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0803.2754 | i | an immersion @xmath6 is _ conformally flat _ if there exists a flat metric in the conformal class of the induced metric @xmath7 : that is there exists a smooth function @xmath8 such that @xmath9 is flat . this condition is equivalent to the weyl tensor of @xmath10 being zero when @xmath11 , and to the schouten tensor @xmath12 being a codazzi tensor when @xmath13 ( i.e. , @xmath14 is a symmetric @xmath15-tensor ) . the history of conformally flat immersions is long , the search for conformally flat submanifolds being a natural task in conformal geometry . the study of conformally flat _ hypersurfaces _ in @xmath0 dates back to cartan @xcite who , demonstrated that the only such hypersurfaces for @xmath11 are the channel hypersurfaces : envelopes of a 1-parameter family of @xmath2-spheres . in particular these have ( at most ) two distinct principal curvatures . for @xmath16 the problem is uninteresting as every surface is conformally flat . for @xmath13 , however , there are more varied conformally flat hypersurfaces : not only are there the channel examples , there also exist hypersurfaces with 3 distinct curvatures . these were first discussed by hertrich jeromin @xcite , who moreover described the link between conformally flat hypersurfaces in @xmath3 , curved flats in the space of circles in @xmath3 , triply orthogonal systems , and guichard nets . the classification of these hypersurfaces , however , remained unknown . given a conformally flat immersion @xmath17 with flat normal bundle , we embed @xmath18 naturally in the _ light - cone _ @xmath19 of isotropic vectors in a lorentzian @xmath20 , and construct a _ flat lift _ @xmath21 : this @xmath22 is immersed , has flat induced metric , and flat normal bundle . since both are flat , the tangent and normal bundle decomposition of the trivial @xmath23-bundle is a _ curved flat _ @xcite in the pseudo - riemannian symmetric space @xmath24 ( the grassmannian of space - like @xmath4-planes in @xmath20 ) . we are thus immediately in the realm of integrable systems . curved flats in @xmath25 with a _ good _ co - ordinate system give rise to terng s @xmath25-system ( @xcite ) , which is constructed as follows : suppose that @xmath26 is the involution of the lie group @xmath27 defined by @xmath28 , where @xmath29 and @xmath30 is the @xmath31 identity matrix . then @xmath32 is the fixed point set of @xmath26 in @xmath27 . let @xmath33 denote the @xmath34-eigenspace decomposition of @xmath35 . @xmath32 then acts on @xmath36 by conjugation . let @xmath37 be a maximal abelian subalgebra in @xmath36 , and @xmath38 a basis of @xmath37 . _ the @xmath25-system defined by @xmath37 _ is the following system for @xmath39 : @xmath40-[a_j,\xi_{x_i}]-[[a_i,\xi],[a_j,\xi]]=0,\quad i\neq j,\ ] ] where @xmath41 is the orthogonal complement of @xmath37 with respect to the killing form . unlike in the riemannian symmetric case , not all maximal abelian subalgebras in @xmath36 are conjugate under @xmath32 ; there are both semi - simple and non - semisimple such subalgebras . we note that two maximal abelian subalgebras in @xmath36 conjugate under @xmath32 give rise to equivalent @xmath25-systems , while two non conjugate maximal abelian subalgebras in @xmath36 give non - equivalent @xmath25 systems . the normal bundle of an immersion is termed _ non - degenerate _ if the dimension of the space of shape operators at each point is equal to the co - dimension . an immersion has _ uniform multiplicity one _ if it has flat normal bundle and distinct curvature normals ( equivalently all curvature distributions have rank one ) . it follows from the definition that an @xmath4-dimensional submanifold in @xmath23 with flat and non - degenerate normal bundle has uniform multiplicity one . we prove that a conformally flat @xmath4-immersion into @xmath18 with uniform multiplicity one gives rise to a flat @xmath4-immersion in the light - cone @xmath19 with flat non - degenerate normal bundle , and that the converse is also true . to study conformally flat @xmath4-immersions in @xmath18 with uniform multiplicity one is thus the same as to study flat @xmath4-immersions in @xmath19 with flat non - degenerate normal bundle . we show that there exist line of curvature co - ordinate systems for these flat immersions , and that their gauss - codazzi equations amount to the @xmath24-system defined by a semi - simple maximal abelian subalgebra @xmath37 . conversely , given a solution to the @xmath25-system defined by @xmath37 , and a null vector @xmath42 we obtain a a conformally flat immersion in @xmath18 with uniform multiplicity one . motivated by definitions in classical differential geometry , we call a diffeomorphism @xmath43 between @xmath4-immersions in space forms with flat normal bundle a _ combescure transform _ if @xmath44 maps principal directions of @xmath45 to those of @xmath46 and they are parallel . a combescure transform @xmath44 is _ christoffel _ if it is orientation reversing . given a solution to the @xmath25-system defined by @xmath37 and a null vector @xmath47 with @xmath48 , we construct a flat @xmath4-immersion @xmath49 in @xmath19 with flat normal and non - degenerate bundle . hence @xmath49 projects to a conformally flat immersion in @xmath18 with uniform multiplicity one . moreover , if @xmath50 and @xmath51 are null vectors with euclidean length @xmath52 , then @xmath53 is a combescure transform . because of the correspondence between solutions of the @xmath25-system and conformally flat @xmath4 immersions in @xmath18 with flat and non - degenerate normal bundle , all the machinery of soliton theory applies : loop - group dressing of solutions to obtain new conformal flats or simply dressing vacuum solutions to obtain more complex explicit conformally flat immersions ; existence results such as cartan - khler and inverse scattering , etc . in particular : 1 . we may dress solutions by special , simple elements , whose action may be calculated explicitly by residues . the action of such elements is seen to be by _ ribaucour transforms _ on conformal flats : corresponding immersions envelop ( have first - order contact with ) a congruence of @xmath4-spheres in such a way that principal curvature directions on the envelopes correspond under the congruence . local analytic conformally flat @xmath4-immersions in @xmath18 are determined by @xmath54 functions of one variable . the cauchy problem for the @xmath25-system with rapidly decaying initial data on a regular line can be solved globally . is _ regular _ if @xmath55 is a injective . ] although the resulting @xmath4 dimensional submanifolds may have cusp singularities , the frame is globally defined and smooth . the moduli space of such immersions has a loop group symmetry . if the normal bundle is degenerate and the curvature distributions @xmath56 ( common eigenspaces of the shape operators ) have constant ranks , then we show that all but one of the @xmath56s have rank one . such submanifolds are thus envelopes of @xmath57-parameter families of @xmath58-spheres . if , in addition , these immersions are assumed to have line of curvature co - ordinates , then the gauss - codazzi equations are the @xmath25-system defined by a non - semisimple maximal abelian subalgebra @xmath37 in @xmath36 . conversely , given a solution of the @xmath25-system defined by @xmath37 , we obtain an @xmath59-parameter family of flat lifts , each of which gives rise to a conformally flat immersion in @xmath18 with flat normal bundle , but not with uniform multiplicity one . when @xmath13 , these give _ channel immersions_. loop group dressing still works , and we can construct channel immersions from any germ of an @xmath60-valued holomorphic map at @xmath61 that satisfies the reality condition associated to @xmath25 . most of the results for conformally flat @xmath4-immersions in @xmath18 with uniform multiplicity one hold for conformally flat @xmath4-immersions in @xmath62 with flat normal bundle , @xmath4 curvature normals such that the orthogonal complement of the subbundle spanned by @xmath4 curvature normals is flat . there exist line of curvature co - ordinates and a correspondence between such immersions and solutions of the @xmath63-system . the paper is organised as follows . in section [ sec : flatlift ] , we generalise hertrich jeromin s work on conformally flat immersions of hypersurfaces in @xmath3 to @xmath4-submanifolds in @xmath18 : we outline the light - cone model and how conformal flat immersions in the sphere correspond to genuine flat immersions in the light - cone , then consider the curvature distributions of corresponding maps , and how their fundamental forms compare . the link between conformal flats and the @xmath25-system is detailed in section [ sec : uk ] , and we explain its generalisation to conformally flat @xmath4-immersions in @xmath62 in section [ sec : codim ] . we give a discussion of the dressing transformations of a negative loop on the space of solutions to the @xmath25-system and their associated conformally flat immersions ; certain dressing transforms are shown to give rise to geometric ribaucour transforms in section [ sec : rib ] . in the final section , we show that solutions of the @xmath25-system defined by a non - semisimple maximal abelian subalgebra give rise to the channel immersions . | . cartan proved that conformally flat hypersurfaces in for have at most two distinct principal curvatures and locally envelop a one - parameter family of-spheres . we prove that the gauss - codazzi equation for conformally flat hypersurfaces in is a soliton equation , and use a dressing action from soliton theory to construct geometric ribaucour transforms of these hypersurfaces . we describe the moduli of these hypersurfaces in and their loop group symmetries . we also generalise these results to conformally flat-immersions in-spheres with flat and non - degenerate normal bundle . research supported in part by nsf advance grant ] research supported in part by nsf grant dms-0707132 ] | . cartan proved that conformally flat hypersurfaces in for have at most two distinct principal curvatures and locally envelop a one - parameter family of-spheres . we prove that the gauss - codazzi equation for conformally flat hypersurfaces in is a soliton equation , and use a dressing action from soliton theory to construct geometric ribaucour transforms of these hypersurfaces . we describe the moduli of these hypersurfaces in and their loop group symmetries . we also generalise these results to conformally flat-immersions in-spheres with flat and non - degenerate normal bundle . research supported in part by nsf advance grant ] research supported in part by nsf grant dms-0707132 ] |
0907.0529 | i | the @xmath0 expansion of qcd @xcite is a valuable tool for studying the nonperturbative dynamics of the strong interactions @xcite . in the limit @xmath2 , the baryon sector of qcd has an exact _ contracted _ @xmath3 spin - flavor - symmetry @xcite . for finite @xmath4 , the contracted spin - flavor - symmetry is broken by effects suppressed by powers of @xmath0 @xcite . the spin - flavor structure of the @xmath0 breaking terms is predicted at each order in the @xmath0 expansion @xcite . the spin - flavor structure of many baryon properties have been derived in a systematic expansion in @xmath0 @xcite , and the results are in excellent agreement with experiment ( for reviews , see @xcite ) . one important application of the baryon @xmath0 expansion is to baryon masses @xcite . by choosing appropriate linear combinations of the baryon masses , one can study coefficients of the baryon mass @xmath0 operator expansion with definite spin and flavor transformation properties . in the case of perturbative @xmath1 flavor - symmetry breaking , the @xmath0 analysis gives a hierarchy of baryon mass relations in powers of @xmath0 and the dimensionless @xmath1 breaking parameter @xmath5 @xcite . the analysis in ref . @xcite showed that the experimentally measured masses of the ground state octet and decuplet baryons exhibit the predicted hierarchy of the combined expansion in @xmath0 and @xmath1 flavor - symmetry breaking . the @xmath0 expansion also has been used to obtain very accurate predictions for the charm and bottom baryon masses @xcite ( to better than 10 mev accuracy ) which have been confirmed by recent experiments . in addition , ref . @xcite derived baryon mass relations which only depend on the @xmath0 expansion and which are valid even if flavor @xmath1 is not a good approximate symmetry , i.e. for large values of the strange quark mass . the predictions of the @xmath0 expansion for baryon masses are in excellent agreement with the experimental values . however , in comparing with the experimental data , one is restricted to only one value of @xmath4 and to the physical quark mass values . testing the predictions as a function of light - quark masses and @xmath4 is now possible with very accurate simulations of baryons using lattice qcd . tremendous progress in lattice qcd has been achieved recently in the simulation of baryon masses using different values of the light - quark masses . extrapolation of baryon masses on the lattice to the physical point has reproduced the experimental values at the @xmath6 level @xcite . the lattice data , however , contain important additional information about the dependence of the baryon masses on the quark masses , which can be utilized . simulations of baryon masses have been performed as a function of @xmath1 flavor - symmetry breaking ranging from small perturbative flavor - symmetry breaking to large nonperturbative flavor - symmetry breaking . there are also lattice simulations at different values of @xmath4 ( though not for baryons ) which are able to test @xmath4 scaling rules @xcite . in this paper , we show that existing lattice simulations ( at @xmath7 ) of the ground state baryon masses already are sufficiently accurate to exhibit and test interesting features of the @xmath0 and @xmath1 flavor - symmetry breaking expansions . still more accurate simulations are needed to test the most suppressed mass combinations of the @xmath0 expansion , but continued improvements in lattice simulations of baryon masses are expected in the short and long term , so it should eventually be possible to test these relations as well . we discuss in this paper how present and future lattice data can be utilized to study the spin and flavor structure of baryon masses . although we do not focus on this point here , it should eventually be possible to test the @xmath4 scaling rules in the baryon sector by lattice simulations which vary the number of colors @xmath4 . an important observation is that the @xmath0 counting rules hold at finite lattice spacing , and so are respected by the lattice results _ including _ the finite lattice spacing corrections dependent on the lattice spacing @xmath8 . thus the discretization corrections are constrained by the @xmath0 expansion . lattice computations of hadron masses are done at varying values of the light - quark masses @xmath9 , @xmath10 and @xmath11 , usually in the isospin limit @xmath12 . the lattice results are then extrapolated to the physical values of @xmath13 . it has not been possible to compute hadron masses for physical values of @xmath13 yet due to the large computational time needed , since @xmath14 . , where @xmath15 is the spatial size of the lattice . ] however , recently , with both algorithmic developments @xcite and large parallel computing machines , there are two groups simulating at or near the physical light - quark mass point @xcite . the light - quark mass dependence of the hadron masses is determined by chiral perturbation theory . there are nonanalytic in @xmath16 contributions from loop corrections which are calculable , as well as analytic terms which depend on low - energy constants ( lecs ) of the chiral lagrangian . in the baryon sector , the leading nonanalytic terms are @xmath17 and @xmath18 . the @xmath17 term , which arises from fig . [ fig : loop ] and is proportional to @xmath19 times axial couplings , radiative corrections . [ fig : loop],width=188 ] is found to be rather large , naively of order a few hundred mev . this term is absent for the pseudo - goldstone boson masses , but is present for other mesons such as the vector mesons @xcite . a surprising feature of recent lattice results is that the baryon masses as a function of @xmath20 do not show a large @xmath21 dependence . fits to the baryon masses also give much smaller values for the baryon axial couplings , and require that the @xmath22 term is almost completely cancelled by @xmath18 and @xmath23 terms . this cancellation must be accidental at the currently accessible lattice quark masses , since the terms have different @xmath16 dependence . an alternative conclusion is that lattice quark masses are too large for @xmath1 chiral perturbation theory to be valid , and that perturbative chiral behavior sets in only for much smaller quark masses than the strange quark mass . this conclusion , however , fails to explain why @xmath1 flavor - symmetry is so evident in baryon phenomenology . the @xmath0 expansion constrains the structure of baryon chiral perturbation theory @xcite . chiral corrections have to respect the spin - flavor structure of the @xmath0 expansion , and this leads to some important restrictions on the form of the chiral loop corrections . for example , the baryon mass , which is of order @xmath4 @xcite , gets nonanalytic corrections proportional to @xmath24 , which are large . this large nonanalytic contribution might lead one to expect that there should be large deviations from the gell - mann okubo mass relations , which were derived assuming that the mass operator was linear in @xmath11 . one can show , however , that the @xmath25 terms are a spin - flavor singlet , and give the same contribution to octet and decuplet baryons , whereas the @xmath24 terms are a spin - singlet flavor - octet , and the @xmath26 terms are spin - singlet flavor-@xmath27 . only the latter terms contribute to the gell - mann okubo mass combinations , so that deviations from these relations are a factor of @xmath28 smaller than naive expectation and consistent with experiment . the small size of the gell - mann okubo relation was recently confirmed for a range of light - quark masses @xcite . for qcd , the cancellation to order @xmath28 arises as a numerical cancellation between octet and decuplet intermediate states @xcite ; to see the parametric form of the cancellation in @xmath0 requires computing the chiral corrections using the @xmath1 flavor representations of baryons containing @xmath4 quarks . the mass relations of ref . @xcite project the baryon masses onto different spin - flavor channels . by studying these mass relations as a function of @xmath16 , one can investigate whether unexpected chiral behavior arises in a particular channel . the organization of this paper is as follows : in sec . ii , the baryon mass relations of the @xmath0 expansion are summarized briefly , and in sec . iii , the lattice simulation data are described . iv presents the results of a computation of @xmath0 mass combinations on the lattice for varying values of @xmath1 flavor - symmetry breaking . v discusses the lattice analysis using heavy baryon chiral perturbation theory @xcite . our conclusions are presented in sec . | baryon mass relations are compared with lattice simulations of baryon masses using different values of the light - quark masses , and hence different values of flavor - symmetry breaking . the validity of baryon mass relations derived without assuming approximate flavor - symmetry also can be tested by lattice data at very large values of the strange quark mass . this scaling is explicitly demonstrated in the present work . | baryon mass relations are compared with lattice simulations of baryon masses using different values of the light - quark masses , and hence different values of flavor - symmetry breaking . the lattice data clearly display both the and flavor - symmetry breaking hierarchies . the validity of baryon mass relations derived without assuming approximate flavor - symmetry also can be tested by lattice data at very large values of the strange quark mass . the expansion constrains the form of discretization effects ; these are suppressed by powers of by taking suitable combinations of masses . this scaling is explicitly demonstrated in the present work . |
1404.1238 | i | this paper considers joint estimation of multiple directed acyclic graph ( dag ) models using integer linear programming ( ilp ) . graphical models are multivariate models in which vertices in a graph @xmath0 represent random variables with edges between the vertices describing conditional independence statements concerning the variables . in many settings the edge structure of the graph is itself uncertain and then an important challenge is to estimate this structure from data . there has been considerable research into structural inference for graphical models over the last decade , including bayesian networks ( bns ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) , gaussian graphical models ( ggms ; * ? ? ? * ; * ? ? ? * ) and discrete graphical models @xcite . many graphical models are based on dags and these are the focus of this paper . in many applications , data @xmath1 are collected on multiple units @xmath2 that may differ with respect to conditional independence structure , such that corresponding dags @xmath3 may be non - identical . at the same time , when the units of study are related , the graphs @xmath3 , while potentially non - identical , are expected to have similarities . it is then natural to ask whether such similarity can be exploited by borrowing strength across the estimation problems indexed by @xmath4 and that is the aim of the present paper . specifically we seek to construct a _ joint _ estimator @xmath5 that estimates a collection of dags together . we contrast such an estimator with the _ independent _ estimator @xmath6 that estimates each dag from the associated data only . the best known class of dag models are bns , but more exotic dag models exist ( e.g. * ? ? ? bns admit an extensive theory of inferred causation that has contributed to their popularity @xcite . structure learning for individual bns is a well - studied problem , with contributions including @xcite . structure learning is np - hard , but an approach that has attracted much recent attention is to cast _ maximum a posteriori _ ( map ) dag estimation as a problem in integer linear programming ( ilp ) , as developed simultaneously by @xcite and subsequently extended in @xcite . in brief , this approach solves a sequence of linear relaxations of the map estimation problem via the introduction of cutting planes and combines this with a branch - and - bound search to produce an optimal solution . if the algorithm terminates , the result is guaranteed to be a global maximum of the posterior distribution and hence inherits theoretical guarantees associated with the map estimator ( see e.g. * ? ? ? ( in this paper , algorithms with this property are termed `` exact '' . ) coupled with powerful techniques from discrete optimisation @xcite , ilps represent an efficient and attractive methodology for structural inference , as demonstrated also by recent empirical results ( e.g. * ? ? ? recent advances in this area are discussed in @xcite . the joint estimation of graphical models has recently received attention , for example @xcite put forward a penalised likelihood formulation that couples together estimation for multiple ( undirected ) ggms . however , joint estimation of multiple dags has so far received relatively little attention . the first discussion of this problem that we are aware of is @xcite ; here a greedy search was used to locate a local maximum of a joint bayesian posterior . @xcite described a markov chain monte carlo ( mcmc ) method for sampling from a joint posterior over graphical structures . however , the generic difficulties associated with stochastic search / sampling in large discrete spaces are well known ; these are exacerbated in the joint case by the size of the joint model space and stochastic search / sampling remains challenging in this setting . the focus of this paper is instead on exact , deterministic algorithms . @xcite proposed an exact algorithm based on bayesian model averaging and belief propagation , under the strong assumption that an ordering of the variables @xmath7 applies simultaneously to all units . at the same time @xcite proposed essentially the same algorithm , applied to the specific class of feed - forward dynamic bns , where an ordering of the variables is implicitly provided by the time index . the algorithmic contributions of the present paper are two - fold . first , we show how to cast exact inference over multiple dags as an ilp problem . we consider map - bayesian estimation for multiple dags and require no restriction on the ordering of the variables . this is done by extending methodology presented in @xcite to the case of multiple dags via a hierarchical bayesian formulation . second , we exploit structural constraints that are imposed by the dags in order to improve computational efficiency . as a illustrative example , our methods currently allow estimation of 10 related dags , each with 10 nodes , in time typically less than one minute on a standard laptop . in addition , we extend previous work by allowing for dependencies between the dags themselves and consider also estimation of this dependency structure . previous work on multiple dags has focused on the special case where the units are exchangeable and all pairs of units undergo an equal amount of regularisation ( including * ? ? ? * ; * ? ? ? * ) . however , in practice , relationships between units ( and their underlying graphical models ) may be complex , e.g. hierarchical , with group and sub - group structure , and such structure may itself be subject to uncertainty . @xcite performed exact inference for non - exchangeable feed - forward dynamic bns for the case where the relationships between units are known _ a priori_. @xcite addressed non - exchangeability in the context of general dag models but did not provide an exact algorithm and , again , assumed that the relationships between units are known at the outset . our approach provides a framework that allows for simultaneous learning of both unit - specific dags and the dependency structure that relates them . however , our empirical results suggest that such simultaneous learning may be extremely challenging in practice . the remainder of the paper is organised as follows : section [ def the model ] introduces a statistical framework for multiple dags and discusses regularisation based on graphical structure . section [ sec estimate ] gives exact , ilp - based estimators for multiple dags . section [ applications ] presents a simulation study and results on fmri data from a multi - subject study . finally we close with a discussion of directions for further research . a companion paper that explores the fmri application in more detail is available as @xcite . | * abstract . * this paper considers the problem of estimating the structure of multiple related directed acyclic graph ( dag ) models . . results are presented on both simulated data and fmri data obtained from multiple subjects . * keywords . * hierarchical models , directed acyclic graphs , non - exchangeability , integer linear programming , joint estimation * author footnote . * chris . | * abstract . * this paper considers the problem of estimating the structure of multiple related directed acyclic graph ( dag ) models . building on recent developments in exact estimation of dags using integer linear programming ( ilp ) , we present an ilp approach for joint estimation over multiple dags , that does not require that the vertices in each dag share a common ordering . furthermore , we allow also for ( potentially unknown ) dependency structure between the dags . results are presented on both simulated data and fmri data obtained from multiple subjects . * keywords . * hierarchical models , directed acyclic graphs , non - exchangeability , integer linear programming , joint estimation * author footnote . * chris . j. oates ( e - mail : _ c.oates@warwick.ac.uk_ ) and jim q. smith ( e - mail : _ j.q.smith@warwick.ac.uk_ ) , department of statistics , university of warwick , coventry , cv4 7al , uk . sach mukherjee ( e - mail : _ sach@mrc-bsu.cam.ac.uk_ ) , mrc biostatistics unit and school of clinical medicine , university of cambridge , cambridge , cb2 0sr , uk . james cussens ( e - mail : _ james.cussens@cs.york.ac.uk_ ) , department of computer science and york centre for complex systems analysis , university of york , york , yo10 5ge , uk . |
1404.1238 | c | this paper introduced a general statistical framework for joint estimation of multiple dags . we considered regularisation based on graphical structure and showed how map estimators in this setting could be characterised as the solutions to ilps that admit efficient exact algorithms based on branching - and - cutting as well as constraint propagation . we believe these are the first exact algorithms for the general case of multiple general dag models ( i.e. with no restriction on ordering of the variables ) . in our general framework for joint learning , we allowed also for dependence between the units themselves , that generalises previous exchangeable formulations . results obtained both from simulations and from an application in neuroscience demonstrate that joint estimation of dags can offer increased statistical efficiency relative to independent estimation . however we observed that the relationships between the units themselves were difficult to estimate from data . in the neuroscience application we illustrated how sophisticated likelihood models such as mdms can be used within our framework . importantly , our methodology requires only that local scores are pre - computed and cached ; it is therefore possible to apply exact algorithms retrospectively , following independent analyses of individual datasets , without the need to recompute local scores . in practice we anticipate that the rationale and appropriateness of joint modelling will need be assessed by domain experts on a case - by - case basis . our computations were performed using the gobnilp package @xcite , that includes sophisticated routines for estimation of individual dags ( see supplementary materials ) . nevertheless our simulation results for large numbers @xmath14 of variables and @xmath39 of units required considerable computational time ( in a worst - case scenario ) . it is becoming commonplace in many applications for the number of units to become very large . it remains to be seen whether our approach can scale up to larger problems . recent work where gobnilp was able to find map dags with 1614 nodes in between 3 and 42 minutes are encouraging @xcite . in practice , as discussed in @xcite , it is very hard to estimate the time required to solve an ilp in advance ; rather this depends in a highly non - trivial way on the details of the problem and is related to the `` phase - transition '' in sat solving . we note that the results of this paper are based on the ilp framework of @xcite for individual dags . an alternative ilp formulation , known as `` characteristic imsets '' , was proposed by @xcite specifically for bns . characteristic imsets are closely related to the essential graph of a bn @xcite and have the property that markov - equivalent bns are score equivalent . in contrast , the approaches we pursued involved a non - unique representation of the essential graph . this flexibility is important when dealing with general dag models including mdms , that are uniquely identifiable from data , but are less natural in the context of bns . at present , the computational performance of characteristic imsets on individual bns is inferior to the approach of @xcite pursued here @xcite . in addition it is currently unclear how prior structural information might be incorporated into that framework . on a related note , it would be interesting to design a joint version of the well - known pc algorithm @xcite that complements score - based estimation . interesting extensions include : 1 . * information sharing for parameters . * in many applications it is reasonable to assume similarity of parameter values @xmath27 between units . at present this appears to be challenging to include within our framework and represents an area for further research . * computation . * the applications discussed in this paper were performed on a single cpu and limited in computational intensity to approximately @xmath133 . since many emerging datasets , not least in biomedical applications , contain many more variables @xmath14 and many more units @xmath39 than we considered here , it would be interesting to investigate strategies for parallel and/or approximate computation that scale better to such regimes . in the other direction , our methodology applies to complex local scores that may be estimated numerically ; it would be interesting to gain a theoretical understanding of how any uncertainty in the estimates for the local scores would impact on the map estimators described here . * new statistical models . * by fixing the network @xmath42 in our general framework we can obtain a number of interesting statistical models : ( i ) taking @xmath42 to be a chain allows us to associate units with a temporal ordering and thereby model time - evolving dag structures . ( ii ) introducing auxiliary dags @xmath150 and enforcing @xmath42 to be bipartite on the partition of auxiliary / non - auxiliary dags produces a mixture model for dags , such that each latent dag is a graphical summary of the various dag structures within its cluster . ( iii ) extending this idea , with @xmath151 we recover a mixture model for dags similar to @xcite . these possibilities are currently being explored , with preliminary results for mixture models presented in @xcite . preliminary indications suggest that imposing additional restrictions on @xmath42 in this way may serve to improve the identification of higher - order structure . more generally , it is straightforward to extend our methods to the case of decomposable undirected models by imposing additional constraints that rule out immoralities . * supplementary materials : * the supplementary materials include additional a discussion of the simulated data and the actual code used to produce the results herein . * acknowledgements : * cjo was supported by the centre for research in statistical methodology ( crism ) epsrc ep / d002060/1 . jc was supported by the medical research council ( project grant g1002312 ) . sm was supported by the uk medical research council and is a recipient of a royal society wolfson research merit award . the authors are grateful to lilia carneiro da costa and tom nichols who collaborated in the analysis of fmri data and to mark bartlett who provided technical support with gobnilp . the authors also thank diane oyen and several other colleagues who provided feedback on an earlier draft . 99 acid , s. , de campos , l.m . : searching for bayesian network structures in the space of restricted acyclic partially directed graphs . _ j. artif . res . _ * 18 * , 445 - 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572 ( 2008 ) . wolsey , l.a . : _ integer programming_. wiley , new york ( 1998 ) . yuan , c. , malone , b. : learning optimal bayesian networks : a shortest path perspective . _ j. artif . res . _ * 48 * , 23 - 65 ( 2013 ) . * supplement to `` exact estimation of multiple directed acyclic graphs '' * chris . j. oates ( e - mail : _ c.oates@warwick.ac.uk_ ) and jim q. smith ( e - mail : _ j.q.smith@warwick.ac.uk_ ) , department of statistics , university of warwick , coventry , cv4 7al , uk . sach mukherjee ( e - mail : _ sach@mrc-bsu.cam.ac.uk_ ) , mrc biostatistics unit and school of clinical medicine , university of cambridge , cb2 0sr , uk . james cussens ( e - mail : _ james.cussens@cs.york.ac.uk_ ) , department of computer science and york centre for complex systems analysis , university of york , yo10 5ge , uk . | building on recent developments in exact estimation of dags using integer linear programming ( ilp ) , we present an ilp approach for joint estimation over multiple dags , that does not require that the vertices in each dag share a common ordering . furthermore , we allow also for ( potentially unknown ) dependency structure between the dags j. oates ( e - mail : _ c.oates@warwick.ac.uk_ ) and jim q. smith ( e - mail : _ j.q.smith@warwick.ac.uk_ ) , department of statistics , university of warwick , coventry , cv4 7al , uk . sach mukherjee ( e - mail : _ sach@mrc-bsu.cam.ac.uk_ ) , mrc biostatistics unit and school of clinical medicine , university of cambridge , cambridge , cb2 0sr , uk . james cussens ( e - mail : _ james.cussens@cs.york.ac.uk_ ) , department of computer science and york centre for complex systems analysis , university of york , york , yo10 5ge , uk . | * abstract . * this paper considers the problem of estimating the structure of multiple related directed acyclic graph ( dag ) models . building on recent developments in exact estimation of dags using integer linear programming ( ilp ) , we present an ilp approach for joint estimation over multiple dags , that does not require that the vertices in each dag share a common ordering . furthermore , we allow also for ( potentially unknown ) dependency structure between the dags . results are presented on both simulated data and fmri data obtained from multiple subjects . * keywords . * hierarchical models , directed acyclic graphs , non - exchangeability , integer linear programming , joint estimation * author footnote . * chris . j. oates ( e - mail : _ c.oates@warwick.ac.uk_ ) and jim q. smith ( e - mail : _ j.q.smith@warwick.ac.uk_ ) , department of statistics , university of warwick , coventry , cv4 7al , uk . sach mukherjee ( e - mail : _ sach@mrc-bsu.cam.ac.uk_ ) , mrc biostatistics unit and school of clinical medicine , university of cambridge , cambridge , cb2 0sr , uk . james cussens ( e - mail : _ james.cussens@cs.york.ac.uk_ ) , department of computer science and york centre for complex systems analysis , university of york , york , yo10 5ge , uk . |
1212.2061 | i | in this article we study differential inclusions of the form @xmath0 where @xmath1 denotes the derivative with respect to time and @xmath2 is assumed to be a maximal monotone relation in time and space . the bounded linear operator @xmath3 , also acting in time and space , is a so - called linear material law , as it was introduced in @xcite . since we are dealing with the autonomous case , the operator @xmath3 and the relation @xmath2 are assumed to commute with the temporal translation operator . we will show that under suitable conditions on @xmath3 and @xmath2 the problem is well - posed , i.e. hadamard s requirements on existence and uniqueness of a solution @xmath4 and its continuous dependence on the given data @xmath5 are satisfied . moreover the issue of causality is addressed , meaning that the behaviour of the solution up to a certain time @xmath6 just depends on the behaviour of the given right hand side up to the same time @xmath6 . + originally this class of problems was discussed in @xcite , where the relation @xmath2 was given by a skew - selfadjoint spatial operator . there , well - posedness and causality were shown under a positive - definiteness constraint on the operator @xmath3 . moreover , the abstract solution theory was applied to several examples of linear equations in mathematical physics . this solution theory was generalized by the author to the case of @xmath2 being a maximal monotone relation in space in @xcite , where @xmath3 was of the particular form @xmath7 we will adopt the proof for the solution theory presented in @xcite to show the well - posedness of problems of the form . the proof mainly relies on the realization of the derivative @xmath1 as a maximal monotone operator on an exponentially weighted @xmath8-space ( see e.g. @xcite or subsection 2.1 of this article ) and the application of a well - known perturbation result for maximal monotone relations ( see ) . for the theory of maximal monotone relations we refer to the monographs @xcite . inclusions of the form cover a large class of possible evolutionary problems , such as integro - differential equations @xcite , delay and neutral differential equations @xcite , certain types of control problems @xcite and equations of mathematical physics involving hysteresis @xcite . + the article is structured as follows . in section 2 we recall the definition of the time - derivative , of linear material laws and some basic facts on maximal monotone relations on hilbert spaces . moreover , we recall the notion of so - called abstract boundary data spaces ( see @xcite ) , which will enable us to formulate evolutionary equations with nonlinear boundary condition as an inclusion of the form . in section 3 the solution theory for is presented . in the remaining part of the article we consider a certain type of partial differential equations with nonlinear boundary conditions . those problems occur for instance in frictional contact problems ( see e.g. @xcite ) in the field of elasticity , where the boundary condition is given by a differential inclusion . in subsection 4.1 we study an abstract nonlinear differential operator @xmath2 , which is given by the restriction of a linear operator to elements , which satisfy the nonlinear boundary condition . we show that under certain constraints on the relation , which occurs in the boundary condition , the operator @xmath2 gets maximal monotone and thus , the solution theory developed in section 3 applies . we illustrate our results in subsection 4.2 . there , in a first example we consider the wave equation with an impedance - type boundary condition . this problem was originally considered in @xcite and we show that it fits in our framework . the second example deals with the equations of visco - elasticity with a frictional boundary condition , which is modelled by a differential inclusion on the boundary . a similar kind of this problem was considered in @xcite for a cylindrical domain and antiplane shear deformations . again we show that the equations are covered by our abstract inclusion . + throughout every hilbert space is assumed to be complex . its inner product and its norm are denoted by @xmath9 and @xmath10 respectively , where the inner product is assumed to be linear with respect to the second and conjugate linear with respect to the first argument . moreover , for a hilbert space @xmath11 and a closed subspace @xmath12 we denote by @xmath13 the orthogonal projection onto @xmath14 is just the canonical embedding . ] then @xmath15 is the orthogonal projector on @xmath16 . | * abstract . * we study an abstract class of autonomous differential inclusions in hilbert spaces and show the well - posedness and causality , by establishing the operators involved as maximal monotone operators in time and space . then the proof of the well - posedness relies on a well - known perturbation result for maximal monotone operators . moreover , we show that certain types of nonlinear boundary value problems are covered by this class of inclusions and we derive necessary conditions on the operators on the boundary in order to apply the solution theory . | * abstract . * we study an abstract class of autonomous differential inclusions in hilbert spaces and show the well - posedness and causality , by establishing the operators involved as maximal monotone operators in time and space . then the proof of the well - posedness relies on a well - known perturbation result for maximal monotone operators . moreover , we show that certain types of nonlinear boundary value problems are covered by this class of inclusions and we derive necessary conditions on the operators on the boundary in order to apply the solution theory . we exemplify our findings by two examples . evolutionary inclusions , well - posedness , causality , maximal monotonicity , impedance type boundary conditions , frictional boundary conditions . 34g25 , 35f30 , 35r20 , 46n20 , 47j35 |
1303.3847 | i | density functional theory ( dft ) and coupled - cluster ( cc ) theory are the two dominant paradigms for the computation of many - electron ground states with complementary capabilities . both theories are built upon the independent - orbital and self - consistent field ( scf ) structure of hartree - fock ( hf ) theory @xcite . dft proves the existence of an exact density functional @xcite for which there are approximations @xcite enabling routine simulation of thousands of electrons @xcite that scale as @xmath1 operations and @xmath3 memory for @xmath2 electrons . cc theory is a systematically improvable hierarchy of methods @xcite indexed by @xmath6 that scale as @xmath7 operations and @xmath8 memory . in practice , dft is limited by accuracy and cc theory is limited by cost . the random phase approximation ( rpa ) is a natural point of convergence in the ongoing development of cc theory and dft . rpa emerges from truncated versions of cc methods as reduced - complexity methods that retain a significant fraction of the cc correlation energy @xcite . rpa also occurs when dft is used to approximate the polarization function in the adiabatic - connection fluctuation - dissipation formula for the correlation energy @xcite . this formula might become a part of more accurate density functionals , but it has a higher cost and complexity @xcite than conventional density functionals . in both cases , a balance of cost and accuracy is being sought with rpa . the electron correlation energy model that we consider in this paper is the rpa correlation energy plus the second - order screened exchange ( sosex ) energy @xcite . rpa+sosex is exact for one electron and to second order in perturbation theory . it agrees with quantum monte carlo benchmarks of the uniform electron gas @xcite to within 0.002 ha / electron @xcite . benchmarks on inhomogeneous systems show mean absolute errors of 0.002 ha / atom for cohesive energies of solids in a 5-solid test set @xcite , 0.008 ha / molecule for atomization energies of molecules in the g2 - 1 test set @xcite , and 0.010 ha / atom for correlation energies of first and second row atoms @xcite . these atomic and molecular rpa+sosex results approach but fail to surpass the accuracy of second - order mller - plesset ( mp2 ) perturbation theory @xcite and the b3lyp density functional @xcite . also , any rpa+sosex energy depends on a mean field choice and is not unique @xcite . the goal of this paper is to further develop rpa+sosex as a compromise between the cost of dft and the accuracy of cc theory . known algorithms for rpa+sosex @xcite and similar models @xcite require @xmath0 operations and either @xmath1 or @xmath9 memory . by utilizing an auxiliary basis set @xcite , fast interaction kernel summation @xcite , and a low - rank approximation of energy denominators @xcite , we design a new algorithm to reduce the cost to @xmath1 operations and @xmath3 memory . we define a precise rpa+sosex total energy with a unique choice of mean field based on brueckner orbitals @xcite that is designed to approximate brueckner coupled - cluster doubles ( bccd ) theory @xcite . this is referred to as brueckner rpa ( brpa ) theory . it is important to be pragmatic about the short - term value of brpa theory and a fast rpa+sosex algorithm . despite its construction , brpa is not a good approximation of bccd . the approximations used to retain the rpa+sosex form are too crude . for a given basis , a fast rpa+sosex calculation will have a large cost relative to conventional dft . with the continued reduction of average errors in density functionals @xcite and a correlation of dft and rpa error outliers @xcite , it is unclear whether the cost is warranted without additional research . the paper proceeds as follows . brpa theory is derived in sec . [ brpatheory ] as a truncation of bccd theory . the scf structure of brpa theory is emphasized . the three main components of a fast rpa+sosex algorithm are introduced in sec . [ fast_components ] . this includes a nonstandard choice of primary and auxiliary basis , which simplifies the cost accounting and algorithm design . in sec . [ rpastructure ] , the tensor structure in existing rpa algorithms @xcite is reviewed and extended to rpa+sosex . the novel structure of rpa+sosex theory enables a significant reduction in the number of its variables . fast and conventional rpa+sosex algorithms are designed in sec . [ fast_design ] as pseudocode . a detailed leading - order cost analysis is given instead of a crude scaling analysis . we study applications to a semiempirical h@xmath5 model in sec . [ applications ] . the accuracy of brpa theory is tested on h@xmath4 , and the scaling of rpa+sosex algorithms is confirmed with h@xmath5 calculations for large @xmath2 . the implications for future work are discussed in sec . [ discussion ] . this includes implementation in established electronic structure codes , basis set convergence problems , and further development of correlation models . we conclude in sec . [ conclusions ] with a brief consideration of rpa as a distinct electronic structure paradigm . | + the random - phase approximation with second - order screened exchange ( rpa+sosex ) is a model of electron correlation energy with two caveats : its accuracy depends on an arbitrary choice of mean field , and it scales as operations and memory for electrons . we derive a new algorithm that reduces its scaling to operations and memory using controlled approximations and a new self - consistent field that approximates brueckner coupled - cluster doubles ( bccd ) theory with rpa+sosex , referred to as brueckner rpa ( brpa ) theory . the algorithm comparably reduces the scaling of second - order mller - plesset ( mp2 ) perturbation theory with smaller cost prefactors than rpa+sosex . within a semiempirical model , we study h dissociation to test accuracy and h rings to verify scaling . | + the random - phase approximation with second - order screened exchange ( rpa+sosex ) is a model of electron correlation energy with two caveats : its accuracy depends on an arbitrary choice of mean field , and it scales as operations and memory for electrons . we derive a new algorithm that reduces its scaling to operations and memory using controlled approximations and a new self - consistent field that approximates brueckner coupled - cluster doubles ( bccd ) theory with rpa+sosex , referred to as brueckner rpa ( brpa ) theory . the algorithm comparably reduces the scaling of second - order mller - plesset ( mp2 ) perturbation theory with smaller cost prefactors than rpa+sosex . within a semiempirical model , we study h dissociation to test accuracy and h rings to verify scaling . |
astro-ph0002070 | c | all the previous section should have made clear that globular clusters can be used for a very wide variety of studies . they can constrain the star formation history of galaxies , in particular on the two or more distinct epochs of star formation in early - type galaxies . they can help explaining the building up of spiral galaxies , and the star formation in violent interactions . they can be useful to study galaxy dynamics at large galactocentric radii . and finally , they provide an accurate distance indicator , independent of cepheids and the distance to the lmc . this makes the study of globular cluster systems one of the most versatile fields in astronomy . extragalactic globular cluster research experienced a boom in the early 90s with the first generation of reliable ccds , and the first imaging from space . we can expect a continuation of the improvement of optical imaging , but more important , the field will benefit from the advancement in near - infrared imaging , and most of all , of the upcoming multi - object spectrographs on 10 m telescopes . the next little revolution in this subject will come with the determination of hundreds of globular cluster abundances around a large number of galaxies . the next 5 years will be an exiting time . first of all , i would like to thank the astronomische gesellschaft for awarding me the ludwig - bierman price . i feel extremely honored and proud . for his constant support , i would like to thank tom richtler , who introduced me to the fascinating subject of globular clusters . for the most recent years , i would like to thank jean brodie for her collaboration and for giving me the first opportunity to use a 10 m telescope to satisfy my curiosity . i am grateful to my current collaborators thomas puzia , claudia maraston , daniel thomas , denis burgarella , veronique buat , sandra chapelon , michael hilker , dante minniti , paul goudfrooij , linda schroder and many others , for helping me to keep up the flame . as usual , i would be lost without karl gebhardt s codes and sharp ideas . i am grateful to klaas de boer and simona zaggia for comments on various points . and last but not least , i am very thankful to steve zepf for a critical reading of the manuscript . | in particular , we address the star - formation histories of early - type galaxies ; globular cluster population in late - type galaxies and their link to early - type galaxies ; star and cluster formation during mergers and violent interactions ; and the kinematics at large radii in early - type galaxies . | _ we present an overview of observational progress in the study of extragalactic globular cluster systems . globular clusters turn out to be excellent tracers not only for the star - formation histories in galaxies , but also for kinematics at large galactocentric radii . their properties can be used to efficiently constrain galaxy formation and evolution . after a brief introduction of the current methods and futures perspectives , we summarize the knowledge gained in various areas of galaxy research through the study of globular clusters . in particular , we address the star - formation histories of early - type galaxies ; globular cluster population in late - type galaxies and their link to early - type galaxies ; star and cluster formation during mergers and violent interactions ; and the kinematics at large radii in early - type galaxies . the different points are reviewed within the context of galaxy formation and evolution . _ finally , we revisit the globular cluster luminosity function as a distance indicator . despite its low popularity in the literature , we demonstrate that it ranks among one of the most precise distance indicators to early - type galaxies , provided that it is applied properly . * extragalactic globular cluster systems * + * a new perspective on galaxy formation and evolution * + markus kissler - patig + european southern observatory + karl - schwarzschild - str . 2 , 85748 garching , germany + mkissler@eso.org , http://www.eso.org/mkissler |
1203.0760 | i | this paper concerns the theory of constant mean curvature ( _ cmc _ for short ) surfaces @xmath3 in @xmath1 . the value @xmath3 is critical in the sense that there is no compact cmc surface for @xmath4 while for @xmath5 there are rotational compact examples . a half - space theorem in @xmath1 ( see @xcite ) proves that for cmc @xmath3 , complete multigraphs are entire graphs over @xmath2 . entire graphs are classified by i. fernndez and p. mira @xcite and their moduli space is modeled on the set of quadratic holomorphic differential @xmath6 defined on the complex plane @xmath7 or the unit disk @xmath8 . the link between @xmath6 and the geometry of the graph is not very well understood . we first deal with complete conformal immersions of the disk @xmath8 , properly immersed into the half - space @xmath9 ( @xmath10 ) , which are entire vertical graphs over @xmath2 . we assume that the third coordinate @xmath11 on any diverging sequence of points in @xmath8 , which means the height function is proper . up to this date , the only simply connected example is a rotational example called the _ hyperboloid _ @xmath12 . in the poincar disk model of @xmath1 see below with polar coordinates @xmath13 , a parametrization of @xmath12 as a graph over @xmath2 is : @xmath14 we describe a family of examples endowed with a structure of infinite dimensional smooth manifold . the manifold structure arises from a suitable _ compactification _ of the mean curvature operator at infinity ( theorem [ thm : compcourb ] ) and is diffeomorphic to a codimension one submanifold of @xmath15 ( theorem [ thm : structg ] ) . this construction comes with a control of the asymptotic behavior in terms of the horizontal ( hyperbolic ) distance from the hyperboloid @xmath12 , namely : for any small @xmath16 such that @xmath17 has unit @xmath18-norm , there exists a cmc-@xmath0 complete entire graph at asymptotic horizontal signed distance @xmath19 from @xmath12 . these graphs are interesting , since any connected complete embedded cmc-@xmath0 surface in @xmath1 which is contained in the half - space @xmath9 and has a proper height function is a vertical entire graph . indeed , apply alexandrov reflection principle to such an immersion with respect to the horizontal slices . namely , reflect through the slice the part of the surface situated below it to obtain a surface which is a bigraph i.e. a graph over each side of the slice . there will be no first point of tangent contact between the initial surface and the part of the bigraph which is not a part of the surface , since there is no compact cmc-@xmath0 surface in @xmath1 . we also prove a half - space property for these entire graphs : let @xmath20 be a cmc-@xmath0 surface which is properly immersed in @xmath1 , lies on one side of a cmc-@xmath0 entire graph @xmath21 in the aforementioned family and is well oriented with respect to @xmath21 . then @xmath20 coincides with @xmath21 up to a vertical translation . the `` well oriented '' assumption is in the sense of l. mazet @xcite and means that if @xmath20 is below @xmath21 , its mean curvature vector points to @xmath21 . we use this result to show an asymptotic rigidity in our family of cmc-@xmath0 entire graphs ( theorem [ thm : rigidinf ] ) . namely , if two graphs in the family are at the same asymptotic horizontal signed distance from the hyperboloid @xmath12 , they coincide up to a vertical translation . in @xmath1 , r. s earp and e. toubiana @xcite construct a one - parameter family of cmc @xmath3 annuli which are rotationally invariant around a vertical geodesic . recently , l. mazet has shown @xcite that for @xmath5 , cmc annuli which are cylindrically bounded around a vertical geodesic are rotational examples . though annuli are not cylindrically bounded for @xmath3 , we prove that in a bounded tubular neighborhood of a rotational example , there are annuli , eventually embedded , which are asymptotic to different rotational examples with different axis : there exist cmc-@xmath0 annuli in @xmath1 with vertical ends , that are asymptotic regarding the horizontal hyperbolic distance to rotational examples with different vertical axis . it means that contrary to the case of embedded minimal surfaces in @xmath22 with finite total curvature and horizontal ends @xcite , the notion of _ axis _ is not relevant in general for cmc-@xmath0 annuli with vertical ends in @xmath1 . let @xmath23 |z| < 1 \rac$ ] be the open unit disk , @xmath24 |z| \leq 1 \rac$ ] its closure and @xmath13 the polar coordinates on @xmath25 . we use two standard models of @xmath1 , which are the minkowski model : @xmath26 where @xmath1 is seen as a subspace of the @xmath27-dimensional minkowski space @xmath28 , and the poincar disk model : @xmath29 the vector field associated to the third coordinate is denoted @xmath30 . in the poincar disk model , the hyperbolic radius @xmath31 of a point @xmath32 is : @xmath33 and we will need the following formula in the proof of proposition [ prop : graphdeform ] : @xmath34 we call vertical graphs in @xmath1 , immersions which are complete graphs over an open subset of the slice @xmath35 , and we call vertical annuli in @xmath1 , immersions which are complete vertical bigraphs . given surfaces @xmath36 in @xmath1 admitting parametrizations in the poincar disk model respectively : @xmath37 the hyperbolic horizontal signed distance @xmath38 between @xmath21 and @xmath39 at height @xmath40 and in the direction @xmath41 is the difference of their hyperbolic radii in the slice @xmath42 and direction @xmath41 : @xmath43 when it exists , the _ asymptotic hyperbolic horizontal signed distance _ between @xmath21 and @xmath39 in the direction @xmath41 is the limit @xmath44 . for any @xmath45 , let @xmath46 be the domain @xmath47 . we consider the set of _ admissible domains _ @xmath48 . the boundary at infinity @xmath49 of @xmath2 is identified with @xmath50 . given @xmath51 , the spaces @xmath52 and @xmath53 , with @xmath54 and @xmath55 , are respectively the usual hlder space and the subspace of functions that are zero on the boundary of @xmath56 . finally , we consider the spaces @xmath57 endowed with the natural scalar product denoted @xmath58 and hilbert norm @xmath59 . | we show that a particular family of complete entire graphs over admits a structure of infinite dimensional manifold with local control on the behaviors at infinity . deforming non degenerate constant mean curvature annuli , we provide a large class of ( non rotational ) examples and construct ( possibly embedded ) annuli without axis , i.e. with two vertical , asymptotically rotational , non aligned ends . | _ we study constant mean curvature surfaces in that admit a compactification of the mean curvature operator . we show that a particular family of complete entire graphs over admits a structure of infinite dimensional manifold with local control on the behaviors at infinity . these graphs also appear to have a half - space property and we deduce a uniqueness result at infinity . deforming non degenerate constant mean curvature annuli , we provide a large class of ( non rotational ) examples and construct ( possibly embedded ) annuli without axis , i.e. with two vertical , asymptotically rotational , non aligned ends . _ _ mathematics subject classification : _ _ 53a10 , 53c42_. |
1009.2741 | i | transport coefficients are essential inputs to describe the space - time evolution of systems not far from equilibrium . during the last years there has been a very active effort to analyze them from both the theoretical and phenomenological points of view in the context of heavy - ion collisions , condensed matter physics , astrophysics and cosmology . the calculation of transport coefficients in quantum field theory at intermediate and strong coupling is still a challenge from both the analytical and the numerical points of view . due to their intrinsic non - perturbative nature , even in weakly interacting theories a resummation of an infinite number of diagrams is needed in order to obtain the leading - order result . in the strongly coupled regime , the most prominent method available is the ads / cft correspondence , although it is only applicable to a limited class of field theories . on the other hand , lattice simulations are still not accurate enough regarding the calculation of spectral densities , and the introduction of a finite quark chemical potential makes things even more difficult because of the sign problem . it was recently conjectured , based on a sum rule for the spectral density of the trace of the energy - momentum tensor in yang - mills theory @xcite , that a maximum of the trace anomaly near the critical temperature might drive a maximum for the bulk viscosity near that temperature . the corresponding sum rule was later corrected in @xcite , and the ansatz for the spectral density used to extract the bulk viscosity questioned @xcite . since the trace anomaly measures the breaking of scale invariance in a system , and the bulk viscosity @xmath1 essentially represents the difficulty for a system to relax back to equilibrium after a scale transformation , it seems in principle reasonable to think that @xmath1 would be maximum when the breaking of scale invariance is maximum . in heavy - ion phenomenology , bulk viscosity has usually been neglected because it is expected to be much smaller than the shear viscosity even at temperatures not very high @xcite . however , as suggested by the analysis of @xcite , non - perturbative phenomena responsible for the main contribution to the trace anomaly near @xmath2 could also produce a significant increase in the bulk viscosity . in this paper i will present an explicit calculation in the massive gross - neveu model in @xmath3 dimensions , where the correlation between trace anomaly and bulk viscosity can be accurately tested . i will not try to give an estimation for the absolute value of @xmath1 in qcd near the phase transition though ; as we will see this model is not suitable for that purpose . there are several works analyzing this issue employing different approaches ( see for instance @xcite and references therein ) , but still the order of magnitude of the bulk viscosity near the crossover temperature is uncertain . in 1 + 1 dimensions , transverse flow of momentum is not possible , and the bulk viscosity is the only viscous coefficient present to linear order in gradients . in this paper i will analyze only finite temperature effects , considering a vanishing fermion chemical potential , thus the thermal conductivity will be zero in this case . therefore , the only constitutive equation relevant for us is . ] @xmath4 with @xmath5 the non - equilibrium expectation value for the spatial component of the energy - momentum tensor , @xmath6 is the pressure in equilibrium , and @xmath7 is the fluid velocity . the bulk viscosity can be in principle calculated perturbatively in field theory @xcite : @xmath8 where @xmath9 is the spectral density corresponding to the thermal propagator @xmath10 . here though , i will use a kinetic theory approach , which should be equivalent to the diagrammatic one in the perturbative ( and dilute ) regime @xcite . this paper is organized as follows . in sections [ sec : mgnvac ] and [ sec : mgntemp ] , i review well known properties of the massive gross - neveu model at zero and finite temperature , and i prove the breaking of integrability in the large-@xmath0 limit when a mass term for the fermion field is explicitly introduced . then in section [ sec : kinetictheory ] , the calculation of the bulk viscosity within kinetic theory is presented . in section [ sec : discussion ] , i comment on sum rules and implications of the previous analysis for other systems . finally in [ sec : conclusions ] i summarize the main conclusions . there is also the appendix [ app : fermions ] , where the result of factorization for fermion loops in @xmath3 dimensions is derived , and appendix [ app : amplitudes ] where the reader can find some details on the calculation of the inelastic scattering amplitude . | a calculation of the bulk viscosity for the massive gross - neveu model at zero fermion chemical potential is presented in the large- limit . the main motivation is , by decreasing the bare mass , to analyze whether there is a correlation between the maximum in the trace anomaly and a possible maximum in the bulk viscosity , as recently conjectured . i also comment on the sum rule for the spectral density in the bulk channel , as well as on implications of this analysis for other systems . | a calculation of the bulk viscosity for the massive gross - neveu model at zero fermion chemical potential is presented in the large- limit . this model resembles qcd in many important aspects : it is asymptotically free , has a dynamically generated mass gap , and for zero bare fermion mass it is scale invariant at the classical level ( broken through the trace anomaly at the quantum level ) . for our purposes , the introduction of a bare fermion mass is necessary to break the integrability of the model , and thus to be able to study momentum transport . the main motivation is , by decreasing the bare mass , to analyze whether there is a correlation between the maximum in the trace anomaly and a possible maximum in the bulk viscosity , as recently conjectured . after numerical analysis , i find that there is no direct correlation between these two quantities : the bulk viscosity of the model is a monotonously decreasing function of the temperature . i also comment on the sum rule for the spectral density in the bulk channel , as well as on implications of this analysis for other systems . |
1207.0644 | i | a relativistic charge in a rectilinear and uniform motion can originate an electromagnetic radiative mechanism crossing the dielectric interface between two different media . the fast dipolar oscillation of the polarization charge , induced on the dielectric interface by the relativistic charge , generates indeed an instantaneous , broad spectral band , radially polarized and highly directional radiation emission , the so called transition radiation @xcite . a photon pulse propagates backward and forward from the dielectric interface according to a double conical spatial distribution whose angular aperture scales down with the lorentz @xmath1 factor of the charge . the higher and steeper the discontinuity of the dielectric constant across the interface , the more intense the radiation emission . thanks to the relativistic origin , transition radiation can be fruitfully exploited for energy detection or mass identification of high energy particles @xcite . transition radiation based diagnostics is commonly used in a particle accelerator to monitor the transverse @xcite or the longitudinal profile of a charged beam @xcite . transition radiators , typically made of a thin metallic foil or a thin polished aluminium coating on dielectric substrates , respond as an ideal conductor over a wide spectral range , from the long wavelength to far beyond the visible optical region @xcite . this experimental situation is supposed in the present work . in particular , the case of a @xmath0 electron bunch colliding at a normal angle of incidence onto a flat ideal conductor surface will be considered in the following . an ideal conductor surface being assimilable to a double layer of charge , transition radiation emission can be thus schematized as the result of the interaction of an incident charge with the conduction electrons of the ideal conductor surface . as the relativistic charge approaches the metallic surface , the charge induced conduction electrons freely move in the transverse plane to maintain the metallic surface equipotential . the sudden and fast dipolar oscillation of the double layer of charge is thus responsible for the emission of the radiation pulse as confirmed by the emission of backward radiation . in the relativistic limit , the electromagnetic field travelling with a relativistic electron can be assimilated to a transverse electromagnetic wave , the so called _ virtual quanta _ field @xcite . in the case of a single relativistic electron , the harmonic component of the electric field at a given wavelength @xmath2 extends indeed in the transverse plane over an efficacious range in the order of @xmath3 , the ratio of the longitudinal to the transverse component being @xmath4 . the _ virtual quanta _ field showing the nature of a quasi - plane wave front , the radiation emission can be formally described as the result of the wave propagation of the _ virtual quanta _ scattered by the metallic surface according to the huygens - fresnel principle @xcite . under the far field approximation @xcite , the harmonic components of the transition radiation field at the observation point can be calculated as the fourier transform of the _ virtual quanta _ field with respect to the spatial coordinates of the radiator surface . the transition radiation energy spectrum can be finally obtained as the flux of the poynting vector . a single and individually radiating electron and a high density electron beam are in general different from the point of view of the features of the spectral and the angular distribution of the emitted radiation . compared to the case of a single and individually radiating electron , bunch - density effects may strongly affect the electromagnetic radiative mechanism by an electron beam . for a fixed energy of the beam , the number and the angular distribution of the photons radiated at a given wavelength may indeed change as a function of the density of the electron beam . in the formula of the radiation energy spectrum , the charge form factor accounts for bunch collective effects . this is , in principle , defined as the square module of the fourier transform of the distribution function of the particle density of the charged beam . aim of the present paper is to verify how well such a theoretical definition of the charge form factor fits into a covariant formulation of the radiation energy spectrum or if , instead , under the covariance and the temporal causality constraints , the radiation energy spectrum shows a dependence on the charge density that goes beyond the above mentioned formal definition of the charge form factor . in fact , the transverse density of the electron beam is a relativistic invariant under a lorentz transformation with respect to the direction of motion of the beam . the invariance of the projection of the electron beam density onto the transverse plane with respect to the direction of motion is expected to leave , on the _ virtual quanta _ field and , consequently , on the radiation field , a covariant mark whose observability , for a given harmonic component of the radiation field , only depends on the transverse dimension of the electron beam compared to the transverse component of the wave - vector which is itself a lorentz invariant with respect to the direction of motion of the beam . in the following , the charge - density - like covariance of the transition radiation energy spectrum by an electron beam will be investigated . first , the formal steps leading from the electromagnetic field of a @xmath0 electron bunch to the transition radiation energy spectrum will be analyzed with regard to the temporal causality constraint . finally , the covariance of the given formula of the electromagnetic field of the @xmath0 electron bunch will be checked by verifying that , under a lorentz transformation from the laboratory to the rest reference frame , the expected formula of the electric field of a charge distribution at rest is obtained . | the case of a electron bunch hitting at a normal angle of incidence a flat metallic surface with arbitrary size and shape will be considered . the covariance feature of the transition radiation energy spectrum deals instead with the lorentz invariance of the projection of the electron spatial density in the transverse plane with respect to the direction of motion of the electron beam . | a charge - density - like covariance is expected to characterize the transition radiation energy spectrum of a electron bunch as far as the charge form factor is intended to account for bunch - density effects in the radiation emission . the beam charge passing from a single electron to a high density electron bunch , the covariance of the transition radiation energy is expected to evolve from a charge - point - like to a charge - density - like one . besides covariance , the radiation energy spectrum is expected to conform to the temporal causality principle : the single electron amplitudes composing the radiation field are expected to propagate from the metallic screen with relative emission phases causally correlated with the temporal sequence of the particle collisions onto the metallic screen . in the present paper , the case of a electron bunch hitting at a normal angle of incidence a flat metallic surface with arbitrary size and shape will be considered . for such an experimental situation , the distribution function of the electron longitudinal coordinates rules the temporal causality constraint into the transition radiation energy spectrum . the covariance feature of the transition radiation energy spectrum deals instead with the lorentz invariance of the projection of the electron spatial density in the transverse plane with respect to the direction of motion of the electron beam . because of the invariance of the electron transverse density under a lorentz transformation with respect to the direction of motion of the electron beam , the single electron radiation amplitudes composing the radiation field show a covariant dependence on the distribution function of the electron transverse coordinates , the relative emission phases of the single electron radiation amplitudes being indeed only a function of the electron longitudinal coordinates because of the temporal causality constraint . as a consequence of the temporal causality and the covariance , both the temporal coherent and incoherent components of the radiation energy spectrum bear the covariant imprinting of the distribution function of the electron transverse coordinates as in the following argued . virtual quanta , coherence , fourier transform , collective effects 41.60.-m , 41.75.-i , 42.25.kb , 42.30.kq |
1207.0644 | c | in the present paper , the issue of the covariant formulation of the radiation energy spectrum and , consequently , of the charge form factor of a @xmath0 electron bunch is argued . the @xmath0 electron bunch is supposed to collide , at a normal angle of incidence , onto a metallic screen @xmath7 which has an arbitrary shape and size ( either finite @xmath31 or infinite @xmath30 ) and is supposed to behave as an ideal conductor in the wavelength region of interest . as the covariance of a charged distribution is expected to evolve from a charge - point - like into a charge - density - like one when passing from a single electron to a @xmath0 electron bunch , the covariance of the radiation energy spectrum as well is expected to behave in the same way when a @xmath0 electron bunch is considered instead of a single electron . in order to verify the covariant formulation of the radiation energy spectrum , the formal steps leading from the virtual quanta field to the transition radiation energy spectrum of a @xmath0 electron bunch have been explicitly derived and checked to meet , first , the temporal causality constraint and , then , to be covariance consistent . finally , the expression of the _ virtual quanta _ field , which , in the present paper and in other papers @xcite , is the starting point to achieve the formula of the radiation energy spectrum , has been checked to transform in a covariant way under a lorentz transformation from the laboratory reference frame ( @xmath45 ) to the reference frame of rest of the @xmath0 electron bunch ( @xmath48 ) . in order to perform such a covariance check , the fourier representations of fields and charged distributions have been suitably extended from the ordinary @xmath41-dimensional space of the spatial coordinates and the conjugate fourier wave - vectors into the lorentz @xmath42-spaces of the space - time and of the conjugate fourier wavevector - frequency in both @xmath45 and @xmath48 . in a temporal causal and covariant formulation of the transition radiation energy spectrum of a @xmath0 electron bunch , the invariance of the projection of the electron density in the transverse plane with respect to the direction of motion manifests itself as a covariant feature of the @xmath0 single electron amplitude composing the radiation field and , consequently , of both the temporal coherent and incoherent components of the transition radiation energy spectrum . | a charge - density - like covariance is expected to characterize the transition radiation energy spectrum of a electron bunch as far as the charge form factor is intended to account for bunch - density effects in the radiation emission . the beam charge passing from a single electron to a high density electron bunch , the covariance of the transition radiation energy is expected to evolve from a charge - point - like to a charge - density - like one . besides covariance | a charge - density - like covariance is expected to characterize the transition radiation energy spectrum of a electron bunch as far as the charge form factor is intended to account for bunch - density effects in the radiation emission . the beam charge passing from a single electron to a high density electron bunch , the covariance of the transition radiation energy is expected to evolve from a charge - point - like to a charge - density - like one . besides covariance , the radiation energy spectrum is expected to conform to the temporal causality principle : the single electron amplitudes composing the radiation field are expected to propagate from the metallic screen with relative emission phases causally correlated with the temporal sequence of the particle collisions onto the metallic screen . in the present paper , the case of a electron bunch hitting at a normal angle of incidence a flat metallic surface with arbitrary size and shape will be considered . for such an experimental situation , the distribution function of the electron longitudinal coordinates rules the temporal causality constraint into the transition radiation energy spectrum . the covariance feature of the transition radiation energy spectrum deals instead with the lorentz invariance of the projection of the electron spatial density in the transverse plane with respect to the direction of motion of the electron beam . because of the invariance of the electron transverse density under a lorentz transformation with respect to the direction of motion of the electron beam , the single electron radiation amplitudes composing the radiation field show a covariant dependence on the distribution function of the electron transverse coordinates , the relative emission phases of the single electron radiation amplitudes being indeed only a function of the electron longitudinal coordinates because of the temporal causality constraint . as a consequence of the temporal causality and the covariance , both the temporal coherent and incoherent components of the radiation energy spectrum bear the covariant imprinting of the distribution function of the electron transverse coordinates as in the following argued . virtual quanta , coherence , fourier transform , collective effects 41.60.-m , 41.75.-i , 42.25.kb , 42.30.kq |
1003.2184 | i | [ s-01 ] the surfaces possessing nontrivial deformations which preserve principal curvatures and directions ( or , equivalently , the shape operator ) were investigated by several authors , see @xcite , @xcite and review with bibliography in @xcite . it is known that surfaces with one family of principal curves being geodesic ( as for parallel curved ( pc ) surfaces recently studied in @xcite@xcite ) represent degenerate case in studying immersions of simply connected surfaces with a prescribed shape operator . recent studying of reconstruction of surfaces by their partially given principal curvatures and directions may be useful for applications of differential geometry to computer graphics , the wavefront analysis in applied optics , etc . in what follows , @xmath1 denotes a @xmath2-regular riemannian 3-space with coordinates @xmath3 , @xmath4 ( @xmath5 ) for some @xmath6 , @xmath7 a @xmath2-regular surface , @xmath8 the coordinate curve , and @xmath9 the _ curvilinear projection_. in the paper we consider surfaces - graphs @xmath10 with _ prescribed curvilinear projection ( onto @xmath0 ) of one field of principal directions_. we show ( see theorem [ t - main0 ] ) that such surfaces depend on two arbitrary functions of one variable , namely , the principal curvatures over @xmath11 which are assumed close enough to corresponding values for @xmath0 . more precisely , we study the following . [ prob-1]given @xmath1 , a vector field @xmath12 transversal to @xmath11 on @xmath0 , functions @xmath13 and @xmath14 of class @xmath15 , * find * a function @xmath16 of class @xmath17 , whose graph @xmath18 in @xmath19 satisfies the conditions : \(i ) _ the projection ( @xmath20 ) onto @xmath0 of the field @xmath21 of principal directions _ _ corresponding to the principal curvature @xmath22 ( of @xmath23 ) coincides with @xmath12 _ , \(ii ) _ the principal curvatures @xmath24 ( of @xmath23 ) over @xmath11 coincide with @xmath25 : @xmath26 _ , \(iii ) _ the values of @xmath16 and @xmath27 at the point @xmath28 of @xmath11 are given_. our approach is based on reducing the * problem [ prob-1 ] * to the cauchy problem for a quasilinear system of pde s which , under certain conditions for data , is hyperbolic and admits a unique local smooth solution . the pc surfaces in @xmath29 represent a special class of solutions when a family of curvature lines projects onto @xmath0 as parallel lines or concentric circles . such surfaces are recovered by an iteration function sequence and using the reconstruction of planar curves by their curvature ( see theorem [ t - rpc2 ] and proposition [ t - rpc1 ] ) . notice that the space of pc surfaces free of umbilics and having the same shape operator depends on arbitrary function of one variable , see @xcite . the structure of the work is the following . * section [ sec : main - res ] * represents main results ( theorems [ t - main0][t - rpc2 ] and corollary [ t - main3 ] for @xmath30 ) . * section [ sec : proofs ] * contains proofs . * section [ sec : appendix ] * contains necessary facts on pc surfaces and examples . | a class of surfaces - graphs in a riemannian 3-space with a prescribed projection of one field of principal directions onto a surface is considered . a problem of determination of such surfaces when both principal curvatures are given over a line in is formulated and studied . the geometric problem is reduced to the cauchy problem for quasilinear pde s which , under certain conditions for data , are hyperbolic and admit a unique solution . such solutions may be found by an iteration function sequence . | a class of surfaces - graphs in a riemannian 3-space with a prescribed projection of one field of principal directions onto a surface is considered . a problem of determination of such surfaces when both principal curvatures are given over a line in is formulated and studied . the geometric problem is reduced to the cauchy problem for quasilinear pde s which , under certain conditions for data , are hyperbolic and admit a unique solution . it is shown that the parallel curved ( pc ) surfaces in space forms provide a special class of global solutions to the geometrical problem with weaker regularity assumptions . such solutions may be found by an iteration function sequence . riemannian space , surface , principal curvature / direction , hyperbolic pde s * ams subject classification : * 53b20 , 53b25 , 53c99 , 53a05 |
1107.4784 | i | conformal blocks , which are defined on the ( punctured ) riemann surfaces , holomorphic in each @xmath1 coordinate except when they meet each other , play an essential role in building correlation functions in two dimensional ( euclidean ) conformal field theories@xcite . they can be best understood as sewing together chiral vertex operators@xcite , which by definition , are not local objects , but the correlation functions are . the later combine both holomorphic and anti - holomorphic conformal blocks in a consistent way to make modular covariant objects . on the sphere , the @xmath2-point conformal block is represented graphically as in fig.[confblock ] , where @xmath3 is the conformal dimension of the primary field inserted at coordinate @xmath1 , and @xmath4 labels the contribution arising from the conformal family descending from a primary field with the conformal dimension @xmath4 . the global conformal invariance is @xmath5 , which may be used to fix three coordinates @xmath6 , @xmath7 and @xmath8 . so the independent variables are @xmath1 , @xmath9 , with the degrees of freedom @xmath10 for the @xmath2-point conformal blocks on the sphere . the calculation of conformal blocks is based on the conformal ward - identities , @xmath11=(z^{n+1}\partial_z + ( n+1)h z^n ) v_h(z).\ ] ] and carried out perturbatively level by level @xcite . in some special cases , the decoupling of the virasoro null vectors can be implemented as differential equations for the conformal blocks . for the general case , recursion relations have been proposed by zamolodchikov@xcite on the meromorphic structures of the conformal blocks either in complex @xmath12-plane or @xmath13-plane . however , in general , the global perspective of the sewing procedure for the conformal blocks was still not fully understood until recently when the agt duality @xcite had been proposed . agt conjecture relates 2d liouville conformal field theories to 4d @xmath14 supersymmetric gauge theories of the @xmath15 type . the main idea is coupling to the liouville field a @xmath16 field field is a gauge symmetry and can be fixed to any desired value . ] , then this system is dual to a @xmath17 superconformal 4d theory . in this case , the partition function by nekrasov instanton counting(nic)@xcite of the 4d @xmath18 theory is to be identified with the conformal blocks of the @xmath16 coupled liouville type . the liouville cft is characterized by a 2d one boson theory with center charge @xmath19 . finally , one can decouple the @xmath20 factor and obtain the instanton partition function of the @xmath21 theory which duals to liouville conformal blocks . liouville interaction breaks down the charge conservation explicitly and leads to the introduction of the screening charges . because of the existence of the screening charges , the conformal blocks of the liouville type is much more complicated than its counterpart of the @xmath16 free boson theory . however , the agt conjecture , if proven true , means that there exists an orthogonal basis upon which the @xmath22 conformal blocks are built . from the above reasoning , there exists a tree - like structure which describes the duality in coupling space of the @xmath14 4d superconformal linear quiver gauge theory . the primary objects for this tree - like structure is the `` bifundmental '' matter coupling , which , if translated correctly , should be represented by the inner products of the bra and ket descendant fields in 2d conformal families sandwiched by a `` primary '' vertex operator at position , say , @xmath23 . such kind of pants - like diagram can be sewed together to form a linear quiver diagram , which , on the 2d cft side , is just the @xmath2-point functions on the sphere for our consideration . of course , in the present context , we mean the @xmath0 2d cft . at first sight , it seems that such duality does not bring in any conveniences . however , the nekrasov instanton counting on the 4d field theory shows a rather compact form for the summands which are completely factorized in `` momentum '' @xmath24 . and the summation is well organized into the combinatorial enumeration of the young tableuax . this simple structure implies liouville theory , in particular , the evaluation of the liouville conformal blocks , could be resolved by embedding it into a bigger system . so one may expect a new construction for the liouville conformal blocks from the corresponding nic . -point conformal block on @xmath25,height=144 ] as pointed out by nakajima@xcite , the instanton counting for @xmath14 gauge theory is equivalent to the hilbert scheme of points on the corresponding seiberg - witten curve ( blow - up riemann surface)@xcite . this can be translated into a topological string description from physicists point of view . by invoking the d4-d0 brane setup@xcite for adhm construction@xcite of the instanton moduli space and the resolving process for ale singularities@xcite , these indicate that the instanton counting is a counting for d0 branes in a toric calabi - yau 3-fold . actually , there are two kinds of d0 branes in the calabi - yau 3-fold , one is the regular d0 brane , which is in regular representation of @xmath26 , the center of the corresponding ade group . it carries no flux and can move freely on the riemann surface . the other is the fractional d0 brane , which is a d2 brane wrapping on a zero - sized two sphere . it is always attached to the ale singularity since it has a nontrivial monodromy while moving around the singularity . it is these fractional d0 branes that resolve the ale singularity , and leave fluxes on the blow - up riemann surface . this property ensures that one can identify these fractional instantons as `` anyons '' on the riemann surface . on the other hand , the regular ones are `` electric charged '' particles on the riemann surface . so the total counting is equvalent to solving the problem of `` electron gas '' system with insertions of anyons at the blow - up singularity on the riemann surface . this point of view is partialy included in dijkgraaf and vafa s article@xcite . for each pants of the pants decomposition for the ( punctured ) riemann surface , one can guess that the instanton partition function can be rewritten as summation over all the intermediate states passing through the sewn holes@xcite . for the interests of the present paper , we concern ourselves only with the special pants diagram that one of the tubes is replaced by the blow - up singularity . then the summand in the instanton partition function represents itself as an inner product of the bra and ket states , sandwiched by the anyonic vertex operator . these bra and ket states should come from the interacting `` electronic '' particles . a candidate description of the `` electronic gas '' system is the integrable system of multiple calogero - sutherland model , each living on a cycle . the whole ( punctured ) riemann surface , can be obtained by sewing together these pants on nonintersecting cycles . there are many efforts on relating the conformal blocks to the nic@xcite from various points of views , and these works confirm the validity of the agt duality . however , the explicit construction for the liouville conformal blocks has remained largely unclear until the recent work @xcite by alba , fateev , litvinov and tarnopolsky . in @xcite , they have put forward the agt duality in a more explicit form [ aflt ] & = & z_bif(|p, ; p , ) , here specifically for a free field realization , @xmath27 with @xmath28 , and @xmath29 is the screening charge in the virasoro sector . the l.h.s . of eq([aflt ] ) is the pants - like ( with one of the tubes labeled by @xmath30 shrinks to a line ) conformal block . the r.h.s . of ( [ aflt ] ) reproduces @xmath31 for the instanton counting , which is given by [ zbif]z_bif(|p, ; p , ) = _ i , j=1 ^ 2 _ sy_i(q - e_y_i , y_j(p_i - p_j|s ) - ) _ ty_j ( e_y_j , y_i(p_j - p_i|t)-),where @xmath32 , @xmath33 and e_y , y(p|s ) p+b^-1(a_y(s)+1)-bl_y(s ) . here @xmath34 and @xmath35 resp . are the arm length and the leg length resp . of the box @xmath36 in the young tableau @xmath37 , defined as @xmath38 @xmath39 and @xmath40 resp . are the @xmath41-th part of the partition @xmath42 and the @xmath43-th part of the transpose partition @xmath44 respectively . ( [ aflt ] ) means that the matrix elements of a special `` chiral vertex operator '' @xmath45 in a suitably chosen basis , can be translated into a 4d theory as an instanton contribution for a special bifundamental contribution of the nic . by sewing together pants - like diagrams one gets any desired duality diagrams in the coupling space of the linear quiver gauge theory . so , the checking of the agt duality reduces to the construction of the states @xmath46 , which we shall call the aflt states@xcite , with @xmath47 the young tableaux . here the @xmath37 s , the partitions of natural numbers , or equivalently represented by young tableaux , are labels for the orthogonal basis for the descendant fields ( verma modules ) in a @xmath0 conformal family from the 2d cft point of view . by definition , the aflt states form a complete set of states for the family members in a given @xmath0 conformal family and the inner products between them , sandwiched by a vertex operator of the particular form , @xmath48 , at position , say , @xmath49 , is factorized exactly as the nic @xmath31 presented on the r.h.s . of ( [ aflt ] ) . the explicit formula , ( [ aflt ] ) , puts strong constraints on the possible forms of the aflt states and make a systematic construction of them unaccessible at first glance . in @xcite , only the explicit form of the state @xmath50 has been found , @xmath51 with @xmath52 the creator @xmath53 s valued jack symmetric function , and @xmath54 the normalization constant . in our opinion , the agt conjecture , written in the form of ( [ aflt ] ) , strongly suggests that the @xmath0 conformal family is a hamiltonian system with @xmath46 the hamiltonian eigenstates . so the construction of the aflt states becomes a quantum mechanical problem of solving the schrodinger equation . put things in this way , we propose a possible form of the hamiltonian @xmath55 and construct its eigenstates explicitly . we shall identify those eigenstates as the aflt states desired . for in all the cases we have checked , ( [ aflt ] ) is verified to be true , using the aflt states we have constructed . we shall present now as the main results of our present paper the explicit form of the hamiltonian @xmath55 along with the complete construction of the aflt states , @xmath46 . more elaborated exposition will come in the subsequent sections . [ main1 ] h&=&h_0+h_i + _ & = & j_-|p_(p).here , @xmath56 are the jack states constructed in terms of the oscillators @xmath57 s or @xmath58 s ( @xmath59 ) solely , @xmath60 the corresponding hamiltonian for the jack symmetric functions , @xmath61 . thus the eigenstate of @xmath62 is just @xmath63 with the eigenvalue @xmath64 . @xmath60 in our formalism is defined to include zero modes @xmath65 also , @xmath66 . it is important that @xmath67 is strictly triangular with respect to the basis vectors of the @xmath62 eigenstates . by `` strictly triangular '' we mean the ( upper or lower ) triangular matrix with zero diagonal entries . it is easy to see that if the interaction term @xmath67 is strictly triangular , then the eigenvalue spectrum of @xmath62 remains unperturbed and @xmath46 in ( [ main1 ] ) well defined for non - degenerate @xmath62 spectrum descending from a mother state @xmath68 for generic values of @xmath24 s . putting things all together , we have [ main2 ] & & h = h_0+h_i , h_0= h^+ + h^- , h_i=_n=1 ^ 2qn a_-n^+a_n^- , + & & h^=(z_z^)^3 + _ n=1^q n a_-n^a_n^ , + & & e_(p)=e_y_1+e_y_2 + 2p(|y_1|-|y_2| ) , e_y=_i ( y^2_ib^-1+(2i-1)y_i b ) , + & & _ ( p)= ( -)^|y_1|b^|y_1|+|y_2|_y_1(2p+(a_y_1 + 1)b^-1-l_y_2b ) _ y_2(2p - a_y_2b^-1 + ( l_y_1 + 1)b ) , + & & _ = j_-|p_(p ) , + & & h_0j_-=e_(p)j_- , h_=e_(p ) _ , -i a^_0=p notice that + 1 ) @xmath50 constructed in @xcite are included in our construction as subcases . + 2 ) the hamiltonian @xmath55 constructed by us , albeit in a disguised form , turns out to coincide up to some trivial factor with @xmath69 , one of the integrals of motion found in a different context in appendix c of @xcite . @xmath69 in @xcite , written in the form of @xmath0 , makes the virasoro symmetry manifest , but is not suitable for solving a perturbation theory with perturbation parameter @xmath70 . the hamiltonian @xmath55 written in terms of the interacting bi - jack polynomial system as in ( [ main2 ] ) , shows virasoro symmetry only implicitly , but makes the perturbation theory exactly solvable as we shall see soon after . the procedure is outlined as follows . on the 2d cft side , the @xmath71 theory can be represented as a theory of two independent scalars @xmath72 and @xmath73 . @xmath72 part is essentially a free theory of timelike oscillators , while the scalar field @xmath74 is spacelike but engaged in a liouville type interaction . the two scalars can be linearly combined to form the `` light - cone '' scalars . @xmath75 and @xmath76 . the labeling @xmath77 of the basis vectors strongly suggests that there exist a bi - jack polynomial structure , plus possibly some interactions between these two sectors . that is , the `` free '' @xmath60 spectrums should be described by @xmath78 and @xmath79 respectively , here @xmath80 denotes jack states related to young tableau @xmath37 . first we construct the `` unperturbed '' energy operator @xmath62 which just sums up the `` energies '' in @xmath81 sectors , @xmath82 . the next thing is to specify the interaction between these two sectors . strictly speaking , @xmath62 does not describe a free theory , since it also contains the interaction terms proportional to @xmath83 . but the new interaction term @xmath67 further mixes the @xmath84 s and the coupling is also a first order in @xmath83 . it is good to see that @xmath67 is strictly triangular with respect to the basis vectors of @xmath62 eigenstates . our method can be easily generalized to wider classes of integrable models , in which the interacting hamiltonian splits into two parts , @xmath85 and @xmath86 , representing respectively the shift of energies and the rotations ( mixings ) of states . the later keeps the eigenvalue spectrum untouched@xcite . besides being triangular , the form of the interaction term is however much restricted , also by the virasoro symmetry . since the total hamiltonian is of the form @xmath0 , an `` interaction energy operator '' @xmath67 is needed to make the `` full energy operator '' @xmath87 the combination of @xmath88 s and @xmath89 s only . once the hamiltonian structure is determined , then the construction of the hamiltonian eigenstate @xmath46 is just a quantum mechanical problem . @xmath62 and @xmath55 share the same eigenvalue spectrum , but only the eigenstates of @xmath55 , represented by @xmath46 s , form a complete set of basis vectors for the @xmath0 conformal family . we have checked by examples the corresponding agt duality formula , ( [ aflt ] ) up to level 4 , and have found that indeed nekrasov instanton counting can be reproduced with this construction , ( [ main2 ] ) . in fact , we have also checked more general cases and all get positive answers . but those more general results will appear elsewhere due to lacking of space to include them in this paper . the insertions of the screening charges play an important role in checking the agt duality . however , in the present work we concern ourselves only with the cases in which the screening charges can be detached away from the vertex operator @xmath45 and moved on to act on the aflt states ( similar to the felder s brst operators)@xcite . the more general cases in which screening charges can not be moved away from @xmath45 will be under our future studies . it is well known that it is possible to map the liouville theory to the analytic continuation of the calogero - sutherland(cs ) model , which was originally and in most cases considered to be a theory with the parameter @xmath90 , while in the liouville case @xmath91 is required . some explanation is given in @xcite . the physical space of the cs model are created by jack polynomials , which are symmetric functions studied in great detail in mathematics and physics literatures@xcite . the integrability of the cs model may be derived in different ways , e.g. , from the knowledge of the hidden @xmath92 symmetry of the model . a recursion relation related to the virasoro singular vectors and an integral representation based on it has appeared recently in @xcite , in which more references can be found on the subjects of the cs model and the jack symmetric functions . it should be stressed again that for @xmath90 , there is no null vectors in the cs model . so the `` null '' vectors are not the true null vectors of the cs model , since the virasoro algebra based on which the null vectors are constructed is not the true conformal algebra of the cs model in that case . but for @xmath91 , yes , there are null vectors in the cs model . it is possible to describe the @xmath93 theory in terms of the jack polynomials considered as analytic continuation from @xmath94 to @xmath95 . there is another hint that the @xmath93 theory has something to do with @xmath91 cs model . it can be found from the nekrasov partition function , in which each term in the summation can be written in the form of the carlsson - okounkov formula@xcite , for the special cases when no screening charges are inserted . carlsson - okounkov formula is a formula for the inner products between the bra jack states and the ket jack states sandwiched with a modified vertex operator . this extraordinary formula is of great help in checking the agt duality with our construction for the orthogonal basis vectors @xmath46 s defined in ( [ aflt ] ) . we notice that the construction we found shares many similarities with the construction of the jack functions themselves . namely , we take the state @xmath96 as the mother state and its descendants are constructed in such a manner that two partitions are `` squeezed '' into other pairs . the squeezing does not change the total level of the two partitions , but does make the inner products of the descendants a triangular form . although the 4d to 2d duality has just begun to be understood , it has been known for sometime that 2d conformal blocks can be equivalently described as insertions of wilson lines in 3d pure chern - simons topological gauge theory . in fact , we can interpret the @xmath2-point conformal block represented by fig.1 as a wilson line insertion inside a three - ball . the path integral in chern - simons - witten gauge theory thus creates a state living on the boundary of the three ball , which is punctured @xmath25 . so it should not be a too big surprise that 2d conformal field theory has something to do with higher dimensional quantum field theories . taking into account that jack symmetric polynomials can be taken as some special limit of the two parameter macdonald symmetric polynomials , one natural guess is that our construction can be generalized to the case of macdonald symmetric polynomials . in that case there should be a 5d to 3d duality . this paper is organized in the following way . our general formalism on the construction of the aflt states is presented in the introduction . in section 2 , we explore the general structure of the @xmath0 conformal family . we found in some cases it is more convenient to work with the bi - jack function basis . section 3 contains the major derivation of our construction . section 4 is the conclusion . and in appendix a the explicit construction of the aflt states up to level 3 is presented . | a complete construction of the aflt states is proposed . with this construction and for all the cases we have checked , the agt conjecture on the equivalence of nekrasov instanton counting ( nic ) to the conformal block has been verified to be true . bao shou , jian - feng wu , and ming yu .5 cm _ institute of theoretical physics , + chinese academy of sciences , beijing , 100190 , china _ | a complete construction of the aflt states is proposed . with this construction and for all the cases we have checked , the agt conjecture on the equivalence of nekrasov instanton counting ( nic ) to the conformal block has been verified to be true . bao shou , jian - feng wu , and ming yu .5 cm _ institute of theoretical physics , + chinese academy of sciences , beijing , 100190 , china _ |
1407.6547 | i | in 2012 , a higgs boson was discovered at the cern large hadron collider ( lhc ) @xcite . its observed properties are consistent with the prediction in the standard model ( sm ) within the current experimental uncertainties @xcite . in addition , so far , no report has been delivered to us for discovery of the other new particles . therefore , it has been found that the sm is a good description for particle physics at the scale of hundred gev , not only in the gauge interactions but also in the sector of electroweak symmetry breaking . although the higgs boson has been discovered and its property has turned out to be sm - like , we know nothing about the structure of the higgs sector . in fact , the minimal higgs sector with one isospin doublet scalar field is just an assumption without any theoretical principle . thus , it is natural to consider a possibility that the higgs sector takes a non - minimal form with additional isospin multiplet scalar fields , such as extra singlet , doublet , triplet and so on . most of these non - minimal higgs sectors can explain current experimental data as well . furthermore , these extended higgs sectors are often introduced in the context of new physics models which try to explain the phenomena beyond the sm ; i.e. , neutrino masses , dark matter , and baryogenesis . therefore , it is very important to experimentally explore the possibility of extended higgs sectors . we then may be able to discriminate new physics models from the property of the higgs sector . for example , extended higgs sectors with multi - doublet scalar fields are introduced in supersymmetric extensions of the sm . they are also motivated to introduce an additional source of cp violation @xcite , and to realize the strong 1st - order phase transition @xcite , both of which are required to have successful electroweak baryogenesis @xcite . singlet scalar fields are often introduced in models with the spontaneously broken @xmath4 gauge symmetry @xcite . the higgs sector with a complex triplet scalar field appears in models that can explain neutrino masses via the seesaw mechanism @xcite . tiny neutrino masses can also be explained via the loop - induced effects of extended scalar sectors @xcite . extended scalar sectors with a discrete symmetry such as @xmath5 can provide a candidate of dark matter @xcite . we here focus on the higgs triplet model ( htm ) @xcite . its higgs sector is composed of an isospin doublet higgs field with a hypercharge as @xmath6 , where @xmath7 is the electric charge and @xmath8 is the third component of the isospin . ] @xmath9 and an isospin triplet higgs field with @xmath10 . in this model , majorana masses of neutrinos are generated via new yukawa interactions among the left - handed lepton doublets and the higgs triplet field ; @xmath11 , where @xmath12 is the vacuum expectation value ( vev ) of the triplet field , and @xmath13 is a matrix in the lagrangian for the yukawa interactions . one of the most characteristic features of the htm is the existence of doubly - charged higgs bosons @xmath0 , in addition to the other additional higgs bosons ; i.e. , singly - charged @xmath14 , cp - even @xmath15 and cp - odd @xmath16 higgs bosons . the discovery of @xmath0 at collider experiments is the direct evidence of the htm . production of these bosons at collider experiments has been studied in refs . . for the decay of @xmath0 , there are three sources ; i.e. , the yukawa interactions with left - handed lepton doublets , electroweak gauge interactions in the gauge - gauge - scalar type , and those in the gauge - scalar - scalar type . they cause the same - sign dilepton decay @xmath17 , the same - sign diboson decay @xmath18 and the cascade decay @xmath19 , respectively decay occurs via the scalar triple couplings , if there is a large mass difference between @xmath0 and @xmath20 . however , such a situation is severely constrained by electroweak precision measurements @xcite . ] . although the dominant decay mode of @xmath0 is determined by parameters in the model , the dilepton decay scenario has been considered as the most promising one for discovery @xcite , because of its cleanness for the detection at colliders . it is also quite appealing that the structure of neutrino mass matrix can be directly tested by measuring the dileptonic branching ratios of @xmath0 @xcite and @xmath20 @xcite , because the branching ratios are predominantly determined by the neutrino yukawa couplings . in this scenario , a sharp peak in the invariant mass distribution of the same - sign dilepton is the characteristic signal of @xmath0 . the experimental searches for @xmath0 in the same - sign dilepton events have been performed at lep @xcite , hera @xcite , tevatron @xcite and the lhc @xcite . assuming that the branching ratio of @xmath0 decay into @xmath21 is 100% , the strongest lower bound on the mass of doubly - charged higgs bosons has been obtained as 459 gev at the lhc @xcite . current bounds have also been set at around 400 gev in several benchmark points for the structure of the neutrino mass matrix @xcite . in this paper , we discuss the direct searches for @xmath0 in the diboson decay scenario , where @xmath0 predominantly decay into same - sign w bosons , at the past , current and future collider experiments , such as lep , the lhc with 7 - 8 tev run and 13 - 14 tev run . the same - sign diboson decay scenario is equally important to the same - sign dilepton decay scenario in the htm . collider phenomenology for this decay mode has been studied in refs . @xcite . in ref . @xcite , the lower limit on the mass of @xmath0 has been derived by using the same - sign dilepton events collected by atlas collaboration at the lhc with 7 tev and 4.7 fb@xmath3 data @xcite . up to our knowledge , this is the first analysis for the constraints on @xmath0 in the diboson scenario . the aim of the present paper is to explain details of the analysis done in ref . @xcite , and to make update on the results by including the qcd correction to the production cross sections . this paper is organized as follows . in section ii , we briefly review the htm . after we define the mass eigenstates for the higgs bosons , we derive the yukawa interaction and the gauge interaction for the triplet - like higgs bosons at the tree level . in section iii , we give expressions for the decay rates of @xmath0 in the all three decay modes . partial decay widths of @xmath0 are evaluated with particular attention to the case of relatively small masses where one or both of the w bosons are forced off - shell . we then show the phase diagram indicating the main decay mode of @xmath0 . next , we evaluate the cross section of @xmath0 productions at the lhc in the leading order ( lo ) and the next - to - leading order ( nlo ) of qcd . in section iv , we exhibit constraints on the mass of @xmath0 in the diboson decay scenario at the lep experiments and also at the lhc . at the lep i experiment , the lower limit on the mass of @xmath0 can be obtained from the total width of the z boson . we also evaluate the expected number of events for the various final states in the process of @xmath22 at the lep ii experiment . we then discuss the mass bound on @xmath0 by using the current lhc limit on the cross section for anomalous production of same - sign dileptons . section v is devoted to our conclusion . in appendix , the cross sections for @xmath0 production at the lhc with various collision energies are collected for reader s convenience . | , the lower limit on the mass of has been set at about 400 gev by atlas and cms collaborations . on the other hand , for the case where the latter decay mode is dominant , no dedicated search has been performed in the past . by taking into account characteristic signals of in the diboson decay scenario at lep and the lhc experiments , we find that the lower mass bound of 60 - 68 gev can be obtained using the same - sign dilepton search performed by atlas collaboration with 4.7 fb data at the collision energy of 7 tev . we give detailed explanations on the decay properties of for relatively small mass cases and also on production cross sections of at the next - to - leading order of qcd at the lhc . | a direct search for doubly - charged higgs bosons is one of the most important probe in the higgs triplet model , which is motivated by generation mechanisms of tiny neutrino masses . there are two major decay modes of ; i.e. , the same - sign dilepton decay and the same - sign diboson decay . for the case where the former decay mode is dominant , the lower limit on the mass of has been set at about 400 gev by atlas and cms collaborations . on the other hand , for the case where the latter decay mode is dominant , no dedicated search has been performed in the past . by taking into account characteristic signals of in the diboson decay scenario at lep and the lhc experiments , we find that the lower mass bound of 60 - 68 gev can be obtained using the same - sign dilepton search performed by atlas collaboration with 4.7 fb data at the collision energy of 7 tev . we also show that the limit can be extended up to about 85 - 90 gev , assuming the integrated luminosity of 20 fb and 7 tev for the collision energy . we give detailed explanations on the decay properties of for relatively small mass cases and also on production cross sections of at the next - to - leading order of qcd at the lhc . |
1601.06502 | i | a _ multiset _ is a generalization of a set in which each element has an associated integer _ multiplicity_. given a possibly infinite set @xmath4 , a set ( resp . multiset ) homomorphic hash function on @xmath4 maps finite subsets of @xmath4 ( resp . finitely - supported multisets on @xmath4 ) to fixed - length hash values , allowing incremental updates : when new elements are added to the ( multi)set , the hash value of the modified ( multi)set can be computed in time proportional to the degree of modification . the incremental update property makes homomorphic hashing a very useful and versatile primitive . it has found applications in many areas of computer security and algorithmics , including network coding @xcite and verifiable peer - to - peer content distribution @xcite , secure internet routing @xcite , byzantine fault tolerance @xcite , _ streaming _ set and multiset equality comparison @xcite , and various aspects of database security , such as access pattern privacy @xcite and integrity protection @xcite . this latter use case provides a simple example of how the primitive is used in practice : one can use homomorphic hashing to verify the integrity of a database with a transaction log , by computing a hash value for each transaction in such a way that the hash of the complete database state is equal to the ( appropriately - defined ) sum of the hashes of all transactions . another observation @xcite is that homomorphic hashing can be used for incremental and parallel hashing of lists , arrays , strings and other similar data structures : for example , the list @xmath5 can be represented as the set @xmath6 , and it suffices to apply the homomorphic hash function to that set . a framework for constructing provably secure homomorphic hash functions ( in some suitably idealized model , such as the random oracle model ) was introduced by bellare and micciancio @xcite , and later extended to the multiset hash setting by clarke et al . @xcite , and revisited by cathalo et al . @xcite . roughly speaking , the framework of bellare and micciancio can be described as follows . to construct a ( multi)set homomorphic hash function on @xmath4 , one can start with a usual hash function @xmath7 from @xmath4 to some additive group @xmath8 , and extend it to finite subsets of @xmath4 ( resp . multisets on @xmath4 ) by setting @xmath9 ( resp . @xmath10 , where @xmath11 is the multiplicity of @xmath12 ) . and in fact , it is clear that all possible homomorphic hash functions arise in that way . note that as in clarke et al . @xcite , and unlike the original framework of bellare and miciancio @xcite , there is no block index @xmath13 included in the hash @xmath14 of each element @xmath12 , because we are hashing unordered sets / multisets , rather than ordered sequences of blocks . assume that the underlying hash function @xmath7 is ideal ( i.e. it behaves like a random oracle ) . then we can ask when the corresponding homomorphic hash function @xmath15 is secure ( collision resistant , say ) . this translates to a knapsack - like number - theoretic assumption on the group @xmath8 , which bellare and micciancio show holds , for example , when the discrete logarithm problem is hard in @xmath8 . concretely , bellare and micciancio and the authors of subsequent works propose a number of possible instantiations for @xmath15 which essentially amount to choosing @xmath16 or @xmath17 for suitable parameters @xmath18 . these concrete instantiations yield simple implementations , but they all suffer from suboptimal output size ( they require outputs of several thousand bits to achieve collision resistance at the @xmath19 security level ) , and their efficiency is generally unsatisfactory . essentially all practical applications of homomorphic hashing in the security literature seem to focus on the case @xmath20 , called ` muhash ` . within bellare and micciancio s framework , constructing a homomorphic hash function amounts to choosing a group @xmath8 where the appropriate number - theoretic assumption holds , together with a hash function to @xmath8 whose behavior is close enough to ideal for the security proof to go through . in this paper , we propose a novel concrete construction of a multiset hash function by choosing @xmath8 as the group of points of a binary elliptic curves , and picking the hash function following the approach of brier et al . @xcite ( which we improve upon slightly ) applied to the binary curve variant of shallue and van de woestijne s encoding function @xcite . we also describe a software implementation of our proposal ( building upon the work of aranha et al . @xcite for binary curve hashing , and using blake2 @xcite as the actual underlying hash function ) and provide extensive performance results showing that our function outperforms existing methods by a large margin on modern cpu architectures ( especially those supporting carry - less multiplication ) . furthermore , choosing an elliptic curve ( with small cofactor ) for the group of hash values solves the `` output size '' problem of homomorphic hashing outright : @xmath21 collision security is achieved with roughly @xmath22-bit long digests . yet , they do not seem to have been used in concrete implementations of homomorphic hashing so far for a more detailed discussion . ] . one can wonder why ; the most likely explanation is that usual methods for hashing to elliptic curves are far too inefficient to make curves attractive from a performance standpoint : almost all such methods require at least one full size exponentiation in the base field of the curve , which will be much more costly by itself than the single multiplication ( in a much larger field ) required by ` muhash`even on curves over fast prime fields at the 128-bit security level @xcite , such an encoding function is over @xmath23 times slower than ` muhash ` at equivalent security on haswell , and over @xmath24 times slower than our construction . only by using binary curves and relatively sophisticated implementation techniques do we avoid that stumbling block and prove that elliptic curves can be competitive . as a result , we achieve a processing speed of over 3 million set elements per second on a intel haswell cpu at the @xmath19-bit security level . speedups are expected with the release of intel broadwell processors and its improved implementation of carry - less multiplication . recently , new developments have been announced regarding the asymptotic complexity of the discrete logarithm problem on binary elliptic curves , particularly by semaev @xcite . these results are somewhat controversial , since they are based on heuristic assumptions that prevailing evidence suggests are unlikely to hold @xcite , and their storage requirements appear to make them purely theoretical anyway @xcite . however , if semaev s claims of an @xmath25 $ ] attack turn out to be correct , the asymptotic security of binary elliptic curve - based ecmh would be reduced . concrete _ security of our construction , on the other hand , would be completely unaffected on curves of up to 300 + bits ( and in particular at the 128-bit security level on gls254 ) , since the claimed attack is worse than generic attacks on such curves . moreover , even if _ actually practical _ @xmath25 $ ] attacks were to be found , ecmh on binary curves is likely to remain attractive , since it mainly competes against ` muhash ` , which is vulnerable to an @xmath26 $ ] subexponential attack . for all these reasons , we believe that ecmh on binary elliptic curves is a safe choice for security - minded practitioners , and that the switch from ` muhash ` to ecmh is entirely justified in view of the considerable performance gain ( which lets designers choose a higher security margin and still come out far ahead ) . | a homomorphic , or incremental , multiset hash function , associates a hash value to arbitrary collections of objects ( with possible repetitions ) in such a way that the hash of the union of two collections is easy to compute from the hashes of the two collections themselves : it is simply their sum under a suitable group operation . in particular , hash values of large collections can be computed incrementally and/or in parallel . homomorphic hashing is thus a very useful primitive with applications ranging from database integrity verification to streaming set / multiset comparison and network coding . unfortunately , constructions of homomorphic hash functions in the literature are hampered by two main drawbacks : they tend to be much longer than usual hash functions at the same security level ( e.g. to achieve a collision resistance of , they are several thousand bits long , as opposed to bits for usual hash functions ) , and they are also quite slow . in this paper , we demonstrate a highly - efficient software implementation of ecmh , which our thorough empirical evaluation shows to be capable of processing over 3 million set elements per second on a intel haswell machine at the 128-bit security level many times faster than previous practical methods . while incremental hashing based on elliptic curves has been considered previously , the proposed method was less efficient , susceptible to timing attacks , and potentially patent - encumbered , and no practical implementation was demonstrated . | a homomorphic , or incremental , multiset hash function , associates a hash value to arbitrary collections of objects ( with possible repetitions ) in such a way that the hash of the union of two collections is easy to compute from the hashes of the two collections themselves : it is simply their sum under a suitable group operation . in particular , hash values of large collections can be computed incrementally and/or in parallel . homomorphic hashing is thus a very useful primitive with applications ranging from database integrity verification to streaming set / multiset comparison and network coding . unfortunately , constructions of homomorphic hash functions in the literature are hampered by two main drawbacks : they tend to be much longer than usual hash functions at the same security level ( e.g. to achieve a collision resistance of , they are several thousand bits long , as opposed to bits for usual hash functions ) , and they are also quite slow . in this paper , we introduce the elliptic curve multiset hash ( ecmh ) , which combines a usual bit string - valued hash function like blake2 with an efficient encoding into binary elliptic curves to overcome both difficulties . on the one hand , the size of ecmh digests is essentially optimal :-bit hash values provide collision resistance . on the other hand , we demonstrate a highly - efficient software implementation of ecmh , which our thorough empirical evaluation shows to be capable of processing over 3 million set elements per second on a intel haswell machine at the 128-bit security level many times faster than previous practical methods . while incremental hashing based on elliptic curves has been considered previously , the proposed method was less efficient , susceptible to timing attacks , and potentially patent - encumbered , and no practical implementation was demonstrated . * keywords : * homomorphic hashing , elliptic curves , efficient implementation , gls254 , pclmulqdq . |
1703.10128 | i | rapid experimental advances in the creation of finite systems of ultracold atoms using few optical traps ( bottom - up ) @xcite or through the use of extended optical lattices ( top - down ) @xcite are promising approaches toward realization of feynman s vision of a quantum simulator @xcite , capable of finding solutions to systems that are otherwise numerically and/or analytically intractable . however , apart from a few double - well investigations @xcite , to date model - independent microscopic studies of multiwell systems providing theoretical insights and guidance to experimental efforts are largely lacking . here we introduce a configuration - interaction ( ci ) schrdinger - equation methodology @xcite for exploring finite plaquette systems assembled from individual optical traps ; for a brief description of the ci method , see appendix [ cime ] . these systems are fundamental stepping stones toward bottom - up realization of large scale checkerboard or square ulracold atom lattices which are promising candidates for emulating the physics underlying @xmath3-wave high - t@xmath4 superconductivity @xcite in optical lattices @xcite . the work described here , demonstrating the feasibility of such exact ci calculations for interatomic contact interactions , can be extended to electronic plaquettes , i.e. , to quantum - dot - arrays , governed by long - range coulomb interactions . the plan of the paper is as follows : in section [ sp ] we explore first the properties of ultracold fermionic atoms ( @xmath0li ) confined in a single square plaquette ( 4-sites ) in the regime of hole pairing , and subsequently analyze the entanglement physics of two coupled plaquettes in the hole - pairing regime . in section [ dp ] , for the `` hole - doped '' coupled plaquettes ( 8-sites , six atoms ) , we analyze the wave function anatomy of the entangled @xcite ( schrdinger - cat ) two almost - degenerate manifolds , comprising : ( @xmath5 ) the ground state ( gs ) and 1st excited ( 1ex ) state , and ( @xmath6 ) the two higher excited states ( 2ex and 3ex ) . the almost - degenerate states have good , but opposite , parities . when symmetry - broken ( sb ) either by superposing @xmath7 the degenerate pair in each manifold , or via offsetting the energies ( tilting ) of the two plaquettes the sb states coming from @xmath5 are characterized by a particle ( `` charge'')-density modulation ( `` wave '' ) , i.e , a cdw portraying the hole - paired , ( 4,2 ) or ( 2,4 ) , components , whereas the sb states originating from @xmath6 remain in a ( 3,3 ) particle distribution ( each of the plaquettes having an unpaired hole ) , but exhibit a spin - polarizarion density modulation ( `` wave '' ) , i.e. , a sdw . for the the double - plaquette case , we further explore effects brought about by breaking the symmetry between two weakly - coupled plaquettes , either by distorting one of the plaquettes , or by tilting and detuning one of the plaquettes with respect to the other . spectral changes caused by increased tunnel coupling ( e.g. , by decreased inter - plaquette distance ) between the two plaquettes are also considered . noteworthy is our finding that the gs @xmath3-wave resonating valence bond ( rvb ) state contributes only partially to the exact many - body wave function i.e. , double occupancies ( referred also as doublons ) need be included . indeed , our microscopic results map properly onto a hubbard hamiltonian ( including extended hubbard models @xcite depending upon the range of the experimental parameters ) , but not onto the double - occupancy - excluding @xmath1-@xmath2 model @xcite . our conclusions agree with recent @xcite observations of doublons in two - dimensional ( 2d ) optical lattices . the rest of the paper comprises the summary ( sec . [ summ ] ) and three appendices , concerning a brief description of the ci method ( appendix [ cime ] ) , the mathematical definitions of single - particle densities and two - body and @xmath8-body conditional probability distributions ( appendix [ defs ] ) , and the specification of the correspondence between spin and rvb functions and ci many - body wave functions ( appendix [ spin ] ) . | a microscopic configuration - interaction ( ci ) methodology is introduced to enable bottom - up schrdinger - equation emulation of unconventional superconductivity in ultracold optical traps . we illustrate the method by exploring the properties ofli atoms in a single square plaquette in the hole - pairing regime , and by analyzing the entanglement ( symmetry - preserving ) and disentanglement physics ( via symmetry - breaking , associated with the separation of charge and spin density waves ) of two coupled plaquettes in the same regime . the single - occupancy rvb states contribute only partially to the exact many - body solutions , and the ci results map onto a hubbard hamiltonian , but not onto the double - occupancy - excluding- one . for the double - plaquette case , effects brought about by breaking the symmetry between two weakly - interacting plaquettes , either by distorting , or by tilting and detuning , one of the plaquettes with respect to the other , as well as spectral changes caused by increased coupling between the two plaquettes , are explored . | a microscopic configuration - interaction ( ci ) methodology is introduced to enable bottom - up schrdinger - equation emulation of unconventional superconductivity in ultracold optical traps . we illustrate the method by exploring the properties ofli atoms in a single square plaquette in the hole - pairing regime , and by analyzing the entanglement ( symmetry - preserving ) and disentanglement physics ( via symmetry - breaking , associated with the separation of charge and spin density waves ) of two coupled plaquettes in the same regime . the single - occupancy rvb states contribute only partially to the exact many - body solutions , and the ci results map onto a hubbard hamiltonian , but not onto the double - occupancy - excluding- one . for the double - plaquette case , effects brought about by breaking the symmetry between two weakly - interacting plaquettes , either by distorting , or by tilting and detuning , one of the plaquettes with respect to the other , as well as spectral changes caused by increased coupling between the two plaquettes , are explored . |
astro-ph0508178 | c | sources qr6 , qr8 , and qr9 show the major properties of stellar wind emission : ( 1 ) they have rising spectral indices in the range of + 0.5 to + 1.0 , ( 2 ) they are unresolved sources at the resolutions observed , which means their physical sizes are @xmath14 0.2 ( 0.009 pc ) at 22.5 ghz and @xmath140.5 ( 0.02 pc ) at 8.5 ghz , and ( 3 ) they are coincident positionally with young , massive stars . therefore , these three radio sources are interpreted to be stellar winds from quintuplet cluster stars . the flux densities of radio detections of stellar winds can be used to calculate mass loss rates ( @xmath23 ) . based on panagia & felli ( 1975 ) and wright & barlow ( 1975 ) , the mass loss rate can be estimated for an isotropic , homogeneous wind with constant velocity , electron density and chemical composition . however , 20 - 25% of early - type stars are known to have a nonthermal ( and variable ) component in their winds ( abbott et al . 1984 ; bieging et al . 1989 ) . the typical spectral index of such sources is in the range of @xmath18@xmath5@xmath70.6 to + 0.3 for frequencies between 5 and 43 ghz ( beiging et al . 1989 ; contreras et al . 1996 ) . contamination of the flux density by the nonthermal component can occur at the lower frequencies . therefore , higher frequency observations ( i.e. , 23 ghz compared to 5 ghz ) may be a better tracer of the thermal component and give more reliable mass - loss rates ( contreras et al . 1996 ) . the mass - loss rate can be estimated from our 22.5 ghz observations using : @xmath24 where s@xmath25 is the flux density of the source at 22.5 ghz , v@xmath26 is the terminal velocity of the wind ( assumed to be 1000 km s@xmath3 ; cotera et al . 1996 ; nagata et al . 1995 ) , and d is the distance to the source ( 8 kpc ) . we have assumed an electron temperature of 10@xmath27 k , z=1 , and a mean molecular weight , @xmath17=2 , due to the enrichment in heavy elements of the late - type wn stars ( leitherer et al . 1997 ) , a classification which figer et al . ( 1999 ) and figer et al . ( 2002 ) has assigned to most of the quintuplet and arches cluster stars . leitherer et al . ( 1997 ) suggest that there is very little variation in the mass - loss rates for wr stars , with an average value of @xmath54 @xmath0 10@xmath1 m@xmath2 yr@xmath3 . cappa et al . ( 2004 ) recently carried out a survey of galactic wr stars and came to similar conclusions concerning the measured radio mass - loss rates ; the average values from their study were 2 - 4 @xmath0 10@xmath1 m@xmath2 yr@xmath3 . values for the mass - loss rates of qr6 , qr8 and qr9 range from 3.2@xmath010@xmath1 m@xmath2 yr@xmath3 to 4.3@xmath010@xmath1 m@xmath2 yr@xmath3 ( see table 4 ) . these values agree well with those found for wr stars described above as well as for other young , massive stellar types . the nature of radio emission from sources qr4 , qr5 and qr7 is less clear than that from qr6 , qr8 and qr9 . although qr4 , qr5 and qr7 show rising spectra and are coincident with near - ir sources , their sizes are more extended ( @xmath131 at 8.5 ghz ) than expected for stellar wind sources . according to rodriguez et al . ( 1980 ) , the angular size of a stellar wind can be related directly to its flux density , independent of mass - loss rate and terminal wind velocity . for the 8.5 ghz flux densities of qr4 , qr5 and qr7 , the expected angular size of the corresponding stellar wind should be @xmath140.05 . the actual sizes of the sources ( 0.3 to 4 ) are clearly larger than the predicted value . however , as mentioned previously ( in @xmath152.2 ) , the quintuplet radio sources are located amidst a very complicated set of extended features , which includes the sickle and pistol h2 regions as well as the linear , nonthermal filaments of the radio arc . although filtering in the _ ( u , v ) _ plane can help by removing the extended structures , the quintuplet sources may still be contaminated by residual background emission , which can make determination of flux densities and sizes difficult . although the errors in size determination are substantial ( 0.3@xmath70.5 ; 20 - 50% ) , the sizes are still much more extended than the predicted values for stellar winds ( e.g. , @xmath140.05 ) . mass - loss rates are calculated for these sources as described above , and the values for qr4 , qr5 and qr7 ( average m of 1.2 @xmath0 10@xmath28 m@xmath2 yr@xmath3 ) are significantly larger than for the stellar wind sources qr6 , qr8 and qr9 ( average m of 3.7@xmath010@xmath1 m@xmath2 yr@xmath3 ) . while values for m of @xmath5 10@xmath28 m@xmath2 yr@xmath3 are not unexpected for some mass - losing stars , they typically correspond to epochs when the star may be experiencing an enhanced rate of mass - loss from its surface ( i.e. , the lbv phase ; white et al . 1994 ) . therefore , the simple interpretation of stellar wind emission may not apply for qr4 , qr5 and qr7 . however , the rising spectra and extended sizes of these sources are consistent with partially - thick free - free emission associated with an inhomogeneous , but spatially unresolved , h2 region . therefore , qr4 , qr5 and qr7 could be inhomogeneous h2 regions or even combinations of stellar wind emission and optically - thin , free - free emission from associated ionized material ( perhaps leftover from the formation of the stars ) . based on mezger & henderson ( 1967 ) , we can estimate the physical properties of these regions of ionized gas , assuming they have become optically - thin at the highest frequency at which they are detected , using the deconvolved sizes of these sources , and using an electron temperature of 10@xmath27 k. table 4 lists the electron density , n@xmath29 , mass of ionized hydrogen , m@xmath30 , emission measure , em , and the production rate of ionizing photons , n@xmath9 for these sources . the values of these h2 properties for qr4 , qr5 and qr7 ( in addition to their linear sizes of 0.03 - 0.05 pc ) are consistent with those for compact and ultra - compact h2 regions ( wood & churchwell 1990 ) . the properties of qr1 - 3 differ from the other radio sources in the quintuplet . as described in @xmath153.1.1 , qr1 - 3 are more spatially extended than qr6 , qr8 and qr9 by at least a factor of 2 . more significantly , qr1 - 3 are the only radio sources in the two clusters without closely associated near - ir counterparts . these sources also define the northern edge of the quintuplet cluster ( 5 - 10 north of qr4 - 7 and the main concentration of stellar members ) . table 4 shows that the spectral indices are close to flat with large errors ( i.e. @xmath18=@xmath70.3@xmath190.2 for @xmath21 for qr1 ) . such spectra can be attributed to optically thin , free - free emission associated with h2 regions . since there are no stellar counterparts to qr1 - 3 , a likely explanation for these sources is that they may represent embedded massive stellar sources ( churchwell 1990 ) . the linear sizes of qr1 - 3 of @xmath50.1 pc are consistent with the upper end of the size distribution of ultra - compact h2 regions in the sample of kurtz , churchwell , & wood ( 1994 ) . as only @xmath31% of ultra - compact h2 regions are detectable at near - ir wavelengths ( hanson , luhman , & rieke 2002 ) , the lack of near - ir emission associated with qr1 - 3 is also consistent with their being ultra - compact h2 regions . as described above , the h2 properties for qr1 - 3 are derived from the radio continuum and listed in table 4 . the production rate of ionizing photons , n@xmath9 , can provide an estimate of the embedded stellar type that is responsible for the associated radio emission ( assuming a single ionizing star ) . values for n@xmath9 range from @xmath32 s@xmath3 . these ionizing fluxes are consistent with embedded b - type massive stars ( @xmath5 b1v - b2v ; @xcite ) . however , a number of galactic ultra - compact h2 regions are observed to be associated with diffuse extended emission @xcite . these studies conclude that typically @xmath33% of the ionizing flux from the embedded stars in uc h2s escapes to the outer envelope . thus , the inferred stellar content from the @xmath34 values determined for sources qr1 - 3 may be an underestimate , in which case they could have spectral types earlier than b. therefore , we tentatively identify qr1 - 3 as ultracompact or compact h2 regions . the proximity of the quintuplet cluster radio sources ( and in particular , sources qr1 - 3 ) to the edge of a molecular cloud identified by serabyn & gsten ( 1991 ) may indicate that these sources are physically associated with this molecular cloud . in the cs ( j=3 - 2 ) image of serabyn & gsten , a finger of emission extends toward the projected location of qr1 - 3 , indicating the presence of moderately dense gas . the apparent physical association between qr1 - 3 and the edge of the molecular cloud suggests that these young stellar objects have recently emerged from the molecular cloud , probably as a result of erosion due to the more evolved massive stars in the quintuplet cluster . indeed , the quintuplet cluster is thought to be the source of ionization for the larger sickle h2 region ( lang , goss & wood 1997 ; figer et al . 1999a ) , which traces the inner edge of the molecular cloud . alternatively , the radio sources which are not interpreted to be stellar winds ( qr1 - 3 , qr4 , qr5 , and qr7 ) may represent clumps of molecular material leftover from the formation of the stars in the cluster that are externally ionized by the quintuplet cluster members . assuming these clumps are located at approximately the projected distances from the center of the cluster , the quintuplet could easily account for their ionization as it is responsible for the ionization of the much larger sickle h2 region ( located 1 or 2.5 pc to the ne ) and the north side of the pistol nebula ( lang et al . 1997 ; figer et al . if any of the quintuplet radio sources are confirmed to be compact or uc h2s , this detection will lead to the study of the earliest stages of stellar evolution in one of the most extreme environments in the milky way . in particular , these objects are located in a region of strong magnetic fields , high turbulent velocities , and extremely high metallicities ( morris & serabyn 1996 ) . figure 5 shows that the radio emission associated with the pistol star has a bipolar morphology centered around the position of the star and has a flat spectral index ( @xmath21 = + 0.1@xmath190.4 ) . one explanation for this source is that it represents the current mass - loss of the star . ( the outer `` pistol - shaped '' nebula is thought to contain the mass associated with an episode of previous mass - loss ( @xmath5 11 m@xmath2 , @xmath55000 years ago ; figer et al . 1998 ) ) . in this case , the integrated flux density would imply a mass - loss rate of 2.7@xmath010@xmath28 m@xmath2 yr@xmath3 . this value is much higher than the average value for wr stellar types ( see @xmath154.1.1 ) , but similarly high mass - loss rates have been detected in other lbv stars during epochs of extreme mass - loss ( e.g. , @xmath4 car ; white et al . 1994 ) and in several wr colliding - wind binary systems ( contreras et al . 1996 ) . however , as for sources qr4 , qr5 , and qr7 , the extended size of the pistol star radio emission is much larger than the expected angular size for a stellar wind source ( @xmath50.1 ) using the methods of rodrguez et al . another problem with the interpretation of the pistol as a stellar wind source is that its double - lobed radio emission is not well modelled by a spherically - symmetric stellar wind surrouding the star . it is possible to calculate a mass - loss rate for the source using the total mass and assuming an expansion velocity for the radio emission . assuming that the radio emission is optically - thin , free - free emission , with t@xmath29@xmath510@xmath27 k , and taking the deconvolved angular sizes of the lobes to be 1@xmath35 , we can estimate the total mass in the lobes to be @xmath5 0.02 @xmath36 . the separation between the pistol star and each of the lobes is @xmath37 cm . assuming that the lobes are expanding at a velocity of 1000 km s@xmath3 , we estimate a mass - loss rate of @xmath38 @xmath36 yr@xmath3 . this value is roughly consistent with the value obtained by using equation ( 1 ) . in addition , it is possible to calculate the ionizing flux required to maintain the bipolar radio source . at a distance of 8 kpc and with the integrated flux density measured , n@xmath9 has a value of @xmath39 s@xmath3 , which can be provided by an early b - type star . however , the two lobes that constitute the radio source are intersecting only a fraction of the 4@xmath40 radians around the star . furthermore , the ionized nebula may be density - bounded , with most of the ionizing flux of the pistol star escaping to the interstellar medium , so the star is likely to be of an earlier spectral type . the physical properties of the radio emission associated with the pistol star are similar to the nebula surrounding the well - known lbv binary system @xmath4 carina ( davidson & humphreys 1997 ) . the bipolar - shaped radio emission around @xmath4 carina has a flat spectral index and a linear size of 0.12 pc ( 10 at 2.5 kpc ; white et al . 1994 ) , similar to the pistol star nebula . in fact , a variety of types of nebular features are detected around other lbv stars and are thought to be the result of stellar wind emission interacting with the surrounding interstellar medium ( nota et al . 1995 ) . in the cases of galactic sources hg car , ag car , wra-751 and he3 - 519 , both extended nebular shells and compact stellar wind emission are detected in the radio continuum ( duncan & white 2002 ) . these authors also calculate substantial mass - loss rates from the central lbv stars ( @xmath135 - 10 @xmath0 10@xmath1 m@xmath2 yr@xmath3 ) . the bipolar morphology of such radio nebulae suggests that the mass - loss is not isotropic ( e.g. , @xmath4 car ) . this type of non - spherical wind emission has been explained by several different mechanisms : ( 1 ) _ internal _ processes , such as the stellar rotation of such massive stars and the resulting mass - loss geometry ( owocki & gayley 1997 ; maeder & maynet 2000 ; smith et al . 2004 ) , and ( 2 ) _ external _ processes , such as the influence of a binary companion on the mass - loss geometry ( iben & livio 1993 ) or expansion of the nebula into a region of high density - contrast interstellar medium ( nota et al . 1995 ) . a comparison of the 8.5 ghz observation of the arches cluster presented here and the observation of lang et al . ( 2001b ) shows that the majority of the sources do not exhibit any strong variability in their flux density . however , moderate variability ( at the level of 12 - 30% ) is detected in several of the arches radio sources previously detected ( ar1 , ar3 , ar4 , ar8 ) and possibly explains the new detection of ar9 . the flux densities of two of the four sources decreased , while the flux density of the other two sources increased . therefore , we can rule out a systematic calibration problem . this moderate variability may indicate ( i ) the presence of a time - variable nonthermal component ( due to shocks in the wind , or colliding winds for a binary system ) and/or ( ii ) variability in the mass - loss rate or wind velocity ( contreras et al . 1996 ; leitherer et al . 1997 ) . nine of the ten arches sources are unresolved in both the 8.5 and 22.5 ghz observations . the major axis of ar1 is resolved at both 8.5 and 22.5 ghz and its deconvolved size is @xmath50.1 . this size is consistent with the expected angular size of 0.06 for an ionized stellar wind source of 1.9 mjy flux density at 8.5 ghz ( rodrguez et al . therefore , all radio sources in the arches cluster exhibit the properties of stellar wind sources ( see @xmath154.1.1 ) . mass - loss rates are also calculated for the arches sources and are listed in table 5 . ar1 has the largest value of m = 2.2 @xmath0 10@xmath28 m@xmath2 yr@xmath3 . such a high value of m is usually attributed to a particular mass - loss epoch during the evolution of a massive star or a colliding wind binary system ( white et al . 1994 ; contreras et al . 1996 ) . the remainder of the arches sources have an average value of @xmath53 @xmath0 10@xmath1 m@xmath2 yr@xmath3 , typical for the types of stars present in the arches and quintuplet clusters ( @xmath154.1.1 ; leitherer et al . 1997 ; cappa et al . 2004 ) . as reported recently by law & yusef - zadeh ( 2004 ) and previously by yusef - zadeh et al . ( 2002 ) , a number of the radio sources in the arches and quintuplet clusters have x - ray counterparts detected with the _ chandra x - ray observatory_. the x - ray emission associated with the arches cluster itself is one of the most luminous x - ray sources in the galactic center region with an x - ray luminosity in the 2 - 10 kev range of @xmath510@xmath41 ergs s@xmath3 ( wang , gotthelf & lang 2002 ) . three x - ray point like sources are detected in the arches cluster : a1s , a1n and a2 ( law & yusef - zadeh 2004 ) . the x - ray sources a1s and a1n can be associated with the radio sources ar4 and ar1 as well as near - ir sources from figer et al . the x - ray emission in these sources is thought to arise from shocks formed in the winds of massive stars ( o- and wr - types ) as are found in the arches and quintuplet clusters ( law & yusef - zadeh 2004 ) . the characteristics of several arches radio sources that have x - ray counterparts suggest that these sources are likely to have shocks in their winds and/or be colliding wind binaries . radio emission from the colliding winds of binary massive stars is well known to have a nonthermal component that flattens the spectral index ( dougherty & williams 2000 ) as well as showing time - variable emission ( abbott et al . 1984 ) . in the case of the radio source ar1 , the spectral index ( @xmath18=+0.3@xmath190.04 ; lang et al . 2001b ) is flatter than the canonical @xmath18=+0.6 value for stellar wind emission and the flux density of the source varies by 12% between epochs . ar4 has a steeper spectral index ( although the errors are larger ) , @xmath18=+0.6@xmath190.2 , but its flux density varied by 30% in the two observed epochs . in the quintuplet cluster , law & yusef - zadeh ( 2004 ) report that two of the radio sources ( qr6 and qr7 ) have probable x - ray counterparts . it is not possible to determine whether the spectral index of qr6 is flattened ( @xmath18=+0.4@xmath190.4 ) or consistent with the value of @xmath18=+0.6 . however , this source was not detected in the results of lang et al . ( 1999 ) , either because of its time - variable nature or because the resolution was too coarse . the possible detection ( 2.6@xmath22 as reported by law & yusef - zadeh 2004 ) of x - ray emission associated with quintuplet source qr7 is interesting as not much is known about the nature of the near - ir counterpart ( @xmath153.1.2 ; figer et al . 1999 ) . the arches and quintuplet clusters are two of the most massive and luminous clusters in our galaxy ( figer et al . 1999a ) . in particular , the stellar density of the arches cluster ( @xmath42 @xmath5 3 @xmath010@xmath43 m@xmath2 pc@xmath44 ) is comparable only to that found in the centers of clusters such as hd in ngc3603 ( moffat et al . 1994 ) and r136 in the large magellanic cloud , associated with the 30 doradus nebula ( massey & hunter 1998 ) . therefore , these clusters should have a profound effect on the surrounding ism . indeed , these clusters are thought to be responsible for the ionization of the adjacent h2 regions , the sickle and the arched filaments ( lang et al . 1997 ; lang et al . 2001a ) . the presence of x - ray emission associated with individual stellar wind sources ( see above ) also indicates that the winds of these massive stars are powerful . in addition to the point - like x - ray sources , both clusters also show diffuse x - ray emission ( yusef - zadeh et al . 2002 ; law & yusef - zadeh 2004 ) . in the arches cluster , the diffuse x - ray emission is apparent for @xmath51 pc surrounding the cluster , and in the quintuplet , the diffuse x - ray emission has a slightly larger extent ( @xmath51.2 pc ) but is not detected with high signal to noise ( law & yusef - zadeh 2004 ) . several authors have suggested that the diffuse emission is related to a `` cluster wind '' triggered by the collective effect of a large number of massive star winds in the cluster core ( cant et al . 2000 ; raga et al . 2001 ; yusef - zadeh et al . such processes can lead to shock acceleration of the wind particles and heating of the gas to high temperatures ( 10@xmath8 to 10@xmath45 k ) . nonthermal radio emission has been detected within a 9 radius ( 0.3 pc ) of the cluster core and is believed to provide additional evidence that particle acceleration occurs here ( yusef - zadeh et al . 2003 ) . the physical parameters of the `` cluster wind '' and the resulting region of x - ray emitting gas depend primarily on the individual mass - loss rates of the stellar wind sources ( stevens & hartwell 2003 ; cant et al . 2000 ; raga et al . the total stellar wind mass injection rate is dependent on the sum of mass - loss rates of individual stars multiplied by the square of the terminal wind velocity . using an estimate for the velocity , predicted by the models in stevens & hartwell ( 2003 ) , or by using the average value of @xmath51000 km s@xmath3 , measured by cotera et al . ( 1996 ) , it is possible to derive the total stellar wind kinetic energy injection rate . the observations presented here provide the most reliable mass - loss rate estimates for the stellar wind sources in the arches and quintuplet clusters . based on the mass - loss rates presented here , the total stellar wind mass injection rate for the arches cluster is @xmath54 @xmath0 10@xmath28 m@xmath2 yr@xmath3 , and for the quintuplet ( including the pistol star as well as sources qr4 , qr5 , and qr7 ) is @xmath57 @xmath0 10@xmath28 m@xmath2 yr@xmath3 . these values for total m are in close agreement with those used by stevens & hartwell ( 2003 ) and raga et al . ( 2001 ) in their calculation of the total stellar wind kinetic energy injection rate . two types of numerical calculations have also been made recently to characterize the x - ray `` cluster wind '' that arises from the arches ( and presumably , the quintuplet ) clusters . cant et al . ( 2000 ) and raga et al . ( 2001 ) present a solution for an adiabatic flow and predict an envelope of x - ray emission of t@xmath55 @xmath010@xmath8 k , extending well beyond ( 10 s of pc ) the cluster core ( 0.2 pc ) . in contrast , silich et al . ( 2004 ) employ radiative cooling in their work , and predict , for the arches cluster , a quasi - adiabatic solution that suggests a much smaller boundary around the cluster of x - ray emission . depending on the values chosen for m and the terminal velocity , silich et al . ( 2004 ) estimate 3 - 7 pc for the extent of the x - ray emitting region . these authors argue that this smaller size for the x - ray emitting region is in part due to the slowing of wind emission by interactions of the wind with cooler surrounding ism material . the arches cluster , in particular , is located in the midst of a dense molecular and ionized environment ( lang et al . 2002 ) and there is a large reservoir of cooler material which may decelerate the wind . morphologically , the arched filaments h2 regions and molecular cloud extend for as much as 20 pc in projection around the cluster . x - ray observations near the arched filaments show that the diffuse emission appears to be confined to the area near the easternmost of the arched filaments ( @xmath14 10 pc ) , as predicted by silich et al . however , the short - integration nature of these observations ( @xmath14 50 ks ) with _ chandra _ and the subsequent low sensitivity to diffuse objects have so far prevented detailed spectral fitting and spatial imaging of the diffuse x - ray emission associated with these clusters ( yusef - zadeh et al . 2002 ; wang , gotthelf & lang 2002 ) . several examples of radio clusters associated with regions of recent star formation have been reported in the literature over the last ten years ( e.g. , garay et al . 1987 ; churchwell et al . 1987 ; becker & white 1988 ; stine & oneal 1998 ; rodrguez et al . 1999 ; gmez et al . 2000 ; lang et al . the compact radio sources in the stellar clusters have a variety of origins : ionized stellar winds , ultracompact h2 regions , externally ionized proplyds , and magnetically active stars with gyrosynchrotron emission are the dominant types of sources . in table 6 we summarize the parameters of the radio sources in these clusters . although more of these clusters should be studied to have a reliable statistical base , some interesting trends are evident . the diameters of the radio clusters are in the 0.2 to 1.0 pc range ( with the exception of the quintuplet cluster with the pistol star included , that reaches up to 2 pc ) . the radio luminosity of the most luminous member is correlated with the bolometric luminosity of the cluster . clearly , the arches and quintuplet cluster have their most luminous radio member with luminosities one to three orders of magnitude larger than typical young clusters in the galactic disk . these clusters of radio sources could be a valuable observational tool to study the stellar population of heavily obscured regions of star formation when combined with high resolution spectroscopic observations . | a multi - frequency , multi - configuration study has been made of the compact radio sources in the galactic center quintuplet and arches stellar clusters using the very large array . we conclude that the three most compact of these sources are produced by stellar wind emission ; thus , mass - loss rates can be derived and have an average value of 3 m yr . the remainder of the sources are likely to be a combination of stellar wind emission and free - free emission from surrounding ionized gas . in three cases , the radio sources have no stellar counterpart and the radio emission is thought to arise from compact or ultra - compact h2 regions . the radio nebula associated with the pistol star resembles the nebula surrounding the lbv star carina and may be related to the stellar wind of the pistol star . several of the sources show moderate variability ( 10 - 30% ) in their flux density , possibly related to a nonthermal component in the wind emission . a number of radio sources in both clusters have x - ray counterparts , which have been interpreted to be the shocked , colliding winds of massive binary systems . | a multi - frequency , multi - configuration study has been made of the compact radio sources in the galactic center quintuplet and arches stellar clusters using the very large array . ten radio sources have been detected in the quintuplet cluster . the majority of these radio sources have rising spectral indices and are positionally coincident with young massive stars that are known to have powerful stellar winds . we conclude that the three most compact of these sources are produced by stellar wind emission ; thus , mass - loss rates can be derived and have an average value of 3 m yr . the remainder of the sources are likely to be a combination of stellar wind emission and free - free emission from surrounding ionized gas . in three cases , the radio sources have no stellar counterpart and the radio emission is thought to arise from compact or ultra - compact h2 regions . if so , these sources would be the first detections of embedded massive stars to be discovered in the galactic center clusters . the radio nebula associated with the pistol star resembles the nebula surrounding the lbv star carina and may be related to the stellar wind of the pistol star . ten compact radio sources are detected in the arches cluster and are interpreted to be stellar wind sources , consistent with previous findings . several of the sources show moderate variability ( 10 - 30% ) in their flux density , possibly related to a nonthermal component in the wind emission . a number of radio sources in both clusters have x - ray counterparts , which have been interpreted to be the shocked , colliding winds of massive binary systems . |
astro-ph0508178 | c | we have made multi - frequency radio observations of both the quintuplet and arches clusters using the vla . our conclusions can be summarized as follows : \(1 ) ten radio sources have been detected in the quintuplet cluster . the majority of these sources have rising spectral indices and are positionally coincident with near - ir sources which have been classified as young , massive stars . sources qr6 , qr8 and qr9 are unresolved and thought to be very compact in size ( @xmath140.01 pc ) ; therefore , these three sources are interpreted to be stellar winds . sources qr4 , qr5 and qr7 are more extended ( 0.3@xmath71.5 ; 0.015 to 0.06 pc ) and likely represent a combination of stellar wind and free - free emission associated with h2 regions . sources qr1 - 3 do not have near - ir counterparts and these are interepreted to be the first detections of embedded massive stars in the quintuplet cluster . \(2 ) an observation of the pistol star shows that its structure is extended and distinctly bipolar , centered on the pistol star near - ir source . the radio emission from this nebula has a flat spectral index . assuming that this emission is optically thin , free - free emission , we can roughly calculate a mass - loss rate of 5 @xmath0 10@xmath28 m@xmath2 yr@xmath3 . the pistol star radio source resembles the nebula that surrounds the luminous blue variable star @xmath4 carina . \(3 ) ten compact radio sources have also been detected in the arches cluster . nine of the sources have spectral indices consistent with a rising spectrum and only one of the sources may be resolved . as found in lang et al . ( 2001b ) , these sources are interpreted to be stellar winds from the young massive stars in the cluster . the average mass - loss rate for these stellar winds is 3 @xmath0 10@xmath1 m@xmath2 yr@xmath3 . \(4 ) both point - like and diffuse x - ray emission has been identified with the arches and quintuplet clusters and thought to be related to the presence of colliding wind binaries in these clusters . the mass - loss estimates from our radio observations here provide the best estimates of one of the parameters used to model the x - ray `` cluster winds '' which are observed to be associated with a number of massive stellar clusters . c.c.l . acknowledges that this material is based on work supported by the national science foundation ( nsf ) under grant no . 0307052 . k.e.j gratefully acknowledges support for this work provided by the nsf through an astronomy & astrophysics postdoctoral fellowship , and nasa through hubble fellowship grant # hf-01173.02-a awarded by the space telescope science institute , which is operated by the association of universities for research in astronomy , inc . , for nasa , under contract nas 5 - 26555 . lcccccc quintuplet&17 46 15.00 & @xmath728 49 35.0&4.9 & bna&june 2002 & 4 + quintuplet&17 46 15.00 & @xmath728 49 35.0&8.5 & bna&june 2002 & 4 + quintuplet&17 46 15.00 & @xmath728 49 35.0&22.5 & bna&sept 2003&2 + quintuplet&17 46 15.00 & @xmath728 49 35.0&22.5 & b&jan . 2004&3 + quintuplet&17 46 15.00&@xmath728 49 35.0&22.5 & dnc , d&june 2004&3.5 + qr1 - 3 sources&17 46 14.60&@xmath728 49 20.0&43.4 & dnc&june 2004&2.5 + qr1 - 3 sources&17 46 14.60&@xmath728 49 20.0&43.4 & d & july 2004&3 + pistol & 17 46 15.24 & @xmath728 50 04.0 & 8.5 & b , bna & nov . 1999 & 4 + arches&17 45 50.41 & @xmath728 49 21.8 & 8.5 & a&jan . 2002&4 + arches&17 45 50.41 & @xmath728 49 21.8 & 22.5 & bna&sept . 2003 & 2.5 + lclccc quintuplet & 4.9 & 1.15 @xmath0 0.85 , pa=62.8&75&10&20.6 + quintuplet & 8.5 & 0.64 @xmath0 0.50 , pa=62.0&55&10&20.6 + quintuplet&22.5 & 0.39 @xmath0 0.22 , pa=52.7&65&15&13.8 + quintuplet & 22.5 & 1.45 @xmath0 0.87 , pa=06.8&75&@xmath7&@xmath7 + quintuplet ( qr1 - 3 ) & 43.4 & 2.54 @xmath0 1.35 , pa=03.3&100&@xmath7&@xmath7 + pistol & 8.5 & 1.17 @xmath0 0.59 , pa=15.2&75&25&8.2 + arches&8.5 & 0.42 @xmath0 0.20 , pa=01.6&25&20&10.3 + arches&22.5 & 0.33 @xmath0 0.20 , pa=47.8&85&10&20.6 + lcccccccc qr1 & 17 46 15.16 & @xmath728 49 22.5 & 2.2@xmath190.4 & 1.9@xmath190.2&1.4@xmath190.3&1.0@xmath190.5&2.5@xmath01.5 & 0.10@xmath00.06 + qr2 & 17 46 14.95 & @xmath728 49 20.2 & 3.6@xmath190.7 & 3.4@xmath190.3 & 2.4@xmath190.4 & 1.4@xmath190.6&4.0@xmath02.5 & 0.17@xmath00.10 + qr3 & 17 46 14.42 & @xmath728 49 19.4 & 1.7@xmath190.4 & 1.5@xmath190.2 & 1.1@xmath190.3 & 0.3@xmath190.2&2.5@xmath02.5 & 0.10@xmath00.10 + qr4 & 17 46 15.06 & @xmath728 49 29.4 & 0.60@xmath190.30 & 1.30@xmath190.30 & ... & ... & 1.4@xmath00.9&0.04@xmath00.03 + qr5 & 17 46 15.11 & @xmath728 49 37.2 & 0.35@xmath190.10 & 0.54@xmath190.10 & 1.50@xmath190.20 & ... & 0.3@xmath00.3 & 0.01@xmath00.01 + qr5ext ... & ... & ... & ... & ... & 2.60@xmath191.00 & 2.80@xmath191.6 & 1.6@xmath01.2 & 0.06@xmath00.05 + qr6 & 17 46 15.14 & @xmath728 49 32.7 & 0.25@xmath190.10 & 0.33@xmath190.10 & 0.48@xmath190.08 & ... & ps & ... + qr7 & 17 46 14.71 & @xmath728 49 40.8 & @xmath140.24 & 0.80@xmath190.20 & ... & ... & 1.3@xmath00.7&0.05@xmath00.03 + qr8 & 17 46 14.04 & @xmath728 49 16.7 & @xmath140.24 & 0.28@xmath190.10 & 0.57@xmath190.08 & ... & ps & ... + qr9 & 17 46 17.98 & @xmath728 49 03.4 & @xmath140.24 & 0.27@xmath190.06 & ... & ... & ps & ... + pistol & 17 46 15.32 & @xmath728 50 04.2 & 3.2@xmath191.0 & 3.4@xmath191.0&3.5@xmath191.0 & 3.4@xmath191.0 & 4.0@xmath02.5 & 0.17@xmath00.10 + lccccccccccc qr1 & @xmath70.3@xmath190.4&@xmath70.3@xmath190.2 & @xmath70.3@xmath190.3&@xmath70.5@xmath190.8 & @xmath7 & 1.5@xmath010@xmath46 & 0.023&2.4@xmath010@xmath43&1.2@xmath010@xmath47 & @xmath7 + qr2 & @xmath70.1@xmath190.4 & @xmath70.4@xmath190.2&@xmath70.3@xmath190.3&@xmath70.8@xmath190.7 & @xmath7 & 0.9@xmath010@xmath46 & 0.065&1.6@xmath010@xmath43 & 2.1@xmath010@xmath47 & @xmath7 + qr3 & @xmath70.2@xmath190.5 & @xmath70.3@xmath190.3&@xmath70.3@xmath190.4&@xmath72.0@xmath191.0 & @xmath7 & 0.9@xmath010@xmath46 & 0.030&1.1@xmath010@xmath43&1.0@xmath010@xmath47 & @xmath7 + qr4 & + 1.4@xmath191.0 & @xmath7 & @xmath7 & @xmath7 & 1.5@xmath010@xmath28 & 2.7@xmath010@xmath46 & 0.008&4.8@xmath010@xmath43&7.9@xmath010@xmath48 & f270s + qr5 & + 0.8@xmath190.6 & + 1.0@xmath190.2 & + 0.9@xmath190.3 & 0.1@xmath190.1 & 1.1@xmath010@xmath28 & 3.0@xmath010@xmath46 & 0.017&7.1@xmath010@xmath43&1.8@xmath010@xmath47 & f241 + qr6 & + 0.5@xmath190.8&+0.4@xmath190.4&+0.4@xmath190.4&@xmath7 & 3.7@xmath010@xmath1 & @xmath7&@xmath7&@xmath7&@xmath7 & f257 + qr7 & + 2.0@xmath190.8&@xmath7&@xmath7&@xmath7 & 9.5@xmath010@xmath1 & 2.7@xmath010@xmath46 & 0.005&4.1@xmath010@xmath43&4.8@xmath010@xmath48 & f231 + qr8 & @xmath13 + 0.3&+0.7@xmath190.4&@xmath13 + 0.6&@xmath7 & 4.3@xmath010@xmath1 & @xmath7&@xmath7&@xmath7&@xmath7 & f320 + qr9 & @xmath13 + 0.2 & @xmath7&@xmath7&@xmath7 & 3.2@xmath010@xmath1 & @xmath7&@xmath7&@xmath7&@xmath7 & f362 + pistol & + 0.1@xmath190.8&+0.0@xmath190.4&+0.1@xmath190.4&+0.0@xmath190.6 & 2.7@xmath010@xmath28 & @xmath7 & @xmath7 & @xmath7 & @xmath7 & f134 + lcccccccc ar1 & 17 45 50.42&@xmath728 49 22.3 & 1.90@xmath190.05&1.49@xmath190.27&@xmath7&2.2@xmath010@xmath28&f6 , n8 , c8 & 0.14 @xmath00.10 + ar2 & 17 45 50.39&@xmath728 49 21.3 & 0.25@xmath190.03&0.60@xmath190.09&+0.9@xmath190.2 & 4.6@xmath010@xmath1 & f8 , n7 , c6&ps + ar3 & 17 45 50.20&@xmath728 49 22.3 & 0.12@xmath190.03&@xmath140.25&@xmath130.8&1.9@xmath010@xmath1&f1 , n4 , c9&ps + ar4 & 17 45 50.47&@xmath728 49 19.5 & 0.34@xmath190.03&0.61@xmath190.09&+0.6@xmath190.2&4.8@xmath010@xmath1&f37 , n10 , c5&ps + ar5 & 17 45 50.57&@xmath728 49 17.5 & 0.16@xmath190.03&@xmath140.25&@xmath130.5&1.9@xmath010@xmath1 & f4 , n11,c2&ps + ar6 & 17 45 49.76&@xmath728 49 26.0 & 0.25@xmath190.03&@xmath140.25&@xmath130.0&3.0@xmath010@xmath1&f19&ps + ar7 & 17 45 50.83&@xmath728 49 26.4 & 0.23@xmath190.03&0.57@xmath190.09&+0.9@xmath190.2&4.3@xmath010@xmath1&f3 , n14 , c11&ps + ar8 & 17 45 50.45&@xmath728 49 31.9 & 0.16@xmath190.03&0.48@xmath190.09&+1.1@xmath190.3&3.7@xmath010@xmath1&f5 , n9&ps + ar9 & 17 45 50.47&@xmath728 49 17.9 & 0.15@xmath190.04&@xmath140.25&@xmath130.5&1.9@xmath010@xmath1&f18&ps + ar10 & 17 45 49.69&@xmath728 49 25.9 & 0.06@xmath190.03&@xmath140.25&@xmath131.5&1.9@xmath010@xmath1&f2 , n1 , c13&ps + lccccc orion & @xmath52@xmath010@xmath43 & 77 & 8.8 & 0.3 & zapata et al . ( 2004 ) + ngc 1579 & @xmath52@xmath010@xmath46 & 16 & 0.6 & 0.3 & stine & oneal ( 1998 ) + ggd 14 & @xmath510@xmath27 & 6 & 0.2 & 0.2 & gmez et al . ( 2002 ) + ngc 1333 & @xmath5120 & 44 & 0.3 & 0.7 & rodrguez et al . ( 1999 ) + ngc 2024 & @xmath55@xmath010@xmath27 & 25 & 3.0 & 0.5 & rodrguez et al . ( 2003 ) + w40 & @xmath55@xmath010@xmath27 & 14 & 2.8 & 0.2 & rodrguez & reipurth ( 2005 ) + quintuplet & @xmath5 10@xmath45 & 10 & 218 & 2.0 & this paper + arches & @xmath510@xmath45 & 10 & 122 & 0.7 & this paper + ngc3603 & @xmath55@xmath0 10@xmath8 & 2 & 147 & 0.7 & moffat et al . ( 2002 ) + ngc3603 & @xmath5 5@xmath0 10@xmath8 & 4 & 687 & 5.0 & mcke et al . ( 2002 ) + 0.22 , pa=52.7 . the rms noise level in this image is 65 @xmath17jy beam@xmath3 and the contours correspond to -3 , 3 , 5 , 7 , 9 and 12 times this level . this image has not been corrected for primary beam attenuation . [ fig3 ] ] ra=0.9 was introduced to align the radio and _ hst _ near - ir images ) . the 8.5 ghz image ( top ) is an inset from figure 2 , with a resolution of 0.64@xmath00.50 , pa=62 ; the 22.5 ghz image ( middle ) is the `` low '' resolution image and has a resolution of 1.45@xmath00.87 , pa=6.8 ; the 43.4 ghz ( bottom ) image has a resolution of 2.54@xmath01.35 , pa=3.3 . contours in each case correspond to -3 , 3 , 5 , 7 , 9 , and 12 times the rms levels of 55 @xmath17jy beam@xmath3 ( 8.5 ghz ) , 75 @xmath17jy beam@xmath3 ( 22.5 ghz ) and 100 @xmath17jy beam@xmath3 ( 43.4 ghz ) . none of these images have been corrected for primary beam attenuation . | ten radio sources have been detected in the quintuplet cluster . the majority of these radio sources have rising spectral indices and are positionally coincident with young massive stars that are known to have powerful stellar winds . if so , these sources would be the first detections of embedded massive stars to be discovered in the galactic center clusters . ten compact radio sources are detected in the arches cluster and are interpreted to be stellar wind sources , consistent with previous findings . | a multi - frequency , multi - configuration study has been made of the compact radio sources in the galactic center quintuplet and arches stellar clusters using the very large array . ten radio sources have been detected in the quintuplet cluster . the majority of these radio sources have rising spectral indices and are positionally coincident with young massive stars that are known to have powerful stellar winds . we conclude that the three most compact of these sources are produced by stellar wind emission ; thus , mass - loss rates can be derived and have an average value of 3 m yr . the remainder of the sources are likely to be a combination of stellar wind emission and free - free emission from surrounding ionized gas . in three cases , the radio sources have no stellar counterpart and the radio emission is thought to arise from compact or ultra - compact h2 regions . if so , these sources would be the first detections of embedded massive stars to be discovered in the galactic center clusters . the radio nebula associated with the pistol star resembles the nebula surrounding the lbv star carina and may be related to the stellar wind of the pistol star . ten compact radio sources are detected in the arches cluster and are interpreted to be stellar wind sources , consistent with previous findings . several of the sources show moderate variability ( 10 - 30% ) in their flux density , possibly related to a nonthermal component in the wind emission . a number of radio sources in both clusters have x - ray counterparts , which have been interpreted to be the shocked , colliding winds of massive binary systems . |
0905.2026 | i | the @xmath11 reaction is one of the critical links in the @xmath12 and @xmath13 branches of the @xmath14chain of solar hydrogen burning [ 13 ] . the total capture rate determined by processes of this chain is sensitive to the cross section @xmath15 ( or the astrophysical @xmath0 factor @xmath6 ) for the @xmath16 reaction and predicted neutrino rate varies as @xmath17^{0.8}$ ] @xcite . despite the impressive improvements in our understanding of the @xmath11 reaction made in the past decades ( see refs [ 411 ] and references therein ) , however , some ambiguities connected with both the extrapolation of the measured cross sections for the aforesaid reaction to the solar energy region and the theoretical predictions for @xmath15 ( or @xmath6 ) still exist and they may influence the predictions of the standard solar model @xcite . experimentally , there are two types of data for the @xmath16 reaction at extremely low energies : i ) six measurements based on detecting of @xmath18-rays capture ( see @xcite and references therein ) from which the astrophysical @xmath0 factor @xmath19 extracted by the authors of those works changes within the range 0.47@xmath200.58 @xmath21 and ii ) six measurements based on detecting of @xmath12 ( see @xcite references therein as well as [ 6 - 11 ] ) from which @xmath19 extracted by the authors of these works changes within the range 0.53@xmath220.63 @xmath23 . all of these measured data have a similar energy dependence for the astrophysical @xmath0 factors @xmath6 . nevertheless , the adaptation of the available energy dependencies predicted in @xcite for the extrapolation of each of the measured data to low experimentally inaccessible energy regions , including @xmath8=0 , leads to a value of @xmath19 that differs from others and this difference exceeds the experimental uncertainty . the theoretical calculations of @xmath19 performed within different methods also show considerable spread [ 12,1419 ] and the result depends on a specific model used . for example , the resonating - group method calculations of @xmath19 performed in ref.@xcite show considerable sensitivity to the form of the effective nn interaction used and the estimates have been obtained within the range of 0.312@xmath20 0.841 @xmath23 . the estimation of @xmath19=0.52@xmath24 0.03 @xmath25 @xcite also should be noted . the latter has been obtained from the analysis of the experimental astrophysical @xmath0 factors @xcite , which were performed within the framework of the standard two - body potential model in the assumption that the dominant contribution to the peripheral reaction comes from the surface and external regions of the nucleus @xmath12 @xcite . at this , in @xcite the contribution from the nuclear interior ( @xmath26 , @xmath27=4 fm ) to the amplitude is ignored . in this case , the astrophysical @xmath0 factor is directly expressed in terms of the nuclear vertex constants ( nvc ) for the virtual decays @xmath28 ( or respective the asymptotic normalization coefficients ( anc ) for @xmath29 ) @xcite . as a result , in ref . @xcite , the nvc - values for the virtual decays @xmath30 and @xmath31 were obtained , which were then used for calculations of the astrophysical @xmath0 factors at @xmath32180 kev , including @xmath8=0 . however , the values of the ancs ( or nvcs ) @xmath33 and the @xmath19 obtained in @xcite may not be enough accurate associated both with the aforesaid assumption in respect to the contribution from the nuclear interior ( @xmath26 ) and with a presence of the spread in the experimental data @xcite used for the analysis . as far available values of these ancs obtained in @xcite , they depend noticeably on a specific model used . therefore , determination of precise experimental values of the ancs for @xmath33(g.s . ) and @xmath29(0.429 mev ) is highly desirable since it has direct effects in the correct extrapolation of the @xmath34 astrophysical @xmath0 factor at solar energies . recently , a modified two - body potential approach ( mtbpa ) was proposed in @xcite for the peripheral direct capture @xmath35 reaction , which is based on the idea proposed in paper @xcite that low - energy direct radiative captures of particle @xmath36 by light nuclei @xmath37 proceed mainly in regions well outside the range of the internuclear @xmath38 interactions . one notes that in mtbpa the direct astrophysical @xmath0 factor is expressed in terms of anc for @xmath39 rather than through the spectroscopic factor for the nucleus @xmath40 in the ( @xmath41 ) configuration as it is made within the standard two - body potential method @xcite . in refs.@xcite , mtbpa has been successfully applied to the radiative proton and @xmath9-particle capture by some light nuclei . therefore , it is of great interest to apply the mtbpa for analysis of the @xmath42 reaction . in this work new analysis of the modern precise experimental astrophysical @xmath0 factors for the direct capture @xmath16 reaction at extremely low energies ( @xmath43 90 kev ) [ 6 - 11 ] is performed within the mtbpa @xcite to obtain @xmath3indirectly determined values both of the ancs ( the nvcs ) for @xmath4(g.s . ) and @xmath5(0.429 mev ) , and of @xmath6 at @xmath44 90 kev , including @xmath8=0 . in this work we quantitatively show that the @xmath45 reaction within the aforesaid energy region is mainly peripheral and one can extract ancs for @xmath46 directly from the @xmath47 reaction where the ambiguities inherent for the standard two -body potential model calculation of the @xmath48 reaction , which is connected with the choice of the geometric parameters ( the radius @xmath49 and the diffuseness @xmath36 ) for the woods saxon potential and the spectroscopic factors @xcite , can be reduced in the physically acceptable limit , being within the experimental errors for the @xmath6 . the contents of this paper are as follows . in section 2 the results of the analysis of the precise measured astrophysical @xmath0 factors for the direct radiative capture @xmath45 reaction is presented ( subsections 2.12.3 ) . the conclusion is given in section 3 . in appendix basic formulae of the modified two - body potential approach to the direct radiative capture @xmath16 reaction are given . | a new analysis of the modern precise measured astrophysical factors for the direct capture reaction [ b.s . new estimates are obtained for theindirectly determined values of the asymptotic normalization constants ( the nuclear vertex constants ) for(g.s . ) and(0.429 mev ) as well as the astrophysical factors at e 90 kev , including=0 . | a new analysis of the modern precise measured astrophysical factors for the direct capture reaction [ b.s . nara singh _ et al . _ , phys.rev.lett . * 93 * , 262503 ( 2004 ) ; d. bemmerer _ et al . _ , phys.rev.lett . * 97 * , 122502 ( 2006 ) ; f.confortola _ et al . _ , phys.rev.c * 75 * , 065803 ( 2007 ) , t.a.d.brown _ et al . _ , phys.rev . c * 76 * , 055801 ( 2007 ) and a di leva , _ et al._,phys.rev.lett . * 102 * , 232502 ( 2009 ) ] populating to the ground and first excited states of is carried out based on the modified two - body potential approach . new estimates are obtained for theindirectly determined values of the asymptotic normalization constants ( the nuclear vertex constants ) for(g.s . ) and(0.429 mev ) as well as the astrophysical factors at e 90 kev , including=0 . the values of asymptotic normalization constants have been used for getting information about the-particle spectroscopic factors for the mirror ()-pair . pacs : 25.55.-e;26.35.+c;26.65.+t + |
0905.2026 | c | the analysis of the modern experimental astrophysical @xmath0 factors , @xmath111 , for the @xmath16 reaction , which were precisely measured at energies @xmath8=92.9 - 1235 kev [ 611 ] , has been performed within the modified two - body potential approach @xcite . the performed scrupulous analysis shows quantitatively that the @xmath11 reaction within the considered energy ranges is mainly peripheral and the parametrization of the direct astrophysical @xmath0 factors in terms of ancs for the @xmath150 is adequate to the physics of the peripheral reaction under consideration . it is shown that the experimental astrophysical @xmath0 factors of the reaction under consideration [ 611 ] can be used as an independent source of getting the information about the ancs ( or nvcs ) for @xmath257 ( or for the virtual decay @xmath258 ) , and the found ancs can predict the experimental astrophysical @xmath0 factors separated for the @xmath235(g.s . ) and @xmath235(0.429 mev ) reactions at low experimentally acceptable energy regions ( 126.5@xmath7 e@xmath7 1203 kev ) obtained from the total experimental astrophysical @xmath0 factors [ 69,11 ] . the new estimation for the weighted means of the ancs for @xmath259 and nvcs for the virtual decay @xmath258 are obtained . also , the values of ancs were used for getting the information about the @xmath9-particle spectroscopic factors for the mirror ( @xmath260)-pair . the obtained values of the ancs were also used for an extrapolation of astrophysical @xmath0 factors at energies less than 90 kev , including @xmath8=0 . in particular , the weighted mean of the branching ratio @xmath256 ( @xmath256=0.43@xmath24 0.01 ) and the total astrophysical @xmath0 factor @xmath19 ( @xmath224(0)=0.613@xmath261 @xmath226 ) obtained here are in agreement with that deduced in [ 711 ] from the analysis the same experimental asprophysical @xmath0 factors . besides , our result for @xmath19 is in an agreement with that @xmath19=0.56 kev @xcite obtained within the microscopical single - channel ( @xmath147 ) cluster model , @xmath19= 0.609 @xmath262 @xcite and @xmath19= 0.621 @xmath262 @xcite obtained within the ( @xmath263)-channel of version of the resonating - group method , but it is noticeably larger than the result of @xmath19=0.516 ( 0.53 ) kev @xcite(@xcite ) obtained within the standard two - body ( @xmath147 ) potential by using @xmath75 potential deduced by a double - folding procedure . the authors are deeply grateful to s. v. artemov , l.d . blokhintsev and a.m. mukhamedzhanov for discussions and general encouragement . the authors thank also d.bemmerer for providing the experimental results of the updated data analysis . the work has been supported by the academy of science of the republic of uzbekistan ( grant no.fa-f2-f077 ) . according to @xcite , for fixed @xmath50 and @xmath51 we can write the astrophysical @xmath0 factor , @xmath98 , for the peripheral direct capture @xmath11 reaction in the following form @xmath264 here , @xmath265 is the anc for @xmath266 , which determines the amplitude of the tail of the @xmath186 nucleus bound state wave function in the ( @xmath267)-channel and is related to the spectroscopic factor @xmath268 for the ( @xmath147)-configuration with the quantum numbers @xmath50 and @xmath51 in the @xmath60 nucleus by the equation @xmath269 @xcite , and @xmath270 , where @xmath271 is the single - particle astrophysical @xmath0 factor @xcite and @xmath78 ( @xmath272 @xcite ) is the single - particle anc , which determines the amplitude of the tail of the single - particle wave function of the bound ( @xmath273 ) state @xmath274(@xmath275 ) @xcite ) and in turn is itself a function of the geometric parameters ( radius of @xmath49 and diffuseness @xmath36 ) of the woods - saxon potential , i.e. @xmath276@xcite . in order to make the dependence of the @xmath87 function on @xmath88 more explicit , in the radial matrix element @xcite entering in the @xmath277 function , we split the space of interaction in two parts separated by the channel radius @xmath278 : the interior part ( @xmath279 , where nuclear forces between the @xmath280-pair are important , and the exterior part ( @xmath281 ) , where the interaction between the @xmath9-particle and @xmath282 is governed by coulomb force only . the contribution from the exterior part of the radial matrix element into the @xmath87 function does not depend on @xmath88 since for @xmath205 the wave function @xmath283 can be approximated by its asymptotic behavior @xcite . consequently , the parametrization of the astrophysical @xmath0 factor in the form ( a1 ) makes one it possible to fix a contribution from the exterior region ( @xmath281 ) , which is dominant for the peripheral reaction , by a model independent way if the ancs @xmath95 are known . it follows from here that the contribution from the interior part of the radial matrix element into the @xmath87 function , which depends on @xmath88 through the fraction @xmath284 @xcite , must exactly determine the dependence of the @xmath277 function on @xmath88 . in eq . ( a1 ) the ancs @xmath95 and the free parameter @xmath88 are unknown . but , for the peripheral @xmath48 reaction the equation ( a1 ) can be used for determination of the ancs . for this aim , obviously the following additional requirements @xcite @xmath285 and @xmath286 must be fulfilled as a function of the free parameter @xmath88 for each energy @xmath8 experimental point from the range @xmath287 and values of the function of @xmath87 from ( a2 ) . the fulfillment of the relations ( a2 ) and ( a3 ) or their violation within the experimental uncertainty for @xmath288 enables one , firstly , to determine an interval for energies e where the dominance of extra - nuclear capture occurs and , secondly , to obtain the value @xmath289 for @xmath290 using the experimental astrophysical @xmath0 factors @xmath66 , precisely measured by authors of refs . [ 611 ] , instead of @xmath98 , i.e. @xmath291 then , the value @xmath289 can be used for extrapolation of the astrophysical @xmath0 factor @xmath98 to the region of experimental inaccessible energies 0@xmath292 by using the obtained value @xmath289 in ( a1 ) . the total astrophysical @xmath0 factor for the @xmath16(g.s.+0.429 mev ) reaction is given by @xmath293 @xmath294\eqno(a6)\ ] ] @xmath295 \eqno(a7)\ ] ] in which @xmath296 and @xmath115 is a branching ratio . one notes that in the two - body potential model the anc @xmath297 is related to the nvc @xmath298 for the virtual decay @xmath299 by the equation @xcite @xmath300 where @xmath301 is the coulomb parameter for the @xmath302 bound state . in ( a8 ) the combinatorial factor taking into account the nucleon s identity is absorbed in @xmath297 and its numerical value depends on a specific model used to describe wave functions of the @xmath303 , @xmath9 and @xmath60 nuclei @xcite . hence , the proportionality factor in ( a8 ) , which relates nvc s and ancs , depends on the choice of nuclear model @xcite . but , as it is noted in @xcite , the nvc @xmath298 is a more fundamental quantity than the anc @xmath297 since the nvc is determined in a model - independent way as the residue of the partial @xmath0-matrix of the elastic @xmath304-scattering at the pole @xmath305 ( @xmath306 is the binding energy of the bound ( @xmath147 ) state of @xmath60 ) @xcite . therefore , it is also of interest to get an information about values of the nvcs from eqs . 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[ cols="^,^,^,^,^,^,^ " , ] | nara singh _ et al . _ , phys.rev.lett . * 93 * , 262503 ( 2004 ) ; d. bemmerer _ et al . _ , phys.rev.lett . * 97 * , 122502 ( 2006 ) ; f.confortola _ et al . _ , phys.rev.c * 75 * , 065803 ( 2007 ) , t.a.d.brown _ et al . _ , phys.rev . c * 76 * , 055801 ( 2007 ) and a di leva , _ the values of asymptotic normalization constants have been used for getting information about the-particle spectroscopic factors for the mirror ()-pair . | a new analysis of the modern precise measured astrophysical factors for the direct capture reaction [ b.s . nara singh _ et al . _ , phys.rev.lett . * 93 * , 262503 ( 2004 ) ; d. bemmerer _ et al . _ , phys.rev.lett . * 97 * , 122502 ( 2006 ) ; f.confortola _ et al . _ , phys.rev.c * 75 * , 065803 ( 2007 ) , t.a.d.brown _ et al . _ , phys.rev . c * 76 * , 055801 ( 2007 ) and a di leva , _ et al._,phys.rev.lett . * 102 * , 232502 ( 2009 ) ] populating to the ground and first excited states of is carried out based on the modified two - body potential approach . new estimates are obtained for theindirectly determined values of the asymptotic normalization constants ( the nuclear vertex constants ) for(g.s . ) and(0.429 mev ) as well as the astrophysical factors at e 90 kev , including=0 . the values of asymptotic normalization constants have been used for getting information about the-particle spectroscopic factors for the mirror ()-pair . pacs : 25.55.-e;26.35.+c;26.65.+t + |
1002.3607 | c | we have detected a moving group of at least four bhb stars in the corner ( toe " ) of the hercules constellation spilling into corona borealis . these stars are coincident in angular position [ @xmath168 ) ] , apparent magnitude ( @xmath1 ) , line - of - sight velocity ( @xmath4 km s@xmath3 , @xmath169 km s@xmath3 , @xmath170 km s@xmath3 ) , and metallicity ( [ fe / h]=@xmath171 ) . we expect that the progenitor of this moving group was a low metallicity globular cluster , with a luminosity like that of one of the smaller globular clusters in the milky way halo . we show that useful surface gravities and metallicities are measured for bhb stars with s / n@xmath172 in sdss dr7 . the mean metallicity of bhb stars in the outer halo is similar to m53 , which has a published metallicity of [ fe / h]=@xmath6 . the metallicity does not appear to change with distance from the sun ( @xmath173 kpc ) . our measurement of the spheroid metallicity is slightly higher than claimed by @xcite and somewhat lower than earlier studies of outer halo stars . the hercules moving group is one of many tidally disrupted stellar associations expected to comprise the spheroid of the milky way and could not have been identified from photometry alone ; more complete spectroscopic surveys are required to identify the component spheroid moving groups , and determine the merger history of our galaxy . we present a statistical technique that allows us to estimate the significance of clumps discovered in multidimensional data . | a new moving group comprising at least four blue horizontal branch ( bhb ) stars is identified at ) . we explore the systematics of sdss dr7 surface gravity metallicity determinations for faint bhb stars , and present a technique for estimating the significance of clumps discovered in multidimensional data . | a new moving group comprising at least four blue horizontal branch ( bhb ) stars is identified at ) . the horizontal branch at magnitude implies a distance of 50 kpc from the sun . the heliocentric radial velocity is km s , corresponding to km s ; the dispersion in line - of - sight velocity is consistent with the instrumental errors for these stars . the mean metallicity of the moving group is [ fe / h] , which is significantly more metal poor than the stellar spheroid . we estimate that the bhb stars in the outer halo have a mean metallicity of [ fe / h]= , with a wide scatter and a distribution that does not change much as a function of distance from the sun . we explore the systematics of sdss dr7 surface gravity metallicity determinations for faint bhb stars , and present a technique for estimating the significance of clumps discovered in multidimensional data . this moving group can not be distinguished in density , and highlights the need to collect many more spectra of galactic stars to unravel the merger history of the galaxy . [ firstpage ] galaxy : structure galaxy : halo methods : statistical . |
0802.2399 | i | _ boundary alignment _ is a critical feature of meshes in many applications . in a boundary aligned mesh the boundary , or some other line , is traced by the sides of high - quality cells , see fig . [ fig : bound_align ] . the definition of a well - shaped cell may be application dependent , but in many cases , cells similar in shape to squares ( for quadrilateral cells ) or equilateral triangles ( for triangles ) are preferred . characteristics of the entire mesh are also important , such as smooth cells - size and cell - shape transitions . [ ptb ] the problem of producing boundary aligned meshes with well - shaped cells has been the subject of extensive research @xcite . still , many popular algorithms are heuristic in nature , and a more general understanding of the subject is called for , especially when quadrilateral meshes are considered . a key difficulty is the problem s _ global _ character : the shape and position of every cell in the mesh is , at least in principle , related to that of any other cell . in a previous work @xcite , we described a relation between the problem of two - dimensional unstructured mesh generation , on both planar and curved surfaces , and another well - known problem , namely the _ inverse poisson _ ( ip ) problem . the ip problem is concerned with reconstructing a _ source distribution _ @xmath0 of the poisson equation @xmath1 , from information on the potential @xmath2 at the boundaries . in that work , the mesh was assumed to be _ conformal _ away from the irregular vertices ( vertices whose degree is different than four ) , like a grid mapped by a conformal mapping . such grids have the property of having square cells in the limit of an increasingly finer grid . under this assumption , the problem of mesh generation was then shown to reduce to an ip problem . the irregular vertices of the mesh correspond to point sources ( delta functions ) of @xmath0 , and @xmath2 is interpreted as the logarithm of the local resolution . this theoretical framework turns the focus to the irregular vertices of the mesh : once their distribution is fixed , the continuum properties of the mesh - local resolution and directionality - are known . note that it is also an explicitly global formulation , since the resolution at any given point is affected , via the function @xmath2 , by the locations of all irregular vertices in the mesh . in this paper the generation of planar quadrilateral meshes is discussed . resting on the results of @xcite , a new ip algorithm is presented , designed to construct source distributions of the appropriate type , which approximate the resolution and mesh directionality inputs at user - specified points , such as at the boundaries . the ip algorithm is then incorporated into a complete mesh generation scheme , which also includes a technique for generating the final mesh . an implementation is described , and shown in example cases to generate boundary aligned meshes , where well - placed point sources create smooth cell transitions and high quality cells . a similar procedure is probably applicable to triangular meshes , but is not discussed in the present work . remeshing of curved surfaces has recently attracted considerable attention ; for a review see @xcite . many of the algorithms receive an input mesh directionality throughout the surface , usually the principal curvature directions of the surface . this setting presents different challenges than those addressed here , since the mesh structure is determined , to a large extent , when mesh directionality is given everywhere in the domain . for example , the locations of critical irregular vertices are dictated by the mesh directionality in the _ vicinity _ of these points . another related subject is surface parameterization , concerned with creating mapping of surfaces to the plane . conformal surface parameterizations are created in @xcite , but boundary alignment is not addressed . the paper is organized as follows : section [ sec : ip_review ] shortly reviews the relevant theoretical background , with emphasis on the relation between the ip problem and unstructured mesh generation . sections [ sec : algorithm ] and [ sec : create_final ] describe the proposed ip algorithm , and the mesh generation scheme . section [ sec : implementation ] describes an implementation of the algorithm , and section [ sec : examples ] gives examples of meshes generated . conclusions and possible directions for future research are discussed in section [ sec : conclusions ] . | away from irregular vertices , the resulting meshes have the properties of nearly conformal grids . the technique is based on a theoretical relation between the present problem , and the inverse poisson ( ip ) problem with point sources . the ip algorithm is incorporated in a complete mesh generation scheme , which also includes an algorithm for creating the final mesh . example results are presented and discussed . | a novel approach to unstructured quadrilateral mesh generation for planar domains is presented . away from irregular vertices , the resulting meshes have the properties of nearly conformal grids . the technique is based on a theoretical relation between the present problem , and the inverse poisson ( ip ) problem with point sources . an ip algorithm is described , which constructs a point - source distribution , whose sources correspond to the irregular vertices of the mesh . both the background theory and the ip algorithm address the global nature of the mesh generation problem . the ip algorithm is incorporated in a complete mesh generation scheme , which also includes an algorithm for creating the final mesh . example results are presented and discussed . |
1509.07090 | i | the question whether there exists an underlying principle to understand the pattern of lepton mixing , which is quite different from the quark mixing , demands the study of neutrino mass matrix as well as the charged lepton one into a deeper level . the smallness of neutrino masses can be well understood by the seesaw mechanism in a natural way . i seesaw mechanism @xcite provides the simplest possibility by extending the standard model ( sm ) with three right - handed ( rh ) neutrinos . an introduction of discrete symmetries into it may reveal the flavor structure of the neutrino and charged lepton mass matrix . for example , a type - i seesaw in conjugation with @xmath8 explains the tribimaximal lepton mixing pattern ( tbm ) @xcite in presence of sm singlet flavon ( charged under @xmath8 ) fields which get vacuum expectation values ( vev)@xcite . however the original approach fails to accommodate the recent observation of non - zero @xmath0@xcite . in @xcite , we have shown that an extension of the altarelli - feruglio ( af ) model @xcite by one additional flavon field can be employed to have a nonzero @xmath0 consistent with the present experimental results . the set - up also constraints the two majorana phases involved in the lepton mixing matrix . the deviation of the tbm pattern is achieved through a deformation of the rh neutrino mass matrix compared to the original one . on the other hand , within the framework of a general type - ii seesaw mechanism ( where both rh neutrinos and @xmath2 triplet higgs are present ) , light neutrino mass depends upon comparative magnitude of the pure type - i ( mediated by heavy rh neutrinos ) and triplet contributions . this interplay is well studied in the literature @xcite . in recent years keeping in mind that @xmath0 is nonzero , efforts have been given to realize leptogenesis @xcite and linking it with @xmath0 in models based on type - ii seesaw@xcite . in this article , we focus on the generation of light neutrino mass matrix through a type - ii seesaw mechanism @xcite . the fields content of the sm is extended with three right - handed neutrinos , one @xmath2 triplet and a set of sm singlet flavon fields . a flavor symmetry @xmath1 is considered . the type - ii seesaw mechanism therefore consists of the conventional type - i seesaw contribution ( @xmath9 ) along with the triplet contribution ( @xmath10 ) to the neutrino mass matrix . here we find the type - i contribution alone can generate the tbm mixing pattern , where the charged lepton mass matrix is a diagonal one . then we have shown that the same flavor symmetry allows us to have a deviation from the conventional type - i contribution , triggered by the @xmath2 triplet s vev . we have found that this deviation is sufficient enough to keep @xmath0 at an acceptable level@xcite . we mostly consider the triplet contribution to the light neutrino mass is subdominat compared to the conventional type - i contribution . we further assume that apart from the flavons ( sm singlets charged under @xmath8 ) involved , there is a @xmath8 singlet ( as well as sm gauge singlet ) field @xmath3 , which gets a complex vacuum expectation value and thereby responsible for spontaneous cp violation@xcite and other extensions of standard model @xcite . ] at high scale @xcite . all other flavons have real vevs and all the couplings involved are considered to be real . it turns out that the magnitude of this complex vacuum expectation value of @xmath3 is responsible for the deviation of tbm by generating nonzero value of @xmath0 in the right ballpark . on the other hand , the phase associated with it generates the dirac cp violating phase in the lepton sector . so in a way , the triplet contribution provides a unified source for cp violation and nonzero @xmath0 . in lepton sector , the other possibilities where cp violation can take place , involves complex vev of higgs triplets@xcite , or when a higgs bi - doublet ( particularly in left - right models ) gets complex vev@xcite or in a mixed situation @xcite . however , we will concentrate in a situation where a scalar singlet @xmath3 present in the theory gets complex vev as in @xcite . we have also studied the lepton asymmetry production through the decay of the heavy triplet involved . the decay of the triplet into two leptons contributes to the asymmetry where the virtual rh neutrinos are involved in the loop . this process is effective when the triplet is lighter than all the rh neutrinos . it turns out that sufficient lepton asymmetry can be generated in this way . on the other hand if the triplet mass is heavier than the rh neutrino masses , the lightest rh neutrino may be responsible for producing lepton asymmetry where the virtual triplet is contributing in the one loop diagram . in @xcite , authors investigated a scenario where the triplet vevs are the sole contribution to the light neutrino mass and a single source of spontaneous cp violation was considered . there , it was shown that the low energy cp violating phase and the cp violation required for leptogenesis both are governed by the argument of the complex vev of that scalar field . the nonzero value of @xmath0 however followed from a perturbative deformation of the vev alignment of the flavons involved . here in our scenario , the tbm pattern is realized by the conventional type - i contribution . therefore in the tbm limit , @xmath0 is zero in our set - up . also there is no cp violating phase in this limit as all the flavons involved in @xmath9 are carrying real vevs , and hence no lepton asymmetry as well . now once the triplet contribution ( @xmath10 ) is switched on , not only the @xmath0 , but also the leptonic cp violation turn out to be nonzero . for generating lepton asymmetry , two triplets were essential in @xcite , while we could explain the lepton asymmetry by a single triplet along with the presence of rh neutrinos . in this case , the rh neutrinos are heavier compared to the mass of the triplet involved . the paper is organized as follows . in section [ sec2 ] , we provide the status of the neutrino mixing and the mass squared differences . then in section [ sec3 ] , we describe the set - up of the model followed by constraining the parameter space of the framework from neutrino masses and mixing in section [ nucons ] . in section [ sec : lep ] , we describe how one can obtain lepton asymmetry out of this construction . finally we conclude in section [ conlc ] . | while the magnitude of its complex vev is responsible for generating , its phase part induces the low energy cp violating phase ( ) and the cp violation required for leptogenesis . hence the triplet contribution , although sub - dominant , plays crucial role in providing a common source for non - zero , and cp - violation required for leptogenesis . | a possible interplay between the two terms of the general type - ii seesaw formula is exercised which leads to the generation of nonzero . the specific flavor structure of the model , guided by the symmetry and accompanied with the standard model singlet flavons , yields the conventional seesaw contribution to produce the tribimaximal lepton mixing which is further corrected by the presence of the triplet contribution to accommodate . we consider the cp symmetry to be spontaneously broken by the complex vacuum expectation value ( vev ) of a singlet field . while the magnitude of its complex vev is responsible for generating , its phase part induces the low energy cp violating phase ( ) and the cp violation required for leptogenesis . hence the triplet contribution , although sub - dominant , plays crucial role in providing a common source for non - zero , and cp - violation required for leptogenesis . we find that the recent hint for close to is somewhat favored in this set - up though it excludes the exact equality with . we also discuss the generation of lepton asymmetry in this scenario . * spontaneous cp violation in lepton - sector : a common origin for , dirac cp phase and leptogenesis * + biswajit karmakar , arunansu sil + _ indian institute of technology guwahati , 781039 assam , india _ |
1411.6098 | i | approaches used to compute vibration - rotation energy levels and wave functions include methods based on perturbation theory , effective hamiltonians and on the use of the variational principle . each of these methods has its advantages and disadvantages when used for larger molecules such as those with more than ten atoms . historically , perturbation theory was the first method employed to treat the many - body anharmonic problem @xcite , see also the review by klein @xcite . second - order perturbation theory is the most common analytic treatment for estimating the vibrational energy levels and including contributions from terms in the hamiltonian beyond the harmonic approximation ( see , for example , refs . @xcite ) as well as infrared intensities @xcite . however , there are drawbacks inherent in this method that prevent extensive use of it in practice . in particular the results obtained depend on the specific form of the hamiltonian and the distribution of degenerate oscillators . polyatomic molecules often show quasi - degeneracies between vibrational energy levels and use of second - order perturbation theory can result in significant errors . furthermore , in practice anharmonic effects in polyatomic molecules are sufficiently large that they are often not converged with a second - order treatment . as a result perturbation theory calculations usually overestimate the anharmonic corrections , sometimes by a factor of two ; see the final columns of tables [ tab : en3 ] and [ tab : en4 ] given below . variational methods for the calculation of anharmonic energy levels were developed independently by a number of authors @xcite . early implementations in general computer codes focused on the use of basis functions for triatomic molecules @xcite . more recent developments have involved the increasing use of the discrete variable representation @xcite and the extension of the work to polyatomic molecules @xcite . the main advantage of this method is that it allows the almost exact calculations of the vibrational - rotational energy levels and wave functions for a given anharmonic potential @xcite . however , variational methods work much better for few - atom systems since the size of the basis set grows factorially with the number of vibrational degrees of freedom in the molecule ; hence the size of the hamiltonian matrices , which must be computed and diagonalized , also grows very rapidly . for this reason , variational calculations for polyatomic molecules are difficult even on modern computers and therefore are not routinely used for molecules with more than six atoms , when only small basis sets can be afforded . the computational errors associated with incomplete basis sets are normally significantly larger than the errors from perturbation approaches ; this point will be discussed further in the following section . the treatment of ro - vibration states adds extra complexity to the calculation . in this case a rotational basis set , usually described with analytical rigid - rotor functions , must be introduced . for high rotational quantum number , @xmath0 , this leads to large hamiltonian matrices even for small polyatomic molecules . however a two - step variational procedure can be used to mitigate the effects of this problem @xcite meaning that it has long been possible to compute rotational excitation up to dissociation for small molecules @xcite . there are a number of modifications of the variational method which facilitates ( ro-)vibrational calculations on larger molecules . for example , the use of vibrational self - consistent field theory by gerber , ratner and others @xcite . bowman , carter and handy @xcite used vibrational configuration interaction ( vci ) to reduce the size of the diagonalization for large molecules . if the molecule has separable degrees of freedom , a significant reduction in the time taken to diagonalize the hamiltonian matrices can be obtained . scribano and benoit @xcite used a hybrid approach where a modification of the vci method take into account the interactions of individual configurations using perturbation theory . similarly the general code multimode @xcite allows such treatments of larger systems @xcite with approximations involving the degree of coupling between vibrational modes . here we propose a new hybrid variational - perturbation theory method based on a physically reasonable division of the large , full , variational hamiltonian matrix into weakly interacting sub - blocks . second - order perturbation theory is used to include cross - interaction effects between these sub - blocks , which are then diagonalized separately . this allows one to replace a single diagonalization of a large hamiltonian matrix by a series of diagonalizations of much smaller sub - matrices . we show that our hybrid method can greatly accelerate the variational procedure by eliminating diagonalization of large matrices without significant loss of precision in the computed vibrational - rotational energy levels and the intensity of transitions between them . this hybrid scheme is able to perform calculations for large molecules containing up to 15 atoms on currently available computers . finally we note that a different version of the hybrid variational - perturbation approach has recently been proposed by fabri _ et al _ @xcite and implemented in the general eckart - watson hamiltonian code dewe @xcite . their hybrid scheme uses a traditional single - state ro - vibrational perturbation method based on the variationally computed vibrational ( @xmath1 ) eigenfunctions as a zero - order solution , where the vibrational problem is solved variationally and the ro - vibrational energies are derived using the second order perturbation theory expressions . this is different from our two - step hybrid scheme , where the perturbation method is an integral part of calculations . we believe that this is the key to extending variational calculations to larger molecules . the method we propose is not dependent on the precise form of the hamiltonian . instead the requirement of the hamiltonian is that it should result in a diagonally dominant matrix representation . for semi - rigid molecules this criterion is naturally satisfied the eckart - watson hamiltonian @xcite as well as number of related hamiltonians @xcite . however the method is unlikely to perform so well for hamiltonians whose emphasis is not on making producing diagonally dominant matrices such those expressed in polyspherical coordinates @xcite . in this work , all calculations are performed using angmol @xcite , a variational program for calculating ro - vibrational spectra of general polyatomic molecules . angmol uses a hamiltonian expressed in curvilinear internal coordinates and an eckart embedding which is discussed briefly in the appendix and extensively elsewhere @xcite . | a procedure for calculation of rotation - vibration states of medium sized molecules is presented . it combines the advantages of variational calculations and perturbation theory . the vibrational problem is solved by diagonalizing a hamiltonian matrix , which is partitioned into two sub - blocks . numerical examples show that the proposed hybrid variational - perturbation method greatly speeds up the variational procedure without significant loss of precision for both vibrational - rotational energy levels and transition intensities . the hybrid scheme can be used for accurate nuclear motion calculations on molecules with up to 15 atoms on currently available computers . molecular rotation | a procedure for calculation of rotation - vibration states of medium sized molecules is presented . it combines the advantages of variational calculations and perturbation theory . the vibrational problem is solved by diagonalizing a hamiltonian matrix , which is partitioned into two sub - blocks . the first , smaller sub - block includes matrix elements with the largest contribution to the energy levels targeted in the calculations . the second , larger sub - block comprises those basis states which have little effect on these energy levels . numerical perturbation theory , implemented as a jacobi rotation , is used to compute the contributions from the matrix elements of the second sub - block . only the first sub - block needs to be stored in memory and diagonalized . calculations of the vibrational - rotational energy levels also employ a partitioning of the hamiltonian matrix into sub - blocks , each of which corresponds either to a single vibrational state or a set of resonating vibrational states , with all associated rotational levels . physically , this partitioning is efficient when the coriolis coupling between different vibrational states is small . numerical perturbation theory is used to include the cross - contributions from different vibrational states . separate individual sub - blocks are then diagonalized , replacing the diagonalization of a large hamiltonian matrix with a number of small matrix diagonalizations . numerical examples show that the proposed hybrid variational - perturbation method greatly speeds up the variational procedure without significant loss of precision for both vibrational - rotational energy levels and transition intensities . the hybrid scheme can be used for accurate nuclear motion calculations on molecules with up to 15 atoms on currently available computers . molecular rotation ; vibration ; variational ; perturbation theory ; infra red spectra ; nuclear motion * acknowledgment * this work was supported by the erc under advanced investigator project 267219 . |
1411.6098 | m | inclusion of rotational motion into the problem adds an extra level of complexity , especially when highly excited rotational states are required . the dimension of the ro - vibrational problem @xmath108 is normally increased by the factor of @xmath109 , where @xmath0 is the total angular momentum quantum number . in the variational approach this leads to matrices which can be several order of magnitude larger than the corresponding pure vibrational ( @xmath1 ) hamiltonian matrices . in order to make the vibrational part of the basis set more compact , we construct the ro - vibrational basis set as a direct product of the rigid - rotor winston s @xmath110 and a set of selected vibrational eigenfunctions of the @xmath1 problem obtained as described above . this approach is sometimes referenced in the literature as the @xmath1 contraction @xcite . let us use @xmath111 as the number of vibrational eigenstates for this purpose . @xmath112 can be significantly reduced compared to the total number of vibrational states , @xmath67 , from the previous step as defined by the size of block 1 . this is because not all of the @xmath67 vibrational states are equally important for the target vibrational bands @xmath113 . indeed , because our expansion of the kinetic energy operator is truncated at the second order ( see appendix ) , many matrix elements with @xmath114 are either exactly zero ( for the harmonic oscillators ) or very small . as an example , consider the situation where we would like to compute a spectrum involving all ro - vibrational energies of hno@xmath33 up to @xmath115 @xmath34 . this range corresponds to @xmath116 = 4 or @xmath113 = 715 . from our experience of the hno@xmath33 molecule , the corresponding ro - vibrational basis set has to include excitations at least up to @xmath117 @xmath34 to reach the convergence of 1 @xmath34 , which means including about @xmath118 = 8000 vibrational levels . then the corresponding rotational contribution may include up to @xmath119 . this situation is common in the exomol project @xcite , which computes spectra of hot molecules . this leads to a set of ro - vibrational hamiltonian matrices with the dimensions @xmath120 ranging from 8000 to @xmath121 = 1 608 000 . in the following we show how to exploit the advantages of the special structure of the ro - vibration hamiltonian matrix in order to simplify this problem . for large polyatomic molecules , the ro - vibrational hamiltonian matrix has a pronounced quasi - block character built around vibrational states . this is due to the fact , see appendix , that the off - diagonal elements @xmath122 between two different vibrational states @xmath123 and @xmath124 always contribute substantially less to the calculated energy levels than the off - diagonal elements within a vibrational state , @xmath125 , where @xmath126 is the quantum number giving the projection of @xmath0 on the body - fixed z - axis ( @xmath127 ) and @xmath123 runs over the vibrational basis set ( @xmath128 ) . physically , this corresponds to small coriolis interactions between different vibrational states , which is achieved by the use of the eckart embedding @xcite . figure [ fig:3 ] illustrates the structure of the ro - vibration matrix for the first four vibrational states of molecules hno@xmath33 ( @xmath129 ) . we can see that most of the off - diagonal @xmath130 elements are zero when @xmath131 reflecting the quadratic differential form of the kinetic energy operator . it can also be seen that the non - zero off - diagonal elements gather near the main diagonal and the matrix has an almost tridiagonal form . the near - diagonal elements @xmath132 are mostly associated with changes in the effective geometry of the rotating molecule through interactions with the corresponding vibrational state @xmath123 . the off - diagonal elements @xmath130 are associated with coriolis coupling between vibrational states @xmath123 and @xmath124 . for the first four vibrational states , @xmath133 ( density of red is proportional to the magnitude of the individual matrix element ) ; b : the contributions from the off - diagonal elements to the final energy levels ( the density of red is proportional to the magnitude of the corresponding contribution ) . the squares along the diagonal depict the pure rotational sub - matrices for vibrational states @xmath123 = 1 , 2 , 3 , 4 respectively.,title="fig:",scaledwidth=80.0% ] + a for the first four vibrational states , @xmath133 ( density of red is proportional to the magnitude of the individual matrix element ) ; b : the contributions from the off - diagonal elements to the final energy levels ( the density of red is proportional to the magnitude of the corresponding contribution ) . the squares along the diagonal depict the pure rotational sub - matrices for vibrational states @xmath123 = 1 , 2 , 3 , 4 respectively.,title="fig:",scaledwidth=80.0% ] + b importantly , the contribution of the off - diagonal elements @xmath130 corresponding to the coupling between different vibrational states is much smaller than that of the off - diagonal elements @xmath134 belonging to the same vibrational state , because of the large energy separation between vibrational levels . this is illustrated in fig . [ fig:3 ] , where the actual contributions from the off - diagonal matrix elements to the final ro - vibrational energy levels for molecules hno@xmath33 are shown . typically contribution from off - diagonal elements corresponding to different vibrational states are very small , less than 0.001 @xmath34 . this feature of the vibration - rotation hamiltonian matrix is common for large molecules and underpins the ro - vibrational version of our hybrid approach , which is schematically represented in fig . [ fig:4 ] . again we use the second - order perturbation theory ( as defined by the jacobi rotation ) to transform the @xmath130 matrix to a block diagonal form @xmath135 built from rotational sub - matrices corresponding to different vibrational states @xmath123 . thus the dimension of each degenerate rotational sub - block is @xmath109 only and we only consider @xmath113 sub - matrices that correspond to @xmath113 vibrational states . in the case of strong resonances between different vibrational states , these states can be combined into one , enlarged sub - matrix and treated together , i.e. for an @xmath136-fold symmetry vibrational degeneracy , the dimension of the sub - block is @xmath137 . for details see the appendix . equation ( [ e : h - nondiag ] ) is a symmetrised version of the standard formula for the single jacobi rotation with respect to the indices @xmath144 and @xmath145 . it should be noted that in the limit @xmath146 , eq . ( [ e : h - nondiag ] ) is identical with the corresponding expression for second - order perturbation theory : @xmath147 the resulting block - diagonal form is then diagonalized for each @xmath123 sub - matrix separately , where @xmath123 indicates either a single vibrational state or a set of strongly interacting vibrational states . thus our algorithm replaces the diagonalization of a huge [ @xmath148 ro - vibrational matrix with a number of diagonalizations of much smaller [ @xmath109 or @xmath149 matrices , where @xmath136 is the number of resonance sub - states . it should be noted that in our algorithm , the ro - vibration wave functions , in contrast to those from an exact diagonalization , do not contain any contribution from other vibrational states . this leads to some errors in the subsequent calculation of the transition dipole moments and intensities of the ro - vibrational transitions @xcite . however , the relative error is rather small for the stronger and most important transitions because of the relatively small perturbation contribution from the off - diagonal ( @xmath150 ) matrix elements @xmath130 . for other states , where intensity stealing is important @xcite , the resonant coupled vibrational states procedure can be used @xcite . finally , we note that our procedure is both very quick and easily parallelized . ; the off - diagonal matrix elements from region 4 are generally smaller and treated using the perturbation theorynote indices @xmath151 and @xmath152 are denoted @xmath153 and @xmath154 , respectively , in the text which distinguished between vibrational and rotational basis functions.,scaledwidth=40.0% ] | the first , smaller sub - block includes matrix elements with the largest contribution to the energy levels targeted in the calculations . numerical perturbation theory , implemented as a jacobi rotation , is used to compute the contributions from the matrix elements of the second sub - block . calculations of the vibrational - rotational energy levels also employ a partitioning of the hamiltonian matrix into sub - blocks , each of which corresponds either to a single vibrational state or a set of resonating vibrational states , with all associated rotational levels . numerical perturbation theory is used to include the cross - contributions from different vibrational states . separate individual sub - blocks are then diagonalized , replacing the diagonalization of a large hamiltonian matrix with a number of small matrix diagonalizations . | a procedure for calculation of rotation - vibration states of medium sized molecules is presented . it combines the advantages of variational calculations and perturbation theory . the vibrational problem is solved by diagonalizing a hamiltonian matrix , which is partitioned into two sub - blocks . the first , smaller sub - block includes matrix elements with the largest contribution to the energy levels targeted in the calculations . the second , larger sub - block comprises those basis states which have little effect on these energy levels . numerical perturbation theory , implemented as a jacobi rotation , is used to compute the contributions from the matrix elements of the second sub - block . only the first sub - block needs to be stored in memory and diagonalized . calculations of the vibrational - rotational energy levels also employ a partitioning of the hamiltonian matrix into sub - blocks , each of which corresponds either to a single vibrational state or a set of resonating vibrational states , with all associated rotational levels . physically , this partitioning is efficient when the coriolis coupling between different vibrational states is small . numerical perturbation theory is used to include the cross - contributions from different vibrational states . separate individual sub - blocks are then diagonalized , replacing the diagonalization of a large hamiltonian matrix with a number of small matrix diagonalizations . numerical examples show that the proposed hybrid variational - perturbation method greatly speeds up the variational procedure without significant loss of precision for both vibrational - rotational energy levels and transition intensities . the hybrid scheme can be used for accurate nuclear motion calculations on molecules with up to 15 atoms on currently available computers . molecular rotation ; vibration ; variational ; perturbation theory ; infra red spectra ; nuclear motion * acknowledgment * this work was supported by the erc under advanced investigator project 267219 . |
1308.2455 | i | the helicity of a vector field is a topological index measuring the _ link _ , _ twist _ and _ writhe _ of the field lines . here we say ` index ' because it is invariant under the action of diffeomorphism groups generated by some ideal dynamics . conventionally , the helicity is defined for three - dimensional vectors ; that the dimension of space is three is , in fact , essential to define a helicity ( because , as to be formulated in a general setting , the helicity is the integral of a 3-form ) . in this work , however , we study four - dimensional vectors in the minkowski space - time . in the relativistic dynamics , the conventional helicity is no longer an invariant . one might connect the non - conservation of the helicity with the topological fact that loops ( 1-cycles ) do not link in four - dimensional space@xcite . however , this is not true ; we will show that the field lines obeying an ideal ( barotropic ) equation of motion is still subject to the topological constraint ; two field lines do link in the spatial subspaces ( temporal cross - sections of the space - time ) , and the linking number conserves . we will formulate a lorentz - covariant helicity , show its invariance , and delineate its relation to the linking number . on the other hand , we will find the reason why the conventional helicity fails to describe the link in the relativistic space - time . we start with a short review of the conventional helicity . let @xmath0 be a three - dimensional vector field defined on a domain @xmath1 . we call @xmath2 the _ vorticity _ of @xmath0 . if @xmath3 has a boundary @xmath4 , we assume @xmath5 where @xmath6 is the unit normal vector onto @xmath4 . we define ( assuming integrability ) @xmath7 and call it the _ helicity _ of @xmath8 ( or , sometimes , of @xmath0 ) on @xmath3 ; the integrand is called the _ helicity density_. the pioneering use of the helicity in the classical field theory was made by woltjer@xcite in order to characterize twisted magnetic field lines ; for the magnetic field @xmath8 and its vector potential @xmath0 , obeying the ideal magnetohydrodynamics ( mhd ) equation ( with the scalar potential @xmath9 ) @xmath10 as well as the boundary condition ( [ bc ] ) , the corresponding _ magnetic helicity _ is a constant of motion . for a fluid , we define the _ fluid helicity _ by putting @xmath11 ( fluid velocity ) and @xmath12 ( vorticity ) . in an ideal fluid , @xmath13 obeys the evolution equation @xmath14 where @xmath15 is the total enthalpy ( we may write @xmath16 with the static enthalpy @xmath17 ; in a homentropic flow , @xmath18 with the density @xmath19 and the pressure @xmath20 ) . if the boundary condition ( [ bc ] ) holds , the fluid helicity is conserved . the helicity is a topological index characterizing the twists of the bundle of field lines ; since it is an integral over a volume , some different geometrical characteristics of field lines are summed up in @xmath21 ; see@xcite . to delineate its topological meaning in the simplest form , let us consider a pair of _ vortex filaments _ ( to be identified as _ pure - state _ differential forms ; see sec.[subsec : pure - state ] ) . suppose that @xmath22 and @xmath23 are a pair of disjoint loops ( closed curves bounding disks ) of class @xmath24 in @xmath25 . the unit tangent vectors on these curves are denoted by @xmath26 and @xmath27 , respectively . the directions of @xmath26 and @xmath27 are arbitrarily chosen , and they are attributed to the loops @xmath22 and @xmath23 as their _ orientations_. we regard both @xmath26 and @xmath27 as @xmath28-measures on the loops @xmath22 and @xmath23 , and consider a pair @xmath29 . for an arbitrary smooth loop @xmath30 that singly links with either @xmath22 or @xmath23 , we find , using stokes s formula ( in the generalized sense for the singular @xmath8 ; see remark[remark : generalize_stokes_formula ] ) and the definition @xmath31 , @xmath32 where @xmath33 is the unit tangential vector on @xmath30 , @xmath34 is the length element on @xmath30 , @xmath35 is a surface bounded by @xmath30 , @xmath6 is the unit normal vector on @xmath35 , and @xmath36 is the surface element on @xmath35 . if @xmath30 and @xmath37 do not link , @xmath38 . we may consider a more complex link of @xmath30 with @xmath37 ; by a homotopical deformation of the path @xmath30 of the integral ( [ stokes ] ) , we find that @xmath39 evaluates the _ linking number _ of the pair of loops @xmath30 and @xmath37 ( the sign of link is determined by the orientations of the loops ) . by the definition , we may write @xmath40 . using ( [ stokes ] ) , we obtain @xmath41 the linking number in ( [ stokes2 ] ) can be evaluated by the _ biot - savart integral_. the relation @xmath42 is inverted as @xmath43 inserting @xmath44 into ( [ biot - savart ] ) , and using the resultant @xmath0 in ( [ helicity_conventional ] ) , we obtain @xmath45 the right - hand - side integral ( @xmath46 ) is called the _ gauss linking number_. the relation ( [ stokes2 ] ) derived by stokes formula gives the proof that @xmath47 evaluates an integer number counting the link of two loops @xmath22 and @xmath23 ( see @xcite for generalization to a higher dimension ; see for the gauss integral for three - component links ) . we will build a topological theory of relativistic field lines around two new constructions ; the first is an appropriate _ relativistic helicity _ in the four - dimensional minkowski space - time , and the second is the notion of _ pure - state vorticities _ by which the helicity reads as the linking number of vortex filaments . in sec.[sec : preliminaries ] , we will start by reviewing the basic equations that describe the relativistic ideal ( barotropic ) dynamics of a plasma ( charged fluid ) . the helicity will be defined for the vorticity of the canonical momentum that combines the mechanical momentum and the electromagnetic field . in sec.[sec : r - helicity ] , we will formulate the relativistic helicity , and show its conservation . section[sec : topology ] is devoted for the delineation of the topological implication of the helicity conservation . to this end , we will consider the link of vortex filaments which are formally the aforementioned @xmath28-measures on co - moving loops . we will justify them as the pure - states of banach algebra , and show that they are the generalized ( weak ) solutions of the equation of motion . for a pair of pure - state vortex filaments , the relativistic helicity evaluates their linking number in the spatial subspace , and its conservation parallels the relativistically corrected kelvin s circulation law . [ remark : generalize_stokes_formula ] to derive ( [ stokes2 ] ) , we evaluated the integral @xmath48 ( so called _ circulation _ ; see lemma[lemma : circulation_law ] ) along the loop @xmath37 on which the @xmath28-measure vorticity @xmath49 is supported , and picked up the contribution from the other @xmath28-measure vorticity on the loop @xmath50 ( @xmath51 ) . however , the integrand @xmath0 is not a continuous function , because it is generated by the @xmath28-measure @xmath8 . here are two mathematical issues pertinent to the use of stokes formula : let us decompose @xmath52 with @xmath53 ( @xmath54 ) . 1 . on the loop @xmath22 , @xmath55 is a smooth function , thus @xmath56 can be evaluated in the classical sense . to relate this integral with the `` source '' @xmath27 of @xmath55 , however , we invoke stokes s formula ( [ stokes ] ) in the generalized sense . to justify ( [ stokes ] ) for a @xmath28-measure vorticity @xmath57 , we first approximate @xmath27 by a smooth vector field @xmath58 that gives the same surface integral @xmath59 independent of @xmath60 , and path the limit of @xmath61 ( @xmath62 ) to define @xmath63 2 . to obtain ( [ stokes2 ] ) , we estimated @xmath64 ( @xmath54 ) , which means that @xmath65 . in the neighborhood of each loop @xmath37 , however , @xmath66 is not continuous . to justify these integrals , we consider a homotopy sequence of loops @xmath67 , and define @xmath68 | a relativistic helicity has been formulated in the four - dimensional minkowski space - time . whereas the relativistic distortion of space - time violates the conservation of the conventional helicity , the newly defined relativistic helicity conserves in a barotropic fluid or plasma , dictating a fundamental topological constraint . the relation between the helicity and the vortex - line topology has been delineated by analyzing the linking number of vortex filaments which are singular differential forms representing the pure states of banach algebra . while the dimension of space - time is four , vortex filaments link , because vorticities are primarily 2-forms and the corresponding 2-chains link in four dimension ; the relativistic helicity measures the linking number of vortex filaments that are proper - time cross - sections of the vorticity 2-chains . a thermodynamic force yields an additional term in the vorticity , by which the vortex filaments on a reference - time plane are no longer pure states . however , the vortex filaments on a proper - time plane remain to be pure states , if the thermodynamic force is exact ( barotropic ) , thus , the linking number of vortex filaments conserves . | a relativistic helicity has been formulated in the four - dimensional minkowski space - time . whereas the relativistic distortion of space - time violates the conservation of the conventional helicity , the newly defined relativistic helicity conserves in a barotropic fluid or plasma , dictating a fundamental topological constraint . the relation between the helicity and the vortex - line topology has been delineated by analyzing the linking number of vortex filaments which are singular differential forms representing the pure states of banach algebra . while the dimension of space - time is four , vortex filaments link , because vorticities are primarily 2-forms and the corresponding 2-chains link in four dimension ; the relativistic helicity measures the linking number of vortex filaments that are proper - time cross - sections of the vorticity 2-chains . a thermodynamic force yields an additional term in the vorticity , by which the vortex filaments on a reference - time plane are no longer pure states . however , the vortex filaments on a proper - time plane remain to be pure states , if the thermodynamic force is exact ( barotropic ) , thus , the linking number of vortex filaments conserves . |
astro-ph0402117 | i | we have created a sample of 123 variable radio sources using two epoch observations of a zero - dec strip toward the south galactic cap . this sample spans the range of radio flux densities from @xmath1 to 1000 mjy . it presents both in size and radio flux density coverage a unique starting point for variability studies of more normal , less agn - dominated galaxies , especially toward the lower flux density limits . we compared both our variable and non - variable samples to the sloan digital sky survey optical data . we found that the quasar fraction of the sample sharply declines as a function of declining radio flux density levels . this is consistent with earlier findings that the radio source population demographics change as one samples at progressively lower flux density levels : agn - dominated systems tend to be found at the brighter radio flux density levels ( @xmath62 mjy ) , whereas star - forming and normal galaxies dominate the counts at sub - mjy flux density levels ( at 1.4 ghz , e.g. , windhorst 2003 ) . our variable sample contains a consistently higher fraction of quasars than the non - variable control sample , independent of radio flux density . while this explains part of our almost @xmath63 higher optical matching rate of the variable sample compared to the non - variable one ( quasars are easier to detect at a given brightness limit than galaxies ) , it does imply that our variable sample contains on average slightly brighter sources ( though not significantly so , see sect . [ apmid ] ) . based on relative number statistics , we estimate that quasars are about 5 times more likely to harbor a variable radio source than galaxies . however , at flux density levels @xmath64 mjy , the majority of ( radio ) variable sources are identified as galaxies . and finally , galaxies and quasars that harbor a variable radio source exhibit , on average , bluer optical colors than hosts of non - variable sources . all of this underlines the fact that both galaxies and quasars can harbor variable radio sources , albeit at different occurrence rates . some of this is obviously due to the beamed nature of the ( variable ) quasars , enhancing the variability both by boosting their brightnesses and shortening the variability timescales . nonetheless , especially toward lower radio flux density limits , a statistically significant study of variability is possible provided one starts with large enough samples . large scale optical surveys like the sdss provide the crucial radio source optical `` environmental '' information . as more sdss data become available on the zero - dec strip , we will investigate this further . the authors like to thank steve croft for useful discussions and careful reading of this paper . wdvs work was performed under the auspices of the u.s . department of energy , national nuclear security administration by the university of california , lawrence livermore national laboratory under contract no . w-7405-eng-48 . the authors also acknowledge support from the national radio astronomy observatory , the national science foundations ( grants ast 00 - 98259 and ast 00 - 98355 ) , the space telescope science institute , and microsoft . this research has made use of the nasa / ipac extragalactic database ( ned ) which is operated by the jet propulsion laboratory , caltech , under contract with the national aeronautics and space administration . lrrrrcrrrrr j000111@xmath46002011 & 00 01 11.19 & @xmath4600 20 11.6 & 33.57 & 23.28 & 1.46 & 15.6 & 0.518 & 18.56 & q & cl + j000257@xmath46002447 & 00 02 57.19 & @xmath4600 24 47.4 & 112.93 & 124.90 & 0.92 & 4.2 & & 19.52 & g & ch + j000549@xmath44005048 & 00 05 49.92 & 00 50 48.1 & 8.87 & 5.13 & 1.77 & 10.1 & & 21.59 & g & ps + j001158@xmath46000208 & 00 11 58.81 & @xmath4600 02 08.2 & 5.95 & 27.34 & 0.22 & 43.2 & & 20.98 & q & ps + j001507@xmath46000801 & 00 15 07.01 & @xmath4600 08 01.8 & 10.54 & 12.67 & 0.85 & 4.3 & 1.704 & 18.23 & q & ps + j001611@xmath46001512 & 00 16 11.07 & @xmath4600 15 12.3 & 827.61 & 1050.26 & 0.80 & 11.2 & 1.575 & 19.70 & q & ps + j002354@xmath44001919 & 00 23 54.58 & 00 19 19.5 & 4.75 & 7.46 & 0.66 & 5.2 & & 22.69 & g & cj + j002738@xmath44000627 & 00 27 38.25 & 00 06 27.6 & 4.56 & 6.91 & 0.68 & 5.4 & & 22.02 & q & ps + j003007@xmath46000007 & 00 30 07.90 & @xmath4600 00 07.4 & 50.52 & 39.32 & 1.30 & 12.1 & & 19.55 & q & ps + j003127@xmath44003959 & 00 31 27.92 & 00 39 59.4 & 9.58 & 12.31 & 0.79 & 6.3 & & 21.80 & q & ps + j003536@xmath46000627 & 00 35 36.30 & @xmath4600 06 27.2 & 17.28 & 14.52 & 1.21 & 6.1 & & 18.94 & g & ps + j003540@xmath46002529 & 00 35 40.22 & @xmath4600 25 29.2 & 4.01 & 6.57 & 0.63 & 6.4 & & 21.82 & g & ps + j003703@xmath44003537 & 00 37 03.36 & 00 35 37.4 & 12.84 & 15.28 & 0.86 & 4.5 & & 21.51 & q & ps + j004332@xmath44002459 & 00 43 32.71 & 00 24 59.8 & 120.94 & 108.53 & 1.13 & 6.1 & 1.127 & 19.20 & q & ps + j004819@xmath44001457 & 00 48 19.11 & 00 14 57.1 & 98.76 & 89.40 & 1.12 & 5.6 & 1.536 & 19.92 & q & ps + j005205@xmath44003538 & 00 52 05.58 & 00 35 38.2 & 34.58 & 81.46 & 0.43 & 38.1 & 0.399 & 16.09 & q & cj + j005212@xmath44000945 & 00 52 12.47 & 00 09 45.2 & 11.92 & 9.77 & 1.24 & 5.5 & & & & ps + j005225@xmath44002627 & 00 52 25.67 & 00 26 28.0 & 12.13 & 9.98 & 1.24 & 5.2 & & 20.75 & g & ps + j005717@xmath46002433 & 00 57 17.01 & @xmath4600 24 33.2 & 89.47 & 114.74 & 0.79 & 11.6 & 2.790 & 19.19 & q & cj + j010525@xmath44001121 & 01 05 25.52 & 00 11 21.7 & 33.97 & 41.73 & 0.83 & 8.3 & & 21.45 & g & ps + j010745@xmath44003952 & 01 07 45.21 & 00 39 52.8 & 10.50 & 8.88 & 1.21 & 4.5 & & & & ps + j010838@xmath44002814 & 01 08 38.56 & 00 28 14.6 & 5.33 & 3.91 & 1.40 & 4.2 & & & & ps + j011106@xmath44000846 & 01 11 06.79 & 00 08 46.1 & 5.25 & 3.43 & 1.57 & 4.9 & & 22.70 & q & ps + j011515@xmath44001248 & 01 15 15.78 & 00 12 48.5 & 46.84 & 43.13 & 1.10 & 4.4 & 0.045 & 14.39 & g & ps + j012213@xmath46001801 & 01 22 13.92 & @xmath4600 18 01.0 & 386.60 & 331.60 & 1.18 & 8.3 & & 20.23 & q & cl + j012528@xmath46000555 & 01 25 28.85 & @xmath4600 05 55.8 & 1333.76 & 1481.35 & 0.91 & 4.8 & 1.076 & 16.47 & q & ps + j012753@xmath44002516 & 01 27 53.70 & 00 25 16.5 & 131.32 & 90.08 & 1.48 & 19.1 & & 20.76 & q & ps + j013457@xmath44003942 & 01 34 57.42 & 00 39 43.0 & 6.09 & 2.87 & 2.18 & 8.7 & & 22.04 & q & ps + j013815@xmath44001446 & 01 38 15.02 & 00 14 46.5 & 42.98 & 51.76 & 0.84 & 8.1 & & 21.64 & g & cj + j015329@xmath46002214 & 01 53 29.75 & @xmath4600 22 14.3 & 17.46 & 14.98 & 1.19 & 5.8 & & 19.07 & q & ps + j015528@xmath44001204 & 01 55 28.47 & 00 12 04.6 & 17.23 & 19.82 & 0.88 & 4.8 & & 22.08 & g & ps + j015950@xmath46002407 & 01 59 50.09 & @xmath4600 24 07.2 & 10.68 & 12.63 & 0.86 & 4.3 & & 21.50 & g & ps + j020141@xmath44003825 & 02 01 41.04 & 00 38 25.5 & 4.50 & 6.35 & 0.73 & 4.7 & & & & ps + j020214@xmath46001748 & 02 02 14.30 & @xmath4600 17 48.3 & 75.12 & 60.00 & 1.27 & 11.5 & & 21.36 & g & ps + j020234@xmath44000301 & 02 02 34.32 & 00 03 01.7 & 30.54 & 39.41 & 0.79 & 10.5 & & 18.42 & g & cl + j020928@xmath46001224 & 02 09 28.85 & @xmath4600 12 24.9 & 4.10 & 2.51 & 1.69 & 4.2 & 0.152 & 16.02 & g & ps + j021202@xmath46002750 & 02 12 02.13 & @xmath4600 27 50.1 & 45.14 & 52.84 & 0.87 & 6.6 & & & & cl + j021301@xmath46001815 & 02 13 01.13 & @xmath4600 18 15.0 & 41.55 & 48.44 & 0.87 & 6.5 & & 21.98 & g & ps + j021553@xmath44001826 & 02 15 53.65 & 00 18 26.9 & 30.90 & 35.82 & 0.88 & 5.6 & & 19.51 & g & ps + j021755@xmath46000935 & 02 17 55.99 & @xmath4600 09 35.9 & 3.96 & 5.85 & 0.70 & 4.4 & & 19.86 & g & cj + j021840@xmath46001515 & 02 18 40.55 & @xmath4600 15 15.9 & 10.06 & 13.17 & 0.78 & 6.4 & 1.171 & 18.81 & q & cx + j022624@xmath44000746 & 02 26 24.61 & 00 07 46.2 & 43.84 & 48.74 & 0.91 & 4.4 & & 22.56 & g & ps + j023105@xmath44000843 & 02 31 05.59 & 00 08 43.5 & 47.60 & 53.86 & 0.90 & 5.0 & 1.338 & 20.00 & q & ps + j025321@xmath44000559 & 02 53 21.04 & 00 05 59.9 & 151.89 & 97.64 & 1.57 & 22.0 & & 22.26 & g & ps + j025333@xmath44002431 & 02 53 33.66 & 00 24 31.7 & 16.86 & 14.86 & 1.15 & 4.6 & & & & ps + j025404@xmath46002628 & 02 54 04.60 & @xmath4600 26 28.7 & 4.39 & 2.71 & 1.67 & 5.2 & & 21.97 & g & ps + j025859@xmath44003618 & 02 58 59.65 & 00 36 18.3 & 29.45 & 23.75 & 1.26 & 9.4 & & 22.28 & q & ps + j025928@xmath46002000 & 02 59 28.52 & @xmath4600 20 00.1 & 243.47 & 225.85 & 1.09 & 4.3 & 2.001 & 17.34 & q & ps + j030137@xmath44001746 & 03 01 37.58 & 00 17 46.1 & 6.86 & 4.45 & 1.58 & 7.0 & & 21.47 & g & cj + j030317@xmath46002644 & 03 03 17.01 & @xmath4600 26 45.0 & 7.41 & 3.68 & 2.06 & 4.9 & & & & cj + j030702@xmath44000651 & 03 07 02.04 & 00 06 51.9 & 5.84 & 8.08 & 0.74 & 5.4 & & & & ps + j030834@xmath44003303 & 03 08 34.31 & 00 33 03.7 & 4.16 & 1.83 & 2.35 & 6.4 & 0.031 & 14.91 & g & ps + j030933@xmath46001901 & 03 09 33.27 & @xmath4600 19 01.2 & 6.31 & 4.58 & 1.41 & 4.6 & & & & ps + j031006@xmath44001549 & 03 10 06.69 & 00 15 49.9 & 1.36 & 5.55 & 0.26 & 11.1 & 0.109 & 18.54 & g & ps + j031118@xmath44000816 & 03 11 18.51 & 00 08 16.4 & 29.30 & 36.54 & 0.81 & 9.1 & & & & ps + j031202@xmath46000442 & 03 12 02.50 & @xmath4600 04 42.5 & 6.42 & 8.59 & 0.77 & 5.2 & 0.038 & 13.50 & g & ps + j031345@xmath46000720 & 03 13 45.04 & @xmath4600 07 20.3 & 3.63 & 6.06 & 0.62 & 6.1 & 2.519 & 20.04 & q & ps + j031353@xmath46000004 & 03 13 53.47 & @xmath4600 00 04.1 & 10.13 & 7.03 & 1.47 & 7.7 & & 21.16 & g & ps + j031357@xmath44003506 & 03 13 57.10 & 00 35 06.9 & 16.54 & 20.23 & 0.83 & 6.7 & & & & ps + j031452@xmath44001346 & 03 14 52.08 & 00 13 46.5 & 6.77 & 10.05 & 0.69 & 7.3 & 3.202 & 19.54 & q & ps + j031609@xmath44000107 & 03 16 09.55 & 00 01 07.9 & 4.02 & 6.03 & 0.69 & 4.6 & & & & ps + j031634@xmath46002039 & 03 16 34.96 & @xmath4600 20 39.6 & 18.17 & 15.63 & 1.18 & 5.5 & & & & ps + j031814@xmath46002948 & 03 18 14.43 & @xmath4600 29 48.9 & 113.20 & 93.91 & 1.22 & 9.7 & & 21.77 & g & ps + j032007@xmath44000354 & 03 20 07.01 & 00 03 54.0 & 12.17 & 10.20 & 1.22 & 5.0 & & 22.10 & g & ps + j212000@xmath46001159 & 21 20 00.72 & @xmath4600 11 59.5 & 42.94 & 48.33 & 0.90 & 4.9 & & & & ps + j212447@xmath44000747 & 21 24 47.34 & 00 07 47.6 & 5.79 & 3.82 & 1.56 & 5.3 & & & & ps + j213206@xmath44003520 & 21 32 06.15 & 00 35 20.1 & 13.65 & 11.95 & 1.16 & 4.1 & & & & ps + j213638@xmath44004154 & 21 36 38.57 & 00 41 54.3 & 4136.68 & 3546.71 & 1.18 & 8.3 & 1.932 & 16.79 & q & ps + j213748@xmath44001219 & 21 37 48.43 & 00 12 19.9 & 41.69 & 36.02 & 1.17 & 7.0 & 1.666 & 17.92 & q & ps + j213804@xmath44000714 & 21 38 04.06 & 00 07 14.7 & 10.31 & 16.33 & 0.64 & 11.6 & & & & cj + j214138@xmath44000319 & 21 41 38.55 & 00 03 19.8 & 11.08 & 14.37 & 0.79 & 6.8 & & & & ps + j214324@xmath44003502 & 21 43 24.37 & 00 35 02.7 & 36.03 & 45.22 & 0.81 & 9.7 & 2.030 & 19.37 & q & ps + j214419@xmath44002055 & 21 44 19.88 & 00 20 55.8 & 10.34 & 7.31 & 1.44 & 7.5 & & & & ps + j214613@xmath44000930 & 21 46 13.31 & 00 09 30.8 & 9.94 & 7.88 & 1.29 & 5.6 & & & & ps + j214807@xmath46000630 & 21 48 07.73 & @xmath4600 06 30.6 & 68.57 & 90.18 & 0.77 & 12.7 & & & & cj + j214811@xmath46001230 & 21 48 11.48 & @xmath4600 12 30.6 & 7.76 & 9.95 & 0.80 & 4.7 & & & & cj + j215349@xmath44003119 & 21 53 49.75 & 00 31 19.6 & 82.50 & 103.65 & 0.81 & 10.4 & & & & ps + j215353@xmath46001339 & 21 53 53.89 & @xmath4600 13 39.5 & 13.48 & 17.27 & 0.79 & 7.3 & & & & ps + j215359@xmath44004412 & 21 53 59.83 & 00 44 12.6 & 9.76 & 6.73 & 1.48 & 7.8 & 1.030 & 19.07 & q & ps + j215733@xmath46000340 & 21 57 33.66 & @xmath4600 03 40.5 & 5.36 & 3.60 & 1.53 & 4.9 & & & & ps + j215949@xmath44005146 & 21 59 49.91 & 00 51 46.7 & 12.29 & 9.21 & 1.36 & 7.8 & & & & ps + j215954@xmath46002150 & 21 59 54.46 & @xmath4600 21 50.1 & 4.07 & 2.46 & 1.71 & 4.7 & 1.960 & 16.98 & q & ps + j220017@xmath46000133 & 22 00 17.37 & @xmath4600 01 33.6 & 7.84 & 6.15 & 1.30 & 4.9 & & & & cl + j220755@xmath46000215 & 22 07 55.25 & @xmath4600 02 15.0 & 78.40 & 61.93 & 1.28 & 11.9 & & & & cj + j220822@xmath44002352 & 22 08 22.88 & 00 23 52.7 & 4.62 & 2.42 & 1.97 & 6.1 & & & & ps + j221001@xmath46001309 & 22 10 01.82 & @xmath4600 13 09.9 & 125.83 & 115.31 & 1.10 & 4.8 & & & & ps + j221031@xmath46001356 & 22 10 31.46 & @xmath4600 13 56.1 & 14.06 & 12.00 & 1.19 & 5.2 & & & & ps + j221909@xmath44003113 & 22 19 09.40 & 00 31 13.4 & 9.13 & 11.70 & 0.80 & 5.3 & & & & cj + j222036@xmath44003334 & 22 20 36.33 & 00 33 34.2 & 15.27 & 12.76 & 1.22 & 5.8 & & & & ps + j222135@xmath46001100 & 22 21 35.00 & @xmath4600 11 00.1 & 14.11 & 8.64 & 1.66 & 12.6 & & & & ps + j222235@xmath44001536 & 22 22 35.87 & 00 15 36.7 & 46.04 & 51.13 & 0.91 & 4.5 & & & & cj + j222704@xmath44004517 & 22 27 04.24 & 00 45 17.5 & 8.13 & 5.57 & 1.49 & 7.0 & & & & ps + j222726@xmath44001059 & 22 27 26.53 & 00 10 59.3 & 6.05 & 4.52 & 1.37 & 4.2 & & & & ps + j222729@xmath44000522 & 22 27 29.08 & 00 05 22.2 & 77.84 & 91.64 & 0.86 & 7.4 & 1.510 & 18.86 & q & cl + j222744@xmath44003450 & 22 27 44.58 & 00 34 50.6 & 22.13 & 30.53 & 0.74 & 12.1 & 1.540 & 19.14 & q & ps + j222758@xmath44003705 & 22 27 58.13 & 00 37 05.2 & 68.69 & 99.12 & 0.70 & 17.3 & & & & ps + j223047@xmath44002756 & 22 30 47.46 & 00 27 56.3 & 8.54 & 13.82 & 0.63 & 11.4 & & & & ps + j224224@xmath44005512 & 22 42 24.14 & 00 55 13.0 & 17.71 & 13.44 & 1.34 & 9.5 & & & & cj + j224331@xmath46001233 & 22 43 31.94 & @xmath4600 12 33.1 & 13.85 & 21.40 & 0.66 & 14.8 & 2.040 & 18.45 & q & ps + j224448@xmath46000619 & 22 44 48.11 & @xmath4600 06 19.8 & 8.24 & 5.58 & 1.51 & 7.3 & & & & ps + j224627@xmath46001214 & 22 46 27.68 & @xmath4600 12 14.2 & 85.55 & 56.00 & 1.55 & 20.9 & & & & ps + j224730@xmath44000006 & 22 47 30.19 & 00 00 06.1 & 464.52 & 183.71 & 2.56 & 43.8 & 0.094 & 18.50 & q & ps + j224922@xmath44001804 & 22 49 22.28 & 00 18 04.6 & 10.41 & 8.51 & 1.25 & 5.3 & & & & ps + j225852@xmath46001857 & 22 58 52.94 & @xmath4600 18 57.3 & 3.45 & 1.28 & 2.80 & 6.0 & & & & ps + j230157@xmath44000351 & 23 01 57.85 & 00 03 52.0 & 3.75 & 7.04 & 0.55 & 7.8 & & 23.23 & q & ps + j230314@xmath44000052 & 23 03 14.85 & 00 00 52.2 & 3.72 & 1.72 & 2.24 & 5.8 & & & & ps + j230655@xmath44003638 & 23 06 55.16 & 00 36 38.1 & 12.99 & 15.19 & 0.87 & 4.3 & & & & ps + j231541@xmath44002936 & 23 15 41.67 & 00 29 36.6 & 21.93 & 13.94 & 1.60 & 15.7 & & 20.81 & q & ps + j231558@xmath46001205 & 23 15 58.64 & @xmath4600 12 05.5 & 3.52 & 6.27 & 0.58 & 7.0 & & 22.85 & g & ps + j231845@xmath46000754 & 23 18 45.81 & @xmath4600 07 54.7 & 3.03 & 5.34 & 0.59 & 5.5 & 0.867 & 19.44 & q & ps + j231856@xmath44001437 & 23 18 56.66 & 00 14 37.7 & 20.16 & 18.34 & 1.12 & 4.2 & 0.030 & 12.76 & g & ps + j231910@xmath44001859 & 23 19 10.33 & 00 18 59.0 & 28.90 & 33.36 & 0.88 & 5.6 & & 22.60 & q & ps + j232038@xmath44003139 & 23 20 38.01 & 00 31 39.8 & 72.69 & 82.95 & 0.89 & 5.7 & 1.911 & 19.03 & q & ps + j232323@xmath44003327 & 23 23 23.95 & 00 33 27.5 & 16.73 & 12.52 & 1.36 & 9.2 & & & & ps + j233448@xmath46001400 & 23 34 48.06 & @xmath4600 14 01.0 & 24.85 & 29.64 & 0.85 & 6.8 & & 21.56 & q & ps + j233822@xmath44001146 & 23 38 22.35 & 00 11 46.6 & 18.19 & 15.44 & 1.20 & 6.0 & & 22.24 & q & ps + j233852@xmath44004843 & 23 38 52.46 & 00 48 43.5 & 11.32 & 9.18 & 1.26 & 5.9 & & & & ps + j234623@xmath44004301 & 23 46 23.73 & 00 43 01.1 & 13.49 & 11.79 & 1.17 & 4.6 & 2.861 & 19.09 & q & ps + j234939@xmath46001315 & 23 49 39.90 & @xmath4600 13 15.2 & 8.46 & 6.97 & 1.24 & 4.2 & 1.267 & 20.22 & q & ps + j235022@xmath44001232 & 23 50 22.41 & 00 12 32.4 & 3.92 & 2.03 & 2.00 & 5.1 & & & & ps + j235050@xmath46002848 & 23 50 50.72 & @xmath4600 28 48.7 & 13.12 & 11.57 & 1.15 & 4.1 & & 19.62 & g & cl + j235409@xmath46001948 & 23 54 09.18 & @xmath4600 19 48.1 & 384.58 & 344.79 & 1.13 & 6.1 & 0.462 & 17.93 & q & ps + j235823@xmath44000213 & 23 58 23.91 & 00 02 13.2 & 8.03 & 10.67 & 0.77 & 5.7 & & & & ps + j001800@xmath44000313 & 00 18 00.79 & 00 03 17.9 & 20.76 & 18.82 & 1.12 & 4.2 & & & & hs + j003246@xmath46001917 & 00 32 46.02 & @xmath4600 19 17.8 & 81.29 & 60.80 & 1.35 & 14.5 & & & & hs + j212058@xmath44000612 & 21 20 59.00 & 00 06 12.7 & 31.17 & 28.05 & 1.13 & 5.3 & & & & hs + j212955@xmath44000758 & 21 29 55.68 & 00 07 59.0 & 1.17 & 3.03 & 0.42 & 4.5 & & & & cl + j235828@xmath44003934 & 23 58 28.77 & 00 39 34.1 & 2.48 & 4.16 & 0.62 & 4.3 & & & & cl + lcc variable & 66 & 53.7% + control 1 & 17 & 13.8% + control 2 & 29 & 23.6% + control 3 & 21 & 17.1% + control 4 & 21 & 17.1% + control 5 & 20 & 16.3% + control 6 & 27 & 22.0% + control 7 & 23 & 18.7% + control 8 & 17 & 13.8% + control 9 & 21 & 17.1% + control 10 & 22 & 17.9% + control 11 & 23 & 18.7% + control 12 & 21 & 17.1% + mean control & 21.8@xmath653.5 & 17.7@xmath652.8% + lcccc variable & 64 & 82.0% & 53.1% & 46.9% + control 1 & 28 & 37.3%&17.9%&82.1% + control 2 & 30 & 40.0%&13.3%&86.7% + control 3 & 21 & 28.0%&14.3%&85.7% + control 4 & 36 & 48.0%&13.9%&86.1% + control 5 & 26 & 34.7%&26.9%&73.1% + control 6 & 32 & 42.7%&18.8%&81.2% + control 7 & 29 & 38.7%&24.1%&75.9% + control 8 & 33 & 44.0% & 9.1%&90.9% + control 9 & 23 & 30.7%&21.7%&78.3% + control 10 & 30 & 40.0%&16.7%&83.3% + control 11 & 31 & 41.3%&29.0%&71.0% + control 12 & 22 & 29.3%&18.2%&81.8% + mean control & 28.4@xmath654.6 & 37.9@xmath656.2% & 18.7@xmath655.9% & 81.3@xmath655.9% + rrr - variable & 0.89@xmath650.13 & 0.40@xmath650.07 + non - variable & 1.37@xmath650.17 & 0.60@xmath650.08 + nominal offset & @xmath460.48 & @xmath460.20 + significance & 2.3 & 1.8 + - variable & 1.81@xmath650.21 & 1.09@xmath650.11 + non - variable & 2.28@xmath650.09 & 1.20@xmath650.05 + nominal offset & @xmath460.47 & @xmath460.11 + significance & 2.0 & 0.9 + | a sample of 123 radio sources that exhibit significant variations at 1.4 ghz on a seven year base - line has been created using first vla b - array data from 1995 and 2002 on a strip at near the south galactic cap . this sample spans the range of radio flux densities from to 1000 mjy . it presents both in size and radio flux density range a unique starting point for variability studies of galaxies and quasars harboring lower luminosity active galactic nuclei ( agn ) . we find , by comparing our variable and non - variable control samples to the sloan digital sky survey the following : 1 ) the quasar fraction of both the variable and non - variable samples declines as a function of declining radio flux density levels ; 2 ) our variable sample contains a consistently higher fraction of quasars than the non - variable control sample , irrespective of radio flux ; 3 ) the variable sources are almost twice as likely to be retrieved from the optical sdss data than the non - variable ones ; 4 ) based on relative numbers , we estimate that quasars are about five times more likely to harbor a variable radio source than are galaxies ; and 5 ) there does not appear to be any significant optical color offset between the two samples , even though the suggestive trend for sources to be bluer when variable has been detected before and may be real . | a sample of 123 radio sources that exhibit significant variations at 1.4 ghz on a seven year base - line has been created using first vla b - array data from 1995 and 2002 on a strip at near the south galactic cap . this sample spans the range of radio flux densities from to 1000 mjy . it presents both in size and radio flux density range a unique starting point for variability studies of galaxies and quasars harboring lower luminosity active galactic nuclei ( agn ) . we find , by comparing our variable and non - variable control samples to the sloan digital sky survey the following : 1 ) the quasar fraction of both the variable and non - variable samples declines as a function of declining radio flux density levels ; 2 ) our variable sample contains a consistently higher fraction of quasars than the non - variable control sample , irrespective of radio flux ; 3 ) the variable sources are almost twice as likely to be retrieved from the optical sdss data than the non - variable ones ; 4 ) based on relative numbers , we estimate that quasars are about five times more likely to harbor a variable radio source than are galaxies ; and 5 ) there does not appear to be any significant optical color offset between the two samples , even though the suggestive trend for sources to be bluer when variable has been detected before and may be real . this leads us to conclude that both radio variability and radio flux density levels , in combination with accurate optical information , are important discriminators in the study of ( radio ) variability of galaxies . the latter start to dominate the source counts belowmjy . in any case , variability appears to be an intrinsic property of radio sources , and is not limited to quasars . radio variability at low flux density levels may offer a unique tool in agn unification studies . |
astro-ph0610043 | r | in our grb model , there are seven adjustable parameters : the @xmath56 spectral power - law indices @xmath179 and @xmath180 , the power - law index @xmath120 of the jet opening - angle distribution , the range of the jet opening angles @xmath181 and @xmath182 , the absolute emitted @xmath13-ray energy @xmath21 , and the detector threshold @xmath86 . as already mentioned , we consider , for simplicity , a flat @xmath31 grb sed , so that @xmath65 , leaving only the bolometric correction factor @xmath55 , which is set equal to 5 . the remaining parameters @xmath181 , @xmath182 , @xmath120 , and @xmath21 are constrained by observations , and the detector properties are used to estimate @xmath86 . besides these parameters , we must also assume a grb comoving rate density @xmath183 , which is also constrained by fitting our model to the observed data . adopting the uniform jet model , we assume that the energy per solid angle is roughly constant within the well - defined jet opening angle ( e.g. , * ? ? ? * ; * ? ? ? * ) . with these assumptions , we investigate the behavior of the redshift and opening angle distributions . before we proceed to compare our model of redshift and jet opening angle distributions with the samples obtained by the and pre- instruments , we need to explore some of the above parameter space . -0.46 in per day per @xmath184 per @xmath0 ] for an assumed @xmath139-function distribution of the jet opening angles , so that @xmath186 , with @xmath187 ( solid - curve ) , using sfr3 and @xmath188 . also shown are the directional event rates with @xmath189 radian , and @xmath190 , @xmath191 , and @xmath192 radians , with @xmath193 . , title="fig:",width=364 ] -2.2 in for illustrative purposes , we utilize sfr3 @xcite for the grb comoving rate density . furthermore , we also assume that @xmath194 , @xmath195 radian ( this assumption implies @xmath95 is a @xmath139-function , see eqs . [ [ eq11 ] ] , [ [ eq12 ] ] , and fig . [ fig5 ] ) , @xmath196 , @xmath197 ergs , and @xmath198 and @xmath199 for the flux sensitivity of detectors for and before , respectively . we choose @xmath29 = 10 s , and @xmath200 ergs because these are the mean values of the grb duration measured with batse assuming that batse grbs are typically at @xmath201 , and the mean beaming corrected @xmath13-ray energy release ( see fig . 1 ) , respectively . we shall use @xmath202 throughout this work . the calculation is normalized to a local ( @xmath203 ) event rate of one grb per day per @xmath204 , so that the normalization of the local rate density is @xmath205 . with the jet opening angle @xmath130 and @xmath206 radians , and the detector threshold @xmath207 , the maximum redshifts from which model grbs could be detected are @xmath208 and @xmath209 , respectivley ( see eq . [ [ eq19 ] ] ) . the directional event rate per unit redshift is shown in figure [ fig5 ] . setting @xmath210 , and @xmath211 , @xmath191 , @xmath192 radians ( the three dashed curves in figure [ fig5 ] ) , we plot the directional event rate per unit redshift for different values of @xmath181 ( see eq . [ [ eq18 ] ] ) . as @xmath212 , the directional event rate per unit redshift of the respective @xmath213-curves approaches the directional event rate per redshift of the @xmath139-function @xmath95 with @xmath206 radian . + 0.05 in -0.46 in , normalized to one event per @xmath214 per day , with @xmath215 , @xmath216 , and @xmath217 , using sfr3 . the thin and thick curves represent the instrument s energy flux sensitivity of @xmath218 ( ) and @xmath219 ( pre- ) , respectively.,title="fig:",width=364 ] -2.2 in in figure [ fig6 ] we examine three different values of the jet opening angle , @xmath220 , @xmath216 , and @xmath127 radians , with two values of flux sensitivity , namely @xmath221 and @xmath222 . as can be seen , the smaller the jet opening angle the larger the maximum redshift from which the grb is detectable . the maximum observable redshift @xmath0 associated with a particular opening angle is indicated by the vertical line in each curve . moreover , figure [ fig6 ] also shows that whatever the lower sensitivity telescope detects will also be detected by a telescope with better sensitivity , because a flat @xmath31 spectrum is considered . thus , being more sensitive than earlier telescopes , would in this model detect a sample of grbs encompassing those detected with instruments before . spectral behavior can be considered in more detailed treatments and will change the relations for maximum distances , though generally with better sensitivity the telescope can view to larger distances , especially when the more sensitive telescope can observe lower energy photons . + 0.1 in -0.46 in per day per @xmath184 per @xmath0 ] with @xmath223 and @xmath224 , @xmath225 , @xmath226 , @xmath227 , @xmath228 , @xmath229 , and @xmath230 , with sfr3 and @xmath231.,title="fig:",width=364 ] -2.2 in by varying the opening angle power - law index parameter @xmath120 over values @xmath225 , @xmath226,@xmath227 , @xmath228 , @xmath229 , and @xmath230 , with @xmath211 , @xmath232 and @xmath231 , while keeping all other assumed parameters as before , figure [ fig7 ] shows that the burst rate per unit redshift is progressively distributed to higher @xmath0 , as expected , since the more negative @xmath120 is , the larger is the fractional number of bursts that occurs at small jet - opening angles @xmath15 ( see also figure [ fig3 ] ) . more importantly , as @xmath233 , the directional event rate per unit redshift reaches an asymptotic limit ( see plots for @xmath234 and @xmath230 in figure [ fig7 ] ) . + .10 in -0.46 in per day per @xmath184 per @xmath0 ] with @xmath223 and @xmath224 for @xmath235 , @xmath236 , and @xmath237 , using sfr3.,title="fig:",width=364 ] -2.2 in the final parameter that we need to explore is the corrected @xmath13-ray energy @xmath21 . by setting @xmath238 , @xmath239 , @xmath240 , and @xmath241 with @xmath211 and @xmath242 , @xmath243 , and @xmath244 , figure [ fig8 ] shows that as the emitted @xmath13-ray energy increases , the bursting rate is shifted to higher @xmath0 , as expected . from @xcite , the pre- jet opening angle distribution extends from @xmath130 to @xmath245 radians . however , other workers have reported that the observed jet - opening angles are as large as @xmath246 ( e.g. , * ? ? ? * ) . hence , in this work we explore different ranges of @xmath15 , ranging from @xmath211 radians to @xmath247 , @xmath131 , and @xmath248 . we also use @xmath202 as the average grb duration in the stationary frame . we first consider sfr3 @xcite to fit the redshift distribution from the and pre- samples , and the opening angle distribution from the pre- sample . a large number of jet opening angles associated with the redshift sample is lacking , so our calculated jet opening angle distribution will be used as a prediction for . as indicated in figure [ fig1 ] , the corrected @xmath13-ray energy release is broadly distributed around the mean value of @xmath249 ; hence we can explore different values of @xmath13-ray energy that best fit the observed redshift and opening angle samples . it is important , however , that the best - fit value is not too far off from the mean value in order for our model to be in accord with observations . later in the analysis we will show that our best fit corrected @xmath13-ray energy @xmath21 is indeed only a factor of 2 larger than the mean . in figures [ fig9 ] , [ fig10 ] and [ fig11 ] , we plot the calculated redshift distribution at three different corrected total @xmath13-ray energies , @xmath197 , @xmath250 , and @xmath251 , respectively . in each panel , each curve represents the calculated cumulative redshift distribution for the power - law index @xmath120 of the jet opening angle distribution . the values @xmath252 , @xmath253 , @xmath254 , and @xmath255 are plotted , from far left to far right , respectively . in each figure , from top to bottom , we have @xmath256 and @xmath257 radians , @xmath256 and @xmath258 radians , and @xmath256 and @xmath259 radians , respectively , as the jet opening angles . the cumulative redshift distribution ( bold solid curves ) of the and pre- samples are plotted for comparison . we do not plot @xmath260 , since there is little variation in the redshift distribution ( see fig . [ fig7 ] ) . the results in figures [ fig9 ] , [ fig10 ] , and [ fig11 ] clearly show that the redshift distributions for different values of @xmath120 from the model do not fit the redshift sample . furthermore , the fits to both the and pre- samples do not correlate together with the associated range of the jet opening angle . in fact it indicates that we need larger burst rates at higher redshifts ( @xmath261 ) . these results rule out the lower sfr , sfr2 , as can be seen in figure [ fig12 ] . it is important to note at this point that the fit to the pre- and redshift distribution should rely on a single comoving rate density of grb with the same physical parameters . if one particular grb rate model can not fit the observed pre- and redshift distribution at the same time , then that grb model has to be eliminated , unless additional complications are introduced into the model , such as the change of grb model parameters with time the comparisons also rule out the possibility that the upper sfr explains the grb data , even though sfr4 has a higher sfr than sfr2 at large redshifts . in figure [ fig13 ] we fit the pre- and redshift samples using three different @xmath13-ray energies , namely @xmath235 , @xmath262 , and @xmath251 , with @xmath252 , @xmath253 , @xmath254 , and @xmath255 using sfr4 . the results for the range of the jet opening angle @xmath256 , and @xmath263 radians show that the calculated redshift distribution between and pre- are anticorrelated , that is , smaller values of @xmath120 , which would improve the fit for grbs , makes the fit for pre- grbs worse . however , with @xmath252 and @xmath264 ergs ( two bottom panels ) , the redshift distributions are marginally acceptable by the model , noting that the kolmogorov - smirnov one - sided test statistic for a sample of 41 and 31 grbs at the 10% level are 0.167 and 0.187 , respectively ( that is , only 10% of randomly chosen samples should exhibit maximum vertical deviations from the model cumulative distribution ; if it is greater than these values , then the model is an inadequate representation of the data at the 90% confidence level ) . only 2.5% of randomly chosen samples should have vertical excursions greater than 0.212 and 0.238 for a sample of 41 and 31 grbs , respectively . when we calculate the opening angle distribution to fit the pre- opening angle sample in figure [ fig14 ] , however , the fit is adequate for @xmath264 ergs and @xmath265 but not for @xmath266 . hence , the upper sfr ( sfr4 ) is statistically eliminated . nevertheless , the results in figure [ fig13 ] suggest that a greater grb burst rate at larger redshifts is required to fit the redshift distributions . the cumulative jet opening angle distribution ( bold solid curves ) of the pre- sample is plotted for comparison , but not for , since the available opening angles for the associated redshift sample are lacking . as a result , we consider a grb comoving source rate density that exhibits positive evolution above redshift @xmath267 . we therefore modify the hopkins & beacom sfr ( sfr3 ) to obtain the comoving rate - density that attains ( 1 ) a constant comoving rate - density at high redshift @xmath0 ( sfr5 ) , and ( 2 ) a monotonically increasing comoving rate - density with increasing redshift ( sfr6 ) in order to provide a high grb burst rate at large @xmath0 ( see caption to figure [ fig4 ] for coefficients of the sfr models ) . in figures [ fig15 ] and [ fig16 ] we begin to see some improvement in fitting the pre- and redshift samples when using sfr5 and sfr6 with @xmath268 . nevertheless , the fitting is still somewhat inconsistent between the pre- redshift distribution and the jet opening angle . for example , in figure [ fig15 ] using sfr5 , the fitting to the pre- and redshift distribution shows a good fit with @xmath269 for a range of jet opening angles between @xmath256 and @xmath259 radians ( two bottom panels ) . however , the result from the fitting between the pre- redshift distribution and the opening angle distribution with @xmath269 is inconsistent ( see figures [ fig15 ] and [ fig16 ] ) . above redshift @xmath270 , the calculated redshift distribution underestimates the fractional number of bursts that should occur at high redshifts @xmath0 . on the other hand , the calculated opening angle distribution overestimates the fractional number of grbs that should occur at small opening angles . recall that if a burst occurs at a small opening angle , then this translates into the possibility of observing this particular burst at a high redshift @xmath0 . hence , the fit with the range of the jet opening angle @xmath256 , and @xmath271 radians is not yet acceptable . in figures [ fig17 ] and [ fig18 ] we see similar results when using sfr6 . however , it is interesting to note that we can resolve this problem if we could increase the fractional number of grbs at both high and low redshifts @xmath0 . as a result , this will provide a self - consistent fitting between the pre- redshift and the jet opening angle distributions , as well as the redshift distribution . this problem can be solved by adjusting the @xmath13-ray energy @xmath21 . using @xmath272 , figures [ fig19 ] and [ fig20 ] , and figures [ fig21 ] and [ fig22 ] show a statistically acceptable fit to the pre- and redshift distribution and the pre- opening angle distribution with @xmath269 , and the range of the jet opening angle @xmath256 and @xmath258 radians for sfr5 and sfr6 , respectively . for sfr5 , the fitting can be improved with @xmath273 , as indicated by the solid curves ( see figures [ fig19 ] and [ fig20 ] ) . the above analyses show that @xmath274 gives the best fit to the pre- redshift and the jet opening angle samples , and the redshift sample . this best - fitted @xmath13-ray energy is interesting since @xcite and @xcite show that the standard - energy reservoir is @xmath275 . however , with a larger sample set , the data from @xcite shows that @xmath21 is broadly distributed between @xmath276 , as shown in figure [ fig1 ] . because we have been able to obtain an acceptable fit to the data with discrete values of @xmath21 and @xmath29 , a more complicated model where a range of values of @xmath21 and @xmath29 were allowed would certainly permit excellent fits to the data . we can utilize our best - fitted parameters to estimate the average beaming factor @xmath277 . the beaming factor average over the opening angle distribution is given by @xmath278 \ , \nonumber \label{eq21}\end{aligned}\ ] ] where @xmath95 is given in equation ( [ eq12 ] ) . using the best - fitted parameters @xmath269 and @xmath273 and the range of the jet opening angles between @xmath256 and @xmath258 , we obtain @xmath279 as the beaming factor for @xmath269 and @xmath273 , respectively . our value of @xmath280 is about a factor of 2 smaller than the value obtained by @xcite . we can also infer the luminosity function from the jet opening angle distribution . from equations ( [ eq8 ] ) and ( [ eq9 ] ) , the jet opening angle distribution is @xmath281 , which is related to luminosity by the expression @xmath282 where @xmath10 is the grb apparent isotropic luminosity , and @xmath95 is the jet opening angle distribution given in equation ( [ eq12 ] ) . since the energy flux is defined as @xmath283 or @xmath284 , then the isotropic luminosity in the bursting frame is @xmath285 , so that @xmath286 . utilizing equation ( [ eq17 ] ) , the jet opening angle in terms of isotropic luminosity is given by @xmath287 substituting equation ( [ eq24 ] ) into equation ( [ eq23 ] ) , we obtain @xmath288 after utilizing the jet opening angle distribution in equation ( [ eq12 ] ) . since our best fit to the pre- redshift and opening angle samples and the redshift sample constrains the jet opening angle distribution power - law index @xmath120 between @xmath289 and @xmath253 for using sfr5 and sfr6 , respectively , our apparent isotropic luminosity function is @xmath7 within the uniform jet model . in the universal structured jet model , the luminosity function is suggested to be @xmath290 ( e.g. , * ? ? ? * ; * ? ? ? * ) ; others have suggested that the luminosity function is @xmath291 ( e.g. , * ? ? ? * ) . for our best fitting model with jet opening angles ranging from @xmath130 to @xmath292 rad , this implies bolometric @xmath13-ray luminosities in the range @xmath293 to @xmath294 ergs s@xmath295 . the results for our best fitting model are shown in figure [ fig23 ] . the fraction of high - redshift grbs in the high - redshift universe implied by our fits using sfr5 and sfr6 is estimated at 8 12% and 2.5 6% at @xmath11 and @xmath12 , respectively . if this fraction is not observed with swift , then it suggests that sfr5 and sfr6 do not continue with positive evolution @xmath296 . indeed , it is likely that the formation rate of grbs at high redshift declines at sufficiently high redshift . we can also derive the fractional number of grbs with @xmath297 from figure [ fig23 ] . we find that @xmath298% of long duration grbs should occur in the low - redshift ( @xmath299 ) universe . | , we find that the data can only be fitted if the comoving rate - density of grb sources exhibits positive evolution to 5 . the mean intrinsic beaming factor of grbs is found to range from 42 , with the average opening half - angle , compared to the pre- average of . within the uniform jet model , the grb luminosity function is , as inferred from our best fit to the opening angle distribution . . the fraction of high - redshift grbs is estimated at 8 12% and 2.5 6% at and , respectively , assuming continued positive evolution of the grb rate density to high redshifts . | a simple physical model for long - duration gamma ray bursts ( grbs ) is used to fit the redshift ( ) and the jet opening - angle distributions measured with earlier grb missions and with . the effect of different sensitivities for grb triggering is sufficient to explain the difference in the distributions of the pre- and samples , with mean redshifts of and , respectively . assuming that the emission properties of grbs do not change with time , we find that the data can only be fitted if the comoving rate - density of grb sources exhibits positive evolution to 5 . the mean intrinsic beaming factor of grbs is found to range from 42 , with the average opening half - angle , compared to the pre- average of . within the uniform jet model , the grb luminosity function is , as inferred from our best fit to the opening angle distribution . because of the unlikely detection of several grbs with , our analysis indicates that low redshift grbs represent a different population of grbs than those detected at higher redshifts . neglecting possible metallicity effects on grb host galaxies , we find that grb occurs every 600,000 yrs in a local spiral galaxy like the milky way . the fraction of high - redshift grbs is estimated at 8 12% and 2.5 6% at and , respectively , assuming continued positive evolution of the grb rate density to high redshifts . |
astro-ph0610043 | c | in this work , we considered whether the differences between the pre- and redshift distributions can be explained with a physical model for grbs that takes into account the different flux thresholds of grb detectors . the model presented here parameterizes the jet opening angle distribution for an assumed flat @xmath31 spectrum , and finds best fit values for the @xmath13-ray energy release for different functional forms of the comoving rate density of grbs , assuming that the properties of grbs do not change with time . adopting the uniform jet model , we assumed that the energy per solid angle is roughly constant within a well - defined jet opening angle . the pre- redshift sample suggests that the range of the jet opening angle is between @xmath300 and @xmath131 radians . we explored three different ranges of jet opening angle with @xmath301 , and @xmath257 , @xmath131 , and @xmath302 radians . the results in 3 show that an intrinsic distribution in the jet opening angles and single values for all other parameters yields a good fit to the pre- and redshift samples , and furthermore provides an acceptable fit to the distribution of opening angles measured with pre- grb detectors . a good fit was only possible , however , by modifying the hopkins & beacom sfr to provide positive evolution of the sfr history of grbs to high redshifts ( sfr5 and sfr6 ; see fig . [ fig4 ] ) . the best fit values were obtained with @xmath303 ergs , @xmath301 , @xmath258 radians , and @xmath304 . the best - fit value for the absolute @xmath13-ray energy released by a grb is about a factor of 2 larger than the mean value of the @xcite sample of pre- grbs . the results of our fitting therefore indicate that grb activity was greater in the past and is not simply proportional to the bulk of the star formation as traced by the blue and uv luminosity density of the universe . this is contrary to the conclusion of @xcite based upon a study of batse size distribution and a small number of grbs with known redshifts in the pre - swift era . they obtain worse fits for sfrs that increase at high redshift . the batse analysis of @xcite gives results that are consistent with the required monotonically increasing grb sfr that we find . @xcite also conclude that a rising grb sfr is implied , and also note that the star formation rate can not increase in proportion to greater past grb activity , as this would result in overproduction of metals . changes in the progenitor population or grb properties , for example , due to changing metallicities with time , might account for the large number of high - redshift grbs observed with swift , but this would introduce a large number of unconstrained parameters . moreover , one would then have to explain why the amati and ghirlanda correlations relating spectral properties and energy releases are insensitive to redshift . assuming that the grb properties do not change with time , we therefore find that the grb source rate density must display positive evolution to at least @xmath165 . why would grb activity be greater in the past ? one possibility suggested by recent studies @xcite on the metallicities of host galaxies of grbs is that grbs derive from progenitor stars with a lower metallicity than the progenitors of type ii sne . in this case , one might expect that the peak of grb activity would occur earlier in the universe than during the peak of star formation activity at @xmath305 3 . nevertheless , the high - redshift fraction of grbs detected with swift will test the validity of sfr5 and sfr6 used to model the grb data . the models that fit the data imply that 8 12% of swift grbs occur at @xmath165 , in accord with the data as shown in figure [ fig23 ] . our model predicts that 2.5 6% of grbs should be detected from @xmath12 . if no grb with @xmath306 is detected by the time @xmath307 swift grbs with measured redshifts are found , then we may conclude that the grb formation rate has begun to decline above @xmath308 . by contrast , if a few @xmath309 grbs have been detected within the next year or two with swift , then this would be in accord with our model and would support the conjecture of @xcite that a second episode of population i and ii star formation took place at @xmath165 , possibly even including some grbs that originate from population iii stars . our model parameters suggest that grbs can be detected with swift to a maximum redshift @xmath310 , even given the limits placed on the range of jet angles , in particular , @xmath311 . the relationship between the galaxies that host grbs and their metallicities remains controversial , with evidence for high - redshift , low - metallicity grb host galaxies @xcite countered by examples of grbs found in galaxies with moderate metallicity @xcite . @xcite find that 5 grbs , or several per cent of the grb population with known redshifts , are found at low ( @xmath312 ) redshift in galaxies that have very low metallicity compared with solar metallicity and with the distribution of metallicities of low - redshift galaxies in the sloan digital sky survey . in 4 out of the 5 low-@xmath0 grbs , the apparent isotropic energy release of these grbs is much smaller than @xmath313 ergs , and so these grbs are unrepresentative of typical long duration grbs . moreover , the large number of these low redshift grbs already means that they are not typical of the sample of grbs considered in this paper , as the model prediction is that @xmath314 0.2% of the classical " grbs should be detected at low redshifts.% can be estimated by considering the fractional volume within @xmath315 compared with @xmath316 , and that the sfr activity at @xmath317 was a factor @xmath318 greater than at @xmath319 . ] these low - redshift grbs may also belong to a completely distinct population of grbs compared to their high - redshift counterparts , as also suggested by the differing luminosity functions of the two populations @xcite . alternatively , the low - redshift grbs could be normal grbs that appear weak because , for example , their jetted emission is observed off - axis ( e.g. * ? ? ? although we have restricted our treatment to the long - duration , soft - spectrum class of grbs , the separation between the short hard and long soft class of grbs is not distinct . moreover , recent observations of nearby grbs show that the nearby ( @xmath320 ) grb 060614 , a long duration grb in terms of its light curve @xcite , lacks a supernova excess @xcite , does not fit neatly into either the short hard or long soft classes @xcite , and could represent a separate population of grbs @xcite . phenomenological studies have also indicated that the long grb population is bimodally distributed in terms of the autocorrelation function of the light curves ( e.g. , * ? ? ? @xcite argues for multiple populations by examining the size distributions of different groups of batse grbs separated according to spectral properties . from an analysis of durations of batse , @xcite identifies an intermediate duration population that displays the softest spectra and comprises @xmath321% of the total batse grbs . that we were able to obtain good fits to the different data sets using a single grb model may indicate that the contribution of sub - populations to the total number of grbs is small . alternately , the model parameters might have been adjusted to take into account the aggregate long duration grbs . this can be tested in future work using a larger swift data set that allows subgroups to be separately studied based on spectra or durations . figure [ fig24]a shows the model integral size distribution ( see eq . [ [ eq18 ] ] ) of grbs predicted by our best fit model . the plots are normalized to the current total number of observed grbs per year from batse , which is 550 bursts per @xmath322 sr exceeding a peak flux of 0.3 photons @xmath323 in the @xmath324 kev band for the @xmath325 s trigger time @xcite . from the model size distribution , we find that @xmath326 to 360 grbs per year should be detected with a batse - type detector over the full sky above an energy flux threshold of @xmath327 , or photon number threshold of @xmath328 . this range of values is determined by the two model fits with sfr6 and sfr5 , respectively . we estimate from our fits that @xmath329 1370 grbs take place per year per @xmath330 sr with a flux @xmath331 , or @xmath332 . the field of view of the bat instrument on swift is 1.4 sr @xcite , implying that swift should detect @xmath333 150 grbs per year ; currently , observes about 100 grbs per year . this minor discrepancy may be a consequence of using a flat @xmath31 grb spectrum in our model . we also show the differential size distribution of batse grbs from the fourth batse catalog @xcite in comparison with our model prediction in figure [ fig24]b . as can be seen , our model gives a good representation of the size distribution of the batse grb distribution within the statistical error bars , except for a slight overprediction of the number of the brightest grbs ( see also * ? ? ? these brightest grbs typically originate from @xmath164 , possibly suggesting a slight reduction in the number of grbs with @xmath203 compared to the grb rate densities considered . below a photon number threshold of @xmath334 in the 50 300 kev band , the observed number of grbs falls rapidly due to the sharp decline in the batse trigger efficiency at these photon fluxes ( see * ? ? ? * ) . the size distribution of the grbs will extend to much lower values , @xmath335 , and we can use our model to predict the peak flux size distribution to fit the data , noting however that the triggering criteria are more complicated than a simple rate trigger , particularly near threshold @xcite . we can derive the burst rate @xmath336 in our local universe by normalizing our best fit models to the batse results of 340 360 grbs per year full sky with peak fluxes exceeding @xmath337 ergs @xmath338 s@xmath295 . comparing this value with the rate calculated from eq . ( [ eq18 ] ) for our best fit models , we obtain local grb rate densities of @xmath339 gpc@xmath340 yr@xmath295 for sfr5 and @xmath341 gpc@xmath340 yr@xmath295 for sfr6 . since the mean volume occupied by an @xmath342 galaxy in the local universe is @xmath343 mpc@xmath340 @xcite , we obtain the local event rate in our galaxy , assumed to be a typical @xmath342 galaxy , to be @xmath344 and @xmath345 events per year per @xmath10 galaxy , for sfr5 and sfr6 , respectively . this implies that about 1 event occurs every 0.6 myrs in our galaxy . comparing this to the rate of type ib / c supernovae of @xmath9 every 360 years in our galaxy @xcite , we estimate that the rate of grbs is only @xmath346% of the rate of sn ib / c supernovae in the galaxy . @xcite find that @xmath347% of grbs could originate from sn ib / c by comparing the radio brightnesses of sn ib / c with that of sn 1998bw associated with grb 980425 . our results fall comfortably within the observational limits . we also find that about 1 out of every 20,000 sne make a grb , compared to the value of 1 out of every @xmath348 @xmath349 found by @xcite . if grbs do occur at the rate of more than once per myr in the milky way , depending on uncertain metallicity effects @xcite , then the earth will intercept the beam of a grb about once every 25 myrs , given the beaming factor @xmath350 obtained in our analysis . if a milky way grb is located at an average distance of 10 kpc , then the average fluence deposited by the grb is on the order of @xmath351 ergs @xmath338 . strong astrobiological effects begin to be important when the grb fluence is @xmath352 @xmath353 ergs @xmath338 , or when the grb occurs within @xmath9 2 kpc with its beam pointing in our direction @xcite . this will happen perhaps a few times per gyr , so that grbs could have significant effects on terrestrial evolution @xcite . the number of jet opening angles measured with follow - up observations of grbs is not yet sufficient to produce a statistically reliable sample . thus our jet opening angle distribution shown in figures [ fig20 ] and [ fig22 ] with @xmath273 and @xmath253 , for sfr5 and sfr6 , respectively , provide a prediction for . with and pre- thresholds of @xmath354 and @xmath355 , our model predicts the mean measured jet opening angle @xmath356 , and @xmath357 rad , respectively , which can be compared with the measured average value of @xmath358 for the pre- sample . thus we expect to detect more faint low - redshift , large opening angle grbs that pre- satellites could not detect ( see also * ? ? ? it is interesting to note that with a better detector sensitivity , the mean jet angle is shifted to a higher angle and not vice versa , as might be expected . this suggests that on average we expect to see more long - duration bursts with larger opening angle ( @xmath359 radian ) , meaning that the mean times for the achromatic breaks in the light curves could be longer than for the pre- grbs when one also takes into account the larger mean redshift of the sample . in conclusion , we have developed a physical model for grbs and obtained parameters to the model by fitting the grb redshift and opening angle distributions . we could only obtain statistically acceptable joint fits if the comoving rate density of grbs increases monotonically to @xmath165 . our fitting results give a rate of @xmath9 grb every 600,000 yrs in our galaxy if metallicity effects do not play a large role in the galaxies in which grbs are formed . in this case , the increase in the grb rate density to high redshifts would have a different cause , for example , an episode of enhanced star formation at @xmath360 @xcite . the opening angle distribution of grbs measured with , and the fraction of low-@xmath0 and high-@xmath0 standard long - duration grbs will test this model . we thank d. band , j. beacom , e. berger , j. bloom , and a. friedman for discussions and correspondence , and the referee for comments , corrections , and useful suggestions . t. l. is funded through nasa _ glast _ science investigation no . dpr - s-1563-y . the work of c. d. d. is supported by the office of naval research . | a simple physical model for long - duration gamma ray bursts ( grbs ) is used to fit the redshift ( ) and the jet opening - angle distributions measured with earlier grb missions and with . the effect of different sensitivities for grb triggering is sufficient to explain the difference in the distributions of the pre- and samples , with mean redshifts of and , respectively . assuming that the emission properties of grbs do not change with time because of the unlikely detection of several grbs with , our analysis indicates that low redshift grbs represent a different population of grbs than those detected at higher redshifts . neglecting possible metallicity effects on grb host galaxies , we find that grb occurs every 600,000 yrs in a local spiral galaxy like the milky way | a simple physical model for long - duration gamma ray bursts ( grbs ) is used to fit the redshift ( ) and the jet opening - angle distributions measured with earlier grb missions and with . the effect of different sensitivities for grb triggering is sufficient to explain the difference in the distributions of the pre- and samples , with mean redshifts of and , respectively . assuming that the emission properties of grbs do not change with time , we find that the data can only be fitted if the comoving rate - density of grb sources exhibits positive evolution to 5 . the mean intrinsic beaming factor of grbs is found to range from 42 , with the average opening half - angle , compared to the pre- average of . within the uniform jet model , the grb luminosity function is , as inferred from our best fit to the opening angle distribution . because of the unlikely detection of several grbs with , our analysis indicates that low redshift grbs represent a different population of grbs than those detected at higher redshifts . neglecting possible metallicity effects on grb host galaxies , we find that grb occurs every 600,000 yrs in a local spiral galaxy like the milky way . the fraction of high - redshift grbs is estimated at 8 12% and 2.5 6% at and , respectively , assuming continued positive evolution of the grb rate density to high redshifts . |
1303.3911 | i | in the present work the equation @xmath0 , \label{i1}\ ] ] is studied , where @xmath1 is a real number , @xmath2 , @xmath3 is a complex - valued continuous function on @xmath4 $ ] satisfying a growth bound @xmath5 at the origin for some @xmath6 , @xmath7 $ ] are complex valued functions , and @xmath8 is a ( complex ) spectral parameter . denote @xmath9 . equations of the form ( [ i1 ] ) appear naturally in many real - world applications after a separation of variables and therefore have received considerable attention ( see , e.g. , @xcite , @xcite , @xcite , @xcite , @xcite ) . in preceding publications equation ( [ i1 ] ) was studied under more restrictive conditions , typically for @xmath3 and @xmath10 being real - valued and @xmath11 . the approach developed in this work does not imply such restrictions and serves both for qualitative study of solutions and spectral problems , as well as for related numerical computation . the main component in the developed approach is a spectral parameter power series ( spps ) representation for the regular solution of ( [ i1 ] ) obtained under the condition that the auxiliary equation @xmath12 possesses a regular solution which does not have zeros on @xmath13 $ ] except at @xmath14 . the spps representation for solutions of nonsingular linear differential equations and its applications in corresponding scattering and spectral problems were studied in @xcite , @xcite , @xcite , @xcite , @xcite , @xcite , @xcite , @xcite , @xcite , @xcite , @xcite , @xcite , @xcite and some other papers . here , in section [ sect2 ] we obtain an analogous result for the perturbed bessel equation ( [ i1 ] ) . the construction and the existence of the required particular solution are addressed in section [ sect3 ] . for example , when @xmath15 , @xmath16 $ ] such nonvanishing on @xmath4 $ ] solution exists . we give an analytic representation for it together with an estimate . let us emphasize that , in general , @xmath17 is allowed to be a complex - valued function , and the existence of such @xmath17 for a complex valued @xmath3 is an open problem . under the assumption that @xmath17 exists we obtain a dispersion ( characteristic ) relation for the sturm - liouville problem for ( [ i1 ] ) on @xmath13 $ ] , @xmath18 ( theorem [ thmspectralproblem ] ) . namely , we construct an entire function @xmath19 in the form of a taylor series such that the set of its zeros coincides with the set of all eigenvalues of the sturm - liouville problem . this immediately implies the discreteness of the spectrum and offers an efficient method for its computation . in practical applications of the spps method it is often convenient to consider not only series with the centre in the origin @xmath20 but also series with the centre at @xmath21 where @xmath22 is some complex number . in section [ sectspshift ] we show that this spectral parameter shifting technique is applicable to equation ( [ i1 ] ) and give necessary details . the spps representation allows us to obtain a result on mapping properties of the transmutation operator corresponding to the operator @xmath23 , which was studied , e.g. , in @xcite , @xcite and @xcite . we show in section [ secttransmut ] how the transmutation operator acts on certain powers of the independent variable . in the case of nonsingular schrdinger operators a result of this kind allowed us to advance in the construction of the transmutation operator itself @xcite and had applications in constructing complete systems of solutions for some partial differential equations @xcite , @xcite , @xcite . in section [ sectnumeric ] we explain the numerical implementation of the developed spps method for solving sturm - liouville problems for ( [ i1 ] ) . first , we consider several known test problems , and comparing the obtained results with the results obtained by the best available codes , as sleign2 , matslise and some others , we show that our method is highly competitive and gives better or at least comparable results on test problems to which other codes are applicable . second , we consider an example which involves a different from zero @xmath24 in ( [ i1 ] ) and a complex spectrum . meanwhile sleign2 and matslise are not applicable to problems admitting complex eigenvalues , our method delivers results which are in excellent agreement with the exact data . | a spectral parameter power series ( spps ) representation for regular solutions of singular bessel type sturm - liouville equations with complex coefficients is obtained as well as an spps representation for the ( entire ) characteristic function of the corresponding spectral problem on a finite interval . it is proved that the set of zeros of the characteristic function coincides with the set of all eigenvalues of the sturm - liouville problem . based on the spps representation a new mapping property of the transmutation operator for the considered perturbed bessel operator is obtained , and a new numerical method for solving corresponding spectral problems is developed . we show that the developed numerical method based on the spps representation is highly competitive in comparison to the best available solvers such as sleign2 , matslise and some other codes and give an example of an exactly solvable test problem admitting complex eigenvalues to which the mentioned solvers are not applicable meanwhile the spps method delivers excellent numerical results . | a spectral parameter power series ( spps ) representation for regular solutions of singular bessel type sturm - liouville equations with complex coefficients is obtained as well as an spps representation for the ( entire ) characteristic function of the corresponding spectral problem on a finite interval . it is proved that the set of zeros of the characteristic function coincides with the set of all eigenvalues of the sturm - liouville problem . based on the spps representation a new mapping property of the transmutation operator for the considered perturbed bessel operator is obtained , and a new numerical method for solving corresponding spectral problems is developed . the range of applicability of the method includes complex coefficients , complex spectrum and equations in which the spectral parameter stands at a first order linear differential operator . on a set of known test problems we show that the developed numerical method based on the spps representation is highly competitive in comparison to the best available solvers such as sleign2 , matslise and some other codes and give an example of an exactly solvable test problem admitting complex eigenvalues to which the mentioned solvers are not applicable meanwhile the spps method delivers excellent numerical results . |
1607.02058 | i | the study of the @xmath9 interaction with coupled channels has been a traditional test field for chiral dynamics in its unitarized version . since the pioneer works of @xcite , many works have been done in this field including also the contribution of the higher order chiral lagrangians ( see recent review in @xcite ) . one of the unexpected results was the finding of two poles for the @xmath4 resonance @xcite , which recently became official in the particle data group @xcite [ see note on the @xmath4 @xcite ] . the other issue that became topical was the possibility of having kaonic nuclei , in particular a bound @xmath7 system . the study of this system has been thoroughly addressed theoretically @xcite ( see review paper in @xcite ) , obtaining a binding energy that varies from a few mev to 100 mev . there are also discrepancies in the width of the state , which varies from 10 to 90 mev . one step forward in the evaluation of the width was given in @xcite , where two - nucleon @xmath10 absorption was explicitly considered . in that study a binding of 1530 mev was found , together with a width of the order of 80 mev . the fact that the width is larger than the binding energy is shared by most of the theoretical approaches . one interesting point of view was to consider this system as a bound state of @xmath11 @xcite . this allows one to make some qualitative pictures that help understanding some of the results obtained when studying the possible formation of this system in different reactions . there have been previous claims of the formation of this state in some experiments , but soon the experimental signals were interpreted in terms of unavoidable conventional mechanisms ( see overview in @xcite ) . yet , the experimental search has continued @xcite with often contradictory views ( see overview in @xcite ) . in this line , very recently , a peak structure was observed near the @xmath0 threshold in the @xmath3 invariant mass spectrum of the in - flight @xmath12 reaction of the j - parc e15 experiment @xcite . according to their analysis , this peak can be described by the breit - wigner formula with mass @xmath13 and width @xmath14 . this structure could be a signal of the @xmath2 bound state with a binding of @xmath15 from the @xmath0 threshold . in this paper we theoretically investigate the origin of the peak structure observed in the j - parc e15 experiment . for this purpose , we take into account two possible mechanisms for producing a peak in the mass spectrum of the @xmath16 reaction . one corresponds to assuming the formation of a @xmath17 resonance that does not not form a bound state with the remaining proton , while the other considers the formation of a bound state of the @xmath7 system . we evaluate the cross section of the @xmath16 reaction assuming the @xmath18 scattering around threshold and the @xmath4 resonance to be described by the chiral unitary approach @xcite , while the description of the @xmath7 bound state is done in terms of the so - called fixed center approximation to the faddeev equation @xcite . as a result , we can unambiguously interpret the experimental spectrum in the scenario of a @xmath7 broad bound state , obtained from the interaction of the @xmath10 with a pair of nucleons @xcite . in addition , in this scenario , we obtain a two peak structure of the mass spectrum near the @xmath19 threshold . the peak below the threshold is the signal of the @xmath7 bound state , while the peak above the threshold originates from the quasi - elastic scattering of the kaon in the first collision emitting a fast nucleon , in processes of the type @xmath20 or @xmath21 . this paper is organized as follows . in sec . [ sec:2 ] we develop our formulation to obtain the cross section of the @xmath16 reaction , describing the details that allow us to obtain the scattering amplitude for the uncorrelated @xmath22 mechanism and for the @xmath19 bound state one . next , in sec . [ sec:3 ] we show our results and discuss the origin of the peak structure observed in the in - flight @xmath23 reaction in the j - parc experiment . section [ sec:4 ] is devoted to the conclusions of this study . | a theoretical investigation is done to clarify the origin of the peak structure observed near the threshold in the in - flight reaction of the j - parc e15 experiment , which could be a signal of the lightest kaonic nuclei , that is , the state . for the investigation one assumes that the resonance is generated after the emission of an energetic neutron from the absorption of the initial , not forming a bound state with the remaining proton . | a theoretical investigation is done to clarify the origin of the peak structure observed near the threshold in the in - flight reaction of the j - parc e15 experiment , which could be a signal of the lightest kaonic nuclei , that is , the state . for the investigation , we evaluate the invariant mass spectrum assuming two possible scenarios to interpret the experimental peak . one assumes that the resonance is generated after the emission of an energetic neutron from the absorption of the initial , not forming a bound state with the remaining proton . this uncorrelated system subsequently decays into the final . the other scenario implies that , after the emission of the energetic neutron , a bound state is formed , decaying eventually into a pair . our results show that the experimental signal observed in the in - flight reaction at j - parc is qualitatively well reproduced by the assumption that a bound state is generated in the reaction , definitely discarding the interpretation in terms of an uncorrelated state . |
cond-mat0002259 | i | the unexpected discovery of high-@xmath0 itinerant ferromagnetism in doped hexaborides@xcite has re - ignited interest in the problem of excitonic ordering near the semiconductor metal transition.@xcite excitonically ordered states are characterized by an off - diagonal order parameter describing pairing between conduction electrons and valence holes . early theoretical work by volkov _ et . al._@xcite anticipated the emergence of ferromagnetism on doping such an excitonic state . these authors considered the limit of nearly nested overlapping conduction and valence bands with _ weak _ repulsive electron - electron interactions . in this limit , the problem can be approximately cast into a form nearly identical to bcs theory , and studied using the techniques of mean - field theory . although this work ( and some subsequent recent studies@xcite ) suffers from the important physical mistake of neglecting the instability to phase separation , ferromagnetism remains nevertheless a generic feature in a corrected treatment.@xcite while the appearance of ferromagnetism in the weak - coupling limit is encouraging , it is far from a conclusive and complete theoretical explanation for the experiments . first , coulomb interactions in the hexaborides are not particularly weak , and most likely are comparable to the fermi energy and band overlap . second , the above explanation appears to hinge on the first - order nature of the excitonic to normal ( e - n ) transition in the bcs limit . while this feature , mathematically analogous to the first - order transition to the normal state due to pair - breaking by an external zeeman field in a superconductor,@xcite is present in the nested mean - field limit , there do not appear to be any general theoretical grounds mandating this behavior more generally . moreover , the universality of the experimental results , now observed in a large number of different compounds ( ca@xmath1la@xmath2b@xmath3 , bab@xmath3 , ca@xmath1ce@xmath2b@xmath3 , srb@xmath3, )@xcite , argues for the robustness of the phenomenon . to determine whether excitonic ferromagnetism is indeed more general than its weak - coupling theoretical basis , we consider here the completely opposite _ strong - coupling _ regime this is not expected to be directly applicable to the hexaborides , as these materials are most likely best described by an intermediate - coupling model . nevertheless , many useful insights are gained from this complementary limit . as usual , the principle assumption of the strong - coupling limit is the dominance of potential over kinetic energy . this is achieved concretely using a tight - binding model ( see eqs . [ tb1]-[nnn ] , in sec . [ sec : model ] ) , in which the conduction and valence bands of the conventional continuum theories are replaced by localized @xmath4 and @xmath5 orbitals , respectively . the analog of band gap in the continuum model is the level splitting @xmath6 . the order parameter characterizing excitonic ordering is then a matrix in spin space : @xmath7 where @xmath8 creates an electron with spin @xmath9 in the @xmath4 orbital , and @xmath10 annihilates an electron with spin @xmath11 in the @xmath5 orbital . excitonically ordered states thus have some partial occupation of the nominally excited @xmath4 states , as a result of coulombic repulsion . in general , @xmath12 is a proper order parameter ( i.e. one which characterizes a spontaneously broken symmetry ) if the @xmath4 and @xmath5 orbitals have different symmetries . in this paper , we consider a `` minimal model '' with this property , comprised of one @xmath4 and one @xmath5 orbital per unit cell see fig . [ fig : orbs ] . this mimics the situation in the hexaborides , for which the conduction and valence states also transform as different representations of the cubic point group@xcite . because of complications arising from orbital degeneracy , however , the appropriate representations for the hexaborides are three - dimensional rather than scalar . we defer the possible complications arising from these additional degrees of freedom to a future investigation . and @xmath13 orbitals , respectively.,width=192 ] as for the more familiar hubbard model ( see , e.g. ref . @xcite ) , the problem simplifies somewhat in the strong - coupling limit . considering first the undoped system ( half - filled = two electrons per unit cell ) , we obtain a novel quantum pseudo - spin model ( eqs . [ tj1]-[ising ] , sec . [ sec : undoped ] ) . within this model , the excitonic insulator ( ei ) appears as an intermediate state separating not a metal and a semiconductor but a _ mott insulator _ and a semiconductor ( or band insulator ) . in some respects , the behavior is argued to be quite similar to that of a quantum spin-@xmath14 xxz antiferromagnet in a magnetic field , with excitonic ordering analogous to xy antiferromagnetism . the `` spins '' of the model , however , can take on _ five _ distinct states per site : one singlet state with both electrons in the lower - energy @xmath5 orbital , and four different spin states with one @xmath4 and one @xmath5 electron . this is in contrast to the two states of a single spin-@xmath14 particle . in the strong coupling limit , this large hilbert space is `` unified '' by several approximate symmetries valid at different energy scales . at the largest energy scales this is an enormous su(4 ) group , corresponding to arbitrary complex rotations of the four components of @xmath12 . the approximate su(4 ) symmetry fully unifies all possible excitonic states , including singlet , triplet , and singlet - triplet coexistences . these are described by the general decomposition @xmath15 where @xmath16 , @xmath17 are the singlet and triplet order parameters , and @xmath18 and @xmath19 are the @xmath20 unit and pauli matrices in spin space , respectively . a system with approximate su(4 ) invariance contains the germ of ferromagnetism , since several possible excitonic states ( those with non - zero @xmath21 and/or @xmath22 ) give rise to net exchange fields , and hence a magnetic moment . su(4 ) symmetry implies that these states are low in energy . at intermediate energies the su(4 ) symmetry reduces to an su(2)@xmath23su(2 ) invariance , which reflects separate spin rotations of the @xmath4 and @xmath5 electrons . the latter is a symmetry of the conventional continuum models of eis , and transforms the order parameter in a `` chiral '' manner : @xmath24 , where @xmath25 and @xmath26 are su(2 ) matrices . finally , further weak interactions reduce this to a simple su(2)@xmath27 symmetry at the ( very ) lowest energies . these symmetry considerations underly the simple physical mechanism for ferromagnetism elucidated here@xcite . the dominant tendency imposed by coulomb interactions is to excitonic ordering . with approximate su(4 ) symmetry , however , the `` orientation '' ( form of @xmath12 ) of the order parameter is nearly free and fixed only by weak `` anisotropy '' terms . in the undoped material , these anisotropies favor a simple paramagnetic triplet state . doping introduces additional exchange energy contributions that modify the anisotropy , causing @xmath12 to `` flop '' into a different orientation with a ferromagnetic moment . in the present model , the excitonic order in the ferromagnet is of non - collinear triplet type , in which @xmath28 as shown in sec . [ sec : discussion ] , in addition to ferromagnetic magnetization , this state has additional spatially - varying local static moments and spin currents transverse to the axis of net magnetization . the transition to this state from the paramagnet is generally first order , and therefore coincides with a jump in the electronic density . since experiments are performed at fixed charge density ( dictated by the concentration of dopant ions ) , the intermediate `` forbidden '' range of dopings can be accommodated only by phase separation . with long - range coulomb interactions included , macroscopic phase separation is impossible , and charge domain formation is expected , as pointed out already in refs . @xcite . the _ detailed _ demonstration of this behavior with doping is non - trivial . as for many other strongly correlated systems , the problem of doping is much more difficult than that of the stoichiometric mott insulator . indeed , as the ei state lies intermediate between band and mott insulators , doping the ei is a sort of interpolation between doping a conventional band insulator and doping an antiferromagnetic insulator . the latter problem is of course at the crux of the physics of high - temperature superconductivity , so that perhaps the experimental and theoretical insights gained in the hexaborides will be helpful more generally . at any rate , doping the ei can be shown by very simple arguments to favor ferromagnetism in strong coupling . essentially , the physics of this behavior is similar to the `` nagaoka effect''@xcite in a doped antiferromagnet ferromagnetic alignment of the excitonic order parameters allows for more coherent propagation of the doped electrons , and hence a lowering of their kinetic energy . this mechanism is actually _ stronger _ in the ei than in the antiferromagnet , because of the global _ coherence _ of the excitonic condensate , and the near degeneracy ( due to approximate su(4 ) symmetry ) of ferromagnetic and paramagnetic states . ( half the bare splitting between @xmath4 and @xmath5 orbitals ) , and @xmath29 , the chemical potential . the shaded region is not analyzed in this paper . thick lines indicate the boundary between the undoped region below ( with two electrons per unit cell ) and the doped region above ( with more than two electrons per unit cell ) . the af@xmath30 and af@xmath31 phases are antiferromagnetic mott insulators with two and three electrons per unit cell , respectively . the bi state is the band insulator . intermediate between the bi and af@xmath30 phases are the excitonic insulator ( ei ) and an insulator with coexisting excitonic and nel order ( ei / af ) . the fm , fm@xmath32 , pfm , and pfm@xmath32 phases are all ferromagnetic metals ( see table [ tab : phases ] for the differences between these states ) , while pm indicates a paramagnetic metallic phase . all the metallic states above exhibit excitonic order.,width=312 ] to provide a concrete demonstration of these ideas , the strong coupling zero temperature phase diagram of the model is calculated in this paper using a `` free fermi gas '' approximation . this approximation captures the most important _ single quasiparticle _ physics of electronic propagation in an excitonically - ordered background , but neglects interactions between these quasiparticles . for simplicity , we also assume a _ fixed amplitude _ , @xmath33 , of the excitonic order parameter . the latter assumption is valid for weak doping , @xmath34 , in which the _ orientation _ of the ordering is of paramount importance . putting together the results of this calculation and the stoichiometric behavior , we arrive at the partial phase diagram in fig . [ partialfig ] . this is in agreement with the general expectations stated above . it should be stressed , however , that this analysis of doping is far from exhaustive . more detailed investigations of both the weak and strong coupling limits are currently underway@xcite the remainder of the paper is structured as follows . in sec . [ sec : model ] , we present a detailed exposition of the ( simplest ) tight - binding model capable of describing excitonically - ordered states , and consider the limit of infinite interaction strength . the bulk of the paper is contained in sec . [ sec : undoped ] , where the model is analyzed for large but finite interactions , focusing on the stoichiometric situation with two valence electrons per unit cell . for this electron density the model is insulating , but can sustain excitonic and other types of ordering . the properties of the model with doping are discussed in sec . [ sec : doping ] . we conclude in sec . [ sec : discussion ] with a clarifying discussion delineating the physical properties of various possible excitonic insulators , and the relation of the results of this paper to the hexaborides . | a tight binding model is introduced to describe the strong interaction limit of excitonic ordering . at stoichiometry excitonic order appears in the pseudo - spin model as in - plane pseudo - magnetism . the u(4 ) symmetry unifies all possible singlet and triplet order parameters describing such states . super - exchange , hunds - rule coupling , and other perturbations act as anisotropies splitting the u(4 ) manifold , ultimately stabilizing a paramagnetic triplet state . a discussion of the physical meaning of various excitonic states and their experimental consequences is also provided . | a tight binding model is introduced to describe the strong interaction limit of excitonic ordering . at stoichiometry , the model reduces in the strong coupling limit to a pseudo - spin model with approximate u(4 ) symmetry . excitonic order appears in the pseudo - spin model as in - plane pseudo - magnetism . the u(4 ) symmetry unifies all possible singlet and triplet order parameters describing such states . super - exchange , hunds - rule coupling , and other perturbations act as anisotropies splitting the u(4 ) manifold , ultimately stabilizing a paramagnetic triplet state . the tendency to ferromagnetism with doping ( observed experimentally in the hexaborides ) is explained as a spin - flop transition to a different orientation of the u(4 ) order parameter . the physical mechanism favoring such a reorientation is the enhanced coherence ( and hence lower kinetic energy ) of the doped electrons in a ferromagnetic background relative to the paramagnet . a discussion of the physical meaning of various excitonic states and their experimental consequences is also provided . |
0707.4135 | i | in polynomial dynamics , _ dynamic rays _ , which foliate the set of escaping points ( points which tend to @xmath6 under iteration ) , were introduced by douady and hubbard as a tool in their famous work on the mandelbrot set @xcite . since then , dynamic rays , and their landing properties in particular , have been an essential ingredient in the success of polynomial dynamics . one of the fundamental results in this area , which goes back to douady , states that each repelling or parabolic ( pre)periodic point of a polynomial with connected julia set is the landing point of at least one ( pre)periodic dynamic ray ( * ? ? ? * theorem 18.11 ) . in the polynomial case , dynamic rays arise naturally as preimages of straight rays under the bttcher isomorphism at @xmath6 . when @xmath0 is a transcendental entire function , @xmath6 is no longer a superattracting fixed point , but rather an essential singularity . hence the _ escaping set _ @xmath7 is no longer open . nevertheless , it has long been known @xcite that for certain classes of entire functions there exist curves in the escaping set which can be seen as analogs of dynamic rays . we shall refer to these curves ( which are also often known as _ hairs _ ) themselves as dynamic rays of @xmath0 to stress the analogy to the polynomial case . ( see definition [ defn_ray ] for a formal definition of dynamic rays , and also below for the special case of _ periodic _ rays that is of main interest to us . ) recently rottenfuer , rckert , rempe and schleicher @xcite proved that , for an entire function that is a composition of finite - order entire functions with bounded sets of singular values , the escaping set consists of dynamic rays and ( some of ) their endpoints . this provides us with a large class of functions where we can study the topology of julia sets by looking at landing properties of dynamic rays . this approach to the study of julia sets and escaping sets has been used with great success in certain families of entire transcendental maps like the exponential family @xmath8 or the cosine family @xmath9 . to state the main question of interest to us , let us begin by formally defining what we mean by a _ periodic ( dynamic ) ray_. [ defn_periodic_rays ] let @xmath0 be an entire function . then a _ periodic ray _ of @xmath0 is an injective curve @xmath10 such that @xmath11 and such that there is an @xmath12 with @xmath13 for all @xmath14 . the minimal such @xmath15 is called the _ period _ of @xmath16 . as usual , we say that @xmath16 _ lands _ at @xmath5 if @xmath17 . a natural question suggested by douady s theorem and the results of @xcite is the following . suppose that @xmath0 is a finite - order entire function whose singular set is bounded , and suppose also that @xmath18 . is every repelling or parabolic periodic point of @xmath0 the landing point of a periodic ray of @xmath0 ? even for exponential and cosine maps , this question is still open . ( for a partial result on exponential maps , compare @xcite . ) however , schleicher and zimmer @xcite obtained a positive answer for exponential maps satisfying certain dynamical assumptions . in this paper , we generalize this statement , under similar conditions , to a much larger class of entire functions . for an entire transcendental function @xmath0 , @xmath1 denotes the set of singular values , @xmath3 the postsingular set , and @xmath2 and @xmath4 the fatou and julia sets of @xmath0 , respectively . ( for definitions , see section [ notations_section ] . ) [ def_gf ] a map @xmath0 is called _ geometrically finite _ if @xmath19 is compact and @xmath20 is finite . we can now state our main result . recall that @xmath0 has _ finite order _ if @xmath21 as @xmath22 . [ thm1 ] let @xmath0 be geometrically finite and assume that @xmath0 has finite order . then , for any repelling or parabolic periodic point @xmath23 of @xmath0 , there is a periodic ray landing at @xmath23 . in fact , the condition that @xmath0 is geometrically finite can be weakened ; what we need is that every iterate of @xmath0 has an `` admissible expansion domain '' at every repelling or parabolic fixed point ( see definition [ dfn_adm_exp_dom ] ) . furthermore , the assumption of finite order is not essential ; we require that `` all periodic rays exist '' . ( see theorem [ maintheorem ] for the precise statement . ) by @xcite , this condition is satisfied by all finite compositions of finite - order functions with bounded sets of singular values . our result implies in particular that each singular value in the julia set of a map @xmath0 to which the theorem applies is the landing point of some periodic dynamic ray . using these rays , one can define a dynamically natural partition of the julia set , as done for exponential and cosine maps in @xcite , which is useful for studying the topological dynamics of @xmath0 in combinatorial terms . we remark that , under more restrictive function - theoretic conditions , our result can be considerably strengthened . indeed , if @xmath24 is a cosine map for which both critical values are strictly preperiodic , then schleicher showed in @xcite that _ every _ point @xmath25 is either on a dynamic ray or the landing point of a dynamic ray . in @xcite we generalize this result to any finite - order _ subhyperbolic _ function @xmath0 such that @xmath4 contains no asymptotic values and the degree of critical points in @xmath4 is uniformly bounded . ( a subhyperbolic entire function is a geometrically finite map without parabolic cycles . ) furthermore , the julia set of such a function can be described as a `` pinched cantor bouquet '' in the sense of @xcite ( see also ( * ? ? ? * definition 1.2 ) ) . here the assumption that there are no asymptotic values in the julia set is essential : indeed , there is _ no _ exponential map @xmath26 for which the asymptotic value belongs to the julia set and every point of @xmath4 is either on a dynamic ray or the landing point of a dynamic ray @xcite . in the case of geometrically finite exponential maps , our theorem is due to schleicher and zimmer @xcite ( although in the case where the singular value is preperiodic some of the details are only sketched ) . this was extended to cosine maps with preperiodic critical values in @xcite . the general strategy of our proof , which we will now describe , follows the same idea as these papers . by passing to a suitable iterate , we can assume that the considered repelling or parabolic periodic point @xmath5 is a fixed point . the idea is to start with _ any _ given curve connecting @xmath5 to infinity , and pull back this curve using iterates of the map @xmath0 . using hyperbolic contraction arguments , we prove that this procedure yields only finitely many different curves up to homotopy . this then allows us to associate a combinatorial object ( a `` periodic external address '' ) to these curves . when we know from the existence theorems of @xcite that there exists a periodic ray corresponding to this address , it then easily follows that this ray lands at @xmath5 . since the `` ad - hoc '' method that was used to obtain hyperbolic contraction estimates in @xcite appears to be difficult to adapt to our more general setting , we develop a rather natural construction using hyperbolic geometry . our argument gives a more streamlined proof even in the established cases . we would like to emphasize that the case when @xmath3 is finite requires less technical constructions than the general setting . this is why we consider this special case separately in the proofs of some results required for the proof of theorem [ thm1 ] ( see section [ sec_3 ] ) . [ ackref ] i would like to thank especially my supervisor , lasse rempe , for his great help and support . i would also like to thank adam epstein , freddie exall and mary rees for helpful and interesting discussions . | a transcendental entire function is called _ geometrically finite _ if the intersection of the set of singular values with the fatou set is compact and the intersection of the postsingular set with the julia set is finite . ( in particular , this includes all entire functions with finite postsingular set . ) | a transcendental entire function is called _ geometrically finite _ if the intersection of the set of singular values with the fatou set is compact and the intersection of the postsingular set with the julia set is finite . ( in particular , this includes all entire functions with finite postsingular set . ) if is geometrically finite , then is either empty or consists of the basins of attraction of finitely many attracting or parabolic cycles . let be a repelling or parabolic periodic point of such a map . we show that , if has finite order , then there exists an injective curve consisting of escaping points of that connects to . ( this curve is called a _ dynamic ray_. ) in fact , the assumption of finite order can be weakened considerably ; for example , it is sufficient to assume that can be written as a finite composition of finite - order functions . |
nlin0410050 | i | the cascade of energy to low wavenumbers in two - dimensional turbulence @xcite , more than other turbulence problems , is suited to the standard methods of statistical field theory and the renormalization group ( rg ) . these methods @xcite , originating in quantum field theory , show that arbitrary short - distance interactions lead to long - distance behavior described by a local , renormalizable action an effective field theory . correlation functions computed from this action contain ultraviolet ( uv ) divergences that can be eliminated by redefining the parameters and fields . the divergences leave their mark , however , in the dependence on the renormalization scale and the resulting anomalous scaling laws . the classic application of statistical field theory is to critical phenomena ( second - order phase transitions ) in condensed matter @xcite . the infrared ( ir ) scale invariance of correlation functions at the transition temperature is explained by a fixed point of the rg flow . the inverse energy cascade of two - dimensional turbulence is likewise believed to be nearly scale invariant , and one might suspect that a similar fixed point is responsible . we will see , however , that no fixed point can reproduce the observed @xmath4 energy spectrum . rather , we will argue that the inverse cascade arises from a nontrivial rg flow and thus is not expected to be completely scale invariant . in the study of turbulence , a deviation from scale invariance ( a dependence of dimensionless physical quantities on scale ) is referred to as intermittency @xcite . while intermittency is recognized as a property of the three - dimensional direct cascade of energy , its existence in the two - dimensional inverse cascade is unsettled . for a nonstationary inverse cascade , in which energy is not dissipated but progresses to ever - lower wavenumbers , intermittency is not observed in numerical simulations @xcite ; a theoretical explanation has been given @xcite . in this paper we deal solely with the stationary regime , where an inverse cascade requires a low - wavenumber energy sink . with few exceptions , simulations of such a cascade confirm the @xmath4 energy spectrum initially predicted @xcite on the basis of scale invariance . but one set of simulations @xcite finds strong intermittency in fourth- and higher - order velocity correlations . other simulations @xcite and experiments @xcite , though , find no significant intermittency . the various studies differ mainly in the precise form of the dissipation terms . the evidence suggests that intermittency in the stationary inverse cascade , permitted by our theory , is at least possibly realized . it is our restricted focus on the inverse cascade that allows us to work with a purely local field theory . the random force that stirs the fluid is correlated over a limited range and is effectively local in a long - distance description . as with the rg treatment of quantum fields and condensed matter , we expect all short - distance details to become irrelevant except as they are manifested in local , renormalizable couplings . the two - dimensional direct cascade of enstrophy to high wavenumbers @xcite thus falls outside our scope . a previous rg analysis of two - dimensional turbulence @xcite is formally similar to ours , but it follows three - dimensional studies by adopting the long - range force correlations necessary for a direct cascade ; even its derivation of the inverse cascade relies on nonlocal forcing . here we apply rg methods in the familiar domain of local field theory , which should allow a physical treatment of the inverse cascade . the explanation of the direct enstrophy cascade may rest on entirely different foundations , such as conformal invariance @xcite . the work most closely aligned with our theoretical approach is due to forster , nelson , and stephen ( fns ) @xcite . at the technical level , our contribution is to add linear friction to fns model a in @xmath5 and compute the rg flow to the next order of perturbation theory , two loops . the inclusion of friction , which dissipates energy at low wavenumbers , makes our theory capable in principle of describing the inverse cascade and its @xmath4 spectrum unlike fns model a , which gives a @xmath6 spectrum corresponding to energy equipartition in @xmath5 . we also note a difference in our viewpoint from that of fns and others @xcite who apply rg methods to turbulence by analogy with critical phenomena . these authors seek a controlled ir - stable fixed point by starting with a logarithmically divergent field theory and then decreasing the dimension of space or the exponent of the stirring - force correlation by @xmath7 . this adds to the @xmath0 function a negative linear term proportional to @xmath7 . with the usual positive one - loop term , there exists an ir - stable fixed point at a coupling that goes to zero with @xmath7 ; the fixed - point theory can then be expanded in @xmath7 instead of the original coupling . like fns , we work in @xmath8 to regulate uv divergences , but we ultimately take @xmath9 , so that the fixed point is trivial . our inverse - cascade model lies not at a fixed point but in the region of large dimensionless coupling . naturally the use of perturbation theory is questionable for strong coupling . our perturbative results will have direct quantitative application only to the extreme ir limit controlled by the trivial fixed point , which is of some interest in itself . nevertheless , we will make reasonable conjectures about the theory s strong - coupling behavior that are consistent with the inverse cascade , bearing in mind the dangers of the nonperturbative regime . besides the concern with the numerical accuracy of extrapolations , there are fundamental difficulties at strong coupling . the anomalous dimensions of operators may be large , and the relevance of terms in the action may differ from the weak - coupling case . furthermore , at strong coupling , there is no simple relation between the couplings in different renormalization schemes , such as the wilsonian cutoff ( useful for physical interpretation ) and minimal subtraction ( convenient for systematic calculation ) . we may hope that these subtleties do not affect the main conclusions even at very strong coupling . at least we know that the theory of critical phenomena in @xmath10 is extrapolated to @xmath11 ( moderate coupling ) with acceptable results . in sec . [ framework ] we describe the basis of our theory and our method of calculation , confirming the one - loop rg flow of fns @xcite . in sec . [ properties ] we present symmetries and other properties of the theory that do not involve a dubious extrapolation to strong coupling . in sec . [ twoloop ] we compute the two - loop term of the @xmath0 function . in sec . [ cascade ] we relate the plausible strong - coupling behavior of the theory to the phenomenology of the inverse cascade . in sec . [ burgers ] , as a test of our methods , we consider a rather different model , the uv - stirred one - dimensional burgers equation . a summary and discussion are presented in sec . [ discussion ] . | rev . a * 16 * , 732 ( 1977 ) ] . with friction , which dissipates energy at low wavenumbers , one expects a stationary inverse energy cascade for strong enough stirring . thus the kolmogorov scaling of the inverse cascade can not be explained by any rg fixed point . an ideal cascade requires a linear function for large , consistent with a pad approximant to the borel transform . the conjecture that the kolmogorov spectrum arises from an rg flow through large is compatible with other results , but the accurate scaling is not explained and the kolmogorov constant is not estimated . the lack of scale invariance should produce intermittency in high - order structure functions , as observed in some but not all numerical simulations of the inverse cascade . | a two - dimensional fluid , stirred at high wavenumbers and damped by both viscosity and linear friction , is modeled by a statistical field theory . the fluid s long - distance behavior is studied using renormalization - group ( rg ) methods , as begun by forster , nelson , and stephen [ phys . rev . a * 16 * , 732 ( 1977 ) ] . with friction , which dissipates energy at low wavenumbers , one expects a stationary inverse energy cascade for strong enough stirring . while such developed turbulence is beyond the quantitative reach of perturbation theory , a combination of exact and perturbative results suggests a coherent picture of the inverse cascade . the zero - friction fluctuation - dissipation theorem ( fdt ) is derived from a generalized time - reversal symmetry and implies zero anomalous dimension for the velocity even when friction is present . thus the kolmogorov scaling of the inverse cascade can not be explained by any rg fixed point . the function for the dimensionless coupling is computed through two loops ; the term is positive , as already known , but the term is negative . an ideal cascade requires a linear function for large , consistent with a pad approximant to the borel transform . the conjecture that the kolmogorov spectrum arises from an rg flow through large is compatible with other results , but the accurate scaling is not explained and the kolmogorov constant is not estimated . the lack of scale invariance should produce intermittency in high - order structure functions , as observed in some but not all numerical simulations of the inverse cascade . when analogous rg methods are applied to the one - dimensional burgers equation using an fdt - preserving dimensional continuation , equipartition is obtained instead of a cascade in agreement with simulations . |
1109.6555 | i | understanding how surfaces may constrain the configuration of biopolymers on mesoscopic scales is important in a number of processes in cell biology . it is particularly relevant in the packing of dna within viral envelopes @xcite . modeling all of the relevant interactions is complicated : one needs to accommodate the competition between polymer elasticity and entropy ; in the case of dna , electric fields will be important . a nice short review in this context is provided in @xcite . recently , simulations treating various features of the confinement process have been performed @xcite ; the former focuses on entropy , the latter on the dramatic effects of friction and the finite transverse dimensions of the confined object . in this paper , we address an aspect of the problem of a fundamental nature that does not appear to have been addressed previously in any detail : how does one characterize the equilibrium states of the three - dimensional elastic energy of a polymer confined within a fixed surface ? while this description of confinement leaves out a lot of the physics that is relevant in biological systems , it presents a well - defined problem exhibiting a striking level of complexity that is worth studying in its own right . the semi - flexible polymer will be modeled as a curve in three - dimensional space parameterized by arc - length @xmath11 . the bending energy associated with a given conformation is quadratic in the frenet - serret curvature along the curve , @xmath12 , @xmath13 curves of fixed length minimizing the unconstrained energy ( [ hamk2 ] ) were first studied in depth by euler . a historical review is provided in ref . @xcite ; a more contemporary approach to the problem is presented in @xcite and reviewed in ref . an alternative framework , which lends itself to adaptation to the confinement problem , was developed in ref . @xcite . in the absence of constraints on a closed loop , there is not a lot to say : the only stable equilibrium is a circle . confinement within a surface , however , will generally oblige the loop to adopt a non - circular shape , increasing its elastic energy . the contact with the surface itself may be complete or it may be partial ; and contrary to one s initial guess , the bound state will not generally follow a surface geodesic ; nor need it be unique . the wrapping of a semi - flexible polymer around a cylinder , a closely related problem relevant in the the winding of dna around histone octamers , was first examined some time ago by nickerson and manning @xcite . their strategy was to look at the independent degrees of freedom of the surface - bound polymer . while focusing directly on these degrees of freedom makes sense , it does not exploit the symmetries of the problem . for , even though the constraint breaks the euclidean invariance of the three - dimensional bending energy , how this occurs is not arbitrary . a variational framework is developed here that involves the unconstrained degrees of freedom , imposing the constraint using a local lagrange multiplier . this multiplier will quantify the loss of euclidean invariance in the constrained system . its value at any point along the loop will be identified as the local normal force that is being transmitted to the surface . the well - known integrability of the euler - lagrange equations for the unconstrained curve is a consequence of the euclidean invariance of its energy . the constrained counterpart generally will not be integrable . in various interesting cases , however , the confining geometry will respect some subgroup of the euclidean group . in particular , the conservation of torque associated with the residual rotational invariance of a sphere permits the euler - lagrange equation to be cast as a quadrature in the geodesic curvature which can be integrated directly . it also provides a direct recipe for the reconstruction of the loop from its curvature data adapted to the conserved torque vector as an axis of symmetry . in particular , we apply this framework to examine the confinement of a closed polymer loop of fixed length @xmath0 within a sphere of radius @xmath14 . in contrast to an open polymer which will wind around a geodesic circle on the sphere when its length exceeds @xmath15 , the closure of the loop is incompatible with a geodesic unless @xmath9 is tuned to be commensurate with an integer multiple of @xmath1 . we show that there exists an infinite number of distinct completely attached states , labeled by a pair of integers , @xmath6 and @xmath10 : @xmath16 , @xmath17 , the number of periods of the polar angle and azimuthal angles in one circuit of the loop respectively . @xmath6 characterizes the dihedral symmetry with respect to the axis of symmetry ; @xmath10 characterizes the number of revolutions about this axis . small loops oscillate symmetrically about a geodesic circle with @xmath5 and an @xmath6-fold symmetry , @xmath18 . the two - fold is the stable ground state . for any finite values of @xmath9 , the higher @xmath6-folds are unstable with respect to decay toward the two - fold . states with @xmath4 are illustrated in figs . [ figure1](a ) - [ figure1](c ) for various normalized values of @xmath9 . + + in the interval @xmath19 $ ] within a unit sphere : ( a ) and ( b ) display increasingly large oscillations about the equator ; ( c ) oscillations develop overhangs ; ( d ) first self - contacts made at the two poles ; ( e ) self - intersecting triple orbit of sphere ; ( f ) orbit collapses to triple cover of the equator . the normalized local confining force @xmath20 is color coded in these figures.,title="fig : " ] beyond some critical size , the loop will exhibit self - intersections on the sphere . when @xmath4 this will occur when @xmath21 where the loop crosses the poles as illustrated in fig . [ figure1](d ) . we suppose that self - intersections are consistent with the physics and do not cost energy . as @xmath9 is increased to @xmath22 , the loop will collapse onto a geodesic circle which it will cover three times , ( see figs . [ figure1](e ) and [ figure1](f ) . the discontinuity in the number of revolutions occurs at an intermediate value of @xmath9 where the loop crosses the poles . this is accompanied by a transition from oscillatory to orbital behavior . as @xmath9 is increased above @xmath23 , the dihedral symmetry of the lowest energy state descending from the @xmath4 ground state will jump to @xmath24 with the reestablishment of oscillatory behavior . the bending energy of this state as well as the total force that gets transmitted to the surface will be determined as a function of @xmath9 . the three- dimensional bending energy decomposes on the surface into a sum of two terms : one is associated with the geodesic curvature , intrinsic to the sphere ; the other is associated with the normal curvature inherited from the surface , constant on a sphere . the latter energy thus counts the number of times the loop is wound within the sphere and it is proportional to @xmath9 . the geodesic energy periodically falls to zero whenever @xmath25 and the loop is geodesic . local maxima , associated with the incommensurability of the loop length with geodesic behavior , are displayed between these values ; we show that their values decrease monotonically with loop size . for large values of @xmath9 , the normal contribution is always dominant . it does not depend on the state in question thus , in this limit , the energy gap between the ground state and excited states disappears . the local force transmitted by the completely attached ground state loop is positive everywhere . surprisingly , the total force does not grow monotonically with loop length except asymptotically , where it grows linearly with @xmath9 and coincides with the naive expression , energy divided by @xmath1 . the change of symmetry as @xmath9 passes through odd integral multiples of @xmath1 , manifests itself in a positive jump in the transmitted force analogous to a euler instability associated with the buckling into oscillations . when @xmath7 , a new set of states appears as oscillations about a doubly covered geodesic circle , @xmath26 with an @xmath6-fold symmetry , @xmath27 ; among these new states , the lowest energy is displayed in the three - fold . also , because the geodesic energy is small when the oscillations are small , these states will have lower energy than the two - fold ground state in a finite band of values of @xmath9 beginning at @xmath28 . there will be similar behavior in an infinite set of bands of values of @xmath9 where the descendant states become geodesic . one thus needs to reassess the stability of the two - fold ground state and its descendants . we provide a heuristic argument for stability by constructing a homotopy that interpolates between the states with @xmath4 , @xmath5 and @xmath29 , @xmath26 . by examining the energy along this homotopy , we show that a steep energy barrier separates the two equilibrium states . the @xmath4 , @xmath5 state and its descendants thus appear to be stable classically . for any @xmath7 , there are two stable states that alternate as the ground state as the length is increased . the paper is organized as follows : in section [ confinedelastica ] , we describe the framework . in particular , the breaking of translational invariance will be quantified by the non - conservation of a vector along the loop . it will be shown that the euler - lagrange equation for the curve can be cast as the vanishing of a linear combination of the unconstrained euler - lagrange derivatives . the constraining force is identified as some other linear combination of these derivatives . in sect . [ spherical ] the confinement of a closed loop by spheres will be considered . in secs . [ weak ] and [ strong ] we analyze loops in the harmonic and non - linear regimes respectively . the equilibrium states of the loop will be identified and the forces they transmit to the surface determined . an assessment of the stability of these states will be provided . we conclude with a discussion and a few suggestions for future work in sec . [ discussion ] . a number of useful definitions , identities and derivations are collected in a set of appendixes . | a variational framework is developed to examine the equilibrium states of a semiflexible polymer that is constrained to lie on a fixed surface . as an application the confinement of a closed polymer loop of fixed length within a spherical cavity of smaller radius , , is considered . it is shown that an infinite number of distinct periodic completely attached equilibrium states exist , labeled by two integers : and , the number of periods of the polar and azimuthal angles respectively . small loops oscillate about a geodesic circle : , is the stable ground state ; states with higher exhibit instabilities . if new states appear as oscillations about a doubly covered geodesic circle ; the state replaces the two - fold as the ground state in a finite band of values of . with increasing , the contribution to energy from geodesic curvature is bounded . in large loops , the energy becomes dominated by a state independent contribution proportional to the loop size ; the energy gap between the ground state and excited states disappears . _ instituto de ciencias nucleares , universidad nacional autnoma de mxico + apdo . postal 70 - 543 , 04510 mxico , df , mexico + _ | a variational framework is developed to examine the equilibrium states of a semiflexible polymer that is constrained to lie on a fixed surface . as an application the confinement of a closed polymer loop of fixed length within a spherical cavity of smaller radius , , is considered . it is shown that an infinite number of distinct periodic completely attached equilibrium states exist , labeled by two integers : and , the number of periods of the polar and azimuthal angles respectively . small loops oscillate about a geodesic circle : , is the stable ground state ; states with higher exhibit instabilities . if new states appear as oscillations about a doubly covered geodesic circle ; the state replaces the two - fold as the ground state in a finite band of values of . with increasing , loop states make a transition from oscillatory and orbital behavior on crossing the poles , returning to oscillation upon collapse to a multiple cover of a geodesic circle ( signaled , respectively , by an increase in and an increase in ) . the force transmitted to the surface does not increase monotonically with loop size , but does asymptotically . it behaves discontinuously where changes . the contribution to energy from geodesic curvature is bounded . in large loops , the energy becomes dominated by a state independent contribution proportional to the loop size ; the energy gap between the ground state and excited states disappears . _ instituto de ciencias nucleares , universidad nacional autnoma de mxico + apdo . postal 70 - 543 , 04510 mxico , df , mexico + _ |
0901.3835 | i | a well - known consequence of the ergodic decomposition theorem is that the space of invariant probability measures of a topological dynamical system , endowed with the weak@xmath0 topology , is a metrizable choquet simplex .. ] the purpose of this paper is to show that every non - empty metrizable choquet simplex arises in this way within the _ logistic family _ of maps @xmath4}$ ] , where for each parameter @xmath5 in @xmath6 $ ] the _ logistic map _ @xmath7 \to [ 0 , 1]$ ] is defined by , @xmath8 to make a more precise statement note that @xmath9 is the unique point in @xmath10 $ ] at which the derivative of @xmath11 vanishes . we call @xmath9 the _ critical point _ of @xmath12 , and its @xmath1-limit set is called the _ post - critical set _ of @xmath11 . it is a compact set that is forward invariant by @xmath11 . the following is our main result . recall that for a compact topological space @xmath13 a continuous map @xmath14 is _ minimal _ , if every forward orbit of @xmath15 is dense in @xmath13 . * main theorem . * _ for each non - empty metrizable choquet simplex @xmath16 there is a parameter @xmath17 $ ] such that the post - critical set of the logistic map @xmath11 is a cantor set , the restriction of @xmath11 to this set is minimal , and such that the space of invariant probability measures supported by this set , endowed with weak@xmath0 topology , is affine homeomorphic to @xmath16 . _ the first result of this kind was shown by downarowicz in @xcite , who showed that every non - empty metrizable choquet simplex arises , up to an affine homeomorphism , as the space of invariant probability measures of a `` minimal cantor system '' ; that is , a dynamical system generated by a minimal homeomorphism of a cantor set . in fact he showed that the minimal cantor system can be taken as a `` 0 - 1 toeplitz flow '' : a special type of subshift of @xmath18 . see also @xcite for a different approach to this result , and see @xcite for an analogous result in the case of actions of @xmath19 . the following corollary is a direct consequence of the main theorem and of the fact that for each non - empty polish space @xmath20 there is a metrizable choquet simplex whose set of extreme points is homeomorphic to @xmath20 , see for example @xcite . [ c : extreme realization ] for each non - empty polish space @xmath20 there is a parameter @xmath17 $ ] such that the post - critical set of the logistic map @xmath11 is a cantor set , the restriction of @xmath11 to this set is minimal , and such that the space of ergodic and invariant probability measures supported by this set , endowed with weak@xmath0 topology , is homeomorphic to @xmath20 . the special case where the polish space @xmath20 is compact and totally disconnected is precisely ( * ? * main theorem ) . the first result in this direction was shown by bruin , who gave an example of a parameter @xmath17 $ ] such that the post - critical set of @xmath12 is a cantor set where @xmath12 is minimal , but not uniquely ergodic ( * ? ? ? * theorem 4 ) . the proof of the main theorem is based on the tools developed by bruin in @xcite , and by bruin , keller and st . pierre in @xcite . one of the interesting features of the main theorem , in constrast with the other realization results mentioned above , is that the systems we consider have a natural differentiable structure . it turns out that , for the parameters @xmath17 $ ] given by ( the proof of ) the main theorem , the invariant measures supported by the post - critical set of @xmath12 correspond precisely to those invariant measures @xmath21 of @xmath12 whose lyapunov exponent @xmath22 vanishes ( * ? ? ? * lemma 21 ) . it also turns out that every invariant probability measure supported on the post - critical set of @xmath12 is an `` equilibrium state of @xmath12 for the potential @xmath23 '' . that is , if for each invariant measure @xmath21 we denote by @xmath24 its measure theoretic entropy , then the supremum @xmath25 is attained at each invariant probability measure supported by the post - critical set of @xmath12 , see ( * ? ? ? * lemma 21 ) . we thus obtain the following corollary of the main theorem . [ c : ergodic theory interval ] for each non - empty metrizable choquet simplex @xmath16 there is a parameter @xmath17 $ ] verifying the conclusions of the main theorem , and such that in addition the space of invariant probability measures of @xmath12 ( resp . equilibrium states of @xmath12 for the potential @xmath26 ) that are of zero lyapunov exponent , endowed with the weak@xmath0 topology , is affine homeomorphic to @xmath16 . this result is in sharp contrast with the fact that for a logistic map there can be at most one ergodic equilibrium state whose lyapunov exponent is strictly positive . for future reference we state an holomorphic version of corollary [ c : ergodic theory interval ] , shown in appendix [ a : ergodic theory ] . for a complex parameter @xmath27 denote by @xmath28 the quadratic polynomial defined by @xmath29 viewed as a dynamical system acting on @xmath30 . [ c : ergodic theory holomorphic ] for each non - empty metrizable choquet simplex @xmath16 there is a parameter @xmath17 $ ] verifying the conclusions of the main theorem , and such that in addition , if we denote by @xmath31 the hausdorff dimension of the julia set of @xmath28 , then the space of invariant probability measures of @xmath28 ( resp . equilibrium states of @xmath28 for the potential @xmath32 ) that are of zero lyapunov exponent , endowed with the weak@xmath0 topology , is affine homeomorphic to @xmath16 . we end this introduction by stating some questions that arise naturally from the main theorem . to do this , for each @xmath17 $ ] we will denote by @xmath33 the post - critical set of @xmath12 . given a compact metrizable topological space @xmath13 and a continuous map @xmath14 , the main theorem implies that there is a parameter @xmath17 $ ] and an affine homeomorphism @xmath34 between the space of invariant probability measures supported on @xmath33 . it is thus natural to ask whether the parameter @xmath17 $ ] and @xmath34 can be chosen in such a way that @xmath34 is induced by a continuous map between @xmath33 and @xmath13 . more precisely , the question is if @xmath17 $ ] and @xmath34 can be chosen in such a way that there is a continuous map @xmath35 such that for each invariant probability measure @xmath21 supported by @xmath33 we have @xmath36 . this type of problem is very well understood in the setting of minimal cantor systems : giordano , putnam , and skau have shown in @xcite that for two minimal cantor systems @xmath37 and @xmath38 there exists a homeomorphism @xmath39 that induces an affine homeomorphism between the corresponding spaces of invariant probability measures , if , and only if , @xmath37 and @xmath40 are `` orbit equivalent '' : there is a homeomorphism between @xmath13 and @xmath41 mapping each orbit of @xmath15 to an orbit of @xmath42 . furthermore , to each minimal cantor system @xmath37 one can associate a dimension group that is a complete invariant for the orbit equivalence relation @xcite : two minimal cantor systems are orbit equivalent if , and only if , the corresponding dimension groups are isomorphic as ordered groups with unit . and taking values in @xmath43 , by the subgroup of functions whose integral with respect to each invariant measure vanishes ; the positive cone is the set of those classes containing a function taking values in @xmath44 , and the unit is the class of the constant function equal to 1 . see for example ( * ? ? ? * theorem 1.13 ) . ] it is thus natural to look for a special class of minimal cantor systems realizing all of the orbit equivalent classes . since the dimension group associated to each toeplitz flow contains the dimension group of an odometer as a subgroup ( * ? ? ? * section 4.1 ) , the class of toeplitz flows is not sufficient to realize all orbit equivalence classes , in spite of the fact that this class realizes all the non - empty metrizable choquet simplices as sets of invariant probability measures @xcite . so the question remains whether minimal post - critical sets of logistic maps realize all orbit equivalence classes . in order to formulate a precise question we will consider natural extensions to stay in the class of minimal cantor systems , and use the generalized odometer associated to a kneading map , see [ ss : unimodal ] , [ ss : kneading odometer ] for definitions . does every orbit equivalence class contain the natural extension of a generalized odometer associated to a kneading map ? does every uniquely ergodic orbit equivalence class contain the natural extension of a generalized odometer associated to a kneading map ? it is well - known that every odometer can be realized , up to a homeomorphism , as the post - critical set of an infinitely renormalizable logistic map , see also @xcite . in [ ss : example ] we give an example of a uniquely ergodic generalized odometer associated to a kneading map , whose natural extension is not orbit equivalent to an odometer , nor to a toeplitz flow . although corollary [ c : extreme realization ] is stronger than ( * ? ? * main theorem ) , we use this last result in the proof of the main theorem to deal with case of finite dimensional choquet simplices . we have stated the main theorem and corollary [ c : extreme realization ] for the logistic family for simplicity . we show that an analogous statement holds for each full family of unimodal maps , as well as for the family of symmetric tent maps . see [ ss : unimodal ] for definitions . in fact , for each infinitely dimensional metrizable choquet simplex we construct kneading map @xmath45 such that the conclusions of the main theorem hold for each unimodal map whose kneading map is @xmath45 , see [ s : doubly resonant kneading ] . furthermore the kneading map satisfies for every @xmath46 the inequality @xmath47 ( part 1 of lemma [ l : doubly resonant kneading ] ) , and therefore every full family of unimodal maps , as well as the family of symmetric tent maps , contains a unimodal map whose kneading map is @xmath45 . for the case of finite dimensional choquet simplices see ( * ? ? ? * remark 1 ) . similarly , it follows from ( * ? ? ? * lemma 21 ) that corollary [ c : ergodic theory interval ] holds for every full family of -unimodal maps . see @xcite for a survey on realization results concerning toeplitz flows . see @xcite for the realization of some concrete simplices as the space of invariant measures of minimal cantor systems . in this section we explain the strategy of the proof of the main theorem and simultaneously describe the organization of the paper . we only deal with infinitely dimensional choquet simplices , the finite dimensional case being covered by ( * ? ? ? * main theorem ) . we use a result of lazar and lindenstrauss that characterizes infinite dimensional metrizable choquet simplices as inverse limits of stochastic matrices , see theorem [ t : choquet as inverse limit ] in [ ss : choquet simplices ] . we describe the logistic maps in the main theorem through their associated `` kneading map '' , see [ ss : unimodal ] for the definition of kneading map and further background on unimodal maps . in fact , the conclusions of the main theorem are valid for each unimodal map having the same kneading map as @xmath12 . to ensure that the post - critical set is a cantor set where the unimodal map is minimal , it is enough to require that the kneading map diverges to @xmath48 ( proposition [ p : continuous unimodal ] ) . in [ ss : doubly resonant kneading ] we introduce a class of kneading maps that diverge to @xmath49 . in [ ss : proof of main theorem ] we state a result describing the space of invariant measures supported on the post - critical set of a unimodal map with a kneading map in this class ( theorem [ t : invariant of doubly resonant ] ) . in [ ss : proof of main theorem ] we also give a proof of the main theorem assuming theorem [ t : invariant of doubly resonant ] . in [ ss : kneading odometer ] we recall the definition and some properties of the generalized odometer associated to a kneading map , that was introduced in @xcite . in [ ss : measure isomorphism ] we show that for a unimodal map whose kneading map @xmath45 is as in [ ss : doubly resonant kneading ] , the space of invariant probability measures supported by the post - critical set is affine homeomorphic to that of the generalized odometer associated to @xmath45 . in turn , this space is affine homeomorphic to the corresponding space of the bratteli - vershik system associated to @xmath45 , introduced in @xcite ; see ( * ? ? ? * proposition 2 ) or theorem [ t : reduction to bratteli ] in [ ss : kneading bratteli - vershik ] . the advantage of this last space is that it can be described explicitly as an inverse limit of some `` transition matrices '' , see [ ss : bratteli - vershik ] . we calculate the transition matrices and some of their products in [ ss : transition matrices ] , and give the proof of theorem [ t : invariant of doubly resonant ] in [ ss : proof of invariant of doubly resonant ] . in [ ss : example ] we give an example of a uniquely ergodic generalized odometer associated to a kneading map whose natural extension is not orbit equivalent to a toeplitz flow . in appendix [ a : ergodic theory ] we give the proof of corollary [ c : ergodic theory holomorphic ] . we are grateful with christian skau for his help with dimension groups of toeplitz flows , and an anonymous referee for his / her suggestions to improve the presentation of the paper . | a well - known consequence of the ergodic decomposition theorem is that the space of invariant probability measures of a topological dynamical system , endowed with the weak topology , is a non - empty metrizable choquet simplex . we show that every non - empty metrizable choquet simplex arises as the space of invariant probability measures on the post - critical set of a logistic map . here , the post - critical set of a logistic map is the-limit set of its unique critical point . | a well - known consequence of the ergodic decomposition theorem is that the space of invariant probability measures of a topological dynamical system , endowed with the weak topology , is a non - empty metrizable choquet simplex . we show that every non - empty metrizable choquet simplex arises as the space of invariant probability measures on the post - critical set of a logistic map . here , the post - critical set of a logistic map is the-limit set of its unique critical point . in fact we show the logistic map can be taken in such a way that its post - critical set is a cantor set where is minimal , and such that each invariant probability measure on this set has zero lyapunov exponent , and is an equilibrium state for the potential . logistic map , post - critical set , invariant measures , choquet simplices , minimal cantor system , generalized odometer 37e05 , 37a99 , 37b10 , 54h20 |
1604.00646 | i | electrons accelerate and radiate radio emission either when they pass near charged particle , or when they propagate in a magnetic field . in the earlier case , the free - free spectral slope @xmath14 is @xmath15 from the radio to @xmath16 . the thermal free - free emission is not viable in objects with a power - law emission with a steeper of a flatter @xmath17 . free - free emission of @xmath18 k gas is also excluded as it over predicts the observed x - ray luminosity ( e.g. lb08 , section 3.6.1 there ) . free - free is a viable mechanism if the gas is cooler than @xmath19k , but this is generally ruled out by the vlbi limits of @xmath11 k on the core emission . thus , electrons in a magnetic field is the only generally viable mechanism . below we briefly review the emission of relativistic electrons with a power - law ( pl ) energy distribution ( synchrotron emission ) , thermal electrons ( cyclotron emission ) , or mildly relativistic thermal electrons ( cyclo - synchrotron emission , hereafter thermal synchrotron emission ) . we provide expressions for the emission , absorption , and radiation transfer used to derive the observed emission . an electron in a magnetic field @xmath20 gyrates around the field lines at a frequency @xmath21 where @xmath22 is the electron charge , @xmath23 is the mass of the electron , @xmath24 is the lorentz factor , and @xmath25 , where @xmath26 is the velocity of the electron . non relativistic electrons ( @xmath27 ) radiate cyclotron emission at @xmath28 . for mildly relativistic electrons ( @xmath29 ) , additional harmonics appear and the emitted spectrum becomes polychromatic with a few additional lines at higher harmonics . as the electrons become more relativistic ( @xmath30 ) , higher harmonics appear . the width of each line becomes wider with increasing harmonic number , and finally for high enough value of @xmath31 and harmonic number @xmath32 , adjacent harmonics merge and a continuous spectrum is generated . the resonant frequencies occur at @xcite . @xmath33 where @xmath34 is the projection of @xmath35 on the direction of the magnetic field , and @xmath36 is the angle between the magnetic field and the observer . the pitch angle @xmath37 is the angle between the direction of motion of the electron and the magnetic field , which gives @xmath38 equation [ eq : omega_s ] is correct when the refraction index of the medium is unity , which is a good approximation for agn coronae , as the frequencies of interest are well above the gyration and langmuir frequencies ( @xmath39 for typical condition assumed ) . we have verified this numerically by solving for the emission spectrum including the corona refractive index . we used zheleznyakov ( 1970 , eqs . 26 & 35 there ) for the emitted energy per harmonic number per solid angle ( erg sec@xmath8 strd@xmath8 ) . the specific emission power ( erg sec@xmath8 strd@xmath8 frequency@xmath8 ) for a unity refraction index is @xcite @xmath40}{2\pi c}\\ \times\delta[s\omega_{\rm 0}-\omega(1-{\beta_{\rm ||}}\cos\alpha ) ] , \label{psingle } \end{gathered}\ ] ] where @xmath41 is the continuum angular frequency , @xmath42 is a bessel function of order @xmath32 , @xmath43 is the first derivative of the bessel function of order @xmath32 , and @xmath44 is @xmath45 equation [ psingle ] provides the emission spectrum of a single electron , with kinetic energy of @xmath46 , propagating at a pitch angle of @xmath37 to the magnetic field . in order to get an expression for the emission per unit frequency per unit volume , i.e. the emission coefficient @xmath47 , we need to multiply eq.[psingle ] by @xmath48 , the number of electrons with velocities in the range of @xmath49 $ ] and a pitch angle in the range of @xmath50 $ ] , and integrate over @xmath51 . we then integrate over @xmath37 , and @xmath36 , by assuming a uniform distribution in both angles , i.e. a random distribution of both the electron velocities and the magnetic field line directions in a given volume , @xmath52 in units of erg sec@xmath8 @xmath53 hz@xmath8 , where @xmath54 is the emission of a single electron . in the next sections we will use @xmath55 and @xmath56 an expression for the absorption coefficient is given by rybicki & lightman ( 2004 , hereafter rl04 , eq . 6.50 there ) . we modify the equation to include the dependence on pitch angle @xmath37 , which gives @xmath57 where e is the kinetic energy of the electron , and @xmath58 is the density of electrons per unit energy per pitch angle . to calculate the radiative transfer we divide the medium into @xmath59 unit volumes ( see below the assumed geometry ) of uniform emission and absorption coefficients , say @xmath60 and @xmath61 . if a ray with an intensity @xmath62 enters this unit volume , and travels a distance of @xmath63 within this volume , then the intensity of the exiting ray is @xmath64 where the first term represents the absorption of the incident intensity , and the second one is the contribution of the emission inside the volume . we assume an optically thick accretion disc which extends from the innermost stable orbit @xmath65 at @xmath66 where @xmath67 is the schwarzschild radius , @xmath68 @xcite . we use @xmath69 for the black hole ( bh ) mass . for the outer boundary of the disc we use @xmath70 pc , which corresponds to @xmath71 . the observed uv spectral energy distribution suggests a maximal temperature @xmath72 @xcite , which commonly implies a thin disc with an inner radius @xmath73 . therefore , in section [ sec : radio_parameters ] we study the effect of a larger @xmath65 on the radio emission spectrum . we assume a blanket - like corona that covers the accretion disc , with a uniform thickness of @xmath74 . the thermal electrons density is assumed to scale with radius as @xmath75 we generally assume @xmath76 @xmath53 , in order to obtain an optical depth for electron scattering of @xmath77 at @xmath78 , as suggested by the x - ray emission ( see paper ii ) . in section 5.3 we explore the dependence of the radio spectrum on the value of @xmath79 . in addition to the thermal electrons , we assume that the corona also has non thermal electrons with a pl energy distribution . the pl distribution may be generated during reconnection events , which may also set the temperature of the thermal component . the energy distribution of the pl and thermal populations are @xmath80 where @xmath81 is a normalization constant , @xmath82 is the power index of the pl distribution ( typically @xmath83 or 3 , see section [ sec : radio_parameters_pl ] ) , and @xmath84 is the normalized temperature @xmath85 , and @xmath86 is the modified bessel function of the second kind . the expression for @xmath87 is the maxwell - juttner distribution , which is the relativistic form of maxwell boltzmann . the energy density of the two populations is assumed to be in equipartition , i.e. @xmath88 which is used to derive the value of @xmath81 . the electron temperature in the corona is generally assumed to be @xmath89 , at all radii . this temperature is naturally expected for a two temperature corona @xcite , for a corona heated by magnetic reconnection @xcite , and also for a pair plasma corona @xcite . in addition , recent _ nustar _ hard x - ray spectroscopy of agn @xcite indeed provide direct evidence for a corona with @xmath90 , i.e. @xmath13k . however , we also explore below the effect of using a lower @xmath91 . we assume that the local value of the magnetic field within the corona is in equipartition with the thermal electrons , i.e. @xmath92 in the numerical calculations we also explore deviations from equipartition , in addition to exploring the effects of different values for @xmath93 , @xmath79 , @xmath82 , and @xmath94 . for this configuration , we assume a spherical symmetry around the bh . the corona extends for a range of radii @xmath95 . the temperature is fixed at @xmath96k , and @xmath97 is given by eq.[eq : nradial ] . the synchrotron emission of both thermal and pl distributions are calculated as described in section [ sec : numerical_scheme ] . | accretion discs in agn may be associated with coronal gas , as suggested by their x - ray emission . we calculate the coronal properties required to produce the observed radio emission in rq agn , either from synchrotron emission of power - law ( pl ) electrons , or from cyclo - synchrotron emission of hot mildly - relativistic thermal electrons . quasars : general . | accretion discs in agn may be associated with coronal gas , as suggested by their x - ray emission . stellar coronal emission includes radio emission , and agn corona may also be a significant source for radio emission in radio quiet ( rq ) agn . we calculate the coronal properties required to produce the observed radio emission in rq agn , either from synchrotron emission of power - law ( pl ) electrons , or from cyclo - synchrotron emission of hot mildly - relativistic thermal electrons . we find that a flat spectrum , as observed in about half of rq agn , can be produced by corona with a disc or a spherical configuration , which extends from the innermost regions out to a pc scale . a spectral break to an optically thin power - law emission is expected around 300 - 1000 ghz , as the innermost corona becomes optically thin . in case of thermal electrons , a sharp spectral cutoff is expected above the break . the position of the break can be measured with vlbi observations , which exclude the cold dust emission , and it can be used to probe the properties of the innermost corona . assuming equipartition of the coronal thermal energy density , the pl electrons energy density , and the magnetic field , we find that the energy density in a disc corona should scale as , to get a flat spectrum . in the spherical case the energy density scales as , and is of the agn radiation energy density . in paper ii we derive additional constraints on the coronal parameters from the gudel - benz relation , , which rq agn follow . [ firstpage ] galaxies : active quasars : absorption lines quasars : general . |
1604.00646 | r | below we present the results of the numerical calculations of the radio emission . the calculations are for the disc and spherical geometries , and for thermal and pl electrons . in all cases we assume an isothermal corona , and a decreasing @xmath188 , i.e. @xmath241 , and @xmath242 ( eq.[eq : tnb_on_r ] ) . the value of @xmath221 is derived from the assumption of equipartition with the thermal electrons ( eq.[eq : mf_eq ] ) . the pl electrons are also assumed to be in equipartition with the thermal electrons ( eq.[eq : pl_eq ] ) . the innermost radius of the corona is assumed to be the last stable orbit for a central bh of a mass of @xmath243 ( eqs.[eq : r0 ] , [ eq : sch_rad ] ) , unless noted otherwise . the outermost radius is taken to be @xmath244 for the thermal distribution , and @xmath245 for the pl electrons , which ensures the spectral break due to transition to optically thick emission occurs at @xmath246 ghz ( see below ) . as noted above , the coronal thickness in the disc configuration is assumed to be constant at @xmath74 . we concentrate below on models which yield a flat spectral slope , as this is the unique signature of a compact emission source , such as a disc corona . as shown below , for some parameters the optically thin emission from the outer parts of the corona , can dominate the emission from the inner parts . in such a case , the overall spectrum is just that of an optically thin source , which is derived also in other more extended emission models . figures [ fig : pl_rings ] and [ fig : thermal_rings ] present typical spectra of thermal and pl distributed electrons within a disc or spherical corona . the values of the density distribution @xmath79 parameters were selected to produce a flat spectrum ( @xmath247 ) in the intermediate frequency range ( 1 - 100 ghz ) , before reaching the steeply falling optically thin limit at higher frequencies , and the steeply rising optically thick emission at lower frequencies . the figures also present the contributions to the total emission of different rings or spherical shells . the emission of thermal electrons from a given ring or shell ( fig.5 ) , covers a smaller range of frequencies , compared to the emission of the pl electrons ( fig.4 ) . both electron distributions produce similar spectral slopes in the locally optically thick emission at @xmath248mhz ( 2 for thermal , 2.5 for pl in a disc ) , but in the locally optically thin part the thermal emission shows a sharp exponential drop , while the pl show only the gradual optically thin falloff with a slope of @xmath249 . the sharp cutoff of the optically thin thermal synchrotron results from the exponential drop in the electron energy ( eq.14 ) , together with the exponential drop in the emission of a single electron ( eq.6.34b in rl04 ) . the slope of the locally optically thick emission in the spherical configuration is steeper than in the disc configuration , as the emission is absorbed by the outer shells . in contrast with the disc case , where the emission from each ring is directly observed . . [ fig : angles ] figure [ fig : angles ] presents the observed disc emission for pl and thermal electrons as a function of @xmath250 , where @xmath251 the disc inclination angle . the spectrum can be roughly divided into three regimes , with different inclination dependence . at low enough frequencies , the whole disc is optically thick , and the luminosity scales as @xmath250 . at high enough frequencies the whole disc is optically thin , the emission becomes isotropic , and the spectra from all inclinations overlap . at intermediate frequencies , the inner part of the disc is optically thick and an outer part of it is optically thin . the transition radius decreases with increasing frequency , until the whole disc emission becomes optically thin . as a result , at intermediate frequencies we get an intermediate inclination dependence . in the thermal case , the exponential drop in the locally optically thin emission , leads to a negligible contribution from the optically thin isotropically emitting part of the disc . as a result , the optically thick inclination dependence remains @xmath250 to almost the highest frequencies . a peculiar behavior occurs for the thermal case at @xmath252hz at high inclination ( lower panel of fig.[fig : angles ] ) , where the luminosity at @xmath253 becomes comparable , and even exceeds the emission at @xmath254 ( for @xmath255hz ) . this results from the adopted geometry , as the highest frequency thermal emission comes mostly from the innermost ring , where the height @xmath74 becomes larger than the radius . most of the emitting area is now in the walls , and the emission becomes stronger at high inclinations , where the projected area of the walls becomes largest . thermal synchrotron emission therefore provides a sensitive measure of the projected surface area of the region where the observed frequency is produced . figure [ fig : pl_parameters ] presents the emission of pl electrons from a disc ( left panels ) or a sphere ( right panels ) . in all cases we assume a pl index @xmath256 , and corona with @xmath120k with a constant thickness , which extends from @xmath257 to @xmath258 for @xmath69 ( which corresponds to @xmath259 cm ) . the upper panels present the dependence of the spectrum on the coronal thermal energy density @xmath260 , where we assume an equipartition with the pl electrons energy density , and the magnetic energy density . as @xmath260 increases , i.e. the normalization of @xmath97 increases , the luminosity increases , as does the transition frequency from optically thick emission ( the 2.5 slope ) to a flat continuum . note also that the disc luminosity is larger than the sphere luminosity by about an order of magnitude . this results from the steeper @xmath261 required in the spherical case , compared to @xmath262 in the disc case , to derive the same flat continuum slope , using the same @xmath93 in both geometries . the second row of panels presents the dependence on @xmath79 ( eq.[eq : nradial ] ) . the values of @xmath79 , noted near each curve , were selected to produce spectral slopes of @xmath263 and @xmath264 for a disc , and @xmath265 and @xmath264 for a sphere , based on the analytical expressions given in eq.[eq : slope_disc_pl ] and eq.[eq : slope_sphere_pl ] . the slopes derived from the numerical solutions are @xmath266 for the disc , and @xmath267 for the sphere , which implies that the analytic estimate is typically accurate to better than 0.1 . the overall trend is that a larger @xmath79 yields a flatter slope , as expected as a larger drop in the density yields a larger drop in the synchrotron emission with radius . since the outer radius contributes at lower frequencies , a higher @xmath79 implies the inner region dominates , leading to a flatter spectral slope . if @xmath79 is low enough ( 0.9 for a disc , 1.7 for a sphere ) , the emission from the outer disc becomes dominant enough , that the optically thin tail it produces at higher frequencies dominates the emission from the inner disc . the observed emission is effectively all produced by emission from a single uniform emitter at the outer disc , rather than by a superposition of emitters at all radii . the spectrum is then a steeply rising optically thick spectrum at low enough frequencies , has a peak where the outer disc becomes optically thin , and shows an optically thin tail at high frequencies , somewhat similar to the spectra observed in ghz peaked radio loud sources @xcite . the @xmath79 values in the spherical case are steeper than in the disc case , for the same spectral slope . this is expected since the change of the synchrotron emissivity with radius depends on the change in the column density with radius . in the disc case @xmath79 also provides the radial dependence of the column density , while in the sphere case the column density scales as @xmath268 . the flat spectrum in the spherical geometry case implies @xmath261 for both pl and thermal electrons . this implies that the gas thermal energy density @xmath269 is a fixed fraction of the radiation energy density @xmath270 . a bolometric luminosity of @xmath271 erg s@xmath8 corresponds to @xmath272 erg s@xmath8 hz@xmath8 at 10 ghz , which is produced by a corona with @xmath273 erg @xmath53 at @xmath274 cm for the pl case ( fig.7 upper right panel ) . this implies a fixed energy density ratio of @xmath275 at all radii . the third row of panels in fig.[fig : pl_parameters ] presents the effect of deviation from equipartition , explored by varying @xmath276 from 0.1 to 100 , while the pl electrons remain in equipartition with the thermal plasma energy density . as expected , the luminosity in the intermediate range of frequencies increases with @xmath276 . in the optically thick range ( low frequencies ) , we get the opposite effect as the luminosity decreases with increasing @xmath276 . that happens because the source function , @xmath277 , where @xmath278 is the emission coefficient ( eq.[eq : p_nu_rl ] ) , and @xmath279 is the absorption coefficient ( eq.[eq : abs ] ) , is @xmath280 . some of the disc models lead to spectra which are optically thick ( @xmath281 ) at @xmath282ghz . such optically thick synchrotron emission is not observed in rq agn , even on mas scales . @xcite . thus , according to our model , the synchrotron source needs to extend up to @xmath283 pc to push the spectral turnover to lower frequencies . in contrast , most spherical corona models presented here become optically thick at frequencies well below 1 ghz for the assumed @xmath284 pc . figure [ fig : thermal_parameters ] explores the dependence of thermal synchrotron emission emission on various model parameters , as done in fig.[fig : pl_parameters ] for the pl electrons . here , however , we explore the dependence on @xmath97 and @xmath91 separately , and not just on @xmath260 , as the synchrotron is produced by the thermal plasma , in contrast with the pl electrons where the thermal plasma is used only to set the equipartition energy density of the pl electrons . the thermal plasma luminosity is significantly weaker than the pl luminosity , for the same parameters , as found for coronally active stars @xcite . another general difference from the pl emission is the sharp drop at @xmath285 hz of the thermal synchrotron , in contrast with the transition to optically thin synchrotron with a slope of @xmath249 . the upper panel in fig.8 shows the sharp dependence of the thermal synchrotron luminosity on @xmath91 , where @xmath140 increases by a factor of 725 when @xmath91 increases by a factor of 5 to @xmath286 k. this is expected since @xmath287 ( eq.[eq : our_nupeak_thermal ] ) , while @xmath288 ( eq.[eq : lnu_thermal_1 ] ) , which gives @xmath289 . the numerical solution presented here yields @xmath290 . at low frequencies , where the emission is optically thick , the spectrum is independent of @xmath97 , @xmath79 , and @xmath276 , and depends only on @xmath91 , as expected since the emission becomes a blackbody emission . the dependence of the emission in the flat part of the spectrum on @xmath291 , and @xmath276 , is similar to the one presented by the pl electrons , as discussed above . the value of @xmath256 , used above for the pl electron energy distribution , is motivated by the fermi acceleration mechanism , and is observed in various systems . steeper values are expected when additional electron cooling processes are taken into account ( longair 1994 ) . the optically thin emission for @xmath256 is @xmath292 . however , steeper spectra are commonly observed in rq agn @xcite , which imply larger values of @xmath82 are present . figure [ fig : pl_3 ] compares some of the earlier spectra with the spectra derived for the @xmath293 case . as expected , the spectra are steeper , keeping the other model parameters fixed . alternatively , higher @xmath79 values are required to derive a flat spectral slopes , as can also be seen from the analytical derivation above ( eq.[eq : slope_disc_pl3 ] ) . a major difference is the drop in @xmath140 by about a factor of 100 , for models with a similar spectral slope . this results from the fact that the integrated electron energy in the @xmath217 case is concentrated near @xmath294 , rather than being evenly spread between @xmath294 and @xmath295 per logarithmic bin in @xmath51 , which is the case for @xmath83 . and @xmath293 . as expected , the spectra in the @xmath293 case are steeper . the @xmath79 values required to derive spectral slopes of approximately @xmath296 and @xmath297 are marked near each curve . note the factor @xmath9 drop in the flat @xmath140 for the @xmath293 case , compared to the @xmath256 case ( see text ) . , width=302 ] the inner disc boundary assumed above is @xmath257 , as expected for a viscous accretion disc around a schwarzschild black hole . however , studies of the observed uv spectral energy distribution of agn , together with theoretical arguments , suggest the thin disc solution my break at a few 10 s of @xmath67 @xcite , below which the accretion flow may change its nature ( e.g. become a low radiative efficiency geometrically thick flow ) . if the corona is formed only above the surface of a thin disc , it may not extend down to @xmath298 . for the specific model parameters used here , @xmath299 shining at the eddington luminosity , we get that @xmath300 from the analytic solution in @xcite . figure [ fig : inner_boundary ] shows the effect of increasing @xmath65 on the observed spectrum from a disc corona of pl electrons . for the sake of completeness , the figure also shows the effect of a larger @xmath65 for a spherical corona . as expected , the frequency of the spectral break from a flat slope to an optically thin slope , decreases with increasing @xmath65 , from @xmath301 thz to a few hundred ghz , as the highest emission frequency originates from the smallest radii ( see fig.[fig : inner_boundary ] ) . one can also estimate the change in the break frequency by applying the analytic estimate for @xmath154 for the innermost disc parameters . one can use this break frequency to deduce the properties of the innermost corona . however , the spectral range above @xmath302 ghz is likely dominated by the rayleigh - jeans tail of the coldest dust emission @xcite , which rises extremely steeply and heavily dominates the emission above @xmath302 ghz . the dust emission may be overcome by vlbi observations , which exclude sources with @xmath303 k , and may be able to overcome the strong background dust emission , and detect the expected synchrotron turnover in the emission of a compact mas size source at the centre . on the observed spectral shape from a pl electron distribution . the values of @xmath65 are noted near each curve for the disc and sphere geometries . as expected , the break frequency from a flat pl emission , to the steeper optically thin emission , decreases with increasing @xmath65 . the spectral range at @xmath304 hz is heavily dominated by dust emission , but it should be excluded by vlbi observations , which measure only the @xmath305 k component . this spectral turnover can be used to probe the coronal @xmath65.,width=302 ] | a spectral break to an optically thin power - law emission is expected around 300 - 1000 ghz , as the innermost corona becomes optically thin . in case of thermal electrons , the position of the break can be measured with vlbi observations , which exclude the cold dust emission , and it can be used to probe the properties of the innermost corona . assuming equipartition of the coronal thermal energy density , the pl electrons energy density , and the magnetic field , we find that the energy density in a disc corona should scale as , to get a flat spectrum . in the spherical case the energy density scales as , and is of the agn radiation energy density . in paper | accretion discs in agn may be associated with coronal gas , as suggested by their x - ray emission . stellar coronal emission includes radio emission , and agn corona may also be a significant source for radio emission in radio quiet ( rq ) agn . we calculate the coronal properties required to produce the observed radio emission in rq agn , either from synchrotron emission of power - law ( pl ) electrons , or from cyclo - synchrotron emission of hot mildly - relativistic thermal electrons . we find that a flat spectrum , as observed in about half of rq agn , can be produced by corona with a disc or a spherical configuration , which extends from the innermost regions out to a pc scale . a spectral break to an optically thin power - law emission is expected around 300 - 1000 ghz , as the innermost corona becomes optically thin . in case of thermal electrons , a sharp spectral cutoff is expected above the break . the position of the break can be measured with vlbi observations , which exclude the cold dust emission , and it can be used to probe the properties of the innermost corona . assuming equipartition of the coronal thermal energy density , the pl electrons energy density , and the magnetic field , we find that the energy density in a disc corona should scale as , to get a flat spectrum . in the spherical case the energy density scales as , and is of the agn radiation energy density . in paper ii we derive additional constraints on the coronal parameters from the gudel - benz relation , , which rq agn follow . [ firstpage ] galaxies : active quasars : absorption lines quasars : general . |
1604.00646 | c | we explore the possible radio emission from either a flat or a spherical magnetized corona , powered by either pl or thermal electrons . we concentrate on flat spectra models , which characterize about half of rq agn , and is the main signature of a compact emission region such as a corona . our main conclusions are as follows : \1 . a flat spectral slope in the 1 - 1000 ghz range requires the superposition of synchrotron emission from the innermost region at @xmath298 to @xmath331 ( @xmath346 pc to 3 pc ) . the radio corona at 1 ghz emission comes from the largest scales , and should be resolved in mas resolution vlbi observations of nearby agn ( closer than @xmath347 mpc ) , the few 100 ghz emission corona overlaps in size the x - ray corona , and its size can only be constrained from its variability timescale . \2 . the synchrotron emission at a given frequency , is produced over a wide range of radii for pl electrons . in contrast , the synchrotron emission of thermal electrons at a given frequency , originates from a narrow range of radii . the synchrotron emission from a disc corona of pl electrons is nearly isotropic , as most of the emission originates from the optically thin outer regions . the emission of thermal electrons from a disc corona shows a @xmath348 dependence , as the observed emission is dominated by the outermost optically thick region . a luminosity of @xmath339 erg s@xmath8 hz@xmath8 can be produced by pl electrons , magnetic field , and a corona , which are all in equipartition , with an energy density which scales roughly as @xmath349 , and @xmath350 erg @xmath53 at @xmath298 . in the spherical corona configuration , the energy density scales as @xmath341 . for pl electrons synchrotron , this equipartition energy density is a constant fraction of @xmath2 of the central source radiation density . thermal synchrotron from @xmath318 k electrons , as suggested by recent hard x - ray observations , under predicts the radio @xmath140 by a factor of @xmath319 . to be a valid mechanism the emitting surface must be significantly larger than assumed here , but it is not yet in contradiction with current vlbi observations . alternatively , one needs @xmath321 k to derive the observed luminosity at @xmath344 ghz . \6 . at @xmath351 - 1000 ghz the innermost corona is expected to become optically thin , and the spectrum is expected to show a spectral break . the position of this break can be used to probe the innermost coronal size . it should display a sharp cutoff , rather than a steeper pl , in case of thermal synchrotron emission . since dust heavily dominates this spectral range , the detection requires vlbi observations , which exclude the low @xmath352 dust emission . additional constraints on the radio and x - ray coronal properties can be derived by including the observed x - ray emission , in particular the enigmatic gudel - benz relation of @xmath353 , which is explored in paper ii clearly , further exploration of the radio emission in rq agn , in particular at the mm range @xcite , will allow to probe directly the distribution of relativistic electrons and magnetic fields closest to the centre . the relation of the mm emission with the x - ray emission , in particular their variabilities @xcite , can provide important insights for the physical mechanisms which power accretion disc coronae in rq agn . | we find that a flat spectrum , as observed in about half of rq agn , can be produced by corona with a disc or a spherical configuration , which extends from the innermost regions out to a pc scale . ii we derive additional constraints on the coronal parameters from the gudel - benz relation , , which rq agn follow . | accretion discs in agn may be associated with coronal gas , as suggested by their x - ray emission . stellar coronal emission includes radio emission , and agn corona may also be a significant source for radio emission in radio quiet ( rq ) agn . we calculate the coronal properties required to produce the observed radio emission in rq agn , either from synchrotron emission of power - law ( pl ) electrons , or from cyclo - synchrotron emission of hot mildly - relativistic thermal electrons . we find that a flat spectrum , as observed in about half of rq agn , can be produced by corona with a disc or a spherical configuration , which extends from the innermost regions out to a pc scale . a spectral break to an optically thin power - law emission is expected around 300 - 1000 ghz , as the innermost corona becomes optically thin . in case of thermal electrons , a sharp spectral cutoff is expected above the break . the position of the break can be measured with vlbi observations , which exclude the cold dust emission , and it can be used to probe the properties of the innermost corona . assuming equipartition of the coronal thermal energy density , the pl electrons energy density , and the magnetic field , we find that the energy density in a disc corona should scale as , to get a flat spectrum . in the spherical case the energy density scales as , and is of the agn radiation energy density . in paper ii we derive additional constraints on the coronal parameters from the gudel - benz relation , , which rq agn follow . [ firstpage ] galaxies : active quasars : absorption lines quasars : general . |
1307.2952 | i | since the launch of modern x - ray space telescopes like and x - ray observatories , x - ray astronomy has entered the epoch of grating observations that can produce spectra with much improved energy resolution . in these spectra narrow absorption / emission lines , which have never been observed before , now have been commonly detected . these x - ray absorption / emission lines can be generated in a variety of astronomical environments , e.g. , stellar coronae ( sc ) , supernova remnants , x - ray binaries ( xrbs ) , galaxies , agn , and interstellar media ( ism ) and intergalactic media ( igm ) ( e.g. , ; @xcite ; @xcite ; @xcite ; ; @xcite ; @xcite ; @xcite ; @xcite ) . these lines carry valuable information about the absorbing / emitting material and thus are powerful diagnostic tools for stellar evolution , explosion mechanism of sne , mass exchange in accretion systems , interplays of different galactic components , feedback of agn , galaxy formation and evolution , evolution of our universe , and so on . clearly , proper identifications of these lines and subsequent scientific derivation and interpretation strongly rely on the accuracy of atomic databases of responsible transitions . there are several available databases of atomic transitions , among which the most commonly referenced four in x - ray community are verner et al . ( 1996 , hereafter v96 ) , nist , xstar ( @xcite ) , and aped ( @xcite ) . while atomic data for ions at high - charge states ( e.g. , he- and h - like ions ) are very accurate in these databases and consistent with observations ( juett et al . 2004 , 2006 , j0406 hereafter ; yao et al . 2009 , y09 hereafter ) , those for k transitions of neutral and mildly ionized metal elements ( e.g. , , , , etc . ) have not been included in any of them . there also exists other serious problems . first , although the statistical errors of the wavelengths of these high - ionized lines can be less than 10 m in astronomical observations , they are still not enough for the study of the low - velocity gases . y09 have measured these high - ionized lines ; however , some lines have large uncertainties , such as k@xmath0 that has a statistical error 6.7 m , equivalent to 125 @xmath1 . in addition , some lines ( e.g. , k@xmath2 ) are too weak to be given statistical errors . second , the methods commonly used ( e.g. , y09 ) to obtain the co - added spectrum introduce biases , which must be corrected and otherwise will result in serious estimation errors of the gas velocities as well as other parameters associated with the gas velocities . finally , some of the line wavelengths are not consistent with each other between the commonly referenced databases . for example , the wavelength of k@xmath0 in nist is 18.6270 but is 18.6288 in v96 . the difference between the two values is @xmath3 @xmath1 , which is a serious problem for the study of the low - velocity gas . therefore , it is essential to make observational identifications of the k - shell transitions of the neutral , low - ionized , and high - ionized metals . since most of k transitions of these low - ionization ions are in the wavelength range of 9.524.0 , within which copious lines of highly ionized metal elements have also been observed , these missing atomic data are important not only for their own rights but also for proper identifications of other lines . recently , several groups have calculated and updated these databases ( e.g. , @xcite ; @xcite ; @xcite ; @xcite ) , but even the centroid wavelengths , the most basic parameters of these lines , are yet to be coincided ( see table 1 ) . in addition , the laboratory measurements are also very uncertain ; e.g. , the wavelength of 1s2p given by stolte et al . ( 1997 ) range from 23.489 to 23.536 . lccccc & 1s2p & @xmath4 & @xmath5 & @xmath6 & @xmath6 + & 1s3p & @xmath7 & @xmath8 & @xmath9 & @xmath7 + & 1s2p & @xmath10 & @xmath11 & @xmath12 & @xmath13 + & 1s3p & @xmath7 & @xmath14 & @xmath7 & @xmath7 + & 1s2p & @xmath15 & @xmath7 & @xmath16 & @xmath7 + & 1s3p & @xmath7 & @xmath7 & @xmath7 & @xmath7 + & 1s3p & @xmath7 & @xmath17 & @xmath17 & @xmath18 + & 1s2p & @xmath19 & @xmath20 & @xmath20 & @xmath21 + & 1s3p & @xmath7 & @xmath7 & @xmath7 & @xmath22 + & 1s2p & @xmath23 & @xmath24 & @xmath24 & @xmath25 + & 1s3p & @xmath7 & @xmath7 & @xmath7 & @xmath26 + & 1s3p & @xmath7 & @xmath7 & @xmath7 & @xmath7 + & 1s2p & @xmath7 & @xmath7 & @xmath7 & @xmath7 + & 1s3p & @xmath7 & @xmath7 & @xmath7 & @xmath7 some galactic x - ray sources are very bright in the soft x - ray band . when the x - ray continuous radiations pass through the ism in different phases , x - ray absorption lines of ions at various charge states are produced and thus are excellent calibration references for these atomic data . for instance , schattenburg & canizares ( 1986 ) used the einstein observations of the crab to obtain the wavelength of the neutral o ( 1s2p of ) , although the result is uncertain ( @xmath27 ) according to modern standard . j0406 analyzed _ chandra _ high energy transmission grating ( hetg ) observations of several galactic xrbs and obtained the wavelengths of some strong lines ( i.e. , 1s2p of , , , , and 1s3p of ) . they found theoretically calculated line centroids of k transitions of low ionized neon and oxygen need to be shifted @xmath28 m to match observed values , which corresponds to @xmath29 shift for oxygen- and neon - absorbers . however , the ism is rotating around the galactic center and thus the observed x - ray lines are not at the rest - frame wavelengths . unfortunately , the above three previous works did not do the galactic rotation correction . it is also necessary to exclude x - ray sources with significant intrinsic absorptions , such as accreting x - ray binaries with winds from either their companions or accretion disks along the line of sight ( los ) . this can be done easily by comparing absorption line and/or total absorption properties of a target with multiple observations , since all x - ray binaries exhibit significant variability . although x - ray absorption lines produced in the ism are indeed ideal sources for calibrating theoretical predictions of atomic transitions , high spectral quality is crucial to accomplishing this important task . the measurements by j0406 are based on relatively poor quality spectra ( with signal - to - noise ratio snr@xmath30 per 10-mspectral bin ) obtained with short observations of several sources ; thus only those strong transitions ( e.g. , 1s2p of , , and ) were relatively well constrained , and others were still of large uncertainties ( @xmath31 m ) or not observed . recently y09 presented an extensive study of ism x - ray absorption lines in the spectrum of cyg x2 observed with the _ chandra_-hetg spectrograph . the high spectral quality not only allowed them to measure most of the transitions listed by j0406 as accurate as @xmath32 m ( 1@xmath33 uncertainty ) but also enabled them to constrain other faint transitions not included and/or misidentified in their list . for instance , y09 detected and measured the 1s3p transition of that could not be revealed in previous poor quality spectra , and they did not confirm the reported line at 23.140 that was misidentified as 1s3p transition of . in this work , our aim is to obtain more accurate k transition data of neutral , low- and high - ionized metals , as well as to find some weak absorption lines ( e.g. , 1s2p of ; 1s3p of ) . we jointly analyze 36 _ chandra_-hetg observations of 11 galactic xrbs and give the most accurate wavelengths for k - shell transitions of neutral , low- and high - ionized atoms . in section 2 we describe our methods of extracting the x - ray spectra and correcting for the galactic rotation . in section 3 we present the results of a joint analysis of all the observational data . in section 4 we estimate systematic uncertainties and make necessary corrections to the detected lines . a discussion and summary are given in section 5 and 6 , respectively . | accurate atomic transition data are important in many astronomical research areas especially in line spectroscopy study . whereas transition data of he - like and h - like ions ( i.e. , ions at high - charge states ) are accurately calculated , that of k transitions of neutral or low - ionized metal elements are still very uncertain . spectroscopy of absorption lines produced in the interstellar medium ( ism ) has been proven to be an effective way to measure the central wavelengths of these atomic transitions . in this work we analyze 36 _ chandra _ _ high energy transmission grating _ observations and search for and measure the ism absorption lines along sight lines of 11 low - mass x - ray binaries . compared to previous studies ( e.g. , yao et al . | accurate atomic transition data are important in many astronomical research areas especially in line spectroscopy study . whereas transition data of he - like and h - like ions ( i.e. , ions at high - charge states ) are accurately calculated , that of k transitions of neutral or low - ionized metal elements are still very uncertain . spectroscopy of absorption lines produced in the interstellar medium ( ism ) has been proven to be an effective way to measure the central wavelengths of these atomic transitions . in this work we analyze 36 _ chandra _ _ high energy transmission grating _ observations and search for and measure the ism absorption lines along sight lines of 11 low - mass x - ray binaries . we correct the galactic rotation velocity to the rest frame for every observation and then use two different methods to merge all the corrected spectra to a co - added spectrum . however the co - added spectra obtained by these methods exhibit biases , either to the observations of high counts or high signal - to - noise ratios of the lines . we make bayesian analysis to several significantly detected lines to obtain the systematic uncertainty and the bias correction of other lines . compared to previous studies ( e.g. , yao et al . 2009 ) , our results improve the accuracy of wavelengths by a factor from two to five and significantly reduce the systematic uncertainties and biases . several weak transitions ( e.g. , 1s2p of and ; 1s3p of and ) are also detected for the first time , albeit with low significance ; future observations with improved accuracy are required to confirm these detections . |
1307.2952 | c | in the section 3 , two different methods are used to merge all the 36 observations . method 1 is very simple : we only need to add the counts of each spectrum channel by channel , i.e. , every spectrum has the same weight . however , method 1 implicitly assumes that all the spectra have the same absorption column densities , i.e. , the same weight . when we merge two spectra that one has a strong absorption line ( e.g. , the k@xmath2 in obsid 6618 ) and the other has a weak absorption line ( e.g. , the k@xmath2 in obsid 8599 ) , the spectrum with weak absorption line can dilute or even wipe out the absorption line in the co - added spectrum , i.e. , reduce the significance of the absorption line . to avoid this problem , we use the significance of the absorption line as the weight to merge the spectra instead ( method 2 ) . the spectra that only contribute to the continuum are abandoned in this method , since the continuum is actually the noise for the absorption lines . from table 4 , we can see that the results obtained by the two methods are consistent with each other for most of the absorption lines , except for some weak lines ( e.g. , k@xmath2 ) that are too weak to be fit with a gaussian function . in addition , several suspicious lines may be faked , as they are only present in the co - added spectrum of method 1 ( k@xmath2 and k@xmath2 ) . the significance levels of the lines in the co - added spectrum obtained with method 2 are not remarkably higher than that obtained with method 1 . this means that the assumptions of both methods 1 and 2 are reasonable for our sample . more high - quality observations of more targets are the key to the determination of the wavelength of these weak transitions . as described in section 4.2 , although both methods 1 and 2 can obtain the spectra with high snrs , however , both @xmath329 and @xmath330 are biased . in addition , @xmath515 and @xmath516 are smaller than the true value , because @xmath207 is not considered . in this sub - section , we emphasize another important source of @xmath207 , i.e. , @xmath207 caused by the uncertainty of the spectral fitting . the snrs of some observations are so low that the fluctuations of the continua will seriously affect the fitting result of the weak lines . we can quantity this uncertainty with simulation . in this simulation , we take the snr of the simulated spectrum as the only variable to test the dependence of the systematic uncertainty on the snrs of the spectra . the snr range is between 1 and 10 , with a step of 0.5 , as shown in figure 10 . for each snr , we make the simulation as follows : 1 . the same as step 1 in section 4.1 , except that the line central wavelength is fixed at 23.5 . 2 . repeat the steps 24 in section 4.1 10000 times to obtain @xmath198 and the error of the @xmath199 . finally , for each snr , we obtain @xmath199 and its error , as shown in figure 10 . we find that @xmath199 increases dramatically as the snr decreases . the relationship between the snr and @xmath199 is similar to the form of a power - law . the reason for this anti - correlation is that the error propagation of the gaussian fitting is nonlinear when the statistical error is large . therefore , the joint analysis of all the observations can reduce the systematic uncertainty due to the improvement of the snrs of the co - added spectra . @xmath207 can be caused by other uncertainties , such as the imprecise galactic rotation correction that depends on several uncertain models ( e.g. , the distribution of the gas inside and above the galactic plane ) . because the parameters of the galactic rotation model are still uncertain , we can not give accurate value of systematic uncertainty caused by this uncertainty . nevertheless , the total systematic uncertainty can be obtained from the distribution of the lines of the 36 observations , as described in section 4.1 . due to the limitation of the snrs , we can not do bayesian analysis to all the lines . however , since the absorption clouds with the similar degree of ionization are assumed to have the same velocities as shown in table 3 , we use @xmath207 of the strongest line to represent the systematic uncertainty of all the lines in the same group , e.g. , @xmath207 ( k@xmath2 ) represents the systematic uncertainty of the low - ionized ne lines . as described above , our results on the wavelengths of k transitions of high - ionized o , ne , and mg are the most accurate so far , because more sources are used to reduce the systematic uncertainty induced by the galactic rotation correction and more observations are used to reduce the statistical error . our results are also unbiased , because we do the bias correction for each line . among all the lines , k@xmath2 is the most accurate , with an error of only 0.7 m , which is equivalent to about 14 @xmath1 . this error is so small that it may be used to measure the low - velocity gas , e.g. , the hot high - ionized gas in galactic halo . for the lines whose wavelengths are consistent between nist and v96 , our results are also consistent with those . for those lines whose wavelengths are inconsistent between nist and v96 , our results are closer to that in nist . compared to y09 , our results are two times more accurate ( e.g. , k@xmath2 ) and are also consistent with theoretical calculations except the k@xmath2 and the k@xmath2 lines ( table 10 ) . in our work , @xmath517 ( k@xmath518 is 7.5 m lower than the value of 12.1339 in v96 . we note that nist does not include this line . the k@xmath2 line is more complicated and it is also 10 m lower than that in nist and v96 . we will discuss this in the next sub - section . unlike the high - ionized elements , in the soft x - ray band , there still exists large discrepancies between the astronomical observations and the theoretical calculations of k transitions of the low - ionized and moderate - ionized elements ( table 1 ) . in fact , some of these lines ( e.g. , 1s3p of and 1s2p of ) have even never been detected . the co - added spectra have excellent snrs that provides us an unique opportunity to detect these weak absorption lines . we are particularly concerned about these transitions and thus search for the signals of these absorption lines around their theoretical values . for the 11 transitions of the low - ionized elements shown in table 8 , @xmath519 and @xmath520 . only @xmath521 ( k@xmath522 , but @xmath523 ( k@xmath524 . despite the low snrs , it is the first detection of the k@xmath2 line so far . several moderate - ionized ions , i.e. , the o , ne , and mg lines shown in table 9 ( except the k@xmath2 that has been detected in y09 ) , also appear for the first time around the theoretical values in the co - added spectra . as shown in table 9 , only @xmath521 ( k@xmath2 and k@xmath525 . however , among the other lines , @xmath521 ( k@xmath2 and k@xmath526 . moreover , the k@xmath2 and k@xmath2 lines are only present in the co - added spectrum of method 1 , which hints that these three lines may be artificial and need to be confirmed by higher quality observations in the future . in y09 , only one of the 2d-3p double lines , i.e. , the 15.01 line was detected , but no statistical error was given . in our work , both 15.01 and 15.26 lines are detected significantly for the first time simultaneously ( figures 3 and 4 ; table 9 ) . all the lines with @xmath527 are shown in figure 11 . by analyzing the co - add spectrum , the wavelengths of all the strong absorption lines can be determined accurately . in our work , several lines , including even the very weak k@xmath65 , are found . however , the wavelength of k@xmath2 is still uncertain . the observations of cyg x2 and xte j1817330 are the most important components of the co - added spectrum , which have enough counts and thus can greatly determine the shape of the co - added spectrum . y09 jointly analyzed four _ chandra_-hetg observations of cyg x2 and found the k@xmath0 line clearly but no k@xmath2 line was detected . this problem can be interpreted by an unknown emission that fills in the absorption ( cabot et al . the spectrum of xte j1817330 is similar to cyg x2 but more complex , which can be caused by not only two absorption lines but also an absorption line plus an emission line . thus , the co - added spectrum can also be well fitted by two different models ( figure 12 ) . for the model of a strong absorption line plus a weak redshifted absorption line , the fitting value of the k@xmath2 line is @xmath528 . however , for the model of an absorption line plus a weak emission line , the fitting value is @xmath529 . both values are very different from the theoretical value of 21.6020 given by nist or 21.6019 given by v96 . we can not distinguish which model is correct due to the limitation of the snr of the spectrum . | 2009 ) , our results improve the accuracy of wavelengths by a factor from two to five and significantly reduce the systematic uncertainties and biases . several weak transitions ( e.g. , 1s2p of and ; 1s3p of and ) are also detected for the first time , albeit with low significance ; future observations with improved accuracy are required to confirm these detections . | accurate atomic transition data are important in many astronomical research areas especially in line spectroscopy study . whereas transition data of he - like and h - like ions ( i.e. , ions at high - charge states ) are accurately calculated , that of k transitions of neutral or low - ionized metal elements are still very uncertain . spectroscopy of absorption lines produced in the interstellar medium ( ism ) has been proven to be an effective way to measure the central wavelengths of these atomic transitions . in this work we analyze 36 _ chandra _ _ high energy transmission grating _ observations and search for and measure the ism absorption lines along sight lines of 11 low - mass x - ray binaries . we correct the galactic rotation velocity to the rest frame for every observation and then use two different methods to merge all the corrected spectra to a co - added spectrum . however the co - added spectra obtained by these methods exhibit biases , either to the observations of high counts or high signal - to - noise ratios of the lines . we make bayesian analysis to several significantly detected lines to obtain the systematic uncertainty and the bias correction of other lines . compared to previous studies ( e.g. , yao et al . 2009 ) , our results improve the accuracy of wavelengths by a factor from two to five and significantly reduce the systematic uncertainties and biases . several weak transitions ( e.g. , 1s2p of and ; 1s3p of and ) are also detected for the first time , albeit with low significance ; future observations with improved accuracy are required to confirm these detections . |
1307.2952 | i | so far , the wavelengths of k alpha absorption lines of neutral , low - ionized , and moderate - ionized o , ne , and mg have not been determined precisely , either in theoretical calculations ( e.g. , v96 ; nist ) , laboratory measurements ( stolte et al . 1997 ) , or in astronomical observations ( j0406 ; y09 ) . in order to obtain the wavelengths of k transitions of o , ne , and mg more accurately , we jointly analyzed 36 _ chandra_-hetg observations of 11 lmxbs at low galactic latitudes in this work . we corrected the galactic rotation velocity to the rest frame for every observation and then used two different methods to merge all the corrected spectra to a co - added spectrum . finally , we fit and obtained the wavelengths of every absorption line in the co - added spectrum obtained by the above two methods ( section 3 ) . both methods give similar and consistent results for most lines , as shown in figure 3 . we noted that the co - added spectra obtained by the usual methods ( methods 1 and 2 ) exhibit biases , which are very important for the calibration of the lines . the co - added spectrum can be biased to the observations of high counts ( method 1 ) or high snrs of the lines ( method 2 ) . we made bayesian analysis to the lines of the k@xmath2 , k@xmath2 , k@xmath2 , k@xmath2 to obtain the systematic uncertainty and the bias correction of the o , ne , and mg lines in all phases . the final results after the bias correction are summarized as follows : \(1 ) for the the neutral , low - ionized , and high - ionized lines , the accuracy of our result is five and two times higher than j0406 and y09 respectively ( table 4 ) . several lines that were not detected ( i.e. , k@xmath2 , k@xmath65 , k@xmath0 , and k@xmath65 ) or too weak to give measurement errors ( i.e. , k@xmath0 , k@xmath65 and k@xmath2 ) in y09 are detected clearly in our work . we also find the moderate - ionized lines of these elements ( k@xmath2 , k@xmath2 , k@xmath2 , k@xmath2 , k@xmath2 , k@xmath2 , k@xmath2 , and k@xmath0 ; table 9 ) whose significances are so low that need to be confirmed in the future . besides the remarkable improvement of the accuracy , it is worth mentioning that all the biases measurements here are corrected ( tables 810 ) . \(2 ) the systematic uncertainty in our measurement mainly comes from the galactic rotation correction and the spectral fitting . the former depends on the galactic model . for the latter , we make simulations to estimate the effect . the total systematic uncertainties are : @xmath530 m and 0.6 mfor low - ionized and high - ionized o , @xmath531 0.4 m and 0.5 m for low - ionized and high - ionized ne , @xmath532 m for mg lines ( table 7 ) . \(3 ) for high - ionized lines of k@xmath0 and k@xmath2 , our results are consistent with that of nist , but 2 m lower than v96 . for k@xmath2 , our result is about 8 m lower than that given by v96 ; nist does not provide the theoretical value . for the moderate - ionized lines , the discrepancy between our measurements and the theoretical calculations are generally between 1 to 80 m . because the statistical errors of these lines are also similar to the discrepancy , the measurements are consistent with the theoretical calculations . jyl thanks dr . yuan liu for helping on improving the draft manuscript . snz acknowledges partial funding support by 973 program of china under grant 2009cb824800 , and by the national natural science foundation of china under grant nos . 11133002 , 10821061 , and 10725313 . * * * a**ppendices * | we correct the galactic rotation velocity to the rest frame for every observation and then use two different methods to merge all the corrected spectra to a co - added spectrum . however the co - added spectra obtained by these methods exhibit biases , either to the observations of high counts or high signal - to - noise ratios of the lines . we make bayesian analysis to several significantly detected lines to obtain the systematic uncertainty and the bias correction of other lines . | accurate atomic transition data are important in many astronomical research areas especially in line spectroscopy study . whereas transition data of he - like and h - like ions ( i.e. , ions at high - charge states ) are accurately calculated , that of k transitions of neutral or low - ionized metal elements are still very uncertain . spectroscopy of absorption lines produced in the interstellar medium ( ism ) has been proven to be an effective way to measure the central wavelengths of these atomic transitions . in this work we analyze 36 _ chandra _ _ high energy transmission grating _ observations and search for and measure the ism absorption lines along sight lines of 11 low - mass x - ray binaries . we correct the galactic rotation velocity to the rest frame for every observation and then use two different methods to merge all the corrected spectra to a co - added spectrum . however the co - added spectra obtained by these methods exhibit biases , either to the observations of high counts or high signal - to - noise ratios of the lines . we make bayesian analysis to several significantly detected lines to obtain the systematic uncertainty and the bias correction of other lines . compared to previous studies ( e.g. , yao et al . 2009 ) , our results improve the accuracy of wavelengths by a factor from two to five and significantly reduce the systematic uncertainties and biases . several weak transitions ( e.g. , 1s2p of and ; 1s3p of and ) are also detected for the first time , albeit with low significance ; future observations with improved accuracy are required to confirm these detections . |
cond-mat0110539 | i | the formation , resonant or phonon - assisted , of qw excitons and their following relaxation towards a final ( quasi- ) equilibrium thermodynamic state at low lattice temperature @xmath2 are the subject of numerous experimental @xcite and theoretical @xcite studies . recently , relaxation thermodynamics has been formulated and developed in order to analyze how bose - einstein statistics of high - density qw excitons influences the phonon - assisted thermalization processes . @xcite the above thermodynamics assumes a strong , dominant exciton - exciton scattering and , therefore , relaxation through quasi - equilibrium thermodynamic states . the relaxation thermodynamics has successfully been applied to model the @xmath8-dependent thermalization and photoluminescence kinetics observed in early experiments @xcite with high - density excitons ( @xmath9 @xmath10 @xmath11 @xmath10 ) in gaas qws . @xcite the long - lived indirect excitons in high - quality gaas / algaas coupled qws provide a unique opportunity for studying quantum degeneracy in a system of two - dimensional bosons . in this case , the long radiative lifetimes of indirect excitons allows the system to cool down to temperatures where the dilute exciton gas becomes statistically degenerate . @xcite the quality of present day gaas / algaas coupled qws has been considerably improved in comparison with those used in the pioneering experiments @xcite one decade ago . furthermore , the very recent magneto - optical experiments @xcite clearly indicate that the in - plane momentum @xmath12 of indirect excitons is a well - defined quantum number in high - quality gaas / algaas coupled qws . thermalization of hot photoexcited excitons down to the temperature of the cold lattice occurs mainly via scattering by thermal bulk longitudinal acoustic ( la ) phonons and is much more efficient for quasi - two - dimensional ( quasi-2d ) systems as compared to bulk semiconductors . this follows from the relaxation of momentum conservation in the @xmath13-direction ( the qw growth direction ) for quasi-2d systems : the ground - state mode @xmath14 , i.e. , the energy state @xmath15 , couples to the continuum energy states @xmath16 , rather than to the single energy state @xmath17 ( @xmath18 is the longitudinal sound velocity and @xmath19 is the in - plane translational mass of excitons ) as occurs in bulk materials . as a result , the la - phonon assisted kinetics of qw excitons becomes dominant at @xmath20 @xmath10 : in this case exciton - exciton scattering can be neglected while bose - einstein ( be ) statistics already strongly influences the relaxation process at low temperatures . @xcite crossover from classical to quantum statistics occurs near the degeneracy temperature @xmath21 , where @xmath22 is the spin degeneracy factor . for @xmath23 @xmath10 the degeneracy temperature of indirect excitons in gaas / algaas coupled qws is @xmath24 k. this estimate refers to @xmath25 , which can be achieved in the @xmath26-type gaas / algaas coupled qws by applying a static magnetic field @xmath27 . @xcite note that the very recent experiments @xcite deal with gaas / algaas coupled qws at extremely low cryostat temperature @xmath28 k. in this paper we study analytically and model numerically the acoustic - phonon - assisted relaxation kinetics of statistically - degenerate qw excitons at low densities . the recent experiments @xcite allow us to visualize , by means of lo - phonon - assisted emission , the la - phonon - assisted kinetics of quasi-2d excitons in znse qws and , in particular , to prove that for @xmath29 @xmath10 the above kinetics indeed occurs through non - equilibrium distributions of qw excitons . in experiments @xcite the concentration of be - degenerate indirect excitons in gaas / algaas coupled qws usually varies from @xmath30 @xmath10 at the very end of an optical excitation pulse to @xmath31 @xmath10 at large delay times @xmath32 ns after the optical excitation . thus , thermalization of the indirect excitons at large delay times can not be described within the relaxation thermodynamics @xcite and does need a separate theoretical analysis . the classical boltzmann kinetic equation has been generalized in order to include quantum statistics by uehling and uhlenbeck . @xcite the relevant quantum kinetic equation for spatially - homogeneous dilute system of statistically - degenerate quasi-2d excitons coupled to bulk la - phonons is @xmath33 \delta(e_{\bf k_{\| } } - e_{{\bf k_{\|}}-{\bf q}_{\| } } - \hbar q v_s ) \nonumber\\ & + & \big [ n_{\bf k_{\| } } n^{\rm ph}_{\bf q } ( 1+n_{{\bf k_{\|}}+{\bf q}_{\| } } ) -(1+n_{\bf k_{\| } } ) ( 1+n^{\rm ph}_{\bf q } ) n_{{\bf k_{\|}}+{\bf q}_{\| } } \big ] \delta(e_{\bf k_{\| } } - e_{{\bf k_{\|}}+{\bf q}_{\| } } + \hbar q v_s ) \big\ } , \label{1.1}\end{aligned}\ ] ] where @xmath34 and @xmath35 are the occupation numbers of exciton in - plane mode @xmath36 and phonon bulk mode @xmath37 , respectively , and @xmath38 is the in - plane projection of @xmath39 . the terms in the first and second square brackets on the right - hand - side ( r.h.s . ) of eq . ( [ 1.1 ] ) describe the stokes and anti - stokes la - phonon - assisted scattering processes , respectively . the acoustical phonons are assumed to be in a thermal equilibrium at the bath temperature @xmath2 . @xcite the matrix element is given by @xmath40^{1/2 } f_z(q_z l_z/2)$ ] , where @xmath41 is the crystal mass density , @xmath42 is the deformation potential of exciton - la - phonon interaction , @xmath43 is the thickness of a qw , and @xmath44 is the volume . the form - factor @xmath45 [ e^{i \chi } / ( 1 - \chi^2/\pi^2)]$ ] refers to an infinite rectangular confinement potential . @xcite the latter function describes the relaxation of the momentum conservation law in the @xmath13-direction and characterizes a spectral band of bulk la - phonons , which effectively interact with qw excitons . note that eq . ( [ 1.1 ] ) is valid only for the kinetic stage of thermalization , i.e. , before a low - temperature collective state of excitons @xcite builds up . the main aim of our work is to study the fundamental features of the acoustic phonon - assisted thermalization kinetics of qw excitons from initial strongly nonequilibrium @xmath46 towards the final equilibrium distribution with well - developed bose - einstein statistics , when @xmath47 . our numerical simulations of the la - phonon - assisted kinetics clearly demonstrate that after the first transient , which lasts a few characteristic scattering times , a slow _ adiabatic stage _ of thermalization builds up ( see fig . [ fig1 ] ) . this stage is characterized by a quasi - equilibrium distribution of high - energy qw excitons with effective temperature @xmath48 and is independent of the initial distribution at @xmath49 . the adiabatic stage lasts many characteristic scattering times and arises due to the need to populate the low - energy in - plane modes with @xmath50 in the presence of effective suppression of the stimulated scattering processes ( e.g. , an intense incoming , stokes flux of excitons into the ground - state mode @xmath51 is nearly compensated by the outgoing , anti - stokes scattering out of the state @xmath36=0 ) . in order to describe the adiabatic stage of relaxation , we find a _ generic solution _ of the quantum kinetic eq . ( [ 1.1 ] ) . while the generic solution we calculate is different from that derived in ref . for the phonon - assisted kinetics of bulk excitons at @xmath52 ( @xmath53 is the critical temperature for bose - einstein condensation of excitons in bulk semiconductors ) , similar to this case it depends only on two control parameters of the system , @xmath54 and @xmath2 . furthermore , the only gross information from a particular shape of the initial distribution at @xmath49 is absorbed by the start time @xmath55 of the adiabatic stage of thermalization . this is shown in the fig . [ fig1](a ) , where we vary the parameters of the initial gaussian distribution . in turn , the first transient depends upon initial distribution and lasts only a few scattering times @xmath56 , as illustrated in fig . [ fig1](b ) . , where @xmath57 , and parameter @xmath58 ( dash - dotted lines ) , @xmath59 ( solid lines ) , and @xmath60 ( dashed lines ) . the gross dependence of evolution upon initial conditions is absorbed by time @xmath61 . ( b ) the first stage of la - phonon - assisted relaxation ( @xmath62 ) for a particular distribution of indirect excitons with @xmath63 . in both plots @xmath64 and @xmath65 . the dimensionless values , time in @xmath66 and energy / temperature in @xmath67 , can easily be rescaled to the dimensional units by using @xmath68 @xmath69ev and @xmath70 ns for gaas / algaas coupled qws , and @xmath71 @xmath69ev and @xmath72 ns relevant to znse qws . ] because at any nonzero bath temperature @xmath73 the occupation number of the ground - state mode @xmath74 is finite , the adiabatic stage ends up with linearized kinetics . the latter kinetics refers to the last , exponential stage of thermalization at @xmath75 and is characterized by @xmath76 . for statistically - degenerate qw excitons the thermalization time @xmath77 reaches its smallest values at bath temperatures @xmath78 . while the above inequality does not usually hold in the experiments with statistically - degenerate indirect excitons in gaas / algaas coupled qws , @xcite we describe a special design of gaas - based microcavities ( mcs ) for optimization of the la - phonon - assisted thermalization kinetics of low - density mc polaritons . in the proposed microcavities with a large positive detuning between the cavity and qw exciton modes , the mc polaritons have radiative lifetimes on a 100 ps - 1 ns time scale , so that the phonon - assisted relaxation towards well - developed bose - einstein statistics with large occupation numbers can optically be visualized . thus , the mc design we discuss is an interesting alternative to the semiconductor microcavities with zero detuning between the cavity mode and qw excitons , where huge nonclassical occupation numbers of the low - energy mc polariton states have recently been observed . @xcite in numerical evaluations we use @xmath79 , @xmath80 cm / s , and @xmath81 ev , relevant to gaas / algaas coupled qws , and @xmath82 , @xmath83 cm / s , and @xmath84 ev , relevant to single znse qws , respectively ( @xmath85 is the free electron mass ) . note that the disorder - induced scattering and localization processes , not included in our model , are relatively strong in up - to - date znse - based qws and require a separate analysis . @xcite however , in the very recent work @xcite the first fabrication of high - quality mgs / znse / mgs qws with less than @xmath86 ml fluctuations of the well width and , therefore , with extremely low inhomogeneous broadening has been reported . in sec . [ sec2 ] , the boltzmann equation is adapted in order to formulate the acoustic - phonon - assisted kinetics of statistically - degenerate qw excitons . we also discuss some approximations for the form - factor @xmath87 , which describes the relaxation of momentum conservation in scattering of qw excitons by bulk la - phonons . in sec . [ sec3 ] , we find the generic solution of the acoustic - phonon - assisted kinetics from a strongly nonequilibrium initial distribution of qw excitons @xmath88 towards a final bose - einstein distribution @xmath89 with large occupation numbers of the low - energy states , @xmath90 , where @xmath91 is the dimensionless energy defined by @xmath57 . the generic solution is independent of a particular shape of the initial distribution and describes the adiabatic stage of thermalization , which starts at @xmath92 in a few characteristic scattering times after @xmath49 . we show that at @xmath93 thermalization of high - energy qw excitons is given by @xmath94 , where @xmath6 is the lowest positive eigenvalue of the relevant linearized kinetics . in sec . [ sec4 ] , the linearized phonon - assisted kinetics of statistically - degenerate qw excitons is formulated and analyzed . we show that the eigenvalues @xmath95 of the linear collision integral form a continuous spectrum , @xmath96 , separated from the nondegenerate eigenvalue @xmath97 , and that the corresponding eigenfunctions @xmath98 have three well - defined isolated critical points . the dependence of the marginal eigenvalue @xmath6 on the bath ( @xmath2 ) and degeneracy ( @xmath54 ) temperatures is studied . in sec . [ sec5 ] , straightforward numerical simulations of the phonon - assisted relaxation kinetics at @xmath99 are compared with the generic solution of the quantum boltzmann equation . we also show that at the beginning of the adiabatic stage of thermalization , at @xmath100 , the population dynamics of the ground - state mode is given by @xmath101 , where the parameters @xmath102 and @xmath103 are calculated analytically . furthermore , we propose a particular design of gaas - based microcavities ( mcs ) , which compromises the efficiency of la - phonon - assisted scattering ( the density of states is @xmath104 ) with the degeneracy temperature @xmath105 , i.e. , allows us to avoid the bottleneck effect in relaxation and , therefore , to optimize the thermalization kinetics of low - density qw excitons . in the appendix , some relationships relevant to the thermalization dynamics of quasi - equilibrated high - energy qw excitons are given . | acoustic - phonon - assisted thermalization kinetics of excitons in quantum wells ( qws ) is developed for small concentrations of particles , , when particle - particle interaction can be neglected while bose - einstein statistics already strongly influences the relaxation processes at low temperatures . in this case the next , adiabatic stage of thermalization usually takes many characteristic scattering times , depends only upon two control parameters , the lattice temperature and the degeneracy temperature , and is characterized by a quasi - equilibrated distribution of high - energy qw excitons with effective temperature . we show that the thermalization law of high - energy particles is given by , where is a marginal value of the continuous eigenvalue spectrum of the linearized kinetics . by analyzing the linearized phonon - assisted kinetics of statistically - degenerate qw excitons , we study the dependence . our numerical estimates refer to high quality gaas and znse qws . finally , we propose a special design of gaas - based microcavities , which considerably weakens the bottleneck effect in relaxation of excitons ( polaritons ) and allows us to optimize the acoustic - phonon - assisted thermalization processes . | acoustic - phonon - assisted thermalization kinetics of excitons in quantum wells ( qws ) is developed for small concentrations of particles , , when particle - particle interaction can be neglected while bose - einstein statistics already strongly influences the relaxation processes at low temperatures . in this case thermalization of qw excitons occurs through nonequilibrium states and is given by the following scenario . during the first transient stage , which lasts a few characteristic scattering times , the correlations with an initial distribution of qw excitons disappear . the next , adiabatic stage of thermalization usually takes many characteristic scattering times , depends only upon two control parameters , the lattice temperature and the degeneracy temperature , and is characterized by a quasi - equilibrated distribution of high - energy qw excitons with effective temperature . we show that the thermalization law of high - energy particles is given by , where is a marginal value of the continuous eigenvalue spectrum of the linearized kinetics . by analyzing the linearized phonon - assisted kinetics of statistically - degenerate qw excitons , we study the dependence . our numerical estimates refer to high quality gaas and znse qws . finally , we propose a special design of gaas - based microcavities , which considerably weakens the bottleneck effect in relaxation of excitons ( polaritons ) and allows us to optimize the acoustic - phonon - assisted thermalization processes . |
1403.2216 | i | the pauli hamiltonian describes a non - relativistic electron with gyromagnetic constant @xmath6 @xmath7 @xmath8 is the vector of pauli matrices and @xmath9 acts on spinors . we use units where @xmath10 . the electric and magnetic fields are determined by the 4-potential @xmath11 : [ fields ] = , = - + a_0 in 1979 aharonov and casher @xcite observed that the pauli operator for static magnetic field in two dimensions , @xmath12 , so @xmath13 , has ( normalizable ) zero energy modes . the zero modes turn into gapped bound states . ] they are gapless ground states and their number @xmath14 , is determined by the total magnetic flux @xmath15 measured in units of quantum flux , [ zero - modes ] d= ceiling of @xmath16 , i.e. the smallest integer @xmath17 . , the total number of zero modes is 6 . since all the fluxons are critical no mode is localized on any one fluxon . if the two clusters are well separated one can choose two of the zero mode to be localized on the cluster of three fluxons and another three modes to be localized on the cluster of four fluxons . the remaining 6-th mode is supported on the two clusters no matter how far they are separated . , width=6 ] we consider a magnetic field @xmath18 localized on a finite number of disjoint fluxons labeled by @xmath19 . the magnetic flux of the @xmath20-th fluxon , @xmath21 , is localized in a region of radius @xmath22 centered at @xmath23 . we _ do not _ assume that @xmath21 is quantized or that all the fluxes @xmath21 are identical . we shall assume w.l.o.g . that @xmath24 . we say that the @xmath20-th fluxon is super - critical if @xmath25 , subcritical if @xmath3 and critical if @xmath26 . the fluxons are viewed as classical parameters and _ not _ as dynamical degrees of freedom : they _ do not _ have a wave function or an equation of motion . ( the dynamical degree of freedom is the electron wave function . ) when the @xmath20-th fluxon is super - critical it can create @xmath27 zero modes which are _ confined _ to it , in the sense that their wave function decays ( as a power law ) over a typical distance @xmath28 determined by the fluxon radius @xmath22 . more interesting are the zero modes which are bound jointly by a number of separate fluxons . we shall call these solutions _ free zero modes_. these states wave functions live in between the fluxons and have typical size determined by the inter - distance @xmath29 . when _ all _ the fluxons are subcritical , @xmath30 _ all _ the zero modes are free : the probability of finding the charge on any of the fluxons is close to zero ( as @xmath31 ) . in general , confined and free modes coexist . the confined modes behave like the charge - flux composites one encounters in the fractional quantum hall effect @xcite , except that here the charge is quantized but the flux is not whereas in the hall effect it is the flux that is quantized and the charge is not . the free modes are a different kettle of fish as the composite involves a single electron jointly bound by several fluxons . as we shall see , these modes can sometimes turn the fluxons into non - abelian anyons @xcite . these new `` topological '' objects are quite interesting . the distinction between confined and free zero modes is meaningful when the radius of the individual fluxons , @xmath22 , is the smallest length scale in the problem , @xmath32 and is sharp for point - like fluxons . the total number of free modes @xmath33 is , as we shall see , [ extended ] 0d_f= max\{0,_a _ a-1}n-1 , _a=_a - d_a we say that the number of free modes is maximal if @xmath34 . this turns out to be the case where the fluxons become non - abelian anyons . if all the fluxons are identical then to have maximal number of free modes leading to interesting representation of the braid group requires [ con ] 1 - 1 n < _ a < 1 ( the case @xmath35 leads to a trivial representation of the pure braid group . ) to study the holonomy of the zero modes we treat the fluxon coordinates , @xmath23 as ( classical ) adiabatic controls . the adiabatic theory we shall need and describe is of interest in its own right , since the weak electric fields generated by the slow motion of the fluxons are important for the adiabatic transport and since the zero modes are gapless ( see sec . [ sec : ad ] for more details ) . adiabaticity means that the characteristic time scale of the controls is large compared with the characteristic time scale of the system . we shall argue that the characteristic time scale in the case of point - like fluxons is set by their mutual distances . this means that points in control space where fluxons collide must be removed : fluxon collisions is like gap closures in gapped systems . both endow control space with an interesting topology which is sine qua non for interesting topological behaviour . the holonomies of braiding point - like fluxes are summarized in : * the berry phase associated with the confined mode on the @xmath20-th super - critical fluxon braided by the fluxon @xmath36 is the aharonov - bohm phase @xmath37 . * the berry phase for a non - degenerate free mode , ( @xmath38 ) , and two fluxons ( @xmath39 ) is _ topological _ ( path independent ) given by @xmath40 . * the berry phase for a single free mode , ( @xmath38 ) and @xmath41 fluxons is abelian but path _ dependent_. in other words , the adiabatic curvature is non trivial , see fig . [ fig : curvature ] . * for @xmath42 and maximal number of free modes , @xmath34 the holonomy is non - abelian and _ topological_. braiding anyon @xmath20 with anyon @xmath43 is associated with the monodromy matrix , _ a = e^-2i_a the eigenvalues of the holonomy matrix are @xmath44 . this is our most significant result . * if , in addition to @xmath34 , all the fluxons carry identical fluxes , then exchanging them make sense and is described by matrices that give the burau representation of the full braid group @xcite : , the fluxons are identical anyons . like ordinary anyons @xcite they have topological braiding and fusion rules . ( the fusion rules are simply flux addition . ) but , unlike ordinary anyons , they are gapless and hence fragile . * @xmath41 and @xmath45 : the holonomy is non - abelian and , in general , path dependent i.e. not topological . | aharonov and casher showed that pauli hamiltonians in two dimensions have gapless zero modes . we study the adiabatic evolution of these modes under the slow motion of fluxons with fluxes . the positions , , of the fluxons are viewed as controls . the holonomies can sometimes be abelian , but in general are not . | aharonov and casher showed that pauli hamiltonians in two dimensions have gapless zero modes . we study the adiabatic evolution of these modes under the slow motion of fluxons with fluxes . the positions , , of the fluxons are viewed as controls . we are interested in the holonomies associated with closed paths in the space of controls . the holonomies can sometimes be abelian , but in general are not . they can sometimes be topological , but in general are not . we analyze some of the special cases and some of the general ones . our most interesting results concern the cases where holonomy turns out to be topological which is the case when all the fluxons are subcritical , , and the number of zero modes is . if it is also non - abelian . in the special case that the fluxons carry identical fluxes the resulting anyons satisfy the burau representations of the braid group . |
0706.1442 | r | in order to check the effects of the np described above , the probability differences @xmath132 ( eq.([e234 ] ) ) for two energy ranges and several baselines , have been calculated . both , the energy ranges and the baselines have been chosen in prospect of the existing , planned , and feasible experiments . in view of @xmath10 and @xmath11 experiments , the first energy range is @xmath133 and the considered baselines are @xmath134 ( @xcite ) . in view of @xmath6 experiments , the second energy range is @xmath135 and the considered baselines are @xmath136 ( @xcite ) . in our numerical calculations of neutrino flavor transitions , we use the realistic prem i @xcite earth density profile model , which assesses the actual matter density @xmath137 and the actual electron fraction @xmath138 , along the neutrino flight path in the earth s interior . then : @xmath139 & = & 7.63\times10^{-5}\,[\frac{\rho}{g / cm^{3}}]\,[\frac{y_{e}}{0.5}]\,[\frac{e_{\nu}}{gev}]\,,\nonumber \\ \label{e301}\\a_{n}\,[ev^{2 } ] & = & 7.63\times10^{-5}\,[\frac{\rho}{g / cm^{3}}]\,[1-y_{e}]\,[\frac{e_{\nu}}{gev}]\,.\nonumber \end{aligned}\ ] ] note here that , for @xmath140 neutrinos pass only the first shell of the earth s crust , with a constant density @xmath141 and @xmath142 , thus @xmath143=1.96\times10^{-4}[e_{\nu}/gev]$ ] and @xmath144=1.0\times10^{-4}[e_{\nu}/gev]$ ] . the @xmath7 oscillation parameters , together with their @xmath0 errors ( @xmath145 c.l . , correlations among parameters are currently considered small ) , are taken from the current global best fit values @xcite : @xmath146\,,\nonumber \\ \sin^{2}(\theta_{12 } ) & = & 0.314\,(1_{-0.15}^{+0.18})\,,\label{e302}\\ \delta m_{atm}^{2 } & = & 2.4\,(1_{-0.26}^{+0.21})\times10^{-3}\,[ev^{2}]\,,\nonumber \\ \sin^{2}(\theta_{23 } ) & = & 0.44\,(1_{-0.22}^{+0.41})\,.\nonumber \end{aligned}\ ] ] in order to implement the effects of heavy neutrinos , as discussed in the previous chapter , the matrix @xmath48 must be introduced . as the number of heavy non - decoupling neutrinos is unknown , we parametrize it in a simplified way , using a single `` effective '' heavy neutrino state . in this way , the number of independent quantities parameterizing the matrix @xmath147 in eq.([e27 ] ) are @xmath148 moduli and @xmath62 cp phases : @xmath62 standard mixing angles ( @xmath63 ) + @xmath12 standard dirac phase ( @xmath64 ) + @xmath62 new small np mixing angles ( @xmath149 ) + @xmath150 new np dirac phases ( @xmath151 ) . then @xmath152 and @xmath153 two sets of @xmath48 parameters , both satisfying present experimental constraints given by eq.([e233 ] ) are discussed below : @xmath154 all calculations have been performed assuming the direct mass scheme only . as nothing is known about values of cp phases , we allow them to vary freely . for both @xmath48 sets , we notice that , the biggest potential of the discovery of the possible presence of any np is pronounced in oscillation channels in which @xmath2 , @xmath3 are not involved at all , that is in @xmath15 and @xmath16 ( including the corresponding antineutrino channels ) . the effects are especially visible for two baselines , @xmath4 and @xmath5 , which , for other reasons , are also considered `` magic '' for future @xmath6 experiments . moreover , comparing numerical results for these two sets of @xmath48 parameters , we can clearly see that , as in eq.([e303].b ) the magnitude of the middle row is @xmath155 times smaller than the corresponding magnitude in eq.([e303].a ) , the np effects for the second @xmath48 set eq.([e303].b ) are smaller by a similar factor ( compare eq.([e304 ] ) below ) , too . in order to find how the uncertainty of the estimation of the @xmath7 oscillation parameters can mimic any possible np effects , we have also performed calculations allowing all @xmath7 oscillation parameters to vary by @xmath1 and @xmath0 ( note here that , the @xmath0 test is also useful in qualitative estimation of the effect of the uncertainties in the earth density profile on the @xmath7 results ) . we have found that in general , in order to give a chance for a discovery of np effects , it is required that @xmath7 neutrino oscillation parameter errors should be diminished to the values expected in the future ( of about @xmath1 ) . however , using neutrinos with energies @xmath156 , together with the `` magic '' baselines mentioned above , it should even be possible to see np effects with today s @xmath7 neutrino oscillation parameter uncertainties ( of about @xmath0 ) . but , this chance depends on the actual magnitudes of the np parameters and the actual precision of the experiment ( which may , of course , be required to be much better than the allowed @xmath7 neutrino oscillation parameter errors ) . in fig.[f1 ] , the probability differences @xmath157 , for the second energy range , and the corresponding baselines set , are shown . the np effects at @xmath158 , when scaled by a factor @xmath159 , can be used as an approximate estimation of the expected results for the first energy range with its considered baselines ( for which we do not show pictures in this paper ) . in the left and middle columns of fig.[f2 ] , the probability differences @xmath160 for the second energy range and the two `` magic '' @xmath6 baselines , are shown . for this oscillation channel , the np effect at @xmath158 is significantly smaller than in the @xmath15 channel at the same distance . this is also the case for this oscillation channel in the first energy range with its considered baselines . hence for this reason , we do not show the corresponding pictures in this paper either . note that , in general , the transition probabilities of both , neutrino and antineutrino , oscillations depend on matter properties in a different way . however , at higher energies the @xmath7 dependence is very much similar , thus any possible uncertainties in the earth density profile cancel in the @xmath7 probability differences ( see the upper row of graphs , in both figures , for @xmath156 ) . in this way , any signal that the probability differences in this energy range are distinctly different from zero will indicate that some np exists . the statistical significance of such signal depends on its actual magnitude . for example , for the first @xmath48 set eq.([e303].a ) , and @xmath5 , there exists a maximum at @xmath161 , in which @xmath162 , while @xmath163 , and thus @xmath164 ( see the middle row of graphs in both figures ) . the neutrino energy dependent magnitude of this effect can well be reproduced just by the largest term in the @xmath165 np correction ( responsible for the effective neutrino mass change , see eq.([e228 ] ) ) , which is not suppressed neither by @xmath116 nor by @xmath122 . taking into account the fact that @xmath166 , one can write ( see eq.([e230 ] ) ) : @xmath167 it should be noted , however , that the above formula does not reproduce the @xmath157 well , in case one of the np feasible cp phases ( @xmath168 ) is equal to @xmath169 ( see the dotted and dashed curves in fig.[f1 ] ) . the reason is that in the above estimation we completely neglect several explicit cp asymmetry bracing terms , for example proportional to @xmath170 , which are relevant in the @xmath157 case , but which never appear in the case of @xmath160 ( see the appendix [ app : approx ] , note also that , these asymmetry terms become dominant at short baselines , and that is why the np effect at @xmath158 is significantly smaller in the @xmath16 channel than in the @xmath15 channel at the same distance ) . from the above formula we can easily learn that , the magnitude of this np effect is linearly proportional to both , the actual matter density through the term @xmath171 , and to the @xmath172 np parameter ( thus , as the value of the @xmath172 parameter that results from the eq.([e303].b ) is @xmath155 times smaller than the one from the eq.([e303].a ) , the np effect for the second @xmath48 set eq.([e303].b ) is smaller by a similar factor , too ) . as already mentioned , one of the easiest channels , from the experimental point of view , is the @xmath14 one , but it will be difficult to observe any np there ( as will be in all channels in which @xmath2 or @xmath3 are involved ) . in the right column of fig.[f2 ] , the probability differences @xmath173 for the second energy range and the longest considered baseline , are shown ( where the biggest effects are expected ) . it can be seen that the np effect is rather miserable in this oscillation channel , regardless of the @xmath48 set given by eq.([e303 ] ) . this conclusion holds also for all another similar oscillation channels and both energy ranges with the corresponding baselines . moreover , these oscillation channels are sensitive to the ( not so very well known ) value of the @xmath174 . in the other channels , without @xmath2 and @xmath3 , the dependence on the @xmath174 is small , giving a better chance to see the np effects . | the possibility of the discovery of such effects in current and future neutrino oscillation experiments is discussed . it appears that the biggest potential of the discovery of the possible presence of any new physics is pronounced in oscillation channels in which , are not involved at all , especially for two baselines , and , which for other reasons are also called `` magic '' for future experiments . | an extension of the new standard model , by introducing a mixing of the low mass `` active '' neutrinos with heavy ones , or by any model with lepton flavor violation , is considered . this leads to non - orthogonal neutrino production and detection states and to modifications of neutrino oscillations in both , vacuum and matter . the possibility of the discovery of such effects in current and future neutrino oscillation experiments is discussed . first order approximation formulas for the flavor transition probabilities in constant density matter , for all experimentally available channels , are given . numerical calculations of flavor transition probabilities for two sets of new physics parameters describing a single `` effective '' heavy neutrino state , both satisfying present experimental constraints , have been performed . two energy ranges and several baselines , assuming both the current ( ) and the expected in future ( ) errors of the neutrino oscillation parameters are considered , keeping their present central values . it appears that the biggest potential of the discovery of the possible presence of any new physics is pronounced in oscillation channels in which , are not involved at all , especially for two baselines , and , which for other reasons are also called `` magic '' for future experiments . |
0706.1442 | c | in the paper , by introducing a mixing of the low mass `` active '' neutrinos with heavy ones , we have investigated a possible new physics ( np ) scenario which is already present at the @xmath8 scale , that means , at energies close to our present - day experimental facilities . in the presented model ( as also in any model with lepton flavor violation ) , the effective mixing matrix is non - unitary , resulting in non - orthogonal neutrino production and detection states . this leads to the modification of the neutrino oscillations in vacuum . additionally , non - standard neutrino interactions with matter particles influence oscillation effects also . first order approximation formulas for the flavor transition probabilities , in constant density matter , for all experimentally available channels , have been given . the possibilities of experimental verification of such model predictions have been discussed in prospect of the existing , planned , and feasible @xmath10 , @xmath11 , and @xmath6 experiments . numerical calculations of flavor transition probabilities for two sets ( satisfying present experimental constraints ) of np parameters which describe a single `` effective '' heavy neutrino state have been performed . they took into account two energy ranges and several baselines , assuming both the current ( @xmath0 ) and the expected in future ( @xmath1 ) errors of today s @xmath7 neutrino oscillation parameters , keeping unchanged their present central values . the realistic prem i earth density profile model has been applied . one of the easiest channels , from the experimental point of view , is the @xmath14 one , but it will be difficult to observe any np there ( as it will be also in all channels in which @xmath2 or @xmath3 are involved ) . it appears that the biggest potential of the discovery of any possible presence of the np is pronounced in oscillation channels in which @xmath2 , @xmath3 are not involved at all , that is in @xmath15 and @xmath16 ( and in the corresponding antineutrino channels ) . the effects are especially visible for the two so called `` magic '' @xmath6 baselines , @xmath4 and @xmath5 . we have also found that in general , in order to give a chance for a discovery of the np effects , it is required that @xmath7 neutrino oscillation parameter errors should be diminished to the values expected in the future ( of about @xmath1 ) . however , using neutrinos with energies @xmath156 together with the `` magic '' baselines mentioned above , it should even be possible with today s @xmath7 neutrino oscillation parameter uncertainties ( of about @xmath0 ) . but , this chance depends on the actual magnitudes of the np parameters and the actual precision of the experiment ( which may , of course , be required to be much better than the allowed @xmath7 neutrino oscillation parameter errors ) . finally , it should be stressed that the full quantitative treatment of the np effects in future facilities should be based on realistic observables related , for example , to the expected numbers of events at the corresponding facilities . however , as stated in the beginning of the chapter [ sec : theory ] , this would require not only the knowledge of the np generated transition probability modifications , but also the np generated modifications of the production and detection neutrino cross sections need to be known . some preliminary studies ( see @xcite ) suggest that the expected effects on the last two terms can be of the same order as these on the first term ( shown in this paper ) . this work has been supported by the polish ministry of science under grant 1p03b04926 . | an extension of the new standard model , by introducing a mixing of the low mass `` active '' neutrinos with heavy ones , or by any model with lepton flavor violation , is considered . first order approximation formulas for the flavor transition probabilities in constant density matter , for all experimentally available channels , are given . two energy ranges and several baselines , assuming both the current ( ) and the expected in future ( ) errors of the neutrino oscillation parameters are considered , keeping their present central values . | an extension of the new standard model , by introducing a mixing of the low mass `` active '' neutrinos with heavy ones , or by any model with lepton flavor violation , is considered . this leads to non - orthogonal neutrino production and detection states and to modifications of neutrino oscillations in both , vacuum and matter . the possibility of the discovery of such effects in current and future neutrino oscillation experiments is discussed . first order approximation formulas for the flavor transition probabilities in constant density matter , for all experimentally available channels , are given . numerical calculations of flavor transition probabilities for two sets of new physics parameters describing a single `` effective '' heavy neutrino state , both satisfying present experimental constraints , have been performed . two energy ranges and several baselines , assuming both the current ( ) and the expected in future ( ) errors of the neutrino oscillation parameters are considered , keeping their present central values . it appears that the biggest potential of the discovery of the possible presence of any new physics is pronounced in oscillation channels in which , are not involved at all , especially for two baselines , and , which for other reasons are also called `` magic '' for future experiments . |
0801.1329 | c | in this paper , we have applied a versatile statistical tool , the maximum likelihood technique , to assess the distribution of stellar mass and the incidence of binaries in the pleiades . we began with a near - infrared catalog of cluster members . our basic assumption was that all cluster members share the same evolutionary age , and that any dispersion in the color - magnitude plane stems from binarity and random photometric errors . we were then able to infer the most probable distribution of masses , both for the cluster as a whole , and as a function of distance from its center . finally , we introduced a simple method for gauging the degree of mass segregation in the cluster . one of our surprising results is the relatively high fraction of binaries . we estimate that 68% of all systems in the cluster are unresolved binaries ; this figure climbs to about 76% if resolved pairs are included . these fractions are significantly higher than the accepted field - star result , so we should scrutinize them carefully . could they stem from an underestimate of the photometric error at faint magnitudes ? since the error in @xmath43 is greater than @xmath15 , we artificially increased the dispersion @xmath135 . we kept @xmath135 at 0.15 until , below which we increased it linearly , reaching at . after redoing the maximum likelihood analysis , the global binary fraction @xmath21 for unresolved pairs is unchanged . another potential difficulty is our neglect of the physical thickness of the cluster . we have assigned all members a distance of 133 pc , although there will naturally be some variation . however , this effect is also relatively small . from section 4.1 , the volumetric number density falls off with radius approximately as a king model with core and tidal radii of 2.1 and 19 pc , respectively . consider the front half of a spherical cluster with such a density distribution . it may readily be shown that the mean distance from the plane of the sky of any cluster member is . for a cluster at mean distance @xmath185 , the induced magnitude spread is , which is 0.04 in our case . although the actual spread in magnitudes is not gaussian , we have added this figure in quadrature to both @xmath135 and @xmath136 , and rerun the analysis . again , the binarity is unaffected . the errors due to both photometry and finite cluster thickness induce a symmetric spread in stellar magnitudes . that is , they scatter as many sources below the fiducial isochrone as above it . thus , they can not reduce the estimated binarity , which stems from an excess of stars above the isochrone . one systematic error that _ would _ affect @xmath21 is an overestimation of the cluster distance . if @xmath185 were lowered , the absolute magnitudes of all sources would decrease equally , as would the inferred @xmath21-value . quantitatively , the distance would have to decrease by about 15 pc to bring the binary fraction down to the field - star result for g - dwarf primaries . an error of this size for the average distance is excluded by current observations , for which the estimated uncertainty is only 1 pc @xcite . since our method relies solely on photometry to assess binarity , we can not distinguish between physically linked pairs and chance alignments . as mentioned in section 3.4 , the resolution limit of our data is @xmath168 , or at the distance of the pleiades . consider a star at a radius @xmath99 from the cluster center . its average number of neighbors within @xmath186 is , where @xmath187 is the projected surface number density of the cluster . since each ring of width @xmath188 contains stars , integration over all members yields the total number of chance alignments : @xmath189 where @xmath0 is the cluster s outer radius , using @xmath187 from figure 10 , we find . thus , chance alignments have no quantitative impact . yet another source of systematic error is the cluster age . we have adopted the lithium - based figure of 125 myr from @xcite . earlier estimates , using the main - sequence turnoff , yielded a range of answers . for example , @xcite found 100 myr . even this minor reduction affects our results , since it lifts the low - mass end of the isochrone toward higher luminosity . for an age of 100 myr , our analysis gives and . the binary fraction is augmented to 0.64 when we include resolved pairs . from figure 3 of @xcite , a 100 myr age corresponds to a lithium edge at , or at the pleiades distance . such a result seems incompatible with the lithium data shown in figure 2 of @xcite , but the total number of observations is relatively small . we conclude that the enhanced binarity is a real effect . what this fact tells us about the origin of open clusters remains to be seen . our next step in addressing this basic issue is to try and account theoretically for the empirical properties just obtained through our statistical analysis . we will ascertain , using direct numerical simulations , the range of initial states that can relax dynamically to the present - day pleiades . such a study will bring us one step closer to understanding the molecular cloud environments that give rise to open clusters . we are grateful to eric huff , who first suggested the use of the maximum likelihood technique for this problem and provided continued insight . we also benefited from conversations with james graham and geoff marcy . the referee s comments helped to improve the final presentation . this research was supported by nsf grant ast-0639743 . | as part of an effort to understand the origin of open clusters , we present a statistical analysis of the currently observed pleiades . starting with a photometric catalog of the cluster , we employ a maximum likelihood technique to determine the mass distribution of its members , including single stars and both components of binary systems . we find that the overall binary fraction for unresolved pairs is 68% . extrapolating to include resolved systems , this fraction climbs to about 76% , significantly higher than the accepted field - star result . finally , we revisit the issue of mass segregation in the pleiades . | as part of an effort to understand the origin of open clusters , we present a statistical analysis of the currently observed pleiades . starting with a photometric catalog of the cluster , we employ a maximum likelihood technique to determine the mass distribution of its members , including single stars and both components of binary systems . we find that the overall binary fraction for unresolved pairs is 68% . extrapolating to include resolved systems , this fraction climbs to about 76% , significantly higher than the accepted field - star result . both figures are sensitive to the cluster age , for which we have used the currently favored value of 125 myr . the primary and secondary masses within binaries are correlated , in the sense that their ratios are closer to unity than under the hypothesis of random pairing . we map out the spatial variation of the cluster s projected and three - dimensional mass and number densities . finally , we revisit the issue of mass segregation in the pleiades . we find unambiguous evidence of segregation , and introduce a new method for quantifying it . |
astro-ph0312263 | c | b0218 + 357 is one of the most promising lens systems for determining the hubble constant with the method of @xcite . this method has , compared with other approaches , the advantage that it is a very simple one - step determination and relies on the understanding of only very little astrophysics . in this way the systematic uncertainties can not only be minimized but ( equally important ) also be estimated much better than in distance ladder methods which have their problems with several astrophysical processes at each step . the only serious uncertainty left is the mass distribution of the lens . in this respect b0218 + 357 is a close to optimal ` golden lens ' . the lensing galaxy seems to be a regular isolated galaxy without contributions from a group or cluster nearby . this not only avoids the inclusions of further parameters to describe the external mass distribution but also allows the assumption that the mass distribution of the galaxy itself is that of an unperturbed smooth ellipsoid . b0218 + 357 provides a wealth of potential constraints for the mass models in the form of highly accurate image positions , a measurable flux density ratio with probably negligible influence by microlensing , substructure on scales resolved by vlbi and a well resolved and structured einstein ring on the scale of the image separation . without using the ring , ` classical ' lens modelling allows us to tightly constrain the ellipticity and mass of the galaxy as long as its position is known . the vlbi structure of several subcomponents along the jet in a radial direction relative to the lens allows us to determine the radial mass distribution with an accuracy not possible in most other lenses ( see e.g. * ? ? ? * and references therein ) . this most important general degeneracy of the lens method can therefore be broken in the case of b0218 + 357 . the only real disadvantage is the small size ( @xmath89 ) of the system . b0218 + 357 is indeed the lens with the smallest image separation known to date . although the images and the lens are detected with optical observations , it was until now not possible to use these observations to accurately measure the galaxy position directly . _ hst_/acs observations obtained recently are currently analysed to allow a first useful direct measurement . the model lens positions from @xcite are probably not too far from the truth and are indeed consistent with our results but their error bars are seriously underestimated . we have shown that , without using the ring , no useful estimate of the galaxy position and thus @xmath4 is possible . previous efforts have not utilized the most striking feature of b0218 + 357 , the beautiful einstein ring shown by radio observations ( e.g. * ? ? ? doing this is much more difficult than using parameters of compact components because the source itself has to be modelled as well , either implicitly or explicitly . the method best suited for this task is lensclean which surprisingly has been used only for very few cases before . this is partly a result of the high numerical demands and partly of serious shortcomings of the original method as it was implemented by @xcite and @xcite . in paper i we discuss the development of our new improved variant of lensclean . one of the most important improvements is the implementation of a new unbiased selection of components . the problem of determining the lens position is almost degenerate because the bright components , which contribute most of the signal , provide no information for this parameter . it is only the relatively weak and fuzzy ring which can be used as constraint . in other words the residuals have a very strong dependency on two directions ( defined by the compact components ) of the five dimensional parameter space but change only weakly in the other three directions as a result of the constraints provided by the ring . finding minima of functions like this is a numerically difficult problem and relies on the very accurate knowledge of the residual function . any numerical noise produces local minima which make the finding of the global minimum difficult . special care was therefore needed to produce the best possible results with the given limited computing resources . the modified lenscleanalgorithm was applied to a 15ghz vla data set resulting in good constraints for all parameters of an _ isothermal _ lens model , including the lens position of @xmath90mas , @xmath91mas relative to the a image . this model was then used to determine the hubble constant from the time - delay of @xcite to be @xmath92 . the error bars are @xmath49 confidence limits which for the hubble constant include the error of the lens position ( 6 per cent ) and the time delay ( 4 per cent ) . the accuracy for all other lens model parameters is much higher as long as the position is fixed . the value for @xmath4 is in agreement with the results from the _ hst _ key project @xcite and the wmap project @xcite but incompatible with the lower distance - ladder results of @xcite , @xcite or @xcite . recently , a series of papers was published by @xcite , see also @xcite , in which it is claimed that a number of gravitational lens time - delays is compatible with the hubble constant from the _ hst _ key project only if the mass concentration is as compact as the light distribution ( @xmath93 in the picture of power - law models ) . our result for b0218 + 357 does not confirm this view and does not give rise to a ` new dark matter problem ' since it is significantly higher than other results from lenses . our isothermal models lead to a value absolutely compatible with the one preferred in @xcite . one could now in a similar way compare with the lower @xmath4 results of @xcite , @xcite or @xcite . however , as long as the local determinations do not agree with each other , we consider such an exercise to be of only limited value . rather than using locally measured values for @xmath4 to constrain lens mass distribution , we prefer a more direct approach , either using the lens effect itself ( see below ) , or by including additional information . the latter approach is followed by thy ` lenses structure and dynamics ' survey lsd @xcite . the general idea is simple : the total mass within the einstein radius is well constrained by lensing , in contrast to the radial mass _ profile_. for the given mass , the stellar velocity dispersion of the lensing galaxy depends strongly on this profile . the more the mass is concentrated in the centre , the deeper the potential well and the higher the velocity dispersion has to be . with some additional assumptions , it is indeed possible to obtain valuable constraints which can then again be included in the lens models to determine @xmath4 . for the important system pg 1115 + 080 , @xcite obtain a power - law index of @xmath94 which increases the otherwise very low estimate of @xmath4 for this system to a value of @xmath95 ( @xmath6 ) . this steep mass profile is quite unusual and moves the galaxy significantly off the fundamental plane . nevertheless it is not as steep as the _ light _ profile and the result for @xmath4 correspondingly lies between the isothermal and constant @xmath96 model results of @xcite . for b1608 + 656 , @xcite are able to constrain the mass profile of the main lensing galaxy quite well and find that it is consistent with isothermal . the resulting hubble constant is @xmath97 ( @xmath98 , inconsistent with the very low mean value of @xmath99 obtained by @xcite for isothermal models for a number of lenses , but consistent with our result for b0218 + 357 . however , b1608 + 656 has the major disadvantage of a second lensing galaxy very close to the primary . the two galaxies might even be interacting which could cause complicated deviations from usual galaxy mass distributions . these examples show that the general picture of lensing constraints for @xmath4 and mass profiles is not as consistent as it appeared not long ago ( e.g. * ? ? ? * ) . in our opinion the only way to resolve the discrepancies is to try and use the best available constraints for all applicable lens systems individually . our work on b0218 + 357 shows this for one example , although there are still many open questions . our alternative way of constraining mass profiles relies on the lens effect alone and has the advantage that additional astrophysics ( galactic dynamics in lsd work ) and corresponding additional model assumptions are not needed , keeping the method simple and clean . it is easy to understand that lenses showing only two or four images of one compact source can not provide sufficient information to constrain the mass profile tightly , especially since quads usually have all images close to the einstein ring so that they probe the potential at only one radius . far better suited are systems with extended sources or at least with substructure in the compact images . b0218 + 357 is a good example which offers both . using our galaxy position as an estimate , non - isothermal power - law models could be constrained with the vlbi substructure . these are not the final results since no self - consistent fitting of all available data has been performed so far , but the weak dependency of the radial mass exponent @xmath26 on the galaxy positions shows that the estimate is nevertheless quite good . we learned that the mass distribution in the lens is close to isothermal ( which would be @xmath3 ) but is slightly shallower ( @xmath100 ) . preliminary calculations with lensclean showed that the shift in the best lens position caused by this small deviation partly compensates for the expected scaling with @xmath72 for a fixed position . we therefore do not expect very significant changes of the result for @xmath4 from the slight non - isothermality in this system . a very conservative error estimate disregarding this compensation effect would be a possible 4 per cent error for @xmath4 . it has to be kept in mind , however , that all results presented so far depend on the assumption of a power - law for the mass profile ( and potential and deflection angle ) . for more general mass models the constraints on @xmath26 are still interesting , but they only measure the _ local _ slope of the profile rather than the global mass profile . it is expected that power - laws can be used as a very good approximation for more general models for limited ranges of radii , and our model fits confirm this view in the case of b0218 + 357 . the lensclean results for the lens position presented here are therefore also valid for other mass profiles , but the value for the hubble constant may well have to be modified slightly . the mass models discussed here do not include external shear because it seems to be very small . the expected true external shear of 12 per cent @xcite would change the result of @xmath4 by about the same amount at most . more difficult to estimate is the possible influence of differences in the ellipticity of the inner and outer parts of the lensing galaxy which can effectively also act as external shear . the possible impact of the remaining higher - order aspects of the mass profile degeneracies and the effects of multi - component mass models ( bulge+disc+halo ) are currently under investigation , especially the aspect of what apparent isothermality in the central part of the galaxy , which should be dominated by the luminous matter in the bulge , means for the true mass distribution . in the future we will avoid the model fitting of vlbi components which are then used as constraints and instead use lensclean itself also on the existing vlbi data that show a wealth of structure in the images ( see fig . [ fig : comp ] ) . to be able to use non - isothermal models with lensclean , we developed the new method lentil for the inversion of the lens equation . with these two methods combined , a simultaneous fit of medium resolution vla data ( sensitive to the lens position ) and the existing high resolution vlbi data of @xcite , which are more sensitive to the radial mass profile , will not only improve the results for all parameters but will , by using self - consistent lens models , also provide realistic error estimates including all uncertainties . to be able to use the 8.4ghz data set it will be necessary to include the extinction and scatter broadening in the a image in the models . this is possible because the 15ghz data are only little affected by extinction so that the combination of the two allows estimates of this effect . since the vlbi constraints for the lens models are mainly given by the positions of subcomponents but not by their size , which is relatively independent of the lens model , the different sizes of components in a and b can be used to obtain better constraints on the scattering measure than the simple estimates from @xcite . the application of lensclean to the vlbi data will also show whether any significant amount of substructure in the mass distribution of the lens is needed to explain all features of the jet . the first results from @xcite seem to be compatible with no substructure , but a quantitative analysis can be done in the future . if significant substructure effects are present , it will also be necessary to correct the medium resolution data for them before using lensclean to determine the galaxy position . clumps of matter close to one of the images could change the magnification significantly which would then mimic a different lens model . the analysis of the vlbi data will show whether this may be the explanation for the ( compared with some other lenses ) relatively high result for @xmath4 . lensclean can also be used to study substructure from vlbi observations of other lenses . it will then be possible to do an analysis similar to e.g. @xcite but without any assumption about the true source structure . to improve the medium resolution side , we recently performed observations of b0218 + 357 at 15ghz using the fibre link between the vla and the vlba telescope pie town . this combination provides a resolution 1.52 times better than the vla alone . the long track observation of the total accessible hour angle range of 14 hours also improves the sensitivity which in combination with the improved resolution can reduce both statistical and ( even more important ) systematical errors . simulations performed before the observations showed that an accuracy of about 12@xmath101 for the lens position should be achievable . this would in comparison to the existing data be an improvement of a factor of @xmath102 . the remaining uncertainty in @xmath4 from the lens position alone will then shrink to below 2 per cent . possible uncertainties caused by calibration errors will also be reduced significantly . this new data set is currently being calibrated and will be analysed with lenscleansoon , alone and in combination with the vlbi data . as a secondary result of lensclean we presented maps of the brightness distribution of the source as well as improved lens plane maps , both produced with a method newly developed together with lensclean in paper i. this allowed the first view of b0218 + 357 s source as it would be seen without the distortion of the lens , but with improved resolution . the ring itself is caused by a bending jet which crosses the tangential caustic of the lens . in the future we will improve these results and also produce source plane polarization maps . the ring shows an interesting radial polarization pattern @xcite and the data will , when projected to the source plane , give us a very detailed and magnified view of the polarization structure in the jet of the source . finally we used lensclean to investigate the puzzling changes of the flux density ratio a@xmath76b with frequency . to test the two theories of either scattering induced extinction caused by the ism in the lensing galaxy or of an effective shift of the source with frequency , we compared our best lensclean models , which were fitted to 15ghz data where the fluxes are expected to be unaffected by propagation effects , with merlin data at 5ghz taking into account a possible relative shift and extinction in the 5ghz data . although the 5ghz data set has some calibration problems , the evidence for a significant extinction ( @xmath103 per cent ) of the a component at 5ghz is very strong . the relative shift of the source between these two frequencies seems to be negligible . this work will be extended in the future by comparing medium resolution data sets at different frequencies in order to study the propagation effects in the lensing galaxy in more detail . observations with merlin at different frequencies will allow us to measure the position dependent faraday rotation and depolarisation , providing invaluable information about the ism of the lensing galaxy . a direct measurement of the galaxy position in b0218 + 357 by _ hst_/acs observations will be available soon . the formal uncertainty of this measurement may be larger than the lensclean model constraints but it will be an absolutely independent and therefore complementary direct measurement . the comparison will either confirm the lens models or give information about deviations of the true mass distribution from the relatively simple models . we will then reach the point were the uncertainty in @xmath4 is no longer dominated by the lens models but by the time - delay uncertainties themselves . a reanalysis of existing monitoring data @xcite and new monitoring campaigns will therefore allow further improvements . with all the constraints available now or soon and the simplicity of the possible mass models , b0218 + 357 has the potential to lead to the most robust measurement of @xmath4 of all time - delay lenses . | b0218 + 357 is one of the most promising systems to determine the hubble constant from time - delays in gravitational lenses . consisting of two bright images , which are well resolved in vlbi observations , plus one of the most richly structured einstein rings , it potentially provides better constraints for the mass model than most other systems . the main problem left until now was the very poorly determined position of the lensing galaxy . after presenting detailed results from classical lens modelling it is , however , compatible with local estimates from the _ hst _ key project and with wmap results , but less prone to systematic errors . we furthermore discuss possible changes of these results for different radial mass profiles and find that the final values can not be very different from the isothermal expectations . maps of the unlensed source plane produced from the best lensclean brightness model show a typical jet structure and allow us to identify the parts which are distorted by the lens to produce the radio ring . finally we use a comparison of observations at different frequencies to investigate the question of possible weakening of one of the images by propagation effects and/or source shifts with frequency . the technical details of our variant of the lensclean method are presented in the accompanying paper i. quasars : individual : jvas b0218 + 357 gravitational lensing distance scale techniques : interferometric | b0218 + 357 is one of the most promising systems to determine the hubble constant from time - delays in gravitational lenses . consisting of two bright images , which are well resolved in vlbi observations , plus one of the most richly structured einstein rings , it potentially provides better constraints for the mass model than most other systems . the main problem left until now was the very poorly determined position of the lensing galaxy . after presenting detailed results from classical lens modelling , we apply our improved version of the lensclean algorithm which for the first time utilizes the beautiful einstein ring for lens modelling purposes . the primary result using isothermal lens models is a now very well defined lens position ofmas relative to the a image , which allows the first reliable measurement of the hubble constant from the time - delay of this system . the result of is very high compared with other lenses . it is , however , compatible with local estimates from the _ hst _ key project and with wmap results , but less prone to systematic errors . we furthermore discuss possible changes of these results for different radial mass profiles and find that the final values can not be very different from the isothermal expectations . the power - law exponent of the potential is constrained by vlbi data of the compact images and the inner jet to be , which confirms that the mass distribution is approximately isothermal ( corresponding to ) , but slightly shallower . the effect on is reduced from the expected 4 per cent decrease by an estimate shift of the best galaxy position of ca . 4mas to at most 2 per cent . maps of the unlensed source plane produced from the best lensclean brightness model show a typical jet structure and allow us to identify the parts which are distorted by the lens to produce the radio ring . we also present a composite map which for the first time shows the rich structure of b0218 + 357 on scales ranging from milli - arcseconds to arcseconds , both in the image plane and in the reconstructed source plane . finally we use a comparison of observations at different frequencies to investigate the question of possible weakening of one of the images by propagation effects and/or source shifts with frequency . the data clearly favour the model of significant ` extinction ' without noticeable source position shifts . the technical details of our variant of the lensclean method are presented in the accompanying paper i. quasars : individual : jvas b0218 + 357 gravitational lensing distance scale techniques : interferometric |
1509.00252 | i | the self - consistent mean - field approach , based on an energy density functional ( edf ) , is a tool of choice to study nuclei in any region of the nuclear chart @xcite . it allows to calculate the properties of the ground state but also of alternative configurations , like shape isomers , or to follow the behaviur of a nucleus along rotational bands or along fission paths . often , one is not directly interested in the total binding energy of a specific nucleus but by its evolution along a series of isotopes or isotones , which can signal structural changes for given neutron or proton numbers . motivated by the needs of the nuclear physics and astrophysics communities , large efforts are underway to push the predictive power of nuclear mass models well below the 500kev level . to reach this goal , the protocols used to adjust the edf s parameters have been revisited . in particular , methods are being developed @xcite to quantify the statistical uncertainty of these parameters . however , besides the errors on observables due to these uncertainties , there is also a numerical error due to the way the scmf equations are solved . one needs to verify that the numerics does not introduce errors that are larger than the maximum error tolerated for mass models . more importantly , these errors should not vary too rapidly from one nucleus to the other , to avoid a spurious behavior of mass differences . the numerical methods used to solve the mean - field equations can be classified according to the way the single - particle wave functions are represented : by coordinate space techniques or by a basis expansion . coordinate space techniques represent the single - particle wave functions in a discretized , finite volume . several discretization techniques exist , utilizing finite difference formulas @xcite , fourier transformations @xcite , b - splines @xcite , wavelets @xcite and the lagrange mesh method @xcite . the second family of numerical representations involves expanding the single - particle wave functions on some chosen ( finite ) set of basis states . usually these basis states are harmonic oscillator ( ho ) eigenstates , although the details often vary . while the origin of numerical errors is quite different for both families of representations , the type of edf does not seem to influence the accuracy of the methods much . the three main families ( relativistic edfs , zero - range skyrme edfs or finite - range edfs ) require similar numbers of basis states to achieve a similar precision ( see e.g. @xcite ) . in what follows , we will limit ourselves to the study of zero - range skyrme edfs . it is the aim of this paper to study the numerical accuracy of a specific implementation of coordinate space techniques : representation on a three - dimensional cartesian mesh of equidistant points . we will focus on two specific techniques : finite - difference ( fd ) formulas and the lagrange mesh ( lm ) method , which are the ones implemented in our codes . as far as we can infer from the tests published in the literature , the accuracy obtained with the other techniques mentioned above is similar to the one obtained within our lm scheme . most of the information relative to the tools that we have developed has been presented for the particular implementation made in the code ` ev8 ` @xcite . more involved implementations have also been used , which differ from ` ev8 ` only in that they impose less symmetries on the nucleus . the presence of these symmetries in general allows for the reduction of the dimension of the problem , e.g. in ` ev8 ` it allows for the reduction of the space by a factor of 1/8 . the article is organized as follows : first we define precisely the quantities that will be used to characterize the accuracy of a mean - field calculation . next , we review the basic ingredients needed to define wave functions on a cartesian mesh and to calculate derivatives and integrals in this representation . we then discuss the main sources of numerical errors : the size of the box in which the nucleus is confined and the step size of the mesh . we discuss the numerical accuracy that can be achieved by comparing energies and radii of doubly magic nuclei with those obtained with a spherical code . finally , we check the convergence of energies , radii and the multipole moments of deformed nuclei by comparing results obtained with decreasing mesh discretization lengths . | background : : mean - field methods based on an energy density functional ( edf ) are powerful tools used to describe many properties of nuclei in the entirety of the nuclear chart . we study this numerical accuracy in detail for a specific numerical choice of representation for the mean - field equations that can accommodate any kind of symmetry breaking . | background : : mean - field methods based on an energy density functional ( edf ) are powerful tools used to describe many properties of nuclei in the entirety of the nuclear chart . the accuracy required on energies for nuclear physics and astrophysics applications is of the order of 500 kev and much effort is undertaken to build edfs that meet this requirement . purpose : : the mean - field calculations have to be accurate enough in order to preserve the accuracy of the edf . we study this numerical accuracy in detail for a specific numerical choice of representation for the mean - field equations that can accommodate any kind of symmetry breaking . method : : the method that we use is a particular implementation of 3-dimensional mesh calculations . its numerical accuracy is governed by three main factors : the size of the box in which the nucleus is confined , the way numerical derivatives are calculated and the distance between the points on the mesh . results : : we have examined the dependence of the results on these three factors for spherical doubly - magic nuclei , neutron - rich , the fission barrier of and isotopic chains around . conclusions : : mesh calculations offer the user extensive control over the numerical accuracy of the solution scheme . by making appropriate choices for the numerical scheme the achievable accuracy is well below the model uncertainties of mean - field methods . |
1402.4220 | i | functional models from analytic functions were developed by sz.nagy-foias @xcite and used for classifying contractive operators on hilbert spaces . a similar approach was used by popescu in @xcite for classifying row contractions by certain multi - analytic operators . these classifying objects were called characteristic functions . in @xcite and more explicitly in @xcite characteristic functions of liftings of row contractions were introduced and it was shown that they are complete invariants for unitary equivalence in a certain class of liftings . here we present an approach which is based on a systematic use of associated functional models which on the one hand exhibits popescu s characteristic functions as special cases of characteristic functions of liftings and which on the other hand fully discloses the additional potentials of our generalization . + let us immediately introduce the two main players . \(1 ) let @xmath1 , the _ full fock space _ over @xmath2 , and let @xmath3 be the standard basis of @xmath4 . denote the left creation operator w.r.t . @xmath5 on @xmath6 by @xmath7 , that is @xmath8 for @xmath9 . let @xmath10 and @xmath11 be hilbert spaces . a linear operator @xmath12 is called _ multi - analytic _ ( cf . @xcite ) if it intertwines @xmath13 for all @xmath14 . a multi - analytic operator @xmath15 is determined by its symbol @xmath16 defined by @xmath17 for all @xmath18 , here @xmath19 denotes the standard basis vector of @xmath20 . we call @xmath15 _ contractive _ if @xmath21 . note that for @xmath22 and @xmath23 a contractive multi - analytic operator corresponds exactly to multiplication with a function in the schur class , i.e. , a bounded analytic function in the open unit disk with supremum norm at most @xmath0 . \(2 ) a @xmath24-tuple @xmath25 of operators on a hilbert space @xmath26 is called a _ row contraction _ if it is a contraction from @xmath27 to @xmath26 or , equivalently , if @xmath28 . if a @xmath24-tuple @xmath29 on a hilbert space @xmath30 can be written in the form @xmath31 for suitable @xmath24-tuples @xmath32 and @xmath33 then @xmath34 is called a _ lifting _ of @xmath35 . the lifting is called _ contractive _ if @xmath34 is still a row contraction and it is called _ minimal _ if @xmath36 is the smallest @xmath34-invariant subspace containing @xmath26 . ( by @xmath34-invariance we mean invariance for all @xmath37 . ) we remark that it presents no particular difficulties to include sequences of operators ( @xmath38 ) but we write all formulas as if @xmath24 is finite . to establish a correspondence between the two main players we start with a detailed discussion of functional models . we use the generalized setting introduced by popescu in @xcite to study row contractions . the case @xmath22 ( single contractions ) is of course of special interest and our results are new also for @xmath22 but these results work just as well for general @xmath24 . this observation is important because from the case @xmath39 there are promising applications to the dynamics of open quantum systems , see @xcite for an introduction to this topic and further references along these lines . the impact of our results on these applications will be worked out elsewhere . in section 2 we prove properties of functional models to be used later . the results about functional models which are new depend on observations about the geometry of an invariant subspace for a row isometry which we derive from a geometric lemma proved in an appendix to this paper . they establish relations between the positions of certain subspaces on the one hand and properties of the symbol @xmath40 of the multi - analytic operator on the other hand . section 3 is the core of the paper . based on the results about the functional model and extending ideas from @xcite , we work out a mapping @xmath41 which , for any given row contraction @xmath35 , maps contractive multi - analytic operators @xmath42 , where @xmath43 is the defect space of @xmath35 , to contractive liftings @xmath34 of @xmath35 . in the converse direction we make use of the theory of the minimal isometric dilation for the contractive lifting @xmath34 to construct a mapping @xmath44 from contractive liftings @xmath34 of a given row contraction @xmath35 to contractive multi - analytic operators @xmath42 . suitably restricted , the maps @xmath41 and @xmath44 become inverses of each other and we obtain a one - to - one correspondence between unitary equivalence classes of minimal contractive liftings and equivalence classes of injective symbols . this justifies to call the corresponding multi - analytic operators characteristic functions of liftings , as has been done already in @xcite . with the theory developed here we provide a complete answer to the open problem posed at the end of section 3 in @xcite , namely to classify the multi - analytic operators which can occur as characteristic functions of liftings . the surprisingly simple answer , all that is needed is the easily checked property of injectivity of the symbol , shows that the connection with liftings is a very natural application of multi - analytic operators and makes it now much easier to develop the applications . we can always study minimal contractive liftings via the corresponding symbols . as an example of such an application we study the factorization of liftings with the help of the corresponding factorization of the characteristic functions ( compare @xcite , @xcite ) . in section 4 we revisit popescu s work in @xcite where he defined a characteristic function for a completely non - coisometric row contraction as a certain multi - analytic operator . we show that this can be considered as a special case of our theory in the sense that popescu s characteristic functions appear as characteristic functions of a special type of liftings and that the property of being a complete invariant for unitary equivalence follows from the corresponding result about liftings in section 3 . in section 5 we look at a class of examples : minimal contractive liftings @xmath45 of a single contraction @xmath46 such that both @xmath46 and @xmath45 have defect equal to @xmath0 . by our theory the unitary equivalence classes of these liftings are in one - to - one correspondence with non - zero schur functions on the unit disk ( up to unimodular complex factors ) . this gives us an opportunity to illustrate many of the previous results by easily computable examples . already in this case in the future there is a lot more to find out about these liftings by making a more systematic use of what is known about schur functions . for example , operator - valued schur functions have been used for commutant lifting , see @xcite and more recently @xcite . related work in different directions is done in @xcite , @xcite , @xcite , @xcite , @xcite , @xcite , @xcite and @xcite . but the explicit parametrization of all minimal contractive liftings achieved here is new and provides an excellent basis for studying applications , for example the dynamics of open quantum systems mentioned earlier . in fact in the special case of the minimal isometric dilation it is long known and well studied how it describes the embedding of open into closed quantum systems , see for example @xcite . let us finish this introduction with a short reminder of the well known theory of the _ minimal isometric dilation _ @xmath47 of a row contraction @xmath48 which appears in several places in this paper . it was first presented in @xcite , with small modifications we use the notation from @xcite . in both papers a lot of additional details can be found . recall that a @xmath24-tuple @xmath49 on a hilbert space @xmath50 is called a _ row isometry _ if @xmath51 or , equivalently , the @xmath52 are isometries with orthogonal ranges . for any row contraction @xmath48 on a hilbert space @xmath53 there exists a row isometry @xmath54 on a bigger hilbert space such that @xmath55 for all @xmath56 . here we use , as in similar cases , the notation @xmath57 for the orthogonal projection onto @xmath58 and the notation @xmath59 for any word @xmath60 with letters @xmath61 to stand for the operator @xmath62 . we call @xmath63 the length of the word . for all @xmath56 means here and later : for all such words of all possible lengths including the empty word @xmath64 of length @xmath65 which corresponds to the identity operator . a row isometry @xmath66 with this property is called an isometric dilation of @xmath67 . it is easily seen that this is actually a very specific example of a contractive lifting in the sense introduced earlier . if we require minimality for an isometric dilation in the sense that the bigger hilbert space is the smallest closed @xmath66-invariant space containing @xmath53 then this determines @xmath66 up to unitary equivalence and we denote it by @xmath47 , on the hilbert space @xmath68 . there is a construction of the minimal isometric dilation analogous to the schffer construction for a single contraction : recall that the operator @xmath69 given by @xmath70 is called the _ defect operator _ , @xmath71 is called the _ defect space _ and the ( hilbert space ) dimension of @xmath72 is called the _ defect _ of @xmath67 . the minimal isometric dilation @xmath47 is then realized in a canonical way on the hilbert space @xmath73 . again @xmath1 is the full fock space over @xmath2 and we note that @xmath47 restricted to @xmath74 is nothing but @xmath75 , with @xmath76 , the @xmath24-tuple of creation operators on @xmath6 . we refer to the row contraction @xmath75 as the _ canonical row shift _ and to @xmath77 as its _ generating wandering subspace_. ( as usual , the subspace is called wandering for @xmath75 because its translates @xmath78 over all words @xmath56 , including the empty word @xmath64 , are orthogonal to each other . ) the minimal isometric dilation @xmath47 is obtained from any isometric dilation @xmath66 by restriction . the unitary equivalence to the canonical construction above is expressed by a canonical unitary from the defect space @xmath72 onto the @xmath66-wandering subspace @xmath79 given by @xmath80 . it can be extended to a canonical unitary from @xmath81 onto @xmath82 ( over all words @xmath56 , including the empty word @xmath64 ) intertwining the two versions of the minimal isometric dilation . compare @xcite and @xcite . | based on a careful analysis of functional models for contractive multi - analytic operators we establish a one - to - one correspondence between unitary equivalence classes of minimal contractive liftings of a row contraction and injective symbols of contractive multi - analytic operators . this allows an effective construction and classification of all such liftings with given defects . | based on a careful analysis of functional models for contractive multi - analytic operators we establish a one - to - one correspondence between unitary equivalence classes of minimal contractive liftings of a row contraction and injective symbols of contractive multi - analytic operators . this allows an effective construction and classification of all such liftings with given defects . popescu s theory of characteristic functions of completely non - coisometric row contractions is obtained as a special case satisfying a szeg condition . in another special case of single contractions and defects equal to all non - zero schur functions on the unit disk appear in the classification . it is also shown that the process of constructing liftings iteratively reflects itself in a factorization of the corresponding symbols . |
0707.4056 | i | according to the nonrelativistic qcd ( nrqcd ) factorization formalism , an annihilation decay rate of a spin - triplet @xmath13-wave ( @xmath14 ) bottomonium @xmath15 in @xmath0 , where @xmath16 is the radial quantum number , unless it is necessary . ] is expressed as an infinite series of nrqcd matrix elements with corresponding short - distance coefficients @xcite . the nrqcd matrix elements , which reflect the long - distance nature of the quarkonium , scale as powers of the bottom - quark velocity @xmath2 in the quarkonium rest frame , which is @xmath17 . at leading order in @xmath2 , the inclusive decay rate of the @xmath15 is dominated by the color - singlet spin - triplet contribution whose nrqcd matrix element is @xmath18 , which is defined in ref . the subscript @xmath19 on the nrqcd four - quark operator @xmath20 denotes that it is a color - singlet operator . thus , at leading order in @xmath2 , the inclusive light - hadronic decay rate of the @xmath15 is expressed in a factorized form : @xmath21 = c_1\ , \frac{\langle \mathcal{o}_1({}^3s_1)\rangle_{\upsilon}}{m_b^2 } , \label{gam - ups}% % ------------------\ ] ] where @xmath22 represents all possible light hadronic final states into which @xmath15 can decay and @xmath23 is the bottom - quark mass . the short - distance coefficient @xmath24 , which is insensitive to the long - distance nature of the @xmath15 , can be calculated perturbatively . the dimensions of the matrix element @xmath25 is @xmath6 so that @xmath24 is dimensionless . at leading order in @xmath1 , the dominant color - singlet contribution to @xmath24 comes from @xmath26 mode , where the three gluons are attached to the bottom - quark line . here , @xmath1 is the strong coupling constant and @xmath27 is the color - singlet @xmath10 pair with spin @xmath28 , orbital angular momentum @xmath29 , and total angular momentum @xmath30 . the leading contribution of order @xmath31 to @xmath24 is known through the orthopositronium decay rate obtained by caswell , lepage , and sapirstein @xcite . the order-@xmath32 corrections to @xmath24 were calculated by mackenzie and lepage @xcite . this result was confirmed recently by campbell , maltoni , and tramontano @xcite . in addition to the three - gluon mode , @xmath24 may include the virtual - photon contribution from @xmath33 . the decay rate is of order @xmath34 , where @xmath35 is the qed coupling constant and @xmath36 is the fractional electric charge of the quark @xmath37 : @xmath38 for an up - type quark and @xmath39 for a down - type quark . that electromagnetic decay rate may appear to be highly suppressed compared to the three - gluon mode of order @xmath31 . however , we can make a rough estimate of the branching fraction @xmath40 $ ] by using the measured branching fractions for @xmath41 $ ] @xcite : @xmath42\approx% n_c\textrm{br}[\upsilon\to e^+e^-]\sum_q e_q^2 $ ] , where the sum is over the four flavors of quarks lighter than the bottom and @xmath43 is the number of colors . according to this estimate , @xmath44\approx 6$]@xmath45 , which may not be negligible . at higher orders in @xmath2 , the nrqcd factorization formula ( [ gam - ups ] ) must include additional contributions from higher fock states which involve color - octet pairs @xmath46 as well as the color - singlet ones , which are suppressed compared to the leading contribution in eq . ( [ gam - ups ] ) . the order-@xmath47 and order-@xmath48 relativistic corrections to the color - singlet contributions were calculated by keung and muzinich @xcite and by bodwin and petrelli @xcite , respectively . some of the color - octet contributions were also calculated in refs . @xcite . because the @xmath15 is heavy enough , the decay products may include a pair of charmed hadrons . however , unlike the light - hadronic decay mode of the @xmath15 , there has been little previous work on open - charm production in @xmath15 decay . in 1978 , fritzsch and streng predicted the branching fraction of the decay of @xmath15 into charm to be a few percents @xcite , where they considered @xmath49 followed by @xmath50 . in 1979 , bigi and nussinov took into account a fusion process @xmath51 of order @xmath52 , in which a pair of virtual gluons create the @xmath53 pair @xcite . in 1992 , argus experiment searched for charm production in direct decays of the @xmath54 to find only an upper limit of @xmath55<0.019 $ ] @xcite . recent runs of the cleo iii experiment have produced a large amount of data samples at the @xmath54 , @xmath56 , and @xmath57 resonances . the @xmath58-factory experiments babar and belle have accumulated data for @xmath56 and @xmath57 provided by initial - state radiation . the belle collaboration has also collected data by running on the @xmath57 resonance . with these high - luminosity data , one can now indeed study open - charm production in @xmath15 decay . very recently , some of the authors have calculated the total production rates and momentum distributions of the charm quark and charmed hadrons , respectively , in the @xmath59-wave bottomonium decays @xmath60 for @xmath61 , 1 , and 2 by using the nrqcd factorization formalism @xcite . in order to calculate the momentum distribution of the charmed hadrons they used the momentum distribution of charmed hadrons measured by the belle collaboration in @xmath62 annihilation @xcite . in this work , as an extension of a previous study @xcite , we consider inclusive charm production in the spin - triplet @xmath13-wave bottomonium decay . at leading order in @xmath2 , the dominant mechanism for the decay is a color - singlet channel @xmath63 followed by @xmath50 . as we have described earlier , the color - singlet mode may have significant virtual - photon contribution from @xmath64 followed by @xmath11 . for inclusive charm production , the virtual - photon contribution may have a larger fraction than that in the inclusive light - hadronic decay because the rate for @xmath65 is suppressed by order @xmath1 compared to that for @xmath66 . we consider the virtual - photon contribution as well as the qcd contributions from @xmath67 and @xmath68 modes . in the current cleo iii analysis on the charmed - hadron ( @xmath69 ) momentum distribution in @xmath15 decay , the virtual - photon contribution is subtracted experimentally @xcite . we therefore also present the results for the qcd contributions after excluding the virtual - photon process . this paper is organized as follows . in sec . [ sec : charm ] , we present the nrqcd factorization formulas for the inclusive charm production rate and charm momentum distribution in the spin - triplet @xmath13-wave bottomonium decay . we also discuss the nrqcd matrix element that appear as a long - distance factor in the factorization formula . we calculate the charm - quark momentum distribution and the total rate for inclusive charm production in the decay in sec . [ sec : rate ] . in sec . [ sec : meson ] , in order to provide a theoretical prediction that can be compared with cleo iii data , we illustrate the charmed - hadron momentum distributions which are obtained by convolving the charm - quark momentum distribution with fragmentation functions for @xmath70 that have been fit to @xmath62 annihilation data . finally , a brief summary of this work is given in sec . [ summary ] . | based on the nrqcd factorization formalism , we calculate the inclusive charm production rate in decay at leading order in the strong coupling constant and the relative velocity of the quark in the quarkonium rest frame . the branching fractions for to charm for , , and are all around% . about% of the branching fraction into charm is from annihilation of the color - singlet pair into . most of the remaining branching fraction is from annihilation of the color - singlet pair decaying into . we also compute the momentum distributions of the charm quark and charmed hadrons in the decays . the virtual - photon contribution to the charm quark momentum distribution | based on the nrqcd factorization formalism , we calculate the inclusive charm production rate in decay at leading order in the strong coupling constant and the relative velocity of the quark in the quarkonium rest frame . the branching fractions for to charm for , , and are all around% . about% of the branching fraction into charm is from annihilation of the color - singlet pair into . most of the remaining branching fraction is from annihilation of the color - singlet pair decaying into . we also compute the momentum distributions of the charm quark and charmed hadrons in the decays . the virtual - photon contribution to the charm quark momentum distribution is concentrated at the end point while the contribution is broad . the momentum distributions for charmed hadrons are obtained by convolving the charm - quark momentum distribution with charm fragmentation functions . this makes the momentum distributions for charmed hadrons softer than that for the charm quark . this effect is particularly significant in the virtual - photon contribution . |
1310.1579 | i | chromosomes , the genetic material of a cell , are duplicated and properly segregated before the cell divides into two daughter cells @xcite . each of the sister chromatids , that result from chromosome replication , is bound to a proteinous structure , called kinetochore ( kt ) that , in turn , is coupled to the plus ends of stiff polar filaments called microtubules ( mt ) @xcite . the negative ends of these mts are located at the poles of the fusiform structure , called mitotic spindle @xcite . the process of chromosome segregation , called mitosis @xcite , is carried out in eukaryotic cells by the dynamic mitotic spindle which is self - organized from its components for this purpose . there are strong indications that molecular motors @xcite are located at the kt - mt interface ( though not in all eukaryotes ) ; these motor proteins , which are capable of generating force by hydrolyzing atp , are believed to generate poleward force or anti - poleward force depending on the family to which they belong . however , polymerizing and depolymerizing mts are also found to make significant contributions to the forces that cause chromosomal movements @xcite . the force - generation capability of depolymerizing mts have been demonstrated _ in - vitro _ however , how a mt maintains contact with the kinetochore while depolymerizing from its tip and how it exerts force on the kt remain challenging open questions in spite of the recent progress in identifying the molecular components of a kt and their spatial organization @xcite , moreover , in most of the eukaryotic organisms multiple mts couple to each kt ; how the ( de-)polymerization of mutiple mts are coordinated is still a mystery . _ budding yeast _ is a much simpler system because each kt is coupled to only a single mt . but , even in budding yeast , the two embraced sister chromatids are coupled to two mts approaching them from the two opposite poles of the spindle . in this paper we theoretically study an even simpler system , namely an isolated kt coupled to a single mt ; this system mimics an _ in - vitro _ experiment @xcite where a single kt - mt attachment was reconstituted using purified extracts from budding yeast . in those _ in - vitro _ experiments the average lifetime was found to vary nonmonotonically with the externally applied load tension . in other words , the kt - mt attachment is stabilized ( instead of getting destabilized ) at sufficiently small load tension . similar tension - induced stabilisation of attachment between e - coli fimbriae and target cells have been interpreted in the past in terms of the concept of `` catch - bond '' @xcite . akiyoshi et al . @xcite developed a 2-state kinetic model that involves four phenomenological rate constants . by assuming the rate constants to vary exponentially with the load tension , they obtained a reasonably good fit of the model predictions with their experimental data . here we do not attempt any quantitative comparison with the experimental data reported in ref.@xcite . instead , our approach allows to derive _ analytical _ expressions for the lifetimes of the kt - mt attachments within the framework of the simple theoretical models that we develop here . these models describe the polymerization- depolymerization of the mt explicitly and capture the effects of its interaction with the coupler by a potential energy landscape that gets modified when the mt is subjected to an external tension . therefore , our results for the mean lifetimes provide deep physical insight into the interplay of opposing forces and competing kinetic processes that , together , determine the stability of the kt - mt coupler . we also make theoretical predictions on the probability distribution of the lifetimes which also indicate the probability of survival of an attachment at least for a time interval @xmath0 after the attachment is established . in principle , our new predictions can be tested by analyzing the data collected in the _ in - vitro _ experiments provided the experiment is repeated sufficiently large number of times . this paper is organized as follows . in sections [ sec - minimalcont ] and [ sec - minimaldisc ] we develop a _ minimal theoretical model _ of a device that couples a single mt with a single kt and subject it to an external tension . space is represented by a continuous variable in section [ sec - minimalcont ] and by a discrete variable in section [ sec - minimaldisc ] . our results on the distribution of the lifetimes of the model kt - mt attachments , derived analytically in section [ sec - minimalcont ] , are compared with the corresponding numerical data , obtained by numerical simulations , in section [ sec - minimaldisc ] . next , in section [ sec - realistic ] , we extend the minimal model by representing the kt - mt interaction by a more realistic potential . calculation of the full distribution of the lifetimes in this case is too difficult to be carried out analytically . therefore , in this case we have directly calculated the mean lifetime of the kt - mt attachments and found that the realistic potential leads to quantitative changes without affecting the qualitative features of the kt - mt lifetimes observed in the minimal model . finally , motivated by recent evidence that the kt - mt coupler in mammalian cells might be a hybrid nano - device , composed of spatially separated active and passive components , we have extended our study also to a simple model of a hybrid coupler in appendix [ sec - hybrid ] . | before a cell divides into two daughter cells , chromosomes are replicated resulting in two sister chromosomes embracing each other . each sister chromosome is bound to a separate proteinous structure , called kinetochore ( kt ) , that captures the tip of a filamentous protein , called microtubule ( mt ) . we theoretically study an even simpler system , namely an isolated kt coupled to a single mt ; this system mimics an _ in - vitro _ experiment where a single kt - mt attachment is reconstituted using purified extracts from budding yeast . experiments provided the experiment is repeated sufficiently large number of times . our theory provides a deep insight into the effects of ( a ) size , ( b ) energetics , and ( c ) stochastic kinetics of the kt - mt coupling on the distribution of the lifetimes of these attachments . | before a cell divides into two daughter cells , chromosomes are replicated resulting in two sister chromosomes embracing each other . each sister chromosome is bound to a separate proteinous structure , called kinetochore ( kt ) , that captures the tip of a filamentous protein , called microtubule ( mt ) . two oppositely oriented mts pull the two kts attached to two sister chromosomes thereby pulling the two sisters away from each other . here we theoretically study an even simpler system , namely an isolated kt coupled to a single mt ; this system mimics an _ in - vitro _ experiment where a single kt - mt attachment is reconstituted using purified extracts from budding yeast . our models not only account for the experimentally observed `` catch - bond - like '' behavior of the kt - mt coupling , but also make new predictions on the probability distribution of the lifetimes of the attachments . in principle , our new predictions can be tested by analyzing the data collected in the _ in - vitro _ experiments provided the experiment is repeated sufficiently large number of times . our theory provides a deep insight into the effects of ( a ) size , ( b ) energetics , and ( c ) stochastic kinetics of the kt - mt coupling on the distribution of the lifetimes of these attachments . |
1310.1579 | i | in this paper , we have presented theoretical models of kt - mt attachment at three levels of detail starting with a minimal version that captures only the key components and essential processes . these models mimic kt - mt attachments reconstituted _ in - vitro _ from purified components extracted from budding yeast @xcite . exploiting the simplicity of the minimal version of the model , we have calculated the distribution of the lifetimes of the kt - mt attachments analytically and compared the results with the corresponding simulation data . equation ( [ main2 ] ) encapsulates the main result of this part of the manuscript . the strength of the minimal model is that the results can be derived analytically . in spite of its simplicity , this minimal model provides deep insight into the roles of opposing forces and competing kinetics in the tension - dependence of the life time of the kt - mt attachments . the second version of the model is an extension of the minimal model ; it incorporates some more realistic features of the interaction between the mt and the coupler . for the second model we have succeeded in calculating analytically only the mean lifetime of the kt - mt attachments under reasonable schemes of approximation . the analytical expression ( [ eq - meant ] ) for the mean life time of the kt - mt attachment in the minimal model has been derived from the corresponding full distribution ( [ main2 ] ) of the life times , the nonmonotonic variation of the mean lifetime of the kt - mt attachments with external tension exhibited over wide range of values of the model parameters in both the models is consistent with the similar trend observed in the _ in - vitro _ experiments @xcite . the realistic potential landscape in the second model does not alter this qualitative trend . we have also demonstrated that the nonmonotonicity of the variation of @xmath58 with @xmath8 depends on the rapidity of decrease of @xmath15 with increasing @xmath8 . by varying @xmath18 , which determines the sharpness of the decrease of @xmath15 , we have shown how the catch - bond - like behavior crosses over to a slip - bond - like behavior . moreover , our analytical calculations also show that , unlike other conventional catch - bonds , the catch - bond - like behavior of the kt - mt attachment arises from the interplay of not only forces derivable from the potential landscape , but also by a competition of the kinetics of polymerization and depolymerization of the mt . in fact , the barrier against the breakdown of the attachment gets contributions from both opposing forces and competing kinetic processes . the slip - bond - like monotonic decrease of @xmath121 with @xmath8 observed in some of the parameter regimes ( whose physical origin has been explained in terms of the nature of @xmath8-dependence of the depolymerization rate @xmath15 ) , might be detectable in experiments under conditions different from those used by akiyoshi et al.@xcite . as expected , we have found that the molecular motors enhance stability or tend to destabilize the kt - mt attachment depending on whether the force it generates opposes or assists the external tension . the latter awaits experimental confirmation . moreover , our analytical predictions on the distribution of the lifetimes will be useful in the future in analyzing the statistical properties of the lifetimes provided the _ in - vitro _ experiment on the reconstituted kt - mt attachment is repeated sufficiently large number of times . in principle , the models presented here can be further extended by including active force generators like molecular motors @xcite . however , so far such active force generators have not been detected at the kt - mt interface in budding yeast although such force - generating motors have been found in other cells including , for example , mammalian cells . but , in contrast to budding yeast , where each kinetochore can attach to only one mt , multiple mts can attach with each kinetochore in a mammalian cell . since modeling of kt - mt attachment in this paper is focussed almost exclusively on budding yeast , we have not discussed the effects of molecular motors in the main text . thus , the extended model in appendix b is not a complete model of kt - mt attachment in mammalian cells ; it is intended to indicate how the effects of molecular motors can be incorporated within the broad theoretical framework developed in this paper . in near future we hope to extend our model to kt - mt attachments in mammalian cells . however , at present , the main hurdle in this modeling is the lack of our current understanding as to how multiple mts attached to the same kt coordinate their kinetics @xcite . the `` sleeve model '' developed originally by hill @xcite was adapted by joglekar and hunt @xcite , to account for the phenomenon of `` directional instability '' @xcite ( also referred to as chromosome oscillation ) observed during mitosis in vertebrates . in those cells each kt can attach with several mt ( typically , up to 35 ) and this feature was also incorporated in the model studied by joglekar and hunt by carrying out computer simulations . our work differs significantly from that of joglekar and hunt @xcite . first , our model is intended to account for the qualititative features of the results obtained from a much simpler _ in - vitro _ system where only one mt is attached to only a single kt that is subjected to an externally applied load force that , in our model , is assumed to alter the rate of mt depolymerization itself . moreover , because of their motivation in studying directional instability , joglekar and hunt @xcite put major emphasis on monitoring the _ spatial _ displacements that characterize this instability . in contrast , the quantity of our main interest is the life _ time _ of the kt - mt attachments . the spatial excursions of the kinetochores was also the main quantity of interest of gardner et al.@xcite who studied the effects of tension - mediated regulation of the kinetics of kinetochore mts by computer simulation of a related model . the scope of the more detailed model studied by scholey et al.@xcite was even broader . our force - balance equations are similar , at least in principle , to the force - balance equations formulated by scholey et al.@xcite . however , instead of the spatial displacements of the kinetochores , the lifetimes of the kt - mt attachments are the main focus of our investigation . most of the earlier models were based on the assumption that the coupler is a stable sleeve or ring . in recent years , a new class of models have been developed on the basis of an altogether different assumption regarding the nature of the coupler . in this scenario @xcite microtubule associated proteins ( maps ) are assumed to make transient attachments with the mt ( i.e. , each map attaches to the mt for a brief duration before getting detached from it ) . the entropic contribution to the force is significant and slight variation in the mean lifetime of the individual transient attachments of the maps can have large effect on the lifetime of the kt - mt attachment . in order to account for their experimental observations , akiyoshi et al . @xcite developed a 2-state kinetic model . this model was motivated by the formal analogy with the models of ` catch - bonds ' in other systems . the growing and shrinking states of the mt were argued to be analogs of the strongly- and weakly - bound states in the models of catch - bonds . in their 4-parameter kinetic model , two parameters were the rates of transitions between the states of growth and shrinkage of the mt . the remaining two parameters were the rates of detachment of the kinetochore from the mt while the latter is in the growing and shrinking phases , respectively . postulating exponential dependence of the type @xmath149 for each of the four rate constants ( n=1,2,3,4 ) , on the external tension @xmath8 , akiyoshi et al . extracted the numerical values of the 8 parameters @xmath150 ( n=1,2,3,4 ) corresponding to the best fit to their experimental data @xcite . in their model , akiyoshi et al.@xcite did not explicitly treat the kinetics of the polymerization and depolymerization , by attachment and detachment of the successive subunits ( more precisely , the @xmath151 tubulin dimers ) , during the growing and shrinking phases of the mt , respectively . therefore , the force - dependence of the four rate constants that provide the best fit to their experimental data do not indicate the corresponding force - dependences of the rates of polymerization and depolymerization of the mt . moreover , the potential energy of interaction between the mt and the coupler as well as the effect of the external tension on this potential energy landscape are not incorporated explicitly in their model . thus , the model of akiyoshi et al.@xcite does not directly demonstrate the interplay of the opposing forces and competing kinetics . since neither any structural features of the coupler , nor the nature of its interaction with the mt enters explicitly into akiyoshi et al.s @xcite model , their model can not be used to study the effects of the size and composition of the coupler or that of the nature of the mt - coupler interaction . neither can it be used to account for the special features of hybrid couplers . in contrast , in this paper we have used our models to study the effects of varying ( i ) the size of the coupler , ( ii ) relative population of active force generators ( molecular motors ) and passive binders , as well as ( iii ) the depth and roughness of the potential of interaction between the coupler and the mt . catch bond formed by the fimbriae of e - coli bacteria with target cell surface and that formed by eukaryotic p - selectin and integrin receptors with their ligands have been studied extensively in recent years @xcite to understand their mechanism . in this paper we have focussed exclusively on the kt - mt attachments to understand the mechanism of its catch - bond - like behavior @xcite using simple theoretical models . one distinct feature of the mt is that , unlike other ligands , these stiff filaments are capable of generating push and pull by their polymerization and depolymerization , respectively . in particular , the pulling force generated by a mt is powered by the hydrolysis of gtp bound to the tubulin subunits . as we have shown in this paper , the competing polymerization and depolymerization of mt gives rise to a unique feature of the catch - bond - like behavior of kt - mt attachments where the stable state can be a non - equilibrium ( local ) steady state rather than a local minimum of the free energy of the ligand - receptor system . we hope the modeling methodology developed here can be usefully adapted to understand the catch - bond - like behavior of depolymerizing actin under load tension observed recently by carrying out _ in - vitro _ experiments with atomic force microscope ( afm ) @xcite . * acknowledgements : * dc thanks frank jlicher for valuable comments . this work has been supported in part by csir ( india ) through a senior research fellowship ( aks ) , by nsf through the grant dms- 1358932 ( bs ) , by mbi , the ohio state university , through the nsf grant dms 0931642 ( bs and dc ) , by iit kanpur through dr . jag mohan garg chair professorship ( dc ) , by serb ( india ) through j.c . bose national fellowship ( dc ) , by dbt ( india ) through a research grant ( dc ) , and by the visitors program of the max - planck institute for the physics of complex systems , dresden ( dc ) . | our models not only account for the experimentally observed `` catch - bond - like '' behavior of the kt - mt coupling , but also make new predictions on the probability distribution of the lifetimes of the attachments . in principle , our new predictions can be tested by analyzing the data collected in the _ in - vitro _ | before a cell divides into two daughter cells , chromosomes are replicated resulting in two sister chromosomes embracing each other . each sister chromosome is bound to a separate proteinous structure , called kinetochore ( kt ) , that captures the tip of a filamentous protein , called microtubule ( mt ) . two oppositely oriented mts pull the two kts attached to two sister chromosomes thereby pulling the two sisters away from each other . here we theoretically study an even simpler system , namely an isolated kt coupled to a single mt ; this system mimics an _ in - vitro _ experiment where a single kt - mt attachment is reconstituted using purified extracts from budding yeast . our models not only account for the experimentally observed `` catch - bond - like '' behavior of the kt - mt coupling , but also make new predictions on the probability distribution of the lifetimes of the attachments . in principle , our new predictions can be tested by analyzing the data collected in the _ in - vitro _ experiments provided the experiment is repeated sufficiently large number of times . our theory provides a deep insight into the effects of ( a ) size , ( b ) energetics , and ( c ) stochastic kinetics of the kt - mt coupling on the distribution of the lifetimes of these attachments . |
1303.7153 | i | vortex states in mesoscopic superconductors have been extensively studied in the past two decades , both theoretically and experimentally.@xcite two main interactions have been found to govern vortex behavior in a mesoscopic system . the first one is the vortex - vortex interaction , which causes vortices to form abrikosov triangular lattices in bulk type - ii superconductors . the second one is the interaction between vortices and sample boundaries , which makes vortex configurations strongly dependent on the size and geometry of mesoscopic samples - whose dimensions are of the order of the penetration depth @xmath3 or the coherence length @xmath4 . for example , in square mesoscopic samples , vortex configurations try to best match the @xmath5 symmetry . when there is only one vortex in the sample ( @xmath6 state where @xmath7 is the winding number or vorticity ) , the vortex always sits in the center of the sample in order to balance the boundary effect from all sides . for the @xmath8 state , two vortices sit on the diagonal such that the vortex - vortex separation is maximal in order to minimize the intervortex interaction . a giant vortex with @xmath8 can be induced when the boundary confinement pushes two single vortices together , as predicted theoretically@xcite and observed experimentally.@xcite for @xmath9 state , because of its incompatability with the four - fold symmetry , the theory predicts that the ground state corresponds to an anti - vortex sitting at the center surrounded by four vortices.@xcite in short , the symmetry of the sample largely determines the vortex configurations when the size of the superconductor is reduced . however , the properties of nanoscale superconductors , whose sizes are of the order of the fermi wavelength @xmath10 , are very different from those of mesoscopic superconductors . this is because the distance between electronic levels becomes comparable to the superconducting energy gap due to quantum confinement @xcite . as a consequence , the number of cooper pairs is suppressed which leads to the quantum - size effect ( qse),@xcite quantum - size cascades,@xcite the shell effect@xcite and inhomogeneous spatial distribution of the order parameter.@xcite the latter is the most important for the present work because it is expected to strongly influence the vortex states in nanoscale superconductors . a similar behavior was shown for an isolated vortex core , where oscillations of there order parameter on the order of the fermi wavelength were predicted . @xcite inhomogeneous superconductivity has been studied in various systems in the last decades and shows more complex behavior than homogeneous ones . it is known that vortices tend to migrate and get pinned in areas where superconductivity is suppressed.@xcite the reason is that it is more favorable energetically for a vortex to suppress the superconducting order parameter in a region where it has already been suppressed , although sometimes vortices can be pinned where the gap is large.@xcite some three - dimensional ( 3d ) samples can also be treated as inhomogeneous systems.@xcite for example for a 3d tip geometry , an asymmetric @xmath6 vortex state can be the ground state because the thick region prevents the vortex from penetrating it.@xcite in multi - layered superconductors , vortices enter first and reside favorably in the weak layers . then , vortices will penetrate into the strong layers only after weak layers become saturated and various vortex clusters and asymmetric vortex states are induced.@xcite also , the fabrication of anti - dots in superconductors results in a spatially varying superconducting energy gap with a barrier at the interfaces . in these systems , the combination of the giant vortex , multi - vortex and anti - vortex states can be found as ground state , which depends strongly on the detailed geometry of the antidots . @xcite for conventional superconductors , @xmath11 ( @xmath12 is the fermi wave vector and @xmath13 is the bcs coherence length ) , systems of size comparable to @xmath10 will not be large enough to host a vortex ( being much smaller than the coherence length ) . however , materials with small coherence lengths , e.g. high - t@xmath14 cuprate superconductors , will have @xmath15 and therefore in such systems it is possible to obtain vortex states in the quantum confinement regime . another such system is a graphene flake deposited on top of a superconductor . because of the proximity effect , cooper pairs will diffuse in graphene@xcite . in graphene the scattering length is large , therefore such a system is in the clean limit . more importantly , near the dirac point , the fermi wavelength is very large and can be easily manipulated by doping . in other words , @xmath16 can be tuned , which will allow for different vortex patterns to be realized in the graphene flake in the quantum confinement regime , but for more accessible sample sizes ( even above 100 nm ) . a similar configuration was also recently proposed by knopnin et . al in ref . yet another system where effects of quantum confinement on vortex matter can be probed systematically are the optically trapped cold gases @xcite , which are nowadays extremely controllable . for studying such nanoscale systems , microscopic bogoliubov - de gennes ( bdg ) theory is required . previous works used the bdg method to focus on isolated single vortex lines@xcite , giant vortices@xcite and to describe the local density of states modifications due to vortex - vortex and vortex - boundary interactions@xcite but are in the mesoscopic limit as opposed to the nanoscale limit considered here . although refs . [ ] and [ ] studied the groud state vortex states in a mesoscopic - nanoscopic crossover region by solving the bdg equations self - consistently , quantum confinement effects do not play any role . recently , we investigated@xcite the vortex states in nano - scale superconducting squares . we found unconventional vortex states in the quantum limit due to shape - induced resonances in the inhomogeneous cooper - pair condensate . vortex - antivortex structures , asymmetric vortex states and vortex clusters were found as ground states over a wide range of parameters . they are distinct from previous results obtained in mesoscopic superconductors using the gl theory . however , there are still several aspects that remained unclear . for example , how does the size of the sample affect the vortex states in nanoscale superconductors ? under which conditions , does one recover the conventional gl results ? why are the antivortex states more stable in the nanoscale limit while giant vortex states are unfavorable ? how do the vortex states change if temperature is increased ? in order to answer these questions , in this paper , we study vortex states in nanoscale superconducting squares systematically . vortex states for different sample sizes , @xmath0 parameters and temperatures @xmath17 are investigated and the stability of the symmetry - induced vortex / anti - vortex molecules is discussed . more unconventional states , very different from the ones obtained within gl theory , are found . this study is therefore fully complementary to what is known for vortex matter in superconductors . the paper is organized as follows . in sec . ii , we introduce the theoretical approach , i.e. the bdg approach for a square geometry . in sec . iii , we present the inhomogeneous superconducting state in the absence of the magnetic field in order to better understand the qse in nano - scale superconductors . in sec . iv , the ground states and metastable states are studied at zero temperature and on the sample size dependence of the vortex states is discussed . in sec . v , the finite temperature ground states are studied . in sec . vi , we discuss the generation of vortex / anti - vortex molecules and study the structure of the vortex core . finally , we summarize our findings in sec . | when the superconducting coherence length is comparable to the fermi wavelength , the shape resonances of the superconducting order parameter have strong influence on the vortex configuration . several unconventional vortex states , including asymmetric ones , giant multi - vortex combinations , and states comprising giant antivortex , were found as ground states and their stability was found to be very sensitive on the value of , the size of the sample , and the magnetic flux . by increasing the temperature and/or enlarging the size of the sample , quantum confinement is suppressed and the conventional mesoscopic vortex states as predicted by the ginzburg - laudau ( gl ) theory are recovered . however , contrary to the gl results we found that the states containing symmetry - induced vortex - antivortex pairs are stable over the whole temperature range . it turns out that the inhomogeneous order parameter induced by quantum confinement favors vortex - antivortex molecules , as well as giant vortices with a rich structure in the vortex core - unattainable in the gl domain . | bogoliubov - de gennes theory is used to investigate the effect of the size of a superconducting square on the vortex states in the quantum confinement regime . when the superconducting coherence length is comparable to the fermi wavelength , the shape resonances of the superconducting order parameter have strong influence on the vortex configuration . several unconventional vortex states , including asymmetric ones , giant multi - vortex combinations , and states comprising giant antivortex , were found as ground states and their stability was found to be very sensitive on the value of , the size of the sample , and the magnetic flux . by increasing the temperature and/or enlarging the size of the sample , quantum confinement is suppressed and the conventional mesoscopic vortex states as predicted by the ginzburg - laudau ( gl ) theory are recovered . however , contrary to the gl results we found that the states containing symmetry - induced vortex - antivortex pairs are stable over the whole temperature range . it turns out that the inhomogeneous order parameter induced by quantum confinement favors vortex - antivortex molecules , as well as giant vortices with a rich structure in the vortex core - unattainable in the gl domain . |
astro-ph9810039 | i | the ccds used in the wide field planetary camera 2 ( `` wfpc2 '' ) of the _ hubble space telescope _ are known to have imperfect charge - transfer efficiency ( `` cte '' ) . this property of the camera has adverse photometric consequences that have proven difficult to characterize quantitatively . the source of the problem appears to be chemical impurities in the crystal lattice of the detectors that trap some photoelectrons during exposure and readout , resulting in an underestimate of the original luminous flux impingent on the camera . these missing electrons eventually diffuse out of the traps and contribute to the background noise of that same exposure or a subsequent one . since the charge packets that represent stellar images recorded at high row numbers must be clocked across a large fraction of the chip during readout , it is to be expected that they will encounter more traps and will lose more electrons than images recorded at lower row numbers . on the other hand , in frames where the diffuse sky brightness is high such as in long exposures through broad - band filters charge traps over the entire detector area can be filled by sky - generated photoelectrons . under these circumstances , the net effect of the charge - transfer inefficiency would be a small overall lowering of the diffuse sky brightness perceived in the astronomical scene , while the apparent projection of the stellar images _ above _ the diffuse sky surface brightness could be comparatively unaffected . an early analysis suggested that in images where the sky brightness is low ( @xmath62030@xmath4e@xmath5/pixel ) a simple `` ramp '' correction proportional to the @xmath0-coordinate position of the star image , amounting to 4% per 800 rows , would approximately correct recorded fluxes for the lost electrons ; in images where the sky was bright ( @xmath7250@xmath4e@xmath5/pixel ) , no correction was needed ; and in cases of moderate sky brightness an intermediate correction was recommended ( holtzman 1995a ) . while holtzman s ramp model corrects for the first - order effects of charge loss during readout , second - order effects may be significant . for instance , one would expect any given charge trap to absorb only a limited number of electrons before their mutual repulsion effectively blocked the ingestion of any additional charge . thus , faint star images consisting of comparatively few photoelectrons should lose a larger fraction of their total charge than images of bright stars containing many photoelectrons . one would therefore expect faint stars to require a larger fractional correction than bright stars near the same row number and in the presence of the same sky brightness . on the other hand , the large electron swarm corresponding to the image of a bright star would occupy a somewhat larger volume within the three - dimensional body of the detector , and might therefore actually enounter more lattice imperfections than would be accessible to smaller electron clouds . consequently , while we might still expect that brighter stars will require smaller magnitude corrections than fainter ones , it may be that the correction will approach zero more slowly than the expected simple exponential dependence as arbitrarily bright stars are considered . in addition , while it has been demonstrated that a given star s electron swarm can lose some charge while being clocked through some number of rows along a ccd s parallel register , it is also possible that said packet may lose yet more charge while being transferred through some number of columns along the serial register . again , this second ramp correction , presumably proportional to the star s @xmath1-coordinate position in the digital image , may also depend upon the sky brightness in the scene and on the brightness of the star image itself . the numerical coefficients of the @xmath0 and @xmath1 ramp corrections may be different , due to the dissimilar morphologies of the parallel and serial registers within the semiconductor material . whitmore & heyer ( 1997 ; wh97 ) have estimated the above first- and second - order effects of the charge - transfer inefficiency in wfpc2 , based upon a special sequence of observations of the globular cluster tauri (= ) . observations were obtained at four pointings of the telescope , with deliberate translations imposed so that a particular piece of the star field could be recorded on the four different chips during the course of the sequence . since the natural coordinate systems of the four detectors are rotated by 90 with respect to each other , this allowed several hundred star images to be used for a purely differential determination of the fraction of charge lost as a function of position on the detector , without a need to know the absolute brightness of any given object . wh97 presented their results in four alternative formulations , of which the most relevant to the present discussion is given by their eqs . 1 , 2d , and 3d : @xmath8 \hbox{\it cts}_{\hbox{\scriptsize observed}}\ ] ] @xmath9 @xmath10 here , _ cts _ represents total wfpc2 data numbers belonging to the stellar image within an aperture 05 in radius , as perceived with the gain = 7@xmath4e@xmath5/dn electronics ; @xmath1 and @xmath0 represent the position of the stellar image in the natural coordinate system of the detector ( pixel ( 1,1 ) being the first one to be read through the amplifier and ( 800,800 ) being the last , with the @xmath1 coordinate varying more rapidly ) ; and @xmath11 is the perceived diffuse sky brightness in the scene , exclusive of recognized stellar images . ( i have opportunistically adopted this version from the four variants presented , because following holtzman ( 1995b ) in reducing wfpc2 data i routinely correct measured magnitudes to their half - arcsecond - aperture equivalents , and because the data files produced by my software contain an estimate of the local sky brightness around and underlying each detected object . ) certain properties of these formulae may be noted . ( 1 ) the correction that is proportional to the @xmath0 position of the star depends only on the diffuse sky brightness in the image , and not on the brightness of the star itself . in contrast , the alternative @xmath0 ramp corrections presented for magnitudes as measured through 02 synthetic apertures depend upon both sky and star brightness . this change in formulation stems from the finding by wh97 that the difference between magnitudes measured in 02 and 05 apertures itself depends upon the stellar brightness : `` we find the cte loss is slightly reduced wthen the larger aperture is used , by about 1% for the bright stars and 3% for the faint stars . '' apparently , in the data studied by wh97 this effect tended to erase the magnitude dependence of the ramp corrections in the larger aperture . however , it should be noted that a simple differencing of the formulae provided by wh97 indicates that perfect cancellation can occur only at a particular value of the sky brightness , and will not be the case in general . ( 2 ) the correction that is proportional to the @xmath1 coordinate depends only on the brightness of the star and not at all on the diffuse brightness of the sky , a result not necessarily to be predicted from the electron - trap model of the charge - transfer inefficiency . ( 3 ) the @xmath0-coordinate ramp is proportional to an exponential function of the logarithm of the sky brightness , so ( a ) the correction tends asymptotically ( but slowly ) to zero as the sky brightness increases without limit , but ( b ) as the sky brightness decreases to zero , the correction @xmath12 this implies a non - physical result for a star of any apparent magnitude detected in the presence of a very low diffuse sky brightness . ( 4 ) the @xmath1-coordinate ramp is a linear function of the logarithm of the star counts , so ( a ) the correction again diverges as the number of star counts approaches zero . however , in this case the divergence is not a serious drawback , since photometry of any stellar image containing that few photoelectrons is unlikely to be of genuine astronomical interest . conversely , ( b ) the correction is identically zero when @xmath13 , and changes sign ( , the lattice imperfections in the serial register appear to act as electron sources rather than electron sinks ) for stellar images containing more than 21,000 dn within a half - arcsecond aperture . this is approximately the amount of flux contained within a star image whose central pixel just equals the digital saturation level of the analog - to - digital converters in wfpc2 when the gain@xmath14e@xmath5/dn electronics are used . therefore , this extrapolation , too , is probably of minimal practical relevance except possibly for very bright stars observed with the high - gain electronics bay . these extrapolations do emphasize , however , that the wh97 corrections are purely empirical constructs , and are not tied directly to a self - consistent physical model of the nature of the charge - transfer inefficiency . ( 5 ) the model correction makes no allowance for the possibility of a charge loss that is independent of position , regardless of whatever dependence it may have on either stellar brightness or sky flux . this feature of the model is a direct consequence of the experimental design , where apparent flux ratios were observed for individual stars recorded at various positions on the chips , without any consideration of ( a ) flux ratios for different stars with externally known magnitude differences , or ( b ) flux ratios for images of individual stars as recorded in frames with different exposure times . charge losses that are independent of position may be expected within the lattice - flaw model : simply by virtue of being formed in the body of the silicon wafer in the first place , the stellar image may lose some charge to traps located near the place of its formation . in addition , any inefficiency inherent in the transfer of charge packets from the top of a parallel register to the serial register , or from the end of the serial register into the on - chip amplifier , would be perceived as a charge loss that is independent of the original location of the stellar image on the detector . a hint that position - independent charge losses _ might _ be significant came from preliminary analyses of wfpc2 images of the globular clusters and palomar 4 ( see w. e. harris 1997 ; stetson 1998 , 1999 ) . a comparison of short - exposure ( @xmath15 s ) images to ground - based photometry of the same objects implied photometric zero - points consistent with those found by holtzman ( 1995b ) from similarly short exposures of the standard field . in contrast , comparison of longer ( @xmath16 s ) exposures of the same fields to the same ground - based photometry implied zero points that were different by approximately 0.05 mag , in the sense that the effective quantum efficiency of the detectors was higher in the longer exposures . a subsequent recalibration of the ground - based photometry , including the addition of many more independent ground - based and hst observations of , suggested that the original estimate of 0.05 mag might in fact have been high ; the additional data suggested that an anomaly of order 0.03 mag on average might be more nearly correct . with the publication of the wh97 corrections , it is apparent that the perceived short - minus - long zero - point difference might just be a reflection of the dependence of the fractional charge loss on the total number of recorded stellar photoelectrons and on the perceived surface flux of the sky . however , it remains possible that some residual position - independent charge loss could occur . in this paper i reinvestigate the photometric consequences of the charge - transfer inefficiency in wfpc2 , based on extensive series of observations of the galactic globular clusters tauri and . in addition to testing the validity of the corrections proposed by wh97 on a much larger data set , i investigate certain aspects of the problem not considered by them . | charge - transfer effects in photometry with wide field planetary camera 2 aboard the _ hubble space telescope _ are investigated by a comparison of wfpc2 observations with groundbased photometry for the galactic globular clusters tauri and . simple numerical formulae describing the fraction of lost signal as functions of position on the detector , stellar brightness , and the diffuse sky brightness recorded in an image are presented , and the resulting corrections are compared to those previously derived by whitmore & heyer ( 1997 , _ instrument science report wfpc2 97 - 08 _ ) . significant lost - charge effects are seen that are proportional to both the coordinate ( , the number of shifts along the parallel register during readout ) and the coordinate ( number of shifts along the serial register ) . the percentage of charge lost decreases as the star brightness or the diffuse sky brightness increases . even with the present set of corrections , additional sources of calibration uncertainty which i am unable identify or characterize with the available data probably limit the _ external _ accuracy of photometry from wfpc2 to of order 12% . = 10000 = 10000 # 1m#1 # 1ngc#1 # 1 and | charge - transfer effects in photometry with wide field planetary camera 2 aboard the _ hubble space telescope _ are investigated by a comparison of wfpc2 observations with groundbased photometry for the galactic globular clusters tauri and . simple numerical formulae describing the fraction of lost signal as functions of position on the detector , stellar brightness , and the diffuse sky brightness recorded in an image are presented , and the resulting corrections are compared to those previously derived by whitmore & heyer ( 1997 , _ instrument science report wfpc2 97 - 08 _ ) . significant lost - charge effects are seen that are proportional to both the coordinate ( , the number of shifts along the parallel register during readout ) and the coordinate ( number of shifts along the serial register ) . a `` typical '' star image ( one containing photoelectrons ) near the center of a `` typical '' intermediate - length exposure ( one with a diffuse sky brightness ofe/pixel , obtained at a camera temperature of 88c ) loses approximately 2% of its electrons to charge traps during readout ; a star in the corner of the image most remote from the readout electronics loses twice that . the percentage of charge lost decreases as the star brightness or the diffuse sky brightness increases . charge losses during the brief period when wfpc2 was operated at a temperature of 76c were approximately 85% greater , but apart from that no significant change in the charge transfer losses with time during the first 3.5 years of wfpc2 s mission is evident , except possibly a weak effect for the very faintest star images . these results are quite similar to those of whitmore & heyer , which were based on a much smaller data set , but there are some differences in detail . even with the present set of corrections , additional sources of calibration uncertainty which i am unable identify or characterize with the available data probably limit the _ external _ accuracy of photometry from wfpc2 to of order 12% . = 10000 = 10000 # 1m#1 # 1ngc#1 # 1 and |
1609.01626 | c | our analysis demonstrates that the chlorine abundance can be measured in solar metallicity stars with temperatures below 3900 k. stars with temperatures above 3900 k are too warm to form the hcl molecule , with the one exception of hd 138481 , at an effective temperature of 3970 k but metal rich at [ fe / h]=+0.2 @xcite . the lowest metallicity star with detectable h@xmath0cl in our sample is hd 35155 at [ fe / h]=0.72 . the majority of stars with h@xmath0cl features in our sample are evolved stars of spectral types m , k , and s. late - type dwarfs are difficult to observe at 3.7 @xmath1 m due to their faint l - band apparent magnitude but the m dwarf bd+68 964 was bright enough to be included . this star has a measurable h@xmath0cl with [ o / fe ] and [ @xmath0cl / fe ] abundances that are consistent with the abundances measured in the evolved giants . the k8 dwarf hd 88230 is also included in our sample but h@xmath0cl is not measurable due this star s effective temperature of 3970 k. the evolutionary state and masses of the stars in our sample with chlorine abundance measurements can be determined from absolute magnitudes computed using hipparcos parallaxes @xcite and 2mass infrared photometry @xcite . the 2mass k@xmath49 magnitude was converted to the johnson k magnitude using methods of @xcite . a bolemetric correction for each star could then be determined from its j - k color and known atmospheric parameter @xcite . the input parameters and results for each star can be found in table [ table::bolcor ] . infrared photometry from iras @xcite , wise @xcite , and 2mass @xcite allowed the identification of any infrared excess due to circumstellar material around each star . stars with possible infrared excesses were then compared to @xcite ; only hd 147923 had j and k band magnitudes that were affected by more than 0.1 mags from the star s intrinsic magnitude @xcite . however , the distance to this star dominates the uncertainty on the luminosity rather than uncertainties in the j and k magnitudes . the stars position in the hertzsprung - russel diagram are compared to models for a star with z=0.017 and y=0.30 @xcite and the results are plotted in figure [ fig::stellarevol ] . the stars in our sample have masses between 1 @xmath53 and 3 @xmath53 and are consistent with the locus of stars on the red giant branch . all but one star is located less than 500 pc from the sun and the sample is therefore representative of stars in the solar neighborhood . l l l l l l bd+68 946 & - & 4.53 @xmath3 0.02 & 4.60 & 0.79 & 6.33 @xmath3 0.01 & 2.51 & -1.64 @xmath3 0.08 + @xmath25 and & 6860 & 60.5 @xmath3 3.0 & 1.85 & 0.89 & 5.69 @xmath3 0.11 & 2.71 & 3.09 @xmath3 0.09 + rz ari & 18191 & 107.8 @xmath3 3.5 & 0.87 & 1.10 & 5.96 @xmath3 0.07 & 2.85 & 3.14 @xmath3 0.08 + @xmath26 cet & 18884 & 76.4 @xmath3 3.0 & 1.82 & 1.10 & 6.17 @xmath3 0.09 & 2.69 & 3.29 @xmath3 0.09 + omicron ori & 30959 & 200@xmath54 28 & 0.66 & 1.15 & 7.10 @xmath3 0.30 & 2.85 & 3.60 @xmath3 0.15 + v1261 ori & 35155 & 288 @xmath3 70 & 2.14 & 1.20 & 5.10 @xmath3 0.53 & 2.85 & 2.80 @xmath3 0.23 + @xmath1 gem & 44478 & 71.0 @xmath3 3.6 & 1.86 & 0.95 & 6.05 @xmath3 0.11 & 2.69 & 3.24 @xmath3 0.09 + v613 mon & 49368 & 495 @xmath3 201 & 2.46 & 1.30 & 5.94 @xmath3 0.88 & 2.85 & 3.13 @xmath3 0.36 + 27 cnc & 71250 & 276 @xmath3 25 & 0.58 & 0.99 & 6.57 @xmath3 0.20 & 2.84 & 3.39 @xmath3 0.11 + @xmath27 vir & 112300 & 60.8 @xmath3 0.8 & -1.19 & 1.08 & 5.04 @xmath3 0.03 & 2.68 & 2.84 @xmath3 0.08 + 83 uma & 119228 & 160.5 @xmath3 5.7 & 0.34 & 1.03 & 5.63 @xmath3 0.08 & 2.68 & 3.08 @xmath3 0.09 + aw cvn & 120933 & 184 @xmath3 6.8 & 0.01 & 1.05 & 6.26 @xmath3 0.08 & 2.69 & 3.33 @xmath3 0.09 + 52 boo & 138481 & 257 @xmath3 17 & 1.21 & 0.85 & 5.78 @xmath3 0.14 & 2.54 & 3.19 @xmath3 0.10 + & 147923 & 1400 @xmath3 1100 & 3.46 & 1.25 & 7.21 @xmath3 1.70 & 2.84 & 3.64 @xmath3 0.70 + @xmath28 aqr & 216386 & 118.1 @xmath3 9.2 & 0.67 & 1.09 & 5.96 @xmath3 0.17 & 2.69 & 3.20 @xmath3 0.10 + 30 psc ( yy psc ) & 224935 & 132.5 @xmath3 10.4 & 0.40 & 1.02 & 5.94 @xmath3 0.17 & 2.85 & 3.13 @xmath3 0.11 + the @xmath0cl/@xmath5cl isotope ratio for the solar system has been measured to be 3.13 @xcite . a survey of hcl in the galaxy from molecular clouds , protoplanetary cores , and evolved stars ( such as irc+10216 ) revealed isotope ratios between 1@xmath55cl/@xmath5cl@xmath25 with most values between 1 and 3 @xcite . the range of values is attributed to different isotope production in type ia and ccsne @xcite . type ia supernova produce yield ratios of @xmath0cl/@xmath5cl @xmath4 3.5 to 5.5 depending on the model parameters , such as ignition , 2d versus 3d models , and model resolution @xcite . ccsne models produces chlorine yields with isotope ratios of 2.32 for z=0.004 , 18 m@xmath6 stars , 1.2 for z=0.02 , 25 m@xmath6 stars , and 1.75 for high energy supernova with z=0.02 and 25 m@xmath6 @xcite . our sample includes only one star , rz ari with a measurable h@xmath5cl feature at 3.7010 @xmath1 m , as shown in fig . [ fig::exspec2 ] . other stars , including the sunspot umbral spectra , were too warm to form an h@xmath5cl feature significantly above the noise level ; the h@xmath5cl feature is stronger at both lower temperature and higher abundance . the isotope ratio in rz ari was determined by comparing the equivalent widths of the two hcl features . the equivalent widths are ew(h@xmath0cl)=(81 @xmath3 6 ) m@xmath16 and ew(h@xmath5cl)=(36 @xmath3 6 ) m@xmath16 . uncertainties are found from the standard deviation of multiple measurements of the equivalent width at different continuum values . the final isotopic abundance ratio is @xmath0cl/@xmath5cl = 2.2 @xmath3 0.4 . this value was also confirmed by spectral synthesis ; this isotope ratio was used to fit the h@xmath5cl feature , shown in figure [ fig::exspec2 ] , with the best fit determined by eye . the isotope ratio for cl for different metallicities in the solar neighborhood has been predicted by @xcite : cl@xmath0/cl@xmath5 = 1.79 at [ fe / h]=0 , and 1.94 at [ fe / h ] = 0.5 . rz ari has an [ fe / h ] of 0.25 . our measured chlorine isotope value of 2.2 @xmath3 0.4 is near one standard deviation of these predicted values . this ratio is also consistent with the range of isotope measurements found in the interstellar medium @xcite and is less than two standard deviations from the solar isotopic ratio . further measurements in cool stars are necessary to explore the dispersion of the isotope ratio in the local solar neighborhood . figure [ fig::cl_vs_o ] compares the @xmath0cl abundance in our sample of stars with the total cl abundance measured in galactic planetary nebula and regions . studies of planetary nebula have found the radial galactic abundance gradients of cl and o to be statistically indistinguishable from one another @xcite and studies of cl in regions found o and cl had identical radial galactic abundance gradients of 0.043 dex kpc@xmath52 @xcite . this indicates both elements , cl and o , are produced in lockstep evolution @xcite . our sample of stars cover too narrow a range of galacto - centric distance to allow us to determine a gradient , but the average a(@xmath0cl ) of our sample is consistent with what is predicted for the solar neighborhood by @xcite and @xcite . the offset between the planetary nebula and stellar abundances shown in figure [ fig::cl_vs_o ] is estimated to be 0.16 dex . this offset is likely due to the difference between the total cl abundance reported in the atomic planetary nebula lines and @xmath0cl abundances measured from the h@xmath0cl molecular feature in our sample . for example the a(@xmath0cl ) abundance in rz ari is 4.82 and the full abundance is a(cl)=4.98 for an isotope ratio of @xmath0cl/@xmath5cl of 2.2 , an increase of 0.16 dex . additionally , the solar isotope ratio is 3.13 which gives a @xmath0cl abundance in the sun of 5.13 , a difference of 0.12 dex between the total abundance and @xmath0cl abundance . therefore while our results show scatter , the stellar abundances follow similar cl to o ratios to those seen in the nebular sources . future observations will use the methodology in this paper to determine cl isotope ratios in a larger sample of cool stars . we present the first direct measurements of chlorine in stellar sources , where cl can be directly compared to the iron abundance . figure [ fig::results ] shows the [ @xmath0cl / fe]cl / fe]=a(@xmath0cl)-a(cl)@xmath6+[fe / h ] ] ratio measured in our stellar sample versus a chemical evolution model from @xcite . the average chlorine abundance of our sample is [ @xmath0cl / fe]=0.10 @xmath3 0.14 . the chemical evolution model predicts that [ @xmath0cl / fe ] to be nearly constant with metallicity , declining by only 0.03 dex from [ fe / h]=0.6 to solar metallicity . our [ @xmath0cl / fe ] ratios are on average higher than the model by 0.16 @xmath3 0.15 dex . the lowest metallicity stars have an average [ @xmath0cl / fe ] ratio of 0.1 dex , about 0.2 dex higher than the solar metallicity but the sample contains too few metal - poor stars to conclude whether or not [ @xmath0cl / fe ] declines with decreasing metallicity . despite the small offset our results are still nearly consistent abundance of chlorine predicted for the solar neighborhood by chemical evolution models . chlorine abundances were also compared with the alpha elements silicon and calcium derived from our spectra . figure [ fig::cl_alpha ] shows our observed abundance ratios versus chemical evolution models ( reproduced from figure 10 ) from @xcite . our measured [ @xmath0cl / si ] and [ @xmath0cl / ca ] ratios are higher than predicted by @xcite by typically 0.4 dex . offsetting each chemical evolution model to match the observed abundance , as shown in fig . 10 , demonstrate that the slope of the fit is consistent with the measured abundances and that the discrepancies do not depend on metallicity in our sample . the bottom panel in figure [ fig::cl_alpha ] shows the [ ca / si ] ratio is also offset compared to the model value . the [ ca / si ] offset was found to be 0.20 dex . we find the [ @xmath0cl / ca ] ratio increases at higher [ fe / h ] . this is expected because [ @xmath0cl / fe ] is predicted to be constant with metallicity while [ ca / fe ] decreases as metallicity increases . the offset between models and chlorine abundance may be due to the additional production of cl by the @xmath29 process , which may affect yields of cl @xcite and is not included in the present model . these offsets between the chemical evolution models and the observed abundances put constraints on the strength of any additional process that produces cl . we are also comparing only @xmath0cl abundances to the cl chemical evolution models . including the @xmath5cl abundance would increase the overall abundance by @xmath40.1 to 0.2 dex depending on the isotope ratio . for example the a(@xmath0cl ) abundance in rz ari is 4.82 and the full abundance is a(cl)=4.98 for an isotope ratio of @xmath0cl/@xmath5cl of 2.2 . to determine if chlorine is produced using the s - process , two low temperature barium - enriched stars were included in our sample . hd 138481 is a class 0.5 ba star and hd 119228 is a class 0.5 ba star @xcite , represented as red diamonds in figure [ fig::results ] . the [ @xmath0cl / fe ] in those stars is similar to normal giants and suggests that additional cl production via the s - process is unlikely . | the average [cl / fe ] abundance in stars with 0.72[fe / h].20 is [cl / fe]=(0.10.15 ) dex . the mean difference between the [cl / fe ] ratios measured in our stars and chemical evolution model values is ( 0.16.15 ) dex . the cl versus o abundances from our sample match cl abundances measured in planetary nebula and regions . in one star where both hcl and hcl could be measured , | chlorine abundances are reported in 15 evolved giants and one m dwarf in the solar neighborhood . the cl abundance was measured using the vibration - rotation 1 - 0 p8 line of hcl at 3.69851 m . the high resolution l - band spectra were observed using the phoenix infrared spectrometer on the kitt peak mayall 4 m telescope . the average [cl / fe ] abundance in stars with 0.72[fe / h].20 is [cl / fe]=(0.10.15 ) dex . the mean difference between the [cl / fe ] ratios measured in our stars and chemical evolution model values is ( 0.16.15 ) dex . the [cl / ca ] ratio has an offset of.35 dex above model predictions suggesting chemical evolution models are under producing cl at the high metallicity range . abundances of c , n , o , si , and ca were also measured in our spectral region and are consistent with f and g dwarfs . the cl versus o abundances from our sample match cl abundances measured in planetary nebula and regions . in one star where both hcl and hcl could be measured , acl/cl isotope ratio of 2.2.4 was found , consistent with values found in the galactic ism and predicted chemical evolution models . |
1306.5029 | i | in the orthogonal range reporting problem , we store a set of points @xmath5 in a data structure so that for an arbitrary range @xmath6\times \ldots\times [ a_d , b_d]$ ] all points from @xmath7 can be reported . due to its importance , one- and multi - dimensional range reporting was extensively studied in computational geometry and database communities . the following situation frequently arises in different areas of computer science : a set of @xmath8-dimensional objects @xmath9 must be preprocessed so that we can enumerate all objects satisfying @xmath10 for arbitrary @xmath11 , @xmath12 . this scenario can be modeled by the orthogonal range reporting problem . the objects in the input set can be distributed into _ categories_. instead of enumerating all objects , we may want to report distinct categories of objects in the given range . this situation can be modeled by the color ( or categorical ) range reporting problem : every point in a set @xmath5 is assigned a color ( category ) ; we pre - process @xmath5 , so that for any @xmath6\times \ldots\times [ a_d , b_d]$ ] the distinct colors of points in @xmath7 can be reported . color range reporting is usually considered to be a more complex problem than point reporting . for one thing , we do not want to report the same color multiple times . in this paper we show that complexity gap can be closed for one - dimensional color range reporting . we describe color reporting data structures with the same space usage and query time as the best known corresponding structures for point reporting . moreover we extend our result to the external memory model . [ [ previous - work . ] ] previous work . + + + + + + + + + + + + + + we can easily report points in a one - dimensional range @xmath13 $ ] by searching for the successor of @xmath14 in @xmath5 , @xmath15 . if @xmath16 is known , we can traverse the sorted list of points in @xmath5 starting at @xmath17 and report all elements in @xmath18 $ ] . we can find the successor of @xmath14 in @xmath5 in @xmath19 time @xcite ; if the universe size is @xmath20 , i.e. , if all points are positive integers that do not exceed @xmath20 , then the successor can be found in @xmath21 time @xcite . thus we can report all points in @xmath18 $ ] in @xmath22 time for @xmath23 . henceforth @xmath3 denotes the number of elements ( points or colors ) in the query answer . it is not possible to find the successor in @xmath24 time unless the universe size @xmath20 is very small or the space usage of the data structure is very high ; see e.g. , @xcite . however , reporting points in a one - dimensional range takes less time than searching for a successor . in their fundamental paper @xcite , miltersen et al . showed that one - dimensional point reporting queries can be answered in @xmath25 time using an @xmath26 space data structure . alstrup et al . @xcite obtained another surprising result : they presented an @xmath1-space data structure that answers point reporting queries in @xmath25 time and thus achieved both optimal query time and optimal space usage for this problem . the data structure for one - dimensional point reporting can be dynamized so that queries are supported in @xmath25 time and updates are supported in @xmath27 time @xcite ; henceforth @xmath28 denotes an arbitrarily small positive constant . we refer to @xcite for further update - query time trade - offs . solutions of the one - dimensional point reporting problem are based on finding an arbitrary element @xmath29 in a query range @xmath30 $ ] ; once such @xmath29 is found , we can traverse the sorted list of points until all points in @xmath30 $ ] are reported . therefore it is straightforward to extend point reporting results to the external memory model . janardan and lopez @xcite and gupta et al . @xcite showed that one - dimensional color reporting queries can be answered in @xmath31 time , both in the static and the dynamic scenarios . muthukrishnan @xcite described a static @xmath1 space data structure that answers queries in @xmath25 time if all point coordinates are bounded by @xmath4 . we can obtain data structures that use @xmath1 space and answer queries in @xmath32 or @xmath33 time using the reduction - to - rank - space technique . no data structure that answers one - dimensional color reporting queries in @xmath34 time was previously known . a dynamic data structure of mortensen @xcite supports queries and updates in @xmath35 and @xmath36 time respectively if the values of all elements are bounded by @xmath4 . recently , the one- and two - dimensional color range reporting problems in the external memory model were studied in several papers @xcite . larsen and pagh @xcite described a data structure that uses linear space and answers one - dimensional color reporting queries in @xmath37 i / os if values of all elements are bounded by @xmath1 . in the case when values of elements are unbounded the best previously known data structure needs @xmath38 i / os to answer a query ; this result can be obtained by combining the data structure from @xcite and reduction of one - dimensional color reporting to three - sided\times [ 0,c]$ ] and @xmath30\times [ c,+\infty]$ ] are three - sdied queries . ] point reporting @xcite . in another recent paper @xcite , chan et al . described a data structure that supports the following queries on a set of points whose values are bounded by @xmath1 : for any query point @xmath39 and any integer @xmath3 , we can report the first @xmath3 colors that occur after @xmath39 . this data structure can be combined with the result from @xcite to answer queries in @xmath2 time . unfortunately , the solution in @xcite is based on the hive graph data structure @xcite . therefore it can not be used to solve the problem in external memory or to obtain a dynamic solution . [ [ our - results . ] ] our results . + + + + + + + + + + + + as can be seen from the above discussion and table [ tab : res ] , there are significant complexity gaps between color reporting and point reporting data structures in one dimension . we show in this paper that it is possible to close these gaps . in this paper we show that one - dimensional color reporting queries can be answered in constant time per reported color for an arbitrarily large size of the universe . our data structure uses @xmath1 space and supports color reporting queries in @xmath2 time . this data structure can be dynamized so that query time and space usage remain unchanged ; the updates are supported in @xmath27 time where @xmath20 is the size of the universe . the new results are listed at the bottom of table [ tab : res ] . our internal memory results are valid in the word ram model of computation , the same model that was used in e.g. @xcite . in this model , we assume that any standard arithmetic operation and the basic bit operations can be performed in constant time . we also assume that each word of memory consists of @xmath40 bits , where @xmath20 is the size of the universe . that is , we make a reasonable and realistic assumption that the value of any element fits into one word of memory . furthermore , we also extend our data structures to the external memory model . our static data structure uses linear space and answers color reporting queries in @xmath41 i / os . our dynamic external data structure also has optimal space usage and query cost ; updates are supported in @xmath42 i / os . [ cols="<,<,^,^,^,^ " , ] | color ( or categorical ) range reporting is a variant of the orthogonal range reporting problem in which every point in the input is assigned a _ color_. while the answer to an orthogonal point reporting query contains all points in the query range , the answer to a color reporting query contains only distinct colors of points in . in this paper we describe an-space data structure that answers one - dimensional color reporting queries in optimal time , where is the number of colors in the answer and is the number of points in the data structure . our result can be also dynamized and extended to the external memory model . | color ( or categorical ) range reporting is a variant of the orthogonal range reporting problem in which every point in the input is assigned a _ color_. while the answer to an orthogonal point reporting query contains all points in the query range , the answer to a color reporting query contains only distinct colors of points in . in this paper we describe an-space data structure that answers one - dimensional color reporting queries in optimal time , where is the number of colors in the answer and is the number of points in the data structure . our result can be also dynamized and extended to the external memory model . |
cs0602092 | i | graphical models such as markov random fields ( mrfs ) are widely used in many application domains , including spatial statistics , statistical signal processing , and communication theory . a fundamental limitation to their practical use is the infeasibility of computing various statistical quantities ( e.g. , marginals , data likelihoods etc . ) ; such quantities are of interest both bayesian and frequentist settings . sampling - based methods , especially those of the markov chain monte carlo ( mcmc ) variety @xcite , represent one approach to obtaining stochastic approximations to marginals and likelihoods . a disadvantage of sampling methods is their relatively high computational cost . for instance , in applications with severe limits on delay and computational overhead ( e.g. , error - control coding , real - time tracking , video compression ) , mcmc methods are likely to be overly slow . it is thus of considerable interest for various application domains to consider less computationally intensive methods for generating approximations to marginals , log likelihoods , and other relevant statistical quantities . variational methods are one class of techniques that can be used to generate deterministic approximations in markov random fields ( mrfs ) . at the foundation of these methods is the fact that for a broad class of mrfs , the computation of the log likelihood and marginal probabilities can be reformulated as a convex optimization problem ( see @xcite for an overview ) . although this optimization problem is intractable to solve exactly for general mrfs , it suggests a principled route to obtaining approximations namely , by relaxing the original optimization problem , and taking the optimal solutions to the relaxed problem as approximations to the exact values . in many cases , optimization of the relaxed problem can be carried out by `` message - passing '' algorithms , in which neighboring nodes in the markov random field convey statistical information ( e.g. , likelihoods ) by passing functions or vectors ( referred to as messages ) . estimating the parameters of a markov random field from data poses another significant challenge . a direct approach for instance , via ( regularized ) maximum likelihood estimation entails evaluating the cumulant generating ( or log partition ) function , which is computationally intractable for general markov random fields . one viable option is the pseudolikelihood method @xcite , which can be shown to produce consistent parameter estimates under suitable assumptions , though with an associated loss of statistical efficiency . other researchers have studied algorithms for ml estimation based on stochastic approximation @xcite , which again are consistent under appropriate assumptions , but can be slow to converge . as illustrated in figure [ figestpred ] , the problem domain of interest in this paper is that of joint estimation and prediction in a markov random field . more precisely , given samples @xmath0 from some unknown underlying model @xmath1 , the first step is to form an estimate of the model parameters . now suppose that we are given a noisy observation of a new sample path @xmath2 , and that we wish to form a ( near)-optimal estimate of @xmath3 using the fitted model , and the noisy observation ( denoted @xmath4 ) . examples of such prediction problems include signal denoising , image interpolation , and decoding of error - control codes . disregarding any issues of computational cost and speed , one could proceed via route a in figure [ figestpred]that is , one could envisage first using a standard technique ( e.g. , regularized maximum likelihood ) for parameter estimation , and then carrying out the prediction step ( which might , for instance , involve computing certain marginal probabilities ) by monte carlo methods . this paper , in contrast , is concerned with the _ computation - limited _ setting , in which both sampling or brute force methods are overly intensive . with this motivation , a number of researchers have studied the use of approximate message - passing techniques , both for problems of prediction @xcite as well as for parameter estimation @xcite . however , despite their wide - spread use , the theoretical understanding of such message - passing techniques remains limited , especially for parameter estimation . consequently , it is of considerable interest to characterize and quantify the loss in performance incurred by using computationally tractable methods versus exact methods ( i.e. , route b versus a in figure [ figestpred ] ) . more specifically , our analysis applies to variational methods that are based _ convex relaxations_. this class includes a broad range of extant methods among them the tree - reweighted sum - product algorithm @xcite , reweighted forms of generalized belief propagation @xcite , and semidefinite relaxations @xcite . moreover , it is straightforward to modify other message - passing methods ( e.g. , expectation propagation @xcite ) so as to `` convexify '' them . at a high level , the key idea of this paper is the following : given that approximate methods can lead to errors at both the estimation and prediction phases , it is natural to speculate that these sources of error might be arranged to partially cancel one another . the theoretical analysis of this paper confirms this intuition : we show that with respect to end - to - end performance , it is in fact beneficial , even in the infinite data limit , to learn the `` wrong '' the model by using _ inconsistent _ methods for parameter estimation . en route to this result , we analyze the asymptotic properties of m - estimators based on convex variational relaxations , and establish a lipschitz stability property that holds for a broad class of variational methods . we show that joint estimation / prediction based on the reweighted sum - product algorithm substantially outperforms a commonly used heuristic based on ordinary sum - product . the remainder of this paper is organized as follows . section [ secbackground ] provides background on markov random fields and associated variational representations , as well as the problem statement . in section [ secconvsurr ] , we introduce the notion of a convex surrogate to the cumulant generating function , and then illustrate this notion via the tree - reweighted bethe approximation @xcite . in section [ secjoint ] , we describe how any convex surrogate defines a particular joint scheme for parameter estimation and prediction . section [ secanal ] provides results on the asymptotic behavior of the estimation step , as well as the stability of the prediction step . section [ secmixgauss ] is devoted to the derivation of performance bounds for joint estimation and prediction methods , with particular emphasis on the mixture - of - gaussians observation model . in section [ secexperiments ] , we provide experimental results on the performance of a joint estimation / prediction method based on the tree - reweighted bethe surrogate , and compare it to a heuristic method based on the ordinary belief propagation algorithm . we conclude in section [ secdiscussion ] with a summary and discussion of directions for future work . | is used to construct an m - estimator for fitting parameters , and to perform approximate marginalization for the prediction step . the key result of this paper is that in the computation - limited setting , using an inconsistent parameter estimator ( i.e. , an estimator that returns the `` wrong '' model even in the infinite data limit ) can be provably beneficial , since the resulting errors can partially compensate for errors made by using an approximate prediction technique . en route to this result , we analyze the asymptotic properties of m - estimators based on convex variational relaxations , and establish a lipschitz stability property that holds for a broad class of variational methods . we show that joint estimation / prediction based on the reweighted sum - product algorithm substantially outperforms a commonly used heuristic based on ordinary sum - product . * inconsistent parameter estimation in markov random fields : benefits in the computation - limited setting * + department of statistics , uc berkeley + technical report 690 | consider the problem of joint parameter estimation and prediction in a markov random field : i.e. , the model parameters are estimated on the basis of an initial set of data , and then the fitted model is used to perform prediction ( e.g. , smoothing , denoising , interpolation ) on a new noisy observation . working under the restriction of limited computation , we analyze a joint method in which the _ same convex variational relaxation _ is used to construct an m - estimator for fitting parameters , and to perform approximate marginalization for the prediction step . the key result of this paper is that in the computation - limited setting , using an inconsistent parameter estimator ( i.e. , an estimator that returns the `` wrong '' model even in the infinite data limit ) can be provably beneficial , since the resulting errors can partially compensate for errors made by using an approximate prediction technique . en route to this result , we analyze the asymptotic properties of m - estimators based on convex variational relaxations , and establish a lipschitz stability property that holds for a broad class of variational methods . we show that joint estimation / prediction based on the reweighted sum - product algorithm substantially outperforms a commonly used heuristic based on ordinary sum - product . * inconsistent parameter estimation in markov random fields : benefits in the computation - limited setting * + department of statistics , uc berkeley + technical report 690 |
cs0602092 | c | key challenges in the application of markov random fields include the estimation ( learning ) of model parameters , and performing prediction using noisy samples ( e.g. , smoothing , interpolation , denoising ) . both of these problems present substantial computational challenges for general markov random fields . in this paper , we have described and analyzed methods for joint estimation and prediction that are based on convex variational methods . our central result is that using inconsistent parameter estimators can be beneficial in the computation - limited setting . indeed , our results provide rigorous confirmation of the fact that using parameter estimates that are `` systematically incorrect '' is helpful in offsetting the error introduced by using an approximate method during the prediction step . in concrete terms , we demonstrated that a joint prediction / estimation method using the tree - reweighted sum - product algorithm yields good performance across a wide range of experimental conditions . although our work has focused on a particular scenario , we suspect that similar ideas and techniques will be useful in related applications of approximate methods for learning and prediction . this work was supported by an alfred p. sloan foundation fellowship , an okawa foundation research fellowship , an intel corporation equipment grant , and nsf grant dms-0528488 . | consider the problem of joint parameter estimation and prediction in a markov random field : i.e. , the model parameters are estimated on the basis of an initial set of data , and then the fitted model is used to perform prediction ( e.g. , smoothing , denoising , interpolation ) on a new noisy observation . working under the restriction of limited computation , we analyze a joint method in which the _ same convex variational relaxation _ | consider the problem of joint parameter estimation and prediction in a markov random field : i.e. , the model parameters are estimated on the basis of an initial set of data , and then the fitted model is used to perform prediction ( e.g. , smoothing , denoising , interpolation ) on a new noisy observation . working under the restriction of limited computation , we analyze a joint method in which the _ same convex variational relaxation _ is used to construct an m - estimator for fitting parameters , and to perform approximate marginalization for the prediction step . the key result of this paper is that in the computation - limited setting , using an inconsistent parameter estimator ( i.e. , an estimator that returns the `` wrong '' model even in the infinite data limit ) can be provably beneficial , since the resulting errors can partially compensate for errors made by using an approximate prediction technique . en route to this result , we analyze the asymptotic properties of m - estimators based on convex variational relaxations , and establish a lipschitz stability property that holds for a broad class of variational methods . we show that joint estimation / prediction based on the reweighted sum - product algorithm substantially outperforms a commonly used heuristic based on ordinary sum - product . * inconsistent parameter estimation in markov random fields : benefits in the computation - limited setting * + department of statistics , uc berkeley + technical report 690 |
1410.2300 | i | flows of realistic mixtures of miscible fluids exhibit several features that make them more difficult to simulate numerically than flows of simple fluids . firstly , the physical properties of the mixture depend on the concentration of the different species composing the mixture . this includes both the density of the mixture at constant pressure , and transport coefficients such as viscosity and mass diffusion coefficients . common simplifying assumptions such as the boussinesq approximation , which assumes a constant density and thus incompressible flow , or assuming constant transport coefficients , are uncontrolled and not appropriate for certain mixtures of very dissimilar fluids . secondly , for liquid mixtures there is a large separation of time scales between the various dissipative processes , notably , mass diffusion is much slower than momentum diffusion . the large schmidt numbers @xmath0 typical of liquid mixtures lead to extreme stiffness and make direct temporal integration of the hydrodynamic equations infeasible . lastly , flows of mixtures exhibit all of the numerical difficulties found in single component flows , for example , well - known difficulties caused by advection in the absence of sufficiently strong dissipation ( diffusion of momentum or mass ) , and challenges in incorporating thermal fluctuations in the description . here we develop a low mach number approach to isothermal binary fluid mixtures that resolves many of the above difficulties , and paves the way for incorporating additional physics such as the presence of more than two species @xcite , chemical reactions @xcite , multiple phases and surface tension @xcite , and others . stochastic fluctuations are intrinsic to fluid dynamics because fluids are composed of molecules whose positions and velocities are random . thermal fluctuations affect flows from microscopic to macroscopic scales @xcite and need to be consistently included in all levels of description . fluctuating hydrodynamics ( fhd ) incorporates thermal fluctuations into the usual navier - stokes - fourier laws in the form of stochastic contributions to the dissipative momentum , heat , and mass fluxes @xcite . fhd has proven to be a very useful tool in understanding complex fluid flows far from equilibrium @xcite ; however , theoretical calculations are often only feasible after making many uncontrolled approximations @xcite , and numerical schemes used for fluctuating hydrodynamics are usually far behind state - of - the - art deterministic computational fluid dynamics ( cfd ) solvers . in this work , we consider binary mixtures and restrict our attention to isothermal flows . we consider a specific equation of state ( eos ) suitable for mixtures of incompressible liquids or ideal gases , but otherwise account for advective and diffusive mass and momentum transport in full generality . recently , some of us developed finite - volume methods for the incompressible equations @xcite . we have also developed low mach number isothermal fluctuating equations @xcite , which eliminate the stiffness arising from the separation of scales between acoustic and vortical modes @xcite . the low mach number equations account for the fact that for mixtures of fluids with different densities , diffusive and stochastic mass fluxes create local expansion and contraction of the fluid . in these equations the incompressibility constraint should be replaced by a `` quasi - incompressibility '' constraint @xcite , which introduces some difficulties in constructing conservative finite - volume techniques @xcite . in section [ sec : equations ] we review the low mach number equations of fluctuating hydrodynamics for a binary mixture of miscible fluids , as first proposed in ref . @xcite . the numerical method developed in ref . @xcite uses an explicit temporal integrator . this requires using a small time step and is infeasible for liquid mixtures due to the stiffness caused by the separation of time scales between fast momentum diffusion and slow mass diffusion . in recent work @xcite , some of us developed temporal integrators for the equations of fluctuating hydrodynamics that have several important advantages . notably , these integrators are semi - implicit , allowing one to treat fast momentum diffusion ( viscous dissipation ) implicitly , and other transport processes explicitly . furthermore , these temporal integrators are constructed to be second - order accurate for the equations of linearized fluctuating hydrodynamics ( lfhd ) , which are suitable for describing thermal fluctuations around stable macroscopic flows over a broad range of length and time scales @xcite . importantly , the linearization of the fluctuating equations is carried out _ automatically _ by the code , making the numerical methods very similar to standard deterministic cfd schemes . finally , specific integrators are proposed in ref . @xcite to handle the extreme separation of scales between the fast velocity and the slow concentration by taking an _ overdamped _ limit of the inertial equations . in this work , we extend the semi - implicit temporal integrators proposed in ref . @xcite for incompressible flows to account for the quasi - incompressible nature of low mach number flows . we apply these temporal integrators to the staggered - grid conservative finite - volume spatial discretization developed in ref . @xcite , and additionally generalize the treatment of advection to allow for the use of monotonicity - preserving higher - order godunov schemes @xcite . our work relies heavily on several prior works , which we will only briefly summarize in the present paper . the spatial discretization we describe in more detail in section [ sub : advection ] is identical to that proposed by donev _ et al _ @xcite , which itself relies heavily on the treatment of thermal fluctuations developed in refs . @xcite . a key development that makes the algorithm presented here feasible for large - scale problems is recent work by some of us @xcite on efficient multigrid - based iterative methods for solving unsteady and steady variable - coefficient stokes problems on staggered grids . our high - order godunov method for mass advection is based on the work of bell _ et al . _ @xcite . the temporal integrators developed in section [ sub : temporaldiscretization ] are a novel approach to low mach number hydrodynamics even in the deterministic context . in high - resolution finite - volume methods , the dominant paradigm has been to use a splitting ( fractional - step ) or projection method @xcite to separate the pressure and velocity updates @xcite . we followed such a projection approach to construct an explicit temporal integrator for the low mach number equations @xcite . when viscosity is treated implicitly , however , the splitting introduces a commutator error that leads to the appearance of spurious or `` parasitic '' modes in the presence of physical boundaries @xcite . there are several techniques to reduce ( but not eliminate ) these artificial boundary layers @xcite , and for sufficiently large reynolds number flows the time step size dictated by advective stability constraints makes the splitting error relatively small in practice . at small reynolds numbers , however , the splitting error becomes larger as viscous effects become more dominant , and projection methods do not apply in the steady stokes regime for problems with physical boundary conditions . methods that do not split the velocity and pressure updates but rather solve a combined stokes system for velocity and pressure have been used in the finite - element literature for some time , and have more recently been used in the finite - volume context for incompressible flow @xcite . here we demonstrate how the same approach can be effectively applied to the low mach number equations for a binary fluid mixture @xcite , to construct a method that is second - order accurate up to boundaries , for a broad range of reynolds numbers including steady stokes flow . we test our ability to accurately capture the static structure factor for equilibrium fluctuation calculations . then , we test our methods deterministically on two variable density and variable viscosity low mach number flows . first , we confirm second - order deterministic accuracy in both space and time for a lid - driven cavity problem in the presence of a bubble of a denser miscible fluid . next , we simulate the development of a kevin - helmholtz instability as a lighter less viscous fluid streams over a denser more viscous fluid . these tests confirm the robustness and accuracy of the methods in the presence of large contrasts , sharp gradients , and boundaries . next we focus on the use of fluctuating low mach number equations to study giant concentration fluctuations . in section [ sec : giantfluct ] we apply our methods to study the development of giant fluctuations @xcite during free diffusive mixing of water and glycerol . we compare simulation results to experimental measurements of the time - correlation function of concentration fluctuations during the diffusive mixing of water and glycerol @xcite . the relaxation times show signatures of the rich deterministic dynamics , and a transition from purely diffusive relaxation of concentration fluctuations at large wavenumbers , to more complex buoyancy - driven dynamics at smaller wavenumbers . we find reasonably - good agreement given the large experimental uncertainties , and observe the appearance of propagative modes at small wavenumbers , which we suggest could be observed in experiments as well . | app . math . and comp . sci . comput . phys . , 16(5):1263 - 1297 , 2014 ] . do not use a fractional time - step approach in the spirit of projection methods , thus avoiding splitting errors and giving full second - order deterministic accuracy even in the presence of boundaries for a broad range of reynolds numbers including steady stokes flow . finally , we apply our algorithms to model the development of giant concentration fluctuations during the diffusive mixing of water and glycerol , and compare numerical results with experimental measurements . we find good agreement between the two , and observe propagative ( non - diffusive ) modes at small wavenumbers ( large spatial scales ) , not reported in published experimental measurements of concentration fluctuations in fluid mixtures . our work forms the foundation for developing low mach number fluctuating hydrodynamics methods for miscible multi - species mixtures of chemically reacting fluids . | continuing on our previous work [ a. donev , a. nonaka , y. sun , t. g. fai , a. l. garcia and j. b. bell , comm . app . math . and comp . sci . , 9 - 1:47 - 105 , 2014 ] , we develop semi - implicit numerical methods for solving low mach number fluctuating hydrodynamic equations appropriate for modeling diffusive mixing in isothermal mixtures of fluids with different densities and transport coefficients . we treat viscous dissipation implicitly using a recently - developed variable - coefficient stokes solver [ m. cai , a. j. nonaka , j. b. bell , b. e. griffith and a. donev , commun . comput . phys . , 16(5):1263 - 1297 , 2014 ] . this allows us to increase the time step size significantly for low reynolds number flows with large schmidt numbers compared to our earlier explicit temporal integrator . also , unlike most existing deterministic methods for low mach number equations , our methods do not use a fractional time - step approach in the spirit of projection methods , thus avoiding splitting errors and giving full second - order deterministic accuracy even in the presence of boundaries for a broad range of reynolds numbers including steady stokes flow . we incorporate the stokes solver into two time - advancement schemes , where the first is suitable for inertial flows and the second is suitable for the overdamped limit ( viscous - dominated flows ) , in which inertia vanishes and the fluid motion can be described by a steady stokes equation . we also describe how to incorporate advanced higher - order godunov advection schemes in the numerical method , allowing for the treatment of ( very ) large peclet number flows with a vanishing mass diffusion coefficient . we incorporate thermal fluctuations in the description in both the inertial and overdamped regimes . we validate our algorithm with a series of stochastic and deterministic tests . finally , we apply our algorithms to model the development of giant concentration fluctuations during the diffusive mixing of water and glycerol , and compare numerical results with experimental measurements . we find good agreement between the two , and observe propagative ( non - diffusive ) modes at small wavenumbers ( large spatial scales ) , not reported in published experimental measurements of concentration fluctuations in fluid mixtures . our work forms the foundation for developing low mach number fluctuating hydrodynamics methods for miscible multi - species mixtures of chemically reacting fluids . # 1 # 1 # 1 # 1#1 # 1#1 # 1#1 # 1|#1| # 1#1 # 1#1 |
1410.2300 | c | we have developed a low mach number algorithm for diffusively - mixing mixtures of two liquids with potentially different density and transport coefficients . in the low mach number setting , the incompressible constraint is replaced by a quasi - compressible constraint that dictates that stochastic and diffusive mass fluxes must create local expansion and contraction of the fluid to maintain a constant thermodynamic ( base ) pressure . we employed a uniform - grid staggered - grid spatial discretization @xcite . following prior work in the incompressible simple - liquid case @xcite , we treated viscosity implicitly without splitting the pressure update , relying on a recently - developed variable - coefficient stokes solver @xcite for efficiency . this approach works well for any reynolds number , including the viscous - dominated overdamped ( zero reynolds number ) limit , even in the presence of nontrivial boundary conditions . furthermore , by using a high - resolution bds scheme @xcite to advect the concentration we robustly handled the case of no mass diffusion ( no dissipation in the concentration equation ) . in our spatial discretization we strictly preserved mass and momentum conservation , as well as the equation of state ( eos ) constraint , by using a finite - volume ( flux - based ) discretization of advective fluxes in which fluxes are computing using extrapolated values of concentration and density that obey the eos . our temporal discretization used a predictor - corrector integrator that treats all terms except momentum diffusion ( viscosity ) explicitly @xcite . we empirically verified second - order spatio - temporal accuracy in the deterministic method . in the stochastic context , establishing the weak order of accuracy is nontrivial in the general low mach number setting . for centered advection our temporal integration schemes can be shown to be second - order accurate for the special case of a boussinesq constant - density ( incompressible ) approximation , or in the overdamped ( inertia - free ) limit . existing stochastic analysis does not apply to the case of bds advection because godunov schemes do not fit a method - of - lines approach , but rather , employ a space - time construction of the fluxes . the presence of nontrivial density differences between the pure fluid components and nonzero mass diffusion coefficient , complicates the analysis even for centered advection , due to the presence of a nontrivial eos constraint on the fluid dynamics . it is a challenge for future work to develop improved numerical analysis of our schemes in both the deterministic and the stochastic setting . in future work , we will demonstrate how to extend the algorithms proposed here to multispecies mixtures of liquids using a generalization of the low mach number constraint . the nontrivial multispecies formulation of the diffusive and stochastic mass fluxes has already been developed by some of us in the compressible setting @xcite . it is also possible to include thermal effects in our formulation , by treating the temperature in a manner similar to the way we treated concentration here . two key difficulties are constructing a spatial discretization that ensures preservation of an appropriately generalized eos , as well as developing temporal integrators that can handle the moderate separation of time scales between the ( typically ) slower heat diffusion and ( typically ) faster momentum diffusion . in particular , it seems desirable to also treat temperature implicitly . such implicit treatment of mass or heat diffusion is nontrivial because it would require solving coupled ( via the eos constraint ) velocity - temperature or velocity - concentration linear systems , and requires further investigation . in the staggered - grid based discretization developed here , we can only employ existing higher - order godunov advection schemes for the cell - centered scalar fields such as concentration and density . it is a challenge for future work to develop comparable methods to handle advection of the staggered momentum field . this would enable simulations of large reynolds number flows . it should be noted , however , that our unsplit approach is most advantageous at small reynolds numbers . a challenge for future work on low mach number fluctuating hydrodynamics is to account for the effects of surface tension in mixtures of immiscible or partially miscible liquids . this can be most straightforwardly accomplished by using a diffuse - interface model , as some of us recently did in the compressible setting for a single - fluid multi - phase system @xcite . one of the key challenges is handling the fourth - order derivative term in the concentration equation in a way that ensures stability of the temporal integrator , as well as developing a consistent discretization of the korteweg stresses on a staggered grid @xcite . the semi - implicit temporal integrators we described here can deal well with a broad range of reynolds or schmidt numbers in the deterministic ( smooth ) setting . in the context of fluctuating hydrodynamics , however , all modes are thermally excited and treatment of viscosity based on a crank - nicolson method ( implicit midpoint rule ) are bound to fail for sufficiently large schmidt numbers ( or sufficiently low reynolds numbers ) . in this work we solved this problem for the case of infinite schmidt , zero reynolds number flows by taking an overdamped limit of the original inertial equations before temporal discretization . it is a notable challenge for the future to develop uniformly accurate temporal integrators that work over a broad range of reynolds or schmidt numbers , including the asymptotic overdamped limit , in the presence of thermal fluctuations . we would like to thank fabrizio croccolo and alberto vailati for sharing their experimental data on water - glycerol mixing , as well as numerous informative discussions . this material is based upon work supported by the u.s . department of energy office of science , office of advanced scientific computing research , applied mathematics program under award number de - sc0008271 and under contract no . de - ac02 - 05ch11231 . additional support for a. donev was provided by the national science foundation under grant dms-1115341 . | , 9 - 1:47 - 105 , 2014 ] , we develop semi - implicit numerical methods for solving low mach number fluctuating hydrodynamic equations appropriate for modeling diffusive mixing in isothermal mixtures of fluids with different densities and transport coefficients . we treat viscous dissipation implicitly using a recently - developed variable - coefficient stokes solver [ m. cai , a. j. nonaka , j. b. bell , b. e. griffith and a. donev , commun . | continuing on our previous work [ a. donev , a. nonaka , y. sun , t. g. fai , a. l. garcia and j. b. bell , comm . app . math . and comp . sci . , 9 - 1:47 - 105 , 2014 ] , we develop semi - implicit numerical methods for solving low mach number fluctuating hydrodynamic equations appropriate for modeling diffusive mixing in isothermal mixtures of fluids with different densities and transport coefficients . we treat viscous dissipation implicitly using a recently - developed variable - coefficient stokes solver [ m. cai , a. j. nonaka , j. b. bell , b. e. griffith and a. donev , commun . comput . phys . , 16(5):1263 - 1297 , 2014 ] . this allows us to increase the time step size significantly for low reynolds number flows with large schmidt numbers compared to our earlier explicit temporal integrator . also , unlike most existing deterministic methods for low mach number equations , our methods do not use a fractional time - step approach in the spirit of projection methods , thus avoiding splitting errors and giving full second - order deterministic accuracy even in the presence of boundaries for a broad range of reynolds numbers including steady stokes flow . we incorporate the stokes solver into two time - advancement schemes , where the first is suitable for inertial flows and the second is suitable for the overdamped limit ( viscous - dominated flows ) , in which inertia vanishes and the fluid motion can be described by a steady stokes equation . we also describe how to incorporate advanced higher - order godunov advection schemes in the numerical method , allowing for the treatment of ( very ) large peclet number flows with a vanishing mass diffusion coefficient . we incorporate thermal fluctuations in the description in both the inertial and overdamped regimes . we validate our algorithm with a series of stochastic and deterministic tests . finally , we apply our algorithms to model the development of giant concentration fluctuations during the diffusive mixing of water and glycerol , and compare numerical results with experimental measurements . we find good agreement between the two , and observe propagative ( non - diffusive ) modes at small wavenumbers ( large spatial scales ) , not reported in published experimental measurements of concentration fluctuations in fluid mixtures . our work forms the foundation for developing low mach number fluctuating hydrodynamics methods for miscible multi - species mixtures of chemically reacting fluids . # 1 # 1 # 1 # 1#1 # 1#1 # 1#1 # 1|#1| # 1#1 # 1#1 |
astro-ph9911476 | i | reverberation mapping , observing the degree and nature of the correlation between continuum and emission line flux variations , is one of the major tools for studying the distribution and kinematics of the gas in the broad line region ( blr ) of active galactic nuclei ( agn ; see reviews by peterson 1993 and by netzer & peterson 1997 ) . about 17 low luminosity agn ( seyfert 1 galaxies ) have been successfully monitored and produce statistically meaningful results ( see wandel , peterson , & malkan 1999 , and references therein ) . best studied among these is the seyfert 1 galaxy , ngc 5548 , which was monitored from the ground for over eight years , and from space for several long periods ( peterson et al . 1999 , and references therein ) . several other seyfert 1s were observed for periods of order 1 year or less , and nine seyfert 1s were studied over a period of eight years ( peterson et al . the measured time lags between the emission lines and the continuum light curves in these objects can be interpreted in terms of the delayed response of a spatially - extended blr to a variable , compact source of ionizing radiation . while the observations do not uniquely determine the geometry of the blr , they give its typical size which , for seyfert 1 galaxies , is of the order of light - days to several light - weeks ( @xmath4@xmath5 cm ) . recent studies have shown that the time lags determined in ngc 5548 for different observing seasons correlate with the seasonal luminosity of the object ( peterson et al . 1999 ) , and have presented evidence for keplerian motions of the blr gas ( peterson & wandel 1999 ) . while seyfert 1 galaxies have been studied successfully using reverberation mapping , few similar studies of the more luminous agn the quasars have been presented . those that have been published have been limited in their success at determining the properties of the quasar blrs . past attempts to monitor quasars spectrophotometrically have generally suffered from temporal sampling and/or flux calibrations that are not sufficient for the determination of the blr size . zheng and collaborators ( zheng & burbidge 1986 , zheng et al . 1987 , and zheng 1988 ) reported results of a monitoring program executed on 30 quasars for several years , with a sampling interval of about one year . they found the emission - line flux to change in response to the continuum changes , but because of the limited time resolution , only an upper limit of about one light - year for the blr size could be deduced . several other groups ( prez , penston , & moles 1989 , korista 1991 , jackson et al . 1992 , and erkens et al . 1995 ) observed samples of quasars for approximately one year sampled with monthly observations . each of these groups reported line variations on time scales of a few months , but because of the short duration of these programs the data were insufficient for more specific conclusions . wisotzki et al . ( 1998 ) monitored the gravitationally lensed quasar he1104@xmath61805 for five years . although the quasar continuum varied considerably , the emission line fluxes appeared to remain constant . researchers have also attempted to use spectra obtained with the _ international ultraviolet explorer ( iue ) _ and reverberation mapping techniques to study agn . in particular , several studies ( gondhalekar et al . 1986 , gondhalekar 1990 , sitko 1990 , obrien & gondhalekar 1991 , and koratkar et al . 1998 ) deduced blr sizes of 0.012 light - years for agn . however , these results are controversial due to insufficient sampling and/or uncertainty regarding the absolute flux calibrations of the iue observations ( e.g. , bohlin & grillmair 1988a , 1988b ) . even the quasar best studied by iue , 3c273 , has yielded disputed results when different researchers have analyzed similar iue data sets . both obrien & harries ( 1991 ) and koratkar & gaskell ( 1991a ) found a measurable and similar lag between continuum and blr variations , while ulrich , courvoisier , & wamsteker ( 1993 ) argue that the line variations reported in the earlier studies were only marginally significant . thus , even in the the best - studied case , past results have proven controversial . to obtain more definite results on the blr size in quasars , we began monitoring a well - defined sub - sample of 28 quasars from the palomar - green ( pg ) sample ( schmidt & green 1983 ) in 1991 march at the wise and steward observatories . results from the first 1.5 years were presented in maoz et al . ( 1994 ; paper i ) where it was shown that most quasars underwent continuum variations ( at 4800 rest wavelength ) with amplitudes of 10% 70% . balmer - line variability , correlated with the continuum variations , was detected in several objects . the preliminary data showed that the typical response time of the emission lines is @xmath76 months . reverberation mapping of such objects therefore requires several years , with sampling intervals of less than a few months . further results , based on four years of monitoring two quasars from our sample , pg0804 + 761 and pg0953 + 414 , were presented in kaspi et al . ( 1996a ; paper the measured time lags between the balmer lines and the continuum variations of the quasars suggested that the blr size grows roughly as the square root of the luminosity of the nucleus . wandel et al . ( 1999 ) confirmed this result using a larger sample of agn . during the period of our spectrophotometric project we also monitored several quasar samples photometrically in the @xmath8 and @xmath9 bands . the broad - band results for radio- and optically - selected samples were presented in netzer et al . ( 1996 ) and giveon et al . ( 1999 ; hereafter paper iii ) , respectively . the optical sample in paper iii consists of 42 pg quasars , including all 28 objects discussed in the present paper . the present paper presents final results from 7.5 years of spectroscopic monitoring of our sample . in 2 we describe the sample , observations , data reduction , and present the light curves . in 3 we perform a time series analysis to determine the response time of the emission - line flux to continuum variations . we estimate the blr size , continuum luminosity , and central mass for each quasar in 4 , and in 5 we discuss the relations among these properties . | correlated variations in the line and continuum emission from active galactic nuclei ( agn ) can be used to determine the size and geometry of the broad emission line regions ( blrs ) . we have spectrophotometrically monitored a well - defined sample of 28 palomar - green quasars in order to obtain measurements of their blrs and to investigate the relationships between quasar luminosity , central black hole mass , and blr size in agn . both the continuum and emission line fluxes of all of the quasars were observed to change during the duration of the observing program . a significant correlation was observed , with the balmer line variations lagging those of the continuum by days ( rest frame ) . mass | correlated variations in the line and continuum emission from active galactic nuclei ( agn ) can be used to determine the size and geometry of the broad emission line regions ( blrs ) . we have spectrophotometrically monitored a well - defined sample of 28 palomar - green quasars in order to obtain measurements of their blrs and to investigate the relationships between quasar luminosity , central black hole mass , and blr size in agn . spectrophotometry was obtained every 14 months for 7.5 years , yielding 2070 observing epochs per object . both the continuum and emission line fluxes of all of the quasars were observed to change during the duration of the observing program . seventeen of the 28 objects were observed with adequate sampling ( independent observing epochs ) to search for correlated variations between the balmer emission lines and the continuum flux . for each of these 17 objects , a significant correlation was observed , with the balmer line variations lagging those of the continuum by days ( rest frame ) . our work increases the available luminosity range for studying the size mass luminosity relations in agn by two orders of magnitude and doubles the number of objects suitable for such studies . combining our results with comparable published data available for seyfert 1 galaxies , we find the blr size scales with the rest - frame 5100 luminosity as . this determination of the scaling of the size of the blr as a function of luminosity is significantly different from those previously published , and suggests that the effective ionization parameter in agn may be a decreasing function of luminosity . we are also able to constrain , subject to our assumption that gravity dominates the motions of the blr gas , the scaling relationship between the mass of the central black holes and the luminosity in agn . we find that the central mass scales with 5100 luminosity as . this is inconsistent with all agn having optical luminosity that is a constant fraction of the eddington luminosity . |
astro-ph9911476 | c | the luminosity relation is presented in fig . [ frvsl ] . the correlation coefficient is 0.827 , and its significance level is 1.7@xmath436 . a linear fit to the points gives @xmath437 ( solid line plotted in fig . [ frvsl ] ) . considering the seyfert nuclei ( @xmath438(5100 )@xmath439 ) , or the pg quasars alone , we find only marginally significant correlations , probably because of the narrow luminosity ranges . a significant correlation emerges only when using the whole luminosity range . the present result is remarkable for two reasons . first , earlier studies of this kind found a smaller power - law index in the blr size agn luminosity relation ( closer to 0.5 , e.g. , koratkar & gaskell 1991b ; wandel et al . a line with this slope was fit to the data and is shown as a dashed line in fig . [ frvsl ] . our result using the combined sample is not consistent with previous results . second , under the assumptions that the shape of the ionizing continuum in agn is independent of @xmath440 , and that all agn are characterized by the same ionization parameter and blr density ( as suggested by the similar line ratios in low- and high - luminosity sources ) , one expects @xmath441 . the theoretical prediction is based on the assumption that the gas distribution , and hence the mean blr size , scales with the strength of the radiation field . our present result suggests that those assumptions should be re - examined . this is also implied from recent models ( e.g. , kaspi & netzer 1999 and references therein ) which show a wide distribution of properties ( such as blr density , column density , and ionization parameter ) across the blr of a single active nucleus . therefore , a range of properties may also exist among different agn , and the above assumptions of uniform ionization parameter and blr density for all agn is likely incorrect . if an effective ionization parameter can be defined , our result suggests that it may be a decreasing function of luminosity . = 8.5 cm our sample allows us to to re - address the issue of the velocity luminosity relation in agn . shuder ( 1984 ) noted that the fwhm of the balmer lines increases with luminosity for a sample of 25 agn . wandel & yahil ( 1985 ) found a correlation coefficient of 0.5 between the full width at zero intensity ( fw0i ) and the 4000 continuum luminosity for a literature compilation of 94 agn . joly et al . ( 1985 ; and references therein ) also reported a weak correlation between the fw0i of and the optical luminosity in a similar collection of objects . boroson & green ( 1992 ) note the fw0i of is quite sensitive to noise and depends strongly on the quality of the feii subtraction . these authors found an anticorrelation coefficient of @xmath60.275 between the fwhm and absolute @xmath442 magnitude , significant at the 99% confidence level , in a sample of 87 pg quasars . in a recent work stirpe , robinson , & axon ( 1999 ) measured velocities for 126 agn and found them to weakly correlate with the luminosity . = 8.5 cm in addressing the issue of the velocity luminosity relation we note that our sample is _ not _ complete , especially with the inclusion of the heterogeneous seyfert sample . the results must therefore be treated with caution . the fwhms from the mean spectrum are plotted against the luminosity in the bottom panel of fig . [ ffvsllog ] . there is no significant correlation between the variables . however , if we omit the data point for the narrow line seyfert 1 , ngc 4051 , which clearly deviates from the other points and highly influences the fit , we do obtain a marginally significant correlation . the correlation coefficient is @xmath60.386 , and its significance level is 2.7@xmath443 . a linear fit gives @xmath444 fwhms from the rms spectrum versus luminosity are plotted in the top panel of fig . [ ffvsllog ] . the correlation coefficient is @xmath60.540 has a significance level of 1.2@xmath445 and the best linear fit to the data is @xmath446 thus , we find significant anticorrelations , in our incomplete sample , between the fwhm of the balmer lines and the luminosity of the objects such that @xmath447 . our result indicates a stronger anticorrelation coefficient when @xmath448 is used . while the above studies used only one epoch for each quasar in their measurements , our study uses the means of the luminosities and velocities for each object over 7.5 years , and averages both and . the pg quasars @xmath431 found in this study agrees with the ones found by boroson & green s ( 1992 ) to within @xmath110% , except for two objects ( pg1351 and pg1704 ) for which we measure significantly narrower lines . if we exclude those two objects , the anticorrelation becomes weaker and its significance decreases . as the correlation we find is opposite to those found by previous studies , the issue of the velocity luminosity relation needs further investigation , which is beyond the scope of this paper . our two velocity estimates produce two mass estimates for each object ( see table [ trlm ] and [ secmass ] ) . our mass estimates based on the determination of the fwhm from the mean spectra are plotted versus luminosity in the bottom panel of fig . [ fmvsl ] . the correlation coefficient between these two parameters is 0.646 and has a significance level of 3.7@xmath449 . a linear fit gives @xmath450 and is plotted as a solid line in the diagram . the mass estimates based on the determination of the fwhm measured from the rms spectra are plotted in the top panel of fig . [ fmvsl ] . we find correlation coefficient of 0.473 with a significance level of 4.7@xmath451 . a linear fit to this relation gives @xmath452 and is also plotted as a solid line . the results of the above two methods are not consistent . moreover , while it is arguable that using the rms spectra to determine @xmath433 is a better method ( see [ secmass ] ) , the mass luminosity correlation based on this measure is less significant . this can perhaps be attributed to the fact that the line fluxes in the rms spectra are weaker and hence the uncertainty in the corresponding fwhm larger . l relation does not agree with the one found by koratkar & gaskell ( 1991b ) of @xmath453 , nor does it agree with the one found by wandel et al . ( 1999 ) of @xmath454 . however , when wandel et al . ( 1999 ) use an unweighted linear fit they find a slope of 0.54 which is in good agreement with our result . we have used a linear regression analysis which takes into account the uncertainties in both coordinates ( see [ sml ] ) . the fact that the scatter in the mass luminosity relation is larger than that of the size luminosity and velocity luminosity relations may indicate that luminosity , rather than mass , is the variable that mainly determines the blr size . in fact , from the individual size and velocity relations we expect : @xmath455 , i.e. , a weak dependence of mass on luminosity , with all agn having similar masses . in practice , we have found a somewhat stronger dependence , @xmath456 , but with a large scatter . l relation can be expressed in terms of the eddington luminosity , @xmath457 . roughly estimating the bolometric luminosity as @xmath458(5100 ) , we obtain an eddington ratio of @xmath459 the eddington limit , based on this rough estimate for @xmath460 , is plotted as a dashed line in fig . [ fmvsl ] . some of the quasars in our sample appear to be approximately at or beyond the eddington limit . according to fig . [ fmvsl ] , the eddington limit traces an envelope in the mass luminosity plane ( if we ignore the two lower right hand points , pg1700 and pg1704 , which have very large errors ) . this depends , somewhat , on the factor chosen to estimate the bolometric correction , i.e. the rather uncertain slope of the unobserved uv continuum . however , if our chosen bolometric correction is realistic , we may be seeing a direct indication that agn energy is generated by gas accretion . wandel ( 1999 ) reviews three classes of agn mass estimation methods . while the x - ray variability method ( using the shortest time scale for global luminosity variations as the light travel time across the schwarzschild radius ) and the accretion disk modeling method ( deriving an accretion disk model that best fits the observed agn continuum ) suggest that the eddington ratio increases with luminosity , wandel ( 1999 ) notes that the kinematic methods ( such as reverberation mapping ) have yet to show a similar trend . our reverberation mapping result ( equation [ eqeddr ] ) indicates for the first time that the eddington ratio increases with luminosity . models that suggest the bulk of the luminosity is due to energy release via mechanisms which radiate up to a set fraction of the eddington luminosity are not consistent with the derived mass luminosity relation . one such model is a geometrically thin , optically thick , accretion disk that requires @xmath461 to be self - consistent ( laor & netzer 1989 ) . our finding suggests that the mass accretion rate grows with luminosity much faster than the central mass , which would mean very different disk properties in low- and high - luminosity sources . our luminosity determination is based on the monochromatic flux at 5100 . in the thin disk model , this monochromatic flux may represents a different fraction of the object s bolometric luminosity in agn of different masses . thus , it is not clear that the thin accretion disk model can be ruled out by the new results . | seventeen of the 28 objects were observed with adequate sampling ( independent observing epochs ) to search for correlated variations between the balmer emission lines and the continuum flux . for each of these 17 objects , this determination of the scaling of the size of the blr as a function of luminosity is significantly different from those previously published , and suggests that the effective ionization parameter in agn may be a decreasing function of luminosity . we are also able to constrain , subject to our assumption that gravity dominates the motions of the blr gas , the scaling relationship between the mass of the central black holes and the luminosity in agn . this is inconsistent with all agn having optical luminosity that is a constant fraction of the eddington luminosity . | correlated variations in the line and continuum emission from active galactic nuclei ( agn ) can be used to determine the size and geometry of the broad emission line regions ( blrs ) . we have spectrophotometrically monitored a well - defined sample of 28 palomar - green quasars in order to obtain measurements of their blrs and to investigate the relationships between quasar luminosity , central black hole mass , and blr size in agn . spectrophotometry was obtained every 14 months for 7.5 years , yielding 2070 observing epochs per object . both the continuum and emission line fluxes of all of the quasars were observed to change during the duration of the observing program . seventeen of the 28 objects were observed with adequate sampling ( independent observing epochs ) to search for correlated variations between the balmer emission lines and the continuum flux . for each of these 17 objects , a significant correlation was observed , with the balmer line variations lagging those of the continuum by days ( rest frame ) . our work increases the available luminosity range for studying the size mass luminosity relations in agn by two orders of magnitude and doubles the number of objects suitable for such studies . combining our results with comparable published data available for seyfert 1 galaxies , we find the blr size scales with the rest - frame 5100 luminosity as . this determination of the scaling of the size of the blr as a function of luminosity is significantly different from those previously published , and suggests that the effective ionization parameter in agn may be a decreasing function of luminosity . we are also able to constrain , subject to our assumption that gravity dominates the motions of the blr gas , the scaling relationship between the mass of the central black holes and the luminosity in agn . we find that the central mass scales with 5100 luminosity as . this is inconsistent with all agn having optical luminosity that is a constant fraction of the eddington luminosity . |
astro-ph9911476 | i | spectrophotometric monitoring of a large , optically - selected quasar sample has shown clear correlations and well - defined time lags between the optical continuum and the balmer - line light curves . while the seyfert 1 galaxies that have been studied in this manner all have optical luminosities @xmath462 1.5@xmath4210@xmath463 ergs s@xmath424 , the new sample allows us to measure time lags in agn with luminosities up to @xmath464 ergs s@xmath424 . our work increases the available luminosity range for studying the size luminosity relations in agn by two orders of magnitude and doubles the number of objects suitable for these studies . we have combined our results for 17 quasars with data for 17 seyfert 1 galaxies having reliable time lag measurements , and derived uniform estimates of blr size , central masses , and luminosities for the combined sample . our main finding is that the blr size scales with the 5100 luminosity as @xmath465 . this is significantly different from wandel et al.s ( 1999 ) analysis and is also in contradiction with simple theoretical expectations , both suggesting @xmath466 . we have also found that the velocity field of the blr scales inversely with the luminosity , @xmath467 . combining the measured with the observed fwhms , we have obtained a mass luminosity relation for agn , @xmath468 , which , however , has a large intrinsic scatter . l correlations are based on two different estimates ( mean and rms ) of the fwhm of the balmer lines and are not consistent with each other , despite the fact that the two measured values for the fwhm are generally consistent . empirically , at least , it is not obvious which method of fwhm measurement is preferable . our results show the usefulness of long - term monitoring of high - luminosity agn . there is a need to expand the luminosity range to include the highest luminosity quasars and this will require some 510 years of observations . follow - up studies are also needed for some of the results obtained here , such as better determinations of the gas distribution in the blr and the exact sed of the quasars . future work using this sample will include the study of time - variable line profiles , lags between continuum bands , the intrinsic baldwin relation , and more . we are grateful to brad peterson for supplying us with the seyfert 1 data and for many enlightening comments and discussions , and to the referee , martin gaskell , for his constructive advice . ari laor is acknowledged for very useful discussions . we thank john dann and the wo staff for their expert assistance with the observations throughout the years . research at the wo is supported by grants from the israel science foundation . monitoring of pg quasars at so was supported by nasa grant nag 5 - 1630 . h.n . & s.k . acknowledge financial support by the the jake adler chair of extragalactic astronomy . s.k . acknowledges financial support by the colton scholarships . | our work increases the available luminosity range for studying the size luminosity relations in agn by two orders of magnitude and doubles the number of objects suitable for such studies . combining our results with comparable published data available for seyfert 1 galaxies , we find the blr size scales with the rest - frame 5100 luminosity as . we find that the central mass scales with 5100 luminosity as . | correlated variations in the line and continuum emission from active galactic nuclei ( agn ) can be used to determine the size and geometry of the broad emission line regions ( blrs ) . we have spectrophotometrically monitored a well - defined sample of 28 palomar - green quasars in order to obtain measurements of their blrs and to investigate the relationships between quasar luminosity , central black hole mass , and blr size in agn . spectrophotometry was obtained every 14 months for 7.5 years , yielding 2070 observing epochs per object . both the continuum and emission line fluxes of all of the quasars were observed to change during the duration of the observing program . seventeen of the 28 objects were observed with adequate sampling ( independent observing epochs ) to search for correlated variations between the balmer emission lines and the continuum flux . for each of these 17 objects , a significant correlation was observed , with the balmer line variations lagging those of the continuum by days ( rest frame ) . our work increases the available luminosity range for studying the size mass luminosity relations in agn by two orders of magnitude and doubles the number of objects suitable for such studies . combining our results with comparable published data available for seyfert 1 galaxies , we find the blr size scales with the rest - frame 5100 luminosity as . this determination of the scaling of the size of the blr as a function of luminosity is significantly different from those previously published , and suggests that the effective ionization parameter in agn may be a decreasing function of luminosity . we are also able to constrain , subject to our assumption that gravity dominates the motions of the blr gas , the scaling relationship between the mass of the central black holes and the luminosity in agn . we find that the central mass scales with 5100 luminosity as . this is inconsistent with all agn having optical luminosity that is a constant fraction of the eddington luminosity . |
1004.0289 | i | crum s seminal paper of 1955 @xcite has played an essential role in elucidating the structure of one - dimensional quantum mechanical systems in general and exactly solvable ones , in particular . throughout this paper , we mean ` exact solvability ' in the schrdinger picture , namely a quantum system is exactly solved when the complete set of eigenvalues and eigenfunctions are known . many exactly solvable quantum mechanical hamiltonians were constructed and investigated by combining shape invariance @xcite and crum s theorem @xcite , or the factorisation method @xcite or the method of the so - called supersymmetric quantum mechanics @xcite . it is interesting to note that most of these shape invariant systems are also solvable in the heisenberg picture @xcite . exactly solvable quantum mechanical systems of one and many degrees of freedom are not only important in their own right but also have fundamental applications in various disciplines of physics / mathematics , _ e.g. _ the fokker - planck equations @xcite and their discretised version , birth and death processes @xcite , to name a few . shape invariance is a sufficient condition for exactly solvable quantum mechanical systems . the number of shape invariant systems , however , was quite limited ; only about a dozen until the recent discovery @xcite of the several types of infinitely many shape invariant hamiltonians @xcite which led to the infinitely many exceptional laguerre , jacobi , wilson and askey - wilson polynomials . many methods were proposed to derive exactly solvable but non - shape invariant quantum mechanical systems from known shape invariant ones @xcite . ( we apologise to those whose work we have missed . ) among them krein - adler s modification @xcite of crum s theorem is the most comprehensive way to generate infinitely many variants of exactly solvable hamiltonians and their eigenfunctions , starting from an exactly solvable one . the derived system is _ iso - spectral _ with the original one except that a finite number of energy levels are deleted . if the original system has polynomial eigenfunctions , as is usually the case , the derived systems have also polynomial eigenfunctions . by construction , these polynomials constitute a complete set of orthogonal functions . but they do not qualify to be called _ exceptional orthogonal polynomials _ @xcite since some members of certain degrees are _ missing _ due to the _ deletion_. the discrete quantum mechanics is a deformation of the ordinary quantum mechanics in the sense that the schrdinger equation is a second order _ difference _ equation instead of differential . in the formulation of odake and sasaki @xcite , the algebraic and analytical structure of quantum mechanics as well as shape invariance and exact solvability are retained in the discrete version . the eigenfunctions of the exactly solvable one - dimensional discrete quantum mechanics are the askey - scheme of hypergeometric orthogonal polynomials and their @xmath2-versions @xcite , _ e.g. _ the continuous hahn , the wilson and the askey - wilson polynomials . these examples are all shape invariant and they are also solvable in the heisenberg picture @xcite . the dynamical symmetry algebra of these algebras are the askey - wilson algebras @xcite and degenerations , which contain the @xmath2-oscillator algebra @xcite . the discrete version of crum s theorem was also established recently @xcite . in this paper we present the discrete quantum mechanics version of adler s modification @xcite of crum s theorem . it allows to generate an infinite variety of exactly solvable discrete quantum hamiltonian systems . the insight obtained from crum s theorems and their modification , in the ordinary and the discrete quantum mechanics , is essential for the recent derivation of the infinite numbers of shape invariant systems and the new exceptional orthogonal polynomials @xcite . we will discuss the main results , the specialisation to the cases of polynomial eigenfunctions and simplest example for various exactly solvable cases ; first for the ordinary quantum mechanics and then for the discrete versions . the reason is two - fold ; firstly to introduce appropriate notion and notation in the familiar cases of the ordinary quantum mechanics . secondly we choose to reveal the underlying logical processes which are not easy to fathom in adler s paper @xcite or in crum s original article @xcite . as seen in the subsequent sections , the logical structures of the associated hamiltonian systems and their modification by _ deletion _ of energy levels are shared by the ordinary and the discrete quantum mechanics . this paper is organised as follows . in section two , adler s modification of crum s theorem is recapitulated in appropriate notation for our purposes . the specialisation to the cases of polynomial eigenfunctions is discussed in some detail . section three provides the discrete quantum mechanics version of the modification of crum s theorem . again the specialisation to the cases of polynomial eigenfunctions is mentioned . appendix gives the simplest examples of the modified hamiltonian systems obtained by deleting the lowest lying @xmath3 excited states for various exactly solvable hamiltonians . appendix a provides three examples from the ordinary quantum mechanics , the harmonic oscillator , the radial oscillator , the darboux - pschl - teller potential . appendix b is for the four examples from the discrete quantum mechanics , the hamiltonians of the meixner - pollaczek , the continuous hahn , the wilson and the askey - wilson polynomials @xcite , which are known to reduce to the hermite , the laguerre and the jacobi polynomials in certain limits , respectively . | crum s theorem in one - dimensional quantum mechanics asserts the existence of an associated hamiltonian system for any given hamiltonian with the complete set of eigenvalues and eigenfunctions . the associated system is iso - spectral to the original one except for the lowest energy state , which is deleted . yukawa institute kyoto + dpsu-10 - 1 + yitp-10 - 15 + * modification of crum s theorem for ` discrete ' quantum mechanics + + * * leonor garca - gutirrez , satoru odake and ryu sasaki * yukawa institute for theoretical physics , + kyoto university , kyoto 606 - 8502 , japan + department of physics , shinshu university , + matsumoto 390 - 8621 , japan | crum s theorem in one - dimensional quantum mechanics asserts the existence of an associated hamiltonian system for any given hamiltonian with the complete set of eigenvalues and eigenfunctions . the associated system is iso - spectral to the original one except for the lowest energy state , which is deleted . a modification due to krein - adler provides algebraic construction of a new complete hamiltonian system by deleting a finite number of energy levels . here we present a discrete version of the modification based on the crum s theorem for the ` discrete ' quantum mechanics developed by two of the present authors . yukawa institute kyoto + dpsu-10 - 1 + yitp-10 - 15 + * modification of crum s theorem for ` discrete ' quantum mechanics + + * * leonor garca - gutirrez , satoru odake and ryu sasaki * yukawa institute for theoretical physics , + kyoto university , kyoto 606 - 8502 , japan + department of physics , shinshu university , + matsumoto 390 - 8621 , japan |
1112.1710 | i | the development of amr @xcite was meant to provide high resolution simulations for much lower computational cost than fixed grid methods would allow . the use of highly parallel systems and the algorithms that go with them were also meant to allow higher resolution simulations to be run faster ( relative to wall clock time ) . the parallelization of amr algorithms , which should combine the cost / time savings of both methods is not straight forward however and there have been many different approaches @xcite . while parallelization of a uniform mesh demands little communication between processors , amr methods can demand considerable communication to maintain data consistency across the unstructured mesh as well as shuffling new grids from one processor to another to balance workload . in this paper we report the development and implementation of new algorithms for the efficient parallelization of amr designed to scale to very large simulations . the new algorithms are part of the astrobear package for simulation of astrophysical fluid multi - physics problems @xcite . the new algorithmic structure described in this paper constitutes the development of version 2.0 of the astrobear code . amr methods come in many varieties . meshes can either be unstructured or semi - structured . semi - structured methods can be further divided into those which allow grids to be of arbitrary size ( patch based ) and those which require grids to be of a fixed size ( block based or cell - based if the block size is 1 ) . with block ( or cell ) based amr , the additional constraints imposed on the structure of the mesh allow for a simpler type of connectivity within a tree . for example in block based amr , any given block will have exactly 8 children or none ( if it is a leaf ) and will have at most 6 face sharing neighbors . with patch based amr , there is no limit to the number of children or neighbors . in addition , the operation of regridding in block based amr is much simpler . as the grid changes a given block will either persist if the physical region continues to require refinement or be destroyed . in patch based amr , a given region may subsequently be better covered by patches of a different shape requiring transfer of data between physically overlapping previous patches and new patches . this adds an additional dimension to the tree structure and increases the complexity of maintaining a distributed tree . for both block ( or cell ) and patch based amr , the actual grid data ( fluid variables etc . ) are always distributed across the various processors . usually some overlap in grid data ( guard / ghost cells ) is desired to allow for frequent access to neighboring values without the need for additional communication . but the metadata that describes the shape and distribution of the grid data is usually stored on every processor . for 100 s or 1000 s of cores , this global tree typically requires less memory than that required for the local grid data and it allows for easy access to data from any part of the domain . for instance after regridding , the entire tree can be updated and stored locally and then finding new neighbors for local cells / blocks / patches can be done without the need for any further communication . for patch - based amr it also allows for better load balancing as each processor can determine from the entire tree which section of data it should be responsible for and can use various knap - sack type algorithms to optimize the degree of interprocessor communication . cell based amr engines typically use an octree data structure to handle the tree metadata and implementations have been developed that support a memory - distributed tree @xcite , or various ways of compressing the global tree @xcite that rely on the simple structure of the tree . for patch based amr , algorithms for implementing a distributed tree have not yet been published though implemented in the package samrai @xcite as well as astrobear . in addition the chombo library @xcite recently has developed a method of compressing the metadata to avoid having to distribute the tree . here we document the distributed tree algorithm used in astrobear 2.0 in which no processor has access to the entire tree but rather each processor is only aware of the amr structure it needs to manage in order to carry out its computations and perform the necessary communications . while currently , this additional memory is small compared to the resources typically available to a cpu , future clusters will likely have much less memory per processor similar to what is already seen in gpu s . additionally each processor only sends and receives the portions of the tree necessary to carry out its communication as opposed to a pruning approach in which every processor receives every new patch , but only keeps those necessary to maintain the local tree . astrobear 2.0 also uses extended ghost cells to decouple advances on various levels of refinement . as we show below this allows for each level s advance to be computed independently on separate threads . such inter - level threading allows for total load balancing across all refinement levels instead of balancing each level independently . independent load balancing becomes especially important for deep simulations ( simulations with low filling fractions but many levels of amr ) as opposed to shallow simulations ( high filling fractions and only a few levels of amr ) . processors with coarse grids can advance their grids simultaneously while processors with finer grids advance theirs . without such a capability , each level would need to have enough cells to be able to be distributed across all of the processors . variations in the filling fractions from level to level can make the number of cells on each level very different . if there are enough cells on the level with the fewest to be adequately distributed , there will likely be far too many cells on the highest level to allow the computation to be completed in a reasonable wall clock time . this often restricts the number of levels of amr that can be practically used . with inter - level threading this restriction is lifted . inter - level threading also allows processors to remain busy while waiting for messages from other processors . in what follows we provide descriptions of the new code and its structure as well as providing tests which demonstrate its effective scaling . in section [ amr_alg ] we review patch based amr . in section [ distributedtree ] we describe in detailt the distributed tree algorithm for patch - based amr , in section [ threading ] we will discuss the inter - level threading of the advance , in section [ loadbalancing ] we will discuss the load balancing algorithm , and in section [ results ] we will present our scaling results and we will conclude in section [ conclusion ] . | this allows for global load balancing instead of level by level load balancing and allows for greater parallelization across both physical space and amr level . threading of level advances can also improve performance by interleaving communication with computation , especially in deep simulations with many levels of refinement . while we see improvements of up to on deep simulations we have employed a distributed tree algorithm that requires processors to only store and communicate local sections of the amr tree structure with neighboring processors . using this distributed approach we are able to get reasonable scaling efficiency ( ) out to 12288 cores and up to 8 levels of amr - independent of the use of threading . | current adaptive mesh refinement ( amr ) simulations require algorithms that are highly parallelized and manage memory efficiently . as compute engines grow larger , amr simulations will require algorithms that achieve new levels of efficient parallelization and memory management . we have attempted to employ new techniques to achieve both of these goals . patch or grid based amr often employs ghost cells to decouple the hyperbolic advances of each grid on a given refinement level . this decoupling allows each grid to be advanced independently . in astrobear we utilize this independence by threading the grid advances on each level with preference going to the finer level grids . this allows for global load balancing instead of level by level load balancing and allows for greater parallelization across both physical space and amr level . threading of level advances can also improve performance by interleaving communication with computation , especially in deep simulations with many levels of refinement . while we see improvements of up to on deep simulations run on a few cores , the speedup is typically more modest ( ) for larger scale simulations . to improve memory management we have employed a distributed tree algorithm that requires processors to only store and communicate local sections of the amr tree structure with neighboring processors . using this distributed approach we are able to get reasonable scaling efficiency ( ) out to 12288 cores and up to 8 levels of amr - independent of the use of threading . |
astro-ph0011556 | r | the h@xmath2 , [ ] @xmath166583 , and [ ] @xmath166716 lines are detected up to nearly @xmath0 kpc on either side of the plane of ngc 4302 . the [ ] @xmath166716 line was not detected at @xmath22 kpc on the west side , though the fainter line at 6731 was detected , due to confusion with noise in that part of the spectra . the flux of the 6716 line at that part of the spectra was inferred from that of the 6731 line assuming the low - density limit of [ ] @xmath23 . the vertical runs of [ ] @xmath8h@xmath2 , [ ] @xmath7h@xmath2 , and [ ] @xmath7 [ ] @xmath24 are shown in figure 3 . the data are averaged over 5 spatial pixels . the [ ] /h@xmath2 and [ ] /h@xmath2 each show a similar rise with @xmath3 , with the [ ] /h@xmath2 rising from about 0.4 in the midplane to 1.0 at @xmath25 kpc on the east side , and to nearly 1.4 at @xmath26 kpc on the west side . the [ ] / [ ] remains fairly constant at about 0.6 for the full range of @xmath3 , with slight increases in the midplane and beyond @xmath27 kpc . in ugc 10288 , the h@xmath2 , [ ] @xmath166583 , and [ ] @xmath166716 lines are all detected well above the midplane , with the [ ] @xmath166583 and h@xmath2 lines being detected up to nearly @xmath1 kpc on the north side for both slit positions . detections of the [ ] @xmath165007 and [ ] @xmath166300 lines are limited to @xmath28 kpc , except for the south side of slit 1 where both lines are detected up to @xmath26 kpc . the vertical runs of [ ] @xmath8h@xmath2 , [ ] @xmath7h@xmath2 , [ ] @xmath7 [ ] @xmath24 , [ ] @xmath10h@xmath2 , and [ ] @xmath29h@xmath2 for both slits are shown in figure 4 . the data are averaged over 5 spatial pixels . for slit 1 , [ ] /h@xmath2 rises from 0.3 at the midplane to 1.6 at @xmath0 kpc on the north side and 1.3 at @xmath0 kpc on the south , while for slit 2 the ratio remains somewhat lower , rising to 1.2 for @xmath26 kpc on either side of the disk . the [ ] / [ ] ratio shows a similar range of values for each slit , varying from 0.8 to 1.4 with the lower values generally occurring in the midplane . these values are significantly higher than those of the other three galaxies . the [ ] /h@xmath2 ratio shows some evidence for an increase with @xmath3 on the south side of either slit , reaching values as high as 0.75 in slit 1 and 0.4 in slit 2 . on the north side of each slit , values reach no higher than 0.5 in slit 1 and 0.4 in slit 2 . such values of [ ] /h@xmath2 are consistent with those observed at comparable @xmath3-heights in ngc 5775 . line ratio data for ngc 5775 , except for the [ ] @xmath29h@xmath2 and extinction corrected [ ] @xmath10h@xmath2 ratios , have been presented previously ( rand 2000 ) . these , along with [ ] @xmath8h@xmath2 , [ ] @xmath7h@xmath2 , and [ ] @xmath7 [ ] @xmath24 ratios are shown in figure 12 . the data are averaged over 10 spatial pixels . an increase with @xmath3 in the extinction corrected [ ] /h@xmath2 for the ne side of slit 1 and the sw side of slit 2 is still indicated by the data , where values range from 0.20.4 in the disk to 1.1 in the halo . the [ /h@xmath2 ratio ranges from about 0.02 in the midplane to 0.06 at @xmath30 kpc in slit 2 , while the [ ] @xmath31 line was not detected for slit 1 . because of the 85 ( irwin 1994 ) inclination of the galaxy , points within 10 of the midplane ( @xmath32 pc in the figures ) on the spatial axis reflect in - plane , highly inclined disk structure rather than true vertical structure . in these lines of sight , regions and areas between them have a greater effect on line ratio variations than extraplanar diffuse gas . this fact explains why some line ratio minima are not at @xmath33 kpc . line ratio data for ngc 891 , previously presented by rand ( 1998 ) , can be seen in figure 13 . in this section we attempt to determine the h@xmath2 emission scale height , @xmath34 , of the galaxies dig halos at the various slit locations . we attempt to model only an exponential halo component , avoiding any disk component where contamination by bright regions could complicate the analysis . we plot the logarithm of h@xmath2 intensity versus @xmath3 for each slit , then fit an exponential component with a least - squares fit except in cases where an exponential is obviously a bad description . plots of logarithmic h@xmath2 intensity vs. @xmath3 , along with exponential fits , for ngc 5775 , ngc 4302 , and ugc 10288 are shown in figures 5 , 6 , and 7 , respectively . results of the fitting procedure are summarized in table 2 . position 1 of ngc 5775 covers the ne filament , dig emission from which has been modeled previously ( crdw ) . the value @xmath35 pc is somewhat consistent with the value of @xmath36 kpc determined from image data . the image data however does include some [ ] contamination . the previously determined value assumes an [ ] /h@xmath2 ratio that does not vary with @xmath3 . since our data shows that this ratio in fact rises with @xmath3 on the ne filament , from 0.5 at @xmath0 kpc to 0.75 at @xmath37 kpc , the value @xmath36 kpc becomes an upper limit on the actual scale height . the dig halos of ngc 4302 and ugc 10288 have been modeled previously as well ( rand 1996 ) . the slit position for ngc 4302 coincides with one of the previously modeled regions , the scale height of which was determined to be @xmath38 pc ( the image is not contaminated by [ ] emission ) . the values determined from spectroscopic data , @xmath39 pc and @xmath40 pc for the east and west sides , are close to those determined from image data . the scale height determined from the image , however , is an average over a 3 kpc radial extent of the disk and thus should not be in exact agreement with the spectroscopically determined value . image data of ugc 10288 revealed very little extraplanar h@xmath2 emission , and thus profiles were not well modeled with exponentials . increased sensitivity of the spectroscopic data have allowed the detection of h@xmath2 emission to greater @xmath3-heights . the resulting h@xmath2 profiles seem to be well modeled by exponentials with small scale heights on the south side , while more complicated structure is present on the north side . ngc 5775 , having the most prominent extraplanar dig emission , also has the dig halo with the greatest scale height . in contrast , ugc 10288 , showing very little extraplanar h@xmath2 emission , has a dig halo of relatively small scale height . ngc 4302 , having a number of extraplanar plumes of h@xmath2 emission , has a dig scale height intermediate between that of ngc 5775 and ugc 10288 . the halo of ngc 891 has been modeled previously ( rkh ) , the scale height of which has been measured to be @xmath41 kpc on the west side and @xmath42 kpc on the east side from imaging data . these results suggest that not only do galaxies such as ngc 5775 and ngc 891 have brighter dig halos , where sensitivity limits allow faint emission to be detected at much greater heights than for a galaxy with a lower h@xmath2 surface brightness , but the halos also have much larger scale heights . this adds further evidence to the notion of a dynamic halo where higher levels of star formation in the disk result in material driven further into the halo ( e.g. rand 1996 ; hoopes et . al . 1999 ) . some of the line ratio behavior observed in ngc 5775 and ugc 10288 , namely the rise in [ ] /h@xmath2 and [ ] /h@xmath2 with @xmath3 , is predicted by the photo - ionization models ( e.g. s94 ; bland - hawthorn 1997 ) . as previously stated , rises in these ratios are expected in such models as the radiation field becomes more dilute with @xmath3 . if the field is in fact becoming more dilute ( low values of the ionization parameter , @xmath9 , which measures the `` diluteness''of the radiation field and is proportional to the ratio of ionizing photon number density to gas density ) with @xmath3 , then the models predict that the [ ] /h@xmath2 ratio should decrease away from the midplane . however , in ngc 5775 , and to a lesser extent ugc 10288 , we see a definite incease in [ ] /h@xmath2 with @xmath3 , indicative of higher excitation conditions above the disk in such models ( high-@xmath9 ) . recent spectroscopy of ngc 5775 by tllmann et . al . ( 2000 ) also reveals a significant rise in [ ] /h@xmath2 with @xmath3 , though these data are not extinction corrected . it is then possible that the [ ] emission originates from a different dig component than that of the [ ] and [ ] . consequently , a secondary mechanism , such as shock ionization , could account for the bright [ ] emission above the plane . in this section we explore the possibility that shock ionization is responsible for some of the dig emission in these galaxies . one could also consider tmls , since they also produce large amounts of [ ] emission . in fact in ngc 891 , tmls do just as well as shocks in fitting the observed data ( rand 1998 ) . our approach is that of rand ( 1998 ) , with some minor changes . we adopt the s94 matter - bounded photoionization model with the lowest terminal hydrogen column density , @xmath43 @xmath44 , of individual gas clumps . the s94 models feature a hard stellar spectrum ( an upper mass cutoff of @xmath12 ) , hardening of the radiation field as it propagates through the ism , and depletion of gas - phase coolants onto dust grains . we do not consider the s94 radiation - bounded model as this model predicts extremely bright [ ] emission . any composite shock model utilizing radiation - bounded models requires a very unusual secondary source of h@xmath2 emission . in the case of ngc 5775 , such a scenario would require a secondary source that accounts for nearly half of the h@xmath2 emission at the midplane ( assuming log @xmath45 at @xmath33 kpc ) , and produces virtually no [ ] or [ ] emission . in addition , such a source must produce significant [ ] and [ ] emission at high-@xmath3 , and still contribute at least 50% of the total h@xmath2 emission . such emission characteristics could be explained with low - speed shocks near the midplane with a shock velocity increasing with @xmath3 . however , the problem is further complicated by the fact that low - speed shocks produce copious amounts of [ ] emission relative to [ ] . in addition , basic energetic arguments render such a scenario , where the secondary source contributes half of the layer s ionization , highly unlikely as the ionizing flux from massive stars exceeds supernovae power output , presumably the initiator of these shocks , by a factor of 6 or 7 ( reynolds 1984 , 1992 ) . in any case , parameters describing these shocks require considerable tweaking to account for the observed run of line ratios . matter - bounded models on the other hand , better match the data before composite modeling , and require a less complicated parameter space describing the secondary source . to obtain line ratios due to shock ionization , we use the models of shull & mckee ( 1979 ; hereafter sm79 ) . line ratios in these models , especially [ ] /h@xmath2 , are highly dependent on shock velocity . other variables affecting line ratios include the preshock gas density and ionization state , abundances , and transverse magnetic field strength . these models assume the gas to be initially neutral at @xmath46 @xmath6 , then penetrated by a precursor ionization front . more appropriate to the case of a partially photo - ionized dig layer , would be a initial density of order @xmath47 @xmath6 , and a high initial ionization fraction . we mainly consider the models with shock velocities of @xmath48 km s@xmath49 and @xmath50 km s@xmath49 in that these higher velocity models are best able to produce large amounts of [ ] emission , while keeping [ ] low relative to [ ] . sm79 also calculate a single model containing depleted abundances with @xmath48 km s@xmath49 , which we also consider in the modeling to allow a comparison between shock and photo - ionization models with depleted abundances . in general , though , we do not consider this analysis as providing a definitive measure of the shock speed in these composite models given the simplicity of the model and the many unrealistic parameters which describe the shocks . for the composite modeling , we assume that some fraction of h@xmath2 emission arises from shock ionization , with that fraction possibly changing with @xmath3 . we still assume a stellar radiation field distinguished by a decrease in ionization parameter , @xmath9 , with @xmath3 . to simplify matters we assume a single shock velocity for each composite model . we attempt to fit the composite model to the data by varying the percent contribution at a given value of @xmath9 ( and thus @xmath3 ) . for ngc 5775 the modeling worked best by allowing shock contributions to begin at log @xmath51 , while models with shocks beginning at log @xmath52 worked best for ugc 10288 . such values are somewhat lower than would be expected for a low-@xmath3 radiation field . however , kinematics of edge - on galaxies indicate that at low-@xmath3 , the line emission we observe is mostly associated with the outer disk regions due to an absorbing dust layer . since star formation is typically concentrated in the inner disk , the radiation field in the outer disk may be relatively dilute , in which case , the value of @xmath9 at low-@xmath3 may be quite low ( rand 1998 ) . we do not carry out a statistical test of the goodness of fit since our goal is to qualitatively assess the feasibility of a secondary source of ionization being able to account for some of the line ratios . although no model reproduces the line ratios to within the errors , we do find that the composite models better reproduce the run of line ratios than photo - ionization models alone . rand ( 1998 ) finds a similar conclusion for ngc 891 , where composite modeling reproduces the run of [ ] /h@xmath17 vs. [ ] /h@xmath2 and [ ] /h@xmath2 , yet is unable to duplicate the [ ] / [ ] ratio at low-@xmath3 . ngc 4302 is not considered due to the lack of critical [ ] @xmath165007 data . for the composite modeling , we consider separately the cases of filamentary and non - filamentary emission . the ne portion of slit 1 crosses the prominent ne h@xmath2 filament , previously identified by crdw as being associated with a shell - like structure and possibly being involved in vigorous disk - halo interactions . in addition , the sw portion of slit 2 crosses a bright plume of h@xmath2 emission immediately adjacent to the very prominent sw h@xmath2 filament , also a region where intense disk - halo activity is suspected . since these two regions appear associated with current cycling of material from disk to halo , it follows that shock ionization may play a greater role in energizing dig in these areas . the sw portion of slit 1 and the ne portion of slit 2 each cross regions with relatively less extended dig emission , where disk - halo interactions appear far less vigorous than regions with more filamentary features . the runs of [ ] /h@xmath2 , [ ] /h@xmath2 , and [ ] /h@xmath2 vs. [ ] /h@xmath2 are shown in figures 8 , 9 , and 10 , respectively . figures 8 and 9 also present filamentary and non - filamentary regions separately . the runs of these ratios in the matter - bounded s94 models , with various values of @xmath9 labeled , are plotted along with predicted line ratios from the sm79 models for @xmath48 km s@xmath49 , @xmath50 km s@xmath49 , and @xmath48 km s@xmath49 with depletions . we find the @xmath48 km s@xmath49 model is best able to reproduce the observed run of line ratios as a secondary source to photoionization . the @xmath50 km s@xmath49 model has difficulty reproducing the run of [ ] /h@xmath2 vs. [ ] /h@xmath2 , and the depleted @xmath48 km s@xmath49 model , though able to duplicate the run of [ ] /h@xmath2 vs. [ ] /h@xmath2 , is unable to reproduce the run of [ ] /h@xmath2 vs. [ ] /h@xmath2 at high-@xmath3 . we find that the run of [ ] /h@xmath2 vs. [ ] /h@xmath2 seen in figure 8a for the filamentary regions is distinctly different from the trend seen in figure 8b for the non - filamentary regions . clearly , for a given value of [ ] /h@xmath2 , the ratio [ ] /h@xmath2 is much greater in dig with a more filamentary morphology . the non - filamentary regions show a run that is very similar to the run observed in ngc 891 ( rand 1998 ) . the run of [ ] /h@xmath2 vs. [ ] /h@xmath2 on the other hand , does not seem to depend strongly on the presence of filamentary dig structure . non - filamentary regions do seem to have greater values of [ ] /h@xmath2 and [ ] /h@xmath2 , in keeping with these ratios anti - correlation with relative h@xmath2 surface brightness ( see also figure 12 ) . all data for the ratio [ ] /h@xmath2 , being detected only for slit 2 to @xmath30 kpc , are plotted in figure 10 . the lack of [ ] /h@xmath2 data for higher values of [ ] /h@xmath2 makes a comparison between various dig morphologies impossible . composite models are plotted in figures 8 , 9 , and 10 as dashed lines joining open hexagons , which mark various values of log @xmath9 . for the filamentary regions , 20% of the h@xmath2 emission at log @xmath51 arises from shock ionization , increasing to 85% at log @xmath53 . this does not mean the majority of extraplanar dig is shock ionized . it does indicate however that at very high-@xmath3 ( log @xmath53 corresponds to a height of about @xmath54 kpc for the filamentary regions and @xmath55 kpc for the non - filamentary ) , it is possible that most of the emission within the filaments arises from shock ionized gas . in the non - filamentary case , shocks contribute 16% of the h@xmath2 emission at log @xmath51 , rising to 27% at log @xmath53 . for the [ ] /h@xmath2 vs. [ ] /h@xmath2 data of figure 10 , the filamentary composite model is plotted over all data points for both filamentary and non - filamentary regions . this model and the non - filamentary model are somewhat successful in explaining the [ ] /h@xmath2 vs. [ ] /h@xmath2 run at low-@xmath3 , in the sense of matching the excess of [ ] /h@xmath2 for a given [ ] /h@xmath2 compared to the s94 model alone . the composite models both match the runs of [ ] /h@xmath2 vs. [ ] /h@xmath2 . the main deficiency in the models is their inability to reproduce the run of [ ] / [ ] at low-@xmath3 . the models do not predict enough [ ] emission for a given value of [ ] . the models perform somewhat better at high-@xmath3 , though the spread in data values at high-@xmath3 make a trend somewhat difficult to discern . again , though models with depleted abundances are better able to produce an excess of [ ] emission relative to [ ] , they do not produce enough [ ] emission . in any case , it is clear that the filamentary regions require a greater contribution from shocks to explain the observed line ratios . the correlation of these features with shells and enhanced 20-cm radio continuum emission is also suggestive of a larger percent contribution to the observed emission in these regions from shock ionization . enhanced [ ] emission can also be produced by other sources of ionization such as tmls . rand ( 1998 ) found that composite photo - ionization / tml models were just as successful in reproducing the runs of observed line ratios in ngc 891 as composite models featuring shocks . tmls are expected to occur at interfaces of gas at different temperatures in the ism , such as in shell walls . it is likely then for ngc 5775 , where shells are associated with the more prominent h@xmath2 filaments , that tmls could also account for the enhanced line ratios . we have performed cursory modeling of the line ratios for ngc 5775 with composite photo - ionization / tml models . using the tml model of slavin , shull , & begelman ( 1993 ) with depleted abundances in the mixing layers of hot and warm gas , a hot gas mixing speed of @xmath56 km s@xmath49 , and a mixed gas temperature of log @xmath57 , we adopt the same approach as the composite photo - ionization / shock modeling . we find that the photo - ionization / tml models work nearly as well as the photo - ionization / shock models , though they are unable to reproduce the run of ratios for the highest values of [ ] /h@xmath2 . a preliminary analysis of the diagnostic diagrams for ugc 10288 has revealed no discernible trend in the run of [ ] /h@xmath2 vs. [ ] /h@xmath2 . to attempt to better reveal any possible trend , line ratios on either side of the disk have been averaged for each slit to establish a run of each ratio vs. @xmath58 . these averaged runs of [ ] /h@xmath2 , [ ] /h@xmath2 , and [ ] /h@xmath2 vs. [ ] /h@xmath2 are shown in figure 11 . the matter - bounded s94 models , with various values of @xmath9 labeled , are plotted along with predicted line ratios from the sm79 models for @xmath48 km s@xmath49 , @xmath50 km s@xmath49 , and @xmath48 km s@xmath49 with depletions . even with the averaging , the [ ] /h@xmath2 vs. [ ] /h@xmath2 run does nt show an obvious trend : slit 1 seems to show a slight trend of an increase in [ ] /h@xmath2 with [ ] /h@xmath2 , while slit 2 data is clustered near the photo - ionization model . for this reason , we attempt to fit the run of [ ] / [ ] instead of [ ] /h@xmath2 vs. [ ] /h@xmath2 . we adopt the @xmath48 km s@xmath49 model with depleted abundances as the other two models do not produce enough [ ] emission relative to [ ] . composite models are plotted in figure 11 as dashed lines joining open hexagons , which mark various values of log @xmath9 . we find that the composite models best agree with the data when shocks begin contributing to emission at log @xmath52 . for the composite model , 20% of the h@xmath2 emission at log @xmath52 arises from a 100 km s@xmath49 shock with depleted abundances , rising to 45% at log @xmath53 ( log @xmath53 corresponds to a height of about @xmath59 pc ) . the composite model successfully replicates the run of [ ] / [ ] at all but very high-@xmath3 where matter bounded photo - ionization models can not explain values of [ ] /h@xmath2 and [ ] /h@xmath2 greater than unity . radiation bounded photo - ionization models feature greater values of these line ratios ; however , predicted runs of [ ] /h@xmath2 can not be reconciled with observations . the run of [ ] /h@xmath2 vs. [ ] /h@xmath2 is also well explained by the composite model . the lack of any obvious trend in the run of [ ] /h@xmath2 vs. [ ] /h@xmath2 makes it difficult to assess whether shocks are a viable contributor to the h@xmath2 emission . however , the high values of [ ] /h@xmath2 in slit 1 do suggest a large contribution from shocks in those regions . it should be noted , though , that the galaxy has a fairly quiescent appearance in the h@xmath2 image with little extraplanar emission , calling into question the notion that shocks are permeating the ism . it would be useful to obtain more [ ] data for this galaxy to better diagnose the possible contributions from secondary ionization sources . nevertheless , a clear departure from a pure photoionization model is indicated by the data . in this section , following the approach of hrt , we attempt to determine whether variations in line ratios with @xmath3 can be explained by a change in gas temperature as one moves off the midplane . a temperature increase with @xmath3 provides a simple explanation of why all ratios of forbidden lines to balmer lines are generally seen to increase with @xmath3 . in fact , hrt find for the reynolds layer that the increase in [ ] /h@xmath2 and [ ] /h@xmath2 , and the constancy of [ ] / [ ] with @xmath3 , up to @xmath14 kpc , can be accounted for by an increase in gas temperature alone , without having to invoke a secondary source of ionization . such a scenario requires additional non - ionizing heating for low - density , high-@xmath3 gas , such as photo - electric heating from dust grains ( reynolds & cox 1992 ) or dissipation of interstellar turbulence ( minter & balser 1997 ) . using line ratio data versus @xmath3 for ngc 4302 , ngc 5775 , ugc 10288 , and ngc 891 , we attempt to test this scenario by determining gas temperatures as well as ionization fractions of constituent elements . for collisionally excited ions , the equation for intensity of emission lines is given by osterbrock ( 1989 ) , @xmath60 where @xmath61 and @xmath62 are the collision strength and energy of the transition , @xmath63 is the statistical weight of the ground level , @xmath64 is the temperature in units of 10@xmath19 k , and @xmath65 is the fraction of downward transitions that produce the emission line . to simplify matters , we assume no change in physical conditions , namely temperature , density , ionization fraction , and gas abundance , along a given line - of - sight through the dig layer . such uniformity of physical parameters within the ism is clearly not accurate when lines - of - sight cross small scale structures such as regions or shocks , where temperature or ionization conditions may change dramatically . the assumption is valid , however , if these parameters are thought of as averages along a given line of sight instead of describing localized conditions within the ism . this assumption , along with the use of the collision strengths provided by aller ( 1984 ) , allows the intensity of [ ] , [ ] , and [ ] emission to be represented , in units of rayleighs , by : @xmath66 @xmath67 and @xmath68 the intensity of h@xmath2 emission is given by , @xmath69 where @xmath70 is the emission measure in units of pc @xmath71 . a further simplification is to assume n@xmath72 and h@xmath72 have approximately equal ionization fractions . this assumption is justified for two reasons . first , photoionization models of s94 predict that n@xmath72/n@xmath73 tracks h@xmath72/h@xmath73 due to similar first ionization potentials ( 14.5 ev versus 13.6 ev ) and a weak charge exchange reaction . second , n is not likely to ionize higher than n@xmath72 in a pure photoionization scenario due to a second ionization potential of 29.6 ev ( hrt ) . a high ionization fraction of o@xmath4 , however , would imply a significant fraction in the n@xmath4 state as the second ionization potential of oxygen is 35.1 ev . thus we will be able to check the validity of this assumption with the [ ] /h@xmath2 data . due to this simplification , of the line ratios available from this data , [ ] /h@xmath2 is the best probe of gas temperature . the forbidden lines are all dependent on their abundance relative to atomic hydrogen . the issue of abundances is a complicated one , the understanding of which is critical for interpreting the line ratios . current evidence suggests that abundances in halo gas may be quite different from gas in the disk . savage and sembach ( 1996 ) find increasing gas - phase abundances in the milky way from disk to halo , possibly as a result of partial grain destruction due to shocks . however , models by sembach et . al . ( 2000 ; hereafter shrk ) predict that the ratios of forbidden lines to balmer lines tend to rise as abundances fall due to decreased cooling efficiency . for this reason , it is unlikely that abundance variations can explain the increase in ratios of forbidden lines to balmer lines . however , observations of the [ ] @xmath163727 line , to complement [ ] @xmath165007 and [ ] @xmath166300 measurements , would be very useful to completely decouple line ratios ( which depend on excitation and temperature ) from variations in abundance . to simplify matters , we adopt abundances that do not vary with @xmath3 . using galactic gas - phase abundances of s , n , and o of ( s / h)@xmath74 ( anders & grevesse 1989 ) , ( n / h)@xmath75 ( meyer , cardelli , & sofia 1997 ) , and ( o / h)@xmath76 ( cardelli & meyer 1997 ) , along with the assumption of equal fractions of n@xmath72 and h@xmath72 , we arrive at expressions relating line ratios to temperature and ionization fraction : @xmath77}{\mbox{h$\alpha$}}=12.2\ t_{4}^{0.426 } e^{-2.18/t_{4}},\ ] ] @xmath78}{\mbox{h$\alpha$}}=14.3 \left(\frac{\mbox{s$^{+}$}/\mbox{s}}{\mbox{h$^{+}$}/\mbox{h}}\right ) t_{4}^{0.307 } e^{-2.14/t_{4}},\ ] ] @xmath78}{[\mbox{\ion{n}{2}}]}=1.2 \left(\frac{\mbox{s$^{+}$}/\mbox{s}}{\mbox{h$^{+}$}/\mbox{h}}\right ) t_{4}^{-0.119 } e^{0.04/t_{4}},\ ] ] and @xmath79}{\mbox{h$\alpha$}}=40 \left(\frac{\mbox{o$^{++}$}/\mbox{o}}{\mbox{h$^{+}$}/\mbox{h}}\right ) t_{4}^{0.52 } e^{-2.87/t_{4}}.\ ] ] the ionization fractions of oxygen and hydrogen are coupled through a charge exchange reaction that allows the ionization state of hydrogen to be determined from the [ ] /h@xmath2 ratio . we use the ratio of emissivities of the [ ] @xmath166300 line to the h@xmath2 line provided by reynolds et . al . ( 1998a ) to obtain , @xmath80}{\mbox{h$\alpha$}}=7.9 \left[\frac{1-\left(\frac{\mbox{h$^{+}$}}{\mbox{h}}\right)}{\left(\frac{\mbox{h$^{+}$}}{\mbox{h}}\right)}\right ] \frac{t_{4}^{1.85}}{1 + 0.605t_{4}^{1.105 } } e^{-2.284/t_{4}}.\ ] ] gas temperatures are derived from the [ ] /h@xmath2 ratio . assuming this ratio accurately reflects temperatures within the medium , line ratios involving the s and o lines are then calculated for a variety of ionization fractions . we can then see whether all the ionization fractions are reasonable and consistent with our initial assumptions . plots of temperature versus @xmath3 and calculated ratios overlayed on measured ratios are shown in figures 3 , 4 , 12 , and 13 for ngc 4302 , ugc 10288 , ngc 5775 , and ngc 891 , respectively . resulting ionization fractions from the temperature modeling are shown in table 3 . to first order , a rise in dig temperature can account for the observed rises in line intensities relative to h@xmath2 emission , as well as the near constancy in the [ ] / [ ] ratio . we find , except in the case of ugc 10288 , the ionization fraction of s@xmath72 relative to h@xmath72 remains very close to 0.5 independent of @xmath3 . this is similar to the result obtained by hrt for the wim . photo - ionization models performed by shrk predict a comparable s@xmath72 ionization fraction for low - density regions . ugc 10288 on the other hand , shows a considerably higher fraction of s@xmath72 relative to h@xmath72 as well as some evidence for an increase with @xmath3 . however , the ionization fraction of h@xmath72 , determined from points relatively close to the midplane , is significantly lower than in the case of ngc 5775 or ngc 891 . as a result , the range in ionization fraction of s@xmath72 in ugc 10288 is nearly equal to the range in values for the other galaxies . this does not explain the observed rise with @xmath3 , however , and the lack of [ ] data at high-@xmath3 makes it impossible to determine whether the ionization state of sulfur changes with @xmath3 . the [ ] /h@xmath2 data , in addition to revealing information about the ionization state of o , is also an excellent test of the validity of some of the assumptions used for this temperature modeling . the data for ngc 5775 and ugc 10288 suggest that a significant fraction of oxygen is in the o@xmath4 state ; at least 15% and in some lines - of - sight nearly 100% relative to h@xmath72 . the second ionization potential of oxygen is 35.1 ev , while that of nitrogen is 29.6 ev . thus one would expect the ionization fractions of the doubly - ionized species of these two elements to roughly track one another in a warm medium . the determined values of ionization fraction of o@xmath4 for these two galaxies imply that a significant percentage of nitrogen atoms are in the n@xmath4 state , particularly at high-@xmath3 for ngc 5775 . the likely presence of significant amounts of n@xmath4 as well as the possibility of its fraction changing with @xmath3 complicates the use of [ ] /h@xmath2 as a temperature indicator . in any event , the high values of o@xmath4 ionization fraction suggest that a temperature increase alone is not able to fully explain the runs of line ratios observed in ngc 5775 and ugc 10288 since pure photo - ionization models predict very low o@xmath4 fractions . one possible explanation for the large ionization fractions of doubly - ionized species is that a secondary ionizing source with emission characteristics similar to shocks or tmls contributes to the layer s ionization . alternatively , the [ ] emitting gas could represent a component that is at a significantly higher temperature than the component responsible for the [ ] emission . ngc 891 on the other hand , shows considerably lower values for the ionization fraction of o@xmath4 relative to h@xmath72 : 13% on either side of the disk and just above 30% in the midplane . the slit runs near a bright filament that is not as prominent as those observed in ngc 5775 . these fractions are slightly larger than those predicted by the shrk model for low - density regions , though still more than an order of magnitude higher than that of the diffuse gas model of shrk . thus it seems the run of line - ratios in ngc 891 is better explained by an increase in gas temperature with @xmath3 than for ngc 5775 . a slightly warmer [ ] emitting component than indicated by the [ ] /h@xmath2 ratio could explain measured values of [ ] /h@xmath2 while keeping the ionization fraction of o@xmath4 at levels predicted by the models . if this is the case , a secondary ionizing source would not be necessary to explain the runs of line ratios in ngc 891 . the run of [ sii]/h@xmath2 in ngc 4302 is well fit by the increase in temperature reflected by the [ nii]/h@xmath2 ratio . it would be interesting to obtain [ oiii]@xmath165007 data for this galaxy to better assess whether line ratio variations can be explained by dig temperature variations . other factors may influence the line ratios such as afore - mentioned abundance gradients ( savage & sembach 1996 ) as well as scattered light due to extraplanar dust . the effects of dust absorption on extraplanar emission have been shown to be significant for a number of dig halos ( howk & savage 1999 ) . modeling of the milky way ( wood & reynolds 1999 ) and ngc 891 ( ferrara et . al . 1996 ) suggest that scattered light makes a minor contribution to diffuse halo h@xmath2 emission that decreases with @xmath3 . in ngc 891 , the contribution to dig emission from scattered light falls to 10% at @xmath81 pc ( ferrara et . al . 1996 ) . in any case , dust - scattered light may be an important consideration for the interpretation of these line ratios . | emission lines are detected up to about kpc in ngc 4302 , and to nearly kpc on the north side of ugc 10288 . for both galaxies , the line ratios [ ] /h and [ ] /h increase with in accordance with dilute photo - ionization models . runs of [ ] / [ ] , and for ugc 10288 , the run of [ ] /h , however , are not explained by the models . scale height determinations of their dig halos are generally lower than those of galaxies with more prominent extraplanar dig features . these data , along with previously presented data for ngc 5775 and ngc 891 , are used to address the issue of how dig halos are energized . composite photo - ionization / shock models are generally better at explaining runs of line ratios in these galaxies than photoionization models alone . composite models for ugc 10288 are successful at reproducing the run of [ ] / [ ] for all but the the highest values of [ ] /h ; however , the run of [ ] /h vs. [ ] /h does not show any discernible trend , making it difficult to determine whether or not shocks contribute to the layer s maintenance . we also examine whether the data can be explained simply by an increase in temperature with in a pure photo - ionization model without a secondary source of ionization . | deep long - slit spectra of the diffuse ionized gas halos of the edge - on spiral galaxies ngc 4302 and ugc 10288 are presented . emission lines are detected up to about kpc in ngc 4302 , and to nearly kpc on the north side of ugc 10288 . for both galaxies , the line ratios [ ] /h and [ ] /h increase with in accordance with dilute photo - ionization models . runs of [ ] / [ ] , and for ugc 10288 , the run of [ ] /h , however , are not explained by the models . scale height determinations of their dig halos are generally lower than those of galaxies with more prominent extraplanar dig features . these data , along with previously presented data for ngc 5775 and ngc 891 , are used to address the issue of how dig halos are energized . composite photo - ionization / shock models are generally better at explaining runs of line ratios in these galaxies than photoionization models alone . models of line ratios in ngc 5775 require a greater contribution from shocks for filamentary regions than for non - filamentary regions to explain the run of [ ] /h . in either case , the [ ] / [ ] ratio is not well fit by the models . composite models for ugc 10288 are successful at reproducing the run of [ ] / [ ] for all but the the highest values of [ ] /h ; however , the run of [ ] /h vs. [ ] /h does not show any discernible trend , making it difficult to determine whether or not shocks contribute to the layer s maintenance . we also examine whether the data can be explained simply by an increase in temperature with in a pure photo - ionization model without a secondary source of ionization . runs of [ ] /h , [ ] /h , and [ ] / [ ] in each of the four galaxies are consistent with such an increase . however , the runs of [ ] /h vs. in ngc 5775 and ugc 10288 require unusually high ionization fractions of o that can not be explained without invoking a secondary ionization source or at the very least a much higher temperature for the [ ] -emitting component than for the [ ] - and [ ] -emitting component . an increase in temperature with is generally more successful at explaining the [ ] /h run in ngc 891 , with the ionization fraction of o remaining relatively low and nearly constant with . |
astro-ph0011556 | c | deep spectroscopy of four edge - on spiral galaxies has allowed physical conditions within their dig halos to be probed , and has provided information on possible sources of ionization of the gas . this study has revealed the following conclusions : \1 . the general trend for the runs of line ratios for ngc 4302 and ugc 10288 is very similar to that of ngc 5775 ( rand 2000 ) and ngc 891 ( rand 1998 ) : an increase in [ ] /h@xmath2 and [ ] /h@xmath2 with [ ] / [ ] remaining nearly constant with @xmath3 , though one slit position for ugc 10288 does show evidence for an increase in [ ] / [ ] with @xmath3 . determinations of halo emission scale height have revealed that the more prominent dig halos also have greater scale height . this is consistent with models of dynamic halos where greater levels of star formation in the disk can push material farther off the midplane . pure photo - ionization models do not explain the runs of line ratios in ngc 5775 and ugc 10288 . composite photo - ionization / shock models can replicate the run of [ ] /h@xmath2 vs. [ ] /h@xmath2 in ngc 5775 , though the run of [ ] /h@xmath2 vs. [ ] /h@xmath2 is still problematic , especially at low-@xmath3 . in addition , we find that the composite models require a greater percent contribution from shocks in filamentary than non - filamentary regions to match the run of [ ] /h@xmath2 . a lack of a discernible trend for the [ ] /h@xmath2 vs. [ ] /h@xmath2 run in ugc 10288 makes it difficult to determine whether shocks contribute to line emission , though the high values of [ ] /h@xmath2 in slit 1 indicate that a shock contribution may be necessary . a composite model , though , can account for the run of [ ] /h@xmath2 vs. [ ] /h@xmath2 . however , if it follows the trend seen in ngc 5775 of more disturbed regions requiring a larger shock contribution , then the lack of a bright , filamentary halo in ugc 10288 may imply that shocks do not play as significant a role in layer s ionization . the [ ] /h@xmath2 vs. [ ] /h@xmath2 run seems to be a key discriminant for shocks . fabry - perot observations of these 3 lines for brighter halos , such as ngc 5775 , are now feasible and would provide much more information on the importance of secondary ionizing sources with local morphology and height . it is important to emphasize that shocks are not the only possible secondary source . other sources such as tmls , x - rays , and ionization by cosmic rays may contribute as well . another scenario that we consider is a pure photoionization model with an increasing temperature with @xmath3 . by making a few reasonably valid assumptions , the run of gas temperature vs. @xmath3 has been determined in each of the four galaxies . we find that an increase in temperature with @xmath3 can explain the general trend of an increase in the ratios of forbidden lines to balmer lines with @xmath3 , including an increase in [ ] /h@xmath2 and the constancy of [ ] / [ ] . by establishing the run of gas temperature , constraints on ionization fractions of sulfur , oxygen , and hydrogen could then be established . the [ ] /h@xmath2 ratios in ngc 5775 and ugc 10288 generally indicate an exceptionally high ionization fraction of o@xmath4 , that invalidates the assumption that n@xmath72 and h@xmath72 have equal ionization fractions , indicating that the line ratios can not be explained by a temperature increase alone . the runs of line ratios in ngc 891 , where the ionization fraction of o@xmath4 is still somewhat high , and ngc 4302 are better explained by an increase in temperature with @xmath3 , though the absence of [ ] @xmath165007 data for ngc 4302 makes a proper assessment difficult . we thank the kpno staff for their help in obtaining the data . we also thank l. m. haffner , r. reynolds , and r. benjamin for comments in the preparation of this paper . this research has made use of the nasa / ipac extragalactic database ( ned ) . this work was partially supported by nsf grant ast-9986113 . aller , l. h. 1984 , physics of thermal gaseous nebulae ( dordrecht : reidel ) anders , e. , & grevesse , n. 1989 , geochim . acta , 53 , 197 bland - hawthorn , j. , freeman , k. c. , & quinn , p. j. 1997 , , 490 , 143 cardelli , j. a. , & meyer , d. m. 1997 , , 477 , l57 collins , j. a. , rand , r. j. , duric , n. , & walterbos , r. a. m. 2000 , , in press ( crdw ) de vaucouleurs , g. , de vaucouleurs , a. , corwin , h. g. , buta , r. j. , paturel , g. , & fouqu , p. 1991 , third reference catalogue of bright galaxies ( new york : springer ) domgrgen , h. , & mathis , j. s. 1994 , , 428 , 647 ferguson , a. m. n. , wyse , r. f. g. , & gallagher , j. s. 1996 , , 112 , 2567 ferrara , a. , bianchi , s. , dettmar , r. -j . , & giovanardi , c. 1996 , , 467 , l69 golla , g. , dettmar , r. -j . , & domgrgen , h. 1996 , , 313 , 439 haffner , l. m. , reynolds , r. j. , & tufte , s. l. 1999 , , 523 , 223 ( hrt ) hoopes , c. g. , walterbos , r. a. m. , & greenawalt , b. e. 1996 , , 112 , 1429 hoopes , c. g. , walterbos , r. a. m. , & rand , r. j. 1999 , , 522 , 669 howk , j. c. , & savage , b. d. 1999 , , 117 , 2077 irwin , j. a. 1994 , , 429 , 618 meyer , d. m. , cardelli , j. a. , & sofia , u. j. 1997 , , 490 , l103 minter , a. h. , balser , d. s. 1997 , , 484 , l133 osterbrock , d. e. 1989 , astrophysics of gaseous nebulae and active galactic nuclei ( mill valley : university science books ) rand , r. j. 1996 , , 462 , 712 rand , r. j. 1997 , , 474 , 129 rand , r. j. 1998 , , 501 , 137 rand , r. j. 2000 , in preparation rand , r. j. , kulkarni , s. r. , & hester , j. j. 1990 , , 352 , l1 ( rkh ) reynolds , r. j. 1984 , , 282 , 191 reynolds , r. j. 1992 , , 392 , l35 reynolds , r. j. 1993 , in aip conf . 278 , back to the galaxy , ed . s. s. holt & f. verter ( new york : aip ) , 156 reynolds , r. j. , & cox , d. p. 1992 , , 400 , 33 reynolds , r. j. , haffner , l. m. , & tufte , s. l. 1999 , , 525 , l21 reynolds , r. j. , hausen , n. r. , tufte , s. l. , & haffner , l. m. 1998a , , 494 , l99 reynolds , r. j. , & tufte , s. l. 1995 , , 439 , l17 reynolds , r. j. , tufte , s. l. , haffner , l. m. , jaehnig , k. , & percival , j. w. 1998b , pasa , 15 , 14 savage , b. d. , & sembach , k. r. 1996 , , 34 , 279 sembach , k. r. , howk , j. c. , ryans , r. s. i. , & keenan , f. p. 2000 , , 528 , 310 ( shrk ) shull , j. m. , & mckee , c. f. 1979 , , 227 , 131 ( sm79 ) slavin , j. d. , shull , j. m. , & begelman , m. c. 1993 , , 407 , 83 sokolowski , j. 1994 , preprint ( s94 ) tllmann , r. , dettmar , r .- j . , soida , m. , urbanik , m. , & rossa , j. 2000 , , in press tully , r. b. 1988 , nearby galaxies catalog ( cambridge : cambridge university press ) wang , j. , heckman , t. m. , & lehnert , m. d. 1997 , , 491 , 114 wood , k. , & reynolds , r. j. 1999 , , 525 , 799 lllcccc ngc 5775 ( slit 1 ) & 14@xmath8253@xmath8357@xmath846 & 03@xmath8532@xmath8640@xmath87 & 24.8 & 32 nw & 6.0 & 2.2 ngc 5775 ( slit 2 ) & & & & 20 se & 5.0 & 2.4 ngc 4302 & 12 21 42.5 & 14 36 05 & 16.8 & 24 s & 4.0 & 9.7 ugc 10288 ( slit 1 ) & 16 14 25.1 & -00 12 27 & 31.5 & 30 e & 6.17 & 2.5 ugc 10288 ( slit 2 ) & & & & 35 w & 5.33 & 3.6 ngc 891 & 02 22 33.1 & 42 20 48 & 9.6 & 100 n & 1.5 & 16.1 lcc ngc 5775 & slit 1 ( ne ) & 2300 ngc 5775 & slit 1 ( sw ) & 2300 ngc 5775 & slit 2 ( ne ) & 2500 ngc 5775 & slit 2 ( sw ) & 1800 ngc 4302 & east & 700 ngc 4302 & west & 550 ugc 10288 & slit 1 ( s ) & 320 ugc 10288 & slit 2 ( s ) & 460 lcccc ngc 5775 ( slit 1 ) & 0.640.95 & 0.300.60 & 0.150.70 & n / a ngc 5775 ( slit 2 ) & 0.620.95 & 0.350.80 & 0.300.75 & 0.650.85 ugc 10288 ( slit 1 ) & 0.601.05 & 0.551.00 & 0.500.80 & 0.450.65 ugc 10288 ( slit 2 ) & 0.600.95 & 0.501.00 & 0.500.80 & 0.300.60 ngc 4302 & 0.661.00 & 0.350.60 & n / a & n / a ngc 891 & 0.651.05 & 0.400.65 & 0.130.30 & 0.550.80 | models of line ratios in ngc 5775 require a greater contribution from shocks for filamentary regions than for non - filamentary regions to explain the run of [ ] /h . in either case , runs of [ ] /h , [ ] /h , and [ ] / [ ] in each of the four galaxies are consistent with such an increase . however , the runs of [ ] /h vs. in ngc 5775 and ugc 10288 require unusually high ionization fractions of o that can not be explained without invoking a secondary ionization source or at the very least a much higher temperature for the [ ] -emitting component than for the [ ] - and [ ] -emitting component . an increase in temperature with is generally more successful at explaining the [ ] /h run in ngc 891 , with the ionization fraction of o remaining relatively low and nearly constant with . | deep long - slit spectra of the diffuse ionized gas halos of the edge - on spiral galaxies ngc 4302 and ugc 10288 are presented . emission lines are detected up to about kpc in ngc 4302 , and to nearly kpc on the north side of ugc 10288 . for both galaxies , the line ratios [ ] /h and [ ] /h increase with in accordance with dilute photo - ionization models . runs of [ ] / [ ] , and for ugc 10288 , the run of [ ] /h , however , are not explained by the models . scale height determinations of their dig halos are generally lower than those of galaxies with more prominent extraplanar dig features . these data , along with previously presented data for ngc 5775 and ngc 891 , are used to address the issue of how dig halos are energized . composite photo - ionization / shock models are generally better at explaining runs of line ratios in these galaxies than photoionization models alone . models of line ratios in ngc 5775 require a greater contribution from shocks for filamentary regions than for non - filamentary regions to explain the run of [ ] /h . in either case , the [ ] / [ ] ratio is not well fit by the models . composite models for ugc 10288 are successful at reproducing the run of [ ] / [ ] for all but the the highest values of [ ] /h ; however , the run of [ ] /h vs. [ ] /h does not show any discernible trend , making it difficult to determine whether or not shocks contribute to the layer s maintenance . we also examine whether the data can be explained simply by an increase in temperature with in a pure photo - ionization model without a secondary source of ionization . runs of [ ] /h , [ ] /h , and [ ] / [ ] in each of the four galaxies are consistent with such an increase . however , the runs of [ ] /h vs. in ngc 5775 and ugc 10288 require unusually high ionization fractions of o that can not be explained without invoking a secondary ionization source or at the very least a much higher temperature for the [ ] -emitting component than for the [ ] - and [ ] -emitting component . an increase in temperature with is generally more successful at explaining the [ ] /h run in ngc 891 , with the ionization fraction of o remaining relatively low and nearly constant with . |
1609.04769 | i | in the last two decades , the study of quantum chromodynamics ( qcd ) bound states has evolved from static , one - dimensional snapshots of quarks and gluons to focus on multidimensional structure and the dynamics of partons . the theoretical framework that has been developed to describe parton dynamics in hadrons involves transverse - momentum - dependent ( tmd ) parton distribution functions ( pdfs ) and fragmentation functions ( ffs ) . in traditional collinear pdfs and ffs , any momentum of the partons transverse to the hadron boost axis is integrated over . in tmd pdfs or ffs , the transverse momentum of the partons is not integrated out and instead remains explicit in the pdf or ff , offering a means of describing the transverse momentum distribution of unpolarized partons within an unpolarized hadron , as well as a variety of spin - momentum correlations when polarized hadrons and/or partons are considered . early theoretical work in tmd pdfs took place in the 1980s by collins , soper , and sterman @xcite , with extensive further development in the 1990s ( see e.g. @xcite ) . however , some theoretical details regarding the definition of tmd pdfs within a perturbative qcd ( pqcd ) framework have only been clarified in the last five years @xcite . we note that due to confinement , the behavior of partons within hadrons is nonperturbative in that it can not be calculated theoretically within pqcd . collinear or tmd pdfs are nonperturbative functions that can be constrained by and/or used to predict high - energy scattering processes within a pqcd framework . in such a framework , the nonperturbative functions such as pdfs as well as ffs factorize from the perturbatively calculable partonic hard scattering cross section and from each other . lattice qcd offers an alternative , complementary approach to pqcd , performing numerical nonperturbative calculations directly . in the past lattice qcd could only calculate moments of pdfs , integrated over parton collinear momentum fraction @xmath12 as well as parton transverse momentum . however , recent developments have demonstrated the potential to go beyond these limitations . these efforts are still in very early stages @xcite . there is already experimental evidence from semi - inclusive deep - inelastic scattering ( sidis ) and drell - yan ( dy ) measurements that several tmd pdfs describing spin - momentum correlations are nonzero @xcite . in addition , there is empirical evidence for nonzero spin - momentum correlations in the process of hadronization from electron - positron annihilation as well as sidis @xcite . furthermore , transverse single - spin asymmetries up to @xmath1340% have been measured in inclusive hadron production in hadronic collisions , indicating large nonperturbative spin - momentum correlations in these processes ( see e.g. @xcite ) . however , these measurements can not probe tmd functions directly because there is no simultaneous observation of perturbative and nonperturbative momentum scales . the recent focus on multidimensional structure and parton dynamics has not only offered richer information on the behavior of partons confined within hadrons , but has moreover brought to light fundamental predictions regarding qcd as a nonabelian gauge - invariant quantum field theory . in particular , the role of color interactions due to soft gluon exchanges with the remnants of the hard scattering have become clearer . because tmd functions preserve more nonperturbative information compared to collinear functions , tmd functions can differ from collinear ones with regards to universality and factorization . for example , the sivers tmd pdf @xcite , a correlation between the proton spin and quark transverse momentum , was shown to possibly be nonzero due to phase interference effects from soft gluon exchanges in sidis @xcite . shortly afterward , @xcite showed that , due to the gauge invariant nature of qcd and the parity and time ( pt ) odd nature of the sivers tmd pdf , the function should be the same magnitude but opposite in sign when measured in drell - yan vs. sidis processes because of the different color flows possible in the initial state vs. final state . twist-2 tmd pdfs that involve one polarization vector are odd under pt transformations , leading to this predicted effect . the nonvanishing nature of the sivers function has already been measured in polarized sidis @xcite ; there is not yet a measurement of this function in polarized drell - yan . a first indication from the drell - yan like w boson production exists @xcite . the results favor a sign - change if tmd evolution effects are small , but at this stage the error bars are still large enough that a definitive statement can not be drawn from this single measurement . it is only for the tmd pdfs odd under pt transformations , where such sign - change behavior is expected , that gluon exchanges can not be completely eliminated via a gauge transformation . in the more complicated qcd process to hadrons , soft gluon exchanges in both the initial and final state are possible , leading to new predicted effects for observables sensitive to a small transverse momentum scale . in such processes , factorization breaking has been predicted @xcite in both polarized and unpolarized interactions . here the nonperturbative objects in the cross section become correlated with one another and can not be factorized into a convolution of tmd pdfs or tmd ffs . however , there are no theoretical claims that the perturbative partonic cross section does not factorize from the nonperturbative physics . similarly to the case of the tmd pdfs that are odd under pt transformations , gluon exchanges that lead to the predicted factorization breaking can not be eliminated via a gauge transformation . it is important to recognize that the ideas behind the predicted sign change of certain tmd pdfs and factorization breaking represent a major qualitative departure from previous purely perturbative approaches that do not account for soft gluon exchanges with remnants of the hard scattering . possibly related effects known as color coherence " have been studied and observed in multijet states in hadronic collisions @xcite , but these types of effects have not been rigorously treated in a tmd framework . in calculations of tmd processes where factorization is predicted to hold , the evolution with the hard scale of the interaction is known to be governed by the collins - soper ( cs ) evolution equation @xcite . note that the cs evolution equation comes directly out of the derivation of tmd factorization @xcite . in contrast to the dglap collinear evolution equations @xcite , which are purely perturbative , the kernel for the cs evolution equation for tmd processes involves the collins - soper - sterman ( css ) soft factor @xcite , which generally contains nonperturbative contributions . the soft factor is understood to be strongly universal , the same for unpolarized and polarized processes , pdfs and ffs , with the only difference being between quarks and gluons @xcite . because lattice calculations of the soft factor are currently not possible , the soft factor must be extracted from parameterizations of experimental measurements within a pqcd framework . for a discussion of the css soft factor and tmd evolution phenomenology , see ref . @xcite . the theoretical expectation from css evolution is that any momentum width sensitive to nonperturbative @xmath14 would grow as the hard scale increases . this can be understood intuitively as a broadening of the phase space for gluon radiation with increasing hard scale . in addition this has been studied and observed in multiple phenomenological analyses of drell - yan and z boson data ( see e.g. @xcite ) , as well as phenomenological analyses of sidis data , where factorization is also predicted to hold ( see e.g. @xcite ) . as mentioned above , because the cs evolution equation comes directly out of the derivation of tmd factorization , it then follows that a promising avenue to investigate factorization breaking effects is by looking for qualitative differences from css evolution in processes where factorization breaking is expected , such as nearly back - to - back dihadron correlations produced in collisions . to have sensitivity to possible factorization breaking and modified tmd evolution effects , a particular observable must be sensitive to a small scale on the order of @xmath15 and measured over a range of hard scales . nearly back - to - back dihadron production has long been used as a proxy for measuring initial - state partonic transverse momentum @xmath14 @xcite , which is defined in fig . [ fig : ktkinematics ] . first used in predictions by ref . @xcite as a method for understanding large differences in hard scattering cross sections between theory and data , nearly back - to - back two - particle and dijet angular correlations have since been used to measure @xmath14 over a large range of center of mass energies @xcite . direct photon - hadron correlations are of particular interest because the photon comes directly from the partonic hard scattering , and thus carries initial - state information without any final - state fragmentation effects . the direct photon approximates the away - side jet energy at leading order ( lo ) while still being directly sensitive to the partonic transverse momentum scale . direct photons also give an interesting comparison to dihadron production because they do not carry color charge , thus , assuming factorization holds , only two tmd pdfs and one tmd ff are necessary in the cross section calculation compared to two tmd pdfs and two tmd ffs in dihadron production . therefore , there should be more avenues for gluon exchange in nearly back - to - back dihadron events when compared to direct photon - hadron events . figure [ fig : ktkinematics ] shows the hard scattering kinematics of a nearly back - to - back dihadron event in the transverse plane . the effect of initial - state @xmath14 and final - state @xmath16 , the transverse momentum of the hadron with respect to the jet axis , can be probed in hadronic collisions by measuring the out - of - plane momentum component @xmath17 with respect to the near - side hadron or direct photon , collectively referred to as the trigger particle . @xmath17 thus quantifies the acoplanarity of the two - particle pair , with @xmath18 signifying exactly back - to - back particle production . using the trigger particle as a proxy for the jet , the 1-dimensional quantity @xmath17 is transverse to the of the trigger particle , @xmath19 , and has a magnitude of : @xmath20 where @xmath21 is the of the associated hadron and @xmath22 is the azimuthal angular separation between the trigger and associated particle as shown in fig . [ fig : ktkinematics ] . reference @xcite has shown that the root mean square of @xmath17 and @xmath14 are related by @xmath23 where @xmath24 and @xmath25 , and quantities with a hat indicate partonic - level quantities . all the quantities on the left side of eq . [ eq : kteqn ] are partonic , while those on the right side can be measured via the correlated away - side hadron . equation [ eq : kteqn ] gives a clear definition for how to relate the root mean square initial - state @xmath14 and final - state @xmath16 to the observable @xmath17 . a diagram showing the hard - scattering kinematics of ( a ) dihadron and ( b ) direct photon - hadron event in the transverse plane . two hard - scattered partons with transverse momenta @xmath26 and @xmath27 [ red lines ] are acoplanar due to the initial - state @xmath28 and @xmath29 from each parton . these result in a trigger and associated jet fragment @xmath19 and @xmath21 with a transverse momentum component perpendicular to the jet axis @xmath30 , which is assumed to be gaussian such that @xmath31 . for ( b ) direct photons only one jet fragment @xmath21 is produced because the direct photon is produced from the hard scattering . the quantity @xmath17 [ blue ] is the transverse momentum component of the away - side hadron perpendicular to the trigger particle axis . , title="fig : " ] + a diagram showing the hard - scattering kinematics of ( a ) dihadron and ( b ) direct photon - hadron event in the transverse plane . two hard - scattered partons with transverse momenta @xmath26 and @xmath27 [ red lines ] are acoplanar due to the initial - state @xmath28 and @xmath29 from each parton . these result in a trigger and associated jet fragment @xmath19 and @xmath21 with a transverse momentum component perpendicular to the jet axis @xmath30 , which is assumed to be gaussian such that @xmath31 . for ( b ) direct photons only one jet fragment @xmath21 is produced because the direct photon is produced from the hard scattering . the quantity @xmath17 [ blue ] is the transverse momentum component of the away - side hadron perpendicular to the trigger particle axis . , title="fig : " ] the relativistic heavy ion collider ( rhic ) is an ideal facility to study nonperturbative factorization breaking effects because they are only predicted in hadronic collisions where at least one final - state hadron is measured , and the measurement has sensitivity to a small initial- and final - state transverse momentum scale . observables of interest are final states where at least one particle has a large , defining a hard scale , at least one final - state hadron is measured , and the observable is also sensitive to initial- and final - state @xmath14 and @xmath16 . at rhic energies , the @xmath9 reach for direct photons and pions is sufficiently large to have separation from the nonperturbative momentum scale . direct photon - hadron and @xmath32-hadron correlations were chosen specifically because of experimental capabilities and because of the differing number of final - state hadrons in the event ; the @xmath32-hadron correlations probe an extra nonperturbative function , assuming factorization , and thus one more gaussian @xmath16 convolution than the direct photon - hadron correlations . | correlations of charged hadrons of gev/ with mesons of gev/ or isolated direct photons of gev/ are used to study nonperturbative effects generated by initial - state partonic transverse momentum and final - state transverse momentum from fragmentation . the nonperturbative behavior is characterized by measuring the out - of - plane transverse momentum component perpendicular to the axis of the trigger particle , which is the high- direct photon or . the gaussian widths and root mean square of are reported as a function of the interaction hard scale to investigate possible transverse - momentum - dependent evolution differences between the-h and direct photon - h correlations and factorization breaking effects . this behavior is in contrast to drell - yan and semi - inclusive deep - inelastic scattering measurements . | dihadron and isolated direct photon - hadron angular correlations are measured in collisions at gev . correlations of charged hadrons of gev/ with mesons of gev/ or isolated direct photons of gev/ are used to study nonperturbative effects generated by initial - state partonic transverse momentum and final - state transverse momentum from fragmentation . the nonperturbative behavior is characterized by measuring the out - of - plane transverse momentum component perpendicular to the axis of the trigger particle , which is the high- direct photon or . nonperturbative evolution effects are extracted from gaussian fits to the away - side inclusive - charged - hadron yields for different trigger - particle transverse momenta ( ) . the gaussian widths and root mean square of are reported as a function of the interaction hard scale to investigate possible transverse - momentum - dependent evolution differences between the-h and direct photon - h correlations and factorization breaking effects . the widths are found to decrease with , which indicates that the collins - soper - sterman soft factor is not driving the evolution with the hard scale in nearly back - to - back dihadron and direct photon - hadron production in collisions . this behavior is in contrast to drell - yan and semi - inclusive deep - inelastic scattering measurements . |
1609.04769 | i | dihadron and direct photon - hadron correlations sensitive to nonperturbative transverse momentum effects have been measured in the phenix experiment at rhic in @xmath34 gev @xmath33 collisions , motivated by the prediction of factorization breaking in such processes @xcite . correlations between @xmath32 or direct photons with charged hadrons were measured . the azimuthal angular separation @xmath22 and out - of - plane transverse momentum component @xmath17 for the correlated pairs were measured . @xmath17 has sensitivity to nonperturbative transverse momentum in the initial state , as well as in the final state when a produced hadron is measured . the @xmath90 and gaussian widths of the @xmath17 distributions were measured from the correlations , and both observables decrease with the hard scale of the interaction @xmath19 . the direct photons exhibit a larger dependence than the @xmath32 triggers on @xmath19 for both @xmath90 and the gaussian widths of @xmath17 . the narrowing of the gaussian widths as a function of @xmath19 indicates that the collins - soper - sterman soft factor can not be driving the evolution , in contrast with drell - yan dilepton production and sidis where factorization is predicted to hold and the widths are empirically known to increase with hard scale . study of the same observables via the pythia event generator , which allows for gluon exchange between partons involved in the hard scattering and the proton remnants , reveals strikingly similar characteristics . the similarity between pythia and the experimental data offers a promising path forward to understand the mechanism in qcd driving the observed evolution in more detail . | dihadron and isolated direct photon - hadron angular correlations are measured in collisions at gev . the widths are found to decrease with , which indicates that the collins - soper - sterman soft factor is not driving the evolution with the hard scale in nearly back - to - back dihadron and direct photon - hadron production in collisions . | dihadron and isolated direct photon - hadron angular correlations are measured in collisions at gev . correlations of charged hadrons of gev/ with mesons of gev/ or isolated direct photons of gev/ are used to study nonperturbative effects generated by initial - state partonic transverse momentum and final - state transverse momentum from fragmentation . the nonperturbative behavior is characterized by measuring the out - of - plane transverse momentum component perpendicular to the axis of the trigger particle , which is the high- direct photon or . nonperturbative evolution effects are extracted from gaussian fits to the away - side inclusive - charged - hadron yields for different trigger - particle transverse momenta ( ) . the gaussian widths and root mean square of are reported as a function of the interaction hard scale to investigate possible transverse - momentum - dependent evolution differences between the-h and direct photon - h correlations and factorization breaking effects . the widths are found to decrease with , which indicates that the collins - soper - sterman soft factor is not driving the evolution with the hard scale in nearly back - to - back dihadron and direct photon - hadron production in collisions . this behavior is in contrast to drell - yan and semi - inclusive deep - inelastic scattering measurements . |
astro-ph0411663 | i | the optical depth of spiral disks has been the topic of many and varied studies since the claim by @xcite that they are transparent . the subject became controversial when @xcite and @xcite argued that disks were virtually opaque . at the cardiff meeting @xcite many possible methods to attack the problem were proposed . the dust disks of spirals may obscure objects in the high redshift universe @xcite or conceal mass in their disks @xcite . an excellent review of the current state of knowledge on the opacity of spiral disks is given by @xcite . early approaches to this subject were presented in @xcite and more recent developments are : + ( 1 ) disks are more opaque in the blue @xcite , ( 2 ) they are practically transparent in the near infrared @xcite , making these bands the best mass - to - luminosity estimator @xcite , ( 3 ) disks are practically transparent in the outer parts but show significant absorption in the inner regions @xcite , ( 4 ) the extinction correlates with galaxy luminosity @xcite , and ( 5 ) spiral arms are more opaque than the disk @xcite the majority of these studies are based on either inclination effects on the light distribution of a large sample of disks or a dust and light model to fit the observed profiles . while there is some agreement on the view that spiral disks are substantially optically thick in their central regions and become optically thin in their outer parts , the exact radial extinction profile remains uncertain . most measurements to date use the disk light itself to measure the extinction , and consequently require an assumption on the relative distribution of dust and light in a spiral disk . the extinction in a disk can be derived from far infrared and sub - mm emission arising from the cold dust in disks . however , these methods make the assumption that the emission characterises the dust in the disk . but the far infrared and sub - mm emission is likely to be dominated by the warmest component of the dust , which tends to be the smaller grains on the outside of molecular clouds facing an energy source @xcite . in this case the far - infrared and sub - mm emission will underestimate the average opacity . @xcite mention this in their estimate of gas - to - dust ratio in spiral disks based on iras observations . to obtain a better characterisation of the absorption in a spiral disk without knowledge of the distribution of stars and dust in the disk , a known background source is needed . @xcite proposed to use an occulted galaxy for this purpose , assuming it has a symmetric light distribution @xcite . @xcite and the companion paper of this work @xcite use the number of distant galaxies in the field as their background source , calibrating this number with simulations following their `` synthetic field method '' ( sfm ) . in @xcite , we describe the details of this method . both the `` occulting galaxy method '' and the sfm have the benefit of not using the disk s own light to measure the extinction . the drawbacks of the keel and white method are the assumption of symmetry of both galaxies and the few suitable pairs available . the sfm is more universally applicable but limited by the poor statistics . it does however not need to assume any distribution of either the disk s light or the absorbers in it . the sfm does require high - resolution images from hst and remains limited to the arms and disk of spiral galaxies due to crowding . in this paper we report radial extinction profiles for spiral disks of different hubble types based on 32 hubble space telescope , wide field planetary camera 2 ( hst / wfpc2 ) fields in 29 galaxies of hubble types sab and later . in section 2 we describe the sample and its selection in detail . the sfm is briefly outlined in section 3 ; a more complete description of the method and the recent improvements we have made to it is given in the companion paper @xcite . in section 4 we discuss the radial profile of average opacity for individual galaxies and , in section 5 , the composite average radial extinction profile for our entire sample . the effects of inclination , spiral arm prominence , and hubble type on the profiles are discussed as well . section 6 discusses the average ( v - i ) color of the field galaxies we found and section 7 , the tentative relation between average surface brightness and opacity . we discuss some of the implications of our results , and end with the conclusions on disk opacity from the numbers and colors of field galaxies seen through them . | this paper presents the first results for a sample of 32 deep hst / wfpc2 archival fields of 29 spirals . the effects of inclination , spiral arms and hubble type on the radial extinction profile are discussed . these results are completely in agreement with earlier work on occulted galaxies . the early type spiral disks in our sample show less extinction than the later types . this grey behaviour is most likely due to the patchy nature of opaque clouds . | dust extinction can be determined from the number of distant field galaxies seen through a spiral disk . to calibrate this number for the crowding and confusion introduced by the foreground image , and developed the `` synthetic field method '' ( sfm ) , which analyses synthetic fields constructed by adding various deep exposures of unobstructed background fields to the candidate foreground galaxy field . the advantage of the sfm is that it gives the average opacity for area of galaxy disk without assumptions about either the distribution of absorbers or of the disk starlight . however it is limited by low statistics of the surviving field galaxies , hence the need to combine a larger sample of fields . this paper presents the first results for a sample of 32 deep hst / wfpc2 archival fields of 29 spirals . the radial profiles of average dust extinction in spiral galaxies based on calibrated counts of distant field galaxies is presented here , both for individual galaxies as well as for composites from our sample . the effects of inclination , spiral arms and hubble type on the radial extinction profile are discussed . the dust opacity of the disk apparently arises from two distinct components ; an optically thicker ( ) but radially dependent component associated with the spiral arms and a relatively constant optically thinner disk ( ) . these results are completely in agreement with earlier work on occulted galaxies . the early type spiral disks in our sample show less extinction than the later types . lsb galaxies and possibly sd s appear effectively transparent . the average color of the field galaxies seen through foreground disks does not appear to change with radius or opacity . this grey behaviour is most likely due to the patchy nature of opaque clouds . the average extinction of a radial annulus and its average surface brightness seem to correlate for the brighter regions . this leads to the conclusion that the brighter parts of the spiral disk , such as spiral arms , are also the ones with the most extinction associated with them . |
cond-mat0005412 | i | linked cluster expansions ( lces ) have a long tradition in statistical physics . originally applied to classical fluids , later to magnetic systems ( @xcite,@xcite,@xcite and references therein ) , they were generalized to applications in particle physics in the eighties @xcite . there they have been used to study the continuum limit of a lattice @xmath10 field theory in 4 dimensions at zero temperature . in @xcite they were further generalized to field theories at finite temperature , simultaneously the highest order in the expansion parameter was increased to 18 . usually the analytic expansions are obtained as graphical expansions . because of the progress in computer facilities and the development of efficient algorithms for generating the graphs , it is nowadays possible to handle of the order of billions of graphs . the whole range from high temperatures down to the critical region becomes available , and thermodynamic quantities like critical indices and critical temperatures are determined with high precision ( the precision is comparable or even better than in corresponding high quality monte carlo results ) @xcite-@xcite . an extension of lces to a finite volume in combination with a high order in the expansion parameter turned out to be a particularly powerful tool for investigating the phase structure of systems with first and second order transitions by means of a finite size scaling analysis @xcite . linked cluster expansions are series expansions of the free energy and connected correlation functions about an ultralocal , decoupled theory in terms of a hopping parameter @xmath11 . the corresponding graphical representation is a sum in terms of connected graphs . the value of @xmath11 parametrizes the strength of interactions between fields at different lattice sites . usually they are chosen as nearest neighbours . in contrast to the ultralocal terms of a generic interaction we will sometimes refer to hopping terms as non - ultralocal . in this paper we develop dynamical linked cluster expansions ( dlces ) . these are linked cluster expansions with hopping parameter terms that are endowed with their own dynamics . such systems are realized in spin glasses with ( fast ) spins and ( slow ) interactions @xcite-@xcite . they also occur in variational estimates for su(n)-gauge - higgs systems , cf . @xcite . like lces they are expected to converge for a large class of interactions . formally dlces amount to a generalization of an expansion scheme from 2-point to point - link - point - interactions . these are interactions between fields associated with two points and with one pair of points called link . the points and links are not necessarily embedded on a lattice , and the links need not be restricted to nearest neighbours . we have developed a new theory in which a generalized notion of connectivity plays a central role . standard notions of equivalence classes of graphs like 1-line irreducible and 1-vertex irreducible graphs have been generalized , and new notions like 1-multiple - line irreducible graphs were defined in order to give a systematic classification . the paper is organized as follows . in sec . 2 we specify the models that admit a dlce . we introduce and explain the idea behind the abstract notions of theory . detailed definitions of , related notions and the computation of weights are given in sect . 4 treats the issue of renormalization in the sense of suitable resummations of graphs . applications to spin glasses are presented in sect . there it is of particular interest that dlces allow for the possibility of avoiding the replica trick . in the quenched limit we derive that dlces must be restricted to a subclass of the corresponding graphical expansion , so - called quenched dlces ( qdlces ) . we also list some examples for models whose phase structure is accessible to qdlces . to these belongs in particular the bond diluted ising model . 6 contains the summary and conclusions . | dynamical linked cluster expansions are linked cluster expansions with hopping parameter terms endowed with their own dynamics . this amounts to a generalization from 2-point to point - link - point interactions . an associated graph theory with a generalized notion of connectivity is reviewed . dynamical linked cluster expansions + for spin glasses hildegard meyer - ortmanns + and + thomas reisz institut fr theoretische physik bergische universitt wuppertal gaustrasse 20 d-42097 wuppertal , germany service de physique thorique de saclay ce - saclay f-91191 gif - sur yvette cedex , france institut fr theoretische physik | dynamical linked cluster expansions are linked cluster expansions with hopping parameter terms endowed with their own dynamics . this amounts to a generalization from 2-point to point - link - point interactions . an associated graph theory with a generalized notion of connectivity is reviewed . we discuss physical applications to disordered systems , in particular to spin glasses , such as the bond - diluted ising model and the sherrington - kirkpatrick spin glass . we derive the rules and identify the full set of graphs that contribute to the series in the quenched case . this way it becomes possible to avoid the vague extrapolation from positive integer to , that usually goes along with an application of the replica trick . dynamical linked cluster expansions + for spin glasses hildegard meyer - ortmanns + and + thomas reisz institut fr theoretische physik bergische universitt wuppertal gaustrasse 20 d-42097 wuppertal , germany service de physique thorique de saclay ce - saclay f-91191 gif - sur yvette cedex , france institut fr theoretische physik universitt heidelberg philosophenweg 16 d-69120 heidelberg , germany |
astro-ph0507219 | i | observations and numerical modeling of galactic stellar bars have been frequently accompanied with basic controversies about their origin and evolution . while modern understanding of bar growth in a live and responsive environment is rooted in the angular momentum redistribution between the inner and outer disks , bulges and dark matter halos ( athanassoula 2003 ) , the efficiency of this process is hardly known and its details are still to be investigated . recent efforts include but are not limited to the issues related to the bar lifetime cycles , gas - star interactions , bar amplitudes and sizes , and bar slowdown ( e.g. , bournaud & combes 2002 ; valenzuela & klypin 2003 ; shen & sellwood 2004 ; weinberg 1985 ; hernquist & weinberg 1992 ; debattista & sellwood 1998 , 2000 ) . the bigger issue of course is how the observational and theoretical aspects of bar evolution fit within the emerging understanding of cosmological galaxy evolution ( e.g. , jogee et al . 2004 ; elmegreen et al . 2004 ) , specifically the bar evolution in triaxial halos ( el - zant & shlosman 2002 ; berentzen , shlosman & jogee 2005 ) . on one hand , early self - consistent models of numerical stellar bars have relied heavily on the ostriker & peebles ( 1973 ) result which emphasized the maclauren sequence parameter @xmath2 the ratio of bulk kinetic - to - gravitational energy as the threshold of bar instability . on the other hand , they revealed robustness of the bars once formed , the bars persisted ( athanassoula 1984 and refs . therein ) . the subsequent increase in the particle number above @xmath3 , switching from 2-d to 3-d models with responsive spheroidal components , and introduction of nonlinear physics tools in the orbital analysis have shown a much more complex bar evolution and morphology than was anticipated originally ( e.g. , review by athanassoula 2002a ) . this refers especially to the numerical confirmation that live halos can indeed drive the bar instability rather than damp it ( athanassoula & misiriotis 2002 ; athanassoula 2003 ) . lastly , it is still unclear to what extent and how closely the numerical bars correspond to their observed counterparts . this issue exacerbates galaxy studies because overall , both theoretically and observationally , the bars appear to be among the most important drivers of galactic evolution across a wide range of spatial scales . in this paper , we have revisited some aspects of a self - consistent evolution of stellar bars originating in _ live _ stellar disks embedded in _ live _ dark matter halos by focusing on dynamical and secular changes disk rotation , and to a secular evolution when they develop on a much longer timescale of @xmath4 rotations . ] in these systems . the evolution of _ numerical _ collisionless bars has been characterized so far in the literature by three distinct phases the initial growth , the rapid vertical buckling and the prolonged quasi - steady regime , i.e. , when the bars preserve their basic parameters ( e.g. , sellwood & wilkinson 1993 ) . however , some indication that bars can grow even in the last phase has been noticed already in low - resolution 3-d models with live halos ( e.g. , sellwood 1980 ) . this ability of the bars to grow over extended period of time due to the momentum exchange with the outer disk and especially with the halo has been confirmed recently and analyzed in self - consistent 3-d simulations ( athanassoula 2003 ) . here we attempt to quantify this secular growth in terms of the bar size and its ellipticity , of the angular momentum exchange , and of the ratio of vertical - to - radial dispersion velocities . moreover , we look into the corollaries of such recurrent growth and find that it leads to additional and substantial 2-d and 3-d structural changes in the bar . we , therefore , discuss the observational consequences of this evolution . early in their growth stage , numerical stellar bars experience a dynamical instability the vertical buckling . the bars thicken profoundly , become more centrally - concentrated and acquire a characteristic peanut / boxy shape when seen edge - on ( combes et al . 1990 ; pfenniger & friedli 1991 ; raha et al . 1991 ; berentzen et al . 1998 ; patsis , skokos & athanassoula 2002a ) , while nearly dissolving the outer half of the bar , beyond the vertical inner lindblad resonance ( martinez - valpuesta & shlosman 2004 ) . this happens in live models with both axisymmetric and mildly triaxial halos ( berentzen , shlosman & jogee 2005 ) . these boxy / peanut shapes are similar to bulge shapes observed in edge - on galaxies ( e.g. , jarvis 1986 ; shaw 1987 ; bureau & freeman 1999 ; merrifield & kuijken 1999 ) , which can be found in nearly half of all edge - on disk galaxies ( ltticke , dettmar & pohlen 2000 ) . although observed in numerical simulations a long time ago ( combes & sanders 1981 ) , the origin of boxy / peanut bulge shapes still has two alternative explanations the well - known firehose instability ( e.g. , toomre 1966 ; raha et al . 1991 ; merritt & sellwood 1994 ) and the resonance heating ( e.g. , combes et al . 1990 ; pfenniger & friedli 1991 ; patsis et al . these two views can be reconciled if buckling is responsible for shortening the secular timescale of particle diffusion out of the disk plane and for accelerating the buildup of boxy / peanut bulges which proceeds on a much shorter dynamical timescale instead ( martinez - valpuesta & shlosman 2004 ) . however , is the buckling really necessary for a buildup of these boxy / peanut shaped bulges ? after all , even _ imposing _ vertical symmetry did not eliminate this effect , albeit the buildup proceeded on a much longer timescale ( friedli & pfenniger 1990 ) . where are the observational counterparts of these asymmetric buckled bars ? due to a particular importance of the buckling instability for the evolution of numerical bars and its plausible connection to the buildup of the pronounced 3-d structure there , we have analyzed the bar behavior during and following this instability . specifically , we find that the bars in a live environment are capable of recurrent growth , that the buckling instability is a recurrent event and that the buildup of the 3-d shape is not necessarily a dynamic phenomenon . much of the analysis of bar evolution and the accompaning instabilities is implemented here by means of nonlinear orbit analysis , because for such strong departures from axial symmetry the epicyclic approximation can not be relied upon being wrong quantitatively it frequently leads to qualitative errors . we shall try to avoid the specific jargon associated with this technique where possible . the angular momentum redistribution in the model is quantified using orbital spectral analysis . in section 2 we provide the details of numerical modeling and analysis . section 3 describes the overall results of secular bar evolution . the 3-d bar orbital structure and the inferred vertical structure in the bar are analyzed in section 4 , and the resonant interaction between the disk and the halo in section 5 . discussion and conclusions are given in sections 6 and 7 . | evolution of stellar bars in disk galaxies is accompanied by dynamical instabilities and secular changes . following the vertical buckling instability , the bars are known to weaken dramatically and develop a pronounced boxy / peanut shape when observed edge - on . using high - resolution-body simulations of stellar disks embedded in live axisymmetric dark matter halos we find that following the initial buckling , the bar resumes its growth from deep inside the corotation radius and follows the ultra - harmonic resonance thereafter . overall , the stellar bars show recurrent behavior in their properties and evolve by increasing their linear and vertical extents , both dynamically and secularly . | evolution of stellar bars in disk galaxies is accompanied by dynamical instabilities and secular changes . following the vertical buckling instability , the bars are known to weaken dramatically and develop a pronounced boxy / peanut shape when observed edge - on . using high - resolution-body simulations of stellar disks embedded in live axisymmetric dark matter halos , we have investigated the long - term changes in the bar morphology , specifically the evolution of the bar size , its vertical structure and exchange of angular momentum . we find that following the initial buckling , the bar resumes its growth from deep inside the corotation radius and follows the ultra - harmonic resonance thereafter . we also find that this secular bar growth triggers a spectacular _ secondary _ vertical buckling instability which leads to the appearance of characteristic boxy / peanut / x - shaped bulges . the secular bar growth is crucial for the recurrent buckling to develop . furthermore , the secondary buckling is milder , persists over a substantial period of time , gyr , and can have observational counterparts . overall , the stellar bars show recurrent behavior in their properties and evolve by increasing their linear and vertical extents , both dynamically and secularly . we also demonstrate explicitly that the prolonged growth of the bar is mediated by continuous angular momentum transfer from the disk to the surrounding halo , and that this angular momentum redistribution is resonant in nature a large number of lower resonances trap disk and halo particles and this trapping is robust , in a broad agreement with the earlier results in the literature . |
astro-ph0507219 | c | to summarize this work , we have studied the long - term stellar bar evolution in a high - resolution self - consistent model of a disk and a responsive halo . we find that a developing bar goes through the vertical buckling instability which weakens it and dissolves its outer half . subsequently , the bar experiences a renewed growth which leads to a recurrent buckling . this evolution is driven by the resonant interaction between the barred disk and the surrounding halo we quantify this effect by means of the spectral analysis of individual orbits in the disk and the halo and show that the halo particles are trapped by numerous lower resonances with the bar and that this trapping is robust . during these periods of recurrent instability , and especially during the slower second buckling , the bar remains vertically asymmetric for a prolonged @xmath1 gyr time interval which in principle can be detected observationally . however , two issues can potentially complicate this detection . first , it is not clear how widespread are the conditions favorable for the recurrent bar growth , although we have detected it in a number of models with different initial conditions . second , while it was shown that a clumpy gaseous component with an isothermal equation of state in the disk will weaken this instability , the effect of a realistic ism was never estimated . a statistically significant sample of ( nearly ) edge - on galaxies is required to test the prediction of a prolonged vertical asymmetry . we also find that the secular bar growth and the triggered buckling instabilities lead to pronounced changes in the bulge shape it grows both radially and vertically , acquiring a peanut , a boxy and finally the x - shaped appearance . while the bar size approximately follows the 4:1 ( ultra - harmonic ) resonance in the disk , the boxy / peanut bulge size appears to be guided by the vertical ilr . concurrently , the bar is going through a structural evolution new families of 3-d periodic orbits appear ( or become more pronounced ) after the bucklings . finally , we find that the bar strength correlates with its pattern speed , in both strengthening and weakening bars . while it was known already that a stellar bar becomes stronger as it slows down , we detect the reverse trend as well bars that weaken during the buckling speed up their tumbling . moreover , the bar size appears to be sensitive to these changes slowing down bars become longer , while speeding up bars shorten . we gratefully acknowledge insightfull discussions with lia athanassoula , ingo berentzen , amr el - zant , johan knapen and alar toomre . this work has been partially supported by grants from nasa / ltsa5 - 13063 , nasa / atp nag 5 - 10823 , hst ar-09546 and 10284 ( i.s . ) , and by nsf ast-0206251 ( c.h . & i.s . ) . i.m . acknowledges support from pparc . simulations and orbital analysis have been performed on a dedicated linux cluster and we thank brian doyle for technical support . athanassoula , e. 1992 , mnras , 259 , 345 athanassoula , e. 2001 , new horizons of computational science , ( eds . ) t.ebisuzaki & j.makino ( dordrecht : kluwer acad . publ . ) , 312 ( astro - ph/0112076 ) athanassoula , e. 2002a , disks of galaxies : kinematics , dynamics & interactions , ( eds . ) e. athanassoula , a. bosma , and r. mujica ( asp conf . 275 ) , 141 binney , j. , tremaine , s. 1987 , galactic dynamics , princeton univ . press bournaud , f. , combes , f. 2002 , a&a , 392 , 83 bureau , m. , freeman , k.c . 1999 , apj , 118 , 126 buta , r. et al . , 1998 , aj , 116 , 1142 combes , f. sanders , r.h . 1981 , a&a , 96 , 164 combes , f. , debbasch , f. , friedli , d. , pfenniger , d. 1990 , a&a , 233 , 82 | we also find that this secular bar growth triggers a spectacular _ secondary _ vertical buckling instability which leads to the appearance of characteristic boxy / peanut / x - shaped bulges . furthermore , the secondary buckling is milder , persists over a substantial period of time , gyr , and can have observational counterparts . we also demonstrate explicitly that the prolonged growth of the bar is mediated by continuous angular momentum transfer from the disk to the surrounding halo , and that this angular momentum redistribution is resonant in nature a large number of lower resonances trap disk and halo particles and this trapping is robust , in a broad agreement with the earlier results in the literature . | evolution of stellar bars in disk galaxies is accompanied by dynamical instabilities and secular changes . following the vertical buckling instability , the bars are known to weaken dramatically and develop a pronounced boxy / peanut shape when observed edge - on . using high - resolution-body simulations of stellar disks embedded in live axisymmetric dark matter halos , we have investigated the long - term changes in the bar morphology , specifically the evolution of the bar size , its vertical structure and exchange of angular momentum . we find that following the initial buckling , the bar resumes its growth from deep inside the corotation radius and follows the ultra - harmonic resonance thereafter . we also find that this secular bar growth triggers a spectacular _ secondary _ vertical buckling instability which leads to the appearance of characteristic boxy / peanut / x - shaped bulges . the secular bar growth is crucial for the recurrent buckling to develop . furthermore , the secondary buckling is milder , persists over a substantial period of time , gyr , and can have observational counterparts . overall , the stellar bars show recurrent behavior in their properties and evolve by increasing their linear and vertical extents , both dynamically and secularly . we also demonstrate explicitly that the prolonged growth of the bar is mediated by continuous angular momentum transfer from the disk to the surrounding halo , and that this angular momentum redistribution is resonant in nature a large number of lower resonances trap disk and halo particles and this trapping is robust , in a broad agreement with the earlier results in the literature . |
astro-ph0112113 | c | it has been clearly shown that , contrary to the belief expressed in many papers and recently by @xcite , atomic diffusion does not necessarily lead to underabundances of metals in pop ii stars . differential radiative accelerations lead to overabundances of fe and some other chemical elements in some turnoff stars . consider the evolution of pop ii stars with no tubulence . as one may see by considering the solid curves in figures [ fig : ab_teff12 ] and [ fig : ab_teff13.5 ] , generalized underabundances by 0.1 dex are expected in the @xmath6 interval from 4600 to 5500 . between 5500 and 6000 , the underabundances are still generalized and increase to 0.3 dex for some species such as . in 12 gyr turnoff stars however ( @xmath67 ) , overabundances by a factor of up to 10 are possible ( e.g. al and ni ) . all calculated species heavier than na may have overabundances . at a given @xmath6 , variations are expected from star to star . at 13.5 gyr , similar but smaller anomalies are expected . the overabundances are sensitive to any left over turbulence below the convection zone . in this paper , the evolution of stars with @xmath21 has been described both with and without turbulence . a 0.1 dex underabundance of metals in turnoff stars as compared to giants has been shown to be the smallest anomaly to be expected ( section [ sec : ab_teff ] ) . star to star variations were seen to be possible around the turnoff , if turbulence is small enough . observations ( see section [ sec : observations ] ) suggest the presence of abundance variations similar to those expected . the comparison to observations is , however , sensitive to the @xmath6 scale . as @xcite concluded at the end of their section 4.2 , higher quality data is probably required to establish the reality of fe abundance variations within m92 . the accurate determination of the abundance of more species is also needed . this may well have implications not only for intrinsic abundance variations in clusters but for internal stellar structure . the effect of varying @xmath34 on the evolution of pop ii stars will be investigated in a forthcoming paper , ( paper iii ) , where comparisons to higher @xmath34 clusters will be undertaken . increasing @xmath34 in pop ii stars will be shown to reduce considerably the expected abundance anomalies . note also that paper ii shows that the present models for @xmath7_0 = -1.31 $ ] accurately reproduce the cmd locations of local population ii subdwarfs having hipparcos parallaxes and metallicities within + /@xmath55 0.2 dex of @xmath7 = -1.3 $ ] . in a number of clusters with higher @xmath34 than m92 , observations suggest that only small variations , if any , are present in turnoff stars ( see for instance @xcite ; @xcite and @xcite ) . furthermore in field halo stars , the li abundance puts strict constraints on any chemical separation . in the companion paper ( paper ii ) we therefore took the cautious approach to use mainly evolutionary models that minimize the effect of atomic diffusion . it has been shown that the use , in complete stellar evolution models , of a relatively simple parametrization of turbulent transport leads to li surface abundances compatible with the li plateau observed in field halo stars ( with a 0.17 dex reduction from the original li abundance ) and small variations in the surface abundances of metals ( a 0.1 dex reduction of the metal abundance in turnoff stars as compared to that in giants in clusters with @xmath21 ) . at the same time , the gravitational settling of he leads to a reduction in the age of globular clusters by some 10% ( see paper ii ) . however simple the parametrization of turbulent transport , it is not understood from first principles . the high level of constancy of li abundance as a function of @xmath6 requires that turbulence mixes to very nearly the same @xmath38 throughout the star evolution and in stars covering a mass interval of approximately 0.6 to 0.8 @xmath3 . as already noted by @xcite , this is not expected in standard stellar models . no convincing hydrodynamic model has been proposed that explains this property . pop ii stars appear to tell us that this is the case , however . mass loss is another physical process that could compete with atomic diffusion and maintain a constant value of li as a function of @xmath6 @xcite . whether , in the absence of turbulent transport , it could be made consistent with the observations of metals in globular cluster turnoff stars is a question that requires further calculations . these may lead to observational tests of the relative importance of mass loss and turbulence in these objects . the number of chemical species that are now included in these calculations and that can be observed makes such tests possible . we thank an anonymous referee for constructive comments that led to significant improvements of this paper . this research was partially supported at the university of victoria and the universit de montral by nserc . we thank the rseau qubcois de calcul de haute performance ( rqchp ) for providing us with the computational resources required for this work . extends his warm thanks to the service dastrophysique at cea - saclay for an enjoyable and productive stay during which part of this research was performed . this work was performed in part under the auspices of the u.s . department of energy , national nuclear security administration , by the university of california , lawrence livermore national laboratory under contract no.w-7405-eng-48 . , r. g. , bonifacio , p. , bragaglia , a. , carretta , e. , castellani , v. , centurion , m. , chieffi , a. , claudi , r. , clementini , g. , dantona , f. , desidera , s. , franois , p. , grundahl , f. , lucatello , s. , molaro , p. , pasquini , l. , sneden , c. , spite , f. , & straniero , o. 2001 , a&a , 369 , 87 , @xmath42 , temperature at the base of the surface convection zone ( @xmath44 ) , mass above the base of the surface convection zone @xmath68 and mass fraction of hydrogen ( @xmath69 ) at the center of stars of 0.5 to 1.0 @xmath3 with @xmath34 = 0.00017 , or @xmath70=-2.31.$ ] all models were calculated with atomic diffusion and radiative accelerations but no turbulent transport . in the black and white figure , are shown 6 of the 15 models that may be seen in the color figure available in the electronic version of the paper . ] with @xmath7=-2.31 $ ] at four epochs identified in the upper part of the figure . the vertical lines give the position of the bottom of the surface convection zone . for other stellar masses , it is the position of the bottom of the convection zone that is most different . the @xmath4 of the various species varies from star to star but much less than the depth of convection zones . gravity is shown in the lower right hand corner and repeated in each panel of the figure . ] with @xmath7=-2.31 $ ] . all models were calculated with atomic diffusion and radiative accelerations but no turbulent transport . in the black and white figure , are shown 6 of the 15 models that may be seen in the color figure available in the electronic version of the paper . as is evident in the color version , a vertical line drawn in each panel of the figure , corresponding to a fixed age , permits one to evaluate the range of surface abundances of a species at that age ( in , e. g. , a globular cluster ) . ] with @xmath7=-2.31 $ ] . the profiles are shown at four different ages , 6.1 , 9.1 , 10.3 and 11.4 gyr . the last is shortly before the star moves to the giant branch . calculations included atomic diffusion and radiative accelerations but no turbulent transport . ] with @xmath7=-2.31 $ ] over the inner 2/3 of the mass of the star at an age of 11.8 gyr . calculations included atomic diffusion and radiative accelerations but no turbulent transport . the zero abundance change occurs around 1/2 the mass of the star at @xmath71 . interior to that point , the abundance of metals is generally larger than the original abundance . the variations of cno are due to their transformation by the cno chain which would lead to a decrease of z at the center if it were not for the settling of the other metals . only mg , fe and ni are shown from the 16 species between ne and ni included in the calculations but all others have very similar behavior . ] models . the nearly horizontal line is the he atomic diffusion coefficient . the other lines are various parametrizations of turbulence . in the upper part of the figure , the corresponding li concentrations are shown , at an age of 10.2 gyr , with the same line identification to give the link between turbulent transport and li burning at @xmath39 . the parameters specifying turbulent transport coefficients are indicated in the name assigned to the model . for instance , in the t5.5d400 - 3 model , the turbulent diffusion coefficient , @xmath14 , is 400 times larger than the he atomic diffusion coefficient at @xmath15 and varying as @xmath16 . ] , @xmath42 , temperature at the base of the surface convection zone ( @xmath44 ) , mass at the base of the surface convection zone ( @xmath68 ) and mass fraction of hydrogen ( @xmath69 ) at the center and of the he , fe and li abundances in 0.8 @xmath3 models both with and without diffusion . see figure [ fig : coefficients ] and table [ tab : parameters ] for a definition of the notation used for models with turbulent transport . ] with @xmath7=-2.31 $ ] and turbulence parametrized by t6.09d400 - 3 and t5.5d400 - 3 . the model with no turbulence is also shown for comparison purposes . the profiles are shown at 11.7 gyr , shortly before the star moves to the giant branch . the same scale is used as for figure [ fig : intern_abundances ] . see figure [ fig : coefficients ] and table [ tab : parameters ] for a definition of the notation used for models with turbulent transport . ] model . the models differ by the boundary conditions and by the value of @xmath19 used in the calculations . see figure [ fig : coefficients ] and table [ tab : parameters ] for a definition of the notation used for models with turbulent transport . ] after 10 gyr of evolution . in the right hand panel , the 0.84 @xmath3 model is replaced by the 0.81 @xmath3 one and it is after 12 gyr of evolution.,title="fig:",scaledwidth=45.0% ] after 10 gyr of evolution . in the right hand panel , the 0.84 @xmath3 model is replaced by the 0.81 @xmath3 one and it is after 12 gyr of evolution.,title="fig:",scaledwidth=45.0% ] star , after 10/12 gyr of evolution for 3 turbulence strengths : atomic diffusion only , the t5.5 and t6.09 turbulence models . , title="fig:",scaledwidth=45.0% ] star , after 10/12 gyr of evolution for 3 turbulence strengths : atomic diffusion only , the t5.5 and t6.09 turbulence models . , title="fig:",scaledwidth=45.0% ] . the continuous line segments link models calculated with atomic diffusion . the dashed and dotted line segments link models calculated with atomic diffusion and , respectively , the t6.09 and t5.5 turbulence models . ] . the continuous line segments link models calculated with atomic diffusion . the dashed and dotted line segments link models calculated with atomic diffusion and , respectively , the t6.09 and t5.5 turbulence models . ] 6000 k , one should remember that , at a given mass and age , a star with the t6.09 turbulence model has a higher @xmath6 ( by some 100 ) than one with atomic diffusion only ( see fig . [ fig : with_without ] ) . references for the observed li abundances are identified on the figure . ] throughout the main sequence and subgiant evolution of 0.84 , 0.8 , 0.77 and 0.75 @xmath3 stars when atomic diffusion is included in the calculations . turbulence is assumed negligible . for the 0.84 @xmath3 star the main sequence starts at @xmath72k where @xmath73 . as evolution proceeds , the fe abundance decreases until an underabundance of @xmath550.4 dex is reached . at that point , the surface convection zone has retracted sufficiently for @xmath4 ( ) to be greater than gravity and the fe abundance increases up to a 1.7 dex overabundance . this occurs at the end of the main sequence evolution ( around the turnoff ) at @xmath74k . as evolution proceeds on the subgiant branch the surface convection zone gets deeper and the fe abundance decreases to 0.1 dex below the original abundance . the fe abundance goes back to its original value when a @xmath6 of 5300@xmath555400 k is reached . similar evolution occurs for the other stars except that it starts at different @xmath6 for each . the li abundance in the 0.84 @xmath3 star has a similar evolution . the li abundance decreases until an underabundance of @xmath551.2 dex is reached . at that point , the surface convection zone has retracted sufficiently for @xmath4 ( ) to be greater than gravity . however @xmath4 ( ) is larger than gravity over a smaller interval than @xmath4 ( ) . it leads to an increase of the li abundance at @xmath75k but to no overabundance . as the subgiant evolution starts , li gets below its main sequence abundance at @xmath76k , creating a loop in the li abundance curve . the li abundance never reaches its original abundance since , before this occurs , the surface convection zone approaches the region of li burning and the li abundance decreases rapidly below 5800 k . ] k for a 0.8 and a 0.78 @xmath3 star with atomic diffusion only . their ages bracket the age of m92 , 13.5 gyr . both stars have very closely the same composition showing that surface abundances are not a sensitive function of mass at @xmath77k , when the star is on the subgiant branch . a 0.78 @xmath3 star with the t6.09 turbulence model is also shown at the same @xmath6 . finally the 0.78 @xmath3 star with no turbulence is shown at 13.5 gyr when it is a turnoff star ( @xmath78k ) . , scaledwidth=60.0% ] lccc h & & 7.646@xmath79 & 7.613@xmath79 + @xmath80he & & 2.352@xmath79 & 2.370@xmath79 + @xmath81c & & 1.727@xmath82 & 1.727@xmath83 + n & & 5.294@xmath84 & 5.294@xmath82 + o & & 9.612@xmath82 & 9.612@xmath83 + ne & & 1.966@xmath82 & 1.966@xmath83 + na & & 3.986@xmath85 & 3.986@xmath84 + mg & & 7.484@xmath84 & 7.484@xmath82 + al & & 1.627@xmath85 & 1.627@xmath84 + si & & 8.072@xmath84 & 8.072@xmath82 + p & & 6.976@xmath86 & 6.976@xmath85 + s & & 4.215@xmath84 & 4.215@xmath82 + cl & & 8.969@xmath86 & 8.969@xmath85 + ar & & 1.076@xmath84 & 1.076@xmath82 + k & & 1.998@xmath86 & 1.998@xmath85 + ca & & 7.474@xmath85 & 7.474@xmath84 + ti & & 3.986@xmath86 & 3.986@xmath85 + cr & & 9.989@xmath86 & 9.989@xmath85 + mn & & 3.890@xmath86 & 3.890@xmath85 + fe & & 7.172@xmath84 & 7.172@xmath82 + ni & & 4.445@xmath85 & 4.445@xmath84 + z & & 1.675@xmath83 & 1.675@xmath87 [ tab : xinit ] lccccc non - diffusive & eddington & 1.687 & @xmath88 & no & no + diffusive & eddington & 1.687 & @xmath88 & yes & no + t5.5 & eddington & 1.687 & @xmath88 & yes & t5.5d400 - 3 + t6.0 & eddington & 1.687 & @xmath88 & yes & t6.0d400 - 3 + t6.09 & eddington & 1.687 & @xmath88 & yes & t6.09d400 - 3 + t6.13 & eddington & 1.687 & @xmath88 & yes & t6.13d400 - 3 + t6.09ks & krishna - swamy & 1.869 & @xmath88 & yes & t6.09d400 - 3 + t6.09ks@xmath19 & krishna - swamy & 2.017 & @xmath88 & yes & t6.09d400 - 3 + non - diffusive@xmath89 & eddington & 1.687 & @xmath89 & no & no + diffusive@xmath89 & eddington & 1.687 & @xmath89 & yes & no + t6.0@xmath89 & eddington & 1.687 & @xmath89 & yes & t6.00d400 - 3 + t6.09@xmath89 & eddington & 1.687 & @xmath89 & yes & t6.09d400 - 3 + [ tab : parameters ] | the effects of radiative accelerations , thermal diffusion and gravitational settling are included . while models were calculated both for and , we concentrate on models with in this paper . it is shown that , as in the case of pop i stars , additional turbulence appears to be present . one series minimizes the spread on the li plateau while another was chosen with turbulence similar to that present in amfm stars of pop i. even when turbulence is adjusted to minimize the reduction of li abundance , there remains a reduction by a factor of at least 1.6 from the original li abundance . independent of the degree of turbulence in the outer regions , gravitational settling of he in the central region reduces the lifetime of pop ii stars by 4 to 7 % depending on the criterion used . the effect on the age of the oldest clusters | evolutionary models have been calculated for pop ii stars of 0.5 to 1.0 from the pre - main sequence to the lower part of the giant branch . rosseland opacities and radiative accelerations were calculated taking into account the concentration variations of 28 chemical species , including all species contributing to rosseland opacities in the opal tables . the effects of radiative accelerations , thermal diffusion and gravitational settling are included . while models were calculated both for and , we concentrate on models with in this paper . these are the first pop ii models calculated taking radiative acceleration into account . it is shown that , at least in a 0.8 star , it is a better approximation not to let diffuse than to calculate its gravitational settling without including the effects of(fe ) . in the absence of any turbulence outside of convection zones , the effects of atomic diffusion are large mainly for stars more massive than 0.7 . overabundances are expected in _ some _ stars with . most chemical species heavier than cno are affected . at 12 gyr , overabundance factors may reach 10 in some cases ( e.g. for al or ni ) while others are limited to 3 ( e.g. for fe ) . the calculated surface abundances are compared to recent observations of abundances in globular clusters as well as to observations of li in halo stars . it is shown that , as in the case of pop i stars , additional turbulence appears to be present . series of models with different assumptions about the strength of turbulence were then calculated . one series minimizes the spread on the li plateau while another was chosen with turbulence similar to that present in amfm stars of pop i. even when turbulence is adjusted to minimize the reduction of li abundance , there remains a reduction by a factor of at least 1.6 from the original li abundance . independent of the degree of turbulence in the outer regions , gravitational settling of he in the central region reduces the lifetime of pop ii stars by 4 to 7 % depending on the criterion used . the effect on the age of the oldest clusters is discussed in an accompanying paper . just as in pop i stars where only a fraction of stars , such as amfm stars , have abundance anomalies , one should look for the possibility of abundance anomalies of metals in some pop ii turnoff stars , and not necessarily in all . expected abundance anomalies are calculated for 28 species and compared to observations of m92 as well as to li observations in halo field stars . |
astro-ph9910098 | i | this study uses fuv , hi , and h@xmath0 observations of dwarf irregular galaxy hoii to trace the interaction between sites of massive star formation and the neutral and ionized components of the surrounding ism and to identify the means by which star formation propagates in this intrinsically simple system . both local and large - scale triggering mechanisms related to massive stars are identified suggesting that feedback from massive stars is a microscopic process operating in all galaxies to a certain degree . \(1 ) in comparison to its optical images , the fuv morphology of hoii is patchy and irregular . massive star formation does not have a smooth spatial distribution ; it is concentrated in several dominate complexes . the fact that massive star formation is not centrally organized has important implications for interpreting images of high redshift galaxies ( z @xmath14 3 ) where the restframe fuv is detected using optical and ir instruments . \(2 ) surface brightness profiles are used to characterize the radial dependence of the azimuthally averaged star formation activity . the fuv profile is generally flat with inflections due to prominent complexes . the b - band profile displays an exponential shape with a central light depression , characteristic of dwarf galaxies . the derived central surface brightness , @xmath49(0 ) , lies in the expected range for dwarf galaxies . comparisons of profiles in the r , b , u and fuv bands indicate that profiles retain the same general shape while inflections due to star formation increase as the wavelength of the profile decreases . the radial continuity of star formation sites demonstrates that the time averaged level of activity in the past correlates with the current activity at the same radius , supporting the theory that star formation in irregular galaxies is dominated by local processes . \(3 ) star formation rates derived from h@xmath0 and fuv photometry are used to characterize the current ( traced by h@xmath0 ) and recent ( traced by fuv ) global star formation activity . comparisons of the two rates allow a rough estimate of the recent star formation history and indicate that the galaxy is sustaining a somewhat recent star formation event . \(4 ) ages for individual regions are derived using b , h@xmath0 , and fuv photometry and show both older , diffuse fuv regions and younger , compact hii regions . the distribution of ages is reconciled with the hi morphology , showing a clear preference of young regions for areas of dense hi and older regions for hi voids . \(5 ) local star formation triggering mechanisms are identified via the relative h@xmath0 and fuv morphologies . observational evidence of the influence that massive stars have on their surroundings includes massive stars inside ionized shells and compact hii regions surrounding aging clusters . causal relationships between massive star formation and secondary sites of star formation are shown . star formation progression patterns are consistent with the sspsf model in that progressively older chains of star formation are seen in clusters . \(6 ) large - scale star formation triggering by hi shells generated by massive stars is seen by comparing the fuv and hi morphologies . given the timescale over which fuv observations trace massive star formation , both secondary sites of star formation and progenitor populations are identified . analysis of the energy available from massive stars inside hi voids indicates that energy deposited into the ism from supernovae and stellar winds are sufficient to account for the observed properties of the hi holes . \(7 ) global kinematical properties may also play a role in the star formation process since differences in the rotation characteristics of the neutral gas disk are coupled with differences in triggering mechanisms . large - scale feedback from massive star formation is shown to operate in regions that lack differential shear in the gas disk . s.g.s . wishes to acknowledge the funding of this study through the alabama space grant consortium project and to thank the university of alabama , nasa / goddard space flight center , and the u. s. naval observatory for their support . funding for the uit project has been through the spacelab office at nasa headquarters under project number 440 - 51 . | the observations trace the interaction between sites of massive star formation and the neutral and ionized components of the surrounding ism in this intrinsically simple system . both local and large - scale triggering mechanisms related to massive star formation are seen , suggesting that feedback from massive stars is a microscopic process operating in all galaxies to a certain degree . the data emphasize the importance of local conditions in regulating star formation from evidence such as massive stars inside ionized shells , compact hii regions surrounding aging clusters , and stars formed in chains of progressing age . surface brightness profiles show current activity correlates with the time averaged level of past star formation at a given radius demonstrating a reliance on local conditions . large - scale triggering by hi shells is supported by observations of progenitor populations as well as secondary sites of star formation associated with their dense rims . analysis of the energy available from massive stars inside hi shells indicates energy deposited into the ism from supernovae and stellar winds is sufficient to account for the hi morphology . ages of individual star forming regions are derived using b , h , and fuv photometry and show both older , diffuse fuv regions and younger , compact hii regions . the distribution of ages is reconciled with the hi morphology , showing a clear preference of young regions for areas of dense hi and old regions for hi voids . global kinematical properties may also play a role in the star formation process since differences in the rotation characteristics of the neutral gas disk correlate with differences in triggering mechanisms . large - scale feedback from massive stars is shown to operate in regions that lack differential shear in the gas disk . | far - ultraviolet ( fuv ) , h , and hi observations of dwarf galaxy holmberg ii are used to investigate the means by which star formation propagates in galaxies lacking global internal triggering mechanisms such as spiral density waves . the observations trace the interaction between sites of massive star formation and the neutral and ionized components of the surrounding ism in this intrinsically simple system . both local and large - scale triggering mechanisms related to massive star formation are seen , suggesting that feedback from massive stars is a microscopic process operating in all galaxies to a certain degree . the data emphasize the importance of local conditions in regulating star formation from evidence such as massive stars inside ionized shells , compact hii regions surrounding aging clusters , and stars formed in chains of progressing age . surface brightness profiles show current activity correlates with the time averaged level of past star formation at a given radius demonstrating a reliance on local conditions . large - scale triggering by hi shells is supported by observations of progenitor populations as well as secondary sites of star formation associated with their dense rims . analysis of the energy available from massive stars inside hi shells indicates energy deposited into the ism from supernovae and stellar winds is sufficient to account for the hi morphology . ages of individual star forming regions are derived using b , h , and fuv photometry and show both older , diffuse fuv regions and younger , compact hii regions . the distribution of ages is reconciled with the hi morphology , showing a clear preference of young regions for areas of dense hi and old regions for hi voids . global kinematical properties may also play a role in the star formation process since differences in the rotation characteristics of the neutral gas disk correlate with differences in triggering mechanisms . large - scale feedback from massive stars is shown to operate in regions that lack differential shear in the gas disk . |
astro-ph0310158 | r | figure [ all ] shows from top to bottom the integral tev flux , the x - ray flux at 10 kev , the 3 - 20 kev x - ray photon index , the v , r , and i band optical data , and the 14.5 ghz and 4.8 ghz radio data . the tev @xmath0-ray data ( fig . [ all]a ) show several strong flares during the first 20 days of the campaign with a flux surpassing 2 crab units on may 1720 ( mjd 5241152414 ) and again roughly two weeks later on june 4 ( mjd 52429 ) . subsequently , the flux leveled off to about 0.3 crab units with the exception of two flares on july 1112 ( mjd 52466 - 52467 ) and july 1415 ( mjd 52469 - 52470 ) with a flux between 1 and 1.5 crab units . holder et al . ( 2002 ) studied the fastest @xmath0-ray flux variability time scales based on the whipple data and found a rapid flux increase with an @xmath33-folding time of 10 hrs . the large `` gaps '' in the @xmath0-ray lightcurves originate from the fact that the cherenkov telescopes are operated during moonless nights only . the 10 kev x - ray flux ( fig . [ all]b , table [ xflux ] ) was strongest on may 18 - 20 ( mjd 52412 - 52414 ) . it slowly decreased by a factor of 18.7 from the maximum on may 20 to a minimum on june 17 ( mjd 52442 ) . as we will discuss in more detail in the next section , the tev @xmath0-ray and x - ray fluxes seem to be correlated , with the notable exception of an `` orphan '' tev @xmath0-ray flare on june 4 , 2002 ( mjd 52429 ) that is not associated with increased x - ray activity . from july 17 ( mjd 52469 ) until the end of the campaign , the x - ray flux stayed at a consistently high level : a factor of 1.7 below the maximum flux observed at the beginning of the campaign , and a factor of 11.5 above the minimum flux measured on june 17 ( mjd 52442 ) . the tev emission level during this `` x - ray plateau state '' is about a factor of 2 lower than at similar x - ray flux levels earlier in the campaign . we analyzed the x - ray flux variability time scale by computing the @xmath33-folding times from the flux changes between observations : @xmath34 @xmath35 @xmath36 with @xmath37 being the time difference between two observations and @xmath38 is the difference of the logarithms of the 10 kev fluxes . the shortest @xmath33-folding times are given in table [ inc ] . we detected faster flux increases than flux decreases : the fastest flux increase has an @xmath33-folding time of @xmath395.9 hrs ; the fastest flux decrease has an @xmath33-folding time of @xmath3915.2 hrs . the 3 - 25 kev photon index @xmath40 ( @xmath41 ) ( fig . [ all]c , table [ xflux ] ) varies between 1.6 and 2.4 . the x - ray photon index and the x - ray flux are clearly correlated , higher flux corresponding to harder energy spectrum . values well below and well above the value of @xmath26 2 show that the low - energy ( presumably synchrotron ) component sometimes peaked above 10 kev , and sometimes below 10 kev . we searched for rapid spectral changes by analyzing photon index variations between _ rxte _ observations , see table [ hard ] . the photon index @xmath40 decreased ( spectral hardening ) by up to 0.09 hr@xmath10 and increased ( spectral softening ) by up to 0.04 hr@xmath10 . as a consequence of synchrotron cooling which is more efficient at higher energies , leptonic models predict that the x - ray emission is harder during the rising phase of a flare than during its decaying phase @xcite . careful inspection of the x - ray lightcurve and the photon indices does not show evidence for such a behavior . while the detection of this effect would impose a constraint on the jet magnetic field and the doppler factor , the non - detection allows large regions in the @xmath4 - @xmath5 plane @xcite . the v , r , and i band optical data ( fig . [ all]d - f ) show flux variations of about 0.1 mag on typical time scales of about 10 days . remarkably , the mean optical brightness increased from the first 4 weeks to the last 2 weeks of the campaign by about 0.1 mag in all three optical bands . both , the optical and the x - ray fluxes increased slowly during the campaign . apart from this joint slow flux increase , we did not find any evidence for a correlation between the optical and the x - ray or the tev @xmath0-ray fluxes . we searched for optical intra - day flux variability by fitting models to the data of individual days . although we performed very long observations of up to @xmath397 hrs per night with small statistical errors of 0.02 mag per 10 minute exposure time , the reduced chi - square values did not show any evidence for statistically significant intra - day flux variability . based on the diurnal brightness averages in the three bands , we computed fastest rise and decay @xmath33-folding times of 0.07 mag day@xmath10 and 0.03 mag day@xmath10 , respectively . within the statistical errors , the v - r and v - i colors stay constant throughout the full campaign . the 14.5 ghz and 4.8 ghz radio data taken with the umrao ( fig . [ all]g - h ) do not show significant flux variations . a fit of a constant flux level to the 14.5 ghz data gives a mean of 0.174@xmath210.004 jy with a chi - square value of 24.5 for 21 degrees of freedom ( chance probability of 27% ) . the mean flux is consistent with the 14.965 ghz flux of 0.18@xmath210.01 jy measured with the vla on may 7 . a fit of a constant flux level to the 4.8 ghz umrao data gives a mean of 0.254@xmath210.016 jy with a chi - square value of 7.1 for 7 degrees of freedom ( chance probability of 42% ) . the mean 4.8 ghz flux is compatible with the 4.885 ghz flux measured on june 7 with the vla of 0.23@xmath210.01 jy , and with 4.85 ghz values of 0.253@xmath210.023 jy and 0.246@xmath210.037 jy reported by gregory & condon ( 1991 ) and becker & white ( 1991 ) , respectively . | -ray telescopes revealed several strong flares , enabling us to sensitively test the x - ray/-ray flux correlation properties . although the x - ray and-ray fluxes seemed to be correlated in general , we found an `` orphan ''-ray flare that was not accompanied by an x - ray flare . while we detected optical flux variability with the boltwood and abastumani observatories , the data did not give evidence for a correlation between the optical flux variability with the observed x - ray and-ray flares . within statistical errors of about 0.03 jy at 14.5 ghz and 0.05 jy at 4.8 ghz , the radio fluxes measured with the university of michigan radio astrophysical observatory ( umrao ) stayed constant throughout the campaign ; the mean values agreed well with the values measured on may 7 and june 7 , 2002 at 4.9 ghz and 15 ghz with the very large array ( vla ) , and , at 4.8 ghz with archival flux measurements . after describing in detail the radio , optical , x - ray and-ray light curves and spectral energy distributions ( seds ) we present initial modeling of the sed with a simple synchrotron self - compton ( ssc ) model . with the addition of another tev blazar with good broadband data | following the detection of strong tev-ray flares from the bl lac object 1es 1959 + 650 with the whipple 10 m cherenkov telescope on may 16 and 17 , 2002 , we performed intensive target of opportunity ( too ) radio , optical , x - ray and tev-ray observations from may 18 , 2002 to august 14 , 2002 . observations with the x - ray telescope _ rxte _ _ ( rossi x - ray timing explorer ) _ and the whipple and hegra _ ( high energy gamma ray astronomy ) _ -ray telescopes revealed several strong flares , enabling us to sensitively test the x - ray/-ray flux correlation properties . although the x - ray and-ray fluxes seemed to be correlated in general , we found an `` orphan ''-ray flare that was not accompanied by an x - ray flare . while we detected optical flux variability with the boltwood and abastumani observatories , the data did not give evidence for a correlation between the optical flux variability with the observed x - ray and-ray flares . within statistical errors of about 0.03 jy at 14.5 ghz and 0.05 jy at 4.8 ghz , the radio fluxes measured with the university of michigan radio astrophysical observatory ( umrao ) stayed constant throughout the campaign ; the mean values agreed well with the values measured on may 7 and june 7 , 2002 at 4.9 ghz and 15 ghz with the very large array ( vla ) , and , at 4.8 ghz with archival flux measurements . after describing in detail the radio , optical , x - ray and-ray light curves and spectral energy distributions ( seds ) we present initial modeling of the sed with a simple synchrotron self - compton ( ssc ) model . with the addition of another tev blazar with good broadband data , we consider the set of all tev blazars to begin to look for a connection of the jet properties to the properties of the central accreting black hole thought to drive the jet . remarkably , the temporal and spectral x - ray and-ray emission characteristics of tev blazars are very similar , even though the masses estimates of their central black holes differ by up to one order of magnitude . |
1607.06848 | i | for @xmath0 , let @xmath1 denote the infinite sector of opening @xmath2 , @xmath21 in the present paper , we are interested in the spectral properties of the associated robin laplacian , to be denoted @xmath4 , which is defined as follows : for @xmath9 , the operator @xmath4 acts in @xmath5 as @xmath22 on the functions @xmath23 satisfying the boundary condition @xmath24 where @xmath25 stands for the outer unit normal . more precisely , @xmath26 is defined as the unique self - adjoint operator in @xmath5 corresponding to the quadratic form @xmath27 where @xmath28 is the one - dimensional hausdorff measure . during the last years , the spectral analysis of robin laplacians attracted a considerable attention . as shown in @xcite , the spectral properties of such operators are sensible to the regularity of the boundary , and the case of smooth domains has been studied in detail @xcite . on the other hand , only partial results are available for domains with a non - smooth boundary , cf . @xcite . the study of the above operator @xmath4 has several interesting aspects from the point of view of the existing results . first , it can be viewed as the simplest non - smooth domain in two dimensions and depending in an explicit way of the single geometric parameter @xmath10 . second , its spectral properties play an important role in the study of more general non - smooth domains in the strong coupling limit , see @xcite , as the sectors @xmath1 exhaust the whole family of possible tangent cones to the boundary in two dimensions . third , the domain @xmath1 and its boundary are non - compact , which may potentially lead to quite unusual spectral properties , such as the presence of an infinite discrete spectrum , and the respective studies in higher dimensions @xcite do not extend directly to the planar case : we will see below that the results are actually quite different . in fact , the only spectral result on @xmath4 available in the existing literature is as follows , see @xcite : @xmath29 and for @xmath30 the value indicated is an eigenvalue with an explicitly known eigenfunction @xmath31 . the aim of the present work is to provide a more detailed spectral analysis . our main results are as follows . first , as the essential spectrum of @xmath4 does not depend on the angle @xmath10 and equals @xmath11 , see theorem [ spec ] , it follows from that the discrete spectrum is non - empty if and only if @xmath30 , i.e. if and only if the sector is strictly smaller than the half - plane . in theorem [ thm31 ] we show that the discrete spectrum is always finite , which is a non - trivial result due to the non - compactness of the boundary . in subsection [ monotone ] we obtain more detailed results : in theorem [ increase ] we show that each individual eigenvalue is a strictly increasing continuous function of the angle @xmath10 and , moreover , that there is just one discrete eigenvalue for @xmath13 , see theorem [ thmpi6 ] . in section [ small1 ] we discuss the behavior of the discrete eigenvalues for small @xmath10 . we show that the @xmath17th eigenvalue @xmath18 behaves as @xmath32 see corollary [ corol43 ] , and , moreover , admits an asymptotic expansion up to any order with respect to the powers of @xmath20 , see theorem [ full ] . the number of discrete eigenvalues becomes arbitrary large and is minorated by @xmath15 , @xmath16 , as @xmath10 approaches @xmath14 , see corollary [ cor42 ] . in theorem [ agmon ] we show that the associated eigenfunctions are localized , in a suitable sense , near the vertex . our proofs are mostly variational and based on the min - max characterization of the essential spectrum and the eigenvalues . the proof of the finiteness of the discrete spectrum uses an idea proposed in @xcite for a different operator involving a similar geometry , while the continuity and the monotonicity of the eigenvalues are established using a suitable change of variables . the asymptotics for small @xmath10 is based on the well - known born - oppenheimer strategy @xcite and , similar to various problems involving small parameters @xcite , on a reduction to a one - dimensional effective operator , which in our case acts in @xmath33 as @xmath34 with a suitable boundary condition at the origin , while an additional work is required due to the singularity of the potential . we remark that the use of an improved hardy inequality @xcite allows one to perform the reduction in a rather direct way , without any preliminary localization argument for the associated eigenfunctions . the properties of the eigenfunctions are studied with the help of the standard agmon - type approach using a suitable decomposition of the domain . we remark that in the present paper we are not discussing the ( non-)existence of eigenvalues embedded into the continuous spectrum . some partial information can be easily obtained , for example , the recent rellich - type result @xcite implies the absence of positive embedded eigenvalues for @xmath35 , but a specific separate study is needed in order to cover all possible cases . this will be discussed elsewhere . | for , let denote the infinite planar sector of opening , and be the laplacian in , , with the robin boundary condition , where stands for the outer normal derivative and . the essential spectrum of does not depend on the angle and equals , and the discrete spectrum is non - empty iff . in this case we show that the discrete spectrum is always finite and that each individual eigenvalue is a continous strictly increasing function of the angle . in particular , there is just one discrete eigenvalue for . as approaches , the number of discrete eigenvalues becomes arbitrary large and is minorated by with a suitable , and theth eigenvalue of behaves as and admits a full asymptotic expansion in powers of . the eigenfunctions are exponentially localized near the origin . | for , let denote the infinite planar sector of opening , and be the laplacian in , , with the robin boundary condition , where stands for the outer normal derivative and . the essential spectrum of does not depend on the angle and equals , and the discrete spectrum is non - empty iff . in this case we show that the discrete spectrum is always finite and that each individual eigenvalue is a continous strictly increasing function of the angle . in particular , there is just one discrete eigenvalue for . as approaches , the number of discrete eigenvalues becomes arbitrary large and is minorated by with a suitable , and theth eigenvalue of behaves as and admits a full asymptotic expansion in powers of . the eigenfunctions are exponentially localized near the origin . |
1006.5928 | i | for a polytope @xmath6 of dimension @xmath4 , the _ face vector _ of @xmath6 is given by @xmath7 where @xmath8 is the number of @xmath9-dimensional faces of @xmath6 . it is an ongoing investigation to describe the face vectors for @xmath4-dimensional convex polytopes . the case @xmath10 are the only cases where this description is complete . a nice theorem by steinitz @xcite states that an integer vector @xmath11 is a face vector for a three dimensional polytope @xmath6 if and only if ( i ) we have the euler relation @xmath12 , ( ii ) @xmath13 and ( iii ) @xmath14 . moreover , the first inequality is tight if and only if @xmath6 is simplicial , the second one is tight if and only if @xmath6 is simple . the only linear equation satisfied by the entries of the face vector is the euler - poincar relation @xmath15 , that holds for all polytopes of dimension @xmath4 . for higher dimensions @xmath16 no such nice description of face vectors has been found yet . in the hope of obtaining a description of a polytope in terms of parameters / entries of a vector , there is the _ flag vector _ @xmath17 , containing @xmath18 entries where for each @xmath19 the entry @xmath20 denotes the number of strict chains ( or _ flags _ ) @xmath21 of faces of @xmath6 where @xmath22 . the entries @xmath20 do satisfy the generalized dehn - sommerville equations which , together with the existence of matching polytopes , implies that the linear span of flag vectors of @xmath4-dimensional polytopes is precisely @xmath23 , the @xmath4-th fibonacci number , as shown in the seminal paper @xcite . one way to encode the information contained in the flag vector of a @xmath4-polytope @xmath6 is to define the @xmath24-index , a linear combination of non - commuting monomials in @xmath25 and @xmath26 of degree @xmath4 in the following way : for @xmath27 let @xmath28 , where @xmath29 if @xmath30 and @xmath31 if @xmath32 . the _ @xmath24-index _ @xmath33 for @xmath6 is then defined as @xmath34 , where the sum is over all @xmath27 and @xmath35 . an elegant consequence of the generalized dehn - sommerville equations for the entries @xmath20 is that each @xmath24-index can be written as a non - commuting polynomial in @xmath36 and @xmath37 . the resulting polynomial called the _ @xmath38-index _ , also denoted @xmath33 and first appeared in @xcite . within the free non - commuting @xmath39-algebra @xmath40 in two variables @xmath25 and @xmath26 , the degree of @xmath41 is set to be 1 and the degree of @xmath42 is set to be two . in this way the number of all homogeneous monomials of degree @xmath4 in @xmath41 and @xmath42 is precisely @xmath23 . hence , the @xmath38-index is the most compact way to encode all the entries of the flag vector of a polytope @xmath6 w.r.t . linear relations of the coefficients of the monomial in @xmath41 and @xmath42 . a lot of current research is therefore directed toward understanding the @xmath38-index for polytopes . although the @xmath38-index is an elegant and compact way to encode the flag vector for a polytope @xmath6 , it can be at times difficult to handle in many explicit computations since algebraic manipulation in @xmath43 can become particularly cumbersome . the purpose of this article is to do the opposite of the above : namely , instead of attempting to compactify the encoding of the flag vector further , then we shall instead relax the constrictions and allow some redundancies among coefficients in order to facilitate direct computations . this will in certain cases allow us to write down explicit ( albeit involved ) formulae from which the entries of the flag vector can be obtained . we will define the _ flag polynomial _ for a polytope , a polynomial in commuting variables that will behave reasonably well when constructing polytopes by known methods i.e. prisms , pyramids , products , free joins and minkowski sums . the article is organized as follows : in section [ sec : basic ] we present our basic definitions and write some direct consequences . in section [ sec : flag - poly ] we state ( but do not prove ) theorem [ thm : flag - main ] , the main result of the section ; a formula for the flag polynomial of a minkowski sum of @xmath0 standard simplices . we then parse through an explicit application of theorem [ thm : flag - main ] and note some consequences . section [ sec : join ] is devoted to the derivation of theorem [ thm : p - acts - flag ] ; a formula for the flag polynomial of a polytope @xmath44 , a polytope obtained when the geometric polytope @xmath6 in @xmath45 acts on the polytopes @xmath46 . this action generalizes the notion of the free join of two polytopes . finally , we note that theorem [ thm : flag - main ] is a corollary of theorem [ thm : p - acts - flag ] . in the last section [ sec : f - poly-2 ] we present two examples on the application of the flag polynomial . in the first example we compute the maximum number of faces of an arbitrary minkowski sum of two simplices . in the latter example we compute the maximum discrepancy between the number of @xmath5-chains of faces of a minkowski sum of two simplices and the number of @xmath5-chains of faces of a simple polytope of the same dimension and on the same number of vertices as the minkowski sum . this article can be viewed as a self contained continuation of @xcite . for related references and history of the significance of minkowski sums of simplices we refer to the introduction of @xcite . also , for self - containment we will present definition [ def : master ] and some terminology from @xcite . for @xmath47 we let @xmath48 = \{1,2,\ldots , r\}$ ] . the _ standard simplex _ @xmath49}$ ] of dimension @xmath50 is given by @xmath51 } = \ { \tilde{x } = ( x_1,\ldots , x_r)\in { { \mbox{$\mathbb r$}}}^r : x_i\geq 0 \mbox { for all $ i$ } , x_1+\cdots+x_r = 1\}. $ ] each subset @xmath52 $ ] yields a _ face _ @xmath53 of @xmath54}$ ] given by @xmath55 } : x_i = 0 \mbox { for } i\not\in f\}. $ ] clearly @xmath53 is itself a simplex embedded in @xmath56 . if @xmath57 is a family of subsets of @xmath48 $ ] , then we can form the _ minkowski sum _ of simplices @xmath58 note that a face of any polytope @xmath59 ( in particular of @xmath54}$ ] ) is given by the set of points that maximize a linear functional @xmath60 for @xmath61 . recall the @xmath3-polynomial of a polytope : let @xmath6 be a @xmath4-dimensional polytope . for each @xmath62 let @xmath63 denote the set of all @xmath9-dimensional faces of @xmath6 and so @xmath64 is the number of @xmath9-dimensional faces of @xmath6 . then @xmath65 is the set of all the faces of @xmath6 except the empty face . @xmath3-polynomial _ of @xmath6 is then given by @xmath66 . [ obs : f - poly - standard ] the @xmath3-polynomial for the standard @xmath67-dimensional simplex @xmath49}$ ] is given by @xmath68 | where . the flag polynomial facilitates many direct computations . to demonstrate this we provide two examples we then compute the maximum discrepancy between the number of-chains of faces of a minkowski sum of two simplices and the number of such chains of faces of a simple polytope of the same dimension and on the same number of vertices . * 2010 msc : * 05a15 , 05a16 , 52b05 , 52b11 . | for a polytope we define the _ flag polynomial _ , a polynomial in commuting variables related to the well - known flag vector and describe how to express the the flag polynomial of the minkowski sum of standard simplices in a direct and canonical way in terms of the _-th master polytope _ where . the flag polynomial facilitates many direct computations . to demonstrate this we provide two examples ; we first derive a formula for the-polynomial and the maximum number of-dimensional faces of the minkowski sum of two simplices . we then compute the maximum discrepancy between the number of-chains of faces of a minkowski sum of two simplices and the number of such chains of faces of a simple polytope of the same dimension and on the same number of vertices . * 2010 msc : * 05a15 , 05a16 , 52b05 , 52b11 . * keywords : * polytope , minkowski sum , master polytope , flag polynomial ,-polynomial . |
0909.0319 | i | courant algebroids were introduced in @xcite as a way to merge the concept of lie bialgebra and the bracket on @xmath4 first discovered by courant @xcite here @xmath5 is a smooth manifold . roytenberg gave an equivalent definition phrased in terms of the dorfman bracket @xcite , which highlighted the relation of courant algebroids to @xmath6-algebras @xcite observed by roytenberg & weinstein @xcite . despite its importance , the subject suffered from the lack of examples for a long time . one important class of courant algebroids , called exact courant algebroids , was discovered by evera @xcite . a courant algebroid @xmath7 is said to be exact if its underlying vector bundle fits into an exact sequence @xmath8 , where @xmath9 is the anchor map . evera proved that the isomorphism classes of exact courant algebroids are classified by a degree-@xmath0 class in the de rham cohomology of @xmath5 : the evera class . furthermore , the structure of transitive courant algebroids courant algebroids with surjective anchors was described independently by vaisman @xcite , evera ( in a private correspondence with weinstein @xcite ) and bressler @xcite . evera and bressler also classified transitive courant algebroids as extensions of transitive lie algebroids @xcite . indeed evera also outlined some nice ideas of classification of transitive courant algebroids in @xcite . in @xcite , the second and third authors introduced the modular class of courant algebroids , a degree 1 characteristic class in the naive cohomology @xmath10 of the courant algebroid @xmath7 . it is natural to ask whether there is a degree 3 characteristic class for a general courant algebroid resembling the severa class , and if so , what role is played by such a class in the classification of courant algebroids . the main purpose of this paper is to answer these questions for regular courant algebroids , that is , courant algebroids with a constant rank anchor . such courant algebroids @xmath7 are particularly easier to handle , for their associated characteristic distributions @xmath11 do not have any singularity . note that characteristic classes of lie algebroids were studied by evens - lu - weinstein @xcite , crainic and fernandes @xcite . it would be interesting to explore if there is any intrinsic connection between these constructions . to any regular courant algebroid , we will construct a degree 3 class , called the characteristic class , in the naive cohomology @xmath12 of the courant algebroid , an analogue of evera s class . note that when @xmath7 is an exact courant algebroid , @xmath12 is isomorphic to the de rham cohomology @xmath1 . however , unlike the exact case , @xmath12 does not classify regular courant algebroids . the classification problem is much subtler in this case . given a regular courant algebroid @xmath7 with anchor @xmath9 , @xmath13 is clearly a regular lie algebroid , i.e. a lie algebroid with a constant rank anchor . it is called the ample lie algebroid of @xmath7 . the kernel @xmath14 of the anchor of @xmath15 is a bundle of quadratic lie algebras , which satisfies certain compatibility conditions . therefore , the lie algebroid @xmath15 is quadratic ( see definition [ def : quad ] ) . the first natural question is whether every quadratic lie algebroid @xmath15 arises in this way . it turns out that there is an obstruction : the first pontryagin class , an element in @xmath16 naturally associated to any quadratic lie algebroid . here @xmath17 is the image of the anchor ( an integrable subbundle of @xmath18 ) and @xmath16 stands for the leafwise de rham cohomology of @xmath17 . this obstruction is similar to the one described by evera and bressler in the transitive case @xcite . to recover a courant algebroid from a given a quadratic lie algebroid @xmath15 with vanishing first pontryagin class , one needs an extra piece of data : a coherent form @xmath19 . this is a closed @xmath0-form on the lie algebroid @xmath15 satisfying certain compatibility conditions . in this case @xmath20 is called a characteristic pair . an equivalence is introduced on characteristic pairs . let @xmath21 denote the cohomology groups of the subcomplex @xmath22 of @xmath23 , where @xmath24 is made of the sections of @xmath25 which are annihilated by all sections of @xmath26 . roughly speaking , two coherent forms @xmath27 and @xmath28 on @xmath15 are equivalent if and only if the class of @xmath29 in @xmath30 is zero . we prove that there is a one - to - one correspondence between regular courant algebroids up to isomorphisms and equivalence classes of characteristic pairs . the inclusion @xmath31 is a lie algebroid morphism . therefore , it induces a morphism @xmath32 . we denote the class of the well - known cartan @xmath0-form of lie theory in @xmath33 by @xmath34 . the main result of the paper can be summarized as the following 1 . there is a natural map from regular courant algebroids to quadratic lie algebroids with vanishing first pontryagin class . 2 . for any quadratic lie algebroid @xmath15 with vanishing first pontryagin class , the isomorphism classes of courant algebroids whose ample lie algebroids are isomorphic to @xmath15 are parametrized by @xmath35 . here @xmath36 is a certain abelian subgroup of @xmath37 and @xmath38 denotes the anchor of @xmath15 . 3 . for any regular courant algebroid @xmath7 , there is a degree 3 characteristic class , which is a cohomology class in @xmath39 . here @xmath15 is the ample lie algebroid of @xmath7 , and @xmath40 is the cartan @xmath0-form . hence , we have the `` exact sequence '' @xmath41 @xmath42 . to this day , little is known about courant algebroid cohomology @xcite : roytenberg computed it for @xmath43 and ginot & grtzmann for transitive and some very special regular courant algebroids @xcite . our result should be useful for computing the courant algebroid cohomology of arbitrary regular courant algebroids . * acknowledgments * the authors thank camille laurent - gengoux , zhang - ju liu , jim stasheff and alan weinstein for useful discussions and comments . in particular , we are indebted to pavol evera , who provides example [ rmk : severa ] , and called their attention to @xcite , which led to the spotting of an error in an early version of the paper . | this intrinsic invariant of the courant algebroid is a degree- class in its naive cohomology . when the courant algebroid is exact , it reduces to the evera class ( in ) . on the other hand , when the courant algebroid is a quadratic lie algebra , it coincides with the class of the cartan-form ( in ) . we also give a complete classification of regular courant algebroids and discuss its relation to the characteristic class . | for any regular courant algebroid , we construct a characteristic class _ la _ chern - weil . this intrinsic invariant of the courant algebroid is a degree- class in its naive cohomology . when the courant algebroid is exact , it reduces to the evera class ( in ) . on the other hand , when the courant algebroid is a quadratic lie algebra , it coincides with the class of the cartan-form ( in ) . we also give a complete classification of regular courant algebroids and discuss its relation to the characteristic class . |
1209.2225 | i | x - ray spectroscopy @xcite is a powerful spectroscopic tool for the elucidation of structural and electronic properties of materials @xcite and ( bio-)molecular systems@xcite . x - ray absorption spectroscopy ( xas ) probes the excitation of core electrons . here , one distinguishes excitations to low - lying unoccupied states ( so - called prepeaks ) , excitations to states close to the ionization threshold ( x - ray absorption near - edge structure , xanes ) , and excitations to continuum states ( extended x - ray absorption fine structure , exafs ) . on the other hand , x - ray emission spectroscopy ( xes ) considers the emission of x - ray radiation after the formation of a core hole . of particular interest are the applications of these techniques to study catalytic mechanisms _ in situ _ ( for examples , see , ref . @xcite ) and to investigate biological or biomimetic systems containing transition metal centers ( see , e.g. , refs . usually , xas and xes are used as fingerprint techniques in such studies to identify the oxidation state , spin state , and local coordination environment of a metal center . this requires either the comparison with spectra measured for model compounds or with theoretical predictions . to be able to extract additional information from x - ray spectroscopic measurements , the development of theoretical methods for the calculation of x - ray spectra is essential @xcite . for exafs spectra , approaches based on scattering theory are well established and make it possible to extract structural parameters such as distances and coordination numbers @xcite . in contrast , for describing prepeaks and the xanes region in xas spectroscopy and for predicting xes spectra , quantum - chemical approaches are usually required . to this end , a wide range of quantum - chemical methods have been developed for describing excitations from core orbitals ( for reviews , see , e.g. , refs @xcite ) . widely used are the static - exchange approximation ( stex ) @xcite , approaches based on transition potential density - functional theory ( dft ) @xcite , and time - dependent density - functional theory ( td - dft ) @xcite . within td - dft , core excitations are accessible by restricting the space of occupied virtual orbital excitations ( restricted - channel approximation ) @xcite , by selectively targeting excitations within a specific energy window @xcite , by using a complex polarization propagator @xcite , or with real - time td - dft methods @xcite . recently , coupled - cluster response theory has also been extended to x - ray spectroscopy @xcite . by combining x - ray spectroscopy with quantum - chemical calculations , it becomes possible to extract information on the electronic structure of molecular systems . for instance , the analysis of the prepeak intensities in ligand k - edge xas spectra of transition metal complexes ( i.e. , excitations from the ligand @xmath0 to metal @xmath1 orbitals ) provides insights into the covalent contributions to metal ligand bonding @xcite . another example is metal k - edge xas , probing excitations from metal @xmath0 to @xmath1 orbitals , which can be used to assign coordination numbers @xcite and to probe details of the metal ligand bonding mechanisms @xcite . recently , we have demonstrated that the prepeaks in fe k - edge xas spectra of ferrocene derivatives are sensitive to subtle differences in the electronic structure at the iron atom , which are induced by substituents at the cyclopentadienyl rings , i.e. , beyond the first coordination shell of the metal center @xcite . such studies are facilitated by high - energy resolution fluorescence detection ( herfd ) techniques , which can resolve the prepeaks with a much higher resolution than conventional xas measurements @xcite . however , beside the challenges posed by the accurate quantum - chemical prediction of the absolute or relative energies of core excitations , for the prepeaks in k - edge xas spectra , theoretical x - ray spectroscopy faces an additional problem . for the calculation of xas intensities for hard x - rays , the well - known dipole approximation , in which the oscillator strengths are proportional to the square of the electric - dipole transition moments is not sufficient . the dipole approximation is based on the assumption that the wavelength of the electromagnetic radiation is large compared to the size of the core orbital . for the high - energy radiation used in hard x - ray spectroscopy , this is not the case anymore . this is particularly important for the prepeaks in metal k - edge xas spectra of transition metal complexes , which are usually dipole forbidden or have a very low dipole intensity due to symmetry . thus , the intensity of these prepeaks is due to contributions that are not included in the dipole approximation @xcite . currently , contributions to xas or xes intensities beyond the dipole approximation , so - called quadrupole intensities , are calculated by including additional contributions that are proportional to the squares of the electric - quadrupole and the magnetic - dipole transition moments @xcite . however , these additional contributions depend on the choice of the origin of the coordinate system . this situation is not satisfactory , as a physical observable should be origin independent . to rectify this , neese and coworkers suggested to choose the origin differently for each excitation such that these additional contributions are minimized @xcite . usually , this is equivalent to placing the origin on the atom at which the excitation takes place . however , in cases where the dipole intensities are very small compared to the quadrupole intensities , this scheme sometimes places the origin far away from the excited core orbital @xcite , which seems unphysical and affects the resulting intensities significantly . moreover , the scheme will also fail for excitations from core orbitals that are delocalized over different atomic centers , a situation which occurs for ligand - edge xas spectra or for metal k - edge spectra in polynuclear transition metal clusters . thus , a theoretical framework for the origin - independent calculation of quadrupole intensities in x - ray spectroscopy would be desirable . here , we show that such a formulation can be obtained if all contributions to the oscillator strengths that are of the same order in the wave vector are included consistently . this work is organized as follows . the theory is presented in section [ sec : theory ] . after introducing the theoretical framework in sections [ sec : radiation ] and [ sec : qm ] , we revisit the multipole expansion of the transition moments in section [ sec : multipole ] . subsequently , in section [ sec : osc - strengths ] this expansion is applied for the calculation of the oscillator strengths and we demonstrate that these become origin - independent if all terms that are of the same order are included consistently . the final equations for the isotropically averaged quadrupole intensities are then derived in section [ sec : average ] . this is followed by a description of our implementation of the resulting formalism within td - dft in section [ sec : compdet ] , before we illustrate its usefulness for two test cases in section [ sec : results ] . finally a summary and concluding remarks are given in sect . [ sec : conclusion ] | for electronic excitations in the ultraviolet and visible range of the electromagnetic spectrum , the intensities are usually calculated within the dipole approximation , which assumes that the oscillating electric field is constant over the length scale of the transition . for the short wavelengths used in hard x - ray spectroscopy , the dipole approximation may not be adequate . have so far been calculated by including contributions depending on the square of the electric - quadrupole and magnetic - dipole transition moments . 1a , 76131 karlsruhe , germany date : = november 27 , 2012 + status : published in _ j. chem . phys . _ * 137 * , 204106 ( 2012 ) . + | for electronic excitations in the ultraviolet and visible range of the electromagnetic spectrum , the intensities are usually calculated within the dipole approximation , which assumes that the oscillating electric field is constant over the length scale of the transition . for the short wavelengths used in hard x - ray spectroscopy , the dipole approximation may not be adequate . in particular , for metal k - edge x - ray absorption spectroscopy ( xas ) , it becomes necessary to include higher - order contributions . in quantum - chemical approaches to x - ray spectroscopy , these so - called quadrupole intensities have so far been calculated by including contributions depending on the square of the electric - quadrupole and magnetic - dipole transition moments . however , the resulting quadrupole intensities depend on the choice of the origin of the coordinate system . here , we show that for obtaining an origin - independent theory , one has to include all contributions that are of the same order in the wave vector consistently . this leads to two additional contributions depending on products of the electric - dipole andelectric - octupole and of the electric - dipole and magnetic - quadrupole transition moments , respectively . we have implemented such an origin - independent calculation of quadrupole intensities in xas within time - dependent density - functional theory , and demonstrate its usefulness for the calculation of metal and ligand k - edge xas spectra of transition metal complexes . * origin - independent calculation of quadrupole intensities in x - ray spectroscopy + * stephan bernadotte , andrew j. atkins , and christoph r. jacob , + karlsruhe institute of technology ( kit ) , + center for functional nanostructures and institute of physical chemistry , + wolfgang - gaede - str . 1a , 76131 karlsruhe , germany date : = november 27 , 2012 + status : published in _ j. chem . phys . _ * 137 * , 204106 ( 2012 ) . + doi : http://dx.doi.org/10.1063/1.4766359 |
1209.2225 | r | to illustrate the origin - independent calculation of quadrupole intensities in x - ray absorption spectroscopy ( xas ) using the theory derived above and to verify our implementation , we consider two test cases . as the first example , we use titaniumtetrachloride ticl@xmath162 ( see fig . [ fig : mol]a for the molecular structure ) and calculate the cl k - edge xas spectrum . this example was considered earlier in refs . @xcite . for such ligand k - edge spectra , the prepeak transitions are dipole - allowed , and the second - order contribution to the oscillator strength should be small compared to the dipole contribution . for the lowest - energy cl k - edge excitation , the different contributions to the isotropically averaged oscillator strengths are calculated using eqs , and are listed in table [ tab : cl ] for different choices of the origin . in addition , we included the oscillator strengths calculated using the approximation of ref . @xcite , i.e. , considering only the electric - dipole electric - dipole , electric - quadrupole electric - quadrupole , and the magnetic - dipole magnetic - dipole contributions [ cf . eq . ] as well as the full second - order oscillator strengths @xmath163 [ cf . eq . ] . the most natural choice for the origin is the chlorine atom from which the @xmath0-electron is excited . in this case , the electric - dipole electric - dipole contribution @xmath164 to the oscillator strength is several orders of magnitude larger than all the second - order contributions , and the approximation of ref . @xcite gives results that are identical to the full second - order oscillator strengths . in this example , the scheme suggested in ref . @xcite to choose the origin such that the sum of the electric - quadrupole electric - quadrupole @xmath165 and the magnetic - dipole magnetic - dipole contributions @xmath166 is minimized leads to an almost identical choice of the origin . thus , this scheme is appropriate here . the situation changes if the origin is not placed at the chlorine atom . to demonstrate this , we moved the origin to the titanium atom . now , the electric - quadrupole electric - quadrupole @xmath165 and the magnetic - dipole magnetic - dipole contributions @xmath166 increase significantly and become several times larger than the dipole oscillator strength @xmath164 . as a consequence , within the approximation of ref . @xcite the oscillator strength increases by more than a factor of two when shifting the origin from the chlorine to the titanium atom . however , also the magnitudes of electric - dipole electric - octupole and the electric - dipole magnetic - quadrupole contributions , @xmath167 and @xmath168 , increase and since these have a negative sign , they exactly cancel the increase of @xmath165 and @xmath166 . thus , the full second - order oscillator strength remains unchanged . in addition , we also shifted the origin away from the molecule by larger amounts . in particular , we used shifts of 10 , 50 , and 100 along the negative @xmath60-direction . here , a similar observation can be made . the electric - quadrupole electric - quadrupole , @xmath165 , and the magnetic - dipole magnetic - dipole , @xmath166 , contributions increase substantially , and for a shift of 100 , the oscillator strength within the approximation of ref . @xcite is four orders of magnitude larger than for the origin at the chlorine atom . on the other hand , when including the electric - dipole electric - octupole and the electric - dipole magnetic - quadrupole contributions , the full second - order oscillator strengths are unchanged , even though the individual contributions differ . cggggg origin & & & & & + & & & & & + @xmath169 & 4.57 10 ^ -4 & 4.57 10 ^ -4 & 4.58 10 ^ -4 & 4.57 10 ^ -4 & 4.58 10 ^ -4 + @xmath170 & 6.04 10 ^ -10 & 7.24 10 ^ -4 & 1.09 10 ^ -2 & 3.65 10 ^ -1 & 1.35 + @xmath171 & 6.61 10 ^ -13 & 1.21 10 ^ -3 & 1.80 10 ^ -2 & 3.58 10 ^ -1 & 2.23 + @xmath172 & -8.94 10 ^ -7 & -3.23 10 ^ -4 & -4.89 10 ^ -3 & -2.46 10 ^ -1 & -6.03 10 ^ -1 + @xmath173 & -9.88 10 ^ -7 & -1.61 10 ^ -3 & -2.40 10 ^ -2 & -4.77 10 ^ -1 & -2.97 + @xmath174 & 4.57 10 ^ -4 & 2.39 10 ^ -3 & 2.94 10 ^ -2 & 7.24 10 ^ -1 & 3.58 + full @xmath175 & 4.55 10 ^ -4 & 4.55 10 ^ -4 & 4.55 10 ^ -4 & 4.55 10 ^ -4 & 4.55 10 ^ -4 + as a second example , we consider vinylferrocene , which is a ferrocene molecule bearing a vinyl substituent at one of the cyclopentadienyl rings ( see fig . [ fig : mol]b for the molecular structure ) . here , we consider the fe k - edge xas spectrum and specifically the lowest - energy ( prepeak ) excitation , which is a @xmath176 transition . in unsubstituted ferrocene , this prepeak excitation is dipole - forbidden for symmetry reasons , and its oscillator strength is solely due to the second - order contributions . in this case , the electric - quadrupole electric - quadrupole and the magnetic - dipole - magnetic - dipole contributions become origin independent ( see section [ sec : shift - equation ] ) , whereas the remaining second - order contributions @xmath167 and @xmath168 vanish . however , in vinylferrocene this symmetry is lost and the lowest - energy transition gains a small dipole oscillator strength ( for a detailed discussion , see ref . @xcite ) . the oscillator strengths and their contributions calculated for the lowest - energy fe k - edge excitation using different choices of the origin are shown in table [ tab : fe ] . first , the most natural choice for the origin is the iron atom . in this case , the electric - dipole electric - dipole and the electric - quadrupole electric - quadrupole contributions to the oscillator strength are comparable in size . the remaining contributions are orders of magnitude smaller . therefore , the oscillator strength calculated with the approximation of ref . @xcite is identical to the full second - order oscillator strength . to investigate the dependence on the origin , we shifted the origin far away from the molecule using a shift of 100 in the negative @xmath60-direction , a shift of 100 in the negative @xmath62-direction , and a shift of 50 in both the negative @xmath60-direction and the negative @xmath62-direction . in all three cases , the electric - quadrupole electric - quadrupole and the magnetic - dipole magnetic - dipole contributions , @xmath165 and @xmath166 , increase by several orders of magnitude compared to the calculation in which the origin is placed at the iron atom . as a result , the oscillator strengths calculated with the approximation of ref . @xcite also increase by up to five orders of magnitude . however , at the same time the two remaining second - order contributions , i.e. , the electric - dipole electric - octupole contribution @xmath167 and the electric - dipole magnetic - quadrupole contribution @xmath168 , assume large negative values and exactly cancel the increase of @xmath165 and @xmath166 such that the total second - order oscillator strength remains origin independent . finally , we used the scheme suggested in ref . @xcite for fixing the origin of the coordinate system , i.e. , we chose the origin such that the sum of @xmath165 and @xmath166 is minimized . in the situation considered here , where the electric - dipole electric - dipole and the electric - quadrupole electric - quadrupole contributions to the oscillator strengths are of similar size , this scheme moves the origin away from the iron atom . the resulting shift is given in the caption of the last column of table [ tab : fe ] . as a consequence , the oscillator strength within the approximation of ref . @xcite decreases by ca . 30 % . again , this decrease is compensated if the remaining second - order contributions are included . thus , the scheme of ref . @xcite can lead to a spurious decrease of the oscillator strength in some cases . previously , we found that this problem is even more severe in cases where the electric - dipole electric - dipole contribution to the oscillator strength is significantly smaller than the quadrupole oscillator strength @xcite . however , if all second - order terms are included consistently the quadrupole oscillator strengths become origin - independent and no special placement of the origin is necessary . cggggg origin & & & & & + & & & & & + & & & & & + & & & & & + @xmath177 & 2.55 10 ^ -6 & 2.55 10 ^ -6 & 2.55 10 ^ -6 & 2.55 10 ^ -6 & 2.55 10 ^ -6 + @xmath178 & 3.09 10 ^ -6 & 8.63 10 ^ -2 & 7.30 10 ^ -2 & 4.47 10 ^ -2 & 1.28 10 ^ -6 + @xmath179 & 1.14 10 ^ -12 & 2.29 10 ^ -2 & 9.04 10 ^ -2 & 5.87 10 ^ -3 & 1.56 10 ^ -7 + @xmath180 & -1.71 10 ^ -8 & -7.83 10 ^ -2 & -4.24 10 ^ -2 & -4.23 10 ^ -2 & 6.33 10 ^ -7 + @xmath181 & -1.52 10 ^ -8 & -3.09 10 ^ -2 & -1.21 10 ^ -1 & -8.19 10 ^ -3 & 9.83 10 ^ -7 + @xmath182 & 5.64 10 ^ -6 & 1.0910 ^ -1 & 1.63 10 ^ -1 & 5.05 10 ^ -2 & 3.99 10 ^ -6 + full @xmath183 & 5.61 10 ^ -6 & 5.61 10 ^ -6 & 5.61 10 ^ -6 & 5.61 10 ^ -6 & 5.61 10 ^ -6 + | * origin - independent calculation of quadrupole intensities in x - ray spectroscopy + * stephan bernadotte , andrew j. atkins , and christoph r. jacob , + karlsruhe institute of technology ( kit ) , + center for functional nanostructures and institute of physical chemistry , + wolfgang - gaede - str . | for electronic excitations in the ultraviolet and visible range of the electromagnetic spectrum , the intensities are usually calculated within the dipole approximation , which assumes that the oscillating electric field is constant over the length scale of the transition . for the short wavelengths used in hard x - ray spectroscopy , the dipole approximation may not be adequate . in particular , for metal k - edge x - ray absorption spectroscopy ( xas ) , it becomes necessary to include higher - order contributions . in quantum - chemical approaches to x - ray spectroscopy , these so - called quadrupole intensities have so far been calculated by including contributions depending on the square of the electric - quadrupole and magnetic - dipole transition moments . however , the resulting quadrupole intensities depend on the choice of the origin of the coordinate system . here , we show that for obtaining an origin - independent theory , one has to include all contributions that are of the same order in the wave vector consistently . this leads to two additional contributions depending on products of the electric - dipole andelectric - octupole and of the electric - dipole and magnetic - quadrupole transition moments , respectively . we have implemented such an origin - independent calculation of quadrupole intensities in xas within time - dependent density - functional theory , and demonstrate its usefulness for the calculation of metal and ligand k - edge xas spectra of transition metal complexes . * origin - independent calculation of quadrupole intensities in x - ray spectroscopy + * stephan bernadotte , andrew j. atkins , and christoph r. jacob , + karlsruhe institute of technology ( kit ) , + center for functional nanostructures and institute of physical chemistry , + wolfgang - gaede - str . 1a , 76131 karlsruhe , germany date : = november 27 , 2012 + status : published in _ j. chem . phys . _ * 137 * , 204106 ( 2012 ) . + doi : http://dx.doi.org/10.1063/1.4766359 |
1209.2225 | c | we have derived origin - independent expressions for calculating xas intensities beyond the dipole approximation . in particular , we have shown that for a consistent formulation , it is necessary to retain all contributions to the oscillator strengths that are of the same order in the wave vector . this differs from the previous approach @xcite , in which the multipole expansion was truncated for the transition moments . here , two additional contributions to the second - order ( quadrupole ) oscillator strengths arise , which are cross - terms depending on products of electric - dipole and electric - octupole transition moments and of electric - dipole and magnetic - quadrupole transition moments , respectively . thus , the origin dependence of the sum of electric - quadrupole electric - quadrupole and magnetic - dipole magnetic - dipole contributions pointed out earlier @xcite is not a fundamental limitation of the use of the multipole expansion . in fact , we could show that to arbitrary order in the wave vector , origin - independent expressions for the oscillator strengths are obtained if all terms of the same order are included consistently . consequently , within the multipole expansion it should always be possible to derive origin - independent expressions for physical observables . an origin - independent formalism for calculating quadrupole intensities is particularly important for studying ligand and metal k - edge xas spectra of transition metal complexes . to this end , we have implemented our theory for calculating xas spectra with td - dft , and applied it to two simple test cases . here , we want to stress that our results do not invalidate any previous results obtained with the formalism of ref . @xcite . on the contrary , our test calculations showed that the two additional contributions are negligible as long as the origin of the coordinate system is placed at the atom where the core excitation occurs . however , with our origin - independent theory , it is no longer necessary to make sure that the origin is chosen appropriately . this is particularly important for cases where the quadrupole intensity is larger than or comparable to the dipole contribution , where the scheme proposed in ref . @xcite might place the origin far away from the relevant core orbital . moreover , it makes it possible to treat excitations from core orbitals that are delocalized over several atoms ( e.g. , for calculating ligand k - edge spectra or metal k - edge spectra in polynuclear transition metal complexes ) without the need to perform a transformation to localized core orbitals . of course , the theory presented here is not limited to td - dft , but can be employed for the calculation of quadrupole intensities in combination with any quantum - chemical method capable of providing the required transition moments . moreover , it is not restricted to xas spectroscopy , but is also applicable for calculating xes intensities , for instance using the approach of ref . finally , we note that it becomes necessary to go beyond the dipole approximation , not only for short wavelengths , such as those employed in hard x - ray spectroscopy , but also for extended molecular systems . for describing the optical response of an extended nanostructure in the visible spectrum , it becomes necessary to go beyond the dipole approximation as well . thus , the origin - independent formalism derived here will also be essential for predicting optical properties of nanostructured materials , such as , for instance , metamaterials @xcite . | this leads to two additional contributions depending on products of the electric - dipole andelectric - octupole and of the electric - dipole and magnetic - quadrupole transition moments , respectively . we have implemented such an origin - independent calculation of quadrupole intensities in xas within time - dependent density - functional theory , and demonstrate its usefulness for the calculation of metal and ligand k - edge xas spectra of transition metal complexes . | for electronic excitations in the ultraviolet and visible range of the electromagnetic spectrum , the intensities are usually calculated within the dipole approximation , which assumes that the oscillating electric field is constant over the length scale of the transition . for the short wavelengths used in hard x - ray spectroscopy , the dipole approximation may not be adequate . in particular , for metal k - edge x - ray absorption spectroscopy ( xas ) , it becomes necessary to include higher - order contributions . in quantum - chemical approaches to x - ray spectroscopy , these so - called quadrupole intensities have so far been calculated by including contributions depending on the square of the electric - quadrupole and magnetic - dipole transition moments . however , the resulting quadrupole intensities depend on the choice of the origin of the coordinate system . here , we show that for obtaining an origin - independent theory , one has to include all contributions that are of the same order in the wave vector consistently . this leads to two additional contributions depending on products of the electric - dipole andelectric - octupole and of the electric - dipole and magnetic - quadrupole transition moments , respectively . we have implemented such an origin - independent calculation of quadrupole intensities in xas within time - dependent density - functional theory , and demonstrate its usefulness for the calculation of metal and ligand k - edge xas spectra of transition metal complexes . * origin - independent calculation of quadrupole intensities in x - ray spectroscopy + * stephan bernadotte , andrew j. atkins , and christoph r. jacob , + karlsruhe institute of technology ( kit ) , + center for functional nanostructures and institute of physical chemistry , + wolfgang - gaede - str . 1a , 76131 karlsruhe , germany date : = november 27 , 2012 + status : published in _ j. chem . phys . _ * 137 * , 204106 ( 2012 ) . + doi : http://dx.doi.org/10.1063/1.4766359 |
1404.0573 | i | we begin by fixing some notation . we assume throughout the paper that @xmath0 is a closed orientable surface of genus @xmath1 . on @xmath0 , there exists a ( hyperbolic ) riemannian metric @xmath2 of constant curvature @xmath3 . the riemannian universal cover of @xmath0 , denoted by @xmath4 , is identified with the poincar disc model , i.e. @xmath5 and @xmath6 , where the geodesics @xmath7 are circle segments meeting @xmath8 orthogonally . let @xmath9 be the distance function on @xmath10 induced by @xmath2 . we write @xmath11 for the group of deck transformations @xmath12 with respect to the covering @xmath13 , which extend to naturally to the `` boundary at infinity '' @xmath14 . let @xmath15 be the set of all oriented , unparametrized geodesics @xmath7 with respect to @xmath2 . we can think of @xmath15 as @xmath16 , associating to @xmath17 its pair of endpoints @xmath18 on @xmath14 . moreover , set @xmath19 we consider any finsler metric @xmath20 , which is assumed to be invariant under @xmath21 ( cf . definition [ def finsler ] ) . we write @xmath22 and @xmath23 for the geodesic with respect to @xmath24 defined by @xmath25 . the main object of interest in this paper are rays and minimal geodesics , that is geodesics @xmath26 , respectively that minimize the length with respect to @xmath24 between any of their points . in @xcite , h. m. morse studied the global behavior of minimal geodesics and showed that minimal geodesics lie in tubes around hyperbolic geodesics @xmath17 of finite width @xmath27 , where @xmath27 is a constant depending only on @xmath24 and @xmath2 . in particular , minimal geodesics @xmath28 have well - defined endpoints @xmath29 at infinity , i.e. on @xmath14 ; the analogous result holds for rays . for @xmath30 set @xmath31 morse studied in particular the behavior of minimal geodesics in @xmath32 , where @xmath33 is fixed under a non - trivial group element @xmath34 ; we will recall these results in subsection [ section periodic ] . however , morse left open the finer structure in the asymptotic directions which are not fixed by an element of @xmath21 and the author is not aware of any other work in the literature in this direction . the purpose of this paper is to fill this gap , i.e. to study the structure of @xmath32 for general @xmath35 . while we are still not able to give the structure for all @xmath33 , we will be able to do it for `` most '' @xmath33 . another novelty in this paper , compared to the work of morse , is the use of finsler instead of riemannian metrics . it was observed by e. m. zaustinsky @xcite , that the results of morse carry over to these much more general systems . moreover , it is known that finsler metrics can be used to describe the dynamics of arbitrary tonelli lagrangian systems in high energy levels , cf . @xcite . the sets @xmath36 are bounded by two particular minimal geodesics . the following lemma is theorem 8 of @xcite . [ bounding geodesics morse ] for @xmath17 , there are two particular non - intersecting , minimal geodesics @xmath37 in @xmath36 , such that all minimal geodesics in @xmath36 lie in the strip in @xmath10 bounded by @xmath37 . as a rule , we will always assume that @xmath38 lies left of @xmath39 with respect to the orientation of @xmath40 . we can now state our first result , saying that for most asymptotic directions @xmath35 , the bounding geodesics @xmath41 of the various @xmath42 do not intersect . the proof makes use of weak kam theory , but does not rely on the group action by @xmath21 . in fact , theorem [ thm countable unstable ] and its corollary [ cor thm countable unstable ] hold for any finsler metric @xmath24 on @xmath10 , which is uniformly equivalent to the norm of @xmath2 ( cf . definition [ def finsler ] ) . [ thm countable unstable ] for all but countably many @xmath35 , the set @xmath43 is a lamination of @xmath10 ( i.e. no curves in @xmath44 intersect transversely ) . it would be desirable to know if the set @xmath44 is a lamination of @xmath10 for all @xmath35 ; we would then gain more insight in the structure of @xmath32 for all @xmath33 , cf . theorem [ thm a ] below . from this we will deduce the following . [ cor thm countable unstable ] for almost all @xmath45 with respect to the lebesgue measure , the set @xmath36 consists of precisely one minimal geodesic . for the statement of our next results , we need the following definitions . a minimal ray @xmath46 is called _ forward unstable _ , if there is no minimal ray @xmath47 with @xmath48 and @xmath49 , which is not a subray of @xmath50 . * instability of minimal geodesics was studied in a paper @xcite of w. klingenberg , and some of our results appear in @xcite . however , @xcite contains errors and the results which we prove here , in particular the existence of unstable geodesics , remained open . our work will be independent of @xcite . * it is easy to see ( cf . lemma [ partial instability bounding ] ) that the unique geodesic in @xmath36 in corollary [ cor thm countable unstable ] is forward ( and backward ) unstable . for @xmath51 define the _ ( forward ) width _ of @xmath52 , respectively by @xmath53 [ bem width def ] due to the morse lemma ( cf . theorem [ morse lemma ] ) , @xmath54 is bounded by the global constant @xmath55 . as we will see , @xmath56 implies that all minimal rays in @xmath32 are forward unstable and in particular that the set @xmath44 in theorem [ thm countable unstable ] is a lamination of @xmath10 , while @xmath57 implies the forward instability of all minimal geodesics in @xmath36 . moreover , @xmath56 implies a uniqueness result for weak kam solutions and busemann functions . hence , the main task in this paper is to show @xmath56 for directions @xmath35 , which are not fixed by elements of @xmath21 . while theorem [ thm countable unstable ] and its corollary did not depend on @xmath21 , the following results rely strongly on the @xmath21-action . assuming a certain behavior of a background geodesic @xmath42 under @xmath21 , we can calculate @xmath54 . [ dense width 0 ] if @xmath58 contains a hyperbolic geodesic , which has a dense forward orbit in the hyperbolic unit tangent bundle of @xmath59 , then @xmath56 . in particular , for almost all @xmath35 with respect to the lebesgue measure , we have @xmath56 . note that theorem [ dense width 0 ] has the following analogue for the case where @xmath0 is the @xmath60-torus @xmath61 ( cf . @xcite for the riemannian , @xcite for the finsler case ) : for all irrational rotation directions ( which have full lebesgue measure in @xmath14 ) , the set of minimal geodesics with this direction is a lamination of the universal cover @xmath62 and the weak kam solutions with irrational directions are unique up to adding constants . for completeness and to exemplify the concept of width , we state the following proposition , which follows directly from the results of morse . [ width periodic intro ] if @xmath63 is an axis of a non - trivial group element @xmath34 ( i.e. @xmath64 ) , then @xmath65 in particular , @xmath66 if and only if there is only one geodesic in @xmath36 . a direct generalization of periodicity is recurrence ; we say that @xmath17 is forward recurrent , if the projection of @xmath67 has a forward recurrent orbit in the hyperbolic unit tangent bundle of @xmath59 . for the case @xmath56 the structure of @xmath32 was explained in remark [ bem width def ] above . if @xmath68 , we have the following results . [ thm intro recurrent ] let @xmath35 and @xmath68 . 1 . there are at most two forward recurrent geodesics in @xmath58 , 2 . if @xmath69 is forward recurrent , then @xmath70 and there is a forward unstable geodesic in @xmath36 , 3 . if @xmath33 satisfies the conclusion of theorem [ thm countable unstable ] ( i.e. if the set of bounding geodesics @xmath44 with direction @xmath33 is a lamination of @xmath10 ) , then there can be at most one forward recurrent direction in @xmath58 and if there is a forward recurrent direction @xmath42 , then @xmath65 if @xmath63 is forward recurrent , then there exists forward unstable minimal geodesic in @xmath36 . if @xmath66 , then all minimal geodesics in @xmath36 are forward unstable by proposition [ w=0 implies unique weak kam ] . if @xmath71 , then the claim follows from theorem [ thm intro recurrent ] ( 2 ) . in mather theory , one studies minimal geodesics @xmath72 ( more generally : action minimizers of lagrangian systems ) , such that @xmath73 is a graph over its projection in @xmath0 . this means that in the universal cover , the curves @xmath74 and @xmath75 are equal or disjoint for all @xmath34 , and @xmath50 is called _ simple_. note that , if a minimal geodesic @xmath50 in @xmath36 is simple , then so is its background geodesic @xmath40 . in this case we have the following result . [ thm simple ] if @xmath58 contains a simple hyperbolic geodesic , which is not the axis of any @xmath76 , then @xmath56 . it is not clear to us whether the conclusion of theorem [ thm countable unstable ] is true for all @xmath35 . if it would hold , then this would clarify the structure of all @xmath32 , as seen in the following theorem . [ thm a ] if @xmath24 is such that for all @xmath35 the set of bounding geodesics @xmath44 with direction @xmath33 defined in theorem [ thm countable unstable ] is a lamination of @xmath10 , then @xmath56 for all @xmath35 that are not fixed under any @xmath76 . the set @xmath44 is a lamination of @xmath10 for all @xmath35 , e.g. , if @xmath24 has no conjugate points , cf . theorem 12.1 in @xcite . in particular , if @xmath24 is riemannian with non - positive curvature , then the only flat strips of uniformly positive width are periodic . by theorem [ thm countable unstable ] , this assumption of theorem [ thm a ] is always `` almost '' true . [ bem high dim ] the main concepts in this paper work in any dimension , in particular the weak kam theory for manifolds of hyperbolic type developed in section [ section weak kam ] and the concept of width in section [ section width ] . e.g. , if in a closed manifold @xmath0 carrying a riemanniam metric @xmath2 of strictly negative curvature there exists a hyperbolic geodesic @xmath40 with respect to @xmath2 , such that in @xmath36 there is only one minimal geodesic , then @xmath56 for almost all @xmath33 in the so - called gromov boundary ( with respect to the lebesgue measure , identifying the gromov boundary with a sphere @xmath77 ) , cf . the arguments for the proof of theorem [ dense width 0 ] . here @xmath54 has to be defined differently , for instance by setting @xmath78 most results , however , are strongest in dimension two , hence we stick to this case to simplicity the exposition . * structure of this paper * in section [ section finsler ] , we recall basic definitions and properties as well as the morse lemma . section [ section weak kam ] is devoted to the proof of theorem [ thm countable unstable ] ; here we study so - called weak kam solutions , which are used to prove theorem [ thm countable unstable ] and its corollary . in section [ section width ] , the concept of width of asymptotic directions is introduced to obtain theorems [ dense width 0 ] , [ width periodic intro ] , [ thm intro recurrent ] , [ thm simple ] and [ thm a ] . in appendix [ appendix ] , we make additional remarks on weak kam solutions in dimension two ; the results of the appendix are , however , not used in this paper . | for finsler metrics ( no reversibility assumed ) on closed orientable surfaces of genus greater than one , we study the dynamics of minimal rays and minimal geodesics in the universal cover . we prove in particular , that for almost all asymptotic directions the minimal rays with these directions laminate the universal cover and that the busemann functions with these directions are unique up to adding constants . | for finsler metrics ( no reversibility assumed ) on closed orientable surfaces of genus greater than one , we study the dynamics of minimal rays and minimal geodesics in the universal cover . we prove in particular , that for almost all asymptotic directions the minimal rays with these directions laminate the universal cover and that the busemann functions with these directions are unique up to adding constants . moreover , using a kind of weak kam theory , we show that for almost all types of minimal geodesics in the sense of morse , there is precisely one minimal geodesic of this type . |
cond-mat0402360 | i | a powerful approach to the quantum many - body problem is density functional theory . hohenberg and kohn @xcite showed that the ( nondegenerate ) ground state of a system of interacting particles is a unique function of the corresponding ground state density @xmath2 . the ground state of the system is thus fully characterized by its density @xmath2 ( ref . ) . for electrons with spin @xmath3 the density @xmath2 must be replaced by a @xmath4 spin density matrix @xmath5 . it is well - known that @xmath5 can be parameterized by the density @xmath6 and ( the magnitude of ) the spin polarization . @xcite the spin polarization at a magnetic field @xmath7 is thus an important quantity for our understanding of many - particle phenomena . recently , the interest in this subject has been renewed because experiments have indicated that the spin polarization of quasi two - dimensional ( 2d ) systems due to a magnetic field @xmath1 parallel to the 2d plane affects the apparent metallic behavior of these systems . @xcite furthermore , the spin polarization is an important parameter for possible applications in the field of spintronics . @xcite in quasi two - dimensional ( 2d ) systems the spin polarization has been studied by applying a magnetic field @xmath1 parallel to the 2d plane . @xcite if the zeeman splitting due to @xmath1 becomes sufficiently large , the minority spin subband is completely depopulated . in such a situation , electron systems are fully spin - polarized . this approach has been used by several groups to study the spin polarization in low - density 2d electron systems . @xcite recently , there has been considerable interest in the spin polarization of quasi 2d hole systems @xcite . hole systems in the uppermost valence band of many common semiconductors such as ge and gaas are different from electron systems due to the fact that the hole states have an effective spin @xmath0 ( ref . ) . usually , hole states in quasi 2d systems are discussed using a basis of angular momentum @xmath0 eigenfunctions with quantization axis perpendicular to the 2d plane . subband quantization yields so - called heavy hole ( hh ) states with @xmath8 component of angular momentum @xmath9 and light hole ( lh ) states with @xmath10 . often it is assumed that the spin degree of freedom of these hh and lh states behaves similar to the spin of @xmath11 electron states . however , recently it has been shown that the spin polarization of quasi 2d hh systems in the presence of an in - plane magnetic field behaves rather different from the more familiar case of @xmath12 electron systems . @xcite for example , the spin polarization of quasi 2d hh systems can change its sign at a finite value of @xmath1 . similar to the case of @xmath12 electron systems , the fundamental object for the characterization of @xmath0 hole systems is the spin density matrix @xmath5 from which we can derive , e.g. , the spin polarization . the dominant character of the occupied eigenstates of hole systems depends on the quantization axis of the underlying @xmath0 basis functions . obviously , all observable quantities such as the spin polarization may not depend on this choice . therefore , it is necessary to formulate the spin density matrix in a way such that observable quantities can be calculated independent of the particular choice for the basis functions that are used . it is the goal of this paper to present such a theory . using the theory of invariants @xcite we will derive an invariant decomposition of the spin density matrix of @xmath0 hole systems that can be interpreted as a multipole expansion . | for hole systems with an effective spin , we present an invariant decomposition of the spin density matrix that can be interpreted as a multipole expansion . the charge density corresponds to the monopole moment and the spin polarization due to a magnetic field corresponds to a dipole moment while heavy hole light hole splitting can be interpreted as a quadrupole moment . for quasi two - dimensional hole systems in the presence of an in - plane magnetic field the spin polarization is a higher - order effect that is typically much smaller than one even if the minority spin subband is completely depopulated . | for hole systems with an effective spin , we present an invariant decomposition of the spin density matrix that can be interpreted as a multipole expansion . the charge density corresponds to the monopole moment and the spin polarization due to a magnetic field corresponds to a dipole moment while heavy hole light hole splitting can be interpreted as a quadrupole moment . for quasi two - dimensional hole systems in the presence of an in - plane magnetic field the spin polarization is a higher - order effect that is typically much smaller than one even if the minority spin subband is completely depopulated . on the other hand , the field can induce a substantial octupole moment which is a unique feature of hole systems . |
1409.8459 | c | in this paper , we have proposed a simple generalization in @xmath1 of the self - dual block renormalization procedure of fernandez - pacheco @xcite , that we have tested for pure and random quantum ising models , with the following conclusions . for the pure models , where the fernandez - pacheco procedure is known to reproduce the exact correlation length exponent @xmath236 @xcite , we have obtained @xmath237 ( to be compared with the 3d classical ising model exponent @xmath5 ) and @xmath238 ( to be compared with the 4d classical ising model mean - field exponent @xmath7 ) . for the random models , where the fernandez - pacheco procedure is known to reproduce exactly the location of the critical point and the critical exponents @xmath239 , @xmath240 and @xmath241 of the infinite disorder fixed point @xcite , we have applied numerically the renormalization rules to two - dimensional samples of linear size @xmath242 , with either random ferromagnetic disorder or spin - glass disorder , both types of disorder leading to the same infinite disorder fixed point : the finite - size correlation exponent @xmath11 coincides with strong disorder renormalization result ( see @xcite and references therein ) , and with the asymmetric block renormalization of ref @xcite , but the activated exponent @xmath9 turns out to be somewhat higher than strong disorder renormalization estimate @xmath243 ( see @xcite and references therein ) . the origin of this difference remains to be clarified . we have also analyzed the rg flows in the disordered and ordered phases , in order to extract the typical correlation length exponent @xmath10 and the analog @xmath235 , and tested the finite - size scaling . in summary , the generalization in @xmath1 of the self - dual block renormalization procedure of fernandez - pacheco @xcite is able to reproduce both the conventional scaling of pure critical points and the activated scaling of infinite disorder fixed points . it would be thus interesting to develop such methods in models governed by strong ( not infinite ) disorder fixed points like the quantum ising model with long - ranged interactions @xcite or the superfluid - insulator transition @xcite , as well as in models where the transition at weak disorder could be in another universality class @xcite . | for the quantum ising chain , the self - dual block renormalization procedure of fernandez - pacheco [ phys . rev . d 19 , 3173 ( 1979 ) ] is known to reproduce exactly the location of the zero - temperature critical point and the correlation length exponent . e 87 , 032154 ( 2013 ) ] have proposed to study the disordered quantum ising model in dimensions by applying the fernandez - pacheco procedure successively in each direction . to avoid the inequivalence of directions of their approach , we propose here an alternative procedure where the directions are treated on the same footing . for the pure model , this leads to the correlation length exponents in ( to be compared with the 3d classical ising model exponent ) and ( to be compared with the 4d classical ising model mean - field exponent ) . for the disordered model in dimension , either ferromagnetic or spin - glass , the numerical application of the renormalization rules to samples of linear size we discuss the similarities and differences with the strong disorder renormalization results . | for the quantum ising chain , the self - dual block renormalization procedure of fernandez - pacheco [ phys . rev . d 19 , 3173 ( 1979 ) ] is known to reproduce exactly the location of the zero - temperature critical point and the correlation length exponent . recently , miyazaki and nishimori [ phys . rev . e 87 , 032154 ( 2013 ) ] have proposed to study the disordered quantum ising model in dimensions by applying the fernandez - pacheco procedure successively in each direction . to avoid the inequivalence of directions of their approach , we propose here an alternative procedure where the directions are treated on the same footing . for the pure model , this leads to the correlation length exponents in ( to be compared with the 3d classical ising model exponent ) and ( to be compared with the 4d classical ising model mean - field exponent ) . for the disordered model in dimension , either ferromagnetic or spin - glass , the numerical application of the renormalization rules to samples of linear size yields that the transition is governed by an infinite disorder fixed point , with the activated exponent , the typical correlation exponent and the finite - size correlation exponent . we discuss the similarities and differences with the strong disorder renormalization results . # 1#2 # 1#2 |
1601.00203 | i | gromov - witten theory has a rich history , both in physics and mathematics . physically , gromov - witten invariants appear in type iia topological string theory as instanton counts associated to interactions between particles . mathematically , they are invariants associated to symplectic manifolds that , roughly speaking , count pseudoholomorphic curves in the manifold . the relationship between these two perspectives is conceptually straightforward : as a string moves in time , it sweeps out a compact riemann surface ( its ` worldsheet ' ) . the amplitudes in string theory encode counts of maps from riemann surfaces into a 3-(complex)-dimensional calabi - yau manifold , and gromov - witten theory assigns invariants to spaces of such maps . in general , counting holomorphic maps from riemann surfaces to a given target space is a difficult problem in enumerative geometry . gromov - witten theory has famously benefited from its connections with string dualities , first with mirror symmetry @xcite , and more recently , large @xmath0 duality @xcite . beginning with @xcite , for toric manifolds , gromov - witten invariants associated to maps of closed surfaces have also been systematically computed using localization @xcite . `` closed '' gromov - witten theory is a natural mathematical counterpart to closed topological string theory , and , in contrast to the `` open '' theory ( i.e. , for maps of riemann surfaces with boundary ) , the moduli spaces involved are rigorously defined . open gromov - witten theory is the subject of this paper . by analogy with the closed case , open gromov - witten theory is a mathematical counterpart to open topological string theory : open strings sweep out compact riemannian surfaces with boundary , and the boundary of the strings are constrained to lie on branes . these boundary constraints are expressed mathematically as lagrangian submanifolds @xmath2 , and the string amplitudes are encoded by counts of holomorphic maps @xmath3 , with the image of the boundary constrained to lie on @xmath4 : @xmath5 . however , as observed in @xcite , there are additional subtleties in adapting the methods of the closed theory to the open case . in particular , even for well - behaved lagrangian boundary cycles , open gromov - witten invariants depend on an additional integral parameter ( in localization , this parameter corresponds to the weights of the torus action ) . in spite of this , the same computational tools of mirror symmetry , large @xmath0 duality , and localization can still be used . in fact , through these string dualities , open gromov - witten theory can be connected to both classical and homological knot theory @xcite . motivated by relationships with the crepant resolution conjecture @xcite , open gromov - witten theory can also be generalized to orbifolds @xcite . this setting has driven a more abstract formulation of open gromov - witten invariants , which has led to a deeper understanding of the a - model . in particular , as will be discussed in detail below , the open invariants contain gamma classes coming from disk terms @xcite . in addition , the open gromov - witten generating function can be obtained from a modification of givental s @xmath6 function @xcite . the primary goal of this paper is to describe a concise and consistent framework for computing open gromov - witten invariants directly , via localization . leveraging the formalism of @xcite , the main tool is a formula for open gromov - witten invariants expressed in terms of local combinatorial data and a gamma class . as expected from @xcite , the construction depends only on the local geometry near a vertex of the moment polytope of @xmath7 . in the case where the associated moduli space of open maps is rigorously defined ( @xcite ) , this formula is proven to be correct . most intriguingly , this result is shown to apply in two unexpected contexts : enumerative invariants associated to the quintic 3-fold , and lagrangian cycles obtained from torus knots appearing in large @xmath0 duality . the author hopes that the approach described herein will lead to a more general construction of open gromov - witten invariants . the paper is organized in the following way . section [ sec : preliminaries ] reviews some general facts about open gromov - witten theory , including deformation theory and localization . most importantly , this section describes how to express an open gromov - witten invariant as the product of a `` disk term '' and an invariant of closed maps . section [ sec : main - results ] contains the proof of the main computational tool of this paper : let @xmath7 be a calabi - yau 3-fold and @xmath2 a lagrangian submanifold . let @xmath8 act on @xmath7 such that the @xmath8 action preserves @xmath4 , and @xmath4 intersects a rigid circle - invariant curve @xmath9 . suppose that @xmath4 can be described in a neighborhood of @xmath10 as the fixed locus of an anti - holomorphic involution . let @xmath11 . then , the genus @xmath12 , 1 boundary component , degree @xmath13 , winding @xmath14 open gromov - witten invariant with lagrangian boundary @xmath4 is @xmath15 where @xmath16 is the disk function @xmath17 here , @xmath18 is the homogeneous iritani gamma class , @xmath19 is the weight of the @xmath8 action along a normal direction to @xmath9 , @xmath20 is the equivariant chern class of the induced representation of @xmath8 at the attachment point of the disk , and @xmath21 is the equivariant class of the image @xmath22 of the disk attachment point . this gamma class formula was encountered previously in @xcite , where the authors study a lagrangian locally described as the fixed locus of the antiholomorphic involution @xmath23 . as will be seen ( lemma [ lem : involution - form ] ) , this result implies the formula above after a change of coordinates . section [ sec : comparisons ] describes how to apply the gamma class formula to several examples where the resulting invariant is already known , and demonstrates that this formula reproduces the expected result . the last two sections study two examples where the assumptions on the local geometry of @xmath4 are not satisfied . section [ sec : quintic ] examines disk enumeration on the quintic 3-fold and finds that a slight modification of this formula again applies . finally , section [ sec : torus - knot - lagrangians ] applies this formula to a novel class of lagrangian cycles motivated by large @xmath0 duality . these lagrangian cycles are obtained from the conormal bundles of torus knots in @xmath24 after the conifold transition , and do not have the same local description required in the above theorem . nevertheless , the main result of this paper is still found to apply to these cycles . the examples in sections [ sec : quintic ] and [ sec : torus - knot - lagrangians ] hint that a version of the theorem may hold for a broader class of lagrangian cycles . the author thanks h. gao , h. jockers , c - c . m. liu , and p. zhou for valuable discussions , and a. brini for helpful corrections . the author is especially grateful to r. cavalieri for the suggestion of this project and many related conversations , and to e. zaslow for guidance and suggestions . | , these disk factors can often be expressed in terms of gamma classes . we verify that this formula encodes the expected invariants obtained from localization by comparing with several examples . we then examine a novel application of this formula to disk enumeration on the quintic 3-fold . finally , motivated by large duality , we show that this formula also unexpectedly applies to lagrangian cycles on constructed from torus knots . | for toric calabi - yau 3-folds , open gromov - witten invariants associated to riemann surfaces with one boundary component can be written as the product of a disk factor and a closed invariant . using the brini - cavalieri - ross formalism , these disk factors can often be expressed in terms of gamma classes . when the lagrangian boundary cycle is preserved by the torus action and can be locally described as the fixed locus of an anti - holomorphic involution , we prove a formula that expresses the disk factor in terms of a gamma class and combinatorial data about the image of the lagrangian cycle in the moment polytope . we verify that this formula encodes the expected invariants obtained from localization by comparing with several examples . we then examine a novel application of this formula to disk enumeration on the quintic 3-fold . finally , motivated by large duality , we show that this formula also unexpectedly applies to lagrangian cycles on constructed from torus knots . |
1305.0980 | m | the evolutionary calculations for giant planets are started at the point where the heavy - element core has a mass of about 1 m@xmath0 , and are carried through the entire formation process as well as the subsequent contraction / cooling phase at constant mass , up to a final age of several gyr . the assumptions and computational procedures were described in detail in previous publications @xcite . the early phase of the formation process is dominated by the accretion of planetesimals onto the core ; during this phase the gaseous envelope has low mass , @xmath13 m@xmath0 , and a low accretion rate compared to that of the core . the latter is given by @xmath14 where @xmath15 is the effective geometrical capture cross section , @xmath16 is the surface density of solid particles ( planetesimals ) in the protoplanetary disk , @xmath17 is the planet s orbital frequency , and @xmath18 is the gravitational enhancement factor , which is obtained from the calculations of @xcite . the planetesimal radius is taken to be 50 km for the cases with a central star of 2 m@xmath11 and 100 km for the cases with a star of 1 m@xmath11 ( see table [ table:1 ] ) . the smaller size , or a reasonable distribution of planetesimal sizes , tends to reduce the formation time but has little effect on the basic results of this paper . if no gaseous envelope is present , then @xmath19 , the radius of the heavy - element core . however , even if the envelope mass is relatively small compared with the core mass , the planetesimals interact with the envelope gas , are slowed down by gas drag , and are subject to ablation and fragmentation . the trajectories of planetesimals through the envelope are calculated @xcite , and the effective @xmath20 is determined . the material that is deposited in the envelope is then allowed to sink to the core , as discussed by @xcite . calculations by @xcite show that this assumption is valid at least for the organic and rock components of the planetesimals . the ices , however , can dissolve in the envelope , so that our ` core mass ' is somewhat overestimated ; the quoted value actually refers to the total excess of heavy - element material , above the solar abundance , in the entire planet . erosion of the core and possible mixing of some core material into the envelope is not considered . this process has been shown to be unlikely for the case of jupiter @xcite , but such estimates have not been extended to the case of planets in the 10 m@xmath1 range . the structure of the hydrogen - helium envelope is calculated according to the differential equations of stellar structure @xcite , which assume hydrostatic equilibrium , a spherically symmetric mass distribution , radiative or convective energy transport , and energy conservation . the energy sources are provided by planetesimal accretion , contraction of the gaseous envelope , and cooling . the additional energy source provided by deuterium burning is included in the later phases of accretion and during the constant - mass final cooling phase , once the mass has exceeded 10 m@xmath1 and internal temperatures exceed @xmath21 k. the full set of equations , supplemented by calculation of the mass accretion rates onto the core and the envelope , and of the planetesimal trajectories , is solved by the henyey method @xcite . at the inner boundary of the envelope the radius is set to @xmath22 , which is determined from its mean density . during the earlier phases of the evolution , when the envelope mass is less than about 0.1 m@xmath1 , the core is assumed to be a mixture of rock and ice with a mean density of @xmath23 . during the later phases , when the pressure at the base of the envelope increases to values above @xmath24 , an aneos equation of state with 50% rock and 50% ice @xcite for the core is used to determine its mean density , which can increase to @xmath25 or higher . in the hydrogen - helium envelope , the equation of state is taken to be given by the tables of @xcite , which take into account the partial degeneracy of the electrons as well as non - ideal effects . the chemical composition is taken to be near - solar , with @xmath26 , and @xmath27 , where @xmath28 are , respectively , the mass fractions of hydrogen , helium , and heavy elements . the tables of course do not include a @xmath29 component , so the @xmath30 component was adjusted upwards to partially compensate . the rosseland mean opacity during the formation phase combines the low - temperature atomic / molecular calculation of @xcite with the interstellar grain opacities of @xcite . the opacity values of the grain component are reduced by a factor 50 to approximately represent the reduction caused by grain growth and settling in the protoplanet @xcite . however , in two of the runs the grain growth and settling are calculated in detail in the temperature range 1001800 k as described in @xcite . the grain size distributions and the opacities are recalculated in every layer at every time step in that temperature range . these opacities are important in regulating the rate at which the envelope can contract , and therefore the rate at which it accretes gas . however , once the envelope is well into the rapid gas accretion phase , at about 0.25 m@xmath1 , the gas accretion rate is limited by the physical properties of the protoplanetary disk near the planet , and the precise values of envelope opacity assume a less - important role . once the planet reaches its final mass , say 12 m@xmath1 , the grains are assumed to settle rapidly and to evaporate in the interior . for the final contraction / cooling phase at constant mass , the molecular opacities of @xcite are used , with solar composition , up to a temperature of 3500 k. at and above that temperature , with any reasonable opacity , the interior is convective . at the outer surface of the envelope , the mass addition rate of gas , during the earlier phases of accretion , is determined by the requirement that the planet radius @xmath31 match the effective accretion radius , which is given by @xcite @xmath32 where @xmath33 is the sound speed in the disk , @xmath34 is the hill sphere radius , and @xmath35 is the total mass of the planet . the constant @xmath36 is determined by three - dimensional numerical simulations which calculate the accretion rate of gas from the protoplanetary disk onto the planet @xcite . as a result , in the limit where @xmath34 is small compared with the bondi accretion radius @xmath37 , @xmath38 . additional boundary conditions at the surface depend on the evolutionary phase . during the early phases when @xmath39 , the density and temperature are set to constant values appropriate for the protoplanetary disk , @xmath40 and @xmath41 , respectively . the density @xmath40 is determined from the assumed value of @xmath16 using a standard gas - to - solid ratio of 70 and @xmath42 , where @xmath43 is the ( gaussian ) disk scale height and @xmath44 is the distance of the planet from the star . however at some point during the rapid gas accretion phase , the mass addition rate required by condition ( [ eq : reff ] ) exceeds the rate at which matter can be supplied by the disk . the disk - limited rates , based on three - dimensional hydrodynamic simulations , are described in the next section . during that phase , the boundary conditions at the actual surface of the planet , whose radius falls well below @xmath45 , are determined through the properties of the accretion shock at this surface , as described in detail by @xcite . the basic assumption is that practically all of the gravitational energy released by the infalling gas is radiated away at the shock ; this energy escapes through the infalling envelope ahead of the shock . this assumption defines the ` cold start ' for planetary evolution . during the final phase of cooling at constant mass , the planet becomes isolated from the disk and the surface boundary conditions change again , to those of a blackbody in hydrostatic equilibrium @xmath46 where @xmath47 is the stefan - boltzmann constant , @xmath48 is the surface temperature , @xmath49 is the total luminosity , and @xmath50 , @xmath51 , and @xmath52 are , respectively , the photospheric values of rosseland mean opacity , pressure , and acceleration of gravity . insolation from the star is not included . significant deuterium burning in the mass range considered begins near the end of the phase of rapid gas accretion . the burning occurs via the reaction @xmath53 with an energy release @xmath54 mev per reaction . the initial deuterium abundance by mass fraction is set to @xmath55 , consistent with the value derived from the local interstellar medium @xcite . the reaction rate ( reactions per second per gram ) is taken from the nuclear astrophysics compilation of reaction rates @xcite : @xmath56\ } , \ ] ] where @xmath57 is the temperature in @xmath58 k , @xmath59 is the density in cgs , @xmath60 is the mass fraction of @xmath5h , and @xmath61 is the mass fraction of @xmath62h ( deuterium ) . this rate is then multiplied by the screening factor , which takes into account ion - ion and ion - electron screening in partially degenerate dense plasmas @xcite . the energy generation @xmath63 , per gram per second , is then obtained , zone by zone , from the rate multiplied by @xmath64 in the appropriate units . to get the change in the deuterium abundance during one time step , it is assumed that the planet interior is fully convective and therefore fully mixed . this assumption is valid for the planets considered during the phase of contraction and cooling , even if no deuterium is burned . the convective velocities of order 10 - 100 @xmath65 , calculated according to the mixing - length approximation , give a mixing time scale far shorter than the d - burning time scale . the reaction rate multiplied by zone mass is integrated over the entire envelope and used to calculate the abundance change . given the central stellar mass @xmath66 , the solid surface density @xmath16 , and the distance of the planet from the star @xmath67 , the isolation mass for the solid material is @xmath68 where @xmath69 is the number of hill - sphere radii on each side of the planetary core from which it is able to capture planetesimals ; @xmath70 in our simulations . once the core mass approaches @xmath71 , the @xmath72 slows down drastically , and beyond that point , gas accretion continues and surpasses the core accretion rate . the core mass increases to a value of about @xmath73 at crossover , when @xmath74 @xcite . this phase of relatively slow accretion rates onto both core and envelope is known as ` phase 2 ' . | after the formation process , which lasts 15 myr and which ends with a ` cold - start ' , low - entropy configuration , the bodies evolve at constant mass up to an age of several gyr . deuterium burning via proton capture is included in the calculation , and we determined the mass , , above which more than 50% of the initial deuterium is burned . | formation of bodies near the deuterium - burning limit is considered by detailed numerical simulations according to the core - nucleated giant planet accretion scenario . the objects , with heavy - element cores in the range 530 m , are assumed to accrete gas up to final masses of 1015 jupiter masses ( m ) . after the formation process , which lasts 15 myr and which ends with a ` cold - start ' , low - entropy configuration , the bodies evolve at constant mass up to an age of several gyr . deuterium burning via proton capture is included in the calculation , and we determined the mass , , above which more than 50% of the initial deuterium is burned . this often - quoted borderline between giant planets and brown dwarfs is found to depend only slightly on parameters , such as core mass , stellar mass , formation location , solid surface density in the protoplanetary disk , disk viscosity , and dust opacity . the values for fall in the range 11.613.6 m , in agreement with previous determinations that do not take the formation process into account . for a given opacity law during the formation process , objects with higher core masses form more quickly . the result is higher entropy in the envelope at the completion of accretion , yielding lower values of . for masses above , during the deuterium - burning phase , objects expand and increase in luminosity by 1 to 3 orders of magnitude . evolutionary tracks in the luminosity versus time diagram are compared with the observed position of the companion to beta pictoris . |
1305.0980 | r | a recent paper on deuterium burning in objects formed through the core - accretion scenario @xcite considered the basic case of a body forming at 5.2 au in a disk around a 1 m@xmath11 star with a solid surface density of @xmath128 and @xmath129 k. their study compared results obtained by varying the following parameters : initial entropy of the object after formation ( hot start vs. cold start ) , helium abundance , metal abundance , initial deuterium mass fraction , @xmath16 , which determines the final planet core mass , and maximum gas accretion rate . their calculations differ from ours in the phase of rapid gas accretion , when disk - limited rates apply . they take that rate to be an arbitrary parameter , while we use the three - dimensional simulations mentioned above ( section [ sec : acc - rates ] ) to determine it . here we concentrate on cold - start models and consider a somewhat different set of parameters : stellar mass , formation position of the planet in the disk , solid surface density @xmath16 , method of computation of the opacity in the planetary envelope during the formation phase , and protoplanetary disk viscosity parameter @xmath86 . the planet s core mass is determined through the calculation itself , and it depends on the first three of these quantities . note that the final core masses found in our calculations fall in the range 4.831 m@xmath0 , while those of @xcite are higher ( 30100 m@xmath0 ) . the formation and evolution are assumed to take place at a fixed orbital radius . the parameters for the runs are given in table [ table:1 ] . the columns in the table give , respectively , the run identifier , the mass of the central star in m@xmath11 , the distance of the planet from the star , the solid surface density @xmath16 , the density @xmath40 at the surface of the planet during the earlier phases when this surface connects with the disk , the temperature @xmath41 at the surface during the same phases , the method of opacity calculation during the formation phase that is , whether it includes the calculation of grain settling and coagulation ( gs ) or not ( ngs) , the value of the viscosity parameter @xmath86 in the disk during the phases of disk - limited gas accretion , and the isolation mass ( equation [ eq : iso ] ) . some results for the six runs are presented in table [ table:2 ] . each run is given two lines , the first for a final planet mass that burns less than half of its deuterium , the second for a nearby mass that burns more than half . the columns give , respectively , the run identification , the final planet mass in m@xmath1 , the total time to reach the final mass ( the formation time ) , the final core mass , the central temperature ( @xmath130 , at the core / envelope interface ) just after formation , the maximum central temperature during d - burning , the central density @xmath131 just after formation , the planet s radiated luminosity @xmath132 just after formation , and the mass fraction of deuterium that remains after 4 gyr of evolution , in units of the initial d mass fraction of @xmath55 . run 1a , with standard parameters of 1 m@xmath11 , 5.2 au , and @xmath133 was originally calculated by @xcite through most of the formation phase , including the detailed calculation of grain opacity ( their run @xmath134 . their run , whose characteristics are listed in that paper , ended at the beginning of disk - limited gas accretion , with a core mass of 16.8 m@xmath0 and an envelope mass of 56.8 m@xmath0 at a total elapsed time of 1 myr . in this work , the run was continued through the disk - limited phase with @xmath135 ( section [ sec : acc - rates ] ) up to the mass range required for deuterium burning . the maximum gas accretion rate was @xmath136 m@xmath0 yr@xmath137 at a total planet mass of 96 m@xmath0 , declining to @xmath138 m@xmath0 yr@xmath137 at 10 m@xmath1 . for several different masses in that range , the accretion was terminated , the opacity was reset in the surface layers to the values given by @xcite , and the evolution was followed at constant mass up to gyr times . the runs were terminated when deuterium burning ceased , and the mass @xmath2 , where 50% of the original deuterium had been burned , was determined . in the run 1a , the total formation time at @xmath2 , up to termination of accretion , was 1.2 myr , well within the lifetime of protoplanetary disks . the planetary luminosity as a function of time for three different final masses in run 1a is shown in figure [ fig:3 ] , where it is compared with that typically obtained in a ` hot - start ' model . in the case of 16 m@xmath1 , just after formation the central temperature @xmath139 k , too low for substantial burning on a short time scale , even though @xmath140 . under these conditions the screening correction factor to the nuclear reaction rate is high , about 88 . consequently , deuterium burning can take place at relatively low temperatures compared to those ( @xmath141 k ) where deuterium burns in solar - mass stars . the central temperature @xmath142 as a function of time ( at the core / envelope interface ) gradually increases as a result of slow deuterium burning and is accompanied by a slight increase in radius . when @xmath142 reaches @xmath143 k , a rapid increase in burning occurs , leading to a peak in luminosity at about @xmath144 years . at the peak about 60% of the deuterium has burned , and @xmath142 is near its maximum of @xmath145 k. at the same time the radius has increased from @xmath146 cm to @xmath147 cm ; then it contracts again after the luminosity peak . at the end of the evolution essentially all the deuterium has burned . a similar process , involving a rapid increase in deuterium burning in the context of a slowly accreting brown dwarf , was studied by @xcite ; he denotes the event a ` deuterium flash ' . the radii for the three masses , as well as for the hot - start case , are shown in figure [ fig : rad ] . the general result that cold - start models result in a radius increase during deuterium burning agrees with the previous results of @xcite . in the case of 12.5 m@xmath1 , right after formation the central temperature is lower , only @xmath148 k , with a central density of @xmath149 and a screening factor of 70 . it takes almost @xmath150 years for rapid deuterium burning to start , at @xmath151 k , with the burning occurring on a much longer time scale than in the case of 16 m@xmath1 . eventually about 98% of the deuterium is burned , and the luminosity peak , which is somewhat lower , is shifted to later times . at the peak , about half of the d has burned , and this point is also close to the maxima in radius and @xmath142 . in the case of 12 m@xmath1 only 6% of the deuterium is burned , and no peak in luminosity appears . at @xmath2 itself , the peak involves only a factor 2 increase in luminosity . in the peaks , the total energy @xmath152 is found to agree closely with the total energy available from d - burning , given by the quantity @xmath153 , where @xmath64 , the energy production per reaction is expressed in ergs , @xmath154 is the mass of a deuterium atom , @xmath35 is the planet mass , @xmath155 is the initial mass fraction of deuterium , and @xmath156 is the fraction of the initial d that burned . the total energy is about @xmath157 ergs for the 12.5 m@xmath1 case . results for two runs whose final masses closely bracket @xmath2 are shown in table [ table:2 ] . the main result of this case is that @xmath158 m@xmath1 with a heavy - element core mass of 16.8 m@xmath0 . by way of comparison , a cold start model with a core , calculated by @xcite with about the same basic parameters ( 1 m@xmath11 , @xmath133 , @xmath159 au ) , with a similar helium mass fraction of 28% , but with some differences in assumptions and computational procedure , gives @xmath160 m@xmath1 . the maximum @xmath142 at the core / envelope interface for @xmath2 in this case is close to @xmath143 k , a very sensitive function of mass . whether significant d - burning occurs depends sensitively on this temperature . if it reaches , say @xmath161 k , practically no d is burned for the corresponding mass of 12.0 m@xmath1 . if it reaches @xmath162 k , practically all ( 98% ) of the d is burned for the corresponding mass of 12.5 m@xmath1 . once the threshold is reached , energy deposition from burning increases the temperature , which increases the reaction rate , as it is proportional to @xmath163 . the resulting expansion leads to a near thermal equilibrium , with the energy produced from d - burning matched closely by the total radiated luminosity . run 1b differs from 1a only with respect to the calculation of the opacity resulting from grains in the protoplanetary envelope during the formation phase . as mentioned above , in run 1a this opacity is obtained through detailed consideration of grain settling and coagulation @xcite . in 1b a table of interstellar grain opacities is used , reduced by a factor of about 50 . the characteristics of this run , up to a mass of about 1 m@xmath1 , are very similar to those listed for run 1sg in @xcite . the crossover mass is 16.16 m@xmath0 , the crossover time is 2.31 myr , and the onset of disk - limited rapid gas accretion occurs at a core mass of 16.8 m@xmath0 and a time of 2.41 myr . note that the evolution time up to this point is 2.4 times longer than in run 1a . note also that the core mass is the same as in run 1a ; the substantial difference in opacity , which can be up to two orders of magnitude in certain ( @xmath164 ) regions , has practically no effect on the core mass . here , the disk - limited accretion rates are used to continue the evolution up to the d - burning mass range . the luminosity as a function of time up to the end of accretion is shown in figure [ fig : lum ] . the results for d - burning after that time show that @xmath165 m@xmath1 , not significantly different from the results of run 1a . as table [ table:2 ] shows , @xmath130 in run 1b , at the same final mass , is slightly higher than that in run 1a , just after formation . correspondingly , @xmath131 is slightly lower . these small differences indicate a slightly higher entropy for 1b after formation , as indicated by the slightly higher luminosity at this point . the increased envelope opacity in 1b as compared with 1a results in slower heat loss and tends to keep internal temperatures higher . however this effect is almost compensated by the fact that the formation time is more than twice as long in 1b . even the slight increase in @xmath130 in 1b as compared with 1a allows @xmath2 to be pushed to a slightly lower mass . run 1c differs from run 1a in that @xmath16 is reduced to @xmath166 , a value only slightly greater than that in a minimum - mass solar nebula @xcite . grain settling and coagulation are included in the opacity calculation . the earlier portions of this run , up to the onset of disk - limited gas accretion , are described in @xcite , their run @xmath167 . the time to reach this point , 3.5 myr , is considerably longer than in run 1a , first , because the core accretion rate is considerably lower , and second , because the lower isolation mass results in reduced luminosity and reduced gas accretion rate during phase 2 @xcite . the crossover mass is 4.09 m@xmath0 , and the core mass at the time of onset of disk - limited accretion is 4.74 m@xmath0 . the calculations were continued up to the point where gas accretion terminated , at which point the core mass was 4.8 m@xmath0 . the total time to reach 12 m@xmath1 was about 4.1 myr , and to 14 m@xmath1 , about 4.5 myr . the peak disk - limited accretion rate was @xmath168 m@xmath0 yr@xmath137 at 0.3 m@xmath1 , a factor of 2.5 lower than in run 1a because of the reduction in @xmath103 by the same factor . by the time the total mass was 5 m@xmath1 the rate was down to @xmath169 m@xmath0 yr@xmath137 , and at 10 m@xmath1 it had further declined to @xmath170 m@xmath0 yr@xmath137 . much of the time during the disk - limited accretion phase was spent in accreting the last 12 m@xmath1 to reach the d - burning point . the luminosity as a function of time for this run , up to the end of accretion , is shown in figure [ fig : lum ] . the luminosity versus time plots for run 1c during the d - burning phase look similar to those for 1a , except in this case m@xmath171 noticeably increases to 13.55 m@xmath1 . the reduced core mass in 1c ( 4.8 m@xmath0 ) as compared to that in 1a ( 16.8 m@xmath0 ) is clearly associated with the difference , in agreement with the results of @xcite . in our calculations , the core equation of state gives a core radius of @xmath172 cm for the run 1c core of mass 4.8 m@xmath0 when the total mass is 12 m@xmath1 . for the core of 16.8 m@xmath0 in run 1a , at the same total mass , the radius is @xmath173 cm . thus , at the core boundary , the gravitational potential is more negative , and the gravity is about 40% greater in 1a than in 1c . the calculated values of @xmath130 are @xmath174 k and @xmath175 k in runs 1a and 1c , respectively . it follows from the equation of hydrostatic equilibrium @xcite that in a convective envelope the adiabatic temperature gradient at the interface should be proportional to the core gravity , so a higher gravity most probably gives a higher temperature . however , this statement is inconclusive . we calculated static models for a planet of 12 m@xmath1 , all with the envelope entropy of run 1c , with core masses ranging from 0 to 15 m@xmath0 . we found practically no difference in @xmath142 as a function of @xmath176 , with @xmath142 _ decreasing _ by less than 1% when the core mass increases from 0 to 15 @xmath177 . the real source of the difference in @xmath130 between runs 1a and 1c is the entropy in the envelope . the lower @xmath130 and higher @xmath131 for 1c as compared with 1a indicate a lower entropy , which is consistent with the fact that the luminosity just after formation is lower by more than a factor 2 in 1c ( table [ table:2 ] ) . the values of entropy just after formation for a planet of 12 m@xmath1 in runs 1a and 1c are , respectively , 8.02 and 7.52 @xmath178 per baryon . the entropy is determined through the physical processes that occur during the entire formation phase ; for example , the formation time for run 1c is almost 4 times longer than that for 1a , and the same opacities were used , which suggests a lower entropy . thus there exists a qualitative understanding of the relation between core mass and @xmath130 , but a quantitative theory , apart from the numerical simulations , is quite difficult . in run 1c , @xmath179 k is the maximum reached for a final mass of 12 m@xmath1 , and it is insufficient for d - burning . in the case of run 1a , the corresponding @xmath130 is much closer to the threshold required for burning . thus the planet with the higher @xmath176 is able to produce significant d - burning at a lower total mass . as table [ table:2 ] shows , in the mass range for run 1c where d - burning begins , just above 13.5 m@xmath1 , @xmath130 is somewhat less ( @xmath180 k ) than in the corresponding mass range for run 1a . however , to compensate , @xmath131 is higher , about @xmath181 , and the screening factor at the center has increased to 160 . again , the lower entropy at formation for 1c , as compared with 1a , a result of various processes associated with the accretion of core and envelope , leads to a higher @xmath2 . the formation phases of runs 2a and 2c , for a central star of 2 m@xmath11 , are illustrated in figure [ fig : lum ] , which gives the luminosity as a function of time , and figure [ fig : mass ] , which gives the core mass , envelope mass , and total mass as a function of time . run 2a differs from 1a in that the planet is placed 9.5 au away from a star of 2 m@xmath11 , in a disk with @xmath182 . in a minimum mass solar nebula , scaled to the mass of this star , the corresponding value would be @xmath183 . the isolation mass , however is quite similar to that in 1a , 12.6 rather than 11.6 m@xmath0 . the opacity during the formation phase of 2a is taken from a table of interstellar grain opacities , reduced by a factor of 50 , as in run 1b . however , the comparison between 1a and 1b showed that these opacities have little effect on @xmath2 . the formation time is longer in 2a than in 1a because of the longer dynamical time at the larger distance , the reduced solid surface density , and the somewhat higher envelope opacity . however , these effects are partially compensated for by the smaller planetesimal size ( 50 km in 2a ; 100 km in 1a ) , which increases the capture cross section @xmath184 , and by the increased gravitational focussing factor @xmath18 at the larger distance . the first luminosity peak for run 2a ( figure [ fig : lum ] ) occurs at @xmath185 yr , with log @xmath49/l@xmath186 , with @xmath187 m@xmath0 , with @xmath188 m@xmath0 , and with @xmath189 m@xmath11 yr@xmath137 . this peak corresponds to the maximum in the accretion rate of solids onto the core . the crossover mass ( figure [ fig : mass ] ) of 17.6 m@xmath0 is reached in @xmath190 years . the second , much higher luminosity peak at @xmath191 yr corresponds to the phase of rapid gas accretion up to a final mass of 15 m@xmath1 . at that time the maximum gas accretion rate is @xmath192 m@xmath0 yr@xmath137 and @xmath193 m@xmath1 . formation is complete , up to 15 m@xmath1 , in @xmath194 yr . the final @xmath195 m@xmath0 is slightly higher than in run 1a . the luminosity as a function of time during the later deuterium - burning phase is shown for five different final masses in run 2a in figure [ fig:4 ] . in the cases of 16 and 15 m@xmath1 , practically all ( @xmath196% ) of the deuterium is burned ; in the case of 13.5 m@xmath1 , about 92% is burned ; for 13.0 m@xmath1 , 75% is burned , and for 12.0 m@xmath1 , just over 50% is burned . thus the value for m@xmath197 m@xmath1 is very close to the values obtained for runs 1a/1b despite substantial differences in assumptions and initial conditions . as discussed in the comparison between runs 1a and 1c , the somewhat larger @xmath176 in 2a as compared to 1a is the main reason for the slightly lower @xmath2 in 2a . after formation , 2a has a slightly higher @xmath130 than 1a and slightly lower @xmath131 , leading to a slightly higher entropy . comparing the luminosity curves for a mass of 16 m@xmath1 in figures [ fig:3 ] and [ fig:4 ] , they look very different but in fact they are consistent . in run 2a ( figure [ fig:4 ] ) the higher @xmath130 ( because of the somewhat higher @xmath176 ) allows d - burning to start earlier than in run 1a , and the value of @xmath49 at the starting point is a factor of 4 higher . in fact the full widths of the two curves are quite similar , the peak values agree to better than a factor 2 , and the integrated luminosities over time of the two curves agree to within 10% . run 2b has exactly the same parameters as 2a except that @xmath86 is reduced by a factor 2.5 , which affects the gas accretion rates during the disk - limited phase . thus the formation time in 2b turns out to be a factor of 1.5 longer at @xmath198 years , but still within the range of observed disk lifetimes . table [ table:2 ] shows that for final mass 12 m@xmath1 the fractions of deuterium burned are in agreement for runs 2a and 2b , within the uncertainties of the calculations . thus @xmath86 has practically no effect on @xmath2 in this case . run 2b has a slightly lower entropy than 2a at 12 m@xmath1 , 8.2 @xmath178 per baryon versus 8.25 , and therefore a slightly higher @xmath2 . thus it appears that the longer time during disk - limited accretion in run 2b has only a weak effect on both @xmath2 and the entropy , at the same @xmath176 . run 2c has the same parameters as 2a except that the solid surface density @xmath16 is increased by a factor 1.5 to @xmath199 . the first luminosity peak ( figure [ fig : lum ] ) occurs at @xmath200 years with log @xmath49/l@xmath201 at @xmath202 m@xmath0 . the crossover mass ( figure [ fig : mass ] ) is reached at @xmath203 yr with a value of 30.7 m@xmath0 . the maximum luminosity in the second peak is above log @xmath49/l@xmath204 , at @xmath205 years and a total mass of 0.62 m@xmath1 . the higher @xmath16 with respect to run 2a results in a markedly higher @xmath206 m@xmath0 and a markedly shorter formation time ( @xmath207 yr at @xmath2 ) . despite these relatively large differences , the value for @xmath2 in 2c is only 2.5% smaller than in 2a . at the end of the formation phase , central temperatures are higher and central densities are lower in 2c as compared with 2a . also , the screening factor is only 14 in 2c compared with 41 for 2a . the entropy at formation , for a final mass of 12 m@xmath1 , is higher in 2c , 9.08 @xmath178/baryon as compared with 8.25 , corresponding to a higher luminosity at that point . the slope in the ( @xmath208 ) diagram between @xmath176= 18.7 and 31 m@xmath0 is -0.024 , a result which differs somewhat from that of @xcite . they obtain a slope ( in the same units ) of -0.01 , although for a different core mass range , 30 to 100 m@xmath0 . plots of luminosity versus time are shown in figure [ fig:5 ] for several different masses in run 2c . as in figure [ fig:4 ] the higher masses give higher peak luminosity at earlier times than the lower masses , and at @xmath2 there is only a very small peak . the @xmath209 curve for 15 m@xmath1 starts at a higher value and reaches a maximum sooner than for the same mass in run 2a , because of the higher internal temperature , but the value of log @xmath49 at the peak is about the same . at 13.7 and 15 m@xmath1 practically all of the initial d is burned . at 12 m@xmath1 , 72% is burned , while at 11.7 m@xmath1 , which is very close to @xmath2 , 61% is burned . plots of @xmath142 versus time , during the deuterium - burning phases , are shown for masses 12 and 15 m@xmath1 in figure [ fig : tem ] , where they are compared with the results from run 2a . the plot shows the effect of varying the core mass at fixed total mass , and of varying the total mass at fixed core mass . for example , for run 2a at 15 m@xmath1 the maximum @xmath142 is @xmath210 k , while for 12 m@xmath1 it is only @xmath211 k and is reached at a much later time . the vertical portions of these curves show the effect of rapid gas accretion from about 1 m@xmath1 to the final mass of either 12 or 15 m@xmath1 . the two nearly horizontal curves are for a total mass of 12 m@xmath1 and core masses of 18.7 ( lower ; run 2a ) and 31 m@xmath0 ( upper ; run 2c ) . the higher core mass results in a higher temperature by a factor of about 1.3 . in the case of the 31 m@xmath0 core , about 75% of the deuterium is burned ; in the 18.7 m@xmath0 core , a little over 50% . note that the d - burning occurs late in the evolution , where small peaks in the temperature are seen . the remaining two curves correspond to a total mass of 15 m@xmath1 , with the same two core masses just mentioned . the d - burning occurs earlier than in the case of the lower total mass , and the higher core mass again gives a higher maximum @xmath142 . in both cases for 15 m@xmath1 practically all the d is burned , and the residual mass fraction is smaller ( @xmath212 ) for the higher core mass as compared to the lower ( @xmath213 ) . in run 2c with 15 m@xmath1 , @xmath142 goes up to about @xmath214 k and there are actually two minor peaks . figure [ fig : llrt ] illustrates in more detail how various quantities vary during this phase . in this case , with a high @xmath130 , nuclear burning starts very early . during most of the phase , the object is not in thermal equilibrium . the first maximum in @xmath142 occurs when about 25% of the d has burned , close to the time of the maximum in the nuclear burning luminosity @xmath215 . here @xmath215 is well above the radiated luminosity @xmath49 , and the extra power goes into expansion , resulting in slight cooling of the interior . when half the deuterium has burned ( @xmath216 yr ) , there is a maximum in luminosity and radius , corresponding to the slight minimum in @xmath142 . then contraction along with a slow decrease in nuclear burning leads to slight heating , and the second maximum occurs when 98% of the d has burned . this maximum corresponds to the time when @xmath215 starts to drop rapidly and to fall well below @xmath49 . beyond that point , even though contraction is occurring , there is insufficient burning to maintain the high temperature , and the object enters its final cooling phase . in contrast , in the case of 12 m@xmath1 , the main d - burning in run 2c takes place at practically constant @xmath142 , radius , and @xmath49 , with a slight maximum in @xmath142 of @xmath217 k at about @xmath144 yr . in this case the configuration is close to thermal equilibrium through most of the d - burning phase . the cross in figure [ fig:5 ] gives the approximate location of the directly imaged companion @xcite to the well - known star beta pictoris . that star , according to http://exoplanet.eu , has a mass of about 1.8 m@xmath11 and an age of 12 ( @xmath218,@xmath219 ) myr . the planet is located between 8 and 15 au from the star @xcite ; thus an approximate comparison can be made with these calculations . beta pic b s position in the ( log @xmath49 , @xmath220 ) diagram is plotted in @xcite where it is shown to fall on a theoretical track with mass 10 m@xmath1 as calculated from a ` hot start ' by @xcite . in http://exoplanet.eu that mass is given as 8 ( @xmath221,@xmath222 ) m@xmath1 . the surface temperature @xmath9 has been estimated from observed near infrared colors @xcite at 1700 k , with considerable uncertainty ( @xmath223 k ) . further infrared and astrometric observations @xcite are essentially in agreement , giving log ( @xmath49/l@xmath224 , @xmath225 k , @xmath226 au , and ` hot - start ' masses in the range 713 m@xmath1 . the bolometric luminosity found by @xcite is in agreement with the above value , and they find ` hot - start ' masses in the range 712 m@xmath1 . in our ` cold - start ' calculations the track for run 2c , 13.7 m@xmath1 , passes close to the object in the ( log @xmath49 , @xmath220 ) diagram , and the calculations give @xmath227 k at an age of 12 myr . our mass 10 m@xmath1 can not possibly provide a fit . the ` hot - start ' models thus would show that the object is a planet , as defined by an object with mass not high enough to burn deuterium . however this particular ` cold - start ' model indicates that beta pictoris b is presently burning deuterium , which , according to the same definition , would classify it as a brown dwarf . as mentioned in section [ sect : intro ] , this definition is not universally agreed upon ; an alternative definition , based on the minimum in the mass distribution of low - mass companions , observed within several au of sunlike stars , places the limit at @xmath228 m@xmath1 . in this case beta pic b would still be a planet . note that in the ` cold - start ' calculations , the fit at 13.7 m@xmath1 with an assumed @xmath229 is not unique ; the companion could also be fit at @xmath182 at a slightly higher mass , about 15.6 m@xmath1 . furthermore , these masses are uncertain and will probably change when the calculations are redone in the future with more detailed model atmospheres . nevertheless , as such they are marginally consistent with the upper limits to the mass of beta pic b derived from radial velocity measurements @xcite . for a planet at 9 au the limit is 12 m@xmath1 ; at 10 au it is 15.4 m@xmath1 . we note also that the luminosity curve for 11.7 m@xmath1 in figure [ fig:5 ] agrees well with the observed luminosity of the directly imaged planet hr 8799 c at the stellar age ( @xmath230 years ) . the observed value is given by @xcite as log @xmath49/l@xmath231 . the agreement of course requires a core mass of @xmath232 m@xmath0 . a hot - start model of about 10 m@xmath1 without a core also agrees . however we do not make a detailed comparison with hr 8799 c , because the metallicity of the star is low ( [ fe / h ] @xmath233 ) and the planet orbits at 43 au , making it highly debatable whether it could have formed by core - nucleated accretion . | are assumed to accrete gas up to final masses of 1015 jupiter masses ( m ) . this often - quoted borderline between giant planets and brown dwarfs is found to depend only slightly on parameters , such as core mass , stellar mass , formation location , solid surface density in the protoplanetary disk , disk viscosity , and dust opacity . the result is higher entropy in the envelope at the completion of accretion , yielding lower values of . for masses above , during the deuterium - burning phase , objects expand and increase in luminosity by 1 to 3 orders of magnitude . | formation of bodies near the deuterium - burning limit is considered by detailed numerical simulations according to the core - nucleated giant planet accretion scenario . the objects , with heavy - element cores in the range 530 m , are assumed to accrete gas up to final masses of 1015 jupiter masses ( m ) . after the formation process , which lasts 15 myr and which ends with a ` cold - start ' , low - entropy configuration , the bodies evolve at constant mass up to an age of several gyr . deuterium burning via proton capture is included in the calculation , and we determined the mass , , above which more than 50% of the initial deuterium is burned . this often - quoted borderline between giant planets and brown dwarfs is found to depend only slightly on parameters , such as core mass , stellar mass , formation location , solid surface density in the protoplanetary disk , disk viscosity , and dust opacity . the values for fall in the range 11.613.6 m , in agreement with previous determinations that do not take the formation process into account . for a given opacity law during the formation process , objects with higher core masses form more quickly . the result is higher entropy in the envelope at the completion of accretion , yielding lower values of . for masses above , during the deuterium - burning phase , objects expand and increase in luminosity by 1 to 3 orders of magnitude . evolutionary tracks in the luminosity versus time diagram are compared with the observed position of the companion to beta pictoris . |
1305.0980 | i | we investigate the boundary between brown dwarfs and giant planets , according to the definition that brown dwarfs can burn the deuterium that is present when they form , and giant planets can not . the main parameters and the results for @xmath2 , the boundary mass at which half of the original deuterium is burned after 4 gyr , are summarized in table [ table:3 ] . the columns give , respectively , the run identification , the stellar mass in m@xmath11 , @xmath44 , the initial disk solid surface density @xmath16 at @xmath44 , the resulting @xmath176 , and @xmath2 . the main cases considered involve a planet / brown dwarf at 5.2 au around a solar - mass star , and a planet / brown dwarf at 9.5 au around a star of 2 m@xmath11 . the table shows that there is only a small variation in the values of @xmath2 , which , however , correlate with the core mass in the sense that the smaller the core mass , the higher the value of @xmath2 . the calculations , taken as a whole , indicate that the envelope entropy , which is a function of initial conditions and which is closely related to the core mass through the accretion processes during the formation phase , is an important factor in determining @xmath2 . however , certain physical processes during formation are shown to have only a small effect . run 1b has the same parameters as run 1a except that the dust opacity during the formation phase is higher by a factor that ranges from 2 to 100 , depending on the depth in the envelope . this difference has a negligible effect on @xmath2 . run 2b has the same parameters as 2a except that the disk viscosity during the phase of disk - limited gas accretion is lower by a factor 2.5 . this difference also has a negligible effect on @xmath2 . however the disk viscosity is important in another respect . if it is significantly lower than the range presented here ( @xmath234 ) , then there will not be time to accrete a planet with mass necessary to burn deuterium during the lifetime of the disk . the gas accretion rate onto a planet of 4 m@xmath1 around a star of 2 m@xmath11 , in a disk with @xmath235 , is reduced by a factor 400 compared with a disk with @xmath236 @xcite , corresponding to less than a jupiter mass in a million years for the initial conditions of run 2b ( formation time about 3 myr ) . of course the minimum viscosity required to build a planet up to about 12 m@xmath1 will depend on parameters such as @xmath16 and @xmath44 . this question is discussed in more detail in appendix [ sec : acc - rates - fit ] . core accretion models , in the cold - start case , are known to have low entropy compared with hot - start models . in @xcite the entropy just after formation for 10 m@xmath1 was found to be 8.2 @xmath178 per baryon for @xmath237 m@xmath0 . the corresponding luminosity at ages of @xmath238 to @xmath144 years was about @xmath239 l@xmath11 , certainly fainter than observed values for directly imaged planets . in this mass range , for the given core mass , the entropy is very insensitive to the planet s total mass , as shown in that paper and confirmed by the present results . however our calculations show that the entropy is quite sensitive to the core mass , as illustrated in figure [ fig : e ] ( a similar effect has been found independently by @xcite for @xmath240 m@xmath0 ) . the points shown are all calculated with the same total mass and the same disk viscosity . all used the reduced interstellar grain opacity , except for the point at @xmath241m@xmath0 , for which the grain - settling opacities were used ( if the interstellar opacities had been used , the formation time would have been considerably longer ) . however the comparison between runs 1a and 1b , which looked at the effect of changing the opacities , showed that the difference in entropy was less than 0.1 @xmath178 per baryon at the same total mass . the effect on the entropy of changing the viscosity ( runs 2a and 2b ) was even smaller . physical effects that do affect the entropy include the planetesimal accretion rate and the rate of contraction of the envelope , both of which affect the internal heating of the envelope . thus the luminosities of newly formed massive planets , depending on formation conditions , can vary by up to two orders of magnitude . information on the runs whose entropies are plotted in the figure is given in table [ table:4 ] . the table is in the same format as table [ table:2 ] and gives the runs in order of decreasing entropy . clearly , for this set of models , a lower entropy is associated with a longer formation time . the luminosity plots for these four cases in figure [ fig : lum ] illustrate the same effect . the combination of @xmath66 , @xmath44 , and @xmath16 determines the isolation mass , and thereby the ultimate core mass , which turns out to be a key factor in determining the entropy of the planet at formation . higher entropy , in particular the higher temperature , favors more rapid nuclear burning , so the higher entropy runs result in lower values of @xmath2 . nevertheless , the range of initial conditions explored here , which is considerable , produces only a small range in @xmath2 , about 11.613.6 m@xmath1 , in agreement with previous independent calculations . we can further conclude , that for cold - start core - accretion models that do burn deuterium , the tracks in the luminosity versus time diagram can potentially provide agreement with the properties of directly - imaged low - mass stellar companions . primary funding for this project was provided by the nasa origins of solar systems program grant nnx11ak54 g ( p. b. , g. d. , j. l. ) . . acknowledges additional support from nasa grant nnx11ad20 p. b. acknowledges additional support from nsf grant ast0908807 . d. s. is supported in part by nasa grants nnh11aq54i and nnh12at89i . the authors are indebted to gilles chabrier for the use of his nuclear screening factors . the 3d hydrodynamical simulations reported in this work were performed using resources provided by the nasa high - end computing ( hec ) program through the nasa advanced supercomputing ( nas ) division at ames research center . thanks los alamos national laboratory for its hospitality . the authors thank the referee dr christoph mordasini for a detailed and constructive review . | the values for fall in the range 11.613.6 m , in agreement with previous determinations that do not take the formation process into account . for a given opacity law during the formation process , objects with higher core masses form more quickly . evolutionary tracks in the luminosity versus time diagram are compared with the observed position of the companion to beta pictoris . | formation of bodies near the deuterium - burning limit is considered by detailed numerical simulations according to the core - nucleated giant planet accretion scenario . the objects , with heavy - element cores in the range 530 m , are assumed to accrete gas up to final masses of 1015 jupiter masses ( m ) . after the formation process , which lasts 15 myr and which ends with a ` cold - start ' , low - entropy configuration , the bodies evolve at constant mass up to an age of several gyr . deuterium burning via proton capture is included in the calculation , and we determined the mass , , above which more than 50% of the initial deuterium is burned . this often - quoted borderline between giant planets and brown dwarfs is found to depend only slightly on parameters , such as core mass , stellar mass , formation location , solid surface density in the protoplanetary disk , disk viscosity , and dust opacity . the values for fall in the range 11.613.6 m , in agreement with previous determinations that do not take the formation process into account . for a given opacity law during the formation process , objects with higher core masses form more quickly . the result is higher entropy in the envelope at the completion of accretion , yielding lower values of . for masses above , during the deuterium - burning phase , objects expand and increase in luminosity by 1 to 3 orders of magnitude . evolutionary tracks in the luminosity versus time diagram are compared with the observed position of the companion to beta pictoris . |
1007.0690 | i | temporal data mining is concerned with finitely many useful patterns in sequential ( symbolic ) data streams @xcite . frequent episode discovery , first introduced in @xcite , is a popular framework for mining patterns from sequential data . the framework has been successfully used in many application domains , e.g. , analysis of alarm sequences in telecommunication networks @xcite , root cause diagnostics from faults log data in manufacturing @xcite , user - behavior prediction from web interaction logs @xcite , inferring functional connectivity from multi - neuronal spike train data @xcite , relating financial events and stock trends @xcite , protein sequence classification @xcite , intrusion detection @xcite , text mining @xcite , seismic data analysis @xcite etc . the data in this framework is a single long stream of events , where each event is described by a symbolic event - type from a finite alphabet and the time of occurrence of the event . the patterns of interest are termed episodes . informally , an episode is a short ordered sequence of event types , and a _ frequent _ episode is one that occurs often enough in the given data sequence . discovering frequent episodes is a good way to unearth temporal correlations in the data . given a user - defined frequency threshold , the task is to efficiently obtain all frequent episodes in the data sequence . an important design choice in frequent episode discovery is the definition of frequency of episodes . intuitively any frequency should capture the notion of the episode occurring many times in the data and , at the same time , should have an efficient algorithm for computing the same . there are many ways to define frequency and this has given rise to different algorithms for frequent episode discovery @xcite . in the original framework of @xcite , frequency was defined as the number of fixed - width sliding windows over the data that contain at least one occurrence of the episode . another notion for frequency is based on the number of _ minimal _ occurrences @xcite . two frequency definitions called _ head frequency _ and _ total frequency _ are proposed in @xcite in order to overcome some limitations of the windows - based frequency of @xcite . in @xcite , two more frequency definitions for episodes were proposed , based on certain specialized sets of occurrences of episodes in the data . many of the algorithms , such as the winepi of @xcite and the occurrences - based frequency counting algorithms of @xcite , employ finite state automata as the basic building blocks for recognizing occurrences of episodes in the data sequence . an automata - based counting scheme for minimal occurrences has also been proposed in @xcite . the multiplicity of frequency definitions and the associated algorithms for frequent episode discovery makes it difficult to compare the different methods . in this paper , we present a unified view of algorithms for frequent episode discovery under all the various frequency definitions . we present a generic automata - based algorithm for obtaining frequencies of a set of episodes and show that all the currently available algorithms can be obtained as special cases of this method . this viewpoint helps in obtaining useful insights regarding the kinds of occurrences tracked by the different algorithms . the framework also aids in deriving proofs of correctness for the various counting algorithms , many of which are not currently available in literature . our framework also helps in understanding the anti - monotonicity conditions satisfied by different frequencies which is needed for the candidate generation step . our general view can also help in generalizing current algorithms , which can discover only serial or parallel episodes , to the case of episodes with general partial orders and we briefly comment on this in our conclusions . the paper is organized as follows . [ sec : overview ] gives an overview of the episode framework and explains all the currently used frequencies in literature . [ sec : algorithms ] presents our generic algorithm and shows that all current counting techniques for these various frequencies can be derived as special cases . [ sec : proof - of - correctness ] gives proofs of correctness for the various counting algorithms utilizing this unified framework . [ sec : candgen ] discusses the candidate generation step for all these frequencies . in sec . [ sec : discussion ] we provide some discussion and concluding remarks . | frequent episode discovery framework is a popular framework in temporal data mining with many applications . over the years many different notions of frequencies of episodes have been proposed along with different algorithms for episode discovery . in this paper we present a unified view of all such frequency counting algorithms . we present a generic algorithm such that all current algorithms are special cases of it . our unified view also helps in obtaining correctness proofs for various algorithms as we show here . we also point out how this unified view helps us to consider generalization of the algorithm so that they can discover episodes with general partial orders . | frequent episode discovery framework is a popular framework in temporal data mining with many applications . over the years many different notions of frequencies of episodes have been proposed along with different algorithms for episode discovery . in this paper we present a unified view of all such frequency counting algorithms . we present a generic algorithm such that all current algorithms are special cases of it . this unified view allows one to gain insights into different frequencies and we present quantitative relationships among different frequencies . our unified view also helps in obtaining correctness proofs for various algorithms as we show here . we also point out how this unified view helps us to consider generalization of the algorithm so that they can discover episodes with general partial orders . |
astro-ph0610481 | i | the origin of ultrahigh energy cosmic rays ( uhecr ) is one of the great remaining mysteries in astrophysics @xcite . the cosmic - ray spectrum has been measured to beyond @xmath1 ev , with a number of events with energy exceeding @xmath2 ev @xcite ; however , it is still debated how such highly energetic particles can be produced . it is now generally accepted that uhecr are of an extragalactic origin @xcite . however , above @xmath3 ev , the process of photopion production ( @xmath4 ) on the cosmic microwave background ( cmb ) is expected to lead to a significant diminution of the cosmic - ray spectrum , the well - known gzk effect @xcite . the relatively short attenuation length associated with the gzk process @xcite necessitates that the observed uhecr arise from local sources . the decay of charged pions produced in this process results in a flux of ultrahigh energy _ neutrinos @xcite . while the observed uhecr spectrum is somewhat insensitive to variations in cosmic source evolution @xcite , the cosmogenic neutrino flux can be greatly enhanced by strong evolution with redshift @xcite , as neutrinos can be produced in larger quantities due to the decreased photopion threshold ( since @xmath5 ) , and can themselves traverse cosmological distances without attenuation . few classes of astrophysical objects can possibly account for the observed cosmic - ray spectrum @xcite . active galactic nuclei ( agn ) have long been considered as possible uhecr sources @xcite . relatively recently , a potential connection between gamma - ray bursts ( grbs ) and uhecr has been explored @xcite . grbs , which are now generally accepted to be related to core - collapse supernovae @xcite , are violent events which release great amounts of energy in the form of gamma - rays ( @xmath6 erg ) @xcite . a number of models have been proposed to utilize their ultra - relativistic environment to accelerate protons to energies @xmath7 ev @xcite . it has also been noted that the source emissivity required to account for @xmath8 ev cosmic rays is comparable to that of gamma - ray bursts @xcite . , normalized to 1 at @xmath9 ) . from top - to - bottom , the metallicity - dependent grb rate density ( this work ) , the quasar ( qso ) evolution model used in refs . @xcite , and the fit to the cosmic star formation history ( sfh ) of ref . models similar to the latter two have been frequently used in cosmic - ray studies . [ fig : cr_evol],width=312 ] just as the core - collapse supernova rate density seems to follow the cosmic star formation history ( sfh ) @xcite , the same might be expected of the cosmological grb rate density @xcite . however , there is mounting evidence that grbs are _ not _ an unbiased tracer of the sfh @xcite . in particular , the host galaxies of grbs have a distinct tendency to be subluminous @xcite and metal - poor @xcite . this has been demonstrated for grb hosts both locally @xcite and at cosmological distances @xcite , which suggests that low metallicity is a key ingredient in the production of a gamma - ray burst @xcite . as we discuss in section [ ratesect ] , a rapidly rotating star , as required in the collapsar model @xcite , can retain much of its original mass and angular momentum if it is metal - poor @xcite . an anti - correlation with metallicity would imply that the cosmological grb rate peaks at a higher redshift @xcite , which now seems to be indicated by swift observations @xcite . simply put : the metallicity of the universe _ decreases _ with redshift , which implies a _ stronger _ evolution of the grb rate density than would be expected from the sfh alone . when grbs are considered as the source of uhecr ( with identical cosmic - ray production per burst ) , the change in the cosmological cosmic - ray emissivity is simply determined by the burst rate history . in the metallicity - biased grb model , this evolution is quite strong , as illustrated in fig . [ fig : cr_evol ] , even exceeding that of models used in past cosmic - ray studies , which have traced quasar ( qso ) luminosities @xcite or the sfh . we examine the effect of enhanced grb rate evolution on the expected flux of cosmogenic neutrinos , the measurement of which may provide the only way to break degeneracies between cosmic - ray models @xcite . in addition to naturally explaining the abundance of high - redshift bursts observed by swift , this strong evolution leads to a measurable neutrino signal , improving the near - term prospects for assessing this scenario with upcoming ultrahigh energy neutrino detectors @xcite . this result would still hold , in general , even if another mechanism is ultimately shown to account for the swift results through enhanced grb evolution . though we focus on the impact on cosmogenic neutrino production ( as dermer and holmes have also recently done for grb rates directly tracing several sfhs @xcite ) , the metallicity - biased evolution model would also have implications for predictions of diffuse neutrino fluxes directly produced in grbs ( e.g. , refs . @xcite ) and their prospects for detection @xcite . | several implications follow from these findings . this necessarily leads , through the gzk process , to an enhancement in the flux of cosmogenic neutrinos , providing a near - term approach for testing the gamma - ray burst cosmic ray connection with ongoing and proposed uhe neutrino experiments . | gamma - ray bursts , which are among the most violent events in the universe , are one of the few viable candidates to produce ultrahigh energy cosmic rays . recently , observations have revealed that grbs generally originate from metal - poor , low - luminosity galaxies and do not directly trace cosmic star formation , as might have been assumed from their association with core - collapse supernovae . several implications follow from these findings . the redshift distribution of observed grbs is expected to peak at higher redshift ( compared to cosmic star formation ) , which is supported by the mean redshift of the swift grb sample , . if grbs are , in fact , the source of the observed uhecr , then cosmic - ray production would evolve with redshift in a stronger fashion than has been previously suggested . this necessarily leads , through the gzk process , to an enhancement in the flux of cosmogenic neutrinos , providing a near - term approach for testing the gamma - ray burst cosmic ray connection with ongoing and proposed uhe neutrino experiments . |
astro-ph0610481 | c | the increase in our understanding of gamma - ray bursts can be traced to the improved capabilities now available to study these phenomena . the ability to quickly and accurately localize a grb has led to the establishment of a grb - supernova connection and allowed for the study of the host galaxies in which these events occur . observations indicate that these hosts tend to be underluminous , star - forming and metal - poor . a connection between a grb and its host galaxy metallicity is not surprising in the context of the collapsar model , which requires rapidly - rotating stars that lack a h / he envelope ( in order to be in accord with supernova observations ) . these requirements can be satisfied by a metal - poor progenitor star . introducing a metallicity bias leads to an accelerated evolution of the cosmic grb rate density ( relative to the sfh ) , which allows for a better fit to recent swift data . in models that attribute ultrahigh energy cosmic rays to gamma - ray bursts , this evolution provides the history of uhecr production in the universe . if grbs are to account for the observed cosmic - ray spectrum , they _ must _ generate a flux of cosmogenic neutrinos . a broad approach which utilizes all available means of observation will continue to be of great utility in unveiling the mysteries of gamma - ray bursts , particularly a relation to uhecr . further observations of grb host galaxies , along with those of core - collapse supernovae ( particularly type ib / ic ) , will provide invaluable information concerning grb progenitors . a systematic study of host metallicities would be important for firmly establishing the grb - metallicity anti - correlation , which , along with more accurate measurements of cosmic metallicity evolution @xcite , would allow for improved grb rate - related calculations . direct observations of neutrinos from grbs ( e.g. , by icecube @xcite or a km@xmath115 mediterranean detector @xcite ) would confirm the acceleration of protons to high ( though not necessarily ultrahigh ) energies . in order to produce a detectable signal , a particularly strong burst , with measured gamma - ray fluence @xmath116 erg @xmath74 ( e.g. , the recent grbs 060928 @xcite and 061007 @xcite , which were observed to have fluences very near this threshold ) is required @xcite . while the needed neutrino telescopes ( which may also reveal the sources of galactic cosmic rays @xcite ) are still under construction , future observations by swift @xcite should reveal similar bursts , in addition to determining the redshift - dependent grb rate with unprecedented precision . cosmogenic neutrinos present a unique tool to examine the grb - as - proton accelerator conjecture . while the cosmic - ray spectrum that will be measured by auger will allow for further assessments of the viability of prospective source models , the combined measurements of neutrinos and uhecr would break degeneracies between the various models , as discussed by seckel and stanev @xcite . it is well - known that strong source evolution can lead to observable cosmogenic neutrino signals ; however , a physically motivated model was lacking . the enhanced cosmogenic neutrino fluxes expected to result from strong grb evolution will allow for this model to be tested in a novel fashion . we have produced this strong evolution naturally through metallicity dependence , however , our general result is also applicable to _ any _ grb model that attempts to explain the observed swift redshift distribution with additional evolution . the sensitivity afforded by anita will allow for near - term examination which may either affirm or , if the expected flux is not found , place substantial constraints upon the model parameters . the lower detection threshold achieved by an arianna - like detector will allow for a realistic opportunity to discriminate between evolution models . measurements of fluxes consistent with that expected from grb evolution would provide compelling evidence for this model . as the relatively young field of particle astrophysics continues to progress , it is quite an exciting possibility that metals , once relegated to the realm of pure astronomy , may , in fact , hold the key to the production of the highest energy particles in the universe . * * acknowledgments.**we would like to thank john beacom for many illuminating discussions ; shinichiro ando , john cairns , charles dermer , kris stanek , floyd stecker , casey watson , and eli waxman for helpful suggestions ; and baha balantekin , jim beatty , and terry walker for useful comments on the manuscript . this work is supported by the national science foundation under career grant no . phy-0547102 to jb , the department of energy grant de - fg02 - 91er40690 , and the ohio state university . | recently , observations have revealed that grbs generally originate from metal - poor , low - luminosity galaxies and do not directly trace cosmic star formation , as might have been assumed from their association with core - collapse supernovae . the redshift distribution of observed grbs is expected to peak at higher redshift ( compared to cosmic star formation ) , which is supported by the mean redshift of the swift grb sample , . if grbs are , in fact , the source of the observed uhecr , then cosmic - ray production would evolve with redshift in a stronger fashion than has been previously suggested . | gamma - ray bursts , which are among the most violent events in the universe , are one of the few viable candidates to produce ultrahigh energy cosmic rays . recently , observations have revealed that grbs generally originate from metal - poor , low - luminosity galaxies and do not directly trace cosmic star formation , as might have been assumed from their association with core - collapse supernovae . several implications follow from these findings . the redshift distribution of observed grbs is expected to peak at higher redshift ( compared to cosmic star formation ) , which is supported by the mean redshift of the swift grb sample , . if grbs are , in fact , the source of the observed uhecr , then cosmic - ray production would evolve with redshift in a stronger fashion than has been previously suggested . this necessarily leads , through the gzk process , to an enhancement in the flux of cosmogenic neutrinos , providing a near - term approach for testing the gamma - ray burst cosmic ray connection with ongoing and proposed uhe neutrino experiments . |
1507.00067 | i | the theory of combinatorial limits has recently attracted a significant amount of attention . this line of research was sparked by limits of dense graphs @xcite , which we focus on here , followed by limits of other structures , e.g. permutations @xcite , sparse graphs @xcite and partial orders @xcite . methods related to combinatorial limits have led to substantial results in many areas of mathematics and computer science , particularly in extremal combinatorics . for example , the notion of flag algebras , which is strongly related to combinatorial limits , resulted in progress on many important problems in extremal combinatorics @xcite . theory of combinatorial limits also provided a new perspective on existing concepts in other areas , e.g. property testing algorithms in computer science @xcite . a convergent sequence of dense graphs can be represented by an analytic object called a _ graphon_. let @xmath4 be the density of a graph @xmath5 in a graphon @xmath2 ( a formal definition is given in section [ sec : limits ] ) . a graphon @xmath2 is said to be _ finitely forcible _ if it is determined by finitely many subgraph densities , i.e. there exist graphs @xmath6 and reals @xmath7 such that @xmath2 is the unique graphon with @xmath8 . finitely forcible graphons appear in many different settings , one of which is in extremal combinatorics . it is known that if a graphon is finitely forcible , then it is the unique graphon which minimizes a fixed finite linear combination subgraph densities , i.e. finitely forcible graphons are extremal points of the space of all graphons . the following conjecture ( * ? ? ? * conjecture 7 ) claims that the converse is also true . let @xmath6 be finite graphs and @xmath9 reals . there exists a finitely forcible graphon @xmath2 that minimizes the sum @xmath10 . finitely forcible graphons are also related to quasirandomness in graphs as studied e.g. by chung , graham and wilson @xcite , rdl @xcite and thomason @xcite . in the language of graph limits , results on quasirandom graphs state that every constant graphon is finitely forcible . a generalization of this statement was proven by lovsz and ss @xcite : every step graphon ( i.e. a multipartite graphon with a finite number of parts and uniform edge densities between its parts ) is finitely forcible . in @xcite , lovsz and szegedy carried out a more systematic study of finitely forcible graphons . the examples of finitely forcible graphons that they constructed led to a belief that finitely forcible graphons must have a simple structure . to formalize this , they introduced the ( topological ) space @xmath11 of typical vertices of a graphon @xmath2 and conjectured the following ( * ? ? ? * conjectures 9 and 10 ) . [ conj - compact ] the space of typical vertices of every finitely forcible graphon is compact . [ conj - dimension ] the space of typical vertices of every finitely forcible graphon has a finite dimension . both conjectures were disproved through counterexample constructions in @xcite . conjecture [ conj - dimension ] is a starting point of our work . analogously to weak regularity of graphs , every graphon has a weak @xmath0-regular partition with at most @xmath12 parts . ( see section [ sec : weak ] for the necessary definitions . ) if the space of typical vertices of a graphon is equipped with an appropriate metric , then its minkowski dimension is linked to the number of parts in its weak regular partitions . in particular , if its minkowski dimension is @xmath13 , then the graphon has weak @xmath0-regular partitions with @xmath14 parts . consequently , if conjecture [ conj - dimension ] were true , the number of parts of a weak @xmath0-regular partitions of a finitely forcible graphon would be bounded by a polynomial of @xmath1 . the number of parts in weak @xmath0-regular partitions of a graphon constructed in @xcite as a counterexample to conjecture [ conj - dimension ] is @xmath15 , which is superpolynomial in @xmath1 , but is much smaller than the general upper bound of @xmath12 . we construct a finitely forcible graphon almost matching the upper bound . [ thm - main ] there exist a finitely forcible graphon @xmath2 and positive reals @xmath16 tending to @xmath17 such that every weak @xmath16-regular partition of @xmath2 has at least @xmath18 parts . as pointed out to us by jacob fox , there is no graphon ( finitely forcible or not ) matching the upper bound for infinitely many values of @xmath0 tending to @xmath17 . in light of this , theorem [ thm - main ] is almost the best possible . [ prop - lower - graphon ] there exist no graphon @xmath2 , positive real @xmath19 and positive reals @xmath16 tending to @xmath17 such that every weak @xmath16-regular partition of @xmath2 has at least @xmath20 parts . the proof of this proposition is sketched at the end of section [ sec : weak ] . we will refer to the graphon @xmath2 from theorem [ thm - main ] as the graphon . is the name of a famous ( and fictitious ) brave czech soldier and , more importantly for us , it is the name of the restaurant where we usually ate lunch during our work on this subject while three of us were visiting the university of west bohemia in pilsen . | lovsz and szegedy conjectured that every finitely forcible graphon , i.e. any graphon determined by finitely many subgraph densities , has a simple structure . we construct a finitely forcible graphon such that the number of parts in any weak-regular partition of is at least exponential in . this bound almost matches the known upper bound for graphs and , in a certain sense , is the best possible for graphons . | graphons are analytic objects representing limits of convergent sequences of graphs . lovsz and szegedy conjectured that every finitely forcible graphon , i.e. any graphon determined by finitely many subgraph densities , has a simple structure . in particular , one of their conjectures would imply that every finitely forcible graphon has a weak-regular partition with the number of parts bounded by a polynomial in . we construct a finitely forcible graphon such that the number of parts in any weak-regular partition of is at least exponential in . this bound almost matches the known upper bound for graphs and , in a certain sense , is the best possible for graphons . |
0812.2630 | i | recently , aqueous solution surfaces of simple electrolytes have attracted much attention for its importance in the broad chemical , environmental and biological processes at the molecular interfaces and membranes.@xcite conventionally , it has been widely believed that the aqueous solution surfaces were empty of ions . this classical picture was based on the macroscopic surface tension measurement of the electrolyte solutions , and was theoretically described by the negative excess of the ions from the thermodynamic gibbs adsorption equation , as well as the microscopic image charge repulsion theory.@xcite recently , new findings in the atmospheric chemical research , direct surface measurement techniques and molecular dynamics ( md ) simulations have challenged this classical picture of ions depletion at the electrolyte solution surface region since gibbs and onsager.@xcite consequently , theoretical simulations and experimental techniques have been employed to provide answers to the two equally important questions about the electrolyte aqueous solution surface . one is how the anions and cations are spatially distributed in the surface region , and the other is how the interfacial water molecules are affected by this ion distribution . to answer the former question requires techniques which can directly measure the presence and profile the depth of the ion distribution at the liquid surface ; while to answer the later requires techniques which can directly measure the presence and change of the interfacial water molecules . the convergence of and the correlation between the two answers may provide a through picture for the understanding of the electrolyte aqueous solution surface . the fact is that the conclusions of most of the recent theoretical and experimental studies converged on the picture that the anions , especially the larger and more polarizable anions , are presented at the interfacial region of various simple inorganic salt solutions . molecular dynamics ( md ) simulation results using the polarizable force fields concluded for significant enrichment of the anions , such as the larger and more polarizable i@xmath4 and br@xmath4 , at the interfacial region.@xcite md simulation also revealed that the anions and cations should be segregated for the electrolytes consist of more polarizable anions at the electrolyte aqueous solution surface , with the double layer structure in which the anions enriched at the top surface layer and the cations resided below.@xcite except for some voices of discontent,@xcite the majority of the experimental studies on the electrolyte aqueous solution surfaces , including the studies with x - ray photoelectron spectroscopy ( xps),@xcite x - ray fluorescence@xcite and nonlinear optical spectroscopy techniques , such as the resonant second harmonic generation ( resonant shg)@xcite and sum - frequency generation vibrational spectroscopy , ( sfg - vs)@xcite lent their support to this new physical picture with the more polarizable anions enriched at the electrolyte aqueous solution surface . since in this new picture the more polarizable anions are expected to be present at the electrolyte aqueous solution surface , while the small non - polarizable hard cations , such as na@xmath3 and k@xmath3 etc . , and anions , such as f@xmath4 , are expected to be repelled from the aqueous solution surface , experimental studies have been focused mainly on either to prove the above conclusions,@xcite or to investigate the specific anion effects on the interfacial water molecules.@xcite therefore , to our best knowledge , so far there has been no systematic investigation on the cation effects on the interfacial water molecules at the electrolyte aqueous solution surfaces.@xcite recently , we developed and employed the surface sensitive non - resonant shg technique to measure the polarization dependent shg response from the interfacial water molecules at the naf , nacl , and nabr salt solution surfaces.@xcite in this study , in order to ensure measurement of small changes in the shg signal , we first developed effective procedures to monitor and remove the impurities in the salt solution samples . then , the quantitative polarization analysis of the measured shg data showed that the average orientation of the interfacial water molecules changed slightly with the increase of the bulk concentration of the naf , nacl and nabr salts from that of the neat air / water interface . the observed significant shg signal increase with the bulk salt concentration was attributed to the overall increase of the thickness of the interfacial water molecular layer , following the order of nabr @xmath1 nacl @xmath2 naf . the absence of the electric - field - induced shg ( efishg ) effect indicated that the electric double layer at the salt aqueous solution surface is much weaker than that predicted from the molecular dynamics ( md ) simulations . these results provided quantitative data to the specific anion effects on the interfacial water molecules of the electrolyte aqueous solution , not only for the larger and more polarizable br@xmath4 anion , but also for the smaller and less polarizable f@xmath4 and cl@xmath4 anions . the most intriguing thing in this study was that we were able to quantitatively measure weak concentration dependent f@xmath4 anion effect of the naf salt on the interfacial water molecules,@xcite even though this f@xmath4 anion effect was expected to be insignificant from the recent md simulations.@xcite this immediately led us to look at whether the f@xmath4 anion effect of the kf salt aqueous solution surface can also be observed . moreover , there is also one known advantage to investigate the kf aqueous solution because this can extend the bulk concentration range of the f@xmath4 anion . the saturation solubility for naf in water at 20@xmath5c is only 0.98 m ( 4.13 g naf per 100 g water ) , while the saturation concentration in kf is 15.4 m ( 89.8 g kf per 100 g water).@xcite surprisingly , we not only observed the significant f@xmath4 anion effect of the kf salt aqueous solution surface up to much higher bulk concentration as expected , we also found that the orientational parameter of the interfacial water molecules went to the opposite direction to that of the naf aqueous solution surface as the bulk concentration of naf and kf salt increase . therefore , we started doing experiment also on the surface of the kcl and kbr salt aqueous solutions , and we observed distinctive specific na@xmath3 and k@xmath3 cation effects on the interfacial water molecules at the naf , nacl , nabr , kf , kcl , and kbr salt solution surfaces . in the following sections , we shall report the non - resonant shg measurement and analysis of the specific na@xmath3 and k@xmath3 cation effects on the interfacial water molecules at these six solution surfaces . we shall show that while the thickness of the interfacial water layer at the surfaces of all the six salt solutions increases as the bulk concentration increase , the orientational parameter d values of the interfacial water molecules changed to opposite directions for the aqueous solutions of the three sodium salts versus the aqueous solutions of the three potassium salts . moreover , the abilities to increase the thickness of the interfacial water layer were found to be in the following order : kbr @xmath1 nabr @xmath1 kcl @xmath1 nacl @xmath2 naf @xmath1 kf . these specific na@xmath3 and k@xmath3 cation effects , as well as the f@xmath4 , cl@xmath4 and br@xmath4 anion effects , indicate that the electrolyte aqueous solution surfaces are likely to be more complex than what the currently prevalent theoretical and experimental understandings can offer . | here we report the polarization dependent non - resonant second harmonic generation ( shg ) measurement of the interfacial water molecules at the aqueous solution of the following salts : naf , nacl , nabr , kf , kcl , and kbr . through quantitative polarization analysis of the shg data , the orientational parameter d,(d= ) value and the relative surface density of the interfacial water molecules at these aqueous solution surfaces were determined . from these results we found that addition of each of the six salts caused increase of the thickness of the interfacial water layer at the surfaces to a certain extent . noticeably , both the cations and the anions contributed to the changes , and the abilities to increase the thickness of the interfacial water layer were in the following order : kbr nabr kcl nacl naf kf . we also found that the orientational parameter d values of the interfacial water molecules changed to opposite directions for the aqueous solutions of the three sodium salts versus the aqueous solutions of the three potassium salts . these findings clearly indicated unexpected specific na and k cation effects at the aqueous solution surface . these effects were not anticipated from the recent molecular dynamics ( md ) simulation results , which concluded that the na and k cations can be treated as small non - polarizable hard ions and they are repelled from the aqueous interfaces . | here we report the polarization dependent non - resonant second harmonic generation ( shg ) measurement of the interfacial water molecules at the aqueous solution of the following salts : naf , nacl , nabr , kf , kcl , and kbr . through quantitative polarization analysis of the shg data , the orientational parameter d,(d= ) value and the relative surface density of the interfacial water molecules at these aqueous solution surfaces were determined . from these results we found that addition of each of the six salts caused increase of the thickness of the interfacial water layer at the surfaces to a certain extent . noticeably , both the cations and the anions contributed to the changes , and the abilities to increase the thickness of the interfacial water layer were in the following order : kbr nabr kcl nacl naf kf . since these changes can not be factorized into individual anion and cation contributions , there are possible ion pairing or association effects , especially for the naf case . we also found that the orientational parameter d values of the interfacial water molecules changed to opposite directions for the aqueous solutions of the three sodium salts versus the aqueous solutions of the three potassium salts . these findings clearly indicated unexpected specific na and k cation effects at the aqueous solution surface . these effects were not anticipated from the recent molecular dynamics ( md ) simulation results , which concluded that the na and k cations can be treated as small non - polarizable hard ions and they are repelled from the aqueous interfaces . these results suggest that the electrolyte aqueous solution surfaces are more complex than the currently prevalent theoretical and experimental understandings . |
1212.4807 | i | since the realization of bose - einstein condensates ( becs ) in 1995 @xcite , there was a significant interest about effects of the disordered potentials on the properties of ultracold quantum gases @xcite . the reason for this is not only because of the unavoidable irregularities in the trapping potential induced by wire imperfections @xcite , but also due to the fact that disorder can be generated and controlled using laser speckles @xcite . it is well known that cold atoms are a promising tool for simulating other physical systems @xcite in the sense of feynman s quantum simulator @xcite . this applies also to the phenomenon of anderson localization , which was originally used to microscopically describe the absence of diffusion in terms of disorder @xcite . it has a clear bec analogue @xcite , which has been directly observed @xcite . also , localization inside becs due to disorder created by atomic impurities on a lattice was studied theoretically @xcite and recently observed experimentally @xcite . for a theoretical analysis of global dirty boson properties , different methods have been used to describe various limits , ranging from the bogoliubov theory @xcite , numerical approaches @xcite , to the parisi replica method @xcite . it turns out that long - range correlations within both the condensate and the superfluid remain , despite the presence of disorder . however , both quantities are depleted due to the localization of fragmented condensates in the local disorder potential minima . for a strong enough disorder in a homogeneous system , the depletion increases to such an extent that even a critical disorder strength exists , above which a bose - glass phase appears , consisting only of localized mini - condensates @xcite . effects of disorder were also studied for harmonically trapped becs @xcite and becs in optical lattices @xcite , while the temperature behavior of dirty boson properties was examined in refs . @xcite . realization of atomic dipolar becs @xcite with long - range anisotropic interaction has generated large interest in the theory of dipolar quantum gases @xcite . increase in the strength of dipolar interaction is possible by substituting atoms with magnetic dipoles by heteronuclear molecules , which have a strong electric dipolar moment in rovibrational ground state @xcite , or by inducing radiative coupling by placing dipoles into a resonator @xcite . dipolar condensates were studied in the case of isotropic disorder @xcite , which yields characteristic anisotropies for both the superfluid density and the sound velocity at zero temperature due to the anisotropy of the dipolar interaction . although a 3d isotropic laser speckle potential has recently been proposed in ref . @xcite , the typical disorder realized in experiments is cylindrically symmetric and , to the best of our knowledge , it has so far been examined only numerically for contact interaction @xcite . therefore , motivated by the experiments with dipolar becs in anisotropic disorder potentials , we develop in this paper a mean - field theory and analytically study the impact of a weak anisotropic disorder on physical properties of a polarized dipolar bec at zero temperature . to this end we proceed as follows . following the approach developed in ref . @xcite , in sec . [ sec : mf ] we calculate the lowest - order corrections of bec properties due to the presence of disorder within a mean - field theory . for the sake of generality we consider an arbitrary two - particle interaction and a general disorder correlation function . in sec . [ sec : diplor ] we specialize the developed formalism to dipolar interaction and a lorentzian - correlated disorder in fourier space . this yields for both the superfluid density and the sound velocity characteristic anisotropies , which should be measurable in an experiment . in sec . [ sec : con ] we present our conclusions and outlook for further related research . finally , the appendix gives analytical results for the condensate depletion and the disorder correction to the chemical potential , while the supplemental material @xcite gives complete analytical results for the disorder correction of both the superfluid density and the sound velocity for the general case of a bec with contact and dipole - dipole interaction in an anisotropic , lorentzian - correlated disorder . | here we study properties of a homogeneous dipolar bose - einstein condensate in a weak anisotropic random potential with lorentzian correlation at zero temperature . to this end we solve perturbatively the gross - pitaevskii equation to second order in the random potential strength and obtain analytic results for the disorder ensemble averages of both the condensate and the superfluid depletion , the equation of state , and the sound velocity . for a pure contact interaction and a vanishing correlation length , | here we study properties of a homogeneous dipolar bose - einstein condensate in a weak anisotropic random potential with lorentzian correlation at zero temperature . to this end we solve perturbatively the gross - pitaevskii equation to second order in the random potential strength and obtain analytic results for the disorder ensemble averages of both the condensate and the superfluid depletion , the equation of state , and the sound velocity . for a pure contact interaction and a vanishing correlation length , we reproduce the seminal results of huang and meng , which were originally derived within a bogoliubov theory around a disorder - averaged background field . for dipolar interaction and isotropic lorentzian - correlated disorder , we obtain results which are qualitatively similar to the case of an isotropic gaussian - correlated disorder . in the case of an anisotropic disorder , the physical observables show characteristic anisotropies which arise from the formation of fragmented dipolar condensates in the local minima of the disorder potential . |
1610.06413 | i | a common practice to provide lifetime enhancement against fatigue and stress - corrosion cracking of metallic parts ( e.g. , critical components of aircraft engines , welds in steel structures , etc . ) is to impart , through surface - conditioning treatments such as shot peening , sand blasting , laser peening , and low plasticity burnishing , a thin surface layer of compressive residual stress on the parts so treated . the protective compressive stress induced by surface conditioning , however , may relax as a result of thermomechanical loadings experienced by the treated part after it is put into service , thus compromising the very purpose of the surface - conditioning treatment . to ensure safety and good performance , a nondestructive technique should be developed so that retention of the protective compressive stress in the treated parts can be monitored in - situ from time to time , thereby providing a basis for deciding whether a treated part should be taken out of service for replacement or re - conditioning treatment . depending on the specific surface - conditioning technique and processing parameters , the layer of compressive residual stress typically starts from the surface and runs to a depth of about 0.3 mm to 1 or 2 mm . the residual stress @xmath0 thus created varies with depth from the surface . at the free surface the principal stress of @xmath0 which has the free surface as principal plane is zero . if another principal stress of @xmath0 is plotted against depth from the surface , the graph typically assumes the shape of a check mark with a long tail ( see fig . 2 in section 5 ) : the principal stress starts negative ( i.e. , compressive ) at the surface and goes through a quick dip , then after a blunt turn at a minimum value ( i.e. , maximum compressive stress ) increases monotonically until it becomes slightly tensile and reaches a maximum , and then decays in a long tail to approximately zero while remaining tensile . for life - prediction purposes , monitoring of not only the surface residual stress but also the profile and depth of penetration of the protective stress layer ( particularly the maximum compressive principal stresses and their locations ) are required , because they all strongly affect the fatigue life and corrosion - crack resistance of the treated part . the presence of stress in a body affects the velocities of elastic waves propagating in it . this phenomenon is called the acoustoelastic effect . there is ample experimental evidence ( see , for example , @xcite ) that the presence of a surface layer of inhomogeneous residual stress in an otherwise homogeneous medium will lead to the dispersion of rayleigh waves , the quantitative data of which can be ascertained by boundary measurements . in this paper we shall explore whether we could monitor the retention of the surface - treatment induced layer of protective compressive stress by measurements of rayleigh - wave dispersion . besides inhomogeneous stress , there are other material characteristics ( e.g. , surface roughness , inhomogeneity in crystallographic texture ) of a treated part that will lead to dispersion of rayleigh waves , often with effects comparable to or stronger than those due to initial stress ( see , e.g. , @xcite ) . should some such characteristics have also changed after the treated part is put in service , other measurements in addition to rayleigh - wave dispersion would be needed to infer the depth profile of the stress . as a first step towards the development of an ultrasonic technique for monitoring stress retention in surface - treated samples , here we will restrict our discussion to the following simple situation : except for the unknown depth - dependent residual stress , all other relevant material parameters are known . one scenario where this could happen is that we have ascertained all relevant material characteristics of the treated sample , including the residual stress imparted by surface - conditioning , before the sample is put into service . after a period of service , the protective residual stress may have suffered from partial or total relaxation , but all other material characteristics of the treated sample remain unchanged after its production . under the theoretical framework of linear elasticity with initial stress @xcite , man et al . @xcite recently presented a general procedure for obtaining a high - frequency asymptotic formula for the dispersion of the phase velocity of rayleigh waves propagating in a vertically - inhomogeneous , prestressed and anisotropic half - space . as a further development , the general procedure given in @xcite was adapted by tanuma et al . @xcite to the case where the incremental elasticity tensor @xmath1 can be written as the sum of an isotropic part @xmath2 and a perturbative part @xmath3 . under a cartesian coordinate system where the material medium occupies the half - space @xmath4 , the perturbative part @xmath5 , the initial stress @xmath6 , and the mass density @xmath7 were assumed to be smooth functions of @xmath8 . moreover , the following linearization assumption ( * ) was made : at the free surface @xmath9 of the material medium the perturbative part @xmath10 and the initial stress @xmath11 are sufficiently small as compared with @xmath2 that for all expressions and formulas which depend on @xmath10 and @xmath11 it suffices to keep only those terms linear in the components of these tensors . under this setting , specific formulas are derived @xcite with which the procedure presented in @xcite can be implemented to compute iteratively each term of a high - frequency asymptotic formula for dispersion relations that pertain to rayleigh waves with various propagation directions . thus for rayleigh waves of sufficiently high frequencies , dispersion curves can be generated by the method developed in @xcite when requisite data on material and stress are given . once we have that capability , the inverse problem of inferring stress retention from rayleigh - wave dispersion can be attacked by an iterative approach . the theory developed in @xcite is meant for applications that include as typical example ultrasonic measurement of stress in metal structural parts , where the perturbative part @xmath3 in the splitting @xmath12 of the incremental elasticity tensor is originated from the presence of crystallographic texture and of the prestress @xmath0 . moreover , the shifts in phase velocities of elastic waves caused by texture and initial stress ( with the latter bounded by the yield surface ) are typically within 2% of their values for the corresponding isotropic medium with @xmath13 , which suggests that linearization assumption ( * ) would be adequate . on the other hand , the theory developed in @xcite does not take into consideration the effects of surface roughness on rayleigh - wave dispersion . several empirical studies ( see e.g. , @xcite ) have shown that if rayleigh - wave dispersion is used for measurement of stress induced by shot - peening or laser - shock peening , the effect of surface roughness on the dispersion curves can not be ignored , for it can totally mask the dispersion due to inhomogeneous stress . surface conditioning by low plasticity burnishing ( lpb ) , however , is different , for lpb leaves a mirror - smooth finish on processed parts . for the dispersion of rayleigh waves which have frequencies suitable for interrogation of the compressive stress induced by lpb treatment , surface roughness is not an issue @xcite . in this paper we will study as prime example the possibility of using the high - frequency formula for rayleigh - wave dispersion developed in @xcite to infer retention of near - surface compressive stress in a thick - plate sample of an aa 7075-t651 aluminum alloy which was surface - treated by low plasticity burnishing ( lpb ) . the plan of this paper is as follows . in section 2 we present , within the context of linear elasticity with initial stress , the constitutive equation of a prestressed medium which is a polycrystalline aggregate of cubic crystallites that carries an orthorhombic texture . details on material parameters and texture coefficients specific to the aluminum sample , which serves as the prime example of our present study , are given in appendix a. in section 3 , after we briefly outline the procedure given in @xcite to arrive at a high - frequency asymptotic formula for rayleigh - wave dispersion , we present a theorem and its corollary which will be instrumental for reducing the inverse problem in question to solving systems of linear equations iteratively . section 4 is devoted to a statement of the inverse problem on monitoring of stress retention and its solution . in section 5 we apply the theory to a specific inverse problem pertaining to the aluminum sample . there after we describe how `` experimental '' data on rayleigh - wave dispersion are simulated over the frequency window from 4 mhz to 70 mhz , we use the third - order approximation of the dispersion relations to infer the depth profile of the residual stress . in our example the inferred and real " stress profiles match well for the range of depth from 0 to 0.7 mm . in section 6 we examine the scenario in which experimental conditions ( e.g. , diffraction errors ) rule out the use of data at frequencies lower than 7 mhz . we use the second - order approximation of and the simulated data on the dispersion relations over the frequency window from 7 mhz to 70 mhz to infer the stress profile . our example shows that the inferred and real " stress profiles match quite well for the range of depth from 0 to 0.5 mm . we end the paper with some closing remarks in section 7 . | our solution of this inverse problem is based on a recently developed algorithm by which each term of a high - frequency asymptotic formula for dispersion relations can be computed for rayleigh waves that propagate in various directions along the free surface of a vertically - inhomogeneous , prestressed , and weakly anisotropic half - space . as a prime example of possible applications we focus on a thick - plate sample of aa 7075-t651 aluminum alloy , which has one face treated by low plasticity burnishing ( lpb ) that induced a depth - dependent prestress at and immediately beneath the treated surface . we model the sample as a prestressed , weakly - textured orthorhombic aggregate of cubic crystallites and assume that by nondestructive and/or destructive measurements we have ascertained everything about the sample , including the lpb - induced prestress , before it is put into service . under the supposition that the prestress be partially relaxed but | herein we study the inverse problem on inferring depth profile of near - surface residual stress in a weakly anisotropic medium by boundary measurement of rayleigh - wave dispersion if all other relevant material parameters of the elastic medium are known . our solution of this inverse problem is based on a recently developed algorithm by which each term of a high - frequency asymptotic formula for dispersion relations can be computed for rayleigh waves that propagate in various directions along the free surface of a vertically - inhomogeneous , prestressed , and weakly anisotropic half - space . as a prime example of possible applications we focus on a thick - plate sample of aa 7075-t651 aluminum alloy , which has one face treated by low plasticity burnishing ( lpb ) that induced a depth - dependent prestress at and immediately beneath the treated surface . we model the sample as a prestressed , weakly - textured orthorhombic aggregate of cubic crystallites and assume that by nondestructive and/or destructive measurements we have ascertained everything about the sample , including the lpb - induced prestress , before it is put into service . under the supposition that the prestress be partially relaxed but other material parameters remain unchanged after the sample undergoes a period of service , we examine the possibility of inferring the depth profile of the partially relaxed stress by boundary measurement of rayleigh - wave dispersion . * keywords * : inverse problem , acoustoelasticity , residual stress , rayleigh waves , dispersion relations , textured media , surface conditioning [ section ] [ thm]corollary [ thm]lemma [ thm]proposition [ thm]definition [ thm]remark |
1408.4323 | i | the general catalog of variable stars lists multiple groups of pulsating variable stars of late spectral type with periods of several tens of days or longer . while historical divisions by period and spectral type were made in the absence of knowledge about stellar evolution , we now recognize that these long period variable ( lpv ) stars are evolved low- and intermediate - mass stars . during the red giant branch ( rgb ) , asymptotic giant branch ( agb ) , and post - agb phases stars are characterized by highly extended stellar envelopes providing the basis for long - period pulsation . the variability time scale of lpvs requires monitoring over years to decades to properly describe the light changes . in the 1990s long term , rapid cadence monitoring of variable stars in selected fields occurred as a byproduct of ground - based photometric surveys searching for microlensing events . this data was transformative , providing large data sets of high quality light curves for stars in the magellanic clouds @xcite . the magellanic cloud variables are at a known distance , and hence have known luminosities , permitting the construction of period - luminosity ( p - l ) diagrams for a large sample of late - type stars . while previously a p - l relation had been known for the large amplitude lpvs , the miras @xcite , period - luminosity relations for all late - type variables are apparent in the lmc data @xcite . in this paper we focus on those lpvs that are luminous with periods of @xmath0100 days or longer but are not miras . in the parlance of the @xcite p - l relation these are sequence c and c @xcite . in the classical variable star nomenclature these are semi - regular ( sr ) variables . the gcvs classification is unfortunately blurred across a large range of temperature and luminosity with sr variables found across a large part of the cool star p - l diagram . here we will discuss those that have agb luminosity . other recent papers have discussed sr variables that are less luminous and are on the rgb @xcite . the improved p - l diagram results in new astrophysical challenges . the c and c variables are pulsating radially in the fundamental mode , a low overtone mode , or a combination of modes . the origin of agb semi - regular behavior , both in cycle length and amplitude , remains unclear , but their light curves can in some cases be reproduced by the combination of several periods ( e.g. * ? ? ? * ; * ? ? ? the major challenge , however , is an explanation of the long secondary periods ( lsp ) found in many of the overtone pulsators . with a typical time scale of several hundreds to thousands of days the secondary periods are significantly longer than the fundamental mode periods and therefore can not be explained by radial pulsation . various solutions have been discussed , but a satisfactory explanation has yet to be found ( e.g. * ? ? ? one route to further understanding of these stars is precision , continuous photometric monitoring . recent studies faced the typical problems of ground - based observations , time series of very limited length , of limited photometric accuracy , and observational gaps resulting from both diurnal and seasonal cycles . a space mission obtaining photometry continuously over a time span of several years is , therefore , clearly relevant . one of the many side benefits of the kepler mission was the ortunity to obtain long and precise time series of variables , including lpvs with uninterrupted sampling extending over months . @xcite presented the first extensive study on the variability of both rgb and agb m giants using the long and continuous light curves provided by kepler . in that paper the authors focused on the global characteristics of the light variations of these cool giants . their study confirmed the presence of several pulsation modes based on the patterns seen in the petersen diagram and it revealed a clear distinction between solar like oscillations and large amplitude , mira - like pulsations . this transition seems to appear at log p(days ) @xmath0 1 . the kepler mission was optimized to search for planet transits , not slow variations spanning weeks to years . as a consequence , the aim of our paper is twofold . in the first part , we investigate the applicability of the kepler data products to the construction of long base line light curves by combining individual data - sets of three months length . we will focus on a small sample of 12 stars , three of which have small amplitudes that are included especially to identify instrument specific effects . the light curves are then used in the second part of the paper to discuss the light change of sr variables on the basis of these uninterrupted and photometrically precise light curves . | photometry for a small sample of nine sr agb stars are examined using a 30 minute cadence over a period of 45 months . while undergoing long period variations of many magnitudes , the light curves are shown to be smooth at the millimagnitude level over much shorter time intervals . the light curves were all well represented by a combination of sinusoids . however , the properties of the sinusoids are time variable with irregular variations present at low level . it is argued that the long secondary period variation seen in many sr variables is intrinsic to the star and linked to multiple mode pulsation . | high precision kepler photometry is used to explore the details of agb light curves . since agb variability has a typical time scale on order of a year we discuss at length the removal of long term trends and quarterly changes in kepler data . photometry for a small sample of nine sr agb stars are examined using a 30 minute cadence over a period of 45 months . while undergoing long period variations of many magnitudes , the light curves are shown to be smooth at the millimagnitude level over much shorter time intervals . no flares or other rapid events were detected on the sub - day time scale . the shortest agb period detected is on the order of 100 days . all the sr variables in our sample are shown to have multiple modes . this is always the first overtone typically combined with the fundamental . a second common characteristic of sr variables is shown to be the simultaneous excitation of multiple closely separated periods for the same overtone mode . approximately half the sample had a much longer variation in the light curve , likely a long secondary period . the light curves were all well represented by a combination of sinusoids . however , the properties of the sinusoids are time variable with irregular variations present at low level . no non - radial pulsations were detected . it is argued that the long secondary period variation seen in many sr variables is intrinsic to the star and linked to multiple mode pulsation . |
gr-qc0311004 | i | since observations show that space is homogeneous at large scales to a very good approximation , homogeneous models have been studied for many decades in classical and quantum cosmology @xcite . they provide insights into the behavior of our universe while avoiding all the complicated field theoretic details of full gravity by their restriction to finitely many degrees of freedom . despite the huge reduction by infinitely many degrees of freedom , a large variety of different models is left which allow a detailed investigation of various non - trivial issues such as : dirac observables , the embedding of symmetric models ( e.g. , isotropic ) into less symmetric ones , dynamics , the approach to classical singularities and cosmological phenomenology . loop quantum cosmological methods for those models have recently been developed @xcite and shown to simplify considerably after a diagonalization of the connection and triad degrees of freedom @xcite . as an example the dynamics of the bianchi i model has been studied and shown to be singularity - free . in this case the hamiltonian constraint resembles that of the full theory @xcite ( albeit it is much simpler ) , and thus the quantization can be regarded as a reliable test . it turned out that the extension from the isotropic case @xcite to this anisotropic case , requires certain features which are in fact present in quantum geometry . in particular , the mechanism of a singularity - free evolution in anisotropic models relies on the fact that the evolution extends through the classical singularity which can be identified with a submanifold in minisuperspace . this requires the classical singularity to lie in the interior of minisuperspace , rather than at the boundary ( including infinity ) . here it is essential that quantum geometry is based on densitized triad variables where in fact the kasner singularity is in the interior ( all densitized triad components being zero ) , while it would be on the infinite boundary in co - triad variables . the isotropic case is not sensitive to this issue since both densitized and co - triad variables have the classical singularity in the interior of minisuperspace ( here , it is essential to use any ( co-)triad rather than metric variables ) ; thus the existence of the extension of the methods to a homogeneous model is non - trivial . an extension to the full theory of the general class of inhomogeneous models seems much more complicated at this point . however , it has been conjectured @xcite that at the classical level , the most general homogeneous behavior is that of the bianchi - ix model and furthermore that it describes the approach to the singularities even of _ inhomogeneous _ models . it is therefore important to study the more general bianchi class a models . homogeneous models other than bianchi i , however , present an additional complication since they have non - zero intrinsic curvature and , as a consequence , their spin connection can not be zero . this is in contrast even to the full theory where the spin connection can be made arbitrarily small locally by choosing appropriate coordinates . in a homogeneous model the freedom of choosing coordinates is restricted to those which preserve ( manifest ) homogeneity , and this implies that the spin connection is a covariant object and that it has to have a certain size in a given model . this has to be taken into account properly when one quantizes the hamiltonian constraint . in particular , in a classical regime only the extrinsic curvature can be assumed to be small , but not necessarily the intrinsic curvature which determines the spin connection . this is essential to understand the semiclassical behavior . since a special treatment is required which is not necessary in the full theory , one has to be more careful when interpreting the results . it introduces more quantization ambiguities which have to be shown not to influence main results . on the other hand , more possibilities for phenomenology emerge which , at least qualitatively , can often be seen to be insensitive to ambiguities . having a more distant relation to the full theory , it is helpful to have a set of admissibility criteria . one such criterion is of course to accommodate a semiclassical approximation to the quantum dynamics . since a semiclassical description is based on continuum geometry while the quantum dynamics is in terms of discrete quantum geometry , this criterion is formulated here in terms of a _ continuum approximation_. a further criterion is provided by the requirement of a _ locally stable evolution _ @xcite . such a requirement arises because the evolution equation derived along the lines of @xcite is usually a high order difference equation which at large volume is well approximated by the second order wheeler dewitt differential equation . hence , one can always construct solutions by choosing initial data , at large volume , to be that provided by solutions of the wheeler dewitt equation . perturbation of such an initial data will generate solutions which will also include surplus solutions of the high order difference equation . generically these will have planck scale oscillations . there is then the possibility that such extra solutions can become dominant under the evolution . in fact these are expected to become dominant as one gets to smaller volume since there are huge differences between the continuum and the discrete formulations in the planck regime . however as one evolves to larger volumes , the perturbations must not grow too rapidly since this would imply domination by solutions with planck scale oscillation even in the classical regime . the requirement we are looking for , local stability , prohibits this behavior by demanding that the local behavior of solutions to the difference equation around a large value of the evolution parameter is not exponentially increasing . in this paper we present a quantization of the hamiltonian constraint for bianchi models with non - zero intrinsic curvature which fulfills this condition and has the correct semiclassical limit . the first quantization given in @xcite was not admissible in this sense ; thus , the quantization given there is valid only for the bianchi i model ( for which we have the same quantization here ) . it turns out that the local stability condition is selective : it requires all roots of a high order polynomial to have unit norm which is not easy to achieve randomly . the selectivity is increased by the fact that the same strategy of quantizing a hamiltonian constraint should work in all homogeneous models , using different procedures in different models would imply that not all of them can be related to the full theory . the stability condition can be side - stepped by quantizing connection components in the constraint by hand such that only a second order difference equation results . while this would eliminate local stability as a selection criterion , it would also imply that the quantization is even more distant to that in the full theory where such a quantization can not be possible . further clues as to the necessary order of the evolution equation can come from studying dirac observables , which will be pursued elsewhere . in section [ s : models ] , we briefly describe the diagonal , homogeneous models addressed in this paper . we recall the classical framework and the corresponding loop quantization , including a quantization of inverse triad components and general aspects of the hamiltonian constraint equation . here we also discuss the two criteria of admissibility of the continuum approximation and local stability . in section [ hami ] , we present a quantization of the hamiltonian constraint . since the hamiltonian constraint now also has a potential term depending on the spin connection , a quantization of the spin connection is required . this is done using the quantization of inverse triad components . we derive the ( partial ) difference equation satisfied by the physical states and show that in the continuum approximation , to the leading order , the approximating differential equation is precisely the wheeler - dewitt equation with a specific factor ordering dictated by the underlying loop quantization . we also show that the requirement of local stability is satisfied . we demonstrate that the singularity avoidance mechanism found in the case of bianchi i @xcite continues to hold for these more general models . we conclude that the proposed quantization of the hamiltonian constraint is satisfactory with respect to all our requirements . in section [ bianchiix ] , we focus attention on the modification of the potential implied by the quantization of the spin connection . the non - trivial and non - perturbative behavior of the inverse triad components may be expected to lead to substantial modifications of the potential at small triad components . in view of the conjectured central role of the bianchi ix potential , we present a brief description of this potential and its implications for the modified approach to classical singularities . | a quantization of the corresponding hamiltonian constraints is presented and shown to lead to a locally stable , non - singular evolution compatible with almost classical behavior at large volume . as an application , the bianchi ix model and its modified behavior close to its classical singularity is explored . | homogeneous cosmological models with non - vanishing intrinsic curvature require a special treatment when they are quantized with loop quantum cosmological methods . guidance from the full theory which is lost in this context can be replaced by two criteria for an acceptable quantization , admissibility of a continuum approximation and local stability . a quantization of the corresponding hamiltonian constraints is presented and shown to lead to a locally stable , non - singular evolution compatible with almost classical behavior at large volume . as an application , the bianchi ix model and its modified behavior close to its classical singularity is explored . |
gr-qc0311004 | c | in this paper we have extended the methods of homogeneous loop quantum cosmology to models with non - zero spin connection . these models have non - trivial intrinsic curvature from the symmetric background which requires a special treatment compared to the bianchi i model as well as compared to the full theory . initially , therefore , part of the guidance from the full theory is lost and there are more ambiguities when one quantizes the hamiltonian constraint . in @xcite two conditions which have to be imposed for a reasonable quantization , have been introduced and discussed in detail . these stem primarily from the requirement that the loop quantization admit physical states which , in a semiclassical regime , have much less sensitivity to variations on the planck scale so that the idealization of the continuum geometric formulation is well justified . this has been made more explicit by the formulation of the continuum approximation . in section [ hami ] we introduced a quantization of the hamiltonian constraint for all diagonalized bianchi class a models , and showed that the two conditions are satisfied . it was important to take the freedom of phases in the triad representation into account which we did by transforming to a new wave function @xmath122 in ( [ stilde ] ) . the transformation was dictated by the requirement of reproducing the wheeler dewitt equation as the leading approximation . it is satisfying to note that the same transformation also appeared in @xcite where it was used in the context of relating the wave functions obtained in the schrdinger quantization based on connection variables and the wave functions of the wheeler dewitt quantization based on the metric variables . since the two sets of basic variables differ significantly only when the spin connection is non - zero which happens when the intrinsic curvature is non - zero , non - trivial phases show up in this situation . the same transformation then appears whether one uses a loop quantization or a schrdinger quantization in the connection formulation and compares with the wheeler dewitt quantization . we also showed that the mechanism for the absence of singularities continues to hold , despite the phases , exactly as in the case of the bianchi - i model @xcite . thus , as the main result of this paper we proved that for all diagonal bianchi class a models there is a loop quantization which satisfies all the conditions for a good continuum limit as well as that of a singularity - free evolution . that such a quantization exists at all , and even one which provides a general form for all the models , is a non - trivial fact and gives further credence to the viability of loop methods and the physical applications obtained so far . in section [ bianchiix ] we carried out a preliminary exploration of the consequences of the quantized spin connection and the correspondingly modified potential . we focused on the bianchi ix case and pointed out that unlike the classical potential which has infinite walls as the singularity is approached , the quantum modified potential has finite walls . since the infinite walls are responsible for the chaotic approach to the singularity , we expect that the finite walls of the modified potential will significantly alter the approach to the classical singularity @xcite . finally a remark about the relevance of the results is in order . since the connection of the quantization discussed here to the full theory is weaker , it is legitimate to question what this quantization may have to do with the full theory . firstly , we observe that the methods adopted as well as the two admissibility criteria work uniformly for all homogeneous models ( including isotropic sub - models ) which do not have a non - zero spin connection . the direct loss of connection to the full theory is only in the models with spin connection . the treatment of the spin connection enters in two distinct places , firstly in using the inverses of triad components and secondly in the modification of holonomies . the first one is quite natural and inherits the ambiguities in the definition of inverses @xcite . the second one has additional ambiguities due to factor ordering . it is already clear that the first type of ambiguity is _ not _ fixed by the two admissibility criteria . our emphasis in this work has been to demonstrate the existence of _ a uniform method _ of quantization which satisfies the two admissibility criteria _ and _ is singularity free . the uniqueness type of issues , while important , requires a systematic classification of possible ambiguities and is beyond the scope of the present work . likewise the general question of how many of the results of a minisuperspace quantization can survive the full theory is also beyond the scope of this paper . while the spin connection is seen to require a careful treatment of the continuum approximation and also to lead to a modified approach to the bkl singularity in the context of homogeneous models , its role in the full theory is more complicated and is an open issue . for instance , the phase factor in @xmath122 which has the form , @xmath263 is not even well - defined in the full theory due to the transformation properties of the spin connection ( which is no longer a covariant object ) . since the spin connection can be made arbitrarily small locally , by choosing appropriate coordinates , one can expect that the diffeomorphism constraint and its algebra with the hamiltonian constraint will play a role in this issue . requirements such as the local stability condition can also be expected to play a role in analyzing this issue , at least with a local version in the absence of a global internal time . m. b. and g. d. are grateful to madhavan varadarajan and naresh dadhich for invitations to and hospitality at the rri , bangalore , and iucaa , pune , where part of the work has been done . m. b. also thanks the imsc , chennai , for hospitality . the work of m. b. and k. v. was supported in part by nsf grant phy00 - 90091 and the eberly research funds of penn state . | homogeneous cosmological models with non - vanishing intrinsic curvature require a special treatment when they are quantized with loop quantum cosmological methods . guidance from the full theory which is lost in this context can be replaced by two criteria for an acceptable quantization , admissibility of a continuum approximation and local stability . | homogeneous cosmological models with non - vanishing intrinsic curvature require a special treatment when they are quantized with loop quantum cosmological methods . guidance from the full theory which is lost in this context can be replaced by two criteria for an acceptable quantization , admissibility of a continuum approximation and local stability . a quantization of the corresponding hamiltonian constraints is presented and shown to lead to a locally stable , non - singular evolution compatible with almost classical behavior at large volume . as an application , the bianchi ix model and its modified behavior close to its classical singularity is explored . |