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

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

a non hypersingular time domain biem for 3 d transient elastodynamic crack analysis a three dimensional 3 d time domain boundary integral equation method biem is presented for transient elastodynamic crack analysis a non hypersingular traction bie formulation is used with the crack opening displacements and their derivatives as unknown quantities a collocation method in conjunction with a time stepping scheme is developed to solve the non hypersingular time domain bies to simplify the analysis and to describe the proper behaviour of the unknown quantities at the crack front a constant spatial shape function is applied for elements away from the crack front while a spatial square root crack tip shape function is adopted for elements near the crack front a linear temporal shape function is used in the time stepping scheme numerical calculations have been carried out for penny shaped and square cracks results for the elastodynamic stress intensity factors are presented as functions of the temporal and the spatial variables for the test examples considered good agreement between the present results and those of other authors is obtained

dynamic modelling of the flat 2 d crack by a semi analytic biem scheme we present an efficient numerical method for solving indirect boundary integral equations that describe the dynamics of a flat two dimensional 2 d crack in all modes of fracture the method is based on a piecewise constant interpolation both in space and time of the slip rate function by which the original equation is reduced to a discrete convolution in space and time of the slip rate and a set of analytically obtained coefficients if the time step interval is set sufficiently small with respect to the spatial grid size the discrete equations decouple and can be solved explicitly this semi analytic scheme can be extended to the calculation of the wave field off the crack plane a necessary condition for the numerical stability of this scheme is investigated by way of an exhaustive set of trial runs for a kinematic problem for the case investigated our scheme is very stable for a fairly wide range of control parameters in modes iii and i whereas in mode ii it is unstable except for some limited ranges of the parameters the use of peirce and siebrits e scheme in time collocation is found helpful in stabilizing the numerical calculation our scheme also allows for variable time steps copyright 2001 john wiley sons ltd

a class of partition functions associated with e_ tau eta gl_3 by izergin korepin analysis recently a class of partition functions associated with higher rank rational and trigonometric integrable models were introduced by foda and manabe we use the dynamical r matrix of the elliptic quantum group e gl3 to introduce an elliptic analog of the partition functions associated with e gl3 we investigate the partition functions of foda manabe type by developing a nested version of the elliptic izergin korepin analysis and present the explicit forms as symmetrization of multivariable elliptic functions we show that special cases are essentially the elliptic weight functions introduced in the works by rimanyi tarasov and varchenko konno and felder rimanyi and varchenko

the nil hecke ring and cohomology of g p for a kac moody group g abstract let g be the group with borel subgroup b associated to a kac moody lie algebra unk with weyl group w and cartan subalgebra unk then h g b has among others four distinguished structures i an algebra structure ii a distinguished basis given by the schubert cells iii a module for w and iv a module for hecke type operators aw for w unk w we construct a ring r which we refer to as the nil hecke ring which is very simply and explicitly defined as a functor of w together with the w module unk alone and such that all these four structures on h g b arise naturally from the ring r

cylinder buckling the mountain pass as an organizing center we revisit the classical problem of the buckling of a long thin axially compressed cylindrical shell by examining the energy landscape of the perfect cylinder we deduce an estimate of the sensitivity of the shell to imperfections key to obtaining this estimate is the existence of a mountain pass point for the system we prove the existence on bounded domains of such solutions for almost all loads and then numerically compute example mountain pass solutions numerically the mountain pass solution with lowest energy has the form of a single dimple we interpret these results and validate the lower bound against some experimental results available in the literature

t equivariant k theory of generalized flag varieties let g be a kac moody group with borel subgroup b and compact maximal torus t analogous to kostant and kumar kostant b kumar s 1986 proc nati acad sci usa 83 1543 1545 we define a certain ring y purely in terms of the weyl group w associated to g and its action on t by dualizing y we get another ring 1 which we prove is canonically isomorphic with the t equivariant ktheory kt g b of gib now kt g b apart from being an algebra over kt pt a t also has a weyl group action and moreover kt g b admits certain operators dw w similar to the demazure operators defined on a t we prove that these structures on kt g b come naturally from the ring y by evaluating the a t module w at 1 we recover k g b together with the above mentioned structures we believe that many of the results of this paper are new in the finite case i e g is a finite dimensional semisimple group over c

the slˆ n k wznw fusion ring a combinatorial construction and a realisation as quotient of quantum cohomology abstract a simple combinatorial construction of the sl n k wznw fusion ring also known as verlinde algebra is given as a byproduct of the construction one obtains an isomorphism between the fusion ring and a particular quotient of the small quantum cohomology ring of the grassmannian gr k k n we explain how our approach naturally fits into known combinatorial descriptions of the quantum cohomology ring by establishing what one could call a boson fermion correspondence between the two rings we also present new recursion formulae for the structure constants of both rings the fusion coefficients and the gromov witten invariants

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

orbital stability investigations for travelling waves in a nonlinearly supported beam abstract we consider the fourth order wave equation t t x x x x f 0 x t r r with a nonlinearity f vanishing at 0 travelling waves x t u x c t satisfy the ode u c 2 u f u 0 on r and for the case f u e u 1 the existence of at least 36 solitary travelling waves was proved in 4 by computer assisted means we investigate the orbital stability of these solutions via computation of their morse indices and using results from 11 and 12 in order to achieve our results we make use of both analytical and computer assisted techniques

a numerical investigation of sign changing solutions to superlinear elliptic equations on symmetric domains in this paper we investigate numerically sign changing solutions of superlinear elliptic equations on symmetric domains based upon the symmetric criticality principle of palais the existence of sign changing solutions which re2ect the symmetry

non symmetric low index solutions for a symmetric boundary value problem abstract we consider the equation u w u 3 on a square domain in r 2 with dirichlet boundary conditions where w is a given positive function that is invariant under all euclidean symmetries of the square this equation is shown to have a solution u with morse index 2 that is neither symmetric nor antisymmetric with respect to any nontrivial symmetry of the square part of our proof is computer assisted an analogous result is proved for index 1

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

time domain boundary element analysis of elastic wave interaction with a crack the mathematical formulation of the problem of transient wave interaction with a crack in a homogeneous isotropic linearly elastic solid has been reduced to the solution of an integral equation over the insonified crack face the integral equation relates the unknown crack opening displacement which depends on time and position to the incident wave field the integral equation has been solved numerically by a time stepping method in conjunction with a boundary element discretization of the crack surface for normal incidence of a longitudinal step stress wave on a penny shaped crack results as functions of time have been obtained for the crack opening displacement the elastodynamic mode i stress intensity factor and the scattered far field

multiple solutions for a semilinear boundary value problem a computational multiplicity proof abstract we prove the first genuine partial differential equation result on a conjecture concerning the number of solutions of second order elliptic boundary value problems with a nonlinearity which grows superlinearly at the proof makes massive use of computer assistance after approximate solutions have been computed by a numerical mountain pass algorithm combined with a newton iteration to improve accuracy a fixed point argument is used to show the existence of exact solutions close to the approximations

quantum k theory of grassmannians we show that equivariant k theoretic 3 point gromov witten invariants of genus zero on a grassmann variety are equal to triple intersec tions computed in the equivariant k theory of a two step flag manifold thus generalizing an earlier result of buch kresch and tamvakis in the process we show that the gromov witten variety of curves passing through 3 general points is irreducible and rational our applications include pieri and giambelli formulas for the quantum k theory ring of a grassmannian which determine the multiplication in this ring we also compute the dual schubert basis for this ring and show that its structure constants satisfy s3 symmetry our for mula for gromov witten invariants can be partially generalized to cominuscule homogeneous spaces by using a construction of chaput manivel and perrin

equivariant k theory and equivariant cohomology for t an abelian compact lie group we give a description of t equivariant k theory with complex coefficients in terms of equivariant cohomology in the appendix we give applications of this by extending results of chang skjelbred and goresky kottwitz macpherson from equivariant cohomology to equivariant k theory

combinatorial properties of symmetric polynomials from integrable vertex models in finite lattice we introduce and study several combinatorial properties of a class of symmetric polynomials from the point of view of integrable vertex models in a finite lattice we introduce the l operator related to the uq sl2 r matrix and construct the wavefunctions and their duals we prove the exact correspondence between the wavefunctions and symmetric polynomials which is a quantum group deformation of the grothendieck polynomials this is proved by combining the matrix product method and an analysis on the domain wall boundary partition functions as applications of the correspondence between the wavefunctions and symmetric polynomials we derive several properties of the symmetric polynomials such as the determinant pairing formulas and the branching formulas by analyzing the domain wall boundary partition functions and the matrix elements of the b operators

a novel derivation of non hypersingular time domain bies for transient elastodynamic crack analysis abstract a novel application of conservation integrals in transient elastodynamic crack analysis is presented to derive non hypersingular time domain boundary integral equations bies the new derivation is based on an elastodynamic conservation integral which is employed to express the displacement gradients in terms of an integral over the surface of the crack bies are obtained by substituting this representation integral into hooke s law and by taking a limit process the bies obtained in this manner are valid for arbitrary crack configurations and they are immediately nonhypersingular the laplace and the fourier transforms of these bies agree with those obtained by other authors by using the conventional derivation in conjunction with regularization techniques numerical examples show that the time domain bies presented here can yield highly accurate results for elastodynamic stress intensity factors

schur polynomials and the yang baxter equation we describe a parametrized yang baxter equation with nonabelian parameter group that is we show that there is an injective map g mapsto r g from rm gl 2 mathbb c times rm gl 1 mathbb c to end v otimes v where v is a two dimensional vector space such that if g h in g then r 12 g r 13 gh r 23 h r 23 h r 13 gh r 12 g here r i j denotes r applied to the i j components of v otimes v otimes v the image of this map consists of matrices whose nonzero coefficients a 1 a 2 b 1 b 2 c 1 c 2 are the boltzmann weights for the non field free six vertex model constrained to satisfy a 1 a 2 b 1 b 2 c 1 c 2 0 this is the exact center of the disordered regime and is contained within the free fermionic eight vertex models of fan and wu as an application we show that with boundary conditions corresponding to integer partitions the six vertex model is exactly solvable and equal to a schur polynomial s times a deformation of the weyl denominator this generalizes and gives a new proof of results of tokuyama and hamel and king

motivic decompositions of twisted flag varieties and representations of hecke type algebras motivated by the motivic galois group and the kostant kumar results on equivariant cohomology of flag varieties we provide a uniform de scription of motivic direct sum decompositions with integer coefficients of versal flag varieties in terms of integer representations of the associated affine nil hecke algebra h more generally we establish an equivalence between the h motivic subcat egory generated by the motive of e b and the category of projective modules of the associated rational algebra d of push pull operators where e is a tor sor for a split semisimple linear algebraic group g over a field k b is a borel subgroup of g h is an algebraic oriented cohomology theory in the sense of levine morel e g chow ring ch or an algebraic cobordism the algebra d can be think of as an integer analogue of the hopf algebra of the h motivic galois group of e b as an application taking h ch and specializing the coefficients to the finite field fp we obtain that p modular projective representations of d h are generated by an irreducible h module corresponding to the generalized rost voevodsky motive for g p

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

level set methods for finding critical points of mountain pass type abstract computing mountain passes is a standard way of finding critical points we describe a numerical method for finding critical points that is convergent in the nonsmooth case and locally superlinearly convergent in the smooth finite dimensional case we apply these techniques to describe a strategy for addressing the wilkinson problem of calculating the distance from a matrix to a closest matrix with repeated eigenvalues finally we relate critical points of mountain pass type to nonsmooth and metric critical point theory

quantum k theory on flag manifolds finite difference toda lattices and quantum groups let x y be homogeneous coordinates on cp a degree d holomorphic map cp 1 cp is uniquely determined up to a constant scalar factor by n 1 relatively prime degree d binary forms f0 x y fn x y omitting the condition that the forms are relatively prime we compactify the space of degree d holomorphic maps cp 1 cp to a complex projective space of dimension n 1 d 1 1 we denote this compactification of the space of maps by cp d this construction defines the compactification d r i 1cp ni 1 di of the space of degree d d1 dr maps from cp 1 to composing degree d holomorphic maps from cp 1 to the flag manifold x with the plucker embedding we embed the space of such maps into d the closure qmd of this space in d is often referred to as the drinfeld s compactification of the space of degree d maps from cp 1 to x and will be called the space of quasimaps following 14 it is a generally speaking singular irreducible projective variety of complex dimension dimx 2d1 2dr the flag manifold is a homogeneous space of the group slr 1 c and of its maximal compact subgroup sur 1 the action of these groups on x lifts naturally to the spaces d and qmd in addition to this action the spaces d and the subspaces qmd carry the circle action induced by the rotation of cp 1 defined by x y 7 x ey thus the product group g s sur 1 and its complex version gc c slr 1 c act on the quasimap spaces we will see later that qmd have g equivariant desingularizations qmd

vertex models tasep and grothendieck polynomials we examine the wavefunctions and their scalar products of a one parameter family of integrable five vertex models at a special point of the parameter the model investigated is related to an irreversible interacting stochastic particle system the so called totally asymmetric simple exclusion process tasep by combining the quantum inverse scattering method with a matrix product representation of the wavefunctions the on off shell wavefunctions of the five vertex models are represented as a certain determinant form up to some normalization factors we find that the wavefunctions are given by grothendieck polynomials which are a one parameter deformation of schur polynomials introducing a dual version of the grothendieck polynomials and utilizing the determinant representation for the scalar products of the wavefunctions we derive a generalized cauchy identity satisfied by the grothendieck polynomials and their duals several representation theoretical formulae for the grothendieck polynomials are also presented as a byproduct the relaxation dynamics such as green functions for the periodic tasep are found to be described in terms of the grothendieck polynomials

constrained mountain pass algorithm for the numerical solution of semilinear elliptic problems a new numerical algorithm for solving semilinear elliptic problems is presented a variational formulation is used and critical points of a c1 functional subject to a constraint given by a level set of another c1 functional or an intersection of such level sets of finitely many functionals are sought first constrained local minima are looked for then constrained mountain pass points the approach is based on the deformation lemma and the mountain pass theorem in a constrained setting several examples are given showing new numerical solutions in various applications

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

numerical variational methods applied to cylinder buckling we review and compare different computational variational methods applied to a system of fourth order equations that arises as a model of cylinder buckling we describe both the discretization and implementation in particular how to deal with a one dimensional null space we show that we can construct many different solutions from a complex energy surface we examine numerically convergence in the spatial discretization and in the domain size finally we give a physical interpretation of some of the solutions found

symmetric functions and wavefunctions of xxz type six vertex models and elliptic felderhof models by izergin korepin analysis we present a method to analyze the wavefunctions of six vertex models by extending the izergin korepin analysis originally developed for domain wall boundary partition functions first we apply the method to the case of the basic wavefunctions of the xxz type six vertex model by giving the izergin korepin characterization of the wavefunctions we show that these wavefunctions can be expressed as multiparameter deformations of the quantum group deformed grothendieck polynomials as a second example we show that the izergin korepin analysis is effective for analysis of the wavefunctions for a triangular boundary and present the explicit forms of the symmetric functions representing these wavefunctions as a third example we apply the method to the elliptic felderhof model which is a face type version and an elliptic extension of the trigonometric felderhof model we show that the wavefunctions can be expressed as one parameter deformations of an elliptic analog of the vandermonde determinant and elliptic symmetric functions we present a method to analyze the wavefunctions of six vertex models by extending the izergin korepin analysis originally developed for domain wall boundary partition functions first we apply the method to the case of the basic wavefunctions of the xxz type six vertex model by giving the izergin korepin characterization of the wavefunctions we show that these wavefunctions can be expressed as multiparameter deformations of the quantum group deformed grothendieck polynomials as a second example we show that the izergin korepin analysis is effective for analysis of the wavefunctions for a triangular boundary and present the explicit forms of the symmetric functions representing these wavefunctions as a third example we apply the method to the elliptic felderhof model which is a face type version and an

a mountain pass method for the numerical solution of semilinear wave equations it is well known that periodic solutions of semilinear wave equations can be obtained as critical points of related functionals in the situation that we studied there is usually an obvious solution obtained as a solution of linear problem we formulate a dual variational problem in such a way that the obvious solution is a local minimum we then find additional non obvious solutions via a numerical mountain pass algorithm based on the theorems of ambrosetti rabinowitz and ekeland numerical results are presented

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

k theoretic analogues of factorial schur p and q functions abstract we introduce two families of symmetric functions generalizing the factorial schur p and q functions due to ivanov we call them k theoretic analogues of factorial schur p and q functions we prove various combinatorial expressions for these functions e g as a ratio of pfaffians a sum over set valued shifted tableaux and a sum over excited young diagrams as a geometric application we show that these functions represent the schubert classes in the k theory of torus equivariant coherent sheaves on the maximal isotropic grassmannians of symplectic and orthogonal types this generalizes a corresponding result for the equivariant cohomology given by the authors we also discuss a remarkable property enjoyed by these functions which we call the k theoretic q cancellation property we prove that the k theoretic p functions form a formal basis of the ring of functions with the k theoretic q cancellation property

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

new solutions of the gelfand problem abstract we propose a computer assisted method for proving existence and obtaining enclosures of solutions of nonlinear elliptic boundary value problems which may be successful in cases where purely analytic methods are not this method is an extension of our previous work interval comput 3 1994 106 121 z angew math mech 76 1996 171 176 on the gelfand problem to nonconvex domains in r 2 here we are able to prove the existence of a solution branch which was unknown so far

positivity in the grothendieck group of complex flag varieties abstract we prove a conjecture of a s buch concerning the structure constants of the grothendieck ring of a flag variety with respect to its basis of schubert structure sheaves for this we show that the coefficients in this basis of the structure sheaf of any subvariety with rational singularities have alternating signs equivalently the class of the dualizing sheaf of such a subvariety is a nonnegative combination of classes of dualizing sheaves of schubert varieties

analytical and numerical results for the fucik spectrum of the laplacian nontrivial solutions of u u vu 0 in rn with zero dirichlet condition on are studied the collection of the pairs v is the so called fucik spectrum f a new variational formulation for parts of f is presented and analyzed based on this formulation a minimization algorithm for the computation of a part of f is developed alternatively an approach using an implicit function argument is discussed analytically and via newton s method also numerically by combining the variational minimization method with newton s method a new bifurcation result is first observed numerically and then proved rigorously by replacing the variational minimization method by the mountain pass algorithm higher curves in f are found numerically several numerical results are discussed including a further example of the previously recorded phenomenon of crossing of fucik curves originating from different eigenvalues

on certain hecke rings abstract we examine rings that embed into the smash product of the group algebra of the weyl group with the field of meromorphic functions on the cartan subalgebra and are generated by elements that satisfy braid relations we prove that every such ring is isomorphic to either the hecke algebra the nil hecke ring or the group algebra of the weyl group

representation theory over tropical semifield and langlands duality recently we propose a class of infinite dimensional integral representations of classical mathfrak gl ell 1 whittaker functions and local archimedean local l factors using two dimensional topological field theory framework the local archimedean langlands duality was identified in this setting with the mirror symmetry of the underlying topological field theories in this note we introduce elementary analogs of the whittaker functions and the archimedean l factors given by ul 1 equivariant symplectic volumes of appropriate kahler ul 1 manifolds we demonstrate that the functions thus defined have a dual description as matrix elements of representations of monoids gl ell 1 mathcal r mathcal r being the tropical semifield we also show that the elementary whittaker functions can be obtained from the non archimedean whittaker functions over mathbb q p by taking the formal limit p 1 hence the elementary special functions constructed in this way might be considered as functions over the mysterious field mathbb q 1 the existence of two representations for the elementary whittaker functions one as an equivariant volume and the other as a matrix element should be considered as a manifestation of a hypothetical elementary analog of the local langlands duality for number fields we would like to note that the elementary local l factors coincide with l factors introduced previously by kurokawa

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

analysis of dynamic crack propagation using a time domain boundary integral equation method abstract presented in this paper is a boundary element method which enables the simulation of dynamic crack propagation under arbitrary loading conditions and without any a priori assumptions regarding the crack path the direction and speed of crack advance are controlled only by a fracture criterion fast crack growth is investigated in an unbounded linear elastic body under plane strain or longitudinal shear for in plane deformation under mixed mode loading the influence of crack closure on the computed crack path is considered by solving the related contact problem the starting point is a non hypersingular time domain traction boundary integral equation a collocation method in conjunction with a time stepping scheme is applied to solve the integral equation numerically the accuracy of the numerical method is checked by comparison with analytical solutions available for simple situations some examples of curved cracks propagating with variable speed in inhomogenous stress fields serve to illustrate the versatility of the method

baxter q operator from quantum k theory abstract we define and study the quantum equivariant k theory of cotangent bundles over grassmannians for every tautological bundle in the k theory we define its one parametric deformation referred to as quantum tautological bundle we prove that the spectrum of operators of quantum multiplication by these quantum classes is governed by the bethe ansatz equations for the inhomogeneous xxz spin chain in addition we prove that each such operator corresponds to the universal elements of quantum group u sl 2 in particular we identify the baxter operator for the xxz spin chain with the operator of quantum multiplication by the exterior algebra tautological bundle the explicit universal combinatorial formula for this operator is found the relation between quantum line bundles and quantum dynamical weyl group is shown

a high linking algorithm for sign changing solutions of semilinear elliptic equations where n is a bounded domain with regular boundary the operator can be replaced by a more general second order uniformly elliptical operator in divergence form the solutions we are interested in nding are of minimax type and sign changing nonlinear elliptic equations of type 1 1 arise naturally in physics engineering and mathematical biology it is known that many such equations have multiple solutions their study has attracted the attention of many pure and applied mathematicians during the past two decades let us assume that f satis es the following regularity and growth conditions a1 f x t is locally lipschitz continuous in