periodic and almost periodic solutions of nonlinear discrete volterra equations with unbounded delay the existence of periodic and almost periodic solutions of nonlinear discrete volterra equations with unbounded delay is obtained by using stability properties of a bounded solution

equivalence between poisson representation and fock space formalism for birth death processes the equivalence between the poisson representation and the fock space formalism both used to derive coarse grained equations from the master equation for non equilibrium systems is explicitly demonstrated

crystal energy functions via the charge in types a and c the ram yip formula for macdonald polynomials at t 0 provides a statistic which we call charge in types a and c it can be defined on tensor products of kashiwara nakashima single column crystals in this paper we prove that the charge is equal to the negative of the energy function on affine crystals the algorithm for computing charge is much simpler and can be more efficiently computed than the recursive definition of energy in terms of the combinatorial r matrix

asymptotically periodic solutions of volterra system of difference equations a volterra system of difference equations of the form x s n 1 a s n b s n x s n email protected email protected i 0nk s p n i x p i where email protected n a s b s x s n r and k s p nxn r s 1 2 r is studied sufficient conditions for the existence of asymptotically periodic solutions of this system are derived

bgg reciprocity for current algebras abstract we study the category i gr of graded representations with finite dimensional graded pieces for the current algebra g c t where g is a simple lie algebra this category has many similarities with the category o of modules for g and in this paper we formulate and study an analogue of the famous bgg duality we recall the definition of the projective and simple objects in i gr which are indexed by dominant integral weights the role of the verma modules is played by a family of modules called the global weyl modules we show that in the case when g is of type s l 2 the projective module admits a flag in which the successive quotients are finite direct sums of global weyl modules the multiplicity with which a particular weyl module occurs in the flag is determined by the multiplicity of a jordan holder series for a closely associated family of modules called the local weyl modules we conjecture that the result remains true for arbitrary simple lie algebras we also prove some combinatorial product sum identities involving kostka polynomials which arise as a consequence of our theorem

asymptotically periodic solutions of second order difference equations abstract in this paper we obtain sufficient conditions for the existence of asymptotically periodic solutions of the second order difference equations of the form r n x n a n f x n b n moreover we give sufficient conditions under which every bounded solution of the above equation is asymptotically periodic our approach allows us to control the degree of approximation

periodic solutions of non linear discrete volterra equations with finite memory in this paper we discuss the existence of periodic solutions of discrete and discretized non linear volterra equations with finite memory the literature contains a number of results on periodic solutions of non linear volterra integral equations with finite memory of a type that arises in biomathematics the summation equations studied here can arise as discrete models in their own right but are as we demonstrate of a type that arise from the discretization of such integral equations our main results are in two parts i results for discrete equations and ii consequences for quadrature methods applied to integral equations the first set of results are obtained using a variety of fixed point theorems the second set of results address the preservation of properties of integral equations on discretizing them the effect of weak singularities is addressed in a final section the detail that is presented which is supplemented using appendices reflects the differing prerequisites of functional analysis and numerical analysis that contribute to the outcomes

predator prey cycles from resonant amplification of demographic stochasticity we present the simplest individual level model of predator prey dynamics and show via direct calculation that it exhibits cycling behavior the deterministic analogue of our model recovered when the number of individuals is infinitely large is the volterra system with density dependent prey reproduction which is well known to fail to predict cycles this difference in behavior can be traced to a resonant amplification of demographic fluctuations which disappears only when the number of individuals is strictly infinite our results indicate that additional biological mechanisms such as predator satiation may not be necessary to explain observed predator prey cycles in real finite populations

chemical langevin equation a path integral view of gillespie s derivation in 2000 gillespie rehabilitated the chemical langevin equation cle by describing two conditions that must be satisfied for it to yield a valid approximation of the chemical master equation cme in this work we construct an original path integral description of the cme and show how applying gillespie s two conditions to it directly leads to a path integral equivalent to the cle we compare this approach to the path integral equivalent of a large system size derivation and show that they are qualitatively different in particular both approaches involve converting many sums into many integrals and the difference between the two methods is essentially the difference between using the euler maclaurin formula and using riemann sums our results shed light on how path integrals can be used to conceptualize coarse graining biochemical systems and are readily generalizable

breakthroughs in the theory of macdonald polynomials in 1998 i g macdonald 1 introduced a remarkable new basis for the space of symmetric functions the elements of this basis are denoted where is a partition and p q are two free parameters the s which are now called macdonald polynomials specialize to many of the well known bases for the symmetric functions by suitable choices of the parameters q and t in fact we can obtain in this manner the schur functions the hall littlewood symmetric functions the jack symmetric functions the zonal symmetric functions the zonal spherical functions and the elementary and monomial symmetric functions

a combinatorial formula for nonsymmetric macdonald polynomials we give a combinatorial formula for the nonsymmetric macdonald polynomials e x q t the formula generalizes our previous combinatorial interpretation of the integral form symmetric macdonald polynomials j x q t we prove the new formula by verifying that it satisfies a recurrence due to knop and sahi that characterizes the nonsymmetric macdonald polynomials

system size expansion using feynman rules and diagrams few analytical methods exist for quantitative studies of large fluctuations in stochastic systems in this article we develop a simple diagrammatic approach to the chemical master equation that allows us to calculate multi time correlation functions which are accurate to any desired order in van kampen s system size expansion specifically we present a set of feynman rules from which this diagrammatic perturbation expansion can be constructed algorithmically we then apply the methodology to derive in closed form the leading order corrections to the linear noise approximation of the intrinsic noise power spectrum for general biochemical reaction networks finally we illustrate our results by describing noise induced oscillations in the brusselator reaction scheme which are not captured by the common linear noise approximation

ω expansion of van kampen through functional integrals we give a functional integral version of the van kampen expansion method for the master equation explicit expressions are given for the generating functional of correlation functions

a rigorous derivation of the chemical master equation it is widely believed that the chemical master equation has no rigorous microphysical basis and hence no a priori claim to validity this view is challenged here through arguments purporting to show that the chemical master equation is exact for any gas phase chemical system that is kept well stirred and thermally equilibrated

a combinatorial formula for macdonald polynomials abstract in this paper we use the combinatorics of alcove walks to give uniform combinatorial formulas for macdonald polynomials for all lie types these formulas resemble the formulas of haglund haiman and loehr for macdonald polynomials of type gl n at q 0 these formulas specialize to the formula of schwer for the macdonald spherical function in terms of positively folded alcove walks and at q t 0 these formulas specialize to the formula for the weyl character in terms of the littelmann path model in the positively folded gallery form of gaussent and littelmann

a probabilistic interpretation of the macdonald polynomials the two parameter macdonald polynomials are a central object of algebraic combinatorics and representation theory we give a markov chain on partitions of k with eigenfunctions the coefficients of the macdonald polynomials when expanded in the power sum polynomials the markov chain has stationary distribution a new two parameter family of measures on partitions the inverse of the macdonald weight rescaled the uniform distribution on cycles of permutations and the ewens sampling formula are special cases the markov chain is a version of the auxiliary variables algorithm of statistical physics properties of the macdonald polynomials allow a sharp analysis of the running time in natural cases a bounded number of steps suffice for arbitrarily large k

an alternative route to the system size expansion the master equation is rarely exactly solvable and hence various means of approximation have been devised a popular systematic approximation method is the system size expansion which approximates the master equation by a generalised fokker planck equation here we first review the use of the expansion by applying it to a simple chemical system the example shows that the solution of the generalised fokker planck equation obtained from the expansion is generally not positive definite and hence cannot be interpreted as a probability density function based on this observation one may also a priori conclude that moments calculated from the solution of the generalised fokker planck equation are not accurate however calculation shows these moments to be in good agreement with those obtained from the exact solution of the master equation we present an alternative simpler derivation which directly leads to the same moments as the system size expansion but which bypasses the use of generalised fokker planck equations thus circumventing the problem with the probabilistic interpretation of the solution of these equations

on some periodic systems of max type difference equations we show that all positive solutions to the system of max type difference equations xn 1 max1 i m1f1ixn ki 1 1 1 xn ki 2 1 2 xn ki l 1 l n xn s 1 xn 2 max1 i m2f2ixn ki 1 2 1 xn ki 2 2 2 xn ki l 2 l n xn s 2 xn l max1 i mlflixn ki 1 l 1 xn ki 2 l 2 xn ki l l l n xn s l n n0 where s l mj ki t j n j t 1 l and for a fixed j i 1 mj and where the functions fji 0 l n0 0 j 1 l i 1 mj satisfy some conditions are eventually periodic with not necessarily prime period s a related result for the corresponding system of min type difference equations is also proved

construction and accuracy of partial differential equation approximations to the chemical master equation the mesoscopic description of chemical kinetics the chemical master equation can be exactly solved in only a few simple cases the analytical intractability stems from the discrete character of the equation and hence considerable effort has been invested in the development of fokker planck equations second order partial differential equation approximations to the master equation we here consider two different types of higher order partial differential approximations one derived from the system size expansion and the other from the kramers moyal expansion and derive the accuracy of their predictions for chemical reactive networks composed of arbitrary numbers of unimolecular and bimolecular reactions in particular we show that the partial differential equation approximation of order q from the kramers moyal expansion leads to estimates of the mean number of molecules accurate to order 2q 3 2 of the variance of the fluctuations in the number of molecules accurate to order 2q 5 2 and of skewness accurate to order q 2 we also show that for large q the accuracy in the estimates can be matched only by a partial differential equation approximation from the system size expansion of approximate order 2q hence we conclude that partial differential approximations based on the kramers moyal expansion generally lead to considerably more accurate estimates in the mean variance and skewness than approximations of the same order derived from the system size expansion

eventually constant solutions of a rational difference equation we describe all the solutions of a rational difference equation from putnam s mathematical competition which are eventually equal to its positive equilibrium email protected 1 as a consequence we give a new elegant and short proof of the fact that the equation has a positive solution which is not eventually equal to one moreover we show that almost all solutions of the equation are not eventually equal to one

matrix product formula for macdonald polynomials we derive a matrix product formula for symmetric macdonald polynomials our results are obtained by constructing polynomial solutions of deformed knizhnik zamolodchikov equations which arise by considering representations of the zamolodchikov faddeev and yang baxter algebras in terms of t deformed bosonic operators these solutions are generalised probabilities for particle configurations of the multi species asymmetric exclusion process and form a basis of the ring of polynomials in n variables whose elements are indexed by compositions for weakly increasing compositions anti dominant weights these basis elements coincide with non symmetric macdonald polynomials our formulas imply a natural combinatorial interpretation in terms of solvable lattice models they also imply that normalisations of stationary states of multi species exclusion processes are obtained as macdonald polynomials at q 1

on combinatorial formulas for macdonald polynomials a recent breakthrough in the theory of type a macdonald polynomials is due to haglund haiman and loehr who exhibited a combinatorial formula for these polynomials in terms of a pair of statistics on fillings of young diagrams ram and yip gave a formula for the macdonald polynomials of arbitrary type in terms of so called alcove walks these originate in the work of gaussent littelmann and of the author with postnikov on discrete counterparts to the littelmann path model in this paper we relate the above developments by explaining how the ram yip formula compresses to a new formula which is similar to the haglund haiman loehr one but contains considerably fewer terms

a littlewood richardson rule for macdonald polynomials macdonald polynomials are orthogonal polynomials associated to root systems and in the type a case the symmetric macdonald polynomials are a common generalization of schur functions macdonald spherical functions and jack polynomials we use the combinatorics of alcove walks to calculate products of monomials and intertwining operators of the double affine hecke algebra from this we obtain a product formula for macdonald polynomials of general lie type

disconjugacy disfocality and oscillation of second order difference equations abstract we obtain necessary and sufficient conditions for the disconjugacy disfocality or oscillation of the second order linear difference equation the conditions are given explicitly in terms of the coefficients in the equation

ls galleries the path model and mv cycles we give an interpretation of the path model of a representation 18 of a complex semisimple algebraic group g in terms of the geometry of its affine grassmannian in this setting the paths are replaced by ls galleries in the affine coxeter complex associated to the weyl group of g to explain the connection with geometry consider a demazure hansen bott samelson desingularization of the closure of an orbit g c t in the affine grassmannian the homology of has a basis given by bia lynicki birula cell s which are indexed by the t fixed points in now the points of can be identified with galleries of a fixed type in the affine tits building associated to g and the t fixed points correspond in this language to combinatorial galleries of a fixed type in the affine coxeter complex we determine those galleries such that the associated cell has a non empty intersection with g c t identified with an open subset of and we show that the closures of the strata associated to ls galleries are exactly the mv cycles 24 which form a basis of the representation v for the langland s dual group g

generalized weyl modules and nonsymmetric q whittaker functions abstract we introduce generalized global weyl modules and relate their graded characters to nonsymmetric macdonald polynomials and nonsymmetric q whittaker functions in particular we show that the series part of the nonsymmetric q whittaker function is a generating function for the graded characters of generalized global weyl modules

periodicity of a class of nonautonomous max type difference equations abstract in this note we prove several general and easily applicable results regarding the periodicity of positive solutions of the following difference equations x n max f 1 x n k 1 n f m x n k m n x n s n n 0 and x n min f 1 x n k 1 n f m x n k m n x n s n n 0 where 1 k 1 k m m s n and where the functions f i 0 n 0 0 1 i m satisfy some additional conditions these results considerably extend and unify numerous results on max type difference equations existing in the literature

an application of darbo s fixed point theorem in the investigation of periodicity of solutions of difference equations using the technique of darbo s fixed point theorem we investigate a nonlinear second order difference equation of the form d r n dx n a nf x n 1 where x n 0 r a r n 0 r and f r r is a continuous function additionally the sturm liouville difference equation is considered as a special case of the above equation sufficient conditions for the existence of an asymptotically periodic solution of this equation are obtained an example illustrates the result next the conditions under which this equation has no asymptotically periodic solutions are presented

periodic solutions of difference equations abstract in this paper we discuss the existence of periodic solutions of periodic difference equations including x n 1 f n x n n z where x and f are d vectors and z denotes the set of integers as well as scalar autonomous difference equations including x n f x n 1 n z first we state some recent results concerning periodic solutions of periodic difference equations then we show the existence of periodic solutions of scalar autonomous difference equations by using schauder s fixed point theorem we illustrate the results with some examples