yang s system of particles and hecke algebras the graded hecke algebra has a simple realization as a certain algebra of operators acting on a space of smooth functions this operator algebra arises from the study of the root system analogue of yang s system of n particles on the real line with delta function potential it turns out that the spectral problem for this generalization of yang s system is related to the problem of finding the spherical tempered representations of the graded hecke algebra this observation turns out to be very useful for both these problems application of our technique to affine hecke algebras yields a simple formula for the formal degree of the generic iwahori spherical discrete series representations

constrained linear quadratic regulation the paper is a contribution to the theory of the infinite horizon linear quadratic regulator lqr problem subject to inequality constraints on the inputs and states extending an approach first proposed by sznaier and damborg 1987 a solution algorithm is presented which requires solving a finite number of finite dimensional positive definite quadratic programs the constrained lqr outlined does not feature the undesirable mismatch between open loop and closed loop nominal system trajectories which is present in the other popular forms of model predictive control mpc that can be implemented with a finite quadratic programming algorithm the constrained lqr is shown to be both optimal and stabilizing the solution algorithm is guaranteed to terminate in finite time with a computational cost that has a reasonable upper bound compared to the minimal cost for computing the optimal solution inherent to the approach is the removal of a tuning parameter the control horizon which is present in other mpc approaches and for which no reliable tuning guidelines are available two examples are presented that compare constrained lqr and two other popular forms of mpc the examples demonstrate that constrained lqr achieves significantly better performance than the other forms of mpc on some plants and the computational cost is not prohibitive for online implementation

random matrix eigenvalue problems in structural dynamics natural frequencies and mode shapes play a fundamental role in the dynamic characteristics of linear structural systems considering that the system parameters are known only probabilistically we obtain the moments and the probability density functions of the eigenvalues of discrete linear stochastic dynamic systems current methods to deal with such problems are dominated by mean centred perturbation based methods here two new approaches are proposed the first approach is based on a perturbation expansion of the eigenvalues about an optimal point which is best in some sense the second approach is based on an asymptotic approximation of multidimensional integrals a closed form expression is derived for a general rth order moment of the eigenvalues two approaches are presented to obtain the probability density functions of the eigenvalues the first is based on the maximum entropy method and the second is based on a chi square distribution both approaches result in simple closed form expressions which can be easily calculated the proposed methods are applied to two problems and the analytical results are compared with monte carlo simulations it is expected that the small randomness assumption usually employed in mean centred perturbation based methods can be relaxed considerably using these methods

a class of parabolic k subgroups associated with symmetric k varieties let g be a connected reductive algebraic group defined over a field k of characteristic not 2 an involution of g defined over k h a k open subgroup of the fixed point group of gk resp hk the set of k rational points of g resp h and gk hk the corresponding symmetric k variety a representation induced from a parabolic k subgroup of g generically contributes to the plancherel decomposition of l2 gk hk if and only if the parabolic k subgroup is split so for a study of these induced representations a detailed description of the hk conjucagy classes of these split parabolic k subgroups is needed in this paper we give a description of these conjugacy classes for general symmetric kvarieties this description can be refined to give a more detailed description in a number of cases these results are of importance for studying representations for real and p adic symmetric k varieties

computational procedure for a fast calculation of eigenvectors and eigenvalues of structures with random properties abstract the inherent uncertainties in geometry material properties etc of engineering structures can be represented by stochastic models where the parameters are described by probabilistic laws results from any analysis based on stochastic models inherit probabilistic information as well which can be used e g for reliability analysis particularly in linear dynamics of structures the calculation and analysis of random eigenvalues and eigenvectors is crucial a very flexible however computationally intensive way to analyze such systems is direct monte carlo simulation mcs in this paper procedures are shown which allow a significant reduction of computational efforts of the simulation using a subspace iteration scheme with optimally selected start vectors the procedures are not restricted to any kind of sampling technique

spectral correspondences for affine hecke algebras we introduce the notion of spectral transfer morphisms between normalized affine hecke algebras and show that such morphisms induce spectral measure preserving correspondences on the level of the tempered spectra of the affine hecke algebras involved we define a partial ordering on the set of isomorphism classes of normalized affine hecke algebras which plays an important role for the langlands parameters of lusztig s unipotent representations

the linear quadratic regulator for periodic hybrid systems abstract the main objective of this paper is to characterize feedback control laws that are optimal with respect to a quadratic cost functional in the framework of linear hybrid systems undergoing time driven periodic jumps namely the so called hybrid linear quadratic regulator lqr problem the optimal solution to the hybrid lqr problem is determined both in the case of finite horizon and infinite horizon optimal control problems by introducing a hybrid periodic extension of the classic differential and difference riccati equations thus leading to the notion of monodromy riccati equation interestingly due to the periodic nature of the discrete time events the computation of the optimal feedback hinges upon the solution of a differential rather than algebraic riccati equation also in the infinite horizon case hence yielding a time varying periodic control law necessary and sufficient conditions that ensure asymptotic stability of the closed loop system are provided and discussed in detail in the case of infinite horizon optimal control problems

unitary representations of affine hecke algebras related to macdonald spherical functions abstract for any reduced crystallographic root system we introduce a unitary representation of the extended affine hecke algebra given by discrete difference reflection operators acting in a hilbert space of complex functions on the weight lattice it is shown that the action of the center under this representation is diagonal on the basis of macdonald spherical functions as an application we compute an explicit pieri formula for these spherical functions

periodic integrable systems with delta potentials in this paper we study root system generalizations of the quantum bose gas on the circle with pair wise delta function interactions the underlying symmetry structures are shown to be governed by the associated graded algebra of cherednik s suitably filtered degenerate double affine hecke algebra acting by dunkl type differential reflection operators we use gutkin s generalization of the equivalence between the impenetrable bose gas and the free fermi gas to derive the bethe ansatz equations and the bethe ansatz eigenfunctions

approximate vibration reanalysis of structures thecombinedapproximationsmethodisdevelopedforvibrationreanalysisofstructures inthesolution process the terms of local series approximations are used as basis vectors in global reduced basis approximations the solution steps are straightforward and the method can be readily used with a general e nite element system the computational effort needed by the method is usually much smaller than the effort needed for complete vibration analysis the efe ciency of the calculations and the accuracy of the results can be controlled by the amount of information considered efe cient solutions are obtained by low order approximations and accurate results can be achievedbyconsideringhigher orderterms numericalexamplesshowthataccurateresultsareachievedefe ciently for very large changes in the design nomenclature b matrix dee ned by eq 21 c matrix dee ned by eq 16 e modulus of elasticity i moment of inertia i identity matrix k stiffness matrix kr reduced stiffness matrix l length m mass matrix mr reduced mass matrix r right hand side vector r displacement vector rb matrix of basis vectors s number of basis vectors u upper triangular matrix y vector of coefe cients coefe cient dee ned by eq 35 angle between vectors 1k change in the stiffness matrix 1m change in the mass matrix kronecker delta eigenvalue circular frequency

lq tracking controls with fixed terminal states and their application to receding horizon controls in this paper finite horizon linear quadratic tracking controls fhlqtc with fixed terminal states are proposed in recursive forms for linear time invariant lti discrete and continuous time systems the proposed fhlqtcs are to minimize finite horizon quadratic cost functions while driving the terminal state exactly to the given fixed reference value the fhlqtcs in this paper are applied to the receding horizon linear quadratic tracking controls rhlqtc with guaranteed stability it is shown through simulation that the proposed fhlqtc moves the terminal state exactly to the given destination and the proposed rhlqtc follows the given reference signal well while guaranteeing stability

efficient modal analysis of systems with local stiffness uncertainties the characterization of the uncertainty in modal quantities of an uncertain linear structural system is essential to the rapid determination of its response to arbitrary loadings although the size of many computational structural models used is extremely large i e thousands of equations the uncertainty to be analyzed is oftentimes localized to very small regions of the model this paper addresses the development of an efficient computational methodology for the modal analysis of linear structural systems with local stiffness uncertainties the newly developed methodology utilizes an enriched basis that consists of the sub spectrum of a nominal structural system augmented with additional basis vectors generated from a knowledge of the structure of the stiffness uncertainty in addition methods for determining bounds on the approximate modal frequencies and mode shapes are discussed numerical results demonstrate that the algorithm produces highly accurate results with greatly reduced computational effort copyright 2009 john wiley sons ltd

orbits and invariants associated with a pair of commuting involutions let be commuting involutions of the connected reductive algebraic group g where and g are defined over a usually algebraically closed field k char k 2 we have fixed point groups h g and k g and an action h k g g where h k g hgk 1 h h k k g g let g h k denote spec o g h k the categorical quotient let a be maximal among subtori s of g such that s s s 1 for all s s there is the associated weyl group w wh k a we show the inclusion a g induces an isomorphism a w g h k in particular the closed h k orbits are precisely those which intersect a the fibers of g g h k are the same as those occurring in certain associated symmetric varieties in particular the fibers consist of finitely many orbits we investigate the structure of w and its relation to other naturally occurring weyl groups and to the action of on the a weight spaces of the relation of the orbit type stratifications of a w and g h k along the way we simplify some of richardson s proofs for the symmetric case and at the end we quickly recover results of berger flensted jensen hoogenboom and matsuki ber57 fj78 hoo84 mat97 for the case k

homology of graded hecke algebras let h be a graded hecke algebra with complex deformation parameters and weyl group w we show that the hochschild cyclic and periodic cyclic homologies of h are all independent of the parameters and compute them explicitly we use this to prove that if the deformation parameters are real the collection of irreducible tempered h modules with real central character forms a q basis of the representation ring of w our method involves a new interpretation of the periodic cyclic homology of nite type algebras in terms of the cohomology of a sheaf over the underlying complex ane variety

on the solution of the riccati differential equation arising from the lq optimal control problem in this paper we consider the matrix riccati differential equation rde that arises from linear quadratic lq optimal control problems in particular we establish explicit closed formulae for the solution of the rde with a terminal condition using particular solutions of the associated algebraic riccati equation we discuss how these formulae change as assumptions are progressively weakened an application to lq optimal control is briefly analysed

fourier transforms on a semisimple symmetric space let g h be a semisimple symmetric space that is g is a connected semisimple real lie group with an involution and h is an open subgroup of the group of xed points for in g the main purpose of this paper is to study an explicit fourier transform on g h in terms of general representation theory the abstract fourier transform of a compactly supported smooth function f 2 c 1 c g h is given by see 5 1 f f zg h f x x dx for h a unitary irreducible representation of g and 2 h an h invariant distribution vector for here dx is the invariant measure on g h thus f isa smooth vector for h depending linearly on our goal is to obtain an explicit version of the restriction of this fourier transform to representations h in the minimal unitary principal series h for g h under the assumption that the center of g is nite in a sequel 9 to this paper it will be proved that a function f 2 c 1 c g h is uniquely determined by the restriction of f to this series a priori it is known that f is determined by f

on the classification of k involutions let g be a connected reductive algebraic group defined over a field k of characteristic not 2 an involution of g defined over k h a k open subgroup of the fixed point group of and gk resp hk the set of k rational points of g resp h the variety gk hk is called a symmetric k variety these varieties occur in many problems in representation theory geometry and singularity theory over the last few decades the representation theory of these varieties has been extensively studied for k r and c as most of the work in these two cases was completed the study of the representation theory over other fields like local fields and finite fields began the representations of a homogeneous space usually depend heavily on the fine structure of the homogeneous space like the restricted root systems with weyl groups etc thus it is essential to study first this structure and the related geometry in this paper we give a characterization of the isomorphy classes of these symmetric k varieties together with their fine structure of restricted root systems and also a classification of this fine structure for the real numbers p adic numbers finite fields and number fields

on form factors in nested bethe ansatz systems we investigate form factors of local operators in a multi component quantum nonlinear schrodinger model a prototype theory solvable by the so called nested bethe ansatz we determine the analytic properties of the infinite volume form factors using the coordinate bethe ansatz solution and establish a connection with the finite volume matrix elements in the two component models we derive a set of recursion relations for the magnonic form factors which are the matrix elements on the nested bethe ansatz states in certain simple cases involving states with only one spin impurity we obtain explicit solutions for the recursion relations

convergence of the time invariant riccati differential equation and lq problem mechanisms of attraction the nature of the attraction of the solution of the time invariant matrix riccati differential equation towards the stabilizing solution of the algebraic riccati equation is studied this is done on an explicit formula for the solution when the system is stabilizable and the hamiltonian matrix has no eigenvalues on the imaginary axis various aspects of this convergence are analysed by displaying explicit mechanisms of attraction and connections are made with the literature the analysis ultimately shows the exponential nature of the convergence of the solution of the riccati differential equation and of the related finite horizon lq optimal state and control trajectories as the horizon recedes computable characteristics are given which can be used to estimate the quality of approximating the solution of a large finite horizon lq problem by the solution of an infinite horizon lq problem

efficient characterization of the random eigenvalue problem in a polynomial chaos decomposition a new procedure for characterizing the solution of the eigenvalue problem in the presence of uncertainty is presented the eigenvalues and eigenvectors are described through their projections on the polynomial chaos basis an efficient method for estimating the coefficients with respect to this basis is proposed the method uses a galerkin based approach by orthogonalizing the residual in the eigenvalue eigenvector equation to the subspace spanned by the basis functions used for approximation in this way the stochastic problem is framed as a system of deterministic non linear algebraic equations this system of equations is solved using a newton raphson algorithm although the proposed approach is not based on statistical sampling the efficiency of the proposed method can be significantly enhanced by initializing the non linear iterative process with a small statistical sample synthesized through a monte carlo sampling scheme the proposed method offers a number of advantages over existing methods based on statistical sampling first it provides an approximation to the complete probabilistic description of the eigensolution second it reduces the computational overhead associated with solving the statistical eigenvalue problem finally it circumvents the dependence of the statistical solution on the quality of the underlying random number generator copyright 2007 john wiley sons ltd

hardy spaces for non compactly causal symmetric spaces and the most continuous spectrum let g h be a semisimple symmetric space then the space l2 g h can be decomposed into a finite sum of series of representations induced from parabolic subgroups of g the most continuous part of the spectrum of l2 g h is the part induced from the smallest possible parabolic subgroup in this paper we introduce hardy spaces canonically related to this part of the spectrum for a class of non compactly causal symmetric spaces the hardy space is a reproducing hilbert space of holomorphic functions on a bounded symmetric domain of tube type containing g h as a boundary component a boundary value map is constructed and we show that it induces a g isomorphism onto a multiplicity free subspace of full spectrum in the most continuous part lmc2 g h of l2 g h we also relate our hardy space to the classical hardy space on the bounded symmetric domain

the riccati equation solution of the linear quadratic problem with constrained terminal state we represent the minimal cost and the optimal feedback for the linear quadratic optimal control problem with linear constraints on the terminal state in terms of a solution of the matrix riccati equation which is obtained as a limit of certain solutions representing the optimal cost for the unconstrained problem with penalized terminal state also we examine the convergence of the solution for the upper limit of the interval tending to infinity and the stabilizing effect of the limit control as a by product of our method we obtain some improvement of the known results for the unconstrained nonautonomous infinite interval problem

formal degrees of unipotent discrete series representations and the exotic fourier transform we introduce a notion of elliptic fake degrees for unipotent elliptic representations of a semisimple p adic group we conjecture and verify in some cases that the relation between the formal degrees of unipotent discrete series representations of a semisimple p adic group and the elliptic fake degrees is given by the exotic fourier transform matrix introduced by lusztig in the study of representations of finite groups of lie type

spectral theory for interacting particle systems solvable by coordinate bethe ansatz we develop spectral theory for the q hahn stochastic particle system introduced recently by povolotsky that is we establish a plancherel type isomorphism result that implies completeness and biorthogonality statements for the bethe ansatz eigenfunctions of the system owing to a markov duality with the q hahn tasep a discrete time generalization of tasep with particles jump distribution being the orthogonality weight for the classical q hahn orthogonal polynomials we write down moment formulas that characterize the fixed time distribution of the q hahn tasep with general initial data the bethe ansatz eigenfunctions of the q hahn system degenerate into eigenfunctions of other not necessarily stochastic interacting particle systems solvable by the coordinate bethe ansatz this includes the asep the asymmetric six vertex model and the heisenberg xxz spin chain all models are on the infinite lattice in this way each of the latter systems possesses a spectral theory too in particular biorthogonality of the asep eigenfunctions which follows from the corresponding q hahn statement implies symmetrization identities of tracy and widom for asep with either step or step bernoulli initial configuration as corollaries another degeneration takes the q hahn system to the q boson particle system dual to q tasep studied in detail in our previous paper 2013 thus at the spectral theory level we unify two discrete space regularizations of the kardar parisi zhang equation stochastic heat equation namely q tasep and asep

hochschild homology of affine hecke algebras abstract let h h r q be an affine hecke algebra with complex possibly unequal parameters q which are not roots of unity we compute the hochschild and the cyclic homology of h it turns out that these are independent of q and that they admit an easy description in terms of the extended quotient of a torus by a weyl group both of which are canonically associated to the root datum r for positive q we also prove that the representations of the family of algebras h r q c come in families which depend analytically on analogous results are obtained for graded hecke algebras and for schwartz completions of affine hecke algebras

espace de schwartz pour la transformation de fourier hypergeometrique resume let r be a reduced root system in an euclidean vector space e w the weyl group of r one determines the hypergeometric fourier transform cf e opdam cuspidal hypergeometric functions preprint of the related schwartz space of w invariant functions on e introduced by tinfou the answer is very natural and quite similar to results of harish chandra the proof requires a theory of the constant term for w invariant functions satisfying the hypergeometric system this is analogous to harish chandra s theory once one has realized that simple difference operators play here the role of some elements of the unipotent radical of a parabolic subalgebra also the fact that the parameter of cuspidal hypergeometric functions might be singular introduces new difficulties this theory being established one can take advantage of the techniques we used for real reductive symmetric spaces cf the introduction to the article by the author formule de plancherel pour les espaces symetriques reductifs ann math 147 1998 417 452 especially the truncation and its consequences

random eigenvalues algebraic methods and structural dynamic models a review of the literature of the study of structural vibration problems with random parameters is provided there have been several approaches to this problem these have been encapsulated in this paper for the benefit of those who need to assess such possibilities in particular the algebraic theory of random variables is delineated here with the application in mind being the determination or estimation of the statistics of the eigenvalues of linear dynamical systems various transformation techniques are summarized and discussed with simple examples

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

on a uniqueness property of supercuspidal unipotent representations abstract the formal degree of a unipotent discrete series character of a simple linear algebraic group over a non archimedean local field in the sense of lusztig 17 is a rational function of q evaluated at q q the cardinality of the residue field the irreducible factors of this rational function are q and cyclotomic polynomials we prove that the formal degree of a supercuspidal unipotent representation determines its lusztig langlands parameter up to twisting by weakly unramified characters for split exceptional groups this result follows from the work of m reeder 28 and for the remaining exceptional cases this is verified in 7 in the present paper we treat the classical families the main result of this article characterizes unramified lusztig langlands parameters which support a cuspidal local system in terms of formal degrees the result implies the uniqueness of so called cuspidal spectral transfer morphisms as introduced in 22 between unipotent affine hecke algebras up to twisting by unramified characters in 23 the essential uniqueness of arbitrary unipotent spectral transfer morphisms was reduced to the cuspidal case

design of l q regulators for state constrained continuous time systems a new methodology to the design of lq regulators for continuous time systems subject to linear state constraints is proposed consisting of two parts in the first one the positive invariance of the polyhedron defined by the constraints in the state space is imposed guaranteeing thereby that the constraints will not be violated furthermore the admissible constrained controllers are parameterized via the determination of the elements which are fixed in the state feedback matrix this parameterization enables in a second part the l q regulator to be obtained from the solution of a parameter optimization problem subject to linear constraints for which it is proposed a specialized feasible directions method

paley wiener theorems for hyperbolic spaces we prove a topological paley wiener theorem for the fourier transform defined on the real hyperbolic spaces soo p q soo p 1 q for p q 2n without restriction to k types we also obtain paley wiener type theorems for l schwartz functions 0 2 for fixed k types

the plancherel decomposition for a reductive symmetric space i spherical functions we prove the plancherel formula for spherical schwartz functions on a reductive symmetric space our starting point is an inversion formula for spherical smooth compactly supported functions the latter formula was earlier obtained from the most continuous part of the plancherel formula by means of a residue calculus in the course of the present paper we also obtain new proofs of the uniform tempered estimates for normalized eisenstein integrals and of the maass selberg relations satisfied by the associated c functions

support properties and holmgren s uniqueness theorem for differential operators with hyperplane singularities let w be a finite coxeter group acting linearly on rn in this article we study the support properties of a w invariant partial differential operator d on rn with real analytic coefficients our assumption is that the principal symbol of d has a special form related to the root system corresponding to w in particular the zeros of the principal symbol are supposed to be located on hyperplanes fixed by reflections in w we show that conv suppdf conv suppf holds for all compactly supported smooth functions f so that conv suppf is w invariant the main tools in the proof are holmgren s uniqueness theorem and some elementary convex geometry several examples and applications linked to the theory of special functions associated with root systems are presented

optimal output transitions for linear systems this article addresses the optimal minimum input energy output transition problem for linear systems the goal is to transfer the output from an initial value y t email protected for all time t t f previous methods solve this output transition problem by transforming it into a state transition problem the initial and final states x t i x t f respectively are chosen and a minimum energy state to state transition problem is solved however the choice of the initial and final states can be ad hoc and the resulting output transition cost input energy may not be minimal the contribution of this article is the solution of the optimal output transition problem an example system with elastic dynamics is studied to illustrate the proposed method simulation results are presented that show substantial reduction of transition costs with the use of the proposed method when compared to the use of minimum energy state to state transitions

an improved computational technique for perturbations of the generalized symmetric linear algebraic eigenvalue problem a technique is developed and presented which improves the accuracy of the predictions for the eigenvalues of the usual first order perturbation scheme for the generalized symmetric linear algebraic eigenvalue problem the improved method very simply replaces the usual b norms appearing in the denominator of the expressions for the first order changes in the eigenvalues by bp norms where bp is the perturbed b matrix with increasing sizes for the perturbations the improved technique produces results whose accuracy approaches those of the usual second order predictions numerical studies are presented to support the analysis

notes on the structure of the function interacting gas intertwining operator in the degenerate affine hecke algebra the wavefunction of the function interacting bose gas on the infinite interval is studied by introducing the intertwining operator of the degenerate affine hecke algebra the non symmetric eigenfunction of the dunkl operator is constructed

invariant differential operators and eigenspace representations on an affine symmetric space let g h be an affine symmetric space of split rank r let d be a preferred polynomial algebra of g invariant differential operators on g h generated by r elements we show that the space of k finite joint eigenfunctions of d on g h form an admissible g k module which is called an eigenspace representation the main content of this paper is description of the algebras of invariant differential operators and determination of the eigenspace representations on g h we also obtain a poisson transform for t spherical eigenfunctions on g h by eisenstein integrals

dynamic analysis using a reduced basis of exact modes and ritz vectors the problem of obtaining an approximate solution for the response of a structure under dynamic forces is addressed the structure is modeled using mass and stiffness matrices and has n degrees of freedom an dimensional approximation is sought in a subspace defined by an arbitrary m orthogonal array a residual vector which is smaller in magnitude the more accurate the approximation is derived the residual contains two terms one is the error involved representing the force vector in the subspace the other is smaller the better the array leads to m approximate solutions of the exact eigenproblem numerical experiments are performed using arrays that are combinations of exact eigenvectors and of ritz vectors in one case when 11 different 10 dimensional bases are used to approximate the solution of a 60 degree of freedom problem it is found that a root mean square measure of relative error is smallest for the basis formed using the 6 lowest exact eigenvectors plus 4 ritz vectors generated from the spatial distribution of the loading

extended kirsch combined method for eigenvalue reanalysis based on the kirsch combined method and the rayleigh quotient a new eigenvalue reanalysis method for modified structures is developed using this method the accuracy of the eigenvalues reanalysis of large modifications of structural parameters is improved as an application of the method a numerical example is given to illustrate the accuracy by comparing the second order perturbation method with the proposed method if the large modifications of structural parameters are made the calculated results show that the proposed method is effective

higgs bundles gauge theories and quantum groups the appearance of the bethe ansatz equation for the nonlinear schrodinger equation in the equivariant integration over the moduli space of higgs bundles is revisited we argue that the wave functions of the corresponding two dimensional topological u n gauge theory reproduce quantum wave functions of the nonlinear schrodinger equation in the n particle sector this implies the full equivalence between the above gauge theory and the n particle sub sector of the quantum theory of the nonlinear schrodinger equation this also implies the explicit correspondence between the gauge theory and the representation theory of the degenerate double affine hecke algebra we propose a similar construction based on the g g gauged wzw model leading to the representation theory of the double affine hecke algebra

the plancherel decomposition for a reductive symmetric space ii representation theory we obtain the plancherel decomposition for a reductive symmetric space in the sense of representation theory our starting point is the plancherel formula for spherical schwartz functions obtained in part i the formula for schwartz functions involves eisenstein integrals obtained by a residual calculus in the present paper we identify these integrals as matrix coefficients of the generalized principal series

the one dimensional kpz equation and its universality class our understanding of the one dimensional kpz equation alias noisy burgers equation has advanced substantially over the past 5 years we provide a non technical review where we limit ourselves to the stochastic pde and lattice type models approximating it

reanalysis of linear and nonlinear structures using iterated shanks transformation reanalysis of structures is a common task in two important fields of structural engineering namely optimization and reliability it implies the solution of the kinematic equations for each new set of data in order to simplify this task several approximate methods have been developed in this paper a simple method for reanalysis applicable to either linear or nonlinear structures is proposed it is based on the improvement of the neumann resolvent series which diverges beyond a certain radius by means of the iterated shanks transformation the method is compared with two relevant proposals intended to improve the convergence of the neumann series namely the pade series and the scaled approximation methods the accuracy and simplicity of the proposed method is demonstrated through some examples concerning linear and nonlinear structures

discrete harmonic analysis on a weyl alcove abstract we introduce a representation of the double affine hecke algebra at the critical level q 1 in terms of difference reflection operators and use it to construct an explicit integrable discrete laplacian on the weyl alcove corresponding to an element in the center the laplacian in question is to be viewed as an integrable discretization of the conventional laplace operator on euclidean space perturbed by a delta potential supported on the reflection hyperplanes of the affine weyl group the bethe ansatz method is employed to show that our discrete laplacian and its commuting integrals are diagonalized by a finite dimensional basis of periodic macdonald spherical functions

a parametrization of the solutions of the finite horizon lq problem with general cost and boundary conditions a generalization of the finite horizon linear quadratic regulator problem is proposed for lti continuous time controllable systems in particular a formulation of the linear quadratic lq problem is considered with affine constraints on the initial and the terminal states and with general quadratic costs in the initial and terminal states the solution presented is simple and attractive from a computational point of view and is based on the solutions of an algebraic riccati equation and of a lyapunov equation that enable all the solutions of the hamiltonian differential equation to be parametrized in closed form

cuspidal discrete series for semisimple symmetric spaces abstract we propose a notion of cusp forms on semisimple symmetric spaces we then study the real hyperbolic spaces in detail and show that there exists both cuspidal and non cuspidal discrete series in particular we show that all the spherical discrete series are non cuspidal

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

spin representations of real reflection groups of noncrystallographic root systems abstract a uniform parametrization for the irreducible spin representations of weyl groups in terms of nilpotent orbits is recently achieved by ciubotaru 2011 this paper is a generalization of this result to other real reflection groups let v 0 r v 0 r be a root system with the real reflection group w we define a special subset of points in v 0 which will be called solvable points those solvable points in the case r crystallographic correspond to the nilpotent orbits whose elements have a solvable centralizer in the corresponding lie algebra then a connection between the irreducible spin representations of w and those solvable points in v 0 is established

analysis of eigenvalues and modal interaction of stochastic systems an eigenvalue spectral analysis of stochastic engineering systems is presented a comparative numerical study between approximations based on monte carlo sampling a taylor series based perturbation approach and the polynomial chaos representation is conducted it is observed that the polynomial chaos representation gives more accurate estimates of the statistical moments than the perturbation method especially for the higher modes the differences of accuracy in the two methods are more pronounced as the system variability increases moreover the chaos expansion gives a more detailed probabilistic description of the eigenvalues and the eigenvectors in addition a method for representing the statistical modal overlapping is presented that characterizes the statistical interaction between the various modes

comments on structural invariant subspaces of singular hamiltonian systems and nonrecursive solutions of finite horizon optimal control problems in this note it is shown that the main result of concerning a characterization of a pair of structural invariant subspaces associated with the extended symplectic system is a particular case of a result presented in and within a more general and rigorous context we also analyse the proof of the main result of and the way such result is used to accomodate the boundary conditions in the solution of a finite horizon linear quadratic optimal control problem

norm formulae for the bethe ansatz on root systems of small rank the norms of the bethe ansatz eigenfunctions for the lieb liniger quantum system of n bosonic particles on a ring with pairwise repulsive delta potential interactions are given by a beautiful determinantal formula first conjectured by gaudin in the early seventies and then proven by korepin about a decade later recently e emsiz formulated a similar conjecture generalizing the gaudin korepin norm formula in terms of the root systems of complex simple lie algebras here we confirm the validity of the conjecture in question for small root systems up to rank 3 thus including the important test case of the exceptional root system g2

discrete series characters for affine hecke algebras and their formal degrees we introduce the generic central character of an irreducible discrete series representation of an affine hecke algebra using this invariant we give a new classification of the irreducible discrete series characters for all abstract affine hecke algebras except for the types e n 1 n 6 7 8 with arbitrary positive parameters and we prove an explicit product formula for their formal degrees in all cases

stationary correlations for the 1d kpz equation we study exact stationary properties of the one dimensional kardar parisi zhang kpz equation by using the replica approach the stationary state for the kpz equation is realized by setting the initial condition the two sided brownian motion bm with respect to the space variable developing techniques for dealing with this initial condition in the replica analysis we elucidate some exact nature of the height fluctuation for the kpz equation in particular we obtain an explicit representation of the probability distribution of the height in terms of the fredholm determinants furthermore from this expression we also get the exact expression of the space time two point correlation function

continuous time inverse quadratic optimal control problem in this paper the problem of finite horizon inverse optimal control ioc is investigated where the quadratic cost function of a dynamic process is required to be recovered based on the observati

lq control for constrained continuous time systems a design method of lq optimal control law is considered for constrained continuous time systems by introducing singular value decomposition for finite time horizon linear systems the sequence of lq sub optimal control laws which converges to the exact solution is obtained based on quadratic programming it is also shown that the nonlinear sub optimal feedback gain is found by the union of affine state functions

asymptotic numerical methods and pade approximants for non linear elastic structures in this paper we apply asymptotic numerical methods for computing non linear equilibrium paths of elastic beam plate and shell structures the non linear branches are sought in the form of asymptotic expansions and they are determined by solving numerically fem several linear problems with a single stiffness matrix a large number of terms of the series can be easily computed by using recurrence formulas in comparison with a more classical step by step procedure the method is rapid and automatic we show with some examples that the choice of the expansion s parameter and the use of pade approximants play an important role in the determination of the size of the domain of convergence

holomorphic h spherical distribution vectors in principal series representations let g h be a semisimple symmetric space the main tool to embed a principal series representation of g into l2 g h are the h invariant distribution vectors if g h is a non compactly causal symmetric space then g h can be realized as a boundary component of the complex crown in this article we construct a minimal g invariant subdomain h of with g h as shilov boundary let be a spherical principal series representation of g we show that the space of h invariant distribution vectors of which admit a holomorphic extension to h is one dimensional furthermore we give a spectral definition of a hardy space corresponding to those distribution vectors in particular we achieve a geometric realization of a multiplicity free subspace of l2 g h mc in a space of holomorphic functions

eigensolution reanalysis of modified structures using epsilon algorithm based on the neumann series expansion and epsilon algorithm a new eigensolution reanalysis method is developed in the solution process the basis vectors can be obtained using the matrix perturbation or the neumann series expansion to construct the vector sequence and then using the epsilon algorithm table to obtain the approximate eigenvectors the approximate eigenvalues are computed from the rayleigh quotients the solution steps are straightforward and it is easy to implement with the general finite element analysis system two numerical examples a 40 storey frame and a chassis structure are given to demonstrate the application of the present method by comparing with the exact solutions and the kirsch method solutions it is shown that the excellent results are obtained for very large changes in the design and that the accuracy of the epsilon algorithm is higher than that of the kirsch method and the computation time is less than that of the kirsch method

support d asymptotiques et croissance des fonctions propres sur un espace symetrique semi simple abstract we associate several distribution boundary values to an eigenfunction with moderate growth on a riemannian symmetric space g k the associated character of the algebra d g k of invariant differential operators is allowed to be non regular we prove results on the support of these boundary values these allow us to recover the theorems of matsuki oshima and oshima on the equivalence between growth of an eigenfunction and limitations on the supports of its boundary values our approach is based on an asymptotic analysis that makes no use of hyperfunction theory

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

scattering theory of discrete pseudo laplacians on a weyl chamber to a crystallographic root system we associate a system of multivariate orthogonal polynomials diagonalizing an integrable system of discrete pseudo laplacians on the weyl chamber we develop the time dependent scattering theory for these discrete pseudo laplacians and determine the corresponding wave operators and scattering operators in closed form as an application we describe the scattering behavior of certain hyperbolic ruijsenaars schneider type lattice calogero moser models associated with the macdonald polynomials

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

rapid reanalysis of the generalized eigenvalue problem of locally modified linear dynamical systems abstract a computationally efficient method for the reanalysis of the damped generalized eigenvalue problems associated with linear dynamical systems with local damping and stiffness modifications is developed no restrictions are imposed on the damping of the system and the modifications in the damping and the stiffness can be independent of each other the application of the developed method to the undamped generalized eigenvalue problem with mass and stiffness modifications is also discussed the proposed method achieves up to three orders of magnitude gains in computational efficiency compared to the conventional solution procedures

generalized springer correspondence and green functions for type b c graded hecke algebras this paper presents a combinatorial approach to the tempered representation theory of a graded hecke algebra h of classical type b or c with arbitrary parameters we present various combinatorial results which together give a uniform combinatorial description of what becomes the springer correspondence in the classical situation of equal parameters more precisely by using a general version of lusztig s symbols which describe the classical springer correspondence we associate to a discrete series representation of h with central character w0c a set w0c of w0 characters where w0 is the weyl group this set w0c is shown to parametrize the central characters of the generic algebra which specialize into w0c using the parabolic classification of the central characters of h t r on the one hand and a truncated induction of weyl group characters on the other hand we define a set w0c for any central character w0c of h t r and show that this property is preserved we show that in the equal parameter situation we retrieve the classical springer correspondence by considering a set u of partitions which replaces the unipotent classes of so2n 1 c and sp2n c and a bijection between u and the central characters of h t r we end with a conjecture which basically states that our generalized springer correspondence determines h t r exactly as the classical springer correspondence does in the equal label case in particular we conjecture that w0c indexes the modules in h t r with central character w0c in the following way a module m in h t r has a natural grading for the action of w0 and the w0 representation m in its top degree is irreducible when m runs through the modules in h t r with central character w0c m runs through w0c moreover still in analogy with the equal parameter case we conjecture that the w0 structure of the modules in h t r can be computed using generalized green functions

eigenvalue error analysis of viscously damped structures using a ritz reduction method the efficient solution of the eigenvalue problem that results from inserting passive dampers with variable stiffness and damping coefficients into a structure is addressed eigenanalysis of reduced models obtained by retaining a number of normal modes augmented with ritz vectors corresponding to the static solutions resulting from the load patterns introduced by the dampers has been empirically shown to yield excellent approximations to the full eigenvalue problem an analysis of this technique in the case of a single damper is presented a priori and a posteriori error estimates are generated and tested on numerical examples comparison theorems with modally truncated models and a markov parameter matching reduced order model are derived these theorems corroborate the heuristic that residual flexibility methods improve low frequency approximation of the system the analysis leads to other techniques for eigenvalue approximation approximate closed form solutions are derived that include a refinement to eigenvalue derivative methods for approximation an efficient newton scheme is also developed a numerical example is presented demonstrating the effectiveness of each of these methods

radon transformation on reductive symmetric spaces support theorems abstract we introduce a class of radon transforms for reductive symmetric spaces including the horospherical transforms and derive support theorems for these transforms a reductive symmetric space is a homogeneous space g h for a reductive lie group g of the harish chandra class where h is an open subgroup of the fixed point subgroup for an involution on g let p be a parabolic subgroup such that p is opposite to p and let n p be the unipotent radical of p for a compactly supported smooth function on g h we define r p g to be the integral of n p n g n h over n p the radon transform r p thus obtained can be extended to a large class of distributions containing the rapidly decreasing smooth functions and the compactly supported distributions for these transforms we derive support theorems in which the support of is partially characterized in terms of the support of r p the proof is based on the relation between the radon transform and the fourier transform on g h and a paley wiener shift type argument our results generalize the support theorem of helgason for the radon transform on a riemannian symmetric space