quantum cohomology of flag manifolds and toda lattices we discuss relations of vafa s quantum cohomology with floer s homology theory introduce equivariant quantum cohomology formulate some conjectures about its general properties and on the basis of these conjectures compute quantum cohomology algebras of the flag manifolds the answer turns out to coincide with the algebra of regular functions on an invariant lagrangian variety of a toda lattice

global attractivity of positive periodic solution to periodic lotka volterra competition systems with pure delay abstract we consider a periodic lotka volterra competition system without instantaneous negative feedbacks i e pure delay systems x i t x i t r i t j 1 n a i j t x j t i j t i 1 2 n we establish some 3 2 type criteria for global attractivity of a positive periodic solution of the system which generalize the well known wright s 3 2 criteria for the autonomous delay logistic equation and thereby address the open problem proposed by both kuang y kuang global stability in delayed nonautonomous lotka volterra type systems without saturated equilibria differential integral equations 9 1996 557 567 and teng z teng nonautonomous lotka volterra systems with delays j differential equations 179 2002 538 561

equivariant quantum schubert polynomials abstract we establish an equivariant quantum giambelli formula for partial flag varieties the answer is given in terms of a specialization of universal double schubert polynomials along the way we give new proofs of the presentation of the equivariant quantum cohomology ring as well as graham positivity of the structure constants in equivariant quantum schubert calculus

global periodic solutions of a generalized n species gilpin ayala competition model abstract in this paper we investigate a generalized n species gilpin ayala competition system with several deviating arguments in periodic environment which is more general and more realistic than the classical lotka volterra competition systems by using the method of coincidence degree a set of easily verifiable sufficient conditions are derived for the existence of at least one strictly positive componentwise periodic solution some new results are obtained as applications we also apply our main results to some special cases of the system we consider here including the classical n species lotka volterra competition systems and n species gilpin ayala competition model which have been studied extensively in the literature some known results are improved and generalized the examples show that our criteria are new general and easily verifiable

quantum cohomology of flag manifolds g b and quantum toda lattices let g be a connected semi simple complex lie group b its borel subgroup t a maximal complex torus contained in b and lie t its lie algebra this setup gives rise to two constructions the generalized nonperiodic toda lattice 28 29 and the flag manifold g b the toda lattice for g b t is the dynamical system on the cotangent bundle t lie t endowed with the canonical holomorphic symplectic form and the holomorphic hamiltonian function we consider in this paper

measure theory and pseudo almost automorphic functions new developments and applications abstract in this work we establish a new concept of weighted pseudo almost automorphic functions using the measure theory we present new results on weighted ergodic functions like completeness and composition theorems the theory of this work generalizes the classical results on weighted pseudo almost periodic and automorphic functions for illustration we provide some applications for evolution equations which include reaction diffusion systems and partial functional differential equations

permanence in nonautonomous lotka volterra system with predator prey the nonautonomous lotka volterra system with m predators and n preys is considered in this paper average conditions are obtained for permanence and global attractivity in the system it is shown that our results are generalization of those of the periodic system

equivariant pieri rules for isotropic grassmannians we give a pieri rule for the torus equivariant cohomology of submaximal grassmannians of lie types b c and d to the authors best knowledge our rule is the first manifestly positive formula beyond the equivariant chevalley formula we also give a simple proof of the equivariant pieri rule for the ordinary type a grassmannian

affine approach to quantum schubert calculus this paper presents a formula for products of schubert classes in the quantum cohomology ring of the grassmannian we introduce a generalization of schur symmetric polynomials for shapes that are naturally embedded in a torus then we show that the coefficients in the expansion of these toric schur polynomials in terms of the regular schur polynomials are exactly the 3 point gromov witten invariants which are the structure constants of the quantum cohomology ring this construction implies three symmetries of the gromov witten invariants of the grassmannian with respect to the groups s3 z nz 2 and z 2z the last symmetry is a certain curious duality of the quantum cohomology which inverts the quantum parameter q our construction gives a solution to a problem posed by fulton and woodward about the characterization of the powers of the quantum parameter q which occur with nonzero coefficients in the quantum product of two schubert classes the curious duality switches the smallest such power of q with the highest power we also discuss the affine nil temperley lieb algebra that gives a model for the quantum cohomology

almost periodic solutions of n species competitive system with feedback controls abstract in this paper the following almost periodic n species competitive system with feedback controls x i t x i t b i t j 1 n a ij t x j t j 1 j i n c ij t x i t x j t d i t u i t u i t r i t e i t u i t f i t x i t i 1 2 n are studied by using the comparison theorem and constructing suitable lyapunov function where some important factors such as the effect of toxic and the age structure are also considered simultaneously some sufficient conditions are obtained for the existence of a unique almost periodic solution of above model examples show that the obtained criteria are new general and easily verifiable

global attractivity of an almost periodic n species nonlinear ecological competitive model abstract by using comparison theorem and constructing suitable lyapunov functional we study the following almost periodic nonlinear n species competitive lotka volterra model x i t x i t r i t j 1 n a i j t x j i j t j 1 n b i j t x j i j t i j t j 1 n c i j t x i i i t x j i j t a set of sufficient conditions is obtained for the existence and global attractivity of a unique positive almost periodic solution of the above model as applications some special competition models are studied again our new results improve and generalize former results examples and their simulations show the feasibility of our main results

existence and global attractivity of an almost periodic ecological model the almost periodic lotka volterra model with m predators and n preys is considered in this paper by constructing suitable lyapunov function some sufficient conditions are obtained for the existence and global attractivity of a unique positive almost periodic solution of this model examples show that our criteria are new general and easily verifiable

a new tableau representation for supersymmetric schur functions abstract we give a new tableau definition for supersymmetric skew schur functions and obtain a number of properties of these functions as easy corollaries

on a periodic multi species ecological model a periodic predator prey model with m predators and n preys is proposed in this paper which can be seen as the modification of the traditional lotka volterra model by using comparison theorem the ultimately bounded region of the system is obtained by using comparison theorem and brouwer fixed point theorem sufficient conditions with guarantee the existence of a positive periodic solution of the system is obtained finally by constructing a suitable lyapunov function some sufficient conditions are obtained for the existence of a unique globally attractive periodic solution of system the results obtained here generalized the main results of j d zhao w c chen global asymptotic stability of a periodic ecological model applied mathematics and computation 147 3 2004 881 892

permanence in nonautonomous multi species predator prey system with feedback controls a nonautonomous multi species predator prey system with feedback controls is proposed in this paper where the competition among the predator species and among the prey species is considered also some important factors such as the effect of toxins and the age structure are also considered average conditions are obtained for permanence and global attractivity in the system the results obtained here generalized the main results of zhao and jiang j d zhao j f jiang permanence in nonautonomous lotka volterra system with predator prey appl math computat 152 2004 99 109

global attractivity of the periodic lotka volterra system a lotka volterra periodic model withm predators andn preys is studied in this paper a set of easily verifiable sufficient conditions that guarantee the existence uniqueness and global attractivity of the positive periodic solutions is obtained finally a suitable example is given to illustrate that the conditions of the main theorem are feasible

the factorial schur function the application of the divided difference of a function to the inhomogeneous symmetric functions factorial schur functions of biedenharn and louck is shown to lead to new relations and simplified proofs of their properties these results include determinantal definitions and the factorial jacobi trudi identities with extensions to skew versions similar properties of a second class of symmetric functions depending on an arbitrary parameter and of importance for generalized hypergeometric functions and series are shown also to be derivable from the divided difference notion slightly extended

positive almost periodic solutions of a class of lotka volterra type competitive system with delays and feedback controls abstract in this paper we study the positive almost periodic solutions for a class of almost periodic lotka volterra type system with delays and feedback controls applying schauder s fixed point theorem a criterion on the existence of the positive almost periodic solution of the system is obtained our new criterion which improves and generalizes some well known results can be easily checked

a littlewood richardson rule for factorial schur functions we give a combinatorial rule for calculating the coefficients in the expansion of a product of two factorial schur functions it is a special case of a more general rule which also gives the coefficients in the expansion of a skew factorial schur function applications to capelli operators and quantum immanants are also given

a different consideraton about the globally asymptotically stable solution of the periodic n competing species problem abstract we consider the volterra lotka equations for n competing species n 2 in which the right hand sides are periodic in time we show that conditions given by k gopalsamy j austral math soc ser b 27 1985 66 72 which imply the existence of a periodic solution with positive components also imply the uniqueness and asymptotic stability of the solution

existence and global stability of positive periodic solution of periodic integrodifferential systems with feedback controls abstract sufficient conditions are obtained for the existence and global stability of a positive periodic solution in some periodic integrodifferential systems with feedback controls by using the technique of coincidence degree and lyapunov functional

equivariant quantum schubert calculus abstract we study the t equivariant quantum cohomology of the grassmannian we prove the vanishing of a certain class of equivariant quantum littlewood richardson coefficients which implies an equivariant quantum pieri rule as in the equivariant case this implies an algorithm to compute the equivariant quantum littlewood richardson coefficients

average conditions for kolmogorov systems in this paper we consider nonautonomous competitive kolmogorov systems which is a generalization of classical lotka volterra competition model applying ahmad and lazer s definitions of lower and upper averages of a function we give the average conditions for the persistence permanence and global attractivity of the system

quantum integrability and generalised quantum schubert calculus abstract we introduce and study a new mathematical structure in the generalised quantum cohomology theory for grassmannians namely we relate the schubert calculus to a quantum integrable system known in the physics literature as the asymmetric six vertex model our approach offers a new perspective on already established and well studied special cases for example equivariant k theory and in addition allows us to formulate a conjecture on the so far unknown case of quantum equivariant k theory

permanence and asymptotic behavior of the n species nonautonomous lotka volterra competitive systems abstract in this paper we study the permanence and global asymptotic behavior for the n species nonautonomous lotka volterra competitive systems where all the parameters are time dependent and asymptotically approach periodic functions respectively some new criteria are established the corresponding results given by peng and chen in 1 are improved and extended

persistence and global stability for discrete models of nonautonomous lotka volterra type abstract consider the following discrete models of nonautonomous lotka volterra type n i p 1 n i p exp c i p a i p n i p j 1 n l 0 m a ij l p n j p k l 1 i n p 0 1 2 n i p n ip 0 p 0 and n i0 0 1 i n where each c i p a i p and a ij l p are bounded for p 0 and inf p 0 a i p 0 a ii 0 p 0 1 i n a ij l p 0 1 i j n 0 l m k 0 0 integers k l 0 1 l m in this paper to the above discrete system we apply the techniques offered by ahmad and lazer nonlinear anal 40 2000 37 49 and establish similar conditions of the persistence and global asymptotic stability of the system

coexistence states for periodic competitive kolmogorov systems abstract we prove the existence of positive periodic solutions for a class of nonautonomous competitive periodic kolmogorov systems which generalize the may leonard model a necessary and sufficient condition is also obtained

positivity in equivariant quantum schubert calculus a conjecture of d peterson proved by w graham states that the structure constants of the t equivariant cohomology of a homogeneous space g p satisfy a certain positivity property in this paper we show that this positivity property holds in the more general situation of equivariant quantum cohomology 1 introduction it is well known that the integral cohomology of the homogeneous space x g p for g a connected semisimple complex lie group and p a parabolic subgroup satisfies a positivity property its structure constants are nonnegative integers equal to the number of intersection points of three schubert varieties in general position whose codimensions add up to the dimension of x recently graham gr has proved a conjecture of peterson p asserting that h x the t equivariant cohomology of x where t c r is a maximal torus in g enjoys a more general positivity property

on a model of competition in periodic environments abstract sufficient conditions are obtained for the existence and global attractivity of a positive periodic solution of the system x i n 1 b i n x i n 1 j 1 m c ij n x j n i 1 2 m n n in which bi n cij n i j 1 2 m n n denote positive periodic sequences with a common period also a method is proposed for the derivation of the discrete system from a continuous time analogue modelling competition in periodic environments with time delays in the stabilizing negative feedback effects

global asymptotic stability of a periodic ecological model the periodic lotka volterra model with m predators and n preys is considered in this paper under the suppose that the intrinsic growth rate of the prey species may be negative while the total intrinsic growth rate in a period are positive sufficient conditions are obtained for existence and global asymptotic stability of the periodic solution of this model

new conditions on the existence and stability of periodic solution in lotka volterra s population system in this paper we revisit the famous periodic lotka volterra competitive system some new and interesting sufficient conditions are obtained to guarantee the existence and global asymptotic stability of the periodic solution in the lotka volterra competitive system our method is based on mawhin s coincidence degree matrix s spectral theory and some new estimation techniques for the priori bounds of unknown solutions to the equation lx lambda nx due to this new method our new results are much different from the known results in the previous literature finally some examples and their simulations show the feasibility of our results

dynamic behaviors of a delay differential equation model of plankton allelopathy in this paper we consider a modified delay differential equation model of the growth of n species of plankton having competitive and allelopathic effects on each other we first obtain the sufficient conditions which guarantee the permanence of the system as a corollary for periodic case we obtain a set of delay dependent condition which ensures the existence of at least one positive periodic solution of the system after that by means of a suitable lyapunov functional sufficient conditions are derived for the global attractivity of the system for the two dimensional case under some suitable assumptions we prove that one of the components will be driven to extinction while the other will stabilize at a certain solution of a logistic equation examples show the feasibility of the main results

schubert classes in the equivariant cohomology of the lagrangian grassmannian abstract let lg n denote the lagrangian grassmannian parametrizing maximal isotropic lagrangian subspaces of a fixed symplectic vector space of dimension 2n for each strict partition 1 k with 1 n there is a schubert variety x let t denote a maximal torus of the symplectic group acting on lg n consider the t equivariant cohomology of lg n and the t equivariant fundamental class of x the main result of the present paper is an explicit formula for the restriction of the class to any torus fixed point the formula is written in terms of factorial analogue of the schur q function introduced by ivanov as a corollary to the restriction formula we obtain an equivariant version of the giambelli type formula for lg n as another consequence of the main result we obtained a presentation of the ring h t lg n

extinction and permanence in nonautonomous competitive system with stage structure in this paper the n species nonautonomous stage structured competitive system is constructed and considered sufficient conditions for its extinction and permanence are obtained results here generalize and unify some previous ones moreover it is concluded that stage structure in this system is one of the important factors that effect the extinction and permanence of species

equivariant quantum cohomology of the odd symplectic grassmannian the odd symplectic grassmannian mathrm ig mathrm ig k 2n 1 parametrizes k dimensional subspaces of mathbb c 2n 1 which are isotropic with respect to a general necessarily degenerate symplectic form the odd symplectic group acts on mathrm ig with two orbits and mathrm ig is itself a smooth schubert variety in the submaximal isotropic grassmannian mathrm ig k 2n 2 we use the technique of curve neighborhoods to prove a chevalley formula in the equivariant quantum cohomology of mathrm ig i e a formula to multiply a schubert class by the schubert divisor class this generalizes a formula of pech in the case k 2 and it gives an algorithm to calculate any multiplication in the equivariant quantum cohomology ring

giambelli formulae for the equivariant quantum cohomology of the grassmannian we find presentations by generators and relations for the equivariant quantum cohomology of the grassmannian for these presentations we also find determinantal formulae for the equivariant quantum schubert classes to prove this we use the theory of factorial schur functions and a characterization of the equivariant quantum cohomology ring

existence uniqueness and stability of positive periodic solution for a nonlinear prey competition model with delays a nonlinear periodic predator prey model with m preys and n m predators and delays is proposed in this paper which can be seen as the modification of the traditional lotka volterra prey competition model sufficient conditions which guarantee the existence of a unique globally attractive positive periodic solution of the system are obtained

persistence and global stability in discrete models of pure delay nonautonomous lotka volterra type abstract consider the persistence and the global asymptotic stability of the following discrete model of pure delay nonautonomous lotka volterra type n i p 1 n i p exp c i p j 1 n l 0 m a ij l p n j p k l p 0 1 2 1 i n n i p n i p 0 p 0 and n i 0 0 1 i n where each ci p and aijl p are bounded for p 0 and inf p 0 l 0 m a ii l p 0 a ij l p 0 i j n 1 i n and k l 0 1 l m in this paper for the above discrete system of pure delay type by improving the former work j math anal appl 273 2002 492 511 which extended the averaged condition offered by s ahmad and a c lazer nonlinear anal 40 2000 37 49 we offer conditions of persistence and considering a lyapunov like discrete function to the above discrete system we establish sufficient conditions of global asymptotic stability