general decay of solutions of a viscoelastic equation abstract in this paper we consider the following viscoelastic equation u t t u 0 t g t u d 0 in a bounded domain and establish a general decay result which is not necessarily of exponential or polynomial type this work generalizes and improves earlier results in the literature

some small unipotent representations of indefinite orthogonal groups abstract we study a family of small unitary representations of indefinite orthogonal groups these representations arise as analytic continuations of the discrete series and were studied extensively by knapp in k3 we complete knapp s analysis by proving that they are irreducible in order to do so we prove that the representations are unipotent and have irreducible associated cycles in which all multiplicities are exactly one moreover we prove that the k type structure of each representation matches up to a shift the k type structure of the ring of functions on the closure a nilpotent k c orbit on p

attractors for wave equations with degenerate memory abstract this paper is concerned with the long time dynamics of a semilinear wave equation with degenerate viscoelasticity u t t u 0 g s div a x u t s d s f u h x defined in a bounded domain of r 3 with dirichlet boundary condition and nonlinear forcing f u with critical growth the problem is degenerate in the sense that the function a x 0 in the memory term is allowed to vanish in a part of when a x does not degenerate and g decays exponentially it is well known that the corresponding dynamical system has a global attractor without any extra dissipation in the present work we consider the degenerate case by adding a complementary frictional damping b x u t which is in a certain sense arbitrarily small such that a b 0 in despite that the dissipation is given by two partial damping terms of different nature none of them necessarily satisfying a geometric control condition we establish the existence of a global attractor with finite fractal dimension

asymptotics of viscoelastic materials with nonlinear density and memory effects abstract this paper is concerned with the nonlinear viscoelastic equation t u t t u t t u u 0 s u t s d s f u h suitable to modeling extensional vibrations of thin rods with nonlinear material density t u t u and presence of memory effects this class of equations was studied by many authors but well posedness in the whole admissible range 0 4 and for f growing up to the critical exponent were established only recently the existence of global attractors was proved in presence of an additional viscous or frictional damping in the present work we show that the sole weak dissipation given by the memory term is enough to ensure existence and optimal regularity of the global attractor a for 4 and critical nonlinearity f

some calculations of the lusztig vogan bijection for classical nilpotent orbits abstract in this manuscript we compute explicitly the lusztig vogan bijection for local systems of some classical special nilpotent orbits using these results we prove a conjecture of achar and sommers on regular functions of some covers of classical nilpotent orbits

cubic unipotent arthur parameters and multiplicities of square integrable automorphic forms l g f g a ld g f g a lcont g f g a into the direct sum of its discrete spectrum and its continuous spectrum the theory of eisenstein series reduces the description of lcont g f g a to that of the discrete spectrum of certain reductive subgroups of g and thus the basic question is the understanding of the discrete spectrum ld g f g a the discrete spectrum has a further orthogonal decomposition

long time behavior of a quasilinear viscoelastic equation with past history abstract this paper is concerned with a class of quasilinear wave equations with memory u t u t t u u t t t t s u s d s u t f u h 0 which was considered by several authors with 0 since 2001 existing results are mainly devoted to global existence energy decay existence with small data and blow up of solutions however uniqueness seems to be an open problem and existence of attractors was no yet considered the objective of the present paper is to provide some results on the well posedness and long time behavior to this equation in a more general setting which includes past history that is by taking in the memory term

a continuous time approach for the asymptotic value in two person zero sum repeated games we consider the asymptotic value of two person zero sum repeated games with general evaluations of the stream of stage payoffs we show existence for incomplete information games splitting games and absorbing games the technique of proof consists of embedding the discrete repeated game into a continuous time game and to use viscosity solution tools

repeated games with lack of information on one side we introduce the dual differential game of a repeated game with lack of information on one side as the natural continuous time version of the dual game introduced by de meyer 1996 a traditional way to study the value of differential games is through discrete time approximations here we follow the opposite approach we identify the limit value of a repeated game in discrete time as the value of a differential game namely we use the recursive structure for the finitely repeated version of the dual game to construct a differential game for which the upper values of the uniform discretization satisfy precisely the same property the value of the dual differential game exists and is the unique viscosity solution of a first order derivative equation which appears implicitely in de meyer and rosenberg 1999 with a limit condition we identify the solution by translating viscosity properties in the primal

long time behavior of a suspension bridge equations with past history in this paper we study a suspension bridge equation with memory effects for the suspension bridge equation without memory there are many classical results existing results mainly devoted to existence and uniqueness of a weak solution energy decay of solution and existence of global attractors however the existence of global attractors for the suspension bridge equation with memory was no yet considered the object of the present paper is to provide some results on the well posedness and long time behavior to the suspension bridge equation when the unique damping mechanism is given by the memory term

quadratic unipotent arthur parameters and residual spectrum of symplectic groups the purpose of this paper is to study certain quadratic unipotent arthur parameters in the sense of moeglin and use them to parametrize a part of the residual spectrum of symplectic groups over number fields coming from the conjugacy class of borel subgroups in particular using certain identities satisfied by local intertwining operators arthur s multiplicity formula is established for them which remarkably enough appears by itself in the corresponding residue of the eisenstein series introduction in this paper we study certain quadratic unipotent arthur

frictional versus viscoelastic damping in a semilinear wave equation in this article we show exponential and polynomial decay rates for the partially viscoelastic nonlinear wave equation subject to a nonlinear and localized frictional damping the equation that models this problem is given by begin eqnarray u tt kappa 0 delta u int 0 t mbox div a x g t s nabla u s ds f u b x h u t 0 mbox in omega times bbb r quad end eqnarray where a b are nonnegative functions a in c 1 overline omega b in l infty omega satisfying the assumption begin eqnarray a x b x geq delta 0 quad forall x in omega end eqnarray and f and h are power like functions we observe that the assumption 0 2 gives us a wide assortment of possibilities from which to choose the functions a x and b x and the most interesting case occurs when one has simultaneous and complementary damping mechanisms taking this point of view into account a distinctive feature of our paper is exactly to consider different and localized damping mechanisms acting in the domain but not necessarily strategically localized dissipations as considered in the prior literature

on special pieces the springer correspondence and unipotent characters let g be a connected reductive algebraic group over the algebraic closure of a finite field bbb f q of good characteristic in this paper we demonstrate a remarkable compatibility between the springer correspondence for g and the parametrization of unipotent characters of g bbb f q in particular we show that in a suitable sense large portions of these two assignments in fact coincide this extends earlier work of lusztig on springer representations within special pieces of the unipotent variety

asymptotic stability and energy decay rates for solutions of the wave equation with memory in a star shaped domain we study the asymptotic stability and give the energy decay rates for solutions of the wave equation with boundary dissipation of the memory type

quantization of special symplectic nilpotent orbits and normality of their closures abstract we study the regular function ring r o for all symplectic nilpotent orbits o with even column sizes we begin by recalling the quantization model for all such orbits by barbasch using unipotent representations with this model we express the multiplicities of fundamental representations appearing in r o by a parabolically induced module finally we will use this formula to give a criterion on the normality of the zariski closure o of o

a gronwall type lemma with parameter and its application to kirchhoff type nonlinear wave equation abstract the paper investigates a new gronwall type lemma with parameter 0 and establishes the uniform in time estimates as an application the paper studies the asymptotic behavior of the kirchhoff type equations with a strong dissipation and proves that the related solution semigroup possesses the global attractor

theoreme de type paley wiener pour les groupes de lie semi simples reels avec une seule classe de conjugaison de sous groupes de cartan abstract in this article we characterize the space of fourier transforms of the smooth compactly supported functions on a semisimple lie group with one conjugacy class of cartan subgroups cf th 6 we get also results on the space of fourier transforms of the elements of the enveloping algebra of the lie algebra of such a group cf th 1 2 and 3

characterizations of character sheaves for complex reductive algebraic groups character sheaves are geometric objects associated to a reductive algebraic group they are closely related to irreducible characters of finite chevalley groups the existence of character sheaves was conjectured by g lusztig in the book characters of reductive groups over finite fields 1984 there he described what one should expect character sheaves to be for complex reductive groups a definition of character sheaves for an arbitrary characteristic quite different from this description was given by him later 1985 it is shown in this thesis in particular that the original 1984 lusztig s suggestion for describing character sheaves is equivalent to the later definition we use alternative characterizations of character sheaves for a connected complex reductive algebraic group g that are obtained in this thesis and also the one conjectured by g lusztig and g laumon that was obtained in 1988 by v ginsburg and independently by i mirkovic and k vilonen these characterizations involve microsupport and microlocalization of a sheaf a microsupport ss cal f of a complex cal f of sheaves on g is a closed conic subset of t sp g g times bf g it tells us about singularities of cal f a microlocalization mu sb m cal f of cal f along a submanifold m of g is a complex of sheaves on the conormal bundle t sbsp m g which allows us to analyze cal f on the infinitesimally small neighbourhood of m theorem let g be a complex connected reductive algebraic group cal f be a g equivariant for conjugation irreducible perverse sheaf on g let cal n be the nilpotent cone of the lie algebra g of g s may denote either g or the set of all semisimple elements of g the following are equivalent i cal f is a character sheaf ii ss cal f vert sb s subset s times cal n iii supp mu sb cal o cal f subset cal o times cal n where cal o subset s is a conjugacy class iv supp mu sb x cal f subset cal n x in s characterization ii when s g is the one given by ginsburg mirkovic and vilonen characterization iii with s being the set of semisimple elements was conjectured by j bernstein for arbitrary characteristic this kind of characterizations essentially simplifies the development and our understanding of the theory currently there is still no such description for positive characteristic an important intermediate result is the description of the microsupport of a doubc constructible sheaf cal f on a manifold x in terms of microlocalizations to the points of this manifold ss cal f overline bigcup limits sb x in x rm supp mu sb x cal f

uniform attractors for a non autonomous viscoelastic equation with a past history abstract this paper investigates the existence of uniform attractors for a non autonomous viscoelastic equation with a past history in a bounded domain r n n 1 by establishing the uniformly asymptotic compactness of the semi process generated by the global solutions our results improve those results by araujo et al 2013

uniform decay of solutions for a quasilinear system of viscoelastic equations abstract in this paper we consider a system of two coupled quasilinear viscoelastic equations in canonical form with dirichlet boundary condition we use the perturbed energy method to show that the dissipations given by the viscoelastic terms are strong enough to ensure uniform decay with exponential and polynomial rates of the solutions energy which extends some existing results for a single equation to the case of a coupled system

upper semicontinuity of pullback attractors for a non autonomous viscoelastic equation this paper is concerned with upper semicontinuity of pullback attractors for a non autonomous viscoelastic equation with a past history with a nonautonomous perturbation in particular under some proper assumptions we prove that the pullback attractor a e t t r of eq 1 1 with e 0 and the global attractor a of eq 1 1 with e 0 satisfy lim e 0 dist x a e t a 0 for any email protected r

dimensions of irreducible modules over w algebras and goldie ranks the main goal of this paper is to compute two related numerical invariants of a primitive ideal in the universal enveloping algebra of a semisimple lie algebra the first one very classical is the goldie rank of an ideal the second one is the dimension of an irreducible module corresponding to this ideal over an appropriate finite w algebra we concentrate on the integral central character case we prove modulo a conjecture that in this case the two are equal our conjecture asserts that there is a one dimensional module over the w algebra with certain additional properties the conjecture is proved for the classical types also modulo the same conjecture we compute certain scale factors introduced by joseph this allows to compute the goldie ranks of the algebras of locally finite endomorphisms of simples in the bgg category mathcal o this completes a program of computing goldie ranks proposed by joseph in the 80 s for integral central characters and modulo our conjecture we also provide an essentially kazhdan lusztig type formula for computing the characters of the irreducibles in the brundan goodwin kleshchev category mathcal o for a w algebra again under the assumption that the central character is integral in particular this allows to compute the dimensions of the finite dimensional irreducible modules the formula is based on a certain functor from an appropriate parabolic category mathcal o to the w algebra category mathcal o this functor can be regarded as a generalization of functors previously constructed by soergel and by brundan kleshchev we prove a number of properties of this functor including the quotient property and the double centralizer property we develop several side topics related to our generalized soergel functor for example we discuss its analog for the category of harish chandra modules we also discuss generalizations to the case of categories mathcal o over dixmier algebras the most interesting example of this situation comes from the theory of quantum groups we prove that an algebra that is a mild quotient of luszitg s form of a quantum group at a root of unity is a dixmier algebra for this we check that the quantum frobenius epimorphism splits

a well posedness result for nonlinear viscoelastic equations with memory abstract we establish an existence uniqueness and continuous dependence result for the weak solutions to the nonlinear viscoelastic equation with hereditary memory on a bounded three dimensional domain t u t t u t t u t u u 0 s u t s d s f u h with dirichlet boundary conditions in particular the parameter belongs to the interval 0 4 the value 4 being critical for the sobolev embeddings while f can reach the critical polynomial order 5

unipotent representations of complex semisimple groups algebraic group over r or c in this paper we restrict attention to c we generalize arthur s definition slightly or perhaps simply make it more precise all of the resulting representations except for a finite set are then unitarily induced from representations of the same kind on proper parabolic subgroups we call the finite set remaining special unipotent representations a precise definition will be given later definition 1 17 our main result theorem iii of this introduction is a character formula for any special unipotent representation of course such a formula can be deduced from the kazhdan lusztig conjecture cf v3 the advantages of theorem iii are that it is in closed form and that it lends itself to verification of some conjectures of arthur in a so let g be a connected complex semisimple lie group and q its lie algebra choose

global attractors for a semilinear hyperbolic equation in viscoelasticity a semilinear partial differential equation of hyperbolic type with a convolution term describing simple viscoelastic materials with fading memory is considered regarding the past history memory of the displacement as a new variable the equation is transformed into a dynamical system in a suitable hilbert space the dissipation is extremely weak and it is all contained in the memory term longtime behavior of solutions is analyzed in particular in the autonomous case the existence of a global attractor for solutions is achieved

exponential and polynomial decay for a quasilinear viscoelastic equation abstract we consider the following nonlinear viscoelastic equation u t u t t u u t t 0 t g t s u s d s 0 in a bounded domain for 0 we show that the dissipation induced by the integral term is strong enough to stabilize the solution this result improves an earlier one given by cavalcanti et al in m m cavalcanti v n d cavalcanti j ferreira existence and uniform decay for nonlinear viscoelastic equation with strong damping math methods appl sci 24 2001 1043 1053

lusztig s canonical quotient and generalized duality abstract we give a new characterization of lusztig s canonical quotient a finite group attached to each special nilpotent orbit of a complex semisimple lie algebra this group plays an important role in the classification of unipotent representations of finite groups of lie type we also define a duality map to each pair of a nilpotent orbit and a conjugacy class in its fundamental group the map assigns a nilpotent orbit in the langlands dual lie algebra this map is surjective and is related to a map introduced by lusztig and studied by spaltenstein when the conjugacy class is trivial our duality map is just the one studied by spaltenstein and by barbasch and vogan which has an image of the set of special nilpotent orbits

variational inequalities system of functional equations and incomplete information repeated games we consider a pair of functional equations obtained by mertens and zamir internat j game theory 1 1971 72 pp 39 64 j math anal appl 60 1977 pp 550 558 to characterize the asymptotic value of a two person zero sum repeated with lack of information on both sides aumann and maschler repeated games with incomplete information mit press cambridge ma 1995 we give a new proof for the convergence of the discounted values of the repeated game and a new characterization of the limit using variational inequalities the same idea allows us to prove existence and uniqueness of a lipschitz solution for the pair of functional equations in a general framework using an auxiliary game the splitting game introduced by sorin a first course on zero sum repeated games preprint

global existence and uniform decay for a nonlinear viscoelastic equation with damping abstract in this paper we investigate a nonlinear viscoelastic equation with linear damping global existence of weak solutions and the uniform decay estimates for the energy have been established

general decay and blow up of solution for a quasilinear viscoelastic problem with nonlinear source abstract in this paper we consider a quasilinear viscoelastic wave equation in canonical form with the homogeneous dirichlet boundary condition we prove that for certain class of relaxation functions and certain initial data in the stable set the decay rate of the solution energy is similar to that of the relaxation function this result improves earlier ones obtained by messaoudi and tatar s a messaoudi n e tatar global existence and uniform stability of solutions for a quasilinear viscoelastic problem math methods appl sci 30 2007 665 680 in which only the exponential and polynomial decay rates are considered conversely for certain initial data in the unstable set there are solutions that blow up in finite time the last result is new since it allows a larger class of initial energy which may take positive values