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1307.3566

i

the lhc experiments have discovered a standard model ( sm)-like higgs boson with @xmath11 gev @xcite . however , it is not clear whether this newly discovered particle is a pure sm higgs boson , or whether it is part of an extension of the sm , which might also include multiple or composite higgs - like states . precise measurements of the higgs couplings and branching ratios will be required to resolve this question . another , more direct way to gain insights about physics beyond the sm ( bsm ) would obviously be to discover more new particles directly . a well - motivated extension of the sm is the minimal supersymmetric sm ( mssm ) , a fully renormalisable theory of high predictive power . in this model , the u(1 ) , su(2 ) and su(3 ) gauge couplings can unify , and a dark matter candidate can be naturally accommodated . furthermore , supersymmetry stabilises the higgs potential with respect to quadratically divergent radiative corrections . in the mssm the higgs mass is no longer a free parameter but becomes strongly constrained . if nature is supersymmetric at the electroweak scale , a large number of new superpartner mass parameters is introduced whose values depend on the unknown mechanism of supersymmetry breaking at high energies . this poses a challenge for any phenomenological analysis . due to the large number of unknown parameters it is very difficult to make reliable predictions for susy background processes , and in many cases the worst background contributions for susy signals originate in fact from susy processes . the experiments at the lhc have not yet found any hints for supersymmetry in squark and gluino searches @xcite , which points towards a rather heavy coloured superpartner spectrum with soft masses above a . however , there are large regions in the mssm parameter space which lead to a sm - like higgs with @xmath12 gev , a light electroweak susy sector , and a rather heavy coloured susy sector beyond the current lhc discovery range . in general , if the electroweak sector is light enough , it should be possible to discover neutralinos and charginos via drell yan production at the lhc @xcite , and the lhc experiments have already put limits on the electroweak - ino parameter space under several simplified model assumptions @xcite . however , in cases where these light states are very close in mass and their decay products are therefore very soft , it will be almost impossible to observe such processes at the lhc . an example is given by scenarios in which the higgsino mass parameter @xmath6 is much smaller than the electroweak gaugino masses @xmath9 and @xmath10 ( current bounds require only @xmath13{gev}$ ] in order to avoid the limits from lep chargino searches @xcite ) . then , the new light states are a chargino and two neutralinos which are higgsino - like and almost mass degenerate . the chargino and the heavier neutralino therefore decay only into extremely soft sm particles and the invisible lightest neutralino . the most `` natural '' versions of the phenomenological mssm which are still not ruled out are indeed characterised by higgsino masses of a few 100 , together with sub- third - generation squarks and a moderately heavy gluino , while the rest of the spectrum can be in the multi - tev range @xcite . models with light higgsinos are also attractive for cosmology , either in combination with gravitino dark matter or with a higgsino - like neutralino as a non - thermally produced lightest supersymmetric particle ( lsp ) ; see e.g. @xcite . while the light higgsino scenario is extremely challenging for the lhc , it may well be observable in the clean environment at an @xmath14 linear collider , for instance , at the international linear collider ( ilc ) @xcite . at the ilc , the striking feature of a precisely known initial state allows to successfully apply the initial - state radiation ( isr ) method , i.e. to select only those signal events that are accompanied by a sufficiently hard photon . in this context , polarised beams help to significantly enhance the signal cross section , at least in the chargino case . the subject of the present work is the phenomenology of light quasi - degenerate higgsinos at the ilc . we study their production and decay , and investigate in how far one can derive the fundamental mssm parameters without assuming any specific susy breaking scheme if only the light higgsinos are kinematically accessible . we study the processes @xmath15 and @xmath16 at @xmath17 gev with polarised beams and initial - state radiation at the ilc . we simulate the detector response for two mssm benchmark points with different @xmath18 gev ) mass splittings between the chargino and neutralinos , leading to different final state sm particles . the uncertainties on masses , mass differences and cross sections as determined by our simulation study are then used as inputs for a fit in order to reconstruct the mssm parameters @xmath9 , @xmath10 , @xmath6 and @xmath19 . our overall goal is to determine both how well the chargino and neutralino masses can be measured at the ilc , and what could be inferred from such a measurement about the underlying susy model . in the following chapters we first give an overview of the theoretical arguments in favour of light higgsinos and the phenomenological aspects of such scenarios in the mssm . the expected sm background will also be discussed , including beam conditions and detector capabilities at the ilc . in section 3 we define two benchmark scenarios on which our study will be based and provide the details for the detector simulation . in section 4 we discuss the expected experimental results at the ilc . in section 5 we present our results on reconstructing the fundamental parameters and the precision we expect to achieve with our analysis strategy .

such light , and states can be almost mass degenerate , and their decays are then difficult to observe at colliders . we perform a parameter fit in the mssm , from which we infer that the higgsino mass parameter can be measured to a precision of about gev . for the electroweak gaugino mass parameters , , which are chosen in the multi - tev range , a narrow region is compatible with the measurements .

in supersymmetric extensions of the standard model , higgsino - like charginos and neutralinos are preferred to have masses of the order of the electroweak scale by naturalness arguments . such light , and states can be almost mass degenerate , and their decays are then difficult to observe at colliders . in addition to the generic naturalness argument , light higgsinos are well motivated from a top - down perspective . for instance , they arise naturally in certain models of hybrid gauge - gravity mediation . in the present analysis , we study two benchmark points which have been derived in the framework of such a model , which exhibit mass differences of in the higgsino sector . for chargino pair and neutralino associated production with initial - state photon radiation , we simulate the detector response and determine how accurately the small mass differences , the absolute masses and the cross sections can be measured at the international linear collider . assuming that 500 fb has been collected at each of two beam - polarisations , we find that the mass differences can be measured to 40300 mev , the cross sections to 25% , and the absolute masses to 1.53.3 gev , where the range of values correspond to the different scenarios and channels . based on these observables we perform a parameter fit in the mssm , from which we infer that the higgsino mass parameter can be measured to a precision of about gev . for the electroweak gaugino mass parameters , , which are chosen in the multi - tev range , a narrow region is compatible with the measurements . for both parameters independently , we can determine a lower bound .

1301.7692

i

warped extra dimensional models have become very popular because they are able to address simultaneously two intriguing issues within the standard model ( sm ) : the hierarchy problem and the mass / flavor problem . they were originally introduced to treat the first issue @xcite in a setup where the sm fields were all localized at one boundary of the extra dimension . later it was realized that by allowing fields to propagate into the bulk , different geographical localization of fields along the extra dimension could help explain the observed masses and flavor mixing among quarks and leptons @xcite . flavor bounds and precision electroweak tests put pressure on the mass scale of new physics in these models @xcite , but extending the gauge groups and/or matter content ( e.g. @xcite ) or by slightly modifying the spacetime warping of the metric ( e.g @xcite ) , it is possible to keep the new physics scale at the tev level at the reach of the large hadron collider ( lhc ) . electroweak symmetry breaking can still happen via a standard higgs mechanism in these scenarios ( although it can also be implemented as as pseudo - nambu - goldstone boson ( pngb ) @xcite or described within the effective theory formalism @xcite ) . as the lhc announced the discovery of a light higgs - like particle of a mass around 125 gev @xcite , it becomes crucial to have a detailed prediction of the properties of the physical higgs particle in these models . the higgs boson itself must be located near the tev boundary of the extra dimension in order to solve the hierarchy problem , and so typically it is assumed to be exactly localized on that boundary ( brane higgs scenario ) . nevertheless , it is possible that it leaks out into the bulk ( bulk higgs scenario ) , and in doing so indirectly alleviate some of the bounds plaguing these models @xcite . the calculation of the production cross section of the brane higgs in these scenarios has been addressed before @xcite but we will pay close attention to the more recent works of @xcite . the towers of fermion kaluza - klein ( kk ) modes will affect significantly the sm prediction and in @xcite it was found that the higgs boson production rate can receive important corrections , either enhancing or suppressing the standard model prediction . the suppression or enhancement depends on the model parameters considered , in particular on the phases appearing in the different yukawa - type operators present in the 5d action . previously , the analysis of @xcite , in which only the first few modes were considered , gave no contribution to the rate from the towers of kk fermions . finally , the analysis of @xcite seems to indicate that with just a few kk modes a substantial effect is obtained , but of opposite sign as the one predicted from summing the infinite tower @xcite . in this work we consider the effects of allowing the higgs boson to propagate in the bulk , with its profile more or less localized towards the ir brane depending on the value of the mass parameter @xmath0 , related to the bulk mass of the 5d higgs field . to keep matters as simple as possible we will set up a model containing a single family of up - type 5d fermions along with a bulk higgs scalar . generalization to a more realistic scenario is straight forward but we prefer to stay as transparent as possible due to the many subtleties involved in the calculation . we first compute the contribution of the complete tower of kk fermions to the higgs production cross section as well as to the tree - level shift happening between the light fermion mass and its yukawa coupling ( leading to flavor violating couplings when considering three fermion families ) . these calculations , as outlined in @xcite , are analytically straightforward and allow us to obtain simple and compact results . we then repeat the same analysis numerically from the point of view of an effective theory in which only the first few kk fermions contribute . we show that for a bulk higgs with a thickness of the order of inverse tev scale , the results obtained are the same as the results obtained by summing the complete kk tower ( i.e. heavier modes decouple ) . moreover , these results are consistent with the predictions obtained in @xcite for the specific case of a brane localized higgs . the two aproaches outlined seem to give different predictions as the bulk higgs is continuously pushed towards the brane . it turns out that in order to maintain the consistency of both approaches we need to include in the analysis the effects of a special type of higher order operators . after these effects are included , we will come back and address in the discussion section the differences among the existing calculations in the literature and stress the importance of including the mentioned higher order operators in the analysis . our paper is organized as follows . in sec . [ sec : setup ] we summarize the simple 5d warped space model used in the calculation . in sec . [ sec : infinite ] we present analytical results for the higgs flavor - changing effects ( [ subsec : infinitefcnc ] ) and production ( [ subsec : infiniteprod ] ) , using the full tower of kk fermions . we use numerical methods to calculate the effects of including just a few kk modes in sec . [ sec : individual ] , both for flavor - changing neutral currents effects ( [ subsec : individualfcnc ] ) and higgs boson production ( [ subsec : individualprod ] ) . we include the effect of the higher order operator in sec . [ sec : ramanop ] and discuss the misalignment between the higgs boson profile and its vacuum expectation value ( vev ) in sec . [ sec : vevmisalignment ] . we discuss the significance of our results , compare them to previous analyses and conclude in sec . [ sec : conclusion ] . we leave some of the details for the appendices [ apen : i ] and [ apen : ii ] .

in the context of warped extra - dimensional models with all fields propagating in the bulk , we address the phenomenology of a bulk scalar higgs boson , and calculate its production cross section at the lhc as well as its tree - level effects on mediating flavor changing neutral currents .

in the context of warped extra - dimensional models with all fields propagating in the bulk , we address the phenomenology of a bulk scalar higgs boson , and calculate its production cross section at the lhc as well as its tree - level effects on mediating flavor changing neutral currents . we perform the calculations based on two different approaches . first , we compute our predictions analytically by considering all the degrees of freedom emerging from the dimensional reduction ( the infinite tower of kaluza klein modes ( kk ) ) . in the second approach , we perform our calculations numerically by considering only the effects caused by the first few kk modes , present in the 4-dimensional effective theory . in the case of a higgs leaking far from the brane , both approaches give the same predictions as the effects of the heavier kk modes decouple . however , as the higgs boson is pushed towards the tev brane , the two approaches seem to be equivalent only when one includes heavier and heavier degrees of freedom ( which do not seem to decouple ) . to reconcile these results it is necessary to introduce a type of higher derivative operator which essentially encodes the effects of integrating out the heavy kk modes and dresses the brane higgs so that it looks just like a bulk higgs . +

1301.7692

c

in this work we have presented the results for the predictions of higgs phenomenology in a toy - model rs setup in which the higgs field is allowed to propagate in the bulk and with a single 5d fermion field . our results can be extended to three families to include full flavor effects , but the generic predictions that we would obtain are expected to be basically the same as the ones presented in @xcite . that is , that in the context of flavor anarchy , where the action parameters are all of the same order but with more or less random values and phases ( with the constraint of obtaining correct sm predictions ) the couplings of the higgs with fermions and gluons and photons can receive important corrections , either enhancing or suppressing the sm predictions . however , the two references mentioned present calculations performed by including the effect of all the kk fermions , technically assuming an infinite cut - off for the model ( where a brane higgs is considered ) . in general , all these scenarios break down at a low cut - off , becoming strongly coupled for both gauge and yukawa interactions . the implicit assumption made in @xcite was that the effects of the heavier modes should decouple quickly , at least for the case of a bulk higgs field . the main motivation to perform the calculations by considering the full infinite fermion kk tower , as well as pushing the higgs into the brane was mainly of technical nature . indeed both the flavor structure of the higgs yukawa couplings as well as the coupling to gluons and photons can be computed analytically with those ingredients . in @xcite , the authors checked analytically that the corrections to the higgs yukawa couplings were actually of the same order for a bulk higgs and a brane higgs . however it was pointed out in @xcite that in the brane higgs case , the effects of the heavier kk fermion modes do not decouple and that they all contribute evenly in the computation of the higgs couplings in the model . on the other hand , we showed in sections [ sec : infinite ] and [ sec : individual ] of this paper that the heavier kk modes in the case of a bulk higgs do decouple very quickly , so that the analytical result obtained by using the infinite kk tower approaches with great precision the numerical result obtained by considering an effective theory with only a few kk fermion modes . moreover , when considering the effective theory with only a few kk modes , one should include in the action all possible operators and in particular the higher derivative ones introduced in section [ sec : ramanop ] . these effects were omitted in @xcite , and as we showed in this work , the importance of these operators increases as the higgs is more and more localized towards the brane . in @xcite , the authors considered an rs setup with a highly localized higgs and the presence of only a few kk fermions and studied the effects on the higgs couplings to gluons and photons , among other observables . in the limit of the sm gauge group ( they did consider an extended gauge group ) they found no significant deviations from the sm predictions . indeed this result is consistent with our findings of section [ sec : individual ] ( no higher derivative operators invoked yet ) , since as it can be seen on figures [ rh ] and [ rl ] , the shift in higgs yukawa couplings and the new effects to higgs - gluon - gluon coupling vanish in the limit of highly localized higgs ( large @xmath0 parameter ) . on the other hand , in @xcite it is claimed that large effects should be present in the case of a brane higgs and with only a few kk modes present in the effective theory ( and no higher derivative operators ) , a result inconsistent with both our findings and those found in @xcite . we can trace the origin of the disagreement in their calculation of the higgs yukawa couplings . those are computed by using the full 5d equations of motion , which as we have said earlier is equivalent to considering the complete tower of kk modes . then , using these couplings , they calculate the @xmath54 radiative coupling but now including only a finite amount of kk fermions . this treatment leads to a highly suppressed top quark yukawa coupling ( due to effects from the infinite kk tower ) and a vanishing contribution to @xmath54 from the loops of kk fermions considered ( one would need the whole tower to obtain a finite effect ) . their end result is a suppressed top quark yukawa coupling and a suppressed @xmath54 coupling ( due to the smaller top quark yukawa ) , predictions which are at odds with the findings of @xcite and of this paper . the procedure of @xcite seems inconsistent because essentially the authors use infinite kk degrees of freedom in one part of their calculation ( the sm quark yukawa couplings computation via equations of motion ) but then they truncate the kk degrees of freedom in order to compute the @xmath54 coupling . in any case , had they included the higher derivative operators introduced in this paper , their results would have changed dramatically since then , the effect to @xmath54 coming from the top quark yukawa loop would remain basically the same , but the effects due to loops of a few kk fermions would dominate the overall effect ( and thus the result would start to become consistent with the findings of @xcite ) . also , the predictions of @xcite should change if one considers the effects of the higher derivative operators introduced in section [ sec : ramanop ] . in that situation , the higgs couplings can receive large corrections , and can be of any sign ( suppression or enhancement ) due to the different phases present in the couplings @xmath119 and @xmath102 . in fact we have found here that for a higgs field in the bulk , our results are more predictive than for a brane higgs field , because the effect of the higher derivativer operators is subdominant for a bulk higgs field . the effects from only the 5d yukawa operators are aligned @xcite , and thus all the kk quarks add up in phase . in that situation we can have definite predictions for the effects caused by a single family of fermions , i.e. it will produce a suppression in the light quark yukawa coupling and an enhancement in higgs boson production ( as well as suppression in the higgs to photons coupling ) @xcite , with the caveat of taking the dimensionless couplings of both yukawa terms and higher derivative operators to be the same ( consistent with the usual assumption that all 5d coefficients have to be of the same order ) . taking into account the three fermion families in conjunction with a bulk higgs field might weaken this prediction due to complicated flavor mixings and structure , but still one should be able to draw a correlation between yukawa couplings and higgs production ( and @xmath126 ) for the case of a bulk higgs field . the parameter space of the bulk higgs scenario can therefore be under a tighter pressure as more and more precise experimental measurements in higgs observables at the lhc become available . in particular if the predicted and correlated deviations of higgs couplings is not clearly observed this should put bounds on the kk scale of the bulk higgs scenario . the situation for a higgs on the brane is different . the higher order derivative terms are now important . each kk tower of light quarks and the top will contribute to the @xmath54 coupling , but their effect depends on arbitrary relative phases ( between @xmath102 and @xmath127 ) , and so one can not make a firm statement about the magnitude and phase of the overall contribution : it can be a suppression or an enhancement , or in between . finally we comment again on the apparent problem of a highly localized higgs scenario ( brane higgs ) in which predictions made from a truncated fermion kk tower are very different from predictions made from an infinite fermion kk tower . this apparent uv - sensitivity can actually be lifted by considering the higher derivative operators described here ( first introduced in @xcite ) . when these are included , the predictions made with a finite kk fermion tower become consistent with the original predictions obtained with an infinite fermion tower . a more esthetic problem with the brane higgs scenario remains , since the definition of the higgs operators seems highly unnatural , if one understands a brane higgs field as a limit of a bulk higgs field . all operators involving higgs fields will have to have a precise and definite dependence on @xmath0 ( a large number ) , which seems quite contrived , specially in a framewrok in which no big numerical hierarchies should arise from fundamental 5d coefficients . in any case , with the ansatz outlined in the text and reviewed in appendix [ apen : ii ] , one can still work consistently with a brane higgs field as a limit case of a bulk higgs field . would like to thank kaustubh agashe for many discussions and comments and specially alex azatov and lijun zhu for their invaluable help and collaboration in the early stages of this work . this work is supported in part by nserc under grant number sap105354 .

first , we compute our predictions analytically by considering all the degrees of freedom emerging from the dimensional reduction ( the infinite tower of kaluza klein modes ( kk ) ) . in the second approach it is necessary to introduce a type of higher derivative operator which essentially encodes the effects of integrating out the heavy kk modes and dresses the brane higgs so that it looks just like a bulk higgs . +

in the context of warped extra - dimensional models with all fields propagating in the bulk , we address the phenomenology of a bulk scalar higgs boson , and calculate its production cross section at the lhc as well as its tree - level effects on mediating flavor changing neutral currents . we perform the calculations based on two different approaches . first , we compute our predictions analytically by considering all the degrees of freedom emerging from the dimensional reduction ( the infinite tower of kaluza klein modes ( kk ) ) . in the second approach , we perform our calculations numerically by considering only the effects caused by the first few kk modes , present in the 4-dimensional effective theory . in the case of a higgs leaking far from the brane , both approaches give the same predictions as the effects of the heavier kk modes decouple . however , as the higgs boson is pushed towards the tev brane , the two approaches seem to be equivalent only when one includes heavier and heavier degrees of freedom ( which do not seem to decouple ) . to reconcile these results it is necessary to introduce a type of higher derivative operator which essentially encodes the effects of integrating out the heavy kk modes and dresses the brane higgs so that it looks just like a bulk higgs . +

1006.4489

i

although the standard single - degenerate ( sd ) chandrasekhar - mass scenario ( see @xcite for a review ) is capable of explaining most of the observed diversity of type ia supernovae ( sne ia ) @xcite via the delayed - detonation @xcite model , it suffers from severe problems in explaining the observed rate of sne ia . in particular , binary evolution population synthesis calculations predict rates which are an order of magnitude too low compared to the observed rate of sne ia ( @xcite , but see @xcite ) . in addition , recent observational studies suggest that chandrasekhar - mass explosions of hydrogen - accreting carbon - oxygen ( c / o ) white dwarfs ( wds ) in sd binary systems can not account for all sne ia : @xcite found that the x - ray flux of nearby elliptical galaxies is significantly weaker than expected for a population of wds accreting hydrogen towards the chandrasekhar - mass needed to explain the observed supernova rate in elliptical galaxies ( see also @xcite ) . moreover , there is growing observational evidence that there are different populations of sne ia @xcite . this has led to a revived interest in alternative explosion mechanisms . here we consider the double detonation scenario applied to sub - chandrasekhar - mass c / o wds . in that scenario a helium - accreting c / o wd explodes below @xmath2 due to a detonation in the accreted helium shell which triggers a secondary core detonation by compressional heating @xcite . this model has some very appealing features . depending on the initial mass of the wd , a wide range of explosion strengths can be realized ( e.g. @xcite ) . moreover , population synthesis studies @xcite predict rates comparable to the observed galactic supernova rate . however , earlier work @xcite found light curves and spectra of such models to be in conflict with the observed spectra and light curves of sne ia . the differences were mainly attributed to the composition of the outer layers . due to the initial helium detonation in the outer shell , the ejecta of sub - chandrasekhar - mass models are surrounded by a layer of helium and its burning products ( which can include iron - peak nuclei ) . this , however , is in apparent contradiction to the layered composition structure of observed sne ia , where the composition changes from iron - group elements in the core to lower mass elements in the outer layers . in a preceding paper @xcite have shown that artificial explosions of `` naked '' sub - chandrasekhar - mass wds can reproduce the observed diversity of sne ia . thus it is natural to ask if somewhat modified properties in the initial helium shells of realistic sub - chandrasekhar - mass models can reduce the negative post - explosion effect of this shell on the observables . in particular , @xcite recently presented new calculations , indicating that detonations might occur for much less massive helium shells than previously thought . @xcite adopted the minimum helium shell masses of @xcite and investigated whether such low - mass helium detonations are capable of triggering a secondary detonation in the c / o core of the wd . in that study , they concluded that as soon as a detonation in the helium shell initiates , a subsequent core detonation is virtually inevitable . for example , they found that even a helium shell mass as low as @xmath3 is sufficient to detonate a c / o wd of @xmath4 . here , we focus on the observable properties of the models presented in @xcite and their comparison to real sne ia . in particular , we investigate whether the low helium shell masses of these models help to alleviate the problems encountered previously when comparing double detonation sub - chandrasekhar - mass models to observed spectra and light curves of sne ia @xcite . the outline of the paper is as follows : in section [ sec : models ] we give a short summary of the models of @xcite before briefly describing details of our radiative transfer simulations in section [ sec : rt ] . in section [ sec : oo ] we present synthetic observables for the @xcite models and compare them to the observed properties of sne ia . the results of this comparison and implications for future work on sub - chandrasekhar - mass double detonations are discussed in section [ sec : discussion ] . finally , we draw conclusions in section [ sec : conclusions ] .

this shell detonation drives a shock front into the carbon - oxygen white dwarf that triggers a secondary detonation in the core . recently , however , it was proposed that detonations might be possible for much less massive helium shells than previously assumed . moreover , it was shown that even detonations of these minimum helium shell masses robustly trigger detonations of the carbon - oxygen core . therefore it is possible that the impact of the helium layer on observables is less than previously thought . here , we present time - dependent multi - wavelength radiative transfer calculations for models with minimum helium shell mass and derive synthetic observables for both the optical and-ray spectral regions .

in the double detonation scenario for type ia supernovae it is suggested that a detonation initiates in a shell of helium - rich material accreted from a companion star by a sub - chandrasekhar - mass white dwarf . this shell detonation drives a shock front into the carbon - oxygen white dwarf that triggers a secondary detonation in the core . the core detonation results in a complete disruption of the white dwarf . earlier studies concluded that this scenario has difficulties in accounting for the observed properties of type ia supernovae since the explosion ejecta are surrounded by the products of explosive helium burning in the shell . recently , however , it was proposed that detonations might be possible for much less massive helium shells than previously assumed . moreover , it was shown that even detonations of these minimum helium shell masses robustly trigger detonations of the carbon - oxygen core . therefore it is possible that the impact of the helium layer on observables is less than previously thought . here , we present time - dependent multi - wavelength radiative transfer calculations for models with minimum helium shell mass and derive synthetic observables for both the optical and-ray spectral regions . these differ strongly from those found in earlier simulations of sub - chandrasekhar - mass explosions in which more massive helium shells were considered . our models predict light curves which cover both the range of brightnesses and the rise and decline times of observed type ia supernovae . however , their colours and spectra do not match the observations . in particular , their colours are generally too red . we show that this discrepancy is mainly due to the composition of the burning products of the helium shell of the models which contain significant amounts of titanium and chromium . using a toy model , we also show that the burning products of the helium shell depend crucially on its initial composition . this leads us to conclude that good agreement between sub - chandrasekhar - mass explosions and observed type ia supernovae may still be feasible but further study of the shell properties is required .

1006.4489

c

in the last section we presented synthetic observables for the sub - chandrasekhar - mass double detonation models with minimum helium shell mass of @xcite . compared to observed sne ia , these models show some promising features : 1 . they predict a wide range of brightnesses that covers the whole range of observed sne ia . 2 . their light curve rise and decline times are for most bands in reasonable agreement with those observed . but despite these positive features the @xcite models can not account for all the properties of observed sne ia since they have peculiar light curves and spectra : 1 . the colours of the models are too red compared to observed sne ia . this is particularly obvious in the evolution of the @xmath1 colour , where all the models are redder than spectroscopically normal sne ia for all epochs . 2 . the model spectra can not reproduce the strong features of intermediate - mass elements typical of sne ia at maximum light . in detail , there are further problems concerning the exact light curve shapes and decline rates as well as a strong viewing - angle dependence which is caused by the point - like ignition of our models . we have argued that all these problems are mainly due to the burning products of the helium shell . moreover , @xcite , have recently shown that detonations of centrally ignited spherically symmetric naked sub - chandrasekhar - mass c / o wds are capable of reproducing the observed diversity of light curves and spectra of sne ia at least to a similar level of agreement as models of the more standard chandrasekhar - mass delayed detonations . this naturally leads us to speculate on whether the helium shell in double detonation models could be altered in some way to reduce its negative impact on the synthetic observables . we first explicitly investigate the extent to which the shortcomings and viewing - angle dependence of our models can indeed be attributed to the helium shell . for that purpose we constructed ( for model 3 ) a toy model which contains only the burning products of the detonation in the initial c / o core but not those of the initial helium shell . for the models of @xcite this can be done in a straightforward manner : since the models use two different sets of tracer particles to simulate the nucleosynthetic yields of core and shell burning respectively ( see section 3.3 of @xcite ) , we obtain such a model by restricting our sph - like reconstruction algorithm ( section [ sec : rt ] ) to the core tracers . properties of this `` core - only '' model ( hereafter 3c ) are listed in table [ tab : modelparas ] . figure [ fig : m03_nohe_lightcurves ] shows band - limited synthetic light curves obtained from our radiative transfer simulations for this model as seen equator - on and from the two polar directions . in contrast to the light curves of model 3 in figure [ fig : m03_lightcurves ] , which are strongly dependent on the viewing angle , the light curves of model 3c show only a moderate line - of - sight dependence . thus , at maximum light we find now only a variation of @xmath292 and @xmath293 magnitudes for @xmath237 and @xmath238 band , respectively . for model 3 these values were significantly larger ( @xmath280 and @xmath281 magnitudes , respectively ) . redder bands show no significant line - of - sight dependence in model 3c . moreover , the light curves of model 3c give an excellent representation of sn 2004eo in the @xmath238 , @xmath239 and @xmath240 bands . @xmath237 and @xmath241 are not in perfect agreement , but still reasonable compared to the agreement between other first principles explosion models and observed sne ia ( e.g. @xcite ) . in particular , the colours of this toy model are now in good agreement with observed sne ia and not too red as it is the case for model 3 . in the nir bands , in contrast , the agreement is no better . however , the nir light curves are much more difficult to model accurately since they require simulations with an extensive atomic data set to properly simulate flux redistribution by fluorescence which strongly affects these bands @xcite . here , however , we have restricted ourselves for computational reasons to a simplified atomic data set ( cd23_gf-5 of @xcite ) with only @xmath294 lines . this has been shown to give reliable results in the optical bands which , for our purposes , are the most important since they are the most different between our toy model 3c and model 3 of @xcite . in the nir bands , in contrast , model 3c and model 3 give rather similar results ( see figure [ fig : compare_m03_lightcurves ] ) . the good agreement between our toy model 3c and observational data does not only hold for band - limited light curves but also for individual spectral features as can be seen from figure [ fig : m03_nohe_spectrum ] which shows a spectrum of model 3c at 3 days before @xmath238-band maximum . compared to sn 2004eo , our toy model succeeds in reproducing the characteristic spectral features of intermediate - mass elements in sne ia . this is highlighted by our colour coding . moreover , it shows an overall flux distribution which is in almost perfect agreement with the observed spectrum of sn 2004eo and we see no strong flux redistribution by titanium ( compare with model 3 in figure [ fig : maxspectra_all ] ) . this confirms our conclusion from section [ sub : oo_los ] that the peculiarities of our model spectra with respect to the observations and their strong viewing - angle dependence are mainly due to the shell material and its compositional asymmetries . it also shows that the off - centre ignition of the secondary detonation in the c / o core causes only a minor viewing - angle dependence which is on the order of the observed variation between sne ia . in light of the discussion above , we are motivated to speculate on how the influence of the helium shell might be reduced . in the sub - chandrasekhar - mass double detonation scenario , the helium shell can not be removed entirely since it is required to trigger the detonation . also the helium shell mass adopted in the @xcite models is already the minimum that might be expected to detonate @xcite . in section [ sec : oo ] , however , we have argued that the differences between our model spectra and observations are not a consequence of the helium itself but of its particular burning products , namely titanium and chromium produced in the outer layers . the yields of these elements are affected by details of the nucleosynthesis in the shell . the degree of burning in the shell material ( and thus its final composition ) can be affected by the initial abundance of heavy nuclei ( e.g. @xmath295c ) which in turn depend strongly on triple-@xmath296 reactions during previous hydrostatic burning and dredge - up phases from the core @xcite . since the time - scale for @xmath296-captures behind the detonation shock front is significantly shorter than that of triple-@xmath296 reactions , such seed - nuclei can limit the @xmath296-chain before reaching nuclear statistical equilibrium . if , for example , in a shell consisting of a mixture of @xmath297he and @xmath295c the number ratio of free @xmath296-particles to @xmath295c - nuclei on average is less than 6 ( corresponding to a mass ratio of 2 ) , the @xmath296-chain will end at @xmath298ar . thus , it is possible that more intermediate - mass elements and less titanium and chromium may be produced . therefore it is interesting to consider how the burning of the helium might be different from that found by @xcite for different initial compositions of the shell . a full study of this goes beyond the scope of this work and will be published in a follow - up study . here , we illustrate the possibility of obtaining better agreement with data for just one example . in the @xcite models it was initially assumed that the shell consisted of pure helium . to demonstrate the sensitivity to the initial composition of the shell , we set up another toy model . for this `` modified '' model ( hereafter 3 m ) , we homogeneously polluted the shell of model 3 with 34% ( by mass ) of @xmath295c and repeated the hydrodynamics and nucleosynthesis calculation ( in the same way as described by @xcite ) ( compared to @xmath299 in model 3 ) to suppress further triple-@xmath296 burning in the shell . as a consequence the density and also the shell mass changes slightly compared to the original model 3 ( cf . table [ tab : modelparas ] ) . ] . we found that a core detonation was still triggered but the different shell burning led to a substantial reduction of the mass of @xmath300ti , @xmath193cr and @xmath192fe in the shell ( nucleosynthetic yields for core and shell of the model are given in table [ tab : modelparas ] ) . since the detonation tables of @xcite are only valid for pure helium , model 3 m is not fully self - consistent . nevertheless , it is a useful toy model to explore the basic effect of a modified shell composition . figure [ fig : compare_m03_lightcurves ] compares the angle - averaged band - limited light curves of this modified model ( 3 m ) , to those of model 3 of @xcite and our core - only toy model ( 3c ) . as can be seen , the modified model 3 m produces light curves very similar to those of model 3c despite having about the same shell mass as model 3 . the most obvious difference between our modified and core - only models occurs in the @xmath237 band : the titanium in the outer layer of model 3 m causes some line blocking , leading generally to a dimmer @xmath237-band magnitude than for model 3c which has no outer layer . compared to model 3 , with its large titanium mass in the shell , however , this effect is much weaker . the @xmath238 band , which is strongly affected in model 3 , shows no significant titanium absorption for model 3 m . another slight difference between model 3 m and 3c occurs after the first peak in the @xmath241 band . comparing the light curves of model 3 m to sn 2004eo , we find qualitatively similar agreement as for model 3c . this generally also holds for the angle - averaged spectrum at 3 days before @xmath238-band maximum , shown in figure [ fig : m03_modified_spectrum ] . compared to model 3c , where the agreement was almost perfect , there are some minor shortcomings . but the model is dramatically improved compared to model 3 of @xcite . the small differences between model 3c and 3 m are again mostly due to the titanium in the outer layers which leads to pronounced absorption troughs bluewards of the caii h and k lines and redwards of @xmath301 . this suggests that model 3 m still over - produced titanium in the shell . interestingly , the enhanced calcium abundance in the outer layers ( cf . table [ tab : modelparas ] ) leads to a stronger caii nir triplet , bringing the model in better agreement with the spectrum of sn 2004eo at the corresponding epoch . therefore some _ calcium _ in the outer shell is an _ improvement _ over model 3c . this suggests that a slight further reduction in the degree of burning so that titanium is further suppressed in favour of calcium ( just one step down the @xmath296-chain ) could lead to very good agreement . in summary , polluting the initial helium shell of model 3 m with @xmath295c significantly improved the agreement between our synthetic spectra and light curves and those observed for sne ia making this model a promising progenitor candidate for sne ia . we stress again that this improvement results only from the change in the composition of the burning products of the helium shell which contains much less titanium and chromium for this model ( the total shell mass stays about the same ) . given that the initial composition of the helium shell depends on several processes including details of the accretion physics and hydrostatic burning phases that might precede the detonation or possible dredge - up of material from the c / o core , this leaves some scope to find sub - chandrasekhar - mass double detonation models which give reasonable agreement with observed sne ia . this , however , must be investigated by future follow - up studies that more fully explore the influence of the initial composition of the helium shell on the burning products and link the initial composition of the helium shell directly to the evolution of progenitor models . moreover , we note that different ignition geometries might also lead to better agreement with observational data . in particular , more symmetric ignition geometries , e.g. ignition in an equatorial ring or simultaneous ignition in multiple points ( as studied by @xcite for the case of more massive helium shells ) , are likely to alleviate the strong viewing - angle dependence found for the point - like ignition of the @xcite models . our results also highlight the strong sensitivity of the radiative transfer to particular elements / ions ( in our case titanium and chromium which represent only a tiny fraction of the ejecta mass yet dominate our conclusions ) . this emphasizes the need for a better description of nuclear reaction rates and continued study of the radiative transfer processes ( and atomic data ) in order to quantify more fully the systematic uncertainties which arise due to the complexity of spectrum formation in supernovae . in particular , we note that almost all the flux redistribution done by titanium and chromium in our models is due to their singly ionized states . since the current ionization treatment of artis neglects non - thermal processes ( see @xcite for more details ) , we can not exclude that the actual ionization state in the helium shell ejecta would be higher due to non - thermal ionization from the radioactive isotopes produced during the helium burning . this could also significantly improve the agreement between our models and observational data , as a numerical experiment with an artificially enforced higher ionization state for titanium and chromium has shown .

we show that this discrepancy is mainly due to the composition of the burning products of the helium shell of the models which contain significant amounts of titanium and chromium . using a toy model , we also show that the burning products of the helium shell depend crucially on its initial composition .

in the double detonation scenario for type ia supernovae it is suggested that a detonation initiates in a shell of helium - rich material accreted from a companion star by a sub - chandrasekhar - mass white dwarf . this shell detonation drives a shock front into the carbon - oxygen white dwarf that triggers a secondary detonation in the core . the core detonation results in a complete disruption of the white dwarf . earlier studies concluded that this scenario has difficulties in accounting for the observed properties of type ia supernovae since the explosion ejecta are surrounded by the products of explosive helium burning in the shell . recently , however , it was proposed that detonations might be possible for much less massive helium shells than previously assumed . moreover , it was shown that even detonations of these minimum helium shell masses robustly trigger detonations of the carbon - oxygen core . therefore it is possible that the impact of the helium layer on observables is less than previously thought . here , we present time - dependent multi - wavelength radiative transfer calculations for models with minimum helium shell mass and derive synthetic observables for both the optical and-ray spectral regions . these differ strongly from those found in earlier simulations of sub - chandrasekhar - mass explosions in which more massive helium shells were considered . our models predict light curves which cover both the range of brightnesses and the rise and decline times of observed type ia supernovae . however , their colours and spectra do not match the observations . in particular , their colours are generally too red . we show that this discrepancy is mainly due to the composition of the burning products of the helium shell of the models which contain significant amounts of titanium and chromium . using a toy model , we also show that the burning products of the helium shell depend crucially on its initial composition . this leads us to conclude that good agreement between sub - chandrasekhar - mass explosions and observed type ia supernovae may still be feasible but further study of the shell properties is required .

1006.4489

c

in this paper we presented synthetic observables for the sub - chandrasekhar - mass double detonation models of @xcite . we found that these models predict light curves which rise and fade on time - scales typical of sne ia . moreover , they produce a large range of brightnesses which covers the whole range of observed sne ia . however , they do not account for all the properties of observed sne ia since they have peculiar spectra and light curves . in particular , their @xmath1 colours are generally too red compared to observed sne ia . this is in contrast to the results of earlier work on models with more massive helium shells @xcite . in addition , our model light curves and spectra show an unreasonably strong viewing - angle dependence due to the point - like ignition of the @xcite models and the resulting ejecta asymmetries . detonation of a pure helium shell leads to a layer containing iron - group elements like titanium and chromium around the core ejecta . these elements have a vast number of optically thick lines in the uv and blue part of the spectrum making them very effective in blocking the flux in these wavelength regions and redistributing it to the red . we used a toy model to show that this layer of titanium and chromium causes the peculiar red colours of our light curves and also the peculiar spectral features . moreover , we found that this toy model reproduces the observed properties of sne ia remarkably well . the toy model also showed that the strong viewing - angle dependence of our models results from the compositional asymmetry in the helium shell ejecta and not from the off - centre ignition of the c / o core . we stress , that the additional energy release in the shell , due to the production of radioactive nuclides during the helium burning , is relatively inconsequential for our models even at @xmath0-ray energies the signatures of the surface @xmath193cr and @xmath192fe are not apparent . instead , in the optical / uv the shell has a strong signature but this is primarily due to the additional opacity in the outer layers which affects the transport of energy from the core to the surface . we conclude that , if the double detonation sub - chandrasekhar - mass model valid for normal sne ia , the properties of the post - burning helium shell material need to be different from those in the @xcite models . since @xcite considered the limit of the least massive helium shells which might ignite a detonation in the helium shell , their models represent the most optimistic case for reducing the influence of the shell material by simply reducing the shell mass . however , we argue that the mass of the helium shell ejecta is not the main problem but rather the peculiar composition including comparably large masses of titanium and chromium . we illustrated this using a second toy model , where the initial composition of the helium shell was polluted with 34% ( by mass ) of @xmath295c . by providing additional seed - nuclei for @xmath296-captures , this leads to burning products with lower atomic number ( i.e. intermediate - mass elements rather than mainly iron - group elements ) . spectra and light curves of this model which has about the same shell mass as model 3 of @xcite show comparably good agreement to observed sne ia as the shell - less toy model . taking into account all these results , we argue that these systems might yet be promising candidates for sn ia progenitors . much more work will be needed to properly investigate this possibility . besides a more detailed description of the excitation / ionization state in the radiative transfer modelling which includes non - thermal effects , we need a better understanding of the initial composition of the helium shell and the incomplete burning processes which take place in this material to reach reliable predictions of the burning products of the helium shell . although only a tiny fraction of the mass , the post - burning composition of the shell material is critical to assessing the viability of the sub - chandrasekhar - mass double detonation scenario .

in the double detonation scenario for type ia supernovae it is suggested that a detonation initiates in a shell of helium - rich material accreted from a companion star by a sub - chandrasekhar - mass white dwarf . earlier studies concluded that this scenario has difficulties in accounting for the observed properties of type ia supernovae since the explosion ejecta are surrounded by the products of explosive helium burning in the shell . our models predict light curves which cover both the range of brightnesses and the rise and decline times of observed type ia supernovae . in particular , their colours are generally too red . this leads us to conclude that good agreement between sub - chandrasekhar - mass explosions and observed type ia supernovae may still be feasible but further study of the shell properties is required .

in the double detonation scenario for type ia supernovae it is suggested that a detonation initiates in a shell of helium - rich material accreted from a companion star by a sub - chandrasekhar - mass white dwarf . this shell detonation drives a shock front into the carbon - oxygen white dwarf that triggers a secondary detonation in the core . the core detonation results in a complete disruption of the white dwarf . earlier studies concluded that this scenario has difficulties in accounting for the observed properties of type ia supernovae since the explosion ejecta are surrounded by the products of explosive helium burning in the shell . recently , however , it was proposed that detonations might be possible for much less massive helium shells than previously assumed . moreover , it was shown that even detonations of these minimum helium shell masses robustly trigger detonations of the carbon - oxygen core . therefore it is possible that the impact of the helium layer on observables is less than previously thought . here , we present time - dependent multi - wavelength radiative transfer calculations for models with minimum helium shell mass and derive synthetic observables for both the optical and-ray spectral regions . these differ strongly from those found in earlier simulations of sub - chandrasekhar - mass explosions in which more massive helium shells were considered . our models predict light curves which cover both the range of brightnesses and the rise and decline times of observed type ia supernovae . however , their colours and spectra do not match the observations . in particular , their colours are generally too red . we show that this discrepancy is mainly due to the composition of the burning products of the helium shell of the models which contain significant amounts of titanium and chromium . using a toy model , we also show that the burning products of the helium shell depend crucially on its initial composition . this leads us to conclude that good agreement between sub - chandrasekhar - mass explosions and observed type ia supernovae may still be feasible but further study of the shell properties is required .

1110.3878

c

in the horndeski s most general scalar - tensor theories we derived conditions for the avoidance of ghosts and laplacian instabilities associated with scalar , tensor , and vector perturbations . the four conditions ( [ qscon ] ) , ( [ cscon ] ) , ( [ qtcon ] ) , and ( [ ctcon ] ) need to be satisfied for the theoretical consistency . vector perturbations do not give rise to any additional conditions to those derived for scalar and tensor perturbations . the horndeski s action covers most of the dark energy models proposed in literature and hence our formulas are general enough to apply them to concrete models with second - order field equations . we proposed new kinetically driven dark energy models described by the functions ( [ gegali ] ) , which cover the covariant galileon as a specific case . for the choice of the powers @xmath61 ( @xmath120 ) given in eq . ( [ power ] ) we showed the existence of the tracker solution along which @xmath339constant . finally the solutions approach a stable de sitter attractor at which @xmath11constant . along the tracker the dark energy equation of state during the matter dominance is found to be @xmath340 . the covariant galileon ( @xmath145 and @xmath146 ) corresponds to @xmath126 during the matter era , which is not favored from the combined data analysis of snia , cmb , and bao . the extended galileon model we proposed can alleviate this problem because @xmath16 can be close to @xmath0 for @xmath178 smaller than @xmath142 . we clarified the theoretically allowed parameter space in which the ghosts and laplacian instabilities are absent . for @xmath145 and @xmath268 we carried out numerical simulations to check the evolution of the background quantities ( like @xmath16 and @xmath301 ) as well as the quantities such as @xmath323 , @xmath272 , @xmath238 , and @xmath341 . as we estimated analytically , the dark energy equation of state for the tracker evolves as @xmath315 ( radiation era ) , @xmath269 ( matter era ) , and @xmath127 ( de sitter era ) , see fig . [ wde ] . for the initial conditions with @xmath242 , @xmath16 starts to evolve from the value estimated by eq . ( [ wdeap ] ) . the approach to the tracker occurs later for smaller initial values of @xmath159 . for the values of @xmath176 and @xmath177 which are inside the allowed parameter space , our numerical simulations show that @xmath323 , @xmath272 , @xmath238 , and @xmath341 remain to be positive in the cosmic expansion history . note that the condition ( [ eq : r2mincond ] ) is important to avoid that @xmath272 becomes negative during the transition from the matter era to the de sitter epoch . while we showed the cosmological evolution for one choice of @xmath178 and @xmath142 , we also confirmed that the analytic estimation is trustable for other values of @xmath178 and @xmath142 . in the limit that @xmath342 the dark energy equation of state for the tracker mimics that in the @xmath343cdm model . it will be of interest to see how the combined data analysis of snia , cmb , and bao places constraints on the tracker solution in the extended galileon models . in order to confront this model with the observations of large scale structure and weak lensing , we also need to study the evolution of matter density perturbations as well as gravitational potentials . we leave these issues for future work .

in the horndeski s most general scalar - tensor theories with second - order field equations , we derive the conditions for the avoidance of ghosts and laplacian instabilities associated with scalar , tensor , and vector perturbations in the presence of two perfect fluids on the flat friedmann - lematre - robertson - walker ( flrw ) background . we clarify the allowed parameter space in which the ghosts and laplacian instabilities are absent and we numerically confirm that such models are indeed cosmologically viable .

in the horndeski s most general scalar - tensor theories with second - order field equations , we derive the conditions for the avoidance of ghosts and laplacian instabilities associated with scalar , tensor , and vector perturbations in the presence of two perfect fluids on the flat friedmann - lematre - robertson - walker ( flrw ) background . our general results are useful for the construction of theoretically consistent models of dark energy . we apply our formulas to extended galileon models in which a tracker solution with an equation of state smaller than is present . we clarify the allowed parameter space in which the ghosts and laplacian instabilities are absent and we numerically confirm that such models are indeed cosmologically viable .

1402.1626

i

_ `` lesson : if you are stupid , do nt get discouraged but try a different way to get the result . '' _ + madan lal mehta the recent developments in the study of @xmath1 susy gauge theories in four dimensions have driven a lot of progress in several fields of theoretical physics . this is particularly the case for the field of random matrix models . the alday - gaiotto - tachikawa ( agt ) correspondence @xcite relates partition functions to summations over young tableaux of a deformed plancherel measure @xcite . this relation has been used extensively to verify the agt proposal in various limits ( see for instance @xcite ) . similarly , the correspondence with quantum integrable systems proposed in @xcite hints for a connection between the standard matrix models methods ( such as loop equations @xcite and collective field theory @xcite ) and the mayer cluster expansion @xcite . the latter is a statistical physics technique employed in @xcite to treat the gauge theory partition functions . an investigation of this possible connection was performed in @xcite . there , the mayer expansion of a grand - canonical ( generalized ) matrix model is compared to the canonical model at large @xmath4 . is the size of the matrix , it is the number of integration variables after diagonalization . it corresponds here to a number of instantons . it should not be mistaken with the rank @xmath5 of the gauge group , which will remain finite in this paper . ] however , the results obtained in @xcite are not directly applicable to susy gauge theories . the goal of this paper is to extend some of these results to a larger class of models relevant to the gauge / integrability correspondence . more precisely , the models considered here are defined as @xmath6 they will be studied in the limit @xmath3 , with a @xmath7 a sum of two terms : @xmath8 the function @xmath9 presents two single poles at @xmath10 with residue @xmath11 . on the other hand , @xmath12 is a symmetric function independent of @xmath0 , and such that @xmath13 is finite . the parameter @xmath0 is assumed to have a small positive imaginary part , and integrations in [ def_cz ] are contour integrals over the real axis . following @xcite , contours are closed in the upper half - plane , and avoid possible singularities at infinity . ] ] furthermore , both @xmath14 and @xmath12 are supposed to have no singularities on @xmath15 . the presence of @xmath9 was neglected in @xcite , and this particular case may be recovered by setting @xmath16 . in the limit @xmath3 , the poles of @xmath9 pinch the integration contour , drastically changing the behavior of the model . physically , @xmath9 gives rise to a strong integration at short distance @xmath17 , which can no longer be treated perturbatively as @xmath18 becomes of order @xmath19 . instanton partition functions of @xmath1 @xmath20 gauge theories reduces after localization to coupled one - dimensional integrals of the form [ def_cz ] with @xmath21 @xcite . this computation is regularized by considering the gauge theories on the @xmath2-background which depends on two equivariant deformation parameters @xmath22 . the euclidean background @xmath23 is recovered in the limit @xmath24 . connections with quantum integrable systems appear when @xmath25 is sent to zero while keeping @xmath26 finite @xcite . this limit is now referred as the nekrasov - shatashvili ( ns ) limit . under the identification @xmath27 , it coincides with the limit where the kernel becomes close to one considered here . the remaining deformation parameter @xmath26 appears in the definition of the function @xmath12 . its expression depends on the matter content of the gauge theory . it is given below in the limit @xmath28 for @xmath1 super - yang - mills ( sym ) with fundamental flavors , and @xmath29 with an adjoint hypermultiplet of mass @xmath30 , @xmath31 let us also mention that setting @xmath32 while keeping @xmath21 , we recover the model proposed by j. hoppe s in @xcite , and further studied in @xcite . this paper is organized as follows . in the first part , we focus on the grand - canonical model and show that the free energy at first order in @xmath0 takes the form of an on - shell effective action . for @xmath21 , we recover the action proposed in @xcite , and the corresponding equations of motion take the form of a tba - like relation . after a brief reminder on bethe equations and thermodynamical bethe ansatz ( tba ) , the identification between the free energy and the yang - yang functional is done . in the second section , we consider the associated canonical model . it is shown that the previous results can be derived from a confinement hypothesis for the eigenvalues of this model . we further study the limit @xmath33 and recover the collective field action of a dyson gas @xcite . eventually , we compare our results to the alternative approach based on the expression of instanton partition functions as sum over young tableaux . appendices gather the most technical details . _ as the present paper was in preparation , we received the preprint @xcite where similar results are derived for nekrasov partition functions . however , the method presented here is different from the one employed in @xcite , and we believe it is interesting in its own right . _

this class is directly relevant to nekrasov partition functions of susy gauge theories on the 4d-background , for which is identified with one of the equivariant deformation parameter . in the nekrasov - shatashvili limit , we show that the free energy is given by an on - shell effective action . the equations of motion take the form of a tba equation . the free energy is identified with the yang - yang functional of the corresponding system of bethe roots . we further study the associated canonical model that takes the form of a generalized matrix model . the collective field theory formulation is recovered . finally , we discuss the connection with the alternative expression of instanton partition functions as sums over young tableaux . apctp pre2014 - 001 confinement and mayer cluster expansions jean - emile bourgine + + _ pohang , gyeongbuk 790 - 784 , republic of korea _

in these notes , we study a class of grand - canonical partition functions with a kernel depending on a small parameter . this class is directly relevant to nekrasov partition functions of susy gauge theories on the 4d-background , for which is identified with one of the equivariant deformation parameter . in the nekrasov - shatashvili limit , we show that the free energy is given by an on - shell effective action . the equations of motion take the form of a tba equation . the free energy is identified with the yang - yang functional of the corresponding system of bethe roots . we further study the associated canonical model that takes the form of a generalized matrix model . confinement of the eigenvalues by the short - range potential is observed . in the limit where this confining potential becomes weak , the collective field theory formulation is recovered . finally , we discuss the connection with the alternative expression of instanton partition functions as sums over young tableaux . apctp pre2014 - 001 confinement and mayer cluster expansions jean - emile bourgine + + _ pohang , gyeongbuk 790 - 784 , republic of korea _

1510.01607

i

in this article , we introduce and investigate a class of finite deterministic automata that recognize the language @xmath1 of reduced words of a finitely generated coxeter system @xmath0 . the definition of these automata is straightforward , requiring only the notion of _ ( right ) weak order @xmath2 _ on @xmath0 @xcite and the related notion of _ garside shadows _ , introduced by m. dyer and the first author in @xcite as an analog of the notion of a garside family in a monoid ; see @xcite and the references therein . for general definitions and properties , we refer the reader to @xcite regarding automata and to @xcite regarding coxeter groups . a _ garside shadow in @xmath0 _ is a subset @xmath3 that contains @xmath4 and is closed under join ( for the right weak order ) and by taking suffixes . in @xcite , the authors show that finite garside shadows exist in any coxeter system @xmath0 . let @xmath5 be a finite garside shadow in @xmath0 . so @xmath6 for any _ bounded _ subset @xmath7 of @xmath5 , i.e. , a subset that has an upper bound . therefore , the following projection from @xmath8 to @xmath9 is well - defined : @xmath10 we denote by @xmath11 the _ length function _ of the coxeter system @xmath0 . [ def : automata ] we define a finite deterministic automaton @xmath12 over the alphabet @xmath4 as follows : * the set of states is @xmath5 ; * the initial state is the identity @xmath13 of @xmath8 , and all states are final ; * the transitions are : @xmath14 whenever @xmath15 . since the intersection of garside shadows is again a garside shadow , there is a smallest garside shadow @xmath16 in @xmath0 . as a first example , the finite automaton built out of the smallest garside shadow @xmath16 for the infinite dihedral group is shown in figure [ fig : infinitedi ] . further examples are given in [ sse : shi ] and in figures [ fig : a1_automaton ] and [ fig : a2_c2_automata ] . our main result is that @xmath12 recognizes the language of reduced words of @xmath0 . [ thm : main ] if @xmath5 is a finite garside shadow in @xmath0 , then the finite deterministic automaton @xmath12 recognizes the language @xmath1 . theorem [ thm : main ] is proved in [ se : auto ] . in [ se : can ] , we show that an inclusion @xmath17 of garside shadows induces a surjective morphism @xmath18 between their associated automata . the smallest garside shadow being finite ( * ? ? ? * corollary 1.2 ) , we are led to the following conjecture . [ conj:1 ] the automaton @xmath19 is the minimal automaton recognizing @xmath1 . using sage @xcite , we checked that conjecture [ conj:1 ] holds for all coxeter groups @xmath8 of rank at most @xmath20 whose corresponding coxeter graph @xmath21 has edge labels less than @xmath22 ; see remark [ rem : sage ] for more details . our initial motivation for this work was to provide a purely combinatorial definition for an automaton that recognizes the language of reduced words . indeed , as we now recall , all previously - defined automata recognizing @xmath1 require the introduction of an auxiliary geometric representation and root system . in 1993 , b. brink and r. howlett @xcite showed that finitely - generated coxeter groups are automatic , in the sense of @xcite , thereby filling a gap in the proof of the `` parallel wall theorem '' of @xcite . for each coxeter system @xmath0 , they provided a _ word - acceptor_that is , a finite automaton that recognizes the language of lexicographically minimal reduced words in @xmath8 . this particular automaton is built using their notion of _ small roots _ , and therefore requires a geometric representation of @xmath0 and its associated root system . in a series of articles @xcite , casselman explains how to perform practical computations in coxeter groups using brink and howlett s word - acceptor . we are often interested in _ all _ reduced words , not only those that are lexicographically - ordered ; see for instance @xcite . in his thesis @xcite , h. eriksson studied a finite deterministic automaton @xmath23 over @xmath4 that recognizes the language @xmath1 . the automaton @xmath23 is called the _ canonical automaton _ in @xcite , and is built using b. brink and r. howlett s technology of small roots . an immediate consequence is that the language @xmath1 is regular , a result we recover in theorem [ thm : main ] . in particular , the generating function for the number of reduced words in @xmath0 with respect to their length is a rational function . for @xmath24 , the canonical automaton was extended , replacing small roots with @xmath25-small roots , in @xcite and @xcite to the @xmath25-canonical automaton @xmath26 . we recall these notions in [ se : can ] , and discuss morphisms between @xmath26 and the automata @xmath12 arising from certain finite garside shadows @xmath5 . in particular , we show in corollary [ cor : minsurj ] that any @xmath25-canonical automaton surjects into the automaton @xmath19 , providing evidence for conjecture [ conj:1 ] . both h. eriksson ( * ? ? ? * theorem 80 ) and p. headley ( * ? ? ? * theorem v.8 ) prove that in type @xmath27 , the canonical automaton @xmath28 is minimal . furthermore , they note that @xmath29 is _ not _ minimal for general affine groups @xmath8 . we conjecture a necessary condition for the canonical automaton to be minimal . the sufficient condition is shown in proposition [ prop : spherical_implies_minimal ] . [ conj:2 ] let @xmath8 be irreducible . then @xmath30 is minimal if and only if @xmath31 , where @xmath32 denotes the set of roots whose support is a finite standard parabolic subgroup . since @xmath29 surjects onto @xmath19 ( corollary [ cor : minsurj ] ) , conjecture [ conj:2 ] implies conjecture [ conj:1 ] for coxeter systems for which @xmath31 . in [ se : min ] , we prove conjecture [ conj:2 ] in the following cases . [ thm : canmin ] conjecture [ conj:2 ] holds in each of the following cases : 1 . @xmath8 is finite . @xmath8 is right - angled , i.e. @xmath33 or @xmath34 for all @xmath35 3 . @xmath21 is a complete graph , i.e. @xmath36 for all @xmath35 . @xmath8 is of type @xmath37 . 5 . @xmath8 has rank @xmath38 . in the first four cases , @xmath31 and @xmath30 is minimal . we also checked that conjecture [ conj:2 ] holds if @xmath8 has rank @xmath20 and @xmath39 for all @xmath40 ; see remark [ rem : sage ] for more details . when @xmath0 is an affine coxeter system , p. headley described a remarkable connection between the canonical automaton and the _ shi arrangement _ @xcite : the states of @xmath23 are in bijection with the ( minimal elements in the ) connected regions of the complement of the shi arrangement for @xmath0 @xcite . the same relationship holds for the states of @xmath41 and the regions of the @xmath25-shi arrangement , as we outline in [ sse : shi ] .

in this article , we introduce and investigate a class of finite deterministic automata that all recognize the language of reduced words of a finitely generated coxeter system . the definition of these automata is straightforward as it only requires the notion of _ weak order _ on and the related notion of _ garside shadows in _ , an analog of the notion of a garside family . then we discuss the relations between this class of automata and the canonical automaton built from brink and howlett s small roots . we end this article by providing partial positive answers to two conjectures : ( 1 ) the automata associated to the smallest garside shadow is minimal ; ( 2 ) the canonical automaton is minimal if and only if the support of all small roots is spherical , i.e. , the corresponding root system is finite .

in this article , we introduce and investigate a class of finite deterministic automata that all recognize the language of reduced words of a finitely generated coxeter system . the definition of these automata is straightforward as it only requires the notion of _ weak order _ on and the related notion of _ garside shadows in _ , an analog of the notion of a garside family . then we discuss the relations between this class of automata and the canonical automaton built from brink and howlett s small roots . we end this article by providing partial positive answers to two conjectures : ( 1 ) the automata associated to the smallest garside shadow is minimal ; ( 2 ) the canonical automaton is minimal if and only if the support of all small roots is spherical , i.e. , the corresponding root system is finite .

1410.4351

i

control of fluid flow has been an important area of research and has many practical applications . the question about controllability of fluid flows has attracted the attention of many researchers , more for incompressible flow but much less for compressible flow . in this paper we are interested in controllability properties of linearized compressible navier - stokes system . for a compressible , isothermal barotropic fluid ( density is a function of only pressure ) , the navier - stokes system in @xmath8 consists of equation of continuity @xmath9 = 0 , \end{array}\ ] ] and the momentum equation @xmath10 \\ [ 3.mm ] \displaystyle = -\nabla p(x , t ) + \mu \bigtriangleup { \bf u}(x , t ) + ( \lambda + \mu ) \nabla [ \mbox{div } { \bf u}(x , t ) ] , \end{array}\ ] ] where @xmath11 is the density of the fluid , @xmath12 denotes the velocity vector in @xmath13 and @xmath14 is an external force field in @xmath15 the pressure satisfies the following constitutive law @xmath16 for some constants @xmath17 , @xmath18 the viscosity coefficients @xmath19 and @xmath20 are assumed to be constant satisfying the following thermodynamic restrictions , @xmath21 , @xmath22 for non - barotropic fluid ( when density is a function of pressure and temperature of the fluid ) , the navier - stokes system consists of the equation of continuity , the momentum equation and an additional thermal energy equation @xmath23(x , t ) + \theta(x , t ) \frac{\partial p}{\partial \theta}(x , t ) \mbox{div } { \bf u}(x , t ) \\ [ 3.mm ] \displaystyle = \kappa \bigtriangleup \theta ( x , t ) + \lambda ( \mbox{div } { \bf u}(x , t))^2 + 2 \mu \sum_{i , j=1}^n \frac{1}{4 } \left(\frac{\partial { \bf u}_i}{\partial x_j } + \frac{\partial { \bf u}_j}{\partial x_i}\right)^2 , \end{array}\ ] ] where @xmath24 denotes the temperature of the fluid , @xmath25 is the specific heat constant and @xmath26 is the heat conductivity constant . for ideal gas , the pressure is given by boyle s law : @xmath27 where @xmath28 is the universal gas constant ( see @xcite ) . in this article , we first consider the compressible navier - stokes system for non - barotropic fluid in a bounded interval @xmath29 linearized around a constant steady state @xmath30 with @xmath31 and @xmath32 more precisely we consider the system @xmath33 where @xmath34 is the characteristic function of an open subset @xmath35 we choose the following initial and boundary conditions for the system : @xmath36 u(0,t ) = 0 , \quad u(l , t ) = 0 \quad \forall \ \ t > 0 , \\ [ 2.mm ] \theta(0,t ) = 0 , \quad \theta(l , t ) = 0 \quad \forall \ \ t > 0 . \end{array}\ ] ] in - , @xmath37 , @xmath38 and @xmath39 are distributed controls . we are interested in the following question : given @xmath40 and @xmath41 can we find interior control functions such that the solution @xmath42 of - satisfies @xmath43 our first main result regarding interior null controllability is the following , [ thm1 ] let @xmath44 i.e. , @xmath45 is a proper subset of @xmath46 let us assume @xmath47 . the system - is not null controllable in any @xmath48 by the interior controls @xmath49 , @xmath50 and @xmath51 acting on density , velocity and temperature equation respectively . the above negative result can be extended to the case of less regular interior controls or boundary control . see remark [ rem : lessregcontrol ] and remark [ rem : bdycontrol ] for more details . our next positive result shows that , if initial density @xmath52 lies in a more regular space then the linearized system is null controllable by velocity and temperature controls acting everywhere in the domain . [ thm2 ] let @xmath53 in . let us denote @xmath54 let us assume @xmath55 . then for any @xmath56 there exist controls @xmath57 and @xmath58 acting everywhere in the velocity and temperature equation respectively , such that the solution of - satisfies @xmath59 the next result shows that the above result is sharp as null controllability can not be achieved by localized interior velocity and temperature controls . [ thm3 ] let @xmath53 in . let @xmath60 i.e. , @xmath61 is a proper subset of @xmath46 let us assume @xmath55 . the system - is not null controllable in any @xmath48 by the interior controls @xmath62 and @xmath63 acting on velocity and temperature equation respectively . the proof of these results relies on the observability inequality . we know that the null controllability of a linear system is equivalent to a certain observability inequality for the solutions of adjoint system ( see @xcite , chapter 2 ) . to prove the negative results , we will construct particular solutions for the adjoint system such that the observability inequality can not hold . in order to do that first we will consider the adjoint system in @xmath64 as a terminal value problem . we will construct highly localized solutions known as `` gaussian beam '' . similar kind of construction has been used for hyperbolic equations by ralston ( @xcite ) and for wave equations by maci and zuazua ( @xcite ) . we will prove that solutions are localized in a small neighbourhood of any @xmath65 thus given an observation set , we can always find an interval away from the observation set such that the solutions are localized in that interval . using this we are able to prove the negative results . to the author s best knowledge these are new results regarding controllability issues of navier - stokes system for non - barotropic fluid . in theorem [ thm3 ] , we proved a positive result when controls acting everywhere in the equation . the question then arises : whether positive results could be obtained by using control supported in a small , but moving region , as in @xcite . rosier and rouchon in @xcite proved that the structurally damped wave equation in one dimension is not null controllable by a boundary control . later on martin , rosier and rouchon in @xcite proved that the same equation in one dimension with periodic boundary conditions , is null controllable with a moving distributed control for sufficiently large time . chaves - silva , rosier and zuazua in @xcite extend the above result to higher dimension . the structure of the system considered by the authors in @xcite , in some sense , is similar to the linearized compressible , barotropic navier - stokes system in one dimension as well as in higher dimension . these issues will be discussed in a future work which is in progress . we have studied in this paper the null controllability of the linearized compressible navier - stokes system only . the `` gaussian beam '' construction is used to show negative results . however one may use other techniques based on the use of nonlinearity ( see @xcite for example ) to achieve controllability results for the full nonlinear system . there have been some results regarding the control of compressible barotropic fluid models in recent years . amosova in @xcite considers compressible navier - stokes system for viscous barotropic fluid in one dimension in lagrangian coordinates in a bounded domain @xmath66 with dirichlet boundary condition . she proves local exact controllability to trajectories for the velocity in any time @xmath56 using a localized interior control on the velocity equation , provided that the initial density is already on the targeted trajectory and initial condition lies in @xmath67 ervedoza , glass , guerrero and puel in @xcite consider the compressible navier - stokes system in one space dimension in a bounded domain @xmath68 . they prove local exact controllability to constant states @xmath69 with @xmath70 using two boundary controls , both for density and velocity , in time @xmath71 when initial condition lies in @xmath72 chowdhury , ramaswamy and raymond in @xcite consider the compressible barotropic navier - stokes system linearized around a constant steady state @xmath73 with @xmath74 in a bounded domain @xmath75 . they proved that the linearized system is not null controllable by a localized control or by boundary control . they also proved that the linearized system is null controllable by an interior control acting everywhere in the velocity equation when initial condition lies in @xmath76 chowdhury in @xcite considers the compressible barotropic navier - stokes system linearized about a constant steady state @xmath77 with @xmath78 in @xmath79 with dirichlet boundary condition and an interior control on the velocity equation acting on open subset @xmath80 he proves that the system is approximately controllable in @xmath81 when @xmath82 he also proves a similar result in two dimension . chowdhury , mitra , ramaswamy and renardy in @xcite consider the compressible barotropic navier - stokes system linearized about a constant steady state @xmath77 with @xmath78 in @xmath83 with periodic boundary condition . they proved that the linearized system is null controllable by a localized velocity control when @xmath84 and initial condition lies in @xmath85 our linearized system - is similar to the linearized system considered by the authors in @xcite . so we expect similar controllability results . their method is based on explicit expression for eigenfunctions and the behaviour of the spectrum of the linearized operator . they proved that there is an accumulation point in the spectrum of the linearized operator . this system behaves very badly with respect to controllability properties and a similar type of controllability behaviour is also observed in @xcite for different types of systems where an accumulation point is present in the spectrum of linearized operator . but the method used in @xcite does not seem to fit very well in our case . in fact one can prove that there is an accumulation point in the spectrum of the linearized operator considered here , for certain boundary condition . but the expressions of eigenvalues and eigenfunctions are complicated . so here we use gaussian beam approach to achieve the negative results . this technique does not require the knowledge of the spectrum and it seems to extend to higher dimension also . the controllability properties are completely different , if we consider compressible navier - stokes system linearized around non null velocity . for barotropic fluid , the system linearized around @xmath86 is not controllable in any time @xmath87 by localized interior control but the system linearized around @xmath88 is null controllable by localized interior control for large time @xmath6 it is interesting to note that there is no accumulation point in the spectrum of the linearized operator in the latter case and better controllability behaviour at least for @xmath87 large enough(see @xcite ) . but the question remains what happens when time @xmath87 is small enough . our results answer this question in the negative . we consider the compressible non - barotropic navier - stokes equation linearized around constant steady state @xmath89 @xmath90 @xmath91 \displaystyle u_t - \frac{\lambda + 2\mu}{{\bar \rho } } u_{xx } + \frac{r { \bar \theta}}{{\bar \rho}}\;\rho_x + \bar v u_x + r \theta_x \ ; = g \chi_{\mathcal{o}_2 } , \mbox { in } ( 0,l ) \times ( 0,t ) , \\[3.mm ] \displaystyle \theta_t ( x , t ) -\frac{\kappa}{{\bar \rho}c_v } \theta_{xx } + \frac{r { \bar \theta}}{c_v } u_x + \bar v \theta_x \ ; = h \chi_{\mathcal{o}_3 } , \mbox { in } ( 0,l ) \times ( 0,t ) , \\[3.mm ] \displaystyle \rho(0 ) = \rho_0,\quad \quad u(0)= u_0 , \mbox { and } \quad \theta(0 ) = \theta_0 , \quad \mbox { in } ( 0,l ) , \\ [ 2.mm ] \displaystyle \rho(0,t ) = 0 , \;\;\ ; u(0,t ) = 0 = u(l , t ) , \;\ ; \forall \ t \in ( 0,t ) , \\ [ 2.mm ] \displaystyle \theta(0,t ) = 0 = \theta(l , t ) , \;\ ; \forall \ t \in ( 0,t ) . \end{array}\ ] ] we prove the following theorem . [ thm1.4 ] let @xmath92 i.e. , @xmath45 is a proper subset of @xmath46 let us assume @xmath47 . if @xmath93 then the system is not null controllable by localized interior controls @xmath94 @xmath50 and @xmath51 acting on density , velocity and temperature equation respectively . as a corollary of the above theorem , one can rule out null controllability of compressible barotropic navier - stokes system linearized around constant steady state @xmath95 in small time @xmath96 using a boundary control or localized interior control . we consider compressible barotropic navier - stokes system linearized around a constant steady state @xmath95 , @xmath97 in @xmath98 or in @xmath99 as in @xcite . * for initial condition belonging to @xmath100 the system with dirichlet boundary condition is not null controllable at any time @xmath40 by a interior control acting only in the velocity equation . the control may act in a non empty open subset of @xmath79 or in the whole domain @xmath101 * the system with periodic boundary condition is not null controllable by interior control localized in @xmath102 acting only in the velocity equation when initial condition lies in @xmath103 and time @xmath104 * for initial condition belonging to @xmath100 the same system is not null controllable by boundary control if time @xmath105 from the above corollary we see that the condition @xmath106 in ervedoza , glass , guerrero and puel ( @xcite ) is natural . next we will show that , our method can be extended to higher dimension also . for simplicity we consider the compressible barotropic navier - stokes system in two - dimensional bounded domain @xmath107 , linearized around a constant steady state solution @xmath108 , @xmath1 @xmath109 \displaystyle { \bf u } - \frac{\mu}{{\bar \rho } } \delta { \bf u } - \frac{\lambda + \mu}{{\bar \rho}}\nabla[\mathrm{div } \ { \bf u } ] + \;a \gamma { \bar \rho}^{\gamma-2 } \nabla \rho \;=\;{\bf g } \chi_{{\mathcal o}_2 } , \mbox { in } \omega \times ( 0,t ) , \\[2.mm ] \displaystyle \rho(0 ) = \rho_0 \quad \mbox{and}\quad { \bf u}(0)= { \bf u}_0 , \quad \mbox { in } \omega , \\ [ 2.mm ] \displaystyle { \bf u } = { \bf 0 } \ \ \mbox { on } \partial \omega \times ( 0,t ) . \end{array}\ ] ] where @xmath110 and @xmath111 are open subsets of @xmath112 we obtain the following negative null controllability result for the system . [ thm1.5 ] let @xmath113 i.e. , @xmath45 is a proper open subset of @xmath112 let us assume that @xmath114 then the system is not null controllable in time any @xmath40 by interior controls @xmath115 and @xmath116 the plan of the paper is as follows . in section 2 , we study the control system linearized around a constant steady state @xmath0 in one dimension . we prove theorem [ thm1 ] , theorem [ thm2 ] and theorem [ thm3 ] here . in section 3 , we study the control system linearized around a constant steady state @xmath117 theorem [ thm1.4 ] is proved here . in section 4 we consider the linearized system in two dimension around constant steady state @xmath7 we prove theorem [ thm1.5 ] here . * acknowledgement * : the author would like to thank prof . sylvain ervedoza for providing important references on gaussian beams . the author also would like to thank him and prof . mythily ramaswamy for very useful discussions which improved the initial version . the author acknowledges the financial support under the project `` pde control '' from the indo french centre for applied mathematics ( ifcam ) .

in this article , we study the null controllability of linearized compressible navier - stokes system in one and two dimension . we first study the one - dimensional compressible navier - stokes system for non - barotropic fluid linearized around a constant steady state . we prove that the linearized system around , with is not null controllable by localized interior control or by boundary control . but the system is null controllable by interior controls acting everywhere in the velocity and temperature equation for regular initial condition . we also prove that the the one - dimensional compressible navier - stokes system for non - barotropic fluid linearized around a constant steady state , with is not null controllable by localized interior control or by boundary control for small time next we consider two - dimensional compressible navier - stokes system for barotropic fluid linearized around a constant steady state we prove that this system is also not null controllable by localized interior control . * key words . * linearized compressible navier - stokes system , null controllability , localized interior control , boundary control , gaussian beam . * ams subject classifications . * 35q30 , 93c20 , 93b05

in this article , we study the null controllability of linearized compressible navier - stokes system in one and two dimension . we first study the one - dimensional compressible navier - stokes system for non - barotropic fluid linearized around a constant steady state . we prove that the linearized system around , with is not null controllable by localized interior control or by boundary control . but the system is null controllable by interior controls acting everywhere in the velocity and temperature equation for regular initial condition . we also prove that the the one - dimensional compressible navier - stokes system for non - barotropic fluid linearized around a constant steady state , with is not null controllable by localized interior control or by boundary control for small time next we consider two - dimensional compressible navier - stokes system for barotropic fluid linearized around a constant steady state we prove that this system is also not null controllable by localized interior control . * key words . * linearized compressible navier - stokes system , null controllability , localized interior control , boundary control , gaussian beam . * ams subject classifications . * 35q30 , 93c20 , 93b05

1111.2398

i

[ [ section ] ] let @xmath6 be a bounded and connected domain in @xmath7 with a smooth boundary . we are going to define a perforated domain @xmath8 by removing a @xmath1-periodic balls out of @xmath6 . for each @xmath9 , let @xmath10 be a ball with center @xmath11 and radius @xmath12 . let @xmath13 and @xmath14 the homogenization of partial differential equations in a perforated domain with dirichlet or neumann boundary value has been studied by many authors . please refer @xcite and @xcite for details . in this paper , we will consider the generalization of the following soft inclusions where the diffusion coefficients are zero on the holes : @xmath15 in @xcite , they show that @xmath16 converges to @xmath17 weakly in @xmath18(strongly in @xmath19 ) , and that the limit @xmath17 satisfies @xmath20 for some constant matrix @xmath21 where @xmath22^n } \chi_{t^{\mathfrak a}}$ ] . their method relies on the energy estimates and compensated compactness to pass the limit in the weak formulation . such energy method can not be applicable to nonlinear equations of non - divergence type since the solutions may have different order of energies , @xcite , even though they satisfy equations in the same class . in this paper , we are going to develop a viscosity method to find the homogenization process of the following semi - linear equation of non - divergence type : @xmath23 where @xmath24 , @xmath25 and @xmath26 . and the equation satisfies the following conditions . * conditions i : * a. [ con - unif - elliptic ] @xmath27 is uniformly elliptic : there are positive constants @xmath28 such that @xmath29 for all @xmath30 and for all @xmath31 . b. [ con - oblique ] @xmath32 satisfies the uniform oblique condition : there is a uniform constant @xmath33 satisfying @xmath34 . and , for the convenience , we also assume @xmath35 . c. [ con - y - periodicity ] @xmath36 , @xmath32 , @xmath37 and @xmath38 are periodic in @xmath39-variable : for every @xmath40 , we have @xmath41 d. [ con - c - decr ] @xmath42 and @xmath37 is non - increasing with @xmath43 variables . e. [ con - alpha ] @xmath44 for every @xmath45 . f. [ con - conti ] @xmath46 and @xmath47 are uniformly continuous with respect to @xmath39 variable . that is , for any given @xmath48 and @xmath49 , @xmath50 throughout this paper , we always assume the conditions i above . our first theorem concerns about the existence of compatibility constant for neumann problem . [ thm - main - c ] consider the equation defined as follow : @xmath51 assume that @xmath52 and @xmath53 is bounded . then , for any given @xmath30 , there is a unique constant @xmath54 that makes the equation has a soution @xmath55 . a. we are going to call @xmath56 a compatibility constant of the equation . b. now suppose that @xmath57 . if @xmath58 for all @xmath30 and the size of halls @xmath59 , then we call @xmath60 and @xmath61 ( or the equation ) satisfies the compatibility condition . we remark that laplace equation equipped the neumann boundary condition satisfies the compatibility condition . we will show it in chapter [ sec - com ] . now let us introduce our main theorem : [ thm - main-1 ] let @xmath2 be a viscosity solution of . suppose that our equation satisfies the conditions i and 1 . the equation satisfies the compatibility condition , 2 . @xmath16 is bounded uniformly on @xmath1 , and @xmath62 on the @xmath63 where @xmath64 and @xmath65 is same in definition [ def - lim - u ] . 3 . @xmath66 for uniform constant @xmath67 in theorem [ thm - cor2-elliptic ] . then , there exists an uniformly elliptic operator @xmath68 . and @xmath16 , solution of , converges to @xmath5 , solution of the equation , uniformly . @xmath69 we will use the condition ( 3 ) to prove the uniformly ellipticity of @xmath68 . hence it can be dropped if @xmath70(laplace case ) and @xmath71(neumann boundary case ) . see chapter [ sec - cor2 and ellipticity ] . and the condition ( 2 ) can be dropped if we can construct a barrier at any boundary points . we will show that such a barrier exists if @xmath6 satisfies exterior sphere condition in chapter [ sec - homogenization ] . [ cor - main-1 ] let @xmath16 be the solution of the equation . assume that satisfies all of the conditions ( 1 ) -(6 ) , condition ( 3 ) in theorem [ thm - main-1 ] and the compatibility condition . assume also that @xmath6 satisfies an exterior sphere condition . then @xmath16 converges uniformly to @xmath17 which is the solution of . [ cor - main-2 ] let @xmath16 be the solution of the equation with the condition . then @xmath16 converges uniformly to @xmath17 which is the solution of if the domain @xmath6 satisfies an exterior sphere condition . finally , we develop the following estimate . it tells us that @xmath16 is almost lipschitz continuous .

in this paper , we consider periodic soft inclusions with periodicity , where the solution , , satisfies semi - linear elliptic equations of non - divergence in with a neumann data on . the difficulty lies in the non - divergence structure of the operator where the standard energy method based on the divergence theorem can not be applied . the main object is developing a viscosity method to find the homogenized equation satisfied by the limit of , called as , as approaches to zero .

in this paper , we consider periodic soft inclusions with periodicity , where the solution , , satisfies semi - linear elliptic equations of non - divergence in with a neumann data on . the difficulty lies in the non - divergence structure of the operator where the standard energy method based on the divergence theorem can not be applied . the main object is developing a viscosity method to find the homogenized equation satisfied by the limit of , called as , as approaches to zero . we introduce the concept of a compatibility condition between the equation and the neumann condition on the boundary for the existence of uniformly bounded periodic first correctors . the concept of second corrector has been developed to show the limit , , is the viscosity solution of a homogenized equation .

1304.6839

i

a natural definition for the laplacian tensor and the signless laplacian tensor of a @xmath0-uniform hypergraph for @xmath3 was introduced in @xcite . see definition 2.2 of this paper . recently , hu , qi and xie @xcite studied the largest laplacian and signless laplacian eigenvalues of a @xmath0-uniform hypergraph , and generalized some classical results of spectral graph theory to spectral hypergraph theory , in particular , when @xmath0 is even . one classical result in spectral graph theory @xcite is that the largest laplacian eigenvalue of a graph is always less than or equal to the largest signless laplacian eigenvalue , and when the graph is connected , the equality holds if and only if the graph is bipartite . in @xcite , it was shown that the largest laplacian h - eigenvalue of a @xmath0-uniform hypergraph is always less than or equal to the largest signless laplacian h - eigenvalue , and when the hypergraph is connected , the equality holds if and only if the hypergraph is odd - bipartite . a @xmath0-uniform hypergraph is called odd - bipartite if @xmath0 is even , and the vertex set of the hypergraph can be divided to two parts , such that each edge has odd number of vertices in each of these two parts . hu , qi and xie @xcite generalized cycles in graphs to hypercycles in @xmath0-uniform hypergraphs . by showing that the largest signless laplacian h - eigenvalue for a hypercycle is computable and an even - order hypercycle is odd - bipartite , they showed that the largest laplacian h - eigenvalue of a hypercycle is computable when @xmath0 is even . another classical result in spectral graph theory @xcite is that the largest laplacian eigenvalue of a graph is always greater than or equal to the maximum degree of that graph , plus one . the lower bound is attained if there exists a vertex adjacent to all the other vertices of that graph . hu , qi and xie @xcite showed that when @xmath0 is even the largest laplacian h - eigenvalue of a @xmath0-uniform hypergraph is always greater than or equal to the maximum degree of that hypergraph , plus @xmath4 , where @xmath5 and @xmath6 . they showed that the lower bound is attained if the hypergraph is a hyperstar . when @xmath0 is odd , the situation is very different . it was shown in @xcite that in this case the largest laplacian h - eigenvalue is always strictly less than the largest signless laplacian h - eigenvalue , and the lower bound of the largest laplacian h - eigenvalue is the maximum degree itself , attained by any hyperstar . the results of @xcite raised the interests to study the laplacian h - eigenvalues of a @xmath0-uniform hypergraph . actually , they posed several questions for further research . first , when @xmath0 is odd , is a hyperstar the only example of a @xmath0-uniform hypergraph whose largest laplacian h - eigenvalue is equal to the maximum degree ? second , can we calculate all laplacian h - eigenvalues for some special @xmath0-uniform hypergraphs , such as hyperstars and hypercycles ? this is useful if one wishes to study the second smallest laplacian h - eigenvalue ( with multiplicity ) of a @xmath0-uniform hypergraph , as the second smallest laplacian eigenvalue of a graph plays a key role in spectral graph theory @xcite . motivated by these questions , we study laplacian h - eigenvalues of some special @xmath0-uniform hypergraphs in this paper . with the same way to generalize stars and cycles to hyperstars to hypercycles , we may generalize an arbitrary graph @xmath7 to a @xmath0-uniform hypergraphs @xmath8 . see definition 2.4 of this paper . we call @xmath8 the @xmath0th power of @xmath7 , hence call it a power hypergraph . in particular , paths are generalized to hyperpaths . we will see that when @xmath0 is even , a power hypergraph is odd - bipartite . we may conclude this for a broader class of @xmath0-uniform hypergraphs . we call such hypergraphs cored hypergraphs . see definition 2.3 of this paper . a power hypergraph is a cored hyoergraph but not vice versa . in particular , we introduce a special subclass of cored hypergraphs , called sunflowers . see definition 3.1 of this paper . a sunflower is not a power hypergraph in general . we show that when @xmath0 is even , a cored hypergraph is odd - bipartite . thus , when @xmath0 is even , the largest laplacian h - eigenvalue and the largest signless laplcian h - eigenvalue of a cored hypergraph is the same . this enhances our understanding on odd - bipartite hypergraphs and their largest laplacian eigenvalues . we will show that the largest laplacian h - eigenvalue of an even - order sunflower is computable . then , when @xmath0 is odd , we will show that the largest laplacian h - eigenvalue of an odd - uniform sunflower , hypercycle and hyperpath is equal to the maximum degree , i.e. , @xmath1 . this shows that for a very broad class of hypergraphs , when @xmath0 is odd , the largest laplacian h - eigenvalue is equal to the maximum degree of the hypergraph . finally , we will compute out all the h - spectra of the class of hyperstars , the hypercycle of size @xmath2 and the hyperpath of length @xmath2 . this will be useful for research on the second smallest laplacian h - eigenvalue of @xmath0-uniform hypergraphs . for discussion on the eigenvectors of the zero laplacian and signless laplacian eigenvalues of a @xmath0-uniform hypergraph , see @xcite . for discussion on eigenvalues of adjacency tensors and the other types of laplacian tensors of @xmath0-uniform hypergraphs , see @xcite and references therein . the rest of this paper is organized as follows . definitions on eigenvalues of tensors and uniform hypergraphs are presented in the next section . cored hypergraphs and power hypergraphs are introduced there . we discuss in section [ sec - ch ] some properties on the cored hypergraphs . an even - uniform cored hypergraph has equality for the largest laplacian and the singless laplacian h - eigenvalues . sunflowers are introduced and investigated in section [ sec - sps ] . we compute out the largest laplacian h - eigenvalues of even - uniform sunflowers and prove that they are equal to the maximum degrees , i.e. , @xmath1 , for odd - uniform sunflowers . we show in section [ sec - ph-1 ] that the largest laplacian h - eigenvalues of odd - uniform hypercycles and hyperpaths are equal to the maximum degrees , i.e. , @xmath1 . we make a conjecture in section [ sec - ph-2 ] that the largest h - eigenvalues of even - uniform power hypergraphs with respect to the same underlying usual graph are strictly decreasing as @xmath0 increasing . this conjecture is proved to be true for hyperstars and hypercycles . in section [ sec - sph ] , we compute out all the h - eigenvalues of hyperstars , the hyperpath of length @xmath2 and the hypercycle of size @xmath2 . some final remarks are made in the last section .

in this paper , we introduce the class of cored hypergraphs and power hypergraphs , and investigate the properties of their laplacian h - eigenvalues . from an ordinary graph , one may generate a-uniform hypergraph , called theth power hypergraph of that graph . power hypergraphs are cored hypergraphs , but not vice versa . we show that the largest laplacian h - eigenvalue of an even - uniform cored hypergraph is equal to its largest signless laplacian h - eigenvalue . especially , we find out these largest h - eigenvalues for even - uniform sunflowers . moreover , we show that the largest laplacian h - eigenvalue of an odd - uniform sunflower , hypercycle and hyperpath is equal to the maximum degree , i.e. , . we also compute out the h - spectra of the class of hyperstars . when is odd , the h - spectra of the hypercycle of size and the hyperpath of length are characterized as well . * key words : * tensor , h - eigenvalue , hypergraph , laplacian , power hypergraph * msc ( 2010 ) : * 05c65 ; 15a18

in this paper , we introduce the class of cored hypergraphs and power hypergraphs , and investigate the properties of their laplacian h - eigenvalues . from an ordinary graph , one may generate a-uniform hypergraph , called theth power hypergraph of that graph . power hypergraphs are cored hypergraphs , but not vice versa . hyperstars , hypercycles , hyperpaths are special cases of power hypergraphs , while sunflowers are a subclass of cored hypergraphs , but not power graphs in general . we show that the largest laplacian h - eigenvalue of an even - uniform cored hypergraph is equal to its largest signless laplacian h - eigenvalue . especially , we find out these largest h - eigenvalues for even - uniform sunflowers . moreover , we show that the largest laplacian h - eigenvalue of an odd - uniform sunflower , hypercycle and hyperpath is equal to the maximum degree , i.e. , . we also compute out the h - spectra of the class of hyperstars . when is odd , the h - spectra of the hypercycle of size and the hyperpath of length are characterized as well . * key words : * tensor , h - eigenvalue , hypergraph , laplacian , power hypergraph * msc ( 2010 ) : * 05c65 ; 15a18

1410.6961

i

in this paper , we establish uniqueness of small solutions to the three - dimensional quintic gross - pitaevskii ( gp ) hierarchy in the scaling - critical sobolev type space . the 3d quintic gp hierarchy is an infinite system of coupled linear equations @xmath2 where @xmath3 , the underlined variables @xmath4 and @xmath5 denote @xmath6-tuples of spacial variables , i.e. , @xmath7 and @xmath8 , and the laplacians are given by @xmath9 and @xmath10 . we assume that for each @xmath11 , @xmath12 is a symmetric marginal density matrix such that @xmath13 and @xmath14 for any permutations @xmath15 and @xmath16 on @xmath17 . the _ contraction operator _ @xmath18 is defined by @xmath19 \gamma^{(k+2)}(t,\underline{x}_{k+2};\underline{x}_{k+2 } ' ) \\ & = \gamma^{(k+2)}(t,\underline{x}_{k},x_j , x_j;\underline{x}_{k}',x_j , x_j ) - \gamma^{(k+2)}(t,\underline{x}_{k},x_j',x_j';\underline{x}_{k}',x_j',x_j ' ) . \end{split}\ ] ] the coupling constant is either @xmath20 or @xmath21 . we call the gp hierarchy _ defocusing _ if @xmath22 , and _ focusing _ if @xmath23 . to define solutions to the gp hierarchy , we introduce the following definitions ( see also @xcite ) . for @xmath24 , we define the homogeneous sobolev space @xmath25 for sequences by @xmath26 where @xmath27 similarly , we define the inhomogeneous sobolev space @xmath28 for sequences by @xmath29 where @xmath30 a sequence @xmath31 is called a _ mild solution _ in @xmath32 ( or @xmath33 ) to the quintic gp hierarchy if it solves the hierarchy of the integral equations @xmath34 where @xmath35 is the free evolution operator . a sequence @xmath36 is called _ admissible _ if for each @xmath11 and @xmath37 , @xmath38 is a non - negative trace class operator on @xmath39 ( subset of @xmath40 functions that satisfy ) and @xmath41 we call a sequence @xmath36 a _ limiting hierarchy _ if there is a sequence @xmath42 of non - negative density matrices on @xmath43 with @xmath44 such that @xmath12 is the weak- * limit of the @xmath6-particle marginals of @xmath45 in the trace class on @xmath39 , that is , @xmath46 in this paper , we consider mild solutions to the gp hierarchy that are admissible or limiting hierarchies . such mild solutions are physically relevant in the theory of derivation of the nonlinear schrdinger equation ( nls ) from the many body linear schrdinger equation ( see section [ subsec : related work ] ) . we now state our main result . [ thm : main thm gp ] suppose that @xmath31 is a mild solution in @xmath47 to the quintic gp hierarchy with initial data @xmath48 , which is either admissible or a limiting hierarchy for each @xmath49 . if @xmath50 for all @xmath51 for @xmath52 sufficiently small , then @xmath31 is the only such solution for the given initial data . the quintic gp hierarchy is closely related to the quintic nls via factorized functions . indeed , one can check that if @xmath53 is a solution to the quintic nls @xmath54 then a sequence of factorized functions , @xmath55 solves the gp hierarchy . in this sense , proving uniqueness for the gp hierarchy is more difficult than it is for the quintic nls . the quintic gp hierarchy was studied by t. chen and pavlovi @xcite for the derivation of the quintic nls as the gross - pitaevskii field limit of a non - relativistic bose gas with @xmath56-particle interactions . as a part of their analysis , the authors proved ( conditional ) uniqueness of solutions to the quintic gp hierarchy in an energy space , that is , a sobolev type space of order 1 , in one and two dimensions . we remark that in all dimensions , proving such uniqueness in an energy space is necessary to derive nls . however , it is an open problem to prove uniqueness in three dimensions . theorem [ thm : main thm gp ] provides an answer for this open problem under a smallness assumption . we remark that the 3d quintic gp hierarchy is scaling - critical in @xmath57 , and that even with our smallness assumption , our theorem is the first uniqueness theorem for the cubic or quintic gp hierarchy in a scaling - critical space . moreover , uniqueness in theorem [ thm : main thm gp ] is unconditional . it remains an open problem to remove the smallness assumption . in the case of the 3d quintic nls , it is known that solutions are unique in the space @xmath58 for @xmath59 , without a smallness assumption @xcite . however , the proof of unconditional uniqueness in the scaling - critical case @xmath60 differs from the proof in the subcritical case @xmath61 . in the case of the 3d quintic gp hierarchy , we also expect that an approach different from the one that we use in the scaling - subcritical case is needed to remove the smallness assumption in the scaling - critical case . currently , the main obstacle to removing the smallness assumption for solutions to the 3d quintic gp hierarchy in the scaling - critical case is the generally infinite cardinality of the support of the measure @xmath62 in the statement of the quantum de finetti theorem , theorem [ thm : strong de finetti ] . to compare scaling - critical and subcritical regimes , we provide a uniqueness theorem for the 3d quintic hartree hierarchy . the 3d quintic hartree hierarchy is also an infinite hierarchy as . however the contraction operator @xmath63 in is replaced by @xmath64 note that the 3d quintic hartree equation is subcritical in @xmath65 if the three - particle interaction potential @xmath66 is less singular than the product of delta functions . this is , if @xmath67 for some @xmath68 . in this case , we can show unconditional uniqueness for the 3d quintic hartree hierarchy without a smallness assumption . [ thm : main thm hartree ] suppose that @xmath67 for some @xmath68 . let @xmath69 be a mild solution to the quintic hartree hierarchy with initial data @xmath48 , which is either admissible or a limiting hierarchy for each @xmath49 . if there exists @xmath52 such that @xmath50 for all @xmath51 , then @xmath31 is the only such solution for the given initial data . the background work in this line goes back to the derivation of schrdinger type equations from interacting particle systems . in the pioneering works by hepp @xcite , spohn @xcite and in a series of more recent breakthroughs by erds , schlein and yau @xcite , the authors derived the cubic nls in @xmath70 . a major ingredient in this derivation is the establishment the uniqueness of solutions to the corresponding gp hierarchy . the proof of uniqueness by erds - schlein - yau requires sophisticated feynman graph expansions . later , klainerman and machedon @xcite rephrased this as a board game argument to provide an alternative approach to prove uniqueness of solutions . however , the result in @xcite is conditional in that the solutions that satisfy an a - priori space - time bound assumption . this assumption is used by kirkpatrick , schlein , and staffilani @xcite in two dimensional settings for compact and non - compact domains . + a recent new proof on the unconditional uniqueness of 3d cubic gp hierarchy was initiated by t.chen , hainzl , pavlovi and seringer @xcite using the _ quantum de finetti theorem_. the quantum de finetti theorem is a quantum analogue of the hewitt - savage theorem in probability theory . the strong version of the quantum de finetti theorem ( see [ thm : strong de finetti ] ) asserts that an infinite sequence of _ admissible _ marginal density matrices can be expressed as an average over factorized states . however , for each @xmath49 , the limiting hierarchies of density matrices do not necessarily satisfy admissibility . in this case , one uses the weak version of the de finetti theorem ( see [ thm : weak de finetti ] ) . this is necessary when working with the bbgky hierarchy approach for the derivation of nls as in @xcite , where one starts with a finite bbgky hierarchy of @xmath71 equations for the bosonic @xmath71-particle system ( see ( 2.1 ) in @xcite ) . in this case , the gp hierarchy of equations is obtained by taking @xmath72 in the finite hierarchy . as part of the derivation , one proves that the weak-@xmath73 limit of solutions @xmath74 to the bbgky hierarchy solve the infinite gp hierarchy . by taking advantage of the quantum de finetti theorems that give an alternative factorized formula for the solutions to the hierarchy , the authors of @xcite established unconditional uniqueness for cubic gp hierarchy at the same regularity level of the corresponding nls . others have also used the de finetti theorem to prove unconditional uniqueness for gp hierarchies in various settings . in @xcite , v. sohinger adapted the method from @xcite to cubic gp hierarchy in a periodic setting . in @xcite , x.chen-smith studied a chen - simon - schrdigner hierarchy . we prove theorem [ thm : main thm gp ] and theorem [ thm : main thm hartree ] in the framework of chen - hainzl - pavlovi - seringer @xcite . due to the linearity of the hierarchy , it suffices to show that solutions solution having a zero initial are the zero solution . in our proof , we iterate the duhamel formula with zero initial data @xmath75 times , resulting in a number of terms that grows factorially in @xmath75 . we reduce the number of terms by the erds - schlein - yau combinatorial argument in klainerman - machedon s formulation @xcite . the quintic version of this combinatoric reduction was used by chen - pavlovic in @xcite . we use it for the 3d quintic gp and hartee hierarchies without modification . next , we apply the quantum de finetti theorem to write each term as an integral sum of factorized states , and reorganize them using a tree - graph structure ( see figure 1 below ) which extends the tree - graph in chen - hainzl - pavlovi - seiringer @xcite . then , we iteratively estimate the @xmath75 integrals . in each step , we apply our multilinear estimates , which can be found in appendix [ sec : multilin estimates ] . finally , we send @xmath76 and find that solutions having zero initial data must be the zero solution . in our previous work @xcite , we proved unconditional uniqueness for the cubic gp hierarchy in a low regularity setting , using a similar approach . in @xcite , our key ingredients were the trilinear estimates @xmath77 , @xmath78 and @xmath79 in lemma 2.6 . these estimates are based on the dispersive estimates @xmath80 and negative order sobolev norm estimates ( lemma a.3 in @xcite ) . in the proof , we applied these estimates to the reorganized integrals iteratively together with multilinear estimates based on strichartz estimates ( @xmath81 , @xmath82 and @xmath83 in lemma 2.6 ) . we remark that the use of dispersive estimates is crucial in obtaining the optimal subcritical low regularity uniqueness theorem . the dispersive estimates do nt work in the scaling - critical space , however . roughly speaking , this is due to the failure of integrability ( in time ) of the bound in . for instance , if one tries to prove uniqueness for the 3d quintic gp hierarchy in @xmath84 by the same approach , one should choose @xmath85 for the multilinear estimate . then , the bound in is not integrable in time . in the present work , instead of using dispersive estimates , we use multilinear estimates ( proposition [ prop : multilinear gp ] and propositions [ prop : multilinear hartree ] ) that are based on by strichartz estimates and a negative order sobolev norm bound . in the case of the hartree hierarchy , we also make use of a convolution estimates of w. beckner @xcite . in order to prove theorem [ thm : main thm gp ] and theorem [ thm : main thm hartree ] at the same time , we define @xmath86 with this notation , we can now combine definitions and of @xmath18 for the gp hierarchy and the hartree hierarchy , respectively , as follows . @xmath87 this paper is organized as follows . in section [ sec : comb ] we present the road map for the proof of the main theorems and reduce the the main theorems to proposition [ prop : zero trace norm ] . we illustrate with an example how to factorize solutions in section [ sec : example factorization ] . in section [ sec : distinguished tree graph ] , we introduce tree graphs to illustrate our decomposition of each factor , and present properties of the associated kernels . the proof of proposition [ prop : zero trace norm ] occupies section [ sec : recursive bounds ] . in appendix [ sec : multilin estimates ] , we prove several multilinear estimates that we use section [ sec : recursive bounds ] .

in this paper , we study solutions to the three - dimensional quintic gross - pitaevskii hierarchy . we prove unconditional uniqueness among all small solutions in the critical space ( which corresponds to on the nls level ) . with slight modifications to the proof , we also prove unconditional uniqueness of solutions to the hartree hierarchy without a smallness condition . our proof uses the quantum de finetti theorem , and is an extension of the work by chen - hainzl - pavlovi - seiringer , and our previous work .

in this paper , we study solutions to the three - dimensional quintic gross - pitaevskii hierarchy . we prove unconditional uniqueness among all small solutions in the critical space ( which corresponds to on the nls level ) . with slight modifications to the proof , we also prove unconditional uniqueness of solutions to the hartree hierarchy without a smallness condition . our proof uses the quantum de finetti theorem , and is an extension of the work by chen - hainzl - pavlovi - seiringer , and our previous work .

1303.6437

i

the * traveling salesman problem ( tsp ) * is one of the best known and most fundamental problems in combinatorial optimization . determining how well it can be approximated in polynomial time is therefore a major open problem , albeit one for which the solution still seems elusive . on the algorithmic side , the best known efficient approximation algorithm for the symmetric case is still a 35-year old algorithm due to christofides @xcite which achieves an approximation ratio of @xmath4 . however , recently there has been a string of improved results for the interesting special case of graphic tsp , improving the ratio to @xmath5 @xcite . for the asymmetric case ( atsp ) , it is not yet known if a constant - factor approximation is even possible , with the best known algorithm achieving a ratio of @xmath6 @xcite . unfortunately , there is still a huge gap between the algorithmic results mentioned above and the best currently known hardness of approximation results for tsp and atsp . for both problems , the known inapproximability thresholds are small constants ( @xmath0 and @xmath1 ( cf . @xcite ) , respectively ) . in this paper , we try to improve this situation somehow by giving modular hardness reductions that slightly improve the hardness bounds for both problems to @xmath2 and @xmath3 , respectively . the latter bound is the first , for more than a decade now , improvement of papadimitriou and vempala bound @xcite for the atsp . the method of our solution differs essentially from that of @xcite and uses some new paradigms of the bounded occurrence optimization which could be also of independent interest in other applications . similarly to @xcite , the hope is that the modularity of our construction , which goes through an intermediate stage of a bounded - occurrence constraint satisfaction problem ( csp ) , will allow an easier analysis and simplify future improvements . indeed , one of the main new ideas we rely on is a certain new variation of the wheel amplifiers first defined by berman and karpinski @xcite to establish inapproximability for 3-regular csps . this construction , which may be of independent interest , allows us to establish inapproximability for a 3-regular csp with a special structure . this special structure then makes it possible to simulate many of the constraints in the produced graph essentially `` for free '' , without using gadgets to represent them . thus , even though for the remaining constraints we mostly reuse gadgets which have already appeared in the literature , we are still able to obtain improved bounds . let us now recall some of the previous work on the hardness of approximation of tsp and atsp . papadimitriou and yannakakis @xcite were the first to construct a reduction that , combined with the pcp theorem @xcite , gave a constant inapproximability threshold , though the constant was not more than @xmath7 for the tsp with distances either one or two . engebretsen @xcite gave the first explicit approximation lower bound of @xmath8 for the problem . the inapproximability factor was improved to @xmath9 by bckenhauer and seibert @xcite , who studied the restricted version of the tsp with distances one , two and three . papadimitriou and vempala @xcite proved that it is -hard to approximate the tsp with a factor better than @xmath10 . presently , the best known approximation lower bound is @xmath0 due to lampis @xcite . the important restriction of the tsp , in which we consider instances with distances between cities being values in @xmath11 , is often referred to as the @xmath12-tsp . the best known efficient approximation algorithm for the @xmath13-tsp has an approximation ratio @xmath14 and is due to berman and karpinski @xcite . as for lower bounds , engebretsen and karpinski @xcite gave inapproximability thresholds for the @xmath12-tsp problem of @xmath15 for @xmath16 and @xmath17 for @xmath18 . more recently , karpinski and schmied @xcite obtained improved inapproximability factors for the @xmath13-tsp and the @xmath19-tsp of @xmath20 and @xmath21 , respectively . for atsp the currently best known approximation lower bound was @xmath1 due to papadimitriou and vempala @xcite . when we restrict the problem to distances with values in @xmath11 , there is a simple approximation algorithm with approximation ratio @xmath22 that constructs an arbitrary tour as solution . blser @xcite gave an efficient approximation algorithm for the @xmath23-atsp with approximation ratio @xmath24 . karpinski and schmied @xcite proved that it is -hard to approximate the @xmath25-atsp and the @xmath19-atsp within any factor less than @xmath26 and @xmath27 , respectively . for the case @xmath18 , engebretsen and karpinski @xcite gave an inapproximability threshold of @xmath28 . * overview : * in this paper we give a hardness proof which proceeds in two steps . first , we start from the max - e3-lin2 problem , in which we are given a system of linear equations mod @xmath29 with exactly three variables in each equation and we want to find an assignment such as to maximize the number of satisfied equations . optimal inapproximability results for this problem were shown by hstad @xcite . we reduce this problem to a special case where variables appear exactly @xmath30 times and the linear equations have a particular structure . the main tool here is a new variant of the wheel amplifier graphs of berman and karpinski @xcite . in the second step , we reduce this @xmath30-regular csp to tsp and atsp . the general construction is similar in both cases , though of course we use different gadgets for the two problems . the gadgets we use are mostly variations of gadgets which have already appeared in previous reductions . nevertheless , we manage to obtain an improvement by exploiting the special properties of the @xmath30-regular csp . in particular , we show that it is only necessary to construct gadgets for roughly one third of the constraints of the csp instance , while the remaining constraints are simulated without additional cost using the consistency properties of our gadgets . this idea may be useful in improving the efficiency of approximation - hardness reductions for other problems . thus , overall we follow an approach unlike that of @xcite , where the reduction is performed in one step , and closer to @xcite . the improvement over @xcite comes mainly from the idea mentioned above , which is made possible using the new wheel amplifiers , as well as several other tweaks . the end result is a more economical reduction which improves the bounds for both tsp and atsp . an interesting question may be whether our techniques can also be used to derive improved inapproximability results for variants of the atsp and tsp ( cf . @xcite,@xcite and @xcite ) or other graph problems , such as the steiner tree problem .

the best up to now known hardness of approximation bounds were for the symmetric case ( due to lampis ) and for the asymmetric case ( due to papadimitriou and vempala ) . we construct here two new bounded occurrence csp reductions which improve these bounds to and , respectively . the latter bound is the first improvement in more than a decade for the case of the asymmetric tsp . one of our main tools , which may be of independent interest , is a new construction of a bounded degree wheel amplifier used in the proof of our results .

in this paper , we study the approximability of the metric traveling salesman problem ( tsp ) and prove new explicit inapproximability bounds for that problem . the best up to now known hardness of approximation bounds were for the symmetric case ( due to lampis ) and for the asymmetric case ( due to papadimitriou and vempala ) . we construct here two new bounded occurrence csp reductions which improve these bounds to and , respectively . the latter bound is the first improvement in more than a decade for the case of the asymmetric tsp . one of our main tools , which may be of independent interest , is a new construction of a bounded degree wheel amplifier used in the proof of our results .

1510.06263

i

the problem of simultaneous localization and mapping ( slam ) has a rich history over the past two decades , which is too broad to cover here , see e.g. @xcite . the extended kalman filter ( ekf ) based slam ( the ekf - slam ) has played an important historical role , and is still used , notably for its ability to close loops thanks to the maintenance of correlations between remote landmarks . the fact that the ekf - slam is inconsistent ( that is , it returns a covariance matrix that is too optimistic , see e.g. , @xcite , leading to inaccurate estimates ) was early noticed @xcite and has since been explained in various papers @xcite . in the present paper we consider the inconsistency issues that stem from the fact that , as only relative measurements are available , the origin and orientation of the earth - fixed frame can never be correctly estimated , but the ekf - slam tends to think " it can estimate them as its output covariance matrix reflects an information gain in those directions of the state space . this lack of observability , and the poor ability of the ekf to handle it , is notably regarded as the root cause of inconsistency in @xcite ( see also references therein ) . in the present paper we advocate the use of the invariant ( i)-ekf to prevent covariance reduction in directions of the state space where no information is available . the invariant extended kalman filter ( iekf ) is a novel methodology introduced in @xcite that consists in slightly modifying the ekf equations to have them respect the geometrical structure of the problem . reserved to systems defined on lie groups , it has been mainly driven by applications to localization and guidance , where it appears as a slight modification of the multiplicative ekf ( mekf ) , widely known and used in the world of aeronautics . it has been proved to possess theoretical local convergence properties the ekf lacks in @xcite , to be an improvement over the ekf in practice ( see e.g. , @xcite and more recently @xcite where the ekf is outperformed ) , and has been successfully implemented in industrial applications to navigation ( see the patent @xcite ) . in the present paper , we slightly generalize the iekf framework , to make it capable to handle very general observations ( such as range and bearing or bearing only observations ) , and we show how the derived iekf - slam , a simple variant of the ekf - slam , allows remedying the inconsistency of ekf - slam stemming from the non - observability of the orientation and origin of the global frame . the issue of ekf - slam inconsistency has been the object of many papers , see @xcite to cite a few , where empirical evidence ( through monte - carlo simulations ) and theoretical explanations in various particular situations have been accumulated . in particular , the insights of @xcite have been that the orientation uncertainty is a key feature in the inconsistency . the article @xcite , in line with @xcite , also underlines the importance of the linearization process , as linearizing about the true trajectory solves the inconsistency issues , but is impossible to implement in practice as the true state is unknown . it derives a relationship that should hold between various jacobians appearing in the ekf equations when they are evaluated at the current state estimate to ensure consistency . a little later , the works of g.p . huang , a.i . mourikis , and s. i. roumeliotis @xcite have provided a sound theoretical analysis of the ekf - slam inconsistency as caused by the ekf inability to correctly reflect the three unobservable degrees of freedom ( as an overall rotation and translation of the global reference frame leave all the measurements unchanged ) . indeed , the filter tends to erroneously acquire information along the directions spanned by those unobservable transformations . to remedy this problem , the above mentioned authors have proposed various solutions , the most advanced being the observability constrained ( oc)-ekf . the idea is to pick a linearization point that is such that the unobservable subspace seen " by the ekf system model is of appropriate dimension , while minimizing the expected errors of the linearization points . our approach , that relies on the iekf , provides an interesting alternative to the oc - ekf , based on a quite different route . indeed , the rationale is to apply the ekf methodology , but using alternative estimation errors to the standard linear difference between the estimate and the true state . any non - linear error that reflects a discrepancy between the true state and the estimate , necessarily defines a local frame around any point , and the idea underlying the iekf amounts to write the kalman jacobians and covariances in this frame . we notice and prove here that an alternative nonlinear error defines a local frame where the unobservable subspace is _ everywhere _ spanned by the same vectors . using this local frame at the current estimate to express kalman s covariance matrix will be shown to ensure the unobservable subspace seen " by the ekf system model is _ automatically _ of appropriate dimension . we thus obtain an ekf variant which automatically comes with consistency properties . moreover , we relate unobservability to the inverse of the covariance matrix ( called information matrix ) rather than on the covariance matrix itself , and we derive guarantees of information decrease over unobservable directions . contrarily to the oc - ekf , and as in the standard ekf , we use here the latest , and thus best , state estimate as the linearization point to compute the filter jacobians . in a nutshell , whereas the key fact for the analysis of @xcite is that the choice of the linearization point affects the observability properties of the linearized state error system of the ekf , the key fact for our analysis is that the choice of the error variable has similar consequences . theoretical results and simulations underline the relevance of the proposed approach . robot - centric formulations such as @xcite , and later @xcite are promising attempts to tackle unobservability , but they unfortunately lack convenience as the position of all the landmarks must be revised during the propagation step , so that the landmarks estimated position becomes in turn sensitive to the motion sensor s noise . they do not provably solve the observability issues considered in the present paper , and it can be noted the oc - ekf has demonstrated better experimental performance than the robocentric mapping filter , in @xcite . in particular , the very recent papers @xcite propose to write the equations of the slam in the robot s frame under a constant velocity assumption . using an output injection technique , those equations become linear , allowing to prove global asymptotic convergence of any linear observer for the corresponding deterministic linear model . this is fundamentally a deterministic approach and property , and as the matrices appearing in the obtained linear model are functions of the observations , the behavior of the filter is not easy to anticipate in a noisy context : the observation noise thus corrupts the very propagation step of the filter . some recent papers also propose to improve consistency through local map joining , see @xcite and references therein . although appealing , this approach is rather oriented towards large - scale maps , and requires the existence of local submaps . but when using submap joining algorithm , inconsistency in even one of the submaps , leads to an inconsistent global map " @xcite . this approach may thus prove complementary , if the iekf slam proposed in the present paper is used to build consistent submaps . note that , the iekf slam can also be readily combined with other measurements such as the gps , whereas the submap approach is tailored for pure slam . from a methodology viewpoint , it is worth noting our approach does not bring to bear estimation errors written in a robot frame , as @xcite . although based on symmetries as well , the estimation errors we use are slightly more complicated . finally , nonlinear optimization techniques have become popular for slam recently , see e.g. , @xcite as one of the first papers . links between our approach , and those novel methods are discussed in the paper s conclusion . the paper is organized as follows . in section [ sec:1 ] , the standard ekf equations and ekf - slam algorithm are reviewed . in section [ sec:2 ] we recall the problem that neither the origin nor the orientation of the global frame are observable , but the ekf - slam systematically tends to think " it observes them , which leads to inconsistency . in section [ sec:22 ] we introduce the iekf - slam algorithm . in section [ sec:3 ] we show how the linearized model of the iekf always correctly captures the considered unobservable directions . in section [ sect::tools ] we derive a property of the covariance matrix output by the filter that can be interpreted in terms of fisher information . in section [ sec:4 ] simulations support the theoretical results and illustrate the benefits of the proposed algorithm . finally , the iekf theory of @xcite is briefly recapped in the appendix , and the iekf slam shown to be an application of this theory indeed . the equations of the iekf slam in 3d are then also derived applying the general theory .

in this paper we address the inconsistency of the ekf - based slam algorithm that stems from non - observability of the origin and orientation of the global reference frame . we prove on the non - linear two - dimensional problem with point landmarks observed that this type of inconsistency is remedied using the invariant ekf , a recently introduced variant of the ekf meant to account for the symmetries of the state space .

in this paper we address the inconsistency of the ekf - based slam algorithm that stems from non - observability of the origin and orientation of the global reference frame . we prove on the non - linear two - dimensional problem with point landmarks observed that this type of inconsistency is remedied using the invariant ekf , a recently introduced variant of the ekf meant to account for the symmetries of the state space . extensive monte - carlo runs illustrate the theoretical results .

1408.0110

i

a polling model is a single - server system in which the server visits @xmath0 queues @xmath1 in cyclic order . customers that arrive at @xmath2 are referred to as type @xmath3 customers . the special feature of the model considered in the present paper is that , within a customer type , we distinguish high and low priority customers . more specifically , we study a polling system which consists of two queues , @xmath4 and @xmath5 . the first of these queues contains customers of two priority classes , high ( @xmath6 ) and low ( @xmath7 ) . the exhaustive , gated and globally gated service disciplines are studied . our motivation to study a polling model with priorities is that the performance of a polling system can be improved through the introduction of priorities . in production environments , e.g. , one could give highest priority to jobs with a service requirement below a certain threshold level . this might decrease the mean waiting time of an arbitrary customer without having to purchase additional resources @xcite . priority polling models also can be used to study traffic intersections where conflicting traffic flows face a green light simultaneously ; e.g. traffic which takes a left turn may have to give right of way to conflicting traffic that moves straight on , even if the traffic light is green for both traffic flows . another application is discussed in @xcite , where a priority polling model is used to study scheduling of surgery procedures in medical emergency rooms . in the computer science community the bluetooth and 802.11 protocols are frequently modelled as polling systems , cf . many scheduling policies that have been considered or implemented in these protocols involve different priority levels in order to improve quality - of - service ( qos ) for traffic that is very sensitive to delays or loss of data , such as voice over wireless ip . the 802.11e amendment defines a set of qos enhancements for wireless lan applications by differentiating between high priority traffic , like streaming multimedia , and low priority traffic , like web browsing and email traffic . although there is quite an extensive amount of literature available on polling systems , only very few papers treat priorities in polling models . most of these papers only provide approximations or focus on pseudo - conservation laws . in @xcite exact mean waiting time results are obtained using the mean value analysis ( mva ) framework for polling systems , developed in @xcite . the mva framework can only be used to find the first moment of the waiting time distribution for each customer type , and the mean residual cycle time . the main contribution of the present paper is the derivation of laplace stieltjes transforms ( lsts ) of the distributions of the marginal waiting times for each customer type ; in particular it turns out to be possible to obtain exact expressions for the waiting time distributions of both high and low priority customers at a queue of a polling system . probability generating functions ( pgfs ) are derived for the joint queue length distribution at polling epochs , and for the steady - state marginal queue length distribution of the number of customers at an arbitrary epoch . the present paper is structured as follows : section [ general ] gathers known results of nonpriority polling models which are relevant for the present study . sections [ gated ] ( gated ) , [ globallygated ] ( globally gated ) , and [ exhaustive ] ( exhaustive ) give new results on the priority polling model . in each of the sections we successively discuss the joint queue length distribution at polling epochs , the cycle time distribution , the marginal queue length distributions and waiting time distributions . the mean waiting times are given at the end of each section . a numerical example is presented in section [ numericalexample ] to illustrate some of the improvements that can be obtained by introducing prioritisation in a polling system .

we study the cycle time distribution , the waiting times for each customer type , the joint queue length distribution at polling epochs , and the steady - state marginal queue length distributions for each customer type . * keywords : * polling , priority levels , queue lengths , waiting times

in this paper we consider a single - server cyclic polling system consisting of two queues . between visits to successive queues , the server is delayed by a random switch - over time . two types of customers arrive at the first queue : high and low priority customers . for this situation the following service disciplines are considered : gated , globally gated , and exhaustive . we study the cycle time distribution , the waiting times for each customer type , the joint queue length distribution at polling epochs , and the steady - state marginal queue length distributions for each customer type . * keywords : * polling , priority levels , queue lengths , waiting times

1406.7649

i

we have derived and modelled distributions of dust and metals in the disc of m31 with the purpose of finding indirect evidence to hopefully distinguish between one of the two competing dust production mechanisms in late - type galaxies : stellar dust production and interstellar grain growth . the data together with simple models point towards interstellar grains growth being the most important mechanism , although firm conclusions can not be drawn due to possible degeneracies between formation and destruction of dust associated with stars . we have computed mean radial dust distributions from the helga dust map based on simple sed models with a varying as well as a fixed emissivity index @xmath0 as well as an overall metal distribution derived using data collected from the literature . in a second step , we used a simple analytical model of the evolution of the dust component in a galaxy and fit this model to the radial dust - to - gas distribution . the dust - to - gas gradient in m31 is steeper than the metallicity gradient , i.e. , there is a clear negative dust - to - metals gradient along the disc , and in such cases our model suggests dust growth must be the dominant dust - formation mechanism in the ism of m31 . taken at face value , our model fit actually suggests the net stellar dust production is almost negligible and the build - up of the dust component in m31 should therefore be dominated by interstellar growth . from the above we conclude that m31 is a strong case for cosmic dust being the result of substantial interstellar grain growth , while the net stellar dust production must be limited . the efficiency of dust production in stars ( most notably sne ) and the grain destruction in the ism may be degenerate , however . consequently , we can only conclude that interstellar grain growth by accretion is _ at least _ as important as stellar dust production channels in building the cosmic dust component in nearby galaxies . however , our result is in line with the recent ( and more detailed ) dust - evolution models of late - type dwarf galaxies by @xcite , which favours low condensation efficiencies in type ii sne combined with substantial grain growth in the ism . thus , it seems worthwhile to construct a detailed model of the radial dust distribution of m31 , which is more independent of the metallicity gradient . we hope to return to this in a future publication .

we derive mean radial dust distributions using a dust map created from _ modified blackbodies are fit to approximately 4000 pixels with a varying , as well as a fixed , dust emissivity index ( ) . an overall metal distribution is also derived using data collected from the literature . we use a simple analytical model of the evolution of the dust in a galaxy with dust contributed by stellar sources and interstellar grain growth , and fit this model to the radial dust - to - metals distribution across the galaxy . our analysis shows that the dust - to - gas gradient in m31 is steeper than the metallicity gradient , suggesting interstellar dust growth is ( or has been ) important in m31 . we argue that m31 helps build a case for cosmic dust in galaxies being the result of substantial interstellar grain growth , while the net dust production from stars may be limited . we note , however , that the efficiency of dust production in stars , e.g. , in supernovae ejecta and/or stellar atmospheres , and grain destruction in the interstellar medium may be degenerate in our simple model . we can conclude that interstellar grain growth by accretion is likely _ at least _ as important as stellar dust production channels in building the cosmic dust component in m31 . [ firstpage ] ism : clouds dust , extinction ism : evolution galaxies : evolution galaxies : individual : m31 galaxies : ism .

in this paper we consider the implications of the distributions of dust and metals in the disc of m31 . we derive mean radial dust distributions using a dust map created from _ herschel _ images of m31 sampling the entire far - infrared peak . modified blackbodies are fit to approximately 4000 pixels with a varying , as well as a fixed , dust emissivity index ( ) . an overall metal distribution is also derived using data collected from the literature . we use a simple analytical model of the evolution of the dust in a galaxy with dust contributed by stellar sources and interstellar grain growth , and fit this model to the radial dust - to - metals distribution across the galaxy . our analysis shows that the dust - to - gas gradient in m31 is steeper than the metallicity gradient , suggesting interstellar dust growth is ( or has been ) important in m31 . we argue that m31 helps build a case for cosmic dust in galaxies being the result of substantial interstellar grain growth , while the net dust production from stars may be limited . we note , however , that the efficiency of dust production in stars , e.g. , in supernovae ejecta and/or stellar atmospheres , and grain destruction in the interstellar medium may be degenerate in our simple model . we can conclude that interstellar grain growth by accretion is likely _ at least _ as important as stellar dust production channels in building the cosmic dust component in m31 . [ firstpage ] ism : clouds dust , extinction ism : evolution galaxies : evolution galaxies : individual : m31 galaxies : ism .

1305.5391

i

the ricci flow , introduced by hamilton , is a geometric flow for metrics on @xmath0-manifolds , and has played a decisive role in the proof of the poincar conjecture and thurston s geometrization conjecture for @xmath1-manifolds . it is natural to then investigate a corresponding problem for contact @xmath1-manifolds . one of way of doing this is to find a cr analogue of the ricci flow on a pseudohermitian @xmath1-manifold ( see section sec : definitions for definitions and basic notions in pseudohermitian geometry ) . recall that a strictly pseudoconvex cr structure on a @xmath1-manifold @xmath2 is given by a cooriented plane field @xmath3 , where @xmath4 is a contact form , together with a compatible complex structure @xmath5 . this gives rise to a natural metric @xmath6 for @xmath2 . given this data , there is a natural connection , the so - called tanaka - webster connection or pseudohermitian connection . we denote the torsion of this connection by @xmath7 , and the webster curvature , a kind of scalar curvature , by @xmath8 . the torsion flow is then the following pde , @xmath9it seems to us that the torsion flow is the right cr analogue of the ricci flow . the torsion flow greatly simplifies if the torsion vanishes . this only happens in very special setups . indeed , cr @xmath1-manifolds with vanishing torsion are @xmath10-contact , meaning that the reeb vector field is a killing vector field for the metric @xmath11 . in general , one can still hope that the torsion flow improves properties of the contact manifold underlying the cr - manifold . the mostly used tools in the study of hamilton s ricci flow @xcite consist of maximum principles . exceptions are formed by hamilton s entropy formula which holds for closed surfaces with positive gaussian curvature @xcite , and also by perelman s entropy formulas @xcite . these formulas can be thought of as monotonicity formulas for integrals of local geometric quantities . in this paper , we try to do the same for the torsion flow by setting up some monotonicity formulas for perelman - type functionals . we conclude this introduction with a brief plan of the paper . * in section [ sec : motivation ] and section [ sec : statement_monotonicity_results ] we motivate the definition of the torsion flow and give more precise statements of our results . * in section [ sec : definitions ] we survey basic notions in cr geometry . * in sections [ section : tanaka_connection_global_frame ] and [ section : homogeneous_contact_manifolds ] we describe cr manifolds with a global coframe and we also define homogeneous cr manifolds . on the latter class the torsion flow reduces to an ode if we start with some appropriate initial conditions . these computations illustrate the behavior of the torsion flow in special cases , and in these cases the torsion flow behaves as can be expected from a ricci - like flow . * finally , in section [ sec : entropy ] we discuss analogues of perelman s entropy formulas . * acknowledgements * we thank jih - hsin cheng for valuable discussions , and we are in particular grateful to his contributions on the entropy formulas , where he suggested the correct formulation of entropy ; he also helped us out with the linearized operator in our attempt to prove short time existence . the research of scc and ctw is supported in part by nsc of taiwan . ovk is supported by nrf grant 2012 - 011755 funded by the korean government . for the basic definitions and notions involved , we refer the reader to section [ sec : definitions ] . consider a closed @xmath12-manifold @xmath2 , with a smooth family of pseudohermitian structures @xmath13 for which @xmath14 is compatible with @xmath15 : this means that @xmath16 forms a hermitian metric on the complex vector bundle @xmath17 . furthermore , @xmath18 induces a metric on all tensor fields . we shall use these metrics and the induced norms without explicitly referring to @xmath18 . throughout the paper , we only consider a fixed contact structure , i.e. @xmath19 is independent of @xmath20 . henceforth , we just write @xmath21 . take a local orthonormal frame @xmath22 , where @xmath23 is the reeb field , @xmath24 is a basis of @xmath25 , and @xmath26 is a basis of @xmath27 . then we write @xmath28 . define @xmath29 , and consider the general flow on @xmath30 given by @xmath31the cr einstein - hilbert functional is defined by @xmath32here @xmath33 is the volume form and @xmath8 denotes tanaka - webster curvature . from variation formulas for the webster curvature and the measure , see the appendix and , it follows that @xmath34if we put @xmath35 and @xmath36 . here @xmath37 denotes the torsion tensor . it is therefore natural to consider the torsion flow on @xmath38 as defined in . unfortunately , we do not know whether a short - time solution to the torsion flow exists in general . for the very special class of homogeneous cr manifolds we define in section [ section : homogeneous_contact_manifolds ] we show a short - time existence result . furthermore , we show the following convergence result . let @xmath39 be a homogeneous contact manifold whose lie algebra is isomorphic to @xmath40 . then there is a unique homogeneous complex structure @xmath41 that is torsion free . moreover , for any choice of homogeneous complex structure @xmath42 , the normalized torsion flow converges to this unique cr - structure @xmath43 . in particular , for any choice of homogeneous complex structure on @xmath44 , the normalized torsion flow converges to the standard cr - structure . in the next section we shall discuss somewhat technical results concerning monotonicity properties of perelman type functionals . as an application of these monotonicity results , we classify torsion breathers and solitons . the classification of torsion solitons is a necessary step in understanding the singularity formation in the torsion flow . indeed , one expects the torsion soliton solutions to model finite time singularities of the torsion flow . in view of the flow and original definition in @xcite , it is natural to define the soliton solutions for torsion flow as follows . \(i ) a family @xmath14 of cr structures on @xmath45 evolving by the torsion flow is called a * breather * if for some @xmath46 and @xmath47 , there is a contact diffeomorphism @xmath48 such that * @xmath49 . * @xmath50 . the cases @xmath51 are called * steady * , * shrinking * or * expanding * breathers , respectively . \(ii ) a breather satisfying the above properties for all pairs of @xmath52 and @xmath53 of real numbers is called a * torsion soliton*. ideas of perelman @xcite ( see also @xcite and @xcite ) can be combined with theorem [ t2 ] , theorem [ t3 ] and theorem [ t4 ] to show the following classification result . [ c51 ] let @xmath54 be a closed pseudohermitian @xmath1-manifold . then \(i ) there is no closed steady torsion soliton other than the one which admits zero tanaka - webster curvature and vanishing pseudohermitian torsion up to a contact transformation . \(ii ) there is no closed expanding torsion soliton other than the one which admits negative tanaka - webster curvature and vanishing pseudohermitian torsion . \(iii ) there is no closed shrinking torsion soliton other than the one which admits positive tanaka - webster curvature and vanishing pseudohermitian torsion . the statements in this section are of a more technical nature : we will derive the cr analogue of perelman s monotonicity formulas for the so - called coupled torsion flows , and in dimension @xmath1 . in section [ sec : f_functional ] we define the cr analogue of perelman s @xmath55-functional by @xmath56with the constraint @xmath57under the flow , this is equivalent to @xmath58therefore , the following coupled torsion flow is natural , @xmath59with @xmath60and @xmath61 [ t2 ] let @xmath54 be a closed @xmath1-dimensional pseudohermitian manifold and @xmath62 be a solution of the coupled torsion flow on @xmath63 then @xmath64the monotonicity formula is strict unless@xmath65that is , up to a contact transformation @xmath66@xmath67 observe that for @xmath68 , @xmath69then the coupled torsion flow on @xmath70 is equivalent to the following system of coupled torsion flows on @xmath71 @xmath72and @xmath73 in section [ sec : wfunctional ] we define two functionals analogous to perelman s @xmath74-functional , namely the @xmath75-functional , @xmath76(4\pi \tau ) ^{-2}e^{-\varphi } d\mu\end{array}\]]and the @xmath77-functional @xmath78(4\pi \tau ) ^{-2}e^{-\varphi } d\mu .\end{array}\ ] ] note that @xmath75 and @xmath77 are invariant under the rescaling @xmath79 and @xmath80 . furthermore , we have @xmath81 for a contact diffeomorphism @xmath82 in view of theorem [ t2 ] , we first study the monotonicity property of @xmath75-functional . by the same discussion as before , the constraint @xmath83is equivalent to another constraint , namely @xmath84under the flow . therefore we consider the following coupled torsion flow : @xmath85with @xmath86and @xmath87 [ t3 ] let @xmath54 be a closed @xmath1-dimensional pseudohermitian manifold and @xmath62 and @xmath88 be a solution of the coupled torsion flow ( [ 2008 - 3 ] ) . then @xmath89the monotonicity formula is strict unless@xmath90that is , up to a contact transformation @xmath66@xmath91 next we study the monotonicity property of @xmath77-functional @xmath78(4\pi \tau ) ^{-2}e^{-\varphi } d\mu .\end{array}\]]by the same discussion as before , the constraint @xmath83is equivalent to @xmath84under the flow . therefore we consider the following coupled torsion flow : @xmath92with @xmath86and @xmath87 [ t4 ] let @xmath54 be a closed @xmath1-dimensional pseudohermitian manifold and @xmath62 and @xmath88 be a solution of the coupled torsion flow ( [ 2008 - 2 ] ) . then @xmath93the monotonicity formula is strict unless@xmath94that is , up to a contact transformation @xmath66@xmath95 note that for @xmath96 we may reparametrize the time @xmath20 by the formula @xmath97 . we have @xmath98 , so the coupled torsion flows and on @xmath70 are equivalent to the following coupled torsion flows on @xmath99 respectively : @xmath100and@xmath101 recall that @xmath102 is called a contact vector field if the lie derivative @xmath103 for some function @xmath104 . such a contact vector field has the form @xmath105 for some smooth function @xmath106 on @xmath2 . furthermore @xmath107so is equivalent to @xmath108with @xmath109 . similar results hold for and .

in this paper we define the torsion flow , a cr analogue of the ricci flow . for homogeneous cr manifolds we give explicit solutions to the torsion flow illustrating various kinds of behavior . we also derive monotonicity formulas for cr entropy functionals . as an application , we classify torsion breathers .

in this paper we define the torsion flow , a cr analogue of the ricci flow . for homogeneous cr manifolds we give explicit solutions to the torsion flow illustrating various kinds of behavior . we also derive monotonicity formulas for cr entropy functionals . as an application , we classify torsion breathers .

1007.1655

i

in this note we complete the work begun in the sequence of papers @xcite , @xcite-@xcite about equilateral triangles , regular tetrahedra , cubes , and regular octahedrons all with vertices having integer coordinates . very often we will refer to this property by saying that the various objects are in @xmath1 . strictly speaking these geometric objects are defined as being more than the set of their vertices that determines them , but for us here , these are just the vertices . so , for instance , an equilateral triangle is going to be a set of three points in @xmath2 for which the euclidean distances between every two of these points are the same . the main purpose of the paper is to take a close look at the regular octahedrons in @xmath2 . the simplest example of a regular octahedron with integer coordinates for its vertices is @xmath3 , [ 1 , 0 , 1 ] , [ 1 , 1 , 0 ] , [ 1 , 1 , 2 ] , [ 1 , 2 , 1 ] , [ 2 , 1 , 1]\},\ ] ] which can be obtained from the usual unit cube in @xmath4 , multiplying the vertices by a factor of two and then taking the coordinates of the centers of the new faces . it turns out that this procedure gives all such octahedrons as shown in @xcite : [ fig1 ] @xmath5 @xmath6 [ oldtheorem ] every regular octahedron in @xmath2 is the dual of a cube that can be obtained ( up to a translation with a vector with integer coordinates ) by doubling a cube in @xmath1 . referring to figure [ fig1 ] ( b ) , we showed that if the regular octahedron @xmath7 is in @xmath2 , then so is the cube @xmath8 and vice versa . this defines a one - to - one correspondence between the classes of cubes ( invariant under integer translations ) and the classes regular octahedra ( invariant under integer translations ) in @xmath2 . in @xcite we determined a sequence of irreducible cubes , one from each of the classes of cubes invariant under integer translations and cube symmetries . for each one of these cubes we can construct as before a regular octahedron , obtaining this way a sequence of irreducible regular octahedrons .

in this paper we describe a procedure for calculating the number of regular octahedrons that have vertices with coordinates in the set . as a result we adapt the method appeared in which was used to find the number of regular tetrahedra with coordinates of their vertices in . [ section ] [ theorem ] proposition [ theorem ] corollary [ theorem ] lemma [ theorem ] definition [ theorem ] problem [ theorem ] conjecture [ theorem ] example [ theorem ] remark # 1#1 # 1#1 # 1*#1 *

in this paper we describe a procedure for calculating the number of regular octahedrons that have vertices with coordinates in the set . as a result , we introduce a new sequence in _ the online encyclopedia of integer sequences _ ( a178797 ) and list the first one hundred terms of it . we adapt the method appeared in which was used to find the number of regular tetrahedra with coordinates of their vertices in . the idea of this calculation is based on the theoretical results obtained in . a new fact proved here helps increasing the speed of all the programs used before . the procedure is put together in a series of commands written for maple . [ section ] [ theorem ] proposition [ theorem ] corollary [ theorem ] lemma [ theorem ] definition [ theorem ] problem [ theorem ] conjecture [ theorem ] example [ theorem ] remark # 1#1 # 1#1 # 1*#1 *

astro-ph0001140

c

in this paper we describe the nog sample , a distance - limited ( @xmath386000 km / s ) and magnitude - limited ( b@xmath114 mag ) sample of 7076 optically - selected galaxies which covers 2/3 of the sky ( @xmath3 ) and has a good completeness in redshift ( 98% ) . we select the nog on the basis of homogenized corrected blue magnitudes in order to minimize systematic effects in galaxy sampling , due to the use of different magnitude systems in different areas of the sky and to galactic and internal extinction . in this sense the nog , which is meant to be the first step towards the construction of a statistically well - controlled optical galaxy sample with homogenized photometric data covering most of the celestial sphere , is in principle designed to offer a largely unbiased view of the galaxy distribution . we identify galaxy systems in the nog by means of both the hierarchical and the percolation _ friends of friends _ methods . after an extensive search in the space of relevant parameters with the guide of available numerical simulations , we choose optimal sets of parameters which allow us to obtain reliable and homogeneous catalogs of loose groups . remarkably , these catalogs turn out to be substantially consistent as far as the distribution of members in groups is concerned . containing about 500 systems ( with at least three members ) , they are among the largest catalogs of groups presently available . although they are drawn from a galaxy sample limited to bright magnitudes , they are useful for studies of the statistical properties of loose groups , since their physical properties were found to be stable , on average , against the inclusion of fainter galaxy members ( ramella et al . 1995a , b ; ramella , focardi & geller 1996 ) . in particular , being extracted from the same galaxy sample , the catalogs allow one to investigate on variations in group properties ( e.g. , velocity dispersion , virial mass and radius ) strictly related to differences in the algorithm adopted . these differences indicate to what extent our knowledge of the location and properties of groups in the nearby universe is inaccurate . previous comparisons between catalogs of groups identified with the h and p algorithms ( pisani et al . 1992 ) were based on catalogs extracted from different galaxy samples . most of the nog galaxies ( @xmath260% ) are found to be members of galaxy pairs ( @xmath2580 pairs comprising @xmath215% of the galaxies ) or groups with at least three members ( @xmath2500 groups comprising @xmath245% of the galaxies ) . about @xmath240% of the galaxies are left ungrouped ( field galaxies ) . though being limited to a depth of 6000 km s@xmath8 , the nog covers interesting regions of prominent overdensities ( in mass and galaxies ) of the nearby universe , such as the `` great attractor '' region and the perseus - pisces supercluster . compared to previous all - sky optical and iras galaxy samples , the nog provides a denser sampling of the galaxy density field in the nearby universe . besides , as expected , the nog delineates overdensity regions with a greater density contrast than iras galaxy samples do . given its high - density sampling and large sky coverage , the nog sample is well suited for mapping the cosmography of the nearby universe beyond the local supercluster and for allowing a comparison of the density field as traced by optical galaxies with that described by iras galaxies ( addressing questions concerning the amount of relative biasing in the galaxy distribution and its possible dependence on scale ) . by virtue of the identification of nog groups , the nog is also well suited for deriving galaxy density parameters on small scales to be used in observational investigations of environmental effects on galaxy properties . environmental studies in which the local galaxy density is decoupled from membership in galaxy systems go beyond the conventional comparison between the properties of cluster and field galaxies and thus can better constrain physical processes responsible for the formation and evolution of galaxies . much of the observed evolution of the properties and populations of galaxies ( e.g. , ellis 1997 ) which has occurred during recent epochs ( @xmath149 ) can be ascribed to interaction of galaxies and their local environment . in a subsequent paper ( see marinoni et al . 1999b for preliminary results ) the nog groups will be used to remove non - linearities in the peculiar velocity field ( e.g. , the velocity dispersion of group members ) on small scales . to correct the redshift distances of field galaxies and groups on large scales , we shall apply models of the peculiar velocity field , following the approach described in paper i. we shall use the locations of individual galaxies and groups calculated in real distance space ( i.e. for distances predicted by different velocity field models ) to calculate the selection function of the nog sample ( see paper ii ) and to reconstruct the galaxy density field . local galaxy density parameters to be used in studies of environmental effects on nearby galaxies will be provided . we wish to thank s. borgani , d. fadda , r. giovanelli , m. girardi , m. hudson , f. mardirossian , m. mezzetti , p. monaco , m. ramella for interesting conversations . c. m. and l. c. are grateful to sissa for its kind hospitality . this research has made use of the lyon - meudon extragalactic database ( leda ) supplied by the leda team at the cral - observatoire de lyon ( france ) and 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 . lrrr n=2 & 587 & 572 & 581 + n=3 & 194 & 221 & 212 + n=4 & 91 & 89 & 93 + n=5 & 55 & 57 & 65 + n=6 & 32 & 33 & 31 + n=7 & 23 & 22 & 20 + n=8 & 13 & 12 & 13 + n=9 & 5 & 10 & 14 + 10@xmath150n@xmath15120 & 39 & 44 & 45 + n@xmath152 & 23 & 19 & 20 +

in this paper we describe the nearby optical galaxy ( nog ) sample , which is a complete , distance - limited ( km / s ) and magnitude - limited ( b ) sample of optical galaxies . the sample covers 2/3 ( 8.27 sr ) of the sky ( ) and appears to have a good completeness in redshift ( 98% ) . we select the sample on the basis of homogenized corrected total blue magnitudes in order to minimize systematic effects in galaxy sampling . most of the nog galaxies (% ) are found to be members of galaxy pairs ( pairs for a total of% of objects ) or groups with at least three members ( groups for a total of% of objects ) . about 40% of galaxies are left ungrouped ( field galaxies ) . compared to previous optical and iras galaxy samples , the nog provides a denser sampling of the galaxy distribution in the nearby universe . given its large sky coverage , the identification of groups , and its high - density sampling , the nog is suited for the analysis of the galaxy density field of the nearby universe , especially on small scales .

in this paper we describe the nearby optical galaxy ( nog ) sample , which is a complete , distance - limited ( km / s ) and magnitude - limited ( b ) sample of optical galaxies . the sample covers 2/3 ( 8.27 sr ) of the sky ( ) and appears to have a good completeness in redshift ( 98% ) . we select the sample on the basis of homogenized corrected total blue magnitudes in order to minimize systematic effects in galaxy sampling . we identify the groups in this sample by means of both the hierarchical and the percolation _ friends of friends _ methods . the resulting catalogs of loose groups appear to be similar and are among the largest catalogs of groups presently available . most of the nog galaxies (% ) are found to be members of galaxy pairs ( pairs for a total of% of objects ) or groups with at least three members ( groups for a total of% of objects ) . about 40% of galaxies are left ungrouped ( field galaxies ) . we illustrate the main features of the nog galaxy distribution . compared to previous optical and iras galaxy samples , the nog provides a denser sampling of the galaxy distribution in the nearby universe . given its large sky coverage , the identification of groups , and its high - density sampling , the nog is suited for the analysis of the galaxy density field of the nearby universe , especially on small scales .

1701.05750

i

bouncing cosmologies , in which the universe collapses to a minimum of expansion before returning to expansion , have a long history @xcite . one of the interesting considerations for such a scenario is whether the matter content of the expanding phase is identical to that of the collapsing phase . in general , one might expect entropy to increase from one cycle to the next @xcite . this implies the production of radiation or the creation of black holes , which can be thought of as the objects that have the highest entropy allowed in nature @xcite . it is therefore of interest to consider whether black holes can persist through a cosmic bounce , as well as what effect they would have on the large - scale dynamics and on each other . one can divide such black holes into two classes : ( i ) those that persist in a universe that recollapses to a big crunch and then bounces into a new expansion phase ; and ( ii ) those that are generated by the high density of matter at the bounce . following ref . @xcite , these objects will be referred to as `` pre - crunch black holes '' ( pcbhs ) and `` big - crunch black holes '' ( bcbhs ) , respectively . it is interesting to consider whether there would be any observational way of distinguishing them from the primordial black holes ( pbhs ) that formed shortly after the bounce , or during the early stages of the standard hot big bang model without a bounce . it is also important to know whether black holes really can persist through a bounce in the way that is being envisaged . the precise properties of a cosmic bounce depend upon the way in which it is generated . many mechanisms have been proposed for this in the long history of the subject . these include classical effects associated with a cosmological constant @xcite , higher - derivative effects @xcite , or other modified theories of gravity @xcite . there are also semi - quantum gravitational effects associated with string theory @xcite , loop quantum gravity @xcite , quantum gravity condensates @xcite , and the `` pre - big - bang '' scenario @xcite . there have also been some claims that scale - invariant fluctuations generated during a cosmological collapse phase could turn into scale - invariant curvature fluctuations after a bounce @xcite , although in a fully quantum gravitational bounce there may be conceptual problems with these approaches @xcite . for a recent review of bouncing cosmologies , see ref . the scalar - field - driven bounces we consider here are probably the simplest models that can accommodate a bounce , and allow us to consider the problem of persistent black holes in an exact and fully relativistic way . besides the mechanism of the bounce itself , there is also the tricky mathematical issue of what is meant by a black hole in a universe that is not eternally expanding . this issue is further complicated by the lack of any asymptotic spatial infinity in most cosmological models . for a pbh that forms after the big bang , it is usually understood that the black hole singularity is part of any future big crunch singularity ( if one should occur ) . however , this can not apply to pcbhs or bcbhs if the cosmological bounce is non - singular . without knowledge of the location of the black hole singularities it is impossible to locate the event horizon , which is defined as the boundary of the past of future null infinity . a more satisfactory approach is therefore to use the concept of the black hole apparent horizon , which is defined as the outermost marginally outer trapped surface ( mots ) @xcite , and will be used throughout this study . a further matter of interest for models of this type concerns what happens to the number density of black holes as the minimum of expansion is approached . one could speculate about the maximum fraction of space that can be filled by pcbhs or about the number of bcbhs that form during the collapsing stage . these quantities would depend on the maximum density of matter that occurs at the bounce , and there are then various interesting constraints that can be imposed on the fraction of the universe s mass density that resides in black holes as a function of mass @xcite . if the black holes are randomly distributed , one would expect a process of hierarchical merging to occur , in which progressively larger apparent horizons form around groups of holes , so that the characteristic hole size steadily increases . alternatively , if the holes were distributed in a uniform fashion then pre - crunch hierarchical merger would be eliminated , and one could investigate the maximum number of black holes that could fit through a given bounce . in this paper , we study this scenario in detail by considering models that contain a regular lattice of black holes and a scalar field at the maximum compression . these restrictions are primarily motivated by mathematical simplicity , as we find that they yield a number of exact solutions to the einstein constraint equations . the associated initial data then prescribe a unique evolution in both the expansion and collapse directions . they therefore constitute a proper cosmological model . although in general the evolution will need to be determined numerically , the initial data itself can be used to analyse a number of the problems outlined above . in particular , the positions of the apparent horizons can be determined and used to calculate the distance between neighbouring black holes . this information can be used to determine the fraction of space that is filled by black holes as a function of the maximum energy density at the bounce . it also unambiguously demonstrates that black holes can persist through a cosmological bounce . the plan of this paper is as follows . in sec . 2 we derive the form of the constraint equations in a time - symmetric scalar field - dominated model , focussing particularly on the geometry of the spatial hypersurface at the bounce . in secs . 3 and 4 we consider black holes in hyperspherical and spatially flat cosmological models , respectively . in sec . 5 we consider black holes in higher - dimensional bouncing cosmologies , a possibility also emphasized in ref . @xcite . in sec . 6 we draw some general conclusions . appendix a discusses solutions for the initial data in which the scalar field is stationary at the maximum compression , these relating to multiple janis - newman - winicour - like black hole solutions . appendix b discusses solutions with a non - vanishing scalar potential .

in this paper we explore the idea that black holes can persist in a universe that collapses to a big crunch and then bounces into a new phase of expansion . we use a scalar field to model the matter content of such a universe near the time of the bounce , and look for solutions that represent a network of black holes within a dynamical cosmology . we derive conditions for the black holes to remain distinct ( i.e. avoid merging ) and hence persist into the new expansion phase .

in this paper we explore the idea that black holes can persist in a universe that collapses to a big crunch and then bounces into a new phase of expansion . we use a scalar field to model the matter content of such a universe near the time of the bounce , and look for solutions that represent a network of black holes within a dynamical cosmology . we find exact solutions to einstein s constraint equations that provide the geometry of space at the minimum of expansion and that can be used as initial data for the evolution of hyperspherical cosmologies . these solutions illustrate that there exist models in which multiple distinct black holes can persist through a bounce , and allow for concrete computations of quantities such as number density . we then consider solutions in flat cosmologies , as well as in higher - dimensional spaces ( with up to nine spatial dimensions ) . we derive conditions for the black holes to remain distinct ( i.e. avoid merging ) and hence persist into the new expansion phase . some potentially interesting consequences of these models are also discussed .

1303.7214

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thermalization of the quark - gluon plasma is one of the most challenging problems in current heavy ion physics . starting with two colliding nuclei in a form of color glass condensate with high gluon occupation @xmath0 below saturation scale @xmath1 @xcite and following the initial impact , a subsequent strong field evolution stage ( likely with instabilities @xcite ) till about the time @xmath2 is then succeeded by a far - from - equilibrium gluon - dominant matter , the glasma @xcite . the evolution of this glasma stage toward a quark - gluon plasma ( qgp ) that is close to local equilibratium and exhibits viscous - hydrodynamic behavior , is indicated by phenomenology to be reached on the order of a fermi over c time ( see e.g. @xcite ) . precisely how this occurs remains to be fully understood . describing the pre - equilibrium evolution with kinetic equations is a very useful approach , based on which the so - called `` bottom - up '' thermalization scenario was developed @xcite . there is however the complication of instability driven by anisotropy that may change this picture ( see e.g. @xcite ) . there are also other kinetic - based approaches , see e.g. @xcite . more recently an alternative thermalization scenario , based on crucial role of high initial overpopulation in the glasma and kinetic evolution dominated by elastic collisions , has been proposed in @xcite . in this scenario , while the initial scale @xmath3 is large compared with @xmath4 and thus the coupling @xmath5 is small , the high occupation @xmath6 elevates the elastic scattering rate to be of the order @xmath7 rather than the usual @xmath8 , and the glasma is essentially an emergent strongly interacting matter with weak coupling albeit large aggregate of constituents . two important scales are introduced to characterize the distribution , the hard cut - off scale @xmath9 beyond which @xmath10 and the soft high - occupation scale @xmath11 below which @xmath6 . while the initial glasma has the two scales overlapping @xmath12 , during the course of thermalization the two scales are separated eventually toward @xmath13 upon thermalization . one particularly nontrivial observation in the elastic - dominant picture is that the high initial overpopulation @xmath14 and the conservation of both energy and particle number will necessarily require the formation of a bose - einstein condensate that absorbs the excess gluons . this has been explicitly shown to occur by numerically solving the elastic kinetic equation derived under small angle approximation @xcite . there have been intensive discussions related to this picture from a variety of approaches , see e.g. @xcite . strong evidences for the formation of such a bose - einstein condensate have been reported for similar thermalization problem in the classical - statistical lattice simulation of scalar field theory @xcite . the case for non - abelian gauge theory is more complicated and still under investigation @xcite . one important question that has not been addressed in the above scenario is the role of inelastic processes . this issue could indeed be critical for at least two reasons ( see discussions in e.g. @xcite ) . first of all the inelastic processes will spoil the particle number conservation , and one might naively argue that the excessive gluons in the overpopulated glasma could simply be eliminated by very fast inelastic collisions . secondly , to make it even worse , the inelastic processes are parametrically at the same order as the elastic processes ( as opposed to naive power counting ) , so there appears no apparent dominance of the elastic over the inelastic and one may indeed worry that the inelastic could efficiently reduce total particle number . in such a situation , an explicit evaluation including both elastic and inelastic collisions becomes mandatory to clarify what will happen after including both types of collisions . to be precise , once the inelastic processes are included , one does not expect any condensation in the ultimate thermal equilibrium because with long enough time the inelastic processes will always remove any excessive particles . the interesting question , instead , is what changes the inelastic collisions bring to the dynamical evolution of the system . in particular , it is found @xcite that with purely elastic scatterings the overpopulated system is driven toward a dynamical onset of condensation in a finite time via critical scaling behavior in the infrared regime . it is extremely interesting to know , upon including the inelastic processes , how such dynamical evolution may be modified and whether the transient off - equilibrium condensation would still occur or not . in this paper , we aim to address this important question by studying the kinetic evolution of a highly overpopulated system starting from glasma - type initial condition with both @xmath15 and @xmath16 scatterings . in section 2 we will derive the inelastic collision kernel under the collinear and small angle approximations using the gunion - bertsch formula for the @xmath17 matrix element . in section 3 we will use numerical solutions as well as analytic analysis to understand the role of the inelastic process for both the global particle number change and the local behavior at small momentum region . finally we will conclude in section 4 . as a first step toward understanding the inelastic contributions and for simplicity and unambiguity , we will focus on the static box case with isotropic distribution in this work and leave the study of expanding case for future work .

we derive the inelastic collision kernel under the collinear and small angle approximations . with both numerics and analytic analysis physics department and center for particle physics and field theory , fudan university , shanghai 200433 , china . 3 . 510a , brookhaven national laboratory , upton , ny 11973 , usa .

in this paper we investigate the role of inelastic collisions in the kinetic evolution of a highly overpopulated gluon system starting from glasma - type initial condition . using the gunion - bertsch formula we derive the inelastic collision kernel under the collinear and small angle approximations . with both numerics and analytic analysis , we show that the inelastic process has two effects : globally changing ( mostly reducing ) the total particle number , while locally at small momentum regime always filling up the infrared modes extremely quickly . this latter effect is found to significantly speed up the emergence of a local thermal distribution in the infrared regime with vanishing local `` chemical potential '' and thus catalyze the onset of dynamical bose - einstein condensation to occur faster ( as compared with the purely elastic case ) in the overpopulated glasma . 1 . physics department and center for exploration of energy and matter , indiana university , 2401 n milo b. sampson lane , bloomington , in 47408 , usa . 2 . physics department and center for particle physics and field theory , fudan university , shanghai 200433 , china . 3 . riken bnl research center , bldg . 510a , brookhaven national laboratory , upton , ny 11973 , usa .

1303.7214

c

in summary , we have studied the kinetic evolution of a highly overpopulated system starting from glasma - type initial condition with the presence of both elastic and inelastic collisions . using the gunion - bertsch formula for the @xmath16 matrix element , we have derived the inelastic collision kernel under the collinear and small angle approximations . putting together the inelastic kernel together with the previously obtained elastic kernel , we have then numerically solved the kinetic evolution for varied choices of parameters . our main finding is that the inelastic process has two effects : globally changing ( mostly reducing ) the total particle number , while locally at small @xmath38 always filling up the infrared regime extremely quickly . the latter effect is shown both from numerics and by analytic analysis . this effect significantly speeds up the emergence of local thermal form near @xmath145 and the vanishing of local `` chemical potential '' @xmath141 as previously found in the purely elastic collision case to lead to the onset of dynamical bose condensation . therefore in our present approach of including the inelastic scattering , we conclude that , contrary to some previously discussed expectations about the role of number non - conserving processes , the inelastic collision actually helps to build up the local `` critical form '' @xmath182 much faster and catalyzes the onset of condensation in the overpopulated glasma . our finding may sound counter - intuitive at first , as the usual conception would suggest that increasing the strength of the inelastic collisions tends to obstruct more effectively the formation of any condensate . it should however be emphasized that the evolution toward onset of bec that has been studied thus far is not the end of the story . our analysis addresses the evolution up to the onset of bec while does not treat the evolution afterwards . as is well known in the bec literature ( see e.g. @xcite ) , in order to describe the kinetic evolution of the system with the presence of condensate , a new set of kinetic equations is needed for an explicit description of the coupled evolution for a condensate plus a regular distribution . efforts are underway to derive these equations , and so far a kinetic study of the stage after bec onset for the glasma system has not been achieved to our best knowledge . however , it appears very plausible that the subsequent evolutions may develop as follows : immediately after onset , the strong ir flux will not cease right away but continue for a while and thus drive the condensate to grow in time ; at certain point , the time would be long enough to allow the inelastic processes to decrease the total number density adequately and cause the condensate to decay thus decreasing in time ; eventually the inelastic processes will be able to remove all excess gluons and lead to the thermal equilibrium state with neither condensate nor any chemical potential . while the detailed understanding of such dynamic processes can only be achieved through solving the new set of kinetic equations , one can reasonably expect that with increasing strength of the inelastic processes the whole evolution would be faster . thus the following overall picture may likely be the case : with increasing strength , the inelastic processes on one hand catalyze the onset of condensation initially , while on the other hand eliminate the fully formed condensate faster , thus limiting the time duration for the presence of condensate to be shorter . a schematic picture of such conjectured full evolution is shown in fig . [ conje ] , which is in line with the usual conception . it is worth mentioning that recent analysis in @xcite has shown that the the @xmath17 inelastic cross section from exact matrix element becomes significantly smaller than that from the gunion - bertsch formula , and amounts to @xmath183 of the @xmath136 cross section . it therefore seems very plausible that a realistic choice of @xmath108 value would be rather modest , which may imply a considerable time window for the condensate to be sizable and play an important role for the evolution . a complete investigation of the evolution including the condensate will be future project to be reported elsewhere .

in this paper we investigate the role of inelastic collisions in the kinetic evolution of a highly overpopulated gluon system starting from glasma - type initial condition . using the gunion - bertsch formula , we show that the inelastic process has two effects : globally changing ( mostly reducing ) the total particle number , while locally at small momentum regime always filling up the infrared modes extremely quickly .

in this paper we investigate the role of inelastic collisions in the kinetic evolution of a highly overpopulated gluon system starting from glasma - type initial condition . using the gunion - bertsch formula we derive the inelastic collision kernel under the collinear and small angle approximations . with both numerics and analytic analysis , we show that the inelastic process has two effects : globally changing ( mostly reducing ) the total particle number , while locally at small momentum regime always filling up the infrared modes extremely quickly . this latter effect is found to significantly speed up the emergence of a local thermal distribution in the infrared regime with vanishing local `` chemical potential '' and thus catalyze the onset of dynamical bose - einstein condensation to occur faster ( as compared with the purely elastic case ) in the overpopulated glasma . 1 . physics department and center for exploration of energy and matter , indiana university , 2401 n milo b. sampson lane , bloomington , in 47408 , usa . 2 . physics department and center for particle physics and field theory , fudan university , shanghai 200433 , china . 3 . riken bnl research center , bldg . 510a , brookhaven national laboratory , upton , ny 11973 , usa .

1402.3694

i

one of the classical problems in discrete geometry , raised by p. erds in 1946 @xcite , is the following : given @xmath2 points in the plane , how many unit distances they may determine ? the key definition related to the question of p. erds is that of a _ unit distance graph_. a graph @xmath1 is a _ unit distance graph _ in @xmath0 if its set of vertices is a finite subset of @xmath0 and the edges are formed by the pairs of vertices which are at unit distance apart . in terms of distance graphs the question is to determine the maximal number of edges in a planar unit distance graph on @xmath2 vertices . in this paper we focus on the questions of this type for _ diameter graphs_. a graph @xmath11 is a _ diameter graph _ in @xmath0 , if @xmath12 is a finite set of diameter 1 , and edges of @xmath1 are formed by the pairs of vertices that are at unit distance apart . diameter graphs arise naturally in the context of the finite version of the famous borsuk s problem ( see , e.g. , @xcite for the survey on borsuk s problem ) , which is stated as follows : is it true that any ( finite ) set of unit diameter in @xmath0 can be partitioned into @xmath13 subsets of strictly smaller diameter ? the finite version is equivalent to the following question concerning diameter graphs : is it true that any diameter graph @xmath1 in @xmath0 satisfies @xmath14 ? a question about diameter graphs analogous to the question from the first paragraph has a simple answer : any set of @xmath2 points in the plane generates at most @xmath2 diameters , or any diameter graph on @xmath2 vertices in the plane has at most @xmath2 edges . this was proved by h. hopf and e. pannwitz in @xcite . interestingly , this result leads to a simple proof of the fact that borsuk s question for finite sets in the plane have a positive answer . indeed , it is easy to derive combinatorially that any graph @xmath1 on @xmath2 vertices with at most @xmath2 edges and such that any of its subgraphs has at least as many vertices as edges satisfies @xmath15 . a. vzsonyi conjectured that any diameter graph in @xmath16 on @xmath2 vertices has at most @xmath17 edges . again , it is easy to see that borsuk s conjecture for finite sets in @xmath16 follows from this statement . vzsonyi s conjecture was proved independently by b. grnbaum @xcite , a. heppes @xcite and s. straszewicz @xcite . an interesting generalization of this result to the case of @xmath18-th diameters was obtained by f. mori and j. pach @xcite . while the maximum number of edges in a diameter graph in @xmath19 is linear in the number of vertices , it becomes quadratic already in @xmath20 . to put the discussion in a more general context , we introduce the following notations . denote by @xmath21 ( @xmath22 ) the maximum number of cliques of size @xmath23 in a diameter ( unit distance ) graph on @xmath2 vertices in @xmath0 . erds @xcite studied @xmath24 and @xmath25 for different @xmath3 . he showed that for @xmath26 we have @xmath27 swanepoel @xcite determined @xmath24 for fixed even @xmath28 and sufficiently large @xmath2 depending on @xmath3 and determined @xmath25 for @xmath26 and sufficiently large @xmath2 . functions @xmath29 @xmath22 for @xmath30 and similar functions were studied in several papers . in particular , the following conjecture was raised in @xcite : [ gypsp ] we have @xmath31 for @xmath32 . this was proved by h. hopf and e. pannwitz for @xmath33 in @xcite and for @xmath34 by z. schur et . al . in @xcite . in the latter paper the authors also proved that @xmath35 . in @xcite p. mori and j. pach progressed towards resolving this conjecture . namely , they showed that schur s conjecture holds in the following special case : [ thph ] given a diameter graph @xmath1 on @xmath2 vertices in @xmath0 , the number of @xmath3-cliques in @xmath1 does not exceed @xmath2 , provided that any two @xmath3-cliques share at least @xmath7 vertices . as it turns out , schur s conjecture and related questions are tightly connected with analogous questions for spherical sets . the spherical analogues were studied in some papers . in particular , in the paper @xcite v. bulankina et al . noted that the statement of theorem [ thph ] holds for spheres of large radii : given a diameter graph @xmath1 on @xmath2 vertices in a @xmath3-dimensional sphere @xmath4 with radius @xmath5 , the number of @xmath3-cliques in @xmath1 does not exceed @xmath2 , provided that any two @xmath3-cliques share at least @xmath7 vertices ( theorem 4 from @xcite ) . moreover , they showed that schur s conjecture holds for @xmath36 for @xmath5 . to be precise , we formulate schur s conjecture for spheres separately : [ gypsph ] any diameter graph @xmath1 on @xmath2 vertices ( and with edges of unit euclidean length ) on a sphere @xmath37 with @xmath5 has at most @xmath2 @xmath3-cliques . in the paper @xcite a. kupavskii studied properties of diameter graphs in @xmath20 . the following theorem completes the description of the quantity @xmath38 for different @xmath23 : [ th10 ] 1 . for @xmath39 we have @xmath40 ( in corollary 3 from @xcite the same was proved for sufficiently large @xmath2 . ) 2 . for all sufficiently large @xmath2 we have @xmath41 3 . ( schur s conjecture in @xmath20 ) for all @xmath42 we have @xmath43 . in @xcite the first author also studied diameter graphs on @xmath44 with @xmath5 . in particular , he showed that an analogue of vzsonyi s conjecture holds for spheres . in the next section we present our main results and discuss related questions . in section [ sec3 ] we introduce the basic objects that are used in the proof . in section [ sec4 ] we present the proofs of the results .

in this paper we prove schur s conjecture in , which states that any diameter graph in on vertices may have at most cliques of size . we obtain an analogous statement for diameter graphs on a sphere of radius . * msc : * 52c10 . + 0.2 cm * keywords : * schur s conjecture , diameter graph , unit simplices in .

in this paper we prove schur s conjecture in , which states that any diameter graph in on vertices may have at most cliques of size . we obtain an analogous statement for diameter graphs on a sphere of radius . the proof rests on the following statement , conjectured by f. mori and j. pach : given two unit regular simplices on vertices in , either they share vertices , or there are vertices such that . the same holds for unit simplices on a-dimensional sphere of radius greater than . * msc : * 52c10 . + 0.2 cm * keywords : * schur s conjecture , diameter graph , unit simplices in .

0911.1267

i

we now have compelling evidence that the rate of expansion of the universe is accelerating , requiring the universe to be dominated by an unknown form of matter , known as ` dark energy , ' characterized by the equation of state @xmath2 . this dark energy could be a cosmological constant unnaturally tuned at the level of 1 part in @xmath3 @xcite , or it may be associated with the potential energy of a scalar field @xcite . if a scalar field drives the accelerated expansion then it must be extremely light , with mass of order @xmath4 . the existence of light scalar fields results in new , long range `` fifth forces , '' which are tightly constrained by experiment @xcite . to avoid these constraints the energy scale controlling the coupling of such scalar fields to matter must be many orders of magnitude larger than the planck scale @xcite . explaining why this coupling is so weak is a major problem for dynamical dark energy models . these problems can be circumvented by allowing the scalar field mass mass to be determined by a _ chameleon _ mechanism @xcite . this makes the field heavy in a dense environment , but light in vacuum . such scalar fields hide from experimental searches for fifth forces @xcite and modifications of gravity @xcite in a novel way : in the interior of a massive object the chameleon is very heavy , and has a correspondingly short interaction length . an observer outside the body feels a scalar force sourced only by a thin shell of matter at the surface of the object . this suppresses unwanted fifth forces @xcite . observations impose few constraints on the strength of the coupling of the scalar field to the standard model , allowing the energy scale of the coupling to be many orders of magnitude below the planck scale @xcite . chameleonic scalar fields have been shown to have a successful classical phenomenology . their equation of state depends on the energy density of the universe . in the early universe the chameleon behaves as an additional matter component , but at late times it has an equation of state with roughly @xmath5 . this enables it to drive an era of accelerated expansion @xcite . weak constraints on the initial conditions ensure that the chameleon does not disrupt the dynamics of the early universe @xcite . in this paper we discuss these theories in the framework of quantum mechanics . this is of interest because the cosmological constant problem is essentially quantum mechanical . there is no profit in replacing the cosmological constant by some other theory which is equally unnatural once quantum corrections are taken into account . the chameleon mechanism makes it mandatory for the dark energy scalar field to interact with conventional matter , and these interactions are potentially very strong . they may help us unravel the microphysics of dark energy if their indirect consequences could be measured @xcite . if the interactions are sufficiently strong , however , then an unambiguous _ direct _ consequence may be observable : new , light scalar quanta must be present in the beam pipe of any particle accelerator , raising the prospect of observing dark energy in the laboratory . fully consistent quantum mechanical realizations of the chameleon have not yet been constructed , and may suffer from the same naturalness and tuning difficulties as quintessence @xcite . in this paper we do not attempt to address naturalness concerns , or the quantum - mechanical construction of the model . instead , supposing such models to exist , we determine constraints which are imposed by experiment . real or virtual chameleon - like particles will certainly be produced in particle collisions if dark energy couples to the electroweak gauge bosons . in this paper , we will provide a microscopic derivation of this coupling for any chameleon - like model . the models which furnish chameleon - like scalar particles at low energy are effective theories , valid below some energy scale @xmath6 . in ref . @xcite , corrections to electroweak precision observables from particles with chameleonic or axionic couplings were analysed , without making a commitment to any specific choice of physics at energies above @xmath6 . it was shown that very weak constraints on the energy scale of the chameleon coupling to matter ensure that the chameleon does not have an observable effect on measurements of the @xmath0-width . for processes involving fermions and @xmath7 gauge bosons , it was shown that large effects could always be absorbed into renormalizations of the fermi constant , @xmath8 , and the gauge boson masses . when higgs processes are included it is no longer clear that large effects can be hidden in this way , potentially allowing higgs production to function as a diagnostic of dark energy physics and its coupling to the standard model . in this paper , we focus on the consequences of such dark energy quanta for higgs production . this is a key target for both the _ large hadron collider _ ( lhc ) and the _ tevatron_. the details of higgs production depend on which couplings occur in the theory . to achieve a successful chameleon phenomenology we assume conformal couplings to matter . this has implications for interactions with gauge boson kinetic terms , which will be described in more detail in [ sec : scalar - corrections ] below . the standard model gauge bosons have conformally invariant kinetic terms and develop no couplings to a scalar field of this type . in this paper we show that violations of conformal symmetry , arising from couplings to matter species , can be communicated to the kinetic term by quantum corrections . therefore , a coupling _ is _ generated at energies below @xmath6 via loops of charged heavy particles . this coupling has important phenomenological consequences which will be briefly recalled . in particular we will show that this coupling induces a coupling to the electromagnetic field and in so doing provides a motivation for the effects probed in quantum laser experiments @xcite . our results apply for any light scalars with the requisite chameleon - like couplings , whatever their origin . what scale , @xmath6 , should be associated with these new degrees of freedom ? a conservative choice would be the gut scale , @xmath9 gev , which may be connected with an early inflationary stage , or a seesaw explanation of neutrino mass @xcite . if so , heavy neutrinos of mass @xmath6 might exist but would be sterile , having no @xmath7 quantum numbers . this would not lead to the required coupling . alternatively , if supersymmetry is realized in nature then @xmath6 could be associated with the scale at which it is spontaneously broken , perhaps of order @xmath10@xmath11 tev . in any supersymmetric completion of the standard model there exist fermionic partners of @xmath12 , @xmath1 , @xmath0 and the higgs , known as gauginos and higgsinos . ( for a review see @xcite and references therein . ) the mass eigenstates of these particles ( `` charginos '' ) would naturally be of order @xmath6 . in this article we remain agnostic about the nature of whatever particles circulate within loops . we give our calculation in a form which can be specialized immediately to the minimal supersymmetric standard model ( mssm ) , but our conclusions are general and do not depend on the details of a specific implementation . in any case , the calculation we describe can be adapted easily to any heavy particle carrying the requisite quantum numbers . the outline of this paper is as follows . in [ sec : axionic ] we introduce our model and show that a coupling between gauge bosons and dark energy is generated at low energy by integrating out heavy particles . some of the phenomenology associated with this coupling is presented . in [ sec : higgs ] we briefly review higgs production at a hadron collider , emphasizing the role of vertices between the electroweak gauge bosons and the higgs . in [ sec : scalar - corrections ] we compute corrections to the higgs production rate arising from the low energy theory written down in [ sec : axionic ] . we show that the effective lagrangian includes new contact interactions between the gauge bosons and the higgs . at scales smaller than @xmath13 these contact interactions resolve into heavy particle loops . we compute the correction to higgs production from both effects . in [ sec : conclusions ] we conclude by arguing that the effect of chameleon - like particles on higgs production is expected to be rather small . throughout this paper , we adopt units in which @xmath14 . we set the reduced planck mass , @xmath15 , to unity . our metric convention is @xmath16 .

we show that a chameleon model with only matter couplings will induce a coupling to photons . in doing so , we derive the first microphysical realization of a chameleonic dark energy model coupled to the electromagnetic field strength . we also study the effect of chameleon fields on higgs production , which is relevant for hadron colliders . we show that , like the tevatron , the lhc will not be able to rule out or observe chameleons through this mechanism , because gauge invariance of the low energy lagrangian suppresses the corrections that may arise . diags

in this paper we study various particle physics effects of a light , scalar dark energy field with chameleon - like couplings to matter . we show that a chameleon model with only matter couplings will induce a coupling to photons . in doing so , we derive the first microphysical realization of a chameleonic dark energy model coupled to the electromagnetic field strength . this analysis provides additional motivation for current and near - future tests of axion - like and chameleon particles . we find a new bound on the coupling strength of chameleons in uniformly coupled models . we also study the effect of chameleon fields on higgs production , which is relevant for hadron colliders . these are expected to manufacture higgs particles through weak boson fusion , or associated production with a or . we show that , like the tevatron , the lhc will not be able to rule out or observe chameleons through this mechanism , because gauge invariance of the low energy lagrangian suppresses the corrections that may arise . diags

hep-ph9407262

i

searching for possible exotic states of nuclear matter is one of the currently most interesting and stimulating subjects in high energy nuclear physics ( see references [ 110 ] ) to create exotic matter such as the quark - gluon plasma ( qgp ) [ 14,5]strange matter @xcite,@xcite , multi@xmath11-matter , pion condensate , kaon condensate @xcite,@xcite it is essential to generate extreme states of nuclear matter regarding density , temperature or energy density @xcite . such extreme states are considered to be produced most probably in high energy heavy ion reactions[1014 ] or in the annihilation of energetic antiprotons(@xmath12)on a nucleus @xcite the possibility suggested by witten @xcite that strange quark - matter might be the lowest energy state of large baryon number , would have many consequences in physics and astrophysics and also cosmology relevance [ 1620 ] . small drops of strange matter(strangelets ) may be natural results of the evaporation of a qgp formed in heavy ion collisions . enhanced strangeness production @xcite in ultrarelativistic heavy ion collisions ( urhic ) is one of the most widely discussed signature for creating qgp @xcite,@xcite . several experiments have been performed to investigate strangeness production in reactions involving relativistic nuclei , particularly at super proton synchrotron ( sps ) at cern : na35 [ 2232 ] , na36 [ 3337],wa85,wa94[3845],[128130],na44 [ 132133 ] , hellios collab.[4650 ] , and at the alternating gradient synchrotron(ags ) at brook- + haven e802 [ 5155 ] , e810 @xcite,@xcite and e814 @xcite,[134 ] . these experiments have measured also the yield of several particle species containing one or two strange quarks ( @xmath13 , @xmath14,@xmath15,@xmath16 and their antiparticles ) or three strange quarks(@xmath17 ) . in all the cases their production relative to pions or negative hadrons and some other ratios are larger in nucleus - nucleus collisions compared to proton - proton . now experiments with truly heavy ion projectile have just started(with a gold beam at bnl)@xcite,@xcite or are under way ( @xmath18 at cern)@xcite . the soft physics data have stimulated a great deal of theoretical activity and in the present introduction it is impossible to give more than a superficial overview of the main types of the monte carlo models used.many models for urhic have been developed : dual partons models(dpm ) [ 5966],@xcite quark gluon string models(qgsm)[6769 ] , venus models for very energetic nuclear scattering [ 7075 ] fritiof model @xcite,@xcite , attila @xcite model , relativistic quantum molecular dynamics ( rqmd ) [ 7983][135138],parton string model(psm)[139 ] , hijet model@xcite . an excelent review and detailed comparision of the models was done by werner @xcite . on the other hand , no unconventional explanation has so far put forward for the observed increase of antyhyperons ( @xmath19,@xmath16 ) . actually , it is the old pomeron exchange picture that is revitalized in some models with a new ingredient that is the pomeron is made of two colored strings.thus the problem of interaction of strings becomes important.this interaction may give rice to the fusion of strings[8588][137,138 ] making a new string with more color charge.these new objects could explain the strangeness enhancement @xcite , @xcite,[137,138]especially @xmath19 enhancement . such models can be seen as an interplay between independent string fragmentation and qgp formation.one version of venus generator also include such interaction @xcite , @xcite . we note also that an interpretation based on a particular dependence of the cross sections for urhic was taken into account @xcite . in this report we have performed a systematic study of strange particle production in proton proton(@xmath20),proton nucleus(@xmath21 ) and nucleus nucleus ( @xmath22 ) interactions at sps cern energies using hijing(heavy ion jet interacting generator ) monte carlo model [ 9197 ] developed at berkeley . the formulation of hijing is guided by the lund fritiof ( @xcite@xcite ) and dpm @xcite models phenomenology for soft nucleus - nucleus reactions at intermediate energies @xmath23 and the successfull implementation of perturbative qcd ( pqcd ) processes in the pythia model @xcite@xcite for hadronic interactions . hijing incorporate the pqcd to multiple jet processes and the nuclear effects such as parton shadowing and jet quenching . hijing provide a link between the physics at intermediate cern sps energies and the highest collider energies at relativistic heavy ion collider(rhic)(@xmath24 and the cern large hadron collider ( lhc ) ( @xmath25 ) . using hijing monte carlo model @xcite,@xcite a detailed discussion and comparison with a wide variety of data in @xmath20,@xmath21 and @xmath22 collisions was reported.however no study on the strangeness production at sps - cern energies in @xmath20,@xmath21 and @xmath22 data exist within this approach . in this paper we present some theoretical calculations for mean multiplicities of strange particle production in the full rapidity range as well as their ratios to negative hadron predicted by hijing model(hij.01 version -see section 2 and 3 for explanation of this version)for @xmath20 , nucleon - nucleon @xmath26,minimum biascollisions @xmath27 and + central collisions@xmath28 at the energy of @xmath29 . also a qualitative prediction for normalised distributions for rapidity , transverse kinetic energy and transverse momenta are done at 200 gev / nucleon in @xmath30 and sau collisions and the results are compared with recent experimental data from na 35 collaboration[140].we note that theoretical calculations are done for full phase space . for giving an idea of abundances and spectra in experiments planned in the near future at cern , we give some predictions for sw and pbpb interactions . we note that the version hij.01 of the model includes neither the formation of qgp nor some collective dynamics on either the hadronic or string level.although the existing trends of hadron distributions are quite satisfactorily approximated by the hijing model @xcite,@xciteand also neutral strange production in @xmath20 and @xmath21 interactions are quite well described , this version is getting less reliable for strange particle produced in heavy colliding systems . the same result is stressed out and in other models qgsm@xcite,@xcite , rqmd@xcite , fritiof@xcite , psm@xcite . a brief description of the hijing monte carlo model and theoretical background are given in section 2.detailed numerical results of this version hij.01 for @xmath31 and @xmath32 reactions especially at cern - sps energies@xmath33 for strangeness production are compared to experimental data and other models predictions like qgsm @xcite,@xcite , dpm @xcite,[141],142],venus @xcite , fritiof @xcite , and rqmd @xcite,[140 ] in section 3 . section 4 concludes with a summary and discussion of future applications .

numerical results for mean multiplicities of neutral strange particles , as well as their ratios to negatives hadrons( ) for p - p , nucleon - nucleon(n - n),p - s , p - ag , p - au(min . bias)collisions and p - au , s - s , s - ag , s - au ( central)collisions are compared to experimental data available from cern experiments and also with recent theoretical estimations given by others models( + and ) . hijing model predictions for coming experiments at cern for s - au , s - w and pb - pb interactions are given . 1truecm = -1truecm = -2truecm

in this report we have made a systematic study of strangeness production in proton - proton(pp),proton - nucleus(pa ) and nucleus- nucleus(aa ) collisions at cern super proton synchroton energies , using + ( version ) . numerical results for mean multiplicities of neutral strange particles , as well as their ratios to negatives hadrons( ) for p - p , nucleon - nucleon(n - n),p - s , p - ag , p - au(min . bias)collisions and p - au , s - s , s - ag , s - au ( central)collisions are compared to experimental data available from cern experiments and also with recent theoretical estimations given by others models( + and ) . neutral strange particle abundances are quite well described for p - p , n - n and p - a interactions , but are underpredicted by a factor of two in a - a interactions for in symmetric collisions(s - s , pb - pb)and for in asymmetric ones(s - ag , s - au , s - w ) . the ratios antistrange / strange are well described for but the ratios for multistrange particles ( ) and their antiparticle , could not be predicted by the model.a detailed analysis in limited rapidity range is required in order to draw some definite conclusion . a qualitative prediction for rapidity , transverse kinetic energy and transverse momenta normalized distributions are performed at 200 gev / nucleon in p - s , s - s , s - ag and s - au collisions in comparison with recent experimental data . hijing model predictions for coming experiments at cern for s - au , s - w and pb - pb interactions are given . the theoretical calculations are estimated in a full phase space . the factor of two can not be explained by considering very central events ( ) and taking into account minijets . new ideas are necessary in order to improve the comparison with experiment for nucleus - nucleus interactions . = 16truecm = 24truecm 1 = = .5truecm = -2truecm 1truecm * dipartamento di fisica `` g.galilei '' + * * istituto di fisica nucleare + * * sezione di padova + * italy * * 2truecm * strangeness production in pp , pa , aa * + 0.3 cm * interactions * + 0.3 cm * at sps energies . hijing approach * + 1truecm * wa94 collaboration * + dipartimento di fisica``g.galilei'',infn sezione di padova , via marzolo 8 , 35131 padova,*italy + * , * romania + * and + dipartamento di fisica `` g.galilei'',infn sezione di padova , via marzolo 8 , 35131 padova,*italy + * * dfpd-94-np-42 8 june 1994 * _ work presented at the international conference `` physics of high energy heavy ion collisions '' , vuosaari(helsinki),finland,17 - 22 june 1994 _ 1truecm = -1truecm = -2truecm

hep-ph9407262

c

in this report , we give a systematic study of strange particle production in pp , pa and aa collisions mainly at sps cern - energies , using hijing monte carlo model developed for high energy collisions . we analyse data from cern experiments ( na35,na36 , + wa85 ) and we give a detailed comparison for mean multiplicities of neutral strange particle for pp , ps , pag , pw , pau , ss , sw and pbpb interactions . one should note that hijing model includes neither hipotheses of qgp . strange particle abundances are quite well described in pp and pa collisions and are underpredicted by a factor of two in nucleus - nucleus interactions , by this version ( theoretical values for hij.01 ) .the ratios for multistrange particles could not be predicted by the model hij.01 .the abundances of @xmath16 and @xmath17 are underpredicted and the abundances of @xmath179 and @xmath180 are overpredicted . more carreful analisis are necessary in limited rapidity interval . we present also some calculations for the ratios @xmath204 and @xmath205 given by hijing model compared with other models predictions like venus and fritiof.all models seems to overpredict @xmath205 ratios and underpredict @xmath204 ratios . rescattering should be important even at sps -energies . a qualitative prediction for rapidity distributions of neutral strange particles in @xmath206 , @xmath207,@xmath208,@xmath209 @xmath210per nucleon are done compared also with recent experimental data[140 ] . the models describes correctly the shapes of the rapidity in central @xmath211 collisions for @xmath212 and @xmath188 but absolute numbers for multiplicities are underestimated by a factor of two for @xmath213 in symmetric collisions and for @xmath214 in asymmetric ones . some discrepancies are seen for @xmath15 rapidity distribution and transverse momentum distributions even for @xmath190 interactions.the factor of two can not be explained taken into account minijets and very central events(@xmath215 ) . the theoretical model predictions hij.01 presented above for central @xmath201,@xmath184 and @xmath202 collisions give an idea for particle abundances in experiments which are under the way or are planned in the near future at cern . the hijing model simulations are however still somewhat too low compared to the data.also the ratios for multistrange baryons could not be described . new ideas are necessary in order to improve the comparison of theoretical calculations with experimental data.the main goal is to increase the rate of neutral strange particle production by a factor of two in nucleus - nucleus interaction at @xmath216 . we remark that other models ( qgsm , venus , dpm and psm ) include now some aditional hypothesis in order to improve the discrepancies ( colective string - string interactions , double or multiple colour exchanges , string fusion ) .

in this report we have made a systematic study of strangeness production in proton - proton(pp),proton - nucleus(pa ) and nucleus- nucleus(aa ) collisions at cern super proton synchroton energies , using + ( version ) . the factor of two can not be explained by considering very central events ( ) and taking into account minijets . new ideas are necessary in order to improve the comparison with experiment for nucleus - nucleus interactions . = 16truecm = 24truecm 1 = = .5truecm = -2truecm 1truecm * dipartamento di fisica `` g.galilei '' + * * istituto di fisica nucleare + * * sezione di padova + * italy * * 2truecm * strangeness production in pp , pa , aa

in this report we have made a systematic study of strangeness production in proton - proton(pp),proton - nucleus(pa ) and nucleus- nucleus(aa ) collisions at cern super proton synchroton energies , using + ( version ) . numerical results for mean multiplicities of neutral strange particles , as well as their ratios to negatives hadrons( ) for p - p , nucleon - nucleon(n - n),p - s , p - ag , p - au(min . bias)collisions and p - au , s - s , s - ag , s - au ( central)collisions are compared to experimental data available from cern experiments and also with recent theoretical estimations given by others models( + and ) . neutral strange particle abundances are quite well described for p - p , n - n and p - a interactions , but are underpredicted by a factor of two in a - a interactions for in symmetric collisions(s - s , pb - pb)and for in asymmetric ones(s - ag , s - au , s - w ) . the ratios antistrange / strange are well described for but the ratios for multistrange particles ( ) and their antiparticle , could not be predicted by the model.a detailed analysis in limited rapidity range is required in order to draw some definite conclusion . a qualitative prediction for rapidity , transverse kinetic energy and transverse momenta normalized distributions are performed at 200 gev / nucleon in p - s , s - s , s - ag and s - au collisions in comparison with recent experimental data . hijing model predictions for coming experiments at cern for s - au , s - w and pb - pb interactions are given . the theoretical calculations are estimated in a full phase space . the factor of two can not be explained by considering very central events ( ) and taking into account minijets . new ideas are necessary in order to improve the comparison with experiment for nucleus - nucleus interactions . = 16truecm = 24truecm 1 = = .5truecm = -2truecm 1truecm * dipartamento di fisica `` g.galilei '' + * * istituto di fisica nucleare + * * sezione di padova + * italy * * 2truecm * strangeness production in pp , pa , aa * + 0.3 cm * interactions * + 0.3 cm * at sps energies . hijing approach * + 1truecm * wa94 collaboration * + dipartimento di fisica``g.galilei'',infn sezione di padova , via marzolo 8 , 35131 padova,*italy + * , * romania + * and + dipartamento di fisica `` g.galilei'',infn sezione di padova , via marzolo 8 , 35131 padova,*italy + * * dfpd-94-np-42 8 june 1994 * _ work presented at the international conference `` physics of high energy heavy ion collisions '' , vuosaari(helsinki),finland,17 - 22 june 1994 _ 1truecm = -1truecm = -2truecm

1307.4044

i

meson decays provide an important tool for exploring the structures of these simplest bound states in qcd , and for studies on non - perturbative behavior of strong interactions . these studies has become a hot topic in recent years . flavourless vector mesons play an important role in hadron physics due to their direct coupling to photons and thus provide an invaluable insight into the phenomenology of electromagnetic couplings to hadrons . thus , a realistic description of vector mesons at the quark level of compositeness would be an important element in our understanding of hadron dynamics and reaction processes at scales where qcd degrees of freedom are relevant . there have been a number of studies@xcite on processes involving strong , radiative and leptonic decays of vector mesons . such studies offer a direct probe of hadron structure and help in revealing some aspects of the underlying quark - gluon dynamics . in this work we study electromagnetic decays of ground state equal mass vector mesons : @xmath10 and @xmath4 ( each comprising of equal mass quarks ) through the process @xmath11 which proceeds through the coupling of quark - anti quark loop to the electromagnetic current in the framework of bethe - salpeter equation ( bse ) , which is a conventional non - perturbative approach in dealing with relativistic bound state problems in qcd and is firmly established in the framework of field theory . from the solutions , we obtain useful information about the inner structure of hadrons which is also crucial in high energy hadronic scattering and production processes . despite its drawback of having to input model dependent kernel , these studies have become an interesting topic in recent years since calculations have satisfactory results as more and more data is being accumulated . we get useful insight about the treatment of various processes using bse due to the unambiguous definition of the 4d bs wave function which provides exact effective coupling vertex ( hadron - quark vertex ) of the hadron with all its constituents ( quarks ) . we have employed qcd motivated bethe - salpeter equation ( bse ) under covariant instantaneous ansatz ( cia)@xcite to calculate this process . cia is a lorentz - invariant generalization of instantaneous ansatz ( ia ) . what distinguishes cia from other 3d reductions of bse is its capacity for a two - way interconnection : an exact 3d bse reduction for a @xmath12 system ( for calculation of mass spectrum ) , and an equally exact reconstruction of original 4d bse ( for calculation of transition amplitudes as 4d quark loop integrals ) . in these studies , the main ingredient is the 4d hadron - quark vertex function @xmath6 which plays the role of an exact effective coupling vertex of the hadron with all its constituents ( quarks ) . the complete 4d bs wave function @xmath13 for a hadron of momentum @xmath14 and internal momentum @xmath15 comprises of the two quark propagators ( corresponding to two constituent quarks ) bounding the hadron - quark vertex @xmath6 . this 4d bs wave function is considered to sum up all the non - perturbative qcd effects in the hadron . now one of the main ingredients in 4d bs wave function ( bsw ) is its dirac structure . the copius dirac structure of bsw was already studied by c.h.l . smith@xcite much earlier . recent studies@xcite have revealed that various mesons have many different dirac structures in their bs wave functions , whose inclusion is necessary to obtain quantitatively accurate observables . it was further noticed that all structures do not contribute equally for calculation of various meson observables@xcite . further , it was amply noted in @xcite that many hadronic processes are particularly sensitive to higher order dirac structures in bs amplitudes . it was further noted in @xcite that inclusion of higher order dirac structures is also important to obtain simultaneous agreement with experimental decay widths for a range of processes such as : @xmath16 , @xmath17 , @xmath18 , etc . for a given choice of parameters . towards this end , to ensure a systematic procedure of incorporating various dirac covariants from their complete set in the bsws of various hadrons ( pseudoscalar , vector etc . ) , we developed a naive power counting rule in ref.@xcite , by which we incorporate various dirac structures in bsw , order - by - order in powers of inverse of meson mass . using this power counting rule we calculated electromagnetic decay constants of vector mesons ( @xmath19 ) using only the leading order ( lo ) dirac structures [ @xmath20 and @xmath21 . however in ref.@xcite , we rigorously studied leptonic decays of unequal mass pseudoscalar mesons @xmath22 and calculated the leptonic decay constants @xmath23 for these mesons employing both the leading order ( lo ) and the next - to - leading order ( nlo ) dirac structures . the contributions of both lo and nlo dirac structures to @xmath23 was worked out . we further studied the relevance of both the lo and the nlo dirac structures to this calculation . in the present paper , we extended these studies to vector mesons and have employed both lo and nlo dirac structures identified according to our power counting rule , to calculate @xmath24 for ground state vector mesons , @xmath25 and in the process we studied the relevance of various dirac structures to calculation of decay constants @xmath24 for vector mesons in the process @xmath16 . we found that contributions from nlo dirac structures are smaller than those of lo dirac structures for all vector mesons . in what follows , we give a detailed discussion of the fit and calculation up to nlo level after a brief review of our framework . the paper is organized as follows : in section 2 we discuss the structure of bsw for vector mesons under cia using the power counting rule we proposed earlier . in section 3 we give the calculation of @xmath24 for vector mesons . a detailed presentation of results and the numerical calculation is given in section 4 . section 5 is relegated to discussion .

in this work we have employed bethe - salpeter equation ( bse ) under covariant instantaneous ansatz ( cia ) to study electromagnetic decays of ground state equal mass vector mesons : , , , and through the process . the electromagnetic decay constants for the above mesons are calculated using the leading order ( lo ) and the next - to - leading order ( nlo ) dirac structures . the relevance of various dirac structures in this calculation is studied . corresponding author .

in this work we have employed bethe - salpeter equation ( bse ) under covariant instantaneous ansatz ( cia ) to study electromagnetic decays of ground state equal mass vector mesons : , , , and through the process . we employ the generalized structure of hadron - quark vertex function which incorporates various dirac structures from their complete set order - by - order in powers of inverse of meson mass . the electromagnetic decay constants for the above mesons are calculated using the leading order ( lo ) and the next - to - leading order ( nlo ) dirac structures . the relevance of various dirac structures in this calculation is studied . corresponding author . shashankbhatnagar@yahoo.com + mahecha@fisica.udea.edu.co

1505.03372

i

approximate bayesian computation ( abc ) now plays an important role in performing ( approximate ) bayesian inference for the parameter of a proposed statistical model ( called the generative model here ) that has an intractable likelihood . despite the intense attention abc has recently received , the approach still suffers from several drawbacks . an obvious disadvantage is the usual necessity to reduce the data to a low dimensional summary statistic . this leads to a loss of information that is difficult to quantify . the second , often less severe but sometimes related , drawback is the computational challenge of achieving stringent matching between the observed and simulated summary statistics . in situations where an alternative parametric model ( referred to as an auxiliary model ) can be formulated that has a tractable likelihood , the methodology known as indirect inference ( ii ) ( see , e.g. , gourieroux , monfort and renault , @xcite and @xcite ) is applicable . ii has been thoroughly examined in the classical framework . most methods differ in the way that observed and simulated data are compared via the auxiliary model . we expand on this later in the article . for the moment , we note that some key references are @xcite , @xcite and @xcite . however , ii has been far less studied in the bayesian paradigm . @xcite developed an abc approach that uses ii to obtain summary statistics . in particular , the estimated parameter of the auxiliary model fitted to the data becomes the observed summary statistic . we adopt a similar naming convention to @xcite and refer to this method as abc ip where `` i '' stands for `` indirect '' and `` p '' stands for `` parameter '' . @xcite also list another method , abc il ( where `` l '' stands for `` likelihood '' ) , which is essentially an abc version of @xcite . this approach follows abc ip in the sense that the parameter estimate of the auxiliary model is again the summary statistic . however , the abc discrepancy is based on the auxiliary likelihood , rather than a direct comparison of the auxiliary parameters . @xcite advocate a slightly different approach to abc with ii , which is effectively an abc version of the classical approach in @xcite . here , @xcite use the score vector based on the auxiliary model as the summary statistic , which is referred to as abc is ( here `` s '' stands for `` score '' ) . the parameter value used in the score is given by the mle of the auxiliary model fitted to the observed data . this approach can be far cheaper from a computational point of view since it avoids an expensive fitting of the auxiliary model to each dataset simulated from the generative model required in abc ip and abc il . throughout the paper , the collection of approaches that use the parametric auxiliary model to form summary statistics is referred to as abc ii methods . an advantage of this approach over more traditional summary statistics is that some insight can be gained on the potential utility of the ii summary statistic prior to the abc analysis . additionally , if the auxiliary model is parsimonious , then the summary statistic can be low - dimensional . @xcite ( see also @xcite ) suggest an alternative approach for combining ii with bayesian inference . this method has similar steps to abc ip and abc il but essentially uses the likelihood of the auxiliary model as a replacement to the intractable generative model likelihood . we note here that this is a fundamentally different approach as it is not a standard abc method . in particular , there is no comparison of summary statistics and no need to choose an abc tolerance . here , we refer to this method as parametric bayesian indirect likelihood ( pbil ) . the focus of this paper is the application of a parametric auxiliary model for the full data , which we refer to as pdbil ( where `` d '' stands for `` data '' ) . however , the ideas in this paper are equally applicable if a parametric model is applied at the summary statistic ( not necessarily obtained using abc ii techniques ) level ( i.e. , some data reduction technique has been applied ; see @xcite , for a review ) . this is referred to as psbil ( where `` s '' stands for `` summary statistic '' ) . we show that the bayesian version of the synthetic likelihood method of @xcite is a psbil method . in the paper , we refer to the collection of abc ii and pbil approaches as pbii methods ( `` bayesian '' `` indirect '' `` inference '' using a `` parametric '' auxiliary model ) . in the process of reviewing these pbii methods , we create a novel framework called bayesian indirect likelihood ( bil ) which encompasses pbii as well as abc methods generally . in particular , if a specific nonparametric auxiliary model is selected ( npbil ) instead of a parametric one ( pbil ) , then the general abc method is recovered . a nonparametric kernel can be applied either at the full data ( npdbil ) or summary statistic ( npsbil ) level . the abc ii approaches are thus a special type of npsbil method where the summary statistic is formed on the basis of a parametric auxiliary model . this framework is shown in figure [ fig : bil ] , which also highlights the methods that this paper addresses . this article does not develop any new algorithms for pbii . however , this paper does make several interesting and useful contributions . firstly , we explore the pdbil method in more detail theoretically , and recognise that it is fundamentally different to abc ii . the behaviour of this method is also substantially different . a technique sometimes applied with classical ii methods is to increase the simulated dataset size beyond that of the observed data in order to reduce the variability of the estimated quantities under the auxiliary model ( see , e.g. , @xcite ; @xcite ; @xcite ) . we demonstrate that pdbil and abc ii behave differently for increasing size of the simulated data . our second contribution is to compare the assumptions required for each pbii approach . our theoretical and empirical results that the pbil method will provide good approximations if the auxiliary model is sufficiently flexible to estimate the likelihood of the true model well based on parameter values within regions of posterior probability . abc ii methods rely on the parameter estimate or the score of the auxiliary model to provide a near - sufficient summary statistic . finally , our creation of the general bil framework provides a clear way to see the connections between pbii and other methods . the paper is organised as follows . in section [ sec : notation ] , the notation used throughout the paper is defined . the abc ii methods are reviewed in section [ sec : abc ii ] . the pbil approach is presented in section [ sec : bil ] . the theoretical developments in this section , which offer additional into the pbil approximation , are new . in addition , this section demonstrates how the synthetic likelihood approach of @xcite is a pbil method on the summary statistic level . section [ sec : abcasbil ] shows how abc can be recovered as a bil method via a nonparametric choice of the auxiliary model . section [ sec : comparebii ] provides a comparison between abc ii and pdbil . the contributions of this article are demonstrated on examples with varying complexity in section [ sec : results ] . the highlights of this section include improved approximate inferences for quantile distributions and a multivariate markov jump process explaining the evolution of parasites within a host . the article concludes with a discussion in section [ sec : discussion ] .

such an approach has been well explored in the classical literature but has received substantially less attention in the bayesian paradigm . the purpose of this paper is to compare and contrast a collection of what we call parametric bayesian indirect inference ( pbii ) methods . one class of pbii methods uses approximate bayesian computation ( referred to here as abc ii ) where the summary statistic is formed on the basis of the auxiliary model , using ideas from ii . another approach proposed in the literature , referred to here as parametric bayesian indirect likelihood ( pbil ) , uses the auxiliary likelihood as a replacement to the intractable likelihood . we show that pbil is a fundamentally different approach to abc ii . we devise new theoretical results for pbil to give extra insights into its behaviour and also its differences with abc ii . furthermore , we examine in more detail the assumptions required to use each pbii method . the results , insights and comparisons developed in this paper are illustrated on simple examples and two other substantive applications . the first of the substantive examples involves performing inference for complex quantile distributions based on simulated data while the second is for estimating the parameters of a trivariate stochastic process describing the evolution of macroparasites within a host based on real data . we create a novel framework called bayesian indirect likelihood ( bil ) that encompasses pbii as well as general abc methods so that the connections between the methods can be established . ,

indirect inference ( ii ) is a methodology for estimating the parameters of an intractable ( generative ) model on the basis of an alternative parametric ( auxiliary ) model that is both analytically and computationally easier to deal with . such an approach has been well explored in the classical literature but has received substantially less attention in the bayesian paradigm . the purpose of this paper is to compare and contrast a collection of what we call parametric bayesian indirect inference ( pbii ) methods . one class of pbii methods uses approximate bayesian computation ( referred to here as abc ii ) where the summary statistic is formed on the basis of the auxiliary model , using ideas from ii . another approach proposed in the literature , referred to here as parametric bayesian indirect likelihood ( pbil ) , uses the auxiliary likelihood as a replacement to the intractable likelihood . we show that pbil is a fundamentally different approach to abc ii . we devise new theoretical results for pbil to give extra insights into its behaviour and also its differences with abc ii . furthermore , we examine in more detail the assumptions required to use each pbii method . the results , insights and comparisons developed in this paper are illustrated on simple examples and two other substantive applications . the first of the substantive examples involves performing inference for complex quantile distributions based on simulated data while the second is for estimating the parameters of a trivariate stochastic process describing the evolution of macroparasites within a host based on real data . we create a novel framework called bayesian indirect likelihood ( bil ) that encompasses pbii as well as general abc methods so that the connections between the methods can be established . ,

1403.0012

i

recently , the poisson point process ( ppp ) has been shown to be a tractable and realistic model of cellular networks @xcite . however , the baseline ppp model predicts the coverage probability of the typical user to be less than 60% if the signal - to - interference - plus - noise ratio ( sinr ) is set to 0 db even if noise is neglected . this is clearly insufficient to provide reasonable user experiences in the network . to improve the user experiences , in cellular systems , the importance of inter - cell interference coordination ( icic ) and intra - cell diversity ( icd ) have long been recognized@xcite . yet , so far , most of the ppp - based cellular analyses lack a careful treatment of these two important aspects of the network , partly due to the lack of a well - established approach to deal with the resulting temporal or spectral correlation @xcite . modeling the cellular network as a homogeneous ppp , this paper explicitly takes into account the temporal / spectral correlation and analyzes the benefits of icic and icd in cellular downlink under idealized assumptions . consider the case where a user is always served by the bs that provides the strongest signal averaged over small - scale fading but not shadowing . for icd , we consider the case where the serving bs always transmit to the user in @xmath0 resource blocks ( rbs ) simultaneously and the user always decodes from the rb with the best sir ( selection combining ) . for icic , we assume under @xmath1-bs coordination , the rbs that the user is assigned are silenced at the next @xmath2 strongest bss . note that both of the schemes create extra load ( reserved rbs ) in the network : icic at the adjacent cells and icd at the serving cell . therefore , it is important to quantify the benefits of icic and icd in order to design efficient systems . the main contribution of this paper is to provide explicit expressions for the coverage probability with @xmath1-bs coordination and @xmath0-rb selection combining . notably , we show that , in the high - reliability regime , where the outage probability goes to zero , the coverage gains due to icic and icd are qualitatively different : icd provides _ order _ gain while icic only offers _ linear _ gain . in contrast , in the high - spectral efficiency regime , where the sir threshold goes to infinity , such order difference does not exist and icic usually offers larger ( linear ) gain than icd in terms of coverage probability . the techniques presented in this paper have the potential to lead to a better understanding of the performance of more complex cooperation schemes in wireless networks , which inevitably involve temporal or spectral correlation . generally speaking , inter - cell interference coordination ( icic ) assigns different time / frequency / spatial dimensions to users from different cells and thus reduces the inter - cell interference . conventional icic schemes are mostly based on the idea of frequency reuse . the resource allocation under cell - centric icic is designed offline and does not depend on the user deployment . while such schemes are advantageous due to their simplicity and small signaling overhead , they are clearly suboptimal since the pre - designed frequency reuse pattern can not cope well with the dynamics of user distribution and channel variation . therefore , there have been significant efforts in facilitating icic schemes , where the interference coordination ( channel assignment ) is based on real user locations and channel conditions and enabled by multi - cell coordination . different user - centric ( coordination - based ) icic schemes in ofdma - based networks are well summarized in the recent survey papers @xcite . conventionally , most of the performance analyses of icic are based on network - level simulation , and the hexagonal - grid model is frequently used @xcite . since real cellular deployments are subject to many practical constraints , recently more and more analyses are based on randomly distributed bss , mostly using the ppp as the model . these stochastic geometry - based models not only provide alternatives to the classic grid models but also come with extra mathematical tractability @xcite . in terms of the treatment of icic , the most relevant papers to this one are @xcite , where the authors analyzed partial frequency reuse schemes using independent thinning . the authors in @xcite considered bs coordination based on clusters grouped by tessellations . different from these papers , this paper focuses on user - centric icic schemes where the spatial correlation of the coordinated cells is explicitly accounted for . it is worth noting that icic is closely related to multi - cell processing ( mcp ) and coordinated multipoint ( comp ) transmission , see @xcite and the references therein . mcp / comp emphasizes the multi - antenna aspects of the cell coordination , while the form of icic considered in this paper does not take into account the use of mimo ( joint transmission ) techniques and thus is not subject to the considerable signaling and processing overheads of typical mcp / comp schemes , which include symbol - level synchronization and joint precoder design @xcite . thus it can be considered as a simple form of mcp / comp that is light on overhead . intra - cell diversity ( icd ) describes the diversity gain achieved by having the serving bs opportunistically assigns users with their best channels . in cellular systems , diversity exists in space , time , frequency and among users@xcite . it is well acknowledged that diversity can significantly boost the network coverage . however , conventional analyses of diversity usually do not include the treatment of interference , _ e.g.,_@xcite . in order to analytically characterize diversity in wireless networks with interference , a careful treatment of interference correlation is necessary , otherwise the results may be misleading . therefore , there have been a few recent efforts in understanding this correlation@xcite . notably , @xcite shows that in an _ ad hoc _ type network , simple retransmission schemes do not result in diversity gain if interference correlation is considered . analyzing the intra - cell diversity ( icd ) under interference correlation , this paper shows that a diversity gain _ can _ be obtained in a cellular setting where the receiver is always connected to the strongest bs , in sharp contrast with the conclusion drawn from _ ad hoc _ type networks in @xcite . the rest of the paper is organized as follows : section [ sec : sysplpspf ] presents the system model and discusses the comparability of icic and icd . sections [ sec : icic ] and [ sec : icd ] derive the coverage probability for the case with icic or icd only , respectively , and provide results on the asymptotic behavior of the coverage probability in the high - reliability as well as high - spectral efficiency regimes . the case with both icic and icd is analyzed in section [ sec : icic&icd ] . we validate our model and discuss fundamental trade - offs between icic and icd in section [ sec : numer ] . the paper is concluded in section [ sec : conclu ] .

modeling cellular base stations ( bss ) as a homogeneous poisson point process ( ppp ) , this paper provides explicit finite - integral expressions for the coverage probability with icic and icd , taking into account the temporal / spectral correlation of the signal and interference . in addition , we show that in the high - reliability regime , where the user outage probability goes to zero , icic and icd affect the network coverage in drastically different ways : icd can provide _ order _ gain while icic only offers _ linear _ gain . in the high - spectral efficiency regime where the sir threshold goes to infinity , the order difference in the coverage probability does not exist ; however a linear difference makes icic a better scheme than icd for realistic path loss exponents .

inter - cell interference coordination ( icic ) and intra - cell diversity ( icd ) play important roles in improving cellular downlink coverage . modeling cellular base stations ( bss ) as a homogeneous poisson point process ( ppp ) , this paper provides explicit finite - integral expressions for the coverage probability with icic and icd , taking into account the temporal / spectral correlation of the signal and interference . in addition , we show that in the high - reliability regime , where the user outage probability goes to zero , icic and icd affect the network coverage in drastically different ways : icd can provide _ order _ gain while icic only offers _ linear _ gain . in the high - spectral efficiency regime where the sir threshold goes to infinity , the order difference in the coverage probability does not exist ; however a linear difference makes icic a better scheme than icd for realistic path loss exponents . consequently , depending on the sir requirements , different combinations of icic and icd optimize the coverage probability .

1612.08401

i

interfaces play an essential role in phonon - mediated heat conduction in solids.@xcite material discontinuities lead to thermal phonon reflections that are manifested on a macroscopic scale as a thermal boundary resistance ( tbr ) , also called kapitza resistance , @xmath0 , that relates the temperature drop at the interface to the heat flux flowing across it . tbr exists at the interface between any dissimilar materials due to differences in phonon states on each side of the interface.@xcite typical interfaces often possess defects or roughness which can lead to additional phonon reflections and hence higher tbr . tbr plays an increasingly important role in applications , particularly as device sizes decrease below the intrinsic mean free paths ( mfps ) of thermal phonons.@xcite at sufficiently small length scales , tbr can dominate the total thermal resistance . for instance , the effective thermal conductivity of a superlattice can be orders of magnitude smaller than that of the constituent materials due to tbr.@xcite this physical effect has been used to realize thermoelectrics with high efficiency@xcite and dense solids with exceptionally low thermal conductivity@xcite . on the other hand , tbr can lead to significant thermal management problems@xcite in applications such as leds@xcite and high power electronics@xcite . numerous works over several decades have investigated the microscopic origin of tbr at solid - solid interfaces , starting with studies performed at low temperatures ( @xmath1 k ) , in which heat is carried predominantly by phonons with frequencies less than 1 thz.@xcite at these low temperatures and for pristine , ordered interfaces , transmission coefficients can be obtained from continuum elastic theory in an analogy with snell s law for light ; this model is known as the acoustic mismatch model ( amm).@xcite the amm was shown to explain the experimentally measured values of tbr at various solid - solid interfaces.@xcite at higher temperatures ( above 1 k ) , heat transport across the interfaces was found to be sensitive to surface roughness . for the limit of completely diffuse scattering in which transmitted and reflected phonons can not be distinguished , swartz constructed the diffuse mismatch model ( dmm).@xcite despite the success of these models at explaining tbr at low temperatures , they generally fail at temperatures larger than 40 k and are unable to account for the atomistic structure of the interface . recent works have focused on remedying these deficiencies . optical methods enable the routine measurement of tbr over a wide range of temperatures for various metal - dielectric interfaces @xcite as well as at interfaces with variable bonding strength . @xcite other works have examined the temperature dependence of the thermal conductivity@xcite in nanocrystalline samples . computational atomistic methods such as molecular dynamics@xcite and atomistic green s functions@xcite have been extensively applied to obtain the transmission coefficients at interfaces with realistic atomic structure . these calculations generally predict the coefficients to decrease with increasing phonon frequency due to reflections of short wavelength phonons by atomistic roughness , a trend that is supported by basic wave physics and indirectly by experiment.@xcite however , a direct determination of the spectral transmission coefficients at an actual interface has not yet been reported . here , we report a metrology to extract the thermal phonon transmission coefficients at a solid interface . our approach , based on combining experimental observations with ab - initio phonon transport modeling , exploits quasiballistic transport near the interface to significantly narrow the possible transmission coefficient profiles at a solid interface compared to the bounds obtained from traditional approaches . applying our approach in conjunction with transmission electron microscopy ( tem ) , we are able to directly link atomic structure to the spectral content of heat crossing the interface . our approach is a useful tool to elucidate the microscopic transport properties of thermal phonons at solid interfaces .

interfaces play an essential role in phonon - mediated heat conduction in solids , impacting applications ranging from thermoelectric waste heat recovery to heat dissipation in electronics . from the microscopic perspective , interfacial phonon transport is described by transmission coefficients that link vibrational modes in the materials composing the interface . our work provides a useful perspective on the microscopic processes governing interfacial heat conduction .

interfaces play an essential role in phonon - mediated heat conduction in solids , impacting applications ranging from thermoelectric waste heat recovery to heat dissipation in electronics . from the microscopic perspective , interfacial phonon transport is described by transmission coefficients that link vibrational modes in the materials composing the interface . however , direct experimental determination of these coefficients is challenging because most experiments provide a mode - averaged interface conductance that obscures the microscopic detail . here , we report a metrology to extract thermal phonon transmission coefficients at solid interfaces using ab - initio phonon transport modeling and a thermal characterization technique , time - domain thermoreflectance . in combination with transmission electron microscopy characterization of the interface , our approach allows us to link the atomic structure of an interface to the spectral content of the heat crossing it . our work provides a useful perspective on the microscopic processes governing interfacial heat conduction .

1305.5884

i

as hetnet provides flexible and efficient topology to boost spectral efficiency , it has recently aroused immense interest in both academia and industry . as illustrated in fig . [ fig : hetnet_2tiers ] , a hetnet consists of a diverse set of regular macro base stations ( bs ) overlaid with low power pico bss . since this overlaid structure may lead to severe interference problem , it is extremely critical to control interference via rrm in hetnet . there has been much research conducted on rrm optimization for traditional cellular networks . in @xcite , the authors considered power and user scheduling in single - carrier cellular networks . in @xcite , the game theoretical approaches are proposed for distributed resource allocation . in @xcite , the authors proposed a dynamic fractional frequency reuse scheme to combat the inter - sector interference under a game - based optimization by each sector . the coordinated multipoint transmission ( comp ) @xcite is another important technique to handle the inter - cell interference . for example , in @xcite , the authors exploited the uplink - downlink duality to do joint optimization of power allocation and beamforming vectors . in @xcite , a wmmse algorithm is proposed to find a stationary point of the weighted sum - rate maximization problem for multi - cell downlink systems . while the above algorithms achieve comparably good performance , they require global channel state information ( csi ) for centralized implementation @xcite or over - the - air iterations and global message passing for distributed implementation @xcite . it is quite controversial whether comp is effective or not in lte systems due to large signaling overhead , signaling latency , inaccurate csit , and the complexity of the algorithm . on the other hand , solutions for traditional cellular networks can not be applied directly to hetnet due to the unique difference in hetnet topology . first , the inter - cell interference in hetnet is more complicated , e.g. , there is _ co - tier interference _ among the pico bss and among the macro bss as well as the _ cross - tier interference _ between the macro and pico bss . furthermore , due to load balancing , some of the mobiles in hetnet may be assigned to a pico bs which is not the strongest bs @xcite and the mobiles in the pico cell may suffer from strong interference from the macro bss . to solve these problems , some eicic techniques , such as the abs control @xcite , have been proposed in lte and lte - a @xcite . in @xcite , the authors analyzed the performance for abs in hetnet under different cell range extension ( re ) biases . however , they focused on numerical analysis for the existing heuristic eicic schemes , which are the baselines of this paper . in @xcite , the authors proposed an algorithm for victim pico user partition and optimal synchronous abs rate selection . however , they used a universal abs rate for the whole network , and as a result , their scheme could not adapt to dynamic network loading for different macro cells . in this paper , we focus on the resource optimization in the downlink of a hetnet without comp . we consider _ dynamic abrb _ for interference control and dynamic user scheduling to exploit _ multi - user diversity_. the abrb is similar to the abs but it is scheduled over both time and frequency domain . unlike @xcite , we do not restrict the abrb rate to be the same for all macro bss and thus a better performance can be achieved . however , this also causes several new technical challenges as elaborated below . * * exponential complexity for dynamic abrb : * optimization of abrb patterns is challenging due to the combinatorial nature and exponentially large solution space . for example , in a hetnet with @xmath0 macro bss , there are @xmath1 different abrb pattern combinations . hence , brute force solutions are highly undesirable . * * complex interactions between dynamic user scheduling and dynamic abrb * : there is complex coupling between the dynamic user scheduling and abrb control . for instance , the abrb pattern will affect the user sets eligible for user scheduling . furthermore , the optimization objective of abrb control depends on user scheduling policy and there is no closed form characterization . * * challenges in rrm architecture : * most existing solutions for resource optimization of hetnet requires global knowledge of csi and centralized implementations . yet , such designs are not scalable for large networks and they are not robust with respect to ( w.r.t . ) latency in backhaul . to address the above challenges , we propose a two timescale control structure where the long term controls , such as dynamic abrb , are adaptive to the large scale fading . on the other hand , the short term control , such as the user scheduling , is adaptive to the local csi within a pico / macro bs . such a multi - timescale structure allows _ hierarchical rrm _ design , where the long term control decisions can be implemented on a rrm server for inter - cell interference coordination . the short - term control decisions can be done locally at each bs with only local csi . such design has the advantages of low signaling overhead , good scalability , and robustness w.r.t . latency of backhaul signaling . while there are previous works on two timescale rrm @xcite , those approaches are heuristic ( i.e. the rrm algorithms are not coming from a single optimization problem ) . our contribution in this paper is a formal study of two timescale rrm algorithms for hetnet based on optimization theory . to overcome the exponential complexity for abrb control , we exploit the sparsity in the _ interference graph _ of the hetnet topology and derive structural properties for the optimal abrb control . based on that , we propose a _ two timescale alternative optimization _ solution for user scheduling and abrb control . the algorithm has low complexity and is asymptotically optimal at high snr . simulations show that the proposed solution has significant performance gain over various baselines . _ notations _ : let @xmath2 denote the indication function such that @xmath3 if the event @xmath4 is true and @xmath5 otherwise . for a set @xmath6 , @xmath7 denotes the cardinality of @xmath6 .

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

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

astro-ph9911292

i

the dominant radio emitting structures at the galactic center ( gc ) are the supernova remnant ( snr)-like shell sagittarius ( sgr ) a east , a three - armed spiral of ionized gas dubbed sgr a west , and , embedded at the center of sgr a west , the galactic dynamical nucleus , sgr a * , thought to be a massive ( @xmath2 ) black hole ( @xcite ; @xcite ; @xcite ) . sgr a east has a major axis length of about 10.5 pc and its center is located 2.5 pc from sgr a * in projection , and probably behind the latter ( @xcite ) . ( 1998 ) have recently determined the intrinsic size of sgr a * to be less than @xmath3 m at @xmath4 mm . also located at the gc is the egret - identified @xmath0-ray source 2eg j1746 - 2852 ( @xcite ) . it has been shown that the high energy ( @xmath5 ) @xmath0-ray emission spectrum of this source is very likely due to the decay of @xmath1 s ( @xcite ; @xcite ) . these pions are produced by p - p collisions which might plausibly take place at either of two shock regions : 1 ) the shock at sgr a * due to gas accretion from ambient winds , or 2 ) the shock produced by the expansion of the snr - like nonthermal shell of sgr a east into the ambient gas of the interstellar medium . thus , _ a priori _ , either or both sgr a * and sgr a east might be the source of the @xmath0-rays which constitute 2eg j1746 - 285 . it has recently been shown , however , that the identification of sgr a * with 2eg j1746 - 28 is disfavored because charged leptons produced in @xmath6 decays would emit too much synchrotron flux in sgr a * s intense magnetic field at ghz frequencies to be consistent with the well - studied radio spectrum of this object ( @xcite ; @xcite ) . on the other hand , given the physical conditions in sgr a east , the putative charged leptons generated there have a distribution that mimics a power - law with index @xmath7 . the synchrotron flux radiated by these charges is consistent with the radio spectrum of sgr a east observed with the vla . in fact , such relativistic electrons and positrons would also radiate by bremsstrahlung and undergo inverse compton scattering in such a way as to self - consistently explain the entire broadband emission spectrum of sgr a east , ranging from ghz frequencies all the way up to the tev energies observed by whipple ( @xcite ) . for the purposes of this paper , then , we shall take it that the egret source 2eg j1746 - 285 is identical with sgr a east ( @xcite ) . we note in passing that the maximum energy attained by the shocked protons at sgr a east , given the energy loss rate via collision in the shock , is @xmath8 ( @xcite ) . regardless of the ultimate identity of the egret source 2eg j1746 - 285 , given that the process producing the high energy emission is pionic , there should be an associated neutrino flux from the gc ( @xcite ) . these neutrinos are due both to direct pion decay ( @xmath9 ) and to the decay of muons to electrons and positrons ( @xmath10 ) , where we take @xmath11 to mean @xmath11 and @xmath12 here ( as we shall often do in the remainder of this paper ) . _ prima facie _ , then , we expect the flavor composition of the neutrino ` beam ' generated at the gc to be essentially @xmath13 @xmath14-like and @xmath15 @xmath16-like by nave channel counting ( c.f . atmospheric neutrinos in the gev energy range ) . note that there is a @xmath17 background produced at the source due to non - pionic processes like charmed hadron decay . this background is , however , small ; see later . of course , in the absence of neutrino flavor oscillations , one would expect to observe g.c . neutrinos at earth with the same flavor composition as that generated at the source . we do not distinguish between @xmath11 and @xmath18 , because present and planned terrestrial detectors do / will not distinguish between the two . there is one interesting proviso to this statement , however : a @xmath19 flux at @xmath20 can be detected by resonant @xmath21 boson production via @xmath22 with the electrons in the detector medium . the resonance energy , however , is just above that attained by neutrinos generated in the processes described above at the gc ( @xcite ; @xcite ; @xcite ) . given our detailed knowledge of the basic physical processes producing the gc @xmath0-rays , we are able to determine an expression for the total neutrino emission at the source , @xmath23 , in terms of the @xmath0-ray emission there , @xmath24 , the numerical power of the proton spectrum at the source , @xmath25 ( such as would result from shock acceleration at either sgr a east or sgr a * ) , and @xmath26 ( @xcite ) . the quantity @xmath25 has been empirically - determined to lie between 2.1 and 2.4 ( @xcite ; @xcite ) , using a procedure to fit the egret spectrum of 2eg j1746 - 2852 with a detailed calculation of the particle cascade using an extensive compilation of pion - multiplicity cross - sections . in the energy range between the @xmath27-resonance ( @xmath28 gev ) and the isr ( intersecting storage rings ) range ( @xmath29 gev ) , simple scaling ( @xcite ) does not adequately take into account the strong dependence of the cross section on the rapidity at lower energy , and the pion distribution is not adequately described by a power - law mimicking the injected relativistic proton distribution between @xmath30 and @xmath31 gev . instead , the distribution steepens in this region and is curved , which is consistent with the suggested spectral shape measured by egret . above about @xmath32 gev , however , the pion distribution settles into the ` asymptotic ' form suggested by scaling , where the power - law index is a direct reflection of the underlying relativistic protons . thus , an egret spectrum with an effective spectral index of @xmath33 below @xmath32 gev is produced by a pion distribution whose power - law index lies in the range @xmath34 above this energy . in other words , a relatively steep and curved @xmath0-ray spectrum below @xmath32 gev is consistent with a flatter neutrino spectrum at tev - energies . the relative normalization between the @xmath0-ray and neutrino distributions is effected at @xmath32 gev where the pions take on a power - law form . we take the neutrino spectrum at earth to be , in general , given by : @xmath35 normalizing to the observed @xmath0-ray flux at earth at @xmath32 gev , one arrives at the following values for the total neutrino flux here ( @xcite ) : @xmath36 for @xmath37 , and @xmath38 for @xmath39 , where we have taken the absolute upper bound to the energy spectrum of g.c . neutrinos to be given by the highest energy ( @xmath40 ) of the shocked protons . ( kinematical calculations show that neutrinos created by the decay of charged pions produced in scattering of a ` beam ' proton off a stationary ` target ' proton can attain energies very close to the ` beam ' proton . ) note that in the above we make the very reasonable assumption that high energy @xmath0 s and @xmath11 s travel to earth equally unimpeded by the ambient matter they encounter ( which has a column number density of barely @xmath41 ) . two factors improve the odds for the detection of the gc neutrino flux above the atmospheric neutrino background . these are 1 ) the effectively point - source nature of the gc , and 2 ) a gc neutrino spectrum that is significantly flatter than that of atmospheric neutrinos ( which goes as @xmath42 ) . if we preliminarily adopt an angular resolution of @xmath43 for the proposed large scale detectors ( @xmath44 effective detector area ) , the condition for the detection of the gc neutrino flux is @xmath45 , where @xmath46 is the solid angle corresponding to the angular resolution of the experiment and @xmath47 is the flux of atmospheric neutrinos per unit solid angle . this condition is fulfilled above a few tev , and the expected event rate from this preliminary analysis is @xmath48 for @xmath39 to @xmath49 for @xmath37 ( @xcite ) . note that a fuller analysis of event rates ( presented later ) must also consider the problems posed by the atmospheric _ muon _ background and earth neutrino opacity . we see therefore that preliminary calculations reveal that there is a well - determined and potentially observable neutrino flux at the earth from the galactic center . we now briefly list the motivations behind this work before going on to consider whether any sort of neutrino oscillation signature might be detectable in the gc signal . the main motivations behind the present work are : 1 . sgr a east is arguably the most thoroughly understood extra - solar astrophysical source of very high energy neutrinos identified to date . it is thus of fundamental importance for the embryonic science of neutrino astronomy . 2 . it is important for general scientific reasons to explore the neutrino oscillation phenomenon in a wide variety of regimes . because of the high energy scales and the very long baselines involved , astrophysical sources such as sgr a east provide a novel regime not investigated in previous and current solar , atmospheric , reactor and accelerator neutrino detection experiments . previous works to have considered propagation effect signatures in galactic and extra - galactic high energy neutrino signals include @xcite . 3 . given that solar and atmospheric neutrino observations have essentially established the existence of neutrino oscillations , it is important to incorporate this propagation effect when examining possible sources for study through neutrino astronomy . neutrino signals from astrophysical sources are an important complement to electromagnetic signals from same , and they will serve to improve our understanding of the dynamics of important astrophysical objects such as supernova remnants , gamma ray bursters and active galactic nucleii .

it has recently been demonstrated that the-ray emission spectrum of the egret - identified , central galactic source 2eg j1746 - 2852 can be well fitted by positing that these photons are generated by the decay of s produced in p - p scattering at or near an energizing shock . such scattering also produces charged pions which decay leptonically . we determine that the next generation of detectors may well find an oscillation signature in the galactic center ( gc ) signal . 3gcm^-3 3cm^-3 # 1#1 #1#2#3#1 , a&a , * # 2 * , # 3 . # 1#2#3#1 , a&a _ ( letters ) _ , * # 2 * , # 3 . # 1#2#3#1 , a&as , # 2 , # 3 # 1#2#3#1 , aj , # 2 , # 3 # 1#2#3#1 , _ apj _ , * # 2 * , # 3 . # 1#2#3#1 , _ apj ( letters ) _ , * # 2 * , # 3 . # 1#2#3#1 , apjs , # 2 , # 3 # 1#2#3#1 , ara&a , # 2 , # 3 # 1#2#3#1 , baas , # 2 , # 3 # 1#2#3#1 , icarus , # 2 , # 3 # 1#2#3#1 , _ mnras _ , * # 2 * , # 3 . # 1#2#3#1 , _ nature _ , * # 2 * , # 3 . # 1#2#3#1 , pasj , # 2 , # 3 # 1#2#3#1 , pasp , # 2 , # 3 # 1#2#3#1 , qjras , # 2 , # 3 # 1#2#3#1 , science , # 2 , # 3 # 1#2#3#1 , soviet astr . , # 2 , # 3 # 1#2#3#1 , soviet astr . lett . , # 2 , # 3 # 1#2#3#4#1 , # 2 , # 3 , # 4

it has recently been demonstrated that the-ray emission spectrum of the egret - identified , central galactic source 2eg j1746 - 2852 can be well fitted by positing that these photons are generated by the decay of s produced in p - p scattering at or near an energizing shock . such scattering also produces charged pions which decay leptonically . the ratio of-rays to neutrinos generated by the central galactic source may be accurately determined and a well - defined and potentially - measurable high energy neutrino flux at earth is unavoidable . an opportunity , therefore , to detect neutrino oscillations over an unprecedented scale is offered by this source . in this paper we assess the prospects for such an observation with the generation of neutrino erenkov telescopes now in the planning stage . we determine that the next generation of detectors may well find an oscillation signature in the galactic center ( gc ) signal . 3gcm^-3 3cm^-3 # 1#1 #1#2#3#1 , a&a , * # 2 * , # 3 . # 1#2#3#1 , a&a _ ( letters ) _ , * # 2 * , # 3 . # 1#2#3#1 , a&as , # 2 , # 3 # 1#2#3#1 , aj , # 2 , # 3 # 1#2#3#1 , _ apj _ , * # 2 * , # 3 . # 1#2#3#1 , _ apj ( letters ) _ , * # 2 * , # 3 . # 1#2#3#1 , apjs , # 2 , # 3 # 1#2#3#1 , ara&a , # 2 , # 3 # 1#2#3#1 , baas , # 2 , # 3 # 1#2#3#1 , icarus , # 2 , # 3 # 1#2#3#1 , _ mnras _ , * # 2 * , # 3 . # 1#2#3#1 , _ nature _ , * # 2 * , # 3 . # 1#2#3#1 , pasj , # 2 , # 3 # 1#2#3#1 , pasp , # 2 , # 3 # 1#2#3#1 , qjras , # 2 , # 3 # 1#2#3#1 , science , # 2 , # 3 # 1#2#3#1 , soviet astr . , # 2 , # 3 # 1#2#3#1 , soviet astr . lett . , # 2 , # 3 # 1#2#3#4#1 , # 2 , # 3 , # 4

cond-mat0207170

r

our investigations were performed on two high - mobility soi structures which were prepared recently in finland . the soi hall bars have identical peak mobilities of @xmath4@xmath5/vs . they were fabricated on commercial unibond wafers produced by the smart - cut process @xcite . the thickness of the si - layer amounts 200 nm and that of the buried oxide layer 400 nm in these wafers . the background doping is n - type of about 10@xmath6@xmath7 which freezes out at low temperature . after a two - step thermal oxidation , the final thickness of the si - layer is 60 nm . the gate oxide thickness is about 90 nm with a dielectric strength of approximately 1v / nm . the electric contacts were phosphorus - implanted to an activated carrier concentration of @xmath8@xmath7 . the al - metallization is 300 nm thick and sintered at 400@xmath9c . a scheme of the final layer sequence is shown in figure [ fig : samplescheme ] . the etched hall bars have a size of 800@xmath10 m times 100@xmath10 m giving a length to width ratio of 8 . all resistivity and hall measurements were performed in a four terminal ac - technique at low frequencies of typically 13 or 17hz . the soi structures were investigated in a @xmath11he cryostat down to 1.4k . most investigations were performed on sample number l2 , sample l1 showed in several measurements practically the same behavior . the soi sample number l2 was intensively investigated in a density range from @xmath12 to @xmath13@xmath14 . at the low density side of this range , the contacts became high ohmic . due to capacitive effects , the measured voltages showed a delay time of more than 100ms before they came into a steady state . thus voltage measurements in dc - technique lead to other values than those with the 13hz ac - technique . we limited our measurements to the range were the ac- and dc - technique gave the same results and performed all investigations in the ac - technique as this gives a better signal - to - noise ratio . at the higher limit of the investigated density range , the gate started to leak and the current between gate and sample would have lead to errors in the resistivity calculations . the investigations were thus kept in the above given density range . from the resistivity and hall measurements , the carrier density @xmath15 and the mobility @xmath10 were calculated assuming the linear drude behavior to be valid . this is not clear a priori as quantum interference effects might give relevant contributions to the conductivity and would alter the necessary evaluation . the possible importance of quantum effects will be discussed later in this work . for now , @xmath15 and @xmath10 should be used as apparent drude parameters for the 2d electron gas . we have evaluated the dependence of mobility @xmath10 on the electron density @xmath15 by varying the gate voltage @xmath16 at a constant temperature of @xmath17k . in fig . [ fig : msisoi ] , the @xmath18-dependence for soi is compared with that for several si - mos samples @xcite with different mobilities . the soi structure has a peak mobility @xmath19 of @xmath20@xmath5/vs at a density of @xmath21@xmath14 . as can be seen from fig . [ fig : msisoi ] , the si - mos samples have lower or higher peak mobilities , depending on the sample quality , but the overall behavior of @xmath18 is very similar for the two sample types . at high densities , the electron wave function is squeezed by the strong electric field of the triangular potential towards the gate - sided si / sio@xmath22 interface and thus interface roughness scattering dominates . at low @xmath15 , the dominating scattering process is caused by impurities . at the transition region between the two scattering mechanisms , the mobility reaches its peak value @xmath19 @xcite . the most typical feature of the metallic state in 2d systems is expressed by the strong resistivity drop towards lower temperatures . this strong drop in @xmath23 is clearly visible in fig . [ fig : rsisoi]b for the soi sample in the range of @xmath24@xmath14 . the maximum decrease in @xmath23 amounts up to about a factor 3.5 for @xmath25@xmath14 . for the two lower density curves with @xmath26 and @xmath27@xmath14 , an insulating behavior is observed . the critical concentration @xmath0 , at which the behavior changes from insulating to metallic , lies at about @xmath28@xmath14 . for comparison , in fig . [ fig : rsisoi]a and [ fig : rsisoi]c the very high and medium mobility si - mos samples @xcite si-15 and si-4/32 with @xmath29 and @xmath30@xmath5/vs , respectively , are shown . in sample si-15 , the strong decrease in @xmath23 is shifted to lower @xmath1 and the maximum decrease amounts up to a factor 7 , in si-4/32 the decrease is at somewhat higher @xmath1 in comparison to the soi structure and the maximum decrease is about a factor 3 . this shows a clear trend of the properties of the metallic regime with the sample quality , i.e. the peak mobility @xmath19 and manifests the similarities between soi and si - mos structures . additionally , in fig . [ fig : rsisoi ] , dashed - dotted lines at @xmath31 , @xmath32 and @xmath33 are shown . 2 figure [ fig : rsisoisingle ] shows a direct comparison of the strong changes in @xmath23 for the four samples si-15 , si-43 , soi - l2 and si-4/32 for the same low-@xmath1 resistivity @xmath34 . the same behavior , as described above , is visible : the higher the peak mobility , the stronger is the change in @xmath23 and the lower is the temperature where the increase starts . for si - mos structures , it was shown that the critical conductance @xmath35 changes systematically with the peak momentum relaxation time @xmath36 , which is a measure of the sample quality @xcite . here @xmath37 denotes the effective conductivity mass which is the transverse mass of the two lower lying valleys of @xmath38@xmath39 and @xmath40 the elementary charge . for the soi structure , @xmath41 is about @xmath42 and is compared in fig . [ fig : gctp ] with the values for several si - mos samples @xcite . it can be seen , that the @xmath41-value for soi coincides quite well with the general behavior of the si - mos samples . as was seen so far , the soi structure behaves very similar to a high mobility si - mos sample . therefore , we ask the question : what are the similarities and differences between the two sample types ? due to the different effective masses @xmath37 perpendicular to the si / sio@xmath22 interfaces and due to strain effects , the six - fold conduction band ( cb ) degeneracy at the @xmath43-points of bulk si is lifted and only the two lower lying valleys with their longitudinal axis perpendicular to the 2d layer are occupied . the si layer in soi is only 60 nm thick , below there is the buried oxide layer , which constitutes also of a thermally grown oxide fabricated before the wafer bonding process . the active si - layer has an n - type background doping of about @xmath44@xmath7 which freezes out at low @xmath1 . in the si - mos samples , the electrons are confined in the surface inversion layer of a nominally p - type doped bulk si crystal . our high mobility si - mos samples were doped with an acceptor density of about @xmath45@xmath7 . due to the applied gate voltage @xmath16 , an approximately triangular potential well results for the electrons at the upper si / sio@xmath22 interface . nevertheless , the extension of the electronic wave function is different for the soi and the si - mos structure due to the different background doping . in the n - type soi structure , the fermi level @xmath46 is pinned underneath the 2d layer between the donor level and the cb edge . the band bending below the 2d layer is therefore quite small . in p - type si - mos , there is a wide depletion layer of about 0.85@xmath10 m below the 2d layer , as @xmath46 is pinned here between the acceptor level and the valence band ( vb ) edge in the undistorted p - type region . the different band bending below the 2d layer has an influence on the extension of the wave functions especially for small electron densities . we have calculated the wave function @xmath47 and the triangular potential @xmath48 by solving the schrdinger and poisson equations in a self consistent way ( with @xmath49 the coordinate perpendicular to the interface layer ) . in order to solve the schrdinger equation , the charge distribution latexmath:[$\rho(z ) \propto among the electrons were considered , thus giving a first order approximation of @xmath47 and @xmath48 . figure [ fig : psiwidth ] shows the width of the wave function versus electron density @xmath15 . the width of @xmath51 is defined as the length where the central 90 percent of the expectation probability @xmath52 is contained , with 5 percent at the left and at the right side ignored . for a high electron density of @xmath53@xmath14 the width is with 7.6 and 9.2 nm nearly the same for both structures . at lower densities , the difference in the width of @xmath51 increases . the depletion width of 0.85@xmath10 m at an acceptor density of @xmath54@xmath7 corresponds to a 2d charge density of @xmath55@xmath14 . thus , at that 2d electron density , the electric field at the si / sio@xmath22 interface is in si - mos twice as high as in the soi structure . as can be seen from fig . [ fig : psiwidth ] , the difference in the wave function extension is indeed large at that density . at still lower density , the width saturates in si - mos , as the field from the conducting 2d electrons is less important than that from the acceptors in the depletion layer . on the opposite , the width in the soi structure further increases , as there is no electric field confinement from other charges than the conducting electrons itself . figure [ fig : psipot ] shows for comparison the potential shape @xmath48 and the wave function @xmath47 at @xmath56@xmath14 . at that density , the confining potential is much steeper in si - mos and thus the energy @xmath57 of the lowest electronic subband is with 31mev much higher than in soi with 14mev . in fig . [ fig : psipot ] , the difference in the width of the wave functions is also clearly visible and is caused by the difference in the potential steepness . for the case that higher subbands @xmath58 are occupied , the differences in their energies and extensions should be even larger . in the considered density range , the width of the wave function is always smaller than the si - layer thickness of the soi structure and thus the electron do not feel the lower si / sio@xmath22 interface towards the buried oxide layer . in order that such an effect could become significant , the si layer would have to be as thin as 20 nm for @xmath59@xmath14 and even thinner for higher densities . we have also investigated the soi structures in perpendicular and parallel ( i.e. in - plane ) magnetic field @xmath60 . figure [ fig : rhobperp ] shows typical magnetoresistivity curves @xmath61 for @xmath60 perpendicular to the 2d layer for different densities at @xmath62k . for the lower densities , the weak localization peak around @xmath63 is clearly visible . at about 1 t , the shubnikov - de haas oscillations set in and gain strongly in amplitude towards higher magnetic fields . at around 4 t , the spin splitting is visible . the behavior in perpendicular magnetic field is again very similar to that of si - mos samples with comparable mobility . for in - plane magnetic field , the relative change @xmath64 of the magnetoresistivity of soi sample l2 is shown in fig . [ fig : rhobpar ] for different densities at @xmath65k . at high densities of about @xmath66@xmath14 , the changes are quite small , whereas for the lower densities there is a strong increase in @xmath67 by up to a factor 4 . at the lowest density of @xmath68@xmath14 , the observed change is again lower than for the previous densities as can be seen from the crossing of the upper curves in fig . [ fig : rhobpar ] . this indicates a saturation of the strong increase in @xmath69 at about @xmath70@xmath14 . an even stronger @xmath71-dependence was observed in high - mobility si - mos structures in the metallic regime @xcite . again the changes were stronger for the lower densities @xmath15 . the increase with @xmath72 was first attributed to the spin polarization of the conducting electrons @xcite and later to the increase in disorder caused by the magnetic field @xcite . a strong change of @xmath71 was also observed in p - type gaas / algaas structures .

it is shown that the electronic conduction in silicon - on - insulator ( soi ) layers exhibits a metallic regime which is very similar to that in high - mobility si - metal oxide semiconductor structures ( mos ) . the peak in the electron mobility versus density , the strong drop in resistivity and the critical concentration for the metal - insulator transition are all consistent . on the basis of our soi data for the temperature and in - plane magnetic field dependence of the resistivity , we discuss several models for the metallic state in two dimensions .

it is shown that the electronic conduction in silicon - on - insulator ( soi ) layers exhibits a metallic regime which is very similar to that in high - mobility si - metal oxide semiconductor structures ( mos ) . the peak in the electron mobility versus density , the strong drop in resistivity and the critical concentration for the metal - insulator transition are all consistent . on the basis of our soi data for the temperature and in - plane magnetic field dependence of the resistivity , we discuss several models for the metallic state in two dimensions . we find that the observed behavior can be well described by the theory on the interaction corrections in the ballistic regime . for the investigated regime , the temperature dependent screening of scattering potentials gives also a good description of the data . 2

1606.09311

i

electron - hole doping asymmetry in mott insulators has been one of the key questions related to the origin of superconductivity in the proximity of the insulating state @xcite . to investigate this , one should prepare an exactly half - filled ( zero - doped ) mott insulator and then inject or extract electrons , preferably by an electrostatic method , in the same sample . however , it is difficult to perform such measurements on high-@xmath0 cuprates for the following three reasons . first , they are mostly either only hole - doped or electron - doped , and the crystallographic structures of the parent materials are often different . second , they require a strong electric field to tune their band filling owing to their high half - filled carrier density ( on the order of 10@xmath1 @xmath2 ) . third , since they are charge - transfer - type insulators , electrons and holes are doped into different electronic orbitals , which may obscure the pure doping asymmetry . in practice , the inverse of the hall coefficient monotonically increases with increasing doping concentration with the sign changing across the zero - doping point @xcite . this is a feature of band insulators rather than mott insulators . in contrast to the high-@xmath0 cuprates , the organic mott insulator @xmath3-(bedt - ttf)@xmath4cu[n(cn)@xmath4]cl @xcite ( abbreviated to @xmath3-cl hereafter ) , where bedt - ttf represents bis(ethylenedithio)tetrathiafulvalene , serves as an appropriate material for examining the electron - hole asymmetry of doped mott insulators because of its relatively low carrier density ( @xmath61.8@xmath710@xmath8 @xmath2 ) and its single electronic orbital nature . indeed , electrons and holes are doped into the same @xmath9 electron band and thus @xmath3-cl can be modeled as a single band hubbard model on an anisotropic triangular lattice ( fig . [ fig.1 ] ) . due to its high controllability of electronic states , @xmath3-cl has been studied intensively in terms of the bandwidth - controlled mott transition between mott insulating and superconducting states at half filling @xcite . the electrostatic method of carrier doping into organic mott insulators with field - effect transistor ( fet ) structure has been implemented by the authors @xcite . however , the limited doping concentration with the fet still forbids the observation of electron - hole asymmetry . here , by fabricating electric - double - layer transistors ( edlts ) based on thin single crystals of @xmath3-cl , we realised for the first time both electron and hole doping into the organic mott insulator . we measured the field effect on transport properties including the sheet resistivity , hall coefficient and resistivity anisotropy to elucidate the doping anisotropy of the electronic state in the doped organic mott insulator . to provide further insights into the fermi surface ( fs ) , we calculated the single - particle spectral functions for an effective model of @xmath3-cl by using cluster perturbation theory ( cpt ) @xcite . the calculations predict that strongly anisotropic suppression of the spectral weight results in the fermi arc state under hole doping , while a relatively uniform spectral weight results in the emergence of a non - interacting - like fs in the electron - doped state . in accordance with the calculations , the experimentally observed hall coefficients and resistivity anisotropy correspond to the pocket formed by the fermi arcs under hole doping and to the non - interacting fs under electron doping .

the doping effect , including the electron - hole doping asymmetry , may be more straightforward in doped organic mott insulators owing to their simple electronic structures . here we investigate the doping asymmetry of an organic mott insulator by carrying out electric - double - layer transistor measurements and using cluster perturbation theory . the calculations predict that strongly anisotropic suppression of the spectral weight results in the fermi arc state under hole doping , while a relatively uniform spectral weight results in the emergence of a non - interacting - like fermi surface in the electron - doped state . in accordance with the calculations , the experimentally observed hall coefficients and resistivity anisotropy correspond to the pocket formed by the fermi arcs under hole doping and to the non - interacting fermi surface under electron doping .

it is widely recognised that the effect of doping into a mott insulator is complicated and unpredictable , as can be seen by examining the hall coefficient in high cuprates . the doping effect , including the electron - hole doping asymmetry , may be more straightforward in doped organic mott insulators owing to their simple electronic structures . here we investigate the doping asymmetry of an organic mott insulator by carrying out electric - double - layer transistor measurements and using cluster perturbation theory . the calculations predict that strongly anisotropic suppression of the spectral weight results in the fermi arc state under hole doping , while a relatively uniform spectral weight results in the emergence of a non - interacting - like fermi surface in the electron - doped state . in accordance with the calculations , the experimentally observed hall coefficients and resistivity anisotropy correspond to the pocket formed by the fermi arcs under hole doping and to the non - interacting fermi surface under electron doping .

0705.3783

i

recent experimental studies @xcite of the dynamics of electrons and holes in graphene ( a single atomic layer of graphite @xcite ) have revealed unusual behaviour related to the dirac nature of its quasiparticles . two dimensional graphene has a honeycomb lattice structure with two atoms per unit cell . its band structure consists of two inequivalent pairs of cones with apex at the brillouin zone corners . for zero chemical potential the lower energy cones are completely filled and the upper empty . in a graphene device an applied gate voltage can be used to introduce electrons in the upper band or , by voltage reversal , holes in the lower band ( cones ) . the quasiparticles in graphene obey dirac @xcite rather than the schrdinger equation and this has profound implications for their dynamics . the unconvetional quantum hall effect was expected theoretically @xcite and recently observed @xcite to have half ( divided by spin and valley degeneracy ) rather than integer filling factors . the predicted phase shift of @xmath3 in the de haas van alphen @xcite and shubnikov de haas @xcite oscillations was also seen experimentally @xcite . another feature related to the dirac - like character of the carriers in graphene seen in the dc measurements is a finite effective cyclotron mass for the massless dirac quasiparticles which varies as the square root of the number of carriers @xcite . in this paper we consider the magneto - optical conductivity of graphene . work without a magnetic field includes the calculations of ando , zheng and suzuura @xcite who considered the effect on the frequency dependent conductivity of short- and long - range scatterers in a self consistent born approximation . more recent work @xcite describes several anomalous properties of the microwave conductivity of graphene with , as well as without magnetic field . these properties are directly related to the dirac nature of the quasiparticles . several analytic formulas for the longitudinal as well as hall ac conductivity are given in the paper @xcite . they also present extensive results for dc properties and preliminary data on the real part of @xmath4 vs @xmath0 in the optical region . another extensive work by peres et al . @xcite on the ac conductivity in graphene treats localized impurities in a self - consistent fashion as well as extended edge and grain boundaries including also effects of electron - electron interactions and self - doping . in this paper we follow most closely ref . @xcite which we extend in several directions . an aim is to provide simpler analytic formulae which should prove useful in the analysis of experiment and check on their accuracy . another is to consider magnetic field as well as chemical potential sweeps possible in graphene field effect transistor devices @xcite . recent work by li et al . in organic metals @xcite has demonstrated that the ac measurements are also possible in such devices . consistent with our aim , we consider impurities only in simplified scattering rate model and neglect real part renormalizations although these could easily be included if wished . while the renormalization effects beyond those included in this simplified model might become important for the interpretation of future experiment , we point out that so far the free quasiparticle model with associated transport lifetime has been remarkably successful in understanding the dc results of refs . @xcite . in sec . [ sec : cond - general ] we relate the magneto - optical conductivity tensor to the dirac fermionic green s function through kubo formula . the general formulas obtained can be greatly simplified and closed form expressions are obtained in two cases . in general the fermionic self - energy can depend on energy and landau level index as well as temperature and value of the external magnetic field @xmath2 . under the assumption that variations with landau level index @xmath5 can be neglected , the sum over transitions between neighboring landau levels can be carried out explicitly and a closed form expression is obtained for conductivity in terms of digamma function . a single integral over an internal frequency remains . the expression obtained is suitable for calculations of the ac conductivity for any value of temperature , chemical potential and magnetic field . its microwave frequency limit has been used to describe properties of graphene in ref . @xcite . in the case when energy dependence of the fermionic self - energy can be neglected , the internal integration over energy can be done and what remains is the sum over the landau level index @xmath5 of lorentzian forms multiplied by thermal factors and algebraic weighting factors . the weighting factors depend on the landau level energies as well as the excitonic gap ( see e.g. refs . @xcite ) should one wish to include this possibility . when we compare numerical results obtained from lorentzian model and from the previous more general expressions in the limit of constant scattering rate , we find good quantitative agreement between the two . this provides support for the analysis of experimental data @xcite . in sec . [ sec : low - b ] we consider the low field limit for the lorentzian model derived in sec . [ sec : lorentzian ] and establish its correspondence with previously known results . in sec . [ sec : cond - diagonal ] we present the numerical results for the real part of the diagonal conductivity as a function of photon energy @xmath0 for fixed value of magnetic field and various values of chemical potential . we also provide results for fixed photon energy while sweeping either chemical potential or external magnetic field which should prove useful in comparing with experiments . the effect of opening of an excitonic gap on the absorption lines is described . in sec . [ sec : weight ] a discussion of optical spectral weight redistribution by the magnetic field is given . section [ sec : cond - hall ] is structured in parallel to sec . [ sec : cond - diagonal ] but deals with the absorptive part of the off - diagonal hall magneto - optical conductivity . discussion and conclusions are given in sec . [ sec : concl ] . some of the algebra needed in this work is found in an [ sec : a ] .

landau level quantization in graphene reflects the dirac nature of its quasiparticles and has been found to exhibit an unusual integer quantum hall effect . in particular the lowest landau level can be thought as shared equally by electrons and holes and this leads to characteristic behaviour of the magneto - optical conductivity as a function of frequency for various values of the chemical potential . we also provide results for magnetic field as well as chemical potential sweeps at selected fixed frequencies which can be particularly useful for possible measurements in graphene .

landau level quantization in graphene reflects the dirac nature of its quasiparticles and has been found to exhibit an unusual integer quantum hall effect . in particular the lowest landau level can be thought as shared equally by electrons and holes and this leads to characteristic behaviour of the magneto - optical conductivity as a function of frequency for various values of the chemical potential . particular attention is paid to the optical spectral weight under various absorption peaks and its redistribution as is varied . we also provide results for magnetic field as well as chemical potential sweeps at selected fixed frequencies which can be particularly useful for possible measurements in graphene . both diagonal and hall conductivity are considered .

1512.03383

i

let @xmath0 be a finite extension of @xmath1 and let @xmath5 . in this article , we use the theory of @xmath2-modules in the lubin - tate setting to construct some classes in @xmath6 , for `` @xmath7-analytic '' representations @xmath3 of @xmath8 . if in addition @xmath3 is crystalline , we describe these classes explicitly using bloch and kato s exponential maps and generalize perrin - riou s period map to the lubin - tate setting . we now describe our constructions in more detail , and introduce some notation which is used throughout this paper . let @xmath7 be a finite galois extension of @xmath1 , with ring of integers @xmath9 and maximal ideal @xmath10 , let @xmath11 be a uniformizer of @xmath9 and let @xmath12 and @xmath13 . let @xmath14 be a lubin - tate formal group @xcite attached to @xmath11 . we fix a coordinate @xmath15 on @xmath14 , so that for each @xmath16 the multiplication - by-@xmath17 map is given by a power series @xmath18(t)=at+{\mathrm{o}}(t^2 ) \in { \mathcal{o}}_f{[\ ! [ t ] \!]}$ ] . let @xmath19 be the attached lubin - tate character . if @xmath0 is a finite extension of @xmath7 , let @xmath20)$ ] and @xmath21 and @xmath22 . let @xmath23 denote the set of power series @xmath24 with @xmath25 such that @xmath26 as @xmath27 and let @xmath28 $ ] , which is a field . it is endowed with a frobenius map @xmath29(t))$ ] and an action of @xmath30 given by @xmath31(t))$ ] . if @xmath0 is a finite extension of @xmath7 , the theory of the field of norms ( @xcite and @xcite ) provides us with a finite unramified extension @xmath32 of @xmath33 . recall @xcite that a @xmath34-module over @xmath32 is a finite dimensional @xmath32-vector space endowed with a compatible frobenius map @xmath35 and action of @xmath36 . we say that a @xmath34-module over @xmath32 is tale if it has a basis in which @xmath37 . the relevance of these objects is explained by the result below ( see @xcite , @xcite ) . theorem there is an equivalence of categories between the category of @xmath7-linear representations of @xmath8 and the category of tale @xmath34-modules over @xmath32 . let @xmath38 denote the set of power series @xmath39 that have a non - empty domain of convergence . the theory of the field of norms again provides us @xcite with a finite extension @xmath40 of @xmath38 . we say that a @xmath34-module over @xmath32 is overconvergent if it has a basis in which @xmath41 and @xmath42 for all @xmath43 . if @xmath44 , every tale @xmath2-module over @xmath32 is overconvergent @xcite . if @xmath45 , this is no longer the case @xcite . let us say that an @xmath7-linear representation @xmath3 of @xmath8 is @xmath7-analytic if for all embeddings @xmath46 , with @xmath47 , the representation @xmath48 is trivial ( as a semilinear @xmath49-representation of @xmath8 ) . the following result is known @xcite . theorem if @xmath3 is an @xmath7-analytic representation of @xmath8 , it is overconvergent . another source of overconvergent representations of @xmath8 is the set of representations that factor through @xmath36 . our first result is the following ( theorem [ varcolmsurj ] ) . theorem a if @xmath3 is an overconvergent representation of @xmath8 , there exists an @xmath7-analytic representation @xmath50 of @xmath8 , a representation @xmath51 of @xmath8 that factors through @xmath36 , and a surjective @xmath8-equivariant map @xmath52 . we next focus on @xmath7-analytic representations . let @xmath53 denote the robba ring , which is the ring of power series @xmath54 with @xmath55 such that there exists @xmath56 such that @xmath57 converges for @xmath58 . we have @xmath59 . the theory of the field of norms again provides us with a finite extension @xmath60 of @xmath53 . if @xmath3 is an @xmath7-linear representation of @xmath8 , let @xmath61 denote the @xmath34-module over @xmath32 attached to @xmath3 . if @xmath3 is overconvergent , there is a well defined @xmath34-module @xmath62 over @xmath40 attached to @xmath3 , such that @xmath63 . we then let @xmath64 , and @xmath65 is the @xmath34-module over @xmath60 attached to @xmath3 . the ring @xmath60 is a free @xmath66-module of degree @xmath67 . this allows us to define @xcite a map @xmath68 that is a @xmath36-equivariant left inverse of @xmath35 , and likewise , if @xmath3 is an overconvergent representation of @xmath8 , a map @xmath69 that is a @xmath36-equivariant left inverse of @xmath35 . the main result of this article is the construction , for an @xmath7-analytic representation @xmath3 of @xmath8 , of a collection of maps @xmath70 having a certain number of properties . for example , these maps are compatible with corestriction : @xmath71 if @xmath72 . another property is that if @xmath44 and @xmath73 ( the cyclotomic case ) , these maps concide with those constructed in @xcite ( and generalized in @xcite ) . if now @xmath74 and @xmath3 is a crystalline @xmath7-analytic representation of @xmath75 , we give explicit formulas for @xmath76 using bloch and kato s exponential maps @xcite . let @xmath3 be as above , let @xmath77 and let @xmath78 . let @xmath79 be a compatible sequence of primitive @xmath80-torsion points of @xmath14 . let @xmath81 denote the positive part of the robba ring , namely the ring of power series @xmath82 with @xmath55 such that @xmath57 converges for @xmath83 . if @xmath84 , we have a map @xmath85\!]}$ ] given by @xmath86 . using the results of @xcite , we prove that there is a natural @xmath34-equivariant inclusion @xmath87 \otimes_f { \mathrm{d}_{\mathrm{cris}}}(v)$ ] . this provides us , by composition , with maps @xmath88 and @xmath89 where @xmath90 is the `` coefficient of @xmath91 '' map . recall finally that we have two maps , bloch and kato s exponential @xmath92 and its dual @xmath93 ( the subscript @xmath94 denotes the dual of @xmath3 twisted by the cyclotomic character , but is merely a notation here ) . the first result is as follows ( theorem [ interdual ] ) . theorem b if @xmath3 is as above and @xmath95 , then @xmath96 let @xmath97 , let @xmath98 if @xmath99 and let @xmath100 be such that @xmath101 . we prove that if @xmath102 , then @xmath103 , and we have the following result ( theorem [ expbk ] ) . theorem c if @xmath3 is as above and @xmath104 , then @xmath105 using theorems b and c , we give in [ bprfan ] a lubin - tate analogue of perrin - riou s `` big exponential map '' @xcite using the same method as that of @xcite which treats the cyclotomic case . it will be interesting to compare this big exponential map with the `` big logarithms '' constructed in @xcite and @xcite . it is also instructive to specialize theorem c to the case @xmath106 , which corresponds to `` lubin - tate '' kummer theory . recall that if @xmath107 is a finite extension of @xmath7 , kummer theory gives us a map @xmath108 . when @xmath107 varies among the @xmath109 , these maps are compatible : the diagram @xmath110 commutes . let @xmath111 denote the set of sequences @xmath112 with @xmath113 and such that @xmath114(x_n)$ ] for @xmath72 . we prove that @xmath111 is big , in the sense that ( if @xmath45 ) the projection on the @xmath115-th coordinate map @xmath116 is onto ( this would not be the case if we did not have the factor @xmath117 in the definition of @xmath111 ) . furthermore , we prove that if @xmath118 , there exists a power series @xmath119 such that @xmath120 for @xmath72 . we have @xmath121 and the following holds ( theorem [ expchif ] ) , where @xmath122 is the basis of @xmath123 corresponding to the choice of @xmath79 . theorem d we have @xmath124 for all @xmath72 . in the cyclotomic case , there is @xcite a power series @xmath125 such that @xmath126 for @xmath72 . we then have @xmath127 , and theorem d is proved in @xcite . in the general lubin - tate case , we do not know whether there is a `` coleman power series '' of which @xmath57 would be the @xmath128 . this seems like a non - trivial question . it would be interesting to compare our results with those of @xcite . the authors of @xcite also construct some classes in @xmath6 , but start from @xmath129 . in another direction , is it possible to extend our constructions to representations of the form @xmath130 with @xmath3 @xmath7-analytic and @xmath51 factoring through @xmath36 , and in particular recover the explicit reciprocity law of @xcite ?

let be a finite extension of . we use the theory of-modules in the lubin - tate setting to construct some corestriction - compatible families of classes in the cohomology of , for certain representations of . if in addition is crystalline , we describe these classes explicitly using bloch - kato s exponential maps . this allows us to generalize perrin - riou s period map to the lubin - tate setting .

let be a finite extension of . we use the theory of-modules in the lubin - tate setting to construct some corestriction - compatible families of classes in the cohomology of , for certain representations of . if in addition is crystalline , we describe these classes explicitly using bloch - kato s exponential maps . this allows us to generalize perrin - riou s period map to the lubin - tate setting .

1312.0913

i

let @xmath4 denote the orientable surface of genus @xmath1 with @xmath5 boundary components , and let @xmath6 denote the _ mapping class group _ of the closed surface @xmath7 , the group of all orientation preserving homeomorphisms of @xmath7 , modulo isotopy ( see farb - margalit @xcite for background ) . a pair of curves on @xmath0 ( shorthand for @xmath7 ) are said to _ fill _ the surface if the complement of their union is a disjoint union of topological disks . let @xmath8 denote the set of all isotopy classes of simple closed curves on @xmath0 . then @xmath9 acts coordinate - wise on the product @xmath10 , and this action descends to an action on the quotient @xmath11.\ ] ] our first result establishes the exponential growth of the @xmath9-orbits of @xmath12 which fill and intersect minimally : [ thm:1 ] there exists a function @xmath13 satisfying the following : let @xmath14 denote the number of @xmath9-orbits of minimally intersecting filling pairs on @xmath0 . then @xmath15 in the statement of theorem [ thm:1 ] , @xmath16 denotes the following growth - rate relation : @xmath17 we compare theorem [ thm:1 ] to theorem @xmath18 of anderson - parlier - pettet @xcite , which says that if @xmath19 is a filling set of simple closed curves on @xmath0 pairwise intersecting no more than @xmath20 times , then @xmath21 thus , when @xmath22 , and indeed we show in section @xmath23 that a minimally intersecting filling pair intersects @xmath24 times if and only if @xmath25 ( when @xmath26 , @xmath27 intersections is minimal ) . given a pair of filling curves @xmath28 , there are potentially many ways of defining a sort of combinatorial `` complexity '' for the pair . one such candidate is to count the number of simple closed curves intersecting the filling pair a small number of times . heuristically , one expects a minimally intersecting filling pair to only `` barely '' fill @xmath0 . therefore , there should be many simple closed curves intersecting the union @xmath29 only once . indeed , we show : [ thm:2 ] let @xmath28 be a filling pair on @xmath30 and define @xmath31 to be the number of simple closed curves intersecting @xmath32 only once . then @xmath33 , with equality if @xmath28 is minimally intersecting . let @xmath34 denote the _ moduli space _ of riemann surfaces of genus @xmath1 , which is identified with the space of all complete , finite volume hyperbolic metrics on @xmath0 up to isometry . the _ systole _ function @xmath35 outputs the length of the shortest closed geodesic for a given hyperbolic metric @xmath36 . @xmath37 has been studied extensively , ( see buser - sarnak @xcite ; parlier @xcite , @xcite ; and schaller @xcite,@xcite , @xcite ) in part due to the fact that it is a _ topological morse function _ @xcite , a generalization of the notion of a classical morse function to functions which are not necessarily smooth , and for which versions of the morse inequalities also hold . thus a careful analysis of the critical points of @xmath37 can yield naturally arising cellular decompositions of @xmath34 . motivated by this program , we initiate the study of another topological morse function @xmath2 , the _ filling pair _ function , which outputs the length of the shortest minimally intersecting filling pair with respect to a metric @xmath38 . here , the length of a pair of curves is simply the sum of the individual lengths . we show : [ thm:3 ] @xmath2 is proper and a topological morse function . for any @xmath39 , @xmath40 where @xmath41 \right)\ ] ] denotes the perimeter of a regular , right - angled @xmath42-gon . furthermore , define @xmath43 then @xmath44 is finite and grows at least exponentially in @xmath1 . if @xmath45 , the injectivity radius of @xmath3 is at least @xmath46 * organization of paper and acknowledgements . * in section @xmath23 , we prove theorem [ thm:1 ] ; in section @xmath47 , we prove theorem [ thm:2 ] ; and in section @xmath27 , we prove theorem [ thm:3 ] . the authors would like to thank yair minsky for carefully reading through a draft of this project and for his many helpful suggestions . the first author would also like to thank dan margalit , chris arettines , subhojoy gupta , babak modami , and jean sun for many enlightening conversations regarding this work .

let denote the closed orientable surface of genus . we construct exponentially many mapping class group orbits of pairs of simple closed curves which fill and intersect minimally , by showing that such orbits are in correspondence with the solutions of a certain permutation equation in the symmetric group . we conclude by initiating the study of a topological morse function over the moduli space of riemann surfaces of genus , which , given a hyperbolic metric , outputs the length of the shortest , minimally intersecting filling pair for the metric .

let denote the closed orientable surface of genus . we construct exponentially many mapping class group orbits of pairs of simple closed curves which fill and intersect minimally , by showing that such orbits are in correspondence with the solutions of a certain permutation equation in the symmetric group . next , we demonstrate that minimally intersecting filling pairs are combinatorially optimal , in the sense that there are many simple closed curves intersecting the pair exactly once . we conclude by initiating the study of a topological morse function over the moduli space of riemann surfaces of genus , which , given a hyperbolic metric , outputs the length of the shortest , minimally intersecting filling pair for the metric . we completely characterize the global minima of , and using the exponentially many mapping class group orbits of minimally intersecting filling pairs that we construct in the first portion of the paper , we show that the number of such minima grow at least exponentially in .

1609.06971

i

this paper investigates the behaviour of the liouville function which is connected to riemann s zeta function , @xmath3 , defined by@xmath8 where @xmath9 is a positive integer and @xmath2 is a complex variable , with the series being convergent for @xmath10 . this function has zeros ( referred to as the trivial zeros ) at the negative even integers @xmath11riemann s hypothesis claims that the nontrivial zeros of the zeta function all occur on the critical line which occurs at @xmath12 it has been shown that there are an infinite number of zeros on the critical line . the claim that these are the only nontrivial zeros has eluded proof to date , and this paper demonstrates that the riemann hypothesis ( rh ) is resolvable by tackling the liouville function s dirichlet series generated by @xmath13 , which is readily rendered in the form@xmath14 where @xmath15 is the liouville function defined by @xmath16 , with @xmath17 being the total number of prime numbers in the factorization of @xmath9 , including the multiplicity of the primes . we would also need the summatory function @xmath18 , which is defined as the partial sum upto @xmath19 terms of the following series : @xmath20 since the function @xmath4 will exhibit poles at the zeros of @xmath21 , we seek to identify where @xmath3 can have zeros by examining the region over which @xmath4 is analytic . by demonstrating that a sufficient condition , derived by littlewood ( 1912 ) , for the rh to be true is indeed satisfied , we show that all the nontrivial zeros of the zeta function occur on the critical line @xmath7 . briefly , our method consists in judiciously partitioning the set of positive integers into sets of sub - series and couching the infinite sum in ( 2 ) into an infinite number of sums over these sub - series , with each such series being uniformly convergent . this method of considering a slowly converging series as a sum of many sub - series was used by the author in problems where neumann series were involved.(eswaran ( 1990 ) ) . in this paper we were able to break up the sum of the liouville function into sums over many sub - series whose behaviour is predictable . it so turns out that one prime number @xmath22 ( and its powers ) which is associated with a particular sub - series controls the behaviour of the sub series . each sub - series is in the form of rectangular functions ( waves ) of unit amplitude but ever increasing periodicity and widths -we call these as ` harmonics ' , so every prime number is thus associated with such harmonic rectangular functions which then play a role in contributing to the value of @xmath18 . it so turns out that if n goes from n to n+1 , the new value of @xmath23 solely depends upon the factorization of n+1 , and the particular harmonic that contributes to the change in @xmath18 is completely determined by this factorization . since factorizations of numbers involve primes which are uncorrelated , we are to deduce that the statistical distribution of @xmath18 when n is large is like that of the cumulative sum of n coin tosses , ( a head contributing + 1 and a tail contributing -1 ) , and thus logically lead to the final conclusion of this paper . we found a new method of factorizing every integer and placing it in an exclusive subset , where it and its other members form an increasing sequence which in turn factorize alternately into odd and even factors ; this method exploited the inherent symmetries of the problem and was very useful in the present context . once this symmetry was recognized , we saw that it was natural to invoke it in the manner in which the sum in ( 2 ) was performed . we may view the sum as one over subsets of series that exhibit convergence even outside the domain of the half - plane @xmath10 . we were rewarded , for following the procedure pursued in this paper , with the revelation that the liouville function ( and therefore the zeta function ) is controlled by innumerable rectangular harmonic functions , whose form and content are now precisely known and each of which is associated with a prime number and all prime numbers play their due role . and in fact all harmonic functions associated with prime numbers below or equal to a particular value n determine the present value of l(n ) . when we are oblivious to the underlying symmetry being alluded to here , we render the summation in ( 2 ) less tractable than necessary . this is precisely what happens when we perform the summation in the usual manner , setting @xmath24 in sequence .

liouville function , multiplicative function , factorization into primes , mobius function , riemann hypothesis this paper investigates the analytic properties of the liouville function s dirichlet series that obtains from the function , where is a complex variable and is the riemann zeta function . furthermore , by showing that a sufficient condition derived by littlewood ( 1912 ) is satisfied , the paper demonstrates that the function is analytic over the two half - planes and . this establishes that the nontrivial zeros of the riemann zeta function can only occur on the critical line . [ firstpage ]

liouville function , multiplicative function , factorization into primes , mobius function , riemann hypothesis this paper investigates the analytic properties of the liouville function s dirichlet series that obtains from the function , where is a complex variable and is the riemann zeta function . the paper employs a novel method of summing the series by casting it as an infinite number of partial sums over sub - series that exhibit a certain symmetry and rapid convergence . in this procedure , which heavily invokes the prime factorization theorem , each sub - series has the property that it oscillates in a predictable fashion , rendering the analytic properties of the dirichlet series determinable . with this method , the paper demonstrates that , for every integer with an even number of primes in its factorization , there is another integer that has an odd number of primes ( multiplicity counted ) in its factorization . furthermore , by showing that a sufficient condition derived by littlewood ( 1912 ) is satisfied , the paper demonstrates that the function is analytic over the two half - planes and . this establishes that the nontrivial zeros of the riemann zeta function can only occur on the critical line . [ firstpage ]

astro-ph0209426

i

disk accretion to a rotating magnetized star is important in a number of astrophysical objects , including t tauri stars ( edwards _ et al . _ 1994 ) , cataclysmic variables ( e.g. , warner 1995 ) , and x - ray pulsars ( e.g. , bildsten _ et al . _ 1997 ) . the accretion of matter to a rotating star with a dipole magnetic field is a complicated problem still only partially solved . the important questions which need to be answered include : ( 1 ) what is the structure of the disk near the magnetized star ? ( 2 ) where is the inner radius of the disk ? ( 3 ) what is the nature of the funnel flows ( ff ) ? for example , which force is dominant in lifting matter to the funnel flow ? ( 4 ) how is the accretion rate influenced by the star s magnetic moment @xmath3 and angular velocity @xmath4 ? ( 5 ) what is the mechanism of angular momentum transport between the star and the disk ? what determines whether star spins - up or spins - down ? ( 6 ) what are the necessary conditions for magneto - centrifugally driven outflows from the disk and/or the star ? many of these questions have been investigated analytically , but the conclusions reached by different authors often differ because the simplifying assumptions are different . for example , regarding question ( 2 ) , some authors conclude that the disk should be disrupted in the region where magnetic and matter stresses are comparable ( e.g. , pringle and rees 1972 - hereafter pr72 ; davidson & ostriker 1973 ; lamb , pethick & pines 1973 ; ghosh , lamb , & pethick 1977 ; scharlemann 1978 ; ghosh & lamb 1979 a , b - hereafter gl79a , b ; camenzind 1990 ; knigl 1991 ; shu it et al . hereafter s94 ) . others , argue that the inner radius of the disk should be farther away , at the corotation radius , because the inner regions of the disk are disrupted by magnetic braking ( ostriker & shu 1995 hereafter os95 ; branderburg & campbell 1998 ; elstner & rdiger 2000 ) . question ( 3 ) is investigated in only a few papers which use the bernoulli integral ( lamb , pethick & pines 1977 ; paatz & camenzind 1996 hereafter pc96 ; li & wilson 1999 ; and koldoba _ et al . the authors agree that the flow should become supersonic ( and slow magnetosonic ) just above the disk . on the other hand opinions differ regarding the driving force which pushes matter up into the ff . li and wilson ( 1999 ) ( see also li et al . 1996 ) propose that the twisting of the magnetic field near the base of the ff should be very large , @xmath5 and the magnetic force should be the main one lifting matter to the ff . here , @xmath6 is the toroidal component of the magnetic field and @xmath7 is the poloidal component . other groups ( e.g. , lamb et al . 1977 ; pc96 ; koldoba et al . 2002 ) argue that the ff should be super - alvnic so that the twisting of the field is small so that the magnetic force is also small . in numerical simulations by ms97 , h97 and gbw99 the magnetospheric accretion was reported , but no clear evidence of funnel flows was presented and no analysis of ffs was performed . a significant part of this paper is devoted to the ffs . another important issue which has been discussed over the past @xmath8 years is question ( 5 ) concerning the transport of angular momentum between the disk and the star . what determines the sign of the torque on the star ? in early papers it was supposed that a star can only be spun - up because matter in a keplerian disk brings positive angular momentum to the star ( e.g. , pr72 ) . later , it was recognized that a star can be spun - down due to the part of the star s magnetic flux which passes through the disk outside of the corotation radius ( gl79b , wang 1995 ) . recently , the idea of `` torqueless '' accretion was proposed where mass but not angular momentum is transported to a star ( e.g. s94 ; os95 ; li , wickramasinghe & rdiger 1996 ; li & wickramasinghe 1997 ) . wang ( 1997 ) presented arguments against this idea , but this still remains an open question . question ( 6 ) regarding magneto - centrifugally driven outflows from the disk has been discussed by a number of authors ( e.g. , camenzind 1990 ; knigl 1991 ; s94 ; os95 ; lovelace , romanova & bisnovatyi - kogan 1995 - hereafter lrbk95 ; fendt , camenzind & appl 1995 ; pc96 ; goodson & winglee 1999 ; bardou & heyvaerts 1999 ; agapitou & papaloizou 2000 ) . for example , s94 proposed that poloidal magnetic flux accumulates near the corotation radius and magnetic winds should blow from this point ( x - point ) . lrbk95 proposed that wind may form from the entire region of the disk outside the corotation radius where the magnetic field threading the disk is open . analytical investigations of disk accretion to a magnetized star are of course limited by the different assumptions made . for this reason robust 2d and 3d simulations are essential to further the understanding of the different phenomena . by robust we mean that the result should not depend on initial conditions , boundary conditions , on grid resolution , and other artificial factors . several 2d mhd simulation studies have been made with different initial conditions aimed at disk / star outflows . in an early work hayashi , shibata , and matsumoto ( 1996 ) ( hereafter hsm96 ) investigated the interaction of a non - rotating star with a keplerian accretion disk and observed the opening of magnetic field lines which initially thread both the star and the disk . they found single event outflows and the corresponding inward collapse of the disk on a dynamical time - scale ( less than one period of rotation of the inner radius of the disk ) with the radial velocity of the disk close to free - fall . this fast evolution was the result of the magnetic braking of the disk by the magnetic field linking the disk and non - rotating star through a conducting corona . this explosive behavior may correspond to some episodic accretion events of actual systems . however , it is important to investigate the possible quiescent behavior of the disk - star systems . miller and stone ( 1997 ; hereafter ms97 ) investigated disk - star interaction for different geometries and stellar magnetic fields using the resistive zeus code . ms97 rotated the corona - which occupies all the space between the star and the disk - with the rotation rate of the star . this decreased the initial magnetic braking ( compared to hsm96 ) , so that they were able to perform simulations for several periods of rotation of the inner radius of the disk . in cases with a relatively weak magnetic field , they got results similar to those of hsm96 . they also found the disk collapsing to the star with velocity @xmath9 , the opening of magnetic field lines , and outflows of matter from the disk . however , in the case of a strong magnetic field , particularly in the case which included a uniform homogeneous vertical magnetic field threading the disk , they observed diminished outflows . instead the matter flowed around the magnetosphere to the star . similar results were obtained by hirose et al . ( 1997 ; hereafter h97 ) . goodson , winglee and bhm ( 1997 ) ( hereafter gwb97 ) and goodson , bhm and winglee ( 1999 ) ( hereafter gbw99 ) did much longer simulations in very large simulation regions . they observed quasi - periodic matter outbursts associated with the quasi - periodic opening of magnetic field lines and matter accretion to the star . the density in the corona was chosen to decrease in a special way , @xmath10 , so that the alfvn speed @xmath11 decreases gradually . this is favorable for the opening of magnetic field lines , and for the generation and propagation of outflows . gwb97 and gbw99 do not investigate cases where the density falls off more slowly with distance . fendt and elstner ( 1999 , 2000 hereafter fe99 , fe00 ) investigated disk - star interaction for thousands of rotations of the inner radius of the disk , and observed the opening of magnetic field lines and outflows . however , they treated the disk as a boundary condition so that they could not take into account the back reaction of the disk on the stellar magnetic field . furthermore , the actual outflow of matter from the disk to the corona may be different from that assumed . the above mentioned simulation studies show that outflows appear either in very non - stationary situations ( hsm96 , ms97 ) or for a special distribution of coronal density and very fast rotation of the star ( gwb97 , gbw99 ) . none of the papers give answer to questions ( 1)-(5 ) . also , it is not clear whether or not outflows exist for more quiescent initial conditions , and for cases where the coronal density falls off slowly with distance . in this paper we investigate disk accretion to a rotating magnetized star and the associated funnel flows . we start from initial conditions which give us the possibility to significantly reduce the initial magnetic braking between the disk and the corona . this allows us to investigate the disk - star interaction and funnel flows for long times and to consider questions ( 1)-(5 ) in detail . in 2 we describe the numerical model , including initial and boundary conditions . we also discuss the evolution of the disk without a magnetic field . in 3 we describe in detail the disk - star interaction for the case of slowly rotating stars and in 4 cases of fast rotating stars . in 5 we consider the dependence of disk - star interaction on the magnetic moment @xmath3 . in 6 we analyze the physics of ffs . in 7 we consider the possibility of outflows . in 8 we apply our simulation results to t tauri stars . in 9 give the conclusions from this work .

magnetohydrodynamic ( mhd ) simulations have been used to study disk accretion to a rotating magnetized star with an aligned dipole moment . dominated regime , more dramatic opening of the magnetic field was observed with the formation of magneto - centrifugally driven outflows . numerical applications of our simulation results are made to t tauri stars . our results are also relevant to cataclysmic variables and magnetized neutron stars in x - ray binaries . = cmmib10 at 10pt

magnetohydrodynamic ( mhd ) simulations have been used to study disk accretion to a rotating magnetized star with an aligned dipole moment . quiescent initial conditions were developed in order to avoid the fast initial evolution seen in earlier studies . a set of simulations was performed for different stellar magnetic moments and rotation rates . simulations have shown that the disk structure is significantly changed inside a radius where magnetic braking is significant . in this region the disk is strongly inhomogeneous . radial accretion of matter slows as it approaches the area of strong magnetic field and a dense ring and funnel flow form at the magnetospheric radius where the magnetic pressure is equal to the total , kinetic plus thermal , pressure of the matter . funnel flows ( ff ) , where the disk matter moves away from the disk plane and flows along the stellar magnetic field , are found to be stable features during many rotations of the disk . the dominant force driving matter into the ff is the pressure gradient force , while gravitational force accelerates it as it approaches the star . the magnetic force is much smaller than the other forces . the funnel flow is found to be strongly sub - alfvnic everywhere . the ff is subsonic close to the disk , but it becomes supersonic well above the disk . matter reaches the star with a velocity close to that of free - fall . angular momentum is transported to the star dominantly by the magnetic field . in the disk the transport of angular momentum is mainly by the matter , but closer to the star the matter transfers its angular momentum to the magnetic field and the magnetic field is dominant in transporting angular momentum to the surface of the star . for slowly rotating stars we observed that magnetic braking leads to the deceleration of the inner regions of the disk and the star spins up . for a rapidly rotating star , the inner regions of the disk rotate with a super - keplerian velocity , and the star spins - down . the average torque is found to be zero when the corotation radius . the evolution of the magnetic field in the corona of the disk depends on the ratio of magnetic to matter energies in the corona and in the disk . most of the simulations were performed in the regime of a relatively dense corona where the matter energy density was larger than the magnetic energy density . in this case the coronal magnetic field gradually opens but the velocity and density of outflowing matter are small . in a test case where a significant part of the corona was in the field dominated regime , more dramatic opening of the magnetic field was observed with the formation of magneto - centrifugally driven outflows . numerical applications of our simulation results are made to t tauri stars . we conclude that our quasi - stationary simulations correspond to the classical t tauri stage of evolution . our results are also relevant to cataclysmic variables and magnetized neutron stars in x - ray binaries . = cmmib10 at 10pt

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influenza a continues to evade eradication resulting in ongoing economic and human cost . yearly epidemics are responsible for 36,000 deaths on average in the united states . three times in the past century global pandemics , including the infamous spanish influenza of 1918 , resulted in catastrophic mortality rates [ @xcite ] . influenza epidemiologists believe a future influenza pandemic is an imminent threat [ @xcite ] . nearly 400 documented transfers [ @xcite ] of the highly virulent and potentially pandemic [ @xcite ] h5n1 strain of avian flu to humans in addition to the recent development of h1n1 swine flu [ @xcite ] bolster the threat . the increasingly relevant necessity of preventing future influenza pandemics requires a clear understanding of the evolutionary mechanisms of influenza as it is the key to vaccine development [ @xcite ] . influenza a is a negative single - stranded rna virus composed of 8 segments that total approximately 13 kb in length and encode 11 proteins . the three largest segments encode polymerases pb1 , pb1-f2 , pb2 and pa all of which are involved in rna transcription and replication . the next three segments code for the two surface glycoproteins haemagglutinin ( ha ) and neuraminidase ( na ) as well as the nucleoprotein ( np ) . the two smallest segments encode the matrix proteins m1 and m2 and the nonstructural proteins ns1 and ns2 [ @xcite , @xcite ] . influenza a research typically focuses on the epitope - rich ha and na segments because they exhibit strong evolutionary selective pressure due to their direct interaction with the host immune system and are the primary determinants of the antigenic variation of influenza [ @xcite ] . the 16 ha and 9 na glycoproteins found in the avian reservoir , referred to as h1 to h16 and n1 to n9 , respectively , characterize and name the subtypes of influenza a [ @xcite ] . the evolutionary history of influenza a involves the interaction of a number of mechanisms including mutation and reassortment . approximately one random sequence mutation every replication cycle combined with the selective pressure on the surface glycoproteins results in an accumulation of point mutations on the ha and na segments termed antigenic drift . the influenza genome also evolves through reassortment in which two subtypes coinfect a single host cell and exchange segments . this exchange of genetic material can lead to an antigenic shift or the creation of a new , potentially lethal , subtype . reassortment between a virus in the avian reservoir and human influenza a resulted in the subtypes responsible for the asian and hong kong influenza pandemics [ @xcite ] and the current swine flu pandemic derives from a triple reassortment event [ @xcite ] . as critically important events in influenza evolution occur at the genome - level , complete genome analysis yields scientific insight that single segments can not afford . for example , @xcite clarify a perplexing question in the evolutionary dynamics of ha by considering the varying histories of each segment . analyzing 156 complete h3n2 viruses over a five - year time span from 1999 to 2004 , @xcite ( @xcite ) discover that while the fujian - variant ha segment has been co - circulating since at least 2000 , the variant only rises to dominance in 2002 after other segments within the influenza genome reassort and provide a synergistic background . this important reassortment event is only understood by studying the influenza genome in its entirety . a more recent study by @xcite emphasizes the importance of incorporating model parameter uncertainty in drawing conclusions about influenza evolution through a bayesian analysis of a truly massive dataset . @xcite compile 687 h3n2 influenza a full genomes sampled from new york over a twelve - year period . @xcite address a host of biologically and clinically relevant questions including : ( 1 ) are reassortment events coincident with shifts in ha antigenicity ? ( 2 ) do certain segments maintain greater genetic diversity ? ( 3 ) are the genetic histories of certain segments correlated ? however , due to a dearth of bayesian massive data techniques , computational constraints force @xcite to partition the data by stratifying on segment , using the data inefficiently , and drawing ad hoc conclusions about potential correlation . this current study rectifies the stratified analyses by fully capitalizing on the hierarchical nature of the influenza data and making formal inference after modeling the complete data in a single bayesian analysis . despite optimized algorithms for missing data integration [ @xcite ] , phylogenetic analysis of dna sequences is lengthy and computationally intensive . massive data measure in the gigabyte to terabyte range [ @xcite ] and are increasingly common [ @xcite , @xcite ] . this pervasiveness is particularly poignant in bayesian models with missing data and especially in bayesian models for stochastic processes where the dimensionality of the missing data can far outweigh the observed data . such is the case in the evolutionary reconstruction of @xcite . one strategy pertinent to massive data inference is stratification [ @xcite , @xcite ] , often undesirable because it comes shackled with the host of difficulties arising from subgroup analysis [ @xcite , @xcite ] . this identifies the direction that @xcite originally follow as they treat each of the eight segments independently . shared computer memory and communication latency between computers limit hopes for considering a proper hierarchical model across segments simultaneously through which to share information and learn about segment - to - segment correlation . even on state - of - the - art equipment , simulating sufficient realizations from posterior distributions conditional only on the data from a single segment , or what we refer to as `` stratified distributions , '' still compels one to devote weeks of computing time per segment . this huge computational investment raises a critical point regarding a massive dataset with hierarchical structure . often researchers perform preliminary analysis stratified by the exchangeable identifiers in the data simply because the statistical tools and computational resources exist for the stratified analysis . in attempting to fit the full hierarchical analysis , the ability to reuse the results from these suboptimal analyses represents a major savings in terms of time and resources and may even be the only feasible option . to this end , we examine the dynamic iteratively reweighting mcmc algorithm ( dyirma ) [ @xcite , @xcite ] . dyirma is based on the meta - analysis technique of using summary statistics from independent studies to infer a single hierarchical model [ @xcite , @xcite ] . instead of summary statistics , however , dyirma combines realizations from independent distributions using importance sampling and markov chain monte carlo ( mcmc ) in an iterative process . importantly , we can adopt dyirma to reuse realizations from preliminary stratified analyses . this desire to not waste intermediate realizations from the stratified analyses is particularly relevant in our influenza example because the realizations themselves require massive computing resources to generate . we further extend the insight of @xcite who combine intermediate realizations from two uncorrelated distributions . our extension is necessary to accommodate correlated sequence data sampled over a span of thirteen years and allows us to entertain a much richer collection of hierarchical models motivated by the science at hand . our scientific aim in this study is to create a joint hierarchical model that addresses the questions raised by @xcite regarding the evolutionary history of influenza a. to this end , the hierarchical model must account for an unknown correlation structure between segments and allow for a flexible time - course in the model response , for which we exploit gibbs variable selection ( gvs ) [ @xcite , @xcite ] to estimate a nonparametric response . the influenza a example illustrates that dyirma is a particularly flexible and valuable approach that reuses realizations via reweighting from computationally expensive distributions in a hierarchical framework . this widely applicable technique can be used to jointly model other independently generated , but in truth correlated , massive datasets . as a preview , the paper continues as follows : section [ sec : stratified ] describes the generation of realizations from the stratified analyses , section [ sec : sampling ] introduces the basic framework used to estimate genealogies . section [ sec : recycle ] relates the machinery necessary to combine these realizations to estimate the joint hierarchical model along with computational concerns . section [ sec : model ] reviews the hierarchical model proposed , prior distributions , mcmc sampling concerns , and various modeling extensions . sections [ sec : theresults ] and [ sec : discuss ] present results and conclude with a discussion .

however , although the initial concession to stratify generally precludes the more sensible analysis using a single joint hierarchical model , we can circumvent this outcome and capitalize on the intermediate realizations by extending the dynamic iterative reweighting mcmc algorithm . in doing so , we reuse the available realizations by reweighting them with importance weights , recycling them into a now tractable joint hierarchical model . we apply this technique to intermediate realizations generated from stratified analyses of 687 influenza a genomes spanning 13 years allowing us to revisit hypotheses regarding the evolutionary history of influenza within a hierarchical statistical framework . , .

massive datasets in the gigabyte and terabyte range combined with the availability of increasingly sophisticated statistical tools yield analyses at the boundary of what is computationally feasible . compromising in the face of this computational burden by partitioning the dataset into more tractable sizes results in stratified analyses , removed from the context that justified the initial data collection . in a bayesian framework , these stratified analyses generate intermediate realizations , often compared using point estimates that fail to account for the variability within and correlation between the distributions these realizations approximate . however , although the initial concession to stratify generally precludes the more sensible analysis using a single joint hierarchical model , we can circumvent this outcome and capitalize on the intermediate realizations by extending the dynamic iterative reweighting mcmc algorithm . in doing so , we reuse the available realizations by reweighting them with importance weights , recycling them into a now tractable joint hierarchical model . we apply this technique to intermediate realizations generated from stratified analyses of 687 influenza a genomes spanning 13 years allowing us to revisit hypotheses regarding the evolutionary history of influenza within a hierarchical statistical framework . , .

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massive galaxies at high redshift have been observed to have properties which can be quite different from their counterparts at similar mass in the local universe . one of the most striking differences is the much smaller sizes observed for high - redshift galaxies . for example , a quiescent galaxy with a stellar mass of @xmath7 m@xmath1 , which has a typical effective radius of 8 kpc in the local universe , has a typical size of only 1.7 kpc at @xmath4 @xcite . early reports of these extremely small sizes based on their spatial extent @xcite have been confirmed by dynamical measurements of their stars @xcite and , more recently , gas @xcite , in addition to higher - resolution hst imaging @xcite . several observational studies have tracked the strong evolution of galaxy size with redshift @xcite , which is present in both star - forming and quiescent galaxies but is particularly dramatic in the latter case . part of the observed trend in median size is due to the continual addition of new , larger galaxies to the massive , quiescent population . star - forming disks have larger sizes at later times @xcite . as these galaxies quench , their appearance in the quiescent population can shift the median size without changing the number density of compact galaxies , a phenomenon known as progenitor bias " @xcite . careful accounting suggests that progenitor bias alone is insufficient to explain the observed size growth , however @xcite , so that the growth of individual galaxies within the quiescent population is also required . recently , some authors have found that compact galaxies may not be as rare in the local universe as previously thought @xcite , suggesting that they must continue to be produced to replace any which size out " of the compact selection . several authors have proposed mechanisms for increasing galaxies sizes after quenching . dry mergers between galaxies of similar masses increase size as @xmath8 @xcite , while minor mergers may increase the sizes even more efficiently , @xmath9 @xcite . in the latter case , it has been suggested that the massive , compact galaxies at high redshift now lie at the centers of the most massive galaxies in the local universe , having accreted a substantial amount of stellar mass since their initial formation @xcite or regrown a star - forming disk @xcite . this idea is supported by the observational fact that massive galaxies interior densities evolve much less strongly than their overall size @xcite . the observed minor merger rate , however , may be too low to be entirely responsible for galaxies growth @xcite . in the absence of mergers and accretion , galaxies may also puff up " due to adiabatic expansion in response to the expulsion of gas from the inner regions from stellar winds and black hole feedback as @xmath10 @xcite . these attempts to explain what happens to high - redshift compact galaxies are hindered by the long timescales of galactic evolution , which limit observations of any individual system to a single evolutionary state . as a result , galaxy populations can only be connected at different epochs in a statistical sense . simulations of galaxy formation , however , offer the opportunity to directly observe " the evolution of individual systems over time . large - volume simulations , moreover , can provide a cosmological context for that evolution and produce populations of galaxies which can be compared against observations . in @xcite , we identified a set of galaxies in the cosmological hydrodynamical simulation illustris which are analogs to observed massive compact ellipticals , with stellar masses of @xmath11 m@xmath1 and stellar half - mass radii of 2 kpc or smaller at @xmath4 . these compact galaxies formed naturally within the cosmological volume alongside non - compact galaxies at similar stellar masses , and we traced their progenitors back to higher redshift to discover how their formation paths differed . we found that the galaxies all reached their compact state through some combination of assembling their stellar mass at very early times when the universe was extremely dense , and/or intense , centralized bursts of star formation usually triggered by gas - rich mergers . both in our work and in that of other simulators ( see e.g. zoom - in simulations by @xcite and @xcite ) , gas - rich environments are required to dissipate angular momentum and produce such compact objects . we now turn our attention to the low - redshift outcomes of these massive , compact galaxies . our aim in this work is to show how a specific population of galaxies which were massive and compact at @xmath4 evolve forward to the present , describe their descendants , and characterize their evolutionary paths . as before , we select a massive , compact population at @xmath4 in illustris , this time tracing the galaxies forward to @xmath5 . in section [ sec : illustris ] , we describe the illustris simulation suite and the merger trees used to connect galaxies at different redshifts . in section [ sec : general ] , we select a population of 35 compact galaxies at @xmath4 and follow them forward to their @xmath5 descendants . in section [ sec : descendants ] , we take a detailed look at four distinct types of descendants and show how common each type of evolutionary track is , and in section [ sec : environment ] we discuss how a galaxy s @xmath4 environment might influence its eventual outcome . finally , in section [ sec : discuss ] we address the likelihood of massive galaxies in the local universe hosting compact cores , explore the different mechanisms for size growth , and discuss the implications for the number density of compact galaxies at low redshift . we summarize the results and conclude in section [ sec : conclusion ] .

massive quiescent galaxies have much smaller physical sizes at high redshift than today . herein , we select a sample of 35 massive , compact galaxies ( m , m/kpc ) at in the cosmological hydrodynamical simulation illustris and trace them forward to to uncover their evolution and identify their descendants . [ firstpage ] galaxies : high - redshift , galaxies : evolution

massive quiescent galaxies have much smaller physical sizes at high redshift than today . the strong evolution of galaxy size may be caused by progenitor bias , major and minor mergers , adiabatic expansion , and/or renewed star formation , but it is difficult to test these theories observationally . herein , we select a sample of 35 massive , compact galaxies ( m , m/kpc ) at in the cosmological hydrodynamical simulation illustris and trace them forward to to uncover their evolution and identify their descendants . by , the original factor of 3 difference in stellar mass spreads to a factor of 20 . the dark matter halo masses similarly spread from a factor of 5 to 40 . the galaxies evolutionary paths are diverse : about half acquire an ex - situ envelope and are the core of a more massive descendant , a third survive undisturbed and gain very little mass , 15% are consumed in a merger with a more massive galaxy , and a small remainder are thoroughly mixed by major mergers . the galaxies grow in size as well as mass , and only% remain compact by . the majority of the size growth is driven by the acquisition of ex - situ mass . the most massive galaxies at are the most likely to have compact progenitors , but this trend possesses significant dispersion which precludes a direct linkage to compact galaxies at . the compact galaxies merger rates are influenced by their environments , so that isolated or satellite compact galaxies ( which are protected from mergers ) are the most likely to survive to the present day . [ firstpage ] galaxies : high - redshift , galaxies : evolution

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given that about half of the compact galaxies from @xmath4 exist at @xmath5 as the cores of very massive galaxies , one can ask the reverse question : how many massive galaxies have cores which were once compact galaxies ? we address this question by tracing all the @xmath5 massive galaxies through the merger tree and calculating @xmath24 at each point on its main progenitor branch . we then define the maximum past compactness " @xmath55 to be the maximum value . ] of @xmath24 when the progenitor had a stellar mass of at least @xmath53 m@xmath1 . in contrast to the analysis thus far , this approach permits a galaxy to be compact at any redshift , rather than specifically tying them to @xmath4 . this allows , for example , a galaxy which forms very early at @xmath58 in a compact state but grows out of the compact population by @xmath4 , or a galaxy which forms via a compact starburst at @xmath39 , both of which would be missed by the preceding analysis . the colors of the points on the mass - size plane in the upper panel of figure [ fig : maxpastcompact ] indicate @xmath55 for all of the massive galaxies at @xmath5 . lines of constant @xmath24 are drawn in the same color scheme . any galaxy whose primary progenitor met the @xmath29 compactness criterion at any point in time is outlined in black . about half of these were compact at exactly @xmath4 and overlap with the sample discussed elsewhere in this paper , while the other half experienced their compact phases at other redshifts . from this scatter plot , there is a clear trend between mass and maximum past compactness : more massive galaxies tend to have more compact progenitors . ( see also @xcite for a discussion of the relationship between mass and compactness in illustris . ) thus , many of the most massive galaxies were indeed once compact galaxies according to our definition but it is interesting to note that this is not universally the case , since there is also considerable scatter in the distribution of @xmath55 at this mass ( and , in fact , at all masses ) . one cause of the spread in @xmath55 at a given mass can be seen in the second panel , which shows the galaxies formation times . here , we have defined formation time " as the median formation redshift of all star particles in the galaxy which formed _ in - situ_. ( this excludes stars acquired via mergers . ) as before , color indicates @xmath55 . in addition to the trend with stellar mass , a trend between formation time and past compactness is clearly visible this is a direct reflection of the fact that galaxies form with smaller sizes at earlier times . thus , the distribution of formation times for galaxies of a given mass also manifests as a distribution of past compactness . the relationship between mass and @xmath55 is further quantified in the lower panel of figure [ fig : maxpastcompact ] . each line shows the cumulative fractional distribution of @xmath55 for the galaxies in a given stellar mass bin . as was suggested by the upper panel , moving to lower masses shifts the distribution to lower past compactness values . for any @xmath24 , we can now see what fraction of galaxies at that mass have ever met that threshold . the orange line denotes the @xmath29 criterion used elsewhere in this paper . approximately 55% of galaxies with @xmath59 m@xmath1 have met this definition of compact , " along with 30% of 6 - 10 @xmath60 m@xmath1 galaxies , 15% of 3 - 6 @xmath60 m@xmath1 galaxies , etc . although a substantial fraction ( 55% ) of very massive galaxies are former compact galaxies , it is important to note that few of these were compact at exactly @xmath4 . rather , their compact phases are spread over @xmath61 , so that only about @xmath62 of galaxies above @xmath63 m@xmath1 at @xmath5 have compact progenitors at @xmath4 . similarly , of the 35 @xmath4 compact galaxies , only 3 ( 9% ) have direct descendants in that mass range at @xmath5 . these fractions demonstrate once again that while there is indeed a strong trend between final stellar mass and past compactness , there is also a large dispersion in galaxies evolutionary tracks that can not be neglected . nearly all of the compact galaxies have grown in size as well as mass since @xmath4 , with very few still remaining in the compact regime at @xmath5 . potential origins of this growth include physical effects such as major mergers @xcite , minor mergers and accretion @xcite , star formation @xcite , and adiabatic expansion due to the expulsion of gas from the galaxy through stellar winds and bh feedback @xcite , as well as numerical effects stemming from the non - constant physical size of the gravitational softening length in illustris . to examine the importance of each of these mechanisms for the size growth of the different descendant types , let us return to the same four example galaxies we discussed in section [ ssec : individs ] . the left - hand panels of figure [ fig : sizeexamples ] show the evolution of their stellar half - mass radii with time . in addition to the black line for total stellar half - mass radius ( which includes all stars bound to the object ) , we also show a red line for the the compact half - mass radius " that is , the half - mass radius including only those stars which belonged to the compact progenitor at @xmath4 . this allows us to disentangle the growth of the compact core from the growth of the galaxy as a whole . in the first ( consumed ) case , both the overall and compact half - mass radii jump up when the compact galaxy falls into the more massive central galaxy . this indicates , as we have seen , that the compact galaxy s stellar mass has been dispersed throughout the descendant . in a similar ( though less extreme ) way , the second ( mixed ) galaxy s compact half - mass radius increases abruptly when it undergoes a disruptive major merger . in these cases , the size growth is primarily driven by merging . conversely , in the third ( core ) case the compact half - mass radius hardly changes despite plenty of mergers and stellar mass growth , while the total half - mass radius greatly increases . the very small change in compact half - mass radius signifies that the core has remained intact , and that the vast majority of the overall size growth is driven by the acquisition of outer layers of stellar material from less - bound galaxies . finally , the galaxy which is undisturbed grows only slightly in size from 2 to 3 kpc over this 10 gyr period . in this case , the stellar mass has changed very little as a small amount of new stars ( @xmath610% , both in - situ and ex - situ ) offset the mass loss from preexisting stars . here , size growth mechanisms such as adiabatic expansion from mass loss and numerical effects start to become significant . in treating the gravitational forces between baryons , illustris uses a gravitational softening length @xmath64 which is a fixed comoving size at redshifts @xmath65 . the increase of the physical scale on which the potential is softened will decrease the depth of the gravitational well in dense regions , artificially inflating the stellar mass profiles as the star particles become less gravitationally bound . it is unlikely that the increasing @xmath64 is the sole culprit , however , since it is capped at a fixed physical size at @xmath16 but the size growth does not completely halt at this time . in the appendix , we estimate how these compact stellar systems will expand adiabatically in response to the changing softening length , stellar mass loss , and gas expulsion . we find that numerical relaxation " of the galaxies @xmath4 profiles to the @xmath5 softening length should only change the half - mass radius by about @xmath62 . adiabatic expansion as a response to stellar mass loss and gas expulsion should produce another @xmath66% growth , and the remaining @xmath67% growth is owed to the small amount of in - situ stars formed and ex - situ stars deposited at larger radii . returning to the compact population as a whole , the right - hand panel of figure [ fig : sizeexamples ] shows the growth in compact half - mass radius since @xmath4 ( @xmath51-axis ) against the ratio between the overall and compact half - mass radii ( @xmath52-axis ) for each @xmath5 descendant . the @xmath51-axis quantity is identical to the ratio between the value of the red lines in the left - hand panel at @xmath5 and @xmath4 , while the @xmath52-axis quantity is the ratio between the black and red lines at @xmath5 . the 30% growth in the compact half - mass radius from the changing softening length , stellar mass loss , and expulsion of gas between @xmath4 and @xmath5 that we estimate in the appendix is marked by the vertical dotted line . @xmath51-values significantly higher than this indicate some disruption of the compact core , like the mergers that caused jumps in the compact half - mass radius in the first and second examples on the left . all of the consumed " galaxies , whose stellar contents have been scattered far and wide , are therefore located on the far right of this panel . a high @xmath52-value indicates the presence of significant ex - situ stellar mass at large radii , so that the core "- type descendants separate from the undisturbed " descendants despite having similar @xmath51-values in many cases . in general , the growth in compact half - mass radius is much smaller than that of the overall half - mass radius . this implies that a majority of the size growth is due to the addition of stellar mass in the outer parts of the galaxies . galaxies which gained little stellar mass typically grow by about 50% or less in compact half - mass radius , while those which experienced some merger events are slightly more perturbed , and those which experienced majorly disruptive mergers grow even more . as we have seen in the previous subsection , nearly all of the compact galaxies grow in size between @xmath4 and @xmath5 even if they experience little growth in mass . of the original 35 compact galaxies , only two remain below the compactness threshold at @xmath5 . shifting the threshold up slightly to account for the 15% change in half - mass radius from numerical effects yields one additional galaxy , for a final survival fraction " of 9% for compact galaxies from @xmath4 to the present day . ( and of these three , one is an extreme case which dives through a massive halo , is stripped of its stellar envelope along the way , and is just escaping out the other side at @xmath5 , having managed to _ decrease _ in both size and mass ! ) the number density of compact galaxies depends not only on the rate at which galaxies _ leave _ the compact population , but also the rate at which they _ enter_. our previous study @xcite found mechanisms for producing compact galaxies which are most efficient in the high - redshift universe when cold , dense gas is abundant . as time increases and the universe becomes more rarefied , we expect that the production rate of compact galaxies should also decrease . in section [ ssec : cores ] , we identified all the galaxies in the simulation which had a massive ( @xmath68 m@xmath1 ) , compact progenitor at any redshift . for each of these galaxies , we have measured the median redshift of the compact phase as well as its total duration . the median redshifts of the compact phases are broadly distributed from @xmath61 , falling off at higher and lower @xmath18 and implying that the compact formation mechanisms indeed no longer operate as efficiently at low redshift . half of the compact phases lasted less than 2 gyr , and only 10% were longer than 5 gyr , indicating that we should expect few massive , compact galaxies to remain in the local universe . both of these exercises ( choosing compact galaxies at @xmath4 and tracing them forward , and finding ex - compact " galaxies at @xmath5 and examining the time and duration of their compact phases ) indicate that the absolute number density of high - mass compact galaxies should decrease at low redshift . physically , this is a reflection of the growing dominance of dissipationless processes ( e.g. dry mergers ) at late times over processes such as disk instability or wet mergers , more common at high redshift , which provide a way to dissipate angular momentum and thus produce more compact stellar populations . studies of compact massive high - redshift galaxies adopt varying initial selection methods , including differences in the compactness criterion ( e.g. the slope and normalization of the power - law cut across the mass - radius plane ) , minimum stellar mass , quiescence ( e.g. cuts in ssfr , uvj , or morphology ) , size estimation ( e.g. major - axis vs. circularized half - light radius , bulge - disk decomposition vs. single - sersic fits ) , and parent surveys ( e.g. sdss , candels , cosmos ) . these differences can make direct comparisons between observational surveys that infer the the mass or size evolution of massive compact galaxies difficult . in this paper , we have primarily focused on the _ physical _ mass and size evolution of compact massive galaxies as realized in a cosmological simulation , which introduces an additional level of complexity when making comparisons . in this subsection , we discuss some of the specific definitions used in this paper and their impact on our conclusions , which may help the reader to place these results in the context of other work , particularly observational work . one question which plagues both simulators and observers is , what is the most meaningful way to talk about size ? in this paper we have opted to use the stellar half - mass radius unambiguously defined as the radius of the sphere containing half of the stellar mass which is gravitationally bound to this object " as our fiducial measurement of size . using the stellar half - mass radius as a proxy for galaxy size has the disadvantages that it is not a directly observable quantity , and that it can be perturbed by the addition of stellar mass at some larger radius when the density profile is steep . however , the half - mass radius has clear physical meaning , can be easily and robustly calculated directly from subfind data , and is related to the circularized half - light radius ( modulo variations in the mass - to - light ratios for differently - aged stellar populations ) . in principle we could have adopted galaxy size estimates by fitting profiles to the stellar light distributions of every galaxy in the simulation and measuring an effective or scale radius ( e.g. , as done in * ? ? ? however , given the complexity and variety of the simulation data the resulting radii would likely be sensitive to several details of the fitting procedure and would likely not yield characteristically different trends from those found and presented in this paper . nevertheless , it could be worthwhile to consider in the future a more even - handed comparison of galaxy size evolution for compact massive galaxies . another choice we have made in the above analysis is a particular selection of compactness threshold value , @xmath29 . we mentioned in section [ sec : general ] that this selection is more severe than that employed by e.g. @xcite . how sensitive are the results to this particular choice of compactness " ? to explore this question , we repeated the analysis with a more comparable threshold of @xmath69 , which yielded a sample of 103 galaxies in the same @xmath11 m@xmath1 mass range . these galaxies are more likely to be star - forming than the original sample of 35 ( only 42% of them have specific star formation rates lower than @xmath70/yr , as opposed to 66% in the original sample ) , so more of them quickly leave the compact regime due to star formation at large radii . when this population is traced forward , we find that they are more likely than the original sample to have descendants in the mixed " or consumed " groups , since they are larger ( less tightly bound ) and hence easier to disrupt via merger . overall , however , we find that the same trends hold as have been described above , and therefore it seems that the qualitative conclusions drawn about the diversity of growth paths for massive compact galaxies is not highly sensitive to how they are initially selected . a more implicit set of assumptions adopted in this paper is the physical / feedback model employed by the illustris simulation . the masses and sizes of the simulated galaxy populations examined in this paper are specifically subject to influence from the finite mass resolution and spatial softening , pressurization of the ism , and agn feedback model . we discuss in some detail the potential effects of gravitational softening ( and its redshift evolution ) in the appendix of this paper . the gravitational softening and ism pressurization adopted in this paper can potentially prevent galaxies from fully collapsing during dissipational events such as gas - rich galaxy mergers . in general , both gravitational softening and our method of ism pressurization risk overestimating the sizes of galaxies and it is therefore possible that some galaxies that would at one time be compact would be missed . thus , our conclusions about the fractions of galaxies which pass through a compact phase might best be viewed as a lower limit . however , the basic convergence study presented in @xcite indicated that the sizes of galaxies are well - converged at the @xmath4 gravitational softening level , so the results should not dramatically change . our ability to further investigate the impact of these simulation traits on our results is limited since we can not easily re - run the illustris simulation volume . reconsidering the conclusions presented in this paper with other simulations ( e.g. eagle , * ? ? ? * ; * ? ? ? * ) or with zoom - in re - simulations of the specific objects studied in this paper or objects selected from other large full - volume simulations may provide additional context for these results in the future . finally , we would like to emphasize that the results herein are primarily focused on very massive galaxies with stellar masses of at least @xmath7 m@xmath1 at @xmath4 , and that caution is warranted when comparing to results in other mass regimes where different physical processes may govern the evolution .

the galaxies grow in size as well as mass , and only% remain compact by . the majority of the size growth is driven by the acquisition of ex - situ mass . the compact galaxies merger rates are influenced by their environments , so that isolated or satellite compact galaxies ( which are protected from mergers ) are the most likely to survive to the present day .

massive quiescent galaxies have much smaller physical sizes at high redshift than today . the strong evolution of galaxy size may be caused by progenitor bias , major and minor mergers , adiabatic expansion , and/or renewed star formation , but it is difficult to test these theories observationally . herein , we select a sample of 35 massive , compact galaxies ( m , m/kpc ) at in the cosmological hydrodynamical simulation illustris and trace them forward to to uncover their evolution and identify their descendants . by , the original factor of 3 difference in stellar mass spreads to a factor of 20 . the dark matter halo masses similarly spread from a factor of 5 to 40 . the galaxies evolutionary paths are diverse : about half acquire an ex - situ envelope and are the core of a more massive descendant , a third survive undisturbed and gain very little mass , 15% are consumed in a merger with a more massive galaxy , and a small remainder are thoroughly mixed by major mergers . the galaxies grow in size as well as mass , and only% remain compact by . the majority of the size growth is driven by the acquisition of ex - situ mass . the most massive galaxies at are the most likely to have compact progenitors , but this trend possesses significant dispersion which precludes a direct linkage to compact galaxies at . the compact galaxies merger rates are influenced by their environments , so that isolated or satellite compact galaxies ( which are protected from mergers ) are the most likely to survive to the present day . [ firstpage ] galaxies : high - redshift , galaxies : evolution

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herein , we have identified a population of 35 massive ( @xmath11 m@xmath1 ) galaxies at @xmath4 in the cosmological hydrodynamical simulation illustris which satisfy the compactness criterion @xmath71 . we have traced these massive , compact galaxies forward in time through the simulation to their @xmath5 descendants at the present day . our most important findings are summarized below : * though the compact galaxies all originated at @xmath4 with the same stellar mass to within a factor of 3 , by @xmath5 their stellar masses differ up to a factor of 20 . similarly , their dm halo masses , originally within a factor of 5 , spread to a factor of 40 . rather than remaining tightly grouped , the descendants stellar masses are dispersed throughout the entire massive galaxy population at @xmath5 . * the compact galaxies can undertake a variety of evolutionary paths to @xmath5 . approximately half survive as the core of a more massive descendant , about 15% are consumed and destroyed in a merger with a much more massive galaxy , about a third are generally undisturbed by mergers , accretion , or star formation activity , and the remaining few are thoroughly mixed by major mergers . these percentages are a weak function of initial stellar mass , with less massive galaxies more likely to be consumed and more massive galaxies more likely to be cores . * very few ( about 10% ) of the galaxies from the @xmath4 compact sample still satisfy our compactness criterion at @xmath5 , suggesting a decrease in the number density of massive , compact galaxies at low redshift in the absence of any replenishing compact formation mechanism . * a compact galaxy s @xmath4 environment has some predictive power for its final evolutionary outcome . compact galaxies in denser environments ( with the exception of satellite galaxies ) are more likely to undergo mergers and gain stellar mass at larger radii , losing their compact identity in the process . those which are isolated at @xmath4 and experience little merger activity are also preferentially isolated at @xmath5 . thus we predict that massive , compact relics in the local universe should predominantly live in underdense environments or as satellites in larger groups . * a galaxy s @xmath5 stellar mass is related to the compactness of its progenitors , such that more massive galaxies are more likely to have once been compact . however , this trend possesses significant dispersion ( partially driven by a spread in formation times ) and it is _ not _ the case that the most massive galaxies at @xmath5 can be directly linked to massive compact galaxies at @xmath4 . while more than half the galaxies with stellar mass @xmath72 m@xmath1 at @xmath5 have a direct compact progenitor at @xmath73 , only 15% have a compact progenitor at exactly @xmath4 . * the dominant source of size growth for the compact galaxies is the addition of ex - situ stellar mass at larger radii through mergers and accretion into the stellar halo , followed by disruptive major mergers , with smaller contributions from ongoing or renewed star formation , adiabatic expansion due to mass loss , and/or numerical effects .

the strong evolution of galaxy size may be caused by progenitor bias , major and minor mergers , adiabatic expansion , and/or renewed star formation , but it is difficult to test these theories observationally . by , the original factor of 3 difference in stellar mass spreads to a factor of 20 . the dark matter halo masses similarly spread from a factor of 5 to 40 . the galaxies evolutionary paths are diverse : about half acquire an ex - situ envelope and are the core of a more massive descendant , a third survive undisturbed and gain very little mass , 15% are consumed in a merger with a more massive galaxy , and a small remainder are thoroughly mixed by major mergers . the most massive galaxies at are the most likely to have compact progenitors , but this trend possesses significant dispersion which precludes a direct linkage to compact galaxies at .

massive quiescent galaxies have much smaller physical sizes at high redshift than today . the strong evolution of galaxy size may be caused by progenitor bias , major and minor mergers , adiabatic expansion , and/or renewed star formation , but it is difficult to test these theories observationally . herein , we select a sample of 35 massive , compact galaxies ( m , m/kpc ) at in the cosmological hydrodynamical simulation illustris and trace them forward to to uncover their evolution and identify their descendants . by , the original factor of 3 difference in stellar mass spreads to a factor of 20 . the dark matter halo masses similarly spread from a factor of 5 to 40 . the galaxies evolutionary paths are diverse : about half acquire an ex - situ envelope and are the core of a more massive descendant , a third survive undisturbed and gain very little mass , 15% are consumed in a merger with a more massive galaxy , and a small remainder are thoroughly mixed by major mergers . the galaxies grow in size as well as mass , and only% remain compact by . the majority of the size growth is driven by the acquisition of ex - situ mass . the most massive galaxies at are the most likely to have compact progenitors , but this trend possesses significant dispersion which precludes a direct linkage to compact galaxies at . the compact galaxies merger rates are influenced by their environments , so that isolated or satellite compact galaxies ( which are protected from mergers ) are the most likely to survive to the present day . [ firstpage ] galaxies : high - redshift , galaxies : evolution

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understanding the formation and pre - main - sequence evolution of massive stars ( @xmath2 m@xmath3 ) is an important goal in modern astrophysics . currently , the formation of low - mass stars via gravitational collapse of a molecular cloud and subsequent disk accretion is , at least conceptually , well understood but it is not clear whether massive stars form through a scaled - up version of the same process or whether more complex , environmental effects are important ( see e.g. bonnell , vine & bate 2004 for a recent discussion ) . since massive stars form inside dense molecular clouds , they are observed primarily at infrared ( ir ) and longer wavelengths . embedded massive ysos are powerful sources of ir h i line emission ( simon et al . 1981 , 1983 ; drew , bunn & hoare 1993 ; bunn , hoare & drew 1995 ; blum et al . 2004 ) with equivalent widths ( ews ) in br@xmath0 of @xmath4 90 ( bunn et al . the lines profiles are complex , displaying both fairly narrow ( full - width - at - half - maximum [ fwhm ] @xmath5 50 100 km s@xmath6 ) line cores and broad wings extending out to @xmath7 km s@xmath6 in extreme cases . in the sample of objects observed by bunn et al . ( 1995 ) , the profiles were always single - peaked but recent observations reported by blum et al . ( 2004 ) include at least one object ( ngc 3576 ) which shows clear evidence of double - peaked profiles . by examining flux ratios across the profiles of lines with differing opacities , bunn et al . ( 1995 ) found evidence that , at least in some sources , the lines form in an accelerating outflow . normal , main - sequence ob stars also show h i emission lines . these are produced in the star s fast ( @xmath8 km s@xmath6 ) spherical wind , but the strength of emission in yso spectra is much greater than in o - stars . if the features in ysos were due to reprocessing of radiation in a spherically symmetric wind , the implied mass - loss rates would be up to @xmath9 m@xmath3 yr@xmath6 ( simon et al . 1983 ) , substantially exceeding those of field stars of comparable spectral type . therefore , to interpret the h i observations it is worthwhile to consider alternatives to formation in a spherically symmetric structure . important observational constraints on the geometry of emitting material in ysos are provided by the first overtone bands of co at 2.3@xmath10 m ( e.g. carr 1989 ; carr et al . 1993 ; chandler et al . 1993 ; chandler , carlstrom & scoville 1995 ) . in particular , both carr et al . ( 1993 ) and chandler et al . ( 1995 ) conclude that models based on an accretion disk surrounding the central object reproduce the co observations of ysos . furthermore , although there are some cases in which co data can be modelled in terms of a stellar wind , in general the disk model encounters fewer difficulties ( see chandler et al . the observations strongly suggest that the co emission originates close to the central star : bik & thi ( 2004 ) and blum et al . ( 2004 ) have recently studied co emission from a range of massive ysos and derived disk radii in the range 0.1 5 au . thus it seems probable that ysos harbour accretion disks containing significant amounts of hot gas fairly close to the central object and it is natural to consider whether the h i emission is associated with such a disk or its environment . hamann & simon ( 1986 ) suggested that powerful recombination line emission of the sort discussed above originates in an outflow associated not with a stellar wind but with mass - loss from a disk . their work focused on mwc 349 , an object whose evolutionary status is unclear but which possesses spectroscopic signatures ( including h i and co emission features ) broadly similar to the embedded massive ysos ( see hamann & simon 1986 ; kraus et al . hollenbach et al . ( 1994 ) developed a model for outflow from young massive stars by photoevaporation of the outer regions of their disks . they were motivated by the need to explain the high frequency of occurrence of ultracompact h ii regions photoevaporation can provide a source of mass input for h ii regions to balance the loss by pressure - driven expansion . although the rate of mass - loss in photoevaporation models is large , it occurs at large radii ( typically @xmath11 au ) and the characteristic flow velocities are low ( comparable to the typical sound speed of 10 50 km s@xmath6 , hollenbach et al . 1994 ) . thus , such a flow does not provide a promising origin for relatively high velocity features , such as the ir h i broad line wings ( bunn et al . 1995 ) . recently , drew , proga & stone ( 1998 ) proposed that the intense radiation field produced by a massive yso may drive mass - loss from the surface of the inner parts of its disk . they performed a hydrodynamical simulation which showed that , in addition to driving a normal hot star wind component , radiation pressure from a massive yso could propel a dense equatorial flow from a surrounding disk with terminal velocity in the same regime as the observed h i linewidths . these results led them to speculate that such a model accounts for the ir h i lines . however , they did not perform the radiative transfer calculations required to confirm this conjecture . to investigate the possibility that massive ysos harbour disk winds which give rise to the observed ir h i line emission , this paper presents the results of new two - dimensional radiative transfer calculations which account for both the complex geometry of a yso disk wind ( starting from , but not limited to , the drew et al . 1998 model ) and the detailed atomic physics of h i line formation . in section 2 , the yso model adopted in this investigation and its parameters are discussed . the radiative transfer calculations have been performed using the monte carlo ( mc ) code described by long & knigge ( 2002 ) after incorporating a sophisticated treatment of h i line formation using the approach described by lucy ( 2002,2003 ) ; the method of calculation and code used are discussed in section 3 . results are presented in section 4 and our conclusions discussed in section 5 .

massive young stellar objects ( ysos ) are powerful infrared h i line emitters . it has been suggested that these lines form in a outflow from a disk surrounding the yso . here , new two - dimensional monte carlo radiative transfer calculations infrared spectra are synthesised for a yso disk wind model based on earlier hydrodynamical calculations .

massive young stellar objects ( ysos ) are powerful infrared h i line emitters . it has been suggested that these lines form in a outflow from a disk surrounding the yso . here , new two - dimensional monte carlo radiative transfer calculations are described which test this hypothesis . infrared spectra are synthesised for a yso disk wind model based on earlier hydrodynamical calculations . the model spectra are in qualitative agreement with the observed spectra from massive ysos , and therefore provide support for a disk wind explanation for the h i lines . however , there are some significant differences : the models tend to overpredict the br/br ratio of equivalent - widths and produce line profiles which are slightly too broad and , in contrast to typical observations , are double - peaked . the interpretation of these differences within the context of the disk wind picture and suggestions for their resolution via modifications to the assumed disk and outflow structure are discussed . [ firstpage ] radiative transfer methods : numerical stars : early - type stars : winds , outflows

1605.04086

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maxwell s equations on the whole three - dimensional space are considered with initial conditions and inhomogeneity having support in a bounded domain that is not required to be convex ( or in a finite collection of such domains ) . the study of such problems leads to transparent boundary conditions , which yield the restriction of the solution to the domain . such boundary conditions are nonlocal in space and time , for both acoustic wave equations and maxwell s equations . there is a vast literature to tackle this problem in general for wave equations : fast algorithms for exact , nonlocal boundary conditions on a ball @xcite , local absorbing boundary conditions @xcite , perfectly matched layers , which were originally considered for electromagnetism in @xcite , and numerical coupling with time - dependent boundary integral operators @xcite . all the above approaches , except the last one , are inadequate for non - convex domains . the local methods fail because waves may leave and re - enter a non - convex domain . inclusion of a non - convex domain in a larger convex domain is computationally undesirable in situations such as a cavity or an antenna - like structure or a far - spread non - connected collection of small domains . the main objective of the present work is to transfer the programme of @xcite from acoustic wave equations to maxwell s equations : to propose and analyze a provably stable and convergent fully discrete numerical method that couples discretizations in the interior and on the boundary , without requiring convexity of the domain . like abboud _ et al . _ @xcite ( and later also @xcite ) for the acoustic wave equation , we start from a symmetrized weak first - order formulation of maxwell s equations . in the interior this is discretized by a discontinuous galerkin ( dg ) method in space @xcite together with the explicit leapfrog scheme in time @xcite . the boundary integral terms are discretized by standard boundary element methods in space and by convolution quadrature ( cq ) in time @xcite . this yields a coupled method that is explicit in the interior and implicit on the boundary . the choice of a cq time discretization of the boundary integral operators is essential for our analysis , and to a lesser extent also the choice of the leapfrog scheme in the interior . however , our approach is not specific to the chosen space discretizations which could , in particular , be replaced by conformal edge elements @xcite . while the general approach of this paper is clearly based on @xcite , it should be emphasized that the appropriate boundary integral formulation requires a careful study of the time - harmonic maxwell s equation . this is based on @xcite , with special attention to the appropriate trace space on the boundary and to the corresponding duality . due to the analogue of green s formula for maxwell s equations , the duality naturally turns out to be an anti - symmetric pairing . the calderon operator for maxwell s equation , which arises in the boundary integral equation formulation of the transparent boundary conditions , differs from the acoustic case to a large extent , and therefore the study of its coercivity property is an important and nontrivial point . similarly to the acoustic case , the continuous - time and discrete - time coercivity is obtained from the laplace - domain coercivity using an operator - valued version , given in @xcite , of the classical herglotz theorem @xcite . both the second and first order formulation of maxwell s equations are used . the spatial semi - discretization of the symmetrized weak first - order formulation of maxwell s equations has formally the same matrix vector formulation as for the acoustic wave equation studied in @xcite , with the same coercivity property of the calderon operator . because of this structural similarity , the stability results of @xcite , which are shown using the matrix vector setting , remain valid for the maxwell case without any modification . on the other hand , their translation to the functional analytic setting differs to a great extent . therefore further care is required in the consistency analysis . in section [ section : recap helmholtz ] we recapitulate the basic theory for maxwell s equation in the laplace domain . based on buffa and hiptmair @xcite , and further on @xcite , we describe the right boundary space , which allows for a rigorous boundary integral formulation for maxwell s equations . then the boundary integral operators are obtained in a usual way from the single and double layer potentials . in section [ section : calderon ] we prove the crucial technical result of the present work , a coercivity property of the calderon operator for maxwell s equation in the laplace domain . this property translates to the continuous - time maxwell s equations later , in section [ subsection : calderon op for maxwell s eqn ] , via an operator - valued herglotz theorem . in section [ section : boundary int form ] we study the interior exterior coupling of maxwell s equations , resulting in an interior problem coupled to an equation on the boundary with the calderon operator . we derive a first order symmetric weak formulation , which is the maxwell analogue of the formulation of @xcite for the acoustic wave equation . together with the continuous - time version of the coercivity property of the calderon operator , this formulation allows us to derive an energy estimate . later on this analysis is transfered to the semi - discrete and fully discrete settings . section [ section : discretization ] presents the details of the discretization methods : in space we use discontinuous galerkin finite elements with centered fluxes in the domain @xcite , coupled to continuous linear boundary elements on the surface . time discretization is done by the leapfrog scheme in the interior domain , while on the boundary we use convolution quadrature based on the second - order backward differentiation formula . an extra term stabilizes the coupling , just as for the acoustic wave equation @xcite . the matrix vector formulation of the semidiscrete problem has the same anti - symmetric structure and the same coercivity property as for the acoustic wave equation , and therefore the stability results shown in @xcite can be reused here . in sections [ section : semidiscrete results ] and [ section : fully discrete results ] we revise the parts of the results and proofs of @xcite where they differ from the acoustic case , which is mainly in the estimate of the consistency error . finally , we arrive at the convergence error bounds for the semi- and full discretizations . to our knowledge , the proposed numerical discretizations in this paper are the first provably stable and convergent semi- and full discretizations to interior exterior coupling of maxwell s equations . we believe that the presented analysis and the techniques , which we share with @xcite , can be extended further : to other discretization techniques for the domain , such as edge element methods @xcite , higher order discontinuous galerkin methods , and different time discretizations in the domain , together with higher order runge kutta based convolution quadratures on the boundary @xcite . for ease of presentation we consider only constant permeability and permittivity . however , it is only important that the permeability and permittivity are constant in the exterior domain and in a neighbourhood of the boundary . in the interior these coefficients may be space - dependent and discontinuous . in the latter case the equations can be discretized in space with the dg method as described in @xcite . in this paper we focus on the appropriate boundary integral formulation and on the numerical analysis of the proposed numerical methods . numerical experiments are intended to be presented in subsequent work . concerning notation , we use the convention that vectors in @xmath0 are denoted by italic letters ( such as @xmath1 ) , whereas the corresponding boldface letters are used for finite element nodal vectors in @xmath2 , where @xmath3 is the ( large ) number of discretization nodes . hence , any boldface letters appearing in this paper refer to the matrix vector formulation of spatially discretized equations . functions defined in the domain @xmath4 are denoted by letters from the roman alphabet , while functions defined on the boundary @xmath5 are denoted by greek letters .

maxwell s equations are considered with transparent boundary conditions , for initial conditions and inhomogeneity having support in a bounded , not necessarily convex three - dimensional domain or in a collection of such domains . the numerical method only involves the interior domain and its boundary . the transparent boundary conditions are imposed via a time - dependent boundary integral operator that is shown to satisfy a coercivity property . the stability of the numerical method relies on this coercivity and on an anti - symmetric structure of the discretized equations that is inherited from a weak first - order formulation of the continuous equations . the method is explicit in the interior and implicit on the boundary .

maxwell s equations are considered with transparent boundary conditions , for initial conditions and inhomogeneity having support in a bounded , not necessarily convex three - dimensional domain or in a collection of such domains . the numerical method only involves the interior domain and its boundary . the transparent boundary conditions are imposed via a time - dependent boundary integral operator that is shown to satisfy a coercivity property . the stability of the numerical method relies on this coercivity and on an anti - symmetric structure of the discretized equations that is inherited from a weak first - order formulation of the continuous equations . the method proposed here uses a discontinuous galerkin method and the leapfrog scheme in the interior and is coupled to boundary elements and convolution quadrature on the boundary . the method is explicit in the interior and implicit on the boundary . stability and convergence of the spatial semidiscretization are proven , and with a computationally simple stabilization term , this is also shown for the full discretization .

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the agt correspondence @xcite implies a relation between the canonical partition function of a and the grand canonical partition function of a generalized matrix model . the former represents a correlator of liouville theory , according to the proposal of dijkgraaf and vafa @xcite , further investigated in @xcite . the latter describes the instanton partition function of a 4d @xmath0 supersymmetric gauge theory in the @xmath1-background , as derived using localization techniques in @xcite . here the term generalized matrix model do not pertain to a matrix origin for the model , but instead refers to a set of models that can be studied using techniques initially developed in the realm of matrix models . among these techniques , the _ topological recursion _ @xcite exploits the invariance of the integration measure to derive a tower of nested equations satisfied by the correlators of the model . these equations , referred as _ loop equations _ , are solved employing methods from algebraic geometry . this technique has recently been extended to a wide spectrum of coupled integrals models in @xcite . in a suitable limit of the , agt - equivalent to the nekrasov - shatashvili ( ns ) limit of the @xmath1-background @xcite , loop equations are no longer algebraic but first order linear differential equations . in this context , the is a natural quantization of the hermitian matrix model , to which it reduces at @xmath2 . the first element of this tower of differential equations has been mapped to the tq relation derived in @xcite that describes the dual susy gauge theory in the ns limit @xcite . it is then natural to ask about the existence of a structure similar to loop equations on the gauge side of the correspondence . but so far , the loop equation technique has not been applied to grand canonical matrix models . on the other hand , the cluster expansion of mayer and montroll @xcite has been successfully employed to derived an effective action relevant to the ns limit @xcite . can we relate this cluster expansion to the topological expansion of a generalized matrix model ? is there an equivalent of the loop equations technique on the grand canonical side ? and more generally , how do canonical and grand canonical coupled integrals relate to each other ? these are the issues we propose to address in these notes . for this purpose , we consider the following grand canonical generalized matrix model , @xmath3 in analogy with the nekrasov partition function , integrals are understood as contour integrals over the real line . the potential @xmath4 and the kernel @xmath5 are free of singularities over the real axis . we propose to study the expansion of @xmath6 when the kernel is close to one . more precisely , we assume the form @xmath7 with @xmath8 an even function , non - vanishing at @xmath9 . although the results of these notes are very general , what we have in mind for the function @xmath8 is typically @xmath10 it is crucial for our considerations that @xmath8 is independent of @xmath11 . in this way , we exclude a class of models more relevant to the study of nekrasov partition functions . for instance , setting @xmath12 , one recovers the model proposed by j. hoppe in @xcite . this model is a one - parameter version of the nekrasov partition function that depends on two @xmath13-background equivariant deformation parameters @xmath14 and @xmath15 @xcite . as @xmath16 , it exhibits a phenomenon referred as _ instanton clustering _ in the context of susy gauge theories @xcite . it corresponds to poles coming from the kernel and pinching the integration contour . such poles should be avoided by a deformation of the contour , picking up the corresponding residues . as a result , terms of the @xmath11-expansions we are considering are reshuffled and the results presented here are no longer valid . these notes are organized as follows . in the second section , we compare the mayer cluster expansion of the grand canonical model with the collective field theory describing the large @xmath17 limit of the canonical model . taking the coupled limit @xmath16 and @xmath18 with @xmath19 fixed , we derive relations between the free energies at first orders . these relations are a consequence of the fact that the grand canonical partition function is the discrete laplace transform of the canonical one . we go on with the study of the canonical loop equations . we show that they relate to graphical identities between generating functions of rooted clusters . such generating functions show up in the mayer expansion and are identified with the multi - point densities . finally , we present a technique to derive directly the grand canonical loop equations . the main results are summarized in the concluding section .

it has recently been applied to the nekrasov instanton partition function of 4d gauge theories . the associated canonical model involves coupled integrations that take the form of a generalized matrix model . it can be studied with the standard techniques of matrix models , in particular collective field theory and loop equations . in the first part of these notes ,

mayer cluster expansion is an important tool in statistical physics to evaluate grand canonical partition functions . it has recently been applied to the nekrasov instanton partition function of 4d gauge theories . the associated canonical model involves coupled integrations that take the form of a generalized matrix model . it can be studied with the standard techniques of matrix models , in particular collective field theory and loop equations . in the first part of these notes , we explain how the results of collective field theory can be derived from the cluster expansion . the equalities between free energies at first orders is explained by the discrete laplace transform relating canonical and grand canonical models . in a second part , we study the canonical loop equations and associate them to similar relations on the grand canonical side . it leads to relate the multi - point densities , fundamental objects of the matrix model , to the generating functions of multi - rooted clusters . finally , a method is proposed to derive loop equations directly on the grand canonical model . apctp pre2013 - 017 notes on mayer expansions and matrix models jean - emile bourgine + + _ pohang , gyeongbuk 790 - 784 , republic of korea _

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in these notes we compared the cluster expansion of a grand canonical model with the standard matrix model treatment of its canonical partition function . at tree level , the grand canonical free energy is given by the minimum of an effective action which is identical to the one provided by the collective field theory approach applied to the canonical model . the correspondence extends to the level of one - loop corrections , where the sum over one - cycle clusters reproduces the expansion of the fredholm determinant computed from the integration over gaussian fluctuations in the collective field theory . the matching of free energies can be explained by the discrete laplace transform relating canonical and grand canonical models . introducing a source term , we were also able to find the relation satisfied by the one - point and two points densities . we continued with the study of canonical loop equations , and realized that a similar set of equations can be derived from the cluster expansions . instead of @xmath137-point connected densities , these equations involve the generating functions of @xmath137-rooted clusters , denoted @xmath177 . using these equations , and the relation between @xmath177-functions and densities derived earlier , we verified that grand canonical densities also obey the canonical loop equations . it implies that the whole loop equation structure is present in the cluster expansion , and takes the form of graphical relations among clusters . finally , we proposed a method to derive directly this set of loop equations within the grand canonical model . our study is restricted to the first two orders in the large @xmath17 and small @xmath11 expansion . the generalization to higher orders still needs to be done , and the general form of loop equations to be worked out . once the full set of equations identified , it may be possible to apply the topological recursion to the grand canonical model . another important point that remains is the description of instantons clustering relevant to the instanton partition function of @xmath0 susy gauge theories . the dual description grand - canonical / canonical may allow a better understanding of this phenomenon . a possible application for this work could be the derivation of the subleading correction in @xmath15 to the partition function , and the investigation of its presumed integrable properties . in this scope , it is tempting to assume that the instanton clustering in sym is entirely described by the effective action [ action_ns ] , and conjecture that the subleading order is given by the associated determinant , @xmath255}}e^{\frac12\int{dx\log\left(e^{\rho(x)}-1\right)}-\frac12\int{dx\log\rho(x)}}\\ & \times\exp\left(\hf g(0)\int{dx(1-e^{\rho(x)})}-\dfrac14\int{dxdy(1-e^{\rho(x)})(1-e^{\rho(y)})g(x - y)^2}\right ) . \end{split}\end{aligned}\ ] ] this proposal is very naive , but it could be tested using the agt correspondence with the representation of liouville correlators . the grand canonical model we studied has a very specific form of interaction and may not be relevant to statistical systems . it would be interesting to consider more physical models . one may also wonder if the topological reduction employed in this context have a matrix model analogue . nevertheless , the results presented here are very general and could be relevant for a large spectrum of problems . they have deep connections with integrable models and the tba equation @xcite . they play a role in the computation of light - like wilson loops at strong coupling in @xmath256 sym @xcite . they may also be applied to the study of 3-points function of scalar operators in this theory @xcite .

we explain how the results of collective field theory can be derived from the cluster expansion . the equalities between free energies at first orders is explained by the discrete laplace transform relating canonical and grand canonical models . in a second part , we study the canonical loop equations and associate them to similar relations on the grand canonical side . finally , a method is proposed to derive loop equations directly on the grand canonical model . apctp pre2013 - 017 notes on mayer expansions and matrix models jean - emile bourgine + + _ pohang , gyeongbuk 790 - 784 , republic of korea _

mayer cluster expansion is an important tool in statistical physics to evaluate grand canonical partition functions . it has recently been applied to the nekrasov instanton partition function of 4d gauge theories . the associated canonical model involves coupled integrations that take the form of a generalized matrix model . it can be studied with the standard techniques of matrix models , in particular collective field theory and loop equations . in the first part of these notes , we explain how the results of collective field theory can be derived from the cluster expansion . the equalities between free energies at first orders is explained by the discrete laplace transform relating canonical and grand canonical models . in a second part , we study the canonical loop equations and associate them to similar relations on the grand canonical side . it leads to relate the multi - point densities , fundamental objects of the matrix model , to the generating functions of multi - rooted clusters . finally , a method is proposed to derive loop equations directly on the grand canonical model . apctp pre2013 - 017 notes on mayer expansions and matrix models jean - emile bourgine + + _ pohang , gyeongbuk 790 - 784 , republic of korea _

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the gauge / gravity correspondence is a fascinating topic in the study of string theory . a certain class of it preserving a conformal symmetry , in which a string theory is defined on an anti - de sitter ( ads ) space and its dual is a conformal field theory ( cft ) , is called the ads / cft correspondence @xcite . a prototypical example is a duality between type iib string theory on the ads@xmath0s@xmath1 background and the four - dimensional @xmath4 @xmath5 super yang - mills theory in the large @xmath6 limit . remarkably , the integrability structure exists behind it and hence one can check the conjectured relations in a rigorous way by employing various integrability techniques ( for a big review , see @xcite ) . in particular , the ads@xmath0s@xmath1 superstring is classically integrable in the sense of kinematical integrability @xcite . the integrability exposed in this case is , however , rarely exceptional and such a good property is not equipped with general examples of the ads / cft ( or gauge / gravity ) correspondence . holographic interpretations in the gauge / gravity correspondence are usually concerned with curved string backgrounds and hence classical motions of a string are described by non - linear equations . but most of the non - linear equations are not integrable and therefore the behavior of classical string solutions should become chaotic . more intriguingly , the holographic counterpart of chaotic strings should exist on the gauge - theory side as well , but it has not been clarified yet . if it has been done , then one could open up a new frontier in the study of the gauge / gravity correspondence . a well - studied example of non - integrable backgrounds is ads@xmath7 @xcite , where @xmath8 is a five - dimensional sasaki - einstein space @xcite has been completed with a supertrace operation @xcite . ] . chaotic string solutions were found in @xcite by computing poincar sections . the chaotic behavior remains even in a near - penrose limit @xcite . similar studies have been done for many backgrounds in the preceding works @xcite . similarly , chaotic motions of d0-branes can also be studied as well as classical strings . the d0-brane dynamics is described by a matrix model proposed by banks , fischler , shenker and susskind ( bfss ) @xcite . chaotic d0-branes in the bfss matrix model were studied in @xcite by following a seminal paper on chaos in a classical yang - mills theory @xcite . in comparison to the bfss case , a matrix model on a pp - wave background , which was proposed by berenstein , maldacena and nastase ( bmn ) @xcite , has a strong advantage that there is no flat direction and all of the trajectories are definitely bounded . classical chaos in the bmn matrix model was shown in @xcite by following @xcite . the chaos at finite temperature was also studied in @xcite in relation to the fast scrambler scenario @xcite . a very recent work @xcite investigated chaotic motions of chiral condensates in a holographic qcd setup @xcite and displayed the dependence of lyapunov exponent on the rank of gauge group @xmath6 and t hooft coupling @xmath9 . a lot of achievements have been obtained for the chaotic behavior of strings and d - branes as introduced above . motivated by this progress , we are concerned here with an application of melnikov s method @xcite in the context of the gauge / gravity correspondence . this method can show analytically the existence of chaos generated by a smale horseshoe , though its applicability is somewhat restricted . in this paper , we will present a string background to which melnikov s method is applicable . this is a brane - wave type deformation of ads@xmath0s@xmath1 presented in @xcite . by employing two reduction anstze , we study two types of coupled pendulum - oscillator systems . then the melnikov function is computed for each of the systems in the standard way of holmes and marsden @xcite . this paper is organized as follows . section 2 gives a concise review of melnikov s method and provides simple examples . in section 3 , we introduce a brane - wave type deformation of ads@xmath0s@xmath1 , and study two anstze . the associated melnikov function is computed for each of the cases . section 4 is devoted to conclusion and discussion . in appendix [ elliptic functions ] , the definitions and useful identities of elliptic functions are summarized . appendix [ apx : calc ] explains a detailed computation of melnikov function for a spinning string ansatz .

it is a powerful technique , though its applicability is somewhat limited . in this paper , we present a solution of type iib supergravity to which melnikov s method is applicable . , we study two types of coupled pendulum - oscillator systems . then the melnikov function is computed for each of the systems by following the standard way of holmes and marsden and the existence of chaos is shown analytically .

melnikov s method is an analytical way to show the existence of classical chaos generated by a smale horseshoe . it is a powerful technique , though its applicability is somewhat limited . in this paper , we present a solution of type iib supergravity to which melnikov s method is applicable . this is a brane - wave type deformation of the adss background . by employing two reduction anstze , we study two types of coupled pendulum - oscillator systems . then the melnikov function is computed for each of the systems by following the standard way of holmes and marsden and the existence of chaos is shown analytically . * melnikov s method in string theory * + yuhma asano , hideki kyono and kentaroh yoshida _school of theoretical physics , dublin institute for advanced studies , + 10 burlington road , dublin 4 , ireland _ +_department of physics , kyoto university , + kitashirakawa oiwake - cho , kyoto 606 - 8502 , japan _

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in this paper , we have considered an application of melnikov s method in the context of the gauge / gravity correspondence . in particular , we have presented a string background to which melnikov s method is applicable . this is a brane - wave type deformation of the ads@xmath0s@xmath1 background . by employing two reduction anstze , we have studied two types of coupled pendulum - oscillator systems . then the melnikov function has been computed for each of the systems and the existence of chaos has been shown analytically . a strong advantage of melnikov s method is that the existence of classical chaos can be shown analytically and the mechanism is explained as a smale s horseshoe . the melnikov functions here have been computed on the string - theory side . one of the most interesting question is what is the gauge - theory interpretation of the melnikov function . this function knows all about the classical chaotic string solutions , including the associated fractal structure . according to the standard ads / cft dictionary , classical string solutions should correspond to composite operators on the gauge - theory side . the information on the fractal structure would be crucial in determining the alignment of component fields in the associated composite operators . hence the melnikov function may play an important role in determining the associated composite operators in some manner . from more technical aspects , it is significant to consider a generalization to include the friction . recall that an external force was induced with the oscillating ansatz . if the friction could be introduced as well , one can realize the system with the energy injection and dissipation . such a system is not a conserved system but dissipative . the chaos in the conserved systems is well described by the kam theorem @xcite and its characteristic is profoundly understood . the chaos in the dissipative systems , however , has a richer structure such as strange attractors . it is really intriguing to realize strange attractors on the string - theory side and consider the physical interpretation of its gauge - theory counter part . it would also be nice to look for other string backgrounds to which melnikov s method is applicable . we have just presented one example here . it is interesting to consider some applications of melnikov s method for black hole solutions in relation to the information loss process . it may be a good issue to clarify the relation between the event horizon and the melnikov function . we hope that melnikov s function would provide a new tool in studying the gauge / gravity correspondence . we are very grateful to daisuke kawai and takeshi matsumoto for useful discussions . the work of y.a . was supported by a dublin institute for advanced studies scholarship . the work of h.k . was supported by the japan society for the promotion of science ( jsps ) . the work of k.y . was supported by the supporting program for interaction - based initiative team studies ( spirits ) from kyoto university and by a jsps grant - in - aid for scientific research ( c ) no.15k05051 . this work was also supported in part by the jsps japan - russia research cooperative program and the jsps japan - hungary research cooperative program .

melnikov s method is an analytical way to show the existence of classical chaos generated by a smale horseshoe . this is a brane - wave type deformation of the adss background . by employing two reduction anstze +_department of physics , kyoto university , + kitashirakawa oiwake - cho , kyoto 606 - 8502 , japan _

melnikov s method is an analytical way to show the existence of classical chaos generated by a smale horseshoe . it is a powerful technique , though its applicability is somewhat limited . in this paper , we present a solution of type iib supergravity to which melnikov s method is applicable . this is a brane - wave type deformation of the adss background . by employing two reduction anstze , we study two types of coupled pendulum - oscillator systems . then the melnikov function is computed for each of the systems by following the standard way of holmes and marsden and the existence of chaos is shown analytically . * melnikov s method in string theory * + yuhma asano , hideki kyono and kentaroh yoshida _school of theoretical physics , dublin institute for advanced studies , + 10 burlington road , dublin 4 , ireland _ +_department of physics , kyoto university , + kitashirakawa oiwake - cho , kyoto 606 - 8502 , japan _

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data with a matrix - valued response for each experimental unit are commonly encountered in contemporary statistical applications . for example , a longitudinal multivariate response can be treated integrally as a matrix - valued variable by designating rows and columns to be time and variates . temporal and spatial data , multivariate growth curve data , image data and data from cross - over designs also generate matrix - valued responses . for example , in a twin cross - over bioassay of insulin by the rabbit blood sugar method ( v@xmath3lund , 1980 ) , each rabbit received two different treatments on two days . blood sugar was measured at hourly intervals for six hours each day . in this case , the response for each rabbit is a @xmath4 matrix , with rows and columns indicating treatments and time respectively . the eeg data ( li et al . 2010 ) is another example that contains temporal and spatial matrix - formed variates from 77 alcoholic subjects and 45 non - alcoholic subjects . the electrical records of each subject form a matrix of dimensions @xmath5 that can be treated as a matrix - valued response variable when we investigate the association between the brain signals and alcoholism . in these examples , the components of the matrix - variates are dependent among rows and columns . this dependence distinguishes longitudinal data in a matrix - valued response from traditional longitudinal modeling of vector - valued responses in which independent units are each measured over time . vectorizing a matrix - valued response , or modeling the row or column vectors separately , typically loses dependency information and fails to capture the data structure . tackling matrix - variates directly can circumvent this issue . research into this topic has gained considerable interest in recent years . ( 2010 ) proposed a class of sufficient dimension reduction ( sdr ) methods , called dimension folding sdr , for data with matrix - valued predictors . pfeiffer et al . ( 2012 ) extended sliced inverse regression ( sir ) to longitudinal predictors . ding and cook ( 2014 ) developed model - based dimension folding methods for matrix - valued predictors . xue and yin ( 2014 ) introduced dimension folding sdr for conditional mean functions . on another track , hung and wang ( 2013 ) , zhou et al . ( 2013 ) , and zhou and li ( 2014 ) extended generalized linear models ( glm ) to matrix- and tensor - valued predictors for analyzing image data . all these methods , however , address data with matrix or tensor - valued predictors . methods for dealing directly with matrix - valued responses are relatively limited in the literature . viroli ( 2012 ) proposed special matrix - variate regressions assuming independent rows for error terms or covariates . yet in many applications such assumptions are violated as dependency can exist among both rows and columns of a matrix - valued response . we recently became aware that li and zhang ( 2015 ) studied tensor response regression where a linear relationship between a tensor - valued response and a predictor vector is considered . their method is more suited for tensor responses and vector - valued predictors . however , our motivation , rationale , model formulation , and estimation algorithms are different from those of li and zhang ( 2015 ) . firstly , our work was motivated by contemporary matrix - variate data and was geared to develop demanding yet limited matrix - variate analysis tools for such data . thus , it is tailored for matrix - valued responses and can handle matrix - valued predictors simultaneously . secondly , our method can fully utilize intrinsic matrix - variate data structure and allow a matrix linear model to reduce number of parameters in matrix - variate regressions . moreover , our method of estimation is different than that used by li and zhang ( 2015 ) . as discussed in section [ sec:7.2 ] this can have important consequences in applications . in this article , we propose matrix - variate regressions in a general framework , where the response @xmath0 on each experimental unit is a random matrix and the predictor @xmath1 can be either a scalar , a vector , or a matrix , treated as non - stochastic in terms of the conditional distribution @xmath2 . neither the rows nor the columns of the matrix - valued variables are required to be independent . thus , the proposed methods can capture intrinsic dependent data structures . moreover , they can reduce the number of parameters and improve efficiency in estimation compared to conventional methods . as matrix - variate responses may have sizable dimensions , extraneous error variation may often occur beyond that anticipated by the model . to allow for such situations , we further propose envelope methods for efficient estimation in matrix - variate regressions . by applying the idea of enveloping , one can achieve dimension reduction in the analysis , by extracting only the useful information and eliminating redundant information from estimation . this can lead to substantial efficiency gains in estimation . the remainder of this article is organized as follows . in section [ sec:2 ] we propose a new class of matrix - variate regressions and connect them with conventional regression models . section [ sec:3.1 ] reviews the idea of enveloping and illustrates it with a real example . section [ sec:3 ] is devoted to the development of envelope methods for matrix - variate regressions . section [ sec:5 ] studies theoretical properties of matrix regression models and envelopes . sections [ sec:7 ] and [ sec:8 ] are devoted to illustrations with simulations and real data analyses . technical details and proofs are included in a supplement .

modern technology often generates data with complex data structures in which both response and explanatory variables are matrix - valued . existing methods in the literature are able to tackle matrix - valued predictors but are rather limited for matrix - valued responses . in this article , we study matrix - variate regressions for such data , where the response on each experimental unit is a random matrix and the predictor can be either a scalar , a vector , or a matrix , treated as non - stochastic in terms of the conditional distribution .

modern technology often generates data with complex data structures in which both response and explanatory variables are matrix - valued . existing methods in the literature are able to tackle matrix - valued predictors but are rather limited for matrix - valued responses . in this article , we study matrix - variate regressions for such data , where the response on each experimental unit is a random matrix and the predictor can be either a scalar , a vector , or a matrix , treated as non - stochastic in terms of the conditional distribution . we propose models for matrix - variate regressions and then develop envelope extensions of these models . under the envelope framework , redundant variation can be eliminated in estimation and the number of parameters can be notably reduced when the matrix - variate dimension is large , possibly resulting in significant gains in efficiency . # 1 * key words : * matrix - valued response ; matrix - variate regression ; reducing subspace ; sufficient dimension reduction .

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a robotic system s decision making is well known to be in need of some hard decision making at times . a most popular example is asimov s laws @xcite , which demonstrate the difficulties to apply logic by robots in practice . a shortened version of these laws is `` 1 . a robot may not allow a human being to come to harm . 2 . a robot must obey the orders given to it by human beings except if the order causes harm to humans . 3 . a robot must protect its own existence as long as such protection does not cause harm to humans . '' assuming these , what would happen to the robot s decision making if a human commands a robot to kill someone , but at the same time threatens to kill himself if the robot does not obey ? in this example the human introduces a contradiction into the logic of the robot . to avoid this the robot may have a complex rule base to provide it with legal and ethical principles and can be equipped by a meta law which says that `` the robot should not allow itself to be dictated by communicated conditions which make its logic contradictory '' . in this example one could say that in legal terms the suicide will remain the sole `` responsibility '' of the threatening person who commands the robot . the problem is not only the imperfection of asimov s robotic laws or that an agent programmer can make mistakes . logical consistency checks are also needed when the robot s perception - based beliefs are wrong . the agent can be programmed to re - examine whether its beliefs may need to be changed as were mistakenly believed to be true or false . this is not unlike enabling the agent to think like poirot , miss marple or sherlock holmes when they are reassessing their initial beliefs or impressions . but there are simpler cases : a robot may decide that the book it sees on the table can not be tom s as that one is in his home . in this paper we address the problem of how a robot can quickly and efficiently resolve inconsistencies in order to make the right decisions . the ability of making fast decisions about logical consistency , and the robot s ability to detect when inconsistency occurs , is an important problem for the future of robotics . it is also of particular importance for logic - based robot control systems , e.g. , @xcite . a typical logic - based robotic system usually contains a belief set , which provides the basis of reasoning for a robot s behaviour @xcite . an inconsistent belief set could lead to a wrong plan causing an unexpected result , e.g. , an unmanned vehicle can hit an obstacle , instead of avoiding it , if it mistakenly believes that any route of avoidance could cause more damage , due to , for instance , mis - perception of the environment . its mis perception could perhaps be corrected if it had been able to combine environmental prior knowledge with current sensing . in a rapidly changing environment bayesian methods can be used to identify and track movements of objects and establish functional relationships , e.g. , @xcite . when faced with balanced probabilities for two hypothetical and competing relationships in the robot s environment , it may need to make a decision based on the application of logic using prior knowledge . discovery of logical inconsistency in geometrical and physical relationships in an environmental model should prompt a robotic agent to revise its perception model of the world . for instance belief - desire - intention ( bdi ) agents should carry out consistency checks in their reasoning cycle in languages such as @xmath0 , @xmath1 and @xmath2 @xcite . in these systems the agent programmer should program logical consistency checks and handling of inconsistencies at design stage of the software . to topic of fast consistency checking by robots has also implications for legal certification of robots . as we humans formulate social and legal behaviour rules in terms of logical implications , the process is likely to be similar for robots and the problem of consistent decisions by robots is an important generic capability . future legal frameworks for certification of robots need to take into account verifiable decision making by robots . consistency checks on a set of logic rules in propositional logic is a textbook problem and has been extended to various types of logic systems in terms of validity , consistency and satisfiability . for instance @xcite provides an authoritative account of the history of logical consistency checking in a propositional logic . relevant methods and algorithms have long been investigated for database systems and rule - based expert systems , e.g. , @xcite , but none has been specifically designed for robotics . query language 4ql @xcite and boolean networks ( bn ) @xcite are very similar to our modelling formalism _ boolean evolution systems_. the former allows a variable to have four values : @xmath3 , @xmath4 , @xmath5 and @xmath6 . the algorithm that computes the unique well - supported model in @xcite can be adapted to check consistency , but it can only deal with one initial evaluation of variables at a time . bn was developed for modelling gene regulatory networks in biology . in bn , a boolean variable can only take either @xmath3 or @xmath4 , while in our formalism , a variable can be initialised as @xmath5 . research on bdi reasoning cycles focuses on runtime detection and resolution of conflicting goals , such as @xcite . no work has been conducted on complex reasoning process , which will be required by autonomous and intelligent robots . for realtime robotic systems it is important to increase solver efficiency to be able to deal with large search spaces with complex reasoning process for both offline and online application . in this respect , the use of binary decision diagram ( bdd ) is very effective by compressing search space through generating a unique and succinct representation of a boolean formula . bdd has been widely adopted for model checking @xcite , and applied successfully to verification of large systems . in this paper we adopt the bdd based symbolic model checking approach @xcite to robotics . to our best knowledge , nothing has been reported on its application on consistency and stability checking of decisions by robots . in this paper we propose a fast method for discovery of inconsistency in a set of logic rules and statements on relationships in a current world model , past actions , planned actions and behaviour rules of a robotic agent . we do not address the problem of how to resolve logical inconsistency , mainly because we hold the view that , to eliminate inconsistencies , a robot can efficiently improve its world model by non - logic based techniques . such techniques can include gathering more perception data , active vision , using alternative action plans or analyzing and deriving spatial temporal models using probabilities . if a single new perception predicate or predicate derived by logic rules of the robot contradicts its otherwise consistent world model , then the robot may apply a set of logic rules to derive a correction of its belief in terms of the predicate . what to derive and analyse for consistency is however a broad topic and lies outside of the scope of this paper . here we focus on fast discovery of inconsistencies which is fundamental for safe operations of autonomous robots . with time it should be a key technical part in the process of legal certification of future autonomous robots . our contribution builds on and develops our past efficient state space generation and parallel computation @xcite methods further . we have previously developed various state space reduction techniques for symbolic model checking via bdds , such as symmetry reduction @xcite and abstraction @xcite . the preliminary results of our techniques have been published in @xcite . in this paper we elucidate the setting for which our techniques are designed and demonstrate their way of using it in robotics . we also extend the techniques to deal with a different semantics and develop a new technique to extract counterexamples efficiently when the system is inconsistent or unstable . the counterexamples are useful for system developers to correct robotic reasoning systems ; they can provide guidance on how to improve the reasoning process of robots . we study the efficiency of the agent s ability to examine the consistency of its beliefs and logic rules and , if inconsistency occurs , generate counterexamples to the rules which can then be used by the robot to resolve inconsistency . our technique can be used both by robot programmers at software design stage and by robots when reasoning . in the former case , system developers can check the logical consistency of reasoning cycles in agent programs at design stage . for each inconsistent check , a counterexample can be produced to help developers understand the source of inconsistency and correct the program . in the latter case , consistency checks are carried out by the robots themselves in realtime and counterexamples are examined to improve reasoning , e.g. , bringing in more sensor data to eliminate ambiguity or bring about alternative decisions about future actions . in section [ sec : system ] we introduce the problem in a robotic framework and its characteristics . in section [ boolevul ] boolean evolution systems are formally represented . in section [ sec : modelling ] , we translate boolean evolution systems into _ transition systems _ which are now widely used in the control systems literature @xcite , which provides the basis of verification . note that in this paper we abstract robotic behaviour to propositional logic to be able to cope with computational complexity of consistency checking . section [ sec : mc ] contains our results on stability of boolean evolution systems in terms of ctl and ltl formulae . an important result states that stability checking can be reduced to a reachability problem which only asks for one fixpoint computation . similarly , consistency checking can be also converted into simple fixpoint computation . section [ sec : case ] presents a case study in a home robotics scenario , which demonstrates the use of uncertain sensory and communication information and a set of rules to satisfy . in section [ sec : exp ] , performance comparison between ctl formulae based solutions and the reachability based algorithms is highlighted and implemented in the symbolic model checker mcmas @xcite . we discuss stability checking under an alternative semantics of evolution in section [ sec : interleaving ] . we conclude the paper in section [ sec : concl ] .

this paper investigates how a robotic agent can use model checking to examine the consistency of its rules , beliefs and actions . a rule set is modelled by a boolean evolution system with synchronous semantics , which can be translated into a labelled transition system ( lts ) . their implementation provides us a computational tool , which can form the basis of efficient consistency checks on - board robots .

most autonomous robotic agents use logic inference to keep themselves to safe and permitted behaviour . given a set of rules , it is important that the robot is able to establish the consistency between its rules , its perception - based beliefs , its planned actions and their consequences . this paper investigates how a robotic agent can use model checking to examine the consistency of its rules , beliefs and actions . a rule set is modelled by a boolean evolution system with synchronous semantics , which can be translated into a labelled transition system ( lts ) . it is proven that stability and consistency can be formulated as computation tree logic ( ctl ) and linear temporal logic ( ltl ) properties . two new algorithms are presented to perform realtime consistency and stability checks respectively . their implementation provides us a computational tool , which can form the basis of efficient consistency checks on - board robots .

1502.03959

i

the discovery of the higgs boson at atlas and cms experiments @xcite made a revolutionary impact on the field of particle physics . it not only confirmed the so - called higgs mechanism for the electroweak symmetry breaking , but also opened a new possibility to perform a precise test of the standard model ( sm ) by studying the properties of the higgs boson . in the sm , the coupling constants of the higgs boson with other particles are well understood using the fact that the masses of quarks , leptons , and weak bosons originate in the vacuum expectation value ( vev ) of the higgs field , resulting in the prediction of the partial decay widths of the higgs boson into various particles . in models with physics beyond the sm ( bsm ) , measurements of the higgs couplings provide even exciting possibilities . in large class of bsm models , there exist new particles at the electroweak to tev scale , which affect the properties of the higgs boson . thus , with the detailed study of the higgs properties at collider experiments , we have a chance to observe a signal of bsm physics . such a study will be one of the major subjects in forthcoming collider experiments , i.e. , the lhc and future @xmath1 colliders like ilc and tlep @xcite . low energy supersymmetry ( susy ) is a well - motivated candidate of bsm physics . compared to the sm , the particle content is enlarged in susy models . even in the minimal setup , i.e. , in the minimal susy sm ( mssm ) , there exist two higgs doublets , @xmath8 and @xmath9 , as well as superparticles . the lightest higgs boson @xmath10 , which plays the role of the `` higgs boson '' discovered by atlas and cms , is a linear combination of the neutral components of @xmath8 and @xmath9 , while there exist other heavier higgses . in the case where the mass scales of the heavier higgses and the superparticles are high enough , the properties of @xmath10 are close to those of the sm higgs boson . on the contrary , if the heavier higgses or superparticles are relatively light , deviations of the higgs properties from the sm predictions may be observed by future collider experiments . with the precise measurement of the partial decay widths ( or branching ratios ) of the higgs boson , information about the heavy higgses and/or superparticles may be obtained even if those heavy particles can not be directly discovered . in this paper , we discuss how low the mass scales of the heavier higgs bosons and superparticles should be to observe a deviation . we evaluate the partial decay widths of the lightest higgs boson in the mssm , taking account of the following phenomenological constraints : higgs mass , flavor constraints of the @xmath11 mesons , stability of the electroweak ( sm - like ) vacuum against the transition to charge and color breaking ( ccb ) vacua , and perturbativity of coupling constants up to a high scale . these constraints exclude large fraction of the parameter region giving rise to a significant deviation . even so , we will see that the deviations of the partial widths from the sm predictions can be of @xmath12 for some of the decay modes in the parameter region allowed by the above - mentioned constraints . in particular , the deviations may be large enough to be observed by future future @xmath1 colliders like ilc and tlep even if superparticles are so heavy that they would not be observed at the lhc . the organization of this paper is as follows . in sec . [ sec : mssm ] , we briefly overview the properties of the higgs bosons in the mssm . we also summarize the phenomenological constraints that are taken into account in our analysis . then , in sec . [ sec : width ] , we calculate the partial decay widths of the lightest higgs boson in the mssm and discuss how large the deviation from the sm prediction can be . [ sec : discussion ] is devoted for conclusions and discussion .

these constraints exclude large fraction of the parameter region where the higgs widths show significant deviation from the standard model predictions . nevertheless , even if superparticles and the heavy higgses are out of the reach of lhc , the deviation may be large enough to be observed at future collider experiments . ut-15 - 02 + ipmu-15 - 0018 + kek - th-1795 + february , 2015 + .75 in * footprints of supersymmetry on higgs decay * .75 in motoi endo , takeo moroi , and mihoko m. nojiri 0.25 in _ department of physics , university of tokyo , tokyo 113 - 0033 , japan _ 0.1 in _ kavli ipmu ( wpi ) , university of tokyo , kashiwa , chiba 277 - 8583 , japan _ 0.1 in _ .5 in

motivated by future collider proposals that aim to measure the higgs properties precisely , we study the partial decay widths of the lightest higgs boson in the minimal supersymmetric standard model with an emphasis on the parameter region where all superparticles and heavy higgs bosons are not accessible at the lhc . taking account of phenomenological constraints such as the higgs mass , flavor constraints , vacuum stability , and perturbativity of coupling constants up to the grand unification scale , we discuss how large the deviations of the partial decay widths from the standard model predictions can be . these constraints exclude large fraction of the parameter region where the higgs widths show significant deviation from the standard model predictions . nevertheless , even if superparticles and the heavy higgses are out of the reach of lhc , the deviation may be large enough to be observed at future collider experiments . ut-15 - 02 + ipmu-15 - 0018 + kek - th-1795 + february , 2015 + .75 in * footprints of supersymmetry on higgs decay * .75 in motoi endo , takeo moroi , and mihoko m. nojiri 0.25 in _ department of physics , university of tokyo , tokyo 113 - 0033 , japan _ 0.1 in _ kavli ipmu ( wpi ) , university of tokyo , kashiwa , chiba 277 - 8583 , japan _ 0.1 in _ kek theory center , ipns , kek , tsukuba , ibaraki 305 - 0801 , japan _ 0.1 in _ the graduate university of advanced studies ( sokendai ) , + tsukuba , ibaraki 305 - 0801 , japan _ .5 in

1502.03959

c

in this paper , we have studied the partial decay widths of the lightest higgs boson in the mssm . taking account of relevant phenomenological constraints , i.e. , higgs mass , flavor constraints , vacuum stability , and the perturbativity of coupling constants up to the gut scale , we have calculated the expected deviations of the partial decay widths from the sm predictions . the partial decay widths are enhanced if the @xmath29-parameter is relatively large . however , such a choice may conflict with some of the phenomenological constraints . in particular , the vacuum - stability condition imposes a stringent constraint on the parameter space . we have found that , with too large @xmath168 , there show up ccb vacua where the down - type higgs field as well as the up - type higgs and stop fields acquire large vevs ; existence of such ccb vacua was not seriously considered in the previous studies . in addition , when @xmath296 is large , non - holomorphic correction to the bottom yukawa interaction becomes so large that the bottom yukawa coupling constant becomes non - perturbative below the gut scale . large value of @xmath296 may also cause too large flavor - violating decay of @xmath11-mesons . by taking these constraints into account , the maximal and minimal possible values of the higgs partial widths are restricted . we found that the deviations of the partial decay widths from the sm predictions can be of @xmath208% for some of the decay modes . in particular , those of @xmath87 and @xmath88 may show significant deviations even if the superparticles are out of the reach of 14tev lhc . in addition , the deviation of @xmath96 may also be sizable if the superparticles are relatively light . we emphasize that , although our scan is limited to some part of the mssm parameter space , we have found the regions where the deviations from the sm predictions are within the reach of proposed @xmath1 colliders even if superparticles would not be observed at the lhc . _ acknowledgements _ : the authors would like to thank yasuhiro shimizu for the collaboration in the early stage of this project . the authors also acknowledge yitp for their hospitality , at which this work was initiated . this work is supported by jsps kakenhi no . 23740172 ( m.e . ) , no . 26400239 ( t.m . ) , and no . 26287039 ( m.m.n . ) . the work is supported by grant - in - aid for scientific research from the ministry of education , science , sports , and culture ( mext ) , japan , no . 23104008 ( t.m . ) and no . 23104006 ( m.m.n . ) , and also by world premier international research center initiative ( wpi initiative ) , mext , japan . g. aad _ et al . _ [ atlas collaboration ] , phys . b * 716 * ( 2012 ) 1 [ arxiv:1207.7214 [ hep - ex ] ] . s. chatrchyan _ et al . _ [ cms collaboration ] , phys . b * 716 * ( 2012 ) 30 [ arxiv:1207.7235 [ hep - ex ] ] . s. dawson , a. gritsan , h. logan , j. qian , c. tully , r. van kooten , a. ajaib and a. anastassov _ et al . _ , arxiv:1310.8361 [ hep - ex ] . y. okada , m. yamaguchi and t. yanagida , prog . theor . phys . * 85 * ( 1991 ) 1 . y. okada , m. yamaguchi and t. yanagida , phys . b * 262 * , 54 ( 1991 ) . j. r. ellis , g. ridolfi and f. zwirner , phys . b * 257 * ( 1991 ) 83 . j. r. ellis , g. ridolfi and f. zwirner , phys . b * 262 * , 477 ( 1991 ) . h. e. haber and r. hempfling , phys . * 66 * ( 1991 ) 1815 . z. skands , b. c. allanach , h. baer , c. balazs , g. belanger , f. boudjema , a. djouadi and r. godbole _ et al . _ , jhep * 0407 * ( 2004 ) 036 [ hep - ph/0311123 ] . k. a. olive et al . 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motivated by future collider proposals that aim to measure the higgs properties precisely , we study the partial decay widths of the lightest higgs boson in the minimal supersymmetric standard model with an emphasis on the parameter region where all superparticles and heavy higgs bosons are not accessible at the lhc . taking account of phenomenological constraints such as the higgs mass , flavor constraints , vacuum stability , and perturbativity of coupling constants up to the grand unification scale , we discuss how large the deviations of the partial decay widths from the standard model predictions can be .

motivated by future collider proposals that aim to measure the higgs properties precisely , we study the partial decay widths of the lightest higgs boson in the minimal supersymmetric standard model with an emphasis on the parameter region where all superparticles and heavy higgs bosons are not accessible at the lhc . taking account of phenomenological constraints such as the higgs mass , flavor constraints , vacuum stability , and perturbativity of coupling constants up to the grand unification scale , we discuss how large the deviations of the partial decay widths from the standard model predictions can be . these constraints exclude large fraction of the parameter region where the higgs widths show significant deviation from the standard model predictions . nevertheless , even if superparticles and the heavy higgses are out of the reach of lhc , the deviation may be large enough to be observed at future collider experiments . ut-15 - 02 + ipmu-15 - 0018 + kek - th-1795 + february , 2015 + .75 in * footprints of supersymmetry on higgs decay * .75 in motoi endo , takeo moroi , and mihoko m. nojiri 0.25 in _ department of physics , university of tokyo , tokyo 113 - 0033 , japan _ 0.1 in _ kavli ipmu ( wpi ) , university of tokyo , kashiwa , chiba 277 - 8583 , japan _ 0.1 in _ kek theory center , ipns , kek , tsukuba , ibaraki 305 - 0801 , japan _ 0.1 in _ the graduate university of advanced studies ( sokendai ) , + tsukuba , ibaraki 305 - 0801 , japan _ .5 in

0807.0619

i

the main result of this paper ( theorem [ main ] ) says that the special fiber of a galois branched cover of the open @xmath0-adic disc is determined by characteristic zero fibers . the motivation for such a theorem comes from the _ global lifting problem for galois covers of curves _ : if @xmath1 is a finite group , @xmath2 is an algebraically closed field of characteristic @xmath3 , and @xmath4 is a finite @xmath1-galois branched cover of smooth projective @xmath2-curves , does there exist a lifting of @xmath5 to a @xmath1-galois branched cover of smooth projective @xmath6-curves , where @xmath6 is a discrete valuation ring of mixed characteristic with residue field @xmath2 ? this problem has been much studied ; see e.g. @xcite , @xcite , @xcite , @xcite , @xcite , @xcite . as a brief survey of the subject , we mention the following results : * ( grothendieck , @xcite ) if @xmath5 is at most tamely ramified , then a lifting exists over @xmath6 , where @xmath6 is any complete dvr with residue field @xmath2 . moreover , the lifting is unique once we fix a lifting of @xmath7 and the branch locus of @xmath5 . * not all wildly ramified covers are liftable : there exist curves of genus @xmath8 in positive characteristic whose automorphism groups are too large to be automorphism groups of genus @xmath9 curves in characteristic zero , by the hurwitz genus bounds . if @xmath10 is such a curve , it is then clear that @xmath11 is not liftable to mixed characteristic . * there are examples of non - liftable @xmath0-elementary abelian covers . see @xcite for an example with @xmath12 for @xmath13 . * oort , sekiguchi , and suwa showed in @xcite that cyclic covers lift if @xmath0 divides @xmath14 at most once . their method was global in nature , involving a group scheme degeneration of kummer theory to artin - schreier theory . * using local methods ( see below ) , green and matignon proved in @xcite that cyclic covers lift if @xmath0 divides @xmath14 at most twice . * pagot has shown in @xcite that klein - four covers always lift . * @xmath15-covers are liftable by @xcite , where the method of differential data is used . * green has shown in @xcite that the @xmath16-cyclic covers that occur as automorphisms of lubin - tate formal groups are liftable . the guiding conjecture in the subject was provided by f. oort ( @xcite , i.7 ) who suggested the cyclic covers always lift . in section 5 , we derive an arithmetic reformulation of a stronger form of this conjecture , which specifies the ring @xmath6 over which the lifting should occur : if @xmath4 is cyclic of order @xmath17 with @xmath18 , then @xmath5 lifts over @xmath19 $ ] , where @xmath20 denotes the witt vectors of @xmath2 . a major breakthrough in the subject came when green and matignon discovered that the obstructions to lifting are not global in nature , but rather local . indeed , using either rigid patching @xcite or deformation theory @xcite , the global lifting problem reduces to a local one , due to the _ ( @xcite section iii , @xcite corollaire 3.3.5 ) _ let @xmath21 be a ramification point for the @xmath1-galois cover @xmath4 , and consider the @xmath22-galois cover @xmath23 obtained by completion , where @xmath22 is the inertia subgroup at @xmath24 . suppose that for each such ramified @xmath21 , the map @xmath25 can be lifted to a @xmath22-galois cover of open @xmath0-adic discs , @xmath26 , where @xmath27)$ ] . then the local liftings , @xmath28 , can be patched together to yield a lifting of @xmath5 to a @xmath1-galois cover of smooth projective @xmath6-curves . since @xmath29 $ ] , we are led to consider the following _ local lifting problem for galois covers of curves _ : given a finite @xmath1-galois extension of power series rings @xmath30|k[[z]]$ ] , does there exist a lifting to a @xmath1-galois extension @xmath31|r[[z]]$ ] , where @xmath6 is a mixed characteristic dvr with residue field @xmath2 ? one could also consider the ( weaker ) _ birational local lifting problem for galois covers of curves _ : given a finite @xmath1-galois extension of laurent series fields @xmath32 , does there exist a lifting to a @xmath1-galois extension of normal rings @xmath33 $ ] ? by a lifting in this case , we mean that @xmath34 is an integral domain ( where @xmath35 is a uniformizer for @xmath6 ) , and the fields @xmath36 and @xmath37 are isomorphic as @xmath1-galois extensions of @xmath38 . in particular , in the birational version of the local problem , we do not require @xmath39 to be smooth : in terms of geometry , this corresponds to allowing the curve @xmath10 to acquire singularities . as mentioned above , the local lifting problem does not always have a positive solution . on the other hand , garuti has shown in @xcite that the birational local lifting problem _ does _ always have a positive solution , so the birational problem is indeed weaker than the local lifting problem . it is clearly advantageous to work with laurent series fields rather than power series rings , however , and the following criterion for good reduction ensures that we may do so without sacrificing the smoothness of our liftings . _ ( @xcite section 5 , @xcite 3.4 ) _ let @xmath40 be a normal integral local ring , which is also a finite @xmath41$]-module . assume moreover that @xmath34 is reduced and @xmath42 is separable . let @xmath43 be the integral closure of @xmath44 , and define @xmath45 . also , setting @xmath46 , denote by @xmath47 the degree of the different of @xmath48\otimes k)$ ] , and by @xmath49 the degree of the different of @xmath42 . then @xmath50 , and if @xmath51 , then @xmath52 $ ] . using this criterion , we obtain the suppose that @xmath30|k[[z]]$ ] is a @xmath1-galois extension of power series rings . then a @xmath1-galois extension of normal integral local rings , @xmath33 $ ] , is a lifting of @xmath30|k[[z]]$ ] if and only if it is a birational lifting of @xmath32 , and @xmath53 . hence , the local lifting problem can be reformulated as follows : given a @xmath1-galois extension @xmath32 , does there exist a @xmath1-galois birational lifting @xmath33 $ ] which preserves the different , i.e. such that @xmath53 ? it is this last formulation of the local lifting problem that provides our motivation for studying galois covers of the open @xmath0-adic disc . in particular , given such a @xmath1-galois branched cover @xmath54 , we are interested in determining geometric and arithmetic properties of the special fiber @xmath55 ( such as irreducibility , separability , and the degree of the different @xmath49 ) from the corresponding properties of the generic fiber @xmath56 and its specializations at various points @xmath57 . our main result ( theorem [ main ] ) provides precisely such a characterization of the special fiber in terms of characteristic zero data . roughly speaking , our result says that the special fiber of a galois cover of the open @xmath0-adic disc `` wants '' to be the field of norms of the characteristic zero fibers , and the degree to which this fails is the phenomenon of inseparability . our work can be regarded as a concrete investigation of the class field theory of the open @xmath0-adic disc , and our main result suggests that the local lifting problem would be answered by a grunwald - wang type theorem for the open disc , with control over the generic different . in section 2 of this paper we review the basic structure of the open @xmath0-adic disc , and then in section 3 we describe the theory of the field of norms due to fontaine and wintenberger , which plays a major role in our main result . section 4 contains the proof of our main theorem characterizing the special fiber of a galois branched cover of the open @xmath0-adic disc in terms of the characteristic zero fibers of the cover . an arithmetic reformulation of the oort conjecture is deduced in section 5 , together with a new proof of this conjecture in the @xmath0-cyclic case . let @xmath58 be a complete discretely valued field with residue characteristic @xmath3 . we make the following notational conventions : * @xmath59 denotes the valuation ring of @xmath58 ; * @xmath60 denotes the maximal ideal of @xmath59 ; * @xmath61 denotes the residue field of @xmath59 ; * @xmath62 denotes the normalized discrete valuation on @xmath58 , so that @xmath63 ; * @xmath64 is the absolute value on @xmath58 induced by @xmath62 , normalized so that @xmath65 . * if @xmath66 is the completion of an algebraic extension of @xmath58 , then we also denote by @xmath62 ( resp . @xmath64 ) the unique prolongation of @xmath62 ( resp . @xmath64 ) to @xmath66 .

motivated by the local lifting problem for galois covers of curves , this paper investigates galois branched covers of the open-adic disc . using the field of norms functor of fontaine and wintenberger , we show that the special fiber of a galois cover is determined by arithmetic and geometric properties of the generic fiber and its characteristic zero specializations . as applications ,

motivated by the local lifting problem for galois covers of curves , this paper investigates galois branched covers of the open-adic disc . using the field of norms functor of fontaine and wintenberger , we show that the special fiber of a galois cover is determined by arithmetic and geometric properties of the generic fiber and its characteristic zero specializations . as applications , we derive a characteristic zero reformulation of the abelian local lifting problem , and give a new proof of the liftability of-cyclic covers .

1111.1668

i

electron - positron pair production by rotation - powered pulsars is believed to be essential to the generation of charged particles in their magnetospheres and winds , as well as to the generation of their coherent pulsed radio emission . pairs can be produced through electromagnetic cascades initiated by accelerated particles and several sites of acceleration have been proposed . near the neutron star surface above the magnetic polar caps ( pcs ) , electrons may be accelerated either in vacuum gaps ( ruderman & sutherland 1975 ) or in space - charge limited flow gaps ( arons & scharlemann 1979 ) . the gamma - ray photons radiated by the electrons will convert to electron - positron pairs by one - photon pair production in the strong magnetic field near the surface , and the pairs radiate synchrotron photons that produce further generations of pairs ( daugherty & harding 1982 ) . in outer gap vacuum accelerators , extending to near the light cylinder , seed photons can produce pairs through interaction with thermal x - ray photons from the hot neutron star surface to create a first generation of particles that accelerate through the gap radiating gamma - ray photons that produce further pairs ( cheng , ho & ruderman 1986 ) . to date , all of the simulations of pulsar pair cascades have assumed pure dipole magnetic fields both to derive the accelerating electric field parallel to the magnetic field and to calculate the spectrum and multiplicity of the pairs . the configuration of neutron star ( ns ) magnetic fields is currently an unsolved problem . the original solution for a rotating magnetic dipole in vacuum ( deutsch 1955 ) shows that retardation causes the field lines near and outside the light cylinder to sweep back in a direction opposite to that of rotation , transitioning to an electromagnetic wave at large distances . the magnetic field sweepback causes the boundary of the open field volume , and the footpoints of field lines on this boundary that define the pc , to shift relative to the magnetic axis opposite to the rotation direction ( arendt & eilek 1998 , dyks & harding 2004 ) . the amount of the shift depends on the magnetic field inclination to the rotation axis , with the largest offsets occurring for oblique rotators . in the last decade , numerical solutions of force - free magnetosphere models assuming ideal mhd ( @xmath4 ) ( contopoulos et al . 1999 ) show an even larger sweepback of the magnetic field near the light cylinder as it transitions to an mhd wind ( spitkovsky 2006 , timokhin 2006 ) , as currents cause significant distortions of the magnetic field . the open field boundary and the pc of a force - free magnetosphere thus have a larger offset than the vacuum magnetosphere . since the vacuum magnetosphere can accelerate particles but is not loaded with charges or currents and the ideal - mhd magnetosphere has charges and currents but can not accelerate particles , a real pulsar magnetosphere lies between these two solutions . in addition to the guaranteed distortions of the dipole field caused by retardation and currents , it is possible that there are also distortions due to multipolar components near the neutron star surface . the emission from millisecond pulsars in particular shows evidence of non - dipolar field structure . the thermal x - ray pulse profiles of some millisecond pulsars ( msps ) show asymmetries which have been successfully modeled by including offsets of the emitting hot spot on the neutron star surface . since the emission likely originates from pc heating , it is argued that msps such as psr j0437 - 4715 ( bognanov et al . 2007 ) and psr j0030 + 0451 ( bogdanov & grindlay 2009 ) have either offset dipoles or offset pcs . modeling of the x - ray pulsations of the nss in low - mass x - ray binaries , the likely progenitors of millisecond pulsars , show possible evidence of even more extreme magnetic field distortions ( lamb et al , 2009 ) , that could result from distortion of the global magnetic field by e.g. the crustal plate tectonics ( ruderman 1991 ) . we have recently shown ( harding & muslimov 2011 , hm11 ) that distortions of a magnetic dipole field that produce offset pcs lead to an increase in the multiplicity of pc pair cascades and to significant shifts of the death line for pair production of curvature radiation photons . we found that even offsets that are small fractions of a neutron star radius can enable high - multiplicity pair cascades in older pulsars that with pure dipole fields would be pair starved . there were a number of previous studies of how multipolar or non - dipolar neutron star fields could increase pair production ( ruderman & sutherland 1975 ; arons 1983 , 1997 ; asseo & khechinashvili 2002 ) , but all of these focussed on the decrease in field line radius of curvature as the primary effect . arons ( 1997 ) , considering the decrease in field line radius of curvature provided by an offset dipole , found that offsets that are large fractions of a stellar radius , ( 0.7 - 0.8)@xmath5 , are required to move the pair death line enough to include the entire radio pulsar population . medin & lai ( 2010 ) proposed an even larger offset of 0.95 @xmath5 , again only considering the decreased field line radius of curvature . kantor & tsygan ( 2003 , 2004 ) considered the space - charge limited flow ( sclf ) @xmath6 solution of a dipole field with the addition of a displaced dipole of smaller magnitude , that effectively produces narrow , bent flux tubes . they take into account both the increase in @xmath6 and decrease in radius of curvature to derive lower death lines for pair production , and find that displacements of @xmath7 for the smaller dipole component can significantly change the death lines for both curvature radiation ( cr ) and inverse compton scattering ( ics ) pair production . our study also takes into account the increase in @xmath6 and particle acceleration energy , which is by far the strongest effect on pair multiplicity of offset dipoles or pcs . the study of hm11 was based on derivation of the electric field parallel to the magnetic field near the pc assuming there is free flow of particles from the neutron star surface , the sclf flow boundary condition ( arons & scharlemann 1979 ) . the resulting current density of primary charges in this case is equal to @xmath8 where @xmath9 is the goldreich - julian charge density . this current distribution differs from the current density distribution demanded by the global ideal - mhd models , so it is presently not clear whether the steady sclf acceleration model will be compatible with a global pulsar magnetosphere . however , more realistic global magnetosphere models with resistivity and dissipation are in development ( li , spitkovsky & tchekhovskoy 2011 ) . the current density distributions of such pulsar magnetospheres with dissipation are significantly different from those with ideal - mhd ( kalapotharakos et al . 2011 ) , with the ability to possibly adjust to the @xmath10 of the pc cascades they rely on for their charge supply . furthermore , timokhin & arons ( 2011 ) have performed particle - in - cell simulations of pc pair cascades and show that sclf acceleration is stable for currents within a 10% range of @xmath11 . in this paper we extend the study of hm11 to include a more general expression for the distorted magnetic field , treating two cases in which the caps from opposite magnetic poles are offset in the same ( symmetric ) or opposite ( asymmetric ) directions . in section 2 , we give the expressions for the magnetic fields in these two cases , as well as the corresponding goldreich - julian charge density , field line equations and parallel electric fields . section 3 describes the pair cascade simulation used to derive the results for pair multiplicity and flux distributions over the pc , and pair spectra given in section 4 . we also examine the changes in the distribution of pc heating luminosity with offset pcs as well as the total pair and heating luminosity .

neutron star magnetic fields may have polar caps ( pc ) that are offset from the dipole axis , through field - line sweepback near the light cylinder or non - symmetric currents within the star . we find that even modest offsets can produce significant increases in pair multiplicity , especially for pulsars that are near or beyond the pair death lines for centered pcs , primarily because of higher accelerating fields .

neutron star magnetic fields may have polar caps ( pc ) that are offset from the dipole axis , through field - line sweepback near the light cylinder or non - symmetric currents within the star . the effects of such offsets on electron - positron pair cascades are investigated , using simple models of dipole magnetic fields with small distortions that shift the pcs by different amounts or directions . using a monte carlo pair cascade simulation , we explore the changes in the pair spectrum , multiplicity and energy flux across the pc , as well as the trends in pair flux and pair energy flux with spin - down luminosity , . we also give an estimate of the distribution of heating flux from returning positrons on the pc for different offsets . we find that even modest offsets can produce significant increases in pair multiplicity , especially for pulsars that are near or beyond the pair death lines for centered pcs , primarily because of higher accelerating fields . pair spectra cover several decades in energy , with the spectral range of millisecond pulsars ( msps ) two orders of magnitude higher than for normal pulsars , and pc offsets allow significant extension of all spectra to lower pair energies . we find that the total pc pair luminosity is proportional to , with for normal pulsars and for msps . remarkably , the total pc heating luminosity for even large offsets increases by less than a factor of two , even though the pc area increases by much larger factors , because most of the heating occurs near the magnetic axis .

1111.1668

r

for a given set of input parameters that include pulsar period @xmath130 , surface magnetic field @xmath20 , inclination angle @xmath117 , and offset degree @xmath16 , direction @xmath23 and symmetry ( case a or b ) , the cascade calculation produces the pair spectrum per primary electron as a function of @xmath82 and surface @xmath83 , @xmath131 , where we will express the pair energy @xmath132 in units of @xmath89 . the pair multiplicity , the number of pairs per primary electron , as a function of @xmath82 and @xmath83 across the pc is [ m+ ] m_+ ( , ) = _ e_min^e_max dn_+(e , , ) de de while the total pair multiplicity from each pc is [ m+pc ] m^pc_+ = _ 0 ^ 2 d_0 ^ 1 m_+ ( , ) _ pc^2r_ns^2d . here , the pc angle varies with @xmath83 , [ sinthetapc ] _ pc = ( r_ns / c)^(1+a)/2 . which comes from setting @xmath133 and @xmath134 in eqn ( [ thetaa ] ) . the total flux of pairs emerging from each pc is [ n+ ] n_+ ( , ) = _ e_min^e_max dn_+(e , , ) de de where [ dotdnde ] dn_+(e , , ) de = dn_+(e , , ) de n_p ( , ) and where @xmath135 is the primary flux over the pc , [ np ] n_p ( , ) = n_gjc r_ns^2 ^2_pc . where @xmath136 is the goldreich - julian number density at the ns surface . in eqn ( [ np ] ) , we use the goldreich - julian charge density given in eqns ( [ rhogj_a ] ) and ( [ rhogj_b ] ) . similarly , the energy flux of pairs from each pc is e_+ ( , ) = _ e_min^e_max dn_+(e , , ) de ede we have computed @xmath131 for a range of parameters to explore the distributions of @xmath137 , @xmath138 and @xmath139 across the pc . our results are dependent on neutron star equation of state ( e0s ) through the @xmath6 dependence on the frame - dragging , specifically through the @xmath140 parameter , and on the neutron star radius , through the pc angle . there have been a number of measurements of neutron star mass for msps that give values well above the canonical @xmath141 and as high as @xmath142 ( demorest et al . we have therefore assumed different eos for normal ( non - recycled ) pulsars and msps . for non - recycled pulsars , we take @xmath143 and radius @xmath144 km , which give a moment of inertia @xmath145 ( lattimer & prakash 2007 ) . we take a rotating ns model ( friedman et al . 1986 ) for msps with @xmath146 , @xmath147 km and @xmath148 . examples of the pair multiplicity and pair flux distributions over the pc are shown in figures 2 and 3 , for a pulsar with @xmath149 s and @xmath150 g , and for case b with @xmath151 for different values of the offset degree @xmath16 . a pulsar with these parameter values lies above but near the cr pair death line for @xmath152 , in which case the cascade produces a modest pair multiplicity . as shown in figure 2 , the maximum pair multiplicity @xmath137 occurs for @xmath153 and is nearly symmetric in @xmath83 , with only a small asymmetry caused by the @xmath83 dependence of @xmath6 . as @xmath16 increases , the peak pair multiplicity increases but only in a small region of the pc toward the offset , in this case at @xmath154 , while the multiplicity decreases in the other side of the pc in the @xmath155 direction . for larger offsets , the region of high multiplicity expands as the pc boundary expands in the direction of the offset . the distribution of pair flux for this same case , shown in figure 3 , is somewhat different from the distribution of pair multiplicity . for @xmath76 , @xmath139 peaks in a small ring around the center of the pc and decreases outside the ring , again with a small asymmetry due to the @xmath83 dependence of @xmath6 . the concentration of @xmath139 closer to the pc center is due to the fact that the lorentz factor of the primary electron is maximum near the magnetic axis because the decrease in radius of curvature of the field lines raises the altitude of the pff ( see figures 4 and 5 of hm98 ) . for @xmath156 , the pair energy flux increases and becomes asymmetric , but with a greater concentration toward the pc center than the multiplicity . figure 4 shows the distribution of pair multiplicity @xmath137 for the case of a millisecond pulsar with @xmath157 ms , @xmath158 g , and for case a with @xmath159 for @xmath16 between 0 and 0.6 . these parameters place the pulsar below the cr pair death line with very small pair multiplicity for @xmath76 , although the distribution of @xmath137 can be seen to show a more pronounced asymmetry toward the ` favorably curved ' field lines at @xmath160 than in the case of the longer period pulsar shown in figure 2 . for pulsars with shorter periods , the second term in @xmath6 depending on @xmath161 is relatively larger since the value of @xmath162 is larger . as @xmath16 increases , the peak pair multiplicity again increases in a small region of the pc toward the offset , but with a stronger asymmetry toward the @xmath160 side of the pc . for millisecond periods there is also less increase of ellipticity of the pc with increasing @xmath16 . the period dependence of the pc ellipticity can easily be seen from theh expression eqn ( [ sinthetapc ] ) for the pc angle . if we define the elongation factor @xmath163 , then @xmath164 is higher for smaller @xmath162 . figure 5 illustrates the effect of inclination angle on the distribution of pair multiplicity and pair flux for a millisecond pulsar with @xmath165 ms , @xmath166 g , and for case a with @xmath159 and @xmath167 . @xmath137 peaks in roughly the same region of the pc for @xmath168 and @xmath169 , but the asymmetry toward favorably curved field lines is is stronger in the case of @xmath169 . both the maximum multiplicity and flux are higher for @xmath169 . the values of peak multiplicity , @xmath170 , for the radio pulsar population in the @xmath130-@xmath171 plane are shown in the contour plots in figures 6 - 8 . figure 6 shows contours of @xmath172 in the case of a centered pc ( @xmath76 ) , where we have assumed the different neutron star eos for normal ( non - recycled ) and msps described above . it can be seen that @xmath173 is high for short periods and higher @xmath171 , reaching a maximum of a few times @xmath174 , and drops sharply towards the death line even on a log scale . about half of the normal pulsar population and most of the msp population lies below the death line , a result that has been noted in a number of previous papers ( e.g. arons 1997 , zhang , harding & muslimov 2000 , hibschman & arons 2001 , harding & muslimov 2002 ) . in figure 7 and 8 , we show contours of @xmath172 for offset pcs with @xmath167 having symmetric ( case a ) and asymmetric ( case b ) offsets . in both symmetric and asymmetric offsets , the regions of high pair multiplicity spread to lower @xmath171 and longer periods . the increases in @xmath173 are most dramatic for that part of the population near and below the pair death line having very low @xmath173 in the centered pc case . the pair death line thus moves down through nearly the entire population of both recycled and non - recycled pulsars for an offset of @xmath167 . for pulsars with high @xmath171 , @xmath173 changes much less with even a large offset , with multiplicity saturating below @xmath175 . this saturation of @xmath173 , noted by hm11 , is caused by several effects . even in the centered pc case , when the magnetic field increases above @xmath176 g pairs are increasingly produced in the low - lying landau states ( baring & harding 2001 ) , which results in fewer synchrotron photons and fewer generations in the cascade . for increasing pc offsets , the resulting increase in accelerating field and particle energy initially produces higher @xmath173 pair cascades , but as the increasingly energetic particles produce higher energy cr photons , the pairs are produced at smaller angles to the magnetic field . this results in higher average pair energies with pairs in lower landau states , further reduces the number of synchrotron photons . there are some significant differences in the @xmath172 contours for cases a and b. the death line in the normal pulsar population is lower for case b and overall values of @xmath173 are a bit higher . for msps , the case b death line moves up for very short periods . the death lines for both @xmath76 and @xmath167 are flatter for the shorter period msps and the death lines at all periods are flatter as for @xmath167 . as noted by hm11 , particle acceleration in pulsars that require a large fraction of the open field voltage to produce pairs is limited by cr reaction . in this regime , the electron lorentz factors to not reach the full voltage drop between the neutron star and the pff , but instead reach a lower steady - state lorentz factor as the acceleration is balanced by the cr loss rate . even for centered pcs , acceleration in msps operates completely in the cr reaction limit ( luo et al . 2000 , harding & al . 2002 ) . the total spectrum of cascade pairs , integrated over the whole pc , is [ n+pc ] dn_+(e)de = _ 0 ^ 2 d_0 ^ 1 dn_+(e , , ) de _ pc^2r_ns^2d . figure 9 shows the integrated pc pair spectra for @xmath177 g and a range of periods typical of non - recycled pulsars , for centered and offset pcs . the spectra exhibit turnovers at low and high energies that depend on period . for centered pcs , the low - energy turnovers occur around lorentz factor @xmath178 and the spectra extend up to @xmath179 , with the range increasing for shorter @xmath130 . for offset pcs , the spectra extend to lower energies , by about a decade for @xmath167 . thus the total pc pair flux increases because pairs can be produced at lower energies . pair spectra for parameters typical of msps are shown in figure 10 . for both centered and offset pcs , the msp pair energies are much higher that those of normal pulsars , by a factor of about 100 . the spectra extend from a low - energy turnover at @xmath180 to a high energy cutoff around @xmath181 for @xmath76 , and from @xmath182 to @xmath183 for @xmath184 . thus the highest - energy pairs are nearly as energetic as the primary electrons ! the large difference between normal and msp pair spectra is due to the difference in field strengths . in lower fields , photons must pair produce at higher energies , decreasing the relative @xmath173 , even though the larger pc sizes of msps give smaller curvature radii . but as is the case for normal pulsars , the increase in @xmath173 for offset pcs results from lower possible pair energies . the pair flux emerging from each pc is [ ndotpair ] n_pair = _ e_min^e_max dn_+(e)de de which we examine as a function of pulsar spin - down luminosity , @xmath0 . figure 11 plots the pc pair flux vs. @xmath0 for the non - recycled pulsar population for different degrees of offset . the lines are least - squares fits to the points for each case , resulting in the following approximate expressions : @xmath185 where @xmath186 . the pair flux is closely proportional to spin - down luminosity for @xmath76 , with the dependence somewhat flattening for offset pcs due to a saturation at high @xmath0 . figure 12 shows the equivalent plot for msps with the least - squares fits giving the following relations between @xmath187 and @xmath0 : @xmath188 the total pair luminosity from each pc is [ lpair ] l_pair = _ e_min^e_max dn_+(e)de ede plots of the pc pair luminosity vs. @xmath0 for non - recycled pulsars are shown in figure 13 , with the least - squared fits yielding the relations : @xmath189 the pair luminosity seems to also be roughly proportional to spin - down luminosity , with the dependence flattening for higher @xmath16 , with the efficiency for pair luminosity being @xmath190 . the @xmath1 vs. @xmath0 for msps is shown in figure 14 , with the corresponding relations from the least - squares fits : @xmath191 the pair luminosity efficiency for msps is much higher than for non - recycled pulsars , with @xmath192 , about an order of magnitude higher . since msps have much lower surface magnetic field strengths , they need to use a higher fraction of the total open - field line voltage to produce pairs , resulting in a higher @xmath193 . msps also have higher @xmath87-ray luminosity efficiencies ( abdo et al . 2009 ) , because the primary particles which radiate the highest energy emission reach higher energies before pairs screen the accelerating field . pair cascades from particle accelerators near the pc will inevitably produce backflowing particles in the process of pair screening of the electric field . the backflowing particles will accelerate through the same potential drop as the primary particles and deposit this energy on the neutron star surface , increasing the surface temperature of the pc . the radiation from the hot pc is predicted to be in the x - ray band ( arons 1981 , hm01 ) , and observations have provided evidence for emission from hot pcs , especially in the case of middle - aged and millisecond pulsars ( zavlin 2007 ) . the predicted x - ray luminosity from heated pcs in the case of sclf accelerators with centered pcs ( hm01 ) roughly agrees with observed luminosities . with the possibility of offset pcs increasing the pair multiplicity near the neutron star surface , it is important to check that the returning positron luminosity does not overheat the pc and violate the x - ray luminosity constraints . for this estimate , we do not perform a detailed screening calculation with pair dynamics to derive the screening scale length , as in hm01 . rather we will compute the maximum returning positron fraction which will give a conservative limit on the positron heating luminosity . the maximum fraction of returning positrons from the pff can be estimated as ( see formula [ 33 ] in hm01 ) = 12 , [ rho_plus ] where @xmath194 is the pff dimensionless altitude and @xmath195 , the primary charge density at the pff , is derived by inserting @xmath196 inside the curly brackets in eqns ( [ rhogj_a ] ) and ( [ rhogj_b ] ) and @xmath197 outside the brackets . the heating flux over the pc from returning positrons can then be estimated as c _ + ( z__pff , , ) ( z__pff , , ) where @xmath198 is the primary electron energy at the pff on the field line with surface @xmath82 and @xmath83 . the reason that we use the the primary energy instead of the potential drop between the surface and the pff to estimate the energy of the returning positrons at the surface is that for some cases the particles become radiation - reaction limited before reaching the pff , so that the potential drop would be an overestimate . figure 15 shows the distribution of positron heating flux over the pc , @xmath199 , for the case of a non - recycled pulsar with @xmath200 s and @xmath150 g , for different @xmath16 values , while figure 16 shows results for a msp with @xmath157 ms and @xmath201 g. it is evident that most of the heating flux is concentrated at the center of the pc near the magnetic axis near @xmath202 , the contrast between the pc center and edges being about a factor of ten . the returning positron fraction is maximum both near @xmath202 and @xmath133 , where the pff is at the highest altitude . however , the @xmath198 is maximum near @xmath202 , causing a maximum heating flux near the pc center . for @xmath156 , most of the offset side of the pc is heated with lower flux than near the pc center . there is also an asymmetry such that the favorably curved field lines have higher heating flux , also because of higher returning positron energies . the substantially lower heating in the outer part of the offset side of the pc , which has a much larger area , keeps the increase in total pc heating due to the offset to a minimum . in fact in figure 15 and 16 , the physical size of the hottest area is comparable for all values of @xmath16 that we examined . thus the most intensely heated area is several times smaller than the canonical pc size , even for large @xmath16 values . in the case of the msp in figure 16 , this effectively smaller heated spot is shifted from the pc center by a smaller fraction of the canonical pc radius than the shift of the whole pc . for the non - recycled pulsar in figure 15 the heated area remains nearly centered while the whole pc is shifted by a larger fraction . the total heating power ( of precipitating positrons ) can be estimated as ( cf . formula [ 61 ] of hm01 ) l_+ c _ s(z__pff ) _ + ( z__pff , , ) ( z__pff , , ) ds , [ ] where the integration is over the area of a sphere cut by the polar flux tube at the radial distance @xmath203 . figure 17 shows the dependence of @xmath204 , as a fraction of spin down luminosity , on pulsar characteristic age , @xmath205 , for different period and offsets . comparing to the results in fig . 7 of hm01 for the @xmath76 case , the @xmath206 here is higher by factors of 2 - 5 since the values we have given here are the upper limits on positron heating rate , while the hm01 results are from numerical computations of the @xmath6 screening . our estimates here nevertheless serve to test the heating of offset pcs against the observational constraints . in the case of both normal pulsars and msps , the increase in heating for @xmath207 is very minimal . even for @xmath167 , the heating luminosity increases by no more than a factor of 2 , and for msps there is almost no increase except in cases where there are different degrees of pair screening for increasing offsets ( e.g. @xmath157 ms at large ages ) . thermal components have been detected from both middle - aged pulsars and msps in the soft x - ray band . in the case of middle - aged pulsars like psr b1055 - 52 and psr b06556 + 14 , typically both hot and cool thermal components appear in the spectra . the hot components , with luminosity @xmath208 , may be due to pc heating and the cool components from neutron star cooling , but in any case hot components provide a limit to any theoretical heating of the pc . for these pulsars , the pc heating efficiency @xmath209 for b0656 + 14 , and @xmath210 for b1055 - 52 ( deluca et al . 2005 ) , which are within a factor of 2 of the our estimated maximum heating efficiencies . in the case of msps , several have measured thermal components , including psr j0437 - 4715 , j2124 - 3358 and j0030 + 0451 . since these pulsars are too old to have significant cooling , their thermal emission is very likely from pc heating . their measured pc heating efficiencies , @xmath211 ( zavlin 2007 ) are a factor of 2 - 3 lower than the maximum estimated efficiencies in figure 17 . we have examined the change in magnetic flux over the pc as @xmath16 increases . for an offset of @xmath207 for a 0.1 s pulsar , the magnetic flux on the offset side of the pc is typically a factor of 2 higher than for the centered pc , while on the side opposite the offset the flux is a factor of 3 lower , and for larger offsets the flux ratio can be several orders of magnitude . for msps , the contrast in flux is much smaller with ratios of only a factor of 4 - 5 in flux across the pc for offset as high as @xmath184 .

pair spectra cover several decades in energy , with the spectral range of millisecond pulsars ( msps ) two orders of magnitude higher than for normal pulsars , and pc offsets allow significant extension of all spectra to lower pair energies . we find that the total pc pair luminosity is proportional to , with for normal pulsars and for msps . remarkably , the total pc heating luminosity for even large offsets increases by less than a factor of two , even though the pc area increases by much larger factors , because most of the heating occurs near the magnetic axis .

neutron star magnetic fields may have polar caps ( pc ) that are offset from the dipole axis , through field - line sweepback near the light cylinder or non - symmetric currents within the star . the effects of such offsets on electron - positron pair cascades are investigated , using simple models of dipole magnetic fields with small distortions that shift the pcs by different amounts or directions . using a monte carlo pair cascade simulation , we explore the changes in the pair spectrum , multiplicity and energy flux across the pc , as well as the trends in pair flux and pair energy flux with spin - down luminosity , . we also give an estimate of the distribution of heating flux from returning positrons on the pc for different offsets . we find that even modest offsets can produce significant increases in pair multiplicity , especially for pulsars that are near or beyond the pair death lines for centered pcs , primarily because of higher accelerating fields . pair spectra cover several decades in energy , with the spectral range of millisecond pulsars ( msps ) two orders of magnitude higher than for normal pulsars , and pc offsets allow significant extension of all spectra to lower pair energies . we find that the total pc pair luminosity is proportional to , with for normal pulsars and for msps . remarkably , the total pc heating luminosity for even large offsets increases by less than a factor of two , even though the pc area increases by much larger factors , because most of the heating occurs near the magnetic axis .

1111.1668

c

we have investigated the effects of offset polar caps on electron acceleration and pair cascades near the neutron star surface , including the cases where the pcs of the two hemispheres are offset symmetrically by the same amount or asymmetrically . the asymmetric offset case would apply to the pcs shifted by retardation and/or currents of the global magnetosphere , while the symmetric case could apply to neutron stars with some interior current distortions that produce multipolar components near the surface . the asymmetric pc offsets have now been shown to be standard in pulsar magnetosphere global geometry . the pcs in the retarded vacuum magnetosphere are shifted in a direction opposite to that of the rotation by an amount that varies from a maximum of 20% of the standard pc radius at @xmath212 to no shift for @xmath213 ( dyks & harding 2004 ) . in terms of our offset parameter @xmath16 , the ratio of offset to standard pc angle is @xmath214 , so that for the vacuum case @xmath215 , where the large values of @xmath216 apply to msps with large @xmath162 . in the force - free ideal - mhd magnetosphere , the offset percentages range from 45% at @xmath212 to 30% at @xmath168 ( bai & spitkovsky 2010 ) , giving a range @xmath217 , again with the larger values of @xmath218 applying to msps . from our calculations , we find that the pair multiplicity and pair flux for offset pcs is distributed very asymmetrically over the pc , with higher values on the side of the pc toward the offset . the regions of peak pair flux and multiplicity occur for @xmath76 at about half the pc radius , symmetrically around the magnetic axis . as @xmath16 increases , small regions of higher peak flux and multiplicity appear toward the offset and grow larger with increasing offset . the increase in peak multiplicity moves the pair death lines downward in @xmath130-@xmath171 space , to encompass nearly the entire radio pulsar population for @xmath167 . since such offset values are higher than those resulting from retardation and currents in pulsar magnetosphere models , they would likely require interior currents that produce large - scale non - dipolar fields near the neutron star surface . examination of the pair spectra reveal that the increase in multiplicity in pulsars with offset pcs comes from the extension of the spectra to lower pair energies . the discovery of pulsed gamma - ray emission from a large number of millisecond pulsars by the large area telescope ( lat ) on the @xmath219 gamma - ray space telescope ( abdo et al . 2009 ) has revealed light curves that are best modeled by narrow radiation gaps in the outer magnetosphere ( venter et al . 2009 , abdo et al . such narrow gaps require screening of the accelerating electric field over most of the magnetosphere by a pair multiplicity that is orders of magnitude larger than standard models of pc acceleration with no offsets are able to produce ( harding & muslimov 2002 ) . since pc offsets that are only small fractions of a stellar radius result in large increases in pair multiplicity , and there is evidence for such offsets in msps , offset pcs may be a viable explanation for the larger - than - predicted pair activity in msps . we find that the integrated pair flux and luminosity from each pc is roughly proportional to the pulsar spin down luminosity , both in the case of non - recycled pulsars and msps with msps being somewhat more efficient in converting spin - down to pair luminosity . for pulsars with offset pcs , both pair flux and luminosity increase with offset but to a greater degree for low spin down power . due to a saturation of the pair multiplicity , pulsars with high @xmath0 have only modest increases in pc pair flux even for large offsets . we estimate that the crab pulsar produces a pair flux from each pc of about @xmath220 in the case of no offset , @xmath221 for @xmath207 and @xmath222 for @xmath167 . the flux from both pcs is still more than an order of magnitude smaller than the pair flux required to account for the radiation from the nebula , which is estimated to be about @xmath223 ( dejager et al . the trend of pair flux and pair luminosity proportional to spin down power may have interesting implications for explaining observed trends in pulsar pulsed and un - pulsed x - ray luminosity that are also proportional to spin down power ( vink et al . such a trend contrasts that of observed @xmath87-ray luminosities which are proportional to @xmath224 ( thompson et al . 1997 , abdo et al . the @xmath87-ray luminosity trend can be understood if the emission is produced by the primary pc current ( proportional to @xmath224 ) accelerated by a constant voltage of about @xmath225 v ( harding 1981 , arons 1996 ) . the x - ray luminosity trend may be understood if both the pulsed emission from the magnetosphere and the unpulsed emission from the pulsar wind nebula ( pwn ) are produced by pairs . it is generally believed that pairs are producing the emission from pwne , and a number of high - energy emission models argue that secondary electron - positron pairs produce the pulsed x rays high in the magnetosphere through synchrotron and/or ics ( cheng et al . 1986 , romani 1996 , takata et al . 2007 , hirotani 2008 , harding et al . 2008 ) . the asymmetric distribution of particle acceleration and pair multiplicity that results from offset pcs should produce asymmetries in observed pulsar emission . the structure and energetics of the proposed slot gaps ( sg ) that form between the boundary of the open magnetic field and the upward curving pff ( arons & scharlemann 1979 ) , and can accelerate particles to high altitude ( muslimov & harding 2004 ) , could be strongly affected by a an offset pc . the particle lorentz factor @xmath87 in the sg , which is expected to reach curvature radiation - reaction limit such that @xmath226 , will be larger on one side of the pc , producing cr emission power proportional to @xmath6 that is larger than for a dipole field . in older pulsars that do not produce enough pair multiplicity and screening to form sgs in centered pcs , sgs may form on only one side of an offset pc . an azimuthal asymmetry of both the radiation power and width of the sg would change both the @xmath87-ray luminosity and the sharp @xmath87-ray peaks that are due to caustics formed by radiation from trailing edge field lines ( dyks & rudak 2003 ) . since the pc offsets that result from retardation and currents in pulsar magnetospheres occur toward the trailing side of the pc , the @xmath87-ray peaks should be enhanced relative to the off - peak emission that is due to emission along the leading - edge field lines . the ratio of @xmath6 , and thus @xmath87-ray flux , between trailing and leading edges of the sg is predicted to be @xmath227 , which can be larger than an order of magnitude for pulsars having short periods . sg model light curves assuming the emission asymmetry predicted in vacuum or non - ideal mhd magnetospheres fit the pulsar light curves measured by @xmath219 significantly better that those of symmetric sgs ( harding et al . 2011 , decesar et al . the asymmetry in pair flux in offset pcs should produce asymmetries in pulsar radio emission if the radio flux is proportional somehow to the pair flux . in fact , evidence for such asymmetries in pulsar radio emission have been observed in the form of partial cone emission ( mitra & rankin et al . 2011 ) . in a future study , we will explore the effect of an offset pcs on the acceleration of particles in the sg at high altitudes and the change to the shapes of @xmath87-ray light curves . we acknowledge support from the nasa astrophysics theory and fundamental physics program , the @xmath219 guest investigator program and the universities space research association . akh also thanks the aspen center for physics . abdo , a. a. et al . 2009 , science , 325 , 848 . abdo , a. a. et al . 2010 , apj , 712 , 957 abdo , a. a. et al . 2010b , apjs , 187 , 460 . arendt & eilek , j. a. 1998 , ( arxiv : astro - 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ph/0349176 ) . de luca , a. , caraveo , p. a , mereghetti , s. , negroni , m. & bignami , g. 2005 , apj , 623 , 1051 . demorest , p. b. , pennucci , t. , ransom , s. m. , roberts , m. s. e. & hessels , j. w. t. 2010 , nature , 467 , 1081 . dermer , c. d. 1990 , apj , 360 , 197 . deutsch , a. j. , 1955 , ann . dastrophys . , 18 , 1 . dyks , j. & harding , a. k. 2004 , apj , 614 , 869 . dyks , j. & rudak , b. 2003 , apj , 598 , 1201 . friedman , j. l. , ipser , j. r. , & parker , l. 1986 , apj , 304 , 115 . harding , a. k. 1981 , apj , 245 , 267 . harding , a. k. , baring , m. g. & gonthier , p. l. 1997 , apj , 476 , 246 . harding , a. k. , & muslimov , a. g. 1998 , apj , 508 , 328 [ hm98 ] . harding , a. k. , & muslimov , a. g. 2001 , apj , 556 , 987 [ hm01 ] . harding , a. k. , & muslimov , a. g. 2002 , apj , 568 , 862 . harding , a. k. , muslimov , a. g. & zhang , b. 2002 , apj , 576 , 366 . harding , a. k. , j. v. stern , j. dyks & m. frackowiak , 2008 , apj , 680 , 1376 . harding , a. k. , & muslimov , a. g. 2011 , apj , 726 , l10 [ hm11 ] . harding , a. k. , decesar , m. e. , miller , m. c. , kalapotharakos , c. & contopoulos , i. 2011 , 2011 fermi symposium proceedings , econf c110509 ( arxiv : astro - ph/0349902 ) . harding , a. k. & preece , r. 1987 , apj , 319 , 939 . hibschman , j. a. & arons , j. 2001 , apj , 560 , 871 . hirotani , k. 2008 , open astronomy ( arxiv:0809.1283 ) kalapotharakos , k. , kazanas , d. , harding , a. k. & contopoulos , i. 2011 , apj , submitted . kantor , e. m. & tsygan , a. i. 2003 , astronomy reports , 47 , 613 . kantor , e. m. & tsygan , a. i. 2004 , astronomy reports , 48 , 1029 . contopoulos , i. , kazanas , d. & fendt , c. 1999 , apj , 511 , 351 . lamb , f. k. et al , 2009 , apj , 706 , 417 . lattimer , j. m. & prakash , m. 2007 , phys . , 442 , 109 . luo , q. ; shibata , s. ; melrose , d. b. 2000 , mnras , 318 , 943 . manchester , r. n. , hobbs , g. b. , teoh , a. & hobbs , m. 2005 , astron . j. , 129 , 1993 medin , z. & lai , d. 2010 , mnras , 406 , 1379 . mitra , d. & rankin , j. 2011 , apj , 727 , 92 . muslimov , a. g. & tsygan , a. i. 1992 , mnras , 255 , 61 muslimov , a. g. & harding , a. k. 2004 , apj , 606 , 1143 romani , r. w. 1996 , apjj , 470 , 469 . ruderman , m. 1991 , apj , 366 , p. 261 . ruderman , m.a . & sutherland , p. g. 1975 , apj , 196 , 51 sokolov , a. a. & ternov , i. m. 1968 , radiation from relativistic electrons , ed . c. w. kilmister ( aip : new york ) spitkovsky , a. apj , 648 , l51 ( 2006 ) li , j. , spitkovsky , a. & tchekhovskoy , a. 2011 , apj , submitted ( arxiv:1107.0979 ) . takata , j. ; chang , h .- k . ; cheng , k. s. 2007 , apj , 656 , 1044 . thompson , d. j. , harding , a.k . , hersen , w. & ulmer , m.p . 1997 , in proc . of the _ 4th compton symposium _ dermer , m.s . strickman & j.d . kurfess ( aip 410 : new york ) , 39 . timokhin , a. mnras , 36 , 1055 ( 2006 ) timokhin , a. & arons , j. 2011 , apj , in prep . tsai , w. y & erber , t. 1974 , phys . d , 10 , 492 . venter , c. ; harding , a. k. ; guillemot , l. 2009 , apj , 707 , 800 . vink , j , mamba , a. & tamazaki , r. 2011 , apj , 727 , 131 . zavlin , v. e. 2007 , astrophysics space sci . , 308 , 297 . zhang , b. , harding , a. k. & muslimov , a. g. 2000 , apj , 531 , l135 . for the symmetric ( case a , top ) and asymmetric ( case b , bottom ) polar cap offsets.,title="fig:",width=529 ] for the symmetric ( case a , top ) and asymmetric ( case b , bottom ) polar cap offsets.,title="fig:",width=529 ] ) as a function of radial distance from the magnetic axis ( in units of neutron star radius ) and magnetic azimuth for @xmath149 s , @xmath150 g and @xmath169 , for different values of offset parameter @xmath16 . , width=793 ] ) as a function of radial distance from the magnetic axis ( in units of neutron star radius ) and magnetic azimuth for @xmath165 ms , @xmath166 g and @xmath229 , for @xmath167 and different values of inclination angle @xmath117 . , width=793 ] and period derivative ( @xmath171 ) , for a centered pc and inclination angle @xmath169 for normal pulsars and @xmath229 for msps . the ns radius @xmath5 , mass @xmath230 ( in solar mass units ) and moment of inertia @xmath231 refer to different ns equations of state used for normal and millisecond pulsars and are described in the text . radio pulsars with measured @xmath171 from the atnf catalog ( manchester et al . 2005 , http://www.atnf.csiro.au/research/pulsar/psrcat ) are plotted as black dots . , width=793 ] ) as a function of radial distance from the magnetic axis ( in units of neutron star radius ) and magnetic azimuth for @xmath200 s , @xmath150 g and @xmath169 , for different values of offset parameter @xmath16 . , width=793 ] ) as a function of radial distance from the magnetic axis ( in units of neutron star radius ) and magnetic azimuth for @xmath157 ms , @xmath232 g and @xmath229 , for different values of offset parameter @xmath16 . , width=793 ]

the effects of such offsets on electron - positron pair cascades are investigated , using simple models of dipole magnetic fields with small distortions that shift the pcs by different amounts or directions . using a monte carlo pair cascade simulation , we explore the changes in the pair spectrum , multiplicity and energy flux across the pc , as well as the trends in pair flux and pair energy flux with spin - down luminosity , .

neutron star magnetic fields may have polar caps ( pc ) that are offset from the dipole axis , through field - line sweepback near the light cylinder or non - symmetric currents within the star . the effects of such offsets on electron - positron pair cascades are investigated , using simple models of dipole magnetic fields with small distortions that shift the pcs by different amounts or directions . using a monte carlo pair cascade simulation , we explore the changes in the pair spectrum , multiplicity and energy flux across the pc , as well as the trends in pair flux and pair energy flux with spin - down luminosity , . we also give an estimate of the distribution of heating flux from returning positrons on the pc for different offsets . we find that even modest offsets can produce significant increases in pair multiplicity , especially for pulsars that are near or beyond the pair death lines for centered pcs , primarily because of higher accelerating fields . pair spectra cover several decades in energy , with the spectral range of millisecond pulsars ( msps ) two orders of magnitude higher than for normal pulsars , and pc offsets allow significant extension of all spectra to lower pair energies . we find that the total pc pair luminosity is proportional to , with for normal pulsars and for msps . remarkably , the total pc heating luminosity for even large offsets increases by less than a factor of two , even though the pc area increases by much larger factors , because most of the heating occurs near the magnetic axis .

gr-qc0402015

i

there is now an extended body of evidence in support of superfluidity in dense , nucleonic matter , such as that which is believed to exist in neutron stars . using conservative estimates one can argue for a fermi temperature on the order of @xmath2 for neutrons in media with supranuclear densities , and that the transition temperature to a superfluid state is about @xmath3 . this is a signficiant fact since it is generally accepted that nascent neutron stars formed from supernovae should cool fairly quickly , and consequently their internal temperatures should pass quickly through the transition value . observational support for such transitions is supplied by the well established glitch phenomenon in pulsars @xcite . these are rapid decreases in the rotational periods followed by a slow recovery @xcite , much too slow to be explained by ordinary fluid viscosity @xcite . the best description is based on superfluid quantized vortices , and how they pin , unpin and then repin as the pulsar s rotation rate evolves @xcite . we present here a fully relativistic formalism to model the rotational properties of superfluid neutron stars within a slow rotation approximation , which is an extension to second order in the rotation rates the previous work of comer and joynt @xcite . the bulk of the studies of neutron star superfluidity have been in the newtonian regime , where the intent is not to be quantitatively descriptive but rather to gain qualitative understandings . however , it is becoming increasingly apparent that general relativity is required to obtain even qualitative understandings . and even if the qualitative does not vary between the two regimes , the number of examples is growing where general relativity can yield factors of two difference from newtonian calculations ( as opposed to `` merely '' @xmath4 to @xmath5 corrections ) . for example , recent modelling of supernovae has revealed that when general relativity is included the range of model parameters that exhibit multiple bounces is significantly smaller than the range found for the newtonian case @xcite . the use of general relativistic hydrodynamics has also led to predictions of the shock radius ( during the shock reheating phase ) being reduced by a factor of two and a corresponding increase by a factor of two for the inflow speed of the material behind the shock @xcite . certainly the need to use general relativity must only be enhanced as supernova remnants become more compact . weber @xcite provides an excellent overview of the many suggestions for the matter content of these remnants , the neutron stars . they range from the traditional neutron / proton / electron models to more exotic configurations with kaon or muon condensates in the core , or hyperons , or even strange stars with absolutely stable u , d , s quark matter . in our study we will be somewhat conservative by considering typical traditional neutron star models , composed of superfluid neutrons , superconducting protons ( with proton fractions on the order of @xmath6 ) , and a highly degenerate relativistic gas of electrons . each species extends from the center of the star all the way to the surface , although in principle one could consider a more realistic scenario wherein the protons and electrons extend out further than the neutrons @xcite and in that way mimic features of a crust . this same technique could be applied in neutron star cores , allowing for the possibility of alternating regions of ordinary fluid and superfluid . the local thermodynamic equilibrium of the matter will be modelled using a relativistic @xmath0 - @xmath1 mean field approach of the type that is attributed to walecka @xcite and discussed in detail for neutron stars by glendenning @xcite . we consider a relativistic approach to be important on two different levels . on the macroscopic level there is the need for general relativity that was discussed above . but on a microscopic level , recall that any fluid approximation for matter has built into it the notion of local , fluid elements . they are small enough that they can be considered to be points with respect to the rest of the star , and yet large enough to contain , say , an avogadro s number of particles . at the densities expected for neutron stars , the local , fermi levels for the nucleons can become high enough that the effective velocities of the nucleons with respect to their fluid elements are relativistic . as for the fluid elements themselves , they can also , in principle , approach speeds near that of light , although in practice ( e.g. for quasinormal mode @xcite or slow rotation calculations @xcite ) they will typically have speeds a few percent of that of light . because of the superfluidity of the neutrons , and superconductivity of the protons , the fluid formalism to be used differs fundamentally from the standard perfect fluid approach in that the neutrons can flow independently of the protons and electrons . there are thus two fluid degrees of freedom in the system , requiring two sets of fluid elements , one set for the neutrons and another for the charged constituents since the electromagnetic interaction very effectively `` ties '' the electrons to the protons . the matter description , therefore , must take into account two fermi levels for the nucleons , and a displacement in momentum space between their respective fermi spheres that will result when one fluid flows with respect to the other . the specification of a local thermodynamic equilibrium for the two fluids requires that the local neutron and `` proton '' ( i.e. a conglomeration of the protons and electrons ) number densities be known as well as the local , relative velocity of the proton fluid elements , say , with respect to those of the neutrons . when the fermi spheres for the nucleons are displaced with respect to each other there results an important effect for neutron star dynamics known as entrainment @xcite . sauls @xcite describes the entrainment effect using quasiparticle language , i.e. just because the neutrons are superfluid and protons superconducting does not mean they no longer feel the strong force . on the contrary , an individual neutron should be understood as being surrounded by a polarization cloud of other neutrons and protons . when this neutron moves , it will be accompanied by this cloud of nucleons . the net effect at the level of the fluid elements is that the momentum of a neutron fluid element is a linear combination of the neutron _ and _ proton number density currents . parameters that are important for entrainment in neutron stars have been calculated in the newtonian regime using a fermi - liquid approach @xcite and in the relativistic regime via a @xmath0 - @xmath1 mean field model @xcite . comer and joynt @xcite find that one of the key parameters that has been much used in superfluid neutron star modelling extends over a much larger range of values than what the newtonian analysis of borumand et al would imply @xcite . comer and joynt also used their formalism to obtain first order rotational corrections to a superfluid neutron star s equilibrium state , which for general relativity means determining the frame - dragging , and the angular momentum . we will extend those calculations to the second order in the rotation rates , and thereby determine rotational corrections to the metric , distribution of particles , the total mass , shape , and the kepler , mass - shedding limit . we construct sequences of equilibrium configurations where members of the sequence have the same relative rotation between the neutrons and protons but are distinguished by their central neutron number densities . we also construct sequences where the central neutron number density is fixed and the relative rotation is allowed to vary ( cf . @xcite for discussions on when to expect a relative rotation ) . one other straightforward , but important , extension here of the work of comer and joynt is a proof that the matter coefficients obtained by them are sufficient for our second order calculations . in order to have a reasonably self - contained document , and to define all the variables , we review in sec . [ review ] the general relativistic superfluid formalism and its application to slowly rotating neutron stars . it is in this section that we prove the matter coefficients obtained by comer and joynt are all that is needed for the extension to second order . in sec . [ meanfield ] we discuss the highlights of the relativistic @xmath0 - @xmath1 model and its mean field limit . we also determine the model s slow rotation limit . in sec . [ results ] we join the slow rotation formalism with the mean field model and produce numerical solutions . after reviewing the main results , the final , concluding section discusses applications beyond those considered here and points out where the formalism should be improved . for convenience we have restated in the appendix results of comer and joynt for the various matter coefficients that are required input for the field equations . we use `` mtw '' @xcite conventions throughout and geometrical units .

neutron stars that are cold enough should have two or more superfluids / supercondutors in their inner crusts and cores . an important consequence of the quasiparticle nature of each condensate is the so - called entrainment effect , i.e. the momentum of a condensate is a linear combination of its own current and those of the other condensates . we present here the first fully relativistic modelling of slowly rotating superfluid neutron stars with entrainment that is accurate to the second - order in the rotation rates . the stars consist of superfluid neutrons , superconducting protons , and a highly degenerate , relativistic gas of electrons . we determine the effect of a relative rotation between the neutrons and protons on a star s total mass , shape , and kepler , mass - shedding limit .

neutron stars that are cold enough should have two or more superfluids / supercondutors in their inner crusts and cores . the implication of superfluidity / superconductivity for equilibrium and dynamical neutron star states is that each individual particle species that forms a condensate must have its own , independent number density current and equation of motion that determines that current . an important consequence of the quasiparticle nature of each condensate is the so - called entrainment effect , i.e. the momentum of a condensate is a linear combination of its own current and those of the other condensates . we present here the first fully relativistic modelling of slowly rotating superfluid neutron stars with entrainment that is accurate to the second - order in the rotation rates . the stars consist of superfluid neutrons , superconducting protons , and a highly degenerate , relativistic gas of electrons . we use a relativistic - mean field model for the equation of state of the matter and the entrainment . we determine the effect of a relative rotation between the neutrons and protons on a star s total mass , shape , and kepler , mass - shedding limit . 00a_0 ^ 0 00b_0 ^ 0 00c_0 ^ 0 00d_0 ^ 0 0.m_*|_o

hep-lat0412040

i

two - dimensional ( @xmath7 ) lattice spin models , besides being interesting on their own right as models of ferromagnets can be viewed as mathematically well defined scheme of the nonperturbative regularization of quantum continuum field theories like @xmath0 principal chiral model or @xmath8 nonlinear sigma model . being a discrete theory lattice models can be simulated on computers by monte - carlo method , and this is nowadays the most important tool of obtaining physical results near the continuum limit . among major analytical methods one could mention the strong coupling expansion and the perturbation theory ( pt ) . the pt is essentially the only analytical tool which provides systematic expansion of different physical quantities in the region of weak coupling , i.e. in the region relevant for the construction of the continuum limit . however , at the best pt is only applicable for studying short - distance quantities and the most important and interesting phenomena , like the mass gap generation can not be described in its frameworks . when attempting to go beyond pt in the region of weak bare coupling , one runs into various mathematical problems , the most important being an absence of any analytical control over exponentially small contributions and , hence the absence of any reliable method to study long - distance physics where nonperturbative effects dominate even if they are exponentially suppressed . these facts impelled people to look for different , though equivalent representations which would allow to study the long - distance physics . one of such popular and promising representations is known as a dual formulation and is based on a certain non - classical change of variables . this formulation deals with dual lattice and appears to be very fruitfull for abelian models @xcite . namely , within such formulation 1 ) have been obtained various important analytical results on the long - distance dynamics of abelian spin fields and 2 ) the topological structure of the vacuum is transparent and can be more easily studied . in the context relevant to this paper we would like to mention the dual of the abelian @xmath4 model @xcite which has been used to prove the existence of a soft phase at low temperatures with power - like decay of the correlation function in two - dimensional @xmath4 model @xcite . in this case the dual of the @xmath4 model is a local theory for certain discrete variables . the conventional dual transformations @xcite for non - abelian models are perfectly well defined in mathematical terms and also lead to a local dual theory for integers which label irreducible representations of the symmetry group . nevertheless , these transformations are not complete if one compares to the abelian case . first of all , the resulting dual variables are not independent but are subject to some constraints known as triangular conditions , and as such they can not really be associated with elements of a dual lattice ( of course , introduction of the dual lattice is not the crucial point but rather matter of convenience ; e.g. , in the abelian case one can work with dual variables on the original lattice ) . it is to say , however that it is not really clear what are true independent degrees of freedom of the duals of non - abelian models . secondly , such formulation is so mathematically involved that today it is even far from obvious if it can be useful for any kind of the study of the model , except for the strong coupling expansion . in particular , the positivity of the boltzmann weight remains unclear ( see discussion in @xcite ) . even the precise definition of what can be termed the dual boltzmann weight is not really obvious , so the possibility of numerical simulations is at present rather uncertain . most importantly , however is that it is not clear how one could proceed in an analytical study of the model at low - temperatures . ] . on the other hand , there exists representation of @xmath0 and @xmath1 spin models in terms of link variables @xcite , and this representation can be formulated directly on the dual lattice . it is a first goal of the present paper to use the link representation for the derivation of an exact dual formulation of non - abelian spin models . here we give two such formulations which appear to be quite different from the conventional formulation mentioned above . in our opinion the most essential advantages of our formulations are that 1 ) one of them can be definitely used for the monte - carlo simulations in @xmath7 and 2 ) they are much more suitable for an analytical investigation of the model in the low - temperature region . the second fact follows from the properties of the link formulation and we refer for the details of why it is so to our papers @xcite . in the last of those papers we have already presented a dual of @xmath7 @xmath5 spin model and proposed approximate representation for the dual partition function at low temperatures . another important feature of one of our formulations is that it carries a close analogy with the abelian @xmath9 model . for instance , for @xmath7 @xmath5 model there are 3 independent degrees of freedom per ( dual ) site . at low temperatures it possesses an analog of the vortex spin - wave representation , etc . the deficiency of our approach is that it can be straightforwardly applied only to models in which spins are elements of the lie group . we do not know any obvious generalization of link formulation to , say , @xmath8 nonlinear sigma models . low - temperature properties of @xmath7 non - abelian models are crucial for construction of their continuum limit . it is commonly recognized that models possess no phase transition , correlation function has exponential decay at any coupling and models are asymptotically free . despite being more than twenty five years old this expectation has not been proven rigorously . on the contrary , certain percolation theory arguments suggest that all non - abelian models have soft low - temperature phase with power - like decay of the correlation function @xcite . it is thus another important motivation of the present investigation , namely to get deeper insight into the nature of the mass gap in non - abelian spin models . for example , in many papers devoted to @xmath7 non - abelian models it is written that `` there is a nonperturbative mass gap generation at arbitrarily small couplings '' . it is not really clear , however what is precise meaning of this `` nonperturbative generation '' . it can not be a simple consequence of the link decorrelation which happens in @xmath10 models . then , one could ask if this `` nonperturbative generation '' follows from the existence of some non - trivial background of defects like vortices of the @xmath4 model or is due to the strong but smooth disorder of non - abelian spins , e.g. like center vortices @xcite . as is well known , the dual formulation of abelian models have been extremely useful in clarifying all these important physical problems @xcite . we think that the dual formulation given here is a good starting point for obtaining reliable analytical results in the low - temperature limit of non - abelian models . this is our next goal to develop a technique within dual formulation of non - abelian models which would allow to investigate them in the limit of the weak coupling . in this context we study here the following two approaches on the simplest example of @xmath5 spin model * we derive an asymptotic expansion of the dual boltzmann weight when @xmath11 uniformly valid in fluctuations of the dual variables . the same procedure is also done for the two - point correlation function . as will be seen , at low temperatures the boltzmann weight converges to a certain gaussian ensemble the fact which ensures a possibility to define an analog of the vortex spin - wave representation for the partition and correlation functions in the semiclassical limit . this step provides a major simplification of the model as @xmath11 but even in this case it can not be yet solved exactly . the formulation obtained permits very simple calculation of the leading perturbative contribution to the correlation function . this contribution results in the power - like decay of the correlation function . * the second approach consists in replacing @xmath5 matrix elements by @xmath6 ones in the vicinity of the identity element of @xmath5 . we shall give a proof that such replacement is valid at low temperatures , compute the asymptotics of the dual boltzmann weight and present simple numerical evidence that practically for all configurations this way of calculations produces very reasonable approximation for the original boltzmann weight . this paper is organized as follows . in the section 2 we review briefly the conventional dual transformations . after introduction of the link formulation we develop original approach to the dual transformations and present two forms of the duals of the principal chiral model . in the section 3 we investigate the low - temperature properties of the dual boltzmann weights of @xmath5 model in two dimensions along the lines described above . some technical details of this investigation are given in the appendix . in the section 4 we summarize our results and outline some perspectives for future investigations . also , we calculate here the leading perturbative contribution to the correlation function .

non - abelian lattice spin models with symmetry group or can be formulated in terms of link variables which are subject to the bianchi constraints . using this representation we derive exact and local dual formulation for the partition function of such models on a cubic lattice in arbitrary dimension . locality means that the dual action is given by a sum over some subset of hypercubes of the dual lattice and the interaction between dual variables ranges over one given hypercube . dual variables are in general discrete - valued and live on-cell of the dual lattice , in close analogy with the model . we use our construction to study in details the dual of principal chiral model in two dimensions . leading terms of the asymptotic expansion of the dual boltzmann factor are computed and it is proven that at low temperatures it converges to a certain gaussian distribution uniformly in all fluctuations of dual variables . this result enables us to define the semiclassical limit of the dual formulation and to determine an analog of the vortex spin - wave representation for the partition function . such representation is used to extract leading perturbative contribution to the correlation function which shows power - like decay at weak coupling . * dual formulations of non - abelian spin models : local representation and low - temperature asymptotics * * o. borisenko , v. kushnir * +

non - abelian lattice spin models with symmetry group or can be formulated in terms of link variables which are subject to the bianchi constraints . using this representation we derive exact and local dual formulation for the partition function of such models on a cubic lattice in arbitrary dimension . locality means that the dual action is given by a sum over some subset of hypercubes of the dual lattice and the interaction between dual variables ranges over one given hypercube . dual variables are in general discrete - valued and live on-cell of the dual lattice , in close analogy with the model . we use our construction to study in details the dual of principal chiral model in two dimensions . we give dual expressions also for two - point correlation function in arbitrary representation and for the free energy of defects . leading terms of the asymptotic expansion of the dual boltzmann factor are computed and it is proven that at low temperatures it converges to a certain gaussian distribution uniformly in all fluctuations of dual variables . this result enables us to define the semiclassical limit of the dual formulation and to determine an analog of the vortex spin - wave representation for the partition function . such representation is used to extract leading perturbative contribution to the correlation function which shows power - like decay at weak coupling . we also present some analytical evidences that the low - temperature limit of the dual formulation is completely described by-like approximation of matrix elements . * dual formulations of non - abelian spin models : local representation and low - temperature asymptotics * * o. borisenko , v. kushnir * +

0806.2739

i

if the dimensions of a device become smaller than the phase coherence length @xmath0 of charge carriers , classical transport theories are not valid any more . instead , carrier dynamics is now governed by quantum mechanics , and the wave - like nature of particles becomes important . in general , the conductance / resistance of such a device does not follow ohm s law . in the regime of coherent quantum transport , the landauer - bttiker formalism @xcite relates the conductance @xmath1 of a device to the total transmission probability @xmath2 of charge carriers through the device , @xmath3 where @xmath4 is the transmission amplitude between different states with transverse quantum numbers @xmath5 and @xmath6 in the left and right lead , respectively . a state with a given transverse quantum number @xmath5 is also called _ channel _ @xmath5 . the problem of calculating the conductance is thus reduced to calculating scattering eigenfunctions @xmath7 for a given energy @xmath8 : @xmath9 where @xmath10 is the hamiltonian of the system . alternatively , the transmission probability can be extracted from the retarded green s function @xmath11 that obeys the equation of motion @xmath12 the fisher - lee relation @xcite then allows to calculate the transmission ( @xmath4 ) and reflection ( @xmath13 ) amplitudes from @xmath14 . in its simplest form , the fisher - lee relation reads @xmath15 and @xmath16 where @xmath17 is the velocity of channel @xmath5 and the integration runs over the cross - section @xmath18 ( @xmath19 ) of the left ( right ) lead . the landauer - bttiker formalism can also deal with multi - terminal systems , but is restricted to linear response , i.e. small bias voltages . in the general case including external bias , the conductance can be calculated using the non - equilibrium green s function formalism ( see , e.g. @xcite ) . except for particularly simple examples , solving eqs . ( [ eq : scattering_eigenstate ] ) and ( [ eq : retarded_greenfunction ] ) exactly is not possible , and therefore a numerical computation is often the method of choice . instead of solving directly a differential equation with its continuous degrees of freedom , such as the schrdinger equation , numerical computations are usually only attempted within a discrete basis set . the differential equation is then replaced by a set of linear equations , and the hamiltonian @xmath10 can be written as a matrix . very often , only few of the matrix elements @xmath20 are nonzero . such _ tight - binding _ representations of the hamiltonian are ubiquitous in quantum transport calculations and can arise from finite differences @xcite , from the finite element method @xcite , from atomic orbitals in empirical tight - binding @xcite or kohn - sham orbitals within density functional theory @xcite . when describing transport , the systems under consideration are open and thus extend to infinity . as a consequence , the tight - binding matrix @xmath10 is infinite - dimensional . however , the conductance calculation can be reduced to a finite problem by partitioning the system into a finite scattering region attached to leads that extend to infinity , as schematically depicted in fig . [ fig : grids](a ) . for the case of two - terminals , the matrix @xmath10 can be written as @xmath21 where @xmath22 is the ( infinite ) hamiltonian of the left ( right ) lead , @xmath23 is the hamiltonian of the scattering region and of finite size . the matrices @xmath24 and @xmath25 represent the coupling between the scattering region and the left and right lead , respectively . in order to reduce the problem size , it is useful to introduce the retarded self - energy @xmath26 , where @xmath27 is the surface green s function of the left ( right ) lead , i.e. the value of the green s function at the interface of the lead disconnected from the scattering region . then , the green s function @xmath28 of the scattering region can be calculated as @xcite @xmath29 reminiscent of eq . ( [ eq : retarded_greenfunction ] ) but with an effective hamiltonian @xmath30 of finite size . this treatment is easily extended to multi - terminal systems . note that it suffices to know the surface green s function of the ( semi-)infinite leads , as in a tight - binding hamiltonian the matrices @xmath31 and @xmath32 have only few nonzero entries . for simple systems , the surface green s function can be calculated analytically @xcite , whereas in more complex situations it must be computed numerically , either by iteration @xcite , or by semi - analytical formulas @xcite . the original infinite - dimensional problem has thus been reduced to a finite size matrix problem that can , in principle , be solved straight - forwardly on a computer . however , for any but rather small problems , the computational task of the direct inversion in eq . ( [ directinversion ] ) is prohibitive . therefore , for two - terminal transport , many algorithms make use of the _ sparsity _ of the hamiltonian matrix in tight - binding representation - in particular that this matrix can be written in block - tridiagonal form : @xmath33 where the index @xmath34 ( @xmath35 ) denotes the blocks in the left ( right ) lead , @xmath36 the blocks within the scattering region , and @xmath37 ( @xmath38 ) the first block in the left ( right ) lead . such a form arises , for example , naturally in the method of finite differences , when grid points are grouped into vertical slices according to their @xmath39-coordinates , as shown in fig . [ fig : grids](b ) , but also applies to any other sparse tight - binding hamiltonian . the block - tridiagonal form of the hamiltonian is the foundation of several quantum transport algorithms for two - terminal systems . the transfer matrix approach applies naturally to block - tridiagonal hamiltonians , but becomes unstable for larger systems . however , a stabilized version has been developed by usuki _ et al . _ @xcite . in the decimation technique @xcite , the hamiltonian of the scattering region is replaced by an effective hamiltonian between the two leads by eliminating internal degrees of freedom . the contact block reduction method @xcite calculates the full green s function of the system using a limited set of eigenstates . the recursive green s function ( rgf ) technique @xcite uses dyson s equation to build up the system s green s function block by block . it has also been adapted to hall geometries with four terminals @xcite and to calculate non - equilibrium densities @xcite . furthermore , the rgf algorithm has been formulated to be suitable for parallel computing @xcite . contains all blocks @xmath40 . ( b ) the green s function @xmath41 is obtained by adding another matrix block . ] of course , there are also other transport techniques not directly based on the block - tridiagonal form of the hamiltonian matrix , such as extracting the green s function from wave packet dynamics @xcite . still , such algorithms are not as widely used as the large class of algorithms , that are directly based on the block - tridiagonal form of the hamiltonian . in order to illustrate the typical computational tasks of this class of algorithms , we briefly explain , as a representative example , the rgf algorithm . the rgf technique is based on dyson s equation @xmath42 ( see , e.g @xcite ) , where @xmath14 denotes the green s function of the perturbed system , @xmath43 that of the unperturbed system and @xmath44 the perturbation . using this equation , the system is built up block by block , as depicted in fig . [ fig : rgf ] . let @xmath45 denote the green s function for the system containing all blocks @xmath40 . then , at energy @xmath8 , the green s function @xmath41 is related to @xmath45 by @xmath46 and @xmath47 starting from @xmath48 , the surface green s function of the right lead , @xmath49 slices are added recursively , until @xmath50 has been calculated . the blocks of the green s function of the full system necessary for transport are then given by @xmath51 and @xmath52 where @xmath53 is the surface green s function of the left lead . @xmath54 and @xmath55 are sufficient to calculate transmission and reflection probabilities via the fisher - lee relation , eqs . and . each step of the algorithm performs inversions and matrix multiplications with matrices of size @xmath56 . since the computational complexity of matrix inversion and multiplications scales as @xmath57 , the complexity of the rgf algorithm is @xmath58 . thus , it scales linearly with the `` length '' n , and cubically with the `` width '' @xmath56 of the system . this scaling also applies to most of the other transport algorithms mentioned above . while for particular cases general transport algorithms , such as the rgf algorithm , can not compete with more specialized algorithms , such as the modular recursive green s function technique @xcite that is optimized for special geometries , they are very versatile and easily adapted to two - terminal geometries provided that the leads are arranged collinearly . amongst other things , this restriction will be lifted by the approach presented in this work . although the block - tridiagonal structure of @xmath10 , eq . ( [ eq : blocktridiagonal ] ) , that arises naturally in many problems appears to have a small `` width '' and thus seems to be quite suitable for transport algorithms , optimizing the block - tridiagonal structure by reordering the matrix @xmath10 may lead to a significant speed - up in the conventional two - terminal algorithms , as we show below . furthermore , such a reordering allows for the application of the established two - terminal algorithms to more complex geometries , such as non - collinear leads or multi - terminal structures , that would otherwise need the development of specialized algorithms . below , we develop a matrix reordering algorithm based on graph partitioning techniques that brings an arbitrary matrix @xmath10 into a block - tridiagonal form optimized for transport calculations . to this end , the paper is organized as follows . in section [ section : algorithm ] we formulate the matrix reordering problem in the language of graph theory and develop the reordering algorithm . in section [ section : examples ] we apply this algorithm to various examples and investigate its performance and the performance of the rgf algorithm for the reordered hamiltonian @xmath10 . we conclude in section [ section : conclusions ] .

numerical quantum transport calculations are commonly based on a tight - binding formulation . a wide class of quantum transport algorithms requires the tight - binding hamiltonian to be in the form of a block - tridiagonal matrix . here , we develop a matrix reordering algorithm based on graph partitioning techniques that yields the optimal block - tridiagonal form for quantum transport . the reordered hamiltonian can lead to significant performance gains in transport calculations , and allows to apply conventional two - terminal algorithms to arbitrary complex geometries , including multi - terminal structures .

numerical quantum transport calculations are commonly based on a tight - binding formulation . a wide class of quantum transport algorithms requires the tight - binding hamiltonian to be in the form of a block - tridiagonal matrix . here , we develop a matrix reordering algorithm based on graph partitioning techniques that yields the optimal block - tridiagonal form for quantum transport . the reordered hamiltonian can lead to significant performance gains in transport calculations , and allows to apply conventional two - terminal algorithms to arbitrary complex geometries , including multi - terminal structures . the block - tridiagonalization algorithm can thus be the foundation for a generic quantum transport code , applicable to arbitrary tight - binding systems . we demonstrate the power of this approach by applying the block - tridiagonalization algorithm together with the recursive green s function algorithm to various examples of mesoscopic transport in two - dimensional electron gases in semiconductors and graphene . , coherent quantum transport , recursive green s function technique , block - tridiagonal matrices , matrix reordering , graph partitioning 72.10.bg , 02.70.-c , 02.10.ox 05c50 , 05c78

1005.3281

i

we have used _ chandra_/letgs observations to study the coronae of the @xmath0 boo binary , and we have compared our results with a previous study of five other moderately active gk dwarfs : @xmath3 eri , 70 oph ab , and 36 oph ab . our findings are summarized as follows : 1 . the two stellar components of @xmath0 boo are resolved for the first time in x - rays . we find that @xmath0 boo a accounts for 88.5% of the counts in the zeroth order image . for comparison , spectral analysis implies an 84.6% contribution to the system s x - ray flux in the rosat - like 0.1 - 2.4 kev bandpass . 2 . with @xmath0 boo b detected for the first time in x - rays , we now know it has an x - ray luminosity comparable to stars known to have strong stellar winds , leading to the conclusion that @xmath0 boo b may account for most of the modest wind observed from the system , with the more active @xmath0 boo a possessing a surprisingly weak wind despite its active corona . this strengthens the case for the presence of a `` wind dividing line '' separating moderately active stars with strong winds and very active stars with weak winds , possibly due to the increasing strength of large scale global magnetic fields that inhibit mass loss . 3 . our emission measure analysis of @xmath0 boo a and b finds evidence for a narrow peak in em at @xmath1 for both stars , a peak also seen for the other moderately active g8-k5 dwarfs in our sample , indicating that the coronal heating mechanism operating on these stars favors this temperature . the emission measure analysis provides us with coronal abundances relative to fe . looking for coronal abundance anomalies requires correcting for stellar photospheric abundances , which in turn involves assuming solar photospheric abundances , since the stellar abundances we use are measured relative to solar . when we simply assume solar abundances from @xcite , for our sample of stars we find that the high - fip element abundances are inconsistent with each other . this inconsistency is decreased significantly for all our stars if we instead assume a photospheric ne abundance consistent with stellar coronal measurements from @xcite , and solar abundances from @xcite for other elements . this is an argument in favor of these latter photospheric abundances , unless there are actual coronal fractionation processes taking place among the high - fip elements that we are erroneously removing . 5 . among low - fip ions , the coronal mg / si ratio is consistently higher than the photospheric ratio for all of our g8-k5 stars , presumably an effect of actual coronal fractionation . an enhancement of mg relative to si in the corona is predicted by many of the models of @xcite , which are designed to explain solar / stellar coronal abundance anomalies . however , although @xcite also finds high mg / si ratios for early g dwarfs , coronal mg / si ratios appear to be _ lower _ than photospheric for m dwarfs @xcite , suggesting a spectral type dependence for this coronal abundance ratio . 6 . with regards to the fip effect , our sample of stars exhibit a range of fip bias , with 70 oph a and @xmath0 boo a having the strongest solar - like fip effect , @xmath3 eri showing little or no fip effect , and 70 oph b possessing a slight inverse fip effect . expanding our sample of stars even further to include early g stars from @xcite and m dwarfs from @xcite , we find that the fip bias correlates beautifully with spectral type , but only if extremely active stars with @xmath2 are excluded . . the tight correlation of fip bias with spectral type suggests to us that the nature and causes of the fip effect can ultimately be traced directly to fundamental surface properties . among normal main sequence stars , these change in a very predictable manner with spectral type , explaining the tight correlation in figure 9 . no such correlation exists for extremely active stars , because many are evolved stars with disparate photospheric characteristics , and in other cases the stellar atmospheres will be affected by rapid rotation or the effects of close binarity . support for this work was provided by the national aeronautics and space administration through chandra award number go8 - 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( pc ) & 6.70 & 6.70 & 1 + radius ( r@xmath8)&0.83 & 0.61 & 2 + p@xmath56 ( days ) & 6.2 & 11.5 & 3 + @xmath57 & 28.86 & 27.97 & 4 + @xmath57 & 28.91 & 28.17 & + @xmath58 ( @xmath59 ) & 5 & 5 & 5 + @xmath60 & 17.92 & 17.92 & 6 + @xmath61}$ ] & @xmath62 & ... & 7 + @xmath63}$ ] & @xmath64 & ... & 7 + @xmath65}$ ] & @xmath66 & ... & 7 + @xmath67}$ ] & @xmath62 & ... & 7 + @xmath68}$ ] & @xmath66 & ... & 7 + lcccc si xiii & 6.648 & 6.99 & @xmath69 & @xmath70 + si xiii & 6.688 & 6.99 & & + si xiii & 6.740 & 6.99 & & + mg xii & 8.419 & 7.11 & @xmath71 & @xmath72 + mg xii & 8.425 & 7.11 & & + mg xi & 9.169 & 6.80 & @xmath73 & @xmath74 + mg xi & 9.231 & 6.80 & & + mg xi & 9.314 & 6.79 & & + ne x & 12.132 & 6.87 & @xmath75 & @xmath76 + ne x & 12.138 & 6.87 & & + fe xvii & 12.264 & 6.62 & @xmath77 & @xmath78 + fe xxi & 12.285 & 6.98 & & + ne ix & 13.447 & 6.58 & @xmath79 & @xmath80 + ne ix & 13.553 & 6.58 & & + ne ix & 13.699 & 6.58 & @xmath81 & @xmath82 + fe xviii & 14.203 & 6.74 & @xmath83 & @xmath84 + fe xviii & 14.208 & 6.74 & & + fe xvii & 15.015 & 6.59 & @xmath85 & @xmath86 + fe xvii & 15.262 & 6.59 & @xmath87 & @xmath88 + o viii & 15.176 & 6.65 & & + fe xix & 15.198 & 6.83 & & + o viii & 16.006 & 6.63 & @xmath89 & @xmath90 + fe xviii & 16.005 & 6.73 & & + fe xviii & 16.072 & 6.73 & & + fe xvii & 16.778 & 6.58 & @xmath91 & @xmath92 + fe xvii & 17.053 & 6.58 & @xmath93 & @xmath94 + fe xvii & 17.098 & 6.58 & & + o vii & 18.627 & 6.34 & @xmath95 & @xmath96 + o viii & 18.967 & 6.59 & @xmath97 & @xmath98 + o viii & 18.973 & 6.59 & & + o vii & 21.602 & 6.32 & @xmath99 & @xmath100 + o vii & 21.807 & 6.32 & @xmath101 & @xmath102 + o vii & 22.101 & 6.31 & @xmath103 & @xmath104 + n vii & 24.779 & 6.43 & @xmath105 & @xmath106 + n vii & 24.785 & 6.43 & & + c vi & 33.734 & 6.24 & @xmath107 & @xmath108 + c vi & 33.740 & 6.24 & & + s xiii & 35.667 & 6.43 & @xmath109 & @xmath110 + si xi & 43.763 & 6.25 & @xmath111 & @xmath112 + si xii & 44.019 & 6.44 & @xmath113 & @xmath114 + si xii & 44.165 & 6.44 & @xmath115 & @xmath116 + si xii & 45.521 & 6.44 & @xmath117 & @xmath118 + si xii & 45.691 & 6.44 & @xmath119 & @xmath120 + fe xvi & 46.661 & 6.43 & @xmath121 & @xmath122 + fe xvi & 46.718 & 6.43 & & + si xi & 49.222 & 6.24 & @xmath123 & @xmath124 + fe xvi & 50.361 & 6.43 & @xmath125 & @xmath126 + fe xvi & 50.565 & 6.43 & @xmath127 & @xmath128 + si x & 50.524 & 6.15 & & + si x & 50.691 & 6.15 & @xmath129 & @xmath130 + si xi & 52.298 & 6.24 & @xmath131 & @xmath132 + fe xv & 52.911 & 6.32 & @xmath133 & @xmath134 + fe xvi & 54.127 & 6.43 & @xmath135 & @xmath136 + fe xvi & 54.710 & 6.43 & @xmath137 & @xmath138 + mg x & 57.876 & 6.22 & @xmath139 & @xmath140 + mg x & 57.920 & 6.22 & & + fe xv & 59.405 & 6.32 & @xmath141 & @xmath142 + fe xvi & 62.872 & 6.42 & @xmath143 & @xmath144 + mg x & 63.295 & 6.21 & @xmath145 & @xmath146 + fe xvi & 63.711 & 6.42 & @xmath147 & @xmath148 + fe xvi & 66.249 & 6.42 & @xmath149 & @xmath128 + fe xvi & 66.357 & 6.42 & & + fe xv & 69.682 & 6.32 & @xmath150 & @xmath151 + fe xv & 69.941 & 6.32 & @xmath152 & @xmath140 + fe xv & 69.987 & 6.32 & & + fe xv & 70.054 & 6.32 & & + mg ix & 72.312 & 6.02 & @xmath153 & @xmath154 + fe xv & 73.472 & 6.32 & @xmath155 & @xmath156 + fe xvi & 76.497 & 6.42 & @xmath157 & @xmath158 + ne viii & 88.082 & 5.96 & @xmath159 & @xmath160 + ne viii & 88.120 & 5.96 & & + fe xviii & 93.923 & 6.68 & @xmath161 & @xmath162 + ne viii & 98.116 & 5.94 & @xmath163 & @xmath164 + ne viii & 98.260 & 5.94 & @xmath165 & @xmath166 + fe xix & 101.550 & 6.80 & @xmath167 & @xmath168 + fe xviii & 103.937 & 6.68 & @xmath169 & @xmath170 + fe xix & 108.355 & 6.79 & @xmath171 & @xmath164 + fe xxii & 117.180 & 7.03 & @xmath172 & @xmath173 + fe xx & 121.845 & 6.88 & @xmath174 & @xmath175 + fe xxi & 128.752 & 6.95 & @xmath176 & @xmath177 + fe xx & 132.840 & 6.88 & @xmath178 & @xmath179 + fe xxiii & 132.906 & 7.12 & & + fe ix & 171.073 & 5.95 & @xmath180 & @xmath181 + cccc @xmath182 & @xmath183 & @xmath184 & 9.4 + @xmath185 & 0.7 & 0.8 & 1 + @xmath186}$ ] & @xmath187 & @xmath188 & @xmath189 + @xmath190}$ ] & @xmath191 & @xmath192 & 1.02 + @xmath193}$ ] & @xmath194 & ... & 0.42 + @xmath195}$ ] & @xmath196 & @xmath197 & 1.33 + @xmath198}$ ] & @xmath199 & @xmath200 & 0.58 + @xmath201}$ ] & @xmath202 & @xmath203 & 0.08 + @xmath204}$ ] & @xmath205 & @xmath206 & 0.05 + @xmath207}$ ] & @xmath208 & ... & @xmath209 +

boo a and b to the system s total x - ray emission now observationally established ( 88.5% and 11.5% , respectively ) , consideration of mass loss measurements for gk dwarfs of various activity levels ( including one for boo ) leads to the surprising conclusion that boo b may dominate the wind from the binary , with boo a s wind being very weak despite its active corona . _ measurements of other moderately active stars with g8-k5 spectral types , all of which exhibit a narrow peak in emission measure near , indicating that the coronal heating process in these stars has a strong preference for this temperature .

on 2008 may 2 , _ chandra _ observed the x - ray spectrum of boo ( g8 v+k4 v ) , resolving the binary for the first time in x - rays and allowing the coronae of the two stars to be studied separately . with the contributions of boo a and b to the system s total x - ray emission now observationally established ( 88.5% and 11.5% , respectively ) , consideration of mass loss measurements for gk dwarfs of various activity levels ( including one for boo ) leads to the surprising conclusion that boo b may dominate the wind from the binary , with boo a s wind being very weak despite its active corona . emission measure distributions and coronal abundances are computed for both stars and compared with _ _ measurements of other moderately active stars with g8-k5 spectral types , all of which exhibit a narrow peak in emission measure near , indicating that the coronal heating process in these stars has a strong preference for this temperature . as is the case for the sun and many other stars , our sample of stars shows coronal abundance anomalies dependent on the first ionization potential ( fip ) of the element . we see no dependence of the degree of `` fip effect '' on activity , but there is a dependence on spectral type , a correlation that becomes more convincing when moderately active main sequence stars with a broader range of spectral types are considered . this clear dependence of coronal abundances on spectral type weakens if the stellar sample is allowed to be contaminated by evolved stars , interacting binaries , or extremely active stars with , explaining why this correlation has not been recognized in the past .

0705.1132

m

once constraints on the principal components of the reionization history have been obtained from cmb polarization data , there are several ways to use those constraints to place limits on observables such as @xmath2 or to test theories of reionization @xcite . we describe some possible applications in this section , and in [ sec : mcmc ] we put these ideas into practice using the 3-year _ wmap _ data and simulated future data . as mentioned in [ sec : intro ] , the constraint on the total optical depth to reionization depends on the assumed model for @xmath4 . the principal component method allows us to explore all globally - averaged ionization histories within a chosen redshift range , @xmath80 . for a given set of eigenmode amplitudes , @xmath91 , equation ( [ eq : mmutoxe ] ) yields the corresponding ionization history which can then be integrated to find the optical depth between any two redshifts @xmath92 and @xmath93 , @xmath94 in particular , the total optical depth to reionization is @xmath95 where @xmath96 for @xmath97 . the principal component approach provides a model - independent way to constrain @xmath2 , so we expect the results to be unbiased and the uncertainty in @xmath2 to accurately reflect the present uncertainty about @xmath4 . moreover , the information about @xmath98 is encapsulated in the first few eigenmodes , i.e. the truncated ionization history of equation ( [ eq : trunc ] ) , @xmath99 , with @xmath100 - 5 for typical fiducial models . the total optical depth and the first principal component amplitude , @xmath101 , are similar quantities in that they are both averages of @xmath4 weighted at high @xmath12 . as we show in the next section , cmb @xmath0-mode polarization data can constrain higher eigenmodes as well ( @xmath102 ) . in particular , the second mode should be the next best constrained quantity since it is constructed to have the smallest variance after @xmath101 . since @xmath103 is related to the difference in @xmath14 between high redshift and low redshift ( see fig . [ fig : eigfn ] ) , we might guess that besides total @xmath2 the data are mainly constraining the fraction of optical depth coming from high redshift versus low redshift . given any set of mode amplitudes defining an ionization history , we can compute a low-@xmath12 optical depth , @xmath104 , and a high-@xmath12 optical depth , @xmath105 , for some choice of intermediate redshift , @xmath106 . note that if either of the redshift ranges @xmath107 $ ] or @xmath108 $ ] is too narrow , constraints on the partial optical depths will be influenced by the physicality priors ( since @xmath109 sets limits on @xmath2 in redshift intervals as discussed in [ sec : phys ] ) and by the uncertainty in modes higher than those included in the chain , which have greater effect on the optical depth in narrower redshift intervals . for these reasons , @xmath106 should be chosen to be not too close to either @xmath110 or @xmath44 . given an appropriately chosen value of @xmath106 , constraints on the principal components can be converted into constraints on the optical depths at high and low redshift . these partial optical depths are observables in the sense that high-@xmath12 and low-@xmath12 optical depth affect the large - scale @xmath0-modes over different ranges of multipoles . for example , compare the two reionization models in figure [ fig : cl ] . both have similar total optical depth , but in one case @xmath2 only comes from @xmath111 , resulting in more power at @xmath112 and less power at @xmath113 than the other model in which the optical depth primarily comes from @xmath114 . besides learning about the relative amount of ionization at @xmath115 and @xmath116 , the partial optical depth constraints also provide a way to empirically set the maximum redshift , @xmath44 . if some set of data are found to place a tight upper bound on @xmath105 for fairly conservative ( i.e. large ) values of @xmath106 and @xmath44 , then for analyses of future data this value of @xmath106 can be used as a new , lower value for @xmath44 since optical depth from higher redshifts is small . this approach assumes that there is essentially no reionization earlier than the original @xmath44 , but as long as this initial maximum redshift is taken to be large the presence or absence of high-@xmath12 ionization can be tested in this way over a wide range of redshifts . we explore this idea further in [ sec : mcmc ] . while we do not examine constraints on specific reionization models in this paper , the principal component approach is well suited to model testing . consider a model of the global reionization history , @xmath117 , parameterized by @xmath118 . this could be a simple toy model ( for example , instantaneous reionization where @xmath118 is a single parameter , the redshift of reionization ) or a more physical model where @xmath118 might include parameters that govern properties of the ionizing sources . for a particular choice of @xmath55 , the principal component amplitudes of the reionization model follow from equation ( [ eq : xetommu ] ) , giving a set of mode amplitudes that depend on the model parameters , @xmath119 . then constraints on the mode amplitudes from cmb polarization data ( defined using the same fiducial history ) can be mapped to constraints on the parameters of the reionization model . applying this method to a model that has as one of its parameters a maximum redshift allows one to obtain constraints on @xmath44 within a class of theoretical models . although obtaining good constraints on model parameters does not necessarily imply validation of the model class , this procedure provides a straightforward method by which different model classes can tested and falsified within a single analysis . generating monte carlo chains to find constraints on @xmath91 as described in the next section can be a somewhat time - consuming process , but it only needs to be done once per redshift range , after which any model of the global reionization history within this range @xmath80 can be tested using the same parameter chains . in cases where constraints from the data turn out to be close to gaussian , the covariance matrix of the principal components can be used in place of the full monte carlo chains , reducing the amount of information needed for model testing from @xmath120 numbers in chains of monte carlo samples to only @xmath121 numbers when @xmath60 eigenmodes are included in the chains [ @xmath60 mean values plus @xmath122 entries in the covariance matrix , @xmath123 . however , near - gaussian constraints on @xmath91 are likely to be possible only for certain realizations of future data , as discussed in the next section .

interpretation of the polarization spectrum usually requires the assumption of a fixed functional form for the evolution , e.g. instantaneous reionization . we develop a model - independent method where a small set of principal components completely encapsulate the effects of reionization on the large - angle-mode polarization for any reionization history within an adjustable range in redshift . using markov chain monte carlo methods constraints on the principal components , which probe the entire reionization history , can test models of reionization , provide model - independent constraints on the optical depth , and detect signatures of high - redshift reionization .

on large angular scales , the polarization of the cmb contains information about the evolution of the average ionization during the epoch of reionization . interpretation of the polarization spectrum usually requires the assumption of a fixed functional form for the evolution , e.g. instantaneous reionization . we develop a model - independent method where a small set of principal components completely encapsulate the effects of reionization on the large - angle-mode polarization for any reionization history within an adjustable range in redshift . using markov chain monte carlo methods , we apply this approach to both the 3-year _ wmap _ data and simulated future data . _ wmap _ data constrain two principal components of the reionization history , approximately corresponding to the total optical depth and the difference between the contributions to the optical depth at high and low redshifts . the optical depth is consistent with the constraint found in previous analyses of _ wmap _ data that assume instantaneous reionization , with only slightly larger uncertainty due to the expanded set of models . using the principal component approach , _ wmap _ data also place a 95% cl upper limit of 0.08 on the contribution to the optical depth from redshifts . with improvements in polarization sensitivity and foreground modeling , approximately five of the principal components can ultimately be measured . constraints on the principal components , which probe the entire reionization history , can test models of reionization , provide model - independent constraints on the optical depth , and detect signatures of high - redshift reionization .

0705.1132

c

observations of the large - scale @xmath0-mode polarization of the cmb in the near future are expected to yield new information about the spatially - averaged reionization history of the universe . the principal components of the reionization history are a promising tool for extracting as much of that information as possible from the data . we have shown that the principal component method can be usefully applied to real , currently available data , and forecasts from simulated data suggest that there is room to substantially improve constraints on the reionization history using this method as measurements of the large - scale @xmath0-modes improve . we find that the key features of the principal component analysis put forward by @xcite continue to apply when we go from the fisher matrix approximation to an exploration of the full likelihood surface using markov chain monte carlo methods . for fairly conservative choices of the maximum redshift of reionization ( @xmath216 - 40 ) , only the first five principal components at most are needed for a complete representation of the @xmath0-mode angular power spectrum to within cosmic variance . to account for arbitrary reionization histories in the analysis of cmb data , only a few additional parameters must be included in chains of monte carlo samples if those parameters are taken to be the lowest - variance principal components of @xmath4 . specific models of reionization can be tested easily by computing their eigenmode amplitudes and comparing with constraints on the eigenmodes from the data . constraints on derived parameters , such as total optical depth or the optical depth from a certain range in redshift , represent other applications of the mcmc constraints on principal components of @xmath4 . often , estimates of the optical depth to reionization are computed assuming instantaneous reionization or some other simple form for @xmath4 . here we extend the analysis of the 3-year _ wmap _ data to allow a more general set of models of the global reionization history . we find that expanding the model space does not significantly widen the uncertainty in @xmath2 beyond the instantaneous reionization value of @xmath217 . robust @xmath2 constraints are important for tests of the dark energy based on the growth of structure since they control the uncertainty on the amplitude of the initial spectrum . moreover , even with current data the principal component constraints are beginning to show the possibility of determining properties of reionization in addition to @xmath2 . by comparing the optical depth from high @xmath12 with that from low @xmath12 , we obtain an upper limit on the contribution to the optical depth from high redshift : @xmath218 at 95% confidence , assuming that there is no significant episode of reionization at @xmath219 . due to the limitations of noise and foreground contamination , only the first two eigenmodes of the reionization history , @xmath101 and @xmath103 , can be determined with present polarization data to any reasonable degree of accuracy . constraints on these two modes come primarily from the main , broad peak in @xmath0-mode power at low @xmath5 from reionization . as measurements of the @xmath0-mode polarization improve , for example from additional _ wmap _ data or through planned future experiments such as _ planck _ , better knowledge of the low - power `` trough '' in @xmath6 between the main reionization peak and the first acoustic peak should enable constraints on the third and higher principal components , up to about @xmath220 for near cosmic variance - limited data . since constraints on these higher modes rely on the ability to identify subtle features in the trough of @xmath6 , the ultimate accuracy to which the eigenmodes can be determined may depend on whether or not the necessary features are well reproduced in the particular random draw of @xmath6 that is available to us . however , even if we are unlucky enough to have a realization in which some of the important features of the spectrum are washed out by randomness , it should still be possible to measure several of the principal components to better accuracy than is currently possible . knowledge of the @xmath221 eigenmodes of @xmath4 , along with improved constraints on @xmath101 and @xmath103 , will allow more stringent tests of reionization models and a better understanding of the global reionization history . : we thank cora dvorkin , gilbert holder , dragan huterer , hiranya peiris , and jochen weller for useful discussions . mjm was supported by a national science foundation graduate research fellowship . wh was supported by the kicp through the grant nsf phy-0114422 , the doe through contract de - fg02 - 90er-40560 and the david and lucile packard foundation . , a. , bernstein , g. , cahn , r. , freedman , w. l. , hewitt , j. , hu , w. , huth , j. , kamionkowski , m. , kolb , e. w. , knox , l. , mather , j. c. , staggs , s. , & suntzeff , n. b. 2006 , arxiv astrophysics e - prints astro - ph/0609591 , l. , hinshaw , g. , komatsu , e. , nolta , m. r. , spergel , d. n. , bennett , c. l. , barnes , c. , bean , r. , dor , o. , halpern , m. , hill , r. s. , jarosik , n. , kogut , a. , limon , m. , meyer , s. s. , odegard , n. , peiris , h. v. , tucker , g. s. , verde , l. , weiland , j. l. , wollack , e. , & wright , e. l. 2006 , arxiv astrophysics e - prints astro - ph/0603450 , d. n. , bean , r. , dor , o. , nolta , m. r. , bennett , c. l. , hinshaw , g. , jarosik , n. , komatsu , e. , page , l. , peiris , h. v. , verde , l. , barnes , c. , halpern , m. , hill , r. s. , kogut , a. , limon , m. , meyer , s. s. , odegard , n. , tucker , g. s. , weiland , j. l. , wollack , e. , & wright , e. l. 2006 , arxiv astrophysics e - prints astro - ph/0603449

on large angular scales , the polarization of the cmb contains information about the evolution of the average ionization during the epoch of reionization . wmap _ data also place a 95% cl upper limit of 0.08 on the contribution to the optical depth from redshifts . with improvements in polarization sensitivity and foreground modeling ,

on large angular scales , the polarization of the cmb contains information about the evolution of the average ionization during the epoch of reionization . interpretation of the polarization spectrum usually requires the assumption of a fixed functional form for the evolution , e.g. instantaneous reionization . we develop a model - independent method where a small set of principal components completely encapsulate the effects of reionization on the large - angle-mode polarization for any reionization history within an adjustable range in redshift . using markov chain monte carlo methods , we apply this approach to both the 3-year _ wmap _ data and simulated future data . _ wmap _ data constrain two principal components of the reionization history , approximately corresponding to the total optical depth and the difference between the contributions to the optical depth at high and low redshifts . the optical depth is consistent with the constraint found in previous analyses of _ wmap _ data that assume instantaneous reionization , with only slightly larger uncertainty due to the expanded set of models . using the principal component approach , _ wmap _ data also place a 95% cl upper limit of 0.08 on the contribution to the optical depth from redshifts . with improvements in polarization sensitivity and foreground modeling , approximately five of the principal components can ultimately be measured . constraints on the principal components , which probe the entire reionization history , can test models of reionization , provide model - independent constraints on the optical depth , and detect signatures of high - redshift reionization .

math0311384

i

during the last 20 years the theory of frames has been growing rapidly , since several new applications have been developed . for example , besides traditional applications as signal processing , image processing , data compression , and sampling theory , frames are now used to mitigate the effect of losses in packet - based communication systems and hence to improve the robustness of data transmission @xcite , and to design high - rate constellations with full diversity in multiple - antenna code design @xcite . to handle these emerging applications of frames new methods have to be developed . one starting point is to first build frames `` locally '' and then piece them together to obtain frames for the whole space . one advantage of this idea is that it would facilitate the construction of frames for special applications , since we can first construct frames or choose already known frames for smaller spaces . and in a second step one would construct a frame for the whole space from them . therefore it is necessary to derive conditions for these components , so that there exists a construction , which yields a frame for the whole space with special properties . various approaches to piecing together familes of vectors to get a frame for the whole space have been done over the years going back to duffin and schaeffer s original work @xcite . one approach used in the wavelet as well as in the gabor case @xcite is to start with non - frame sequences and piece them together to build frames for the whole space . another is to build frames locally and piece them together orthogonally to get frames . we refer to heil and walnut @xcite for an excellent introduction to these methods and gabor frames in general . recently , another approach was introduced by fornasier @xcite . fornasier uses subspaces which are quasi - orthogonal to construct local frames and piece them together to get global frames . in this paper we will formulate a general method for piecing together local frames to get global frames . the importance of this approach is that it is both necessary and sufficient for the the construction of global frames from local frames . some of these results are generalizations of fornasier s work @xcite although they were done before his papers became available to us . another motivation comes from the theory of c*-algebras . just recently casazza , christensen , lindner , and vershynin @xcite proved that the so - called `` feichtinger conjecture '' is equivalent to the weak bourgain - tzafriri conjecture . the feichtinger conjecture states that each bounded frame is a finite union of riesz basic sequences . then , casazza and vershynin @xcite showed that the kadison - singer problem is equivalent to the strong bourgain - tzafriri conjecture and that these two problems have a positive solution if and only if both the feichtinger conjecture and the @xmath0-conjecture have positive solutions . the @xmath0-conjecture states : for every @xmath1 , every unit norm riesz basis is a finite union of @xmath2-unconditional basic sequences . a unit norm sequence @xmath3 is a @xmath2-basic sequence if for every sequence of scalars @xmath4 we have @xmath5 to attack these problems it is important to know into which components we can divide a frame . as we will see in this paper , the necessary divisions will form a frame of subspaces for the space . at this time , it is not even known how to divide a frame into two infinite frame sequences . in this paper we want to answer the following two questions , which relate to the two different motivations : * let @xmath6 be a collection of closed subspaces in a hilbert space @xmath7 in which we want to decompose our function , where each subspace @xmath8 is equipped with a weight @xmath9 , which indicate its importance . when can we find frames for @xmath8 for each @xmath10 so that the collection of all of them is a frame with special properties for the whole space @xmath7 ? * let @xmath11 be a frame for a hilbert space @xmath7 , and let @xmath12 be a partition of @xmath13 so that @xmath14 is a frame sequence for each @xmath15 . which relations exist between the closed linear spans of @xmath14 , @xmath15 ? we start our consideration by giving a brief review of the definitions and basic properties of frames and bases and stating some notation in section [ review ] . in section [ fos ] it will turn out that both questions above lead to the definition of a frame of subspaces . in the first subsection we will state the definition of a frame of subspaces for a given family of closed subspaces @xmath6 in a hilbert space and a family of weights @xmath16 . then it is shown that this definition leads to some answers to the above questions ( see theorem [ transfer_frame ] ) , since it shows that frames of subspaces behave as a link between local components of a frame and the global structure . this will also enlighten the advantage of our approach , since now we can choose the frames for the single subspaces @xmath8 arbitrarily and always get a frame for the whole hilbert space by just collecting them together . thus it differs from previous approaches and is a generalization of the approach of fornasier @xcite . it will turn out that frames of subspaces behave as a generalization of frames . we first give a definition of completeness of a family of subspaces and show that the relation between this property and the notion of a frame of subspaces is similar to the relation between the definition of completeness of a sequence and a frame . further in subsection [ frame_prop ] we introduce an analysis and a synthesis operator , a frame operator , and a dual frame of subspaces for a given frame of subspaces and prove that they behave in an analogous way as the corresponding objects in abstract frame theory . we even obtain a reconstruction formula using these ingredients ( proposition [ reconstruction ] ) . the next subsection deals with parseval frames of subspaces , which share several properties with parseval frames . finally in subsection [ roti ] we show that using the theory of frames of subspaces we can construct several useful resolutions of the identity . section [ riesz_dec ] deals with riesz decompositions , which are a generalization of the notion of riesz bases to our general setting . we further define minimality for a family of subspaces and show that it behaves as expected . also exactness is defined in a canonical way . however , it will turn out that this property is much weaker than exactness of a frame ( compare theorem [ riesz_minimal_exact ] ) . some constructions are given in section [ constr ] . here we first state some results which help constructing frames of subspaces . an extended example concerning the situation of gabor frames is added . in subsection [ constr_frame_riesz ] we then show how to construct frames and riesz frames using a frame of subspaces . finally , section [ harmonic_fos ] deals with harmonic frames of subspaces . these are a generalization of harmonic frames , which distinguish themselves by having an easy construction formula . in both the finite and the infinite dimensional cases we give the definition of a harmonic frame of subspaces , state some results , and give examples , e.g. , subspaces coming from gabor systems and subspaces coming from multiresolution analysis , for their occurance .

one approach to ease the construction of frames is to first construct local components and then build a global frame from these . in this paper we will show that the study of the relation between a frame and its local components leads to the definition of a frame of subspaces . we introduce this new notion and prove that it provides us with the link we need . it will also turn out that frames of subspaces behave as a generalization of frames . in particular , we can define an analysis , a synthesis and a frame operator for a frame of subspaces , which even yield a reconstruction formula . also concepts such as completeness , minimality , and exactness are introduced and investigated . we show that wavelet subspaces coming from multiresolution analysis belong to this class .

one approach to ease the construction of frames is to first construct local components and then build a global frame from these . in this paper we will show that the study of the relation between a frame and its local components leads to the definition of a frame of subspaces . we introduce this new notion and prove that it provides us with the link we need . it will also turn out that frames of subspaces behave as a generalization of frames . in particular , we can define an analysis , a synthesis and a frame operator for a frame of subspaces , which even yield a reconstruction formula . also concepts such as completeness , minimality , and exactness are introduced and investigated . we further study several constructions of frames of subspaces , and also of frames and riesz frames using the theory of frames of subspaces . an important special case are harmonic frames of subspaces which generalize harmonic frames . we show that wavelet subspaces coming from multiresolution analysis belong to this class .

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to answer users changing requirements , software systems must continuously evolve @xcite . having developers to understand a program s source code which may include thousands of lines of code , and focus on those parts on which they want to perform their maintenance activities , is the main reason of 60% to 90% of the overall software maintenance and evolution costs @xcite . this is mainly due to the fact that original developers of the software system may no longer be available in the team , and thus , any maintenance tasks in this phase can be highly error prone . hence , program comprehension and understanding the rationale behind the code are the most critical and time consuming steps for developers during software maintenance and evolution @xcite , being that they spend substantial efforts on reading and finding relative parts of the code rather than applying the modification @xcite . however , using _ code summaries _ can be a solution to this difficulty @xcite . a code summary is a brief description about the functionality and the purpose of a section of the source code . code summaries can help programmers find the relevant parts of the source code to their maintenance tasks much easier and faster @xcite . mcburney et al . @xcite pointed out that the code authors use more development details and low level implementation informations . meanwhile , readers are those who want to understand the code ; therefore , they must be able to deduce the concepts from the low level details . ignoring the way which a reader would write a summary is the most important defect in the existing source code summarization approaches . _ crowdsourcing _ is a novel problem - solving approach which is a way to outsource different tasks to a crowd of people through open call ( e.g. , via the internet ) instead of traditional suppliers @xcite . in recent years , crowdsourcing has attracted significant attentions to support a wide range of software engineering activities like requirements engineering , design , coding , testing , and evolution and maintenance @xcite . however , to the bests of our knowledge , using a crowd of developers to write summaries for a piece of a code is a novel idea in the context of program comprehension which we will present in this article . more specifically , here we explain our crowdsourcing platform , named _ crowdsummarizer _ , that applies the concepts of crowdsourcing , gamification , and natural language processing to motivate developers to write high level summaries for the methods of a java program . furthermore , crowdsummarizer continuously learns a set of weights and sentence templates from the set of methods and their corresponding summaries which have been collected from the crowd so far , and uses them to also automatically generate the summaries . this helps developers to save time and not necessarily wait for the crowd to write summaries for their methods . we have implemented crowdsummarizer as a web - based game as well as an eclipse plugin that not only automatically generates natural language summaries for the methods of a java program , but also submits those methods to crowdsummarizer s website for the summarization by the crowd . we have used this implementation in an empirical study with 149 developers of various levels of experience to generate summaries for 128 methods with different properties from 11 widely used open - source java applications . the results of the study showed that crowdsummarizer is applicable for developers to use it in practice . in addition , the results of another empirical study performed with the help of 14 experts illustrated that crowdsummarizer can automatically generate accurate and comprehensible summaries . our main contributions of this article include : 1 . presenting a novel approach for generating natural language summaries for a java program and its methods using a topic modelling approach and power of the crowd . 2 . developing a crowdsourcing platform which uses the gamification elements to encourage users engagement . evaluation of the applicability and usefulness of crowdsummarizer platform as well as the quality of our automatically generated summaries . a complete implementation of our approach for java language as an eclipse plug - in . the remainder of this article is organized as follows . we rst provide a motivating example , used through- out this article ( section [ motivating - example ] ) . we then introduce the details of crowdsummarizer technique ( section [ proposed_approach ] ) and its implementation ( section [ sec : impl ] ) . we describe an example which crowdsummarizer try to solve it ( section [ exp ] ) . next , we present two empirical evaluation to evaluate the applicability of the crowdsummarizer platform , and different aspects of summaries generated for the 78 java methods from the 11 open - source java applications ( section [ sec : eval ] ) . finally , we discuss several observations on collected summaries from the crowdsummarizer ( section [ discussion ] ) ; compare the technique with related work ( section [ sec : relatedwork ] ) ; and conclude ( section [ sec : conclusion ] ) .

however , generating code summaries can be a challenging task . to mitigate this problem , in this article , we introduce _ crowdsummarizer _ , a code summarization platform that benefits from the concepts of crowdsourcing , gamification , and natural language processing to automatically generate a high level summary for the methods of a java program .

one of the first steps to perform most of the software maintenance activities , such as updating features or fixing bugs , is to have a relatively good understanding of the program s source code which is often written by other developers . a code summary is a description about a program s entities ( e.g. , its methods ) which helps developers have a better comprehension of the code in a shorter period of time . however , generating code summaries can be a challenging task . to mitigate this problem , in this article , we introduce _ crowdsummarizer _ , a code summarization platform that benefits from the concepts of crowdsourcing , gamification , and natural language processing to automatically generate a high level summary for the methods of a java program . we have implemented crowdsummarizer as an eclipse plugin together with a web - based code summarization game that can be played by the crowd . the results of two empirical studies that evaluate the applicability of the approach and the quality of generated summaries indicate that crowdsummarizer is effective in generating quality results .

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this article introduced crowdsummarizer , a novel approach for automated generation of natural language summaries for java programs and their methods . to this end , crowdsummarizer employs the concepts of crowdsourcing , gamification , and natural language processing . furthermore , it applies the lda technique to identify the latent topics in a java program , and to classify its methods into those topics . we have implemented crowdsummarizer as an eclipse plugin that works with a web - based code summarization game that can be played by the crowd . the results of an empirical study conducted by 149 developers with different levels of experience illustrated that crowdsummarizer is applicable and motivating for developers to use it in practice . moreover , the results of another empirical study done with the help of 14 experts indicated that developers are satisfied with the quality of automatically generated summaries in terms of accuracy and comprehensibility . in future , crowdsummarizer can be easily extended to include other programming languages like c++ as well . in addition , we plan to work on other text retrieval techniques to extract keywords from the method . we also intend to generate natural language summaries for classes using the crowd and the crowdsummarizer . finally , we want to use dynamic analysis in combination of static and linguistic information .

we have implemented crowdsummarizer as an eclipse plugin together with a web - based code summarization game that can be played by the crowd . the results of two empirical studies that evaluate the applicability of the approach and the quality of generated summaries indicate that crowdsummarizer is effective in generating quality results .

one of the first steps to perform most of the software maintenance activities , such as updating features or fixing bugs , is to have a relatively good understanding of the program s source code which is often written by other developers . a code summary is a description about a program s entities ( e.g. , its methods ) which helps developers have a better comprehension of the code in a shorter period of time . however , generating code summaries can be a challenging task . to mitigate this problem , in this article , we introduce _ crowdsummarizer _ , a code summarization platform that benefits from the concepts of crowdsourcing , gamification , and natural language processing to automatically generate a high level summary for the methods of a java program . we have implemented crowdsummarizer as an eclipse plugin together with a web - based code summarization game that can be played by the crowd . the results of two empirical studies that evaluate the applicability of the approach and the quality of generated summaries indicate that crowdsummarizer is effective in generating quality results .

1403.4290

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an important and challenging task in computational modeling is the solution of inverse problems , which convert noisy and indirect observational data into useful characterizations of the unknown parameters of a numerical model . in this process , statistical methods bayesian methods in particular play a fundamental role in modeling various information sources and quantifying the uncertainty of the model parameters @xcite . in the bayesian framework , the unknown parameters are modeled as random variables and hence can be characterized by their posterior distribution . markov chain monte carlo ( mcmc ) methods @xcite provide a powerful and flexible approach for sampling from posterior distributions . the bayesian framework has been applied to inverse problems in various fields , for example , geothermal reservoir modeling @xcite , groundwater modeling @xcite , ocean dynamics @xcite , remote sensing @xcite , and seismic inversion @xcite . to generate a sufficient number of samples from the posterior distribution , mcmc methods require sequential evaluations of the posterior probability density at many different points in the parameter space . each evaluation of the posterior density involves a solution of the forward model used to define the likelihood function , which typically is a computationally intensive undertaking ( e.g. , the solution of a system of pdes ) . in this many - query situation , one way to address the computational burden of evaluating the forward model is to replace it with a computationally efficient surrogate . surrogate models have been applied to inverse problems in several settings ; for example , @xcite employed generalized polynomial chaos expansions , @xcite employed gaussian process models , and @xcite used projection - based reduced - order models . in this work , we also focus on projection - based reduced - order models ( although the data - driven strategy underlying our approach should be applicable to other types of surrogate models ) . a projection - based reduced - order model reduces the computational complexity of the original or `` full '' forward model by solving a projection of the full model onto a reduced subspace . for the model reduction methods we consider , the construction of the reduced subspace requires evaluating the full model at representative samples drawn from the parameter space . the solutions of the full model at these samples are referred to as snapshots @xcite . their span defines the reduced subspace , represented via an orthogonal basis . the quality of the reduced - order model relies crucially on the choice of the samples for computing the snapshots . to construct reduced - order models targeting the bayesian solution of the inverse problem , we employ existing projection - based model reduction techniques . our innovation is in a data - driven approach that adaptively selects samples from the posterior distribution for the snapshot evaluations . this approach has two distinctive features : 1 . we integrate the reduced - order model construction process into an adaptive mcmc algorithm that simultaneously samples the posterior and selects posterior samples for computing the snapshots . during the sampling process , the numerical accuracy of the reduced - order model is adaptively improved . the approximate posterior distribution defined by the reduced - order model is used to increase the efficiency of mcmc sampling . we either couple the reduced - order model and the full model together to accelerate the sampling of the full posterior distribution , or directly explore the approximate posterior distribution induced by the reduced - order model . in the latter case , sampling the approximate distribution yields a biased monte carlo estimator , but the bias can be controlled using error indicators or estimators . compared to the classical offline approaches that build the reduced - order model before using it in the many - query situation , the motivation for collecting snapshots during posterior exploration is to build a reduced - order model that focuses on a more concentrated region in the parameter space . because the solution of the inverse problem is unknown until the data are available , reduced - order models built offline have to retain a level of numerical accuracy over a rather large region in the parameter space , which covers the support of the posterior distributions for all the possible observed data sets . for example , samples used for computing the snapshots are typically drawn from the prior distribution @xcite . in comparison , our data - driven approach focuses only on the posterior distribution resulting from a particular data set . as the observed data necessarily increase the information divergence of the prior distribution from the posterior distribution @xcite , the support of the posterior distribution is more compact than that of the prior distribution . figure [ fig : prior_post ] illustrates a simple two - dimensional inference problem , where the prior distribution and the posterior distribution are represented by the blue and red contours , respectively . the left plot of figure [ fig : prior_post ] shows 50 randomly drawn prior samples for computing the snapshots , each of which has a low probability of being in the support of the posterior . in comparison , the samples selected by our data - driven approach , as shown in the right plot of figure [ fig : prior_post ] , are scattered within the region of high posterior probability . by retaining numerical accuracy only in a more concentrated region , the data - driven reduced - order model requires a basis of lower dimension to achieve the same level of accuracy compared to the reduced - order model built offline . for the same reason , the data - driven model reduction technique can potentially have better scalability with parameter dimension than the offline approach . we note that goal - oriented model reduction approaches have been developed in the context of pde - constrained optimization @xcite , in which the reduced - order model is simultaneously constructed during the optimization process . in these methods , the snapshots for constructing the reduced basis are only evaluated at points in the parameter space that are close to the trajectory of the optimization algorithm . the remainder of this paper is organized as follows . in section 2 , we outline the bayesian framework for solving inverse problems and discuss the sampling efficiency and monte carlo error of mcmc . in section 3 , we introduce the data - driven model reduction approach , and construct the data - driven reduced - order model within an adaptive delayed acceptance algorithm to speed up mcmc sampling . we also provide results on the ergodicity of the algorithm . section 4 analyzes some properties of the data - driven reduced - order model . in section 5 , we discuss a modified framework that adaptively constructs the reduced - order model and simultaneously explores the induced approximate posterior distribution . we also provide an analysis of the mean square error of the resulting monte carlo estimator . in section 6 , we demonstrate and discuss various aspects of our methods through numerical examples . section 7 offers concluding remarks .

one of the major challenges in the bayesian solution of inverse problems governed by partial differential equations ( pdes ) is the computational cost of repeatedly evaluating numerical pde models , as required by markov chain monte carlo ( mcmc ) methods for posterior sampling . the proposed technique has two distinctive features . first , the model reduction strategy is tailored to inverse problems : the snapshots used to construct the reduced - order model are computed adaptively from the posterior distribution .

one of the major challenges in the bayesian solution of inverse problems governed by partial differential equations ( pdes ) is the computational cost of repeatedly evaluating numerical pde models , as required by markov chain monte carlo ( mcmc ) methods for posterior sampling . this paper proposes a data - driven projection - based model reduction technique to reduce this computational cost . the proposed technique has two distinctive features . first , the model reduction strategy is tailored to inverse problems : the snapshots used to construct the reduced - order model are computed adaptively from the posterior distribution . posterior exploration and model reduction are thus pursued simultaneously . second , to avoid repeated evaluations of the full - scale numerical model as in a standard mcmc method , we couple the full - scale model and the reduced - order model together in the mcmc algorithm . this maintains accurate inference while reducing its overall computational cost . in numerical experiments considering steady - state flow in a porous medium , the data - driven reduced - order model achieves better accuracy than a reduced - order model constructed using the classical approach . it also improves posterior sampling efficiency by several orders of magnitude compared to a standard mcmc method .

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we have introduced a new data - driven model reduction approach for solving statistical inverse problems . our approach constructs the reduced - order model using adaptively - selected posterior samples to compute the snapshots . the reduced - order model construction process is integrated into the posterior sampling , to achieve simultaneous posterior exploration and model reduction . based on the data - driven reduced - order model , we have also developed two mcmc algorithms to sample the posterior distribution more efficiently than standard full - model mcmc algorithms . the full target algorithm aims to accelerate sampling of the full posterior distribution by coupling the full and approximate posterior distributions together . the @xmath62-approximate algorithm samples an approximate posterior distribution , and attempts to reduce the mean squared error of the resulting monte carlo estimator , compared to a standard mcmc algorithm . both algorithms adaptively construct the reduced - order model online through the mcmc sampling . the full target algorithm preserves ergodicity with respect to the true posterior . the @xmath62-approximate algorithm does not sample the full posterior , but can provide further speedups for some problems . in the case studies , we have demonstrated that both algorithms are able to accelerate mcmc sampling of computationally expensive posterior distributions by up to two orders of magnitude , and that the sampling accuracy of the @xmath62-approximate algorithm is comparable to that of a reference full - model mcmc . we have also used the first case study to show the numerical accuracy of the data - driven reduced - order model , compared to a reduced - order model that is built offline with respect to the prior distribution . in this example , for the same number of reduced basis vectors , the posterior - averaged output error of the data - driven reduced - order model is several orders of magnitude smaller than that of the reduced - order model built with respect to the prior . furthermore , we have demonstrated the impact of the amount of information carried in the observed data on the dimensionality of the reduced basis . for solving statistical inverse problems , these results suggest that a data - driven reduced - order model is preferable to a reduced - order model built with respect to the prior , especially when the data are informative . even though our approach is designed for constructing projection - based reduced - order models , the concept of building posterior - oriented surrogate models can be generalized to other approximation approaches such as gaussian process regression and generalized polynomial chaos . the authors thank florian augustin , tan bui - thanh , omar ghattas , and jinglai li for many helpful comments and discussions . this work was supported by the united states department of energy applied mathematics program , awards de - fg02 - 08er2585 and de - sc0009297 , as part of the diamond multifaceted mathematics integrated capability center . cui t , fox c , osullivan mj . bayesian calibration of a large - scale geothermal reservoir model by a new adaptive delayed acceptance metropolis hastings algorithm . _ water resource research _ 2011 ; * 47*:w10521 . higdon d , lee h , holloman c. markov chain monte carlo - based approaches for inference in computationally intensive inverse problems . _ bayesian statistics 7 _ , bernardo jm , bayarri mj , berger jo , dawid ap , heckerman d , smith afm , west m ( eds . ) . oxford university press , oxford 2003 ; 181197 . haario h , laine m , lehtinen m , saksman e , tamminen j. markov chain monte carlo methods for high dimensional inversion in remote sensing . _ journal of the royal statistical society : series b ( statistical methodology ) _ 2004 ; * 66*:591608 . martin j , wilcox lc , burstedde c , ghattas o. a stochastic newton mcmc method for large - scale statistical inverse problems with application to seismic inversion . _ siam journal on scientific computing _ 2012 ; * 34*(3):a1460a1487 . bayarri mj , berger j , kennedy m , kottas a , paulo r , sacks j , cafeo j , lin c , tu j. predicting vehicle crashworthiness : validation of computer models for functional and hierarchical data . _ journal of the american statistical association _ 2009 ; * 104*:929943 . galbally d , fidkowski k , willcox ke , ghattas o. nonlinear model reduction for uncertainty quantification in large scale inverse problems . _ international journal for numerical methods in engineering _ 2008 ; * 81*(12):15811608 . arian e , fahl m , sachs ew . trust - region proper orthogonal decomposition for flow control . _ technical report icase-2000 - 25 _ , institute for computer applications in science and engineering , hampton , va 2000 . barrault m , maday y , nguyen nc , patera at . an `` empirical interpolation '' method : application to efficient reduced - basis discretization of partial differential equations . _ comptes rendus mathematique _ 2004 ; * 339*(9):667672 . haasdonk b , dihlmann m , , ohlberger m. a training set and multiple bases generation approach for parameterized model reduction based on adaptive grids in parameter space . comput . model . _ 2011 ; * 17*:423442 . patera at , rozza g. _ reduced basis approximation and a posteriori error estimation for parametrized partial differential equations_. mit pappalardo monographs in mechanical engineering ( to appear ) , copyright mit ( 20062007 ) , 2007 .

this paper proposes a data - driven projection - based model reduction technique to reduce this computational cost . posterior exploration and model reduction are thus pursued simultaneously . the data - driven reduced - order model achieves better accuracy than a reduced - order model constructed using the classical approach . it also improves posterior sampling efficiency by several orders of magnitude compared to a standard mcmc method .

one of the major challenges in the bayesian solution of inverse problems governed by partial differential equations ( pdes ) is the computational cost of repeatedly evaluating numerical pde models , as required by markov chain monte carlo ( mcmc ) methods for posterior sampling . this paper proposes a data - driven projection - based model reduction technique to reduce this computational cost . the proposed technique has two distinctive features . first , the model reduction strategy is tailored to inverse problems : the snapshots used to construct the reduced - order model are computed adaptively from the posterior distribution . posterior exploration and model reduction are thus pursued simultaneously . second , to avoid repeated evaluations of the full - scale numerical model as in a standard mcmc method , we couple the full - scale model and the reduced - order model together in the mcmc algorithm . this maintains accurate inference while reducing its overall computational cost . in numerical experiments considering steady - state flow in a porous medium , the data - driven reduced - order model achieves better accuracy than a reduced - order model constructed using the classical approach . it also improves posterior sampling efficiency by several orders of magnitude compared to a standard mcmc method .

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the fluctuations in the cosmic microwave background ( cmb ) radiation are theoretically very well understood , allowing precise and unambiguous predictions for a given cosmological model ( bond 1996 , hu and dodelson 2002 ) . the measurement of cmb anisotropy with the ongoing wilkinson microwave anisotropy probe ( wmap ) and the upcoming planck surveyor , has ushered in a new era of precision cosmology . such data rich experiments , with increased sensitivity , high resolution and ` full ' sky measurements pose a stiff challenge for current analysis techniques to realize the full potential of precise determination of cosmological parameters . the analysis techniques must not only be computationally fast to contend with the huge size of the data , but , the higher sensitivity also limits the simplifying assumptions that can then be invoked to achieve the desired speed without compromising the final accuracy . as experiments improve in sensitivity , the inadequacy in modeling the observational reality start to limit the returns from these experiments . the current effort is to push the boundary of this inherent compromise faced by the current cmb experiments that measure the anisotropy in the cmb temperature and polarization . accurate estimation of the angular power spectrum , @xmath3 , is inarguably the foremost concern of most cmb experiments . the extensive literature on this topic has been summarized ( hu and dodelson 2002 , bond 1996 , efstathiou 2004 ) . for gaussian , statistically isotropic cmb sky , the @xmath4 that corresponds to the covariance that maximizes the multivariate gaussian pdf of the temperature map , @xmath5 , is the maximum likelihood ( ml ) solution . different ml estimators have been proposed and implemented on cmb data of small and modest sizes ( gorski 1994 , gorski et al . 1994 , 1996 , 1997 , tegmark 1997 , bond et al . while it is desirable to use optimal estimators of @xmath4 that obtain ( or iterate toward ) the ml solution for the given data , these methods are usually limited by the computational expense of matrix inversion that scales as @xmath6 with data size @xmath7 ( borrill 1999 , bond et al . various strategies for speeding up ml estimation have been proposed , such as , exploiting the symmetries of the scan strategy ( oh et al . 1999 ) , using hierarchical decomposition ( dore et al . 2001 ) , iterative multi - grid method ( pen 2003 ) , etc . variants employing linear combinations of @xmath5 such as @xmath8 on set of rings in the sky can alleviate the computational demands in special cases ( challinor et al . 2002 , van leeuwen et al . 2002 , wandelt & hansen 2003 ) . other promising ` exact ' power estimation methods have been recently proposed(knox et al . 2001 , jewell et al . 2004 , wandelt 2004 ) . however there also exist computationally rapid , sub - optimal estimators of @xmath4 . exploiting the fast spherical harmonic transform ( @xmath9 ) , it is possible to estimate the angular power spectrum @xmath10 rapidly ( yu & peebles 1969 , peebles 1974 ) . this is commonly referred to as the pseudo-@xmath4 method ( wandelt et al . analogous approach employing fast estimation of the correlation function @xmath11 have also been explored ( szapudi et al . 2001 , szapudi et al . it has been recently argued that the need for optimal estimators may have been over - emphasized since they are computationally prohibitive at large @xmath12 . sub - optimal estimators are computationally tractable and tend to be nearly optimal in the relevant high @xmath12 regime . moreover , already the data size of the current sensitive , high resolution , ` full sky ' cmb experiments , such as wmap have been compelled to use sub - optimal pseudo-@xmath4 and related methods ( bennett et al . 2003 , hinshaw et al . 2003 ) . on the other hand , optimal ml estimators can readily incorporate and account for various systematic effects , such as noise correlations , non - uniform sky coverage and beam asymmetries . a hybrid approach of using ml estimation of @xmath4 for low @xmath12 for low resolution map and pseudo-@xmath4 like estimation for large @xmath12 where it is nearly optimal has been suggested ( efstathiou 2004 ) and has even been employed by the recent analysis of the wmap-3 year data ( hinshaw et al . the systematic correction to the pseudo-@xmath4 power spectrum estimate arising from non - uniform sky coverage has been studied and implemented for cmb temperature ( hivon et al . 2002 ) and polarization ( brown at al 2005 ) . the _ leading _ order systematic bias due to noncircular beam has been studied by us in an earlier publication ( mitra et al . 2004 ) . here we extend the results in a thorough analytical approach to include all the significant perturbation orders and combine the effect of non - uniform sky coverage . it has been usual in cmb data analysis to assume the experimental beam response to be circularly symmetric around the pointing direction . however , real beam response functions have deviations from circular symmetry . even the main lobe of the beam response of experiments are generically non - circular ( non - axisymmetric ) since detectors have to be placed off - axis on the focal plane . ( side lobes and stray light contamination add to the breakdown of this assumption ) . for highly sensitive experiments , the systematic errors arising from the beam non - circularity become progressively more important . dropping the circular beam assumption leads to major complications at every stage of the analysis pipeline . the extent to which the non - circularity of the beam affects the step of going from the time - stream data to sky map is very sensitive to the scan - strategy . the beam now has an orientation with respect to the scan path that can potentially vary along the path . this introduces an additional complication - that the beam function is inherently time dependent and difficult to deconvolve . even after a sky map is made , the non - circularity of the effective beam affects the estimation of the angular power spectrum , @xmath4 , by coupling the power at different multipoles , typically , on scales beyond the inverse angular beam - width . mild deviations from circularity can be addressed by a perturbation approach ( souradeep & ratra 2001 , fosalba et al . 2002 ) and the effect of non - circularity on the estimation of cmb power spectrum can be studied ( semi ) analytically ( mitra et al . figure [ clerr ] shows the predicted level of bias due to noncircular beams in our formalism for elliptical beams compared to the noncircular beam corrections computed in the recent data release by wmap ( hinshaw et al . 2007 ) . to avoid contamination of the primordial cmb signal by galactic emission , the region around the galactic plane is masked from maps . if the galactic cut is small enough , then the coupling matrix will be invertible , and the two - point correlation function can be determined on all angular scales from the data within the uncut sky ( mortlock et al . hivon et al . ( 2002 ) present a technique ( master ) for fast computation of the power spectrum accounting for the galactic cut , but restricted to circular beams . in our present work , we present analytical expressions for the bias matrix of the pseudo-@xmath13 estimator for the incomplete sky coverage , _ using a noncircular beam_. in section [ form ] we show a heuristic approach to the analytic form of the bias matrix taking into account the above mentioned effects . we have shown that our estimation matches with the existing results in different limits in section [ sec : limit ] . in section [ num ] we outline the numerical implementation of our approach , estimate the computational cost and suggest a potential algorithmic route to reducing the cost . the discussion and conclusion of this work is given in section [ disc ] .

the analysis techniques must not only be computationally fast to contend with the huge size of the data , but , the higher sensitivity also limits the simplifying assumptions which can then be invoked to achieve the desired speed without compromising the final precision goals . while maximum likelihood is desirable , the enormous computational cost makes the suboptimal method of power spectrum estimation using pseudo - c unavoidable for high resolution data . the approach is perturbative in the distortion of the beam from non - circularity , allowing for rapid computations when the beam is mildly non - circular . we numerically implement our method for _ non - rotating beams_. we present preliminary estimates of the computational cost to evaluate the bias for the upcoming cmb anisotropy probes ( ) , with angular resolution comparable to the planck surveyor mission .

over the last decade , measurements of the cmb anisotropy has spearheaded the remarkable transition of cosmology into a precision science . however , addressing the systematic effects in the increasingly sensitive , high resolution , ` full ' sky measurements from different cmb experiments pose a stiff challenge . the analysis techniques must not only be computationally fast to contend with the huge size of the data , but , the higher sensitivity also limits the simplifying assumptions which can then be invoked to achieve the desired speed without compromising the final precision goals . while maximum likelihood is desirable , the enormous computational cost makes the suboptimal method of power spectrum estimation using pseudo - c unavoidable for high resolution data . the debiasing of the pseudo - c needs account for non - circular beams , together with non - uniform sky coverage . we provide a ( semi)analytic framework to estimate bias in the power spectrum due to the effect of beam non - circularity and non - uniform sky coverage including incomplete / masked sky maps and scan strategy . the approach is perturbative in the distortion of the beam from non - circularity , allowing for rapid computations when the beam is mildly non - circular . we advocate that it is computationally advantageous to employ ` soft ' azimuthally apodized masks whose spherical harmonic transform die down fast with . we numerically implement our method for _ non - rotating beams_. we present preliminary estimates of the computational cost to evaluate the bias for the upcoming cmb anisotropy probes ( ) , with angular resolution comparable to the planck surveyor mission . we further show that this implementation and estimate is applicable for rotating beams on equal declination scans and possibly can be extended to simple approximations to other scan strategies . cosmic microwave background

0909.1717

i

an analysis using three - particle correlations that are directly sensitive to the -violation effects in heavy - ion collisions has been presented for au+au and cu+cu collisions at @xmath0=200 and 62 gev . the results are reported for different particle charge combinations as a function of collision centrality , particle separation in pseudorapidity , and particle transverse momentum . qualitatively the results agree with the magnitude and gross features of the theoretical predictions for local -violation in heavy - ion collisions , except that the signal persists to higher transverse momenta than expected @xcite . the particular observable used in our analysis is -even and might be sensitive to non - parity - violating effects . so far , with the systematics checks discussed in this paper , we have not identified effects that would explain the observed same - charge correlations . the observed signal can not be described by the background models that we have studied ( hijing , hijing+@xmath14 , urqmd , mevsim ) , which span a broad range of hadronic physics . a number of future experiments and analyses are naturally suggested by these results . one of them is the study of the correlation dependence on the energy of the colliding ions . the charge separation effect is expected to depend strongly on the formation of a quark - gluon plasma @xcite , and the signal might be greatly suppressed or completely absent at an energy below that at which a quark - gluon plasma can be formed . improved theoretical calculations of the expected signal and potential physics backgrounds in high energy heavy ion collisions are essential to understand whether or not the observed signal is due to local strong parity violation , and to further experimental study of this phenomenon .

we report measurements of this observable using the star detector in au+au and cu+cu collisions at=200 and 62 gev . the results are presented as a function of collision centrality , particle separation in rapidity , and particle transverse momentum . a signal consistent with several of the theoretical expectations

parity - odd domains , corresponding to non - trivial topological solutions of the qcd vacuum , might be created during relativistic heavy - ion collisions . these domains are predicted to lead to charge separation of quarks along the orbital momentum of the system created in non - central collisions . to study this effect , we investigate a three particle mixed harmonics azimuthal correlator which is a -even observable , but directly sensitive to the charge separation effect . we report measurements of this observable using the star detector in au+au and cu+cu collisions at=200 and 62 gev . the results are presented as a function of collision centrality , particle separation in rapidity , and particle transverse momentum . a signal consistent with several of the theoretical expectations is detected in all four data sets . we compare our results to the predictions of existing event generators , and discuss in detail possible contributions from other effects that are not related to parity violation .

1402.4886

i

the well known @xmath2-queens problem asks for the number of ways to place @xmath2 nonattacking queens on an @xmath1[d : n ] chessboard . a more general question separates the number of pieces from the size of the board ; that is the @xmath0-queens problem , which asks for the number of ways to place @xmath0[d : q ] nonattacking queens on an @xmath1 board . this paper is part iii of a series @xcite in which we develop a general method for solving such questions and apply it not only to queens but to other pieces of the type called `` riders '' , whose moves have unlimited distance . we convert the chess problem into a geometry problem : moves become hyperplanes in @xmath3 ; the @xmath1 board becomes the set of @xmath4-fractional lattice points in the unit square ; and the number of nonattacking configurations becomes a linear combination of the numbers of @xmath0-tuples of these lattice points that lie in subspaces determined by the move hyperplanes . the ultimate goal is to produce a usable general formula for any such problem , but that is probably impossibly difficult because it requires complete knowledge of configuration theorems in the real plane . still , we provide important information about the nature of such a formula . we proved in part i that in each such problem the number of solutions is a quasipolynomial function of @xmath2that means it is given by a cyclically repeating sequence of polynomials as @xmath2 varies and that the coefficient of each power of @xmath2 is ( up to a factor ) a polynomial function of @xmath0 . in part ii we narrowed our focus to the square board and found , for instance , that the coefficients of the very highest powers of @xmath2 do not vary periodically with @xmath2 , or at most have period 2 . we briefly review the essentials from parts i and ii in section [ essentials ] . this part is the capstone of the series . in it we repeatedly apply the theory from parts i and ii to gain detailed information on the counting functions for a well - behaved family of pieces that we name `` partial queens''those rider pieces whose moves are a subset of those of the queen . first , we explicitly calculate the coefficients of the four highest powers of @xmath2 for partial queens . surprisingly , we are also able to prove formulas for the _ periodic _ parts of the coefficients of the next two highest powers of @xmath2 ( but not for the nonperiodic parts ) . we further establish the highest and lowest powers of @xmath0 and their coefficients within the coefficient of each power of @xmath2 . these results combine to form our main theorem , theorem [ p : hvdiag ] . the proof of this theorem , which comprises the remainder of section [ coeffs ] , requires us to determine the counting quasipolynomials for all low - codimensional subspaces in the lattice of intersections of move hyperplanes . in section [ two - three ] , we further apply our theory to calculate the number of nonattacking placements of two and three partial queens on a square board of any size and the number of combinatorially distinct nonattacking configurations of three partial queens , which turns out to be determined by the number , not the kind , of moves . a deeper reason we study this set of pieces is that our ultimate object is to find out what factors control properties of the counting formula , such as the period ( the length of a repeating cycle in the cyclically repeating polynomials ) , the periods of the individual coefficients of powers of @xmath2 , formulas for the coefficients in terms of the set of moves of the piece , or anything that will let us predict aspects of the counting functions by knowing the moves of the piece under consideration . for this partial queens can be a valuable test set of pieces , not as hard as general riders but varied enough to suggest patterns for counting functions . indeed , it was the formulas and their proofs for partial queens that led us to several of the general properties proved in parts i and ii . in part iv , next in the series , counting function properties that we establish here ( supplemented by a theorem in part v ) will let us prove some of the specific formulas for bishops and queens derived empirically through extensive computation , or brilliantly intuited , by kotovec in @xcite . we conclude part iii with questions related to our results and methods , and we append a dictionary of notation .

parts i and ii showed that the number of ways to place nonattacking queens or similar chess pieces on an square chessboard is a quasipolynomial function of in which the coefficients are essentially polynomials in . we explore this function for partial queens , which are pieces like the rook and bishop whose moves are a subset of those of the queen . we compute the five highest - order coefficients of the counting quasipolynomial , which are constant ( independent of ) , and find the periodicity of the next two coefficients , which depend on the move set . for two and three pieces we derive the complete counting functions and the number of combinatorially distinct nonattacking configurations . the method , as in parts i and ii , is geometrical , using the lattice of subspaces of an inside - out polytope .

parts i and ii showed that the number of ways to place nonattacking queens or similar chess pieces on an square chessboard is a quasipolynomial function of in which the coefficients are essentially polynomials in . we explore this function for partial queens , which are pieces like the rook and bishop whose moves are a subset of those of the queen . we compute the five highest - order coefficients of the counting quasipolynomial , which are constant ( independent of ) , and find the periodicity of the next two coefficients , which depend on the move set . for two and three pieces we derive the complete counting functions and the number of combinatorially distinct nonattacking configurations . the method , as in parts i and ii , is geometrical , using the lattice of subspaces of an inside - out polytope .

0710.5598

r

for the oxygen atoms of ppo and @xmath23 . the temperatures are , from left to right , @xmath24 , @xmath25 , @xmath26 , @xmath27 , @xmath28 , @xmath29 , @xmath30 , and @xmath31 . the dashed lines are von schweidler fits , see eq . ( [ eq_von ] ) . the inset shows the time temperature superposition of the same data . the time constants @xmath32 from kww interpolations of the @xmath1 relaxation were used for the scaling of the time axis . the dashed line is a kww fit of the @xmath1 relaxation ( @xmath33).__,width=311 ] it is well established for the @xmath1 relaxation of polymer melts that the time dependence differs from a simple exponential behavior and the temperature dependence does not obey an arrhenius law.@xcite therefore , we first demonstrate that the studied peo and ppo models show these key features . while the incoherent intermediate scattering function @xmath34\}\rangle\ ] ] provides us with information about translational motion , the orientational autocorrelation function @xmath35 ^ 2 - 1\rangle\ ] ] yields insights into rotational motion . specifically , the scattering function @xmath36 depends on the translational displacements @xmath37 $ ] of the atoms during the time interval @xmath38 , where the absolute value of the scattering vector , @xmath39 , determines the length scale on which dynamics is probed . throughout this contribution , we focus on the translational motion of the oxygen atoms , but we ensured that qualitatively similar findings are obtained for the carbon atoms . the orientational correlation function @xmath40 depends on the angular displacements @xmath41 during the time interval @xmath38 . we study the reorientation of the c - h bonds and @xmath42 is the unit vector describing the direction of a c - h bond at time @xmath43 . @xmath44h nmr stimulated - echo experiments were used to measure @xmath45 for c - d bonds of deuterated ppo.@xcite mimicking the experimental situation , we restrict the analysis to c - h bonds in the methylene groups to avoid effects from fast threefold methyl group jumps . finally , in eqs . ( [ eq_sqt ] ) and ( [ eq_f2 ] ) , the brackets @xmath46 denote the average over various time origins @xmath47 and over all atoms or bonds belonging to the considered atomic or bond species . first , we use these correlation functions to ascertain the temperature dependence of the translational and rotational motion associated with the @xmath1 relaxation . to address the translational aspect , we calculate @xmath48 using @xmath49 and @xmath50 for peo and ppo , respectively . these values of the momentum transfer correspond to the respective position of the first maximum of the intermolecular oxygen - oxygen pair distribution functions.@xcite in fig.[fig1 ] , we see that the incoherent intermediate scattering functions @xmath48 for the oxygen atoms of ppo show a pronounced temperature dependence , in particular at low temperatures . to quantify the slowdown of the structural relaxation , we extract temperature dependent translational and rotational correlation times according to @xmath51 and @xmath52 . figure [ fig2 ] shows the results for peo and ppo . for both models , @xmath53 and @xmath54 exhibit a comparable temperature dependence that can not be described by an arrhenius law , as expected for polymer melts . rather , a vogel - fulcher - tammann ( vft ) law,@xcite @xmath55 enables good interpolations of the data . vft fits to @xmath53 ( @xmath54 ) yield @xmath56 ( @xmath57 ) for peo and @xmath56 ( @xmath58 ) for ppo . the results for ppo are in reasonable agreement with @xmath59 obtained in light scattering and dielectric spectroscopy studies.@xcite , the mean cluster size @xmath60 and the mean string length @xmath61 are a maximum at @xmath62 , @xmath63 and @xmath64 , respectively . moreover , @xmath65 and @xmath52 . the solid and dashed lines are vft interpolations of @xmath66 ( ppo : @xmath56 , peo : @xmath56 ) and @xmath67 ( ppo : @xmath58 , peo : @xmath57).__,width=302 ] inspecting the shape of the scattering functions in fig . [ fig1 ] , we see nonexponential decays . as expected for glass - forming liquids,@xcite the time dependence in the @xmath1-relaxation regime is well described by a kohlrausch - williams - watts ( kww ) function , or , equivalently , stretched exponential function @xmath68\;\;\;\;(0\!\leq\!\beta\!\leq\!1).\ ] ] here , @xmath32 and @xmath69 quantify the time scale and the stretching , respectively . fitting @xmath48 to a kww function , we find stretching parameters @xmath70 for peo and @xmath71 for ppo , independent of temperature in the studied temperature ranges . for ppo , the time - temperature superposition is further demonstrated in the inset of fig . clearly , the curves coincide in the @xmath1-relaxation regime when the time axis is scaled with the correlation time @xmath32 . kww interpolations of @xmath45 yield stretching parameters @xmath72 for peo and @xmath73 for ppo , consistent with stretching parameters obtained from @xmath44h nmr stimulated - echo experiments near @xmath0.@xcite thus , with respect to both temperature and time dependence , the structural relaxation of the studied polymer models resembles that of peo and ppo melts , confirming the quality of the used force fields.@xcite we note that , for ppo at @xmath74 , the translational and rotational correlation functions give evidence for some deviations from time - temperature superposition in the early stages of the decay , where the curves decrease significantly faster than expected from the kww behavior , see fig . [ fig1 ] . since dielectric spectroscopy studies demonstrated that ppo exhibits a johari - goldstein secondary relaxation , which starts to separate from the primary relaxation at @xmath75 , @xcite we attribute these deviations to the onset of this secondary process . of the oxygen atoms . ( a ) correlation times @xmath76 resulting from kww fits of the @xmath1-relaxation regime for the indicated values of @xmath77 . the solid lines are mct power laws @xmath78 where @xmath79 and @xmath80 . ( b ) temperature dependence of @xmath81 , using @xmath82 for @xmath83 , @xmath84 for @xmath23 , and @xmath85 for @xmath86 . the solid lines are linear interpolations @xmath87.__,width=302 ] the time temperature superposition observed for the @xmath1 relaxation of the peo and ppo models is in agreement with mct.@xcite therefore , we now check whether further predictions of this theory are fulfilled . according to mct , the particles of glass - forming liquids are trapped in cages formed by their neighbors for some time until an escape from these cages is possible during the structural relaxation . as a consequence of this trapping , a plateau regime , or , equivalently , @xmath69-relaxation regime , preceding the @xmath1-relaxation regime develops when the temperature is decreased towards the critical temperature @xmath2 . quantitatively , mct predicts a power - law divergence of the @xmath1-relaxation time at the critical temperature @xmath2 : @xmath88 another key prediction of mct is the factorization theorem for the @xmath69-relaxation regime . it states that , for such times , all correlation functions , in particular , the incoherent intermediate scattering functions for different values of the momentum transfer , can be written as @xmath89 here , the plateau value @xmath90 and the amplitude @xmath91 depend on the value of @xmath77 , while the @xmath69 correlator @xmath92 is independent of the observable . if the factorization theorem is obeyed , the curves @xmath93 for different values of the momentum transfer will collapse onto a master curve , provided the times @xmath94 and @xmath95 are chosen inside the @xmath69-relaxation regime.@xcite hence , calculation of @xmath96 allows one to check the factorization theorem without invoking a fitting procedure . @xmath92 can be expanded for times close to the central @xmath69-relaxation time @xmath97 . the expansion for @xmath98 leads to the von schweidler law @xmath99 showing that a power law characterized by the universal von schweidler exponent @xmath100 describes the initial stages of the decay from the plateau . for temperatures @xmath101 , the amplitude @xmath102 decreases upon cooling according to @xmath103 finally , within mct , @xmath100 and @xmath104 are related via the exponent parameter @xmath105 : @xmath106 to check these mct predictions for the ppo model , we analyze the scattering functions @xmath48 of the oxygen atoms for various values of the momentum transfer @xmath77 . first , we fit a kww function to the scattering functions in the @xmath1-relaxation regime . figure [ fig3](a ) shows the temperature dependent correlation times @xmath107 resulting from these fits for three values of @xmath77 . a mct power law , see eq . ( [ eq_mct ] ) , with @xmath108 and @xmath109 nicely describes all data at @xmath110 . these findings are in reasonable agrement with @xmath111 and @xmath112 from experimental studies.@xcite however , there are deviations from the mct power law at the lowest temperature @xmath113 . at the present , it is not clear whether these deviations are related to the possible onset of the johari - goldstein secondary relaxation near @xmath2 . qualitatively similar deviations from the mct predictions were reported for a chemically realistic model of polybutadiene in the vicinity of the critical temperature.@xcite . the functions @xmath96 were obtained from the data @xmath48 for the oxygen atoms according to eq . ( [ eq_r ] ) . the values of the momentum transfer @xmath77 are indicated and times @xmath114 and @xmath115 were used.__,width=311 ] next , we fit the von schweidler law to the scattering functions @xmath48 . due to a possible interference of the johari - goldstein secondary relaxation , we exclude the data for the lowest temperature @xmath116 from further mct analysis . the von schweidler law enables a good interpolation of the decays in the late-@xmath69/ early-@xmath1 relaxation regime , see fig . [ fig1 ] . for @xmath50 , the von schweidler fits yield @xmath117 and @xmath118 independent of temperature . according to eq . ( [ eq_exp ] ) , this value of the von schweidler exponent @xmath100 translates into @xmath119 , in agreement with @xmath109 determined from the temperature dependent correlation times @xmath107 . to check the validity of eq.([eq_hq ] ) , we use the exponents @xmath120 and @xmath119 and plot the temperature dependence of @xmath81 in fig . [ fig3](b ) . in harmony with the mct prediction , there is a linear relationship and , by extrapolation , @xmath81 vanishes at the critical temperature @xmath79 , extracted from the temperature dependence of the @xmath1 relaxation . however , the mct predictions are not fulfilled when including the scattering functions for other values of @xmath77 into the analysis . von schweidler fits for @xmath121 and @xmath122 yield @xmath123 and @xmath124 , respectively , corresponding to @xmath125 and @xmath126 . thus , there is no universal von schweidler exponent and , for high and low values of @xmath77 , the values of @xmath104 calculated from the von schweidler exponents deviate from @xmath80 resulting from the temperature dependence of the @xmath1 relaxation . to demonstrate the violation of the factorization theorem independent of any fitting routine , we show @xmath96 for different values @xmath77 in fig . the data was calculated from @xmath48 for the oxygen atoms of ppo at @xmath127 . clearly , the data do not collapse onto a master curve in the @xmath69-relaxation regime , indicating the violation of the factorization theorem . qualitatively similar results were observed at all studied temperatures . other models of the glass transition focus on the heterogeneity and the cooperativity of the dynamics.@xcite the importance of these effects was demonstrated in md simulation studies on various glass - forming liquids.@xcite here , we investigate the relevance of heterogeneity and cooperativity for the first time for chemically realistic polymer models . for various model - glass formers , it was found that the distribution of scalar particle displacements @xmath128 deviates from a gaussian at intermediate times @xmath38 between ballistic and diffusive motion . these deviations , which are a first indication for the existence of heterogeneous dynamics , can be quantified by the non - gaussian parameter @xmath129 ^ 4\rangle}{\langle[\mathbf{r}(\tilde{t}_0\!+\!t)\!-\!\mathbf{r}(\tilde{t}_0)]^2\rangle^2}-1.\ ] ] figure [ fig5 ] shows @xmath130 for the oxygen atoms of ppo at various temperatures . we see that the non - gaussian parameter exhibits a maximum in the late-@xmath69/ early-@xmath1 relaxation regime , consistent with findings for various model - glass formers.@xcite when the temperature is decreased , the position of the maximum , @xmath62 , shifts to longer times . in fig.[fig2](a ) , we see that the temperature dependence of @xmath62 is somewhat weaker , but still comparable to that of the @xmath1-relaxation time . the maximum @xmath131 increases upon cooling , in particular near the critical temperature @xmath2 , but the values are relatively small . more precisely , at comparable temperatures near @xmath2 , @xmath132 was reported for the other studied models of glass - forming liquids , except for the silicon atoms in silica showing @xmath133 , see ref . for a detailed comparison . for peo , the non - gaussian parameter is even smaller than for ppo . this results in some ambiguities when extracting the maximum positions so that we refrain from discussing the temperature dependence of @xmath62 for peo . for the oxygen atoms of ppo at various temperatures ( @xmath24 , @xmath25 , @xmath26 , @xmath27 , @xmath28 , @xmath29 , @xmath30 , @xmath31).__,width=321 ] to ascertain the spatial heterogeneity of the peo and ppo dynamics , we demonstrate that highly mobile oxygen atoms form clusters larger than expected from random statistics . following previous studies,@xcite we characterize the particle mobility in a time interval @xmath38 by the scalar displacement and select the 5% most mobile oxygen atoms for further analysis . then , we define a cluster as a group of the most mobile oxygen atoms that reside in the first neighbor shells of each other . for both polymers , we use the position of the first minimum of the respective intermolecular oxygen - oxygen pair distribution function as criterion for the extension of the neighbor shell . based on the probability distribution @xmath134 of finding a cluster of size @xmath135 for a time interval @xmath38 , we calculate the weight - averaged mean cluster size @xmath136 this quantity measures the average size of a cluster to which one of the most mobile oxygen atoms belongs . previously , it was shown that the conclusions resulting from such analysis are unchanged when the fraction of highly mobile particles is varied in a meaningful range.@xcite of highly mobile oxygen atoms for peo at temperatures @xmath24 , @xmath25 , @xmath26 , @xmath27 , @xmath28 , and @xmath29 . when the particles used for the analysis are chosen irrespective of their mobilities , a mean cluster size of @xmath137 results , as indicated by the dashed line . _ , width=302 ] the mean cluster size @xmath138 is displayed for peo at various temperatures in fig.[fig6 ] . it is evident that @xmath138 shows a maximum , which increases upon cooling . by contrast , a time and temperature independent small size @xmath137 characterizes the clusters for the case of random statistics , i.e. , when 5% of the oxygen atoms are chosen for analysis irrespective of their mobilities . these results clearly demonstrate the existence of spatially heterogeneous dynamics . the transient nature of the clusters can be quantified , when we determine the times @xmath139 at which @xmath138 is a maximum . in fig . [ fig2 ] , we see for peo and ppo that the clusters are largest in the @xmath1-relaxation regime at all studied temperatures . for ppo , we observe that the single maximum of @xmath138 at high temperatures splits into two peaks near @xmath2 , see fig . this can be taken as another hint that a primary and a secondary relaxation coexist at sufficiently low temperatures . in fig . [ fig6 ] , a closer inspection of the data for the highest temperatures reveals that @xmath138 increases at long times in the diffusive regime . this increase is the mere consequence of the chain connectivity . at sufficiently long times , the displacements of the individual atoms are largely determined by the displacement of the center - of - mass of the respective polymer chain and , hence , those oxygen atoms are highly mobile that belong to chains , showing the largest center - of - mass displacements from the statistical distribution . due to their spatial proximity along the chain , these atoms form extended clusters so that @xmath138 increases when the single particle displacements start to become dominated by the center - of - mass displacements . by contrast , the maximum at shorter times is not due to chain connectivity . of highly mobile oxygen atoms for ppo at temperatures @xmath24 , @xmath25 , @xmath26 , @xmath27 , @xmath28 , @xmath29 , @xmath30 , and @xmath31 . when the oxygen atoms used for the analysis are chosen irrespective of their mobilities , a mean string length of @xmath140 results , as indicated by the bottommost dashed line . the upmost dashed line is the mean cluster size @xmath138 of highly mobile oxygen atoms for ppo at @xmath74.__,width=311 ] it has been shown that cooperative string - like motion is an important channel for the relaxation of highly mobile particles in simple model - glass formers.@xcite therefore , we investigate the relevance of string - like motion for the chemically realistic peo and ppo models . following these previous studies , we construct strings by connecting any two oxygen atoms @xmath141 and @xmath142 if the condition @xmath143<\delta\ ] ] holds for the atomic positions at two different times and set @xmath144 to about 55% of the intermolecular oxygen - oxygen distance . then , this condition means that one oxygen atom has moved and an another oxygen atom has occupied its position . we checked that our conclusions are not altered when @xmath144 is varied in a meaningful range . using the above criterion , we determine the probability @xmath145 of finding a string of length @xmath146 for a time interval @xmath38 and calculate the weight - averaged mean string length @xmath147 in analogy with eq . ( [ eq_sw ] ) . figure [ fig7 ] shows @xmath147 for ppo at various temperatures . evidently , the strings grow and shrink in time and they are substantially longer than that resulting from random statistics . upon cooling , the position of the maximum , @xmath148 , shifts to longer times and the height of the maximum increases , where the growth is particularly prominent in the vicinity of @xmath2 . thus , string - like motion is an important phenomenon at sufficiently low temperatures . @xmath147 and @xmath138 show a similar behavior . in particular , both quantities exhibit a two - peak signature at @xmath74 the temperature dependence of @xmath148 and @xmath139 is compared in fig . [ fig2 ] . for peo and ppo , the strings , like the clusters , are largest in the @xmath1-relaxation regime . by contrast , simulation studies on simple glass - forming liquids , including a bead - spring polymer model , found that @xmath148 and @xmath139 are located in the late-@xmath69/ early-@xmath1 relaxation regime , i.e. , at significantly shorter times.@xcite we conclude that the peo and ppo models do show spatially heterogenous and cooperative dynamics , however the characteristics of these effects differ from that for simple model - glass formers . . ( a ) typical trajectory @xmath149 of an occo dihedral angle during a time interval of @xmath150 . the straight lines mark the discontinuous trajectory @xmath151 resulting from mapping of the trajectory @xmath149 onto the conformational states @xmath152 , @xmath38 , and @xmath153 . ( b ) probability distributions of the occo ( solid line ) and cocc ( dashed line ) dihedral angles , @xmath154 and @xmath155 , respectively . _ , width=311 ] local conformational dynamics are of central importance for the structural relaxation of polymers.@xcite in the following , we investigate the torsional motion of the peo model by analyzing the angular trajectories @xmath149 and @xmath156 , describing the time evolution of the occo and cocc dihedral angles , respectively . figure [ fig9 ] presents a typical trajectory @xmath149 . comparison with the probability distribution of the dihedral angle , @xmath154 , shows that the torsional motion is comprised of well defined transitions between the @xmath157 ( @xmath152 ) , @xmath158 ( @xmath38 ) , and @xmath159 ( @xmath153 ) states , suggesting that a discretization is useful for an analysis of the conformational dynamics . therefore , we map the trajectories @xmath160 ( @xmath161 ) onto discrete sequences @xmath162 of the conformational states , see fig.[fig9 ] . in this way , we eliminate effects from librational motions , which do not lead to structural relaxation . then , analysis of the discrete sequences @xmath162 enables a straightforward characterization of the relevant torsional motion , e.g. , in terms of waiting times @xmath163 and back - jump probabilities @xmath164 , see below . , defined as @xmath165 , and the rotational correlation times @xmath166 , see fig . [ fig2 ] , together with vft interpolations . moreover , the mean waiting times of the occo dihedrals in the @xmath38 state are displayed . while all escape processes from the @xmath38 state are considered for the calculation of @xmath167 , @xmath168 ( @xmath169 ) characterizes explicitly the forward ( backward ) jumps @xmath170 ( @xmath171 ) . the solid lines are arrhenius fits ( @xmath167 : @xmath172 , @xmath173 : @xmath174 , @xmath169 : @xmath175 ) . finally , we present the mean times @xmath176 needed for an occo dihedral to visit each of the three conformational states at least once . calculated values of @xmath176 are shown as dotted and dashed lines . while @xmath177 transitions were neglected when calculating the former data , these transitions were taken into account when computing the latter data , see text for details.__,width=311 ] to study the time scale of the conformational relaxation , we use the discrete sequences @xmath178 and determine the probabilities @xmath179 of finding a dihedral in the same conformational state at two times separated by a time interval @xmath38 . due to the finite number of conformational states , these probabilities exhibit a finite and temperature dependent plateau value @xmath180 . to remove this effect , we calculate the torsional correlation functions @xmath181 and determine torsional correlation times @xmath182 according to @xmath183 . in fig . [ fig8 ] , we compare the torsional correlation time @xmath184 with the rotational correlation time @xmath54 , characterizing the reorientation of the c - h bond vectors . we see that both time constants show a comparable temperature dependence , implying that the conformational and the structural relaxations are related , consistent with results for other polymer models.@xcite some deviations in the temperature dependence are expected since , in particular at high temperatures , librational motions affect the rotational correlation times more than the torsional correlation times obtained from analysis of the discrete jump sequences . moreover , the conformational dynamics of several dihedral species render the orientation of a given c - h bond time dependent and , hence , the absolute values of the rotational and torsional correlation times should differ . next , we study the waiting times @xmath163 in the conformational states , i.e. , the time intervals between two subsequent conformational transitions . we find that the waiting times differ between the occo and cocc dihedral angles and , for each dihedral species , they depend on the conformational state . therefore , we separately determine the waiting times for each dihedral species and conformational state . the temperature dependent mean waiting time of the occo dihedrals in the @xmath38 state , @xmath185 , is included in fig . we see that the mean waiting time follows an arrhenius law with a small activation energy @xmath172 . qualitatively similar results are observed for all other dihedral species and conformational states . hence , the temperature dependence of the mean waiting times is much weaker than that of the rotational and torsional correlation times , indicating that the longer residence times in the conformational states at lower temperatures are not sufficient to explain the strong slowdown of the @xmath1 relaxation upon cooling , in harmony with results for other all - atom polymer models.@xcite characterizing the waiting times of the occo dihedrals in the @xmath38 state . results for peo at various temperatures are compared.__,width=311 ] closer insights into the nature of the conformational dynamics are available from the probability distributions of the waiting times , @xmath186 . figure [ fig10 ] shows the distributions @xmath187 , which characterize the waiting times of the occo dihedrals in the @xmath38 state . in particular at the lower temperatures , there are substantial deviations from an exponential waiting - time distribution , indicating that a markov process does not apply to the conformational dynamics . at short waiting times , we see a nonexponential decay , which exhibits a weak temperature dependence . at long waiting times , there is a nearly exponential and more temperature dependent decay . qualitatively similar waiting - time distributions are observed for all dihedral species and conformational states , suggesting that the distributions are comprised of two contributions governing the behavior at short and long waiting times , respectively . and @xmath188 characterizing the waiting times of the occo and cocc dihedrals in the @xmath38 state , respectively . results for peo at @xmath189 are shown . for both dihedral species , we distinguish the waiting times prior to a forward jump ( @xmath170 ) from that prior to a backward jump ( @xmath171).__,width=311 ] in general , one can distinguish between forward and backward jumps . let @xmath190 be a sequence of three subsequently visited conformational states . then , a forward and a backward jump in the state @xmath191 are associated with @xmath192 and @xmath193 , respectively . unlike backward jumps , forward jumps enable visiting all three conformational states and , hence , they may be the cornerstone of conformational relaxation . since transitions @xmath194 are rare for the occo and cocc dihedrals at the studied temperatures , the behavior in the @xmath38 state is of particular importance for the exploration of all conformational states . figure [ fig11 ] shows the probability distributions @xmath195 characterizing the waiting times in the @xmath38 state during the forward ( @xmath170 ) and backward ( @xmath171 ) sequences , respectively . we see that the results for the two dihedral species are comparable . thus , it is not important that the @xmath38 state is the majority state of @xmath196 , whereas it is the minority state of @xmath197 , see fig . [ fig9 ] . however , the waiting times strongly depend on the direction of the jumps . for both dihedral species , the forward jumps are characterized by nearly exponential waiting - time distributions and , hence , the propensity to perform such transition does not depend on the jump history . the backward jumps exhibit a more complex behavior . in a semilogarithmic representation , the slope of the curves decreases with increasing waiting time until it becomes constant , indicative of an exponential behavior at sufficiently long waiting times . in the latter regime , the slope is comparable for the forward and backward jumps . these findings imply that both uncorrelated and correlated conformational transitions occur . in some cases , the time and the direction of a jump are independent of the history , resulting in an exponential behavior and in similar rates of forward and backward jumps . in other cases , the dihedrals have a high tendency to return to the previous conformational state almost immediately , i.e. , to perform a correlated backward jump , leading to the observed nonexponentiality at short waiting times . one can imagine that this proneness to a correlated backward jump depends on the jump history . for example , it may become weaker when time elapses after the previous transition . also , a correlated backward jump may be less likely when a large number of forward - backward jumps have already taken place , i.e. , when previous unsuccessful attempts have paved the way for a successful transition to a new state , see below . characterizing the waiting times of the occo dihedrals in the @xmath38 state prior to a forward jump . results for peo at various temperatures are shown . the inset displays the temperature dependence of the probability @xmath198 that a forward jump @xmath199 is followed by a direct backward jump @xmath200 to the initial dihedral state.__,width=311 ] to further investigate the role of the forward jumps for the conformational and structural relaxations , we analyze their temperature dependent behavior . figure [ fig12 ] shows the distributions @xmath187 characterizing the waiting times of the occo dihedrals in the @xmath38 state prior to a forward jump at various temperatures . evidently , all waiting - time distributions are nearly exponential , confirming that the point in time of a forward jump is independent of the history . in fig . [ fig8 ] , we compare the temperature dependent mean waiting times @xmath201 and @xmath202 characterizing the forward and the backward jumps in the @xmath38 state , respectively . we see that the forward jumps exhibit a higher temperature dependence than the backward jumps . specifically , the activation energies amount to @xmath174 for the former and @xmath175 for the latter . however , the temperature dependence of @xmath203 is still much weaker than that of the torsional and rotational correlation times , corroborating our previous conclusion that the longer waiting times at lower temperatures do not provide an explanation for the slowdown of the @xmath1 relaxation upon cooling . these findings show that it is not sufficient to study single events , but it is necessary to correlate the times and the directions of two or more conformational transitions . first , we analyze the directions of two consecutive jumps . specifically , we determine the back - jump probability @xmath204 in the @xmath38 state , i.e. , the probability that a transition @xmath199 is followed by a transition @xmath200 . in fig . [ fig12 ] , the temperature dependent back - jump probabilities @xmath204 of the occo dihedrals are presented . we see a substantial increase of @xmath204 upon cooling . thus , when the temperature is decreased , both longer waiting times and higher back - jump probabilities contribute to the slower exploration of the conformational states . in other words , the increase of the back - jump probabilities leads to an additional delay of the conformational relaxation and , hence , to deviations from an arrhenius behavior . to study the exploration of the dihedral states in more detail , we determine the times @xmath205 needed for the dihedrals to visit all three conformational states . specifically , we start from a randomly chosen time origin and define @xmath205 as the time elapsing until a given dihedral has first visited each of the three conformational states at least once . in fig . [ fig9 ] , we include the mean times @xmath176 resulting from the conformational dynamics of the occo dihedrals at various temperatures . unlike the mean waiting times , the exploration times @xmath176 do not follow an arrhenius law . however , their temperature dependence is still weaker than that of the torsional correlation times , implying that the relation between the exploration process and the conformational relaxation is not straightforward , but subtle . we note that previous work on polybutadiene did not relate the exploration process to the primary , but rather to the secondary relaxation.@xcite the question arises whether knowledge of the back - jump probabilities and of the mean waiting times prior to forward and backward jumps is sufficient to calculate the exploration times . to tackle this question , we assume that , at the randomly chosen time origin , an occo dihedral occupies one of the @xmath206 states , being the majority states , and calculate the average time needed for the dihedral to visit each conformational state at least once . first , we neglect direct transitions between the @xmath206 states due to their rareness . then , a forward jump in the @xmath38 state is necessary to visit all dihedral states . prior to a forward jump , the occo dihedral can perform @xmath207 backward jumps in the @xmath38 state . thus , in general , sequences @xmath208 lead to the exploration of all conformational states . provided the back - jump probability in the @xmath38 state is independent of the history , the probability of finding a sequence with @xmath207 backward jumps in the @xmath38 state is given by @xmath209 . then , the average time @xmath210 to move from one of the @xmath206 states to the other can be written as @xmath211 here , @xmath212 is the time needed for a forward - backward jump sequence and @xmath213 is the sum of the times elapsing prior to the first and the last conformational transition . when we assume that the waiting times can depend on the jump direction , but are otherwise independent of the history , @xmath212 and @xmath213 are determined by the mean waiting times @xmath214 and @xmath215 and by the mean waiting time @xmath216 in the @xmath217 states , which is found to be essentially independent of the jump direction . specifically , @xmath218 and @xmath219 . the factor 1/2 in the latter equation is a consequence of the fact that , on average , the randomly chosen time origin lies in the middle of the waiting time before the first jump . utilizing the knowledge of the mean waiting times and of the back - jump probabilities , we calculate the times @xmath210 according to eq . ( [ eq_t3 ] ) . in fig.[fig9 ] , we see that the temperature dependence of @xmath210 deviates from an arrhenius law due to the higher back - jump probabilities at lower temperatures . while the calculated and the actual exploration times are similar at high temperatures , the former show a weaker temperature dependence . therefore , we dropped the assumption that direct transitions between the @xmath206 states can be neglected and recalculated the mean times @xmath210 . in fig . [ fig9 ] , it is evident that considering the transitions between the @xmath206 states has hardly any effect . therefore , we refrain from specifying the equations for this case . also , we expect that the assumption to start in one of the @xmath206 states is not crucial , in particular at low temperatures , where the occupation of the @xmath158 state is small . the deviations between the calculated and the actual exploration times rather show that the assumption of history independent waiting times and back - jump probabilities is not justified , further illustrating the complexity of the conformational dynamics . of finding @xmath135 conformational transitions of the occo dihedrals during a time interval @xmath38 . we used @xmath220 and @xmath221 for peo at @xmath222 ( @xmath223 ) and @xmath224 ( @xmath225 ) , respectively . thus , the ratio between the time interval and the mean waiting time amounts to @xmath226 for both temperatures . the solid line is the poisson distribution calculated using the values of @xmath38 and @xmath163 at @xmath224 , i.e. , @xmath227 . _ , width=311 ] finally , we analyze the torsional motion of the occo dihedrals on a longer time scale @xmath228 , where @xmath229 is the mean waiting time resulting from all conformational transitions of this dihedral species . we calculate the probability distribution @xmath230 of finding @xmath135 conformational transitions during a time interval @xmath38 . figure [ fig12 ] shows the distributions @xmath230 obtained for @xmath228 at @xmath222 and @xmath224 . the data for the higher temperature resemble the poisson distribution for the used ratio @xmath231 and , hence , a markov process approximates the conformational dynamics . by contrast , substantial deviations from the poisson distribution are obvious for the lower temperature . specifically , @xmath230 is much broader , indicating that a large fraction of dihedrals performs less or more transitions than expected for the case of uncorrelated jump events . thus , at sufficiently low temperatures , pronounced dynamical heterogeneities do not only govern the structural relaxation , see sec . [ sec_shd ] , but also the conformational relaxation . furthermore , it is evident from fig . [ fig12 ] that it is more probable to find an even than an uneven number of conformational transitions in a given time interval . this effect , which is most pronounced for small @xmath135 and low temperatures , is another consequence of the existence of correlated forward - backward jumps . it implies that , at a given time , each occo dihedral has a preferred conformational state . after the exit of its preferred state , a dihedral tends to return to this state within a very short period of time performing a correlated backward jump . then , on average , the dihedrals spend much longer times in the preferred state than in the unpreferred state of the forward - backward sequence , resulting in a higher probability of finding an even number of jumps , consistent with the observation . at @xmath222 , the difference between even and uneven @xmath135 is small for @xmath232 and , hence , the information about the preferred state vanishes after about 30 conformational transitions , i.e. , after about 15 forward - backward jump sequences . we have performed md simulations to investigate the translational and rotational motion of chemically realistic polymer models . for the studied peo and ppo models , the structural relaxation shows the typical properties of polymer melts , i.e. , its time dependence differs from a single exponential function and its temperature dependence deviates from an arrhenius law , in harmony with results in the literature.@xcite specifically , for both polymer models , a kww function well interpolates the translational and rotational correlation functions in the @xmath1-relaxation regime , time temperature superposition is obeyed , and a vft law describes the temperature dependent @xmath1-relaxation times . the stretching and the temperature dependence of the correlation functions are consistent with results from experimental work , confirming the quality of the used force fields . recently , the applicability of mct to the structural relaxation of chemically realistic polymer models was controversially discussed.@xcite here , we have performed a mct analysis for the ppo model using the incoherent intermediate scattering functions of the oxygen atoms . on the one hand , the analysis shows that a mct power law with a critical temperature @xmath108 well describes the temperature dependent correlation times in the range @xmath233 . also , for a momentum transfer @xmath77 corresponding to the intermolecular oxygen - oxygen distance , further mct predictions are fulfilled . on the other hand , the temperature dependence of the @xmath1-relaxation time deviates from a mct power law in the immediate vicinity of @xmath2 and the factorization theorem , being a central mct prediction for the @xmath69-relaxation regime , is violated . we conclude that mct captures several aspects of the structural relaxation at appropriate temperatures and length scales , however it does not provide a complete description since energy barriers against the torsional motion affect the dynamical behavior . furthermore , we have studied the heterogeneity and the cooperativity of the structural relaxation . for the peo and ppo models , we have demonstrated that highly mobile oxygen atoms aggregate into transient clusters , indicating the spatially heterogeneous nature of the dynamics . furthermore , we have shown that cooperative string - like motion facilitates the translational displacements of the highly mobile oxygen atoms at intermediate times between ballistic motion and diffusive motion . both clusters and strings increase in size upon cooling and , hence , spatial heterogeneity and cooperativity are prominent aspects of the molecular dynamics in particular at low temperatures , in harmony with results for various models of glass - forming liquids.@xcite concerning the transient nature of these phenomena , we have found that the mean cluster size @xmath60 and the mean string length @xmath61 are a maximum at comparable times @xmath234 in the @xmath1-relaxation regime for peo and ppo . in this respect , the present findings differ from previous results . specifically , the clusters and the strings were reported to be largest at significantly earlier times in the late-@xmath69/ early-@xmath1 relaxation regime for models of atomic liquids,@xcite water@xcite , silica,@xcite and a bead - spring polymer.@xcite we conclude that the existence of spatially heterogeneous and cooperative dynamics are common to a broad variety of glass - forming liquids . however , the characteristics of these phenomena differ among the materials and , hence , relating the properties of the heterogeneity and the cooperativity of the molecular dynamics to the respective features of the structural relaxation may yield interesting insights into the glass transition phenomenon . for the peo model , we have demonstrated that a straightforward study of the conformational relaxation is possible , when mapping the continuous trajectories of the dihedral angles onto discrete sequences of the dihedral states . various results have indicated a complex nature of the conformational dynamics . in particular , for both dihedral species and for all dihedral states , the probability distributions of the waiting times @xmath163 , i.e. , of the time intervals between two subsequent conformational transitions , strongly deviate from an exponential function , indicating that a markov process does not apply to the conformational dynamics . to analyze the origin of this behavior , we have discriminated between backward and forward jumps , i.e. , we have distinguished whether or not the dihedrals are in the same conformational state after exactly two transitions . this analysis revealed that the nonexponential waiting - time distributions are a consequence of correlated forward - backward jumps , which are an important aspect of the conformational dynamics at sufficiently low temperatures . consistent with previous results for all - atom polymer models,@xcite we have observed that the mean waiting times show a weaker temperature dependence than the time constants of the conformational and structural relaxations . hence , it is not possible to explain the glassy slowdown on the basis of single events , but it is necessary to correlate the times and the directions of several consecutive conformational transitions . analyzing the directions of subsequent transitions , we have shown that the probability of backward jumps increases upon cooling and , hence , the exploration of the conformational states is slower at lower temperatures due to both longer waiting times and higher back - jump probabilities . to obtain insights into the times of consecutive transitions , we considered probability distributions @xmath235 of finding @xmath135 transitions in a time interval @xmath38 . in this way , we have revealed that the conformational dynamics resembles a poisson process at high temperatures . however , when the temperature is decreased , dynamical heterogeneities become important for the conformational relaxation and the dihedrals start having a preferred conformational state at a given time . after an exit of the preferred state state , the dihedrals show a high tendency to return to this state via a correlated backward jump . at the studied temperatures , the dihedrals remember this preferred state for up to about 30 conformational transitions . in other words , the conformational states are not sampled according to their statistical weights in the early stages of the conformational relaxation .

the temperature dependence of these relaxation processes deviates from an arrhenius law for both polymers . we demonstrate that mode - coupling theory captures some aspects of the glassy slowdown , but it does not enable a complete explanation of the dynamical behavior . moreover , the transitions between the conformational states cease to obey poisson statistics . in particular , we show that , at sufficiently low temperatures , correlated forward - backward motion is an important aspect of the conformational relaxation , leading to strongly nonexponential distributions for the waiting times of the dihedrals in the various conformational states .

performing molecular dynamics simulations for all - atom models , we characterize the conformational and structural relaxations of poly(ethylene oxide ) and poly(propylene oxide ) melts . the temperature dependence of these relaxation processes deviates from an arrhenius law for both polymers . we demonstrate that mode - coupling theory captures some aspects of the glassy slowdown , but it does not enable a complete explanation of the dynamical behavior . when the temperature is decreased , spatially heterogeneous and cooperative translational dynamics are found to become more important for the structural relaxation . moreover , the transitions between the conformational states cease to obey poisson statistics . in particular , we show that , at sufficiently low temperatures , correlated forward - backward motion is an important aspect of the conformational relaxation , leading to strongly nonexponential distributions for the waiting times of the dihedrals in the various conformational states .

hep-ex9801024

i

this paper describes a search for photonic events with large missing energy in @xmath11 collisions at centre - of - mass energies of 130 , 136 , 161 and 172 gev . two different search topologies are used . cross - section measurements and search results from single - photon and acoplanar - photons topologies at @xmath59-@xmath60 gev@xcite and @xmath61 gev@xcite have been previously published . those results have also been used to search for excited neutrinos with photonic decays at @xmath59-@xmath60 gev @xcite and @xmath61 gev @xcite . in the analyses presented in this paper , both the single and acoplanar - photons search techniques are based on those previously published by opal , but in each case the kinematic acceptance of the analysis has been extended to lower energy and more forward angles , and the efficiency has been increased by allowing for the possibility of photon conversions . these results supersede our previous results . the single - photon and acoplanar - photons search topologies presented here are designed to select events with one or more photons and significant missing transverse energy , indicating the presence of at least one neutrino - like invisible particle which interacts only weakly with matter . results on photonic events without missing energy are presented in a separate paper.@xcite . the single - photon search topology is sensitive to neutral events in which there are one or two photons and missing energy , which within the standard model are expected from the @xmath1 process . measurements of single - photon production have been made in @xmath11 collisions at the @xmath62 and at lower energies @xcite . results from centre - of - mass energies significantly above the @xmath62 mass have also been reported @xcite . the expected visible energies are sufficiently large at present centre - of - mass energies that doubly radiative neutrino pair production can lead to two photons being detected ; the experimental topology therefore includes such cases . the acoplanar - photons search topology is designed to select neutral events with two or more photons and significant missing transverse energy , which within the standard model are expected from the @xmath2 process . the selection is designed to retain acceptance for events with a number of photons , @xmath63 , greater than two if the system formed by the three most energetic photons shows evidence for significant missing transverse energy . these photonic final - state topologies are sensitive to new physics of the type @xmath64 and @xmath65 where @xmath66 is neutral and decays radiatively ( @xmath67 ) and @xmath5 is stable and only weakly interacting . for the general case of massive @xmath66 and @xmath5 this includes conventional supersymmetric processes@xcite @xmath68 . in this context it has been emphasised @xcite that the radiative branching ratio of the @xmath69 may be large . there is particularly good sensitivity for the special case of @xmath70 that applies both to the production of excited neutrinos @xmath71 and to supersymmetric models in which the lightest supersymmetric particle ( lsp ) is a light gravitino(kev ) . ] , and @xmath72 is the next - to - lightest supersymmetric particle ( nlsp ) which decays to a gravitino and a photon , ( @xmath73 ) . in this case , the branching ratio of this decay of the @xmath72 is naturally large . such a signature has been discussed in @xcite and more recently in @xcite for a no - scale supergravity model and in @xcite for a model with gauge - mediated supersymmetry breaking ; in each case , @xmath74 production cross - sections of order 1 pb are predicted at these centre - of - mass energies , for @xmath75 gev . other types of new physics to which these search topologies are sensitive include the production of invisible particles made visible through initial - state radiation and the production of an invisible particle in association with a photon . the acoplanar - photons search topology also has sensitivity to the production of two particles , one invisible , or with an invisible decay mode , and the other decaying into two photons . such events might arise from the production of a higgs - like particle , @xmath76 : @xmath77 , followed by s@xmath78@xmath79 , @xmath80 . results for this model searching for the hadronic and leptonic @xmath81 decays appear elsewhere@xcite . this paper will first briefly describe the detector , the data sample and the monte carlo samples used , including a discussion of event generators for @xmath82 . the event selection for each search topology will then be described , followed by cross - section measurements for @xmath1 and @xmath2 and comparisons with standard model expectations . implications of these results on the possibility of new physics processes of the type @xmath64 or @xmath83 , @xmath67 will be discussed .

photonic events with large missing energy have been observed in collisions at centre - of - mass energies of 130 , 136 , 161 and 172 gev using the opal detector at lep . results are presented based on search topologies designed to select events with a single photon and missing transverse energy or events with a pair of acoplanar photons . in both search topologies , no evidence is observed for new physics contributions to these final states .

photonic events with large missing energy have been observed in collisions at centre - of - mass energies of 130 , 136 , 161 and 172 gev using the opal detector at lep . results are presented based on search topologies designed to select events with a single photon and missing transverse energy or events with a pair of acoplanar photons . in both search topologies , cross - section measurements are performed within the kinematic acceptance of the selection . these results are compared with the expectations from the standard model processes ( single - photon ) and ( acoplanar - photons ) . no evidence is observed for new physics contributions to these final states . upper limits on and + are derived for the case of stable and invisible . these limits apply to single and pair production of excited neutrinos ( ) , to neutralino production ( ) , and to supersymmetric models in which and is a light gravitino . for the latter scenario , the results of the acoplanar - photons search are used to provide model - dependent lower limits on the mass of the lightest neutralino . plus 1pt minus 1pt plus 1pt european laboratory for particle physics cern - ppe/97 - 132 + 6 october 1997 * search for anomalous production of photonic + events with missing energy in collisions + at * the opal collaboration ( to be submitted to z. phys . c ) the opal collaboration k.ackerstaff , g.alexander , j.allison , n.altekamp , k.j.anderson , s.anderson , s.arcelli , s.asai , s.f.ashby , d.axen , g.azuelos , a.h.ball , e.barberio , r.j.barlow , r.bartoldus , j.r.batley , s.baumann , j.bechtluft , c.beeston , t.behnke , a.n.bell , k.w.bell , g.bella , s.bentvelsen , s.bethke , s.betts , o.biebel , a.biguzzi , s.d.bird , v.blobel , i.j.bloodworth , j.e.bloomer , m.bobinski , p.bock , d.bonacorsi , m.boutemeur , s.braibant , l.brigliadori , r.m.brown , h.j.burckhart , c.burgard , r.brgin , p.capiluppi , r.k.carnegie , a.a.carter , j.r.carter , c.y.chang , d.g.charlton , d.chrisman , p.e.l.clarke , i.cohen , j.e.conboy , o.c.cooke , c.couyoumtzelis , r.l.coxe , m.cuffiani , s.dado , c.dallapiccola , g.m.dallavalle , r.davis , s.de jong , l.a.del pozo , k.desch , b.dienes , m.s.dixit , m.doucet , e.duchovni , g.duckeck , i.p.duerdoth , d.eatough , j.e.g.edwards , p.g.estabrooks , h.g.evans , m.evans , f.fabbri , a.fanfani , m.fanti , a.a.faust , l.feld , f.fiedler , m.fierro , h.m.fischer , i.fleck , r.folman , d.g.fong , m.foucher , a.frtjes , d.i.futyan , p.gagnon , j.w.gary , j.gascon , s.m.gascon-shotkin , n.i.geddes , c.geich-gimbel , t.geralis , g.giacomelli , p.giacomelli , r.giacomelli , v.gibson , w.r.gibson , d.m.gingrich , d.glenzinski , j.goldberg , m.j.goodrick , w.gorn , c.grandi , e.gross , j.grunhaus , m.gruw , c.hajdu , g.g.hanson , m.hansroul , m.hapke , c.k.hargrove , p.a.hart , c.hartmann , m.hauschild , c.m.hawkes , r.hawkings , r.j.hemingway , m.herndon , g.herten , r.d.heuer , m.d.hildreth , j.c.hill , s.j.hillier , p.r.hobson , a.hocker , r.j.homer , a.k.honma , d.horvth , k.r.hossain , r.howard , p.hntemeyer , d.e.hutchcroft , p.igo-kemenes , d.c.imrie , m.r.ingram , k.ishii , a.jawahery , p.w.jeffreys , h.jeremie , m.jimack , a.joly , c.r.jones , g.jones , m.jones , u.jost , p.jovanovic , t.r.junk , j.kanzaki , d.karlen , v.kartvelishvili , k.kawagoe , t.kawamoto , p.i.kayal , r.k.keeler , r.g.kellogg , b.w.kennedy , j.kirk , a.klier , s.kluth , t.kobayashi , m.kobel , d.s.koetke , t.p.kokott , m.kolrep , s.komamiya , t.kress , p.krieger , j.von krogh , p.kyberd , g.d.lafferty , r.lahmann , w.p.lai , d.lanske , j.lauber , s.r.lautenschlager , j.g.layter , d.lazic , a.m.lee , e.lefebvre , d.lellouch , j.letts , l.levinson , s.l.lloyd , f.k.loebinger , g.d.long , m.j.losty , j.ludwig , d.lui , a.macchiolo , a.macpherson , m.mannelli , s.marcellini , c.markopoulos , c.markus , a.j.martin , j.p.martin , g.martinez , t.mashimo , p.mttig , w.j.mcdonald , j.mckenna , e.a.mckigney , t.j.mcmahon , r.a.mcpherson , f.meijers , s.menke , f.s.merritt , h.mes , j.meyer , a.michelini , g.mikenberg , d.j.miller , a.mincer , r.mir , w.mohr , a.montanari , t.mori , u.mller , s.mihara , k.nagai , i.nakamura , h.a.neal , b.nellen , r.nisius , s.w.oneale , f.g.oakham , f.odorici , h.o.ogren , a.oh , n.j.oldershaw , m.j.oreglia , s.orito , j.plinks , g.psztor , j.r.pater , g.n.patrick , j.patt , r.perez-ochoa , s.petzold , p.pfeifenschneider , j.e.pilcher , j.pinfold , d.e.plane , p.poffenberger , b.poli , a.posthaus , c.rembser , s.robertson , s.a.robins , n.rodning , j.m.roney , a.rooke , a.m.rossi , p.routenburg , y.rozen , k.runge , o.runolfsson , u.ruppel , d.r.rust , r.rylko , k.sachs , t.saeki , w.m.sang , e.k.g.sarkisyan , c.sbarra , a.d.schaile , o.schaile , f.scharf , p.scharff-hansen , j.schieck , p.schleper , b.schmitt , s.schmitt , a.schning , m.schrder , h.c.schultz-coulon , m.schumacher , c.schwick , w.g.scott , t.g.shears , b.c.shen , c.h.shepherd-themistocleous , p.sherwood , g.p.siroli , a.sittler , a.skillman , a.skuja , a.m.smith , g.a.snow , r.sobie , s.sldner-rembold , r.w.springer , m.sproston , k.stephens , j.steuerer , b.stockhausen , k.stoll , d.strom , r.strhmer , p.szymanski , r.tafirout , s.d.talbot , s.tanaka , p.taras , s.tarem , r.teuscher , m.thiergen , m.a.thomson , e.von trne , e.torrence , s.towers , i.trigger , z.trcsnyi , e.tsur , a.s.turcot , m.f.turner-watson , p.utzat , r.van kooten , m.verzocchi , p.vikas , e.h.vokurka , h.voss , f.wckerle , a.wagner , c.p.ward , d.r.ward , p.m.watkins , a.t.watson , n.k.watson , p.s.wells , n.wermes , j.s.white , b.wilkens , g.w.wilson , j.a.wilson , t.r.wyatt , s.yamashita , g.yekutieli , v.zacek , d.zer-zion school of physics and space research , university of birmingham , birmingham b15 2tt , ukdipartimento di fisica dell universit di bologna and infn , i-40126 bologna , italyphysikalisches institut , universitt bonn , d-53115 bonn , germanydepartment of physics , university of california , riverside ca 92521 , usacavendish laboratory , cambridge cb3 0he , uk ottawa - carleton institute for physics , department of physics , carleton university , ottawa , ontario k1s 5b6 , canadacentre for research in particle physics , carleton university , ottawa , ontario k1s 5b6 , canadacern , european organisation for particle physics , ch-1211 geneva 23 , switzerlandenrico fermi institute and department of physics , university of chicago , chicago il 60637 , usafakultt fr physik , albert ludwigs universitt , d-79104 freiburg , germanyphysikalisches institut , universitt heidelberg , d-69120 heidelberg , germanyindiana university , department of physics , swain hall west 117 , bloomington in 47405 , usaqueen mary and westfield college , university of london , london e1 4ns , uktechnische hochschule aachen , iii physikalisches institut , sommerfeldstrasse 26 - 28 , d-52056 aachen , germanyuniversity college london , london wc1e 6bt , ukdepartment of physics , schuster laboratory , the university , manchester m13 9pl , ukdepartment of physics , university of maryland , college park , md 20742 , usalaboratoire de physique nuclaire , universit de montral , montral , quebec h3c 3j7 , canadauniversity of oregon , department of physics , eugene or 97403 , usarutherford appleton laboratory , chilton , didcot , oxfordshire ox11 0qx , ukdepartment of physics , technion - israel institute of technology , haifa 32000 , israeldepartment of physics and astronomy , tel aviv university , tel aviv 69978 , israelinternational centre for elementary particle physics and department of physics , university of tokyo , tokyo 113 , and kobe university , kobe 657 , japanbrunel university , uxbridge , middlesex ub8 3ph , ukparticle physics department , weizmann institute of science , rehovot 76100 , israeluniversitt hamburg / desy , ii institut fr experimental physik , notkestrasse 85 , d-22607 hamburg , germanyuniversity of victoria , department of physics , p o box 3055 , victoria bc v8w 3p6 , canadauniversity of british columbia , department of physics , vancouver bc v6 t 1z1 , canadauniversity of alberta , department of physics , edmonton ab t6 g 2j1 , canadaduke university , dept of physics , durham , nc 27708 - 0305 , usaresearch institute for particle and nuclear physics , h-1525 budapest , p o box 49 , hungaryinstitute of nuclear research , h-4001 debrecen , p o box 51 , hungaryludwigs - maximilians - universitt mnchen , sektion physik , am coulombwall 1 , d-85748 garching , germany and at triumf , vancouver , canada v6 t 2a3 and royal society university research fellow and institute of nuclear research , debrecen , hungary and department of experimental physics , lajos kossuth university , debrecen , hungary and department of physics , new york university , ny 1003 , usa

hep-ex9801024

c

we have searched for photonic events with large missing energy in two different and complementary topologies in data taken with the opal detector at lep , at centre - of - mass energies in the region of 130 - 172 gev . in the single - photon selection which requires at least one photon with @xmath121 in the region @xmath122 or one photon with @xmath123 in the region @xmath126 ( @xmath129 a total of 138 events are observed in the data compared to the koralz prediction for the contribution from the standard model process @xmath1 of @xmath233 events and an expected non - physics background of @xmath185 events . the corresponding cross - sections for @xmath1 are @xmath234 , @xmath235 , @xmath236 and @xmath237 pb for @xmath189 = 130 , 136 , 161 and 172 gev , respectively , in agreement with the standard model expectations . we derive upper limits on the cross - section times branching ratio for the process @xmath64 , @xmath67 for the general case of massive @xmath66 and @xmath5 . the limits vary from 0.31 to 1.8 pb in the region of interest of the @xmath175 plane and include the special case of @xmath70 , where the limit varies between 0.36 and 0.76 pb for the @xmath99 mass range from @xmath178 to 172 gev . the acoplanar - photons selection requires at least two photons with scaled energy @xmath125 within the polar angle region @xmath238 or at least two photons with energy @xmath239 gev with one satisfying @xmath128 and the other satisfying @xmath238 . from the combined data sample 11 events are selected . the koralz prediction for the number of events from @xmath2 is @xmath208 . the cross - section for this process is measured at each centre - of - mass energy ( see table [ tab : g2_nevt ] ) . due to the uncertainties in the current modelling of the standard model background , @xmath2 , all limits from the acoplanar - photons analysis were calculated without taking into account the background estimate . based on a variety of kinematic distributions , all observed events appear consistent with @xmath2 . we derive 95% cl upper limits on @xmath4 ranging from 0.18 to 0.80 pb for the general case of massive @xmath66 and @xmath5 , and from 0.35 to 0.50 pb for the special case of @xmath70 . for the single - photon and acoplanar - photons search topologies , the general case of massive @xmath66 and @xmath5 is relevant to the supersymmetry models in which @xmath171 and @xmath172 , with @xmath98 and @xmath72 stable . the special case of @xmath70 is of particular interest for single and pair production of excited neutrinos and for supersymmetric models in which the lsp is a light gravitino and the @xmath72 is the nlsp which decays as @xmath240 . for the latter scenario , the results of the acoplanar - photons search are used to place model - dependent lower limits on the @xmath72 mass . comparison with the model predictions of lopez and nanopoulos@xcite permits exclusion of that model for @xmath241 gev . a similar model from babu , kolda and wilczek@xcite is excluded for @xmath242 gev . the results of these searches have also been used to place limits on the production of excited neutrinos@xcite .

upper limits on and + are derived for the case of stable and invisible . these limits apply to single and pair production of excited neutrinos ( ) , to neutralino production ( ) , and to supersymmetric models in which and is a light gravitino . for the latter scenario , the results of the acoplanar - photons search are used to provide model - dependent lower limits on the mass of the lightest neutralino . plus 1pt minus 1pt plus 1pt european laboratory for particle physics cern - ppe/97 - 132 + 6 october 1997 * search for anomalous production of photonic + events with missing energy in collisions + at * the opal collaboration ( to be submitted to z. phys .

photonic events with large missing energy have been observed in collisions at centre - of - mass energies of 130 , 136 , 161 and 172 gev using the opal detector at lep . results are presented based on search topologies designed to select events with a single photon and missing transverse energy or events with a pair of acoplanar photons . in both search topologies , cross - section measurements are performed within the kinematic acceptance of the selection . these results are compared with the expectations from the standard model processes ( single - photon ) and ( acoplanar - photons ) . no evidence is observed for new physics contributions to these final states . upper limits on and + are derived for the case of stable and invisible . these limits apply to single and pair production of excited neutrinos ( ) , to neutralino production ( ) , and to supersymmetric models in which and is a light gravitino . for the latter scenario , the results of the acoplanar - photons search are used to provide model - dependent lower limits on the mass of the lightest neutralino . plus 1pt minus 1pt plus 1pt european laboratory for particle physics cern - ppe/97 - 132 + 6 october 1997 * search for anomalous production of photonic + events with missing energy in collisions + at * the opal collaboration ( to be submitted to z. phys . c ) the opal collaboration k.ackerstaff , g.alexander , j.allison , n.altekamp , k.j.anderson , s.anderson , s.arcelli , s.asai , s.f.ashby , d.axen , g.azuelos , a.h.ball , e.barberio , r.j.barlow , r.bartoldus , j.r.batley , s.baumann , j.bechtluft , c.beeston , t.behnke , a.n.bell , k.w.bell , g.bella , s.bentvelsen , s.bethke , s.betts , o.biebel , a.biguzzi , s.d.bird , v.blobel , i.j.bloodworth , j.e.bloomer , m.bobinski , p.bock , d.bonacorsi , m.boutemeur , s.braibant , l.brigliadori , r.m.brown , h.j.burckhart , c.burgard , r.brgin , p.capiluppi , r.k.carnegie , a.a.carter , j.r.carter , c.y.chang , d.g.charlton , d.chrisman , p.e.l.clarke , i.cohen , j.e.conboy , o.c.cooke , c.couyoumtzelis , r.l.coxe , m.cuffiani , s.dado , c.dallapiccola , g.m.dallavalle , r.davis , s.de jong , l.a.del pozo , k.desch , b.dienes , m.s.dixit , m.doucet , e.duchovni , g.duckeck , i.p.duerdoth , d.eatough , j.e.g.edwards , p.g.estabrooks , h.g.evans , m.evans , f.fabbri , a.fanfani , m.fanti , a.a.faust , l.feld , f.fiedler , m.fierro , h.m.fischer , i.fleck , r.folman , d.g.fong , m.foucher , a.frtjes , d.i.futyan , p.gagnon , j.w.gary , j.gascon , s.m.gascon-shotkin , n.i.geddes , c.geich-gimbel , t.geralis , g.giacomelli , p.giacomelli , r.giacomelli , v.gibson , w.r.gibson , d.m.gingrich , d.glenzinski , j.goldberg , m.j.goodrick , w.gorn , c.grandi , e.gross , j.grunhaus , m.gruw , c.hajdu , g.g.hanson , m.hansroul , m.hapke , c.k.hargrove , p.a.hart , c.hartmann , m.hauschild , c.m.hawkes , r.hawkings , r.j.hemingway , m.herndon , g.herten , r.d.heuer , m.d.hildreth , j.c.hill , s.j.hillier , p.r.hobson , a.hocker , r.j.homer , a.k.honma , d.horvth , k.r.hossain , r.howard , p.hntemeyer , d.e.hutchcroft , p.igo-kemenes , d.c.imrie , m.r.ingram , k.ishii , a.jawahery , p.w.jeffreys , h.jeremie , m.jimack , a.joly , c.r.jones , g.jones , m.jones , u.jost , p.jovanovic , t.r.junk , j.kanzaki , d.karlen , v.kartvelishvili , k.kawagoe , t.kawamoto , p.i.kayal , r.k.keeler , r.g.kellogg , b.w.kennedy , j.kirk , a.klier , s.kluth , t.kobayashi , m.kobel , d.s.koetke , t.p.kokott , m.kolrep , s.komamiya , t.kress , p.krieger , j.von krogh , p.kyberd , g.d.lafferty , r.lahmann , w.p.lai , d.lanske , j.lauber , s.r.lautenschlager , j.g.layter , d.lazic , a.m.lee , e.lefebvre , d.lellouch , j.letts , l.levinson , s.l.lloyd , f.k.loebinger , g.d.long , m.j.losty , j.ludwig , d.lui , a.macchiolo , a.macpherson , m.mannelli , s.marcellini , c.markopoulos , c.markus , a.j.martin , j.p.martin , g.martinez , t.mashimo , p.mttig , w.j.mcdonald , j.mckenna , e.a.mckigney , t.j.mcmahon , r.a.mcpherson , f.meijers , s.menke , f.s.merritt , h.mes , j.meyer , a.michelini , g.mikenberg , d.j.miller , a.mincer , r.mir , w.mohr , a.montanari , t.mori , u.mller , s.mihara , k.nagai , i.nakamura , h.a.neal , b.nellen , r.nisius , s.w.oneale , f.g.oakham , f.odorici , h.o.ogren , a.oh , n.j.oldershaw , m.j.oreglia , s.orito , j.plinks , g.psztor , j.r.pater , g.n.patrick , j.patt , r.perez-ochoa , s.petzold , p.pfeifenschneider , j.e.pilcher , j.pinfold , d.e.plane , p.poffenberger , b.poli , a.posthaus , c.rembser , s.robertson , s.a.robins , n.rodning , j.m.roney , a.rooke , a.m.rossi , p.routenburg , y.rozen , k.runge , o.runolfsson , u.ruppel , d.r.rust , r.rylko , k.sachs , t.saeki , w.m.sang , e.k.g.sarkisyan , c.sbarra , a.d.schaile , o.schaile , f.scharf , p.scharff-hansen , j.schieck , p.schleper , b.schmitt , s.schmitt , a.schning , m.schrder , h.c.schultz-coulon , m.schumacher , c.schwick , w.g.scott , t.g.shears , b.c.shen , c.h.shepherd-themistocleous , p.sherwood , g.p.siroli , a.sittler , a.skillman , a.skuja , a.m.smith , g.a.snow , r.sobie , s.sldner-rembold , r.w.springer , m.sproston , k.stephens , j.steuerer , b.stockhausen , k.stoll , d.strom , r.strhmer , p.szymanski , r.tafirout , s.d.talbot , s.tanaka , p.taras , s.tarem , r.teuscher , m.thiergen , m.a.thomson , e.von trne , e.torrence , s.towers , i.trigger , z.trcsnyi , e.tsur , a.s.turcot , m.f.turner-watson , p.utzat , r.van kooten , m.verzocchi , p.vikas , e.h.vokurka , h.voss , f.wckerle , a.wagner , c.p.ward , d.r.ward , p.m.watkins , a.t.watson , n.k.watson , p.s.wells , n.wermes , j.s.white , b.wilkens , g.w.wilson , j.a.wilson , t.r.wyatt , s.yamashita , g.yekutieli , v.zacek , d.zer-zion school of physics and space research , university of birmingham , birmingham b15 2tt , ukdipartimento di fisica dell universit di bologna and infn , i-40126 bologna , italyphysikalisches institut , universitt bonn , d-53115 bonn , germanydepartment of physics , university of california , riverside ca 92521 , usacavendish laboratory , cambridge cb3 0he , uk ottawa - carleton institute for physics , department of physics , carleton university , ottawa , ontario k1s 5b6 , canadacentre for research in particle physics , carleton university , ottawa , ontario k1s 5b6 , canadacern , european organisation for particle physics , ch-1211 geneva 23 , switzerlandenrico fermi institute and department of physics , university of chicago , chicago il 60637 , usafakultt fr physik , albert ludwigs universitt , d-79104 freiburg , germanyphysikalisches institut , universitt heidelberg , d-69120 heidelberg , germanyindiana university , department of physics , swain hall west 117 , bloomington in 47405 , usaqueen mary and westfield college , university of london , london e1 4ns , uktechnische hochschule aachen , iii physikalisches institut , sommerfeldstrasse 26 - 28 , d-52056 aachen , germanyuniversity college london , london wc1e 6bt , ukdepartment of physics , schuster laboratory , the university , manchester m13 9pl , ukdepartment of physics , university of maryland , college park , md 20742 , usalaboratoire de physique nuclaire , universit de montral , montral , quebec h3c 3j7 , canadauniversity of oregon , department of physics , eugene or 97403 , usarutherford appleton laboratory , chilton , didcot , oxfordshire ox11 0qx , ukdepartment of physics , technion - israel institute of technology , haifa 32000 , israeldepartment of physics and astronomy , tel aviv university , tel aviv 69978 , israelinternational centre for elementary particle physics and department of physics , university of tokyo , tokyo 113 , and kobe university , kobe 657 , japanbrunel university , uxbridge , middlesex ub8 3ph , ukparticle physics department , weizmann institute of science , rehovot 76100 , israeluniversitt hamburg / desy , ii institut fr experimental physik , notkestrasse 85 , d-22607 hamburg , germanyuniversity of victoria , department of physics , p o box 3055 , victoria bc v8w 3p6 , canadauniversity of british columbia , department of physics , vancouver bc v6 t 1z1 , canadauniversity of alberta , department of physics , edmonton ab t6 g 2j1 , canadaduke university , dept of physics , durham , nc 27708 - 0305 , usaresearch institute for particle and nuclear physics , h-1525 budapest , p o box 49 , hungaryinstitute of nuclear research , h-4001 debrecen , p o box 51 , hungaryludwigs - maximilians - universitt mnchen , sektion physik , am coulombwall 1 , d-85748 garching , germany and at triumf , vancouver , canada v6 t 2a3 and royal society university research fellow and institute of nuclear research , debrecen , hungary and department of experimental physics , lajos kossuth university , debrecen , hungary and department of physics , new york university , ny 1003 , usa

1601.03906

i

bivariate extreme value copulas are characterized by the pickands dependence functions , these are functions @xmath3 \to{\mathbb{r}}$ ] which satisfy the conditions : 1 . _ boundary conditions _ : @xmath4 , @xmath5 $ ] . convexity condition _ : @xmath6 is convex . in the presence of the _ convexity condition _ , the above _ boundary conditions _ are simply saying that the lines @xmath7 and @xmath8 , @xmath9 $ ] , are both _ support lines _ of @xmath6 at @xmath10 and @xmath11 respectively , and it follows that they can be equivalently replaced by 1 . _ endpoint conditions _ : @xmath12 and @xmath13 , @xmath14 ( see , e.g. , roberts and varberg @xcite , page 14 , problem k ) . note also that in the case where @xmath6 is twice differentiable the _ convexity condition _ can be replaced by @xmath15 , for all @xmath5 $ ] . let @xmath16 be the space of pickands dependence functions . for @xmath17 , the copula is @xmath18 and @xmath19 has a density if @xmath20 is absolutely continuous on @xmath21 $ ] . bivariate extreme value copulas may also be parameterized by the so - called spectral measure @xmath22 , that is , a positive measure on @xmath21 $ ] such that @xmath23}w h(\mathrm{d}w)=1=\int_{[0,1]}(1-w ) h(\mathrm{d}w)$ ] . let @xmath24 denote the space of spectral measures . the one - to - one correspondence between @xmath16 and @xmath24 , is given by @xmath25}\bigl\{(1-t)w\bigr\ } \vee\bigl \{t(1-w)\bigr\ } h(\mathrm{d}w),\qquad t\in[0,1],\ ] ] and @xmath26)=1+a'_+(w)$ ] , for all @xmath27 , where @xmath28 is the right derivative of @xmath6 , see beirlant _ _ @xcite . in the past literature , parametric models for the function @xmath6 have been studied for instance in tawn @xcite , coles and tawn @xcite , joe _ et al . _ @xcite , ledford and tawn @xcite and dupuis and tawn @xcite . the modeling of @xmath22 has also been considered , as it offers some advantages , especially when extensions to the multivariate case are desired . inference within parametric families of spectral measures can be found in boldi and davison @xcite , coles and tawn @xcite , coles and tawn @xcite , de haan _ et al . _ @xcite , einmahl _ et al . _ @xcite , joe _ et al . _ @xcite , ledford and tawn @xcite and smith @xcite . polynomial splines models have also been proposed by guillotte _ _ @xcite and fougres _ et al . _ @xcite . in view of the infinite - dimensional nature of the space of pickands functions , inference is often done nonparametrically for maximum flexibility , see for instance capra _ @xcite , hall and tajvidi @xcite , fils - villetard _ et al . _ @xcite and recently bcher _ et al . _ @xcite , to cite only a few . however , the nonparametric estimators do not usually satisfy the properties of pickands functions for finite samples and are often modified to do so . this can be cumbersome , see , for instance , fils - villetard _ et al . _ @xcite and the references therein . one alternative to the nonparametric approaches is based on having a series of nested models indexed by @xmath29 , @xmath1 . each model is parametric but @xmath29 is not bounded . within each model , a genuine estimator of the pickands dependence function is proposed . in this spirit , we consider modeling the pickands function @xmath6 using polynomials on @xmath21 $ ] . the latter are natural candidates here , the quadratic case being known as the symmetric mixed model , while the cubic case corresponds to the asymmetric mixed model , see beirlant _ et al . _ @xcite , pages 308 and 309 , and the references therein . an attempt at characterizing the space of pickands polynomials for higher degrees was made in klppelberg and may @xcite , page 1472 , theorem 2.5 . it is stated there that a polynomial ( in the power basis ) @xmath30 , @xmath31 $ ] , is a pickands dependence function if and only if its coefficients satisfy the four following conditions : @xmath32 it turns out that these conditions fail to be sufficient for @xmath33 . in fact , a counterexample dates back to beirlant _ et al . _ @xcite , page 308 : the polynomial @xmath34 satisfies the above conditions but is not convex . we will return to this example in section [ sect : characterization ] . first , section [ sect : bernstein ] is a short preliminary section giving the essential definitions and general mathematical results concerning the bernstein basis used throughout the paper . the reason for this choice of basis here is , essentially , its ability to reflect geometric and algebraic particularities of this problem on the coefficients . this will be reinforced , with an example and pointers to the text in the concluding remarks of section [ sect : conclusion ] . in section [ sect : characterization ] , the aim is to characterize , for each polynomial having degree at most @xmath29 , @xmath1 , the domain in which the coefficients of the polynomial ( in the bernstein basis ) must belong . we are looking for a constructive solution leading ultimately to a parameterization , which could be directly applicable in the search of the maximum likelihood estimate , for instance . we approach the problem by exploiting the one - to - one correspondence between the pickands function and the related spectral measure mentioned above . for polynomials , this boils down to the link between the pickands function and its second derivative ( a nonnegative polynomial ) . the solution therefore exploits the characterization of nonnegative polynomials given by lukcs , see karlin and shapley @xcite . we subsequently refer to this solution as being the solution to the _ full model_. in section [ sect : approximation ] , the focus is on characterizing , for each degree @xmath1 , the space of polynomials that can be obtained from a bernstein approximation of a pickands dependence function . we call this model the _ submodel_. we try to give an answer to the question : what is the gap between the _ full model _ and the _ submodel _ ? now in section [ sect : quality ] , we use tools from approximation theory and probability for obtaining accurate bounds for measuring the closeness between the space of pickands dependence functions and the one obtained from the _ submodel_. note that bernstein approximations of copulas have appeared in the past literature and some of their properties have been studied in sancetta and satchell @xcite and sancetta and satchell @xcite . finally , in section [ sect : simulations ] , we present the results of a simulation comparing the maximum likelihood estimator from the full model with the one from the submodel and a version of the popular nonparametric `` cfg '' estimator in capra _ our estimators should be easily implemented , and could become part of standard extreme value statistics packages , such as the evd package in r. the existing packages offer only polynomial models of degree at most three , this may be explained by the absence of a successful characterization for higher degrees in the previous literature .

pickands dependence functions characterize bivariate extreme value copulas . in this paper , we study the class of polynomial pickands functions . we provide a solution for the characterization of such polynomials of degree at most , , and show that these can be parameterized by a vector in belonging to the intersection of two ellipsoids . we give necessary and sufficient conditions for which a polynomial pickands function is in fact a bernstein approximation of some pickands function . approximation results of pickands dependence functions by polynomials are given . finally , inferential methodology is discussed and comparisons based on simulated data are provided . ./style / arxiv - general.cfg

pickands dependence functions characterize bivariate extreme value copulas . in this paper , we study the class of polynomial pickands functions . we provide a solution for the characterization of such polynomials of degree at most , , and show that these can be parameterized by a vector in belonging to the intersection of two ellipsoids . we also study the class of bernstein approximations of order of pickands functions which are shown to be ( polynomial ) pickands functions and parameterized by a vector in belonging to a polytope . we give necessary and sufficient conditions for which a polynomial pickands function is in fact a bernstein approximation of some pickands function . approximation results of pickands dependence functions by polynomials are given . finally , inferential methodology is discussed and comparisons based on simulated data are provided . ./style / arxiv - general.cfg

1505.00047

m

our key idea is to adapt the pce to the dynamics of the system . consider for concreteness the following sde : @xmath43,\end{aligned}\ ] ] where @xmath44 is a general function of @xmath20 ( and possibly other deterministic or stochastic parameters ) , @xmath29 is a brownian motion , @xmath45 is a constant , and @xmath46 is an initial condition with a prescribed probability density . we assume that a solution exists and is unique on @xmath30 $ ] . as the system evolves , a fixed polynomial chaos basis adapted to @xmath46 and @xmath47 may not be optimal to represent the solution @xmath48 for long times . moreover , the dimension of the representation of @xmath47 increases linearly with time @xmath40 , which renders the pce method computationally intractable even for moderate values of @xmath40 . we will therefore introduce an increasing sequence of restart times @xmath49 and construct a new pce basis at each @xmath50 based on the solution @xmath51 and all additional random variables that need to be accounted for . a very similar methodology with @xmath52 was considered earlier in @xcite . when random forcing is present and satisfies the markov property as in the above example , the restarting strategy allows us to `` forget '' past random variables that are no longer necessary and focus on a significantly smaller subset of random variables that influence the future evolution . as an example of application , we will show that the restarting strategy allows us to capture the invariant measure of the above sde , when such a measure exists . to simplify notation , we present our algorithm on the above scalar sde with @xmath44 a function of @xmath20 , knowing that all results also apply to systems of sdes with minor modifications ; see sections [ sec : higherdim ] and [ sec : numeric ] . first , we notice that the solution @xmath48 of is a random process depending on the initial condition and the paths of brownian motion up to time @xmath40 : @xmath53 therefore , recalling the expansion for @xmath54 , the solution at time @xmath40 can be seen as a nonlinear functional of @xmath46 and the countably infinite variables @xmath14 . as previously noticed in @xcite , the solution @xmath48 can be represented as a linear chaos expansion in terms of the polynomials in itself and therefore , for sufficiently small later times @xmath55 , @xmath56 , the solution @xmath57 can be efficiently captured by low order chaos expansions in @xmath48 everything else being constant . moreover , the solution @xmath57 on the interval @xmath58 depends on @xmath59}$ ] and not on values of @xmath60 outside of this interval . this significantly reduces the number of random variables in @xmath14 that need to be accounted for . this crucial observation clearly hinges upon the markovian property of the dynamics . hence , dividing the time horizon into small pieces and iteratively employing pce offer a possible way to alleviate both curse of dimensionality and long - term integration problems . we decompose the time horizon @xmath30 $ ] into @xmath12 subintervals ; namely @xmath61,[t_1,t_2],\ldots,[t_{n-1},t]$ ] where @xmath62 . the idea is then to employ polynomial chaos expansion in each subinterval and restart the approximation depending on the distributions of both @xmath63 and @xmath51 at each @xmath50 , @xmath64 ; see figure [ fig : prop_chaos ] . here , @xmath63 denotes the gaussian random variables required for brownian forcing on the interval @xmath65 $ ] . throughout the paper , we utilize the term @xmath66 to represent orthonormal chaos basis involving its arguments . in order to establish evolution equations for chaos basis @xmath67 , triple products @xmath68 $ ] are needed . this procedure basically corresponds to computing the coefficients and indices in the hermite product formula . the probability distribution of @xmath51 does not belong to any classical family such as the askey family in general . the construction of the orthogonal polynomials in @xmath51 is therefore fairly involved computationally and is based on the general pce results mentioned earlier in the section . at each time step , we compute the moments of @xmath51 using its previously obtained chaos representation and incorporate them in a modified gram schmidt method . this is a computationally expensive step . armed with the orthogonal basis , we then compute the triple products in @xmath51 to perform the necessary galerkin projections onto spaces spanned by this orthogonal basis and thus obtain evolution equations for the deterministic expansion coefficients . letting @xmath69 , equations for the coefficients @xmath70 of @xmath71 are given in the general form @xmath72 ds + \sigma \ , { \mathbb{e}}\left[t_{\alpha}({\boldsymbol{\xi}}_j , u_j ) \ , d{w_s}\right ] , \end{aligned}\ ] ] where @xmath73 . before integration of these expansion coefficients in time , @xmath74 is represented in terms of its own orthogonal polynomials , which provides a description of the initial conditions on that interval . we then perform a high order time integration . cumulants of the resulting solution are then computed to obtain relevant statistical information about the underlying distribution . the proposed methodology does not require the computation of any probability density function ( pdf ) and rather depends only on its moments at each restart time . the statistical information contained in such moments provides useful data for decision making in modeling and , for determinate distributions , moments characterize the distribution uniquely . this aspect will be essential in the proof of convergence in section [ sec : conv ] . comparing our method with probability distribution function methods , e.g. , the fokker planck equation , we note that it essentially evolves the coefficients of orthogonal polynomials of the projected solution in time instead of evolving the pdf . let @xmath75 represents the nonlinear chaos expansion mapping @xmath76 to @xmath77 . our scheme may be demonstrated by figure [ fig : prop_chaos ] . mathematically , we have @xmath78 in simulations , @xmath63 is truncated with finite dimension @xmath79 . let also @xmath80 denote the maximum degree of the polynomials in the variables @xmath81 . here , to simplify , @xmath82 and @xmath25 are chosen independently of @xmath83 . the multi - indices having @xmath82 dimensions and degree @xmath25 belong to the set @xmath84 we now present our algorithm , called _ dynamical generalized polynomial chaos _ ( dgpc ) , in one dimension for concreteness ; see also section [ sec : higherdim ] . + * algorithm 1d * : dgpc + - decompose time domain @xmath30 = [ t_0,t_1 ] \cup \ldots.\cup [ t_{n-1},t]$ ] + - initialize degrees of freedom @xmath85 + - compute coefficients / indices in hermite product formula for @xmath86 + - integration in time : + - time step @xmath87 : obtain random variable @xmath74 + - calculate moments @xmath88 $ ] + - construct orthogonal polynomials @xmath89 + - compute triple products @xmath90 $ ] + - perform galerkin projection onto @xmath91 + - set up initial conditions for the coefficients @xmath92 + - evolve the expansion coefficients @xmath92 + - calculate cumulants + - postprocessing + several remarks are in order . first , since our stochastic forcing has identically distributed independent increments , we observe that at each subinterval the distribution of @xmath63 is the same . computing and storing ( in sparse format ) the coefficients for the product formula only once drastically reduces the computational time . at each iteration , the random variable @xmath93 is projected onto a finite dimensional chaos space and the next step is initialized with the projected random variable . for a @xmath94-dimensional sde system , this projection leads to @xmath95 terms in the basis for the subinterval @xmath96 $ ] . therefore , the total number of degrees of freedom used in @xmath30 $ ] becomes @xmath97 ; see also section [ sec : cost ] . we emphasize that small values of @xmath85 are utilized in each subinterval such that computations can be carried out quickly . the idea of iteratively constructing chaos expansions for arbitrary measures was considered earlier in @xcite . in @xcite , an iterative method is proposed to solve coupled multiphysics problems , where a dimension reduction technique is used to exchange information between iterations while allowing the construction of pc in terms of arbitrary ( compactly supported ) distributions at each iteration . a similar iterative procedure for hermite pc was suggested for stochastic partial differential equations ( spdes ) with brownian forcing in the conclusion section of @xcite without any mathematical construction or numerical example . we also stress an important difference between our approach and @xcite : our scheme solely depends on the statistical information , i.e. , moments , which are available through chaos expansion , whereas @xcite either requires the pdf of @xmath74 at @xmath50 or uses mappings to transform @xmath74 back to the original input random variables , which in our problem is high dimensional , including the @xmath42 variables . we now describe the implementation of our algorithm . provided that the distribution of the initial condition @xmath46 is known , several methods allow us to compute moments : analytic formulas , quadrature methods , or monte carlo sampling . also , in the case of limited data , moments can be generated from raw data sets . throughout the paper , we assume that the moments of the initial condition can be computed to some large finite order so that the algorithm can be initialized . we first provide the hermite product formula , which establishes the multiplication of two pce for random variables @xmath20 and @xmath98 . the chaos expansion for the product becomes @xmath99 with @xmath100^{1/2 } , $ ] @xcite . as our applications include nonlinear terms , this formula will be used repeatedly to multiply hermite chaos expansions . to compute moments of the random variable @xmath74 ( recall ) at time step @xmath50 , @xmath101 , we use its chaos representation and previously computed triple products recursively as follows . due to the markov property of the solution , the normalized chaos basis in @xmath102 and @xmath103 becomes a tensor product , and thus we can write @xmath104 where @xmath105 and @xmath106 . then for an integer @xmath107 , @xmath108 here , @xmath109 and @xmath110 . hence , using , the coefficients @xmath111 , where @xmath112 and @xmath113 , can be calculated using the expression @xmath114 \ , ( u_j^{m-1})_{\gamma-\theta+\alpha , l}\,(u_j)_{\theta+\alpha , k},\end{aligned}\ ] ] which allows us to compute moments by simply taking the first coefficient , i.e. @xmath115 = ( u_j^m)_{\boldsymbol{0},0}$ ] . even though the coefficients @xmath116 can be computed offline , the triple products @xmath117 $ ] must be computed at every step as the distribution of @xmath77 evolves . given the real random variable @xmath74 obtained by the orthogonal projection of the solution @xmath20 onto homogeneous chaos at @xmath50 , the gram schmidt orthogonalization procedure can be used to construct the associated orthogonal polynomials of its continuous distribution . for theoretical reasons , we assume that the moment problem for @xmath74 is uniquely solvable ; i.e. , the distribution is nondegenerate and uniquely determined by its moments @xcite . to construct orthogonal polynomials , the classical gram schmidt procedure uses the following recursive relation @xmath118 where the coefficients are given by @xmath119 \left({\mathbb{e}}[t^2_l(u_j)]\right)^{-1}$ ] . note that @xmath19 denotes the lebesgue stieltjes integral with respect to distribution of @xmath74 . we observe that the coefficient @xmath120 requires moments up to degree @xmath121 as @xmath122 is at most of degree @xmath123 . thus , normalization would need first @xmath124 moments for orthonormal polynomials up to degree @xmath125 . after the construction of orthonormal polynomials of @xmath74 , the basis @xmath126 can be generated by tensor products since @xmath127 and @xmath74 are independent . moreover , triple products @xmath128 $ ] , where @xmath129 can be found by simply noting that this expectation is a triple sum involving moments up to order @xmath130 . hence , the knowledge of moments yields not only the orthonormal set of polynomials but also the triple products required at each iterative step . in our numerical computations , we prefer to utilize a modified gram schmidt algorithm as the classical approach is overly sensitive to roundoff errors in cases of high dimensionality @xcite . we refer the reader to @xcite for other algorithms , e.g. , stieltjes and modified chebyshev methods . when the algorithm is reinitialized at the restart point @xmath50 , it needs to construct the initial condition in terms of chaos variables for the next iteration on @xmath65 $ ] . in other words , we need to find the coefficients @xmath131 in terms of @xmath74 and @xmath127 . to this end , we observe that the first two polynomials in @xmath74 are given by @xmath132 and @xmath133 ) , $ ] where @xmath134 is the standard deviation of the random variable @xmath74 . notice that @xmath135 , because we assumed in the preceding section that the distribution was nondegenerate . hence , the initial condition can be easily found by the relation @xmath136 t_0(u_j ) + \sigma_{u_j } t_1(u_j ) = \sum_{\alpha \in \mathcal{j } , \ , k \in \mathbb{n}_0 } ( u_j)_{\alpha , k } \ , t_{\alpha}({\boldsymbol{\xi}}_{j } ) \,t_k(u_{j}).\end{aligned}\ ] ] we now extend our algorithm to higher dimensions , and for concreteness , we consider two dimensions . let @xmath137 be a two - dimensional random variable obtained by projection of the solution @xmath20 onto homogeneous chaos space at time @xmath138 . the two - dimensional case requires the calculation of the mixed moments @xmath139 , l , m \in \mathbb{n } \cup \{0 \}$ ] . in the case of independent components @xmath140 , the moments become products of marginal moments and the basis @xmath141 is obtained by tensor products of the one - dimensional bases . for a nonlinear system of equations , the components of the solution typically do not remain independent as time evolves . therefore , we need to extend the procedure to correlated components . denoting by @xmath142 and @xmath143 the corresponding chaos expansions on @xmath65 $ ] so that @xmath144 and @xmath145 , we compute the mixed moments by the change of variables formula @xmath146= { \mathbb{e}}_{(v_1,v_2)}[v^l_1 \ , v^m_2 ] = { \mathbb{e}}_{({\boldsymbol{\xi}}_{j},u_{j } ) } \left[z_{j+1,1}^l \ , z_{j+1,2}^m \right],\end{aligned}\ ] ] where @xmath147 represents the expectation with respect to the joint distribution induced by the subscript . note that the latter integral in can be computed with the help of previously computed triple products of @xmath63 and @xmath77 . incidentally , similar transformation methods were used previously in @xcite . since the components may not be independent , the tensor product structure is lost but the construction of orthogonal polynomials is still possible . based on the knowledge of marginal moments , we first compute the tensor product @xmath141 . however , this set is not orthogonal with respect to the joint probability measure of @xmath148 . therefore , we further orthogonalize it via a gram schmidt method . note that this procedure requires only mixed moments @xmath149 $ ] , which have already been computed in the previous step of the algorithm . it is worth noting that the resulting set of orthogonal polynomials is not unique as it depends on the ordering of monomials @xcite . in applications , we consider the same ordering used for the set of multi - indices . finally , calculation of triple products and initial conditions can be extended in an obvious way . in this section , we discuss the computational costs of our algorithm using a vector - valued version of the sde in @xmath150 for a fixed @xmath151 . for each scalar sde , we assume that the nonlinearity is proportional to @xmath125th order monomial with @xmath152 . furthermore , we compare costs of dgpc and hermite pc in the case where both methods attain a similar order of error . throughout this section , @xmath85 will denote the dimension and the degree in @xmath153 for hermite pc . here @xmath14 represents the truncation of @xmath94-dimensional brownian motion ; therefore , @xmath154 . the time interval is fixed and given by @xmath155 $ ] . we also let @xmath156 @xmath157 be the degrees of freedom for the simple truncation so that the dimension of truncated multi - indices @xmath158 becomes @xmath159 . for the dgpc method , we divide the interval into @xmath160 identical subintervals so that @xmath161 . we now slightly change the notation of previous sections and denote by @xmath162 the dimension and degree of polynomials of @xmath163 used in dgpc approximation for each @xmath64 . with additional @xmath94 variables at each restart , the dimension of the multi - index set for dgpc becomes @xmath164 due to the tensor product structure . further , let @xmath165 and @xmath166 denote the time step and global order of convergence for the time integration method employed in hermite pc , respectively . with the above notation , we summarize the computational costs in table [ table : compcost ] . all terms should be understood in big @xmath167 notation . r | p6.5cm|p3.9 cm flops & dgpc & hermite pc + offline & @xmath168 & @xmath169 + time evolution & @xmath170 $ ] & @xmath171 $ ] + moments & @xmath172 $ ] & + gram schmidt & @xmath173 & + triple products & @xmath174 & + initials & @xmath175 & + the estimates in table [ table : compcost ] are obtained as follows . triple products for @xmath14 with @xmath176 can be calculated by the equation @xmath177 & = { \mathbb{e}}\left [ \prod_{i=1}^k h_{\alpha_i } ( \xi_i ) \prod_{j=1}^k h_{\beta_j } ( \xi_j ) \prod_{k=1}^k h_{\gamma_k } ( \xi_k ) \right]= \prod_{i=1}^k { \mathbb{e}}[h_{\alpha_i}(\xi_i)h_{\beta_i}(\xi_i)h_{\gamma_i}(\xi_i)].\end{aligned}\ ] ] thus , the offline cost is of order @xmath178 assuming that one - dimensional triple products can be read from a precomputed table . since a time discretization with error @xmath179 is employed for hermite pc , then for dgpc in each subinterval , time steps of order @xmath180 should be used to attain the same global error . without nonlinearity , the total cost in hermite pc for evolution of a @xmath94-dimensional system becomes @xmath181 . due to the functional @xmath182 , the cost should also include the computation of @xmath183 at each time integration step . this requires additional @xmath184 work ( see the computation of moments below ) . hence , the corresponding total cost of evolution becomes the sum of these costs . the coefficients of the @xmath185th power of a single projected variable @xmath74 are given by @xmath186 , \quad 2 \leq k \leq 3n_*,\ ] ] assuming that , after each multiplication , the variable is projected onto the pc basis . here , @xmath187 . thus , the computation of marginal moments of all variables requires @xmath188 calculations at each restart time in dgpc . since we use the same ordering for multidimensional moments as @xmath189 , computing joint moments further needs @xmath190 amount of work . the gram schmidt procedure costs are cubic in the size of the gram matrix , which has dimension @xmath191 in dgpc . each triple product @xmath192 , \alpha,\beta,\gamma \in \mathcal{j}_{d , n_*}$ ] , is a triple sum ; therefore , the total cost is proportional to @xmath193 . also , initial conditions can be computed by orthogonal projection onto the dgpc basis , which costs @xmath164 per dimension . note that although we employed simple truncation for multi - indices , sparse truncation techniques can be utilized to further reduce the costs of both computing moments and evolution ; see @xcite . we now discuss the error terms in both algorithms . we posit that the hermite pce converges algebraically in @xmath25 and @xmath82 and that the error at time @xmath40 satisfies @xmath194 and @xmath195 depending on the sde . here @xmath196 denotes that @xmath197 for @xmath198 with @xmath199 . this assumption enforces an increase in the degrees of freedom @xmath200 if one wants to maintain the accuracy in the long term . algebraic convergence in @xmath82 and the term @xmath201 stems from the error bound . we do not show errors resulting from time discretization in since we already fixed those errors to the same order in both methods . incidentally , even though we assumed algebraic convergence in @xmath25 , exponential numerical convergence , depending on the regularity of the solution in @xmath14 , is usually observed in the literature @xcite . although it is usually hard to quantify the constants @xmath202 , this may be done for simple cases . for instance , for the ornstein uhlenbeck ( ou ) process , using the exact solution and fourier basis of cosines , we can obtain the parameters @xmath203 and @xmath204 for short times . the parameter @xmath205 does not play a role in this case since the sde stays gaussian and hence we take @xmath206 . we also note that for complex dynamics , @xmath205 may depend on @xmath25 ; see @xcite in the case of the stochastic burgers equation and advection equations . now , we fix the error terms in to the same order @xmath207 , @xmath56 , by choosing @xmath208 . since dgpc uses truncated @xmath14 of dimension @xmath209 and polynomials of degree @xmath210 in each subinterval of size @xmath211 , the @xmath18 error at @xmath212 has the form @xmath213 of accuracy , we can choose @xmath214 from table [ table : compcost ] , we observe that to minimize costs for dgpc , we can take @xmath215 and @xmath216 and maximize @xmath12 according to the previous equation as @xmath217 with these choices of parameters , the total computational cost for dgpc is of algebraic order in @xmath82 and in general , is dominated by the computation of moments . for hermite pc with large @xmath82 and @xmath218 ( or @xmath219 in the case of exponential convergence in @xmath25 ) , both offline and evolution stages include the cubic term @xmath220 . both costs increase exponentially in @xmath82 . therefore , the asymptotic analysis suggests that dgpc can be performed with substantially lower computational costs using frequent restarts ( equation ) in nonlinear systems driven by high dimensional random forcing . we now consider the convergence properties of our scheme as degrees of freedom tend to infinity . there is an extensive literature on the moment problem for probability distributions ; see @xcite and their references . we provide the relevant background in the analysis of dgpc . _ ( hamburger moment problem ) _ for a probability measure @xmath221 on @xmath222 , the moment problem is uniquely solvable provided moments of all orders @xmath223 , @xmath224 exist and they uniquely determine the measure . in this case , the distribution @xmath221 is called _ there are several sufficient conditions for a one - dimensional distribution to be determinate in hamburger sense : these are compact support , exponential integrability and carleman s moment condition @xcite . for instance , gaussian and uniform distributions are determinate , whereas the lognormal distribution is not . the moment problem is intrinsically related to the density of associated orthogonal polynomials . indeed , if the cumulative distribution @xmath225 of a random variable @xmath20 is determinate , then the corresponding orthogonal polynomials constitute a dense set in @xmath226 and therefore also in @xmath227 @xcite . additionally , finite dimensional distributions on @xmath150 with compact support are determinate ; see @xcite . now , let @xmath228 denote an independent collection of general random variables , where each @xmath229 ( not necessarily identically distributed ) has finite moments of all orders and its cumulative distribution function is continuous . under these assumptions , @xcite proves the following theorem . [ thm : conv_ernst ] for any random variable @xmath230 , gpc converges to @xmath205 in @xmath18 if and only if the moment problem for each @xmath229 , @xmath231 , is uniquely solvable . this result generalizes the convergence of hermite pce to general random variables whose laws are determinate . notably , to prove @xmath18 convergence of chaos expansions , it is enough to check one of the determinacy conditions mentioned above for each one - dimensional @xmath229 . we now consider the relation between determinacy and distributions of sdes . consider the following diffusion @xmath232,\end{aligned}\ ] ] where @xmath54 is @xmath94-dimensional brownian motion , @xmath233 , and @xmath234 are globally lipschitz and satisfy the usual linear growth bound . these conditions imply that the sde has a unique strong solution with continuous paths @xcite . determinancy of the distribution of @xmath235 is established by the following theorem ; see @xcite for details . [ thm : det_sde ] the law of the solution of is determinate if @xmath236 is finite . consider the solution of the sde written as @xmath237 where @xmath238 is the exact evolution operator mapping the initial condition @xmath46 and @xmath239 to the solution @xmath235 at time @xmath240 . here , with a slight change of notation , @xmath241 represents brownian motion on the interval @xmath242 $ ] . note that future solutions @xmath243 , @xmath244 , are obtained by @xmath245 , where @xmath246 denotes brownian motion on the interval @xmath247 $ ] . even though the distribution of the exact solution @xmath235 is determinate under the hypotheses of thm . [ thm : det_sde ] , it is not clear that this feature still holds for the projected random variables . to address this issue , we introduce , for @xmath248 , a truncation function @xmath249 @xmath250 where @xmath251 \subset { \mathbb{r}}^d$ ] , and such that @xmath252 decays smoothly and sufficiently slowly on @xmath253 such that the lipschitz constant @xmath254 of @xmath255 equals @xmath256 ; i.e. , for @xmath257 , the euclidean distance in @xmath150 , @xmath258 . such a function is seen to exist by appropriate mollification for each coordinate using the continuous , piecewise smooth , and odd function @xmath259 defined by @xmath260 on @xmath261 $ ] , and extended by @xmath262 outside @xmath263 $ ] . this truncation is a theoretical tool that allows us to ensure that all distributions with support properly restricted on compact intervals remain determinate ; see @xcite . we can now introduce the following approximate solution operators for @xmath248 and @xmath264 : @xmath265 where @xmath266 denotes the orthogonal projection onto polynomials in @xmath20 and @xmath241 , and @xmath159 is the total number of degrees of freedom used in the expansion . throughout this section , we use a linear indexing to represent polynomials in both @xmath20 ( restricted on @xmath267 ) and the random variables @xmath14 . the main assumptions we imposed on the solution operator @xmath268 are summarized as follows : a. @xmath269 , where @xmath270 and @xmath271 with @xmath272 . [ item : zero ] b. @xmath273 , where @xmath274 , @xmath275 , and @xmath271 with @xmath272 . [ item : one ] c. for @xmath56 and @xmath276 , there is @xmath277 so that @xmath278 , where @xmath279 . [ item : two ] assumption [ item : zero ] is a stability estimate ensuring that the chain remains bounded in @xmath280th norm starting from a point in @xmath150 ( here @xmath262 without loss of generality owing to [ item : one ] ) . assumption [ item : one ] is a lipschitz growth condition controlled by a constant @xmath281 . these assumptions involve the @xmath282 norm , and hence are slightly stronger than control in the @xmath18 sense . note that @xmath283 also satisfies assumption [ item : one ] with the same constant @xmath281 since @xmath284 . finally , assumption [ item : two ] is justified by the weierstrass approximation theorem for the @xmath20 variable in @xmath267 and by the cameron martin theorem to handle the @xmath14 variable . consider now the following versions of the markov chains : @xmath285 where @xmath286 ( possibly random and independent of the variables @xmath14 ) and @xmath287 is a sequence of i.i.d . gaussian random variables representing brownian motion on the interval @xmath288 $ ] . we also use the notation @xmath289 to denote brownian motion on the interval @xmath290 $ ] . then we have the first result . [ lemma : rapprox ] assume [ item : zero ] , [ item : one ] , and [ item : two ] hold , and let @xmath291 be finite . suppose also that @xmath292 . then , for each @xmath56 , there exists @xmath248 depending on @xmath12 , @xmath293 , and @xmath294 such that @xmath295 for each @xmath296 . let @xmath56 be fixed . using properties [ item : zero ] and [ item : one ] , we observe that @xmath297 where for all @xmath83 , @xmath127 and @xmath241 are identically distributed . then , by induction , we obtain @xmath298 where the constant @xmath299 is bounded . the last inequality indicates that @xmath280th norm of the solution grows with @xmath83 , possibly exponentially , but remains bounded . for @xmath300 , we note that @xmath301 since @xmath74 is independent of future forcing . by markov s inequality and , we get @xmath302 let @xmath303 be the indicator function of the set @xmath304 . using and , we compute the following error bound @xmath305 , \quad 1 \leq j \leq n,\\ \nonumber & \leq 4 \ , { \mathbb{e } } [ \mathbbm{1 } _ { \ { u_{j } \in a_r^c \ } } |u_{j } |^2 ] \leq 4 \ , ( { \mathbb{e}}|u_j|^p ) ^{2/p } \ , { \mathbb{p}}(u_j \in a_r^c)^{1/q } , \quad p=2+\epsilon , q= 1 + 2\epsilon^{-1 } , \\ \label{eq : ujrbound } & \leq c_{s , n , c_0 } \ , r^{-2/q}.\end{aligned}\ ] ] then , using property [ item : one ] gives @xmath306 taking expectation with respect to remaining measures and using yields @xmath307 setting @xmath308 in entails @xmath309 since @xmath310 , we also have @xmath311 . thus , we can choose @xmath312 large enough such that @xmath313 for each @xmath296 . based on the preceding lemma , we prove that the chain converges to the solution of the chain in the setting of a finite number of restarts . [ thm : convfinite ] under the assumptions of lemma [ lemma : rapprox ] , for each @xmath56 , there exists @xmath314 and then @xmath315 such that @xmath316 for each @xmath296 . let @xmath317 be fixed . by the previous lemma , choose @xmath276 such that @xmath318 for @xmath296 . then , using calculations similar to those above , by [ item : one ] we get @xmath319 where @xmath320 and @xmath321 . then by [ item : two ] , we choose @xmath322 sufficiently large so that @xmath323 the last inequality can be rewritten as @xmath324 therefore , @xmath325 for each @xmath326 . consider the setting of the solution operator to the sde given in . as @xmath293 increases to @xmath327 , the random variable @xmath328 may converge in distribution to a limiting random variable @xmath329 , whose distribution is the invariant measure of the evolution equation . although there are many efficient ways to analyze and compute such invariant measures ( see for instance @xcite ) , we wish to show that our iterative algorithm also converges to that invariant measure as degrees of freedom tend to infinity . in other words , our pce - based method allows us to remain accurate for long - time evolutions something that can not be achieved without restarts . now we iterate each discrete chain in , and for an arbitrary @xmath330 . in order to ensure long - time convergence , we need a stricter condition than [ item : one ] and impose instead the following contraction condition 1 . @xmath273 , where @xmath331 , @xmath332 @xmath272 , and @xmath274 . [ item : onemod ] we first prove the following result about the existence and uniqueness of an invariant measure for the original chain . [ lemma : invariant ] under conditions [ item : zero ] and [ item : onemod ] , there exists a unique invariant measure @xmath333 of the chain with bounded @xmath280th moment . moreover , if @xmath310 , then @xmath334 is bounded uniformly in @xmath83 . for @xmath335 ( and using that @xmath336 ) , we compute @xmath337 where @xmath338 is chosen so that @xmath339 and @xmath340 . thus , there exists a lyapunov function @xmath341 with a constant @xmath342 . this in turn implies the existence of an invariant measure @xmath333 with finite @xmath280th moment for the process ; see ( * ? ? ? * corollary 4.23 ) . uniqueness also follows from assumption [ item : onemod ] ; see ( * ? ? ? * theorem 4.25 ) . let @xmath343 be a random variable with law @xmath333 and independent of @xmath46 . consider the chain @xmath344 started from @xmath343 . note @xmath345 for all @xmath330 . then , from the bound @xmath346 we conclude that @xmath347 . the following theorem establishes the exponential convergence of the pc chain to the chain as time @xmath83 increases . assume [ item : zero ] , [ item : onemod ] and [ item : two ] hold , and @xmath310 . then , for each @xmath56 , there exists @xmath276 independent of @xmath83 , and then @xmath348 , such that @xmath316 for all @xmath349 . let @xmath317 be given . by lemma [ lemma : invariant ] , @xmath334 is bounded uniformly in @xmath83 . the inequality holds with a constant independent of @xmath83 . then , choosing @xmath350 so that @xmath351 in implies that there exists a constant @xmath352 independent of @xmath83 such that @xmath353 . as in the proof of theorem [ thm : convfinite ] , we choose @xmath350 so that @xmath354 . then , with @xmath348 large enough , is replaced by @xmath355 therefore , @xmath356 this concludes our proof of convergence of the dgpc method to the invariant measure @xmath333 . let us conclude this section with the following remark . it would be desirable to obtain that the chain ( corresponding to @xmath357 , or equivalently no support truncation ) @xmath358 remains determinate . we were not able to do so and instead based our theoretical results on the assumption that the true distributions of interest were well approximated by compactly supported distributions . in practice , the range of @xmath159 has to remain relatively limited for at least two reasons . first , large @xmath159 rapidly involves very large computational costs ; and second , the determination of measures from moments becomes exponentially ill - posed as the degree of polynomials @xmath25 increases . for these reasons , the support truncation has been neglected in the following numerical section since , heuristically , for large @xmath359 and limited @xmath25 , the computation of ( low order ) moments of distributions with rapidly decaying density at infinity is hardly affected by such a support truncation .

their main challenge is that the computational cost becomes prohibitive when the dimension of the parameters modeling the stochasticity is even moderately large . we present a restart procedure that precisely allows pce to depend only on that information . we present theoretical results of convergence for our dynamical generalized polynomial chaos ( dgpc ) method .

polynomial chaos expansions ( pce ) allow us to propagate uncertainties in the coefficients of differential equations to the statistics of their solutions . their main advantage is that they replace stochastic equations by systems of deterministic equations . their main challenge is that the computational cost becomes prohibitive when the dimension of the parameters modeling the stochasticity is even moderately large . we propose a generalization of the pce framework that allows us to keep this dimension as small as possible in favorable situations . for instance , in the setting of stochastic differential equations ( sdes ) with markov random forcing , we expect the future evolution to depend on the present solution and the future stochastic variables . we present a restart procedure that precisely allows pce to depend only on that information . the computational difficulty then becomes the construction of orthogonal polynomials for dynamically evolving measures . we present theoretical results of convergence for our dynamical generalized polynomial chaos ( dgpc ) method . numerical simulations for linear and nonlinear sdes show that it adequately captures the long - time behavior of their solutions as well as their invariant measures when the latter exist . keywords : polynomial chaos , stochastic differential equations , uncertainty quantification , markov processes

1505.00047

c

polynomial chaos expansions ( pce ) applied to the simulation of evolution equations suffer from two drawbacks : in the presence of stochastic forcing , the dimension of randomness is too large , and the long - time solution may not be sparsely represented in a fixed pce basis @xcite . in the setting of markovian random forcing , we have proposed a restart method that addresses the two aforementioned drawbacks . such restarts of the pce , which we called here dynamical generalized polynomial chaos ( dgpc ) , allow us to both to keep the number of random variables small and to obtain a solution that remains reasonably sparse in the evolving basis of orthogonal polynomials , as was done earlier in @xcite . following @xcite , we applied dgpc to the long - time numerical simulation of various stochastic differential equations ( sdes ) . to demonstrate the ability of the algorithm to reach long - time solutions , we computed invariant measures for sdes that admit one and found a very good agreement between dgpc and other standard methods such as monte carlo - based simulations . we also presented a simple theoretical justification for such a convergence . the main computational difficulty in dgpc , as in most gpc - based methods , is the estimation of the orthogonal polynomials of evolving arbitrary measures . our method , as in @xcite , is based on estimating moments of the multivariate distribution of interest and estimating orthogonal polynomials by a gram schmidt procedure . the bottleneck in such estimations is the cost of computing moments . this is similar to the cost of solving differential equations by pce methods , where moment estimations are also the most costly step . however , in dgpc , such estimations can not be performed offline . from a theoretical point of view , we need to ensure that the evolving measures remain _ determinate _ so that their orthogonal polynomials span square integrable functionals . since distributions with compact support are determinate , our theoretical results have been applied in the setting where sde solutions are well approximated by compactly supported distributions . in this connection , we note that the calculation of orthogonal polynomials from knowledge of moments is an ill - posed problem , which is a serious impediment to gpc methods in general . extension of the method to larger systems than those considered here , including to systems obtained by solving stochastic partial differential equations ( spdes ) is challenging . the method works reasonably well for low dimensional sdes and it needs modifications for larger systems . however , the general idea remains applicable . to keep the number of uncertain variables in check , the method has to be coupled with a compression technique such as karhunen loeve projection at each restart time . moreover , sparse truncation techniques further help to reduce costs . the analysis of extensions of dgpc to larger systems , faster methods to compute orthogonal polynomials , and adaptive restart procedures is the subject of ongoing research .

polynomial chaos expansions ( pce ) allow us to propagate uncertainties in the coefficients of differential equations to the statistics of their solutions . we propose a generalization of the pce framework that allows us to keep this dimension as small as possible in favorable situations . for instance , in the setting of stochastic differential equations ( sdes ) with markov random forcing , we expect the future evolution to depend on the present solution and the future stochastic variables . the computational difficulty then becomes the construction of orthogonal polynomials for dynamically evolving measures . numerical simulations for linear and nonlinear sdes show that it adequately captures the long - time behavior of their solutions as well as their invariant measures when the latter exist . keywords : polynomial chaos , stochastic differential equations , uncertainty quantification , markov processes

polynomial chaos expansions ( pce ) allow us to propagate uncertainties in the coefficients of differential equations to the statistics of their solutions . their main advantage is that they replace stochastic equations by systems of deterministic equations . their main challenge is that the computational cost becomes prohibitive when the dimension of the parameters modeling the stochasticity is even moderately large . we propose a generalization of the pce framework that allows us to keep this dimension as small as possible in favorable situations . for instance , in the setting of stochastic differential equations ( sdes ) with markov random forcing , we expect the future evolution to depend on the present solution and the future stochastic variables . we present a restart procedure that precisely allows pce to depend only on that information . the computational difficulty then becomes the construction of orthogonal polynomials for dynamically evolving measures . we present theoretical results of convergence for our dynamical generalized polynomial chaos ( dgpc ) method . numerical simulations for linear and nonlinear sdes show that it adequately captures the long - time behavior of their solutions as well as their invariant measures when the latter exist . keywords : polynomial chaos , stochastic differential equations , uncertainty quantification , markov processes

0910.2612

c

for a large class of supersymmetric extensions of the standard model gluinos will be sufficiently stable to form color - neutral non - relativistic boundstates . the nnlo potential between color - octet constituents forming color singlets has been derived . just as in leading and next - to - leading order the coulombic part of the nnlo potential is related to the quarkonium potential through the substitution @xmath169 in the overall factor . the transformation of the short distance terms involving an additional factor @xmath170 is only slightly more complicated and discussed in section [ sec : properties ] . based on this potential , predictions for the binding energy and the wave function at the origin of the lowest - lying boundstates and the first radial excitation have been derived . for gluino masses between @xmath152 and @xmath171 the @xmath91 binding energy ( in the pole mass scheme ) varies between @xmath172 and @xmath173 , the @xmath174 energy difference between @xmath175 and @xmath176 . for the case of gluinos being lighter than squarks the gluino decay rate is significantly smaller than the level spacing and the boundstates are thus well separated from the continuum . also the wave function at the origin , which enters bound state production and decay rates , has been evaluated in nnlo . for @xmath121 the decay rate of gluinonia into two gluon jets is evaluated in nlo and shown to dominate all other modes . for top squarks with masses comparable to @xmath33 the decay mode into @xmath177 may reach several percent , the decay through virtual quark loops into two photons is too small to be detectable . we evaluate the boundstate production cross section again in nlo . large positive corrections of around @xmath178 arise from the nnlo terms in the wave function at the origin and around @xmath179 from the nlo terms in the cross section . the production rate depends strongly on the gluino mass , with up to @xmath142 events for gluino masses close to the lower limit around @xmath144 ( for an integrated luminosity of @xmath140 ) dropping quickly to @xmath141 events only for @xmath180 . using standard assumptions for the dijet - mass resolution and assuming that gluon and quark jets can be separated , a signal - to - background ratio between @xmath181 and @xmath164 percent might be conceivable . for favorable gluino- and squark - mass combinations gluinonia might thus be detectable . * acknowledgements * m.s . thanks alexander penin for discussions about @xmath182 . this work was supported by the bmbf through grant no . 05h09vke . _ note added _ : after completion of this work we received a preprint by k. hagiwara and h. yokoya where the production of unstable gluinos in the threshold region is considered @xcite .

predictions for energy levels , production and decay rate of gluinonia , non - relativistic boundstates of gluinos , are presented . the potential between color - octet constituents is derived in next - to - next - to leading order and one - loop qcd corrections are derived for the production cross section and the decay rate into gluon jets . in addition we evaluate the decay rate into top quarks and into two photons . the signal - to - background ratio is estimated for the dominant decay mode and found to be around . for relatively light gluinos the bound states thus might be detectable .

predictions for energy levels , production and decay rate of gluinonia , non - relativistic boundstates of gluinos , are presented . the potential between color - octet constituents is derived in next - to - next - to leading order and one - loop qcd corrections are derived for the production cross section and the decay rate into gluon jets . in addition we evaluate the decay rate into top quarks and into two photons . the signal - to - background ratio is estimated for the dominant decay mode and found to be around . for relatively light gluinos the bound states thus might be detectable . pacs numbers : 12.38.bx 12.60.jv 14.80.ly

1509.01921

i

multicomponent fractional quantum hall ( fqh ) systems feature rich physics of strongly interacting electrons . in contrast to strong electronic correlations in various ( material dependent ) crystal settings , in fqh systems the kinetic energy is effectively removed from the problem due to the landau quantization in strong magnetic fields . this makes the multicomponent fqh systems a simpler ( better controlled ) setting that is amenable to the investigations of fundamental physics of strongly correlated electrons . one of the most prominent examples of multicomponent fqh systems is the quantum hall bilayer ( qhb ) at total filling factor one . the filling factor @xmath1 is defined as a product of electron density @xmath2 and the magnetic area @xmath3 , where @xmath4 is the emergent length scale in the magnetic field ( magnetic length " ) @xcite . alternatively , the filling factor represents the ratio of the number of electrons @xmath5 to the number of magnetic flux quanta @xmath6 through the system . in the case of the @xmath7 qhb studied here , there are two quantum wells ( layers " ) that we label by @xmath8 , and @xmath9 . note that throughout this work we assume the ordinary electron spin to be fully polarized by the external magnetic field . because of the two - fold layer degree of freedom , in general we need to distinguish the interaction between electrons in the same layer ( intra - layer " ) versus electrons in opposite layers ( inter - layer " ) . the difference between these two interaction strengths can be tuned experimentally via the parameter called the bilayer distance @xmath10 , which makes the physics in this system much richer compared to an ordinary system of electrons with spin . qhb system has indeed attracted much attention since the time of the theoretical prediction @xcite and experimental detection @xcite of the ordered state of excitons ( bosons ) that forms when @xmath11 , despite the fact that the fundamental charge carriers in the system are fermions . perhaps equally remarkable and interesting is the physics at larger ( intermediate ) distances , typically @xmath12 . there , an intermediate phase of paired composite fermions , introduced in ref . [ see eqs . ( [ fpairing1 ] ) and ( [ sdispairing ] ) below ] , is believed to exist . on the other hand , an exact diagonalization study of the system in the torus geometry @xcite also pointed out to the existence of pseudospin state(s ) with spiraling order in the intermediate region . here we reconcile these two descriptions of the intermediate region by considering the underlying bosonic representation of the composite fermion paired state in the long distance limit , and by performing extensive exact diagonalization studies on the torus . this paper is organized as follows . in section [ sec : review ] we start with an overview of the qhb physics from the point of view of model wave functions . we first discuss the limits of small and large bilayer distances which are well - understood and correspond to the exciton superfluid and the product of two decoupled ( composite ) fermi liquid states , respectively . we also review the theoretical evidence for a paired phase in the regime of intermediate bilayer distances , and briefly motivate and summarize our contributions . in section [ sec : picture ] we provide a simple physical picture for the formation of the putative intermediate state . section [ sec : numerics ] contains results of exact diagonalization studies of the qhb on the torus . in section [ sec : boson ] we discuss the torus results from the composite boson viewpoint , and give arguments for meron deconfinement in the intermediate region . we conclude in section [ sec : conc ] with a short summary and discussion . appendices contain further resullts on the intermediate phase .

an exact diagonalization study in the torus geometry pointed out the existence of state(s ) with pseudospin spiraling order . here we reconcile these two descriptions of the intermediate state by considering the underlying bosonic representation of the composite fermion paired state in the long distance limit , and by performing extensive exact diagonalizations on the torus . the fractionally charged vortex excitations of the xy magnetically ordered state constitute some of the topological sectors .

quantum hall bilayer phase diagram with respect to interlayer distance bears a remarkable similarity with phase diagrams of strongly correlated systems as a function of doping , with magnetic ordering on the one end and fermi - liquid - like behavior on the other . moreover , it has been suggested that a bcs correlated state of composite fermions with-wave pairing may exist in the intermediate region . in the same region , an exact diagonalization study in the torus geometry pointed out the existence of state(s ) with pseudospin spiraling order . here we reconcile these two descriptions of the intermediate state by considering the underlying bosonic representation of the composite fermion paired state in the long distance limit , and by performing extensive exact diagonalizations on the torus . we argue that the spiraling states belong to the manifold of degenerate ground state(s ) , and are a consequence of bose condensation of the quasiparticles ( with critical algebraic correlations ) at non - zero momenta in the two pseudospin states . the spiraling states , generated in this way as spin - textures , can be identified with meron - antimeron constructions . thus , merons the fractionally charged vortex excitations of the xy magnetically ordered state constitute some of the topological sectors . it follows that merons are deconfined in the intermediate state , and allow for a smooth transition between the magnetically ordered and fermi - liquid - like phases , in which they are bound in pairs .

1103.3323

i

quantum phases of matter with exotic types of order have continued to emerge over the past decades . examples include fractional quantum hall states@xcite , 1d haldane phase@xcite , chiral spin liquids,@xcite @xmath0 spin liquids,@xcite non - abelian fractional quantum hall states,@xcite quantum orders characterized by projective symmetry group ( psg),@xcite topological insulators@xcite , etc . why are there different orders ? what is a general framework to understand all these seemingly very different phases ? how to classify all possible phases and identify new ones ? much effort has been devoted to these questions , yet the picture is not complete . first , we want to emphasize that quantum phase is a property of a class of hamiltonians , not of a single hamiltonian . we call such a class of hamiltonian an h - class . for an h - class , of a certain dimension and with possible symmetry constraints , we ask whether the hamiltonians in it are separated into different groups by phase transition and hence form different phases . two hamiltonians in an h - class are in the same / different phase if they can / cannot be connected _ within the h - class _ without going through phase transition . we see that without identifying the class of hamiltonians under consideration , it is not meaningful to ask which phase a hamiltonian belongs to . two hamiltonians can belong to the same / different phases if we embed them in different h - classes . for an h - class with certain symmetry constraints , one mechanism leading to distinct phases is symmetry breaking.@xcite starting from hamiltonians with the same symmetry , the ground states of them can have different symmetries , hence resulting in different phases . this symmetry breaking mechanism for phases and phase transitions is well understood.@xcite however , it has been realized that systems can be in different phases even without breaking any symmetry . such phases are often said to be ` topological ' or ` exotic ' . however , the term ` topological ' in literature actually refers to two different types of quantum order . the first type has ` intrinsic ' topological order . this type of order is defined for the class of systems without any symmetry constraint , which corresponds to the original definition of ` topological order'.@xcite that is , it refers to quantum phases in an h - class which includes all local hamiltonians ( of a certain dimension ) . if we believed that landau symmetry breaking theory describes all possibles phases , this whole h - class would belong to the same phase as there is no symmetry to break . however , in two and three dimensions , there are actually distinct phases even in the h - class that has no symmetries . these phases have universal properties stable against any small local perturbation to the hamiltonian . to change these universal properties , the system has to go through a phase transition . phases in this class include quantum hall systems@xcite , chiral spin liquids,@xcite @xmath0 spin liquids,@xcite quantum double model@xcite and string - net model@xcite . ground states of these systems have ` long - range entanglement ' as discussed in . the ` topological ' quantum order of the second type is ` symmetry protected ' . the class of systems under consideration have certain symmetry and the ground states have only short - range entanglement@xcite , like in the symmetry breaking case . however unlike in symmetry breaking phases , the ground states have the same symmetry as the hamiltonian and , even so , the ground states can be in different phases . this quantum order is protected by symmetry ; as according to the discussion in , if the symmetry constraint on the class of systems is removed , all short - range entangled states belong to the same phase . only when symmetry is enforced , can short range entangled states with the same symmetry belong to different phases . examples of this type include the haldane phase@xcite and topological insulators.@xcite phases in these two classes share some similarities . for example , they both are beyond landau symmetry breaking theory . also quantum hall systems and topological insulators both have stable gapless edge states.@xcite however , the latter requires symmetry protection while the former does not . despite the similarities , these two classes of topological phases are fundamentally different , as we can see from quantities that are sensitive to long - rang entanglement . for example , ` intrinsic ' topological order has a robust ground state degeneracy that depends on the topology of the space.@xcite the ground states with ` intrinsic ' topological order also have non - zero topological entanglement entropy@xcite while ground states in ` symmetry protected ' topological phases are short range entangled and therefore have zeros topological entanglement entropy . also , the low energy excitations in ` symmetry protected ' topological phases do not have non - trivial anyon statistics , unlike in ` intrinsic ' topological phases . @xcite in the following discussion , we will use ` topological phase ' to refer only to the first type of phases ( ` intrinsic ' topological phases ) . for the second type , we will call them ` symmetry protected topological ' ( spt ) phases , as in . similar to the quantum orders characterized by psg,@xcite different spt phases are also characterized by the projective representations of the symmetry group of the hamiltonian.@xcite the psg and projective representations of a symmetry group can be viewed as a ` fractionalization ' of the symmetry . thus , we may say that different spt phases are caused by ` symmetry fractionalization ' . long - rang entanglement , symmetry fractionalization , and symmetry breaking represent three different mechanisms to separate phases and can be combined to generate a very rich quantum phase diagram . [ fig : phase ] shows a phase diagram with possible phases generated by these three mechanisms . in order to identify the kind of quantum order in a system , we first need to know whether topological orders exist . that is , whether the ground state has long range entanglement . next , we need to identify the symmetry of the system ( of the hamiltonian and allowed perturbation ) . then we can find out whether all or part of the symmetry is spontaneously broken in the ground state . if only part is broken , what is the spt order due to the fractionalization of the unbroken symmetry . combining these data together gives a general description of a quantum phase . most of the phases studied before involve only one of the three mechanisms . examples where two of them coexist can be found in which combine long range entanglement ( the intrinsic topological order ) and symmetry breaking and in which combine long range entanglement and symmetry fractionalization . in fact , the psg provides a quite comprehensive framework for symmetry fractionalization in topologically ordered states.@xcite based on this general understanding of quantum phases , we address the following question in this paper : what quantum phases exist in one - dimensional gapped spin systems . the systems we consider can have any finite strength finite range interactions among the spins . in , we gave a partial answer to this question . we first showed that one - dimensional gapped spin systems do not have non - trivial topological order . so to understand possible 1d gapped phases , we just need to understand symmetry fractionalization and symmetry breaking in short - range entangled states . in other words , quantum phases are only different because of symmetry breaking and symmetry fractionalization . in , we then considered symmetry fractionalization , and gave a classification of possible spt phases with time reversal , parity and on - site unitary symmetry respectively . in this paper , we complete the classification by considering spt phases of combined time reversal , parity and/or on - site unitary symmetry and finally incorporate the possibility of symmetry breaking . we find that in 1d gapped spin system with on - site symmetry of group @xmath1 , the quantum phase are basically labeled by : + ( 1 ) the unbroken symmetry subgroup @xmath2 + ( 2 ) projective representation of the unbroken part of on - site unitary and anti - unitary symmetry respectively + ( 3 ) ` projective ' commutation relation between representations of unbroken symmetries . + here the projective representation and the ` projective ' commutation relation represent the symmetry fractionalization . parity is not an on - site symmetry and its spt phases are not characterized by projective representations . the classification involving parity does not fall into the general framework above , but proceeds in a very similar way , as we will show in section [ cs ] . actually , ( 2 ) and ( 3 ) combined gives the projective representation of @xmath2 and if parity is present , it should be treated as an anti - unitary @xmath0 element . our result is consistent with those obtained by schuch et al@xcite . our discussion is based on the matrix product state representation@xcite of gapped 1d ground states . the matrix product formalism allows us to directly deal with interacting systems and its entangled ground state . in particular , spt order of a system can be identified directly from the way matrices in the representation transform under symmetries.@xcite moreover , symmetry breaking in entangled systems can be represented in a nice way using matrix product states.@xcite the traditional understanding of symmetry breaking in quantum systems actually comes from intuition about classical systems . for example , in the ferromagnetic phase of classical ising model , the spins have to choose between two possible states : either all pointing up or all pointing down . both states break the spin flip symmetry of the system . however , for quantum system , the definition of symmetry breaking becomes a little tricky . in ferromagnetic phase of quantum ising model , the ground space is two fold degenerate with basis states @xmath3 , @xmath4 . each basis state breaks the spin flip symmetry of the system . however , quantum systems can exist in any superposition of the basis states and in fact the superposition @xmath5 is symmetric under spin flip . what is meant when a _ quantum _ system is said to be in a symmetry breaking phase ? can we understand symmetry breaking in a quantum system without relying on its classical picture ? in matrix product representation , the symmetry breaking pattern can be seen directly from the matrices representing the ground state . if we choose the symmetric ground state in the ground space and write it in matrix product form , the matrices can be reduced to a block diagonal ` canonical form ' . the ` canonical form ' contains more than one block if the system is in a symmetry breaking phase . if symmetry is not broken , it contains only one block.@xcite hence , the ` canonical form ' of the matrices give a nice illustration of the symmetry breaking pattern of the system . this relation will be discussed in more detail in section [ sb ] . our classification is focused on 1d interacting spin systems , however it also applies to 1d interacting fermion systems as they are related by jordan - wigner transformation . as an application of our classification result , we study quantum phases(especially spt phases ) in gapped 1d fermion systems . our result is consistent with previous studies.@xcite the paper is organized as follows : in section [ spt ] , we review the previous classification results of spt phases with time reversal , parity and on - site unitary symmetry respectively . we also introduce notations for matrix product representation ; in section [ cs ] , we present classification result of spt phases with combined time reversal , parity and/or on - site unitary symmetry ; in section [ sb ] , we incorporate the possibility of symmetry breaking ; in section [ fermion ] , we apply classification results of spins to the study of phases in 1d fermion systems ; and finally we conclude in section [ conclude ] .

quantum phases with different orders exist with or without breaking the symmetry of the system . recently , a classification of gapped quantum phases which do not break time reversal , parity or on - site unitary symmetry has been given for 1d spin systems in [ x. chen , z .- c . gu , and x .- wen , phys . rev . b * 83 * , 035107 ( 2011 ) ; arxiv:1008.3745 ] . it was found that , such symmetry protected topological ( spt ) phases are labeled by the projective representations of the symmetry group which can be viewed as a symmetry fractionalization . in this paper , we extend the classification of 1d gapped phases by considering spt phases with combined time reversal , parity , and/or on - site unitary symmetries and also the possibility of symmetry breaking . we clarify how symmetry fractionalizes with combined symmetries and also how symmetry fractionalization coexists with symmetry breaking . in this way we find that in general , symmetry fractionalization , symmetry breaking and long range entanglement(present in 2 or higher dimensions ) represent three main mechanisms to generate a very rich set of gapped quantum phases . as an application of our classification

quantum phases with different orders exist with or without breaking the symmetry of the system . recently , a classification of gapped quantum phases which do not break time reversal , parity or on - site unitary symmetry has been given for 1d spin systems in [ x. chen , z .- c . gu , and x .- g . wen , phys . rev . b * 83 * , 035107 ( 2011 ) ; arxiv:1008.3745 ] . it was found that , such symmetry protected topological ( spt ) phases are labeled by the projective representations of the symmetry group which can be viewed as a symmetry fractionalization . in this paper , we extend the classification of 1d gapped phases by considering spt phases with combined time reversal , parity , and/or on - site unitary symmetries and also the possibility of symmetry breaking . we clarify how symmetry fractionalizes with combined symmetries and also how symmetry fractionalization coexists with symmetry breaking . in this way , we obtain a complete classification of gapped quantum phases in 1d spin systems . we find that in general , symmetry fractionalization , symmetry breaking and long range entanglement(present in 2 or higher dimensions ) represent three main mechanisms to generate a very rich set of gapped quantum phases . as an application of our classification , we study the possible spt phases in 1d fermionic systems , which can be mapped to spin systems by jordan - wigner transformation .

cond-mat0702495

c

the bipolaronic extension @xcite of the bcs theory towards the strong interaction between electrons and ion vibrations proved that the cooper pairing in momentum space @xcite and the ogg - schafroth real - space pairing @xcite are two extreme limits of the same problem . for a very strong electron - phonon coupling , polarons become self - trapped on a single lattice site and bipolarons are on - site singlets . in the holstein model of the zero - range epi their mass appears only in the second order of polaron hopping @xcite , so that on - site bipolarons are very heavy . this estimate led some authors to the conclusion that the formation of itinerant small polarons and bipolarons in real materials is unlikely @xcite , and high - temperature bipolaronic superconductivity is impossible @xcite . in fact , we have demonstrated here that small but light bipolarons could exist for realistic values of the finite - range epi with high - frequency optical phonons in staggered ladder systems . small light bipolarons are an essential precursor to high - temperature superconductivity , since the bose - einstein condensate has transition temperature that is inversely proportional to mass , and wavefunctions may not overlap . such bipolarons are easily formed on lattices with triangular plaquettes in the presence of extremely large on - site coulomb repulsion , and persist to large epi . this conclusion is backed up by analytics in the anti - adiabatic approximation in the presence of large intersite coulomb attraction . another important conclusion is that the triplet - singlet exchange energy is of the first order in the hopping integral , and triplet bipolarons are heavier than singlets in certain lattice structures at variance with simple intuitive expectations . we summarise the types of lattices where light `` crab '' bipolarons may be formed in figure [ fig : pictorial ] , contrasting with the traditional holstein bipolarons ( bottom ) and describe the required effective interaction in figure [ fig : effinteraction ] demonstrating the underlying physics of such bipolarons . our ctqmc simulations lead us to believe that the following recipe is worth investigating to look for very high - temperature superconductivity : ( a ) the parent compound should be an ionic insulator with light ions to form high - frequency optical phonons , ( b ) the structure should be quasi two - dimensional to ensure poor screening of high - frequency c - axis polarized phonons , ( c ) a triangular lattice is required in combination with strong , on - site coulomb repulsion to form the small superlight crab bipolaron ( d ) moderate carrier densities are required to keep the system of small bipolarons close to the dilute regime . many of these conditions are already met in the cuprates .

recent angle - resolved photoemission spectroscopy ( arpes ) has identified that a finite - range frhlich electron - phonon interaction ( epi ) with c - axis polarized optical phonons is important in cuprate superconductors , in agreement with an earlier proposal by alexandrov and kornilovitch . the estimated unscreened epi is so strong that it could easily transform doped holes into mobile lattice bipolarons in narrow - band mott insulators such as cuprates . applying a continuous - time quantum monte - carlo algorithm ( ctqmc ) we compute the total energy , effective mass , pair radius , number of phonons and isotope exponent of lattice bipolarons in the region of parameters where any approximation might fail taking into account the coulomb repulsion and the finite - range epi . such light small bipolarons are a necessary precursor to high - temperature bose - einstein condensation in solids . another surprising result is that the triplet - singlet exchange energy is of the first order in the hopping integral and triplet bipolarons are heavier than singlets in certain lattice structures at variance with intuitive expectations . finally , we identify a range of lattices where superlight small bipolarons may be formed , and give estimates for their masses in the anti - adiabatic approximation .

recent angle - resolved photoemission spectroscopy ( arpes ) has identified that a finite - range frhlich electron - phonon interaction ( epi ) with c - axis polarized optical phonons is important in cuprate superconductors , in agreement with an earlier proposal by alexandrov and kornilovitch . the estimated unscreened epi is so strong that it could easily transform doped holes into mobile lattice bipolarons in narrow - band mott insulators such as cuprates . applying a continuous - time quantum monte - carlo algorithm ( ctqmc ) we compute the total energy , effective mass , pair radius , number of phonons and isotope exponent of lattice bipolarons in the region of parameters where any approximation might fail taking into account the coulomb repulsion and the finite - range epi . the effects of modifying the interaction range and different lattice geometries are discussed with regards to analytical strong - coupling / non - adiabatic results . we demonstrate that bipolarons can be simultaneously small and light , provided suitable conditions on the electron - phonon and electron - electron interaction are satisfied . such light small bipolarons are a necessary precursor to high - temperature bose - einstein condensation in solids . the light bipolaron mass is shown to be universal in systems made of triangular plaquettes , due to a novel crab - like motion . another surprising result is that the triplet - singlet exchange energy is of the first order in the hopping integral and triplet bipolarons are heavier than singlets in certain lattice structures at variance with intuitive expectations . finally , we identify a range of lattices where superlight small bipolarons may be formed , and give estimates for their masses in the anti - adiabatic approximation .

0909.2644

i

motivated by the recent discovery of massive planets on wide orbits , we explore the requirements for making gas giants at large separations from their host star via gravitational instability , hereafter , gi . in particular , we consider the formation mechanism for the system hr 8799 which contains three @xmath0@xmath1 objects orbiting at distances between @xmath0 30 and 70 au @xcite . the standard core accretion model for planet formation , already strained in the outer solar system , has difficulty explaining the presence of these objects . while gi is an unlikely formation mechanism for close in planets @xcite , for more widely separated planets , or sub - stellar companions , the viability gi - driven fragmentation deserves further investigation . in the inner regions of a protoplanetary disk , gas can not cool quickly enough to allow a gravitationally unstable disk to fragment into planets @xcite . for this reason , core accretion in which solid planetesimals collide and grow into a massive core which then accretes a gaseous envelope has emerged as the preferred mechanism for forming planets at stellar separations @xmath2 au . planets at wider separations have only recently been discovered by direct imaging around the a - stars hr 8799 @xcite , fomalhaut @xcite , and possibly beta pic . searches at large radii surrounding solar - type stars have yet to turn up similar companions @xcite . standard core accretion models can not form these planets , though further investigation is warranted . in favor of this possibility , all three systems show some evidence of processes related to core accretion : all have infrared excess due to massive debris disks at large radii . this is at least partially a selection effect as these systems were targeted due to the disks presence . nevertheless , these debris disks are composed of reprocessed grains from collisions of planetesimals . the disks long lifetimes prohibit a primordial origin for small grains they are removed quickly by radiation pressure and poynting - roberston drag @xcite and so must be regenerated from collisions between larger bodies that formed through the coagulation of solids at early times . therefore , planetesimal formation , a necessary ingredient in core accretion models , has taken place ( e.g. , * ? ? ? * ; * ? ? ? in addition , other a - stars host planets at 12 au @xcite which , although they have a distinct semi - major axis distribution from planets orbiting g and m stars , are likely formed by core accretion . if future surveys demonstrate that this distribution extends smoothly to wide separation planets , then simplicity would argue against a distinct formation mechanism for the wide giants . yet the standard core accretion model faces a serious problem at large distances . the observed lifetimes of gas disks are short , at most a few myr @xcite . in contrast , typical core accretion times increase with radius and exceed 10 myr beyond 20 au . whether or not this theoretical difficulty can be overcome will require careful modeling of the interactions between planetesimals and the young gas disk . could wide orbit planets have formed at smaller radii , and migrated outwards ? @xcite , have investigated the possibility of forming the hr 8799 system via scattering in the absence of dynamically important gas , but find that putting three massive planets into such closely spaced yet wide orbits is unlikely . @xcite have suggested that under favorable circumstances outward migration in resonances might be feasible . alternatively , the core of a giant planet could be scattered outward by a planet , or migrate outward before accreting its gas envelope , either by interactions with the gas disk ( type iii migration ; e.g. @xcite ) or with planetesimals embedded in the gas ( capobianco , duncan , & levison , in prep ) . neither mechanism has yet been shown to move a core to such large distances , though this possibility has not been ruled out . given these difficulties , it is natural to search for other formation mechanisms , and gi @xcite stands out as a promising alternative . if any planets form by gi , the recently discovered directly imaged planets are the most likely candidates @xcite . in this paper , we examine this possibility in more detail , considering the expected mass scale of fragments and the effect of global disk evolution on the formation process . the inferred masses for the hr 8799 planets are close to the deuterium burning limit of @xmath3 @xcite . for simplicity , we take this as a the dividing line between planets and brown dwarfs and we refer to the hr 8799 objects as planets throughout . however , there is no reason for a given formation mechanism to function only above or below this threshold , and in fact , we will argue that if the hr 8799 planets formed by gi , their histories are more akin to those of higher - mass brown dwarfs than to lower - mass planets . to constrain gi as a mechanism for wide giant planet formation , we set the stage by describing the hr 8799 system in [ sec - hr8799 ] . we review the standard requirements for fragmentation in [ idealgi ] and discuss the initial mass scale of fragments in [ sec - masses ] . in [ sec - giaccrete ] we show that under typical disk conditions , fragments will continue to accrete to higher masses . we then discuss important global constraints on planet formation provided by star formation models , disk evolution timescales , and migration mechanisms in [ sec - starform ] , and [ sec - migration ] . we compare predictions of our analysis with the known wide substellar companions and exoplanets in [ sec - observe ] , suggesting that future observations will provide a definitive answer to the formation mechanism for hr 8799 . in the appendices we re - examine the heating and cooling properties of disks that are passively and actively heated , with special attention to the implications for irradiated disks , which become increasingly relevant for more massive stars .

recent direct imaging discoveries suggest a new class of massive , distant planets around a stars . these widely separated giants have been interpreted as signs of planet formation driven by gravitational instability , but the viability of this mechanism is not clear cut . in this paper , we first discuss the local requirements for fragmentation and the initial fragment mass scales . under more typical conditions , disk - born objects will likely grow well above the deuterium burning planetary mass limit . their absence would suggest planet formation by a different mechanism such as core accretion , which is consistent with the debris disks detected in these systems .

recent direct imaging discoveries suggest a new class of massive , distant planets around a stars . these widely separated giants have been interpreted as signs of planet formation driven by gravitational instability , but the viability of this mechanism is not clear cut . in this paper , we first discuss the local requirements for fragmentation and the initial fragment mass scales . we then consider whether the fragment s subsequent growth can be terminated within the planetary mass regime . finally , we place disks in the larger context of star formation and disk evolution models . we find that in order for gravitational instability to produce planets , disks must be atypically cold in order to reduce the initial fragment mass . in addition , fragmentation must occur during a narrow window of disk evolution , after infall has mostly ceased , but while the disk is still sufficiently massive to undergo gravitational instability . under more typical conditions , disk - born objects will likely grow well above the deuterium burning planetary mass limit . we conclude that if planets are formed by gravitational instability , they must be the low mass tail of the distribution of disk - born companions . to validate this theory , on - going direct imaging surveys must find a greater abundance of brown dwarf and m - star companions to a - stars . their absence would suggest planet formation by a different mechanism such as core accretion , which is consistent with the debris disks detected in these systems .

1111.7219

i

the remarkable speed and multiplexing capability of optics makes it very attractive for information processing . these features have enabled the telecommunications revolution of the past decades . however , so far they have not been exploited insomuch as computation is concerned . the reason is that optical nonlinearities are very difficult to harness : it remains challenging to just demonstrate optical logic gates , let alone compete with digital electronics@xcite . this suggests that a much more flexible approach is called for , which would exploit as much as possible the strenghts of optics without trying to mimic digital electronics . reservoir computing @xcite , a recently introduced , bio - inspired approach to artificial intelligence , may provide such an opportunity . here we report the first experimental reservoir computer based on an opto - electronic architecture . as nonlinear element we exploit the sine nonlinearity of an integrated mach - zehnder intensity modulator ( a well known , off - the - shelf component in the telecommunications industry ) , and to store the internal states of the reservoir computer we use a fiber optics spool . we report results comparable to state of the art digital implementations for two tasks of practical importance : nonlinear channel equalization and speech recognition . reservoir computing , which is at the heart of the present work , is a highly successful method for processing time dependent information . it provides state of the art performance for tasks such as time series prediction @xcite ( and notably won a financial time series prediction competition@xcite ) , nonlinear channel equalization @xcite , or speech recognition @xcite . for some of these tasks reservoir computing is in fact the most powerful approach known at present . the central part of a reservoir computer is a nonlinear recurrent dynamical system that is driven by one or multiple input signals . the key insight behind reservoir computing is that the reservoir s response to the input signal , i.e. , the way the internal variables depend on present and past inputs , is a form of computation . experience shows that in many cases the computation carried out by reservoirs , even randomly chosen ones , can be extremely powerful . the reservoir should have a large number of internal ( state ) variables . the exact structure of the reservoir is not essential : for instance , in some works the reservoir closely mimics the interconnections and dynamics of biological neurons in a brain @xcite , but many other architectures are possible . to achieve useful computation on time dependent input signals , a good reservoir should be able to compute a large number of different functions of its inputs . that is , the reservoir should be sufficiently high - dimensional , and its responses should not only depend on present inputs but also on inputs up to some finite time in the past . to achieve this , the reservoir must operate at the threshold of dynamical instability which ensures that the system exhibits a `` fading memory '' , meaning that it will gradually forget previous inputs . reservoir computing is a versatile and flexible concept . this follows from two key points : 1 ) many of the details of the nonlinear reservoir itself are unimportant except for the dynamic regime ( threshold of instability ) which can be tuned by some global parameters ; and 2 ) the only part of the system that is trained is a linear output layer . because of this flexibility , reservoir computing is amenable to a large number of experimental implementations . thus proof of principle demonstrations have been realized in a bucket of water @xcite and using an analog vlsi chip @xcite , and arrays of semiconductor amplifiers have been considered in simulation @xcite . however , it is only very recently that an analog implementation with performance comparable to digital implementations has been reported : namely , the electronic implementation presented in @xcite . our experiment is based on a similar architecture as that of @xcite , namely a single non linear node and a delay line . the main differences are the type of non linearity and the desynchronisation of the input with respect to the period of the delay line . these differences highlight the flexibility of the concept . the performance of our experiment on two benchmark tasks , isolated digit recognition and non linear channel equalization , is comparable to state of the art digital implementations of reservoir computing . compared to@xcite , our experiment is almost 6 orders of magnitude faster , and a further 2 - 3 orders of magnitude speed increase should be possible with only small changes to the system . the flexibility of reservoir computing and its success on hard classification tasks makes it a promising route for realizing computation in physical systems other than digital electronics . in particular it may provide innovative solutions for ultra fast or ultra low power computation . in supplementary material 1 we describe reservoir computing in more detail and provide a road map for building high performance analog reservoir computers . in the discussion we discuss how to measure the speed of experimental reservoir computers . understanding how physical systems can process information is a major scientific challenge . the tremendous progress that has been accomplished during the past century has given rise to whole new fields of science , such as computer science and digital ( silicon ) electronics , or quantum information . information processing by classical analog systems is a case apart because remarkable examples are found in the biological sciences ( e.g. the brain ) , but our understanding is still very incomplete . thus , we understand at the basic level many aspects of how information is processed using biochemical reactions in cells , or how information is processed by neurons in a brain , to take two examples . but how these elementary processes are organized at a higher level , and what is the large scale architecture of these systems , still escapes us . understanding these issues would be of great conceptual interest . it could also have huge technological repercussions , as it could completely change the way we build information processing machines . that this tremendous scope for progress exists is illustrated by the approximately 6 orders of magnitude gap in energy consumption between a brain and a present day silicon computerwatt . for a presentation of the simulation , see @xcite ] . so far most work on information processing in analog systems has been based on imitating biological systems . this has given rise to the field of artificial neural networks . more abstract approaches have also been developed , such as hidden markov models , or support vector machines . reservoir computing @xcite provides an alternative line of attack . the interest of reservoir computing is that 1 ) it proposes architectures that are quite different from those studied up to now ; and 2 ) despite a relatively recent history -the first papers on the topic date from 2002- , its performances are comparable to , and sometimes even exceed , those of other approaches to machine learning . a deeper understanding of reservoir computing could provide new insights into how analog classical systems , and in particular biological systems , process information . additionally it could give rise to novel approaches for implementing computation and enable new applications . the aim of the present text is not to duplicate existing reviews of reservoir computing @xcite . rather , we wish to present the subject from the point of view of the physicist or engineer who wishes to build an experimental reservoir computer . we will argue that reservoir computing provides a quite detailed , but also quite flexible , road map towards building physical systems that process information with architectures very different from those used up to now .

reservoir computing is a recently introduced , highly efficient bio - inspired approach for processing time dependent data . we report an opto - electronic implementation of reservoir computing based on a recently proposed architecture consisting of a single non linear node and a delay line . we illustrate its performance on tasks of practical importance such as nonlinear channel equalization and speech recognition , and obtain results comparable to state of the art digital implementations .

reservoir computing is a recently introduced , highly efficient bio - inspired approach for processing time dependent data . the basic scheme of reservoir computing consists of a non linear recurrent dynamical system coupled to a single input layer and a single output layer . within these constraints many implementations are possible . here we report an opto - electronic implementation of reservoir computing based on a recently proposed architecture consisting of a single non linear node and a delay line . our implementation is sufficiently fast for real time information processing . we illustrate its performance on tasks of practical importance such as nonlinear channel equalization and speech recognition , and obtain results comparable to state of the art digital implementations .

1506.07040

i

evolution equations often contain some structural information reflecting inherent physical properties such as positivity of solutions , conservation laws , and entropy dissipation . numerical schemes should be designed in such a way that these structural features are preserved on the discrete level in order to obtain accurate and stable algorithms . in the last decades , concepts of structure - preserving schemes , geometric integration , and compatible discretization have been developed @xcite , but much less is known about the preservation of entropy dissipation and large - time asymptotics . entropy - stable schemes were derived by tadmor already in the 1980s @xcite in the context of conservation laws , thus without ( physical ) diffusion . later , entropy - dissipative schemes were developed for ( finite - volume ) discretizations of diffusion equations in @xcite . in @xcite , a finite - volume scheme which preserves the gradient - flow structure and hence the entropy structure is proposed . all these schemes are based on the implicit euler method and are of first order ( in time ) only . higher - order time schemes with entropy - dissipating properties are investigated in very few papers . a second - order predictor - corrector approximation was suggested in @xcite , while higher - order semi - implicit runge - kutta ( dirk ) methods , together with a spatial fourth - order central finite - difference discretization , were investigated in @xcite . in @xcite , multistep time approximations were employed , but they can be at most of second order and they dissipate only one entropy and not all functionals dissipated by the continuous equation . in this paper , we remove these restrictions by investigating time - discrete runge - kutta schemes of order @xmath0 for general diffusion equations . we stress the fact that we are interested in the analysis of entropy - dissipating schemes by `` translating '' properties for the continuous equation to the semi - discrete level , i.e. , we study the stability of the schemes . however , we will not investigate convergence , stiffness , or computational issues here ( see e.g. @xcite ) . more precisely , we consider time discretizations of the abstract cauchy problem @xmath1 = 0 , \quad t>0 , \quad u(0)=u^0,\ ] ] where @xmath2 is a ( differential ) operator defined on @xmath3 and @xmath4 is a banach space with dual @xmath5 . in this paper , we restrict ourselves to diffusion operators @xmath6 $ ] defined on some sobolev space with solutions @xmath7 , which may be vector - valued . a typical example is @xmath6={\operatorname{div}}(a(u)\na u)$ ] defined on @xmath8 with domain @xmath9 , where @xmath10 is a smooth function ( see section [ sec.de ] ) . equation often possesses a lyapunov functional @xmath11=\int_\omega h(u)dx$ ] ( here called _ entropy _ ) , where @xmath12 , such that @xmath13 = \int_\omega h'(u)\pa_t u dx = -\int_\omega h'(u)a[u ] dx \le 0,\ ] ] at least when the _ entropy production _ dx$ ] is nonnegative , here , @xmath15 is the derivative of @xmath16 and @xmath17 $ ] is interpreted as the inner product of @xmath18 and @xmath6 $ ] in @xmath19 . furthermore , if @xmath16 is convex , the convex sobolev inequality @xmath14dx \ge \kappa h[u]$ ] for some @xmath20 may hold @xcite , which implies that @xmath21 and hence exponential convergence of @xmath11 $ ] to zero with rate @xmath22 . the aim is to design a higher - order time - discrete scheme which preserves this entropy - dissipation property . to this end , we propose the following semi - discrete runge - kutta approximation of : given @xmath23 , define @xmath24 , \quad i=1,\ldots , s,\ ] ] where @xmath25 are the time steps , @xmath26 is the uniform time step size , @xmath27 approximates @xmath28 , and @xmath29 denotes the number of runge - kutta stages . since the cauchy problem is autonomous , the knots @xmath30 are not needed here . in concrete examples ( see below ) , @xmath27 are functions from @xmath31 to @xmath19 . if @xmath32 for @xmath33 , the runge - kutta scheme is explicit , otherwise it is implicit and a nonlinear system of size @xmath34 has to be solved to compute @xmath35 . we assume that scheme is solvable for @xmath36 . given @xmath12 , we wish to determine conditions under which the functional @xmath37 = \int_\omega h(u^k(x))dx\ ] ] is dissipated by the numerical scheme , @xmath38 + \tau\int_\omega a[u^k]h'(u^k)dx \le h[u^{k-1 } ] , \quad k\in{{\mathbb n}}.\ ] ] in many examples ( see below ) , @xmath39h'(u^k)dx\ge 0 $ ] and thus , the function @xmath40 $ ] is decreasing . such a property is the first step in proving the preservation of the large - time asymptotics of the numerical scheme ( see remark [ rem.exp ] ) . our main results , stated on an informal level , are as follows : a. we determine an abstract condition under which the discrete entropy - dissipation inequality holds for sufficiently small @xmath41 . this condition is made explicit for special choices of @xmath42 and @xmath16 , yielding entropy - dissipative implicit or explicit runge - kutta schemes of any order . b. numerical experiments for the porous - medium equation indicate that @xmath43 may be chosen independent of the time step @xmath44 , thus yielding discrete entropy dissipation for all discrete times . c. we show that for runge - kutta schemes of order @xmath45 , the abstract condition in ( i ) is exactly the criterion of liero and mielke @xcite to conclude geodesic 0-convexity of the entropy . in particular , it is related to the bakry - emery condition @xcite . let us describe the main results in more detail . we recall that the runge - kutta scheme is consistent if @xmath46 and @xmath47 . furthermore , if @xmath48 , it is at least of order two ( * ? ? ? we introduce the number @xmath49 which takes only three values : @xmath50 the _ first main result _ is an abstract entropy - dissipation property of scheme for entropies of type . [ thm.ed ] let @xmath51 , let @xmath2 be frchet differentiable with frchet derivative @xmath52:x\to x'$ ] at @xmath53 , and let @xmath54 be the runge - kutta solution to . suppose that @xmath55(a[u^k])+h''(u^k)(a[u^k])^2\big)dx > 0.\ ] ] then there exists @xmath41 such that for all @xmath56 , @xmath57 + \tau\int_\omega a[u^k]h'(u^k)dx \le h[u^{k-1}].\ ] ] we assume that the solutions to are sufficiently regular such that the integral can be defined . in the vector - valued case , @xmath58 is the hessian matrix and we interpret @xmath59)^2 $ ] as the product @xmath60^\top h''(u^k)a[u^k]$ ] . for runge - kutta schemes of order @xmath45 ( for which @xmath61 ) , the integral corresponds exactly to the second - order time derivative of @xmath62 $ ] for solutions @xmath63 to the _ continuous _ equation . observe that the entropy - dissipation estimate is only _ local _ , since the time step restriction depends on the time step @xmath44 . for implicit euler schemes ( and convex entropies @xmath16 ) , it is known that @xmath43 can be chosen independent of @xmath44 . for general runge - kutta methods , we can not prove rigorously that @xmath43 stays bounded from below as @xmath64 . however , our numerical experiments in section [ sec.num ] indicate that inequality holds for sufficiently small @xmath65 uniformly in @xmath44 . [ rem.exp ] if the convex sobolev inequality @xmath39h'(u^k)dx\ge \kappa h[u^k]$ ] holds for some constant @xmath20 and if there exists @xmath66 such that @xmath67 for all @xmath68 , we infer from that for @xmath69 , @xmath70 \le ( 1+\kappa\tau)^{-k}h[u^0 ] = \exp(-\eta\kappa t^k)h[u^0 ] , \quad\mbox{where } \eta = \frac{\log(1+\kappa\tau)}{\kappa\tau } < 1,\ ] ] which implies exponential decay of the discrete entropy with rate @xmath71 . this rate converges to the continuous rate @xmath22 as @xmath72 and therefore , it is asymptotically sharp . theorem [ thm.ed ] can be generalized to a larger class of entropies , namely to so - called _ first - order entropies _ @xmath73 = \int_\omega |\na f(u^k)|^2 dx,\ ] ] where , for simplicity , we consider only the scalar case with @xmath74 . an important example is the fisher information with @xmath75 . [ thm.ed2 ] let @xmath76 , let @xmath2 be frchet differentiable , and let @xmath54 be the runge - kutta solution to . assume that the boundary condition @xmath77 on @xmath78 is satisfied . furthermore , suppose that @xmath79|^2 - c_{\rm rk}\delta f(u^k)f'(u^k)da[u^k](a[u^k ] ) \\ & { } - \delta f(u^k)f''(u^k)(a[u^k])^2\big)dx > 0 . \end{aligned}\ ] ] then there exists @xmath41 such that for all @xmath56 , @xmath80 + \tau\int_\omega a[u^k]f'(u^k)dx \le f[u^{k-1}].\ ] ] the key idea of the proof of theorem [ thm.ed ] ( and similarly for theorem [ thm.ed2 ] ) is a concavity property of the difference of the entropies at two consecutive time steps with respect to the time step size @xmath81 . to explain this idea , let @xmath82 be fixed and introduce @xmath83 . clearly , @xmath84 . a formal taylor expansion of @xmath85-h[v(\tau)]$ ] yields @xmath70-h[u^{k-1 } ] = g(\tau ) = g(0 ) + \tau g'(0 ) + \frac{\tau^2}{2}g''(\xi^k),\ ] ] where @xmath86 . a computation , made explicit in section [ sec.meth ] , shows that @xmath87h'(u^k)dx$ ] and @xmath88 . now , if @xmath89 , there exists @xmath41 such that @xmath90 for @xmath91 $ ] and in particular @xmath92 . consequently , @xmath93 , which equals . the definition of @xmath94 assumes implicitly that is _ backward _ solvable . we prove in proposition [ prop.back ] below that this property holds if the operator @xmath42 is a smooth self - mapping on @xmath4 . [ rem.tauk]since @xmath54 is expected to converge to the stationary solution , @xmath95 . thus , in principle , for larger values of @xmath44 , we expect that @xmath43 becomes smaller and smaller , thus restricting the choice of time step sizes @xmath81 . however , practically , the situation is better . for instance , for the implicit euler scheme , if @xmath16 is convex , we obtain @xmath70 - h[u^{k-1 } ] \le \int_\omega h'(u^k)(u^k - u^{k-1})dx = -\tau\int_\omega h'(u^k)a[u^k]dx\ ] ] for _ any _ value of @xmath65 . moreover , for other ( higher - order ) runge - kutta schemes , the numerical experiments in section [ sec.num ] indicate that there exists @xmath66 such that @xmath90 holds for all @xmath96 $ ] uniformly in @xmath68 . in this situation , inequality holds for all @xmath97 , and thus our estimate is global . in fact , the function @xmath98 is numerically even nonincreasing in some interval @xmath99 $ ] but we are not able to prove this analytically . the _ second main result _ is the specification of the abstract conditions and for a number of examples : a quasilinear diffusion equation , porous - medium or fast - diffusion equations , a linear diffusion system , and the fourth - order derrida - lebowitz - speer - spohn equation ( see sections [ sec.de]-[sec.dlss ] for details ) . for instance , for the porous - medium equation @xmath100 we show that the runge - kutta scheme scheme satisfies @xmath70 + \tau\beta\int_\omega(u^k)^{\alpha+\beta-2}|\na u^k|^2 dx \le h[u^{k-1 } ] , \quad\mbox{where } h[u]=\frac{1}{\alpha(\alpha+1)}\int_\omega u^{\alpha+1 } dx,\ ] ] for @xmath56 and all @xmath101 belonging to some region in @xmath102 ( see figure [ fig.0th ] below ) . for @xmath103 , we write @xmath11=\int_\omega u(\log u-1)dx$ ] . in one space dimension and for runge - kutta schemes of order @xmath45 , this region becomes @xmath104 , which is the same condition as for the continuous equation ( except the boundary values ) . furthermore , the first - order entropy is dissipated for runge - kutta schemes of order @xmath45 , in one space dimension , @xmath80 + \tau c_{\alpha,\beta}\int_\omega ( u^k)^{\alpha+\beta-2}(u^k)_{xx}^2 dx \le f[u^{k-1 } ] , \quad\mbox{where } f[u]=\int_\omega ( u^{\alpha/2})_x^2 dx,\ ] ] for @xmath56 and all @xmath101 belonging to the region shown in figure [ fig.1st ] below , and @xmath105 is some constant . this region is smaller than the region of admissible values @xmath101 for the continuous entropy . the borders of that region are indicated in the figure by dashed lines . the proof of the above results , and namely of @xmath89 , is based on systematic integration by parts @xcite . the idea of the method is to formulate integration by parts as manipulations with polynomials and to conclude the inequality @xmath89 from a polynomial decision problem . this problem can be solved directly or by using computer algebra software . our _ third main result _ is the relation to geodesic 0-convexity of the entropy and the bakry - emery approach when @xmath61 ( runge - kutta scheme of order @xmath45 ) . liero and mielke formulate in @xcite the abstract cauchy problem as the gradient flow @xmath106dh[u ] , \quad t>0 , \quad u(0)=u^0,\ ] ] where the onsager operator @xmath107 $ ] describes the sum of diffusion and reaction terms . for instance , if @xmath6={\operatorname{div}}(a(u)\na u)$ ] , we can write @xmath6={\operatorname{div}}(a(u)h''(u)^{-1}\na h'(u))$ ] and thus , identifying @xmath18 and @xmath108 $ ] , we have @xmath107\xi={\operatorname{div}}(a(u)h''(u)^{-1}\na\xi)$ ] . it is shown in @xcite that the entropy @xmath109 is geodesic @xmath110-convex if the inequality @xmath111k[u]\xi\rangle - \frac12\langle\xi , dk[u]a[u]\xi\rangle \ge \lambda\langle\xi , k[u]\xi\rangle\ ] ] holds for all suitable @xmath112 and @xmath113 . we will prove in section [ sec.meth ] that @xmath114 hence , if @xmath115 then with @xmath116 is satisfied for @xmath117 and @xmath118 , yielding geodesic @xmath119-convexity for the semi - discrete entropy . moreover , if @xmath120 then we obtain geodesic @xmath110-convexity . since @xmath121/dt$ ] and @xmath122/dt^2 $ ] in the continuous setting , the inequality @xmath120 can be written as @xmath123 \ge -\lambda \frac{dh}{dt}[u],\ ] ] which corresponds to a variant of the bakry - emery condition @xcite , yielding exponential convergence of @xmath11 $ ] ( if @xmath67 for all @xmath44 ) . thus , our results constitute a first step towards a _ discrete bakry - emery approach_. the paper is organized as follows . the abstract method , i.e. the proof of backward solvability and of theorems [ thm.ed ] and [ thm.ed2 ] , is presented in section [ sec.meth ] . the method is applied in the subsequent sections to a scalar diffusion equation ( section [ sec.de ] ) , the porous - medium equation ( section [ sec.pme ] ) , a linear diffusion system ( section [ sec.sys ] ) , and the fourth - order derrida - lebowitz - speer - spohn equation ( section [ sec.dlss ] ) . finally , section [ sec.num ] is devoted to some numerical experiments showing that @xmath98 is negative in some interval @xmath99 $ ] .

semi - discrete runge - kutta schemes for nonlinear diffusion equations of parabolic type are analyzed . conditions are determined under which the schemes dissipate the discrete entropy locally . the dissipation property is a consequence of the concavity of the difference of the entropies at two consecutive time steps . the concavity property is shown to be related to the bakry - emery approach and the geodesic convexity of the entropy . the abstract conditions are verified for quasilinear parabolic equations ( including the porous - medium equation ) , a linear diffusion system , and the fourth - order quantum diffusion equation . numerical experiments for various runge - kutta finite - difference discretizations of the one - dimensional porous - medium equation show that the entropy - dissipation property is in fact global .

semi - discrete runge - kutta schemes for nonlinear diffusion equations of parabolic type are analyzed . conditions are determined under which the schemes dissipate the discrete entropy locally . the dissipation property is a consequence of the concavity of the difference of the entropies at two consecutive time steps . the concavity property is shown to be related to the bakry - emery approach and the geodesic convexity of the entropy . the abstract conditions are verified for quasilinear parabolic equations ( including the porous - medium equation ) , a linear diffusion system , and the fourth - order quantum diffusion equation . numerical experiments for various runge - kutta finite - difference discretizations of the one - dimensional porous - medium equation show that the entropy - dissipation property is in fact global .

1604.06697

i

sorting a sequence of elements of some totally ordered universe remains one of the most fascinating and well - studied topics in computer science . moreover , it is an essential part of many practical applications . thus , efficient sorting algorithms directly transfer to a performance gain for many applications . one of the most widely used sorting algorithms is quicksort , which has been introduced by hoare in 1962 @xcite and is considered to be one of the most efficient sorting algorithms . for sorting an array , it works as follows : first , it chooses an arbitrary pivot element and then rearranges the array such that all elements smaller than the pivot are moved to the left side and all elements larger than the pivot are moved to the right side of the array this is called _ partitioning_. then , the left and right side are both sorted recursively . although its average number of comparisons is not optimal @xmath3 vs. @xmath4 for mergesort , its over - all instruction count is very low . moreover , by choosing the pivot element as median of some larger sample , the leading term @xmath5 for the average number of comparisons can be made smaller even down to @xmath6 when choosing the pivot as median of some sample of growing size @xcite . other advantages of quicksort are that it is easy to implement and that it does not need extra memory except the recursion stack of logarithmic size ( even in the worst case if properly implemented ) . a major drawback of quicksort is its quadratic worst - case running time . nevertheless , there are efficient ways to circumvent a really bad worst - case . the most prominent is introsort ( introduced by musser @xcite ) which is applied in gcc implementation of ` std:\!:sort ` : as soon as the recursion depth exceeds a certain limit , the algorithm switches to heapsort . another deficiency of quicksort is that it suffers from branch mispredictions ( or branch misses ) in an essential way . on modern processors with long pipelines ( 14 stages for intel haswell , broadwell , skylake processors for the older pentium 4 processors even more than twice as many ) every branch misprediction causes a rather long interruption of the execution since the pipeline has to be filled anew . in @xcite , kaligosi and sanders analyzed the number of branch mispredictions incurred by quicksort . they examined different simple branch prediction schemes ( static prediction and 1-bit , 2-bit predictors ) and showed that with all of them , quicksort with a random element as pivot causes on average @xmath7 branch mispredictions for some constant @xmath8 ( resp . @xmath9 , @xmath10 ) . in particular , in quicksort with random pivot element , every fourth comparison is followed by a mispredicted branch . the reason is that for partitioning , each element is compared with the pivot and depending on the outcome either it is swapped with some other element or not . since for an optimal pivot ( the median ) , the probability of being smaller the pivot is @xmath11 , there is no way to predict these branches . kaligosi and sanders also established that choosing skewed pivot elements ( far off the median ) might even decrease the running time because it makes branches more predictable . this also explains why , although theoretically larger samples for pivot selection were shown to be superior , in practice the median - of three variant turned out to be the best . in @xcite , the skewed pivot phenomenon is confirmed experimentally . moreover , in @xcite , precise theoretical bounds on the number of branch misses for quicksort are given establishing also theoretical superiority of skewed pivots under the assumption that branch mispredictions are expensive . in @xcite brodal and moruz proved a general lower bound on the number of branch mispredictions given that every comparison is followed by a conditional branch which depends on the outcome of the comparison . in this case there are @xmath12 branch mispredictions for a sorting algorithm which performs @xmath13 comparisons . as elmasry and katajainen remarked in @xcite , this theorem does not hold anymore if the results of comparisons are not used for conditional branches . indeed , they showed that every program can be transformed into a program which induces only a constant number of branch misses and whose running time is linear in the running time of the original program . however , this general transformation introduces a huge constant factor overhead . still , in @xcite and @xcite elmasry , katajainen and stenmark showed how to efficiently implement many algorithms related to sorting with only few branch mispredictions . they call such programs _ lean_. in particular , they present variants of mergesort and quicksort suffering only very little from branch misses . their quicksort variant ( called tuned quicksort , for details on the implementation , see @xcite ) is very fast for random permutations however , it does not behave well with duplicate elements because it applies lomuto s uni - directional partitioner ( see e.g. @xcite ) . another development in recent years is multi - pivot quicksort ( i.e.several pivots in each partitioning stage @xcite ) . it started with the introduction of yaroslavskiy s dual - pivot quicksort @xcite which , surprisingly , was faster than known quicksort variants and , thus , became the standard sorting implementation in oracle java 7 and java 8 . concerning branch mispredictions all these multi - pivot variants behave essentially like ordinary quicksort @xcite ; however , they have one advantage : every data element is accessed only a few times ( this is also referred to as the number of _ scans _ ) . as outlined in @xcite , increasing the number of pivot elements further ( up to 127 or 255 ) , leads to super scalar sample sort , which has been introduced by sanders and winkel @xcite . super scalar sample sort not only has the advantage of few scans , but also is based on the idea of avoiding conditional branches . indeed , the correct bucket ( the position between two pivot elements ) can be found by converting the results of comparisons to integers and then simply performing integer arithmetic . in their experiments sanders and winkel show that super scalar sample sort is approximately twice as fast as quicksort ( ` std:\!:sort ` ) when sorting random integer data . however , super scalar sample sort has one major draw - back : it uses a linear amount of extra space ( for sorting @xmath2 data elements , it requires space for another @xmath2 data elements and additionally for more than @xmath2 integers ) . in the conclusion of @xcite , kaligosi and sander raised the question : _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ however , an in - place sorting algorithm that is better than quicksort with skewed pivots is an open problem . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ ( here , in - place means that it needs only a constant or logarithmic amount of extra space . ) in this work , we solve the problem by presenting our block partition algorithm , which allows to implement quicksort without any branch mispredictions incurred by conditional branches based on results of comparisons ( except for the final insertionsort also there are still conditional branches based on the control - flow , but their amount is relatively small ) . we call the resulting algorithm blockquicksort . our work is inspired by tuned quicksort from @xcite , from where we also borrow parts of our implementation . the difference is that by doing the partitioning block - wise , we can use hoare s partitioner , which is far better with duplicate elements than lomuto s partitioner ( although tuned quicksort can be made working with duplicates by applying a check for duplicates similar to what we propose for blockquicksort as one of the further improvements in section [ sec : improvments ] ) . moreover , blockquicksort is also superior to tuned quicksort for random permutations of integers . [ [ our - contributions ] ] our contributions * we present a variant of the partition procedure that only incurs few branch mispredictions by storing results of comparisons in constant size buffers . * we prove an upper bound of @xmath14 branch mispredictions on average , where @xmath15 for our proposed block size ( theorem [ thm : few_branches ] ) . * we propose some improvements over the basic version . * we implemented our algorithm with an ` stl`-style interface . * we conduct experiments and compare blockquicksort with ` std:\!:sort ` , yaroslavskiy s dual - pivot quicksort and super scalar sample sort on random integer data it is faster than all of these and also katajainen et al.s tuned quicksort . [ [ outline ] ] outline section [ sec : prelims ] introduces some general facts on branch predictors and mispredictions , and gives a short account of standard improvements of quicksort . in section [ sec : partition ] , we give a precise description of our block partition method and establish our main theoretical result the bound on the number of branch mispredictions . finally , in section [ sec : experiments ] , we experimentally evaluate different block sizes , different pivot selection strategies and compare our implementation with other state of the art implementations of quicksort and super scalar sample sort .

since the work of kaligosi and sanders ( 2006 ) , it is well - known that quicksort which is commonly considered as one of the fastest in - place sorting algorithms suffers in an essential way from branch mispredictions . , we avoid conditional branches based on outcomes of comparisons at all ( except for the final insertionsort ) . moreover , we prove that for a static branch predictor the average total number of branch mispredictions is at most for some small depending on the block size when sorting elements .

since the work of kaligosi and sanders ( 2006 ) , it is well - known that quicksort which is commonly considered as one of the fastest in - place sorting algorithms suffers in an essential way from branch mispredictions . we present a novel approach to address this problem by partially decoupling control from data flow : in order to perform the partitioning , we split the input in blocks of constant size ( we propose 128 data elements ) ; then , all elements in one block are compared with the pivot and the outcomes of the comparisons are stored in a buffer . in a second pass , the respective elements are rearranged . by doing so , we avoid conditional branches based on outcomes of comparisons at all ( except for the final insertionsort ) . moreover , we prove that for a static branch predictor the average total number of branch mispredictions is at most for some small depending on the block size when sorting elements . our experimental results are promising : when sorting random integer data , we achieve an increase in speed ( number of elements sorted per second ) of more than 80% over the gcc implementation of c++ ` std:\!:sort ` . also for many other types of data and non - random inputs , there is still a significant speedup over ` std:\!:sort ` . only in few special cases like sorted or almost sorted inputs , ` std:\!:sort`can beat our implementation . moreover , even on random input permutations , our implementation is even slightly faster than an implementation of the highly tuned super scalar sample sort , which uses a linear amount of additional space .

1604.06697

r

we ran thorough experiments with implementations in c++ on different machines with different types of data and different kinds of input sequences . if not specified explicitly , the experiments are run on an intel core i5 - 2500k cpu ( 3.30ghz , 4 cores , 32 kb l1 instruction and data cache , 256 kb l2 cache per core and 6 mb l3 shared cache ) with 16 gb ram and operating system ubuntu linux 64bit version 14.04.4 . we used gnu s ` g++ ` ( 4.8.4 ) ; optimized with flags ` -o3 -march = native ` . for time measurements , we used ` std:\!:chrono:\ ! : high_resolution_clock ` , for generating random inputs , the mersenne twister pseudo - random generator ` std:\!:mt19937 ` . all time measurements were repeated with the same 20 deterministically chosen seeds the displayed numbers are the average of these 20 runs . moreover , for each time measurement , at least 128 mb of data were sorted if the array size is smaller , then for this time measurement several arrays have been sorted and the total elapsed time measured . our running time plots all display the actual time divided by the number of elements to sort on the ` y`-axis . we performed our running time experiments with three different data types : * ` int ` : signed 32-bit integers . * ` vector ` : 10-dimensional array of 64-bit floating - point numbers ( ` double ` ) . the order is defined via the euclidean norm for every comparison the sums of the squares of the components are computed and then compared . * ` record ` : 21-dimensional array of 32-bit integers . only the first component is compared . the code of our implementation of blockquicksort as well as the other algorithms and our running time experiments is available at https://github.com/weissan/blockquicksort . [ [ sec : blocksize_experiments ] ] different block sizes figure [ fig : blocksizes ] shows experimental results on random permutations for different data types and block sizes ranging from 4 up to @xmath76 . we see that for integers only at the end there is a slight negative effect when increasing the block size . presumably this is because up to a block size of @xmath77 , still two blocks fit entirely into the l3 cache of the cpu . on the other hand for ` vector ` a block size of 64 and for ` record ` of 8 seem to be optimal with a considerably increasing running time for larger block sizes . as a compromise we chose to fix the block size to 128 elements for all further experiments . an alternative approach would be to choose a fixed number of bytes for one block and adapt the block size according to the size of the data elements . [ [ sec : skewed ] ] skewed pivot experiments [ sec : pivot_experiments ] means that @xmath78-th element is chosen as pivot of an array of length @xmath2 . ] we repeated the experiments from @xcite with skewed pivot for both the usual hoare partitioner ( ` std:\!:__unguarded_partition ` , from the gcc implementation of ` std:\!:sort ` ) and our block partition method . for both partitioners we used our tuned quicksort loop . the results can be seen in figure [ fig : skewed_pivots ] : classic quicksort benefits from skewed pivot , whereas blockquicksort works best with the exact median . therefore , for blockquicksort it makes sense to invest more effort to find a good pivot . [ [ sec : pivot_selection ] ] different pivot selection methods we implemented several strategies for pivot selection : * median - of - three , median - of - five , median - of - twenty - three , * median - of - three - medians - of - three , median - of - three - medians - of - five , median - of - five - medians - of - five : first calculate three ( resp . five ) times the median of three ( resp . five ) elements , then take the pivot as median of these three ( resp . five ) medians , * median - of-@xmath79 . all pivot selection strategies switch to median - of - three for small arrays . moreover , the median - of-@xmath79 variant switches to median - of - five - medians - of - five for arrays of length below 20000 ( for smaller @xmath2 even the number of comparisons was better with median - of - five - medians - of - five ) . the medians of larger samples are computed with ` std:\!:nth_element ` . despite the results on skewed pivots figure [ fig : skewed_pivots ] , there was no big difference between the different pivot selection strategies as it can be seen in figure [ fig : pivot_method ] . . ] as expected , median - of - three was always the slowest for larger arrays . median - of - five - medians - of - five was the fastest for ` int ` and median - of-@xmath79 for ` vector ` . we think that the small difference between all strategies is due to the large overhead for the calculation of the median of a large sample and maybe because the array is rearranged in a way that is not favorable for the next recursive calls . we compare variants of blockquicksort with the gcc implementation of ` std:\!:sort ` ( which is known to be one of the most efficient quicksort implementations see e.g. @xcite ) , yaroslavskiy s dual - pivot quicksort @xcite ( we converted the java code of @xcite to c++ ) and an implementation of super scalar sample sort @xcite by hbschle - schneider . for random permutations and random values modulo @xmath79 , we also test tuned quicksort @xcite and three - pivot quicksort implemented by aumller and bingmann from @xcite ( which is based on @xcite ) for other types of inputs we omit these algorithms because of their poor behavior with duplicate elements . [ [ branch - mispredictions-1 ] ] branch mispredictions we experimentally determined the number of branch mispredictions of blockquicksort and the other algorithms with the _ chachegrind _ branch prediction profiler , which is part of the profiling tool _ _ valgrind__. the results of these experiments on random ` int ` data can be seen in figure [ fig : branch_misses ] the ` y`-axis shows the number of branch misprediction divided the the array size . we only display the median - of - three variant of blockquicksort since all the variants are very much alike . we also added plots of blockquicksort and tuned quicksort skipping final insertionsort ( i.e. the arrays remain partially unsorted ) . we see that both ` std:\!:sort`and yaroslavskiy s dual - pivot quicksort incur @xmath80 branch mispredictions . the up and down for super scalar sample sort presumably is because of the variation in the size of the arrays where the base case sorting algorithm ` std:\!:sort`is applied to . for blockquicksort there is an almost non - visible @xmath6 term for the number of branch mispredictions . indeed , we computed an approximation of @xmath81 branch mispredictions . thus , the actual number of branch mispredictions is still better then our bounds in theorem [ thm : few_branches ] . there are two factors which contribute to this discrepancy : our rough estimates in the mentioned results , and that the actual branch predictor of a modern cpu might be much better than a static branch predictor . also note that approximately one half of the branch mispredictions are incurred by insertionsort only the other half by the actual block partitioning and main quicksort loop . finally , figure [ fig : branch_misses ] shows that katajainen et al.s tuned quicksort is still more efficient with respect to branch mispredictions ( only @xmath64 ) . this is no surprise since it does not need any checks whether buffers are empty etc . moreover , we see that over @xmath82 of the branch misses of tuned quicksort come from the final insertionsort . [ [ running - time - experiments ] ] running time experiments in figure [ fig : random_int ] we present running times on random ` int ` permutations of different blockquicksort variants and the other algorithms including katajainen s tuned quicksort and aumller and bingmann s three - pivot quicksort . the optimized blockquicksort variants need around 45ns per element when sorting @xmath83 elements , whereas ` std:\!:sort`needs 85ns per element thus , there is a speed increase of 88% ( i.e. the number of elements sorted per second is increased by 88% ) . and @xmath79 . ] the same algorithms are displayed in figure [ fig : modulo_int ] for sorting random ` int`s between @xmath84 and @xmath79 . here , we observe that tuned quicksort is much worse than all the other algorithms ( already for @xmath85 it moves outside the displayed range ) . all variants of blockquicksort are faster than ` std:\!:sort ` the ` duplicate check ` ( dc ) version is almost twice as fast . figure [ fig : powers ] presents experiments with data containing a lot of duplicates and having specific structures thus , maybe coming closer to `` real - world '' inputs ( although it is not clear what that means ) . since here tuned quicksort and three - pivot quicksort are much slower than all the other algorithms , we exclude these two algorithms from the plots . the array for the left plot contains long already sorted runs . this is most likely the reason that ` std:\!:sort`and yaroslavskiy s dual - pivot quicksort have similar running times to blockquicksort ( for sorted sequences the conditional branches can be easily predicted what explains the fast running time ) . the arrays for the middle and right plot start with sorted runs and become more and more erratic ; the array for the right one also contains a extremely high number of duplicates . here the advantage of blockquicksort avoiding conditional branches can be observed again . in all three plots the check for duplicates ( dc ) established a considerable improvement . of ` int ` with duplicates : _ left _ : @xmath86 = i \mod { \left\lfloor\mathinner{\sqrt{n } } \right\rfloor}$ ] ; _ middle _ : @xmath86 = i^2 + n/2\mod n$ ] ; _ right _ : @xmath86 = i^8 + n/2 \mod n$ ] . since @xmath2 is always a power of two , the value @xmath87 occurs approximately @xmath88 times in the last case . ] in figure [ fig : recordvector ] , we show the results of selected algorithms for random permutations of ` vector ` and ` record ` . we conjecture that the good results of super scalar sample sort on ` record`s are because of its better cache behavior ( since ` record ` are large data elements with very cheap comparisons ) . more running time experiments also on other machines and compiler flags can be found in appendix [ app : more_experiments ] . [ [ more - statistics ] ] more statistics table [ tab : more_stats ] shows the number of branches taken / branch mispredicted as well as the instruction count and cache misses . although ` std:\!:sort`has a much lower instruction count than the other algorithms , it induces most branch misses and ( except tuned quicksort ) most l1 cache misses (= l3 refs since no l2 cache is simulated ) . blockquicksort does not only have a low number of branch mispredictions , but also a good cache behavior one reason for this is that insertionsort is applied during the recursion and not at the very end . .instruction count , branch and cache misses when sorting random ` int ` permutations of size @xmath89 . all displayed numbers are divided by the number of elements . [ cols="^,^,^,^,^,^,^",options="header " , ]

our experimental results are promising : when sorting random integer data , we achieve an increase in speed ( number of elements sorted per second ) of more than 80% over the gcc implementation of c++ ` std:\!:sort ` . also for many other types of data and non - random inputs moreover , even on random input permutations , our implementation is even slightly faster than an implementation of the highly tuned super scalar sample sort , which uses a linear amount of additional space .

since the work of kaligosi and sanders ( 2006 ) , it is well - known that quicksort which is commonly considered as one of the fastest in - place sorting algorithms suffers in an essential way from branch mispredictions . we present a novel approach to address this problem by partially decoupling control from data flow : in order to perform the partitioning , we split the input in blocks of constant size ( we propose 128 data elements ) ; then , all elements in one block are compared with the pivot and the outcomes of the comparisons are stored in a buffer . in a second pass , the respective elements are rearranged . by doing so , we avoid conditional branches based on outcomes of comparisons at all ( except for the final insertionsort ) . moreover , we prove that for a static branch predictor the average total number of branch mispredictions is at most for some small depending on the block size when sorting elements . our experimental results are promising : when sorting random integer data , we achieve an increase in speed ( number of elements sorted per second ) of more than 80% over the gcc implementation of c++ ` std:\!:sort ` . also for many other types of data and non - random inputs , there is still a significant speedup over ` std:\!:sort ` . only in few special cases like sorted or almost sorted inputs , ` std:\!:sort`can beat our implementation . moreover , even on random input permutations , our implementation is even slightly faster than an implementation of the highly tuned super scalar sample sort , which uses a linear amount of additional space .

1412.2433

i

the ability to learn the social path lengthto other social network users can often help individuals make informed trust and access control decisions . for instance , if attendees at a large convention could easily find other attendees with whom they share social links ( e.g. , a linkedin connection ) , this could help them decide who to chat or meet up with . similarly , travelers and commuters could more consciously decide with whom to interact , share rides , etc . in general , discovering the social path lengthbetween users is beneficial in many interesting scenarios , such as estimating the _ familiarity _ to a location ( which can in turn be used for context - based security @xcite ) , as well as for routing in delay - tolerant ad - hoc mobile networks @xcite and anonymous communications @xcite . * problem statement . * the widespread adoption of online social networks ( osns ) makes it appealing to measure the length of a path between two nodes , e.g. , to use this information as a signal of reciprocal trust and/or social interest . today , a facebook user can see the number of common friends with another user , while linkedin also displays the social path length . however , as popular osns are centralized systems , so are the features they offer to discover social paths . as such , they do not particularly adapt well to mobile environments where social interactions are tied to physical proximity , thus severely limiting the feasibility of many applications scenarios users may not always be able to connect to the internet or willing to reveal their location and/or interests to the provider . relying on centralized systems to learn social path lengths implies that , whenever alice queries a server for the social path lengthto bob , the server learns that alice is interested in bob , their frequency of interactions , and their locations . this prompts the need for decentralized and privacy - preserving techniques for social path lengthestimation . users should only learn if they have any common friends , without having to reciprocally reveal the identities of friends that they do not share , and discover the length of the social path between them ( for paths longer than two ) , without learning which users are in the path . * technical roadmap . * our work builds on common friends @xcite , a system supporting privacy - preserving common friend discovery on mobile devices : building on a cryptographic primitive called private set intersection ( psi ) @xcite , it allows mutual friends to be discovered by securely computing the intersection of friendship _ capabilities _ , which are periodically distributed from a common friendsuser to all the friends who are also using it . however , besides being limited to social paths of length two , common friendsonly discovers the subset of the mutual friends who are _ already _ in the system , thus suffering from an inherent bootstrapping problem . this paper introduces social pal , the first system that supports the privacy - preserving estimation of the social path lengthbetween any two social network users . we introduce the notion of _ ersatz nodes _ , created for users that are direct friends of one or more users of social palbut are not part of the system . we guarantee that two users of social palwill be able to discover _ all _ their common friends in the osn ( i.e. , all paths of length two ) . we then present a hash chain - based protocol that supports the ( private ) discovery of social paths longer than two , and demonstrate its effectiveness by means of extensive simulations . our work is not limited to designing protocols : we also fully implement social paland deploy a scalable server architecture and an android client library enabling developers to seamlessly integrate it into their applications . * contributions . * in summary , we make the following contributions : 1 . we present an efficient privacy - preserving estimation of social paths of arbitrary length ( section [ sec : social ] ) . 2 . we state and prove several properties of social palincluding the fact that , for two users and : ( i ) social palwill find all common friends of and , including those who are not using it , and ( ii ) for each discovered path between and , social palallows each party to compute the _ exact _ length of the path ( section [ sec : analysis ] ) . 3 . using samples of the facebook graph , we empirically show that even when only 40% of users use the system , social palwill discover more than 70% of all paths , and over 40% with just 20% of the users ( section [ sec : evaluation ] ) . 4 . we support facebook and linkedin integration and release the implementation of a scalable server - side architecture and a modular android client library , allowing developers to easily integrate social palin their applications ( section [ sec : implementation ] ) . we build two android apps : a friend radar displaying the social path lengthto nearby users , and a tethering app enabling users to securely share their internet connection with people with whom they share mutual friends ( see section [ sec : apps ] ) .

this paper presents social pal , a system supporting the privacy - preserving discovery of arbitrary - length social paths between any two social network users . e.g. , discovering 70% of all paths with only 40% of the users . we implement social palusing a scalable server - side architecture and a modular android client library , allowing developers to seamlessly integrate it into their apps .

social relationships are a natural basis on which humans make trust decisions . online social networks ( osns ) are increasingly often used to let users base trust decisions on the existence and the strength of social relationships . while most osns allow users to discover the length of the social path to other users , they do so in a centralized way , thus requiring them to rely on the service provider and reveal their interest in each other . this paper presents social pal , a system supporting the privacy - preserving discovery of arbitrary - length social paths between any two social network users . we overcome the bootstrapping problem encountered in all related prior work , demonstrating that social palallows its users to find all paths of length two and to discover a significant fraction of longer paths , even when only a small fraction of osn users is in the social palsystem e.g. , discovering 70% of all paths with only 40% of the users . we implement social palusing a scalable server - side architecture and a modular android client library , allowing developers to seamlessly integrate it into their apps . [ distributed applications ]

1412.2433

c

this paper presented social pal a system geared to privately estimate the social path length between two social network users . we demonstrated its effectiveness both analytically and empirically , showing that , for any two osn users , social paldiscovers all social paths of length two and a significant portion of longer paths . using different samples of the facebook graph , we showed that even when only 20% of the osn users use the system , we discover more than 40% of all paths between any two users , and 70% with 40% of users . we also implemented a scalable server - side architecture and a modular client library bringing social palto the real world . our deployment supports facebook and linkedin integration and allows developers to easily incorporate it in their projects . social palcan be used in a number of applications where , by relying on the ( privacy - preserving ) estimation of social path length , users can make informed trust and access control decisions . * acknowledgments . * we thank minas gjoka and michael sirivianos for sharing the facebook datasets , swapnil udar for helping with the ` spotshare`implementation , and jussi kangasharju , pasi sarolahti , cecilia mascolo , panos papadimitratos , and narges yousefnezhad for providing feedback on the paper . simon eberz suggested the idea of signed bloom filters discussed in section [ subsec : privacy - consideration ] . this work was partially supported by the academy of finland s `` contextual security '' project ( 274951 ) , the ec fp7 precious project ( 611366 ) , and the eit ict labs .

social relationships are a natural basis on which humans make trust decisions . we overcome the bootstrapping problem encountered in all related prior work , demonstrating that social palallows its users to find all paths of length two and to discover a significant fraction of longer paths , even when only a small fraction of osn users is in the social palsystem

social relationships are a natural basis on which humans make trust decisions . online social networks ( osns ) are increasingly often used to let users base trust decisions on the existence and the strength of social relationships . while most osns allow users to discover the length of the social path to other users , they do so in a centralized way , thus requiring them to rely on the service provider and reveal their interest in each other . this paper presents social pal , a system supporting the privacy - preserving discovery of arbitrary - length social paths between any two social network users . we overcome the bootstrapping problem encountered in all related prior work , demonstrating that social palallows its users to find all paths of length two and to discover a significant fraction of longer paths , even when only a small fraction of osn users is in the social palsystem e.g. , discovering 70% of all paths with only 40% of the users . we implement social palusing a scalable server - side architecture and a modular android client library , allowing developers to seamlessly integrate it into their apps . [ distributed applications ]

1109.2239

i

a central goal in systems neuroscience is to determine what underlying principles might shape sensory processing in the nervous system . several coding optimization principles have been proposed , including redundancy reduction @xcite , and information maximization @xcite , which have both enjoyed some successes in predicting the behavior of real neurons@xcite . closely related to these notions of coding efficiency is the principle of sparseness @xcite , which posits that few neurons are active at any given time ( population sparseness ) , or that individual neurons are responsive to few specific stimuli ( lifetime sparseness ) . sparseness is an appealing concept , in part because it provides a simple code for later stages of processing and it is in principle more quickly and easily modifiable by simple learning rules compared with more distributed codes involving many simultaneously active units @xcite . there is some experimental evidence for sparse coding in the cortex @xcite , but there are also reports of dense neural activity @xcite and mixtures of both @xcite as well . compounding this , it is not obvious what absolute standard should be used to assess the degree of sparseness in cortex , but it is notable that the relative level of sparseness of cortical responses to natural images increases when a larger fraction of the visual field is covered by the stimulus @xcite , as a result of inhibitory interneuronal connections @xcite . interestingly , the correlations between the neuronal activities also decreases when a larger area is stimulated , as a result of these inhibitory connections @xcite . in a landmark paper , olshausen and field @xcite reproduced several qualitative features of the receptive fields ( rfs ) of neurons in primary visual cortex ( vi ) without imposing any biological constraints other than their hypothesis that cortical representations simultaneously minimize the average activity of the neural population while maximizing fidelity when representing natural images . however , agreement with measured v1 simple cell receptive fields was not perfect @xcite . recently , a more sophisticated version of olshausen and field s algorithm @xcite has been developed that is capable of minimizing the number of active neurons rather than minimizing the average activity level across the neural population . this algorithm , called the sparse - set coding ( ssc ) network @xcite , learns the full set of physiologically observed rf shapes of simple cells in v1 , which include small unoriented features , localized oriented features resembling gabor wavelets , and elongated edge - detectors . we note that , under certain conditions @xcite not necessarily satisfied by the natural image coding problem , minimizing the average activity across the neural population ( @xmath0-norm minimization ) , as is done by olshausen and field s original sparsenet algorithm , can be equivalent to minimizing the number of active units ( @xmath1-norm minimization ) , as is achieved by rehn and sommer s ssc algorithm . the ssc model is the only sparse coding algorithm that has been shown to learn , from the statistics of natural scenes alone , rfs that are in quantitative agreement with those observed in v1 . it has also been found that sufficiently overcomplete representations ( 4 times more model neurons than image pixels ) that minimize the @xmath0 norm can display the same qualitative variety of rf shapes , but these have not been quantitatively compared with physiologically measured rfs @xcite . unfortunately , the lack of work on biophysically realistic sparse coding models has left in doubt whether v1 could actually employ a sparse code for natural scenes . indeed , it is not clear how olshausen s original algorithm @xcite , the highly overcomplete @xmath0-norm minimization algorithm @xcite , or that of rehn and sommer @xcite , could be implemented in the cortex . rather than employing local network modification rules such as the synaptic plasticity that is thought to underly learning in cortex @xcite , all three of these networks rely on learning rules requiring that each synapse has access to information about the receptive fields of many other , often distant , neurons in the network . furthermore , both the ssc sparse coding model that has successfully reproduced the full diversity of v1 simple cell rfs @xcite as well as the @xmath0-norm minimization algorithm that achieved qualitatively similar rfs @xcite involve non - spiking computational units : continuous - valued information is shared between units while inference is being performed . in cortex , however , information is transferred in discrete , stereotyped pulses of electrical activity called action potentials or spikes . particularly for a sparse coding model with few or no spikes elicited per stimulus presentation , approximating spike trains with a graded function may not be justified . spiking image processing networks have been studied @xcite , but none of them have been shown to learn the full diversity of v1 rf shapes using local plasticity rules . it remains to be demonstrated that sparse coding can be achieved within the limitations imposed by biological architecture , and thus that it could potentially be an underlying principle of neural comptutation . here we present a biologically - inspired variation on a network originally due to fldik @xcite that performs sparse coding with spiking neurons . our model performs learning using only synaptically local rules . using the fact that constraints imposed by such mechanisms as homeostasis and lateral inhibition cause the units in the network to remain sparse and independent throughout training , we prove mathematically that it learns to approximate the optimal linear generative model of the input , subject to constraints on the average lifetime firing rates of the units and the temporal correlations between the units firing rates . this is the first demonstration that synaptically local plasticity rules are sufficient to account for the observed diversity of v1 simple cell rf shapes , and the first rigorous derivation of a relationship between synaptically local network modification rules and the twin properties of sparseness and decorrelation . finally , we describe several emergent properties of our image coding network , in order to elucidate some experimentally testable hallmarks of our model . interestingly , we observe a lognormal distribution of inhibitory connection strengths between the units in our model , when it is trained on natural images ; such a distribution has previously been observed in the excitatory connections between neurons in rat v1 @xcite , but the inhibitory connection strength distribution remains unknown .

sparse coding algorithms trained on natural images can accurately predict the features that excite visual cortical neurons , but it is not known whether such codes can be learned using biologically realistic plasticity rules . we further prove , mathematically , that sparseness and decorrelation are the key ingredients that allow for synaptically local plasticity rules to optimize a cooperative , linear generative image model formed by the neural representation . finally , we discuss several interesting emergent properties of our network , with the intent of bridging the gap between theoretical and experimental studies of visual cortex .

sparse coding algorithms trained on natural images can accurately predict the features that excite visual cortical neurons , but it is not known whether such codes can be learned using biologically realistic plasticity rules . we have developed a biophysically motivated spiking network , relying solely on synaptically local information , that can predict the full diversity of v1 simple cell receptive field shapes when trained on natural images . this represents the first demonstration that sparse coding principles , operating within the constraints imposed by cortical architecture , can successfully reproduce these receptive fields . we further prove , mathematically , that sparseness and decorrelation are the key ingredients that allow for synaptically local plasticity rules to optimize a cooperative , linear generative image model formed by the neural representation . finally , we discuss several interesting emergent properties of our network , with the intent of bridging the gap between theoretical and experimental studies of visual cortex .

1504.06474

i

sparse matrix - vector multiplication ( spmv ) is perhaps the most widely - used non - trivial sparse basic linear algebra subprogram ( blas ) in computational science and modeling . the operation multiplies a sparse matrix @xmath0 of size @xmath1 by a dense vector @xmath2 of size @xmath3 and gives a dense vector @xmath4 of size @xmath5 . despite its simplicity at the semantic level , an efficient spmv implementation is generally hard , because @xmath0 s sparsity structure can be very irregular and unpredictable . compared to cpus , co - processors ( e.g. , gpus and xeon phi ) promise much higher peak floating - point performance and memory bandwidth . thus a lot of research has focused on accelerating spmv on co - processors . one straightforward way on utilizing co - processors is to develop all - new sparse matrix formats ( e.g. , hyb @xcite , cocktail @xcite , jad @xcite , esb @xcite , bccoo @xcite and brc @xcite ) for specific hardware architectures . the experimental results showed that these formats can provide performance improvement for various spmv benchmarks . however , the completely new formats bring several new problems . the first one is backward - compatibility . when the input data are stored in basic formats such as compressed sparse row ( csr ) , a format conversion is required for using the new format based spmv . in practice , fusing a completely new format into well - established toolkits ( e.g. , petsc @xcite ) for scientific software is not a trivial task @xcite because of the format conversion . moreover , kumbhar @xcite pointed out that once an application ( in particular a non - linear solver ) needs repeated format conversion after a fixed small number of iterations , the new formats may degrade overall performance . furthermore , langr and tvrdk @xcite demonstrated that isolated spmv performance is insufficient to evaluate a new format . thus more evaluation criteria , such as format conversion cost and memory footprint , must be taken into consideration . secondly , when the spmv operation is used with other sparse building blocks ( e.g. , sparse matrix - matrix multiplication @xcite ) that require basic storage formats , using the all - new formats is less feasible . to leverage the spmv performance and the practicality , liu and vinter proposed the csr5 format @xcite to extend the basic csr format . the experimental results showed that the csr5 format delivers excellent spmv performance , but merely needs very short format conversion time ( a few spmv operations ) and very small extra memory footprint ( around 2% of the csr data ) . because the csr5 format shares data with the csr format , the csr - based sparse blas routines can efficiently work with the csr5 format . however , when a solver only needs a few iterations , the csr5 may not deliver speedups , compared to using the basic csr data . thereofore , improving performance of spmv using the most widely supported csr format has also gained plenty of attention @xcite . most of the related work @xcite has focused on improving row block method for the csr - based spmv . however , these newly proposed approaches are not highly efficient . the main reason is that co - processors are designed for load balanced high throughput computation , which is not naturally offered by the row pointer information of the csr format . on the other hand , using segmented sum method for the csr - based spmv has been proposed by @xcite and been implemented in libraries cudpp @xcite and modern gpu @xcite for nvidia gpus . unlike the row block methods , the segmented sum algorithms evenly partition an input matrix @xmath0 for nearly perfect load balancing , and thus may be suitable for a co - processor implementation . but unfortunately , this method can not recognize empty rows and requires more costly global operations . these extra overheads may offset performance gain of load balanced segmented sum and degrade overall spmv efficiency . recently , heterogeneous processors ( which are also known as heterogeneous chip multiprocessors ) have been designed @xcite and implemented @xcite . compared to homogeneous processors such as cpus or gpus , heterogeneous processors can deliver improved overall performance and power efficiency @xcite , while sufficient heterogeneous parallelisms exist . the main characteristics of heterogeneous processors include unified shared memory and fast communication among different types of cores ( e.g. , cpu cores and gpu cores ) . practically , heterogeneous system architecture ( hsa ) @xcite , opencl @xcite and cuda @xcite have supplied toolkits for programming heterogeneous processors . our work described in this paper particularly focuses on accelerating csr - based spmv on cpu - gpu heterogeneous processors . the main idea of our spmv algorithm is first speculatively executing spmv on a heterogeneous processor s gpu cores targeting high throughput computation , and then locally re - arranging resulting vectors by the cpu cores of the same chip for low - latency memory access . to achieve load balanced first step computation and to utilize both cpu and gpu cores , we improved the conventional segmented sum method by generating auxiliary information ( e.g. , segment descriptor ) at runtime and recognizing empty rows on - the - fly . compared to the row block methods for the csr - based spmv , our method delivers load balanced computation to achieve higher throughput . compared with the classic segmented sum method for the csr - based spmv , our approach decreases the overhead of global synchronization and removes pre- and post - processing regarding empty rows . this paper makes the following contributions : * we propose a fast csr - based spmv algorithm that efficiently utilizes different types of cores in emerging cpu - gpu heterogeneous processors . * we develop an speculative segmented sum algorithm by generating auxiliary information on - the - fly and eliminating costly pre- and post - processing on empty rows . * we evaluate our csr - based spmv algorithm on a widely - adopted benchmark suite and achieve stable spmv performance independent of the sparsity structure of input matrix . on a benchmark suite composed of 20 matrices with diverse sparsity structures , our approach greatly outperforms the row block methods for the csr - based spmv running on gpu cores of heterogeneous processors . on an intel heterogeneous processor , the experimental results show that our method obtains up to 6.90x and on average 2.57x speedup over an opencl implementation of the csr - vector algorithm in cusp running on its gpu cores . on an amd heterogeneous processor , our approach delivers up to 16.07x ( 14.43x ) and on average 5.61x ( 4.47x ) speedup over the fastest single ( double ) precision csr - based spmv algorithm from paralution and an opencl version of cusp running on its gpu cores . on an nvidia heterogeneous processor , our approach delivers up to 5.91x ( 6.20x ) and on average 2.69x ( 2.53x ) speedup over the fastest single ( double ) precision csr - based spmv algorithm from cusparse and cusp running on its gpu cores . the paper is organized as follows . we first introduce background knowledge about the csr format , the csr - based spmv algorithms and heterogeneous processors in section 2 . then we describe our csr - based spmv algorithm in section 3 . moreover , we give and analyze our experimental results in section 4 . we review the related approaches in section 5 .

, we propose a compressed sparse row ( csr ) format based spmv algorithm utilizing both types of cores in a cpu - gpu heterogeneous processor . on three heterogeneous processors from intel , amd and nvidia , using 20 sparse matrices as a benchmark suite , the experimental results show that our method obtains significant performance improvement over the best existing csr - based spmv algorithms .

sparse matrix - vector multiplication ( spmv ) is a central building block for scientific software and graph applications . recently , heterogeneous processors composed of different types of cores attracted much attention because of their flexible core configuration and high energy efficiency . in this paper , we propose a compressed sparse row ( csr ) format based spmv algorithm utilizing both types of cores in a cpu - gpu heterogeneous processor . we first speculatively execute segmented sum operations on the gpu part of a heterogeneous processor and generate a possibly incorrect results . then the cpu part of the same chip is triggered to re - arrange the predicted partial sums for a correct resulting vector . on three heterogeneous processors from intel , amd and nvidia , using 20 sparse matrices as a benchmark suite , the experimental results show that our method obtains significant performance improvement over the best existing csr - based spmv algorithms . sparse matrices , sparse matrix - vector multiplication , compressed sparse row , speculative execution , segmented sum , heterogeneous processor

1307.1021

i

models of spontaneous wave function collapse @xcite explicitly modify the schrdinger equation . new terms are added to the usual quantum hamiltonian which describe the nonlinear interaction between any system and a classical noise field . the effect of the new terms is to collapse the wave function in space . they are engineered in such way to give only small deviations from the schrdinger dynamics for microscopic systems . however , their effect grows with the size of the system such that for macroscopic objects the collapse of the wave function is dominant . in this way the possibility of having superpositions of different macroscopic states , like those described in the schrdinger s cat gedanken experiment , is avoided , and the quantum measurement problem vanishes . the deviation from the standard schrdinger dynamics implies that collapse models make predictions which differ from quantum mechanical ones . therefore , they can be tested experimentally @xcite . so far the process of spontaneous emission of electromagnetic radiation from charged particles sets the strongest upper bound on the collapse parameters @xcite among all possible tests . the idea is very simple . charged particles are always subject to a random motion due to the interaction with the noise causing the collapse . therefore they emit radiation even when according to standard quantum mechanics there should not be any . this problem has been considered several times in the literature @xcite because of a discrepancy in the theoretical formulas , which have been derived using different techniques which are summarized in the introductory section of @xcite . in a nutshell , the situation is the following . when the emission rate is computed by using standard perturbation theory , two terms are present at the first perturbative order : the first one , as expected , is proportional to the fourier component of the correlation function of the noise field computed at the same frequency as that of the emitted photon . the second term is instead proportional to the zero fourier component of the noise field s correlation . this latter term is unexpected and seems unphysical . the analysis carried out in @xcite shows that a way to avoid such an unphysical term is to use wave packets instead of plane waves and to confine the noise field to a bounded region . this means that standard perturbation theory can not be naively applied to collapse models but leaves open the question of the precise ( mathematical ) origin of the unphysical term . in @xcite the same problem was considered within the mathematically simpler qmupl ( quantum mechanics with universal position localizations ) model @xcite , which allows for an analytical treatment of the radiation process . it was shown that the undesired term is associated to an exponentially vanishing function of time . at first perturbative order it survives simply because the exponential is approximated by 1 . this result is true if the particle is bounded no matter how weakly . in this paper we apply the same methodology used for the qmupl model to the structurally different csl model . we consider the csl model of a charged particle bounded by a harmonic potential . apart from the dipole approximation , we solve the equation of motion for the particle s electrodynamic part of the hamiltonian _ exactly _ while we treat the interaction with the noise field perturbatively . we show that the unphysical term does not appear even when considering plane waves and without bounding the noise to a finite region . our analysis implies that the unphysical term , which is present in the standard first perturbative order , is canceled when higher order terms are included in the perturbative series . it appears fictitiously in the first perturbative analysis . we will comment on the picture which emerges from this analysis and on the relation between with the result of @xcite .

each particle interacts with a noise field that induces the collapse of its wave function . as a consequence of this interaction , when the particle is electrically charged , it radiates . as discussed in , the formula for the emission rate , to first perturbative order , contains two terms : one is proportional to the fourier component of the noise field at the same frequency as that of the emitted photon and one is proportional to the zero fourier component of the noise field . for this purpose , the electrodynamic part of the equation of motion is solved exactly while the part due to the noise is treated perturbatively .

spontaneous photon emission in the continuous spontaneous localization ( csl ) model is studied one more time . in the csl model each particle interacts with a noise field that induces the collapse of its wave function . as a consequence of this interaction , when the particle is electrically charged , it radiates . as discussed in , the formula for the emission rate , to first perturbative order , contains two terms : one is proportional to the fourier component of the noise field at the same frequency as that of the emitted photon and one is proportional to the zero fourier component of the noise field . as discussed in previous works , this second term seems unphysical . in , it was shown that the unphysical term disappears when the noises is confined to a bounded region and the final particle s state is a wave packet . here we investigate the origin of the unphysical term and why it vanishes according to the previous prescription . for this purpose , the electrodynamic part of the equation of motion is solved exactly while the part due to the noise is treated perturbatively . we show that the unphysical term is connected to exponentially decaying function of time which dies out in the large time limit , however , approximates to 1 in the first perturbative order in the electromagnetic field .

1011.0748

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how a specific trading mechanism effects on the price formation is one of the essential queries to understand the process and outcomes of exchanging assets under a given concrete rule . to investigate the issue , a lot of studies concerning the micro - structure of markets have been done in various research fields @xcite . recently , lots of on - line trading services on the internet were constructed by several major banks such as the sony bank @xcite . as the result , one can gather a lot of trading data sets to investigate the statistical properties extensively . as such studies , several authors focused on the fact that the sony bank uses a trading system in which foreign currency exchange rates change according to a first - passage process ( fpp ) @xcite . automatic forex trading systems such as the sony bank are now popular in japan where many investors use a scheme called _ carry trade _ by borrowing money in a currency with low interest rate and lending it in a currency offering higher interest rates . with these demands in mind , several studies have been done to investigate the stochastic process and made a model of it to reproduce the fpp in order to provide useful information for customers @xcite . the data sets of the sony bank rate @xcite are composed of time index and trading rate at that time . as we explained , a huge number of market data are reduced to a small amount of it , namely , the number of the sony bank rate is reduced by the first - passage process and unfortunately , the market rates behind the sony bank rate are not available for us . as well - known , there are several data sets whose price are determined by the so - called _ double - auction _ system . in the double - auction market , each trader ( investor ) posts his ( or her ) selling price or buying price in market order of a specific commodity with its volume to the market . then , the market maker determines the minimum price of buying orders , what we call _ ask _ , and the maximum price of selling orders , the so - called _ bid _ at each trading time and discloses these prices to the public . then , the difference between the bid and the ask is referred to as _ spread _ or _ bid - ask spread_. in market rates available for traders ( on the web for instance ) , there are two types of bid - ask spread , that is , ` constant ' or ` distributed ' , and which type of spread is disclosed depends on the market makers ( securities companies ) . results of market making , especially , statistical properties of the bid - ask spread might have an impact on the market and several studies have been done to reveal the relationship between the properties of bid - ask spread and behaviour of the market @xcite . for instance , madhavan , richardson and roomans @xcite proposed a phenomenological model to explain the price dynamics of double - auction market in market order , however , their model is apparently limited to the case in which the bid - ask spread remains constant during the price dynamics . therefore , much more extensive studies including empirical data analysis seem to be needed to investigate to what extent the model proposed by madhavan , richardson and roomans can explain the behaviour of market with _ stochastic _ bid - ask spread through some relevant quantity . in this paper , we investigate statistical properties of double - auction markets with bid - ask spread through the dynamical quantities such as response function . we first attempt to examine the so - called _ madhavan - richardson - roomans model _ ( mrr for short ) to simulate the stochastic process of the price - change in empirical data sets ( say , eur / jpy or usd / jpy exchange rates ) in which the bid - ask spread fluctuates in time . we find that the mrr theory apparently does not simulate so much as the qualitative behaviour ( ` non - monotonic ' behaviour ) of the response function calculated from the data sets . it is possible for us to show that a linear relationship @xmath3 between auto - correlation function @xmath2 and response function @xmath0 holds for the mrr model . namely , these two macroscopic quantities are related each other and the relationship should be explained from the microscopic view point as statistical physics provides the microscopic foundation of thermodynamics . moreover , we find that the linear relationship @xmath3 is apparently broken down in order - driven double - auction markets with fluctuating bid - ask spread . this fact tells us that on the analogy of physics , the phenomenological mrr model for the constant spread might be regarded as ` thermodynamics ' which usually deals with the macroscopic quantities such as price , auto - correlation , response functions and the relationship between them . it does not need to consider the detail behaviour of microscopic ingredients such as traders . in this paper , we show that the phenomenological model is apparently limited and fails to reproduce the dynamical quantities @xmath7 efficiently . hence , here we attempt to construct a kind of ` statistical mechanics ' in finance , which provides a microscopic foundation of phenomenological theory such as the mrr model . for this end , we utilize the minority game with a finite market history length having the distributed bid - ask spread to reproduce similar behaviour of macroscopic dynamical quantities as the empirical evidence shows . in our minority game modeling , we first fix each decision component ( buying : @xmath8 , selling : @xmath9 ) in their look - up tables before playing the game ( in this sense , the decision components are ` quenched variables ' in the literature of disordered spin systems such as spin glasses @xcite ) . we also consider the case in which a certain amount of traders update their decision components according to the macroscopic market history ( they ` learn ' from the behaviour of markets ) so as to be categorized into two groups with a finite probability ( in this sense , the components are now regarded as ` annealed variables ' ) . namely , at each round of the game , if the number of sellers is smaller / greater than that of buyers , a fraction of traders , what we call _ optimistic group_/_pessimistic group _ , is more likely to rewrite their own decision components from @xmath9/@xmath8 to @xmath8/@xmath9 . we find that the minority game modeling with the adaptive look - up table reproduces the non - linear relationship @xmath4 ( @xmath5 stands for a non - linear function leading to ` @xmath6 -shapes ' ) more effectively than fixed ( frozen ) look - up table does . this paper is organized as follows . in the next section [ sec : data ] , we explain our data sets and their format . we investigate their statistical properties . in the next section [ sec : data_analysis ] , we evaluate two relevant quantities , namely , the auto - correlation function @xmath2 and the response function @xmath0 , which are our key quantities to discuss the double - auction markets , for our data sets . we find that a linear relationship @xmath3 holds for the data sets having a constant bid - ask spread , however , the relation is broken for the data with a stochastic spread . in section [ sec : mrr ] , we introduce the so - called mrr model as a phenomenological model and derive the @xmath2 and @xmath0 . the difference between the mrr theory and the empirical evidence , the origin of the difference is discussed . in section [ sec : mg ] , we introduce and modify the minority game with a finite market history length and apply to explain the typical behaviour of the response function for the data with stochastic bid - ask spread . in section [ sec : adapt ] , we reveal that the minority game modeling with the adaptive ( ` annealed ' ) look - up table reproduces the non - linear relationship @xmath4 ( @xmath5 stands for a non - linear function leading to ` @xmath6-shapes ' ) more effectively than fixed ( ` quenched ' ) look - up table does . in section [ sec : discuss ] , we comment on the possible extensions of our approach . the last section is summary .

statistical properties of order - driven double - auction markets with bid - ask spread are investigated through the dynamical quantities such as response function . we first attempt to utilize the so - called _ madhavan - richardson - roomans model _ ( mrr for short ) to simulate the stochastic process of the price - change in empirical data sets ( say , eur / jpy or usd / jpy exchange rates ) in which the bid - ask spread fluctuates in time . we find that the mrr theory apparently fails to simulate so much as the qualitative behaviour ( ` non - monotonic ' behaviour ) of the response function ( denotes the difference of times at which the response function is evaluated ) calculated from the data .

statistical properties of order - driven double - auction markets with bid - ask spread are investigated through the dynamical quantities such as response function . we first attempt to utilize the so - called _ madhavan - richardson - roomans model _ ( mrr for short ) to simulate the stochastic process of the price - change in empirical data sets ( say , eur / jpy or usd / jpy exchange rates ) in which the bid - ask spread fluctuates in time . we find that the mrr theory apparently fails to simulate so much as the qualitative behaviour ( ` non - monotonic ' behaviour ) of the response function ( denotes the difference of times at which the response function is evaluated ) calculated from the data . especially , we confirm that the stochastic nature of the bid - ask spread causes apparent deviations from a linear relationship between the and the auto - correlation function , namely , . to make the microscopic model of double - auction markets having stochastic bid - ask spread , we use the minority game with a finite market history length and find numerically that appropriate extension of the game shows quite similar behaviour of the response function to the empirical evidence . we also reveal that the minority game modeling with the adaptive ( ` annealed ' ) look - up table reproduces the non - linear relationship ( stands for a non - linear function leading to `-shapes ' ) more effectively than the fixed ( ` quenched ' ) look - up table does .

1011.0748

i

in this paper , statistical properties of double - auction markets with bid - ask spread were investigated through the response function . we first attempted to utilize the so - called _ madhavan - richardson - roomans model _ ( mrr for short ) to simulate the stochastic process of the price - change in empirical data sets ( say , eur / jpy or usd / jpy exchange rates ) in which the bid - ask spread fluctuates in time . we found that the mrr theory apparently does not simulate so much as the qualitative behaviour ( ` non - monotonic ' behaviour ) of the response function calculated from the data . especially , we were confirmed that the stochastic nature of the bid - ask spread causes apparent deviations from a linear relationship between the @xmath0 and the auto - correlation function @xmath2 , namely , @xmath3 . to make the microscopic model of double - auction markets having stochastic bid - ask spread , we utilized the minority game with a finite market history length and found numerically that appropriate extension of the game shows quite similar behaviour of the response function to the empirical evidence . we also revealed that the minority game modeling with the adaptive ( ` annealed ' ) look - up table reproduces the non - linear relationship @xmath4 ( @xmath5 stands for a non - linear function leading to ` @xmath6-shapes ' ) more effectively than fixed ( ` quenched ' ) look - up table does . of course , there are still gaps between the theoretical prediction and the empirical evidence . we should modify our modeling of the double - auction market and figure out the micro - macro relationship in the market much more quantitatively .

especially , we confirm that the stochastic nature of the bid - ask spread causes apparent deviations from a linear relationship between the and the auto - correlation function , namely , . to make the microscopic model of double - auction markets having stochastic bid - ask spread , we use the minority game with a finite market history length and find numerically that appropriate extension of the game shows quite similar behaviour of the response function to the empirical evidence . we also reveal that the minority game modeling with the adaptive ( ` annealed ' ) look - up table reproduces the non - linear relationship ( stands for a non - linear function leading to `-shapes ' ) more effectively than the fixed ( ` quenched ' ) look - up table does .

statistical properties of order - driven double - auction markets with bid - ask spread are investigated through the dynamical quantities such as response function . we first attempt to utilize the so - called _ madhavan - richardson - roomans model _ ( mrr for short ) to simulate the stochastic process of the price - change in empirical data sets ( say , eur / jpy or usd / jpy exchange rates ) in which the bid - ask spread fluctuates in time . we find that the mrr theory apparently fails to simulate so much as the qualitative behaviour ( ` non - monotonic ' behaviour ) of the response function ( denotes the difference of times at which the response function is evaluated ) calculated from the data . especially , we confirm that the stochastic nature of the bid - ask spread causes apparent deviations from a linear relationship between the and the auto - correlation function , namely , . to make the microscopic model of double - auction markets having stochastic bid - ask spread , we use the minority game with a finite market history length and find numerically that appropriate extension of the game shows quite similar behaviour of the response function to the empirical evidence . we also reveal that the minority game modeling with the adaptive ( ` annealed ' ) look - up table reproduces the non - linear relationship ( stands for a non - linear function leading to `-shapes ' ) more effectively than the fixed ( ` quenched ' ) look - up table does .

1311.3666

i

the cosmic microwave background reveals the earliest observable structure in the universe , which dates back to the epoch of recombination at @xmath17 . the next most distant observable structure represents the population of galaxies and quasars at @xmath18 . the interval between @xmath19 and @xmath16 contains many landmark events , such as the formation of the first stars and galaxies , the growth of the first massive black holes and the reionization of the neutral hydrogen in the intergalactic medium ( igm ) . measurements by the _ wilkinson microwave anisotropy probe _ give a @xmath20 lower bound to the epoch of reionization of @xmath21 @xcite but do not pin down when reionization began or ended . another probe of the ionized state of the igm are spectra of high redshift quasars . already , the analysis of spectra of @xmath22 quasars suggest that the epoch of reionization ended around @xmath23 @xcite . while in recent years the evidence is pointing toward galaxies providing the required uv flux to reionize the universe ( e.g. * ? ? ? * ; * ? ? ? * ) , the identification and study of quasars up to the highest redshifts is very important for several reasons : ( 1 ) absorption line studies of cosmologically distributed intervening material allow the determination of the baryonic content and physical conditions ( metallicity , temperature , ionization state ) of the universe during the epoch of reionization ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) ; ( 2 ) the number density of high redshift quasars provide constraints on the mechanisms that are required to seed and grow @xmath24@xmath10 supermassive black holes less than a gyr after the big bang ( e.g. , * ? ? ? * ) ; and ( 3 ) high redshift quasars are generally located in luminous , massive host galaxies . by studying the host galaxies we get insight into the formation of galaxies in the early universe ( e.g. , * ? ? ? * ; * ? ? ? * ) . in the last decade more than sixty @xmath25 quasars have been discovered in various different surveys ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? the vast majority of these quasars have been discovered in optical surveys , in particular the sloan digital sky survey ( sdss ; e.g. , * ? ? ? the canada france high@xmath1 quasar survey ( cfhqs ; * ? ? ? * ) . the limitation of optical surveys that use silicon based ccd detectors is that they can not find quasars beyond @xmath26 as the sources become too faint in the reddest optical band of the survey ( @xmath1-band ) due to absorption by the intervening ly@xmath27 forest . to overcome this limit large area surveys in near - infrared bandpasses are necessary . one of the first wide - field near - infrared surveys reaching a depth that allowed a search for the highest redshift quasars was the uk infrared telescope infrared deep sky survey ( ukidss ) large area survey ( las ; * ? ? ? * ) . using the ukidss las , @xcite discovered the first quasar beyond @xmath28 , ulas j1120 + 0641 at @xmath29 . this one @xmath15 quasar already provides constraints on the neutral fraction of hydrogen up to @xmath29 @xcite , on the metallicity of the igm @xcite , and on the formation of dust and stars in the host galaxy @xcite . this shows that more quasars at @xmath28 are needed . here we present the first @xmath28 quasars discovered in the visible and infrared survey telescope for astronomy ( vista ) kilo - degree infrared galaxy ( viking ) survey . in section [ surveydata ] we give a short description of the viking survey . in section [ candidateselection ] we discuss the selection of high - redshift quasar candidates in the viking data , followed by the details of the imaging and spectroscopic observations of quasar candidates in section [ followup ] . we continue with an overview of the results of the follow - up observations in section [ results ] . in section [ properties ] we characterize the newly discovered quasars . in sections [ catcomp][colcomp ] we assess various sources of incompleteness and the effect on our quasar search . we discuss the implications of our results on the quasar luminosity function at @xmath15 in section [ qsolf ] . we conclude with a summary in section [ discussion ] . all magnitudes are given in the ab system . for magnitudes obtained with vista we use the vega to ab conversions provided on the cambridge astronomical survey unit ( casu ) website : @xmath30 , @xmath31 , @xmath32 , @xmath33 and @xmath34 . an @xmath35-dominated cosmology with @xmath36 , @xmath37 and @xmath38kms@xmath39mpc@xmath39 @xcite is adopted .

only one quasar has been discovered , namely the quasar ulas j1120 + 0641 , using near - infrared imaging . here we report the discovery of three new quasars in 332deg of the visible and infrared survey telescope for astronomy kilo - degree infrared galaxy ( viking ) survey , thus extending the number from 1 to 4 . the newly discovered quasars have redshifts of , , and .

studying quasars at the highest redshifts can constrain models of galaxy and black hole formation , and it also probes the intergalactic medium in the early universe . optical surveys have to date discovered more than 60 quasars up to , a limit set by the use of the-band and ccd detectors . only one quasar has been discovered , namely the quasar ulas j1120 + 0641 , using near - infrared imaging . here we report the discovery of three new quasars in 332deg of the visible and infrared survey telescope for astronomy kilo - degree infrared galaxy ( viking ) survey , thus extending the number from 1 to 4 . the newly discovered quasars have redshifts of , , and . the absolute magnitudes are between 26.0 and 25.5 , 0.61.1mag fainter than ulas j1120 + 0641 . near - infrared spectroscopy revealed the emission line in all three objects . the quasars are powered by black holes with masses of . in our probed redshift range of we can set a lower limit on the space density of supermassive black holes ofmpc . the discovery of three quasars in our survey area is consistent with the quasar luminosity function when extrapolated to . we do not find evidence for a steeper decline in the space density of quasars with increasing redshift from to .

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we used the viking near - infrared imaging survey to search for quasars above a redshift @xmath57 . we analyzed 332deg@xmath4 of viking data down to a median 7@xmath209 @xmath42-band depth of @xmath212 . we employed two sets of color criteria to select potential quasars at high redshift . our conservative color criteria is sensitive to quasars in the redshift range @xmath179 and with these criteria we selected 43 quasar candidates . our extended color criteria probed an increased redshift range of @xmath11 and selected an additional 42 objects . after we obtained optical imaging of 45 quasar candidates with the ntt , 6 objects remained having colors consistent with high redshift quasars . spectroscopic follow - up with vlt / fors2 of four of these sources identified three previously unknown quasars with redshifts of @xmath114 . the fourth object was a foreground star . the newly discovered quasars have the following characteristics : 1 . the absolute magnitudes of the new quasars are in the range of @xmath213 , which is about 12mag fainter than the @xmath22 quasars discovered in the sdss main survey @xcite and roughly similar to the quasars at @xmath22 found in the deep stripe of the sdss southern survey @xcite . the only other @xmath15 quasar published so far , j1120 + 0641 at @xmath29 , has an absolute magnitude of @xmath214 @xcite , 0.61.1mag brighter than the viking quasars presented in this article . 2 . near - infrared spectroscopy revealed that the new viking quasars exhibit emission lines commonly seen in lower redshift quasars , including @xmath1171397+]@xmath1171402 , @xmath1171549 , ] @xmath1171909 , and @xmath1172799 . furthermore , j23483054 shows continuum absorption blueward of the emission line and could be a bal quasar . 3 . using the infrared spectra , @xcite determined the mass of the central black hole powering the quasars . by fitting a gaussian to the line and measuring the continuum luminosity , they estimate central black hole masses of @xmath215@xmath10 . these black hole masses are very similar to those powering quasars at @xmath22 that were discovered in the sdss main survey ( e.g. , * ? ? ? * ; * ? ? ? * ) , but generally more massive than those of the @xmath22 quasars discovered in the cfhqs @xcite . ignoring any incompleteness in our quasar search we can set a lower limit to the density of supermassive black holes in the early universe . the discovery of three black holes with masses @xmath216@xmath10 in the 2.9comoving gpc@xmath182 probed in this study implies a minimum density of @xmath12mpc@xmath13 . to derive the space density of quasars in the volume probed by the viking data , we computed the efficiency of our selection criteria in order to correct for the incompletenesses in our quasar search . the volume density of quasars at @xmath11 that we derive agrees well with the extrapolated @xmath14 quasar luminosity function . this includes a decline in the space density of quasars of a factor 3 per unit redshift , which was previously obtained by measuring the density of bright quasars between @xmath198 . our results do not support a steeper decline in the space density of quasars with redshift . these findings imply that we should discover a significant number of quasars beyond @xmath57 in the remainder of the viking area . the total area that will be covered by viking will be 1500deg@xmath4 . if we ignore any incompleteness , we expect 1819 quasars between @xmath179 and @xmath821 quasars at @xmath11 in the viking area down to @xmath217 . with a larger number of quasars we will be able to constrain the evolution of the quasar luminosity function at @xmath15 . these new quasars are prime targets for various follow - up studies . in future articles we will study the emission lines that can be seen in the near - infrared spectra and examine the implications on the chemical enrichment near the black holes . the optical spectra will be used to derive constraints on the column density of neutral hydrogen along the line of sight . furthermore , the southern declination of the sources makes them ideal targets for alma observations to study the galaxies that host these quasars . funding for sdss - iii has been provided by the alfred p. sloan foundation , the participating institutions , the national science foundation , and the u.s . department of energy office of science . the sdss - iii web site is http://www.sdss3.org/. sdss - iii is managed by the astrophysical research consortium for the participating institutions of the sdss - iii collaboration including the university of arizona , the brazilian participation group , brookhaven national laboratory , university of cambridge , carnegie mellon university , university of florida , the french participation group , the german participation group , harvard university , the instituto de astrofisica de canarias , the michigan state / notre dame / jina participation group , johns hopkins university , lawrence berkeley national laboratory , max planck institute for astrophysics , max planck institute for extraterrestrial physics , new mexico state university , new york university , ohio state university , pennsylvania state university , university of portsmouth , princeton university , the spanish participation group , university of tokyo , university of utah , vanderbilt university , university of virginia , university of washington , and yale university . , g. b. , caldwell , m. , ward , a. k. , et al . 2006 , in society of photo - optical instrumentation engineers ( spie ) conference series , vol . 6269 , society of photo - optical instrumentation engineers ( spie ) conference series , m. j. , lewis , j. , hodgkin , s. , et al . 2004 , in society of photo - optical instrumentation engineers ( spie ) conference series , vol . 5493 , society of photo - optical instrumentation engineers ( spie ) conference series , ed . p. j. quinn & a. bridger , 411 , j. r. , irwin , m. , & bunclark , p. 2010 , in astronomical society of the pacific conference series , vol . 434 , astronomical data analysis software and systems xix , ed . y. mizumoto , k .- morita , & m. ohishi , 91 , j. r. , irwin , m. j. , hodgkin , s. t. , et al . 2005 , in astronomical society of the pacific conference series , vol . 347 , astronomical data analysis software and systems xiv , ed . p. shopbell , m. britton , & r. ebert , 599 , a. , goldoni , p. , royer , f. , et al . 2010 , in society of photo - optical instrumentation engineers ( spie ) conference series , vol . 7737 , society of photo - optical instrumentation engineers ( spie ) conference series , r. a. , burgasser , a. j. , bernstein , r. a. , et al . 2008 , in society of photo - optical instrumentation engineers ( spie ) conference series , vol . 7014 , society of photo - optical instrumentation engineers ( spie ) conference series , r. a. , burgasser , a. j. , bochanski , j. j. , et al . 2010 , in society of photo - optical instrumentation engineers ( spie ) conference series , vol . 7735 , society of photo - optical instrumentation engineers ( spie ) conference series

near - infrared spectroscopy revealed the emission line in all three objects . the quasars are powered by black holes with masses of . in our probed redshift range of we can set a lower limit on the space density of supermassive black holes ofmpc . we do not find evidence for a steeper decline in the space density of quasars with increasing redshift from to .

studying quasars at the highest redshifts can constrain models of galaxy and black hole formation , and it also probes the intergalactic medium in the early universe . optical surveys have to date discovered more than 60 quasars up to , a limit set by the use of the-band and ccd detectors . only one quasar has been discovered , namely the quasar ulas j1120 + 0641 , using near - infrared imaging . here we report the discovery of three new quasars in 332deg of the visible and infrared survey telescope for astronomy kilo - degree infrared galaxy ( viking ) survey , thus extending the number from 1 to 4 . the newly discovered quasars have redshifts of , , and . the absolute magnitudes are between 26.0 and 25.5 , 0.61.1mag fainter than ulas j1120 + 0641 . near - infrared spectroscopy revealed the emission line in all three objects . the quasars are powered by black holes with masses of . in our probed redshift range of we can set a lower limit on the space density of supermassive black holes ofmpc . the discovery of three quasars in our survey area is consistent with the quasar luminosity function when extrapolated to . we do not find evidence for a steeper decline in the space density of quasars with increasing redshift from to .

math0107002

i

the spectral scale was introduced by the present authors and nik weaver in @xcite and further developed by the authors in @xcite . it is defined for any finite set of self - adjoint operators in a finite von neumann algebra . the main theme in @xcite and @xcite is that full spectral information about real linear combinations of such operators is contained in the spectral scale and that much of this information is reflected by the geometry of the spectral scale . in the present paper we view the spectral scale from another perspective and thereby show that full information about the @xmath10numerical range of a finite dimensional operator is also contained in its spectral scale . thus , in this restricted case at least , two rather different sets of data are combined into one three dimensional , compact , convex set . although we restrict consideration in this paper to the finite dimensional world , we shall write this paper using the language of operators ( rather than matrices ) to emphasize how easily many of the concepts generalize to infinite dimensional situations . the notation developed here will be used throughout the rest of the paper . let @xmath0 denote a linear operator on an @xmath1dimensional complex hilbert space @xmath2 and let @xmath3 denote the normalized trace on @xmath4 , the algebra of all linear operators on @xmath2 . write @xmath17 for the subalgebra of @xmath4 generated by @xmath18 and the identity @xmath19 of @xmath4 and set @xmath5 and @xmath20 . the spectral scale was defined in @xcite and @xcite via the map @xmath21 defined by the formula @xmath22 and the spectral scale @xmath23 was defined as @xmath24 it is convenient for the present paper to view the second and third real coordinates as a single complex number . the definition of @xmath21 then becomes @xmath25 and we now define the spectral scale @xmath26 of @xmath0 by the formula @xmath27 thus , we now view @xmath7 as a subset of @xmath28 , rather than as a subset of @xmath29 . as shown in , @xmath30 , where for any self - adjoint subalgebra @xmath31 of @xmath4 , @xmath32 as noted above , the geometry of the spectral scale reflects spectral data for real linear combinations of @xmath33 , i.e. matrix pencil information . information of this sort is widely valued as documented in @xcite the basis for this current work is the observation that @xmath7 also essentially contains the @xmath10-numerical range of @xmath0 for each @xmath34 . since we shall show that several of the currently known properties of the numerical range generalize to the @xmath10numerical range , let us begin by reviewing these properties . recall that the * numerical range * of @xmath0 is by definition @xmath35 we use @xmath36 to denote the spectrum of @xmath0 . the following statements hold . 1 . @xmath37 is a compact , convex subset of @xmath38 . 2 . if @xmath39 and @xmath40 then @xmath41 $ ] . 3 . if @xmath0 is a normal operator , then @xmath37 is the convex hull of its eigenvalues . @xmath37 is a line segment if and only if @xmath42 , where @xmath12 and @xmath43 are complex numbers and @xmath44 is self - adjoint . the boundary of @xmath37 is the union of a finite number of analytic arcs . if @xmath45 and @xmath12 lies in the relative boundary of @xmath37 , then @xmath12 is a reducing eigenvalue of @xmath0 . the boundary of @xmath37 is contained in the real zero set of an algebraic curve . assertion @xmath46 is known as the _ toeplitz hausdorff theorem _ ( see @xcite and @xcite ) . a proof of this and the next 3 assertions can be found in @xcite . the assertion in ( 5 ) was proved independently by agler ( * ? ? ? * theorem 4.1 ) and narcowich ( * ? ? ? * corollary 3.5 ) . finally assertions ( 6 ) and ( 7 ) are due to kippenhahn @xcite . observe that if @xmath37 is a line segment , then it follows from parts ( 2 ) and ( 4 ) of theorem 0.1 that the endpoints of @xmath37 are reducing eigenvalues of @xmath0 . also , @xmath47 if and only if @xmath48 . in fact if @xmath37 is a line segment and @xmath49 is a vector in @xmath2 such that @xmath50 lies on an end point of @xmath37 , then @xmath51 and @xmath52 . to see this , note that by part ( 4 ) of theorem 0.1 , it suffices to show this when @xmath37 is contained in @xmath53 and @xmath54 . in this case we get that @xmath55 and so it has a positive square root . hence , we get @xmath56 and so @xmath57 . the numerical range has a * corner * at @xmath12 if @xmath37 has dimension two and there is more than one tangent line of support for @xmath37 at @xmath12 . the presence of a corner in the numerical range signals the fact that @xmath0 enjoys a special structure . further , it turns out that corners in @xmath37 must be * lineal * in the sense that there are two tangent lines of support which intersect the boundary in line segments of positive length . since we shall show that an analogous results hold for the @xmath10-numerical range , we now discuss this in more detail . recall that a point on the boundary of a convex subset of @xmath58 is said to be * singular * if the the boundary curve is not differentiable at this point . if @xmath37 has dimension two and the boundary of @xmath37 is singular at @xmath12 , then the following statements hold . 1 . @xmath37 has a corner at @xmath12 . 2 . @xmath12 is a reducing eigenvalue for @xmath0 . @xmath12 is lineal . let us begin by presenting a proof of the assertion in ( 1 ) . rotating and translating if necessary , we may assume that @xmath59 , @xmath37 lies in the upper half plane and that the positive imaginary axis intersects the interior of @xmath37 . in this case we may find a convex function @xmath60 defined on an open interval containing 0 whose graph gives a portion of the boundary of @xmath37 which contains 0 . since 0 is a singular point of the boundary , @xmath60 is not differentiable at 0 and since @xmath60 is convex , it follows that @xmath61 thus , the lines though @xmath62 with slopes @xmath63 are tangent to @xmath37 . kippenhahn established assertion ( 2 ) in ( * ? ? ? * satz 13 ) . the proof offered below seems to be new . it is convenient to rotate once more so that @xmath64 and @xmath37 lies in the right half plane so that the corner has the form shown below where both @xmath65 are tangent to @xmath37 at 0 , @xmath66 has positive slope and @xmath67 has negative slope as shown below . .2 in ( 2,2)(-2.5,0 ) ( -1.5,0)(1,0)2 ( -1.3,-1.25)(0,1)2.5 ( -1.3,0)(1,1)1 ( -1.3,0)(1,-1)1 ( .55,-.03)@xmath68 ( -1.33,1.32)@xmath69 ( -.28,1)@xmath66 ( -.28,-1.1)@xmath67 ( -1.34,-.04)@xmath70 ( -1.75,.05)@xmath71 1.8 in since @xmath37 is contained in the right half plane , we get @xmath72 and since there is a unit vector @xmath49 such that @xmath73 , we get @xmath74 by the remark following theorem 0.1 . we may now select @xmath75 so small that @xmath76 is also contained in the right half plane . arguing as above , we get that @xmath77 and so @xmath78 and therefore @xmath79 . hence , @xmath12 must be a reducing eigenvalue for @xmath0 . assertion @xmath80 is due to lancaster ( * ? ? ? * corollary 4 ) . observe that the proof above shows that if @xmath81 is any compact , convex , two dimensional subset of @xmath58 , then the boundary of of @xmath81 is singular at @xmath12 if and only if @xmath81 has a corner at @xmath12 . the @xmath10numerical range of @xmath0 is defined by the formula @xmath82 observe that when @xmath83 we have @xmath84 so that @xmath85 consists of a single point . as the results in this paper make clear it is natural to also include the case where @xmath86 and to define @xmath87 . since we have @xmath88 , this notion is a generalization of the standard numerical range . let us now review the basic properties of the @xmath10-numerical range . if @xmath89 , then the following statements hold . 1 . @xmath9 is a compact , convex subset of @xmath38 . 2 . @xmath90 . we have @xmath91 . if @xmath44 is self - adjoint with eigenvalues @xmath92 , @xmath93 and @xmath94 then @xmath95 $ ] 5 . if @xmath96 denotes the sum of the @xmath10 largest eigenvalues of @xmath15 , then the line @xmath97 is tangent to @xmath9 . the assertion in @xmath46 is due to berger who introduced the @xmath10numerical range in his thesis @xcite . a proof may be found in ( * ? ? ? * problem 167 ) . the assertion in @xmath98 follows from part @xmath46 and a simple calculation . assertion @xmath80 follows from the fact that @xmath99 . the proof of assertion @xmath100 is straight forward . for assertion @xmath101 observe that @xmath102 so that @xmath103 for some real @xmath104 . further , since @xmath96 is the sum of the @xmath10 largest eigenvalues of @xmath15 , if @xmath105 , then @xmath106 . let us now describe our results in more detail . the key to understanding the role played by the @xmath10-numerical range in the spectral scale is the notion of an _ isotrace slice _ of the spectral scale . if @xmath107 , then the * isotrace slice * of @xmath7 at @xmath108 is by definition @xmath109 we prove in theorem 1.3 that if @xmath110 and we define the map @xmath111 from @xmath38 to @xmath28 by @xmath112 , then @xmath111 is an affine map that is a bijection from @xmath9 onto @xmath113 . thus , we may view the @xmath10-numerical range as a subset of the spectral scale . it is also shown in section 1 ( theorem 1.1 ) that the extreme points of @xmath7 lie on the isotrace slices of the form @xmath113 for @xmath114 and so @xmath7 is the convex hull of this finite collection of sets . in section 2 we present some examples ( and pictures ) of various spectral scales . section 3 contains our results on corners in the @xmath10-numerical range . it is shown that several of the known facts about corners on the boundary of @xmath37 generalize to @xmath9 by using the additional structure provided by the spectral scale . for example , in theorem 3.4 , we show that a singularity on the boundary of @xmath9 always occurs at an isolated extreme point of @xmath9 and such points correspond to central projections in the algebra @xmath17 , i.e. reducing subspaces for @xmath0 . we also show that @xmath0 is normal if and only if @xmath9 is a polygon for @xmath115 ( theorem 3.5 ) . sections 4 and 5 are devoted to establishing that the boundary of the @xmath10-numerical range is the finite union of line segments and curved real analytic arcs ( theorem 5.4 ) . in section 4 we review some classical background which is required for our analysis and then use this to derive our results in section 5 . section 6 contains some open question , stated as conjectures . as our results show , the spectral scale provides a new way to study @xmath1-tuples of self - adjoint finite dimensional operators . in the case under study here , when there are just two operators so that @xmath7 is a subset of three dimensional real euclidean space ( or @xmath116 ) , we may actually visualize @xmath7 as shown in the examples and pictures in section 2 . these pictures were created using a matlb program written by jeff duzak as part of an reu research project supervised by the second author . readers can contact the second author for a copy of this program which is quite useful for testing conjectures .

suppose that is a linear operator acting on an-dimensional complex hilbert space , and let denote the normalized trace on . set and , and write for the the spectral scale of with respect to . we show that contains full information about , the-numerical range of for each . for example , we show in theorem 3.4 that the point is a singular point on the boundary of if and only if is an isolated extreme point of . in this case we show in theorem 3.5 , that is normal if and only if is a polygon for each . finally , it is shown in theorem 5.4 that the boundary of the-numerical range is the finite union of line segments and curved real analytic arcs .

suppose that is a linear operator acting on an-dimensional complex hilbert space , and let denote the normalized trace on . set and , and write for the the spectral scale of with respect to . we show that contains full information about , the-numerical range of for each . we then use our previous work on spectral scales to prove several new facts about . for example , we show in theorem 3.4 that the point is a singular point on the boundary of if and only if is an isolated extreme point of . in this case , where is a central projection in in the algebra generated by , and the identity . we show in theorem 3.5 , that is normal if and only if is a polygon for each . finally , it is shown in theorem 5.4 that the boundary of the-numerical range is the finite union of line segments and curved real analytic arcs .

1307.7785

i

coevolution of the dynamics and topology of networks is widely observed in diverse systems from cellular biology to social networks @xcite . in the brain , the spiking dynamics of neurons depends on how they are connected . on the other hand , the connectivity can be modified by the spiking activity . the connections ( synapses ) between neurons in many brain areas are modified according to the spike - timing - dependent plasticity ( stdp ) rule @xcite . a most common stdp rule for excitatory neurons is as follows @xcite : the connection strength ( synaptic weight ) from neurons a to b is strengthened if a spikes before b ( long - term potentiation , or ltp ) , and weakened if a spikes after b ( long - term depression , or ltd ) . the amount of modification decreases with the time difference between the spikes of a and b. the connection between two neurons is lost if the synaptic weight is reduced below a threshold ; conversely , it can be established through consistent parings of the spikes of the neurons . studies that use stdp in spiking neural networks have shown a number of emergent properties @xcite . in this paper , we show that synfire chain connectivity @xcite , in which subsequent groups of neurons are connected into a feedforward network that supports sequential spiking of the neurons , emerges through coevolution of the spiking activity and the connectivity across many presentations of a training stimulus to a subset of neurons ( training neurons ) . sequential spiking of neurons is observed in a number of brain areas @xcite . some of the strongest experimental evidence for synfire chains producing spike sequences is from zebra finch premotor nucleus hvc ( proper name ) @xcite . projection neurons in hvc spike sequentially at precise times relative to the learned song @xcite . consistent with the synfire chain dynamics , cooling hvc uniformly slows down the song @xcite , and the sub - threshold membrane potentials of neurons rapidly depolarize 5 - 10 ms before they spike @xcite . it is well - established that synfire chains robustly produce spike sequences @xcite . however , how neurons are wired into synfire chains is not well - understood . an intriguing possibility is that synfire chains self - organize through activity - driven synaptic plasticity . earlier studies using stdp or similar hebbian rules resulted in short chains with a few groups @xcite . the most likely reason is that these rules are prone to producing unstable growth of connections @xcite . two recent studies introduced additional homeostatic synaptic rules to limit such instability , and showed that long synfire chains can form @xcite . the key idea behind both studies is to restrict the connectivity of the network after a certain amount of growth has occurred . fiete et al achieved this by limiting the total synaptic weight in and out of every neuron @xcite . however , a study using large scale simulations and mean - field analysis suggested that the regulation of the total synaptic weights does not prevent unstable network growth @xcite . in @xcite ( jun - jin model ) , we took a different approach , and imposed an axon remodeling rule that limits the number @xmath0 of strong connections , defined as those with synaptic weights above a threshold , that one neuron can maintain . reaching the limit leads to pruning of all weak connections from the neuron . there are two additional features of the jun - jin model . first , the model includes an gradual , activity - independent decay of synaptic weights , which we call potentiation decay . second , an activation threshold switches synapses on or off depending on the magnitude of the synaptic weights . this rule allows the active connections between neurons to form or disappear as the synaptic weights are modified . simulations of 1000 leaky integrate - and - fire neurons showed that synfire chains emerge from initially random active connections when 6 to 40 training neurons are intermittently activated by external inputs for many trials . the number of neurons in each group roughly equals to @xmath0 ( set to 10 in the simulations ) , and is not affected by the number of training neurons except for the first 2 - 3 groups @xcite . in this paper , we perform an in - depth analysis of the jun - jin model . we address unresolved fundamental questions such as what determines the lengths of the emergent chains and how the length distributions are influenced by the total number of the neurons ( network size ) . we establish that , when the network is randomly active , the synaptic plasticity rules in this model allow the network connectivity to fluctuate , but the synaptic weights remain in a statistically stationary distribution . this ensures that the chain formation does not depend on specific initial network connectivity . we demonstrate that in between training trials , sequential spikes can spontaneously emerge in the forming chain . this noise - induced re - activation of the chain creates connections from neurons outside of the chain to those in the chain , and plays a critical role in determining the length distributions of the final chains . most notably , there is an upper limit for the mean chain length as the network size becomes large . we show that slow potentiation decay leads to short chains with narrow length distributions , while fast potentiation decay leads to long chains with a wide length distributions . we compare the results of network simulations to a lottery - type stochastic process in which neurons are selected iteratively to enter a chain , and the chain stops growing when a loop is formed , either by selecting neurons already in the chain or by selecting neurons connected to the chain . the distribution of chain lengths from the network simulations fits well with distributions generated by the lottery process . the analysis of this simple growth model shows that the rate of potentiation decay influences the chain length distributions by controlling the emergence of the connections to the growing chain . these connections also lead to a finite limit for the mean chain length as the network size increases .

synfire chains are thought to underlie precisely - timed sequences of spikes observed in various brain regions and across species . how they are formed is not understood . here potentiation decay is the gradual , activity - independent reduction of synaptic weights over time . we show that potentiation decay enables a dynamic and statistically stable network connectivity when neurons spike spontaneously . we demonstrate that chain length distributions depend on the potentiation decay . fast potentiation decay leads to long chains with wide distributions , while slow potentiation decay leads to short chains with narrow distributions .

synfire chains are thought to underlie precisely - timed sequences of spikes observed in various brain regions and across species . how they are formed is not understood . here we analyze self - organization of synfire chains through the spike - timing dependent plasticity ( stdp ) of the synapses , axon remodeling , and potentiation decay of synaptic weights in networks of neurons driven by noisy external inputs and subject to dominant feedback inhibition . potentiation decay is the gradual , activity - independent reduction of synaptic weights over time . we show that potentiation decay enables a dynamic and statistically stable network connectivity when neurons spike spontaneously . periodic stimulation of a subset of neurons leads to formation of synfire chains through a random recruitment process , which terminates when the chain connects to itself and forms a loop . we demonstrate that chain length distributions depend on the potentiation decay . fast potentiation decay leads to long chains with wide distributions , while slow potentiation decay leads to short chains with narrow distributions . we suggest that the potentiation decay , which corresponds to the decay of early long - term potentiation of synapses ( e - ltp ) , is an important synaptic plasticity rule in regulating formation of neural circuity through stdp .

1307.7785

c

in large recurrent networks with stdp , axon - remodeling and an activity - independent potentiation decay of synapses , we observed emergence of long , stereotypical sequences of spikes . the sequences are produced by stable synfire chain topologies that self - organize via a stochastic growth process . we studied the distributions of synfire chain lengths and concluded that the rate of potentiation decay in our synaptic plasticity model primarily controls the shape of the distributions . the chains develop in response to a stimulus presented to the network in a dynamic ground state , in which the distribution of synaptic weights is invariant to synaptic modifications due to spontaneous activity on the network . this network state would not exist without the potentiation decay . synfire chain growth in our network model results from a global response of the connectivity among the neurons to a stimulus that targets only a small subset of the population , the training neurons . repeated stimulations of the training neurons leads to iterative growth of a synfire chain embedded in the network . this result was expected based on previous work @xcite . however , what was not expected , but what we observed , is global response of the connectivity as the chain develops . as the sequence begins to emerge , neurons in the pool are increasingly likely to target the neurons in the chain . we suggest this process of targeting the strongly connected neurons in the chain is loosely analogous to preferential attachment in other complex networks @xcite . in contrast to other systems with preferential attachment , a scale - free distribution does not emerge from training because of the topological constraint imposed on the network by axon remodeling . the complex response of synapses throughout the network illustrates co - evolution of spike activity ( the emerging sequence ) and synaptic topology ( preferential targeting ) . we expect this observation generalizes to other recurrent network models with stdp in which spike sequences emerge . since pre - post synaptic strengthening is a common feature of stdp models , other neurons will attach to sequence members when a sequence is initiated by spontaneous activity . we believe our insight may explain the observation of neuron clustering @xcite and small - world network degree distributions @xcite in other studies where the number of strong connections a neuron can make is unconstrained . the coevolution of the network activity and network connectivity in response to an external stimulus is reflected in the spectrum of length distributions of the synfire chains . when the potentiation decay is too slow to sufficiently reduce the weights of connections from pool neurons onto a partially formed chain , the variation of chain lengths is reduced . when potentiation decay is fast , the number of preferential connections is reduced and the synfire chain has an opportunity to grow longer . we contrast our mechanism for synfire chain development with other studies in which chains emerge in a recurrent network , such as in fiete , et . al . @xcite . in fiete s model , the synaptic plasticity rules are designed in such a way that each neuron ( or group of neurons receiving correlated external input ) must connect to one other neuron ( or neuron group ) that is not already targeted . the selection of target is random , which leads to multiple closed loops where every neuron ( or neuron group ) is incorporated into a loop . the distribution of chain lengths in this model follows a power law . hence , short chains are more numerous than long chains . in contrast , the distribution of chain lengths in our model is close to a skewed gaussian . there are typical chain lengths , and short chains are rare . in our model , not every neuron is part of the chain . we introduced a growth model that incorporates preferential targeting to confirm the general form of the length distributions of the chains . the growth model is verified with corresponding simulations of networks producing single - neuron chains . the model illustrates tuning of the length distribution through the potentiation decay rate . furthermore , it predicts that the mean chain length approaches a constant in the limit of large network size . simulations of the more complex process of synfire chain growth confirm the same saturation effect ( fig . [ fig_synfire_sims ] ) . this is in contrast to the case of @xmath49 , for which the mean length diverges as @xmath30 . any small preferential targeting probability @xmath31 limits the mean length as @xmath29 . this result indicates that chain size is bounded softly , even in the limit of very large networks . it would be interesting to confirm this plateau effect in recurrent networks larger than those we were able to simulate . in at least one case we know of @xcite , a much larger network has been simulated . however , chains did not emerge upon externally stimulating the network in this study . contrasting the result of this study with our own , we have validated the iterative recruitment of synfire chain groups using a power - law stdp rule @xcite instead of the additive ltp / multiplicative ltd model ( see eqs . ( [ ltp ] ) and ( [ ltd ] ) ) introduced by @xcite . additionally , we observe the growth process is unaffected when setting the number of allowed strong connections to larger values ( 50 instead of 5 used in the simulations presented in the results ) , demonstrated also in @xcite . key differences that may account for the emergence of chains in our model are , dually , the vast restructuring of the network connections allowed by imposing an activation threshold on each synaptic weight , and also restricting the influence of a single neuron by imposing the axon - remodeling rule . before training , networks are initialized to a dynamic ground state . the distribution of synaptic weights in a ground state network is stationary while neurons are spontaneously active . synapses are activated and silenced by random activity , and the average flux of weights across the active threshold is zero . our synaptic dynamics model is distinguished from others in two ways . one , we impose an activation threshold on a synaptic weight between every pair of neurons in the network . the picture that emerges is one in which neurons are actively connecting and disconnecting to other neurons in the population freely and on a relatively short time scale , minutes to hours . a number of imaging studies support this fast restructuring of network connectivity patterns @xcite . the time scale of emergence and subsequent withdrawl of dendritic spines can be as short as ten minutes and has been linked to synaptic activity @xcite . network rewiring is not permitted in all but a few network growth models that have been proposed @xcite . instead , it is much more common to select _ a priori _ the postsynaptic targets of each neuron . we argue that this modeling choice neglects an important feature of biological networks and places limits the emergent topology of the network . two , we subject all synapses to an activity - independent decay . we propose that this is related to the widely observed decay of the early phase long - term potentiation ( e - ltp ) . the time scale of the potentiation decay is several hours @xcite , much longer than the length of an individual training trial . the role of the potentiation decay is to avoid the accumulation of random potentiation of synaptic weights known to destabilize the network dynamics @xcite . the combination of these two rules yields a robust spectrum of stationary network states . as a final note , we compare our network rewiring rules with a similar approach taken by iglesias et al @xcite . in this study a recurrent network is initialized with all - to - all connectivity and network connections are eliminated via stdp . this modeling choice is also notable in that final network connectivity is not limited by only modifying weights between specific pairs of neurons . however , it is unclear from the results of this study whether sequences emerge after pruning . the stability of the ground state network indicates the rules of our model encode a homeostatic mechanism @xcite . several other models of homeostatic mechanisms have been proposed recently , including a sliding modification threshold based on post - synaptic firing history @xcite , a dependence on fluctuations of post - synaptic membrane potential @xcite and heterosynaptic plasticity that limits the total weight targeting a single neuron @xcite . an overlooked mechanism that we propose is activity - independent , multiplicative rescaling of weights , as we have implemented here , potentiation decay or decay of e - ltp . this form of ltp returns synaptic efficacies to the baseline within 3 hours @xcite and is independent of protein synthesis . only through repeated potentiation , e - ltp can turn into the late phase ltp ( l - ltp ) , which is maintained by protein synthesis and can last over days and weeks @xcite . in our model , consistently potentiated synapses turn into supersynapses whose decay is much weaker than other active synapses . the supersynapses can be considered in the l - ltp state . emergence of the synfire chain relies on stabilization of the small percentage of supersynapses , while there are many weaker , more transient synapses . this long - tail synaptic weight distribution is consistant with physiological observation @xcite and appear in other theoretical studies @xcite . the functional role of e - ltp decay is largely ignored in the ltp literature . our model suggests that the e - ltp decay may be crucial in stabilizing synaptic weight distribution against random accumulations of ltp through spontaneous activity . moreover , the e - ltp decay can be important in the formation of functional networks through stdp . the time scale of the potentiation decay in our model is congruent with time scales of e - ltp decay , which can be seen with an order of magnitude estimate . if we assume learning occurs on a scale of tens of days and @xmath1 training trials are necessary for synaptic chains to crystalize , this places the time scale corresponding to our decay parameter @xmath15 , which we vary from 0.9 to 0.99 , in a range of @xmath50 and @xmath1 seconds . it will be interesting to test these ideas by manipulating the e - ltp decay constants in developing or learning brains _ in vivo_. a natural extension of this work is to construct a growth model in such a way that more complex asymptotic synaptic topologies emerge . many learned motor behaviors can be complex . for example , stochastic ordering of distinct elements of a behavior is one kind of behavioral complexity . the song of the bengalese finch can be described by this type of stochastic process @xcite . a single synfire chain can not capture this complexity since they produce only a single spike sequence ; multiple chain or branching chains would be necessary . one possible scenario for growing multiple chains in the same network would be to have distinct sets of training neurons . however , it turns out that preferential targeting of sequence members within the network prohibits the development of distinct synaptic structures . we implemented two distinct training groups of 5 neurons in a network of 2500 neurons . a training neuron set that is excited at the beginning of each learning trial . which of the two sets is selected at random with equal probability . in fig . [ two_training ] we display the resulting growth . the chains develop several groups individually , but ultimately they merge to a single chain . initially , the training neurons seed two disjoint sequences that recruit targets iteratively . emergence of synfire groups embeds two disjoint sequences in the network . these sequences are occasionally activated by spontaneous activity . therefore , neurons in the pool will target the partial chains preferentially . in particular , neurons at the end of one of the chains may target a neurons in the other chain with elevated probability . once one chain reliably activates neurons in the other chain , they will merge . merging occurs reliably each time we simulated two training groups . other growth mechanisms must be present , or the chains must be encoded within distinct populations of neurons . this conclusion is consistent with other studies , such as @xcite .

we analyze self - organization of synfire chains through the spike - timing dependent plasticity ( stdp ) of the synapses , axon remodeling , and potentiation decay of synaptic weights in networks of neurons driven by noisy external inputs and subject to dominant feedback inhibition . periodic stimulation of a subset of neurons leads to formation of synfire chains through a random recruitment process , which terminates when the chain connects to itself and forms a loop . we suggest that the potentiation decay , which corresponds to the decay of early long - term potentiation of synapses ( e - ltp ) , is an important synaptic plasticity rule in regulating formation of neural circuity through stdp .

synfire chains are thought to underlie precisely - timed sequences of spikes observed in various brain regions and across species . how they are formed is not understood . here we analyze self - organization of synfire chains through the spike - timing dependent plasticity ( stdp ) of the synapses , axon remodeling , and potentiation decay of synaptic weights in networks of neurons driven by noisy external inputs and subject to dominant feedback inhibition . potentiation decay is the gradual , activity - independent reduction of synaptic weights over time . we show that potentiation decay enables a dynamic and statistically stable network connectivity when neurons spike spontaneously . periodic stimulation of a subset of neurons leads to formation of synfire chains through a random recruitment process , which terminates when the chain connects to itself and forms a loop . we demonstrate that chain length distributions depend on the potentiation decay . fast potentiation decay leads to long chains with wide distributions , while slow potentiation decay leads to short chains with narrow distributions . we suggest that the potentiation decay , which corresponds to the decay of early long - term potentiation of synapses ( e - ltp ) , is an important synaptic plasticity rule in regulating formation of neural circuity through stdp .

1205.2138

r

the nucleus @xmath100120 is predicted to be a spherical doubly magic nucleus in cdft @xcite . its potential energy surface in the @xmath101 plane is shown in fig . [ pes-2d-6panel ] . it is interesting to compare it with the pes of the nucleus @xmath102pu shown in fig.[pes-2d ] . these two pes s are representative examples of typical pes s in actinides and superheavy nuclei . the gross structure of these two pes s is defined by the fact that the total energy is generally increasing when moving away from the @xmath103 axis ; so it looks like a canyon . however , there are local structures inside the canyon which define the differences between the two mass regions with respect to the impact of triaxiality on the inner and outer fission barriers . in @xmath102pu , a large hill is located at the axial shape @xmath104 inside a canyon . as a consequence , the fission path from the normal deformed minimum initially proceeds along the @xmath105 axis , then bypasses the axial @xmath68=0.5 hill via a path with @xmath106 , and then proceeds along the bottom of the canyon on an axially symmetric path again . as a result of this bypass , the inner fission barrier heights of the actinides are lowered by @xmath107 mev due to triaxiality @xcite . however , the calculated outer fission barriers of the actinides are not affected by triaxiality @xcite . nucleus obtained with the nl3 * parametrization . the black solid , red dashed and blue dot - dashed lines display the deformation energy curves for the axially symmetric , triaxial and axial octupole deformed solutions . we show the deformation energy curves for the last two solutions only in the range of @xmath68 values where it is lower in energy than the deformation energy curve of the axially symmetric solution . note that the tr - b fission path via saddle at @xmath108 ( see text for details ) is not shown since this saddle is lower than the one of the ax fission path only by 0.2 mev ( see table [ tab5 ] ) ; the results of ref.@xcite suggests that spontaneous fission exploiting this path is less likely than the one along the ax fission path.,width=302 ] the properties of the pes of the nucleus @xmath100120 , defining the fission pathes , are completely opposite to the case of @xmath102pu since there are two triaxial and one axial hills inside the pes canyon of @xmath100120 . two triaxial hills are located at moderate deformations ( @xmath109 ) , while the axial hill is superdeformed ( @xmath110 ) . the fission path ( shown by red dashed line in fig.[pes-2d-6panel ] ) starts at a spherical shape , then proceeds between two triaxial hills ( @xmath11130@xmath112 ) and bypasses the axial hill at @xmath1130.75 via a @xmath114 path . the @xmath115-softness of the pes , which exists between the two triaxial hills , has only a minor effect on the height of the inner fission barrier ; the triaxial solution is lower than the axial one by 100 - 200 kev at @xmath116 deformations ( see fig . [ pot ] ) . however , this figure shows that _ along this fission path _ the height of inner fission barrier is not affected by triaxiality . the second fission path shown by a solid red line starts at spherical shape , and proceeds along the axially symmetric @xmath117 axis , via a saddle point at ( @xmath118 , @xmath119 ) and then along the first fission path after second minimum . the existence of the valley between the walls of the canyon and triaxial hills at ( @xmath120 ) is a necessary condition for the existence of this path . this type of fission path exists in a number of nuclei , so for convenience we will call it as triaxial tr - b fission path . the unusual physical feature of this second path is the fact that initially the nucleus has to be squeezed along the axis of symmetry , thus creating an oblate nucleus with quadrupole deformation @xmath121 . this is contrary to the usual picture of fission where the prolate or near - prolate nucleus is stretched out along the path of increasing quadrupole deformation . note that the saddle of this fission path is only by 0.2 mev lower than the saddle of the first fission path ( see table [ tab5 ] ) and in addition it is much longer . therefore , considering a dynamical calculation taking into account the action integral along the entire fission path @xcite this path will probably not contribute much to the fission probability . contrary to the actinides , the triaxiality has a considerable impact on the shape and the height of outer fission barrier which is lowered by @xmath122 3 mev in @xmath100120 nucleus ( see fig . [ pot ] ) . note , that in this nucleus , the lowering of outer fission barrier due to octupole deformation is substantially smaller than the one due to triaxiality . it is also evident that the landscape of the pes and the existence of saddle points and valleys depends on the proton and neutron numbers . this is clearly seen in fig . [ pes-2d-6panel ] and in table [ tab5 ] . in the @xmath123 isotopes , the increase of neutron number up to @xmath124 and beyond leads to the emergence of an axial hill at @xmath125 ; this is clearly visible in the nucleus @xmath126120 in fig . [ pes-2d-6panel ] . its existence in high-@xmath127 and neutron - rich nuclei leads to the shift of the first fission path in the deformation space . in lighter systems this path proceeds via the axially symmetric saddle as , for instance , in the nucleus @xmath100120 in fig . [ pes-2d-6panel ] . for heavier systems it starts at spherical shape mev relatively to the superdeformed minimum ) which suggests that these states are extremely unstable against fission . thus , we start the discussion from spherical / weakly(normal)-deformed minima which have better survival probability against fission . ] , proceeds along the axially symmetric @xmath117 axis up to @xmath128 and then via a saddle point at ( @xmath129 ) , located between a triaxial ( @xmath130 ) below . ] and an axial @xmath131 hill , to the axially symmetric superdeformed minimum . this type of fission path exists in a number of nuclei . we will label it as triaxial tr - a for convenience . in addition , there is a second triaxial path tr - b via a saddle at @xmath132 , which is similar to the one present in the nucleus @xmath100120 . the structure of the pes of the nucleus @xmath133116 is similar to the one of the nucleus @xmath126120 in fig . [ pes-2d-6panel ] . the lowering of neutron number below @xmath124 in the nuclei under study leads to the situation that the fission path via the axially symmetric inner saddle ( called as ax below ) becomes energetically favored . this is clearly seen in the pes of the @xmath134116 , @xmath135112 and @xmath136112 nuclei in fig . [ pes-2d-6panel ] and in table [ tab5 ] . it is clear that the landscape of the pes of superheavy nuclei in the region of the inner fission barrier is more complicated than in the case of the actinides . the energetically favored fission path in the majority of these nuclei proceeds through an axially symmetric saddle ( see fig . [ pes-2d-6panel ] and table [ tab5 ] ) . only in a few nuclei ( @xmath137120 , @xmath138 and @xmath133116 ) , triaxial fission path tr - a is lower in energy than axial ax fission path . the triaxial fission path tr - b exists in the majority of the nuclei apart from neutron - poor @xmath139 nuclei and @xmath140 nuclei ( see fig . [ pes-2d-6panel ] and table [ tab5 ] ) . the energy of the saddle point of this fission path is lower than the one of the first fission path ( either ax or tr - a ) in a number of nuclei . however , the results of calculations within the mm method @xcite suggest that although triaxiality lowers the static fission barriers , it plays a minor role in spontaneous fission of superheavy nuclei with @xmath141 . this is because a fission path via an oblate shape and triaxial saddles is substantially longer as compared to the axially symmetric path which leads to a significant reduction of the penetration probability . however , this complexity of the pes and the presence of two fission pathes for the inner fission barriers calls for finding the dynamical path along which the fission process takes place in cdft . work in this direction is in progress . figs . [ pot3a ] and [ pot3b ] show deformation energy curves for several isotopes with charge numbers @xmath142 and 116 for three classes of cdft models . experimental estimates of the inner fission barrier heights were obtained for these nuclei in ref . the potential energy structure of these nuclei is similar to the one seen in @xmath100120 ( fig . [ pes-2d-6panel ] ) ; the presence of two triaxial hills at moderate deformations ( @xmath143 ) defines two possible fission pathes : one between the hills ( either ax or ax - tr pathes ) and another between the hill and the walls of the pes canyon ( tr - b path ) . the @xmath115-softness of the pes , which exists between the two triaxial hills , defines whether the fission path passing between these two hills is fully axially symmetric ( ax path ) or has some degree of triaxiality ( ax - tr path ) . the ax - tr path is generally similar to the ax path but differs from it by the fact that moderate triaxiality @xmath144 appears at the saddle and along the shoulder of inner fission barrier . although the heights of the discussed hills in the region of inner fission barrier depends on the parametrization , the observed features of the potential energy surfaces and fission pathes are rather independent of the parametrization ( fig . [ pes-2d - dif - param ] ) . tables [ tab5 ] and [ tab7 ] show that the saddle point along this path has triaxiality only in the case of the two nuclei @xmath145114 . however , even in these cases the ax saddle is only 50 - 150 kev higher than the tr - a saddle . thus , similarly to the @xmath100120 nucleus triaxiality only marginally affects the inner fission barriers . note that the ground states are somewhat deformed in these nuclei ( see figs . [ pot3a ] and [ pot2]a ) . for these nuclei triaxiality has a considerable impact on the shape and height of the outer fission barriers ; the decrease of the heights of the outer fission barriers due to triaxiality is typically in the range of 1.5 - 2.0 mev and it depends on the particle number and on the rmf parametrization . note that the outer fission barriers are affected also by octupole deformation . in the @xmath140 nuclei , the triaxial saddle is lower in energy than the octupole saddle . the situation is reversed in the @xmath146 nuclei . among the different classes of cdft models , the dd - me2 parametrization always gives the highest values for the inner and outer fission barrier heights . they are ( on average ) by 1 and 1.5 mev higher than the ones obtained in the nl3 * and the dd - pc1 parametrizations . the heights and the shapes of the inner fission barriers are very similar for the nl3 * and dd - pc1 parametrizations . the outer fission barriers also come close to each other in these two parametrizations in axial [ a ] , triaxial [ t ] and octupole [ o ] ( for @xmath140 nuclei ) calculations . however , with increasing of @xmath127 from 114 to 116 the difference between the results of octupole calculations increases ; the difference between the energies of the octupole saddles obtained in the nl3 * and dd - pc1 parametrizations reaches 2 mev in the @xmath147 nuclei ( see fig . [ pot3a ] ) . nuclei as a function of neutron number . ( panel ( b ) ) the energies of axial ( large solid symbols , thick lines ) and triaxial ( either ax - tr or tr - a ; shown by small open symbols and thin lines ) saddles with respect to the spherical / weakly(normal)-deformed minima in these nuclei as a function of neutron number . note that the same type of symbols is used for axial and triaxial saddles in a given isotope chain.,width=302 ] there is only one experimental work @xcite where estimates on the heights of inner fission barriers in superheavy nuclei with @xmath148 112 , 114 , and 116 have been obtained . unfortunately , experimentally the fission barriers are accessible only indirectly and a model - dependent analysis is used to obtain these quantities , which causes an ambiguity in the comparison with theoretical results . even in the actinide region where the fission barrier heights were extracted from a number of independent experiments with high statistics ( see , for example , ref . @xcite ) , a typical uncertainty in the experimental values , as suggested by the differences among various compilations , is of the order of @xmath149 mev @xcite . these uncertainties are expected to be higher in superheavy nuclei since the estimates of ref . @xcite are based on experimental data represented by low statistics and on a method which differs from the methods used in the analysis of fission barrier heights in actinides . in addition , there is no independent confirmation of the inner fission barrier height estimates of ref . the interpretation of experimental data based on cross sections in terms of fission barrier heights becomes even more complicated when the fission path has a double hump structure , which according to many calculations may be the case in superheavy nuclei . the widening of the barrier due to the second hump ( or its remnant ) would require the lowering of the inner fission barrier height ; this possibility has not been taken into account in the analysis of ref . based on this discussion it is clear that the level of confidence of fission barrier height estimates for superheavy nuclei is significantly lower than the one for the actinides . according to ref . @xcite , the estimated lower limits for fission barrier heights in even - even @xmath150 and 116 nuclei shown in fig . [ pot3a ] and [ pot3b ] are 5.5 , 6.7 and 6.4 mev , respectively . our results for the heights of inner fission barrier in these nuclei along the ax and ax - tr fission pathes are always smaller than the experimental data by @xmath151 mev . the saddle along the tr - b fission path is somewhat lower in energy ( by 0.4 - 0.7 mev ) than the saddle along the ax - tr path in @xmath145116 nuclei ( see table [ tab7 ] ) . however , as discussed above ( based on the results of ref . @xcite ) it is unlikely that the fission predominantly proceeds along the tr - b path considering especially ( i ) that the gain in saddle point energy as compared with the ax - tr path is small and ( ii ) the tr - b path is substantially longer as compared with the ax - tr path . considering the discussion above , it is not clear at this moment how serious the discrepancy _ between calculations and experimental estimates _ is . however , the results of the calculations suggest one possible way to increase the heights of inner fission barriers . potential energy surfaces in the ground state region of these nuclei are extremely soft ( see figs . [ pot3a ] and [ pot3b ] ) . for such nuclei , the correlations beyond mean field taken into account , for example , by the generator coordinate method can lower the energy of the ground state by a few mev without affecting much the barrier top , thus effectively increasing the height of inner fission barrier . systematic calculations of inner fission barriers have been performed for even - even @xmath0 nuclei with @xmath152 using the nl3 * parametrization . this is the same parametrization which has been used in systematic calculations of fission barriers in actinides @xcite and which provides 0.76 mev average deviation from experiment for the heights of inner fission barriers in these nuclei . for a given proton number @xmath127 a sequence of nine even - even nuclei is selected such that the middle nucleus of this sequence roughly corresponds to an experimentally observed nucleus ( or a nucleus which is expected to be observed in near future ) . this crudely outlines the region which may be experimentally studied within the next one or two decades . the results of these calculations are summarized in table [ tab5 ] . based on the results of ref . @xcite , it is reasonable to neglect the path tr - b since it is substantially longer as compared with other pathes which leads to significant reduction of the penetration probability . in fig . [ pot2]b we consider the evolution of the heights of the inner fission barriers as a function of the neutron number @xmath153 . the @xmath154 and 116 isotope chains show a generally increasing trend for the barrier heights with increasing neutron number for @xmath155 . [ pot2]a suggests that the origin of this trend can be traced back to the deformation of the ground state . for small neutron numbers these nuclei are deformed in the ground state . however , they gradually become spherical when approaching @xmath124 because there is a spherical shell gap at this neutron number ( see , for example , fig . 28 in ref . the negative shell correction energy at the ground state is larger in absolute value in the vicinity of the @xmath124 spherical shell gap than at lower neutron numbers , where the ground state is deformed and characterized by a larger level density in the vicinity of the fermi level . the level density ( and , as a consequence , the shell correction energy ) at the saddle point of the inner fission barrier does not change so drastically as the one at the ground state . as a result , the heights of inner fission barriers , which are defined as the energy differences between the binding energies of the ground state and saddle point , show the observed features . note that in these nuclei the energies of the ax and ax - tr saddles differ by at most 200 kev , the difference between the energies of the ax and tr - a saddles does not exceed 400 kev . the nuclei with @xmath148 118 and 120 have , with the exception of the nucleus @xmath156 , spherical minima due to the presence of the @xmath123 spherical shell gap . the nucleus @xmath157 has the highest value for the fission barrier among all nuclei under investigation . this is connected with its doubly magic nature in cdft . in the isotope chain with @xmath123 , moving away from the @xmath158 shell closure , shell effects connected with spherical shape become less pronounced and this leads to a decrease of the inner fission barrier , because the barrier height in these nuclei is defined with respect to the spherical ground state . apart from the lightest two and the heaviest @xmath159 isotopes , the fission barrier heights of the @xmath159 chain are nearly constant as a function of neutron number and they are close to 4.5 mev . note that in these nuclei the energies of the ax and ax - tr saddles differ only by 100 - 400 kev . however , the difference between the energies of the ax and tr - a saddles can reach 2 mev ( as in the case of the nucleus @xmath160120 , see table [ tab5 ] ) . it is important to mention that the valley between the axial @xmath161 and triaxial @xmath162 hills leading to a tr - a saddle is rather shallow . its depth with respect to the tip of the triaxial hill varies between 100 and 300 kev . the latter value is obtained , for example , in @xmath163120 nucleus , see fig . [ 302-valley ] . & @xmath164 & ( @xmath68,@xmath115 ) & @xmath165 & ( @xmath68,@xmath115 ) + 1 & 2 & 3 & 4 & 5 & 6 & 7 + + @xmath160120 & 5.11 & 0.20 & 3.04 & ( 0.34,26@xmath112 ) & 2.23 & ( 0.48,22@xmath112 ) + @xmath166120 & 5.00 & 0.20 & 3.46 & ( 0.33,26@xmath112 ) & 2.02 & ( 0.50,22@xmath112 ) + @xmath126120 & 5.46 & 0.21 & 4.51 & ( 0.33,26@xmath112 ) & 3.30 & ( 0.51,21@xmath112 ) + @xmath163120 & 5.37 & 0.19 & 5.00 & ( 0.33,25@xmath112 ) & 3.93 & ( 0.50,24@xmath112 ) + @xmath133120 & 5.35 & 0.32 & 4.93 * & ( 0.31,11@xmath112 ) & 4.08 & ( 0.50,26@xmath112 ) + @xmath167120 & 5.35 & 0.33 & 5.02 * & ( 0.33,12@xmath112 ) & 4.27 & ( 0.49,28@xmath112 ) + @xmath168120 & 5.67 & 0.34 & 5.57 * & ( 0.35,10@xmath112 ) & 4.92 & ( 0.50,27@xmath112 ) + @xmath169120 & 5.27 & 0.34 & & & 4.73 & ( 0.50,27@xmath112 ) + @xmath100120 & 5.82 & 0.36 & & & 5.61 & ( 0.50,27@xmath112 ) + + @xmath126118 & 4.97 & 0.21 & 3.92 & ( 0.33,26@xmath112 ) & 3.03 & ( 0.51,26@xmath112 ) + @xmath163118 & 5.07 & 0.23 & 4.57 & ( 0.33,26@xmath112 ) & 3.41 & ( 0.51,25@xmath112 ) + @xmath133118 & 5.18 & 0.28 & 4.94 * & ( 0.27,15@xmath112 ) & 3.37 & ( 0.51,26@xmath112 ) + @xmath167118 & 5.07 & 0.32 & 4.72 * & ( 0.32,12@xmath112 ) & 3.67 & ( 0.50,27@xmath112 ) + @xmath168118 & 5.00 & 0.32 & 4.73 * & ( 0.33,10@xmath112 ) & 3.87 & ( 0.50,26@xmath112 ) + @xmath169118 & 4.95 & 0.32 & 4.82 * & ( 0.33,10@xmath112 ) & 4.12 & ( 0.50,25@xmath112 ) + @xmath100118 & 4.75 & 0.34 & & & 4.35 & ( 0.50,23@xmath112 ) + @xmath136118 & 4.15 & 0.36 & & & 4.08 & ( 0.49,24@xmath112 ) + @xmath170118 & 3.28 & 0.36 & & & & + + @xmath133116 & 4.83 & 0.26 & 4.48 & ( 0.35,25@xmath112 ) & 3.13 & ( 0.52,22@xmath112 ) + @xmath167116 & 4.78 & 0.28 & 4.57 & ( 0.30,15@xmath112 ) & 3.29 & ( 0.51,23@xmath112 ) + @xmath168116 & 4.53 & 0.32 & 4.32 * & ( 0.32,11@xmath112 ) & 3.35 & ( 0.51,24@xmath112 ) + @xmath169116 & 4.37 & 0.32 & 4.22 * & ( 0.33,7@xmath112 ) & 3.40 & ( 0.50,25@xmath112 ) + @xmath100116 & 3.98 & 0.33 & 3.93 * & ( 0.34,7@xmath112 ) & 3.39 & ( 0.48,26@xmath112 ) + @xmath136116 & 3.39 & 0.34 & & & 3.47 & ( 0.48,27@xmath112 ) + @xmath170116 & 3.55 & 0.34 & & & & + @xmath134116 & 3.43 & 0.36 & & & & + @xmath171116 & 3.40 & 0.37 & & & & + + @xmath168114 & 4.65 & 0.28 & 4.43 * & ( 0.31,15@xmath112 ) & 3.42 & ( 0.49,24@xmath112 ) + @xmath169114 & 4.19 & 0.29 & 4.14 * & ( 0.31,10@xmath112 ) & 3.08 & ( 0.49,25@xmath112 ) + @xmath100114 & 3.81 & 0.32 & & & 3.18 & ( 0.48,25@xmath112 ) + @xmath136114 & 3.40 & 0.34 & & & 3.38 & ( 0.48,26@xmath112 ) + @xmath170114 & 3.52 & 0.32 & & & & + @xmath134114 & 3.32 & 0.33 & & & & + @xmath171114 & 3.07 & 0.38 & & & & + @xmath172114 & 3.08 & 0.42 & & & & + @xmath173114 & 3.20 & 0.42 & & & & + & @xmath164 & ( @xmath68,@xmath115 ) & @xmath165 & ( @xmath68,@xmath115 ) + 1 & 2 & 3 & 4 & 5 & 6 & 7 + + @xmath100112 & 3.94 & 0.30 & & & 3.64 & ( 0.48,27@xmath112 ) + @xmath136112 & 3.48 & 0.31 & & & 3.44 & ( 0.47,26@xmath112 ) + @xmath170112 & 3.07 & 0.31 & & & & + @xmath134112 & 2.90 & 0.32 & & & & + @xmath171112 & 2.65 & 0.32 & & & & + @xmath172112 & 2.35 & 0.41 & & & & + @xmath173112 & 2.70 & 0.43 & & & & + @xmath135112 & 3.06 & 0.44 & & & & + @xmath174112 & 3.34 & 0.44 & & & & + & @xmath175 & ( @xmath68,@xmath115 ) & @xmath165 & ( @xmath68,@xmath115 ) + + @xmath169116 & 3.64 & 0.31 & 3.47 * & ( 0.32,10@xmath112 ) & 2.58 & ( 0.48,27@xmath112 ) + @xmath100116 & 3.63 & 0.32 & 3.55 * & ( 0.32,5@xmath112 ) & 2.83 & ( 0.49,26@xmath112 ) + @xmath136114 & 3.42 & 0.34 & & & & + @xmath170114 & 3.38 & 0.32 & & & & + @xmath134112 & 3.02 & 0.31 & & & & + @xmath171112 & 2.75 & 0.31 & & & & + + @xmath169116 & 5.36 & 0.32 & 5.20 * & ( 0.32,8@xmath112 ) & 4.45 & ( 0.48,25@xmath112 ) + @xmath100116 & 4.77 & 0.32 & 4.68 * & ( 0.32,5@xmath112 ) & 4.24 & ( 0.48,27@xmath112 ) + @xmath136114 & 4.33 & 0.34 & & & & + @xmath170114 & 4.18 & 0.32 & & & & + @xmath134112 & 3.62 & 0.31 & & & & + @xmath171112 & 3.12 & 0.31 & & & & + it is interesting to compare the current results with the ones obtained in other models . the results of skyrme dft calculations of ref . @xcite for the @xmath124 isotones show that the impact of triaxiality on the inner fission barrier is small for @xmath140 , but increases with increasing z ( see fig . 4 in ref . the lowering of the inner fission barrier due to triaxiality is around 2 mev for @xmath123 and exceeds 3 mev for @xmath176 . in the etfsi model calculations @xcite the inner fission barriers are lowered due to triaxiality in the nuclei @xmath177 and @xmath178 by 0.5 and 1.1 mev , respectively . in the macroscopic+microscopic calculations of ref . @xcite the largest reduction of the inner barrier height due to triaxiality is about 2 mev and it appears in the region around @xmath179 , @xmath180 ( see fig . 2 in ref . @xcite ) . as discussed in ref . @xcite on the example of actinide nuclei , the reduction of the inner fission barrier height due to triaxiality is caused by the level densities in the vicinity of the fermi level which are lower at triaxial shape as compared with axial one . the different location of the `` magic '' shell gaps in superheavy nuclei in the macroscopic+microscopic model ( at @xmath139 , @xmath124 ) , in skyrme dft ( predominantly at @xmath176 , @xmath124 ) and in cdft ( at @xmath123 , @xmath158 ) results in different deformed single - particle structures at the deformations typical for the saddles of the inner fission barrier . this is one of the sources of the differences in the predictions of different models . we have to keep in mind , however , that we use in this investigation a seniority zero pairing force with a fixed cutoff energy of @xmath181 mev . it has been shown in ref . @xcite that pairing correlations play an important role for the calculation of the fission barriers and that even if the strength of the pairing force is adjusted to experimental gap parameters at the ground state , the range of the pairing force has an influence on the height of the barrier . this means for zero range forces and for the seniority zero forces that the barriers depend on the cutoff energy . it also has been shown in ref . @xcite that the parameter set dd - me2 in connection with the finite range gogny force d1s in the pairing channel produces in axial symmetric calculations in most of the actinide nuclei inner barriers which are too high as compared with experiment ( see fig . 9 of ref . @xcite ) and that it produces for the superheavy elements with @xmath148 112 , 114 , and 116 , where experimental estimates are available @xcite in axially symmetric calculations barriers which are in rather good agreement with those values ( see fig . 10 of ref . @xcite ) . of course , so far , there exist no relativistic triaxial calculations with the finite range gogny force in the pairing channel and full gogny calculations @xcite are hard to compare because of the different spin - orbit force used in this model . we have , however to keep in mind , that triaxiality reduces the barriers in the actinides , but not in the superheavy elements with @xmath148 112 , 114 , and 116 . therefore , at the moment , we can only conclude that for a final comparison with the experimental data we have to wait for full rhb calculations or at least rmf+bcs calculations with the finite range gogny force in the pairing channel . investigations in this direction are in progress . it is well known that reflection - asymmetric ( octupole deformed ) shapes become important at the second fission barrier and beyond ( see refs . @xcite and references therein ) . our calculations indicate that triaxiality can play a similarly important role at the outer fission barriers . usually , the triaxiality of these barriers is not mentioned in the publications . to our knowledge , it is only discussed in ref . @xcite that non - axial degrees of freedom play an important role in the description of outer fission barriers of actinides . [ ofb ] compares the energies of outer barrier saddle points as obtained in axial reflection symmetric ( a ) , triaxial ( t ) and axial reflection asymmetric [ octupole deformed ] ( o ) calculations . one can see that the inclusion of triaxiality or octupole deformation always lowers the outer fission barrier . the underlying shell structure clearly defines which of the saddle points ( triaxial or octupole deformed ) is lower in energy . for example , the lowest saddle point is obtained in triaxial calculations in proton - rich nuclei ( @xmath182 ) . on the contrary , the lowest saddle point is obtained in octupole deformed calculations in neutron - rich nuclei ( @xmath183 ) . note that the decrease of the saddle point energy by octupole deformation or triaxiality reaches 3 mev in some nuclei . thus , one can conclude that due to the structure of the pes in the fission path valley , we observe in the superheavy region an opposite situation as compared to actinide nuclei where the triaxiality has no impact on the outer fission barriers . our results also suggest that in some superheavy nuclei [ mostly in the nuclei where the deformation energy curves of the a and o calculations are similar in energy ] the combination of two deformations ( triaxiality and odd - multipole deformations ) may be important in the definition of the fission path for @xmath184 . the cdft calculations with both deformations included are at present not yet possible , but require further investigations . nuclei relative to spherical / weakly(normal)-deformed minima as a function of neutron number . the results of calculations of type a ( axial ) , t ( triaxial ) and o ( octupole ) obtained with the nl3 * parametrization are presented.,width=302 ] fig . [ oct - second ] shows that some nuclei are superdeformed in the ground state . a summary for such nuclei is given in table [ sd - summary ] . whether these states are stable or metastable should be defined by the height and the width of the outer fission barrier . the current calculations show that in many nuclei this barrier is appreciable in the axial calculations of type a. however , the inclusion of triaxial or octupole deformation decreases this barrier substantially so it is around 2 mev in the majority of the nuclei . this low barrier would translate into a high penetration probability for spontaneous fission , such that most likely these superdeformed states are metastable . calculations of the spontaneous fission half - lives from spherical / weakly - deformed and superdeformed minima are needed in order to decide which of these minima is more stable against fission . table [ sd - summary ] shows that the saddle point obtained in octupole deformed calculations ( o ) is the lowest in energy in most of the cases . only in proton - rich @xmath169120 and @xmath185118 nuclei , the saddle point of triaxial calculations ( t ) is the lowest . a superdeformed minimum exists also in the doubly magic nucleus @xmath100120 at a low excitation energy of approximately 0.6 mev ( see fig.[pot ] ) . this is definitely not an artifact of the model under consideration , since similar minimum exists also in axial relativistic hartree - bogoliubov calculations with the dd - me2 parametrization with gogny d1s forces in the pairing channel ( see fig . 8 in ref.@xcite ) . its calculated excitation energy depends on the actual strength of pairing and varies between 0 and 2 mev . it is interesting to compare our results for the structure of the outer fission barriers with those obtained in the axial rmf+bcs model ( with and without reflection symmetry ) of ref . @xcite which employed the nl3 and nl - z2 parametrizations . similar to our calculations , the superdeformed minima exist in the calculations of ref . @xcite ( see fig . 5 in this reference ) without octupole deformation . however , in the calculations of ref . @xcite with the nl - z2 parametrization , the heights of outer fission barriers with respect to the sd minimum are lower by approximately 2 mev as compared to our calculations . as a consequence , the inclusion of octupole deformation completely eliminates the outer fission barriers leading to results contradicting ours . on the other hand , the results of the calculations of ref . @xcite ( see fig . 6 in this reference ) for a few selected nuclei based on the nl3 parametrization are very similar to ours , namely , the ground state is superdeformed and the outer fission barrier has a height of approximately 2.5 mev . this result is not surprising considering that the nl3 * parametrization is very similar to nl3 @xcite . however , the results of the calculations for a few selected @xmath150 and 116 nuclei shown in figs . [ pot3a ] and [ pot3b ] reveal that the outer fission barriers survive in the presence of octupole and triaxial deformation not only in the nl3 * parametrization but also in the dd - me2 and dd - pc1 parametrizations . on average , the height of outer fission barrier in these nuclei is around 2 mev . on the contrary , the calculations with the nl - z2 parametrization in ref . @xcite show that octupole deformation kills the outer fission barriers in these nuclei ( see fig . 5 in this reference ) . but for the nuclei @xmath167120 and @xmath166120 . note that for simplicity only the axial solution ( a ) is shown at @xmath186.,width=302 ] but for the nucleus @xmath163120 . in order to show the potential energy surface between the axial @xmath187 and triaxial ( @xmath162 ) hills in greater details the energy difference between two neighboring equipotential lines is set to 0.1 mev and only one fourth of deformation plane of fig . [ pes-2d-6panel ] is shown.,width=302 ] & @xmath188 & @xmath189 + 1 & 2 & 3 & 4 & 5 + + @xmath190 & 4.84 & 4.52 & 2.75 & * 1.62 * + @xmath191 & 4.26 & 4.36 & 3.00 & * 1.82 * + @xmath192 & 3.48 & 4.49 & 3.57 & * 2.07 * + @xmath193 & 2.89 & 5.30 & 3.65 & * 2.55 * + @xmath194 & 2.60 & 6.18 & 3.82 & * 2.61 * + @xmath195 & 1.99 & 5.71 & 3.38 & * 2.25 * + @xmath196 & 1.72 & 4.87 & 2.94 & * 2.52 * + @xmath197 & 0.62 & 4.63 & * 2.24 * & 2.71 + + @xmath198 & 2.68 & 2.41 & 2.14 & * 1.31 * + @xmath199 & 2.92 & 3.58 & 2.42 & * 1.42 * + @xmath200 & 1.78 & 5.16 & 3.40 & * 2.20 * + @xmath201 & 1.70 & 5.34 & 3.54 & * 2.34 * + @xmath202 & 1.48 & 5.17 & 3.15 & * 2.26 * + @xmath203 & 1.15 & 4.99 & 2.82 & * 2.35 * + @xmath204 & 0.69 & 4.58 & * 2.28 * & 2.59 + @xmath205 & 0.40 & 4.03 & * 1.61 * & 2.60 + + @xmath206 & 0.92 & 2.76 & 2.60 & * 1.75 * + @xmath207 & 0.83 & 3.98 & 2.95 & * 1.88 * +

systematic calculations of fission barriers allowing for triaxial deformation are performed for even - even superheavy nuclei with charge number using three classes of covariant density functional models . the outer fission barriers are considerably affected by triaxiality and octupole deformation . general trends of the evolution of the inner and the outer fission barrier heights are discussed as a function of the particle numbers .

systematic calculations of fission barriers allowing for triaxial deformation are performed for even - even superheavy nuclei with charge number using three classes of covariant density functional models . the softness of nuclei in the triaxial plane leads to an emergence of several competing fission pathes in the region of the inner fission barrier in some of these nuclei . the outer fission barriers are considerably affected by triaxiality and octupole deformation . general trends of the evolution of the inner and the outer fission barrier heights are discussed as a function of the particle numbers .