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

braid structure and raising lowering operator formalism in sutherland model we algebraically construct the fock space of the sutherland model in terms of the eigenstates of the pseudomomenta as basis vectors for this purpose we derive the raising and lowering operators which increase and decrease eigenvalues of pseudomomenta the operators exchanging eigenvalues of two pseudomomenta have been known all the eigenstates are systematically produced by starting from the ground state and multiplying these operators by it

strict lyapunov functions for time varying systems with persistency of excitation we study the stability of the origin for a class of linear time varying systems with a drift that may be divided in two parts under the action of the first a function of the trajectories is guaranteed to converge to zero under the action of the second the solutions are restricted to a periodic orbit hence by assumption the systems trajectories are bounded our main results focus on two generic case studies that are motivated by common nonlinear control problems model reference adaptive control control of nonholonomic systems tracking control problems to name a few then based on the standing assumption that the systems dynamics is persistently excited we construct a time dependent lyapunov function that has a negative definite derivative our main statements may be regarded as off the shelf tools of analysis for linear and nonlinear time varying systems

intertwining operators associated to the group for any finite reflection group g on an euclidean space there is a parametrized commutative algebra of differential difference operators with as many parameters as there are conjugacy classes of reflections in g there exists a linear isomorphism on polynomials which intertwines this algebra with the algebra of partial differential operators with constant coefficients for all but a singular set of parameter values containing only certain negative rational numbers this paper constructs an integral transform implementing the intertwining operator for the group s3 the symmetric group on three objects for parameter value the transform is realized as an absolutely continuous measure on a compact subset of m2 r which contains the group as a subset of its boundary the construction of the integral formula involves integration over the unitary group u 3 associated to any finite reflection group g on an euclidean space there is a parametrized commutative algebra of differential difference operators with as many parameters as there are conjugacy classes of reflections in g it has been shown that there exists a linear isomorphism on polynomials which intertwines this algebra with the algebra of partial differential operators with constant coefficients for all but a singular set of parameter values this singular set contains only negative values and is closely linked to the zero set of the poincare series of g this paper constructs an integral transform implementing the intertwining map for the group s3 the symmetric group on three objects for positive parameter values previously this had been done only for the group z2 acting by sign change on r where the transform is a classical fractional integral the transform in this paper has its origin in the adjoint action of the unitary group u 3 on the linear space of real diagonal 3 x 3 matrices the complexification of the maximal torus this will lead to a transform realized as an absolutely continuous measure on a certain compact subset of m2 r here is a concise statement of the main result rephrased from formulas 5 1 5 6 the operator v intertwines the differential difference operators received by the editors august 22 1994 1991 mathematics subject classification primary 22e30 33c80 secondary 33c50 20b30

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

one dimensional double hecke algebra and gaussian sums we introduce and describe one dimensional cyclotomic gauss selberg sums generalizing the classical gaussian sums they correspond to irreducible self dual unitary spherical representations of the one dimensional double affine hecke algebra

shape sensitivity analysis and optimal design of disks and plates with strong discontinuities of kinematic fields abstract disks plates or beams subjected to loading and initial strains or displacements are assumed to be composed of portions interconnected by hinge or slip and dilatancy lines admitting discontinuities in displacements or slopes the shape of such lines and their stiffness properties are subjected to variation the sensitivity analysis is first discussed for an arbitrary functional of generalized stress strain displacement and boundary traction the variation of complementary and potential energies is considered as a particular case of a general derivation for sensitivity the optimal design problem is then considered and the relevant optimally conditions are derived the general theory is illustrated by examples of sensitivity analysis and optimal design with respect to shape position and stiffness of discontinuity lines for disks plates and beams numerical aspects of sensitivity analysis are also discussed

integrability and huygens principle on symmetric spaces the explicit formulas for fundamental solutions of the modified wave equations on certain symmetric spaces are found these symmetric spaces have the following characteristic property all multiplicities of their restricted roots are even as a corollary in the odd dimensional case one has that the huygens principle in hadamard s sense for these equations is fulfilled we consider also the heat and laplace equations on such a symmetric space and give explicitly the corresponding fundamental solutions heat kernel and green s function this continues our previous investigations 16 of the spherical functions on the same symmetric spaces based on the fact that the radial part of the laplace beltrami operator on such a space is related to the algebraically integrable case of the generalised calogero sutherland moser quantum system in the last section of this paper we apply the methods of heckman and opdam to extend our results to some other symmetric spaces in particular to complex and quaternian grassmannians

a consistent finite element technique for recovery of distributed reactions and surface tractions an effective technique is presented for recovering surface tractions and distributed reaction forces from finite element displacement solutions of structural mechanics problems nodal values of surface tractions and assumed trial functions are used to represent the distribution of surface stresses equations obtained from the familiar galerkin formulation are used to compute the nodal traction values the method is demonstrated to be efficient and highly accurate even in the presence of jump conditions or singularities extensions of the method are possible for recovering secondary solution variables in non structural finite element applications

integrable spin chain with reflecting end a new integrable spin chain of the haldane shastry type is introduced it is interpreted as the inverse square interacting spin chain with a reflecting end the lattice sites of this model consist of the square roots of the zeros of the laguerre polynomial using the exchange operator formalism the integrals of motion for the model are explicitly constructed

lyapunov conditions for finite time stability of time varying time delay systems in this paper we develop the lyapunov razumikhin method to finite time stability fts and finite time contractive stability ftcs of time delay systems several lyapunov based sufficient conditions for establishing these fts properties are obtained then the theoretical results are applied to fts and ftcs for a class of linear time varying ltv time delay system the efficiency of the proposed criteria is illustrated by three numerical examples where a stabilizing memoryless controller for ftcs of a second order ltv system with time delay is proposed

razumikhin type theorems for time delay systems with persistent impulses abstract the main results of the paper are generalizations of the razumikhin classical theorems for stability analysis of impulsive time delay systems by employing lyapunov razumikhin method and impulsive control theory several razumikhin type theorems for uniform stability and global exponential stability are obtained the significance and novelty of the results lie in that the criteria for stability of impulsive time delay systems admit the existence of persistent impulses and unbounded time varying delays two examples are provided to show the effectiveness of the proposed results

global well posedness and scattering for a class of nonlinear dunkl schrodinger equations in this paper we introduce a class of nonlinear schrodinger equations associated with the dunkl operators in this regard we study local and global well posedness and the scattering theory associated with these equations

loubignac s iterative method in finite element elastostatics finite element models provide only approximate results under most loadings a beam modelled by constant strain triangles and loaded by pure moment is a case in point yet if the iterative method advanced by loubignac et al 1 is applied to this model the exact stress field will be generated the iterative method derives from a parent formulation that can be solved without iteration 2 as shown in this paper the parent formulation contains the sum of two stiffness matrices one is produced by the customary finite element shape functions the other is produced by finite difference expressions for strain gradients that are not contained in the shape functions loubignac s method produces a stress field that is continous across interelement boundaries other methods that seek to impose interelement continuity are noted in the concluding section

fueter s theorem and its generalizations in dunkl clifford analysis in this paper we give a construction of dunkl monogenic and dunkl harmonic functions starting from holomorphic functions in the plane this construction has the advantage of not needing dunkl s intertwining operator or dunkl spherical harmonics to this end we study vekua type systems and prove a version of fueter s theorem in the case of finite reflection groups important examples such as a dunkl monogenic gaussian distribution or a cauchy kernel will be given at the end

harmonic analysis associated with the jacobi dunkl operator on π 2 π 2 we consider a differential difference operator 1 2 1 2 on 2 2 the eigenfunction of this operator equal to 1 at zero is related to the jacobi polynomials and to their derivatives we give a laplace integral representation for this function called the jacobi dunkl polynomial next we study the harmonic analysis associated with the operator

a characterization of integral input to state stability the notion of input to state stability iss is now recognized as a central concept in nonlinear systems analysis it provides a nonlinear generalization of finite gains with respect to supremum norms and also of finite l sup 2 gains it plays a central role in recursive design coprime factorizations controllers for nonminimum phase systems and many other areas in this paper a newer notion that of integral input to state stability iiss is studied the notion of iiss generalizes the concept of finite gain when using an integral norm on inputs but supremum norms of states in that sense generalizing the linear h sup 2 theory it allows one to quantify sensitivity even in the presence of certain forms of nonlinear resonance we obtain several necessary and sufficient characterizations of the iiss property expressed in terms of dissipation inequalities and other alternative and nontrivial characterizations

some new criteria on p th moment stability of stochastic functional differential equations with markovian switching this note gives some new razumikhin type theorems on th moment stability of stochastic functional differential equations with markovian switching sfdewms by using auxiliary ordinary differential equation the main results of this note allow the diffusion operator associated with the underlying sfdewms of the lyapunov function along a solution of the system to be not always negative an example is provided to illustrate the effectiveness of the proposed results

operational rules and a generalized hermite polynomials in this paper we use operational rules associated with three operators corresponding to a generalized hermite polynomials introduced by szego to derive as far as we know new proofs of some known properties as well as new expansions formulae related to these polynomials

stabilizability of systems with exponential dichotomy in this paper we introduce the concept of controllability into a closed subspace for time varying linear systems various characterizations are given and the dual relation is discussed this concept is used to present a necessary and sufficient condition for the stabilizability of systems with exponential dichotomy

on asymptotic stability of linear time varying systems this paper is concerned with asymptotic stability analysis of linear time varying ltv systems with the help of the notion of stable functions some differential lyapunov inequalities dlis based necessary and sufficient conditions are derived for testing asymptotic stability exponential stability and uniformly exponential stability of general ltv systems with the help of the concept of non uniformly exponential stability a class of upper triangular ltv systems is carefully investigated based on the developed stability analysis approaches a couple of numerical examples with some of them borrowed from the literature is provided to illustrate the effectiveness of the proposed theoretical results

robustness of stability of time varying linear systems abstract this paper introduces the concept of stability radius for time varying linear systems invariance properties of the stability radius are analysed for the group of bohl transformations we also explore the relationship between the stability radius the norm of a certain perturbation operator and the solvability of a nonstandard differential riccati equation as an application we construct robust lyapunov functions and show how they can be used to analyze robustness with respect to nonlinear perturbations

integrability of difference calogero moser systems a general class of n particle difference calogero moser systems with elliptic potentials is introduced besides the step size and two periods the hamiltonian depends on nine coupling constants we prove the quantum integrability of the model for n 2 and present partial results for n 3 in degenerate cases rational hyperbolic or trigonometric limit the integrability follows for arbitrary particle number from previous work connected with the multivariable q polynomials of koornwinder and macdonald liouville integrability of the corresponding classical systems follows as a corollary limit transitions lead to various well known models such as the nonrelativistic calogero moser systems associated with classical root systems and the relativistic calogero moser system

support theorem for the fundamental solution to the schrodinger equation on certain compact symmetric spaces abstract in this paper we construct the fundamental solution to the schodinger equation on a compact symmetric space with even root multiplicities using shift operators of heckman and opdam next we prove that the support of the fundamental solution becomes a lower dimensional subset at a rational time whereas its support and its singular support coincide with the whole symmetric space at an irrational time moreover we also show that generalized gauss sums appear in the expression of the fundamental solution

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

lyapunov functions for persistently excited cascaded time varying systems application to consensus we present some results on stability of linear time varying systems with particular structures such systems appear in diverse problems which include the analysis of adaptive systems persistently excited observers and consensus of systems interconnected through time varying links the originality of our statements rely in the fact that we provide smooth strict lyapunov functions hence our proofs are constructive and direct moreover we establish uniform global exponential stability with explicit stability and decay estimates for illustration we address a brief but representative case study of consensus of lagrangian systems interconnected through unreliable links

degenerate principal series representations and their holomorphic extensions let x h l be an irreducible real bounded symmetric domain realized as a real form in an hermitian symmetric domain d g k the intersection s of the shilov boundary of d with x defines a distinguished subset of the topological boundary of x and is invariant under h it can be realized as s h p for certain parabolic subgroup p of h we study the spherical representations of h induced from p we find formulas for the spherical functions in terms of the macdonald hypergeometric function this generalizes the earlier result of faraut koranyi for hermitian symmetric spaces d we consider a class of h invariant integral intertwining operators from the representations on l2 s to the holomorphic representations of g restricted to h we construct a new class of complementary series for the groups h so n m su n m with n m 2 and sp n m with n m 1 we realize them as discrete components in the branching rule of the analytic continuation of the holomorphic discrete series of g su n m su n m su n m and su 2n 2m respectively

stability analysis of linear time varying time delay systems by non quadratic lyapunov functions with indefinite derivatives abstract this paper presents a stability analysis of linear time varying ltv time delay systems by using non quadratic lyapunov functions and functionals two types of sufficient conditions are proposed for testing several different kinds of stability such as asymptotic stability exponential stability and uniformly exponential stability for ltv systems without delay then by constructing suitable non quadratic lyapunov functions functionals that are respectively the time varying weighted l 1 and l norms of the state variables and by using properties of the uniformly asymptotically stable uas function and the recently established improved razumikhin and krasovskii stability theorems both delay dependent and delay independent conditions are proposed for testing uniformly exponential stability of a class of ltv time delay systems the time derivatives of the non quadratic lyapunov functions functionals along the solutions are allowed to be indefinite namely to take both negative and positive values numerical examples show that the non quadratic lyapunov functions functionals based methods are more efficient than the existing ones that are based on quadratic functions functionals

spherical dunkl monogenics and a factorization of the dunkl laplacian in this paper we consider and study a factorization of the dunkl laplacian in terms of spherical coordinates this allows for the construction of a direct sum decomposition of spherical dunkl harmonics by explicit representation in spherical coordinates of dunkl harmonics one obtains explicit projection operators from dunkl harmonics to inner resp outer dunkl monogenics concrete examples of spherical dunkl monogenics will be given at the end

riesz transform and riesz potentials for dunkl transform analogues of riesz potentials and riesz transforms are defined and studied for the dunkl transform associated with a family of weight functions that are invariant under a reflection group the lp boundedness of these operators is established in certain cases

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

integrability in the theory of schrodinger operator and harmonic analysis the algebraic integrability for the schrodinger equation in n and the role of the quantum calogero sutherland problem and root systems in this context are discussed for the special values of the parameters in the potential the explicit formula for the eigenfunction of the corresponding sutherland operator is found as an application the explicit formula for the zonal spherical functions on the symmetric spacessu 2n spn type a ii in cartan notations is presented

uniform global asymptotic stability of a class of adaptively controlled nonlinear systems we give a new explicit global strict lyapunov function construction for the error dynamics for adaptive tracking control problems under an appropriate persistency of excitation condition we then allow time varying uncertainty in the unknown parameters in this case we construct input to state stable lyapunov functions under suitable bounds on the uncertainty provided the regressor also satisfies an affine growth condition this lets us quantify the effects of uncertainties on both the tracking and the parameter estimation we illustrate our results using rossler systems

on elliptic calogero moser systems for complex crystallographic reflection groups abstract to every irreducible finite crystallographic reflection group i e an irreducible finite reflection group g acting faithfully on an abelian variety x we attach a family of classical and quantum integrable systems on x with meromorphic coefficients these families are parametrized by g invariant functions of pairs t s where t is a hypertorus in x of codimension 1 and s g is a reflection acting trivially on t if g is a real reflection group these families reduce to the known generalizations of elliptic calogero moser systems but in the non real case they appear to be new we give two constructions of the integrals of these systems an explicit construction as limits of classical calogero moser hamiltonians of elliptic dunkl operators as the dynamical parameter goes to 0 implementing an idea of v buchstaber g felder and a veselov 1994 bfv and a geometric construction as global sections of sheaves of elliptic cherednik algebras for the critical value of the twisting parameter we also prove algebraic integrability of these systems for values of parameters satisfying certain integrality conditions

shift operators for the quantum calogero sutherland problems via knizhnik zamolodchikov equation we give a natural interpretation of the shift operators for calogero sutherland quantum problem via kz equation using matsuo cherednik mappings the explicit formulas for the inversions of these mappings and versions of shift operators for kz equations are also found as an application we show that the shift operator can be described via a factorization problem for an appropriate quantum integral discriminant of the calogero system

intertwining relations for the matrix calogero like models supersymmetry and shape invariance we investigate the intertwining relations for n particle calogero like models with internal degrees of freedom starting from the well known dunkl polychronakos operators we construct a new type of local without exchange operation differential operator these operators intertwine the matrix hamiltonians corresponding to irreducible representations of the permutation group sn in particular cases this method allows us to construct a new class of exactly solvable dirac like equations and a new class of matrix models with shape invariance we establish a connection with the approach of multi dimensional supersymmetric quantum mechanics

construction of strict lyapunov krasovskii functionals for time varying time delay systems abstract for stability analysis of time varying time delay systems it is known that time derivatives time shifts in the discrete time setting of the lyapunov krasovskii functionals lkfs for both internal stability and input to state stability iss can be indefinite which can improve the resulting stability conditions in some cases however the non strict lkfs may be insufficient and inconvenient to use in practice in this paper based on the non strict iss lkfs for time varying time delay systems and with the help of uniformly asymptotically stable uas and uniformly exponentially bounded ueb scalar functions three classes of strict iss lkfs are constructed such that their time derivatives are strictly negative definite along the trajectories of the considered system a general construction of strict iss lkfs by using the positive definite and uniformly bounded solution to a scalar lyapunov differential equation is established which includes the proposed three classes of strict iss lkfs as special cases the approaches are also extended to deal with a lyapunov inequality whose right hand side is indefinite and nonlinear both continuous time and discrete time systems are considered the effectiveness of the proposed methods is illustrated by some examples borrowed from the literature

yang baxter equation in long range interacting systems we consider the su p spin chains with long range interactions and the spin generalization of the calogero sutherland models we show that their properties derive from a transfer matrix obeying the yang baxter equation we obtain the expression of the conserved quantities of the dynamical models and we diagonalise them in the spin chain case we establish the connection between the degeneracies of the spectrum and the representation theory of the yangians we use a correspondence with the dynamical models to diagonalise the hamiltonian finally we extend the previous results to the case of a trigonometric r matrix

differential difference operators associated to reflection groups there is a theory of spherical harmonics for measures invariant under a finite reflection group the measures are products of powers of linear functions whose zero sets are the mirrors of the reflections in the group times the rotation invariant measure on the unit sphere in rn a commutative set of differential difference operators each homogeneous of degree 1 is the analogue of the set of first order partial derivatives in the ordinary theory of spherical harmonics in the case of r2 and dihedral groups there are analogues of the cauchy riemann equations which apply to gegenbauer and jacobi polynomial expansions the analysis of orthogonality structures for polynomials in several variables is a problem of vast dimensions this paper is part of an ongoing program to establish a workable theory for one particular class the underlying structure is based on finite coxeter groups these are finite groups acting on euclidean space generated by reflections in the zero sets of a collection of linear functions the roots the weight functions for the orthogonality are products of powers of these linear functions restricted to the surface of the unit sphere in addition the weight function is required to be invariant under the action of the group the resulting theory has strong similarities to the theory of spherical harmonics this was established in previous papers of the author 3 4 5 most notably a homogeneous polynomial is orthogonal to all polynomials of lower degree if and only if it is annihilated by a certain second order differential difference operator ordinary partial differentiation acts as an endomorphism on ordinary harmonic functions the use of such operators and their adjoints leads to recurrence formulas and orthogonal decompositions for harmonic polynomials in this paper we construct a commutative set of first order differentialdifference operators associated to the second order operator previously mentioned received by the editors june 1 1987 1980 mathematics subject classification 1985 revision primary 33a45 33a65 20h15 secondary 20c30 42c10 51f15

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

two step asymptotics of scaled dunkl processes dunkl processes are generalizations of brownian motion obtained by using the differential difference operators known as dunkl operators as a replacement of spatial partial derivatives in the heat equation special cases of these processes include dyson s brownian motion model and the wishart laguerre eigenvalue processes which are well known in random matrix theory it is known that the dynamics of dunkl processes is obtained by transforming the heat kernel using dunkl s intertwining operator it is also known that under an appropriate scaling their distribution function converges to a steady state distribution which depends only on the coupling parameter as the process time t tends to infinity we study scaled dunkl processes starting from an arbitrary initial distribution and we derive expressions for the intertwining operator in order to calculate the asymptotics of the distribution function in two limiting situations in the first one is fixed and t tends to infinity approach to the steady s

riesz transforms and fractional integration for orthogonal expansions on spheres balls and simplices abstract this paper studies the hardy littlewood sobolev hls inequality and the riesz transforms for fractional integration associated to weighted orthogonal polynomial expansions on spheres balls and simplexes with weights being invariant under a general finite reflection group on r d the sharp index for the validity of the hls inequality is determined and the l p boundedness of the riesz transforms is established in particular our results extend a classical inequality of muckenhoupt and stein on conjugate ultraspherical polynomial expansions our idea is based on a new decomposition of the dunkl laplace beltrami operator on the sphere and some sharp asymptotic estimates of the weighted christoffel functions

properties of some families of hypergeometric orthogonal polynomials in several variables limiting cases are studied of the koornwinder macdonald multivariable generalization of the askey wilson polynomials we recover recently and not so recently introduced families of hypergeometric orthogonal polynomials in several variables consisting of multivariable wilson continuous hahn and jacobi type polynomials respectively for each class of polynomials we provide systems of difference or differential equations recurrence relations and expressions for the norms of the polynomials in terms of the norm of the constant polynomial

littlewood paley operators associated with the dunkl operator on r we consider a vector version of l p multipliers for the dunkl transform on r and we prove l p inequalities for the littlewood paley operators

input to state stability of nonlinear systems based on an indefinite lyapunov function abstract this study investigated the input to state stability iss and integral iss iiss of nonlinear systems and discovered new sufficient conditions for them these conditions are more relaxed than others in that they employ an indefinite lyapunov function rather than a negative definite one thus a nonlinear time varying system satisfying them has uniform asymptotic stability two numerical examples show their effectiveness and advantages

macdonald polynomials and algebraic integrability we construct explicitly nonpolynomial eigenfunctions of the difference operators by macdonald in the case t qk k z this leads to a new more elementary proof of several macdonald conjectures proved first by cherednik we also establish the algebraic integrability of macdonald operators at t qk k z generalizing the result of etingof and styrkas our approach works uniformly for all root systems including the bcn case and related koornwinder polynomials moreover we apply it for a certain deformation of the an root system where the previously known methods do not work

on construction of lyapunov functions for scalar linear time varying systems abstract it is known that the construction of lyapunov functions for scalar linear time varying systems is related with solutions to the scalar lyapunov differential equation whose solution involves both improper integrals and double integrals and thus are not easy to compute in general this paper establishes a systematic method for constructing lyapunov functions for scalar linear time varying systems the constructed lyapunov functions involve an integral of the system parameter with a weighting function over a finite interval explicit conditions are imposed on the weighting function and the integral interval such that the lyapunov function is both positive definite and uniformly bounded and its time derivative is negative definite as a result constructive solutions to the associated scalar lyapunov differential equations are also obtained the established method includes some existing ones as special cases examples demonstrate the effectiveness of the proposed methods

symplectic reflection algebras calogero moser space and deformed harish chandra homomorphism to any finite group sp v of automorphisms of a symplectic vector space v we associate a new multi parameter deformation h of the algebra v smash product of with the polynomial algebra on v the parameter runs over points of r where r number of conjugacy classes of symplectic reflections in the algebra h called a symplectic reflection algebra is related to the coordinate ring of a poisson deformation of the quotient singularity v this leads to a symplectic analogue of mckay correspondence which is most complete in case of wreath products if is the weyl group of a root system in a vector space and v then the algebras h are certain rational degenerations of the double affine hecke algebra introduced earlier by cherednik

the importance of the selberg integral it has been remarked that a fair measure of the impact of atle selberg s work is the number of mathematical terms that bear his name one of these is the selberg integral an n dimensional generalization of the euler beta integral we trace its sudden rise to prominence initiated by a question to selberg from enrico bombieri more than thirty years after its initial publication in quick succession the selberg integral was used to prove an outstanding conjecture in random matrix theory and cases of the macdonald conjectures it further initiated the study of q analogues which in turn enriched the macdonald conjectures we review these developments and proceed to exhibit the sustained prominence of the selberg integral as evidenced by its central role in random matrix theory calogero sutherland quantum many body systems knizhnik zamolodchikov equations and multivariable orthogonal polynomial theory

continuous stress fields by the finite element difference method standard finite elements yield a stress field that is discontinuous across interelement boundaries in this paper results given by standard finite elements are postprocessed by an efficient iterative scheme that makes use of nodal strains finite difference operations and interpolations that augment and may differ from the shape functions used to generate the element stiffness matrix various types of interpolation are considered each leads to a continuous stress field that is more accurate than the unprocessed stress field

on the estimation of the consensus rate of convergence in graphs with persistent interconnections abstractthe aim of the current article is to establish myriad convergence rate estimates to consensus for time varying graphs with persistent interaction several novel analysis methodologies for consensus protocols employing the notions of persistence of excitation and lyapunov functions are provided the estimates are compared with each other and existing literature numerical simulations on test examples are illustrated to support the theoretical findings

a unified razumikhin type criterion on input to state stability of time varying impulsive delayed systems abstract this paper discusses the problem of input to state stability iss of time varying impulsive delayed systems by introducing a switching parameter and using the notion of average impulsive interval a unified razumikhin type criterion on iss which is simultaneously effective for stabilizing impulses and destabilizing impulses is derived the condition which requires the coefficient of the estimated upper bound of the derivative of a lyapunov function to be constant in the existing results on iss of impulsive systems is weakened the results in this paper allow the coefficient of the derivative of a lyapunov function to be time varying function which can be both positive and negative and may even be unbounded furthermore the impulsive intervals of an impulsive sequence are allowed to have arbitrarily small lower bound and large enough upper bound simultaneously as a by product a unified criterion on iss for time varying impulsive delay free systems is also presented two examples are presented to illustrate the effectiveness of our results

intertwining operators for a degenerate double affine hecke algebra and multivariable orthogonal polynomials several properties of the multivariable hermite and laguerre polynomials associated with the quantum calogero models are investigated by using the operators that intertwine representations of a degenerate version of the double affine hecke algebra as applications raising operators and shift operators for the polynomials are constructed in a unified manner

uncertainty principles for jacobi expansions abstract in this paper an uncertainty principle for jacobi expansions is derived as a generalization of that for ultraspherical expansions by rosler and voit indeed a stronger inequality is proved which is new even for fourier cosine or ultraspherical expansions a complex base of exponential type on the torus z c z 1 related to jacobi polynomials is introduced which are the eigenfunctions both of certain differential difference operators of the first order and the second order an uncertainty principle related to such exponential base is also proved

boundary k matrix elliptic dunkl operator and quantum many body systems the infinite dimensional representation of the boundary k operator the solution of the reflection equation is considered the trigonometric k matrix is studied as a degenerate case of the elliptic operator a method to construct the elliptic dunkl operator is proposed and the relationship with the quantum many body problem is also discussed

fundamental solutions for partial differential equations with reflection group invariance the dunkl differential difference operator associated with a finite reflection group is used to extend the weyl heisenberg algebra the subalgebra of mastersymmetries graded by the characters of the coxeter group w is constructed it provides us with a new class of intertwining operators for appropriate w invariant differential equations these intertwining operators turn out to be suitable for constructing fundamental solutions the remarkable example is given by the iterated wave operator with a calogero type potential for which the fundamental solution can be derived through the relevant intertwining operator in an explicit form it is found that under appropriate conditions such an equation satisfies huygens s principle in the sense that its fundamental solution possesses a nontrivial inner lacuna

fermionic limit of the calogero sutherland system we present a construction of an integrable model as a projective type limit of calogero sutherland models of n fermionic particles when n tends to infinity explicit formulas for limits of dunkl operators and of commuting hamiltonians by means of vertex operators are given we present a construction of an integrable model as a projective type limit of calogero sutherland models of n fermionic particles when n tends to infinity explicit formulas for limits of dunkl operators and of commuting hamiltonians by means of vertex operators are given

multi level finite element solution algorithms based on multiplicative and additive correction procedures owing to the failure of the finite element analyst to employ a properly refined computational model the accuracy of preliminary finite element computations is often low thus it is useful to introduce a computational procedure for improving the results obtained from a preliminary finite element solution such a procedure is presented in this paper in this procedure the solution error components are decomposed into two parts one of the error components is assumed to have a long period variation the other component is assumed to have a short period variation multiplicative and additive correction procedures are introduced to iteratively eliminate the two error components the multiplicative and additive correction procedures are implemented using multi level solution techniques

application of the zienkiewicz zhu error estimator for plate and shell analysis the global error in energy norm is computed by summing all the elemental errors the method is well suited for adaptive mesh refinement as the error contribution for each element is known demonstration examples using uniform and adaptive refinement schemes for both the plate and shell analysis show the effectiveness of the method proposed

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

orthogonal polynomials on the sphere with octahedral symmetry for any finite reflection group g acting on rn there is a family of g invariant measures h2 dw where h is a certain product of linear functions whose zero sets are the reflecting hyperplanes for g on the unit sphere and an associated partial differential operator lh f a fh fa h a is the laplacian analogously to spherical harmonics there is an orthogonal with respect to h2 do decomposition of homogeneous polynomials that is if p is of degree n then i n 21 p x j ix12jpn 2 x 0 where lh p 0 and deg p i for each i but with the restriction that p and p must all be g invariant the main topic is the hyperoctahedral group with h x x ix2 i n u x xj i j the special case n 2 leads to jacobi polynomials a detailed study of the case n 3 is made an important result is the construction of a third order differential operator that maps polynomials associated to h with indices a i to those associated with a 2 fi 1 introduction regular figures have long been objects of fascination we study here the interplay between them and the theory of orthogonal polynomials this will also touch on finite groups generated by reflections and their polynomial invariants as well as a generalization of spherical harmonics the underlying situation throughout will consist of spaces of homogeneous polynomials on rn that satisfy a second order differential equation and whose restrictions to the unit sphere have an orthogonality relation with respect to a weight function and of a finite reflection group for which all these objects are invariant the weight measure h2 x dw x on the unit sphere s in r3 where h x xlx2x3 a x 23 3 a 2 dw is surface area and the associated polynomials will be studied most intensely some of the highlights of the theory are as follows define the differential operator lhf f fah received by the editors april 19 1982 and in revised form december 15 1982 1980 mathematics subject classification primary 33a45 33a65 secondary 20h15 this research was partially supported by nsf grant mcs 81 02581 d 984 american mathematical society 0002 9947 84 1 00 25 per page

brief strict lyapunov functions for time varying systems uniformly asymptotically stable periodic time varying systems for which is known a lyapunov function with a derivative along the trajectories non positive and negative definite in the state variable on non empty open intervals of the time are considered for these systems strict lyapunov functions are constructed

double affine hecke algebras and macdonald s conjectures in this paper we prove the main results about the structure of double affine hecke algebras announced in c1 c2 the technique is based on the realization of these algebras in terms of demazure lusztig operators bgg d l2 ls c3 and rather standard facts from the theory of affine weyl groups in particular it completes the proof partially published in c2 of the macdonald scalar product conjecture see ml 12 6 including the famous macdonald constant term conjecture the q t case we mainly follow the opdam paper 0 where the macdonald mehta conjectures in the degenerate differential case were deduced from certain properties of the heckman opdam operators ho and the existence of the shift operators heckman s interpretation of these operators via the so called dunkl operators see he and also c5 was important to our approach we note that the ho operators are closely related to the so called quantum many body problem calogero sutherland moser olshanetsky perelomov the conformal field theory knizhnik zamolodchikov equations the harmonic analysis on symmetric spaces harish chandra helgason etc and last but not the least the classic theory of the hypergeometric functions establishing the connection between the difference counterparts of heckman opdam operators introduced in c4 and the macdonald theory ml m2 including the construction of the difference shift operators is the main thrust of this paper once the connection is established it is not very difficult to calculate the scalar squares of the macdonald polynomials and to prove the constant term conjecture from his fundamental paper m3 to simplify the exposition we discuss the reduced root systems only and impose the relation q tk for k e z to avoid infinite products in the definition of macdonald s pairing the purpose of this work is to present a concrete application of the new technique arbitrary q t can be handled in a

jack laurent symmetric functions we develop the general theory of jack laurent symmetric functions which are certain generalizations of the jack symmetric functions depending on an additional parameter p0

bispectrality for the quantum ruijsenaars model and its integrable deformation an elementary construction of the eigenfunctions for the quantum rational ruijsenaars model with integer coupling parameter is presented as a by product we establish the bispectral duality between this model and the trigonometric calogero moser model in particular this gives a new way for calculating jack polynomials we propose also a certain one parameter deformation of the ruijsenaars model proving its integrability and bispectrality the generalizations related to other root systems and difference operators by macdonald are considered