asymptotic behavior of partial and false theta functions arising from jacobi forms and regularized characters motivated by recent developments in the representation theory of vertex algebras and conformal field theory we prove several asymptotic results for partial and false theta functions arising from jacobi forms as the modular variable tends to 0 along the imaginary axis and the elliptic variable z is unrestricted in the complex plane we observe that these functions exhibit stokes phenomenon the asymptotic behavior of these functions sharply differs depending on where the elliptic variable z is located within the complex plane we apply our results to study the asymptotic expansions of regularized characters and quantum dimensions of the 1 p singlet w algebra modules important in logarithmic conformal field theory this in particular recovers and extends several results from the work of t creutzig et al int math res not 2016 e print arxiv 1411 3282 pertaining to regularized quantum dimensions

linear response theory for thermally driven quantum open systems this note is a continuation of our recent paper v jaksic y ogata and c a pillet the green kubo formula and onsager reciprocity relations in quantum statistical mechanics commun math phys in press where we have proven the green kubo formula and the onsager reciprocity relations for heat fluxes in thermally driven quantum open systems in this note we extend the derivation of the green kubo formula to heat and charge fluxes and discuss some other generalizations of the model and results of v jaksic y ogata and c a pillet the green kubo formula and onsager reciprocity relations in quantum statistical mechanics commun math phys in press

false theta functions and the verlinde formula abstract we discover new analytic properties of classical partial and false theta functions and their potential applications to representation theory of w algebras and vertex algebras in general more precisely motivated by clues from conformal field theory first we are able to determine modular like transformation properties of regularized partial and false theta functions then after suitable identification of regularized partial false theta functions with the characters of atypical modules for the singlet vertex algebra w 2 2 p 1 we formulate a verlinde type formula for the fusion rules of irreducible w 2 2 p 1 modules

holomorphic differentials thermostats and anosov flows we introduce a new family of thermostat flows on the unit tangent bundle of an oriented riemannian two manifold suitably reparametrised these flows include the geodesic flow of metrics of negative gauss curvature and the geodesic flow induced by the hilbert metric on the quotient surface of divisible convex sets we show that the family of flows can be parametrised in terms of certain weighted holomorphic differentials and investigate their properties in particular we prove that they admit a dominated splitting and we identify special cases in which the flows are anosov in the latter case we study when they admit an invariant measure in the lebesgue class and the regularity of the weak foliations

hamiltonian reformulation and pairing of lyapunov exponents for nose hoover dynamics the nose hamiltonian is adapted leading to a derivation of the nose hoover equations of motion which does not involve time transformations and in which the degree of freedom corresponding to the external reservoir is treated on the same footing as those of the rest of the system in this form it is possible to prove the conjugate pairing rule for lyapunov exponents of this system

the liapunov dimension of strange attractors abstract many papers have been published recently on studies of dynamical processes in which the attracting sets appear quite strange in this paper the question of estimating the dimension of the attractor is addressed while more general conjectures are made here particular attention is paid to the idea that if the jacobian determinant of a map is greater than one and a ball is mapped into itself then generically the attractor will have positive two dimensional measure and most of this paper is devoted to presenting cases with such jacobians for which the attractors are proved to have non empty interior

well posedness and large time behaviour for the non cutoff kac equation with a gaussian thermostat we consider here a kac equation with a gaussian thermostat in the case of a non cutoff cross section under the sole assumptions of finite mass and finite energy for the initial data we prove the existence of a global in time solution for which mass and energy are preserved then via fourier transform techniques we show that this solution is smooth unique and converges to the corresponding stationary state

local stability of dynamical processes in random media abstract a particular type of random dynamical process is considered in which the stochasticity is introduced through randomly fluctuating parameters a method of local multipliers is developed for treating the local stability of such dynamical processes corresponding to infinite dimensional dynamical systems the method is illustrated by several examples by the random diffusion equation random wave equation and random schrodinger equation the evolution equation for the density matrix of a quasiopen statistical system subject to the action of random surrounding is considered the stationary solutions to this equation are found to be unstable against arbitrary small finite random perturbations the notion of random structural stability is introduced

six variable generalization of ramanujan s reciprocity theorem and its variants abstract by virtue of shukla s well known bilateral 8 8 summation formula and watson s transfor mation formula we extend the four variable generalization of ramanujan s reciprocity theorem due to andrews to a six variable one some novel variants of ramanujan s reciprocity theorem and q series identities are presented

dyson s crank of a partition holds he was thus led to conjecture the existence of some other partition statistic which he called the crank this unknown statistic should provide a combinatorial interpretation of p lln 6 in the same way that 1 1 and 1 2 treat the primes 5 and 7 in 4 5 one of us was able to find a crank relative to vector partitions as follows for a partition 7r let 7r be the number of parts of ir and cr n be the sum of the parts of ir or the number ir is partitioning with the convention 0 for the empty partition 0 of 0 let

chaotic principle an experimental test abstract the chaotic hypothesis discussed in gallavotti and cohen 1995 is tested experimentally in a simple conduction model besides a confirmation of the hypothesis predictions the results suggest the validity of the hypothesis in the much wider context in which as the forcing strength grows the attractor ceases to be an anosov system and becomes an axiom a attractor a first text of new predictions is also attempted

entropy production for a class of inverse srb measures we study the entropy production for inverse srb measures for a class of hyperbolic folded repellers presenting both expanding and contracting directions we prove that for most such maps we obtain strictly negative entropy production of the respective inverse srb measures moreover we provide concrete examples of hyperbolic folded repellers where this happens

on q series identities for false theta series abstract we prove several infinite families of q series identities for false theta functions and related series these identities are motivated by considerations of characters of modules of vertex operator superalgebras and of quantum dilogarithms we also obtain closely related modular identities of the gollnitz gordon andrews type as a byproduct of our identities we establish several identities for the rogers dilogarithm function coming from multi q hypergeometric series with double poles

a shooting approach to layers and chaos in a forced duffing equation abstract we study equilibrium solutions for the problem u t e 2 u xx u 3 u cos x u x 0 t u x 1 t 0 using a shooting method we find solutions for all nonzero e for small e we add to the solutions found by previous authors especially angenent mallet paret and peletier and hale and sakamoto and also give new elementary ode proofs of their results among the new results is the existence of internal layer type solutions considering the ode satisfied by equilibria but on an infinite interval we obtain chaos results for 0 3 2 2 3 and 0 1 4 we also consider the problem of bifurcation of solutions as increases

takiff superalgebras and conformal field theory a class of non semisimple extensions of lie superalgebras is studied they are obtained by adjoining to the superalgebra its adjoint representation as an abelian ideal when the superalgebra is of affine kac moody type a generalization of sugawara s construction is shown to give rise to a copy of the virasoro algebra and so presumably to a conformal field theory evidence for this is detailed for the extension of the affinization of the superalgebra its highest weight irreducible modules are classified using spectral flow the irreducible supercharacters are computed and a continuum version of the verlinde formula is verified to give non negative integer structure coefficients interpreting these coefficients as those of the grothendieck ring of fusion partial results on the true fusion ring and its indecomposable structures are deduced

smooth dynamics and new theoretical ideas in nonequilibrium statistical mechanics this paper reviews various applications of the theory of smooth dynamical systems to conceptual problems of nonequilibrium statistical mecanics we adopt a new point of view which has emerged progressively in recent years and which takes seriously into account the chaotic character of the microscopic time evolution the emphasis is on nonequilibrium steady states rather than the traditional approach to equilibrium point of view of boltzmann the nonequilibrium steady states in presence of a gaussian thermostat are described by srb measures in terms of these one can prove the gallavotti cohen fluctuation theorem one can also prove a general linear response formula and study its consequences which are not restricted to near equilibrium situations at equilibrium one recovers in particular the onsager reciprocity relations under suitable conditions the nonequilibrium steady states satisfy the pairing theorem of dettmann and morriss the results just mentioned hold so far only for classical systems they do not involve large size i e they hold without a thermodynamic limit

statistical properties of dynamical systems with some hyperbolicity this paper is about the ergodic theory of attractors and conservative dynamical systems with hyperbolic properties on large parts though not necessarily all of their phase spaces the main results are for discrete time systems to put this work into context recall that for axiom a attractors the picture has been fairly complete since the 1970 s see s1 b r2 since then much progress has been made on two fronts there is a general nonuniform theory that deals with properties common to all diffeomorphisms with nonzero lyapunov exponents o p1 ka ly and there are detailed analyses of specific kinds of dynamical systems including for example billiards 1 dimensional and henon type maps s2 bsc hk j bc2 by1 statistical properties such as exponential decay of correlations are not enjoyed by all diffeomorphisms with nonzero lyapunov exponents the goal of this paper is a systematic understanding of these and other properties for a class of dynamical systems larger than axiom a this class will not be defined explicitly but it includes some of the much studied examples by looking at regular returns to sets with good hyperbolic properties one could give systems in this class a simple dynamical representation conditions for the existence of natural invariant measures exponential mixing and central limit theorems are given in terms of the return times these conditions can be checked in concrete situations giving a unified way of proving a number of results some new and some old among the new results are the exponential decay of correlations for a class of scattering billiards and for a positive measure set of henon type maps

on differentiability of srb states for partially hyperbolic systems consider a one parameter family of diffeomorphisms f e such that f 0 is an anosov element in a standard abelian anosov action having sufficiently strong mixing properties let e be any u gibbs state for f e we prove theorem 1 that for any c function a the map e e a is differentiable at e 0 this implies corollary 2 2 that the difference of birkhoff averages of the perturbed and unperturbed systems is proportional to e we apply this result corollary 3 3 to show that a generic perturbation of the time one map of geodesic flow on the unit tangent bundle over a surface of negative curvature has a unique srb measure with good statistical properties

a shooting approach to the lorenz equations we announce and outline a proof of the existence of a homoclinic orbit of the lorenz equations in addition we develop a shooting technique and two key conditions which lead to the existence of a one to one correspondence between a set of solutions and the set of all infinite sequences of 1 s and 3 s

the leading root of the partial theta function abstract i study the leading root x 0 y of the partial theta function 0 x y n 0 x n y n n 1 2 considered as a formal power series i prove that all the coefficients of x 0 y are strictly positive indeed i prove the stronger results that all the coefficients of 1 x 0 y after the constant term 1 are strictly negative and all the coefficients of 1 x 0 y 2 after the constant term 1 are strictly negative except for the vanishing coefficient of y 3

nonmeromorphic operator product expansion and c2 cofiniteness for a family of w algebras we prove the existence and associativity of the nonmeromorphic operator product expansion for an infinite family of vertex operator algebras the triplet algebras using results from p z tensor product theory while doing this we also show that all these vertex operator algebras are c2 cofinite

on the triplet vertex algebra w p abstract we study the triplet vertex operator algebra w p of central charge 1 6 p 1 2 p p 2 we show that w p is c 2 cofinite but irrational since it admits indecomposable and logarithmic modules furthermore we prove that w p is of finite representation type and we provide an explicit construction and classification of all irreducible w p modules and describe block decomposition of the category of ordinary w p modules all this is done through an extensive use of zhu s associative algebra together with explicit methods based on vertex operators and the theory of automorphic forms moreover we obtain an upper bound for dim a w p finally for p prime we completely describe the structure of a w p the methods of this paper are easily extendable to other w algebras and superalgebras

some theorems on the rogers ramanujan continued fraction in ramanujan s lost notebook in his first two letters to g h hardy and in his notebooks ramanujan recorded many theorems about the rogers ramanujan continued fraction in his lost notebook he offered several further assertions the purpose of this paper is to provide proofs for many of the claims about the rogers ramanujan and generalized rogers ramanujan continued fractions found in the lost notebook these theorems involve among other things modular equations transformations zeros and class invariants

ramanujan s lost notebook iv stacks and alternating parity in partitions this paper is centered around the partition function oe n the number of partitions of n in which the parts arranged in ascending order alternate in parity starting with the smallest part odd briefly oe n counts the number of odd even partitions of n the first few examples for the computation of oe n are n relevant partitions of n oe n 1 1 1 2 0 3 3 1 2 2 4 0 5 5 1 4 2 6 1 2 3 1 7 7 1 6 3 4 3 the generating function for oe n turns out to be

partial theta functions i beyond the lost notebook it is shown how many of the partial theta function identities in ramanujan s lost notebook can be generalized to infinite families of such identities key in our construction is the bailey lemma and a new generalization of the jacobi triple product identity by computing residues around the poles of our identities we find a surprising connection between partial theta function identities and garrett ismail stanton type extensions of multisum rogers ramanujan identities

vertex algebras for s duality we define new deformable families of vertex operator algebras mathfrak a mathfrak g psi sigma associated to a large set of s duality operations in four dimensional supersymmetric gauge theory they are defined as algebras of protected operators for two dimensional supersymmetric junctions which interpolate between a dirichlet boundary condition and its s duality image the mathfrak a mathfrak g psi sigma vertex operator algebras are equipped with two mathfrak g affine vertex subalgebras whose levels are related by the s duality operation they compose accordingly under a natural convolution operation and can be used to define an action of the s duality operations on a certain space of vertex operator algebras equipped with a mathfrak g affine vertex subalgebra we give a self contained definition of the s duality action on that space of vertex operator algebras the space of conformal blocks in the derived sense i e chiral homology for mathfrak a mathfrak g psi sigma is expected to play an important role in a broad generalization of the quantum geometric langlands program namely we expect the s duality action on vertex operator algebras to extend to an action on the corresponding spaces of conformal blocks this action should coincide with and generalize the usual quantum geometric langlands correspondence the strategy we use to define the mathfrak a mathfrak g psi sigma vertex operator algebras is of broader applicability and leads to many new results and conjectures about deformable families of vertex operator algebras

the metric entropy of diffeomorphisms part i characterization of measures satisfying pesin s entropy formula soit m une variete de riemann compacte soit f m m un diffeomorphisme et soit m une mesure de probabilite de borel f invariante sur m on identifie les mesures pour lesquelles l inegalite de margulis et ruelle qui relie l entropie aux exposants de lyapunov atteint l egalite on demontre une formule valable pour toutes les mesures invariantes

dynamical system models of transport chaos characteristics the macroscopic limit and irreversibility the escape rate formalism and the thermostating algorithm describe relaxation towards a decaying state with absorbing boundaries and a steady state of periodic systems respectively it has been shown that the key features of the transport properties of both approaches if modeled by low dimensional dynamical systems can conveniently be described in the framework of multibaker maps in the present paper we discuss in detail the steps required to reach a meaningful macroscopic limit the limit involves a sequence of coarser and coarser descriptions projections until one reaches the level of irreversible macroscopic advection diffusion equations the influence of boundary conditions is studied in detail only a few of the chaos characteristics possess a meaningful macroscopic limit but none of these is sufficient to determine the entropy production in a general non equilibrium state 2003 elsevier b v all rights reserved

dynamical ensembles equivalence in fluid mechanics abstract an attempt to put together various theoretical mathematical or experimental results recently developed in apparently unrelated subjects namely ruelle s 1980 approach to turbulence the body of nose hoover type of molecular dynamics experiments evans and morriss 1990 evans et al 1993 bonetto et al 1996 mathematical results on lyapunov exponents the pairing rule dressler 1988 and dettman and morriss 1995 and experimental results on them sarman et al 1992 bohr et al 1996 theoretical as well as mathematical results on fluctuations frisch and parisi 1984 multifractality and gallavoti and cohen 1995 chaotic hypothesis the key idea that we try to clarify is that of dynamical ensembles as a generalization of the classical equilibrium ensembles arguing that they should be identified with the srb distributions and that they share several properties with the classical ensembles most of the results invoked here did not deal directly with the navier stokes equations and yet they seem to have a lot to do with them as we shall argue here the discussion will focus on the navier stokes and dissipative euler equations with the aim of proposing several experiments apt to test the equivalence of dynamical ensembles and the chaotic hypothesis the ideas developed to a great extent from the efforts put in interpreting the experimental results in bonetto et al 1996

vector valued higher depth quantum modular forms and higher mordell integrals abstract we introduce vector valued higher depth quantum modular forms and investigate examples coming from characters of representations of vertex algebras these are expressed as rank two false theta series generalizing unary false theta series studied from several standpoints we also discuss certain double integrals which may be viewed as the obstruction to modularity of the depth two quantum modular forms we then find explicit formulas for the double error integrals in the form reminiscent of the classical mordell integral

positivity of entropy production we discuss the positivity of the mean entropy production for stochastic systems driven from equilibrium as it was defined in refs 7 and 8 non zero entropy production is closely linked with violation of the detailed balance condition this connection is rigorously obtained for spinflip dynamics we remark that the positivity of entropy production depends on the choice of time reversal transformation hence on the choice of the dynamical variables in the system of interest

higher rank partial and false theta functions and representation theory abstract we study higher rank jacobi partial and false theta functions generalizations of the classical partial and false theta functions associated to positive definite rational lattices in particular we focus our attention on certain kostant s partial theta functions coming from ade root lattices which are then linked to representation theory of w algebras we derive modular transformation properties of regularized higher rank partial and false theta functions as well as kostant s version of these modulo conjectures in representation theory as an application we compute regularized quantum dimensions of atypical and typical modules of narrow logarithmic w algebras associated to rescaled root lattices this paper substantially generalize our previous work 19 pertaining to 1 p singlet w algebras the sl 2 case results in this paper are very general and are applicable in a variety of situations

an expectation formula with applications abstract in this paper we derive an expectation formula of a random variable having distribution w x q as applications of the expectation formula we give a transformation formula and an expansion of sears 2 3 transformation formula

the t coefficient method to partial theta function identities and ramanujan s 1ψ1 summation formula this paper is concerned with a uniform approach the t coefficient method to basic hypergeometric series by virtue of this method some new and elementary proofs for four partial theta function identities due to alladi and berkovich 2004 8 andrews and warnaar 2007 4 berkovich 5 and warnaar 2003 2 as well as for ramanujan s 1 1 summation formula are obtained

extensions of ramanujan s reciprocity theorem and the andrews askey integral abstract ramanujan s reciprocity theorem may be considered as a three variable extension of jacobi s triple product identity using the method of q partial differential equations we extend ramanujan s reciprocity theorem to a seven variable reciprocity formula using the same method the andrews askey integral formula is extended to a q integral formula which has seven parameters with base q

modular group representations and fusion in logarithmic conformal field theories and in the quantum group center the sl 2 representation on the center of the restricted quantum group open image in new window at the primitive 2pth root of unity is shown to be equivalent to the sl 2 representation on the extended characters of the logarithmic 1 p conformal field theory model the multiplicative jordan decomposition of the open image in new window ribbon element determines the decomposition of into a pointwise product of two commuting sl 2 representations one of which restricts to the grothendieck ring this restriction is equivalent to the sl 2 representation on the 1 p characters related to the fusion algebra via a nonsemisimple verlinde formula the grothendieck ring of open image in new window at the primitive 2pth root of unity is shown to coincide with the fusion algebra of the 1 p logarithmic conformal field theory model as a by product we derive q binomial identities implied by the fusion algebra realized in the center of open image in new window

conformally symplectic dynamics and symmetry of the lyapunov spectrum a generalization of the hamiltonian formalism is studied and the symmetry of the lyapunov spectrum established for the resulting systems the formalism is applied to the gausssian isokinetic dynamics of interacting particles with hard core collisions and other systems

resonances for axiom bf a flows etant donne un flot d axiome a sur m et des fonctions lisses b c m r on montre que la fonction de correlation en temps bc pour un etat de gibbs a une transformee de fourier bc mesomorphe dans une bande on etudie les residus des poles de bc

on the dual nature of partial theta functions and appell lerch sums abstract in recent work hickerson and the author demonstrated that it is useful to think of appell lerch sums as partial theta functions this notion can be used to relate identities involving partial theta functions with identities involving appell lerch sums in this sense appell lerch sums and partial theta functions appear to be dual to each other this duality theory is not unlike that found by andrews between various sets of identities of rogers ramanujan type with respect to baxter s solution to the hard hexagon model of statistical mechanics as an application we construct bilateral q series with mixed mock modular behaviour in subsequent work we see that our bilateral series are well suited for computing radial limits of ramanujan s mock theta functions

proof of lyapunov exponent pairing for systems at constant kinetic energy we present a proof that a system consisting of any finite number of particles that move under the action of a scalar potential at constant kinetic energy exhibits conjugate pairing of lyapunov exponents that is the lyapunov exponents come in pairs which sum to the same constant this result generalizes previous results because it is independent of the size of the system s1063 651x 96 51206 7

differentiation of srb states abstract let f be a diffeomorphism of a manifold m and f a generalized srb state for f if supp f is a hyperbolic compact set we show that the map is differentiable in a suitable functional setup and we compute the derivative when supp f is an attractor the derivative is given by where x is the vector field this formula can be extended to time dependent situations and also at least formally to nonuniformly hyperbolic situations the above results will find their use in the study of the onsager reciprocity relations and the fluctuation dissipation formula of nonequilibrium statistical mechanics

measure theoretic degrees and topological pressure for non expanding transformations we consider invariant sets of saddle type for non invertible smooth maps f and equilibrium measures associated to holder potentials on we define a notion of measure theoretic asymptotic degree of f with respect to the measure on the fractal set in our case the equilibrium measure is the unique linear functional in c tangent to the pressure function p c r at in particular for the measure of maximal entropy 0 of f we obtain the asymptotic degree of f which represents the average rate of growth of the number of n preimages of x that remain in when n notice that in general is not totally invariant for f to this end we will obtain first a formula for the jacobians of the probability with respect to arbitrary iterates fm m 2 we then show that a formula for the topological pressure p that holds in the expanding case is no longer true on saddle sets in the saddle case we find a new formula for the pressure involving weighted sums on preimage sets we also apply the asymptotic degrees together with various pressure functionals in order to obtain estimates for the hausdorff dimension of stable slices through certain sets of full measure in the fractal in the end we give also some concrete examples on saddle folded sets

fluctuation of the entropy production for the lorentz gas under small external forces in this paper we study the physical and statistical properties of the periodic lorentz gas with finite horizon driven to a non equilibrium steady state by the combination of non conservative external forces and deterministic thermostats a version of this model was introduced by chernov eyink lebowitz and sinai and subsequently generalized by chernov and the third author non equilibrium steady states for these models are srb measures and they are characterized by the positivity of the steady state entropy production rate our main result is to establish that the entropy production in this context equal to the phase space contraction satisfies the gallavotti cohen fluctuation relation the main tool needed in the proof is the family of anisotropic banach spaces introduced by the first and third authors to study the ergodic and statistical properties of billiards using transfer operator techniques

on the asymptotic behavior of unimodal rank generating functions abstract in a recent paper j lovejoy and the second author conjectured that ranks for four types of unimodal like sequences satisfy certain inequalities in this paper we prove these conjectures asymptotically for this we use wright s circle method and analyze the asymptotic behavior of certain general partial theta functions

a review of the hydrodynamic lyapunov modes of hard disk systems a review of the results obtained for hard disk fluids confined to a quasi one dimensional qod system is presented one of the main achievements in recent years has been determining the hydrodynamic lyapunov modes hlms as covariant stable and unstable manifolds defined by k vector analogues of the zero exponent subspace of conserved quantities the tangent space dynamics allows an interpretation of the hlms as reduced hydrodynamic fields over the perturbations and is expanded upon in this paper the time evolution of these modes is governed by the dynamics of conjugate pairs of stable and unstable directions each pair is seen to interact in a subspace almost completely separated from other vectors the modes undergo a rotation until they are collinear with the stable or unstable manifold the angle distributions between these covariant stable and unstable hlms are also determined and tangencies are not observed for a generic chaotic trajectory this article is part of a special issue of journal of physics a mathematical and theoretical devoted to lyapunov analysis from dynamical systems theory to applications

probabilistic approach to pattern selection abstract the problem of pattern selection arises when the evolution equations have many solutions whereas observed patterns constitute a much more restricted set an approach is advanced for treating the problem of pattern selection by defining the probability distribution of patterns then the most probable pattern naturally corresponds to the largest probability weight this approach provides the ordering principle for the multiplicity of solutions explaining why some of them are more preferable than others the approach is applied to solving the problem of turbulent photon filamentation in resonant media

ergodic properties of classical dissipative systems i we consider a class of models in which a hamiltonian system a with a finite number of degrees of freedom is brought into contact with an infinite heat reservoir b we develop the formalism required to describe these models near thermal equilibrium using a combination of abstract spectral techniques and harmonic analysis we investigate the singular spectrum of the liouvillean l of the coupled system a b we provide a natural set of conditions which ensure that the spectrum of l is purely absolutely continuous except for a simple eigenvalue at zero it then follows from the spectral theory of dynamical systems koopmanism that the system a b is strongly mixing from a probabilistic point of view we study a new class of random processes on finite dimensional manifolds non markovian ornstein uhlenbeck processes the paths of such a process are solutions of a random integro differential equation with gaussian noise which is a natural generalization of the well known langevin equation in this context we establish that under appropriate conditions the ou process is strongly mixing even when the langevin equation has memory and is driven by non white noise

the green kubo formula and power spectrum of reversible markov processes as is known the entropy production rate of a stationary markov process vanishes if and only if the process is reversible in this paper we discuss the reversibility of a stationary markov process from a functional analysis point of view it is shown that the process is reversible if and only if it has a symmetric markov semigroup equivalently a self adjoint infinitesimal generator applying this fact we prove that the green kubo formula holds for reversible markov processes by demonstrating that the power spectrum of each reversible markov process is lorentz typed we show that it is impossible for stochastic resonance to occur in systems with zero entropy production

one dimensional gibbs states and axiom a diffeomorphisms on etudie la mecanique statistique en equilibre de systemes reticules classiques a une dimension a interactions a decroissance exponentielle pour de tels systemes les transformees de fourier des fonctions de correlation de paires sont meromorphes dans une bande et les residus des poles peuvent s exprimer en termes de distributions de gibbs on applique ces resultats aux diffeomorphismes de l axiome a de smale

on decay of correlations in anosov flows there is some disagreement about the meaning of the phrase chaotic flow however there is no doubt that mixing anosov flows provides an example of such systems anosov systems were introduced and extensively studied in his classical memoir a among other things he proved the following fact known now as anosov alternative for flows either every strong stable and strong unstable manifold is everywhere dense or the flow gt is a suspension over an anosov diffeomorphism by a constant roof function if the first alternative holds gt is mixing with respect to every gibbs measure see pp2 now the natural question is what is the estimated rate of mixing this is certainly one of the simplest questions concerning correlation decay in continuous time systems nevertheless the only results obtained until recently dealt with the case when the system discussed had an additional algebraic structure the easier case of anosov diffeomorphisms can be treated by the methods of thermodynamic formalism of sinai ruelle and bowen b2 namely one uses markov partitions to construct an isomorphism between the diffeomorphism and a subshift of finite type and then proves that all such subshifts are exponentially mixing this method would succeed also for flows if any suspension over a subshift of finite type had exponentially decaying correlations however the simplest example suspensions with locally constant roof functions never have such a property ri one can use the above observation to produce examples of axiom a flows with arbitrary slow correlation decay it became clear therefore that some additional geometric properties should be taken into account in recent work chernov chl ch2 has employed a uniform nonintegrability condition to get a subexponential estimate for correlation functions for geodesic flows on surfaces of variable negative curvature his method relies on the technique of markov approximation developed in chl the aim of this paper is to combine geometric considerations of chernov with the thermodynamic formalism approach the later method seems to be more appropriate than markov approximations since it gives simpler

differentiating the absolutely continuous invariant measure of an interval map f with respect to f let the map f 1 1 1 1 have a c i m absolutely continuous f invariant measure with respect to lebesgue let be the change of corresponding to a perturbation x f f 1 of f formally we have for differentiable a

a note on moment integrals and some applications abstract in this paper two moment integrals are given by the method of transformation in addition generalizations of sears s transformation are obtained by moment integrals moreover certain q mehler formulas for rogers szego polynomials are gained by moment integrals besides an open problem of trilinear generating function is deduced by moment integrals at last generalizations of u n 1 type kalnins miller transformation are achieved by moment integrals

microscopic origins of irreversible macroscopic behavior time asymmetric behavior as embodied in the second law of thermodynamics is observed in individual macroscopic systems it can be understood as arising naturally from time symmetric microscopic laws in accord with the ideas of thompson maxwell and boltzmann alternative explanations based on equating irreversible macroscopic behavior with the mixing type of behavior already present in the time evolution of ensembles probability distributions of certain systems having only a few degrees of freedom are in my opinion unnecessary misguided and misleading

tribasic integrals and identities of rogers ramanujan type some integrals involving three bases are evaluated as infinite products using complex analysis many special cases of these integrals may be evaluated in another way to find infinite sum representations for these infinite products the resulting identities are identities of rogers ramanujan type some integer partition interpretations of these identities are given generalizations of the rogers ramanujan type identities involving polynomials are given again as corollaries of integral evaluations

ramanujan s lost notebook and the virasoro algebra by using the theory of vertex operator algebras we gave a new proof of the famous ramanujan s modulus 5 modular equation from his lost notebook p 139 in r furthermore we obtained an infinite list of q identities for all odd moduli thus we generalized the result of ramanujan

nonsemisimple fusion algebras and the verlinde formula we find a nonsemisimple fusion algebra open image in new window associated with each 1 p virasoro model we present a nonsemisimple generalization of the verlinde formula which allows us to derive open image in new window from modular transformations of characters

on dyson s crank conjecture and the uniform asymptotic behavior of certain inverse theta functions in this paper we prove a longstanding conjecture by freeman dyson concerning the limiting shape of the crank generating function we fit this function in a more general family of inverse theta functions which play a key role in physics

hamiltonian derivation of a detailed fluctuation theorem we analyze the microscopic evolution of a system undergoing a far from equilibrium thermodynamic process explicitly accounting for the degrees of freedom of participating heat reservoirs we derive a hybrid result similar in form to both the fluctuation theorem and a statement of detailed balance we relate this result to the steady state fluctuation theorem and to a free energy relation valid far from equilibrium

hardy petrovitch hutchinson s problem and partial theta function in 1907 m petrovitch initiated the study of a class of entire functions all whose finite sections i e truncations are real rooted polynomials he was motivated by previous studies of e laguerre on uniform limits of sequences of real rooted polynomials and by an interesting result of g h hardy an explicit description of this class in terms of the coefficients of a series is impossible since it is determined by an infinite number of discriminant inequalities one for each degree however interesting necessary or sufficient conditions can be formulated in particular j i hutchinson has shown that an entire function p x a 0 a 1 x a n x n with strictly positive coefficients has the property that all of its finite segments a i x i a i 1 x i 1 a j x j have only real roots if and only if a i 2 a i 1 a i 1 4 for i 1 2 in the present paper we give sharp lower bounds on the ratios a i 2 a i 1 a i 1 i 1 2 for the class considered by m petrovitch in particular we show that the limit of these minima when i infinity equals the inverse of the maximal positive value of the parameter for which the classical partial theta function belongs to the laguerre polya class l pi we also explain the relation between newton s and hutchinson s inequalities and the logarithmic image of the set of all real rooted polynomials with positive coefficients

expansion exponents for nonequilibrium systems abstract local expansion exponents for nonequilibrium dynamical systems described by partial differential equations are introduced these exponents show whether the system phase volume expands contracts or is conserved in time the ways of calculating the exponents are discussed the principle of minimal expansion provides the basis for treating the problem of pattern selection the exponents are also defined for stochastic dynamical systems the analysis of the expansion exponent behavior for quasi isolated systems results in the formulation of two other principles the principle of asymptotic expansion tells that the phase volumes of quasi isolated systems expand at asymptotically large times the principle of time irreversibility follows from the asymptotic phase expansion since the direction of time arrow can be defined by the asymptotic expansion of phase volume

the number dependence of the maximum lyapunov exponent we examine the number dependence of the largest lyapunov exponent for nonlinear dynamical systems in one two and three cartesian dimensions our results suggest that the largest lyapunov exponent diverges logarithmically with system size independently of the number of cartesian dimensions and the interaction potential

chaotic hypothesis fluctuation theorem and singularities the chaotic hypothesis has several implications which have generated interest in the literature because of their generality and because a few exact predictions are among them however its application to physics problems requires attention and can lead to apparent inconsistencies in particular there are several cases that have been considered in the literature in which singularities are built in the models for instance when among the forces there are lennard jones potentials which are infinite in the origin and the constraints imposed on the system do not forbid arbitrarily close approach to the singularity even though the average kinetic energy is bounded the situation is well understood in certain special cases in which the system is subject to gaussian noise here the treatment of rather general singular systems is considered and the predictions of the chaotic hypothesis for such situations are derived the main conclusion is that the chaotic hypothesis is perfectly adequate to describe the singular physical systems we consider ie deterministic systems with thermostat forces acting according to gauss principle for the constraint of constant total kinetic energy isokinetic gaussian thermostats close and far from equilibrium near equilibrium it even predicts a fluctuation relation which in deterministic cases with more general thermostat forces ie not necessarily of gaussian isokinetic nature extends recent relations obtained in situations in which the thermostatting forces satisfy gauss principle this relation agrees where expected with the fluctuation theorem for perfectly chaotic systems the results are compared with some recent works in the literature

reversibility coarse graining and the chaoticity principle we describe a way of interpreting the chaotic principle of gc1 more extensively than it was meant in the original works mathematically the analysis is based on the dynamical notions of axiom a and axiom b and on the notion of axiom c that we introduce arguing that it is suggested by the results of an experiment bgg on chaotic motions physically we interpret a breakdown of the anosov property of a time reversible attractor replaced as a control parameter changes by an axiom a property as a spontaneous breakdown of the time reversal symmetry the relation between time reversal and the symmetry that remains after the breakdown is analogous to the breakdown of t invariance while tcp still holds

non equilibrium statistical mechanics of anharmonic chains coupled to two heat baths at different temperatures we study the statistical mechanics of a finite dimensional non linear hamiltonian system a chain of anharmonic oscillators coupled to two heat baths described by wave equations assuming that the initial conditions of the heat baths are distributed according to the gibbs measures at two different temperatures we study the dynamics of the oscillators under suitable assumptions on the potential and on the coupling between the chain and the heat baths we prove the existence of an invariant measure for any temperature difference i e we prove the existence of steady states furthermore if the temperature difference is sufficiently small we prove that the invariant measure is unique and mixing in particular we develop new techniques for proving the existence of invariant measures for random processes on a non compact phase space these techniques are based on an extension of the commutator method of hormander used in the study of hypoelliptic differential operators

logarithmic conformal field theory beyond an introduction this article aims to review a selection of central topics and examples in logarithmic conformal field theory it begins with the remarkable observation of cardy that the horizontal crossing probability of critical percolation may be computed analytically within the formalism of boundary conformal field theory cardy s derivation relies on certain implicit assumptions which are shown to lead inexorably to indecomposable modules and logarithmic singularities in correlators for this a short introduction to the fusion algorithm of nahm gaberdiel and kausch is provided while the percolation logarithmic conformal field theory is still not completely understood there are several examples for which the formalism familiar from rational conformal field theory including bulk partition functions correlation functions modular transformations fusion rules and the verlinde formula has been successfully generalized this is illustrated for three examples the singlet model related to the triplet model symplectic fermions and the fermionic bc ghost system the fractional level wess zumino witten model based on at related to the bosonic ghost system and the wess zumino witten model for the lie supergroup related to at and 1 the bershadsky polyakov algebra and the feigin semikhatov algebras these examples have been chosen because they represent the most accessible and most useful members of the three best understood families of logarithmic conformal field theories the logarithmic minimal models the fractional level wess zumino witten models and the wess zumino witten models on lie supergroups excluding in this review the emphasis lies on the representation theory of the underlying chiral algebra and the modular data pertaining to the characters of the representations each of the archetypal logarithmic conformal field theories is studied here by first determining its irreducible spectrum which turns out to be continuous as well as a selection of natural reducible but indecomposable modules this is followed by a detailed description of how to obtain character formulae for each irreducible a derivation of the action of the modular group on the characters and an application of the verlinde formula to compute the grothendieck fusion rules in each case the genuine fusion rules are known so comparisons can be made and favourable conclusions drawn in addition each example admits an infinite set of simple currents hence extended symmetry algebras may be constructed and a series of bulk modular invariants computed the spectrum of such an extended theory is typically discrete and this is how the triplet model arises for example moreover simple current technology admits a derivation of the extended algebra fusion rules from those of its continuous parent theory finally each example is concluded by a brief description of the computation of some bulk correlators a discussion of the structure of the bulk state space and remarks concerning more advanced developments and generalizations the final part gives a very short account of the theory of staggered modules the simplest class of representations that are responsible for the logarithmic singularities that distinguish logarithmic theories from their rational cousins these modules are discussed in a generality suitable to encompass all the examples met in this review and some of the very basic structure theory is proven then the important quantities known as logarithmic couplings are reviewed for virasoro staggered modules and their role as fundamentally important parameters akin to the three point constants of rational conformal field theory is discussed an appendix is also provided in order to introduce some of the necessary but perhaps unfamiliar language of homological algebra

chaotic hypothesis onsager reciprocity and fluctuation dissipation theorem it is shown that the chaoticity hypothesis recently introduced in statistical mechanics which is analogous to ruelle s principle for turbulence implies the onsager reciprocity and the fluctuation dissipation theorem in various reversible models for coexisting transport phenomena

on simplicity of vacuum modules we find necessary and sufficient conditions of irreducibility of vacuum modules over affine lie algebras and superalgebras from this we derive conditions of simplicity of minimal w algebras moreover in the case of the virasoro and neveu schwarz algebras we obtain explicit formulas for the vacuum determinants

a shooting approach to chaos in the lorenz equations abstract we prove that under certain conditions the lorenz equations support a form of chaos the conditions have been verified for a particular set of parameter values showing that there is a natural 1 1 correspondence between a set of solutions and the set of all sequences on countably many symbols the computing required was modest

ramanujan s lost notebook v euler s partition identity on examine deux identites de ramanujan etroitement liees a l identite d euler relative aux partitions

on the definition of entropy production via examples we present a definition of entropy production rate for classes of deterministic and stochastic dynamics the point of departure is a gibbsian representation of the steady state path space measure for which the density is determined with respect to the time reversed process the gibbs formalism is used as a unifying algorithm capable of incorporating basic properties of entropy production in nonequilibrium systems our definition is motivated by recent work on the gallavotti cohen local fluctuation theorem and it is illustrated via a number of examples