quantum cohomology of the springer resolution abstract let g denote a complex semi simple simply connected group and b its associated flag variety we identify the equivariant quantum differential equation for the cotangent bundle t b with the affine knizhnik zamolodchikov connection of cherednik and matsuo this recovers kim s description of quantum cohomology of b as a limiting case a parallel result is proven for resolutions of the slodowy slices extension to arbitrary symplectic resolutions is discussed

generalized macdonald ruijsenaars systems abstract we consider the polynomial representation of double affine hecke algebras dahas and construct its submodules as ideals of functions vanishing on the special collections of affine planes this generalizes certain results of kasatani in types a n c n c n we obtain commutative algebras of difference operators given by the action of invariant combinations of cherednik dunkl operators in the corresponding quotient modules of the polynomial representation this gives known and new generalized macdonald ruijsenaars systems thus in the cases of dahas of types a n and c n c n we derive chalykh sergeev veselov operators and a generalization of the koornwinder operator respectively together with complete sets of quantum integrals in the explicit form

the globally weak attraction and global semi stability of lienard system this paper obtains some necessary and sufficient conditions for the globally weak attraction and global semi stability of the zero solution for lienard equation x h y f x y g x

a unification of knizhnik zamolodchikov and dunkl operators via affine hecke algebras some generalizations of the lusztig lascoux schutzenberger operators for affine hecke algebras are considered as corollaries we obtain lusztig s isomorphisms from affine hecke algebras to their degenerate versions a natural interpretation of the dunkl operators and a new class of differential difference operators generalizing dunkl s ones and the knizhnik zamolodchikov operators from the two dimensional conformal field theory

a geometric property of banach spaces related to the fixed point property abstract in this note we give a sufficient condition for the fixed point property for nonexpansive mappings in banach spaces which we call the amcp this property is more general than normal structure and the space c 0 james space as well as every banach space endowed with a suitable schauder basis also have the amcp on the other hand a renorming of the tsirelson space and also 1 lack the amcp

a universal infinite dimensional modulus for normed spaces and applications abstract we define an infinite dimensional modulus which can be simultaneously considered as a measure of nearly uniform convexity and nearly uniform smoothness we compute this modulus in some classical banach spaces and we show some basic properties of this modulus and some applications to prove that a banach space satisfies the w fixed point property for non expansive mappings

fixed point property of direct sums abstract for a uniformly convex space z we show that z direct sums x 1 x n z of banach spaces x 1 x n with r a x i 1 a for some a 0 1 have the fixed point property for nonexpansive mappings as a direct consequence the result holds for all direct sums with being strictly convex the same result is extended to all direct sums x y of spaces x and y with property m whenever 1 the permanence of properties that are sufficient for the fixed point property are obtained for z direct sums and then for direct sums such properties include the properties r x 2 wnus c nj a x 2 ukk and nuc

elliptic ruijsenaars operators and functional equations we study mutually commutative difference operators introduced by ruijsenaars which are regarded as elliptic analogs of macdonald operators and consist of the jacobi theta functions and shift operators these operators are constructed in terms of r operators due to shibukawa and ueno or root algebras introduced by cherednik which give rise to a set of functional equations we investigate solutions of these functional equations for generalizations of elliptic ruijsenaars operators

ruijsenaars commuting difference operators and invariant subspace spanned by theta functions we study a family of mutually commutative difference operators introduced by ruijsenaars the conjugations of these operators with an appropriate function give the hamiltonians of some relativistic quantum systems these operators can be regarded as elliptic analogs of the macdonald operators and their coefficients consist of the jacobi theta functions we show that these operators act on the space of meromorphic functions on the cartan subalgebra of affine lie algebras and that the space spanned by characters of a fixed positive level is invariant under the action of these operators

essential self adjointness of the elliptic ruijsenaars models we study the elliptic ruijsenaars models associated with arbitrary root systems which are difference analogs of the calogero moser model we give a dense subspace in the space of square integrable functions invariant under the action of the weyl group on a torus as a domain of its hamiltonian and prove its essential self adjointness by using perturbation theory it is also clarified that these models consist of pure point spectrum

on a generalized james constant we introduce a generalized james constant j a x for a banach space x and prove that if j a x 3 a 2 for some a 0 1 then x has uniform normal structure the class of spaces x with j 1 x 2 is proved to contain all u spaces and their generalizations for the james constant j x itself we show that x has uniform normal structure provided that j x 1 5 2 improving the previous known upper bound at 3 2 finally we establish the stability of uniform normal structure of banach spaces

the general uq sl 2 invariant xxz integrable quantum spin chain the boundary conditions are constructed for the general xxz quantum integrable spin chain such that the resulting system is invariant under the quantum algebra uq sl 2 action the result generalizes that due to pasquier and saleur 1990 obtained for spin 1 2 and spin 1

the c function expansion of a basic hypergeometric function associated to root systems we derive an explicit c function expansion of a basic hypergeometric function associated to root systems the basic hypergeometric function in question was constructed as an explicit series expansion in symmetric macdonald polynomials by cherednik in case the associated twisted ane root system is reduced its construction was extended to the nonreduced case by the author it is a meromorphic weyl group invariant solution of the spectral problem of the macdonald q dierence operators the c function expansion is its explicit expansion in terms of the basis of the space of meromorphic solutions of the spectral problem consisting of q analogs of the harish chandra series we express the expansion coecients in terms of a q analog of the harish chandra c function which is explicitly given as product of q gamma functions the c function expansion shows that the basic hypergeometric function formally is a q analog of the heckmanopdam hypergeometric function which in turn specializes to elementary spherical functions on noncompact riemannian symmetric spaces for special values of the parameters

degenerations of sklyanin algebra and askey wilson polynomials a new trigonometric degeneration of the sklyanin algebra is found and the functional realization of its representations in space of polynomials in one variable is studied a further contraction gives the standard quantum algebra uq sl 2 it is shown that the degenerate sklyanin algebra contains a subalgebra isomorphic to algebra of functions on the quantum sphere su 2 so 2 q12 the diagonalization of general quadratic form in its generators leads in the functional realization to the difference equation for askey wilson polynomials

fixed point property for banach algebras associated to locally compact groups abstract in this paper we investigate when various banach algebras associated to a locally compact group g have the weak or weak fixed point property for left reversible semigroups we proved for example that if g is a separable locally compact group with a compact neighborhood of the identity invariant under inner automorphisms then the fourier stieltjes algebra of g has the weak fixed point property for left reversible semigroups if and only if g is compact this generalizes a classical result of t c lim for the case when g is the circle group t

the continuation of solutions for certain nonlinear differential equations and its applications we give some sufficient conditions for the continuation of solutions for some nonlinear differential equations as an application we obtain a new criterion for the oscillation of solutions of the lienard equation

weak fixed point property of reduced fourier stieltjes algebra and generalization of baggett s theorem abstract in this paper we show that if the reduced fourier stieltjes algebra b g of a second countable locally compact group g has either weak fixed point property or asymptotic center property then g is compact as a result we give affirmative answers to open problems raised by fendler and et al in 2013 we then conclude that a second countable group with a discrete reduced dual must be compact this generalizes a theorem of baggett we also construct a compact scattered hausdorff space for which the dual of the scattered c algebra c lacks weak fixed point property the c algebra c provides a negative answer to a question of randrianantoanina in 2010 in addition we prove a variant of bruck s generalized fixed point theorem for the preduals of von neumann algebras furthermore we give some examples emphasizing that the conditions in bruck s generalized conjecture bgc can not be weakened any more

boundary conditions for integrable quantum systems a new class of boundary conditions is described for quantum systems integrable by means of the quantum inverse scattering r matrix method the method proposed allows the author to treat open quantum chains with appropriate boundary terms in the hamiltonian the general considerations are applied to the xxz and xyz models the nonlinear schrodinger equation and toda chain

the quantum strip liouville theory for open strings the quantum group structure of 2d gravity recently put forward by one of us j l g is used to study quantum gravity on the strip the boundary conditions previously studied by a neveu and this author become easy to implement when one introduces the universal family of chiral operators associated withu q sl 2 a general formula for inverse powers of the metric tensor operator is thereby derived it contains a new universal matrixa acting in representation space which obeys identities involving ther matrix the clebsch gordon coefficients and the co products ofu q sl 2 the physical meaning of these identities is to ensure that these powers of the metric are local and closed by fusion

the spectral problem for the q knizhnik zamolodchikov equation and continuous q jacobi polynomials the spectral problem for theq knizhnik zamolodchikov equations for uq widehat sl 2 0 q 1 at arbitrary non negative levelk is considered the case of two point functions in the fundamental representation is studied in detail the scattering states are given explicitly in terms of continuousq jacobi polynomials and thes matrix is derived from their asymptotic behavior the level zeros matrix is closely connected with the kink antikinks matrix for the spin tfrac 1 2 xxz antiferromagnet an interpretation of the latter in terms of scattering on quantum symmetric spaces is discussed in the limit of infinite level we observe connections with harmonic analysis onp adic groups with the primep given byp q 2

theta function solutions of the quantum knizhnik zamolodchikov bernard equation for a face model we consider the quantum knizhnik zamolodchikov bernard equation for a face model with elliptic weights the sos model we provide explicit solutions as theta functions on the so called combinatorial line in which the model is equivalent to the three colour model these solutions are shown to be eigenvectors of the transfer matrix with periodic boundary conditions

an algebraic approach to macdonald koornwinder polynomials rodrigues type formula and inner product identity we study macdonald koornwinder polynomials in the context of double affine hecke algebras nonsymmetric macdonald koornwinder polynomials are constructed by use of raising operators provided by a representation theory of the double affine hecke algebra associated with a2l 2 type affine root system this enables us to evaluate diagonal terms of scalar products of the nonsymmetric polynomials algebraically the macdonald koornwinder polynomials are expressed by linear combinations of the nonsymmetric counterparts we show a new proof of the inner product identity of the macdonald koornwinder polynomials without opdam cherednik s shift operators

on the euler sequence spaces which include the spaces lp and l ii abstract by e p r we mean the space of all sequences whose euler transforms of order r are in the sequence spaces l p and l see b altay f basar m mursaleen on the euler sequence spaces which include the spaces l p and l i inform sci 2005 in press where 1 p in the present paper we essentially characterize the classes e p r l e 1 r l p and e p r f of infinite matrices for 1 p and give the characterizations of some other matrix mappings from the space e p r to the euler riesz difference etc sequence spaces by means of a given basic lemma we devote the final section of the paper to examining some geometric properties of the space e p r

jackson integrals of jordan pochhammer type and quantum knizhnik zamolodchikov equations we show that theq difference systems satisfied by jackson integrals of jordan pochhammer type give a class of the quantum knizhnik zamolodchikov equation for u q widehat mathfrak s mathfrak l 2 in the sense of frenkel and reshetikhin

new connection formulae for the q orthogonal polynomials via a series expansion of the q exponential using a realization of the q exponential function as an infinite multiplicative series of the ordinary exponential functions we obtain new nonlinear connection formulae of the q orthogonal polynomials such as q hermite q laguerre and q gegenbauer polynomials in terms of their respective classical analogues

a general formula for soliton form factors in the quantum sine gordon model a general formula for soliton form factors of several local fields in the quantum sine gordon model is presented

some estimates for the weakly convergent sequence coefficient in banach spaces we present relations between the weakly convergent sequence coefficient of a banach space and other coefficients some estimates are sharp

extended trigonometric cherednik algebras and nonstationary schrodinger equations with delta potentials we realize an extended version of the trigonometric cherednik algebra as affine dunkl operators involving heaviside functions we use the quadratic casimir element of the extended trigonometric cherednik algebra to define an explicit nonstationary schrodinger equation with delta potential we use coordinate bethe ansatz methods to construct solutions of the nonstationary schrodinger equation in terms of generalized bethe wave functions it is shown that the generalized bethe wave functions satisfy affine difference knizhnik zamolodchikov equations as functions of the momenta the relation to the vector valued root system analogs of the quantum bose gas on the circle with delta function interactions is indicated

connection problems for quantum affine kz equations and integrable lattice models cherednik attached to an affine hecke algebra module a compatible system of difference equations called quantum affine knizhnik zamolodchikov kz equations in the case of a principal series module we construct a basis of power series solutions of the quantum affine kz equations relating the bases for different asymptotic sectors gives rise to a weyl group cocycle which we compute explicitly in terms of theta functions

on james and von neumann jordan constants and sufficient conditions for the fixed point property abstract in this paper we prove that a banach space x and its dual space x have uniform normal structure if c nj x 1 3 2 the garcia falset coefficient r x is estimated by the c nj x constant and the weak orthogonality coefficient introduced by b sims finally we present an affirmative answer to a conjecture by l maligranda concerning the relation between the james and c nj x constants for a banach space

quantum knizhnik zamolodchikov equations and affine root systems quantum difference knizhnik zamolodchikov equations s1 fr are generalized for ther matrices from ch1 with the arguments in arbitrary root systems and their formal counterparts in particular qkz equations with certain boundary conditions are introducted the self consistency of the equations from fr and the cross derivative integrability conditions for ther matrix kz equations from ch2 are obtained as corollaries a difference counterpart of the quantum many body problem connected with macdonald s operators is defined as an application

the fixed point property in banach spaces with the nus property abstract in this paper we show that the weak nearly uniform smooth banach spaces have the fixed point property for nonexpansive mappings

lectures on affine hecke algebras and macdonald s conjectures this paper gives a review of cherednik s results on the representation theoretic approach to macdonald polynomials and related special functions macdonald polynomials are a remarkable 2 parameter family of polynomials which can be associated to every root system as special cases they include the schur functions the q jacobi polynomials and certain spherical functions on real and p adic symmetric spaces they have a number of elegant combinatorial properties which however are extremely difficult to prove in this paper we show that a natural setup for studying these polynomials is provided by the representation theory of hecke algebras and show how this can be used to prove some of the combinatorial identities for macdonald polynomials

relationships between properties that imply the weak fixed point property a number of banach space properties have previously been shown to imply the weak fixed point property we investigate various connections between these properties

on the euler sequence spaces which include the spaces l p and l i in the present paper we introduce the euler sequence space e p r consisting of all sequences whose euler transforms of order r are in the space p of non absolute type which is the bk space including the space p and prove that the spaces e p r and p are linearly isomorphic for 1

some functions that generalize the askey wilson polynomials we determine all biinfinite tridiagonal matrices for which some family of eigenfunctions are also eigenfunctions of a second order q difference operator the solution is described in terms of an arbitrary solution of a q analogue of gauss hypergeometric equation depending on five free parameters and extends the four dimensional family of solutions given by the askey wilson polynomials there is some evidence that this bispectral problem for an arbitrary order q difference operator is intimately related with some q deformation of the toda lattice hierarchy and its virasoro symmetries when tridiagonal matrices are replaced by the schroedinger operator and q 1 this statement holds with toda replaced by kdv in this context this paper determines the analogs of the bessel and airy potentials

new connection formulae for some q orthogonal polynomials in q askey scheme new nonlinear connection formulae of the q orthogonal polynomials such as continuous q laguerre continuous big q hermite q meixner pollaczek and q gegenbauer polynomials in terms of their respective classical analogues are obtained using a special realization of the q exponential function as infinite multiplicative series of ordinary exponential function

connection coefficients for basic harish chandra series basic harish chandra series are asymptotically free meromorphic solutions of the system of basic hypergeometric difference equations associated to root systems the associated connection coefficients are explicitly computed in terms of jacobi theta functions we interpret the connection coefficients as the transition functions for asymptotically free meromorphic solutions of cherednik s root system analogs of the quantum knizhnik zamolodchikov equations they thus give rise to explicit elliptic solutions of root system analogs of dynamical yang baxter and reflection equations applications to quantum c functions basic hypergeometric functions reflectionless difference operators and multivariable baker akhiezer functions are discussed

the fixed point property under renorming in some classes of banach spaces abstract assume that y is a banach space such that r y 2 where r is garcia falset s coefficient and x is a banach space which can be continuously embedded in y we prove that x can be renormed to satisfy the weak fixed point property w fpp on the other hand assume that k is a scattered compact topological space such that k 0 and c k is the space of all real continuous functions defined on k with the supremum norm we will show that c k can be renormed to satisfy r c k 2 thus both results together imply that any banach space which can be continuously embedded in c k k as above can be renormed to satisfy the w fpp these results extend a previous one about the w fpp under renorming for banach spaces which can be continuously embedded in c 0 furthermore we consider a metric in the space p of all norms in c k which are equivalent to the supremum norm and we show that for almost all norms in p in the sense of porosity c k satisfies the w fpp

homoclinic orbits in generalized lienard systems abstract this paper is devoted to the investigation on the existence of homoclinic orbits of the planar system of lienard type x h y f x y g x here h y is strictly increasing but is not imposed h sufficient conditions are given for a positive orbit of the system starting at a point on the curve h y f x to approach the origin without intersecting the x axis the obtained theorems include previous results as special cases our results are applied to a concrete system and their sharpness are improved

affine hecke algebra macdonald polynomials and quantum many body systems the macdonald operators associated with the classical root systems are constructed based on the infinite dimensional representation for solutions of the yang baxter equation and the reflection equation

integrable connections related to zonal spherical functions we define a system of differential equations of first order for a function valued in the group algebra of the weyl group associated with an arbitrary root system this is equivalent to the system of differential equations given by heckman and opdam which is a deformation of the system satisfied by the zonal spherical function of the riemannian symmetric spaceg k of non compact type when the root system is a n type our equation is related to the knizhnik zamolodchikov equation in conformal field theory

uniformly nonsquare banach spaces have the fixed point property for nonexpansive mappings abstract it is shown that if the modulus x of nearly uniform smoothness of a reflexive banach space satisfies x 0 1 then every bounded closed convex subset of x has the fixed point property for nonexpansive mappings in particular uniformly nonsquare banach spaces have this property since they are properly included in this class of spaces this answers a long standing question in the theory

properties ua 2 and wa 2 in orlicz spaces and some of their consequences abstract we introduce a new geometric property a 2 and we show that it is equivalent to its uniform version u a 2 next we introduce a new property u a 2 and we show that if a separable banach space x has this property then both x and x have the weak fixed point property we also prove that a uniformly gateaux differentiable banach space has property u a 2 and that if x has property u a 2 then x has the ukk property criteria in order that orlicz spaces have the properties u a 2 u a 2 and nus are given

boundary quantum knizhnik zamolodchikov equations and bethe vectors solutions to boundary quantum knizhnik zamolodchikov equations are constructed as bilateral sums involving off shell bethe vectors in case the reflection matrix is diagonal and only the 2 dimensional representation of mathcal u q widehat mathfrak sl 2 is involved we also consider their rational and classical degenerations

macdonald polynomials from sklyanin algebras a conceptual basis for the p adics quantum group connection we establish a previously conjectured connection betweenp adics and quantum groups we find in sklyanin s two parameter elliptic quantum algebra and its generalizations the conceptual basis for the macdonald polynomials which interpolate between the zonal spherical functions of related real andp adic symmetric spaces the elliptic quantum algebras underlie thez n baxter models we show that in then limit the jost function for the scattering offirst level excitations in the 1 1 dimensional field theory model associated to thez n baxter model coincides with the harish chandra likec function constructed from the macdonald polynomials associated to the root systema 1 the partition function of thez 2 baxter model itself is also expressed in terms of this macdonald harish chandrac function albeit in a less simple way we relate the two parametersq andt of the macdonald polynomials to the anisotropy and modular parameters of the baxter model in particular thep adic regimes in the macdonald polynomials correspond to a discrete sequence of xxz models we also discuss the possibility of q deforming euler products

dynamical weyl groups and applications abstract following a preceding paper of tarasov and the second author we define and study a new structure which may be regarded as the dynamical analog of the weyl group for lie algebras and of the quantum weyl group for quantized enveloping algebras we give some applications of this new structure

elliptic quantum many body problem and double affine knizhnik zamolodchikov equation the elliptic matrix quantum olshanetsky perelomov problem is introduced for arbitrary root systems by means of an elliptic version of the dunkl operators its equivalence with the double affine generalization of the knizhnik zamolodchikov equation in the induced representations is established

non archimedean strings and bruhat tits trees it is shown that the bruhat tits tree for thep adic linear groupgl 2 is a natural non archimedean analog of the open string world sheet the boundary of the tree can be identified with the field ofp adic numbers we construct a lattice quantum field theory on the bruhat tits tree with a simple local lagrangian and show that it leads to the freund olson amplitudes for emission processes of the particle states from the boundary

nearly uniformly noncreasy banach spaces we introduce and study the class of nearly uniformly noncreasy banach spaces it is proved that they have the weak fixed point property a stability result for this property is obtained

a hierarchic array of integrable models motivated by harish chandra theory starting from a simple cdd pole s matrix a hierarchy of new s matrices involving ever higher in the sense of barnes gamma functions are constructed these new s matrices correspond to scattering of excitations in ever more complex integrable models from each of these models new ones are obtained either by q deformation or by considering the selberg type euler products of which they represent the infinite place a hierarchic array of integrable models is thus obtained a remarkable diagonal link in this array is established though many entries in this array correspond to familiar integrable models the array also leads to new models setting up this array led to new results on the q gamma function and on the q deformed bloch wigner function

a solution of the quantum knizhnik zamolodchikov equation of type cn we construct a solution of cherednik s quantum knizhnik zamolodchikov equation associated with the root system of type c n this solution is given in terms of a restriction of a q jordan pochhammer integral as its application we give an explicit expression of a special case of the macdonald polynomial of the c n type finally we explain the connection with the representation of the hecke algebra

neocompact sets and the fixed point property abstract we use a newly introduced concept of neocompactness to study problems from metric fixed point theory in particular we give a sufficient condition for a superreflexive banach space x to have the fixed point property and obtain shorter proofs of some well known results in that theory