @xmath195consider functions @xmath196 defined as@xmath197first, using ([3.17]) and ([3.25]), we single out in @xmath198 the part, which is linear with respect to the vector function @xmath199. then@xmath200@xmath201@xmath202by ([3.25]) @xmath203 h^{\prime } \right\ } \overline{a}\]]@xmath204 \int\limits_{k}^{\overline{k}}h^{\prime } \left (x,\tau \right) d\tau \cdot \overline{a } \label{3.261}\]]@xmath205 \overline{a}.\]]hence,@xmath206 \overline{l\left (p\right) } % h^{\prime } \]]@xmath207 \overline{l\left (p\right) } \int\limits_{k}^{\overline{k}}h^{\prime } \left (x,\tau \right) d\tau \label{3.27}\]]@xmath208where @xmath209 depends nonlinearly on the vector function @xmath210. also, by ([3.220])-([3.24]) and the cauchy - schwarz inequality@xmath211to explain the presence of the multiplier 1/2 " at @xmath212 in (3.28), we note that it follows from ([3.260]) that the term @xmath213 in ([3.261]) contains the term @xmath214 which is included in ([3.27]) already, as well as terms@xmath215we now show how do we estimate the third term in ([3.280]), since estimates of two other terms are simpler. we use the so - called cauchy - schwarz inequality with @xmath216@xmath217where @xmath218 is the scalar product in @xmath219 hence,@xmath220thus, choosing appropriate numbers @xmath221 we obtain the term @xmath222 in (3.28). the second term in the right hand side of ([3.28]) is obtained similarly. analogously, using ([3.250])-([3.25]), we obtain@xmath223 h^{\prime } \right\ } } \cdot l\left (p\right)\]]@xmath224 \int\limits_{k}^{\overline{k}}h^{\prime } \left (x,\tau \right) d\tau } \cdot l\left (p\right) \label{3.29}\]]@xmath225where @xmath226 depends nonlinearly on the vector function @xmath227 and similarly with ([3.28])@xmath228 it is clear from ([3.25]), ([3.27])-([3.30]) that the linear with respect to the vector function @xmath227 part of @xmath229 consists of the sum of the first two lines of ([3.27]) with the first two lines of ([3.29]). we denote this linear part as @xmath230 then @xmath231thus, using ([3.20]) and ([3.25]), we obtain@xmath232@xmath233consider the expression @xmath234@xmath235it follows from ([3.17]), ([3.220]), ([3.27]) and ([3.29]) that @xmath236 is a bounded linear functional. hence, by riesz theorem, there exists unique element @xmath237 such that @xmath238, \forall h\in h. \label{3.33}\]]it follows from ([3.28]) and ([3.30])-([3.33]) that@xmath239 = o\left (\left\vert h\right\vert _ { h}^{2}\right).\]]thus, the frecht derivative @xmath240 of the functional @xmath170 at the point @xmath241 exists and @xmath242 note that @xmath243. \label{3.35}\]]hence, using ([3.28]), ([3.30])-([3.34]) and lemma 3.1, we obtain@xmath244@xmath245@xmath246@xmath247\]]@xmath248choose the number @xmath249 }, r\right) > 1 $] so large that @xmath250 then, using ([3.35]) and ([3.36]), we obtain with a new generic constant @xmath66 for all @xmath181@xmath251 Global convergence of the gradient projection method using theorem 4.1, we establish in this section the global convergence of the gradient projection method of the minimization of the functional @xmath252 as to some other versions of the gradient method, they will be discussed in follow up publications. first, we need to prove the lipschitz continuity of the functional @xmath254 with respect to @xmath241. * theorem 5.1*. _ let conditions of theorem 3.1 hold. then the functional _ @xmath178 _ _ is lipschitz continuous on the closed ball _ _ @xmath169 _ _ in other words,__@xmath255 * proof*. consider, for example the first line of ([3.27]) for @xmath256 and denote it @xmath257 we define @xmath258 similarly. both these expressions are linear with respect to @xmath259 denote @xmath260 we have@xmath261@xmath262 \label{5.2}\]]@xmath263 h^{\prime }.\]]it is clear from ([3.17]) that @xmath264 hence, using ([3.35]), ([5.2]) and cauchy - schwarz inequality, we obtain@xmath265@xmath266the rest of the proof of ([5.1]) is similar. @xmath74 theorem 5.2 claims the existence and uniqueness of the minimizer of the functional @xmath170 on the set @xmath268 * theorem 5.2*. _ let conditions of theorem 4.1 hold. then for every _ _ there exists unique minimizer _ _ @xmath269 _ _ of the functional _ _ @xmath170 _ _ on the set _ _ @xmath169 _ _ furthermore,__@xmath270 \geq 0,\forall y\in \overline{b\left (r\right) }. \label{5.3}\]] * proof*. this theorem follows immediately from the above theorem 4.1 and lemma 2.1 of @xcite. @xmath74 let @xmath271 be the operator of the projection of the space @xmath158 on the closed ball @xmath272 let @xmath273 and let @xmath274 be an arbitrary point of @xmath164. consider the sequence of the gradient projection method,@xmath275 * theorem 5.3. * _ let conditions of theorem 4.1 hold. then for every _ @xmath181 _ _ there exists a sufficiently small number _ _ @xmath276 }, \left\vert p_{1}\right\vert _ { c\left [\underline{k},\overline{k}\right] }, r,\lambda \right) \in \left (0,1\right) $] _ and a number _ @xmath277 _ _ such that for every _ _ @xmath278 _ _ the sequence ([5.4]) converges to the unique minimizer _ _ @xmath279 _ _ of the functional _ _ @xmath280 _ _ on the set _ _ @xmath281 _ _ and _ _ @xmath282 * proof*. this theorem follows immediately from the above theorem 4.1 and theorem 2.1 of @xcite. @xmath74 as it was pointed out in section 3.2, following one of the main concepts of the regularization theory @xcite, we assume the existence of the exact solution @xmath133 of our imsp with the exact, i.e. noiseless, data @xmath283 in ([2.8]). below the superscript @xmath284 " denotes quantities generated by @xmath285 the level of the error @xmath142 was introduced in our data in ([3.14]). in particular, it follows from (3.7), ([3.70]) and ([3.14]) that@xmath286 }, \left\vert p_{1}-p_{1}^{\ast } \right\vert _ { c\left [\underline{k},% \overline{k}\right] } \leq c_{3}\delta, \label{5.6}\]]where the number @xmath287 depends only on listed parameters. thus, in this section we show that the gradient projection method delivers points in a small neighborhood of the function @xmath288 and, therefore, of the function @xmath289 the size of this neighborhood is proportional to @xmath290 it is convenient to indicate in this section dependencies of the functional @xmath291 from @xmath292 and @xmath293 hence we write in this section @xmath294 * theorem 5.4*. _ assume that conditions of theorem 4.1 hold. also, let the exact function _ @xmath295 _ _ then the following accuracy estimates hold for each _ _ @xmath181@xmath296@xmath297__where _ _ @xmath279 _ _ is the minimizer of the functional _ _ @xmath298 _ _, which is guaranteed by theorem 5.2 and _ _ @xmath299__is the corresponding reconstructed coefficient (section 3.1). in addition, let _ _ _ be the sequence ([5.4]) of the gradient projection method, where _ _ @xmath301 _ _ is an arbitrary point of _ _ @xmath302 _ _ and numbers _ _ @xmath303 _ _, _ _ @xmath304 _ _ and _ _ @xmath305 _ _ are the same as in theorem 5.3. _ be the corresponding sequence of reconstructed coefficients (section 3.1). then the following estimates hold__@xmath307@xmath308 * proof*. obviously@xmath309using ([3.15]), ([3.170]), ([3.17]), ([5.6]) and ([5.11]), we obtain@xmath310@xmath311 j_{\lambda } \left (p^{\ast } p_{0}^{\ast }, p_{1}^{\ast }, v^{\ast } \right) \label{5.12}\]]@xmath312 } ^{2}+\left\vert p_{1}-p_{1}^{\ast } \right\vert _ { c% \left [\underline{k},\overline{k}\right] } ^{2}+\left\vert v_{\alpha \left (\delta \right) } -v^{\ast } \right\vert _ { c^{1}\left [0,1\right] } ^{2}\right) \leq c_{2}\delta ^{2}.\]]by theorems 4.1 and 5.2@xmath313 \label{5.13}\]]@xmath314by ([5.3]) and ([5.12])@xmath315 \leq 0,j_{\lambda } \left (p^{\ast }, p_{0},p_{1},v_{\alpha \left (\delta \right) } \right) \leq c_{2}\delta ^{2}.\]] hence, ([5.13]) implies ([5.7]). since the function @xmath316 is obtained from the functions @xmath279 and @xmath317 as described in the end of section 3.1, then ([5.7]) implies ([5.8]). next, ([5.9]) follows from (5.5) and ([5.7]). finally, ([5.10]) follows from that procedure of section 3.1 and ([5.8]). @xmath74 * remark 5.1*. therefore, theorem 5.4 ensures the global convergence property of our method, see the definition in introduction. Numerical studies since the theory of sections 3 - 5 is the main focus of this paper, we omit some details of the numerical implementation, both in this and next sections. we now briefly describe our numerical steps for both computationally simulated and experimental data. to minimize the functional @xmath318 we have written the derivatives of the operator @xmath319 via finite differences with the step size @xmath320. also, we have written integrals with respect to @xmath26 in discrete forms, using the trapezoidal rule, with the step size @xmath321 the differentiation of the data @xmath28 with respect to @xmath26, which we need in our method (see ([3.70])), was performed using finite differences with the step size @xmath321 we have not observed any instabilities after the differentiation, probably because the number @xmath322 is not too small. similar conclusions were drawn in works @xcite where similar differentiations were performed, including cases with experimental data next, we have minimized the corresponding discrete version of @xmath168 with respect to the values of the function @xmath323 at those grid points. initially we have used the gradient projection method. however, we have observed in our computations that the regular and simpler gradient method provides practically the same results. hence, all computational results below are obtained via the gradient method. the starting point of this method was @xmath324 and a specific ball @xmath325 was not used. the latter means that computational results are less pessimistic ones than our theory is. the step size of the gradient method @xmath326 was used. we have observed that this step size is the optimal one for our computations. the computations were stopped after 5000 iterations. based on our above theory, we have developed the following algorithm : 1. find the tail function @xmath327 via minimizing the functional ([3.12]). 2. minimize the functional ([3.20]). let @xmath328 be its minimizer. 3. calculate the function @xmath329 see ([3.16]) and (3.170). 4. compute @xmath330 5. compute the function @xmath331 see (2.1) and ([3.5]), @xmath332 in this algorithm, unlike the previous globally convergent algorithms, @xcite, we do not need to update the tail function @xmath327. first, we reconstruct the spatially distributed dielectric constant from computationally simulated data, which is generated by solving the problem ([2.4]), ([2.6]) via the 1-d analog of the lippmann - schwinger equation @xcite : @xmath333here and thereafter, we have use @xmath334 in all our computations. keeping in mind our desired application to imaging of flash explosive - like targets, we have chosen in our numerical experiments the true test coefficient @xmath335 as : @xmath336where @xmath337 is the location of the center of our target of interest and @xmath338 is its width. hence, the inclusion / background contrast in ([6.0]) is 7. for our numerical experiments we have chosen in ([6.0]) @xmath339 figure [fig : u0_abs] displays a typical behavior of the modulus of the simulated data @xmath340 at the measurement point @xmath341. one can observe that @xmath342 next, @xmath340 changes too rapidly for @xmath343 hence, the interval @xmath344 $] seems to be the optimal one, and we indeed observed this in our computations. hence, we choose for our study @xmath345 and @xmath346. we note that even though the above theory of the choice of the tail function @xmath327 works only for sufficiently large values of @xmath347 the notion sufficiently large " is relative, see, e.g. ([6.20]). besides, it is clear from section 7 that we actually work in the gigahertz range of frequencies, and this can be considered as the range of large frequencies in physics., scaledwidth=40.0%] next, having the values of @xmath348, we calculate the function @xmath349 in ([2.8]) and introduce the random noise in this function @xmath350where @xmath351 and @xmath352 are random numbers, uniformly distributed on @xmath353. the next important question is about the choice of an optimal parameter @xmath354 indeed, even though theorem 4.1 says that the functional @xmath170 is strictly convex on the closed ball @xmath164 for all @xmath355 in fact, the larger @xmath356 is, the less is the influence on @xmath168 of those points @xmath357 which are relatively far from the point @xmath358 where the data are given. hence, we need to choose such a value of @xmath359 which would provide us satisfactory images of inclusions, whose centers @xmath337 are as in ([6.2]) : @xmath360 $]. let @xmath361 be the discrete @xmath362 norm of the gradient of the above described discrete version of the functional @xmath363 figure [fig : gnorm] displays the dependencies of this norm on the number of iteration of the gradient method for different values of @xmath356. we have observed in our computations that these dependencies are very similar for targets satisfying ([6.0]), ([6.2]) with different values of target / background contrasts. one can see that the process diverges at @xmath364, which is to be expected, since convexity of @xmath365 is not guaranteed. also, we observe that the larger @xmath366 is, the faster the process converges. we have found that the optimal value of @xmath356 for targets satisfying ([6.2]) is @xmath367. we also apply a post - processing procedure after step 5 of the above algorithm. more precisely, we smooth out the function @xmath368 ([c]) using a simple averaging procedure over two neighboring grid points. next, the resulting function @xmath369 is truncated as @xmath370the function @xmath371 in ([6.1]) is considered as our reconstructed coefficient @xmath372 norm of the gradient of the functional @xmath373 for different @xmath374,scaledwidth=40.0%] the computational results @xmath375 for different values of @xmath337 are shown in figure [fig : results]. one can see that the proposed algorithm accurately reconstructs both locations and values of the coefficient @xmath335. similar accuracy was obtained for other target / background contrasts in ([6.0]) varying from 2 to 10. Numerical results for experimental data we use here the same experimental data as ones used in klibloc, kuzh, ieee, where these data were treated by the tail functions method. thus, it is worth to test the new method of this paper on the same data set. in @xcite the wave propagation process was modeled by a 1-d hyperbolic equation, the laplace transform with respect to time was applied to the solution of this equation and then the tail functions method was applied to the corresponding imsp. in @xcite the process was modeled by imsp ([2.8]) and the tail functions method was applied to this imsp. the data in @xcite and in @xcite were obtained after applying laplace and fourier transforms respectively to the original time dependent data. we have observed a substantial mismatch of amplitudes between computationally simulated and experimental data. hence, we have calibrated experimental data here via multiplying them by the calibration factor @xmath376 just as in @xcite. (11,0.3) (10.5,2) (7,2.5) arc (270:200:0.5) (5.5,4.5) arc (20:90:0.8) (-1,5) arc (90:180:0.8) (-3,0.3) (-2.7,0) (-1.2,0) arc (180:0:1.2) (6.8,0) arc (180:0:1.2) (10.7,0) (11,0.3) ; (0,0) circle (1) ; (8,0) circle (1) ; (4.3,5) (4,6) (3.7,5) (4.3,5) ; (5,6.5) (5,7.5) (4,7.3) (2, 6.3) (2, 5.7) (3,5.5) (5,6.5) ; (3,5.5) (3,6.5) (5,7.5) ; (3,6.5) (2,6.3) ; (3.5,6.2) circle (0.2) ; (4,6.5) circle (0.2) ; (4.5,6.8) circle (0.2) ; (-5,-1) rectangle (30,-6) ; (20,-1.5) rectangle (25,-3) ; at (22.5,-2.25) target ; (5.5,6) arc (300:350:0.7) ; (7.5,5.0) arc (300:350:1.5) ; (9.5,4) arc (300:350:2.5) ; (11.5,3) arc (300:350:3.5) ; (13.5,2) arc (300:350:4.5) ; (15.5,1) arc (300:350:5.5) ; (17.5,0) arc (300:350:6.5) ; (19.5,-1) arc (300:350:7.5) ; our experimental data were collected in the field by the forward looking radar of the us army research laboratory @xcite. the schematic diagram of data collection is presented on figure [fig : setup]. the device has two sources placed on the top of a car. sources emit pulses. the device also has 16 detectors. detectors measure backscattering time resolved signal, which is actually the voltage. pulses of only one component of the electric field are emitted and the same component is measured on those detectors. the time step size of measurements is 0.133 nanosecond and the maximal amplitudes of the measured signal are seen about 2 nanoseconds, see figure 5. since 1 nanosecond corresponds to the frequency of 1 gigahertz @xcite, then the corresponding frequency range is in gigahertz, which are considered as high frequencies in physics. the car moves and the time dependent backscattering signal is measured on distances from 20 to 8 meters from the target of interest. the collected signals are averaged. users know horizontal coordinates of each target with a very good precision : to do this, the ground positioning system is used. two kinds of targets were tested : ones located in air and ones buried on the depth of a few centimeters in the ground. . the horizontal axis is time in nanoseconds.,scaledwidth=40.0%] while it is assumed both in ([2.1]) and ([6.1]) that @xmath377 we had one target buried in the ground, in which @xmath378 this target was a plastic cylinder. it was shown on page 2944 of @xcite that, using the original time dependent date, one can figure out that inside the target @xmath379 hence, in this case we replace ([c]) and ([6.1]) with@xmath380@xmath381 suppose that a target occupies a subinterval @xmath382 in fact, we estimate here the ratio of dielectric constants of targets and backgrounds for @xmath383. thus, actually our computed function @xmath384 in ([6.1]) and ([7.2]) is an estimate of the function @xmath385@xmath386where @xmath387 is the spatially distributed dielectric constant of that target. using ([6.1]), ([7.20]), (7.2) and ([7.3]), we define the computed target / background contrast in the dielectric constant as@xmath388, \\ \min c_{comp}\left (x\right), \text { if } c_{comp}\left (x\right) \leq 1,\forall x\in \left [0,1\right].% \end{array}% \right. \label{7.4}\]]finally, we introduce the number @xmath389 which is our estimate of the dielectric constant of a target, @xmath390 we have chosen the interval @xmath391 $] as @xmath392 = \left [\underline{k},\overline{k}\right]. \label{7.1}\]]the considerations for the choice ([7.1]) were similar with ones for the case of simulated data in section 6.2. we had experimental data for total five targets. the background was air in the case of targets placed in air with @xmath393 and it was sand with @xmath394 $] @xcite in the case of buried targets. two targets, bush and wood stake, were placed in air and three targets, metal box, metal cylinder and plastic cylinder, were buried in sand. figures [fig : exp_res] display some samples of calculated images of targets. dielectric constants of targets were not measured in experiments. so, the maximum what we can do at this point is to compare our computed values of @xmath395 with published ones. this is done in table [tab1], in which @xmath396 is a published value. as to the metallic targets, it was established numerically in @xcite that they can be approximated as dielectric targets with large values of the dielectric constant,@xmath397.\]]published values of dielectric constants of sand, wood and plastic can be found in @xcite. as to the case when the target was a bush, we took the interval of published values from @xcite. bush was the most challenging target to image. this is because bush is obviously a significantly heterogeneous target..summary of estimated dielectric constants @xmath398. [cols="<,^,^,^,^,^",options="header ",] for the engineering part of this team of coauthors (ln and as), the depth of burial of a target is not of an interest here since all depths are a few centimeters. it is also clear that it is impossible to figure out the shape of the target, given so limited information content. on the other hand, the most valuable piece of the information for ln and as is in estimates of the dielectric constants of targets. therefore, table [tab1] is the most interesting piece of the information from the engineering standpoint. indeed, one can see in this table that values of estimated dielectric constants @xmath398 are always within limits of @xmath399 as it was pointed out in section 1, these estimates, even if not perfectly accurate, can be potentially very useful for the quite important goal of reducing the false alarm rate. this indicates that the technique of the current paper might potentially be quite valuable for the goal of an improvement of the false alarm rate. the above results inspire ln and as to measure dielectric constants of targets in the future experiments. our team plans to treat those future experimental data by the numerical method of this publication. Concluding remarks we have developed a new globally convergent numerical method for the 1-d inverse medium scattering problem ([2.8]). unlike the tail function method, the one of this paper does not impose the smallness condition on the size of the interval @xmath391 $] of wave numbers. the method is based on the construction of a weighted cost functional with the carleman weight function in it. the main new theoretical result of this paper is theorem 4.1, which claims the strict convexity of this functional on any closed ball @xmath400 for any radius @xmath1, as long as the parameter @xmath401 of this functional is chosen appropriately. global convergence of the gradient method of the minimization of this functional to the exact solution is proved. numerical testing of this method on both computationally simulated and experimental data shows good results. h. ammari, y. t. chow, and j. zou, _ phased and phaseless domain reconstructions in the inverse scattering problem via scattering coefficients _, siam journal on applied mathematics, 76 (2016), pp. 10001030. a. b. bakushinskii, m. v. klibanov, and n. a. koshev, _ carleman weight functions for a globally convergent numerical method for ill - posed cauchy problems for some quasilinear pdes _, nonlinear analysis : real world applications, 34 (2017), pp. 201224. m. v. klibanov, n. a. koshev, j. li, and a. g. yagola, _ numerical solution of an ill - posed cauchy problem for a quasilinear parabolic equation using a carleman weight function _, journal of inverse and ill - posed problems, 24 (2016), pp. 761776. m. v. klibanov, d.- nguyen, l. h. nguyen, and h. liu, _ a globally convergent numerical method for a 3d coefficient inverse problem with a single measurement of multi - frequency data _, (2016), https://arxiv.org/abs/1612.04014. m. v. klibanov, l. h. nguyen, a. sullivan, and l. nguyen, _ a globally convergent numerical method for a 1-d inverse medium problem with experimental data _, inverse problems and imaging, 10 (2016), pp. 10571085. a. v. kuzhuget, l. beilina, m. v. klibanov, a. sullivan, l. nguyen, and m. a. fiddy, _ blind backscattering experimental data collected in the field and an approximately globally convergent inverse algorithm _, inverse problems, 28 (2012), p. 095007. a. v. kuzhuget, l. beilina, m. v. klibanov, a. sullivan, l. nguyen, and m. a. fiddy, _ quantitative image recovery from measured blind backscattered data using a globally convergent inverse method _, ieee transactions on geoscience and remote sensing, 51 (2013), pp. 29372948. nguyen, m. v. klibanov, l. h. nguyen, a. e. kolesov, m. a. fiddy, and h. liu, _ numerical solution of a coefficient inverse problem with multi - frequency experimental raw data by a globally convergent algorithm _, (2016), https://arxiv.org/abs/1609.03102. l. nguyen, d. wong, m. ressler, f. koenig, b. stanton, g. smith, j. sichina, and k. kappra, _ obstacle avoidance and concealed target detection using the army research lab ultra - wideband synchronous impulse reconstruction (uwb sire) forward imaging radar _, 2007, p. 65530h. m. sini and n. t. thnh, _ regularized recursive newton - type methods for inverse scattering problems using multifrequency measurements _, esaim : mathematical modelling and numerical analysis, 49 (2015), pp. 459480. n. t. thnh, l. beilina, m. v. klibanov, and m. a. fiddy, _ imaging of buried objects from experimental backscattering time - dependent measurements using a globally convergent inverse algorithm _, siam journal on imaging sciences, 8 (2015), pp. 757786. _ dielectric constant table _, https://www.honeywellprocess.com/library/marketing/tech-specs/dielectric constant table.pdf[https://www.honeywellprocess.com/library/marketing/tech-specs/dielectric constant table.pdf]

a new numerical method is proposed for a 1-d inverse medium scattering problem with multi - frequency data. this method is based on the construction of a weighted cost functional. the weight is a carleman weight function (cwf). in other words, this is the function, which is present in the carleman estimate for the undelying differential operator. the presence of the cwf makes this functional strictly convex on any a priori chosen ball with the center at @xmath0 in an appropriate hilbert space. convergence of the gradient minimization method to the exact solution starting from any point of that ball is proven. computational results for both computationally simulated and experimental data show a good accuracy of this method. * key words * : global convergence, coefficient inverse problem, multi - frequency data, carleman weight function * 2010 mathematics subject classification : * 35r30.

1703.08158

Introduction the @xmath3-transformation has been extensively studied since the seminal paper of rnyi in 1957. there is a huge amount of literature on the map itself and on the associated symbolic dynamics. over the past decade, people became interested in the @xmath2-transformation, changing the slope of the map from positive to negative. various studies have focused on the similarities and differences between the two maps from several points of view. this paper fits in this last line of research. the paper compares two aspects of the @xmath2-shift to the @xmath3-shift. for the @xmath3-shift it is known that a prefix code always exists. the paper first investigates whether or not the same is true for the @xmath2-shift. by @xmath1-shift (respectively @xmath3-shift) we understand the closure of the set of expansions in base @xmath14 (respectively @xmath15). the conclusion is stated in theorem[t1], which says that a prefix code exists in the negative case only under certain conditions, namely if and only if @xmath3 is bigger than the golden ratio and the orbit of the left endpoint of the domain of the @xmath2-transformation is not periodic with odd period. it turns out that the discovered prefix codes are recurrent. even though the codes can be complicated, the zeta functions apparently have a very simple form (see theorem [t2]) and it can be noted the similarities with that of the positive parameter determined in @xcite. rnyi introduced the @xmath15-expansion of positive real numbers in @xcite : for a fixed real @xmath16, all non - negative real number @xmath17 has one representation in base @xmath6. he defined the @xmath15-transformation @xmath18 from @xmath19 into @xmath19 by @xmath20 where @xmath21 denotes the largest integer less than @xmath17. we can find a sequence of positive integers @xmath22 such that @xmath23 where the integer @xmath24 is taken in @xmath25, @xmath26 and @xmath27 is the smallest positive integer for which @xmath28 belongs to the interval @xmath19. various topics about @xmath15-expansion have been studied. let @xmath29 be an integer strictly bigger than @xmath30. it is well - known that any number can be represented without a sign in base @xmath31 thanks to digits of the alphabet @xmath32. in 2009, s. ito and t. sadahiro in @xcite generalized this approach for all negative base (integer or not). they defined a @xmath2-transformation @xmath33, the map from the interval @xmath34 into itself such that @xmath35 the expansion in base @xmath4 of a real @xmath17 (denoted by @xmath36) is given by the following algorithm : * if @xmath17 belongs to @xmath37, @xmath38 where @xmath39 * if @xmath40, one finds the smallest integer @xmath27 for which one has @xmath41. in this case, the expansion is @xmath42, where @xmath43 in both cases, @xmath44. if there is no mixup, we often denote @xmath45 by @xmath46. let @xmath47 be an alphabet. consider two words on @xmath48, @xmath49 and @xmath50. we denote by @xmath51, the relation such that : @xmath52 if there exists an integer @xmath53 such that @xmath54 and @xmath55. the relation @xmath56 is called * alternating lexicographic order*. this definition can be extended on infinite words on @xmath48. in the following, we set @xmath57 with @xmath58, @xmath59, @xmath60 and @xmath61 where @xmath62. the * @xmath1-shift * is the sub - shift @xmath63 on the alphabet @xmath48 such that @xmath64 Coded negative beta-shift let us start by giving the definitions of the main terms used throughout this paper. let @xmath65 be an alphabet. a word on @xmath66 is a concatenation of elements of @xmath66. let @xmath67 be a word on @xmath48, we call sub - word (or factor) of @xmath68 all word which appears in @xmath68. a language @xmath69 on @xmath66 is a set of words on @xmath66 (or the set of finite sequences on @xmath66). a language @xmath70 is extendable if for all word @xmath71 in @xmath70, there exist two letters @xmath72 and @xmath29 in @xmath48 such that @xmath73 belongs to @xmath70. it is said transitive if @xmath74, there exists @xmath75 such that @xmath76. let @xmath77 be a finite alphabet. one endows @xmath78 (or @xmath79) with the topology product of the discrete topology on @xmath66. let @xmath80 be the map from @xmath81 (or @xmath82) into itself defined by @xmath83. the closed @xmath84-invariant subsets of @xmath78 are called sub - shifts. let @xmath85 be a sub - shift on an alphabet @xmath66. the pair @xmath86 is called symbolic dynamical system. @xmath87 will be denoted by @xmath88 whenever there is no risk of mixup. a language @xmath89 of a dynamical system @xmath88 is the set of factors of words of @xmath88. the system is transitive if its language is transitive. a code @xmath90 on an alphabet @xmath48 is a language such that, for any equality @xmath91 with @xmath92 one has @xmath93 and @xmath94. let @xmath95 be a language on @xmath66. in the following, @xmath96 denotes the concatenations of words of @xmath95. we call prefix code a language @xmath97 for which no word is the beginning of another. @xmath98 where @xmath99 is the empty word. if in @xmath97, no word ends by another, then the language @xmath97 is a suffix code. the symbolic dynamical system @xmath88 is said coded by the prefix code @xmath100 if @xmath101 is the set of factors of words contained in @xmath102. let @xmath70 be a language on an alphabet @xmath48. the radius @xmath103 of the power series @xmath104 is called radius of convergence of @xmath70. a prefix code @xmath97 is said recurrent positive if @xmath105 let @xmath6 be a real number strictly bigger than 1 and @xmath106 the associated @xmath3-shift. let @xmath107 be the expansion of 1 in base @xmath6. @xmath108 is coded by the prefix code @xmath109 defined as follows : @xmath110 hence, all @xmath6-shifts are coded. it is one of the differences between @xmath15-shifts and @xmath1-shifts. for a negative parameter @xmath111, one of the natural and important question is whether the @xmath2-shift is coded. in this section, we shall provide some contributions to this question. in fact, the @xmath1-shifts, as defined above are not always coded. furthermore, when it is coded, to find an appropriate code for its language can be difficult according to the complexity of the expansion @xmath112. we distinguish two cases : for all @xmath113, @xmath114 (the code is simple) and @xmath115 for some @xmath113 (in this case, it is not easy to determine a language which codes the @xmath2-shift). [t1] let @xmath6 be a real number strictly bigger than 1. the associated @xmath2-shift @xmath116 is coded if only if @xmath6 is bigger than or equal to the golden ratio and @xmath112 is not periodic with odd period. [lem1] let @xmath117 and @xmath118 the @xmath1-expansion of @xmath5. if for all integer @xmath113, @xmath119, then @xmath6 is bigger than or equal to the golden ratio. let @xmath120 be the golden ratio. to prove this lemma, it is enough to determine the @xmath121-expansion of @xmath122. in fact, if we consider two real numbers @xmath6 and @xmath123 strictly bigger than 1 then, @xmath124 we obtain the equivalence above thanks to lemma 3 of @xcite. @xmath125 we assume that @xmath126 and we set @xmath127. then, @xmath128 that means there exists @xmath27 such that @xmath129 and @xmath130. thus, @xmath131 and @xmath27 is even. so, there exists @xmath132 such that @xmath133 and @xmath134. thus, if @xmath135 for all @xmath113, @xmath136. [prop1] let @xmath6 be a real number strictly bigger than 1. we denote by @xmath116 the associated @xmath2-shift, @xmath118 the @xmath1-expansion of @xmath137. if @xmath118 is periodic with odd period or @xmath138 then, @xmath116 is not transitive. * assume @xmath138. by lemma[lem1], there exists @xmath132 such that @xmath139 in the admissible words, the possible longest sequence of zeros has length @xmath140. @xmath141 then, we can not add at right of 1 a long sequence of zeros. thus, @xmath116 is not transitive for @xmath126. * assume that @xmath142. consider a word @xmath143 such that @xmath144 then, @xmath145 there exists an integer @xmath146, @xmath147 such that @xmath148 for @xmath149 and @xmath150. that is @xmath151. in other words, @xmath152 in fact, @xmath153 implies that @xmath154 if @xmath155 is even and @xmath156 if @xmath155 is odd. + but, @xmath157. thus, @xmath158. hence, it follows that @xmath159. thus, for all @xmath160 of the form @xmath161, with @xmath162 (we have @xmath163) there does not exist @xmath164 such that @xmath165. this implies that @xmath116 is not transitive. it is well - known that all coded system is transitive (see @xcite). that is, a non transitive system can not be coded. thus, the previous proposition allows us to observe that for all @xmath15 such that @xmath166 is periodic with odd period or @xmath6 less than the golden ratio @xmath120, @xmath167 is not a coded system. we denote by @xmath168 the set of words @xmath169, @xmath170, @xmath171 the concatenations of such words and @xmath172 the admissible words of @xmath171. @xmath173 let @xmath174, @xmath175, @xmath176 and @xmath97 be the sets of words defined by : @xmath177 on one hand, if @xmath6 is less than or equal to the golden ratio, @xmath178 ; however if @xmath179, by construction, @xmath97 is a prefix code on the alphabet @xmath47. one essential property of @xmath97 is the fact that after a word of this language, we can add any admissible sequence. so, we distinguish two types of admissible words in @xmath180 : the ones starting by a word of @xmath97 and those starting by a word of @xmath181 and ending by a word of the type @xmath182. @xmath183 * if for all @xmath184, @xmath185 then, @xmath186 ; that is all concatenations of words of @xmath168 is admissible and @xmath187 * suppose there exists an integer @xmath113 such that @xmath115. so, the expansion @xmath112 is defined thanks to sequences of positive integers @xmath188 (increasing) and @xmath189 such that : @xmath190 + in @xmath112, the word @xmath191 follows @xmath192. if @xmath193, @xmath112 is periodic with odd period. if @xmath194 is not periodic with odd period, @xmath195 satisfies both following conditions : @xmath196 and @xmath197. in this case, all concatenations of words of @xmath168 is not admissible. + let @xmath6 be the algebraic integer satisfying @xmath198 ; @xmath199 the sequence @xmath188 is finite : @xmath200, @xmath201, and thus we have @xmath202, @xmath203, and @xmath204. + we set @xmath205. @xmath206 is the set of concatenations of words of two types : @xmath207 for some integer @xmath113 (we suppose that @xmath208), and @xmath209 where @xmath210, @xmath211 satisfies and @xmath212. + moreover, @xmath213 with @xmath208. note that if @xmath214 for some integer @xmath113, we have both conditions @xmath215 and @xmath216. it follows that if @xmath217, we have @xmath218 thus, the words of @xmath97 are of the form @xmath219 where @xmath220 satisfies or and @xmath221 or @xmath222 with @xmath223, @xmath224 and @xmath225. [r1] from theorem 3 of @xcite, if @xmath226 (that is @xmath227), all sequence between @xmath228 and @xmath229 can not be the expansion of @xmath230, for some @xmath0 except @xmath231. then, there exists @xmath232 such that @xmath233. indeed, @xmath234 \rightarrow (d_i)_{i \geq 1 } \prec d_1\overline{(d_1 - 1)0}.\]] so, we can find @xmath235 such that @xmath236 or @xmath237 belongs to @xmath238. [p2] if @xmath239 and @xmath118 is not periodic with odd period, then for all @xmath27, @xmath240. from remark[r1], @xmath175 contains at least one word @xmath241 such that @xmath242. if @xmath119, @xmath243 for all @xmath244 and @xmath245 with @xmath233. thus, for all @xmath170, @xmath246 is the beginning of a word of @xmath97. we assume satisfied and @xmath247 is non - periodic with odd period. suppose that for all @xmath248, @xmath249 is not the beginning of a word of @xmath97. then, for all @xmath250, @xmath251 and @xmath252 are consecutive (with respect to the alternating order). that is @xmath253. note that there does not exist an integer @xmath53 such that @xmath254, otherwise @xmath255 for all @xmath256 with @xmath242. if @xmath258 is even, @xmath259, otherwise @xmath260 so @xmath261 moreover, @xmath262 implies that @xmath263. since @xmath264 it follows that, @xmath265 then, @xmath266. that is @xmath267 + 1 $]. if @xmath268 is odd, @xmath269, otherwise @xmath270 then, @xmath271 let @xmath272 and @xmath273 be the words defined by : @xmath274 it is easy to see that @xmath275 and @xmath273 are consecutive (with respect to the alternating order). moreover, @xmath118 is a concatenation of two words : @xmath275 and its successor @xmath273. @xmath276 since @xmath277 and @xmath278 is even, the minimum (with respect to the alternating order) of sequences @xmath279 over the alphabet @xmath280 is @xmath281 and the maximum is @xmath282. from proposition 9 and theorem 3 of @xcite, the unique sequence between @xmath281 and @xmath283 which is the @xmath2-expansion of @xmath284, for some @xmath285 is @xmath286 if @xmath287 or @xmath283 if @xmath288. this is absurd, since @xmath289 is supposed to be non - periodic with odd period and @xmath290 (@xmath194 starts by @xmath275). then the fact that for all @xmath291, @xmath246 is not the beginning of a word of @xmath97 is false. from proposition 9 of @xcite, when @xmath292, the sub - shift of infinite words for which all sub - words is bigger than @xmath293 and the sub - shift of infinite words for which all sub - words is bigger than @xmath282 have the same entropy. note @xmath294 the map from @xmath280 into @xmath295 defined by @xmath296 and @xmath297. the limit of non - periodic infinite words @xmath194 such that @xmath298 and @xmath299 for all @xmath300 (not necessary the @xmath2-expansion of @xmath301 for some @xmath226) is @xmath302. @xmath303 in fact, if @xmath304, then there exists @xmath305 such that @xmath306. if @xmath307, @xmath308, @xmath309. @xmath310 in this case, the @xmath2-expansion of @xmath311 is between @xmath312 and @xmath313. @xmath314 if we replace 1 by 2 and 0 by 1, we recognize a relative form of the sequence a026465 of the encyclopedia of sloane. its elements count the number of consecutive identical symbols in the thue - morse sequence @xmath315 defined by : @xmath316 [l2] let @xmath6 be a real number bigger than the golden ratio @xmath120 and @xmath317 non - periodic with odd period. then, the @xmath2-shift @xmath167 is coded by @xmath97. suppose @xmath285. then, @xmath97 is a prefix code. we want to show that @xmath318 where @xmath319 denotes the set of concatenations of words of @xmath97. from, @xmath320. we have to show that @xmath321. let @xmath322. if @xmath323, then, @xmath324 with @xmath325 and @xmath326. by construction of @xmath97, we have the following successive implications : @xmath327 hence the result. from the proof of proposition [p2], if @xmath179 and @xmath328, then @xmath329 and the word @xmath330 is an intransitive word. the @xmath2-shift, as defined in @xcite, is not coded, but contains a coded sub - shift : the dynamical system of words for which all sub - word is bigger (with respect to the alternating order) than @xmath331. moreover, both systems have the same entropy. if one considers the definition of the @xmath1-shift given in the introduction in the case where @xmath6 is integer, that is, the expansion of @xmath5 is periodic with period 1, one obtains @xmath332 but in this special case, the definition given by ito and sadahiro is @xmath333 in fact, all sequence @xmath334 of @xmath167 satisfies, for all @xmath27 @xmath335 with @xmath336 and @xmath337. using lemma 6 of @xcite, @xmath338 and @xmath339. in this form, @xmath167 is coded. generally, the symbolic dynamical system @xmath167 contains a sub - shift coded by a prefix code. this sub - shift defines the support of the maximal entropy measure. in the particular case for which @xmath112 is periodic with odd period @xmath340, we introduce the sub - shift @xmath341 defined by : @xmath342 with @xmath343 using lemma 6 of @xcite, we see easily that @xmath344 in the special case for which @xmath112 is periodic with odd period @xmath340, @xmath341 contains the coded subset of the system. if @xmath345, one identifies the coded subset of the system by @xmath341. all real has a representation in @xmath341 since @xmath346 (see the proof of proposition 8 of @xcite). we can choose as @xmath2-shift, the set @xmath341 instead of @xmath116. in fact, the sequence @xmath347 plays the role of @xmath348 in base 10. for instance, consider a real number @xmath17 taken in the open interval @xmath349, we have @xmath350 now, we are ready to prove theorem [t1]. to prove theorem [t1], it is enough to use lemma [l2] and proposition [prop1]. throughout the rest of this paper, we focus our interest in the study of @xmath351 instead of @xmath116. we redefine the language @xmath97 using as sequence of reference @xmath352. in other words, if @xmath118 is periodic with odd period, we replace this sequence by @xmath347. also, we modify a little bit the sets @xmath353 and @xmath97. @xmath354 let @xmath6 be a real number bigger than 1. in the previous paragraph, we constructed a prefix code of the @xmath2-shift. but, if a code is recurrent positive, it is more interesting above all if the system is intrinsically ergodic with entropy @xmath355. we can say more about recurrent positive prefix code. start by giving the following definition : let @xmath211 be a symbolic dynamical system and @xmath315 a word of @xmath356. we call cylinder of support @xmath357 and we denote by @xmath358 $] the set of words @xmath359 such that @xmath360 so, in the one - side symbolic dynamical system, the cylinder @xmath361 $] is the set of words starting by @xmath315. consider a symbolic dynamical system @xmath211. in fact, the existence of a recurrent positive prefix code @xmath97 implies that there exists a probability measure on the space @xmath362, closure of the set of concatenations of words of the code @xmath97 (@xmath363). if @xmath355 denotes the entropy of @xmath364 endowed with the shift, this probability @xmath365 is defined by : @xmath366) = \frac{1}{\beta^{\vert x \vert } }, \text { for all $ x \in { \mathfrak{c}}$}.\]] @xmath367 $] ; for @xmath368, @xmath369, @xmath370 \cap [y] = \varnothing $]. thus, @xmath371) \\ & = \sum\limits_{x \in { \mathfrak{c } } } \frac{1}{\beta^{\vert x \vert } } \\ & = 1.\end{aligned}\]] the entropy @xmath372 of the probability @xmath373 is : @xmath374)\log \nu([x]) \\ & = \left (\sum\limits_{x\in { \mathfrak{c } } } \frac { \vert x \vert } { \beta^{\vert x \vert } } \right) \log \beta.\end{aligned}\]] this expression exists since @xmath375 when the code is recurrent positive. thus, the maximal entropy measure @xmath376 is given by : @xmath377 see for example @xcite and @xcite for more explanations. in subsection 2.2, we have seen that there exist two types of admissible words in @xmath378 : the ones starting by a word of @xmath97 and those starting by a word of @xmath379 and ending by a sequence of the form @xmath380. in fact, @xmath381 is partitioned into sub - languages coded by optional alphabets of words to be specified in the following. thus, admissible words of @xmath180 are concatenations of words of @xmath382 and (or) words of @xmath383 completed eventually at right by a string of the type @xmath384. * in the case @xmath385 for all @xmath386, we have two alphabets of words : @xmath353 and @xmath97 given respectively in and. note that in this case, the words of @xmath387 are factors of elements of @xmath97. that is @xmath388. * we assume satisfied. in this case, we know that @xmath389. if there is no ambiguity, we set @xmath390. + [rem2] the word @xmath391 over the alphabet @xmath392 is admissible if and only if @xmath393, with @xmath394. + we set @xmath395 @xmath396 @xmath397 @xmath398 + [corollary 1] for all @xmath399, @xmath400, @xmath401 with @xmath402, @xmath403 + it is enough to see that for @xmath404, for all @xmath405, @xmath406 if @xmath27 is odd, @xmath407 is even and then, @xmath408. + if @xmath27 is even, @xmath409 can be seen as a concatenation of words @xmath410 and @xmath411. so, @xmath412 + from remark [rem2], all concatenation of @xmath413 is not admissible. let construct suitable alphabets for the writing of admissible products of @xmath413. + [rem3] let @xmath414. from remark [rem2], @xmath210 with @xmath415. + if @xmath211 belongs to one of the alphabets, @xmath416 is admissible since all concatenations of words of the same alphabet are contained in @xmath381. thus, @xmath417. + we set @xmath418 and for all @xmath113, @xmath419 let @xmath420 be the set defined as follows : @xmath421 from remark [rem3], @xmath422. + [rem4] let @xmath423 and @xmath424, with @xmath425. then, @xmath426. now, we know the different sub - languages necessary to construct words of @xmath180 : @xmath97, @xmath427, @xmath420 for all @xmath428 if the corrected @xmath2-expansion of @xmath284 satisfies ; or @xmath97 and @xmath353 if does not hold. using the formal power series, the following result establishes the link between @xmath180 and its different sub - languages. [th2] let @xmath6 be a real number (@xmath429), @xmath180 the language of the corrected @xmath2-shift, @xmath430, @xmath100, @xmath431 and @xmath420 be the sets defined in,,, and respectively. in the meaning of the alternating order, we have the following relations : @xmath432 if occurs, or @xmath433 if @xmath434, @xmath435 and where @xmath347 is the corrected @xmath2-expansion of @xmath436. in the following, @xmath437, @xmath438, @xmath439 and @xmath440 denote respectively the number of words of length @xmath27 in @xmath180, @xmath97, @xmath431 (if hold) or @xmath441 (if there is no integer @xmath113 such that @xmath442) and the set @xmath443. note that a word of @xmath97 can be extended at right by any word of @xmath180. therefore, at right of a word of @xmath427 (if occurs for @xmath444) or a word of @xmath168 (if does not occur) can be added any word of the type @xmath445 (with @xmath446). indeed, from @xmath447 let explain a little bit the equation above. the coefficient of the formal power series @xmath448 count admissible strings starting by a word of @xmath97. denote by @xmath449 the formal power series for which the coefficients count the words of @xmath450 which start by @xmath451 with @xmath452. so, the coefficients of @xmath453 count all words of @xmath454. @xmath455 this equation means that in @xmath456, we have admissible strings starting by words of @xmath427 and those belonging to @xmath450. we set @xmath457 from corollary [corollary 1] and remark [rem4], @xmath458 and for all @xmath459, @xmath460 from, and, we have @xmath461 thus, to obtain, it is enough to multiply by @xmath462 and use. in the disk @xmath463 of center 0 and radius @xmath464 and assume @xmath347 not periodic with odd period, @xmath465 with @xmath466 when @xmath467 for all @xmath113. let @xmath468 be the number of words of length @xmath27 in @xmath381. from @xcite, we know that the formula for the factor complexity of the corrected @xmath1-shift (or the language @xmath378) is given by : @xmath469 with @xmath470, @xmath471 and @xmath472. the corollary above is due to theorem [th2] and the fact that in the sense of power formal series, @xmath473 the expansion @xmath474 is supposed to be periodic with odd period @xmath475. all word @xmath359 in the @xmath2-shift @xmath116 satisfies @xmath476 according to lemma 1 of @xcite, the formula for the factor complexity of the @xmath2-shift is : @xmath477 where @xmath478 denotes the number of words of length @xmath27 in the language of the @xmath2-shift. this implies that @xmath479 the least root in modulus of the left power series of is @xmath11. then, the support of the maximal entropy measure is coded by a recurrent positive code. from proposition [p2], @xmath100 codes the @xmath2-shift when @xmath6 is greater than the golden ration. that is, in this case, @xmath480 and @xmath481 for all integer @xmath184, and @xmath482. however, if @xmath6 is less than the golden ration, both inclusions can not hold. indeed, @xmath483 and then @xmath484. the support of the maximal entropy measure is coded by @xmath431 (in this case @xmath485) or by @xmath420 for some @xmath113 (and @xmath486). recall that the morphism @xmath487 on @xmath488 is given by @xmath489, @xmath490 and define the sequences @xmath491 and @xmath492 by @xmath493, @xmath494 and for @xmath495, @xmath496, @xmath497. from lemma 2 of @xcite, @xmath498 is odd and @xmath499 is even. in fact, @xmath500 moreover, note that there is no word between @xmath501 and @xmath502. indeed, from the definition of @xmath503, we observe easily that @xmath504 and @xmath505 let @xmath506 be the real number such that @xmath507 @xmath508 is the largest number satisfying : @xmath509 that is, @xmath508 is the largest root of @xmath510, where @xmath511. the sequence @xmath512 decreases until 1 (see proposition 5 and 6 of @xcite) and we have @xmath513. the following proposition gives us a writing of @xmath514 for @xmath6 less than the golden ratio. [prop7] let @xmath6 be a real number such that @xmath515. then, there exists a sequence of integers @xmath516 and @xmath517 such that @xmath518 or @xmath519 let @xmath6 be a real number and suppose @xmath515. since @xmath520 decreases until 1, there exists an integer @xmath27 such that @xmath521. so, @xmath522 this means @xmath523 it is easy to see that @xmath524 and there is no word between @xmath503 and @xmath525. in an infinite word, @xmath526 is followed by @xmath527 or by @xmath503. we obtain by using the fact that @xmath528. to obtain, we interpret the fact that @xmath529. s. ito and t. sadahiro determined the unique @xmath530-invariant measure with maximal entropy on @xmath531. note that the structure of the one - side (right) @xmath2-shift endowed with the shift @xmath84 is transported to @xmath532 endowed with the @xmath2-transformation. for @xmath6 taken in the open interval delimited by 1 and the golden ratio, we know that the system is not coded, and then the support of the measure of maximal entropy is a coded subsystem strictly included in the @xmath2-shift. the images by @xmath533 of subsystems non included in the support of the intrinsic ergodic measure correspond to gaps on @xmath534. this phenomenon of gaps has been closely studied by l. liao and w. steiner in @xcite. the gaps on @xmath532 are the intervals @xmath535 defined as followed : @xmath536 with @xmath537, @xmath538 and @xmath539. note that @xmath540 from, @xmath541 and @xmath542 thus, the @xmath2-expansions of reals of gaps start by @xmath543 or @xmath544. if @xmath6 belongs to @xmath545, @xmath514 satisfies. as given in @xmath546 thus, we have @xmath547 and @xmath548 if @xmath113 odd or @xmath549 if @xmath113 even (with @xmath550). in fact, @xmath427 or one of @xmath420 codes the support of the maximal entropy measure on @xmath167 endowed with the shift @xmath551. that is, the support is the closure of @xmath552 or @xmath553 for some @xmath554. [l5] let @xmath6 be a real number such that @xmath555. then @xmath431 codes the support of the maximal entropy measure and @xmath556. observe that @xmath557. from in proposition [prop7], @xmath558 and also, @xmath559 and @xmath560 belong to @xmath431. the proof of lemma is done thanks to the fact that for all integer @xmath113, we can always find two integers @xmath561 and @xmath562 such that @xmath563. then @xmath564 for all @xmath113. for @xmath565, @xmath534 contains one gap, the interval @xmath566, and all @xmath2-expansion in this set starts by 0, it follows that @xmath567 and @xmath568 are subsets of the language of the maximal entropy measure support on @xmath63. since @xmath564 for all @xmath113, it follows that @xmath556. [l6] let @xmath15 be a real number such that for all @xmath27 in @xmath569, @xmath570, the support of the maximal entropy measure is coded by @xmath571. note that @xmath572, @xmath573 (with @xmath574). therefore, @xmath575 and satisfies. that is @xmath576 we obtain the same result as in lemma [l5] by changing the alphabet @xmath577 to @xmath578. so the language which codes the support of the maximal entropy of the @xmath84-invariant measure contains @xmath579 thus, the right choice is @xmath580. for @xmath581, @xmath582. but this inclusion can not hold for @xmath574. each set @xmath100, @xmath431, @xmath583, @xmath584 defines a class of words forbidden in the language of the support of the maximal entropy measure. these words are : @xmath585 and @xmath586 with @xmath587 and @xmath588. in the one side right @xmath2-shift, we add @xmath589 and @xmath590 (with @xmath591). it easy to see that one of these sequences appears in the expansion of a real taken in a gap. Lap counting function the lap counting function of a continuous map @xmath592 whose consists of a finite number of monotone segments (called laps) is the formal power series @xmath593 in which @xmath594 and for all @xmath595, @xmath596 counts the number of laps of the iterate @xmath597. this function was been introduced by milnor and thurston. it is another approach to computing the zeta function. let @xmath117. in this section, we give lap - counting function of the @xmath2- transformation and its classical properties. in the following, @xmath598 denotes the one - side right @xmath2-shift (@xmath2-representations of real belonging to @xmath534). [th3] let consider a real @xmath117 and @xmath599 the associated @xmath2- transformation. we set @xmath600. then, the lap counting function @xmath601 of @xmath530 is given by : @xmath602 where @xmath603 is defined in. [rem5] for a fixed real non integer @xmath0, the graph of @xmath33 consists of @xmath604 segments. indeed, we can see @xmath37 as union of @xmath604 intervals @xmath605 defined by @xmath606, @xmath607 $] with @xmath608, and @xmath609 $]. [lem3] the @xmath2-transformation @xmath599 is affine on each interval @xmath605. moreover, @xmath610 if only if @xmath611 starts by @xmath53. let @xmath612. @xmath613 we set @xmath614. we know that @xmath615 (see @xcite). so, we have proved that for all @xmath616, @xmath617 and @xmath618, @xmath619. note that, if @xmath620, @xmath621. in this case, the graph of the @xmath2- transformation is given by @xmath6 segments. from the previous lemma, @xmath599 has @xmath622 laps if @xmath623 and @xmath6 laps otherwise. @xmath624 with @xmath625 and @xmath626. it is easy to see that @xmath627 is the maximum (with respect to the alternating order) in the family of words of @xmath106 starting by @xmath53. throughout the rest of this section, we set @xmath628. [rem6] let @xmath592 be the map from @xmath116 to @xmath534 defined by @xmath629. @xmath592 is increasing in the sense of the alternating order (see @xcite). it is easy to verify that the words @xmath630 and @xmath631 have same image by @xmath592. indeed, @xmath632 we have the same result if in the remark [rem6], we replace @xmath633 by @xmath634. [lem4] let @xmath628. it is the interval of real numbers of @xmath534 for which the @xmath2-expansions begins by the admissible word @xmath635. let @xmath636. for @xmath637, @xmath638. from the previous corollary, @xmath639 that is @xmath640 begins by @xmath641 furthermore, consider the set of @xmath2-expansions of reals of @xmath642. * if @xmath643 ends by a word of the type @xmath644 with @xmath645 and @xmath646, the @xmath2-expansions of @xmath647 endpoints are @xmath648 and @xmath649. in fact, in the @xmath2-shift, @xmath650 is one of the endpoints @xmath2-representations starting by @xmath641. but, this word can not be an expansion. from remark[rem6], @xmath649 and @xmath650 have the same image by @xmath592. * we assume that @xmath651 for all integer @xmath113 and @xmath643 ends by a sequence of the type @xmath652, we set @xmath653. thus, the word @xmath654 is a endpoint @xmath2-expansion. the maximum in @xmath167 starting by @xmath652 is @xmath655 if @xmath53 is odd and @xmath656 if @xmath53 is even. this last word is not a @xmath2-expansion. we replace it by @xmath657 * now, we assume that @xmath643 ends by a sequence of the type @xmath652, @xmath53 even and @xmath658, then, both infinite words @xmath659 and @xmath660 are the endpoints @xmath2-expansions. + this last case implies that @xmath112 satisfies. in the language of @xmath116, we know that @xmath192 is always followed by @xmath661. thus, if @xmath662, the endpoints of @xmath663 have expansions @xmath664 and @xmath665. that is, for all @xmath666 and @xmath667 in @xmath668, such that @xmath669, @xmath670 suppose @xmath317 periodic with odd period @xmath340. there exists @xmath113 such that @xmath671 and @xmath672. (c) allows us to say that @xmath673 is reduced to the singleton @xmath674 }. it is not an interval. in fact, in an infinite admissible word, @xmath169 is always followed by itself. [rem7] let @xmath6 be a real number strictly bigger than 1 and @xmath381 the language of the words @xmath675 such that @xmath676 from the previous corollary, one has for all fixed integer @xmath677, @xmath678 rather than counting intervals, one can just counts the words of the language associated @xmath63. this approach allows to obtain the laps of @xmath679, for all @xmath170. let @xmath0. we recall that @xmath600. from the formula for the factor complexity of the corrected @xmath1-shift given in section 2 (we can also see in @xcite), and from lemma [lem3], lemma [lem4], and remark [rem7], one has @xmath680 in fact, @xmath681 counts the number of words of length @xmath27 in the language of @xmath351. then, by simple calculus in the open unit disk except in @xmath11, @xmath682 hence the result. the following figures represent the graphs of @xmath683, @xmath684 and @xmath685 respectively in @xmath686. for @xmath687,width=377,height=113] for @xmath687,width=377,height=113] for @xmath687,width=377,height=113] we set @xmath688 and we verify easily that @xmath689, @xmath690 and @xmath691. the laps of @xmath692 correspond to the number of oblique segments in the different graphs. these oblique segments allow to determine the number of words of length @xmath113 in the @xmath693-shift language. * for @xmath694, we have three laps (oblique segments) and three admissible words of length 1 : 0, 1 and 2 (@xmath689). * if @xmath695, @xmath696 ; there are 8 oblique lines and then 8 admissible words of length 2 : 21, 22, 10, 11, 12, 00, 01, 02. * when @xmath697, @xmath698 ; we count twenty laps, then there exist 20 admissible words of length @xmath699 : 211, 210, 222, 221, 102, 101, 100, 112, 111, 110, 122, 121, 002, 001, 000, 012, 011, 010, 022, 021. Zeta function the notion of dynamical zeta function was been introduced by m. artin and b. mazur in 1965. we consider a diffeomorphism @xmath700 on a compact space such that all of its iterates @xmath701 have isolated fixed points. the zeta function associated to @xmath700 is given by : @xmath702 where @xmath703 counts the number of fixed points of @xmath704, by analogy with the geometric zeta function. in 1994, lepold flatto, jeffrey lagarias and bjorn poonen (see @xcite) dealt with the zeta function of the @xmath6-transformation. they consider the application from @xmath19 to @xmath19 defined by : @xmath705 where @xmath706 denotes the fractional part of @xmath17. the associated zeta function is : @xmath707 where @xmath708 counts the number of fixed points of @xmath709. in other words, @xmath708 is the number of periodic admissible sequences @xmath710 with period @xmath53. we denote by @xmath711 the expansion of @xmath17 in base @xmath6. @xmath712 and then, @xmath713 since the expansion is unique for each number and all sub - words of an admissible word is an expansion. after the introduction of the @xmath2-expansion in 2009 by ito and sadahiro in @xcite, in the following sentences, we focus our study on the determination of the zeta functions of the @xmath2-transformation and the one of the @xmath2-shift endowed with the shift. let @xmath211 be a symbolic dynamical system. suppose @xmath211 coded by @xmath714. the code @xmath714 is said _ exhaustive _ if all periodic word @xmath715 can be written uniquely as : @xmath716 where @xmath717 @xmath718 and @xmath719 have the same orbit. [t3] let @xmath211 be a coded system defined by an exhaustive prefix code @xmath714. then, if @xmath720 counts the number of periodic words of period @xmath721 in @xmath211 the associated zeta function is defined by @xmath722 where @xmath723 counts the number of words of length n in @xmath714. let @xmath724 be the sequence of positive integers such that @xmath725 counts the number of periodic words of length @xmath27 having the same orbit than a product of @xmath53 pieces of the exhaustive prefix code. thus, @xmath726. in the sense of formal power series, we rewrite @xmath727 as : @xmath728 let @xmath729 k pieces of @xmath714 with @xmath730 and @xmath731 two integers such that @xmath732 where @xmath733 is minimal, that is @xmath734 denotes the smallest word (in size) with orbit @xmath735. @xmath736 then, @xmath737 thus, @xmath738 @xmath733 minimal, @xmath739, the integer @xmath740 counts the periodic words resulting to the circular permutations of concatenations of @xmath53 monotonic piecewises of the code. then, regardless of the commutative property of the multiplication in @xmath741, and in the sense of formal power series, @xmath742 but @xmath743 that is, @xmath744 hence, @xmath745 then, we obtain @xmath746. theorem [t3] traduce an important property of coded systems : the density of the set of periodic points. let @xmath6 be a real number, @xmath16. let @xmath747 be the @xmath15-shift and @xmath107 the expansion of 1 in base @xmath15. we assume @xmath6 not to be a simple @xmath15-number and we set @xmath748. the @xmath15-shift @xmath747 is coded by @xmath749 which is an exhaustive prefix code. the integer @xmath750 counts the number of pieces of length @xmath751 in @xmath752. thus, the zeta function associated to @xmath747 is given by : @xmath753 we consider a real number @xmath0. recall that the @xmath1-transformation @xmath599 denotes the map from @xmath754 into itself defined by : @xmath755 the aim of this section is to determine the zeta function @xmath756 of the map @xmath599. we know that each number has a @xmath2-representation in @xmath757. moreover, it is easy to see that a real @xmath17 is a fixed point of @xmath758 if only if the @xmath2-representation of @xmath17 in @xmath757 is periodic with period dividing @xmath53. [t2] let @xmath6 be a real number strictly greater than 1, @xmath118 the @xmath2-expansion of @xmath759 and @xmath760 the zeta function of the @xmath2-transformation. then, in the ball of radius @xmath11 and center 0, * if @xmath633 is not periodic @xmath761 * if @xmath633 is periodic with period @xmath53, @xmath762 if we consider a real @xmath429 and @xmath763 the @xmath3-expansion of 1, the zeta function of the @xmath3-shift, determined by leopold flatto, jeffrey lagarias and bjorn poonen in @xcite, is given by @xmath764 so, we remark some similarities between this zeta function and that of the @xmath2-shift given in the previous theorem. for instance, @xmath464 is a pole of these both functions. [lem2] let @xmath429 and @xmath127 periodic with period @xmath765. @xmath766 then, @xmath767. suppose @xmath768. since @xmath769 @xmath770 furthermore, @xmath771 since @xmath772 and @xmath768, it follows that @xmath773 thus, @xmath774. this is absurd. then, @xmath775. as consequence, @xmath776 (with @xmath777) is not periodic. then, in addition to periodic words listed in the previous paragraphs, if @xmath317 is periodic with even period, we should take account of circular permutations of @xmath778. for each integer @xmath53, there are @xmath779 words. let @xmath780 be the set of @xmath2-expansions of real numbers which belongs to @xmath534. we know that the correspondence @xmath781 is one to one. in fact, each real number has one and only one @xmath2-expansion. moreover, @xmath780 is invariant by the shift (all infinite sub - words of a @xmath2-expansion is a @xmath2-expansion). let @xmath782 and @xmath783. then, @xmath784. so, if @xmath17 is a fixed point of @xmath785, then @xmath45 is periodic with period dividing @xmath27. the number of fixed points of @xmath786 equals the number of periodic orbits in @xmath787 with period dividing @xmath27. note that in @xmath788, the sequences which are not @xmath2-expansions end by @xmath789 (with @xmath790). when @xmath633 is not periodic with odd period @xmath791 is not periodic. in this case, the periodic orbits of @xmath792 belong to @xmath780. when @xmath633 is periodic with odd period, @xmath789 is periodic. the periodic orbits of @xmath792 belong to @xmath780 except @xmath789. in section 2, we distinguished in @xmath793 three types of admissible concatenations and then three types of periodic words : * at first, there are the concatenations in @xmath97 ; * secondly, the concatenations of words of @xmath431 ; * and finally, there are products of words of @xmath794, for all positive integer @xmath113. in other hands, the periodic orbits of @xmath351 are circular permutations of sequences of @xmath795, if @xmath796 non periodic. for the periodic case with period @xmath797, we add the circular permutations of @xmath798. the sets @xmath799, @xmath800 and @xmath801 are coded by @xmath97, @xmath427 and @xmath794 respectively. let @xmath802, @xmath803, @xmath804 and @xmath805 be the number of periodic orbits of period dividing @xmath27 in @xmath351, @xmath799, @xmath800 and @xmath801 respectively. if @xmath796 is not periodic, @xmath806 if @xmath807 is periodic with period @xmath779, @xmath808 if @xmath807 is periodic with period odd @xmath809, @xmath810 thus, @xmath756 is product of elementary zeta functions of @xmath799, @xmath800, @xmath801 and also @xmath811 (if @xmath807 is periodic with period @xmath812). from theorem [t3], @xmath813 if @xmath194 is not periodic ; or @xmath814 if @xmath194 is periodic with period @xmath53. we conclude thanks to. the previous theorem can be proved also just using the lap - counting function. indeed, from the equation and the formula of the factor complexity, the coefficients of the power series expansions of the fractional functions @xmath815, @xmath816 and @xmath817 for all @xmath113, count the fixed points of iterates of @xmath599 except the orbit of the left end point of @xmath534 when its expansion is periodic. furthermore, [ef21] and [ef22] provide us an interesting information on the influence of gaps in the interval on the @xmath2-transformation zeta function for @xmath818. they correspond to factors in the denominator of the zeta function. [r7] from section [s3] and theorem [t2], we have the following relation between the zeta - function of the @xmath2-transformation and its lap - counting function : @xmath819 let @xmath820 be the golden ratio : @xmath821, @xmath822. the zeta function is given by two types of periodic words : the sequences of zero and words @xmath49 such that @xmath823. @xmath824, @xmath825. moreover, the @xmath121-shift and the @xmath121-transformation have the same zeta function, since @xmath826 is not periodic. thus @xmath827 moreover, @xmath828. then, @xmath829 we verify easily that @xmath830. the previous remark allows to observe that the relation between the zeta function of the @xmath2-transformation and its lap - counting function differs a little bit from that of the @xmath3-transformation and the associated lap - counting function. indeed, denote by @xmath831 and @xmath832 the zeta function and the lap - counting function of the @xmath3-transformation, according to @xcite, @xmath833 if @xmath6 is not simple @xmath3-number. consider a real number @xmath0. we have seen that when @xmath514 is not periodic with odd period, @xmath789 is not periodic and for a periodic orbit @xmath834, @xmath835 in other words, all periodic word is a @xmath2-expansion. and then, @xmath116 and @xmath599 have the same zeta function. however, if @xmath118 is periodic with odd period @xmath733, @xmath836 is periodic too (with period @xmath837). but, it is not an expansion in base @xmath4. let @xmath720 counts the number of fixed points of @xmath838 and @xmath839 the number of periodic words with period dividing @xmath27 in @xmath116. we have the following result @xmath840 @xmath841 counts the circular permutations of the sequences @xmath842. we denote by @xmath843 the zeta function of the @xmath1-shif. then, @xmath844 in short, @xmath845, @xmath846 Conclusion finally, we have seen that for a real number @xmath117, the associated @xmath2-shift is coded if only if @xmath847 and the @xmath2-expansion of @xmath848 is not periodic with odd period. the non - coded case is due to the existence of intransitive words in the language of the system. in the periodic case with odd period, shall we say @xmath849, the word @xmath850 is intransitive. for @xmath851, @xmath852 is an intransitive word, with @xmath853, where @xmath854, @xmath855, @xmath856, and @xmath857. moreover, the zeta - functions of the @xmath2- shift and @xmath3- shift (determined by leopold flatto, jeffrey c. lagarias and bjorn poonen in @xcite) have some similarities : @xmath464 is a pole for these both functions. however, if we consider @xmath351 as @xmath1-shift, the table above changes a little bit : @xmath351 is coded if only if @xmath6 is greater than or equal to the golden ratio @xmath120. but, if @xmath6 less than the golden ratio, the systems @xmath858 or @xmath351 contains coded sub - shift with maximal entropy @xmath859. jeffrey c. lagarias. number theory zeta functions and dynamical zeta functions. in _ spectral problems in geometry and arithmetic (iowa city, ia, 1997) _, volume : 237 of _ contemp. math. _ pages 4586, 1999.

given a real number @xmath0, we study the associated @xmath1-shift introduced in @xcite. we compares some aspects of the @xmath2-shift to the @xmath3-shift. when the expansion in base @xmath4 of @xmath5 is periodic with odd period or when @xmath6 is strictly less than the golden ratio, the @xmath1-shift, as defined in @xcite can not be coded because its language is not transitive. this intransitivity of words explains the existence of gaps in the interval. we observe that an intransitive word appears in the @xmath2-expansion of a real number taken in the gap. furthermore, we determine the zeta function @xmath7 of the @xmath2-transformation and the associated lap - counting function @xmath8. these two functions are related by @xmath9. we observe some similarities with the zeta function of the @xmath3-transformation. the function @xmath7 is meromorphic in the unit disk, is holomorphic in the open disk @xmath10, has a simple pole at @xmath11 and no other singularities @xmath12 such that @xmath13. we also note an influence of gaps (@xmath3 less than the golden ratio) on the zeta function. in factors of the denominator of @xmath7, the coefficients count the words generating gaps.

1701.00774

Introduction in @xcite the problem of how to give an explicit combinatorial definition of weak higher categories was solved, and the development of a conceptual framework for their further analysis was begun. in the aftermath of this, the expository work of other authors, most notably street @xcite and leinster @xcite, contributed greatly to our understanding of these ideas. the central idea of @xcite is that the description of any @xmath0-dimensional categorical structure @xmath2, may begin by starting with just the underlying @xmath0-globular set, that is, the sets and functions @xmath3^{t } \ar@<-1ex>[l]_{s } & { x_2 } \ar@<1ex>[l]^{t } \ar@<-1ex>[l]_{s } & { x_3 } \ar@<1ex>[l]^{t } \ar@<-1ex>[l]_{s } & {... } \ar@<1ex>[l]^{t } \ar@<-1ex>[l]_{s } & { x_n } \ar@<1ex>[l]^{t } \ar@<-1ex>[l]_{s}}\]] satisfying the equations @xmath4 and @xmath5, which embody the the objects (elements of @xmath6), arrows (elements of @xmath7) and higher cells of the structure in question. at this stage no compositions have been defined, and when they are, one has a globular set with extra structure. in this way the problem of defining an n - categorical structure of a given type is that of defining the monad on the category @xmath8 of @xmath0-globular sets whose algebras are these structures. as explained in the introduction to @xcite, this approach works because the monads concerned have excellent formal properties, which facilitate their explicit description and further analysis. the @xmath0-operads of @xcite can be defined from the point of view of monads : one has the monad @xmath9 on @xmath8 whose algebras are strict @xmath0-categories, and an @xmath0-operad consists of another monad @xmath1 on @xmath8 equipped with a cartesian monad morphism @xmath10. the algebras of this @xmath0-operad are just the algebras of @xmath1. strict @xmath0-categories are easily defined by iterated enrichment : a strict @xmath11-category is a category enriched in the category of strict @xmath0-categories via its cartesian product, but are too strict for the intended applications in homotopy theory and geometry. for @xmath12 the strictest structure one can replace an arbitrary weak @xmath13-category with and not lose vital information is a gray category, which is a category enriched in @xmath14 using the gray tensor product of 2-categories instead of its cartesian product @xcite. this experience leads naturally to the idea of trying to define what the higher dimensional analogues of the gray tensor product are, so as to set up a similar inductive definition as for strict @xmath0-categories, but to capture the appropriate semi - strict @xmath0-categories, which in the appropriate sense, would form the strictest structure one can replace an arbitrary weak @xmath0-category with and not lose vital information. crans in @xcite attempted to realise this idea in dimension 4, and one of our main motivations is to obtain a theory that will deliver the sort of tensor product that crans was trying to define explicitly, but in a conceptual way that one could hope to generalise to still higher dimensions. our examples([ex : gray]) and ([ex : crans]) embody the progress that we have achieved in this direction in this paper. in @xcite the theory of the present paper is used to show that the _ funny tensor product _ of categories which is what one obtains by considering the gray tensor product of @xmath15-categories but ignoring what happens to 2-cells generalises to give an analogous symmetric monoidal closed structure on the category of algebras of any higher operad. from these developments it seems that a conceptual understanding of the higher dimensional analogues of the gray tensor product is within reach. fundamentally, we have two kinds of combinatorial objects important for the description and study of higher categorical structures @xmath0-operads and tensor products. in @xcite a description of the relationship between tensor products and @xmath0-operads was begun, and @xmath11-operads whose algebras involve no structure at the level objects were canonically related with certain lax tensor products on @xmath8. under this correspondence the algebras of the @xmath11-operad coincide with categories enriched in the associated lax tensor product. sections([sec : eg - lmc])-([sec:2-functoriality]) of the present paper continue this development by studying, for a given category @xmath16, the passage @xmath17 where @xmath18 is the category of graphs enriched in @xmath16, in a systematic way. this analysis culminates in section([sec:2-functoriality]) where the above assignment is seen as the object part of a 2-functor @xmath19 where @xmath20 is a sub 2-category of the 2-category of lax monoidal categories, and @xmath21 is as defined by the formal theory of monads @xcite. from this perspective, one is able to describe in a more efficient and general way, many of the previous developments of higher category theory in the globular style. for instance, in section([ssec : induction]) we give a short and direct explicit construction of the monads @xmath9 for strict @xmath0-categories from which all their key properties are easily witnessed. in sections([ssec : general - op - mult]) and ([ssec : induction]) we give shorter and more general proofs of some of the main results of @xcite. in section([ssec : monmonad - distlaw]) using a dual version of our 2-functor @xmath22 and the formal theory of monads @xcite, we obtain a satisfying general explanation for how it is that monad distributive laws arise in higher category theory see @xcite @xcite. in sections([ssec : tci]) and ([ssec : tcii]) we apply our theory to simplifying many aspects of @xcite. the correspondence between @xmath11-operads and certain lax monoidal structures on @xmath8 given in @xcite, associates to the 3-operad @xmath23 for gray categories, a lax tensor product on the category of 2-globular sets. however the gray tensor product itself is a tensor product of 2-categories. any lax monoidal structure on a category @xmath16 comes with a `` unary '' tensor product, which rather than being trivial as is the usual experience with non - lax tensor products, is in fact a monad on @xmath16. for the lax tensor product induced by @xmath23, this is the monad for 2-categories. in section([sec : lift - mult]) we solve the general problem of lifting a lax monoidal structure, to a tensor product on the category of algebras of the monad defined by its unary part. this result, theorem([thm : lift - mult]), is the main result of the paper, and provides also the sense in which these lifted tensor products are unique. in practical terms this means that in order to exhibit a given tensor product on some category of higher dimensional structures as arising from our machinery, it suffices to exhibit an operad whose algebras are categories enriched in that tensor product. in this way, one is able see that the usual gray tensor product and that of crans, do so arise. moreover applying this lifting to the lax tensor products on @xmath8 associated to general @xmath11-operads (over @xmath24), one exhibits the structures definable by @xmath11-operads as enriched categories whose homs are some @xmath0-dimensional structure. in this way the globular approach is more closely related to some of the inductive approaches to higher category theory, such as that of tamsamani @xcite. in section([ssec : a - infinity]) we describe two applications of the lifting theorem. in theorem([thm : a - infinity - app]) we construct a tensor product of @xmath25-algebras. as explained in @xcite the problem of providing such a tensor product is of relevance to string theory, and it proved resistant because of the negative result @xcite which shows that no `` genuine '' tensor product can exist. however this result does not rule out the existence of a _ lax _ tensor product, which is what we were able to provide in theorem([thm : a - infinity - app]). it is possible to see the identification by joyal and street @xcite, of braided monoidal categories as monoidal categories with a multiplication as an instance of theorem([thm : a - infinity - app]). another instance is our second application given in corollary([cor : coh - bm2c]), namely an analogous result to that of joyal and street but for braided monoidal 2-categories. a weak @xmath0-category is an algebra of a _ contractible _ @xmath0-operad. in section([sec : contractibility]) we recall this notion, give an analogous notion of contractible lax monoidal structure and explain the canonical relationship between them. in this paper we operate at a more abstract level than in much of the previous work on this subject. in particular, instead of studying monads on the category of @xmath0-globular sets, or even on presheaf categories, we work with monads defined on some category @xmath18 of enriched graphs. as our work shows, the main results and notions of higher category theory in the globular style can be given in this setting. so one could from the very beginning start not with @xmath24 as the category of @xmath26-categories, but with a nice enough @xmath16. for all the constructions to go through, such as that of @xmath9, the correspondence between monads and lax tensor products, their lifting theorem, as well as the very definition of weak @xmath0-category, it suffices to take @xmath16 to be a locally c - presentable category in the sense defined in section([ssec : lcpres]). proceeding this way one obtains then the theory of @xmath0-dimensional structures enriched in @xmath16. that is to say, the object of @xmath0-cells between any two @xmath27-cells of such a structure would be an object of @xmath16 rather than a mere set. some alternative choices of @xmath16 which could perhaps be of interest are : (1) the ordinal @xmath28=\{0<1\}$] (for the theory of locally ordered higher dimensional structures), (2) simplicial sets (to obtain a theory of higher dimensional structures which come together with a simplicial enrichment at the highest level), (3) the category of sheaves on a locally connected space, or more generally a locally connected grothendieck topos, (4) the algebras of any @xmath0-operad or (5) the category of multicategories (symmetric or not). the point is, the theory as we have developed it is actually _ simpler _ than before, and the generalisations mentioned here come at _ no _ extra cost. Enriched graphs and lax monoidal categories given a topological space @xmath2 and points @xmath29 and @xmath30 therein, one may define the topological space @xmath31 of paths in @xmath2 from @xmath29 to @xmath30 at a high degree of generality. in recalling the details let us denote by @xmath32 a category of `` spaces '' which is complete, cocomplete and cartesian closed. we shall write @xmath33 for the terminal object. we shall furthermore assume that @xmath32 comes equipped with a bipointed object @xmath34 playing the role of the interval. a conventional choice for @xmath32 is the category of compactly generated hausdorff spaces with its usual interval, although there are many other alternatives which would do just as well from the point of view of homotopy theory. let us denote by @xmath35 the reduced suspension of @xmath2, which can be defined as the pushout @xmath36 { i{\times}x}="tr " [d] { \sigma{x}.}="br " [l] { 1{+}1}="bl " " tl"(:"tr":"br",:"bl":"br ") " br " [u(.3)l(.3)] (: @{-}[r(.15)],:@{-}[d(.15)])}\]] writing @xmath37 for the category of bipointed spaces, that is to say the coslice @xmath38, the above definition exhibits the reduced suspension construction as a functor @xmath39 in a sense this functor is the mother of homotopy theory applying it successively to the inclusion of the empty space into the point, one obtains the inclusions of the @xmath27-sphere into the @xmath0-disk for all @xmath40, and its right adjoint @xmath41 is the functor which sends the bipointed space @xmath42, to the space @xmath31 of paths in @xmath2 from @xmath29 to @xmath30. this adjunction @xmath43 is easy to verify directly using the above elementary definition of @xmath44 as a pushout, and the pullback square @xmath45 { x^i}="tr " [d] { x^{1{+}1}}="br " [l] { 1}="bl " " tl"(:"tr":"br"^{x^i},:"bl":"br"_-{(a, b) }) " tl " [d(.3)r(.3)] (: @{-}[l(.15)],:@{-}[u(.15)])}\]] where @xmath46 is the inclusion of the boundary of @xmath34. the collection of spaces @xmath31 is our first example of an enriched graph in the sense of [def : enriched - graph] let @xmath16 be a category. graph @xmath2 enriched in @xmath16 _ consists of an underlying set @xmath6 whose elements are called _ objects _, together with an object @xmath31 of @xmath16 for each ordered pair @xmath47 of objects of @xmath2. the object @xmath31 will sometimes be called the _ hom _ from @xmath29 to @xmath30. a morphism @xmath48 of @xmath16-enriched graphs consists of a function @xmath49 together with a morphism @xmath50 for each @xmath47. the category of @xmath16-graphs and their morphisms is denoted as @xmath18, and we denote by @xmath51 the obvious 2-functor @xmath52 with object map as indicated. the 2-functor @xmath51 is the mother of higher category theory in the globular style applying it successively to the inclusion of the empty category into the point (ie the terminal category), one obtains the inclusion of the category of @xmath27-globular sets into the category of @xmath0-globular sets. in the case @xmath53 this is the inclusion with object map @xmath54 {... } = " m " [r] { x_{n{-}1}}="r " " r":@<1ex>"m":@<1ex>"l " " r":@<-1ex>"m":@<-1ex>"l " } } & \mapsto & { \xygraph{{x_0}="l " [r] {... } = " m " [r] { x_{n{-}1}}="r " [r] { \emptyset}="rr " " rr":@<1ex>"r":@<1ex>"m":@<1ex>"l " " rr":@<-1ex>"r":@<-1ex>"m":@<-1ex>"l " } } \end{array}\]] and when @xmath55 this is the functor @xmath56 which picks out the empty set. thus there is exactly one @xmath57-globular set which may be identified with the empty set. it is often better to think of @xmath51 as taking values in @xmath58. by applying the endofunctor @xmath51 to the unique functor @xmath59 for each @xmath16, produces the forgetful functor @xmath60 which sends an enriched graph to its underlying set of objects. this manifestation @xmath61 has a left adjoint which we shall denote as @xmath62 for reasons that are about to become clear. the functor @xmath62 is a variation of the grothendieck construction. to a given functor @xmath63 it associates the category @xmath64 with objects triples @xmath65 where @xmath29 is an object of @xmath1, and @xmath66 is an ordered pair of objects of @xmath67. maps are just maps in @xmath1 which preserve these base points in the obvious sense. it is interesting to look at the unit and counit of this 2-adjunction. given a category @xmath16, @xmath68 is the category of bipointed enriched graphs in @xmath16. the counit @xmath69 sends @xmath42 to the hom @xmath31. when @xmath16 has an initial object @xmath70 has a left adjoint given by @xmath71. given a functor @xmath63 the unit @xmath72 sends @xmath73 to the enriched graph whose objects are elements of @xmath67, and the hom @xmath74 is given by the bipointed object @xmath65. consider the case where @xmath75 and @xmath76 is the representable @xmath77. then @xmath64 may be regarded as the category of endo - cospans of the object @xmath26, that is to say the category of diagrams @xmath78 and a point of @xmath73 is now just a map @xmath79. when @xmath1 is also cocomplete one can compute a left adjoint to @xmath80. to do this note that a graph @xmath2 enriched in @xmath64 gives rise to a functor @xmath81 where @xmath6 is the set of objects of @xmath2. for any set @xmath82, @xmath83 is defined as the following category. it has two kinds of objects : an object being either an element of @xmath82, or an ordered pair of elements of @xmath82. there are two kinds of non - identity maps @xmath84 where @xmath66 is an ordered pair from @xmath82, and @xmath83 is free on the graph just described. a more conceptual way to see this category is as the category of elements of the graph @xmath85 { z}="r " " l":@<1ex>"r " " l":@<-1ex>"r"}\]] where the source and target maps are the product projections, as a presheaf on the category @xmath86 { = } [r(1.25)] { \xybox{\xygraph{0 [r] 1 " 0":@<1ex>"1":@<1ex>@{<-}"0 " } } } * \frm{-}}\]] and so there is a discrete fibration @xmath87. the functor @xmath88 sends singletons to @xmath75, and a pair @xmath66 to the head of the hom @xmath89. the arrow map of @xmath88 encodes the bipointings of the homs. one may then easily verify let @xmath75, @xmath77 and @xmath1 be cocomplete. then @xmath90 has left adjoint given on objects by @xmath91. in the exposition thus far we have focussed on building an analogy between the reduced suspension of a space and the graphs enriched in a category. now we shall bring these constructions together. as we have seen already to each space @xmath2 one can associate a canonical topologically enriched graph whose homs are the path spaces of @xmath2. denoting this enriched graph as @xmath92, the assignment @xmath93 is the object map of the composite right adjoint in @xmath94 { \ca g(\top_{\bullet})}="m " [r] { \ca g\top}="r " " l":@<-1.5ex>"m"_-{\eta}|{}="b1":@<-1.5ex>"l"|{}="t1 " " t1":@{}"b1"|{\perp } " m":@<-1.5ex>"r"_-{\ca gh}|{}="b2":@<-1.5ex>"m"_-{\ca g\sigma}|{}="t2 " " t2":@{}"b2"|{\perp}}.\]] as explained by cheng @xcite, this functor @xmath95 is a key ingredient of the trimble definition of weak @xmath0-category. the important properties of @xmath51 are apparent because of the close connection between @xmath51 and the @xmath96 construction. a very mild reformulation of the notion of @xmath16-graph is the following : a @xmath16-graph @xmath2 consists of a set @xmath6 together with an @xmath97-indexed family of objects of @xmath16. together with the analogous reformulation of the maps of @xmath18, this means that we have a pullback square @xmath98_{(-)_0=\ca gt_v } \save \pos?(.3)="lpb " \restore \ar[r]^- { } \save \pos?(.3)="tpb " \restore & { \fam{v } } \ar[d]^{\fam(t_v) } \save \pos?(.3)="rpb " \restore \\ { \set } \ar[r]_-{(-)^2 } \save \pos?(.3)="bpb " \restore & { \set } \pos " rpb " ; " lpb " * * @ { } ;?! { " bpb";"tpb"}="cpb " * * @ { } ;? * * @{- } ; " tpb " ; " cpb " * * @ { } ;? * * @{-}}}\]] in @xmath99, and thus a cartesian 2-natural transformation @xmath100. from @xcite theorem(7.4) we conclude [prop : gfam2fun] @xmath51 is a familial 2-functor. in particular it follows from the theory of @xcite that @xmath51 preserves conical connected limits as well as all the notions of `` grothendieck fibration '' which one can define internal to a finitely complete 2-category. moreover the obstruction maps for comma objects are right adjoints. see @xcite for more details on this part of 2-category theory. we shall not use these observations very much in what follows. more important for us is [lem : g - em - object] @xmath51 preserves eilenberg - moore objects. given a monad @xmath101 on a category @xmath16, we shall write @xmath102 for the category of @xmath101-algebras and morphisms thereof, and @xmath103 for the forgetful functor. we shall denote a typical object of @xmath102 as a pair @xmath104, where @xmath2 is the underlying object in @xmath16 and @xmath105 is the @xmath101-algebra structure. from @xcite the 2-cell @xmath106, whose component at @xmath104 is @xmath107 itself has a universal property exhibiting @xmath102 as a kind of 2-categorical limit called an _ eilenberg - moore object_. see @xcite or @xcite for more details on this general notion. the direct proof that for any monad @xmath101 on a category @xmath16, the obstruction map @xmath108 is an isomorphism comes down to the obvious fact that for any @xmath16-graph @xmath109, a @xmath110-algebra structure on @xmath109 is the same thing as a @xmath101-algebra structure on the homs of @xmath109, and similarly for algebra morphisms. let us recall the notions of lax monoidal category and category enriched therein from @xcite. for a category @xmath16, the free strict monoidal category @xmath111 on @xmath16 has a very simple description. an object of @xmath111 is a finite sequence @xmath112 of objects of @xmath16. a map is a sequence of maps of @xmath16 there are no maps between sequences of objects of different lengths. the unit @xmath113 of the 2-monad @xmath114 is the inclusion of sequences of length @xmath33. the multiplication @xmath115 is given by concatenation. a _ lax monoidal category _ is a lax algebra for the 2-monad @xmath114. explicitly it consists of an underlying category @xmath16, a functor @xmath116, and maps @xmath117 for all @xmath82, @xmath118 from @xmath16 which are natural in their arguments, and such that @xmath119 ^ -{u\ope\limits_i } \ar[d]_{1 } \save \pos?(.4)="domeq " \restore & { e_1\ope\limits_iz_i } \ar[dl]^{\sigma } \save \pos?(.4)="codeq " \restore \\ { \ope\limits_iz_i } \pos " domeq " ; " codeq " * * @ { } ;? * { = } } } } ; (4,0)*{\xybox{\xymatrix @c=1.5em { { \ope\limits_i\ope\limits_j\ope\limits_kz_{ijk } } \ar[r]^-{{\sigma}\ope\limits_k } \ar[d]_{\ope\limits_i\sigma } \save \pos?="domeq " \restore & { \ope\limits_{ij}\ope\limits_kz_{ijk } } \ar[d]^{\sigma } \save \pos?="codeq " \restore \\ { \ope\limits_i\ope\limits_{jk}z_{ijk } } \ar[r]_-{\sigma } & { \ope\limits_{ijk}z_{ijk } } \pos " domeq " ; " codeq " * * @ { } ;? * { = } } } } ; (8,0)*{\xybox{{\xymatrix @c=1em { { \ope\limits_ie_1z_i } \ar[dr]_{\sigma } \save \pos?(.4)="domeq " \restore & { \ope\limits_iz_i } \ar[d]^{1 } \save \pos?(.4)="codeq " \restore \ar[l]_-{\ope\limits_iu } \\ & { \ope\limits_iz_i } } \pos " domeq " ; " codeq " * * @ { } ;? * { = } } } } \endxy\]] in @xmath16. as in @xcite we use either of the expressions @xmath120 as a convenient yet precise short - hand for @xmath121, and we refer to the endofunctor of @xmath16 obtained by observing the effect of @xmath122 on singleton sequences as @xmath123. the data @xmath124 is called a _ multitensor _ on @xmath16, and @xmath125 and @xmath126 are referred to as the unit and substitution of the multitensor respectively. given a multitensor @xmath124 on @xmath16, a _ category enriched in @xmath122 _ consists of @xmath127 together with maps @xmath128 for all @xmath40 and sequences @xmath129 of objects of @xmath2, such that @xmath130{e_1x(x_0,x_1)}^-{u } : [d]{x(x_0,x_1)}="bot"^{\kappa }, : " bot"_{\id }) } } } [r(5)][d(.15)] { \xybox{\xygraph{!{0;(2.75,0):(0,.5) : : } { \ope\limits_i\ope\limits_jx(x_{(ij)-1},x_{ij }) } (: [r]{\ope\limits_{ij}x(x_{(ij)-1},x_{ij})}^-{\sigma } : [d]{x(x_0,x_{mn_m})}="bot"^{\kappa},:[d]{\ope\limits_ix(x_{(i1)-1},x_{in_i})}_{\ope\limits_i\kappa } : " bot"_-{\kappa})}}}}\]] commute, where @xmath131, @xmath132 and @xmath133. since a choice of @xmath46 and @xmath134 references an element of the ordinal @xmath135, the predecessor @xmath136 of the pair @xmath137 is well - defined when @xmath46 and @xmath134 are not both @xmath33. with the obvious notion of @xmath122-functor (see @xcite), one has a category @xmath138 of @xmath122-categories and @xmath122-functors together with a forgetful functor @xmath139 the notation we use makes transparent the analogy between multitensors and monads, and categories enriched in multitensors and algebras for a monad. in particular the unit and subtitution for @xmath122 provide @xmath123 with the unit and multiplication of a monad structure. moreover, any object of the form @xmath140 is canonically an @xmath123-algebra, as is the hom of any @xmath122-category, and the substitution maps of @xmath122 are @xmath123-algebra morphisms (see @xcite lemma(2.7)). thus in a sense, any multitensor @xmath124 on a category @xmath16 is aspiring to be a multitensor on the category @xmath141 of @xmath123-algebras, but of course there is no meaningful way to regard @xmath125 as living in @xmath141, except in the boring situation when @xmath123 is the identity monad, that is, when @xmath125 is an identity natural transformation. the multitensors with @xmath125 the identity are called _ normal_. [def : lift] let @xmath124 be a multitensor on a category @xmath16. a _ lift _ of @xmath124 is a normal multitensor @xmath142 on @xmath141 together with an isomorphism @xmath143 which commutes with the forgetful functors into @xmath144. in @xcite we explained how to associate normalised @xmath145-operads and @xmath0-multitensors, which are multitensors on the category of @xmath0-globular sets. in the present paper we shall explain why any @xmath0-multitensor has a canonical lift. Multitensors from monads at an abstract level much of this paper is about the interplay between the theory of monads on categories of enriched graphs, and the theory of multitensors. it is time to be more precise about which monads on @xmath18 we are interested in. [def : nmnd] let @xmath16 be a category. a monad _ over @xmath24 _ on @xmath18 is a monad on @xmath146 in the 2-category @xmath58. that is, a monad @xmath147 on @xmath18 is over @xmath24 when the functor @xmath101 does nt affect the object sets, in other words @xmath148 for all @xmath127 and similarly for maps, and moreover the components of @xmath149 and @xmath150 are identities on objects. in this section we will describe how such a monad, in the case where @xmath16 has an initial object denoted as @xmath151, induces a multitensor on @xmath16 denoted @xmath152. let us describe this multitensor explicitly. first we note that @xmath153 enables us to regard any sequence of objects @xmath112 of @xmath16 as a @xmath16-graph. the object set is @xmath154, @xmath155 for @xmath156, and all the other homs are equal to @xmath151. then we define @xmath157 and the unit as @xmath158 before defining @xmath159 we require some preliminaries. given objects @xmath160 of @xmath16 where @xmath156, and @xmath161 denote by @xmath162 the obvious subsequence inclusion in @xmath18 : the object map preserves successor and @xmath163, and the hom maps are identities. now given objects @xmath118 of @xmath16 where @xmath164 and @xmath132, one has a map @xmath165 given on objects by @xmath166 and @xmath167 for @xmath164, and the hom map between @xmath168 and @xmath46 is @xmath169. with these definitions in hand we can now define the components of @xmath159 as @xmath170 ^ -{\{t\tilde{\tau}\}_{0,k } } & & { t(z_{11},...,z_{kn_k})(0,n_{\bullet }) } \ar[rr]^-{\mu_{0,n_{\bullet } } } & & { \tbar\limits_{ij}z_{ij}}}.\]] from now until the end of ([ssec : tbar]) we shall be occupied with the proof of [thm : tbar] let @xmath16 be a category with an initial object @xmath151 and @xmath147 be a monad over @xmath24 on @xmath18. then @xmath152 as defined in ([eq : tbar])-([eq : mubar]) defines a multitensor on @xmath16. in principle one could supply a proof of this result immediately by just slogging through a direct verification of the axioms. instead we shall take a more conceptual approach, and along the way encounter various ideas that are of independent interest. for most of the time we will assume a little more of @xmath16 : that it has finite coproducts, to enable our more conceptual approach. in the end though, we will see that only the initial object is necessary. we break up the construction of @xmath152 into three steps. first in ([ssec : nmonad->monmonad]), we describe how @xmath171 acquires a monoidal structure and @xmath147 induces a monoidal monad @xmath172 on @xmath171. then we see that this monoidal monad induces a lax monoidal structure on @xmath171, which in turn can be transferred across an adjunction to obtain @xmath152. these last two steps are very general : they work at the level of the theory of lax algebras for an arbitrary 2-monad (which in our case is @xmath114). so in ([ssec : laxalg - const1]) and ([ssec : laxalg - const2]) we describe these general constructions, and in ([ssec : tbar]) we finish the proof of theorem([thm : tbar]). finally in ([ssec : path - like]) we present a condition on @xmath101 which ensures that @xmath101-algebras and @xmath173-categories may be identified. from now until just before the end of ([ssec : tbar]) we shall assume that @xmath16 has finite coproducts. we now describe some consequences of this. first, the functor @xmath174 which sends an enriched graph to its set of objects becomes representable. we shall denote by @xmath26 the @xmath16-graph which represents @xmath175. it has one object and its unique hom is @xmath151. the second consequence is that @xmath171 inherits a natural monoidal structure and any normalised monad @xmath147 on @xmath18 can then be regarded as a monoidal monad @xmath172. the explanation for this begins with the observation that the representability of the underlying set functor @xmath175 enables a useful reformulation of the category @xmath171 as the category of endocospans of @xmath26 as in section([ssec : enriched - graphs]). the usefulness of this is that such cospans can be composed, thus endowing @xmath171 with a canonical monoidal structure. the presence of @xmath151 in @xmath16 enables one to compute coproducts in @xmath18. the coproduct @xmath2 of a family @xmath176 of @xmath16-graphs has object set given as the disjoint union of the object sets of the @xmath177, @xmath178 when @xmath107 and @xmath179 are objects of @xmath177, and all the other homs are @xmath151. with finite coproducts available one can also compute pushouts under @xmath26, that is the pushout @xmath180 of maps @xmath181 ^ -{y } \ar[l]_-{x } & y}\]] in @xmath18 is described as follows. the object set of @xmath180 is the disjoint union of the object sets of @xmath2 and @xmath182 modulo the identification of @xmath107 and @xmath179, and let us write @xmath183 for this special element of @xmath180. the homs of @xmath180 are inherited from @xmath2 and @xmath182 in almost the same way as for coproducts. that is if @xmath29 and @xmath30 are either both objects of @xmath2 or both objects of @xmath182 and they are not both @xmath183, then their hom @xmath184 is taken as in @xmath2 or @xmath182. the hom @xmath185 is the coproduct @xmath186. otherwise this hom is @xmath151. note that in the special case where the homs @xmath187 and @xmath188 are both @xmath151, one only requires the initial object in @xmath16 to compute this pushout. given a sequence @xmath189 of doubly - pointed @xmath16-graphs, one defines their _ join _ @xmath190 where @xmath191 is the colimit of @xmath192_-{b_1 } \ar[r]^-{a_2 } & {... } & 0 \ar[l]_-{b_{n-1 } } \ar[r]^-{a_n } & { x_n}}\]] in @xmath18 which can be formed via iterated pushouts under @xmath26. this defines a monoidal structure on @xmath171 whose tensor product we denote as @xmath193. given a functor @xmath194 over @xmath24, one has a functor @xmath195 whose object map is indicated on the right in the previous display. when both @xmath16 and @xmath196 have an initial object, one defines for each sequence of doubly - pointed @xmath16-graphs a map @xmath197 as follows. write @xmath198 for the components of the colimit cocones ([eq : join]). using the unique map @xmath199, there is a unique map @xmath200 such that @xmath201. by the unique characterisation of these maps, they assemble together to provide the coherence 2-cell @xmath202 ^ - { * } \ar[d]_{mt_{\bullet } } \save \pos?="dom " \restore & { \ca gv_{\bullet } } \ar[d]^{t_{\bullet } } \save \pos?="cod " \restore \\ { m\ca gw_{\bullet } } \ar[r]_- { * } & { \ca gw_{\bullet } } \pos " dom " ; " cod " * * @ { }? (.35) \ar@{=>}^{\tau }? (.65)}}\]] for a lax monoidal functor, and for @xmath203 ^ -{s } & { \ca gv } \ar[r]^-{t } & { \ca gw}}\]] over @xmath24 one has @xmath204 as monoidal functors. moreover any natural transformation @xmath205 over @xmath24 defines a monoidal natural tranformation @xmath206. in fact, denoting by @xmath207 the 2-category whose objects are categories with initial objects, a 1-cell @xmath208 in @xmath207 is a functor @xmath194 over @xmath24, and a 2-cell between these is a natural tranformation also over @xmath24, we have defined a 2-functor @xmath209 applying @xmath210 to monads gives [prop : nmnd->monmnd] if @xmath16 has finite coproducts and @xmath147 is a monad over @xmath24 on @xmath18, then @xmath172 is a monoidal monad on @xmath171, whose monoidal structure is given by pushout - composition of cospans. finally we note that the functor @xmath211 has a left adjoint @xmath212 which we shall now describe. given @xmath213 the underlying @xmath16-graph of @xmath214, which we shall denote as @xmath215, has object set @xmath216 and the distinguished pair is @xmath217. as for the homs, @xmath218 is just @xmath82 itself, and all the other homs are @xmath151. formally it is the composite @xmath219 ^ -{ml } & { m\ca gv_{\bullet } } \ar[r]^- { * } & { \ca gv_{\bullet } } \ar[r]^-{u } & { \ca gv}}\]] where @xmath220 is the obvious forgetful functor, which enables us to view a sequence of objects of @xmath16 as a @xmath16-graph, as in ([ssec : defnmonad]) above. now and in ([ssec : laxalg - const2]) let @xmath221 be a 2-monad on a 2-category @xmath222. suppose that we are given a monad _ in _ @xmath223. let us write @xmath224 for the underlying lax @xmath225-algebra, @xmath226 for the lax @xmath225-algebra endomorphism of @xmath16, and @xmath46 and @xmath227 for the unit and multiplication respectively. then one obtains another lax @xmath225-algebra structure on @xmath16 with one cell part given as the composite @xmath228 ^ -{e } & v \ar[r]^-{f } & v}\]] and the 2-cell data as follows @xmath229 { sv}="tr " [d] { v}="br " [l] { v}="bl " " tl":"tr"^-{\eta}:"br"^{e}:"bl"^{f } " tl":"br"|{1 } " tl":"bl"_{1 } " tl " [d(.35)r(.6)] : @{=>}[r(.2)]^{\iota } " tl " [d(.7)r(.2)] : @{=>}[r(.2)]^{i } } } } [r(3)] { \xybox{\xygraph{{s^2v}="l1 " [d] { sv}="l2 " [d] { sv}="l3 " [r] { v}="m " [r] { v}="r3 " [u] { v}="r2 " [u] { sv}="r1 " " l1":"l2"_{se}:"l3"_{sf}:"m"_-{e}:"r3"_-{f } " l1":"r1"^-{\mu}:"r2"^{e}:"r3"^{f } " l2":"r2"^-{e}:"m"_{f } " l1 " [d(.5)r(.85)] : @{=>}[r(.3)]^{\sigma } " l2 " [d(.5)r(.5)] : @{=>}[r(.3)]^{\phi } " m " [u(.2)r(.5)] : @{=>}[r(.3)]^{m}}}}}\]] the verification of the lax algebra axioms is an easy exercise that is left to the reader. now suppose we are given a lax s - algebra @xmath224 together with an adjunction @xmath230 w " v":@<-1.2ex>"w"_-{r}:@<-1.2ex>"v"_-{l } " v":@{}"w"|-{\perp}}\]] with unit @xmath125 and counit @xmath231. one can then induce a lax @xmath225-algebra structure on @xmath196. the one - cell part is given as the composite @xmath232 ^ -{sl } & { sv } \ar[r]^-{e } & v \ar[r]^-{r } & w}\]] and the 2-cell data as follows @xmath233 { v}="m " [r] { sv}="r " [l(.5)d] { v}="br " [l] { w}="bl " " m " [u] { sw}="t " " l":"t"^-{\eta}:"r"^-{sl}:"br"^-{e}|{}="e":"bl"^-{r } " l":"m"^-{l}:"br"_{1}|{}="onev " " m":"r"^-{\eta } " l":"bl"_{1}|{}="onew " " m":@{}"t"|(.4)*{= } " onew":@{}"onev"|(.35){}="d1"|(.65){}="c1 " " d1":@{=>}"c1"^-{u } " onev":@{}"e"|(.2){}="d2"|(.8){}="c2 " " d2":@{=>}"c2"^-{\iota } } } } [r(4)] { \xybox { \xygraph{!{0;(1,0):(0,.8) : : } { s^2w}="l1 " [dl] { s^2v}="l2 " [d] { sv}="l3 " [d] { sw}="l4 " [r] { sv}="ml " [r] { v}="mr " [r] { w}="r4 " [u] { v}="r3 " [u] { sv}="r2 " [ul] { sw}="r1 " " l1":"l2"_{s^2l}:"l3"_{se}:"l4"_{sr}:"ml"_{sl}:"mr"_{e}:"r4"_{r } " l1":"r1"^-{\mu}:"r2"^{sl}:"r3"^{e}:"r4"^{r } " l3":"ml"^{1}|{}="onemv " " mr":"r3"^{1}|{}="onev " " l2":"r2"^-{\mu } " l1 " [d(2)r(.35)] : @{=>}[r(.3)]^{\sigma } " l3 " [d(.7)r(.15)] : @{=>}[r(.25)]^{sc } " r4 " [u(.25)l(.35)] { = } " l1 " [d(.5)r(.5)] { = } } } } } \]] the reader will easily verify that the lax @xmath225-algebra axioms for @xmath196 follow from those of @xmath16 and the triangle identities of the adjunction. let us now put together ([ssec : nmonad->monmonad])-([ssec : laxalg - const2]). given a monad @xmath147 on @xmath18 over @xmath24 such that @xmath16 has finite coproducts, we obtained the monoidal monad @xmath172 on @xmath171 in proposition([prop : nmnd->monmnd]). in other words @xmath172 is a monad in @xmath234. applying ([ssec : laxalg - const1]) for @xmath235 gives us a multitensor on @xmath171, and then applying ([ssec : laxalg - const1]) to this last multitensor and the adjunction @xmath236 { v}="w " " v":@<-1.2ex>"w"_-{\varepsilon}:@<-1.2ex>"v"_-{l } " v":@{}"w"|-{\perp}}\]] gives us a multitensor on @xmath16 which we denote as @xmath152. we shall now unpack this multitensor to see that it does indeed agree with that of theorem([thm : tbar]). the one cell part @xmath173 is the composite @xmath219 ^ -{ml } & { m\ca gv_{\bullet } } \ar[r]^- { * } & { \ca gv_{\bullet } } \ar[r]^-{t_{\bullet } } & { \ca gv_{\bullet } } \ar[r]^-{\varepsilon } & v}\]] which agrees with equation([eq : tbar]) and one also easily reconciles equation([eq : etabar]) for the unit. as for the substitution unpacking the @xmath159 of our conceptual approach gives the following composite @xmath237 { mv}="p12 " [dr] { m\ca gv_{\bullet}}="p22 " [l(3)] { m^2\ca gv_{\bullet}}="p21 " [dl] { m\ca gv_{\bullet}}="p31 " [d] { m\ca gv_{\bullet}}="p41 " [d] { mv}="p51 " [r(1.25)d] { m\ca gv_{\bullet}}="p52 " [r(1.25)d] { \ca gv_{\bullet}}="p53 " [r(1.25)u] { \ca gv_{\bullet}}="p54 " [r(1.25)u] { v}="p55 " [u] { \ca gv_{\bullet}}="p42 " [u] { \ca gv_{\bullet}}="p32 " " p11":"p12"^-{\mu}|{}="mv1 " " p21":"p22"^-{\mu}|{}="mv2 " " p31":"p32"^-{*}|{}="mv3 " " p51":"p52"_-{ml}:"p53"_-{*}:"p54"_-{t_{\bullet}}:"p55"_-{\varepsilon } " p11":"p21"_-{m^2l}:"p31"_-{m*}:"p41"_-{mt_{\bullet}}:"p51"_-{m\varepsilon } " p12":"p22"^{ml}:"p32"^{*}:"p42"^ { t_{\bullet}}:"p55"^{\varepsilon } " p41":@/^{2pc}/"p52"^{1}|{}="mh1 " " p32":@/_{2.5pc}/"p53"_{t_{\bullet}}|{}="mh2 " " p42":@/_{.5pc}/"p54"_{1}|{}="mh3 " " mv1":@{}"mv2"|*{=}:@{}"mv3"|*{\iso } " p51":@{}"mh1"|{}="mid1":@{}"mh2"|{}="mid2":@{}"mh3"|{}="mid3":@{}"p55"|*{= } " mid1 " [l(.15)] : @{=>}[r(.3)]^{mc } " mid2 " [l(.15)] : @{=>}[r(.3)]^{\tau } " mid3 " [l(.15)] : @{=>}[r(.3)]^{\mu_{\bullet}}}\]] which we shall now unpack further. the counit @xmath231 of the adjunction @xmath238 is described as follows. for a given doubly - pointed @xmath239-graph @xmath42, the corresponding counit component @xmath240 is specified by insisting that the hom map between @xmath26 and @xmath33 is @xmath241. define @xmath242 by @xmath243 { t\{(0,t(x_{11},...,x_{1n_1}),n_1)*... *(0,t(x_{k1},...,x_{kn_k}),n_k)\}}="tl " [d(.4)r] { t^2(x_{11},......,x_{kn_k})}="tr " [d(.6)] { t(x_{11},......,x_{kn_k})}="br " " bl"(:"tl"^{t\{c*... *c\}}:"tr"_-{t\tau}:"br"^{\mu},:"br"_-{\tilde{\mu } }) " bl":@{}"tr"|*{=}}\]] and then @xmath159 is the effect of @xmath242 on the hom between @xmath26 and @xmath244. but one may easily verify that the composite @xmath245 is just @xmath246 described in ([ssec : defnmonad]). this completes the proof of theorem([thm : tbar]) for the case where @xmath16 has finite coproducts. the general case is obtained by observing that only the joins of doubly - pointed @xmath16-graphs @xmath42 such that @xmath247 and @xmath248 are actually used in the construction of the multitensor and in its axioms, and these only require an initial object in @xmath16. we shall now give a condition on a normalised monad @xmath101 which ensures that categories enriched in @xmath173 are the same thing as @xmath101-algebras. let @xmath249 and consider a sequence @xmath250 of objects of @xmath2 such that @xmath251 and @xmath252. define the @xmath16-graph @xmath253 and we have a map @xmath254 given on objects by @xmath255 for @xmath256, and the effect on the hom between @xmath168 and @xmath46 is the identity for @xmath156. [def : path - like] let @xmath16 be a category with an initial object and @xmath196 be a category with all small coproducts. a functor @xmath194 over @xmath24 is _ path - like _ when for all @xmath249, the maps @xmath257 for all @xmath40 and sequences @xmath250 such that @xmath251 and @xmath252, form a coproduct cocone in @xmath196. a normalised monad @xmath147 on @xmath258 is _ path - like _ when @xmath101 is path - like in the sense just defined. [ex : cat - monad - path - like] let @xmath259 and @xmath101 be the free category endofunctor of @xmath260. for any graph @xmath2 and @xmath261, the hom @xmath262 is by definition the set of paths in @xmath2 from @xmath29 to @xmath30. each path determines a sequence @xmath250 of objects of @xmath2 such that @xmath251 and @xmath252, by reading off the objects of @xmath2 as they are visited by the given path. conversely for a sequence @xmath250 of objects of @xmath2 such that @xmath251 and @xmath252, @xmath263 identifies the elements @xmath264 with those paths in @xmath2 from @xmath29 to @xmath30 whose associated sequence is @xmath107. thus @xmath101 is path - like. [prop : pl - alg<->cat] let @xmath16 have small coproducts and @xmath147 be a path - like monad on @xmath18 over @xmath24. then @xmath265. let @xmath2 be a @xmath16-graph. to give an identity on objects map @xmath266 is to give maps @xmath267. by path - likeness these amount to giving for each @xmath40 and @xmath250 such that @xmath268 and @xmath269, a map @xmath270 since @xmath271, that is @xmath272. when @xmath273, for a given @xmath274, @xmath107 can only be the sequence @xmath275. the naturality square for @xmath149 at @xmath276 implies that @xmath277, and the definition of @xmath278 says that @xmath279. thus to say that a map @xmath266 satisfies the unit law of a @xmath101-algebra is to say that @xmath29 is the identity on objects and that the @xmath280 described above satisfy the unit axioms of a @xmath173-category. to say that @xmath29 satisfies the associative law is to say that for all @xmath274, @xmath281 ^ -{\{\mu_x\}_{y, z } } \ar[d]_{tx_{y, z } } & { tx(y, z) } \ar[d]^{a_{y, z } } \\ { tx(y, z) } \ar[r]_-{a_{y, z } } & { x(y, z)}}\]] commutes. given @xmath250 from @xmath2 with @xmath268 and @xmath269, and @xmath282 from @xmath283 with @xmath284 and @xmath285, consider the composite map @xmath286 ^ -{t\overline{w}_{0,k } } & { t^2x^*x(0,n) } \ar[r]^-{t\overline{x}_{0,n } } & { t^2x(y, z)}}\]] and note that by path - likeness, and since the coproduct of coproducts is a coproduct, all such maps for @xmath107 and @xmath287 such that @xmath268 and @xmath269 form a coproduct cocone. precomposing ([eq : assoc1]) with ([eq : copr]) gives the commutativity of @xmath288 ^ -{\overline{\mu } } \ar[d]_{\tbar\limits_ia } & { \tbar\limits_{ij}x(x_{ij-1},x_{ij }) } \ar[d]^{a_x } \\ { \tbar\limits_ix(x_{w_{i-1}},x_{w_i }) } \ar[r]_-{a_w } & { x(y, z)}}\]] and conversely by the previous sentence if these squares commute for all @xmath107 and @xmath287, then one recovers the commutativity of ([eq : assoc1]). this completes the description of the object part of @xmath265. let @xmath289 and @xmath290 be @xmath101-algebras and @xmath291 be a @xmath16-graph morphism. to say that @xmath292 is a @xmath101-algebra map is a condition on the maps @xmath293 for all @xmath274, and one uses path - likeness in the obvious way to see that this is equivalent to saying that the @xmath294 are the hom maps for a @xmath173-functor. the proof of proposition([prop : pl - alg<->cat]) is not new : exactly the same argument was used in the second half of the proof of theorem(7.6) of @xcite, although in that case the setting was far less general. the real novelty is the generality of definition([def : path - like]) which is crucial for section([sec : lift - mult]). Monads from multitensors the general way of obtaining a monad from a multitensor, which is the topic of this section, applies to multitensors which conform to [def : dist - mult] let @xmath16 be a category with small coproducts. then a multitensor @xmath124 is _ distributive _ when the functor @xmath122 preserves coproducts in each variable. that is to say, for each @xmath40, the functor @xmath295 obtained by observing @xmath122 s effect on sequences of length @xmath0, preserves coproducts in each of its @xmath0 variables. the first step in associating a monad to a distributive multitensor is to identify the bicategory @xmath296 which has the property that monads in @xmath296 are exactly distributive multitensors in the sense of definition([def : dist - mult]). there is also a useful reformulation of the notion of monad on @xmath18 over @xmath24 : as a monad in another 2-category @xmath297 where @xmath298 denotes the 2-comonad on @xmath99 induced by the adjunction @xmath299 of section([ssec : enriched - graphs]). our monad - from - multitensor construction is then achieved by means of a pseudo functor (homomorphism of bicategories) @xmath300 which as a pseudo functor sends monads to monads. in fact for a distributive multitensor @xmath122, @xmath301 is path - like and so @xmath302. the objects of @xmath296 are categories with coproducts. a morphism @xmath303 in @xmath296 is a functor @xmath304 which preserves coproducts in each variable. a 2-cell between @xmath122 and @xmath305 is simply a natural transformation between these functors @xmath306. vertical composition of 2-cells is as for natural transformations. the horizontal composite of @xmath307 and @xmath308, denoted @xmath309, is defined as a left kan extension @xmath310 ^ -{\mu } \ar[d]_{mf } \save \pos?="dom " \restore & { mv } \ar[d]^{e \comp f } \save \pos?="cod " \restore \\ { mw } \ar[r]_-{e } & { x } \pos " dom " ; " cod " * * @ { }? (.35) \ar@{=>}^{l_{e, f } }? (.65)}}\]] of @xmath311 along @xmath150. computing this explicitly gives the formula @xmath312 where @xmath164 and @xmath132 on the right hand side of this formula, and we denote by @xmath313 ^ -{\opc\limits_{ij } } & { \opeof\limits_{ij}x_{ij}}}\]] and also by @xmath314, the corresponding coproduct inclusion. the definition of horizontal composition is clearly functorial with respect to vertical composition of 2-cells. [prop : dist] @xmath296 is a bicategory and a monad in @xmath296 is exactly a distributive multitensor. it remains to identify the coherences and check the coherence axioms. the notation we have used here for the coproduct inclusions matches that used in section(3) of @xcite. the proof of the first part of proposition(3.3) of @xcite interpretted in our present more general setting, is the proof that @xmath296 is a bicategory. the characterisation of distributive multitensors as monads in @xmath296 is immediate from the definitions and our abstract definition of horizontal composition in terms of kan extensions. what in @xcite was called @xmath315 is here the hom @xmath316. because of the adjunction @xmath299 and the definition of the comonad @xmath298, a normalised monad on @xmath18 is the same thing as a monad on @xmath16 in the kleisli 2-category @xmath297 of the (2-)comonad @xmath298. the reason this is sometimes useful is that it expresses how our monads over @xmath24 only involve information at the level of homs. the validity of this reformulation is most plainly seen by realising that the factorisation of @xmath51 as an identity on objects 2-functor followed by a 2-fully - faithful 2-functor, can be realised as @xmath317 ^ -{r } & { \kl { \ca g_{\bullet } } } \ar[r]^-{j } & { \cat/\set}}\]] where @xmath318 is the right adjoint part of the kleisli adjunction for @xmath298. thus applying @xmath210 sends a monad in @xmath297 on @xmath16 to a monad on @xmath18 over @xmath24, and the definition of @xmath51 and the 2-fully - faithfulness of @xmath210 ensures that any such monad arises uniquely in this way. it is the identity on objects. given a distributive @xmath304, @xmath301 is defined as a left kan extension @xmath319 ^ - { * } \ar[d]_{m\varepsilon } \save \pos?="dom " \restore & { \ca g_{\bullet}v } \ar[d]^{\gamma{e } } \save \pos?="cod " \restore \\ { mv } \ar[r]_-{e } & { w } \pos " dom " ; " cod " * * @ { }? (.35) \ar@{=>}^{\gamma_e }? (.65)}}\]] of @xmath320 along @xmath321. we shall now explain why this left kan extension exists in general, give a more explicit formula for @xmath301 in corollary([cor : explicit - gamma]), and then with this in hand it will become clear why @xmath22 is a pseudo - functor. of course one could just define @xmath301 via the formula in corollary([cor : explicit - gamma]). we chose instead to give the above more abstract definition, because it will enable us to attain a more natural understanding of why @xmath22 produces _ path - like _ monads from distributive multitensors in ([ssec : distmult->monad]). conceptually, the reason why the left kan extension @xmath322 involves only coproducts in @xmath196 is that @xmath321 is a local left adjoint. a functor @xmath323 is a _ local left adjoint _ when @xmath324 is a local right adjoint. this is equivalent to asking that for all @xmath73 the induced functor @xmath325 between coslices is a left adjoint. we shall explain in lemma([lem : spancomp - lra]) why and how pullback - composition of spans can be seen as a local right adjoint, and the statement that @xmath321 is a local left adjoint is just the dual of this because @xmath321 is defined as pushout - composition of cospans in @xmath18. in lemma([lem : lla - ple]) we give a formula for computing the left kan extension along a local left adjoint, and this will then be applied to give our promised more explicit description of @xmath22 s one - cell map. [lem : spancomp - lra] let @xmath1 be a category with finite products and let @xmath29 be an object of @xmath1 such that @xmath326 also has finite products. then the one - cell part @xmath327 of the monoidal structure on @xmath328 given by span composition is a local right adjoint. note that for all @xmath40 the slices @xmath329, where @xmath330 is the @xmath0-fold cartesian product of @xmath29, have finite products, so the statement of the lemma makes sense and all the limits we mention in this proof exist. in general a functor out of a coproduct of categories is a local right adjoint iff its composite with each coproduct inclusion is a local right adjoint. thus it suffices to show that @xmath0-fold composition of spans @xmath331 is a local right adjoint for all @xmath40. the case @xmath332 may be exhibited as a composite @xmath333 ^ -{1_a } & { a / a } \ar[r]^-{\delta _! } & { a / a{\times}a}}\]] of local right adjoints and thus is a local right adjoint. the case @xmath273 may be regarded as the identity. it suffices to verify the case @xmath334 because with this in hand an easy induction will give the general case. the pullback composite of spans as shown on the left @xmath335 b ([dl] { a}="a1 ", [dr] { a}="a2 "), [dr] c [dr] { a}="a3 ") " d":"b"_{p}:"a1"_{w } " d":"c"^{q}:"a3"^{z } " b":"a2"^{x } " c":"a2"_{y } " d " [d(.3)l(.3)] [d(.1)r(.1)]:@{-}[d(.2)r(.2)]:@{-}[u(.2)r(.2)] } } [r(4)] * \xybox{\xygraph{!{0;(1.2,0):(0,.7) : : } d [r] { b{\times}c}="bc " [d] { a^4}="a4 " [l] { a^3}="a3 " [d] { a^2}="a2 " " d"(:@{.>}"a3"_{(wp, xp = yq, zq)}(:@{.>}"a2"_{(\pi_1,\pi_3)},:"a4"_-{a{\times}\delta{\times}a }), : " bc"^-{(p, q)}:"a4"^{(w, x){\times}(y, z) } " d " [d(.3)] [r(.1)]:@{-}[r(.2)]:@{-}[u(.2)]}}}\]] may be constructed as the dotted composite shown on the right in the previous display, and so binary composition of endospans of @xmath29 is encoded by the composite functor @xmath336 ^ -{\textnormal{prod}_{(a^2,a^2) } } & { a/(a^4) } \ar[r]^-{(a{\times}\delta{\times}a)^ * } & { a/(a^3) } \ar[r]^-{(\pi_1,\pi_3) _! } & { a / a{\times}a}}\]] where @xmath337 denotes the (right adjoint) functor @xmath338 which sends a pair to its cartesian product (and @xmath339 is the slice of @xmath337 over the pair @xmath340). the constituent functors of this last composite are clearly all local right adjoints. recall @xcite @xcite that local right adjoints can be characterised in terms of generic factorisations. the dual characterisation is as follows. a functor @xmath323 is a local left adjoint iff for all @xmath341 the components of the comma category @xmath342 have terminal objects. a map @xmath343 which is terminal in its component of @xmath342 is said to be _ cogeneric _, and by definition any @xmath76 can be factored as @xmath344 ^ -{fh } & { fc } \ar[r]^-{g } & b}\]] where @xmath345 is cogeneric. this factorisation is unique up to unique isomorphism and is called the _ cogeneric factorisation _ of @xmath76. [lem : lla - ple] let @xmath323 be a local left adjoint. suppose that @xmath1 has a set @xmath346 of connected components, each of which has an initial object, and that @xmath109 is locally small. for a functor @xmath347 where @xmath348 has coproducts, the left kan extension @xmath349 of @xmath23 along @xmath292 exists and is given by the formula @xmath350 where @xmath351 denotes the initial object of the component @xmath352, and @xmath353 ^ -{fh_f } & { fa_f } \ar[r]^-{g_f } & b}\]] is a chosen cogeneric factorisation of @xmath354. by the general formula for computing left kan extensions as colimits, it suffices to identify the given formula with the colimit of @xmath355 ^ -{p } & a \ar[r]^-{g } & d}\]] where @xmath356 is the obvious projection. this follows since the components of @xmath342 are indexed by pairs @xmath357, where @xmath352 and @xmath354, and the component of @xmath342 corresponding to @xmath357 has terminal object given by @xmath358, which is mapped by @xmath356 to @xmath359. in the case of @xmath193 note the initial objects of the components of @xmath360 are of the form @xmath361 and a map @xmath362 is a map @xmath363 which amounts to a sequence of elements of @xmath2 of length @xmath145 starting at @xmath29 and finishing at @xmath30. the reader will easily verify that @xmath364 & { x^*x } \ar[r]^-{\overline{x } } & { x}}\]] is a cogeneric factorisation of the map associated to @xmath250. so we have given a conceptual explanation of @xmath365 and @xmath276 which were used in ([ssec : path - like]), as well as completed the proof of [cor : explicit - gamma] for @xmath16 and @xmath196 with coproducts and @xmath304, the defining left kan extension of @xmath301 exists and we have the formula @xmath366 we shall identify @xmath22 with the composite @xmath367, which amounts to identifying @xmath368 with its mate @xmath369 by the adjunction @xmath299. then for @xmath127 and @xmath261, we have for each @xmath40 and for each sequence @xmath250 such that @xmath251 and @xmath252 a coproduct inclusion @xmath370 by corollary([cor : explicit - gamma]). the components of the coherence natural transformations for @xmath22 will be identities on objects. from the definition of the unit @xmath371 in @xmath296, one has that for @xmath261 the coproduct inclusion @xmath372 is an isomorphism. thus we have an isomorphism @xmath373. given distributive @xmath374 and @xmath304, @xmath375 and @xmath261, we define the hom maps of @xmath376 for @xmath377, as the unique isomorphism such that for all @xmath378 where @xmath164 and @xmath132, @xmath379 and @xmath380, the diagram @xmath381{(\opeof\limits_{ij})x(x_{ij-1},x_{ij})}="c " [d][r(.5)]{\gamma(e{\comp}f)x(a, b)}="e ", [r]{\ope\limits_i\gamma(f)x(x_{i-1},x_i)}="b " [d]{\gamma(e)\gamma(f)x(a, b)}="d ") " a":"b"^-{\ope\limits_i\opc\limits_j}:"d"^-{c_{x_{i\bullet}}}:"e"^-{\gamma_2 } " a":"c"_-{\opc\limits_{ij}}:"e"_-{c_{x_{ij}}}}\]] in @xmath196 commutes, where @xmath382 and @xmath383 for @xmath384. we have selected the notation so as to match up with the development of @xcite section(4), and the proof of the first part of proposition(4.1) of _ loc. _ interpretted in the present context gives [prop : gamma - psfunctor] the coherences @xmath385 just defined make @xmath22 into a pseudo - functor. given a monad @xmath101 on @xmath18 over @xmath24, and a set @xmath82, one obtains by restriction a monad @xmath386 on the category @xmath387 of @xmath16-graphs with fixed object set @xmath82. let us write @xmath388 for the functor labelled as @xmath22 in @xcite. then for a given distributive multitensor @xmath122, our present @xmath22 and @xmath388 are related by the formula @xmath389 where the @xmath33 on the right hand side of this equation indicates a singleton. in other words we have just given the `` many - objects version '' of the theory presented in @xcite section(4). having just established the machinery to convert distributive multitensors on @xmath16 to monads on @xmath18 over @xmath24, we shall now relate the enriched categories to the algebras. this involves two things : seeing that the normalised monads constructed from distributive multitensors are path - like, and understanding the relationship between @xmath22 and the construction of section([sec : monad->mult]) of multitensors from monads. [lem : gamma - path - like] let @xmath16 and @xmath196 have coproducts and @xmath304 preserve coproducts in each variable. then @xmath390 is path - like. the condition that @xmath194 is path - like can be expressed more 2-categorically. @xmath391 { 1}="tr " [d] { \ca gv_{\bullet}}="mr " [l] { m\ca gv_{\bullet}}="ml " [d] { } = " bl " [r] { w}="br " " tl"(:"ml"_{p}:"mr"^-{*},:"tr":"mr"^{(a, x, b)}:"br"^{t }) " tl " [d(.5)r(.35)] : @{=>}[r(.3)]^{\lambda } } } } [r(4)d(.1)] * { \xybox{\xygraph{!{0;(1.5,0):(0,.67) : : } { * /(a, x, b)}="tl " [r] { 1}="tr " [d] { \ca gv_{\bullet}}="mr " [l] { m\ca gv_{\bullet}}="ml " [d] { mv}="bl " [r] { w}="br " " tl"(:"ml"_{p}(:"bl"_{m\varepsilon}:"br"_-{e},:"mr"^-{*}),:"tr":"mr"^{(a, x, b)}:"br"^{\gamma{e } }) " tl " [d(.5)r(.35)] : @{=>}[r(.3)]^{\lambda } " ml " [d(.5)r(.35)] : @{=>}[r(.3)]^{\gamma_e}}}}}\]] writing @xmath392 for 2-cell part of the comma object, @xmath101 is path - like iff the 2-cell on the left exhibits @xmath393 as a colimit. to see this recall from ([ssec : gamma]) that the set of components of @xmath394 may be regarded as the set of sequences of objects of @xmath6 starting at @xmath29 and finishing at @xmath30, that each of these components has a terminal object, and that @xmath395 is terminal in the component corresponding to the sequence @xmath250 the situation for a given @xmath122 is depicted on the right in the previous display, and by definition this composite 2-cell is a colimit. thus it suffices to show that the component of the 2-cell @xmath322 at @xmath396 is invertible for all sequences @xmath250 from @xmath6. the component of @xmath322 at a general @xmath397 is the coproduct inclusion @xmath398 ^ -{c_w } & & { \coprod\limits_{(z_0,...,z_m) } \ope\limits_i y(z_{i-1},z_i)}}\]] corresponding to the sequence @xmath399 where @xmath400. in the case of @xmath401, @xmath402, and for summands corresponding to sequences @xmath183 different from @xmath287, we will have @xmath403 for some @xmath46. by the distributivity of @xmath122 those summands will be @xmath151, whence @xmath404 will be invertible. given a distributive multitensor @xmath124 note that one can apply @xmath22 to it and then @xmath405 to the result. one has @xmath406 where the @xmath407 in the sum are elements of @xmath154 and @xmath408 and @xmath409. unless the sequence @xmath410 is just an in - order list @xmath411 of the elements of @xmath154, at least one of the homs @xmath412 must be @xmath151 making that summand @xmath151 by the distributivity of @xmath122. thus the coproduct inclusion @xmath413 is invertible. moreover using the explicit description of the multitensor @xmath414 one may verify that this isomorphism is compatible with the units and substitutions, and so we have [lem : gamma - bar] if @xmath124 is a distributive multitensor on a category @xmath16 with coproducts, then one has an isomorphism @xmath415 of multitensors. together with lemma([lem : gamma - path - like]) and proposition([prop : pl - alg<->cat]) this implies [cor : gamma - alg - ecat] if @xmath124 is a distributive multitensor on a category @xmath16 with coproducts, then one has @xmath302 commuting with the forgetful functors into @xmath18. we now describe the sense in which @xmath22 and @xmath405 are adjoint. first let us note that equation([eq : tbar]) defining the construction @xmath405 in section([ssec : defnmonad]) may be seen as providing functors @xmath416 for all @xmath417 in @xmath99. we have abused notation slightly by denoting by @xmath18 (resp. @xmath258) the category of @xmath16-enriched graphs together with its forgetful functor into @xmath24. in ([ssec : defnmonad]) we considered only the case @xmath418 and when @xmath419 is part of a monad, but equation([eq : tbar]) obviously makes sense in this more general context. in order to relate this with @xmath22 we make [def : ndist] let @xmath16 and @xmath196 have coproducts. a functor @xmath420 over @xmath24 is _ distributive _ when @xmath421 preserves coproducts in each variable. we denote by @xmath422 the full subcategory of @xmath423 consisting of the distributive functors from @xmath18 to @xmath258. [prop : pl - adjoint - char] let @xmath16 and @xmath196 be categories with coproducts. then we have an adjunction @xmath424 { \ndist(v, w)}="nd " " d":@<1ex>"nd"^-{\gamma_{v, w}}|{}="t":@<1ex>"d"^-{\overline{(-)}_{v, w}}|{}="b " " t":@{}"b"|-{\perp}}\]] whose unit is invertible. a distributive @xmath420 is in the image of @xmath425 iff it is path - like. by lemma([lem : gamma - bar]) applying @xmath22 does indeed produce a distributive functor, so @xmath425 is well - defined and one has an isomorphism of @xmath426 with the identity. for @xmath127 and @xmath261 one has @xmath427 induced by the hom - maps of @xmath263, giving @xmath428 natural in @xmath101, and so by @xcite lemma(2.6) to establish the adjunction it suffices to show that @xmath429 is inverted by @xmath405. to this end note that when @xmath430 for @xmath431, the above summands are non - initial iff the sequence @xmath432 is the sequence @xmath433, by the distributivity of @xmath101. the characterisation of path - likeness now follows too, since this condition on a given @xmath101 is by definition the same as the invertibility of @xmath434. an immediate consequence of proposition([prop : pl - adjoint - char]) and proposition([prop : gammae - basic])([geb1]) below is the following result. a direct proof is also quite straight forward and is left as an exercise. [cor : pl->copr - pres] let @xmath16 and @xmath196 be categories with coproducts. if @xmath420 over @xmath24 is distributive and path - like, then it preserves coproducts. Categorical properties preserved by @xmath22 [[ssec : intro - reexpress]] let us now regard @xmath22 as a pseudo - functor @xmath435 that is to say, we take for granted the inclusion of @xmath297 in @xmath58. in this section we shall give a systematic account of the categorical properties that @xmath22 preserves. the machinery we are developing gives an elegant inductive description of the monads @xmath436 for strict @xmath0-categories, provides explanations of some of their key properties, and gives a shorter account of the central result of @xcite on the equivalence between @xmath0-multitensors and @xmath145-operads. let @xmath392 be a regular cardinal. an object @xmath346 in a category @xmath16 is _ connected _ when the representable functor @xmath437 preserves coproducts, and @xmath346 is _ @xmath392-presentable _ when @xmath437 preserves @xmath392-filtered colimits. the object @xmath346 is said to be _ small _ when it is @xmath392-presentable for some regular cardinal @xmath392. a category @xmath16 is _ extensive _ when it has coproducts and for all families @xmath438 of objects of @xmath16, the functor @xmath439 is an equivalence of categories. a more elementary characterisation is that @xmath16 is extensive iff it has coproducts, pullbacks along coproduct coprojections and given a family of commutative squares @xmath440 ^ -{c_i } \ar[d]_{f_i } & x \ar[d]^{f } \\ { y_i } \ar[r]_-{d_i } & y}\]] where @xmath441 such that the @xmath442 form a coproduct cocone, the @xmath443 form a coproduct cocone iff these squares are all pullbacks. it follows that coproducts are disjoint and the initial object of @xmath16 is strict. another sufficient condition for extensivity is provided by [lem : easy - ext] if a category @xmath16 has disjoint coproducts and a strict initial object, and every @xmath444 is a coproduct of connected objects, then @xmath16 is extensive. the proof is left as an easy exercise. note this condition is not necessary : there are many extensive categories whose objects do nt decompose into coproducts of connected objects, for example, take the topos of sheaves on a space which is not locally connected. a category is _ lextensive _ when it is extensive and has finite limits. there are many examples of lextensive categories : grothendieck toposes, the category of algebras of any higher operad and the category of topological spaces and continuous maps are all lextensive. denoting the terminal object of a lextensive category @xmath16 by @xmath33, the representable @xmath445 has a left exact left adjoint @xmath446 which sends a set @xmath82 to the copower @xmath447. this functor enables one to express coproduct decompositions of objects of @xmath16, _ internal to @xmath16 _ because to give a map @xmath448 in @xmath16 is the same thing as giving an @xmath34-indexed coproduct decomposition of @xmath2. the lextensive categories in which every object decomposes into a sum of connected objects are characterised by the following well - known result. [prop : lext - decompose - charn] for a lextensive category @xmath16 the following statements are equivalent : 1. every @xmath444 can be expressed as a coproduct of connected objects.[copcon] 2. the functor @xmath449 has a left adjoint.[pi0] the most common instance of this is when @xmath16 is a grothendieck topos. the toposes @xmath16 satisfying the equivalent conditions of proposition([prop : lext - decompose - charn]) are said to be _ locally connected_. this terminology is reasonable since for a topological space @xmath2, one has that @xmath2 is locally connected as a space iff its associated topos of sheaves is locally connected in this sense. a set @xmath450 of objects of @xmath16 is a _ strong generator _ when for all maps @xmath48, if @xmath451 is bijective for all @xmath452 then @xmath76 is an isomorphism. a locally small category @xmath16 is _ locally @xmath392-presentable _ when it is cocomplete and has a strong generator consisting of small objects. finally recall that a functor is _ accessible _ when it preserves @xmath392-filtered colimits for some regular cardinal @xmath392. the theory of locally presentable categories is one of the high points of classical category theory, and this notion admits many alternative characterisations @xcite @xcite @xcite. for instance locally presentable categories are exactly those categories which are the @xmath24-valued models for a limit sketch. grothendieck toposes are locally presentable because each covering sieve in a grothendieck topology on a category @xmath453 gives rise to a cone in @xmath454, and a sheaf is exactly a functor @xmath455 which sends these cones to limit cones in @xmath24. that is to say a grothendieck topos can be seen as the models of a limit sketch which one obtains in an obvious way from any site which presents it. just as locally presentable categories generalise grothendieck toposes, the following notion generalises locally connected grothendieck toposes. [def : lcc] a locally small category @xmath16 is _ locally c - presentable _ when it is cocomplete and has a strong generator consisting of small connected objects. just as locally presentable categories have many alternative characterisations we have the following result for locally c - presentable categories. its proof is obtained by applying the general results of @xcite in the case of the doctrine for @xmath392-small connected categories, which is `` sound '' (see @xcite), and proposition([prop : lext - decompose - charn]). [thm : conn - gabulm] for a locally small category @xmath16 the following statements are equivalent. 1. @xmath16 is locally c - presentable.[lc1] 2. @xmath16 is cocomplete and has a small dense subcategory consisting of small connected objects.[lc2] 3. @xmath16 is a full subcategory of a presheaf category for which the inclusion is accessible, coproduct preserving and has a left adjoint.[lc4] 4. @xmath16 is the category of models for a limit sketch whose distingished cones are connected.[lc5] 5. @xmath16 is locally presentable and every object of @xmath16 is a coproduct of connected objects.[lc6] 6. @xmath16 is locally presentable, extensive and the functor @xmath456 has a left adjoint.[lc7] [ex : lc - groth - toposes] by theorem([thm : conn - gabulm])([lc7]) a grothendieck topos is locally connected in the usual sense iff its underlying category is locally c - presentable. just as with locally presentable categories, locally c - presentable categories are closed under many basic categorical constructions. for instance from theorem([thm : conn - gabulm])([lc6]), one sees immediately that the slices of a locally c - presentable category are locally c - presentable from the corresponding result for locally presentable categories. another instance of this principle is the following result. [thm : acc - monad] if @xmath16 is locally c - presentable and @xmath101 is an accessible coproduct preserving monad on @xmath16, then @xmath102 is locally c - presentable. first we recall that colimits in @xmath102 can be constructed explicitly using colimits in @xmath16 and the accessibility of @xmath101 (see for instance @xcite for a discussion of this). by definition we have a regular cardinal @xmath392 such that @xmath101 preserves @xmath392-filtered colimits and @xmath16 is locally @xmath392-presentable. defining @xmath457 to be the full subcategory of @xmath16 consisting of the @xmath392-presentable and connected objects, @xmath458 is a monad with arities in the sense of @xcite. one has a canonical isomorphism @xmath459 ^ -{v^t(i,1) } \ar[d]_{u } \save \pos?="d " \restore & { \psh { \theta}_t } \ar[d]^{\res_j } \save \pos?="c " \restore \\ v \ar[r]_-{v(i_0,1) } & { \psh { \theta}_0 } \pos " d";"c " * * @ { } ;? * { \iso}}\]] in the notation of @xcite. thus @xmath460 is accessible since @xmath461 creates colimits, and @xmath101 and @xmath462 are accessible. by the nerve theorem of @xcite @xmath460 is also fully faithful, it has a left adjoint since @xmath102 is cocomplete given by left extending @xmath46 along the yoneda embedding, and so we have exhibited @xmath102 as conforming to theorem([thm : conn - gabulm])([lc4]). [ex : lc - noperad - algebras] an @xmath0-operad for @xmath463 in the sense of @xcite, gives a finitary coproduct preserving monad on the category @xmath8 of @xmath0-globular sets, and its algebras are just the algebras of the monad. thus the category of algebras of any @xmath0-operad is locally c - presentable by theorem([thm : acc - monad]). at the object level, to apply @xmath22 is to apply @xmath51, so we shall now collect together many of the categorical properties that @xmath51 preserves. for @xmath16 with an initial object @xmath151, we saw in section([ssec : nmonad->monmonad]) how to construct coproducts in @xmath18 explicitly. from this explicit construction, it is clear that the connected components of a @xmath16-graph @xmath2 may be described as follows. objects @xmath29 and @xmath30 of @xmath2 are in the same connected component iff there exists a sequence @xmath129 of objects of @xmath2 such that for @xmath156 the hom @xmath464 is non - initial. moreover @xmath2 is clearly the coproduct of its connected components, coproducts are disjoint and the initial object of @xmath18, whose @xmath24 of objects is empty, is strict. thus by lemma([lem : easy - ext]) we obtain [prop : gv - ext] if @xmath16 has an initial object then @xmath18 is extensive and every object of @xmath18 is a coproduct of connected objects. given finite limits in @xmath16 it is straight forward to construct finite limits in @xmath18 directly. the terminal @xmath16-graph has one object and its only hom is the terminal object of @xmath16. given maps @xmath465 and @xmath466 in @xmath18 their pullback @xmath180 can be constructed as follows. objects are pairs @xmath467 where @xmath29 is an object of @xmath1 and @xmath231 is an object of @xmath346 such that @xmath468. the hom @xmath469 is obtained as the pullback of @xmath470 ^ -{f } & { b(fa_1,fa_2) } & { c(c_1,c_2) } \ar[l]_-{g}}\]] in @xmath16. thus one has [prop : gv - lext] if @xmath16 has finite limits then so does @xmath18. if in addition @xmath16 has an initial object, then @xmath18 is lextensive and every object of @xmath16 is a coproduct of connected objects. as for cocompleteness one has the following result due to betti, carboni, street and walters. [prop : gv - cocomp]@xcite if @xmath16 is cocomplete then so is @xmath18 and @xmath174 is cocontinuous. we now turn to local c - presentability. first we require a general lemma which produces a dense subcategory of @xmath18 from one in @xmath16 in a canonical way. [lem : gv - dense] let @xmath450 be a full subcategory of @xmath16 and suppose that @xmath16 has an initial object. define an associated full subcategory @xmath471 of @xmath18 as follows : * @xmath472. * @xmath473. if @xmath450 is dense then so is @xmath471. for a regular cardinal @xmath392, if the objects of @xmath450 are @xmath392-presentable then so are those of @xmath471. given functions @xmath474 natural in @xmath475, we must show that there is a unique @xmath48 such that @xmath476. the object map of @xmath76 is forced to be @xmath477, and naturality with respect to the maps @xmath478 ^ -{0 } & (d) & 0 \ar[l]_-{1}}\]] ensures that the functions @xmath479 amount to @xmath477 together with functions @xmath480 natural in @xmath452 for all @xmath261. by the adjointness @xmath238 these maps are in turn in bijection with maps @xmath481 natural in @xmath452 for all @xmath261, and so by the density of @xmath450 one has unique @xmath482 in @xmath16 such that@xmath483. thus @xmath477 and the @xmath482 together form the object and hom maps of the unique desired map @xmath76. since any colimit in @xmath18 is preserved by @xmath175 @xcite, one can easily check directly that @xmath26 is @xmath392-presentable for all @xmath392. let @xmath452 be @xmath392-presentable. one has natural isomorphisms @xmath484 exhibiting @xmath485 as a coproduct of functors that preserve @xmath392-filtered colimits, and thus is itself @xmath392-filtered colimit preserving. [cor : gv - lc] if @xmath16 is locally presentable then @xmath18 is locally c - presentable. immediate from theorem([thm : conn - gabulm])([lc2]), lemma([lem : gv - dense]) and proposition([prop : gv - cocomp]). in @xcite kelly and lack proved that if @xmath16 is locally presentable then so is @xmath18 by an argument almost identical to that given here. the only difference is that in their version of lemma([lem : gv - dense]), their @xmath471 differs from ours only in that they use @xmath486 where we use @xmath26, and they instead prove that @xmath471 is a strong generator given that @xmath450 is. we have given the above proof because the present form of lemma([lem : gv - dense]) is more useful to us in section([ssec : gamma - pres12-lra]). next we shall see that @xmath51 preserves toposes. first a lemma of independent interest. [lem : abstract - wreath] let @xmath453 be a category and @xmath487 be a lextensive category. consider the category @xmath488 constructed from @xmath453 as follows. there is an injective on objects fully faithful functor @xmath489 and @xmath488 has an additional object @xmath26 not in the image of @xmath490. moreover for each @xmath491 one has maps @xmath492 and for all @xmath493 one has the equations @xmath494 and @xmath495. then @xmath496 $] is equivalent to the full subcategory of @xmath497 $] consisting of those @xmath2 such that @xmath6 is a copower of @xmath33, the terminal object of @xmath487. let us write @xmath498 for the full subcategory of @xmath497 $] described in the statement of the lemma. we shall describe the functors @xmath499 \rightarrow \ca f } & & { (-)^{- } : \ca f \rightarrow \ca g[\op { \c},\ca e] } \end{array}\]] which provide the desired equivalence directly. given @xmath500 $] define @xmath501 and @xmath502 in the obvious way this definition is functorial in @xmath2 and @xmath346. conversely given @xmath503 choose a set @xmath504 such that @xmath505. such a set is determined uniquely up to isomorphism. then define the homs of @xmath506 via the pullbacks @xmath507 _ { } \save \pos?(.3)="lpb " \restore \ar[r]^- { } \save \pos?(.3)="tpb " \restore & { yc_+ } \ar[d]^{(y\sigma, y\tau) } \save \pos?(.3)="rpb " \restore \\ { 1 } \ar[r]_-{(a, b) } \save \pos?(.3)="bpb " \restore & { y0{\times}y0 } \pos " rpb " ; " lpb " * * @ { } ;?! { " bpb";"tpb"}="cpb " * * @ { } ;? * * @{- } ; " tpb " ; " cpb " * * @ { } ;? * * @{-}}}\]] in @xmath487 for all @xmath508. note that since @xmath487 is lextensive and hence distributive, @xmath509 is itself the coproduct of copies of @xmath33 indexed by such pairs @xmath47. the natural isomorphisms @xmath510 come from extensivity. [cor : gv - topos] 1. if @xmath16 is a presheaf topos then so is @xmath18.[g6] 2. if @xmath16 is a grothendieck topos then @xmath18 is a locally connected grothendieck topos.[g7] applying lemma([lem : abstract - wreath]) in the case where @xmath453 is small and @xmath511 one obtains the formula @xmath512 and thus ([g6]). since a grothendieck topos is a left exact localisation of a presheaf category, the 2-functoriality of @xmath51 together with ([g6]), corollary([cor : gv - lc]) and example([ex : lc - groth - toposes]), implies that to establish ([g7]) it suffices to show that @xmath51 preserves left exact functors between categories with finite limits. this follows immediately from the explicit description of finite limits in @xmath18 given in the proof of proposition([prop : gv - lext]). the construction @xmath513 in the previous proof is easily discovered by thinking about why, as pointed out in section([ssec : enriched - graphs]), applying @xmath51 successively to the empty category does one produce the categories of @xmath0-globular sets for @xmath40. the construction @xmath513 is just the general construction on categories, which when applied successively to the empty category produces the categories @xmath514 for @xmath40, presheaves on which are by definition @xmath0-globular sets. first we note that by the explicit description of coproducts, the terminal object and pullbacks of enriched graphs, and the formula of corollary([cor : explicit - gamma]), one has [prop : gammae - basic] let @xmath16 and @xmath196 have coproducts and @xmath304 be distributive. 1. @xmath301 preserves coproducts.[geb1] 2. if @xmath122 preserves the terminal object then so does @xmath301. if @xmath122 preserves pullbacks then so does @xmath301. the precise conditions under which @xmath22 preserves and reflects cartesian natural transformations are identified by proposition([prop : gamma - cart]). first we require a lemma which generalises lemma(7.4) of @xcite, whose proof follows easily from the explicit description of pullbacks in @xmath18 discussed in ([ssec : gamma - pres]). [lem : pb - hom] suppose that @xmath16 has pullbacks. given a commutative square (i) @xmath515_{f } \save \pos?="d " \restore \ar[r]^-{h } & x \ar[d]^{g } \save \pos?="c " \restore \\ y \ar[r]_-{k } & z \pos " d";"c " * * @ { } ;? * { \textnormal{i } } } } } ; (1,0)*-!l{\xybox{\xymatrix{{w(a, b) } \ar[d]_{f_{a, b } } \save \pos?="d " \restore \ar[r]^-{h_{a, b } } & { x(ha, hb) } \ar[d]^{g_{ha, hb } } \save \pos?="c " \restore \\ { y(a, b) } \ar[r]_-{k_{a, b } } & { z(ha, hb) } \pos " d";"c " * * @ { } ;? * { \textnormal{ii } } } } } * * @ { }? * { } \end{xy}}\]] in @xmath18 such that @xmath477 and @xmath516 are identities, one has for each @xmath517 commuting squares (ii) as in the previous display. the square (i) is a pullback iff for all @xmath517, the square (ii) is a pullback in @xmath16. [prop : gamma - cart] let @xmath16 have coproducts, @xmath196 be extensive, and @xmath122 and @xmath307 be distributive. then @xmath518 is cartesian iff @xmath519 is cartesian. let @xmath48 be in @xmath18, @xmath261 and @xmath432 be a sequence of objects of @xmath2 such that @xmath251 and @xmath252. for each such @xmath520 it suffices by lemma([lem : pb - hom]), to show that the square (ii) in the commutative diagram @xmath521 { \gamma{e}x(a, b)}="i12 " [d] { \gamma{e}y(fa, fb)}="i22 " [l] { \ope\limits_iy(fx_{},fx_i)}="i21 " " i11 " [ul] { \gamma{f}x(a, b)}="o11 " " i12 " [ur] { \opf\limits_ix(x_{},x_i)}="o12 " " i21 " [dl] { \opf\limits_iy(fx_{},fx_i)}="o21 " " i22 " [dr] { \gamma{f}y(fa, fb)}="o22 " " i11":"i12"^-{c_{x_i}}:"i22"^{\gamma{e}(f)_{a, b}}|{}="l3 " " i11":"i21"_{\ope\limits_if}|{}="l2":"i22"_-{c_{x_i } } " o11":"o12"^-{c_{x_i}}:"o22"^{\gamma{f}(f)_{a, b}}|{}="l4 " " o11":"o21"_{\opf\limits_if}|{}="l1":"o22"_-{c_{x_i } } " i11":"o11"^{\phi } " i21":"o21"^{\phi } " i12":"o12"^{\gamma\phi } " i22":"o22"^{\gamma\phi } " l1":@{}"l2"|(.4)*{(iii)}:@{}"l3"|*{(i)}:@{}"l4"|(.7)*{(ii)}}\]] is a pullback. the square (i) and the largest square are pullbacks since @xmath196 is extensive and the @xmath231-maps are coproduct inclusions, and (iii) is a pullback since @xmath522 is cartesian. thus for all such @xmath432 the composite of (i) and (ii) is a pullback. the result follows since the @xmath523 form coproduct cocones and @xmath196 is extensive. conversely suppose that we have @xmath524 in @xmath16 for @xmath156. then by the isomorphisms @xmath525 and @xmath526, and their naturality with respect to @xmath522 one may identify the naturality square on the left @xmath527 ^ -{\phi_{x_i } } \ar[d]_{\ope\limits_if_i } & { \opf\limits_i } \ar[d]^{\opf\limits_if_i } \\ { \ope\limits_iy_i } \ar[r]_-{\phi_{y_i } } & { \opf\limits_iy_i } } } } ; (1,0)*-!l{\xybox{\xymatrix{{\gamma(e)(x_1,...,x_n)(0,n) } \ar[r]^-{\gamma(\phi)_{0,n } } \ar[d]_{\gamma(e)(f_1,...,f_n)_{0,n } } & { \gamma(f)(x_1,...,x_n)(0,n) } \ar[d]_{\gamma(f)(f_1,...,f_n)_{0,n } } \\ { \gamma(e)(y_1,...,y_n)(0,n) } \ar[r]_-{\gamma(\phi)_{0,n } } & { \gamma(f)(y_1,...,y_n)(0,n) } } } } * * @ { }? * { } \end{xy}}\]] with that on the right in the previous display, which is cartesian by lemma([lem : pb - hom]) and since @xmath528 is cartesian, and so @xmath522 is indeed cartesian. @xmath22 s compatibility with the bicategory structure of @xmath296 is expressed in [prop : gamma - cart2] 1. let @xmath304 and @xmath374 be distributive and @xmath196 be extensive. if @xmath122 and @xmath292 preserve pullbacks then so does @xmath309.[cart2 - 1] 2. let @xmath529 and @xmath530 be distributive and pullback preserving, and @xmath196 be extensive. if @xmath531 and @xmath532 are cartesian then so is @xmath533.[cart2 - 2] ([cart2 - 1]). suppose that for @xmath156 we have pullback squares @xmath534_{h_i } \save \pos?(.3)="lpb " \restore \ar[r]^-{f_i } \save \pos?(.3)="tpb " \restore & { b_i } \ar[d]^{k_i } \save \pos?(.3)="rpb " \restore \\ { d_i } \ar[r]_-{g_i } \save \pos?(.3)="bpb " \restore & { c_i } \pos " rpb " ; " lpb " * * @ { } ;?! { " bpb";"tpb"}="cpb " * * @ { } ;? * * @{- } ; " tpb " ; " cpb " * * @ { } ;? * * @{-}}}\]] in @xmath220. then for each partition @xmath535 of @xmath0 we have a commutative diagram @xmath536 { (\opeof\limits_i)b_i}="i12 " [d] { (\opeof\limits_i)d_i}="i22 " [l] { (\opeof\limits_i)c_i}="i21 " " i11 " [ul] { \ope\limits_i\opf\limits_ja_{ij}}="o11 " " i12 " [ur] { \ope\limits_i\opf\limits_jb_{ij}}="o12 " " i21 " [dl] { \ope\limits_i\opf\limits_jc_{ij}}="o21 " " i22 " [dr] { \ope\limits_i\opf\limits_jd_{ij}}="o22 " " i11":"i12"^-{(\opeof\limits_i)f_i}:"i22"^{(\opeof\limits_i)k_i}|{}="l3 " " i11":"i21"_{(\opeof\limits_i)h_i}|{}="l2":"i22"_-{(\opeof\limits_i)g_i } " o11":"o12"^-{(\ope\limits_i\opf\limits_j)f_{ij}}:"o22"^{(\ope\limits_i\opf\limits_j)k_{ij}}|{}="l4 " " o11":"o21"_{(\ope\limits_i\opf\limits_j)h_{ij}}|{}="l1":"o22"_-{(\ope\limits_i\opf\limits_j)g_{ij } } " o11":"i11"^{c_{n_i } } " o21":"i21"^{c_{n_i } } " o12":"i12"^{c_{n_i } } " o22":"i22"^{c_{n_i } } " l1":@{}"l2"|(.35)*{(i)}:@{}"l3"|*{(ii)}:@{}"l4"|(.7)*{(iii)}}\]] in @xmath196 in which the @xmath231-maps are coproduct inclusions, the coproducts being indexed over the set of all such partitions. for the outer square @xmath164 and @xmath132. we must show that (ii) is a pullback. the squares (i) and (iii) are pullbacks because @xmath196 is extensive. the large square is a pullback because @xmath122 and @xmath292 preserve pullbacks. thus the composite of (i) and (ii) is a pullback, and so the result follows by the extensivity of @xmath196. ([cart2 - 2]). given @xmath537 for @xmath156 and @xmath535 we have a commutative diagram @xmath536 { (\opeof\limits_i)b_i}="i12 " [d] { (\opeofpr\limits_i)b_i}="i22 " [l] { (\opeofpr\limits_i)a_i}="i21 " " i11 " [ul] { \ope\limits_i\opf\limits_ja_{ij}}="o11 " " i12 " [ur] { \ope\limits_i\opf\limits_jb_{ij}}="o12 " " i21 " [dl] { \opepr\limits_i\opfpr\limits_ja_{ij}}="o21 " " i22 " [dr] { \opepr\limits_i\opfpr\limits_jb_{ij}}="o22 " " i11":"i12"^-{(\opeof\limits_i)f_i}:"i22"^{\psi{\comp}\phi } " i11":"i21"_{\psi{\comp}\phi}:"i22"_-{(\opeofpr\limits_i)f_i } " o11":"o12"^-{(\ope\limits_i\opf\limits_j)f_{ij}}:"o22"^{\psi\phi } " o11":"o21"_{\psi\phi}:"o22"_-{(\opepr\limits_i\opfpr\limits_j)f_{ij } } " o11":"i11"^{c_{n_i } } " o21":"i21"^{c_{n_i } } " o12":"i12"^{c_{n_i } } " o22":"i22"^{c_{n_i}}}\]] in @xmath196, to which we apply a similar argument as in ([cart2 - 1]) to demonstrate that the inner square is cartesian. let @xmath16 be lextensive. then by proposition([prop : gamma - cart2]) the monoidal structure of @xmath316 restricts to pullback preserving @xmath538 and cartesian transformations between them. a multitensor @xmath124 on @xmath16 is _ cartesian _ when @xmath122 preserves pullbacks and @xmath125 and @xmath126 are cartesian. by slicing over @xmath122 one obtains a monoidal category @xmath539, whose objects are cartesian transformations @xmath540. to give @xmath541 a monoid structure is to give @xmath1 the structure of a cartesian multitensor such that @xmath542 is a cartesian multitensor morphism. such an @xmath541 is called an _ @xmath122-multitensor_. the category of @xmath122-multitensors is denoted @xmath543. [ex : non - sig - operads] let us denote by @xmath544 the multitensor on @xmath24 given by finite products. by the lextensivity of @xmath24, @xmath544 is an lra multitensor and thus cartesian. a @xmath544-multitensor is the same thing as a non - symmetric operad in @xmath24. for given a cartesian multitensor map @xmath545, one obtains the underlying sequence @xmath546 of sets of the corresponding operad as @xmath547. one uses the cartesianness of the naturality squares corresponding to the maps @xmath548 to recover @xmath122 from the @xmath549. similarly the multitensor structure of @xmath122 corresponds to the unit and substitution maps making the @xmath549 into an operad. given a cartesian normalised monad @xmath101 on @xmath18, one obtains a monoidal category @xmath550, whose objects are cartesian transformations @xmath551 over @xmath24. explicitly, to say that a general collection, which is a cartesian transformation @xmath551, is over @xmath24, is to say that the components of @xmath542 are identities on objects maps of @xmath16-graphs. the tensor product of @xmath550 is obtained via composition and the monad structure of @xmath101, and a monoid structure on @xmath541 is a cartesian monad structure on @xmath1 such that @xmath542 is a cartesian monad morphism. such an @xmath541 is called a _ @xmath101-operad over @xmath24_. the category of @xmath101-operads over @xmath24 is denoted as @xmath552. by the results of this subsection @xmath22 induces a strong monoidal functor @xmath553 and we shall now see that this functor is a monoidal equivalence. applying this equivalence to the monoids in the respective monoidal categories gives the promised general equivalence between multitensors and operads over @xmath24 in corollary([cor : mult - nop - equiv]) below. [lem : transfer - dpl] let @xmath16 be a lextensive category and @xmath101 be a cartesian monad on @xmath18 over @xmath24. let @xmath551 be a collection over @xmath24. 1. if @xmath101 is distributive then so is @xmath1.[tdpl1] 2. if @xmath101 is path - like then so is @xmath1.[tdpl2] ([tdpl1]) : given an @xmath0-tuple @xmath554 of objects of @xmath16 and a coproduct cocone @xmath555 where @xmath156, we must show that the hom - maps @xmath556 form a coproduct cocone. for @xmath557 we have a pullback square @xmath558 { a(x_1,...,x_i,...,x_n)(0,n)}="tr " [d] { t(x_1,...,x_i,...,x_n)(0,n)}="br " [l] { t(x_1,...,x_{ij},...,x_n)(0,n)}="bl " " tl"(:"tr"^-{a(x_1,...,c_j,...,x_n)_{0,n}}:"br"^{\alpha},:"bl"_{\alpha}:"br"_-{t(x_1,...,c_j,...,x_n)_{0,n } }) " tl " [r(.1)d(.4)] { \xybox{\xygraph{!{0;(0.2,0) : } (: @{-}[u], : @{-}[l])}}}}\]] and by the distributivity of @xmath101 and lemma([lem : pb - hom]), the @xmath559 form a coproduct cocone, and thus so do the @xmath560 by the extensivity of @xmath16. + ([tdpl2]) : given @xmath127, @xmath261 and a sequence @xmath129 of objects of @xmath2 such that @xmath251 and @xmath252, we have the map @xmath561 and we must show that these maps, where the @xmath562 range over all sequences from @xmath29 to @xmath30, form a coproduct cocone. by the path - likeness of @xmath101 we know that the maps @xmath257 form a coproduct cocone, so we can use the cartesianness of @xmath542, lemma([lem : pb - hom]) and the extensivity of @xmath16 to conclude as in ([tdpl1]). [prop : coll - equiv] let @xmath16 be lextensive and @xmath563 be a distributive cartesian multitensor on @xmath16. then @xmath564 is a monoidal equivalence @xmath565. the functor @xmath564 is the result of applying the functor @xmath566 of proposition([prop : pl - adjoint - char]) over @xmath122. thus by proposition([prop : pl - adjoint - char]), proposition([prop : gamma - cart]) and lemma([lem : transfer - dpl]), @xmath564 is an equivalence. [cor : mult - nop - equiv] let @xmath16 be lextensive and @xmath563 be a distributive cartesian multitensor on @xmath16. then applying @xmath564 gives @xmath567. local right adjoint monads, especially defined on presheaf categories, are fundamental to higher category theory. indeed a deeper understanding of such monads is the key to understanding the relationship between the operadic and homotopical approaches to the subject @xcite. we will now understand the conditions under which @xmath22 preserves local right adjoints. first we require two lemmas. [lem : partial - adjoint] let @xmath568 be a functor, @xmath16 be cocomplete, @xmath220 be a small dense full subcategory of @xmath196, and @xmath569 be a partial left adjoint to @xmath318, that is to say, one has isomorphisms @xmath570 natural in @xmath571 and @xmath444. defining @xmath572 as the left kan extension of @xmath212 along the inclusion @xmath573, one has @xmath574. denoting by @xmath575 the canonical forgetful functor for @xmath576 and recalling that @xmath577, one obtains the desired natural isomorphism as follows @xmath578(lp,\textnormal{const}(x)) } & { \iso } & { \lim_{f{\in}i / y } v(l(\textnormal{dom}(f)),x) } \\ & { \iso } & { \lim_f w(\textnormal{dom}(f),rx) } & { \iso } & { \ca b(y, rx) } \end{array}\]] for all @xmath444. [lem : lra - dense] let @xmath208 be a functor, @xmath16 be cocomplete and @xmath196 have a small dense subcategory @xmath220. then @xmath101 is a local right adjoint iff every @xmath579 with @xmath580 admits a generic factorisation. if in addition @xmath16 has a terminal object denoted @xmath33, then generic factorisations in the case @xmath581 suffice. for the first statement (@xmath582) is true by definition so it suffices to prove the converse. the given generic factorisations provide a partial left adjoint @xmath583 to @xmath584 where @xmath34 is the inclusion of @xmath220. now @xmath585 is a small dense subcategory of @xmath586, and so by the previous lemma @xmath212 extends to a genuine left adjoint to @xmath587. in the case where @xmath16 has @xmath33 one requires only generic factorisations in the case @xmath581 by the results of @xcite section(2). the analogous result for presheaf categories, with the representables forming the chosen small dense subcategory, was discussed in @xcite section(2). [prop : gammae - lra] let @xmath16 and @xmath196 be locally c - presentable and @xmath304 be distributive. if @xmath304 is a local right adjoint then so is @xmath301. let @xmath450 be a small dense subcategory of @xmath196 consisting of small connected objects. by lemma([lem : lra - dense]) and lemma([lem : gv - dense]) it suffices to exhibit generic factorisations of maps @xmath588 where @xmath225 is either @xmath26 or @xmath589 for some @xmath452. in the case where @xmath225 is @xmath26 the first arrow in the composite @xmath478 & { \gamma{e}0 } \ar[r]^-{\gamma{e}t } & { \gamma{e}1}}\]] is generic because @xmath26 is the initial @xmath196-graph with one object (and @xmath590 here is the unique map). in the case where @xmath591, to give @xmath76 is to give a map @xmath592 in @xmath16, by corollary([cor : explicit - gamma]) since @xmath348 is connected. since @xmath122 is a local right adjoint, @xmath549 is too and so one can generically factor @xmath593 to obtain @xmath594 ^ -{g'_f } & { \ope\limits_iz_i } \ar[r]^-{\ope\limits_it } & { e_n1}}\]] from which we obtain the generic factorisation @xmath595 ^ -{g_f } & { \gamma{e}z } \ar[r]^-{\gamma{e}t } & { \gamma{e}1}}\]] where @xmath596, the object map of @xmath358 is given by @xmath166 and @xmath597, and the hom map of @xmath358 is @xmath598 composed with the coproduct inclusion. first note that while it is a very different thing for @xmath304 to preserve coproducts compared with preserving coproducts in each variable, the situation is simpler for @xmath392-filtered colimits, where @xmath392 is any regular cardinal. note that @xmath599 preserves @xmath392-filtered colimits in each variable iff @xmath292 preserves @xmath392-filtered colimits. for given a connected category @xmath346, the colimit of a functor @xmath600 constant at say @xmath2 is of course @xmath2, and since @xmath392-filtered colimits are connected, one can prove @xmath601 by keeping all but the variable of interest constant. for the converse it is sufficient to prove that @xmath292 preserves colimits of chains of length less than @xmath392, and this follows by a straight forward transfinite induction. since @xmath111 is a sum of @xmath602 s, from the connectedness of @xmath392-filtered colimits it is clear that @xmath122 preserves @xmath392-filtered colimits iff each @xmath603 does, and so we have proved [lem : multi - lambda] for @xmath304 the following statements are equivalent for any regular cardinal @xmath392. 1. @xmath122 preserves @xmath392-filtered colimits in each variable. @xmath122 preserves @xmath392-filtered colimits. @xmath603 preserves @xmath392-filtered colimits for all @xmath40. as already mentioned, colimits in @xmath18 for a cocomplete @xmath16 were calculated in @xcite. let us spell out transfinite composition in @xmath18. given an ordinal @xmath392 and a @xmath392-chain @xmath604 in @xmath18 with colimit @xmath2, one may consider the induced @xmath392-chain @xmath605 in @xmath24. this will have colimit @xmath606 because @xmath392-filtered colimits and products commute in @xmath24 and @xmath607 is cocontinuous. for @xmath377 let us denote by @xmath608 the full subcategory of the category of elements of ([chain2]), consisting of those elements which are sent to @xmath47 by the universal cocone. we shall call this the _ @xmath47-component _ of the chain ([chain1]). now @xmath608 is of course no longer a chain, but one may easily verify that it is @xmath392-filtered. by the explicit description of colimits in @xmath24, the @xmath608 are just the connected components of the category of elements of ([chain2]). to pairs @xmath609 which are elements of @xmath608, one may associate the corresponding hom @xmath610, and in this way build a functor @xmath611. the hom @xmath31 is the colimit of this functor. [prop : gamma - accessible] let @xmath16 and @xmath196 be cocomplete, @xmath304 be distributive and @xmath392 be a regular cardinal. if @xmath122 preserves @xmath392-filtered colimits in each variable then @xmath301 preserves @xmath392-filtered colimits. it suffices to show @xmath301 preserves colimits of @xmath392-chains. consider the chain ([chain1]) in @xmath18. for all @xmath40 one has an @xmath145-ary version of ([chain2]), that is involving @xmath145-fold instead of binary cartesian products in @xmath24. these of course also commute with @xmath392-filtered colimits. similarly one obtains a @xmath392-filtered category @xmath612 and a functor @xmath613 applying @xmath301 does nothing at the object level. let us write @xmath614 for the @xmath47-component of the chain obtained by applying @xmath301 to ([chain1]), and @xmath615 for the corresponding functor into @xmath196. from the explicit description of @xmath22 s effect on homs of corollary([cor : explicit - gamma]), one sees that @xmath615 is the coproduct of the composites @xmath616 ^ -{f_{x_0,...,x_n } } & { v^n } \ar[r]^-{e_n } & w}.\]] over all sequences @xmath129 starting at @xmath29 and finishing at @xmath30. by lemma([lem : multi - lambda]) the colimits of the @xmath617 are preserved by the @xmath549, and so by the explicit description of colimits of @xmath392-chains in @xmath258, the colimit of ([chain1]) is indeed preserved by @xmath301. let us recall the construction @xmath618. given a monad @xmath147 on @xmath16 a category with products, one has a multitensor @xmath619 defined by @xmath620 and the unit and substitution is induced in the obvious way from @xmath149 and @xmath150. when @xmath16 is lextensive, @xmath101 is a local right adjoint, and @xmath149 and @xmath150 are cartesian, it follows that @xmath619 is a local right adjoint its unit and multiplication are also cartesian. when @xmath101 preserves coproducts and the cartesian product for @xmath16 is distributive, then @xmath619 is a distributive multitensor. if in addition finite limits and filtered colimits commute in @xmath16 (which happens when, for example @xmath16 is locally finitely presentable), then @xmath619 is finitary. moreover by proposition(2.8) of @xcite one has @xmath621 where the enrichment on the right hand side is with respect to cartesian products. thus one can consider the following inductively - defined sequence of monads * put @xmath622 equal to the identity monad on @xmath24. * given a monad @xmath9 on @xmath623, define the monad @xmath624 on @xmath625. recalling that @xmath623 is the category of @xmath0-globular sets. @xcite[thm : enhopi - main - theorem] for @xmath40, @xmath626 is the strict @xmath0-category monad on @xmath0-globular sets. this monad is coproduct preserving, finitary and local right adjoint. by ([eq : tcross - cat]) and corollary([cor : gamma - alg - ecat]), one has @xmath627 and so by definition @xmath628 is the category of strict @xmath0-categories and strict @xmath0-functors between them. by the remarks at the beginning of this section and corollary([cor : gv - topos]) @xmath629 will produce a distributive, finitary, local right adjoint multitensor on a presheaf category when it is fed a coproduct preserving, finitary, local right adjoint monad on a presheaf category. by corollary([cor : gv - topos]), proposition([prop : gammae - basic]), proposition([prop : gamma - accessible]), proposition([prop : gammae - lra]) and proposition([prop : gamma - cart2]), @xmath22 will produce a coproduct preserving, finitary, local right adjoint monad on a presheaf category when it is fed a distributive, finitary, local right adjoint multitensor on a presheaf category. thus the monads @xmath626 are indeed coproduct preserving, finitary and local right adjoint for all @xmath40. the objects of @xmath630 @xmath0-operads over @xmath24 were in @xcite @xcite called `` normalised '' @xmath0-operads. many @xmath0-categorical structures of interest, such as weak @xmath0-categories, can be defined as algebras of @xmath0-operads over @xmath24. objects of @xmath631 are called @xmath0-multitensors. these are a nice class of lax monoidal structures on the category of @xmath0-globular sets. by corollary([cor : mult - nop - equiv]) and theorem([thm : enhopi - main - theorem]) one obtains @xcite[cor : nnp1op - nmult] for all @xmath40, applying @xmath22 gives @xmath632. that is to say, @xmath22 exhibits @xmath145-operads over @xmath24 and @xmath0-multitensors as the same thing, and under this correspondence, the algebras of the operad correspond to the categories enriched in the associated multitensor by corollary([cor : gamma - alg - ecat]). The 2-functoriality of the monad-multitensor correspondence up to this point @xmath22 has been our notation for the process @xmath633 and @xmath405 has been our notation for the reverse construction. for the most complete analysis of these constructions one must acknowledge that they are the object maps of 2-functors in two important ways. this 2-functoriality together with the formal theory of monads @xcite gives a satisfying explanation of how it is that monad distributive laws arise naturally in this subject (see @xcite). as the lax - algebras of a 2-monad @xmath114 (see section([ssec : lmc])), lax monoidal categories form a 2-category @xmath234. see @xcite for a complete description of the 2-category of lax algebras for an arbitrary 2-monad. explicitly a lax monoidal functor between lax monoidal categories @xmath634 and @xmath635 consists of a functor @xmath636, and maps @xmath637 natural in the @xmath177 such that @xmath638 { f_1hx}="r " [dl] { he_1x}="b " " l"(:"r"^-{u_{hx}}:"b"^{\psi_x},:"b"_{hu_x }) } } } [r(5)] { \xybox{\xygraph{!{0;(2,0):(0,.5) : : } { \opf\limits_i\opf\limits_jhx_{ij}}="tl " [r] { \opf\limits_ih\ope\limits_jx_{ij}}="tm " [r] { h\ope\limits_i\ope\limits_jx_{ij}}="tr " [l(.5)d] { h\ope\limits_{ij}x_{ij}}="br " [l] { \opf\limits_{ij}hx_{ij}}="bl " " tl " (: @<1ex>"tm"^-{\opf\limits_i\psi}:@<1ex>"tr"^-{\psi\ope\limits_j}:"br"^{h\sigma},:"bl"_{\sigma{h}}:@<1ex>"br"_-{\psi})}}}}\]] commute for all @xmath2 and @xmath639 in @xmath16. a monoidal natural transformation between lax monoidal functors @xmath640 consists of a natural transformation @xmath641 such that @xmath642 { h\ope\limits_ix_i}="tr " [d] { k\ope\limits_ix_i}="br " [l] { \opf\limits_ikx_i}="bl " " tl " (: @<1ex>"tr"^-{\psi}:"br"^{\phi\ope\limits_i},:"bl"_{\opf\limits_i\phi}:@<1ex>"br"_-{\kappa})}\]] commutes for all @xmath177. [def:2cat - distmult] the 2-category @xmath20 of distributive multitensors, is defined to be the full sub-2-category of @xmath234 consisting of the @xmath634 such that @xmath16 has coproducts and @xmath122 is distributive. for any 2-category @xmath222 recall the 2-category @xmath643 from @xcite of monads in @xmath222. another way to describe this very canonical object is that it is the 2-category of lax algebras of the identity monad on @xmath222. explicitly the 2-category @xmath644 has as objects pairs @xmath645 where @xmath16 is a category and @xmath101 is a monad on @xmath16. an arrow @xmath646 is a pair consisting of a functor @xmath636 and a natural transformation @xmath647 satisfying the obvious 2 axioms : these are just the `` unary '' analogues of the axioms for a lax monoidal functor written out above. for example, any lax monoidal functor @xmath648 as above determines a monad functor @xmath649. a monad transformation between monad functors @xmath650 consists of a natural transformation @xmath641 satisfying the obvious axiom. for example a monoidal natural transformation @xmath522 as above is a monad transformation @xmath651. in fact as we are interested in monads over @xmath24, we shall work not with @xmath644 but rather with @xmath652. an object @xmath645 of this latter 2-category is a category @xmath16 equipped with a functor into @xmath24, together with a monad @xmath101 on @xmath16 which `` acts fibre - wise '' with respect to this functor. that is @xmath101 s object map does nt affect the underlying object set, similarly for the arrow map of @xmath101, and the components of @xmath101 s unit and multiplication are identities on objects in the obvious sense. an arrow @xmath646 of @xmath652 is a pair @xmath648 as in the case of @xmath644, with the added condition that @xmath653 s components are the identities on objects, and similarly the 2-cells of @xmath652 come with an extra identity - on - object condition. we shall now exhibit the 2-functor @xmath19 which on objects is given by @xmath654. let @xmath655 be a lax monoidal functor between distributive lax monoidal categories. then for @xmath127 and @xmath261, we define the hom map @xmath656 to be the composite of @xmath657 { \coprod\limits_{a = x_0,...,x_n = b } h\ope\limits_ix(x_{i-1},x_i)}="m " " l":@<1ex>"m"^-{\coprod \psi}}\]] and @xmath658 s coproduct preservation obstruction map. it follows easily from the definitions that @xmath659 as defined here satisfies the axioms of a monad functor. moreover given a monoidal natural transformation @xmath660, it also follows easily from the definitions that @xmath661 is a monad transformation. it is also straight - forward to verify that these assignments are 2-functorial. by proposition([prop : pl - adjoint - char]) and corollary([cor : pl->copr - pres]) we understand objects of the image of @xmath22 and we collect this information in [prop : image - gamma - objects] for @xmath16 a category with coproducts, a monad @xmath662 over @xmath24 is in the image of @xmath22 iff @xmath101 is distributive and path - like. moreover any such @xmath101 automatically preserves coproducts. one recovers the distributive multitensor @xmath122 such that @xmath663 as @xmath664. since the construction @xmath405 is itself obviously 2-functorial, the arrows and 2-cells in the image of @xmath22 may also be easily characterised. [prop : image - gamma-1 - 2-cells] 1. let @xmath634 and @xmath635 be distributive lax monoidal categories. a monad functor of the form @xmath665 is in the image of @xmath22 iff @xmath666 for some @xmath667.[gamma - char1] 2. let @xmath668 be lax monoidal functors between distributive lax monoidal categories. a monad transformation @xmath669 is in the image of @xmath22 iff it is of the form @xmath670.[gamma - char2] by definition monad functors and transformations in the image of @xmath22 have the stated properties, so we must prove the converse. given @xmath671 such that the components of @xmath653 are the identities on objects, one recovers for @xmath672 from @xmath16, the corresponding lax monoidal functor coherence map as the hom map from @xmath26 to @xmath0 of the component @xmath673. that is to say, we apply @xmath405 to the appropriate monad functors to prove ([gamma - char1]), and we do the same to the appropriate monad transformations to obtain ([gamma - char2]). [def : plmnd] we denote by @xmath674 the following 2-category. its objects are monads @xmath675 over @xmath24 such that @xmath16 has coproducts and @xmath101 is distributive and path - like. its arrows are arrows @xmath676 of @xmath652, and its 2-cells are 2-cells @xmath677 of @xmath652. thus from the proof of proposition([prop : image - gamma-1 - 2-cells]) we have [cor:2eq - mult - mon] @xmath22 and @xmath405 provide a 2-equivalence @xmath678. lax algebras of a 2-monad organise naturally into _ two _ different 2-categories depending on whether one takes lax or oplax algebra morphisms. so in particular one has the 2-category @xmath679 of lax monoidal categories, _ _ op__lax - monoidal functors between them and monoidal natural transformations between those. the coherence @xmath653 for an oplax @xmath655 goes in the other direction, and so its components look like this : @xmath680 the reader should easily be able to write down explicitly the two coherence axioms that this data must satisfy, as well as the condition that must be satisfied by a monoidal natural transformation between oplax monoidal functors. similarly there is a dual version @xmath681 of the 2-category @xmath643 of monads in a given 2-category @xmath222 discussed above @xcite. an arrow @xmath646 of @xmath682 consists of a functor @xmath636 and a natural transformation @xmath683 satisfying the two obvious axioms. an arrow of @xmath682 is called a monad opfunctor. as before @xmath684 differs from @xmath652 in that all the categories involved come with a functor into @xmath24, and all the functors and natural transformations involved are compatible with these forgetful functors. we now describe the dual version of the 2-functoriality of @xmath22 discussed in sections([ssec : gamma-2-functor]) and ([ssec : gamma - image]). when defining the one - cell map of @xmath22 in section([ssec : gamma-2-functor]) we were helped by the fact that the coproduct preservation obstruction went the right way : see the definition of the monad functor @xmath685 above. this time however we will not be so lucky, and for this reason we must restrict ourselves in the following definition to coproduct preserving oplax monoidal functors. [def : opdistmult] the 2-category @xmath686 is defined to be the locally full sub-2-category of @xmath679 consisting of the distributive lax monoidal categories, and the oplax monoidal functors @xmath648 such that @xmath658 preserves coproducts. we denote by @xmath687 the following 2-category. its objects are monads @xmath675 over @xmath24 such that @xmath16 has coproducts and @xmath101 is distributive and path - like. its arrows are arrows @xmath676 of @xmath684 such that @xmath658 preserves coproducts. its 2-cells are 2-cells @xmath677 of @xmath684. we now define @xmath688 with object map @xmath654 as before, and the rest of its definition is obtained by modifying the earlier definition of @xmath22 in what should now be the obvious way. the proof of the following result is obtained by a similar such modification of the proof of corollary([cor:2eq - mult - mon]). [cor:2eq - mult - mon - dual] @xmath22 and @xmath405 provide a 2-equivalence @xmath689 as explained in @xcite the assignment @xmath690 is in fact the object map of a strict 3-functor. just exploiting 2-functoriality here and corollaries([cor:2eq - mult - mon]) and ([cor:2eq - mult - mon - dual]) one immediately obtains [thm : monmon - distlaw] @xmath22 and @xmath405 provide two 2-equivalences of 2-categories : 1. @xmath691.[mdl1] 2. @xmath692.[mdl2] the meaning of this result is understood by understanding what the objects of the 2-categories involved are, that is to say, what monads are in each of the 2-categories @xmath20, @xmath686, @xmath674 and @xmath687. a very beautiful observation of @xcite is that to give a monad on @xmath645 in @xmath643 is to give another monad @xmath225 on @xmath16, together with a distributive law @xmath693. similarly to give a monad on @xmath645 in @xmath681 is to give another monad @xmath225 on @xmath16, together with a distributive law @xmath694 in the other direction. thus the 2-categories @xmath695 and @xmath696 really have the same objects : such an object being a pair of monads on the same category and a distributive law between them. thus both @xmath697 and @xmath698 are 2-categories whose objects are monad distributive laws between monads defined on categories of enriched graphs, with some extra conditions. on the other hand a monad in the 2-category @xmath234 of lax monoidal categories and lax monoidal functors is also a well - known thing, and such things are usually called _ monoidal monads_. similarly an opmonoidal monad @xmath101 on a monoidal category @xmath16, that is to say a monad on @xmath16 in @xmath679, comes with the extra data of coherence maps @xmath699 that are compatible with the monad structure. if for instance @xmath700 is just cartesian product, then the product obstruction maps for @xmath101 endow it with an opmonoidal structure in a unique way. by definition the objects of @xmath701 are monoidal monads defined on distributive lax monoidal categories, and the objects of @xmath702 are coproduct preserving opmonoidal monads defined on distributive lax monoidal categories. thus the meaning of theorem([thm : monmon - distlaw]) is that it exhibits these kinds of monoidal and opmonoidal monads as being equivalent to certain kinds of distributive laws. we shall spell this out precisely in corollaries([cor : monmon - distlaw]) and ([cor : opmonmon - distlaw]) below. let @xmath634 be a lax monoidal category and @xmath101 be a monad on @xmath16. in section([ssec : laxalg - const1]) we saw that when @xmath101 is a monoidal monad, that is to say one has coherence maps @xmath703 making the underlying endofunctor of @xmath101 a lax monoidal functor and the unit and multiplication monoidal natural transformations, then one has another multitensor on @xmath16 given on objects by @xmath704, and with unit and substitution given by the composites @xmath705 { tx}="m " [r(1.5)] { te_1x}="r " " l":"m"^-{\eta}:"r"^-{tu } } } } [r(5)d(.1)] { \xybox{\xygraph{!{0;(2,0) : } { t\ope\limits_it\ope\limits_j x_{ij}}="l " [r] { t^2\ope\limits_i\ope\limits_j x_{ij}}="m " [r] { t\ope\limits_{ij}x_{ij}}="r " " l":@<1ex>"m"^-{t{\tau}e}:@<1ex>"r"^-{\mu\sigma}}}}}\]] in particular if @xmath122 is distributive and @xmath101 preserves coproducts, then this new multitensor @xmath706 is also distributive. if instead @xmath101 has the structure of an opmonoidal monad, with the coherences @xmath707 going in the other direction, then in the same way one can construct a new multitensor @xmath708 on @xmath16 which on objects is defined by given by @xmath709. once again if @xmath122 is distributive and @xmath101 coproduct preserving, then @xmath708 is a distributive multitensor. in particular when @xmath122 is cartesian product, @xmath708 is the multitensor @xmath619 of section([ssec : induction]). with regards to monoidal monads, unpacking what theorem([thm : monmon - distlaw])([mdl1]) says at the object level gives [cor : monmon - distlaw] let @xmath634 be a distributive lax monoidal category and @xmath101 be a monad on @xmath16. to give maps @xmath710 making @xmath101 into a monoidal monad on @xmath634, is the same as giving a monad distributive law @xmath711 whose components are the identities on objects. in the case where @xmath101 preserves coproducts one may readily verify that @xmath712 as monads, and so by corollary([cor : gamma - alg - ecat]) one understands what the algebras of this composite monad @xmath713 are. [cor : algebras - gtgammae] if in the situation of corollary([cor : monmon - distlaw]) @xmath101 also preserves coproducts, then @xmath714. similarly, one can unpack what theorem([thm : monmon - distlaw])([mdl2]) says at the object level, witness @xmath715 and use corollary([cor : gamma - alg - ecat]) to conclude [cor : opmonmon - distlaw] let @xmath634 be a distributive lax monoidal category and @xmath101 be a coproduct preserving monad on @xmath16. to give maps @xmath716 making @xmath101 into an opmonoidal monad on @xmath634, is the same as giving a monad distributive law @xmath717 whose components are the identities on objects. moreover @xmath718. [ex : cheng1] from the inductive description of @xmath436 of section([ssec : induction]) and corollary([cor : opmonmon - distlaw]) one obtains a distibutive law @xmath719 for all @xmath0, between monads on @xmath623, and the composite monad @xmath720. thus we have recaptured the decomposition of @xcite of the strict @xmath0-category monad into a `` distributive series of monads ''. pursuing the idea of the previous example, we shall now begin to recover and in some senses generalise cheng s analysis and description @xcite of the trimble definition of weak @xmath0-category. from @xcite example(2.6) non - symmetric operads in the usual sense can be regarded as multitensors. here we shall identify a non - symmetric operad @xmath721 in a braided monoidal category @xmath16, with the multitensor @xmath722 it generates. recall that one object @xmath122-categories for @xmath122 a non - symmetric operad are precisely algebras of the operad @xmath122 in the usual sense. if @xmath700 is cartesian product, then the projections @xmath723 are the components of a cartesian multitensor map @xmath724. conversely such a cartesian multitensor map exhibits @xmath122 as an operad via @xmath725 for all @xmath40. let @xmath16 be a distributive category and @xmath101 a coproduct preserving monad on @xmath16. let us denote by @xmath726 a non - symmetric operad in @xmath102. the `` @xmath429 '' is meant to denote the @xmath101-algebra actions, that is @xmath727 is the @xmath101-algebra structure, and so @xmath122 denotes the underlying operad in @xmath16. since @xmath728 preserves products it is the underlying functor of a strong monoidal functor @xmath729 between lax monoidal categories. since the composites @xmath730 { te_n \times \prod\limits_i tx_i}="m " [r] { e_n \times \prod\limits_itx_i}="r " " l":@<1ex>"m"^-{\textnormal{prod. obstn.}}:@<1ex>"r"^-{\varepsilon_n \times \id}}\]] form the components of an opmonoidal structure for the monad @xmath101, we find ourselves in the situation of corollary([cor : opmonmon - distlaw]) and so obtain [prop : monad - trimble] let @xmath16 be a distributive category, @xmath101 a coproduct preserving monad on @xmath16 and @xmath726 a non - symmetric operad in @xmath102. then one has a distributive law @xmath731 between monads on @xmath18, and isomorphisms @xmath732 of categories over @xmath18. this result has an operadic counterpart. [prop : operad - trimble] let @xmath16 be a lextensive category, @xmath101 a cartesian and coproduct preserving monad on @xmath16, @xmath733 a @xmath101-operad and @xmath726 a non - symmetric operad in @xmath734. then the monad @xmath735, whose algebras by proposition([prop : monad - trimble]) are @xmath726-categories, has a canonical structure of a @xmath736-operad. with @xmath737 we must exhibit a cartesian monad map @xmath738. we have the cartesian multitensor map @xmath739 which exhibits the multitensor @xmath726 as a non - symmetric operad, thus @xmath740 is also a cartesian multitensor map, and since @xmath16 is lextensive @xmath22 sends this to a cartesian monad morphism. the required cartesian monad map is thus @xmath741. recall the path - space functor @xmath742 discussed in section([ssec : enriched - graphs]). to say that a non - symmetric topological operad @xmath1 acts on @xmath180 is to say that @xmath180 factors as @xmath94 { \enrich a}="m " [r] { \ca g(\top)}="r " " l":"m"^-{p_a}:"r"^-{u^a}}\]] the main example to keep in mind is the version of the little intervals operad recalled in @xcite definition(1.1). as this @xmath1 is a contractible non - symmetric operad, @xmath1-categories may be regarded as a model of @xmath1-infinity spaces. since @xmath180 is a right adjoint, @xmath743 is also a right adjoint by the dubuc adjoint triangle theorem. a product preserving functor @xmath744 into a distributive category, may be regarded as the underlying functor of a strong monoidal functor @xmath745 between lax monoidal categories. applying @xmath22 to this gives us a monad functor @xmath746 with underlying functor @xmath747, which amounts to giving a lifting @xmath748 as indicated in the commutative diagram @xmath749 { \enrich a}="tm " [r] { \enrich { qa}}="tr " [d] { \ca gv}="br " [l] { \ca g(\top)}="bl " " tl"(:"tm"^-{p_a}(:"tr"^-{\overline{q}}:"br"^{u^{qa}},:"bl":"br"_-{\ca g(q)}),:"bl"_{p})}\]] and so we have produced another product preserving functor @xmath750 where @xmath751 and @xmath752. the functor @xmath748 is product preserving since @xmath753 is and @xmath754 creates products. the assignment @xmath755 in the case where @xmath1 is as described in @xcite definition(1.1), is the inductive process lying at the heart of the trimble definition. in this definition one begins with the connected components functor @xmath756 and defines the category @xmath757 of `` trimble 0-categories '' to be @xmath24. the induction is given by @xmath758 and so this definition constructs not only a notion of weak @xmath0-category but the product preserving @xmath759 s to be regarded as assigning the fundamental @xmath0-groupoid to a space. applying proposition([prop : monad - trimble]) to this situation produces the monad on @xmath0-globular sets whose category of algebras is @xmath760 as well as its decomposition into an iterative series of monads witnessed in @xcite section(4.2). applying proposition([prop : operad - trimble]) and the inductive description of @xmath436 of section([ssec : induction]) exhibits these monads as @xmath0-operads. Lifting multitensors applied to the normalised @xmath13-operad for gray categories @xcite, the results of the section([sec : reexpress]) produce a lax monoidal structure @xmath122 on the category of @xmath15-globular sets whose enriched categories are exactly gray categories. for this example it turns out that @xmath123 is @xmath761, and so providing a lift of @xmath122 in the sense of definition([def : lift]) amounts to the construction of a tensor product of @xmath15-categories whose enriched categories are gray categories, that is to say, an abstractly constructed gray tensor product. by the main result of this section theorem([thm : lift - mult]), _ every _ @xmath0-multitensor has a lift which is unique given certain properties. while the proof of theorem([thm : lift - mult]) is fairly abstract, and the uniqueness has the practical effect that in the examples we never have to unpack an explicit description of the lifted multitensors provided by the theorem, we provide such an unpacking in section([ssec : explicit - lifting]) anyway. this enables us to give natural conditions when the construction of the lifted multitensor is simpler. doing all this requires manipulating some of the transfinite constructions that arise in monad theory, and we give a self - contained review of these in the appendix. in appendix [sec : dubuc] we recall an explicit description, for a given monad morphism @xmath762 between accessible monads on a locally presentable category @xmath16, of the left adjoint @xmath763 to the canonical forgetful functor @xmath764 induced by @xmath522. the key point about @xmath763 is that it is constructed via a transfinite process involving only _ connected _ colimits in @xmath16. the importance of this is underscored by [lem : concol - pathlike] let @xmath16 be a category with an initial object, @xmath196 be a cocomplete category, @xmath210 be a small connected category and @xmath765\]] be a functor. suppose that @xmath292 sends objects of @xmath210 to normalised functors, and arrows of @xmath210 to natural transformations whose components are identities on objects. * then the colimit @xmath766 of @xmath292 may be chosen to be normalised.[cpl1] given such a choice of @xmath767 : * if @xmath768 is path - like for all @xmath557, then @xmath767 is also path - like.[cpl2] * if @xmath768 is distributive for all @xmath557, then @xmath767 is also distributive.[clp3] colimits in @xmath769 $] are computed componentwise from colimits in @xmath258 and so for @xmath127 we must describe a universal cocone with components @xmath770 we demand that the @xmath771 are identities on objects. this is possible since the @xmath772 form the constant diagram on @xmath6 by the hypotheses on @xmath292. for @xmath261 we choose an arbitrary colimit cocone @xmath773 in @xmath196. one may easily verify directly that since @xmath210 is connected, the @xmath771 do indeed define a univeral cocone for all @xmath2 in order to establish (1). since the properties of path - likeness and distributivity involve only colimits at the level of the homs as does the construction of @xmath767 just given, (2) and (3) follow immediately since colimits commute with colimits in general. with these preliminaries in hand we are now ready to present the monad version of the multitensor lifting theorem, and then the lifting theorem itself. [lem : mnd - lift - mult] let @xmath16 be a locally presentable category, @xmath318 be a coproduct preserving monad on @xmath16, @xmath225 be an accessible and normalised monad on @xmath18, and @xmath774 be a monad morphism whose components are identities on objects. denote by @xmath101 the monad induced by @xmath775. * one may choose @xmath763 so that @xmath101 becomes normalised. given such a choice of @xmath763 : * if @xmath225 is path - like then @xmath101 is path - like. * if @xmath225 is distributive then @xmath101 is distributive. let @xmath392 be the regular cardinal such that @xmath225 preserves @xmath392-filtered colimits. to verify that @xmath776 is a normalised monad one must verify : (i) @xmath101 is normalised, and (ii) the components of the unit @xmath777 are identities on objects. since @xmath778 is a retraction of @xmath779, it will then follow that the components of @xmath778 are also identities on objects. but @xmath101 is normalised iff @xmath780 is normalised, and @xmath781, so for (i) it suffices to show that one can choose @xmath763 making @xmath782 normalised. this follows by a transfinite induction using the explicit description of @xmath782 of section([sec : dubuc]) and lemma([lem : concol - pathlike]). for the initial step note that @xmath783 can be chosen to be normalised, because @xmath784 is a componentwise - identity on objects natural transformation between normalised functors, since the monads @xmath225 and @xmath785 are normalised. thus the coequaliser defining @xmath783 is a connected colimit involving only normalised functors and componentwise - identity on objects natural transformations, and so @xmath783 can be taken to be normalised by lemma([lem : concol - pathlike]). for the inductive steps the argument is basically the same : at each stage one is taking connected colimits of normalised functors and componentwise identity on objects natural transformations, so that by lemma([lem : concol - pathlike]) one stays within the subcategory of @xmath786 $] consisting of such functors and natural transformations. moreover using lemma([lem : concol - pathlike]) @xmath101 will be path - like if @xmath225 is. as for (ii) it suffices to prove that the components of @xmath787 are identities on objects. writing @xmath788 for transfinite composite constructed as part of the definition of @xmath763 (note that @xmath789 by definition) recall from the end of section([sec : dubuc]) that one has a commutative square @xmath790 ^ -{\ca g\rho } \ar[d]_{{\phi}u^m } & { u^{\ca gr } } \ar[d]^{\ca g(u^r)\eta^t } \\ { su^{\ca gr } } \ar[r]_-{q } & { u^s\phi_{!}}}\]] where @xmath791 is the 2-cell datum for @xmath318 s eilenberg - moore object, which we recall is preserved by @xmath51. now @xmath791 is componentwise the identity objects since @xmath792 is and @xmath791 is a retraction of it, @xmath793 is the identity on objects by definition, and @xmath794 is by construction, so the result follows. recall from definition([def : lift]) that a _ lift _ of @xmath124 is a normal multitensor @xmath142 on @xmath141 together with an isomorphism @xmath143 which commutes with the forgetful functors into @xmath144. when in addition @xmath305 is distributive, we say that it is a _ distributive lift _ of @xmath122. recall from @xcite that for any category @xmath16, the functor @xmath795 which sends a monad @xmath101 on @xmath16 to the forgetful functor @xmath103, is fully - faithful. [thm : lift - mult] let @xmath124 be a distributive multitensor on @xmath16 a locally presentable category, and let @xmath122 be accessible in each variable. then @xmath122 has a distributive lift @xmath305, which is unique up to isomorphism. write @xmath796 for the distributive multitensor on @xmath16 whose unary part is @xmath123 and whose non - unary parts are constant at @xmath151. there is an obvious inclusion @xmath797 of multitensors and one clearly has @xmath798 applying lemma([lem : mnd - lift - mult]) with @xmath799, @xmath800 and @xmath801 one produces a path - like, normalised and distributive monad @xmath101 on @xmath802, because @xmath225 is accessible by proposition([prop : gamma - accessible]). thus one has a distributive multitensor @xmath173 on @xmath141. applying proposition([prop : pl - alg<->cat]) to @xmath173, and corollary([cor : gamma - alg - ecat]) to @xmath122, gives @xmath803 in view of the monadicity of @xmath804. that is to say, @xmath173 is a distributive lift of @xmath122. as for uniqueness suppose that @xmath142 is a distributive lift of @xmath122. then by corollary([cor : gamma - alg - ecat]) and proposition([prop : pl - adjoint - char]), @xmath805 is a distributive monad on @xmath144 and one has @xmath806 commuting with the forgetful functors into @xmath144. by the fully - faithfulness of @xmath807 recalled above, one has an isomorphism @xmath808 of monads, and thus by proposition([prop : pl - adjoint - char]) an isomorphism @xmath809 of multitensors. applying this result to any normalised @xmath145-operad @xmath1, exhibits its algebras as categories enriched in the algebras of some @xmath0-operad. the @xmath0-operad is @xmath810, and tensor product over which one enriches is @xmath811. in cases where we already know what our tensor product ought to be, the uniqueness part of theorem([thm : lift - mult]) ensures that it is. an instance of this is [ex : gray] in @xcite the normalised 3-operad @xmath23 whose algebras are gray categories was constructed. as we have already seen, @xmath812 is a lax monoidal structure on @xmath813 whose enriched categories are gray categories, and @xmath814 is the operad for strict 2-categories. note that the usual gray tensor product is symmetric monoidal closed and thus distributive. thus by theorem([thm : lift - mult]) @xmath815 is the gray tensor product. in other words, the general methods of this paper have succeeded in producing the gray tensor product of @xmath15-categories from the operad @xmath23. more generally given a distributive tensor product @xmath700 on the category of algebras of an @xmath0-operad @xmath109, and a normalised @xmath145-operad @xmath1 whose algebras are the categories enriched in @xmath109-algebras, theorem([thm : lift - mult]) exhibits @xmath700 as the more generally constructed @xmath811. [ex : crans] in @xcite sjoerd crans explicitly constructed a tensor product on the category of gray - categories. this explicit construction was extremely complicated. it is possible to exhibit the crans tensor product as an instance of our general theory, by rewriting his explicit constructions as the construction of the 4-operad whose algebras are teisi in his sense. the multitensor @xmath122 associated to this 4-operad has @xmath123 equal to the 3-operad for gray categories. thus theorem([thm : lift - mult]) constructs a lax tensor product of gray categories whose enriched categories are teisi. since the tensor product explicitly constructed by crans is distributive, the uniqueness of part of theorem([thm : lift - mult]) ensures that it is indeed @xmath305, since teisi are categories enriched in the crans tensor product by definition. honestly writing the details of the 4-operad of example([ex : crans]) is a formidable task and we have omitted this here. in the end though, such details will not be important, because such a tensor product (or more properly a biclosed version thereof) will only be really useful once it is given a conceptual definition. let @xmath16 be a symmetric monoidal model category which satisfies the conditions of @xcite or the monoid axiom of @xcite. in this case the category of pruned @xmath0-operads of @xcite can be equipped with a monoidal model structure @xcite. so we can speak of cofibrant @xmath0-operads in @xmath16. for @xmath273 let us fix a particular cofibrant and contractible @xmath33-operad @xmath1. the algebras of @xmath816 can be called _ @xmath25-categories enriched in @xmath16_. up to homotopy the choice of @xmath816 is not important. so we can speak of _ the _ category of @xmath25-categories. for @xmath334 we denote by @xmath817 a cofibrant contractible @xmath15-operad in @xmath16. let @xmath818 be the corresponding multitensor on @xmath18. one can always choose @xmath817 in such a way that its unary part is @xmath816. as in @xcite for an arbitary multitensor @xmath122, one object @xmath122-categories are called @xmath122-monoids. similarly one object @xmath25-categories are called @xmath25-monoids. [thm : a - infinity - app] 1. there is a distributive lift @xmath819 of @xmath109 to the category of @xmath25-categories. 2. @xmath819 restricts to give a multitensor @xmath346 on the category of @xmath25-monoids. the category of @xmath346-monoids is equivalent to the category of algebras of @xmath820 and therefore is quillen equivalent to the category of the algebras of the little squares operad. the first statement is a direct consequence of theorem([thm : lift - mult]). the second statement follows from the fact that @xmath819 is the cartesian product on the object level. the last statement follows from the theorem(8.6) of @xcite. applying this result to the case @xmath821 with its folklore model structure one recovers [cor : js - braided] [joyal - street] the category of braided monoidal categories is equivalent to the category of monoidal categories equipped with multiplication. the previous corollary is proved by joyal and street @xcite by a direct application of a `` categorified '' eckmann - hilton argument. the following analogous result for @xmath15-categories appears to be new. [cor : coh - bm2c] the category of braided monoidal @xmath15-categories is equivalent to the category of gray - monoids with multiplication. apply theorem([thm : a - infinity - app]) with @xmath822 equipped with the gray tensor product and lack s folklore model structure for 2-categories @xcite. thus theorem([thm : a - infinity - app]) should be considered as an @xmath823-generalisation of the above corollaries. we believe it sheds some light on the problem of defining the tensor product of @xmath25-algebras initiated by @xcite. as explained in the introduction, the negative result of @xcite shows that there is no hope to get an `` honest '' tensor product of such algebras. thus the multitensor @xmath346 constructed in theorem([thm : a - infinity - app]) is genuinely lax, and exhibits laxity as a way around the aforementioned negative result. in future work we will generalise this theorem to arbitrary dimensions. let us now instantiate the constructions of section([sec : dubuc]) to produce a more explicit description of the lifted multitensor @xmath305. beyond mere instantiation this task amounts to reformulating everything in terms of hom maps which live in @xmath16, because in our case the colimits being formed in @xmath18 at each stage of the construction are connected colimits diagrams whose morphisms are all identity on objects. moreover these fixed object sets are of the form @xmath154 for @xmath40. + + * notation*. we shall be manipulating sequences of data and so we describe here some notation that will be convenient. a sequence @xmath824 from some set will be denoted more tersely as @xmath825 leaving the length unmentioned. similarly a sequence of sequences @xmath826 of elements from some set will be denoted @xmath827 the variable @xmath46 ranges over @xmath164 and the variable @xmath134 ranges over @xmath132. triply - nested sequences look like this @xmath828, and so on. these conventions are more or less implicit already in the notation we have been using all along for multitensors. see especially section([ssec : lmc]) and @xcite. we denote by @xmath829 the ordinary sequence obtained from the @xmath244-tuply nested sequence @xmath830 by concatenation. in particular given a sequence @xmath825, the set of @xmath827 such that @xmath831 is just the set of partitions of the original sequence into doubly - nested sequences, and will play an important role below. this is because to give the substitution maps for a multitensor @xmath122 on @xmath16, is to give maps @xmath832 for all @xmath833 and @xmath834 from @xmath16 such that @xmath835. + + the monad map @xmath762 is taken as @xmath836 where @xmath837 is the inclusion of the unary part of the multitensor @xmath122. note the notational abuse we regard write @xmath123 for the multitensor on @xmath16 obtained from @xmath122 by ignoring (ie setting to constant at @xmath151) the non - unary parts, but also as the monad on @xmath16 and so @xmath838 as monads. the role of @xmath104 in @xmath839 is played by sequences @xmath840 of @xmath123-algebras regarded as objects of @xmath841 as in section([ssec : defnmonad]). the transfinite induction produces for each ordinal @xmath227 and each sequence of @xmath123-algebras as above of length @xmath0, morphisms @xmath842 in @xmath18 which are identities on objects, and thus we shall now evolve this notation so that it only records what s going on in the hom between @xmath26 and @xmath0. by the definition of @xmath225 we have the equation on the left @xmath843 and the equation on the right is a definition. because of these definitions and that of @xmath225 we have the equation @xmath844 the data for the hom maps of the @xmath845 thus consists of morphisms @xmath846 in @xmath16 whenever one has @xmath847 as sequences of @xmath123-algebras. to summarise, the output of the transfinite process we are going to describe is, for each ordinal @xmath227, the following data. for each sequence @xmath840 of @xmath123-algebras, one has an object @xmath848 and morphisms @xmath849 of @xmath16 where @xmath847. + + * initial step*. first we put @xmath850, @xmath851, and then form the coequaliser @xmath852 { \ope\limits_ix_i}="m " [r] { \opeone\limits_i(x_i, x_i)}="r " " l":@<2ex>"m"^-{\sigma } " l":"m"_-{\ope\limits_ix_i}:@<1ex>"r"^-{q^{(0)}_{(x_i, x_i)}}}\]] in @xmath16 to define @xmath853. put @xmath854 and @xmath855. + + * inductive step*. assuming that @xmath845, @xmath856 and @xmath857 are given, we have maps @xmath858 { \ope\limits_i\opempone\limits_{jk}}="r " " l":"r"^-{\ope\limits_iv^{(m) } } } } } & & { \xybox{\xygraph{!{0;(2,0) : } { \ope\limits_i\ope\limits_j\opem\limits_k}="l " [r] { \ope\limits_{ij}\opem\limits_k}="m " [r] { \ope\limits_{ij}\opempone\limits_k}="r " " l":"m"^-{{\sigma}\opem\limits_k}:"r"^-{q^{(m) } } } } } \end{array}\]] and these are used to provide the parallel maps in the coequaliser @xmath859 { \coprod\limits_{\con(x_{ij},x_{ij})=(x_i, x_i) } \ope\limits_i\opempone\limits_j(x_{ij},x_{ij})}="m " [d] { \opemptwo\limits_i (x_i, x_i)}="r " " l":@<-2ex>"m " " l":@<2ex>"m":"r"^-{(v^{(m{+}1)}_{(x_{ij},x_{ij})})}}\]] which defines the @xmath860, the commutative diagram @xmath861 { e_1\opempone\limits_i (x_i, x_i)}="il " [r(4)] { \coprod\limits_{\con(x_{ij},x_{ij})=(x_i, x_i) } \ope\limits_i\opempone\limits_j(x_{ij},x_{ij})}="ir " [ru] { \opemptwo\limits_i (x_i, x_i)}="r " " l":"il"_-{u}:"ir":"r"^(.35){v^{(m{+}1)}_{(x_i, x_i)}}:@{<-}"l"_-{q^{(m{+}1)}_{(x_i, x_i)}}}\]] in which the unlabelled map is the evident coproduct inclusion defines @xmath862, and @xmath863. + + * limit step*. define @xmath864 as the colimit of the sequence given by the objects @xmath865 and morphisms @xmath866 for @xmath867, and @xmath868 for the component of the universal cocone at @xmath869. @xmath870 { \colsum\limits_{\con(x_{ij},x_{ij})=(x_i, x_i) } \ope\limits_i\oper\limits_j(x_{ij},x_{ij})}="tm " [r] { \colim_{r{<}m } \oper\limits_i (x_i, x_i)}="tr " [d] { \opem\limits_i (x_i, x_i)}="br " [l] { \coprod\limits_{\con(x_{ij},x_{ij})=(x_i, x_i) } \ope\limits_i\opem\limits_j(x_{ij},x_{ij})}="bm " [l] { \coprod\limits_{\con(x_{ijk},x_{ijk})=(x_i, x_i) } \ope\limits_i\ope\limits_j\opem\limits_k(x_{ijk},x_{ijk})}="bl " " tl":@<1ex>"tm"^-{\sigma^{(<{m})}}:@<1ex>"tr"^-{v^{(<{m }) } } " tl":@<-1ex>"tm"_-{(ev)^{(<{m})}}:@<-1ex>@{<-}"tr"_-{u^{(<{m }) } } " bl":"bm"_-{\mu}:@{<-}"br"_-{uc } " tl":"bl"^{o_{m,2 } } " tm":"bm"^{o_{m,1 } } " tr":@{=}"br"}\]] as before we write @xmath871 and @xmath872 for the obstruction maps, and @xmath231 denotes the evident coproduct injection. the maps @xmath873, @xmath874, @xmath875 and @xmath876 are by definition induced by @xmath877, @xmath878, @xmath879 and @xmath880 for @xmath867 respectively. define @xmath845 as the coequaliser of @xmath881 and @xmath882, @xmath883 and @xmath884. + + instantiating corollary([cor : explicit - phi - shreik]) to the present situation gives [cor : lifted - obj] let @xmath16 be a locally presentable category, @xmath392 a regular cardinal, and @xmath122 a distributive @xmath392-accessible multitensor on @xmath16. then for any ordinal @xmath227 with @xmath885 one may take @xmath886 where the action @xmath887 is given as the composite @xmath888 { \opempone\limits_i (x_i, x_i)}="m " [r] { \opem\limits_i (x_i, x_i)}="r " " l":"m"^-{v^{(m)}}:"r"^-{(q^{(m)})^{-1}}}\]] as an explicit description of the object map of the lifted multitensor @xmath305 on @xmath141. in corollaries ([cor : phi - shreik - simple]) and ([cor : vexp - simple]), in which the initial data is a monad map @xmath762 between monads on a category @xmath16 together with an algebra @xmath104 for @xmath114, we noted the simplification of our constructions when @xmath225 and @xmath889 preserve the coequaliser @xmath890 { sx}="m " [r] { q_1x}="r " " l":@<-1ex>"m"_-{sx } " l":@<1ex>"m"^-{\mu^ss(\phi)}:"r"^-{q_0}}\]] in @xmath16, which is part of the first step of the inductive construction of @xmath763. in the present situation the role of @xmath16 is played by the category @xmath18, the role of @xmath225 is played by @xmath891, and the role of @xmath104 played by a given sequence @xmath840 of @xmath123-algebras, and so the role of the coequaliser ([eq : monad - coeq]) is now played by the coequaliser @xmath892 { \gamma e(x_i)}="m " [r] { q_1}="r " " l":@<-1ex>"m"_-{sx } " l":@<1ex>"m"^-{\mu^ss(\phi)}:"r"^-{q^{(0)}}}\]] in @xmath18. here we have denoted by @xmath893 the @xmath16-graph with objects @xmath154 and homs given by @xmath894 taking the hom of ([eq : monad - coeq2]) between @xmath26 and @xmath0 gives the coequaliser @xmath895 { \ope\limits_ix_i}="m " [r] { \opeone\limits_i(x_i, x_i)}="r " " l":"m"_-{\ope\limits_ix_i } " l":@<2ex>"m"^-{\sigma}:@<1ex>"r"^-{q^{(0)}}}\]] in @xmath16 which is part of the first step of the explicit inductive construction of @xmath305. we shall refer to ([eq : mult - coeq]) as the _ basic coequaliser associated to the sequence @xmath840 _ of @xmath123-algebras. note that all coequalisers under discussion here are reflexive coequalisers, with the common section for the basic coequalisers given by the maps @xmath896. the basic result which expresses why reflexive coequalisers are nice, is the @xmath897-lemma, which we record here for the reader s convenience. a proof can be found in @xcite. [lem:3by3] * @xmath897-lemma*. given a diagram @xmath898 ^ -{f_1 } \ar@<-1ex>[r]_{g_1 } \ar@<1ex>[d]^{b_1 } \ar@<-1ex>[d]_{a_1 } & b \ar[r]^-{h_1 } \ar@<1ex>[d]^{b_2 } \ar@<-1ex>[d]_{a_2 } & c \ar@<1ex>[d]^{b_3 } \ar@<-1ex>[d]_{a_3 } \\ d \ar@<1ex>[r]^-{f_2 } \ar@<-1ex>[r]_-{g_2 } & e \ar[r]^-{h_2 } & f \ar[d]^{c } \\ & & h}\]] in a category such that : (1) the two top rows and the right - most column are coequalisers, (2) @xmath899 and @xmath900 have a common section, (3) @xmath901 and @xmath902 have a common section, (3) @xmath903, (4) @xmath904, (5) @xmath905 and (6) @xmath906 ; then @xmath907 is a coequaliser of @xmath903 and @xmath904. if @xmath908 is a functor which preserves connected colimits of a certain type, then it also preserves these colimits in each variable separately, because for a connected colimit, a cocone involving only identity arrows is a universal cocone. the most basic corollary of the @xmath897-lemma says that the converse of this is true for reflexive coequalisers. [cor:3by3] let @xmath908 be a functor. if @xmath292 preserves reflexive coequalisers in each variable separately then @xmath292 preserves reflexive coequalisers. and this can be proved by induction on @xmath0 using the @xmath897-lemma in much the same way as @xcite lemma(1). the most well - known instance of this is [cor:3by3 - 2]@xcite let @xmath239 be a biclosed monoidal category. then the @xmath0-fold tensor product of reflexive coequalisers in @xmath239 is again a reflexive coequaliser. in particular note that by corollary([cor:3by3]) a multitensor @xmath122 preserves (some class of) reflexive coequalisers iff it preserves them in each variable separately. returning to our basic coequalisers an immediate consequence of the explicit description of @xmath891 and corollary([cor:3by3]) is [lem : reformulate - simplifying - conditions] let @xmath122 be a distributive multitensor on @xmath16 a cocomplete category, and @xmath840 a sequence of @xmath123-algebras. if @xmath122 preserves the basic coequalisers associated to all the subsequences of @xmath840, then for all @xmath909, @xmath910 preserves the coequaliser ([eq : monad - coeq2]). and applying this lemma and corollary([cor : phi - shreik - simple]) gives [cor : lifted - obj - simple] let @xmath16 be a locally presentable category, @xmath392 a regular cardinal, @xmath122 a distributive @xmath392-accessible multitensor on @xmath16 and @xmath840 a sequence of @xmath123-algebras. if @xmath122 preserves the basic coequalisers associated to all the subsequences of @xmath840, then one may take @xmath911 where the action @xmath29 is defined as the unique map such that @xmath912. note in particular that when the sequence @xmath840 of @xmath123-algebras is of length @xmath332 or @xmath273, the associated basic coequaliser is absolute. in the @xmath332 case the basic coequaliser is constant at @xmath913, and when @xmath273 the basic coequaliser may be taken to be the canonical presentation of the given @xmath123-algebra. thus in these cases it follows from corollary([cor : lifted - obj - simple]) that @xmath914 and @xmath915. reformulating the explicit description of the unit in corollary([cor : vexp - simple]) one recovers the fact from our explicit descriptions, that the unit of @xmath305 is the identity, which was of course true by construction. to complete the task of giving a completely explicit description of the multitensor @xmath305 we now turn to unpacking its substitution. so we assume that @xmath122 is a distributive @xmath392-accessible multitensor on @xmath16 a locally presentable category, and fix an ordinal @xmath227 so that @xmath885, so that @xmath305 may be constructed as @xmath916 as in corollary([cor : lifted - obj]). by transfinite induction on @xmath917 we shall generate the following data : @xmath918 and @xmath919 whenever @xmath847, such that @xmath920 { \operpone\limits_{ij}\opem\limits_k}="tr " [d] { \opem\limits_{ijk}}="br " [l] { \ope\limits_i\opem\limits_{jk}}="bl " " tl":"tr"^-{v^{(r)}e^{(m)}}:"br"^-{\sigma^{(r{+}1)}}:@{<-}"bl"^-{(q^{(m)})^{-1}v^{(m)}}:@{<-}"tl"^-{\ope\limits_i\sigma^{(r)}}}\]] commutes. + + * initial step*. define @xmath921 to be the identity and @xmath922 as the unique map such that @xmath923 by the universal property of the coequaliser @xmath853. + + * inductive step*. define @xmath924 as the unique map such that @xmath925 using the universal property of @xmath926 as a coequaliser. + + * limit step*. when @xmath917 is a limit ordinal define @xmath927 as induced by the @xmath928 for @xmath929 and the universal property of @xmath930 as the colimit of the sequence of the @xmath931 for @xmath929. then define @xmath932 as the unique map such that @xmath933 using the universal property of @xmath879 as a coequaliser. + + the fact that the transfinite construction just specified was obtained from that for corollary([cor : induced - monad - very - explicit]), by taking @xmath934 and looking at the homs, means that by corollaries ([cor : induced - monad - very - explicit]) and ([cor : vexp - simple]) one has [cor : induced - substitution - very - explicit] let @xmath16 be a locally presentable category, @xmath392 a regular cardinal, @xmath122 a distributive @xmath392-accessible multitensor on @xmath16 and @xmath840 a sequence of @xmath123-algebras. then one has @xmath935 as an explicit description of the substitution of @xmath305. if moreover @xmath122 preserves the basic coequalisers of all the subsequences of @xmath840, then one may take @xmath936 as the explicit description of the substitution. recall @xcite @xcite that when @xmath222 has eilenberg - moore objects, the one and 2-cells of the 2-category @xmath643 admit another description. given monads @xmath645 and @xmath937 in @xmath222, and writing @xmath103 and @xmath938 for the one - cell data of their respective eilenberg - moore objects, to give a monad functor @xmath939, is to give @xmath658 and @xmath940 such that @xmath941. this follows immediately from the universal property of eilenberg - moore objects. similarly to give a monad 2-cell @xmath942 is to give @xmath943 and @xmath944 commuting with @xmath728 and @xmath945. note that eilenberg - moore objects in @xmath58 are computed as in @xmath99, and we shall soon apply these observations to the case @xmath946. suppose we have a lax monoidal functor @xmath655, that @xmath16 and @xmath196 are locally presentable, and that @xmath122 and @xmath292 are accessible. then we obtain a commutative diagram @xmath947 { \ca gv^{e_1}}="tm " [r] { \ca gv}="tr " [d] { \ca gw}="br " [l] { \ca gw^{f_1}}="bm " [l] { \enrich f}="bl " " tl":"tm":"tr " " bl":"bm":"br " " tl":"bl " " tm":"bm " " tr":"br"}\]] of forgetful functors in @xmath58. applying the previous paragraph to the left - most square gives a monad morphism @xmath948, and then applying @xmath405 to this gives the lax monoidal functor @xmath949 between the induced lifted multitensors. arguing similarly for monoidal transformations and monad 2-cells, one finds that the assignment @xmath950 is 2-functorial. let @xmath951 be an @xmath0-multitensor. in terms of the previous paragraph, this is the special case @xmath952, @xmath953, @xmath954. the lifted multitensor corresponding to @xmath955 is just cartesian product for strict @xmath0-categories. one has a component of @xmath956 for each sequence @xmath957 of strict @xmath0-categories, and since @xmath958 as a right adjoint preserves products, this component may be regarded as a map @xmath959 of @xmath123-algebras. if in particular @xmath123 is itself @xmath9 and @xmath960, then these components of @xmath956 give a canonical comparison from the lifted tensor product @xmath305 of @xmath0-categories to the cartesian product. for instance, when @xmath122 is the multitensor corresponding to the 3-operad for gray categories, then @xmath956 gives the well - known comparison map from the gray tensor product of 2-categories to the cartesian product, which we recall is actually a componentwise biequivalence. returning to the general situation, it is routine to unpack the assignment @xmath961 as in section([ssec : explicit - lifting]) and so obtain the following 1-cell counterpart of corollary([cor : lifted - obj - simple]). [cor : free - lift-1cell] let @xmath655 be a lax monoidal functor such that @xmath16 and @xmath196 are locally presentable, and @xmath122 and @xmath292 are accessible. let @xmath554 be a sequence of objects of @xmath16. then the component of @xmath962 at the sequence @xmath963 of free @xmath123-algebras is just @xmath964. Contractibility let @xmath16 be a category and @xmath965 a class of maps in @xmath16. denote by @xmath966 the class of maps in @xmath16 that have the right lifting property with respect to all the maps in @xmath965. that is to say, @xmath48 is in @xmath966 iff for every @xmath967 in @xmath965, @xmath542 and @xmath968 such that the outside of @xmath969 { x}="tr " [d] { y}="br " [l] { b}="bl " " tl " (: " tr"^-{\alpha}:"br"^{f},:"bl"_{i}(:"br"_-{\beta},:@{.>}"tr"|{\gamma}))}\]] commutes, then there is a @xmath970 as indicated such that @xmath971 and @xmath972. an @xmath973 is called a _ trivial @xmath965-fibration_. the basic facts about @xmath966 that we shall use are summarised in [lem : basic - tf] let @xmath16 be a category, @xmath965 a class of maps in @xmath16, @xmath210 a set and @xmath974 a family of maps in @xmath16. 1. @xmath966 is closed under composition and retracts.[tfib1] 2. if @xmath16 has products and each of the @xmath975 is a trivial @xmath965-fibration, then @xmath976 is also a trivial @xmath965-fibration.[tfib2] 3. the pullback of a trivial @xmath965-fibration along any map is a trivial @xmath965-fibration.[tfib3] 4. if @xmath16 is extensive and @xmath977 is a trivial @xmath965-fibration, then each of the @xmath975 is a trivial fibration.[tfib4] 5. if @xmath16 is extensive, the codomains of maps in @xmath965 are connected and each of the @xmath975 is a trivial @xmath965-fibration, then @xmath977 is a trivial @xmath965-fibration.[tfib5] ([tfib1])-([tfib3]) is standard. if @xmath16 is extensive then the squares @xmath978 { \coprod_jx_j}="tr " [d] { \coprod_jy_j}="br " [l] { y_j}="bl " " tl " (: " tr":"br"^{\coprod_jf_j},:"bl"_{f_j}:"br")}\]] whose horizontal arrows are the coproduct injections are pullbacks, and so ([tfib4]) follows by the pullback stability of trivial @xmath965-fibrations. as for ([tfib5]) note that for @xmath967 in @xmath965, the connectedness of @xmath109 ensures that any square as indicated on the left @xmath979 { \coprod_jx_j}="tr " [d] { \coprod_jy_j}="br " [l] { b}="bl " " tl " (: " tr":"br"^{\coprod_jf_j},:"bl"_{i}:"br ") } } } [r(4)] { \xybox{\xygraph{!{0;(1.5,0):(0,.666) : : } { s}="tl " [r] { x_j}="tr " [d] { y_j}="br " [l] { b}="bl " " tl " (: " tr":"br"^{f_j},:"bl"_{i}:"br")}}}}\]] factors through a unique component as indicated on the right, enabling one to induce the desired filler. let @xmath980 be functors and @xmath965 be a class of maps in @xmath16. a natural transformation @xmath981 is a _ trivial @xmath965-fibration _ when its components are trivial @xmath965-fibrations. note that since trivial @xmath965-fibrations in @xmath16 are pullback stable, this reduces, in the case where @xmath196 has a terminal object @xmath33 and @xmath522 is cartesian, to the map @xmath982 being a trivial @xmath965-fibration. given a category @xmath16 with an initial object, and a class of maps @xmath965 in @xmath16, we denote by @xmath983 the class of maps in @xmath18 containing the maps @xmath984 where @xmath985. the proof of the following lemma is trivial. [lem : ind - tf] let @xmath16 be a category with an initial object and @xmath965 a class of maps in @xmath16. then @xmath48 is a trivial @xmath983-fibration iff it is surjective on objects and all its hom maps are trivial @xmath965-fibrations. in particular starting with @xmath986 the category of globular sets and @xmath987 the empty class of maps, one generates a sequence of classes of maps @xmath988 of globular sets by induction on @xmath0 by the formula @xmath989 since @xmath990 may be identified with @xmath991, and moreover one has inclusions @xmath992. more explicitly, the set @xmath988 consists of @xmath145 maps : for @xmath993 one has the inclusion @xmath994, where @xmath244 here denotes the representable globular set, that is the `` @xmath244-globe '', and @xmath995 is the @xmath244-globe with its unique @xmath244-cell removed. one defines @xmath996 to be the union of the @xmath988 s. note that by definition @xmath997. there is another version of the induction just described to produce, for each @xmath40, a class @xmath998 of maps of @xmath999. the set @xmath1000 consists of the functions @xmath1001 so @xmath1002 is the class of bijective functions. for @xmath40, @xmath1003. as maps of globular sets, the class @xmath1004 consists of all the maps of @xmath1005 together with the unique map @xmath1006. let @xmath1007. an @xmath0-operad @xmath1008 is _ contractible _ when it is a trivial @xmath998-fibration. an @xmath0-multitensor @xmath1009 is _ contractible _ when it is a trivial @xmath998-fibration. by the preceeding two lemmas, an @xmath145-operad @xmath1010 over @xmath24 is contractible iff the hom maps of @xmath1011 are trivial @xmath1004-fibrations. as one would expect an @xmath145-operad over @xmath24 is contractible iff its associated @xmath0-multitensor is contractible. this fact has quite a general explanation. recall the 2-functoriality of @xmath22 described in section([ssec : gamma-2-functor]) and that of the lifting described in section([ssec : functoriality - lifting]). [prop : contractible] let @xmath655 be a lax monoidal functor between distributive lax monoidal categories, and @xmath965 a class of maps in @xmath196. suppose that @xmath196 is extensive, @xmath658 preserves coproducts and the codomains of maps in @xmath965 are connected. then the following statements are equivalent * @xmath653 is a trivial @xmath965-fibration. * @xmath1012 is a trivial @xmath983-fibration. and moreover when in addition @xmath16 and @xmath196 are locally presentable and @xmath122 and @xmath292 are accessible, these conditions are also equivalent to * the components of @xmath1013 at sequences @xmath1014 of free @xmath123-algebras are trivial @xmath965-fibrations. for each @xmath127 the component @xmath1015 is the identity on objects and for @xmath261, the corresponding hom map is obtained as the composite of @xmath1016 and the canonical isomorphism that witnesses the fact that @xmath658 preserves coproducts. in particular note that for any sequence @xmath112 of objects of @xmath16, regarded as @xmath16-graph in the usual way, one has @xmath1017 thus @xmath1018 follows from lemmas([lem : basic - tf]) and ([lem : ind - tf]). @xmath1019 follows immediately from corollary([cor : free - lift-1cell]). [cor : contractible] let @xmath1007, @xmath1010 be an @xmath1020-operad over @xmath24 and @xmath951 be the corresponding @xmath0-multitensor. tfsae : 1. @xmath1010 is contractible. @xmath951 is contractible. the components of @xmath1021 of section([ssec : functoriality - lifting])([eq : mult->prod]) are trivial @xmath998-fibrations of @xmath0-globular sets, when the @xmath840 are free strict @xmath0-categories. by induction one may easily establish that the codomains of the maps in any of the classes : @xmath988, @xmath998, @xmath996 are connected so that proposition([prop : contractible]) may be applied. [ex : gray - contractible] applying this last result to the 3-operad @xmath23 for gray - categories, the contractibility of @xmath23 is a consequence of the fact that the canonical 2-functors from the gray to the cartesian tensor product are identity - on - object biequivalences. continuing the discussion from section([ssec : tci]), we now explain why the operads which describe trimble @xmath0-categories " are contractible. this result appears in @xcite as theorem(4.8) and exhibits trimble @xmath0-categories as weak @xmath0-categories in the batanin sense. let us denote by @xmath1022 the set of inclusions @xmath1023 of the @xmath0-sphere into the @xmath0 disk for @xmath40. as we remarked in section([ssec : enriched - graphs]) these may all be obtained by successively applying the reduced suspension functor @xmath126 to the inclusion of the empty space into the point. as we recalled in section([ssec : tci]), a basic ingredient of the trimble definition is a version of the little intervals operad which acts on the path spaces of any space. a key property of this operad is that it is contractible a topological operad @xmath1 being contractible when for each @xmath0 the unique map @xmath1024 is in @xmath1025. this is equivalent to saying that the cartesian multitensor map @xmath1026 is a trivial @xmath1022-fibration. a useful fact about the class trivial @xmath1022-fibrations is that it gets along with the construction of path - spaces in the sense of [lem : tf - path - spaces] if @xmath48 is a trivial @xmath1022-fibration then so is @xmath50 for all @xmath1027. to give a commutative square as on the left in @xmath1028 { x(a, b)}="tr " [d] { y(fa, fb)}="br " [l] { d^n}="bl " " tl"(:"tr":"br",:"bl":"br ") } } } & & & { \xybox{\xygraph{!{0;(2,0):(0,.5) : : } { s^n}="tl " [r] { (a, x, b)}="tr " [d] { (fa, y, fb)}="br " [l] { d^{n{+}1}}="bl " " tl"(:"tr":"br",:"bl":"br ") } } } \end{array}\]] is the same as giving a commutative square in @xmath37 as on the right in the previous display, by @xmath43. the square on the right admits a diagonal filler @xmath1029 since @xmath76 is a trivial @xmath1022-fibration, and thus so does the square on the left. we shall write @xmath1030 for the forgetful functor for each @xmath0. the relationship between trivial fibrations of spaces and of globular sets is expressed in [prop : pres - tf] if @xmath48 is a trivial @xmath1022-fibration then @xmath1031 is a trivial @xmath1032-fibration. we proceed by induction on @xmath0. having the right lifting property with respect to the inclusions @xmath1033 ensures that @xmath76 surjective and injective on path components, and thus is inverted by @xmath1034. for the inductive step we assume that @xmath1035 sends trivial @xmath1022-fibrations to trivial @xmath998-fibrations and suppose that @xmath76 is a trivial @xmath1022-fibration. then so are all the maps it induces between path spaces by lemma([lem : tf - path - spaces]). but from the inductive definition of @xmath1036 recalled in section([ssec : tci]), we have @xmath1037 and so @xmath1038 is a morphism of @xmath11-globular sets which is surjective on objects (as argued already in the @xmath332 case) and whose hom maps are trivial @xmath998-fibrations by induction. thus the result follows by lemma([lem : ind - tf]). in section([ssec : tci]) we exhibited @xmath760 as the algebras of an @xmath0-operad by a straight - forward application of two abstract results propositions([prop : monad - trimble]) and ([prop : operad - trimble]). we now provide a third such result relating to contractibility. [prop : tf - formal] given the data and hypotheses of proposition([prop : operad - trimble]) : @xmath16 is a lextensive category, @xmath101 a cartesian and coproduct preserving monad on @xmath16, @xmath733 a @xmath101-operad and @xmath726 a non - symmetric operad in @xmath734. suppose furthermore that a class @xmath965 of maps of @xmath16 is given, and that the non - symmetric operad @xmath1039 and the @xmath101-operad @xmath653 are trivial @xmath965 fibrations. then the @xmath736-operad @xmath1040 of proposition([prop : operad - trimble]) is a trivial @xmath1041-fibration. by definition this monad morphism may be written as the composite @xmath1042. since @xmath653 is a trivial @xmath965-fibration so is @xmath1043 by lemma([lem : basic - tf]), and thus @xmath1044 is a trivial @xmath1041-fibration by proposition([prop : contractible]). since @xmath542 is a trivial @xmath965-fibration, @xmath1045 is a trivial @xmath1041-fibration again by proposition([prop : contractible]), and so the result follows since trivial fibrations compose. starting with a contractible topological operad @xmath1 which acts on path spaces, proposition([prop : pres - tf]) ensures that @xmath1046 will be a contractible non - symmetric operad of @xmath0-globular sets. then proposition([prop : tf - formal]) may be applied to give, by induction on @xmath0, the contractibility of the @xmath0-operad defining trimble @xmath0-categories. Acknowledgements we would like to acknowledge clemens berger, richard garner, andr joyal, steve lack, joachim kock, jean - louis loday, paul - andr mellis, ross street and dima tamarkin for interesting discussions on the substance of this paper. we would also like to acknowledge the centre de recerca matemtica in barcelona for the generous hospitality and stimulating environment provided during the thematic year 2007 - 2008 on homotopy structures in geometry and algebra. the first author would like to acknowledge the financial support on different stages of this project of the scott russell johnson memorial foundation, the australian research council (grant no.dp0558372) and luniversit paris 13. the second author would like to acknowledge the support of the anr grant no. the third author would like to acknowledge the laboratory pps (preuves programmes systmes) in paris, the max planck institute in bonn and the institut des hautes tudes scientifique in bures sur yvette for the excellent working conditions he enjoyed during this project, as well as macquarie university for the hospitality he enjoyed in august of 2008. 10 j. adamek, f. borceux, s. lack, and j. rosicky. a classification of accessible categories., 175:730, 2002. j. adamek and j. rosicky.. number 189 in london math soc. lecture notes. cambridge university press, 1994. m. barr and c. wells.. number 12 in tac reprints. theory and applications of categories, 2005. m. batanin. monoidal globular categories as a natural environment for the theory of weak @xmath0-categories., 136:39103, 1998. m. batanin. symmetrisation of @xmath0-operads and compactification of real configuration spaces., 211:684725, 2007. m. batanin, c. berger, and d - c. higher braided operads and the stabilisation hypothesis. in preparation. m. batanin and m. weber. algebras of higher operads as enriched categories. to appear in applied categorical structures available at http://www.pps.jussieu.fr/ weber/, 2008. c. berger and i. moerdijk. axiomatic homotopy theory for operads., 78:805831, 2003. r. betti, a. carboni, r. street, and r.f.c. variation through enrichment., 29:109127, 1983. e. cheng. iterated distributive laws. arxiv:0710.1120v1, 2007. e. cheng. comparing operadic theories of @xmath0-category. arxiv:0809.2070v2, 2008. s. crans. a tensor product for gray categories., 5:1269, 1999. e. dubuc.. number 145 in slnm. springer verlag, 1970. p. gabriel and f. ulmer.. number 221 in slnm. springer verlag, 1971. r. gordon, a.j. power, and r. street.. number 117 in mem. ams, 1995. johnstone.. academic press new york, 1977. a. joyal and r. street. braided tensor categories., 102(1):2078, 1993. kelly. a unified treatment of transfinite constructions for free algebras, free monoids, colimits, associated sheaves, and so on., 22, 1980. kelly and s. lack. is locally presentable or locally bounded if @xmath16 is so., 8:555575, 2001. s. lack. on the monadicity of finitary monads., 140:6573, 1999. s. lack. codescent objects and coherence., 175:223241, 2002. s. lack. a quillen model structure for 2-categories., 26:171205, 2002. s. lack and r.h. the formal theory of monads ii., 175:243265, 2002. t. leinster.. lecture note series. london mathematical society, 2003. m. makkai and r. par., volume 104 of _ contemp. _ ams, 1989. m. markl and s. shnider. associahedra, cellular @xmath196-construction and products of @xmath25-algebras., 358:23532372, 2006 no.6. s. saneblidze and r. umble. diagonals on the permutahedra, multiplihedra and associahedra., 6:363411, 2004 no.1. s. schwede and b.e. algebras and modules in monoidal model categories., (3) 80:491511, 2000 no.2. r. street. the formal theory of monads. 2:149168, 1972. r. street. the role of michael batanin s monoidal globular categories., 230:99116, 1998. z. tamsamani. sur les notions de @xmath0-categorie et n - groupoide non - strictes via des ensembles multi - simpliciaux., 16:5199, 1999. m. weber. generic morphisms, parametric representations, and weakly cartesian monads., 13:191234, 2004. m. weber. familial 2-functors and parametric right adjoints., 18:665732, 2007. m. weber. yoneda structures from 2-toposes., 15:259323, 2007. m. weber. free products of higher operad algebras. arxiv:0909.4722, 2009. Coequalisers in categories of algebras in these appendices we review some of the transfinite constructions in monad theory that we used in section([sec : lift - mult]). an earlier reference for these matters is @xcite. however due to the technical nature of this material, and our need for its details when we come to making our constructions explicit, we feel that it is appropriate to give a rather thorough account of this background. let @xmath101 be a monad on a category @xmath16 that has filtered colimits and coequalisers and let @xmath1047 { (b, b)}="r " " l":@<-1ex>"r"_-{g } " l":@<1ex>"r"^-{f}}\]] be morphisms in @xmath102. we shall now construct morphisms @xmath1048 starting with @xmath1049 by transfinite induction on @xmath0, such that for @xmath0 large enough @xmath1050 is the coequaliser of @xmath76 and @xmath345 in @xmath102 when @xmath101 is accessible. the initial stages of this construction are described in the following diagram. @xmath1051 { t^2b}="tb " [r] { t^2q_1}="t0 " [r] { t^2q_2}="t1 " [r] { t^2q_3}="t2 " [r] { t^2q_4}="t3 " [r] {... } = " t4 " " ta " [d] { ta}="ma " [r] { tb}="mb " [r] { tq_1}="m0 " [r] { tq_2}="m1 " [r] { tq_3}="m2 " [r] { tq_4}="m3 " [r] {... } = " m4 " " ma " [d] { a}="ba " [r] { b}="bb " [r] { q_1}="b0 " [r] { q_2}="b1 " [r] { q_3}="b2 " [r] { q_4}="b3 " [r] {... } = " b4 " " ta":@<-1ex>"tb"_-{t^2 g } " ta":@<1ex>"tb"^-{t^2f}:"t0"^-{t^2q_0}:"t1"^-{t^2q_1}:"t2"^-{t^2q_2}:"t3"^-{t^2q_3}:"t4 " " ma":@<-1ex>"mb"_-{tg } " ma":@<1ex>"mb"^-{tf}:"m0"^-{tq_0}:"m1"^-{tq_1}:"m2"^-{tq_2}:"m3"^-{tq_3}:"m4 " " ba":@<-1ex>"bb"_-{g } " ba":@<1ex>"bb"^-{f}:"b0"_-{q_0}:"b1"_-{q_1}:"b2"_-{q_2}:"b3"_-{q_3}:"b4 " " ta":@<-1ex>"ma"_-{\mu}:@<-1ex>"ba"_-{a } " ta":@<1ex>"ma"^-{ta}:@<1ex>@{<-}"ba"^-{\eta } " tb":@<-1ex>"mb"_-{\mu}:@<-1ex>"bb"_-{b } " tb":@<1ex>"mb"^-{tb}:@<1ex>@{<-}"bb"^-{\eta } " t0":"m0"^-{\mu}:@{<-}"b0"^-{\eta } " t1":"m1"^-{\mu}:@{<-}"b1"^-{\eta } " t2":"m2"^-{\mu}:@{<-}"b2"^-{\eta } " t3":"m3"^-{\mu}:@{<-}"b3"^-{\eta } " tb":"m0"^-{tv_0 } " mb":"b0"^-{v_0 } " t0":"m1"^-{tv_1 } " m0":"b1"^-{v_1 } " t1":"m2"^-{tv_2 } " m1":"b2"^-{v_2 } " t2":"m3"^-{tv_3 } " m2":"b3"^-{v_3}}\]] * initial step*. define @xmath1052 to be the identity, @xmath1053 to be the coequaliser of @xmath76 and @xmath345, @xmath1054 and @xmath1055. note also that @xmath1056. + + * inductive step*. assuming that @xmath1057, @xmath1058 and @xmath1059 are given, we define @xmath1060 to be the coequaliser of @xmath1061 and @xmath1062, @xmath1063 and @xmath1064. one may easily verify that @xmath1065, and that @xmath1066 could equally well have been defined as the coequaliser of @xmath1067 and @xmath1068. + + * limit step*. define @xmath1069 as the colimit of the sequence given by the objects @xmath1070 and morphisms @xmath1071 for @xmath1072, and @xmath1050 for the component of the universal cocone at @xmath1073. @xmath1074 { \colim_{m{<}n } tq_m}="tm " [r] { \colim_{m{<}n } q_m}="tr " [d] { q_n}="br " [l] { tq_n}="bm " [l] { t^2q_n}="bl " " tl":@<1ex>"tm"^-{\mu_{<{n}}}:@<1ex>"tr"^-{v_{<{n } } } " tl":@<-1ex>"tm"_-{(tv)_{<{n}}}:@<-1ex>@{<-}"tr"_-{\eta_{<{n } } } " bl":"bm"_-{\mu}:@{<-}"br"_-{\eta } " tl":"bl"_{o_{n,2 } } " tm":"bm"^{o_{n,1 } } " tr":@{=}"br"}\]] we write @xmath1075 and @xmath1076 for the obstruction maps measuring the extent to which @xmath101 and @xmath1077 preserve the colimit defining @xmath1069. we write @xmath1078, @xmath1079, @xmath1080 and @xmath1081 for the maps induced by the @xmath1082, @xmath1083, @xmath1084 and @xmath1085 for @xmath1072 respectively. the equations @xmath1086 follow easily from the definitions. define @xmath1057 as the coequaliser of @xmath1087 and @xmath1088, @xmath1089 and @xmath1090. + + * stabilisation*. we say that the sequence _ stabilises at @xmath0 _ when @xmath1058 and @xmath1091 are isomorphisms. in the case @xmath332 one may easily show that stabilisation is equivalent to just @xmath1053 being an isomorphism, which is the same as saying that @xmath1092. [stable - limit] if @xmath0 is a limit ordinal and @xmath1075 and @xmath1076 are invertible, then the sequence stabilises at @xmath0. let us write @xmath1093, @xmath1094 for the colimit cocones. first we contemplate the diagram @xmath1074 { \colim_{m{<}n } tq_m}="tm " [r] { \colim_{m{<}n } q_m}="tr " [d] { q_n}="br " [l] { tq_n}="bm " [l] { t^2q_n}="bl " [d] { t^2q_{n{+}1}}="bbl " [r] { tq_{n{+}1}}="bbm " [r] { q_{n{+}1}}="bbr " [d] { q_{n{+}2}}="bbbr " " tl":@<1ex>"tm"^-{\mu_{<{n}}}:@<1ex>"tr"^-{v_{<{n } } } " tl":@<-1ex>"tm"_-{(tv)_{<{n}}}:@<-1ex>@{<-}"tr"_-{\eta_{<{n } } } " bl":"bm"_-{\mu}:@{<-}"br"_-{\eta } " bbl":"bbm"_-{\mu}:@{<-}"bbr"_-{\eta } " tl":"bl"_{o_{n,2 } } " tm":"bm"^{o_{n,1 } } " tr":@{=}"br " " bl":"bbl"_{t^2q_n } " bm":"bbm"^{tq_n } " br":"bbr"^{q_n}:"bbbr"^{q_{n{+}1 } } " bl":"bbm"_-{tv_n}:"bbbr"_-{v_{n{+}1 } } " bm":"bbr"_-{v_n}}\]] and in general one has @xmath1095 to prove this note that from the definitions of @xmath1071 and @xmath1058 and the naturality of the @xmath1096 in @xmath227, one may show easily that @xmath1097, and from this last equation and all the definitions it is easy to show that @xmath1098 for all @xmath1072 from which ([eq : succ - limit]) follows. suppose that @xmath1075 and @xmath1076 are isomorphisms. then define @xmath1099 as the unique map such that @xmath1100. it follows easily that @xmath1101. from ([eq : succ - limit]) and the invertibility of @xmath1076 it follows easily that @xmath1102 and so there is a unique @xmath1103 such that @xmath1104, from which it follows easily that @xmath1105. [really - stable] if the sequence stabilises at @xmath0 then it stabilises at any @xmath1106, and moreover one has an isomorphism of sequences between the given sequence @xmath1107 and the following one : @xmath1108 {... } = " p2 " [r] { q_n}="p3 " [r] { q_n}="p4 " [r] {... } = " p5 " " p1":"p2"^-{q_0}:"p3":"p4"^-{\id}:"p5"^-{\id}}\]] we show for @xmath1106 that @xmath1071 and @xmath1109 are isomorphisms, and provide the component isomorphisms @xmath1110 of the required isomorphism of sequences, by transfinite induction on @xmath227. we define @xmath1111 to be the identity when @xmath1112. in the initial step @xmath1113, @xmath1071 and @xmath1109 are isomorphisms by hypothesis and we define @xmath1114. in the inductive step when @xmath1106 is a non - limit ordinal, we must show that @xmath1115 is an isomorphism and define @xmath1116. the key point is that @xmath1117 because with this equation in hand one defines @xmath1118 as the unique morphism satisfying @xmath1119 using the universal property of @xmath1120, and then it is routine to verify that @xmath1121. so for the inductive step it remains to verify ([eq : key]). but we have @xmath1122 and so ([eq : key]) follows since @xmath1071 is an isomorphism. in the case where @xmath227 is a limit ordinal, we have stabilisation at @xmath1123 established whenever @xmath1124 by the induction hypothesis. thus the colimit defining @xmath1070 is absolute (ie preserved by all functors) since its defining sequence from the position @xmath0 onwards consists only of isomorphisms. thus @xmath1071 and @xmath1109 are isomorphisms by lemma([stable - limit]). by induction, the previously constructed @xmath1125 s provide a cocone on the defining diagram of @xmath1070 with vertex @xmath1069, thus one induces the isomorphism @xmath1111 compatible with the earlier @xmath1125 s and defines @xmath1126. [coeq - when - stable] if the sequence stabilises at @xmath0 then @xmath1127 is a @xmath101-algebra and @xmath1128 is the coequaliser of @xmath76 and @xmath345 in @xmath102. the unit law for @xmath1127 is immediate from the definition of @xmath1058 and the associative law is the commutativity of the outside of the diagram on the left @xmath1129 { tq_n}="tr " [d] { q_{n{+}1}}="mr " [d] { q_n}="br " [l] { q_{n{+}1}}="bm " [l] { tq_n}="bl " [u] { tq_{n{+}1}}="ml " " bm " [u(.75)] { q_{n{+}2}}="mm " [u(.75)] { tq_{n{+}1}}="tm " " tl":"tr"^-{\mu}:"mr"^-{v_n}:"br"^-{q_n^{-1}}:@{<-}"bm"^-{q_n^{-1}}:@{<-}"bl"^-{v_n}:@{<-}"ml"^-{tq_n^{-1}}:@{<-}"tl"^-{tv_n } " tr":"tm"^-{tq_n}:"mm"_-{v_{n{+}1}}:"bm"_-{q_{n{+}1}^{-1 } } " ml":"mm"_-{v_{n{+}1}}:"mr"^-{q_{n{+}1}^{-1 } } } } } [r(5)u(.1)] { \xybox{\xygraph{{tb}="tl " [r(2)] { tq_n}="tr " [d] { q_{n{+}1}}="m " [d] { q_n}="br " [l(2)] { b}="bl " " tl":"tr"^-{tq_{{<}n}}:"m"^-{v_n}:"br"^-{q_n^{-1}}:@{<-}"bl"^-{q_{{<}n}}(:@{<-}"tl"^-{b},:"m"^-{q_{{<}n{+}1 } }) " m":@{}"tl"|{(i)})}}}}\]] the regions of which evidently commute. the commutativity of the outside diagram on the right exhibits @xmath1130 as a @xmath101-algebra map, and this follows immediately from the commutativity of the region labelled (i). the equational form of (i) says @xmath1131 and we now proceed to prove this by transfinite induction on @xmath0. the case @xmath332 is just the statement @xmath1055. the inductive step comes out of the calculation @xmath1132 and since @xmath1133. the case where @xmath0 is a limit ordinal is the commutativity of the outside of @xmath1134 { b}="b " [r(3)] { q_{n{+}1}}="qnp1",[dr] { tb}="tb2 " [r(3)] { \col \, tq_m}="ctq " [ur] { tq_n}="tqn ") " tb"(:"b"^-{b}:"qnp1"^-{q_{{<}n{+}1}},:"tb2"_-{\id}|-{}="mla":@/_{1pc}/"ctq"_-{q'_{0,n}}|-{}="mma":"tqn"_-{o_{n,1}}|-{}="mra":"qnp1"_-{v_n } " b":@{}"ctq"|*{tb}="m ") " m " (: @{}"b"|(.6)*{q_n}="ptl",:@{}"qnp1"|(.6)*{tq_n}="ptr",:@{}"mra"|(.6)*{\col \, tq_m}="pmr",:@{}"mma"|(.6)*{\col \, t^2q_m}="pb",:@{}"mla"|(.6)*{t^2b}="pml ") " b":"ptl"_(.6){q_{{<}n}}:"ptr"^-{\eta}:"qnp1"^(.4){v_n } " tb":"pml"^-{\eta}:"m"^-{tb}(:"ptr"^(.4){tq_{{<}n}},:"pmr"^-{q'_{0,n } }) " pml":"pb"^-{q''_{0,n}}:"pmr"^-{(tv)_{{<}n}}:"ptr"^-{o_{n,1 } } " pml":"tb2"^-{\mu } " pb":"ctq"^-{\mu_{{<}n}}}\]] the regions of which evidently commute. thus @xmath1130 is indeed a @xmath101-algebra map. to see that @xmath1130 is a coequaliser let @xmath1135 such that @xmath1136. for each ordinal @xmath227 we construct @xmath1137 such that @xmath1138 for all @xmath227 by transfinite induction on @xmath227. when @xmath1073 we define @xmath1139 and @xmath1140 as unique such that @xmath1141. the equation @xmath1142 is easily verified. for the inductive step we note that the commutativity of @xmath1143 { tq_n}="tl " [r(2)] { tc}="tr " [dr] { c}="mr " [dl] { tc}="br " [l(2)] { tq_{n{+}1}}="bl " [ur] { t^2c}="m " " ml":"tl"^-{\mu}:"tr"^-{th_n}:"mr"^-{c}:@{<-}"br"^-{c}:@{<-}"bl"^-{th_{n{+}1}}:@{<-}"ml"^-{tv_n } " m"(:@{<-}"ml"_-{t^2h_n},:"tr"^-{\mu},:"br"^-{tc})}\]] and the universal property of @xmath1144 ensures there is a unique @xmath1145 such that @xmath1146. when @xmath227 is a limit ordinal it follows from all the definitions that @xmath1147 for all @xmath1148, and so @xmath1149 and so by the universal property of @xmath1144 there is a unique @xmath1150 such that @xmath1138. the sequence of @xmath1151 s just constructed is clearly unique such that @xmath1139 and @xmath1138. it follows immediately that @xmath1152 is a @xmath101-algebra map, and that @xmath1153. conversely given @xmath1154 such that @xmath1155, one constructs @xmath1156 as @xmath1157, and it follows easily that @xmath1158, @xmath1159 and @xmath1160 whence @xmath1161 and so @xmath1162. from these results we recover the usual theorem on the construction of coequalisers of algebras of accessible monads. [thm : coeq - talg] let @xmath16 be a category with filtered colimits and coequalisers, @xmath101 be a monad on @xmath16 and @xmath1047 { (b, b)}="r " " l":@<-1ex>"r"_-{g } " l":@<1ex>"r"^-{f}}\]] be morphisms in @xmath102. if @xmath101 is @xmath392-accessible for some regular cardinal @xmath392, then @xmath1050 as constructed above is the coequaliser of @xmath76 and @xmath345 in @xmath102 for any ordinal @xmath0 such that @xmath1163. take the smallest such ordinal @xmath0 it is necessarily a limit ordinal, and @xmath101 and @xmath1077 by hypothesis preserve the defining colimit of @xmath1069. thus by lemmas([stable - limit]) and ([coeq - when - stable]) the result follows in this case, and in general by lemmas([really - stable]) and ([coeq - when - stable]). finally we mention the well - known special case when the above transfinite construction is particularly simple, that will be worth remembering. [prop : simple - coeq] let @xmath16 be a category with filtered colimits and coequalisers, @xmath101 be a monad on @xmath16 and @xmath1047 { (b, b)}="r " " l":@<-1ex>"r"_-{g } " l":@<1ex>"r"^-{f}}\]] be morphisms in @xmath102. if @xmath101 and @xmath1077 preserve the coequaliser of @xmath76 and @xmath345 in @xmath16, then the sequence @xmath1164 stabilises at @xmath33. denoting by @xmath1165 the unique map such that @xmath1166, @xmath1167 is the coequaliser of @xmath76 and @xmath345 in @xmath102. refer to the diagram in @xmath16 above that describes the first few steps of the construction of @xmath1164. since @xmath1053 and @xmath1168 are epimorphisms, the @xmath101-algebra axioms for @xmath1169 follow from those for @xmath1170, and @xmath1053 is a @xmath101-algebra map by definition. thus @xmath287 is the coequaliser in @xmath16 of @xmath1171 and @xmath1172, and since @xmath1168 is an epimorphism it is also the coequaliser of @xmath1173 and @xmath1174, but so is @xmath1066, and so @xmath1175 is the canonical isomorphism between them. to see that @xmath1176 is also invertible, apply the same argument with the composite @xmath1177 in place of @xmath1053. the result now follows by lemma([coeq - when - stable]). Monads induced by monad morphisms suppose that @xmath16 is locally presentable, @xmath1178 and @xmath1179 are monads on @xmath16, and @xmath1180 is a morphism of monads. then one has the obvious forgetful functor @xmath1181 and when @xmath225 is accessible, @xmath804 has a left adjoint which we denote as @xmath763. the general fact responsible for the existence of @xmath763, and which in fact gives a formula for it in terms of coequalisers in @xmath734, is the dubuc adjoint triangle theorem @xcite : for an algebra @xmath1182 of @xmath114, one has the reflexive coequaliser @xmath1183 { (sx,\mu^s_x)}="m " [r] { \phi_!(x, x)}="r " " l":@<2ex>"m"^-{\mu^s_xs(\phi_x)}:"l"|-{s\eta^m_x}:@<-2ex>"m"_-{sx}:"r"^-{q_{(x, x)}}}\]] in @xmath734. putting this together with section([sec : coequalisers]) an explicit description of the composite @xmath782 is given as follows. we construct morphisms @xmath1184 @xmath1185 starting with @xmath1186 by transfinite induction on @xmath0. + + * initial step*. define @xmath1052 to be the identity, @xmath1053 to be the coequaliser of @xmath1187 and @xmath1188, @xmath1054 and @xmath1055. note also that @xmath1189. + + * inductive step*. assuming that @xmath1057, @xmath1058 and @xmath1059 are given, we define @xmath1060 to be the coequaliser of @xmath1190 and @xmath1191, @xmath1192 and @xmath1064. + + * limit step*. define @xmath1193 as the colimit of the sequence given by the objects @xmath1194 and morphisms @xmath1071 for @xmath1072, and @xmath1050 for the component of the universal cocone at @xmath1073. @xmath1195 { \colim_{m{<}n } sq_m}="tm " [r] { \colim_{m{<}n } q_m}="tr " [d] { q_n}="br " [l] { sq_n}="bm " [l] { s^2q_n}="bl " " tl":@<1ex>"tm"^-{\mu_{<{n}}}:@<1ex>"tr"^-{v_{<{n } } } " tl":@<-1ex>"tm"_-{(sv)_{<{n}}}:@<-1ex>@{<-}"tr"_-{\eta_{<{n } } } " bl":"bm"_-{\mu}:@{<-}"br"_-{\eta } " tl":"bl"_{o_{n,2 } } " tm":"bm"^{o_{n,1 } } " tr":@{=}"br"}\]] we write @xmath1075 and @xmath1076 for the obstruction maps measuring the extent to which @xmath225 and @xmath889 preserve the colimit defining @xmath1193. we write @xmath1196, @xmath1197, @xmath1080 and @xmath1198 for the maps induced by the @xmath1199, @xmath1200, @xmath1084 and @xmath1201 for @xmath1072 respectively. define @xmath1057 as the coequaliser of @xmath1087 and @xmath1202, @xmath1203 and @xmath1090. + + instantiating theorem([thm : coeq - talg]) to the present situation gives [cor : explicit - phi - shreik] suppose that @xmath16 is a locally presentable category, @xmath114 and @xmath225 are monads on @xmath16, @xmath1180 is a morphism of monads, and @xmath104 is an @xmath114-algebra. if moreover @xmath225 is @xmath392-accessible for some regular cardinal @xmath392, then for any ordinal @xmath0 such that @xmath1163 one may take @xmath1204 as an explicit definition of @xmath1205 and the associated coequalising map in @xmath734 coming from the dubuc adjoint triangle theorem. [cor : phi - shreik - simple] suppose that under the hypotheses of corollary([cor : explicit - phi - shreik]) that @xmath225 and @xmath889 preserve the coequaliser of @xmath1206 and @xmath1188 in @xmath16. then the sequence @xmath1164 stabilises at @xmath33, and writing @xmath1207 for the unique map such that @xmath1208, one may take @xmath1209 as an explicit definition of @xmath1205 and the associated coequalising map in @xmath734. here is a degenerate situation in which corollary([cor : phi - shreik - simple]) applies. since @xmath1210 we have @xmath1211, but another way to view this isomorphism as arising is to apply the corollary in the case where @xmath104 is a free @xmath114-algebra, say @xmath1212, for in this case one has the dotted arrows in @xmath1213 { smz}="m " [r] { sz}="r " " l":@<1.5ex>"m"^-{(\mu^sm)(s{\phi}m) } " l":@<-1.5ex>"m"^-{s\mu^m } " l":@{<.}@<-3.5ex>"m"_-{sm\eta^m } " m":"r"^-{\mu^ss(\phi) } " m":@{<.}@<-2ex>"r"_-{s\eta^m}}\]] exhibiting @xmath1214 as a split coequaliser, and thus absolute. let us denote by @xmath776 the monad on @xmath839 induced by the adjunction @xmath775. while a completely explicit description of this monad is unnecessary for the proof of theorem([thm : lift - mult]), we will require such a description in section([ssec : explicit - lifting]) when we wish to give an explicit description of the `` lifted '' multitensors that this theorem provides for us. let @xmath104 be in @xmath839, suppose @xmath225 is @xmath392-accessible and fix an ordinal @xmath0 such that @xmath1163. then by corollary([cor : explicit - phi - shreik]) one may take @xmath1215 as the definition of the endofunctor @xmath101. note that @xmath1216 is just a more refined notation for @xmath1205. referring to the diagram @xmath1217 { mx}="tm " [r] { x}="tr " [d] { q_n}="br " [l] { sx}="bm " [l] { smx}="bl " " tl":@<-1ex>"tm"_-{mx } " tl":@<1ex>"tm"^-{\mu^m_x}:"tr"^-{x } " bl":@<-1ex>"bm"_-{sx } " bl":@<1ex>"bm"^-{\mu^s_xs(\phi_x)}:"br"_-{q_{{<}n } } " tl":"bl"_{\phi_{mx } } " tm":"bm"^{\phi_x } " tr":@{.>}"br"^{\eta^t_{(x, x)}}}\]] one may define the underlying map in @xmath16 of @xmath1218 as the unique map making the square on the right commute. this makes sense since the top row is a coequaliser in @xmath16. via the evident @xmath114-algebra structures on each of the objects in this diagram, one may in fact interpret the whole diagram in @xmath839 with the top row now being the canonical presentation coequaliser for @xmath104, and this is why @xmath1218 is an @xmath114-algebra map. the proof that @xmath1218 possesses the universal property of the unit of @xmath775 is straight forward and left to the reader. as for @xmath1219, it is induced from the following situation in @xmath734 : @xmath1220 { (sq_n,\mu^s)}="m " [r(4)] { (q_n(q_n, a\phi),a(q_n, a\phi))}="r " [dl] { (q_n(x, x),a(x, x))}="b " " l":@<-1ex>"m"_-{\mu^ss(\phi) } " l":@<1ex>"m"^-{s(a(x, x)\phi)}(:"r"^-{(q_{{<}n})_{(q_n, a\phi)}}:@{.>}"b"^(.35){\mu^t_{(x, x)}},:"b"_{a(x, x)})}\]] since by definition @xmath1219 underlies an @xmath225-algebra map, to finish the proof that our definition really does describe the multiplication of @xmath101, it suffices by the universal property of @xmath777 to show that @xmath1221 is the identity, and this is easily achieved using the defining diagrams of @xmath778 and @xmath777 together. the data of @xmath101 is still not quite explicit enough for our purposes. what remains to be done is to describe @xmath777 and (especially) @xmath778 in terms of the transfinite data that gives @xmath1193. so we shall for each ordinal @xmath227 provide @xmath1222 and @xmath1223 in @xmath16 such that @xmath1224, by transfinite induction on @xmath227. + + * initial step*. define @xmath1225 to be the identity, and @xmath1226 and @xmath1227 as the unique morphisms such that @xmath1228 by the universal properties of @xmath107 and @xmath1053 (as the evident coequalisers) respectively. + + * inductive step*. define @xmath1229 and @xmath1230 as the unique map satisfying @xmath1231 using the universal property of @xmath1144 as a coequaliser. + + * limit step*. when @xmath227 is a limit ordinal define @xmath1232 and @xmath1233 as the maps induced by the @xmath1234 and @xmath1235 for @xmath1236 and the universal property of @xmath1194 as the colimit of the sequence of the @xmath1237 for @xmath1236. then define @xmath1238 and @xmath1230 as the unique map satisfying @xmath1231 using the universal property of @xmath1144 as a coequaliser. + + the fact that the induction just given was obtained by unpacking the descriptions of @xmath777 and @xmath778 of the previous paragraph in terms of the transfinite construction of the endofunctor @xmath101 (ie the @xmath1194), is expressed by [cor : induced - monad - very - explicit] suppose that @xmath16 is a locally presentable category, @xmath114 and @xmath225 are monads on @xmath16, @xmath1180 is a morphism of monads, and @xmath104 is an @xmath114-algebra. if moreover @xmath225 is @xmath392-accessible for some regular cardinal @xmath392, then for any ordinal @xmath0 such that @xmath1163 one may take @xmath1239 as constructed above as an explicit description underlying endofunctor, unit and multiplication of the monad generated by the adjunction @xmath775. [cor : vexp - simple] under the hypotheses of corollary([cor : induced - monad - very - explicit]), if for @xmath1240, @xmath225 and @xmath889 preserve the coequaliser of @xmath1206 and @xmath1188 in @xmath16, then one may take @xmath1241 with @xmath287 as constructed in corollary([cor : phi - shreik - simple]).

one of the open problems in higher category theory is the systematic construction of the higher dimensional analogues of the gray tensor product. in this paper we continue the work of @xcite to adapt the machinery of globular operads @xcite to this task. the resulting theory includes the gray tensor product of 2-categories and the crans tensor product @xcite of gray categories. moreover much of the previous work on the globular approach to higher category theory is simplified by our new foundations, and we illustrate this by giving an expedited account of many aspects of cheng s analysis @xcite of trimble s definition of weak @xmath0-category. by way of application we obtain an `` ekmann - hilton '' result for braided monoidal 2-categories, and give the construction of a tensor product of @xmath1-infinity algebras. [multiblock footnote omitted] [multiblock footnote omitted] [multiblock footnote omitted]

0909.4715

introduction reticulation processes refer to the transfer of genetic material between living organisms in a non - reproduction manner. horizontal gene transfer is believed to be a highly significant reticulation process occurring between single - cell organisms (doolittle and bapteste 2007 ; treangen and rocha 2011). other reticulation processes include introgression, recombination and hybridization (fontaine et al. 2015 ; mcbreen and lockhart 2006 ; marcussen et al. 2014). in the past two decades, phylogenetic networks have often been seen for the modeling and visualization of reticulation processes (gusfield 2014 ; huson et al. 2011). galled trees, galled networks, reticulation visible networks are three of the popular classes of phylogenetic networks introduced to study the combinatorial and algorithmic perspectives of phylogenetics (wang et al. 2001 ; gusfield et al. 2004 ; huson and kloepper 2007 ; huson et al. 2011). reticulation visible networks include galled trees and galled networks. they are tree - based (gambette et al. 2015). the tree - based networks are introduced by francis and steel (2015) recently. it is well known that the number of internal nodes in a phylogenetic tree with @xmath0 leaves is @xmath4. in contrast, an arbitrary phylogenetic network with 2 leaves can have as many internal nodes as possible. therefore, one interesting research problem is how large a phylogenetic network in a particular class can be. for example, it is well known that a tree - child network with @xmath0 leaves has @xmath2 non - leaf nodes at most. a regular network with @xmath0 leaves has @xmath5 nodes at most (willson 2010). to investigate whether or not the tree containment problem is polynomial time solvable, surprisingly, gambette et al. (2015) proved that a reticulation visible network with @xmath0 leaves has at most @xmath6 non - leaf nodes. the class of nearly - stable networks was also introduced in their paper. they also proved the existence of a linear upper bound on the number of reticulation nodes in a nearly - stable network. in the present paper, we establish the tight upper bound for the size of a network defined by a visibility property using a sub - tree technique that was introduced in gambette et al. the rest of this paper is divided into six sections. section [sec : basic] introduces concepts and notation that are necessary for our study. recently, bordewich and semple (2015) proved that there are at most @xmath2 reticulation nodes in a reticulation visible network. in section [sec : stable], we present a different proof of the @xmath2 tight bound for reticulation visible networks. section [sec : galled] proves that there are at most @xmath1 reticulation nodes in a galled networks with @xmath0 leaves. section [sec : nearlystable] and [sec : stablechild] establish the tight upper bounds for the sizes of nearly - stable and stable - child networks, respectively. in section [conc], we conclude the work with a few remarks. basic concept an acyclic digraph is a simple connected digraph with no directed cycles. let @xmath7 be an acyclic digraph and let @xmath8 and @xmath9 be two nodes in @xmath10. if @xmath11, it is called an _ outgoing _ edge of @xmath8 and _ incoming edge _ of @xmath9 ; @xmath8 and @xmath9 are said to be the tail and head of the edge. the numbers of incoming and outgoing edges of a node are called its _ indegree _ and _ outdegree _, respectively. @xmath10 is said to be _ rooted _ if there is a unique node @xmath12 with indegree 0 ; @xmath12 is called the _ root _ of @xmath10. note that in a rooted acyclic digraph there exists a directed path from the root to every other node. for @xmath13, @xmath14 denotes the digraph with the same node set and the edge set @xmath15. for @xmath16, @xmath17 denotes the digraph with the node set @xmath18 and the edge set @xmath19. if @xmath20 and @xmath21 are subdigraphs of @xmath10, @xmath22 denotes the subdigraph with the node set @xmath23 and the edge set @xmath24. a _ phylogenetic network _ on a finite set of taxa, @xmath25, is a rooted acyclic digraph in which each non - root node has either indegree 1 or outdegree 1 and there are exactly @xmath26 nodes of outdegree 0 and indegree 1, called _ leaves _, that correspond one - to - one with the taxa in the network. in a phylogenetic network, a node is called a _ tree node _ if it is either the root or a node having indegree one ; it is called a _ reticulation node _ if its indegree is greater than one. note that leaves are tree nodes and a tree node may have both indegree and outdegree one. a non - leaf node is said to be _ internal_. a phylogenetic network without reticulation nodes is simply a _ phylogenetic tree_. for a phylogenetic network @xmath27, we use the following notation : * @xmath28 : the root of @xmath27. * @xmath29 : the set of nodes. * @xmath30 : the set of tree nodes. * @xmath31 : the set of reticulation nodes. * @xmath32 : the set of edges. * @xmath33 : the set of leaves. for two nodes @xmath34 in @xmath29, if @xmath35, @xmath8 is said to be a _ parent _ of @xmath9 and, equivalently, @xmath9 is a _ child _ of @xmath8. in general, if there is a directed path from @xmath8 to @xmath9, @xmath8 is an _ ancestor _ of @xmath9 and @xmath9 is a _ descendant _ of @xmath8. we sometimes say that @xmath9 is below @xmath8 when @xmath8 is an ancestor of @xmath9. let @xmath36 and @xmath37 be two simple paths from @xmath8 to @xmath9 in @xmath27. we use @xmath38 and @xmath39 to denote their node sets, respectively. they are _ internally disjoint _ if @xmath40. finally, a phylogenetic network is _ binary _, if its root has outdegree 2 and indegree 0, all internal nodes have degree 3, and all the leaves have indegree one. here, we are interested in how large a binary phylogenetic network can be. in the rest of the paper, a binary phylogenetic work is simply called a network and a phylogenetic tree a tree. for sake of convenience for discussion, we also add an open edge entering the root of a network. a node @xmath9 in a network is _ visible _ (or stable) with respect to a leaf @xmath41 if @xmath9 is in every path from the network root to @xmath41. we say @xmath9 visible if it is visible with respect to some leaf in the network. [prop1] let @xmath27 be a network and @xmath42 a subnetwork of @xmath27 with the same root and leaves as @xmath27. then, a node is visible in @xmath42 if it is visible in @xmath27. equivalently, a node is not visible in @xmath27 if it is not visible in @xmath42. suppose @xmath43 is visible with respect to a leaf @xmath41 in @xmath27. for each path @xmath36 from @xmath44 to @xmath41, since it is also a path from @xmath28 to @xmath41 in @xmath27, it must pass through @xmath9. thus, @xmath9 is also visible with respect to the same leaf in @xmath42. _ reticulation visible _ networks are networks in which reticulation nodes are all visible (huson et al. they are also called stable networks by gambette et al. (2015). a network is _ galled _ if every reticulation node @xmath45 has an ancestor @xmath46 such that there are two disjoint tree paths from @xmath46 to @xmath45 (huson and kloepper 2007). here, a path is tree path if its internal nodes are all tree nodes in the network. galled networks are reticulation visible and are also known as level-1 networks. _ nearly - stable _ networks are networks in which for every pair of nodes @xmath8 and @xmath9, either @xmath8 or @xmath9 is visible if @xmath47 is an edge (gambette et al. 2015). _ stable - child _ networks are networks in which every node has a visible child. tree - based networks comprise another interesting class of networks that is introduced recently (francis and steel 2015). a network is _ tree - based _ if it can be obtained from a tree with the same leaves by the insertion of a set of edges between different edges in the tree. [stable_1] (gambette et al. 2015) for every reticulation - visible network @xmath27, there exists a subset of edges @xmath48 such that @xmath49 contains exactly an incoming edge for each reticulation node and @xmath50 is a subtree with the same leaves as @xmath27. theorem [stable_1] indicates that every reticulation visible network is tree - based. however, nearly - stable networks and stable - child networks are not necessarily tree - based. we finish this section by presenting a technical lemma that will frequently be used in establishing the tight upper bound on the size of a network in each of the four classes defined above. [lemma22] let @xmath27 be a network, @xmath51, and @xmath52 be a finite set of reticulation nodes below @xmath8. if each @xmath53 has a parent @xmath54 such that either (a) @xmath54 is below another @xmath55 in @xmath52, or (b) there is a path from @xmath56 to @xmath54 that avoids @xmath8, then there exists a path from @xmath56 to @xmath41 avoiding @xmath8 for every leaf @xmath41 below a reticulation node @xmath53. let @xmath57. assume @xmath41 is below some @xmath58. then, there is a path @xmath59 from @xmath60 to @xmath41 that avoids @xmath8. since @xmath52 is finite and @xmath27 is acyclic, there exists a series of reticulation nodes, @xmath61 such that : \(i) each @xmath62 has a parent @xmath63 below @xmath64 for @xmath65, and \(ii) the node @xmath66 has a parent @xmath67 such that there is a path @xmath68 from @xmath28 to @xmath67 that avoids @xmath8. since @xmath63 is below @xmath64, there exists a path @xmath69 from @xmath64 to @xmath70 for each @xmath71. since @xmath27 is acyclic and @xmath64 is below @xmath8, the path @xmath69 avoids @xmath8. concatenating these paths, we obtain the following path @xmath72 from @xmath73 to @xmath41 that avoids @xmath8. reticulation visible networks gambette et al. (2015) proved that there are at most @xmath74 reticulation nodes in a reticulation visible network with @xmath0 labeled leaves. on the other hand, there are as many as @xmath2 reticulations in the reticulation visible network in figure [fig1]. so, what is the tight upper bound on the number of reticulation nodes? interestingly, @xmath2 is the tight upper bound, which was independently proved by bordewich and semple (2015) using the induction approach. here, we present an alternative proof to illustrate our approach. are of degree 3 [fig1],scaledwidth=80.0%] given a reticulation visible network @xmath27 with @xmath0 leaves, we let @xmath49 be a set of edges such that @xmath50 is a subtree with the same root and leaves as @xmath27 (theorem [stable_1]). since @xmath50 has @xmath0 leaves, there are exactly @xmath4 nodes of degree 3. thus, there are @xmath75 paths whose internal nodes are of degree 2, starting at a degree-3 node and terminating at either another node of degree 3 or a leaf. let these @xmath75 paths be @xmath76. the edges of @xmath50 not in @xmath77 make up a path @xmath78 that contains the root @xmath56 (figure [fig1]). if @xmath56 is of degree 2, @xmath78 passes through @xmath28 and terminates at a degree-3 node. if @xmath28 is of degree 3, @xmath78 is simply the open edge entering @xmath56. altogether, these @xmath79 paths are called the _ trivial paths _ of @xmath50. note that @xmath80. it is not hard to see that, for each edge in @xmath49, its head and tail are both found in these trivial paths. an edge @xmath81 is called a _ cross _ edge if @xmath82 and @xmath83 for @xmath84 ; it is called _ non - cross _ edge otherwise. the facts in the following proposition appear in the proof of theorem 1 in gambette et al (2015). [31] * no two cross edges @xmath85 have their heads in the same trivial path in @xmath50. * for each non - cross edge @xmath47 such that @xmath86 for some @xmath87, there are at least one cross edge @xmath88 such that @xmath89 is between @xmath8 and @xmath9 in @xmath90. for a cross edge @xmath47 such that @xmath82 and @xmath83(@xmath91), we say @xmath47 _ leaves _ @xmath90 and _ enters _ @xmath92. for a non - cross path @xmath47 and a cross edge @xmath93, if @xmath8 and @xmath9 are in @xmath94 and @xmath89 is a node between @xmath8 and @xmath9 in @xmath90, we say @xmath47 _ jumps over _ @xmath93. it is trivial to see that no cross edge enters the trivial path @xmath78. proposition [31] suggests that @xmath49 contains at most @xmath75 cross edges and thus at most @xmath75 non - cross edges. by theorem [stable_1], @xmath95. to obtain the tight upper bound @xmath96 for @xmath97, we define the cost @xmath98 of a cross edge @xmath85 as : @xmath99 we will charge the cost of a cross edge to the trivial path it enters and call it the _ weight _ of the trivial path. if no cross edge enters a trivial path, the weight of this trivial path is set to be 0. by proposition [31], the weight of a trivial path is at most 2. we use @xmath100 to denote the weight of a trivial path @xmath90, @xmath101. . @xmath102 are the three trivial paths incident to a degree-3 node @xmath103 ; @xmath104 is the cross edge ending at @xmath105 in @xmath106 and the non - cross edge @xmath107 jumps over @xmath104 for @xmath108. here, @xmath109 is not drawn. (* a *) @xmath110 and @xmath111 are both between @xmath103 and @xmath112 in @xmath113. (* b *) @xmath110 is between @xmath103 and @xmath112 in @xmath113, but @xmath111 is below @xmath112 in @xmath113. (* c *) the node @xmath110 is below @xmath112 and @xmath114 is below @xmath115. this case is impossible to occur, as there is a directed cycle. [fig2],scaledwidth=95.0%] for an internal node @xmath103 of degree 3 in @xmath50, we use @xmath116 to denote the trivial path entering @xmath103 and @xmath117 to denote the two trivial paths leaving @xmath103. [stable_3] let @xmath103 be a degree-3 node in @xmath50. \(i) if @xmath118 and @xmath119, then @xmath120. \(ii) if @xmath121, then @xmath122 and @xmath123. for sake of simplicity, we let @xmath124 and use @xmath125 to denote the unique path from @xmath126 to @xmath127 for a node @xmath126 and a descendant @xmath127 of @xmath126 in @xmath128. @xmath27 and @xmath128 have the same root and leaves. the common root of @xmath27 and @xmath128 is written @xmath73. \(i) assume that @xmath118 and @xmath119. then, there exists a cross edge @xmath129 entering @xmath130 and a non - cross edge @xmath131 jumping over @xmath129 for each @xmath132. we shall prove that @xmath120 by showing that no cross edge enters @xmath116. assume @xmath110 is between @xmath103 and @xmath112 in @xmath113. when @xmath111 is below @xmath110 in @xmath133 (figure [fig2]a), there are two cases. if @xmath114 is in the path @xmath134 or below it, then, @xmath135 does not pass @xmath111. if @xmath114 is not below @xmath103, @xmath135 does not pass @xmath103 and so @xmath111. therefore, by lemma [lemma22], there is path from @xmath73 to every leaf below @xmath112 that does not pass @xmath111. for any leaf @xmath41 not below @xmath112 in @xmath128, @xmath136 avoids @xmath111. hence, @xmath111 is a reticulation node in @xmath27, but not visible. this is a contradiction. when @xmath111 is below @xmath112 in @xmath128 (figure [fig2]b), @xmath111 is below @xmath112 as a reticulation node. since @xmath137 does not pass @xmath112, by lemma [lemma22], there is a path from @xmath73 to a leaf below @xmath112 that does not pass @xmath112. for any leaf @xmath41 not below @xmath112 in @xmath128, @xmath136 avoids @xmath112. hence, @xmath112 is not visible, a contradiction. we have proved that @xmath110 is not between @xmath103 and @xmath112 in the tree path @xmath113. by symmetry, @xmath114 is not between @xmath103 and @xmath115 in @xmath134. assume there is a cross edge enters @xmath116. let @xmath45 be the lowest reticulation node in @xmath116. then, @xmath110 and @xmath114 are both not in @xmath138. otherwise, either @xmath111 or @xmath139 is between @xmath45 and @xmath103, contradicting that @xmath45 is the lowest reticulation node in @xmath116. combining this fact with that @xmath140 is not between @xmath103 and @xmath141 in @xmath128 for @xmath142, we conclude that either @xmath140 is below @xmath141 or there is a path from @xmath73 to @xmath140 not passing @xmath45 for each @xmath142. hence, by lemma [lemma22], @xmath45 is not visible with respect to any leaf below @xmath45. for any leaf @xmath41 not below @xmath45 in @xmath128, the tree path @xmath136 avoids @xmath45. hence, @xmath45 is not visible, a contradiction. we have proved that @xmath120. \(ii) if @xmath121, then @xmath103 is an ancestor of any other degree-3 node in @xmath50. since @xmath27 is acyclic, there does not exist @xmath143 such that @xmath144 for some @xmath145 and @xmath146. hence, @xmath147. assume on the contrary the weights of @xmath134 and @xmath113 are both 2. then, @xmath140 is not between @xmath103 and @xmath141 for @xmath142, proved above. if @xmath110 or @xmath114 is in @xmath78, the lowest reticulation in @xmath78 is not visible, a contradiction. otherwise, @xmath110 is below @xmath112 and @xmath114 is below @xmath115, implying a cycle in @xmath27 (figure [fig2]c). this is a contradiction. hence, either @xmath134 or @xmath113 has weight less than 2. @xmath148 let @xmath27 be a reticulation visible network with @xmath0 leaves. then, @xmath149 let @xmath150 denote the set of @xmath151 internal nodes of degree 3 in @xmath50. note that any trivial path other than @xmath78 starts with a node in @xmath150 define : @xmath152 for @xmath153. clearly, @xmath154 s are pairwise disjoint and hence @xmath155 when @xmath156, @xmath157. by proposition [stable_3], @xmath158. let @xmath159 be the start node of @xmath160 for each @xmath156. again, by proposition [stable_3], @xmath161. it is clear that under the map @xmath162, at most two nodes in @xmath163 are mapped to the same node in @xmath164, and different nodes in @xmath163 are mapped to different nodes in @xmath165. thus, @xmath166. since @xmath147, the inequality implies that @xmath167\\ & = & |v_1|+2|v_2|+3|v_3|+ (3|v_4|+|v_4|) \\ & \leq & 2(|v_0|+|v_1|) + 3(|v_2|+|v_3|+|v_4|)\\ & \leq & 3(|v_0|+|v_1|+|v_2|+|v_3|+|v_4|)\\ & = & 3(n-1), \end{array}\]] where the first inequality is derived from the substitution of @xmath168 for @xmath169. galled networks galled networks form a subclass of reticulation visible networks (huson et al. 2011). in this section, we shall show that there are at most @xmath1 reticulations in a galled network with @xmath0 leaves. given that the galled network shown in figure [fig3]a has exactly @xmath1 reticulations, @xmath1 is the tight bound on the number of reticulation nodes in a galled network with @xmath0 leaves. : there is a non - cross edge @xmath47 in @xmath49 such that @xmath8 and @xmath9 are in @xmath78, and there is a cross - edge edge @xmath170 and a non - cross edge @xmath109 both ending at a node in a trivial path other than @xmath78, where @xmath171 is not drawn. in (* b *) and (* c *), solid straight and curve arrows represent edges and paths in @xmath50, respectively ; round dot arrows represent edges in @xmath49, respectively [fig3],scaledwidth=80.0%] [gn_1] for a galled network @xmath27 with @xmath0 leaves, @xmath172 let @xmath27 be a galled network with @xmath0 leaves and let @xmath173. since @xmath27 is reticulation visible, by theorem [stable_1], there is a set of edges @xmath49 such that (a) @xmath49 contains exactly one incoming edge for each reticulation node and (b) @xmath50 is a subtree with the same leaves as @xmath27. we use the same notation as in section 3. @xmath78 denotes the trivial path whose first edge is the open edge entering @xmath73 ; @xmath174 denote the other @xmath75 trivial paths in @xmath50. we prove the result by showing that @xmath49 does not contain any non - cross edges and only one cross - edge can end at a node in each @xmath90 for @xmath145. if @xmath78 contains only the open edge entering @xmath73, there is no edge in @xmath49 that enters @xmath78. we first prove that this fact is also true even if @xmath78 contains other edges below @xmath73. since @xmath27 is acyclic and there is a directed path from the end of @xmath78 to a node in @xmath90 for any @xmath145, there is no cross edge @xmath81 such that @xmath8 is in @xmath90 and @xmath9 is in @xmath78. if there is a non - cross edge @xmath175 such that @xmath176 are in @xmath78 (figure [fig3]b), we let @xmath89 be the other child of @xmath8 in @xmath78. then, @xmath89 must be a tree node such that @xmath88, where @xmath177 is a reticulation node in some trivial path @xmath90, @xmath145. (if @xmath89 is a reticulation, it is not visible, a contradiction.) since @xmath27 is galled and @xmath177 is a reticulation node, there exist two paths @xmath178 and @xmath179 from a common tree node to @xmath177 in @xmath27 such that (i) they are internally disjoint and (2) @xmath177 is the only reticulation node in them. note that no edges in @xmath49 other than @xmath93 can appear in @xmath178 and @xmath179. otherwise, either @xmath178 and @xmath179 contains another reticulation node. thus, @xmath180 is a subtree of @xmath50. this implies that one of @xmath178 and @xmath179 is the single edge @xmath93 and the other is @xmath181, the unique path from @xmath89 to @xmath177 in the tree @xmath50. this is impossible, as the reticulation node @xmath9 is in @xmath181. we have shown that there is no edge in @xmath49 that enters @xmath78. next, we show that there is at most one edge in @xmath49 that enters @xmath90 for each @xmath182. assume that @xmath183 and @xmath184 are two edges in @xmath49 such that @xmath139 is below @xmath111 in some @xmath90 (@xmath145) (figure [fig3]c). then, @xmath185 must be a non - cross edge and @xmath186 is also below @xmath111. (otherwise, @xmath111 is not visible.) again, by fact (2) in proposition [31], there is a cross edge @xmath187 such that @xmath89 is between @xmath186 and @xmath139 in @xmath90 and @xmath177 is in @xmath188, @xmath189. since @xmath177 is a reticulation node and @xmath27 is galled, there are two internally disjoint paths @xmath178 and @xmath179 from a common tree node to @xmath177 in which any nodes other than @xmath177 are a tree node. if @xmath190 contains an edge in @xmath49 other than @xmath93, the head of the edge is a reticulation node and appears in either @xmath178 or @xmath179, a contradiction. hence, @xmath191 is a subtree of @xmath128. without loss of generality, we may assume @xmath178 contains @xmath93. that is, @xmath93 is the last edge of @xmath178. note that @xmath192 are all nodes in @xmath90, ordered from top to bottom. if @xmath178 contains more than one edge in @xmath128, it must pass through @xmath111, a contradiction. if @xmath178 is equal to @xmath93, then @xmath179 must pass through @xmath139, a contradiction. therefore, there is at most one edge in @xmath49 whose head is in each trivial path @xmath90, @xmath145. in summary, there are @xmath75 trivial paths other than @xmath78 and there is at most one edge in @xmath49 entering each of them. hence, @xmath193 nearly-stable network in this section we will give a tight bound for the number of reticulations in a nearly - stable network. the class of nearly - stable networks is different from the class of reticulation visible networks, but surprisingly the tight upper bound is also @xmath2. the network shown in figure [fig4]a is an example for a nearly - stable network with @xmath2 reticulations. we need the following fact, proved by gambette et al. (2015). [ns_1] let @xmath27 be a nearly - stable network with @xmath0 leaves. there exists a set @xmath49 of edges such that (a) @xmath50 is a reticulation visible subnetwork over the same leaves as @xmath27, and (b) @xmath49 contains exactly one incoming edge for each reticulation node that is not visible in @xmath27. with four leaves. it has nine reticulations (shaded circles), five of which are not visible. the round dot edges are those removed to obtain the reticulation visible network @xmath42 in (* b *). the dashed paths in @xmath42 are the cross and non - cross paths removed to obtain a subtree with the same leaves as @xmath27. (* c *) and (* d *) are two cases considered in the proof of the part (c) in lemma [ns_2] : a non - cross path from @xmath8 and @xmath9 contains a tree node @xmath194 of @xmath27, and it contains a reticulation node @xmath194 of @xmath27 [fig4],scaledwidth=90.0%] let @xmath49 be the set of edges satisfying the two properties in proposition [ns_1] and let @xmath195 (figure [fig4]a). the edges in @xmath49 are said to be _ ns - edges_. we remark that @xmath42 is a subdivision of a binary reticulation visible network. that is, the reticulation visible network can be obtained from @xmath42 by replacing some paths whose internal nodes are of degree 2 with directed edges with the same orientation. hence, @xmath42 contains degree-2 nodes if @xmath49 is not empty. for a path @xmath36, we use @xmath196 to denote the set of its internal nodes. since @xmath42 is a subdivision of a binary reticulation visible network with the same leaves as @xmath27, by theorem [stable_1], there is a set @xmath197 of paths in @xmath198 such that (i) each path @xmath199 is from a degree-3 tree node to a visible reticulation node in @xmath42 and its internal nodes are all of degree-2 in @xmath42, and (ii) @xmath200 is a subtree with the same leaves as @xmath27. let @xmath201. @xmath128 is obtained from the removal of the internal nodes and edges of the paths in @xmath202. we can classify the paths in @xmath197 as _ cross paths _ and _ non - cross paths _ accordingly as in section 3 (figure [fig4]b). [ns_2] let @xmath27 be a nearly - stable network and let @xmath49, @xmath42, @xmath128 and @xmath197 be defined above. \(a) every internal node in a path in @xmath197 is not visible in @xmath27. \(b) each cross path in @xmath197 consists of either a single edge or two edges in @xmath27. \(c) each non - cross path in @xmath197 is simply an edge in @xmath27. \(d) if @xmath36 is a cross path in @xmath197 from @xmath89 to @xmath177 and @xmath178 is a non - cross path in @xmath197 from @xmath8 to @xmath9 such that @xmath89 is between @xmath8 and @xmath9 (figure [fig4]b), then @xmath36 and @xmath178 are both a single edge in @xmath27. \(e) every two distinct paths in @xmath197 are node disjoint. we remark that @xmath203 denotes the unique path from @xmath177 to @xmath204 for any two nodes @xmath177 and @xmath204 in @xmath128. \(a) let @xmath36 be a path in @xmath197 and let @xmath204 be an internal node of it. for any leaf @xmath205, the unique path @xmath136 does not pass @xmath204 in @xmath128. hence, @xmath204 is not visible in @xmath27. \(b) if there are two or more internal nodes in a path in @xmath197, by (a), they are consecutive and not visible in @xmath27, contradicting that @xmath27 is nearly - stable. \(c) we use @xmath73 to denote the root of @xmath27, which is also the root of @xmath42 and @xmath128. let @xmath36 be a non - cross path between @xmath8 and @xmath9, where @xmath8 and @xmath9 are in some path @xmath90 in @xmath128. note that @xmath206 is a sub - path of @xmath90 and is internally disjoint from @xmath36. by fact (2) in proposition [31], there is an internal node @xmath89 in @xmath206 that is the start node of a cross path @xmath207 in @xmath197. first, any node @xmath204 between @xmath8 and @xmath9 in @xmath206 is not visible. this is because for any network leaf @xmath41 not below @xmath9 in @xmath128, @xmath208 does not pass through @xmath204, and for any network leaf @xmath41 below @xmath9 in @xmath128, @xmath209 is a path not passing through @xmath204. therefore, @xmath89 must be the unique internal node of @xmath206. that is, @xmath89 is the child of @xmath8 and the parent of @xmath9 in @xmath206. assume that @xmath36 is not an edge in @xmath27. by (a), there is a unique degree-2 node @xmath194 between @xmath8 and @xmath9 in @xmath36. we consider the following two cases. if @xmath210 (figure [fig4]c), then the other outgoing edge @xmath211 had been removed to obtain @xmath42. that is, @xmath212. by the definition of @xmath49, @xmath126 is a reticulation node and not visible in @xmath27. that @xmath194 and @xmath126 are both not visible contradicts that @xmath27 is nearly - stable. if @xmath213 (figure [fig4]d), then the other incoming edge @xmath214 had been removed to obtain @xmath42. note that @xmath215 and @xmath216, as @xmath89 has degree 3 in @xmath42. in addition, @xmath127 is not an internal node of a path in @xmath197. (otherwise, by (a), @xmath194 and @xmath127 are both not visible). so @xmath127 is a node in @xmath128. clearly @xmath127 is not below @xmath9 and hence not below @xmath89 in @xmath128. (otherwise @xmath27 has a cycle.) hence, @xmath217 does not pass through @xmath8. consider a network leaf @xmath57. if it is not below @xmath9, then @xmath208 does not pass through @xmath8. if @xmath41 is below @xmath9, then @xmath218 is a path not passing through @xmath8 in @xmath27. therefore, @xmath8 is not visible. that @xmath8 and @xmath89 are both not visible in @xmath27 contradicts that @xmath27 is nearly - stable. \(d) by the proof of (c), @xmath178 is a single edge in @xmath27 and @xmath89 is the only node in @xmath36 and not visible. thus @xmath36 must be an edge in @xmath27. (otherwise by (a) @xmath89 and its child in @xmath36 are not visible, contradicting that @xmath27 is nearly - stable.) \(e) it can be easily derived from the definition of the cross path. let @xmath219 be a cross path from @xmath89 to @xmath177. then, @xmath177 is a visible reticulation node in @xmath27. it may have as many as two reticulation parents that are not visible. let @xmath220, where @xmath221 is the set of all parents of @xmath177 and @xmath222 is the set of all reticulation nodes that are not visible in @xmath27. @xmath223 or @xmath224. define the cost of @xmath225 as : @xmath226 where @xmath224 is used to count @xmath177 and the other child of @xmath89 which is a visible reticulation node if there is a non - cross edge jumping over @xmath89. as in section [sec : stable], we let @xmath78 denote the trivial path whose first edge is the incoming edge to @xmath73 and let @xmath227 denote the other @xmath75 trivial paths in @xmath128. we charge the cost of a cross path to the trivial path @xmath90 in @xmath128 in which the cross path enters and call it the weight of @xmath90. the weight of @xmath90 is denoted by @xmath100. if a trivial path does not contain any end node of the cross paths in @xmath197, its weight is set to be 0. each visible reticulation node contributes to at least one unit of weight. by the definition of nearly - stable networks, any reticulation node that is not visible must have a visible reticulation node as its child, and by the proof of lemma [ns_2] (c), any reticulation node that is not visible in @xmath27 must be in some @xmath228, @xmath177 being the end node of a cross path, so it also contributes to at least one unit weight. therefore, @xmath229. to bound this, we first establish a useful lemma. as in section [sec : stable], we use @xmath116 to denote the trivial path entering @xmath103 and @xmath134, @xmath113 to denote the trivial paths leaving @xmath103 for a node @xmath103 of degree 3 in @xmath128. . solid arrows and curves represent the edges and paths in @xmath128, while square dot arrows and curves represent the removed edges and paths. @xmath230 is the reticulation child of @xmath103 in a trivial path @xmath134 leaving @xmath103. @xmath231 is the child of @xmath103 in the trivial path @xmath113. the path from @xmath114 to @xmath112 is a cross path entering @xmath112. (* a *) a cross path from @xmath110 to @xmath115 and @xmath232 is a node between @xmath103 and @xmath115, where @xmath115 is in @xmath134. (* b *) the unique tree node @xmath231 between @xmath103 and @xmath112 is also a parent of @xmath230 in @xmath27 [fig5],scaledwidth=50.0%] [ns_33] let @xmath130 be a trivial path defined above and let @xmath233 be a cross path from @xmath140 to @xmath234, where @xmath234 is in @xmath130 and @xmath235. define @xmath236. \(a) the tree path @xmath237 consists of either a single edge or two edges in @xmath130. if @xmath237 includes two edges, the internal node is not visible. \(b) if there exists a node @xmath238 between @xmath103 and @xmath234 in @xmath130 and @xmath238 is a reticulation node in @xmath27, then no cross - path enters the other trivial path @xmath239. \(c) assume that @xmath233 contains an internal node that is a reticulation node in @xmath27. if there is a cross - path @xmath240 from @xmath241 to @xmath242 such that @xmath242 is in @xmath239, then @xmath140 is not in @xmath243. note that @xmath244 for @xmath142. without loss of generality, we may assume that @xmath245 and @xmath246, that is @xmath247 and @xmath248. \(a) let @xmath232 be a node between @xmath103 and @xmath115 in @xmath134 (figure [fig5]a) and let @xmath41 be a leaf in @xmath27. if @xmath41 is not below @xmath115 in @xmath128, the path @xmath249 does not pass through @xmath232. let @xmath41 be a leaf below @xmath115 in @xmath128. since @xmath110 is not in @xmath134 in @xmath128, the tree path @xmath250 does not pass @xmath232. by lemma [lemma22] there is a path from @xmath73 to @xmath41 that avoids @xmath232. therefore, @xmath232 is not visible. since @xmath27 is nearly - stable, there is at most one node in @xmath251, as each internal node is not visible. \(b) suppose on the contrary, there is a cross path @xmath252 from @xmath114 to @xmath112 entering @xmath113, where @xmath112 is in @xmath113. by (a), @xmath112 is a child of @xmath103 or there is a unique node @xmath231 between @xmath103 and @xmath112 in @xmath113. we first show that @xmath103 is not visible in @xmath27. if @xmath112 is a child of @xmath103 or there is a node @xmath231 in @xmath133 such that @xmath231 is a reticulation node in @xmath27, @xmath103 has two reticulation children in @xmath27. by lemma [lemma22], @xmath103 is not visible. if @xmath113 contains a node @xmath231 between @xmath103 and @xmath112 in @xmath27, @xmath253 must not be an edge in @xmath27. otherwise, as shown in figure [fig5]b, @xmath230 and @xmath231 are then not visible, contradicting that @xmath27 is nearly - stable. let @xmath254 be the edge removed from @xmath230 in the process of transforming @xmath27 to @xmath42. since @xmath255, either @xmath256 is below @xmath112 or there is a path from @xmath73 to @xmath256 that avoids @xmath103. since @xmath114 is in another trivial path and there is no node between @xmath103 and @xmath230 in @xmath134, @xmath89 is either below @xmath230 or the path @xmath135 does not pass @xmath103. since the reticulation nodes @xmath257 are below @xmath103 and satisfy the condition in lemma [lemma22], there is a path from @xmath73 to @xmath41 that avoids @xmath103 for any leaf @xmath41 below @xmath230 or @xmath112. for any leaf @xmath41 below neither @xmath230 nor @xmath112, it is not below @xmath103 and the path @xmath136 does not pass through @xmath103. therefore, @xmath103 is also not visible. the fact that @xmath103 and @xmath230 are both not visible contradicts that @xmath27 is nearly - stable. this implies that there is no cross path entering @xmath113. \(c) if @xmath112 is the child of @xmath103 in @xmath113, the case is trivial. assume that there is an internal node @xmath231 between @xmath103 and @xmath112 in @xmath113. by the fact (a), @xmath231 is not visible. if @xmath258, then @xmath110 and its child in @xmath259 are both not visible, contradicting @xmath27 is nearly - stable network. , while square dot arrows and curves represent the removed edges and paths. the path from @xmath110 to @xmath115 is the cross path ending at a node in a trivial path leaving @xmath103 [fig6],scaledwidth=90.0%] [ns_4] for an internal node @xmath103 of degree @xmath260 in @xmath128, \(a) @xmath261 and @xmath262. \(b) if @xmath263, then @xmath264, where @xmath265. \(c) if @xmath266 and @xmath119, then @xmath120. moreover, assume @xmath267 is the degree-3 ancestor of @xmath103 such that @xmath268. then @xmath269. \(d) if @xmath270, then @xmath271 and @xmath123. \(a) we only prove that @xmath261. if there is no non - cross edge jumping over the start node of the cross path entering @xmath134, by eqn. ([weight_def]), the weight of @xmath134 is at most 3. if there is a non - cross edge jumping over the start node @xmath110 of the cross - path @xmath272 ending at a node @xmath115 in @xmath134, by the fact (d) of lemma [ns_2], @xmath272 is equal to the single edge @xmath273. therefore, @xmath115 has at most one reticulation parent, which is in @xmath134 if exists. by eqn. ([weight_def]), @xmath261. \(b) assume @xmath274. then, there is a cross path @xmath272 from @xmath110 to @xmath115 where @xmath115 is in @xmath134. if there is no non - cross edge jumping over @xmath110, by eqn. ([weight_def]), @xmath115 has two reticulation parents (figure [fig6]a). if there is a non - cross edge jumping over @xmath110, by the fact (d) of lemma [ns_2], @xmath272 is equal to a single edge @xmath273, and by eqn. ([weight_def]), @xmath115 has one reticulation parent @xmath230 in @xmath134 (figure [fig6]b). by the fact (b) of lemma [ns_33], there is no cross path that enters @xmath113, implying @xmath275. \(c) assume @xmath266 and @xmath119. let @xmath233 be the cross path from @xmath140 to @xmath234, with @xmath234 in @xmath276. since @xmath119, by the fact (b) of lemma [ns_33], there is no reticulation node between @xmath103 and @xmath234 for each @xmath277. hence, for each @xmath277, either the parent of @xmath234 in @xmath233 is a reticulation node and not visible (figure [fig6]c), or there is a non - cross edge @xmath278 jumping over @xmath279 (figure [fig6]e). by the facts (a) and (b) of lemma [ns_33], either @xmath234 is the child of @xmath103 in @xmath130 or there is a tree node @xmath238 between @xmath103 and @xmath234 in @xmath130 for @xmath265. assume that there is a tree node @xmath238 between @xmath103 and @xmath234 in @xmath130, @xmath265. let @xmath236. if @xmath240 has an internal node that is a reticulation, by the fact (c) of lemma [ns_33], @xmath280. if there is a non - cross edge jumping over @xmath241, by the fact (d) of lemma [ns_2], that @xmath281 implies that the endpoints of the non - cross edge are also between @xmath103 and @xmath234 this is impossible, as there is only @xmath238 between @xmath103 and @xmath234. therefore, @xmath280. similarly, @xmath282. we have proved that for @xmath283, @xmath140 is not between @xmath103 and @xmath242. thus, @xmath140 is either below @xmath242 or there is a path from @xmath73 to @xmath140 that does not pass @xmath103. therefore, by lemma [lemma22], there is a path from @xmath73 to @xmath41 not passing through @xmath103 for any leaf below either @xmath115 or @xmath112. for any leaf @xmath41 below neither @xmath115 nor @xmath112, since it is not below @xmath103 in @xmath128, @xmath136 does not contain @xmath103. therefore, @xmath103 is not visible. this also implies that @xmath115 and @xmath112 are children of @xmath103. assume @xmath267 is the start node of @xmath116 and @xmath268. we further prove that @xmath116 consists of only an edge @xmath284 in @xmath27. assume on the contrary there are nodes between @xmath267 and @xmath103 in @xmath285. we consider the parent @xmath204 (@xmath286) of @xmath103 in the trivial path @xmath285. if @xmath204 is a reticulation node, that @xmath103 is not visible implies that @xmath204 is also not visible, a contradiction. hence @xmath204 must be a tree node in @xmath27. we consider the following two cases. * case 1*. @xmath204 is equal to @xmath140 or equal to the other parent of the internal node of @xmath233 for some @xmath287. without loss of generality, we may assume @xmath247 (figure [fig7]a and b). this implies that there is no non - cross edge jumping over the cross path @xmath272 and there is a reticulation node @xmath288 in @xmath272. when @xmath289, let @xmath290 be the edge removed from @xmath288 in the first stage. since @xmath291 is a parent of @xmath288, if @xmath291 is below @xmath204, it must be below @xmath112. when @xmath292, @xmath110 is below @xmath112 if it is below @xmath204. similarly, @xmath114 is below @xmath115 and thus below @xmath288 if it is below @xmath204. the set of reticulation nodes @xmath293 and @xmath204 and satisfy the condition in lemma [lemma22], so there is path from @xmath73 to @xmath41 that avoids @xmath204 for any leaf @xmath41 below @xmath288 and @xmath112. if @xmath41 is below neither @xmath288 nor @xmath112, it is not below @xmath204, and @xmath136 does not pass through @xmath204. hence, @xmath204 is not visible. * case 2*. @xmath204 is neither @xmath140 nor the other parent of the internal node of @xmath233 for each @xmath283 (figure [fig7]c). in this case, for each @xmath283, @xmath234 is either below @xmath242 or there is a path from @xmath73 to @xmath140 that avoids @xmath204. applying lemma [lemma22] on the set of reticulations @xmath294 and @xmath204, we conclude that there is a path from @xmath73 to @xmath41 that avoids @xmath204 for any leaf @xmath41 below @xmath115 or @xmath112. clearly, for any leaf @xmath41 not below @xmath115 or @xmath112, @xmath208 avoids @xmath204. therefore, @xmath204 must be not visible. that @xmath204 and @xmath103 are two consecutive nodes and not visible contradicts that @xmath27 is nearly - stable. after proving that the path @xmath285 is actually an edge @xmath284, we now prove that @xmath269. assume on the contrary @xmath295. then, the child @xmath296 of @xmath267 in @xmath297 must be a reticulation node (figure [fig7]d). then, the set of reticulations @xmath298 and @xmath267 satisfy the conditions in lemma [lemma22], so there exists a path from @xmath73 to @xmath41 that does not pass through @xmath267 for any leaf @xmath41 below @xmath267 in @xmath128. for any leaf @xmath41 not below @xmath267, the tree path @xmath136 does not pass through @xmath267. hence, @xmath267 is not visible. that @xmath267 and @xmath103 are not visible contradicts that @xmath27 is nearly - stable network. \(d) if @xmath270, then @xmath103 is an ancestor of any degree-3 node in @xmath128. since @xmath27 is acyclic, there does not exist any cross path @xmath299 from @xmath89 to @xmath177, such that @xmath300 while @xmath301 for @xmath145. hence @xmath147. if the weight of @xmath134 and @xmath113 are both 2, and if @xmath279 is the start node of the cross path @xmath302 that enters @xmath106 for @xmath303, either @xmath110 or @xmath114 is a node in @xmath78. following the proof of fact (c), we conclude that @xmath103 and its parent in @xmath78 are both not visible, a contradiction. . the path from @xmath279 to @xmath304 is a cross path ending at @xmath304 in a trivial path @xmath106 leaving @xmath103 for @xmath108. (* a *) @xmath110 is an internal node in @xmath116. (* b *) @xmath204 is the other parent of the unique internal node of the cross path from @xmath110 to @xmath115. (* c *) neither @xmath110 nor @xmath114 is equal to @xmath204 and has a common child with @xmath204. (* d *) @xmath267 is the parent of @xmath103 and has a reticulation node @xmath296 as the other child. solid arrows and curves represent the edges and paths in @xmath128, round dot arrows represent edges in @xmath49 that were removed to form @xmath42, and square dot arrows and curves represent the edges and paths that were removed to transform @xmath42 to @xmath128. [fig7],scaledwidth=90.0%] let @xmath27 be a nearly - stable network with @xmath0 leaves. then, @xmath305 let @xmath150 denote the set of internal nodes of degree 3 in @xmath128, and let @xmath306 for any @xmath307, we define @xmath159 to be the start node of the trivial path @xmath160 that enters @xmath9. by proposition [ns_4] (c) and (d), that @xmath308 implies @xmath309. additionally, there are at most two different nodes @xmath310 and @xmath311 in @xmath163 such that @xmath312, as there are only two trivial paths leaving a degree-3 tree node in @xmath128 ; for different @xmath310 and @xmath311, if @xmath313 and @xmath314 are in @xmath315, then @xmath316. taken together, the two facts imply that @xmath317. since @xmath147, @xmath318\\ & = & |v_1|+2|v_2|+3|v_3|+4|v_4| \\ & \leq & 2|v_0|+2|v_1|+3|v_2|+3|v_3|+3|v_4|\\ & \leq & 3(|v_0|+|v_1|+|v_2|+|v_3|+|v_4|)\\ & = & 3(n-1). \end{array}\]] stable-child network the stable - child network shown in figure [fig8]a has as many as @xmath3 reticulation nodes. in this section, we shall prove that a stable - child network can have @xmath3 reticulation nodes at most. we first transform a stable - child network to a reticulation visible network and then to a binary tree with the same leaves by removing some edges into reticulations nodes. if it is not visible : (* b *) @xmath45 and its child @xmath319 have a common parent. (* c *) @xmath45 has a parent @xmath320 and a sibling @xmath55 under @xmath320 such that @xmath55 and @xmath320 has a common parent @xmath89. (* d *) neither (* b *) nor (* c *) is true. (* e *) both (* b *) and (* c *) occur at the same time [fig8],scaledwidth=60.0%] [sc_1] let @xmath27 be a stable - child network. there is a set of edges @xmath49 such that (1) @xmath50 is a subdivision of a reticulation visible network over the same leaves as @xmath27, and (2) @xmath49 contains exactly an incoming edge for a reticulation node if it is not visible in @xmath27. for a reticulation node @xmath45 that is not visible in @xmath27, its unique child must be a visible reticulation node. furthermore, since each node has a visible child, its parents both have a visible child other than @xmath45 and are both a tree node. to transform @xmath27 into a reticulation visible network, we will delete one or two edges around a reticulation if it is not visible. for each reticulation node @xmath45 that is not visible, we consider the following three cases. if @xmath45 and its unique child @xmath319 have a common parent @xmath320 (figure [fig8]b), then @xmath321 is removed. if @xmath45 and its child @xmath319 do not have a common parent, but @xmath45 has a reticulation sibling @xmath55 such that the parent @xmath89 of the common parent @xmath320 of @xmath45 and @xmath55 is the other parent of @xmath55 (figure [fig8]c), @xmath322 and @xmath323 are then deleted at the same time. when neither occurs (figure [fig8]d), we arbitrarily select an incoming edge of @xmath45 to remove. the edges removed in the above process is called _ sc - edges_. each sc - edge is from a tree node to a reticulation node. a sc - edge is _ concealed _ if the head is a visible reticulation node ; it is _ revealed _ otherwise. note that a concealed sc - edge is deleted only when the case shown in figure [fig8]c is satisfied. therefore, a concealed sc - edge jumps over the associated revealed sc - edge that is removed at the same time. it is not hard to see that the sc - edges that are removed when different reticulation nodes are considered are different. let @xmath49 be the set of sc - edges. first, we only deleted an incoming edge for each reticulation node and did not delete the incoming edge for each tree node, so the resulting network @xmath50 is connected. second, @xmath50 has the same leaves as @xmath27. the reasons for this include that (i) we do not remove any outgoing edge of a reticulation node, and (ii) for any tree node @xmath103, if an outgoing edge of it is removed, the other outgoing edge enters another tree node and thus has never not been removed. now, we show that @xmath50 is a subdivision of a binary reticulation visible network. since we had deleted an incoming edge for a reticulation node if it is not visible, all the remaining reticulation nodes are visible. @xmath50 is reticulation visible. we can also show that there are no two internally disjoint paths from a common tree node to a common reticulation node in which each internal node is of degree 2, implying that @xmath50 is a subdivision of a binary reticulation visible network. assume on the contrary there are two internally disjoint path @xmath259 and @xmath324 between @xmath8 and @xmath9 such that their internal nodes are all of degree 2. if neither @xmath259 nor @xmath324 is a single edge, then the two children of @xmath8 in @xmath259 and @xmath324 are both not visible, contradicting that @xmath27 is a stable - child network. therefore, either @xmath259 or @xmath324 is a single edge from @xmath8 and @xmath9. without loss of generality, we may assume that @xmath324 is equal to the edge @xmath175. according to the three rules which we used to remove the edges in @xmath49, if an incoming edge of a node is removed, its child in @xmath50 is visible in @xmath27. this implies that @xmath50 does not contain a path consisting of two or more degree-2 nodes that are not visible in @xmath27. therefore, @xmath259 has exactly one internal node @xmath177. if @xmath177 is a tree node in @xmath27, then, we removed an outgoing edge of @xmath177 according to either the second or third case. in the former case, we remove @xmath325 at the same time. in the later case, @xmath47 does not exist in @xmath27. this is impossible. when @xmath177 is a reticulation node, we removed an incoming edge of it. again, the edge @xmath47 does not exist in @xmath27 in each possible case, a contradiction. we have proved that @xmath50 is a subdivision of a binary network. let @xmath326 be the subnetwork obtained after the removal of the edges in @xmath49. @xmath42 is a subdivision of a reticulation visible network. by theorem [stable_1], there exist a set of paths @xmath197 such that (i) @xmath327 is a subtree of @xmath27 with the same leaves and (ii) all the internal nodes in each path in @xmath197 are of degree 2. again, we use @xmath328 to denote the trivial paths in @xmath128, where @xmath78 denotes the trivial path starting with @xmath28. as in the last section, a path in @xmath197 is called a non - cross path if its start and end nodes are both in @xmath188 for some @xmath277 ; it is called a cross path otherwise. [sc_2] let @xmath36 be a path in @xmath197. \(a) every internal node in @xmath36 is not visible in @xmath27. \(b) if @xmath36 is a non - cross path, it is simply an edge. \(c) if @xmath36 is a cross path and ends at a node @xmath177 in the trivial path @xmath188, every node between the start node of @xmath188 and @xmath177 in @xmath128 is not visible in @xmath27. \(d) if @xmath36 is a cross path and there is a non - cross path @xmath178 jumping over it, then either @xmath36 is an edge or the start node of @xmath36 is the parent of the end node of @xmath178 in @xmath128. \(a) and (b) are essentially the restatement of the fact (a) and (c) in lemma [ns_2]. \(c) let @xmath204 be a node between the start node of @xmath188 and @xmath177 in @xmath128. for any leaf @xmath41 that is not below @xmath177 in @xmath128, @xmath249 is a path that does not pass @xmath177 and hence @xmath204. for any leaf @xmath41 below @xmath177 in @xmath128, the path @xmath329 avoids @xmath204, as @xmath89 is the start node of @xmath36 in a trivial path different from @xmath188. hence, @xmath204 is not visible in @xmath27. \(d) by (b), @xmath178 is simply an edge @xmath47 in @xmath42. let @xmath36 start at a node @xmath89. if neither @xmath330 nor @xmath36 is a single edge, the two children of @xmath89 in @xmath36 and @xmath331 are both not visible, contradicting that @xmath27 is stable - child. let @xmath45 be a reticulation node and not visible in @xmath27. then, a revealed sc - edge @xmath332 was removed from @xmath45 to obtain @xmath42 from @xmath27. we define the cost @xmath333 of @xmath45 to be : @xmath334 recall that @xmath335. we can define the cost of a cross path @xmath219 from @xmath89 to @xmath177 as follows : @xmath336 we further charge the cost of @xmath225 to the trivial path @xmath90 to which @xmath177 belongs and call it the weight of @xmath90, written @xmath100. if there is no cross path entering @xmath90, the weight of @xmath90 is set to be 0. as for nearly - stable networks, we have that @xmath337 for an internal node @xmath103 with degree-3 in @xmath128, we still use @xmath134 and @xmath338 to denote the trivial paths leaving @xmath103 and @xmath116 to denote the trivial path entering @xmath103. [sc_3] for each internal node @xmath103 of degree 3 in @xmath128, \(a) @xmath339, @xmath142. \(b) @xmath340. \(c) if @xmath266 and @xmath341, then @xmath271. moreover, assume @xmath267 is the start node of @xmath116 and @xmath268. then @xmath342. \(d) if @xmath270, then @xmath271 and @xmath343. \(a) we will only prove that @xmath344. let @xmath272 denote the cross path sending at a node @xmath115 in @xmath134. let @xmath110 be the start node of @xmath272 in a trivial path different from @xmath134. note that @xmath345. if there is no non - cross edge jumping over @xmath110, then there are at most 2 elements in @xmath346, and each element can have two unit cost at most. thus, by eqn. ([scformula]), @xmath347. if there is a non - cross edge @xmath183 jumping over @xmath110. by the fact (b) in lemma [sc_2], @xmath348 is empty or a singleton. moreover, by the fact (d) in lemma [sc_2] (d), either @xmath272 is an edge, or @xmath349 is an edge in @xmath128. if @xmath272 is an edge, then @xmath350. if @xmath349 is an edge, @xmath351. both implies that @xmath352. therefore, by eqn. ([scformula]), @xmath353. we remark that, if the parent of @xmath115 in @xmath128 is not a reticulation node in @xmath27, then @xmath354, and therefore @xmath355. equality holds only if there is a non - cross edge jumping over @xmath177. \(b) if @xmath356 and @xmath357, we assume that the cross path @xmath302 ending at a node @xmath115 in @xmath106 starts at @xmath279 for @xmath108. by the fact (c) of lemma [sc_2], every internal node in @xmath358 and @xmath359 is not visible. if there is a node between @xmath103 and @xmath304 for each @xmath360, then the two children of @xmath103 are not visible in @xmath27, contradicting that @xmath27 is stable - child. so @xmath103 is the parent of either @xmath115 or @xmath112. without loss of generality, we may assume that @xmath103 is the parent of @xmath115 in @xmath128. by the remark in the end of the proof of (a), @xmath355 and hence @xmath361. \(c) assume that @xmath266 and @xmath267 be the start node of @xmath116 such that @xmath268. if @xmath341, by (a), the weights of @xmath134 and @xmath113 are both not zero. hence, there is a cross path @xmath302 ending at a node @xmath304 in @xmath106 and starting at a node @xmath279 in a trivial path different from @xmath106 for each @xmath362. we first show that @xmath279 either (i) below @xmath363, or (ii) neither in @xmath116 nor below @xmath103 for @xmath364 and @xmath224. without loss of generality, we assume @xmath364. * case 1*. @xmath110 is in @xmath359 (figure [fig9]a). if there is a non - cross edge @xmath183 jumping over @xmath110, @xmath111 is either in @xmath365 or @xmath111 is below @xmath112 in @xmath113. the former implies that @xmath111 is not visible, whereas the latter implies that @xmath112 is not visible. this contradicts that both @xmath111 and @xmath112 are visible in @xmath42. since @xmath110 is an internal node between @xmath103 and @xmath112 in @xmath128, then @xmath103 is the parent of @xmath115 in @xmath128. by the fact (a) and (c) of lemma [sc_2], each internal node in @xmath272 or @xmath365 is not visible. thus, @xmath366 or @xmath367 is an edge. otherwise, the two children of @xmath110 are not visible, contradicting @xmath27 is stable - child network. if @xmath366 is an edge in @xmath27, then @xmath368 and hence @xmath343, a contradiction. if @xmath367 is an edge, then @xmath369. since @xmath110 is not a reticulation, by the remark at the end of the proof of (a), @xmath370. therefore, @xmath343. this is impossible. * case 2*. @xmath110 is a node in the path @xmath116. without loss of generality, we can assume @xmath110 is lower than @xmath114 if @xmath114 is also in @xmath116. we claim that @xmath103 and all the internal nodes in @xmath371 are not visible. let @xmath8 be either @xmath103 or an internal node in @xmath371. if @xmath114 is in @xmath372, the case is symmetric to case 1. so there are two cases to consider : either @xmath114 is below @xmath115 (figure [fig9]b), or @xmath114 is not below @xmath110 (figure [fig9]c). in both cases, the reticulation node set @xmath373 are below the node @xmath8, and satisfies the condition in lemma [lemma22], so @xmath8 is not visible with respect to each leaf @xmath41 below either @xmath115 or @xmath112. for any leaf @xmath41 not below @xmath115 or @xmath112, the path @xmath208 avoids @xmath8. hence @xmath8 is not visible in @xmath27. there are some observations from this result. first, there is no non - cross edge @xmath183 jumping over @xmath110, otherwise @xmath111 is not visible. second, a child of @xmath110 in @xmath116 is not visible, and so the cross path @xmath272 is simply an edge. otherwise by the fact (a) of lemma [sc_2], the two children of @xmath110 are both not visible in @xmath27. by (b) of lemma [sc_2], either @xmath374 or @xmath375 is an edge in @xmath128. if @xmath103 is the parent of @xmath115 in @xmath128, then @xmath376 according to eqn. ([scformula]). by (a), @xmath343. if @xmath103 is the parent of @xmath112, @xmath369 and @xmath370 according to the remark at the end of the proof of (a). taken together, both facts imply that @xmath343, which contradicts the assumption that @xmath341. * case 3*. @xmath110 is below @xmath112 and @xmath114 is below @xmath115 (figure [fig9]d). this case is impossible since there is a cycle in @xmath27, contradicting @xmath27 is acyclic. to sum up, @xmath341 implies that either (i) or (ii) is true. but in both cases, if we let @xmath8 be either @xmath103 or any internal node in @xmath116, the set of reticulations @xmath294 are below @xmath8 and satisfies the condition in lemma [lemma22]. therefore, @xmath103 and any internal node of @xmath116 are not visible. there are two observations from this result. first, @xmath271 because there is no cross path that ends at @xmath116. (otherwise the cross path enters @xmath116 at a reticulation that is not visible in @xmath27.) second, the child of @xmath267 in @xmath268 is not visible. clearly, @xmath377 if there is no cross path that ends at @xmath297. assume there is a cross path @xmath378 from @xmath379 to @xmath380 with @xmath380 in @xmath297. by (c) of lemma [sc_2], each internal node in @xmath381 is not visible. but the child of @xmath267 in @xmath285 is not visible. hence, @xmath382 is an edge in @xmath128. then, by the remark in the end of proof of (a), @xmath342. (d) if @xmath270, then @xmath103 is an ancestor of any degree-3 node in @xmath128. since @xmath27 is acyclic, there does not exist any cross path @xmath383 such that @xmath384 for @xmath145 and @xmath385. hence, @xmath147. if @xmath341, then, every node in @xmath78 is not visible in @xmath27, shown in (c). this contradicts that @xmath78 contains the network root @xmath28 and @xmath28 is visible with respect to each leaf in @xmath27. . (* a *) @xmath110 is between @xmath103 and @xmath112. (* b *) @xmath110 is the trivial path @xmath116 entering @xmath103, and @xmath114 is below @xmath115. (* c *) @xmath110 is in @xmath116, and @xmath114 is not below @xmath115. (* d *) @xmath110 is below @xmath114 and @xmath114 is also below @xmath115. this is impossible in a network [fig9],scaledwidth=80.0%] let @xmath27 be a stable - child network with @xmath0 leaves. then, @xmath386 let @xmath150 denote the set of the @xmath151 internal nodes of degree 3 in @xmath128. define @xmath387 by proposition [sc_3] (b), @xmath388. hence, @xmath389 and thus @xmath390. let @xmath159 be the start node of the trivial path entering @xmath9 in @xmath128. by proposition [sc_3] (c) and (d), if @xmath391, then @xmath392 for each @xmath393. by proposition [sc_3] (c), under the mapping @xmath162, at most two nodes in @xmath394 are mapped to the same node in @xmath164, and only one node can be mapped to @xmath395. thus, @xmath396 since @xmath147, the above inequality implies that @xmath397\\ & = & \sum_{i=0}^{10 } i |v_i| \\ & \leq & |v_1|+2|v_2|+3|v_3|+4|v_4| + 7 \sum_{i=5}^{10 } |v_i| + |v_8| + 2|v_9| + 3|v_{10}|) \\ & \leq & |v_1|+2|v_2|+3|v_3|+4|v_4| + 7 \sum_{i=5}^{10 } |v_i| + 6 |v_0| + 3\sum_{i=1}^4 |v_i| \\ & = & 6|v_0|+4|v_1|+5|v_2|+6|v_3|+7|v_4| + 7 \sum_{i=5}^{10 } |v_i| \\ & \leq & 7\sum_{i=0}^{10 } |v_i|\\ & = & 7(n-1). \end{aligned}\]] conclusion we have established the tight upper bounds for the sizes of galled, nearly - stable, and stable - child networks. since the number of internal tree nodes is equal to the number of leaves plus the number of reticulation nodes in a binary network, we can summarize our results in table [table1]. without question, these tight bounds provide insight to the study of combinatorial and algorithmic aspects of the network classes defined by visibility property..the tight upper bounds on the sizes of binary networks with @xmath0 leaves defined by visibility property. the bound for reticulation visible network is found in bordewich and semple(2015). [table1] [cols="<,^,^",options="header ",] gambette p, gunawan adm, labarre a, vialette s, zhang l (2015) locating a tree in a phylogenetic network in quadratic time, in proc. of the 19th intl conf. res. in comput. (recomb), pp. 96107, warsaw, poland.

phylogenetic networks are mathematical structures for modeling and visualization of reticulation processes in the study of evolution. galled networks, reticulation visible networks, nearly - stable networks and stable - child networks are the four classes of phylogenetic networks that are recently introduced to study the topological and algorithmic aspects of phylogenetic networks. we prove the following results. * a binary galled network with @xmath0 leaves has at most @xmath1 reticulation nodes. * a binary nearly - stable network with @xmath0 leaves has at most @xmath2 reticulation nodes. * a binary stable - child network with @xmath0 leaves has at most @xmath3 reticulation nodes.

1510.00115

Introduction galaxy clusters, the largest gravitationally - bound structures in the universe, are ideal cosmological tools. accurate measurements of their masses provide a crucial observational constraint on cosmological models. several dynamical methods have been available to estimate cluster masses, such as (1) optical measurements of the velocity dispersions of cluster galaxies, (2) measurements of the x - ray emitting gas, and (3) gravitational lensing. good agreements between these methods have been found on scales larger than cluster cores. however, joint measurements of lensing and x - rays often identify large discrepancies in the gravitational masses within the central regions of clusters by the two methods, and the lensing mass has always been found to be @xmath0 times higher than the x - ray determined mass. this is the so - called `` mass discrepancy problem '' (allen 1998 ; wu 2000). many plausible explanations have been suggested, e.g., the triaxiality of galaxy clusters (morandi et al. 2010), the oversimplification of the strong lensing model for the central mass distributions of clusters (bartelmann & steinmetz 1996), the inappropriate application of the hydrostatic equilibrium hypothesis for the central regions of clusters (wu 1994 ; wu & fang 1997), or the magnetic fields in clusters (loeb & mao 1994). recently richard et al. (2010) present a sample of @xmath1 strong lensing clusters taken from the local cluster substructure survey (locuss), among which @xmath2 clusters have x - ray data from chandra observations (sanderson et al. they show that the x - ray / lensing mass discrepancy is @xmath3 at @xmath4 significance clusters with larger substructure fractions show greater mass discrepancies, and thus greater departures from hydrostatic equilibrium. on the other hand, lensing observations of the bullet cluster 1e0657 - 56 (clowe et al. 2006), combined with earlier x - ray measurements (markevitch et al. 2006), clearly indicate that the gravitational center of the cluster has an obvious offset from its baryonic center. furthermore, recent studies (shan et al. 2010) of lensing galaxy clusters reveal that offset between the lensing center and x - ray center appears to be quite common, especially for unrelaxed clusters. among the recent sample of 38 clusters of shan et al. (2010), @xmath5 have been found to have offsets greater than @xmath6, and @xmath7 clusters even have offsets greater than @xmath8. motivated by such observations, we propose to investigate galaxy cluster models where the center of the dark matter (dm) halo does not coincide with the center of the x - ray gas (see figure 1). [fig : offset] if the x - ray center of a cluster has an offset from its lensing (gravitational) center, then the x - rays and lensing are indeed measuring different regions of the cluster. given the same radius, the lensing is measuring the dm halo centered at the gravitational center (shown by the dark blue sphere in figure 1), while the x - rays are measuring the sphere of the halo that is offset from the _ true _ gravitational center (shown by the red circle in figure 1). in this case, there will always be a _ natural _ discrepancy between the lensing and x - ray measured masses or specifically, the x - ray mass will always be lower than the lensing mass, just as the long - standing `` mass discrepancy problem '' has indicated. in this paper, we investigate the lensing - x - ray mass discrepancy caused by the offsets between dm and x - ray gas. to check our predictions, we compile a sample of @xmath9 clusters with good lensing and x - ray measurements. we conclude that such `` offset '' effect should not be ignored in our dynamical measurements of galaxy clusters. a flat @xmath10cdm cosmology is assumed throughout this paper, where @xmath11=0.3, @xmath12=0.7, and @xmath13. Mass discrepancy as a result of the dark matter-baryon offset we model our galaxy cluster with a fiducial model as the following : (1) the dm halo is modeled by the navarro - frenk - white (nfw) profile (navarro et al. 1997) with concentration @xmath14 and scaled radius @xmath15, (2) the gas distribution is modeled by a @xmath16 model with @xmath17, the cluster core radius @xmath18, and the gas fraction @xmath19, (3) the mass density of the bcg is described by a singular isothermal sphere (sis) with a velocity dispersion of @xmath20. the projected mass within a sphere of radius @xmath21 is @xmath22 r'\, dr'\, d{\theta},\end{aligned}\]] where @xmath23 is the 2-d radius from the halo center, @xmath24 is the 2-d radius from the x - ray gas center, @xmath25 is the 2-d offset between the halo center and x - ray center, and @xmath26, @xmath27, and @xmath28 are the projected mass densities of the dm halo, the gas and the bcg, respectively. for a given radius @xmath29, the gravitational mass measured by lensing @xmath30 can be given by @xmath31 (as shown by the dark blue sphere in figure 1), while the projected mass measured by x - rays @xmath32 is described by @xmath33 (the mass within the red circle in figure 1). we now calculate the mass ratio @xmath34, or equivalently, @xmath35. figure 2 shows the mass ratio as a function of the 2-d offset @xmath25, for a typical rich cluster. the solid curves are the mass ratio with the fiducial model, the dashed and dotted curves are the mass ratio with the nfw concentration @xmath36 and @xmath37 (top left), the cluster core radius @xmath38 and @xmath39 (top right), the @xmath16 index @xmath40 and @xmath41 (bottom left), the gas fraction @xmath42 and @xmath43, respectively. for these cases, the three curves from top to bottom are for the three measuring radii @xmath44, respectively. from figure 2 we have the following conclusions : \(1) the lensing measured mass @xmath30 is always higher than the x - ray measured mass @xmath32. for typical values of offset @xmath45 and @xmath46, @xmath47, comparable to the ratio found in early studies (allen 1998 ; wu 2000 ; richard et al. 2010). \(2) the `` offset effect '' we are reporting here should contribute significantly to the long - standing `` mass discrepancy problem ''. \(3) the ratio of @xmath48 increases with offset @xmath25. \(4) @xmath48 depends very strongly on @xmath29. here @xmath29 acts like the arc radius @xmath49 in strong lensing, i.e., we only measure the enclosed mass within a small region of @xmath50. when @xmath29 is very small, the offset effect is most prominent and gives large @xmath48. increasing @xmath29 will reduce @xmath48. when @xmath29 is very large (compared with @xmath25), the offset effect will be `` smeared out '', and the @xmath30-@xmath32 discrepancy introduced by the offset will vanish. \(5) the mass ratio is very sensitive to the nfw concentration, and it increases dramatically with @xmath51. \(6) the mass ratio increases with the core radius, and decreases with @xmath16 index and gas fraction. however, the mass ratio is not very sensitive to the gas model. [fig : theory] and the measuring radius @xmath29. the solid curves are the mass ratio for the fiducial model with @xmath14, @xmath52, @xmath53, @xmath54, and @xmath55. the dashed and dotted curves are for the nfw concentration @xmath36 and @xmath37 (top left), the cluster core radius @xmath38 and @xmath39 (top right), the @xmath16 index @xmath40 and @xmath41 (bottom left), the gas fraction @xmath42 and @xmath43, respectively. the three dotted (dashed, solid as well) curves from top to bottom in one panel correspond to @xmath56, respectively.,title="fig:",width=336] Comparison with observational data to compare with our theoretical predictions, we compile a sample of @xmath9 clusters with @xmath57 arc - like images, which have both strong lensing and x - ray measurements. the clusters and their lensing and x - ray data are listed in table 1. for the @xmath58 arcs that have no redshift information, we estimate their lensing masses @xmath59 by assuming the mean redshifts of @xmath60 and @xmath61, respectively. the x - ray data are taken from tucker et al. (1998), wu (2000), bonamente et al. (2006), and references therein. the offsets between lensing and x - ray centers are taken from shan et al. (2010). the clusters in our table are classified as relaxed (with cooling flow) and unrelaxed (which are dynamically unmature), from their x - ray morphologies. the definition has been used in the literature by allen (1998), wu (2000), baldi et al. (2007), and dunn & fabian (2008). _ mass from strong lensing. _ assuming a spherical matter distribution, one can calculate the gravitational mass of a galaxy cluster projected within a radius of @xmath49 on the cluster plane as @xmath62 where @xmath63 is the critical surface mass density, @xmath64, @xmath65 and @xmath66 are the angular diameter distances to the cluster, to the background galaxy, and from the cluster to the galaxy, respectively. the above equation is actually the lensing equation for a cluster lens of spherical mass distribution with a negligible small alignment parameter for the distant galaxy within @xmath49. the values of @xmath30 within the arc radius @xmath49 are listed in table 1. allen (1998) pointed out that the use of more realistic, elliptical mass models can reduce the masses within the arc radii by up to @xmath67, though a value of @xmath68 is more typical. however, such corrections are still not very significant compared with the large discrepancies between the lensing and x - ray determined masses. we will discuss it in more detail in the next section. _ mass from x - rays. _ assuming that the intra - cluster gas is isothermal and in hydrostatic equilibrium, the cluster mass @xmath69 enclosed within a radius @xmath70 can be easily calculated from @xmath71 where @xmath72 is the gas temperature, @xmath73 the gas number density, @xmath74 the proton mass, and @xmath75 the mean molecular weight. here we assume that the gas follows the conventional @xmath16 model, i.e., @xmath76. in order to compare the mass measured by x - rays with the lensing result, we need to convert this @xmath69 (i.e., 3-d) into the projected mass @xmath32 (see e.g. wu 1994) : @xmath77 where @xmath78 the mass ratio @xmath48 are listed in table 1. figure 3 shows the relation between the mass ratios @xmath79 and the (scaled) offsets for our sample of @xmath9 clusters (@xmath57 arc images). it should be pointed out that the @xmath9 clusters in our sample have quite different sizes and masses. this can be seen from the wide range of the cluster temperatures from @xmath80 to @xmath81. therefore, it is useful to compare the offsets of the clusters on the same scale. we realize that the m - t relation of clusters scales as@xmath82 (e.g., nevalainen et al. 2000 ; xu et al. 2001), and that @xmath83, where r is the size of the cluster. therefore, in figure 3, instead of using the physical offset @xmath25, we use a scaled offset which is characterized by @xmath84. from figure 3, the mass ratios @xmath48 exhibits large dispersions roughly ranging from @xmath85 to @xmath86. many clusters have large error bars. it appears that relax clusters (marked by crosses) have smaller @xmath48 ratios. the fact that @xmath87 is consistent with our theoretical predictions, and the ratio of @xmath88 is also roughly consistent with our predictions as plotted in figure 2. however, no strong correlation has been found between the offset and mass discrepancies. we notice that many clusters in the sample have very small offset values smaller than the errors in lensing and x - ray measurements which are typically a few arcseconds. so these offset values are not robustly measured themselves, and we thus remove these data points and only focus on clusters with large offsets of @xmath89, as has been suggested in shan et al. this leaves a sub - sample of only @xmath90 arc images. the dashed line in figure 3 shows a @xmath91 fit to this sub - sample, which satisfies @xmath92 with a reasonable @xmath93. we can find @xmath94 increasing slightly with @xmath25. [fig : offset - ratio] clusters (@xmath57 arc images). the x - label show the scaled offset between dm and baryons. the squares denote unrelaxed clusters, and crosses the relaxed clusters. the dashed line shows a @xmath91 fit satisfying @xmath95 with @xmath93 for the clusters with offset larger than @xmath6.,title="fig : "] Discussion and conclusions as has been reported by shan et al. (2010), it might be fairly common in galaxy clusters that the x - ray center has an obvious offset from the gravitational center. we have explored the dynamical consequences of this lensing - x - ray offset and tried to attribute such an effect to the long - standing `` mass discrepancy problem '' in galaxy clusters. our theoretical model predicts that such an offset effect will always result in a larger @xmath30 than @xmath96, with a typical mass ratio @xmath97, which is consistent with observations. to test our model, we have compiled a sample of @xmath9 clusters, and studied in detail their lensing and x - ray properties and obtained their lensing and x - ray masses, @xmath30 and @xmath32. the lack of strong correlation between @xmath48 and the offset @xmath25 suggests that the problem is more complicated. as we have found in section 2, @xmath48 is not only a function of @xmath25, but also depends very strongly on @xmath29 (or the arc radius @xmath49). apparently, each cluster in our sample has quite different @xmath49. \(1) the central regions of clusters may be still undergoing dynamical relaxation, and the x - ray gas may not be in good hydrostatic equilibrium. therefore, large errors could be induced in the x - ray measurement of cluster cores, especially for unrelaxed clusters. \(2) the spherical models are too simple to reflect the real mass distribution of clusters. the use of more realistic mass model could reduce the lens mass within the arc radius by up to @xmath67, though values of @xmath98 are more typical (bartelmann 1995 ; allen 1998). \(3) the presence of substructures may complicate our simple spherical lens model, and hence could be a main source of uncertainties in @xmath30. the absence of the secondary arc - like images in most arc - cluster systems may indicate the limitations of the spherical mass distribution in the central regions of clusters. it should be noted that the mass ratios we obtained here are slightly higher than allen (1998) and wu (2000) because they unfortunately used a hubble constant of @xmath99. the use of @xmath13 here will of course make the mass discrepancy problem more pronounced. it should be noted that the gas represents only a @xmath100 perturbation due to the small ratio of gas - to - dm in the central region, likewise the offset of the gas is only a small perturbation (less than @xmath100) to the otherwise concentric matter density or potential. it is unlikely to create a factor of two difference in the lensing - derived enclosed masses within an arc. to illustrate the lensing effect of the offset perturbation and triaxiality, we show the critical curves in figure 4. the solid curves indicate the critical curve of circular nfw plus @xmath16 model without offset, the dotted curves indicate the critical curve of elliptical nfw plus @xmath16 model with offset @xmath101. the square and cross denote the center of dark matter and the hot gas, respectively. for the nfw profile, @xmath102 ; for the @xmath16 model, @xmath103. we also introduce the triaxiality with the ellipticity @xmath104 and position angle @xmath105. we also assume the lens and source redshifts @xmath106, @xmath107. we can see that the predicted critical curves (dotted lines) have very similar sizes as the predicted critical curves for a benchmark model (solid lines) with the same mass dm and gas mass but in concentric spheres. [fig : offset & triallity] model without offset. the dotted curves indicate the critical curve of elliptical nfw & @xmath16 model with offset @xmath101. the square and cross denote the center of dark matter and the x - ray gas, respectively. for the nfw profile, @xmath102 ; for the @xmath16 model, @xmath103. the ellipticity and position angle are @xmath104 and @xmath105. the lens and source redshifts are @xmath106, @xmath107.,title="fig:",width=302] early studies have suggested that statistically unrelaxed clusters have larger mass discrepancies than relaxed clusters (allen 1998 ; wu 2000 ; richard et al. 2010). as shan et al. (2010) have reported, the clusters with large offset of @xmath89 are all unrelaxed clusters. if such offsets exist and are big, then they must come into play in our dynamical studies of galaxy cluster, and should not be ignored, especially for unrelaxed clusters. allen, s.w., 1998, mnras, 296, 392 baldi, a., et al., 2007, apj, 666, 835 bartelmann, m.,1995, a&a, 299, 11 bartelmann, m., & evrard, a. e., 1996, mnras, 283,431 bonamente, m., joy, m., laroque, s., carlstrom, j., reese, e., & dawson, k., 2006, apj, 647, 25 bradac, m., et al., 2006, apj, 652, 937 dunn, r. j. h., & fabian, a. c., 2008, mnras, 385, 757 gioia, i. m., shaya, e. j., le fevre, o., falco, e. e., luppino, g. a., & hammer, f., 1998, apj, 497, 573 jee, m.j., et al., 2007, apj, 661, 728 kneib, j.- p., mellier, y., fort, b., & mathez, g., 1993, a&a, 273, 367 limousin, m., et al., 2007, apj, 668, 643 loeb, a., & mao, s., 1994, apj, 435, 109 markevitch, m., 2006, xru, conf, 723 morandi, a., pedersen, k., & limousin, m., 2010, apj, 713, 491 morandi, a., pedersen, k., & limousin, m., 2010, arxiv : astro - ph/1001.1656 navarro, j.f., frenk, c.s., & white, s.d.m., 1997, apj, 490, 493 nevalainen, j., markevitch, m., & forman, w., 2000, apj, 532, 694 newbury, & fahlman, 1999, arxiv : astro - ph/9905254 richard, j., et al., 2010, mnras, 404, 325 sand, d. j., treu, t., ellis, r. s., & smith g. p., 2005, apj, 627, 32 sanderson, a.j.r., edge, a.c., & smith, g.p., 2009, mnras, 398, 1698 shan, h.y., qin, b., fort, b., tao, c., wu, x.- zhao, h.s., 2010, mnras, accepted smith, g. p., kneib, j., smail, i., mazzotta, p., ebeling, h., & czoske, o., 2005, mnras, 359, 417 tucker, w., et al., 1998, apj, 496, 5 wu, x.- p., 1994, apj, 436, 115 wu, x.- p., & fang, l.- z., 1997, apj, 483, 62 wu, x.- p., 2000, mnras, 316, 299 xu, h., jin, g., & wu, x.-, 2001, apj, 553, 78 lllllllllllllll cluster & @xmath108 & & @xmath109 & @xmath49 & @xmath30 & ref. a@xmath110 & kt & @xmath16 & @xmath111 & @xmath32 & @xmath48 & class@xmath112 & ref.b@xmath110 + & & (arcsec) & (kpc) & & (mpc) & @xmath113 & & (kev) & & (mpc) & @xmath113 & & & + 1e0657 - 56 & 0.296 & 47.4 & 209.2 & 3.24 & 0.25 & 4.37 & 3,4 & @xmath114 & @xmath115 & @xmath116 & @xmath117 & @xmath118 & u & 12 + a68@xmath119 & 0.255 & 14.3 & 56.7 & 1.60 & 0.04 & 0.13 & 11 & @xmath120 & @xmath121 & @xmath122 & @xmath123 & @xmath124 & u & 2 + & & & & 1.60 & 0.10 & 0.80 & & & & & @xmath125 & @xmath126 & & + & & & & 2.63 & 0.11 & 0.94 & & & & & @xmath127 & @xmath128 & + & & & &... & 0.211 & 4.49(3.54)@xmath129 & & & & & @xmath130 & @xmath131 & + & & & & 0.86 & 0.28 & 7.49 & & & & & @xmath132 & @xmath133 & + & & & &... & 0.27 & 7.26(5.72)@xmath129 & & & & & @xmath134 & @xmath135 & + & & & & 1.27 & 0.32 & 8.66 & & & & & @xmath136 & @xmath137 & + & & & &... & 0.12 & 1.54(1.22)@xmath129 & & & & & @xmath138 & @xmath139 & + a267 & 0.230 & 9.62 & 35.3 &... & 0.12 & 1.48(1.20)@xmath129 & 11 & @xmath140 & @xmath141 & @xmath142 & @xmath143 & @xmath144 & u & 2 + a370@xmath119 & 0.375 & 19.9 & 102.7 & 1.30 & 0.41 & 13.1 & 7 & @xmath145 & @xmath146 & @xmath147 & @xmath148 & @xmath149 & u & 13 + & & & & 0.72 & 0.19 & 4.09 & & & & & @xmath150 & @xmath151 & + a697 & 0.282 & 3.07 & 13.1 &... & 0.12 & 1.51(1.15)@xmath129 & 10 & @xmath152 & @xmath153 & @xmath154 & @xmath155 & @xmath156 & u & 2 + a773@xmath119 & 0.217 & 6.43 & 22.6 & 0.65 & 0.11 & 1.39 & 11 & @xmath157 & @xmath153 & @xmath158 & @xmath159 & @xmath160 & u & 2 + & & & & 0.40 & 0.21 & 7.08 & & & & & @xmath161 & @xmath162 & + & & & &... & 0.25 & 6.50(5.36)@xmath129 & & & & & @xmath163 & @xmath164 & + & & & &... & 0.23 & 5.41(4.45)@xmath129 & & & & & @xmath165 & @xmath166 & + & & & & 1.11 & 0.213 & 4.34 & & & & & @xmath161 & @xmath167 & + & & & & 0.40 & 0.16 & 4.14 & & & & & @xmath168 & @xmath169 & + & & & &... & 0.04 & 0.18(0.15)@xmath129 & & & & & @xmath170 & @xmath171 & + & & & & 0.49 & 0.23 & 7.42 & & & & & @xmath165 & @xmath172 & + a963@xmath119 & 0.206 & 7.10 & 24.0 &... & 0.057 & 0.35(0.29)@xmath129 & 11 & @xmath173 & @xmath174 & @xmath175 & @xmath176 & @xmath177 & r & 13 + & & & & 0.71 & 0.09 & 0.87 & & & & & @xmath178 & @xmath179 & + a1689 & 0.183 & 0.60 & 1.85 &... & 0.20 & 4.5(3.8)@xmath129 & 8 & @xmath180 & @xmath181 & @xmath182 & @xmath183 & @xmath184 & r & 13 + a1835 & 0.252 & 1.61 & 6.33 &... & 0.17 & 2.82(2.23)@xmath129 & 11 & @xmath185 & @xmath186 & @xmath187 & @xmath188 & @xmath189 & r & 13 + a1914 & 0.171 & 11.3 & 32.9 &... & 0.10 & 1.16(1.01)@xmath129 & 10 & @xmath190 & @xmath191 & @xmath192 & @xmath193 & @xmath194 & u & 2 + a2204@xmath119 & 0.151 & 1.20 & 3.15 &... & 0.025 & 0.08(0.07)@xmath129 & 10 & @xmath195 & @xmath196 & @xmath197 & @xmath198 & @xmath199 & r & 2 + & & & &... & 0.01 & 0.013(0.012)@xmath129 & & & & & @xmath200 & @xmath201 & + a2163 & 0.203 & 44.0 & 146.9 & 0.73 & 0.07 & 0.58 & 1 & @xmath202 & @xmath203 & @xmath204 & @xmath205 & @xmath206 & u & 13 + a2218@xmath119 & 0.176 & 19.1 & 56.9 & 1.03 & 0.28 & 8.60 & 11 & @xmath207 & @xmath208 & @xmath209 & @xmath210 & @xmath211 & u & 13 + & & & & 0.70 & 0.09 & 0.89 & & & & & @xmath212 & @xmath213 & + & & & & 2.52 & 0.09 & 0.82 & & & & & @xmath212 & @xmath214 & + a2219@xmath119 & 0.228 & 11.3 & 41.2 &... & 0.09 & 0.79(0.64)@xmath129 & 11 & @xmath215 & @xmath216 & @xmath217 & @xmath218 & @xmath219 & u & 13 + & & & &... & 0.12 & 1.54(1.26)@xmath129 & & & & & @xmath220 & @xmath221 & + a2259 & 0.164 & 16.3 & 45.9 & 1.48 & 0.04 & 0.13 & 10 & @xmath222 & @xmath223 & @xmath224 & @xmath225 & @xmath226 & u & 2 + a2261@xmath119 & 0.224 & 1.31 & 4.72 &... & 0.12 & 1.49(1.22)@xmath129 & 10 & @xmath227 & @xmath228 & @xmath229 & @xmath230 & @xmath231 & r & 2 + & & & &... & 0.11 & 1.26(1.03)@xmath129 & & & & & @xmath232 & @xmath233 & + a2390 & 0.228 & 6.00 & 21.9 & 0.91 & 0.20 & 3.8 & 10 & @xmath234 & @xmath235 & @xmath236 & @xmath237 & @xmath238 & u & 13 + cl0024 & 0.395 & 13.2 & 70.4 & 1.68 & 0.26 & 4.7 & 6 & @xmath239 & @xmath240 & @xmath241 & @xmath242 & @xmath243 & u & 13 + ms0440 & 0.190 & 1.50 & 4.89 & 0.53 & 0.10 & 1.23 & 5,10 & @xmath244 & @xmath245 & @xmath246 & @xmath247 & @xmath248 & r & 13 + ms0451 & 0.550 & 12.1 & 76.8 &... & 0.23 & 7.6(3.5)@xmath129 & 10 & @xmath249 & @xmath250 & @xmath251 & @xmath252 & @xmath253 & u & 13 + ms1008 & 0.360 & 5.43 & 27.3 &... & 0.30 & 9.2(6.2)@xmath129 & 1 & @xmath254 & @xmath255 & @xmath256 & @xmath257 & @xmath258 & r & 13 + ms1358 & 0.329 & 2.79 & 13.2 & 4.92 & 0.14 & 1.24 & 10 & @xmath259 & @xmath260 & @xmath261 & @xmath262 & @xmath263 & r & 13 + ms1455 & 0.258 & 2.77 & 11.1 &... & 0.11 & 1.22(0.96)@xmath129 & 9,10 & @xmath264 & @xmath265 & @xmath266 & @xmath267 & @xmath268 & u & 13 + ms2053 & 0.580 & 10.5 & 69.1&3.15 & 0.16 & 1.41 & 10 & @xmath269 & @xmath265 & @xmath270 & @xmath271 & @xmath272 & u & 2 + ms2137 & 0.313 & 5.70 & 26.1 &... & 0.10 & 0.99(0.72)@xmath129 & 9,10 & @xmath273 & @xmath274 & @xmath275 & @xmath276 & @xmath277 & r & 13 + pks0745 & 0.103 & 6.82 & 12.9&0.43 & 0.05 & 0.42 & 10 & @xmath278 & @xmath279 & @xmath280 & @xmath281 & @xmath282 & r & 13 + rxj1347 & 0.451 & 2.81 & 16.2&0.81 & 0.28 & 8.9 & 1 & @xmath283 & @xmath284 & @xmath285 & @xmath286 & @xmath287 & r & 13 +

recent studies of lensing clusters reveal that it might be fairly common for a galaxy cluster that the x - ray center has an obvious offset from its gravitational center which is measured by strong lensing. we argue that if these offsets exist, then x - rays and lensing are indeed measuring different regions of a cluster, and may thus naturally result in a discrepancy in the measured gravitational masses by the two different methods. here we investigate theoretically the dynamical effects of such lensing - x - ray offsets, and compare with observational data. we find that for typical values, the offset alone can give rise to a factor of two difference between the lensing and x - ray determined masses for the core regions of a cluster, suggesting that such `` offset effect '' may play an important role and should not be ignored in our dynamical measurements of clusters. [firstpage] dark matter - gravitational lensing - x - rays : galaxies : clusters

1006.3484

Introduction nuclear fragmentation resulting from heavy ion collsions is a complex phenomenon. the role of equilibration and dynamics has not yet been determined as a plethora of approaches have been investigated. examples of approaches are evaporative pictures@xcite, percolation models@xcite, lattice gas models, and dynamical models based on boltzmann simulations@xcite. in this paper we consider the statistical approach@xcite where one considers sampling all configurations of non - interacting clusters. recently, chase and mekjian@xcite derived relations which allow the exact calculation of the canonical partition function for such a system. by eliminating the need for computationally intensive monte carlo procedures and associated approximations, this technique allows a deeper insight into the thermodynamic principles which drive the statistics of fragmentation. in the next section we present the recursive technique of chase and mekjian and review the thermodynamic properties, some of which have already been presented in the literature. we emphasize that the surface energy is the most important parameter in determining the fragmentation and phase transition properties of the model. in the three subsequent sections, we present extensions of the model which are necessary for serious modeling of nuclear systems : excluded volume, coulomb effects, and isospin degrees of freedom. in section [micro_sec] we show how a microcanonical distribution may be generated from the canonical distribution. The model for completeness, we present an outline of the model, which is based on the work of chase and mekjian@xcite. the expressions used here are based on a picture of non - interacting liquid drops. mekjian and lee had also applied similar recursion relations@xcite to a more algebraically motivated fragmentation model that was not based on a liquid - drop picture. we consider that there are @xmath0 nucleons which thermalize in a volume @xmath1 much larger than @xmath2 where @xmath3 is the ground state volume of a nucleus of @xmath0 nucleons. these nucleons can appear as monomers but also as composites of @xmath4 nucleons. the canonical partition function of this system can be written as @xmath5 where @xmath6 is the partition function of a single composite of size @xmath7, @xmath8 is the number of such composites and the sum goes over all the partitions which satisfy @xmath9. a priori this appears to be a horrendously complicated problem but @xmath10 can be computed recursively via the formula, @xmath11 here @xmath12 is 1. it is this formula and the generalisation of this to more realistic case (see later) that makes this model so readily soluble. all properties of the system are determined by the partition functions of indepedent particles. the recursive formula above allows a great deal of freedom in the choice of partition functions for individual fragments, @xmath6. any function of temperature, density and @xmath0 is allowed. however, explicit dependence on the configuration of the remainder of the system is outside the scope of this treatment. for the illustrative purposes of this section, we assume the form, @xmath13 the first part is due to the kinetic motion of the center of mass of the composite in the volume @xmath1 and the second part is due to the internal structure. following the choice of reference@xcite we assume the form @xmath14 here @xmath15 is the volume energy per nucleon(=16 mev), @xmath16 is the surface tension which is a function of the temperature @xmath17. the origin of the different terms in eq. ([bondorf_fe_eq]) is the following : @xmath18 is the ground state energy of the composite of @xmath19 nucleons, and the last term in the exponential arises because the composite can be not only in the ground state but also in excited states which are included here in the fermi - gas approximation. following reference @xcite the value of @xmath20 is taken to be 16 mev. lastly the temperature dependence of @xmath16 in ref@xcite is @xmath21^{5/4}$] with @xmath22 mev and @xmath23 mev. any other dependence could be used including a dependence on the average density. upon calculation, the model described above reveals a first order phase transition. in figure [cv_fig] the specific heat at constant volume, @xmath24, is displayed as a function of temperature for systems of size, @xmath25, @xmath26 and @xmath27. the sharp peak represents a discontinuity in the energy density, which sharpens for increasingly large systems. the usual picture of a liquid - gas phase transition gives a discontinuity in the energy density when pressure is kept constant rather than when the volume is kept constant. to understand this result we consider a system divided into one large cluster and many small clusters. the pressure and free energy may then be approximated as @xmath28 where @xmath29 is the number of clusters. the bulk term depends only on the temperature and not on the way in which the nucleons are partioned into fragments. we have neglected the surface energy term which is proportional to @xmath30. in this limit, @xmath31 and @xmath32 become @xmath33 the bulk term depends only on the temperature and is therefore continuous across the phase transition. thus, a spike in @xmath32 is equivalent to a spike in @xmath34 since both are proportional to @xmath35. it is difficult to make a connection between this approach and the standard maxwell construction, since here interactions between particles enter only through the surface term. intrinsic thermodynamic quantities may be calculated in a straightforward manner. for instance the pressure and chemical potentials may be calculated through the relations, @xmath36 calculations of @xmath37 and @xmath38 are displayed in figure [mup_fig] as a function of density for a system of size @xmath39. both the pressure and chemical potential remain roughly constant throughout the region of phase coexistence. of particular note is that the pressure actually falls in the coexistence region due to finite size effects. we now make some comments about influences of various factors in eq. ([bondorf_fe_eq]). the bulk terms, @xmath40, are not affected by the free energy, thus they may be ignored when calculating fragmentation observables. their influence with respect to intrinsic thermodynamic quantities is of a trivial character. the surface term @xmath16 is completely responsible for determining all observables related to fragmentation and therefore all aspects of the phase transition. aside from the system size @xmath0, fragmentation is determined by two dimensionless parameters. the first is the specific entropy, @xmath41 and the second is the surface term @xmath42. at a given temperature the free energy @xmath43 of @xmath0 nucleons should be minimized. with the surface tension term, @xmath44 is minmised if the whole system appears as one composite of @xmath0 nucleons but the entropy term encourages break up into clusters. at low temperatures the surface term dominates while at high temperatures entropy prevails and the system breaks into small clusters. the mass distribution may be calculated given the partition function. @xmath45 the mass distribution is displayed in figure [massdist_fig] for three temperatures, 6.0, 6.25 and 6.5 mev which are centered about the transition temperature of 6.25 mev. the distributions have been multiplied by @xmath7 to emphasize the decomposition of the system. the mass distribution changes dramatically in this small temperature range. the behavior is reminiscent of that seen in the percolation or lattice gas models@xcite. Excluded volume the volume used to define to the partition functions of individual fragments, @xmath6 given in eq. ([bondorf_fe_eq]), should reflect only that volume in which the fragments are free to move. hahn and stcker suggested using @xmath46 to incorporate the volume taken up by the nuclei. by inspecting eq. ([bondorf_fe_eq]) on can see that this affects the partion function by simply mapping the density or volume used to plot observables. more realistically, the excluded volume could depend upon the multiplicity. nonetheless, in rather complicated calculations not reported here, it was found that for the purpose of obtaining @xmath47 diagrams in the domain of interest in this paper, it is an acceptable approximation to ignore the multiplicity dependence of the excluded volume@xcite. incorporating a multiplicity dependence would be outside the scope of the present model, as it would represent an explicit interaction between fragments. however, one could add an @xmath4-dependence to the volume term to account for the difficulty of fitting fragments of various sizes into a tight volume. this might affect the model in a non - trivial fashion. we like to remind the reader that the parameter @xmath48 in the van der waals eos : @xmath49 also has its roots in the excluded volume. but there @xmath48 plays a crucial role. we could not for example set @xmath48=0 without creating an instability at high density. furthermore, the phase transition disappears when @xmath4 is set to zero. Coulomb effects it has been understood that the coulomb effects alter the phase structure of nuclear matter@xcite. although explicit coulomb interactions are outside the scope of this treatment, they may be approximated by considering a screened liquid drop formula for the coulomb energy as has been used by bondorf and donangelo@xcite. the addition to the internal free energy given in eq. ([bondorf_fe_eq]) is @xmath50 this form implies a jellium of uniform density that cancels the nucleons positive charge when averaged over a large volume. this may be more physically motivated for the modeling of stellar interiors where the electrons play the role of the jellium. we display @xmath31, both with and without coulomb terms for an @xmath51 system in figure [cvcoulomb_fig]. coulomb forces clearly reduce the temperature at which the transition occurs. for sufficiently large systems, coulomb destroys the transition as large drops become unstable to the coulomb force. Conservation of isospin the recursive approach employed here is easily generalized to incorporate multiple species of particles. if there exist a variety of particles with conserved charges @xmath52, @xmath53, one can write a recursion relation for each charge@xcite. @xmath54 where @xmath55 is the net conserved charge of type @xmath56 and @xmath57 is the charge of type @xmath56 carried by the fragment noted by @xmath19. for the nuclear physics example, one would wish to calculate @xmath58 where @xmath59 and @xmath60 were the conserved neutron and proton numbers. to find @xmath61 one must know @xmath62 for all @xmath63 or @xmath64. to accomplish this one must use both recursion relations. Obtaining the microcanonical distribution in nuclear collisions, one does not have access to a heat bath, but one can vary the excitation energy. a microcanonical treatment is therefore more relevant for practical calculations, particularly given the existence of a first order phase transition which occupies an infinitesimal (in the limit of large @xmath0) range of temperatures in a canonical calculation, but a finite range of energies in a microcanonical ensemble. the relevant partition function for a microcanonical ensemble is the density of states, @xmath65 where the sum over @xmath56 represents the sum over all many - body states. although @xmath66 is easily calculable given the recursion relations discussed in the previous sections, one must perform the integral over @xmath67 numerically. the true solution for the density of states would be ill - defined given the discreet nature of quantum spectra which can not be combined with a delta function. however, if one defines the density of states in a finite region of size @xmath68, the density of states becomes well - behaved even for discreet spectra. for that reason we more pragmatically define the density of states as @xmath69 one might have considered replacing the delta function by a lorentzian rather than by a gaussian, but this would be dangerous given that the density of states usually rises exponentially for a many - body system. the finite range @xmath68 used to sample the density of states might correspond to the range of excitation energies sampled in an experimental binning. in the limit @xmath70, @xmath71 approaches the density of states. as an example of a quantity one may wish to calculate with a microcanonical approach, we consider the average multiplicity of a fragment of type @xmath19 in a system whose total energy is within @xmath68 of @xmath44. @xmath72 where @xmath73 is the number of particles of species @xmath19 within the fragment @xmath56. the integration over @xmath67 clearly provides an added numerical challenge that increases for small @xmath68. for the purposes of generating a mass distribution, one must perform this integration for every species. it might be worthwile to consider estimating the integrals over @xmath67 with the saddle point method, although one should be wary of taking derivatives of @xmath74 with respect to @xmath67 in the phase transition region. microcanonical quantities might also be calculated in a completely different manner by discreetizing the energy. for instance one might measure energies in units of 0.1 mev. one might then treat energy on the same footing as any other conserved charge. one may then write recursion relations for @xmath75, the number of ways to arrange @xmath0 nucleons with net energy @xmath44, where @xmath44 is an integer. @xmath76 here, @xmath77 is the number of ways of arranging a fragment of type @xmath19 with net energy @xmath78. all other relavant microcanonical quantities may be calculated in a similar manner. since one needs to calculate @xmath59 at all energies @xmath79 less than the targeted energy @xmath44, and must sum over all energies less than @xmath79 to obtain @xmath80, the length of the calculation is proportional to @xmath81. typically, nuclear decays occur with on the order of a gev of energy deposited in a nucleus. therefore, these calculations may become numerically cumbersome unless the energy is discreetized rather coarsely. Summary the recursive techniques discussed here have several attractive features. they are easy to work with, incorporate characteristics of nuclear composites and appear to have standard features of liquid - gas phase transitions. in the present forms these models are resricted to low densities. for modeling nuclear disintegration this is not a serious problem, although for completeness it would be nice to be able to modify the model so that it can be extended to higher density. in this paper we have studied thermal properties of the model, and we emphasize the importance of the surface term in determining these properties. we can associate the discontinuity in the energy density with temperature to the discontinuity in the number of clusters. in addition, we have seen that including coulomb effects lowers the temperature at which the fragmentation transition occurs and reduces the sharpness of the phase transition. we have also presented an extension of the formalism for the inclusion of isospin degrees of freedom. for comparing to nuclear physics experiments, development of the microcanonical approaches presented here is of greatest importance. it remains to be seen whether the microcanonical formalisms are tenable, as they have yet to be implemented. p. chomaz, ann. france * 21 *, 669 (1996) burgio, ph. chomaz and j. randrup, phys. 69 *, 885 (1992). h. feldmeier and j. schnack, prog. particle nucl. phys. * 39 *, 393 (1997). a. ohnishi and j. randrup, phys. b * 394 *, 260 d. kiderlen and p. danielewicz, nucl. a * 620 *, 346 (1997). s. pratt, c. montoya and f. ronning, phys. b*349 *, 261 (1995). j. randrup and s. koonin, nucl. a * 356 *, 321 (1981). gross, rep. phys. * 53 *, 605 (1990). bondorf, a.s. botvina, a.s. iljinov, i.n. mishustin and k. sneppen, physics reports * 257 *, 133 - 221 (1995). lee and a.z. mekjian, phys. c*47 *, 2266 (1993). lee and a.z. mekjian, phys. c*45 *, 1284 (1992) lee and a.z. mekjian, phys. c*50 *, 3025 (1994). bondorf, a.s. botvina, a.s. ijilinov, i.n. mishustin and k. sneppen, phys. rep. * 257*,133(1995)

the statistical model of chase and mekjian, which offers an analytic solution for the canonical ensemble of non - interacting fragments, is investigated for it s thermodynamic behavior. various properties of the model, which exhibits a first - order phase transition, are studied. the effects of finite particle number are investigated. three extensions of the model are considered, excluded volume, coulomb effects and inclusion of isospin degrees of freedom. a formulation of a microcanonical version of the model is also presented. 2.0 cm

nucl-th9903007

Introduction in the standard model (sm), the only source of cp violation is the kobayashi - maskawa phase @xcite, localized in the unitarity triangle (ut) of the cabibbo - kobayashi - maskawa (ckm) matrix @xcite. thanks to the precise measurements at the current @xmath3-factories, cp violation could be established in @xmath4 @xcite, leading to a precise measurement of @xmath5, where the current world average yields @xcite @xmath6. the extractions of the other two angles @xmath7 and @xmath8 are expected mainly through cp violation in the charmless @xmath3 decays, such as @xmath9 and similar modes @xcite. the current @xmath3-factories measurements have been averaged to yield @xcite : @xmath10 on the theoretical side, the analysis is challenging due to the need to know the ratio of penguin - to - tree amplitude contributing to this process. in this talk, we present the result of @xcite, where a transparent method of exploring the ut through the cp violation in @xmath11, combined with the `` gold - plated '' mode @xmath4 has been proposed. a model independent lower bound on the ckm parameters as functions of @xmath12 and @xmath1 is derived. our estimate of the hadronic parameters are carried out in qcd factorization (qcdf) and confronted to other approaches. Basic formulas the time - dependent cp asymmetry in @xmath0 decays is defined by @xmath13 @xmath14 with @xmath15, and @xmath16 and @xmath8 are ckm angles which are related to the wolfenstein parameters @xmath17 and @xmath18 in the usual way @xcite. the penguin - to - tree ratio @xmath19 can be written as @xmath20. the real parameters @xmath21 and @xmath22 defined in this way are pure strong interaction quantities without further dependence on ckm variables. for any given values of @xmath21 and @xmath22 a measurement of @xmath12 and @xmath23 defines a curve in the (@xmath17, @xmath18)-plane, expressed respectively through @xmath24}{((1-\bar\rho)^2+\bar\eta^2) (r_b^2+r^2 + 2 r\bar\rho \cos\phi)}\end{aligned}\]] @xmath25 the penguin parameter @xmath26 has been computed in @xcite in the framework of qcdf. the result can be expressed in the form @xmath27 } { a_1+a^u_4 + r^\pi_\chi a^u_6 + r_a[b_1+b_3 + 2 b_4]},\]] where we neglected the very small effects from electroweak penguin operators. a recent analysis gives @xcite @xmath28 where the error includes an estimate of potentially important power corrections. in order to obtain additional insight into the structure of hadronic @xmath3-decay amplitudes, it will be also interesting to extract these quantities from other @xmath3-channels, or using other methods. in this perspective, we have considered them in a simultaneous expansion in @xmath29 and @xmath30 (@xmath31 is the number of colours) in ([rqcd]). expanding these coefficients to first order in @xmath29 and @xmath30 we find that the uncalculable power corrections @xmath32 and @xmath33 do not appear in ([rqcd]), to which they only contribute at order @xmath34. using our default input parameters, one obtains the central value @xcite : @xmath35, which seems to be in a good agreement with the standard qcdf framework at the next - to - leading order. as a second cross - check, one can extract @xmath21 and @xmath22 from @xmath36 and @xmath37, leading to the central value @xcite @xmath38, in agreement with the above results to its experimental value @xmath39.], although their definitions differ slightly from @xmath40 (see @xcite for further discussions). Ut through cp violation observables it is possible to fix the ut by combining the information from @xmath12 with the value of @xmath1, well known from the `` gold - plated '' mode @xmath41. the angle @xmath16 of the ut is given by @xmath42 the current world average @xcite @xmath6, implies @xmath43 given a value of @xmath44, @xmath17 is related to @xmath18 by @xmath45. the parameter @xmath17 may thus be eliminated from @xmath12 in ([srhoeta]), which can be solved for @xmath18 to yield @xmath46,\nonumber \end{aligned}\]] with @xmath47 the two observables @xmath44 (or @xmath1) and @xmath12 determine @xmath18 and @xmath17 once the theoretical penguin parameters @xmath21 and @xmath22 are provided. the determination of @xmath18 as a function of @xmath12 is shown in fig. [fig : etabspp], which displays the theoretical uncertainty from the penguin parameters @xmath21 and @xmath22 in qcdf. since the dependence on @xmath22 enters in ([etataus]) only at second order, it turns out that its sensitivity is rather mild in contrast to @xmath21. in the determination of @xmath18 and @xmath17 described here discrete ambiguities do in principle arise, however they are ruled out using the standard fit of the ut (see @xcite for further discussions). after considering the implications of @xmath12 on the ut, let s explore now @xmath23. since @xmath23 is an odd function of @xmath22, it is therefore sufficient to restrict the discussion to positive values of @xmath22. a positive phase @xmath22 is obtained by the perturbative estimate in qcdf, neglecting soft phases with power suppression. for positive @xmath22 also @xmath23 will be positive, assuming @xmath48, and a sign change in @xmath22 will simply flip the sign of @xmath23. in contrast to the case of @xmath12, the hadronic quantities @xmath21 and @xmath22 play a prominent role for @xmath23, as can be seen in ([crhoeta]). this will in general complicate the interpretation of an experimental result for @xmath23. the analysis of @xmath23 becomes more transparent if we fix the weak parameters and study the impact of @xmath21 and @xmath22. an important application is a test of the sm, obtained by taking @xmath17 and @xmath18 from a sm fit and comparing the experimental result for @xmath23 with the theoretical expression as a function of @xmath21 and @xmath22. in fig. [fig : cpipi], a useful representation is obtained by plotting contours of constant @xmath23 in the (@xmath21, @xmath22)-plane, for given values of @xmath17 and @xmath18. within the sm this illustrates the correlations between the parameters @xmath40 and observable @xmath23. as it has been shown in @xcite, a bound on the parameter @xmath23 exists, given by @xmath49 with @xmath50 and where the maximum occurs at @xmath51. if @xmath52, no useful upper bound is obtained. however, if @xmath53, then @xmath54 is maximized for @xmath55, yielding the general bound @xmath56. for the conservative bound @xmath57, @xmath58 this implies @xmath59. the bound on @xmath23 can be strengthened by using information on @xmath22, as well as on @xmath60, and employing ([barc]). then @xmath58 and @xmath61 gives @xmath62. Model independent bound on the ut as has been shown in @xcite, the following inequality can be derived from ([etataus]) for @xmath63 @xmath64 this bound is still _ exact _ and requires no information on the phase @xmath22. assuming now @xmath65, we have @xmath66 and @xmath67 we emphasize that this lower bound on @xmath18 depends only on the observables @xmath44 and @xmath12 and is essentially free of hadronic uncertainties. since both @xmath21 and @xmath22 are expected to be quite small, we anticipate that the lower limit ([etabound]) is a fairly strong bound, close to the actual value of @xmath18 itself (see @xcite for further details). we also note that the lower bound ([etabound]) represents the solution for the unitarity triangle in the limit of vanishing penguin amplitude, @xmath68. in other words, the model - independent bounds for @xmath18 and @xmath17 are simply obtained by ignoring penguins and taking @xmath69 when fixing the unitarity triangle from @xmath12 and @xmath1. let us briefly comment on the second solution for @xmath18, which has the minus sign in front of the square root in ([etataus]) replaced by a plus sign. for positive @xmath12 this solution is always larger than ([etataus]) and the bound ([etabound]) is unaffected. for @xmath70 the second solution gives a negative @xmath18, which is excluded by independent information on the ut (for instance from @xmath71). because we have fixed the angle @xmath16, or @xmath44, the lower bound on @xmath18 is equivalent to an upper bound on @xmath72. the constraint ([etabound]) may also be expressed as a lower bound on the angle @xmath8 or a lower bound on @xmath73 (see @xcite for further details). in figs. [fig : etabound], we represent the lower bound on @xmath18 as a function of @xmath12 for various values of @xmath1. from fig. [fig : etabound] we observe that the lower bound on @xmath18 becomes stronger as either @xmath12 or @xmath1 increase. in fig. [fig : utbound] we illustrate the region in the @xmath74 plane that can be constrained by the measurement of @xmath1 and @xmath12 using the bound in ([etabound]). we finally note that the condition @xmath75, which is crucial for the bound, could be independently checked @xcite by measuring the mixing - induced cp - asymmetry in @xmath76, the u - spin counterpart of the @xmath77 mode @xcite. Summary in this talk, we have proposed strategies to extract information on weak phases from cp violation observables in @xmath0 decays even in the presence of hadronic contributions related to penguin amplitudes. assuming knowledge of the penguin pollution, an efficient use of mixing - induced cp violation in @xmath0 decays, measured by @xmath12, can be made by combining it with the corresponding observable from @xmath2, @xmath1, to obtain the unitarity triangle parameters @xmath17 and @xmath18. the sensitivity on the hadronic quantities, which have typical values @xmath78, @xmath79, is very weak. in particular, there are no first - order corrections in @xmath22. for moderate values of @xmath22 its effect is negligible. concerning our penguin parameters, namely @xmath21 and @xmath22, they were investigated systematically within the qcdf framework. to validate our theoretical predictions, we have calculate these parameters in the @xmath29 and @xmath30 expansion, which exhibits a good framework to control the uncalculable power corrections, in the factorization formalism. as an alternative proposition, we have also considered to extract @xmath21 and @xmath22 from other @xmath3 decay channels, such as @xmath36 and @xmath37, relying on the su(3) argument. using these three different approaches, we found a compatible picture in estimating these hadronic parameters. Acknowledgements i thank the organizers for their invitation and i am very grateful to gerhard buchalla for the extremely pleasant collaboration. this work is supported by the dfg under contract bu 1391/1 - 2. m. kobayashi and t. maskawa, prog. * 49 * (1973) 652. n. cabibbo, phys. * 10 * (1963) 531. b. aubert _ et al. _ [babar collaboration], phys. rev. lett. * 87 * (2001) 091801. k. abe _ et al. _ [belle collaboration], phys. * 87 * (2001) 091802. heavy flavor averaging group, + http://www.slac.stanford.edu/xorg/hfag/ g. buchalla and a. s. safir, hep - ph/0406016 ; a. s. safir, hep - ph/0311104. g. buchalla and a. s. safir, phys. * 93 * (2004) 021801 l. wolfenstein, phys. rev. lett. * 51 * (1983) 1945 ; a. j. buras, _ et al. d * 50 * (1994) 3433. m. beneke _ et al. _, b*606*(2001)245 ckmfitter working group, lp2003 update, sep. 2003, http://ckmfitter.in2p3.fr

we study the implication of the time - dependent cp asymmetry in @xmath0 decays on the extraction of weak phases taking into account the precise measurement of @xmath1, obtained from the `` gold - plated''mode @xmath2. predictions and uncertainties for the hadronic parameters are investigated in qcd factorization. furthermore, independent theoretical and experimental tests of the factorization framework are briefly discussed. finally, a model - independent bound on the unitarity triangle from cp violation in @xmath0 and @xmath2 is derived.

hep-ph0409031

Introduction the open connectome project (located at http://openconnecto.me) aims to annotate all the features in a 3d volume of neural em data, connect these features, and compute a high resolution wiring diagram of the brain, known as a connectome. it is hoped that such work will help elucidate the structure and function of the human brain. the aim of this work is to automatically annotate axoplasmic reticula, since it is extremely time consuming to hand - annotate them. specifically, the objective is to achieve an operating point with high precision, to enable robust contextual inference. there has been very little previous work towards this end @xcite. axoplasmic reticula are present only in axons, indicating the identity of the surrounding process and informing automatic segmentation. Procedure the brain data we are working with was color corrected using gradient - domain image - stitching techniques @xcite to adjust contrast through the slices. we use this data as the testbed for running our filters and annotating axoplasmic reticula. the bilateral filter @xcite is a non - linear filter consisting of one 2d gaussian kernel @xmath0, which decays with spatial distance, and one 1d gaussian kernel @xmath1, which decays with pixel intensity : @xmath2_p = \frac{1}{w_p}\sum_{q\in s}g_{\sigma_{s}}(||p - q||)g_{\sigma_{r}}(i_p - i_q)i_q,\\ & \hspace{4mm}\textrm{where } w_p = \sum_{q\in s}g_{\sigma_{s}}(||p - q||)g_{\sigma_{r}}(i_p - i_q) \end{split}\]] is the normalization factor. this filter smooths the data by averaging over neighboring pixels while preserving edges, and consequently important detail, by not averaging over pixels with large intensity difference. applying this filter accentuates features like axoplasmic reticula in our data. even with a narrow gaussian in the intensity domain, the bilateral filter causes some color bleeding across edges. we try to undo this effect through laplacian sharpening. the laplacian filter computes the difference between the intensity at a pixel and the average intensity of its neighbors. therefore, adding a laplacian filtered image to the original image results in an increase in intensity where the average intensity of the surrounding pixels is less than that of the center pixel, an intensity drop where the average is greater, and no change in areas of constant intensity. hence, we use the 3x3 laplacian filter to highlight edges around dark features such as axoplasmic reticula. we use a morphological region growing algorithm on our filtered data to locate and annotate axoplasmic 26.5 mm @xmath3 26.5 mm @xmath3 26.5 mm @xmath3 26.5 mm @xmath3 26.5 mm 26.5 mm 2 reticula. we implement this by iterating over the filtered image and looking for dark pixels, where a dark pixel is defined as a pixel with value less than a certain specified threshold. when a dark pixel is found, we check its 8-neighborhood to determine if the surrounding pixels are also below the threshold. then, we check the pixels surrounding these, and we do this until we find only high intensity pixels, or until we grow larger than the diameter of an axoplasmic reticula. the thresholds we use in our algorithm are biologically motivated and tuned empirically. finally, we track our annotations through the volume to verify their correctness and identify axoplasmic reticula that were missed initially. for each slice, we traverse the annotations and check if an axoplasmic reticulum is present in the corresponding xy - location (with some tolerance) in either of the adjacent slices. if a previously annotated axoplasmic reticulum object is present, we confirm the existing annotation. otherwise, the adjacent slice locations are checked for axoplasmic reticula with a less restrictive growing algorithm, and new annotations are added in the corresponding slice. if no axoplasmic reticulum object is found in either of the adjacent slices, then we assume the annotation in the current slice to be incorrect, and delete it. Results and future work we qualitatively evaluated our algorithm on 20 slices from the kasthuri11 dataset, and quantitatively compared our results against ground truth from a neurobiologist. our algorithm annotates axoplasmic reticulum objects with 87% precision, and 52% recall. these numbers are approximate since there is inherent ambiguity even among expert annotators. our current algorithm is designed to detect transverally sliced axoplasmic reticula. in future work, we plan to extend our morphological region growing algorithm to also find dilated axoplasmic reticula, and to incorporate a more robust tracking method such as kalman or particle filtering. additionally, our algorithm can be adapted to annotate other features in neural em data, such as mitochondria, by modifying the morphological region growing algorithm.

* _ abstract _ : * in this paper, we present a new pipeline which automatically identifies and annotates axoplasmic reticula, which are small subcellular structures present only in axons. we run our algorithm on the kasthuri11 dataset, which was color corrected using gradient - domain techniques to adjust contrast. we use a bilateral filter to smooth out the noise in this data while preserving edges, which highlights axoplasmic reticula. these axoplasmic reticula are then annotated using a morphological region growing algorithm. additionally, we perform laplacian sharpening on the bilaterally filtered data to enhance edges, and repeat the morphological region growing algorithm to annotate more axoplasmic reticula. we track our annotations through the slices to improve precision, and to create long objects to aid in segment merging. this method annotates axoplasmic reticula with high precision. our algorithm can easily be adapted to annotate axoplasmic reticula in different sets of brain data by changing a few thresholds. the contribution of this work is the introduction of a straightforward and robust pipeline which annotates axoplasmic reticula with high precision, contributing towards advancements in automatic feature annotations in neural em data. + 2

1404.4800

Introduction compact x - ray sources exhibit a wide range of temporal variabilities (from milliseconds to years). perhaps none of these is as exotic and diverse as the x - ray temporal variability observed from the black hole microquasar grs 1915 + 105 (castro - tirado, brandt, & lund 1992 ; greiner, morgan, & remillard 1996 ; morgan, remillard & greiner 1997 ; muno, morgan & remillard 1999). this object is one of two known galactic x - ray sources that exhibit superluminal radio jets (mirabel & rodrigues 1994). the combination of relativistic constraints and radio measurements at hi indicate that the source lies behind the sagittarius arm at a distance of 12.5 @xmath3 kpc (mirabel & rodrigues 1994). interstellar extinction limits optical / ir studies to weak detections at wavelengths less than 1 micron (mirabel et al. the source is suspected to be a black hole binary because of its spectral and temporal similarities with the other galactic x - ray source with superluminal radio jets, gro j1655 - 40 (zhang et al 1994), which has a binary mass function indicative of a black hole system (bailyn et al. 1995). estimates for the mass of the compact object in grs 1915 + 105 range from 7 to 33 @xmath4. even with the uncertainty in distance, its peak x - ray luminosity is unusually high, i.e., @xmath5 ergs / sec, which is around the eddington luminosity for a @xmath6 @xmath4 object. in spite of several attempts it has proven especially illusive to interpret the x - ray light curves of grs1915. it is not yet clear that even the basic time scales exhibited by the variability have been successfully explained. belloni et al. (1997a, b) accounted for the observations with an empirical model in which the inner disk region `` disappears '' in the low count rate state, and is then replenished on a viscous time scale. the parameters of their model are : the inner disk radius, @xmath7 ; the corresponding effective temperature of the disk @xmath8, and an ad - hoc non - thermal power law (which is possibly produced in the disk corona). although no detailed physical model for the instability was given, very interesting patterns of behavior for @xmath7 and @xmath8, as well as several other observables, were deduced from the data, and the shakura - sunyaev viscosity parameter was found to be unexpectedly low (which may mean that the standard viscosity prescription is invalid for this source). the rather small values of @xmath7 found by these authors can be used to discriminate between different models of the accretion flow in grs 1915 (see appendix and [sect : previous]). abramowicz, chen & taam (1995) suggested a model for the low frequency quasi - periodic oscillations (qpo) observed in selected x - ray binaries, in which a corona above the _ standard accretion _ disk leads to a mild oscillatory behavior. with some modifications, this model could reasonably be expected to account for at least some of the temporal variability in grs 1915 + 105 as well (taam, chen & swank 1997). however, it appears to us that the analysis of abramowicz et al. (1995) and taam et al. (1997) contains an error in the heating / cooling equation for the disk which, when corrected, constrains their model to have the same stability characteristics as a standard shakura - sunyaev disk, and is therefore unlikely to explain the grs 1915 + 105 observations (see appendix a). we show more generally in appendix [sect : geometry] that neither a hot central region, nor an advection - dominated flow, nor a `` slim '' accretion disk are compatible with the observations of grs 1915 + 105. (`` slim '' accretion disk theory was developed in the most detail by abramowicz et al. 1988 ; it is similar to a standard thin shakura - sunyaev disk, except for the energy equation, which incorporates the radial advection of energy into the black hole.) in this paper, we attempt to undertake a more systematic study of the variability patterns in grs 1915 + 105 within the context of the `` cold disk@xmath9hot corona '' picture. in [sect : framework] we present a general discussion that will guide us in our selection of a novel (though somewhat ad - hoc) prescription for the viscosity in cases where the radiation pressure is substantial ( [sect : prescription]). the details of our numerical algorithm to solve the time - dependent disk equations with the use of this new viscosity prescription are given in [sect : code]. in [sect : results], we present the results of our time - dependent disk calculations. the calculated light curves are found to agree qualitatively with many observational features of grs 1915 + 105. in particular, the characteristic cycle times and duty cycles are in reasonable agreement with the observations. moreover, the trend in the cycle time with the average accretion rate, @xmath10 has the correct sense. however, there are important disagreements as well. we therefore introduce a more elaborate model in [sect : jet], where, in accordance with the observations (e.g., mirabel & rodrigues 1994), the inner disk is allowed to expel some of its energy in the form of a non - steady jet. we assume that the ejected energy is not observed in x - rays, but rather that it ultimately produces radio emission. we show that this more elaborate model agrees with the grs 1915 + 105 observations much better, perhaps indicating that we are finally developing a zeroth order understanding of the geometry and the most important processes in this enigmatic source. in [sect : previous] we discuss our results in the light of the earlier work on grs 1915, and in [sect : discussion] we summarize our conclusions. Limit cycle behavior in grs1915+105 figure 1 shows four examples of typical x - ray light curves from grs 1915 obtained with the rxte satellite (see, e.g., morgan, remillard, & greiner 1997). it appears that the source undergoes a limit - cycle type of instability ; the cycle times in panels (a) (d) are @xmath1 2400, 60, 1200, and 800 sec, respectively. within the cycles shown in panels (c) and (d) there appear yet other quasi - regular oscillations of still shorter time scale. the shortest time scale over which a substantial change in the x - ray flux occurs is @xmath1 5 sec. we will associate the longer cycles with the viscous time scales of the inner accretion disk, whereas the more rapid behavior will be related to the thermal time scale. the x - ray spectrum of grs 1915 + 105 varies systematically with x - ray intensity ; usually the spectral hardness is strongly correlated (or anti - correlated) with intensity (e.g., belloni et al. 1997a, b ; taam et al. 1997 ; muno et al. more specifically, the spectrum is typically found to be composed of a multi - temperature disk component, hereafter the thermal component ", and a power - law component with a varying spectral index. finally, the source exhibits a wide range of qpos with central frequencies in the range of 0.01 - 10 hz. the amplitude and frequency of the qpos appear to be strongly correlated with the spectral state of grs 1915 + 105 (chen et al. 1997 ; muno et al. 1999). the idea of using a modified viscosity law to test time - dependent phenomena in accretion disks around black holes is not new (e.g., taam & lin 1984 ; chen & taam 1994 ; and further references below in this section). but let us examine in global terms how the various disk configurations relate to the behavior of grs 1915 + 105 in particular. in geometrically thin disks the advection of energy is not important. the viscous time scale is much longer than both the thermal and hydrostatic time scales (see frank et al. 1992), unless @xmath11 which is unlikely for this source (belloni et al. 1997b). in this context, then, the equations for vertical hydrostatic equilibrium and energy balance between viscous heating and radiative losses should be valid. it is customary and instructive to plot the disk effective temperature @xmath12 versus the column density @xmath13 to investigate the stability properties of disks. in the one zone limit, the equation for hydrostatic equilibrium in the vertical direction is @xmath14 where @xmath15 is the mass of the black hole, @xmath16 is the scale height and @xmath17 is the radial distance from the central object. the gas pressure is given by @xmath18 (for hydrogen rich material), where @xmath19 is the mid - plane temperature, @xmath20 is the proton mass, and the radiation pressure is @xmath21. the energy balance equation is given by @xmath22 where @xmath23 is the vertical radiation flux, @xmath24 is the optical depth of the disk, @xmath25, and @xmath26 is the radiative opacity (assumed to be dominated by electron scattering opacity), @xmath27 is the stress tensor (see, e.g., shapiro & teukolsky, 14.5), and @xmath28 is the keplerian angular frequency. as we said earlier, in the standard shakura - sunyaev formulation, the stress tensor is proportional to the total pressure, @xmath29 we can now combine equations ([eq1]) ([ss]) to eliminate the variable h and thereby derive an expression to generate the @xmath12@xmath13 curve for any given viscosity. in figure ([fig : scurve]) we show several illustrative @xmath12@xmath13 curves to help motivate our selection of @xmath0. the dashed curve represents the track that the standard shakura - sunyaev theory with @xmath30 const produces. such a model is thermally and viscously unstable when the slope of the @xmath12@xmath13 curve becomes negative (e.g., frank et al. 1992), which might be a desirable result in the view of the instabilities observed in grs 1915 + 105. however, the observed light curves require a quasi - stable accretion mode for the high - luminosity state as well, which is _ not predicted _ by the standard theory (see also the end of app. b). this suggestion (already discussed by belloni et al. 1997a, b and taam et al. 1997) is motivated by the fact that in some cases the upper luminosity state of the source lasts even longer than the low state, which is presumably the stable gas - pressure - dominated branch. a @xmath12-@xmath13 relation that has a stable high - luminosity state is shown in figure ([fig : scurve], solid curve). it has the characteristic `` s - shape '', familiar from the studies of the thermal ionization instability in accretion disks (e.g., meyer & meyer - hofmeister 1981, bath & pringle 1982, cannizzo, chen & livio 1995, and frank et al. 1992, 5.8) or from studies of radiation - pressure driven instabilities by, e.g., taam & lin (1984) and chen & taam (1994). it is well known that parts of an s - curve that have a negative slope are unstable. in short, when the @xmath12-@xmath13 relation produces an s - curve, the unstable region of the disk oscillates between the two stable branches of the solution, which gives rise to the high and low luminosity states. so far, we have not considered flows where advection is important. as adafs (advection dominated accretion flows) will be discussed in [sect : adaf], here we concentrate on the slim accretion disks (abramowicz et al. these flows have a stable high @xmath10 solution even with the standard viscosity prescription due to the additional cooling (energy transport) via advection. however, the s - curves produced by these models have the upper stable branch for @xmath31 only (see, e.g., see fig. 2 in chen & taam 1993). in other words, if one were to position the upper stable branch of the slim accretion disks on our figure ([fig : scurve]), then it would look similar to the corresponding stable branch of the solid curve in the figure, except that it would be positioned higher by a factor of about @xmath32 (at a given @xmath13). thus, if one assumes that an unstable part of the disk makes a transition from the lower bend in the s - curve upwards to the stable branch, and that @xmath13 remains approximately constant during this process, then the accretion rate in the inner disk must increase by as much a factor of a few hundred to a thousand to reach the stable advective regime. while this may not be a problem in terms of the luminosity output, since most of the energy is advected inwards and the luminosity of the disk saturates at @xmath33 (see fig. 1 in szuszkiewicz, malkan & abramowicz 1996), the light curves produced will always be very spiky and the duty cycle of the high state is always much smaller than the values of @xmath34 that are often seen in grs 1915 + 105. the physical cause of the spiky behavior is the vast difference between the accretion rates in the high and low states. as we will discuss later in [sect : results], an outburst starts because `` too much '' mass (column density @xmath13) has been accumulated above the maximum possible stable value of @xmath35 (i.e., @xmath13 at the lower bend of the s - curve in figure [fig : scurve]). if the high stable state accretion rate is very much larger than the low state accretion rate, it then takes only a very short time in the high state to remove this excess mass from the inner disk. in other words, it would not be possible to maintain mass conservation if the accretion rate persisted at the high state for very long. this is indeed seen from the temperature curves computed for slim disks by szuszkiewicz & miller (1998, see their figures 7 - 9). therefore, we believe that slim disks are not likely to adequately account for the behavior of grs 1915 + 105, at least not if one seeks a theory capable of explaining all of the variability in this source. The basic model the geometry in our unstable disk model is that of a standard shakura - sunyaev configuration (except for the viscosity law) overlayed with a hot corona. here we describe the set of equations that we use to follow the temporal evolution of this disk ; these equations are applicable to any local viscosity law. a discussion of the viscosity itself is deferred until [sect : prescription]. the standard euler equations for conservation of mass and angular momentum in the disk (see, e.g., frank et al. 1992) can be combined in the usual way to yield the equation describing the evolution of the surface density : @xmath36 \right\}\,, \label{eq4}\]] where all of the variables, except for @xmath37, were defined in the previous section. here @xmath37 is the viscosity that is related to @xmath0 by @xmath38 = @xmath39. the time - dependent energy equation includes the heating and cooling terms of equation ([eq2]), but must also be able to take into account large radial gradients of temperature, and therefore includes a number of additional terms. the form of the energy equation that we use follows the formalism of abramowicz et al. (1995) with some modifications, and is given by @xmath40 \nonumber \\ = f^{+ } - f^{- } - { 2\over r}\, { \partial(r f_r h)\over \partial r } + j , \label{eq5}\end{aligned}\]] where @xmath41, @xmath42 is the ratio of the radiation to the gas pressure, @xmath43 is the ratio of specific heats (@xmath44) and @xmath45 is given in abramowicz et al. (the radial velocity @xmath46 is given by the standard expression (eq. 5.7 of frank et al. 1992) : @xmath47. \label{vr}\]] the terms on the left hand side of equation ([eq5]) represent the full time derivative (e.g., @xmath48) of the gas entropy, while the terms on the right are the viscous heating, the energy flux in the vertical direction, the diffusion of radiation in the radial direction, and the viscous diffusion of thermal energy. following cannizzo (1993 ; and references cited therein), we take @xmath49 $] to be the radial energy flux carried by viscous thermal diffusion, where @xmath50 is the specific heat at constant pressure. @xmath51 is the accretion disk heating rate per unit area, and is given by @xmath52 the radiation flux in the radial direction is @xmath53 the overall cooling rate (larger than simply that from radiative diffusion) in the vertical direction is given by @xmath54 where @xmath55 is the fraction of the power that is transferred to the disk surface by mechanisms other than the usual radiation diffusion (cf. abramowicz et al. 1995, svensson & zdziarski 1994). equations ([heatr]) and ([fvert]) differ from those used by abramowicz et al. (1995) for reasons that are detailed in appendix b. we note that in the most general sense, the quantity @xmath56 should be thought of as a factor correcting the vertical energy transport in the standard accretion disk theory. in our formulation of the problem, it is computationally irrelevant whether this fraction of energy is deposited in the corona and radiated as a non - thermal power - law component, or it contributes to the blackbody disk flux, since we are concerned only with the bolometric luminosity of the disk for now. the latter case could be realized if this additional energy transport were to deposit its energy just below the photosphere of the disk. of course, if our model produces light curves that are in reasonable agreement with those of grs 1915 + 105, then a future study, one that would carefully delineate the different spectral components, will be warranted. equations ([eq4]) and ([eq5]), together with the equation of hydrostatic equilibrium ([eq1]), yield a closed set of coupled, time - dependent equations for evolving the variables t and @xmath13. for solving these equations we employ a simple explicit scheme with a variable time step. the time step is chosen to be always a fraction of the smallest thermal time scale in the disk (i.e., in its inner part). we also use a fixed grid in @xmath17. the spacing is uniform in @xmath57, and we take 100 to 300 radial bins. we will often refer to the radial coordinate in a dimensionless form, i.e., @xmath58, where @xmath59 is the schwarzschild radius. the outer boundary of the disk is chosen at a large radius, @xmath60 few hundred to a thousand, such that the disk at @xmath61 is always gas - pressure dominated. thus, the outer boundary conditions are given by @xmath62 and @xmath63 for @xmath64, where @xmath65 and @xmath66 are the temperature and the column density in the shakura - sunyaev formulation at that radius for the given @xmath67, @xmath10 and @xmath68, where @xmath69 is the viscosity parameter for a gas - dominated disk (see below). the inner boundary of the disk is fixed at @xmath70, and the boundary conditions there are @xmath71 and @xmath72. in this section we present our modified viscosity law that we anticipate will account for the unstable behavior of grs 1915 + 105. the shakura - sunyaev viscosity prescription postulates that the viscous stress tensor @xmath27 is given by equation ([ss]), where the parameter @xmath0 is defined as @xmath73 and where @xmath74 and @xmath75 are the turbulent eddy velocity and scale - length, respectively, @xmath76 is the sound speed, and @xmath77 is the volume averaged magnetic field energy density. obviously, this approach is useful only when @xmath0 remains approximately constant, i.e., independent of the local thermodynamic variables in the disk, which is probably the case in a number of situations. whenever the physical state of the accreting gas changes considerably, it is hard to see why @xmath0 would not change. one confirmation of this is the well - known thermal (ionization) instability of accretion disks in dwarf nova systems (e.g., meyer & meyer - hofmeister 1981 ; bath & pringle 1982 ; cannizzo, chen & livio 1995 ; and frank et al. 1992 ; 5.8). it has been shown in this case that the @xmath0-parameter needs to be larger on the hot stable branch than it is on the low branch of the solution. some more recent work suggests that, in addition, @xmath0 should have a power - law dependence on @xmath78, e.g., @xmath79, in order to explain the observed exponential luminosity decline of the dwarf nova outbursts (cannizzo et al. 1995 ; vishniac & wheeler 1996). further, based on theoretical considerations, different authors have adopted several variants on the prescription for @xmath27 when @xmath80. for example, lightman & eardley (1974 ; hereafter le74), and stella & rosner (1984 ; hereafter sr84) suggested that @xmath81 where @xmath69 is a constant. their physical reasoning was based on the expectation that chaotic magnetic fields should be limited by the gas pressure only. in addition, sakimoto & coronitti (1989 ; hereafter sc89) showed that even if one assumes that magnetic fields are effectively generated by radiation - dominated disks, magnetic buoyancy quickly expels these fields from the disks. since the disk viscosity may be produced to a large extent by magnetic fields, this result implies that the @xmath0-parameter must decrease when the radiation pressure becomes dominant. an attractive feature of this viscosity prescription (eq. [le]) is the fact that it is stable even when @xmath80, as indicated by the positive slope generated by this model for the track in @xmath12@xmath13 parameter space (see the dotted curve in fig. [fig : scurve]). at the same time, however, this viscosity prescription (to which we will refer as the lightman - eardley prescription) has no unstable region at all, and thus fails to reproduce the unstable behavior observed in grs 1915 + 105. of course, the conclusions of le74, sr84 and sc89 should be regarded as qualitative, at best, since it is not yet feasible to describe the turbulence and magnetic fields immersed in a radiation - dominated fluid in a quantitative and model - independent way. in this paper, we adopt the view that it may still be possible to couple the radiation to the particles through collisions and thereby allow the radiation pressure to contribute to the viscosity when @xmath82 is not too great. for example, sr84 argued that the energy density in the chaotic magnetic fields can not exceed the particle energy density, since it is the latter that generates the magnetic fields in the first place. however, as long as @xmath83, the magnetic field pressure does not exceed the gas pressure (because then @xmath84), so that the sr84 arguments are superfluous. to cast our suggestion into a mathematical form, we have adopted a viscosity prescription intermediate between those of shakura - sunyaev and lightman - eardley. our viscosity law is proportional to the total pressure for small to moderate values of @xmath42, while for large values of @xmath85, it becomes proportional to the gas pressure. we have devised the following simple formula for the viscosity prescription that smoothly joins these two limits, such that @xmath0 is given by @xmath86 where @xmath87 is an adjustable parameter. the s - curve plotted in figure ([fig : scurve]) was generated using this viscosity prescription with @xmath88. Results there are essentially 5 free parameters of the model that need to be specified and/or systematically varied to ascertain the sensitivity of the results to their value : (_ i _) the black hole mass, @xmath15, (_ ii _) the average accretion rate, @xmath10, (_ iii _) the viscosity parameter, @xmath69, for the gas - dominated region, (_ iv _) @xmath87, the critical ratio @xmath82, and (_ v _) @xmath67, the fraction of the luminosity transported vertically by mechanisms other than radiative diffusion. we adopt a mass of @xmath89, which is typical of the measured masses of galactic black hole candidates in binary systems (see, e.g., barret et al. 1996). for purposes of illustration, we chose a value of @xmath69 equal to @xmath90, since it leads to viscous time scales comparable to those observed. similarly, we selected a value of @xmath91, in part because it produces a reasonable s - curve. however, we checked a range of other values as well (from @xmath92 to @xmath93), and found that @xmath94 reproduces the grs 1915 + 105 observations most closely. we present a computed light curve and variations of the disk parameters through one complete cycle in figures ([fig : feqo]) and ([fig : temp_tau_distr]), respectively, for one set of illustrative parameters : @xmath95 and @xmath96. in figure ([fig : temp_tau_distr]), panels (a), (b), (c) and (d) show the evolution of the disk effective temperature @xmath97, the ratio of radiation to gas pressure (@xmath85), the disk thomson optical depth @xmath24 and the radial flow velocity @xmath98. note that since we are currently concerned only with the luminosity integrated over all photon energies, we define @xmath97 to include the emissivity from _ both _ the corona and the disk, i.e., @xmath99, where @xmath100 is given by equation ([fvert]). in each panel, the series of curves are for temporal increments of @xmath101 sec. the curve with the smallest luminosity @xmath102 (where @xmath103) in the upper left panel of figure ([fig : temp_tau_distr]) corresponds to the state of the disk right after the end of an outburst. an outburst starts in the innermost region of the disk, when a few zones at the smallest radii quickly make a transition to the upper stable branch of the solution (high state). although this is not obvious from fig. ([fig : temp_tau_distr]), the viscosity of the high state is larger than it is in the low state, because @xmath104, and so even though @xmath0 decreases, the increase in @xmath76 and @xmath16 leads to a larger @xmath37. the larger viscosity allows the gas to dispose of its angular momentum faster, thus allowing a faster inflow of the gas into the black hole. because of angular momentum conservation, the angular momentum of the gas that plummets into the black hole is transferred to larger @xmath17, where it produces an excess of angular momentum, and thus some matter actually flows to larger @xmath17, which is seen in panel (d) of figure ([fig : temp_tau_distr]) as positive spikes in the radial velocity distribution. a heating wave is initiated and propagates from the inner disk to larger @xmath17 (fig. [fig : temp_tau_distr]a). this wave is often referred to as either a ` density wave'or a ` transition wave '. as the wave propagates outward, the material on the inside of the wave becomes hot and shifts into the high viscosity state. it then rapidly loses its angular momentum and is transferred into the innermost disk, where the material continuously plummets into the hole. since the innermost stable region is a `` bottle - neck '' for the accreting gas, some material builds up there, which explains the bump in @xmath13 in that region as seen in figure ([fig : temp_tau_distr]c) during the outburst. when the density wave reaches @xmath105, the instability saturates, since the disk in that region is always gas - pressure dominated (for the chosen @xmath10 and @xmath67). as the density wave dies away, the outermost unstable regions are cleared of the excess mass and cool down. they are now in the low viscosity state, and thus the rate of mass inflow in the innermost region of the disk drops. an inward propagating cooling wave is initiated and the outburst dies out as the last excess mass in the innermost disk sinks into the hole. finally, a new cycle begins with the accumulation of new excess mass (since the local accretion rate is smaller than the average rate). we now systematically explore the disk behavior with the new viscosity prescription, in order to eventually compare the results with the gross temporal behavior of grs 1915 + 105. we adopted an energy transport fraction @xmath67 = 0.9. as with the other parameters, we also tested a wide range of values of @xmath67, and found that @xmath106 is the most appropriate due to the following considerations. the threshold for the onset of the instability is about @xmath107 0.26 for @xmath108 0.9, and scales as : @xmath109^{-9/8 } \label{dmo}\]] (see svensson & zdziarski 1994 and note that their definition of @xmath10 differs from ours by a factor that accounts for the standard disk radiative efficiency, i.e., @xmath110). it seems clear observationally that @xmath111 should be larger by a factor of at least @xmath112 than the value of @xmath113 corresponding to the transition from the gas- to radiation pressure - dominated regime in the standard theory. indeed, if we assume that the maximum luminosity of @xmath114 erg sec@xmath115 observed in grs 1915 + 105 corresponds to the eddington luminosity for this source, then the instability seems to exist only for @xmath116. in addition, galactic black hole candidates (gbhcs) with a lower persistent luminosity, or weaker transient behavior, have not shown such violent instabilities as grs 1915 + 105 does, and yet many of them are as bright as @xmath117 (see, e.g., barret, mcclintok & grindlay 1996). we will discuss the parameter @xmath67 further in [sect : gross]. in figure ([fig : lcurves]) we show the evolution of the integrated disk luminosity as a function of time for the nominal values of the parameters listed above and for @xmath118 @xmath119, 0.31, 0.5, and 1.5. in agreement with equation ([dmo]), we find that for values of @xmath10 below 0.26 the disk is stable, because there is no radiation - pressure dominated zone in the disk. we also find, in general, that as the value of @xmath10 increases, the duty cycle (i.e., the fraction of time that the source spends in the high state) increases, from about 5% at @xmath118 0.26, to approximately 50% when @xmath118 1.5. similarly, the ratio of maximum to minimum luminosity through the cycle grows with increasing @xmath10. the time for the disk to complete one of its cycles (defined as the cycle time) also increases with @xmath10, except for very near the minimum value of @xmath118 0.26 where the instability first sets in ; here the cycle time actually decreases with increasing @xmath10. the reason is that there seems to exist a `` critical excess mass '' in the inner disk region, such that the instability will appear only when the mass in the disk exceeds this value. the rate for building up the excess mass is @xmath120, where @xmath111 is the maximum stable accretion rate given by equation ([dmo]), and therefore a small increase in @xmath10 above @xmath111 can bring about a substantial decrease in the cycle time. for larger accretion rates, the cycle time starts to increase with increasing accretion rate, because a larger region of the disk is unstable, and it takes longer to clear the excess mass during the outburst. we will now compare our results shown in figure ([fig : lcurves]) with observations of grs 1915 + 105. we should acknowledge at the start that it seems a daunting task to try to explain with a single model the diverse, unstable behavior exhibited by this source, but we expect to benefit from a comparison of even the gross properties of our model with the observations. panel (a) of figure ([fig : lcurves]) is similar to figure ([fig : data]a) in that in both figures there exists a long phase of mass accumulation (the low state). furthermore, an examination of the grs 1915 + 105 data reveals that the cyclic limiting behavior disappears when the average count rate is lower than @xmath121 counts / sec, so that there indeed exists a threshold accretion rate below which the instability does not operate. in simulations, the spiky nature of the outburst is explained by the fact that only a very narrow region in @xmath17 within the disk is unstable, and it takes very little time to get rid of the excess mass in that region. the differences with fig. (1a) are mainly due to the fact that an outburst in the data does not seem to completely clear the excess mass, and that more oscillations follow with a gradually declining amplitude, whereas in the simulations the hot state persists until all the excess mass is swallowed by the black hole. panel (b) of fig. ([fig : data]) is also somewhat similar to panels (b) and (c) of the simulations (fig. [fig : lcurves]). the progression of time scales and the average luminosity (i.e., roughly speaking, the count rate) from pattern (a) to (b) in fig. ([fig : data]) is similar to that from panel (a) to (b) in fig. ([fig : lcurves]) as well. the physical reason for the cycle time getting shorter with increasing @xmath10 is that the excess mass can build up faster for larger accretion rates. next, panel (c) in figure ([fig : data]) is most interesting from the point of view that the high state is long lasting and is clearly stable at least for the first half. panel (d) of the simulations can account for some of the properties of this variability pattern. in particular, the duty cycle increased in both the simulations and the data as the accretion rate increased. the cycle time did not increase nearly enough in the simulations, however. it could be made longer by choosing a smaller @xmath69, but then the time scales in all the other panels would increase as well, which does not appear acceptable. also, a general trend seen in the first three panels (a - c) in both the data and the simulations is that the minimum of each light curve shows a slow gradual rise in luminosity before the instability sets in. this is a clear indication of the disk accumulating mass until it reaches a global instability. we note that panel (d) of figure ([fig : data]) seems to be rather different from the other three variability examples. if the high state is again defined as the one with larger count rates, then a peculiarity of panel (d) is that the count rate first decreases and then increases by the end of a high state episode. this is the opposite of what is seen for the other states, such as that shown in panel (c) of fig. (1). from a theoretical point of view, this is a highly significant observation. in all of our simulations, we observed that as the outburst progresses, the amount of mass in the inner disk region builds up because of the greater and greater inflow of mass from larger radii. once in a given (high or low) stable state, the local disk luminosity is proportional to the local column density @xmath13, which can be seen from figure ([fig : scurve]). therefore, in an outburst, the luminosity decreases only after the outer disk cools down and the influx of mass stops, leading to a decrease in @xmath13 in the inner disk. thus, the outburst profile in the simulations is such that the second time derivative @xmath122 is always negative, not positive as seen in panel (d) of figure 1. we do not see a clear explanation for this disagreement in the context of the current model, but we will show that there might be a natural cause for this phenomenon if a jet is allowed ( [sect : jet]). finally, we note that in a number of the panels in figure ([fig : lcurves]), the first (and sometimes also the second) peak in the light curve is of slightly different intensity or duration than the peaks that come later in the sequence. this is simply a consequence of the disk adjusting to its quasi - steady cyclic pattern after starting from the imposed initial conditions that are given by the shakura - sunyaev solution corresponding to the mean value of @xmath10. let us now discus the rather extreme value for @xmath67 invoked here, which requires that as much as @xmath123 of the accretion power is carried out of the disk by processes other than the usual radiative diffusion. if this process is magnetic buoyancy or some other mhd process heating the corona, then there appears to be a contradiction with the observations of grs 1915 + 105. the blackbody component in the spectrum of this source is generally small in the low state, but it can be dominant in the outburst state (muno, morgan & remillard 1999). one could argue here that as much as @xmath124 of the x - rays emitted by the corona towards the disk may be reprocessed into the soft disk radiation, (e.g., magdziarz & zdziarski 1995) if the reflection / reprocessing takes place in a _ neutral medium _ (neutral in the sense of high - z elements like oxygen and iron, that are important for the reflection spectrum). thus, in principle, the blackbody flux could exceed that in the non - thermal component by the ratio @xmath125 (see eqs. 15 in haardt & maraschi 1991 with their parameters @xmath126 and @xmath127). the rather high inferred disk temperature in grs 1915 + 105, e.g., @xmath128 kev (belloni et al. 1997a), rules out the possibility of neutral reflection, however. as found by nayakshin & dove (1998) and nayakshin (1998), the integrated albedo of the reflected spectrum in the case of a strongly ionized disk can be much higher than the standard 0.1 - 0.2 of the neutral reflector (i.e., @xmath129 in eqs. 15 of haardt & maraschi 1991), and thus it seems difficult to have @xmath67 as large as 0.9 given the large amount of soft power observed from the disk. a possible way out of this dilemma is that the high value of @xmath67 does not necessarily represent the energy flux from the disk into the corona. indeed, equation ([fvert]) states that the energy transport out of the disk may be @xmath56 times faster than that given by the standard diffusion of radiation. convection of energy in the vertical direction (e.g., bisnovatiy - kogan & blinnikov 1977 ; goldman & wandel 1995 ; and references therein) is one physical mechanism that can speed up the transfer of energy out of the disk. mhd waves dissipating their energy before they reach the corona could be another. in addition, there is no proof that the vertical averaging procedure (i.e., a one zone approximation) used in the standard accretion disk theory does not lead to a substantial underestimation of the vertical radiation flux out of the disk. for example, if @xmath130, where @xmath131 few, then @xmath111 becomes @xmath132 (see sz94), and thus one may have @xmath111 as large as that observed in grs 1915 + 105 due to a faster or an additional disk cooling mechanism rather than due to a transfer of most of the disk power into the corona. it is notable that panel (c) of fig. ([fig : data]) shows rapid chaotic oscillations as fast as @xmath1 tens of seconds at the end of the high state, whereas our simulations do not show a similar behavior. furthermore, note that the rise / fall time scales are shorter in the data than they are in our model. it appears that the heating of the unstable region in grs1915 happens on a time scale that is very much shorter than the cycle time. by contrast, the outermost part of the unstable region of the simulated disk becomes unstable only after @xmath133 the duration of the hot phase, that is of the order of the disk viscous time for the high luminosity cases. one explanation here could be that these fast oscillations are failed attempts by the disk to make a state transition from the high to the low state. further, during these fast oscillations the disk is always brighter than it is in the low state, which could be interpreted as an indication that only a part of the disk (most likely the innermost region) in grs 1915 + 105 takes part in these rapid oscillations, whereas the rest of the unstable region is still on the upper stable branch of the s - curve. for this to be true, the inner disk must be able to decouple to some extent from the rest of the unstable disk, and be variable on a much shorter time scale than the outer disk. to test this idea, we have carried out several simulations in which the @xmath0-parameter is a decreasing function of radius. our hope is that in this case, since @xmath0 in the inner disk is larger than that in the outer disk, the time scale for oscillations in the inner disk may be much shorter than the overall viscous time, which has the dependence @xmath134 (e.g., frank et al. 1992). in figure ([fig : variable_alpha]) we show one such simulation, in which the viscosity parameter was chosen to be @xmath135 $]. this functional form allows the @xmath0-parameter to be roughly @xmath136 in the inner disk region (@xmath137), and to be @xmath138 for @xmath139. the presence of these two time scales is obvious just from a perusal of the resulting light curves. oscillations of the inner disk produce the precursor seen before each major outburst in fig. ([fig : variable_alpha]). its relative magnitude is small because the amount of excess mass stored in the inner disk is small compared to that in the outer disk. furthermore, in the simulations the inner disk decouples from the outer one only at the beginning of the outburst, not at the end as is seen in the grs 1915 + 105 data. this is due to the fact that once the outer disk makes a transition to the high state, the large mass supply to the inner disk forces the latter to go into the high state as well and remain there. in summary, varying the @xmath0-parameter with radial distance does not appear to be a viable explanation for the rapid chaotic oscillations seen in the data which are superposed on the more regular long time scale disk evolution. an alternative way of explaining the fast oscillations could have been provided if the heating / cooling fronts stall and are reflected back as cooling / heating fronts. this behavior was observed in simulations of the classical thermal ionization disk instability by cannizzo (1993, see text above his equation 5), where it happened to be an unwanted result. however, we have not been able to see such stalled transition fronts (except for the case presented in fig. [fig : variable_alpha]), which should not be too surprising, due to the fact that the underlying physics of the ionization instability and the one explored in this paper are vastly different, and one does not expect a direct correspondence between these two instabilities. although we will not present any light curves, we should mention that we have also attempted to allow the fraction @xmath67 to be a function of @xmath85, since the transition from @xmath140 to @xmath141 means a substantial change in the physical conditions in the disk. our hope was that the freedom in choosing @xmath142 might help to reproduce the disk flickering. however, all our attempts in this regard (with @xmath67 decreasing or increasing across the transition in @xmath85) have been unsuccessful. Plasma ejection from the inner disk although our model is able to reproduce a number of the general trends seen in the light curve of grs 1915 + 105, the rise / fall time scales are always shorter in the data than they are in the simulations. the fast rises and falls are perhaps the reason why the profile of an outburst in some of the actual light curves of grs 1915 + 105 (e.g., panel (c) of fig. [fig : data]) is reminiscent of a square - wave like shape rather than the rounded shapes that an outburst has in the simulations (see fig. [fig : lcurves]d). the difference between our simulations and the data appears important enough to us to require the addition of a new feature to our model and test whether it can bring the theory closer to the observations. the following considerations have guided us in choosing the additional ingredient in our model. the speed with which the hot state propagates outward in the accretion disk is equal to the speed of the transition wave, which was studied analytically by, e.g., meyer (1984) and numerically by menou, hameury & stehle (1998). both of these studies find that the propagation speed of the transition wave is @xmath143. the time scale for the transition front to traverse an unstable region of size r is @xmath144, i.e., the transitions in the disk actually take @xmath145 times longer than the thermal time scale of the outermost part of the unstable disk region. this is the reason why the rises and falls take too long in the simulations compared to the data. however, notice that during the initial stage of the transition from the low to the high state, the luminosity rises quite rapidly in the simulations (see fig [fig : lcurves]d), at a rate probably as fast as that seen in the data. now, if some physical process were to `` cut '' the simulated light curves at, say, @xmath146, so that anything emitted by the disk above this value _ is not seen _ by the observer, then the outburst profile would be much more consistent with the data. to find a process able to `` cut '' the light curve, we should recall that powerful plasma ejections are known to occur in grs 1915 + 105 (e.g., mirabel and rodriguez 1994). recently, eikenberry et al. (1998), mirabel et al. (1997) and fender & pooley (1998) have shown that there exists a strong link between the x - ray emission and emission in the infrared and radio frequencies. the radio and infrared emission is attributed to plasma ejection events taking place in the innermost part of the disk. the minimum power deposited at the base of the jet (that may consist of individual ejection events) was found (e.g., gliozzi, bodo & ghisellini 1999) to be as high as @xmath147 few @xmath148 erg sec@xmath115 for the event first discovered by mirabel and rodriguez (1994). during other observations (e.g., eikenberry et al. 1998), the radio fluxes were a factor of @xmath149 lower than the ones observed by mirabel & rodriguez (1994), which still may require @xmath150 few @xmath151 erg sec@xmath115 ejected into the jet, which is as large as the largest x - ray luminosities produced by the source. thus, if one were to model these plasma ejections in the framework of our model, one would have to allow for a significant portion of the accretion _ energy _ to be diverted into the jet. from here on, we will therefore assume that the total accretion disk power @xmath152 consists of two parts : the first is the x - ray luminosity @xmath153, that is equal to the sum of the thermal disk and non - thermal coronal luminosities ; and the second part @xmath154 is the `` jet power '', i.e., the energy ejected into the jet. for simplicity we further assume that the jet power is not seen in x - rays, and appears only in the radio or other non - x - ray wavebands. as before, the locally produced total disk power is given by equation ([fvert]), where we will again suppose that @xmath155. the jet luminosity is then @xmath156 where @xmath157 is the fraction of power that escapes from the disk into the jet. this parameter (@xmath158) may be a constant, or it may be a function of local disk conditions or a function of some global variable, e.g., the total disk power @xmath152. since the importance of advection relative to the radiative cooling is roughly given by the ratio @xmath159, and since according to our discussion in appendix [sect : geometry] @xmath160 in this source, the advection of energy is not important in our model. thus, the observed x - ray power @xmath153 will consist from the disk power (fraction @xmath161 of the total disk power) and the coronal luminosity which is given by an equation analogous to equation ([lj]) except with a parameter @xmath162 instead of @xmath158 inside the integral. to eliminate any ambiguities of our approach, we note that the accretion power is divided among the disk, the corona, and the jet as @xmath161, @xmath163 and @xmath164, respectively. we should also mention that the amount of mass carried away by the jet is small compared to that accreted into the black hole, so we neglect the former in the mass conservation (eq. [eq4]). the point here is rather simple : if the ultimate source of the jet power is the underlying accretion disk, then to produce one relativistic proton in the jet, many protons in the disk must transfer their gravitational energy (which is small compared with their rest mass) to that one jet proton. these protons will therefore have to sink into the black hole in order to send one proton into the jet. more precisely, the mass outflow rate is @xmath165, where @xmath166 few is the terminal bulk lorenz factor of the jet. this is to be compared with the accretion rate through the disk, that is equal to @xmath167, where @xmath168 is the `` radiative '' efficiency of the disk and @xmath152 is the total disk power. accordingly, @xmath169 since @xmath170. this estimate is quite conservative. for example, if the jet power is dominated by magnetic fields or by electron - positron pairs, then the mass carried away in the jet is even smaller. a similar consideration allows us to neglect the angular momentum outflow from the disk into the jet. we point out however that the latter approximation depends on the model of the jet ; if the jet particles actually gain a considerable amount of the angular momentum as they are streaming away from the disk, then the torque exerted by the jet on the disk may become non - negligible. we plan to explore these effects in future work. eikenberry et al. (1998) found that the ejection events were absent in the low state and started at the onset and during the high state. we thus will accept a parameterization of @xmath158 such that it is close to zero when the disk is in the low state, and it is relatively large (but still smaller than unity, of course) when the disk is in the high state. we now explore three different illustrative prescriptions for the function @xmath158 to show the possible effects of the disk power being diverted into a jet. let us assume that the fraction @xmath158 is described by the following (somewhat arbitrarily chosen) function : @xmath171 where @xmath172 is the total power of the source in terms of its eddington luminosity. this equation is qualitatively reasonable since one expects particles to be ejected from the disk when the luminosity @xmath102 approaches (and then exceeds) the eddington limit. while the radiation may not be the ultimate driving power of the jet, the radiation pressure can cause more matter to be ejected and possibly produce stronger jets. we show in figure ([fig : steady1]) the resulting x - ray light curve (solid line) and the total disk luminosity @xmath152 (represented by a dashed curve ; scaled down by the factor of 3 to fit the figure) for an accretion rate @xmath173. note that the overall luminosity curve is the same as we would have obtained with our basic model for the same choice of @xmath67 and @xmath10, but with no jet ejections (see [sect : basic] & [sect : results]). the shape of the outburst indeed becomes more like a square - wave, and thus the rises and falls appear to be sharper than they were for the model with @xmath174, thus making it possible to reproduce this feature of the observed light - curves (in particular, panel (c) of fig. [fig : data]). as a second example, we test the following prescription for the jet power : @xmath175 (this prescription, in contrast to the one given by eq. [[fj1]], was chosen among several that we tested to reproduce the panels (c) and (d) of fig. [[fig : data]] simultaneously as described in [sect : test3]). the corresponding x - ray light curve and the total luminosity of the system are shown in figure ([fig : steady2]). one notices that the profile of the outburst in x - rays is now inverted with respect to the actual disk power. when the disk produces most of the energy output, the x - ray light curve actually has a local minimum, since most of the energy is ejected into the jet during that time. this effect may be responsible for the `` strange '' shape of the outburst seen in panel (d) of fig ([fig : data]). as a final example we present a sequence of light curves produced by our model with a fixed prescription for the jet energy fraction @xmath158 and with the accretion rate being the only parameter that is varied. the functional dependence of @xmath158 is again given by eq. ([fj]). further, motivated by the fact that the x - ray emission can fluctuate wildly when it is in the high state (see fig. 1), whereas it does not fluctuate as much in the low luminosity state, we allow the fraction @xmath67 carried away from the disk other than by the usual radiation diffusion to fluctuate around some mean. numerically, the @xmath67-factor will now contain a _ variable part @xmath176, such that @xmath177 : @xmath178 where @xmath179 is the constant part fixed at the previously selected value of @xmath180, @xmath181 is a random number distributed uniformly between -1 and + 1 (a new value of @xmath181 is randomly chosen every 3 seconds), and @xmath182 is a function of radius @xmath183 such that it is unity in the inner disk and it approaches zero for @xmath184. the latter radial dependence is introduced simply to avoid complications with the outer boundary condition at @xmath185, where we set the disk structure to be that given by the gas - dominated shakura - sunyaev configuration with a constant @xmath186 (see [sect : code]). the shape of the function @xmath187 is almost completely irrelevant for the resulting light curves as long as @xmath188 is large, since most of the disk power is liberated in the inner disk region. finally, although the second term on the right hand side of the equation ([ft]) appears to be complicated and model - dependent, its presence is not required and serves only to make the light curves look somewhat more random. none of the final conclusions of this paper depend on the particular choice for the random part in @xmath67. figure ([fig : xandj]) shows the resulting x - ray light curves for four different accretion rates. it is worthwhile comparing figure ([fig : xandj]) with figure ([fig : lcurves]), in which the plasma ejections and random fluctuations in @xmath67 were absent. the case with the accretion rate just above @xmath111 (i.e., panel c in fig. [fig : lcurves]) has not been affected as strongly as that with the higher accretion rate ; this is of course due to the fact that little energy is ejected in the jet in the former case compared with the latter. yet panels (a) and (c) do look more chaotic, which is entirely due to the fluctuating part in the fraction @xmath67 (see eq. ([ft]). analyzing panels (b) and (d) of this figure, one notices that it turns out to be possible to reproduce both the sharp square - wave like pattern (c) and the `` inverted '' pattern (d) of figure 1 with the _ same _ prescription for @xmath158. it is the same process, i.e., the plasma ejection, that allows our model to better reproduce the data (panels [c] and [d] of fig. 1) ; the only difference being that in panel (d) the ejected fraction of the disk energy is much greater than in panel (b). for the completeness of presentation, we also plot the jet power @xmath154 with a dotted curve in fig. ([fig : jet1]) corresponding to the simulation presented in panel (b) of fig. ([fig : xandj]). given the relatively good agreement between the model and the observations of grs 1915 + 105, we believe that we now have a better handle on the key characteristics of accretion in this source. while we feel less confident about the origin of the s - curve in the @xmath189 space, we think that the geometry of the accretion flow in grs 1915 + 105 is that of a geometrically thin and optically thick flow (see [sect : geometry]). most of the x - ray luminosity is produced in an optically thin corona, that can cover the entire inner disk or consist of localized transient magnetic flares (e.g., haardt, maraschi & ghisellini 1994 ; nayakshin & melia 1997 ; nayakshin 1998). in the latter case, the fraction @xmath67 of the power transported from the disk into the corona and the jet must be thought of as a time averaged variable. further, we believe that plasma ejection events must be an integral part of the accretion process, and are likely to be the result of an excessively large radiation pressure. we will discuss this issue in a broader context in [sect : discussion]. Comparison to previous work belloni et al. (1997a, b) were the first to fit the spectrum in grs 1915 + 105 with a two - component model, consisting of a multi - temperature disk - blackbody with a variable inner disk radius plus a power - law. very importantly, these authors found that most of the complex spectral variability can be explained by a rapid change in the disk inner radius @xmath7 and a corresponding change in the temperature of the disk at @xmath7. they have also shown a correlation between the disk filling time scale and the inferred radius of the inner disk. our results are qualitatively similar to the framework of the disk instability discussed by belloni et al. for grs 1915 + 105 since in the low state the inner disk is quite dim (up to a factor of 10 - 30, see figure [fig : dim]) compared with what it should have been if the disk were stable. thus, the inner part of the disk can be largely unobservable and can be said to be `` missing ''. at the same time, in the high state, the inner disk is brighter than its equilibrium luminosity by typically a factor of a few, which may make the outer part of the disk (that is stable and so has just the `` normal '' flux) seem comparatively dim. under these conditions, it may be non - trivial to distinguish observationally between a disk where the temperature is a continuous function of r (@xmath190) and a disk where the temperature is lower than expected in the outer part of the disk, since the outer disk carries a small fraction of the overall luminosity. in addition, the inner disk is never completely empty of mass in our model, although the difference between the inner disk surface density in the high versus that in the low state can be a factor of @xmath1 few to 10 (see fig. [fig : temp_tau_distr]c). note that @xmath7 as defined by belloni et al. should be identified with the largest radius reached by the heating wave in our simulations. more specifically, during the high state, @xmath7 should be defined as the radius of innermost stable orbit, whereas during the low state it is approximately equal to the largest radius in the disk in which @xmath191 (i.e., @xmath192 for fig. [fig : dim]). our values of @xmath7 are in general much larger than those obtained by belloni et al. the value of @xmath193 inferred from the observations could be larger if one were to fit the grs 1915 + 105 emission with a more complicated (and likely more realistic) spectrum than a multi- temperature blackbody appropriate for a shakura - sunyaev disk. moreover, for the high accretion rates observed in this source, the disk may become effectively optically thin and thus radiate as a modified (rather than a pure) blackbody (see discussion in taam et al. since blackbody emission produces the maximum flux for a given temperature, the modified emission would require a larger @xmath7. finally, the values of @xmath7 obtained by us would presumably be smaller for a kerr black hole. we plan to address this question in the future. belloni et al. (1997b) showed that the outburst duration is proportional to the duration of the preceding quiescent state, but there is no correlation between the former and the quiescent time after the burst. profiles of outbursts in our model are rather regular, so we believe we see a correlation among these three quantities in contrast to the belloni et al. results. on the other hand, in many observations (other than the one presented in belloni et al. 1997b) grs 1915 + 105 does not show a good correlation between the duration of the outburst and the preceding quiescent phase (t. belloni 1999, private communication). Discussion we have systematically analyzed the physical principles underlying the behavior of grs 1915 + 105, and have arrived at several important conclusions regarding the nature of the time - dependent accretion flow in this system. as discussed in appendix [sect : geometry], geometrically thick advection - dominated flows are unacceptable for this source, since they would produce burst - like instabilities with a very small duty cycle compared to those seen in the data, for accretion rates smaller than the eddington value. to produce outbursts as long as @xmath194 seconds, geometrically thick accretion flows must have implausibly small values of @xmath0. however, if the accretion rate is highly super - eddington, then a thick adaf disk could yield outbursts with a reasonable duty cycle (@xmath34). but since the viscous time scale for geometrically thick disks is as short as the thermal one, their light curves would be rather smooth which again contradicts the data (see, e.g., panels c & d of fig. it is also unclear how the observation (e.g., belloni et al. 1997a, b ; muno, morgan & remillard 1999) that the disk extends down to the innermost stable orbit during the high state can be reconciled with the structure of a geometrically thick, hot and optically thin flow. for these reasons, the standard cold accretion disk with a modified viscosity law, a corona, and plasma ejections seems to be the only reasonable choice. as we have shown in this paper, this geometry indeed allows one to obtain rather good general agreement between the theory and observations, and to understand particular features of the light curves in terms of fundamental physical processes. nevertheless, a critical reader may question whether the relatively large number of parameters introduced by us in this work allows us to pin - point actual values of these parameters. we believe that the answer to this question is `` yes '', because spectral constraints, used very sparingly here, may prove to be quite restrictive in future studies. for example, we have shown only the total luminosity light curves in this paper, whereas the data also contain a wealth of information about the spectral evolution in grs 1915 + 105 (belloni et al. 1997 ; muno et al. 1999). if future observations also provide more examples of the disk - jet connection (e.g., eikenberry et al. 1998), then we have a means of constraining the dependence of both @xmath158 and the total fraction @xmath67 on the disk luminosity. further, one should also attempt to explore the fact that other transient and persistent black hole sources do not exhibit instabilities similar to grs 1915 + 105, at least not to the same extreme degree. thus, putting all these observational and theoretical constraints together may be quite effective in limiting the theoretical possibilities for the instability and the coronal and jet activity. as an example of the need for a detailed study of the spectra and qpos in the context of our model, we point out the following. from the results of muno et al. (1999) it appears that qpos are present when the power - law component dominates the spectrum, which usually happens in the low state (`` low '' means the lower count rate). further, these authors find that for the burst profiles similar to the one shown in fig. 1(d), the qpos are present during the `` high '' state, which is opposite to most of the other cases. if qpos indeed track the presence of a vigorous corona, this would imply that although our panel (d) in fig. ([fig : xandj]) looks similar to fig. 1(d), it does not represent the actual situation very well. namely, the data seem to imply that the higher count rate states actually have lower accretion rates through the inner disk than the lower luminosity states (because excess luminosity escapes into the jet). it is thus possible that plasma ejections occur during the sharp dips in fig 1(d) rather than during the longer phases (e.g., from @xmath195 to @xmath1 960 sec in this figure). the `` m - shape '' of the outbursts in fig. ([fig : data]d) may then be understood because these would be the low states as tracked by the accretion rate, and the low states of the three other panels (a - c) of fig. ([fig : data]) show similar shapes albeit with lower count rates. although we have not yet made a detailed study of this suggestion, we can probably accommodate it within our model, but only if we allow the parameters in our viscosity law to depend on the disk luminosity. _ note, however, that in either case we must require plasma ejections from the inner disk in order to understand panel (d) of fig. (1). _ the discussion above of course bears on the particular choice of the viscosity law that we have made in this paper. we emphasize that our model is still rather empirical and we have not identified the actual physics of the viscosity law in grs 1915 + 105. the real physics might well be yet more complex, and in fact, the existence of the s - curve might be due to other than radiation pressure instabilities, perhaps involving an effect that we do not currently understand. however, based on the results presented in this paper, we feel confident that there is an s - curve in this source, and that this curve can be modeled approximately by the viscosity law given by equation ([eq3]). a clear shortcoming of our work is the use of a non - relativistic disk around a non - spinning black hole. we plan to improve this in our future work. we expect that this will change the `` reasonable '' values of the parameters that we used, such as the fraction @xmath67, and the inner disk radius, of course, but that it will not affect the general nature of our conclusions. the authors thank t. belloni, r. taam, m. muno, e. morgan, r. remillard, e. vishniac and d. kazanas for useful discussions. this work was supported in part by nasa grants nag5 - 8239 and nag5 - 4057, under the astrophysics theory program.

during the past two years, the galactic black hole microquasar grs 1915 + 105 has exhibited a bewildering diversity of large amplitude, chaotic variability in x - rays. although it is generally accepted that the variability in this source results from an accretion disk instability, the exact nature of the instability remains unknown. here we investigate different accretion disk models and viscosity prescriptions in order to provide a basic explanation for the exotic temporal behavior in grs 1915 + 105. we discuss a range of possible accretion flow geometries. based on the fact that the overall cycle times are very much longer than the rise / fall time scales in grs 1915, we rule out the geometry of advection dominated accretion flow (adaf) or a hot quasi - spherical region plus a cold outer disk for this source. a cold disk extending down to the last inner stable orbit plus a hot corona above it, on the other hand, is allowed. we thus concentrate on geometrically thin (though not necessarily standard) shakura - sunyaev type disks (shakura & sunyaev 1973 ; hereafter ss73). we argue that x - ray observations clearly require a quasi - stable accretion disk solution at high accretion rates where radiation pressure begins to dominate, which excludes the standard @xmath0-viscosity prescription. to remedy this deficiency, we have therefore devised a modified viscosity law that has a quasi - stable upper branch, and we have developed a code to solve the time - dependent equations to study such an accretion disk. via numerical simulations, we show that the model does account for several gross observational features of grs 1915 + 105, including its overall cyclic behavior on time scales of @xmath1 100 - 1000 s. on the other hand, the rise / fall time scales are not short enough, no rapid oscillations on time scales @xmath2 10 s emerge naturally from the model, and the computed cycle - time dependence on the average luminosity is stronger than is found in grs 1915 + 105. we then consider, and numerically test, several effects as a possible explanation for the residual disagreement between the model and the observations. a hot corona with the energy input rate being a function of the local cold disk state and a radius - dependent @xmath0-parameter do _ not _ appear to be promising in this regard. however, a more elaborate model that includes the cold disk, a corona, and plasma ejections from the inner disk region allows us to reproduce several additional observed features of grs 1915 + 105. we conclude that the most likely structure of the accretion flow in this source is that of a cold disk with a modified viscosity prescription, plus a corona that accounts for much of the x - ray emission, and unsteady plasma ejections that occur when the luminosity of the source is high. the disk is geometrically thin due to the fact that most of the accretion power is drained by the corona and the jet.

astro-ph9905371

Introduction the natural shape of an isolated self - gravitating fluid is axially symmetric. for this reason, exact axial symmetric solutions of einstein field equations are good candidates to model astrophysical bodies in general relativity. in the last decades, several exact solutions were studied as possible galactic models. static thin disk solutions were first studied by @xcite and @xcite, where they considered disks without radial pressure. disks with radial pressure and with radial tension had been considered by @xcite and @xcite, respectively. self - similar static disks were studied by @xcite, and @xcite. moreover, solutions that involve superpositions of black holes with static disks were analyzed by @xcite and @xcite. also, relativistic counter - rotating thin disks as sources of the kerr type metrics were found by @xcite. counter - rotating models with radial pressure and dust disks without radial pressure were studied by @xcite, and @xcite, respectively ; while rotating disks with heat flow were studied by @xcite. furthermore, static thin disks as sources of known vacuum spacetimes from the chazy - curzon metric @xcite and zipoy - voorhees @xcite metric were obtained by @xcite. also, @xcite found an infinite number of new relativistic static solutions that correspond to the classical galactic disk potentials of kuzmin & toomre @xcite and mestel & kalnajs @xcite. stationary disk models including electric fields @xcite, magnetic fields @xcite, and both electric and magnetic fields @xcite had been studied. in the last years, exact solutions for thin disks made with single and composite halos of matter @xcite, charged dust @xcite and charged perfect fluid @xcite were obtained. for a survey on relativistic gravitating disks, see @xcite and @xcite. most of the models constructed above were found using the metric to calculate its energy momentum - tensor, i.e. an inverse problem. several exact disk solutions were found using the direct method that consists in computing the metric for a given energy momentum tensor representing the disk @xcite. in a first approximation, the galaxies can be thought to be thin, what usually simplifies the analysis and provides very useful information. but, in order to model real physical galaxies the thickness of the disks must be considered. exact axially symmetric relativistic thick disks in different coordinate systems were studied by @xcite. also, different thick disks were obtained from the schwarzschild metric in different coordinates systems with the displace, cut, fill, and reflect " method @xcite. the applicability of these disks models to any structure found in nature lays in its stability. the study of the stability, analytically or numerically, is vital to the acceptance of a particular model. also, the study of different types of perturbations, when applied to these models, might give an insight on the formation of bars, rings or different stellar patterns. moreover, a perturbation can cause the collapse of a stable object with the posterior appearance of a different kind of structure. an analytical treatment of the stability of disks in newtonian theory can be found in @xcite, @xcite and references therein. in general, the stability of disks in general relativity is done in two ways. one way is to study the stability of the particle orbits along geodesics. this kind of study was made by @xcite transforming the rayleigh criterion of stability @xcite into a general relativistic formulation. using this criterion, the stability of orbits around black holes surrounded by disks, rings and multipolar fields were analyzed @xcite. also, this criterion was employed by @xcite to study the stability of the isotropic schwarzschild thin disk, and thin disks of single and composite halos. the stability of circular orbits in stationary axisymmetric spacetimes was studied by @xcite and @xcite. moreover, the stability of circular orbits of the lemos - letelier solution @xcite for the superposition of a black hole and a flat ring was considered by @xcite and @xcite. also, @xcite analyzed the stability of several thin disks without radial pressure or tension studying their velocity curves and specific angular momentum. another way of studying the stability of disks is perturbing its energy momentum tensor. this way is more complete than the analysis of particle motions along geodesics, because we are taking into account the collective behavior of the particles. however, there are few studies in the literature performing this kind of perturbation. a general stability study of a relativistic fluid, with both bulk and dynamical viscosity, was done by @xcite. he considered the coefficients of the perturbed variables as constants, i.e. local perturbations. usually, this condition is too restrictive. stability analysis of thin disks from the schwarzschild metric, the chazy - curzon metric and zipoy - voorhees metric, perturbing their energy momentum tensor with a general first order perturbation, were made by @xcite, finding that the thin disks without radial pressure are not stable. moreover, stability analysis of the static isotropic schwarzschild thick disk as well as the general perturbation equations for thick disks were studied by @xcite. in newtonian gravity, models for globular clusters and spherical galaxies were developed by @xcite and @xcite. in the case of disk galaxies, important thick disk models were obtained by miyamoto and nagai @xcite from the prior work of @xcite and @xcite about thin disks galaxies. miyamoto and nagai thickened - up " toomre s series of disk models and obtained pairs of three - dimensional potential and density functions. also, @xcite obtained a family of three - dimensional axisymmetric mass distribution from the higher order plummer models. the miyamoto - nagai potential shares many of the important properties of actual galaxies, especially the contour plots of the mass distribution which are qualitatively similar to the light distribution of disk galaxies @xcite. recently, two different extensions of the miyamoto - nagai potential appeared in the literature : a triaxial generalization @xcite which has as a particular case the original axially symmetric model, and a relativistic version @xcite which has as a newtonian limit the same original model. in order to have a general relativistic physical model for galaxies, we must consider, first of all, the thickness of the disk and its stability under perturbations of the fluid quantities. the purpose of this work is to study numerically the stability of the general relativistic miyamoto - nagai disk under a general first order perturbation. the perturbation is done in the temporal, radial, axial and azimuthal components of the quantities involved in the energy momentum tensor of the fluid. in the general thick disk case @xcite, the number of unknowns is larger than the number of equations. this opens the possibility of performing several types of combinations of the perturbed quantities. in this manuscript we search for perturbations in which a perturbation in a given direction of the pressure creates a perturbation in the same direction of the four velocity. the energy momentum perturbation considered in this manuscript is treated as test matter ", so it does not modified the background metric obtained from the solution of einstein equations. the article is organized as follows. in sec. [sec2], we present the general perturbed conservation equations for the thick disk case. the energy momentum tensor is considered diagonal with all its elements different from zero. also, in particular, we discuss the perturbations that will be considered in some detail in the next sections of this work. in sec. [sec3], we present the thick disk model whose stability is analyzed, i.e. the general relativistic miyamoto - nagai disk. the form of its energy density and pressures, as well as, the restrictions that the thermodynamic quantities must obey to satisfy the strong, weak and dominant energy conditions are shown. later, in sec. [sec4], we perform the perturbations to the general relativistic miyamoto - nagai disk ; in particular we study its stability. finally, in sec. [sec5], we summarize our results. Perturbed equations the thick disk considered is a particular case of the general static - axially - symmetric metric @xmath0 where @xmath1, @xmath2 and @xmath3 are functions of the variables (@xmath4). (our conventions are : @xmath5, metric signature + 2, partial and covariant derivatives with respect to the coordinate @xmath6 denoted by @xmath7 and @xmath8, respectively.) in its rest frame, the energy momentum tensor of the fluid @xmath9 is diagonal with components (-@xmath10), where @xmath11 is the total energy density and (@xmath12) are the radial, azimuthal and axial pressures or tensions, respectively. so, in this frame of reference, the energy momentum tensor can be written as @xmath13 where @xmath14, @xmath15, @xmath16, and @xmath17 are the four vectors of the orthonormal tetrad @xmath18 which satisfy the orthonormal relations. note that with the above definitions, the timelike four velocity of the fluid is @xmath14 and the quantities @xmath15, @xmath16, and @xmath17 are the spacelike principal directions of the fluid. furthermore, the energy momentum tensor satisfies einstein field equations, @xmath19. moreover, the quantities involved in the energy momentum tensor and the coefficients of the perturbed conservation equations are functions of the coordinates (@xmath4) only. let us consider a general perturbation @xmath20 of a quantity @xmath21 of the form @xmath22 where @xmath23 is the unperturbed quantity and @xmath24 is the perturbation. replacing ([perturb]) for each quantity in the energy momentum tensor ([tmunu]) and calculating the perturbed energy momentum equations, @xmath25, we obtain @xmath26 @xmath27 \nonumber \\ & & + \delta u^\theta [i k_\theta (\rho u^t + \xi_2 p_\theta y^\theta)] \nonumber \\ & & + \delta u^z [{ \rm f}(t, z,\rho u^t) + \xi_{3,z } p_z z^z + \xi_3 { \rm f}(t, z, p_z z^z)] \nonumber \\ & & + \delta \rho (-i w u^t u^t) = 0,\end{aligned}\]] @xmath28 @xmath29 \nonumber \\ & & + \delta \rho (u^t u^t \gamma^r_{tt }) + \delta p_r { \rm g}(r, r, x^r x^r) \nonumber \\ & & + \delta p_\theta (y^\theta y^\theta \gamma^r_{\theta\theta }) + \delta p_z (z^z z^z \gamma^r_{zz }) = 0,\end{aligned}\]] @xmath30 @xmath31 + \delta p_\theta (k_\theta y^\theta y^\theta) = 0,\end{aligned}\]] @xmath32 @xmath33 \nonumber \\ & & + \delta \rho (u^t u^t \gamma^z_{tt }) + \delta p_r (x^r x^r \gamma^z_{rr }) \nonumber \\ & & + \delta p_{\theta } (y^\theta y^\theta \gamma^z_{\theta\theta }) + \delta p_z { \rm g}(z, z, z^z z^z) = 0.\end{aligned}\]] where @xmath34 and @xmath35 are the christoffel symbols. in finding eqs. ([t])-([z]) we assumed that the perturbed energy momentum tensor does not modify the background metric. also, we disregard terms of order greater or equal to @xmath36. for details see @xcite. besides the four equations furnished by the energy momentum conservation equations, @xmath37, there is another important conservation equation, the equation of continuity, @xmath38 where @xmath39 is the proper number density of particles. the proper number density of particles @xmath39, and the total energy density @xmath11 are related through the relation, @xmath40 where @xmath41 is the constant mean baryon mass and @xmath42 the internal energy density. multiplying eq. ([varepsilon]) by @xmath14, performing the covariant derivative (@xmath8) and using eq. ([continuity]), we obtain that @xmath43 now, from the relation @xmath44 and the energy momentum tensor ([tmunu]), we obtain an expression for @xmath45. substituting this last expression into eq. ([rhovar]) we finally arrive to @xmath46 which is a first order differential equation for @xmath42. therefore, with @xmath42 given by ([pdevar]) the equation of continuity ([continuity]) is satisfied. for this reason, the continuity equation can be omitted in our stability analysis because, in principle, we can always find a solution for @xmath42. hereafter, the contribution of @xmath47 and @xmath42 to the total energy density are taken into account in @xmath11. in the case in which the internal energy density of the fluid is given, the equation of continuity must be considered. the thermodynamic properties of the system can be obtained from observations or theoretically, e.g. from the fokker - planck equation, where we obtain the particle distribution function of the disk. solving the three dimensional fokker - planck equation is not an easy task, but some progress in newtonian gravity had been done @xcite. the four equations, ([t])-([z]), contain seven independent unknowns, say @xmath48. so, at this point, the number of unknowns are greater than the number of equations. this opens the possibility to perform different kind of perturbations. in this article we are interested in perturbations in which the velocity perturbation in a certain direction leads to a pressure perturbation in the same direction. for example, if we perturbed the axial component of the velocity, @xmath49, then we must perturb @xmath50. with the above criterion, and without imposing any extra conditions to the individual perturbations, only four perturbations combinations are allowed and will be considered in our thick disk model. furthermore, we perform the perturbation @xmath51 with the extra imposed condition @xmath52. in this particular case, the system of equations reduces to a second order partial differential equation. General relativistic miyamoto-nagai galaxies a static general relativistic version of the miyamoto - nagai disk was constructed by @xcite by making a correspondence between the general isotropic line element in cylindrical coordinates and the miyamoto - nagai model @xcite. these general relativistic disks are obtained with ([metric]) and the specializations, @xmath53 where @xmath54, @xmath55 is the mass of the disk, and (@xmath56) are constants that control the shape of the density curves. with this metric, the energy density and pressures for the general relativistic miyamoto - nagai disk are @xmath57}{4 \pi \xi^3 [\frac{1}{2}+ \chi]^5 }, \label{rho } \\ & & p_r = p_\theta = \frac{b^2 \left[a r^2 + (a + \xi)^2 (a+ 2\xi) \right]}{16 \pi \xi^3 [\frac{1}{2 } + \chi]^5 [-\frac{1}{2 } + \chi] }, \label{pr } \\ & & p_z = \frac{b^2 (a+ \xi)^2}{8 \pi \xi^2 [\frac{1}{2 } + \chi]^5 [-\frac{1}{2 } + \chi] }, \label{pz } \end{aligned}\]] where @xmath58 and @xmath59. without losing generality we set @xmath60 in eqs. ([rho])-([pz]). to satisfy the strong energy condition (gravitational attractive matter) we must have that the `` effective newtonian density '' @xmath61. the weak energy condition requires @xmath62 and the dominant energy condition requires @xmath63, @xmath64 and @xmath65. the parameters used in this article satisfy all energy conditions. furthermore, the level curves show that it is physically acceptable. we remark that these are not the only parameters in which the level curves are physically acceptable. in the next section we apply the selected perturbations of sec. [sec2] to the general relativistic miyamoto - nagai disk mentioned above and study its stability. Perturbations before applying the different kinds of perturbations to the general relativistic miyamoto - nagai disk we must do some considerations. note that the general relativistic miyamoto - nagai disk is infinite in the radial and axial directions. we want to study the stability of a finite disk. so, in order to achieve this requirement we need a cutoff in the radial coordinate. in eqs. ([rho]), ([pr]) and ([pz]), we see that the thermodynamic quantities decrease rapidly enough to define a cutoff in both coordinates. the radial cutoff @xmath66 and the axial cutoff @xmath67 are set by the following criterion : the energy density within the disk formed by the cutoff parameters has to be more than 90% of the infinite thick disk energy density. the above criterion, and the parameters used in the article, leads to a radial cutoff of @xmath68 units and an axial cutoff of @xmath69 units. the other 10% of the energy density that is distributed from outside the cutoff parameters to infinity can be treated, if necessary, as a perturbation in the outermost boundary condition. we start perturbing the four velocity in its components @xmath74 and @xmath75. from the physical considerations mentioned in sec. [sec2] we also expect variations in the thermodynamic quantities @xmath76 and @xmath77. the set of equations ([t])-([z]) reduces to a second order ordinary differential equation for the perturbation @xmath73, say @xmath78 where @xmath79 are functions of (@xmath80), see appendix [fmna]. for this particular case we have, @xmath81. note that in eq. ([edo2mna]) the coordinate @xmath82 only enters as a parameter. moreover, the equation for @xmath73 is independent of the parameter @xmath83, but @xmath83 needs to be different from zero to reach that form. the second order equation ([edo2mna]) is solved numerically with two boundary conditions, one at @xmath84 and the other at the radial cutoff. at @xmath84 we set the perturbation @xmath73 to be @xmath85 10% of the unperturbed pressure @xmath77 ([pr]). in the outer radius of the disk we set @xmath86 because we want our perturbation to vanish when approaching the edge of the disk and, in that way, to be in accordance with the applied linear perturbation. we say that our perturbations are valid if their values are lower, or of the same order of magnitude, than the 10% values of its unperturbed quantities. in fig. [figmna], we present the amplitude profile of the radial pressure perturbation in the plane @xmath87 for different values of the parameters @xmath88 and @xmath89. as in the newtonian case, the less the ratio @xmath90, the flatter is the mass distribution. we see that the perturbation @xmath73 for (@xmath91) decreases rapidly with @xmath75 and has oscillatory behavior. at first sight, the perturbation @xmath73 appears to be stable for all @xmath75, but in order to make a complete analysis we have to compare at each radius the values of the perturbations with the values of the radial pressure. for this purpose, we included in the same graph a profile of the 10% value of @xmath77. we see that the perturbations of @xmath73 for different values of @xmath92 are always lower or, at least, of the same order of magnitude when compared to these 10% values. in the flatter case (@xmath93), the perturbation @xmath73 shows the same qualitative behavior, but the amplitudes of the oscillations are slightly higher. in both cases the amplitudes are well below the 10% values of @xmath77. if we consider a very flat galaxy (@xmath94) with @xmath95 we found that some modes are not stable in a small region near the center of the disk, from @xmath96 to @xmath97, because the perturbation amplitude is bigger than the 10% value of @xmath77 and our general linear perturbation is no longer valid. we also performed stability analyses for the physical radial velocity perturbation @xmath98 and the physical azimuthal velocity perturbation @xmath99. note that our four velocity @xmath14 ([tetrad]) has only components in the temporal part, so we do not have values of @xmath100 and @xmath101 to make comparisons with the perturbed values @xmath102 and @xmath103. for that reason we compared, in first approximation, the amplitude profiles of these perturbations with the value of the escape velocity in the newtonian limit. in the newtonian limit of general relativity, @xmath104, we have the well known relation @xmath105. so, the newtonian escape velocity @xmath106 can be written as @xmath107, see @xcite. with this criterion, the perturbations @xmath108 and @xmath103 are stable because their values are always well below the escape velocity value. recall that the perturbation @xmath73 does not depend on the parameter @xmath83, but the perturbations @xmath102 and @xmath103 do. we performed numerical solutions for the perturbations @xmath102 and @xmath103 with different values of the frequency @xmath83, and we find that when we increase the value of @xmath83 the perturbations become more stable. in this subsection we set the value of the parameter @xmath87. we performed the same analysis for different values of the parameter @xmath109, and we found that the perturbations show the same qualitative behavior. therefore, we can say that the general relativistic miyamoto - nagai disk shows some not - stable modes for very flat galaxies, e.g. (@xmath94). otherwise the disk is stable under perturbations of the form presented in this subsection. nevertheless, if we treat the 10% of the energy density as a perturbation in the outermost radius of the disk by setting @xmath110, where @xmath111 of @xmath112, the qualitative behavior of the mode profiles is the same. in the case of flat galaxies, when they present not stable modes, more complex structures like rings, bars or halos can be formed. moreover, if we set the frequency @xmath113 we obtain the same equation for the perturbation @xmath73, say ([edo2mna]). in this case, the real part of the general perturbation diverges with time and the perturbation is not stable. these last considerations can be applied to every perturbation in the following subsections. in this subsection we perturb the four velocity in its components @xmath74 and @xmath82, and we expect variations in the thermodynamic quantities @xmath76 and @xmath114. the set of equations ([t])-([z]) reduces to a second order ordinary differential equation for the perturbation @xmath50 given by @xmath115 where @xmath79 are functions of (@xmath80), see appendix [fmnb]. note that in eq. ([edo2mnb]) the coordinate @xmath75 only enters as a parameter. like the previous case, eq. ([edo2mnb]) is independent of the parameter @xmath83, but in order to reach that form we must have @xmath83 different from zero. the second order equation ([edo2mnb]) is solved numerically with two boundary conditions, one in @xmath116 and the other in @xmath117. at @xmath87 we set the perturbation @xmath50 to be @xmath118 of the unperturbed pressure @xmath114 ([pz]). in the outer plane of the disk we set @xmath119 because we want our perturbation to vanish when approaching the edge of the disk, and in that way, to be in accordance with the linear perturbation applied. in fig [figmnb1], we present the amplitude profiles of the axial pressure perturbation, the physical axial velocity perturbation @xmath120 and the physical azimuthal velocity perturbation for @xmath121 and different values of the parameters @xmath88 and @xmath89. for comparison reasons, we included in the graphs the amplitude profile that corresponds to 10% of the value of @xmath114 and the escape velocity profile. note that for (@xmath91) some modes of the axial pressure perturbation are above the 10% profile of @xmath114, e.g. the modes with @xmath122 and @xmath123. in these cases we can say that the mode with @xmath122 is not stable and that the mode with @xmath123 is near the validity criterion used for the perturbations. these modes are also present in the flatter galaxy (@xmath93) and have the same behaviors. the mode @xmath123 is actually not stable. this can be seen in the azimuthal velocity perturbation profiles, where its amplitude is greater than the escape velocity. note that in the velocity perturbation graphs the mode @xmath122 is also not stable. the azimuthal pressure perturbation, not depicted in fig [figmnb1], has all the modes well below the 10% profile of @xmath76, and therefore is stable. the perturbations @xmath50 and @xmath71 do not depend on the parameter @xmath83, but the perturbations @xmath124 and @xmath103 do. we performed numerical solutions for the perturbations @xmath124 and @xmath103 with different values of the frequency @xmath83, and we find that when we increase the value of @xmath83 the perturbations become more stable. we have performed the same above analysis for different values of the parameter @xmath125, and we found that the qualitative behavior is the same. we see from fig. [figmnb1] that the not stable modes are more pronounced for the flatter galaxy. furthermore, for very flat galaxies some modes like @xmath126 become not stable. in general, for not stable modes, more complex structures like rings, bars or spiral arms may be formed. in this subsection we perturb the radial component of the four velocity, the radial pressure and the energy density of the fluid. the set of equations ([t])-([z]) reduces to a second order ordinary differential equation for the perturbation @xmath73 of the form ([edo2mna]). the forms of the functions @xmath79 are given in appendix [fmnc]. in this case, the coordinate @xmath82 only enters as a parameter. due to the fact that we are not considering perturbations in the azimuthal axis, the coefficients of the second order ordinary differential equation do not depend on the wavenumber @xmath92. this second order equation is solved numerically with the same boundary conditions described in sec. [perturb1]. in fig. [figmnc] we present the amplitude profiles for different perturbation modes of the radial pressure in the plane @xmath87 for different values of the parameters @xmath88 and @xmath89. we see in the graph that the perturbation profiles decrease rapidly in few units of @xmath75. also, the values of the radial velocity perturbation and energy density perturbation, not depicted, are well below the escape velocity and the 10% energy profile, respectively. we performed the above analysis for different values of @xmath109 and we found that the quantities involved have the same qualitative behavior. from these results, we can say that the general linear perturbation applied is highly stable and, for that reason, the perturbations do not form more complex structures. in this subsection we perturb the axial component of the four velocity, the axial component of the pressure and the energy density of the fluid. the set of equations ([t])-([z]) reduces to a second order ordinary differential equation for the perturbation @xmath50 of the form ([edo2mnb]). the functions @xmath79 are given in appendix [fmnd]. note that, like in sec. [perturb2], the coordinate @xmath75 only enters as a parameter. in this case, we are not considering azimuthal perturbations and therefore the quantities involved do not depend on the parameter @xmath92. the second order equation is solved following the procedure of sec. [perturb2]. in fig. [figmnd] we present the amplitude profiles of the axial pressure perturbation and the physical axial velocity perturbation, for @xmath121 and for different values of the parameters @xmath88 and @xmath89. we see that the axial pressure perturbation modes for (@xmath91) are always of the some order of magnitude or lower when compared to the 10% profile. in the flatter case (@xmath93), note that the amplitude of the mode @xmath95 is greater in some region of the domain. this fact is reflected in the axial velocity perturbation profile where the mode @xmath95 have a strange behavior. all of the modes, including the mode with @xmath95, are stable because they are well below the escape velocity, which is not depicted. the modes that correspond to the energy density perturbation are all stable. for highly flat galaxies the mode @xmath95 is not stable and may form more complex structures. for higher values of the parameter @xmath83 the modes are more stable. we performed the above analysis for different values of the parameter @xmath125 and we found that the quantities involved have the same qualitative behavior. in this subsection we perturb the radial component of the four velocity, the axial component of the four velocity, the radial pressure and the axial pressure. as we said in sec. [sec2], we need an extra condition to set the number of unknowns equal to the number of equations. in this case, we set @xmath128. therefore, the set of equations ([t])-([z]) reduces to a second order partial differential equation for the pressure perturbation @xmath129, say @xmath130 where (@xmath131) are functions of (@xmath132), see appendix [fmne]. the partial differential equation ([pde2mne]) is solved numerically with four boundary conditions, at @xmath133, @xmath134, @xmath84 and @xmath135. they are different ways in which we can set the boundary conditions in order to simulate various kinds of pressure perturbations. here, we treat only the case when we have a pressure perturbation at @xmath84 and along the @xmath82 axis, i.e. some kind of a rod perturbation. we set the value of the rod pressure perturbation to be 10% of the axial pressure. we set the values of the other boundary conditions equal to zero because we want the perturbation to vanish when approaching the edge of the disk. we choose the 10% of the value of the axial pressure instead of the radial pressure because it has the lowest value near @xmath84. in that way, the perturbation values are also below the 10% values of the radial pressure and the general linear perturbation is valid. in fig. [figmne], we present the perturbation amplitudes for the pressure, the physical radial velocity and the physical axial velocity, for @xmath95 and for different values of the parameters @xmath88 and @xmath89. we see in the pressure perturbation graph that the perturbation rapidly decays to values near zero when we move out from the center of the disk. this behavior is the same for every galaxy considered. in the velocity perturbations profiles we can see a phenomenon that is more clear in the flatter galaxy. note that in the lower domain of the disk [-5,0) the axial velocity perturbation is positive and in the upper domain (0,5] the axial velocity perturbation is negative. this means that due to the linear perturbation the disk tries to collapse to the plane @xmath87. now, if we look to the radial velocity perturbation graph, we note that the upper and lower parts depart from the center of the disk due to the positive radial perturbation. so, with these considerations, we may say that the disk tends to form some kind of ring around the center of the disk. this phenomenon is greater for highly flat galaxies and lower for more spherical systems. Conclusions in this article we studied the stability of the recently proposed general relativistic miyamoto - nagai model [@xcite] by applying a general first order perturbation. we can say that the stability analysis performed is more complete than the stability analysis of particle motion along geodesics because we have taken into account the collective behavior of the particles. however, this analysis can be said to be incomplete because the energy momentum perturbation tensor of the fluid is treated as a test fluid and does not alter the background metric. this is a second degree of approximation to the stability problem in which the emission of gravitational radiation is considered. the different stability analyses made to the general relativistic miyamoto - nagai disk show that this disk is stable for higher values of the wave number @xmath92 and the frequency @xmath83. for lower values of @xmath92 and @xmath83 the disk presents not - stable modes that may form more complex structures like rings, bars or halos, but in order to study them we need a higher order perturbation formalism. in general, not - stable modes appear more for flatter galaxies and less for spherical systems. Acknowledgments m.u. and p.s.l. thanks fapesp for financial support ; p.s.l. also thanks cnpq. Functions @xmath136, @xmath137 and @xmath138 of section 4.1 the general form of the functions (@xmath139) appearing in the second order ordinary differential equation ([edo2mna]) is given by @xmath140 where @xmath141, @xmath142 and @xmath143 are @xmath144 in eqs. ([a1]) and ([a2]), we denote the coefficients of eq. ([t]) by @xmath145, the coefficient of eq. ([r]) by @xmath146, the coefficient of eq. ([varphi]) by @xmath147, the coefficient of eq. ([z]) by @xmath148, e.g., the first term in ([t]) has the coefficient @xmath149 multiplied by the factor @xmath150, the second term has the coefficient @xmath151 multiplied by the factor @xmath72, etc. the explicit form of the above equations is obtained replacing the fluid variables (@xmath10) of the isotropic schwarzschild thick disk. Functions @xmath136, @xmath137 and @xmath138 of section 4.2 the general form of the functions (@xmath139) appearing in the second order ordinary differential equation ([edo2mnb]) is given by @xmath152 where @xmath141, @xmath142 and @xmath143 are @xmath153 and the meaning of the coefficients (@xmath154) is explained in appendix [fmna]. Functions @xmath136, @xmath137 and @xmath138 of section 4.3 the general form of the functions (@xmath139) is given by @xmath155 where @xmath141, @xmath142 and @xmath143 are @xmath156 and the meaning of the coefficients (@xmath157) is explained in appendix [fmna]. Functions @xmath136, @xmath137 and @xmath138 of section 4.4 the general form of the functions (@xmath139) is given by @xmath158 where @xmath141, @xmath142 and @xmath143 are @xmath159 and the meaning of the coefficients (@xmath157) is explained in appendix [fmna]. Functions @xmath136, @xmath137, @xmath138, @xmath160 and @xmath161 of section 4.5 the general form of the functions (@xmath131) appearing in the partial second order differential equation ([pde2mne]) is given by semerk o., 2002, gravitation : following the prague inspiration, to celebrate the 60th birthday of jiri bik, edited by semerk o., podolsky j., zofka m., world scientific, singapure, 111, available at http://xxx.lanl.gov/abs/gr-qc/0204025.

the stability of a recently proposed general relativistic model of galaxies is studied in some detail. this model is a general relativistic version of the well known miyamoto - nagai model that represents well a thick galactic disk. the stability of the disk is investigated under a general first order perturbation keeping the spacetime metric frozen (no gravitational radiation is taken into account). we find that the stability is associated with the thickness of the disk. we have that flat galaxies have more not - stable modes than the thick ones i.e., flat galaxies have a tendency to form more complex structures like rings, bars and spiral arms. [firstpage] relativity galaxies : kinematics and dynamics

0707.4010

Introduction topological phase of condensed matter systems is a quantum many - body state with nontrivial momentum or real space topology in the hilbert spaces @xcite. recent newly discovered topological superconductor (tsc) has spawned considerable interests since this kind of topological phase supports the emergence of majorana fermion (mf) @xcite which is a promising candidate for the fault - tolerant topological quantum computation (tqc) @xcite. there are several proposals for hosting mfs in tsc, for example, chiral @xmath1-wave superconductor @xcite, cu - doped topological insulator @xmath2 @xcite, superconducting proximity devices @xcite and noncentrosymmetric superconductor (ncs) @xcite. the signatures of mfs have also been reported in the superconducting insb nanowire @xcite, @xmath3 @xcite and topological insulator josephson junction @xcite. to obtain a readily manipulated majorana platform for tqc, more experimental confirmations and theoretical proposals are therefore highly desirable. in this paper, we study the topological phase and majorana fermion at the edge and in the vortex core of the @xmath0-wave dresselhaus (110) spin - orbit (so) coupled ncs. it is found that the asymmetric so interaction plays a crucial role in realizing topological phases in the ncs. although the rashba so coupled ncs has been previously investigated @xcite, the dresselhaus (110) so coupled ncs is relatively less discussed theoretically @xcite. interestingly, we find that there is a novel semimetal phase in the dresselhaus ncs, where the energy gap closes in the whole region and different kinds of flat andreev bound states (abss) emerge. we demonstrate that these flat abss support the emergence of mfs analytically and numerically. it is known that the chern number is not a well - defined topological invariant in the gapless region, however, we find that the topologically different semimetal phases in this gapless region can still be distinguished by the pfaffian invariant of the particle - hole symmetric hamiltonian. several authors have proposed the flat abss in the ncs @xmath4 with high order so couplings @xcite, @xmath5-wave superconductor, @xmath6-wave superconductor and @xmath7-wave superconductor @xcite. instead, our proposal for hosting the flat abss is an @xmath0-wave dresselhaus (110) so coupled ncs in an in - plane magnetic field which is more flexible than the previous proposals where one needs to apply a magnetic field in the @xmath8 direction to the materials @xcite. our proposal is experimentally more feasible. the flat dispersion implies a peak in the density of states (dos) which is clearly visible and has an experimental signature in the tunneling conductance measurements @xcite. the zero - bias conductance peak has been observed in recent experiments on the insb nanowire @xcite and @xmath3 @xcite and argued to be due to the flat abs. thus if the majorana fermion exists in the dresselhaus ncs, the flat abs and the zero - bias conductance peak in the dos predicted here should be detectable. the paper is organized as follows. the model for @xmath0-wave ncs with dresselhaus (110) so coupling is given in sec. [model]. the phase diagrams and topological invariants of this model are discussed in sec. the numerical and analytical solutions to the majorana fermions at the edge of the system are demonstrated in sec. [mfatedge]. the majorana fermions in the vortex core of the system are numerically shown in sec. [mfinvortex]. finally, we give a brief summary in sec. [summary]. Model we begin with modeling the hamiltonian in a square lattice for the two dimensional @xmath0-wave ncs with dresselhaus (110) so interaction in an in - plane magnetic field, which is given by @xmath9 : @xmath10,\\ \end{split}\]] where @xmath11 denotes the creation (annihilation) operator of the electron with spin @xmath12 at site @xmath13. @xmath14 is the hopping term with hopping amplitude @xmath15 and chemical potential @xmath16. @xmath17 is the zeeman field induced by the in - plane magnetic field with components @xmath18. @xmath19 is the dresselhaus (110) so coupling and @xmath20 is the @xmath0-wave superconducting term with gap function @xmath21. we assume @xmath22 throughout this paper. in the momentum space, the hamiltonian is @xmath23 with @xmath24, where @xmath25, @xmath26 is the wave vector in the first brillouin zone and the bogoliubov - de gennes (bdg) hamiltonian is @xmath27 where @xmath28, @xmath29 and @xmath30 are the pauli matrices operating on the particle - hole space and spin space, respectively. the nontrivial topological order in the dresselhaus ncs is characterized by the existence of gapless edge state and majorana fermion. below we shall demonstrate these features in the hamiltonian eq. ([eq1]). Phase diagrams and topological invariants for comparison, we first briefly summarize the known results of the @xmath0-wave rashba ncs, in which the dresselhaus (110) so coupling @xmath19 in the hamiltonian eq. ([eq1]) is replaced by the rashba so coupling @xmath31 $] and the in - plane magnetic field is replaced by a perpendicular magnetic field @xcite. as usual, we can use the chern number to characterize the nontrivial momentum space topology of the rashba ncs. the chern number defined for the fully gapped hamiltonian is @xmath32, where @xmath33 is the strength of the gauge field @xmath34, where @xmath35 is the eigenstates of the hamiltonian. the integral is carried out in the first brillouin zone and the summation is carried out for the occupied states. as long as the topological quantum transition does not happen, the chern number remains unchanged. since the topological quantum transition happens when the energy gap closes, the phase boundary can be depicted by studying the gap - closing condition of the hamiltonian. in the phase diagram of the rashba ncs as shown in the fig. ([fig1]a), we find that the gap closes in some lines and the chern number is attached to each region of the phase diagram. however, in the present case, we shall show that the phase diagram of the dresselhaus ncs has a gapless region that makes the chern number ill - defined. to see this, we diagonalize the bdg hamiltonian eq. ([eq2]) in the periodic boundary conditions of the @xmath36 and @xmath8 directions, then the energy spectrum is @xmath37, where @xmath38 and @xmath39. therefore, we can find that the energy gap closes at @xmath40 which leads to the following gap - closing conditions : @xmath41, @xmath42. after some straightforward calculations, we find that when @xmath43, @xmath44 ; when @xmath45, @xmath46. finally, the gap closes at @xmath47 or @xmath48 subjected to @xmath49. therefore, we can find that the gap closes in the regions from a to g as shown in the fig. ([fig1]b). the number of gap - closing points at @xmath43, @xmath50 and @xmath45, @xmath51 are also shown as a pair @xmath52. later we shall derive a relation between the number of gap - closing points in the first brillouin zone and the topological invariant of the hamiltonian. interestingly, different from the phase diagram of the rashba ncs in the fig. ([fig1]a), where the gap closes in some boundary lines and each gapped region between them has a distinct chern number, the phase diagram of the dresselhaus ncs has a gapless area from a to g as shown in the fig. ([fig1]b), which means that the system is in the semimetal phase in the whole region. inside the gapless region, it is well known that the chern number is not well - defined. however, several other topological invariants which are obtained from symmetry analysis of the hamiltonian can still be used to characterize the topologically different semimetal phases in the gapless region. for the hamiltonian eq. ([eq2]), we enumerate several symmetries as follow : (i) particle - hole symmetry, @xmath53 ; (ii) partial particle - hole symmetry, @xmath54 and (iii) chiral symmetry, @xmath55, where @xmath56, @xmath57 and @xmath58 is the complex conjugation operator. we can define the pfaffian invariant @xcite for the particle - hole symmetric hamiltonian as @xmath59\text{pf}[\mathcal{h}(\mathbf{k_{4}})\sigma_{x}]}{\text{pf}[\mathcal{h}(\mathbf{k_{2}})\sigma_{x}]\text{pf}[\mathcal{h}(\mathbf{k_{3}})\sigma_{x}]}\right\ }, \end{split}\]] where @xmath60, @xmath61, @xmath62 and @xmath63 are the four particle - hole symmetric momenta in the first brillouin zone of the square lattice. similarly, the pfaffian invariant @xcite for the partial particle - hole symmetric system is @xmath64}{\text{pf}[\mathcal{h}(0,k_{y})\sigma_{x}]}\right\}. \end{split}\]] for the chiral symmetry, if we take the basis where @xmath65 is diagonal, @xmath66, then the hamiltonian becomes off - diagonal, @xmath67. using this @xmath68, we can define the winding number @xcite as @xmath69. \end{split}\]] the pfaffian invariant @xmath70 can be used for identifying topologically different semimetal phases of the hamiltonian eq. ([eq2]). it is easy to check that @xmath71 and @xmath72 in the phase diagram of the dresselhaus ncs as shown in the fig. ([fig1]b). therefore, the semimetal phases in the region of a, b, c, d and the region of e, f, g are topologically inequivalent. as for the other two topological invariants @xmath73 and @xmath74, below we shall show that they can be used to determine the range of edge states in the edge brillouin zone. -wave (a) rashba and (b) dresselhaus ncs. the parameters are @xmath75 and @xmath76. in (b), @xmath77. the chern number in different regions is indicated in (a). the number of gap - closing points at @xmath43, @xmath50 and @xmath45, @xmath51 in different regions are also shown as a pair @xmath78 in (b).,width=302] Majorana fermions at the edge of the system to demonstrate the novel properties in the semimetal phase of the dresselhaus ncs, we study the andreev bound states and majorana fermions at the edge and in the vortex core of it. we now first turn to study the abss of the dresselhaus ncs. by setting the boundary conditions of @xmath36 direction to be open and @xmath8 to be periodic, we diagonalize the hamiltonian eq. ([eq2]) in the cylindrical symmetry and get the edge spectra of the hamiltonian. interestingly, although the gap closes in the semimetal phase from region a to g as shown in the fig. ([fig1]b), there exist dispersionless abss at the edge of the system. the two topologically different semimetal phases in the region a and e are depicted in the fig. ([fig2]a) and ([fig2]b), respectively. we would like to study the number and range of the flat abss in these two different semimetal phases. by the pfaffian invariant eq. ([pfky]) or winding number eq. ([wky]), the range where the flat abss exist in the edge brillouin zone can be exactly obtained as shown in the fig. ([fig2]c) and ([fig2]d). the number of flat abss is half of the number of gap - closing points in the first brillouin zone. from the hamiltonian in the chiral basis, we can see that the gap closes when @xmath79. in the complex plane of @xmath80, a winding number can be assigned to each gap - closing point @xmath81 as @xmath82, where @xmath83 is a contour enclosing the gap - closing point. due to the particle - hole symmetry, we find that @xmath84, therefore, the gap - closing points with opposite winding number are equal in number. the function @xmath85 in the region a and e are shown in the fig. ([fig2]e) and ([fig2]f). as long as the projection of opposite winding number gap - closing points does not completely overlap in the edge brillouin zone, there will be flat abss connecting them @xcite. therefore, the number of flat abss is @xmath86 and it is easy to check that @xmath70 is the parity of @xmath87, @xmath88. the corresponding dos of these two different semimetal phases are shown in the fig. ([fig2]g) and ([fig2]h). we find that there is a peak at zero energy which is clearly visible in the tunneling conductance measurements. therefore, all of these flat abss have clear experimental signature in the tunneling conductance measurements and the mfs predicted at the edge of the dresselhaus ncs should be experimentally observable. as for the robustness of the flat abss against disorder or impurity, we can discuss it from the topological point of view. as long as the disorder or impurity does not break the symmetries of hamiltonian eq. ([eq2]), these flat abss will be protected by the three topological invariants mentioned above. [cols="^ ",] Summary in summary, we have investigated the topological phase and majorana fermion in the @xmath0-wave dresselhaus (110) so coupled ncs. we find that there is a gapless region appearing in the phase diagram of the dresselhaus ncs. we observe that there exist flat andreev bound states which host majorana fermions in the gapless region. the chemical compound insb has the largest dresselhaus (110) so coupling @xcite which makes it promising to observe the mfs at the edge or in the vortex core of the system. we can fabricate an insb (110) quantum well, in contact with an @xmath0-wave superconducting aluminum and couple it to an in - plane magnetic field. thus we can apply the tunneling conductance measurements to detect the zero - bias conductance peak of the system. 34ifxundefined [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.49.405 [* *, ()] link:\doibase 10.1103/physrevlett.61.2015 [* *, ()] link:\doibase 10.1103/physrevlett.96.106802 [* *, ()] link:\doibase 10.1143/jpsj.77.031007 [* *, ()] link:\doibase 10.1103/physrevlett.95.226801 [* *, ()] link:\doibase 10.1103/physrevlett.95.146802 [* *, ()] link:\doibase 10.1103/physrevlett.98.106803 [* *, ()] link:\doibase 10.1103/physrevb.61.10267 [* *, ()] link:\doibase 10.1103/physrevb.82.134521 [* *, ()] @noop * *, () link:\doibase 10.1103/physrevlett.100.096407 [* *, ()] link:\doibase 10.1103/physrevlett.105.177002 [* *, ()] link:\doibase 10.1016/s0003 - 4916(02)00018 - 0 [* *, ()] link:\doibase 10.1103/physrevlett.108.107005 [* *, ()] @noop * *, () @noop * *, () link:\doibase 10.1103/physrevb.81.125318 [* *, ()] link:\doibase 10.1103/physrevb.86.161108 [* *, ()] @noop * *, () link:\doibase 10.1103/physrevlett.107.217001 [* *, ()] link:\doibase 10.1103/physrevlett.109.056803 [* *, ()] link:\doibase 10.1103/physrevlett.104.040502 [* *, ()] link:\doibase 10.1103/physrevb.84.060504 [* *, ()] link:\doibase 10.1103/physrevb.84.020501 [* *, ()] link:\doibase 10.1103/physrevb.83.224511 [* *, ()] @noop link:\doibase 10.1103/physrevlett.105.097002 [* *, ()] link:\doibase 10.1103/physrevb.82.184525 [* *, ()] link:\doibase 10.1103/physrevlett.109.150408 [* *, ()] link:\doibase 10.1103/physrevb.86.094512 [* *, ()] link:\doibase 10.1103/physrevb.79.094504 [* *, ()] link:\doibase 10.1016/0031 - 9163(64)90375 - 0 [* *, ()] link:\doibase 10.1103/physrevb.82.155327 [* *, ()]

the asymmetric spin - orbit interactions play a crucial role in realizing topological phases in noncentrosymmetric superconductor (ncs). we investigate the edge states and the vortex core states in the @xmath0-wave ncs with dresselhaus (110) spin - orbit coupling by both numerical and analytical methods. in particular, we demonstrate that there exists a novel semimetal phase characterized by the flat andreev bound states in the phase diagram of the @xmath0-wave dresselhaus ncs which supports the emergence of majorana fermions. the flat dispersion implies a peak in the density of states which has a clear experimental signature in the tunneling conductance measurements and the majorana fermions proposed here should be experimentally detectable.

1209.0930

Introduction x - ray studies of fairly normal " galaxies, with high - energy emission not obviously dominated by a luminous active galactic nucleus (agn), have recently been extended to cosmologically interesting distances in the deep field (cdf) surveys, which have now reached 1 ms of exposure (cdf - n : hornschemeier et al. 2001, hereafter paper ii ; brandt et al. 2001b, hereafter paper v ; cdf - s : tozzi et al. 2001 ; p. rosati et al., in prep.). galaxies with @xmath8 are detected in appreciable numbers at 0.52 kev fluxes below @xmath9 erg @xmath6 s@xmath7 (e.g., paper ii) ; the cdf - n survey goes almost two orders of magnitude fainter, detecting significant numbers of normal galaxies among the population of x - ray sources making the diffuse x - ray background (xrb ; paper ii ; a.j. barger et al., in prep.). these normal galaxies contribute as much as 510% of the xrb flux in the 0.52 kev band. the bulk of the energy density of the xrb is certainly explained by agn, but the investigation of the typical " galaxy, whether its x - ray emission is dominated by a population of x - ray binaries, hot interstellar gas, or even a low - luminosity agn, is an equally important function of deep x - ray surveys. normal galaxies are likely to be the most numerous extragalactic x - ray sources in the universe and are expected to dominate the number counts at 0.52 kev fluxes of @xmath10@xmath11 erg @xmath6 s@xmath7 (ptak et al. 2001). the cdf - n has reached the depths necessary to detect individually many normal [@xmath12 ; @xmath13 is from 0.52 kev] @xmath14 galaxies to @xmath15, corresponding to a look - back time of @xmath16 gyr (@xmath17 km s@xmath7 mpc@xmath7, @xmath18, and @xmath19 are adopted throughout this paper). reaching larger look - back times presents the exciting possibility of detecting the bulk x - ray response to the heightened star - formation rate at @xmath203 (e.g., madau et al. 1996). one thus expects the x - ray luminosity per unit @xmath2-band luminosity to be larger at @xmath211 in the past due to the increased energy output of x - ray binary populations at @xmath203 ; this x - ray emission represents a fossil record " of past epochs of star formation (e.g., ghosh & white 2001 ; ptak et al. therefore, measurements of the x - ray luminosities of typical galaxies can constrain models of x - ray binary production in galaxies. while x - ray emission from individual galaxies is not easily detected at @xmath22, it is possible to estimate the emission at their extremely faint flux levels using statistical methods such as stacking, a technique implemented successfully on the cdf - n survey data in several previous studies. these include the detection of x - ray emission from the average @xmath21 bright (@xmath23) galaxy in the hubble deep field - north () described in brandt et al. (2001a, hereafter paper iv) and a study of x - ray emission from @xmath244 lyman break galaxies identified in the (brandt et al. 2001c, hereafter paper vii). encouraged by the success of these analyses, we extend here the study of normal galaxies to the entire plus flanking fields region, now concentrating on galaxies at @xmath25 to complement the study of @xmath26 galaxies performed in paper vii. we focus on this redshift range due to the extensive spectroscopic redshift coverage (cohen et al. 2000 and references therein) and superb imaging which has allowed a comprehensive galaxy morphology study (van den bergh, cohen, & crabbe 2001). the cdf - n data provide extremely deep x - ray coverage over this area (see figure 7 of paper v for the exposure map of this region) ; the point - source detection limits in this region of the cdf - n survey in the 0.52 kev and 28 kev bands are @xmath27 erg @xmath6 s@xmath7 and @xmath28 erg @xmath6 s@xmath7, respectively. in this study, we place observational constraints on the evolution of the ratio of x - ray luminosity to @xmath2-band luminosity of normal " spiral galaxies up to @xmath29 ; this ratio is an indicator of the current level of star formation in a galaxy (e.g., david, jones, & forman 1992 ; shapley et al. we also place constraints on the fraction of the diffuse xrb explained by galaxies lingering just below the cdf - n detection threshold, and thus the contribution to the xrb by normal galaxies. Galaxy samples spectroscopic redshifts for the galaxies are drawn from the catalogs of cohen et al. (2000), cohen (2001), and dawson et al. (2001) in the range @xmath30. spectroscopic redshift determination is difficult in the range @xmath31 due to the absence of strong features in the observed - frame optical band and the lack of the lyman break feature useful to identify higher redshift objects. we have therefore used the deep photometric redshift catalog of fernndez - soto, lanzetta, & yahil (1999) for the redshift interval @xmath32, which allows some overlap in redshift space with the spectroscopic catalogs for cross - checking. the spectroscopic catalogs cover the entire hdf - n plus a substantial fraction of the flanking fields region, whereas the photometric catalog only covers the hdf - n. we shall refer to these two samples as the spectroscopic sample " and the photometric sample " throughout the rest of this letter. for the spectroscopic sample, the @xmath33-band magnitude was used to filter the sources by optical luminosity, as this is best matched to rest - frame @xmath2 over most of the redshift range under consideration here. the @xmath33 magnitudes are those given in barger et al. (1999) for the hawaii flanking fields area. for the photometric sample, the f814w (hereafter @xmath34) magnitudes of fernndez - soto et al. (1999) were used. we chose galaxies which had no x - ray detection within 40 in the 0.58 kev (full), 0.52 kev (soft) and 28 kev (hard) bands down to a wavdetect (freeman et al. 2002) significance threshold of @xmath35 in the restricted acis grade set of paper iv. this low detection threshold ensures that our study does not include sources with x - ray emission just below the formal detection limits of paper v. we have attempted to construct a sample of galaxies similar to spiral galaxies in the local universe. to accomplish this, we have used the morphological classes of van den bergh et al. (2001) for galaxies from @xmath36 in the hdf - n and the flanking fields. to simplify the morphological filtering, we have cast objects in the van den bergh et al. (2001) catalog into the following four classes : (1) e / s0 " and e ", (2) merger ", (3) sa"sc ", including proto - spirals and spiral / irregulars, and (4) irr ", peculiar " and/or tadpole ". we then filtered the catalog to keep only classes (2) and (3). filtering the photometric sample is more difficult due to the faintness of many of the sources and problems due to morphological evolution with redshift. we have used the spectral energy distribution (sed) classifications of fernndez - soto et al. (2001) to exclude all galaxies of type e ". comparison of the source lists reveals that, within the area covered by both, @xmath37 70% of galaxies identified through the two methods are in common. since the evolution of x - ray properties with redshift is of interest, we have made an effort to study objects with comparable optical luminosities at different redshifts. this is particularly important due to the non - linear relationship between x - ray luminosity and @xmath2-band luminosity for some types of spiral galaxies (@xmath38 ; e.g. fabbiano & shapley 2001). using the value of @xmath39 in the @xmath40-band as determined by blanton et al. (2001) for a large sample of galaxies in the sloan digital sky survey, we determined the value of @xmath39 in the @xmath2-band. the sloan filter @xmath40 is best matched to @xmath2 ; the resulting value of @xmath39 in the @xmath2 band is @xmath41. to ensure that our results are not sensitively dependent upon the galaxy sed used to determine the optical properties, we have used both the sa and sc galaxy seds of poggianti et al. (1997) to calculate @xmath33 and @xmath34 vs. @xmath42 for an @xmath39 galaxy using the synthetic photometry package synphot in iraf. these calculations are shown in figure [sample_definition]. note the close similarity between the sa and sc tracks ; this is because the @xmath33 band corresponds to rest - frame @xmath2 in the middle of our redshift range. also plotted in figure 1 are the 151 galaxies in the spectroscopic sample with spiral or merger morphology having @xmath43 and the 651 galaxies in the photometric sample with sed class other than e " having @xmath44. these galaxies were filtered by optical flux to lie within 1.5 mags of the @xmath39 galaxy tracks discussed above ; the galaxy samples constructed assuming sa and sc seds were identical (or nearly so) for all redshifts up to @xmath45. galaxies meeting the optical magnitude filter were then divided by redshift into several bins ; these bins were constructed to ensure that there were @xmath46 galaxies per bin. the number of galaxies, median redshift, median look - back time and median optical magnitude for each bin are listed in table 1. in figure 1, we mark all the objects in the sc sed sample with colors indicating the different redshift bins. table 1 also includes the number of galaxies rejected from each redshift bin due to the presence of an x - ray detection within 40 ; this exclusion radius ensures that our results will not be adversely affected by the wings of the psf of very bright x - ray sources. these galaxies satisfied both the optical magnitude and morphology filtering constraints and were rejected only due to x - ray detection. this exclusion criterion is very conservative, however, considering that our astrometry is accurate to @xmath476 in the area under consideration (see paper v). to allow for the off - nuclear nature of some of the x - ray sources found in normal galaxies (e.g., paper iv), we consider galaxies to be highly confident x - ray detections if the x - ray source is within 15 of the galaxy s center this matching radius is also well matched to the chandra psf. off - axis for 0.52 kev and the 83% encircled - energy radius for 0.58 kev.] we therefore also give the number of galaxies having an x - ray detection within 15 in table 1. Stacking procedure and results the x - ray imaging data at each position were stacked in the same manner as in paper vii, keeping the 30 pixels whose centers fall within an aperture of radius 15. the detection significance in each band was assessed by performing 100,000 monte - carlo stacking simulations using local background regions as in paper vii. a source is considered to be significantly detected if the number of counts over background exceeds that of 99.99% of the simulations. no single source in the stacking sample appeared to dominate the distribution, demonstrating the effectiveness of our selection criteria. stacking of the galaxies in the redshift bins described in table [sample_table] and figure [sample_definition] resulted in significant detections in the soft band for all of the redshift bins up to @xmath48 (see table 2). we also stacked galaxies in the redshift bin @xmath49, but there was not a significant detection. the results for the two different spiral galaxy sed samples are nearly or exactly identical except for the detection in the highest redshift bin (@xmath50). we adopt a @xmath51 power law for the calculation of x - ray fluxes and luminosities, assuming that these galaxies are similar to spiral galaxies in the local universe and have their x - ray emission dominated by x - ray binaries (e.g., kim, fabbiano, & trinchieri 1992). while there were several cases of significant detections in the full band, there were no highly significant detections in the hard band. given the variation of effective area and background rate with energy, the signal - to - noise ratio for sources with the assumed spectrum is highest in the soft band and lowest in the hard band, so this behavior is expected. the flux level of the soft - band detections for the spectroscopic sample is (56)@xmath52 erg @xmath6 s@xmath7. the corresponding rest - frame 0.52 kev luminosities for the average galaxy are @xmath53 erg s@xmath7 for the lowest redshift bin and @xmath54 erg s@xmath7 for the highest redshift bin. for the photometric sample, the soft - band flux level of the detections is (35)@xmath52 erg @xmath6 s@xmath7. we also give fluxes for the less significant detections in the full band for those redshift bins having highly significant soft - band detections. we have investigated the properties of the sources which were rejected from the stacking samples due to an x - ray detection at or near the position of the galaxy (the numbers of such sources are given in the last column of table 1). there are only 15 distinct galaxies with x - ray detections within 15. of these 15 sources, three are broad - line agn, which are clearly not normal galaxies. one object has a photometric redshift which differs significantly from its spectroscopic redshift. since the optical properties of this object at its spectroscopic redshift place it outside our sample boundaries, we have rejected it. one object is very near another x - ray source which has been positively identified with a narrow - line agn. thus, there are a total of 10 normal " galaxies positively identified with x - ray sources within this sample, consituting a small minority of the galaxies under study here. the worst case is in the lowest redshift bin where 15% of the galaxies had x - ray detections. figure [sbhistogram] shows a histogram of @xmath55 values calculated for both the stacking samples and the individually x - ray detected galaxies. the individually x - ray detected galaxy set does possibly contain some lower - luminosity agn, including the agn candidate (see papers ii and iv). with the exception of the objects in the redshift bin with median @xmath56, figure [sbhistogram] shows that typically the x - ray luminosities of the individually detected objects are on average much higher than those of the stacked galaxies ; they are sufficiently more luminous as to appear atypical of the normal galaxy population. for the lowest redshift bin, it is plausible that our results are moderately biased by the exclusion of the x - ray detected sources. in 4, we therefore also give results which include the individually x - ray detected objects in the sample average for this lowest redshift bin. for additional comparison, we have considered the radio properties of the individually detected galaxies and the stacked galaxies using the catalogs of richards et al. (1998) and richards (2000). the percentage of radio detections among the individually x - ray detected galaxies is higher than that among the stacked galaxies (@xmath57% vs. @xmath585%). this possible difference between the two populations is significant at the 93% level as determined using the fisher exact probability test for two independent samples (see siegel & castellan 1988). due to the x - ray luminosity difference and possible difference in radio properties, and to the fact that these galaxies constitute a small minority of those under study, we are confident we have not biased our determination of the properties of the typical galaxy by omitting these x - ray detected objects from further consideration. Discussion in figure [lxlb]a, we plot the x - ray - to - optical luminosity ratio @xmath59 for each stacked detection, where @xmath13 is calculated for the rest - frame 0.52 kev band. we have also plotted our approximate sensitivity limit in figure [lxlb]a, which is simply the corresponding @xmath60 x - ray luminosity detection limit achieved for a 30 ms stacking analysis divided by @xmath61. we do not expect to detect galaxies having less x - ray emission per unit @xmath2-band luminosity than this value. for comparison with the local universe, we also plot the mean @xmath59 for spiral galaxies of comparable @xmath62 from the sample of shapley et al. this sample includes 234 spiral galaxies observed with _ einstein _ and excludes agn where the x - ray emission is clearly dominated by the nucleus. the galaxies in the shapley et al. (2001) sample all have @xmath63 ; median redshift is @xmath64. in figure [lxlb]b, we plot @xmath59 versus @xmath65 ; the values up to @xmath66 are consistent with what is observed in the shapley et al. sample for objects with comparable @xmath65, although they are toward the high end of what is observed. this is consistent with figure [lxlb]a, which shows the average @xmath59 derived from stacking being somewhat higher than that for the shapley et al. galaxies of comparable optical luminosity. there is a slight increase (factor of 1.6) in the average @xmath67 from the local universe to @xmath68. for the highest redshift bin (@xmath50), the results become somewhat sensitive to the galaxy sed one assumes for determining the optical properties. we suspect that an sc galaxy sed is more appropriate at this epoch due to the higher prevalence of star formation. adopting this sed, we find that @xmath69 has increased somewhat more substantially (@xmath70 times) at @xmath50. one may also constrain star - formation models using only the x - ray properties of these galaxies. the average x - ray luminosity of the spiral galaxies in the shapley et al. (2001) sample having the same range of @xmath65 as used in this study is @xmath71 erg s@xmath7 (converted to 0.52 kev). the average galaxy in our stacking sample has a luminosity @xmath72 times higher at @xmath68. this increases to @xmath73 times higher at @xmath74. the @xmath68 value is most likely affected by some bias due to the exclusion of legitimate normal galaxies in the lowest redshift bin (see 3). if we include these individually x - ray detected objects, then the average galaxy in our stacking sample has a luminosity @xmath75 times higher at @xmath68, consistent with the predictions of ghosh & white (2001) that the x - ray luminosity of the typical sa - sbc spiral galaxy should be @xmath75 times higher at @xmath76. including the x - ray detected galaxies does not significantly affect our results for the interval @xmath77 (the difference is @xmath78%). however, since the x - ray luminosities of the x - ray detected objects with @xmath74 are substantially higher (by an order of magnitude) than the x - ray stacking averages (see figure 2), it is not appropriate to include these objects in the calculation of the average x - ray luminosity. we thus find a smaller increase in the average x - ray luminosity of galaxies at @xmath79 than the increase by a factor of @xmath80 predicted by ghosh & white (2001). we find that the average x - ray luminosities of galaxies have not evolved upward by more than a factor of @xmath73 by @xmath74, regardless of exclusion or inclusion of x - ray detected objects. the range of average 0.52 kev fluxes for the spiral galaxies studied here is (@xmath46@xmath5 erg @xmath6 s@xmath7. these x - ray fluxes are consistent with independent predictions made by ptak et al. (2001) that galaxies of this type will be detected at 0.52 kev flux levels of @xmath81 erg @xmath6 s@xmath7 (converted from their 210 kev prediction assuming a @xmath82 power law). assuming a 0.52 kev xrb flux density of @xmath83 erg @xmath6 s@xmath7 deg@xmath84 (garmire et al. 2001), we have identified @xmath85% of the soft xrb as arising from spiral galaxies not yet individually detected in deep surveys. many of these objects should be sufficently bright to be detected with acis exposures of @xmath86 ms, which should be achievable in the next several years of the mission. we thank alice shapley for useful discussions and sharing data. we gratefully acknowledge the financial support of nasa grant nas 8 - 38252 (gpg, pi), nasa gsrp grant ngt5 - 50247 (aeh), nsf career award ast-9983783 (wnb, dma, feb), and nsf grant ast99 - 00703 (dps). this work would not have been possible without the enormous efforts of the entire team. ccccccc + 0.400.75 & sc & 0.635 & 6.34 & 21.73 & 29 & 12/6 + 0.750.90 & sc & 0.821 & 7.39 & 21.83 & 38 & 2/2 + 0.901.10 & sc & 0.960 & 8.05 & 22.29 & 30 & 6/4 + + + 0.501.00 & sc & 0.920 & 7.87 & 22.99 & 37 & 9/4 + 1.001.50 & sc & 1.200 & 8.97 & 24.28 & 64 & 6/2 + 1.001.50 & sa & 1.240 & 9.09 & 24.69 & 80 & 8/2 + rrrrrrrrrcrrc + 0.400.75 & sc & 49.6 & 31.7 & 99.99 & @xmath87 & 25.41 & 25.44 & 1.52 & 0.65 & 2.94 & 1.26 & 0.67 + 0.750.90 & sc & 58.0 & 36.4 & @xmath87 & @xmath87 & 33.05 & 33.09 & 1.36 & 0.57 & 4.92 & 2.07 & 1.33 + 0.901.10 & sc & 54.4 & 25.9 & @xmath87 & @xmath87 & 26.54 & 26.56 & 1.60 & 0.51 & 8.44 & 2.68 & 1.40 + + + 0.501.00 & sc & 42.0 & 33.5 & 99.77 & @xmath87 & 34.51 & 34.54 & 0.95 & 0.51 & 4.51 & 2.40 & 1.12 + 1.001.50 & sc & 44.8 & 38.5 & 98.94 & @xmath87 & 59.67 & 59.73 & 0.58 & 0.34 & 5.34 & 3.06 & 0.79 + 1.001.50 & sa & 45.6 & 34.2 & 98.29 & 99.95 & 74.61 & 74.68 & 0.48 & 0.24 & 4.70 & 2.36 & 0.81 + . the blue curves give @xmath33 vs. @xmath42 for @xmath39 sa and sc galaxies (the lower curve at higher redshift is for the sa galaxy). galaxies without a colored circle were not included in the stacking sample because they either were not within the range of optical luminosity specified or because an x - ray detection was found within 40. (b) the black filled circles are the photometric redshift sample of fernndez - soto et al. (1999), excluding the e "- type galaxies. the blue curves give @xmath34 vs. @xmath42 for @xmath39 sa and sc galaxies (the lower curve at higher redshift is for the sa galaxy). [sample_definition]] as a function of redshift for the stacking samples. the redshift error bars indicate the full extent of the redshift bin ; the data points are at the median redshift value for that bin. the solid line indicates the 2@xmath88 x - ray sensitivity limit normalized by @xmath89. the dashed lines above and below the solid line indicate the effect of decreasing and increasing the optical luminosity by one magnitude, respectively. objects which have less x - ray luminosity per unit @xmath2-band luminosity than this are not expected to be detected in the current data. the error bar on the shapley et al. (2001) data point indicates the dispersion of values in this sample. (b) @xmath59 vs. @xmath65 for both the shapley et al. (2001) sample (open circles) and the stacked detections presented here. the error bars on @xmath59 in both figures were calculated following the numerical method described in 1.7.3. of lyons (1991). the solid line in (b) indicates @xmath89 ; again the dashed lines correspond to one magnitude fainter and brighter than @xmath89.

we present a statistical x - ray study of spiral galaxies in the hubble deep field - north and its flanking fields using the chandra deep field north 1 ms dataset. we find that @xmath0 galaxies with @xmath1 have ratios of x - ray to @xmath2-band luminosity similar to those in the local universe, although the data indicate a likely increase in this ratio by a factor of @xmath33. we have also determined that typical spiral galaxies at @xmath1 should be detected in the 0.52 kev band in the flux range (@xmath46@xmath5 erg @xmath6 s@xmath7. 1_heao-1 _

astro-ph0110094

Introduction the cdf collaboration has reported an excess in the production of two jets in association with a @xmath1 boson production @xcite from data collected at the tevatron with a center - of - mass energy of 1.96 tev and an integrated luminosity of 4.3 fb@xmath6. the @xmath1 boson is identified through a charged lepton (electron or muon) with large transverse momentum. the invariant mass of the dijet system is found to be in the range of 120 - 160 gev. the @xmath0dijet production has a few pb cross - section which is much larger than standard model (sm) expectation. the dijet system may be interpreted as an unidentified resonance with mass around 150 gev which predominantly decays into two hadron jets. this leads to the speculation that a beyond sm new particle has been found. at present the deviation from the sm expectation is only at 3.2@xmath7 level. the excess needs to be further confirmed. on the theoretical side, our understanding of the parton distributions and related matter still have room for improvement to make sure that the excess represents genuine new physics beyond the sm @xcite. nevertheless studies of new particle explanation has attracted much attention. several hypothetic particles beyond sm have been proposed to explain the cdf @xmath0dijet excess, such as leptophobic @xmath8 model @xcite, technicolor @xcite, colored vector, scalar @xcite, quasi - inert higgs bosons @xcite and the other possibilities @xcite. common to all of these models is that the new particle must decay predominantly into hadrons (dijet). we note that a class of particles which can naturally have this property. these are those scalars which are colored and couple to quarks directly. in order for these scalars to be considered as a possible candidate, it must satisfy constraints obtained from existing experimental data. colored particles which couple to two quarks have been searched for at the tevatron and the lhc. if the couplings to quarks / gluon are the same as the qcd coupling, the color triplet diquark with a mass in the range @xmath9 gev is excluded at the tevatron @xcite, and the mass intervals, @xmath10 gev, @xmath11 tev and @xmath12 tev are excluded at the lhc @xcite whereas the lhc data is limited for @xmath13 gev. the color sextet diquarks with electric charge, @xmath14, @xmath15, @xmath16, are excluded for their masses less than 1.8, 1.9, 2.7 tev, respectively @xcite. the color octet vectors / scalars which interact with quarks / gluon by qcd coupling are excluded for @xmath17 tev @xcite. if their couplings to quarks / gluon are smaller than the qcd coupling the constraints are weaker. some aspects of colored scalars relevant to the cdf @xmath0dijet data have been considered recently @xcite. in this work we carry out a systematic study to investigate the possibility of colored scalar bosons @xmath18 as the new particle explaining the cdf excess through @xmath19 production followed by @xmath18 decays into two hadron jets. at the tree level, there are several new scalar bosons which can have renormalizable couplings to two quarks (or a quark and an anti - quark). a complete list of beyond sm scalars which can couple to sm fermions at the tree level @xcite and some of the phenomenology have been studied before @xcite. the required production cross section and the mass from @xmath0dijets excess put constraints on model parameters. some possible scenarios are ruled out when confronted with other existing data, such as data from flavor changing neutral current (fcnc) processes. we find that without forcing of the yukawa couplings to be some special forms most of the scalars, except the @xmath5, are in trouble with fcnc data. we, however, do find that some other cases can be made consistent with all data by tuning their couplings providing a possible explanation for the @xmath0dijet excess from cdf. justification of such choices may have a realization in a flavor model, which is beyond the scope of the present work of phenomenology. these colored scalars also have interesting signatures at the relativistic heavy ion collider (rhic) and the large hadron collider (lhc) which can be used to further distinguish different models. the paper is organized as follows. in section [sec : cs], we study possible colored - scalars which can couple to two quark (or a quark and a anti - quark), and determine their yukawa couplings by requiring that the colored scalar with a mass of 150 gev to explain the cdf @xmath0dijet excess data. in section [sec : fcnc], we study the constraints from fcnc data on colored scalar couplings. in section [sec : rhic], we give some implications for the rhic and the lhc. finally, we summarize our results in sec. [sec : sum]. Colored scalars and the cdf @xmath0dijet excess scalar bosons which have color and have renormalizable yukawa couplings to two quarks or a pair of a quark and an anti - quark can be easily determined by studying bi - products of two quarks @xcite. the quarks transform under the sm @xmath20 as : @xmath21 the following bi - products, @xmath22 and @xmath23. here @xmath24 and @xmath25 are generation indices. with these quantum numbers, we can have the following quark bi - products @xmath26 where the superscript `` @xmath27 '' indicates charge conjugation. for those scalars which only couple to right - handed quarks, the contribution to @xmath1 associated production will be small because they do not directly couple to @xmath1 boson. to have large @xmath1 associated production for the cdf excess, we therefore consider the following colored scalars which can couple to left - handed quarks @xmath28 where @xmath29 is a color index, @xmath30 is the @xmath31 generator normalized as @xmath32, and @xmath33 (@xmath34) is a generator of the symmetric tensor (@xmath35, @xmath36). the color component fields, @xmath37, @xmath38 and @xmath39, are defined by having the kinetic energy term normalized properly. we denote the component fields of @xmath40 as follows : @xmath41 where @xmath42 are the @xmath40 indices. for neutral @xmath43, the physics component can be separated according to their parity property with @xmath44 and @xmath45. for @xmath19 production by @xmath46 collision, the leading contributions are from the @xmath47-channel and @xmath48-channel tree diagrams as shown in fig. [feyn]. -channel and @xmath48-channel process in @xmath49 production. solid, wavy and dashed line represents quark or anti - quark, @xmath50 and @xmath18, respectively. [feyn]] for @xmath51, @xmath52 and @xmath53 the @xmath48-channel production can exist in addition to the diagram of @xmath47-channel quark exchange. one needs to know how the colored scalars couple to quarks and the @xmath1 boson. we list the yukawa couplings in the quark mass eigenstate basis in the following, @xmath54 where @xmath55. the flavor space is described as @xmath56, @xmath57 and @xmath58, where @xmath59 is a ckm quark mixing matrix, and @xmath60 (@xmath618@xmath62, (6,3), (6,1), (3,3), (3,1)) are the coupling matrix in flavor space. @xmath63 and @xmath64 are symmetric, and, @xmath65 and @xmath66 are anti - symmetric, under the exchange of flavor indices @xmath24 and @xmath25. the diquark couplings are @xmath67 where @xmath68 is @xmath40 triplet, and @xmath69 is @xmath40 singlet. the electroweak gauge interactions are given by @xmath70 where @xmath71 is the covariant derivative. the electroweak gauge interactions of the colored scalars are obtained from the following : @xmath72 where @xmath73 is the sine of the weinberg angle @xmath74, and @xmath75. in the above color indices are suppressed and interaction with gluons are omitted. for @xmath51 color - octet and @xmath52, @xmath76 diquarks, the dominant contributions to the @xmath19 production are from @xmath18 couplings to the first generation, @xmath77. for @xmath78, @xmath53 diquarks, on the other hand, because the yukawa coupling matrix is anti - symmetric in generation space, the dominant contribution would come from @xmath79 term (which includes @xmath80 and @xmath81 quark coupling suppressed by cabibbo mixing). in general different component in @xmath18 can have different masses. in order to avoid the contribution to @xmath82 parameter, we assume that all the components have the same masses for simplicity. since interactions and the masses of the colored scalars are fixed, the only unknowns parameters, the yukawa couplings, can be determined by requiring the colored scalars to explain the cdf @xmath0dijet data. we consider the different type of colored scalars separately. for the fit, we use madgraph / madevent @xcite and pythia @xcite for the particle - level event - generation, and pgs for the fast detector simulation. jets are defined in cone algorithm with @xmath83. we apply the same kinematical cuts as those denoted in ref. the reconstructed jet momenta are rescaled so that the dijet invariant - mass has correct peak at the resonance masses. the simulation result for the case of color - octet scalars @xmath51 with @xmath84 coupling is shown in fig. inclusive @xmath49 production cross - section at the tevatron is estimated to be 2 pb (without multiplying @xmath85-factor). for other cases, we obtained similar distribution. dijets events at the tevatron. cdf data taken from ref. @xcite is shown with our mc simulation results for the color - octet scalar model with @xmath86 (solid) and the standard - model @xmath87 contribution (dashed). [cdf]] we list the central values of yukawa couplings for each case in the following, @xmath88 we see that the yukawa couplings are of order @xmath89 for color - octet scalars, but close to @xmath90 for diquark scalars. the sizeable yukawa couplings for the diquark scalars come from the fact that the tevatron is a @xmath91 collider therefore the production of diquark must pick up one sea - quark whose distribution function is suppressed. the large values for @xmath92 and @xmath93 couplings are required since the production cross - section is suppressed due to the cabibbo mixing suppressed couplings to @xmath80 and @xmath81 quarks and the suppressed @xmath48 or @xmath27-quark parton distribution inside a proton. the difference between them is mainly due to being a triplet or a singlet under @xmath40. two comments are in order about the sizeable colored scalar yukawa couplings which may cause problems in decay widths and constraints from direct resonant search for these scalers, at experiments such as at the ua2 @xcite. first, the decay widths of these scalars are less than 1 gev for color - octet cases, and a few to several gev for diquark scalars, where the flavour structure of the yukawa couplings of the scalars to quarks is assumed to those determined in the next section. these decay widths are small enough to regard the width of the observed dijet resonance as the consequence of the resolution of the jet momentum measurements. second, inclusive production of the scalars which couple to quarks are constrained by the two - jet invariant mass spectrum measurement in the ua2 experiment @xcite. for @xmath94 gev, the cross - section times the branching ratio to two jets is excluded for @xmath95 pb. the couplings in eq.(8) provide values for @xmath96 in pb as @xmath97.\]] from the above values we see that the couplings in eq.(8) can not be excluded by the ua2 measurement. the cross section for @xmath78 is on the border of the constraint. we can estimate the @xmath98dijet production cross - section at the tevatron. for the couplings in eq. ([central - value]), the @xmath99+@xmath18 production cross - sections are estimated to be @xmath100.\]] the largest @xmath101 is about 0.7 pb, which is 23% of @xmath102 within the sm estimation in leading - order. this fraction is similar to @xmath103. therefore, although there have been no statistically significant signal on the diboson production in @xmath104 mode at the tevatron yet, @xmath105 production should be more carefully studied. Fcnc constraints on colored scalars from the previous section we see that the yukawa couplings of these colored scalars to the first and second generations are much larger than that of the usual higgs in order to explain the cdf @xmath0dijets excess. therefore we need to check whether such large yukawa couplings are consistent with data. we now study constraints from new fcnc interactions by colored scalars which may induce sizeable meson - antimeson mixing. we consider each case separately in the following. some phenomenological studies of the octet - doublet scalar can be found in @xcite. here we study the constraints from the mixing of mesons for large yukawa coupling to the first generation of quarks. for @xmath51 couples with @xmath106 and @xmath107, we have @xmath108 if @xmath109 is not diagonal, exchange of @xmath110 will induce large fcnc effects at tree level, such as @xmath111-@xmath112 mixing, making the model inconsistent. even if @xmath109 is diagonal, exchange of @xmath113 at loop level can also induce fcnc interaction which may result in too large @xmath114-@xmath115 and @xmath116-@xmath117 mixings. to minimize possible fcnc interaction, we will work with a special case where @xmath118 (where @xmath119 is a unit matrix) for illustration (@xmath120). -@xmath115 mixing induced by @xmath109 coupling. dashed line represents the octet - doublet scalar propagation. [fcnc]] in the case where only @xmath109 coupling is turned on (@xmath121), the @xmath114-@xmath115 mixing operator @xmath122 is induced by @xmath1-@xmath18 and @xmath18-@xmath18 box diagrams shown in fig. [fcnc]. in order to show the @xmath18 contribution, we define the following quantities : @xmath123 where @xmath124, @xmath125 (@xmath126 is the mass of @xmath18 boson), and @xmath127. the loop functions @xmath128 and @xmath129 are given by @xmath130 we obtain the @xmath18 contribution of @xmath114-@xmath115 mixing amplitude as @xmath131 where @xmath132 is a kaon decay constant, @xmath133 is a kaon mass, and @xmath134 is a bag parameter from the matrix element of the @xmath122 operator between @xmath85 mesons @xcite. here the @xmath135 and @xmath136 terms are from the @xmath1-@xmath18 box diagram while the @xmath137 term is from the @xmath18-@xmath18 box diagram. the kaon mass difference is obtained by @xmath138. inserting the value @xmath139 given in eq. ([central - value]) (under the current assumption, @xmath140), we find that the @xmath135 term gives dominant contribution, which corresponds to the @xmath1-@xmath18 box diagram with charm quark mass insertions. the short distance sm contribution has uncertainty which mainly comes from the charm mass and qcd correction. for the @xmath114-@xmath115 system, the short distance sm contribution with the next - to - leading order qcd correction can fill roughly 80% of the experimental result, @xmath141 gev @xcite. though the long distance contribution is hard to be estimated, the total mixing amplitude in sm can be consistent with the experiment. we exhibit the ratio of the leading order @xmath18 contribution and the short distance sm contribution, which is free from the hadronic uncertainty, @xmath142 here we see that the contributions from the octet scalar is at 12% of the short distance sm contribution of the mixing amplitude for the value @xmath143 suggested by the @xmath0dijet excess, and therefore, consistent with the experimental result of kaon mass difference. the imaginary part of the mixing amplitude gives indirect cp violation in @xmath114-@xmath115 mixing. we find that @xmath144 is less than 2%, and thus it is consistent with experiments. the mixing amplitudes of @xmath145-@xmath146 are obtained just by replacing @xmath147, @xmath132, @xmath133 and @xmath134 properly, and they are found to be at the level of less than 1% of the standard model prediction as well as the experimental result. when @xmath51 couples to @xmath148 and @xmath107, we have @xmath149 in this case, to avoid large tree level fcnc, one is forced to have @xmath150 to be diagonal. also similar to the previous case to avoid potential large one loop fcnc, we make our illustration, of the form (@xmath151) @xmath152 with @xmath153. in this case, the @xmath18 contribution is @xmath154 where @xmath155 and @xmath156 operators are the bag parameters from the matrix elements of the operators, @xmath157 and @xmath158 between @xmath85 mesons, respectively @xcite. the contribution is small (@xmath159% of the experimental value) for the value for the yukawa coupling chosen from @xmath0dijet excess. the situation is the same for the @xmath160-@xmath161 mixing amplitude. the @xmath111-@xmath112 mixing amplitude is obtained by exchanging @xmath162, and replacing @xmath163 and @xmath164 in the expressions of @xmath114-@xmath115 mixing. the mixing amplitude of @xmath111-@xmath112 induced by the @xmath165 coupling is found to be very small at the level of less than @xmath166 compared to the short distance sm contribution. on the other hand, @xmath111-@xmath112 mixing amplitude induced by the @xmath167 coupling receives a large @xmath1-@xmath18 box contribution (corresponding to the @xmath135 term), which is comparable to the short distance sm one. however, the short distance sm contribution of @xmath111-@xmath112 mixing is tiny compared to the experimental result, @xmath168 it is expected that long distance contribution in the sm can produce the experimental value. for our purpose, it is therefore safe to say that @xmath84 coupling required by the cdf @xmath0dijet data can satisfy constraints from @xmath111-@xmath112 mixing data note that the octet - doublet scalar with the form @xmath169 can decay into @xmath170, giving 20% of @xmath171-jet pair fraction in @xmath49 events. one can also try to keep both @xmath109 and @xmath84 simultaneously non - zero. but there is a large contribution to @xmath172 (@xmath173) amplitude proportional to @xmath174 (@xmath175). this combination must be small resulting in one of the @xmath176 to be much smaller than the other. this virtually goes back to the previous two cases studied. one may be able to forbid one of the @xmath177 couplings by some discrete @xmath178 symmetry, such as @xmath179 and @xmath180 and @xmath181 to eliminate @xmath182. but to have @xmath183 proportional to unit chosen earlier, this raises a question how natural the choice is. while this may be achievable by some flavor symmetry to enforce the special texture form, such endeavor is beyond scope of this work and we will confine ourselves to phenomenological study only. we conclude that the cases with @xmath51 couples to either @xmath106 only or @xmath148 only is a phenomenologically viable model. now let us study if the color sextet or triplet diquarks are allowed. some phenomenological studies of the color sextet and triplet scalars can be found in @xcite. the sextet diquark @xmath52 with yukawa couplings required to explain cdf @xmath0dijet excess will lead to too large mixing in @xmath111-@xmath112 and @xmath114-@xmath115 in contradiction with data. from eq. ([di - int]) one can see that exchange of @xmath185 at tree level can generate a mixing amplitude for @xmath111-@xmath112 if @xmath186. the constraint is estimated as @xmath187. tree level mixing for @xmath188 is also generated by @xmath189 exchange. these mixing contributions can be eliminated by letting @xmath190 by choosing @xmath191, where @xmath192 is a cabibbo mixing angle. however, under the exact cancellation of the tree level contributions, the loop level contributions are too large. one then has to arrange cancellation between the tree and one loop contributions. this may represents a problem of fine tuning. although this appears quite unnatural and harder to realize for building a model compared to the octet case, from purely phenomenological point of view it is not ruled out yet. the other diquarks @xmath193 do not induce the tree - level meson - antimeson mixing, but can be generated at the 1-loop level through the box diagram. because the diquark @xmath78 is an @xmath40 singlet and @xmath65 is anti - symmetric. the cdf @xmath0dijet excess requires a large value of @xmath92. for illustration, let us consider a simple case with @xmath195 and @xmath196 in the @xmath197 basis : @xmath198 the contribution to @xmath114-@xmath115 mixing amplitude from @xmath78 is @xmath199 for @xmath200 and @xmath201 gev as required by the cdf @xmath0dijet data, it gives twice that of the short distance sm contribution constructively, due to the enhancement factor @xmath202. the coupling @xmath203 can also cause an excess of strangeless charm decay, such as @xmath204. the interaction generates a strangeless charm decay operator, @xmath205,\]] where we use @xmath206 the contribution interfere with the standard model amplitude at 40% (including the color suppressed process) for @xmath207 and @xmath208 gev, which contradict with the experimental result of the branching ratio @xcite : br(@xmath209. we conclude that @xmath78 is problematic to explain the cdf @xmath0dijets excess, though the quantities can be adjusted by choosing the possible couplings to right - handed quarks. in the case of @xmath53, a similar analysis as in the previous section can be done by supposing @xmath211 and @xmath212 in the @xmath197 basis. in this case, @xmath1-@xmath18 box diagram for the @xmath114-@xmath115 mixing vanishes due to the color anti - symetricity, @xmath213 and only @xmath18-@xmath18 box diagram contributes. as a result we have : @xmath214 for @xmath215 and @xmath216 gev which is chosen from the @xmath0dijet excess, the box contribution is the same size of the short distance sm contribution. the coupling can also contribute to the strangeless charm decay width about 20%. while those quantities may be allowed within hadronic uncertainty, they nevertheless push this scenario to the allowed boundary. in the case of @xmath76 diquark, the diquark coupling is a symmetric matrix, @xmath218 the @xmath1-@xmath18 box contribution also vanishes due to the color anti - symmetricity. the contribution to @xmath114-@xmath115 mixing amplitude is @xmath219 where @xmath220 if we take @xmath221, @xmath222 and @xmath208 gev, the @xmath18 contribution is twice as much as the short distance sm contribution. however, @xmath223 is a symmetric matrix, and thus, one can choose @xmath224 and @xmath225 to be zero to eliminate the flavor changing process. (under the choice, @xmath226.) therefore, the mixing amplitudes can be consistent with experiments. there is no contribution to strangeless charm decay in this choice. we note that the color triplet bosons, @xmath76 and @xmath53, can also have a leptoquark coupling @xmath227 in general, and it causes a severe problem of inducing too rapid nucleon decays. one can avoid the rapid proton decays by introducing a symmetry @xcite, allowing a milder baryon number violating process, such as neutron - antineutron oscillations which can be tested at near future experiments @xcite. we summarize the results in table [table1] for the yukawa couplings of the colored scalars and fcnc constraints. we conclude that there are scenarios which are consistent with fcnc data. other ways of distinguishing these scenarios should be studied. in the next section, we will study possible signatures at the rich and lhc..list of eligibility from the fcnc constraints of the couplings to explain the cdf @xmath1+dijets. [cols="^,^,^,^,^,^,^",options="header ",] the triplet diquark @xmath76 can couple with right - handed quarks by yukawa - type interaction, @xmath228 the coupling @xmath229 is generally independent from the couplings to the left - handed quarks. although the @xmath49 production cross - section is unchanged by the right - handed quark coupling, the single @xmath18 production cross - section can be increased. in fig. [rhic], we also show how the cross - section would change by introducing the coupling to right - handed quarks for @xmath76 case. assuming the same size coupling @xmath230 to the first generation quarks, @xmath231 scatterings give the same size cross - section as the @xmath232 scatterings, as easily expected. couplings to right - handed quarks are also possible for @xmath194 case, but forbidden in color - octet, @xmath233 and @xmath210 cases. note that @xmath49 production cross - section has no dependence on the couplings to the right - handed quarks, but @xmath105 production cross - section has small dependence on the couplings to the right - handed quarks, because @xmath234 couplings are smaller than the @xmath235 couplings. at the rhic, using the polarization of the proton beam @xcite, it is possible to test the chiral structure of the diquark couplings to quarks. the partonic spin asymmetry, defined as @xmath236 where the subscripts describe the parton s helicity (chirality), is found to be @xmath237 for the case we consider. thus, it can probe the ratio of the left - handed coupling @xmath238 which is fixed by the cdf @xmath0dijet excess, and the right - handed coupling @xmath229 which is unknown yet. using the knowledge of the polarized parton distribution functions of quarks in valence distribution regions, it is possible to extract the partonic spin asymmetry from the hadronic observables. however, the detailed study is beyond the scope of this paper. and @xmath239 gev. @xmath240 pb@xmath6 of the integrated luminosity is assumed. background event (dashed) is estimated by the 2@xmath2412 qcd processes without @xmath85-factor correction. signal events in @xmath76 case are estimated without couplings to right - handed quarks (solid), and with couplings to right - handed quarks with @xmath242 (dotted). [rhic]] at the lhc, @xmath1+@xmath18 or @xmath99+@xmath18 process followed by @xmath243 decay can be the signal again. the production cross - sections at the lhc with @xmath244 tev are also listed in table [table2]. the expected major backgrounds are similar to those at the tevatron ; @xmath245jets, @xmath246 and single - top production. detailed studies for the signal - to - background analysis at the lhc can be found in refs. @xcite, for example. the @xmath1+@xmath18 or @xmath99+@xmath18 processes have large cross - sections as can be seen from table [table2], especially for the diquark - type models. following the study in ref. @xcite, by taking into account the qcd @xmath0jets background, the @xmath247 signal in the @xmath76 case can be seen with the signal - to - background ratio of @xmath248 for the events with @xmath249 [gev]. assuming the total detection efficiency to be @xmath250, an expected integrated luminosity for the @xmath251 discovery is @xmath252 [fb@xmath6] in this case. for the color - octet scalar cases, the signal - to - background ratio is estimated to be @xmath254, therefore a better understanding of the background events is needed to find the signal. Summary we have studied the possibility of explaining the cdf @xmath0dijet excess by introducing colored scalar @xmath18 bosons. being colored scalars, through coupling to two quarks, they naturally decay into dijet which provides one of the key feature of the @xmath0dijet excess. there are several colored scalars, @xmath255, @xmath256, @xmath257, which can have tree level renormalizable yukawa couplings with two quarks. not all of them can successfully explain the @xmath0dijet excess. because the @xmath0dijet excess requires a sizable coupling to the first generation of quarks compared to the higgs couplings to them, the sizable couplings must also be consistent with other existing experimental data. we have analyzed fcnc constraints from meson - antimeson data. we find that without forcing of the yukawa couplings to be some special texture forms most of the scalars, except the @xmath258, are in trouble with fcnc data. we, however, find that the @xmath259, @xmath260 and @xmath261 can be made consistent with all data. while we confined our study to phenomenological implications of these colored particle, we note that a concrete realization of their coupling is harder to achieve. even one finds a flavor symmetry to forbid certain entries of the yukawa matrices, for example the off diagonal entries of @xmath109 for the octet, it is often the case that they are induced at loop level. in this sense, all the scenarios discussed here are to be considered as fine tuned until a concrete realization is achieved. the candidate of the color triplet scalar is an @xmath40 singlet, and it also produces @xmath98dijet excess at about 1/4 of the @xmath262 process, which is not observed as a bump around 150 gev yet. we also studied some predictions for the diquark signal at @xmath263 colliders, the rhic and the lhc. if the cdf excess is the diquark origin, it may be confirmed at the early lhc study. the rhic experiment can help to distinguish the diquarks. after finishing this work, the cdf reported an updated analysis @xcite using data collected through to november 2010 corresponding to an integrated luminosity of 7.3 fb@xmath6. their results are consistent with their early analysis @xcite and increased the significance to 4.1@xmath7. recently d0 collaboration also reported their results of an analysis @xcite with an integrated luminosity of 4.3 fb@xmath6. they did not find similar @xmath0diget excess. although d0 was also looking at similar excess, the methodology differs in some way which may be potentially important cause for differences. we are not in a position to decide which one may be correct which has to be settled among the experimental groups. we think that a study of implications of the cdf results is still worthy. our results are not altered by the new cdf data. p. chiappetta, j. layssac, f. m. renard and c. verzegnassi, phys. d * 54 *, 789 (1996) ; k. s. babu, c. f. kolda and j. march - russell, phys. rev. d * 54 *, 4635 (1996) ; v. d. barger, k. m. cheung and p. langacker, phys. b * 381 *, 226 (1996). m. r. buckley, d. hooper, j. kopp and e. neil, phys. d * 83 *, 115013 (2011) ; f. yu, phys. d * 83 *, 094028 (2011) ; x. -p. wang, y. -k. wang, b. xiao, j. xu and s. -h. zhu, phys. d * 83 *, 117701 (2011) ; k. cheung and j. song, phys. lett. * 106 *, 211803 (2011) ; s. jung, a. pierce and j. d. wells, phys. d * 84 *, 055018 (2011) ; m. buckley, p. fileviez perez, d. hooper and e. neil, phys. b * 702 *, 256 (2011) ; s. chang, k. y. lee and j. song, arxiv:1104.4560 [hep - ph] ; j. e. kim and s. shin, arxiv:1104.5500 [hep - ph] ; f. del aguila, j. de blas, p. langacker and m. perez - victoria, phys. d * 84 *, 015015 (2011). q. -h. cao, m. carena, s. gori, a. menon, p. schwaller, c. e. m. wagner and l. -t. wang, jhep * 1108 *, 002 (2011) ; a. e. nelson, t. okui and t. s. roy, phys. d * 84 *, 094007 (2011) [arxiv:1104.2030 [hep - ph]]. g. zhu, phys. b * 703 *, 142 (2011) ; b. dutta, s. khalil, y. mimura and q. shafi, arxiv:1104.5209 [hep - ph]. c. kilic and s. thomas, phys. d * 84 *, 055012 (2011) ; r. sato, s. shirai and k. yonekura, phys. b * 700 *, 122 (2011) ; l. a. anchordoqui, h. goldberg, x. huang, d. lust and t. r. taylor, phys. b * 701 *, 224 (2011) ; h. b. nielsen, arxiv:1104.4642 [hep - ph] ; b. bhattacherjee and s. raychaudhuri, arxiv:1104.4749 [hep - ph] ; k. s. babu, m. frank and s. k. rai, phys. rev. lett. * 107 *, 061802 (2011). t. aaltonen _ et al. _ [cdf collaboration], phys. d * 79 *, 112002 (2009).

the recent data on @xmath0dijet excess reported by cdf may be interpreted as the associated production of a @xmath1 and a new particle of mass about 150 gev which subsequently decays into two hadron jets. we study the possibility of explaining the @xmath0dijet excess by colored scalar bosons. there are several colored scalars which can have tree level renormalizable yukawa couplings with two quarks, @xmath2, @xmath3, @xmath4. if one of these scalars has a mass about 150 gev, being colored it can naturally explain why the excess only shows up in the form of two hadron jets. although the required production cross section and mass put constraints on model parameters and rule out some possible scenarios when confronted with other existing data, in particular fcnc data, we find that there are strong constraints on the yukawa couplings of these scalars. without forcing the couplings to be some special texture forms most of the scalars, except the @xmath5, are in trouble with fcnc data. we also study some features for search of these new particles at the rhic and the lhc and find that related information can help further to distinguish different models.

1105.2699

Introduction the discovery of topological insulators in 2d and 3d has recently attracted a great deal of attention @xcite (reference therein). the appearance of the gapless edge states within the bulk gap is generally believed to be a signature of the topological insulator. generally speaking, a topological insulator has gapless robust edge states while it s energy spectrum is gapped in the bulk. the bulk energy gap closes if a system is deformed adiabatically to a topologically nonequivalent system. remarkably, topological phase is not restricted to two and three dimensional systems. of special importance in the context of localization and topological phase in 1d is the aubry and andre (aa) model @xcite. a mapping of the 1d aa model to topologically - nontrivial 2d quantum hall (qh) system was constructed and used to topologically classify 1d quasicrystals @xcite. topological equivalence between crystal and quasicrystal band structures @xcite and between the fibonacci quasicrystal and the harper model were shown @xcite. recently, the parameter range for a off - diagonal gapless aa model that can not be mapped onto a qh system was explored @xcite. + the periodic table of topological insulators has been constructed only for hermitian hamiltonians. this is because of the fact that hermiticity was required in quantum mechanics to guarantee the reality of the spectrum. however, it was shown in 1998 that hermiticity requirement is replaced by the analogous condition of @xmath0 symmetry, where @xmath1 and @xmath2 operators are parity and time reversal operators, respectively @xcite. in 2010, a non - hermitian system was experimentally realized in an optical system @xcite. it is well known that a @xmath0 symmetric non - hermitian hamiltonian admits real spectrum as long as non - hermitian degree is below than a critical number. therefore, a natural question arises. does there exist a topologically nontrivial system described by non - hermitian hamiltonian? this problem was considered by some authors @xcite. hu and hughes @xcite and esaki et al. @xcite studied non - hermitian generalization of topologically insulating phase at almost the same time. the main finding of these papers is that topological states do nt exist in the @xmath0 symmetric region because of the existence of complex energy eigenvalues. in @xcite, they considered dirac - type hamiltonians and concluded that the appearance of the complex eigenvalues is an indication of the non - existence of the topological insulator phase in non - hermitian models. they discussed that finding non - hermitian topological phases without dirac - type hamiltonians could be possible. esaki et al. considered tight binding honeycomb lattice with imaginary onsite potentials and non - hermitian generalizations of the luttinger hamiltonian and kane - mele model @xcite. they found that the zero energy edge states are robust against small non - hermitian perturbation, while these states decay because of the imaginary part of eigenvalues. in 2012, ghosh studied topological phase in some non - hermitian system by changing the metric of the hilbert space i.e. modifying the inner product @xcite. schomerus considered a one dimensional non - hermitian tight binding lattice with staggered tunneling amplitude and showed that the system admits complex spectrum @xcite. another attempt has recently been made to find non - hermitian hamiltonian admitting topological insulator phase @xcite. zhu, lu and chen specifically considered non - hermitian su - schrieffer - heeger model with two conjugated imaginary potential located at the edges of the system @xcite. they found that zero energy modes are unstable and corresponding energy eigenvalues are not real in the topologically nontrivial region. in all of these papers, no example of topological edge states for a non - hermitian system with a real spectrum has been found. the existence of topological phase transition for @xmath0 symmetric non - hermitian systems is still an important question. in the present paper, we consider a non - hermitian generalization of the off - diagonal aubry andre model. in contrast to general belief, we find topological edge states in the @xmath0 symmetric region, i.e. topological edge states with real spectrum. this is the first example in the literature where topological states are compatible with @xmath0 symmetry. Model we consider tight binding description of a @xmath0 symmetric lattice @xcite. let us begin with the 1d off - diagonal aa model. this is a 1d tight binding model with modulated tunneling parameter. in addition, suppose two non - hermitian impurities are inserted in the system. the impurities are balanced gain and loss materials. we require that the non - hermitian impurities are arranged at symmetrical sites with respect to the center of the lattice, i.e., particles are injected on the @xmath3-th site and removed from the @xmath4-th site, where @xmath5 is the number of lattice sites. the hamiltonian reads @xmath6 where the parameter @xmath7 represents non - hermitian degree, @xmath8 is unmodulated tunneling amplitude, @xmath9 and @xmath10 denote the creation and annihilation operators of fermionic particles on site @xmath11, respectively. the first term in the hamiltonian is the kinetic energy from the nearest - neighbor tunneling and the last term is the non - hermitian potential due to two non - hermitian impurities. the constant @xmath12 is the strength of the modulation and @xmath13 controls the periodicity of the modulation. the modulation is periodical if @xmath13 is rational and quasi - periodical if it is irrational. as it is usual on the topic of topological insulator in 1d, the modulation phase @xmath14 is an additional degree of freedom. + below, we will find energy spectrum of this hamiltonian numerically. we adopt open boundary conditions with @xmath15 and @xmath16 being the two edge sites. note that the translational invariance of the system is broken for open boundary conditions. before going into the details, let us study the @xmath0 symmetry of the hamiltonian qualitatively. it is clear that the system has no parity symmetry when the tunneling is modulated quasi - periodically. therefore, the hamiltonian is not @xmath0 symmetric when @xmath13 is irrational. consider now periodical modulation of tunneling. for finite number of @xmath17, the particles on the two ends of the chain may not have the same tunneling to their nearest neighbors. furthermore, particles injected on site @xmath18 and to be lost from site @xmath19 may feel different tunneling depending on @xmath13 and @xmath17. therefore, we say that the system with edges does nt generally possess global @xmath0 symmetry in the usual sense. for example, for @xmath20, the system is invariant under usual @xmath0 operation when @xmath21 but not when @xmath22 and @xmath23. it is well known in the theory of @xmath0 symmetric quantum theory that a commuting anti - linear operator can be found for a non - hermitian hamiltonian if the spectrum is real @xcite. the associated anti - linear symmetry can be interpreted as @xmath0 symmetry. we will show that our system admits real spectrum for some particular cases. + [figcem], @xmath24 and @xmath25 when @xmath26 and @xmath27 sites. topological zero energy edge states appear in the whole region of @xmath14 when the site number is odd., title="fig:",width=170],, @xmath24 and @xmath25 when @xmath26 and @xmath27 sites. topological zero energy edge states appear in the whole region of @xmath14 when the site number is odd., title="fig:",width=170] [fig2], @xmath24, @xmath26 sites and @xmath28. observe that topological zero energy states appear in the unbroken @xmath29 symmetric phase if @xmath30. note that the spectrum would be real valued when @xmath31 if @xmath32.,title="fig:",width=162],, @xmath24, @xmath26 sites and @xmath28. observe that topological zero energy states appear in the unbroken @xmath29 symmetric phase if @xmath30. note that the spectrum would be real valued when @xmath31 if @xmath32.,title="fig:",width=166],, @xmath24, @xmath26 sites and @xmath28. observe that topological zero energy states appear in the unbroken @xmath29 symmetric phase if @xmath30. note that the spectrum would be real valued when @xmath31 if @xmath32.,title="fig:",width=162],, @xmath24, @xmath26 sites and @xmath28. observe that topological zero energy states appear in the unbroken @xmath29 symmetric phase if @xmath30. note that the spectrum would be real valued when @xmath31 if @xmath32.,title="fig:",width=166] [fig1], @xmath24, @xmath27 sites and @xmath28. at this value of the non - hermitian degree, topological zero energy edge states appear in the unbroken @xmath29 symmetric phase only if the non - hermitian impurities are placed either on the neighbors of the edges, i.e., @xmath33 or the neighbors of the center of the lattice, i.e. @xmath34. if @xmath35 or @xmath36, the spectrum is not real anymore. contrary to the case with @xmath35, the spectrum is complex in both topologically trivial and nontrivial regions when @xmath36.,title="fig:",width=162],, @xmath24, @xmath27 sites and @xmath28. at this value of the non - hermitian degree, topological zero energy edge states appear in the unbroken @xmath29 symmetric phase only if the non - hermitian impurities are placed either on the neighbors of the edges, i.e., @xmath33 or the neighbors of the center of the lattice, i.e. @xmath34. if @xmath35 or @xmath36, the spectrum is not real anymore. contrary to the case with @xmath35, the spectrum is complex in both topologically trivial and nontrivial regions when @xmath36.,title="fig:",width=166],, @xmath24, @xmath27 sites and @xmath28. at this value of the non - hermitian degree, topological zero energy edge states appear in the unbroken @xmath29 symmetric phase only if the non - hermitian impurities are placed either on the neighbors of the edges, i.e., @xmath33 or the neighbors of the center of the lattice, i.e. @xmath34. if @xmath35 or @xmath36, the spectrum is not real anymore. contrary to the case with @xmath35, the spectrum is complex in both topologically trivial and nontrivial regions when @xmath36.,title="fig:",width=162],, @xmath24, @xmath27 sites and @xmath28. at this value of the non - hermitian degree, topological zero energy edge states appear in the unbroken @xmath29 symmetric phase only if the non - hermitian impurities are placed either on the neighbors of the edges, i.e., @xmath33 or the neighbors of the center of the lattice, i.e. @xmath34. if @xmath35 or @xmath36, the spectrum is not real anymore. contrary to the case with @xmath35, the spectrum is complex in both topologically trivial and nontrivial regions when @xmath36.,title="fig:",width=166] our aim is to look for a topologically nontrivial phase with real spectrum for our non - hermitian hamiltonian. since the @xmath0 symmetry is broken for quasi - periodical tunneling, we consider periodical tunneling parameter. let us start with @xmath24, where the tunneling alternates at fixed @xmath14 between the two values @xmath37 and @xmath38. hence, the corresponding spectrum has two energy bands. suppose first that the system is hermitian, @xmath39. in this case, our model is reduced to the well known su - schreier - heeger model (ssh) @xcite. we plot the energy spectrum as a function of the phase @xmath14 in the fig.1. as can be seen, the system has maximum band gap at @xmath40 and the bulk gap closes and reopens as @xmath14 changes. remarkably, zero energy states appear in the spectrum. it is well known that these zero - energy states are localized around the two edges of the system @xcite. this is a signature of topologically nontrivial phase. the region of nontrivial topological phase depends on whether the number of lattice sites is odd or even. if @xmath5 is odd, zero energy edge states appear for all @xmath41. however, this is not case if @xmath5 is even. the zero energy edge state exists at @xmath42 and varying @xmath41 leads to the topological phase transitions that occur at @xmath43 and @xmath44. the region @xmath45 corresponds to topologically trivial phase. let us now study the existence of edge states if the two non - hermitian impurities are introduced into the system. we are looking for topological edge states with real energy eigenvalues. we expect that the effect of impurities on the real part of energy spectrum is not appreciable provided @xmath17 is large and @xmath7 is not much bigger than the tunneling parameter @xmath46. to check this idea and study the reality of the spectrum, we present numerical calculation for odd and even number of lattice sites. we see that zero energy states exist even in the presence of the non - hermitian impurities. however, we are not interested in zero energy states in broken @xmath0 symmetric region. we are looking for such states in the unbroken @xmath0 symmetric region. the key point here is the site position of particle injection and removal. more precisely, where we inject and remove particles has almost nothing to do with the real part of spectrum but plays a vital role on the reality of the spectrum. this can be seen clearly in the fig.2 @xmath47 and fig.3 @xmath48. let us start our discussion with the case of @xmath49, i.e., the particles are injected to the one edge and lost from the other edge. in this case, the @xmath0 symmetric phase is spontaneously broken and the spectrum becomes complex in the topologically nontrivial region whereas the spectrum is real as long as @xmath7 is below than a critical number in the topologically trivial region. as a result, we say that the spectrum is complex in the whole region of the phase @xmath50 (in the region @xmath45) when @xmath17 is an odd (even) number. secondly, let us now suppose that the gain / loss occurs at @xmath51, where @xmath52 are positive integers (@xmath53 for @xmath24). it is remarkable that the spectrum becomes real in the whole region of the phase provided that non - hermitian degree @xmath7 is below than a critical number, @xmath54. the critical numbers for even and odd values of @xmath17 are almost the same and slightly changes if we increase @xmath17. however, it decreases with the modulation strength @xmath55 and vanishes if @xmath56. at fixed @xmath5 and @xmath55, @xmath57 decreases also with increasing @xmath58. the maximum value of @xmath54 occurs when the non - hermitian impurities are on the nearest neighbors of the edges, i.e., at @xmath33. (@xmath59 for @xmath60 and decreases with increasing @xmath12). as a final step, we numerically confirm that the zero energy states are localized around the two edges of the lattice. the main finding of this paper is that our system has nontrivial topological edge states in the unbroken @xmath0 symmetric region. to this end, we note that @xmath54 is either very small or zero when @xmath3 is not equal to @xmath61. the only exception is the one when the impurities are on the two symmetrical neighboring sites with respect to the center of the lattice. in this case, @xmath62 when @xmath26 and @xmath63 when @xmath27. if we change @xmath17, the critical value @xmath57 changes drastically. + we show that the position of non - hermitian impurities has dramatic effect on the reality of the spectrum for @xmath24. we can extend our previous analysis for other values of @xmath64, where @xmath65 is a positive integer. in this case, energy spectrum is split into @xmath66 bands. we stress that topological zero energy edge modes do nt exist when @xmath66 is an odd number. below we first study the reality of the spectrum when @xmath66 is an odd number and then discuss topological phase when @xmath66 is an even number. as an example consider @xmath67, where there are three bands that are crossed as the phase @xmath14 sweeps from zero to @xmath68. in the presence of impurities, the real part of the spectrum is almost left unchanged provided @xmath7 is not much bigger than tunneling amplitude. compared to the case with @xmath69, the impurity lattice sites necessary for the reality of the spectrum are shifted according to the formula @xmath70, where the critical number @xmath57 decreases with increasing @xmath61. we note that the site number @xmath17 plays also a role on the reality of the spectrum. to understand better, suppose that particles on the one edge of the system feel tunneling with amplitude @xmath37. depending on the number @xmath17, particles on the other edge of the lattice feel either the same tunneling or one of following tunnelings with amplitudes @xmath71, @xmath72. so, there are three different configurations at fixed @xmath73 and we should calculate the spectrum separately for three different consecutive numbers. as an example, we take @xmath74, @xmath26 and @xmath27 and @xmath36. we numerically see that the spectrum is real when @xmath27 unless @xmath75 (at @xmath60) and @xmath0 symmetry is broken since the spectrum is complex for @xmath74 and @xmath26. analogously, we expect a similar picture if @xmath76 since the same configuration is obtained. therefore, the spectrum is real if n equals to.@xmath77. and @xmath78, where the critical number @xmath57 slightly changes with @xmath17. suppose now that @xmath66 is an even number. we are mainly interested in this case since topological zero energy states are available. let us study the reality of the spectrum. as an example, consider @xmath79, where the impurity lattice sites necessary for the reality of the spectrum are given by @xmath80. the spectrum becomes real if @xmath81. and the critical values @xmath54 changes slightly with @xmath17 at fixed @xmath12. the critical value @xmath82 for @xmath79 (at @xmath60 and @xmath83) and decreases with @xmath66. as a result, we say that topological zero energy states exist when @xmath66 is even and we show that the corresponding spectrum becomes real under certain conditions. + it is worthwhile to point out that the topological phase in the unbroken @xmath0 symmetric phase can be broken by the next - nearest - neighbor tunneling. we introduce next - nearest - neighbor tunneling to the off - diagonal @xmath0 symmetric aa model model. the new hamiltonian reads @xmath84 where @xmath85 is given by ([mcabjs4]) and @xmath86 is the next - nearest - neighbor tunneling constant. suppose the next - nearest - neighbor tunneling parameter is site independent. without any loss of generality we assume the next - nearest - neighbor tunneling is constant. the energy of the edge states are not zero anymore and the particle - hole symmetry is lost since next - nearest - neighbor tunneling contributes the energy spectrum perturbatively. introducing non - hermitian impurities into the system breaks the @xmath0 symmetry spontaneously in the presence of next - nearest - neighbor tunneling. although edge states are still robust as a direct result of the topological nature, complex eigenvalues appear in the system. fortunately, the imaginary part of energy eigenvalues are very small in a broad range of @xmath7 if @xmath87. for example, for the parameters @xmath88, @xmath89, @xmath90, @xmath24, @xmath33 and @xmath27, the imaginary part of energy eigenvalues are at the order of @xmath91 when @xmath92 and @xmath93 when @xmath94. + [fig000], @xmath95 @xmath96. the site number @xmath17 changes the particular values of the phase @xmath14 where the bands are crossed.,title="fig:",width=170],, @xmath95 @xmath96. the site number @xmath17 changes the particular values of the phase @xmath14 where the bands are crossed.,title="fig:",width=170] so far, we have omitted quasi - periodical modulation of the tunneling parameter since the system is not @xmath0 symmetric when @xmath13 is irrational. here, we prove numerically that the spectrum is complex for this case. in the absence of non - hermitian impurities, the energy spectrum is broken into fractal set of bands and gaps due to the quasi periodical nature of the tunneling parameter. as shown in the fig.4, the bands remains almost the same but they are connected by the edge states as @xmath14 shifts adiabatically from zero to @xmath97. the states within the band gap are localized states either on the right or the left edge of the lattice, while the states within the bands are extended. in the presence of non - hermitian potential, we numerically check that the real part of the spectrum is almost unchanged as long as the non - hermitian degree is weak. however, we see that complex eigenvalues appear whenever @xmath7 is different from zero. we repeat our calculations for different values of site number @xmath3 and we see no signature of nontrivial topological phase with real spectrum. + as an another example where quasi - periodicity breaks the @xmath0 symmetry, let us consider a non - hermitian generalization of the diagonal aubry andre model @xmath98 where we suppose @xmath13 is irrational. for this hamiltonian, tunneling parameter is constant but the onsite potential is quasi - periodic. it is well known that the hermitian model exhibits a localization transition at @xmath99 when the modulation is quasi - periodic. all bulk eigenstates are extended for @xmath100 and localized for @xmath101. the energy spectrum as a function of @xmath14 consists of fractal set of bands and within the band gap edge states appear. in the presence of non - hermitian impurities, the spectrum becomes complex. we perform extensive numerical calculation but we did not find topologically nontrivial phase with real spectrum. + we have discussed that topological zero energy edge states appear in our system when @xmath102. to understand the topological origin of the edge states for @xmath24, let us rewrite the hamiltonian (1) in the majorana basis, where the annihilation operators at site @xmath103 and @xmath104 are defined by @xmath105, @xmath106 @xcite. the two new operators @xmath107 and @xmath108 are two species of majorana fermions at site @xmath18. therefore the hamiltonian (1) without the non - hermitian term becomes @xmath109, where @xmath110. this hamiltonian describes two uncoupled identical majorana chains and the system has @xmath111 topological index when @xmath112. the majorana physics is most easily understood in the limit @xmath113. in this case, the system has two unpaired majorana operators in each uncoupled chains, @xmath114, @xmath115 and @xmath116, @xmath117. so, zero energy majorana fermions localized at the edges appear. however, non - hermitian interaction term includes a coupling term between these unpaired majorana operators if non - hermitian impurities are placed at the edges, @xmath35. therefore the corresponding eigenvalues become complex valued. if the non - hermitian impurities are placed such that @xmath33, then the non - hermitian term of the hamiltonian (1) in the majorana basis contains a coupling term @xmath118. since it does nt contain unpaired majorana operators, the system admits real energy eigenvalues if the non - hermitian degree is smaller than a critical value. + to sum up, we have studied the non - hermitian generalization of aubry andre model. in this paper, we have shown a novel topological aspect for the non - hermitian generalization of the aubry andre model. we have shown that the system can undergo a topological phase transition in the @xmath0 symmetric region by varying the lattice position of the non - hermitian impurities. we think photonic crystals could be used to observe the topological edge states for our model. our system is the first example in the literature and it is worth to look for other topologically nontrivial @xmath0 symmetric systems with real spectra. another important task is to extend the periodic table of topological insulator to non - hermitian hamiltonians. 0 m. z. hasan and c. l. kane, rev. * 82 * 3045 (2010). s. aubry and g. andre, ann.. soc. * 3 * 133 (1980). y. e. kraus, y. lahini, z. ringel, m. verbin, and o. zilberberg, phys. * 109 * 106402 (2012). lang and s. chen, phys. b * 86 * 205135 (2012). kevin a. madsen, emil j. bergholtz, and piet w. brouwer, phys. b * 88 * 125118 (2013). y. e. kraus and o. zilberberg, phys. lett. * 109 * 116404 (2012). sriram ganeshan, kai sun, and s. das sarma, phys. lett. * 110 * 180403 (2013). li - jun lang and shu chen, j. phys. b : at. mol. 47 * 065302 (2014). mor verbin, oded zilberberg, yaacov e. kraus, yoav lahini, and yaron silberberg, phys. 110 * 076403 (2013). ryan barnett, phys. a * 88 * 063631 (2013). indubala i. satija and gerardo g. naumis, phys. b * 88 * 054204 (2013). huaiming guo, phys. lett. a * 378 * 1316 (2014). itzhack dana, phys. b * 89 * 205111 (2014). c. m. bender and s. boettcher, phys lett. * 80 *, 5243 (1998). c. e. ruter, k. g. makris, r. el - ganainy, d. n. christodoulides, m. segev and d. kip, nat. phys. * 6 *, 192 (2010). yi chen hu, taylor l. hughes, phys. rev. b * 84 * 153101 (2011). kenta esaki, masatoshi sato, kazuki hasebe, and mahito kohmoto, phys. b * 84 * 205128 (2011). pijush k. ghosh, j. phys. : matter * 24 * 145302 (2012). henning schomerus, opt. lett. * 38 * 1912 (2013). baogang zhu, rong lu, and shu chen, phys. rev. a * 89 * 062102 (2014). julia m. zeuner, et. al., arxiv 1408.2191. c. yuce, phys. lett. a * 378 * 2024 (2014). andrey a. sukhorukov, sergey v. dmitriev, sergey v. suchkov, and yuri s. kivshar, opt. * 37 * 2148 (2012). o. bendix, r. fleischmann, t. kottos, b. shapiro, phys. lett. * 103 * 030402 (2009). w. h. hu, l. jin, y. li, and z. song, phys. a * 86 * 042110 (2012). j. wu and x.- t. xie, phys. a * 86 * 032112 (2012). giuseppe dellavalle and stefano longhi, phys. a * 87 * 022119 (2013). o. bendix, r. fleischmann, t. kottos, b. shapiro, j. phys. theor. * 43 * 265305 (2010). c. m. bender, m. v. berry, a. mandilara, j. phys. a : math. theor. * 35 * l467 (2002). w. p. su, j. r. schrieffer, a. j. heeger, phys. lett. * 42 * 1698 (1979).

in this work, we consider a tight binding lattice with two non - hermitian impurities. the system is described by a non - hermitian generalization of the aubry andre model. we show for the first time that there exists topologically nontrivial edge states with real spectra in the @xmath0 symmetric region. non - hermitian hamiltonian, @xmath0 symmetry, topological phase

1502.07160

Introduction: statement of results, sketch of proof and plan of the paper we study the problem to find solutions of _ non - homogeneous _ double - confluent heun equations that are monodromy eigenfunctions. our study is motivated by applications to nonlinear equations modeling the josephson effect in superconductivity. the main results, their existence and uniqueness (theorems [noncom] and [xi=0]) are stated in subsection 1.1. applications to monodromy eigenfunctions and eigenvalues of homogeneous double confluent heun equations and to nonlinear equations modeling josephson effect are presented in subsection 1.1, sections 4 and 5. each eigenfunction is the product of a monomial @xmath13 and a function @xmath14 holomorphic on @xmath15. the heun equation is equivalent to recurrence relations on the laurent coefficients of the function @xmath16. the proofs of the above - mentioned results are based on studying the latter recurrence relations. we prove existence and uniqueness theorem [cons] for converging solutions of a more general class of recurrence relations (stated in subsection 1.2 and proved in section 2). its proof is based on ideas from hyperbolic dynamics and a fixed point argument for appropriate contracting mapping. we consider the family of double confluent heun equations @xmath17 @xmath18 our goal is to study existence of its eigenfunctions with a given eigenvalue @xmath19, @xmath5 : solutions of the form @xmath20 to do this, we study the non - homogeneous equations of type @xmath21 one of our main results is the next theorem [noncom] for every @xmath22, @xmath23 there exist holomorphic functions @xmath24 on a neighborhood of zero, @xmath25 such that the functions @xmath26 satisfy equations ([nonhom]) for appropriate @xmath27, @xmath28. the functions @xmath29 are unique up to constant factor (depending on the parameters), and they are entire functions : holomorphic on @xmath30. for every sign index @xmath31 the corresponding vector @xmath32 is uniquely defined up to scalar factor depending on parameters. the above constant factors can be chosen so that both @xmath29 and @xmath33 depend holomorphically on @xmath22 and @xmath24 are real - valued in @xmath34 for real parameter values. [cordon] let @xmath22. the corresponding equation ([heun]) has a monodromy eigenfunction with eigenvalue @xmath19, @xmath5, if and only if the corresponding vectors @xmath35 are proportional : @xmath36 theorem [noncom] will be proved in the next subsection and section 2. the corollary will be proved in the next subsection. the explicit formulas for the functions @xmath29 and @xmath33, together with an explicit form for equation ([schivka]) will be given in section 3. [xi=0] for every @xmath37 and @xmath38 there exists a unique function @xmath39 (up to constant factor) holomorphic on a neighborhood of zero such that @xmath40. the latter constant @xmath41 depends only on the parameters @xmath42. theorem [xi=0] will be proved in the next subsection. theorem [xi=0] is closely related to the question of the existence of a solution holomorphic at 0 of equation ([heun]) (such a solution is automatically entire, i.e., holomorphic on @xmath30). this question was studied by v.m.buchstaber and s.i.tertychnyi in @xcite. the existence of a solution @xmath43 from theorem [xi=0] and explicit expressions for @xmath43 and the corresponding function @xmath44 (analytic in @xmath38) were given in @xcite. the existence result implies that if @xmath45, then the homogeneous equation ([heun]), i.e., @xmath0 has a solution holomorphic on @xmath30. a conjecture stated by v.m.buchstaber and s.i.tertychnyi in loc. cit. said that the converse is true : if equation @xmath46 has a holomorphic solution at 0, then @xmath45. this conjecture was studied in loc. cit and @xcite, where it was reduced to a series of conjectures on polynomial solutions of auxiliary heun equations and modified bessel functions of the first kind. all these conjectures were solved in @xcite. as the next corollary shows, theorem [xi=0] implies the conjecture of buchstaber and tertychnyi immediately, without using neither polynomial solutions, nor modified bessel functions. [cxi] (*??? * theorem 3.5) equation ([heun]) has an entire solution, if and only if @xmath45, where @xmath44 is the function from loc. cit., introduced in (*??? * formula (31), p. 337) ; see also formula ([xil]) in subsection 4.3 below. let @xmath45. then the function @xmath43 from theorem [xi=0] is an entire solution of equation ([heun]) : @xmath0. conversely, let equation ([heun]) have a solution @xmath43 holomorphic at 0. if @xmath47, then there exists a holomorphic function @xmath48 on a neighborhood of zero satisfying the equation @xmath49, by theorem [xi=0]. this together with the uniqueness statement of theorem [xi=0] implies that @xmath50 up to constant factor, hence @xmath51. the contradiction thus obtained proves the corollary. equation ([heun]) is equivalent to the recurrence relations @xmath52 which can be written in the matrix form @xmath53 @xmath54 [remnoncom] a function @xmath55 satisfies equation ([nonhom]) for some @xmath56, if and only if its taylor coefficients @xmath57 satisfy ([recur]), or equivalently, ([mat1]) for @xmath58. similarly, a function @xmath59, @xmath60 satisfies ([nonhom]), if and only if its coefficients satisfy ([recur]) for @xmath61. * of corollary [cordon]. * let @xmath62 be a solution of equation ([heun]) having type ([multi]). then @xmath63 the coefficients @xmath57 satisfy ([recur]) for all @xmath64. this together with the above remark implies that the functions @xmath26 satisfy ([nonhom]). the corresponding expressions @xmath65 should cancel out, since @xmath43 is a solution of the homogeneous equation. this implies ([schivka]). conversely, let @xmath26 be solutions of ([nonhom]), and let ([schivka]) hold. then we can normalize the latter solutions by constant factors (not both vanishing simultaneously) so that @xmath66. then the function @xmath43 given by ([defef]) is a solution of equation ([heun]). this proves the corollary. as it is shown below, theorem [noncom] is implied by the following general theorem [cons] consider recurrence relations @xmath67 in unknown sequence @xmath57, where sequences @xmath68 numerated by @xmath69 satisfy the following conditions : @xmath70 @xmath71 then there exists a unique series @xmath72 with @xmath57 satisfying ([rec]) for @xmath73 and having positive converging radius. it converges on all of @xmath30. theorem [cons] will be proved in the next section. [run] in the series @xmath74 from theorem [cons] for every @xmath75 the two neighbor coefficients @xmath57, @xmath76 do not vanish simultaneously : hence, they present a point @xmath77. each pair of neighbor coefficients @xmath78 determines a unique sequence satisfying ([rec]). both statements follow from the fact that for every @xmath73 the coefficient @xmath79 is expressed as a linear combination of @xmath80 and @xmath57 by ([rec]), since @xmath81. hence, if some two neighbor coefficients @xmath82 vanish, then all the coefficients vanish, and the series is zero, a contradiction. [conv] let @xmath83, then for every @xmath84 such that @xmath85 for every @xmath86, @xmath87 there exists and unique nonzero one - sided series @xmath88 (up to multiplicative constant) converging on some punctured disk centered at 0 that satisfies recurrence relations ([recur]) (or equivalently, ([mat1])) for @xmath89. similarly, for every @xmath84 such that @xmath90 there exists and unique one - sides series @xmath91 (up to multiplicative constant) that satisfies recurrence relations ([recur]) for @xmath92 and converges outside some disk centered at 0. both series converge on the whole punctured complex line @xmath15. [exkko] let in the conditions of theorem [conv] one have @xmath93 (cf. ([defu])). then its statements hold for all @xmath84, since inequalities ([kko]) hold for all @xmath94. otherwise, if either @xmath95, or @xmath96, then the statements of theorem [conv] @xmath97 @xmath98 theorem [conv] together with remark [remnoncom] and the first statement of example [exkko] imply theorem [noncom]. * of theorems [conv] and [noncom]. * the coefficients @xmath99 of recurrence relations ([recur]) satisfy the conditions of theorem [cons] for @xmath100 (@xmath92). indeed, the asymptotics ([condi]) is obvious. inequalities @xmath81 follow from ([kko]) (respectively, ([kko2])). this together with theorem [cons] proves theorem [conv], and hence, theorem [noncom]. * of theorem [xi=0]. * let @xmath101, @xmath37. inequalities ([kko]) hold for @xmath102. therefore, there exists a unique series @xmath103 converging on a neighborhood of the origin, whose coefficients satisfy ([recur]) for @xmath73, and it converges on all of @xmath30 (theorem [conv]). the system of relations ([recur]) for @xmath73 is equivalent to the statement that @xmath104. this proves theorem [xi=0]. our results are motivated by applications to the family @xmath105 of nonlinear equations, which arises in several models in physics, mechanics and geometry : in a model of the josephson junction in superconductivity (our main motivation), see @xcite ; in planimeters, see @xcite. here @xmath106 is a fixed constant, and @xmath107 are the parameters. set @xmath108 the variable change @xmath109 transforms ([josbeg]) to a non - autonomous ordinary differential equation on the two - torus @xmath110 with coordinates @xmath111 : @xmath112 the graphs of its solutions are the orbits of the vector field @xmath113 on @xmath114. the _ rotation number _ of its flow, see @xcite, is a function @xmath115 of parameters. it is given by the formula @xmath116 where @xmath117 is an arbitrary solution of equation ([jostor]). the _ phase - lock areas _ are the level subsets of the rotation number in the @xmath107-plane with non - empty interior. they have been studied by v.m.buchstaber, o.v.karpov, s.i.tertychnyi et al, see @xcite and references therein. each phase - lock area is an infinite chain of adjacent domains separated by _ adjacency points_. the description of their coordinates as solutions of analytic functional equations was conjecturally stated by v.m.bushstaber and s.i.tertychnyi in @xcite and proved by the authors of the present paper in @xcite. namely, the family of non - linear equations was reduced in @xcite to two families of second order linear differential equations of double confluent heun type : equation ([heun]) and the equation @xmath118 the latter equation is obtained by writing equation ([heun]) in terms of the parameter @xmath119 and changing sign at @xmath120. in the case, when @xmath121, buchstaber and tertychnyi have shown that the adjacencies correspond exactly to those parameter values, for which equation ([heun]) has a non - trivial holomorphic solution at 0 (which is automatically an entire solution : holomorphic on @xmath30) ; see the statement in (*??? *, paragraph 2) and the proof in (*??? * theorem 3.3 and subsection 3.2). they have explicitly constructed a family of holomorphic solutions for parameters satisfying an explicit functional equation @xmath122, see corollary [cxi]. they have conjectured that the latter functional equation describes the adjacencies completely. they have reduced this conjecture to another one saying that if equation ([heun2 *]) has a polynomial solution, then equation ([heun]) does not have an entire solution. later they have reduced their second conjecture on polynomial solutions to the third one saying that appropriate determinants formed by modified bessel functions of the first type do not vanish on the positive semiaxis. the latter conjecture together with the other ones were proved in @xcite. the above - mentioned conjecture of buchstaber and tertychnyi on functional equation describing the adjacencies follows from corollary [cxi] and their correspondence to entire solutions of heun equations. v.m.buchstaber and s.i.tertychnyi have constructed symmetries of double confluent heun equation ([heun]) @xcite. the symmetry @xmath123, which is an involution of its solution space, was constructed in @xcite. it corresponds to the symmetry @xmath124 of the nonlinear equation ([josbeg]) ; the latter symmetry was found in @xcite. in @xcite they have found new nontrivial symmetries in the case, when @xmath125 and equation ([heun2 *]) does not have polynomial solutions. everywhere in the paper by _ solution @xmath126 (or @xmath127) of linear recurrence relation @xmath128 we mean a (one- or two - sided) sequence of complex numbers @xmath57 satisfying the relation for all @xmath100 (respectively, @xmath92). (here one may have two - sided infinite sequences.) if in addition, the power series @xmath74 converges on some annulus centered at 0 (for all the relations under consideration, this would automatically imply convergence on all of @xmath15) then the formal solution under question is called simply _ a solution _ : the adjective `` converging '' is omitted for simplicity. in section 3 we write down explicit formulas for solutions of recurrence relations ([recur]) using the proof of theorem [cons]. then in section 4 we deduce explicit functional equations satisfied by monodromy eigenvalues of double confluent heun equations (explicit versions of corollary [cordon]). in section 5 we apply results of sections 3 and 4 to phase - lock areas in the model of josephson effect. it is known that the ratio of the monodromy eigenvalues of equation ([heun]) equals @xmath129 and their product equals @xmath130. this together with results of section 4 implies an explicit functional equation satisfied by non - integer level curves of the rotation number as a function of parameters (theorem [rhonon] in subsection 5.1). the union of boundaries of the phase - lock areas coincides with the set where the monodromy has multiple eigenvalue and is described by the condition that the monodromy of equation ([heun]) has eigenvalue @xmath131. this together with results of sections 3, 4 implies an explicit description of the boundaries (corollary [cboun] in subsection 5.2). open problems on phase - lock areas and possible approaches to some of them using the above description of boundaries are discussed in subsections 5.5 and 5.6. the following new result will be also proved in section 5 using results of section 4. [poteig0] let @xmath132, @xmath133, @xmath134, @xmath135, @xmath136, @xmath137, @xmath138. the double confluent heun equation ([heun2 *]) corresponding to the latter @xmath7, @xmath8 and @xmath120 has a polynomial solution, if and only if @xmath139, @xmath140 and @xmath141, and in addition, the point @xmath107 lies in the boundary of the phase - lock area number @xmath142 and is not an adjacency. in other terms, the points @xmath143 corresponding to equations ([heun2 *]) with polynomial solutions lie in boundaries of phase - lock areas and are exactly their intersection points with the lines @xmath144, @xmath141, except for the adjacencies. v.m.buchstaber and s.i.tertychnyi have shown in @xcite that if a point @xmath143 corresponds to equation ([heun2 *]) with a polynomial solution, then @xmath120, @xmath142 are integers, @xmath141 and @xmath140. for every initial condition @xmath145 there exists a unique sequence @xmath146 satisfying recurrence relations ([rec]), by remark [run]. but in general, the series @xmath147 may diverge. we have to prove that it converges for appropriately chosen unknown initial condition. to do this, we use the following trick : we run the recursion in the opposite direction, `` from infinity to zero ''. that is, take a big @xmath64 and a given `` final condition '' @xmath148. then the inverse recursion gives all @xmath149, @xmath150. it appears that the initial condition @xmath145 we are looking for can be obtained as a limit of the initial conditions @xmath151 obtained by the above inverse recursion (after rescaling), as @xmath152 ; the only condition on @xmath153 is that its projectivization @xmath154=(a_k : a_{k+1})\in\cp^1 $] should avoid some small explicitly specified `` bad region '', which contracts to the point @xmath155, as @xmath152. the projectivized inverse recursion @xmath156=(a_{k}:a_{k+1})\mapsto [q_{k-1}]=(a_{k-1}:a_k)\]] defined by ([rec]) can be considered as the dynamical system @xmath157 where for every @xmath158 and @xmath159 one has @xmath160 it appears that for every @xmath64 large enough @xmath161 has a strongly attracting fixed point tending to @xmath162 and a strongly repelling fixed point tending to @xmath155, as @xmath152. this together with the ideas from basic theory of hyperbolic dynamics implies that the fixed point @xmath163 of the transformation @xmath164 should have a unique unstable manifold : an invariant sequence @xmath165)$] converging to @xmath166. we show that a solution @xmath167 of recurrence relations ([rec]) gives a converging taylor series @xmath74 on some neighborhood of zero, if and only if @xmath168 $] for all @xmath64, and then the series converge everywhere. this will prove theorem [cons]. the existence and uniqueness of the above - mentioned unstable manifold is implied by the following discrete analogue of the classical hadamard perron theorem on the unstable manifold of a dynamical system at a hyperbolic fixed point. [metric] let @xmath169 be a sequence of complete metric spaces with uniformly bounded diameters. for brevity, the distance on each of them will be denoted @xmath170. let @xmath171 be a sequence of uniformly contracting mappings : there exists a @xmath7, @xmath172 such that @xmath173 for every @xmath174 and @xmath58. then there exists a unique sequence of points @xmath175 such that @xmath176 for all @xmath58. one has @xmath177 and the convergence is uniform in @xmath178 : for every @xmath179 there exists some @xmath180 such that for every @xmath181 and every @xmath182 one has @xmath183. if in addition the spaces @xmath184 coincide with one and the same space @xmath43 and the fixed points of the mappings @xmath161 tend to some @xmath185, as @xmath152, then @xmath186 the proof repeats the argument of the classical proof of hadamard perron theorem. consider the space @xmath187 of all sequences @xmath188, @xmath189, equipped with the distance @xmath190 the transformation @xmath191 is a contraction. therefore, it has a unique fixed point, which is exactly the sequence we are looking for. the second statement of the theorem on the uniform convergence of compositions to @xmath192 follows from the uniform convergence of iterations of the contracting map @xmath164 to its fixed point. in the last condition of theorem [metric] statement ([limxk]) follows by the above fixed point argument in the subspace in @xmath187 of the sequences @xmath193 tending to @xmath194, as @xmath152 : this is a complete @xmath164-invariant metric subspace in @xmath187, and hence, @xmath164 has a fixed point there, which coincides with the previous sequence @xmath193 by uniqueness. theorem [metric] is proved. Proof of theorem recurrence relations ([rec]) can be written in the matrix form @xmath195 consider the inverse matrices @xmath196 and their projectivizations @xmath197 acting on the projective line @xmath198 with homogeneous coordinates @xmath199. let us introduce the affine coordinate @xmath200 on it. for every @xmath201 we denote @xmath202 [pkconv] the transformations @xmath161 converge to @xmath203 uniformly on every closed disk @xmath204, @xmath201, as @xmath152. their inverses converge to @xmath205 uniformly on the complement of every disk @xmath206. the image of a vector @xmath207 with @xmath208 under the matrix @xmath209 is the vector @xmath210 recall that @xmath211, see ([condi]), hence, @xmath212 for all @xmath64 large enough. the latter asymptotics and formula together imply that @xmath213 uniformly on @xmath204 and prove the first statement of the proposition. let us prove its second statement. for every fixed @xmath201 and every @xmath64 large enough (dependently on @xmath214) one has @xmath215, by the first statement of the proposition. the image of a vector @xmath207 under the matrix @xmath216 is @xmath217. this together with ([condi]) implies that @xmath218 uniformly on @xmath219, as @xmath152 ; or equivalently, @xmath220 uniformly on @xmath221. the proposition is proved. * of theorem [cons]. * let @xmath222, @xmath223 denote the closed disk @xmath224 equipped with the euclidean distance. there exist a @xmath172 and a @xmath225 such that for every @xmath226 one has @xmath227 and the mapping @xmath228 is a @xmath7-contraction : @xmath229. this follows from the first statement of the proposition and schwarz lemma. the fixed point of the mapping @xmath161 tends to 0, as @xmath152, by uniform convergence (proposition [pkconv]). this together with theorem [metric] implies that there exists a unique sequence @xmath230 such that @xmath231 for all @xmath226 and @xmath232. the latter sequence corresponds to a unique sequence @xmath233 (up to multiplicative constant) such that @xmath234 ; one has @xmath235 for every @xmath236. the sequence @xmath167 satisfies relations ([rec]) for @xmath226, which are equivalent to the equalities @xmath231. it extends to a unique sequence @xmath146 satisfying ([rec]) for @xmath73, as in remark [run]. in addition, @xmath237, i.e., @xmath238, as @xmath239, by ([limxk]) and since the attracting fixed points of the mappings @xmath161 converge to 0, by proposition [pkconv]. therefore, the series @xmath240 converges on the whole complex line @xmath30. the existence is proved. now let us prove the uniqueness. let, by contradiction, there exist a series @xmath241 satisfying relations ([rec]), having a positive convergence radius and not coinciding with the one constructed above. then there exists a @xmath226 such that @xmath242, i.e., @xmath243. for every @xmath244 one has @xmath245, that is, @xmath246, as @xmath247, by the second statement of proposition [pkconv]. hence the series diverges everywhere : has zero convergence radius. the contradiction thus obtained proves theorem [cons]. Explicit formulas for solutions and the coefficients @xmath33 here we present explicit formulas for the unique converging series from theorem [conv] solving recurrence relations ([recur]). first in subsection 3.1 we provide a general method for writing them, which essentially repeats and slightly generalizes the method from (*??? *, pp. 337338). then we write them for @xmath248, and afterwards for @xmath249. here we consider a solution of general recurrence relations ([rec]) from theorem [cons]. let @xmath250, @xmath251, @xmath252 be the coefficients in ([rec]). let @xmath197 be the projectivizations of the transformations @xmath209, see ([lak]). let @xmath74 be a solution to ([rec]). recall that we denote @xmath253, in the standard coordinate @xmath254 on @xmath255 one has @xmath256. we have @xmath176, and for every @xmath64 the infinite product @xmath257 converges to @xmath192. more precisely, @xmath258, as @xmath259 uniformly on compact subsets in @xmath260, as in the proof of theorem [cons]. one can then deduce that there exists a number sequence @xmath261 such that for every @xmath64 the infinite matrix product @xmath262 converges to a rank 1 matrix @xmath263 such that @xmath264 this would be an explicit formula for the solution @xmath74. however the infinite product of the matrices @xmath209 themselves diverges, since their terms @xmath265 tend to infinity : one has to find a priori unknown normalizing constants @xmath261. to construct a converging matrix product explicitly, we will consider a rescaled sequence @xmath57, that is @xmath266 rewriting relations ([rec]) in terms of the new sequence @xmath267 yields @xmath268 @xmath269 the matrices @xmath270 converge to the projector @xmath271 our goal is to choose the above rescaling so that the infinite products @xmath272 converge : then the limit is a one - dimensional operator @xmath273 with @xmath274 being generated by the vector @xmath275. it appears that one can achieve the latter convergence by appropriate choice of normalizing constants @xmath153. we use the following sufficient conditions of convergence of products of almost projectors @xmath270. [lemkp] let @xmath276 be either a finite dimensional, or a hilbert space. let @xmath277 be a sequence of bounded operators that tend (in the norm) to an orthogonal projector @xmath278. let @xmath279 then the infinite product @xmath280 converges in the norm, and @xmath281. one has @xmath282, as @xmath152, in the operator norm, and @xmath283 for every @xmath64 large enough. fix a @xmath64 and set @xmath284 for @xmath285 ; @xmath286. one has @xmath287 the latter equality implies that @xmath288 this implies that @xmath289 now one has @xmath290 @xmath291 @xmath292 here we have used the equality @xmath293, which holds for all the projectors. the two latter formulas together with ([intk]) imply that @xmath294 the latter right - hand side being a converging series in @xmath295, the sum of the left - hand sides in @xmath295 converges and so does @xmath296, as @xmath297, in the operator norm. this also implies that the norm distance of each @xmath296 to the limit @xmath298, and in particular, @xmath299 is bounded from above by @xmath300. one has @xmath301, as @xmath297 uniformly in @xmath64, and also @xmath301, as @xmath302 so that @xmath303. this implies that @xmath273 and @xmath270 converge to the same limit @xmath304 in the operator norm, as @xmath152. for every @xmath305 one has @xmath306. hence, @xmath307, as @xmath297. therefore, @xmath308 and @xmath309. let @xmath310 be such that for every @xmath226 one has @xmath311. let us show that @xmath283 for these @xmath64. indeed, suppose the contrary : @xmath274 is strictly bigger than @xmath312. note that @xmath313 (orthogonal decomposition), since @xmath304 is an orthogonal projector. therefore, there exists a vector @xmath314 such that @xmath315. hence, @xmath316 while @xmath317, since @xmath304 is a projector. the contradiction thus obtained proves the lemma. * addendum to lemma [lemkp]. * _ let in lemma [lemkp] the operators @xmath318 depend holomorphically on some parameters so that the series @xmath319 converges uniformly on compact subsets in the parameter space. then the infinite products @xmath273 are also holomorphic in the parameters. _ the above proof implies that the sequence @xmath296 converges uniformly on compact subsets in the parameter space. this together with the weierstrass theorem implies the holomorphicity of the limit. [clem] let @xmath320 then the infinite product @xmath280 converges, and the right column of the limit product matrix @xmath273 vanishes. in the case, when @xmath321 for all @xmath64, the limit matrix @xmath273 has rank 1 for all @xmath64 : its kernel is generated by the vector @xmath275. in the case, when @xmath322 depend holomorphically on some parameters and the convergence of the corresponding series is uniform on compact sets, the limit @xmath273 is also holomorphic. * addendum 1 to corollary [clem]. * _ in the conditions of the corollary set @xmath323 then the sequence @xmath267 is a solution of recurrence relations ([bmk]) such that @xmath324, and one has _ @xmath325 * of corollary [clem]. * this is the direct application of the lemma and its addendum for the norm induced by appropriate scalar product : the latter product should make the matrix @xmath326 an orthogonal projector. the kernel @xmath274 contains the kernel @xmath312, which is generated by the vector @xmath275 ; @xmath283, i.e., @xmath327 for all @xmath64 large enough, by the lemma. in particular, the right column in each @xmath273 vanishes. now it remains to note that @xmath327 for all @xmath64, since the matrices @xmath270 are all non - degenerate : @xmath321. the corollary is proved. * of addendum 1. * consider the affine chart @xmath328 with the coordinate @xmath329 centered at @xmath330. the projectivizations @xmath161 of the linear operators @xmath331 converge to @xmath330 uniformly on compact subsets in @xmath30. hence, for every @xmath201 there exist a @xmath332 and an @xmath172 such that for every @xmath236 one has @xmath333, and @xmath161 is a @xmath334-contraction of the disk @xmath335, as in the proof of theorem [cons] in the previous section. this together with theorem [metric] implies that there exists a sequence @xmath336, @xmath337, @xmath338, as @xmath152, such that @xmath231 and @xmath339 converges to @xmath192 uniformly on compact subsets in @xmath30, as @xmath259. convergence at @xmath162 implies that @xmath340. moreover, @xmath341, since @xmath342 and the matrix @xmath270 has lower raw @xmath343. the two last statements together imply that the sequence @xmath344 satisfies recurrence relations ([bmk]). one has @xmath345, since @xmath346. this proves the addendum. [coral] consider recurrence relations ([rec]). let @xmath347 be a sequence such that the rescaling @xmath348 transforms ([rec]) to ([bmk]). let the corresponding matrices @xmath270 from ([bmk]) be the same, as in ([mkdk]). let @xmath267 be the same, as in ([setck]). then the sequence @xmath349 is a solution of relations ([rec]) such that the series @xmath240 converges on all of @xmath30. the sequence @xmath167 is a solution of ([rec]), by construction and addendum 1. one has @xmath350, since the above sequence rescaling transforms ([rec]) to ([bmk]). therefore, @xmath351, by the latter statement and since @xmath324, as was shown above. this implies the convergence of the series @xmath240 on @xmath30 and proves the corollary. here we give explicit formulas for the solution @xmath74 of recurrence relations ([recur]) converging, as @xmath248. * case 1) : @xmath352 (i.e., the conditions of theorem [noncom] hold). * let us invert matrix relation ([mat1]). we get @xmath353 @xmath354 to obtain an explicit formula for solution of relation ([recur]), we will use results of subsection 3.1. to do this, we reduce equation ([mat2]) to a similar equation with the matrix in the right - hand side converging to a projector. this is done by renormalizing the sequence @xmath57 by multiplication by appropriate constants depending on @xmath75. namely, set @xmath355 recall that the symbol @xmath356 is called the _ pochhammer symbol_. translating relations ([mat2]) in terms of the sequence @xmath267 yields @xmath357 @xmath358 @xmath359 the infinite matrix product @xmath360 converges and depends analytically on @xmath361 whenever the denominators in its definition do not vanish, by corollary [clem]. [th24] let @xmath352. set @xmath362 @xmath363 the coefficients @xmath57 satisfy recurrence relations ([recur]), for all @xmath64, and the series @xmath364 converges on all of @xmath30. the sequence @xmath267 satisfies relations ([mat3]), and @xmath324, as @xmath152, by addendum 1 to corollary [clem]. this implies that @xmath57 satisfy ([recur]). the series ([f+z]) converges on all of @xmath30, by corollary [coral]. this proves the theorem. * case 2) : some of the numbers @xmath365 or @xmath366 is an integer. * set @xmath367 note that now the product @xmath368 can be equal to zero, and thus, the sequence @xmath57 defined by ([ak]) is not necessarily well - defined. let us modify the above rescaling coefficients relating @xmath57 and @xmath267 as follows. for every @xmath369 set @xmath370 set @xmath371 the sequence @xmath167 satisfies ([mat1]), if and only if the sequence @xmath372 satisfies ([mat3]). the above formulas remain valid with the same matrices @xmath270, which are well - defined for @xmath373 : the denominators in its fractions do not vanish. therefore, the infinite product @xmath280 is well - defined for the same @xmath64 in the case under consideration. [th25] let @xmath365, @xmath295, @xmath374 be as above, @xmath270 be as in ([mat3]), @xmath280, @xmath375 @xmath376 @xmath377 the sequence @xmath378 satisfies recurrence relations ([recur]) for @xmath379. the series @xmath380 converges on all of @xmath30. the sequence @xmath267 satisfies relations ([mat3]) for @xmath373, by corollary [clem]. therefore, @xmath57 satisfy relations ([mat1]), which are equivalent to ([recur]), see the previous discussion. formula ([ako2]) is equivalent to relation ([recur]) for @xmath381. the denominator @xmath382 in ([ako2]) does not vanish. in the case, when @xmath383, this is obvious. in the case, when @xmath96, one has @xmath384, by ([kogeq]). the series @xmath385 converges on @xmath30, as in the previous subsection. the theorem is proved. here we give explicit formulas for the solution @xmath386 of recurrence relations ([recur]) with @xmath249. set @xmath387 relation ([mat1]) in new variables @xmath388 and @xmath389 takes the matrix form @xmath390 writing the latter equation with permuted order of vector components (we place @xmath391 having smaller indices above) yields the same equation with the new matrix obtained from @xmath392 by permutation of lines and columns : @xmath393 @xmath394 let us renormalize the sequence @xmath389 : set @xmath395 translating equation ([mat5]) in terms of the sequence @xmath396 yields @xmath397 @xmath398 @xmath399 [conv-] let @xmath400. let the matrices @xmath401 be as above, @xmath402 @xmath403 the sequence @xmath167 satisfies recurrence relations ([recur]), and the series @xmath404 converges on @xmath30. the above matrix product converges, and the sequence @xmath396 satisfies equation ([csm]), by corollary [clem]. this implies that the corresponding sequence @xmath389 satisfies ([hamm]), the sequence @xmath57 satisfies ([recur]) and the series @xmath405 converges, as in the previous subsection. this proves the theorem. * case 2) : some of the numbers @xmath365 or @xmath366 is an integer. * let @xmath406 the above pochhammer symbol may be not well - defined in the case, when @xmath407, @xmath408. we use the inequalities @xmath409 @xmath410 which follow immediately from ([koleq]). the sequence rescaling @xmath411 is well - defined and invertible for all @xmath412, by ([ineqm]). it differs from the previous one by multiplication by constant independent on @xmath388, and hence, transforms ([hamm]) to ([csm]), as above. the matrices @xmath401 are well - defined for @xmath413 : the denominators in their fractions do not vanish, by ([ineqm]). let @xmath414 be their products ([tmprod]) defined for @xmath413. [thko-] let @xmath401 and @xmath414 be the same, as in ([tmprod]), @xmath415 @xmath416 the sequence @xmath417 satisfies recurrence relations ([recur]), and the series @xmath418 converges on @xmath30. the proof of theorem [thko-] repeats the proof of theorem [conv-] with obvious changes. let @xmath55 and @xmath421 be the functions from ([f+z]) and ([f - z]) constructed in the two previous subsections, case 1). then @xmath422 @xmath423 where @xmath424 and @xmath425 are the same, as in ([ak]) ; @xmath426 where @xmath427 and @xmath428 are the same, as in ([cm]). the left - hand side in ([diffop]) with index `` @xmath429 '' is a taylor series with coefficients at @xmath430 being equal to the left - hand side of the corresponding recurrence relation ([recur]). the latter relation holds for all @xmath58, by construction. this implies ([diffop]) with @xmath431, @xmath432 being equal to the left - hand sides of relations ([recur]) for @xmath433 and @xmath434 respectively. this implies ([d+f]). the proof for the index `` @xmath435 '' is analogous. Application: monodromy eigenvalues here we study the eigenfunctions of the monodromy operator of heun equation ([heun]). this is the operator acting on the space of germs of solutions at a point @xmath436 by analytic extension along a positive circuit around zero. each monodromy eigenfunction with eigenvalue @xmath19 has the form of a series @xmath437 converging on @xmath15. here we write down an explicit analytic equation on those @xmath365, for which the latter solution @xmath438 of equation ([heun]) exists, i.e., there exists a bi - infinite sequence @xmath439 satisfying recurrence relations ([recur]) such that the the bi - infinite series @xmath440 converges on @xmath15. we consider different cases, but the method of finding the above @xmath365 is general for all of them. the coefficients @xmath57 with @xmath248 should form a unique converging series (up to multiplicative constant) that satisfies recurrence relations ([recur]). similarly, its coefficients with @xmath249 should form a unique converging series satisfying ([recur]). finally, the above positive and negative parts of the series should paste together and form a solution of heun equation. in the simplest, non - resonant case, when @xmath400, the pasting equation is given by ([schivka]). the coefficients @xmath57, @xmath73 satisfying ([recur]) for @xmath58 and forming a converging series are given by formulas ([ak]) ; the sequence @xmath441, @xmath442, satisfying ([recur]) for @xmath443 and forming a converging series is given by formula ([cm]). it appears that substituting the above - mentioned formulas for @xmath57 to formulas ([d+f]) and ([d - f]) for @xmath33 and then substituting the latter formulas to ([schivka]) yields a rather complicated pasting equation. to obtain a simpler formula, we proceed as follows. in the non - resonant case we extend the sequence @xmath444 to @xmath433 by putting appropriate @xmath445 instead @xmath428 (we get @xmath446) so that the longer sequence thus obtained satisfies ([recur]) also for @xmath434. similarly, we extend the sequence @xmath441 to @xmath434 by putting appropriate @xmath447 instead of @xmath424 (we get @xmath448) in order to satisfy equation ([recur]) for @xmath433. the positive and negative series thus constructed paste together to a converging bi - infinite series @xmath440 satisfying ([recur]) (after their rescaling by multiplicative constants), if and only if @xmath449 we obtain an explicit expression for equation ([aoal]). in what follows, we use the two next propositions. [monprod] the determinant of the monodromy operator of heun equation ([heun]) equals @xmath450. the monodromy matrix is the product of the formal monodromy matrix @xmath451 and a pair of unipotent matrices : the inverse to the stokes matrices, cf. * formulas (2.15) and (3.2)). therefore, its determinant equals @xmath450. another possible proof would be to use the formula for wronskian of two linearly independent solutions of equation ([heun]) from (*??? *, proof of theorem 4). it shows that the wronskian equals @xmath452 times a function holomorphic on @xmath15, and hence, it gets multiplied by @xmath453 after analytic continuation along a positive circuit around zero. recall (*??? *, lemma 1) that the transformation @xmath454 : @xmath455 is an involution acting on the space of solutions of equation ([heun]). [propdiez] let the monodromy operator of heun equation have distinct eigenvalues. then the involution @xmath456 permutes the corresponding eigenfunctions. the involution under question is a composition of transformation of a function to its linear combination with its derivative, the variable change @xmath457 and multiplication by @xmath452. let now @xmath43 be a monodromy eigenfunction with eigenvalue @xmath458. the composition of the first and second operations transforms @xmath43 to a function whose monodromy extension along positive circuit around the origin multiplies it by @xmath459 : the second operation inverses the direction of the circuit. the multiplication by @xmath452 multiplies the above result of analytic extension by @xmath450. therefore, @xmath460 is a monodromy eigenfunction with the eigenvalue @xmath461. it coincides with the second monodromy eigenvalue, since it is found by the condition that @xmath462, see proposition [monprod]. this proves the proposition. in this case the denominators in formulas ([mat3]) and ([defsm]) for the matrices @xmath270 and @xmath401 respectively are nonzero for all integer @xmath64 and @xmath388, and hence, the matrices are well - defined together with the infinite products @xmath280, @xmath463. [eight] let @xmath352. equation ([heun]) has a monodromy eigenfunction with eigenvalue @xmath19, @xmath5, if and only if @xmath464 let @xmath55 be a converging series satisfying ([recur]) for @xmath58. recall that @xmath465 set @xmath466 the sequence @xmath446 satisfies ([recur]) for @xmath73, by theorem [th24]. recall that @xmath467 set @xmath468 the sequence @xmath469 satisfies ([recur]) for all @xmath470, by theorem [conv-]. substituting the above formulas for @xmath424, @xmath471, @xmath428, @xmath472 to pasting equation ([aoal]) yields ([paste]). the theorem is proved. note that in the case under consideration one has @xmath473. [p1res] let @xmath473. then heun equation ([heun]) has a solution of type @xmath474 with @xmath365 satisfying the assumption of the subsection and the function @xmath14 being holomorphic on @xmath15, if and only if it has a solution holomorphic on @xmath15, i.e., corresponding to @xmath101. in this case the monodromy eigenvalues are 1 and @xmath475. let the above solution @xmath43 exist. then it is a monodromy eigenfunction with the eigenvalue @xmath19. the other eigenvalue equals @xmath476, by proposition [monprod]. exactly one eigenvalue equals one, by assumption. the monodromy eigenfunction corresponding to unit eigenvalue is holomorphic on @xmath15. conversely, a solution holomorphic on @xmath15 is a solution @xmath43 as above with @xmath101. the last statement of the proposition follows from the existence of unit eigenvalue and proposition [monprod]. this proves proposition [p1res]. let @xmath473. a solution @xmath43 as in proposition [p1res] exists, if and only if the recurrence relations ([recur]) with @xmath101 : @xmath477 have a solution @xmath439 such that the series @xmath440 converges on @xmath15. [propeq] every semiinfinite sequence @xmath478 satisfying equations ([recur0]) for @xmath479 (without convergence condition) satisfies the relation @xmath480 equation ([a12]) coincides with ([recur0]) for @xmath481. [corel] let @xmath482. a solution @xmath439 to ([recur0]) with the series @xmath74 converging on @xmath15 exists, if and only if the unique semiinfinite sequence @xmath483 solving ([recur0]) for @xmath484 with series @xmath485 converging on @xmath15 satisfies relation ([a12]). let @xmath486 be a semiinfinite solution of recurrence relations ([recur0]) for @xmath484. note that for every @xmath61 its coefficient @xmath57 is uniquely determined as a linear combination of the two previous ones @xmath487 and @xmath488, see ([recur0]) written for @xmath484. the same holds in the opposite sense : for every @xmath489 the coefficient @xmath57 is expressed as a linear combination of the coefficients @xmath76 and @xmath490 by ([recur0]), since @xmath482. the two latter statements together imply that @xmath491 and @xmath427 do not both vanish. therefore, the above negative semiinfinite series can be extended to positive @xmath64 as a (may be just formal) two - sided solution of ([recur0]) only in the case, when relation ([a12]) holds. let us show that in this case it does extend to a true (not just formal) two - sided solution. note that @xmath492, by relation ([a12]) and since @xmath491, @xmath427 do not vanish both and @xmath493. equation ([recur0]) with @xmath481 has zero multiplier at @xmath428 and hence, holds for arbitrary @xmath428. the same equation with @xmath433 yields @xmath494 this is a linear non - homogeneous equation on the pair @xmath145. hence, its solutions form a line @xmath495 that does not pass through the origin : @xmath492. the pairs @xmath145 extendable to semiinfinite solutions in positive @xmath64 are all proportional (uniqueness of solution up to constant factor and since for every @xmath75 the coefficient @xmath76 is uniquely determined by @xmath57 and @xmath488 via relations ([recur0])). hence, they form a line @xmath496 through the origin. let us choose @xmath145 to be the intersection of the above lines @xmath496 and @xmath497, provided they are not parallel (the case of parallel lines is discussed below). then the pair @xmath145 extends to a semiinfinite solution of relations ([recur0]) in positive @xmath64, by construction. the complete laurent series @xmath498 thus constructed is a solution to equations ([recur0]) and hence, to heun equation ([heun]). * case, when @xmath496 and @xmath497 are parallel. * in this case @xmath145 defines a solution to ([recur0]) with positive @xmath64, if and only if @xmath499. this solution extends to negative @xmath64 by putting @xmath500 for @xmath443, since relation ([a011]) for @xmath481 is equivalent to ([a12]). finally we obtain a _ converging taylor series _ satisfying ([recur0]) and hence, presenting a solution of heun equation ([heun]) holomorphic on @xmath30. this implies that @xmath501. indeed, if @xmath473, then the corresponding heun equation has no entire solution, or equivalently, the corresponding recurrence relations ([recur]) have no solution @xmath167 with @xmath500 for all @xmath443. this follows from the fact that if @xmath473, then for every @xmath64 the coefficients at @xmath79 in the corresponding relation ([recur]) do not vanish : hence, no two neighbor coefficients @xmath57 and @xmath76 of a solution vanish simultaneously. the contradiction thus obtained shows that the case of parallel lines is impossible, if @xmath473. corollary [corel] is proved. let @xmath473, @xmath502 @xmath503 heun equation ([heun]) has a solution holomorphic on @xmath15, if and only if @xmath504 set @xmath505 @xmath506 the sequence @xmath507 satisfies recurrence relations ([recur0]) for @xmath484, as in subsection 3.3, and the series @xmath508 converges on @xmath30 : here we have rewritten the formulas from subsection 3.3 for @xmath101. one has @xmath509 by definition. substituting the latter formulas and @xmath119 to ([a12]) yields ([pastreson]). this together with corollary [corel] proves the theorem. # 1#1 recall that we study the existence of solution ([solb]) of heun equation ([heun]). in the case under consideration @xmath95, and without loss of generality we can and will consider that @xmath101. in this case a solution we are looking for is holomorphic on @xmath15 and presented by a laurent series @xmath511 converging on @xmath15. without loss of generality we will also consider that @xmath512. one can achieve this by applying the transformation @xmath513 which is an isomorphism of the solution space of equation ([heun]) (written in terms of the parameter @xmath119) onto the solution space of the same equation with @xmath119 replaced by @xmath514, see (*??? *, formula (39)) : @xmath515 it sends solutions of equation ([heun]) holomorphic on @xmath15 onto solutions of equation ([heun2]) holomorphic on @xmath15. [talt] let @xmath125, @xmath121. equation ([heun]) with @xmath516 has a solution holomorphic on @xmath15, if and only if its monodromy is unipotent. this happens, if and only if equation ([heun]) satisfies one of the two following incompatible statements : \1) either it has an entire solution, i.e., holomorphic on @xmath30 ; 2) or the corresponding equation ([heun2]) has a nontrivial polynomial solution. theorem [talt] will be proved below. the sets of parameter values for which statements 1) or 2) hold were already described in @xcite. let us recall this description. to do this, consider the following matrices @xmath270, @xmath273 and numbers @xmath57, @xmath517 : @xmath518 @xmath519 @xmath520 [thol] a heun equation ([heun]) with @xmath521, @xmath121 has an entire solution, if and only if @xmath122. theorem [thol] is equivalent to corollary [cxi]. it was partly proved and conjectured in (*??? *, theorem 2) and proved completely in (*??? * subsection 3.1, theorem 3.5). for completeness of presentation let us give its direct proof without using results of loc. cit. * of theorem [thol]. * the above matrices @xmath270 and numbers @xmath57 coincide with those from ([mat3]) and ([ak01]) respectively constructed for recurrence relations ([recur]) with @xmath101, @xmath516, @xmath522 here @xmath523. (if @xmath524, then @xmath525, but the corresponding sequence @xmath57 from ([ak01]) remains the same, as in ([xil]), up to constant factor.) this together with theorem [th25] implies that the sequence @xmath167 satisfies ([b=0]) for @xmath73 and the series @xmath526 converges on @xmath30. therefore, @xmath104, and the latter constant is the left - hand side of the relation ([b=0]) corresponding to @xmath433 : that is, @xmath527. this together with the uniqueness of an entire function @xmath43 for which @xmath104 (theorem [xi=0]) implies the statement of theorem [thol]. to describe equations ([heun2]) with polynomial solutions, consider the three - diagonal matrix @xmath528 @xmath529 [tpol] @xcite a heun equation ([heun2]) with @xmath125, @xmath121 has a polynomial solution, if and only if the three - diagonal matrix @xmath530 has zero determinant. * of theorem [talt]. * let equation ([heun]) have a solution @xmath511 holomorphic on @xmath15. equivalently, its series converges on @xmath15 and the coefficients @xmath57 satisfy recurrence relations ([b=0]). for @xmath531 and @xmath481 respectively these relations take the form @xmath532 @xmath533 in particular, they do not contain @xmath534, @xmath535 therefore, given a solution holomorphic on @xmath15 of heun equation ([heun]), its laurent coefficients @xmath57 with @xmath536 should form a vector @xmath537 satisfying equations ([-l]), ([-1]) and the @xmath538 recurrence equations ([b=0]) for intermediate @xmath539. in other terms, the latter vector should be in the kernel of the three - diagonal @xmath540- matrix @xmath541 of equations ([b=0]) with @xmath542 : its line number @xmath543 consists of the coefficients of the @xmath64-th relation ; the coefficient at @xmath544 stands at the column number @xmath545. [trconj] let @xmath541 be the latter matrix, and let @xmath546 be the transposed matrix ([defh]). one has @xmath547 the proposition follows from definition. * case 1). there exists a solution of heun equation ([heun]) holomorphic on @xmath15 with @xmath548 for all @xmath549, and @xmath550. * then the series @xmath240 is an entire solution, i.e., holomorphic on @xmath30 : it satisfies relations ([b=0]) for all @xmath64 by assumption and ([-1]). it is known that in this case each solution of equation ([heun]) is holomorphic on @xmath15 and its laurent series does not contain monomials @xmath551, @xmath549 (*??? * lemma 3, statement 6). * case 2). there exists a solution of heun equation holomorphic on @xmath15 with @xmath552 for some @xmath553. * in this case the three - diagonal matrix @xmath541 of relations ([b=0]) with @xmath554 has zero determinant, by the above arguments. hence, the matrix @xmath530, whose transposed is conjugated to @xmath541 (proposition [trconj]), also has zero determinant. therefore, equation ([heun2]) has a polynomial solution (theorem [tpol]). it is known that if ([heun2]) has a polynomial solution, then the corresponding equation ([heun]) does not have entire solution (*??? * theorem 3.10). therefore, cases 1) and 2) are incompatible. theorem [talt] is proved. Applications to phase-lock areas in the model of josephson effect here we apply the above results to the family of nonlinear equations ([josbeg]) : @xmath555 we fix an arbitrary @xmath132 and consider family ([jos]) depending on two variable parameters @xmath107. the variable change @xmath556 transforms ([jos]) to differential equation ([jostor]) on the two - torus @xmath110 with coordinates @xmath111. its solutions are tangent to the vector field @xmath557 on the torus. the _ rotation number _ of the equation ([jos]) is, by definition, the rotation number of the flow of the field ([josvect]), see @xcite. it is a function @xmath558 of parameters. (normalization convention : the rotation number of a usual circle rotation equals the rotation angle divided by @xmath559.) the @xmath560-axis will be called the _ abscissa, _ and the @xmath561-axis will be called the _ ordinate. _ * : phase - lock areas and arnold tongues. * the behavior of phase - lock areas for small @xmath561 demonstrates the arnold tongues effect @xcite. the phase - lock areas are called `` arnold tongues '' in (*??? * definition 1.1). recall that the rotation number of system ([jos]) has the physical meaning of the mean voltage over a long time interval. the segments in which the phase - lock areas intersect horizontal lines correspond to the shapiro steps on the voltage - current characteristic. - the boundary of each phase - lock area @xmath564 consists of two analytic curves, which are the graphs of two functions @xmath565 (see @xcite ; this fact was later explained by a.v.klimenko via symmetry, see @xcite) ; - each phase - lock area is an infinite chain of bounded domains going to infinity in the vertical direction, each two subsequent domains are separated by one point, the separation points lying outside the horizontal @xmath560-axis are called the _ adjacency points _ (or briefly _ adjacencies) _, see fig.1 ; in the present section we obtain functional equations satisfied by non - integer level curves @xmath568 of the rotation number (subsection 5.1) and the boundaries of the phase - lock areas (subsection 5.2) using relation of equation ([jos]) to heun equation ([heun]) (recalled below) and the results on monodromy eigenvalues of heun equations from the previous section. the above - mentioned functional equations will be written in the complement to the adjacencies and the algebraic set of the parameters corresponding to the existence of a polynomial solution of equation ([heun2]). afterwards we discuss open problems and possible approaches to them using the same results on heun equations. [[reduction - to - double - confluent - heun - equations.- entire - and - polynomial - solutions - special - boundary - points - of - phase - lock - areas]] reduction to double confluent heun equations. entire and polynomial solutions : special boundary points of phase - lock areas ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ set @xmath569 the complexified equation ([jos]) is equivalent to the riccati equation @xmath570 the latter is the projectivization of the following linear equation in vector function @xmath571 with @xmath572, see (*??? * subsection 3.2) : @xmath573 this reduction to a system of linear equations was earlier obtained in slightly different terms in @xcite. it is easy to check that a function @xmath574 is the component of a solution of system ([tty]), if and only if the function @xmath575 satisfies equation ([heun]) with @xmath516 and @xmath576 [tadj] (see (*??? * theorems 3.3, 3.5)). for every @xmath132, @xmath121 a point @xmath133 with @xmath577, @xmath578 is an adjacency for family of equations ([jos]), if and only if @xmath125 and the corresponding equation ([heun]) with @xmath516 and @xmath8, @xmath7 as in ([josvect]) and ([param]) has a nontrivial entire solution, i.e., if and only if equation @xmath122 holds ; @xmath517 is the same, as in ([xil]). [rktriv] the statement of theorem [tadj] holds, if and only if the monodromy operator of heun equation ([heun]) with @xmath516 is trivial. this follows from (*??? * theorem 3) and (*??? * proposition 3.2). [roteig] let @xmath132, @xmath133, and let @xmath138 denote the corresponding rotation number. if @xmath107 does not lie in the interior of a phase - lock area, then the monodromy operator of the corresponding heun equation ([heun]) with @xmath516, @xmath579 has eigenvalues @xmath580 and @xmath581. in particular, the latter pair of eigenvalues is the same for all other points @xmath582 with the same @xmath560 and @xmath583. let @xmath458, @xmath584 denote the eigenvalues of the above monodromy operator of heun equation. the point @xmath107 does not lie in the interior of a phase - lock area. if @xmath585, then the monodromy of the corresponding riccati equation is an elliptic mbius transformation conformally conjugated to the rotation by angle @xmath586. therefore, it has two fixed points with multipliers @xmath587. the latter multipliers are ratios of the eigenvalues @xmath588, and without loss of generality we consider that @xmath589. on the other hand, @xmath590, by proposition [monprod]. this implies that the eigenvalues under question are equal to @xmath591, @xmath592. in the case, when @xmath566, the point @xmath107 lies in the boundary of a phase - lock area and the monodromy of the riccati equation is parabolic, the monodromy of the heun equation has multiple eigenvalue given by same (now coinciding) formulas. the correct sign should be the same for all the points @xmath107 in the complement of the parameter plane to the union of the interiors of phase - lock areas, by path connectivity of the latter complement and continuity. the sing is `` @xmath429 '' at each adjacency, since the corresponding monodromy is trivial (remark [rktriv]). hence, it is `` @xmath429 '' everywhere. this proves the proposition. [poteig] let @xmath132, @xmath133, @xmath134, @xmath135, @xmath136, @xmath137, @xmath138. the double confluent heun equation ([heun2]) corresponding to the latter @xmath7, @xmath8 and @xmath120 has a polynomial solution, if and only if @xmath139, @xmath593 and @xmath594, and in addition, the point @xmath107 lies in the boundary of the phase - lock area number @xmath142 and is not an adjacency. in other terms, the points @xmath143 corresponding to equations ([heun2]) with polynomial solutions lie in boundaries of phase - lock areas and are exactly their intersection points with the lines @xmath144, @xmath141, except for the adjacencies, see fig. it is known that every point @xmath143 corresponding to equation ([heun2]) with a polynomial solution lies in the boundary of the phase - lock area number @xmath142, and one has @xmath139, @xmath144, @xmath141 (*??? * corollary 6 and theorem 5). in addition, @xmath107 is not an adjacency (*??? * theorem 3.10). let us prove the converse : if @xmath107 satisfy all the latter statements, then the corresponding equation ([heun2]) has a polynomial solution. indeed, if @xmath595 lies in the boundary of the phase - lock area number @xmath142, @xmath125 and @xmath140, then the monodromy of heun equation ([heun]) is unipotent, by proposition [roteig] : the corresponding eigenvalues are equal to @xmath596. let us now suppose that @xmath107 is not an adjacency, or equivalently, equation ([heun]) does not have an entire solution. then equation ([heun2]) has a polynomial solution, by theorem [talt]. theorem [poteig] is proved. for given @xmath132 and @xmath597 set @xmath598}=\sqcup_{v\equiv\pm r(mod 2)}l_{v}.\]] we consider that @xmath599 : then each @xmath600, @xmath601 is an analytic curve, the graph of an analytic function @xmath602. here we write down analytic equations defining the set @xmath603}$] in the complement @xmath604}$], where @xmath605}=\cup_{\pm}\ { (b, a)\in\rr^2 \ | \ l=\frac b{\omega}\equiv\pm r(mod 2)\}.\label{neqm2}\]] for every @xmath606}$] the corresponding heun equation ([heun]) has a monodromy eigenfunction of the type @xmath607 by proposition [roteig]. on has @xmath608, if @xmath609}$]. therefore, the analytic subset @xmath603}\setminus\sigma_{[r]}\subset(\rr^2\setminus\sigma_{[r]})$] is described by equation ([paste]). let us write it down explicitly. the corresponding matrices @xmath270, @xmath273, @xmath401, @xmath414, see ([mat3]) and ([defsm]) are @xmath610 @xmath611 @xmath503 a point @xmath613}$] is contained in @xmath603}$], if and only if @xmath617. or equivalently, some of the corresponding monodromy eigenvalues @xmath618 equals @xmath619. the latter statement is equivalent to ([pasterho]), by theorem [eight]. this proves theorem [rhonon]. [pjcell] a point in the parameter space of equation ([jos]) lies in the boundary of a phase - lock area, if and only if the monodromy of the corresponding heun equation ([heun]) is parabolic : has multiple eigenvalue. the point under question lyes in the boundary of a phase - lock area, if and only if the flow mapping of the vector field ([josvect]) for the period @xmath559 (restricted to the coordinate @xmath620-circle) is parabolic : has a fixed point with unit derivative. the period mapping is the restriction to the unit circle of the monodromy of the corresponding riccati equation : the projectivized monodromy. parabolicity of the projectivization of a two - dimensional linear operator is equivalent to its own parabolicity. the proposition is proved. the monodromy matrix is the product of the formal monodromy matrix @xmath451 and a pair of unipotent matrices : the inverse to the stokes matrices at 0, cf. * formulas (2.15) and (3.2)). therefore, if the monodromy of a heun equation ([heun]) is a multiplication by scalar number, then the stokes matrices are trivial, and the monodromy coincides with the formal one. hence, both monodromies are scalar and given by the above diagonal matrix with unit eigenvalue. thus, they are trivial. the proposition is proved. the condition saying that the monodromy has multiple eigenvalue is equivalent to the statement that it has eigenvalue @xmath621, by proposition [monprod]. this is equivalent to the statement that there exists a multivalued solution @xmath622 of heun equation with @xmath623 : a monodromy eigenfunction with the above eigenvalue. heun equations ([heun]) satisfying the latter statement will be described below by using the following proposition. afterwards we deduce immediately the description of boundaries of phase - lock areas. the proposition follows from the fact that the involution @xmath456 sends monodromy eigenfunctions to eigenfunctions (proposition [propdiez]). * case 1 : @xmath625, @xmath626 : the monodromy eigenvalue equals @xmath627. * then the monodromy operator of heun equation ([heun]) is a jordan cell, by proposition [pjcell2]. consider the matrices @xmath628 the corresponding eigenfunction @xmath43 has the form @xmath632 equation ([heun]) is equivalent to recurrence equations ([recur]) with @xmath626 : @xmath633 the series @xmath14 should converge on @xmath15. the above matrices @xmath270 and @xmath273 coincide with those constructed in ([mat3]), and they are well - defined for all @xmath94. therefore, the coefficients @xmath57, @xmath75 are given by formulas ([ak]) up to common constant factor, by theorem [conv] : @xmath634 now we will use the condition of (anti-) invariance @xmath635 (proposition [dinvar]), which takes the form @xmath636 or equivalently, @xmath637 the free (zero power) term of the latter equation is equivalent to the relation @xmath638 which is in its turn equivalent to ([r0eql]), by ([a011]). therefore, existence of the above solution @xmath43 implies ([r0eql]). let us prove the converse : each equation ([r0eql]) implies the existence of a solution ([eheun]) of heun equation. to do this, consider the action of the transformation @xmath456 on the _ formal _ series ([eheun]) (with @xmath16 not necessarily converging). it sends formal solutions of heun equation (equivalently, formal solutions of ([l2])) to formal solutions. (the proof of symmetry of heun equation under the transformation @xmath456 uses only leibniz differentiation rule and remains valid for formal series.) the space of formal solutions is two - dimensional, and it is identified with the space of its initial conditions @xmath639. the transformation @xmath456 is its involution. its eigenvalues are equal to @xmath640, and the corresponding eigenspaces are defined by initial conditions that satisfy ([eql]). therefore, both eigenspaces are one - dimensional and are exactly characterized by equations ([eql]), since both equations ([eql]) are nontrivial. thus, _ a formal solution @xmath439 of recurrence relations ([l2]) is @xmath456-(anti)-invariant, if and only if its coefficients @xmath427, @xmath428 satisfy ([eql]) with the corresponding sign. _ fix the one - sided solution @xmath641 of recurrence relations ([l2]) for @xmath75. it satisfies ([eql]), by ([r0eql]). the sequence @xmath642 extends uniquely to a two - sided formal solution @xmath439 of ([l2]) (a priori, not necessarily presenting a converging series for @xmath249), since the coefficients at @xmath79 in ([l2]) do not vanish. the latter formal solution should be @xmath456-(anti-)invariant, by ([eql]) and the previous statement. hence, @xmath643 by ([dinv]). the series @xmath644 converges on @xmath15 : it is bounded from above by converging series @xmath645, by the latter formula. this together with the above argument proves the theorem. we are looking for a double - infinite solution @xmath652 of heun equation ([heun]) with @xmath16 holomorphic on @xmath15. that is, with @xmath57 satisfying recurrence relations ([recur]) for @xmath647, which take the form @xmath653 the above matrices @xmath270 and @xmath273 coincide with those constructed in ([mat3]), and they are well - defined for all @xmath94. therefore, the coefficients @xmath57, @xmath75 are given by formulas ([ak]) up to common constant factor, by theorem [conv]. in particular, @xmath654 the condition of (anti-) invariance under the involution @xmath456 of the solution takes the form @xmath655 or equivalently, @xmath656 the free term (zero power term) of the latter equation is @xmath657 which is equivalent to ([r1eql]). the rest of proof of theorem [heun5] is analogous to the proof of theorem [heun4]. [cboun] let @xmath658, @xmath615, @xmath121, @xmath516, @xmath578, @xmath659. the point @xmath107 lies in the boundary of a phase - lock area, if and only if one of the following four incompatible statements holds : if one of the above statements holds, then @xmath107 lies in the boundary of a phase - lock area, by proposition [pjcell] and theorems [talt], [heun4], [heun5]. conversely, let @xmath107 lie in the boundary of a phase - lock area. the monodromy of the corresponding heun equation ([heun]) is parabolic, by proposition [pjcell]. it is unipotent, if and only if some of the two incompatible statements 1) or 2) holds, by theorems [talt] and [tadj]. otherwise, the monodromy has jordan cell type with eigenvalue @xmath663. therefore, one of the statements 3) or 4) holds, by theorems [heun4] and [heun5]. statements 3) and 4) are incompatible : they correspond to heun equation ([heun]) with monodromy having multiple eigenvalue @xmath664 or @xmath665 respectively. this proves the corollary. [newex] let @xmath125. for given @xmath658 and @xmath516 the corresponding heun equation ([heun]) has a monodromy eigenfunction with eigenvalue @xmath666, if and only if the corresponding point @xmath133 lies in the boundary of a phase - lock area with a rotation number @xmath667. for @xmath125 the monodromy has unit determinant (proposition [monprod]). therefore, if it has eigenvalue @xmath666, then its other eigenvalue is also @xmath666. hence, the point @xmath107 lies in the boundary of the phase - lock area number @xmath138 (proposition [pjcell]). thus, @xmath668 for both signs, by proposition [roteig]. the latter equality holds if and only if @xmath667. conversely, if a point @xmath107 with @xmath669 lies in the boundary of a phase - lock area, and @xmath558 satisfied the above equality, then the monodromy eigenvalues are equal to @xmath666, by proposition [monprod]. the proposition is proved. here we state conjectures that are motivated by numerical simulations and theoretical results of @xcite. in what follows for every @xmath566 we denote @xmath670 the next five conjectures are due to the first author (v.m.buchstaber) and s.i.tertychnyi. [nwrem] it was shown in (*??? * theorems 1.2, 3.17) that for every @xmath566 the abscissa of each adjacency in @xmath671 equals @xmath674, @xmath125, @xmath675 ; @xmath676 if @xmath677 ; @xmath678 if @xmath679. [graph] conjecture [c1] implies conjecture [c2]. indeed, the points @xmath107 with @xmath695 large enough of the phase - lock area @xmath671, @xmath566 lie close to @xmath696, i.e., they are separated from the line @xmath694 by @xmath697. this follows from the fact that its boundary consists of graphs of two functions @xmath698 and @xmath699, as @xmath700 (follows from results of @xcite). each one of conjectures [c1], [c2] together with (*??? * theorems 1.2, 3.17) (see remark [nwrem]) imply conjecture [garl]. * a possible strategy for conjecture [c1]. * if the boundary of the phase - lock area with rotation number @xmath692 intersects the line @xmath696, then the intersection points correspond to parabolic monodromy operator of jordan cell type with both eigenvalues equal to -1 (proposition [newex]). that is, some of equations ([r0eql]) or ([r1eql]) should hold at each intersection point. [c3] let @xmath180, and let the parameter @xmath8 satisfy some of equations ([r0eql]) if @xmath625, or ([r1eql]) if @xmath701. then the double - valued eigenfunction @xmath43 from proposition [newex] gives a periodic solution of the corresponding riccati equation having rotation number between 0 and @xmath120. * a possible strategy for conjecture [c2]. * we know that for @xmath702 the statements of conjecture [c1] and hence conjecture [c2] hold (chaplygin theorem argument, see (*??? * lemma 4) and (*??? * proposition 3.4)). the adjacencies of a phase - lock area with rotation number @xmath142 can not lie on lines @xmath703 with @xmath704, see (*??? * theorem 3.17) ; this also follows from proposition [newex]. suppose that for a certain `` critical '' value @xmath705 the boundary of the phase - lock area number @xmath706 moves from the right to the left, as @xmath106 decreases to @xmath707, and touches the line @xmath703 at some point @xmath107, as @xmath708. then there are two possibilities for the corresponding heun equation : - the associated heun equation ([heun2]) (equation ([heun]) with @xmath120 replaced by @xmath514) has a polynomial solution. but this case is forbidden by buchstaber tertychnyi result (*??? * theorem 4), which states that then the corresponding rotation number can not be greater than @xmath120. - the point @xmath107 an adjacency : heun equation ([heun]) has a solution holomorphic on @xmath30. this together with the above - mentioned known fact that the boundaries of phase - lock areas are graphs of functions (remark [graph]) implies that both boundary components of the phase - lock area with rotation number @xmath709 are tangent to the line @xmath703 at the point @xmath107. a possible approach to conjecture [c4] could be studying equations ([r0eql]) and ([r1eql]) defining the boundaries and to see what happens with them when the `` non - resonant '' parameters approach the resonant ones. a first step is done below. [[description - of - boundaries - of - phase - lock - areas - near - adjacencies.- relation - to - conjecture - c2]] description of boundaries of phase - lock areas near adjacencies. relation to conjecture [c2] ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ let us write down equation ([r0eql]) on the boundaries in a neighborhood of a line @xmath713, @xmath714. let us recall the formulas for the corresponding matrices : @xmath715 equation ([r0eql]) for the boundaries is @xmath716 note that the matrices @xmath270 are analytic in a neighborhood of the line @xmath717 except for the matrix @xmath718, which has pole of order one along the latter line. one has @xmath719 set @xmath720 one has @xmath721 @xmath722 by ([xil]) and since the matrices @xmath723, @xmath724 coincide with the matrices @xmath270, @xmath273 preceding ([xil]) with @xmath717. therefore, @xmath725 now equation ([r0eql]) can be rewritten as @xmath726 taking into account asymptotics ([rzeta]) one gets asymptotic form of equation ([r0l0]) : @xmath727 now let us consider the case, when @xmath728, and let us write down equation ([r1eql]) in a neighborhood of the line @xmath713. the corresponding matrices from ([mkrk1]) are @xmath729 set @xmath730 analogously to the above calculations, we get asymptotic relation ([rzeta]). together with ([r1eql]), it implies @xmath731 conjecture [c5] together with the above remark would imply that for every @xmath125 at every adjacency in the line @xmath703 at least one boundary component of the corresponding phase - lock area (depending on the above - chosen sign) is transversal to the line @xmath703. conjecture [c5] implies that no adjacency can be born from a tangency of both boundary components with a line @xmath717, @xmath736, by transversality (the above statement). in other words, it implies conjecture [c4], and hence, conjecture [c2], by proposition [prconj2]. * open question 6. * study the degeneracy of equation ([pasterho]) on non - integer level curves of rotation number, as the latter number tends to an integer value. the level curves should degenerate to boundaries of phase - lock areas. how to retrieve equations ([r0as]) and ([r1as]) on the boundaries and equation @xmath122 on the adjacencies from asymptotics of degenerating equation ([pasterho])? d.a.filimonov, v.a.kleptsyn, i.v.schurov, v.m.buchstaber and s.i.tertychnyi have done numerical simulations studying what happens to the phase - lock areas, as @xmath689. they have observed that after appropriate rescaling of the variables @xmath107, the phase - lock areas tend to open sets (which we will call the _ limit rescaled phase - lock areas _) whose components partition the whole plane somewhat similarly to a chess table turned by @xmath737, see fig. 5 for @xmath738. * open question 7. * study what happens with the phase - lock areas, as @xmath689. study the separation curves and adjacencies of the limit rescaled phase - lock areas. to do this, it would be helpful to study the asymptotics of the functions @xmath739 in variable @xmath8 with fixed @xmath106 and their zeros, as @xmath689. arnold, v. i. _ geometrical methods in the theory of ordinary differential equations, _ second edition. grundlehren der mathematischen wissenschaften [fundamental principles of mathematical sciences], 250. springer - verlag, new york, 1988. buchstaber, v.m. ; karpov, o.v. ; tertychnyi, s.i. _ on properties of the differential equation describing the dynamics of an overdamped josephson junction, _ russian math. surveys, * 59:2 * (2004), 377378. buchstaber, v.m. ; karpov, o.v. ; and tertychnyi, s.i. _ peculiarities of dynamics of a josephson junction shifted by a sinusoidal shf current _ (in russian). radiotekhnika i elektronika, * 51:6 * (2006), 757762. buchstaber, v.m. ; karpov, o.v. ; tertychnyi, s.i. _ the system on torus modeling the dynamics of josephson junction, _ uspekhi mat. nauk, * 67*:1(403) (2012), 181182 ; english transl. : russ. surveys, * 67 * (2012), 178180. buchstaber, v.m. ; tertychnyi, s.i. _ holomorphic solutions of the double confluent heun equation associated with the rsj model of the josephson junction, _ theoret. and math. phys., * 182:3 * (2015), 329355. ilyashenko, yu.s. ; filimonov, d.a. ; ryzhov, d.a. _ phase - lock effect for equations modeling resistively shunted josephson junctions and for their perturbations, _ funct. analysis and its appl. * 45 * (2011), no. 3, 192203.

we study a family of double confluent heun equations that are linearizations of nonlinear equations on two - torus modeling the josephson effect in superconductivity. they have the form @xmath0, where @xmath1 is a family of differential operators of order two acting on germs of holomorphic functions in one complex variable. they depend on parameters @xmath2, @xmath3, @xmath4. we show that for every @xmath5 and @xmath6 satisfying a certain `` non - resonance condition '' and every parameter values @xmath7, @xmath8 there exists a unique entire function @xmath9 (up to multiplicative constant) such that @xmath10 for some @xmath11. the latter @xmath12 are expressed as functions of the parameters. this result has several applications. first of all, it gives the description of those parameter values for which the monodromy operator of the corresponding heun equation has given eigenvalues. this yields the description of the non - integer level curves of the rotation number of the family of equations on two - torus as a function of parameters. in the particular case, when the monodromy is parabolic (has multiple eigenvalue), we get the complete description of those parameter values that correspond to the boundaries of the phase - lock areas : integer level sets of the rotation number, which have non - empty interiors.

1609.00244

Introduction in the past decade, observations of type ia supernovae (sne ia ; riess et al. 1998 ; schmidt et al. 1998 ; perlmutter et al. 1999), cosmic microwave background (cmb) fluctuations (bennett et al. 2003 ; spergel et al. 2003, 2007), and large - scale structures (lss ; tegmark et al. 2004, 2006) have been used to explore cosmology extensively. these observations are found to be consistent with the so - called concordance cosmology, in which the universe is spatially flat and contains pressureless matter and dark energy, with fractional energy densities of @xmath6 and @xmath7 (davis et al. observations of sn ia provide a powerful probe in the modern cosmology. phillips (1993) found that there is an intrinsic relation between the peak luminosity and the shape of the light curve of sne ia. this relation and other similar luminosity relations make sne ia standard candles for measuring the geometry and dynamics of the universe. however, the maximum redshift of the sne ia which we can currently use is only about 1.7, whereas fluctuations of the cmb provide the cosmological information from last scattering surface at @xmath8. therefore, the earlier universe at higher redshift may not be well - studied without data from standardized candles in the `` cosmological desert '' from the sne ia redshift limit to @xmath9. gamma - ray bursts (grbs) are the most intense explosions observed so far. their high energy photons in the gamma - ray band are almost immune to dust extinction, whereas in the case of sn ia observations, there is extinction from the interstellar medium when optical photons propagate towards us. moreover, grbs are likely to occur in the high - redshift range up to at least @xmath10 (krimm et al. 2006) ; higher redshift grbs up to @xmath11 should have already been detected, although none have been identified (lamb & reichart 2000 ; bromm & loeb 2002, 2006 ; lin, zhang, & li 2004). thus, by using grbs, we may explore the early universe in the high redshift range which is difficult to access by other cosmological probes. these advantages make grbs attractive for cosmology research. grb luminosity / energy relations are connections between measurable properties of the prompt gamma - ray emission and the luminosity or energy. in recent years, several empirical grb luminosity relations have been proposed as distance indicators (see e.g. ghirlanda et al. 2006a ; schaefer (2007) for reviews), such as the luminosity - spectral lag (@xmath12-@xmath13) relation (norris, marani, & bonnell 2000), the luminosity - variability (@xmath12-@xmath14) relation (fenimore & ramirez - ruiz 2000 ; reichart et al. 2001), the isotropic energy - peak spectral energy ((@xmath15-@xmath16) relation (i.e., the so - called amati relation, amati et al. 2002), the collimation - corrected energy - peak spectral energy (@xmath17-@xmath16) relation (i.e., the so - called ghirlanda et al. 2004a), the @xmath12-@xmath16 relation (schaefer 2003a ; yonetoku et al. 2004), and two multi - variable relations. the first of these multiple relations is between @xmath15, @xmath16 and the break time of the optical afterglow light curves @xmath18 (i.e., the so - called liang - zhang relation, liang & zhang 2005) ; the other is between luminosity, @xmath16 and the rest - frame `` high - signal '' timescale (@xmath19) (firmani et al. 2006). several authors have made use of these grb luminosity indicators as standard candles at very high redshift for cosmology research (e.g. schaefer 2003b ; dai et al. 2004 ; ghirlanda et al. 2004b ; firmani et al. 2005, 2006, 2007 ; liang & zhang 2005 ; xu et al. 2005 ; wang & dai 2006 ; see also e.g. ghirlanda et al. 2006a and schaefer (2007) for reviews). schaefer (2003b) derived the luminosity distances of nine grbs with known redshifts by using two grb luminosity relations to construct the first grb hubble diagram. dai et al. (2004) considered the ghirlanda relation with 12 bursts and proposed another approach to constrain cosmological parameters. liang & zhang (2005) constrained cosmological parameters and the transition redshift using the @xmath15-@xmath16-@xmath20 relation. more recently, schaefer (2007) used five grb relations calibrated with 69 grbs by assuming two adopted cosmological models to obtain the derived distance moduli for plotting the hubble diagram, and joint constraints on the cosmological parameters and dark energy models have been derived in many works by combining the 69 grb data with sne ia and the other cosmological probes, such as the cmb anisotropy, the baryon acoustic oscillation (bao) peak, the x - ray gas mass fraction in clusters, the linear growth rate of perturbations, and the angular diameter distances to radio galaxies (wright 2007 ; wang et al. 2007 ; li et al. 2008 ; qi et al. 2008 ; daly et al. 2008). however, an important point related to the use of grbs for cosmology is the dependence on the cosmological model in the calibration of grb relations. in the case of sn ia cosmology, the calibration is carried out with a sample of sne ia at very low redshift where the luminosities of sne ia are essentially independent of any cosmological model (i.e., at @xmath21, the luminosity distance has a negligible dependence on the choice of the model). however, in the case of grbs, the observed long - grb rate falls off rapidly at low redshifts, and some nearby grbs may be intrinsically different (e.g., grb 980425, grb 031203 ; norris 2002 ; soderberg et al. 2004 ; guetta et al. 2004 ; liang & zhang 2006a). therefore, it is very difficult to calibrate the relations with a low - redshift sample. the relations of grbs presented above have been calibrated by assuming a particular cosmological model (e.g. the @xmath3cdm model). in order to investigate cosmology, the relations of standard candles should be calibrated in a cosmological model - independent way. otherwise, the circularity problem can not easily be avoided. many of the works mentioned above treat the circularity problem with a statistical approach. a simultaneous fit of the parameters in the calibration curves and the cosmology is carried out to find the optimal grb relation and the optimal cosmological model in the sense of a minimum scattering in both the luminosity relations and the hubble diagram. firmani et al. (2005) has also proposed a bayesian method to get around the circularity problem. (2008b) presented another new method to deal with the problem, using markov chain monte carlo (mcmc) global fitting analysis. instead of a hybrid sample over the whole redshift range of grbs, takahashi et al. (2003) first calibrated two grb relations at low redshift (@xmath22), where distance - redshift relations have been already determined from sn ia ; they then used grbs at high redshift (@xmath23) as a distance indicator. bertolami & silva (2006) considered the use of grbs at @xmath24 calibrated with the bursts at @xmath25 as distance markers to study the unification of dark energy and dark matter in the context of the generalized chaplygin gas model. schaefer (2007) made a detailed comparison between the hubble diagram calibrated only with 37 bursts at @xmath26 and the one calibrated with the data from all 69 grbs to show that the results from fits to the grb hubble diagram calibrated with the hybrid sample are robust. these calibration methods above carried out with the sample at low redshifts have been derived from the @xmath3cdm model. however, we note that the circularity problem can not be circumvented completely by means of statistical approaches, because a particular cosmology model is required in doing the joint fitting. this means that the parameters of the calibrated relations are still coupled to the cosmological parameters derived from a given cosmological model. in principle, the circularity problem can be avoided in two ways (ghirlanda et al. 2006a) : (1) through a solid physical interpretation of these relations that would fix their slope independently from cosmological model, or (2) through the calibration of these relations by several low - redshift grbs. recently the possibility of calibrating the standard candles using grbs in a low - dispersion in redshift near a fiducial redshift has been proposed (lamb et al. 2005 ; ghirlanda et al. liang & zhang (2006b) elaborated this method further based on bayesian theory. however, the grb sample available now is far from what is needed to calibrate the relations in this way. as analyzed above, due to the lack of low - redshift grbs, the methods used in grb luminosity relations for cosmology are different from the standard hubble diagram method used in sn ia cosmology. in this work, we present a new method to calibrate the grb relations in a cosmological model - independent way. in the case of sn ia cosmology, the distance of nearby sne ia used to calibrate the luminosity relations can be obtained by measuring cepheid variables in the same galaxy, and with other distance indicators. thus, cepheid variables have been regarded as the first - order standard candles for calibrating sne ia, the second - order standard candles. it is obvious that objects at the same redshift should have the same luminosity distance in any cosmology. for calibrating grbs, there are so many sne ia (e.g., 192 sne ia used in davis et al. 2007) that we can obtain the distance moduli (also the luminosity distance) at any redshift in the redshift range of sne ia by interpolating from sn ia data in the hubble diagram. furthermore, the distance moduli of sne ia obtained directly from observations are completely cosmological model independent. therefore, in the same sense as cepheid variables and sne ia, if we regard sne ia as the first order standard candles, we can obtain the distance moduli of grbs in the redshift range of sne ia and calibrate grb relations in a completely cosmological model - independent way. then if we further assume that these calibrated grb relations are still valid in the whole redshift range of grbs, just as for sne ia, we can use the standard hubble diagram method to constrain the cosmological parameters from the grb data at high redshift obtained by utilizing the relations. the structure of this paper is arranged as follow. in section 2, we calibrate seven grb luminosity / energy relations with the sample at @xmath27 obtained by interpolating from sne ia data in the hubble diagram. in section 3, we construct the hubble diagram of grbs obtained by using the interpolation methods and constrain cosmological parameters. conclusions and a discussion are given in section 4. The calibration of the luminosity relations of gamma-ray bursts we adopt the data for 192 sne ia (riess et al. 2007 ; wood - vasey et al. 2007 ; astier et al. 2007, davis et al. 2007), shown in figure 1. it is clear that for grbs in the redshift range of sne ia, there are enough data sn ia points and the redshift intervals of the neighboring sn ia data points are also small enough, to be able to use interpolation methods to obtain the distance moduli of grbs. since there is only one sn ia point at @xmath28 (the redshift of sn1997ff is @xmath29), we initially exclude it from our sn ia sample used to interpolate the distance moduli of grbs in the redshift range of the sn ia sample (191 sne ia data at @xmath27). as the distance moduli of sne ia data in the hubble diagram are obtained directly from observations, with the sample at @xmath27 by interpolating from the hubble diagram of sne ia, we can therefore calibrate grb luminosity relations in a completely cosmology - independent way. in this section, we calibrate seven grb luminosity / energy relations with the sample at @xmath27, i.e., the @xmath13-@xmath12 relation, the @xmath14-@xmath12 relation, the @xmath12-@xmath16 relation, the @xmath17-@xmath16 relation, the @xmath30-@xmath12 relation, where @xmath30 is the minimum rise time in the grb light curve (schaefer, 2007), the @xmath15-@xmath16 relation, and the @xmath15-@xmath16-@xmath20 relation. the isotropic luminosity of a burst is calculated by @xmath31 where @xmath32 is the luminosity distance of the burst and @xmath33 is the bolometric flux of gamma - rays in the burst. the isotropic energy released from a burst is given by @xmath34 where @xmath35 is the bolometric fluence of gamma - rays in the burst at redshift @xmath36. the total collimation - corrected energy is then calculated by @xmath37 where the beaming factor, @xmath38 is @xmath39 with the jet opening angle (@xmath40), which is related to the break time (@xmath20). a grb luminosity relation can be generally written in the form @xmath41 where @xmath42 and @xmath43 are the intercept and slope of the relation respectively ; @xmath44 is the luminosity (@xmath12 in units of @xmath45) or energy (@xmath46 or @xmath17 in units of @xmath47) ; @xmath48 is the grb parameters measured in the rest frame, e.g., @xmath49, @xmath50, @xmath51, @xmath52, @xmath53, for the 6 two - variable relations above. we adopt the data for these quantities from schaefer (2007). for the one multi - variable relation (i.e., @xmath15-@xmath54-@xmath20), the calibration equation is @xmath55 where @xmath56 and @xmath57 are @xmath58, @xmath59 respectively, and @xmath60 and @xmath61 are the slopes of @xmath56 and @xmath57 respectively. the error of the interpolated distance modulus of a grb must account for the original uncertainties of the sne ia as well as for the uncertainties from the interpolation. when the linear interpolation is used, the error can be calculated by @xmath62 ^ 2\epsilon_{\mu, i}^2+[(z - z_{i})/(z_{i+1}-z_i)]^2\epsilon_{\mu, i+1}^2)^{1/2},\]] where @xmath63 is the error of the interpolated distance modulus, @xmath64 is the interpolated distance modulus of a source at redshift @xmath36, @xmath65 and @xmath66 are errors of the sne, @xmath67 and @xmath68 are the distance moduli of the sne at nearby redshifts @xmath69 and @xmath70 respectively. when the cubic interpolation is used, the error can be calculated by @xmath71 where @xmath65, @xmath66, @xmath72, and @xmath73 are errors of the sne ; and @xmath67, @xmath68, @xmath74, and @xmath75 are the distance moduli of the sne at nearby redshifts @xmath69, @xmath70, @xmath76, and @xmath77 : @xmath78/[(z_{i+1}-z_{i})(z_{i+2}-z_{i})(z_{i+3}-z_{i})]$] ; @xmath79/[(z_{i}-z_{i+1})(z_{i+2}-z_{i+1})(z_{i+3}-z_{i+1})]$] ; @xmath80/[(z_{i}-z_{i+2})(z_{i+1}-z_{i+2})(z_{i+3}-z_{i+2})]$] ; @xmath81/[(z_{i}-z_{i+3})(z_{i+1}-z_{i+3})(z_{i+2}-z_{i+3})]$]. are obtained by interpolating from sn ia data (_ black circles _ : the cubic interpolation method ; _ black stars _ : the linear interpolation method) ; and the 42 gbrs at @xmath28 (_ blue circles _) are obtained with the five relations calibrated with the sample at @xmath27 using the cubic interpolation method. the curve is the theoretical distance modulus in the concordance model (@xmath82, @xmath83, @xmath84), and the vertical dotted line represents @xmath85., scaledwidth=47.0%] we determine the values of the intercept (@xmath42) and the slope (@xmath43) with their 1-@xmath5 uncertainties calibrated with the grb sample at @xmath27 by using two interpolation methods (the linear interpolation methods and the cubic interpolation method). for the 6 two - variable relations we use the same method (the bisector of the two ordinary least - squares) as used in schaefer (2007), and for the one multi - variable relation, the multiple variable regression analysis is used. the bisector of the two ordinary least - squares (isobe et al., 1990) does not take the errors into account ; but the use of weighted least - squares, taking into account the measurable uncertainties, results in almost identical best fits. however, taking into account the measurable uncertainties in the regression, when they are smaller than the intrinsic error, indeed will not change the fitting parameters significantly (for further discussion, see schaefer 2007). the calibration results are summarized in table 1, and we plot the grb data at @xmath27 with the distance moduli obtained by using the two interpolation methods from sn ia data in figure 1 for comparison. in the previous treatment, the distance moduli of grbs are obtained by assuming a particular cosmological model with a hybrid sample in the whole redshift range of grbs to calibrate the relations. therefore, for comparison in table 1 we also list the results calibrated with the same sample (@xmath27) assuming the @xmath3cdm model (@xmath86) or the riess cosmology (@xmath87, riess et al. 2004), which were used to calibrate the five grb relations in schaefer (2007). the calibration results obtained by using the interpolation methods are carried out with the 27 grbs at @xmath27, namely, 13, 19, 25, 12, 24, 12, and 12 grbs for the @xmath13-@xmath12, @xmath14-@xmath12, @xmath12-@xmath16, @xmath17-@xmath16, @xmath30-@xmath12, @xmath15-@xmath16, and @xmath15-@xmath16-@xmath20 relations respectively;-@xmath16 relation.] therefore, the uncertainties of the results are somewhat larger than those with the hybrid grb sample. uncertainty for the five grb relations with 69 grbs (@xmath88) obtained by assuming the @xmath3cdm model in schaefer (2007) are also given in table 1, for comparison.] [cols="<,^,^,^,^,^,^,^,^,^,^,^ ",] the linear correlation coefficients of the calibration with the sample at @xmath27 using the cubic interpolation methods are -0.88, 0.65, 0.89, 0.94, -0.75, and 0.68 for the above 6 two - variable relations respectively, and 0.94 for @xmath15 vs. the combined variable (@xmath89 + b_2\log[t_{\rm b}/(1+z)]$]) for the @xmath15-@xmath16-@xmath20 relation, which shows that these correlations are significant. from table 1 and figure 1, we find that the calibration results obtained using the linear interpolation methods are almost identical to the results calibrated by using the cubic interpolation method. we also find the results obtained by assuming the two cosmological models with the same sample differ only slightly from, but are still fully consistent with, those calibrated using our interpolation methods. the reason for this is easy to understand, since both cosmological models are fully compatible with sn ia data. nevertheless, it should be noticed that the calibration results obtained using the interpolation methods directly from sn ia data are completely cosmology independent. The hubble diagram of gamma-ray bursts if we further assum grb luminosity / energy relations do not evolve with redshift, we are able to obtain the luminosity (@xmath12) or energy (@xmath15 or @xmath90) of each burst at high redshift (@xmath28) by utilizing the calibrated relations. therefore, the luminosity distance (@xmath91) can be derived from equation (1@xmath923). the uncertainty of the value of the luminosity or energy deduced from a grb relation is @xmath93 where @xmath94, @xmath95 and @xmath96 are 1-@xmath5 uncertainty of the intercept, the slope and the grb measurable parameters, and @xmath97 is the systematic error in the fitting that accounts for the extra scatter of the luminosity relations. the value of @xmath97 can be estimated by finding the value such that a @xmath98 fit to the calibration curve produces a value of reduced @xmath98 of unity (schaefer 2007). a distance modulus can be calculated as @xmath99 (where @xmath91 is in mpc). the propagated uncertainties will depend on whether @xmath33 or @xmath35 is used : @xmath100^{1/2},\]] or @xmath101^{1/2},\]] and @xmath102^{1/2}.\]] we use the same method as used in schaefer (2007) to obtain the best estimate @xmath64 for each grb which is the weighted average of all available distance moduli. the derived distance modulus for each grb is @xmath103 with its uncertainty @xmath104, where the summations run from 1 to 5 over the five relations used in schaefer (2007) with available data. we have plotted the hubble diagram of the 69 grbs obtained using the interpolation methods in figure 1. the 27 grbs at @xmath27 are obtained by using two interpolation methods directly from sne data. the 42 grb data at @xmath28 are obtained by utilizing the five relations calibrated with the sample at @xmath27 using the cubic interpolation method. then the cosmological parameters can be fitted by the minimum @xmath105 method. the definition of @xmath105 is @xmath106 in which @xmath107 is the theoretical value of distance modulus, @xmath108 is the observed distance modulus with its error @xmath109. the theoretical value of the distance modulus (@xmath110) depends on the theoretical value of luminosity distance. for the @xmath3cdm model, the luminosity distance is calculated by @xmath111 }, \\\]] where @xmath112, and @xmath113@xmath114 is @xmath115 for @xmath116, @xmath117 for @xmath118, and @xmath48 for @xmath119. for the dark energy model with a constant equation of state (@xmath120), the luminosity distance in a flat universe is @xmath121 here we adopt @xmath122. figure 2@xmath42 shows the joint confidence regions for (@xmath123) in the @xmath3cdm model from the 42 grb data (@xmath28) obtained by utilizing the five relations calibrated with the sample at @xmath27 using the cubic interpolation method. the 1-@xmath5 confidence region is @xmath124 with @xmath125 for 40 degrees of freedom. for a flat universe, we obtain @xmath1 and @xmath2, which is consistent with the concordance model (@xmath126, @xmath127) in the 1-@xmath5 confidence region. from equation (14), it is obvious that for the @xmath3cdm model, the theoretical value of the luminosity distance mainly depends on @xmath128 at higher redshift, and strongly depends on @xmath129 at lower redshift. therefore, we can find that the shape of the likelihood contour of grbs at higher redshift is almost vertical to the horizontal axis (@xmath128) compared to that of sne ia at lower redshift. to understand the shape of the confidence regions in the (@xmath130) plane, firmani et al. (2007) explored the behavior of the luminosity distance @xmath32 at different redshifts @xmath36 in a given cosmological parameter space. the stripe where @xmath32 varies by @xmath131 at @xmath132, shown in the (@xmath130) plane, was almost vertical to the @xmath128 axis, which corresponds roughly to the typical redshifts (@xmath133) of the grb sample at @xmath28. if we add the one sn ia point (sn1997ff at @xmath29) at @xmath28 into our sn ia sample used to interpolate the distance moduli of grbs, we can calibrate the grb relations with 36 grbs at @xmath134 by using the cubic interpolation method. we find that the fitting results from the 33 grb data (@xmath135) obtained by utilizing the relations calibrated with the sample at @xmath134 using the cubic interpolation method is @xmath136 for a flat universe (figure 2@xmath43), which is consistent with the result from the 42 grb data (@xmath28) obtained using the cubic interpolation method. however, if the relations are calibrated with the grb sample at @xmath27 by assuming a particular cosmological model, we find that the fitting results the 42 grb data (@xmath28) obtained assuming the @xmath3cdm model or the riess cosmology are @xmath137 (figure 2@xmath138) or @xmath139 (figure 2@xmath140) for a flat universe, which are systematically higher than @xmath141 beyond a 1-@xmath5 deviation. ) in the @xmath3cdm model from the data for 42 grbs (@xmath28) obtained by utilizing the five relations calibrated with the sample at @xmath27 using the cubic interpolation method. the plus sign indicates the best fit values. the contours correspond to 1, 2, and 3-@xmath5 confidence regions, and the dashed line represents the flat universe. the 1-@xmath5 confidence region are @xmath124. for a flat universe prior, @xmath1 and @xmath2. (_ b _) joint confidence regions for (@xmath142) from the data for 33 grbs (@xmath135) obtained by utilizing the relations calibrated with the sample at @xmath134 by using the cubic interpolation method. for a flat universe prior, @xmath136. (_ c _) the joint confidence regions for (@xmath142) from the data for 42 grbs (@xmath28) obtained by utilizing the relations calibrated with the sample at @xmath27, assuming a @xmath3cdm model. for a flat universe prior, @xmath137. (_ d _) the joint confidence regions for (@xmath143) from the 42 grbs data (@xmath28) obtained by utilizing the relations calibrated with the sample at @xmath27, assuming a riess cosmology. for a flat universe prior, @xmath144.,scaledwidth=47.0%] figure 3 shows likelihood contours in the (@xmath145) plane in the dark energy model with a constant @xmath120 for a flat universe from the 42 grb data (@xmath28) obtained by utilizing the five relations calibrated with the sample at @xmath27 using the cubic interpolation method. the best - fit values are @xmath146. for a prior of @xmath126, we obtain @xmath4, which is consistent with the cosmological constant in a 1-@xmath5 confidence region. ) plane in the dark energy model with a constant @xmath120 for a flat universe from the 42 grbs data (@xmath28) obtained by utilizing the five relations calibrated with the sample at @xmath27 using the cubic interpolation method. the plus sign indicates the best - fit values (@xmath147). the contours correspond to 1, 2, and 3-@xmath5 confidence regions. for a prior of @xmath148 (_ vertical line _), @xmath149 (_ solid part of vertical line_).,scaledwidth=47.0%] Summary and discussion with the basic assumption that objects at the same redshift should have the same luminosity distance, we can obtain the distance modulus of a grb at given redshift by interpolating from the hubble diagram of sne ia at @xmath27. since the distance modulus of sn ia is completely cosmological model independent, the grb luminosity relations can be calibrated in a completely cosmology independent way. instead of a hybrid sample in the whole redshift range of grbs used in most previous treatment, we choose the grb sample only in the redshift range of sne ia to calibrate the relations. we find there are not significant differences between the calibration results obtained by using our interpolation methods and those calibrated by assuming one particular cosmological model (the @xmath3cdm model or the riess cosmology) with the same sample at @xmath27. this is not surprising, since the two cosmological models are consistent with the data of sne ia. therefore, the grb luminosity relations calibrated from the cosmological models should not be far from the true ones. however, we stress again that the luminosity relations we obtained here are completely cosmological model independent. in order to constrain the cosmological parameters, we have applied the calibrated relations to grb data at high redshift. since our method does not depend on a particular cosmological model, when we calibrate the parameters of grb luminosity relations, the so - called circularity problem can be completely avoided. we construct the grb hubble diagram and constrain cosmological parameters by the minimum @xmath105 method as in sn ia cosmology. from the 42 grbs data (@xmath28) obtained by our interpolation method, we obtain @xmath1 and @xmath2 for the flat @xmath3cdm model, and for the dark energy model with a constant equation of state @xmath4 for a flat universe, which is consistent with the concordance model within the statistical error. our result suggests the the concordance model (@xmath150, @xmath126, @xmath127) which is mainly derived from observations of sne ia at lower redshift is still consistent with the grb data at higher redshift up to @xmath10. for the calibration of sne ia, the luminosity relations could evolve with redshift, in such a way that local calibrations could introduce biases (astier et al., 2006 ; schaefer, 2007). however, riess et al. (2007) failed to reveal the direct evidence for sn ia evolution from analysis of the @xmath23 sample - averaged spectrum. it is possible that some unknown biases of sn ia luminosity relations may propagate into the interpolation results and thus the calibration results of grb relations. nevertheless, the distance moduli of sn ia are obtained directly from observations, and therefore the observable distance moduli of sne ia used in our interpolation method to calibrate the relations of grbs are completely independent of any given cosmological model. recently, there have been discussions of possible evolution effects and selection bias in grb relations. li (2007) used the amati relation as an example to test the possible cosmic evolution of grbs and found that the fitting parameters of the relation vary systematically and significantly with the mean redshift of the grbs. however, ghirlanda et al. (2008) found no sign of evolution with redshift of the amati relation and the instrumental selection effects do not dominate for grbs detected before the launch of the _ swift _ satellite. massaro et al. (2008) investigated the cosmological relation between the grb energy index and the redshift, and presented a statistical analysis of the amati relation searching for possible functional biases. tsutsui et al. (2008) reported a redshift dependence of the lag - luminosity relation in 565 baste grbs. oguri & takahashi (2006) and schaefer (2007) discussed the gravitational lensing and malmquist biases of grbs and found that the biases are small. butler et al. (2007) claimed that the best - fit of the amati relation presented in the _ swift _ sample is inconsistent with the best - fit pre-_swift _ relation at @xmath151 significance. butler et al. (2008) showed that the ghirlanda relation is effectively independent of the grb redshifts. nevertheless, further examinations should be required for considering grbs as standard candles to cosmological use. because of the small grb sample used to constrain the cosmology, the present method provides no more accurate cosmological parameter constraints than the previous works where the grb and sn ia samples were used jointly. for example, wang, dai, & zhu (2007) presented constraints on the cosmological parameters by combining the 69 grb data with the other cosmological probes to get @xmath152. however, our main point is not on improvement of the `` nominal statistical '' error ; rather we emphasize that our method avoids the circularity problem more clearly than previous cosmology dependent calibration methods. therefore, our results for these grb relations are less dependent on prior cosmological models. Acknowledgements we thank hao wei, pu - xun wu, rong - jia yang, zi - gao dai, and en - wei liang for kind help and discussions. we also thank the referee for constructive suggestions and yi - zhong fan for valuable comments and the mention of the reference (takahashi k. et al. this project was in part supported by the ministry of education of china, directional research project of the chinese academy of sciences under project kjcx2-yw - t03, and by the national natural science foundation of china under grants 10521001, 10733010, and 10725313. amati, l. et al. 2002, a&a, 390, 81 astier, p. et al. 2006, a&a, 447, 31 bennett, c. l. et al. 2003, apjs, 148, 1 bertolami, o. & silva, p. t. 2006, mnras, 365, l1149 bromm, v. & loeb, a. 2002, apj, 575, 111 bromm, v. & loeb, a. 2006, apj, 642, 382 butler, n. r. et al. 2007, apj, 671, 656 butler, n. r. et al. 2008, apj, submitted (arxiv : 0802.3396) dai, z. g., liang, e. w., & xu, d. 2004, apj, 612, l101 davis t. m. et al. 2007, apj, 666, 716 daly, r. a. et al. 2008, apj, 677, 1 fenimore, e. e. & ramirez - ruiz, e. 2000, preprint(arxiv : astro - ph/0004176) firmani, c., avila - reese, v., ghisellini, g., & ghirlanda, g. 2006, mnras, 372, l28 firmani, c., avila - reese, v., ghisellini, g., & ghirlanda, g. 2007, rmxaa, 43, 203 firmani, c., ghisellini, g., ghirlanda, g., & avila - reese, v. 2005, mnras, 360, l1 ghirlanda, g., ghisellini, g., & lazzati, d. 2004a, apj, 616, 331 ghirlanda, g., ghisellini, g., & firmani, c. 2006a, new j. phys, 8, 123 ghirlanda, g. et al. 2004b, apj, 613, l13 ghirlanda, g. et al. 2006b, a&a, 452, 839 ghirlanda, g. et al. 2008, mnras, 387, 319 guetta, d., et al. 2004, apj, 615, l73 isobe, t. et al. 1990, apj, 364, 304 krimm, h. et al. 2006, in gamma - ray bursts in the swift era, eds s. s. holt, n. gehrels, and j. a. nousek (aip conf. 836), pp. 145 - 148. lamb, d. q. & reichart, d. e. 2000, apj, 536, 1 lamb, d. q. et al. 2005, preprint (astro - ph/0507362) li, h., su, m., fan, z., dai, z., & zhang, x. 2008a, phys. b, 658, 95 li, h. et al. 2008b, apj, 680, 92 li, l. x. 2007, mnras, 379, 55 liang, e. w. & zhang, b. 2005, apj, 633, 603 liang, e. w. & zhang, b. 2006a, apj, 638, l67 liang, e. w. & zhang, b. 2006b, mnras. 369 l37 lin j. r., zhang s. n., & li t. p. 2004, apj, 605, 819 massaro, f. et al. 2008, in aip conf. 1000, gamma - ray bursts 2007 (melville : aip), 84 norris, j. p., marani, g. f., & bonnell, j. t. 2000, apj, 534, 248 norris, j. p. 2002, apj, 579, 386 oguri, m. & takahashi, k. 2006, phys. d, 73, 123002 perlmutter, s. et al. 1999, apj, 517, 565 phillips, m. 1993, apj, 413, l105 qi, s., wang, f. y., & lu, t. 2008, a&a, 483, 49 riechart, d. e., lamb, d. q., fenimore, e. e., ramirez - ruiz, cline, t. l., & hurley, k. 2001, apj, 552, 57. 1998, aj, 116, 1009 riess, a. g. et al. 2004, aj, 607, 665 riess, a. g. et al. 2007, apj, 659, 98 schaefer, b. e. 2003a, apj, 583, l71 schaefer, b. e. 2003b, apj, 583, l67 schaefer, b. e. 2007, apj, 660, 16 schmidt, b. p. et al. 1998, apj, 507, 46 soderberg, a. m., et al. 2004, nature, 430, 648 spergel, d. n. et al. 2003, apjs, 148, 175 spergel, d. n. et al. 2007, apjs, 170, 377 takahashi k. et al. 2003, preprint (astro - ph/0305260) tegmark, m. et al. 2004, phys. rev. d, 69, 103501 tegmark, m. et al. 2006, phys. d, 74, 123507 tsutsui, r. et al. 2008, mnras, 386, l33 wang, f. y. & dai, z. g. 2006, mnras, 368 l371 wang, f. y., dai, z. g., & zhu, z. h. 2007, apj, 667, 1 wood - vasey, w. m. et al. 2007, apj, 666, 694 wright, e. l. 2007, apj, 664, 633 xu, d., dai, z. g., & liang. e. w. 2005, apj, 633, 603 yonetoku, d. et al. 2004, apj, 609, 935

an important concern in the application of gamma - ray bursts (grbs) to cosmology is that the calibration of grb luminosity / energy relations depends on the cosmological model, due to the lack of a sufficient low - redshift grb sample. in this paper , we present a new method to calibrate grb relations in a cosmology - independent way. since objects at the same redshift should have the same luminosity distance and since the distance moduli of type ia supernovae (sne ia) obtained directly from observations are completely cosmology independent, we obtain the distance modulus of a grb at a given redshift by interpolating from the hubble diagram of sne ia. then we calibrate seven grb relations without assuming a particular cosmological model and construct a grb hubble diagram to constrain cosmological parameters. from the 42 grbs at @xmath0, we obtain @xmath1, @xmath2 for the flat @xmath3cdm model, and for the dark energy model with a constant equation of state @xmath4, which is consistent with the concordance model in a 1-@xmath5 confidence region.

0802.4262

Introduction almost forty years ago exotic, apparently hybrid and unexpected events, dubbed centauros, were observed in cosmic ray (cr) experiments in emulsion chambers in chacaltaya by lattes and collaborators @xcite. those events were very different from what is commonly observed in crs, exhibiting a large number of hadrons and a small number of electrons and gammas, which suggests the presence of very few rapid - gamma - decaying hadrons. so, a possible imbalance in the number of neutral to charged pions could be envisaged. the nature and reality of centauro events started a long debate, that includes the reexamination of the original emulsion chamber plates, and is still not resolved @xcite. nevertheless, centauro events were certainly an experimental motivation for the development of the theory of disoriented chiral condensates (dccs) that started in the early 1990s @xcite. for a detailed review, see ref. @xcite. assuming that a given nuclear system could be heated above the critical (crossover) transition region for chiral symmetry restoration, i.e. for temperatures of the order of @xmath1mev @xcite, then quenched to low temperatures, the chiral condensate initially melted to zero could grow in any direction in isospin space. besides the vacuum (stable) direction, it could build up as in a metastable, misaligned pseudo - vacuum state, and later decay to the true, chirally broken vacuum. the fact that dccs could be formed in high - energy heavy ion collisions stimulated several theoretical advances and experimental searches @xcite. most likely the temperatures achieved in current heavy ion experiments are high enough to produce an approximately chirally symmetric quark - gluon plasma, and the following rapid expansion can cool the system back to the vacuum @xcite, so that the dynamics of chiral symmetry restoration and breakdown can be described in a quench scenario @xcite, so that the evolution of the order parameter can be much affected by an explosive behavior that naturally leads to large fluctuations and inhomogeneities @xcite. since, by assumption, the order parameter for chiral symmetry breaking, i.e. the chiral condensate, is misaligned with respect to the vacuum direction (associated with the @xmath2-direction in effective models for strong interactions) in a dcc, this would be a natural candidate to explain the excessive production of hadrons unaccompanied by electrons and photons, suggesting the suppression of neutral pions with respect to charged pions. regardless of the outcome of the debate on the nature of centauro events, dcc formation seems to be a quite natural phenomenon in the theory of strong interactions. however, given its symmetric nature (in isospin space), it should be washed out by standard event averaging methods. so far, there has been no evidence from colliders or cr experiments. motivated by the possibility of attaining much higher statistics in current ultra - high energy cosmic ray (uhecr) experiments than in the past, so that an event - by - event analysis for very high - energy collisions can in principle be performed, we consider possible signatures of dcc production in cr air showers. if dccs are formed in high - energy nuclear collisions in the atmosphere, the relevant outcome from the primary collision are very large event - by - event fluctuations in the neutral - to - charged pion fraction, and this could affect the nature of the subsequent atmospheric shower. very preliminary, yet encouraging results were presented in ref. @xcite. in this paper we search for fingerprints of dcc formation in two different observables of uhecr showers. we present simulation results for the depth of the maximum (@xmath0) and number of muons on the ground, evaluating their sensitivity to the neutral - to - charged pion fraction asymmetry produced in the primary interaction. to model the effect from the presence of a dcc, we simply modify the neutral - to - charged pion fraction, assuming that the system follows the same kinematics, as will be detailed below. although this is certainly a very crude description of the dynamics of the primary collision, we believe it captures the essential features that have to be tested in order to verify the feasibility of detecting dccs in uhecr showers. this paper is organized as follows. in section ii we briefly review some characteristics of dccs, especially the baked - alaska scenario and the inverse square root distribution of the neutral pion fraction. in section iii the method for the simulation is presented. we use corsika @xcite, a program for detailed simulation of extensive air showers initiated by high - energy cosmic ray particles. in section iv we show and discuss our results. section v contains our conclusions. Dcc features and the neutral pion fraction distribution it is widely believed that for high enough energy densities, achieved e.g. by increasing dramatically the temperature, strong interactions becomes approximately chiral (it would be an exact symmetry only if current quarks were strictly massless), so that the chiral condensate, which is the order parameter for that transition, essentially vanishes. on the other hand, for low temperatures the chiral condensate acquires a non - vanishing value and breaks spontaneously the chiral symmetry of (massless) qcd @xcite. in a given model, one can construct an effective potential for the chiral condensate degrees of freedom and study the mechanism of chiral symmetry restoration and breakdown. if we restrict our analysis to two flavors of light quarks, up and down, that can be easily accomplished by the linear sigma model coupled to quarks @xcite. in that case, the effective degrees of freedom are pions, @xmath3, and the sigma, @xmath2. in the high - temperature limit all field expectation values vanish, whereas in the vacuum one has @xmath4 and @xmath5, where @xmath6 is the pion decay constant. the physical picture we have in mind is a very high - energy heavy ion collision that will create a hot quark - gluon plasma where chiral symmetry is approximately restored. as the plasma is quenched to low temperatures by expansion, the system will evolve to the vacuum emitting a large number of pions. however, the evolution can proceed along many different paths in chiral space before it finally reaches the true vacuum, i.e. it can `` roll '' into different directions, and the ratio of neutral to charged pions produced depends strongly on the chosen metastable state in each event. in other words, the misalignment of the vacuum is reflected in the distribution of produced pions, generating a coherent state and an anomaly in the ratio of charged to neutral pions. this effect will be, of course, washed out by event averages. a more intuitive physical picture for the formation of dccs, the _ baked - alaska _ scenario, was proposed by bjorken and collaborators @xcite. consider a high - multiplicity collision at high energy. before hadronization, most of the energy released at the collision point is carried away by the primary partons at nearly the speed of light. this hot and thin expanding shell isolates the relatively cold interior from the outer vacuum. as the state evolves, the interior cools down and what was before a global minimum becomes a local maximum. the symmetry is spontaneously broken and one of the pseudo - vacua should be chosen, with a slight preference for the true vacuum. however, if the lifetime of the shell is short enough, the quark condensate in the interior might be rotated from its ordinary direction, since it costs relatively little energy, @xmath7 mev/@xmath8 @xcite. when the hot shell hadronizes, the reorientation induced by the external contact is reflected in the produced pions, which will be coherent and present fluctuations in an event - by - event analysis. assuming that the correlated region is large enough to be described semiclassically, one can use the linear sigma model with explicit symmetry breaking for the description of the dynamics @xcite. quantitatively, one can represent the shell of this fireball as the source of these excitations of pions : @xmath9 where @xmath10 is the isospin index. a spherically expanding shell is represented by @xmath11 with initial radius @xmath12, where @xmath13 is the time where the expansion starts. after the hadronization, the currents @xmath14 vanish and the fields @xmath3 decay towards the vacuum into freely propagating pions. so, the pion emission is characterized by the state @xmath15 where @xmath16 is a normalization factor, and the sum is over isospin directions @xmath17. the creation operator of a pion with momentum @xmath18 and isospin component @xmath10 is @xmath19 and @xmath20 is the 4-dimensional fourier transform of the source @xmath21 at @xmath22. the number of pions follows a poisson distribution with average @xmath23 and the number of pions produced per unit phase space is approximately given by @xcite @xmath24 statistically, one expects that the magnitude and the chiral orientation of the source @xmath25 will fluctuate event by event for each mode. let us assume that the vacuum orientation is tilted into one of the pion directions, i.e. : @xmath26 so that all relevant modes of chiral condensate point in the same direction @xmath27 in isospin space : @xmath28 in generic models of production the neutral pion fraction, defined as @xmath29 is a binomial distribution with average @xmath30. in this way, in the limit of large numbers @xmath31, the probability of all pions being charged is very small. however, if pions are the product of a dcc decay, this probability is not negligible. in fact, if there is no privileged isospin direction, the vector @xmath27 can point in any direction within the unit sphere. then : @xmath32 where @xmath33 is the angle between the unit vector @xmath27 and the @xmath34 direction. so, one finds the following well - known distribution for the neutral pion fraction @xcite : @xmath35 the probability of less than 10@xmath36 of pions be neutral, for instance, is @xmath37. Dcc simulations the conditions of a high temperature initial state followed by a rapid cooling stage are both possible to happen in heavy ion collisions at the very high energies like those accessible at rhic and to be reached at the lhc, as well as in uhecr collisions in the top of the atmosphere. the large aperture of a detector like that of the pierre auger observatory @xcite (combining shower sampling at ground level and longitudinal shower profile reconstruction) has been providing high quality data and unprecedented statistics in the field @xcite. therefore, even though the formation of a dcc is probably rare, we believe it is worth studying the implications of such events to the physics of showers generated by uhecr. the neutral pion fraction distribution, eq. ([sqrt]), is at the basis of our strategy to search for dcc fingerprints in uhecr showers. therefore, any investigation to measure the impact caused by the presence of a dcc should assess the sensitivity of a given observable with respect to the neutral pion fraction produced in the primary interaction. this fraction determines the initial distribution of particles between the electromagnetic and hadronic components of the showers. in this paper, we consider two observables which are usually measured by uhecr detectors : the slant depth in the atmosphere (defined as the integral of the atmosphere density along the shower axis @xmath38 and expressed in units of g/@xmath39) at which the shower reaches its maximum development, @xmath0, and the number of muons on the ground @xmath40. it is known that the parameter @xmath0 is affected by the first interaction cross - section and its associated multiplicity and inelasticity @xcite. if dccs really exist, the conditions for them to be produced should include not only high energy densities, but the regions where such densities are achieved should not be small as well, since dccs are considered `` macroscopic '' space - time regions where the chiral parameter is not oriented in the same direction as the vacuum. with those requirements in mind, we have chosen to work with fe initiated showers at @xmath41 ev. then, central collisions are privileged over peripheral ones by selecting events with a large number of participating nucleons (@xmath42) which, in turn, should lead to a higher multiplicity in the first interaction. for all simulations presented in this work, we have used corsika 6.617 @xcite with the interaction models sibyll 2.1 @xcite and gheisha 2002d @xcite, for high and low energy processes, respectively. the dcc - like shower simulation chain is as follows : large @xmath43 collisions are selected and their first secondaries (mostly pions and kaons) separated ; after, some of the neutral pions in this sample are converted into charged ones ; the resulting particle list is then inserted back into corsika to proceed with usual cascade development through the atmosphere. such a procedure was performed for several @xmath44 fractions and 2 different zenith angles. the first interaction slant depth (@xmath45) distribution for dcc - like showers will therefore be the same as for normal fe initiated showers in central collisions. this is a valid assumption, since the dcc formation process takes place during a subsequent cooling stage of the initial hot plasma, with the first interaction cross section being the same as in a fe - nucleus collision. we believe that even though the simulation approach adopted here is simplified, the essential features of the process are being taken into account. for comparison, proton initiated showers as well as normal fe initiated ones were also generated. for the normal fe case, we have produced both a sample rich in central collisions and another sample with all the centralities. for each shower we extract the value of @xmath0 and the number of muons on the ground. Discussion of results from now on we shall identify our dcc - like fe initiated showers by fe+dcc. [xmax] shows the @xmath0 distribution of a vertical shower (@xmath46) corresponding to an extremely asymmetric situation (@xmath47), that is, where all the initially produced @xmath44 s are converted into @xmath48. for the distribution of eq. ([sqrt]), sharply peaked at @xmath47, the probability for less than 1% of @xmath44 s being produced is 10%. four types of showers : normal fe (central collision : fe - central), fe (all centralities), fe+dcc and proton initiated are shown. on can clearly see that both samples generated from a central collision have smaller than average @xmath0, since the higher the multiplicity in the first interaction the faster is the subsequent cascade development in the atmosphere. and there is essencially no difference between the fe - central and the fe - dcc samples in terms of @xmath0, which might indicate that the early stage where the @xmath44 population is depleted together with the higher interaction probability due to the large multiplicity allow for a complete recovery of these particles in the subsequent interactions. nonetheless, it is clear that one should look at low @xmath0 events when searching for dccs signatures, and this property is independent of the initial @xmath44 fraction. another feature which can be appreciated in fig. [xmax] is the smaller @xmath0 fluctuations for fe - dcc and fe - central as compared to those of proton and fe with all centralites. this is to be expected, since the fluctuations in @xmath0 have basically two components : the the ones in the first interaction slant depth @xmath45 (which, in turn, depends on the interaction cross - section) and those introduced by the cascade growth up to the maximum. the latter depends on the initial shower size (the multiplicity), in average being smaller for high multiplicity events, i.e. those found in the central collisions. the muon number distributions on the ground for the same showers described above are shown in fig. [mu_dist]. as expected, increasing the number of initial charged pions will lead to a corresponding larger density of muons on the ground as the products of the charged pions decay. as one combines both pieces of information (@xmath0 and @xmath40), one finds a good separation between the fe - dcc sample and all the other populations, as can be seen from fig. [scatterplot]. when collisions with all the centralities are allowed, iron showers exhibit a positive correlation between @xmath0 and @xmath40. the correlation is such that showers reaching maximum development at increasingly larger slant depths produce, in turn, a higher muon density at ground level, due to the decrease in the attenuation in the atmosphere from @xmath0 to the ground. whereas for fe - central showers one sees no correlation between @xmath0 and @xmath40, for the case of fe - dcc showers, instead, a small anti - correlation of about @xmath49 g/@xmath50/ @xmath51 muons is present distribution provides @xmath52 g/@xmath50/muon.], as can be seen from fig. [scatterplot]. such an inverted correlation can be understood remembering the additional correlation between @xmath0 and the multiplicity in the first interaction, with low @xmath0 events corresponding, in average, to high multiplicity. as we convert neutral pions to charged ones, high multiplicity (small @xmath0) events show a larger muon density on the ground as compared to low multiplicity ones (large @xmath0). since the number of muons on the ground depend both on the @xmath44 fraction @xmath53 and on the zenith angle, due to atmospheric attenuation, we decided to perform a more systematic study by varying those parameters. applying a linear discriminant analysis (lda), similar to what has been done in @xcite we have determined, as a function of @xmath53, a power of discrimination defined as @xmath54, with the classification error @xmath55, in the 2 populations case (@xmath56 and @xmath57) given by : @xmath58 where @xmath59 (@xmath60) are the number of events from population @xmath56 (@xmath57) misclassified as @xmath57 (@xmath56) and @xmath61 (@xmath62) is total number of events in population @xmath56 (@xmath57). as training samples we have used half of the simulated showers, with the other half being used for the determination of the power of discrimination. [merit] (top, middle and bottom left) shows again the two - dimensional scatter plots in the @xmath63 plane for the same 4 populations analyzed above and 5 values of @xmath44 fraction @xmath64 and @xmath65 degrees. the merit factor as a function of @xmath53 is shown at the bottom right plot of fig. [merit]. one sees that for any value of @xmath53, proton showers can always be reasonably separated from fe - dcc, with merit factors above 97% in the whole interval of @xmath53. for fe showers, of course, as we approach the @xmath66 fraction, the separation power gets weaker. fe - central showers are easier to separate from the fe (all centralities) signals only at small values of @xmath53, where the excess of muons at ground is large enough to segregate the fe - dcc population. fe - dccs showers in which the first interaction produces a large amount of @xmath44 are very poor in muons and would also be easily recognized. however, even if dccs exist in nature, such large-@xmath53 events are very unlikely to happen taken into account the distribution of eq. ([sqrt]). a more robust estimation of the power of discrimination would have to incorporate a priori information on the population frequencies for proton and fe, as well as the dcc occurrence frequency. nonetheless, since neither the cosmic rays chemical composition at the highest energies (@xmath67 ev) nor the full dynamics of dcc formation are known, flat priors on these variables are the best one can do so far. Conclusions the recent increase in quality and statistics of uhecr data brings the possibility to look for exotic phenomena as well as rare phenomena within the standard model. among the latter, dccs have been predicted about twenty years ago, yet never detected due to their elusive character. in this paper, using air shower simulations, we searched for fingerprints of dccs formed in ultra - high - energy (@xmath68ev) central iron collisions in the upper atmosphere. in particular, we studied the influence of the dcc formation on the observables @xmath0 and @xmath40, via the asymmetry in the neutral - to - charged pion ratio in the primary collision. for comparison, we considered also regular air showers generated by protons and iron. since dccs are expected to be formed in central collisions, which lead to smaller than average values of @xmath0 (a difference of about @xmath69g/@xmath50 as compared to iron, for instance), one should concentrate searches within this region of shower depth. for the same reason, if dcc events are present they should yield small fluctuations in @xmath0. however, based only on the behavior of this observable one can not distinguish between an iron shower and a central collision and one produced in the presence of a dcc. the formation of dccs is associated with large event - by - event fluctuations in the ratio of neutral to charged pions, @xmath53. in particular, for @xmath53 large or small as compared to @xmath70, one should expect a sizable change in the muon density on the ground, especially for vertical showers. this fact was noticed in the preliminary study of ref. @xcite. in the extreme case of @xmath47, where dcc events lead to muon - rich showers, we showed that this signature distinguishes between the cases of iron (even for central collisions) and a dcc event. for large @xmath53, one can also separate these two cases due to the large depletion of muons on the ground. even in this case, the signature is not contaminated by proton events. for vertical showers, there is a clear anti - correlation between @xmath0 and @xmath40. this comes about since there is a correlation between @xmath0 and the first interaction multiplicity, as was discussed in the previous section. this behavior is not expected in a regular iron shower. in fact, for iron, due to atmospheric attenuation, one would expect a positive correlation. even though the analysis presented here is very simplified, it has the advantage of providing a setup that is totally under control for simulations, and we believe it contains the essential ingredients of the phenomena considered. nevertheless, a more realistic study, that should contain a description of the dynamics of dcc formation, especially its dependence on energy, is certainly necessary. Acknowledgments we thank f. catalani and j. takahashi for useful discussions. we also thank v. de souza for computer time at the usp - so carlos cluster [fapesp, grant 2008/04259 - 0]. this work was partially supported by by capes, cnpq, faperj, fapesp and fujb / ufrj. c. m. g. lattes _ et al. _, prog. suppl. * 47 *, 1 (1971) ; c. m. g. lattes, y. fujimoto and s. hasegawa, phys. rept. * 65 *, 151 (1980). c. r. a. augusto, s. l. c. barroso, v. kopenkin, m. moriya, c. e. navia and e. h. shibuya, phys. d * 59 *, 054001 (1999). e. gladysz - dziadus, phys. nucl. * 34 *, 285 (2003) [fiz. chast. atom. yadra * 34 *, 565 (2003)]. v. kopenkin, y. fujimoto and t. sinzi, phys. d * 68 *, 052007 (2003). a. ohsawa, e. h. shibuya and m. tamada, phys. d * 70 *, 074028 (2004) ; v. kopenkin and y. fujimoto, phys. rev. d * 73 *, 082001 (2006). r. engel, nucl. proc. suppl. * 151 *, 437 (2006). a. a. anselm and m. g. ryskin, phys. b * 266 *, 482 (1991). j. d. bjorken, slac - pub-5673 (1991) ; int. j. mod. phys. a * 7 *, 4189 (1992). j. p. blaizot and a. krzywicki, phys. d * 46 *, 246 (1992). k. rajagopal and f. wilczek, nucl. b * 399 *, 395 (1993) ; nucl. b * 404 *, 577 (1993). b. mohanty and j. serreau, phys. rept. * 414 *, 263 (2005). a. bazavov _ et al. _, phys. d * 80 *, 014504 (2009). p. stankus, d. silvermyr, s. sorensen and v. greene (eds.), proceedings of quark matter 2009 - the 21st international conference on ultrarelativistic nucleus - nucleus collisions, nucl. a * 830 *, 1c-985c (2009). l. p. csernai and i. n. mishustin, phys. rev.lett. * 74 *, 5005 (1995) ; a. abada and j. aichelin, phys. lett. * 74 *, 3130 (1995) ; a. abada and m. c. birse, phys. rev. d * 55 *, 6887 (1997). i. n. mishustin and o. scavenius, phys. lett. * 83 *, 3134 (1999) ; i. n. mishustin, phys. lett. * 82 *, 4779 (1999) ; i. n. mishustin and o. scavenius, nucl. phys. a * 638 *, 519 (1998). o. scavenius and a. dumitru, phys. lett. * 83 *, 4697 (1999) ; o. scavenius, a. dumitru, e. s. fraga, j. t. lenaghan and a. d. jackson, phys. d * 63 *, 116003 (2001). k. paech, h. stoecker and a. dumitru, phys. c * 68 *, 044907 (2003) ; k. paech and a. dumitru, phys. b * 623 *, 200 (2005). c. e. aguiar, e. s. fraga and t. kodama, j. phys. g * 32 *, 179 (2006) ; b. g. taketani and e. s. fraga, phys. d * 74 *, 085013 (2006) ; e. s. fraga and g. krein, phys. b * 614 *, 181 (2005). t. p. viana, e. s. fraga and j. r. t. de mello neto, int. j. mod. e * 16 *, 2872 (2007). d. heck, g. schatz, t. thouw, j. knapp and j. n. capdevielle, report fzka 6019 (1998). forschungszentrum karlsruhe ; + http://wwwik.fzk.de/corsika/physics description / corsika phys.html. k. yagi, t. hatsuda and y. miake, camb. monogr. part. phys. nucl. cosmol. * 23 *, 1 (2005). m. gell - mann and m. levy, nuovo cim. * 16 *, 705 (1960). j. d. bjorken, k. l. kowalski and c. c. taylor, slac - pub-6109 (1993).

although the generation of disoriented chiral condensates (dccs), where the order parameter for chiral symmetry breaking is misaligned with respect to the vacuum direction in isospin state, is quite natural in the theory of strong interactions, they have so far eluded experiments in accelerators and cosmic rays. if dccs are formed in high - energy nuclear collisions, the relevant outcome are very large event - by - event fluctuations in the neutral - to - charged pion fraction. in this note we search for fingerprints of dcc formation in observables of ultra - high energy cosmic ray showers. we present simulation results for the depth of the maximum (@xmath0) and number of muons on the ground, evaluating their sensitivity to the neutral - to - charged pion fraction asymmetry produced in the primary interaction.

1006.2415

Introduction quantum systems can be correlated in ways that supersede classical descriptions. however, there are degrees of non - classicality for quantum correlations. for simplicity, we consider only bipartite correlations, with the two, spatially separated, parties being named alice and bob as usual. at the weaker end of the spectrum are quantum systems whose states can not be expressed as a mixture of product - states of the constituents. these are called non - separable or entangled states. the product - states appearing in such a mixture comprise a local hidden state (lhs) model for any measurements undertaken by alice and bob. at the strongest end of the spectrum are quantum systems whose measurement correlations can violate a bell inequality @xcite, hence demonstrating (modulo loopholes @xcite) the violation of local causality @xcite. this phenomenon commonly known as bell - nonlocality @xcite is the only way for two spatially separated parties to verify the existence of entanglement if either of them, or their detectors, can not be trusted @xcite. we say that a bipartite state is bell - local if and only if there is a local hidden variable (lhv) model for any measurements alice and bob perform. here the ` variables'are not restricted to be quantum states, hence the distinction between non - separability and bell - nonlocality. in between these types of non - classical correlations lies epr - steering. the name is inspired by the seminal paper of einstein, podolsky, and rosen (epr) @xcite, and the follow - up by schrdinger @xcite, which coined the term `` steering '' for the phenomenon epr had noticed. although introduced eighty years ago, as this special issue celebrates, the notion of epr - steering was only formalized eight years ago, by one of us and co - workers @xcite. this formalization was that epr - steering is the only way to verify the existence of entanglement if one of the parties conventionally alice @xcite or her detectors, can not be trusted. we say that a bipartite state is epr - steerable if and only if it allows a demonstration of epr - steering. a state is not epr - steerable if and only if there exists a hybrid lhv lhs model explaining the alice bob correlations. since in this paper we are concerned with steering, when we refer to a lhs model we mean a lhs model for bob only ; it is implicit that alice can have a completely general lhv model. the above three notions of non - locality for quantum states coincide for pure states : any non - product pure state is non - separable, eps - steerable, and bell - nonlocal. however for mixed states, the interplay of quantum and classical correlations produces a far richer structure. for mixed states the logical hierarchy of the three concepts leads to a hierarchy for the bipartite states : the set of separable states is a strict subset of the set of non - epr - steerable states, which is a strict subset of the set of bell - local states @xcite. although the epr - steerable set has been completely determined for certain classes of highly symmetric states (at least for the case where alice and bob perform projective measurements) @xcite, until now very little was known about what types of states are steerable even for the simplest case of two qubits. in this simplest case, the phenomenon of steering in a more general sense i.e. within what set can alice steer bob s state by measurements on her system has been studied extensively using the so - called steering ellipsoid formalism @xcite. however, no relation between the steering ellipsoid and epr - steerability has been determined. in this manuscript, we investigate epr - steerability of the class of two - qubit states whose reduced states are maximally mixed, the so - called t - states @xcite. we use the steering ellipsoid formalism to develop a deterministic lhs model for projective measurements on these states and we conjecture that this model is optimal. furthermore we obtain two sufficient conditions for t - states to be epr - steerable, via suitable epr - steering inequalities @xcite (including a new asymmetric steering inequality for the spin covariance matrix). these sufficient conditions touch the necessary condition in some regions of the space of t - states, and everywhere else the gap between them is quite small. the paper is organised as follows. in section 2 we discuss in detail the three notions of non - locality, namely bell - nonlocality, epr - steerability and non - separability. section 3 introduces the quantum steering ellipsoid formalism for a two - qubit state, and in section 4 we use the steering ellipsoid to develop a deterministic lhs model for projective measurements on t - states. in section 5, two asymmetric steering inequalities for arbitrary two - qubit states are derived. finally in section 6 we conclude and discuss further work. [sec_lhsmodel]epr-steering and local hidden state models two separated observers, alice and bob, can use a shared quantum state to generate statistical correlations between local measurement outcomes. each observer carries out a local measurement, labelled by @xmath1 and @xmath2 respectively, to obtain corresponding outcomes labelled by @xmath3 and @xmath4. the measurement correlations are described by some set of joint probability distributions, @xmath5, with @xmath1 and @xmath2 ranging over the available measurements. the type of state shared by alice and bob may be classified via the properties of these joint distributions, for all possible measurement settings @xmath1 and @xmath2. the correlations of a _ bell - local _ state have a local hidden variable (lhv) model @xcite, @xmath6 for some ` hidden'random variable @xmath7 with probability distribution @xmath8. hence, the measured correlations may be understood as arising from ignorance of the value of @xmath7, where the latter locally determines the statistics of the outcomes @xmath3 and @xmath4 and is independent of the choice of @xmath1 and @xmath2. conversely, a state is defined to be bell_-nonlocal _ if it has no lhv model. such states allow, for example, the secure generation of a cryptographic key between alice and bob without trust in their devices @xcite. in this paper, we are concerned with whether the state is _ steerable _ ; that is, whether it allows for correlations that demonstrate epr - steering. as discussed in the introduction, epr - steering by alice is demonstrated if it is _ not _ the case that the correlations can be described by a hybrid lhv lhs model, wherein, @xmath9 where the local distributions @xmath10 correspond to measurements on local quantum states @xmath11, i.e., @xmath12.\]] here @xmath13 denotes the positive operator valued measure (povm) corresponding to measurement @xmath2. the state is said to be _ steerable _ by alice if there is _ no _ such model. the roles of alice and bob may also be reversed in the above, to define steerability by bob. comparing eqs. ([lhv]) and ([lhs]), it is seen that all nonsteerable states are bell - local. hence, all bell - nonlocal states are steerable, by both alice and bob. in fact, the class of steerable states is strictly larger @xcite. moreover, while not as powerful as bell - nonlocality in general, steerability is more robust to detection inefficiencies @xcite, and also enables the use of untrusted devices in quantum key distribution, albeit only on one side @xcite. by a similar argument, a separable quantum state shared by alice and bob, @xmath14, is both bell - local and nonsteerable. moreover, the set of separable states is strictly smaller than the set of nonsteerable states @xcite. it is important that epr - steerability of a quantum state not be confused with merely the dependence of the reduced state of one observer on the choice of measurement made by another, which can occur even for separable states. the term ` steering'has been used with reference to this phenomenon, in particular for the concept of ` steering ellipsoid ', which we will use in our analysis. epr - steering, as defined above, is a special case of this phenomenon, and is only possible for a subset of nonseparable states. we are interested in the epr - steerability of states for all possible _ projective _ measurements. if alice is doing the steering, then it is sufficient for bob s measurements to comprise some tomographically complete set of projectors. it is straightforward to show in this case that the condition for bob to have an lhs model, eq. ([lhs]), reduces to the existence of a representation of the form @xmath15 = \sum_\lambda p(\lambda)\, p(1|e,\lambda) \,\rho_b(\lambda), \\ \label{reduced_p } p_e & = \operatorname{tr } [\rho e\otimes i] = \sum_\lambda p(\lambda) p(1|e,\lambda).\end{aligned}\]] here @xmath16 is any projector that can be measured by alice ; @xmath17 is the probability of result ` @xmath18'and @xmath19 is the corresponding probability given @xmath7 ; @xmath20 is the reduced state of bob s component corresponding to this result ; and @xmath21 $] denotes the partial trace over alice s component. note that this form, and hence epr - steerability by alice, is invariant under local unitary transformations on bob s components. determining epr - steerability in this case, where alice is permitted to measure any hermitian observable, is surprisingly difficult, with the answer only known for certain special cases such as werner states @xcite. however, in this paper we give a strong necessary condition for the epr - steerability of a large class of two - qubit states, which we conjecture is also sufficient. this condition is obtained via the construction of a suitable lhs model, which is in turn motivated by properties of the ` quantum steering ellipsoid' @xcite. properties of this ellipsoid are therefore reviewed in the following section. [sec_qse]the quantum steering ellipsoid an arbitrary two - qubit state may be written in the standard form @xmath22 here @xmath23 denote the pauli spin operators, and @xmath24,~ b_j=\operatorname{tr}[\rho\, \i\otimes\sigma_j],~t_{jk}=\operatorname{tr}[\rho \,\sigma_j\otimes\sigma_k].\]] thus, @xmath25 and @xmath26 are the bloch vectors for alice and bob s qubits, and @xmath27 is the spin correlation matrix. if alice makes a projective measurement on her qubit, and obtains an outcome corresponding to projector @xmath16, bob s reduced state follows from eq. ([reduced]) as @xmath28 } { \operatorname{tr}[\rho \,e \otimes \i] }.\]] we will also refer to @xmath20 as bob s ` steered state '. to determine bob s possible steered states, note that the projector @xmath16 may be expanded in the pauli basis as @xmath29, with @xmath30. this yields the corresponding steered state @xmath31, with associated bloch vector @xmath32 where @xmath33 is the associated probability of result @xmath34, @xmath35 = \frac 12 (1+{\boldsymbol}a\cdot{\boldsymbol}e),\]] called @xmath17 previously. in what follows we will refer to the vector @xmath36 rather than its corresponding operator @xmath16. the surface of the steering ellipsoid is defined to be the set of steered bloch vectors, @xmath37, and in ref. @xcite it is shown that interior points can be obtained from positive operator - valued measurements (povms). the ellipsoid has centre @xmath38 and the semiaxes @xmath39 are the roots of the eigenvalues of the matrix @xmath40 the eigenvectors of @xmath41 give the orientation of the ellipsoid around its centre @xcite. thus, the general equation of the steering ellipsoid surface is @xmath42 with @xmath43 being the displacement vector from the centre @xmath44. entangled states typically have large steering ellipsoids the largest possible being the bloch ball, which is generated by every pure entangled state @xcite. in contrast, the volume of the steering ellipsoid is strictly bounded for separable states. indeed, a two - qubit state is separable if and only if its steering ellipsoid is contained within a tetrahedron contained within the bloch sphere @xcite. thus, the separability of two - qubit states has a beautiful geometric characterisation in terms of the quantum steering ellipsoid. no similar characterisation has been found for epr - steerability, to date. however, for non - separable states, knowledge of the steering ellipsoid matrix @xmath41, its centre @xmath45, and bob s bloch vector @xmath46 uniquely determines the shared state @xmath47 up to a local unitary transformation on alice s system @xcite, @xcite and so is sufficient, in principle, to determine the epr - steerability of @xmath47. in this paper we find a direct connection between epr - steerability and the quantum steering ellipsoid, for the case that the bloch vectors @xmath48 and @xmath46 vanish. [sec_tstate]necessary condition for epr-steerability of t-states let @xmath49 be a singular value decomposition of the spin correlation matrix @xmath27, for some diagonal matrix @xmath50 and orthogonal matrices @xmath51. noting that any @xmath52 is either a rotation or the product of a rotation with the parity matrix @xmath53, it follows that @xmath27 can always be represented in the form @xmath54, for proper rotations @xmath55, where the diagonal matrix @xmath56 may now have negative entries. the rotations @xmath57 and @xmath58 may be implemented by local unitary operations on the shared state @xmath47, amounting to a local basis change. hence, all properties of a shared two - qubit state, including steerability properties in particular, can be formulated in a representation in which the spin correlation matrix has the diagonal form @xmath59 $]. it follows that if the shared state @xmath47 has maximally - mixed reduced states with @xmath60, then it is completely described, up to local unitaries, by a diagonal @xmath27, i.e. one may consider @xmath61 without loss of generality. such states are called t - states @xcite. they are equivalent to mixtures of bell states, and hence form a tetrahedron in the space parameterised by @xmath62 @xcite. entangled t - states necessarily have @xmath63, and the set of separable t - states forms an octahedron within the tetrahedron @xcite. the t - state steering ellipsoid is centred at the origin, @xmath64, and the ellipsoid matrix is simply @xmath65, as follows from eqs. ([qse_centre]) and ([qse_mat]) with @xmath66. the semiaxes are @xmath67 for @xmath68, and are aligned with the @xmath69-axes of the bloch sphere. thus, the equation of the ellipsoid surface in spherical coordinates @xmath70 is @xmath71, with @xmath72 we find a remarkable connection between this equation and the epr - steerability of t - states in the following subsection. without loss of generality, consider measurement by alice of hermitian observables on her qubit. such observables can be equivalently represented via projections, @xmath73, with @xmath74. the probability of result ` @xmath18'and the corresponding steered bloch vector are given by eqs. ([qsteerb]) and ([pe]) with @xmath60, i.e., @xmath75 hence, letting @xmath76 denote the bloch vector corresponding to @xmath77 in eq. ([reduced]), then from eqs. ([reduced]) and, it follows there is an lhs model for bob if and only if there is a representation of the form @xmath78 for all unit vectors @xmath36. noting further that @xmath76 can always be represented as some mixture of unit vectors, corresponding to pure @xmath11, these conditions are equivalent to the existence of a representation of the form @xmath79 with integration over the bloch sphere. thus, the unit bloch vector @xmath80 labels both the local hidden state and the hidden variable. given lhs models for bob for any two t - states, having spin correlation matrices @xmath81 and @xmath82, it is trivial to construct an lhs model for the t - state corresponding to @xmath83, for any @xmath84, via the convexity property of nonsteerable states @xcite. our strategy is to find _ deterministic _ lhs models for some set of t - states, for which the result ` @xmath18'is fully determined by knowledge of @xmath80, i.e., @xmath85. lhs models can then be constructed for all convex combinations of t - states in this set. to find deterministic lhs models, we are guided by the fact that the steered bloch vectors @xmath86 are precisely those vectors that generate the surface of the quantum steering ellipsoid for the t - state @xcite. we make the ansatz that @xmath87 is proportional to some power of the function @xmath88 in eq. ([ell_eqn]) that defines this surface, i.e., @xmath89^m \equiv n_t\,\left[{\boldsymbol}n^\intercal t^{-2}{\boldsymbol}n\right]^{m/2}\]] for @xmath90, where @xmath91 is a normalisation constant. further, denoting the region of the bloch sphere, for which @xmath92 by @xmath93 $], the condition in eq. ([peint]) becomes @xmath94 } p({\boldsymbol}n) { \mathop{}\!\mathrm{d}}^2{\boldsymbol}n = \frac12 $]. we note this is automatically satisfied if @xmath95 is a hemisphere, as a consequence of the symmetry @xmath96 for the above form of @xmath87. hence, under the assumptions that (i) @xmath87 is determined by the steering ellipsoid as per eq. ([pform]), and (ii) @xmath93 $] is a hemisphere for each unit vector @xmath36, the only remaining constraint to be satisfied by a deterministic lhs model for a t - state is eq. ([beint]), i.e., @xmath97 } \left [{ \boldsymbol}n^\intercal t^{-2 } { \boldsymbol}n\right]^{m/2}\,{\boldsymbol}n { \mathop{}\!\mathrm{d}}^2{\boldsymbol}n = \frac12 t{\boldsymbol}e,\]] for some suitable mapping @xmath98 $]. extensive numerical testing, with different values of the exponent @xmath99, show that this constraint can be satisfied by the choices @xmath100= \{{\boldsymbol}n : { \boldsymbol}nt^{-1 } { \boldsymbol}e\geq 0\},\]] for a two - parameter family of t - states. assuming the numerical results are correct, it is not difficult to show, using infinitesimal rotations of @xmath36 about the @xmath101-axis, that this family corresponds to those t - states that satisfy @xmath102 fortunately, we have been able to confirm these results analytically by explicitly evaluating the integral in eq. ([con]) for @xmath103 (see appendix a). an explicit form for the corresponding normalisation constant @xmath91 is also given in appendix a, and it is further shown that the family of t - states satisfying eq. ([surface]) is equivalently defined by the condition @xmath104 this may be interpreted geometrically in terms of the harmonic mean radius of the ` inverse'ellipsoid @xmath105 being equal to @xmath106. equation ([surface]) defines a surface in the space of possible @xmath27 matrices, plotted in fig. 1(a) as a function of the semiaxes @xmath107 and @xmath108. as a consequence of the convexity of nonsteerable states (see above), all t - states corresponding to the region defined by this surface and the positive octant have local hidden state models for bob. also shown is the boundary of the separable t - states (@xmath109 @xcite), in red, which is clearly a strict subset of the nonsteerable t - states. the green plane corresponds to the sufficient condition @xmath110 for epr - steerable states, derived in sec. 5 below. it follows that a necessary condition for a t - state to be epr - steerable by alice is that it corresponds to a point above the sandwiched surface shown in fig. note that this condition is in fact symmetric between alice and bob, since their steering ellipsoids are the same for t - states. because of the elegant relation between our lhs model and the steering ellipsoid, and other evidence given below, we conjecture that this condition is also _ sufficient _ for epr - steerability. . * top figure (a) * : the red plane separates separable (left) and entangled (right) t - states. the sandwiched blue surface corresponds to the necessary condition for epr - steerability generated by our deterministic lhs model in sec. 4b : all t - states to the left of this surface are not epr - steerable. we conjecture that this condition is also sufficient, i.e., that all states to the right of the blue surface are epr - steerable. for comparison, the green plane corresponds to the sufficient condition for epr - steerability in eq. ([linsuff]) of section 5a : all t - states to the right of this surface are epr - steerable. only a portion of the surfaces are shown, as they are symmetric under permutations of @xmath111. * bottom figure (b) * : cross section through the top figure at @xmath112, where the necessary condition can be determined analytically (see sec. the additional black dashed curve corresponds to the non - linear sufficient condition for epr - steerability in eq..,title="fig:",width=336]. * top figure (a) * : the red plane separates separable (left) and entangled (right) t - states. the sandwiched blue surface corresponds to the necessary condition for epr - steerability generated by our deterministic lhs model in sec. 4b : all t - states to the left of this surface are not epr - steerable. we conjecture that this condition is also sufficient, i.e., that all states to the right of the blue surface are epr - steerable. for comparison, the green plane corresponds to the sufficient condition for epr - steerability in eq. ([linsuff]) of section 5a : all t - states to the right of this surface are epr - steerable. only a portion of the surfaces are shown, as they are symmetric under permutations of @xmath111. * bottom figure (b) * : cross section through the top figure at @xmath112, where the necessary condition can be determined analytically (see sec. the additional black dashed curve corresponds to the non - linear sufficient condition for epr - steerability in eq..,title="fig:",width=259] when @xmath113 we can solve eq. ([surface]) explicitly, because the normalisation constant @xmath91 simplifies. the corresponding equation of the @xmath108 semiaxis, in terms of @xmath114, is given by @xmath115 ^{-1 } & u < 1, \\ \left[1 - \frac { \sqrt{1-u^{-2}}}{2(u^2 - 1)}\ln\frac{|1-\sqrt{1-u^{-2}}|}{1+\sqrt{1-u^{-2 } } } \right] ^{-1 } & u > 1, \end{array } \right.\end{aligned}\]] and @xmath116 for @xmath117. 1(b) displays this analytic epr - steerable curve through the t - state subspace @xmath118, showing more clearly the different correlation regions. the symmetric situation @xmath119 corresponds to werner states. our deterministic lhs model is for @xmath120 in this case, which is known to represent the epr - steerable boundary for werner states @xcite. thus, our model is certainly optimal for this class of states. Sufficient conditions for epr-steerability in the previous section a strong necessary condition for the epr - steerability of t - states was obtained, corresponding to the boundary defined in eq. ([surface]) and depicted in fig. 1. while we have conjectured that this condition is also sufficient, it is not actually known if all t - states above this boundary are epr - steerable. here we give two sufficient general conditions for epr - steerability, and apply them to t - states. these conditions are examples of epr - steering inequalities, i.e., statistical correlation inequalities that must be satisfied by any lhs model for bob @xcite. thus, violation of such an inequality immediately implies that alice and bob must share an epr - steerable resource. our first condition is based on a new epr - steering inequality for the spin covariance matrix, and the second on a known nonlinear epr - steering inequality @xcite. both epr - steering inequalities are further of interest in that they are asymmetric under the interchange of alice and bob s roles. suppose alice and bob share a two - qubit state with spin covariance matrix @xmath121 given by @xmath122 and that each can measure any hermitian observable on their qubit. we show in appendix [app : cov] that, if there is an lhs model for bob, then the singular values @xmath123, @xmath124, @xmath125 of the spin covariance matrix must satisfy the linear epr - steering inequality @xmath126 from @xmath127, and using @xmath60 and @xmath128 for t - states, it follows immediately that one has the simple _ sufficient _ condition @xmath129 for the epr - steerability of t - states (by either alice or bob). the boundary of t - states satisfying this condition is plotted in figs. 1 (a) and (b), showing that the condition is relatively strong. in particular, it is a tangent plane to the necessary condition at the point corresponding to werner states (which we already knew to be a point on the true boundary of epr - steerable states). however, in some parameter regions a stronger condition can be obtained, as per below. suppose alice and bob share a two - qubit state as before, where bob can measure the observables @xmath130, @xmath131 on his qubit, with @xmath132, for any @xmath133 $], and alice can measure corresponding hermitian observables @xmath134 on her qubit, with outcomes labelled by @xmath135. it may then be shown that any lhs model for bob must satisfy the epr - steering inequality @xcite @xmath136,\end{aligned}\]] where @xmath137 denotes the probability that alice obtains result @xmath138, and @xmath139 is bob s corresponding conditional expectation value for @xmath130 for this result. as per the first part of sec. 4a, we may always choose a representation in which the spin correlation matrix @xmath27 is diagonal, i.e., @xmath140 $], without loss of generality. making the choices @xmath141 and @xmath142 in this representation, then @xmath143 and @xmath139 are given by @xmath33 and the third component of @xmath144 in eqs. ([pe]) and ([qsteerb]), respectively, with @xmath145. hence, the above inequality simplifies to @xmath146,\end{aligned}\]] where @xmath147 and @xmath148 are the third components of alice and bob s bloch vectors @xmath48 and @xmath46. for t - states, recalling that @xmath149, the above inequality simplifies further, to the nonlinear inequality @xmath150 hence, since similar inequalities can be obtained by permuting @xmath151, we have the _ sufficient _ condition @xmath152 for the epr - steerability of t - states. the boundary of t - states satisfying this condition is plotted in fig. 1(b) for the case @xmath112. it is seen to be stronger than the linear condition in eq. ([linsuff]) if one semiaxis is sufficiently large. the region below both sufficient conditions is never far above the smooth curve of our necessary condition, supporting our conjecture that the latter is the true boundary. Recapitulation and future directions in this paper we have considered steering for the set of two - qubit states with maximally mixed marginals (` t - states'), where alice is allowed to make arbitrary projective measurements on her qubit. we have constructed a lhv lhs model (lhv for alice, lhs for bob), which describes measurable quantum correlations for all separable, and a large portion of non - separable, t - states. that is, this model reproduces the steering scenario, by which alice s measurement collapses bob s state to a corresponding point on the surface of the quantum steering ellipsoid. our model is constructed using the steering ellipsoid, and coincides with the optimal lhv lhs model for the case of werner states. furthermore, only a small (and sometimes vanishing) gap remains between the set of t - states that are provably non - steerable by our lhv lhs model, and the set that are provably steerable by the two steering inequalities that we derive. as such, we conjecture that this lhv lhs model is in fact optimal for t - states. proving this, however, remains an open question. a natural extension of this work is to consider lhv lhs models for arbitrary two - qubit states. how can knowledge of their steering ellipsoids be incorporated into such lhv lhs models? investigations in this direction have already begun, but the situation is far more complex when alice and bob s bloch vectors have nonzero magnitude and the phenomenon of `` one - way steering '' may arise @xcite. finally, our lhv lhs models apply to the case where alice is restricted to measurements of hermitian observables. it would be of great interest to generalize these to arbitrary povm measurements. however, we note that this is a very difficult problem even for the case of two - qubit werner states @xcite. nevertheless, the steering ellipsoid is a depiction of all collapsed states, including those arising from povms (they give the interior points of the ellipsoid) and perhaps this can provide some intuition for how to proceed with this generalisation. sj would like to thank david jennings for his early contributions to this project. sj is funded by epsrc grant ep / k022512/1. this work was supported by the australian research council centre of excellence ce110001027 and the european union seventh framework programme (fp7/2007 - 2013) under grant agreement n@xmath153 [316244]. Details of the deterministic lhs model the family of t - states described by our deterministic lhs model in sec. 4b corresponds to the surface defined by either of eqs. ([surface]) and ([surface2]). this is a consequence of the following theorem, proved further below. for any full - rank diagonal matrix @xmath27 and nonzero vector @xmath154 one has @xmath155 note that substitution of eq. ([re]) into constraint ([con]) immediately yields eq. ([surface]) via the theorem (with @xmath156). further, taking the dot product of the integral in the theorem with @xmath157, multiplying by @xmath91, and integrating @xmath157 over the unit sphere, yields (reversing the order of integration) @xmath158 whereas carrying out the same operations on the righthand side of the theorem yields @xmath159. equating these immediately implies the equivalence of eqs. ([surface]) and ([surface2]) as desired. an explicit analytic formula for the normalisation constant @xmath91 is given at the end of this appendix. first, define @xmath160 ; that is, @xmath161 and @xmath162 noting @xmath157 in the theorem may be taken to be a unit vector without loss of generality, we will parameterise the unit vectors @xmath80 and @xmath157 by @xmath163 with @xmath164 $] and @xmath165. thus, @xmath166. further, without loss of generality, it will be assumed that @xmath167 points into the northern hemisphere, so that @xmath168. then @xmath169 $] and @xmath170. the surface of integration is a hemisphere bounded by the great circle @xmath171. in the simple case where @xmath172, the boundary curve has the parametric form @xmath173 for @xmath174. hence, the boundary curve in the generic case can be constructed by applying the orthogonal operator @xmath175, that rotates @xmath167 from @xmath176 to @xmath177, to the vector @xmath178. that is, @xmath179 and the boundary curve has the form @xmath180 for the purposes of integrating over the hemisphere, it is convenient to vary @xmath181 from @xmath182 to @xmath183 and @xmath184 from @xmath182 to its value @xmath185 on the boundary curve. from the above expression for the boundary, and using @xmath186 and @xmath187, it follows that @xmath188 and @xmath189. the last equation be rearranged to read @xmath190, and after squaring both sides this equation solves to give @xmath191^{1/2}}.\]]now, @xmath192 assumes its maximum value when @xmath193, which according to the relation @xmath188 and the fact that @xmath194 $] should correspond to @xmath195. so we take the upper sign in the last equation, yielding @xmath196^{1/2}}\\ & = \frac{-\sin \alpha\cos (\phi -\beta)}{[\cos ^{2}\alpha+\sin ^{2}\alpha\cos ^{2}(\phi -\beta)]^{1/2 } }.\end{aligned}\]] it follows immediately that @xmath197^{1/2}},\end{aligned}\]] with the choice of sign fixed by the fact that @xmath198 and (by assumption) @xmath168. the surface integral for @xmath199 in eq. ([qv]) can now be written in the form : @xmath200 to evaluate the the third component of @xmath199, note that the integral over @xmath184, @xmath201can be evaluated explicitly by making the substitution @xmath202, as @xmath203 for any @xmath204, yielding @xmath205 after inserting the expressions for @xmath206 and @xmath207 derived earlier, we have @xmath208 we now need to integrate the last expression over @xmath181. introducing new constants @xmath209 the full surface integral simplifies to a form that may be evaluated by mathematica (or by contour integration over the unit circle in the complex plane) : @xmath210the indeterminate sign here is fixed by examining the case @xmath211 and @xmath212, for which @xmath213 and the integrand reduces to @xmath214, which integrates to give @xmath215. so, unsurprisingly, we choose the positive sign. this yields the third component of surface integral to be @xmath216_3 = \frac{\pi \cos \alpha}{c[ab\cos ^{2}\alpha+c(a\sin ^{2}\beta+b\cos ^{2}\beta)\sin ^{2}\alpha]^{1/2}}.\end{aligned}\]] the integrals over @xmath184 in the remaining two components of @xmath199 in eq. ([qvnew]) are unfortunately not so straightforward. however, there is a simple trick that allows us to calculate both surface integrals explicitly, and that is to differentiate the integrals with respect to the parameters @xmath217 and @xmath218. since the only dependence on @xmath217 and @xmath218 comes through the function @xmath219, this eliminates the need to integrate over @xmath184. in fact we only need to differentiate with respect to one of these parameters, choose @xmath217. to see this, note that @xmath220where @xmath221 can be evaluated by making use of the equations and. in fact, @xmath222^{1/2}}\right)\end{aligned}\]] @xmath223^{3/2}}.\end{aligned}\]] inserting the last two equations and the expressions for @xmath224 and @xmath225 into the integrals above, and using the constants @xmath226 and @xmath227 defined earlier, then gives : @xmath228^{2}}{\mathop{}\!\mathrm{d}}\phi \nonumber \\ \label{xy1 } ~&=&\cos ^{2}\alpha\int_{0}^{2\pi } \frac{(\sin \beta\sin \phi \cos \phi + \cos \beta\cos ^{2}\phi, \sin \beta\sin ^{2}\phi + \cos \beta\sin \phi \cos \phi) } { (l \cos ^{2}\phi + m \sin ^{2}\phi + 2n \sin \phi \cos \phi) ^{2}}{\mathop{}\!\mathrm{d}}\phi.\end{aligned}\]] consequently, there are three separate integrals we need to evaluate and these can be done in mathematica (or by complex contour integration) : @xmath229 using the values we have for @xmath230 we substitute these back into equation and integrate over @xmath217 to obtain @xmath231_1 = \pi\int \frac{\cos ^{2}\alpha(m\cos \beta - n\sin \beta) } { (lm - n^2)^{3/2}}d\alpha\end{aligned}\]] @xmath232^{1/2}},\end{aligned}\]] and @xmath233_2 = \pi\int \frac{\cos ^{2}\alpha(l\sin \beta -n\cos \beta) } { (lm - n^2)^{3/2}}d\alpha\\ \label{qv2 } & = \frac{b^{-1}\pi \sin\alpha\sin\beta}{[ab \cos^2\alpha + c\sin^2\alpha(b\cos^2\beta + a\sin^2\beta)]^{1/2}}.\end{aligned}\]] the absence of integration constants can be confirmed by noting that these expressions vanish for @xmath211 i.e., when the vector @xmath234 is aligned with the @xmath101-axis as they should by symmetry. note the denominators of eqs. and simplify to @xmath235. combining this with eqs. ([qv3]) and ([qv1])-([qv2]), we have @xmath236 and so setting @xmath237, the theorem follows as desired. finally, the normalisation constant @xmath91 in eq. ([surface]) may be analytically evaluated using mathematica. under the assumption that @xmath238, denote @xmath239. we find @xmath240+a(b+c)k[c]+ib(c - a)(e[a_1,b]-e[a_2,b])+ic(a+b)(f[a_1,b]-f[a_2,b])\right\}\big),\end{aligned}\]] [app:cov]epr-steering inequality for spin covariance matrix to demonstrate the linear epr - steering inequality in eq. ([lin]), let @xmath253 denote some dichotomic observable that alice can measure on her qubit, with outcomes labelled by @xmath135, where @xmath157 is any unit vector. we will make a specific choice of @xmath253 below. define the corresponding covariance function @xmath254 if there is an lhs model for bob then, noting that one may take @xmath255 in eq. ([lhs]) to be deterministic without loss of generality, there are functions @xmath256 such that @xmath257\,[{\boldsymbol}n(\lambda)-{\boldsymbol}b]\cdot{\boldsymbol}v$], where @xmath258, and the hidden state @xmath11 has corresponding bloch vector @xmath76. now, the bloch sphere can be partitioned into two sets, @xmath259\cdot{\boldsymbol}v \geq 0\}$] and @xmath260\cdot{\boldsymbol}v < 0\}$], for each value of @xmath7. hence, noting @xmath261, @xmath262 is equal to @xmath263\,[{\boldsymbol}n(\lambda)-{\boldsymbol}b]\cdot{\boldsymbol}v \right.\\ & ~ & \left. + \int_{s_-(\lambda) } { \mathop{}\!\mathrm{d}}^2{\boldsymbol}v\, [\alpha_{{\boldsymbol}v}(\lambda) - \bar{\alpha}_{{\boldsymbol}v}]\,[{\boldsymbol}n(\lambda)-{\boldsymbol}b]\cdot{\boldsymbol}v \right\}\end{aligned}\]] @xmath264\,[{\boldsymbol}n(\lambda)-{\boldsymbol}b]\cdot{\boldsymbol}v \right.\\ & ~ & -\left. \int_{s_-(\lambda) } { \mathop{}\!\mathrm{d}}^2{\boldsymbol}v\, [1 + \bar{\alpha}_{{\boldsymbol}v}]\,[{\boldsymbol}n(\lambda)-{\boldsymbol}b]\cdot{\boldsymbol}v \right\}\\ & = & \sum_\lambda p(\lambda) \int { \mathop{}\!\mathrm{d}}^2{\boldsymbol}v\, |[{\boldsymbol}n(\lambda)-{\boldsymbol}b]\cdot{\boldsymbol}v|\end{aligned}\]] @xmath265\cdot{\boldsymbol}v\\ & = & \sum_\lambda p(\lambda) |{\boldsymbol}n(\lambda)-{\boldsymbol}b|\,\int { \mathop{}\!\mathrm{d}}^2{\boldsymbol}v\, |{\boldsymbol}v\cdot { \boldsymbol}w(\lambda)|,\end{aligned}\]] where @xmath266 denotes the unit vector in the @xmath267 direction, and the last line follows by interchanging the summation and integration in the second term of the previous line. the integral in the last line can be evaluated for each value of @xmath7 by rotating the coordinates such that @xmath268 is aligned with the @xmath101-axis, yielding @xmath269. hence, the above inequality can be rewritten as @xmath270^{1/2 } \\ \label{cav } & \leq & \frac{1}{2}\sqrt{1-{\boldsymbol}b\cdot{\boldsymbol}b},\end{aligned}\]] where the second and third lines follow using the schwarz inequality and @xmath271, respectively. note, by the way, that the first inequality is tight for the case @xmath272\cdot{\boldsymbol}v\right)$]. now, making the choice @xmath273 with @xmath274, one has from eqs. ([cjk]) and ([cv]) that @xmath275 combining with eq. ([cav]) immediately yields the epr - steering inequality @xmath276 finally, this inequality may similarly be derived in a representation in which local rotations put the spin covariance matrix @xmath121 in diagonal form, with coefficients given up to a sign by the singular values of @xmath121 (similarly to the spin correlation matrix @xmath27 in sec. 4a). since @xmath277 is invariant under such rotations, eq. ([lin]) follows. a. j. bennet, d. a. evans, d. j. saunders, c. branciard, e. g. cavalcanti, h. m. wiseman, and g. j. pryde, `` arbitrarily loss - tolerant einstein - podolsky - rosen steering allowing a demonstration over 1 km of optical fiber with no detection loophole, '' phys. x * 2 *, 031003 (2012). c. branciard, e. g. cavalcanti, s. p. walborn, v. scarani, and h. m. wiseman, `` one - sided device - independent quantum key distribution : security, feasibility, and the connection with steering, '' phys. a * 85 *, 010301 (2012). for ellipsoids of separable states, there is a further ambiguity in the ` chirality'of alice s local basis, that is, we may determine @xmath278 up to a local unitary and a partial transpose on alice s system @xcite.

the question of which two - qubit states are steerable (i.e. permit a demonstration of epr - steering) remains open. here , a strong necessary condition is obtained for the steerability of two - qubit states having maximally - mixed reduced states, via the construction of local hidden state models. it is conjectured that this condition is in fact sufficient. two provably sufficient conditions are also obtained, via asymmetric epr - steering inequalities. our work uses ideas from the quantum steering ellipsoid formalism, and explicitly evaluates the integral of @xmath0 over arbitrary unit hemispheres for any positive matrix @xmath1.

1411.1517

Introduction superconducting josephson junction qubit is one of the most promising candidates for implementing quantum computation @xcite. single qubit coherent oscillations in superconducting qubits have been demonstrated experimentally @xcite and furthermore two qubit coupling and entanglement have been performed in charge @xcite, flux @xcite and phase @xcite qubits. scalable quantum computing requires controllable and selective coupling between two remote as well as nearest neighbor qubits. recently much theoretical efforts have been devoted on the study about the controllable coupling of charge @xcite, charge - phase @xcite and flux qubits @xcite. for flux qubits the controllable coupling schemes use inductive coupling, but it is too weak to perform efficient two - qubit gate operations. hence, while in superconducting charge qubit two - qubit coherent oscillations and cnot gate operations were experimentally observed @xcite, only spectroscopy measurement was done for inductively coupled flux qubit @xcite. in this study thus we suggest a scheme to give both strong and tunable coupling between two phase - coupled flux qubits. the phase - coupling scheme, which we previously proposed @xcite, has been realized in a recent experiment @xcite. the controllable coupling scheme using phase - coupled qubits with threading ac magnetic field was also studied theoretically @xcite. further, there have been studies about somewhat different phase - coupling schemes @xcite. two current states of a flux qubit are characterized by the induced loop current related with the phase differences across josephson junctions in the qubit loop. if we try to couple two flux qubits using mutual inductance, the coupling strength @xmath0 @xcite will be too weak to perform the discriminating cnot gate operations @xcite, since the mutual inductance @xmath1 and the induced currents of the left (right) qubit @xmath2 is very small. even though the induced currents of flux qubits are weak, the phase differences across josephson junctions @xmath3 are as large as @xmath4. hence, if two flux qubits are coupled by the phase differences between two josephson junctions of different qubits, we can achieve a strong coupling of the order of josephson coupling energy @xmath5 of the josephson junctions in the connecting loop whose typical value is as large as up to about 200ghz. introducing two dc - squid s interrupting the connecting loop as shown in fig. [coup3jjs] we can control the coupling between phase - coupled flux qubits. the control fluxes, @xmath6 and @xmath7, threading two dc - squid loops must be in _ opposite _ directions in order to give rise to the controllable coupling. when two fluxes are in the _ same _ direction, the change of control fluxes induces an additional current flowing in the connecting loop, causing the shift of qubit states as well as the change of coupling strength. such a dilemma also persists in the case of _ one _ dc - squid loop in connecting loop. however, if the control fluxes are in opposite directions, we have found that the additional currents coming from two dc - squid s are cancelled each other and thus the coupling strength can be tunable remaining the qubit states unchanged. Phase-coupling of flux qubits the three - josephson junctions qubits @xcite in fig. [coup3jjs] has two current states ; if the qubit current @xmath8, it is diamagnetic while, if @xmath9, paramagnetic. introducing the notation @xmath10 @xmath11 for diamagnetic (paramagnetic) current state of a qubit in pseudo spin language, there can be four current states of coupled qubits, @xmath12, @xmath13, @xmath14 and @xmath15, of which we show one of the same current states, @xmath12, and one of the different current states, @xmath14, in fig. [coup3jjs]. the phase @xmath16 and @xmath17 of the josephson junctions of the three - josephson junctions qubits have different values if two qubits are in different states. then the phase difference @xmath18 induces the phases @xmath19 in the josephson junctions of dc - squid loops. if we neglect small kinetic inductance, the boundary conditions of the left (right) qubit and the connecting loop can be approximately written as @xmath20 where @xmath21 is total flux and @xmath22 with the external flux @xmath23 and the unit flux quantum @xmath24 is dimensionless reduced flux threading the left (right) qubit. here @xmath25 with the self inductance @xmath26 and the induced current @xmath2 of qubit loop is the induced flux of each qubit and @xmath27 that of the connecting loop and @xmath28 and @xmath29 are integers. we consider that the external fluxes @xmath30 and @xmath31 threading the qubit loops are also in opposite directions, since they are connected in a twisted way in the scalable design of ref.. however, for just two qubit coupling, we can choose the directions of external fluxes threading the qubit loops arbitrary. actually there is no external flux in the connecting loop, but the phase difference @xmath32 in the boundary condition of eq. ([loopbc]) plays the role of effective flux in the connecting loop, @xmath33 when two qubits are in different current state, i.e., one is diamagnetic and the other paramagnetic, the value of @xmath34 becomes @xmath35. since the induced flux of flux qubit is so weak as @xmath36, large value of @xmath37 in the phase - coupled flux qubits can give a strong coupling compared to the inductive coupling scheme. the hamiltonian of the coupled qubits can be given by @xmath38 which describes dynamics of a particle with effective mass @xmath1 in the effective potential @xmath39 with @xmath40. the kinetic part of the hamiltonian comes from the charging energy of the josephson junctions such as @xmath41 where @xmath42 and @xmath43 is the capacitance of the josephson junctions of the left (right) qubit loop and the connecting loop, respectively. the number of excess cooper pair charges on josephson junction @xmath44 @xmath45 is conjugate to the phase difference @xmath46 such as @xmath47=i$], where @xmath48, @xmath49 and @xmath50 the capacitance of the josephson junctions. here we introduce the canonical momentum @xmath51 and the effective mass @xmath52 @xmath53 to obtain the kinetic part of the hamiltonian. the effective potential of the coupled qubits is composed of the inductive energy of loops and the josephson junction energy terms ; @xmath54 here @xmath55 is the inductive energy of loops with the current of the right qubit @xmath56, left qubit @xmath57 and connecting loop @xmath58. @xmath59 is the energy of the josephson junctions in two qubit loop and @xmath60 that of the connecting loop with josephson coupling energies @xmath61 and @xmath5. introducing another rotated coordinate @xmath76 and using the boundary conditions in eq. ([switchbc]) to get @xmath77 the josephson junction energy of the connecting loop @xmath78 can also be written as @xmath79\cos[\phi'_p+\pi(f'_l - f'_r)],\nonumber\end{aligned}\]] where we set @xmath80 and @xmath81. Coupled-qubit states in effective potential first of all we consider the case that the control fluxes, @xmath6 and @xmath7, have opposite directions such that @xmath82 note that the boundary conditions in eq. ([switchbc]) already have opposite signs. in order to obtain the effective potential as a function of @xmath83 and @xmath84, we reexpress @xmath85 as @xmath86 using the boundary conditions in eq. ([loopbc]) and the expression in eq. ([phi1]). here @xmath87 can be written as @xmath88 neglecting small induced flux @xmath89 in the connecting loop. depending on whether @xmath90 is even or odd, the results will be quantitatively different, but qualitatively the same. here and after, thus, we choose @xmath90 is even and specifically @xmath91, @xmath92 and @xmath93 for simplicity and then @xmath85 becomes @xmath94 and the energy of josephson junctions in connecting loop in eq. ([uswi]) @xmath95.\end{aligned}\]] since the induced energy @xmath55 can be negligible, the total effective potential @xmath96 in eq. ([usum]) is given by the sum of the energies in eqs. ([uqubit]) and ([uswidiff]), @xmath97 the lowest energy level of @xmath98 in @xmath99 plane can be obtained by setting the remaining variable @xmath100 in eq. ([uswidiff]) as @xmath101 when @xmath102. we plot the effective potential @xmath103 in fig. [potential] (a) with four local minima. the value of local minima of case (i) can be obtained from @xmath104 and we have found that two local minima have the same value, @xmath105, for equal pseudo - spin state with @xmath106. similarly we get @xmath107 of case (ii) from @xmath108 for different pseudo - spin state. as a result, we obtain @xmath109 where @xmath110 is the value of @xmath111 at local minima of the same spin states, @xmath112, and @xmath113 the value of @xmath114 of the different spin states, @xmath115 with @xmath116 thus the energy of the same spin states, @xmath117, is lower than that of different spin states, @xmath118, as shown in fig. [potential](a). here we set @xmath119, @xmath120 and @xmath70. for the same spin states we have two solutions, @xmath121, corresponding to two local minima, @xmath122 and @xmath123. when @xmath124, @xmath125 and @xmath126 and thus @xmath127 using @xmath128 from the boundary condition in eq. ([qubitbc]) with @xmath129. since the loop currents of both qubits then @xmath130 are diamagnetic as can be seen from fig. [coup3jjs](a), this coupled - qubits state can be represented as @xmath12 as shown in fig. [potential](a). on the other hand, when @xmath131, @xmath132 then the qubit current @xmath133 corresponds to the paramagnetic current states, @xmath13. we would like to note that, since the external fluxes @xmath30 and @xmath31 threading left and right qubit loops are already in opposite directions, diamagnetic (paramagnetic) currents of both qubits in @xmath12 (@xmath13) state are also in opposite directions..the values of phase differences of coupled qubits states in several coordinates with @xmath119, @xmath120, @xmath70 and @xmath102. [cols="^,^,^,^",options="header ",] for different spin states, two solutions are also obtained for @xmath134. when @xmath135, @xmath136 for left and right qubit respectively, which corresponds to the state, @xmath14 in fig. [potential] (a). in the same way @xmath137 corresponds to the state @xmath15. hence we can identify four stable states, @xmath138 and @xmath15, with energies @xmath139 and @xmath140 at the local minima of @xmath103 as shown in fig. [potential] (a). even though above four states are stable states in @xmath141 plane, it can be unstable in the other dimensions if they are saddle points. thus we need represent the effective potential @xmath98 in @xmath142 plane. from the expression of @xmath143 in eq. ([ump]) and the relation @xmath144 in eq. ([rel]), we can get @xmath145 and, following similar procedure as in the @xmath141 plane, we obtain the effective potential as shown in fig. [potential] (b), where we can again see local minima. in figs. [potential] (b), for the states @xmath12 and @xmath13 of case (i), we can get the values @xmath146 and, for case (ii), the values @xmath147 using eqs. ([rel]), ([i]) and ([ii]) and the values of @xmath83 in each case. as a result, we are able to identify the spin states at local minima of figs. [potential] (b) from fig. [potential] (a) with above values and confirm the stability of the states in both planes. in table [table] we summarize the values of the phase differences for four states, @xmath148, of coupled qubits in several coordinates. actually we obtained higher energy states in fig. [potential] (a), but found that they are unstable in @xmath142 plane. Tunable coupling of flux qubits the hamiltonian of coupled qubits can be written as @xmath149 where @xmath150 and @xmath151 with @xmath152 and @xmath153 is the @xmath154 identity matrix. first two terms are qubit terms, the third is coupling term and last two terms are tunnelling terms which come from the quantum fluctuation described by the kinetic term of the hamiltonian. then the coupling constant @xmath155 of the coupled qubits is given by @xcite @xmath156 in fig. [contpot] we plot the energies of coupled - qubits for various @xmath157 with @xmath120, @xmath119 and @xmath70 in @xmath158 plane. when @xmath102 in fig. [contpot](a), the energies @xmath139 of the same spin states, @xmath12 and @xmath13, are lower than @xmath140, of the different spin states, @xmath14 and @xmath15. the positions of four local minima are shown in table [table]. as increases @xmath157, the energy difference @xmath159 becomes smaller (upper panel in (b)) and finally @xmath160 at @xmath161 in (c). since @xmath162 and @xmath163, the coupling strength can be written as @xmath164 therefore the coupling strength between two flux qubits changes as varying the control fluxes @xmath157 threading the dc - squid loop in the connecting loop. from eq. ([umin]) the coupling constant @xmath155 can be represented as a function of @xmath157 by @xmath165, which gives @xmath166 in fig. [twodiff](a) we plot the energies @xmath167 and @xmath168 as a function of @xmath157, where @xmath169. when @xmath102, @xmath155 is of the order of @xmath5 so that we can obtain a sufficiently strong coupling. by adjusting @xmath157 the coupling strength can be tuned from strong coupling to zero at @xmath161. the coupling strength @xmath168 in eq. ([j]) depends on @xmath170 as well as @xmath5. when @xmath171 is small, @xmath168 is proportional to @xmath5 and @xmath172. recently the phase - coupling scheme has been experimentally implemented @xcite, where four - josephson junctions qubits are employed instead of usual three - josephson junctions qubits. in that experiment the josephson junction energy @xmath173 of fourth junction is large so that the value of @xmath170 is about @xmath174. as a result, the experiment exhibits rather small coupling strength. the current of connecting loop can be written as @xmath175, which gives the relations, @xmath176 and then @xmath177 with integer @xmath178 using the boundary conditions in eq. ([switchbc]). then, using eq. ([phi2]) and the effective flux @xmath34 in eq. ([rel]), the current @xmath179 is given by @xmath180 this current - phase relation can be considered as the josephson junction type relation, @xmath181, with the effective josephson coupling energy, @xmath182, of two dc - squid s in the connecting loop @xmath183 and the phase difference @xmath184. the coupling constant in eq. ([j]) also can be represented by the effective josephson coupling energy, @xmath182. thus the large phase difference, @xmath185, and the josephson coupling energy, @xmath182, induce the current in the connecting loop and the coupling energy of the phase - coupled qubits. for the same spin states, @xmath12 and @xmath13, the current of connecting loop @xmath58 becomes zero, since @xmath186 and thus @xmath187. for a different spin states @xmath14 with @xmath102, @xmath188 and we have @xmath189 from @xmath190 and the relation in eq. ([k]). then weak current @xmath58 in the connecting loop flows satisfying current conservation condition between left qubit and connecting loop such that @xmath191 for @xmath102. when @xmath157 approaches @xmath192, the effective josephson coupling energy @xmath193 and thus the current @xmath58 in connecting loop become zero, which means that the coupling between two qubits is switched off. now we want to explain the case that two control fluxes are in the same directions and the case that there is only single dc - squid in connecting loop. if two control fluxes are in the same direction such as @xmath194 the josephson junction energy of the connecting loop becomes @xmath195. \nonumber\\\end{aligned}\]] similar procedure as in the case of opposite directions of control fluxes shows that the same spin states, @xmath12 and @xmath13, have equal energy such as @xmath196 for @xmath197. for different spin states, @xmath14 and @xmath15, the energies @xmath198 and @xmath199 are obtained at two local minima @xmath200 which can be derived from eq. ([swisame]). since the states, @xmath14 and @xmath15, have different sign for @xmath113, the second term produces the energy difference @xmath201 where @xmath113 is again one of the values of @xmath114 for the different spin states. figure [contpot](b) (lower panel) for @xmath202 shows that, when two control fluxes are in the same direction, the energies @xmath198 and @xmath199 are different while @xmath203. the energy levels of @xmath204 are plotted in fig. [twodiff](b). in this case the effective fluxes @xmath205 and @xmath206 applied to left and right qubits in the hamiltonian of eq. ([hcoup]) become different each other, @xmath207, as @xmath157 increases from zero. for the different current state in fig. [coup3jjs](b), if the control fluxes @xmath6 and @xmath7 threading the dc - squid loops are in the same direction, the increased current @xmath58 in the connecting loop will flow through the left and right qubit loops. thus the qubit states are influenced by additional effective fluxes, which will makes the two - qubit operations difficult. however, if two control fluxes @xmath6 and @xmath7 are in opposite directions, the energies of different spin states remains equal to each other, @xmath208, as shown in fig. [twodiff](a). this means that the additional currents coming from two dc - squid s are cancelled each other and total additional current induced by the control fluxes @xmath6 and @xmath7 is vanishing in the connecting loop. as a result, the net effect is just renormalizing the coupling constant @xmath155 of the coupled qubit system. we also calculated energies of coupled qubit states with single dc - squid loop whose boundary conditions become @xmath209 instead of those in eqs. ([loopbc]) and ([switchbc]). then we get the josephson junction energies of the dc - squid, @xmath210 which gives results similar to those of two dc - squid s with fluxes in the same direction such that @xmath211 for @xmath197 and @xmath212 as shown in fig. [twodiff](c). hence the behaviors of one dc - squid in the connecting loop are qualitatively the same as those of two dc - squid s with fluxes in the same direction. therefore we need two control fluxes threading dc - squid s in opposite directions to cancel the additional currents in the connecting loop for obtaining the controllable coupling. in order to obtain the controllable coupling both the qubit operating flux, @xmath213, and control flux, @xmath214, of the left qubit become in opposite direction to those of the right qubit, @xmath31 and @xmath7 as shown in fig. [coup3jjs]. in real experiments it will be very hard to apply magnetic fluxes of different directions simultaneously. we have previously suggested a scalable design for phase - coupled flux qubits @xcite, where an arbitrary pair of qubits are coupled in a twisted way. thus just applying all magnetic fluxes in the same direction makes automatically the effect of fluxes in opposite directions, removing the experimental difficulty. the recent experiment on the phase - coupled flux qubits without dc - squid loop @xcite has shown that the coupled qubit states are in quantum mechanically superposed regime. the dc - squid loops in the connecting loop of the present tunable coupling scheme may cause a decoherence effect on the coupled qubit states. a recent study argued that the dc - squid based oscillator should be the main source of the decoherence of the flux qubits @xcite. for the scalable design in ref., however, the decoherence from two dc - squid s can be reduced. since two dc - squid s are connected in a twisted way, the fluctuations from tank circuit or flux lines can be cancelled each other. in realistic implementation of qubit operations, operating external fluxes are slightly different from the co - resonance point, @xmath70, and moreover we can not any more neglect small kinetic inductance and induced fluxes. hence we confirmed the results in this study by numerical calculation using the exact boundary conditions similar to those in eqs. ([qubitbc])@xmath215([switchbc]), current - phase relation @xmath216 and current conservation conditions @xcite. Summary controllable coupling between two phase - coupled flux qubits can be achieved by using two dc - squid s in the connecting loop with threading fluxes in opposite directions. we analytically show at co - resonance point (@xmath70) that the coupling strength of the phase - coupled flux qubits can be adjusted by varying the threading fluxes @xmath157 from @xmath217 to @xmath192 ; it can be as strong as o@xmath218 and zero in switching - off limit. when either two control fluxes are in the same directions or there is only one dc - squid in the connecting loop, the coupled qubits can not be described by the coupling hamiltonian. in slightly different parameter regimes of experimental implementations numerical calculations can be done to obtain exact results. 1 y. makhlin, g. schn, and a. shnirman, rev. phys. * 73 *, 357 (2001) ; a. galindo and m. a. martn - delgado, _ ibid. _ * 74 *, 347 (2002). y. nakamura, yu. a. pashkin, and j. s. tsai, nature * 398 *, 786 (1999). y. yu, s. han, x. chu, s. chu, and z. wang, science * 296 *, 889 (2002). i. chiorescu, y. nakamura, c. j. p. m. harmans, and j. e. mooij, science * 299 *, 1869 (2003). a. pashkin, t. yamamoto, o. astafiev, y. nakamura, d. v. averin, and j. s. tsai, nature * 421 *, 823 (2003). t. yamamoto, yu. a. pashkin, o. astafiev, y. nakamura, and j. s. tsai, nature * 425 *, 941 (2003). a. izmalkov, m. grajcar, e. iichev, th. wagner, h.- meyer, a.yu. smirnov, m. h. s. amin, alec maassen van den brink, and a.m. zagoskin, phys.. lett. * 93 *, 037003 (2004). m. grajcar, a. izmalkov, s. h. w. van der ploeg, s. linzen, e. iichev, th. wagner, u. hubner, h.- meyer, alec maassen van den brink, s. uchaikin, and a. m. zagoskin, phys. b * 72 *, 020503(r) (2005). j. b. majer, f. g. paauw, a. c. j. ter haar, c. j. p. m. harmans, and j. e. mooij, phys. rev. lett. * 94 *, 090501 (2005). a. j. berkley _ et al. _, science * 300 *, 1548 (2003). d. v. averin and c. bruder, phys. 91 *, 057003 (2003) ; j. q. you, j. s. tsai, and f. nori, phys b * 68 *, 024510 (2003). a. blais, alec maassen van den brink, and a. m. zagoskin, phys. lett. * 90 *, 127901 (2003). b. l. t. plourde _ et al. _, b * 70 *, 140501(r) (2004). p. bertet, c. j. p. m. harmans, and j. e. mooij, phys. b * 73 *, 064512 (2006). y.- x. liu, l. f. wei, j. s. tsai, and f. nori, phys. * 96 *, 067003 (2006). a. o. niskanen, y. nakamura, and j. s. tsai, phys. b * 73 *, 094506 (2006). j. e. mooij _ et al. _, science * 285 *, 1036 (1999) ; caspar h. van der wal _ et al. _, science * 290 *, 773 (2000). t. p. orlando _ et al. _, phys. rev. b * 60 *, 15398 (1999). m. d. kim, d. shin, and j. hong, phys. rev. b * 68 *, 134513 (2003).

we propose a scheme for tunable coupling of phase - coupled flux qubits. the phase - coupling scheme can provide a strong coupling strength of the order of josephson coupling energy of josephson junctions in the connecting loop, while the previously studied inductive coupling scheme can not provide due to small mutual inductance and induced currents. we show that, in order to control the coupling, we need _ two _ dc - squid s in the connecting loop and the control fluxes threading the dc - squid s must be in _ opposite _ directions. the coupling strength is analytically calculated as a function of the control flux at the co - resonance point.

cond-mat0602604

Introduction one of the most interesting topic of quantum physics in recent days is the characterization of universal properties of bosonic many - body system in the unitary regime @xcite. by using the feshbach resonance the two - body scattering length @xmath1 is tuned to very large values. the unitary regime is characterized by simple universal laws. for weakly interacting dilute bose gas, the gas like state becomes unstable as @xmath1 increases @xcite. however the efimov effect in quantum three - body systems leads to different concept of universality. efimov effect appears in the three - body level (@xmath0=3) where the attractive two - body interaction is such that the scattering length is much larger than the range of the interaction. under such condition, a series of weakly bound and spatially extended states of efimov character appears in the system. although the efimov character and ultracold behaviour of fermi gas is well understood, the exhaustive study of bosonic system with large scattering length are few. helium trimer @xmath2 is a well studied quantum three - body system in this direction @xcite, its first excited state is theoretically claimed as of efimov state, however no experimental observation is still reported. whereas the recent experimental observations of efimov phenomena in ultracold gases has drawn interest in the study of universality in few - body quantum systems @xcite but the extension of efimov physics for larger system (@xmath3) is not straightforward. there are several studies in this direction which predicted the universality of the system @xcite. though predictions and conclusions made in these works are qualitatively similar quantitative differences exist. this necessitates further study of universal properties of bosonic cluster state having efimov character. + in this work we consider few - bosonic clusters of @xmath4rb atoms interacting with van der waals interaction. our motivation also comes from recent experiments of weakly bound molecules created from ultracold bose gas. utilizing the feshbach resonance the effective interatomic interaction can be essentially tuned to any desired value. for weakly interacting dilute systems, the efimov state appears at unitary (@xmath5). our motivation is to study the near threshold behaviour of weakly bound three - dimensional clusters. to characterize this delicate system we prescribe two - body correlated basis function for the many - body cluster interacting through shape - dependent two - body van der waals potential. we expect that our present study will explore the generic behaviour of three - dimensional bosonic cluster near the unitary. the usage of realistic potential with a short range repulsive core and long - range attractive tail @xmath6 may give qualitative conclusion as before but different quantitative behaviours are expected. the paper is organized as follows. in sec. ii we discuss the many - body hamiltonian and numerical calculation. iii considers the results and exhibit the signature of universal cluster states with efimov character. iv concludes with a summary. Formalism we approximately solve the many - body schrdinger equation by potential harmonic expansion method (phem). we have successfully applied phem to study different properties of bose einstein condensate @xcite and atomic clusters @xcite. the method has been described in detail in our earlier works @xcite. we briefly describe the method below for interested readers. we consider a system of @xmath7 @xmath4rb atoms, each of mass @xmath8 and interacting via two - body potential. the hamiltonian of the system is given by @xmath9 here @xmath10 is the two - body potential and @xmath11 is the position vector of the @xmath12th particle. it is usual practice to decompose the motion of a many - body system into the motion of the center of mass where the center of mass coordinate is @xmath13 and the relative motion of the particles in center of mass frame. for atomic clusters, the center of mass behaves like a free particle in laboratory frame and we set its energy zero. hence, we can eliminate the center of mass motion by using standard jacobi coordinates, defined as @xcite @xmath14 and obtain the hamiltonian for the relative motion of the atoms @xmath15 here @xmath16 is the sum of all pair - wise interactions expressed in terms of jacobi coordinates. the hyperspherical harmonic expansion method (hhem) is an _ ab - initio _ complete many - body approach and includes all possible correlations. the hyperspherical variables are constituted by the hyperradius @xmath17 and @xmath18 hyperangular variables which are comprised of @xmath19 spherical polar angles @xmath20 associated with @xmath21 jacobi vectors and @xmath22 hyperangles @xmath23 given by their lengths. however the calculation of potential matrix elements of all pairwise potentials becomes a formidable task and the convergence rate of the hyperspherical harmonic expansion becomes extremely slow for @xmath24, due to rapidly increasing degeneracy of the basis. thus hhem is not suitable for the description of large diffuse atomic clusters. but for a diffuse cluster like rb - cluster, only two - body correlation and pairwise interaction are important. therefore we can decompose the total wave function @xmath25 into two - body faddeev component for the interacting @xmath26 pair as @xmath27 it is important to note that @xmath28 is a function of two - body separation (@xmath29) and the global @xmath30 only. therefore for each of the @xmath31 interacting pair of a @xmath0 particle system, the active degrees of freedom is effectively reduced to only four, _ viz. _, @xmath29 and @xmath30 and the remaining irrelevant degrees of freedom are frozen. since @xmath25 is decomposed into all possible interacting pair faddeev components, _ all two - body correlations _ are included. thus the physical picture for a given faddeev component is that when two particles interact, the rest of the particles behave as inert spectators.thus the effect of two - body correlation comes through the two - body interaction in the expansion basis. it is to be noted that @xmath28 is symmetric under the exchange operator @xmath32 for bosonic atoms and satisfy the faddeev equation @xmath33\phi_{ij } = -v(\vec{r}_{ij})\sum_{kl > k}^{n}\phi_{kl } \label{eq.faddeev - eqn}\]] where @xmath34 is the total kinetic energy operator. applying the operator @xmath35 on both sides of eq. ([eq.faddeev-eqn]), we get back the original schrdinger equation. since we assume that when (@xmath36) pair interacts the rest of the bosons are inert spectators, the total hyperangular momentum and the orbital angular momentum of the whole system is contributed by the interacting pair only. next the @xmath26th faddeev component is expanded in the set of potential harmonics (ph) (which is a subset of hyperspherical harmonic (hh) basis and sufficient for the expansion of @xmath37) appropriate for the (@xmath36) partition as @xmath38 @xmath39 denotes the full set of hyperangles in the @xmath40-dimensional space corresponding to the @xmath26 interacting pair and @xmath41 is called the ph. it has an analytic expression : @xmath42 @xmath43 is the hh of order zero in the @xmath44 dimensional space spanned by @xmath45 jacobi vectors ; @xmath46 is the hyperangle between the @xmath47-th jacobi vector @xmath48 and the hyperradius @xmath30 and is given by @xmath49 = @xmath50. for the remaining @xmath22 noninteracting bosons we define hyperradius as @xmath51 such that @xmath52. the set of @xmath18 quantum numbers of hh is now reduced to _ only _ @xmath53 as for the @xmath22 non - interacting pair @xmath54 and for the interacting pair @xmath55, @xmath56 and @xmath57. thus the @xmath40 dimensional schrdinger equation reduces effectively to a four dimensional equation with the relevant set of quantum numbers : energy @xmath58, orbital angular momentum quantum number @xmath59, azimuthal quantum number @xmath8 and grand orbital quantum number @xmath60 for any @xmath0. substituting eq. ([eq.faddeev-comp]) in eq. ([eq.faddeev-eqn]) and projecting on a particular ph, a set of coupled differential equation for the partial wave @xmath61 is obtained @xmath62u_{kl}(r) + \displaystyle{\sum_{k^{\prime}}}f_{kl}v_{kk^{\prime}}(r) f_{k^{\prime}l } u_{k^{\prime}l}(r) = 0&\\ \end{array } \label{eq.cde}\]] + where @xmath63, @xmath64, @xmath65 and @xmath66. + @xmath67 is a constant and represents the overlap of the ph for interacting partition with the sum of phs corresponding to all partitions @xcite. the potential matrix element @xmath68 is given by @xmath69 we do not require the additional short - range correlation function @xmath70 as mentioned in ref. Results it is already pointed out that the universal properties of ultracold dilute atomic gas in the unitary regime is characterized when the two - body scattering length @xmath1 is tuned to very large values by using the feshbach resonance. although the unitary fermi gas has been largely investigated both experimentally and theoretically @xcite, the bosonic unitary regime is a formidable challenge in the many - body theories. even though the range of the interaction is small compared with the particle separation, interatomic correlations are very important and the standard mean - field theories are inadequate. + the interaction strength of sufficiently dilute atomic cloud is parameterized by a single parameter - the @xmath71-wave scattering length. however for our present study to explore the generic behaviour near the unitary, we consider the van der waals potential characterized by two parameters : the cutoff radius of the repulsive hard core @xmath72 and the strength of the long - range tail @xmath73. thus keeping @xmath73 fixed, it is possible to tune the value of @xmath72. solving the two - body schrdinger equation it is possible to calculate the scattering length for each choice of @xmath72. we solve the zero - energy two - body schrdinger equation for the two - body wave function @xmath70 as @xmath74 where @xmath75 for @xmath76 and @xmath77 for @xmath78. the asymptotic form of @xmath70 is @xmath79, @xmath80 is the normalization constant. the solution of two - body equation shows that the value of @xmath1 changes from negative to positive passing through an infinite discontinuity. in fig. [fig.as], we plot the zero - energy scattering length @xmath1 as a function of @xmath72. at each discontinuity, one extra node in the two - body wave function appears which corresponds to one extra two - body bound state. however for our present study we fix @xmath72 such that it corresponds to zero node in the two - body wave function. we impose the constraint just to avoid the formation of the molecules, otherwise when @xmath1 sufficiently increases, the rate of three - body collisions will increase which deplete the density by forming molecules. + with the above set of parameters we solve the set of coupled differential equations (cdes) by hyperspherical adiabatic approximation (haa) @xcite. in haa, we assume the hyperradial motion is slow compared to the hyperangular motion. thus the solution of the hyperangular motion is obtained by diagonalizing the potential matrix including the diagonal hypercentrifugal repulsion for a fixed value of @xmath30. the cde is then decoupled approximately into a single uncoupled differential equation @xmath81\zeta_{0}(r)=0\hspace*{.1 cm }, \label{eq.eaa}\]] which is known as extreme adiabatic approximation (eaa) and the lowest eigenvalue @xmath82 is the effective potential in which the hyperradial motion takes place. the above equation is solved to obtain the energy and wave function with appropriate boundary conditions on @xmath83. + in fig. [fig.gndstate], we plot the calculated bosonic cluster ground state energies in the negative scattering length near the unitary for different cluster sizes with @xmath0 = 3,4,5,6,7 as a function of the scattering length @xmath1 which represents the universal properties of the bosonic cluster energy at large @xmath84. it is to be noted that the effective interaction of the bosonic cluster is determined by @xmath85. with increase in particle number, the number of interacting pair @xmath86 also increases and the energy becomes more negative as expected. for two particles the infinite scattering length corresponds to a bound state at zero energy. for three particles the efimov effect appears at @xmath87 or @xmath88 = 0, infinitely many three - body bound states with smaller binding energy and larger radii will appear. moving in the opposite by decreasing the attraction, these states cease to be bound one by one. we have calculated the spectrum of bosonic clusters and in fig. [fig.excited] we plot @xmath89 as a function of the state number @xmath90 of the negative energy states. the radii of efimov states @xmath91 as a function of state number @xmath90 is shown in fig. [fig.rav]. [fig.excited], we observe that for each of the @xmath0-body systems there is a series of bound states with exceedingly small energies. it is seen that these series of states show exponential dependence upon the state number as @xmath89 @xmath92 @xmath93. the exponential fits give the numbers as @xmath94 = 0.448, @xmath95 = 0.278, @xmath96 = 0.198, @xmath97 = 0.149. whereas in fig. [fig.rav], we observe that the spatial extension of efimov states is much larger than the interaction range and the r. m. s. radii of efimov states are well reproduced with the exponential @xmath98 @xmath92 @xmath99 where the fitted parameters are @xmath100 = 0.18, @xmath101 = 0.12, @xmath73 = 0.09 and @xmath102 = 0.068. the ratio @xmath103 = 0.41 for @xmath0 = 4, 0.43 for @xmath0 = 5, 0.46 for @xmath0 = 6 and 0.47 for @xmath0 = 7, is close to the value of 0.5 as reported in ref. @xcite for trapped bosons. finally we analyse the structural properties of the cluster states by calculating the pair - correlation function @xmath104 which determines the probability of finding the @xmath26 pair of particles at a relative separation @xmath105. [fig.correlation] presents the pair correlation function for @xmath106 at unitarity. @xmath104 is considered as a more effective quantity in the description of structural properties as the interatomic interaction plays a crucial role. when atoms try to form clusters, due to the attractive part of van der waals interaction, the short range hard core repulsion has the effect of repulsion, thus @xmath104 is zero for @xmath105 smaller than the hard core radius @xmath72. we calculate @xmath104 by @xmath107 where @xmath108 is the many - body wave function and the integral over the hypervolume excludes the integration over @xmath105. the position of the maximum is shifted to larger @xmath105 with increase in @xmath0 and peak height reduces. however we do not observe any structure in the correlation function. it says that the extremely diffuse cluster behaves just like diffuse liquid blob as observed in earlier work @xcite. it is already mentioned that while the universal behaviours of the trimer are quite well understood, much less is known about the larger systems. in this context the investigation of correlations between energies of three and four - particle systems is indeed required. the earlier studies in this direction are mainly focused on the tjon line which refers to the approximately linear correlation between the energies of three - nucleon and four - nucleon systems @xcite. it is expected that the bosonic cluster energy close to the unitarity, for different cluster states should follow the generalized tjon line. it says that the energies are linearly correlated to each other and a two - parameter relation is maintained. @xmath109 in fig. [fig.tjon], we present the energy ratio @xmath110 as a function of @xmath111 for different cluster sizes @xmath0 = 4,5,6. solid lines show linear fits of the form @xmath112. the fitting parameters are summarized in table [table - tjon]. we refer the approximate linear fitting of the energy ratios of clusters as the generalized tjon line. we observe that the values of the fitting parameters gradually decreases with increasing @xmath0 and this is consistent with earlier finding @xcite..values of fitting parameters of tjon line. [cols="<,^,^",options="header ",] [table - tjon] this definitely opens the possibilities of future investigations of how the behaviour of the generalized tjon lines are related in the description of the universal properties of diffuse bosonic clusters. cc + (a) @xmath113 & + & + (b) @xmath114 & + + (c) @xmath115 & + Conclusion the physics of weakly bound few - body systems and their universal behaviour near the unitary is a challenging research area in recent days. the recent experimental observation of efimov phenomena in ultracold bose gases has renewed the interest in universal few - body physics. the theoretical study of three - dimensional bosonic cluster with more than three particles is also challenging and the numerical treatment becomes complicated with @xmath24. the cluster is weakly bound as the kinetic and potential energy nearly cancel. it needs to include interatomic correlation. in the present study we utilize two - body correlated basis function for the study of @xmath0-boson systems. use of realistic van der waals potential presents the actual feature of such delicate systems. we calculate the energy spectrum of @xmath0-body cluster with @xmath0 upto 7 atoms. at large scattering length, which is much larger than the range of interaction, the ground state energy of @xmath0-body cluster shows universal behaviour. next, to exhibit the efimov like character of the energy states we present the exponential dependence on the state number. we also calculate the r.m.s radii of the spatially extended systems and also shows their exponential dependence on the state number. calculation of two - body pair correlation exhibit the expected feature and does not show any structure. it says that the weakly interacting cluster behaves just like a single quantum stuff. we also calculate the energy correlation between two clusters differing by one atom and shows that they maintain a two parameter linear relation. we refer the tjon line as the characteristic of universal behaviour of bosonic cluster. bc would like to thank the university of south africa (unisa) for the financial support of visit where part of work was done. bc also acknowledges financial support of dst (india) under the research grant sr / s2/cmp-0126/2012.

we study three - dimensional bosonic cluster interacting through van der waals potential at large scattering length. we use faddeev - type decomposition of the many - body wave function which includes all possible two - body correlations. at large scattering length, a series of efimov - like states appear which are spatially extended and exhibit the exponential dependence on the state number. we also find the existence of generalized tjon lines for @xmath0- body clusters. signature of universal behaviour of weakly bound clusters can be observed in experiments of ultracold bose gases.

1504.01746

Introduction several decades ago, @xcite postulated that the absorption lines observed in the spectra of distant qsos are due to the extended gaseous halos of intervening galaxies. since that time, astronomers have identified the galaxies associated with the observed absorption (e.g., *??? * ; *??? * ; *??? *) and have used qso spectroscopy of various rest - frame ultraviolet transitions to constrain the nature of baryonic processes in the outer regions of galaxies, e.g., feedback, accretion, and cooling @xcite. these transitions include ly@xmath9 and lines that probe cooler, photoionized gas @xcite, and and doublets that trace warmer, more diffuse material @xcite. however, while the apparent brightness of qsos enables high signal - to - noise, high spectral resolution datasets, they severely outshine the galaxies projected nearby and therefore limit follow - up analysis, especially at small impact parameters. furthermore, qsos are too rare in the sky to probe numerous individual galaxies with multiple sightlines. therefore, constraints on the covering fraction (@xmath10) and spatial distribution of halo gas must be statistical (e.g., *??? * ; *??? the @xmath11 doublet in particular has been studied extensively, as it is easily accessible at optical wavelengths at redshifts @xmath12 and traces cool gas (with temperature @xmath13 ; *??? *) in a broad range of neutral hydrogen column densities (@xmath14@xmath15 ; *??? * ; *??? *). in spite of this, the physical origins of the gas giving rise to absorption remain obscure several studies (e.g., *??? *) suggest an infall origin for this gas ; for example, @xcite present an infall model with which they reproduce the observed frequency distribution function for systems as well as their clustering properties @xcite. other studies @xcite suggest instead that these systems arise in gas which has been blown out of star - forming galaxies via superwinds. in support of this latter scenario, a recent study by @xcite of kinematics in coadded deep2 spectra of a sample of star - forming galaxies at @xmath16 reveals frequent, perhaps even ubiquitous, outflows of cool gas. @xcite also detect very high velocity outflows in @xmath17 post - starburst galaxies traced by absorption. such winds may redistribute absorbing gas, possibly to a galaxy s halo, although the distances to which the winds extend remain uncertain. in one of the only studies addressing this issue, @xcite observes cool outflowing gas several kiloparsecs from the nuclei of a sample of @xmath18 ultraluminous infrared galaxies. in principle, galaxy spectra can also probe the halo gas of foreground galaxies along the sightline (e.g., *??? * ; *??? this novel technique offers several advantages over qso - galaxy pair studies (d. koo et al. 2009, in preparation). the projected number density of galaxies on the sky is much greater than that of qsos ; therefore the use of galaxies can vastly increase the number of potential background probes for a given galaxy halo. moreover, many galaxies are extended sources which provide the opportunity to study gas along multiple lines of sight through a given foreground halo, including their own (e.g., *??? * ; *??? integral field unit (ifu) spectrographs may then be used to study the morphology of halo absorption and the spatial distribution of outflowing gas. close transverse pairs of galaxies may be identified in large spectroscopic or photometric surveys so that a targeted search for foreground absorption in the background galaxy spectra can be performed. this technique has been used at @xmath19 to study absorbing gas traced by transitions such as near lyman break galaxies by @xcite ; but it has not yet been used at lower redshifts where higher resolution imaging and spectroscopy can be obtained for each galaxy. in the process of carrying out a spectroscopic survey to measure outflow properties in galaxies at @xmath20, we identified a close transverse pair of galaxies in which the spectroscopy of the more distant object (@xmath2) allows the detection (in absorption) of gas in the environs of a foreground galaxy (at @xmath0 and at impact parameter @xmath21 kpc). this foreground absorber shows signs of recent merger activity, has a stellar continuum consistent with that of a post - starburst galaxy, and is host to a low - luminosity agn. here we analyze the spectrum of the background galaxy to examine halo gas in the foreground system. we discuss our observations of the galaxy pair and data reduction in [sec.thepair]. analysis of the luminous components of the galaxies and the foreground halo absorption is given in [sec.analysis]. the possible origins of the observed cool halo gas are discussed in [sec.discussion], and we conclude in [sec.conclusions]. we adopt a @xmath22 cosmology with @xmath23, @xmath24, and @xmath25. where it is not explicitly written, we assume @xmath26. magnitudes quoted are in the ab system. The galaxy pair our targeted galaxy pair is located in the goods - n field (great observatories origins survey ; *??? *) and has been imaged by the hst advanced camera for surveys in 4 optical bands (f435w, f606w, f775w and f850lp, or @xmath27, @xmath28, @xmath29 and @xmath30). galaxy properties derived in previous studies and references are given in table [tab.photinfo]. we obtained spectroscopy of both galaxies using the low resolution imaging spectrometer (lris) on keck 1 @xcite on 2008 may 30 - 31 ut. we used the @xmath31 grism blazed at 4000 on the blue side and the @xmath31 grating blazed at 7500 on the red side with the d560 dichroic. this setup affords a fwhm resolution ranging between @xmath6 and @xmath32 and wavelength coverage between @xmath33 and @xmath34 . two sets of spectra were obtained. we first used a slitmask to place a @xmath35 slitlet across the center of the background galaxy oriented ne and collected @xmath36 exposures with fwhm @xmath37 seeing. we also placed a @xmath38-wide longslit across both galaxies and obtained @xmath39 exposures with @xmath40 seeing (see figure [fig.im]). the data were reduced using the xidl lowreduxxavier / lowredux/] data reduction pipeline. the pipeline includes bias subtraction and flat - fielding, slit finding, wavelength calibration, object identification, sky subtraction, cosmic ray rejection, and flux calibration. vacuum and heliocentric corrections were applied. because the separation between the galaxies is only 2.8, we used a spatially narrow gaussian centered on the foreground object to weight the extracted spectrum (with @xmath41 and @xmath42 for the blue and red sides, respectively). this limits contamination of the foreground spectrum by emission from the background object to @xmath43. using a wider extraction window (@xmath44) does slightly change the equivalent widths (ews) of absorption lines (e.g., h9,) measured for the foreground galaxy, but it does not affect the results of our stellar population modeling (discussed in [sec.fg]). we derive redshifts for the two galaxies using idl code adapted for use in the deep2 survey (s. faber et al. 2009, in preparation ; *??? *) from the publicly available programs developed for the sdss.] in brief, this code calculates the minimum @xmath45 value as a function of the lag between an observed spectrum and a linear combination of three templates. one template is an artificial emission line spectrum with lines broadened for consistency with an instrumental resolution of @xmath46, one is the coadded spectrum of thousands of absorption - line dominated galaxies @xcite, and one is an a star spectrum. the spectral region which includes interstellar uv absorption lines was not used to constrain the fits. each galaxy redshift has also been confirmed by eye. we find that the redshifts of the background and foreground galaxies are @xmath47 and @xmath48, respectively. the errors are the formal @xmath49 uncertainties in the fitted values. these redshifts are within @xmath50 of the redshifts obtained in the team keck treasury redshift survey (tkrs ; *??? *) for these objects. we define the systemic velocity of each galaxy to be at the corresponding fitted redshift. figures [fig.im] and [fig.fullspec] show hst imaging of the galaxy pair and our lris spectrum of the background galaxy, respectively. uv absorption lines in the rest - frame of the background object are marked in red and are offset to negative velocity from systemic, indicating the presence of an outflow. emission due to, as well as emission near the absorption lines, is also evident. the same lines in absorption are marked at the systemic velocity of the foreground galaxy in blue ; these marks coincide with strong absorption in the spectrum, and are presumably due to gas associated with the @xmath51 foreground galaxy. we measured ews of absorption and emission features after normalizing the spectra to the continuum level. this level was determined via a linear fit to the continuum around each feature of interest. we selected different continuum regions for each transition with widths between 8 and 72 . we used a feature - finding code described in @xcite to identify and measure the boxcar ew in both emission and absorption lines with @xmath52 (table [tab.ew]). the foreground galaxy halo / disk absorption has rest equivalent width @xmath53 and is classified as an ultrastrong " absorber @xcite. @xmath54 values this large are rare in surveys of absorption in qso sightlines ; e.g., absorbers with @xmath55 make up only 3% of the population of all absorbers with @xmath56 @xcite. although the number of known galaxy - qso absorber pairs with impact parameters less than @xmath57 is quite small, only two of these have @xmath55 . (we are not including the galaxy - absorber pairs from the sample of @xcite here, as they lack spectroscopic confirmation of the redshifts of the associated galaxies.) this system also exhibits @xmath58 and is classified as a strong " (@xmath59 ) absorber @xcite. the @xmath60 of this halo / disk gas is among the largest measured for the absorbers in the @xcite study. we find no evidence of emission (e.g., [] @xmath61) due to another galaxy at the redshift of the foreground absorber in either the slitmask or the longslit spectrum. furthermore, the other objects identified in the hst image are significantly fainter and lie at larger impact parameters than the previously identified foreground galaxy. we conclude that the foreground absorption is associated with this galaxy located at @xmath62 from the center of the background galaxy, where the center is determined from quantitative morphological analysis of the @xmath29-band image by j. lotz (2008, private communication), using the method described in @xcite. Analysis of the galaxies the background galaxy is one of the brightest galaxies in the @xmath27-band in the tkrs @xcite at its redshift and is among the bluest objects in the blue cloud " in the galaxy color - magnitude diagram (cmd), as shown in the left panel of figure [fig.cmd]. we derive a star formation rate (sfr) of @xmath63 using the luminosity of the h@xmath64 line measured in @xcite and assuming a @xcite dust attenuation curve with effective v - band optical depth @xmath65 as derived from the stellar population modeling discussed below. we assume case b recombination to calculate an h@xmath9 luminosity and apply the calibration of @xcite to convert this to sfr. @xcite measure an oxygen abundance of @xmath66, close to the solar abundance value of 8.72. our lris spectrum reveals a weak [] @xmath67 emission line, which indicates the object is host to an agn (e.g., *??? agn activity may therefore affect the sfr estimate and the estimate of the oxygen abundance. because the [] emission is weak, a gaussian fit to the line profile results in a rest - frame dispersion of @xmath68 ; this value is within @xmath69 of the instrumental velocity dispersion at this wavelength. (all other velocity dispersions reported are calculated by performing a nonlinear least - squares fit of a gaussian to the appropriate emission line and its surrounding continuum and subtracting the instrumental velocity dispersion from the fitted gaussian @xmath70 in quadrature. continuum regions were chosen to extend at least 7 from line center.) we also observe strong emission that results in a p - cygni line - profile (figure [fig.fullspec]), as well as a series of emission features near the resonance lines which we identify as fine - structure transitions from excited states of. to our knowledge, this is the first reported case of _ narrow _ fine - structure emission from any extragalactic object. these emission features may be related to agn activity, but perhaps could be attributed to some other physical mechanism. their origin will be discussed in a future work. to estimate the contribution of the agn to the continuum emission, we used @xcite stellar population synthesis models to generate a grid of synthetic starburst galaxy spectra. all models include an old stellar population, specifically a 7 gyr - old single - burst stellar population (ssp7) with solar metallicity, and the stellar population of an 100 myr - old ongoing burst with constant star formation rate. these particular choices of old and bursting stellar populations are motivated by @xcite, who use them to identify post - starburst galaxies via stellar population modeling. to each of these spectra, we add a featureless power - law continuum with a variable normalization to model the agn contribution to the galaxy continuum. we also allow for a variable amount of dust attenuation, parametrized by @xmath71. (details of the fitting are discussed in [sec.fg] and k. rubin et al. 2009, in preparation.) the spectrum is fit well by models with a range in @xmath72 that combine a strong starburst with a power - law component that contributes less than 30% of the emission at 4200 in the rest frame. we conclude that the continuum emission is consistent with a galaxy spectrum that is dominated by intense star formation, rather than agn activity. because of its high surface brightness and strong continuum blueward of observed wavelength @xmath73 , this galaxy is an excellent and rare candidate for probing gas in foreground halos. at the same time, its spectrum may be used to probe cool gas (via low - ionization absorption) associated with its own ism and halo but foreground to its bright star - forming regions. as noted previously (see [sec.thepair]), the spectrum of tkrs4389 exhibits evidence for a substantial outflow of cool gas. a quantitative analysis of the outflow properties (e.g., velocity, @xmath10, limits on the outflow column density, and mass outflow rate) of a large sample of galaxies including this one will be undertaken by k. rubin et al. (2009, in preparation). vs. @xmath74 for tkrs galaxies with @xmath75 (left - hand panel) and @xmath76 (right - hand panel). background (tkrs4389) and foreground (tkrs4259) galaxies are marked with a large circle and square, respectively. the solid line divides the red sequence " from the blue cloud " as given in @xcite. the foreground galaxy lies in the so - called green valley " and may evolve to the bright end of the red sequence (e.g., *??? [fig.cmd],width=172] and @xmath77 ; red). the smoothed error values in the data are shown with the dotted line. emission lines and the central regions of balmer absorption lines in the galaxy spectrum were masked out prior to fitting (masked regions are shown in green), as the model spectra do not include line emission from agn or regions. bad pixels are also masked out. models with 600myr @xmath78 1.4gyr successfully reproduce the strong balmer, k and g band absorption in the data. [fig.specmodel]] kpc. the systemic velocity is marked with vertical dotted lines. the dashed line marks the continuum level. the rest ew of @xmath4 absorption due to the foreground halo is @xmath5 . we note that these absorption profiles are asymmetric about the deepest part of each line. the lines are heavily saturated (see [sec.psb_scenario]) and have depths of @xmath79% of the continuum level. this is indicative of a @xmath10 for absorbing clouds of @xmath80. [fig.fgspecstamps]] the proximity of this galaxy pair allows us to analyze the halo properties of the foreground galaxy at an impact parameter of @xmath3 kpc. to understand these properties in context, we first present an analysis of the luminous components of the foreground galaxy. tkrs4259 is a massive (@xmath8, where @xmath81 is the stellar mass ; @xcite), relatively luminous galaxy, located in the green valley " between the red sequence and the blue cloud in the @xmath82 vs. @xmath74 cmd (figure [fig.cmd] ; *??? although a gini / m20 analysis (j. lotz, 2008, private communication) indicates tkrs4259 is an early - type galaxy, it shows signs of a disturbed morphology or dust lane in all of the hst / acs bands (i.e., a low surface brightness tidal feature ; figure [fig.im]). both its location in the cmd and its morphology indicate the galaxy may be transitioning from a previously star - forming galaxy to a fully quenched object. tkrs galaxies of similar @xmath74 and redshift have approximately solar oxygen abundance or greater @xcite. the [], @xmath83, and [] emission - line luminosities for this galaxy were measured by @xcite and are included in table [tab.photinfo]. we also detect narrow [] @xmath67 emission with rest - frame velocity dispersion @xmath84 (see table [tab.ew]), indicating this galaxy hosts an agn. before calculating a sfr based on the @xmath83 or [] luminosities, we investigate the possible agn contribution to this lower ionization line emission. the line luminosity ratio [] @xmath85 ; the galaxy falls just below the [] @xmath86 ratio dividing liners (at lower values) and seyferts (at higher values) in the system of spectral classification described in, e.g., @xcite, @xcite and @xcite. unfortunately, we do not have spectral coverage of [] and @xmath87 and can not strictly classify this galaxy as a liner. however, because [] /h@xmath9 is correlated with metallicity in star - forming galaxies, and because there is a tight relation between mass and metallicity in these objects (e.g., *??? *), we may use @xmath81 as a proxy for [] /h@xmath9. this is demonstrated in figure 2 of @xcite, which shows a correlation between absolute @xmath88-band magnitude and both [] /h@xmath9 and [] /h@xmath64 in blue cloud galaxies. our foreground galaxy lies far from the locus of star - forming galaxies in [] /h@xmath64 - @xmath81 space, with a @xmath89 times higher than in galaxies with similar [] /h@xmath64 ratios in the tkrs. it also satisfies the agn criterion of r. yan et al. (2009, in preparation), who find that @xmath82 color can serve as a proxy for [] /h@xmath9 as well. we conclude that liner or agn activity dominates the observed line emission. in addition, @xcite measure an x - ray luminosity of @xmath90 for this object. this luminosity is low enough to be produced by star formation ; however, based on our analysis of the optical spectrum, we attribute the x - ray emission to weak agn activity. we can not, however, rule out ongoing star formation based on this analysis alone. the upper limit on the sfr assuming all @xmath83 emission is due to star formation (and assuming the largest @xmath71 from the range derived from stellar population modeling, @xmath91, as discussed below), is @xmath92. we measure a velocity dispersion of @xmath93 for this line, and note that a line of this width may arise from an agn, as narrow emission from agn has been shown to have approximately the same velocity dispersion as the bulges of the agn host galaxies @xcite. in order to further constrain the rates of current and recent star formation, we generate synthetic starburst spectra as described in [sec.bg]. as above, all models include a 7.0 gyr - old single - burst stellar population (ssp7) of mass @xmath94 with solar metallicity (@xmath95) and an 100 myr - old burst with constant sfr which generates a mass in stars @xmath96. we generate a similar set of models with supersolar metallicity (@xmath97). the strength of the burst is parametrized by @xmath98, where @xmath99 (e.g., *??? *). we create models with @xmath100 and compare them to the lris spectrum of the foreground galaxy at 5620 @xmath101 7550 (see figure [fig.specmodel]). we do not fit the portion of the spectrum blueward of this range, as it falls on the opposite side of the d560 dichroic, and because it lacks spectral features useful for distinguishing star formation histories. the portion redward of this range is quite noisy and is subject to atmospheric absorption. for each model, we find the best - fit value of @xmath102, with an attenuation curve parametrized in @xcite, by performing a @xmath45 minimization. the models do not include nebular or agn line emission, so emission lines in the data are masked out prior to fitting. we find that both solar and super - solar metallicity model spectra can not simultaneously match the strengths of the balmer absorption and the k and g band absorption in the data. the addition of a featureless power - law (qso) component (@xmath103 with @xmath104 ; *??? *) does not improve the fits significantly. these results suggest that there is no ongoing star formation in this galaxy. a strictly passively evolving model is ruled out by the spectrum as well. we also wish to investigate alternative star formation histories. one possible star formation history that suggests itself, based on the location of the galaxy in the cmd as well as its morphology, is one in which the galaxy evolved passively for several billion years and then experienced a burst of star formation, which has since ceased. we again generate a suite of solar metallicity models, all with a ssp which formed 7 gyr ago. a burst with constant sfr lasting 100 myr is added to each model at a time @xmath105, where @xmath105 varies between 1.8 gyr ago and 200 myr ago. as described below, this span in burst age encompasses the range in ages for models which provide a good fit to the data. models have values @xmath106. we allow for the addition of a qso power - law continuum component as described above. we find the best - fit values of @xmath71 and the fraction of light contributed by the power - law component by minimizing @xmath45 for each model in our grid. we perform a visual inspection of the fit of each model, and find that models with intermediate @xmath105 values and larger values of @xmath98 are most successful at fitting the deep balmer absorption and k and g band absorption strengths in the data. the best - fit model has a high value of @xmath98 (@xmath107 ; larger values give similar results) and a 1 gyr - old burst ; this model is shown in figure [fig.specmodel]. the best - fit value of @xmath108, and the fraction of the total light contributed by the power - law component at 4200 in the rest - frame is @xmath109. models with @xmath110, with @xmath111 and @xmath112, or with @xmath113 can not simultaneously match the balmer, k and g band absorption strengths and thus yield unacceptable fits. models with @xmath114 yield acceptable fits if @xmath98 is large enough at older ages, with a range in fitted @xmath71 values between 0 and 1.8. we conclude that a starburst occurred in the galaxy between 1.4gyr and 600myr ago. the results are similar for @xmath97 models. + in summary, we find that the foreground galaxy is host to a low luminosity agn and has a stellar continuum consistent with that of a post - starburst galaxy. it has an asymmetric morphology and is located in the green valley " in the cmd. we note that the morphology of this object in the @xmath115 band is in general consistent with that of simulated merger remnants @xmath116 after the final coalescence @xcite. these results can be explained by a scenario in which the galaxy recently experienced a merger which triggered a starburst. the starburst then ceased, perhaps because the cool gas supply was exhausted or expelled. in this case, the galaxy will migrate to the red sequence. + analyzing our spectrum of the background galaxy (tkrs4389) which lies at an impact parameter of @xmath117, we identify strong,, and absorption from gas associated with the foreground galaxy. a subset of these features is shown in figure [fig.fgspecstamps] with velocities relative to the systemic velocity of tkrs4259. the spectrum shown was obtained with the slitlet marked with the solid lines in figure [fig.im] ; however, the spectrum taken with the longslit (marked with dashed lines in figure [fig.im]) exhibits foreground absorption line profiles with similar depths and shapes as those shown. the measured @xmath4 ew is extreme : @xmath53 . the oscillator strengths of the lines in the doublet have a ratio 2:1, and as the optical depth of the lines increases and they saturate, the ew ratio of the doublet decreases from 2:1 to 1:1. the ew ratio of the lines in our system is @xmath118, indicating a high degree of saturation. in the case of heavily saturated lines, the @xmath54 value is indicative of extreme kinematics, i.e., a large velocity width of @xmath119 @xcite. at such a low impact parameter, it is possible that our line of sight is probing gas in the outer disk of the galaxy. however, even if the galaxy were edge - on (it is not), the predicted differential rotation of a disk at this impact parameter would be of the order a few tens of @xmath120. we conclude that the gas dynamics are dominated by non - rotational motions and that the majority of gas is extraplanar, i.e., tracing material in the halo of tkrs4259. because the lines are strongly saturated, the depth of the lines depends only on the fraction of the background light source that is covered by absorbing gas (@xmath10) and the instrument resolution. the deepest parts of the line profiles from the spectra taken with slits at _ both _ orientations reach @xmath79% of the continuum level, which places a lower limit @xmath121. the relatively low spectral resolution smears the profiles, and the deepest parts of the lines may in fact appear at much lower normalized flux levels when observed at higher resolution. this constraint on @xmath10 is particularly striking given the large beam size provided by the background galaxy : the distance between the two brightest knots in this galaxy is @xmath122 kpc at @xmath0. this may suggest that the absorbing gas complex almost completely covers the background source and has a large velocity dispersion at all the locations probed. this is consistent with previous results from @xcite, who report velocity coherence in absorbing clouds on @xmath123 kpc scales. we leave detailed modeling of the effect of such a large beam size on the observed absorption to a future work (although see @xcite for an analysis of the effects of grb vs. qso beam size on absorption system studies). Discussion here we examine several possible scenarios for the origin of the gas in the foreground galaxy halo. in the galaxy formation scenario proposed by @xcite and @xcite, @xmath124 clouds condense out of the hot gas in galaxy halos as it cools. the velocity dispersion of these clouds is approximately that of the galaxy halo ; we may therefore estimate the expected velocity dispersion in -absorbing clouds given a halo mass for the galaxy. using the relation between halo mass and stellar mass derived in @xcite via the abundance matching " technique, we estimate that the halo mass (@xmath125) of the foreground galaxy is @xmath126 for @xmath127. using the relations between @xmath125 and halo virial velocity given in @xcite and assuming a singular isothermal sphere for the halo density profile, we find that the expected fwhm of the line - of - sight velocity distribution for cool clouds in a halo this massive is @xmath128. this dispersion could easily produce a @xmath129 @xcite, consistent with the measured @xmath54 for this galaxy. the thermal stability analysis of the hot gas surrounding galaxies performed by @xcite suggests that in the case of an isolated halo more massive than that of the milky way (@xmath130 ; *??? *), a smoothly stratified hot gaseous corona is stable to thermal perturbations and thus will not condense into cool clouds. however, this study does not account for the effects of gas inflow from the igm ; nor does the assumption of a smooth hot gaseous halo likely apply in this case, given that our host galaxy is a merger remnant. the three - dimensional cosmological simulations of @xcite show that even in halos well above the transition " mass (@xmath131), in which hot, virialized atmospheres develop via shock - heating during gas accretion, cold filaments of gas from the intergalactic medium (igm) can penetrate deep into the hot halo. at high redshifts (@xmath132), these cold flows reach the central galaxy and fuel star formation. at @xmath133, in the case of a @xmath134 halo, the filaments penetrate to only half the virial radius, but can fragment into dense, cold clouds via shocks, cooling instabilities, or other mechanisms. as these clouds are virialized within the halo, they will acquire a large velocity dispersion and produce a large @xmath135, as discussed above. however, we do not consider cold inflow from the igm to be the most likely origin of the cool halo gas we observe. first, our detection of @xmath136 @xmath4 halo absorption and @xmath137 k halo absorption in this system (as noted in [sec.thepair] ; see also figure [fig.fgspecstamps] and table [tab.ew]) may indicate that the absorbing gas is enriched with metals and dust. we can not measure the metallicity or dust content of this system directly, as we lack the necessary spectral coverage. however, strong absorption systems in the redshift range @xmath138 have been shown to significantly redden background qsos @xcite ; furthermore, the degree of reddening increases with @xmath135. @xcite find that for absorbers with @xmath139 at @xmath51, the mean rest - frame @xmath140 (see their equation 18). they also find that absorbers with @xmath56 have an average dust - to - metals ratio close to those measured in the milky way ism. similar results for the mean dust - to - metals ratio of strong absorbers have been reported by @xcite, who additionally show that dust content increases with larger @xmath60 (see also *??? as noted above, the absorption we observe has an @xmath60 value as large as the strongest absorbers in these studies. these findings suggest that there may be a substantial amount of dust in this system. second, we do not resolve individual gas clouds because of our low spectral resolution (@xmath141) ; however, we note that the absorption line profiles in figure [fig.fgspecstamps] exhibit asymmetry. @xcite find that cool gas clouds distributed such that the gas density is proportional to the total matter density in an isothermal spherical halo tend to produce a symmetric velocity profile, although this depends on the number of absorbing clouds in the halo and the particular sightline observed. both the likely high dust content of the absorption system and the line profile asymmetry suggest an alternate origin for the observed cool gas. we next consider whether the observed halo gas could have plausibly originated in the ism of the host galaxy or progenitor galaxies which was ejected during a previous starburst, merger or luminous agn phase. if this phase occurred @xmath137 gyr ago (as is likely ; see [sec.psb_scenario]), cool outflows would have had to reach speeds of only @xmath142 to reach a projected distance of @xmath3 kpc today, where @xmath143 is the angle between the plane of the sky and the path taken by the gas. cool outflows commonly reach speeds in excess of @xmath144 in starburst galaxies @xcite and have even been observed in post - starburst galaxies at intermediate redshifts @xcite. cool gas which has been tidally stripped from the progenitor galaxies of this merger remnant may also give rise to enriched halo absorption with large @xmath54, provided that the velocity spread in the stripped clouds is large enough @xcite. the magellanic stream, for instance, has a velocity dispersion of @xmath145 @xcite, and so could not produce the observed @xmath54 in the absence of other clouds. additionally, anisotropic ejection of gas clouds or tidal streams could easily produce asymmetric absorption line profiles. we conclude that cool ism which was driven or stripped out of the galactic star - forming regions during a past starburst, merger or agn phase is a plausible origin for this gas. if in reality the gas originated in a cold inflow or halo condensation as discussed in [sec.multiphase], our finding that the galaxy experienced a starburst @xmath137 gyr ago is simply a coincidence, and is not necessarily related to the observed absorbing gas. we note that it is unlikely that there is ongoing expulsion of ism, as there is little current star formation activity in the galaxy, and the observed agn activity is weak. therefore, in this scenario, once the cool gas was ejected or stripped, it is likely to have remained cold for approximately 1 gyr, i.e., since the starburst activity. the analysis of @xcite suggests that ejected _ hot _ gas could not have cooled through condensation, although again, the assumptions made in that analysis may not apply in this case. cool gas clouds ejected or stripped from the ism and moving through a hot medium may be subject to the kelvin - helmholtz instability. @xcite showed that such disturbances on the surface of a cloud can not completely disrupt the cloud unless its cooling time is longer than its sound crossing time. following @xcite, we assume an isothermal halo with temperature @xmath146 and with gas density profile @xmath147, where @xmath148 is the cosmic baryon fraction @xcite, and @xmath149 is the maximum rotation velocity of the halo (equal to the virial velocity in this case ; see @xcite for details). we assume that the cool clouds are at @xmath150 and that they immediately come into pressure equilibrium with the surrounding hot gas as soon as they enter the halo. the sound crossing time for a cloud is given by the ratio of the cloud radius to the cloud sound speed (@xmath151). to calculate the cooling time of the cloud, we use equation 33 from @xcite. we find that the sound crossing time exceeds the cooling time by at least one order of magnitude for clouds with masses @xmath152 out to the halo virial radius (@xmath153 kpc). we expect that these clouds will be stable against the kelvin - helmholtz instability. we also note that the clouds at the upper end of this mass range are evaporated by conduction on timescales of @xmath154 gyr, where the evaporation timescale is given by equation 35 in @xcite. if we assume that the hot gas profile does not change substantially with time, we can conclude that cold clouds driven into the halo from the ism during a starburst or merger occurring @xmath137 gyr ago should remain cold today, and indeed for at least 1 gyr more. outflow during the past starburst, merger or luminous agn event may have been powered by a variety of mechanisms. the galaxy evolution model advocated by (*??? * and references therein) suggests that in a gas rich major merger gas is driven to the center of the galaxies once they have coalesced, fueling powerful starburst activity and black hole growth and igniting a luminous qso. feedback from supernovae explosions is well known to drive outflows in starburst galaxies (e.g., *??? * ; *??? *) and could have contributed all of the power needed to accelerate the cool ism to velocities of @xmath155. simulations of major mergers @xcite show that they can generate a shock - heated gas wind, possibly forcing cool gas into the halo. in addition, gas from the ism that was tidally stripped during a merger event may have been left behind at large impact parameters. finally, a luminous agn could have contributed to driving cool gas into the galactic halo and to the quenching of star formation. in the @xcite scenario, the phase in which the remaining ism material is blown away and the qso is revealed is expected to occur within a few hundred myr of the merger itself. the phase in which the qso has faded and the merger remnant reddens and exhibits a post - starburst stellar continuum occurs @xmath156 after the merger. our results on the star formation history, colors, and morphology of this galaxy fit well into this scenario. however, we note that we have the strongest evidence that the galaxy is post - starburst ; the occurrence of a quasar phase is purely speculative. we also consider whether clouds ejected from the ism would have a large spread in velocities of @xmath157 1 gyr after the starburst event has ceased. to determine this, we construct a toy model of the behavior of the outflowing gas inside the halo potential. to start, we follow @xcite and assume that the velocity (@xmath158) of the outflowing gas as a function of distance from the starburst region (@xmath159) is given by @xmath160, where @xmath161 is the halo velocity dispersion, @xmath162 is the lower limit on the starburst luminosity needed to expel a large fraction of the galaxy s gas when it is optically thick, and @xmath163 is the initial radius of the outflow. this equation applies until the terminal velocity is reached, @xmath164. for @xmath165, @xmath166 kpc and @xmath167, the gas is driven to radii of @xmath168 kpc during a 100 myr - long starburst. the results are insensitive to the choice of @xmath169 as long as @xmath170 ; however, @xmath171 and @xmath163 must have values such that the gas does not fall back onto the galaxy in 1 gyr. if @xmath166 kpc, values of @xmath172 are viable ; if @xmath163 is doubled, slightly lower values of @xmath173 yield cloud distances and velocities high enough so that they remain in the halo for the necessary amount of time. we then assume that the starburst ends abruptly, and that the clouds from the wind are now subject only to the gravitational potential of the halo, and neglect the effect of drag forces from the surrounding hot halo gas. by integrating the equations of motion for clouds with distances @xmath174 kpc and with initial velocities given by the above equation for @xmath158, we calculate the final positions and velocities after 1 gyr. a bipolar geometry for the outflow with an opening angle of @xmath175 is assumed, consistent with measurements of the opening angle of the outflow cones traced by h@xmath9 and co emission in m82 (e.g., *??? * ; *??? *), such that our line of sight intersects both outflow cones and is parallel to the axis passing through the centers of the cones. we calculate the line - of - sight velocities for the clouds that are at a projected distance of @xmath176 kpc after 1 and 2 gyr. the velocity spread is well above @xmath177 at both time increments. while this model is quite rudimentary, it indicates that clouds driven out of a galaxy 1 gyr ago may have the large velocity dispersion that is observed, and that they will maintain this large dispersion for at least another 1 gyr. from all of the evidence discussed above, we conclude that the absorption system is most likely gaseous fragments of the ism from the merger remnant we observe. this gas may have been ejected by a wind driven by a starburst, merger, or agn, or tidally stripped from the progenitor galaxies. if indeed these clouds originated in the ism, this is one of the first measurements of the radial extent of such outflows or tidally stripped gas in the distant universe and suggests that it reaches far beyond the stellar disk. because the clouds we observe are likely dusty (see [sec.multiphase]), this mechanism may be responsible for distributing the diffuse dust detected in galaxy halos to projected separations much larger than the scales of galactic disks @xcite. a complementary method which has recently been used to study the host galaxies of absorption systems measures the clustering strength of a large sample of absorbers with respect to that of luminous red galaxies (lrgs), to determine the mean halo masses in which absorbers of varying strengths are found. these studies have shown that there is a weak anticorrelation between @xmath135 and the bias of systems at @xmath179. this has been interpreted as an anticorrelation with mean absorber halo mass, significant at the 1@xmath70 level ; i.e., absorbers with 1.0 @xmath180 1.5 have mean halo mass @xmath181, while @xmath182 absorbers have mean halo mass @xmath183 (@xcite ; see also @xcite and @xcite). the number of @xmath184 absorbers in these studies is relatively small ; however, the @xmath185 we infer for our foreground absorber, while higher than the mean halo mass of strong absorbers quoted above, is well within the uncertainties in the estimates of the mean halo masses for _ both _ weak and strong absorbers (see figure 7c, *??? one explanation proposed by @xcite for the origin of strong absorbers suggests that cool gas is driven into galaxy halos via starburst - driven winds. these authors propose that winds are more likely to remove gas from galactic disks in shallower potential wells, and therefore expect strong absorbers to be preferentially found in lower - mass halos. @xcite show, however, that the clustering of strong absorbers is _ unbiased _ with respect to dark matter halos and thus strong absorption does not occur preferentially at low masses. furthermore, the present study provides one example of strong absorption which is most likely due to past outflow or merger activity in a high mass (@xmath186) halo. our result implies that while galactic winds may be an important source of cool halo gas, they should not necessarily be the dominant contributor in only low mass halos, and indeed _ must _ not occur preferentially in lower mass halos if they are to fully explain the bias results of @xcite. an alternative picture is inspired by simulations of, e.g., @xcite and @xcite, which provide evidence for the transition " halo mass, above which cold gas is shock heated as it is accreted onto a halo as discussed in [sec.multiphase]. @xcite find that an anticorrelation between @xmath125 and @xmath135, as well as the observed frequency distribution of absorbers, can be reproduced in a halo occupation model which includes such a transition mass, so that the most massive halos are mostly too hot to produce large @xmath54 while lower mass halos contain more cool gas and can give rise to high @xmath54 systems. as shown in @xcite, cold flows can partially penetrate through the hot gas in massive halos at @xmath187 and form cold clouds, which may even then fuel further star formation. while our galaxy is unusual in that it is post - starburst, the existence of an _ enriched _ @xmath188 system in a halo with a mass well above the transition mass " is consistent with this picture, and if the observed gas originated in an outflow, implies that (1) the effects of star formation on halo gas are observed well into the mass regime in which hot gas accretion dominates the total gas accretion, or (2) there is a spread in transition mass for different galaxy halos. Concluding remarks using multiwavelength analysis of the luminous components of a galaxy at @xmath0 in concert with spectroscopy of a bright, close transverse background galaxy at @xmath2, we demonstrate that the cool gas traced by absorption in the foreground galaxy halo at impact parameter @xmath189 kpc most likely originated in the ism of the host galaxy or its progenitors. this galaxy has little ongoing star formation and exhibits only weak agn activity ; however, it experienced a starburst @xmath137 gyr ago, which may have been triggered by a merger event. the cool halo gas could have easily been driven or stripped away from the star - forming regions during this past violent phase, and could have survived in the hot halo since that time. our rudimentary toy model for the motions of clouds driven into the halo by a past starburst suggests that these clouds would have a large velocity dispersion @xmath190 today, consistent with our observations. we emphasize that the analysis of the luminous components of this galaxy achieves an unprecedented level of detail in the context of studies of distant absorption - selected galaxies with @xmath191 . such detail is possible because of the rich dataset available in goods, including the galaxy redshifts from tkrs @xcite and our lris spectroscopy. we were able to identify the post - starburst and age - date it using this dataset ; previous works could not necessarily find such a galaxy, and thus present a less complete view of the host galaxies of strong absorbers. @xcite searched for @xmath87 emission near @xmath192 absorbers at @xmath133 to investigate a starburst origin for absorption ; this method can not be used to identify host galaxies without @xmath87 emission, such as post - starbursts with weak agn activity. @xcite investigated environments of @xmath193 absorbers by imaging the corresponding qso fields ; however, they lacked spectroscopic confirmation of galaxy - absorber associations and have not constrained the star formation history of absorber hosts. our study only begins to demonstrate the power of using background galaxies to probe foreground halo gas. several large spectroscopic surveys have recently concluded or are nearing completion that can be used to identify pairs suitable for a targeted search for foreground absorption. spectroscopic followup with an instrument such as lris requires that the background source be bright and blue so that a continuum signal - to - noise ratio of @xmath194 at @xmath195 can be acquired in a reasonable exposure time (i.e., @xmath196 in less than 4 hours). this @xmath197 is needed to achieve a @xmath198 ew detection limit of 0.5 for a absorber at @xmath179. there are @xmath199 galaxy pairs with redshifts measured as part of the deep2 survey of the extended groth strip (egs) with angular separations @xmath200 @xcite which meet these requirements. surveys such as combo-17 @xcite and zcosmos @xcite will provide pair candidates in numbers on the same order as deep2, as will the forthcoming vimos vlt surveys @xcite. the primus surveyeisenste / primus / home.html] @xcite will take galaxy spectra over 10 square degrees and should enable selection of over 1500 galaxy pairs. the pan - starrs survey, which will image 1200 square degrees in 4 filters to @xmath201, will provide an unprecedented number of photometric redshifts. using the average sampling rate and the average redshift success rate of the egs portion of the deep2 survey @xcite to scale the number of pair candidates to that expected from a photometric survey, and additionally scaling by the 2400-fold increase in sky coverage with pan - starrs, we expect that @xmath202 galaxy pairs will be available for spectroscopic followup. the use of galaxies as background probes is complementary to analysis of grb sightlines, which have a negligible beam size but similarly allow for close analysis of the absorber host @xcite. in good seeing conditions and with extended background galaxies, one may probe multiple sightlines through a given foreground halo. size scales smaller than @xmath203 kpc at @xmath204 are not currently accessible because of seeing limitations. however, with adaptive optics and ifus on 10m - class telescopes and beyond, spatially extended background galaxies will enable study of the morphology and kinematics of halo absorption in concert with detailed analysis of the luminous components of absorber hosts. the authors wish to thank c. tremonti for freely providing her stellar population modeling code, k. cooksey for providing her absorption feature - finding code and for a careful reading of the manuscript, and s. patel for providing idl code to produce color images. we thank james bullock, hsiao - wen chen, sara ellison, jenny graves, bill mathews, crystal martin, brice mnard, david rosario, jeremy tinker, chris thom and luke winstrom for interesting and helpful discussions during the analysis of these results. the authors wish to recognize and acknowledge the very significant cultural role and reverence that the summit of mauna kea has always had within the indigenous hawaiian community. we are most fortunate to have the opportunity to conduct observations from this mountain. , j. g., cromer, j., & southard, jr., s. 1994, in astronomical society of the pacific conference series, vol. 61, astronomical data analysis software and systems iii, ed. d. r. crabtree, r. j. hanisch, & j. barnes, 469+ , m., et al. 2003, in society of photo - optical instrumentation engineers (spie) conference series, vol. 4834, society of photo - optical instrumentation engineers (spie) conference series, ed. p. guhathakurta, 161172 , a. m. 1995, in astronomical society of the pacific conference series, vol. 80, the physics of the interstellar medium and intergalactic medium, ed. a. ferrara, c. f. mckee, c. heiles, & p. r. shapiro, 478+ lccc z & 0.69425 & 0.47285 + @xmath74 & @xmath205 & @xmath206 + @xmath207 & @xmath208 & 0.95 + @xmath209 (kpc) & 6.15 & 6.85 + @xmath210 & @xmath211 & @xmath212 + @xmath213)$] @xmath214 & @xmath215 & @xmath216 + @xmath217 @xmath214 & @xmath218 & @xmath219 + @xmath220)$] @xmath214 & @xmath221 & @xmath222 + @xmath223 @xmath214 & @xmath224 & @xmath225 + @xmath226 & 0.58/@xmath227 & 0.56/@xmath228 + lccc & & & + 2344 & @xmath229 & @xmath230 & + 2374 & @xmath231 & @xmath232 & + 2382 & @xmath233 & @xmath234 & + 2586 & @xmath235 & @xmath236 & + 2600 & @xmath237 & @xmath238 & + 2796 & @xmath239 & @xmath240 & + 2803 & @xmath241 & @xmath242 & + 2852 & @xmath243 & @xmath244 & + 3934 & @xmath245 & @xmath246 & + 3969 & no coverage & @xmath247 & + [0.5ex] + [-1.5ex] & & & + @xmath248 $] 2324 - 2329 / * 2328 & @xmath249 & & no coverage + * 2365 & @xmath250 & & no coverage + * 2383 & @xmath251 & & no coverage + * 2396.1/2396.3 & @xmath252 & & no coverage + * 2612 & @xmath253 & & no coverage + * 2626/2629/2631/2632 & @xmath254 & & no coverage + 2796 & @xmath255 & & @xmath256 + 2803 & @xmath257 & & @xmath258 + @xmath259@xmath248 $] 3345 & @xmath260 & & @xmath261 + @xmath259@xmath248 $] 3426 & @xmath262 & & @xmath263 + @xmath259@xmath248 $] 3727 & @xmath264 & & @xmath265 + @xmath259@xmath248 $] 3869 & @xmath266 & & @xmath267 + @xmath268 & @xmath269 & & @xmath270 + @xmath271 & no coverage & & @xmath272 + @xmath259@xmath248 $] 4960 & no coverage & & @xmath273 + @xmath259@xmath248 $] 5008 & no coverage & & @xmath274 +

we study the cool gas around a galaxy at @xmath0 using keck / lris spectroscopy of a bright (@xmath1) background galaxy at @xmath2 at a transverse distance of @xmath3 kpc. the background galaxy spectrum reveals strong,,, and absorption at the redshift of the foreground galaxy, with a @xmath4 rest equivalent width of @xmath5 , indicative of a velocity width exceeding @xmath6. because the background galaxy is large (@xmath7 kpc), the high covering fraction of the absorbing gas suggests that it arises in a spatially extended complex of cool clouds with large velocity dispersion. spectroscopy of the massive (@xmath8) host galaxy reveals that it experienced a burst of star formation about 1 gyr ago and that it harbors a weak agn. we discuss the possible origins of the cool gas in its halo, including multiphase cooling of hot halo gas, cold inflow, tidal interactions, and galactic winds. we conclude the absorbing gas was most likely ejected or tidally stripped from the interstellar medium of the host galaxy or its progenitors during the past starburst event. adopting the latter interpretation , these results place one of only a few constraints on the radial extent of cool gas driven or stripped from a galaxy in the distant universe. future studies with integral field unit spectroscopy of spatially extended background galaxies will provide multiple sightlines through foreground absorbers and permit analysis of the morphology and kinematics of the gas surrounding galaxies with a diverse set of properties and environments.

0907.0231

Introduction the visionary who first thought of using the spin polarization of a single electron to encode a binary bit of information has never been identified conclusively. folklore has it that feynman mentioned this notion in casual conversations (circa 1985), but to this author s knowledge there did not exist concrete schemes for implementing spintronic logic gates till the mid 1990s. encoding information in spin may have certain advantages. first, there is the possibility of lower power dissipation in switching logic gates. in charge based devices, such as metal oxide semiconductor field effect transistors, switching between logic 0 and logic 1 is accomplished by moving charges into and out of the transistor channel. motion of charges is induced by creating a potential gradient (or electric field). the associated potential energy is ultimately dissipated as heat and irretrievably lost. in the case of spin, we do not have to _ move _ charges. in order to switch a bit from 0 to 1, or vice versa, we merely have to toggle the spin. this may require much less energy. second, spin does not couple easily to stray electric fields (unless there is strong spin - orbit interaction in the host material). therefore, spin is likely to be relatively immune to noise. finally, it is possible that spin devices may be faster. if we do not have to move electrons around, we will not be limited by the transit time of charges. instead, we will be limited by the spin flip time, which could be smaller. Spintronic classical (irreversible) logic in 1994, we proposed a concrete scheme for realizing a classical universal logic gate (nand) using three spins placed in a weak magnetic field @xcite. by `` three spins '', we mean the spin orientations of three conduction band electrons, each confined in a semiconductor quantum dot. the system is shown schematically in fig. exchange interaction is allowed only between nearest neighbor spins (second nearest neighbor interaction is considered too weak to have any effect). because of the magnetic field, the spin orientation in any quantum dot becomes a _ binary variable_. the spin polarization is either along the magnetic field, or opposite to the field. to understand this, consider the hamiltonian of an isolated dot : @xmath0 where @xmath1 is the unperturbed hamiltonian in the absence of the magnetic field, @xmath2 is the magnetic field, @xmath3 is the land g - factor of the quantum dot material, @xmath4 is the bohr magneton, and @xmath5 is the pauli spin matrix. if the magnetic field is directed along the z - direction, then @xmath6 diagonalizing the above hamiltonian yields the eigenspinors (1,0) and (0,1) which are + z and -z polarized spins. therefore, the spin orientation is a binary variable ; it is either parallel or anti - parallel to the applied magnetic field. in the presence of exchange interaction between two electrons confined to two _ separate _ potentials (such as two different quantum dots), the anti - ferromagnetic ordering, or the singlet state, (i.e. two neighboring spins are anti - parallel) is preferred over the ferromagnetic ordering, or triplet state (two spins are parallel) @xcite. we will assume that the tendency to preserve this anti - ferromagnetic ordering is _ stronger _ than the tendency for all spins to line up along the magnetic field. this merely requires that the exchange splitting energy @xmath7 (energy difference between triplet and singlet states) exceed the zeeman splitting energy @xmath8. we ensure this by reducing the potential barrier between neighboring dots to enhance the exchange, while at the same time, making the magnetic field sufficiently weak to reduce the zeeman energy. under this scenario, the ground state of the array has the spin configuration shown in fig. we will call `` upspin '' the spin orientation directed along the magnetic field and `` downspin '' the opposite orientation. we encode logic 1 in the upspin state. furthermore, we will consider the two edge dots in fig. 1(a) as input ports to a logic gate, and the middle dot as the output port. it is obvious that when the two inputs are logic 1, the output will be logic 0 when the system reaches ground state (anti - ferromagnetic ordering). next, consider the situation when the two inputs are logic 0 (see fig. the output must be logic 1 in order to conform to the anti - ferromagnetic ordering. however, there is a subtle issue. 1(b) is actually _ not _ the ground state of the system, fig. this is because of the weak magnetic field. the difference between fig. 1(a) and fig. 1(b) is that in the former case, _ two _ spins are aligned parallel to the magnetic field, while in the latter, _ two _ spins are aligned anti - parallel to the magnetic field. therefore, if the system is left in the state of fig. 1(b), it must ultimately decay to the state in fig. 1(a), according to the laws of thermodynamics. but that may take a very long time because of three reasons. first, the system must emit some energy carrying entity to decay. this entity is most likely a phonon. however, phonon emissions in quantum dots are suppressed by the `` phonon bottleneck '' effect @xcite. second, phonons do not couple easily to spin unless we have a strongly pyroelectric material as the host. finally, if spins flip one at a time (all three spins flipping simultaneously is very unlikely), then in order to access the state in fig 1(a), the state in fig. 1(b) will have to go through a state where two neighboring spins will be parallel. such a state is much higher in energy than either fig. 1(a) or fig. therefore, fig. 1(a) and fig. 1(b) are separated by an energy barrier, making fig. 1(b) a long lived metastable state. as long as the input bit rate is high enough so that inputs change much more rapidly than the time it takes for the metastable state to decay to the global ground state of fig. 1(a), we need not worry about this issue. what happens if one of the inputs is logic 1, and the other is logic 0 as shown in fig. 1(c)? here the magnetic field comes in handy to break the tie. in this case, logic 1 is preferred as the output since the all other things being equal, a spin would prefer to line up parallel to the magnetic field, rather than anti - parallel. thus, when either input is logic 0, the ouput is logic 1. we have realized the truth table in table 1..truth table of a spintronic nand gate [cols="^,^,^",options="header ",] the reader will recognize that this is the truth table of a nand gate, which is one of two universal boolean logic gates. since we can realize a nand, we can realize any arbitrary boolean logic circuit (combinational or sequential) by connecting nand gates. a number of different logic devices (half adders, flip - flops, etc.) were designed and illustrated in ref. @xcite. these devices have been extensively studied by others @xcite using time independent simulations. the time - independent simulations address the steady state behaviors and therefore do not directly reveal a serious problem with these devices that was already recognized in ref. @xcite. in the following section, we explain this problem. Problem: lack of unidirectionality at the time these logic gates were proposed, it was also realized that they have a severe shortcoming that precludes their use in pipelined architectures @xcite. to understand the nature of the problem, consider three inverters (not gates) in series. a single not gate is the simplest device ; it is realized by two exchange coupled spins, one of which is the input and the other is the output. because of the anti - ferromagnetic ordering, the output is always the logic complement of the input. 2(a) shows three conventional inverters in series and fig. 2(b) shows the corresponding spintronic realization. the input to the first inverter is logic 1 and the output of the last inverter is logic 0, as it should be. but now, let us suddenly change the input at the first inverter to logic 0 at time @xmath9 = 0. the situation at time @xmath9 = 0 + is shown in fig. we expect that ultimately the output of the last inverter will become logic 1. unfortunately, this can not happen. in fig. 2(c), the second spin from the left finds its left neighbor asking it to flip (because of the exchange interaction that enforces anti - ferromagnetic ordering between two neighboring spins) while its right neighbor is asking it to stay put because of the same exchange interaction. both influences from the left and from the right are exactly equally strong and the second cell is stuck in a logic indeterminate state that it can not get out of @xcite. rolf landauer later termed it a metastable state that prevents decay to the desired ground state @xcite. in fact, if we take the external magnetic field into account, then there is a preference for the second cell to actually _ not _ flip in response to the input since there is a slight preference for the upspin state because of the external magnetic field. in this case, the logic signal can not propagate from the input to the output and the circuit simply does not work! similar situations were examined in ref. the real problem is that exchange interaction is _ bidirectional _ which can not ensure _ unidirectional _ flow of logic signal from the input to the output of the logic device. this unidirectionality is a required attribute of any logic device (for the five necessary requirements of a classical logic device, see ref. @xcite). we can think of desperate measures to enforce the unidirectionality. for example, we can claim that if we hold the input at the first inverter (leftmost cell in fig. 2(c)) to logic 0, and do not let go, then the second cell which is equally likely to follow its left neighbor and right neighbor, will have no option but to ultimately follow its left neighbor since it is adamant and persistent (we are not letting go of the input). this will happen in spite of the magnetic field since the exchange energy is larger than the zeeman splitting. in this case, we are trying to enforce unidirectionality via the input signal itself (note that the input device does indeed break the inversion symmetry of the system in fig. this possible remedy was studied theoretically in ref. @xcite which reached the conclusion that it does not always work. in fact, the process of logic signal propagation under this scenario is inefficient thermally assisted random walk and the final logic state, if reached, can be destroyed by thermal fluctuations. the idea of using the input device to enforce unidirectionality was also implicitly used in the experiment of ref. @xcite. while this may work for a few cells (as it did in ref. @xcite), it will obviously not work for a large number of cells since the influence of the input decays with increasing distance from the input. ultimately, the remote cells that are far from the input, will not feel the input s effect and remain stuck in metastable states, producing wrong answers to simple logic problems. in ref. @xcite, one solution that was offered to break this impasse was to progressively increase the distance between cells. this makes the influence of the left neighbor always stronger than that of the right neighbor since the strength of the exchange interaction has an exponential dependence on the separation between neighboring cells. this is not an elegant solution since ultimately the exchange splitting energy will become smaller than the zeeman splitting energy, at which point the paradigm will no longer work. in 1996, we proposed a more elegant solution @xcite. this was inspired by the realization that in charge coupled device (ccd) arrays, there is no inherent unidirectionality, yet charge is made to propagate from one device to the next unidirectionally. this is achieved by _ clocking_. we mentioned that unidirectionality can be imposed in time or space @xcite and clocking imposes unidirectionality in time. however, a cursory examination revealed that normal clocking will not work in our case. say, we put gate pads on the barriers between neighboring cells. initially, all the the barriers are high and opaque so that there is no overlap between the wavefunctions of adjacent electrons and hence no exchange interaction between neighboring spins. now, we lower the first barrier by applying a positive potential to gate 1 as shown in fig. this allows the wavefunctions of electrons on either side of the gate pad to leak out into the barrier, overlap. and cause an exchange interaction. exchange causes the second spin to assume a polarization anti - parallel to the input spin orientation since the singlet state is lower in energy than the triplet. in other words, the second cell switches. at this point, if we let go of the input, raise the first barrier back up, and lower the second barrier by applying a positive potential to gate 2, then either the third cell switches in response to the second (which is shown in the upper branch), or the second cell switches in response to the third (which is shown in the lower branch). the upper branch is the desired state, but it is equally likely that the lower branch will result since both branches obey the ant - ferromagnetic ordering between the two exchange coupled cells (cell 2 and 3). therefore, a simple sequential clock will not work. what is required is that both gate 1 and gate 2 have positive potentials while the input is applied. now the first three cells assume the correct polarizations as shown in fig. then, the input is removed, gate 1 is returned to zero potential and positive potentials are applied to gates 2 and 3. this causes the first four cells to assume the correct polarization, and so on. this situation is shown in fig. 2(f) and is the desired configuration. thus by lowering _ two adjacent barriers _ pairwise at the same time, we can propagate the input state through a linear array. in other words, we will need a _ three - phase clock _, a single phase will not work. the three clock pulse trains for a three - phase clock are shown in fig. each train is phase shifted from the previous one by @xmath10 radians. such a situation is not unusual since charge coupled device arrays also need a multi - phase clock (push clock, drop clock) to work @xcite. while, multi - phase clocking can make these devices work, it is hardly an attractive solution since one needs to fabricate gate pads between every two cells. the separation between the cells may need to be @xmath11 5 nm in order to have sufficient exchange coupling. aligning a gate pad to within a space of 5 nm is a major lithography challenge. furthermore, the gate potentials are lowered and raised by moving charges into and out of the gate pads, leading to considerable energy dissipation that completely negates the advantage of using spins. therefore, these devices present interesting physics, but at this time, do not appear to be serious candidates for practical applications. Using spin as a qubit so far, we have discussed the use of spin in classical irreversible logic gates. these logic gates dissipate a minimum of @xmath12 amount of energy per bit flip @xcite. let us assume that we can make quantum dots with a density of 10@xmath13 @xmath14. quantum dots self assembled by electrochemical techniques in our own lab (and in many other labs) can achieve this density today. we show a raw atomic force micrograph of quantum dots self assembled in our lab in fig. 3. the dark areas are the dots and the surrounding light areas are the barriers. the dot diameter in this micrograph is 50 nm and the dot density is close to 10@xmath15 @xmath14. by using slightly different synthesis conditions, we can actually achieve densities exceeding 10@xmath13 @xmath14. let us now assume that we can flip the spin in a quantum dot in 1 psec. then the minimum power that will be dissipated per unit area will exceed @xmath16 /(1 psec) = 3 kw/@xmath17 (actually most of the power will be dissipated in the clock cycles, which we have ignored). this dissipation is at least 30 times more than what the pentium iv chip dissipates @xcite. although removal of 1 kw/@xmath17 of heat from a chip was demonstrated almost two decades ago, removing 3 kw/@xmath17 from a chip is still a major challenge in heat sinking. the obvious way to overcome (or, rather, circumvent) this challenge is to develop reversible logic gates that are not constrained by the landauer @xmath12 barrier. in 1996, we devised a logically and physically reversible quantum inverter using two exchange coupled spins @xcite. this device is very similar to the single electron parametron idea @xcite and can be viewed as a single spin parametron. since either spin could exist in a coherent superposition of two orthogonal spin states (call them `` upspin '' and `` downspin '' states), this would also be a `` qubit ''. later, loss and divincenzo devised a universal quantum gate using two exchange coupled spins in two closely spaced quantum dots @xcite. recently, experimental demonstration of coherent transfer of electron spins between quantum dots coupled by conjugated molecules has been demonstrated, opening up real possibilities in this area @xcite. Spintronic quantum gates the idea of using a single electron or nuclear spin to encode a qubit and then utilizing this to realize a universal quantum gate, has taken hold @xcite. the motivation for this is the realization that spin coherence times in solids is much larger than charge coherence time. charge coherence times in semiconductors tend to saturate to about 1 nsec as the temperature is lowered @xcite. this is presumably due to coupling to zero point motion of phonons which can not be eliminated by lowering temperature @xcite. on the other hand, electron spin coherence times of 100 nsec in gaas at 5 k has already been reported @xcite and much higher coherence times are expected for nuclear spins in silicon @xcite. therefore, spin is obviously the preferred vehicle to encode qubits in solids. using spin to carry out all optical quantum computing has also appeared as a viable and intriguing idea @xcite. the advantage of the all - optical scheme over the electronic scheme is that we do not have to read single electron spins _ electrically _ to read a qubit. electrical read out is extremely difficult @xcite, although some schemes have been proposed for this purpose @xcite. recently, some experimental progress has been made in this direction @xcite, but reading a single qubit in the solid state still remains elusive,. the difficult part is that electrical read out requires making contacts to individual quantum dots, which is an engineering challenge. in contrast, optical read out does not require contacts. the qubit is read out using a quantum jump technique @xcite which requires monitoring the fluorescence from a quantum dot. recently, it has been verified experimentally that the spin state of an electron in a quantum dot can be read by circularly polarized light @xcite. therefore, optical read out appears to be a more practical approach. Conclusion in this article we have provided a brief history of the use of single electron spin in computing. we have indicated where and why spin may have an advantage over charge in implementing the type of devices and architectures discussed here. s. bandyopadhyay and v. p. roychowdhury, proceedings of the international conference on superlattices and microstructures, liege, belgium, 1996, also in superlat. 3, 411416 (1997). t. calarco, a. datta, p. fedichev, e. pazy and p. zoller, phys. rev. a, vol. 68, no. 1, 012310 - 1 012310 - 21 (2003) ; e. pazy, e. biolattia, t. calarco, i. d amico, p. zanardi, f. rossi and p. zoller, europhys. 62, 175 (2003).

this article reviews the use of single electron spins to compute. in classical computing schemes, a binary bit is represented by the spin polarization of a single electron confined in a quantum dot. if a weak magnetic field is present, the spin orientation becomes a binary variable which can encode logic 0 and logic 1. coherent superposition of these two polarizations represent a qubit. by engineering the exchange interaction between closely spaced spins in neighboring quantum dots, it is possible to implement either classical or quantum logic gates.

cond-mat0404659

Introduction circumstellar material holds clues about the mass - loss history of massive stars. indeed, as the winds interact with the interstellar medium (wind - blown bubbles, bow shocks), they leave a characteristic signature that depends on the wind properties. moreover, the material ejected during short eruptive phases is visible as nebulae around massive stars. the analysis of these features reveals which material was ejected and in which quantity. with the recent reduction in mass - loss rates, these episodes of enhanced mass - loss have gained more attention, as they seem more crucial than ever in the evolution of massive stars. another reason to study the close environment of massive stars is to better understand the evolution of supernova remnants (snrs). indeed, the famous rings of sn1987a may only be understood if one considers the previous mass - loss episodes of the progenitor. morphology is not the only snr parameter which is affected, as the snr dynamics in an homogeneous medium or in winds and circumstellat ejecta is not identical. for its study, the ir provides several key diagnostics. continuum emission in this range is provided by heated dust, which may have a range of temperatures depending of the framework (very close hot features, large, old, and cool bubbles). in addition, ir lines probe the many phases of the material : molecules (e.g. pahs) for the neutral material, ionized metals for hii regions,... this summary of sps5 - part iii examines each case of circumstellar environment in turn, and concludes with the potential offered by current and future facilities. Blue supergiants circumstellar structures around bsgs have been predominantely identified as bow shocks around runaway stars. originally discovered with iras (e.g. van buren & mccray, 1988, apj, 329, l93), such structures have also been seen with msx and wise (peri et al. 2012). a more general survey of bsgs, i.e. not targeting runaway stars, with objects selected from crowther et al. (2006) and przybilla et al. (2010), reveals ir material around six of the 45 targets at 22@xmath0 m with wise, also mostly in the form of bow shocks (wachter, in prep). several examples of bipolar nebulae around bsgs are also known (e.g. sher 25, smartt et al. 2002 ; hd 168625, smith 2007). however, this material could have also been ejected during an lbv phase, since lbvs can exhibit bsg spectra, and we will therefore concentrate on the bow shocks. emission (greyscale) of vela x-1 with pacs 70@xmath0 m emission contours shown on top. _ right : _ colour composite image of bow shock of bd+43@xmath13654 (wise 12@xmath0 m in blue, pacs 70@xmath0 m in green, and pacs 160@xmath0 m in red). the direction of proper motion is indicated by the arrow in both cases. from cox (in prep.).,title="fig:",width=226] emission (greyscale) of vela x-1 with pacs 70@xmath0 m emission contours shown on top. _ right : _ colour composite image of bow shock of bd+43@xmath13654 (wise 12@xmath0 m in blue, pacs 70@xmath0 m in green, and pacs 160@xmath0 m in red). the direction of proper motion is indicated by the arrow in both cases. from cox (in prep.).,title="fig:",width=226] runaway stars have large stellar velocities (above 30kms@xmath2) resulting from dynamical interactions in (dense) clusters or from a supernova explosion in a binary system. these stars can thus travel at supersonic speeds through the local medium giving rise to `` bow shocks '' as their stellar winds interact with the surrounding medium, which has been previously ionised by stellar photons from the hot star (weaver 1977). the occurrence of such bow shocks has been shown to depend primarily on the ism conditions (huthoff & kaper 2002). for example, even a runaway star may travel at subsonic speeds in the tenuous interior of a superbubble, where the sound speed can be as much as 100kms@xmath2, hence no (detectable) bow shock will be produced in that case. the filling factor of ism with @xmath3kms@xmath2 is 20% and 75% of o - stars have velocities @xmath410kms@xmath2, so the expected fraction of o - stars with bow shocks is @xmath515%. this is remarkably similar to the values derived from iras and wise observations (noriega - crespo et al. 1997, peri et al. 2012). once formed, the size, shape and morphology of a bow shock depends on both stellar (wind kinetic energy and stellar velocity) and interstellar parameters (density and temperature). in particular the ratio @xmath6 indicates whether or not instabilities are likely to develop (dgani et al. 1996), and the stand - off distance between the star and the apex of the shock is determined from the pressure balance between the stellar wind and the ism (see analytical results by wilkin 1996 and simulations by e.g. comeron & kaper 1998, blondin & koerwer 1998). independent estimates of the wind parameters can thus be inferred from bow shocks, which serves as a useful check for atmosphere models, but the values are sensitive to the ism properties, which are not always known with precision. m of the dust thermal emission obtained by kervella et al. north is up, east to the left, and the field of view is given in the upper right corner of each image., width=529] currently, a small survey with herschel - pacs of 5 runaways with known bow - shocks is ongoing : @xmath7cam, @xmath8oph, @xmath9cma, vela x-1 and bd+43@xmath13654 (cox et al., in preparation). for vela x-1, the peak emission of the dust emission is co - spatial with the most prominent h@xmath7 arc seen in the supposed direction of space motion (fig. [cox]) : it is concluded that the outer shock is radiative, but the inner shock is adiabatic, though some h@xmath7 emission possibly related to (part of) the inner termination shock is also detected. from the analysis of its `` puffed - up '' bow shock (fig. [cox]), the mass - loss rate of bd+43@xmath13654 (o4if) was found to be 10@xmath10m@xmath11yr@xmath2 : this is very high (by 2 orders of magnitude) in view of current mass - loss rate estimates of such stars, but the exact value strongly depends on ism density, which need to be refined. the dust temperature, @xmath5 45 k, is compatible with heating by stellar photons only, suggesting there is no additional shock - heating of grains. the thickness of a bow shock (@xmath5 1 pc) suggests a mach number close to unity, implying a ism temperature of 10@xmath12 10@xmath13 k. Red supergiants circumstellar structures on scales of a few arcseconds or less around rsgs have been revealed through interferometric techniques (e.g. monnier et al. stencel et al. (1988, 1989) reported the iras detection of resolved shells with typical radii of a few arcminutes around rsgs for a significant fraction (25%) of their sample. however, higher resolution spitzer images fail to confirm several of these extended structures (wachter, in prep), indicating that a systematic survey is needed to ascertain the occurrence of large scale circumstellar shells around rsgs. m to 192@xmath0 m. multitude of molecular lines have been detected. from matsuura et al. (in prep.).,width=453] a few (famous) cases have however been studied in depth. one of these is betelgeuse, a cool (3600k), large (700 r@xmath11), rather massive (1015 m@xmath11), luminous (@xmath14 l@xmath11), and nearby (150 pc) star. because of its distance, betelgeuse can be probed on almost all scales, providing a unique panorama of stellar surroundings (fig. [betel]). space - based and interferometric instruments (e.g. hst, iota / ionic and vlti / pionier) revealed the photosphere, notably the expected non - uniformities due to large convection cells. adaptive optics imaging in the near - ir (e.g. naco) and (radio or ir) interferometers unveiled the properties of the internal, compact molecular envelope (110 r@xmath15). precursors of dust have been found there, as well as an extended `` plume '' reaching 6r@xmath15 and maybe linked to a hot spot on the photosphere. high - res imaging (e.g. vlt / visir) shows the envelope at intermediate scales (10100 r@xmath15), where the dust forms (a possible signature of silicates has been found). at these small and intermediate scales, betegeuse presents a complex circumstellar envelope (with knots and filaments) at all wavelengths, which implies an inhomogeneous spatial distribution of the material lost by the star. finally, at the largest scale, ir imagers such as herschel unveil the cool external envelope (100 - 10000 r@xmath15), where a bow shock with the ism is detected (cox et al. 2012). m (blue ; blue contours) and 22 @xmath0 m (red ; yellow contours) of ws1, discovered and initially characterized by gvaramadze et al. the contours for each band help illuminate the morphology of the nebular material, which has an overall se - nw elongation, reminiscent of bipolar structure. _ bottom left : _ optical photometric monitoring since discovery show both the r and i light curves have brightened by about 1 magnitude over the last year. arrows indicate times when same night spectroscopy were secured. _ right panel : _ optical spectroscopic monitoring indicates evolution to cooler temperature with near disappearance of the he i 5876 @xmath16 and 6678 @xmath16 and changes in the @xmath17 line profile. figures from stringfellow et al. (in preparation).,width=377] herschel has also probed the envelope of other red supergiants (groenewegen et al. turning in particular to the case of vy cma (matsuura et al., in prep.), the potential of ir spectroscopy is obvious. herschel - spire reveals a rich spectrum, with a dust continuum and hundreds of lines dues to molecules (one third linked to water, others to co, cs, sio,...), which constrain the envelope s properties. for example, the isotopic ratio @xmath18c/@xmath19c is found to be 6.5, in agreement with observations of other rsgs but at odds with theoretical predictions which are four times higher at least. very strong emission of submm molecular lines can be explained if a temperature gradient is present in the envelope, e.g. because of dust formation at a certain radius. Luminous blue variables because of their spectacular eruptions, lbvs are the most well - known cases of massive stars with ejecta. it is not yet certain, however, at what stage (bsg? after a rsg phase or not?) this material is ejected, and how (multiple events?). lbvs are rare : in the list of clark et al. (2005), there are only 12 confirmed and 23 candidate lbvs. ir has played a key role in recent years. the search, through surveys like mipsgal, of round - shaped nebulae with luminous central stars resulted in the discovery of many new nebulae : 62 shells in wachter et al. (2010), 115 shells and bipolar nebulae in gvaramadze et al. (2010), 416 structures in mizuno et al. many of these nebulae are preferentially detected with spitzer 24@xmath0 m band, indicating relatively cold material. identifying shell - like structures is only the first step. to ascertain a clbv status, the central object needs to be studied spectroscopically. this was done for many of these new detections (c.f., gvaramadze et al. 2010 ; wachter et al. 2010 ; stringfellow et al. 2012a, b, and in preparation). the classification does not rely on the presence of a particular line, but rather on the morphological resemblance of the spectra to spectra of known lbvs - while not 100% perfect (some peculiar o and wr stars display similar features), this method has the advantage of being simple and rather robust. a more definitive answer can be provided through photometric and/or spectroscopic monitorings. indeed, as their name indicate, lbvs should be _ variable_. near - simultaneous photometric and spectroscopic monitoring in the optical (and ir) of about a dozen newly identified candidate lbvs has revealed that ws1 (discovered by gvaramadze et al. 2012) is indeed a bona fide lbv, presently displaying what appears to be s dor type variability as shown in fig. [guy] (stringfellow et al. 2012, in preparation). ir is also useful in revealing details of particular objects. for example, a herschel survey of lbvs undertaken at lige yielded as first result a characterization of the surroundings of wray 15 - 751 (vamvatira - nakou et al., submitted). ir photometry can only be explained if the star evolves at constant luminosity and dust grains are fe - rich. images also revealed the presence of a second shell, about 4 times larger than the previously known one, which most probably results from an older eruptive event. considering both structures, there is about 0.075 m@xmath11 of dust in the system. ionized gas is responsible for several forbidden lines observed in the herschel - pacs range (fig. [chloi]), which allow a n / o abundance of about 7 times solar and a mass of ionized gas of 12 m@xmath11 (20 times that of dust), to be derived. dust can be well studied in the ir, so this range may provide clues on where dust come from in galaxies. two examples of such feedback were presented in the session : @xmath20car and sn1987a. the latter was observed with herschel at 100 - 500@xmath0 m wavelengths, and 0.4 - 0.7 m@xmath11 of dust was detected - mostly silicates and amorphous carbon (matsuura et al. it is thought that this dust come from the explosion, but the role played by previous mass - loss episodes, in particular the lbv phase, is not yet clear. for example, about 0.12 m@xmath11 of dust was detected, thanks to 30@xmath0 m minitao observations, in the famous lbv @xmath20car. up to 80% of that dust belonged to the torus, hence may not be related to the big 1843 event. Wolf-rayet stars only a few percentage (4 - 6%) of wolf - rayet stars displays surrounding nebulosities in the wise survey, and most are found around wn stars (wachter, in prep). the morphological classification scheme of wr nebulae proposed by chu (1981) has been revised in this meeting by guerrero et al. (toal et al, in prep.) using _ wise _ ir images and sdss or super cosmos sky survey h@xmath7 images for 35 nebulae associated with wrs. two _ wise _ bands were particularly used : the one at 12@xmath0 m, which encompasses pah lines and lines of low excitation ions, and that at 22@xmath0 m, to which thermal emission from dust and lines of he i as well as high excitation ions contribute. three phases are defined. in the first one, wr nebulae appear as complete shells or bubbles. it corresponds to the star just entering the wr stage, when its powerful wind sweeps up the previous slow and dense winds (from e.g. lbv or rsg stages). the second phase is the clumpy phase. at that point, the nebulae display knots of gas and dust connected by partial shells and arcs. it corresponds to an age of a few 10@xmath13yr, when instabilities break down the swept - up shell. the stellar motion through the ism has an impact on the morphology, for example one - sided arc may be sometimes seen. finally, the mixed nebular phase ends the cycle, with no definite morphology nor always a 1-to-1 correspondence between optical and ir images. it corresponds to the last stage, when the circumstellar nebula begins to dissolve into the ism. Studying the close environment of massive stars the close environment of massive stars is the `` missing link '' between the star itself and the large circumstellar features. it plays a key role in understanding the mass - loss, but it is also difficult to probe directly. emission lines arising in the wind and circumstellar material are a classical way to study this region, as well as near - ir excess linked to disk - like features. in this context, be and b[e] stars are targets of choice, and surprises are frequent. for example, graus et al. (2012) found three new sgb[e] in the smc : they display typical spectra, with forbidden lines, but the line strengths as well as the ir excess appear reduced compared to usual objects of this class. it suggests that either the disks have less material or less dust than usual, or maybe that these stars are transitional objects. another case intriguingly shows the opposite situation : cd@xmath2149@xmath13441 displays forbidden lines and appreciable ir excess, but is a main - sequence be star away from any star - forming region (lee and chen 2009). a possibility may be that this star is in fact a weak b[e], rather than a classical be. the environment close to the star can also be studied, directly, by means of interferometry, which is usually performed at long wavelengths. most optical / ir interferometric measurements rely on the measurements of `` visibilities '', which are directly linked to the size of the object. recently, several massive stars, including nine lbvs, were observed with the vlti (groh et al., in prep.). amongst these, hd316285 (fig. [groh]) : the recorded visibilities implied a size of 0.002 for the source of continuum radiation, and 0.004 for the source of the br@xmath22 line. a cmfgen fit to the spectrum yields a stellar model with which one can estimate the wind size in ir, and it agrees well with vlti observations. the asymetric shape of the measured differential phases (red vs blue side) favors a prolate shape for the rotating star wind, but it could also be explained by clumps or binarity. since the latter imply in time variability, a monitoring will be needed to ascertain the exact nature of the asymmetry. = 0.19, rather than 0.25 as von zeipel predicted in 1924.,width=377] while visibilities provide valuable data, `` real '' images are always more impressive. interferometric instruments such as michigan infrared combiner (mirc) and pionier are beginning to provide such data. mirc was the first to image altair (monnier et al. 2007) and several other rapidly rotating stars as shown in fig. it has also imaged circumstellar disks and multi - object systems. for example, the disk contribution of @xmath23sco was shown to remain stable during the periastron in 2011 (che et al. 2012), and the mass - exchange in the @xmath24lyr system can be clearly imaged (zhao et al. 2008), as well as the 3 components of algol (baron et al. 2012) or the disk of the eclipsing companion of @xmath25aur (kloppenborg et al., 2010). Conclusion this session has demonstrated the usefulness of studies in the ir in studying the environment of massive stars. recent advances in this domain are notably provided by surveys, as they enable discovery of new objects to study, thereby improving the census of nebular features associated with hot stars. furthermore, ir diagnostics unveil the properties of these neighbouring nebulosities : morphology, temperature, composition, density are the necessary keys paving the way of a better understanding of the mass - loss in massive stars. yn acknowledges comments from augusto daminelli and support from fnrs and prodex herschel / xmm - integral contracts. nljc thanks fwo and prodex - herschel for financial support. jhg is supported by an ambizione fellowship of the swiss national science foundation. cdl has been supported financially by grant nsc-101 - 2922-i-008 - 120 of the national science council of taiwan.

the last part of sps5 dealt with the circumstellar environment. structures are indeed found around several types of massive stars, such as blue and red supergiants, as well as wrs and lbvs. as shown in the last years , the potential of ir for their study is twofold : first, ir can help discover many previously unknown nebulae, leading to the identification of new massive stars as their progenitors ; second, ir can help characterize the nebular features. current and new ir facilities thus pave the way to a better understanding of the feedback from massive stars.

1210.3986

Introduction this paper presents data for the last two patches (c and d) of the sky observed by the public eso imaging survey (eis), being carried out in preparation for the first year of regular operation of vlt. the i - band data reported here covers a total area of 12 square degrees, down to @xmath6, corresponding to two patches probing separated regions of the sky, 6 square degrees each. the present work complements earlier papers in the series (nonino 1998 ; paper i, prandoni 1998 ; paper iii) and completes the presentation of the data accumulated by the eis observations carried out in the period july 1997-march 1998 as part of the wide - angle imaging survey originally described by renzini and da costa (1997) and in paper i. the primary science goal for surveying patches c and d was to search for and produce a list of distant galaxy cluster candidates that would complement those of the other two patches (a and b) reported earlier (olsen 1998a, b : paper ii and v), providing vlt targets nearly year - round. patches c and d were also selected to overlap with the ongoing 92 cm westerbork survey in the southern hemisphere (wish) being carried out in the region @xmath7 and @xmath8. originally, the eis observations were expected to be carried out in two passbands (v and i). however, because of time constraints and the prospect of supplementing the eis observations at the ntt with the new wide - field imager for the 2.2 m eso / mpia telescope, preference was given to increase the area covered by the i - band observations, more suitable for identifying distant clusters with @xmath9 (see paper v). this decision allowed the full coverage of the selected patches, yielding a total coverage of 12 square degrees. combined with the data for patches a and b the eis i - band data covers a total area of about 17 square degrees, currently the largest available survey of its kind in the southern hemisphere. the goal of the present paper is to describe the characteristics of the i - band observations of patches c and d. in section 2, the observations, calibration and the quality of the data are described. in section 3, the object catalogs extracted from the images are examined and compared with data from the other patches and other data sets to comparable depth. concluding remarks are presented in section 4. Observations and data reduction the observations of patches c and d were carried out over several months in the period november (december for patch d) 1997 to march 1998, using the red channel of the emmi camera on the 3.5 m new technology telescope (ntt) at la silla. the red channel of emmi is equipped with a tektronix 2046 @xmath10 2046 chip with a pixel size of 0.266 arcsec and a useful field - of - view of about @xmath11. the observations were carried out as a series of overlapping 150 sec exposures, with each position on the sky being sampled at least twice, using the wide - band filter wb829#797 described in paper i, and for which the color term relative to the cousins system is small. the data for patches c and d consist of 1348 frames but only 1203 were accepted for final analysis, discarding 145 frames obtained in poor seeing condition (@xmath12 arcsec). the frames actually accepted have a seeing in the range 0.5 to 1.6 arcsec, considerably better than the data available for patches a and b obtained at the peak of el nio. figure [fig : seeing] shows the seeing distribution of all observed frames in each patch. for comparison the figure also shows the seeing distribution of the accepted frames, with the vertical lines in each panel indicating the median seeing and the quartiles of the distribution. from the figure one finds that the median seeing for both patches is sub - arcsec (@xmath13 arcsec) with only 25% of the area covered by frames with a seeing larger than 1 arcsec. the good quality of the observations can also be seen from figure [fig : limiso] which shows the @xmath14 limiting isophote within 1 arcsec for each patch. apart from one subrow in patch c, in both cases the limiting isophote is typically @xmath15 25.3 @xmath16 mag arcsec@xmath17. the two - dimensional distributions of the seeing and limiting isophote are shown in figures [fig : seeing_cont] and [fig : limiso_cont]. comparison with similar distributions presented in earlier papers (paper i and iii) shows that the data for patches c and d are significantly better. note that for each patch tables are available listing the position of each accepted frame, its seeing, limiting isophote and photometric zero - point and can be found at `` http://www.eso.org/eis ''. in late february 1998, a realignment of the secondary mirror was carried out by the ntt team in an attempt to minimize the image distortions seen in the upper part, especially the upper - right corner, of the emmi frames. some frames for patch c and most of the frames in patch d were observed with the new setup of the ntt. examination of the point spread function for these frames showed no significant improvement in the quality of the images. this points out the need to introduce a position - dependent estimator for the point - spread function to assure uniformity in the star / galaxy separation across the frame. this is particularly important for images observed under good seeing conditions. in fact, examining the uniformity of the classification as a function of position on the chip it is found that there is a 10% increase in the density of galaxies at the upper edge of the chip, due to misclassifications, significantly larger than that seen in paper i. in the last three runs (january - march) it was also noticed faint (at the @xmath18 level of the background noise) linear features aligned along the east - west direction (perpendicular to the readout axis) associated with moderately bright stars located in the lower half of the ccd not previously seen. the cause for the these features are at the present time unclear but are probably due to the electronic of the old - generation ccd controller of emmi, when used in a dual - port readout mode. these affects two - thirds of the patch c frames and essentially all the patch d frames. these light trails occur randomly in the patch and there is no obvious way of correcting for them a priori. an important consequence of this problem is that it leads to a localized increase in the detection of low - surface brightness objects over a range of magnitudes (typically @xmath19) which can have a significant impact in the cluster detection algorithm (scodeggio 1998, paper vii). this is unfortunate because both patches c and d are located at lower galactic latitudes (@xmath20) with almost an order of magnitude larger density of stars than the previous patches. the photometric calibration of the patch was carried out, as described in papers i and iii, by determining a common zero - point for all frames from the solution of a global least - squares fit to all the relative zero - points, constraining their sum to be equal to zero. the absolute zero - point was determined by a simple zero - point offset determined from the common zero - point of all frames observed in photometric conditions. there are 340 and 290 such frames, covering about 80% and 60% of the surveyed area, in patches c and d, respectively (see figure [fig : overlaps]). the zero - points for these frames were determined using a total of 10 fields containing of the order of 45 standard stars taken from landolt (1992 a, b), observed in 10 nights for patch c and in 11 nights for patch d. altogether 215 independent measurements of standards in the three passbands were used in the calibration. comparison with external data suggests that a zero - point offset provides an adequate photometric calibration for the entire patch. in order to check the photometric calibration and the uniformity of the zero - points, strips from the denis survey (epchtein 1996) crossing the surveyed area the regions of overlap of these data are shown in figure [fig : overlaps], which shows that there are five strips crossing patch c and two strips crossing patch d. in the figure the regions observed under photometric conditions are also indicated. comparison of this figure with their counterparts presented in papers i and iii, clearly shows that the data for patches c and d are of superior quality, with a much larger fraction of frames taken under photometric conditions. in order to investigate possible systematic errors in the photometric zero - point over the scale of the patch, the eis catalogs were compared with object catalogs extracted from the denis strips that cross the survey regions (see figure [fig : overlaps]). comparison of the catalogs allows one to investigate the variation of the zero - point over the patch. the results are shown in figure [fig : denis]. the domain in which the comparison can be made is relatively small because of saturation of objects in eis at the bright end (@xmath21) and the shallow magnitude limit of denis (@xmath22). still, within the two magnitudes where comparison is possible one finds a roughly constant zero - point offset of less than 0.02 mag for both strips and a scatter of @xmath23 mag that can be attributed to the errors in the denis magnitudes (deul 1998). Data evaluation in order to evaluate the quality of the data simple statistics computed from the object catalogs extracted from the images are compared in this section with model predictions and other data sets. the catalogs derived from individual frames are used to generate the even, odd and best seeing catalogs, described in earlier papers. the spatial distribution of stars and galaxies, defined using similar star / galaxy classification criteria as in previous papers of the series, are shown in figures [fig : pc_visu] and [fig : pd_visu] down to @xmath24 and @xmath25 for stars and galaxies, respectively. the latter corresponds roughly to the completeness limit of the object catalog. this limit was established using the object catalog extracted from the co - addition of images of a reference frame taken periodically during the observations of a patch. note that because of the much better seeing star / galaxy classification is possible down to @xmath26 and the completeness is about 0.5 mag deeper. some improvement in the classification is expected from a new estimator being implemented in sextractor based on a position - dependent psf fitting scheme currently being tested. this new version should also improve the uniformity of the classification across the chip. the distribution of the stars and galaxies shown in figures [fig : pc_visu] and [fig : pd_visu] is remarkably homogeneous and considerably better than those seen in the previous eis patches due to the much better observing conditions. this is true except for a small region of about 0.2 square degrees in patch c which has been removed, as indicated in figure [fig : pc_visu]. the only problem seen with the galaxy catalogs in these patches is the presence of several relatively thin linear features clearly seen at high resolution (see eis release page). these features are a consequence of the electronic problem mentioned above and are not easily corrected for at the image level. in order to evaluate the data the general properties of the extracted object catalogs are investigated and compared with model predictions and other data sets. note that patches c and d are located at lower galactic latitude and the number of stars is considerably larger. in addition, the seeing is considerably better than in previous patches. therefore, it is of interest to re - evaluate the overall performance of the eis pipeline reduction under these new conditions. figure [fig : star_counts], shows the comparison of the star counts for patches c and d derived using the stellar sample extracted from the object catalogs, with the predicted counts based on a galactic model composed of an old - disk, a thick disk and a halo. the star - counts have been computed using the model described by mndez and van altena (1996), using the standard parameters described in their table 1 and an @xmath27 of 0.015 and 0.010 for patches c and d, respectively. it is important to emphasize that no attempt has been made to fit any of the model parameters to the observed counts. the model is used solely as a guide to evaluate the data. as can be seen there is a good agreement at bright magnitudes (@xmath28), but the observed counts show an excess at fainter magnitudes (@xmath29). even though it is unlikely that this excess is due to misclassified galaxies at these relatively bright magnitudes, a better agreement can be achieved if a higher stellarity index is assumed. on the other hand, it is also possible that the model underestimates the contribution of the thick - disk which makes a significant contribution in this magnitude range. the steep drop in the stellar counts beyond @xmath30 is partially due to the relatively high stellarity index adopted, which was chosen to minimize the losses of galaxies. by adopting a stellarity index of 0.5 the drop in the counts may be avoided down to @xmath31. however, at these magnitudes and this value of the stellarity index contamination by galaxies may be significant. another potential problem at these faint magnitudes is the misclassification of stars as a consequence of the distortion effects in emmi, that can have some impact for images taken in good seeing conditions. in order to evaluate the depth of the galaxy samples, galaxy counts in patches c and d are compared with those of previous patches in figure [fig : gal_counts]. there is a remarkable agreement among the counts derived for the different patches, indicating that the identification of galaxies has not been affected by the observations at lower galactic latitudes. the galaxy counts obtained from the different patches have been combined to compute the mean galaxy counts and the variance. this is also shown in figure [fig : gal_counts] where it is compared to other ground - based counts (postman 1998) and those from hdf (williams 1996), appropriately converted to the cousins system (see paper iii). as can be seen the eis galaxy counts agree extremely well with the ground - based data covering comparable area over the entire magnitude range down to @xmath32 and with the bright end of the hdf counts. the excellent internal and external agreement of the i - band galaxy counts serves as a confirmation of the reliability of the eis galaxy catalogs. extraction from co - added images should allow reaching about 0.5 mag deeper. one way of examining the overall uniformity of the galaxy catalogs is to use the two - point angular correlation function, @xmath33, as departures from uniformity should affect the correlation function especially at faint magnitudes. the latter should be sensitive to artificial patterns, especially to the imprint of the individual frames, or possible gradients in the density over the field, which could result from large - scale gradients of the photometric zero - point. note that any residual effect due to the improper association of objects in the border of overlapping frames would lead to a grid pattern (see the weight map in the eis release page) that could impact the angular correlation function. figure [fig : w] shows @xmath33 obtained for different magnitude intervals for both patches, using the estimator proposed by landy & szalay (1993). the calculation has been done over the entire area of patch d and most of the area of patch c, with only one subrow (10 consecutive frames) removed according to the discussion above (see section [obs]). for comparison, @xmath33 computed for the other patches are also shown (papers i and iii) from which the cosmic variance can be evaluated directly from the data. as can be seen there is a remarkable agreement for all the magnitude intervals considered. moreover, the larger contiguous area of patches c and d allows to estimate the angular correlation function out to @xmath34 degree. in all cases @xmath33 is well described by a power law @xmath35 with @xmath36 in the range 0.7 - 0.8. note that for patch b the results refer to the galaxy sample obtained after removing the foreground cluster (see paper iii). in particular, there is no evidence for any underlying pattern associated with the overlap of different frames. the effect on @xmath33 was evaluated by carrying out simulations by adding to the observed galaxy distribution a grid pattern with different density contrast. it was found that for high contrast this would lead to local depressions in the angular correlation function on scales of half the size of the diagonal of the grid and its multiples, with the depth of depression depending on the relative density. none such features are seen further indicating the uniformity of the derived galaxy catalogs. finally, note that the good agreement of @xmath33 for the different patches confirms that the observed small - scale linear features associated with the faint light trails, mentioned in section [obs], have very little impact in the angular correlation function. as shown in paper iii the dependence of the amplitude of the correlation function on the limiting magnitude of the sample is consistent with earlier estimates based on significantly smaller areas and the recent results reported by postman (1998). these results show that the eis galaxy catalogs are spatially uniform and form a homogeneous data set independent of the patch, yielding reproducible results. finally, note that even though a single power - law with a slope between 0.7 - 0.8 gives a reasonable fit for the correlation computed in all magnitude bins, there is some indication that for fainter samples (@xmath37) the angular correlation function may be better represented by two distinct power - laws. on small scales (@xmath38) the slope remains the same while on larger scales it becomes gradually flatter. a similar behavior is seen in the @xmath33 computed for all four patches. this flattening seems to be consistent with earlier claims by campos (1995) and neuschaefer and windhorst (1995) using significantly smaller samples, and more recently by postman (1998) with a sample of similar size to eis but covering a single contiguous area. Summary one year after the first observations, the full data set accumulated by eis is being made public in the form of astrometrically and photometrically calibrated pixel maps and object catalogs extracted from individual images. in addition, separate papers have presented derived catalogs listing candidate targets for follow - up work. the eis data set consists of about 6000 science and calibration frames, totaling 96 gb of raw data and over 200 gb of reduced images and derived products. all the information regarding these frames are maintained in a continuously growing database. together with the science archive group a comprehensive interface has been built to provide users with a broad range of products and information regarding the survey. from the verification of the object catalogs and their comparison against model predictions and other observations, it has been found that the extracted catalogs are reliable and uniform. when all patches are included, the combined eis galaxy catalog contains about one million galaxies and it is by far the largest data set of faint galaxies currently available in the southern hemisphere. the star counts show a good agreement with current galactic models, especially at high - galactic latitudes, and the galaxy counts agree remarkably well with other ground - based observations as well as with the counts derived from hdf. the data from the different patches seem to be rather homogeneous, as strongly suggested from measurements of the angular two - point correlation function which should be sensitive to large - scale gradients in a patch or to relative offsets of the photometric zero - points for the different patches. as expected eis - wide has provided large samples (50 to over 200 candidates) of distant clusters of galaxies (olsen 1998a, b, scodeggio 1998) and of potentially interesting point sources (zaggia 1998), more than adequate for the first year of observations with vlt, the main goal of eis. some of the targets can also be observed nearly year round. in order to expedite the delivery of the products all the results refer to single exposure frames as discussed in the previous papers of the series. even though co - addition has been done for all the patches some problems have been uncovered during the verification of the object catalogs extracted from them and require further work. however, the samples already public are sufficiently deep and large for programs to be conducted in the first year of operation of the vlt. the results obtained from the co - added images will become available before the vlt proposal deadline. this paper completes the first phase of eis which will now focus on the deep observations of the hdf - south (@xmath39) and axaf deep (@xmath40) fields. the results presented so far show the value of a public survey providing the community at large with the basic data and tools required to prepare follow - up observations at 8-m class telescopes. the experience acquired by eis in pipeline processing, data archiving and mining will now be transferred to the pilot survey, a deep wide - angle imaging survey to be conducted with the wide - field camera mounted on the eso / mpia 2.2 m telescope. we thank all the people directly or indirectly involved in the eso imaging survey effort. in particular, all the members of the eis working group for the innumerable suggestions and constructive criticisms, the eso archive group and the st - ecf for their support. we also thank the denis consortium for making available some of their survey data. the denis project development was made possible thanks to the contributions of a number of researchers, engineers and technicians in various institutes. the denis project is supported by the science and human capital and mobility plans of the european commission under the grants ct920791 and ct940627, by the french institut national des sciences de lunivers, the education ministry and the centre national de la recherche scientifique, in germany by the state of baden - wurttemberg, in spain by the dgicyt, in italy by the consiglio nazionale delle richerche, by the austrian fonds zur frderung der wissenschaftlichen forschung und bundesministerium fr wissenschaft und forschung, in brazil by the fundation for the development of scientific research of the state of so paulo (fapesp), and by the hungarian otka grants f-4239 and f-013990 and the eso c & ee grant a-04 - 046. our special thanks to the efforts of a. renzini, vlt programme scientist, for his scientific input, support and dedication in making this project a success. finally, we would like to thank eso s director general riccardo giacconi for making this effort possible in the short time available.

this paper presents the i - band data obtained by the eso imaging survey (eis) over two patches of the sky, 6 square degrees each, centered at @xmath0, @xmath1, and @xmath2, @xmath3. the data are being made public in the form of object catalogs and, photometrically and astrometrically calibrated pixel maps. these products together with other useful information can be found at `` http://www.eso.org/eis ''. the overall quality of the data in the two fields is significantly better than the other two patches released earlier and cover a much larger contiguous area. the total number of objects in the catalogs extracted from these frames is over 700,000 down to @xmath4, where the galaxy catalogs are 80% complete. the star counts are consistent with model predictions computed at the position of the patches considered. the galaxy counts and the angular two - point correlation functions are also consistent with those of the other patches showing that the eis data set is homogeneous and that the galaxy catalogs are uniform. 1@xmath5 # 1

astro-ph9807334

[sec:introduction]introduction the presence of a surfactant film at a fluid - fluid interface alters the dynamics of the interface. this is manifested in behavior of the interfacial waves, induced either externally or by thermal fluctuations @xcite. the interfacial dynamics can be probed by measuring the light scattered on such surface waves (see the review by earnshaw @xcite). the scattering of light on surface waves is a powerful tool for probing the properties of surfactant films at fluid interfaces @xcite, and a variety of systems have been recently investigated using this method (e.g. refs @xcite, see also the review by cicuta and hopkinson @xcite). recently, the application of surfactant films to modify the interfacial properties has been extended to the systems in which one of the fluids is in liquid - crystalline phase (e.g. liquid crystal colloids @xcite). the presence of a liquid crystal as one of the fluids complicates the problem of probing the interfacial properties by studying the dynamics of the surface waves for the following reasons. firstly, there are additional degrees of freedom in the bulk of the liquid crystal phase due to its anisotropy. secondly, the interaction with the surfactant film is more complicated due to anisotropic anchoring. finally, the surfactant film in the anisotropic field created by the neighboring liquid crystal can itself show anisotropic behavior, even if it behaves as an two - dimensional isotropic fluid at the boundary between isotropic fluids. a promising new direction for chemical and biological sensing devices has recently emerged which utilizes the properties of surfactant films self - assembled on the interface between water and a nematic liquid crystal. the surfactant film induces preferred orientation of the nematic director @xcite. the adsorption of chemical or biological molecules at such interface can then lead to reorientation of the nematic director, enabling detection by an imaging system @xcite. in these methods, easy detection is limited to the systems in which adsorption changes anchoring properties of the interface with respect to the adjacent liquid crystal phase quite considerably. namely, the equilibrium anchoring angle should change in magnitude. the range of application of these systems could be made significantly broader, however, if a method were used that was sensitive to changes in the anchoring properties of the interface that did not necessarily result in nematic director reorientation. for example, the anchoring orientation may remain unchanged @xcite, the adsorption only changing the strength of the anchoring. if a small amount of an analyte is present in the water it may be adsorbed at the surfactant layer, provided the surfactant molecules possess appropriate chemical properties. generally, such adsorption will result in a change in the elastic and viscous properties of the interface. hence sensitive experiments which are able to determine the interfacial properties will allow much more detailed experimental insight into the properties of the interaction between the surfactants and the analyte than has hitherto been available, and experimental study of surface waves is a possible technique for this purpose. the theoretical description of surface waves at interfaces between nematic and isotropic liquids was made back in 1970s @xcite. the results demonstrated that the spectrum of surface waves has a more complicated structure than in the isotropic case, and allows the use surface scattering experiments to determine properties of nematic interfaces @xcite. since then, several theoretical and experimental advances have been made, and presently these systems remain a subject of investigation @xcite. the present paper presents a theoretical study of the dispersion of the surface waves at a monomolecular surfactant film between an isotropic liquid (e.g. water) and a nematic liquid crystal.the main distinguishing features of such interfaces, are (i) the anchoring induced by the surfactant layer, (ii) the curvature energy of the interface, (iii) reduction of surface tension due to surfactant, and (iv) the anisotropy of the surface viscoelastic coefficients. we base our treatment on the mechanical model for anisotropic curved interfaces by rey @xcite, which takes into account anchoring and bending properties of the surfactant. we consider the case of the insoluble surfactant film that is in its most symmetric phase (isotropic two - dimensional fluid), and induces homeotropic (normal to the surface) orientation of the director. the paper is organized as follows. the continuum model used in the rest of the paper is set up in section [sec : model]. in section [sec : dispersion] the dispersion relation for surface waves is derived. in section [sec : modes] the numerical solution of the dispersion relation is solved with typical values of material parameters, and dispersion laws for different surface modes are analyzed in absence of the external magnetic field, and the influence of the magnetic field is discussed in section [sec : field]. the explicit form of the dispersion relation is written in appendix [app : dispersion]. [sec:model]the model in this section we formulate the model of the surfactant - laden interface between an isotropic liquid and a nematic liquid crystal, used in the present paper, and write down the governing equations. we base our treatment upon the models of the nematic - isotropic interface by rey @xcite, and well known hydrodynamic description of isotropic liquids @xcite and nematic liquid crystals @xcite. we consider the case when the surfactant film induces homeotropic (normal to the surface) orientation of the nematic director, which is usually true in a range of the surfactant concentrations @xcite. this case is the simplest to analyze, and, at the same time, the most important for biosensing applications where the direct change in anchoring angle can not be always observed. we include optional external magnetic field in our study and limit our analysis by considering the direction of the magnetic field that does not change equilibrium orientation of the nematic director. we assume that the system is far enough from any phase transitions both in the surfactant film @xcite and in the nematic phase @xcite. thus we avoid complications related to the fluctuations of the nematic and surfactant order parameters and the divergence of viscoelastic parameters near phase transitions. the surfactant films can exhibit rich phase behavior @xcite, and the form of the surface stress tensor depends upon the symmetry of the interface. however, this does not normally influence much the dispersion laws of the surface modes compared to the isotropic case @xcite. in the present paper we assume that the surfactant film is in the most symmetric phase (isotropic two - dimensional fluid). although the symmetry of the film should break in presence of the adjacent liquid - crystalline bulk phase, the film remains isotropic in equilibrium if the anchoring of the nematic is homeotropic, and symmetry breaking can occur only due to fluctuations of the director field. if we introduce the order parameter for the film, the corresponding anisotropic contributions to the interfacial stress tensor would be of higher order in the fluctuations of the dynamic variables than is required in our linearized treatment, so such contributions can be omitted. we consider a surfactant layer at an interface between nematic and isotropic liquids to be macroscopically infinitely thin. we assume that the surfactant film is insoluble and newtonian. this means that the model is applicable to systems in which the interchange of surfactant molecules between the interface and adjacent bulk fluids is small, and the relaxation of the orientation of surfactant molecules is fast compared to relaxation of surface waves. we also assume heat diffusion to be sufficiently fast so that the system is in thermal equilibrium. we do not consider systems where other effects, such as polarity, are important. we shall choose coordinate system in such a way that the unperturbed interface lies at a plane @xmath0, the half - space @xmath1 is occupied by the uniaxial nematic liquid crystal, and the half - space @xmath2 is filled by the isotropic liquid. other details of the geometry used in the present paper are summarized in appendix [app : geometry]. the central equations in the present section are the conditions for the balance of forces (eq. ([eq : forcebalance]) and torques (eq. ([eq : torquebalance]) at the interface. the explicit form of these equations depends upon the chosen macroscopic model, and the rest of this section is devoted to formulation of the model used in the present paper. the interfacial force balance equation is the balance between the interfacial force and the bulk stress jump : @xmath3 here @xmath4 is the force per unit area exerted by the interfacial stress @xmath5, @xmath6 is the force per unit area exerted by the isotropic fluid, @xmath7 is the force per unit area exerted by the nematic liquid crystal, the subscript @xmath8 indicates that the bulk stress fields in the isotropic liquid, @xmath9, and in the nematic, @xmath10, are evaluated at the interface, @xmath11 is the unit vector normal to the interface and directed into the isotropic liquid. the interfacial torque balance equation can be cast as @xmath12 where @xmath13 is the interfacial torque arising due to surface interactions, @xmath14 is the torque exerted upon the interface by the adjacent nematic liquid crystal. the explicit model for surface and bulk stresses and torques that enter eqs ([eq : forcebalance]) and (eq. ([eq : torquebalance]) is expanded in the remainder of this section. in this and the following subsections we summarize the equations for the surface stress tensor @xmath5 and surface torque vector @xmath13. we represent these quantities as a sum of corresponding non - dissipative (elastic) and dissipative (viscous) contributions : @xmath15 @xmath16 to describe the non - dissipative contributions in the surface stress tensor, @xmath17, and surface torque vector, @xmath18, we use the equilibrium model proposed by rey @xcite, which is summarized below. rey considered the interface with the helmholtz free energy per unit mass @xmath19 of the form @xmath20 where @xmath21 is the surface mass density, @xmath22 is the second fundamental tensor of the interface (see appendix [app : geometry]). the corresponding differential was written as @xmath23 where @xmath24_{\mathbf k,\mathbf b}\]] is the interfacial tension, @xmath25 is the tangential component of the capillary vector (@xmath26 is the surface projector), and @xmath27 is the bending moment tensor. the elastic surface stress tensor was found to be @xmath28 where the tangential surface molecular field is given by @xmath29 @xmath30 is surface gradient operator, @xmath31 denotes variational derivative with respect to @xmath11. the elastic contribution to surface torque was written as @xmath32 where @xmath33 is the surface couple stress, @xmath34 is the levi - civita tensor, and @xmath35 is the surface alternator tensor. the viscous properties of interfaces between an isotropic fluid and a nematic liquid crystal were considered in detail by rey @xcite, and the results are summarized below. the forces and fluxes that contribute to the dissipation function @xmath36 were identified as follows : @xmath37 where @xmath38 and @xmath39 are, correspondingly, symmetric and antisymmetric parts of the surface viscous stress tensor @xmath40, @xmath41 and @xmath42 are the components of the surface viscous molecular field tangential and normal to the surface, @xmath43\]] is the surface rate - of - deformation tensor (@xmath44 denotes the transposed tensor), @xmath45\]] is the surface vorticity tensor, @xmath46 is surface velocity, @xmath47 and @xmath48 are the total time derivatives of the components @xmath49 and @xmath50 of the nematic director field @xmath51, tangential and normal to the surface, correspondingly. generally, presence of the surfactant film at the interface complicates the form of the entropy production due to additional internal degrees of freedom of the surfactant, and to the anisotropy of the adjacent nematic liquid. however, if the surfactant film that is in its isotropic liquid phase and favors homeotropic anchoring of the nematic, the resulting anisitropic terms in the entropy production introduce corrections to the hydrodynamic equations of higher order than linear, and therefore can be neglected in the linearized treatment. since this is the case we are considering, we shall adopt the form of the entropy production ([eq : entropyproduction]) in our model and use the form of the viscous contribution to the surface stress tensor derived by rey @xcite, which is given by @xmath52,\end{aligned}\]] where @xmath53 is the surface jaumann (corrotational) derivative @xcite of the tangential component of the director @xmath54, and @xmath55, @xmath56 are nine independent surface viscosity coefficients. in the isotropic case @xmath57, the expression for the surface viscous stress tensor reduces to the viscous stress tensor of boussinesq - schriven surface fluid @xcite with the interfacial shear viscosity @xmath58 given by @xmath59 and dilatational viscosity @xmath60 given by @xmath61 the surface viscous torque, corresponding to eq. ([eq : entropyproduction]), is given by @xcite @xmath62 where the surface viscous molecular field @xmath63 is @xmath64 the viscosity coefficients @xmath65 can be expressed in terms of quantities @xmath66. we shall need only the expression for the tangential rotational viscosity : @xmath67 to calculate explicitly the interfacial tension @xmath68 (eq. ([eq : def - tau])), the tangential component of the capillary vector @xmath69 (eq. ([eq : def - xi])), and the bending moment tensor @xmath70 (eq. ([eq : def - m])), we need to know the dependence of the surface free energy @xmath19 on the orientation of the interface given by unit normal vector @xmath11, and on its curvature described by second fundamental tensor @xmath22. for small deviations of @xmath11 and @xmath22 from equilibrium, we can expand the free energy in powers of these quantities and truncate the series. the result can be represented as @xmath71 each of the contribution described below. the contribution @xmath72 corresponds to the surface tension @xmath73 of the equilibrium interface (flat interface, adjacent nematic director normal to the interface) : @xmath74 the anchoring contribution to the surface free energy density, @xmath75, describes the energetics of the preferred alignment direction of the nematic director relative to the interface. for the homeotropic equilibrium anchoring, it can be written in terms of @xmath54 as follows : @xmath76 such expansion applied to the widely used rapini - papoular form of the anchoring free energy density @xcite @xmath77 shows that these definitions of the anchoring strength coefficient have opposite signs : @xmath78 we shall use @xmath79 as the anchoring strength coefficient to ensure that it is positive in the case of the homeotropic anchoring being considered. the third contribution to the surface free energy density, @xmath80, is caused by finite interface thickness, and is related to the difference of the curvature of a surfactant film from the locally preferred (spontaneous) value. the widely used form of this contribution is the helfrich curvature expansion @xcite @xmath81 here the geometry of the interface is described by the mean curvature @xmath82 and the gaussian curvature @xmath83, and the material parameters characterizing the interface are the bending rigidity @xmath84, the saddle - splay (or gaussian) rigidity @xmath85, and the spontaneous curvature @xmath86. the term @xmath87 guarantees that the curvature energy of a flat interface (@xmath88, @xmath89) is zero. to complete the description of the interface, we need the continuity equation for the surfactant concentration @xmath90. for insoluble surfactants, the continuity equation reads : @xmath91 we shall extend the description of the dependence of the interfacial tension upon the concentration of surfactant, presented by buzza @xcite, to other parameters characterizing the interface (surface tension @xmath73, anchoring strength @xmath79, bending rigidity @xmath84, saddle - splay rigidity @xmath85, spontaneous curvature @xmath86, and surface viscosities @xmath66, @xmath92). for small deviation @xmath93 of the surfactant concentration @xmath90 from its equilibrium value @xmath94, these coefficients can be written in form @xmath95 and similarly for other quantities. casting surface velocity @xmath46 as the time derivative of the small surface displacement @xmath96, @xmath97 we obtain from the continuity equation eq. ([eq : continuitys]) that @xmath98 this allows us to represent the material parameters of the interface as @xmath99 @xmath100 @xmath101 @xmath102 in these formulas @xmath103, @xmath104, @xmath105, @xmath106 are, correspondingly, the interfacial tension, anchoring strength, bending rigidity, and spontaneous curvature in the unperturbed interface, @xmath107 is the static dilatational elasticity, @xmath108, @xmath109, and @xmath110 are coefficients in the first order term of the expansion of anchoring strength, bending rigidity, and spontaneous curvature in powers of (@xmath111). there are similar expansions for gaussian rigidity @xmath85 and surface viscosities @xmath66, @xmath92. magnetic field @xmath112 in the isotropic and nematic regions satisfies maxwell equations @xcite @xmath113 @xmath114 neglecting magnetization of the interface, the boundary conditions read @xmath115 @xmath116 here the magnetization of the isotropic liquid is @xmath117 where @xmath118 is the magnetic permeability of the isotropic liquid, the magnetization of the uniaxial nematic liquid crystal is @xcite @xmath119 where @xmath120 is the difference of the longitudinal and transversal magnetic permeabilities of the nematic : @xmath121 we assume both the isotropic liquid and the nematic liquid crystal are incompressible, so that their densities @xmath122 and @xmath123, are constant. the linearized equations for the incompressible isotropic liquid are well known @xcite. they are the continuity equation @xmath124 and navier - stokes equations @xmath125 where the hydrodynamic stress tensor is given by @xmath126 where @xmath127 is the shear viscosity of the isotropic liquid, @xmath128 is the unit tensor, @xmath129\]] is the strain rate tensor. we assume the non - slip boundary condition for the velocities of bulk fluids adjacent to the interface, which means the equality of the velocity of surfactant, @xmath46, and that of the bulk fluids at an interface, @xmath130 : @xmath131 to describe the dynamics of the nematic liquid crystal that is far from the isotropic - nematic transition and has small deviations from its equilibrium state, we shall use the linearized form of the eriksen - leslie theory @xcite. the linearized equations for the incompressible nematic liquid crystal are the continuity equation ([eq : continuity]), the equation for the velocity @xmath132 and the equation for the director @xmath133 here @xmath134\]] is the antisymmetric vorticity tensor, @xmath135 is the reactive material parameter, @xmath136 is orientational viscosity, @xmath137 is the molecular field which, assuming frank form of the elastic free energy of a nematic liquid crystal in magnetic field @xcite @xmath138 ^ 2 \\ & + & \frac{k_3}2\left[\mathbf n\times\left(\nabla\times\mathbf n\right)\right]^2 -\frac12\chi_a(\mathbf n\cdot\boldsymbol{\mathcal h})^2,\end{aligned}\]] has the linearized form @xmath139 where @xmath140\right\}+ \\ n_0\times\left[\mathbf n_0\times \left(\nabla\times\delta\mathbf n\right)\right]\right\ } + \chi_a(\mathbf n\cdot\boldsymbol{\mathcal h})\boldsymbol{\mathcal h},\end{aligned}\]] @xmath141, @xmath142, and @xmath143 are the splay, twist, and bend frank elastic constants, correspondingly. the stress tensor can be represented as a sum of reactive and viscous (dissipative) contributions, @xmath144 the linearized form of the reactive part is @xmath145 the linearized viscous stress tensor of incompressible nematic is @xmath146 the quantities @xmath147, @xmath148, @xmath149, @xmath136, and @xmath135 can be expressed through more commonly used leslie viscosity coefficients @xcite. note that equating @xmath150 recovers the viscous stress tensor @xmath151 of the isotropic incompressible fluid (last term in eq. ([eq : sigmai])). [sec:dispersion]dispersion relation the aim of this section is to construct the dispersion relation for the surface waves on the basis of the model set up above. we consider a surface wave with frequency @xmath152 and wavevector @xmath153 propagating along @xmath154 axis, and solve force balance equation, eq. ([eq : forcebalance]), and torque balance equation, eq. ([eq : torquebalance]) using linearized form of the hydrodynamic equations written in section [sec : model]. in order to linearize the hydrodynamic equations, we represent pressure @xmath155 and the nematic director @xmath156, where @xmath157 is the position in space, @xmath158 is time, in form @xmath159 @xmath160 where @xmath161 and @xmath162 are the deviations of pressure and director from their equilibrium values @xmath163 and @xmath164, correspondingly. the velocity @xmath165 is itself the deviation from zero equilibrium velocity. homeotropic anchoring corresponds to @xmath166 for small deviations from the equilibrium, we shall use the hydrodynamic equations linearized in @xmath167, @xmath161, and @xmath162. we shall assume these quantities to be independent of the coordinate @xmath168 (@xmath169) and vanish at @xmath170. the magnetic field can be also represented as @xmath171, where @xmath172 is the equilibrium value, and the deviation @xmath173 can be found from the linearized form of the maxwell equations ([eq : rotmaxwell]), ([eq : divmaxwell]). the terms in the final equations, containing @xmath173, are of higher order than linear, so we shall use only the equilibrium value, and skip the ` 0'subscript, so that @xmath174. substituting the interfacial free energy density ([eq : fs]) into eqs ([eq : def - tau]), ([eq : def - xi]), ([eq : def - m]), and ([eq : def - hse]), we find the contributions up to the first order in @xmath96 (and its derivatives) and @xmath54 into surface tension @xmath175 bending moment tensor @xmath176\mathbf i_s -\bar\kappa\mathbf b,\]] tangential component of the capillary vector @xmath177 and tangential surface molecular field @xmath178 + \nabla_s\cdot\left(\bar\kappa\mathbf b\right).\]] the non - vanishing components of the surface viscous stress tensor ([eq : sigmasv]) are @xmath179 the total interfacial force @xmath180 can be found by substituting eqs ([eq : sigmas]), ([eq : sigmase]), and ([eq : expl - tau])([eq : expl - sigmasv]) into eq. ([eq : def - forces]), and has components @xmath181 where @xmath182 to write the explicit form of the force balance equations ([eq : forcebalance]), we also need the expressions for the components of the force ([eq : def - forcei]) exerted by the isotropic fluid, @xmath183 and the components of the force ([eq : def - forces]) exerted by the nematic liquid crystal, @xmath184_{z=-0 }, \\ f^n_y&=&\left[\frac{1+\lambda}2h_y-\nu_3\partial_zv_y\right]_{z=-0 }, \\ f^n_z&=&\left(p-2\nu_1\partial_zv_z\right)_{z=-0},\end{aligned}\]] the hydrodynamic fields @xmath167, @xmath185, @xmath51 in the bulk isotropic and nematic liquids are found by solution of the hydrodynamic expressions. the explicit formulas are presented in appendices [app : isotropic] and [app : nematic]. next we introduce fourier transforms in the @xmath154 coordinate and in time as @xmath186 @xmath187 @xmath188 (for brevity we shall henceforth omit arguments of the transformed functions). performing fourier - transform of the force balance equation ([eq : forcebalance]), and substituting @xmath189, we obtain balance equations for the force components in form @xmath190c_i^{n\vert}+ \\ \nonumber + i\omega\frac{1+\lambda}2 \sum_{i=1}^3\left[k_3\left(m_i^{n\vert}\right)^2-k_1q^2+\chi_a\mathcal h^2\right] b_i^\vert c_i^{n\vert } \\ \label{eq : balancevx } = 0,\quad\end{aligned}\]] @xmath191 b_i^\bot c_i^{n\bot } \\ \label{eq : balancevy } = 0,\end{aligned}\]] @xmath192 where @xmath193 is the complex dilatational modulus, @xmath194 is defined in appendix [app : isotropic] by eq. ([eq : mi]), and the quantities @xmath195, @xmath196, @xmath197, @xmath198, @xmath199, @xmath200, and @xmath201 are defined in appendix [app : nematic] by eqs ([eq : mvert]), ([eq : mbot]), ([eq : cvert]), ([eq : cbot]), ([eq : bvert]), ([eq : bbot]), and ([eq : a]), correspondingly. to write the interfacial torque balance equation ([eq : torquebalance]), we cast the torque exerted upon the interface by the nematic liquid crystal, @xmath14, and the interfacial torque arising due to surface interactions, @xmath13, entering the interfacial torque balance equation ([eq : torquebalance]), in form @xmath202 and @xmath203 where the molecular field from the bulk @xmath204_s\]] has linearized components @xmath205 @xmath206 and the surface molecular field @xmath207 can be represented as a sum of elastic (@xmath208) and viscous (@xmath63) contributions @xmath209 given by eqs ([eq : def - hse]) and ([eq : def - hsv]), correspondingly, and can be represented in components as @xmath210 @xmath211 @xmath212 @xmath213 then the surface torque balance equations can be written as @xmath214 @xmath215 or, substituting the expressions ([eq : solvz]), ([eq : solnx]) and ([eq : solny]), @xmath216c^{n\vert}_i&=&0,\end{aligned}\]] @xmath217 the interfacial force balance equations (eqs ([eq : balancevx])([eq : balancevz])) and the interfacial torque balance equations (eqs ([eq : balancenx])([eq : balanceny])) form, with account of eqs ([eq : cbot]) and ([eq : cvert]), a homogeneous system of linear algebraic equations in @xmath218, @xmath219, @xmath220, @xmath221 and @xmath222. the dispersion relation is obtained from the condition of existence of a solution to these equations, i.e. the requirement for the determinant @xmath223 of the matrix of coefficients for this system to be zero @xmath224 the equations ([eq : balancevy]) and ([eq : balanceny]) in @xmath219 and @xmath222 decouple from the equations ([eq : balancevx]), ([eq : balancevz]) and ([eq : balancenx]) in @xmath218, @xmath220 and @xmath225. therefore, the matrix of coefficients is block - diagonal, and the dispersion relation (eq. ([eq : d])) is equivalent to a pair of relations for @xmath226 and @xmath168 directions : @xmath227 @xmath228 where @xmath229 is the determinant of the @xmath230 matrix @xmath231 of coefficients for the equations ([eq : balancevx]), ([eq : balancevz]) and ([eq : balancenx]), and @xmath232 is the determinant of the @xmath233 matrix of coefficients for the equations ([eq : balancevy]) and ([eq : balanceny]). the explicit form of the dispersion relations is presented in appendix [app : dispersion] and can be readily used for the numerical analysis of surface modes. [sec:modes]surface modes in this section the dispersion equation, which is presented in appendix [app : dispersion], is solved numerically, and surface modes of different types are analyzed. for simplicity, we assume the density of the isotropic liquid, @xmath122, to be small enough to be neglected (e.g. nematic surfactant air interface). we also assume that the magnetic field is absent. the surface modes can be easily classified at low wavevectors @xmath234. expansion of the dispersion relation in powers of the wavevector @xmath234 is a straightforward exercise in algebra, and the resulting modes are described below. firstly, there is a transverse capillary mode, which has the dispersion law similar to that in the case of an isotropic liquid - liquid interface @xcite : @xmath235 the principal contribution to this mode at large wavelengths arises due to the restoring influence of surface tension @xmath236, and the predominant motion is in the direction normal to the interface (@xmath237). the differences from the isotropic case, related to anisotropy of viscous dissipation in the nematic, appear in higher orders in @xmath234. the dilatational (or compressional) mode with predominant motion in the direction along wave propagation (@xmath154) arises in presence of surfactant layer due to the restoring force provided by the dilatational elastic modulus @xmath238. the dispersion law for this mode can be written as @xmath239^{1/3 } + o\left(q^{4/3}\right),\]] where the miesowicz viscosity @xmath240 is given by @xcite @xmath241 the difference from the dispersion law for the dilatational mode in the case of a surfactant film at the interface between isotropic fluids, given by @xcite @xmath242 arises due to anisotropy of viscous dissipation in nematic. a new mode, specific to the nematic, is driven by relaxation of the director field to equilibrium due to anchoring at the interface and has the disperion law @xmath243 such relaxation is present even in absence of motion of the interface (e.g. when the interface is solid), so that @xmath244 does not vanish at @xmath245. for nematic - isotropic interfaces, the corresponding motion of the interface is induced by backflow effects. finally, behavior of the in - plane shear mode, with motion in @xmath168-direction, is also governed by relaxation of the nematic director due to anchoring. the corresponding dispersion law @xmath246 appears to be different from the isotropic case, where the damping of the in - plane shear mode in absence of anchoring is governed by the surface viscosity @xmath58 @xcite. gravity @xmath247, so far neglected in our analysis, becomes important at wavevectors @xmath248 and can be taken into account by adding the hydrostatic pressure term @xmath249 to eq. ([eq : m22]), which corresponds to the additional contribution @xmath250 to the vertical component of the force, eq. ([eq : fsz]). the resulting dispersion law for transversal mode is given by expression @xmath251 which describes well - known gravity waves @xcite. in the opposite case of large wavevectors, the curvature energy becomes important. analysis of eqs ([eq : fsz]) and ([eq : tsx]) yields the characteristic values of @xmath234 @xmath252 and @xmath253 below which one can neglect in the dispersion relation the terms containing bending rigidity @xmath84 and its derivatives with respect to surfactant concentration, given by @xmath254. usually @xmath255, and the range of @xmath234 in which both gravity and curvature contributions become small, given by @xmath256 is rather wide. for typical values @xmath257kg / m@xmath258, @xmath259m / s@xmath260, @xmath261j / m@xmath260, @xmath262j, the equation ([eq : good - q]) reads @xmath263@xmath264@xmath265, which includes the range of wavevectors typically probed by surface light scattering experiments. to obtain the dispersion laws for surface modes at larger values of the wavevector @xmath234, the dispersion equation must be solved numerically. the numerical solution presented below uses the following typical values of the material parameters when it is not indicated otherwise. for the nematic liquid crystal we use the parameters of 4-@xmath266-pentyl-4-cyanobiphenyl (5cb) at 26@xmath267c @xcite : the density @xmath268kg / m@xmath269, the elastic constants @xmath270n, @xmath271n, @xmath272n, the leslie viscosities @xmath273kg/(m@xmath274s), @xmath275kg/(m@xmath274s), @xmath276kg/(m@xmath274s), @xmath277kg/(m@xmath274s), @xmath278kg/(m@xmath274s), @xmath279kg/(m@xmath274s). the viscosity coefficient used in the present paper can be calculated from the leslie equations @xcite and equal @xmath280kg/(m@xmath274s), @xmath281kg/(m@xmath274s), @xmath282kg/(m@xmath274s), @xmath283kg/(m@xmath274s), @xmath284. we use the value of the bending rigidity @xmath285j which is typical for surfactant layers @xcite. for other parameters we use the following typical values : @xmath286kg / s, @xmath287, @xmath288n / m, @xmath289n / m, @xmath290j / m@xmath291, @xmath292j / m@xmath291. dispersion law @xmath293 for different surface modes in absence of gravity, obtained by solution of the dispersion relation ([eq : dvert]) with the values of the parameters given in the text. numbers 1, 2, 3 denote transverse, dilatational, and nematic director relaxation modes, correspondingly. prime and double prime denote real (solid line) and imaginary (dashed line) parts of @xmath152, correspondingly.] the dispersion law @xmath293 for different surface modes in absence of gravity, obtained by solution of the dispersion relation ([eq : dvert]) with the values of the parameters given above, is presented in figure [fig : nogravity]. at low @xmath234 the dispersion of for modes 1, 2, 3, as denoted figure [fig : nogravity], is in good agreement with approximate formulas ([eq : omegac]), ([eq : omegad]), and ([eq : omegan]), correspondingly. the noticeable discrepancy in behavior of capillary and dilatational modes appears at @xmath294@xmath265, and the damping of surface waves becomes large at larger @xmath234, which is qualitatively similar to the case of the interface between isotropic liquids. the results presented in figure [fig : nogravity] suggest that in the typical range of @xmath234 probed by surface light scattering experiments (@xmath295@xmath296@xmath265), the approximate expressions ([eq : omegac]), ([eq : omegad]) do not describe well the dispersion curves, and accurate solution of the dispersion equation should be used instead. dispersion law @xmath293 for different surface modes in presence of gravity @xmath297m / s@xmath260, obtained by solution of the dispersion relation ([eq : dvert]) with the values of the parameters given in the text. numbers 1, 2, 3 denote transverse, dilatational, and nematic director relaxation modes, correspondingly. prime and double prime denote real (solid line) and imaginary (dashed line) parts of @xmath152, correspondingly. vertical dotted line corresponds to the value of @xmath298 given by eq. ([eq : qg]).] figure [fig : gravity] presents the dispersion law @xmath293 for different surface modes obtained by solution of the dispersion relation ([eq : dvert]) in presence of gravity @xmath297m / s@xmath260. in agreement with the discussion above, the influence of gravity on the dispersion laws is small at @xmath299, where @xmath298 is given by eq. ([eq : qg]). dependence of the real (solid line) and imaginary (dashed line) parts of the frequency of the mode 1 (as defined on figure [fig : nogravity]) upon the bending rigidity @xmath300, calculated at @xmath301@xmath265 in absence of gravity. vertical line corresponds to the value of @xmath300 that satisfies eq. ([eq : qkappa]).] if the bending rigidity @xmath84 is large, its influence becomes noticeable, as it is demonstrated in figure [fig : bending]. for @xmath302, typical for surfactant films, the value of @xmath303, given by eq. ([eq : qkappa]), corresponds to wavelength close to atomic scales, and curvature energy can be neglected in typical surface light scattering experiments, in agreement with the discussion above. dependence of the real (solid line) and imaginary (dashed line) parts of the frequency @xmath152 of the mode 3 (as defined on figure [fig : nogravity]), normalized by @xmath304 (see eq. ([eq : omegan])), upon the anchoring strength @xmath305, calculated at @xmath306@xmath265 in absence of gravity.] dependence of the real (solid line) and imaginary (dashed line) parts of the frequency @xmath152 of the in - plane shear mode, normalized by @xmath304 (see eq. ([eq : omegas])), upon the anchoring strength @xmath305, calculated at @xmath306@xmath265 in absence of gravity.] the dispersion law for the modes governed by relaxation of the nematic director field in @xmath154 and @xmath168 directions due to anchoring of the nematic director at the interface, obtained by numerical solution of the dispersion equation with the values of the parameters given above, are well described by the equations ([eq : omegan]) and ([eq : omegas]). however, as the anchoring strength becomes smaller, other mechanisms start to take over, as demonstrated in figures [fig : xanchoring] and [fig : yanchoring]. [sec:field]influence of magnetic field in this section we discuss how the surface modes described in section [sec : modes] are altered in presence of the external magnetic field directed normally to the surface (along @xmath237 axis). the external magnetic field effectively acts on the nematic molecules as an additional molecular field (see eq. ([eq : h - star])), and the primary counteracting mechanism is provided by orientational shear relaxation. thus we may expect the influence of the magnetic field become noticeable at @xmath307 dependence of the real (solid lines) and imaginary (dashed lines) parts of the frequencies of the modes 1 and 2 (as defined on figure [fig : nogravity]) upon the magnetic field, calculated at @xmath308@xmath265 in absence of gravity. vertical dotted line corresponds to @xmath309 (see eq. ([eq : hstar])).] the results of the numerical solution of the dispersion equation in presence of magnetic field, presented in figure [fig : field], confirm that noticeable change in dispersion of capillary and dilatational modes arises only around the value of the field given by eq. ([eq : hstar]). the change due to magnetic field in modes governed by anchoring is found to be negligibly small. at low @xmath234 the dispersion of a capillary mode in strong magnetic field is different from the law ([eq : omegac]) and is given by @xmath310 the frequency of this mode becomes sensitive to the anchoring properties of the interface, because the nematic director tends to be oriented along the field rather than to be advected with the nematic liquid. the practical use of this effect is, however, limited, because at short wavelengths extremely large magnetic field is required, and at long wavelengths gravity becomes dominating (eq. ([eq : omegag]). in principle, magnetic field can also influence surface waves through change in the properties of the interface (e.g. surface tension) due to the magnetization of the surfactant. separate study is required to estimate the magnitude of this effect. [sec:conclusion]conclusion we have obtained the dispersion relation for the surface waves at a surfactant - laden nematic isotropic interface for the case when the surfactant film induces homeotropic (normal to the surface) orientation of the director, and the surfactant film is in the isotropic two - dimensional fluid phase. we have analyzed the dispersion law of different surface modes analytically in long wavelength limit, and numerically in broader range of wave vectors, using typical values of the material parameters. at long wavelengths the dispersion of capillary, dilatational (or compression), in - plane shear, and director relaxation modes is described by equations ([eq : omegac]) (or ([eq : omegag])), ([eq : omegad]), ([eq : omegas]), and ([eq : omegan]), correspondingly. at smaller wavelength, the solution of the full dispersion relation should be used. gravity influences the transversal mode at small wavevectors (eq. ([eq : qg])), and curvature energy of surfactant can be neglected if wavevector is not too large (eq. ([eq : qkappa])). for all modes, the influence of the external magnetic field directed normally to the interface is small. the influence of the magnetic field should be more pronounced if the direction of the field does not coincide with equilibrium nematic director. in this case the dispersion law for surface modes may be expected to be quantitatively different due to anisotropy of viscous dissipation in nematic, and different anchoring energy. the results of the present paper can be readily extended to the case of arbitrary direction of the external field and to other types of nematic anchoring. other possible developments, which may increase the range of accessible systems and conditions, is the extension of the results to wider range of the states of the surfactant film, and the study of the effects which may be caused by the phase transitions in the surfactant film and bulk liquid crystal. dependence of the dispersion waves upon the parameters of the interface suggests the surface light scattering on a surfactant - laden nematic - isotropic interface as a potential method for determining of the properties of surfactant - laden nematic - isotropic interfaces, and as a possible candidate for a chemical or biological sensing technique. i thank prof. c. m. care for fruitful discussion of the results, and prof. p. d. i. fletcher for the discussion about surfactant - laden nematic - isotropic interfaces which instigated this work. [app:geometry]differential geometry of the interface the geometrical description we use is similar to that of that presented in works @xcite and @xcite. we choose the plane @xmath0 to coincide with the unperturbed interface, the half - space @xmath1 to be occupied by the uniaxial nematic liquid crystal, and the half - space @xmath2 to be filled by the isotropic liquid. let the position of a fluid particle at the interface be @xmath311, where @xmath312 is its position on the undeformed interface (@xmath0), and @xmath313 is the displacement vector with components @xmath314. we shall use @xmath315 and @xmath316 as surface coordinates and denote them as @xmath317, @xmath318 and other greek indices taking values 1 and 2. the position @xmath319 of fluid particles at the interface in 3d space can be cast as @xmath320 the surface tangent base vectors @xmath321, corresponding to the chosen surface coordinates, can be written in terms of the components of the displacement vectors : @xmath322 and @xmath323 the surface metric tensor @xmath324 has determinant @xmath325 the corresponding reciprocal base vectors @xmath326 and metric tensor @xmath327 take form @xmath328 @xmath329 @xmath330 the base and reciprocal base vectors satisfy @xmath331 we write the unit vector @xmath11, normal to the interface and directed into the isotropic liquid, as @xmath332 we shall also define the dyadic surface idem factor @xmath333 the surface gradient operator @xmath334 and the second fundamental tensor @xmath335 the mean curvature @xmath82 and gaussian curvature @xmath83 are given by @xmath336 @xmath337 other useful identities include the surface projection @xmath54 of a nematic director field @xmath51, eqs ([eq : n]) and ([eq : n0]), @xmath338 and its surface divergence @xmath339 [app:isotropic]bulk solution for isotropic liquid this appendix presents the solution to the linearized hydrodynamic equations in bulk isotropic liquid, obtained by kramer @xcite. substitution of eq. ([eq : vfourier]) into eq. ([eq : continuity]) yields @xmath340 substituting eqs ([eq : vfourier]) and ([eq : pfourier]) into eqs ([eq : navier - stokes])([eq : s]), we obtain @xmath341 \tilde v_x&=&iq\tilde p, \\ \label{eq : kramer - vy } \left[i\omega\rho^i+\eta\left(q^2-\partial_z^2\right)\right] \tilde v_y&=&0, \\ \label{eq : kramer - vz } \left[i\omega\rho^i+\eta\left(q^2-\partial_z^2\right)\right] \tilde v_z&=&-\partial_z\tilde p,\end{aligned}\]] where equation ([eq : kramer - vy]) is decoupled from other equations. the general solution to eqs ([eq : continuity - fourier])([eq : kramer - vz]) vanishing at @xmath342 can be written as @xmath343 @xmath344 @xmath345 @xmath346 with @xmath347 the quantities @xmath348, @xmath349, and @xmath350 are functions of @xmath234 and @xmath152 and are determined by the boundary conditions at the interface as follows : @xmath351 @xmath352 @xmath353 where the superscript @xmath354 indicates that the values of the corresponding dynamic variables are taken at @xmath355. [app:nematic]bulk solution for nematic liquid crystal in this appendix the solution is presented to the linearized hydrodynamic equations in bulk nematic liquid crystal. for the equilibrium director along @xmath237 axis (eq. ([eq : n0])) the fourier - transform, similar to equations ([eq : vfourier])([eq : nfourier]), of the linearized molecular field (eq. ([eq : h])), @xmath356, has non - zero components @xmath357 @xmath358 substituting them into eqs ([eq : eriksenleslie]), ([eq : sigman])([eq : sigmanv]), we obtain the following linear differential equations, @xmath359\tilde v_x-\frac{1-\lambda}2\partial_z\tilde h_x = iq\tilde p,\]] @xmath360\tilde v_y -\frac{1-\lambda}2\partial_z\tilde h_y=0,\]] @xmath361 \tilde v_z - iq\frac{1+\lambda}2\tilde h_x=-\partial_z\tilde p,\]] which are analogous to eqs ([eq : kramer - vx])([eq : kramer - vz]) for isotropic liquids. equation ([eq : dndt]) for the director after fourier transform gives two equations, @xmath362 and @xmath363 where @xmath364 and @xmath365 are given by eqs ([eq : hx]) and ([eq : hy]). thus we have six linear differential equations (eqs ([eq : continuity - fourier]) and ([eq : kramer - vxn])([eq : kramer - nyn])) for six dynamic variables (pressure, three components of velocity, and two components of director). equations ([eq : kramer - vyn]) and ([eq : kramer - nyn]) for @xmath366 and @xmath367 decouple from the others, their general solution vanishing at @xmath368 can be cast as @xmath369 @xmath370 where @xmath371},\]] @xmath372 and @xmath373, @xmath374, are the roots of the quadratic equation @xmath375 where @xmath376 @xmath377 @xmath378 the general solution to the equations ([eq : continuity - fourier]), ([eq : kramer - vxn]), ([eq : kramer - vzn]), and ([eq : kramer - nxn]) vanishing at @xmath368 can be cast as @xmath379 @xmath380 @xmath381 @xmath382 where @xmath383 b^\vert_i+ \\ + \left. \left[i\omega\rho^n+\nu_3q^2 -\left(2\nu_1-\nu_3\right)\left(m^{n\vert}_i\right)^2\right] \right\},\end{aligned}\]] @xmath384 @xmath385 and @xmath386, @xmath387, are the roots of the cubic equation @xmath388 where @xmath389 @xmath390 q^2\right\}k_3,\end{aligned}\]] @xmath391 \\ \nonumber -\left\{i\omega\rho^n -\left[\frac{1+\lambda^2}2\gamma_1 - 2\left(\nu_1+\nu_2-\nu_3\right)\right]q^2 \right\ } \\ \nonumber \times\left(k_1q^2-\chi_a\mathcal h^2\right) \\ -\left[i\omega\rho^n -\left(\frac{1-\lambda^2}4\gamma_1-\nu_3\right)q^2 \right]k_3q^2,\end{aligned}\]] @xmath392\left(k_1q^2-\chi_a\mathcal h^2\right)q^2.\end{aligned}\]] the quantities @xmath197 and @xmath198 are functions of @xmath234 and @xmath152 and are determined by the boundary conditions at the interface as @xmath393 @xmath394/\delta,\end{aligned}\]] where @xmath395 and expressions for @xmath396, @xmath397, and @xmath398 are obtained from eqs ([eq : cbot]) and ([eq : cvert]) by cyclic permutation of subscript indices. [app:dispersion]explicit form of dispersion relation to write the explicit form of the dispersion relations ([eq : dvert]) and ([eq : dbot]), we recast equations ([eq : cbot]) and ([eq : cvert]) in form @xmath399 @xmath400 where @xmath401 @xmath402 is given by eq. ([eq : delta]), @xmath403 and @xmath404 are given by eqs ([eq : bbot]) and ([eq : bvert]), @xmath196 and @xmath195 are given by eqs ([eq : mvert]) and ([eq : mbot]), correspondingly. then the dispersion relation ([eq : dbot]) can be written as @xmath405 where @xmath406 is @xmath407 matrix of coefficients for equations ([eq : balancevy]) and ([eq : balanceny]) @xmath408 with the following components : @xmath409b^\bot_i\right\}l^{\left(vy\right)}_i,\end{aligned}\]] @xmath410b^\bot_i\right\}l^{\left(ny\right)}_i,\end{aligned}\]] @xmath411 @xmath412 the dispersion relation ([eq : dvert]) can be written as @xmath413 where @xmath414 is @xmath415 matrix of coefficients for equations ([eq : balancevx]), ([eq : balancevz]), and ([eq : balancenx]) @xmath416 with the following components : @xmath417 l^{\left(vx\right)}_i + i\omega\frac{1+\lambda}2\times \\ \times\sum_{i=1}^3\left[k_3\left(m_i^{n\vert}\right)^2-k_1q^2+\chi_a\mathcal h^2\right]b_i^\vert l^{\left(vx\right)}_i,\end{aligned}\]] @xmath418 l^{\left(vz\right)}_i + i\omega\frac{1+\lambda}2\times \\ \times\sum_{i=1}^3\left[k_3\left(m_i^{n\vert}\right)^2-k_1q^2+\chi_a\mathcal h^2\right]b_i^\vert l^{\left(vz\right)}_i,\end{aligned}\]] @xmath419 l^{\left(nx\right)}_i+ \\ \nonumber { } + i\omega\frac{1+\lambda}2\times \\ \times\sum_{i=1}^3\left[k_3\left(m_i^{n\vert}\right)^2-k_1q^2+\chi_a\mathcal h^2\right]b_i^\vert l^{\left(nx\right)}_i,\end{aligned}\]] @xmath420 @xmath421 @xmath422 @xmath423\right\}l^{\left(vx\right)}_i,\end{aligned}\]] @xmath424\right\}l^{\left(vz\right)}_i,\end{aligned}\]] @xmath425\right\}l^{\left(nx\right)}_i.\end{aligned}\]] note that gravity @xmath247 has been incorporated into the dispersion relation by adding the hydrostatic pressure term @xmath249 to @xmath426 (eq. ([eq : m22])). by setting @xmath427, and setting to zero quantities @xmath141, @xmath142, @xmath143, @xmath135, @xmath136, @xmath428, and @xmath120, specific to nematic, and neglecting curvature contributions by setting to zero @xmath300 and @xmath254, the dispersion relation is reduced to the well studied form for the case of isotropic liquids @xcite.

a theoretical study is presented of surface waves at a monomolecular surfactant film between an isotropic liquid and a nematic liquid crystal for the case when the surfactant film is in the isotropic two - dimensional fluid phase and induces homeotropic (normal to the interface) orientation of the nematic director. the dispersion relation for the surface waves is obtained, and different surface modes are analyzed with account being taken of the anchoring induced by the surfactant layer, the curvature energy of the interface, and the anisotropy of the viscoelastic coefficients. the dispersion laws for capillary and dilatational surface modes retain structure similar to that in isotropic systems, but involve anisotropic viscosity coefficients. additional modes are related to relaxation of the nematic director field due to anchoring at the interface. the results can be used to determine different properties of nematic - surfactant - isotropic interfaces from experimental data on surface light scattering.

cond-mat0703646

Introduction in 1985, hoare s paper _ a couple of novelties in the propositional calculus _ @xcite was published. in this paper the ternary connective @xmath0 is introduced as the _ conditional_. with @xmath1 and @xmath2 denoting programs and @xmath3 a boolean expression.] a more common expression for a conditional statement @xmath4 is @xmath5 but in order to reason systematically with conditional statements, a notation such as @xmath6 is preferable. in a conditional statement @xmath6, first @xmath2 is evaluated, and depending on that evaluation result, then either @xmath1 or @xmath7 is evaluated (and the other is not) and determines the evaluation value. this evaluation strategy is a form of _ short - circuit _ evaluation. in @xcite, hoare proves that propositional logic is characterized by eleven equational axioms, some of which employ constants @xmath8 and @xmath9 for the truth values @xmath10 and @xmath11. in 2011, we introduced _ proposition algebra _ in @xcite as a general approach to the study of the conditional : we defined several _ valuation congruences _ and provided equational axiomatizations of these congruences. the most basic and least identifying valuation congruence is _ free _ valuation congruence, which is axiomatized by the axioms in table [cp]. ''' '' @xmath12 ''' '' [cp] these axioms stem from @xcite and define the conditional as a primitive connective. we use the name @xmath13 (for conditional propositions) for this set of axioms. interpreting a conditional statement as an if - then - else expression, axioms @xmath14 are natural, and axiom (distributivity) can be clarified by case analysis : if @xmath15 evaluates to @xmath10 and @xmath16 as well, then @xmath17 determines the result of evaluation ; if @xmath15 evaluates to @xmath10 and @xmath16 evaluates to @xmath11, then @xmath18 determines the result of evaluation, and so on and so forth. in section [sec : free] we characterize free valuation congruence with help of _ evaluation trees _ : given a conditional statement, its evaluation tree directly represents all its evaluations (in the way a truth table does in the case of propositional logic). two conditional statements are equivalent with respect to free valuation congruence if their evaluation trees are equal. evaluation trees are simple binary trees, proposed by daan staudt in @xcite (that appeared in 2012). free valuation congruence identifies less than the equivalence defined by hoare s axioms in @xcite. for example, the atomic proposition @xmath19 and the conditional statement @xmath20 are not equivalent with respect to free valuation congruence, although they are equivalent with respect to _ static _ valuation congruence, which is the valuation congruence that characterizes propositional logic. a valuation congruence that identifies more than free and less than static valuation congruence is _ repetition - proof _ valuation congruence, which has an axiomatization that comprises two more (schematic) axioms, one of which reads @xmath21 and thus expresses that if @xmath19 evaluates to @xmath11, a consecutive evaluation of @xmath19 also evaluates to @xmath11, so the conditional statement at the @xmath16-position will not be evaluated and can be replaced by any other. as an example, @xmath22, and the left - hand and right - hand conditional statements are equivalent with respect to repetition - proof valuation congruence, but are not equivalent with respect to free valuation congruence. in section [sec : rp] we characterize repetition - proof valuation congruence by defining a transformation on evaluation trees that yields _ repetition - proof _ evaluation trees : two conditional statements are equivalent with respect to repetition - proof valuation congruence if, and only if, they have equal repetition - proof evaluation trees. although this transformation on evaluation trees is simple and natural, our proof of the mentioned characterization |which is phrased as a completeness result| is non - trivial and we could not find a proof that is essentially simpler. valuation congruences that identify more conditional statements than repetition - proof valuation congruence are contractive, memorizing, and static valuation congruence, and these are all defined and axiomatized in @xcite. in sections @xmath23, each of these valuation congruences is characterized using a transformation on evaluation trees : two conditional statements are c - valuation congruent if, and only if, their c - transformed evaluation trees are equal. these transformations are simple and natural, and only for static valuation congruence we use a slightly more complex transformation. in section [sec : conc] we discuss the general structure of the proofs of these results, which are all based on normalization of conditional statements. the paper ends with a brief digression on short - circuit logic, an example on the use of repetition - proof valuation congruence, and some remarks about side effects. a spin - off of our approach can be called `` basic form semantics for proposition algebra '' : for each valuation congruence c that we consider (including the case c = free), two conditional statements are c - valuation congruent if, and only if, they have equal c - basic forms, where c - basic forms are obtained by a syntactic transformation of conditional statements, which is a form of normalization. Evaluation trees for free valuation congruence consider the signature @xmath24 with constants @xmath25 and @xmath9for the truth values @xmath10 and @xmath11, respectively, and constants @xmath19 for atomic propositions, further called _ atoms _, from some countable set @xmath26. we write @xmath27 for the set of closed terms, or _ conditional statements _, over the signature @xmath28. given a conditional statement @xmath6, we refer to @xmath2 as its _ central condition_. we define the _ dual _ @xmath29 of @xmath30 as follows : @xmath31 observe that @xmath13 is a self - dual axiomatization : when defining @xmath32 for each variable @xmath17, the dual of each axiom is also in @xmath13, and hence @xmath33 a natural view on conditional statements in @xmath34involves short - circuit evaluation, similar to how we consider the evaluation of an `` @xmath35 '' expression. the following definition is taken from @xcite. [def : trees] the set @xmath36 of * evaluation trees over @xmath26 with leaves in @xmath37 * is defined inductively by @xmath38 the operator @xmath39 is called * post - conditional composition over @xmath19*. in the evaluation tree @xmath40, the root is represented by @xmath19, the left branch by @xmath41 and the right branch by @xmath42. we refer to trees in @xmath36 as evaluation trees, or trees for short. post - conditional composition and its notation stem from @xcite. evaluation trees play a crucial role in the main results of @xcite. in order to define our `` evaluation tree semantics '', we first define the _ leaf replacement _ operator, ` replacement'for short, on trees in @xmath36 as follows. let @xmath43 and @xmath44. the replacement of @xmath25 with @xmath42 and @xmath45 with @xmath46 in @xmath41, denoted @xmath47,\]] is defined by @xmath48&= y,\\ { \ensuremath{{\sf f}}}[{\ensuremath{{\sf t}}}\mapsto y,{\ensuremath{{\sf f}}}\mapsto z]&= z,\\ (x'{{\raisebox{1pt}{\footnotesize$\;\underline{\triangleleft}~$}}}a{{\raisebox{1pt}{\footnotesize$~\underline{\triangleright}\;$}}}x'')[{\ensuremath{{\sf t}}}\mapsto y,{\ensuremath{{\sf f}}}\mapsto z] & = x'[{\ensuremath{{\sf t}}}\mapsto y,{\ensuremath{{\sf f}}}\mapsto z]{{\raisebox{1pt}{\footnotesize$\;\underline{\triangleleft}~$}}}a{{\raisebox{1pt}{\footnotesize$~\underline{\triangleright}\;$}}}x''[{\ensuremath{{\sf t}}}\mapsto y,{\ensuremath{{\sf f}}}\mapsto z].\end{aligned}\]] we note that the order in which the replacements of leaves of @xmath41 is listed is irrelevant and we adopt the convention of not listing identities inside the brackets, e.g., @xmath49=x[{\ensuremath{{\sf t}}}\mapsto { \ensuremath{{\sf t}}},{\ensuremath{{\sf f}}}\mapsto z]$]. furthermore, repeated replacements satisfy the following equation : @xmath50\big)\;[{\ensuremath{{\sf t}}}\mapsto y_2,{\ensuremath{{\sf f}}}\mapsto z_2]\\ & = x[{\ensuremath{{\sf t}}}\mapsto y_1[{\ensuremath{{\sf t}}}\mapsto y_2,{\ensuremath{{\sf f}}}\mapsto z_2],~ { \ensuremath{{\sf f}}}\mapsto z_1[{\ensuremath{{\sf t}}}\mapsto y_2,{\ensuremath{{\sf f}}}\mapsto z_2]].\end{aligned}\]] we now have the terminology and notation to define the interpretation of conditional statements in @xmath34 as evaluation trees by a function @xmath51 (abbreviating short - circuit evaluation). [def : se] the * short - circuit evaluation function * @xmath52 is defined as follows, where @xmath53 : @xmath54.\end{aligned}\]] [ex : fr] the conditional statement @xmath55 yields the following evaluation tree : @xmath56\\ & = ({ \ensuremath{{\sf f}}}{{\raisebox{1pt}{\footnotesize$\;\underline{\triangleleft}~$}}}a{{\raisebox{1pt}{\footnotesize$~\underline{\triangleright}\;$}}}{\ensuremath{{\sf t}}})[{\ensuremath{{\sf t}}}\mapsto se(a)]\\ & = { \ensuremath{{\sf f}}}{{\raisebox{1pt}{\footnotesize$\;\underline{\triangleleft}~$}}}a{{\raisebox{1pt}{\footnotesize$~\underline{\triangleright}\;$}}}({\ensuremath{{\sf t}}}{{\raisebox{1pt}{\footnotesize$\;\underline{\triangleleft}~$}}}a{{\raisebox{1pt}{\footnotesize$~\underline{\triangleright}\;$}}}{\ensuremath{{\sf f}}}).\end{aligned}\]] a more pictorial representation of this evaluation tree is the following, where @xmath57 yields a left branch and @xmath58 a right branch : @xmath59 \node (a) { $ a$ } child { node (b1) { $ { \ensuremath{{\sf f}}}$ } } child { node (b2) { $ a$ } child { node (d1) { $ { \ensuremath{{\sf t}}}$ } } child { node (d2) { $ { \ensuremath{{\sf f}}}$ } } } ; \end{tikzpicture}\]] _ end example. _ as we can see from the definition on atoms, evaluation continues in the left branch if an atom evaluates to @xmath10 and in the right branch if it evaluates to @xmath11. we shall often use the constants @xmath25 and @xmath9 to denote the result of an evaluation (instead of @xmath10 and @xmath11). [def : eval] let @xmath30. an * evaluation * of @xmath1 is a pair @xmath60 where @xmath61 and @xmath62, such that if @xmath63, then @xmath64 (the empty string) and @xmath65, and otherwise, @xmath66 where @xmath67 is a complete path in @xmath68 and * for @xmath69, if @xmath70 is a left child of @xmath71 then @xmath72, and otherwise @xmath73, * if @xmath74 is a left child of @xmath75 then @xmath76, and otherwise @xmath77. we refer to @xmath78 as the * evaluation path * and to @xmath74 as the * evaluation result*. so, an evaluation of a conditional statement @xmath1 is a complete path in @xmath68 (from root to leaf) and contains evaluation values for all occurring atoms. for instance, the evaluation tree @xmath79 from example [ex : fr] encodes the evaluations @xmath80, @xmath81, and @xmath82. as an aside, we note that this particular evaluation tree encodes all possible evaluations of @xmath83, where ` & & ` is the connective that prescribes _ short - circuit conjunction _ (we return to this connective in section [sec : conc]). in turn, each evaluation tree gives rise to a _ unique _ conditional statement. for example [ex : fr], this is @xmath84 (note the syntactical correspondence). [def : basic] * basic forms over @xmath26 * are defined by the following grammar @xmath85 we write @xmath86 for the set of basic forms over @xmath26. the * depth * @xmath87 of @xmath88 is defined by @xmath89 and @xmath90. the following lemma s exploit the structure of basic forms and are stepping stones to our first completeness result (theorem [thm:1]). [la:2.5] for each @xmath30 there exists @xmath91 such that @xmath92. first we establish an auxiliary result : if @xmath93 are basic forms, then there is a basic form @xmath94 such that @xmath95. this follows by structural induction on @xmath2. the lemma s statement follows by structural induction on @xmath1. the base cases @xmath96 are trivial, and if @xmath97 there exist by induction basic forms @xmath98 such that @xmath99, hence @xmath100. now apply the auxiliary result. [la:2.6] for all basic forms @xmath1 and @xmath2, @xmath101 implies @xmath102. by structural induction on @xmath1 the base cases @xmath103 are trivial. if @xmath104, then @xmath105 and @xmath106 with @xmath107, so @xmath108 and @xmath109. by induction we find @xmath110, and hence @xmath102. [def : freevc] * free valuation congruence *, notation @xmath111, is defined on @xmath34 as follows : @xmath112 [la : x] free valuation congruence is a congruence relation. let @xmath113 and assume @xmath114, thus @xmath115. then @xmath116= se(q)[{\ensuremath{{\sf t}}}\mapsto se(p'),{\ensuremath{{\sf f}}}\mapsto se(r)]=se(p'{{\raisebox{0pt}{\footnotesize$\;\triangleleft~$}}}q{{\raisebox{0pt}{\footnotesize$~\triangleright\;$}}}r)$], and thus @xmath117. the two remaining cases can be proved in a similar way. [thm:1] for all @xmath118, @xmath119 we first prove @xmath120. by lemma [la : x], @xmath111 is a congruence relation and it easily follows that all @xmath13-axioms are sound. for example, soundness of axiom follows from @xmath121\\ & = \big(se(r)[{\ensuremath{{\sf t}}}\mapsto se(q),{\ensuremath{{\sf f}}}\mapsto se(s)]\big)\;[{\ensuremath{{\sf t}}}\mapsto se(p),{\ensuremath{{\sf f}}}\mapsto se(u)]\\ & = se(r)[{\ensuremath{{\sf t}}}\mapsto se(q)[{\ensuremath{{\sf t}}}\mapsto se(p),{\ensuremath{{\sf f}}}\mapsto se(u)], { \ensuremath{{\sf f}}}\mapsto \,se(s)[{\ensuremath{{\sf t}}}\mapsto se(p),{\ensuremath{{\sf f}}}\mapsto se(u)]]\\ & = se(r)[{\ensuremath{{\sf t}}}\mapsto se(p{{\raisebox{0pt}{\footnotesize$\;\triangleleft~$}}}q{{\raisebox{0pt}{\footnotesize$~\triangleright\;$}}}u),{\ensuremath{{\sf f}}}\mapsto se(p{{\raisebox{0pt}{\footnotesize$\;\triangleleft~$}}}s{{\raisebox{0pt}{\footnotesize$~\triangleright\;$}}}u)]\\ & = se((p{{\raisebox{0pt}{\footnotesize$\;\triangleleft~$}}}q{{\raisebox{0pt}{\footnotesize$~\triangleright\;$}}}u){{\raisebox{0pt}{\footnotesize$\;\triangleleft~$}}}r{{\raisebox{0pt}{\footnotesize$~\triangleright\;$}}}(p{{\raisebox{0pt}{\footnotesize$\;\triangleleft~$}}}s{{\raisebox{0pt}{\footnotesize$~\triangleright\;$}}}u)).\end{aligned}\]] in order to prove @xmath122, let @xmath123. according to lemma [la:2.5] there exist basic forms @xmath124 and @xmath125 such that @xmath126 and @xmath127, so @xmath128. by soundness (@xmath120) we find @xmath129, so by lemma [la:2.6], @xmath130. hence, @xmath131. a consequence of the above results is that for each @xmath30 there is a _ unique _ basic form @xmath124 with @xmath126, and that for each basic form, its @xmath51-image has exactly the same syntactic structure (replacing @xmath132 by @xmath57, and @xmath133 by @xmath58). in the remainder of this section, we make this precise. [def : bf] the * basic form function * @xmath134 is defined as follows, where @xmath53 : @xmath135.\end{aligned}\]] given @xmath136, the auxiliary function @xmath137:{\ensuremath{\textit{bf}_a}}\to{\ensuremath{\textit{bf}_a}}$] for which post - fix notation @xmath138 $] is adopted, is defined as follows : @xmath139&=q,\\ { \ensuremath{{\sf f}}}[{\ensuremath{{\sf t}}}\mapsto q, { \ensuremath{{\sf f}}}\mapsto r]&=r,\\ (p_1{{\raisebox{0pt}{\footnotesize$\;\triangleleft~$}}}a{{\raisebox{0pt}{\footnotesize$~\triangleright\;$}}}p_2)[{\ensuremath{{\sf t}}}\mapsto q, { \ensuremath{{\sf f}}}\mapsto r] & = p_1[{\ensuremath{{\sf t}}}\mapsto q, { \ensuremath{{\sf f}}}\mapsto r]{{\raisebox{0pt}{\footnotesize$\;\triangleleft~$}}}a{{\raisebox{0pt}{\footnotesize$~\triangleright\;$}}}p_2[{\ensuremath{{\sf t}}}\mapsto q, { \ensuremath{{\sf f}}}\mapsto r].\end{aligned}\]] (the notational overloading with the leaf replacement functions on valuation trees is harmless). so, for given @xmath140, the auxiliary function @xmath141 $] applied to @xmath88 (thus, @xmath142 $]) replaces all @xmath25-occurrences in @xmath1 by @xmath2, and all @xmath9-occurrences in @xmath1 by @xmath7. the following two lemma s imply that @xmath143 is a normalization function. [la : bf] for all @xmath30, @xmath144 is a basic form. by structural induction. the base cases are trivial. for the inductive case we find @xmath145 $], so by induction, @xmath144, @xmath146, and @xmath147 are basic forms. furthermore, replacing all @xmath25-occurrences and @xmath9-occurrences in @xmath146 by basic forms @xmath144 and @xmath147, respectively, yields a basic form. [la:2.12] for each basic form @xmath1, @xmath148. by structural induction on @xmath1. [def : freevca] the binary relation @xmath149 on @xmath34is defined as follows : @xmath150 [la : rephrase] the relation @xmath149 is a congruence relation. let @xmath113 and assume @xmath151, thus @xmath152. then @xmath153= { \ensuremath{\mathit{bf}}}(q)[{\ensuremath{{\sf t}}}\mapsto { \ensuremath{\mathit{bf}}}(p'),{\ensuremath{{\sf f}}}\mapsto { \ensuremath{\mathit{bf}}}(r)]={\ensuremath{\mathit{bf}}}(p'{{\raisebox{0pt}{\footnotesize$\;\triangleleft~$}}}q{{\raisebox{0pt}{\footnotesize$~\triangleright\;$}}}r)$], and thus @xmath154. the two remaining cases can be proved in a similar way. before proving that @xmath13 is an axiomatization of the relation @xmath149, we show that each instance of the axiom satisfies @xmath149. [la : nieuw] for all @xmath155, @xmath156 by definition, the lemma s statement is equivalent with @xmath157\big)\;[{\ensuremath{{\sf t}}}\mapsto{\ensuremath{\mathit{bf}}}(q_1),{\ensuremath{{\sf f}}}\mapsto{\ensuremath{\mathit{bf}}}(q_2)]\\ \label{eq : bf } & = { \ensuremath{\mathit{bf}}}(p)[{\ensuremath{{\sf t}}}\mapsto{\ensuremath{\mathit{bf}}}(q_1{{\raisebox{0pt}{\footnotesize$\;\triangleleft~$}}}p_1{{\raisebox{0pt}{\footnotesize$~\triangleright\;$}}}q_2),{\ensuremath{{\sf f}}}\mapsto{\ensuremath{\mathit{bf}}}(q_1{{\raisebox{0pt}{\footnotesize$\;\triangleleft~$}}}p_2{{\raisebox{0pt}{\footnotesize$~\triangleright\;$}}}q_2)].\end{aligned}\]] by lemma [la : bf], @xmath144, @xmath158,and @xmath159 are basic forms. we prove by structural induction on the form that @xmath144 can have. if @xmath160, then @xmath161\big)&\;[{\ensuremath{{\sf t}}}\mapsto{\ensuremath{\mathit{bf}}}(q_1),{\ensuremath{{\sf f}}}\mapsto{\ensuremath{\mathit{bf}}}(q_2)] { \ensuremath{\mathit{bf}}}(p_1)[{\ensuremath{{\sf t}}}\mapsto{\ensuremath{\mathit{bf}}}(q_1),{\ensuremath{{\sf f}}}\mapsto{\ensuremath{\mathit{bf}}}(q_2)] \end{aligned}\]] and @xmath162 & = { \ensuremath{\mathit{bf}}}(q_1{{\raisebox{0pt}{\footnotesize$\;\triangleleft~$}}}p_1{{\raisebox{0pt}{\footnotesize$~\triangleright\;$}}}q_2)\\ & = { \ensuremath{\mathit{bf}}}(p_1)[{\ensuremath{{\sf t}}}\mapsto{\ensuremath{\mathit{bf}}}(q_1),{\ensuremath{{\sf f}}}\mapsto{\ensuremath{\mathit{bf}}}(q_2)].\end{aligned}\]] if @xmath163, then follows in a similar way. the inductive case @xmath164 is trivial (by definition of the last defining clause of the auxiliary functions @xmath141 $] in definition [def : bf]). [thm:1a] for all @xmath118, @xmath165 we first prove @xmath120. by lemma [la : rephrase], @xmath149 is a congruence relation and it easily follows that arbitrary instances of the @xmath13-axioms @xmath14 satisfy @xmath149. by lemma [la : nieuw] it follows that arbitrary instances of axiom also satisfy @xmath149. in order to prove @xmath122, assume @xmath166. according to lemma [la:2.5], there exist basic forms @xmath124 and @xmath125 such that @xmath126 and @xmath127, so @xmath128. by @xmath120 it follows that @xmath167, which implies by lemma [la:2.12] that @xmath130. hence, @xmath131. [cor:1] for all @xmath30, @xmath168 and @xmath169. by lemma [la : bf] and lemma [la:2.12], @xmath170, thus @xmath168. by theorem [thm:1a], @xmath171, and by theorem [thm:1], @xmath169. Evaluation trees for repetition-proof valuation congruence in @xcite we defined _ repetition - proof _ @xmath13 as the extension of the axiom set @xmath13 with the following two axiom schemes, where @xmath19 ranges over @xmath26 : @xmath172 we write @xmath173 for this extension. these axiom schemes characterize that for each atom @xmath19, a consecutive evaluation of @xmath19 yields the same result, so in both cases the conditional statement at the @xmath16-position will not be evaluated and can be replaced by any other. note that and are each others dual. we define a proper subset of basic forms with the property that each propositional statement can be proved equal to such a basic form. [def:3.1] * rp - basic forms * are inductively defined : * @xmath25 and @xmath9 are rp - basic forms, and * @xmath174 is an rp - basic form if @xmath175 and @xmath176 are rp - basic forms, and if @xmath177 is not equal to @xmath25 or @xmath9, then either the central condition in @xmath177 is different from @xmath19, or @xmath177 is of the form @xmath178. it will turn out useful to define a function that transforms conditional statements into rp - basic forms, and that is comparable to the function @xmath143. [def:3.2] the * rp - basic form function * @xmath179 is defined by @xmath180 the auxiliary function @xmath181 is defined as follows : @xmath182 for @xmath53, the auxiliary functions @xmath183 and @xmath184 are defined by @xmath185 and @xmath186 thus, @xmath187 maps a conditional statement @xmath1 to @xmath144 and then transforms @xmath144 according to the auxiliary functions @xmath188, @xmath189, and @xmath190. [la:3.3] for all @xmath53 and @xmath88, @xmath191 and @xmath192. by structural induction on @xmath1. the base cases @xmath103 are trivial. for the inductive case @xmath193 we have to distinguish the cases @xmath194 and @xmath107. if @xmath194, then @xmath195 and @xmath196 follows in a similar way. if @xmath107, then @xmath197, and hence @xmath191 the second pair of equalities can be derived in a similar way. in order to prove that for all @xmath30, @xmath198 is an rp - basic form, we use the following auxiliary lemma. [la:3.4] for all @xmath53 and @xmath88, @xmath199 and @xmath200. fix some @xmath53. we prove these inequalities by structural induction on @xmath1. the base cases @xmath103 are trivial. for the inductive case @xmath193 we have to distinguish the cases @xmath194 and @xmath107. if @xmath194, then @xmath201 and @xmath202 follows in a similar way. if @xmath107, then @xmath197, and hence @xmath199 and @xmath200. [la:3.5] for all @xmath30, @xmath198 is an rp - basic form. we first prove an auxiliary result : @xmath203 this follows by induction on the depth @xmath87 of @xmath1. if @xmath204, then @xmath103, and hence @xmath205 is an rp - basic form. for the inductive case @xmath206 it must be the case that @xmath207. we find @xmath208 which is an rp - basic form because * by lemma [la:3.4], @xmath209 and @xmath210 are basic forms with depth smaller than or equal to @xmath211, so by the induction hypothesis, @xmath212 and @xmath213 are rp - basic forms, * @xmath212 and @xmath213 both satisfy the following property : if the central condition (if present) is @xmath19, then the outer arguments are equal. we show this first for @xmath212 by a case distinction on the form of @xmath2 : 1. if @xmath214, then @xmath215, so there is nothing to prove. 2. if @xmath216, then @xmath217 and thus by lemma [la:3.3], + @xmath218. 3. if @xmath219 with @xmath107, then @xmath220, so + @xmath221 and there is nothing to prove. + the fact that @xmath213 satisfies this property follows in a similar way. this finishes the proof of . the lemma s statement now follows by structural induction : the base cases (comprising a single atom @xmath19) are again trivial, and for the inductive case, @xmath222 for some basic form @xmath94 by lemma [la : bf], and by auxiliary result , @xmath223 is an rp - basic form. the following, rather technical result is used in proposition [prop:3.7] and lemma [la:3.8]. [la:3.6] if @xmath224 is an rp - basic form, then @xmath225 and @xmath226. we first prove an auxiliary result : @xmath227 we prove both equalities by simultaneous induction on the structure of @xmath2 and @xmath7. the base case, thus @xmath228, is trivial. if @xmath229 and @xmath230, then @xmath2 and @xmath7 are rp - basic forms with central condition @xmath19, so @xmath231 and the equality for @xmath7 follows in a similar way. if @xmath229 and @xmath232, then @xmath233, and the result follows as above. all remaining cases follow in a similar way, which finishes the proof of . we now prove the lemma s statement by simultaneous induction on the structure of @xmath2 and @xmath7. the base case, thus @xmath228, is again trivial. if @xmath229 and @xmath230, then by auxiliary result , @xmath234 and by induction, @xmath235. hence, @xmath236, and @xmath237 and the equalities for @xmath7 follow in a similar way. if @xmath229 and @xmath232, the lemma s equalities follow in a similar way, although a bit simpler because @xmath238. for all remaining cases, the lemma s equalities follow in a similar way. [prop:3.7] for all @xmath30, @xmath198 is an rp - basic form, and for each rp - basic form @xmath1, @xmath239. the first statement is lemma [la:3.5]. for the second statement, it suffices by lemma [la:2.12] to prove that for each rp - basic form @xmath1, @xmath205. this follows by case distinction on @xmath1. the cases @xmath103 follow immediately, and otherwise @xmath104, and thus @xmath240. by lemma [la:3.6], @xmath241 and @xmath242, hence @xmath205. [la:3.8] for all @xmath88, @xmath243. we apply structural induction on @xmath1. the base cases @xmath103 are trivial. assume @xmath104. by induction @xmath244. we proceed by a case distinction on the form that @xmath175 and @xmath176 can have : 1. if @xmath245 with @xmath246, then @xmath247 and @xmath248, and hence @xmath249, and thus @xmath243. 2. if @xmath250 and @xmath251 with @xmath107, then @xmath248 and by auxiliary result in the proof of lemma [la:3.5], @xmath252 and @xmath253 are rp - basic forms. we derive @xmath254 3. if @xmath255 with @xmath107 and @xmath256, we can proceed as in the previous case, but now using axiom scheme and the identity @xmath247, and the fact that @xmath257 and @xmath258 are rp - basic forms. if @xmath250 and @xmath256, we can proceed as in two previous cases, now using both and, and the fact that @xmath252 and @xmath258 are rp - basic forms. + [thm:3.9] for all @xmath30, @xmath259. by theorem [thm:1a] and corollary [cor:1] we find @xmath260. by lemma [la:3.8], @xmath261, and @xmath262. [def:3.10] the binary relation @xmath263 on @xmath34is defined as follows : @xmath264 [thm:3.11] for all @xmath118, @xmath265. assume @xmath266. by theorem [thm:3.9], @xmath267. in @xcite the following two statements are proved (theorem 6.3 and an auxiliary result in its proof), where @xmath268 is a binary relation on @xmath269 : 1. for all @xmath118,@xmath270. 2. for all rp - basic forms @xmath1 and @xmath271. by lemma [la:3.5] these statements imply @xmath272, that is, @xmath273. assume @xmath273. by lemma [la:2.12], @xmath274. by theorem [thm:1a], @xmath275. by theorem [thm:3.9], @xmath266. so, the relation @xmath263 is axiomatized by @xmath173, and is thus a congruence. with this observation in mind, we define a transformation on evaluation trees that mimics the function @xmath187, and prove that equality of two such transformed trees characterizes the congruence that is axiomatized by @xmath173. [def:3.12] the unary * repetition - proof evaluation function * @xmath276 yields * repetition - proof evaluation trees * and is defined by @xmath277 the auxiliary function @xmath278 is defined as follows (@xmath53) : @xmath279 for @xmath53, the auxiliary functions @xmath280 and @xmath281 are defined by @xmath282 and @xmath283 [ex : rp] let @xmath284. we depict @xmath68 (as in example [ex : fr]) and the repetition - proof evaluation tree @xmath285 : @xmath286{ll } \begin{array}[t]{l } \\[-4 mm] \begin{tikzpicture } [ level distance=7.5 mm, level 1/.style={sibling distance=30 mm }, level 2/.style={sibling distance=15 mm }, baseline=(current bounding box.center)] \node (a) { $ a$ } child { node (b1) { $ { \ensuremath{{\sf f}}}$ } } child { node (b2) { $ a$ } child { node (d1) { $ { \ensuremath{{\sf t}}}$ } } child { node (d2) { $ { \ensuremath{{\sf f}}}$ } } } ; \end{tikzpicture } \end{array } & \qquad \begin{array}[t]{l } \\[-4 mm] \qquad \begin{tikzpicture } [ level distance=7.5 mm, level 1/.style={sibling distance=30 mm }, level 2/.style={sibling distance=15 mm }, baseline=(current bounding box.center)] \node (a) { $ a$ } child { node (b1) { $ { \ensuremath{{\sf f}}}$ } } child { node (b2) { $ a$ } child { node (d1) { $ { \ensuremath{{\sf f}}}$ } } child { node (d2) { $ { \ensuremath{{\sf f}}}$ } } } ; \end{tikzpicture } \\[-1 mm] \end{array}\end{array}\]] _ end example. _ the similarities between @xmath287 and the function @xmath187 can be exploited : [la:3.13] for all @xmath53 and @xmath288, @xmath289 and + @xmath290. by structural induction on @xmath41 (cf. the proof of lemma [la:3.3]). we use the following lemma in the proof of this section s last completeness result. [la:3.14] for all @xmath88, @xmath291. we first prove an auxiliary result : @xmath292 we prove the first equality of by structural induction on @xmath1. the base cases @xmath103 are trivial. for the inductive case @xmath207, let @xmath293. we have to distinguish the cases @xmath194 and @xmath107. if @xmath194, then @xmath294 if @xmath107, then @xmath295 the second equality can be proved in a similar way, and this finishes the proof of . the lemma s statement now follows by a case distinction on @xmath1. the cases @xmath103 follow immediately, and otherwise @xmath207, and thus @xmath296 finally, we relate conditional statements by means of their repetition - proof evaluation trees. * repetition - proof valuation congruence *, notation @xmath297, is defined on @xmath34 as follows : @xmath298 the following characterization result immediately implies that @xmath297 is a congruence relation on @xmath269 (and hence justifies calling it a congruence). [prop:3.16] for all @xmath118, @xmath299. in order to prove @xmath120, assume @xmath300, thus @xmath301. by corollary [cor:1], @xmath302, so by lemma [la:3.14], @xmath303. by lemma [la:2.6] and auxiliary result (see the proof of lemma [la:3.5]), it follows that @xmath304, that is, @xmath273. in order to prove @xmath122, assume @xmath273, thus @xmath304 and @xmath303. by lemma [la:3.14], @xmath302. by corollary [cor:1], @xmath305 and @xmath306, so @xmath301, that is, @xmath307. we end this section with the completeness result we were after. [thm:3.17] for all @xmath118, @xmath308 combine theorem [thm:3.11] and proposition [prop:3.16]. Evaluation trees for contractive valuation congruence in @xcite we introduced @xmath309, _ contractive _ @xmath13, as the extension of @xmath13 with the following two axiom schemes, where @xmath19 ranges over @xmath26 : @xmath310 these schemes prescribe contraction for each atom @xmath19 for respectively the _ true_-case and the _ false_-case (and are each others dual). it easily follows that the axiom schemes and are derivable from @xmath309, so @xmath309 is also an axiomatic extension of @xmath173. again, we define a proper subset of basic forms with the property that each propositional statement can be proved equal to such a basic form. [def:4.1] * cr - basic forms * are inductively defined : * @xmath25 and @xmath9 are cr - basic forms, and * @xmath174 is a cr - basic form if @xmath175 and @xmath176 are cr - basic forms, and if @xmath177 is not equal to @xmath25 or @xmath9, the central condition in @xmath177 is different from @xmath19. it will turn out useful to define a function that transforms conditional statements into cr - basic forms, and that is comparable to the function @xmath143 (see definition [def : bf]). [def:4.2] the * cr - basic form function * @xmath311 is defined by @xmath312 the auxiliary function @xmath313 is defined as follows : @xmath314 for @xmath53, the auxiliary functions @xmath315 and @xmath316 are defined by @xmath317 and @xmath318 thus, @xmath319 maps a conditional statement @xmath1 to @xmath144 and then transforms @xmath144 according to the auxiliary functions @xmath320, @xmath321, and @xmath322. [la:4.3] for all @xmath53 and @xmath88, @xmath323 and @xmath324. fix some @xmath53. we prove these inequalities by structural induction on @xmath1. the base cases @xmath103 are trivial. for the inductive case @xmath193 we have to distinguish the cases @xmath194 and @xmath107. if @xmath194, then @xmath325 and @xmath326 follows in a similar way. if @xmath107, then @xmath327, and hence @xmath323 and @xmath324. [la:4.4] for all @xmath30, @xmath328 is a cr - basic form. we first prove an auxiliary result : @xmath329 this follows by induction on the depth @xmath87 of @xmath1. if @xmath204, then @xmath103, and hence @xmath330 is a cr - basic form. for the inductive case @xmath206 it must be the case that @xmath207. we find @xmath331 which is a cr - basic form because * by lemma [la:4.3], @xmath332 and @xmath333 are basic forms with depth smaller than or equal to @xmath211, so by the induction hypothesis, @xmath334 and @xmath335 are cr - basic forms, * by definition of the auxiliary functions @xmath321 and @xmath322, the central condition of @xmath332 and @xmath333 is not equal to @xmath19, hence @xmath336 is a cr - basic form. this completes the proof of . the lemma s statement now follows by structural induction : the base cases (comprising a single atom @xmath19) are again trivial, and for the inductive case, @xmath337 for some basic form @xmath94 by lemma [la : bf], and by , @xmath338 is a cr - basic form. the following, somewhat technical lemma is used in proposition [prop:4.6] and lemma [la:4.7]. [la:4.5] if @xmath224 is a cr - basic form, then @xmath339 and @xmath340. by simultaneous induction on the structure of @xmath2 and @xmath7. the base case, thus @xmath228, is again trivial. if @xmath341 and @xmath342, then @xmath343 and thus @xmath344 and @xmath345. moreover, @xmath346 and @xmath347 have no central condition @xmath3, hence @xmath348 and @xmath349, and thus @xmath350 the equalities for @xmath7 follow in a similar way. if @xmath351 and @xmath352, the lemma s equalities follow in a similar way, and this is also the case if @xmath214 and @xmath353. with lemma [la:4.5] we can easily prove the following result. [prop:4.6] for each @xmath30, @xmath328 is a cr - basic form, and for each cr - basic form @xmath1, @xmath354. the first statement is lemma [la:4.4]. for the second statement, it suffices by lemma [la:2.12] to prove that @xmath330. we prove this by case distinction on @xmath1. the cases @xmath103 follow immediately, and otherwise @xmath104, and thus @xmath355. by lemma [la:4.5], @xmath356 and @xmath357, hence @xmath330. [la:4.7] for all @xmath88, @xmath358. we apply structural induction on @xmath1. the base cases @xmath103 are trivial. assume @xmath104. by induction @xmath359. furthermore, by auxiliary result in the proof of lemma [la:4.4], @xmath360 is a cr - basic form, and by lemma [la:4.5], @xmath361 we derive @xmath362 [thm:4.8] for all @xmath30, @xmath363. by theorem [thm:1a] and corollary [cor:1], @xmath364, and by lemma [la:4.7], @xmath365, and @xmath366. [def:4.9] the binary relation @xmath367 on @xmath34is defined as follows : @xmath368 [thm:4.10] for all @xmath118, @xmath369. assume @xmath370. then, by theorem [thm:4.8], @xmath371. in @xcite the following two statements are proved (theorem 6.4 and an auxiliary result in its proof), where @xmath372 is a binary relation on @xmath269 : 1. for all @xmath118,@xmath373. 2. for all cr - basic forms @xmath1 and @xmath374. by lemma [la:4.4], these statements imply @xmath375, that is, @xmath376. assume @xmath376. by lemma [la:2.12], @xmath377. by theorem [thm:1a], @xmath378. by theorem [thm:4.8], @xmath370. hence, the relation @xmath367 is axiomatized by @xmath379, and is thus a congruence. we now define a transformation on evaluation trees that mimics the function @xmath319, and prove that equality of two such transformed trees characterizes the congruence that is axiomatized by @xmath379. [def:4.12] the unary * contractive evaluation function * @xmath380 yields * contractive evaluation trees * and is defined by @xmath381 the auxiliary function @xmath382 is defined as follows (@xmath53) : @xmath383 for @xmath53, the auxiliary functions @xmath384 and @xmath385 are defined by @xmath386 and @xmath387 as a simple example we depict @xmath388 and the contractive evaluation tree @xmath389 : @xmath286{ll } \begin{array}[t]{l } \\[-4 mm] \begin{tikzpicture}[level distance=7.5 mm, level 1/.style={sibling distance=30 mm }, level 2/.style={sibling distance=15 mm }, level 3/.style={sibling distance=7.5 mm }] \node (a) { $ a$ } child { node (b1) { $ a$ } child { node (c1) { $ a$ } child { node (d1) { $ { \ensuremath{{\sf t}}}$ } } child { node (d2) { $ { \ensuremath{{\sf f}}}$ } } } child { node (c2) { $ { \ensuremath{{\sf f}}}$ } } } child { node (b2) { $ { \ensuremath{{\sf f}}}$ } } ; \end{tikzpicture } \end{array } & \qquad \begin{array}[t]{l } \\[-4 mm] \qquad \begin{tikzpicture}[level distance=7.5 mm, level 1/.style={sibling distance=30 mm }, level 2/.style={sibling distance=15 mm }, level 3/.style={sibling distance=7.5 mm }] \node (a) { $ a$ } child { node (c1) { $ { \ensuremath{{\sf t}}}$ } } child { node (b2) { $ { \ensuremath{{\sf f}}}$ } } ; \end{tikzpicture } \\[8 mm] \end{array}\end{array}\]] the similarities between the evaluation function @xmath390 and the function @xmath319 can be exploited, and we use the following lemma in the proof of the next completeness result. [la:4.13] for all @xmath88, @xmath391. we first prove the following auxiliary result : @xmath392 we prove the first equality by structural induction on @xmath1. the base cases @xmath103 are trivial. for the inductive case @xmath207, let @xmath293. we have to distinguish the cases @xmath194 and @xmath107. if @xmath194, then @xmath393 if @xmath107, then @xmath394 the second equality can be proved in a similar way, and this finishes our proof of . the lemma s statement now follows by a case distinction on @xmath1. the cases @xmath103 follow immediately, and otherwise @xmath207, and thus @xmath395 finally, we relate conditional statements by means of their contractive evaluation trees. [def:4.14] * contractive valuation congruence *, notation @xmath396, is defined on @xmath34as follows : @xmath397 the following characterization result immediately implies that @xmath396 is a congruence relation on @xmath269 (and hence justifies calling it a congruence). [prop:4.15] for all @xmath118, @xmath398. in order to prove @xmath120, assume @xmath399, thus @xmath400. by corollary [cor:1], @xmath401, so by lemma [la:4.13], @xmath402. by lemma [la:2.6] and auxiliary result (see the proof of lemma [la:3.5]), it follows that @xmath403, that is, @xmath376. in order to prove @xmath122, assume @xmath376, thus @xmath403 and @xmath402. by lemma [la:4.13], @xmath401. by corollary [cor:1], @xmath305 and @xmath306, so @xmath400, that is, @xmath404. our final result in this section is a completeness result for contractive valuation congruence. [thm:4.16] for all @xmath118, @xmath405 combine theorem [thm:4.10] and proposition [prop:4.15]. Evaluation trees for memorizing valuation congruence in @xcite we introduced @xmath406, _ memorizing @xmath13 _, as the extension of @xmath13 with the following axiom : @xmath407 axiom expresses that the first evaluation value of @xmath16 is memorized. more precisely, a memorizing evaluation " is one with the property that upon the evaluation of a compound propositional statement, the first evaluation value of each atom is memorized throughout the evaluation. we write @xmath406 for the set @xmath408 of axioms. replacing the variable @xmath16 in axiom by @xmath409 and/or the variable @xmath410 by @xmath411 yields all other memorizing patterns : @xmath412 furthermore, if we replace in axiom @xmath410 by @xmath45, we find the _ contraction law _ @xmath413 and replacing @xmath16 by @xmath409 then yields the dual contraction law @xmath414 hence, @xmath406 is an axiomatic extension of @xmath379. we define a proper subset of basic forms with the property that each propositional statement can be proved equal to such a basic form. [def:5.1] let @xmath415 be a subset of a. * mem - basic forms over @xmath415 * are inductively defined : * @xmath25 and @xmath9 are mem - basic forms over @xmath415, and * @xmath416 is a mem - basic form over @xmath415 if @xmath417 and @xmath1 and @xmath2 are mem - basic forms over @xmath418. @xmath1 is a * mem - basic form * if for some @xmath419, @xmath1 is a mem - basic form over @xmath415. note that if @xmath26 is finite, the number of mem - basic forms is also finite. it will turn out useful to define a function that transforms conditional statements into mem - basic forms. [def:5.2] the * mem - basic form function * @xmath420 is defined by @xmath421 the auxiliary function @xmath422 is defined as follows : @xmath423 for @xmath53, the auxiliary functions @xmath424 and @xmath425 are defined by @xmath426 and @xmath427 thus, @xmath428 maps a conditional statement @xmath1 to @xmath144 and then transforms @xmath144 according to the auxiliary functions @xmath429, @xmath430, and @xmath431. we will use the following equalities. [la:5.3] for all @xmath432 with @xmath433 and @xmath88, @xmath434 by structural induction on @xmath1. the base cases @xmath103 are trivial. for the inductive case @xmath435 we have to distinguish three cases : 1. if @xmath436, then equality follows by @xmath437 and equality follows by @xmath438 equalities and can be proved in a similar way. 2. if @xmath439, then equality follows by @xmath440 and equality follows by @xmath441 equalities and can be proved in a similar way. 3. if @xmath442, then equality follows by @xmath443 equalities @xmath444 can be proved in a similar way. [la:5.4] for all @xmath53 and @xmath88, @xmath445 and @xmath446. fix some @xmath53. we prove these inequalities by structural induction on @xmath1. the base cases @xmath103 are trivial. for the inductive case @xmath193 we have to distinguish the cases @xmath194 and @xmath107. if @xmath194, then @xmath447 and @xmath448 follows in a similar way. if @xmath107, then @xmath449 and @xmath450 follows in a similar way. [la:5.5] for all @xmath30, @xmath451 is a mem - basic form. we first prove an auxiliary result : @xmath452 this follows by induction on the depth @xmath87 of @xmath1. if @xmath204, then @xmath103, and hence @xmath453 is a mem - basic form. for the inductive case @xmath206 it must be the case that @xmath207. we find @xmath454 which is a mem - basic form because by lemma [la:5.4], @xmath455 and @xmath456 are basic forms with depth smaller than or equal to @xmath211, so by the induction hypothesis, @xmath457 is a mem - basic form over @xmath458 and @xmath459 is a mem - basic form over @xmath460 for suitable subsets @xmath458 and @xmath460 of @xmath26. notice that by definition of @xmath430 and @xmath431 we can assume that the atom @xmath19 does not occur in @xmath461. hence, @xmath462 is a mem - basic form over @xmath463, which completes the proof of . the lemma s statement now follows by structural induction : the base cases (comprising a single atom @xmath19) are again trivial, and for the inductive case, @xmath464 for some basic form @xmath94 by lemma [la : bf], and by , @xmath465 is a mem - basic form. the following lemma is used in proposition [prop:5.7] and lemma [la:5.8]. [la:5.6] if @xmath224 is a mem - basic form, then @xmath466 and @xmath467. assume @xmath224 is a mem - basic form over @xmath415. by definition, @xmath2 and @xmath7 are mem - basic forms over @xmath468. we prove both pairs of equalities simultaneously by induction on the structure of @xmath2 and @xmath7. the base case, thus @xmath228, is trivial. if @xmath341 and @xmath342, then @xmath469 and @xmath470. moreover, the @xmath98 are mem - basic forms over @xmath471, hence @xmath472 and @xmath473, and thus @xmath474 the equalities for @xmath7 follow in a similar way. if @xmath351 and @xmath352, the lemma s equalities follow in a similar way, and this is also the case if @xmath214 and @xmath353. with lemma [la:5.6] we can easily prove the following result. [prop:5.7] for each @xmath30, @xmath451 is a mem - basic form, and for each mem - basic form @xmath1, @xmath475. the first statement is lemma [la:5.5]. for the second statement, it suffices by lemma [la:2.12] to prove that @xmath453. we prove this by case distinction on @xmath1. the cases @xmath103 follow immediately, and otherwise @xmath104, so @xmath476. by lemma [la:5.6], @xmath477 and @xmath478, hence @xmath453. [la:5.8] for all @xmath88, @xmath479. we apply structural induction on @xmath1. the base cases @xmath103 are trivial. assume @xmath104. by induction @xmath480. furthermore, by auxiliary result in the proof of lemma [la:5.5], @xmath481 is a mem - basic form, and @xmath482 are mem - basic forms over @xmath483, and thus @xmath484 we derive @xmath485 [thm:5.9] for all @xmath30, @xmath486. by theorem [thm:1a] and corollary [cor:1], @xmath487, and by lemma [la:5.8], @xmath488, and @xmath489. [def:5.10] the binary relation @xmath490 on @xmath34is defined as follows : @xmath491 [thm:5.11] for all @xmath118, @xmath492. assume @xmath493. then, by theorem [thm:5.9], @xmath494. in @xcite the following two statements are proved (theorem 8.1 and lemma 8.4), where @xmath495 is a binary relation on @xmath269 : 1. for all @xmath118,@xmath496. 2. for all mem - basic forms @xmath1 and @xmath497. by lemma [la:5.5] these statements imply @xmath498, that is, @xmath499. assume @xmath499. by lemma [la:2.12], @xmath500. by theorem [thm:1a], @xmath501. by theorem [thm:5.9], @xmath493. hence, the relation @xmath490 is axiomatized by @xmath406 and is thus a congruence. we define a transformation on evaluation trees that mimics the function @xmath428, and prove that equality of two such transformed trees characterizes the congruence that is axiomatized by @xmath406. [def:5.12] the unary * memorizing evaluation function * @xmath502 yields * memorizing evaluation trees * and is defined by @xmath503 the auxiliary function @xmath504 is defined as follows (@xmath53) : @xmath505 for @xmath53, the auxiliary functions @xmath506 and @xmath507 are defined by @xmath508 and @xmath509 as a simple example we depict @xmath510 and the memorizing evaluation tree @xmath511 : @xmath512 \node (a) { $ a$ } child { node (b1) { $ b$ } child { node (c1) { $ a$ } child { node (d1) { $ { \ensuremath{{\sf t}}}$ } } child { node (d2) { $ { \ensuremath{{\sf f}}}$ } } } child { node (c2) { $ { \ensuremath{{\sf f}}}$ } } } child { node (b2) { $ { \ensuremath{{\sf f}}}$ } } ; \end{tikzpicture } \end{array } & \qquad \begin{array}{l } \qquad \begin{tikzpicture}[level distance=7.5 mm, level 1/.style={sibling distance=30 mm }, level 2/.style={sibling distance=15 mm }, level 3/.style={sibling distance=7.5 mm }] \node (a) { $ a$ } child { node (b1) { $ b$ } child { node (c1) { $ { \ensuremath{{\sf t}}}$ } } child { node (c2) { $ { \ensuremath{{\sf f}}}$ } } } child { node (b2) { $ { \ensuremath{{\sf f}}}$ } } ; \end{tikzpicture } \\[8 mm] \end{array}\end{array}\]] the similarities between @xmath513 and the function @xmath428 will of course be exploited. [la:5.13] for all @xmath432 with @xmath433 and @xmath288, * @xmath514, * @xmath515, * @xmath516, * @xmath517. by structural induction on @xmath41 (cf. the proof of lemma [la:5.3]). we use the following lemma s in the proof of our next completeness result. [la:5.14] for all @xmath53 and @xmath88, @xmath518 we first prove an auxiliary result : @xmath519 fix some @xmath53. we prove by structural induction on @xmath1. the base cases @xmath103 are trivial. for the inductive case @xmath193 we have to distinguish the cases @xmath194 and @xmath107. if @xmath194, then @xmath520 and if @xmath107, then @xmath521 this finishes the proof of . we now prove the lemma s equalities. fix some @xmath53. we prove the first equality by induction on @xmath87. the base case @xmath204, thus @xmath103, is trivial. for the inductive case @xmath206, it must be the case that @xmath193. we have to distinguish the cases @xmath194 and @xmath107. if @xmath194, then @xmath522 if @xmath107, then @xmath523 the second equality can be proved in a similar way. [la:5.15] for all @xmath88, @xmath524. by a case distinction on @xmath1. the cases @xmath103 follow immediately, and otherwise @xmath207, and thus @xmath525 [def:5.16] * memorizing valuation congruence *, notation @xmath526, is defined on @xmath34as follows : @xmath527 the following characterization result immediately implies that @xmath526 is a congruence relation on @xmath269 (and hence justifies calling it a congruence). [prop:5.17] for all @xmath118, @xmath528. for @xmath120, assume @xmath529, thus @xmath530. by corollary [cor:1], @xmath531 so by lemma [la:5.15], @xmath532 by lemma [la:2.6], it follows that @xmath533, that is, @xmath499. in order to prove @xmath122, assume @xmath499, thus @xmath533. then @xmath534 and by lemma [la:5.15], @xmath535 by corollary [cor:1], @xmath530, that is, @xmath536. we end this section with a completeness result for memorizing valuation congruence. [thm:5.18] for all @xmath118, @xmath537 combine theorem [thm:5.11] and proposition [prop:5.17]. Evaluation trees for static valuation congruence the most identifying axiomatic extension of @xmath13 we consider can be defined by adding the following axiom to @xmath406 : @xmath538 so, the evaluation value of each atom in a conditional statement is memorized, and by axiom , no atom @xmath19 can have a side effect because @xmath539 for all @xmath30. we write @xmath540 for the set of these axioms, thus @xmath541 observe that the duality principle also holds in @xmath540, in particular, @xmath542. a simple example on @xmath540 illustrates how the order of evaluation of @xmath17 and @xmath16 can be swapped : @xmath543 equation can be derived as follows : @xmath544 the following lemma is a direct consequence of axiom . [la:6.1] for all @xmath118, @xmath545. @xmath546 in @xcite we defined @xmath547 as the extension of @xmath13 with the following two axioms : @xmath548 axiom expresses how the order of evaluation of @xmath410 and @xmath16 can be swapped, and (as explained section [sec : mem]) the contraction law expresses that the evaluation result of @xmath16 is memorized. because we will rely on results for @xmath547 that are proven in @xcite, we first prove the following result. [prop:3] the axiom sets @xmath549 and @xmath540 are equally strong. we show that all axioms in the one set are derivable from the other set. we first prove that the axiom is derivable from @xmath549 : @xmath550 where the contraction law , that is @xmath551, is derivable from @xmath549 : replace @xmath16 by @xmath409 in . hence @xmath552. furthermore, if we take @xmath553 in axiom we find @xmath554, hence @xmath555. in order to show that @xmath556 recall that the contraction law is derivable from @xmath406 (see section [sec : mem]). so, it remains to be proved that @xmath557 and with equation we can easily derive this axiom from @xmath540 : @xmath558 given a finite, ordered subset of atoms we define a proper subset of basic forms with the property that each propositional statement over these atoms can be proved equal to such a basic form. [def:6.3] let @xmath559 be the set of strings over @xmath26 with the property that each @xmath560 contains no multiple occurrences of the same atom. for the empty string, thus @xmath561.] * st - basic forms over @xmath562 * are defined as follows : * @xmath25 and @xmath9 are st - basic forms over @xmath563. * @xmath416 is an st - basic form over @xmath564 if @xmath1 and @xmath2 are st - basic forms over @xmath565. @xmath1 is an * st - basic form * if for some @xmath560, @xmath1 is an st - basic form over @xmath78. for example, an st - basic form over @xmath566 has the following form : @xmath567 with @xmath568. for @xmath569, there exist @xmath570 different st - basic forms over @xmath78. it will turn out useful to define a function that transforms conditional statements to st - basic forms. therefore, given @xmath562 we consider terms in @xmath571, where @xmath415 is the finite subset of @xmath26 that contains the elements of @xmath78. if @xmath64, then @xmath572 and the st - basic forms over @xmath563 are @xmath25 and @xmath9. [def:6.4] the * alphabet function * @xmath573 returns the set of atoms of a string in @xmath574 : @xmath575 [def:6.5] let @xmath562. the conditional statement @xmath576 is defined as @xmath577 the * st - basic form function * @xmath578 is defined by @xmath579 so, for each @xmath562, @xmath580 is an st - basic form over @xmath78 in which the constant @xmath25 does not occur, e.g., @xmath581 [la:6.6] let @xmath562. for all @xmath30, @xmath582. by induction on the structure of @xmath78. if @xmath64, then @xmath583 and @xmath584 if @xmath585 for some @xmath586 and @xmath53, then @xmath587, and hence @xmath588 [la:6.7] let @xmath562. for all @xmath589, @xmath590 is an st - basic form. we use two auxiliary results : @xmath591.\end{aligned}\]] this easily follows by induction on the structure of @xmath78. @xmath592)=e^\rho[{\ensuremath{{\sf f}}}\mapsto\ell_a({\ensuremath{\mathit{bf}}}(p))]\\[1 mm] \text{and~}~{r}_a((e^\rho[{\ensuremath{{\sf f}}}\mapsto{\ensuremath{\mathit{bf}}}(p)])=e^\rho[{\ensuremath{{\sf f}}}\mapsto{r}_a({\ensuremath{\mathit{bf}}}(p))], \end{array}\end{aligned}\]] both equalities follow easily by induction on the structure of @xmath565. we prove the lemma s statement by induction on the structure of @xmath78. if @xmath64, then @xmath1 contains no atoms. hence, @xmath593. if @xmath160 then @xmath594 which is an st - basic form over @xmath563, and similarly for the case @xmath163. if @xmath585 for some @xmath586 and @xmath53, we derive @xmath595) & & \text{by~\eqref{aux : s1}}\\ & = { \ensuremath{\mathit{mf}}}((e^\rho{{\raisebox{0pt}{\footnotesize$\;\triangleleft~$}}}a{{\raisebox{0pt}{\footnotesize$~\triangleright\;$}}}e^\rho)[{\ensuremath{{\sf f}}}\mapsto{\ensuremath{\mathit{bf}}}(p)])\\ & = { \ensuremath{\mathit{mf}}}(e^\rho[{\ensuremath{{\sf f}}}\mapsto{\ensuremath{\mathit{bf}}}(p)]{{\raisebox{0pt}{\footnotesize$\;\triangleleft~$}}}a{{\raisebox{0pt}{\footnotesize$~\triangleright\;$}}}e^\rho[{\ensuremath{{\sf f}}}\mapsto{\ensuremath{\mathit{bf}}}(p)])\\ & = { \ensuremath{\mathit{mf}}}(\ell_{a}(e^\rho[{\ensuremath{{\sf f}}}\mapsto{\ensuremath{\mathit{bf}}}(p)])){{\raisebox{0pt}{\footnotesize$\;\triangleleft~$}}}a{{\raisebox{0pt}{\footnotesize$~\triangleright\;$}}}{\ensuremath{\mathit{mf}}}({r}_{a}(e^\rho[{\ensuremath{{\sf f}}}\mapsto{\ensuremath{\mathit{bf}}}(p)])) \\ & = { \ensuremath{\mathit{mf}}}(e^\rho[{\ensuremath{{\sf f}}}\mapsto\ell_{a}({\ensuremath{\mathit{bf}}}(p))]){{\raisebox{0pt}{\footnotesize$\;\triangleleft~$}}}a{{\raisebox{0pt}{\footnotesize$~\triangleright\;$}}}{\ensuremath{\mathit{mf}}}(e^\rho[{\ensuremath{{\sf f}}}\mapsto{r}_{a}({\ensuremath{\mathit{bf}}}(p))]) & & \text{by~\eqref{aux : s2}}\\ & = { \ensuremath{\mathit{mf}}}(e^\rho[{\ensuremath{{\sf f}}}\mapsto{\ensuremath{\mathit{bf}}}(\ell_{a}({\ensuremath{\mathit{bf}}}(p)))]){{\raisebox{0pt}{\footnotesize$\;\triangleleft~$}}}a{{\raisebox{0pt}{\footnotesize$~\triangleright\;$}}}{\ensuremath{\mathit{mf}}}(e^\rho[{\ensuremath{{\sf f}}}\mapsto{\ensuremath{\mathit{bf}}}({r}_{a}({\ensuremath{\mathit{bf}}}(p)))]) & & \text{by lemma~\ref{la:2.12}}\\ & = { \ensuremath{\mathit{mf}}}({\ensuremath{\mathit{bf}}}({\ensuremath{{\sf t}}}{{\raisebox{0pt}{\footnotesize$\;\triangleleft~$}}}e^\rho{{\raisebox{0pt}{\footnotesize$~\triangleright\;$}}}\ell_{a}({\ensuremath{\mathit{bf}}}(p)))){{\raisebox{0pt}{\footnotesize$\;\triangleleft~$}}}a{{\raisebox{0pt}{\footnotesize$~\triangleright\;$}}}{\ensuremath{\mathit{mf}}}({\ensuremath{\mathit{bf}}}({\ensuremath{{\sf t}}}{{\raisebox{0pt}{\footnotesize$\;\triangleleft~$}}}e^\rho{{\raisebox{0pt}{\footnotesize$~\triangleright\;$}}}{r}_{a}({\ensuremath{\mathit{bf}}}(p)))) & & \text{by~\eqref{aux : s1}}\\ & = { \ensuremath{\mathit{sbf}}}_\rho(\ell_{a}({\ensuremath{\mathit{bf}}}(p))){{\raisebox{0pt}{\footnotesize$\;\triangleleft~$}}}a{{\raisebox{0pt}{\footnotesize$~\triangleright\;$}}}{\ensuremath{\mathit{sbf}}}_\rho({r}_{a}({\ensuremath{\mathit{bf}}}(p))),\end{aligned}\]] where the penultimate equality follows by applied to @xmath565 because @xmath596 and @xmath597 are conditional statements in @xmath598 (thus, not containing @xmath19). by induction, @xmath599 and @xmath600 are st - basic forms over @xmath565. hence, @xmath590 is an st - basic form over @xmath78. [la:6.8] let @xmath562. for all @xmath589, @xmath601. by lemma [la:6.6], @xmath582. by theorem [thm:5.9], @xmath602, hence @xmath601. [thm:6.9] let @xmath562. for all @xmath603, @xmath604. assume @xmath605. then, by lemma [la:6.8], @xmath606, and by proposition [prop:3], @xmath607. in @xcite the following two statements are proved (theorem 9.1 and an auxiliary result in its proof), where @xmath608 is a binary relation on @xmath269 : 1. for all @xmath118,@xmath609. 2. for all st - basic forms @xmath1 and @xmath610. by lemma [la:6.7] these statements imply @xmath611. assume @xmath611, and thus @xmath612. by theorem [thm:5.11], @xmath613, and by lemma [la:6.6] this implies @xmath605. [def:6.11] let @xmath562. the binary relation @xmath614 on @xmath615 is defined as follows : @xmath616 [thm:6.12] let @xmath562. for all @xmath603, @xmath617. assume @xmath605. by lemma [la:6.8], @xmath606, and by theorem [thm:6.9], @xmath618. assume @xmath618, and thus @xmath611. by theorem [thm:6.9], @xmath619. hence, the relation @xmath614 is a congruence on @xmath615 that is axiomatized by @xmath540. we define a transformation on evaluation trees that mimics the function @xmath620 and prove that equality of two such transformed trees characterizes the congruence that is axiomatized by @xmath540. [def:6.13] let @xmath562. the unary * static evaluation function * @xmath621 yields * static evaluation trees * and is defined as follows : @xmath622 where @xmath580 is defined in definition [def:6.5]. [ex : st] let @xmath623. we depict @xmath68 at the left - hand side. the static evaluation tree @xmath624 is depicted in the middle, and the static evaluation tree @xmath625 is depicted at the right - hand side : @xmath626 \node (a) { $ a$ } child { node (b1) { $ b$ } child { node (c1) { $ a$ } child { node (d1) { $ { \ensuremath{{\sf t}}}$ } } child { node (d2) { $ { \ensuremath{{\sf f}}}$ } } } child { node (c2) { $ { \ensuremath{{\sf f}}}$ } } } child { node (b2) { $ { \ensuremath{{\sf t}}}$ } } ; \end{tikzpicture } \end{array } & \qquad \begin{array}{l } \qquad \begin{tikzpicture}[level distance=7.5 mm, level 1/.style={sibling distance=15 mm }, level 2/.style={sibling distance=7.5 mm }, level 3/.style={sibling distance=7.5 mm }] \node (a) { $ a$ } child { node (b1) { $ b$ } child { node (c1) { $ { \ensuremath{{\sf t}}}$ } } child { node (c2) { $ { \ensuremath{{\sf f}}}$ } } } child { node (b2) { $ b$ } child { node (c3) { $ { \ensuremath{{\sf t}}}$ } } child { node (c4) { $ { \ensuremath{{\sf t}}}$ } } } ; \end{tikzpicture } \\[8 mm] \end{array } & \qquad \begin{array}{l } \qquad \begin{tikzpicture}[level distance=7.5 mm, level 1/.style={sibling distance=15 mm }, level 2/.style={sibling distance=7.5 mm }, level 3/.style={sibling distance=7.5 mm }] \node (a) { $ b$ } child { node (b1) { $ a$ } child { node (c1) { $ { \ensuremath{{\sf t}}}$ } } child { node (c2) { $ { \ensuremath{{\sf t}}}$ } } } child { node (b2) { $ a$ } child { node (c3) { $ { \ensuremath{{\sf f}}}$ } } child { node (c4) { $ { \ensuremath{{\sf t}}}$ } } } ; \end{tikzpicture } \\[8 mm] \end{array } \end{array}\]] the two different static evaluation trees correspond to the different ways in which one can present truth tables for @xmath1, that is, the different possible orderings of the valuation values of the atoms occurring in @xmath1 : @xmath627 _ end example. _ the reason that @xmath628 is defined relative to some @xmath562 that covers the alphabet of @xmath1 is that in order to prove completeness of @xmath540 (and @xmath547), we need to be able to relate conditional statements that contain different sets of atoms, such as for example @xmath629 and that have equal static evaluation trees. for @xmath630 and @xmath631 the static evalution trees for example are @xmath632 and @xmath633, thus @xmath634 \node (a) { $ a$ } child { node (b1) { $ b$ } child { node (c1) { $ { \ensuremath{{\sf f}}}$ } } child { node (c2) { $ { \ensuremath{{\sf f}}}$ } } } child { node (b2) { $ b$ } child { node (c3) { $ { \ensuremath{{\sf f}}}$ } } child { node (c4) { $ { \ensuremath{{\sf f}}}$ } } } ; \end{tikzpicture } \end{array } & \qquad \begin{array}{l } \qquad \begin{tikzpicture}[level distance=7.5 mm, level 1/.style={sibling distance=30 mm }, level 2/.style={sibling distance=15 mm }, level 3/.style={sibling distance=7.5 mm }] \node (a) { $ b$ } child { node (b1) { $ a$ } child { node (c1) { $ { \ensuremath{{\sf f}}}$ } } child { node (c2) { $ { \ensuremath{{\sf f}}}$ } } } child { node (b2) { $ a$ } child { node (c3) { $ { \ensuremath{{\sf f}}}$ } } child { node (c4) { $ { \ensuremath{{\sf f}}}$ } } } ; \end{tikzpicture } \end{array } \end{array}\]] the similarities between @xmath635 and the function @xmath620 can be exploited and lead to our final completeness result. [def:6.15] let @xmath560. * static valuation congruence over @xmath78 *, notation @xmath636, is defined on @xmath615 as follows : @xmath637 the following characterization result immediately implies that for all @xmath562, @xmath636 is a congruence relation on @xmath615. [prop:6.14] let @xmath562. for all @xmath603, @xmath638 we have to show @xmath639 and this immediately follows from proposition [prop:5.17]. [thm:6.15] let @xmath560. for all @xmath603, @xmath640 combine theorem [thm:6.12] and proposition [prop:6.14]. Conclusions in @xcite we introduced proposition algebra using hoare s conditional @xmath641 and the constants @xmath8 and @xmath45. we defined a number of varieties of so - called _ valuation algebras _ in order to capture different semantics for the evaluation of conditional statements, and provided axiomatizations for the resulting valuation congruences : @xmath642 (four axioms) characterizes the least identifying valuation congruence we consider, and the extension @xmath406 (one extra axiom) characterizes the most identifying valuation congruence below propositional logic, while static valuation congruence, axiomatized by adding the simple axiom @xmath554 to @xmath406, can be seen as a characterization of propositional logic. in @xcite we introduced an alternative valuation semantics for proposition algebra in the form of _ hoare - mccarthy algebras _ (hma s) that is more elegant than the semantical framework provided in @xcite : hma - based semantics has the advantage that one can define a valuation congruence without first defining the valuation _ equivalence _ it is contained in. in this paper, we use staudt s evaluation trees @xcite to define free valuation congruence as the relation @xmath111 (see section [sec : free]), and this appears to be a relatively simple and stand - alone exercise, resulting in a semantics that is elegant and much simpler than hma - based semantics @xcite and the semantics defined in @xcite. by theorem [thm:1], @xmath111 coincides with `` free valuation congruence as defined in @xcite '' because both relations are axiomatized by @xmath642 (see (*??? * thm.4.4 and thm.6.2)). the advantage of `` evaluation tree semantics '' is that for a given conditional statement @xmath1, the evaluation tree @xmath68 determines all relevant evaluations, so @xmath123 is determined by evaluation trees that contain no more atoms than those that occur in @xmath1 and @xmath2, which corresponds to the use of truth tables in propositional logic. in section [sec : rp] we define repetition - proof valuation congruence @xmath297 on @xmath269 by @xmath307 if, and only if @xmath300, where @xmath643 and @xmath644 is a transformation function on evaluation trees. it is obvious that this transformation is `` natural '', given the axiom schemes @xmath645 and that are characteristic for @xmath173. the equivalence on @xmath269 that we want to prove is @xmath646 by which @xmath297 coincides with `` repetition - proof valuation congruence as defined in @xcite '' because both are axiomatized by @xmath173 (see (*??? * thm.6.3)). however, equivalence implies that @xmath297 is a _ congruence _ relation on @xmath34 and we could not find a direct proof of this fact. we chose to simulate the transformation @xmath287 by the transformation @xmath187 on conditional statements and to prove that the resulting equivalence relation @xmath263 is a congruence axiomatized by @xmath173. this is theorem [thm:3.11], the proof of which depends on (*??? * thm.6.3)) _ and _ on theorem [thm:3.9], that is, @xmath647 in order to prove equivalence (which is theorem [thm:3.17]), it is thus sufficient to prove that @xmath263 and @xmath297 coincide, and this is proposition [prop:3.16]. the structure of our proofs on the axiomatizations of the other valuation congruences that we consider is very similar, although the case for static valuation congruence requires a slightly more complex proof (below we return to this point). moreover, these axiomatizations are incremental : the axiom systems @xmath173 up to and including @xmath540 all share the axioms of @xmath13, and each succeeding system is defined by the addition of either one or two axioms, in most cases making previously added axiom(s) redundant. given some @xmath562, this implies that in @xmath615, @xmath648 where all these inclusions are proper if @xmath649, and thus @xmath650, and thus @xmath651. we conclude that repetition - proof evaluation trees and the valuation congruence @xmath297 provide a full - fledged, simple and elegant semantics for @xmath173, and that this is also the case for contractive evaluation trees and the valuation congruence @xmath396, and memorizing evaluation trees and the valuation congruence @xmath526. static valuation congruence over @xmath615 for some @xmath562, coincides with any standard semantics of propositional logic in the following sense : @xmath652 where @xmath653 and @xmath654 refer to hoare s definition @xcite : @xmath655 let @xmath53. the fact that @xmath656 identifies more than @xmath526 is immediately clear : @xmath657 while it is easy to see that @xmath658. our proof that @xmath540, and thus @xmath547 is an axiomatization of static valuation congruence is slightly more complex than those for the other axiomatizations because evaluation of a conditional statement @xmath1 does not enforce a canonical order for the evaluation of its atoms, and therefore such an ordering as encoded by a static evaluation tree should be fixed beforehand. to this purpose, we use some @xmath562. a spin - off of our approach can be called `` basic form semantics for proposition algebra '' : for each valuation congruence c considered, two conditional statements are c - valuation congruent if, and only if, they have equal c - basic forms, where c - basic forms are obtained by a syntactic transformation of conditional statements, which is a form of normalization. it is cumbersome, but not difficult to prove that given some @xmath562, the function @xmath620 is a normalization function on @xmath615. in addition to lemma [la:6.7] this requires proving that for each st - basic form @xmath1 over @xmath78, @xmath659. we conclude with a brief digression on _ short - circuit logic _, which we defined in @xcite (see @xcite for a quick introduction), and an example on the use of @xmath173. familiar binary connectives that occur in the context of imperative programming and that prescribe short - circuit evaluation, such as ` & & ` (in c called `` logical and ''), are often defined in the following way : @xmath660 independent of the precise syntax of @xmath1 and @xmath2, hence, @xmath661. in a similarly way, negation can be defined by @xmath662. short - circuit logic focuses on the question @xmath663 a first approach to this question is to adopt the conditional as an auxiliary operator, as is done in @xcite, and to answer question using definitions of the binary propositional connectives as above and the axiomatization for the valuation congruence of interest in proposition algebra (or, if `` mixed conditional statements '' are at stake, axiomatizations for the appropriate valuations congruences). an alternative and more direct approach to question is to establish axiomatizations for short - circuited binary connectives in which the conditional is _ not _ used. for free valuation congruence, an equational axiomatization of short - circuited binary propositional connectives is provided by staudt in @xcite, where @xmath664 $] and @xmath665 $] (and @xmath51 is also defined for short - circuited disjunction), and the associated completeness proof is based on decomposition properties of such evaluation trees. for repetition - proof valuation congruence it is an open question whether a finite, equational axiomatization of the short - circuited binary propositional connectives exists, and an investigation of repetition - proof evaluation trees defined by such connectives might be of interest in this respect. we conclude with an example on the use of @xmath173 that is based on (*??? * ex.4). [ex : rp2] let @xmath26 be a set of atoms of the form ` (e==e') ` and ` (n = e) ` with @xmath666 some initialized program variable and @xmath667 arithmetical expressions over the integers that may contain @xmath666. assume that ` (e==e') ` evaluates to @xmath10 if @xmath668 and @xmath669 represent the same value, and ` (n = e) ` always evaluates to _ true _ with the effect that @xmath668 s value is assigned to @xmath666. then these atoms satisfy the axioms of @xmath173. follow from @xmath173, e.g., @xmath670. we note that a particular consequence of @xmath173 in the setting of short - circuit logic is @xmath671 (cf. example [ex : rp]), and that example [ex : rp2] is related to the work of wortel @xcite, where an instance of _ propositional dynamic logic _ @xcite is investigated in which assignments can be turned into tests ; the assumption that such tests always evaluate to _ true _ is natural because the assumption that assignments always succeed is natural.] notice that if @xmath666 has initial value 0 or 1, @xmath672 and @xmath673 evaluate to different results, so the atom ` (n = n+1) ` does not satisfy the law @xmath674, by which this example is typical for the repetition - proof characteristic of @xmath173. _ end example. _ we finally note that all valuation congruences considered in this paper can be used as a basis for systematic analysis of the kind of _ side effects _ that may occur upon the evaluation of short - circuited connectives as in example [ex : rp], and we quote these words of parnas @xcite : _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ `` most mainline methods disparage side effects as a bad programming practice. yet even in well - structured, reliable software, many components do have side effects ; side effects are very useful in practice. it is time to investigate methods that deal with side effects as the normal case. '' _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ bergstra, j.a. and ponse, a. (2012). proposition algebra and short - circuit logic. in f. arbab and m. sirjani (eds.), _ proceedings of the 4th international conference on fundamentals of software engineering _ (fsen 2011), tehran. volume 7141 of lecture notes in computer science, pages 15 - 31. springer - verlag.

proposition algebra is based on hoare s conditional connective, which is a ternary connective comparable to if - then - else and used in the setting of propositional logic. conditional statements are provided with a simple semantics that is based on evaluation trees and that characterizes so - called free valuation congruence : two conditional statements are free valuation congruent if, and only if, they have equal evaluation trees. free valuation congruence is axiomatized by the four basic equational axioms of proposition algebra that define the conditional connective. valuation congruences that identify more conditional statements than free valuation congruence are repetition - proof, contractive, memorizing, and static valuation congruence. each of these valuation congruences is characterized using a transformation on evaluation trees : two conditional statements are c - valuation congruent if, and only if, their c - transformed evaluation trees are equal. these transformations are simple and natural, and only for static valuation congruence a slightly more complex transformation is used. also, each of these valuation congruences is axiomatized in proposition algebra. a spin - off of our approach is `` basic form semantics for proposition algebra '' : for each valuation congruence c considered, two conditional statements are c - valuation congruent if, and only if, they have equal c - basic forms, where c - basic forms are obtained by a syntactic transformation of conditional statements, which is a form of normalization. conditional composition, evaluation tree, proposition algebra, short - circuit evaluation, side effect

1504.08321

Introduction over the past 70 years, there have been multiple attempts to dynamically model the movement of polymer chains with brownian dynamics @xcite, which have more recently been used as a model for dna filament dynamics @xcite. one of the first and simplest descriptions was given as the rouse model @xcite, which is a bead - spring model @xcite, where the continuous filament is modelled at a mesoscopic scale with beads connected by springs. the only forces exerted on beads are spring forces from adjacent springs, as well as gaussian noise. hydrodynamic forces between beads and excluded volume effects are neglected in the model in favour of simplicity and computational speed, but the model manages to agree with several properties of polymer chains from experiments @xcite. other models exist, for example the zimm model introduces hydrodynamic forces @xcite between beads, or bending potentials can be introduced to form a wormlike chain and give a notion of persistence length @xcite, see, for example, review article @xcite or books @xcite on this subject. most of the aforementioned models consider the filament on only a single scale. in some applications, a modeller is interested in a relatively small region of a complex system. then it is often possible to use a hybrid model which is more accurate in the region of interest, and couple this with a model which is more computationally efficient in the rest of the simulated domain @xcite. an application area for hybrid models of polymer chains is binding of a protein to the dna filament, which we study in this paper. the model which we have created uses rouse dynamics for a chain of dna, along with a freely diffusing particle to represent a binding protein. as the protein approaches the dna, we increase the resolution in the nearby dna filament to increase accuracy of our simulations, whilst keeping them computationally efficient. in this paper we use the rouse model for analysis due to its mathematical tractability and small computational load. such a model is applicable to modelling dna dynamics when we consider relatively low resolutions, when hydrodynamic forces are negligible and persistence length is significantly shorter than the kuhn length between each bead @xcite. the situation becomes more complicated when we consider dna modelling at higher spatial resolutions. inside the cell nucleus, genetic information is stored within strands of long and thin dna fibres, which are separated into chromosomes. these dna fibres are folded into structures related to their function. different genes can be enhanced or inhibited depending upon this structure @xcite. folding also minimises space taken up in the cell by dna @xcite, and can be unfolded when required by the cell for different stages in the cell cycle or to alter gene expression. the folding of dna occurs on multiple scales. on a microscopic scale, dna is wrapped around histone proteins to form the nucleosome structure @xcite. this in turn gets folded into a chromatin fibre which gets packaged into progressively higher order structures until we reach the level of the entire chromosome @xcite. the finer points of how the nucleosome packing occurs on the chromatin fibre and how these are then packaged into higher - order structures is still a subject of much debate, with long - held views regarding mesoscopic helical fibres becoming less fashionable in favour of more irregular structures in vivo @xcite. in the most compact form of chromatin, many areas of dna are not reachable for vital reactions such as transcription @xcite. one potential explanation to how this is overcome by the cell is to position target dna segments at the surface of condensed domains when it is needed @xcite, so that transcription factors can find expressed genes without having to fit into these tightly - packed structures. this complexity is not captured by the multiscale model of protein binding presented in this paper. however, if one uses the developed refinement of the rouse model together with a more detailed modelling approach in a small region of dna next to the binding protein, then such a hybrid model can be used to study the effects of microscopic details on processes over system - level spatial and temporal scales. when taking this multiscale approach, it is necessary to understand the error from including the less accurate model in the hybrid model and how the accuracy of the method depends on its parameters. these are the main questions studied in this paper. the rest of the paper is organized as follows. in section [secmrbs], we introduce a multi - resolution bead - spring model which generalizes the rouse model. we also introduce a discretized version of this model which enables the use of different timesteps in different spatial regions. in section [section3], we analyze the main properties of the multi - resolution bead - spring model. we prove two main lemmas giving formulas for the diffusion constant and the end - to - end distance. we also study the appropriate choice of timesteps for numerical simulations of the model and support our analysis by the results of illustrative computer simulations. our main application area is studied in section [section4] where we present and analyze a dna binding model. we develop a method to increase the resolution in existing segments on - the - fly using the metropolis - hastings algorithm. in section [secdiscussion], we conclude our paper by discussing possible extensions of the presented multiscale approach (by including more detailed models of dna dynamics) and other multiscale methods developed in the literature. Multi-resolution bead-spring model we generalize the classical rouse bead - spring polymer model @xcite to include beads of variable sizes and springs with variable spring constants. in definition [defmrbs], we formulate the evolution equation for this model as a system of stochastic differential equations (sdes). we will also introduce a discretized version of this model in algorithm [algoneiter], which will be useful in sections [section3] and [section4] where we use the multi - resolution bead - spring model to develop and analyze multiscale models for dna dynamics. [defmrbs] let @xmath0 be a positive integer. a multi - resolution bead - spring polymer of size @xmath1 consists of a chain of @xmath1 beads of radius @xmath2, for @xmath3, connected by @xmath4 springs which are characterized by their spring constants @xmath5, for @xmath6. the positions @xmath7 $] of beads evolve according to the system of sdes (for @xmath3) @xmath8 where @xmath9 is the frictional drag coefficient of the @xmath10-th bead given by stokes theorem, @xmath11 is the solvent viscosity, @xmath12 $] is a wiener process, @xmath13 is absolute temperature, @xmath14 is boltzmann s constant and we assume that each spring constant @xmath5 can be equivalently expressed in terms of the corresponding kuhn length @xmath15 by @xmath16 we assume that the behaviour of boundary beads (for @xmath17 and @xmath18) is also given by equation @xmath19 simplified by postulating @xmath20 and @xmath21 _ [figmrbeadspring]] in figure [figmrbeadspring], we schematically illustrate a multi - resolution bead - spring polymer for @xmath22. the region between the @xmath23-th and the @xmath24-th bead is described with the highest resolution by considering smaller beads and springs with larger spring constants (or equivalently with smaller kuhn lengths). the scalings of different parameters in definition [defmrbs] are chosen so that we recover the classical rouse model @xcite if we assume @xmath25 and @xmath26. then equation ([sdedef]) simplifies to @xmath27 where @xmath28, @xmath29 and we again define @xmath20 and @xmath30 in equations for boundary beads. in the polymer physics literature @xcite, the rouse model ([sderouse]) is equivalently written as @xmath31 where random thermal noises @xmath32 exerted on the beads from brownian motion are characterized by the moments @xcite @xmath33 where @xmath34 and @xmath35. for the remainder of this paper, we will use the sde notation as given in ([sderouse]), because we will often study numerical schemes for simulating polymer dynamics models. the simplest discretization of ([sderouse]) is given by the euler - maruyama method @xcite, which uses the finite timestep @xmath36 and calculates the position vector @xmath37 of the @xmath10-th bead, @xmath38, at discretised time @xmath39 by @xmath40 for @xmath41, where @xmath42 is normally distributed random variable with zero mean and unit variance (i.e. @xmath43) for @xmath44. in order to discretize the multi - resolution bead - spring model, we allow for variable timesteps. [defvartimestep] let @xmath45 and let @xmath46, @xmath47 be positive integers such that @xmath48 or @xmath49 for @xmath50. let us assume that at least one of the values of @xmath46 is equal to 1. we define @xmath51 for @xmath6 and we call @xmath52 a timestep associated with the @xmath10-th spring. definition [defvartimestep] specifies that all timesteps must be integer multiples of the smallest timestep @xmath36. the timesteps associated with two adjacent springs are also multiples of each other. the time evolution of the multi - resolution bead - spring model is computed at integer multiples of @xmath36. one iteration of the algorithm is shown in algorithm [algoneiter]. the position of the @xmath10-th bead is updated at integer multiples of @xmath53 by calculating the random displacement due to brownian motion, with displacement caused by springs attached to the bead also updated at integer multiples of the timesteps associated with each spring, i.e. @xmath54 or @xmath55 considering the situation that all beads, springs and timesteps are the same, then one can easily deduce the following result. update positions of internal beads which are connected to two springs : + update of the first bead : + update of the last bead : + [lemconsnum] let @xmath56, @xmath57, @xmath58 and @xmath45 be positive constants and @xmath0 be an integer. consider a multi - resolution bead - spring polymer of size @xmath1 with @xmath59, @xmath60, for @xmath3, and @xmath61, for @xmath6. let the timesteps associated with each spring be equal to @xmath36, i.e. @xmath62 in definition @xmath63 then algorithm @xmath64 is equivalent to the euler - maruyama discretization of the rouse model given as equation @xmath65. lemma [lemconsnum] shows that the multi - resolution bead - spring model is a generalization of the rouse model. in the next section, we will study properties of this model which will help us to select the appropriate parameter values for this model and use it in multiscale simulations of dna dynamics. Macroscopic properties and parameterizations of multi-resolution bead-spring models we have formulated a multiscale rouse model which varies the kuhn lengths throughout the filament, but we would like to keep properties of the overall filament constant regardless of the resolution regime being considered for the filament. we consider a global statistic for the system to be _ consistent _ if the expected value of the statistic is invariant to the resolution regime being considered for the filament. we consider the _ self diffusion constant _ and _ root mean squared (rms) end - to - end distance _ as two statistics we wish to be consistent in our system, which can be ensured by varying the bead radius and the number of beads respectively. the precise way to vary these properties will be explored in this section. the _ self diffusion constant _ is defined as @xmath66 where @xmath67 is the _ centre of mass _ of the polymer chain at time @xmath39, which is defined by @xmath68 definition ([eq : com]) is an extension to the definition given by doi and edwards @xcite for the centre of mass of a continuous chain on only one scale. if all beads have the same radius @xmath69 (i.e. if @xmath59 for @xmath70), then equation ([eq : com]) simplifies to the centre of mass definition for the classical rouse model. in this case, the self diffusion constant is given by @xcite @xmath71 where @xmath1 is the number of beads. this result explains the, on the face of it, counterintuitive scaling of equation ([eq : com]) with @xmath2. if we suppose that each bead had the same density, then the mass of each bead would be proportional to its volume, i.e. to @xmath72. however, in definition ([eq : com]), we have used weights @xmath2 instead of @xmath73 because beads do not represent physical bead objects like nucleosomes, but representations of the filament around it, so the bead radius scales with the amount of surrounding filament, which is linear in bead radius in this formulation. if we consider dna applications, we could imagine each bead as a tracker for individual base pairs at intervals of, say, thousands of base pairs away from each other along the dna filament. the filament in the model is then drawn between adjacent beads. this linear scaling with @xmath2 can also be confirmed using equation ([eq : rousediff]) for the classical rouse model. if we describe the same polymer using a more detailed model consisting of twice as many beads (i.e. if we change @xmath1 to @xmath74), then we have to halve the bead radius (i.e. change @xmath69 to @xmath75) to get a polymer model with the same diffusion constant ([eq : rousediff]). in particular, the mass of a bead scales with @xmath69 (and not with @xmath76). in the next lemma, we extend result ([eq : rousediff]) to a general multi - resolution bead - spring model. [lemdg] let us consider a multi - resolution bead - spring polymer of size @xmath1 and a set of timesteps associated with each spring satisfying the assumptions of definitions @xmath77 and @xmath78. then the self diffusion constant of the polymer evolution described by algorithm @xmath64 is given by @xmath79 algorithm @xmath64 describes one iteration of our numerical scheme. multiplying the steps corresponding to the @xmath10-th bead by @xmath2 and summing over all beads, we obtain how @xmath80 changes during one timestep @xmath36. since @xmath9, tension terms cancel after summation and the evolution rule for @xmath80 simplifies to @xmath81 where @xmath82 $] and function @xmath83 is defined for positive integers @xmath84 and @xmath85 by @xmath86 let us denote by @xmath87 the least common multiple of @xmath88 every bead is updated in algorithm @xmath64 at integer multiples of @xmath89. we can eliminate function @xmath90 from equation ([rgequation]) if we consider the evolution of @xmath91 when time @xmath39 is evaluated at integer multiples of @xmath89. we obtain the evolution rule @xmath92 for @xmath93 $], where we used the fact that the sum of normally distributed random variables is again normally distributed. dividing equation ([rgequation2]) by @xmath94, we obtain @xmath95.\]] using definition ([dgdef]), we obtain ([eq : sdc]). the formula ([eq : sdc]) is a generalization of equation ([eq : rousediff]) obtained for the rouse model. it is invariant to the resolutions provided that the mass of the filament @xmath94 remains constant through selection of the number of beads and bead radius, therefore the self diffusion constant is consistent. we define the _ end - to - end vector _ @xmath96 from one end of the filament to the other @xcite. an important statistic to consider related to this is the _ root mean squared (rms) end - to - end distance _ of the filament @xmath97. the expected value of the long - time limit of the rms end - to - end distance, denoted @xmath98, for the classical rouse model is given by @xcite @xmath99 we generalize this result in the following lemma. [lemrms] let us consider a multi - resolution bead - spring polymer of size @xmath1 satisfying the assumptions of definition @xmath77. then @xmath100 and the long - time limit of the rms end - to - end distance is given by @xmath101 equations ([sdedef]) describe a system of @xmath102 linear sdes. however, the sdes corresponding to different spatial dimensions are not coupled. we therefore restrict our investigation to the behaviour of the first coordinates of each vector in ([rmsbond]). let us arrange the differences of the first coordinates of subsequent beads into the @xmath103-dimensional vector @xmath104.\]] then sdes ([sdedef]) can be rewritten to the system of sdes for @xmath105 in the matrix form @xmath106 where @xmath107 is a three - diagonal matrix given by @xmath108 @xmath109 is a two - diagonal matrix given by @xmath110 and @xmath111 is @xmath1-dimensional noise vector @xmath112^t. $] the stationary covariance matrix, defined by @xmath113 is the solution of lyapunov equation @xcite @xmath114. it can be easily verified that the unique solution of this equation is diagonal matrix @xmath115 with diagonal elements @xmath116 @xmath117 multiplying this result by 3 (the number of coordinates), we obtain ([rmsbond]). the end - to - end distance can be rewritten as @xmath118 substituting into ([eqrmsend]), using ([rmsbond]) and the fact that the stationary covariance matrix @xmath119 is diagonal, we obtain ([eqrmsend]). lemmas [lemdg] and [lemrms] describe theoretical results which have been derived under slightly different assumptions. lemma [lemdg] is formulated as a property of algorithm [algoneiter], but the same result, equation ([eq : sdc]), also holds when we calculate the self - diffusion coefficient of the sde formulation of the multi - resolution bead - spring model ([sdedef]). algorithm [algoneiter] is designed in such a way that all force terms corresponding to springs cancel when the evolution equation for @xmath120 is derived (see equation ([rgequation])). in particular, lemma [lemdg] holds for any choices of the lengths of timesteps associated with different springs. on the other hand, lemma [lemrms] describes the property of the sde formulation ([sdedef]). if we use a discretized version of ([sdedef]), then we introduce a discretization error. this error can be made smaller by choosing smaller timesteps. in this section, we show that the smallest timesteps are only required in the regions with the highest spatial resolution. we define a family of optimal multi - resolution (omr) models designed to have macroscopic properties invariant to resolution regime. [defomr] let us consider a bead - spring polymer consisting of @xmath1 beads of radius @xmath69 connected by @xmath103 springs with the kuhn length @xmath121 and simulated by @xmath65 with timestep @xmath36. let us divide the polymer into @xmath122 regions containing @xmath123, @xmath124, consecutive springs, i.e. @xmath125 the first region contains springs indexed by @xmath126 and the @xmath84-th region, @xmath127, contains springs indexed by @xmath128. let us associate with each region an integer resolution @xmath129, where @xmath130 or @xmath131, for @xmath127, and @xmath132 for @xmath133, with at least one region in resolution @xmath134 which is the region with the finest detail. larger values of @xmath135 represent coarser representations of the filament. we define the omr model as the multi - resolution bead - spring model which consists of @xmath136 regions of consecutive beads and springs. in the @xmath84-th region, we have @xmath137 springs with kuhn length @xmath138 and associated time steps @xmath139 given by @xmath140 where @xmath141 is the radius of beads which are connected to two springs which have the same kuhn length @xmath138. we assume that the bead radius of beads on region boundaries sharing springs with kuhn lengths @xmath142 and @xmath138 is @xmath143, for @xmath144 moreover, we assume that the bead radius of the first and last bead of the polymer chain are equal to @xmath145 and @xmath146, respectively. substituting scalings ([sjscalings]) into ([eq : com]), and using ([nm1assum]), we obtain that the omr model satisfies @xmath147 substituting into ([eq : sdc]), we deduce that the omr model has the same self - diffusion constant as the original detailed model (given by ([eq : rousediff])). considering the limit @xmath148, we can use lemma [lemrms] and scalings ([sjscalings]) to derive the expected rms end - to - end distance for the filament : @xmath149 which is again independent of the choice of resolutions @xmath129, @xmath124. as the kuhn length and bead radius vary across resolutions, it is important to consider the numerical stability of the model @xcite. we choose timesteps to be sufficiently small so that solutions do not grow exponentially large. in discretized equations of algorithm [algoneiter], drift terms appear in the form @xmath150, which is proportional in the @xmath84-th region of the omr model to @xmath151 using scalings ([sjscalings]) and assuming that @xmath152 is of the same order as the kuhn length @xmath138, we obtain that the size of ([driftscal]) scales with @xmath129. assuming that @xmath36 is chosen in the original fine scale model so that @xmath153 is small compared to the kuhn length @xmath121, then the drift term of the omr model, given by ([driftscal]) is also small compared to @xmath154 the characteristic lengthscale of the omr model in the @xmath84-th region, @xmath155 next, we compare the number of calculations made by the original detailed single - scale rouse model with the omr model. the @xmath84-th region has @xmath137 springs simulated with timestep @xmath139. using scalings ([sjscalings]), we obtain that we use @xmath156-times fewer calculations in the @xmath84-th region by advancing fewer beads over larger timesteps. assuming that the computational intensity of the simulation of the detailed model in each region is proportional to the size of the region, @xmath157, we can quantify the fraction of computational time which is spent by the omr model (as compared to the detailed model) by @xmath158 for example, if we coarse - grained the detailed model everywhere using the integer resolution @xmath159, then ([nm1assum]) and ([fraccost]) implies that we speed up our simulations by the factor of 64. in this section we show that simulations of the omr method match the original single - scale rouse model. we also compare this to analytic results predicted from equations ([eq : sdc]) and ([eq : rmse2e]) for the rms end - to - end distance and the self diffusion constant of a filament in an equilibrium state. for the detailed model, we choose the parameters : @xmath160 where the kuhn length is chosen to be longer than the persistence length of dna @xcite, and the other parameters are chosen arbitrarily. for the remainder of this paper, we shall use @xmath161, @xmath162 and @xmath163 the viscosity of water. we compare two resolution regimes for the same system, with the single scale model considering the full system in high resolution and a multiscale model considering the middle @xmath164 of the filament in high resolution and the remainder in low resolution. the corresponding parameters of the omr model are given in table [tab : consistencyparams]. the omr model contains 69 beads connected by 68 springs, while the original detailed model is given by 501 beads connected by 500 springs.._parameters of the omr model system used to demonstrate consistency. _ [tab : consistencyparams] [cols="^,^,^,^",options="header ",] at different initial starting distances the model runs until the protein is either bound or has escaped the filament. in figure [fig : transcriptionresults], we present the probability of binding of the protein to the filament, @xmath165, as a function of the initial distance @xmath166 $] of the protein from the middle bead of the filament. we estimate @xmath165 as a fraction of simulations which end up with the protein bound to dna. each data point in figure [fig : transcriptionresults] represents the value of @xmath165 estimated from @xmath167 independent realizations of the process. if @xmath168, then the protein is immediately bound to dna, i.e. @xmath169 for @xmath168. if @xmath170, then the probability of binding is nonzero, because the initial placement, @xmath171, is the distance of the protein from the centre of the filament. in particular, the minimum distance from protein to filament is less than or equal to the initial placement distance, @xmath171, and the simulations (with the possibility of binding) take place even if @xmath170., depending on starting distance, @xmath171, from the filament for the single - scale (black points) and omr (blue line) models. error bars give a 95% confidence interval based on the wilson score interval for binomial distributions @xcite. _ [fig : transcriptionresults]] due to computational constraints of the single - scale model we consider a selection of initial distances at points @xmath172 m, @xmath173 (black points), where error bars give a 95% confidence interval based on the wilson score interval for binomial distributions @xcite. we run simulations for more initial distances, @xmath174 m, @xmath175 (blue line), using the computationally efficient omr model and present our results as the blue line in figure [fig : transcriptionresults]. we see that @xmath165 is very similar between the single - scale and omr models. the model also succeeds in reducing computational time. for @xmath167 simulations with the protein starting @xmath176 from the middle bead, with parameters given in table [tab : transcriptionparmas], the omr model represented a 3.2-times speedup compared to the detailed model, with only a 3-times resolution difference. we expect for larger resolution differences to see greater improvements in speed. Discussion in this paper we have extended basic filament modelling techniques to multiple scales by developing omr methods. we have presented an mcmc approach for increasing the resolution along a static filament segment, as well as an extension to the rouse model to dynamically model a filament which considers multiple scales. the bead radius, as well as the number of beads associated with each resolution, is altered to maintain consistency with the end - to - end distance and diffusion of a filament across multiple scales, as well as the timestep to ensure numerical convergence. we have then illustrated the omr methodology using a simple model of protein binding to a dna filament, in which the omr model gave similar results to the single - scale model. we have also observed a 3.2-times speed - up in computational time on a model which considers only a 3-times increase in resolution, which illustrates the use of the omr approach as a method to speed up simulations whilst maintaining the same degree of accuracy as the more computationally intensive single - scale model. the speed - up in computational time could be further increased by replacing brownian dynamics based on time - discretization ([eq : particlediffuse]) by event - based algorithms such as the fpkmc (first passage kinetic monte carlo) and gfrd (green s function reaction dynamics) methods @xcite. when considering the zooming out of the dna binding model, note that it is generally possible to zoom in and out repetitively, as long as the dynamics are such that we can generate a high resolution structure independent from the previous one (i.e., once we zoom out, the microscopic structure is completely forgotten). however, particularly in the case of chromatin, histone modification and some dna - binding proteins may act as long - term memory at a microscopic scale below the scales currently considered. to reflect the effect of the memory, some properties of the microscopic structure should be maintained even after zooming out. fractal dimension may serve as a candidate of indices @xcite, which can be also estimated in living cells by single - molecule tracking experiments @xcite. the omr method could be applied to modern simulations of dna and other biological polymers which use the rouse model @xcite in situations where certain regions of the polymer require higher resolutions than other regions. the model considered in this report uses rouse dynamics, which is moderately accurate given its simplicity, but as we zoom in further towards a binding site, then we will need to start to consider hydrodynamic forces and excluded volume effects acting between beads. models which include hydrodynamic interactions such as the zimm model @xcite have previously been used to look at filament dynamics @xcite. therefore it is of interest to have a hybrid model which uses the rouse model in low resolutions and the zimm model in high resolutions. the combination of different dynamical models might give interesting results regarding hierarchical structures forming as we move between resolutions. as we go into higher resolutions, strands of dna can be modelled as smooth @xcite, unlike the fjc model where angles between beads are unconstrained. the wormlike chain model of kratky and porod @xcite, implemented via algorithm by hagermann and zimm @xcite, gives a non - uniform probability distribution for the angles between each bead. allison @xcite then implements the zimm model dynamics on top of the static formulation to give bending as well as stretching forces. another interesting open multiscale problem is to implement this at higher resolutions, with the rouse model at lower resolutions, in order to design a hybrid model. to introduce even more realism, we would see individual histones and consider forces between these as in the model of rosa and everaers @xcite which includes lennard - jones and fene forces between beads. as we approach an atomistic level, it may be interesting to consider a molecular dynamics approach to modelling the dna filament. coarser brownian dynamics models can be estimated from molecular dynamics models either analytically @xcite or numerically @xcite, depending on the complexity of the molecular dynamics model. a variety of structure - based coarse - grained models have been used for chromatin (e.g. @xcite), also with transcription factors @xcite. multiscale modelling techniques (e.g. @xcite with iterative coarse - graining), as well as adaptive resolution models (e.g. @xcite for solvent molecules), have been developed. we expect these studies will connect with polymer - like models at a certain appropriate length and time scale. on top of this, models for the target searching process by proteins such as transcription factors could be improved (for example, by incorporating facilitated diffusion under crowded environment @xcite). the need for developing and analyzing multiscale models of dna which use one of the above detailed simulation approaches for small parts of the dna filament is further stimulated by recent experimental results. chromosome conformation capture (3c)-related techniques, particularly at a genome - wide level using high - throughput sequencing (hi - c @xcite), provide the three - dimensional structure of the chromosomes in an averaged manner. moreover, recent imaging techniques have enabled us to observe simultaneously the motion and transcription of designated gene loci in living cells @xcite. simulated processes could be compared with such experimental results. recent hi - c experiments also revealed fine structures such as loops induced by dna - binding proteins @xcite. to develop more realistic models, information about the binding sites for these proteins may be utilized when we increase the resolution in our scheme. s. shinkai, t. nozaki, k. maeshima, and y. togashi. dynamic nucleosome movement provides structural information of topological chromatin domains in living human cells. biorxiv doi:10.1101/059147, 2016.

a multi - resolution bead - spring model for polymer dynamics is developed as a generalization of the rouse model. a polymer chain is described using beads of variable sizes connected by springs with variable spring constants. a numerical scheme which can use different timesteps to advance the positions of different beads is presented and analyzed. the position of a particular bead is only updated at integer multiples of the timesteps associated with its connecting springs. this approach extends the rouse model to a multiscale model on both spatial and temporal scales, allowing simulations of localized regions of a polymer chain with high spatial and temporal resolution, while using a coarser modelling approach to describe the rest of the polymer chain. a method for changing the model resolution on - the - fly is developed using the metropolis - hastings algorithm. it is shown that this approach maintains key statistics of the end - to - end distance and diffusion of the polymer filament and makes computational savings when applied to a model for the binding of a protein to the dna filament. polymer dynamics, dna, rouse model, brownian dynamics, multiscale modelling 60h10, 60j70, 82c31, 82d60, 92b99

1607.08062

Introduction cataclysmic variables (cvs) are short - period binaries containing a white dwarf (wd) primary (with mass @xmath2) and a low mass main sequence secondary (with mass @xmath3). the secondary fills its roche lobe and transfers mass to the wd through the inner lagrangian (@xmath4) point. the main features of the orbital period distribution of cvs with hydrogen rich donors are the lack of systems in the 2 - 3 hr period range (the so - called period gap) and the sharp cut off of the distribution at around 77 minutes, as can be seen in figure [combined] (upper frame ; e.g. ritter & kolb 1998). so far theoretical models have been unable to reproduce the precise position of the observed short - period cut - off and observed shape of the cv orbital period distribution near this cut - off. this is summarised in figure [combined]. systems that evolve under the influence of gravitational radiation (gr ; kraft et al. 1962) as the only sink of orbital angular momentum (am) reach a minimum period at @xmath5 minutes (figure[combined], middle frame ; paczyski 1971 ; kolb & baraffe 1999).the probability of finding a system within a given period range is proportional to the time taken to evolve through this region. we thus have n(p), for the number @xmath6 of systems found within a given orbital period range around @xmath7, and @xmath8 is the secular period derivative at this period. we thus expect an accumulation of systems (a spike) at @xmath9 where @xmath10 (figure [combined], lower frame), while no such spike is present in the observed distribution (figure[combined], upper frame). the orbital period evolution reflects the radius evolution of the mass donor, which in turn is governed by two competing effects. mass transfer perturbs thermal equilibrium and expands the star. thermal relaxation reestablishes thermal equilibrium and contracts the star back to its equilibrium radius. the minimum period occurs where the two corresponding time scales, the mass transfer time @xmath11 and the thermal (kelvin - helmholtz) time @xmath12 are about equal (e.g. paczyski 1971 ; king 1988). if @xmath13 then the star is able to contract in response to mass loss, but if @xmath14 the star will not shrink rapidly enough and will become oversized for its mass. the position of the minimum period is therefore affected by the assumed mass transfer rate, and in particular by the assumed rate of orbital angular momentum (am) losses. in this paper we investigate ways to increase the period minimum by increasing the mass transfer rate, and investigate ways to `` hide '' the spike by introducing a spread of @xmath9 values in the cv population. in particular, we study the effect of a form of consequential am loss (caml) where the am is lost as a consequence of the mass transferred from the secondary, i.e. @xmath15 (see e.g. webbink 1985). in section [theory] we outline our general model assumptions and introduce the prescription for caml. in section [sec22] we present detailed calculations of the long - term evolution of cvs, and in section [comptest] we compare the observed short period cv period distribution with various theoretically synthesized model distributions based on the calculations in section 2. Theoretical versus observed minimum period in this section we investigate possible solutions to the mismatch between the theoretical and observed minimum orbital period in cvs. the orbital am loss rate @xmath16 of a cv can be written as the sum of two terms, = _ sys+_caml, where @xmath17 denotes the `` systemic '' am loss rate, such as gravitational wave radiation, that is independent of mass transfer, while @xmath18 is an explicit function of the mass transfer rate. we have = 0 and _ caml0_20 we consider the general case in which the caml mechanism, along with nova mass ejections, causes a fraction of the transferred mass to leave the system. this fraction may be greater than unity as the primary may lose more mass during a nova outburst than was accreted since the last outburst. we employ a generic prescription of the effect of a caml mechanism, thus avoiding the need to specify its physical nature. possible caml mechanisms include a magnetic propeller, i.e. a system containing a rapidly spinning magnetic wd where some of the transferred material gains angular momentum from the wd spin by interaction with the wd s magnetic field (see e.g. wynn, king & horne 1997), and an accretion disc wind (see e.g. livio & pringle 1994). our caml prescription largely follows the notation of king & kolb (1995). the am is assumed to be lost via mass loss that is axis - symmetrical with respect to an axis a fixed at the wd centre but perpendicular to the orbital plane. we define @xmath19 as the total fraction of mass lost from the secondary that leaves the system. we assume further that a fraction @xmath20 (@xmath21) of the transferred mass leaves the system with some fraction @xmath22 of the angular momentum it had on leaving the @xmath4 point. we also consider mass that is lost from the system via nova mass ejections, which over the long term can be considered as an isotropic wind from the primary (see e.g. kolb et al. this material will carry away the specific orbital angular momentum of the primary and will account for the fraction (@xmath23) of the mass loss. we thus obtain _ caml = b^2_2 +, where we define @xmath24 as the caml efficiency. for comparison with king & kolb (1995) we equate this to [eq : jdotcaml] _ caml = j, > 0, and obtain [eq : nufinal] = (1+q)()^2 +. for our calculations shown below we use the approximation 1-+-,^3=. this is an adaptation of the expression given in kopal (1959) and is accurate to within 1% over the range @xmath25. in this subsection we present calculations of the long - term evolution of cvs as they approach and evolve beyond the period minimum. for the computations we used the stellar code by mazzitelli (1989), adapted to cvs by kolb & ritter (1992). some of these evolutionary sequences are the basis for the theoretical cv period distributions we present in section [comptest] below. we calculated the evolution of individual systems that are subject to caml according to equations [eq : jdotcaml] and [eq : nufinal]. we chose @xmath26 and initial donor mass @xmath27, with a range of caml efficiencies @xmath28 as shown in figure [fig : fullcaml]. the systems initially evolve from longer periods towards the period bounce (right to left) at almost constant mass transfer rate. the minimum period increases with increasing caml efficiency to a maximum of around 70 min for @xmath29. mass transfer stability sets an upper limit on the caml efficiency. an obvious upper limit is 1, where all the angular momentum of the transferred material is ejected from the system. although the ejected material may carry more angular momentum than was transferred (as in the case of a propeller system where additional angular momentum is taken from the spin of the wd) this does not affect the net loss of orbital angular momentum. the maximum caml efficiency still compatible with mass transfer stability could be smaller than unity. the stability parameter @xmath30 which enters the expression for steady - state mass transfer, equation [eq : stab] (e.g. king & kolb 1995) must be greater than zero ; this defines an upper limit on @xmath31. [eq : stab] -_2=m_2 () a plot of @xmath30 against @xmath32 for an initially marginally stable system (@xmath33, @xmath34 and @xmath35) is given in figure [fig : dq]. the system initially exhibits cycles of high mass transfer rate @xmath36 (@xmath30 close to 0) and very low mass transfer rate @xmath37. the high states are short lived, on the order of @xmath38 years (see figure [fig : highmdot]). the system finally stabilizes with @xmath39. at around @xmath40 @xmath30 starts to decrease further but always remains positive, settling at a value around @xmath41. the tidal deformation of the secondary may have an effect on the period minimum. calculations by renvoiz, baraffe, kolb & ritter (2002), [see also kolb 2002] using 3dimensional sph models suggest that the secondary is deformed in the non - spherical roche lobe such that its volume equivalent radius is around 1.06 times that of the same star in isolation. we mimic this effect in our 1-dimensional stellar structure code by multiplying the calculated radius by a deformation factor @xmath42 before the mass transfer rate is determined from the difference between the radius and the roche lobe radius via -_2=_0(-). here @xmath43 is the mass transfer rate of a binary in which the secondary just fills its roche potential and @xmath44 is the photospheric pressure scale height of the secondary (see e.g. ritter 1988). figure [fig : barraffe] shows the effect on the minimum period and mass transfer rate for systems with various deformation factors @xmath42, ranging from 1 (no deformation) to 1.24. the mass transfer rate is seen to decrease with increasing deformation. this can be understood from the functional dependence on orbital period and donor mass in the usual quadrupole formula for the am loss rate due to gravitational radiation (see e.g. landau & lifschitz 1958). although the quadrupole formula is strictly valid only if both components are point masses, rezzolla, ury & yoshida (2001) found that the gr rate obtained using a full 3-dimensional representation of the donor star differs from the point mass approximation by less than a few percent. it can be seen from the figure that with the deformation factor 1.06 the minimum period increases from around 65 min to around 69 min, consistent with renvoiz et al (2002) for geometrical effects alone. a deformation factor of around 1.18 was required to raise the minimum period to the observed value of @xmath45 min. this is somewhat larger than the intuitive expectation () ^=()^=1.12 from kepler s law and roche geometry. in our calculations we consider the simple case in which only the geometrical deformation effects are taken into account. the inclusion of the thermal effects considered by renvoiz et al (2002) have the likely effect of reducing @xmath9, possibly by around 2% compared to the case with purely geometrical effects one possible physical mechanism that could cause a deformation factor above the value of 1.06 is magnetic pressure inside the star, as suggested by dantona (2000). we note that patterson (2000) claims to find observational evidence for `` bloated '' secondaries in short period cvs. on the basis of donor mass estimates from the observed superhump excess period he finds that the donors have @xmath46 larger radii than predicted from 1 dimensional., non deformed stellar models if gravitational radiation is the only am sink. even if true, this observation can not distinguish between an intrinsic deformation of the donor star or the non - equilibrium caused by orbital am losses in excess of the gr rate. Parent distributions versus observations to test the statistical significance of the theoretically predicted accumulation of systems near the period minimum (`` period spike '') we calculated the period distributions of model populations for various assumptions about evolutionary parameters. for each parameter a series of evolutionary tracks were generated, typically around 20. as systems evolve after the minimum period a point is reached (typically when @xmath47 falls below @xmath48) where numerical fluctuations in @xmath47 become so large that the henyey scheme no longer converges. the stellar code uses tables to interpolate / extrapolate the opacities and equation of state for each iteration, and in this region the extrapolations become very uncertain. to extend the tracks we used a semi - analytical method as follows. the tracks were terminated at a value of @xmath49, where @xmath50 is the mass transfer rate at the minimum period for the track. the radius of the star for the final part of the track is approximated by r_2=r_0m_2^, where @xmath51 and @xmath52 are assumed to be constant. the values of @xmath51 and @xmath52 were determined from the final few data points for each track. (@xmath52 takes a typical value of around 0.15 for systems beyond the period bounce.) to generate the extension to the track we then calculated @xmath7 from the roche lobe condition, and @xmath53 by assuming stationarity as in section [minpnumeric] (see figure [combined], middle frame for an example of an extended track). we weight the chances of observation to the brighter systems by assuming , 1.0. for the detection probability. we tested the calculated model parent distributions for various values of the free parameter @xmath54 against the observed cv period distribution. a k - s (kolmogorov - smirnov) test is insensitive to the differences between the parent distributions. the greatest difference in the cumulative distribution functions (cdfs) of the observed and modelled distributions occur at the boundaries of the cdfs, i.e. in the least sensitive region for the k - s test (press et al 1992). we thus decided to use the following modified @xmath55 test. for each parent distribution 10000 model samples each containing 134 systems were generated. ; ritter & kolb 1998, internal update june 2001, as of july 2002 the number of systems in this period range is now 152 though this does alter the values given by the @xmath55 test, the trends and hence the results remain unaltered] each sample was tested against the model parent distribution using a @xmath55 test, with 1, 2 and 4 minute bins. this range bridges the need for good resolution and significance of the @xmath55 test which requires a minimum number of cvs per bin. the observed period distribution was tested against the model parent distribution also, giving the reduced @xmath55 value @xmath56. the fraction @xmath22 of generated samples with a reduced @xmath55 value less than @xmath56 was used as a measure of the significance level of rejecting the hypothesis that the observed distribution is drawn from the parent distribution. in the following we quote the rejection probability pr=@xmath22. kolb & baraffe (1999) noted that the observed distribution of non - magnetic cvs (figure [fig : mnmcvs], middle frame), and the observed distribution of magnetic cvs (figure [fig : mnmcvs], lower frame) show no significant difference below the period gap. to test and quantify this we compared these distributions for @xmath57 min, giving a reduced @xmath55 probability of 0.1213. hence we can not rule out that the distributions are drawn from the same underlying parent distribution. this is borne out by the results of comparing both distributions with a parent distribution that is flat in @xmath7 (see also table [tab : tabcomb], entries f and g) which give similar rejection probabilities (pr=0.709 and pr=0.781, respectively). we thus find no significant difference between the two distributions. in the following we therefore test models against the combined magnetic and non - magnetic distribution of observed systems. the lack of any distinct features in the combined observed period distribution (figure [fig : mnmcvs], upper frame) does indeed suggest an essentially flat distribution for the underlying parent distribution. the flat distribution gives pr = 0.552 (for the 1 minute bin width, see table [tab : tabcomb]). we use this value as a benchmark for the models discussed below. [cols="^,^,^,^,^,^,^ ",] king, schenker & hameury (2002) constructed a (nearly) flat period distribution by superimposing individual idealized pdfs with different bounce periods @xmath58 according to a suitably tailored weighting. for the double box - shaped idealised pdfs modelled on the pdf shown in our figure [combined] (lower frame) the required weighting is @xmath59 $] (@xmath60 is the observed minimum period). this weighting function effectively mirrors the shape of the sharply peaked individual pdfs. king et al. (2002) found that the range @xmath61 is sufficient to wash out the period spike. it is clear that this procedure involves a certain degree of fine - tuning for @xmath62 if the shape of the input pdf is given. such a fine - tuning must surprise as the two functions involved presumably represent two very different physical effects. we applied the weighting @xmath62 quoted in king et al. (2002) to our non - idealized model pdfs that involve the caml efficiency and the deformation factor as a means to vary @xmath58. the weighting produced a marginally worse fit (@xmath63 versus @xmath64 ; 1 minute binning) for the caml pdfs compared to the parent population based on a flat caml efficiency spectrum we discussed earlier. in part this is due to the fact that the upper limit on @xmath31 does not allow a big enough range of @xmath58. in the case of the deformation factor pdfs the fit marginally improved (@xmath65 versus @xmath66 ; 1 minute binning, @xmath67). it is possible to optimise the fit by adding systems with deformation factors up to 1.42, and by using the weighting @xmath68 $], but this still gives the fairly large value @xmath69 (see also figure [fig : kings]). however, such a parent population is inconsistent with the observed distribution for longer periods. as can be seen from figure [fig : barraffe] systems that are subject to larger deformation factors would evolve into the period gap, hence the gap would be populated in this model. for completeness we show in figure [fig : grsum1] the result of the superposition suggested by king, schenker & hameury (2002) if realistic rather than idealised pdfs are used. this model assumes additional systemic am losses (@xmath70 ; no caml, no deformation factor, @xmath71) as the control parameter for varying @xmath58, and the weighting as in king et al. the pronounced feature just above 2 hrs orbital period is the result of the adiabatic reaction of the donor stars at turn - on of mass transfer (see e.g. ritter & kolb 1992). such a feature is absent in the observed distribution. if deformation effects are taken into account the additional am losses required to wash out the @xmath9 spike would cover a similar range but at a smaller magnitude. the resulting period distribution would be similar to the one shown in figure [fig : grsum1] Discussion we have investigated mechanisms that could increase the bounce period for cvs from the canonical theoretical value @xmath0 min to the observed value @xmath72 min, and ways to wash out the theoretically predicted accumulation of systems near the minimum period (the period spike). unlike king, schenker & hameury 2002 we focussed on effects other than increased systemic angular momentum (am) losses, i.e. we assume that gravitational radiation is the only systemic sink of orbital am. we find that even a maximal efficient consequential am loss (caml) mechanism can not increase the bounce period sufficiently. as the real cv population is likely to comprise systems with a range of caml efficiencies we would in any case expect to have a distribution of systems down to @xmath0 min, rather than the observed sharp cut - off. we considered donor stars that are `` bloated '' due to intrinsic effects, such as the tidal deformation found in 3-dim. sph simulations of roche - lobe filling stars. an implausibly large deformation factor of around 1.18 is needed to obtain a bounce period of @xmath45 min. a possible alternative identification of @xmath9 as an age limit rather than a period bounce (king & schenker 2002) would limit the donor mass in any cv in a cv population dominated by hydrogen rich, unevolved systems to @xmath73. any system with donor mass much less than this would either have an orbital period less than 78 minutes or would have already evolved beyond the period minimum. there are indeed systems with suspected @xmath74 ; good candidates are wz sge (@xmath75 ; patterson et al 1998) and oy car (@xmath76 ; pratt et al. 1999). it is also possible that systems die or fade before reaching the period bounce, and hence become undetectable as cvs. the fact that the very different groups of non - magnetic and magnetic cvs show almost identical values of @xmath9 (see figure [fig : mnmcvs]) strongly suggests that the physical cause for the potential fading would have to be rooted in the donor stars or the evolution rather than the accretion physics or emission properties of the systems. even if the bounce period problem is ignored we find in all synthesized model populations (except for the age limit model) a pronounced remaining feature due to the accumulation of systems near the bounce. we employ a modified @xmath55 test to measure the `` goodness '' of fit against the observed sample. an f - test (press et al 1992) was also applied to the majority of @xmath77 models and the same general trends observed. none of our synthesised model populations fits as well as the distribution which is simply flat in orbital period (rejection probability @xmath78). only models where brighter systems carry a far greater weight than expected in a simple magnitude limited sample (selection factor @xmath79 with @xmath80 rather than @xmath81) achieve similar values for @xmath82. however, most of our models with @xmath83 canot be rejected unambiguously on the basis of this test. models designed to `` wash out '' the period spike by introducing a large spread of the caml efficiency do generally better than population models based on donor stars that are subject to a large spread of intrinsic deformation factors. for all models the rejection probability decreases if the full wd mass spectrum is taken into account, as this introduces an additional spread in the bounce period. model populations where all cvs form at long orbital periods (chiefly above the period gap) give a much better fit than models that include newborn cvs with small donor mass. adding these systems to the population introduces a general increase of the orbital period distribution towards short periods, thus making the period spike more pronounced. this suggests that most cvs must have formed at long periods and evolved through the period gap to become short - period cvs. this is consistent with independent evidence that cv secondary stars are somewhat evolved (baraffe & kolb 2000 ; schenker et al. 2002 ; thorstensen et al 2002). recently, king, schenker & hameury (2002) constructed a flat orbital period distribution by superimposing idealised pdfs that describe subpopulations of cvs with a fixed initial donor mass and initial wd mass, but different bounce periods. this superposition required a strongly declining number of systems with increasing bounce periods. we repeated this experiment with a realistic pdf, but failed to obtain a markedly improved fit. in conclusion, we find that the period minimum problem and the period spike problem remain an open issue. it is possible to construct cv model populations where the period spike is washed out sufficiently so that it can not be ruled out unambiguously on the basis of an objective statisticial test against the observed cv period distribution. Acknowledgements we thank graham wynn, andrew king and isabelle baraffe for useful discussions. andrew conway and chris jones who gave advice on the statistical analysis. we also thank andrew norton for a critical reading of the paper and the referee jean - marie hameury for useful comments.

we investigate if consequential angular momentum losses (caml) or an intrinsic deformation of the donor star in cvs could increase the cv bounce period from the canonical theoretical value @xmath0 min to the observed value @xmath1 min, and if a variation of these effects in a cv population could wash out the theoretically predicted accumulation of systems near the minimum period (the period spike). we are able to construct suitably mixed cv model populations that a statisticial test can not rule out as the parent population of the observed cv sample. however, the goodness of fit is never convincing, and always slightly worse than for a simple, flat period distribution. generally, the goodness of fit is much improved if all cvs are assumed to form at long orbital periods. the weighting suggested by king, schenker & hameury (2002) does not constitute an improvment if a realistically shaped input period distribution is used. binaries : close stars : evolution stars : mass - loss novae, cataclysmic variables.

astro-ph0212125

I. introduction having been discovered more than a century ago @xcite, whistler waves become one of the most important waves in plasmas. such waves (also known as helicons in solid state plasmas) are low - frequency (lf) (in comparison with the electron - cyclotron frequency, @xmath0) right - hand circularly polarized (rcp) electromagnetic (em) waves guided almost along the external magnetic field in dense plasmas. because of the increase of their group velocity with the frequency, @xmath1 (see, e.g. ref. @xcite), the lf waves arrive later giving rise a whistling down - effect observed at ground level. stenzel in his classic paper stenzel demonstrated experimentally the creation of magnetic field - aligned density perturbations excited by the ponderomotive force exerted by the em whistlers. whistler waves are important not only in space plasmas due to wave - particle interactions, but also in laboratory plasmas as helicons for efficient plasma production as well as in dense astrophysical environments whistlertheory1,whistlertheory2,mi, whistlertheory3(nearsonic),whistlertheory4,whistlertheory5,whistlerparametricinstability. on the other hand, large amplitude whistlers propagating in a magnetized plasma can initiate a great variety of nonlinear effects, e.g., three - wave interactions, parametric instabilities @xcite, modulational instability and the subsequent soliton formation whistlertheory1,whistlertheory2,mi. the latter which, in turn, causes local electron density enhancement or depletion in plasmas, are considered as a basis for understanding laser energy deposition in pellets @xcite, pulsar radiation interaction with the ambient magnetosphere ambientmagnetosphere, whistler wave propagation in solar winds solarwind etc. recent laboratory experiment @xcite and observations from the freja satellite @xcite show the clear evidence for the formation of whistler envelope solitons accompanied by plasma density cavities. moreover, electrons in van allen radiation belts can be accelerated to mev energies within a short period by large amplitude whistlers @xcite. the latter have recently been observed by the cluster spacecraft @xcite, the stereos cattell and the themis @xcite. furthermore, laboratory experiments @xcite and theoretical confirmation @xcite have demonstrated the existence of propagating whistler spheromaks with fields exceeding the ambient magnetic field. whistlers also contribute to fast magnetic reconnection and plasma dynamics in two - beam laser - solid density plasma interaction experiments @xcite. recently, there has been a notably growing interest in investigating various quantum plasma effects in view of some experimental progresses in nanoscale plasmas @xcite, ultracold plasmas @xcite, spintronics @xcite and plasmonics @xcite. on the other hand, superdense quantum plasmas are omnipresent in compact astrophysical objects, e.g., the interior of massive white dwarfs, interior of jupitors, magnetars etc. @xcite, as well as in the next generation intense laser - solid density plasma interaction experiments @xcite. in dense plasmas, degenerate electrons follow fermi - dirac pressure law, and there are typically quantum force associated with the bohm de broglie potential, which produce wave dispersion at nanoscales quantum1,quantum2,quantum3. furthermore, the effects of the electron spin manifests itself in terms of a magnetic dipole force, as well spin precession, which can be exploited by transforming the pauli equation to fluid - like variables @xcite. more elaborate kinetic models has also been developed @xcite. hence the dynamics of electrons in fermi degenerate plasmas will be affected not only by the lorentz force, but also by the effects of quantum statistical pressure, the bohm force as well as the effects due to intrinsic spin of electrons. we ought to mention that in a dense magnetized plasma there also exist spin waves, which can be excited by intense neutrino fluxes. thus, nonlinear theories of em waves, in particular whistlers in magnetized dense plasmas need to be developed in its own right accounting for all these quantum effects. recently, the theory of the ponderomotive force in plasmas has been extended to account for the contribution from the intrinsic spin of electrons @xcite. it has been demonstrated that an em pulse can induce a spin - polarized plasma by this spin - ponderomotive force (spf). such force could also play an important role in the propagation of lf em waves, e.g., whistlers, alfvn waves. our objective here is to present a theoretical study of modulated whistler wave packets interacting nonlinearly with background lf density perturbations that are reinforced by the classical ponderomotive force (cpf) @xcite as well as the spf @xcite. the role of the ion motion as well as the dispersive effects due to charge separation and the electron tunneling are also taken into account. we will include the field aligned velocity perturbation (free electron streaming) associated with the lf motion, and in addition, generalize the related classical results that exist in the literature (see, e.g., refs. whistlertheory1,whistlertheory2). the obtained results could be useful for understanding the propagation of localized em whistlers which may emanate in the interior of magnetized white dwarfs, magnetars as well as in the next generation intense laser - solid density plasma experiments. Ii. nonlinear evolution equations let us consider the propagation of nonlinearly coupled em whistlers and ion - acoustic (ia) density perturbations along a constant magnetic field @xmath2 in a quantum electron - ion plasma where any equilibrium drift velocity is zero. in the modulational representation, the high - frequency (hf) em wave field for the rcp whistlers is given by @xmath3c.c., where @xmath4 is the slowly varying (both in space and time) envelope of the whistler wave electric field and c.c. stands for the complex conjugate. also, @xmath5 @xmath6 represents the whistler wave frequency (number). the basic equations for the evolution of nonlinear whistlers then read spin1,spinponderomotive, spinevolutioneqn. @xmath7 @xmath8 @xmath9 where @xmath10 @xmath11 @xmath12 denote the number density, mass and velocity of electrons respectively, @xmath13 is the magnetic field and @xmath14 is the electron thermal pressure. also, @xmath15 is the spin angular momentum with its absolute value @xmath16 @xmath17where @xmath18 is the electron @xmath19-factor and @xmath20 is the bohr magneton. the equations ([b1])-([b3]) are then closed by the following maxwell equations with @xmath21.@xmath22@xmath23the equations ([b1])-([b3]) represent the nonrelativistic evolution of spin@xmath24 electrons, and are applicable even when different states with spin - up and spin - down (relative to the magnetic field) can be well represented by a macroscopic average. this may, however, occur in the regimes of very strong magnetic fields (or a very low temperature regimes), where generally the electrons occupy the lowest energy spin states. on the other hand, for a time - scale longer than the spin - flip frequency, the macroscopic spin state is well - described by the thermodynamic equilibrium spin configuration, and in this case the above model can still be applied. however, such case in which the macroscopic spin state will be attenuated by a factor decreasing the effective value of @xmath25 below @xmath26, will not be considered further in the present work. as a consequence, our studies will be focused on the regime of strong magnetic fields and high density plasmas. taking the curl of eq. ([b2]) and using eqs. ([b3])-([b5]) we readily obtain the following evolution equation for whistlers. @xmath27 \notag \\ & -\frac{2\mu } { e\hbar } \frac{\partial } { \partial t}\left [\frac{1}{n_{e}}\nabla \times \left (\nabla \times n_{e}\mathbf{s}\right) \right] + \frac{1}{m_{e}\mu _ { 0}n_{e}}\nabla \times \left [\left (\nabla \times \mathbf{b}\right) \times \mathbf{b}\right] + \frac{2\mu } { m_{e}\hbar } \nabla \times \left (s^{a}\nabla b_{a}\right) \notag \\ & -\frac{\varepsilon _ { 0}}{m_{e}n_{e}}\nabla \times \left (\frac{\partial \mathbf{e}}{\partial t}\times \mathbf{b}\right) -\frac{2\mu } { m_{e}\hbar n_{e}}\nabla \times \left [\left (\nabla \times n_{e}\mathbf{s}\right) \times \mathbf{b}\right]. \label{ev}\end{aligned}\]] in the linear theory, the whistler frequency @xmath5 and the wave number @xmath28 are related by the following linear dispersion relation in the nonrelativistic limit (see for details, ref. @xcite). @xmath29 where @xmath30 is the refractive index, @xmath31 is the frequency due to the plasma magnetization current and @xmath32 is the electron skin depth with @xmath33 denoting the electron (ion) plasma frequency. also, @xmath34is the electron - cyclotron frequency and @xmath35 is the electron spin - precession frequency. the nonlinear dynamics of whistler wave envelopes under the modulation of electron density perturbations associated with the lf ia fluctuations and of the nonlinear frequency - shift caused by the magnetic field aligned free streaming of electrons with flow speed @xmath36 can be described by the following nonlinear schrdinger (nls)-like equation which is obtained from the em wave equation ([ev]) as @xmath37where @xmath38 and the group speed, @xmath39 [see eq.@xmath40(11) in ref. @xcite] and the group dispersion, @xmath41 of whistlers are given by @xmath42 @xmath43.\]] the nonlinear frequency shift @xmath44 is given by@xmath45, \label{3}\]]where @xmath46and @xmath47 is the relative perturbed density. by disregarding the spin contribution one can recover the previous results @xcite. note that the term @xmath48 representing the doppler shift due to the plasma streaming along the external magnetic field, is no longer negligible, but may be comparable to the other nonlinear terms, and can thus change the sign of the nonlinearity as well. more precisely, both @xmath49 and @xmath44 will change their sign depending on the frequency range to be considered as well as the contribution from the spin correction terms. later, we will see that the change of sign is important for the formation of localized wave packets at different whistler frequencies. the quantities @xmath50 and @xmath51 are related to each other by the electron continuity equation. @xmath52 note that the ponderomotive force due to the em whistlers usually drives the lf (compared to the whistler wave frequency @xmath5) density perturbations which propagate along the field lines with low - phase speed (compared to the electron thermal speed). thus, the lf electrostatic modulation also satisfies the electron momentum equation @xmath53where @xmath54 is the lf part of the wave electric field and @xmath55 @xmath56 is the fermi speed relevant for a high density plasma @xcite. here @xmath57and @xmath58 is the boltzmann constant. the term @xmath59 is the quantum correction associated with the bohm de broglie potential. the ponderomotive force contributions are proportional to the constants @xmath60 and @xmath61 where@xmath62 in which the first terms appear due to cpf @xcite and the second ones @xmath63 are due to the spf spinponderomotive. the equations for the cold ion motion involved in the lf ia perturbations are @xmath64@xmath65 @xmath66 eliminating @xmath67 @xmath68 @xmath69 and disregarding the term @xmath70 we obtain from eqs. ([5]), ([7])-([9]) the driven wave equation for lf perturbations of the boussinesq - type as @xmath71where @xmath72and @xmath73 @xmath74 is the ion - acoustic speed and @xmath75 is the fermi screening length for electrostatic oscillations. thus, we have a set of three coupled equations, namely ([2]), ([4]) and ([10]), modified from previous results by the spf and quantum tunneling, which describes the nonlinear coupling of electron whistler waves with the field aligned electrostatic density fluctuations. these equations can be recast by normalizing the variables according to @xmath76 @xmath77, in which case we obtain@xmath78@xmath79and@xmath80 n \notag \\ & = \lambda _ { 1}\frac{\partial ^{2}}{\partial z^{2}}\left (1+\frac{\partial ^{2}}{\partial t^{2}}\right) |e|^{2}-\lambda _ { 2}\frac{\partial ^{2}}{\partial z\partial t}\left (1+\frac{\partial ^{2}}{\partial t^{2}}\right) @xmath81 @xmath82 @xmath83 @xmath84 is the quantum coupling parameter, @xmath85and @xmath86 equations ([11])-([13]) contain the main results of the present work. in particular, previous results @xcite can be recovered by disregarding the spin contribution @xmath87 as well as the particle dispersion @xmath88 and considering, e.g., the isothermal equation of state (relevant for low or moderate density plasmas). Iii. stationary localized solutions in this section we will investigate the properties of nonlinear whistlers by solving numerically the eqs. ([11])-([13]) in the stationary frame @xmath89 (where @xmath90 @xmath91. we will consider the parameter regimes for the density and the magnetic field for which the nonrelativistic fluid model is valid and spf is comparable to the cpf. we will also see that the case in which spf dominates over the cpf may correspond to the strongly magnetized superdense plasmas where relativistic treatement may be necessary. however, before going further to such discussions let us first consider the particular case in which the dispersion due to charge separtion (quasineutrality) is negligibe. the latter can be justified even when the spin effects dominate, i.e., @xmath92 @xcite. from the scaling@xmath93we find that the quasineutrality limit @xmath94 holds in nonrelativistic (@xmath95) plasmas as long as @xmath96and @xmath97 @xmath98. however, we will see that in a specific parameter regime, such restrictions can be valid for very lf (@xmath99 @xmath100) whistler modes. in this case, @xmath101 t and @xmath102 @xcite with @xmath103 moreover, when @xmath104 the contribution from the term @xmath105 @xmath106 can be smaller than that @xmath105 @xmath107since @xmath108 thus, in the quasineutral regime, we obtain from eqs. ([11])-([13]) the following nls equation.@xmath109together with @xmath110then we can write eq. ([3]) as @xmath111, where @xmath112 is defined as @xmath113where @xmath114 physically, the electrons experience a longitudinal force exerted by the front of the whistler pulse, and thereby gain a net energy. the electrons gain energy during the rising front of the pulse, but then slows down by the backward ponderomotive - like force. moreover, electrons can approach the group velocity of the whistler when it reaches the pulse peak at the center. from eq. ([22]), we find that this can be possible for @xmath115, which may happen for a whistler frequency satisfying @xmath116 and for high density (@xmath117m@xmath118) and strongly magnetized (@xmath119 t) plasmas. in this case, the fermi speed may exceed the group speed (@xmath120). on the other hand, corresponding to the parameters as in fig. 1 below, @xmath121 and @xmath122 so that @xmath123 @xmath124 and @xmath125 again, note that slow electrons can freely move along the direction of the external magnetic field. the finite velocity perturbations would then induce an additional density change in order to maintain the conservation of particles (equation of continuity) under localized disturbances. consequently, the total density variation in the frequency - shift becomes @xmath126 where @xmath127 ) and ([19]) with associated electric field @xmath128 (upper panel) and density perturbation @xmath50 (lower panel) for @xmath129 (solid line) and @xmath130 (dashed line). the other parameter values are @xmath131m@xmath118, @xmath132 t, @xmath133, @xmath134 @xmath135m / s@xmath136, width=288,height=288] ) and ([19]) with associated electric field @xmath128 (upper panel) and density perturbation @xmath50 (lower panel) for @xmath137 t (solid line) and @xmath138 t (dashed line). the other parameter values are @xmath131m@xmath118, @xmath129, @xmath139 the corresponding @xmath90 values are @xmath140 (solid line) and @xmath141 (dashed line)., width=288,height=288] clearly, @xmath142 changes sign whenever the third term @xmath87 in eq. ([delta]) dominates over the other two terms. now, for lf propagation of whistlers, @xmath143and as in the previous section, @xmath144 @xmath145 is smaller compared to @xmath146 when the spin contribution dominates. thus, in the quasineutral lf regime, the density and velocity perturbations are positive and negative according as the whistler wave propagation is subsonic or supersonic [see eq. ([22])]. furthermore, localized bright (dark) envelope solutions of eq. ([21]) exist through the modulational instability (stability) when @xmath147 for lf waves @xmath148 when @xmath149, @xmath150 @xmath151 and @xmath152 according as @xmath153 hence, a possible final state of the mi could be a supersonic (subsonic) bright (dark) soliton - like structure in a quasineutral spin quantum plasma. equation ([21]) has an exact soliton solution (when @xmath112 and @xmath154 have the same sign) of the form @xmath155, \]]where @xmath156 are constants. the other particular cases, namely the quasistationary lf density response (i.e., @xmath157 for which @xmath158 @xcite and the case of unidirectional propagation (near sonic envelope) in which the quasineutrality is not a valid assumption @xcite will not be discussed here as those cases are not so relevant to the parameter regimes to be considered, instead we will focus on our main eqs. ([11])- ([13]). thus, we look for stationary solutions of eqs. ([11])- ([13]) in the stationary frame @xmath89. here we assume @xmath159 to be of the form @xmath160where @xmath128 is a real function and @xmath161 is a real constant. ([11])- ([13]) reduce to @xmath162@xmath163where @xmath164 we numerically solve the equations ([18]) and ([19]) by newton method with the boundary conditions @xmath165 @xmath166 @xmath167 @xmath168 as @xmath169. we consider the density and magnetic field strength to vary as @xmath170m@xmath118 and @xmath119 t. figure 1 illustrates the existence of double - hump localized whistler envelope accompanied with a density depletion for a set of parameters : @xmath171m@xmath172 @xmath173 t, @xmath174 and @xmath175 the corresponding frequencies are @xmath176s@xmath177 @xmath178 and @xmath179s@xmath180 also, @xmath181 @xmath182 m and @xmath183 m/s. thus, the whistlers have negative group dispersion with @xmath184 from the dispersion relation we obtain @xmath185m@xmath177 which corresponds to whistlers with a wavelength of @xmath186 m, and the group speed is @xmath187 m/s. furthermore, the nonlinear frequency shift is obtained as @xmath188. the density depletion is observed quite small due to large group velocity (compared to the sound speed) of the whistler waves. in another illustration (fig. 2) with a higher magnetic field, we observe a dark - soliton - like structure correlated with a density hump. the amplitude of the solitary pulse decreases as the magnetic field increases. in fig. 3 we have presented the solitary structures when the density is very high (@xmath189m@xmath118). this basically corresponds to the case when @xmath190 however, in this case one must note that the fermi speed is close to or can even be larger than the speed of light in vacuum and so, nonrelativistic quantum fluid model may no longer be appropriate. the quantum parameter @xmath191 has no significant role for the regime considered here, as can be seen that @xmath192 mainly dominates in the term @xmath193 [eq. ([19])], because of large group velocity (@xmath194). in order that @xmath191 can be comparable to @xmath90, one might have to consider relatively higher densities @xmath195m@xmath196 and weakly magnetized @xmath197 plasmas. however, in this case the coefficient @xmath198 (@xmath199) will be much larger than the other coefficients, which might prevent any hope for localized solution. as shown in fig. 4, one can excite a nondiverging whistler with a positive group dispersion in other regime, e.g., @xmath200 @xmath201 @xmath202m@xmath118 and @xmath173 t for which @xmath203 @xmath204 @xmath205 m/s, @xmath206, @xmath207 m/s, @xmath208 m/s. this basically corresponds to oscillatory pulse associated with a field - aligned density hump (@xmath209 ) and ([19]) with associated electric field @xmath128 (upper panel) and density perturbation @xmath50 (lower panel) for @xmath210 (solid line) and @xmath211 (dashed and dotted ine). the other parameter values are @xmath132 t, @xmath212m@xmath118 (for solid and dashed line) and @xmath213m@xmath118 (for dotted line), @xmath214. the values of @xmath90 are @xmath215 (for solid and dashed line) and @xmath216 (for dotted line)., width=288,height=288] ) and ([19]) with associated electric field @xmath128 (upper panel) and density perturbation @xmath50 (lower panel) for @xmath217 @xmath132 t, @xmath218m@xmath118, @xmath219. the other parameters are @xmath220 @xmath221 m/s., width=288,height=288] Iv. growth rate of instability nonlinear interaction of the hf pump em whistlers @xmath222 with lf electrostatic field aligned perturbations @xmath223 gives rise upper and lower side bands with frequency and wave numbers respectively @xmath224 and @xmath225 the latter interacts with the pump and thus produces a lf ponderomotive force which eventually reinforces the lf electrostatic oscillations. when all the perturbations are aligned along the external magnetic field, the parametric interactions of em waves can be described from eqs. ([11])- ([13]) by the following dispersion relation. @xmath226 \notag \\ & = 4v_{g}e_{0}^{2}\left (1-\omega ^{2}\right) \left (\lambda _ { 1}k+\lambda _ { 2}\omega \right) \left (k\gamma _ { 1}+\zeta \omega \right), \label{23}\end{aligned}\]] where @xmath227 and @xmath228 @xmath28 have been normalized by @xmath100 and @xmath229 respectively. some simplification can be in order. note that under the quasineutrality assumption, the coefficient of @xmath230 @xmath231 and the term @xmath232as well as the term @xmath233 in the coefficient of @xmath234 will not appear. also, for lf propagation of whistlers @xmath235 @xmath236 is smaller and thus being neglected. moreover, the ratio of the term @xmath237 in the coefficient of @xmath238 (which appears due to the parallel electron streaming @xmath51) and the constant term @xmath239 scales as @xmath240 and we need @xmath241 for spin effects to be dominant. thus, in this case the dispersion relation reduces to @xmath242where @xmath243 in which @xmath244 etc. are being normalized. clearly, mi sets in for modulation wave numbers satisfying @xmath245 or @xmath246 for highly dense medium and small @xmath247 the growth rate of instability @xmath248 is then given by @xmath249 hence, in the long - wavelength limit (@xmath250 maximum growth rate of instability can be achieved, and is roughly proportional to the pump wave electric field @xmath251 and @xmath252 for parameters as in fig. 1, we obtain @xmath253 it basically restricts the characteristic length - scale to a certain value for the formation of envelope solitons through mi. V. discussion and conclusion in the present investigation focusing on whistler waves we point out that the spin contribution is substantial when @xmath254 i.e., when @xmath255 and @xmath256 this corresponds to the case in which the magnetic field strength, @xmath257 and the particle density is very high, i.e., @xmath258m@xmath118 for which the magnetic field is non - quantizing and does not affect the thermodynamic properties of electrons. however, in such regimes, the fermi velocity may approach or exceed the whistler group velocity (close to @xmath259 in the present study), and so the nonrelativistic quantum fluid model may no longer be appropriate to consider. in the present work, we have considered @xmath119 t and the density to vary in the range @xmath260m@xmath118 in order that the nonrelativistic fluid model is valid to some extent. moreover that @xmath261 and the terms due to spin magnetization current together with the spf are comparable to the classical counter parts. furthermore, in this regime the velocity of electrons remains much smaller than the whistler group velocity (@xmath120). since the whistler group speed is much higher than the ia speed, whistler solitons are not significantly affected by the particle dispersion associated with the bohm potential as well as the fermi - dirac pressure, though the length scale of excitation is of the order of the compton wavelength. however, those effects reduce the plasma characteristic wavelength of excitation. such effects can be more significant in some other regimes when @xmath262 and/ or for possible excitation of the ion wakefields at nanoscales. note that since degenerate electrons follow the fermi - dirac pressure law (where the fermi temperature is density dependent), the cold plasma limit can not be recovered from the present study unless one consideres, e.g., isothermal equation of state to be relevant for low or moderate density plasmas. furthermore, @xmath263 means that one approaches the higher density regimes and @xmath264 is the case when one simply disregards the quantum tunneling effect. the parameter regimes considered here can be achievable in the magnetized white dwarfs (@xmath117m@xmath118) as well as in the next generation intense laser - solid density plasma experiments (@xmath265m@xmath118), in x - ray free electron lasers, and in plasmonic devices. one can, in principle, go beyond the parameter regimes considered here (since there is no specific theoretical limit for the density), however, we have to be careful about those parameter values for the excitation of localized whistlers and for spin - ponderomotive force to have a role. the latter may dominantly accelerate the ions by separating the electric charges and building up a high electric field. however, plasma can sustain such high electric fields, and so it remains an attractive medium for particle acceleration, which is still a most important areas of research works in both laboratory and astrophysical plasmas. in conclusion, we have presented a new set of nonlinear equations which governs the dynamics of modulated whistlers interacting with the field - aligned electrostatic lf density perturbations due to ia fluctuation, in a magnetized spin quantum plasma. both the classical as well as the spin - induced ponderomotive force has been considered to modify the local plasma density in a self - consistent manner. numerical simulation of the governing equations in the stationary frame [eqs. ([18]) and ([19])] reveals the existence of supersonic stationary envelope solitons characterized by a single or double hump whistler wave electric fields thatare trapped in a self - created density cavity. this happens for wave frequency satisfying @xmath1 and when the whistler has negative group dispersion. when the whistler frequency is smaller than @xmath266 and the group dispersion is positive at higher densities, one can excite a nondiverging whistler wave, i.e., a ducted whistler. the latter corresponds to a field aligned density hump with @xmath267 furthermore, the whistler solitons with density dips and humps can occur depending on the consideration of the frequency regime as well as the magnetic field strength and/ or the particle density. we ought to mention that our present investigation on the nonlinear propagation of em whistlers might play an important role in studies of beat - wave particle accelerators @xcite as well as in the problem of radio - frequency electron - cyclotron - resonance heating @xcite of plasmas where the driver, instead of being a laser, is a whistler wave. apm is grateful to the kempe foundations, sweden, for support through grant no. mm was supported by the european research council under contract no. 204059-qpqv and the swedish research council under contract no. 2007 - 4422. m. marklund and g. brodin in _ new aspects of plasma physics _ : proceedings of the 2007 ictp summer college on plasma physics, edited by p. k. shukla, l. stenflo, and b. eliasson (aip, world scentific, london, 2008) ;

the nonlinear propagation of electromagnetic (em) electron - cyclotron waves (whistlers) along an external magnetic field, and their modulation by electrostatic small but finite amplitude ion - acoustic density perturbations are investigated in a uniform quantum plasma with intrinsic spin of electrons. the effects of the quantum force associated with the bohm potential and the combined effects of the classical as well as the spin - induced ponderomotive forces (cpf and spf respectively) are taken into consideration. the latter modify the local plasma density in a self - consistent manner. the coupled modes of wave propagation is shown to be governed by a modified set of nonlinear schrdinger - boussinesq - like equations which admit exact solutions in form of stationary localized envelopes. numerical simulation reveals the existence of large - scale density fluctuations that are self - consistently created by the localized whistlers in a strongly magnetized high density plasma. the conditions for the modulational instability (mi) and the value of its growth rate are obtained. possible applications of our results, e.g., in strongly magnetized dense plasmas and in the next generation laser - solid density plasma interaction experiments are discussed.

1009.0990

Proof of theorem[thm:irsconverse] we first classify the multiplicity - free permutation characters given by the actions of symmetric groups on their conjugacy classes. for this we shall need the following lemma, ultimately due to frobenius, which implies that multiplicity - free permutation characters only come from permutation actions with relatively high degrees of homogeneity. [lemma : tworow] let @xmath95 be a permutation group acting on @xmath96. let @xmath5 be the permutation character of the action of @xmath1 on the cosets of @xmath37. let @xmath97 be the number of orbits of @xmath37 on @xmath98-subsets of @xmath96. if @xmath99 then @xmath100 we shall also need the forms of young s rule and pieri s rule given in the proposition below. note that pieri s rule follows from young s rule if we conjugate by the sign character, so there is no need for us to use the littlewood richardson rule. (for a proof of young s rule see (*??? * chapter 17). the modular version of young s rule proved by james in this reference will be useful to us later see theorem [thm : jamesyoung] in 3 below.) _ proof. _ that @xmath5 is multiplicity - free in cases (i) and (ii) follows from young s rule, while case (iii) is given by the @xmath101 and @xmath102 cases of the theorem of inglis, richardson and saxl. (as saxl notes in @xcite, the @xmath101 case of this theorem dates back at least to thrall : see (*??? * theorem iii).) now suppose that @xmath5 is multiplicity - free. applying lemma [lemma : tworow] with the character @xmath103 shows that @xmath104, and hence @xmath105 has either @xmath26 or @xmath30 orbits on @xmath106. similarly, applying lemma [lemma : tworow] with the character @xmath107 shows that @xmath108 and hence @xmath109 has at most @xmath32 orbits on the @xmath30-subsets of @xmath106. suppose first of all that @xmath105 is transitive on @xmath106. then @xmath29 must have cycle type @xmath110 for some @xmath111 and @xmath112 such that @xmath113. the centralizer @xmath109 is permutation isomorphic to the wreath product. it is not hard to see that the number of orbits of @xmath114 on unordered pairs from @xmath106 is @xmath115 comparing with, this shows that if @xmath5 is multiplicity - free then @xmath116. now suppose that @xmath105 has @xmath30 orbits on @xmath106. the previous paragraph counts the number of orbits of @xmath109 on unordered pairs with both elements lying in a single orbit of @xmath109 on @xmath106. it is clear that there is exactly one orbit involving unordered pairs of the form @xmath117 with @xmath118 and @xmath119 taken from different orbits of @xmath109. we leave it to the reader to check that these remarks imply that either @xmath120 and @xmath29 has cycle type @xmath121, or @xmath122 and @xmath29 has cycle type @xmath123. to finish the proof we must show that if @xmath29 has cycle type @xmath124 or @xmath123 then @xmath5 is not multiplicity - free, even though it contains @xmath107 only once. the simplest way to do this seems to be to count degrees. let @xmath125 be the sum of the degrees of all the irreducible characters of @xmath1. we shall show that @xmath126 whenever @xmath127. this leaves only three cases to be analysed separately. it follows from the theorem of inglis, richardson and saxl that @xmath125 is the number of elements of @xmath1 of order at most @xmath30 (of course this result can also be seen in other ways, for example via the frobenius schur count of involutions, or the robinson schensted correspondence). from this it follows that @xmath128 for @xmath129 and hence that @xmath130 for @xmath129. these results imply that @xmath131 let @xmath132 be the degree of @xmath5. a short inductive argument using the last inequality shows that @xmath133 for all @xmath134. now, provided that @xmath134, we have@xmath135 which is the other inequality we require. when @xmath136, one finds that @xmath137 and @xmath138, and so the degree - counting approach also works in this case. the remaining two cases can be checked by hand ; one source for the required character tables is (*??? * appendix i.a). one finds that if @xmath29 has cycle type @xmath139 then @xmath5 contains @xmath140 twice, while if @xmath29 has cycle type @xmath141 then @xmath5 contains both @xmath142 and @xmath143 twice.@xmath144 for @xmath145, one can show by direct calculation that if the permutation character of @xmath1 acting on the conjugacy class of a non - identity element @xmath29 is multiplicity - free, then @xmath29 has one of the cycle types in the table below. note that if @xmath146 then all non - identity classes appear. l|l @xmath2 & cycle types + @xmath30 & @xmath147 ''' '' + @xmath32 & @xmath148, @xmath149 + @xmath150 & @xmath151, @xmath152, @xmath153, @xmath154 + @xmath59 & @xmath155, @xmath156, @xmath157, @xmath158 + @xmath60 & @xmath159, @xmath160, @xmath161, @xmath162 we are now ready to prove theorem [thm : irsconverse]. let @xmath163 and let @xmath24. let @xmath25 be a fixed - point - free permutation, let @xmath7 be a @xmath26-dimensional character of @xmath27, and let @xmath164. if @xmath7 is the trivial character then @xmath165 is merely the permutation character of @xmath1 acting on the conjugacy class of @xmath1 containing @xmath29, so the result follows from proposition [prop : conj]. we may therefore assume that @xmath166 and that @xmath35. since @xmath167 if @xmath165 is multiplicity - free, then @xmath168 must also be multiplicity - free. if @xmath169 is not transitive on @xmath170 then we have seen that @xmath171 it now follows from pieri s rule that @xmath165 contains @xmath172 at least twice. hence, @xmath109 acts transitively, and by proposition [prop : conj] and the table above, either @xmath29 is a fixed - point - free involution in @xmath34, or @xmath29 has cycle type @xmath149, @xmath154 or @xmath173 with @xmath174, @xmath150 or @xmath60 respectively. if @xmath29 is a fixed - point - free involution then the theorem of inglis, richardson and saxl states that @xmath165 is multiplicity - free. if @xmath29 is a @xmath32-cycle then it follows from pieri s rule that @xmath5 is multiplicity - free. if @xmath29 is a @xmath150-cycle then @xmath175 which contains @xmath176 twice. similarly, if @xmath29 has cycle type @xmath173 then @xmath177 which contains @xmath178 twice. this completes the proof of theorem [thm : irsconverse]. Proof of theorem[thm:main] a very large step towards classifying the multiplicity - free permutation characters of symmetric groups was made by saxl in @xcite. in this paper saxl gives a list of subgroups of @xmath1 for @xmath179, which he proves contains all subgroups @xmath37 such that the permutation character of @xmath1 acting on the cosets of @xmath37 is multiplicity - free. our contribution is to prune his list of the unwanted subgroups. there are several interesting features that still remain for us to discover, and to obtain the most uniform result, we must assume that @xmath36. since we shall frequently need to refer to it, we give a verbatim statement of saxl s theorem from @xcite. there is a minor error in case (v), to which the groups @xmath180 for @xmath181 should be added. (it follows from the argument at the bottom of page 342 of saxl s paper that these groups should be considered for inclusion, and in fact both give rise to multiplicity - free characters.) we now examine each of saxl s cases in turn. the most interesting case (iii) is left to the end, and we consider case (iv) together with (ii) and (iii). we shall frequently need the well - known result (see for example @xcite) that if @xmath4 is a partition then @xmath182 where @xmath183 is the conjugate partition to @xmath4. (recall that @xmath183 is the partition defined by @xmath184 ; the diagram of @xmath183 is obtained from the diagram of @xmath4 by reflecting it in its main diagonal.) we shall also frequently use the fact that if @xmath185 and @xmath186 is multiplicity - free, then @xmath89 is also multiplicity - free. if @xmath187 or @xmath188 then the subgroups from this case clearly give multiplicity - free characters. they contribute to our case (a1). if then it follows from together with young s rule and pieri s rule that @xmath189 hence, for @xmath55 in this range, @xmath190 is multiplicity - free unless @xmath31 or @xmath191. when @xmath31 it is easily seen that @xmath192 is not multiplicity - free, while if @xmath37 is one of the other two index @xmath30-subgroups of @xmath193, namely @xmath194 or @xmath195, then @xmath89 _ is _ multiplicity - free. this gives the remaining groups in our case (a1) and the groups in case (a2). if @xmath2 is even then we have already dealt with all the groups from saxl s case (i). if @xmath47 is odd then we still have to deal with the subgroups of @xmath196 properly containing @xmath197. a calculation similar to shows that all these groups give multiplicity - free characters ; they appear in our case (b2). there are three index @xmath30 subgroups of @xmath49, namely @xmath51, @xmath198 and one other, which we shall denote by @xmath199. figure 2 overleaf shows the lattice of subgroups we must consider ; note that they are in bijection with the subgroups of the dihedral group of order @xmath200. [fig : sk2] (16,26) (9.1,10.1)(1,1)5.8 (6.9,10.1)(-1,1)5.8 (8,10.1)(0,1)5.8 (6.9,24)(-1,-1)5.8 (8,24)(0,-1)5.8 (9.1,24)(1,-1)5.8 (8,6.8)(0,-1)5 (-1,15.5)(-1,-2)2.9 (-3,11.5)(1,-2)0.9 (17.5,15.5)(1,-2)2.9 (19.5,11.5)(-1,-2)0.9 (-2,6.8)(1,-1)5.8 (17.5,6.8)(-1,-1)5.8 (3,8.1)@xmath201 (14.8,16.5)@xmath51 (7.275,16.5)@xmath199 (-7,16.5)@xmath198 (5.8,25)@xmath49 (-6,8.1)@xmath202 (17,8.1)@xmath203 (4.8,0)@xmath204 from we know that @xmath205 is not multiplicity - free. however, it turns out that every subgroup of @xmath49 which properly contains @xmath204 does give a multiplicity - free permutation character. these groups appear in our case (b1). for later use we give their permutation characters in full. we shall need the character @xmath206 of @xmath49 defined by the composition of maps @xmath207 ; note that @xmath208. example 2.3 in @xcite tells us that @xmath209 given, it follows from the known decomposition of @xmath210 that @xmath211 using and together with young s rule and pieri s rule we find that @xmath212 similar calculations give the permutation characters induced from the index @xmath150 subgroups : @xmath213 where in the last line @xmath214 to decide which of these characters remain multiplicity - free when induced from @xmath215 to @xmath216, and so should be taken from saxl s case (iv), we first note that @xmath217 contains @xmath218 twice. (in the second induction above, young s rule may be replaced with the ordinary branching rule : see (*??? * chapter 9).) hence, if @xmath37 is any subgroup of @xmath51, then @xmath219 is not multiplicity - free. similarly one shows that the permutation character induced from @xmath202 is not multiplicity - free, while the characters induced from @xmath198 and @xmath199 are. this gives the remaining groups in our case (b2). we now turn to saxl s case (v). the subgroups @xmath220 for @xmath221 are each @xmath222-homogeneous. (it follows from young s rule and lemma 4 that this is a necessary condition for the induced characters @xmath223 to be multiplicity - free for every @xmath2.) they are : the @xmath30-transitive frobenius group @xmath224 ; the @xmath32-transitive subgroup @xmath225 ; and the @xmath32-transitive but @xmath150-homogeneous subgroup @xmath226. here @xmath69 denotes the split extension of @xmath227 given by the order @xmath32 frobenius twist @xmath228. calculation using young s rule shows that the permutation characters @xmath229 are always multiplicity - free. a nice way to obtain these equations uses the outer automorphism of @xmath230 : if @xmath231 is a point stabiliser in @xmath230, then @xmath10 is mapped under an outer automorphism of @xmath230 to a subgroup permutation isomorphic to @xmath232. inspection of the character table of @xmath230 shows that the constituent @xmath233 of @xmath234 is mapped to the constituent @xmath235 of @xmath236. since @xmath237 is conjugate in @xmath238 to @xmath64, the character induced from @xmath64 can then be obtained by restriction. the remaining character from saxl s case (v) is @xmath239 which is always multiplicity - free. we outline one way to obtain this equation. one easily checks that @xmath240, so by , @xmath3 appears in @xmath241 if and only if @xmath242 appears. by lemma [lemma : tworow], none of @xmath243, @xmath244, or their conjugates, appears in @xmath5. from the equation @xmath245 and frobenius reciprocity one sees that @xmath246 where @xmath247 if @xmath248, and @xmath249 otherwise. (since @xmath69 is @xmath150-homogeneous, it does not matter which subgroup @xmath250 we choose.) clearly @xmath251. the group @xmath227 has a unique conjugacy class of elements of even order ; these are involutions, and since @xmath227 is sharply @xmath32-transitive, they must act with cycle type @xmath252. hence @xmath253. it follows from the identity @xmath254 for @xmath255, that the only new even order that appears when we extend @xmath227 to @xmath69 is @xmath60. therefore no @xmath150-cycles appear in the cycle decomposition of any element of @xmath69, and @xmath256. hence, apart from @xmath257 and @xmath258, the only hook character to appear in @xmath5 is @xmath259. we now have @xmath260 where @xmath165 has degree @xmath261. with the exception of @xmath262 (which has degree @xmath263) and the pair @xmath264, @xmath265 (each of degree @xmath266), all the irreducible characters of @xmath267 that are still eligible to appear in @xmath165 have too high a degree. if @xmath262 appears four times, then we would have @xmath268 ; the required character values may be computed by hand, or found in (*??? * appendix i.a). however, @xmath69 contains no elements of order @xmath59, so clearly @xmath269. equation follows. it is straightforward to check using the formulae @xmath270 and pieri s rule that, provided @xmath271, the characters @xmath272 and @xmath273 are also multiplicity - free. this gives the groups in our case (e). it remains to deal with saxl s case (iii) : subgroups of of index at most @xmath150. by the theorem of inglis, richardson and saxl, @xmath274 is multiplicity - free. moreover, this character is still multiplicity - free if we induce up to @xmath216, since @xmath275 where the sum is over all partitions @xmath4 of @xmath276 with exactly one odd part. we therefore take @xmath70 from saxl s case (iv). this gives our case (c). it now only remains to look at the proper subgroups of @xmath70. let @xmath10 be the unique normal subgroup of @xmath70 of index @xmath30 in the base group @xmath277. a straightforward argument shows that, provided @xmath278, the group @xmath279 is the only subgroup of @xmath70 of index @xmath150. this subgroup is normal in @xmath70, and the quotient group @xmath280 is isomorphic to @xmath281. it follows that there are three subgroups of index @xmath30 in @xmath70, namely @xmath282, @xmath283, and one other, which we shall denote by @xmath284. the subgroup lattice is shown in figure 3 below. [fig : s2k] (16,17) (9.1,1.1)(1,1)5.8 (6.9,1.1)(-1,1)5.8 (8,1.1)(0,1)5.8 (6.9,15)(-1,-1)5.8 (8,15)(0,-1)5.8 (9.1,15)(1,-1)5.8 (5.8,-0.9)@xmath279 (14.8,7.5)@xmath283 (7.275,7.5)@xmath284 (-3.75,7.5)@xmath282 (5.8,16)@xmath70 it is easy to check that the subgroup @xmath282 is equal to @xmath285. hence @xmath286 and so @xmath287 where the sums are over all partitions @xmath4 of @xmath288 with only even parts. from now on, we shall say that a partition all of whose parts are even is _ even_. we see from that @xmath289 fails to be multiplicity - free if and only if there is an even partition @xmath4 whose conjugate @xmath183 is also even. if @xmath55 is even then @xmath290 is such a partition, while if @xmath55 is odd then it is clear that no such partition can exist. this gives case (d) of theorem [thm : main]. suppose that @xmath55 is odd. if we induce the character @xmath289 up to @xmath216, then we obtain the constituent @xmath291 twice : once by adding a node to the even partition @xmath292, and once by adding a node to the partition @xmath290, whose conjugate @xmath293 is even. the group @xmath282 is therefore not included in those coming from saxl s case (iv). to complete the proof of theorem [thm : main], it suffices to show that if , then neither of the permutation characters induced from the other two index @xmath30 subgroups of @xmath70 is multiplicity - free. in order to describe the constituents of these permutation characters we shall use the following notation : if @xmath294 is a partition of @xmath55 with distinct parts, let @xmath295 = 2[a_1, \ldots, a_r]$] denote the partition @xmath296 of @xmath288 whose leading diagonal hook lengths are @xmath297, and such that @xmath298 for @xmath299. for instance, figure 4 overleaf shows @xmath300 $]. = (5,5,4,2)$]] we can now state the following lemma, which is the analogue of in 2.2. before proving lemma [lemma : twistedperm] we use it to complete the proof of theorem [thm : main]. by the first statement in the lemma we have @xmath301 hence, @xmath302 fails to be multiplicity - free if and only if there is an even partition of the form @xmath295 $]. now, the partition @xmath303 $] is even if and only if the following conditions hold : @xmath304 is odd and @xmath305 for all @xmath306, and, if @xmath98 is odd, then @xmath307. it follows, on setting @xmath308, that @xmath302 fails to be multiplicity - free if and only if there is a strictly decreasing sequence of positive integers @xmath309 such that either @xmath310 one now shows, by looking at the possible values of @xmath55 mod @xmath150, that provided @xmath311, at least one of these equations has a solution. for example, if @xmath312 with @xmath313, then one can solve the second equation by taking @xmath314, @xmath315, @xmath316 and @xmath317. the bound on @xmath55 is strict : when @xmath318, neither equation is soluble, and hence the permutation character @xmath319 is multiplicity - free. finally we consider the subgroup @xmath284. it is easy to check that @xmath320 hence @xmath321 by and lemma [lemma : twistedperm], we see that @xmath322 fails to be multiplicity - free if and only there is a partition @xmath295 $] whose _ conjugate _ is even. the partition @xmath303 $] has an even conjugate if and only if @xmath304 is even and @xmath305 for all @xmath306, and @xmath98 is even. it follows, on setting @xmath323, that @xmath322 fails to be multiplicity - free if and only if there is a strictly decreasing sequence of positive integers @xmath324 such that @xmath325 by a very similar argument to before, we now find that @xmath322 is not multiplicity - free if @xmath326. again this bound is strict. it is easy to see that @xmath327 is the unique non - trivial constituent of @xmath328. to proceed further, we adapt the proof of the decomposition of @xmath329 attributed to james and saxl in (*??? * example 2.2). given a partition @xmath4, we define the _ rank _ of @xmath4 to be the maximum integer @xmath98 such that @xmath330. (thus the partition @xmath331 $] has rank @xmath98.) let @xmath332. to prove the lemma, we must show that @xmath333}$], where the sum is over all partitions @xmath334 of @xmath55 with distinct parts. by an easy application of mackey s lemma (see (*??? * theorem 3.3.4)) we have @xmath335 it follows by induction that @xmath336 } { \big\uparrow}^{s_{2k-1}}\]] where the sum is over all partitions @xmath334 of @xmath337 with distinct parts. we now calculate, again using mackey s lemma, that @xmath338 where in the sums @xmath339 runs over a set of representatives for the double cosets of @xmath340 and @xmath51 in @xmath215. it follows from pieri s rule that @xmath341 contains either @xmath342 or @xmath343 with positive multiplicity. from we see that the latter character can not occur in @xmath341, while @xmath342 can occur at most once. thus @xmath341 contains @xmath344}$] exactly once. suppose now that @xmath345 is a constituent of @xmath341. if @xmath296 has rank @xmath32 or more, it follows immediately from that @xmath346 $] for some @xmath334. the rank @xmath26 and rank @xmath30 possibilities need a little more care, but in close analogy with saxl s argument, one can rule out the appearance of any unwanted characters by using the known occurrence of @xmath344}$]. finally, suppose that @xmath347 is a partition of @xmath55 with distinct parts and that @xmath348}$] does not appear in @xmath341. then, if @xmath349 is the partition obtained from @xmath350 $] by removing a node from row @xmath98, @xmath351 does not appear in @xmath352, in contradiction to.@xmath353 working through the cases in theorem [thm : main] we get a complete list of all the irreducible characters of symmetric groups that can be obtained as a constituent of a multiplicity - free permutation representation. [cor] let @xmath36 and let @xmath4 be a partition of @xmath2. the irreducible character @xmath345 is a constituent of a multiplicity - free permutation character of @xmath1 if and only if (at least) one of : @xmath296 has at most two rows or at most two columns ; @xmath354 for some @xmath118 with @xmath355 ; @xmath356 where @xmath45 and @xmath357 ; @xmath296 has at most one row of odd length ; @xmath296 has columns all of even length and @xmath358 mod @xmath150 ; @xmath296 can be obtained by adding nodes to one of the following partitions @xmath359 subject to the restriction that all added nodes are in different columns ; @xmath296 can be obtained by adding nodes to one of the following partitions @xmath360 subject to the restriction that all added nodes are in different rows.@xmath353 cases (1) and (2) give the characters coming from cases (a1), (a2) and (b1) of theorem [thm : main]. if @xmath53 then it follows from the explicit calculations in 2.2 that the groups in case (b2) contribute the further characters with labels @xmath361, @xmath362 for @xmath363 and @xmath364 for @xmath365. the first family is subsumed by case (4) ; the others form case (3). the remaining cases are straightforward : case (4) comes directly from (c), case (5) from (d) and cases (6) and (7) from (e). the following immediate corollary of theorem [thm : main] is also of interest. let @xmath36. suppose that @xmath37 is a subgroup of @xmath1 such that the permutation character of @xmath1 acting on the cosets of @xmath37 is multiplicity - free. if the permutation character of @xmath366 acting on the cosets of @xmath37 is also multiplicity - free then _ either _ @xmath367 _ or _ @xmath45 is even and either @xmath54 or @xmath37 is a transitive subgroup of @xmath49 of index at most @xmath30 Proof of theorem[thm:mod] we begin by collecting the background results we need for the proof of theorem [thm : mod]. we shall distinguish between inner tensor products, denoted @xmath368, and outer tensor products, denoted @xmath369. [lemma : sgnfilt] let @xmath83 be a field. if @xmath86 is a module for @xmath84 with a specht filtration then @xmath370 has a filtration by the duals of specht modules. in particular, if @xmath371 is a self - dual module for @xmath84 with a specht filtration, then @xmath372 also has a specht filtration. _ by (*?? * theorem 8.15), if @xmath4 is any partition then @xmath373 (this is the modular version of above.) since the functor sending an @xmath84-module @xmath86 to @xmath374 is clearly exact, this is all we need to prove the lemma.@xmath353 [thm : jamesyoung] let @xmath83 be a field and let @xmath375. if @xmath4 is a partition of @xmath55 then @xmath376 has a specht filtration, with the specht factors given by young s rule. similarly @xmath377 has a specht filtration, with the specht factors given by pieri s rule. _ the first statement follows from james modular version of young s rule (*??? * corollary 17.14). the second may be deduced from the first by using . @xmath353 since the functor sending an @xmath378-module @xmath86 to @xmath379 is exact, it follows from theorem [thm : jamesyoung] that if @xmath86 is an @xmath378-module with a specht filtration then @xmath379 also has a specht filtration, with the specht factors given by repeated applications of young s rule. naturally there is a similar result for @xmath380. it remains to state two results concerning summands of permutation modules. both of these have a slightly technical flavour, but neither is at all difficult to apply. [lemma : selfdual] let @xmath83 be a field of prime characteristic @xmath93. if @xmath37 is a subgroup of @xmath1 such that the permutation character @xmath381 is multiplicity - free then all the summands of @xmath91 are self - dual. _ for simplicity, we assume that @xmath83 is the field with @xmath93 elements. let @xmath86 be an indecomposable direct summand of @xmath91. let @xmath382 denote the ring of @xmath93-adic integers. since @xmath86 is a direct summand of a permutation module, we may lift @xmath86 to a @xmath383-module @xmath384 such that @xmath384 is a direct summand of @xmath385 and @xmath386. (see @xcite for an outline of this lifting process.) suppose that @xmath86 is not self - dual. then the lifted module @xmath384 is not self - dual either. since @xmath387 is self - dual, we may find a summand @xmath388 of @xmath387 such that @xmath389. as @xmath384 and @xmath388 are non - isomorphic, they are distinct summands of @xmath387. but @xmath384 and @xmath390 have the same ordinary character. this contradicts our assumption that the character @xmath381 is multiplicity - free.@xmath353 this lemma deals with the assertions about duality in theorem [thm : mod]. it may also be used to replace the reference to the author s d. phil thesis (*??? * theorem 6.5.1) in the proof of theorem 4 of @xcite. finally, we shall often be in the position of knowing that a permutation module @xmath91 has a specht filtration, and wishing to prove that the same result holds for each of its summands. since in theorem [thm : mod] we assume that our ground field @xmath83 is algebraically closed and of characteristic @xmath391, we may use the homological algebra approach developed by hemmer and nakano in @xcite. (for an alternative, slightly less technological approach, see the remark attributed to s. donkin at the end of 1 of @xcite.) we are now ready to prove theorem [thm : mod]. by the work of hemmer and nakano @xcite, when @xmath88, the multiplicities of the factors in a specht filtration are well - defined. hence it suffices to show that each of the permutation modules in theorem [thm : mod] has a specht filtration, with the specht factors given by its ordinary character. we start with the modules coming from case (a) of theorem [thm : main]. suppose that @xmath392. since the ground field @xmath83 has odd characteristic, @xmath393 it follows from lemma [lemma : sgnfilt] and theorem [thm : jamesyoung] that each of the four summands has a specht filtrations. proposition [prop : summands] now guarantees that _ any _ indecomposable summand of @xmath394 has a specht filtration. this deals with all the subgroups appearing in case (a), and also the subgroups of @xmath196 in case (b2). now suppose that @xmath45. we first note that @xmath395 where @xmath206 is the @xmath26-dimensional representation of @xmath49 defined in 2.2. hence both of the summands have a specht filtration. (here it is essential that @xmath83 has odd characteristic : see 4.2 for an example when @xmath83 has characteristic @xmath30.) it follows that @xmath396 and @xmath397 both have specht filtrations. moreover, if @xmath37 is a subgroup of @xmath49 of index @xmath150 then @xmath398 is in every case a direct summand of @xmath399, and so has a specht filtration. hence if @xmath37 is any subgroup of @xmath215 such that @xmath46, then @xmath400 has a specht filtration. by theorem [thm : jamesyoung], @xmath401 also has a specht filtration. these remarks deal with all the remaining subgroups in case (b). in cases (c) and (d), a considerable amount of work is done for us by theorem 2 of @xcite, which implies that @xmath402 has a specht filtration. by theorem [thm : jamesyoung], @xmath403 also has a specht filtration. the argument giving in 2.4 shows that @xmath404 we may now apply lemma [lemma : sgnfilt] and proposition [prop : summands] to deduce that the summands of the left - hand side have specht filtrations. it only remains to deal with the permutation modules from case (e). by our usual arguments, together with the remark following theorem [thm : jamesyoung], it suffices to show that for @xmath63, @xmath60 or @xmath61, @xmath405 has a specht filtration. this is immediate if @xmath406, as then every @xmath378-module is a direct sum of specht modules. the other cases turn out to be surprisingly easy. if @xmath407 and @xmath408 or @xmath409 then the specht modules corresponding to the non - trivial ordinary characters in @xmath410 are simple and projective. hence @xmath411 when @xmath407 and @xmath412 we observe that since @xmath69 does not contain any elements of order @xmath59, @xmath413 is projective. it is well known (see @xcite) that any projective module for a symmetric group has a specht filtration. the only case left is when @xmath414 and @xmath412. one sees from that each irreducible ordinary character appearing in @xmath415 lies in a different @xmath416-block of @xmath267, and that the only non - projective constituents are the trivial and sign representations. hence @xmath417 this completes the proof of theorem [thm : mod]. Two counterexamples [[section]] let @xmath83 be an algebraically closed field of characteristic @xmath32. we shall show that the module @xmath418 is a counterexample to the conjecture that every permutation module has a specht filtration. it would be interesting to collect further examples of permutation modules over fields of odd characteristic which do not have specht filtrations at the moment, it is far from clear how common they are. the shortest demonstration that the author has been able to discover hinges on the simple module @xmath419. (see (*??? * definition 11.2) for the definition of the @xmath420.) one can show, either with the help of computer algebra, or more lengthily by _ ad hoc _ arguments (see below), that @xmath371 contains a submodule isomorphic to @xmath419. it follows from the table of decomposition numbers of @xmath267 in characteristic @xmath32 (see (*??? * p145)) that there are only four specht modules which (a) have @xmath419 as a composition factor, and (b) do not also have other composition factors that are absent in @xmath371. they are @xmath421 by a standard result (see (*??? * corollary 12.2)), @xmath419 only appears at the top of @xmath422. by, @xmath419 appears in the socle of @xmath423 if and only if @xmath424 appears in the top of @xmath425 ; this is also ruled out by (*??? * corollary 12.2). the same argument works for @xmath426. finally, one can use the long exact sequence @xmath427 given by the maps @xmath428 constructed by hamernik in (*??? * p449) to show that @xmath429 contains @xmath419 in its top, but not in its socle. (hamernik works only with symmetric groups of prime degree, but it is easy to generalise this part of his work to deal with hook - specht modules for @xmath84 whenever the characteristic of @xmath83 divides @xmath2 : see @xcite.) thus none of the candidate specht modules contains @xmath419 as a submodule. the result follows. it remains to show that @xmath419 appears in the socle of @xmath371. for this we shall need the following lemma, which is of some independent interest. [lemma : endo] let @xmath83 be an algebraically closed field, let @xmath37 be a finite group and let @xmath371 be an indecomposable @xmath430-module such that @xmath431, @xmath83 appears exactly twice as a composition factor of @xmath371. then @xmath432 is @xmath30-dimensional. let @xmath433. by a corollary of fitting s lemma (see (*??? * lemma 1.4.5)), @xmath7 is either nilpotent or invertible. if @xmath7 is nilpotent then @xmath434 is isomorphic to a proper submodule of @xmath371 ; by (b) this can only happen if @xmath435 and @xmath7 is, up to a scalar, the map @xmath436 defined by @xmath437 if @xmath7 is invertible, then it has a non - zero eigenvalue @xmath438. since @xmath439 is not invertible, it must be a scalar multiple of @xmath436. hence @xmath440. [lemma:522] let @xmath83 be an algebraically closed field of characteristic @xmath32. the simple module @xmath419 lies in the socle of @xmath441. _ proof_. by basic clifford theory, it is equivalent to show that @xmath442 lies in the socle of @xmath443 ; for an introduction to the clifford theory needed to relate representations of the alternating and symmetric groups, see (*?? * chapter 5). it follows from in 2.3 that @xmath444 where @xmath445 is one of the two irreducible constituents of @xmath446. (the labelling of this pair of characters is essentially arbitrary, and we do not need to be any more precise here.) from this, one can use decomposition numbers of @xmath267 to show that the composition factors of @xmath447 are @xmath448 let @xmath449 be a sylow @xmath32-subgroup of @xmath69 and let @xmath450 be a sylow @xmath32-subgroup of @xmath451 containing @xmath449. note that @xmath452. it follows from mackey s lemma that @xmath453 where @xmath339 runs over a set of representatives for the double cosets of @xmath69 and @xmath450 in @xmath451. if @xmath454 is any subgroup of @xmath450 then, by frobenius reciprocity @xmath455 hence each summand of the right - hand side of is indecomposable with a @xmath26-dimensional socle. the dimension of @xmath456 is @xmath457 which is divisible by @xmath452. it follows that if @xmath86 is any indecomposable summand of @xmath447 (considered as a @xmath458-module) then @xmath86 has dimension divisible by @xmath32. since the ordinary character @xmath459 is multiplicity - free with three irreducible constituents, the @xmath460-permutation module @xmath461 has a @xmath32-dimensional endomorphism algebra. by (*??? * theorem 3.11.3), the endomorphism algebra @xmath462 is also @xmath32-dimensional. the @xmath416-dimensional simple module @xmath463 and the @xmath464-dimensional simple module @xmath465 can not appear as summands of @xmath447. since they each appear but once as compositions factors of @xmath447, they must lie in its middle loewy layer. if, in addition, @xmath442 does not appear in the socle of @xmath447, then @xmath447 must be indecomposable, with top and socle both isomorphic to the trivial module @xmath83. but then lemma [lemma : endo] implies that @xmath466 is @xmath30-dimensional, a contradiction. hence @xmath467 contains @xmath442. @xmath144 _ remark : _ a small extension of this argument shows that @xmath468 is indecomposable, with the loewy layers shown below. @xmath469 [[section-1]] we now consider @xmath94. it is easy to show that this module has the loewy layers @xmath470 by in 2.2, the ordinary character associated to @xmath94 is @xmath471. it is known that although the trivial module is a composition factor of @xmath472, it does not appear in either the top or socle of @xmath472 (see (*??? * example 24.5(iii))). it follows that there is no specht filtration of @xmath94 with the factors @xmath473 and @xmath472. one can, however, exploit the outer automorphism of @xmath230, which sends the specht module @xmath474 to @xmath475 and leaves @xmath476 fixed, to show that there is a short exact sequence @xmath477 thus @xmath94 has a specht filtration, but the specht factors required are not those indicated by the associated ordinary character. it is left to the reader to formulate any of the many conjectures to which this module is a counterexample. Acknowledgements i should like to thank michael collins for asking me the question that lead to the work of 2, and an anonymous referee for his or her very helpful comments on earlier versions of this paper., on multiplicity - free permutation representations, in : finite geometries and designs (proc. conf., chelwood gate, 1980), vol. 49 of _ london math. lecture note ser. _, cambridge university press, 1981, pp.

building on work of saxl, we classify the multiplicity - free permutation characters of all symmetric groups of degree @xmath0 or more. a corollary is a complete list of the irreducible characters of symmetric groups (again of degree @xmath0 or more) which may appear in a multiplicity - free permutation representation. the multiplicity - free characters in a related family of monomial characters are also classified. we end by investigating a consequence of these results for specht filtrations of permutation modules defined over fields of prime characteristic. _ keywords : _ multiplicity - free, symmetric group, permutation character, monomial character, specht filtration _ email address : _ ` wildon@maths.ox.ac.uk ` _ affiliation : _ mathematics department, swansea university, singleton park, swansea, sa2 8pp _ address for correspondence : _ department of mathematics, university of bristol, university walk, bristol, bs8 1tw. _ tel : _ 07747 636959 in this paper we prove three theorems on the multiplicity - free representations of symmetric groups. these theorems have interesting consequences for the permutation actions of symmetric groups, and for the theory of specht filtrations of permutation modules, while also being of interest in their own right. our notation is standard. let @xmath1 denote the symmetric group of degree @xmath2, and let @xmath3 denote the ordinary irreducible character of @xmath1 canonically labelled by the partition @xmath4 of @xmath2. (for an account of the character theory of the symmetric group see fulton & harris (*??? * chapter 4), or james @xcite. we shall use james lecture notes as the main source for the deeper results we need.) a character @xmath5 of @xmath1 is said to be _ multiplicity - free _ if @xmath6 for all partitions @xmath4 of @xmath2. if @xmath7 is a character of a subgroup of @xmath1 then we write @xmath8 for the character of @xmath1 induced by @xmath7. dually, the arrow @xmath9 denotes restriction. later we shall extend this notation from characters to their associated representations. if @xmath10 is a subgroup of @xmath11 then @xmath12 is the wreath product of @xmath10 with @xmath13, acting as a subgroup of @xmath14 (as defined in @xcite). finally, let @xmath15 denote the alternating group of degree @xmath2. our first two theorems are motivated by a result of inglis, richardson and saxl @xcite which shows that every irreducible character of a symmetric group is a constituent of a multiplicity - free monomial character. [thm : irs] let @xmath16 and let @xmath17 be a fixed - point - free involution in the symmetric group @xmath18. if @xmath19 where @xmath20 then @xmath21 where the sum is over all partitions @xmath4 of @xmath2 with precisely @xmath22 odd parts. our theorem [thm : irsconverse] shows that conversely, these characters are nearly the only ones of their type that are multiplicity - free. [thm : irsconverse] let @xmath23, let @xmath24 and let @xmath25 be a fixed - point - free permutation. let @xmath7 be a @xmath26-dimensional character of @xmath27. the monomial character @xmath28 is multiplicity - free if and only if _ either _ @xmath29 is a @xmath30-cycle and @xmath31 _ or _ @xmath29 is a @xmath32-cycle and @xmath33 _ or _ @xmath29 is a fixed - point - free involution in @xmath34 and @xmath35. we prove this theorem in 2 below. the proof is straightforward, and will help to introduce the techniques used in the remainder of the paper. in light of the theorem of inglis, richardson and saxl, it is very natural to ask whether every irreducible character of a symmetric group is a constituent of a multiplicity - free _ permutation _ character. our second theorem, which builds on work of saxl @xcite, gives the classification needed to show that this is very far from the case. [thm : main] let @xmath36. the permutation character of @xmath1 acting on the cosets of a subgroup @xmath37 is multiplicity - free if and only if one of : @xmath38 or @xmath39 or @xmath40 or @xmath41 or @xmath42 ; @xmath43 where @xmath44 ; @xmath45 and @xmath46 ; @xmath47 and either @xmath48 or @xmath37 is a subgroup of @xmath49 of index @xmath50 other than @xmath51 ; @xmath52 or @xmath53 and @xmath54 ; @xmath52 where @xmath55 is odd and @xmath56 ; @xmath57 or @xmath58 where @xmath55 is either @xmath59, @xmath60 or @xmath61, or @xmath62 where @xmath63 or @xmath60, and @xmath64 is the frobenius group of order @xmath65 acting on @xmath59 points, @xmath66 is @xmath67 in its natural projective action on @xmath60 points and @xmath68 is @xmath69 in its natural projective action on @xmath61 points. it seems unavoidable that cases (a) and (b) have a slightly fiddly statement. the part of the proof that leads to these cases is, however, the most routine. the reader may wish to refer ahead to figure 2 in 2.2 which shows the subgroups of @xmath49 that appear in case (b). if @xmath45 then the subgroup @xmath70 in case (c) is the centralizer of a fixed - point - free involution in @xmath71. it may also be helpful to recall that @xmath72 is permutation isomorphic to the weyl group of type @xmath73 in its action on the vectors @xmath74 spanning the root space for. (see (*??? * chapter 3) for more details.) under the natural embedding @xmath75, the weyl group of type @xmath76 acts on @xmath74 as a subgroup of index @xmath30 in @xmath73 ; it is this subgroup which appears in case (d) of theorem [thm : main]. we define the subgroups in case (e) more fully in 2.3 below. a corollary of theorem [thm : main] is a complete list of the irreducible characters of @xmath1 for @xmath36 which may appear in a multiplicity - free permutation representation (see corollary [cor] below). the reader will see that there are very few such characters ; in particular, if @xmath77, then @xmath78 never appears in such a representation. , was first stated and proved in the author s d. phil thesis (*??? * chapter 4).] since a permutation character of a symmetric group is multiplicity - free if and only if all the orbitals in the corresponding permutation action are self - paired (see @xcite), theorem [thm : main] also serves to classify such actions. while theorem [thm : main] is stated for @xmath36, the proof given in 2 below gives a complete classification for all @xmath77. the predicted list of subgroups has been checked using the computer algebra package magma @xcite to be correct for @xmath79. the same check has been made for the list of irreducible characters in corollary [cor]. one could easily use magma to generate the full list of subgroups of symmetric groups of degree @xmath80 which give multiplicity - free permutation characters, but we shall not pursue this possibility here. the author recently learned of parallel work by c. godsil and k. meagher @xcite, to appear in annals of combinatorics. their paper gives a complete classification of multiplicity - free permutation characters of every degree. when @xmath77, their results are in agreement with theorem 2. our third theorem concerns the permutation modules whose ordinary characters appear in theorem [thm : main]. note that, by (*??? * theorem 3.5), these are exactly the permutation modules whose centralizer algebra is abelian. to understand the statement of this theorem the reader will need to know a little about specht modules : see (*??? * chapters 4, 5) for an introduction. we recall here that if @xmath81 is the integral specht module for @xmath82 labelled by the partition @xmath4 of @xmath2 then, regarding the entries of the representing matrices as rational numbers, @xmath81 affords the ordinary irreducible character @xmath3. if, instead, we regard the entries as elements of a field @xmath83 of prime characteristic, then the resulting module for @xmath84, denoted simply @xmath85, is usually no longer irreducible indeed, determining the composition factors of specht modules in prime characteristic is one of the main unsolved problems in modular representation theory. we say that an @xmath84-module @xmath86 has a _ specht filtration _ if there is a chain of submodules @xmath87 such that each successive quotient is isomorphic to a specht module. [thm : mod] let @xmath83 be an algebraically closed field of prime characteristic @xmath88 and let @xmath36. if @xmath37 is a subgroup of @xmath1 such that the ordinary permutation character @xmath89 is multiplicity - free, then each summand of the permutation module @xmath90 is a self - dual module with a specht filtration. the specht module @xmath85 for @xmath84 appears in a specht filtration of @xmath91 if and only if @xmath3 is a constituent of the ordinary character @xmath89. the author first suspected the existence of this theorem after reading paget s paper @xcite. s main result is that the permutation modules coming from case (c) of theorem [thm : main] have a specht filtration, with the expected specht factors. it is a simple matter to adapt her work to deal with case (d). in 4 we show that if @xmath83 is an algebraically closed field of characteristic @xmath32 then @xmath92 does not have a specht filtration. this gives an interesting example of a permutation module in odd prime characteristic without a specht filtration. using more sophisticated techniques, mikaelian has constructed a family of examples of such modules for fields of characteristic @xmath88. the existence of such modules is a clear indication that results such as theorem [thm : mod] can not be obtained by any routine ` reduction mod @xmath93'argument. a preliminary investigation has shown that the situation in characteristic @xmath30 is still more complicated. this is to be expected on theoretical grounds : see @xcite and @xcite for an introduction to the general theory of specht filtrations. to demonstrate the difficulties of working in characteristic @xmath30 we end 4 by showing that although the module @xmath94 has a specht filtration, it _ does not _ have a specht filtration with the specht factors indicated by its ordinary character.

0903.2864

Introduction 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. The model 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] General evolution 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). The late action of a black-hole binary 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] Mass segregation 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] Velocity distributions 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] Binaries 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 ",] Scaling of previous timescale results 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. Summary 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. Acknowledgments 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

1208.4880

Introduction the properties of the relativistic fermi gas (rfg) model of the nucleus @xcite have inspired the idea of superscaling. in the rfg model, the responses of the system to an external perturbation are related to a universal function of a properly defined scaling variable which depends upon the energy and the momentum transferred to the system. the adjective universal means that the scaling function is independent on the momentum transfer, this is called scaling of first kind, and it is also independent on the number of nucleons, and this is indicated as scaling of second kind. the scaling function can be defined in such a way to result independent also on the specific type of external one - body operator. this feature is usually called scaling of zeroth - kind @xcite. one has superscaling when the three kinds of scaling are verified. this happens in the rfg model. the theoretical hypothesis of superscaling can be empirically tested by extracting response functions from the experimental cross sections and by studying their scaling behaviors. inclusive electron scattering data in the quasi - elastic region have been analyzed in this way @xcite. the main result of these studies is that the longitudinal responses show superscaling behavior. the situation for the transverse responses is much more complicated. the presence of superscaling features in the data is relevant not only by itself, but also because this property can be used to make predictions. in effect, from a specific set of longitudinal response data @xcite, an empirical scaling function has been extracted @xcite, and has been used to obtain neutrino - nucleus cross sections in the quasi - elastic region @xcite. we observe that the empirical scaling function is quite different from that predicted by the rfg model. this indicates the presence of physics effects not included in the rfg model, but still conserving the scaling properties. we have investigated the superscaling behavior of some of these effects. they are : the finite size of the system, its collective excitations, the meson exchange currents (mec) and the final state interactions (fsi). the inclusion of these effects produce scaling functions rather similar to the empirical one. our theoretical universal scaling functions, @xmath3, and the empirical one @xmath4, have been used to predict electron and neutrino cross sections. Superscaling beyond rfg model the definitions of the scaling variables and functions, have been presented in a number of papers @xcite therefore we do not repeat them here. the basic quantities calculated in our work are the electromagnetic, and the weak, nuclear response functions. we have studied their scaling properties by direct numerical comparison (for a detailed analysis see ref. @xcite). we present in fig. [fig : fexp] the experimental longitudinal and transverse scaling function data for the @xmath0c, @xmath2ca and @xmath5fe nuclei given in ref. @xcite for three values of the momentum transfer. we observe that the @xmath6 functions scale better than the @xmath7 ones. the @xmath7 scaling functions of @xmath0c, especially for the lower @xmath8 values, are remarkably different from those of @xmath2ca and @xmath5fe. the observation of the figure, indicates that the scaling of first kind, independence on the momentum transfer, and of zeroth kind, independence on the external probe, are not so well fulfilled by the experimental functions. these observations are in agreement with those of refs. @xcite. , and transverse, @xmath7, scaling functions obtained from the experimental electromagnetic responses of ref. @xcite. the numbers in the panels indicate the values of the momentum transfer in mev / c. the full circles refer to @xmath0c, the white squares to @xmath2ca, and the white triangles to @xmath5fe. the thin black line in the @xmath6 panel at 570 mev / c, is the empirical scaling function obtained from a fit to the data. the thick lines show the results of our calculations when all the effects beyond the rfg model have been considered. the full lines have been calculated for @xmath0c, the dotted lines for @xmath1o, and the dashed lines for @xmath2ca. the dashed thin lines show the rfg scaling functions.,height=604] to quantify the quality of the scaling between a set of @xmath9 scaling functions, each of them known on a grid of @xmath10 values of the scaling variable @xmath11, we define the two indexes : @xmath12 \, - \, \min_{\alpha=1,\ldots, m } \left [f_\alpha(\psi_i) \right] \right\ } \,, \label{eq : delta}\]] and @xmath13 \, - \, \min_{\alpha=1,\ldots, m } \left [f_\alpha(\psi_i) \right] \right\ } \label{eq : erre}\]] where @xmath14 is the largest value of the @xmath15. the two indexes give complementary information. the @xmath16 index is related to a local property of the functions : the maximum distance between the various curves. since the value of this index could be misleading if the responses have sharp resonances, we have also used the @xmath17 index which is instead sensitive to global properties of the differences between the functions. since we know that the functions we want to compare are roughly bell shaped, we have inserted the factor @xmath18 to weight more the region of the maxima of the functions than that of the tails..[tab : rdelta]values of the @xmath16 and @xmath17 indexes, for the experimental scaling functions of fig. [fig : fexp]. [cols="^,^,^ ",] in tab. [tab : rdelta] we give the values of the indexes calculated by comparing the experimental scaling functions of the various nuclei at fixed value of the momentum transfer. we consider that the scaling between a set of functions is fulfilled when @xmath19 0.096 and @xmath20 0.11. these values have been obtained by adding the uncertainty to the values of @xmath17 and @xmath16 for @xmath6 at 570 mev / c. from a best fit of this last set of data we extracted an empirical universal scaling function @xcite represented by the thin full line in the lowest left panel of fig. [fig : fexp]. this curve is rather similar to the universal empirical function given in ref. @xcite. let s consider now the scaling of the theoretical functions. the thin dashed lines of fig. [fig : fexp] show the rfg scaling functions. the thick lines show the results of our calculations when various effects beyond the rfg are introduced, _ i.e. _ : nuclear finite size, collective excitations, final state interactions, and, in the case of the @xmath7 functions, meson - exchange currents. we have studied the effects of the nuclear finite size, by calculating scaling functions within a continuum shell model. at q=700 mev / c, these scaling functions are very similar to those of the rfg model. at lower values of the momentum transfer, the shell model scaling functions show sharp peaks, produced by the shell structure, not present in the rfg model. we found that shell model scaling functions fulfill the scaling of first kind, the most likely violated, down to 400 mev / c. we have estimated the effects of the collective excitations by doing continuum rpa calculations with two different residual interactions@xcite. the rpa effects become smaller the larger is the value of the momentum transfer. at @xmath21 600 mev / c, the rpa effects are negligible if calculated with a finite - range interaction. collective excitations breaks scaling properties, but we found that scaling of first kind is satisfied down to about 500 mev / c. the presence of the mec violates the scaling of the transverse responses. we included the mec by using the model of ref. @xcite. in our calculations only one - pion exchange diagrams are considered, including those with virtual excitation of the @xmath22. in our model mec effects start to be relevant for @xmath23 600 mev / c. we found that mec do not destroy scaling in the kinematic range of our interest. the main modification of the shell model scaling functions, are produced by the fsi, we have considered by using the model developed in ref. we obtained scaling functions very different from those predicted by the rfg model, and rather similar to the empirical ones. in any case, the fsi do not heavily break the scaling properties. we found that the scaling of first kind is conserved down to @xmath8=450 mev / c. the same type of scaling analysis applied to @xmath24 reaction leads to very similar results @xcite. Superscaling predictions to investigate the prediction power of the superscaling hypothesis, we compared responses, and cross sections, calculated by using rpa, fsi and eventually mec, with those obtained by using @xmath3 and @xmath25. we show in fig. [fig : ee_xsect] double differential electron scattering cross sections calculated with complete model (full) and those obtained with @xmath3 (dashed lines) and @xmath26 (dotted lines). these results are compared with the data of refs. @xcite. c data @xcite have been measured at a scattering angle of @xmath27=37.5@xmath28, the @xmath1o data @xcite at @xmath27=32.0@xmath28 and the @xmath2ca data @xcite at @xmath27=45.5@xmath28. the full lines show the results of our complete calculations. the cross sections obtained by using @xmath3 are shown by the dashed lines, and those obtained with @xmath4 by the dotted lines.,height=566] the excellent agreement between the results of the full calculations and those obtained by using @xmath3, indicates the validity of the scaling approach in this kinematic region where the @xmath8 values are larger than 500 mev / c. the differences with the cross sections obtained by using the empirical scaling functions, reflect the differences between the various scaling functions shown in fig. [fig : fexp]. the disagreement with the experimental data is probably due to the fact that our models do not consider the excitation of the real @xmath22 resonance, and the pion production mechanism. o. in all the panels the full lines show the result of our complete calculation, the dashed (dotted) lines the result obtained with our universal (empirical) scaling function. the results shown in panels (a), (b) and (c) have been obtained for neutrino energy of 300 mev. panel (a) : double differential cross sections calculated for the scattering angle of 30@xmath28 as a function of the nuclear excitation energy. panel (b) : cross sections integrated on the scattering angle, always as a function of the nuclear excitation energy. panel (c) : cross sections integrated on the nuclear excitation energy, as a function of the scattering angle. panel (d) : total cross sections, as a function of the neutrino energy.,height=566] the situation for the double differential cross sections is well controlled, since all the kinematic variables, beam energy, scattering angle, energy of the detected lepton, are precisely defined, and consequently also energy and momentum transferred to the target nucleus. this situation changes for the total cross sections which are of major interest for the neutrino physics. the total cross sections are only function of the energy of the incoming lepton, therefore they consider all the scattering angles and of the possible values of the energy and momentum transferred to the nucleus, with the only limitation of the global energy, and momentum, conservations. this means that, in the total cross sections, kinematic situations where the scaling is valid and also where it is not valid are both present. we show in the first three panels of fig. [fig : nue] various differential charge - exchange cross sections obtained for 300 mev neutrinos on @xmath1o target. in the panel (a) we show the double differential cross sections calculated for a scattering angle of 30@xmath28, as a function of the nuclear excitation energy. the values of the momentum transfer vary from about 150 to 200 mev / c. this is not the quasi - elastic regime where the scaling is supposed to hold, and this explains the large differences between the various cross sections. the cross sections integrated on the scattering angle are shown as a function of the nuclear excitation energy in the panel (b) of the figure, while the cross sections integrated on the excitation energy as a function of the scattering angle are shown in the panel (c). the first three panels of the figure illustrate in different manner the same physics issue. the calculation with the scaling functions fails in reproducing the results of the full calculation in the region of low energy and momentum transfer, where surface and collective effects are important. this is shown in panel (b) by the bad agreement between the three curves in the lower energy region, and in panel (c) at low values of the scattering angle, where the @xmath8 valued are minimal. total charge - exchange neutrino cross sections are shown in panel (d) as a function of the neutrino energy @xmath29. the scaling predictions for neutrino energies up to 200 mev are unreliable. these total cross sections are dominated by the giant resonances, and more generally by collective nuclear excitation. we have seen that these effects strongly violate the scaling. at @xmath29 = 200 mev the cross section obtained with our universal function is still about 20% larger than those obtained with the full calculation. this difference becomes smaller with increasing energy and is about the 7% at @xmath29 = 300 mev. this is an indication that the relative weight of the non scaling kinematic regions becomes smaller with the increasing neutrino energy.

superscaling analysis of electroweak nuclear response functions is done for momentum transfer values from 300 to 700 mev / c. some effects, absent in the relativistic fermi gas model, where the superscaling holds by construction, are considered. from the responses calculated for the @xmath0c, @xmath1o and @xmath2ca nuclei, we have extracted a theoretical universal superscaling function similar to that obtained from the experimental responses. theoretical and empirical universal scaling functions have been used to calculate electron and neutrino cross sections. these cross sections have been compared with those obtained with a complete calculation and, for the electron scattering case, with the experimental data.

nucl-th0612072

Introduction a geodesic flow in a given direction on a translation surface induces on a transverse segment an interval exchange map. dynamic of such transformations has been extensively studied during these last thirty years providing applications to billiards in rational polygons, to measured foliations on surfaces, to teichmller geometry and dynamics, etc. interval exchange transformations are closely related to abelian differentials on riemann surfaces. it is well known that the continued fractions encode cutting sequences of hyperbolic geodesics on the poincar upper half - plane. similarly, the rauzy - veech induction (analogous to euclidean algorithm) provides a discrete model for the teichmller geodesics flow (@xcite). using this relation h. masur in @xcite and w. a. veech in @xcite have independently proved the keane s conjecture (unique ergodicity of almost all interval exchange transformations). using combinatorics of rauzy classes, kontsevich and zorich classified the connected components of strata of the moduli spaces of abelian differentials @xcite. more recently, avila, gouzel and yoccoz proved the exponential decay of correlations for the teichmller geodesic flow also using a renormalization of the rauzy - veech induction (see @xcite). avila and viana used combinatorics of rauzy - veech induction to prove the simplicity of the essential part of the lyapunov spectrum of the teichmller geodesic flow on the strata of abelian differentials (see @xcite). recently bufetov and gurevich proved the existence and uniqueness of the measure of maximal entropy for the teichmller geodesic flow on the moduli space of abelian differentials @xcite. avila and forni proved the weak mixing for almost all interval exchange transformations and translation flows @xcite. these examples show that rauzy - veech induction which was initially elaborated to prove ergodicity of interval exchange transformations and ergodicity of the teichmller geodesic flow is, actually, very efficient far beyond these initial problems. however, all the aforementioned results concern only the moduli space of abelian differentials. the corresponding questions for strata of strict quadratic differentials (i.e. of those, which are not global squares of abelian differentials) remain open. note that the (co)tangent bundle to the moduli space of curves is naturally identified with the moduli space of _ quadratic _ differentials. from this point of view, the strata of abelian differentials represent special orbifolds of high codimension in the total space of the tangent bundle. our interest in teichmller dynamics and geometry of the strata of strict quadratic differentials was one of the main motivations for developing rauzy - veech induction for quadratic differentials. natural generalizations of interval exchange transformations were introduced by danthony and nogueira in @xcite (see also @xcite) as cross sections of measured foliations on surfaces. they introduced the notion of linear involutions, as well as the notion of rauzy induction on these maps. studying lyapunov spectrum of the teichmller geodesic flow kontsevich and zorich have performed series of computer experiments with linear involutions corresponding to quadratic differentials @xcite. these experiments indicated appearance of attractors for the rauzy - veech induction, as well as examples of generalized permutations such that the corresponding linear involutions are minimal for a domain of parameters of positive measure, and non minimal for a complementary domain of parameters also of positive measure (examples of this type are presented in figure [rauzy : class:2ii] and figure [fig : rauzy:2] in appendix @xmath2). but at this point, there was no combinatorial explanation. thus, in order to generalize technique of rauzy - veech induction to quadratic differentials in a consistent way it was necessary to find combinatorial criteria allowing to identify generalized permutations, which belong to attractors and those ones, which represents cross sections of vertical foliations of quadratic differentials. it was also necessary to distinguish those generalized permutation which give rise to minimal linear involution, and to specify the domains of appropriate parameters. in this paper we establish corresponding combinatorial criteria, which enable us to develop technique of rauzy - veech induction for quadratic differentials. partial results in this direction were obtained by the second author in @xcite. we also study relations between combinatorics, geometry and dynamics of linear involutions. to compare similarities and differences between linear involutions corresponding to abelian and to quadratic differentials let us first briefly review the situation in the classical case. an interval exchange transformation is encoded by a combinatorial data (permutation @xmath3 on @xmath4 elements) and by a continuous data (lengths @xmath5 of the intervals). recall that the keane s property (see below) is a criterion of `` irrationality '' (which, in particular, implies minimality) of an interval exchange transformation. this property is satisfied for almost all parameters @xmath6 when the permutation @xmath3 is irreducible (i.e. @xmath7, while when @xmath3 is reducible, the corresponding interval exchange map is _ never _ minimal. on the other hand the irrational interval exchange maps are precisely those that arise as cross sections of minimal vertical flows on well chosen transverse intervals. the rauzy - veech induction consists in taking the first return map of an interval exchange transformation to an appropriate smaller interval. this induction can be viewed as a dynamical system on a finite - dimensional space of interval exchange maps. the behavior of an orbit of the induction provides important information on dynamics of the interval exchange transformation representing the starting point. this information is especially useful when all iterates are well defined and when the length of the underlying subintervals tends to zero. an interval exchange transformation satisfying the latter conditions is said to have _ property_. for a given irreducible permutation @xmath3, the subset of parameters @xmath6 which give rise to interval exchange transformations satisfying keane s property contains all irrational parameters, and so it is a full lebesgue measure subset. moreover, for the space of interval exchange transformations with irreducible combinatorial data, the renormalized induction process is recurrent with respect to the lebesgue measure (and even ergodic by a theorem of veech). note that the corresponding invariant measure has infinite total mass. in this paper we use the definition of _ linear involution _ be the involution of @xmath8 given by @xmath9. a linear involution is a map @xmath10, from @xmath8 into itself, of the form @xmath11, where @xmath12 is an involution of @xmath8 without fixed point, continuous except in finitely many points, and which preserves the lebesgue measure. in this paper we will only consider linear involutions with the additional condition. the derivative of @xmath12 is @xmath13 at @xmath14 if @xmath14 and @xmath15 belong to the same connected component, and @xmath13 otherwise ; see also convention [convention : oriented].] proposed by danthony and nogueira (see @xcite). as above, a _ linear involution _ is encoded by a combinatorial data (`` generalized permutation '') and by continuous data. a generalized permutation of type @xmath16 (with @xmath17) is a two - to - one map @xmath18 to an alphabet @xmath19. a generalized permutation is called _ irreducible _ if there exists a linear involution associated to this generalized permutation, which represents an appropriate cross section of the vertical foliation of some quadratic differential. a generalized permutation is called _ dynamically irreducible _ if there exists a minimal linear involution associated to this generalized permutation. it is easy to show that any irreducible generalized permutation is dynamically irreducible ; the converse is not true in general as we will see. irreducible and dynamically irreducible generalized permutations can be characterized by natural criteria expressed in elementary combinatorial terms. the corresponding criteria are stated as definitions [def : irred] and definition [def : dyn : irr] respectively. consider a dynamically irreducible generalized permutation @xmath3. the parameter space of normalized linear involutions associated to @xmath3 is represented by a hyperplane section of a simplex. we describe an explicit procedure which associates to each generalized permutation @xmath3 an open subset in the parameter space defined by a system of linear inequalities determined by @xmath3. this subset is called the set of _ admissible parameters_. when @xmath3 is irreducible, the set of admissible parameters coincides with entire parameter space ; in general it is smaller. the next result gives a more precise statement than theorem @xmath2 in the dynamically irreducible case. @xmath20 1. if @xmath3 is not dynamically irreducible, or if @xmath3 is dynamically irreducible, but @xmath6 does not belong to the set of admissible parameters, the linear involution @xmath21 is not minimal. if @xmath3 is dynamically irreducible, then for almost all admissible parameters @xmath6 the linear involution @xmath21 satisfies the keane s property, and hence is minimal. since the rauzy - veech induction commutes with dilatations, it projectivizes to a map @xmath22 on the space of normalized linear involutions ; we shall call this map the _ renormalized rauzy - veech induction_. let t be a linear involution on the unit interval and let us consider a sequence @xmath23 of iterates by the renormalized rauzy - veech induction @xmath22. 1. if @xmath10 has the keane s property, then there exists @xmath24 such that @xmath25 is irreducible for all @xmath26. the renormalized rauzy - veech induction, defined on the set @xmath27, is recurrent. having a generalized permutation @xmath3 we can define one or two other generalized permutations @xmath28 and @xmath29 reflecting the possibilities for the image of the rauzy - veech induction @xmath30. these combinatorial rauzy operations define a partial order in the set of irreducible permutations represented by an oriented graph. rauzy class _ is a connected component of this graph. note that geometry of the rauzy graphs is very different and more complicated than in the case of `` true '' permutations since for some irreducible generalized permutations one of the rauzy operations might be not defined. from theorem @xmath31 we will deduce that a rauzy class is an equivalence class for the equivalence relation given by these combinatorial operations (see proposition [prop : classes]). in analogy with the case of the `` true '' permutations, we introduce one more combinatorial operation on generalized permutations and define _ extended rauzy classes _ as minimal subsets of irreducible generalized permutations invariant under these corresponding three operations. the moduli spaces of abelian differentials and of quadratic differentials are stratified by multiplicities of the zeroes of the corresponding differentials. we denote a stratum of the moduli space of strict quadratic differentials (with at most simple poles) by @xmath32, where @xmath33 are the multiplicities of the zeroes (@xmath34 corresponds to a pole). extended rauzy classes of irreducible generalized permutations are in one - to - one correspondence with connected components of strata in the moduli spaces of quadratic differentials. historically, extended rauzy classes where used to prove the non - connectedness of some strata of abelian differentials. for permutations of a small number of elements, it is easy to construct explicitly the subset of irreducible permutations and then using the rauzy operations to decompose it into a disjoint union of extended rauzy classes. using this approach veech proved that the minimal stratum in genus @xmath35 has two connected components and arnoux proved that the minimal stratum in genus @xmath36 has three connected components (for abelian differentials). having established an explicit combinatorial criterion of irreducibility of a generalized permutation (namely theorem @xmath2) one can apply theorem @xmath37 to classify the connected components of all strata of quadratic differentials of sufficiently small dimension. this justifies, in particular, the following experimental result of zorich. each of the following four exceptional strata of quadratic differentials @xmath38 and @xmath39 contains exactly two connected components. note that a theorem of the second author @xcite classifies all connected components of all other strata of meromorphic quadratic differentials with at most simple poles. these strata are either connected, or contain exactly two connected components one of which being hyperelliptic. the same theorem @xcite proves that each of the remaining four exceptional strata might have at most two connected components. however, the only currently available proof of the fact they are disconnected is the one based on explicit calculation of the extended rauzy classes and corresponds to the theorem of zorich. it would be interesting to have an algebraic - geometrico proof of the last theorem ; namely a topological invariant as in the kontsevich - zorich s classification @xcite. note also that a paper of zorich @xcite gives explicits representatives elements for each extended rauzy class. see also @xcite for programs concerning calculations of these rauzy classes. in section [sec : iem] we recall basic properties of flat surfaces, moduli spaces and interval exchange maps. in particular we recall the rauzy - veech induction and its dynamical properties. we relate these properties to irreducibility. + in section [sec : giem] we recall the definition of a linear involution and give basics properties. then in section [combinatoric] we define a combinatorial notion of irreducibility, and prove the first part of theorem @xmath2. the main tool we use to prove this theorem is the presentation proposed by marmi, moussa and yoccoz which appears in @xcite + in section [sec : dyn] we introduce the keane s property for the linear involutions and prove the second part of theorem @xmath2, that is theorem @xmath40. for that we prove that @xmath10 satisfies the keane s property if and only if the rauzy - veech induction is always well defined and the length parameters tends to zero. then if @xmath10 does not satisfy the keane s property we show that there exists @xmath24 such that @xmath41 is dynamically reducible which then implies that @xmath10 is also dynamically reducible. + in section [dyn : rauzy : veech], we study the dynamics of the renormalized rauzy - veech map on the space of the linear involutions, and prove theorem @xmath31. for that we use the teichmller geometry and the finiteness of the volume of the strata proved by masur and veech (see @xcite). + section [rauzy : classes : giem] is devoted to a proof of theorem @xmath37 on extended rauzy classes ; we present a result of zorich based on an explicit calculation of these classes in low genera. + in the appendix we present some explicit rauzy classes as illustration of the problems which appear in the general case. we also give a property concerning the extended rauzy classes. we thank anton zorich for useful discussions. we thank arnaldo nogueira for comments and remarks on a preliminary version of this text. + this work was partially supported by the anr `` teichmller projet blanc '' anr-06-blan-0038. Background in this section we review basic notions concerning flat surfaces, moduli spaces and interval exchange maps. for general references see say @xcite and @xcite. in this paper we will mostly follows notations presented in the paper @xcite, or equivalently @xcite. a _ flat surface _ is a (real, compact, connected) genus @xmath42 surface equipped with a flat metric (with isolated conical singularities) such that the holonomy group belongs to @xmath43. here holonomy means that the parallel transport of a vector along a long loop brings the vector back to itself or to its opposite. this implies that all cone angles are integer multiples of @xmath3. equivalently a flat surface is a triple @xmath44 such that @xmath45 is a topological compact connected surface, @xmath46 is a finite subset of @xmath45 (whose elements are called _ singularities _) and @xmath47 is an atlas of @xmath48 such that the transition maps @xmath49 are translations or half - turns : @xmath50, and for each @xmath51, there is a neighborhood of @xmath52 isometric to a euclidean cone. therefore we get a _ quadratic differential _ defined locally in the coordinates @xmath53 by the formula @xmath54. this form extends to the points of @xmath46 to zeroes, simple poles or marked points (see @xcite). we will sometimes use the notation @xmath55 or simply @xmath45. observe that the holonomy is trivial if and only if there exists a sub - atlas such that all transition functions are translations or equivalently if the quadratic differentials @xmath56 is the global square of an abelian differential. we will then say that @xmath45 is a translation surface. for @xmath57, we define the moduli space of abelian differentials @xmath58 as the set of pairs @xmath59 modulo the equivalence relation generated by : @xmath60 if there exists an analytic isomorphism @xmath61 such that @xmath62. + for @xmath63, we also define the moduli space of quadratic differentials @xmath64 as the moduli space of pairs @xmath65 (where @xmath56 is not the global square of any abelian differential) modulo the equivalence relation generated by : @xmath66 if there exists an analytic isomorphism @xmath61 such that @xmath67. the moduli space of abelian differentials (respectively quadratic differentials) is stratified by the multiplicities of the zeroes. we will denote by @xmath68 (respectively @xmath69) the stratum consisting of holomorphic one - forms (respectively quadratic differentials) with @xmath70 zeroes (or poles) of multiplicities @xmath71. these strata are non - connected in general (for a complete classification see @xcite in the abelian differentials case and @xcite in the quadratic differentials case). the linear action of the @xmath72-parameter subgroup of diagonal matrices @xmath73 on the flat surfaces presents a particular interest. it gives a measure - preserving flow with respect to a natural measure @xmath74, preserving each stratum of area one flat surfaces. this flow is known as the _ teichmller geodesic flow_. masur and veech proved the following theorem. the teichmller geodesic flow acts ergodically on each connected component of each stratum of the moduli spaces of area one abelian and quadratic differentials (with respect to a finite measure in the lebesgue class). this theorem was proved by masur @xcite and veech @xcite for the @xmath68 case and for the @xmath75 case. + the ergodicity of the teichmller geodesic flow is proved in full generality in @xcite, theorem @xmath76. the finiteness of the measure appears in two @xmath77 preprints of veech : dynamical systems on analytic manifolds of quadratic differentials i, ii (see also @xcite p.445). these preprints have been published in @xmath78 @xcite. in this section we recall briefly the theory of interval exchange maps. we will show that, under simple combinatorial conditions, such transformations arise naturally as poincar return maps of measured foliations and geodesic flows on translation surfaces. moreover we will present the rauzy - veech induction and its geometric and dynamical properties (see @xcite for more details). let @xmath79 be an open interval and let us choose a finite subset @xmath80 of @xmath81. its complement is a union of @xmath82 open subintervals. an interval exchange map is a one - to - one map @xmath10 from @xmath83 to a co - finite subset of @xmath81 that is a translation on each subinterval of its definition domain. it is easy to see that @xmath10 is precisely determined by the following data : a permutation @xmath84 that encodes how the intervals are exchanged (expressing that the k - th interval, when numerated from the left to the right, is sent by @xmath10 to the place @xmath85), and a vector with positive entries that encodes the lengths of the intervals. following marmi, moussa, yoccoz @xcite, we denote these intervals by @xmath86, with @xmath19 a finite alphabet. the length of the intervals is a vector @xmath87, and the combinatorial data is a pair @xmath88 of one - to - one maps @xmath89. then @xmath84 is a one - to - one map from @xmath90 into itself given by @xmath91. we will usually represent such a permutation by a table : @xmath92 [example : permutation] let us consider the following alphabet @xmath93 with @xmath94. then we define a permutation @xmath3 as follows. @xmath95 in this section we introduce the notion of winner and loser, following the terminology of the paper of avila, gouzel and yoccoz @xcite. for @xmath21 we define the _ type _ @xmath96 of @xmath10 by @xmath97. we will then say that @xmath98 is the winner and @xmath99 is the loser. then we define a subinterval @xmath100 of @xmath81 by removing the loser of @xmath81 as follows. @xmath101 the rauzy - veech induction @xmath102 of @xmath10 is defined as the first return map of @xmath10 to the subinterval @xmath100. it is easy to see that this is again an interval exchange transformation, defined on @xmath4 letters (see e.g. @xcite). we now see how to compute the data of the new map. there are two cases to distinguish depending on whether @xmath10 is of type @xmath103 or @xmath72 ; the combinatorial data of @xmath30 only depends on @xmath3 and on the type of @xmath10. this defines two maps @xmath104 and @xmath105 by @xmath106, with @xmath96 the type of @xmath10. 1. @xmath10 has type @xmath103 ; equivalently the winner is @xmath107. + in that case, we define @xmath108 by @xmath109 where @xmath110. in an equivalent way @xmath111. then @xmath112 where @xmath113 and @xmath114 we have @xmath115 if @xmath116 and @xmath117. 2. @xmath10 has type @xmath72 ; equivalently the winner is @xmath118. + in that case, we define @xmath108 by @xmath119 where @xmath120. in an equivalent way @xmath121. then @xmath122 where @xmath123 and @xmath124 we have @xmath115 if @xmath125 and @xmath126. let @xmath93 be an alphabet. let us consider the permutation @xmath3 of example [example : permutation]. then @xmath127 we stress that the rauzy - veech induction is well defined if and only if the two rightmost intervals do not have the same length i.e. @xmath128. in the next, we want to study the rauzy - veech induction as a dynamical system defined on the space of interval exchange transformations. thus we want the iterates of the rauzy - veech induction on @xmath10 to be always well defined. we also want this induction to be a good renormalization process, in the sense that the iterates correspond to inductions on subintervals that tend to zero. this leads to the definition of reducibility and to the keane s property. we will say that @xmath88 is reducible if there exists @xmath129 such that @xmath130 is invariant under @xmath91. this means exactly that @xmath10 splits into two interval exchange transformations. we will say that @xmath10 satisfies the keane s property (also called the infinite distinct orbit condition or i.d.o.c. property), if the orbits of the singularities of @xmath131 by @xmath10 are infinite. this ensures that @xmath3 is irreducible and the iterates of the rauzy - veech induction are always well defined. if the @xmath132 are rationally independent vectors, that is @xmath133 for all nonzero integer vectors @xmath134, then @xmath10 satisfies the keane s property (see @xcite). however the converse is not true. note that if @xmath10 satisfies the keane s property then @xmath10 is minimal. let @xmath21 be an interval exchange map. let us denote by @xmath135 the length of the interval associated to the symbol @xmath136 for the @xmath70-th iterate of @xmath10 by @xmath137 ; we denote @xmath138 if it is well defined. the following are equivalent. 1. @xmath10 satisfies the keane s property. the rauzy - veech induction @xmath137 is always well - defined and for any @xmath136, the length of the intervals @xmath139 goes to zero as @xmath70 tends to infinity. as we will see this situation is very similar in the case of linear involutions. if we want to study the rauzy - veech induction as a dynamical system on the space of interval exchange maps, it is useful to consider the rauzy - veech renormalisation on the projective space of lengths parameters space. the natural associated object is the renormalized rauzy - veech induction defined on length one intervals : @xmath140 given a permutation @xmath3, we can define two other permutations @xmath141 with @xmath142. conversely, any permutation @xmath143 has exactly two predecessors : there exist exactly two permutations @xmath144 and @xmath145 such that @xmath146. note that @xmath3 is irreducible if and only if @xmath141 is irreducible. thus the relation generated by @xmath147 is a partial order on the set of irreducible permutations ; we represent it as a directed graph @xmath148. we call rauzy classes the connected components of this graph. the above relation is an equivalence relation on the set of permutations. in particular, the equivalent class of a permutation is the rauzy class. the key remark is the following : if @xmath149 then there exists @xmath150 such that @xmath151. now assume there exists an oriented path in @xmath148 joining @xmath3 and @xmath152, _ i.e. _ there exist @xmath153 such that @xmath154. then there exists @xmath155 such that @xmath156. iterating this argument, there exist @xmath157 such that @xmath158. thus there is an oriented path in @xmath148 that joins @xmath152 and @xmath3. we will see that there is an analogous proposition in the case of generalized permutations although the situation is much more complicated. here we describe the construction of a _ suspension _ over an interval exchange map @xmath10, that is a flat surface for which @xmath10 is the first return map of the vertical flow on a well chosen segment. let @xmath21 be an interval exchange transformation. a _ suspension data _ for @xmath10 is a collection of vectors @xmath159 such that : 1. 2. @xmath161. 3. @xmath162. given a suspension datum @xmath163, we consider the broken line @xmath164 on @xmath165 defined by concatenation of the vectors @xmath166 (in this order) for @xmath167 with starting point at the origin (see figure [fig : suspension : iet]). similarly, we consider the broken line @xmath168 defined by concatenation of the vectors @xmath169 (in this order) for @xmath167 with starting point at the origin. if the lines @xmath164 and @xmath168 have no intersections other than the endpoints, then we can construct a translation surface @xmath45 as follows : we can identify each side @xmath170 on @xmath164 with the side @xmath170 on @xmath168 by a translation (in the general case, we must use the veech zippered rectangle construction, see section [subsec : zip]). let @xmath171 be the horizontal interval defined by @xmath172. then the interval exchange map @xmath10 is precisely the one defined by the first return map to @xmath81 of the vertical flow on @xmath45. [] [] @xmath173 [] [] @xmath174 [] [] @xmath175 [] [] @xmath176 we have not yet discussed the existence of such a suspension datum for a general interval exchange map. a necessary condition for @xmath10 to have suspension data is that @xmath3 is irreducible. indeed, if we have @xmath177 such that @xmath178, and let @xmath179 be a collection of complex numbers, then : @xmath180 so the imaginary part of this number can not be both positive and negative, and @xmath163 is not a suspension data for @xmath10. if @xmath3 is irreducible, the existence of a suspension data is given by masur and veech independently (see @xcite page 174 and @xcite formula @xmath181 page 207). we explain the construction here. first, let us remark that @xmath88 is irreducible if and only if @xmath182 of course if @xmath3 is irreducible, then so is @xmath183, therefore @xmath184 let us define a collection of complex number @xmath179 as follows : @xmath185 then following and, the collection @xmath186 is a suspension data over @xmath10. here we describe an alternative construction of the suspension over an interval exchange transformation that works for _ any _ suspension data, namely the so called zippered rectangles construction due to veech @xcite. let @xmath21 be an interval exchange map, and let us assume that @xmath3 is irreducible. let @xmath163 be any suspension over @xmath10. then we define @xmath187 by @xmath188 for each @xmath189 let us consider a rectangle @xmath190 of width @xmath191 and of height @xmath192 based on @xmath193. the zippered rectangle construction is the translation surface @xmath194 where @xmath195 is the following equivalence relation : we identify the top and the bottom of these rectangles by @xmath196 for @xmath197. then we `` zip '' the vertical boundaries of these rectangles that are adjacent (see figure [fig : zip : iet] ; see also @xcite for a more precise description). [] [] @xmath2 [] [] @xmath40 [] [] @xmath31 [] [] @xmath37 we can define the rauzy - veech induction on the space of suspensions, as well as on the space of zippered rectangles. let @xmath21 be an interval exchange map and let @xmath163 be a suspension over @xmath10. then we define @xmath198 as follows. we define @xmath199 (the standard rauzy - veech induction). if @xmath200 is the winner for @xmath201 then @xmath202 [rk : moduli] since @xmath203 is obtained from @xmath204 by `` cutting '' and `` gluing '', these two surfaces differ by an element of the mapping class group, hence they define the same point in the moduli space (see figure [fig : rauzy : suspension] for an example). [] [] @xmath173 [] [] @xmath174 [] [] @xmath175 [] [] @xmath176 [] [] @xmath205 [] [] @xmath206 [] [] @xmath207 [] [] @xmath208 hence the new suspension data are @xmath209, @xmath210, @xmath211 and @xmath212.] if @xmath31 is a rauzy class, we define @xmath213 we have thus defined the rauzy - veech map on the space @xmath214. it is easy to check that it defines an almost everywhere invertible map : if @xmath215 then every @xmath216 has exactly one preimage for @xmath137. we define the quotient @xmath217 of @xmath214 by the equivalence relation generated by @xmath218. the zippered rectangle construction, provides a mapping @xmath219 from @xmath220 to a stratum @xmath221 of the moduli space of abelian differentials (see remark [rk : moduli]). observe that @xmath71 can be calculated in terms of @xmath222. one can also show that @xmath220 is connected and so the image belongs to a connected component of a stratum. we will denote by @xmath223 the natural lebesgue measure on @xmath214 i.e. @xmath224, were @xmath225 is the counting measure on @xmath31 and @xmath226 is the lebesgue measure. the mapping @xmath227 preserves @xmath223, so it induces a measure, denoted again by @xmath223 on @xmath220. there is natural action of the matrix @xmath228 on @xmath214 by @xmath229, where @xmath230 acts on @xmath231 linearly. this action preserves the measure @xmath223 on @xmath214 and commutes with @xmath227, so it descends to a 1-parameter action on @xmath232 called the _ teichmller flow_. since the action of @xmath230 on @xmath220 preserves the area of the corresponding flat surface, the teichmller flow also acts on the subset @xmath233 corresponding to area one surfaces, and preserves the measure @xmath234 induced by the measure @xmath223 on that subset. note also that @xmath235 is a fundamental domain of @xmath214 for the relation @xmath195 and the poincar map of the teichmller flow on latexmath:[\[\mathcal{s}=\{(\pi,\zeta);\ \pi \textrm { irreducible},\ induction on suspensions. one can show (see @xcite) that the mapping @xmath219 is a finite covering from @xmath237 onto a subset of full measure in a connected component of a stratum and the measure @xmath223 projects to the measure @xmath74 defined in section [sec : moduli : spaces]. moreover the action of @xmath230 is equivariant with respect to @xmath219, that is @xmath238. hence if we restrict to area one surfaces, the result of masur and veech (finiteness of the measure) implies that the measure @xmath234 is finite on @xmath233. the renormalized rauzy - veech induction is recurrent on @xmath239. veech proved a stronger result, that is the ergodicity of @xmath230 (on the level of @xmath240 for any rauzy class @xmath31), which implies the ergodicity of the teichmller flow for abelian differentials (see @xcite). he also proved that the induced measure on @xmath241 is always infinite. Linear involutions let @xmath45 be a (compact, connected, oriented) flat surface with @xmath1 linear holonomy and let @xmath242 be a horizontal segment with a choice of a positive vertical direction (or equivalently, a choice of left and right ends). we consider the first return map @xmath243 of vertical geodesics starting from @xmath242 in the positive direction. any vertical geodesic which start from @xmath242 and does nt hit a singularity will intersect @xmath242 again. therefore, the map @xmath244 is well defined outside a finite number of points @xmath80 (called singular points) that correspond to vertical geodesics that stop at a singularity before intersecting again the interval @xmath242. the set @xmath245 is a finite union of open intervals @xmath246 and the restriction of @xmath244 on each of these intervals is of the kind @xmath247. the map @xmath244 alone does not properly correspond to the dynamics of vertical geodesics since when @xmath248 on the interval @xmath249, then @xmath250, and @xmath251 does not correspond to the successive intersections of a vertical geodesic with @xmath242 starting from @xmath252. to fix this problem, we have to consider @xmath253 the first return map of the vertical geodesics starting from @xmath242 in the negative direction. now if @xmath248 then the successive intersections with @xmath242 of the vertical geodesic starting from @xmath252 will be @xmath254 we get a dynamical system on @xmath255. following danthony and nogueira (see @xcite) we will call such a dynamical system a linear involution. we recall here the definition that we have restricted to our purpose. [def : giem] let @xmath242 be an open interval and let @xmath256 be two disjoint copies of @xmath242. a linear involution on @xmath242 is a map @xmath257, where : * @xmath258 is a smooth involution without fixed point defined on @xmath259, where @xmath80 is a finite subset of @xmath260. * if @xmath261 and @xmath262 belong to the same connected component of @xmath263 then the derivative of @xmath258 at @xmath219 is @xmath13 otherwise the derivative of @xmath258 at @xmath219 is @xmath72. * @xmath264 is the involution @xmath265. [convention : oriented] in this paper, we are interested with non oriented measured foliations defined on _ oriented _ surfaces. observe that the orientability of the surface @xmath45 forces the second condition on the derivative of @xmath10 in definition [def : giem]. the previous definition is motivated by the following remark. the first return map of the vertical geodesic foliation on a horizontal segment @xmath242 in a flat surface @xmath45 defines a linear involution in the following way. choose a positive vertical direction in a neighborhood of @xmath242, and replace @xmath242 be two copies of @xmath242 as in figure [ttilde]. we denote by @xmath266 the one on the top and by @xmath267 the one on the bottom. then we consider the first return map on @xmath8 of vertical geodesics, where a geodesic starting from @xmath266 is taken in the positive vertical direction, and a geodesic starting from @xmath268 is taken in the negative direction. we obtain a map @xmath258 and it is easy to check that @xmath258 satisfies the condition of definition [def : giem]. then it is clear that the map @xmath269 encodes the successive intersections of a vertical geodesic with @xmath242. recall that interval exchange maps are encoded by combinatorial and metric data : these are a permutation and a vector with positive entries. we define an analogous object for linear involutions. [def2] let @xmath270 be an alphabet of @xmath4 letters. a _ generalized permutation _ of type @xmath16, with @xmath271, is a two - to - one map @xmath272. we will usually represent such generalized permutation by the table : @xmath273 a generalized permutation @xmath274 defines an involution @xmath275 without fixed points by the following way @xmath276 note that a permutation defines in a natural way a generalized permutation. we now describe how a linear involution naturally defines a generalized permutation. let @xmath10 be a linear involution and let @xmath258 be the corresponding involution as in definition [def : giem]. the domain of definition of @xmath258 is a finite union @xmath277 of open intervals, where @xmath278 are subintervals of @xmath266 and @xmath279 are subintervals of @xmath268. since @xmath258 is an isometric involution without fixed point, each @xmath249 is mapped isometrically to a @xmath280, with @xmath281, hence @xmath258 induces an involution without fixed point @xmath282 on @xmath283. as in section [sec : iem], we choose a name @xmath284 to each pair @xmath285 and we get a generalized permutation in the sense of the above definition which is defined up to a one - to - one map of @xmath19. [example1] in view of figure [ex : giem], let us consider the following alphabet @xmath93 with @xmath286. then we define a generalized permutation @xmath3 as follows. @xmath287 in an equivalent way, we can define an involution without fixed point in order to define @xmath3. @xmath288 we represent @xmath274 by the following table @xmath289 one can check that the discrete datum associated to the linear involution described in figure [ex : giem] is the generalized permutation @xmath3. note that @xmath3 is a `` true '' permutation on @xmath4 letters if and only if @xmath290 and for any @xmath291, @xmath292. in this case (if @xmath293) : @xmath294 conversely, let @xmath274 be a generalized permutation of type @xmath16 and let @xmath275 be the associated involution. if @xmath274 is not a `` true '' permutation, then an obvious necessary and sufficient condition for @xmath274 to come from a linear involution is that there exist at least two indices @xmath295 and @xmath296 such that @xmath297 and @xmath298. [conv : normal] from now, unless explicitly stated (in particular in section [suff : cond]), we will always assume that generalized permutations will satisfy the following convention. there exist at least two indices @xmath295 and @xmath296 such that @xmath297 and @xmath298. let @xmath299 be a collection of positive real numbers such that @xmath300 it is easy to construct a linear involution on the interval @xmath301 with combinatorial data @xmath302. as in section [sec : iem], we will denote by @xmath303 a linear involution. we recall the _ rauzy - veech induction _ on linear involutions introduced by danthony and nogueira (see @xcite p. 473). let @xmath303 be a linear involution on @xmath301, with @xmath274 of type @xmath16. if @xmath304, then the rauzy - veech induction @xmath102 of @xmath10 is the linear involution obtained by the first return map of @xmath10 to @xmath305 as in the case of interval exchange maps, the combinatorial data of the new linear involution depends only on the combinatorial data of @xmath10 and whether @xmath306 or @xmath307. as before, we say that @xmath10 has type @xmath103 or type @xmath72 respectively. the corresponding combinatorial operations are denoted by @xmath308 for @xmath142 respectively. note that if @xmath274 is a given generalized permutation, the subsets @xmath309 and @xmath310 can be empty because @xmath311 or because of the linear relation on the @xmath312 that must be satisfied. we first describe the combinatorial rauzy operations @xmath313. let @xmath275 be the associated involution to @xmath274. 1. map @xmath314. + @xmath315 if @xmath316 and if @xmath317 then we define @xmath318 to be of type @xmath16 and such that : @xmath319 @xmath315 if @xmath320, and if there exists a pair @xmath321 included in @xmath322 then we define @xmath323 to be of type @xmath324 and such that : @xmath325 @xmath315 otherwise @xmath326 is not defined.. + @xmath315 if @xmath328 and if @xmath317 then we define @xmath329 to be of type @xmath16 such that : @xmath330 @xmath315 if @xmath331 and if there exists a pair @xmath321 included in @xmath332 then @xmath329 is of type @xmath333 and : @xmath334 @xmath315 otherwise @xmath335 is not defined. we now describe the rauzy - veech induction @xmath30 of @xmath10 : * if @xmath303 has type @xmath103, then @xmath336, with @xmath337 if @xmath338 and @xmath339. * if @xmath303 has type @xmath72, then @xmath340, with @xmath337 if @xmath341 and @xmath342. let us consider the permutation of example [example1], namely @xmath343. then @xmath344 [ex2] let us consider the permutation @xmath274 defined on the alphabet @xmath345 by @xmath346. then @xmath347 and @xmath348 is not defined. indeed, consider any linear involution with @xmath274 as combinatorial data. then we must have @xmath349 therefore we necessarily have @xmath350 and @xmath351 never happens. [ex3] consider the permutation @xmath274 defined on the alphabet @xmath352 by @xmath353. then @xmath354 is not defined for any @xmath96. indeed, consider any linear involution with @xmath274 as combinatorial data. then we must have @xmath355, hence the rauzy - veech induction of @xmath10 is not defined for any parameters. in the case of interval exchange maps, one usually define the rauzy - veech induction only for irreducible combinatorial data. here we have not yet defined irreducibility. however, it will appear in section [combinatoric] that some interesting phenomena with respect to rauzy - veech induction appear also in the reducible case. in the next section we will define a notion of irreducibility which is equivalent to have a suspension data. it is easy to see that a generalized permutation @xmath274 such that neither @xmath356 nor @xmath348 is defined is necessarily reducible. however, the permutation @xmath274 of example [ex2] is irreducible (see definition [def : irred] and theorem [cns]) while @xmath348 is not defined. starting from a linear involution @xmath10, we want to construct a flat surface and a horizontal segment whose corresponding first return maps @xmath357 of the vertical foliation give @xmath10. such surface will be called a _ suspension _ over @xmath10, and the parameters encoding this construction will be called _ suspension data_. let @xmath10 be a linear involution and let @xmath358 be the lengths of the corresponding intervals. let @xmath359 be a collection of complex numbers such that : 1. 2. @xmath361 3. @xmath362 4. @xmath363. the collection @xmath364 is called a _ suspension data _ over @xmath10. we will also speak in an obvious manner of a suspension data for a generalized permutation. let @xmath164 be a broken line (with a finite number of edges) on the plane such that the edge number @xmath365 is represented by the complex number @xmath366, for @xmath367, and @xmath168 be a broken line that starts on the same point as @xmath164, and whose edge number @xmath368 is represented by the complex number @xmath369 for @xmath370 (figure [figure : suspension : data]). if @xmath164 and @xmath168 only intersect on their endpoints, then @xmath164 and @xmath168 define a polygon whose sides comes by pairs and for each pair the corresponding sides are parallel and have the same length. then identifying these sides together, one gets a flat surface. it is easy to check that the first return map of the vertical foliation on the segment corresponding to @xmath242 in @xmath45 defines the same linear involution as @xmath10, so we have constructed a suspension over @xmath10. we will say in this case that @xmath163 defines a _ suitable polygon_. the broken lines @xmath164 and @xmath168 might intersect at other points (see figure [figure : wrong : polygon]). however, we can still define a flat surface by using an analogous construction as the zippered rectangles construction. we now give a sketch of this construction (see e.g. @xcite for the case of interval exchange maps, or section [subsec : zip]). this construction is very similar to the usual one, although its precise description is very technical. still, for completeness, we give an equivalent but rather implicit formulation. we first consider the previous case when @xmath164 and @xmath168 define a suitable polygon. for each pair of interval @xmath371 on @xmath242, the return time @xmath372 of the vertical foliation starting from @xmath373 and returning in @xmath374 is constant. this value depends only on the generalized permutation and on the imaginary part of the suspension data @xmath163. there is a natural embedding of the open rectangle @xmath375 into the flat surface @xmath45 and this surface is obtained from @xmath376 by identifications on the boundaries of the @xmath377. identifications for the horizontal sides @xmath378 $] are given by the linear involution and identifications for the vertical sides only depend on the generalized permutation and of @xmath379. .] for the general case, we construct the rectangles @xmath377 using the same formulas. identifications for the horizontal sides are straightforward. identifications for the vertical sides, that do not depends on the horizontal parameters, will be well defined after the following lemma. let @xmath163 be a suspension data for a linear involution @xmath10, and let @xmath274 be the corresponding generalized permutation. there exists a linear involution @xmath380 and a suspension data @xmath381 for @xmath382 such that : * the generalized permutation associated to @xmath382 is @xmath274. * for any @xmath383 the complex numbers @xmath384 and @xmath385 have the same imaginary part. * the suspension data @xmath381 defines a suitable polygon. we can assume that @xmath386 (the negative case is analogous and there is nothing to prove when the sum is zero). it is clear that @xmath387 otherwise there would be no possible suspension data. if @xmath388, then we can shorten the real part of @xmath389, keeping conditions (1)(4) satisfied, and get a suspension data @xmath381 with the same imaginary part as @xmath163, and such that @xmath390. this last condition implies that @xmath381 defines a suitable polygon. if @xmath331, then condition @xmath391 implies that @xmath392 is necessary bigger than @xmath393. however, we can still change @xmath163 into a suspension data @xmath381, with same imaginary part, and such that @xmath394 is very close to @xmath395. in that case, @xmath381 also defines a suitable polygon. see @xcite, lemma 2.1 for more details. we have therefore defined the zippered rectangle construction for any suspension data. note that we have not yet discussed the existence of a suspension data. this will be done in the upcoming section. this notion is natural. see @xcite and the following proposition. [prop : irr : natural] let @xmath45 be a flat surface with no vertical saddle connections and let @xmath242 be a horizontal interval attached to a singularity on the left. let @xmath396 be the vertical leaf passing through the right endpoint of @xmath242, we assume that @xmath396 meets a singularity before returning to @xmath242, in positive or negative direction. let @xmath21 be the linear involution given by the cross section on @xmath242 of the vertical flow. there exists a suspension data @xmath163 such that @xmath204 defines a surface isometric to @xmath45. see the construction given in the proof of proposition @xmath397 in @xcite. we can define the rauzy - veech induction on the space of suspensions, as well as on the space of zippered rectangles. let @xmath303 be a linear involution and let @xmath163 be a suspension over @xmath10. then we define @xmath398 as follows. * if @xmath303 has type @xmath103, then @xmath399, with @xmath400 if @xmath338 and @xmath401. * if @xmath303 has type @xmath72, then @xmath402, with @xmath400 if @xmath341 and @xmath403. we can show that @xmath404 is a suspension over @xmath30 and defines a surface isometric to the one corresponding to @xmath405. as in the case of interval exchange maps we consider the renormalized rauzy - veech induction defined on lengths one intervals : @xmath140 one can define obviously the corresponding renormalized rauzy - veech induction on the suspensions data by contracting the imaginary parts by a factor @xmath406 which preserves the area of the corresponding flat surface. Geometry of generalized permutations in this section we give a necessary and sufficient condition for a generalized permutation to admit a suspension ; this will prove the first part of theorem @xmath2. let us first introduce some notations to make clear the definition. _ notation : _ if @xmath407 is an alphabet, we will denote by @xmath408 the set with multiplicities @xmath409 of cardinal @xmath410, and we will use analogous notations for subsets of @xmath19. we will also call _ top _ (respectively _ bottom _) the restriction of a generalized permutation @xmath274 to @xmath411 (respectively @xmath412) where @xmath16 is the type of @xmath274. _ notation : _ let @xmath413 be (possibly empty) unordered subsets of @xmath19 or @xmath408. we say that a generalized permutation @xmath274 of type @xmath16 is decomposed if @xmath414 and there exist @xmath415 and @xmath416 such that * @xmath417 * @xmath418 * @xmath419 * @xmath420. the sets @xmath421, and @xmath422 will be referred as top - left, top - right, bottom - left and bottom - right corners respectively. we do not assume that @xmath423, or @xmath424. [def : irred] we will say that @xmath274 is _ reducible _ if @xmath274 admits a decomposition @xmath425 where the subsets @xmath426 are not all empty and one of the following statements holds 1. no corner is empty 2. exactly one corner is empty and it is on the left. 3. exactly two corners are empty and they are either both on the left, either both on the right. a permutation that is not reducible is _ irreducible_. the main result of this section is the next theorem which, being combined with proposition [prop : irr : natural] implies first part of theorem @xmath2. we make clear that in this section, we only speak of suspensions given by the construction of section [susp : giem]. [cns] let @xmath21 be a linear involution. then @xmath10 admits a suspension @xmath163 if and only if the underlying generalized permutation @xmath3 is irreducible. note that the existence or not of a suspension is independent of the length data @xmath6. one can see that this reducibility notion is not symmetric with respect to the left / right, contrary to the case of interval exchange maps. therefore, the choice of attaching a singularity on the left end of the segment in the construction of section [susp : giem] is a real choice. this will have an important consequence in terms of extended rauzy classes. [rem : strongly : irreducible] in the usual case of interval exchange maps, one can always choose @xmath163 in such a way that @xmath427 (_ i.e. _ there is a singularity on the left and on the right of the interval). here it is not always possible. more precisely one can show that @xmath10 admits such a suspension with this extra condition if and only if for any decomposition of @xmath274 as in equation @xmath428 above, all the corners are empty. [cn] a reducible generalized permutation does not admit any suspension data. let us consider @xmath3 a reducible generalized permutation. it is convenient to introduce some notations. let us assume that there exists a suspension @xmath163 over @xmath3. then we define @xmath429 the real number @xmath430 ; we define @xmath431 if the set @xmath2 is empty. finally we define @xmath432, @xmath433 and @xmath4 in an analogous manner for @xmath434 and @xmath37. we also define @xmath435. we distinguish three cases following definition [def : irred]. _ i- no corner is empty. _ + then the following inequalities hold @xmath436 subtracting the second one from the first one, and the fourth one from the third one, we get : @xmath437 which is a contradiction. _ ii- exactly one corner is empty, and it is on the left. _ + we can assume without loss of generality that it is the top - left one. that means that @xmath2, @xmath40 are empty, and @xmath31, @xmath37 are nonempty. therefore the following inequalities holds : @xmath438 subtracting the third inequality from the second one, we get @xmath439, which contradicts the first one. _ iii- exactly two corners are empty. _ + if they are both on the left side, then we have @xmath40 and @xmath31 empty and @xmath37 non empty. this implies that @xmath440 is both positive and negative, which is impossible. + if they are both on the right side, it is similar. if the two corners forming a diagonal were empty, then it is easy to see that all the corners would be empty, hence this case does nt occur by assumption. the proposition is proven. in this section, we will not necessarily assume that generalized permutations satisfy convention [conv : normal], since for technical reasons, some intermediary results of this section must be stated for an arbitrary generalized permutation. we will have to work only on the imaginary part of the @xmath441 in order to built a suspension. hence, in order to simplify the notations we will use the following ones. we will use this vocabulary only in this section. [def : pseudo] a _ pseudo - suspension _ is a collection of real numbers @xmath442 such that : * for all @xmath443. * for all @xmath444. * @xmath445 a pseudo - suspension is _ strict _ if all the previous inequalities are strict except for the extremal ones. a _ vanishing index _ on the top (respectively bottom) of a pseudo - suspension is an integer @xmath446 (respectively @xmath447) such that @xmath448 (respectively @xmath449). a pseudo - suspension @xmath450 is _ better _ than @xmath451 if the set of vanishing indices of @xmath452 is strictly included into the set of vanishing indices of @xmath451. we will say that @xmath274 is _ strongly irreducible _ if for any decomposition of @xmath274 as in @xmath428 of definition [def : irred], all the corners are empty. of course strong irreducibility implies irreducibility. the following lemma is obvious and left to the reader. [lm : strongly : irred] let @xmath274 be generalized permutation satisfying convention [conv : normal] that admits a strict pseudo - suspension. then @xmath274 admits a suspension @xmath163 with @xmath453. let us assume that @xmath3 is any irreducible permutation. one has to find a suspension @xmath163 over @xmath3. we will first assume that @xmath3 is strongly irreducible and we will show that @xmath3 admits such a suspension with the extra equality @xmath453. this corresponds to a special case of proposition [max : donc : reducible : star]. we will then relax the condition on the irreducibility of @xmath3 and prove our main result. note that one can extend the proof of proposition [cn] to show that if @xmath163 is a suspension data such that @xmath454, then @xmath3 is strongly irreducible. from lemma [lm : strongly : irred] we have reduced the problem to the construction of a strict pseudo - suspension. as we have seen in section [sec : iem], in the case of true permutations, there is an explicit formula, due to masur and veech, that gives a suspension when the permutation is irreducible. we will first build a pseudo - suspension @xmath455 by extending this formula to generalized permutations. this will not give in general a strict pseudo - suspension. let @xmath456 be a generalized permutation. we can decompose @xmath19 into three disjoint subsets * the subset @xmath457 of elements @xmath458 such that @xmath459 contains exactly one element in @xmath411 and one element in @xmath412. the restriction of @xmath274 on @xmath460 defines a true permutation. * the subset @xmath461 of elements @xmath458 such that @xmath462 contains exactly two elements in @xmath411 (and hence no elements in @xmath412). * the subset @xmath463 of elements @xmath458 such that @xmath462 contains exactly two elements in @xmath412 (and hence no elements in @xmath411). the next lemma is just a reformulation of the construction of a suspension data in section [susp : data : iem] [solmv] let @xmath3 be a true permutation defined on @xmath464, then the integers @xmath465 for @xmath466 define a pseudo - suspension over @xmath3. furthermore, we have : @xmath467 recall that we do not assume any more that a generalized permutation satisfies convention [conv : normal]. [solmv2] let @xmath274 be a generalized permutation of type @xmath468 and @xmath275 the associated involution. there exists a collection of real numbers @xmath469 with @xmath470 for all @xmath471 and such that @xmath472 we will construct from @xmath473 a new permutation @xmath474 on @xmath4 symbols. let us consider the `` mirror symmetry '' @xmath475 of @xmath476 as follows. in tabular representation @xmath476 is @xmath477 ; @xmath475 is of type @xmath478 and its tabular representation is @xmath479. then @xmath474 is in tabular representation @xmath480 with @xmath481 is obtained from @xmath482 by removing the second occurrence of each letter. for instance, if @xmath483 then @xmath484 and @xmath485. it is easy to check that @xmath474 is reducible if and only if there exists @xmath471 such that @xmath486. moreover the solution of lemma [solmv] gives the desired collection of numbers @xmath487. we define the pseudo - suspension @xmath455 over @xmath3 by the collection of real numbers given by * the solutions given by lemma [solmv] and lemma [solmv2] for the restrictions of @xmath274 on @xmath460 and on @xmath488. * the solution of lemma [solmv2] for the restriction of @xmath274 on @xmath489, taken with opposite sign. [lm : dec] let @xmath490 be any vanishing index on the top of @xmath455. setting @xmath491 and @xmath492, there exists @xmath493 and @xmath494 such that the generalized permutation @xmath274 decomposes as @xmath495 with @xmath496 and with one of the following properties : either @xmath497 or there exist @xmath498 such that @xmath499 and @xmath500. there is an analogous decomposition for vanishing indices in @xmath412 but with different subsets @xmath501 @xmath502, @xmath503 and @xmath504 _ a priory_. it follows from lemmas [solmv] and [solmv2]. [flip] if @xmath451 is a pseudo - suspension of @xmath505 then @xmath506 is a pseudo - suspension of @xmath507 and @xmath451 is a pseudo - suspension of the generalized permutation @xmath508. + hence we can `` flip '' the generalized permutation @xmath274 by top / bottom or left / right without loss of generality. in the next two lemmas, we denote by @xmath451 a pseudo - suspension that is better than @xmath455 and maximal (i.e. there is no better pseudo - suspensions). [lemmea] let @xmath509 and @xmath510 be the two first top and bottom vanishing indices for @xmath451 (possibly @xmath511). let @xmath512 and @xmath513. then either @xmath514 or @xmath515 or @xmath516. [] [] @xmath517 [] [] @xmath518 [] [] @xmath519 [] [] @xmath520 better than @xmath451.,title="fig : "] we assume that neither @xmath2 nor @xmath501 is empty. lemma [lm : dec] implies that one of this set is a subset of the other one. without loss of generality, we can assume that @xmath521. let us assume @xmath522 ; we will get a contradiction. so there exist @xmath523 in @xmath460 such that @xmath524. but by definition of @xmath2 and @xmath501, we also have @xmath525. the definition of @xmath510 implies that there exists @xmath526 such that, for @xmath527, the following inequality holds : @xmath528 now we replace @xmath529 (respectively @xmath530) by @xmath531 (respectively @xmath532) and get a vector @xmath450, see figure [exemple1]. we have * @xmath533 for @xmath534. * @xmath535 for @xmath536. * @xmath537 for @xmath538. * @xmath539 for @xmath540 (since @xmath541). hence, @xmath450 is a pseudo - suspension better than @xmath451, contradicting its maximality. therefore @xmath542 and the lemma is proven. [lemmeb] let @xmath509 and @xmath510 be the first and last top vanishing indices of @xmath274. let @xmath543 and @xmath544. then either @xmath545 or @xmath546 or @xmath547. moreover if there exist @xmath548 in @xmath549 such that @xmath550 then @xmath551. we sketch the proof here. we assume that there exist @xmath552 and @xmath553 in @xmath549 such that @xmath550. if there exists @xmath554 such that @xmath555, then we set : @xmath556 then is is easy to see that, for @xmath96 small enough, @xmath450 is a pseudo - suspension and is better than @xmath451, contradicting its maximality. remark [flip] implies that the same statement is true for @xmath557 ; hence, we can assume that @xmath558. we conclude using the same argument as the one of the proof of the previous lemma [lemmea]. [max : donc : reducible : star] let @xmath274 be a strongly irreducible generalized permutation. let @xmath451 be any pseudo - suspension which is better than @xmath455 and maximal. then @xmath451 is a strict pseudo - suspension. let us assume that @xmath451 is not strict. from lemmas [lemmea] and [lemmeb] and remark [flip] we have the following decomposition of @xmath274. @xmath559 with @xmath560, @xmath561 and @xmath562 by assumption, and with the condition that either @xmath563 are equal, or at least one of them is empty (and similar statement for the pair @xmath564) ; and the condition that if @xmath558 then they are either equal, or at least one of them is empty, otherwise one of them is @xmath565 (and similar statements for @xmath566). by convention from now on, we will keep the notation @xmath40 or @xmath31 only when they are not equal to @xmath565 or @xmath567, and therefore subsets of @xmath461 or @xmath463. let us note that if there is no vanishing index in @xmath568 or in @xmath569, the corresponding right corner is just empty. but if @xmath451 is not strict, then there exists at least a pair of nonempty corners in the top or in the bottom. if there is a vanishing index on the top, then the two corresponding corners are non - empty. then it is easy to see that either there is a corner with only @xmath570 or @xmath37, or the corners are respectively @xmath571 or @xmath572, with @xmath573 nonempty. in this case lemma [lemmea] implies that there must be a vanishing index in @xmath574. since there must be a vanishing index in the top, or in the bottom, the previous argument implies that either @xmath274 is not strongly irreducible, or there is one corner that only consists of one set @xmath575 or @xmath37. thanks to remark [flip], we assume that this is the top - left corner ; this leads to the two next cases. the general idea of the next part of the proof is first to remove the cases that correspond to not strongly irreducible permutations, and then show that the other cases correspond to a non - maximal pseudo - solution. _ first case : the top - left corner is @xmath40. _ + there is necessary a vanishing index in @xmath568, and hence the top - right corner is not empty. it also does not contains all @xmath565, hence it is necessary @xmath40, @xmath37 or @xmath572. recall that @xmath3 is assumed to be strongly irreducible, so the top - right corner is not @xmath40. if the bottom - right corner were @xmath37, the generalized permutation @xmath274 would decompose as @xmath576 or @xmath577 which are not strongly irreducible. hence the bottom - right corner is not @xmath37. this also implies that @xmath463 can not be empty. let us assume that there are no vanishing indices in the bottom line. we choose any element @xmath578, @xmath579, and @xmath580 and change @xmath581 by @xmath582, @xmath583 by @xmath584 and @xmath585 by @xmath586. if @xmath96 is small enough, then the new vector @xmath450 is better than @xmath451, which contradicts its maximality. so, the bottom admits vanishing indices ; then the bottom - left corner can be @xmath587 or @xmath588. let us discuss these cases in details. * @xmath31 : the bottom - right corner is @xmath31, @xmath37 or @xmath589. in the first and second cases, @xmath274 is not strongly irreducible. if for instance, the top - right is @xmath572, then @xmath274 decomposes as @xmath590 and therefore @xmath274 is not strongly irreducible. the other case is similar. * @xmath567 or @xmath591 : in that case, the bottom - right corner is necessary @xmath37 and we have already proved that @xmath3 is not strongly irreducible in this situation. * @xmath2 or @xmath592 : we construct a better pseudo - suspension @xmath452. + let @xmath593 be the smallest index such that @xmath594 and let @xmath595 be the largest one. let @xmath509 be the first vanishing index. there exists @xmath596 such that @xmath597 otherwise the top - line would have a decomposition as @xmath598, and @xmath274 would be not strongly irreducible. let @xmath599 be the first index in @xmath489 (see figure [exemple2]). + now we define @xmath450 in the following way : @xmath600 in the extremal case @xmath601, the following arguments will work similarly if we define @xmath602 by @xmath603. we have @xmath604 here @xmath605 is the difference between the number of indices in @xmath606 smaller than or equal to @xmath108, and number of indices in @xmath607 smaller than or equal to @xmath108. this value is _ always _ greater than or equal to zero for @xmath608, and is strictly greater than zero when @xmath108 is the first vanishing index. + similarly @xmath609 is the difference between the number of indices in @xmath610 that are in @xmath611, and number of indices in @xmath612 that are in @xmath611. this value might be positive. let @xmath613 be respectively the first and last bottom vanishing indices. we have the following facts : * * @xmath614 otherwise the bottom - left corner is @xmath31. * * @xmath615 otherwise the bottom - right corner is @xmath37. + hence it is easy to check that @xmath609 can be strictly positive only for @xmath616 or @xmath617. + then if @xmath96 is small enough, @xmath450 is a pseudo - suspension, and is better than @xmath451 (see figure [exemple2]), which contradicts the maximality of @xmath451. better than @xmath451.] _ second case : the top - left corner is @xmath2. _ + we assume that there are no corners @xmath40 or @xmath31, since this case has already been discussed. let us assume that there is no vanishing index in the bottom line. then, according to lemma [lemmea], @xmath618 ; therefore the top - right corner is @xmath565 or @xmath40. if @xmath463 is empty, then @xmath274 decomposes as @xmath619 so @xmath274 is not strongly irreducible. if @xmath463 is not empty, we choose any element @xmath620, @xmath621, @xmath622, and replace @xmath623 by @xmath624, @xmath581 by @xmath625, and @xmath583 by @xmath626. this new pseudo - suspension we have constructed is better that the old one for @xmath96 small enough. if there are vanishing indices in the bottom, then the bottom - left corner belongs to the list : @xmath627 or @xmath567. * @xmath2 : the permutation @xmath274 is then obviously not strongly irreducible. * @xmath592 : the bottom - right corner is necessary @xmath37 or @xmath628. if the top - right corner where @xmath37, then @xmath274 would be not strongly irreducible. in particular, that means @xmath461 is not empty. hence there exists @xmath629 such that @xmath630 and @xmath631. then we choose any index @xmath620 and any index @xmath632, and set : @xmath633 then @xmath450 is better than @xmath451 for @xmath634 small enough, which contradicts its maximality. * @xmath635, or @xmath567. the bottom - right corner is necessary @xmath37. if @xmath461 is empty, then the top - right corner is also @xmath37, and therefore @xmath274 is not strongly irreducible. if @xmath461 is not empty, then we choose @xmath620, @xmath621 and @xmath579, and set : @xmath636 and @xmath450 is better than @xmath451. the proposition is proved. we now have all necessary tools for proving our main result. we only have to prove the sufficient condition. we consider a pseudo - suspension @xmath451 better than @xmath455 and maximal for this property. we can assume that @xmath637 is not strongly irreducible (i.e. at least one corner is non empty in the decomposition) otherwise the theorem follows from lemma [lm : strongly : irred] and proposition [max : donc : reducible : star]. let us consider a decomposition of @xmath274 as @xmath638 where @xmath639 is maximal. note that @xmath640 defines a generalized permutation which is not strongly irreducible by assumption. note also that @xmath641 does not necessary satisfy convention [conv : normal], even if @xmath274 satisfies that convention. we define @xmath642 ; from proposition [max : donc : reducible : star], the restriction of @xmath451 to @xmath643 is strict for @xmath641. since @xmath274 is irreducible, there is one or two empty corners in the decomposition. * if only one corner is empty, then it is on the right. so we can assume that @xmath274 decomposes as : @xmath644 with @xmath645 strongly irreducible. + let @xmath509 be the first vanishing index in the top line of @xmath274 and @xmath510 be the first vanishing index of the second line. consider @xmath646 the first index such that @xmath647. then @xmath648 otherwise there would be a subdecomposition of @xmath274 as @xmath649 and @xmath274 would be reducible. now let @xmath650 and let @xmath579. we set : @xmath651 if @xmath96 is small enough, then @xmath450 satisfies : * * for all @xmath652. * * for all @xmath653. + and then, we can deduce from @xmath450 a suspension over @xmath274. * if two corner are empty, then we can assume that @xmath274 decomposes as : @xmath654 with @xmath655 irreducible. now we choose @xmath578 and @xmath579, and then set : @xmath656 then @xmath450 defines a suspension over @xmath274 for @xmath96 small enough. the theorem is proven. Irrationality of linear involutions for an interval exchange map @xmath21 either the underlying permutation is reducible and then the transformation is never minimal or @xmath3 is irreducible and @xmath10 has the keane s property (and hence is minimal) for almost every @xmath6 (see section [sec : iem]). furthermore @xmath10 admits a suspension if and only if @xmath3 is irreducible. + hence the combinatorial set for which the dynamics of @xmath10 is good coincides with the one for which the geometry is good. as we will see, the situation is more complicated in the general case. in this section we prove theorem @xmath40 and the second half of theorem @xmath2. a linear involution has a connection (of length @xmath657) if there exist @xmath658 and @xmath659 such that * @xmath14 is a singularity for @xmath131. * @xmath660 is a singularity for @xmath10. a linear involution with no connection is said to have the keane s property. note that, by definition of a singularity, if we have a connection of length @xmath657 starting from @xmath14, then @xmath661 is not a singularity for @xmath10. we first prove the following proposition : [no : connection] let @xmath10 be a linear involution. the following statements are equivalent. 1. @xmath10 satisfies the keane s property. @xmath662 is well defined for any @xmath663 and the lengths of the intervals @xmath664 tends to @xmath103 as @xmath70 tends to infinity. moreover in the above situation the transformation @xmath10 is minimal. we denote by @xmath664 the length parameters of the map @xmath665, by @xmath666, @xmath667, @xmath668 the combinatorial data, and by @xmath669 the subinterval of @xmath242 corresponding to @xmath665. let us assume that @xmath10 has no connection. then all the iterates of @xmath10 by the rauzy - veech induction are well defined. indeed it is easy to see that @xmath10 has the keane s property if and only if its image @xmath30 by the rauzy - veech induction is well defined and has the keane s property. hence if @xmath10 has the keane s property, then by induction, all its iterates by @xmath227 are well defined and have the keane s property. now we have to prove that @xmath664 goes to zero as @xmath70 tends to infinity. let @xmath670 be the subset of elements @xmath671 such that @xmath672 decreases an infinite number of time in the sequence @xmath673, and let @xmath674 be its complement. repeating the arguments for the proposition and corollary 1 and 2 of section 4.3 in @xcite, we have that : * for @xmath70 large enough, the permutation @xmath666 can be written as : @xmath675 with @xmath676 * for all @xmath677, @xmath678 tends to zero. if @xmath679, then the proposition is proven. so we can assume that @xmath670 is a strict subset of @xmath19. note that @xmath670 can not be empty. therefore, we must have @xmath680 for some @xmath681 and @xmath682. this means that @xmath683 has a connection of length zero, hence @xmath10 has a connection. this contradicts the hypothesis. so we have proven that if @xmath10 has no connections, then the sequence @xmath673 of iterates of @xmath10 by the rauzy - veech induction is infinite and all length parameters of @xmath684 tend to zero when @xmath70 tends to infinity. now we assume that @xmath10 has a connection. so, there exists @xmath685 in @xmath8 which is a singularity of @xmath131, and such that its sequence @xmath686 of iterates by @xmath10 is finite, with @xmath687 a singularity of @xmath10. we denote by @xmath688 the projections of @xmath689 on @xmath242. let @xmath690 be the element of @xmath691 whose corresponding projection to @xmath242 is minimal. we have @xmath692. if for all @xmath663, the map @xmath683 is well defined and @xmath693, then @xmath669 does not tend to zero, and hence there exists @xmath694 such that @xmath135 does not tend to zero. hence we can assume that there exists a maximal @xmath24 such that @xmath695 is well defined, and @xmath696 contains @xmath697. we want to show that @xmath698 is not defined. assume that @xmath698 is defined, then @xmath699. since @xmath695 is an acceleration of @xmath10, there must exists an iterate of @xmath690 by @xmath10, say @xmath700 which is a singularity for @xmath695. either @xmath701 is in @xmath702, either it is its right end. however, @xmath702 does not contain @xmath697, and @xmath703. therefore, we must have @xmath704, so @xmath690 is a singularity for @xmath695. we prove in the same way that @xmath690 is also a singularity for @xmath705. this implies that we are precisely in the case when the rauzy - veech induction is not defined. hence we have proven that if @xmath10 has a connection, then either the sequence @xmath706 is finite, either the length parameters do not all tend to zero. this proves the first part of the proposition. now let @xmath10 be a linear involution on @xmath242 satisfying the keane s property. recall that @xmath10 is defined on the set @xmath255. let us consider the first return map @xmath244 on @xmath707. by definition of @xmath244, one has for each @xmath14 some return time @xmath708 such that @xmath709. but @xmath10 is piecewise linear thus for any @xmath710 in a small neighborhood of @xmath14, the return time @xmath711. since the derivative of @xmath10 is @xmath72 if @xmath14 and @xmath15 belong to the same connected component and @xmath13 otherwise, the derivative of @xmath244 is necessary @xmath72. hence @xmath244 is an _ interval exchange _ map. obviously @xmath244 has no connexion since it is an acceleration of @xmath10, hence @xmath244 is minimal. similarly, the first return map @xmath253 of @xmath10 on @xmath267 is also minimal. since @xmath10 satisfies convention [conv : normal], any orbit of @xmath10 is dense on @xmath707 and @xmath267 therefore @xmath10 is minimal. the proof is complete. [def : dyn : irr] let @xmath303 be a linear involution. we will say that @xmath6 is _ admissible _ for @xmath3 (or @xmath10 has admissible parameters) if none of the following assertions holds : 1. [def:1] @xmath274 decomposes as @xmath712 + with @xmath713 and @xmath714 non empty in the two first cases. [def:2] there is a decomposition of @xmath274 as @xmath715, with (up to switching the top and the bottom of @xmath274) @xmath716 and the length parameters @xmath6 satisfy the following inequality @xmath717 a generalized permutation @xmath274 will be called _ dynamically irreducible _ if the corresponding set of admissible parameters is nonempty. the set of admissible parameters of a generalized permutation is always open. [rm : th : un] these two combinatorial notions of reducibility were introduced by the second author (see @xcite). observe that if @xmath6 is not admissible for @xmath274, then @xmath21 have a connection of length @xmath103 or @xmath72 depending on cases or of definition [def : dyn : irr], and is never minimal. more precisely there exist two invariant sets of positive measure. + one can also note that if @xmath3 is irreducible then @xmath3 is dynamically irreducible (the set of admissible parameters being the entire parameters space). the length parameters for @xmath10 can not be linearly independent over @xmath718 since they must satisfy a nontrivial relation with integer coefficients. a linear involution @xmath719 is said to have _ irrational parameters _ if @xmath720 generates a @xmath718-vector space of dimension @xmath721. almost all linear involutions have irrational parameters, and this property is preserved by the rauzy - veech induction. if @xmath3 is dynamical reducible, the non minimality comes from remark [rm : th : un]. conversely let us assume that @xmath3 is a dynamical irreducible permutation and let @xmath21 be a linear involution with irrational parameters and @xmath6 admissible for @xmath3. we still denote by @xmath664 the length parameters of @xmath722 and by @xmath666, @xmath667, @xmath668 the combinatorial data. the proof has two steps : first we show using proposition [no : connection] that if @xmath10 does not have the keane s property, then there exists @xmath24 such that @xmath695 does not have admissible parameter (case of definition [def : dyn : irr]). then we show that in this case @xmath6 is not admissible for @xmath274. this will imply the theorem. _ first step : _ we assume that the sequence is finite. then there exists @xmath695 that admits no rauzy - veech induction. since @xmath723 is irrational then either @xmath724, or @xmath725 belongs to the only pair @xmath726 on the top of the permutation and @xmath727 belongs to the only pair @xmath728 on the bottom of the permutation. in each case, @xmath695 does not have admissible parameter (case ). now we assume that the lengths parameters do not all tend to zero. as in the proof of proposition [no : connection], for @xmath70 large enough, the generalized permutation @xmath666 decomposes as : @xmath729 with @xmath730, for some @xmath731 and some @xmath732 and @xmath733. recall that @xmath734 the map @xmath683 has irrational parameters, therefore @xmath666 must decompose as : @xmath735 so @xmath684 does not have admissible parameter (case ). _ second step : _ it is enough to prove that if @xmath736 does not have admissible parameter, then so is @xmath10. we can assume without loss of generality that the combinatorial rauzy - veech transformation is @xmath104. we denote by @xmath737 the data of @xmath10 and by @xmath738 the data of @xmath382. if @xmath739 decomposes as : @xmath740 let us consider @xmath741 the last element of the top line. its twin @xmath742 is on the bottom - right corner, but is not @xmath743. we denote by @xmath744. then it is clear that we obtain @xmath274 by removing @xmath745 from that place and putting it at the right - end of the bottom line. then @xmath10 does not have admissible parameter (case ). now we assume that @xmath739 decomposes as : @xmath746 if @xmath742 is on the bottom line, the situation is analogous to the previous case. if not, then we denote by @xmath747 and @xmath748, and we get @xmath274 by removing @xmath745 from @xmath749 and putting it on the right - end of the bottom line. if this place is in the top - right corner, then clearly, @xmath10 does not have admissible parameter (case ). however, it might be the last element of the top - left corner. in that case, setting @xmath750, the generalized permutation @xmath274 decomposes as : @xmath751 with @xmath752 and @xmath753, hence @xmath10 does not have admissible parameter (case ). now we assume that @xmath739 decomposes as @xmath754 then we obtain @xmath274 from @xmath739 by removing an element on the top - left corner or on the bottom - right corner, and putting it at the right - end of the bottom line. then @xmath10 does not have admissible parameter (case ). the other cases are similar. Dynamics of the renormalized rauzy-veech induction as we have seen previously, there are two notions of irreducibility for a linear involution. * `` geometrical irreducibility '' as stated in section [combinatoric], that we just called irreducibility. * dynamical irreducibility as stated in section [sec : dyn]. in this section, we first prove that the set of irreducible linear involutions in an attractor for the renormalized rauzy - veech induction. then we show that (analogously to the case of interval exchange transformations) the renormalized rauzy - veech induction is recurrent for almost all irreducible linear involutions. we can find a non - zero pseudo - suspension @xmath755 (see definition [def : pseudo]) otherwise it is easy to show that @xmath10 does not have admissible parameter (case (1)). for all @xmath383, we denote by @xmath170 the complex number @xmath756. then, as in section [sec : red12], we consider a broken line @xmath164 which starts at @xmath103, and whose edge number @xmath365 is represented by the complex number @xmath366, for @xmath757. then we consider a broken line @xmath168, which starts on the same point as @xmath164, and whose edge number @xmath368 is represented by the complex number @xmath369 for @xmath370. is the first return map of the vertical foliation on a union of saddle connections.] _ special case : _ we assume that @xmath164 and @xmath168 only intersect on their endpoints. then they define a flat surface @xmath45, and @xmath10 appears as a first return map of the vertical foliation on a segment @xmath242 which is a union of horizontal saddle connections (see figure [special : case]). after @xmath70 steps of the rauzy - veech induction, the resulting linear involution @xmath683 is the first return map of the vertical flow of @xmath45 on a shorter segment @xmath669, which is adjacent to the same singularity as @xmath242. since @xmath10 has no connection, then the length of @xmath669 tends to zero when @xmath70 tends to infinity by the first part of proposition [no : connection]. hence for @xmath70 large enough, @xmath683 is the first return map of the vertical flow of @xmath45 on a segment, adjacent to a singularity, and with no singularities in its interior. with our construction of @xmath45, it is clear that any vertical saddle connection would intersect @xmath242 and would give a connexion on @xmath45. since @xmath10 has no connection, the surface @xmath45 has no vertical saddle connection (note that this is not true in general for a first return map on a transverse segment). according to proposition [prop : irr : natural], @xmath758 admits a suspension and hence theorem [cns] implies that @xmath25 is irreducible. the theorem is proven for that case. as a first return map on a regular segment of a surface @xmath759.] _ general case : _ the two broken lines @xmath164 and @xmath168 might have other intersection points. we first show this still defines a flat surface. we consider the line @xmath760 that starts at the complex number @xmath761. then we join the first points of @xmath762 and @xmath168 by a vertical segment, and do the same for their ends points (see figure [general : case]). this defines a polygon and the non vertical sides come by pairs, so we can glue them as previously. there are two vertical segments left. we decompose each vertical segment into a pair of vertical segments of the same length and glue them together (see the figure). this creates a pole for each initial segment. we denote by @xmath759 the resulting flat surface. the first return map of the vertical flow on the horizontal segment @xmath763 joining the two poles is @xmath10. the surface @xmath759 has two vertical saddle connections of length @xmath96 starting from the poles, but there is no other vertical saddle connection on @xmath759 since @xmath10 has no connections. when @xmath96 tends to zero, the two vertical saddle connections are the only ones that shrink to zero. hence there is no loop that shrink to zero. furthermore, the initial pseudo - suspension is nonzero, so the area of @xmath759 is bounded from below. hence, the surface @xmath759 does not degenerate when @xmath96 tends to zero and so there exists a sequence @xmath764 that tends to zero such that @xmath765 tends to a surface @xmath45. the segment @xmath766 corresponding to the limit of @xmath767, as @xmath108 tends to infinity, might be very complicated and the first return map on @xmath242 is not well defined. the transformation @xmath683 is the first return map of the vertical flow of @xmath768 on a shortest horizontal segment @xmath769, adjacent to one of the poles. if @xmath70 is large enough, then the segment @xmath770 corresponding to the limit of @xmath771 has no singularity on its interior. since the surgery corresponding to contracting @xmath96 does not change the the vertical foliation, the first return map of the vertical foliation of @xmath45 on @xmath669 is precisely @xmath683. as in the special case, the surface @xmath45 does not have any vertical saddle connection, so the generalized permutation corresponding to @xmath683 is irreducible and the proposition is proven. the following lemma is analogous to proposition 9.1 in @xcite. [lem : sing] let @xmath10 be a linear involution on @xmath301 with no connection and let @xmath772 be a singularity for @xmath10. let @xmath773 be the subinterval corresponding to the linear involution @xmath683. there exists @xmath150 such that @xmath774. since @xmath10 has no connection, there exists a first @xmath150 such that @xmath775. so @xmath776, and @xmath14 is still a singularity for @xmath777. we obtain @xmath684 from @xmath777 by considering the first return map on the largest subinterval @xmath778 whose right endpoint corresponds to a singularity of @xmath777. so @xmath774. let @xmath779 be an irreducible generalized permutation, and let @xmath31 be the set of generalized permutations that can be obtained by iterations of the maps @xmath104 and @xmath105 (when possible). we define @xmath780. we have defined the rauzy - veech map on the space @xmath214. it defines an almost everywhere invertible map : if @xmath781 then @xmath204 has exactly one preimage for @xmath137. we define the quotient @xmath782 of @xmath214 by the equivalence relation generated by @xmath783. one will denote by @xmath223 the natural lebesgue measure on @xmath214 i.e. @xmath784, where @xmath226 is the natural lebesgue measure on the hyperplane @xmath785, and @xmath786 is the counting measure. the mapping @xmath227 preserves @xmath223, so it induces a measure, again denoted by @xmath223 on @xmath782. the matrix @xmath230 acts on @xmath214 by @xmath229, where @xmath230 acts on @xmath231 linearly. this action preserves the measure @xmath223 on @xmath214 and commutes with @xmath227, so it descends to a measure preserving flow on @xmath787 called the teichmller flow. if @xmath204 is a suspension data, we denote by @xmath788 the length of the corresponding interval, _ i.e. _ @xmath789. the subset @xmath790 is a fundamental domain of @xmath214 for the relation @xmath195 and the first return map of the teichmller flow on @xmath791 is the renormalized rauzy - veech induction on suspensions. the zippered rectangle construction provides a finite covering @xmath792 from @xmath782 to a subset of full measure in a connected component of a stratum @xmath793 of the moduli space of quadratic differentials. the degree of this cover is @xmath794 where @xmath795 is the dimension of the stratum. moreover @xmath796 (@xmath42 is the genus of the surfaces). let @xmath45 be a (generic) flat surface in @xmath793 with no vertical and no horizontal saddle connection. consider a horizontal separatrix @xmath797 adjacent to a given singularity @xmath798. we call _ admissible _ a segment @xmath242 adjacent to @xmath798, such that the vertical geodesic passing through the right endpoint of @xmath242 meets a singularity before returning to @xmath242, in positive or negative direction. then proposition [prop : irr : natural] implies that there exists a corresponding suspension datum @xmath163 such that @xmath799. conversely, any @xmath163 such that @xmath799 is obtained by this construction. now let @xmath800 be two admissible segments, and let @xmath801 be the corresponding suspension data. one can assume without loss of generality that @xmath802 and their left endpoint is the singularity @xmath798. let @xmath803 be the linear involutions corresponding to @xmath800. the right endpoint of @xmath804 corresponds to a singularity of @xmath253. hence there exists @xmath663 such that @xmath805, and therefore @xmath806. so we have proven that for each separatrix @xmath797 adjacent to a singularity, there is only one preimage of @xmath45 by the mapping @xmath792. so @xmath792 is a finite covering. the degree of @xmath792 is obvious by construction. if @xmath807 is the number of possible choices of horizontal separatrices then the degree of @xmath792 is @xmath794 (choices of labels and the choice of the intervals @xmath255). for each singularity, one has @xmath808 separatrices. thus @xmath809 the proposition is proven. the subset @xmath810 corresponding to surfaces of area 1 is a finite ramified cover of a connected component of a stratum of quadratic differentials, and the corresponding lebesgue measures are proportional. by theorem 0.2 in @xcite the volume of the moduli space of quadratic differentials is finite, and so, @xmath810 has finite measure. hence the teichmller geodesic flow on @xmath782 is recurrent for the lebesgue measure. recall that the rauzy - veech renormalization for suspensions @xmath811 is the cross section of the teichmller geodesic flow on @xmath241 ; therefore the rauzy - veech renormalization for suspension is recurrent. we have @xmath812, and the rauzy - veech induction commutes with the projection @xmath813. so, for almost all parameters @xmath6, the sequence @xmath814 is recurrent. note that the proof of theorem @xmath31 does not use the fact that a linear involution satisfying the keane s property is minimal. we can use this theorem to give an alternative proof of the minimality of such map. let @xmath10 be a linear involution with the keane s property. from theorem @xmath31, there exists @xmath815 such that @xmath816 is the cross section of the vertical foliation on a flat surface with no vertical saddle connection. any infinite vertical geodesic on such a surface is dense (see e.g. @xcite). thus @xmath662 is minimal and so is @xmath10. Rauzy classes as we have seen previously, the irreducible generalized permutations are an attractor for the rauzy - veech induction. in this section, we prove that there is no smaller attractor. we also prove theorem @xmath37. we first define the rauzy classes and then the extended rauzy classes. given a permutation @xmath3, we can define _ at most _ two other permutations @xmath141 with @xmath142 when @xmath817 is well defined. the relation @xmath818 generates a partial order on the set of generalized permutations ; we represent it as a directed graph @xmath148, and as for permutations, we will call rauzy classes the connected components of this graph. in the case of interval exchanges, the periodicity of the maps @xmath104 and @xmath105 gives an easy proof of the fact that the above relation is an equivalence relation (proposition of section [rauzy : classes : iem]). here the argument fails because these maps are not always defined, and it may happen that @xmath819 is well defined, but not @xmath820. however, the corresponding statement is still true. [prop : classes] the above partial order is an equivalence relation on the set of _ irreducible _ generalized permutations. let @xmath274 and @xmath739 be two generalized permutations. assume that there is a sequence of maps @xmath104 and @xmath105 that sends @xmath274 to @xmath739. if @xmath821, then for any parameters @xmath822, there exist parameters @xmath823 such that @xmath824. iterating this argument, there exists @xmath825 and @xmath24 such that @xmath826. but for any @xmath6 in a sufficiently small neighborhood @xmath827 of @xmath828, the generalized permutation corresponding to @xmath829 is @xmath152. recall that renormalized rauzy - veech induction map is recurrent (theorem @xmath31) thus one can find @xmath830 such that the sequence @xmath814 come back in a neighborhood of @xmath302 infinitely many time. furthermore, @xmath831. thus @xmath814 gives a sequence of generalized permutations that reach @xmath641 and then reach @xmath274. so, it gives a combination of the maps @xmath104 and @xmath105 that sends @xmath739 to @xmath274. this proves the proposition. let @xmath832. we define the symmetric permutation @xmath52 of @xmath833 by @xmath834. if @xmath274 is a generalized permutation of type @xmath16 defined over an alphabet @xmath19 of @xmath4 letters, we define the generalized permutation @xmath835 to be of type @xmath836 by @xmath837 we start from an irreducible generalized permutation @xmath274 and we construct the subset of irreducible generalized permutation that can be obtained from @xmath274 by some composition of the maps @xmath838, and @xmath52. the quotient of this set by the equivalence relation generated by @xmath839 for any bijective map @xmath264 from @xmath270 onto @xmath270 is called the _ extended rauzy class _ of @xmath274. the quotient by the equivalence relation generated by @xmath840 means that we look at generalized permutations defined up to renumbering. this is needed for technical reasons in the proof of theorem @xmath37. in opposite to the case of interval exchange maps, the definition of irreducibility we gave in section [combinatoric] is not invariant by the map @xmath52 : for instance, the generalized permutation @xmath841 is irreducible while @xmath842 is reducible. so an extended rauzy class is obtained after considering the set of generalized permutations obtained from @xmath274 by the extended rauzy operations, and intersecting this set by irreducible generalized permutations. the results from the previous section shows that our definition of irreducibility is the good one with respect to the rauzy - veech induction, but we see that the convention of the `` left - end singularity '' is a real choice. let @xmath10 be a linear involution defined on an interval @xmath301. recall that rauzy - veech induction applied on @xmath10 consists in considering the first return map on @xmath843, where @xmath844 is the maximal element of @xmath845 that corresponds to a singularity of @xmath10. in terms of generalized permutation, this corresponds to the @xmath817 mapping. one can consider the first return map of @xmath10 on the interval @xmath846, where @xmath847 is the minimal element of @xmath845 that corresponds to a singularity of @xmath10. in terms of generalized permutations, this corresponds to the the conjugaison of @xmath848 map. we will call this the `` rauzy - veech induction of @xmath10 by cutting on the left of @xmath242 '', while the usual rauzy - veech induction will on the opposite called the `` rauzy - veech induction of @xmath10 by cutting on the right of @xmath242 ''. let @xmath475 be an irreducible generalized permutation. the corresponding set of suspension data is connected (even convex), so the set of surfaces constructed from a suspension data, using the zippered rectangle construction, belongs to a connected component of the moduli space of quadratic differentials. it is also open and invariant by the action of the teichmller geodesic flow, hence it is a subset of full measure by ergodicity. let @xmath849 be a generalized permutation that corresponds to the same connected component of the moduli space. then there exists a surface @xmath45 and two segments @xmath850 and @xmath851, each one being adjacent to a singularity @xmath852 and @xmath853, such that for each @xmath365, the linear involution @xmath854 given by the first return maps on @xmath249 has combinatorial data @xmath482. we can assume that @xmath45 has no vertical saddle connection. we recall that each @xmath249 has an orientation so that the corresponding singularity @xmath855 is in its left end. consider the vertical separatrix @xmath797 starting from @xmath853, in the positive direction and let @xmath856 be its first intersection point with @xmath857. applying the usual rauzy - veech induction for @xmath858, the map @xmath859 is a first return map of the vertical flow on a subinterval @xmath860, adjacent to @xmath853. if @xmath70 is large enough, then @xmath859 is isomorphic to the first return map on the subinterval @xmath861, of the same length as @xmath862. we assume first that @xmath863, hence this first return map is consistent with the positive direction on @xmath850. we now have to apply rauzy - veech inductions (on the right and on the left) on @xmath253 until we get a first return map on @xmath864 with corresponding generalized permutation @xmath865. since @xmath865 is by construction, up to renumbering the alphabet, in the same rauzy class as @xmath849, we will therefore find some composition of the maps @xmath817, @xmath866 that send @xmath475 to @xmath849. note that @xmath867 might not correspond a priory to some singularities of @xmath253, so naive rauzy - veech induction on @xmath850 might miss the interval @xmath864. but @xmath868 or @xmath869 is a singularity, so we can cut the interval on the left until @xmath856 is the left end, this will eventually occurs because of lemma [lem : sing]. then after cutting on the right @xmath867 will become the right end of the corresponding interval. if @xmath870, then similarly, by cutting on the right and then on the left, we get two linear involutions corresponding to first returns maps that only differ by a different choice of orientation. hence we have found some composition of the maps @xmath817, @xmath866 that send @xmath475 to some @xmath865, such that @xmath871 is in the same rauzy class as @xmath849. hence we have proved that if two irreducible generalized permutations correspond to the same connected component, then they are in the same extended rauzy class. to prove the converse, we must consider a slightly more general kind of suspensions that do not necessary corresponds to a singularity on the left. the corresponding `` extended '' suspension data satisfy 1. 2. @xmath873 3. @xmath874 4. @xmath363 for some @xmath875 (the case @xmath876 corresponds to suspension data as seen previously). then we can extend the zippered rectangle construction to these extended suspension data. as in the usual case, the space of extended suspension data corresponding to a generalized permutation is convex, so the set of surfaces corresponding to a given generalized permutation belong to a connected component of stratum. then it is easy to see that if @xmath143 is obtained from @xmath3 by the map @xmath104, @xmath105 or @xmath52, then the corresponding connected component is the same. historically, extended rauzy classes have been used to prove the non connectedness of some stratum of abelian differentials (see for instance @xcite). for this case, some topological invariants were found by kontsevich and zorich @xcite (hyperellipticity and spin structure). for the case of quadratic differentials, all non - connected components (except four special cases) are distinguished by hyperellipticity @xcite. for the four `` exceptional ones '', the only known proof up to now is an explicit computation of the corresponding extended rauzy classes. theorem @xmath37, which is now formally proven complete the proof of the following the strata @xmath877, @xmath878, @xmath879 and @xmath880 are non connected. the generalized permutations @xmath881 and @xmath882 are irreducible. the corresponding suspension surfaces belong to the stratum @xmath883. according to zorich s computation, these two permutations do not belong to the same extended rauzy classes (see table [table : rauzy] in the appendix). hence the stratum @xmath883 is not connected. in fact this stratum has precisely two connected components corresponding to the two extended rauzy classes. we have similar conclusions for other strata with the following generalized permutations. for the stratum @xmath884 one can consider the generalized permutations @xmath885 for the stratum @xmath886 one can consider the generalized permutations @xmath887 for the stratum @xmath888 one can consider the generalized permutations @xmath889 the theorem is proven. Computation of the rauzy classes here we give explicit examples of _ reduced _ rauzy classes (_ i.e. _ up to the equivalence @xmath839, for any permutation @xmath264 of @xmath19). it is easy to see that there is only one rauzy class filled by (irreducible) generalized permutations defined over @xmath35 letters. in that case the rauzy class contains @xmath36 generalized permutations and a permutation is irreducible if and only if it is dynamically irreducible. thus there is no interesting phenomenon in this `` simple '' case. if we consider a slightly more complicated case, for instance permutations defined over @xmath36 letters we get some interesting phenomenon. figure [rauzy : class:2ii] illustrates such a rauzy class. it corresponds to the stratum @xmath890. the generalized permutations @xmath891 and @xmath892 are not formally in the rauzy class since they are reducible, but we can see there is concretely the `` attraction '' phenomenon. as we can see the (reduced) rauzy classes for generalized permutations are in general much more complicated than the one for usual permutation since the vertex are either of valence one or of valence two. in figure [fig : rauzy:2] we present a more complicated case with an `` unstable '' set of permutations. we end this section with an explicit calculation of the cardinality of the rauzy classes of the four exceptional strata (performed with anton zorich s software @xcite). . [table : rauzy] representatives elements for the special strata. [cols="^,^,^ ",] [] [] @xmath893 [] [] @xmath894 [] [] @xmath895 [] [] @xmath896 [] [] @xmath897 [] [] @xmath898 [] [] @xmath899 [] [] @xmath900 [] [] @xmath901 [] [] @xmath902 [] [] @xmath903 [] [] @xmath904 [] [] @xmath905 [] [] @xmath906 [] [] @xmath907 [] [] @xmath908 [] [] @xmath909 [] [] @xmath910 [] [] @xmath911 [] [] @xmath912 [] [] @xmath913 [] [] @xmath914 [] [] @xmath915 [] [] @xmath916 [] [] @xmath917 [] [] @xmath918 [] [] @xmath919 [] [] @xmath920 [] [] @xmath921 [] [] @xmath922 [] [] @xmath923 [] [] @xmath924 [] [] @xmath925 [] [] @xmath926 [] [] @xmath927 [] [] @xmath928 [] [] @xmath929 [] [] @xmath930 [] [] @xmath931 [] [] @xmath932 [] [] @xmath933 [] [] @xmath934 [] [] @xmath935 [] [] @xmath936 [] [] @xmath937 [] [] 0 [] [] 1.,title="fig : "] [] [] 0 [] [] 1 an example of a rauzy class. the corresponding stratum is @xmath938. there are @xmath939 permutations in the whole `` class '' and @xmath940 permutations in the `` good '' rauzy class. the @xmath941 remaining permutations belong to the reducible part (@xmath942 permutations) and the `` unstable '' part (@xmath943 permutations). note that there is no smaller attractor set : the three irreducible permutations belong to the same rauzy class. let us also note that the extended rauzy class has @xmath944 elements., width=491] An other definition of the extended rauzy class in section [rauzy : classes : giem], we have defined an extended rauzy class by considering the set of generalized permutations obtained from an irreducible permutation @xmath274 by the extended rauzy operations. this set is not in general a subset of the irreducible generalized permutations, therefore we must intersect it with the set of irreducible generalized permutations to get an extended rauzy class. one could also define an extended rauzy class in the following way : it is a minimal subset of the irreducible generalized permutations stable by the operations @xmath104, @xmath105, and @xmath52. it is equivalent to say that we forbid the operation @xmath52 for @xmath143 such that @xmath945 is reducible. for the purpose of this section, let us call this new class a _ weakly extended rauzy class_. a priory, an extended rauzy class is a union of weakly extended rauzy classes. we will prove : all we have to prove is that if two irreducible generalized permutations @xmath946 and @xmath947 correspond to the same connected component of a stratum of quadratic differentials, then we can join them (up to relabelling) by a combination of the maps @xmath104, @xmath105, and @xmath52, such that _ all _ the corresponding intermediary generalized permutations are irreducible. recall that if @xmath3 is irreducible, then so are @xmath28 and @xmath29 (when defined). the idea is now to modify the proof of theorem @xmath37, by using the three following elementary remarks. let @xmath163 be a suspension datum over an irreducible generalized permutation @xmath3 (of type @xmath16). 1. in remark [rem : strongly : irreducible] we gave a condition in order to have @xmath427. equivalently if a decomposition of @xmath3 holds then there is no empty corner. it is obvious to check that, under this condition, @xmath835 is irreducible. 2. let us assume that the two lines joining the end points of @xmath164 and the end points of @xmath168 do not have any other intersection point with @xmath164 and @xmath168. then applying to @xmath441 the matrix @xmath948 for a suitable @xmath949, we get a new suspension data @xmath950 over @xmath10 with @xmath951. hence @xmath835 is irreducible. 3. let @xmath952 minimize the value @xmath953. lemma [lem : sing] implies that there exists @xmath150 such that @xmath683 is the first return map of @xmath10 to the subinterval @xmath954. let us consider @xmath955. by construction @xmath956 satisfies the previous condition, hence @xmath957 is irreducible. let us now prove the proposition. let @xmath946 and @xmath947 be two generalized permutations in the same extended rauzy class. the proof of theorem @xmath37 asserts that there exists a surface @xmath45 and two segments @xmath850 and @xmath851, each one being adjacent to a singularity @xmath852 and @xmath853, such that for each @xmath365, the linear involution @xmath854 given by the first return maps on @xmath249 has combinatorial datum @xmath482. we can assume that @xmath45 has no vertical saddle connection. the previous remark implies that, up to replacing @xmath253 by some @xmath958 for some well chosen @xmath24, one can and do assume that @xmath959 is irreducible. let @xmath960 be the suspension over @xmath253 that corresponds to the surface @xmath45, then up to applying to @xmath163 the matrix @xmath948 for a suitable @xmath949 (which does not change the vertical foliation), we can assume that @xmath427. for @xmath70 large enough, @xmath859 is isomorphic to the first return map on a subinterval @xmath864 of @xmath850, with @xmath868 or @xmath869 a singularity of @xmath253. let @xmath952 that minimizes the value @xmath961 and let @xmath962 be the corresponding point. if @xmath963 then we also have @xmath964 (since @xmath867 can be chosen arbitrarily close to @xmath856). we then apply the rauzy - veech induction to @xmath253 until we get a first return map on @xmath965. if @xmath966 then we also have @xmath967. by definition @xmath163 is a suspension data over @xmath968 (i.e. we are `` rotating by @xmath969 '' the polygon and the linear involution @xmath253). we apply the rauzy - veech induction on @xmath968 until we get a a first return map on @xmath970 that contains @xmath971. the result is a linear involution @xmath972 such that @xmath945 is irreducible, and a suspension @xmath950 over @xmath973. as before we can assume that @xmath974 and then @xmath975 is a suspension over @xmath976 that corresponds to a first return map of @xmath253 on the subinterval @xmath977. moreover the sequence of generalized permutations joining @xmath946 to @xmath978 corresponding to our description consists entirely of irreducible elements. iterating this argument, there will be a step where the point @xmath979 minimizing the value @xmath980 is precisely @xmath856 (because the surface admits a finite number of vertical separatrices starting from the singularities). the same argument produces a sequence of irreducible generalized permutations joining @xmath981 to @xmath947. continued fraction algorithms for interval exchange maps : an introduction , _ frontiers in number theory, physics and geometry _ volume 1 : on random matrices, zeta fonctions and dynamical systems. springer verlag (2006). cole de physique des houches, 2003.

interval exchange maps are related to geodesic flows on translation surfaces ; they correspond to the first return maps of the vertical flow on a transverse segment. the rauzy - veech induction on the space of interval exchange maps provides a powerful tool to analyze the teichmller geodesic flow on the moduli space of abelian differentials. several major results have been proved using this renormalization. danthony and nogueira introduced in @xmath0 a natural generalization of interval exchange transformations, namely the linear involutions. these maps are related to general measured foliations on surfaces (orientable or not). in this paper we are interested by such maps related to geodesic flow on (orientable) flat surfaces with @xmath1 linear holonomy. we relate geometry and dynamics of such maps to the combinatorics of generalized permutations. we study an analogue of the rauzy - veech induction and give an efficient combinatorial characterization of its attractors. we establish a natural bijection between the extended rauzy classes of generalized permutations and connected components of the strata of meromorphic quadratic differentials with at most simple poles, which allows, in particular, to classify the connected components of all exceptional strata.

0710.5614

Introduction general relativity in spacetime dimension @xmath3 larger than four supports black brane solutions that, unlike in lower dimensions, are not uniquely characterized by their asymptotic charges (mass, spin, gauge charges). an example of this situation is the kaluza - klein black hole, a solution of the einstein equations consisting of a black hole embedded in a compactified spacetime, for instance @xmath4. because of the lack of uniqueness in @xmath5, this system exhibits a range of phases, characterized by the horizon topology, as the period @xmath6 of the @xmath7 is varied. for @xmath6 much larger than the horizon length scale, the horizon topology is @xmath8 corresponding to an isolated black hole. as @xmath6 becomes of order @xmath9 one finds uniform and non - uniform black string phases with horizon topology @xmath10. there is evidence to support the conjecture that uniform string decays @xcite proceed via a topology changing phase transition into a black hole final state (see @xcite for reviews). other proposals for the final state of the unstable black string can be found in @xcite. understanding the dynamics of the black hole / black string phase transition is important for a variety of reasons. apart from being a toy model for studying the physics of topology change in higher dimensional general relativity, it is also relevant for its connection to gauge / gravity duality in string theory @xcite. also, the kaluza - klein black hole plays a role in the phenomenology of scenarios where gravity is strong at the tev scale, and production of higher dimensional black holes at the lhc becomes a possibility. there does not exist an analytic solution of the einstein equations describing a black hole in the background @xmath0 with @xmath11 (however, see @xcite ; for @xmath12, a closed form metric can be found in ref. @xcite). for generic values of the ratio @xmath13 one must resort to numerical simulations in order to find solutions. these have been carried out in @xcite. here, we will consider the asymptotic region of the phase diagram in which the parameter @xmath14 is much less than unity, and analytic solutions can be found perturbatively. although this region of parameter space is likely to be far from where the black hole / black string transition is expected to take place, it is a region that can be mapped out analytically. these perturbative calculations provide a useful test of the numerical simulations, and by extrapolation, may give qualitative information on the full phase diagram of solutions. the @xmath1 corrections to the thermodynamics of a small black hole in the background @xmath4 have been calculated in ref. @xcite to leading order for arbitrary @xmath3, and in ref. @xcite to order @xmath2 for @xmath15. in ref. @xcite, the order @xmath1 corrections were calculated by employing a specialized coordinate system @xcite for the entire spacetime. alternatively, the approach taken in @xcite is to split the spacetime into a region near the black hole where the solution is the @xmath3-schwarzschild metric, @xmath16 weakly perturbed by compactification, and a far region in which the metric can be parametrized in terms of asymptotic multipole moments (see ref. @xcite for a systematic discussion of this procedure). these two solutions are then patched together in an overlap region, yielding a relation between the short distance parameters (the scale @xmath9 of the @xmath3-dimensional schwarzschild metric) and the mass @xmath17 and tension @xmath18 as measured by an observer far from the black hole. this behavior can be traced to the short distance singularities of the @xmath3-dimensional flat space green s function. a prescription for handling such divergences at leading order in @xmath1 can be found in @xcite.]. as discussed in @xcite, all thermodynamic quantities relevant to the phase diagram can be calculated given the asymptotic charges " @xmath19. here, we propose a different method for calculating the phase diagram in the perturbative region @xmath20, based on the effective field theory approach applied to extended gravitational systems developed in @xcite. since in the @xmath20 limit there is a large hierarchy between the short distance scale @xmath9 and the compactification size, it is natural to integrate out ultraviolet modes at distances shorter than @xmath9 to obtain an effective lagrangian describing the dynamics of the relevant degrees of freedom at the scale @xmath6. in the resulting eft, the scale @xmath9 only appears in the wilson coefficients of operators in the action constructed from the relevant modes. ignoring horizon absorption @xcite and spin @xcite, these long wavelength modes are simply the metric tensor @xmath21 coupled to the black hole worldline coordinate @xmath22. the couplings of the particle worldline to the metric can be obtained by a fairly straightforward matching calculation, although one expects that all operators consistent with symmetries (diffeomorphism invariance, worldline reparametrizations) are present. although clearly there are some similarities between the eft approach and the matched asymptotics of @xcite, there are several advantages to formulating the @xmath1 expansion in the language of an eft : * in the eft, it is possible to disentangle the terms in the perturbative expansion that arise from the finite extent of the black hole, which scale like integer powers of @xmath13, versus post - newtonian " corrections due to the non - linear terms in the einstein - hilbert lagrangian that scale like integer powers of @xmath23 and are therefore also equivalent to powers of @xmath1. * the eft has manifest power counting in @xmath1. this means that it is possible to determine at what order in the expansion effects from the finite size of the black hole horizon first arise. as we will show in the next section, the first finite size correction, which in the eft manifests itself through a non - minimal coupling of the black hole worldline to the riemann tensor arises at order @xmath24. for a fixed @xmath3 the finite size effects, for example the tidal distortion of the black hole horizon, will contribute to the thermodynamic variables at order @xmath25 relative to the leading order result. thus the finite horizon effects become as large as @xmath2 as @xmath26. this also indicates that the results of refs. @xcite are not sensitive to the specific structure of the kaluza - klein black hole, but rather reflect the thermodynamics of structureless point particles. * in the eft, calculations can be carried out using the standard tools of field theoretic perturbation theory. in particular, the perturbative expansion has a diagrammatic interpretation in terms of standard feynman diagrams. ultraviolet divergences that arise in feynman integrals can be dealt with using a standard regulator (e.g, dimensional regularization) and absorbed into the coefficients of local operators. there is no impediment to renormalizing the theory to all orders in @xmath1. as an example of this procedure we calculate in sec. [sec : pt] the @xmath27 corrections to the asymptotic mass and tension of the kaluza - klein black hole. our results are organized as follows. in sec. [sec : eft] we formulate the eft and derive the power counting rules for @xmath20. using this power counting we analyze the relative contribution of an arbitrary finite size worldline operator. in sec. [sec : pt] we use the eft to calculate the @xmath27 corrections to the asymptotic charges @xmath17 and @xmath18 for arbitrary @xmath3 and use these results in sec. [sec : thermo] to work out the corresponding corrections to the thermodynamic relations. in this section we also compare our analytic formulas to the results of numerical simulations @xcite for @xmath15 and @xmath28. The effective field theory we consider an isolated black hole in a background spacetime of the form @xmath0. coordinates on @xmath29 are denoted by @xmath30 and @xmath31 labels circumference along @xmath7. coordinates on @xmath3-dimensional spacetime are denoted @xmath32. the period of the @xmath7 factor as measured by an observer at @xmath33 is @xmath6. in order to determine the phase diagram of this system, it is sufficient to calculate the moments of the kaluza - klein black hole that appear in the first non - trivial corrections to the asymptotic metric. by the symmetries of the background, the non - vanishing terms are, to leading order as @xmath34, @xmath35 the coefficients @xmath36 and @xmath37 are related to the asymptotic mass @xmath17 and tension @xmath18 by the relations @xcite, @xmath38 the constant @xmath39 is defined such that the newton potential between two masses in uncompactified @xmath3-dimensional space is @xmath40. in the limit @xmath41, these quantities can be calculated in perturbation theory. one method is to solve the einstein equations perturbatively, using the matched asymptotic techniques of @xcite. another possibility is to first integrate out the black hole, replacing the spacetime in the vicinity of the horizon with an effective lagrangian for the black hole worldline coupled to gravity. including all terms with up to two derivatives (we will be more specific about the expansion parameter in this expansion below) this lagrangian takes the form @xmath42 -m_0\int ds + c_e\int ds e_{\alpha\beta } e^{\alpha\beta } + c_b\int ds b_{\alpha_1\cdots\alpha_{d-2 } } b^{\alpha_1\cdots\alpha_{d-2 } } + \cdots.\]] here, as in any other eft, we have simply written down all terms compatible with diffeomorphism invariance and worldline reparametrization invariance. in this equation, @xmath43 and the tensors @xmath44, and @xmath45 are the electric and magnetic components of the riemann tensor. @xmath46 note that if the black hole is in a ricci flat background then operators involving the ricci tensor can be removed by field redefinitions of the metric. in this case the components of @xmath47 are sufficient to specify the riemann tensor. all coefficients of operators in eq. ([eq : efts]) scale like powers of @xmath48 and @xmath9, given by @xmath49 in a way that we can be fixed by matching to the full schwarzschild solution (see below). starting from eq. ([eq : efts]), we use the background field method @xcite to calculate @xmath17 and @xmath18. we decompose the metric tensor into a long wavelength non - dynamical background field @xmath50 and a short wavelength graviton field @xmath51 @xmath52 and do the path integral over @xmath53, holding the black hole worldline to some fixed value @xmath22, @xmath54 = \int { \cal d } h_{\alpha\beta } \exp i \left(s[{\bar g}+h, x] + s_{gf}[{\bar g},h]\right),\]] where @xmath55 is a suitable gauge fixing term. it is convenient to choose @xmath55 to be compatible with background field diffeomorphisms, for example @xmath56 with @xmath57. to calculate eq. ([eq : pi]) it is sufficient to linearize about flat space, @xmath58 and to take @xmath59. the relation between @xmath60 and @xmath61 can be read off the linear terms in @xmath62 $] with no derivatives @xmath63 = -{1\over 2 } m\int dt { \bar h}_{00 } + { 1\over 2 } \tau l\int dt { \bar h}_{zz } + \cdots.\]] because of eq. ([eq : pi]), these two terms are simply the sum of feynman diagrams like those of fig. [eb], fig. [m2] and fig. wavy internal lines denote the propagator for the graviton @xmath53, which given our form for @xmath55 is @xmath64 with @xmath65 $], and @xmath66 is the kaluza - klein representation of the propagator on flat @xmath0. the solid lines denote the black hole worldline. there are no propagators associated with such lines. an external line denotes an insertion of a factor of @xmath67. finally, the vertices are constructed from the @xmath68-graviton terms in the expansion of eq. ([eq : efts]) about flat space. diagrams that become disconnected by the removal of the particle worldline do not contribute to the terms in @xmath62 $]. note also that if we treat eq. ([eq : tad]) as an effective source term in the einstein equations, we recover the relation eq. ([eq : ab]) between the metric coefficients @xmath69 and the thermodynamic charges. each feynman diagram in the eft contributes a definite power of @xmath1 to the terms in eq. ([eq : tad]). counting powers of @xmath1 is straightforward. given that the only scale in the propagator is @xmath6 we assign @xmath70 and thus @xmath71 so that we assign the scaling @xmath72. we assign no power counting factors to @xmath67. power counting relative to the action for the free background graviton, the parameter @xmath73 that counts graviton loops is @xmath74 this means that @xmath75 and for example the terms @xmath76 obtained by expanding eq. ([eq : efts]), scale as @xmath77. therefore the diagram in fig. [m2](b) gives a contribution to @xmath62 $] that scales like (@xmath78 denotes a time ordered vev in the free graviton theory), @xmath79 so that by eq. ([eq : lo]) it gives a contribution to @xmath17 that is suppressed by a single power of @xmath1 relative to the leading order result @xmath80. likewise, the @xmath81 vertex in the gravitational action is @xmath82 so that fig. [m2](a) scales like @xmath83. in general, a given diagram scales as @xmath84, where @xmath85 and @xmath86, the latter bound saturated by diagrams containing no internal graviton loops. in order to power count the worldline operators with more derivatives, we first need to fix the dependence of the coefficients on @xmath48 and @xmath9. this is done by matching the effective lagrangian of eq. ([eq : efts]) to the full black hole theory, described by the schwarzschild metric of eq. ([eq : bhmetric]). as in any other eft, the matching procedure consists of adjusting the couplings in the effective lagrangian so that observables calculated in the eft agree with those of the full theory. a convenient observable to match to is the @xmath87-matrix element for low energy elastic graviton scattering off the black hole geometry. in the full theory, this is obtained by solving the linearized wave equation for the graviton field in the schwarzschild metric. after separation of variables this boils down to solving a radial equation that generalizes the regge - wheeler equation describing perturbations of four - dimensional schwarzschild black holes to @xmath3-dimensions. the explicit form of this equation can be found in @xcite (see also @xcite). since the only scale in the full theory is @xmath9 we expect the amplitude to take the form @xmath88 where @xmath89 is the energy of the incident graviton, @xmath90 are spin labels, @xmath91 is the scattering angle and @xmath92 is a calculable function. in the eft, the scattering cross section receives contributions from insertions of all the couplings in eq. ([eq : efts]). in particular, the two - derivative operators give rise to terms in the amplitude that go like @xmath93,\]] where @xmath94 are functions whose specific form is not important for our purposes here. thus @xmath95 are non - zero only if @xmath92 has a term in the low energy limit that scales like @xmath96. if this is the case then we find that @xmath97. after expanding about flat space, we have for @xmath98, @xmath99, @xmath100 the first contribution to the tadpoles in @xmath62 $] due to an insertion of @xmath101 is from the diagram in fig. [eb]. according to our power counting rules @xmath102 implying that the @xmath20 thermodynamics is not sensitive to the structure of the black hole until order @xmath103, which for @xmath15 is one order beyond the second order results of @xcite and becomes @xmath27 as @xmath26. more generally, a worldline operator with @xmath104 derivatives and @xmath105 factors of the graviton scales like @xmath106 and it gives a contribution to the charges @xmath17 and @xmath18 that is order @xmath107 (@xmath108). Asymptotic charges as an application of the eft method, we now compute the @xmath27 corrections to the quantities @xmath60 that govern the thermodynamics of the kaluza - klein black hole. according to the power counting rules established in the previous section, the relevant diagrams are those of fig. [m2] and fig. finite size effects do not come in at this order. the first corrections to the mass and tension of the system arise from the two diagrams in fig. the diagram in fig. [m2](a) gives a contribution to the background field effective action which, using the feynman rules of the eft, is of the form @xmath109 ^ 2 v_{k_\perp n}({\bar h}),\]] where the vertex function is @xmath110 { \bar h}_{\mu\nu}.\]] since we are only interested in the terms of eq. ([eq : tad]) we have set @xmath111 to a constant, in which case the momentum flowing into the diagram vanishes and the calculation of the integral simplifies. for the @xmath112 term, eq. ([eq : v]) gives @xmath113 where we have used @xmath114 where @xmath115 is the riemann zeta function. note that this integral is actually ultraviolet divergent. the divergence renormalizes the point particle mass and can be absorbed by a shift in @xmath116. we use dimensional regularization to deal with this. since we are interested in @xmath11, the divergent part of the integral is simply set to zero by the regulator. for the @xmath117 term we have @xmath118 here we have used the additional integral @xmath119 ^ 2 = \left(2-{d\over 2}\right) i_0(l).\]] since the source is at rest, fig. [m2](b) only gives a contribution to the @xmath112 tadpole @xmath120 combining these results we find to first order in @xmath1 @xmath121 where we have defined @xmath122. this reproduces the results of @xcite. it is convenient to consider separately the corrections to the @xmath123, @xmath124 terms in the effective action. for the @xmath124 terms, the diagrams in fig. [m3](a), fig. [m3](c), and fig. [m3](e) do not contribute. it is straightforward to derive the feynman rules necessary to calculate the diagrams of fig. we will simply write down the results of evaluating each diagram. for the @xmath123 tadpoles in the effective action, one finds that evaluating the diagrams at zero external momentum gives rise to no new integrals : the integration over the internal momenta factorizes into the square of the integrals @xmath125 of the previous section. the results are @xmath126 for the tadpole terms @xmath124, we find from fig. [m3](b) @xmath127 the contribution of graphs fig. [m3](d), fig. [m3](f) to the tadpole @xmath123 does not factorize into the integrals of the form @xmath125. however their sum does, @xmath128 thus the @xmath129 terms in the effective action are @xmath130\right|_{\ell\lambda^2 } = -{1\over 4}m_0{\hat\lambda}^2 \int dt { \bar h}_{00 } -{1\over 2 } m_0 (d-3) { \hat\lambda}^2 \int dt { \bar h}_{zz}.\]] Thermodynamics we have found, from the diagrams in fig. [m2] and fig. [m3] @xmath131 to obtain observables which can be tested against the numerical data of @xcite, we must eliminate the unphysical bare mass parameter @xmath116 from these two equations. this gives, @xmath132 which agrees with the results of @xcite when @xmath15. we may then relate the asymptotic charge to the thermodynamics quantities via smarr s relation @xmath133 (see ref. @xcite) which, using eq. ([eq : tvm]), gives @xmath134 as a function of @xmath135. as @xmath136 the entropy @xmath87 simply becomes the entropy of an isolated @xmath3-dimensional black hole. this scales like the area of the black hole, @xmath137. thus for @xmath20 we expect @xmath138 the function @xmath139 can be obtained from the relation @xmath140 together with the formula for @xmath134 that follows from smarr s law. we finds @xmath141 and @xmath142\zeta^2(d-3) \left({2 g_n m\over l^{d-3}}\right)^2+\cdots,\]] where @xmath143 and @xmath144 are the entropy and temperature of an uncompactified black hole, @xmath145 we may then compare with the numerical results of kudoh and wiseman @xcite in the special cases of five and six dimensions which are shown in fig. [fig : data](a) and fig. [fig : data](b) respectively in units where the entropy of the uncompactified @xmath3-dimensional black hole is @xmath146.]. the difference between the numerical data and the analytical results grows with @xmath87, but it is difficult to gauge the relevance of this deviation without some measure of the errors in the numerical computation. as a crude measure of convergence of the perturbative expansion we plot in fig. [fig : conv] the ratio of the @xmath27 to the @xmath147 terms in the series expansion for @xmath148 versus @xmath87. to @xmath147 terms in the perturbative expansion of @xmath149 versus @xmath87 for @xmath15 (dashed line) and @xmath28 (solid line).,width=302] Conclusions in this paper, we have used eft methods to determine the qualitative structure of the thermodynamics of kaluza - klein black holes when their radius is much smaller than the compactification scale. using the power counting in the eft, we find that the asymptotic charges @xmath19 are related in the regime @xmath150 by an expansion of the form @xmath151 where @xmath152 is analytic about zero and @xmath153. for @xmath154, we find @xmath155,\]] which agrees with the @xmath156 results of @xcite in @xmath3 spacetime dimensions and with the @xmath157 @xmath15 results of @xcite calculated in perturbation theory about the full uncompactifed schwarzschild background. thus our results indicate that the existing analytical tests of the numerics for @xmath158 only probe the thermodynamics of point particles and are not sensitive to the dynamics of the black hole horizon. it would be interesting to repeat the numerical simulations for large dimension where the phase diagram is more sensitive to the structure of the horizon. for instance in @xmath159, the first finite size effect comes in at order @xmath160, @xmath161, and is distinguishable from terms in the black hole thermodynamics that can be reproduced by a minimal point particle action @xmath162. + + we thank h. kudoh and t. wiseman for making the numerical results of @xcite available to us, and t. wiseman for helpful discussions. y - zc and wg are supported in part by the department of energy under grant de - fg02 - 92er40704. izr is supported in part by the department of energy under grants doe - er-40682 - 143 and deac02 - 6ch03000. r. gregory and r. laflamme, `` black strings and @xmath104-branes are unstable, '' phys. rev. lett. * 70 *, 2837 (1993) [arxiv : hep - th/9301052]. r. gregory and r. laflamme, `` the instability of charged black strings and @xmath104-branes, '' nucl. b * 428 *, 399 (1994) [arxiv : hep - th/9404071]. b. kol, `` the phase transition between caged black holes and black strings : a review, '' arxiv : hep - th/0411240. t. harmark and n. a. obers, `` phases of kaluza - klein black holes : a brief review, '' arxiv : hep - th/0503020. g. t. horowitz and k. maeda, `` fate of the black string instability, '' phys. lett. * 87 *, 131301 (2001) [arxiv : hep - th/0105111]. s. s. gubser, `` on non - uniform black branes, '' class. * 19 *, 4825 (2002) [arxiv : hep - th/0110193]. o. aharony, j. marsano, s. minwalla and t. wiseman, `` black hole - black string phase transitions in thermal 1 + 1 dimensional supersymmetric yang - mills theory on a circle, '' class. * 21 *, 5169 (2004) [arxiv : hep - th/0406210]. o. aharony, j. marsano, s. minwalla, k. papadodimas, m. van raamsdonk and t. wiseman, `` the phase structure of low dimensional large @xmath163 gauge theories on tori, '' arxiv : hep - th/0508077. t. harmark and n. a. obers, `` black holes on cylinders, '' jhep * 0205 *, 032 (2002) [arxiv : hep - th/0204047]. r. c. myers, `` higher dimensional black holes in compactified space - times, '' phys. d * 35 *, 455 (1987). e. sorkin, b. kol and t. piran, phys. d * 69 *, 064032 (2004) [arxiv : hep - th/0310096]. h. kudoh and t. wiseman, `` properties of kaluza - klein black holes, '' prog. phys. * 111 *, 475 (2004) [arxiv : hep - th/0310104]. h. kudoh and t. wiseman, `` connecting black holes and black strings, '' phys. lett. * 94 *, 161102 (2005) [arxiv : hep - th/0409111]. t. harmark, phys. d * 69 *, 104015 (2004) [arxiv : hep - th/0310259]. d. gorbonos and b. kol, `` matched asymptotic expansion for caged black holes : regularization of the post - newtonian order, '' class. * 22 *, 3935 (2005) [arxiv : hep - th/0505009]. d. karasik, c. sahabandu, p. suranyi and l. c. r. wijewardhana, `` analytic approximation to 5 dimensional black holes with one compact dimension, '' phys. rev. d * 71 *, 024024 (2005) [arxiv : hep - th/0410078]. d. gorbonos and b. kol,a dialogue of multipoles : matched asymptotic expansion for caged black holes, " jhep * 0406 *, 053 (2004) [arxiv : hep - th/0406002]. t. harmark and n. a. obers, `` new phase diagram for black holes and strings on cylinders, '' class. * 21 *, 1709 (2004) [arxiv : hep - th/0309116]. b. kol, e. sorkin and t. piran, `` caged black holes : black holes in compactified spacetimes. i : theory, '' phys. d * 69 *, 064031 (2004) [arxiv : hep - th/0309190]. w. d. goldberger and i. z. rothstein, `` an effective field theory of gravity for extended objects, '' arxiv : hep - th/0409156. w. d. goldberger and i. z. rothstein, `` dissipative effects in the worldline approach to black hole dynamics, '' arxiv : hep - th/0511133. r. a. porto, `` post - newtonian corrections to the motion of spinning bodies in nrgr, '' arxiv : gr - qc/0511061. b. dewitt, in _ relativity, groups and topology, proceedings of the les houches summer school of theoretical physics, _ eds. c. dewitt and b. dewitt, gordon and breach, 1964. h. kodama and a. ishibashi, `` a master equation for gravitational perturbations of maximally symmetric black holes in higher dimensions, '' prog. phys. * 110 *, 701 (2003) [arxiv : hep - th/0305147]. a. ishibashi and h. kodama, `` stability of higher - dimensional schwarzschild black holes, '' prog. phys. * 110 *, 901 (2003) [arxiv : hep - th/0305185].

we study the thermodynamics of small black holes in compactified spacetimes of the form @xmath0. this system is analyzed with the aid of an effective field theory (eft) formalism in which the structure of the black hole is encoded in the coefficients of operators in an effective worldline lagrangian. in this effective theory , there is a small parameter @xmath1 that characterizes the corrections to the thermodynamics due to both the non - linear nature of the gravitational action as well as effects arising from the finite size of the black hole. using the power counting of the eft we show that the series expansion for the thermodynamic variables contains terms that are analytic in @xmath1, as well as certain fractional powers that can be attributed to finite size operators. in particular our operator analysis shows that existing analytical results do not probe effects coming from horizon deformation. as an example , we work out the order @xmath2 corrections to the thermodynamics of small black holes for arbitrary @xmath3, generalizing the results in the literature.

hep-th0602016

Introduction the mauritius radio telescope (mrt) @xcite is a fourier synthesis, t - shaped non - coplanar array operating at 151.5 mhz. the telescope was built to fill the gap in the availability of deep sky surveys at low radio frequencies in the southern hemisphere. the aim of the survey with mrt is to contribute to the database of southern sky sources in the declination (@xmath3) range @xmath4 to @xmath5, covering the entire right ascension (@xmath6), with a synthesised beam of @xmath7 and an expected point source sensitivity (1-@xmath0) of @xmath8 mjy beam@xmath9. the _ zenith angle _ @xmath10 is given by @xmath11, where, @xmath12 @xmath13 is the latitude of mrt. mrt has been designed to be the southern - sky equivalent of the cambridge 6c survey at 151.5 mhz @xcite. the next generation radio telescopes, like the low frequency array (lofar) and the murchison widefield array (mwa), that are being built are low frequency arrays ; clearly indicating a renewed interest in metre - wavelength astronomy. the key astrophysical science drivers include acceleration, turbulence and propagation in the galactic interstellar medium, exploring the high redshift universe and transient phenomenon, as well as searching for the redshifted signature of neutral hydrogen from the cosmologically important epoch of reionisation (eor). the surveys made using such arrays will provide critical information about foregrounds which will also provide a useful database for both extragalactic and galactic sources. mrt survey at 151.5 mhz is a step in that direction and, in addition, will provide the crucial sky model for calibration. imaging at mrt is presently done only on the meridian to minimise the problems of non - coplanarity. a two - dimensional (2-d) image in @xmath6-@xmath14 coordinates is formed by stacking one - dimensional (1-d) images on the meridian at different sidereal times. images of @xmath15 a steradian @xmath16 of the southern sky, with an rms noise in images of @xmath17 mjy beam@xmath9 (1-@xmath0), were produced by @xcite. a suite of programs developed in - house was used to reduce @xmath18 hours of the survey data (a quarter of the total @xmath19 hours observed over a span of @xmath20 years). the deconvolved images and a source catalogue of @xmath21 sources were published by @xcite. systematics in positional errors were found when the positions of sources common to mrt catalogue and the molonglo reference catalogue (mrc) @xcite were compared. @xcite treated the systematics in errors in @xmath6 and @xmath14 independently. by estimating two separate 1-d least - squares fits for errors in @xmath6 and @xmath14 the systematics were corrected only in the source catalogue. however, errors remained in the images which impede usefulness of mrt images for multi - wavelength analysis of sources. in addition, the source of errors was not investigated. at mrt, the visibility data is processed through several complex stages of data reduction specific to the array, especially, arising due to its non - coplanarity @xcite. it was therefore decided to correct for errors in the image domain and avoid re - processing the visibility data. this paper describes the application of 2-d homography, a technique ubiquitous in the computer vision and graphics community, to correct the errors in the image domain. homography is used to estimate a transformation matrix (which includes rotation, translation and non - isotropic scaling) that accounts for positional errors in the linearly gridded 2-d images. in our view, this technique will be of relevance to the new generation radio telescopes where, owing to huge data rates, only images after a certain integration would be recorded as opposed to raw visibilities @xcite. this paper also describes our investigations tracing the positional errors to errors in the array geometry used for imaging. our hypothesis on the array geometry, its subsequent confirmation endorsed by re - estimation of the array geometry and its effect on the images are also described. the rest of the paper is organised as follows. section [s : poserror] compares positions of sources common to mrt catalogue and mrc. the 2-d homography estimation is briefly described in section [s : homography]. section [s : scheme] presents the correction scheme and typical results. the re - estimation of mrt array geometry is described in section [s : arraygeometry]. finally, we summarise and present our conclusions in section [s : conclusions]. Positional errors the positions of sources common to mrt catalogue and mrc were compared. we used mrc because of its overlap with mrt survey, its proximity in frequency compared to other reliable catalogues available and, comparable resolution @xmath22. moreover, for sources of listed flux density @xmath23 jy (at 408 mhz) the catalogue is reported to be substantially complete and, the reliability is reported to be 99.9% @xcite. for our further discussions, errors in mrc source positions are considered random, without any systematics. about 400 bright sources common to the two catalogues and with flux density at 151.5 mhz greater than 5 jy (@xmath24-@xmath0) were identified and their positions were compared. the sources were labelled as common if they lie within @xmath25 of each other. since mrc has a source density of @xmath26 source deg@xmath27, the chances of considering two unrelated sources as common are extremely low. a flux threshold of 15-@xmath0 ensures a source population abundant to reliably estimate homography (explained in next section). the positional errors in @xmath6 and @xmath3 show no systematics as a function of @xmath6 (refer _ first rows _ of fig. [f : sourcecomparison]a and [f : sourcecomparison]b). for visualisation, the errors are shown in percentages of mrt beamwidths. the errors in @xmath6 and @xmath3 show a linear gradient as a function of @xmath14. the errors in @xmath6, plotted against @xmath14, reach @xmath28 of the mrt beamwidth (refer _ second row _ of fig. [f : sourcecomparison]a). whereas, the errors in @xmath29, plotted against @xmath14, are significant and reach @xmath30 of mrt beamwidth. (refer _ second row _ of fig. [f : sourcecomparison]b). histograms in fig. [f : sourcecomparison]c and fig. [f : sourcecomparison]d show the distribution of errors in @xmath6 and @xmath3, respectively. the histogram of errors in @xmath3 shows a broader spread compared to errors in @xmath6. re - imaging, to correct for errors in the images, would involve re - reducing the @xmath31 hours of observed data. owing to the complexity involved it was decided to correct for the positional errors in the images, thus avoiding re - processing. the 2-d homography estimation technique was employed for correcting positional errors in images and is discussed in detail in the following section. 2-d homography the 2-d planar homography is a non - singular linear relationship between points on planes. given two sets of @xmath32 corresponding image points in projective coordinates, @xmath33, homography maps @xmath34 to the corresponding @xmath35 @xcite. where, @xmath36. the homography sought here is a non - singular @xmath37 matrix @xmath38 such that : @xmath39= \left [\begin{array}{ccc } h^{}_{11}&h^{}_{12}&h^{}_{13}\\ h^{}_{21}&h^{}_{22}&h^{}_{23}\\ h^{}_{31}&h^{}_{32}&h^{}_{33 } \end{array}\right] \left [\begin{array}{c } x^{}_{k}\\y^{}_{k}\\1\end{array}\right]. \label{eq : inhomog}\]] where, @xmath40 and @xmath41 represent @xmath42 of @xmath32 corresponding mrt and mrc sources, respectively. in equation [eq : inhomog], @xmath43 and @xmath44 are referred to as the _ homogeneous coordinates _ and are always represented one dimension higher than the dimension of the problem space. this is a commonly used representation in computer graphics. the simple reason is that with a @xmath45 matrix one can only _ rotate _ a set of 2-d points around the origin and _ scale _ them towards or away from the origin. a @xmath45 matrix is incapable of _ translating _ a set of 2-d points. the homogeneous coordinates allow one to express a translation as a multiplication. a single @xmath46 matrix, with homogeneous coordinates, can account for rotation, scaling and translation of 2-d coordinates. for example, from equation [eq : inhomog], @xmath47. notice, @xmath48 (representing translation in @xmath6-dimension) is simply being added to the normal dot product @xmath49 that together represents rotation and scaling. in homogeneous coordinates, the 2-d problem space is a plane hovering in the third dimension at a unit distance. a general homography matrix, for projective transformation, has 8 degrees - of - freedom (dof). for our system, both errors in @xmath6 and @xmath3 have only @xmath14-dependency. therefore, a less general, 2-d affine transformation is sufficient. a 2-d affine transformation (two rotations, two translations and two scalings) requires 6-dof @xcite, therefore in @xmath38, @xmath50 and @xmath51 are zero. since each 2-d point provides two independent equations, a minimum of 3 point correspondences are necessary to constrain @xmath38 in the affine space. a set of @xmath32 such equation pairs, contributed by @xmath32 point correspondences, form an over - determined linear system : @xmath52 @xmath53\mbox {, } \]] @xmath54^{t}\mbox{and, } \]] @xmath55^{t}.\]] in equation [e : system], @xmath56 represents transpose of a matrix. this system can be solved by least squares - based estimators. at this stage it is useful to consider the effect of using (@xmath57)-coordinates to represent the brightness distribution on the celestial sphere. ideally, it is the directional cosines @xmath58, with respect to the coordinates of the array, which represent the spherical coordinates in the sky. therefore, the image coordinates in which homography should in principle be estimated are @xmath59. however, at mrt, for 1-d imaging on the meridian : @xmath60 therefore, @xmath14 is a natural choice for one of the coordinates and is indeed used in the present case. on the meridian, the directional cosine @xmath61 is zero. for small errors, @xmath62, in @xmath61, i.e. close to the meridian : @xmath63 @xmath64 [e : sinza] here, @xmath65 is the error in @xmath6. equation [e : sinza] shows that an error in @xmath61 will lead to an error in @xmath6 with a @xmath66-dependence. the 2-d images of mrt are 1-d images on the meridian made at different sidereal times and stacked. therefore, positional errors both in @xmath6 and @xmath3 do not show systematics as a function of @xmath6 (_ first rows _ of figs. [f : sourcecomparison]a and [f : sourcecomparison]b). we preferred @xmath42-representation because all mrt images were already generated in this coordinate system. this choice compelled us to seek solutions for errors in @xmath6 as a function of @xmath14 rather than @xmath66. we plotted errors in @xmath6 against both @xmath66 and @xmath14 and obtained separate linear least - squares fits. the rms of residuals in both fits is @xmath67 of the beamwidth in @xmath6. however, the rms of difference between the fitting functions @xmath66 and @xmath14 in the @xmath3 range of mrt @xmath68 is only @xmath69 of the beamwidth in @xmath6. therefore, the random errors in the source positions are larger than the errors introduced by the preferred @xmath42-coordinates for @xmath34 and @xmath35. in @xmath34 and @xmath35, the @xmath6 ranges from 18 hours to 24 hours and the @xmath14 ranges from @xmath70 to @xmath71 (corresponding to the declination range of @xmath72 to @xmath73). moreover, in matrix @xmath74 (refer equation [e : system]) there are entries of 1 s & 0 s. such a matrix is ill - conditioned and in the presence of noise in the source positions, the solution for an over - determined system may diverge from the correct estimate @xcite. the effect of an ill - conditioned matrix is that it amplifies the divergence. a normalisation (or pre - conditioning) is therefore required. to obtain a good estimate of the transformation matrix we adopted the normalisation scheme proposed by @xcite. the normalisation ensures freedom on arbitrary choices of scale and coordinate origin, leading to algebraic minimisation in a fixed canonical frame. the homography matrix @xmath75 is estimated from normalised coordinates by the least - squares method using singular value decomposition (svd). the matrix is then denormalised to obtain @xmath38. the scheme is briefly described below : 1. * normalisation of @xmath76 : * compute a transformation matrix @xmath77, consisting of a translation and scaling, that takes points @xmath34 to a new set of points @xmath78 such that the centroid of the points @xmath78 is the coordinate origin @xmath79, and their average distance from the origin is @xmath80. * normalisation of @xmath81 : * compute a similar transformation matrix @xmath82, transforming points @xmath35 to @xmath83. * estimate homography : * estimate the homography matrix @xmath75 from the normalised correspondences @xmath84 using the algorithm described earlier in the main section. * denormalisation : * the final homography matrix is given by : @xmath85 The correction scheme [f : scheme] shows the block schematic of the correction scheme. at mrt, the full declination range for each sidereal hour range is divided into 4 _ zones _ (refer _ second row _ in fig. [f : sourcecomparison]a or [f : sourcecomparison]b). each zone is imaged with different delay settings to keep the bandwidth decorrelation to @xmath86. therefore, the 6 sidereal hours of images under consideration, have 24 images (@xmath87). using the population of common sources, there are four possible alternatives to correct mrt images by computing : 1. 24 homography matrices - one for each image. 2. 6 matrices - one for each sidereal hour. 3. 4 matrices - one for each declination zone. 4. a single homography matrix for the entire steradian. in principle, bright sources in each image (@xmath88) can be used to independently estimate a homography matrix. our earlier experiments to correct each image independently showed that the homography matrices were similar. the plots of errors in @xmath6 and @xmath3 plotted against @xmath6 and @xmath14 (refer to fig. [f : sourcecomparison]a and [f : sourcecomparison]b) indicate that the errors are independent of the four delay zones and the range of @xmath6. this implies that estimating a single homography matrix for the entire source population should suffice in representing the errors. the homography matrix estimated using @xmath89 common sources (described in section [s : poserror]) is : @xmath90. \label{eq : matrixvalues}\]] in the estimated homography matrix, @xmath91 indicates there is no correction required in @xmath6 as a function of @xmath6. @xmath92 indicates mrt images should be corrected in @xmath6 with a @xmath14 dependence. the estimated correction is up to @xmath93 of the beam in @xmath6, at the extreme ends of the @xmath14 range. similarly, @xmath94 indicates that there is no correction required in @xmath14, as a function of @xmath6. however, @xmath95 indicates that mrt images should be compressed in @xmath14 by a factor of 0.9990 (which is @xmath1 part in 1000). the values of @xmath48 and @xmath96 indicate that the zero cross - overs of errors in both @xmath6 and @xmath14 plotted against @xmath14 are close to the @xmath14 of the calibration source (mrc1932 - 464) used for imaging. using equation [eq : inhomog], the homography matrix is used to project each pixel from the images to a new position, effectively correcting for positional errors in images. [f : sourcecomparisoncorrect] shows positional errors in @xmath3 after homography - based correction. a comparison of these plots with fig. [f : sourcecomparison] demonstrate that homography has removed the systematics and the residual errors are within 10% of the beamwidth for sources above 15-@xmath0, as expected. [f : scatterplot]a and fig. [f : scatterplot]b show scatter plots of errors in @xmath3 against errors in @xmath6 before and after correction, respectively. for visualisation, the errors are represented in percentages of respective mrt beamwidths. notice, after correction (refer fig. [f : scatterplot]b) the scatter is almost circular as opposed to elliptical before correction (refer fig. [f : scatterplot]a). the rms before correction is @xmath97 of the beamwidth. after correction, the rms is reduced to @xmath98 of the beamwidth and, the systematic errors have been removed. [f : contourplots]a and [f : contourplots]b show mrt contours before and after correction, respectively, overlaid on sumss (sydney university molonglo sky survey) image @xcite, for a source around @xmath99. the corrected mrt image contours in fig. [f : contourplots]b overlap with the source in sumss image. [f : contourplots]c and [f : contourplots]d show similar comparison for a source around @xmath100. notice fig. [f : sourcecomparison]d, since the errors around @xmath100 are within 10% of the beamwidth, the contours in both figs. [f : contourplots]c and [f : contourplots]d show a good overlap as expected and homography has not applied perceivable correction to images at this declination. we have overlaid mrt contours on a number of extended sources at 843 mhz reported by @xcite. [f : contourplotsextsources] shows a typical overlay of mrt contours on sumss image of a region around the cluster abell 3667. the overlay is perceivably satisfactory. the 2-d homography corrected the positional errors in the image domain. for imaging the remaining @xmath101 steradians of mrt survey, @xmath102 hours of data has to be reduced. ideally, for imaging the new regions, one would like to trace the source of these errors and correct them in the visibilities. in the following section we discuss how we traced the source of errors and corrected them in the visibility domain. Array geometry: hypothesis & re-estimation this section describes our _ expansion - compression _ hypothesis for the source of errors in our images. the subsequent corrections we estimated and applied to eliminate the errors are also described. for meridian transit imaging, @xmath103. the brightness distribution in the sky as a function of @xmath14 and the complex visibilities measured for different values of the north - south (ns) baseline vector component @xmath104 form a fourier pair @xcite. a scaling error of @xmath105 in @xmath106 will result in a scaling factor of @xmath107 in the @xmath104-component of the baseline vector. by positional error analysis it is clear that mrt images are stretched (_ expanded _) in declination, i.e., @xmath108 @xmath109 [e : expcomp] note, for images the 2-d homography estimated a correction (_ compression _) factor, @xmath107, of 0.9990. this cued to the hypothesis that we have compressed the north - south baseline vectors. equation [e : expcomp]b means, a baseline distance of @xmath110 m in the ns arm was wrongly measured as @xmath111 m (1 part in 1000). similarly, a @xmath14-dependent correction in @xmath6 cued to possible @xmath104-component in the east - west (ew) baseline vectors. next, we describe the re - estimation of array geometry. we begin with a brief description of the mode of observations with mrt. mrt has 32 fixed antennas in the ew arm and 15 movable antenna trolleys in the ns arm. for measuring visibilities, the 15 ns trolleys are configured by spreading them over 84 m with an inter - trolley spacing of 6 m (to avoid shadowing of one trolley by another). mrt measures different fourier components of the brightness distribution of the sky in 63 different configurations (referred to as _ allocations _) to sample ns baselines every 1 m. therefore, effectively, there are 945 antenna positions (63 allocations * 15 antennas / allocation) in the ns arm and a total of 30,240 (945 * 32) visibilities are used for imaging. a small error in a measuring scale of relatively shorter length is likely to build up systematically while establishing the geometry of longer baselines. this effect would be observed in the instrumental phases estimated using different calibrators. in principle, the instrumental phases estimated using two calibrators at different declinations, for a given baseline, should be the same, allowing for temporal variations in the instrumental gains. a non - zero difference in these estimates may be due to positional errors of the baseline or positions of calibrators. as mentioned earlier, our analysis of positional error in sources and the homography matrix cued to positional errors in baselines (or antenna positions). the simple principle of astrometry @xcite was used to estimate errors in antenna positions and is discussed below. the observed visibility phase, @xmath112, in a baseline with components @xmath113, due to calibrator @xmath114 with direction cosines @xmath115, is given by : @xmath116 where, @xmath117 represents true instrumental phases, @xmath118 represents ew antennas and @xmath119 represents ns antennas. for meridian transit imaging equation [e : obsphasebasiceqn] becomes : @xmath120 the instrumental phases, @xmath121, estimated using the measured geometry are given by : @xmath122 here, @xmath123 and @xmath124 are errors in the assumed baseline vectors. @xmath121 are phases of complex baseline gains obtained in the process of calibration. equation [eq : calphaserelation] has three unknowns. to reduce the number of unknowns, one can eliminate the true instrumental phases by taking a difference @xmath125 between the instrumental phases estimated using two calibrators. this difference gives : @xmath126 } \nonumber \\ \lefteqn{\hspace{22 mm } + \delta w^{}_{ij } \left[\cos\left(za^{\mathcal{s}^{}_{1}}\right) - \cos\left(za^{\mathcal{s}^{}_{2}}\right)\right]. } \label{eq : diff1}\end{aligned}\]] note, the @xmath127-components of the baseline vectors are short and non - cumulative measurements. therefore, in principle, one can consider @xmath124 as zero - mean random errors with no systematics. equation [eq : diff1] in that case can be written as : @xmath128. \label{eq : simplediff1}\]] describing the system in terms of errors in antenna positions, as opposed to errors in baseline positions, equation [eq : simplediff1] becomes : @xmath129. \label{eq : diffantenna1}\]] this equation is also not sufficient to solve for errors in the antenna positions as we have two unknowns and one equation. we set up another equation using a third calibrator source, @xmath130, spaced away in declination from @xmath131 and @xmath132 : @xmath133. \label{eq : diffantenna2}\]] the equations [eq : diffantenna1] and [eq : diffantenna2] are a linear set of equations for one baseline. for the measurements in 63 allocations, the set of equations can be formulated in a matrix form and solved by svd - based least - squares estimator : @xmath134 where, the _ measurement vector _ @xmath135 is to be determined. here, @xmath136. the measurement vector gives @xmath137 and @xmath138 estimates for 32 ew and 945 ns antenna locations, respectively. observation vector _ @xmath139 consists of two sub - matrices, @xmath140 and @xmath141, formed using the left - hand - side of equations [eq : diffantenna1] and [eq : diffantenna2], respectively. here, @xmath142, i.e., the total number of visibilities measured for imaging. therefore, @xmath143. the _ data matrix _ @xmath144. each row in the data matrix has only two non - zero elements, corresponding to a baseline formed by one ew and one ns antenna, making it very sparse. the observation vector is constructed from the gain tables of the array obtained using calibrators mrc0407 - 658 (@xmath145), mrc0915 - 118 (@xmath146) and mrc1932 - 464 (@xmath147). the sensitivity per baseline at mrt is @xmath148 jy for a 1 mhz bandwidth and an integration time of one second. it takes @xmath149 minutes of time for sources at @xmath150 to transit a 2@xmath151 primary beamwidth of elements in the east - west array. this leads to a sensitivity per baseline (including the non - uniform weighting due to primary beam) of @xmath152 jy. the flux density of these three calibrators as seen by mrt is @xmath153 jy ; strong to get reliable calibration. further, the calibrators are unresolved and isolated from confusing sources and have well known measured positions @xcite. a plot of typical phase differences obtained using the pair of calibrators @xmath154 is shown in fig. [f : phasediff]. fig. [f : error_estimate]a shows the estimated errors in 945 ns antenna positions. the errors show a gradient of 1 part in 1000 along the ns arm. this matches with the linear gradients in the phase differences estimated from the calibrators. the estimates in fig. [f : error_estimate]b show alignment errors of the 32 antennas in the ew arm along the ns - direction. the fit shows a gradient of about 2 part in 10,000. this indicates that the ew arm is mis - aligned from the true ew - direction. at one extreme end (1 km from the centre of the array) of the ew arm the error is @xmath155 m, equivalent to an angular distance of @xmath156 from the centre of the array. this is the source of a small @xmath14-dependent error in @xmath6 that was observed in both positional error analysis and the homography matrix. further, our simulation of the synthesised beam in @xmath6 with old ew antenna positions and the corrected ew antenna positions indeed confirm this @xmath14-dependent error in @xmath6. using the new antenna positions we have re - imaged one hour from the steradian and have also imaged a completely new steradian. we find no systematics in positional errors thus endorsing our re - estimated array geometry. Conclusions the homography - based correction was able to correct for systematics in positional errors in the image domain and the errors are within 10% of the beamwidth for sources above 15-@xmath0. the corrected images of one steradian are available for download at _ http://www.rri.res.in / surveys / mrt_. positional error analysis showed that uncorrected mrt images are stretched in declination by @xmath1 part in 1000. this translates to a compression of the ns baseline vector, in the visibility domain. the analysis also showed a @xmath14-dependent error in @xmath6. this cued towards possible errors in our estimation of the array geometry. by formulating a linear system, using instrumental phases estimated from three well separated calibrators whose positions are well known, the array geometry was re - estimated. the estimated error in the @xmath104-component of the ns baseline vectors is about 1 mm / m. in other words, the error is about half a wavelength at 150 mhz (1 m) for a 1 km baseline. the estimates also show a small (2 part in 10,000) @xmath104-component in the purely ew baseline vectors. this indicates that the ew arm is mis - aligned and inclined at an angle of @xmath2, to the true ew direction. these estimates match with the observed stretching of mrt images shown by both the positional error analysis and the homography matrix. using the new antenna positions we have re - imaged one hour from the steradian and have also imaged a completely new steradian. we find no systematics in positional errors. this endorses our re - estimated array geometry. re - imaging one steradian starting from visibilities would have been a very time consuming exercise. development of 2-d homography - based correction enabled us to correct for the positional errors in the image domain. in our view, this new technique will be of relevance to the new generation radio telescopes where, owing to huge data rates, only images after a certain integration would be recorded as opposed to raw visibilities. Acknowledgement soobash daiboo acknowledges a phd bursary from the south african square kilometer array project. the authors would like to thank the anonymous referee for the constructive comments and suggestions

a steradian of the southern sky has been imaged at 151.5 mhz using the mauritius radio telescope (mrt). these images show systematics in positional errors of sources when compared to source positions in the molonglo reference catalogue (mrc). we have applied two - dimensional homography to correct for systematic positional errors in the image domain and thereby avoid re - processing the visibility data. positions of bright (above 15-@xmath0) point sources, common to mrt catalogue and mrc, are used to set up an over - determined system to solve for the homography matrix. after correction the errors are found to be within 10% of the beamwidth for these bright sources and the systematics are eliminated from the images. this technique will be of relevance to the new generation radio telescopes where, owing to huge data rates, only images after a certain integration would be recorded as opposed to raw visibilities. it is also interesting to note how our investigations cued to possible errors in the array geometry. the analysis of positional errors of sources showed that mrt images are stretched in declination by @xmath1 part in 1000. this translates to a compression of the baseline scale in the visibility domain. the array geometry was re - estimated using the astrometry principle. the estimates show an error of @xmath1 mm / m, which results in an error of about half a wavelength at 150 mhz for a 1 km north - south baseline. the estimates also indicate that the east - west arm is inclined by an angle of @xmath2 to the true east - west direction. [firstpage] surveys techniques : image processing astrometry techniques : interferometric telescope catalogues

1006.2015

Introduction cross sections of high - energy nuclear reactions are expressed in terms of nuclear parton distribution functions (npdfs), so that precise npdfs are essential for finding any new phenomena in the high - energy reactions. recently, this topic is becoming important in heavy - ion collisions for investigating properties of quark - hadron matters @xcite and also in neutrino reactions for investigating neutrino - oscillation physics @xcite. determination of precise npdfs is valuable for studying various phenomena in heavy - ion reactions such as color glass condensate @xcite, @xmath8 suppression @xcite, and parton - energy loss @xcite. the npdf studies should be also important for heavy - ion collisions at lhc (large hadron collider) @xcite. in neutrino oscillation experiments, most data are taken at small @xmath7 (@xmath91 gev@xmath10). we could approach such a kinematical region from the high - energy deep inelastic one by using quark - hadron duality @xcite. however, there are still unresolved issues in neutrino deep inelastic scattering. for example, an anomalous @xmath11 value was reported in the neutrino - iron scattering by the nutev collaboration @xcite. it could be related to a nuclear modification difference between the parton distributions @xmath12 and @xmath13 @xcite because the iron target is used in the nutev measurements. there is also an issue that nuclear corrections are different from the ones expected from electron and muon scattering experiments according to recent nutev data @xcite. in these high - energy nuclear reactions, nucleonic pdfs rather than the nuclear ones are often used in calculating cross sections by neglecting nuclear modifications although it is well known that nuclear corrections could be as large as 20% in medium - size nuclei @xcite. these nuclear modifications have been experimentally investigated mainly by the measurements of structure - function ratios @xmath0 and drell - yan cross - section ratios @xmath1. physical mechanisms of the nuclear corrections are, for example, summarized in ref. @xcite. in the small-@xmath6 region, the npdfs become smaller than the corresponding nucleonic ones, which is called shadowing. there are depletions at medium @xmath6, which is related to the nuclear binding mechanism and possibly to a nucleonic modification inside a nuclear medium @xcite. at large @xmath6, the nucleon s fermi motion gives rise to positive corrections. because the pdfs are related to the nonperturbative aspect of quantum chromodynamics (qcd), theoretical calculations have been done by lattice qcd or phenomenological models. however, such calculations are not accurate enough at this stage. one would like to have accurate npdfs, which are obtained in a model - independent way, for calculating precise nuclear cross sections. we should inevitably rely on experimental data for determining them. studies of nucleonic pdfs have a long history with abundant experimental data in a wide kinematical region @xcite. however, determination of npdfs is still at a premature stage with the following reasons. first, available experimental data are limited. the experiments of the hadron - electron ring accelerator (hera) provided data for structure functions at small @xmath6 in a wide range of @xmath7 ; however, such data do not exist for nuclei. because of final - state interactions, hadron - production data may not be suitable for the npdf determination, whereas they are used in the nucleonic analysis. second, the analysis technique is not established. parametrization studies for the npdfs started only recently. the npdfs are expressed in terms of a number of parameters which are then determined by a @xmath14 analysis of the nuclear data. however, it is not straightforward to find functional forms of mass - number (@xmath15) and bjorken-@xmath6 dependencies in the npdfs. furthermore, higher - twist effects could be important in the small-@xmath7 region. a useful parametrization was investigated in ref. @xcite by analyzing @xmath16 structure functions and drell - yan data ; however, the distributions were obtained by simply assigning appropriate parameter values by hand in the versions of 1998 and 1999. the first @xmath14 analysis was reported in ref. @xcite, and then uncertainties of the npdfs were obtained @xcite. all of these analyses are done in the leading order (lo) of the running coupling constant @xmath2. a next - to - leading - order (nlo) analysis was recently reported @xcite. the lo @xmath14 analysis with the uncertainties was also investigated in the 2007 version of ref. there are related studies on the nuclear shadowing @xcite and a global analysis of structure functions @xcite. in this way, the parametrization studies have been developed recently for the npdfs, and they are not sufficient. here, we extend our studies in refs. @xcite by focusing on the following points : * nlo analysis with npdf uncertainties together with a lo one, * roles of nlo terms on the npdf determination by comparing lo and nlo results, * better determination of @xmath6 and @xmath15 dependence, * nuclear modifications in the deuteron by including @xmath17 data, * flavor asymmetric antiquark distributions. this article is organized as follows. in sec. [analysis], our analysis method is described for determining the npdfs. analysis results are explained in sec. [results]. nuclear modifications in the deuteron are discussed in sec. [deuteron]. the results are summarized in sec. [summary]. [analysis] analysis method the optimum npdfs are determined by analyzing experimental data of the @xmath16 structure functions and drell - yan cross sections for nuclear targets. details of our analysis method are described in refs. @xcite, so that only the outline is explained in the following. the parton distribution functions are expressed by two variables @xmath6 and @xmath7. the variable @xmath7 is defined @xmath18 with the virtual photon momentum @xmath19 in the lepton scattering, and the scaling variable @xmath6 is given by @xmath20 with the nucleon mass @xmath21 and the energy transfer @xmath22. the variables for the drell - yan process are momentum fractions, @xmath23 and @xmath24 for partons in the projectile and target, respectively, and @xmath7 defined by the dimuon mass as @xmath25. in our analysis, the npdfs are expressed in terms of corresponding nucleonic pdfs multiplied by weight functions : @xmath26 the functions @xmath27 and @xmath28 indicate type-@xmath29 npdf and nucleonic pdf, respectively, and @xmath30 is a weight function which indicates a nuclear modification for the type-@xmath29 parton distribution. the function @xmath30 generally depends on not only @xmath6 and @xmath15 but also the atomic number @xmath31. it should be noted that this expression sacrifices the large-@xmath6 (@xmath32) nuclear distributions. finite distributions exist even at @xmath33 in nuclei, whereas the distributions should vanish in the nucleon. flavor symmetric antiquark distributions are assumed for @xmath34, @xmath35, and @xmath36 in the previous analyses @xcite. from the violation of the gottfried sum rule and drell - yan measurements, it is now well known that these antiquark distributions are different @xcite. it is more natural to investigate modifications from the flavor asymmetric antiquark distributions in the nucleon. in this work, flavor asymmetric antiquark distributions are used in nuclei, and the distribution type @xmath29 indicates @xmath12, @xmath13, @xmath34, @xmath35, @xmath36, and @xmath37 : @xmath38 whereas the relation @xmath39 is assumed in refs. the number of flavor is three (four) at @xmath40 (@xmath41) @xcite. the bottom and top quark distributions are neglected. the strange - quark distributions are assumed to be symmetric (@xmath42) although there are recent studies on possible asymmetry @xmath43 @xcite. the charm - quark distributions are created by @xmath7 evolution effects @xcite. as for the nucleonic pdfs in the lo and nlo, the mrst (martin, roberts, stirling, and thorne) parametrization of 1998 is used @xcite in this analysis, where the charm - quark mass is @xmath44=1.35 gev and scale parameters are @xmath45=0.174 and 0.300 gev for the lo and nlo, respectively. in our previous analysis @xcite, the mrst-2001 version was employed. since the nlo gluon distribution is negative at @xmath46 and @xmath471 gev@xmath10 in the 2001 version, we use the 1998 parametrization in this work. negative gluon distributions in nuclei could affect our analysis inappropriately in the shadowing region. furthermore, other researchers may use our npdfs at small @xmath6 (@xmath48) with @xmath491 gev@xmath10 for calculating cross sections, for example, in lhc experiments. we tested various pdfs of the nucleon, but overall results are not significantly changed. since we are interested in obtaining the distributions at @xmath501 gev@xmath10 in comparison with other distributions and also our previous distributions, we decided to use the mrst distributions. in the analyses of the cteq (coordinated theoretical / experimental project on qcd phenomenology and tests of the standard model) collaboration, the initial scale @xmath7=(1.3)@xmath10 gev@xmath10 is used. the nuclear modification is assumed to have the following functional form : @xmath51 where @xmath52, @xmath53, @xmath54, @xmath55, @xmath56, and @xmath57 are parameters. the parameter @xmath53 controls the shadowing part, @xmath54, @xmath55, and @xmath56 determine a minute functional form, and @xmath57 is related to the fermi - motion part at large @xmath6. the parameter @xmath57 is fixed at @xmath57=0.1 because it can not be determined from a small number of data in the fermi - motion part. as it will be shown in the result section, the antiquark and gluon modifications can not be determined from the present data at @xmath58. if a large @xmath57, for example @xmath57=1, is taken, the antiquark and gluon distributions could become unrealistically large at large @xmath6. in order to avoid such an issue, @xmath59 is used. the overall @xmath15 dependence in eq. ([eqn : wi]) is taken @xmath60 simply by considering nuclear volume and surface contributions @xcite. there are three constraints for the parameters by the nuclear charge @xmath31, baryon number @xmath15, and momentum conservation @xcite : @xmath61, \nonumber \\ a & = \int dx \, \frac{a}{3 } \left [ u_v^a (x, q_0 ^ 2) + d_v^a (x, q_0 ^ 2) \right], \label{eqn:3conserv } \\ a & = \int dx \, a \, x \big [ u_v^a (x, q_0 ^ 2) + d_v^a (x, q_0 ^ 2) \nonumber \\ & + 2 \, \big \ { \bar u^a(x, q_0 ^ 2) + \bar d^a(x, q_0 ^ 2) + \bar s^a(x, q_0 ^ 2) \big \ } + g^a (x, q_0 ^ 2) \big]. \nonumber\end{aligned}\]] we selected three parameters, @xmath62, @xmath63, and @xmath64, which are determined by these conditions. following improvements are made from the previous analysis @xcite. first, the parametrization of @xmath6 dependence is modified. the meaning of the parameters @xmath54, @xmath55, and @xmath56 is not obvious, so that it is difficult to limit the ranges of these parameters in the analysis. here, we take @xmath6 points (@xmath65, @xmath66) of extreme values for the function @xmath67 as the parameters instead of @xmath54 and @xmath55. they are related with each other by @xmath68 then, the values of @xmath65 and @xmath66 become transparent at least for the valence - quark distributions. from the measurements of the ratios @xmath69, where @xmath70 indicates the deuteron, we have a rough idea that the extreme values should be @xmath71 and @xmath72. the drell - yan measurements indicate @xmath73 for the antiquark distributions. the value @xmath74 and the extreme values for the gluon distribution are not obvious. second, the @xmath15 dependence @xmath75 is too simple. from gross nuclear properties, the leading @xmath15 dependence could be described by @xmath76. in order to describe more details, the parameters @xmath13, @xmath77, @xmath78, and @xmath79 are taken to be @xmath15 dependent : @xmath80 because the extreme values @xmath65 and @xmath66 are almost independent of @xmath15 according to the @xmath16 data and also from the previous analyses @xcite, they are assumed to be independent of @xmath15. in addition, the parameters @xmath81 and @xmath82 are fixed in the antiquark and gluon distributions as follows. because the gluon distributions can not be well determined from the present data, the parameter @xmath83 is taken @xmath84 as assumed in the previous analysis @xcite. it means that @xmath82 and @xmath85 are related by @xmath86 from eq. ([eqn : bici]). there are still six parameters for the antiquark distributions and they should be too many in comparison with four parameters for the valence - quark distributions. we decided to fix the parameter @xmath81, which is sensitive to the gluon shadowing ratio to the antiquark one because of the momentum conservation. we found that the gluon shadowing can not be well determined even in the nlo analysis ; therefore, the value of @xmath81 is taken so that the gluon shadowing is similar to the antiquark shadowing. after all, the following twelve parameters are used for expressing the nuclear modifications : @xmath87 these parameters are determined by the following global analysis. most of used experimental data are explained in ref. first, the data for @xmath69 are from european muon collaboration (emc) @xcite, the slac (stanford linear accelerator center)-e49, e87, e139, and e140 collaborations @xcite, the bologna - cern - dubna - munich - saclay (bcdms) collaboration @xcite, the new muon collaboration (nmc) @xcite, the fermilab (fermi national accelerator laboratory)-e665 collaboration @xcite, and the hermes @xcite. second, the ratios @xmath88 (@xmath89) are from the nmc @xcite. third, the drell - yan data are from the fermilab - e772 @xcite and e866/nusea @xcite collaborations. additional data to the hkn04 (hirai, kumano, nagai in 2004) analysis are the ones for the deuteron - proton ratio @xmath5. these deuteron data are added because precise nuclear modifications are needed for the deuteron which is, for example, used in heavy - ion experiments at the relativistic heavy ion collider (rhic) @xcite. the @xmath5 data are taken from the measurements by the emc @xcite, the bcdms @xcite, the fermilab - e665 @xcite, and the nmc @xcite. these data are used for extracting information on the flavor asymmetric antiquark distributions @xmath90 in the nucleon @xcite. therefore, the flavor asymmetric antiquark distributions in eq. ([eqn : wpart]) are essential for a successful fit and for extracting information on modifications in the deuteron. because the dglap (dokshitzer - gribov - lipatov - altarelli - parisi) evolution can be applied only in the perturbative qcd region, the data with small @xmath7 values can not be used in the analysis. however, the data in a relatively small-@xmath7 region (@xmath7=1@xmath913 gev@xmath10) are valuable for determining antiquark distributions at small @xmath6 (@xmath92). as a compromise of these conflicting conditions, only the data with @xmath93 gev@xmath10 are used in the analysis. however, one should note that the data in the range, @xmath7=1@xmath913 gev@xmath10, may contain significant contributions of higher - twist effects which are not considered in our leading - twist analysis. the parameters are determined by fitting experimental data for the ratios of the structure functions @xmath94 and drell - yan cross sections. the total @xmath14 @xmath95 is minimized to obtain the optimum parameters. the ratio @xmath96 indicates experimental data for @xmath0 and @xmath97, and @xmath98 is a theoretical ratio calculated by the parametrized npdfs. the initial scale @xmath99 is taken @xmath99=1 gev@xmath10, and the distributions in eq. ([eqn : wpart]) are evolved to experimental @xmath7 points to calculate the @xmath14 by the dglap evolution equations @xcite. all the calculations are done in the lo or nlo, and the modified minimal subtraction (@xmath100) scheme is used in the nlo analysis. the structure function @xmath101 is expressed in terms of the npdfs and coefficient functions : @xmath102 \nonumber \\ & + c_g(x,\alpha_s) \otimes g^a (x, q^2) \bigg\ }, \label{eqn : f2}\end{aligned}\]] where @xmath103 and @xmath104 are the coefficient functions @xcite, and @xmath105 is a quark charge. the symbol @xmath106 denotes the convolution integral : @xmath107 the proton - nucleus drell - yan cross section is given by the summation of @xmath108 annihilation and compton processes @xcite : @xmath109 they are expressed in terms of the pdfs and subprocess cross sections : @xmath110 \nonumber \\ & \ \times [q_i (y_1,q^2) \bar q_i^a (y_2,q^2) + \bar q_i (y_1,q^2) q_i^a (y_2,q^2)], \label{eqn : dyqqbar}\end{aligned}\]] @xmath111 \nonumber \\ & \ \ \ + \frac{d\hat\sigma_{qg}}{dq^2 dx_f } [q_i (y_1,q^2) + \bar q_i (y_1,q^2)] g^a (y_2,q^2) \bigg]. \label{eqn : dyqg}\end{aligned}\]] the cross sections @xmath112 and @xmath113 indicate subprocess cross sections for @xmath108 annihilation processes in the lo and nlo, respectively. the @xmath114 indicates the cross section for @xmath115 and @xmath116 processes. the nlo expressions of these cross sections are, for example, found in ref. effects of possible parton - energy loss in the drell - yan process @xcite are neglected in this analysis. [cols="^,^,^,^,^,^,^",options="header ",] using these expressions for the structure functions @xmath94 and drell - yan cross sections, we calculate the theoretical ratios @xmath98 in eq. ([eqn : chi2]). the total @xmath14 is minimized by the cern program library minuit. from this analysis, an error matrix which is the inverse of a hessian matrix, is obtained. npdf uncertainties are estimated by using the hessian matrix as @xmath117 ^ 2=\delta \chi^2 \sum_{i, j } \left (\frac{\partial f^a(x,\xi)}{\partial \xi_i } \right) _ { \xi=\hat{\xi } } h_{ij}^{-1 } \left (\frac{\partial f^a(x,\xi)}{\partial \xi_j } \right) _ { \xi=\hat{\xi } } , \label{eq : dnpdf}\]] where @xmath118 is the hessian matrix, @xmath119 is a parameter, and @xmath120 indicates the optimum parameter set. the @xmath121 value determines the confidence region, and it is calculated so that the confidence level @xmath122 becomes the one-@xmath123-error range (@xmath124) for a given number of parameters (@xmath125) by assuming the normal distribution in the multiparameter space. in the analysis with the twelve parameters, it is @xmath126. this hessian method has been used for estimating polarized pdfs and fragmentation functions @xcite as well as nuclear pdfs in our previous version @xcite. the details of the uncertainty analysis are discussed in refs. @xcite as well as in the nucleonic pdf articles @xcite. [results] results determined parameters are listed in table [table : parameters] for both lo and nlo. three parameters are fixed by the constraints from baryon - number, charge, and momentum conservations in eq. ([eqn:3conserv]), and we chose @xmath62, @xmath63, and @xmath64 for these parameters. the values of the obtained parameters and their errors are similar in the lo and nlo. however, the errors indicate that there are slight nlo improvements in comparison with the lo results. c@c@c @c@c nucleus & reference & # of data & @xmath14 & @xmath14 + & & & (lo) & (nlo) + d / p & @xcite & 290 & 375.5 & 322.5 + @xmath127he / d & @xcite & 35 & 60.9 & 51.8 + li / d & @xcite & 17 & 36.9 & 36.4 + be / d & @xcite & 17 & 39.3 & 53.0 + c / d & @xcite & 43 & 105.8 & 78.7 + n / d & @xcite & 162 & 136.3 & 121.7 + al / d & @xcite & 35 & 45.5 & 44.9 + ca / d & @xcite & 33 & 43.3 & 34.1 + fe / d & @xcite & 57 & 108.0 & 97.4 + cu / d & @xcite & 19 & 12.1 & 13.2 + kr / d & @xcite & 144 & 115.1 & 115.9 + ag / d & @xcite & 7 & 12.5 & 9.1 + sn / d & @xcite & 8 & 13.3 & 14.1 + xe / d & @xcite & 5 & 2.2 & 2.3 + au / d & @xcite & 19 & 55.6 & 32.3 + pb / d & @xcite & 5 & 5.7 & 4.5 + @xmath69 total & & 606 & 792.4 & 709.3 + be / c & @xcite & 15 & 12.6 & 11.9 + al / c & @xcite & 15 & 5.0 & 5.1 + ca / c & @xcite & 39 & 29.9 & 29.1 + fe / c & @xcite & 15 & 8.0 & 8.3 + sn / c & @xcite & 146 & 204.0 & 172.1 + pb / c & @xcite & 15 & 15.7 & 12.2 + c / li & @xcite & 24 & 67.4 & 64.9 + ca / li & @xcite & 24 & 69.0 & 65.3 + @xmath88 total & & 293 & 411.6 & 369.0 + c / d & @xcite & 9 & 9.3 & 8.1 + ca / d & @xcite & 9 & 5.8 & 13.8 + fe / d & @xcite & 9 & 12.6 & 17.9 + w / d & @xcite & 9 & 27.8 & 29.6 + fe / be & @xcite & 8 & 3.3 & 3.6 + w / be & @xcite & 8 & 14.9 & 12.1 + drell - yan total & & 52 & 73.8 & 85.1 + total & & 1241 & 1653.3 & 1485.9 + (@xmath14/d.o.f.) & & & (1.35) & (1.21) + each @xmath14 contribution is listed in table [tab : chi2]. the values suggest that medium and large nuclei should be well explained by the current lo and nlo parametrizations. however, small nuclei are not so well reproduced. the lo fit (@xmath14/d.o.f.=1.35) is better than the previous analysis with @xmath14/d.o.f.=1.58 @xcite, which is partly due to the introduction of new parameters for the @xmath15 dependence in eq. ([eqn : more - a]). if the @xmath14 values of the lo analysis are compared with the ones in table iii of ref. @xcite, we find that much improvements are obtained for the nuclear ratios, @xmath128, @xmath129, @xmath130, @xmath131, @xmath132, @xmath133, @xmath134, @xmath135, @xmath136, @xmath137, and @xmath138, whereas the fit becomes worse for @xmath139 and @xmath140. the @xmath14/d.o.f. is further reduced in the current nlo analysis. according to table [tab : chi2], the nlo results with @xmath14/d.o.f.=1.21 reproduce the data better than the lo ones with @xmath14/d.o.f.=1.35, especially in the following data sets : @xmath141, @xmath142, @xmath143, @xmath144, @xmath145, @xmath146, @xmath147, @xmath133, @xmath134, and @xmath148, whereas it becomes worse in @xmath149, @xmath136, and @xmath139. the deuteron - proton data @xmath5 are added in this analysis to the data set of the previous version @xcite, and they should provide a valuable constraint on pdf modifications in the deuteron. because the @xmath5 data are sensitive to @xmath150 asymmetry @xcite, flavor asymmetric antiquark distributions should be used in our analysis. if the flavor symmetric distributions are used as initial ones, the fit produces a significantly larger @xmath14. (color online) comparison with experimental ratios @xmath151 and @xmath5. the rational differences between experimental and theoretical values [@xmath152 are shown. the nlo parametrization is used for the theoretical calculations at the @xmath7 points of the experimental data. theoretical uncertainties in the nlo are shown at @xmath7=10 gev@xmath10 by the shaded areas.,title="fig:",width=159] (color online) comparison with experimental ratios @xmath151 and @xmath5. the rational differences between experimental and theoretical values [@xmath152 are shown. the nlo parametrization is used for the theoretical calculations at the @xmath7 points of the experimental data. theoretical uncertainties in the nlo are shown at @xmath7=10 gev@xmath10 by the shaded areas.,title="fig:",width=147] + (color online) comparison with experimental ratios @xmath151 and @xmath5. the rational differences between experimental and theoretical values [@xmath152 are shown. the nlo parametrization is used for the theoretical calculations at the @xmath7 points of the experimental data. theoretical uncertainties in the nlo are shown at @xmath7=10 gev@xmath10 by the shaded areas.,title="fig:",width=159] (color online) comparison with experimental ratios @xmath151 and @xmath5. the rational differences between experimental and theoretical values [@xmath152 are shown. the nlo parametrization is used for the theoretical calculations at the @xmath7 points of the experimental data. theoretical uncertainties in the nlo are shown at @xmath7=10 gev@xmath10 by the shaded areas.,title="fig:",width=147] (color online) comparison with experimental data of @xmath153. the ratios @xmath154 are shown. the theoretical ratios and their uncertainties are calculated in the nlo. the notations are the same as fig. [fig : rd].,title="fig:",width=159] (color online) comparison with experimental data of @xmath153. the ratios @xmath154 are shown. the theoretical ratios and their uncertainties are calculated in the nlo. the notations are the same as fig. [fig : rd].,title="fig:",width=147] + (color online) comparison with drell - yan data of @xmath155. the ratios @xmath154 are shown. the theoretical ratios and their uncertainties are calculated in the nlo. the theoretical ratios are calculated at the @xmath7 points of the experimental data. the uncertainties are estimated at @xmath7=20 and 50 gev@xmath10 for the the @xmath156 type and @xmath157 one, respectively.,title="fig:",width=159] (color online) comparison with drell - yan data of @xmath155. the ratios @xmath154 are shown. the theoretical ratios and their uncertainties are calculated in the nlo. the theoretical ratios are calculated at the @xmath7 points of the experimental data. the uncertainties are estimated at @xmath7=20 and 50 gev@xmath10 for the the @xmath156 type and @xmath157 one, respectively.,title="fig:",width=147] + the fit results of the nlo are compared with the used data in figs. [fig : rd], [fig : ra], and [fig : dy] for the ratios @xmath69, @xmath88, and @xmath158, respectively. the rational differences between experimental and theoretical values @xmath154, where @xmath159 is @xmath151, @xmath88, or @xmath158, are shown. for the theoretical values, the nlo results are used and they are calculated at the experimental @xmath7 points. the uncertainty bands are also shown in the nlo, and they are calculated at @xmath7=10 gev@xmath10 for the structure function @xmath16 and at @xmath7=20 or 50 gev@xmath10 for the drell - yan processes. the scale @xmath7=10 gev@xmath10 is taken because the average of all the @xmath16 data is of the order of this value. the scale is @xmath7=50 gev@xmath10 for the drell - yan ratios of the @xmath157 type, and the lower scale 20 gev@xmath10 is taken for the ratio of the @xmath156 type because experimental @xmath7 values are smaller. these figures indicate that the overall fit is successful in explaining the used data. we notice that the @xmath14 values, 53.0, 64.9, and 29.6 in the nlo, are especially large for @xmath160, @xmath161, and @xmath140 in comparison with the numbers of their data, 17, 24, and 9, according to table [tab : chi2]. these large @xmath14 values come from deviations from accurate e139, nmc, and e772 data ; however, such deviations are not very significant in figs. [fig : rd], [fig : ra], and [fig : dy]. there are general tendencies that medium- and large - size nuclei are well explained by our parametrization, whereas there are slight deviations for small nuclei. because any systematic deviations are not found from the experimental data, our analyses should be successful in determining the optimum nuclear pdfs. ratio @xmath162 and the drell - yan ratio @xmath136. in the upper figures, the theoretical curves and uncertainties are calculated at @xmath7=10 gev@xmath10 for the @xmath16 ratio and at @xmath7=50 gev@xmath10 for the drell - yan ratio. the dashed and solid curves indicate lo and nlo results, and the lo and nlo uncertainties are shown by the dark- and light - shaded bands, respectively. the lower figures indicate the ratios @xmath154 where @xmath159 indicates @xmath144 or @xmath136. here, the theoretical ratios are calculated at the experimental @xmath7 points. for comparison, the lo curves and their uncertainties are also shown by @xmath163 and @xmath164.,title="fig:",width=160] ratio @xmath162 and the drell - yan ratio @xmath136. in the upper figures, the theoretical curves and uncertainties are calculated at @xmath7=10 gev@xmath10 for the @xmath16 ratio and at @xmath7=50 gev@xmath10 for the drell - yan ratio. the dashed and solid curves indicate lo and nlo results, and the lo and nlo uncertainties are shown by the dark- and light - shaded bands, respectively. the lower figures indicate the ratios @xmath154 where @xmath159 indicates @xmath144 or @xmath136. here, the theoretical ratios are calculated at the experimental @xmath7 points. for comparison, the lo curves and their uncertainties are also shown by @xmath163 and @xmath164.,title="fig:",width=154] + next, actual data are compared with the lo and nlo theoretical ratios and their uncertainties for the calcium nucleus as an example in fig. [fig : f2-dy]. in the upper figures, the theoretical curves and the uncertainties are calculated at fixed @xmath7 points, @xmath7=10 gev@xmath10 and 50 gev@xmath10 for the @xmath16 and the drell - yan, respectively, whereas the experimental data are taken at various @xmath7 values. the rational differences @xmath154 are shown together with the difference between the lo and nlo curves, @xmath163, in the lower figures. the comparison suggests that both lo and nlo parametrizations should be successful in explaining the @xmath6 dependence of the calcium data. it is noteworthy that the nlo error band of the @xmath16 ratio becomes slightly smaller in comparison with the lo one at small @xmath6 ; however, magnitudes of both uncertainties are similar in the region, @xmath165. the nlo improvement is not clearly seen in the drell - yan ratio @xmath136 in the range of @xmath166. there are discrepancies between the theoretical curves and the @xmath162 data at @xmath167 ; however, they are simply due to @xmath7 differences. if the theoretical ratios are calculated at the same experimental @xmath7 points, they agree as shown in the @xmath162 part of fig. [fig : rd]. these lo and nlo results indicate that the available data are taken in the limited @xmath6 range without small-@xmath6 data, and they are not much sensitive to nlo corrections. this fact leads to a difficulty in determining nuclear gluon distributions because the gluonic effects are typical nlo effects through the coefficient functions and in the @xmath7 evolution equations. one of possible methods for determining the gluon distribution in the nucleon is to investigate @xmath7 dependence of the structure function @xmath16 @xcite. because @xmath7 dependent data exist in the @xmath168 ratios, it may be possible to find nuclear modifications of the gluon distribution. -dependent data of @xmath169 by the hermes collaboration. the dashed and solid curves indicate lo and nlo results, and the nlo uncertainties are shown by the shaded bands.,scaledwidth=41.0%] -dependent data of @xmath170 by the nmc. the notations are the same as fig. [fig : krd - q2].,scaledwidth=41.0%] the lo and nlo parametrization results are compared with @xmath7 dependent data for @xmath169 and @xmath170 measured by the hermes and nmc collaborations, respectively, in figs. [fig : krd - q2] and [fig : snc - q2]. the uncertainties are shown by the shaded bands in the nlo. because the lo and nlo uncertainties are similar except for the small-@xmath6 region, the lo ones are not shown in these figures. they are compared later in this subsection. the results indicate that overall @xmath7 dependencies are well explained by our parametrizations in both lo and nlo. the comparison suggests that the experimental data are not accurate enough to probe the details of the @xmath7 dependence. furthermore, @xmath7 dependencies of the hermes and nmc results are different. the hermes ratio @xmath171 tends to decrease with increasing @xmath7 at @xmath6=0.035, 0.045, and 0.055, whereas the nmc ratio @xmath133 increases with @xmath7 at the same @xmath6 points, although the nuclear species are different. this kind of difference together with inaccurate @xmath7-dependent measurements makes it difficult to extract precise nuclear gluon distributions within the leading - twist dglap approach. it is reflected in large uncertainties in the gluon distributions as it becomes obvious in sec. [npdfs]. in our previous versions @xcite, the experimental shadowing in @xmath133 is underestimated at small @xmath6 (@xmath172) partly because of an assumption on a simple @xmath15 dependence. as shown in fig. [fig : snc - q2], the shadowing is still slightly underestimated at @xmath173 ; however, the deviations are not as large as before. if the experimental errors and the npdf uncertainties are considered, our parametrization is consistent with the data. dependence of the ratio @xmath162 is compared in the lo and nlo at @xmath6=0.001, 0.01, 0.01, and 0.7. the dashed and solid curves indicate lo and nlo results, and lo and nlo uncertainties are shown by the dark- and light - shaded bands, respectively.,scaledwidth=40.0%] the nlo uncertainties are compared with the lo ones in fig. [fig : cad - q2] for the ratio @xmath162. the lo and nlo ratios and their uncertainties are shown at @xmath6=0.001, 0.01, 0.1, and 0.7. the differences between both uncertainties are conspicuous at small @xmath6 (= 0.001 and 0.01) ; however, they are similar at larger @xmath6. the lo and nlo slopes are also different at small @xmath6. these results indicate that the nlo effects become important at small @xmath6 (@xmath174), and the determination of the npdfs is improved especially in this small-@xmath6 region. because the nlo contributions are obvious only in the region, @xmath167, it is very important to measure the @xmath7 dependence to pin down the nlo effects such as the gluon distributions. the possibilities are measurements at future electron facilities such as erhic @xcite and elic @xcite. nuclear modifications of the pdfs are shown for all the analyzed nuclei and @xmath175 at @xmath7=1 gev@xmath10 in fig. [fig : npdfs - all]. it should be noted that the modifications of @xmath12 are the same as the ones of @xmath13 in isoscalar nuclei, but they are different in other nuclei. the modifications increase as the nucleus becomes larger, and the dependence is controlled by the overall @xmath75 factor and the @xmath15 dependence in eq. ([eqn : more - a]). the extreme values (@xmath65, @xmath66) are assumed to be independent of @xmath15 in our current analysis as explained in sec. [paramet], so that they are the same in fig. [fig : npdfs - all]. although the oxygen data are not used in our global analysis, its pdfs are shown in the figure because they are useful for an application to neutrino oscillation experiments @xcite. our code is supplied at the web site in ref. @xcite for calculating the npdfs and their uncertainties at given @xmath6 and @xmath7. (@xmath176, @xmath13, @xmath177, and @xmath37) are shown in the nlo for all the analyzed nuclei and @xmath175 at @xmath7=1 gev@xmath10. as the mass number becomes larger in the order of d, @xmath127he, li,..., and pb, the curves deviate from the line of unity (@xmath178).,width=302] = 1 gev@xmath10. the dashed and solid curves indicate lo and nlo results, and lo and nlo uncertainties are shown by the dark- and light - shaded bands, respectively.,title="fig:",width=151] = 1 gev@xmath10. the dashed and solid curves indicate lo and nlo results, and lo and nlo uncertainties are shown by the dark- and light - shaded bands, respectively.,title="fig:",width=151] as examples of medium and large nuclei, we take the calcium and lead and show their distributions and uncertainties at @xmath7=1 gev@xmath10 in fig. [fig : npdfs - ca - pb]. because the deuteron is a special nucleus and it needs detailed explanations, its results are separately discussed in sec. [deuteron]. the figure indicates that valence - quark distributions are determined well in the wide range, @xmath179 because the uncertainties are small. it is also interesting to find that the lo and nlo uncertainties are almost the same. there are following reasons for these results. the valence - quark modifications at @xmath58 are determined by the accurate measurements of @xmath16 modifications. the antishadowing part in the region, @xmath180, is also determined by the @xmath16 data because there is almost no nuclear modification in the antiquark distributions according to the drell - yan data. if the valence - quark distributions are obtained at @xmath181, the small-@xmath6 behavior is automatically constrained by the baryon - number and charge conservations in eq. ([eqn:3conserv]). because of these strong constraints, the lo and nlo results are not much different. in the near future, the jlab (thomas jefferson national accelerator facility) measurements will provide data which could constraint the nuclear valence - quark distribution especially at medium and large @xmath6 @xcite. furthermore, future neutrino measurements such as the miner@xmath22a experiment at fermilab @xcite and the one at a possible neutrino factory @xcite should provide important information at small @xmath6 (@xmath182). the antiquark distributions are also well determined except for the large-@xmath6 region, @xmath4, because there is no accurate drell - yan data and the structure functions @xmath94 are dominated by the valence - quark distributions. the lo and nlo uncertainties are similar except for the large-@xmath6 region. because of gluon contributions in the nlo, the antiquark shadowing modifications are slightly different between the lo and nlo. possible large-@xmath6 drell - yan measurements such as j - parc (japan proton accelerator research complex) @xcite and fermilab - e906 @xcite should be valuable for the nuclear antiquark distributions in the whole-@xmath6 range. in the similar energy region, there is also the gsi - fair (gesellschaft fr schwerionenforschung -facility for antiproton and ion research) project @xcite. the gluon distributions contribute to the @xmath16 and drell - yan ratios as higher - order effects. therefore, they should be determined more accurately in the nlo analysis than the lo one. such tendencies are found in fig. [fig : npdfs - ca - pb] because the nlo uncertainties are smaller than the lo ones in both carbon and lead. however, these nlo improvements are not as clear as the cases of polarized pdf @xcite and fragmentation functions @xcite. it is because the @xmath7-dependent data are not accurate enough to probe such higher - order effects as discussed in sec. [q2-dependence]. in order to fix the gluon distributions, accurate measurements are needed for the scaling violation of @xmath0 @xcite. the gluon distributions should be also probed by production processes of such as charged hadrons @xcite, heavy flavor @xcite, @xmath183 @xcite, low - mass dilepton @xcite, and direct photon @xcite. there is a recent study on the gluon shadowing from the hera diffraction data @xcite. nuclear gluon distributions play an important role in discussing properties of quark - hadron matters in heavy - ion reactions, so that they need to be determined experimentally. = 100 gev@xmath10.,width=302] we also show the nuclear modifications at @xmath7=100 gev@xmath10 for the calcium in fig. [fig : w - ca-100]. the nuclear modifications are not very different from those at @xmath7=1 gev@xmath10 for the valence - quark distributions. however, the shadowing corrections become smaller in the antiquark and gluon distributions in comparison with the ones at @xmath7=1 gev@xmath10, and the modifications tend to increase at medium and large @xmath6. the distribution functions themselves and their uncertainties are shown in figs. [fig : npdf - ca] and [fig : npdf - pb] for the calcium and lead, respectively. both lo and nlo distributions are shown. here, the uncertainties from the nucleonic pdfs are not included in the uncertainty bands. the calcium is an isoscalar nucleus, so that @xmath184 and @xmath185 are equal to @xmath186 and @xmath187. however, they are different in the lead nucleus because of the neutron excess. in order to see nuclear modification effects, we show the distributions without the nuclear modifications. for example, the distribution @xmath188 is shown in the figure of @xmath184 by using the mrst distributions for @xmath12 and @xmath13. although the uncertainties are large in the antiquark and gluon distributions at medium and large @xmath6, they are not very conspicuous in figs. [fig : npdf - ca] and [fig : npdf - pb] because the distributions themselves are small. = 1 gev@xmath10. no modification " indicates, for example, the distribution @xmath188 in the figure of @xmath184.,width=275] = 1 gev@xmath10. no modification " indicates, for example, the distribution @xmath188 in the figure of @xmath184.,width=275] , is shown for the proton, lithium, aluminum, iron, and lead at @xmath7=1 gev@xmath10. in the isoscalar nuclei, the distributions vanish (@xmath189).,width=226] the flavor asymmetric antiquark distributions are assumed in this analysis as they are defined in eq. ([eqn : wpart]). from this definition, it is obvious that @xmath185 and @xmath187 are equal in isoscalar nuclei such as carbon and calcium. in fig. [fig : ub - db], the ratio @xmath190 is shown for the proton (p), lithium (li), aluminum (al), iron (fe), and lead (pb) at @xmath7=1 gev@xmath10. because the nuclear corrections are assumed to be equal for the antiquark distributions at @xmath99 in eq. ([eqn : wpart]), they are almost independent of nuclear species except for the isoscalar nuclei. it is interesting to investigate possible nuclear modifications on the distribution @xmath191 at the future facilities @xcite. there are noticeable differences between our nlo analysis results and the ones in ref. @xcite especially in the strange - quark and gluon modifications. these differences come from various sources. first, the analyzed experimental data sets are slightly different. second, the strange - quark distributions are created by the dglap evolution by assuming @xmath192 at the initial @xmath7 scale, and the charm distributions are neglected in ref. these differences lead to the discrepancies of the gluon modifications. the determined npdfs and their uncertainties can be calculated by using our code, which is supplied on our web site @xcite. by providing a kinematical condition for @xmath6 and @xmath7 and also a nuclear species, one can calculate the npdfs. it is explained in appendix [library]. if one needs analytical expressions of the npdfs at the initial scale @xmath99, one may read instructions in appendix [appen - a]. Nuclear modifications in deuteron nuclear densities are usually independent of the mass number, which indicates that the average nucleon separation is constant in nuclei. however, the deuteron is a special nucleus in the sense that its radius is about 4 fm, which is much larger than the average nucleon separation in ordinary nuclei (@xmath193 fm). because it is a dilute system, nuclear modifications are often neglected. in fact, corrections to nucleonic structure functions and pdfs are small, namely within a few percentages according to theoretical estimates @xcite even if they are taken into account. c@c@c @c reference & @xmath7 (gev@xmath10) & @xmath6 & modification (%) + @xcite & 4 & 0.015 & 1.1 + @xcite & 4 & 0.010 & 2.5 + @xcite & 4 & 0.010 & 2.0 + @xcite & 4 & 0.010 & 0.4@xmath910.8 + @xcite & 4 & 0.010 & 1@xmath912 (3.5) + theoretical modifications in the deuteron depend much on models. the shadowing of ref. @xcite, which was used in the mrst analysis, is 1.1% at @xmath6=0.015 and @xmath7=4 gev@xmath10 ; however, other model calculations are different, 0.5@xmath913.5% at @xmath194 as shown in table [tab : d - modification]. furthermore, the modifications at medium @xmath6 could be as large as or larger than the small-@xmath6 shadowing (for example, 5% at @xmath195 according to ref. @xcite). such deuteron corrections are becoming important recently although the magnitude itself may not be large. for example, precise nuclear modifications need to be taken into account for investigating quark - hadron matters in heavy - ion collisions such as deuteron - gold reactions in comparison with deuteron - deuteron ones at rhic @xcite. they are also valuable for discussing gottfried - sum - rule violation and flavor - asymmetric antiquark distributions because deuteron targets are used @xcite. in our previous versions on the npdfs @xcite, the data of @xmath5 are not included in the used data set. obtained nuclear modifications tend to be large in comparison with the theoretical model estimations. for example, the hkn04 analysis @xcite indicates about 6 and 8% corrections in antiquark and gluon distributions, respectively, at small @xmath6 (@xmath196) with @xmath7=1 gev@xmath10 and about 5% in valence - quark ones at medium @xmath6 (@xmath197). they are possibly overestimations in the sense that typical theoretical models have corrections within the order of a few percentages. in order to obtain reasonable deuteron modifications from experimental data, we added @xmath5 measurements into the data set in our @xmath14 analysis. these deuteron data were already included in the analysis results in sec. [results]. in addition to the analysis of sec. [results], two other analyses have been made by modifying eq. ([eqn : wi]) : @xmath198 an additional factor @xmath199 is introduced. the analysis results in sec. [results] correspond to the @xmath199=1 case. the other analyses have been made by taking @xmath199=0 and by taking it as a free parameter. we call them analyses 1, 2, and 3, respectively : * analysis 1 : @xmath199=1, * analysis 2 : @xmath199=0, * analysis 3 : @xmath199=free parameter. the @xmath14 values of these analyses are listed in table [tab : chi2-d]. c@c@c @c@c nucleus & # of data & @xmath14 (@xmath200) & @xmath14 (@xmath201) & @xmath14 (free) + d / p & 290 & 322.5 & 282.6 & 284.9 + @xmath69 & 606 & 709.3 & 704.8 & 703.8 + @xmath88 & 293 & 369.0 & 381.6 & 375.3 + drell - yan & 52 & 85.1 & 84.5 & 85.4 + total & 1241 & 1485.9 & 1453.4 & 1449.4 + (@xmath14/d.o.f.) & & (1.21) & (1.18) & (1.18) + the deuteron modifications are terminated in the analysis 2 by taking @xmath201. although such an assumption does not seem to make sense, we found a smaller @xmath14 value from the @xmath5 data than the one of the first analysis as shown in table [tab : chi2-d]. there are three major reasons for this result. first, the deuteron modifications obtained by the overall @xmath15 dependence in eq. ([eqn : wi]) are too large because the deuteron is a loosely bound system which is much different from other nuclei. smaller modifications are expected from the large nucleon separation. second, deuteron data are used for determining the pdfs in the nucleon " @xcite by considering nuclear shadowing modifications of ref. the modifications are calculated in a vector - meson - dominance mechanism and the shadowing in the deuteron is about 1% at @xmath202 according to this model. if this shadowing is not a realistic correction, the nucleonic pdfs of the mrst should partially contain nuclear effects at small @xmath6. the medium- and large-@xmath6 regions are not corrected, so that some deuteron effects could be also included in the nucleonic pdfs in these regions. however, the corrections are not experimentally obvious in such @xmath6 regions as we find in the actual data of fig. [fig : f2dp]. in this way, the nucleonic pdfs could contain some deuteron modification effects. third, the nuclear effects could be absorbed into the @xmath150 asymmetry because it is determined partially by the @xmath5 data. these are the possible reasons why analysis 2 produces the smaller @xmath14 value. . the solid, dotted, and dashed curves indicate the nlo results of the analyses 1 (@xmath200), 2 (@xmath201), and 3 (@xmath199=free), respectively. the shaded bands indicate uncertainties of the analysis-3 curves.,width=302] in the third analysis, the additional parameter @xmath199 is determined from the global analysis. as mentioned, the internucleon separation is exceptionally large in the deuteron. it leads to small nuclear corrections, which are much smaller than a smooth @xmath15-dependent functional form, as calculated in various models @xcite in table [tab : d - modification]. these models could be used for estimating an appropriate value for @xmath199. however, we try to determine the nuclear pdfs without relying on specific theoretical models. the modification parameter @xmath199 is determined from the experimental data, and our analysis 3 indicates @xmath203. if the diffuse deuteron system is considered, the 70% reduction may make sense. nonetheless, it should be noted that this factor may not reflect a realistic modification because it is likely that the deuteron effects are contained in the nucleonic pdfs. we show actual comparisons with the experimental data for @xmath5 by the nmc in fig. [fig : f2dp]. three global analysis results are shown, and the shaded areas indicate uncertainty bands in analysis 3. the figure suggests that all the analyses are successful in explaining the data. however, as indicated in the @xmath14 reductions in the analyses 2 and 3, it is clear that their curves are closer to the experimental data at small @xmath6 such as @xmath6=0.005 and 0.008. there is a tendency that deviations from the data become larger as the deuteron modifications are increased. all the analysis results are more or less similar in the medium- and large-@xmath6 regions. all the curves of the analysis 1 (@xmath200) and 2 (@xmath201) are within the uncertainty bands of the analysis 3 (@xmath199=free), although the analysis-1 curves are at the edges of the error bands at small @xmath6. it means that all these analysis are consistent with each other and they explain the experimental data. = 1 gev@xmath10. the solid and dashed curves are obtained by analyses 1 and 3, respectively, and their uncertainties are shown by the shaded bands.,width=302] modifications of the pdfs are shown for the deuteron in fig. [fig : npdfs - d] at @xmath7=1 gev@xmath10. the results of analyses 1 and 3 are shown with their uncertainty estimation. there is no deuteron modification in analysis 2 as obvious from the definition in eq. ([eqn : wi - d]). the uncertainty bands of analysis 3 shrink at the points, where the nuclear modifications vanish (@xmath204), for example, in the figure of the valence - quark modification @xmath205. this is caused by the error from the parameter @xmath199. its contributions to the uncertainties are large and the derivative @xmath206 is proportional to @xmath207, which vanishes at the same points as the function @xmath208 in eq. ([eqn : wi - d]). it leads to the gourd - shaped uncertainty band in @xmath205 because other terms are small. such an error shape does not appear in analysis 1 because the error term of @xmath199 does not exist. here, the derivative @xmath209 is also a cubic polynomial and vanishes at three @xmath6 points. however, @xmath210 and @xmath211 are quadratic functions, which do not vanish at the same @xmath6 points, and their contributions to the uncertainties are of the same order of the @xmath212 and @xmath213 terms. this is the reason why such a gourd - shaped function does not appear in the uncertainties of the @xmath200 analysis. the antiquark shadowing is about 2% at @xmath214 and the valence - quark modification is about 1% in the analysis 1 according to fig. [fig : npdfs - d]. we should note that the antiquark shadowing is reduced about 30% at @xmath7=4 gev@xmath10 as shown in fig. [fig : cad - q2] in comparing it with the theoretical values in table [tab : d - modification]. in analysis 3, the antiquark shadowing becomes 0.5%, which could be slightly smaller than the theoretical ones. both corrections (2% and 0.5%) are slightly different from the assumed correction (1% at @xmath6=0.01 and @xmath7=4 gev@xmath10) in the mrst fit. the uncertainty bands depend on the initial functional form or assignment of the parameters in the @xmath14 analysis. the uncertainties are generally larger in analysis 3, and the line of @xmath204 is within the bands. it suggests that precise deuteron modifications can not be determined at this stage. the modifications in fig. [fig : npdfs - d] may not be seriously taken because the deuteron effects could be partially included in the nucleonic pdfs. it is difficult to judge what the realistic deuteron modifications are at this stage. actual modifications are possibly in between these analysis results. in order to obtain realistic modifications, the nucleonic pdfs should be determined by considering the deuteron modifications, for example, of our analysis results. then, using new nucleonic pdfs, we redetermine our nuclear pdfs including the ones in the deuteron. realistic deuteron modifications should be obtained by repeating this step. [summary] summary nuclear pdfs have been determined by the global analyses of experimental data for the ratios of the @xmath16 structure functions and drell - yan cross sections. the uncertainties of the determined npdfs are estimated by the hessian method. the first important point is that the uncertainties were obtained in both lo and nlo so that we can discuss the nlo improvement on the determination. we found slight nlo improvements for the antiquark and gluon distributions at small @xmath6 (@xmath215) ; however, they are not significant at larger @xmath6. accurate experimental measurements, especially on the @xmath7 dependence at small @xmath6, should be useful for determining higher - order effects such as the nuclear gluon distributions. the valence - quark distributions are well determined. the antiquark distributions are also determined at @xmath3 ; however, they have large uncertainties at @xmath4. the gluon modifications are not precisely determined in the whole @xmath6 region. future measurements are needed to determine accurate nuclear distributions. nuclear modifications were discussed for the deuteron in comparison with the experimental data of @xmath5. however, it is difficult to find accurate modifications at this stage because deuteron effects could be partially contained in the nucleonic pdfs. an appropriate nucleonic pdf analysis is needed in addition to accurate measurements on the ratio @xmath5. our npdfs and their uncertainties can be calculated by using the codes in ref. @xcite. m.h. and s.k. were supported by the grant - in - aid for scientific research from the japanese ministry of education, culture, sports, science, and technology. was supported by japan society for the promotion of science. [appen-a] parameters @xmath62, @xmath63, and @xmath64 the determined parameters are listed in table [table : parameters]. there are other parameters, @xmath62, @xmath63, and @xmath64, which are automatically calculated from the tabulated values by the conservation conditions of nuclear charge, baryon number, and momentum in eq. ([eqn:3conserv]). because they depend on nuclear species, namely on @xmath15 and @xmath31, their calculation method is explained in the following. the flavor asymmetric antiquark distributions in eq. ([eqn : wpart]) are used in this analysis, whereas the flavor symmetric ones are used in the previous versions in refs. @xcite, so that relations are slightly different from the ones in appendix of ref. one can obtain values of these parameters for any nuclei by calculating the following integrals : @xmath216, \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! & \nonumber \\ i_8 & = \int dx \frac{x}{(1-x)^{\beta_g } } g(x), & \end{aligned}\]] where @xmath217 and @xmath218 is @xmath219 here, the parameters @xmath13, @xmath77, @xmath78, and @xmath220 depend on @xmath15 according to eq. ([eqn : more - a]). it is noteworthy that the integrals @xmath221, @xmath222, and @xmath223 depend on @xmath15. using these integrals, one obtains parameter values by the conservations in eq. ([eqn:3conserv]) : @xmath224 @xmath225 @xmath226 . \label{eqn : ag}\end{aligned}\]] from these values together with the parameters in table [table : parameters] and the nucleonic pdfs at @xmath7=1 gev@xmath10 of the mrst parametrization @xcite, one obtains the npdfs at @xmath7=1 gev@xmath10 for a given nucleus. if distributions are needed at different @xmath7, one needs to evolve the npdfs by using own evolution code. if one does not have such an evolution code, one had better use the code in appendix b. numerical values of the npdfs can be obtained for given @xmath6, @xmath7, and @xmath15. in particular, if one is interested in estimating the uncertainties of the npdfs, this practical code needs to be used. [library] code for calculating nuclear parton distribution functions and their uncertainties codes for calculating nuclear pdfs and their uncertainties can be obtained from the web site @xcite. a general guideline for usage is explained in appendix b of ref. @xcite, and the conditions are the same in the current version, hkn07 (hirai, kumano, nagai in 2007). therefore, the details should be found in ref. the npdfs can be calculated for nuclei at given @xmath6 and @xmath7, for which the kinematical ranges should be in @xmath227 and 1 gev@xmath228 gev@xmath10. input parameters for running the code are explained in the beginning of the file npdf07.f. a sample program, sample.f, is supplied as an example for calculating the nuclear pdfs and uncertainties. the uncertainties of the npdfs are estimated by using the hessian matrix and grid data for derivatives with respect to the parameters. the usage is explained in the sample program. j. adams _ et al. lett. * 97 *, 152302 (2006) ; b. a. cole _ et al. _, hep - ph/0702101 ; l. c. bland _ et al. j. c * 43 *, 427 (2005) ; v. guzey, m. strikman, and w. vogelsang, phys. lett. b * 603 *, 173 (2004) ; p. jacobs and x.- wang, prog. part. phys. * 54 *, 443 (2005). m. sakuda, nucl. phys. * b112 *, 109 (2002) ; e. a. paschos and j. y. yu, phys. d * 65 *, 033002 (2002). j. jalilian - marian and y. v. kovchegov, prog. part. phys. * 56 *, 104 (2006) ; e. iancu and r. venugopalan, hep - ph/0303204, in _ quark gluon plasma 3 _, edited by r. c. hwa and x. n. wang, world scientific (2003). f. arleo and v.- n. tram, hep - ph/0612043. t. renk and j. ruppert, phys. c * 72 *, 044901 (2005) ; phys. b * 646 *, 19 (2007). a. accardi _ et al. _, hep - ph/0308248, subgroup report, 3rd workshop on hard probes in heavy ion collisions, geneva, switzerland, oct. 7 - 11, 2002. a. bodek and u. k. yang, nucl b112 *, 70 (2002). for summary, see w. melnitchouk, r. ent, and c. keppel, phys. rept. * 406 * (2005) 127. g. p. zeller _ et al. _, phys. lett. * 88 *, 091802 (2002) ; erratum, _ ibid. _ * 90 *, 239902 (2003). s. kumano, phys. d * 66 *, 111301 (2002) ; m. hirai, s. kumano, and t.- h. nagai, phys. d * 71 *, 113007 (2005). s. j. brodsky, i. schmidt, and j.- j. yang, phys. d * 70 *, 116003 (2004). k. j. eskola and h. paukkunen, jhep, * 0606 *, 008 (2006). j. f. owens _ et al. _, d * 75 *, 054030 (2007). after submitting this paper for publication, we noticed the work on nuclear pdfs in the iron, i. schienbein _ et al., _, arxiv:0710.4897v1 [hep - ph]. for summary, see d. f. geesaman, k. saito, and a. w. thomas, ann. nucl. part. sci. * 45 *, 337 (1995) ; n. armesto, j. phys. g * 32 *, r367 (2006). c. j. benesh, t. goldman, and g. j. stephenson, phys. c * 68 *, 045208 (2003) ; h. mineo _ et al. _, nucl. phys. * a735 *, 482 (2004) ; i. c. cloet, w. bentz, and a. w. thomas, phys. rev. lett. * 95 *, 052302 (2005) ; phys. b * 642 *, 210 (2006) ; g. a. miller, eur. j. a * 31 *, 578 (2007). http://durpdg.dur.ac.uk / hepdata/. k. j. eskola _ _, nucl. phys. * b535 *, 351 (1998) ; eur. phys. j. c * 9 *, 61 (1999) ; jhep, * 0705 *, 002 (2007). m. hirai, s. kumano, and m. miyama, phys. d * 64 *, 034003 (2001). m. hirai, s. kumano, and t.- h. nagai, phys. c * 70 *, 044905 (2004). d. de florian and r. sassot, phys. d * 69 *, 074028 (2004). n. armesto, eur. j. c * 26 *, 35 (2002) ; n. armesto _ et al. j. c * 29 *, 531 (2003). j. qiu and i. vitev, phys. 93 *, 262301 (2004). l. frankfurt, v. guzey, and m. strikman, phys. d * 71 *, 054001 (2005). s. a. kulagin and r. petti, nucl. * a765 *, 126 (2006). s. kumano, phys. rept. * 303 *, 183 (1998) ; g. t. garvey and j.- c. peng, prog. * 47 *, 203 (2001). et al. _, jhep * 04 *, 089 (2007) ; d. mason, pp.165 - 168 in proceedings of the 14th international workshop on deep inelastic scattering, world scientific (2007). a. d. martin, r. g. roberts, w. j. stirling, and r. s. thorne, eur. j. c * 4 *, 463 (1998). i. sick and d. day, phys. b * 274 *, 16 (1992). l. l. frankfurt, m. i. strikman, and s. liuti, phys. lett. * 65 *, 1725 (1990). j. ashman _ et al. _, b * 202 *, 603 (1988). m. arneodo _ et al. _, b333 *, 1 (1990). j. ashman _ et al. _, z. phys. c * 57 *, 211 (1993). lett. * 50 *, 1431 (1983). et al. _, lett. * 51 *, 534 (1983). et al. _, lett. * 60 *, 2591 (1988). et al. _, d * 49 *, 4348 (1994). g. bari _ et al. * 163b *, 282 (1985). a. c. benvenuti _ et al. _, b * 189 *, 483 (1987). p. amaudruz _ et al. _, b441 *, 3 (1995) ; m. arneodo _ _, _ ibid. _ * b441 *, 12 (1995). m. r. adams _ et al. lett. * 68 *, 3266 (1992). m. r. adams _ et al. _, z. phys. c * 67 *, 403 (1995). a. airapetian _ b * 475 *, 386 (2000) ; erratum, _ ibid. _ * 567 *, 339 (2003). m. arneodo _ et al. _, b481 *, 3 (1996). m. arneodo _ _, nucl. phys. * b481 *, 23 (1996). d. m. alde _ et al. _, lett. * 64 *, 2479 (1990). m. a. vasiliev _ lett. * 83 *, 2304 (1999). j. j. aubert _ et al. _, b293 *, 740 (1987). a. c. benvenuti _ b * 237 *, 599 (1990). m. r. adams _ et al. lett. * 75 *, 1466 (1995). m. arneodo _ et al. _, * b487 *, 3 (1997). m. miyama and s. kumano, comput. * 94 *, 185 (1996) ; m. hirai, s. kumano, and m. miyama, comput. 108 *, 38 (1998) ; * 111 *, 150 (1998) ; s. kumano and t.- h. nagai, j. comput. phys. * 201 *, 651 (2004). r. k. ellis, w. j. stirling, and b. r. webber, _ qcd and collider physics _, cambridge university press (1996). p. j. sutton, a. d. martin, r. g. roberts, and w. j. stirling, phys. d * 45 *, 2349 (1992). duan, n. liu, and z.- y. yan, eur. j. c * 50 *, 585 (2007). m. hirai, s. kumano, and n. saito, phys. d * 69 *, 054021 (2004) ; * 74 *, 014015 (2006). m. hirai, s. kumano, t.- h. nagai, and k. sudoh, phys. d * 75 *, 094009 (2007) ; m. hirai, s. kumano, m. oka, and k. sudoh, arxiv:0708.1816 [hep - ph]. j. pumplin _ et al. _, phys. d * 65 *, 014013 (2002) ; jhep, * 0207 *, 012 (2002) ; a. d. martin _ et al. _, j. c * 28 *, 455 (2003) ; * 35 *, 325 (2004). k. prytz, phys. b * 311 *, 286 (1993). a. deshpande, r. milner, r. venugopalan, and w. vogelsang, ann. nucl. part. sci. * 55 *, 165 (2005). our npdf codes can be obtained from http://research.kek.jp/people/kumanos/nuclp.html. j. arrington _ et al. _, nucl - ex/0701017. for the miner@xmath22a project, see http://minerva.fnal.gov/. s. kumano, hep - ph/0310166, aip conf. proc. * 721 *, 29 (2004). p04 proposal at http://j-parc.jp/nuclpart/proposal_e +.html ; s. kumano, nucl. phys. * a782 *, 442 (2007). p906 proposal at http://p25ext.lanl.gov/e866/e866.html. li and x.- wang, phys. b * 527 *, 85 (2002). b. z. kopeliovich and a. v. tarasov, nucl. phys. * a710 *, 180 (2002). l. frankfurt _ et al. _, jhep, * 0308 *, 043 (2003). g. fai, j. qiu, and x. zhang, phys. c * 71 *, 014901 (2005). f. arleo and t. gousset, arxiv:0707.2944 [hep - ph]. k. tywoniuk _ _, arxiv:0705.1596 [hep - ph]. b. badelek and j. kwiecinski, phys. rev. d * 50 *, r4 (1994). v. r. zoller, z. phys. c * 54 *, 425 (1992). v. barone _ et al. _, z. phys. c * 58 *, 541 (1993). w. melnitchouk and a. w. thomas, phys. d * 47 *, 3783 (1993) ; w. melnitchouk, a. w. schreiber, and a. w. thomas, phys. b * 335 *, 11 (1994). g. piller, w. ratzka, and w. weise, z. phys. a * 352 *, 427 (1995).

nuclear parton distribution functions (npdfs) are determined by global analyses of experimental data on structure - function ratios @xmath0 and drell - yan cross - section ratios @xmath1. the analyses are done in the leading order (lo) and next - to - leading order (nlo) of running coupling constant @xmath2. uncertainties of the npdfs are estimated in both lo and nlo for finding possible nlo improvement. valence - quark distributions are well determined, and antiquark distributions are also determined at @xmath3. however, the antiquark distributions have large uncertainties at @xmath4. gluon modifications can not be fixed at this stage. although the advantage of the nlo analysis, in comparison with the lo one, is generally the sensitivity to the gluon distributions, gluon uncertainties are almost the same in the lo and nlo. it is because current scaling - violation data are not accurate enough to determine precise nuclear gluon distributions. modifications of the pdfs in the deuteron are also discussed by including data on the proton - deuteron ratio @xmath5 in the analysis. a code is provided for calculating the npdfs and their uncertainties at given @xmath6 and @xmath7 in the lo and nlo.

0709.3038

Introduction one of the oldest problems of algebra is the equation solvability problem over a given algebraic structure. nowadays, many such classical problems arise in a new perspective, namely to consider their computational complexity. in this paper we investigate the complexity of the equation solvability problem over finite groups and rings. the _ equation solvability problem _ over a finite group @xmath0 asks whether or not two group expressions (i.e. products of variables and elements of @xmath0) can attain the same value for some substitution over @xmath0. in other words, for the equation solvability problem, one needs to find if there exists at least one substitution satisfying the equation. another interesting problem is whether or not _ all _ substitutions satisfy the equation. the _ equivalence problem _ over a finite group @xmath0 asks whether or not two group expressions @xmath1 and @xmath2 are equivalent over @xmath0 (denoted by @xmath3), that is whether or not @xmath1 and @xmath2 determine the same function over @xmath0. first burris and lawrence @xcite investigated the complexity of the equivalence problem over finite groups. they proved that if a group @xmath0 is nilpotent or @xmath4, the dihedral group for odd @xmath5, then the equivalence problem for @xmath0 has polynomial time complexity. they conjectured that the equivalence problem for @xmath0 is in polynomial time if @xmath0 is solvable, and conp - complete otherwise. horvth and szab @xcite confirmed the conjecture for @xmath6, where @xmath7 and @xmath8 are abelian groups such that the exponent of @xmath7 is squarefree and @xmath9. later horvth @xcite generalized this result to semidirect products @xmath10, where @xmath7 and @xmath11 are abelian groups (here @xmath12 denotes the centralizer of @xmath7 in @xmath13). horvth, lawrence, mrai and szab @xcite proved the conp - complete part of the conjecture. but the complexity of the equivalence problem over many solvable, not nilpotent groups is not determined, yet. three of the smallest groups, for which this complexity is not known, are @xmath14, @xmath15 and a non - commutative group of order @xmath16. see @xcite for a more comprehensive list. even less is known about the equation solvability problem. goldmann and russel @xcite proved that if @xmath0 is nilpotent then the equation solvability problem over @xmath0 is solvable in polynomial time, while if @xmath0 is not solvable, then the equation solvability problem is np - complete. little is known for solvable, not nilpotent groups. horvth proved in (*??? * corollary 2) that the equation solvability problem over @xmath17 is solvable in polynomial time for certain groups @xmath18, where @xmath19 or @xmath20 or @xmath21 and @xmath13 is commutative. note that all results for both the equivalence and the equation solvability problem over solvable, not nilpotent groups are about groups @xmath22, where @xmath23 is abelian. one of the groups of small order, for which the equation solvability problem is unknown, is the group @xmath24. here, @xmath25 denotes the noncommutative group of @xmath26 upper unitriangular matrices over @xmath27. horvth explicitly asks in (*??? * problem 4) the complexity of the equivalence and equation solvability problems over this group. the group @xmath24 is isomorphic to a special subgroup of the @xmath28 upper triangular matrices over @xmath29. motivated by the definition of pattern groups from @xcite, we call a group @xmath22 a _ semipattern _ group, if @xmath23 is a subgroup of the group of upper unitriangular matrices, and @xmath13 is a subgroup of the diagonal matrices. we give the precise definition of semipattern groups in section [spcs]. the main result of the paper is the following. [fotetel] the equation solvability problem over semipattern groups is solvable in polynomial time. the group @xmath24 defined in (*??? * problem 4) is in fact a semipattern group, thus theorem [fotetel] answers horvth s question completely. further, from theorem [fotetel] the equivalence problem over semipattern groups is solvable in polynomial time, as well. indeed, it is known that for a group @xmath17 if the equation solvability problem is solvable in polynomial time, then the equivalence problem is solvable in polynomial time, as well. in the proof of theorem [fotetel] we reduce the solvability of the input equation over a matrix group over a finite field to the solvability of a system of equations over the same field. then we apply some results over finite rings. therefore, we summarize the known results over rings. the _ equation solvability problem _ over a finite ring @xmath30 asks whether or not two polynomials can attain the same value for some substitution over @xmath30. the _ equivalence problem _ over a finite ring @xmath30 asks whether or not two polynomials are equivalent over @xmath30 i.e. if they determine the same function over @xmath30. the complexity of these questions was completely characterized in the past two decades. hunt and stearnes @xcite investigated the equivalence problem for finite commutative rings. later burris and lawrence @xcite generalized their result to non - commutative rings. they proved that the equivalence problem for @xmath31 is solvable in polynomial time if @xmath31 is nilpotent, and is conp - complete otherwise. the proof of burris and lawrence reduces the satisfiability (sat) problem to the equivalence problem by using long products of sums of variables. nevertheless, if we expand this polynomial into a sum of monomials then the length of the new polynomial may become exponential in the length of the original polynomial. such a change in the length suggests that the complexity of the equivalence problem might be different if the input polynomials are restricted to be written as sums of monomials. this motivated lawrence and willard @xcite to introduce the _ sigma equivalence _ and _ sigma equation solvability problems _, where the input polynomials are given as sums of monomials. lawrence and willard conjectured that if the factor by the jacobson radical is commutative then the sigma equivalence problem is solvable in polynomial time, and is conp - complete otherwise. szab and vrtesi proved the conp - complete part of the conjecture in @xcite. horvth confirmed the conjecture for commutative rings in @xcite. the polynomial part of this conjecture is completely proved in the manuscript @xcite. most of the results for the [sigma] equation solvability problem follow from the corresponding result for the [sigma] equivalence problem. in particular, from the argument of szab and vrtesi follows that if the factor by the jacobson radical is not commutative then the sigma equation solvability problem is np - complete. horvth, lawrence and willard @xcite proved that if this factor is commutative then the sigma equation solvability problem is solvable in polynomial time. thus, the sigma equation solvability problem is completely characterized. for the general equation solvability, arguments of burris and lawrence from @xcite yield that if the ring is not nilpotent then the problem is np - complete. horvth in @xcite proved that the equation solvability problem is solvable in polynomial time otherwise. [nr] if @xmath30 is a finite, nilpotent ring then the equation solvability problem over @xmath30 is solvable in polynomial time. horvth uses ramsey s theorem in the proof of theorem [nr]. he defines a number @xmath32 that depends only on the ring @xmath31. then he proves that the image of every polynomial can be obtained by substituting @xmath33 into all but @xmath32-many variables. thus one can decide whether or not @xmath34 is solvable over @xmath31 in @xmath35 time. however, this number @xmath36 is huge in the size of the ring. in fact, @xmath32 is greater than @xmath37, where the height of the tower in the exponent is the nilpotency class of @xmath31. horvth specifically asks in (*?? * problem 3) whether or not this number @xmath32 can be decreased. in the second half of the paper we give a new proof of theorem [nr]. our algorithm is much more efficient than horvth s. wilson @xcite characterizes nilpotent rings with the help of special kind of nilpotent matrix rings. we can decide the equation solvability problem over these special matrix rings similarly as we do over semipattern groups in theorem [fotetel]. in particular, we show that over a nilpotent ring @xmath31 we can decide in @xmath38 time whether or not @xmath34 is solvable, thus providing a partial answer to problem 3 in @xcite, as well. we mention that in a completely independent way, krolyi and szab also found a way to decrease the exponent @xmath32 in @xcite. in section [bev] we summarize the notations, definitions and theorems, that we use in the paper. in particular, in section [spcs] we review the definition of pattern groups, then we define semipattern groups. in section [2.3] we discuss the generalization of the equation solvability problem over rings for systems of equations. we are going to apply these results in order to prove theorems [fotetel] and [nr]. in section [nil] we lay the groundwork for the proof of theorem [nr]. in section we prove theorem [fotetel]. we use ideas of this proof in section [3.], where we prove theorem [nr]. Preliminaries let @xmath39 denote the finite field of @xmath40 elements. let us consider the group @xmath41 of @xmath42 upper triangular matrices, that is those matrices whose elements under the diagonal are zero and the elements in the diagonal are non - zero : @xmath43 here the group operation is the matrix multiplication. let the @xmath42 identity matrix denoted by @xmath44. let @xmath45 denote the @xmath42 matrix whose elements are all zero except for the @xmath46 element in the @xmath47 row, which is 1. let @xmath48. let @xmath49 thus, @xmath50 contains all those upper triangular matrices, where every element in the diagonal is @xmath51, and every element whose position is not occurring in p has to be @xmath52. if @xmath50 is a subgroup of @xmath41, then we call @xmath50 a _ pattern _ group. for more details on pattern groups, see e.g. @xcite. let @xmath53 be subgroups of @xmath54 and let @xmath55 be the set of @xmath42 matrices over @xmath39 whose @xmath47 element in the diagonal is from @xmath56 @xmath57 : @xmath58 if @xmath50 is a pattern group then @xmath59 is a subgroup of @xmath41. then we call @xmath59 a _ semipattern group _ and we denote such a group by @xmath60. further, we note that @xmath61 and @xmath62. the group @xmath24 defined in (*??? * problem 4) is in fact a semipattern group : @xmath63 let @xmath64 be a commutative, unital ring, @xmath65 @xmath66 be subsets of @xmath64. for nonnegative integers @xmath67, let @xmath68, @xmath69, @xmath70, @xmath71 be pairwise disjoint sets of variables. we say that _ @xmath72 is solvable over @xmath64 for substitutions from @xmath73 _ (and write @xmath74 @xmath75 is solvable over @xmath31) if there exist @xmath76, @xmath77, @xmath78, @xmath79 such that the two polynomials attain the same value on this substitution : @xmath80 for proving theorem [fotetel], we will directly apply the following result of horvth @xcite. [egyrszmo] let @xmath39 be a finite field. let @xmath81 be subgroups of @xmath54. let @xmath82 $] be a polynomial, written as a sum of monomials. then it can be decided whether the system of equations @xmath83 is solvable over @xmath39 in @xmath84 time. to prove theorem [nr] we are going to use the following from @xcite. [er] let @xmath85 $] be polynomials, written as sums of monomials. then it can be decided whether the system of equations @xmath86 is solvable over @xmath87 in @xmath88 time. the complexity of the equation solvability problem over finite nilpotent rings is known. horvth @xcite proved that the equation solvability problem is solvable in polynomial time. we give a new algorithm in section [3.] that is much more efficient than horvth s. in this part we show that we can characterize the complexity of the equation solvability problem over nilpotent rings using the sigma equation solvability problem over special kind of nilpotent matrix rings. hence we can apply the same ideas that we used in the proof of theorem [fotetel]. horvth proved theorem [nr] using ramsey s theory. he defined a number @xmath32 that depends only on the ring @xmath31. then he proved that the image of every polynomial can be obtained by substituting @xmath52 into all but @xmath32-many variables. thus one can decide whether or not @xmath34 is solvable over @xmath31 in @xmath89 time. but the number @xmath36 is huge in the size of the ring. let @xmath90 be the characteristic of @xmath31 and @xmath91 be the nilpotency class of @xmath31. let furthermore @xmath92. then @xmath32 is greater than @xmath93, where the height of the tower in the exponent is @xmath91. we give a new proof of theorem [nr] in section [3.]. our algorithm is much more efficient than horvth s. we prove that @xmath94 time is enough. first, we show that we can handle the equation solvability problem over nilpotent rings using the sigma equation solvability problem over special kind of nilpotent matrix rings. if @xmath31 is a nilpotent ring then the complexity of the _ general _ equation solvability problem over @xmath31 is the same as the complexity of the _ sigma _ equation solvability over @xmath31. indeed, we can rewrite every polynomial @xmath95 over @xmath31 as a sum of monomials in @xmath96 time, where @xmath91 is the nilpotency class of @xmath31. now, @xmath97. hence, at the cost of an extra @xmath98 factor in the exponent, we may assume that every input polynomial is given as a sum of monomials. furthermore it is enough to consider the equation solvability problem over nilpotent rings with prime power characteristic, because every ring is a direct sum of rings of prime power characteristic, and the equation solvability problem can be handled componentwise. wilson @xcite characterizes finite nilpotent rings with prime power characteristic. [wil] let @xmath31 be a finite nilpotent ring with characteristic @xmath99, and let @xmath31 have an independent generating set consisting of @xmath100 generators over @xmath101. then @xmath31 is a homomorphic image of a ring @xmath102 of matrices over @xmath101 where every entry on or below the main diagonal is a multiple of @xmath103. in fact for nilpotent rings it is enough to consider the equation solvability problem over such a matrix ring @xmath104, since if the equation solvability problem is solvable in polynomial time for @xmath104, then it is solvable in polynomial time for a factor @xmath105, as well. indeed, let @xmath106 be a polynomial over @xmath107 and @xmath95 be any polynomial over @xmath104 whose factor by @xmath108 is @xmath106. then @xmath109 is solvable over @xmath107 if and only if @xmath110 is solvable over @xmath104 for some @xmath111. this gives an extra @xmath112 factor to the running time. however, @xmath113, and we have @xmath114, @xmath115. thus, @xmath116, which only depends on @xmath31, and thus can be forgotten about. thus it is enough to consider the sigma equation solvability problem over such a matrix ring of matrices over @xmath87 where every entry on or below the main diagonal of each matrix is a multiple of @xmath103. we can handle such matrix rings similarly as we do with the semmipattern groups. hence we can use ideas of the proof described in section [2.]. we give a new, efficient algorithm that decides the equation solvability problem over nilpotent matrix rings in section [3.]. Equation solvability problem over semipattern groups in this section we consider the equation solvability problem over semmipattern groups. first we characterize the multiplication of matrices from @xmath41 in lemma [mxszor]. we use this formula in our algorithm. [mxszor] let n be a natural number. for every @xmath117 let @xmath118 let @xmath119 then * for every @xmath120 we have @xmath121 * for every @xmath122 we have + @xmath123 the length of is @xmath124, and in particular is polynomial in @xmath125. the lemma can be proved by induction on @xmath125. however, instead of giving the technical induction proof, we explain how one can arrive at this formula. let us consider the @xmath46 element of the @xmath47 row @xmath126 of the matrix @xmath127 @xmath128. we can express this element @xmath126 with a sum of some appropriate products. in every such product we multiply one element from each matrix @xmath129 @xmath130. (in the notation the index @xmath131 appears as the last index.) furthermore the index of the column of a term must equal with the index of the row of the following term in every such product. (therefore a term of the form @xmath132 or @xmath133 follows the term of the form @xmath132 or @xmath134.) the row index of the first term is @xmath135, the column index of the last term is @xmath136. the matrices @xmath137 are upper triangular matrices, hence the row index of every term of every product is less than or equal to the column index. (thus for every term of the form @xmath133, that are above the diagonal, we have @xmath138.) thus the @xmath46 element of the @xmath47 row @xmath126 of the matrix @xmath139 is a sum of products of @xmath125 terms such that * the row index of the first term is @xmath135, the column index of the last term is @xmath136 ; * the column index of every term equals to the row index of the next term ; * the row index of a term is at most the row index of the next term. notice, that every @xmath125-term product is uniquely determined by those terms where the row index differs from the column index. let two such consecutive terms of the product be @xmath140 and @xmath141. (here @xmath142 and @xmath143.) the product of the terms between these two terms have row and column index @xmath144 and their last index is more than @xmath145 and less than @xmath146. thus between the terms @xmath140 and @xmath141 can only be the product @xmath147. thus formula is proved. now, we calculate the length of formula . formula is a sum of products. the length of every product is @xmath125, thus we need to calculate the number of products. notice, that every product is uniquely determined by the column indices of the terms. more exactly we need to know the column indices of the first @xmath148 terms, because the column index of the last term is @xmath136. we can choose these indices from the set @xmath149. the order of the selected elements does not matter. we only need to determine how many indices are equal to @xmath135 or to @xmath150, etc, or to @xmath151. thus we need to choose @xmath148 element from a set of @xmath152 elements such that repetitions are allowed. hence the number of products is @xmath153 since every product has length @xmath125, the length of @xmath126 is @xmath154 thus the length of @xmath126 is at most @xmath155 for every index @xmath156. let @xmath60 be a semipattern group. let @xmath157 be a polynomial over @xmath60. thus @xmath158 can indicate a constant or a variable over @xmath60 @xmath159. of course @xmath158 and @xmath160 can indicate the same constant or variable. let @xmath161 if @xmath162 indicates constant then @xmath163 is a constant in @xmath164 and @xmath165 is a constant in @xmath39 @xmath166. if @xmath162 is a variable, then @xmath163 is a variable, that we can substitute from @xmath56, and @xmath165 is a variable that we can substitute from @xmath39. furthermore @xmath167 if and only if @xmath168 (for every @xmath169) and @xmath170 for every @xmath122. we can rewrite the polynomial @xmath171 with this notation as @xmath172 after multiplying these matrices using lemma [mxszor] we obtain @xmath173 where @xmath174 the polynomial @xmath171 can attain the unit matrix for a substitution if and only if @xmath175 attains @xmath51 (@xmath176) and @xmath177 attains @xmath52 for the same substitution (@xmath122). thus @xmath178 is solvable over @xmath60 if and only if the system of equations @xmath179 is solvable over @xmath39. this is a system of equations over @xmath39 where the polynomials are given as sums of monomials. hence we can decide the solvability of this system of equations in polynomial time by theorem [egyrszmo]. the rewriting of @xmath171 over @xmath60 into the system of equations @xmath180 over @xmath39 can be done in @xmath124 time by lemma [mxszor], and the length of each equation is @xmath181. the number of equations is @xmath182, which does not depend on @xmath125, only on the group @xmath60. by theorem [egyrszmo] one can decide whether this system has a solution in @xmath183 time. The complexity of equation solvability problem over nilpotent matrix rings let @xmath104 be a ring of @xmath184 matrices over @xmath87 where every entry on or below the main diagonal of each matrix in @xmath104 is a multiple of @xmath103. let @xmath171 be a polynomial over @xmath104 given as a sum of monomials. let @xmath185 denote a monomial in the sum. let @xmath186 if @xmath187 indicates constant then @xmath188 are constants in @xmath87. if @xmath187 is a variable, then @xmath188 are variables, that we can substitute from @xmath87. furthermore, @xmath167 if and only if @xmath189 and @xmath190 for every @xmath191. first @xmath195 is a polynomial given as a sum of monomials over @xmath101. in every such monomial we multiply one term from each matrix @xmath158 @xmath196. hence the length of every monomial is @xmath125. there are at most @xmath197 monomials in every polynomial @xmath195 according to the usual multiplication of matrices. however, every nonzero monomial contains at most @xmath198 terms from on or below the main diagonal, since the characteristic of @xmath101 is @xmath99. therefore every nonzero monomial is of the form @xmath199 where @xmath200 and @xmath201. notice that @xmath202 holds as well. indeed, every term @xmath203 is from above the main diagonal, hence @xmath204. therefore @xmath205 and thus @xmath206. similarly @xmath207. hence the length of every nonzero monomial is at most @xmath208. in particular, if @xmath209 then every monomial in @xmath195 equals @xmath52. therefore there are at most @xmath210 monomials in every polynomial @xmath195 and thus @xmath211. the input polynomial @xmath171 is a sum of at most @xmath212 monomials @xmath213. let @xmath214 then @xmath215 is the sum of at most @xmath212-many polynomials @xmath195 for every @xmath216. thus @xmath217, as @xmath218 and @xmath100 depend only on the ring @xmath31. the polynomial @xmath171 can attain the zero matrix for a substitution if and only if @xmath215 attains zero for the same substitution for every @xmath219. thus @xmath220 is solvable over @xmath104 if and only if the system of equations @xmath221 is solvable over @xmath101. this is a system of equations over @xmath222 where the polynomials are given as sums of monomials. hence we can decide the solvability of this system of equations in polynomial time by theorem [er]. the rewriting of @xmath171 over @xmath104 into the system of equations @xmath223 can be done in @xmath224 time, and @xmath225. the number of equations is @xmath226, which does not depend on @xmath212, only on the ring @xmath104. by theorem [er] one can decide whether this system has a solution in @xmath227 time. finally, we explain how this result can be applied to determine if an equation @xmath34 is solvable over an _ arbitrary _ nilpotent ring @xmath31 of characteristic @xmath228. let @xmath104 be the matrix ring as in theorem [wil], and let @xmath229. further, let @xmath171 denote the polynomial in @xmath104 corresponding to @xmath95. notice, that @xmath230 was the characteristic of @xmath31 in theorem [wil], hence @xmath231. furthermore @xmath232 and @xmath233. therefore @xmath234. hence, one can decide if @xmath235 is solvable over @xmath236 in @xmath237 time. thus, one can decide if the equation @xmath34 is solvable over @xmath31 in @xmath238 time. now, @xmath239, which only depends on @xmath31. therefore, if @xmath95 is given as a _ sum of monomials _, then @xmath240, and one can decide @xmath241 over an arbitrary nilpotent ring @xmath31 in @xmath242 time, as well. if, however, @xmath95 is an arbitrary polynomial over @xmath31, then after rewriting it as a sum of monomials we have @xmath243, giving an extra @xmath244 factor in the exponent. thus, one can decide @xmath241 over an arbitrary nilpotent ring @xmath31 in @xmath245 time. mikael goldmann and alexander russell. the complexity of solving equations over finite groups. in _ proceedings of the 14th annual ieee conference on computational complexity _, pages 8086, atlanta, georgia, 1999.

the complexity of the equation solvability problem is known for nilpotent groups, for not solvable groups and for some semidirect products of abelian groups. we provide a new polynomial time algorithm for deciding the equation solvability problem over certain semidirect products, where the first factor is not necessarily abelian. our main idea is to represent such groups as matrix groups, and reduce the original problem to equation solvability over the underlying field. further, we apply this new method to give a much more efficient algorithm for equation solvability over nilpotent rings than previously existed.

1603.05788

Introduction. since its discovery, in 1980, the quantum hall (qh) effect @xcite has been a very rich source of interesting problems related to topological and correlation effects in condensed matter systems. the qh effect is observed in two - dimensional electron gases@xcite (2deg) exposed to a strong magnetic field perpendicular to the plane of the gas. at appropriate electron densities, the 2deg forms an incompressible liquid. the qh effect manifests itself in the precise and universal quantization of the hall conductance. this behavior originates from an interplay between the landau quantization of the orbital motion of electrons @xcite and interaction effects, which leads to the formation of a bulk energy gap. as a consequence an incompressible state is formed in the bulk of the 2deg. at the edge of a 2deg exhibiting the qh effect there exist, however, gapless chiral modes that are the quantum analogue of classical skipping orbits.@xcite in the presence of strong coulomb interactions, these modes can be viewed as collective edge plasmon modes. remarkably, it has been predicted@xcite that, at fractional fillings of the landau levels, besides the collective modes the edge states of the 2deg also describe quasi - particles with _ fractional charges _ and fractional statistics. @xcite for instance, in qh liquids with filling factor @xmath3, where @xmath4 is an odd integer, laughlin quasi - particle excitations have an electric charge @xmath5, where @xmath6 is the elementary electric charge. such excitations can be scattered between opposite edges at narrow constrictions forming quantum point contacts (qpc), thus contributing to a backscattering current. [frol - simpl] the fractional charge of laughlin quasi - particles has been confirmed, experimentally, in measurements of the shot noise of weak backscattering currents.@xcite the quasi - particle charge in these experiments is inferred from the fano factor of noise, which is the ratio of the noise power to the average backscattering current. although, at present, there is a consensus on the interpretation of the experimental results, this type of measurement does not, in general, represent a direct test of the fractional charge of quasi - particles. @xcite indeed, the fano factor of noise is not universal and may be reduced or enhanced for various reasons.@xcite for instance, the so called `` charge fractionalization '' in nonchiral one - dimensional systems@xcite is a property of collective modes that has nothing to do with the existence of fractionally charged quasi - particles. nevertheless, this phenomenon reduces the fano factor of noise at relatively high frequencies.@xcite a direct measurement of the quasi - particle charge should rely on its definition as a coupling constant in the interaction hamiltonian coupling matter to the electromagnetic field and on the quantum nature of quasi - particles. the most appealing approach is to make use of the aharonov - bohm (ab) effect,@xcite which relies on the fact that the interference of quasi - particles is affected by a magnetic flux. following a commonly used formulation of this effect, we consider a gedanken interference experiment shown in the upper panel of fig. [frol - simpl]. quasi - particles of charge @xmath7 traverse two beam splitters and follow paths that enclose a _ singular _ magnetic flux @xmath8. the relative phase between the two amplitudes, for the upper and lower path, is shifted by an amount of @xmath9, which leads to ab oscillations in the current from the source to the drain as a function of the flux @xmath8 with quasi - particle period @xmath2. the quasi - particles can then be detected by differentiating their contribution to the ab effect from electron oscillations with period @xmath0. ab oscillations with quasi - particle periods larger than the flux quantum have been observed in a number of recent experiments@xcite involving qh interferometers of the fabry - prot type. in these interferometers, chiral edge states of a qh system form a loop. the magnetic flux through the loop is varied either by changing the strength of the homogeneous magnetic field, or by deforming the edge of the qh system with the help of a modulation gate. the theoretical interpretation of the experimental results as demonstrating a quasi - particle ab effect appears obvious. nevertheless, it remains the subject of a theoretical debate for the following reasons. first of all, oscillations of the current through a fabry - prot interferometer as a function of the magnetic flux could originate from the coulomb blockade@xcite rather than from interference effects. second, the original gedanken formulation of the ab effect outlined above requires a singular magnetic flux threading through the region of the plane not accessible to quasi - particles. this is not the case in the experiments, see ref. [], where quasi - particles are directly affected by the magnetic field. third, and more importantly, the gedanken formulation of the ab effect itself leads to a paradox : in any electronic system, including fractional qh systems, ab oscillations should have an electronic period @xmath0, according to the byers - yang theorem.@xcite this is so, because, after adiabatic insertion of a flux quantum through a hole in the sample, an electronic system relaxes to its initial state, since the flux quantum can be removed by a single - valued gauge transformation. the first two problems can in principle be solved by using a qh interferometer of a different type, namely the electronic analog of an optical mach - zehnder (mz) interferometer.@xcite very recently, ab oscillations in these interferometers utilizing qh liquids at integer filling factors have become the subject of intensive experimental studies @xcite and theoretical discussions.@xcite a typical electronic mz interferometer is sketched, schematically, in the lower panel of fig. [frol - simpl]. in this interferometer the 2deg is confined to a region with a shape topologically equivalent to an annulus, i.e., a so called corbino disk. chiral edge states are used in place of optical beams and two qpcs (shown by dashed lines) play the role of optical beam splitters. ohmic contacts are attached to the inner and outer edge of the corbino disk and serve as a source and drain for the current. although mz interferometers are more difficult to manufacture, they have an advantage over fabry - prot interferometers : first, modes in two edge channels of the interferometer propagate in the same direction and without backscattering, because they are interrupted by ohmic contacts. as a result, the coulomb blockade can be avoided in these systems by strongly coupling the edge states to ohmic contacts. second, a singular magnetic flux can, at least in principle, be inserted in the hole of the corbino disk, i.e., in the region not accessible to electrons. therefore, electronic mz interferometers might be thought to be a solid - state implementations of the gedanken experiment shown in the upper panel of fig. [frol - simpl]. they may thus represent an ideal system for addressing the third problem mentioned above, namely the paradox associated with the byers - yang theorem. there have been several theoretical attempts to resolve this paradox. @xcite in early work [], thouless and gefen have considered the energy spectrum of a qh liquid confined to a corbino disk and weakly coupled to ohmic contacts. they have found that as a result of weak quasi - particle tunneling between the inner and the outer edge of the corbino disk, the energy spectrum, and consequently, `` any truly thermodynamic quantity '' is a periodic function of the magnetic flux with the electronic period @xmath0. although thouless and gefen have made an important first step towards understanding the ab effect in qh interferometers, their analysis is rather qualitative, and some of their statements concerning the tunneling rates and currents are not firmly justified.@xcite the results of their work are difficult to interpret at the level of effective theories. but, more importantly, they can not easily be generalized to the situation where the magnetic flux is varied with the help of a gate voltage. this situation has stimulated further interest in the fractional ab effect in mz interferometers. more recently, a number of authors (see refs. []) have proposed that the correct description of the mz interferometers should take into account the presence of additional quantum numbers in a qh state. in the thermodynamic limit, averaging the quasi - particle current over these quantum numbers is claimed to restore the electronic ab periodicity. these quantum numbers are usually introduced ad hoc, with minimal justification. depending on the particular theoretical discussion, they are represented either in terms of so called klein factors in the tunneling operators,@xcite or as additional phase shifts induced by quasi - particles localized at the inner edge of the corbino disk. @xcite we are aware of only one attempt to theoretically justify the introduction of klein factors in mz interferometers : in their recent work [], ponomarenko and averin implement a resummation of electron tunneling processes between the inner and outer edge and claim that there is a duality to weak quasi - particle tunneling, where the klein factors arise naturally. however, their analysis is done entirely at the level of an effective theory, where weak quasi - particle and weak electron tunneling are the two fixed points of a renormalization group flow. there is no guarantee that microscopic considerations will yield the same result. the results of the works [] can be summarized as follows : they resolve the byers - yang paradox by stating that ab oscillations with quasi - particle periods can not be observed in the steady - state current through an mz interferometer. in other words, these works do not discriminate between the effects of a singular flux, modulation gate, and a homogeneous magnetic field. our findings do not agree with this conclusion. in previous work,@xcite we have argued that quasi - particle ab oscillations should be observable if the magnetic flux is varied by deforming the edge of the qh system, or, alternatively, by changing the strength of the homogeneous magnetic field. we have proposed to use this effect as a spectroscopic tool for experimental investigations of scaling dimensions of quasi - particles at different filling factors. in the present paper we construct the theoretical basis for this effect. our predictions can be verified experimentally by pinching off the qpcs and comparing the regimes of weak electron tunneling and of weak quasi - particle tunneling, where the periods of ab oscillations should differ from each other by a factor of @xmath4. our theoretical arguments are presented on three different levels : by considering a microscopic wave function for the ground state of an mz interferometer, deriving an effective theory from it, and describing the relaxation of the interferometer to a stationary state with a master equation. [mz] at the level of the microscopic theory we arrive at the rather surprising conclusion that the ab effect, in its original formulation,@xcite is not exhibited by qh interferometers. in order to clarify this important claim, we first remark that the original formulation of the effect relies on a single - particle picture, where a particle wave travels through the interferometer, as shown in the upper panel of fig. [frol - simpl]. while traveling through the upper and lower paths of the interferometer, the particle wave accumulates a phase difference containing three contributions : a dynamical and a kinematic one, and the ab phase differences. this picture is quite appealing and is therefore commonly used in the physics literature. however, for the description of the ab effect, this picture is redundant, because at low energies, the dynamical phase can be neglected. instead, it is more appropriate to consider the relative phase of the wave function overlaps at the two beam splitters. moreover, in correlated systems, such as qh interferometers, the single - particle description is, strictly speaking, not applicable. in particular, the naive (gedanken) formulation of the ab effect may lead to serious misconceptions and problems, one of them being the byers - yang paradox ; (see also the discussion in sec. [conseq]). thus, we propose to investigate many - particle wave function overlaps. to this end, we consider a microscopic model of an mz interferometer schematically shown in fig. a qh liquid at filing factor @xmath3 is confined to a region between two circles of radii @xmath10 and @xmath11. by separating the centers of the two circles, we deliberately break the axial symmetry in order to avoid specific geometry effects and to exhibit the general character of our results. the inner and outer edges of the qh liquid are connected to ohmic contacts at points @xmath12 and @xmath13 via strong electronic tunneling. weak backscattering at two qpcs is modeled by quasi - particle tunneling between points @xmath14, @xmath15, and @xmath16, @xmath17, as indicated in fig. [mz] by dashed lines. the state of the qh liquid is described by a laughlin - type wave function, see eqs. ([laugh]) and ([w - ring]). we generalize the laughlin wave function, in order to take into account the deformation of the qh edge caused by the modulation gate, eq. ([w - modg]). exactly the same procedure can be applied in order to describe a qh liquid of arbitrary shape. this will allow us to choose @xmath18 and @xmath19 in order to describe realistic qpcs. finally, the effect of a singular flux threading through the hole in the interferometer is described by the wave function in eq. ([w - magn]). it turns out that the wave function overlap ([tun - matr]) at a point @xmath20, i.e., the matrix element of a quasi - particle tunneling operator ([tun - micro]), is a single - valued function of @xmath20, for any number, @xmath21, of quasi - particles in the interferometer. this implies that the commonly used picture,@xcite according to which every quasi - particle carries a `` statistical phase tube '' seen by other quasi - particles as an additional phase shift, does not apply _ naively _ to mz interferometers. more importantly, in the presence of a singular magnetic flux @xmath8, the laughlin wave function ([w - magn]) contains a flux - dependent factor, which depends _ only on the distance _ between electrons and the flux tube. in other words, the ab phase from the gauge field is canceled, locally, by the phase induced by the physical deformation of the laughlin wave function in response to the singular flux. this fact is independent of the shape of the qh system and leads to an exact cancellation of the relative ab phase in the tunneling amplitudes ([h - tun - f]). thus, we arrive at the rather remarkable result that the ab effect, in its original formulation,@xcite does not exist in qh interferometers. however, in response to local edge deformations induced by a modulation gate, as described by eq. ([w - modg]), the wave function and, consequently, the tunneling amplitudes acquire a relative phase proportional to the total change in the homogeneous magnetic flux through the qh liquid enclosed by the interferometer. this effect leads to oscillations in the current with quasi - particle period @xmath2. note that this phenomenon, which we propose to call a _ quasi - particle ab effect _, does _ not _ violate the byers - yang theorem. in order to connect these findings to the low - energy effective theory of the qh effect,@xcite we consider small incompressible deformations of the ground state of an mz interferometer, as shown in fig. [def], below. the microscopic wave function resulting from such deformations, given by eqs. ([wave-1]) and ([omega]), is parameterized by an infinite set of variables, @xmath22. we invoke the classical plasma analogy @xcite and follow the steps of ref. [] in order to project the microscopic hamiltonian onto the subspace of these deformations. after the projection, the variables @xmath22 turn into oscillator operators, @xmath23, with canonical commutation relations ([commut - mic]). these operators describe _ gapless _ plasmon excitations at the edge of the qh liquid. the projected hamiltonian ([h - edge]) contains the oscillator part with a linear spectrum and the coulomb charging energy, which depends on the number of quasi - particles, @xmath21, and the number of electrons, @xmath24, in the qh system. the projected tunneling operators take the form of vertex operators ([h - tunnel]). interestingly, we do not find any trace of klein factors@xcite in the tunneling amplitudes. moreover, contrary to earlier suggestions,@xcite the tunneling hamiltonians at different points in space do not commute with each other (see appendix [app - comm]). we show that the low - energy theory so derived agrees well with the effective theory of wen and one of us,@xcite generalized so as to take into account the finite size of the qh system and the effects of a modulation gate and of a singular magnetic flux. at the level of the effective theory, the three contributions to the overall phase shift in the current oscillations, the dynamical, kinematic, and ab phase mentioned above, acquire the following interpretation.@xcite the dynamical phase has its origin in temporal fluctuations of the charge density at the edge described with the help of the oscillator operators, @xmath23. these fluctuations suppress the ab effect at high temperatures but can be neglected at low energies. the kinematic phase can be viewed as a contribution of zero modes. in order to interpret the ab phase, we note that the effective theory in refs. [] and [] arises as a boundary contribution to a topological chern - simons theory@xcite describing the bulk of the 2deg in the presence of an external electromagnetic field. within this theory, the ab phase is picked up by a wilson loop [see eq. ([two - tun])] along the interferometer contour. there are two contributions to this phase : one is proportional to the gauge field describing the singular magnetic flux ; the other one comes from the charge accumulated at the inner edge as a result of an adiabatic variation of the singular flux. it turns out that these two contributions cancel each other exactly, so that the total phase due to a singular magnetic flux vanishes. in the particular gauge we use, the cancellation is local in space and can be viewed as screening of the singular flux by the qh liquid. we propose to call this effect _ topological screening _, in order to emphasize its independence of the particular sample geometry. having constructed the low - energy theory of an isolated qh system, we proceed to analyze the quasi - particle transport at qpcs and the electron transport at ohmic contacts. we apply the tunneling approximation to describe both processes as rare transitions that change the numbers, @xmath25 and @xmath26, of electrons at the inner and outer edge of the corbino disk and the number, @xmath27, of quasi - particles at the inner edge, where @xmath28. the dynamics of the mz interferometer on a long time scale is described by a master equation for the probability to find the system in a state corresponding to given values of @xmath29. in order to model ohmic contacts, we consider the limit of strong electron tunneling. as a result, the density matrix in the electronic sector approaches the product of two equilibrium distributions determined by the ohmic reservoirs ([rho - eq]). then we sum over the electronic numbers @xmath25 and @xmath26 and obtain the master equation, see ([master - eq - qp]) and ([t - sum]), for the density matrix in the quasi - particle sector indexed by @xmath27. finally, solving this master equation, we find the current through the mz interferometer as a function of the magnetic flux, thereby establishing the _ main results _ of our paper. namely, we find that the current oscillates as a function of the singular magnetic flux with the electronic period @xmath0, and these oscillations are exponentially suppressed with temperature and system size. this implies that they can be viewed as a coulomb blockade effect. in contrast, as a function of the magnetic flux modulation due to a gate, the current oscillates with the quasi - particle period @xmath2, and these oscillations survive in the thermodynamic limit. thereby, the byers - yang paradox is resolved. an important result of our analysis is to predict the screening of the magnetic flux through the hole in a corbino disk by the qh liquid. since this effect has a topological character, i.e., does not depend on the distribution of the flux inside the hole and the sample geometry, our prediction also holds in a homogeneous magnetic field. thus, the effect of topological screening can be tested, in principle, in an experiment with an mz interferometer by changing the area of the hole with a modulation gate but preserving the area of the qh liquid inside the interferometer, as shown in fig. [exper1]. we predict that, after the application of a gate voltage, the periodicity of the current as a function of the homogeneous magnetic field will not change, contrary to a naive expectation that the period depends on the total area of the interferometer. this is because the flux through the hole is screened and the ab phase is proportional to the area of the 2deg enclosed by the interferometer paths. the last fact leads also to another important prediction : in a strongly disordered system stuck on a qh plateau, the topological screening leads to a linear dependence of the ab period (measured in ab oscillations, as a function of the homogeneous magnetic field) on the filling factor. this is because by changing the filling factor we also change the area of the qh liquid inside the interferometer, even if the interfering paths are fixed. we stress that our predictions also apply to a qh system at the @xmath30 plateau, where the measurements can be done more easily (see also the discussion in sec. [conseq]). our paper is organized as follows. in sec. [s - wave - func] we construct the ground state wave function of an mz interferometer in the fractional qh effect regime and discuss the effects of a singular magnetic flux and of a modulation gate. in sec. [s - project], by projecting the microscopic hamiltonian and tunneling operators onto the subspace of states corresponding to small incompressible deformations of a qh liquid, we derive the low - energy theory of the mz interferometer. then, in sec. [s - effective], we compare the description derived from our microscopic theory with the one provided by the effective chern - simons theory. in sec. [s - out - of - eq] we use the low - energy effective hamiltonian in order to find the density matrix of the mz interferometer strongly coupled to ohmic reservoirs and to evaluate the charge current in the presence of a singular magnetic flux and of a modulation gate. we then discuss physical consequences of topological screening and propose simple experiments to test our predictions in sec. [conseq]. finally, in appendix [app - comm], we discuss the commutation relations for the quasi - particle operators and for tunneling operators, which are important ingredients of our theory. Microscopic description of the mz interferometer in this section we construct the many - particle wave functions of the ground state and of gapless excited states of an mz interferometer. we do this step by step, starting from the laughlin wave function and manipulating it, in order to arrive at a realistic model of the interferometer. we first present the most important results before we prove them, in sec. [s - incompr], using the classical plasma analogy. @xcite the variational wave function, proposed by laughlin in ref. [] and later justified by haldane and rezayi in ref. [], describes an approximate ground state, @xmath31, of a qh system with @xmath24 electrons at filling factor @xmath3 : @xmath32 here @xmath33 denotes a set of complex coordinates @xmath34 describing the position of the @xmath35 electron, @xmath36, and @xmath37 is the magnetic length. it is known @xcite that the wave function ([laugh]) describes a circular droplet of a qh liquid of constant density @xmath38 and of radius @xmath39. in the next step, we add to the state ([laugh]) a macroscopic number, @xmath21, of laughlin quasi - particles @xcite at the point @xmath40 : @xmath41 in sec. [s - incompr] we explicitly show that the wave function ([w - ring]) describes a qh state of constant electron density @xmath42 inside a corbino disk, as shown in fig. the inner hole of the disk is centered at @xmath40 and has a radius @xmath43, while the outer radius of the disk is given by @xmath44. we stress that the sample geometry so obtained is not symmetric under rotations around the axis through the origin perpendicular to the sample plane, for @xmath45. we deliberately break the axial symmetry in order to avoid accidental effects of symmetry and to come closer to a realistic model of an interferometer (see fig. [frol - simpl], lower panel). additional small incompressible deformations of the qh liquid disk may be described as follows. we note that all states of the lowest landau level can be described by holomorphic functions of electron coordinates. we therefore look for a wave function of the form @xcite @xmath46\langle \underbar{z}\,|n, m\rangle, \label{wave-1}\]] where the function @xmath47 \label{omega}\]] is analytic inside the corbino disk (shown in fig. [mz]), and @xmath48 denotes a set of parameters @xmath22, @xmath49. in sec. [s - incompr] we show that the shape of the deformed disk is given by the solution of a two - dimensional electrostatic problem, with @xmath50 playing the role of an external potential. we utilize the wave function ([wave-1]) in two ways. first of all, small incompressible deformations are known to be the gapless excitations of the qh state.@xcite therefore, in order to describe the low - energy physics, we will use the wave functions ([wave-1]) to project the microscopic hamiltonian of the mz interferometer onto the subspace corresponding to incompressible deformations. second, we investigate the effect a modulation gate located near one of the arms of the interferometer. if a negative potential is applied to such a gate, the 2deg is depleted locally. in sec. [s - incompr] we show that the following choice of wave function @xmath51 \langle \underbar{z}\,|n, m\rangle, \label{w - modg}\]] with @xmath52, describes the local deformation of a corbino disk near the point @xmath53, @xmath54 on its outer edge. this deformation is parameterized by the total magnetic flux @xmath8 through the depleted region and the location, @xmath53, of the gate. finally, after the adiabatic insertion of a singular magnetic flux @xmath8 through the hole in the corbino disk at the point @xmath40, the wave function is multiplied by the phase factor @xmath55 $]. at the same time, the wave function is deformed by the spectral flow in order to preserve its single - valuedness. this effect is described by the additional multiplier @xmath56. the overall effect of a singular flux on the wave function can thus be represented by replacing the original wave function by @xmath57 it is important to note that the function ([w - magn]) is single - valued, and it describes an incompressible deformation of the initial state ([w - ring]). the classical plasma analogy@xcite has proven to be an efficient method in the analysis of qh states.@xcite it relies on the important observation that the norm of the laughlin wave function may be written as the partition function of an ensemble of @xmath24 classical particles interacting via the two - dimensional (logarithmic) coulomb potential. in the large-@xmath24 limit the evaluation of the partition function reduces to solving a two - dimensional electrostatic problem. here we apply this method directly to the wave function ([wave-1]). we write @xmath58 where the inverse temperature of the plasma is @xmath4 and the energy is given by the expression : @xmath59 \label{e}\end{gathered}\]] introducing the microscopic density operator @xmath60, we can formally write @xmath61 where @xmath62 and normal ordering is assumed in the first term on the right hand side of eq. ([ener]) in order to remove the self - interaction contribution. this representation makes it obvious, that the partition function ([z1]) describes a gas of charged particles interacting via the 2d coulomb potential that are confined by the external potential @xmath63. the first term in eq. ([extphi]) describes the interaction with a neutralizing homogeneous background charge of density @xmath64. the second term can be viewed as describing a repulsion from a macroscopic charge @xmath65 at the point @xmath40. finally, the last term describes the effect of an external (chargeless, since @xmath66) potential on the particles in the gas. the next step is to approximate the integral over coordinates in eq. ([z1]) by a functional integral@xcite over the density @xmath67 : @xmath68}. \label{z2}\]] after this approximation, which can be justified in the large-@xmath24 limit,@xcite the evaluation becomes straightforward. we note that the energy of the plasma is a quadratic function of the density. hence the average density @xmath69 is given by the solution of the saddle - point equation @xmath70, which reads @xmath71 important consequences of this simple equation are the following ones. first, it implies that the total potential vanishes in the region where @xmath72, i.e. where the 2deg is not fully depleted. in other words, the coulomb plasma is a `` perfect metal '' that completely screens the external potential @xmath73. applying the laplacian to ([eom-1]), we find that @xmath74, i.e., the coulomb plasma is distributed homogeneously to screen the background charge. this confirms that the wave function ([wave-1]) describes an incompressible deformation of the qh droplet. in particular, the wave function ([w - ring]) describes the approximate ground state of an mz interferometer, shown in fig. indeed, the plasma analogy suggests that the hole in the corbino disk is formed symmetrically around the point @xmath40, where the macroscopic charge @xmath65 is located. it serves to screen this charge, so that the total potential vanishes in the region occupied by the 2deg. because of perfect screening, the shape of the outer edge is, however, independent of the position of the hole and displays the symmetry of boundary conditions in the background charge distribution (see first term in eq. ([extphi])). we now investigate the effect of the potential @xmath50 perturbatively. let us denote by @xmath75 the region to which the qh system is confined. we search for the solution of eq. ([eom-1]) in the form @xmath76, for @xmath77, and @xmath78 otherwise. thus we can rewrite eq. ([eom-1]) as @xmath79 introducing a small deformation, @xmath80, and taking into account that the integral over the undeformed corbino disk, @xmath81, cancels the first two terms in eq. ([extphi]), we arrive at the following result @xmath82 in polar coordinates (see fig. [def]), the boundaries of the deformed disk can be parameterized as @xmath83, where @xmath84, and @xmath85 are the 1d charge densities accumulated at the inner and outer edge due to the deformation. because of perfect screening in the two - dimensional coulomb plasma, one can solve eq. ([eom-2]) independently for each edge. using the series expansion @xmath86, for @xmath87, and the explicit expression ([omega]) for the potential @xmath50, we solve equation ([eom-2]) by power series. the result can be presented in the form of fourier series : [sol - def] @xmath88 these series show how the microscopic wave function ([wave-1]) determines the shape of the deformed corbino disk. finally, we analyze the effects of a modulation gate and of a singular magnetic flux, as illustrated in fig. [mod]. according to the result ([sol - def]), the deformation described by the wave function ([w - modg]) has the following form : @xmath89 where @xmath90 is the argument of the position, @xmath53, of the modulation gate. this function correctly captures the effects of the modulation gate : the local depletion of the 2deg at the point @xmath91 and the homogeneous expansion of the qh liquid due to its incompressibility. it is easy to check that the flux through the depleted area under the modulation gate, @xmath92, is indeed equal to @xmath8. in the presence of a singular magnetic flux, the plasma energy contains an additional term @xmath93 in the language of the coulomb plasma it describes the addition of a charge @xmath94 in the hole of the interferometer at the point @xmath40. this homogeneously shifts the edges of the corbino disk by an amount @xmath95, @xmath84, as illustrated in fig. [mod]. having found the set of states ([wave-1]) describing incompressible deformations of a qh liquid, we proceed to construct operators generating the subspace of such low - energy states when applied to the undeformed ground - state and finding their commutation relations. first of all, let us introduce zero - mode operators changing the number of electrons @xmath24 and quasi - particles @xmath21 in the system @xmath96 [zeromodedef] states with different numbers of electrons @xmath24 are obviously orthogonal, while the overlaps of wave functions with different numbers of quasi - particles @xmath21 are strongly suppressed in the large-@xmath24 limit.@xcite taking this observation into account, one derives from the definitions ([zeromodedef]) the following commutation relations for the operators of zero modes@xcite @xmath97 = e^{i\phi }, \ ; \ ; [n, e^{i\phi_n }] = e^{i\phi_n}. \label{zeromodecom}\]] next, we introduce deformation operators @xmath98, @xmath84, for any @xmath99, acting from the right as [ak][defa] @xmath100 the states ([wave-1]) are coherent states under the action of these operators. in order to find commutation relations for these operators, we need to evaluate scalar products of the states ([wave-1]). we start with the norm of a wave function, which is given by the square root of the partition function of the coulomb plasma, ([z1]) and ([z2]), and evaluate the `` free energy '' @xmath101 considering the potential @xmath50 as a perturbation, we obtain @xmath102 where the constant @xmath103 is the contribution from the unperturbed state and from the determinant of the gaussian integral. we evaluate this integral with the help of the solution ([sol - def]) and present the result as a bilinear form in the coefficients @xmath22, @xmath104 \\ -\sum_{k\geq k'>0}k'c_k^{k'}[z_0^{k - k'}t_kt_{-k'}+{\rm c.c. }], \label{free - en}\end{gathered}\]] where @xmath105 are binomial coefficients. the holomorphic structure of this bilinear form allows us to extend the result for the norm @xmath106 to the scalar products @xmath107. next, we define differential operators, via their matrix elements, as follows : @xmath108. a straightforward calculation then yields : @xmath109 \langle n, m,\underbar{t}\,|n, m,\underbar{t}\,'\rangle. \label{interm}\end{gathered}\]] using definition ([defa]), we may write @xmath110. substituting this equation in ([interm]) we finally express the adjoint operators acting on the states @xmath111, via the parameters @xmath22, as @xmath112 repeating exactly the same calculations for the operators @xmath113, we obtain : @xmath114 using expressions ([ak]), ([a - ku]) and ([a - kd]) for the operators @xmath98 and their adjoints and the relation @xmath115 = \delta_{kk'}$], we obtain the commutation relations : @xmath116 = \frac{1}{m}\delta_{kk'}\delta_{ss'}. \label{commut - mic}\]] similarly, one finds that @xmath117 = [a_{ks},a_{k's'}] = 0 $]. thus the deformation operators, @xmath98 and @xmath118, introduced in ([ak]) satisfy canonical commutation relations, and the subspace of incompressible deformations has a natural fock space structure with respect to these operators. Projection onto the low-energy subspace starting from the microscopic model, we now explicitly derive the low - energy effective theory of an interferometer. for this purpose, we project the microscopic hamiltonian and tunneling operators of quasi - particles between the two edges onto the low - energy subspace constructed above. the projection of these operators is defined by @xmath119, where the orthogonal projection @xmath120 is written as : @xmath121 where the norm @xmath122, in the large @xmath24 limit, is given by the integral ([z2]). we first implement the projection procedure in the absence of an external magnetic flux. in sec. [s - flux] we then consider situations where a singular flux tube is inserted and a modulation gate voltage is applied. the microscopic hamiltonian for @xmath24 electrons, restricted to the lowest landau level, is given by the expression @xmath123 where @xmath124 is the potential of the screened 3d coulomb interaction and @xmath125 is the confining potential, which forces electrons to form the interferometer. note that we have omitted the kinetic energy operator, since, acting on the lowest landau level, it gives a constant contribution, @xmath126, where @xmath127 is the cyclotron frequency. the projection of the microscopic hamiltonian onto the subspace of incompressible deformations @xmath128 is given by @xmath129 where @xmath130, and the potential energy contribution reads @xmath131 we first consider diagonal matrix elements @xmath132 in ([edge - energy]). they can be rewritten in terms of the electronic density in the deformed state as follows : @xmath133 in the large-@xmath24 limit and for the long - range coulomb interaction we approximate this function as @xmath134, neglecting the `` exchange '' contribution. thus, we may rewrite the energy of a deformation in terms of the average density as @xmath135 next, we propose to express the projected hamiltonian ([edge - matr]) in terms of the deformation operators ([ak]). to this end, we consider small deformations of the state ([w - ring]) and take into account the fact that the density is constant, @xmath76, for @xmath77. writing the deformed region as @xmath136, one can expand the integral ([edge - energy-2]) in the small deformation @xmath137 and evaluate the correction term with the help of the result ([sol - def]) : @xmath138. \label{edge - spect}\end{gathered}\]] where @xmath139 is the energy of a qh system confined to an undeformed corbino disk, and the last two terms originate from the deformation @xmath137. the excitation spectra, @xmath140, @xmath84, are determined by the two - body interaction and by the confining potential : @xmath141. \label{spectr}\]] using again the holomorphic structure of the bilinear form ([edge - spect]) to extend this result to off - diagonal matrix elements, we arrive at the projected hamiltonian ([edge - matr]) in the following form : @xmath142 we further assume that the potential @xmath143 describes coulomb interactions screened at a distance @xmath144. in the low - energy limit, i.e., for @xmath145, the deformation energy in ([spectr]) is then linear as a function of the mode number : @xmath146, where the constants @xmath147 are the group velocities of edge excitations. the energy of the undeformed state takes the following form as a function of the number of electrons @xmath24 and the number of quasi - particles @xmath21 : @xmath148 replacing the mode number with the wave vector @xmath149, we arrive at the final expression for the edge hamiltonian @xmath150 we conclude this section with the following important remark. equation ([spectr]) contains two terms : the first term is the drift velocity, proportional to the boundary electric field, @xmath151, while the second one is proportional to the `` coulomb logarithm '' @xmath152 the ultraviolet cutoff in ([velo]) is determined by correction terms in a @xmath153-expansion of the two - point density correlation function. in fact, result ([velo]) coincides with an expression proposed earlier in ref. [] on the basis of the classical electrostatic picture. the tunneling hamiltonian for an mz interferometer may be written as a sum of tunneling operators at the left and right qpc : @xmath154. \nonumber\]] the tunneling operator is an operator annihilating a quasi - particle at a point @xmath155 on one edge and recreating it at a point @xmath20 on the other edge : @xmath156 where @xmath157 and @xmath158 are quasi - particle operators. as for the edge hamiltonian, the projection of the tunneling operator, @xmath159, is expressed in terms of matrix elements in deformed states @xmath160. these matrix elements can be found by inserting a complete set of intermediate states : @xmath161 in other words, we define the tunneling operator as an operator whose matrix elements are overlaps of wave functions with quasi - particles located on the inner and the outer edge. the wave functions with insertion of a quasi - particle, @xmath162 are constrained by the condition that the annihilation of @xmath4 quasi - particles at the same point is equivalent to the annihilation of an electron. the electron operator is defined by @xmath163, which leads to the result : @xmath164 where we omitted a combinatorial factor, because it can be absorbed into the tunneling amplitudes. a quasi - particle at the point @xmath20 is therefore described by the following wave function : @xmath165 expression ([qp - def]) differs from laughlin s quasi - particle definition only by a constant factor. we show in appendix [app - comm] that the quasi - particle operator, obtained by the projection of @xmath166 defined by ([qp - def]) onto the low - energy subspace, satisfies all the physical requirements (locality, charge and statistics). in order to evaluate the matrix elements ([tun - matr]), we first assume that @xmath167, as in the previous section, and then generalize our findings. the product @xmath168 in eq. ([qp - def]) can be rewritten as @xmath169 $]. thus, expanding the logarithms in power series on the outer edge, @xmath170, and on the inner edge, @xmath171, we arrive at the following expression for the matrix elements ([tun - matr]) : @xmath172 \\ \times\int \prod_id^2z_i (z^*_i - z^*_0)\exp\left\ { -\sum_{k>0}\left [\frac{z_i^k}{k\xi^k}+ \frac{(\xi^*-z^*_0)^k}{k(z^*_i - z^*_0)^k}\right]\right\ } \langle n, m,\underbar{t}\,|\underbar{z}\,\rangle \langle \underbar{z}\,|n, m',\underbar{t}\,\rangle. \label{hren}\end{gathered}\]] here we have assumed for simplicity that tunneling points are close to each other and have set @xmath173. taking into account that @xmath174 and @xmath175, we pull out of the integral the power series in @xmath176. then we use the fact that @xmath177 to rewrite the expression ([hren]) in the following form : @xmath178 \\ \times\exp\left\{-\sum_{k>0}\left[\frac{1}{k\xi^k}\frac{\partial}{\partial t_k}+\frac{(\xi^*-z_0^*)^k}{k}\frac{\partial}{\partial t_{-k}^*}\right]\right\}\langle n, m+1,\underbar{t}\,|n, m',\underbar{t}\,\rangle. \label{hren-2}\end{gathered}\]] one can immediately see that the matrix element ([hren-2]) vanishes unless @xmath179. finally, using expression ([free - en]) for the logarithm of the norm @xmath180 and definition ([ak]), we can write the projection of the tunneling operator ([tun - micro]) in terms of the deformation operators as follows : @xmath181, \label{h - tunnel}\]] where all the constant multiplicative factors are absorbed into a prefactor @xmath182, and we have introduced the following notations : [bsefield] @xmath183,\]] @xmath184.\]] to complete our description of the low - energy physics of the interferometer we must find the projection of the operators of electron tunneling from the quantum hall edges to the ohmic contacts. this can be done by applying the technique used above to the electron annihilation operator ([el - def]). the result of the projection is given by @xmath185\\ \times\exp\big[i\phi_n + mn\ln\xi_u+ m\ln(\xi_u - z_0)\big] \label{elec - u}\end{gathered}\]] for tunneling from the outer edge to the upper ohmic contact (see fig. [mz] for notations), while the tunneling operator on the inner edge is given by @xmath186 \\ \times\exp\big[i\phi_n + im\phi + m\ln(\xi_d - z_0)\big], \label{elec - d}\end{gathered}\]] to conclude this section we note that one can find the projected charge density operators at the edges of a qh system, @xmath187 and @xmath188, by rewriting the result ([sol - def]) directly in terms of the operators ([ak]) : @xmath189 [rho - edge - mic] here the homogeneous contributions describe the charge accumulation caused by the variation of quantum numbers @xmath21 and @xmath24. expressions ([rho - edge - mic]) lead to the following commutation relations @xmath190 = \pm \frac{1}{mr_s } \delta(\theta - \theta_s){\cal a}(\xi),\]] where the angles @xmath191 and @xmath192 parametrize the position of the tunneling point in coordinates of the inner and outer edge. these commutation relations show that the tunneling operator ([h - tunnel]) creates a pair of point - like charges of magnitude @xmath193. similarly, one can check that the operators ([elec - u]) and ([elec - d]) create a unit charge at the corresponding edge. expressions ([h - tunnel]-[rho - edge - mic]) complete the projection procedure (see also appendix [app - comm]). the resulting low - energy theory agrees well with the effective theory of refs. [] and []. having outlined a self - consistent procedure for the projection of observables onto low - energy states, we next consider the projection of the edge hamiltonian and of the tunneling operators in the situation where a magnetic flux is threading through the interferometer. first, we consider the situation where the flux is varied by application of a modulation gate voltage. the wave function of the interferometer in this situation is given by eq. ([w - modg]), and the edge density in this state is given by eq. ([dens - g]). similarly to eq. ([wave-1]), we define deformations of the ground state ([w - modg]) as follows : @xmath194\langle \underbar{z}\,|n, m\rangle, \label{w - def - g}\end{gathered}\]] where @xmath52 describes the local deformation of a corbino disk near the point @xmath53, @xmath54 on its outer edge. the scalar products of these states can be deduced from eq. ([free - en]). they are given by the following expression : @xmath195, \label{free - gate}\end{gathered}\]] where @xmath196, for @xmath99, and @xmath197, for @xmath198. using the result ([free - gate]), one concludes that the algebra of deformation operators ([commut - mic]) does not change after the application of a gate voltage. next, we find that the edge density in the state ([w - def - g]) coincides with the result ([sol - def]), but with @xmath22 replaced by @xmath199. this does not affect the energy of small deformations, which is still bilinear in the coefficients @xmath22. therefore, the only effect of the modulation gate on the edge hamiltonian, up to irrelevant constants, is given by : @xmath200 we can also find the projection of the tunneling operators by applying the shift @xmath201 to formula ([h - tunnel]), which yields @xmath202\\\times \exp \big[-i\phi + n\ln\xi + i\varphi_u(\xi) - i\varphi^\dag_d(\xi^*-z^*_0)\big]. \label{tun - gate}\end{gathered}\]] thus, we note that, after an application of a gate voltage, the tunneling operators @xmath203 at the left qpc and @xmath204 at the right qpc acquire a relative phase @xmath205 $] proportional to the flux @xmath8 through the region of 2deg depleted by the modulation gate. using the explicit form of the function @xmath206, we arrive at the following expression : @xmath207 where @xmath208 is the length of the outer arm of the interferometer (see fig. [e - b]). we will show in sec. [s - gate] that only the topological part @xmath209 of this phase enters tunneling rates, while the geometry - dependent part cancels exactly with the zero - mode contribution. the situation where the flux is changed via the insertion of a singular flux tube is described by the wave function ([w - magn]). in sec. [s - incompr] we have shown that the insertion of a singular flux is equivalent to the insertion of a point - like charge (in classical plasma language) and that it leads to a homogeneous shift of the edges. thus, we find the projected edge hamiltonian by replacing @xmath21 with @xmath210 (to take into account the shift of the edges), which yields @xmath211 the wave functions of the deformed states in the presence of a singular flux @xmath8 differ from ([wave-1]) only by the factor @xmath212. thus the tunneling operator can be found with the help of the same calculations as in sec. [s - tunnel], which yields an expression that differs from eq. ([h - tunnel]) only by a real prefactor : @xmath213 \label{h - tun - f}\end{gathered}\]] we conclude that, in contrast to the situation where a modulation gate is introduced, the tunneling operators at different qpcs do not acquire any relative phases due to the singular flux. this is interpreted as an exact cancellation of the phase shift caused by the flux itself and the phase shift caused by charge accumulation at the edges. in sec. [s - effective], we show that this fact is universal, i.e., it does not depend on the specific wave function ([w - ring]) of the interferometer. Effective theory in this section we show that the low - energy effective theory, derived, in the last section, from the specific wave function ([wave-1]), can be formulated in a universal gauge - invariant form. in order to do so, we start from the topological chern - simons theory@xcite in the bulk of a qh system and construct a boundary action so that the total action is local and gauge invariant. then we show that the gauge - invariant edge hamiltonian coincides with the one derived from microscopic theory, see eq. ([h - edge]), and tunneling amplitudes coincide with those given by eq. ([h - tunnel]). finally, we show that the independence of the tunneling amplitudes on a singular flux is a gauge - invariant effect, caused by topological screening of a magnetic flux by the chern - simons field. we begin by recalling the construction of the effective low - energy theory of an infinitely extended qh liquid at filling factor @xmath3. such a liquid is described by a conserved current, @xmath214. the continuity equation, @xmath215, is solved by introducing potentials @xmath216, @xmath217 here and below, we use units where @xmath218, and adopt the einstein summation convention, unless specified otherwise. the current is invariant under the gauge transformations @xmath219. by counting dimensions, it is easy to see that the gauge invariant action for the potential @xmath220 given by @xmath221 = \frac{m}{4\pi}\int d^3r \epsilon^{\mu\nu\lambda}b_\mu\partial_\nu b_\lambda \label{bulk - act}\]] has zero dimension, while all other possible terms have _ lower _ dimensions, i.e., are _ irrelevant _ at large distance and low energy scales. for example, the maxwell - like term has dimension @xmath222. next, the interaction with an external electromagnetic field, described by a vector potential @xmath223, is given by the term : @xmath224 = \int d^3r a_\mu j^\mu = \frac{1}{2\pi}\int d^3r a_\mu \epsilon^{\mu\nu\lambda}\partial_\nu b_{\lambda}. \nonumber\end{gathered}\]] integrating out the fields @xmath225, we arrive at an effective action for the electromagnetic field in the chern - simons form : @xmath226 = \frac{1}{4\pi m } \int d^3r \epsilon^{\mu\nu\lambda}a_\mu\partial_\nu a_\lambda. \nonumber\]] the average current @xmath227/\delta a_\mu$] is then given by hall s law : @xmath228 where @xmath229 is the hall conductivity. thus, we conclude that, for an infinitely extended liquid, the action ([bulk - act]) correctly describes the qh effect at @xmath3. however, in the situation where a qh liquid is confined to a finite region @xmath75, the effective action for the qh liquid in the presence of an external electromagnetic field, @xmath230 = \frac{1}{4\pi m}\int_{d}d^3r\epsilon^{\mu\nu\lambda}\big[2a_\mu+ mb_\mu\big]\partial_\nu b_\lambda, \label{s - bulk}\]] is not gauge invariant.@xcite namely, one easily sees that, under a gauge transformation @xmath231, @xmath232, the action ([s - bulk]) transforms as @xmath233\to s[a, b]+\delta s[a, b]$] with : @xmath234 = \frac{1}{4\pi}\int_{d}d^3r \epsilon^{\mu\nu\lambda } \big[2\partial_\mu \alpha+m\partial_\mu\beta \big]\partial_\nu b_\lambda \\ = \frac{1}{4\pi}\int_{\partial d}d^2r [2\alpha+m\beta]\epsilon^{\mu\nu}\partial_\mu b_\nu \label{anomal}\end{gathered}\]] where @xmath235 is the restriction of the bulk field @xmath220 to the boundary @xmath236. a physical reason for the gauge anomaly is the fact that in a qh liquid confined to a finite region the bulk hall current ([j - bulk]) is not conserved. consequently, the electric charge may accumulate at the edge of the sample. in order to restore the gauge invariance of the effective theory, we take into account the boundary degrees of freedom. it is easy to see that the boundary action @xmath237 = \frac{m}{4\pi}\int_{\partial d } d^2r [d_t\phi d_x\phi - h(d_x\phi) + \epsilon^{\mu\nu}b_\mu\partial_\nu\phi], \label{s - edge1}\end{gathered}\]] which is similar to the action found in ref. [] from a quantization of the hydrodynamics of a charged liquid, cancels the gauge anomaly ([anomal]), provided one assumes that the edge field @xmath238 transforms as @xmath239 and the covariant derivative is given by the expression @xmath240 note that the commutation relations for the field @xmath241 are determined by the first term in the canonical action ([s - edge1]) : @xmath242 = \frac{2\pi i}{m}\delta(x - y). \label{commut - eff}\]] however, the precise form of the boundary hamiltonian density @xmath243 is not fixed by the effective theory. the only requirement is that the hamiltonian density should be a positive definite function of @xmath244. the simplest possible expression @xmath245, justified in the case of small edge deformations, leads to the chiral edge dynamics with a linear dispersion law. finally, the expression for the charge density at the edge of a qh system may be found by evaluating the derivative @xmath246 of the total action with respect to the boundary field. the result reads : @xmath247 note that, after integrating out the fields @xmath216, the action constructed above leads, for a trivial topology, to the effective theory that has been considered in ref. having constructed a gauge invariant low - energy action for an incompressible qh liquid, we proceed to analyze the spectrum of local excitations. the action ([bulk - act]) arises in the context of a topological field theory, where excitations are described by wilson lines. @xcite for instance, a general local excitation at the point @xmath248 may be written as : @xmath249 where @xmath250 is a constant. the statistical phase of two excitations of the type ([op - bulk]) is determined by braiding the corresponding wilson lines. @xcite considering two excitations labeled by @xmath251 and @xmath252, one arrives, after a simple calculation of braiding, at the following expression for the statistical phase : @xmath253. moreover, if one defines the charge operator as an integral over a space - like plane @xmath254, then the charge of the excitation ([op - bulk]) is given by @xmath255. the above expressions for the statistical phase and the charge show that the excitation ([op - bulk]) with @xmath256 has the quantum numbers of an electron, while for @xmath257 in ([op - bulk]) this operator has the quantum numbers of a laughlin quasi - particle. an operator that creates a laughlin quasi - particle at the point @xmath20 and a laughlin quasi - hole at the point @xmath155 has the form of a wilson line between these points : @xmath258 note, however, that this operator is not gauge invariant. according to the gauge transformation of edge fields, eq. ([gaugetransform]), the _ gauge - invariant _ operator of tunneling between two edge states, described by the fields @xmath259 and @xmath260 correspondingly, @xcite may be written as @xmath261\big)e^{-i\phi_d(x) }, \label{tun - ef}\]] where the 1d and 2d coordinates are related via the expression @xmath262 for the outer edge and @xmath263 for the inner edge of the mz interferometer ; (see fig. [e - b]). the tunneling operator ([tun - ef]) creates a pair of local charges of the value @xmath264 at the edges, which may be checked by evaluation of the commutator of this operator with the charge density operator ([rho - edge]) with the help of eq. ([commut - eff]). to compare the tunneling operator ([tun - ef]) with the microscopic one, eq. ([h - tunnel]), we first note that, assuming the tunneling path is short, one may set @xmath265 and neglect the wilson line contribution in eq. ([tun - ef]). next, we introduce a normal mode expansion for the edge densities @xmath266, @xmath84 : [rho - a] @xmath267, \\ \!\!\!2\pi\rho_u(x) = \frac{m+mn}{mr_u } + \sum_{k>0}\sqrt{\frac{k}{r_u}}[a_{ku}e^{ikx } + { \rm h.c.}\,],\end{aligned}\]] where the summation ranges over the wave vector @xmath268. here the operators @xmath98 and @xmath269 obey the canonical commutation relations ([commut - mic]). this guarantees that the commutation relation ([commut - eff]) is satisfied, when zero modes are taken into account. moreover, the edge hamiltonian @xmath270, \label{h - edge - rho}\]] expressed in terms of the operators @xmath98 and @xmath269, coincides with the microscopically derived hamiltonian ([h - edge]). finally, in this section and later in this paper, the fields @xmath271 naturally contain zero - mode contributions.@xcite in order to find these contributions we need to know the vacuum distribution of the field @xmath216 at the edges for a state with given values of zero modes @xmath21 and @xmath24. this field describes the homogeneous distribution of the qh liquid in the interferometer and thus satisfies the equation @xmath272. typically, for a fixed geometry, one subtracts the background contribution from the chern - simons field and sets @xmath273. however, in the present case of a corbino disk geometry, the field @xmath274 varies, depending on the numbers @xmath21 and @xmath24. we find the field by taking into account the change in the background charge @xmath42 caused by the variation of the quantum numbers @xmath21 and @xmath24. to this end, we note that the field @xmath275 may be viewed as an `` electric field '' that satisfies gauss law with the constant charge density @xmath42. the variation of the number @xmath21 leads to a shift of the edges of the corbino disk, and to an accumulation of a background charge @xmath276 in the form of two homogeneous rings. solving the 2d electrostatic problem we find that @xmath277 on the inner edge and @xmath278 \label{b - solution}\]] on the outer one. we use the distribution ([b - solution]), the normal mode expansion ([rho - a]), and the relation ([rho - edge]) between the edge densities and edge fields to arrive at the following expressions [phi - a] @xmath279/\sqrt{kr_u},\end{gathered}\]] @xmath280/\sqrt{kr_d},\end{gathered}\]] which should be compared to eq. ([bsefield]). we conclude that the tunneling operator ([tun - ef]), in the absence of an external electromagnetic field @xmath223, coincides with the microscopic expression ([h - tunnel]) if one sets @xmath265. it is important to note that the contributions of zero modes to the edge fields are strongly affected by the specific geometry of an mz interferometer. namely, the phase shift of the tunneling amplitude caused by the zero mode @xmath21 is not constant along the outer edge, provided @xmath45. in this section we show that the effects of a singular flux and of a modulation gate found in sec. [s - flux] can be reformulated in terms of the gauge - invariant low - energy effective theory. we start from the situation where the modulation gate voltage is applied in order to deform the outer edge of a mz interferometer. this situation can be described at the level of the effective theory by adding a term in the hamiltonian, that describes the interaction of the potential @xmath281 created by the modulation gate with charge density accumulated at the edge. thus, the total hamiltonian of the outer edge is given by the expression : @xmath282. \nonumber\]] then we find the ground - state expectation value of the charge density by minimizing the hamiltonian with the constraint @xmath283. for the situation where the gate is located at the point @xmath284, we approximate the potential as @xmath285. the expectation value of the density is then given by @xmath286.\nonumber\]] the parameter @xmath287 can be expressed in terms of the flux @xmath8 through the region depleted by the gate, @xmath288, so that the expression for the accumulated charge density acquires the form @xmath289. \label{vac - rho}\]] finally, we redefine the charge density on the outer edge by subtracting the vacuum contribution ([vac - rho]) : @xmath290. as a consequence, the edge hamiltonian ([h - edge - rho]) acquires an additional term : @xmath291. \label{ham - gate - eff}\]] substituting into this equation the normal mode expansion ([phi - a]), we arrive at the result ([h - edge]) with the vacuum energy @xmath139 given by eq. ([ham - gate]), in full agreement with the microscopic theory. next, we investigate the effect of the modulation gate on the tunneling operator ([tun - ef]). we will show in the next section that, to lowest order in the tunneling hamiltonian, the ab contribution to the current is determined by the product of two such operators taken at two qpcs. therefore, only a relative phase of the tunneling operators has a physical meaning. we express this product in terms of the edge densities with the help of eq. ([rho - edge]) : @xmath292\big) \\ \times \exp \big(-2\pi i\int_{x_l}^{x_r } dx[\rho_u(x)+\rho_d(x)]\big). \label{two - tun}\end{gathered}\]] here @xmath293 is the interference contour, @xmath294, @xmath295 are positions of the qpcs, and the integral of densities extends over the arms of the interferometer (see fig. [e - b]). then, as explained above, we subtract the vacuum charge density ([vac - rho]) accumulated due to the interaction with the modulation gate. the product ([two - tun]) of tunneling operators then acquires a phase shift @xmath296. using the explicit expression for @xmath297 in the situation where the gate is located at a point @xmath284 between the qpcs, we arrive at the result ([add - phas]). thus, we see that calculations based on the gauge - invariant effective theory fully agree with the microscopic derivation of the tunneling operators. it remains to investigate the effect of a singular magnetic flux. to this end, we minimize the action ([s - edge1]) with respect to the edge densities in the presence of an external electromagnetic field. this leads to the following equations of motion : @xmath298 where @xmath84, and @xmath299 is the component of the electric field along the edge of the sample. it follows from this equation that, in equilibrium, @xmath300, and for @xmath301, the accumulated charges are always homogeneously distributed along the edges : @xmath302. moreover, integrating eq. ([eom - rho]) over the coordinate @xmath303 and subtracting the equation for the inner edge (@xmath304) from the equation for the outer one (@xmath305), we arrive at the following equation : @xmath306 where @xmath8 is the flux through the interferometer. here we have used maxwell s equations and restored physical units. thus, we conclude that, after the adiabatic insertion of the singular flux, the quantum number @xmath21 is shifted by @xmath307. this, in turn, leads to an effective hamiltonian of the form : @xmath308 ^ 2. \nonumber\]] using again the normal mode expansion ([phi - a]), we arrive at the hamiltonian ([h - edge]), with the vacuum energy @xmath139 given by eq. ([e0]), in agreement with the microscopic theory. finally, one can see from the expansion ([phi - a]) that the tunneling operator ([tun - ef]) for @xmath309 does not depend on @xmath21. therefore, it does not change after the insertion of the singular flux. this effect may be interpreted as an exact cancellation of two contributions to the phase of expression ([two - tun]). the first contribution arises from the gauge - invariant integral @xmath310 $] of the bulk fields over the interference contour @xmath293 upon variation of the singular flux. the second one is proportional to the charge @xmath311 $] accumulated along the arms of the interferometer. physically, this means that the phase shift of an edge excitation along a closed contour @xmath293 caused directly by a variation of the vector potential @xmath223 is screened by the phase shift caused by the concomitant reconstruction of the qh liquid. this kind of _ topological _ screening is the reflection, on the effective - theory level, of the fact that after the insertion of the singular flux the microscopic wave function ([w - magn]) undergoes a deformation in order to preserve its single - valuedness. Interferometer out of equilibrium we have shown above that the ab effect, in its original formulation, does not exist in qh interferometers, because the tunneling operators do not depend on the singular magnetic flux threading the corbino disk. this, however, does not imply that the current through the interferometer is independent of the flux. in this section we demonstrate that it oscillates as a function of @xmath8 with the electronic period @xmath0, in agreement with the byers - yang theorem. these oscillations originate from the coulomb blockade effect and hence vanish in the thermodynamic limit@xcite and in the limit of strong coupling to ohmic contacts. in contrast, oscillations of the current as a function of the modulation gate voltage originate from quantum interference. therefore they do not vanish in the thermodynamic limit. we consider a mz interferometer, strongly coupled to ohmic contacts, with weak quasi - particle tunneling between inner and outer edge (see fig. [relax]). the strong coupling of electrons to ohmic contacts guarantees that the inner and outer edge states are in equilibrium with the metallic reservoirs, with corresponding electro - chemical potentials @xmath312 and @xmath313. the charge current between ohmic contacts, arising as a response to the potential difference @xmath314, is due to weak quasi - particle tunneling at the qpcs. it is then convenient to introduce a new notation for zero modes. we denote by @xmath26 and @xmath25 the numbers of electrons at the outer and inner edge of the interferometer, respectively. the number of quasi - particles localized on the inner edge is denoted by @xmath315 : @xmath316 the electron quantum numbers, @xmath26 and @xmath25, change due to tunneling at the ohmic contacts while the quantum number @xmath27 changes by @xmath317 when a quasi - particle tunnels from one edge to the other one, as illustrated in fig. [relax]. without loss of generality, we consider the zero modes to be classical variables and derive a master equation for the probability distribution functions. quantum coherence manifests itself in oscillations of the quasi - particle tunneling rates as functions of the magnetic flux @xmath8. these oscillations originate from the interference of the two quasi - particle tunneling amplitudes at the left and right qpc. formally, these oscillations stem from the @xmath8-dependent phase factor in the tunneling operators. in this section we derive the master equation that describes weak quasi - particle tunneling, and we find the tunneling rates using the effective theory constructed above. the strong coupling to ohmic contacts implies that, after every event of quasi - particle tunneling, the edge states relax to the equilibrium state described by the probability distribution function @xmath318 where @xmath319 is the inverse temperature, @xmath320 is the electrochemical potential of the @xmath321 ohmic contact, and @xmath322 is the partition function. the probability @xmath323 depends on the number @xmath27 of quasi - particles via the ground - state energy @xmath324, given by eq. ([ener - zero]). importantly, one does not need to specify the precise form of coupling to the ohmic contacts, because the only role of this coupling is to equilibrate the edge states.@xcite to lowest order in quasi - particle tunneling, the full distribution function of zero modes may be written as @xmath325 where @xmath326 is the probability to find the system in a state with @xmath27 quasi - particles, and @xmath323 plays the role of a conditional probability to find the interferometer in a state with @xmath26 and @xmath25 electrons at the edges, for a given number @xmath27 of quasi - particles. considering quasi - particle tunneling as a weak process that changes the number @xmath27, we may describe it with the master equation @xmath327 where @xmath328 and @xmath329 are the rates of transition from the state with @xmath27 quasi - particles to the state with @xmath330 and @xmath331 quasi - particles, respectively. these rates are given by the expression : @xmath332 where @xmath333 are the rates of quasi - particle tunneling between two states with fixed numbers of electrons, @xmath26 and @xmath25. at time scales much larger than the characteristic times of tunneling, the interferometer reaches a steady state regime with @xmath334. it is easy to see that in this regime the following quantity is independent of @xmath27 @xmath335 this quantity is in fact the charge current that we are looking for. this follows from the expression for the current @xmath336 and from the periodic boundary condition, @xmath337, which can be verified directly using equations ([rho - eq]) and ([rho - fact]). note that the detailed balance equation @xmath338 is satisfied only if @xmath339. thus, the mz interferometer in a non - equilibrium steady - state regime represents an interesting example of a system with broken detailed balance. we evaluate the tunneling rates @xmath333 to leading order in the tunneling hamiltonian @xmath340, where @xmath341. a straightforward calculation, based on the fermi golden rule, gives the following expression : @xmath342 and a similar expression for @xmath343. here the operator @xmath344 projects onto states with given numbers @xmath345 and @xmath27, and the operator @xmath346 is the equilibrium density matrix for the oscillators. next, we outline some further steps, the details of which may be found in refs. [] and []. we write the equations of motion for the tunneling operators @xmath347, @xmath295, by evaluating the commutator of the operator ([h - tunnel]) with the hamiltonian ([h - edge]), and solve these equations in order to find the time evolution of @xmath348. as a result, we arrive at the following expression for the tunneling rates taking the form of a sum of an incoherent and coherent part : @xmath349, \label{tun - rates-2}\end{gathered}\]] here @xmath350 is the energy difference between final states with @xmath351 quasi - particles and an initial state with @xmath27 quasi - particles, and the functions @xmath352^{1/m}$], @xmath353, are the equilibrium correlation functions of the oscillator modes. we recall that @xmath354 is the total number of electrons in the corbino disk, and @xmath14 and @xmath16 are the 2d coordinates of the qpcs. note that the magnetic flux enters the tunneling rates ([tun - rates-2]), in particular, via the energy difference @xmath355. in the next two sections we use eqs. ([ham - gate]) and ([e0]) for the energy @xmath139 in order to investigate the flux dependence of the tunneling rates in different situations. in the case where the flux through the interferometer is varied with the help of a modulation gate we use eq. ([ham - gate]) to evaluate the energy difference between the final and initial state : @xmath356 in this situation, an additional flux dependence is due to the tunneling operators @xmath357 and @xmath358 at the left and right qpcs, which acquire a relative phase @xmath359 given by eq. ([add - phas]), which, for convenience, we rewrite here : @xmath360 $]. this phase enters the interference term of the tunneling rates ([tun - rates-2]). next, we note that in the thermodynamic limit@xcite the sum over the electronic numbers in eq. ([tun - rates-2]) and in the expression for the partition function @xmath322 can be well approximated by an integral, so that we arrive at the expressions @xmath361 it is convenient to shift the integration variables : @xmath362 and @xmath363. important consequence of this transformation are the following ones : first of all, the energy of zero modes simplifies, @xmath364, where @xmath353. second, the energy difference becomes independent of the number @xmath27 and of the flux : @xmath365. finally, the ab phase difference @xmath366 takes a topological value, @xmath367, because its geometry - dependent part cancels exactly with the contribution from the phase @xmath368 in the interference term of the rates ([tun - rates-2]). using the new expressions for @xmath139, @xmath355, and @xmath359, we find that the integral ([tun - therm]) is gaussian with the saddle point located, in the thermodynamic limit, at @xmath369 substituting this values into eq. ([t - sum]), we obtain the following expressions for tunneling rates : @xmath370, \label{tun - rates-3}\end{gathered}\]] with @xmath354 in the second term given by @xmath371 as follows from the expressions ([saddle - gate]). thus, we arrive at a first important conclusion : the rates do not depend on @xmath27, @xmath372. this implies, according to eq. ([curr - d]), that the current is equal to @xmath373), i.e., it coincides with the expression for the current obtained in ref. []. we therefore do not evaluate the integral in ([tun - rates-3]) and refer the reader to [], where the current has been thoroughly analyzed in the linear and non - linear regimes. second, we note that the rates ([tun - rates-3]) depend on the magnetic flux. thus we conclude, that the ab effect survives in the thermodynamic limit if the flux is varied with the help of a modulation gate. one can see that the current has the _ quasi - particle _ periodicity. this result does _ not _ violate the byers - yang theorem, because the flux through the interferometer caused by the application of a modulation gate can not be gauged away. finally, we note that tunneling rates ([tun - rates-3]) depend not only on the voltage bias @xmath374, but also on @xmath312 and @xmath313 separately, via the total number of electrons in the interferometer, @xmath375, which is given by expression ([totaln]). this seems to violate the gauge invariance. in fact, this is not the case, because we assumed, for simplicity, that the coulomb interaction is screened by a nearby metallic gate, so that the inner and outer edges of the corbino disk do not interact. this situation is realized in a number of recent experiments, e.g., in ref. the additional metallic gate plays a role of the third electrode, which independently controls the number of electrons in the system. the potentials @xmath312 and @xmath313 are then measured with respect to this gate. we have to admit that the degree of screening depends on details of an experimental situation. we stress, however, that these details do not affect the generality of our conclusions concerning the periodicity of the ab oscillations. here we show that, in the situation where the flux through the interferometer is varied by inserting a singular flux tube, the electronic periodicity of the current is restored. in this case the energy difference between the states with @xmath376 and @xmath27 quasi - particles is given by @xmath377.\]] we may now pass to the thermodynamic limit and replace the sum over @xmath345 by an integral to arrive at expression ([tun - therm]). it is then easy to see that, in the present case, after shifting variables, @xmath378, the integral becomes independent of the flux @xmath8. this is because, here, the magnetic flux enters the tunneling rates ([tun - rates-2]) only via the energy of zero modes, @xmath139, in contrast to the situation where the flux is varied with the help of a modulation gate. we conclude that there is no ab effect associated with a singular flux, and in order to find the flux dependence of the tunneling rates we need to take into account terms that are small in the thermodynamic limit. to this end, we use an identity@xcite that allows to rewrite a sum over integer numbers as an integral, @xmath379 along the contour shown as a red line in fig. [pol]. using this identity and performing the shift of variables @xmath380, we arrive at the following expression for transition rates : @xmath381 where we have omitted unimportant constants, and @xmath333 are the tunneling rates ([tun - rates-2]). we stress that after the shift of variables indicated above, the tunneling rates @xmath333 are seen _ not _ to depend on the number @xmath27 and the flux @xmath8. next, we use the analyticity of the rates @xmath382 as functions of the variables @xmath26 and @xmath25 to deform the integration contour in ([integr-2]) to the new position, @xmath383, as shown in fig. [pol]. assuming that @xmath384 is large, we expand the cotangent in the integral ([integr-2]) and keep the leading harmonic : @xmath385.\nonumber\end{gathered}\]] substituting this expansion into eq. ([integr-2]), we find that the first term reproduces our earlier result for the leading order contribution in the thermodynamic limit. for the second, sub - leading, term, the saddle point is located far from the real axis : @xmath386 because @xmath387. this justifies the expansion of the cotangent in the thermodynamic limit. using the saddle point values of @xmath345 we arrive at the following expression for the tunneling rates ([t - sum]) : @xmath388 \\ \times\exp\left[2\pi i \big(\frac{\mu_s r_s}{mv_s}+\frac{l}{m } - \frac{\phi}{m\phi_0}\big)\right], \label{tun - rates-4}\end{gathered}\]] where the first term, @xmath389, is the rate ([tun - rates-3]) taken at @xmath390, and the functions @xmath391 of @xmath312 and @xmath313 are given by expressions similar to those in eq. ([tun - rates-3]). these functions, the exact form of which is not important, are independent of @xmath27. as expected, the tunneling rates ([tun - rates-4]) contain oscillatory contributions in @xmath8 that are exponentially suppressed in the thermodynamic limit.@xcite these oscillations may be interpreted as resulting from the coulomb blockade effect. finally, we note that the flux enters the transition rates ([tun - rates-4]) solely in the combination @xmath392. a shift of the flux @xmath8 by one flux quantum @xmath0 may be then compensated by the shift @xmath393. the quasi - particle current is given by expression ([curr - d]), where the probabilities satisfy eqs. ([conserv]) with the constraint @xmath394. all these equations are periodic in @xmath27 with period equal to @xmath4 and invariant under the replacement @xmath395. this implies that the average current has the electronic periodicity, @xmath396, in agreement with the byers - yang theorem. for example, the solution of eqs. ([conserv]) and ([curr - d]) for @xmath397 gives the average current @xmath398 which is explicitly periodic function of @xmath8 with the electronic period @xmath0. at the same time, each tunneling rate @xmath399 has a quasi - particle periodicity. therefore, the quasi - particle periodicity with respect to a singular flux may in principle be observed via, e.g., the current noise measurements at finite frequencies. the last fact does not violate the byers - yang theorem, because this theorem applies only to a stationary state and to long - time measurements. Discussion the effect of topological screening plays a central role in resolving the byers - yang paradox. we have shown, both on the microscopic and on the effective theory levels, that it leads to a cancellation of the total ab phase due to the singular magnetic flux tube threading through the interferometer s loop and of the phase shift accumulated as a result of the physical displacement of the state. it is this last fact that makes interferometers based on qh systems to stand out from the variety of other electronic interferometers, where the ab effect is regarded nowadays as a simple textbook physics. we then conclude that the ab effect, in its original formulation, does not exist in qh interferometers. to further illuminate an important role of topological screening, we will consider here a simple example of a qh liquid at filling factor @xmath400, where our theory also applies. according to a commonly used single - particle picture, electrons in @xmath400 state drift along the equipotential lines. one may then consider a qh interferometer to be an open system, where single - particle orbits form a loop as shown in the upper panel of fig. [frol - simpl], and apply the scattering theory. since the position of the orbits does not change, being fixed by the equipotential lines, the only effect of the singular magnetic flux is to shift the phase of single - particle wave functions, which leads to the overall ab phase shift @xmath401 in scattering amplitudes. finally, using the landauer - bttiker formula,@xcite one easily finds that the average current should oscillate as a function of the singular flux with the period @xmath0, in strong contradiction with the effect of topological screening. addressing this problem, we first reiterate that (i) the qh effect has a topological character, therefore the naive open systems approach may not be applicable to qh interferometers, and (ii) topological screening stems from the cooperative action of many electrons, so that the single - particle picture may not correctly describe the effect of insertion of the singular flux. as we demonstrate below, the single - particle picture is consistent with the microscopic description of the @xmath400 state, particularly with the many - particle laughlin wave function ([laugh]), only in the case of an axially symmetric corbino disk (i.e., for @xmath402). however, in this case the ab phase shift cancels locally with the phase shift caused by the physical displacement of the orbits after the insertion of the singular flux, in full agreement with the effect of topological screening. one may argue, though, that a realistic interferometer is not symmetric : it is a small part of a larger qh system. therefore, we will focus now on the case of an asymmetric corbino disk (see fig. [singl]) and show that the single - particle description violates the pauli principle. indeed, in a qh system, the eigenstates of the non - interacting hamiltonian are the single - particle orbits that drift along the equipotential lines, shown in fig. [singl] as thin black lines. two separatrices (blue line @xmath403 and red line @xmath404) split the orbits into three groups with different angular momenta with respect to the hole in the corbino disk, i.e., with respect to @xmath405. initially, in a ground state all the single - particle orbits inside the corbino disk are filled. the adiabatic insertion of the singular flux through the hole at the point @xmath40 does not affect the orbits between two separatrices, because those orbits do not encircle the flux. at the same time, the orbits that belong to the other two groups encircle the flux, and therefore, shift inwards or outwards in order to maintain single valuedness. this leads to the accumulation or depletion of the charge density in the regions along the lines @xmath403 and @xmath404. thus we conclude that the single - particle picture, based on the eigenstates of a non - interacting hamiltonian, fails to correctly describe the insertion of the singular flux, because it leads to compressible deformations in the bulk of the 2deg. the correct approach consists of using the single - particle basis which corresponds to unconfined orbits @xmath406. these are not the eigenstates of a non - interacting hamiltonian in the presence of a confining potential. nevertheless, before the insertion of the singular flux two bases are equivalent to each other and related by a unitary transformation. after the insertion of the flux, the correct singe - particle basis leads to the wave function ([w - magn]), which describes _ incompressible _ deformations. in the end, we would like to mention that the insertion of a singular flux is an idealization introduced in this paper in order to formulate the effect of topological screening. such a procedure is not easily realizable in experiment. therefore, in what follows we consider some physical consequences of the effect of topological screening in qh systems where a homogeneous magnetic field is varied. we note that even in this case the total magnetic flux through the hole in the mz interferometer is screened and gives no contribution to the ab phase, because the homogeneous flux may be viewed as a distributed singular flux. this screening effect can, in principle, be tested in an experiment with a mz interferometer at filling factor @xmath407. namely, one should apply a voltage to an additional gate in order to significantly change the size of the hole in the interferometer, without affecting the total area of the qh liquid enclosed by the interfering paths, as shown in fig. [exper1]. we predict that due to the effect of topological screening the period of ab oscillations as a function of the homogeneous magnetic field will remain unchanged, despite the strong variation of the geometrical area of the interferometer after the application of the gate voltage. next, we consider the effect of topological screening in qh systems with a long - range disorder. if the filling factor is slightly different from the value @xmath3, but the qh system is still at the corresponding plateau of the hall conductivity, then there are several unoccupied localized states in the bulk. this implies, that the incompressible qh liquid in the bulk of the 2deg contains several holes, which result from the fluctuations of the disorder potential. due to the effect of topological screening, the flux through these holes does not contribute to the ab phase. therefore, the ab phase is proportional only to the total area of the incompressible qh liquid in the interferometer @xmath408, and not to the geometrical area @xmath409. this should lead to a linear dependence of the period of ab oscillations on the magnetic field : @xmath410 in addition, in samples with a strong disorder, so that the fluctuating potential exceeds the cyclotron gap, such a behavior should be present in a range of magnetic fields that is larger than the width of a single plateau. we think that the linear magnetic field dependence ([linear]) might have already been observed in a number of experiments on fabry - prot interferometers.@xcite Conclusion recently, the physics of ab oscillations in electronic interferometers has become a subject of a debate. a number of works have claimed that only the electronic periodicity may be observed in mz interferometers based on qh states at fractional filling factors @xmath3. we have briefly reviewed those papers in the introduction, and in more detail, in the appendix c of our earlier paper []. here we recall that the main argument against the observability of ab oscillations with a longer, quasi - particle periods is based on the byers - yang theorem, which states that the steady - state current through the interferometer oscillates with the electronic period @xmath0 as a function of the singular magnetic flux @xmath8 threading through the interferometer s loop. different proposed models have in common that they do not differentiate between three methods of the variation of the flux, namely, by inserting the singular flux tube, by applying a modulation gate voltage, and by varying a homogeneous magnetic field. all these models rely on the effective theory approach. in our earlier paper we have criticized those works and argued that the average current may oscillate with the quasi - particle period @xmath2 in response to the modulation gate voltage, and yet this behavior does not violate the byers - yang theorem (see appendix c of ref. []). however, the arguments based solely on the effective theory seem to be insufficient. therefore, in the present paper we refine our approach and justify it on the microscopic, effective, and kinetic theory levels. we consider the laughlin variational wave function of the @xmath3 state, which is known to have extremely large overlap with the exact wave function, and construct the space of gapless incompressible deformations of the qh liquid of the mz interferometer. we proceed by projecting the microscopic hamiltonian on the subspace of such excitations and arrive at the low - energy theory of the interferometer in the presence of a singular magnetic flux, or in the case of the applied modulation gate voltage. we derive the effective theory for a mz interferometer from the chern - simons theory and find essentially perfect agreement with the results of the low - energy projection. finally, using the effective theory, we develop the kinetic theory approach to a mz interferometer in a non - equilibrium current - carrying state, and find ab oscillations in the average current. we confirm that the model of ref. [] is correct. our results are summarized below. first of all, we predict the effect of topological screening of a magnetic flux threading through a hole in an incompressible qh liquid, which manifest itself in the cancellation of the total ab phase of the wave function due to the flux and of the phase shift accumulated as a result of the physical displacement of the wave function. on the microscopic level, this effect appears naturally as a consequence of the single - valuedness of the laughlin wave function. on the effective theory level, it arises as a cancellation of the vector potential associated with the magnetic flux and of the chern - simons field induced by the reconstruction of the wave function in response to the insertion of the flux tube. this effect has a global, topological character, because it does not depend on the choice of the gauge. as a result, the ab effect, in its original formulation, can not be observed in qh systems in principle. taking into account this situation is crucial for resolving the byers - yang paradox. nevertheless, a qh system in general, and an electronic mz interferometer in particular, will respond to the insertion of the singular flux in the form of periodic oscillations in the average steady - state current. these oscillations result from the redistribution of the charge between edge states, and from the charge quantization. however, these oscillations are associated with the coulomb blockade effect, should have an electronic period @xmath0, and will vanish in the thermodynamic limit, where the mz interferometer may be considered an open quantum system. thereby, our theory conforms to the byers - yang theorem. on the other hand, we predict that when the total magnetic flux @xmath8 though the interferometer s loop is varied with the help of a modulation gate attached to one arm of the interferometer, the average current will oscillate as a function of this flux with the quasi - particle period @xmath2. such a striking difference between two behaviors may be interpreted as if the modulation gate only `` couples '' to the local quasi - particle charge @xmath7 at the edge of a qh system, while the singular flux `` couples '' to the global charge of laughlin quasi - particles, which is equal to zero. we admit that to fully resolve the controversy concerning the nature of the quasi - particle interference it is important to experimentally confirm or disprove ab oscillations with quasi - particle periods in electronic mz interferometers. however, we understand that this experiment, which has to be done at fractional filling factors, might be not an easy task. therefore, we suggest, as an intermediate step, to directly demonstrate the effect of topological screening by carrying out specific measurements, as explained in sec. [conseq] (see fig. [exper1]). importantly, by doing such measurements one may observe the effect of topological screening even at integer filling factors, e.g., at @xmath400. we further predict that in strongly disordered qh systems, as a result of topological screening, the period of ab oscillations should linearly depend on the magnetic field across the plateau of the hall conductivity. our additional remarks concern the properties of quasi - particle operators and of the tunneling hamiltonians. first we note that we directly construct quasi - particle operators by projecting quasi - particle excitations of the laughlin wave function onto the low - energy subspace. we then demonstrate that so constructed operators have anyonic commutation relations and create local excitations of the charge @xmath7. however, our calculations show that it is incorrect to consider such excitations as free particles propagating from one ohmic contact to another and carrying a `` statistical phase tube ''. instead, they are created by tunneling perturbations locally at qpcs. it is convenient to represent tunneling operators in term of wilson lines, which connect end points of the tunneling paths. then it becomes clear why tunneling operators are insensitive to the number @xmath21 of quasi - particles localized at the hole of the corbino disk, contrary to what some earlier papers suggest. this is because two wilson lines that describe tunneling at two qpcs may be exchanged without winding the hole of the corbino disk. finally, we find no evidence of klein factors in tunneling operators. therefore, taken at two spatial points, they do not generally commute, except for a fabry - prot interferometer, as explained in the appendix [app - comm]. we conclude by saying that the proposed effective theory of an electronic mz interferometer can be generalized to states with other filling factors than @xmath1, in particular, to the states with non - abelian statistics of excitations. moreover, our results are easily generalizable to systems with a different geometry, e.g., to electronic fabry - prot interferometers. although, the physics in general will remain the same, detailed considerations may bring new interesting results. in view of our new findings concerning the nature of the quasi - particle interference, it is very interesting to reconsider the effects of quasi - particle exchange and statistics. we thank c. w. j. beenakker, v. cheianov, and a. koroliuk for valuable discussions. this work has been supported by the swiss national science foundation. Commutation relations for quasi-particle operators here we discuss commutation relations for quasi - particle operators and for tunneling operators ([h - tunnel]) at spatially separated points. first of all, we show that the microscopic theory proposed in sec. [s - project] leads to typical anyonic commutation relations@xcite for the quasi - particle operators. to this end, we apply the projection procedure ([projection - def]) to the quasi - particle operator ([qp - def]) and carry out calculations similar to those that lead to the expression ([h - tunnel]) for tunneling operators. this procedure gives us the following expressions for the low - energy projection of quasi - particle operators note, that the operator ([qp - def]) has zero charge, i.e., it creates a local excitation with the charge @xmath7 together with the homogeneous density along the edge with total charge @xmath414. therefore, after the low - energy projection we have multiplied the quasi - particle operators by the factor @xmath415 in order to cancel the contribution of the homogeneous density. thus, the resulting operators ([qp - proj]) indeed create local excitations with the charge @xmath7. comparing equations ([phi - a]) and ([qp - proj]), we see that so defined fields agree with whose derived in sec. [s - effective] at the effective theory level. and to similar expressions for the adjoint operators. here we define @xmath417 with @xmath295 at the outer edge, and @xmath418 at the inner edge. the commutation relations ([commut - qp]) show that quasi - particles ([qp - proj]) are anyons@xcite with the statistical phase @xmath419. note, that the phase factors on the right hand side of the expressions ([commut - qp]) are @xmath4-valued functions on a circle. this is consistent with the fact that quasi - particle operators are also multi - valued. interestingly, straightforward calculations show that the operators at the opposite edges do not commute : @xmath420 in other words, a quasi - particle on the outer edge feels the presence of another quasi - particle on the inner edge. this is because the creation of a quasi - particle on the inner edge shifts quantum numbers @xmath24 and @xmath21. the change of this number is a _ topological _ effect which leads to a reconstruction of the wave function and changes the chern - simons field ([b - solution]). next, we focus on the commutation relations for the operators of quasi - particle tunneling. first of all, we would like to emphasize that at the microscopical level two tunneling operators do commute, as they are just polynomials. however, being projected onto low - energy subspace, they do not necessary commute, if taken at different spatial points. the direct calculation for the operators ([h - tunnel]) with the help of eqs. ([zeromodecom]) and ([commut - mic]) leads to the following expression : @xmath421\mathcal{a}(\xi_r)\mathcal{a}(\xi_l)\]] in the thermodynamic limit, @xmath422, which describes an open mz interferometer, this leads to a simpler expression : @xmath423 where the statistical phase takes the topological value. for a fabry - prot interferometer, setting @xmath424 and @xmath425 and vice versa (see fig. [e - b]), we find that tunneling operators commute @xmath426 = 0 $]. we stress that the contribution of the commutator of zero modes is crucial in the above calculations. the zero modes in part related to the bulk chern - simons field (see the discussion at the end of sec. [qptunneling]) and naturally take into account the topology of the interferometer. finally, we note that the results ([commut - qp]) and ([commut - mz]) provide a microscopic justification of the model of the mz interferometer that we proposed in ref. we have argued there that tunneling operators, derived from the effective theory of the qh edge states, do not need to commute, because the presence of a gap in the spectrum of bulk excitations of the incompressible qh liquid makes them nonlocal. importantly, we do not see any evidence of the klein factors@xcite in tunneling operators ([h - tunnel]), when expressed in terms of the quasi - particle operators ([qp - proj]). very recently, unexpected values of quasi - particle charges, determined via shot noise measurements, have been reported in m. dolev, y. gross, y. c. chung, m. heiblum, v. umansky, and d. mahalu, arxiv:0911.3023. these results may indicate that the fano factor of a weak backscattering current is not determined solely by the quasi - particle charge. the use of the term `` charge fractionalization '' in this context is somewhat unfortunate, because, in contrast to the quasi - particle fractionalization, the corresponding process is completely classical in nature. in fact, it is very similar to the displacement current in electrical circuits. j. a. simmons, h. p. wei, l. w. engel, d. c. tsui, and m. shayegan, phys. lett. * 63 *, 1731 (1989) ; j. a. simmons, s. w. hwang, d. c. tsui, h. p. wei, l. w. engel, and m. shayegan, phys. rev. b * 44 *, 12933 (1991) ; f. e. camino, w. zhou and v.j. goldman, phys. lett. * 95 *, 246802 (2005) ; f. e. camino, w. zhou and v.j. goldman, phys. b * 72 *, 075342 (2005) ; f. e. camino, w. zhou and v.j. goldman, phys. * 98 *, 076805 (2007) ; f. e. camino, w. zhou and v.j. goldman, phys. b * 74 *, 115301 (2006) ; r. l. willett, l. n. pfeiffer, k. w. west, pnas * 106 *, 8853 (2009) ; r. l. willett, l. n. pfeiffer, k. w. west,, arxiv:0911.0345. i. neder, m. heiblum, y. levinson, d. mahalu, and v. umansky, phys. * 96 *, 016804 (2006) ; i. neder, f. marquardt, m. heiblum, d. mahalu, and v. umansky, nature physics * 3 *, 534 (2007) ; e. bieri, _ correlation and interference experiments with edge states _, phd thesis, university of basel (oct. 2007) ; e. bieri, m. weiss, o. goktas, m. hauser, c. schonenberger, and s. oberholzer, phys. rev. b 79, 245324 (2009) ; p. roulleau, f. portier, d.c. glattli, p. roche, a. cavanna, g. faini, u. gennser, and d. mailly, phys. b * 76 *, 161309(r) (2007) ; p. roulleau, f. portier, d.c. glattli, p. roche, a. cavanna, g. faini, u. gennser, and d. mailly, phys. * 100 *, 126802 (2008) ; l.v. litvin, h.- tranitz, w. wegscheider, and c. strunk, phys. b * 75 *, 033315 (2007) ; l.v. litvin, a. helzel, h.- tranitz, w. wegscheider, and c. strunk, phys. b * 78 *, 075303 (2008). b. blok, x.g. wen, phys. b * 43 *, 8337 (1991) ; i. kogan, a.m. perelomov and g.w. semenoff, phys. rev. b * 45 *, 12084 (1992) ; v. gurarie, c. nayak, nucl. b * 506 *, 685 (1997) ; r. de gail, n. regnault, and m. o. goerbig, phys. b * 77 *, 165310 (2008). to be precise, the operators ([zeromodedef]) are not unitary if @xmath24 and @xmath21 are finite. however, in the thermodynamic limit (@xmath428) these operators become unitary and satisfy the commutation relations ([zeromodecom]). note that the states at the inner and outer edge have opposite chiralities. at the same time, we choose the coordinate @xmath303 to run counter - clockwise at the both edges. this leads to the opposite signs of the commutators ([commut - eff]) of fields @xmath259 and @xmath260, and to different signs in the expressions for edge densities @xmath429 and @xmath430 in terms of these fields, as e.g., in eqs. ([rho - edge - mic]). the zero modes (@xmath238, @xmath431, @xmath21 and @xmath24 in our notations) and the oscillator modes are usually treated separately in the literature on the qh edge theory (see, e.g., ref. []). here we follow our previous works @xcite and combine them in a single - field operator, because this simplifies the commutation relations and the edge hamiltonian. the notion of the thermodynamic limit for a mz interferometer requires some clarification. indeed, this type of interferometers are typically considered to be open quantum systems, as illustrated in the upper panel of fig. [frol - simpl]. in reality, however, electronic mz interferometers based on a qh liquid are finite size systems. this is because they are connected to an ohmic contact, located inside the interferometer (see the lower panel of fig. [frol - simpl]). nevertheless, it is reasonable to assume that the coupling of a mz interferometer to ohmic contacts is strong. therefore, unless specified otherwise, we model this situation by taking the limit @xmath387, where @xmath353, i.e., assuming that the level spacing at the both edges is small compared to temperature.

interference of fractionally charged quasi - particles is expected to lead to aharonov - bohm oscillations with periods larger than the flux quantum @xmath0. however, according to the byers - yang theorem, observables of an electronic system are invariant under adiabatic insertion of a quantum of singular flux. we resolve this seeming paradox by considering a _ microscopic _ model of an electronic mach - zehnder interferometer made from a quantum hall liquid at filling factor @xmath1. such interferometers have the shape of a corbino disk and utilize quantum hall edge states in place of optical beams and quantum point contacts as beam splitters connecting different edge channels. an approximate ground state of such an interferometer is described by a laughlin type wave function, and low - energy excitations are incompressible deformations of this state. we construct a low - energy effective theory by projecting the state space of the liquid onto the space of such incompressible deformations and show that the theory of the quantum hall edge so obtained is a generalization of a chiral conformal field theory. amplitudes of quasi - particle tunneling in this theory are found to be insensitive to the magnetic flux threading through the hole in the corbino disk. this behavior is a consequence of _ topological screening _ of the singular flux by the quantum hall liquid. we describe strong coupling of the edges of the liquid to ohmic contacts and the resulting quasi - particle current through the interferometer with the help of a master equation. as a function of the singular magnetic flux, the current oscillates with the electronic period @xmath0 , i.e., our theory conforms to the byers - yang theorem. these oscillations, which originate from the coulomb blockade effect, are suppressed with increasing system size. in contrast, when the magnetic flux through the interferometer is varied with a modulation gate, current oscillations have the quasi - particle period @xmath2 and survive in the thermodynamic limit.

1005.5703

Introduction let @xmath0 be an oriented surface of finite type, i.e. @xmath0 is a closed surface of genus @xmath1 from which @xmath2 points, so - called _ punctures _, have been deleted. we assume that @xmath3, i.e. that @xmath0 is not a sphere with at most @xmath7 punctures or a torus with at most @xmath8 puncture. we then call the surface @xmath0 _ nonexceptional_. since the euler characteristic of @xmath0 is negative, the _ teichmller space _ @xmath9 of @xmath0 is the quotient of the space of all hyperbolic metrics on @xmath0 under the action of the group of diffeomorphisms of @xmath0 which are isotopic to the identity. the smooth fibre bundle @xmath10 over @xmath9 of all _ holomorphic quadratic differentials _ of area one can naturally be viewed as the unit cotangent bundle of @xmath9 for the _ teichmller metric_. the _ teichmller geodesic flow _ @xmath5 on @xmath10 commutes with the action of the _ mapping class group _ @xmath11 of all isotopy classes of orientation preserving self - homeomorphisms of @xmath0. thus this flow descends to a flow on the quotient @xmath12, again denoted by @xmath5. for a quadratic differential @xmath13 define the _ unstable manifold _ @xmath14 to be the set of all quadratic differentials whose vertical measured geodesic lamination is a multiple of the vertical measured geodesic lamination for @xmath15. then @xmath16 is a submanifold of @xmath10 which projects homeomorphically onto @xmath9 @xcite. similarly, define the _ strong stable manifold _ @xmath17 to be the set of all quadratic differentials whose horizontal measured geodesic lamination coincides with the horizontal measured geodesic lamination of @xmath15. the sets @xmath16 (or @xmath17) @xmath18) define a foliation of @xmath10 which is invariant under the mapping class group and hence projects to a singular foliation on @xmath4 which we call the _ unstable foliation _ (or the _ strong stable foliation _). there is a @xmath5-invariant probability measure @xmath19 on @xmath4 in the lebesgue measure class which was discovered by masur @xcite (see also the papers @xcite of veech which contain a similar construction). this measure is absolutely continuous with respect to the strong stable and the unstable foliation, it is ergodic and mixing, and its metric entropy equals @xmath6 (note that we use a normalization for the teichmller flow which differs from the one in @xcite). recently avila, gouzel and yoccoz @xcite established that the lebesgue measure @xmath20 on the moduli space of _ abelian _ differentials is even exponentially mixing, i.e. exponential decay of correlations for hlder observables holds. the proof of this result uses the fact, independently due to athreya @xcite, that the teichmller flow is _ exponentially recurrent _ to some fixed compact set, i.e. there is a compact set @xmath21 and a number @xmath22 such that @xmath23 for all @xmath24\}<e^{-ct}/c$] for every @xmath25. the main goal of this paper is to construct a subshift of finite type @xmath26 and a borel suspension @xmath27 over @xmath26 which admits a semi - conjugacy @xmath28 into the teichmller flow. the images under @xmath28 of the flow invariant measures on @xmath27 induced by gibbs equilibrium states for the shift @xmath26 define an uncountable family of @xmath5-invariant probability measures on @xmath4 including the lebesgue measure. we summarize their properties as follows. there is an uncountable family of @xmath5-invariant probability measures on @xmath4 including the lebesgue measure which are mixing, absolutely continuous with respect to the strong stable and the unstable foliation and which moreover are exponentially recurrent to a compact set. denote by @xmath29 the teichmller metric on @xmath9. for a fixed point @xmath30 and a number @xmath31, the _ poincar series _ of the mapping class group with exponent @xmath32 and basepoint @xmath33 is defined to be the series @xmath34 the _ critical exponent _ of @xmath11 is the infimum of all numbers @xmath35 such that the poincar series with exponent @xmath32 converges @xcite. for a compact subset @xmath36 of @xmath4 and for @xmath37 let @xmath38 be the number of periodic orbits for the teichmller geodesic flow on @xmath4 of period at most @xmath39 which intersect @xmath36. we show. 1. the critical exponent of the mapping class group equals @xmath6, and the poincar series diverges at @xmath6. 2. for every sufficiently large compact subset @xmath36 of @xmath4 we have @xmath40 theorem 3 is an improvement of a recent result of bufetov @xcite, with a different proof. much earlier, veech @xcite established a counting result for _ all _ periodic orbits of the teichmller flow. namely, denote by @xmath41 the number of all periodic orbits for the teichmller flow on @xmath4 of period at most @xmath39. veech showed the existence of a number @xmath42 such that @xmath43 as a corollary of theorem c, we deduce that we can take @xmath44 which is still far from the conjectured value @xmath6. on the other hand, in @xcite we constructed for a surface @xmath0 of genus @xmath45 with @xmath46 punctures and @xmath47 and for every compact subset @xmath36 of @xmath4 a periodic orbit for @xmath5 which does not intersect @xmath36. thus if @xmath47 then for every compact subset @xmath36 of @xmath4 we have @xmath48 for all sufficiently large @xmath39 (depending on @xmath36). recently eskin and mirzakhani announced that the asymptotic growth of periodic orbits is indeed @xmath6. the organization of the paper is as follows. in section 2 we review some properties of train tracks and geodesic laminations needed in the sequel. in section 3 we use the _ curve graph _ of @xmath0 to show that the restriction of the teichmller flow to any invariant compact subset @xmath36 of @xmath4 is a hyperbolic flow in a topological sense. in section 4 we use train tracks to construct a special subshift of finite type @xmath26. in section 5 we construct a bounded roof function @xmath49 for our subshift of finite type and a semi - conjugacy of the suspension of @xmath26 with roof function @xmath49 into the teichmller flow. this is used in section 6 to show theorem 1. in section 7 we calculate the critical exponent for @xmath11, and section 8 is devoted to the proof of the second part of theorem 2. Train tracks and geodesic laminations in this section we summarize some results and constructions from @xcite which will be used throughout the paper (compare also @xcite). let @xmath0 be an oriented surface of genus @xmath1 with @xmath2 punctures and where @xmath3. geodesic lamination _ for a complete hyperbolic structure on @xmath0 of finite volume is a _ compact _ subset of @xmath0 which is foliated into simple geodesics. a geodesic lamination @xmath19 is called _ minimal _ if each of its half - leaves is dense in @xmath19. thus a simple closed geodesic is a minimal geodesic lamination. a minimal geodesic lamination with more than one leaf has uncountably many leaves and is called _ minimal arational_. every geodesic lamination @xmath19 consists of a disjoint union of finitely many minimal components and a finite number of isolated leaves. each of the isolated leaves of @xmath19 either is an isolated closed geodesic and hence a minimal component, or it _ spirals _ about one or two minimal components @xcite. a geodesic lamination is _ maximal _ if its complementary regions are all ideal triangles or once punctured monogons. a geodesic lamination _ fills up @xmath0 _ if its complementary regions are all topological discs or once punctured topological discs. a geodesic lamination @xmath19 is called _ complete _ if @xmath19 is maximal and can be approximated in the _ hausdorff topology _ by simple closed geodesics. a _ measured geodesic lamination _ is a geodesic lamination @xmath19 together with a translation invariant transverse measure. such a measure assigns a positive weight to each compact arc in @xmath0 with endpoints in the complementary regions of @xmath19 which intersects @xmath19 nontrivially and transversely. the geodesic lamination @xmath19 is called the _ support _ of the measured geodesic lamination ; it consists of a disjoint union of minimal components. the space @xmath50 of all measured geodesic laminations on @xmath0 equipped with the weak@xmath51-topology is homeomorphic to @xmath52. its projectivization is the space @xmath53 of all _ projective measured geodesic laminations_. there is a continuous symmetric pairing @xmath54, the so - called _ intersection form _, which extends the geometric intersection number between simple closed curves. the measured geodesic lamination @xmath55 _ fills up @xmath0 _ if its support fills up @xmath0. the projectivization of a measured geodesic lamination which fills up @xmath0 is also said to fill up @xmath0. a _ maximal generic train track _ on @xmath0 is an embedded 1-complex @xmath56 whose edges (called _ branches _) are smooth arcs with well - defined tangent vectors at the endpoints. at any vertex (called a _ switch _) the incident edges are mutually tangent. every switch is trivalent. through each switch there is a path of class @xmath57 which is embedded in @xmath58 and contains the switch in its interior. in particular, the branches which are incident on a fixed switch are divided into `` incoming '' and `` outgoing '' branches according to their inward pointing tangent at the switch. the complementary regions of the train track are trigons, i.e. discs with three cusps at the boundary, or once punctured monogons, i.e. once punctured discs with one cusp at the boundary. we always identify train tracks which are isotopic (see @xcite for a comprehensive account on train tracks). a maximal generic train track or a geodesic lamination @xmath59 is _ carried _ by a train track @xmath58 if there is a map @xmath60 of class @xmath57 which is isotopic to the identity and maps @xmath59 into @xmath58 in such a way that the restriction of the differential of @xmath61 to the tangent space of @xmath59 vanishes nowhere ; note that this makes sense since a train track has a tangent line everywhere. we call the restriction of @xmath61 to @xmath59 a _ carrying map _ for @xmath59. write @xmath62 if the train track or the geodesic lamination @xmath59 is carried by the train track @xmath58. a _ transverse measure _ on a maximal generic train track @xmath58 is a nonnegative weight function @xmath63 on the branches of @xmath58 satisfying the _ switch condition _ : for every switch @xmath64 of @xmath58, the sum of the weights over all incoming branches at @xmath64 is required to coincide with the sum of the weights over all outgoing branches at @xmath64. the train track is called _ recurrent _ if it admits a transverse measure which is positive on every branch. we call such a transverse measure @xmath63 _ positive _, and we write @xmath65. the space @xmath66 of all transverse measures on @xmath58 has the structure of an euclidean cone. via a carrying map, a measured geodesic lamination carried by @xmath58 defines a transverse measure on @xmath58, and every transverse measure arises in this way @xcite. thus @xmath66 can naturally be identified with a subset of @xmath50 which is invariant under scaling. a maximal generic train track @xmath58 is recurrent if and only if the subset @xmath66 of @xmath50 has nonempty interior. a _ tangential measure _ @xmath63 for a maximal generic train track @xmath58 associates to every branch @xmath67 of @xmath58 a nonnegative weight @xmath68 such that for every complementary triangle with sides @xmath69 we have @xmath70 (indices are taken modulo three). the space @xmath71 of all tangential measures on @xmath58 has the structure of an euclidean cone. the maximal generic train track @xmath58 is called _ transversely recurrent _ if it admits a tangential measure @xmath63 which is positive on every branch @xcite. there is a one - to - one correspondence between the space of tangential measures on @xmath58 and the space of measured geodesic laminations which _ hit @xmath58 efficiently _ (we refer to @xcite for an explanation of this terminology). with this identification, the pairing @xmath72 defined by @xmath73 is just the intersection form on @xmath50 @xcite. a maximal generic train track @xmath58 is called _ complete _ if it is recurrent and transversely recurrent. a half - branch @xmath74 in a complete train track @xmath58 incident on a switch @xmath75 of @xmath58 is called _ large _ if every embedded arc of class @xmath57 containing @xmath75 in its interior passes through @xmath74. a half - branch which is not large is called _ small_. a branch @xmath67 in a complete train track @xmath58 is called _ large _ if each of its two half - branches is large ; in this case @xmath67 is necessarily incident on two distinct switches, and it is large at both of them. a branch is called _ small _ if each of its two half - branches is small. a branch is called _ mixed _ if one of its half - branches is large and the other half - branch is small (for all this, see @xcite p.118). there are two simple ways to modify a complete train track @xmath58 to another complete train track. first, we can _ shift _ @xmath58 along a mixed branch to a train track @xmath76 as shown in figure a below. if @xmath58 is complete then the same is true for @xmath76. moreover, a train track or a lamination is carried by @xmath58 if and only if it is carried by @xmath76 (see @xcite p.119). in particular, the shift @xmath76 of @xmath58 is carried by @xmath58. note that there is a natural bijection of the set of branches of @xmath58 onto the set of branches of @xmath76. second, if @xmath77 is a large branch of @xmath58 then we can perform a right or left _ split _ of @xmath58 at @xmath77 as shown in figure b. note that a right split at @xmath77 is uniquely determined by the orientation of @xmath0 and does not depend on the orientation of @xmath77. using the labels in the figure, in the case of a right split we call the branches @xmath78 and @xmath79 _ winners _ of the split, and the branches @xmath80 are _ losers _ of the split. if we perform a left split, then the branches @xmath80 are winners of the split, and the branches @xmath81 are losers of the split. the split @xmath76 of a train track @xmath58 is carried by @xmath58, and there is a natural choice of a carrying map which maps the switches of @xmath76 to the switches of @xmath58. there is a natural bijection of the set of branches of @xmath58 onto the set of branches of @xmath76 which maps the branch @xmath77 to the diagonal @xmath82 of the split. the split of a complete train track is maximal, transversely recurrent and generic, but it may not be recurrent. in the sequel we denote by @xmath83 the set of isotopy classes of complete train tracks on @xmath0. The curve graph in this section we relate teichmller geodesics and the teichmller flow to the geometry of the _ curve graph _ @xmath84 of @xmath0. this curve graph is a metric graph whose vertices are the free homotopy classes of _ essential _ simple closed curves on @xmath0, i.e. curves which are neither contractible nor freely homotopic into a puncture of @xmath0. in the sequel we often do not distinguish between an essential simple closed curve and its free homotopy class whenever no confusion is possible. two such curves are connected in @xmath84 by an edge of length one if and only if they can be realized disjointly. since our surface @xmath0 is nonexceptional by assumption, the curve graph @xmath84 is connected. any two elements @xmath85 of distance at least 3 _ jointly fill up @xmath0 _, i.e. they decompose @xmath0 into topological discs and once punctured topological discs. by bers theorem, there is a number @xmath86 such that for every complete hyperbolic metric on @xmath0 of finite volume there is a _ pants decomposition _ of @xmath0 consisting of @xmath87 simple closed geodesics of length at most @xmath88. moreover, for a given number @xmath89, the diameter in @xmath84 of the (non - empty) set of all simple closed geodesics on @xmath0 of length at most @xmath90 is bounded from above by a universal constant only depending on @xmath90 (and on the topological type of @xmath0). as in the introduction, denote by @xmath10 the bundle of quadratic differentials of area one over teichmller space @xmath9. by possibly enlarging our constant @xmath86 as above we may assume that for every @xmath13 there is an essential simple closed curve on @xmath0 whose _ @xmath15-length _, i.e. the minimal length of a representative of the free homotopy class of @xmath32 with respect to the singular euclidean metric defined by @xmath15, l at most @xmath88 (see @xcite and lemma 2.1 of @xcite). thus we can define a map @xmath91 by associating to a quadratic differential @xmath15 a simple closed curve @xmath92 whose @xmath15-length is at most @xmath88. by lemma 2.1 of @xcite, if @xmath93 is any other choice of such a map then we have @xmath94 for all @xmath13 where @xmath29 is the distance function on @xmath84. similarly we obtain a map @xmath95 by associating to a hyperbolic metric @xmath45 on @xmath0 of finite volume a simple closed curve @xmath79 of @xmath45-length at most @xmath88. by the discussion in @xcite there is a constant @xmath96 such that @xmath97 let @xmath98 be the canonical projection. by lemma 2.2 of @xcite (or see the work of rafi @xcite), we have. [simple2] there is a constant @xmath99 such that @xmath100 for all @xmath13. for a quadratic differential @xmath13 the _ strong stable manifold _ @xmath101 is defined as the set of all quadratic differentials of area one whose horizontal measured geodesic lamination coincides _ precisely _ with the horizontal measured geodesic lamination of @xmath15. the _ strong unstable _ manifold @xmath102 is defined as @xmath103 where @xmath104 is the _ flip _ @xmath105. the map @xmath106 which associates to a quadratic differential its horizontal projective measured geodesic lamination restricts to a homeomorphism of @xmath102 onto the open contractible subset of @xmath53 of all projective measured geodesic laminations @xmath63 which together with @xmath107 _ jointly fill up @xmath0 _, i.e. are such that for every measured geodesic lamination @xmath108 we have @xmath109 (note that this makes sense even though the intersection form @xmath110 is defined on @xmath50 rather than on @xmath53). by definition, the manifolds @xmath111 @xmath112 define continuous foliations of @xmath10 which are invariant under the action of @xmath11 and under the action of the _ teichmller geodesic flow _ @xmath5. hence they descend to singular foliations on @xmath4 which we denote by the same symbol. veech @xcite showed that there is a family @xmath113 of distance functions on strong stable manifolds in @xmath4 such that for a set @xmath114 of quadratic differentials which has full mass with respect to the natural _ lebesgue measure _ on @xmath4 @xcite and every @xmath115 there is a constant @xmath116 such that @xmath117 the following proposition is a modified version of this result. namely, define a closed invariant subset @xmath36 for a continuous flow @xmath118 on a topological space @xmath119 to be a _ topological hyperbolic set _ if there is a distance @xmath29 on @xmath36 defining the subspace topology and there is a constant @xmath120 such that every point @xmath121 has an open neighborhood @xmath122 in @xmath119 which is homeomorphic to a product @xmath123 and such that the following holds. 1. for all @xmath124, all @xmath125 such that @xmath126 @xmath127 and all @xmath128 we have @xmath129. 2. for all @xmath130, @xmath131 with @xmath132 @xmath127 there is a number @xmath133 $] such that @xmath134 for all @xmath128. we show. [hyperbolic] every @xmath5-invariant compact subset @xmath36 of @xmath4 is a topological hyperbolic set for @xmath5. let as before @xmath86 be such that for every complete hyperbolic metric on @xmath0 of finite volume there is a pants decomposition consisting of pants curves of length at most @xmath88. choose a smooth function @xmath135 $] with @xmath136\equiv 1 $] and @xmath137. for each @xmath138, the number of essential simple closed curves @xmath32 on @xmath0 whose hyperbolic length @xmath139 (i.e. the length of a geodesic representative of its free homotopy class) does not exceed @xmath140 is bounded from above by a universal constant not depending on @xmath141, and the diameter of the subset of @xmath84 containing these curves is uniformly bounded as well. thus we obtain for every @xmath138 a finite borel measure @xmath142 on @xmath84 by defining @xmath143 where @xmath144 denotes the dirac mass at @xmath79. the total mass of @xmath142 is bounded from above and below by a universal positive constant, and the diameter of the support of @xmath142 in @xmath84 is uniformly bounded as well. moreover, the measures @xmath142 depend continuously on @xmath138 in the weak@xmath51-topology. this means that for every bounded function @xmath145 the function @xmath146 is continuous. the curve graph @xmath147 is a hyperbolic geodesic metric space @xcite and hence it admits a _ gromov boundary _ @xmath148. for @xmath149 there is a complete distance function @xmath144 on @xmath150 of uniformly bounded diameter and there is a number @xmath151 such that @xmath152 for all @xmath153. for @xmath138 define a distance @xmath154 on @xmath148 by @xmath155 clearly @xmath154 depends continuously on @xmath141, moreover the metrics @xmath154 are equivariant with respect to the action of @xmath11 on @xmath9 and @xmath150. let @xmath156 be the set of all hyperbolic structures whose _ systole _ is at least @xmath157. let @xmath98 be the canonical projection and denote by @xmath158 the collection of all quadratic differentials whose orbit under the teichmller flow @xmath5 projects to a teichmller geodesic entirely contained in @xmath159. by theorem 2.1 of @xcite and lemma [simple2], there is a constant @xmath160 only depending on @xmath157 such that for every @xmath161 the curve @xmath162 is a _ @xmath163-quasi - geodesic _ in @xmath84, i.e. we have @xmath164 for @xmath165 define a distance function @xmath166 on @xmath167 as follows. recall that the gromov boundary @xmath148 of @xmath84 can be identified with the set of all (unmeasured) minimal geodesic laminations @xmath19 on @xmath0 which fill up @xmath0, i.e. which decompose @xmath0 into topological discs and once punctured topological discs @xcite. if @xmath168 then the horizontal measured geodesic lamination of @xmath169 is _ uniquely ergodic _, i.e. it admits a unique transverse measure up to scale, and its support @xmath170 is minimal and fills up @xmath0 @xcite. moreover, the @xmath163-quasi - geodesic @xmath171 converges in @xmath172 to @xmath170. in particular, for every @xmath173 there is a natural injection @xmath174, and this map is continuous (compare the discussion of the topology on @xmath148 in @xcite). its image is a locally compact subset of @xmath148. for @xmath175, a _ chain _ from @xmath176 to @xmath177 is a finite collection @xmath178. for such a chain @xmath79, write @xmath179 and let @xmath180 note that by the properties of the distance functions @xmath154 @xmath181 the function @xmath166 is a distance function on @xmath167 which does not depend on @xmath75. locally near @xmath75, the map @xmath182 is uniformly bilipschitz with respect to @xmath166 and the distance function @xmath183 on @xmath148. the resulting family @xmath166 of metrics on the intersection with @xmath184 of strong unstable manifolds is invariant under the action of the mapping class group. moreover, it is continuous in the following sense. if @xmath185 and if @xmath186 then @xmath187. similary we obtain a continuous @xmath11-invariant family @xmath188 of distance functions on the intersections with @xmath184 of the strong stable manifolds. since for every @xmath189 the assignment @xmath190 is a @xmath163-quasi - geodesic in @xmath84 where @xmath160 only depends on @xmath157, we obtain from hyperbolicity of @xmath84, the definition of the metrics @xmath144, lemma 3.2 and inequality ([lipschitz]) above the existence of numbers @xmath191 only depending on @xmath157 such that the flow @xmath5 satisfies the following contraction property. 1. @xmath192 for all @xmath193. @xmath194 for all @xmath195. now the assignment which associates to a quadratic differential @xmath15 its horizontal and its vertical measured foliation is continuous and therefore for every @xmath161 there is an open neighborhood @xmath196 of @xmath15 in @xmath10, an open neighborhood @xmath197 of @xmath15 in @xmath17, an open neighborhood @xmath198 of @xmath15 in @xmath102, a number @xmath22 and a homeomorphism @xmath199 which associates to a quadratic differential @xmath200 the triple @xmath201. here @xmath202 equals the quadratic differential in @xmath203 and @xmath204 is such that @xmath205, and @xmath206 equals the quadratic differential in @xmath207. the homeomorphism @xmath208 then defines a box metric @xmath209 on @xmath210 by @xmath211 let @xmath212 be the canonical projection ; the map @xmath213 is open. the projection @xmath214 of @xmath184 to @xmath4 is compact. choose a finite covering of @xmath215 by open sets @xmath216 which are images under @xmath213 of open subset @xmath217 of the above form @xmath218. for every @xmath110 let @xmath219 be the box metric on @xmath220 and define a distance @xmath221 on @xmath222 by @xmath223. let @xmath224 be a partition of unity for this covering and define a distance @xmath49 on @xmath215 by @xmath225. the distance function @xmath49 has the properties required in the definition of a topological hyperbolic set. the proposition now follows from the fact that @xmath226 for @xmath227 and that @xmath228 contains every compact @xmath5-invariant subset of @xmath4. A symbolic system in this section we use train tracks to construct a subshift of finite type which we use in the following sections to study the teichmller geodesic flow. define a _ numbered complete train track _ to be a complete train track @xmath58 together with a numbering of the branches of @xmath58. since a mapping class which preserves a train track @xmath229 as well as each of its branches is the identity (compare the proof of lemma 3.3 of @xcite), the mapping class group @xmath11 acts _ freely _ on the set @xmath230 of all isotopy classes of numbered complete train tracks on @xmath0. if a complete train track @xmath76 is obtained from @xmath58 by a single shift, then a numbering of the branches of @xmath58 induces a numbering of the branches of @xmath76 in such a way that the branch with number @xmath110 is large (or mixed or small) in @xmath58 if and only if this is the case for the branch with number @xmath110 in @xmath76. define the _ numbered class _ of a numbered train track @xmath58 to consist of all numbered train tracks which can be obtained from @xmath58 by a sequence of numbered shifts. the mapping class group naturally acts on the set of all numbered classes. we have. [freeaction] the action of @xmath11 on the set of numbered classes is free. let @xmath231 be a numbered complete train track and let @xmath232 be such that @xmath233 is contained in the numbered class of @xmath58. we have to show that @xmath234 is the identity. for this note that since @xmath233 can be obtained from @xmath58 by a sequence of numbered shifts, there is a natural bijection between the complementary components of @xmath58 and the complementary components of @xmath233. these complementary components are trigons, i.e. topological discs with three sides of class @xmath57 which meet at the cusps of the component, and once punctured monogons, i.e. once punctured discs with a single side whose endpoints meet at the cusp of the component. thus if @xmath234 is not the identity then with respect to the natural identification of complementary components, @xmath234 induces a nontrivial permutation of the complementary components of @xmath58. now for each fixed complementary component @xmath235 of @xmath58, the train track @xmath58 can be modified with a sequence of shifts to a train track @xmath76 with the property that the boundary of the complementary component @xmath236 of @xmath76 corresponding to @xmath235 consists entirely of small and large branches. but this just means that the complementary component @xmath235 is determined by the small and large branches in its boundary. since @xmath233 is contained in the numbered class of @xmath58, the map @xmath234 preserves the large and the small branches of @xmath58 and hence it maps @xmath235 to itself. this implies that @xmath234 is the identity and shows the lemma. define a _ numbered combinatorial type _ to be an orbit of a numbered class under the action of the mapping class group. thus by definition, the set of numbered combinatorial types equals the quotient of the set of numbered classes under the action of the mapping class group. let @xmath237 be the set of all numbered combinatorial types. if the numbered combinatorial type defined by a numbered train track @xmath58 is contained in a subset @xmath238 of @xmath237, then we say that @xmath58 is _ contained _ in @xmath238 and we write @xmath239. if the complete train track @xmath76 can be obtained from a complete train track @xmath58 by a single split, then a numbering of the branches of @xmath58 naturally induces a numbering of the branches of @xmath76 and therefore such a numbering defines a _ numbered split_. in particular, we can define a _ split _ of a numbered class @xmath33 to be a numbered class @xmath240 with the property that there are representatives @xmath241 of @xmath242 such that @xmath76 can be obtained from @xmath58 by a single numbered split in this sense. if @xmath76 is obtained from @xmath58 by a single split at a large branch with number @xmath77 then a large branch @xmath243 in @xmath58 is large in @xmath76 as well. thus we can define a _ full split _ of a numbered class @xmath33 to be a numbered class @xmath240 with the property that there are representatives @xmath241 of @xmath242 such that @xmath76 can be obtained from @xmath58 by splitting @xmath58 at each large branch once. numbered splitting and shifting sequence _ is a sequence @xmath244 of numbered complete train tracks such that for each @xmath110, the numbered train track @xmath245 can be obtained from a shift of @xmath246 by a single numbered split. similarly, a _ full numbered splitting and shifting sequence _ is a sequence @xmath244 of numbered complete train tracks such that for each @xmath110, the numbered train track @xmath245 can be obtained from a shift of @xmath246 by a single full numbered split. we say that a numbered combinatorial type @xmath247 is _ splittable _ to a numbered combinatorial type @xmath248 if there is a numbered train track @xmath249 which can be connected to a numbered train track @xmath250 by a full numbered splitting and shifting sequence. recall from @xcite the definition of a train track in _ standard form _ for some _ framing _ (or _ marking _ in the terminology of masur and minsky @xcite) of our surface @xmath0. each such train track is complete. twist connector _ in a train track to be an embedded closed trainpath of length @xmath251 consisting of a large branch and a small branch. we call a train track @xmath58 to be in _ special standard form _ if it satisfies the following properties. 1. every large branch of @xmath58 is contained in a twist connector. the union of all twist connectors in @xmath58 is the pants decomposition @xmath252 of our framing. @xmath58 carries a complete geodesic lamination @xmath19 whose minimal components are the pants curves of @xmath0 and such that every pair of pants of our decomposition contain precisely three leaves of @xmath19 spiraling about mutually distinct pairs of boundary curves of the pair of pants. we have. [numberedtype] there is a set @xmath253 of numbered combinatorial types with the following properties. 1. for all @xmath254, @xmath33 is splittable to @xmath240. if @xmath58 is contained in @xmath238 and if @xmath244 is any full numbered splitting and shifting sequence issuing from @xmath255 then @xmath246 is contained in @xmath238 for all @xmath256. 3. for every train track @xmath59 in special standard form for some framing of @xmath0 there is a numbering of the branches of @xmath59 such that the resulting numbered train track is contained in @xmath238. let @xmath231 be any numbered train track in special standard form for some framing of @xmath0 and let @xmath253 be the set of numbered combinatorial types of all complete train tracks which can be obtained from @xmath58 by a full numbered splitting and shifting sequence. by construction, the set @xmath238 has property (2) in the lemma. let @xmath257 be any numbered train track which contains a twist connector @xmath79 consisting of a large branch @xmath77 and a small branch @xmath67. then the train track @xmath258 obtained from @xmath257 by a split at @xmath77 with @xmath67 as a winner is contained in the numbered class which is obtained from @xmath257 by exchanging the numbers of @xmath77 and @xmath67. this means the following. let @xmath259 be the large branches of @xmath257 different from @xmath77 and assume that the numbered train track @xmath59 is obtained from @xmath257 by one split at each large branch @xmath259. then there is a numbered combinatorial type obtained from the numbered combinatorial type of @xmath58 by a full numbered split and which is obtained from the type of @xmath59 by a permutation of the numbering. let @xmath260 and let @xmath261 be a numbered train track representing @xmath176. it follows from the considerations in @xcite that there is a numbered splitting sequence connecting @xmath59 to a numbered train track @xmath262 which can be obtained from a point in the @xmath11-orbit of @xmath58 by a sequence of shifts and by a permutation of the numbering. our above consideration shows that @xmath59 can also be connected with a full numbered splitting and shifting sequence to a numbered train track which can be obtained from a point in the @xmath11-orbit of @xmath58 by a permutation of the numbering. since the collection of all permutations of the numbering of @xmath58 obtained in this way clearly forms a group, the train track @xmath59 can be connected to a train track of the same numbered combinatorial type as @xmath58 by a full numbered splitting and shifting sequence. thus @xmath238 has the first property stated in the lemma. the third property follows from @xcite in exactly the same way. let @xmath263 be the cardinality of a set @xmath253 as in lemma [numberedtype] and number the @xmath264 elements of @xmath238 in an arbitrary order. we identify each element of @xmath238 with its number. define @xmath265 if the numbered combinatorial type @xmath110 can be split with a single full numbered split to the numbered combinatorial type @xmath266 and define @xmath267 otherwise. the matrix @xmath268 defines a _ subshift of finite type_. the phase space of this shift is the set of biinfinite sequences @xmath269 with the property that @xmath270 if and only if @xmath271 for all @xmath110. every biinfinite full numbered splitting and shifting sequence @xmath272 contained in @xmath238 defines a point in @xmath273. vice versa, since by lemma [freeaction] the action of @xmath11 on the set of numbered classes of train tracks is free, a point in @xmath273 determines an @xmath11-orbit of biinfinite full numbered splitting and shifting sequences which is unique up to replacing the train tracks in the sequence by shift equivalent train tracks. we say that such a numbered splitting and shifting sequence _ realizes _ @xmath274. the shift map @xmath275 acts on @xmath273. for @xmath276 write @xmath277 ; the shift @xmath278 is _ topologically transitive _ if for all @xmath279 there is some @xmath276 such that @xmath280. namely, if we define a finite sequence @xmath281 of points @xmath282 to be _ admissible _ if @xmath271 for all @xmath110 then @xmath283 equals the number of all admissible sequences of length @xmath46 connecting @xmath110 to @xmath266 @xcite. the following observation is immediate from the definitions. [shift] the shift @xmath26 is topologically transitive. let @xmath284 be arbitrary. by lemma [numberedtype], there is a nontrivial finite full numbered splitting and shifting sequence @xmath285 connecting a train track @xmath286 of numbered combinatorial type @xmath287 to a train track @xmath288 of numbered combinatorial type @xmath266. this splitting and shifting sequence then defines an admissible sequence @xmath289 connecting @xmath110 to @xmath266. the subshift of finite type defined by the matrix @xmath268 is _ topologically mixing _ if there is some @xmath276 such that @xmath280 for all @xmath284. the next simple lemma will imply that the shift defined by numbered splitting and shifting sequences as above is topologically mixing. [mixing] let @xmath26 be a topologically transitive subshift of finite type for the finite alphabet @xmath290 and with transition matrix @xmath268. if there are @xmath291 such that @xmath292 then @xmath26 is topologically mixing. let @xmath26 be a topologically transitive subshift of finite type as in the lemma with @xmath293-transition matrix @xmath268 and such that for some @xmath279 we have @xmath292. assume to the contrary that @xmath26 is not topologically mixing. following @xcite, there is then a number @xmath294 and there is a partition of the set @xmath290 into disjoint subsets @xmath295 such that for @xmath296 we have @xmath265 only if @xmath297. let @xmath279 be as above and let @xmath298 be such that @xmath296 where indices are taken modulo @xmath299. from our assumption @xmath265 we conclude that @xmath297, on the other hand it follows from @xmath300 that also @xmath301. in other words, @xmath302 mod @xmath299 and therefore @xmath303 which contradicts our assumption that the shift is not topologically mixing. this shows the lemma. [submix] the subshift of finite type defined by the set @xmath238 of numbered combinatorial types of complete numbered train tracks on @xmath0 is topologically mixing. by lemma [mixing], it is enough to find some numbered combinatorial types @xmath284 such that our transition matrix @xmath268 satisfies @xmath292. however, such numbered combinatorial types of train tracks are shown in figure c below. namely, the train tracks @xmath304 and @xmath305 are numbered shift equivalent, and the same is true for the train tracks @xmath306 and @xmath307. to complete the proof of the corollary, simply observe that any large branch in a complete train track @xmath58 incident on a switch @xmath75 with the property that at least one of the branches of @xmath58 which is incident on @xmath75 and small at @xmath75 is _ not _ a small branch defines a configuration as in figure c. in particular, every large branch of a train track in special standard form for some framing of @xmath0 is contained in such a configuration (see @xcite, h06a). Symbolic dynamics for the teichmller flow in this section we relate the subshift of finite type @xmath26 constructed in section 4 to the extended teichmller flow. as in section 2, for @xmath231 denote by @xmath66 the convex cone of all transverse measures on @xmath58. recall that @xmath66 coincides with the space of all measured geodesic laminations which are carried by @xmath58. in particular, if @xmath76 is shift equivalent to @xmath58 then we have @xmath308. the projectivization of @xmath66 can naturally be identified with the space @xmath309 of all projective measured geodesic laminations which are carried by @xmath58. then @xmath309 is a _ compact _ subset of the compact space @xmath53 of all projective measured geodesic laminations on @xmath0. if @xmath310 is any numbered splitting and shifting sequence then @xmath311 and hence @xmath312 is well defined and non - empty. thus by the considerations in section 4, every sequence @xmath270 determines an orbit of the action of @xmath11 on the space of compact subsets of @xmath53. we call the sequence @xmath270 _ uniquely ergodic _ if @xmath312 consists of a single _ uniquely ergodic _ point, whose support @xmath19 fills up @xmath0 and admits a unique transverse measure up to scale. let @xmath313 be the set of all uniquely ergodic sequences. we define a function @xmath314 as follows. for @xmath315 choose a full numbered splitting and shifting sequence @xmath272 which realizes @xmath274. by the definition of a uniquely ergodic sequence there is a distinguished uniquely ergodic measured geodesic lamination @xmath63 which is carried by each of the train tracks @xmath246 and such that the maximal weight disposed by this measured geodesic lamination on a _ large _ branch of @xmath286 equals one. note that if @xmath257 is shift equivalent to @xmath286 then the maximal weight that @xmath63 disposes on a large branch of @xmath257 equals one as well, i.e. this normalization only depends on the numbered class of @xmath286. define @xmath316 by the requirement that the maximal mass that the measured geodesic lamination @xmath317 disposes on a large branch of @xmath318 equals one. by equivariance under the action of the mapping class group, the number @xmath319 only depends on the sequence @xmath274. in other words, @xmath49 is a function defined on @xmath320. we have. [roofcont] the function @xmath314 is continuous. let @xmath315 and let @xmath321. by the definition of the topology on our shift space it suffices to show that there is some @xmath322 such that @xmath323 whenever @xmath324 is such that @xmath325 for @xmath326. for this let @xmath244 be a numbered full splitting and shifting sequence which realizes @xmath274. then @xmath244 defines a measured geodesic lamination @xmath19 which is carried by @xmath318 and such that the maximal weight which is disposed by @xmath19 on any large branch of @xmath318 equals one. by definition, @xmath319 equals the logarithm of the maximal weight which is disposed by @xmath19 on a large branch of @xmath286. now the space @xmath327 of all transverse measures for @xmath318 equipped with the topology as a family of weights on the branches of @xmath318 coincides with a closed subset of the set of all measured geodesic laminations equipped with the weak@xmath51 topology. thus there is a neighborhood @xmath122 of @xmath19 in the space @xmath50 of all measured geodesic laminations such that for every @xmath328 the logarithm of the maximal weight of a large branch of @xmath318 defined by @xmath329 is contained in the interval @xmath330 and that the logarithm of the maximal weight that @xmath329 disposes on a large branch of @xmath286 is contained in the interval @xmath331. on the other hand, for every @xmath332 the set @xmath333 of all measured geodesic laminations which are carried by @xmath334 is a closed cone in @xmath50 containing @xmath19 and we have @xmath335 for @xmath336 and @xmath337. as a consequence, there is some @xmath338 such that @xmath339. this implies that the value of @xmath49 on the intersection with @xmath320 of the cylinder @xmath340 for @xmath341 is contained in @xmath342 which shows the lemma. the next lemma gives additional information on the function @xmath49. [bounded] 1. the function @xmath314 is nonnegative and uniformly bounded from above. 2. there is a number @xmath160 such that for every @xmath315 we have @xmath343 3. if the support of a measured geodesic lamination defined by @xmath315 is maximal then @xmath344. we have to show that there is a number @xmath345 such that @xmath346 for every @xmath315. to show the first inequality, choose a numbered full splitting and shifting sequence @xmath272 which realizes @xmath274. we assume that @xmath318 is obtained from @xmath286 by a single split at each large branch of @xmath286. using the above notations, let @xmath347 be the uniquely ergodic measured geodesic lamination defined by @xmath244. then @xmath347 is carried by each of the train tracks @xmath246, and the maximal weight of a large branch of @xmath286 for the transverse measure @xmath63 on @xmath286 defined by @xmath347 equals one. if @xmath348 is the transverse measure on @xmath318 defined by the measured geodesic lamination @xmath347 then for every large branch @xmath77 of @xmath286 we have @xmath349 where @xmath80 are the neighbors of the diagonal @xmath82 of our split in @xmath318 which are the images of the losing branches of the split under the natural identification of the branches of @xmath286 with the branches of @xmath318. moreover, we have @xmath350 for every branch @xmath351 of @xmath286 and the corresponding branch of @xmath352, and the measure @xmath348 is nonnegative. in particular, the @xmath348-weight of any branch of @xmath318 does not exceed the @xmath63-weight of the corresponding branch of @xmath286. since for every transverse measure @xmath329 on a complete train track @xmath58 the maximum of the @xmath329-weights of the branches of @xmath58 is assumed on a large branch @xcite we conclude that our function @xmath49 is non - negative. let @xmath259 be the large branches in @xmath286 with @xmath353 and let @xmath354 be the branches of @xmath286 which are incident on the endpoints of @xmath355 and which are winners of the split. let moreover @xmath356 be the losing branches of the split chosen in such a way that the branch @xmath357 is incident on the same endpoint of @xmath355 as @xmath358. denote by @xmath359 the branches in @xmath318 corresponding to @xmath354. if there is a large branch in @xmath318 of weight one with respect to the measure @xmath348 then necessarily this branch is among the branches @xmath360, so assume that @xmath361 is such a branch. then the @xmath63-weight of the branch @xmath362 vanishes and hence @xmath344 if the support of the measured geodesic lamination @xmath347 is maximal. this shows the third part of our lemma. if @xmath347 is any uniquely ergodic measured geodesic lamination which fills up @xmath0 then the subtrack @xmath363 of @xmath286 of all branches of positive @xmath63-weight is _ large _, i.e. its complementary components are all topological discs or once punctured topological discs. a branch @xmath362 of @xmath286 with @xmath364 is contained in the interior of a complementary component of @xmath363 which contains both branches @xmath365 in its boundary. by the considerations in @xcite, the number of consecutive splits of @xmath286 of this form at a large branch contained in the side of a complementary region of @xmath59 is bounded from above by the number of branches contained in this side. this just means that there is a universal number @xmath160 such that @xmath366 and complete the proof of the second part of our lemma. on the other hand, by the switch condition for transverse measures on a train track @xmath58, the maximum of the @xmath348-weights of the two winning branches @xmath81 of a split of @xmath286 at a large branch @xmath77 is not smaller than half the @xmath63-weight of @xmath77. since the maximum of the weight of a transverse measure on a complete train track @xmath58 is assumed on a large branch, our function @xmath49 is uniformly bounded. this completes the proof of the lemma. following @xcite we call a finite admissible sequence @xmath367 _ tight _ if for one (and hence every) numbered splitting and shifting sequence @xmath368 realizing @xmath274 the natural carrying map @xmath369 maps every branch @xmath67 of @xmath288 _ onto _ @xmath286. by the definition of @xmath238, by lemma [numberedtype] and by the considerations in section 5 of @xcite, tight finite admissible sequences exist. call a biinfinite sequence @xmath370 _ normal _ if every finite admissible sequence occurs in @xmath371 infinitely often in forward and backward direction. the next lemma follows from the considerations in @xcite. for this we define a measured geodesic lamination @xmath19 on @xmath0 to be _ recurrent _ if for one (and hence every) teichmller geodesic @xmath372 defined by a quadratic differential with horizontal lamination @xmath19 there is a compact subset @xmath36 of moduli space with the property that the intersection of the projection of @xmath372 to moduli space with the compact @xmath36 is unbounded. by the results of masur @xcite, a recurrent lamination is uniquely ergodic and its support fills up @xmath0. we have. [normal] let @xmath270 be normal and let @xmath373 be a numbered splitting and shifting sequence which realizes @xmath274 ; then @xmath312 consists of a single recurrent projective measured geodesic lamination. let @xmath270 be normal and let @xmath374 be a numbered splitting and shifting sequence defined by @xmath274 and the choice of a numbered train track @xmath286 of numbered combinatorial type @xmath375. then for every @xmath110 there is a natural projective linear map @xmath376. following @xcite, this map can be described as follows. let @xmath377 be the number of branches of a complete train track on @xmath0. if the train track @xmath245 is obtained from @xmath246 by a single split at a large branch @xmath77 then every measured geodesic lamination @xmath19 which is carried by @xmath245 defines both a transverse measure @xmath378 on @xmath245 and a transverse measure @xmath379 on @xmath246. if @xmath81 are the losers of the split connecting @xmath246 to @xmath245 and if @xmath82 is the diagonal branch of the split, then we have @xmath380, i.e. the convex cone @xmath381 of transverse measures on @xmath245 is mapped to the convex cone @xmath382 of transverse measures on @xmath246 by the product @xmath383 of two elementary @xmath384-matrices. for a column vector @xmath385 write @xmath386. viewing a measure in the cone @xmath387 as a point in @xmath388 we denote by @xmath389 the intersection of @xmath387 with the sphere @xmath390. the set @xmath389 can be identified with the projectivization @xmath391 of @xmath387 and is homeomorphic to a closed ball in @xmath392. for all @xmath393 the projective linear map @xmath394 can naturally be identified with the map @xmath395 (see @xcite). the jacobian of this map is bounded from above by one at every point of @xmath396 @xcite. following the reasoning in @xcite, it is enough to show that there is a number @xmath397 and there are infinitely many @xmath266 such that for each of the transformations @xmath398 @xmath399 which is given by products @xmath400 of @xmath401 elementary @xmath384-matrices, every column vector is added to every other column vector at least once. namely, in this case the jacobians of the projectivizations of the maps @xmath402 are bounded from above by a number @xmath403 which is independent of @xmath266. then the jacobians of the projectivized transformations @xmath404 tend to zero as @xmath405 uniformly on @xmath389. moreover, the maps @xmath406 are uniformly quasiconformal and therefore their pointwise dilatations tend to zero as well (see @xcite). in particular, the intersection @xmath407 consists of a single point @xmath19. moreover, for every @xmath110 the point @xmath19 is contained in the _ interior _ of the convex polyhedron @xmath391 and hence the transverse measure defined by @xmath19 on @xmath246 is positive on every branch of @xmath246 (compare the beautiful argument in @xcite). now for every tight admissible sequence @xmath408 and for every numbered splitting and shifting sequence @xmath409 which realizes @xmath410, the transformation @xmath411 has the above property. as a consequence, if @xmath412 is normal and if @xmath413 is a splitting and shifting sequence which realizes @xmath414 then @xmath407 consists of a single point @xmath19. since @xmath312 contains with every @xmath415 every projective measured geodesic lamination with the same support, the projective measured geodesic lamination @xmath19 is necessarily uniquely ergodic. we are left with showing that it also fills up @xmath0. for this call a geodesic lamination @xmath19 _ complete _ if @xmath19 is maximal and if moreover @xmath19 can be approximated in the hausdorff topology by simple closed geodesics. the space @xmath416 of _ complete _ geodesic laminations carried by @xmath246 is a compact non - empty subset of the compact space @xmath417 of all complete geodesic laminations on @xmath0 (compare @xcite) and consequently @xmath418 is non - empty. every projective measured geodesic lamination whose support is contained in a lamination from the set @xmath36 is contained in @xmath312. since the intersection @xmath312 consists of the unique point @xmath19, the laminations from the set @xmath36 contain a single minimal component @xmath419 which is the support of @xmath19. if @xmath419 does not fill up @xmath0 then there is a simple closed curve @xmath79 on @xmath0 which intersects a complete geodesic lamination @xmath420 in a finite number of points. by the considerations in @xcite we can find a sequence @xmath421 of simple closed curves on @xmath0 which approximate @xmath329 in the hausdorff topology such that the intersection numbers @xmath422 are bounded from above by a universal constant @xmath120. moreover, for every @xmath332 there is some @xmath423 such that the train track @xmath334 carries the curve @xmath424. as a consequence, there is a splitting and shifting sequence connecting @xmath334 to a train track @xmath425 which contains @xmath424 as a _ vertex cycle _ @xcite. this means that the transverse measure on @xmath425 defined by a carrying map @xmath426 spans an extreme ray in the convex cone @xmath427 of all transverse measures on @xmath425. let @xmath84 be the _ curve graph _ of @xmath0 and let @xmath428 be any map which associates to a complete train track @xmath429 a vertex cycle of @xmath59. it was shown in @xcite that there is a number @xmath430 such that the image under @xmath431 of every splitting and shifting sequence @xmath432 is an _ unparametrized @xmath163-quasi - geodesic _ in @xmath84. this means that there is an nondecreasing map @xmath433 such that the assignment @xmath434 is a uniform quasi - geodesic. since the curve graph is hyperbolic and since the distance in @xmath84 between the curves @xmath79 and @xmath424 is uniformly bounded @xcite, this implies that the diameter of the image under @xmath431 of the splitting and shifting sequence @xmath413 is _ finite_. however, the sequence @xmath270 is normal by assumption and hence the sequence @xmath413 contains subsequences whose images under the map @xmath61 are unparametrized quasi - geodesics in @xmath84 with endpoints of arbitrarily large distance. this is a contradiction and implies that the support of @xmath19 fills up @xmath0 as claimed. in other words, the set of normal points in @xmath273 is contained in the set @xmath320 of uniquely ergodic points. since normal points are dense in @xmath273, the same is true for uniquely ergodic points. recall from section 2 that for every @xmath229 the cone @xmath71 of all tangential measures on @xmath58 can be identified with the space of all measured geodesic laminations which hit @xmath58 efficiently. if @xmath435 is obtained from @xmath229 by a single split at a large branch @xmath77 and if @xmath235 is the matrix which describes the transformation @xmath436 then there is a natural transformation @xmath437 given by the transposed matrix @xmath438. this transformation maps the tangential measure on @xmath58 determined by a measured geodesic lamination @xmath329 hitting @xmath58 efficiently to the tangential measure on @xmath76 defined by the same lamination (which hits @xmath76 efficiently as well, compare @xcite). denote by @xmath439 the space of projective measured geodesic laminations which hit @xmath58 efficiently. as in the proof of lemma [normal] we observe. [fill] let @xmath270 be normal and let @xmath272 be a numbered splitting and shifting sequence which realizes @xmath274 ; then @xmath440 consists of a single uniquely ergodic projective measured geodesic lamination which fills up @xmath0. we call the sequence @xmath270 _ doubly uniquely ergodic _ if @xmath274 is uniquely ergodic as defined above and if moreover for one (and hence every) numbered splitting and shifting sequence @xmath441 which realizes @xmath274 the intersection @xmath442 consists of a unique point. by lemma [normal] and lemma [fill], every normal sequence is doubly uniquely ergodic and hence the borel set @xmath443 of all doubly uniquely ergodic sequences @xmath270 is dense. moreover, for each such sequence @xmath444 and every numbered splitting and shifting sequence @xmath272 which realizes @xmath274 there is a unique pair @xmath445 of uniquely ergodic measured geodesic laminations which fill up @xmath0 and which satisfy the following additional requirements. 1. @xmath347 is carried by each of the train tracks @xmath246 and the total mass disposed by @xmath347 on the large branches of @xmath286 equals one. @xmath446 hits each of the train tracks @xmath246 efficiently. 3. @xmath447. in particular, by equivariance under the action of the mapping class group, every sequence @xmath444 determines a quadratic differential of total area one over a point in moduli space. let @xmath448 be the set of all area one quadratic differentials whose horizontal and vertical measure foliations are both uniquely ergodic and fill up @xmath0. then @xmath449 is a @xmath5-invariant borel subset of @xmath4. a _ suspension _ for the shift @xmath278 on the subspace @xmath450 of all doubly uniquely ergodic sequences in the phase space @xmath273 with roof function @xmath451 is the space @xmath452\mid (x_i)\in { { \mathcal{d}}{\mathcal{u}}}\}/\sim$] where the equivalence relation @xmath453 identifies the point @xmath454 with the point @xmath455. there is a natural flow @xmath118 on @xmath27 defined by @xmath456 (for @xmath128) where @xmath322 is such that @xmath457. a _ semi - conjugacy _ of @xmath458 into a flow space @xmath459 is a (continuous) map @xmath460 such that @xmath461 for all @xmath462 and all @xmath463. we call a semi - conjugacy @xmath28 _ countable - to - one _ if the preimage of any point is at most countable. as an immediate consequence of lemma [bounded] we obtain. [semi] there is a countable - to - one semi - conjugacy @xmath28 of the borel suspension @xmath27 for the shift @xmath278 on @xmath464 with roof function @xmath49 to the teichmller geodesic flow. the image of @xmath28 is the @xmath5-invariant subset @xmath448. the existence of a countable - to - one semi - conjugacy @xmath28 as stated in the corollary is clear from the above considerations. we only have to show that this semi - conjugacy is continuous and that its is image equals precisely the set @xmath449. continuity of @xmath28 follows as in the proof of lemma [roofcont]. to show that the image of @xmath28 is all of @xmath449 let @xmath445 be any pair of distinct uniquely ergodic geodesic laminations with @xmath447. then every leaf of @xmath347 intersects every leaf of @xmath446 transversely. by the results from section 2 of @xcite, there is a complete train track @xmath58 which carries @xmath347 and which hits @xmath446 efficiently. for a suitable numbering of the branches of @xmath58 and up to modifying @xmath58 by a sequence of shifts the results of section 4 show that we can choose our train track in such a way that it is contained in the set @xmath238. by construction, this means that there is an infinite full splitting and shifting sequence @xmath272 such that the intersection @xmath312 consists of a unique point which is just the class of @xmath347 and that the intersection @xmath465 consists of a unique point which is just the class of @xmath446. this shows that our map @xmath28 maps @xmath464 onto @xmath449. a quadratic differential @xmath13 of area one defines a marked piecewise euclidean metric on our surface @xmath0. every essential closed curve @xmath372 can be represented by a geodesic with respect to this metric. if @xmath466 are the horizontal and vertical measured geodesic laminations for @xmath15 then the length of @xmath372 with respect to the metric defined by @xmath15 is contained in the interval @xmath467 $]. recall that a point in teichmller space can be viewed as a (marked) hyperbolic metric on @xmath0 and that there is a number @xmath86 only depending on the topological type of our surface such that for every @xmath30 there is a pants decomposition for @xmath0 which consists of pants curves of hyperbolic length at most @xmath88. rafi @xcite showed that for every quadratic differential @xmath15 of area one on the riemann surface corresponding to @xmath33 the @xmath15-lengths of the pants curves of this short pants decomposition (i.e. the lengths of the @xmath15-geodesic representatives) are bounded from above by a universal constant @xmath468. for every @xmath321 there is a constant @xmath469 such that if @xmath33 is contained in the _ @xmath157-thick part _ @xmath470 of all marked hyperbolic metrics without closed geodesics of length less than @xmath157 then the hyperbolic metric and the @xmath15-metric are @xmath471-bilipschitz equivalent (compare e.g. @xcite for this well known fact). however, for points in the thin part of teichmller space there may be curves which are short in the @xmath15-metric but which are long in the hyperbolic metric. the next lemma gives some information on the curves which become short for the quadratic differential metric along a teichmller geodesic defined by a quadratic differential @xmath472. for its formulation, recall from @xcite the definition of a _ vertex cycle _ for a train track @xmath58. such a vertex cycle is a simple closed curve @xmath79 which is carried by @xmath58 and such that the transverse measure defined by @xmath79 spans an extreme ray in the convex polygon of all transverse measures on @xmath58. for simplicity, we identify @xmath273 with a subset of our suspension space. we have. [shortcurves] let @xmath444 and let @xmath272 be a full numbered splitting and shifting sequence which realizes @xmath274. let @xmath473 and let @xmath474 be a lift of @xmath15 defined by @xmath244. then there is a vertex cycle on @xmath286 of uniformly bounded @xmath475-length. let @xmath444 and let @xmath272 be a full numbered splitting and shifting sequence which realizes @xmath274. let @xmath445 be the pair of transverse measured geodesic laminations which are defined by the lift @xmath475 of @xmath476 which is determined by @xmath244. by construction, the maximal weight that is disposed by @xmath477 on a large branch of @xmath286 is one and hence the total weight of the transverse measure on @xmath286 defined by @xmath347 is uniformly bounded. thus by corollary 2.3 of @xcite, the intersection number between @xmath347 and any vertex cycle of @xmath286 is uniformly bounded. by the results of @xcite, the number of vertex cycles for a complete train track @xmath58 is uniformly bounded. since the space @xmath66 of transverse measures on @xmath58 is spanned by the vertex cycles, there is a decomposition @xmath478 where @xmath479 and @xmath480 are the vertex cycles of @xmath286. now the number of vertex cycles is uniformly bounded and hence we have @xmath481 for a universal number @xmath321 and at least one @xmath110. then the weight that the tangential measure @xmath446 disposes on any branch @xmath67 contained in the image of @xmath480 is bounded from above by @xmath482. since a vertex cycle runs through every branch of @xmath286 at most twice @xcite, the intersection @xmath483 is bounded from above by twice the total weight of the measure @xmath446 on the branches of @xmath286 in the image of @xmath480. now @xmath484 and hence by our choice of @xmath480 this total weight is uniformly bounded. but this means that @xmath485 is uniformly bounded and shows the lemma. now let @xmath486 be a periodic point for the teichmller geodesic flow. then @xmath487 as is well known and therefore there is some @xmath444 such that the orbit of the suspension flow through @xmath15 is mapped by the semi - conjugacy @xmath28 onto the @xmath5-orbit of @xmath15. since our semi - conjugacy @xmath28 is countable - to - one this orbit of the suspension flow need not be closed. however we have. [closedorbit] let @xmath444 be a point which is mapped by @xmath28 to a periodic point of the teichmller flow. then the closure of the set @xmath488 is contained in @xmath464. the lemma is a consequence of a general observation about the piecewise euclidean metric defined by a quadratic differential @xmath489 and its relation to the preimages of @xmath15 in @xmath273. for this let @xmath444 and let @xmath490. let @xmath272 be a numbered splitting and shifting sequence which realizes @xmath274 and let @xmath474 be the lift of @xmath15 defined by @xmath244. let @xmath466 be the measured geodesic laminations defined by the quadratic differential @xmath475. then the maximal weight that @xmath347 disposes on a large branch of @xmath286 equals one. by lemma [bounded] we know that there is a number @xmath160 such that @xmath491. we claim that there is a constant @xmath263 only depending on @xmath15 such that @xmath492 for a small number @xmath493, assume that @xmath494 for large @xmath264. let @xmath495 be the subgraph of @xmath286 of all branches whose weight with respect to the measure @xmath63 defined by @xmath347 is at least @xmath496. it follows from the considerations in the proof of lemma [bounded] that @xmath59 carries a simple closed curve @xmath79. moreover, the @xmath63-weight of every branch @xmath497 which is incident on a switch in @xmath79 and which is not contained in @xmath79 is close to zero. by the considerations in lemma 2.5 of @xcite, this implies that @xmath498 is very small (depending on @xmath157). now if @xmath329 is the tangential measure on @xmath286 defined by the vertical measured geodesic lamination of @xmath15 then @xmath499 and therefore @xmath500 is uniformly bounded. as a consequence, the @xmath15-length of @xmath79 is uniformly bounded, and @xmath501 is very small. since the measured geodesic laminations @xmath466 fill up @xmath0 there is a unique point @xmath463 such that @xmath79 is _ balanced _ with respect to @xmath502. this means that the intersection numbers @xmath503 conicide and are not smaller than half of the @xmath502-length of @xmath79. in other words, if @xmath504 is small then the minimum over all @xmath463 of the @xmath502-lengths of @xmath79 is very small. on the other hand, if the orbit of @xmath15 under the teichmller geodesic flow is periodic then for sufficiently small @xmath321 it does not intersect the subset of teichmller space of hyperbolic metrics whose systole is smaller than @xmath157. but then the @xmath502-length of any simple closed curve on @xmath0 is uniformly bounded from below independent of @xmath505. this shows our claim. let @xmath428 be a map which associates to a complete train track on @xmath0 one of its vertex cycles. by our above consideration and the results from @xcite, the assignment @xmath506 is an @xmath507-quasi - geodesic for some @xmath508 depending only on @xmath15. however, this means that there is a constant @xmath509 such that for each @xmath266 the sequence @xmath510 has the following property. let again @xmath244 be a full numbered splitting and shifting sequence which realizes @xmath274. then for each @xmath46, the distance between @xmath511 and @xmath512 in @xmath84 is at least three. now there are only finitely many full numbered splitting and shifting sequences of length @xmath513 with this property. in particular, by the considerations in the proof of lemma [fill], there is a universal constant @xmath468 with the following property. let @xmath514 be any such sequence and let @xmath329 be _ any _ measured geodesic lamination which is carried by @xmath288 and such that the maximal weight disposed by @xmath329 on a large branch of @xmath58 equals one. then the maximal weight disposed by @xmath329 on a large branch of @xmath286 is at least @xmath515. this then implies that the function @xmath49 is bounded from below on @xmath488 by a positive constant and that moreover the assignment @xmath516 is a uniform quasi - geodesic in @xmath84. as a consequence, the sequence @xmath517 is contained in a compact subset of @xmath464. this shows the lemma. Bernoulli measures for the teichmller flow masur and veech @xcite constructed a probability measure in the lebesgue measure class on the space @xmath4 of area one quadratic differentials over moduli space. this measure is invariant, ergodic and mixing under the teichmller flow @xmath5, and it gives full measure to the space of quadratic differentials whose horizontal and vertical measured geodesic laminations are uniquely ergodic and fill up @xmath0. moreover, the measure is absolutely continuous with respect to the strong stable and the unstable foliation. in this section we use the results from section 5 to construct an uncountable family of @xmath5-invariant probability measures on @xmath4 including the lebesgue measure which give full measure to the set of quadratic differentials with uniquely ergodic horizontal and vertical measured geodesic laminations. these measures are ergodic and mixing, with exponential recurrence to a fixed compact set, and they are absolutely continuous with respect to the strong stable and the unstable foliation. this completes the proof of theorem 1 from the introduction. we begin with an easy observation on @xmath5-invariant probability measures on @xmath4. [finiteinv] let @xmath63 be a @xmath5-invariant borel probability measure on @xmath4. then @xmath518. since @xmath519 is a @xmath5-invariant borel subset of @xmath4 it is enough to show that there is no @xmath5-invariant borel probability measure @xmath63 on @xmath4 with @xmath520. for this assume to the contrary that such a measure exists. note that @xmath4 can be represented as a countable union of compact sets and hence there is some compact subset @xmath36 of @xmath4 with @xmath521. now by a result of masur @xcite, for every @xmath522 there is a number @xmath523 such that @xmath524 for every @xmath525. the thus defined function on @xmath526 can be chosen to be measurable. then there is some @xmath25 such that @xmath527. by invariance of @xmath63 under the teichmller flow we have @xmath528 and therefore @xmath529 which is a contradiction. the lemma follows. recall from section 5 the construction of the semi - conjugacy @xmath28 from our subshift of finite type @xmath26 with roof function @xmath49 onto the @xmath5-invariant set @xmath449. since the roof function @xmath49 on @xmath530 is uniformly bounded and essentially positive, every @xmath278-invariant borel probability measure on @xmath273 which gives full mass to @xmath464 defines a finite invariant measure for the @xmath49-suspension flow. the image of this measure under the semi - conjugacy @xmath28 is a finite @xmath5-invariant borel measure on @xmath4 which we may normalize to have total mass one. this means that the map @xmath28 induces a map @xmath531 from the space @xmath532 of @xmath278-invariant borel probability measures on @xmath273 which give full mass to @xmath464 into the space @xmath533 of @xmath5-invariant probability measures on @xmath4. we equip both spaces with the weak@xmath51-topology. we have. [continuous] the map @xmath531 is continuous. since @xmath273 is a compact metrizable space, the space of all probability measures on @xmath273 is compact. thus we only have to show that whenever @xmath534 in @xmath532 then @xmath535. but this just means that for every continuous function @xmath536 on @xmath4 we have @xmath537 which is obvious since @xmath28 is continuous. we also have. [dense] the image of @xmath531 is dense. there is a one - to - one correspondence between the space of all locally finite @xmath11-invariant borel measures on the product @xmath538 which give full measure to the pairs of distinct uniquely ergodic points and the space of all @xmath5-invariant finite borel measures on @xmath4. now every closed teichmller geodesic gives rise to such a measure which is a sum of dirac measures supported on the endpoints of all lifts of the teichmller geodesic to teichmller space. here the pair of endpoint is the pair in @xmath538 which consists of the horizontal and the vertical projective measured geodesic lamination of a quadratic differential defining a lift of the closed teichmller geodesic to teichmller space. since the pairs of endpoints of teichmller geodesics are known to be dense in @xmath538 the set of @xmath5-invariant borel probability measures which can be approximated by weighted point masses on closed geodesics is dense in @xmath533. as a consequence, it is enough to show that every measure @xmath539 which is supported on a single closed geodesic is contained in the image of @xmath531. to see that this is indeed the case, let @xmath15 be a periodic point for the teichmller flow. up to replacing @xmath15 by its image under the time-@xmath505-map of the flow for some small @xmath505, we may assume that there is some @xmath444 with @xmath540. denote as before by @xmath458 the suspension flow defined by the roof function @xmath49. for @xmath25 define @xmath541 where by abuse of notation we view @xmath274 as a point in @xmath27. by lemma [closedorbit], the supports of these measures are all contained in a fixed compact subset of @xmath464, viewed as a subset of @xmath27. therefore there is a sequence @xmath542 going to infinity such that the measures @xmath543 converge as @xmath405 weakly to a @xmath118-invariant measure @xmath63. however, it follows immediately from the discussion in the proof of lemma [closedorbit] that the support of @xmath63 is contained in the preimage of the closed orbit @xmath15 under the map @xmath28. then @xmath544 is a borel probability measure on @xmath4 which is supported on the @xmath5-orbit of @xmath15. this shows the lemma. let @xmath264 be the cardinality of the set @xmath238 of numbered combinatorial types as in lemma [numberedtype] and let @xmath545 be any @xmath293-matrix with non - negative entries @xmath546 such that @xmath547 if and only if @xmath548 (where the matrix @xmath268 is defined in the paragraph preceding lemma [shift]). we require that @xmath549 is _ stochastic _, i.e. that @xmath550 for all @xmath110. by corollary [submix] and the definitions, there is a number @xmath551 such that @xmath552. thus by the perron frobenius theorem, there is a unique positive _ probability vector _ @xmath553 with @xmath554 and such that @xmath555 by our normalization. then the probability vector @xmath163 together with the stochastic matrix @xmath252 defines a markov probability measure @xmath329 for the subshift @xmath26 of finite type which is invariant under the shift and gives full measure to the set of normal points. the @xmath329-mass of a cylinder @xmath556 @xmath557 equals @xmath558, and the @xmath329-mass of @xmath559 is @xmath560. note that the identity @xmath561 for all @xmath266 is equivalent to invariance of @xmath329 under the shift. since by corollary [submix] our shift @xmath26 is topologically mixing, the shift @xmath562 is equivalent to a bernoulli shift (see p.79 of @xcite). in particular, it is ergodic and strongly mixing, with exponential decay of correlations, and it gives full measure to the subset @xmath563 of normal points. we call a shift invariant probability measure on @xmath273 of this form a _ bernoulli measure_. the support of every bernoulli measure @xmath63 on @xmath273 is all of @xmath273, and @xmath63 is invariant and ergodic under the action of the shift @xmath278. the flow @xmath118 on the borel suspension @xmath27 with roof function @xmath49 constructed in section 4 preserves the borel measure @xmath564 given by @xmath565 where @xmath566 is the lebesgue measure on the flow lines of our flow. since by lemma [bounded] the roof function @xmath49 on @xmath567 is bounded, the measure @xmath564 is finite and moreover ergodic under the action of the flow @xmath118. the push - forward @xmath568 of the measure @xmath564 by the semi - conjugacy @xmath28 defined in corollary [semi] is a finite @xmath5-invariant ergodic measure on @xmath4 which gives full mass to the quadratic differentials with uniquely ergodic horizontal and vertical measured geodesic laminations which fill up @xmath0. we call its normalization to a probability measure on @xmath4 a _ bernoulli measure _ for the teichmller geodesic flow @xmath5. by construction, a bernoulli measure for @xmath5 is absolutely continuous with respect to the strong stable and the unstable foliation. [mixing2] a bernoulli measure for the teichmller geodesic flow is mixing. let @xmath569 be the borel set of all projective measured geodesic laminations on @xmath0 which are uniquely ergodic and whose support fills up @xmath0 and denote by @xmath570 the diagonal in @xmath571. the set @xmath572 of quadratic differentials in @xmath10 with uniquely ergodic horizontal and vertical measured geodesic lamination which fill up @xmath0 is (non - canonically) homeomorphic to @xmath573 by choosing a borel map which associates to a pair of transverse projective measured geodesic laminations in @xmath574 a point on the teichmller geodesic determined by the pair and extending this map in such a way that it commutes with the natural @xmath575-actions. with this identification, the teichmller geodesic flow acts on @xmath573 via @xmath576. in particular, there is a natural orbit space projection @xmath577 which is equivariant with respect to the natural action of the mapping class group on @xmath578 and the diagonal action of @xmath11 on @xmath579. let @xmath63 be a bernoulli measure for the teichmller geodesic flow. then @xmath63 lifts to a @xmath5-invariant @xmath11-invariant radon measure on @xmath10, and this radon measure disintegrates to a radon measure @xmath580 on @xmath581 which is invariant under the diagonal action of @xmath11 and which gives full mass to the invariant subset @xmath582. since @xmath63 is ergodic under the teichmller geodesic flow, the measure @xmath583 is ergodic under the action of @xmath11. assume to the contrary that the measure @xmath63 is not mixing. then there is a continuous function @xmath234 on @xmath4 with compact support and @xmath584 so that @xmath585 does not converge weakly to zero as @xmath586. we follow @xcite and conclude that there is a non - constant function @xmath208 which is the almost sure limit of cesaro averages of @xmath234 of the form @xmath587, i.e. for positive and negative times. replacing @xmath208 by the function @xmath588 for sufficiently small @xmath157 guarantees that there is a subset @xmath589 of full @xmath580-measure so that the lift @xmath590 to @xmath10 of the resulting function is well defined and continuous on the lines in @xmath591. the periods of the function @xmath590 define a measurable @xmath11-invariant map of @xmath592 into the set of closed subgroups of @xmath575 which is constant by ergodicity. since @xmath593 by assumption, the set @xmath594 where this group is of the form @xmath595 for some @xmath596 has full measure. the radon measure @xmath580 on @xmath581 is absolutely continuous with respect to the product foliation of @xmath597. this means that @xmath580 is locally of the form @xmath598 where @xmath599 is a measure on the space of projective measured geodesic laminations whose measure class is invariant under the action of @xmath11 and where @xmath536 is a positive measurable function. since the action of @xmath11 on @xmath53 is minimal, the measures @xmath600 are of full support. now the horizontal measured geodesic lamination of a typical point for the measure @xmath63 on @xmath4 is uniquely ergodic and fills up @xmath0 and hence if @xmath601 are typical points for @xmath63 on the same strong stable manifold in @xmath4 then @xmath602 for a suitable choice of a distance function @xmath29 on @xmath4 @xcite. from continuity of the function @xmath234 and absolute continuity of @xmath63 with respect to the strong stable and the unstable manifolds we deduce that @xmath603. similarly we argue for the strong unstable manifolds (see the nice exposition of this argument in @xcite). as a consequence, there is an @xmath11-invariant subset @xmath604 of full @xmath583-mass such that for every @xmath605 and every @xmath606 the function @xmath590 is constant almost everywhere along the strong stable manifold @xmath17 and along the strong unstable manifold @xmath102. as in @xcite we define @xmath607 by absolute continuity, the set @xmath608 has full measure with respect to @xmath580. choose a density point @xmath609 for @xmath580 and let @xmath610 be small disjoint neighborhoods of @xmath611 in @xmath53 with the property that for every @xmath612 and every @xmath613 the projective measured geodesic laminations @xmath614 together fill up @xmath0. such neighborhoods are well known to exist. then every pair @xmath615 determines a unique teichmller geodesic. cross ratio _ function @xmath616 as follows. choose a quadratic differential @xmath13 whose horizontal measured geodesic lamination @xmath347 is in the class @xmath617 and whose vertical measured geodesic lamination @xmath446 is in the class @xmath618. for @xmath613, the pair @xmath619 defines a teichmller geodesic whose corresponding one - parameter family of quadratic differentials intersects @xmath17 in a unique point @xmath620. similarly, the family of quadratic differentials defined by the pair @xmath621 intersects @xmath622 in a unique point @xmath623, the family of quadratic differentials defined by the pair @xmath624 intersects @xmath625 in a unique point @xmath626, and finally the family of quadratic differentials defined by @xmath627 intersect @xmath628 in a unique point @xmath629. the value @xmath630 does not depend on the choice of the quadratic differential @xmath15 defining the pair @xmath627 the resulting function @xmath631 is continuous and satisfies @xmath632. recall that there is a natural action of the group @xmath633 on @xmath10 where the diagonal subgroup of @xmath633 acts as the teichmller geodesic flow. each @xmath633-orbit can naturally and equivariantly be identified with the unit tangent bundle of the hyperbolic plane together with its usual action by isometries. in particular, for every @xmath634 the orbit through @xmath177 of the unipotent subgroup of @xmath633 of all upper triangular matrices of trace two is an embedded line in @xmath635, and the orbit through @xmath177 of the unipotent subgroup of all lower triangular matrices of trace two is contained in @xmath636. thus the @xmath633-orbit through the above quadratic differential @xmath15 defines two embedded line segments @xmath637 containing @xmath611 in their interior so that the restriction of the function @xmath59 to @xmath638 coincides with the restriction of the usual dynamical cross ratio on the space @xmath639 of oriented geodesics in the hyperbolic plane. in particular, the restriction of the function @xmath59 to @xmath638 is not constant in no neighborhood of @xmath627. on the other hand, if we write @xmath640 then @xmath641 are dense in @xmath610. since our function @xmath59 is continuous and not constant in any neighborhood of @xmath627 we conclude that for every @xmath321 there are points @xmath642 such that @xmath643. however, by our choice of @xmath644, the function @xmath590 is constant along the manifolds @xmath645 and @xmath628 and therefore @xmath646. in other words, the function @xmath590 has arbitrarily small periods which is a contradiction to our assumption that the set of periods of @xmath590 equals @xmath595 for some @xmath596. this completes the proof of our proposition. we also obtain a control of the return time to a suitably chosen compact subset of @xmath4 for a flow line of the teichmller geodesic flow @xmath5 on @xmath4 which is typical for a bernoulli measure @xmath63 on @xmath4. the following observation is a version of theorem 2.15 of @xcite for all bernoulli measures (see also @xcite). [returntime] there is a compact subset @xmath36 of @xmath4 and for every bernoulli measure @xmath63 for the teichmller geodesic flow @xmath5 there is a number @xmath647 and a constant @xmath648 such that @xmath649,\phi^sq\not\in k\}\leq ce^{-\epsilon t}.\]] let @xmath650 be an admissible sequence with the additional property that there is some @xmath651 such that the sequences @xmath652 and @xmath653 are both tight. by lemma 5.2 of @xcite there is a compact subset @xmath36 of @xmath4 such that for every numbered splitting and shifting sequence @xmath654 which realizes @xmath650 the following holds. let @xmath655 be a measured geodesic lamination which is carried by @xmath656 and which defines a transverse measure on @xmath286 for which the sum of the masses of the large branches of @xmath286 equals one. let @xmath657 be a measured geodesic lamination which hits @xmath286 efficiently and such that @xmath658. then the quadratic differential @xmath659 with horizontal measured geodesic lamination @xmath19 and vertical measured geodesic lamination @xmath329 is contained in the lift of @xmath36 to @xmath10. let @xmath63 be any bernoulli measure on the shift space @xmath26. then @xmath63 is exponentially mixing and hence there are constants @xmath660 such that @xmath661. the corollary now follows from this observation and the fact that the roof function @xmath49 on @xmath567 which defines our borel suspension which is semi - conjugate to the teichmller flow is uniformly bounded. we conclude this section with a description of the @xmath11-invariant @xmath5-invariant measure @xmath19 in the lebesgue measure class on the unit cotangent bundle @xmath10 of teichmller space which is invariant under the natural action of the group @xmath633 and which projects to a probability measure on @xmath4. this measure induces a @xmath11-invariant radon measure @xmath20 on @xmath662. using the notations from section 4, let @xmath238 be the collection of all numbered combinatorial types as in lemma 4.2. for every @xmath663 choose a numbered train track @xmath664 of combinatorial type @xmath266. let @xmath665 be the space of all projective measured geodesic laminations which are carried by @xmath664 and let @xmath666 be the space of all projective measured geodesic laminations which hit @xmath664 efficiently. note that @xmath667 are closed disjoint subsets of @xmath53 with dense interior. define a _ full split _ of @xmath664 to be a complete train track @xmath59 which can be obtained from @xmath664 by splitting @xmath664 at each large branch precisely once. since splits at different large branches of @xmath664 commute, the numbering of the branches of @xmath664 induces a numbering of the branches of @xmath59. thus the train tracks @xmath668 obtained from @xmath664 by a full split are numbered. we obtain a decomposition @xmath669 into borel sets @xmath670 where @xmath671 consists of the set of all projective measured geodesic laminations which are carried by @xmath672 (@xmath673). for @xmath674 the intersection @xmath675 is contained in a hyperplane of @xmath53 (with respect to the natural piecewise linear structure which defines the lebesgue measure class) and hence it has vanishing lebesgue measure. if @xmath676 is the combinatorial type of the numbered train track @xmath672 then define @xmath677 and define @xmath678 for @xmath679. note that we have @xmath680 for all @xmath64 and @xmath681. by invariance of the measure @xmath20 under the action of the mapping class group, the probabilities @xmath682 only depend on the combinatorial type @xmath663 and the choice of a representative in the shift equivalence class of @xmath266. consider the random walk on the space @xmath683 of all numbered classes defined by the @xmath11-invariant transition probabilities @xmath549. it follows from our above consideration that the exit boundary of this random walk can be viewed as a probability measure on @xmath53 in the lebesgue measure class. by invariance under the action of the mapping class group, the shift invariant measure @xmath63 on the subshift @xmath684 of finite type with alphabet @xmath238 and transition matrix defined by the above full splits is contained in the lebesgue measure class. therefore the measure on @xmath538 defined by the biinfinite random walk with the above transition probabilities coincides with @xmath20 up to scale. we use this observation to obtain a new (and simpler) proof of theorem 2.15 of @xcite and of the main theorem of @xcite. [return] let @xmath19 be the @xmath5-invariant probability measure on @xmath4 in the lebesgue measure class. there is a compact subset @xmath36 of @xmath4 and a number @xmath321 such that @xmath685 for every @xmath686\}\leq e^{-\epsilon t}/\epsilon.$] even though a priori it is not clear whether our subshift of finite type is topologically mixing (or even transitive), we can apply the same consideration as above. namely, since set of uniquely ergodic minimal and maximal measured geodesic laminations has full lebesgue measure, there is a finite sequence @xmath687 with @xmath688 for all @xmath110 and there is a number @xmath551 such that for every full splitting and shifting sequence @xmath654 which realizes @xmath414 in the above sense, the sequences @xmath689 and @xmath690 are tight in the sense described before lemma 4.6. as in the proof of lemma 5.2, from this observation the proposition follows (compare also the discussion in @xcite). The critical exponent of @xmath11 in this section we derive the first part of theorem c from the introduction. namely, we show that the critical exponent of the mapping class group equals @xmath6. for the proof of this result we continue to use the assumptions and notations from section 2 - 6. in particular, we denote by @xmath29 the teichmller distance on @xmath9. fix a point @xmath30. the poincar series at @xmath33 with exponent @xmath31 is defined to be the series @xmath34 the critical exponent of @xmath11 is the infimum of all numbers @xmath35 such that the poincar series with exponent @xmath32 converges. note that this critical exponent does not depend on the choice of @xmath33. we first give a lower bound for this critical exponent. [diverge] the poincar series diverges at the exponent @xmath691. let @xmath692 be a forward recurrent point for the teichmller geodesic flow and let @xmath693 be a lift of @xmath694. denoting again by @xmath695 the canonical projection, we may assume that @xmath696 is not fixed by any nontrivial element of @xmath11. in particular, there is a number @xmath37 such that the images under the elements of @xmath11 of the ball of radius @xmath697 about @xmath33 are pairwise disjoint. using the notations from proposition 3.3 of @xcite and its proof, let @xmath698 be the set of all projective measured geodesic laminations which contain a minimal sublamination which fills up @xmath0. let @xmath148 be the gromov boundary of the curve graph.. then there is a natural continuous, surjective and closed map @xmath699. for @xmath13 let @xmath700 be the image under the map @xmath701 of the lebesgue measure @xmath702 on the strong unstable manifold @xmath102. let @xmath703 be the preimage under @xmath704 of the subset @xmath705 of all recurrent projective measured geodesic laminations determined by @xmath694. by this we mean that for @xmath706 the point @xmath694 is contained in the @xmath707-limit set of the projection to @xmath4 of any flow line of the teichmller flow with horizontal lamination contained in the class @xmath329. for @xmath115 and @xmath708 denote by @xmath709 the closed @xmath710-ball of radius @xmath90 about @xmath711 we showed in proposition 3.3 of @xcite that there is a number @xmath712 and there is an open neighborhood @xmath122 of @xmath713 in @xmath10 with the following properties. 1. the projection of @xmath122 to @xmath9 is contained in the ball of radius @xmath39 about @xmath714. 2. there is a number @xmath120 such that for all @xmath715 we have @xmath716. the family @xmath717 of all pairs @xmath718 for some @xmath719 is a vitali relation for @xmath720. by the proof of proposition 3.3 of @xcite, there is an open relative compact subset @xmath721 of @xmath722 with the property that @xmath723 is an open neighborhood of @xmath724 in @xmath150. let @xmath31 be the diameter of the projection of @xmath721 to @xmath9. by 3) above there is a countable set @xmath725 such that the sets @xmath726 @xmath727 are pairwise disjoint, are contained in @xmath728 and cover @xmath720-almost all of @xmath728. moreover, for each @xmath110 there is some @xmath729 and some @xmath730 such that @xmath731. by 2) above we have @xmath732, and 1) above implies that @xmath733. from this together with the transformation rule for the measures @xmath702 under the action of the teichmller flow we obtain with @xmath691 that @xmath734 where @xmath735 is a fixed constant. let @xmath736. since the closed sets @xmath737 @xmath727 are pairwise disjoint by assumption, every point @xmath738 is contained in at most one of these sets. as a consequence, the family @xmath739 forms a covering relation for @xmath235 which is fine at every point of @xmath235. using again proposition 3.3 of @xcite and its proof, this covering relation is a vitali relation for the measure @xmath720. in particular, we can find a countable subset @xmath740 @xmath741 as above such that the sets @xmath742 are pairwise disjoint, are contained in @xmath728 and cover @xmath720-almost all of @xmath728. summing over @xmath743 yields that @xmath744 inductively we conclude that @xmath745 for every @xmath746. in other words, the poincar series diverges at the exponent @xmath141. with the same method we obtain. [converge] for every @xmath321 the poincar series converges at @xmath747. for @xmath30 let @xmath748 be the set of all unit area quadratic differentials which project to @xmath33. the sets @xmath749 @xmath750 form a foliation of @xmath10 of dimension @xmath751, and the same is true for the sets @xmath752 for some fixed @xmath128. let @xmath753 be the conformal density of dimension @xmath754 in the lebesgue measure class. for @xmath13 and for @xmath746 write @xmath755. let @xmath30 be a point which is not fixed by any nontrivial element of @xmath11. there is a number @xmath37 such that the images under the elements of @xmath11 of the balls of radius @xmath756 about @xmath33 are pairwise disjoint. for @xmath757 and @xmath746 let @xmath758 be the set of all points @xmath759 which project into the closed ball of radius @xmath39 about @xmath33. using continuity and compactness, for sufficiently large @xmath760, say for all @xmath761, and for all @xmath757 the @xmath762-mass of @xmath763 is bounded from below by a universal constant @xmath468 (compare the discussion in the proof of proposition 3.3 of @xcite). for an integer @xmath263 let @xmath764. we claim that there is a constant @xmath22 such that for all @xmath765 the cardinality of the set @xmath766 is bounded from above by @xmath767. then for every @xmath321 we have @xmath768 which shows our lemma. to show our claim, let @xmath769, let @xmath770 and let @xmath771. let moreover @xmath772 be such that @xmath773. we claim that if @xmath774 then the sets @xmath775 and @xmath776 are disjoint. to see this assume otherwise. since the restriction of the projection @xmath704 to @xmath749 is a homeomorphism, there is then a quadratic differential @xmath777 with @xmath778. now @xmath779 and therefore we have @xmath780 and hence @xmath781. by our choice of @xmath39, this implies that @xmath782. the @xmath783-mass of the set @xmath784 is bounded from below by @xmath67 and consequently by the definition of the measure @xmath762 the @xmath785-mass of @xmath784 is bounded from below by @xmath786. since by our above considerations the sets @xmath775 @xmath787 are pairwise disjoint, their number does not exceed a universal constant times @xmath788. from this our above claim and hence the lemma is immediate. as an immediate corollary of lemma [diverge] and lemma [converge] we obtain. [exponent] the critical exponent of @xmath11 equals @xmath6. by lemma [diverge], the poincar series of @xmath11 diverges at @xmath691 and hence the critical exponent of @xmath11 is not smaller than @xmath141. on the other hand, lemma [converge] shows that for every @xmath321 the poincar series converges at @xmath747 and hence the critical exponent is not larger that @xmath747. together the corollary follows. * remark : * shortly after this work was completed, a preprint of athreya, bufetov, eskin and mirzakhani @xcite appeared which contains precise asymptotics for the poincar series. Counting closed teichmller geodesics for @xmath37 let @xmath41 be the number of periodic orbits for the teichmller flow on @xmath4 of period at most @xmath39. veech @xcite showed that there is a number @xmath42 such that @xmath789. similarly, for @xmath37 and a compact set @xmath790 let @xmath38 be the cardinality of the set of all periodic orbits for the teichmller geodesic flow of period at most @xmath39 which intersect @xmath36. by the results of @xcite, if @xmath47 then for every compact subset @xmath790 and every sufficiently large @xmath39 we have @xmath791. the next proposition shows the second part of theorem c from the introduction (see also @xcite for a related result). for this we denote by @xmath792 the moduli space of @xmath0. choose a framing @xmath61 for @xmath0 and let @xmath796 be the collection of all train tracks in standard form for @xmath61. let @xmath797 and recall that there is a constant @xmath263 such that every complete train track on @xmath0 is at distance at most @xmath264 from a train track in the collection @xmath798 in the _ train track complex _ of @xmath0 (see @xcite for the definition of the train track complex and its basic properties). recall from section 3 the definition of the map @xmath799. there is a number @xmath800 such that @xmath801 for all @xmath802 (where we denote by @xmath29 the distance in the train track complex as well as the distance in @xmath9, see @xcite and lemma 3.4). 1. @xmath59 is contained in the @xmath11-orbit of @xmath805. @xmath805 can be connected to @xmath59 by a splitting and shifting sequence. the distance in the train track complex between @xmath59 and @xmath419 is bounded from above by @xmath79. as a consequence, there is a number @xmath508 with the following property. let @xmath806 ; then there is some @xmath807 with @xmath808 and there is a train track @xmath809 in standard form for @xmath61 which can be connected to the train track @xmath810 by a splitting and shifting sequence. write @xmath811 and for each @xmath812 choose a tight splitting and shifting sequence connecting @xmath246 to a train track @xmath813 for some @xmath814 ; such a sequence exists by @xcite and section 4 (see @xcite and the paragraph before lemma 4.6 for the definition of a tight sequence). for @xmath806 extend a splitting and shifting sequence connecting @xmath815 to @xmath816 by the splitting and shifting sequence connecting @xmath817 to @xmath818. then the distance between @xmath819 and @xmath820 is bounded from above by a universal constant. moreover, by the results of @xcite, the element @xmath821 is pseudo - anosov and its axis passes through a fixed compact neighborhood of @xmath822. in other words, there is a universal constant @xmath648 and for every @xmath806 there is a pseudo - anosov element @xmath823 with @xmath824 whose axis projects to a closed geodesic @xmath372 in the moduli space with the following properties. let @xmath827 be the preimage of @xmath826 under the projection @xmath828. since by lemma [diverge] the poincar series diverges at the exponent @xmath6 and since moreover the number of elements @xmath829 with @xmath830 is bounded from above by a universal constant not depending on @xmath45, our above consideration implies that @xmath831. we are left with showing that @xmath832 for every compact subset @xmath36 of @xmath4. for this let @xmath833 be any compact subset of @xmath834 with dense interior and let @xmath835 be a relatively compact fundamental domain for the action of @xmath11 on the lift @xmath836 of @xmath833. let @xmath113 be the diameter of @xmath837. let @xmath838 be a point which is not fixed by any element of @xmath11. let @xmath806 be a pseudo - anosov element whose axis projects to a closed geodesic @xmath372 in moduli space which intersects @xmath833. then there is a point @xmath839 which lies on the axis of a conjugate of @xmath45 which we denote again by @xmath45 for simplicity. by the properties of an axis, the length @xmath840 of our closed geodesic @xmath372 equals @xmath841. on the other hand, we have @xmath842 by the definition of @xmath113 and the choice of @xmath843. therefore, if we denote by @xmath21 the preimage of @xmath844 under the natural projection and if we define @xmath845 for @xmath37 to be the number of all @xmath806 with @xmath846, then we have @xmath847 since by corollary [exponent] the critical exponent of the poincar series equals @xmath6, we conclude that @xmath848 as claimed. for a point @xmath849 define @xmath850 to be the length of the shortest closed geodesic on the hyperbolic surface @xmath33. for a periodic geodesic @xmath851\to { \rm mod}(s)$] write @xmath852\}$]. we have. let @xmath708 be sufficiently small that for every complete hyperbolic metric @xmath141 on @xmath0 of finite volume and every closed geodesic @xmath79 for the metric @xmath141 of length at most @xmath90, the length of every closed @xmath141-geodesic which intersects @xmath79 nontrivially is bigger than @xmath855. via an appropriate choice of the constant @xmath151 in the statement of the lemma, it is enough to show the lemma for closed geodesics @xmath372 in moduli space with @xmath856. thus let @xmath851\to { \rm mod}(s)$] be such a closed geodesic parametrized proportional to arc length. via reparametrization, assume that @xmath857. let @xmath858 be a lift of @xmath372 to @xmath9 which is the axis of the pseudo - anosov element @xmath806 and let @xmath79 be an essential simple closed curve on @xmath859 of length @xmath856. we claim that the length of the geodesic arc @xmath860 $] is at least @xmath861. namely, the function @xmath862 which associates to @xmath463 the length @xmath863 of the geodesic representative of the simple closed curve @xmath79 with respect to the hyperbolic metric @xmath864 is smooth, and the derivative of its logarithm does not exceed the length of the tangent vector of @xmath865 (see @xcite). thus if the length of @xmath860 $] is smaller than @xmath866 then the length of the curve @xmath79 with respect to the hyperbolic structure @xmath867 is smaller than @xmath90 for @xmath868. by our choice of @xmath90, for every @xmath513 there are at most @xmath87 distinct simple closed geodesics of length at most @xmath90 for the hyperbolic metric @xmath867. however, since for each @xmath110 the length of @xmath869 on the hyperbolic surface @xmath870 equals @xmath871, the length of @xmath872 on @xmath873 does not exceed @xmath90. as a consequence, there is some @xmath874 such that the curves @xmath79 and @xmath872 coincide. this contradicts the fact that @xmath45 is pseudo - anosov. the lemma follows. for @xmath31 let @xmath876 be the set of all hyperbolic metrics on @xmath0 up to isometry whose systole is at least @xmath32, i.e. @xmath876 is the projection of the subset @xmath877 defined in section 3. by proposition [compactret], there is a number @xmath321 such that for every closed geodesic @xmath372 in moduli space of length at most @xmath39 there is a point @xmath878 whose distance to @xmath372 is not bigger than @xmath879. namely, every @xmath138 can be connected to @xmath159 with a geodesic of length at most @xmath880 where @xmath648 is a universal constant (see @xcite). let @xmath746 be the diameter of @xmath881. for @xmath882 there is a lift @xmath865 of @xmath372 to @xmath9 which passes through a point @xmath176 of distance at most @xmath883 to @xmath33. let @xmath806 be the pseudo - anosov map preserving @xmath865 which defines @xmath372. we then have @xmath884 and hence our corollary follows from lemma [converge] and proposition [compactret]. to conclude, we reduce the counting problem for closed geodesics in moduli space to a counting problem for pseudo - anosov elements with small translation length in the curve graph. for this we fix again a point @xmath30. [counting] for every @xmath321 there is a number @xmath885 with the following property. let @xmath30 and let @xmath372 be a closed geodesic in moduli space of length @xmath840. let @xmath886 be the conjugacy class in @xmath11 defined by @xmath372. if @xmath806 is such that @xmath887 then @xmath888. let @xmath889 and let @xmath372 be a closed geodesic in moduli space of length @xmath840. let @xmath890 be sufficiently small that no two simple closed geodesics with respect to a complete hyperbolic metric of finite volume on @xmath0 of length at most @xmath90 can intersect. let @xmath891 be the compact subset of moduli space of all hyperbolic metrics whose systole is at least @xmath90. let @xmath30 be a point which projects to a point in @xmath36 and let @xmath892 be such that @xmath893. assume that @xmath372 is parametrized in such a way that @xmath857. by the triangle inequality and the choice of @xmath45, there is a constant @xmath120 not depending on @xmath372 and an element @xmath892 such that with @xmath894 we have @xmath895. thus if @xmath896 then we have @xmath897. let @xmath79 be a curve of length @xmath871 on @xmath859. with @xmath708 as above, let @xmath898 be the smallest number such that the length of @xmath79 on @xmath899 equals @xmath90 ; then @xmath900. by our choice of @xmath871, there is a curve @xmath901 on @xmath902 of length at most @xmath871, and this curve is necessarily disjoint from @xmath79. as a consequence, we have @xmath903 in the curve graph @xmath84. we repeat this consideration as follows. let @xmath904 be the smallest number such that the length of @xmath901 on @xmath905 is at least @xmath90. as before, we have @xmath906. thus after @xmath907 steps we obtain a curve @xmath908 on @xmath909 whose distance to @xmath79 in the curve graph is at most @xmath264 and whose distance to @xmath910 is at most one. from this the lemma follows. [conjug] for every @xmath321 there is a number @xmath911 with the following property. let @xmath912 be the set of conjugacy classes of pseudo - anosov elements whose translation length on the curve graph is at most @xmath913. for @xmath37 let @xmath914 be the number of all elements in @xmath915 which correspond to closed geodesics in moduli space of length at most @xmath39 ; then @xmath916 r. canary, d. epstein, p. green, _ notes on notes of thurston _, in `` analytical and geometric aspects of hyperbolic space '', edited by d. epstein, london math. lecture notes 111, cambridge university press, cambridge 1987. u. hamenstdt, _ train tracks and the gromov boundary of the complex of curves _, in `` spaces of kleinian groups '' (y. minsky, m. sakuma, c. series, eds.), london math. notes 329 (2006), 187207.

let @xmath0 be an oriented surface of genus @xmath1 with @xmath2 punctures and @xmath3. we construct an uncountable family of probability measures on the space @xmath4 of area one holomorphic quadratic differentials over the moduli space for @xmath0 containing the usual lebesgue measure. these measures are invariant under the teichmller geodesic flow @xmath5, and they are mixing, absolutely continuous with respect to the stable and unstable foliation and exponentially recurrent to a compact set. we show that the critical exponent of the mapping class group equals @xmath6. moreover, this critical exponent coincides with the logarithmic asymptotic for the number of closed teichmller geodesics in moduli space which meet a sufficiently large compact set.

math0607386

Introduction gliese 229b is not only the first brown dwarf recognized as genuine @xcite, but it is also the brightest and best - studied t dwarf known. with an effective temperature of @xmath5k, it lies squarely between the latest l dwarfs (@xmath6k, @xcite) and the giant planets of the solar system (@xmath7k). indeed, its near infrared spectrum shows the strong h@xmath8o absorption bands characteristic of very - low mass stars and the strong ch@xmath9 bands seen in the spectra of jupiter, saturn and titan. the transitional nature of the spectrum of gl 229b is remarkable and hints at the spectral appearance of extrasolar giant planets which have effective temperatures in the range 200 1600@xmath10k @xcite. a wealth of data on gl 229b has accumulated since its discovery five years ago. broad band photometry from @xmath11 through @xmath12 and an accurate parallax @xcite allow an accurate determination of its bolometric luminosity. spectroscopic observations @xcite covering the range from 0.8 to 5.0@xmath13 m have revealed a very rapidly declining flux shortward of 1@xmath13 m, the unmistakable presence of ch@xmath9, h@xmath8o, and cs, and demonstrated the _ absence _ of the crh, feh, vo and tio features characteristic of late m and early l dwarfs. finally, noll, geballe & marley (1997) and @xcite have detected co with an abundance well above the value predicted by chemical equilibrium, a phenomenon also seen in the atmosphere of jupiter. model spectra for gl 229b @xcite reproduce the overall energy distribution fairly well and all agree that 1) @xmath5k, 2) compared to gaseous molecular opacity, the dust opacity is small if not negligible in the infrared, 3) the gravity of gl 229b is poorly constrained at present. the rapid decline of the flux at wavelengths shortward of 1@xmath13 m is interpreted as caused by an absorbing haze of complex hydrocarbons (griffith, yelle & marley 1998) or alternatively by the pressure - broadened red wing of the k i resonance doublet at 0.77@xmath13 m (tsuji, ohnaka & aoki 1999 ; burrows, marley & sharp 1999). in this paper, we present new high - resolution spectra in the @xmath0, @xmath1, and @xmath2 bands. with the inclusion of the `` red '' spectrum of @xcite, we analyze each part of the spectrum separately to obtain independent measures of the h@xmath8o abundance of gl 229b broadly interpreted as the metallicity index to detect for the first time the presence of nh@xmath3 in its spectrum, and to estimate the nh@xmath3 abundance at two different depths in the atmosphere. our results are expressed in terms of the surface gravity which can not be determined from the data presented here. nevertheless, we identify a reduced set of acceptable combinations of @xmath14 and gravity, using the observed bolometric luminosity of gl 229b @xcite. the observations and the near infrared spectra are presented in 2. section 3 shows how an accurate parallax, a well - sampled spectral energy distribution and evolutionary models greatly reduce the possible range of combinations of @xmath14 and gravity without having to resort to spectrum fitting. the synthetic spectrum calculation and our method of analysis are described in 4. the results concerning several molecules of interest which are at least potentially detectable are presented in 5, followed by a discussion in 6. finally, a summary of the results and directions for future study are given in 7. Spectroscopic observations spectra of gl 229b in selected narrow intervals in the @xmath0, @xmath1, and @xmath2 windows were obtained at the 3.8 m united kingdom infrared telescope (ukirt) in 1998 january, using the facility spectrometer cgs4 @xcite and its 150 l / mm grating. details of the observations are provided in table 1. these are among the highest resolution spectra obtained of any t dwarf. the spectra were obtained in the standard stare / nod mode with the @xmath15 wide slit of the spectrometer oriented at a position angle of @xmath16, nearly perpendicular to the line connecting gl 229a and gl 229b. the southward - going diffraction spike of gl 229a together with scattered light from that star, which is 10 magnitudes brighter than gl 229b, contaminated the array rows near and to the southwest of those containing the spectrum of gl 229b. the contamination on the gl 229b rows was determined by interpolation and was subtracted ; typically it was comparable or somewhat smaller than the signal from gl 229b. in order to remove telluric absorption features, spectra of the a0v star bs 1849 were measured just prior to gl 229b. in all cases the match in airmasses was better than five percent and hence in the ratioed spectra residual telluric features are small compared to the noise level. wavelength calibration was achieved by observations of arc lamps and is in all cases better than one part in @xmath17 (@xmath18). the spectra shown in this paper have been slightly smoothed, so that the resolving powers are lower than those in table 1 by approximately 25 percent. they also have been rebinned to facilitate coadding like spectra and joining adjacent spectral regions. the error bars can be judged by the point - to - point variations in featureless portions of the spectra, the signal - to - noise ratios at the continuum peaks are approximately 40 in the @xmath2 band, 25 in the @xmath1 band, and 30 in the @xmath0 band. the flux calibration of each spectrum is approximate as no attempt was made to match the photometry of gl 229b. while we identify the spectra by their corresponding standard photometric infrared bandpass, their wavelength coverages are much narrower than the @xmath19 filters and typically corresponds to the peak flux of gl 229b in each bandpass. in the @xmath0 band spectrum (fig. 1), nearly all features are caused by h@xmath8o. the short wavelength end of the spectrum shows the red side of a ch@xmath9 band, which is responsible for the features seen shortward of @xmath20 m. two lines of neutral potassium are easily detected near @xmath21 m. no other lines of alkali metals fall within the wavelength coverage of our observations. the @xmath1 band spectrum (fig. 2) is relatively rich in molecular opacity sources. all the features seen in the spectrum are either due to h@xmath8o (@xmath22 m) or part of a very strong ch@xmath9 absorption band (@xmath23 m). features seen between 1.59 and 1.6@xmath13 m can not presently be ascribed with certainty and are due to either h@xmath8o or ch@xmath9. while the opacities of nh@xmath3 and h@xmath8s are not negligible in this part of the spectrum, neither molecule forms distinctive spectral features. their presence can not be directly ascertained from these data, mainly because their opacity is weaker than that of h@xmath8o and ch@xmath9 and because of significant pressure broadening (see 5.3). the @xmath2 band flux emerges in an opacity window between a strong h@xmath8o band and a strong ch@xmath9 band (fig. spectral features are caused by h@xmath8o at shorter wavelengths (@xmath24 m) and by ch@xmath9 at longer wavelengths (@xmath25 m). several distinctive features of nh@xmath3 are expected at the blue end of this spectrum and models predict a single absorption feature of h@xmath8s at @xmath26 m. all features seen in figures 1 3 are unresolved blends of numerous molecular transitions. a spectral resolution at least 10 times higher would be required to resolve the intrinsic structure of the spectrum of gl 229b. Effective temperature and gravity while synthetic spectra have been fairly successful at reproducing the unusual spectrum of gl 229b @xcite, the entire spectral energy distribution has not yet been modeled satisfactorily. limitations in the opacity databases are partly responsible for the remaining discrepancies between synthetic spectra and the data (see 4.2). these shortcomings have impeded the determination of @xmath14 and of the gravity @xmath27 in particular. on the other hand, the bolometric luminosity of gl 229b is now well determined. combining spectroscopic and photometric data from 0.82 to 10@xmath13 m with the parallax, @xcite found @xmath28. with new @xmath29 photometry, @xcite found @xmath30. a recalibration using the hst photometry of @xcite gives @xmath31. evolutionary models @xcite allow us to find a family of @xmath32 models with a given @xmath33. figure 4 shows the cooling tracks of solar metallicity brown dwarfs in terms of the surface parameters @xmath14 and @xmath27. models with the bolometric luminosity of gl 229b fall within the band running through the center of the figure. using a very conservative lower limit for the age of gl 229a of 0.2@xmath10gyr @xcite, we find @xmath34k. on the other hand, the gravity remains poorly determined with @xmath35, corresponding to a mass range of 0.015 0.07@xmath36. while the upper range is very close to the lower main sequence mass limit, gl 229b s status as a brown dwarf is secure. a star at the edge of the main sequence would be much hotter with @xmath37k ; well outside of fig. 4. in the remainder of this paper, the discussion focuses on three atmosphere models which span the range of allowed solutions (table 2 and fig. 4) : @xmath38 (870, 4.5), (940, 5.0) and (1030, 5.5), which we label models a, b, and c, respectively. these constraints on @xmath14 and @xmath27 from cooling sequences are quite firm. we find the same result, within the error bar on @xmath33, from several cooling sequences which predate @xcite. the latter were computed with different input physics such as the equation of state and surface boundary conditions derived from grey and non - grey atmosphere models using several opacity tabulations. in section 5, we show that these three models can fit the spectra only if they have different metallicities, ranging from [m / h]@xmath39 to @xmath40. the evolution of brown dwarfs is sensitive to the metallicity through the atmospheric opacity which controls the rate of cooling. we find that for the range of interest here, the effect of a reduced metallicity on our determination of @xmath14, @xmath27, and the cooling age is smaller but comparable to that of the uncertainty on the value of @xmath33. we choose to ignore it for simplicity. Method of analysis our analysis is based on the atmosphere models of brown dwarfs and extrasolar giant planets described in @xcite. briefly, the atmospheres are in radiative / convective equilibrium and the equation of radiative transfer is solved with the k - coefficient method. the chemical equilibrium is treated as in @xcite. gas phase opacities include rayleigh scattering, the collision - induced opacity of h@xmath8 and the molecular opacities of h@xmath8o, ch@xmath9, nh@xmath3, h@xmath8s, ph@xmath3, and co, as well as the continuum opacities of h@xmath41 and h@xmath42. the molecular line opacity database is described in more detail in 4.2. atomic line opacity is not included. because of the relatively large gravity of gl 229b, pressure broadening of the molecular lines plays an important role in determining the @xmath43 profile of the atmosphere and in shaping the spectrum. the line - by - line broadening theory we use is described in @xcite. the strong continuum opacity source responsible for the rapid decrease of the flux of gl 229b shortward of 1.1@xmath13 m is included following the haze model of @xcite. details of the haze model and of our fitting procedure are given in 5.1. the @xmath43 structures of these atmosphere models are shown in fig. the profiles intersect each other because both @xmath14 and the gravity vary between the models. the inflexion point at @xmath44 signals the top of the convection zone. using the same monochromatic opacities used to compute the k - coefficients, high - resolution synthetic spectra are generated from the atmospheric structures by solving the radiative transfer equation with the feautrier method on a frequency grid with @xmath45@xmath46. spectra with resolution lower than @xmath47@xmath46 can then be generated for comparison with data. an unusual aspect of t dwarf atmospheres is the great variation of the opacity with wavelength. these atmospheres are strongly non - grey and the near infrared spectrum is sculpted by strong absorption bands of h@xmath8o and ch@xmath9. most of the flux emerges in a small number of relatively transparent opacity windows. the concept of photosphere becomes rather useless since the level at which the spectrum is formed depends strongly on the wavelength. figure 6 shows the depth of the `` photosphere '' @xmath48 in both temperature and pressure as a function of wavelength for model b (@xmath49k, @xmath50). in the @xmath51, @xmath0, and @xmath1 bands, and, to a lesser extent, in the @xmath2 and @xmath52 bands, the atmosphere is very transparent and can be probed to great depths. for @xmath53 m, the spectral energy distribution approaches a planck function with @xmath54k. figure 6 reveals that spectroscopy between 0.8 and 12@xmath13 m can probe the atmosphere from @xmath54k down to a depth where @xmath55k, corresponding to a range of 6 pressure scale heights! this provides an exceptional opportunity to study the physics of the atmosphere of a brown dwarf over an extended vertical range. the top of the convection zone for model b (located at @xmath56k) is below the `` photosphere '' at all wavelengths and is not directly observable, however. given the range of acceptable values of @xmath14 and @xmath27 ( 3), we can determine the metallicity of gl 229b and the abundance of several key molecules by fitting synthetic spectra to the observations, for each of the three models. the precision of our results is determined by the reliability of the models, the noise level in the data and, most significantly, by the limitations of the molecular line lists used to compute their opacities. the latter point requires a detailed discussion. the opacities of ch@xmath9 and nh@xmath3 are computed from line lists obtained by combining the hitran @xcite and geisa @xcite databases, which are complemented with recent laboratory measurements and theoretical calculations. further details are provided in @xcite. the resulting line lists for these two molecules are very nearly complete for @xmath57k, and their degree of completeness decreases rapidly at higher temperatures where absorption from excited level become important. furthermore, the line list of ch@xmath9 is limited to @xmath58 m. we extend the ch@xmath9 opacity to shorter wavelengths with the laboratory measurements of @xcite which provide the absorption coefficient averaged over intervals of 5@xmath10@xmath46 between 1 and 5@xmath13 m at @xmath59k. we use @xcite opacities for @xmath60 m and the line list for @xmath61 m. this puts the transition from one tabulation to the other in a strong h@xmath8o absorption band and obliterates any discontinuity in ch@xmath9 opacity at the transition. for @xmath62 m, the tabulation of @xcite gives the absorption coefficient of ch@xmath9 determined from spectroscopic observations of the giant planets at 0.0004@xmath13 m intervals. because of the low temperatures found in the atmospheres of giant planets, the karkoschka ch@xmath9 opacities are appropriate for @xmath63k. to our knowledge, this compilation of nh@xmath3 and ch@xmath9 opacity is the most complete presently available. as can be seen in figure 6, the temperature in the atmosphere of gl 229b is everywhere greater than 300@xmath10k. for ch@xmath9, we compute temperature - dependent line opacity (which is incomplete above 300@xmath10k) from the population of excited levels determined by the boltzmann formula for @xmath61 m, and use temperature independent opacity at shorter wavelengths. while the synthetic spectra computed reproduce the fundamental band of ch@xmath9 very well (centered at @xmath64 m), the match with the 1.6 and 2.3@xmath13 m bands is rather poor. even though ch@xmath9 is a very prominent molecule in the spectrum of gl 229b, the current knowledge of its opacity is not adequate for a quantitative analysis of its spectral signature. for this reason, we have essentially ignored the regions in our spectra where ch@xmath9 is prominent. unfortunately, this prevents us from estimating the abundance of ch@xmath9 and therefore of carbon in gl 229b. ammonia shows significant absorption in both the @xmath1 and @xmath2 band spectra. the line list for nh@xmath3 starts at @xmath65 m. because the line list does not include transitions from highly excited levels which occur at @xmath66k, the nh@xmath3 opacity we compute at a given wavelength is strictly a lower limit to the actual opacity. except for the collision - induced absorption by h@xmath8, the most important molecular absorber in gl 229b is h@xmath8o, for which the opacity is now relatively well understood. we use the most recent and most extensive ab initio line list (@xcite, @xmath67 transitions). this line list is essentially complete for @xmath68k. as a demonstration of the equality of this database, we find that the h@xmath8o features computed with this line list correspond extremely well in frequency with the observed features of gl 229b (_ e.g. _ figs. 7 to 9). however, we find noticeable discrepancies in the relative strengths of h@xmath8o features which we attribute to the calculated oscillator strength of the transitions ( 5.1). this effect can also be seen in fig. 1c of @xcite. at high resolution, the distribution of molecular transitions in frequency and strength is nearly random, and the inaccuracies in oscillator strengths we have found limit the accuracy of model fitting in a fashion similar to noise. this `` opacity noise '' is at least as significant as the noise intrinsic to our data. Results from fitting the spectra we have constructed a grid of synthetic spectra for the three models shown in fig. 4 with metallicity @xmath69 } \le 0.1 $] in steps of 0.1. we use these modeled spectra to fit four distinct spectral regions (the `` red '', @xmath0, @xmath1, and @xmath2 spectra) separately to determine the metallicity as a function of gravity with an internal precision of @xmath70 dex. for the purpose of fitting the data, the synthetic spectra were renormalized to the observed flux at a selected wavelength in each spectral region. the model spectra show some distortions in the overall shape of the spectrum which are probably due to remaining uncertainties in the @xmath71 profile of the atmosphere, the inadequate ch@xmath9 opacities, and possible effects of dust opacity. considering the additional problems with the strength of the h@xmath8o features, we elected to do all fits `` by eye, '' except where otherwise noted. we discuss the fitting of each spectral interval below. in the interest of brevity, we present a detailed discussion of fits obtained only with the model of intermediate gravity (model b). the best fits obtained with models a and c are very nearly identical to those with model b. the results are summarized in table 2. the `` red '' spectrum extends from 0.83 to @xmath72 m and is formed deep in the atmosphere where @xmath73k. the spectra of @xcite and @xcite reveal two lines of cs i (at 0.852 and 0.894@xmath13 m) and a strong h@xmath8o band but _ not _ the bands of tio and vo common to late m dwarfs and early l dwarfs (fig. 7). refractory elements, such as ti, fe, v, ca, and cr, are expected to be bound in condensed compounds in a low - temperature atmosphere such as that of gl 229b and therefore are not available to form molecular bands @xcite. with the exception of the strong, unidentified feature at 0.9874@xmath13 m, all features between 0.89 and 1.0@xmath13 m can be attributed to h@xmath8o. an overlap of a weak band of h@xmath8o and a weaker ch@xmath9 band causes the small depression at 0.894@xmath13 m which was tentatively tentatively attributed to ch@xmath9 by @xcite and @xcite. features below 0.89@xmath13 m can not be identified at present. the two cs i lines are not included in our model. the flux from gl 229b is also observed to decrease very rapidly toward shorter wavelengths @xcite, which, in the absence of the strong tio and vo bands, is evidently caused by the presence of a missing source of opacity in the atmosphere. spectra computed with molecular opacities only (but excluding tio, vo, feh, etc) predict visible fluxes which are grossly overestimated @xcite but the detailed sequence of absorption features of the spectrum are well reproduced, indicating that the short wavelength flux is suppressed by a _ opacity source. we fit the red spectrum of gl 229b between 0.82 and 1.15@xmath13 m to obtain the metallicity. we have recalibrated the published spectrum @xcite using the hst photometry of @xcite. we model the continuum opacity with a layer of condensates following the approach of @xcite. condensates are expected in the atmosphere of gl 229b on the basis of chemical equilibrium calculations @xcite and can provide the required opacity. alternatively, tsuji, ohnaka & aoki (1999) and burrows, marley, & sharp (1999) attribute this rapid decline to the pressure broadened red wing of the 0.77@xmath13 m k i resonance doublet. the first optical spectrum of a t dwarf (sdss 1624 + 0029) shows that the latter explanation is correct @xcite. the nature of this opacity source is not very important for the determination of the metallicity, however, as long as the proper continuum opacity background is present in the calculation. the dust opacity is computed with the mie theory of scattering by spherical particles and is determined by the vertical distribution of the particles, their grain size distribution, and the complex index of refraction of the condensate. the cloud model of @xcite is described by 1) the vertical density profile of the condensate, taken as : @xmath74 where @xmath75 is the number density of condensed particles, @xmath76 is the ambient gas pressure, and the cloud layer is bound by @xmath77 2) the size distribution of the particles @xmath78 ^ 2,\]] where @xmath79 is the diameter of the particles ; and 3) the complex index of refraction of the condensate @xmath80 the parameters @xmath81, @xmath82, @xmath83, @xmath84, the function @xmath85, and the metallicity of the atmosphere are free parameters. such a multi - parameter fit of the observed spectrum is not unique. furthermore, arbitrarily good fits of the `` continuum '' flux level can be obtained by adjusting the imaginary part of the index of refraction since its wavelength dependence is weakly constrained _ a priori_. our results for the three models are qualitatively similar to those of @xcite. typical values of the fitted dust parameters are @xmath86@xmath87bar@xmath88, @xmath89bar, @xmath90 m, and @xmath91, with the imaginary part of the index of refraction decreasing from @xmath92 at 0.8@xmath13 m to 0.01 at 1.10@xmath13 m. these parameters applied to model b with a metallicity of [m / h]=@xmath93 result in the fit shown in figure 7. the metallicity of the atmosphere [m / h] is largely independent of the dust parameters, however, as it is constrained by the amplitude of the features in the h@xmath8o band which we fit between 0.925 and 0.98@xmath13 m. a larger metallicity results in a larger amplitude of the features inside the band. we obtain the same value of [m / h] as long as a good fit of the `` continuum '' flux level is obtained, regardless of the particular values of dust parameters. the lower panel of fig. 7 clearly shows differences in the relative strengths of the absorption features in the h@xmath8o band between the synthetic and observed spectra. similar differences also occur in the @xmath0, @xmath1, and @xmath2 bands. the same differences are found for all three atmospheric profiles and point to inaccuracies in the oscillator strength of the ab initio line list of h@xmath8o @xcite. the metallicity is fitted to give the best overall fit of these features with a precision of @xmath70 dex. strictly speaking, this procedure gives the h@xmath8o abundance, or [o / h] rather than [m / h]. for solar metallicity, the condensation of silicates deep in the atmosphere of gl 229b will reduce the amount of oxygen available to form h@xmath8o by @xmath94% @xcite which implies that @xmath95 } = { \rm [o / h] } + 0.07.\]] the correction, which decreases for subsolar metallicities, is smaller than our fitting uncertainty and will be ignored hereafter. the @xmath0 band spectrum probes the most transparent window of the spectrum of gl 229b and is formed at great depth where @xmath96k and @xmath97bar (fig. 6). our spectrum contains almost exclusively h@xmath8o features, with the exception of ch@xmath9 absorption for @xmath98 m and of two prominent k i lines (fig. since we have elected to ignore ch@xmath9 bands and our synthetic spectra do not include alkali metal lines, we fit the @xmath0 spectrum between 1.215 and 1.298@xmath13 m to determine the metallicity from the depth of the h@xmath8o absorption features. figure 8 shows the effect of the metallicity on the spectrum (top panel) and our best fit (bottom panel) for model b. the flux level in the @xmath0 spectrum varies by a factor of 3 and our best fit shows distortions in the general shape of the spectrum. the distortions may be caused by a combination of uncertainties in the @xmath43 profile of the atmosphere, problems with the h@xmath8o opacities or a small amount of dust opacity in the infrared spectrum of gl 229b (not modeled). since the fit is based on the depth of the features, we ignore these distortions and fit the logarithm of the flux rather than the flux, as shown in fig. 8. as in the red spectrum ( 5.1), we find a remarkable correspondence of spectral features between the observed and modeled spectra although the model spectrum is somewhat less successful at reproducing the relative strengths of the h@xmath8o features. while the @xmath1 band spectrum falls between the red side of a strong h@xmath8o band (for @xmath22 m) and the blue side of a prominent ch@xmath9 band (for @xmath99 m), nh@xmath3 and h@xmath8s have non - negligible opacity in this wavelength interval which also includes a band of co. the strong ch@xmath9 band is responsible for the turnover of the flux at 1.59@xmath13 m. in this wavelength range, the ch@xmath9 opacity is described in our calculation by the @xcite laboratory measurements which are restricted to @xmath59k. as a consequence, the ch@xmath9 band comes in at 1.61@xmath13 m in the synthetic spectra, which results in strong departures of the mean flux level between data and models for @xmath100 m. this limits our analysis of the @xmath1 band spectrum to wavelengths shorter than 1.58@xmath13 m. in this wavelength range, the spectrum is formed deep in the atmosphere, where @xmath101k and @xmath102 bar (fig. 6). within this spectral region, we determine the abundance of nh@xmath3 as a function of the metallicity, but can not untangle the two. we also comment on the presence of h@xmath8s and co. while nh@xmath3 absorption can significantly affect the slope of the spectrum shortward of 1.56@xmath13 m (fig. 9), there is no distinctive feature at this spectral resolution (@xmath103) to provide an unambiguous detection of this molecule. the detection of co at 4.7@xmath13 m well above the equilibrium abundance @xcite suggests the possibility that nh@xmath3 may also depart from its chemical equilibrium abundance @xcite. we therefore vary the abundance of nh@xmath3 by reducing its chemical equilibrium abundance _ as computed for a given metallicity _ uniformly throughout the atmosphere by constant factor. we found that reduction factors of 1, 0.5, 0.25, and 0 provide an adequate grid of nh@xmath3 abundances given the s / n ratio of the data and the residual problems with the h@xmath8o opacity. the effect of varying the metallicity on the @xmath1 band spectrum is shown in fig. 10 for model b. all spectra in figure 10 were computed with the equilibrium abundance of nh@xmath3. in this case, we find a best fitting metallicity of [m / h]@xmath104. figures 9 and 10 show that varying the abundance of nh@xmath3 and varying the metallicity have very similar effects on the synthetic spectrum. in the absence of any distinctive feature of nh@xmath3, it is not possible to determine both the metallicity and the nh@xmath3 abundance separately. for each value of [m / h], we can adjust the nh@xmath3 abundance to obtain a good fit, higher metallicities requiring lower nh@xmath3 abundances. the best fitting solutions are given in table 2. these fits are nearly indistinguishable from each other although the higher metallicity fits are marginally better. it is in the @xmath1 band that we find the poorest match in the detailed features of the data and the models. while the fit is determined by matching the relative amplitudes of the features, the two - parameter fit (metallicity and nh@xmath3 abundance) we have performed amounts to little more than fitting the slope of the spectrum. as an internal check on our fitting procedure and precision, we have have verified that our fits indeed have the same slope as the data by plotting data and models at a very low spectral resolution which eliminates all absorption features. hydrogen sulfide (h@xmath8s) has non - negligible opacity over most of the wavelength range of our fit to the @xmath1 band. figure 11 shows two spectra computed with and without h@xmath8s opacity. its opacity is weaker than that of nh@xmath3 however, and there are no distinctive features which would allow a positive identification. since the h@xmath8s features are fairly uniformly distributed in strength and wavelength, fitting the spectrum with either the chemical equilibrium abundance of h@xmath8s or with no h@xmath8s at all only has a small - to - negligible effect on the determination of the nh@xmath3 abundance for a given metallicity (table 2). for the low gravity model a, the nh@xmath3 abundance determined from spectra without h@xmath8s is about half of the value found with the chemical equilibrium abundance of h@xmath8s. the difference decreases at higher gravities and is negligible for model c. chemical equilibrium calculations @xcite indicate that h@xmath8s is present in gl 229b with an abundance essentially equal to the elemental abundance of sulfur in the atmosphere (see 6.3 for further discussion). unfortunately, we are unable to ascertain the presence of h@xmath8s in the atmosphere of gl 229b at present. the discovery of co in the 4.7@xmath13 m spectrum of gl 229b with an abundance about 3 orders of magnitude higher than predicted by chemical equilibrium revealed the importance of dynamical processes in its atmosphere @xcite. the 4.7@xmath13 m spectrum probes the atmosphere at the 2 3 bar level where @xmath105k (fig. 6). at this level, chemical equilibrium calculations predict a co abundance of @xmath106 while @xcite found @xmath107. the second overtone band of co falls within the @xmath1 band and, in principle, could provide a determination of @xmath108 at a deeper level of the atmosphere, where @xmath109 bar and @xmath110k (fig. 6). figure 12 shows a comparison of the data with synthetic spectra computed with various amounts of co for model b with solar metallicity. the first two spectra are computed with the chemical equilibrium abundance of co (@xmath111 at 14@xmath10bar) and without co (@xmath112). these two spectra are very nearly identical. a third spectrum is computed in the unrealistic limit where all the carbon in the atmosphere is in the form of co (@xmath113), which represents the maximum possible co enhancement. as we found for nh@xmath3 and h@xmath8s, there is no distinctive spectral signature of co at this resolution (@xmath103). our extreme case represents a flux reduction of @xmath114 at best. we are unable to constrain the co abundance with our data. since the @xmath115 band of co is @xmath116 times weaker than the fundamental band at 4.7@xmath13 m, obtaining a useful co abundance from @xmath1 band spectroscopy will be a difficult undertaking. of our three near infrared spectra, the @xmath2 band spectrum is formed highest in the atmosphere : @xmath117 1020@xmath10k and @xmath118bar (fig. this is the same level as is probed with 4.7@xmath13 m spectroscopy. as in the @xmath1 band, the spectrum contains mainly h@xmath8o features on the blue side (@xmath119 m) and a strong ch@xmath9 band appears at longer wavelengths. there are also several nh@xmath3 features for @xmath120 m and the models predict an isolated feature of h@xmath8s (fig. 3). figure 13 compares a spectrum computed for model b with [m / h]@xmath104 with the entire @xmath2 band spectrum. there is an excellent agreement in the structure of the spectrum even though the overall shape is not very well reproduced. for @xmath120 m, the model predicts strong features of nh@xmath3 which we discuss in the next section. beyond 2.12@xmath13 m is a ch@xmath9 band which is too weak in the model. the structure within the modeled band is remarkably similar to the observed spectrum, however. this much better agreement of the ch@xmath9 band than we obtained in the @xmath1 band is due to two factors. first, in this band the ch@xmath9 opacity is computed from a line list, and can therefore be computed as a function of temperature rather than at a fixed value of 300@xmath10k. second, the lower temperature where the band is formed (fig. 6) reduces the effect of the incompleteness of the ch@xmath9 line list above 300@xmath10k. nevertheless, we do not consider the ch@xmath9 features here and limit our analysis to @xmath121 m. the @xmath2 band spectrum provides a unique opportunity : once the metallicity is determined by matching the depth of the h@xmath8o features between 2.05 and 2.10@xmath13 m, the abundance of nh@xmath3 can be obtained by fitting its features below 2.05@xmath13 m. the fit of the metallicity is shown in fig. 14 for model b, which shows [m / h]=0 and @xmath40 (top panel) and our best fit, [m / h]=@xmath93 (bottom panel). values for models a and c are given in table 2. all features in this region are due to h@xmath8o and, as we found in the red spectrum and in the @xmath0 and @xmath1 bands, the oscillator strengths of the h@xmath8o line list do not reproduce the relative strength of the observed features very well. in the wavelength range shown in fig. 15, the spectrum consists of a few nh@xmath3 features on a background of h@xmath8o absorption. synthetic spectra predict seven strong nh@xmath3 features in this spectrum, three of which are clearly present (2.033, 2.037 and 2.046@xmath13 m), one is absent (2.041@xmath13 m) and three appear to be missing (2.026, 2.029 and 2.031@xmath13 m). this constitutes an ambiguous detection of nh@xmath3 in gl 229b. the determination of the abundance of nh@xmath3 from the @xmath2 band spectrum is hampered by the limited accuracy of the oscillator strengths of the h@xmath8o line list and by the incompleteness of our nh@xmath3 line list for temperatures above 300@xmath10k. the effect of the former can be seen in the trio of features at 2.026, 2.029 and 2.031@xmath13 m, which overlap h@xmath8o absorption features. even after removing all nh@xmath3 opacity, these features are still too strong in the calculated spectrum (top panel of fig. the top panel of fig. 15 as well as fig. 13 show that for model b, the abundance of nh@xmath3 derived from the chemical equilibrium for the adopted metallicity of [m / h]@xmath104 is too high. we therefore consider a depletion in nh@xmath3 in the atmosphere of gl 229b at the level probed by the @xmath2 band spectrum. following the approach used in fitting the @xmath1 band spectrum, we express this depletion as a fraction of the chemical equilibrium abundance of nh@xmath3 for the metallicity obtained independently from the amplitude of the h@xmath8o features in the @xmath2 band. this fraction is applied uniformly throughout the atmosphere for the computation of the synthetic spectrum. given the ambiguous presence of nh@xmath3 in the @xmath2 band, we have determined the optimal nh@xmath3 abundance by minimizing the @xmath122 of the spectral fit for @xmath123 m. this gives a depletion factor of @xmath124 with no nh@xmath3 present being an acceptable fit. restricting the fit to the region where nh@xmath3 features are clearly observed (@xmath125) gives a similar result but favors a finite value for the nh@xmath3 depletion of @xmath126. the results are summarized in table 2. the lower panel of figure 15 shows the model b fit obtained by reducing the nh@xmath3 abundance throughout the atmosphere to @xmath127% of its chemical equilibrium value _ for the adopted metallicity_. with this significant degree of depletion, the model reproduces the three detected features (2.033, 2.037 and 2.046@xmath13 m) extremely well and makes the 2.041@xmath13 m feature consistent with the observations. because our nh@xmath3 line list is incomplete at high temperatures, the opacity which we compute is strictly a lower limit to the actual nh@xmath3 opacity at any wavelength. if the incompleteness is significant for the features found in the @xmath2 band spectrum, then the nh@xmath3 abundance is actually lower still. it appears extremely unlikely that the errors in the oscillator strength of the h@xmath8o transitions would conspire to mimic the depletion of nh@xmath3 which we find. for example, if we imagine that there is no nh@xmath3 depletion (dotted curve in the top panel of fig. 14), the residuals between the data and the fitted spectrum for @xmath120 m would be much larger than the typical mismatch which we find in h@xmath8o features in all four spectra presented here. we consider the depletion of nh@xmath3 in the @xmath2 band spectrum to be firmly established. we are not able to establish the presence of h@xmath8s from our @xmath1 band spectrum ( 5.3.2). throughout the @xmath2 band, the h@xmath8s opacity is generally overwhelmed by h@xmath8o and ch@xmath9 absorption. however, there is a peak in the opacity of h@xmath8s which is about one order of magnitude higher than all other opacity maxima (fig. our synthetic spectra indicate that this feature is strong enough to become visible in the midst of the background of h@xmath8o and ch@xmath9 features. figure 17 shows the relevant portion of the @xmath2 band spectrum with synthetic spectra for all three models (a, b, and c) using the fitted metallicity (table 2). all three panels are remarkably similar. the strength of the predicted feature is well above the noise level of the data, and taken at face value, figure 17 indicates a probable depletion of h@xmath8s by more than a factor of 2. on the other hand, chemical equilibrium calculations indicate that the h@xmath8s abundance should be very near the elemental abundance of sulfur, even in the presence of vertical transport and condensation @xcite. since there is no reason to expect a significant depletion of h@xmath8s, the discrepancy is probably due to remaining uncertainties in the opacities. our h@xmath8s line list is based on an ab initio calculation (r. wattson, priv. which hasnt been compared to laboratory data in this part of the spectrum. the strong feature centered at 2.1084@xmath13 m is a blend of three strong lines from three different bands of h@xmath8s. possible errors in the position or strength of these lines could significantly reduce the amplitude of the feature in our synthetic spectra. the limitations of the background opacity of h@xmath8o and ch@xmath9 may also be responsible for the observed mismatch. nevertheless, it is desirable to look for this feature at a higher resolution and a higher s / n ratio as an absence of sulfur in gl 229b would be a most intriguing result. given our determination of @xmath14 and [m / h] in terms of the surface gravity, we can obtain the abundance of co from the 45@xmath13 m spectra of @xcite and @xcite which is consistent with our results. for each model a, b, and c and using the metallicity given in table 2, we computed synthetic spectra with various co abundances. the latter is varied freely without imposing stoichiometric constraints. the synthetic spectra are binned to the wavelength grid of the data and fitted to the data by a normalization factor adjusted to minimize the @xmath122. the @xmath122 of the fitted spectra shows a well - defined minimum as a function of the co abundance, @xmath108 (table 3). the uncertainty on the co abundance is obtained by generating synthetic data sets by adding a gaussian distribution of the observed noise to the best fitting model spectrum. after doing the same analysis on 1000 synthetic data sets, we obtain a (non - gaussian) distribution of values of @xmath108. the uncertainties given in table 3 correspond to the 68% confidence level. the @xcite spectrum gives co abundances which are 0.1 dex higher than those obtained from the @xcite spectrum ; which is well within the fitting uncertainty. @xcite found a co mole fraction of 50 to 200 ppm (@xmath128) by assuming a h@xmath8o abundance of 300 ppm (effectively, [m / h]@xmath129), which agrees well with our result for model a which has [m / h]@xmath130. our results are also consistent with those of @xcite who find co abundances of @xmath131 and @xmath132 ppm (@xmath133 and @xmath134) for [m / h]=@xmath135 and 0, respectively, using the data of @xcite. since this part of the spectrum contains only h@xmath8o and co features, our fitting procedure is sensitive only to the co to h@xmath8o abundance ratio. chemical equilibrium calculations show that the h@xmath8o abundance scales linearly with the metallicity at the level probed with 45@xmath13 m spectroscopy (_ i.e. _ all the gas phase oxygen is in h@xmath8o). accordingly, the co abundance we find scales with the metallicity of the model. as shown in figure 18, the co abundance determined from the 4.7@xmath13 m band corresponds approximately to the co / ch@xmath9 transition in chemical equilibrium. the observations definitely exclude the very - low chemical equilibrium abundance of co. the fact that the extreme case where all carbon is co provides an acceptable fit to the data (while we know that a good fraction of the carbon is in ch@xmath9) is due to the rather noisy spectra. Discussion it is highly desirable to restrict the surface gravity @xmath27 of gl 229b to an astrophysically useful range. since @xmath33 is known, a determination of @xmath27 fixes @xmath14, the radius, the mass, and the age of gl 229b (fig. 4), as well as the metallicity and the abundances of important molecules such as co and nh@xmath3. the large uncertainty on @xmath27 results in a large uncertainty in the mass of gl 229b and on the age of the system determined from cooling tracks. while a dynamical determination of the mass may be possible in a decade or so @xcite, a spectroscopic determination might be obtained much sooner. unfortunately, it is not possible to constrain the gravity better than @xmath136 with the data and models presently available. our high resolution spectroscopy does not allow us to choose between models a, b and c (table 2) as an increase in gravity can be compensated by an increase in metallicity to lead to an identical fit. the gravity sensitivity of the @xmath2 band synthetic spectrum models reported by @xcite occurs for a fixed metallicity only. similarly, @xcite and @xcite report that the spectral energy distribution of gl 229b models is fairly sensitive to the gravity. but this is true only for a fixed metallicity. for the three models indicated in fig. 4, and using the metallicity we have determined for each (table 2) the gravity dependence of the infrared colors is @xmath137 and @xmath138 and @xmath139 (table 4). this dependence is very weak in the light of the uncertainty in the photometry of gl 229b. more problematic is the fact that the synthetic @xmath140 disagrees with the photometry. furthermore, the incomplete ch@xmath9 opacities used in the spectrum calculation almost certainly result in an inaccurate redistribution of the flux in the near infrared opacity windows which determines the broad band colors. an example of this effect on the @xmath1 band flux can be seen in fig. we conclude that a photometric determination of the gravity is not possible at present. an alternative approach to determining the gravity of gl 229b is through a study of the pressure - broadened molecular lines of its spectrum. the spectrum of gl 229b is formed of a forest of unresolved molecular lines maily due to h@xmath8o, ch@xmath9, and nh@xmath3. because of the limitations of the ch@xmath9 and nh@xmath3 opacity data bases, a detailed study of molecular features is best performed in spectral domains where these two molecules do not contribute significantly to the opacity. spectroscopic observations with a resolution of @xmath141 can reveal the shape of individual h@xmath8o lines in regions where they are relatively sparse, _ e.g. _ from 2.08 to 2.105@xmath13 m. our determination of the metallicity of gl 229b, with an uncertainty of @xmath70, is given in table 2 for the three gravities considered. we find an excellent agreement between our three independent determinations of [m / h] for each gravity and conclude that gl 229b is likely to be depleted in heavy elements, _ e.g. _ oxygen. the metallicity is near solar at high gravity and decreases significantly for lower gravities. in their analysis of the `` red '' spectrum, @xcite found a h@xmath8o abundance between 0.3 and 0.45 of the solar value for a @xmath142k, @xmath50 model and they adopt a value of 0.25 in @xcite. metal depletion in gl 229b is consistent with the analysis of the 0.985 1.02@xmath13 m feh band in the spectrum of the primary star gl 229a by schiavon, barbuy & singh (1997) who find [fe / h]@xmath143. the infrared colors of gl 229a imply that it is slightly metal - rich, however @xcite. the relative metallicity of the components of this binary system may have been affected by their formation process. if the pair formed from the fragmentation of a collapsing cloud (like a binary star system), the two objects should share the same composition. if the brown dwarf formed like a planet, from accretion within a dissipative keplerian disk around the primary, the selective accretion of solid phase material could lead to an _ enrichment _ in heavy elements compared to the primary star, as is observed in the gaseous planets of the solar system. the low mass of the primary (@xmath144) and the large semi - major axis and eccentricity (@xmath145au and @xmath146, @xcite) suggests that the binary formation process, and therefore equal metallicities, are more plausible. a more detailed study of the metallicity of gl 229a is desirable to better understand the history of this system. the results of our analysis of the metallicity and abundances of several molecules in the atmosphere of gl 229b are summarized in fig each panel corresponds to a different model (see table 2) and shows the abundances of important molecules as a function of depth in the atmosphere based on chemical equilibrium calculations including condensation cloud formation. the chemistry of these abundant molecules is fairly simple. the abundance of h@xmath8o is uniformly reduced by @xmath147% by silicate condensation. except for a small depletion for @xmath148 due to the condensation of na@xmath8s, all sulfur is found in h@xmath8s. the other molecules shown are not affected by condensation. nitrogen is partitioned between n@xmath8 and nh@xmath3, with the latter being favored at lower temperatures and higher pressures. nh@xmath3 dominates near the surface and rapidly transforms into n@xmath8 at the higher temperatures found deeper in the atmosphere. deep in the atmosphere, the higher pressures cause a partial recombination of nh@xmath3 and the ratio of the nh@xmath3 to n@xmath8 abundances increases slowly with depth. in a similar fashion, all elemental carbon is found in ch@xmath9 at the surface but co starts to form at higher temperatures and rapidly becomes the most abundant carbon - bearing molecule. the formation of co consumes h@xmath8o, as can be seen in fig. 18. in each panel, a dotted box indicates the co abundance we have determined from the 4.7@xmath13 m spectrum, the location of the box along the ordinate shows the level probed at this wavelength (fig. 6). as originally discussed by @xcite and @xcite, the co abundance is @xmath149 orders of magnitude larger than the chemical equilibrium value. stochiometric constraints imply that this also results in a significant reduction of the ch@xmath9 abundance at the 870 950@xmath10k level in gl 229b. in model b, the equilibrium abundance of ch@xmath9 at 900@xmath10k is @xmath150, which corresponds to the abundance of elemental carbon. the co abundance determined from the 4.7@xmath13 m band is @xmath151 (with a large error bar). conservation of the total number of carbon atoms then requires that the ch@xmath9 abundance be @xmath152, a full factor of 2 below the equilibrium abundance. if we use the lowest co abundance allowed by our analysis, a 25% reduction of ch@xmath9 relative to its equilibrium abundance at 900@xmath10k results. depletion of ch@xmath9 at this depth is readily accessible spectroscopically in the 1.6 and 3.3@xmath13 m bands, and may also affect the 2.3@xmath13 m band if the non - equilibrium co abundance persists at higher levels (fig. after h@xmath8o, ch@xmath9 is the most important near infrared molecular absorber in gl 229b. accurate modeling of the spectrum demands a careful treatment of the non - equilibrium ch@xmath9 abundance. similarly, solid boxes show the nh@xmath3 abundance determined from our @xmath1 and @xmath2 band spectra. while the @xmath1 band abundance is in excellent agreement with the equilibrium value, there is a clear depletion of nh@xmath3 in the @xmath2 band. the @xmath1 and @xmath2 band abundances are marginally consistent with each other but it appears that the nh@xmath3 abundance decreases upwards through the atmosphere. processes which take place faster than the time scale of key chemical reactions can drive the composition of the mixture away from equilibrium. the case of co / ch@xmath9 chemistry has been well - studied in the atmosphere of jupiter where an overabundance of co is also observed. carbon monoxyde is a strongly bound molecule and the conversion of co to ch@xmath9 through the (schematic) reaction @xmath153 proceeds relatively slowly, while the reverse reaction is much faster. vertical transport, if vigorous enough, can carry co - rich gas from deeper levels upwards faster than the co to ch@xmath9 reaction can take place. the co / ch@xmath9 ratio at any level is fixed (`` quenched '') by the condition @xmath154 where @xmath155 is the chemical reaction time scale and @xmath156 the dynamical transport time scale. wherever @xmath157 in the presence of a vertical gradient in the equilibrium abundance, non - equilibrium abundances will result. as discussed by @xcite, @xcite, @xcite (and references therein), this naturally explains the very high co abundance observed at the 900@xmath10k level. in this picture, co - rich gas would be carried upward from @xmath158k. convection is the most obvious form of vertical transport in a stellar atmosphere but in gl 229b the convection zone remains about 3 pressure scale heights below the level where co is observed (shaded area in fig. perhaps convective overshooting can transport co to the observed level. @xcite propose `` eddy diffusion '' as a slower, yet adequate transport mechanism. the eddy diffusion (or mixing) time scale is constrained by the poorly known co abundance and the somewhat uncertain chemical pathway between co and ch@xmath9. from @xcite, we infer that @xmath159 to 10 years and could be much smaller. in analogy to the co / ch@xmath9 equilibrium, a low nh@xmath3 abundance can be explained by vertical transport which can quench the nh@xmath3/n@xmath8 ratio at a value found in deeper layers in the atmosphere. as it is carried upward, n@xmath8 is converted to nh@xmath3 by the reaction @xmath160 the n@xmath8 molecule is very strongly bound, however, and this reaction proceeds extremely slowly at low temperatures, much more slowly than reaction (2). thermochemical kinetic calculations of the chemical lifetime for conversion of n@xmath8 to nh@xmath3 were performed as described in @xcite. the time scale for reaction (4) along the @xmath71 profiles of models a, b, and c and for @xmath161bar is given by @xmath162 where @xmath155 is in year, @xmath163 is the temperature in k, and @xmath27 is the surface gravity in cm / s@xmath164. this time scale assumes that the n@xmath8 conversion occurs in the gas phase although it could possibly be shortened by catalysis on the surface of grains. the time scale increases very steeply with decreasing temperature. at the level probed in the @xmath1 band, @xmath165 and @xmath166 and in the @xmath2 band, @xmath167 and @xmath168! at some intermediate level (which depends on @xmath27), the time scale for the conversion of n@xmath8 into nh@xmath3 becomes longer than the age of gl 229b. in view of the relatively very short mixing time scale inferred from the co abundance, it follows that the nh@xmath3 abundance in the @xmath1 and @xmath2 bands is _ entirely _ determined by non - equilibrium processes, _ not _ by reaction (4). at depths where @xmath169 (which corresponds to the top of the convection zone), @xmath170yr and the reaction proceeds fast enough to establish chemical equilibrium between nh@xmath3 and n@xmath8. we therefore expect that the n@xmath8/nh@xmath3 ratio will be quenched at its value at @xmath171, throughout the remainder of the atmosphere. while the time scale for eddy diffusion increases in the upper levels of the atmosphere, the extremely long @xmath155 ensures that the nh@xmath3/n@xmath8 ratio remains unchanged, regardless of how slowly the vertical mixing proceeds. for model b, this corresponds to @xmath172, in perfect agreement with the abundance found in the @xmath1 band (fig. 18b). while the abundance of nh@xmath3 determined from the @xmath2 band spectrum is not very precise, it is marginally consistent with vertical mixing figure 18 shows that it is more likely that the @xmath2 band abundance is smaller than found in the @xmath1 band, however. on the other hand, the abundances shown in fig. 18 and table 2 are depletion factors which were applied uniformly to the chemical equilibrium abundance profile of nh@xmath3, which has a large vertical gradient. for consistency with the mixed atmosphere picture, we have therefore redetermined nh@xmath3 abundances using a constant abundance throughout the atmosphere and found @xmath173 and @xmath174 from the @xmath2 and @xmath1 band spectra, respectively (model b). the former is in good agreement with our simple prediction while the @xmath1 band value is now rather high. the discrepancy between the @xmath1 and @xmath2 band results thus persists in this new analysis. changes in [m / h] within the @xmath70 uncertainty have little effect either. we consider the possibility that this vertical gradient in the nh@xmath3 abundance may be caused by a different non - equilibrium process such as the photolysis of nh@xmath3 by the uv flux from the primary star. ammonia is a relatively fragile molecule which is easily dissociated by uv photons : @xmath175 with a photodissociation cross section of @xmath176cm@xmath164 per molecule, optical depth unity for the photodissociation of nh@xmath3 is reached at pressures of a few millibars in gl 229b. photodissociation of the much more abundant h@xmath8 molecules does not effectively shield nh@xmath3 from incoming uv photons since the two molecules absorb over different wavelength ranges. photodissociation of nh@xmath3 therefore represents a net sink of nh@xmath3 which occurs at the very top of the atmosphere. we can estimate a lower limit to the time scale of photodissociation of nh@xmath3 by assuming that each photodissociating photon results in the destruction of a nh@xmath3 molecule. the incident photon flux is @xmath178 where photons causing dissociation of nh@xmath3 are between @xmath179 and @xmath180, @xmath181 is the flux at the surface of the primary star, @xmath182 the radius of the primary star and @xmath79 the separation of the binary system. for nh@xmath3, we have @xmath183 and @xmath184 (moses, priv. comm.). the primary star has a dm1 spectral type, with @xmath185erg@xmath10@xmath186s@xmath88@xmath46 and @xmath187 cm @xcite. the binary separation is @xmath188au @xcite. this results in @xmath189@xmath186s@xmath88. photodissociation will affect significantly the nh@xmath3 abundance when @xmath190 where @xmath84 is the column density of nh@xmath3. because the incident flux of uv photons is fairly low, this condition is satisfied only at pressures below a few microbars, _ i.e. _ very high in the atmosphere. in the region of interest, photodissociation destroys a very small fraction of nh@xmath3 during one mixing time scale and therefore has little effect on the abundance of nh@xmath3. photolysis of nh@xmath3 can not explain the relatively low nh@xmath3 abundance we find in the @xmath2 band spectrum. we believe that the difference between the @xmath1 and @xmath2 band determinations of the nh@xmath3 abundance arise from the limitations of the nh@xmath3 opacity data used for the calculation of the synthetic spectra. as discussed above, the incompleteness of the nh@xmath3 line list for @xmath66k results in upper limits for the nh@xmath3 abundances obtained by fitting the data. since this effect increases with temperature, we expect that the abundance determined from the @xmath1 band spectrum is overestimated relative to the one obtained from the @xmath2 band spectrum, which is what we observe. until nh@xmath3 opacities become available for @xmath191 1200@xmath10k, we will not be able to quantify this effect. figures 6 and 18 show that ammonia offers a third window of opportunity for a determination of its abundance. the region between 8.3 and 14.4@xmath13 m is rich in strong nh@xmath3 features, the two strongest being at 10.35 and 10.75@xmath13 m (fig. this spectral region probes a higher level in the atmosphere (@xmath192k) where the hot bands of nh@xmath3 which are missing from opacity data bases are less problematic than in the @xmath1 and @xmath2 bands. in this spectral region, the spectrum is very sensitive to the nh@xmath3/h@xmath8o ratio, especially for nh@xmath3 abundances below 25% of the equilibrium value (fig. this would allow a good determination of the degree of nh@xmath3 depletion in the upper levels of the atmosphere. we anticipate that 10@xmath13 m spectroscopy should reveal a nh@xmath3 abundance of @xmath193% of its equilibrium value. Conclusion with the availability of extensive photometric, astrometric, and spectroscopic data, our picture of the atmosphere of gl 229b is gradually becoming more exotic and more complex. the initial discovery of ch@xmath9 in its spectrum set it appart and has prompted the creation of a new spectral class, the t dwarfs. h@xmath8o, co, cs i, and k i have also been detected. there is good evidence that the rapid decrease of the flux at visible wavelengths is caused by unprecedently broad lines of atomic alkali metals @xcite. the presence of condensates may also play a role in shaping the spectrum of gl 229b. it is unfortunate that the surface gravity of gl 229b remains poorly constrained. we have not been able to further restrict the allowed range with our new @xmath0, @xmath1, and @xmath2 spectroscopy. as a result, all our results are expressed as a function of gravity. this is the most significant obstacle to further progress in elucidating the astrophysics of this t dwarf. the surface gravity can probably be determined from the study of the pressure broadened shape of molecular lines. we have found good evidence for the presence of nh@xmath3 in the spectrum of gl 229b, which was expected from chemical equilibrium calculations. we have been able to determine its abundance at two different levels in the atmosphere, and we find a significant deviation from chemical equilibrium. a similar situation has been found with co previously @xcite and this abundance pattern can be explained by vertical mixing in the atmosphere. the extent of the convection zone is not sufficient to account for the abundances we find and the mixing may be due to overshooting or to less efficient eddy diffusion. we find that nh@xmath3 photolysis is not important in shaping the spectrum of gl 229b. because nh@xmath3 can be observed in three different bands corresponding to three distinct depths in the atmosphere, an accurate determination of its abundance in each band provides information on the time scale of mixing as a function of depth. this is an unusual and powerful diagnostic tool which can provide valuable clues for modeling the vertical distributions of possible condensates. in principle, any absorber with a large abundance gradient through the visible part of the atmosphere can be used to infer the details of the mixing process. among detected and abundant molecules, only co and nh@xmath3 satisfy this criterion. chemical equilibrium calculations with rainout of condensates @xcite show that we can expect significant vertical gradients in the abundances of atomic k, rb, cs, and na as they become bound in molecules (kcl, rbcl, cscl and na@xmath8s, respectively) in the cooler, upper reaches of the atmosphere. cesium and potassium have been detected in the spectrum of gl 229b, and resonance doublets of k i and na i appear to shape the visible spectrum. however, the chemical timescales for alkali metals are so short that they should always remain in thermodynamic equilibrium (lodders 1999). therefore, they can not serve as probes of vertical mixing in gl 229b. further progress in understanding the atmosphere of gl 229b requires better opacities for ch@xmath9 and nh@xmath3, and, to a lesser extent, of h@xmath8o. a more accurate determination of the co abundance from 4 5@xmath13 m spectroscopy is very desirable and will require higher signal - to - noise spectroscopy than is currently available. similarly, 10@xmath13 m spectroscopy to determine the nh@xmath3 abundance for @xmath194bar, while difficult, is important. the issue of vertical mixing and departures from chemical equilibrium gains importance when we consider that the observed departure of co from chemical equilibrium implies a significantly reduced ch@xmath9 abundance, by conservation of the abundance of elemental carbon. similarly, our results imply that nh@xmath3 absorption in the 10@xmath13 m region is reduced. because ch@xmath9 is a significant absorber in the near infrared, as is nh@xmath3 in the 10@xmath13 m range, departures from equilibrium must be taken into account when accurate modeling of the atmosphere and spectrum of gl 229b is desired. this new level of complexity compounds the exoticism and the challenges posed by t dwarfs. the astrophysics of gl 229b is far richer than has been originally anticipated. gl 229b is currently the only t dwarf known to be in a binary system. there is no evidence that the illumination from the primary star has a significant effect on the state of its atmosphere and gl 229b is most likely typical of isolated t dwarfs. it remains the brightest and by far the best studied of the seven t dwarfs currently known, but the list should expand to several dozens during the next 2 3 years @xcite. the existing body of work on gl 229b points to the most rewarding observations to conduct on t dwarfs. the possibility of studying trends in the physics of t dwarf atmospheres as a function of effective temperature is a fascinating prospect. we thank t. guillot for sharing programs which were most useful to our analysis, j. moses for invaluable information regarding the photolysis of nh@xmath3 in giant planets, and k. noll and b. oppenheimer for sharing their data. we are grateful to the staff at the united kingdom infrared telescope, which is operated by the joint astronomy center hawaii on behalf of the uk particle physics and astronomy research council. this work was supported in part by nsf grants ast-9318970 and ast-962487 and nasa grants nag5 - 4988 and nag5 - 4970. work by b. fegley and k. lodders is supported by grant nag5 - 6366 from the nasa planetary atmospheres program. allard, f., hauschildt, p.h., baraffe, i. & chabrier, g. 1996,, 465, l123 allard, f. 1998, in brown dwarfs and extrasolar planets, asp. series vol 134, rebolo, r., martin, e.l. and zapatero osorio, m.r., eds., p370 allard, f., alexander, d.r., tamanai, a. & hauschildt, p.h. 1998, in brown dwarfs and extrasolar planets, asp. series vol 134, rebolo, r., martin, e.l. and zapatero osorio, m.r. anders, e., & grevesse, n. 1989, geo. chem. 53, 197 bohren, c.f. & huffman, d.r. 1983, absorption and scattering of light by small particles, (wiley : new york) burgasser, a.j., kirkpatrick, j.d., brown, m.e., reid, i.n., gizis, j.e., dahn, c.c., monet, d.g., beichman, c.a., liebert, j., cutri, r.m., & skrutskie, m.f. 1999,, 552, l65 burrows, a., marley, m.s., hubbard, w.b., lunine, j.i., guillot, t., saumon, d., freedman, r.s., sudarsky, d., & sharp, c. 1997,, 491, 856 burrows, a., hubbard, w.b., lunine, j.i., marley, m.s., & saumon, d. 1999a, in protostars and planets iv, v., mannings, a. boss, s. russell, eds., (university of arizona press : tucson) in press. burrows, a., marley, m.s., & sharp, c. 1999b, submitted to, astro - ph9908078 burrows, a., & sharp, c.m. 1999,, 512, 843 fegley, b. jr., & lodders, k. 1994, icarus, 110, 117 fegley, b. jr., & lodders, k. 1996,, 472, l37 fegley, b. jr., & prinn, r.g. 1988,, 324, 621 geballe, t.r., kulkarni, s.r., woodward, c.e. & sloan, g.c. 1996,, 467, l101 golimowski, d.a., burrows, c.j., kulkarni, s.r., oppenheimer, b.r., & brukardt, r.a. 1998,, 115, 2579 grevesse, n., & noels, a. 1993, in origin and evolution of the elements, prantzos et al. (cambridge university press : cambridge), p15. griffith, c.a., yelle, r.a. & marley, m.s. 1998, science, 282, 2063 griffith, c.a. & yelle, r.a. 1999,, 519, l85 guillot, t. 1999, science, 286, 72 guillot, t., saumon, d., burrows, a., hubbard, w.b., lunine, j.i., marley, m.s., & freedman, r.s. 1996, in _ `` astronomical and biochemical origins and the search for life in the universe, '' _ eds. cosmovici, s. bowyer and d. werthimer, editrice compositori, 343 husson, n., bonnet, b., scott, n.a., & chedin, a. 1994,, 48, 509 hubbard, w.b., burrows, a., lunine, j.i., & saumon, d. 1995, in elementary processes in dense plasmas, s. ichimaru, and s. ogata, eds., (addison - wesley : reading), p227 jones, h.r.a. & tsuji, t. 1997,, 480, l39 karkoschka, e. 1994, icarus, 111, 174 kirkpatrick, j.d., reid, i.n., liebert, j., cutri, r.m., nelson, b., beichman, c.a., dahn, c.c., monet, d.g., gizis, j.e. & skrutskie, m.f. 1999,, 519, 834 leggett, s.k. 1992,, 82, 351 leggett, s.k., allard, f. berriman, g., dahn, c.c. & hauschildt, p.h. 1996,, 104, 117 leggett, s.k., toomey, d.w., geballe, t.r. & brown, r.h. 1999,, 517, l139 liebert, j., reid, i.n., burrows, a., burgasser, a.j., kirkpatrick, j.d. & gizis, e. 2000,, in press (astro - ph/0003068) lodders, k. 1999,, 519 793 marley, m.s., saumon, d., guillot, t., freedman, r.s., hubbard, w.b., burrows, a., & lunine, j.i. 1996, science, 272, 1919 matthews, k., nakajima, t., kulkarni, s.r. & oppenheimer, b.r. 1996,, 112, 1678 mountain, c.m., robertson, d.j., lee, t.j. & wade, r. 1990, proc. spie, 1235, 25 nakajima, t., oppenheimer, b.r., kulkarni, s.r., golimowski, d.a., matthews, k., & durrance, s.t. 1995,, 378, 463 noll, k.s., geballe, t.r. & marley, m.s. 1997,, 489, l87 oppenheimer, b.r., kulkarni, s.r., matthews, k. & nakajima, t. 1995, science, 270, 1478 oppenheimer, b.r., kulkarni, s.r., matthews, k. & van kerkwijk, m.h. 1998,, 502, 932 partridge, h. & schwenke, d.w. (1997) j. chem. 106, 4618 perryman et al. 1997,, 323, l49 rothman, l.s., 1998,, 60, 665 schiavon, r.p., barbuy, b., & singh, p.d. 1997,, 484, 499 schultz, a.b., 1998,, 492, l181 strauss, m.a. et al. 1999,, 552, l61 strong, k., taylor, f.w., calcutt, s.b., remedios, j.j. & ballard, j. 1993, jqsrt, 50 363 tsuji, t., ohnaka, k., & aoki, w. 1996a,, 305, l1 tsuji, t., ohnaka, k., aoki, w. & nakajima, t. 1996b,, 308, l29 tsuji, t., ohnaka, k., & aoki, w. 1999,, 520, l119 19980125 & 2.10 - 2.18 & 3200 & 80 & 1.37 & 1.39 + 19980125 & 2.02 - 2.10 & 3100 & 93 & 1.40 & 1.35 + 19980126 & 1.53 - 1.61 & 2350 & 53 & 1.46 & 1.39 + 19980126 & 1.25 - 1.30 & 3000 & 53 & 1.33 & 1.32 + 19980126 & 1.20 - 1.25 & 2900 & 53 & 1.41 & 1.40 + 19980127 & 2.02 - 2.10 & 3100 & 93 & 1.36 & 1.32 + ccrccccccrr a & 4.5 & 870 & @xmath40 & @xmath195 & @xmath93 & @xmath40 & & 0 & 0.25 & @xmath196 + & & & & & @xmath197 & & & 0.25 & 0.5 & + & & & & & @xmath198 & & & 0.5 & 1 & + b & 5.0 & 940 & @xmath93 & @xmath199 & @xmath200 & @xmath93 & & 0 & @xmath201 & @xmath202 + & & & & & @xmath203 & & & 0.25 - 0.5 & 0.5 & + & & & & & @xmath204 & & & @xmath205 & 1 & + c & 5.5 & 1030 & @xmath203 & 0.1 & 0.0 & @xmath200 & & 0.5 & 0.5 & @xmath202 + & & & & & @xmath206 & & & 1 & 1 & + ccrcccc a & 4.5 & 870 & @xmath40 & @xmath210 & @xmath211 & 2.25 + b & 5.0 & 940 & @xmath93 & @xmath212 & @xmath213 & 2.26 + c & 5.5 & 1030 & @xmath200 & @xmath214 & @xmath215 & 2.17 + & @xmath216 & @xmath217 & @xmath218 +

we present new, high resolution, infrared spectra of the t dwarf gliese 229b in the @xmath0, @xmath1, and @xmath2 bandpasses. we analyze each of these as well as previously published spectra to determine its metallicity and the abundances of nh@xmath3 and co in terms of the surface gravity of gl 229b, which remains poorly constrained . the metallicity increases with increasing gravity and is below the solar value unless gl 229b is a high - gravity brown dwarf with @xmath4. the nh@xmath3 abundance is determined from both the @xmath1 and the @xmath2 band spectra which probe two different levels in the atmosphere. we find that the abundance from the @xmath2 band data is well below that expected from chemical equilibrium, which we interpret as strong evidence for dynamical transport of nh@xmath3 in the atmosphere. this is consistent with the previous detection of co and provides additional constraints on the dynamics of the atmosphere of this t dwarf. # 1 10^#1 # 1 # 1(#1_1, ,#1_n) # 1d#1_1 d#1_n 1nu^n_1 # 1#1 _ 0 2h2 _ 2 _ # 1#2 # 1_k_#1 # 1

astro-ph0003353

Introduction densely - grafted chains on nonadsorbing substrate surfaces form the so - called `` polymer brush''@xcite. these systems find various important applications @xcite, e.g. as lubricants @xcite, for colloid stabilization @xcite, for tuning of adhesion and wetting properties @xcite, for improving the biocompatibility of drugs @xcite, as protective coatings preventing protein adsorption (`` nonfouling '' surfaces) in a biological milieu @xcite, microfluidic chips for biomolecule separation @xcite, etc. the theoretical description of the conformations of macromolecules in these polymer brushes and their dynamics has been an active topic of research hitherto (e.g., @xcite ; for early reviews see @xcite. also the interaction of the brushes with either the solvent molecules (e.g. @xcite) or globular proteins @xcite and/or other nanoparticles (e.g., @xcite) has found much recent attention. however, in many situations of interest there will also occur free polymer chains in the solution, interacting with the polymers of the brush. this interaction has received relatively less attention, apart from the case where a polymer brush interacts with a dense polymer melt @xcite. the latter case is particularly interesting because there is very little interpenetration of the grafted chains in the melt and the free chains in the brush even if their chemical nature is identical (`` wetting autophobicity '' @xcite). in contrast, scaling theory @xcite, self - consistent field @xcite and simulation @xcite have predicted partial penetration of free chains into moderately dense brushes of identical chemical nature in semi - dilute solutions when the monomer volume fraction in solution approaches that of the brush, and this behavior has been confirmed experimentally @xcite. of course, when the polymer solution is very dilute, the brush provides a free energy barrier for penetration of free chains into it and this limits the grafting density that can be achieved when one prepares a brush by grafting chains from solution @xcite (see also some attempts to model this process by simulations @xcite). similarly, since typically the energy won by the chain end when it gets grafted is of the order of @xmath22 only @xcite, there is a nonzero probability that brush chains get released from the grafting substrate surface and are subsequently expelled from the brush @xcite. however, most cases studied so far refer to the situation that (apart from chain end effects) the chains in the bulk and those in the solution are identical. it is interesting, therefore, to consider the more general situation when the grafted chains and those in the bulk differ in their chemical nature. then the problem of compatibility (traditionally modelled by introducing a flory - huggins @xmath1-parameter @xcite) between the two types of chains arises. then, there is also no reason to assume that the length @xmath23 of the grafted chains, and the length @xmath0 of the free chains are equal. such situations (in particular, when the grafted and the free chains attract each other, @xmath24) are of great interest for modern applications such as protein adsorption, antifouling surfaces @xcite, etc. however, to the best of our knowledge, no systematic study of the effects of the various parameters (@xmath25 and monomer concentration of the free chains @xmath26) on the amount of absorption and the penetration kinetics has been reported so far. the present paper presents simulation and density functional theory (dft) results in an effort to fill this gap. in section [model] we describe the model and comment on some simulation aspects ; section [theory] summarizes our theoretical approach which includes both static and dynamic versions (ddft) of dft as well as self - consistent field theory (scft) calculations. the numerical results are described in section [results_sec] while section [summary] contains a summary and discussion. at grafting density @xmath27 and free chains of length @xmath28 at equilibrium : (left) absorption at @xmath29, and (right) expulsion at @xmath30. [snapshots_fig],title="fig : "] at grafting density @xmath27 and free chains of length @xmath28 at equilibrium : (left) absorption at @xmath29, and (right) expulsion at @xmath30. [snapshots_fig],title="fig : "] Model and simulation aspects we have used a coarse grained off - lattice bead spring model@xcite to describe the polymer chains in our system. as far as for many applications in a biological context rather short grafted chains are used @xcite, we restrict ourselves to length @xmath31 of the grafted chains. the polymer brush consists of linear chains of length @xmath23 grafted at one end to a flat structureless surface. the effective bonded interaction is described by the fene (finitely extensible nonlinear elastic) potential, @xmath32\]] with @xmath33. thus the equilibrium bond length between nearest neighbor monomers is @xmath34. here and in what follows we use the maximal extension of the bonds,@xmath35, as our unit length while the potential strength is measured in units of thermal energy @xmath36 where @xmath37 is the boltzmann constant. the nonbonded interactions between brush and free chain segments are described by the morse potential, @xmath38 - 2\exp[-\alpha(r - r_{min})]\;,\]] with @xmath39 standing for the strength of brush - brush, @xmath40, polymer - polymer, @xmath41, and brush - polymer, @xmath42 interactions. in our present study we take typically @xmath43, @xmath44 (that is, in the good solvent regime with only excluded volume interactions). for @xmath45 the free chains are absorbed in the brush whereas for @xmath46 the polymer brush ejects them into the bulk. note that we define here the compatibility parameter @xmath1 simply as @xmath47, and do not include the coordination number (which is done when one uses lattice models @xcite. the size of the container is @xmath48. the polymer chains are tethered to grafting sites which constitute a triangular periodic lattice on the substrate whereby the closest distance between grafting sites is @xmath35. thus the largest grafting density @xmath49 involves @xmath50 brush segments, if the polymer chains are anchored at distance @xmath35, and @xmath51, i.e., @xmath52 brush segments, if the lattice constant, i.e., the distance between adjacent head monomers on the surface is equal to @xmath53. note that @xmath54 corresponds to a simulation where the monomer density in the brush near the wall is close to the density of a polymer melt while @xmath55 would correspond to a rather concentrated polymer solution. for the chain model, @xmath56 corresponds to good solvent conditions since the theta - point for a (dilute) solution of polymers described by the model, eqs. [fene]-[morse] has been estimated@xcite as @xmath57. in all our simulations we use brushes formed by polymer chains consisting of @xmath31 effective monomers whereas the number of free chains @xmath58 of length @xmath0 (where @xmath0 spans the interval @xmath59) is taken such that the total number of free chain segments remains constant and is equal to @xmath60. for a certain length @xmath61, however, we also change the concentration of free chains in the container by varying their number @xmath58 in the interval @xmath62. thus, the volume fraction @xmath63 of @xmath64-free chains is varied between @xmath65. note that, as usual, solvent molecules are not explicitly included @xcite but work which includes solvent explicitly @xcite would yield very similar results. for a dense brush with polymer chains of lengths @xmath31 statistical averages were derived from typically @xmath66 monte carlo steps (mcs) per monomer. the monte carlo algorithm consists of attempted moves whereby a monomer is chosen at random and one attempts to displace it to a new randomly chosen position @xmath67 regarding the old position. we use periodic boundary conditions in the @xmath68 directions and impenetrable hard walls in the @xmath69 direction. two typical configurations of the polymer brush with free chains of length @xmath28, are shown in fig. [snapshots_fig] for the case of good, @xmath29, and poor, @xmath30, compatibility with the polymer brush. Theory we employ classical dft to compute density profiles of free and grafted polymer chains. theory has been discussed in detail in previous publications, so here we briefly summarize its most important aspects. the starting point of the dft treatment is the expression for the grand free energy, @xmath70, as a functional of the density profiles of free and grafted chains, @xmath71 and @xmath72, respectively (@xmath73, where @xmath74 are the positions of the chain segments). the functional @xmath70 is related to the helmholtz free energy functional, @xmath75, via a legendre transform:@xcite @xmath76= f[\phi_o({\textbf}{r}_o),\phi_p({\textbf}{r}_p)] + \sum_{\alpha = o, p}\int d{\textbf}{r}_{\alpha } \phi_{\alpha}({\textbf}{r}_{\alpha})v_{\alpha}({\textbf}{r}_{\alpha }), \label{omega}\]] where @xmath77 is the external field, which in the present case is due to the hard - sphere like interaction of the polymer segments with the hard wall, @xmath78, where @xmath79 for @xmath80 and @xmath81 otherwise, with analogous expression holding for @xmath82. additionally, the innermost (@xmath83) bead of each grafted chain is tethered to the wall via a grafting potential @xmath84=\delta(z_1)$], where @xmath85. note that the chemical potential of both free and grafted chains is absent from the second term of eq. ([omega]) because the dft calculations are performed at a fixed number of both free and grafted segments in order to mimic the mc simulations : @xmath86 and @xmath87. in the above, @xmath88 is the box length and @xmath89 is the wall area. the helmholtz free energy functional is separated into ideal and excess parts,@xcite with the former given by : @xmath90= \sum_{\alpha = o, p}\left\ { \int d{\textbf}{r}_{\alpha } \phi_{\alpha}({\textbf}{r}_{\alpha }) [\ln \phi_{\alpha}({\textbf}{r}_{\alpha})-1] + \beta \int d{\textbf}{r}_{\alpha } \phi_{\alpha}({\textbf}{r}_{\alpha }) v_b({\textbf}{r}_{\alpha})\right\}. \label{fideal}\]] where the bonding energy @xmath91 for the grafted chains is taken as follows : @xmath92= \prod_{i=1}^{n-1}\frac{\delta(|{\textbf}{r}_i-{\textbf}{r}_{i+1}|-b_l)}{4\pi b_{l}^{2 } }, \label{vbond}\]] with a similar expression for free chains, with @xmath23 replaced by @xmath0. this bonding potential constrains adjacent segments to a fixed separation @xmath93. the excess part of the helmholtz free energy is written as a sum of repulsive (hard chain) and attractive terms, with the former computed in the weighted density approximation and the latter obtained within mean - field approach, using eqs. (12)-(17) from ref. @xcite ; for the sake of brevity we do not reproduce these equations here. the minimization of the grand free energy functional with respect to @xmath72 yields the equilibrium density distribution for the grafted chains which can be integrated over grafting and bonding delta - functions to obtain the following result for the density profile of the @xmath94th segment of the grafted chains:@xcite @xmath95 where @xmath96, \label{ipz}\]] with @xmath97 the two propagators in eq. ([phipi]), @xmath98 and @xmath99 move from the free (@xmath100) and the tethered (@xmath83) ends of the chain, respectively. they are computed via recursive relations given by eqs. (23)-(25) of ref. @xcite. the normalization constant @xmath101 in eq. ([phipi]) is chosen to ensure that the @xmath94th segment density profile is normalized to @xmath102. the total segment density profile for the grafted chains is given by : @xmath103 the equilibrium density profile for the segments of the free chains can be obtained in a similar way, by minimizing the grand free energy functional with respect to @xmath71 and integrating over bond - length constraining delta functions. the dft equations described above are solved simultaneously to obtain the segment density profiles for free and grafted chains. the equations are solved iteratively using picard algorithm,@xcite with the step size along the @xmath69 coordinate taken to be 0.0325. the above procedure yields equilibrium segment density profiles for a given set of interaction potentials. in addition to the equilibrium structural properties, we have also studied the kinetics of the adsorption of free chains into the brush, following a switch of the interaction potential between free and grafted segments from repulsive to attractive. to this end, we have employed the ddft method, which is a dynamical generalization of the dft approach.@xcite mc simulations have indicated that the segment density profiles of the grafted chains are essentially independent of the strength of the attraction between free and grafted segments. accordingly, in our ddft calculations we take @xmath104 to be time independent and focus on the time dependence of the free chain density, @xmath105. the time evolution of the segment density profile of free chains is given by the following equation:@xcite @xmath106 where @xmath107 is the non - equilibrium local chemical potential, and dimensionless time @xmath19 is defined according to @xmath108, where @xmath109 is the mobility coefficient. initial density profile @xmath110 corresponds to the equilibrium distribution of free chains at a repulsive brush, i.e. @xmath30. at @xmath111, the brush - free polymer attraction is instantaneously `` switched on '', i.e. we set @xmath112. the time - dependent polymer density profile is then propagated according to the eq. ([phipzt]), with the time - dependent chemical potential given by : @xmath113 where @xmath114 is obtained by substituting the time - dependent density @xmath105 into the expression for @xmath115 (and likewise for the propagators @xmath116 and @xmath117. we solve eq. ([phipzt]) using crank - nicholson scheme.@xcite note that eq. ([phipzt]) has the form of a continuity equation with the flux (current density) given by @xmath118. the fact that the ddft method propagates @xmath105 via a continuity - type equation guarantees the conservation of the total number of segments in the system, which is consistent with the simulation set - up. in order to compare the results of the ddft approach with kinetic mc data, we set the mobility coefficient @xmath109 equal to unity and adjust the conversion factor between ddft dimensionless time and kinetic mc number of steps for one particular set of parameters @xmath0 and @xmath26. comparisons for all other values of @xmath0 and @xmath26 are performed using the same conversion factor, while assuming @xmath109 to be inversely proportional to both @xmath0 and @xmath26. with the goal of shedding further light on the thermodynamic aspects of the adsorption process, we have also performed self - consistent field theory (scft) calculations of the structural properties as a function of the interaction strength between the segments of the brush and the free chains (in scft approach this interaction is characterized by the parameter @xmath1 which is calculated in the standard fashion from the corresponding potential well - depths : @xmath119). the main motivation behind carrying out scft calculations is the fact that this approach provides a more straightforward way to decompose the free energy into entropic and energetic components, thereby providing a complementary (to dft) view of the adsorption process. the basic equations of the scft method are well known,@xcite and will not be reproduced here for the sake of brevity. once again, the density profiles for free and grafted chains are written in terms of the propagators, the only major difference from the dft approach being that instead of the equation of state, one employs the incompressibility constraint to set up the equations for the density profiles, which are once again solved iteratively using picard s method. for example, the equation for the density profile of the grafted chain segments takes the form : @xmath120 where the normalization constant @xmath121 is obtained from the grafting density @xmath102, and @xmath122, with @xmath123. the hard core potential @xmath124 is independent of the segment type and serves as a lagrange multiplier enforcing the incompressibility condition, meaning that the lattice space is completely filled and no segment overlap occurs. the density profile of the free chain segments is obtained in a similar way. once the profiles are calculated, one can easily obtain excess entropy and energy of the free chains (relative to pure unmixed components) as follows @xcite : @xmath125 @xmath126 where @xmath127 is the bulk volume fraction of free chains. Results in fig. [phi_l_fig] we show the density profiles of the free chains, @xmath128, of length @xmath0 for an attractive, @xmath29, and a neutral, @xmath129, brush along with the monomer density profile of the brush itself, @xmath104. , (shaded area) and of free chains, @xmath128, (thick lines) of length @xmath0, (given as parameter) at two grafting densities : @xmath27 (upper row), and @xmath130 (lower row). (a) and (c) illustrate good compatibility between brush and free chains, @xmath131 while (b) and (d) demonstrate a case of bad compatibility, @xmath129. thin solid lines in (a) and (b) denote results from the dft calculation. the densities in (a) are normalized so as to reproduce the correct ratio of brush to free chains concentrations @xmath132 and @xmath26 (the absolute particle concentration @xmath133 is indicated in the alternative @xmath134axis. for the sake of better visibility, in (b), (c), and (d) the density of all species is normalized to unit area. [phi_l_fig],title="fig : "], (shaded area) and of free chains, @xmath128, (thick lines) of length @xmath0, (given as parameter) at two grafting densities : @xmath27 (upper row), and @xmath130 (lower row). (a) and (c) illustrate good compatibility between brush and free chains, @xmath131 while (b) and (d) demonstrate a case of bad compatibility, @xmath129. thin solid lines in (a) and (b) denote results from the dft calculation. the densities in (a) are normalized so as to reproduce the correct ratio of brush to free chains concentrations @xmath132 and @xmath26 (the absolute particle concentration @xmath133 is indicated in the alternative @xmath134axis. for the sake of better visibility, in (b), (c), and (d) the density of all species is normalized to unit area. [phi_l_fig],title="fig : "] +, (shaded area) and of free chains, @xmath128, (thick lines) of length @xmath0, (given as parameter) at two grafting densities : @xmath27 (upper row), and @xmath130 (lower row). (a) and (c) illustrate good compatibility between brush and free chains, @xmath131 while (b) and (d) demonstrate a case of bad compatibility, @xmath129. thin solid lines in (a) and (b) denote results from the dft calculation. the densities in (a) are normalized so as to reproduce the correct ratio of brush to free chains concentrations @xmath132 and @xmath26 (the absolute particle concentration @xmath133 is indicated in the alternative @xmath134axis. for the sake of better visibility, in (b), (c), and (d) the density of all species is normalized to unit area. [phi_l_fig],title="fig : "], (shaded area) and of free chains, @xmath128, (thick lines) of length @xmath0, (given as parameter) at two grafting densities : @xmath27 (upper row), and @xmath130 (lower row). (a) and (c) illustrate good compatibility between brush and free chains, @xmath131 while (b) and (d) demonstrate a case of bad compatibility, @xmath129. thin solid lines in (a) and (b) denote results from the dft calculation. the densities in (a) are normalized so as to reproduce the correct ratio of brush to free chains concentrations @xmath132 and @xmath26 (the absolute particle concentration @xmath133 is indicated in the alternative @xmath134axis. for the sake of better visibility, in (b), (c), and (d) the density of all species is normalized to unit area. [phi_l_fig],title="fig : "] our mc simulation results indicate that at fixed segment concentration, @xmath133, the brush profile, @xmath104, is virtually insensitive to @xmath0, whereupon we keep only one such profile in the graphs. the most striking feature which may be concluded from fig. [phi_l_fig] is, somewhat counter - intuitively, the strong increase of absorption with growing length of the absorbed free chains @xmath0. evidently, both at moderate, @xmath27, and high, @xmath130, grafting density, the longer polymers are entirely placed , (shaded area) and of free chains, @xmath128, (thick lines) of length @xmath61 for different free chain concentration (number of free chains @xmath135) (given as parameter) at two grafting densities : @xmath27 (upper row), and @xmath130 (lower row). thin lines in (a) denote dft results. the inset in (b) shows the change in the density profile of brush monomers for two values of free chain concentration : @xmath136. in (a), (c) and (d) @xmath132 stays practically constant. [dens_nobst_fig],title="fig : "], (shaded area) and of free chains, @xmath128, (thick lines) of length @xmath61 for different free chain concentration (number of free chains @xmath135) (given as parameter) at two grafting densities : @xmath27 (upper row), and @xmath130 (lower row). thin lines in (a) denote dft results. the inset in (b) shows the change in the density profile of brush monomers for two values of free chain concentration : @xmath136. in (a), (c) and (d) @xmath132 stays practically constant. [dens_nobst_fig],title="fig : "] +, (shaded area) and of free chains, @xmath128, (thick lines) of length @xmath61 for different free chain concentration (number of free chains @xmath135) (given as parameter) at two grafting densities : @xmath27 (upper row), and @xmath130 (lower row). thin lines in (a) denote dft results. the inset in (b) shows the change in the density profile of brush monomers for two values of free chain concentration : @xmath136. in (a), (c) and (d) @xmath132 stays practically constant. [dens_nobst_fig],title="fig : "], (shaded area) and of free chains, @xmath128, (thick lines) of length @xmath61 for different free chain concentration (number of free chains @xmath135) (given as parameter) at two grafting densities : @xmath27 (upper row), and @xmath130 (lower row). thin lines in (a) denote dft results. the inset in (b) shows the change in the density profile of brush monomers for two values of free chain concentration : @xmath136. in (a), (c) and (d) @xmath132 stays practically constant. [dens_nobst_fig],title="fig : "] inside the polymer brush whereas the much more mobile short species @xmath137 remain uniformly distributed in the bulk above the brush end. since some of the absorbing chains with larger @xmath0 get stuck inside the brush, their density profiles could not smoothen sufficiently for the time of the simulation run. therefore, we observe rather large statistical fluctuations in @xmath128. for repulsive brushes all species are largely expelled from the brush whereby the situation is reversed as far as the free chain length @xmath0 is concerned. in the very dense brush @xmath138, the brush profile displays the typical oscillations near the grafting surface suggesting some layering immediately in the vicinity of the grafting wall - fig. [phi_l_fig]c, d. in all graphs one observes pronounced depletion effects at the upper container wall, opposing the brush. however, the inhomogeneity of @xmath128 near the wall at @xmath139 has no effect on @xmath128 in the region of the polymer brush, the flat part of @xmath128 in between the brush and the confining wall at @xmath139 is broad enough to eliminate any finite - size effects associated with the finite linear dimension of the simulation box in @xmath140direction. one should note also the good agreement between simulation and dft results. in fact, the thin lines, indicating the latter, may hardly be distinguished from the monte carlo data (thick lines) in fig. [phi_l_fig]a, b. the only significant discrepancy between theory and simulation is observed in the brush profile in the vicinity of the grafting wall, where dft approach overestimates the oscillations. this discrepancy is likely due to the fact that in the dft method the bond lengths are constraint via delta - functions to a constant value of @xmath141, while in the simulations the bonds are allowed to vibrate under fene potential, eq. ([fene]). for @xmath61, fig. [dens_nobst_fig] shows a qualitatively similar behavior of the density profiles for the cases of gradually increasing free chain concentration (indicated by the number of free chains @xmath142 as parameter). expectedly, for @xmath143 (which corresponds to monomer concentration @xmath144) and @xmath27, the free chains are present in the bulk over the brush as the brush interior is then entirely filled. however, when the brush - free chain attraction increases to @xmath145, the mc data (not shown here) indicate complete absorption of the free chains into the brush with virtually no free chains in the bulk above the polymer brush even at the highest concentration of @xmath146. with increasing grafting density and/or free chain concentration, the agreement between dft and mc deteriorates somewhat, with the theory underestimating the degree of penetration of free chains into the brush (see discussion of fig. [ads_amount_fig] below), which is likely due to the simple tarazona s weighting function employed in our dft calculations. it is well known that at higher densities it would be more appropriate to use weighting functions from the fundamental measure theory.@xcite indeed, precisely such approach has been recently used to study adsorption and retention of spherical particles in polymer brushes @xcite. with polymerization index @xmath0 of the free chains for two grafting densities. empty symbols denote dft results. the case @xmath147 refers to the _ critical _ degree of brush - polymer compatibility (cf. section [crit_sect]). (b) mean squared radius of gyration, @xmath148, and end - to - end distance, @xmath149, parallel and perpendicular, @xmath150 to the grafting plane against length of the free chains @xmath0 at grafting density @xmath27. dashed lines denote the observed slope @xmath151. only for the longest free chains with @xmath61 a marked deviation from the standard scaling behavior may be detected. [rg_fig],title="fig : "] with polymerization index @xmath0 of the free chains for two grafting densities. empty symbols denote dft results. the case @xmath147 refers to the _ critical _ degree of brush - polymer compatibility (cf. section [crit_sect]). (b) mean squared radius of gyration, @xmath148, and end - to - end distance, @xmath149, parallel and perpendicular, @xmath150 to the grafting plane against length of the free chains @xmath0 at grafting density @xmath27. dashed lines denote the observed slope @xmath151. only for the longest free chains with @xmath61 a marked deviation from the standard scaling behavior may be detected. [rg_fig],title="fig : "] next, we present mc and dft results for the absorbed amount of free chains as a function of degree of polymerization and concentration. the absolute absorbed amount is defined as the number of polymer segments located `` inside the brush '', namely, in the region @xmath152, where the cutoff distance @xmath153 is defined in such a way that 99% of the brush segments are located in the region @xmath152. the _ relative _ absorbed amount @xmath154 is defined as the ratio of the absolute absorbed amount to the total number of free chain segments. in fig. [rg_fig]a one may observe the steep increase in @xmath154 with growing polymer length @xmath0 both for brushes with @xmath51 and @xmath3 when @xmath131. indeed, as indicated also in fig. [phi_l_fig], as soon as @xmath155, the adsorbed amount saturates at nearly @xmath156. a much more gradual growth of @xmath154 is found for the _ critical _ attraction @xmath147 (see below). in fig. [rg_fig]a one sees again that dft results for the absorbed amount of polymers as a function of the absorbate polymerization index (shown here for the case of lower grafting density) are in good agreement with mc data, with the exception of the intermediate - length chains, where dft overestimates the adsorbed amount somewhat. with free chain concentration @xmath133 for @xmath61 and two grafting densities : @xmath27 (circles), and @xmath130 (squares). full symbols correspond to polymer absorption with @xmath131 and empty symbols denote expulsion @xmath30. (a) absorbed fraction vs. @xmath133, (b) total number of absorbed monomers against @xmath133. [ads_amount_fig],title="fig : "] with free chain concentration @xmath133 for @xmath61 and two grafting densities : @xmath27 (circles), and @xmath130 (squares). full symbols correspond to polymer absorption with @xmath131 and empty symbols denote expulsion @xmath30. (a) absorbed fraction vs. @xmath133, (b) total number of absorbed monomers against @xmath133. [ads_amount_fig],title="fig : "] especially interesting is the observation, fig. [rg_fig]b, that the conformations of the absorbed chains inside the brush practically do not change with respect to those of the free chains in the bulk - the scaling behavior of the parallel and perpendicular components of the end - to - end (squared) distance @xmath157 and radius of gyration, @xmath158, is demonstrated in logarithmic coordinates by straight lines whereby the value of the flory exponent @xmath159. due to the short lengths of the free chains used here this value is slightly larger than what is expected for very long chains (namely @xmath160). only the absorbed chains that are longer than the polymers of the brush, @xmath161, indicate deviations from the scaling law of single polymers with excluded - volume interactions : the parallel component @xmath162 slightly exceeds, and the perpendicular component, @xmath163, falls below the straight line suggesting that the original shape of the @xmath61 coil flattens parallel to the grafting plane. fig. [ads_amount_fig] displays the dependence of absorbed amount of polymers on the concentration for the highest polymerization index studied, @xmath61. one sees that for both grafting densities the total number of absorbed monomers increases with concentration, while the relative absorbed amount decreases. dft results (again presented for the case of lower grafting density) fall below mc data at higher concentrations, illustrating the aforementioned observation that dft underestimates the degree of penetration of free chains into the brush at higher concentrations. at the `` critical '' strength of attraction @xmath164 for different lengths @xmath0. (b) scft results for the variation of energy (solid lines) and entropy @xmath165 (symbols) of free chains of length @xmath0 with changing attraction @xmath166 to the polymer brush. arrow indicates the intersection point of energy, @xmath167, which coincides with the position of the minima in @xmath165. all energy values are multiplied by @xmath168 for better visibility. in the inset the entropy @xmath165 for chains with @xmath61 in the brush (full squares) and in the bulk (empty squares) is displayed against @xmath42. [crit_fig],title="fig : "] at the `` critical '' strength of attraction @xmath164 for different lengths @xmath0. (b) scft results for the variation of energy (solid lines) and entropy @xmath165 (symbols) of free chains of length @xmath0 with changing attraction @xmath166 to the polymer brush. arrow indicates the intersection point of energy, @xmath167, which coincides with the position of the minima in @xmath165. all energy values are multiplied by @xmath168 for better visibility. in the inset the entropy @xmath165 for chains with @xmath61 in the brush (full squares) and in the bulk (empty squares) is displayed against @xmath42. [crit_fig],title="fig : "] as a remarkable feature of polymer absorption in a brush we find the existence of a _ critical _ degree of compatibility @xmath16 between the grafted and free chains. [crit_fig]a displays brush and free chain density profiles for various polymer chain lengths at the critical value of the brush - polymer attraction strength (@xmath164 for mc simulations and @xmath169 for scft calculations). while simulation and theoretical results differ quantitatively, there is a striking qualitative similarity in that the density profiles, irrespective of the length @xmath0 of the free chains, all intersect in two single points (inside and outside the brush). the dft approach produces exactly the same behavior albeit for a smaller @xmath170 (not shown here). [crit_fig]b shows scft results for the excess entropy and for the internal energy per monomer (given by eqs. ([entropy]) and ([energy]), respectively) as a function of @xmath166. one notes immediately that all the energy curves intersect in a single point, corresponding to @xmath171, while all the entropy curves pass through a minimum at this point. furthermore, the entropic curves corresponding to the polymer segments located `` inside '' and `` outside '' the brush (as defined earlier) intersect at the same value of @xmath166 as shown in the inset of fig. [crit_fig]b. while at @xmath172 there exists thus a distance @xmath69 from the grafting plane where the local concentration of polymer solutions is independent of polymer length @xmath0, provided @xmath173 is kept constant for all @xmath0, the value of @xmath172 itself is expected to depend on the concentration and/or the size of the grafted chains @xmath23. we performed scft calculations to see how the `` critical '' value of @xmath172 changes with @xmath133 and @xmath23 within a broad range : @xmath174 and @xmath13. we find that it increases as @xmath175 with increasing free chain concentration @xmath133, and decreases as @xmath176 with increasing length @xmath23 of the grafted chains (in the latter case, the grafting density is adjusted such that the typical scaling variable for grafted polymers @xmath177 is kept constant). here we present our simulation and theoretical results for the kinetics of polymer adsorption / desorption into, or out of the brush. with elapsed time @xmath178 after an instantaneous change of the interaction between brush and free chains. here @xmath27 and the averaging was performed over @xmath179 cycles. the inset shows the filling kinetics for different size @xmath0 of free chains. the total number of free chain monomers @xmath60 was kept constant. (b) absorption time @xmath19 against polymer length @xmath0 for @xmath27 displays three distinct regimes i (@xmath9), ii (@xmath17, and iii (@xmath180) (shaded areas). [kin_fig],title="fig : "] with elapsed time @xmath178 after an instantaneous change of the interaction between brush and free chains. here @xmath27 and the averaging was performed over @xmath179 cycles. the inset shows the filling kinetics for different size @xmath0 of free chains. the total number of free chain monomers @xmath60 was kept constant. (b) absorption time @xmath19 against polymer length @xmath0 for @xmath27 displays three distinct regimes i (@xmath9), ii (@xmath17, and iii (@xmath180) (shaded areas). [kin_fig],title="fig : "] fig. [kin_fig]a shows the variation of the absorbed relative amount, @xmath154, with elapsed time @xmath178 following an instantaneous switch of the interaction between brush and free chains. as expected, the expulsion of the adsorbate from the brush after an instantaneous switching off of brush - polymer attraction proceeds much faster than the absorption kinetics. the latter, as is visible from the inset to fig. [kin_fig]a, proceeds through an initial steep increase toward a saturation plateau of @xmath154 whereby the small species absorb faster than those with larger @xmath0. from the intersection of the tangent to the initial steep growth of @xmath154 and the saturation value one may determine the characteristic time of absorption @xmath19 as function of @xmath0 - fig. [kin_fig]b. the results are presented for all values of free chain lengths, and one sees that ddft results are again in good agreement with kinetic mc data. this also holds in fig. [kin_fig]b where indeed the theory is in good agreement with simulations for @xmath31. for the case of longer grafted chains (@xmath23=64, @xmath181=0.2), no simulations were performed and only against elapsed time @xmath178 after the onset of absorption for different concentration of free chains with @xmath61. the log - log plot shows that @xmath154 grows by power law @xmath182. the measured slopes @xmath183 are plotted in the inset against the number of free chains @xmath58. one finds @xmath184. (b) the same as in (a) but at the `` critical '' attraction @xmath147 and fixed @xmath185 where @xmath186. the exponent @xmath187 (inset). [dens_fig],title="fig : "] against elapsed time @xmath178 after the onset of absorption for different concentration of free chains with @xmath61. the log - log plot shows that @xmath154 grows by power law @xmath182. the measured slopes @xmath183 are plotted in the inset against the number of free chains @xmath58. one finds @xmath184. (b) the same as in (a) but at the `` critical '' attraction @xmath147 and fixed @xmath185 where @xmath186. the exponent @xmath187 (inset). [dens_fig],title="fig : "] theoretical predictions are shown. nonwithstanding, for both values of @xmath23, one clearly sees three regimes in the dependence of @xmath19 on @xmath0. in the first regime, the absorption time grows fast and essentially linearly with @xmath0 (up to @xmath188 for the shorter brush and @xmath189 for the longer one). by analyzing the data presented in fig. [rg_fig]b, one sees that this initial linear regime corresponds to the situation when @xmath11 of absorbed chains is less than or equal to the average distance between the grafting points. as @xmath0 (and, consequently, @xmath11) is increased beyond the aforementioned values, one enters the second regime where the growth of @xmath19, while still nearly linear, is markedly slower. we interpret this slowing down as a halmark of an increased friction of the penetrating coils when their radius of gyration exceeds the size of the cavities in the polymer brush. this regime extends up to the point where the lengths of free and grafted chains become equal. beyond this point, for @xmath190, the third regime is sets in, where the absorption time is essentially independent of the free chain length. one might see therein an indication of a change in the mechanism of free chain penetration into the brush with thickness @xmath191 whereby additionally the coil flattens inside the grafted layer due to gain in absorption energy. [dens_fig]a displays simulation and theoretical results for the absorption kinetics for @xmath31, @xmath61, and several values of the concentration @xmath192. both mc and ddft data show that at early and intermediate times the time dependence of the absorbed amount follows a power law @xmath182. the corresponding effective exponent @xmath183 is decreasing as the concentration increases (see inset), although the value of @xmath154 at the beginning of the intermediate time regime is larger for larger values of @xmath58. this result is somewhat counter - intuitive, as one would expect the driving force for absorption (and, hence, the absorption rate) to increase with increasing concentration of free chains. a slowing down of absorption kinetics with growing size @xmath0 and concentration @xmath133 of the free chains has been experimentally observed @xcite in a porous medium (activated carbon) which resembles in certain aspects the polymer brush. in fig. [dens_fig]b we show the variation of the absorbed amount, @xmath186, for the critical attraction @xmath193 - see [crit_sect]. we point out that this well pronounced power law increase of @xmath154 was observed only at this particular value of @xmath42 whereas for @xmath194 where most of our kinetic measurement were performed, no simple @xmath195 relationship was found - cf. [kin_fig]a. thus, in a sense, the particular kinetics of absorption underlines the special role of the critical compatibility between brush and free chains. of free chains with time elapsed after a quench from @xmath196 to @xmath197 from dft data. the time is given in logarithmic coordinates. here the mean concentration @xmath198 and the time unit corresponds to 25000 mcs. the polymer brush is located at @xmath199. (b) variation of the flux of free chains into the polymer brush with time for two concentrations @xmath200 i.e., @xmath201, and @xmath61. [kin_3d_fig],title="fig : "] of free chains with time elapsed after a quench from @xmath196 to @xmath197 from dft data. the time is given in logarithmic coordinates. here the mean concentration @xmath198 and the time unit corresponds to 25000 mcs. the polymer brush is located at @xmath199. (b) variation of the flux of free chains into the polymer brush with time for two concentrations @xmath200 i.e., @xmath201, and @xmath61. [kin_3d_fig],title="fig : "] in order to shed further light on the observed behavior, fig. [kin_3d_fig] shows ddft results for the time - dependent density profile @xmath105 (@xmath58=24, left panel) and flux @xmath202 (@xmath58=8 and 32, right panel). in the left panel, one observes two `` ridges '' in @xmath105 at all times a principal ridge, initially located in the bulk above the brush, moves gradually inside, while another (smaller) ridge is located near the opposite (bare) wall and gradually disappears still moving in the bulk. from the right panel, one can see that for higher concentration (@xmath58=32) the flux prevails over the lower concentration one inside the brush and at shorter times (thereby explaining higher initial values of @xmath154 seen in fig. [dens_fig] for larger values of @xmath58), while the situation is reversed outside the brush at longer times. the latter behavior is presumably due to higher mobility at lower concentrations and explains the decrease of slope @xmath183 with @xmath58 seen in the inset of fig. [dens_fig]. Discussion in this work we studied a scarcely explored yet important aspect of oligomer and linear macromolecule absorption in a polymer brush - the case of (more or less) good compatibility between species in the bulk and grafted chains. starting from oligomers (mono- and dimers) and going up to chain lengths @xmath0 which exceed twice the length @xmath23 of the grafted chains, we have determined the conformation of the absorbed species, the absorbed amount @xmath154, and absorption kinetics (the propagation rate into the polymer brush) at different concentration of the free chains for two cases of moderately to very dense polymer brushes. in addition, by combining monte carlo simulations with dft and scft calculations, we have substantially broadened the range of lengths of the grafted chains to @xmath13 in order to test more comprehensively our findings. the most salient, and - to some extent - unexpected features of linear chain absorption in a polymer brush that we find are : * the dramatic increase in adsorbed amount @xmath7 with _ growing _ chain length @xmath0, and * the significant slowdown of absorption kinetics with growing concentration (i.e., with the increase of the starting gradient in density) of the free chains besides these static and dynamic properties of polymer absorption in brushes, we find that both the absorbed macromolecules and the brush itself largely retain their structure and conformation, as seen in quantities like @xmath203 and the monomer density profile @xmath104, for different length @xmath0 and concentration @xmath26 of the free chains, and different strength @xmath42 of attraction to the grafted chains. in particular, the degree to which the brush profile @xmath104 is affected by absorption is found to be much less that anticipated in some earlier theoretical predictions @xcite. nontheless, even within these small changes we observe a slight contraction of @xmath104 at small absorbed amounts @xmath154 while @xmath104 gradually attains its extension roughly to that corresponding to zero concentration of free chains with growing @xmath154. an interesting finding which still needs deeper understanding is the observed existence of a critical compatibility @xmath204 (i.e., brush - oligomer attraction @xmath205). at @xmath206 we find both in mc as well as in dft / scft that the energy of all absorbed species has a value independent of their size @xmath0 whereas their entropy experiences a minimum. the critical attraction @xmath205 is manifested by the existence of unique distance from the grafting plane where all monomer density profiles of the free chains intersect. moreover, at @xmath205 the kinetics of free chain absorption into the brush follows a clear cut power law with exponent @xmath207. undoubtedly, much more work is needed until all these fascinating new features are fully understood. last not least, we emphasize the finding of three distinct regimes in the kinetics of free chain absorption as far as the size of the free chains @xmath0 is concerned. in the first regime the characteristic time for absorption @xmath19 grows rapidly with oligomer length @xmath0 as long as the oligomer size @xmath208 remains smaller than the separation between grafting sites. the second regime is marked by a slower increase of @xmath19 with @xmath0 and ends roughly at @xmath209. the third regime of absorption kinetics holds for @xmath180 (i.e., the penetrating free chain can not accommodate within the brush) and is characterized by a nearly constant @xmath19 as far as length @xmath0 is concerned. interestingly, this rich kinetic behavior has been experimentally observed in absorption in porous media @xcite. Acknowledgments one of us, (a. m.), acknowledges support under grant no. bi314/22. another, (s. a. e.), acknowledges support from the alexander von humboldt foundation, germany. s. alexander, j. physique (paris) * 38 *, 983 (1977). g. de gennes, macromolecules, * 13 *, 1069 (1980). a. m. skvortsov, a. a. gorbunov, i. v. pavlushkov, e. b. zhulina, o. v. borisov, and v. a. pryamitzyn, polym. 30 *, 1706 (1988). t. cosgrove, t. heath, b. van lahr, f. leermakers, and j. m. h. scheutjens, macromolecules * 20 *, 1692 (1988). s. t. milner, t. a. witten, and m. e. cates, macromolecules * 21 *, 2610 (1988). m. muthukumar and j. s. ho, macromolecules * 22 *, 965 (1989). m. murat and g. s. grest, macromolecules * 22 *, 4054 (1989). p. y. lai and k. binder, j. chem. phys. * 95 *, 9288 (1991). s. t. milner, science * 251 *, 905 (1991). a. halperin, m. tirrell, and t. p. lodge, adv. sci. * 100 *, 33 (1992). i. szleifer and m. a. carignano, adv. chem. phys. * 94 *, 165 (1996). j. klein, ann. * 26 *, 581 (1996). g. s. grest and m. murat, in _ monte carlo and molecular dynamics simulations in polymer science _ k. binder, oxford univ. press, new york, 1995, pp. 476 - 578. g. s. grest, adv. sci. * 138 *, 149 (1999). l. leger, e. raphael, and h. hervet, adv. sci. * 138 *, 185 (1999)., eds. r. c. advincula, w. j. brittain, k. c. caster, and j. rhe, wiley - vch, weinheim, 2004. d. h. napper, _ polymeric stabilization of colloidal dispersions _, academic press, london, 1983. h. r. brown, mat. res. soc. bull. * 21 *, 24 (1996) g. storm, s. o. belliot, t. daemen, and d. d. lasic, adv. drug. deliv. 17 *, 31 (1995). a. hucknall, a. j. simmick, r. t. hill, a. chieboli, a. garcia, m. s. johannes, r. l. clarck, s. zaucher, and b. d. ratner, biointerfaces * 4 *, fa50 (2009). a. j. wang, j. j. xu, and h. y. chen, j. chromatography, a * 1147 *, 120 (2007). j. u. kim and b. o. oshaughnessy, macromolecules * 39 *, 413 (2006). d. i. dimitrov, a. milchev, k. binder, j. chem. phys. * 127 *, 084905 (2007). a. halperin, g. fragneto, a. schollier, and m. sferrazza, langmuir, * 23 *, (2007). s. a. egorov, j. chem. phys. * 129 *, 064901 (2008). j. yaneva, d. i. dimitrov, a. milchev and k. binder, j. colloid interface sci. * 336 *, 51 (2009). d. trombly and v. ganesan, j. polym. b * 47 *, 2566 (2009). d. dukes, y. li, s. lewis, b. benicevicz, l. schadler, and s. k. kumar, macromolecules * 43 *, 1564 (2010). e. p. currie, j. van der gucht, o. v. borisov, and m. a. cohen - stuart, langmuir * 14 *, 5740 (1998). k. chen and y. a. ma, j. phys. b * 109 *, 17617 (2005). j. u. kim and m. w. matsen, macromolecules * 41 *, 246 (2008). a. milchev, d. i. dimitrov and k. binder, polymer * 49 *, 3611 (2009). s. gupta, m. agraval, p. uhlmann, f. simon, u. oetel, and m. stamm, macromolecules * 41 *, 8152 (2008). a. p. gast, l. leibler, macromolecules, * 19 *, 686 (1986). h. r. brown, k. chare, and v. r. deline, macromolecules * 23 *, 3383 (1990). t. witten, l. leibler, and p. pincus, macromolecules, * 23 *, 824 (1990). e. b. zhulina, o. v. borisov, l. brombacher, macromolecules, * 24 *, 4679 (1991) c. m. wijmans, e. b. zhulina, and g. j. fleer, macromolecules, * 27 *, 3238 (1994) ; c. m. wijmans and b. j. factor, macromolecules, * 29 *, 4406 (1996). m. aubouy, e. raphael, macromolecules, * 27 *, 5182 (1994) ; j. i. martin and z. g. wang, j. phys. chem. * 99 *, 2833 (1995). m. p. pepin and m. d. whitmore, j. chem. * 114 *, 8181 (2001). i. borukhov and l. leibler, macromolecules * 35 *, 5171 (2002) h. huang, a. cammers, and l. s. penn, macromolecules, * 39 *, 7064 (2006). r. yerushalmi - rosen, j. klein and l. j. fetters, science * 263 *, 793 (1994). c. j. clarke, r. a. l. jones, j. l. edwards, k. r. shull, and j. penfold, macromolecules * 28 *, 2042 (1995). g. reiter, p. auroy, and l. auvray, macromolecules * 29 *, 2150 (1996). m. mller and l. g. macdowell, europhys. lett. * 55 *, 221 (2001). s. jain, v. v. ginzburg, p. jog, j. weinhold, r. srivastava, and w. g. chapman, j. chem. phys. * 131 *, 044908 (2009). f. pierce, d. perahia, and g. grest, macromolecules * 42 *, 7969 (2009) p. y. lai, j. chem. phys. * 98 *, 669 (1999). a. kopf, j. baschnagel, j. wittmer, and k. binder, macromolecules * 29 *, 1433 (1996). j. wittmer, a. johner, j. f. joanny, and k. binder, j. chem. phys. * 101 *, 4397 (1994). p. flory, _ principles of polymer chemistry _, cornell university press, ithaca, 1953. m. borowko, w. rzysko, s. sokolowski, and t. staszewski, j. phys. b * 113 *, 4763 (2009) milchev a, paul w, binder k, j. chem. phys. * 99 *, 4786 (1993). milchev a, binder k. macromol. theory simul. * 5 *, 915 (1994) s. semal, m. vou, m. j. de ruijter, j. dehuit, and j. de coninck, j. phys. b * 103 *, 4854 (1999). a. milchev, k. binder, macromolecules * 29 *, 343 (1996). a. milchev, k. binder, j. chem. phys. * 114 *, 8610 (2001). a. yethiraj and c. e. woodward, j. chem. phys. * 102 *, 5499 (1995). c. e. woodward, j. chem. phys. * 94 *, 3183 (1991). h. d. do, d. d. do, and i. prasetyo, aiche j. * 47 *, 2515 (2001). j. g. e. m. fraaije, j. chem. 99 *, 9202 (1993). h. xu, h. liu, and y. hu, chem. * 62 *, 3494 (2007). b. m. steels, j. koska, and c. a. haynes, j. chromatography b, * 743 *, 41 (1991). r. roth, j. phys. * 22 *, 063102 (2010).

the absorption of free linear chains in a polymer brush was studied with respect to chain size @xmath0 and compatibility @xmath1 with the brush by means of monte carlo (mc) simulations and density functional theory (dft) / self - consistent field theory (scft) at both moderate, @xmath2, and high, @xmath3, grafting densities using a bead - spring model. different concentrations of the free chains @xmath4 are examined. contrary to the case of @xmath5 when all species are almost completely ejected by the polymer brush irrespective of their length @xmath0, for @xmath6 we find that the degree of absorption (absorbed amount) @xmath7 undergoes a sharp crossover from weak to strong (@xmath8) absorption, discriminating between oligomers, @xmath9, and longer chains. for a moderately dense brush, @xmath2, the longer species, @xmath10, populate predominantly the deep inner part of the brush whereas in a dense brush @xmath3 they penetrate into the `` fluffy '' tail of the dense brush only. gyration radius @xmath11 and end - to - end distance @xmath12 of absorbed chains thereby scale with length @xmath0 as free polymers in the bulk. using both mc and dft / scft methods for brushes of different chain length @xmath13 , we demonstrate the existence of unique _ critical _ value of compatibility @xmath14. for @xmath15 the energy of free chains attains the _ same _ value, irrespective of length @xmath0 whereas the entropy of free chain displays a pronounced minimum. at @xmath16 all density profiles of absorbing chains with different @xmath0 intersect at the same distance from the grafting plane. the penetration / expulsion kinetics of free chains into the polymer brush after an instantaneous change in their compatibility @xmath1 displays a rather rich behavior. we find three distinct regimes of penetration kinetics of free chains regarding the length @xmath0 : i (@xmath9), ii (@xmath17), and iii (@xmath18), in which the time of absorption @xmath19 grows with @xmath0 at a different rate. during the initial stages of penetration into the brush one observes a power - law increase of @xmath20 with power @xmath21 whereby penetration of the free chains into the brush gets _ slower _ as their concentration rises.

1003.3435

Introduction in the last years wireless communication systems coped with the problem of delivering reliable information while granting high throughput. this problem has often been faced resorting to channel codes able to correct errors even at low signal to noise ratios. as pointed out in table i in @xcite, several standards for wireless communications adopt binary or double binary turbo codes @xcite and exploit their excellent error correction capability. however, due to the high computational complexity required to decode turbo codes, optimized architectures (e.g. @xcite, @xcite) have been usually employed. moreover, several works addressed the parallelization of turbo decoder architectures to achieve higher throughput. in particular, many works concentrate on avoiding, or reducing, the collision phenomenon that arises with parallel architectures (e.g. @xcite). although throughput and area have been the dominant metrics driving the optimization of turbo decoders, recently, the need for flexible systems able to support different operative modes, or even different standards, has changed the perspective. in particular, the so called software defined radio (sdr) paradigm made flexibility a fundamental property @xcite of future receivers, which will be requested to support a wide range of heterogeneous standards. some recent works (e.g. @xcite, @xcite, @xcite) deal with the implementation of application - specific instruction - set processor (asip) architectures for turbo decoders. in order to obtain architectures that achieve both high throughput and flexibility multi - asip is an effective solution. thus, together with flexible and high throughput processing elements, a multi - asip architecture must feature also a flexible and high throughput interconnection backbone. to that purpose, the network - on - chip (noc) approach has been proposed to interconnect processing elements in turbo decoder architectures designed to support multiple standards @xcite, @xcite, @xcite, @xcite, @xcite, @xcite. in addition, noc based turbo decoder architectures have the intrinsic feature of adaptively reducing the communication bandwidth by the inhibition of unnecessary extrinsic information exchange. this can be obtained by exploiting bit - level reliability - based criteria where unnecessary iterations for reliable bits are avoided @xcite. in @xcite, @xcite, @xcite ring, chordal ring and random graph topologies are investigated whereas in @xcite previous works are extended to mesh and toroidal topologies. furthermore, in @xcite butterfly and benes topologies are studied, and in @xcite binary de - bruijn topologies are considered. however, none of these works presents a unified framework to design a noc based turbo decoder, showing possible complexity / performance trade - offs. this work aims at filling this gap and provides two novel contributions in the area of flexible turbo decoders : i) a comprehensive study of noc based turbo decoders, conducted by means of a dedicated noc simulator ; ii) a list of obtained results, showing the complexity / performance trade - offs offered by different topologies, routing algorithms, node and asip architectures. the paper is structured as follows : in section [sec : system_analysis] the requirements and characteristics of a parallel turbo decoder architecture are analyzed, whereas in section [sec : noc] noc based approach is introduced. section [sec : topologies] summarizes the topologies considered in previous works and introduces generalized de - bruijn and generalized kautz topologies as promising solutions for noc based turbo decoder architectures. in section [sec : ra] three main routing algorithms are introduced, whereas in section [sec : tnoc] the turbo noc framework is described. section [sec : routing_algo_arch] describes the architecture of the different routing algorithms considered in this work, section [sec : results] presents the experimental results and section [sec : concl] draws some conclusions. System requirement analysis a parallel turbo decoder can be modeled as @xmath0 processing elements that need to read from and write to @xmath0 memories. each processing element, often referred to as soft - in - soft - out (siso) module, performs the bcjr algorithm @xcite, whereas the memories are used for exchanging the extrinsic information @xmath1 among the sisos. the decoding process is iterative and usually each siso performs sequentially the bcjr algorithm for the two constituent codes used at the encoder side ; for further details on the siso module the reader can refer to @xcite. as a consequence, each iteration is made of two half iterations referred to as interleaving and de - interleaving. during one half iteration the extrinsic information produced by siso @xmath2 at time @xmath3 (@xmath4) is sent to the memory @xmath5 at the location @xmath6, where @xmath7 and @xmath8 are functions of @xmath2 and @xmath3 derived from the permutation law (@xmath9 or interleaver) employed at the encoder side. thus, the time required to complete the decoding is directly related to the number of clock cycles necessary to complete a half iteration. without loss of generality, we can express the number of cycles required to complete a half iteration (@xmath10) as @xmath11 where @xmath12 is the total number of trellis steps in a data frame, @xmath13 is the number of trellis steps processed by each siso, @xmath14 is the siso output rate, namely the number of trellis steps processed by a siso in a clock cycle, and @xmath15 is the interconnection structure latency. thus, the decoder throughput expressed as the number of decoded bits over the time required to complete the decoding process is @xmath16 where @xmath17 is the clock frequency, @xmath18 is the number of iterations, @xmath19 for binary codes and @xmath20 for double binary codes. when the interconnection structure latency is negligible with respect to the number of cycles required by the siso, we obtain @xmath21 thus, to achieve a target throughput @xmath22 and satisfactory error rate performance, a proper number @xmath23 of iterations should be used. the minimum @xmath0 (@xmath24) to satisfy @xmath22 with @xmath23 iterations can be estimated from ([eq : tapprox]) for some asip architectures available in the literature. if we consider @xmath25, as in @xcite, @xcite, @xmath0 ranges in [5, 37] to achieve @xmath26 mb / s (see table [tab : pasip]). it is worth pointing out that the @xmath27 values in table [tab : pasip] represent the average numbers of cycles required by the siso to update the soft information of one bit (see table vi in @xcite and table i in @xcite). moreover, @xmath28 strongly depends on the internal architecture of the siso and in general tends to increase with the code complexity. as a consequence, several conditions can further increase @xmath0, namely 1) interconnection structures with larger @xmath15 ; 2) higher @xmath29 values ; 3) higher @xmath22 ; 4) higher @xmath23 ; 5) lower clock frequency. thus, we consider as relevant for investigation a slightly wider range for @xmath0 : @xmath30..parallelism degree required to obtain @xmath26 mb / s for @xmath25 with some asip architectures available in the literature [cols="^,^,^,^,^,^,^ ",] the area and the percentage are not really zero, but they are negligible compared with the i m and lm contribution to the total area. the most important conclusions that can be derived from results in table [tab : wimax_results] and [tab : mhoms_results] are : 1. the asp - ft routing algorithm is the best performing solution both in terms of throughput and area when @xmath31. 2. the routing memory overhead of the asp - ft algorithm (see fig. [fig : node] (b)) becomes relevant as @xmath14 decreases and ssp solutions become the best solutions mainly for @xmath32 and @xmath33. 3. in most cases topologies with @xmath34=4 achieve higher throughput with lower complexity overhead than topologies with @xmath34=2 when @xmath35. 4. in most cases, generalized de - bruijn and generalized kautz topologies are the best performing topologies. as a significant example, in fig. [fig : r1_asp - ft], we show the experimental results obtained with @xmath31 and asp - ft routing algorithm for the wimax interleaver with @xmath36 (a) and the circular shifting interleaver with @xmath37 (b). each point represents the throughtput and the area obtained for a certain topology with a certain parallelism degree @xmath0. results referred to the same @xmath0 value are bounded into the same box and a label is assigned to each point to highlight the corresponding topology, namely topologies are identified as r - ring, h - honeycomb, t - toroidal mesh, k - generalized kautz with the corresponding @xmath34 value (k2, k3, k4). as it can be observed, generalized kautz topologies with @xmath38 (k4) are always the best solutions to achieve high throughput with minimum area overhead. in fig. [fig : tar_tot] significant results extracted from table [tab : wimax_results] and [tab : mhoms_results] are shown in graphical form. in particular, for @xmath31 the asp - ft routing algorithm is the best solution, whereas for @xmath39 ssp routing algorithms, implemented as in fig. [fig : node] (c), tend to achieve the same performance as the asp - ft routing algorithm with lower complexity overhead (see fig. [fig : tar_tot] (a) and (b) for the wimax interleaver, @xmath36 and fig. [fig : tar_tot] (c) and (d) for the circular shifting interleaver, @xmath37). an interesting phenomenon that arises increasing the interleaver size is the performance saturation that can be observed in the table [tab : mhoms_results] for @xmath40 topologies, namely the throughput tends to saturate and increasing @xmath14 has the effect of augmenting the area with a negligible increase or even with a decrease of throughput. as an example, the generalized kautz topology with @xmath41 and asp - ft routing algorithm achieves more than 180 mb / s with @xmath31, @xmath42, @xmath43. however, the solution with the smallest area is the one obtained with @xmath43. the throughput flattening of low @xmath34 topologies can be explained by observing that high values of @xmath14 tend to saturate the network. furthermore, high values of @xmath14 lengthen the input fifos as highlighted in table [tab : percentage], where the total area of the network is given as the breakdown of the building blocks, namely the input fifos, the crossbars (cb), the output registers, the routing algorithm / memory (ra / m), the identifier memory (i m) and the location memory (lm) is given for some significant cases : the highest throughput (light - gray), the highest area (mid - gray), and lowest area (dark - gray) points for each @xmath34 value in table [tab : mhoms_results]. Conclusions in this work a general framework to design network on chip based turbo decoder architectures has been presented. the proposed framework can be adapted to explore different topologies, degrees of parallelism, message injection rates and routing algorithms. experimental results show that generalized de - bruijn and generalized kautz topologies achieve high throughput with a limited complexity overhead. moreover, depending on the target throughput requirements different parallelism degrees, message injection rates and routing algorithms can be used to minimize the network area overhead. a. giulietti, l. v. der perre, and m. strum, `` parallel turbo coding interleavers : avoiding collisions in accesses to storage elements, '' _ iet electronics letters _, vol. 38, no. 5, pp. 232234, feb 2002. m. j. thul, f. gilbert, and n. wehn, `` optimized concurrent interleaving architecture for high - throughput turbodecoding, '' in _ ieee international conference on electronics, circuits and systems _, 2002, pp. 10991102. c. neeb, m. j. thul, and n. wehn, `` network - on - chip - centric approach to interleaving in high throughput channel decoders, '' in _ ieee international symposium on circuits and systems _, 2005, pp. 17661769. h. moussa, o. muller, a. baghdadi, and m. .jezequel, `` butterfly and benes - based on - chip communication networks for multiprocessor turbo decoding, '' in _ design, automation and test in europe conference and exhibition _, 2007, pp. 654659. s. benedetto, d. divsalar, g. montorsi, and f. pollara, `` soft - input soft - output modules for the construction and distributed iterative decoding of code networks, '' _ european transactions on telecommunications _, vol. 9, no. 2, pp. 155172, mar / apr 1998. o. muller, a. baghdadi, and m. jezequel, `` asip - based multiprocessor soc design for simple and double binary turbo decoding, '' in _ design, automation and test in europe conference and exhibition _, 2006, pp. 13301335. o. muller, a. baghdadi, and m. jezequel, `` exploring parallel processing levels for convolutional turbo decoding, '' in _ ieee international conference on information and communication technologies : from theory to applications _, 2006, pp.

this work proposes a general framework for the design and simulation of network on chip based turbo decoder architectures. several parameters in the design space are investigated, namely the network topology, the parallelism degree, the rate at which messages are sent by processing nodes over the network and the routing strategy. the main results of this analysis are : i) the most suited topologies to achieve high throughput with a limited complexity overhead are generalized de - bruijn and generalized kautz topologies ; ii) depending on the throughput requirements different parallelism degrees, message injection rates and routing algorithms can be used to minimize the network area overhead.

0909.1876

Introduction monte carlo methods @xcite appeared about sixty years ago with the need to evaluate numerical values for various complex problems. these methods evolved and were applied early to quantum problems, thus putting within reach exact numerical solutions to non - trivial quantum problems @xcite. many improvements of these methods followed, avoiding critical slowing down near phase transitions and allowing to work directly in the continuous imaginary time limit @xcite. in recent years, interest in methods that work in the canonical ensemble with global updates yet allow access to green functions has intensified @xcite. however, a method that works well for a given hamiltonian often needs major modifications for another. for example, the addition of a 4-site ring exchange term in the bosonic hubbard model required special developments for a treatment by the stochastic series expansion algorithm @xcite, as well as by the wordline algorithm @xcite. this can result in long delays. it is, therefore, advantageous to have at one s disposal an algorithm that can be applied to a very wide class of hamiltonians without requiring any changes. in a recent publication @xcite, the stochastic green function (sgf) algorithm was presented, which meets this goal. the algorithm can be applied to any lattice hamiltonian of the form @xmath3 where @xmath1 is diagonal in the chosen occupation number basis and @xmath2 has only positive matrix elements. this includes all kinds of systems that can be treated by other methods presented in ref.@xcite, for instance bose - hubbard models with or without a trap, bose - fermi mixtures in one dimension, heisenberg models... in particular hamiltonians for which the non - diagonal part @xmath2 is non - trivial (the eigen - basis is unknown) are easily treated, such as the bose - hubbard model with ring exchange @xcite, or multi - species hamiltonians in which a given species can be turned into another one (see eq.([twospecies]) and fig. [density] and [momentum] for a concrete example). systems for which it is not possible to find a basis in which @xmath1 is diagonal and @xmath2 has only positive matrix elements are said to have a `` sign problem '', which usually arises with fermionic and frustrated systems. as other qmc methods, the sgf algorithm does not solve this problem. the algorithm allows to measure several quantities of interest, such as the energy, the local density, local compressibility, density - density correlation functions... in particular the winding is sampled and gives access to the superfluid density. equal - time n - body green functions are probably the most interesting quantities that can be measured by the algorithm, by giving access to momentum distribution functions which allow direct comparisons with experiments. all details on measurements are given in ref.@xcite. in addition the algorithm has the property of being easy to code, due in part to a simple update scheme in which all moves are accepted with a probability of 1. despite of such generality and simplicity, the algorithm might suffer from a reduced efficiency, compared to other algorithms in situations where they can be applied. the purpose of this paper is to present a `` directed '' update scheme that (i) keeps the simplicity and generality of the original sgf algorithm, and (ii) enhances its efficiency by improving the sampling over the imaginary time axis. while the sgf algorithm is not intended to compete with the speed of other algorithms, the improvment resulting from the directed update scheme is remarkable (see section v). but what makes the strength of the sgf method is that it allows to simulate hamiltonians that can not be treated by other methods or that would require special developments (see eq.([twospecies]) for a concrete example). the paper is organized as follows : we introduce in section ii the notations and definitions used in ref.@xcite. in section iii, we propose a simplification of the update scheme used in the original sgf algorithm, and determine how to satisfy detailed balance. a generalization of the simplified update scheme is presented in section iv, which constitutes the directed updated scheme. finally section v shows how to determine the introduced optimization parameters, and presents some tests of the algorithm and a comparison with the original version. Definitions and notations in this section, we recall the expression of the `` green operator '' introduced in the sgf algorithm, and the extended partition function which is considered. although not required for understanding this paper, we refer the reader to ref.@xcite for full details on the algorithm. as many qmc algorithms, the sgf algorithm samples the partition function @xmath4 the algorithm has the property of working in the canonical ensemble. in order to define the green operator, we first define the `` normalized '' creation and annihilation operators, @xmath5 where @xmath6 and @xmath7 are the usual creation and annihilation operators of bosons, and @xmath8 is the number operator. from ([normalizedoperators]) one can show the following relations for any state @xmath9 in the occupation number representation, @xmath10 with the particular case @xmath11. appart from this exception, the operators @xmath12 and @xmath13 change a state @xmath9 by respectively creating and annihilating one particle, but they do not change the norm of the state. using the notation @xmath14 to denote two subsets of site indices @xmath15 and @xmath16 with the constraint that all indices in subset @xmath17 are different from the indices in subset @xmath18 (but several indices in one subset may be equal), we define the green operator @xmath19 by @xmath20 where @xmath21 is a matrix that depends on the application of the algorithm @xcite. in order to sample the partition function ([partitionfunction]), an extended partition function @xmath22 is considered by breaking up the propagator @xmath23, and introducing the green operator between the broken parts, @xmath24 defining the time dependant operators @xmath25 and @xmath26, @xmath27 and working in the occupation number basis in which @xmath1 is diagonal, the extended partition function takes the form @xmath28 where the sum @xmath29 implicitly runs over complete sets of states @xmath30. we will systematically use the labels @xmath31 and @xmath32 to denote the states appearing on the left and the right of the green operator, and use the notation @xmath33 to denote the diagonal energy @xmath34. we will also denote by @xmath35 and @xmath36 the time indices of the @xmath2 operators appearing on the left and the right of @xmath19. as a result, the extended partition function is a sum over all possible configurations, each being determined by a set of time indices @xmath37 and a set of states @xmath38, @xmath39, @xmath40,@xmath41, @xmath42. the algorithm consists in updating those configurations by making use of the green operator. assuming that the green operator is acting at time @xmath43, it can `` create '' a @xmath2 operator (that is to say a @xmath2 operator can be inserted in the operator string) at the same time, thus introducing a new intermediate state, then it can be shifted to a different time. while shifting, any @xmath2 operator encountered by the green operator is `` destroyed '' (that is to say removed from the operator string). assuming a left (or right) move, creating an operator will update the state @xmath44 (or @xmath41), while destroying will update the state @xmath41 (or @xmath44). when a diagonal configuration of the green operator occurs, @xmath45, such a configuration associated to the extended partition function ([extendedpartitionfunction]) is also a configuration associated to the partition function ([partitionfunction]). measurements can be done when this occurs (see ref.@xcite for details on measurements). next section presents a simple update scheme that meets the requirements of ergodicity and detailed balance. Simplified update scheme before introducing the directed update, we start by simplifying the update scheme used in the original sgf algorithm. we will assume in the following that a left move of the green operator is chosen. in the original version, the green operator @xmath26 can choose to create or not on its right a @xmath2 operator at time @xmath43. then a time shift @xmath46 to the left is chosen for the green operator with an exponential distribution in the range @xmath47. if an operator is encountered while shifting the green operator, then the operator is destroyed and the move stops there. as a result, four possible situations can occur during one move : 1. no creation, shift, no destruction. 2. creation, shift, no destruction. 3. no creation, shift, destruction. 4. creation, shift, destruction. it appears that the first possibility `` no creation, no destruction '' is actually useless, since no change is performed in the operator string. the idea is to get rid of this possibility by forcing the green operator to destroy an operator if no creation is chosen a further simplification can be done by noticing that the last possibility `` creation, destruction '' is not necessary for the ergodicity of the algorithm, and can be avoided by restricting the range of the time shift after having created an operator. therefore we replace the original update scheme by the following : we assume that the green operator is acting at time @xmath43 and that the operator on its left is acting at time @xmath35. the green operator @xmath26 chooses to create or not an operator on its right at time @xmath43. if creation is chosen, then a time shift @xmath46 of the green operator is chosen to the left in the range @xmath48, with the probability distribution defined below. if no creation is chosen, then the green operator is directly shifted to the operator on its left at time @xmath35, and the operator is destroyed. as a result only two possibilities have to be considered : 1. creation, shift. 2. shift, destruction. figure [simplfiedupdatescheme] shows the associated organigram. section iii.b explains how detailed balance can be satisfied with this simplified update scheme. when updating the configurations according to the chosen update scheme, we need to generate different transitions from initial to final states with probabilities that satisfy detailed balance. in this section we propose a choice for these probabilities, and determine the corresponding acceptance factors. we denote the probability of the initial (final) configuration by @xmath49 (@xmath50). we denote by @xmath51 the probability of the transition from configuration @xmath17 to configuration @xmath52, and by @xmath53 the probability of the reverse transition. finally we denote by @xmath54 the acceptance rate of the transition from @xmath17 to @xmath52, and by @xmath55 the acceptance rate of the reverse transition. the detailed balance can be written as @xmath56 we will make use of the metropolis solution @xcite, @xmath57 with @xmath58 we will use primed (non - primed) labels for states and time indices to denote final (initial) configurations. we consider here the case where a left move is chosen, an operator is created on the right of the green operator at time @xmath43, and a new state is chosen. then a time shift to the left is chosen for the green operator in the range @xmath59. it is important to note that @xmath60 and @xmath61 correspond to the time indices of the operators appearing on the left and the right of the green operator after the new operator has been inserted, that is to say at the moment where the time shift needs to be performed. thus we have @xmath62 and @xmath63. the probability of the initial configuration is the boltzmann weight appearing in the extended partition function ([extendedpartitionfunction]) : @xmath64 the probability of the final configuration takes the form : @xmath65 it is important here to realize that the green operator only inserted on its right the operator @xmath66, before being shifted from @xmath61 to @xmath67. therefore we have the equalities @xmath68, @xmath69, @xmath70, and @xmath71. the probability @xmath51 of the transition from the initial configuration to the final configuration is the probability @xmath72 of a left move, times the probability @xmath73 of a creation, times the probability @xmath74 to choose the new state @xmath75, times the probability @xmath76 to shift the green operator by @xmath77, knowing that the states on the left and the right of the green operator at the moment of the shift are @xmath78 and @xmath79 : @xmath80 the probability of the reverse transition is simply the probability @xmath81 of a right move, times the probability of no creation, @xmath82 : @xmath83\]] from the original version of the sgf algorithm, we know that choosing the time shift with an exponential distribution is a good choice, because it cancels the exponentials appearing in the probabilities of the initial ([initial]) and final ([final]) configurations, avoiding exponentially small acceptance factors. however a different normalization must be used here, since the time shift is chosen in the range @xmath84 instead of @xmath47. the suitable solution is : @xmath85 it is straightforward to check that the above probability is correctly normalized and well - defined for any real value of @xmath86, the particular case @xmath87 reducing to the uniform distribution @xmath88 (note that @xmath89 is always a positive number). for the probability @xmath74 to choose the new state @xmath75, the convenient solution is the same as in the original version : @xmath90 putting everything together, the acceptance factor ([metropolis2]) becomes @xmath91\big[1-e^{-(\tau_l^\prime-\tau_r^\prime)(v_r^\prime - v_l^\prime)}\big]}{v_r^\prime - v_l^\prime},\end{aligned}\]] where we have used the notation @xmath92 to emphasize that this acceptance factor corresponds to a creation. it is also important for the remaining of this paper to note that @xmath92 is written as a quantity that depends on the initial configuration, times a quantity that depends on the final configuration. we consider here the case where a left move is chosen, and the operator on the left of the green operator is destroyed. this move corresponds to the inverse of the above `` creation, shift '' move. thus, the corresponding acceptance factor @xmath93 is obtained by inverting the acceptance factor @xmath92, exchanging the initial time @xmath43 and final time @xmath67, and switching the direction. however @xmath94 represents an absolute time shift, so @xmath35 and @xmath36 do not have to be exchanged. we get @xmath95\big[1-e^{-(\tau_l-\tau_r)(v_l - v_r)}\big] } \\ & \times & \frac{\big\langle\psi_l^\prime\big|\hat\mathcal g\big|\psi_r^\prime\big\rangle p(\rightarrow^\prime)p_\rightarrow^\dagger(\tau^\prime)}{\big\langle\psi_l^\prime\big|\hat\mathcal t\hat\mathcal g\big|\psi_r^\prime\big\rangle},\end{aligned}\]] which is written as a quantity that depends on the initial configuration, times a quantity that depends on the final configuration. we will use here the short notation @xmath96, @xmath97, and @xmath98 to denote respectively the quantities @xmath99, @xmath100, and @xmath101. as in ref. @xcite, we have some freedom for the choice of the probabilities of choosing a left or right move, @xmath72 and @xmath102, and the probabilities of creation @xmath73 and @xmath103. a suitable choice for those probabilities can be done in order to accept all moves, resulting in an appreciable simplification of the algorithm. for this purpose, we impose the acceptance factor @xmath92 (or @xmath104) to be equal to the acceptance factor @xmath93 (or @xmath105). this allows to determine the probabilities @xmath73 and @xmath103, @xmath106 and the acceptance factors @xmath107 and @xmath108 take the form @xmath109 with @xmath110 finally we can impose the acceptance factors @xmath111 and @xmath112 to be equal. this implies @xmath113 defining @xmath114, we are left with a single acceptance factor, @xmath115 which is independent of the chosen direction, and independent of the nature of the move (creation or destruction). thus all moves can be accepted by making use of a proper reweighting, as explained in ref. the appendix shows how to generate random numbers with the appropriate exponential distribution ([exponentialdistribution]). although the above simplified update scheme works, it turns out to have a poor efficiency. this is because of a lack of `` directionality '' : the green operator has, in average, a probability of @xmath116 to choose a left move or a right move. therefore the green operator propagates along the operator string like a `` drunk man '', with a diffusion - like law. the basic creation and destruction processes correspond to the steps of the random walk. this suggests that the efficiency of the update scheme can be improved if one can force the green operator to move in the same direction for several iterations. next section presents a modified version of the simplified update scheme, which allows to control the mean length of the steps of the random walk, that is to say the mean number of creations and destructions in a given direction. the proposed directed update scheme can be considered analogous to the `` directed loop update '' used in the stochastic series expansion algorithm @xcite, which prevents a worm from going backwards. however the connection should not be pushed too far. indeed the picture of a worm whose head is evolving both in space and imaginary time accross vertices is obvious in a loop algorithm. in such algorithm, a creation (or an annihilation) operator which is represented by the head of a worm is propagated both in space and imaginary time, while an annihilation (or a creation) operator represented by the tail of the worm remains at rest. the loop ends when the head of the worm bites the tail. such a worm picture is not obvious in the sgf algorithm : instead of single creation or annihilation operators, it is the full green operator over the whole space that is propagated only in imaginary time. this creates open worldlines, thus introducing discontinuities. these discontinuities increase or decrease while propagating in imaginary time. all open ends of the worldlines are localized at the same imaginary time index. therefore it is actually not possible to draw step by step a worm whose head is evolving in space and imaginary time until it bites its tail. Directed update scheme we present in this section a directed update scheme which is obtained by modifying slightly the simplified update scheme, thus keeping the simplicity and generality of the algorithm. assuming that a left move is chosen, the green operator chooses between starting the move by a creation or a destruction. after having created (or destroyed) an operator, the green operator can choose to keep moving in the same direction and destroy (or create) with a probability @xmath117 (or @xmath118), or to stop. if it keeps moving, then a destruction (or creation) occurs, and the green operator can choose to keep moving and create (or destroy) with a probability @xmath118 (or @xmath117)... and so on, until it decides to stop. if the last action of the move is a creation, then a time shift is chosen. the organigram is represented in figure [directedupdatescheme]. in order to satisfy detailed balance, in addition to the acceptance factors @xmath92 and @xmath93, we need to determine new acceptance factors of the form @xmath119 and @xmath120. we first determine the new expressions of @xmath92 and @xmath93 resulting from the directed update scheme. for @xmath92, the previous probability @xmath51 has to be multiplied by the probability to stop the move after having created, @xmath121. the previous probability @xmath53 has to be multiplied by the probability to stop the move after having destroyed, @xmath122. we get for @xmath92 and @xmath93 the new expressions : @xmath123}{\big\langle\psi_l\big|\hat\mathcal g\big|\psi_r\big\rangle p(\leftarrow)p_\leftarrow^\dagger(\tau) } \\ & \times & \frac{p(\rightarrow^\prime)\big[1-p_\rightarrow^\dagger(\tau^\prime)\big]\big[1-e^{-(\tau_l^\prime-\tau_r^\prime)(v_r^\prime - v_l^\prime)}\big]}{\big[1-p_\leftarrow^{kd}(\tau^\prime)\big]\big(v_r^\prime - v_l^\prime\big) } \\ \nonumber q_\leftarrow^d & = & \frac{\big[1-p_\rightarrow^{kd}(\tau)\big]\big(v_l - v_r\big)}{p(\leftarrow)\big[1-p_\leftarrow^\dagger(\tau)\big]\big[1-e^{-(\tau_l-\tau_r)(v_l - v_r)}\big] } \\ & \times & \frac{\big\langle\psi_l^\prime\big|\hat\mathcal g\big|\psi_r^\prime\big\rangle p(\rightarrow^\prime)p_\rightarrow^\dagger(\tau^\prime)}{\big\langle\psi_l^\prime\big|\hat\mathcal t\hat\mathcal g\big|\psi_r^\prime\big\rangle\big[1-p_\leftarrow^{kc}(\tau^\prime)\big]},\end{aligned}\]] we consider here the case where a left move is chosen, an operator is created on the right of the green operator, and a new state is chosen. then the operator on the left of the green operator is destroyed. using the superscripts @xmath124 to denote intermediate configurations between initial and final configurations, the sequence is the following 1. @xmath125 2. @xmath126 3. @xmath127, where we have @xmath128, @xmath129, @xmath130, and @xmath131. the probability of the transition from the initial configuration to the final configuration is the probability @xmath72 to choose a left move, times the probability @xmath73 to create an operator at time @xmath43, times the probability @xmath132 to choose the new state @xmath133, times the probability @xmath134 to keep moving and destroy, times the probability @xmath135 to stop the move after having destroyed : @xmath136\]] the probability of the reverse move is exactly symmetric : @xmath137\]] it is important to notice that, when in the intermediate configuration @xmath7, the time @xmath138 of the operator to the left of the green operator is equal to @xmath35, and the time @xmath139 of the operator to the right of the green operator is equal to @xmath43. thus the acceptance factor takes the form @xmath140}{\big\langle\psi_l\big|\hat\mathcal g\big|\psi_r\big\rangle p(\leftarrow)p_\leftarrow^\dagger(\tau) } \\ \nonumber & \times & \frac{e^{-\big(\tau_l^a-\tau_r^a\big)v_r^a}p_\rightarrow^{kd}(a)}{e^{-\big(\tau_l^a-\tau_r^a\big)v_l^a}p_\leftarrow^{kd}(a) } \\ & \times & \frac{\big\langle\psi_l^\prime\big|\hat\mathcal g\big|\psi_r^\prime\big\rangle p(\rightarrow^\prime)p_\rightarrow^\dagger(\tau^\prime)}{\big\langle\psi_l^\prime\big|\hat\mathcal t\hat\mathcal g\big|\psi_r^\prime\big\rangle\big[1-p_\leftarrow^{kc}(\tau^\prime)\big]},\end{aligned}\]] and is written as a quantity that depends on the initial configuration, times a quantity that depends on the intermediate configuration @xmath7, times a quantity that depends on the final configuration. it is useful for the remaining of the paper to define the intermediate acceptance factor, @xmath141 we consider here the case where a left move is chosen, the operator on the left of the green operator is destroyed, then an operator is created on its right, and a new state is chosen. finally a time shift is chosen. the sequence of configurations is the following 1. @xmath125 2. @xmath142 3. @xmath127, where we have @xmath143, and @xmath144. the probability of the transition from the initial configuration to the final configuration is the probability @xmath72 to choose a left move, times the probability @xmath145 of no creation, times the probability @xmath146 to keep moving and create, times the probability @xmath74 to choose the new state @xmath75, times the probability @xmath121 to stop the move after having destroyed, times the probability @xmath76 to shift the green operator by @xmath77 : @xmath147p_\leftarrow^{kc}(a)p_\leftarrow(\psi_r^\prime) \\ & \times & \big[1-p_\leftarrow^{kd}(\tau^\prime)\big]p_\leftarrow^{l^\prime r^\prime}(\tau^\prime-\tau_r^\prime)\end{aligned}\]] the probability of the reverse move is exactly symmetric : @xmath148p_\rightarrow^{kc}(a)p_\rightarrow(\psi_l) \\ & \times & \big[1-p_\rightarrow^{kd}(\tau)\big]p_\rightarrow^{lr}(\tau_l-\tau)\end{aligned}\]] the acceptance factor takes the form @xmath149\big(v_l - v_r\big)}{p(\leftarrow)\big[1-p_\leftarrow^\dagger(\tau)\big]\big[1-e^{-(\tau_l-\tau_r)(v_l - v_r)}\big] } \\ \nonumber & \times & \frac{\big\langle\psi_l^a\big|\hat\mathcal g\hat\mathcal t\big|\psi_r^a\big\rangle p_\rightarrow^{kc}(a)}{\big\langle\psi_l^a\big|\hat\mathcal t\hat\mathcal g\big|\psi_r^a\big\rangle p_\leftarrow^{kc}(a) } \\ & \times & \frac{p(\rightarrow^\prime)\big[1-p_\rightarrow^\dagger(\tau^\prime)\big]\big[1-e^{-(\tau_l^\prime-\tau_r^\prime)(v_r^\prime - v_l^\prime)}\big]}{\big[1-p_\leftarrow^{kd}(\tau^\prime)\big]\big(v_r^\prime - v_l^\prime\big)},\end{aligned}\]] and is written as a quantity that depends on the initial configuration, times a quantity that depends on the intermediate configuration @xmath7, times a quantity that depends on the final configuration. it is useful for the remaining of the paper to define the intermediate acceptance factor, @xmath150 we consider here the case where a left move is chosen, an operator is created on the right of the green operator, then the operator on its left is destroyed, then a second operator is created on its right. finally, a time shift of the green operator is performed. the sequence of configurations is the following 1. @xmath125 2. @xmath126 3. @xmath151 4. @xmath152, considering the intermediate configurations @xmath7 and @xmath153 between the intial and final configurations, it is easy to show that the corresponding acceptance factor can be written @xmath154 we consider here the case where a left move is chosen, the operator on the left of the green operator is destroyed, then an operator is created on its right. finally a second operator on the left of green operator is destroyed. the sequence of configurations is the following 1. @xmath155 2. @xmath156 3. @xmath157 4. @xmath127, considering the intermediate configurations @xmath7 and @xmath153 between the intial and final configurations, it is easy to show that the corresponding acceptance factor can be written @xmath158 it is straighforward to show that the acceptance factors of the form @xmath159, @xmath160, @xmath161 (or @xmath162, @xmath163, @xmath164) can be expressed as products of the acceptance factor @xmath92 (or @xmath93) and the intermediate factors @xmath165 and @xmath166. in the same manner, the acceptance factors of the form @xmath167, @xmath168, @xmath169 (or @xmath170, @xmath171, @xmath172) can be expressed as products of the acceptance factor @xmath173 (or @xmath174) and the intermediate factors @xmath165 and @xmath166. here again it is possible to take advantage of the freedom that we have for the choice of the probabilities @xmath72, @xmath175, @xmath118, and @xmath117 (or @xmath102, @xmath176, @xmath177, and @xmath178). a proper choice of these probabilities can be done in order to allow us to accept all moves, simplicity and generality being the leitmotiv of the sgf algorithm. for this purpose, we impose to all acceptance factors corresponding to left (or right) moves to be equal. this requires the intermediate acceptance factors @xmath165 and @xmath166 (or @xmath179 and @xmath180) to be equal to 1. this is realized if @xmath181 where @xmath182 and @xmath183 are optimization parameters belonging to @xmath184. by tuning these parameters, the mean length of the steps of the green operator can be controlled. note that we have explicitly excluded @xmath185 from the allowed values for these optimization parameters. this is necessary for the green operator to have a chance to end in a diagonal configuration, @xmath45. indeed, the choice @xmath186 would systematically lead to values of @xmath185 for the probabilities @xmath187 and @xmath188 for diagonal configurations. therefore the green operator would never stop in a diagonal configution, and no measurement could be done. it is important here to note that the quantities @xmath96, @xmath97, and @xmath98 are evaluated between the states on the left and the right of the green operator that are present at the moment where those quantities are needed, as well as for the times indices @xmath189 and @xmath190 and the potentials @xmath191 and @xmath192. all acceptance factors corresponding to a given direction of propagation become equal if we choose for the creation probabilities : @xmath193(v_l - v_r)}{\big[1-p_\rightarrow^{kc}\big]\big[1-e^{-(\tau_l-\tau_r)(v_l - v_r)}\big] } } \\ & & p_\rightarrow^\dagger(\tau)=\frac{\big\langle\hat\mathcal t\hat\mathcal g\big\rangle}{\big\langle\hat\mathcal t\hat\mathcal g\big\rangle+\big\langle\hat\mathcal g\big\rangle\frac{\big[1-p_\leftarrow^{kd}\big](v_r - v_l)}{\big[1-p_\leftarrow^{kc}\big]\big[1-e^{-(\tau_l-\tau_r)(v_r - v_l)}\big]}},\end{aligned}\]] finally, all acceptances factors become independant of the direction of propagation if we choose @xmath194 and @xmath195 with @xmath196\frac{\big\langle\hat\mathcal g\hat\mathcal t\big\rangle}{\big\langle\hat\mathcal g\big\rangle}+\frac{\big[1-p_\rightarrow^{kd}\big](v_l - v_r)}{\big[1-e^{-(\tau_l-\tau_r)(v_l - v_r)}\big] } \\ r_\rightarrow(\tau)=\big[1-p_\leftarrow^{kc}\big]\frac{\big\langle\hat\mathcal t\hat\mathcal g\big\rangle}{\big\langle\hat\mathcal g\big\rangle}+\frac{\big[1-p_\leftarrow^{kd}\big](v_r - v_l)}{\big[1-e^{-(\tau_l-\tau_r)(v_r - v_l)}\big]}.\end{aligned}\]] as a result all moves can be accepted again, ensuring the maximum of simplicity of the algorithm. we still have some freedom for the choice of the optimization parameters @xmath182 and @xmath183. this is discussed in next section. Test and optimization of the algorithm from the central limit theorem, we know that the errorbar associated to any measured quantity must decrease as the square root of the number of measurements, or equivalently, the square root of the time of the simulation. therefore it makes sense to define the efficiency @xmath197 of a qmc algorithm by @xmath198 where @xmath199 represents the set of all optimization parameters of the algorithm, @xmath200 is the measured quantity of interest, @xmath201 is the time of the simulation, and @xmath202 is the errorbar associated to the measured quantity @xmath200. this definition ensures that @xmath197 is independent of the time of the simulation. as a result, the larger @xmath197 the more efficient the algorithm. in the present case we have @xmath203, while @xmath204 for the original sgf algorithm. it is useful here to realize that, by symmetry, the mean values of @xmath118 and @xmath177 (and @xmath117 and @xmath178) must be equal. therefore we define @xmath205 and @xmath206. it seems reasonable to impose a condition of uniform sampling, @xmath207. this condition can be satisfied by adjusting dynamically the values of @xmath182 and @xmath183 during the thermalization process. for this purpose we introduce a new optimization parameter @xmath208 and apply the following algorithm from time to time while thermalizing (we start with @xmath209) : @xmath210 thus we are left with the optimization parameter @xmath211. in order to determine the optimal value, we have considered 2 different hamiltonians @xmath212 and @xmath213, and evaluated the efficiency of the algorithm while scanning @xmath211. the first hamiltonian we have considered describes free hardcore bosons and is exactly solvable, @xmath214 where the sum runs over pairs of first neighboring sites and @xmath215 is the hopping parameter. the second hamiltonian is highly non - trivial and describes a mixture of atoms and diatomic molecules, with a special term allowing conversions between the two species @xcite, @xmath216 where @xmath217 and @xmath218 (@xmath219 and @xmath220) are the creation and annihilation operators of atoms (molecules), @xmath221, @xmath222, @xmath223, @xmath224, and @xmath225 are respectively the hopping parameter of atoms, the hopping parameter of molecules, the atomic onsite interaction parameter, the molecular onsite interaction parameter, and the inter - species interaction parameter. the conversion term is tunable via the parameter @xmath226 and does not conserve the number @xmath227 of atoms or the number @xmath228 of molecules. however the total number of particles @xmath229 is conserved and is the canonical constraint. the parameter @xmath230 allows to control the ratio between the number of atoms and molecules. the application of the sgf algorithm to the hamiltonian ([twospecies]) is described in details in ref.@xcite. the changes coming with the directed update scheme are completely independent of the chosen hamiltonian. the following table shows the mean number of creations and destructions in one step, @xmath231, and the relative efficiency @xmath232 of the algorithm applied to @xmath212 at half filling, for which we have measured the energy @xmath233, the superfluid density @xmath234, and the number of particles in the zero momentum state @xmath235 : . relative efficiency of the algorithm applied to @xmath212 at half filling for the energy, the superfluid density, and the number of particles in the zero momentum state. [cols="^,^,^,^,^",options="header ",] while the best value of @xmath211 depends on the hamiltonian which is considered and the measured quantity, it appears that a good compromise is to choose @xmath211 between @xmath236 and @xmath237. the improvment of the efficiency is remarkable. in the following, we illustrate the applicability of the algorithm to problems with non - uniform potentials, by adding a parabolic trap to the hamiltonian ([twospecies]) : @xmath238 the parameters @xmath239 and @xmath240 allow to control the curvature of the trap associated to atoms and molecules, respectively, and @xmath31 is the number of lattice sites. the inclusion of this term in the algorithm is trivial since only the values of the diagonal energies @xmath241 and @xmath242 are changed. figures ([density]) and ([momentum]) show the density profiles and momentum distribution functions obtained for a system with @xmath243 lattice sites initially loaded with @xmath244 atoms and no molecules, and the parameters @xmath245, @xmath246, @xmath247, @xmath248, @xmath249, @xmath250, @xmath251, @xmath252, @xmath253, and @xmath254. the presented results have been obtained by performing @xmath255 updates for thermalization, and @xmath256 updates with measurements (an update is to be understood as the occurence of a diagonal configuration). the time of the simulation is about 8 hours on a cheap 32 bits laptop with 1ghz processor, with an implementation of the algorithm involving dynamical structures with pointers (see ref.@xcite). ) to the hamiltonian ([twospecies]). the errorbars are smaller than the symbol sizes, and are the biggest in the neighborhood of site indices 23 and 47 where they equal the size of the symbols., scaledwidth=45.0%] ) to the hamiltonian ([twospecies]). the errorbars are smaller than the symbol sizes, and are the biggest for @xmath257 where they equal the size of the symbols., scaledwidth=45.0%] Conclusion we have presented a directed update scheme for the sgf algorithm, which has the properties of keeping the simplicity and generality of the original algorithm, and improves significantly its efficiency. i would like to express special thanks to peter denteneer for useful suggestions. this work is part of the research program of the `` stichting voor fundamenteel onderzoek der materie (fom), '' which is financially supported by the `` nederlandse organisatie voor wetenschappelijk onderzoek (nwo). '' we describe here how to generate numbers with the appropriate exponential distribution ([exponentialdistribution]). assuming that we have at our disposal a uniform random number generator that generates a random variable @xmath258 with the distribution @xmath259 for @xmath260, we would like to find a function @xmath52 such that the random variable @xmath261 is generated with the distribution @xmath262 where @xmath46 and @xmath263 are the parameters of the exponential distribution. because of the relation @xmath261, the probability to find @xmath264 in the range @xmath265 must be equal to the probability to find @xmath258 in the range @xmath266. this implies the condition @xmath267 with @xmath268. thus we have @xmath269 taking the anti - derivative with respect to @xmath270 on both sides of the equation, we get @xmath271 where @xmath272 is a constant. this constant and the correct sign are determined by imposing the conditions @xmath273 and @xmath274. as a result, if @xmath270 is a realization of @xmath258, then a realization of @xmath264 is given by @xmath275.\]] 10 nicholas metropolis and s. ulam, journal of the american statistical association, number 247, volume 44 (1949). handscomb, proc. 58, 594 (1962). kalos, phys. 128, 1791 (1962). r. blankenbecler, d.j. scalapino and r.l. sugar, phys. d 24, 2278 (1981). g.g. batrouni and r.t. scalettar, phys. b * 46 *, 9051 (1992). w. von der linden, phys. rep. 220, 53 (1992). evertz, g. lana and m. marcu, phys. 70, 875 - 879 (1993). ceperley, rev. 67, 279 (1995). beard and u.- j. wiese, phys. 77 5130 (1996). `` quantum monte carlo methods in physics and chemistry '', ed. m.p. nightingale and c.j. umrigar, nato science series c 525, kluwer academic publishers, dordrecht, (1999). sandvik, j. phys. a * 25 *, 3667 (1992) ; phys. rev. b * 59 *, 14157 (1999). n.v. prokofev, b.v. svistunov, and i.s. tupitsyn, jetp lett. * 87 *, 310 (1998). m. rigol, a. muramatsu, g.g. batrouni, and r.t. scalettar, phys. lett. * 91 *, 130403 (2003). k. van houcke, s.m.a. rombouts, and l. pollet, phys. e * 73*,056703 (2006). rousseau, phys. e * 77 *, 056705 (2008). sandvik, s. daul, r.r.p. singh, and d.j. lett. * 89 *, 247201 (2002). rousseau, r.t. scalettar, and g.g. batrouni, phys. b * 72 *, 054524 (2005). n. metropolis, a.w. rosenbluth, m.n. metropolis, a.h. teller, and e. teller, j. chem. phys. * 21 *, 1087 (1953). olav f. syljuasen, anders w. sandvik, phys. e * 66 *, 046701 (2002). rousseau and p.j.h. denteneer, phys. a * 77 *, 013609 (2008).

in a recent publication we have presented the stochastic green function (sgf) algorithm, which has the properties of being general and easy to apply to any lattice hamiltonian of the form @xmath0, where @xmath1 is diagonal in the chosen occupation number basis and @xmath2 has only positive matrix elements. we propose here a modified version of the update scheme that keeps the simplicity and generality of the original sgf algorithm, and enhances significantly its efficiency.

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