multiplicity one theorems the archimedean case let g be one of the classical lie groups gln 1 r gln 1 c u p q 1 o p q 1 on 1 c so p q 1 son 1 c and let g 0 be respectively the subgroup gln r gln c u p q o p q on c so p q son c embedded in g in the standard way we show that every irreducible casselman wallach representation of g 0 occurs with multiplicity at most one in every irreducible casselman wallach representation of g similar results are proved for the jacobi groups gln r nh2n 1 r gln c n h2n 1 c u p q nh2p 2q 1 r sp2n r nh2n 1 r sp2n c nh2n 1 c with their respective subgroups gln r gln c u p q sp2n r and sp2n c

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

orbits connecting critical points of differential equations depending on a parameter a sufficient and necessary condition for a planar critical point to be isolated as an invariant set is given using the concept of an isolated invariant set some existence criteria of orbits connecting two critical points bifurcating from a single critical point for planar differential equations depending on a parameter are given

the local langlands conjecture for gsp 4 we prove the local langlands conjecture for gsp4 f where f is a nonarchimedean local eld of characteristic zero

some local global non vanishing results of theta lifts for symplectic orthogonal dual pairs following the approach of b roberts we characterize the non vanishing of the global theta lift for symplectic orthogonal dual pairs in terms of its local counterpart in particular we replace the temperedness assumption present in robert s work by a certain weaker assumption and apply our results to small rank similitude groups among our applications is a certain instance of langlands functorial transfer of a non generic cuspidal automorphic representation of gsp 4 to gl 4

the local converse theorem for so 2n 1 and applications in this paper we characterize irreducible generic representations of so 2n 1 k where k is a p adic field by means of twisted local gamma factors the local converse theorem as applications we prove that two irreducible generic cuspidal automorphic representations of so 2n 1 a where a is the ring of adeles of a number field are equivalent if their local components are equivalent at almost all local places the rigidity theorem and prove the local langlands reciprocity conjecture for generic supercuspidal representations of so 2n 1 k

on vertices of simple modules for symmetric groups of small degrees abstract we present some algorithmic methods for the computation of vertices of indecomposable and simple modules over group algebras in prime characteristic furthermore we apply these to the simple modules of the symmetric groups and determine the vertices of all simple modules of the symmetric groups of degree at most 14 and 15 in characteristic 2 and 3 respectively with one exception

on the local converse theorem and the descent theorem in families in this paper we prove an analogue of jacquet s conjecture on the local converse theorem for ell adic families of co whittaker representations of mathrm gl n f where f is a finite extension of mathbb q p and ell ne p we also prove an analogue of jacquet s conjecture for a descent theorem which asks for the smallest collection of gamma factors determining the subring of definition of an ell adic family these two theorems are closely related to the local langlands correspondence in ell adic families

defect zero p blocks for finite simple groups we classify those finite simple groups whose brauer graph or decomposition matrix has a p block with defect 0 completing an investigation of many authors the only finite simple groups whose defect zero p blocks remained unclassified were the alternating groups an here we show that these all have a p block with defect 0 for every prime p 5 this follows from proving the same result for every symmetric group sn which in turn follows as a consequence of the t core partition conjecture that every non negative integer possesses at least one t core partition for any t 4 for t 17 we reduce this problem to lagrange s theorem that every non negative integer can be written as the sum of four squares the only case with t 17 that was not covered in previous work was the case t 13 this we prove with a very different argument by interpreting the generating function for t core partitions in terms of modular forms and then controlling the size of the coefficients using deligne s theorem nee the weil conjectures we also consider congruences for the number of p blocks of sn proving a conjecture of garvan that establishes certain multiplicative congruences when 5 p 23 by using a result of serre concerning the divisibility of coefficients of modular forms we show that for any given prime p and positive integer m the number of p blocks with defect 0 in sn is a multiple of m for almost all n we also establish that any given prime p divides the number of p modularly irreducible representations of sn for almost all n 1996 american mathematical society

generalized whittaker models and the bernstein center let f be a non archimedean local field and g the group of f points of a connected reductive algebraic group defined over f let b be a minimal f parabolic subgroup of g with levi component t and unipotent radical u we let be a smooth character of u nondegenerate in a certain sense we consider the theory of smooth whittaker functions on g via the structure of the representation c ind of g which is compactly induced by we prove a finiteness result for these generalized whittaker models corresponding to the uniqueness property for classical whittaker models on quasi split groups in many cases including the one where g is quasi split we describe the ring of g endomorphisms of c ind in terms of the bernstein center this has consequences for the structure of the category of smooth representations of g it gives an algebraic technique for finding improved and more general versions of results on classical whittaker functions originally obtained by more analytic methods

a theory of miyawaki liftings the hilbert siegel case the miyawaki liftings are defined by the pullbacks of ikeda liftings recently ikeda and yamana extended the theory of ikeda liftings to hilbert siegel modular forms in this paper using their results we establish a theory of miyawaki liftings both locally and globally in the local theory we describe the miyawaki liftings for almost tempered unitary representations explicitly in the global theory we discuss the non vanishing of the miyawaki liftings using seesaw identities and the global gan gross prasad conjecture as an application of local miyawaki liftings we prove a new case of the local gan gross prasad conjecture

une preuve simple des conjectures de langlands pour gl n sur un corps p adique let f be a finite extension of p for each integer n 1 we construct a bijection from the set f 0 n of isomorphism classes of irreducible degree n representations of the absolute weil group of f onto the set f 0 n of isomorphism classes of smooth irreducible supercuspidal representations of gl n f those bijections preserve epsilon factors for pairs and hence we obtain a proof of the langlands conjectures for gl n over f which is more direct than harris and taylor s our approach is global and analogous to the derivation of local class field theory from global class field theory we start with a result of kottwitz and clozel on the good reduction of some shimura varieties and we use a trick of harris who constructs non galois automorphic induction in certain cases

the heights of irreducible brauer characters in 2 blocks of the symmetric groups abstract we prove that there exists a unique irreducible brauer character of height zero in any 2 blocks of the symmetric group this generalizes the theorem of p fong and g d james that the dimension of every non trivial 2 modular simple module of the symmetric group is even

equivalence classes of second order ordinary differential equations with only a three dimensional lie algebra of point symmetries and linearisation there are seven equivalence classes of second order ordinary differential equations possessing only three lie point symmetries and hence not linearisible by means of a point transformation we examine the representatives of these classes for linearisibility by means of other types of transformation in particular we compare the potential for linearisibility and the possession of the painleve property the complete symmetry group is realised in the standard algebra for each of the equivalence classes

multiplicity one theorems for fourier jacobi models for every genuine irreducible admissible smooth representation pi of the metaplectic group widetilde rm sp 2n over a p adic field and every smooth oscillator representation omega psi of widetilde rm sp 2n we prove that the tensor product pi otimes omega psi is multiplicity free as a smooth representation of the symplectic group rm sp 2n similar results are proved for general linear groups and unitary groups

caracterisation de la correspondance de langlands locale par les facteurs e de paires soitf un corps local non archimedien a corps residuel fini la conjecture de langlands locale prouvee par laumon rapoport et stuhler quandf est de caracteristique non nulle associe a une classe d isomorphisme de representations irreductibles du groupe de weil def une classe d isomorphisme de representations admissibles irreductibles supercuspidales d un groupe lineaire surf le groupe gl n f si est de dimensionn nous prouvons dans cet appendice qu il existe au plus une telle correspondance qui preserve les facteurs e de paires et coincide en dimension 1 avec l application donnee par la theorie du corps de classes

the gan gross prasad conjecture for u n u n abstract we prove the gan gross prasad conjecture for u n u n under some local conditions using a relative trace formula we deduce some new cases of the gan gross prasad conjecture for u n 1 u n from the case of u n u n

a proof of langland s conjecture on plancherel measures complementary series of p adic groups analysis of p adic reductive groups our first result theorem 7 9 proves a conjecture of langlands on normalization of intertwining operators by means of local langlands root numbers and l functions at least when the group is quasi split and the inducing representation is generic assuming two natural conjectures in harmonic analysis of p adic groups we also prove the validity of the conjecture in general theorem 9 5 as our second result we obtain all the complementary series and special representations of quasi split p adic groups coming from rank one parabolic subgroups and generic supercuspidal represen

trajectories joining critical points of nonlinear parabolic and hyperbolic partial differential equations abstract consider the following nonlinear dirichlet boundary value problems d t u t x lu t x f u t x t 0 x u t x 0 t 0 x 1 1 d t d t u t x lu t x l d t u t x f u t x 0 t 0 x u t x 0 t 0 x 1 2 lu x f u x x u x 0 x 1 3 in all of these equations f r r is a locally lipschitzian asymptotically linear function with positive asymptotic slope f 0 0 and l is a self adjoint negativedefinite and strongly elliptic second order differential operator on a smooth domain in rn the solutions of 1 1 and 1 2 generate semiflows which are not pointdissipative and whose equilibria are determined by solutions of 1 3 in this paper using an extension due to the present author of conley s morse index theory to noncompact spaces we prove not only the existence of positive solutions of 1 3 a result shown earlier by peitgen and schmitt using different methods but also show the existence of nonconstant heteroclinic orbits of 1 1 and 1 2 joining two sets of equilibria

stability of the local gamma factor arising from the doubling method let g be either a symplectic group or an orthogonal group over a local non archimedean field f of characteristic zero we consider the local gamma factor associated to an irreducible admissible representation of g by the doubling method of piatetski shapiro and rallis ps r g ps r l r denote this gamma factor by s where is a character of f and is a fixed non trivial character of f in this paper we prove

on icosahedral artin representations ii we prove that some new infinite families of odd two dimensional icosahedral representations of the absolute galois group of inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i are modular and hence satsify the artin conjecture we also give an account of work of ramakrishna on lifting mod l galois representations to characteristic zero

trajectories joining critical points in this paper we shall be concerned with the existence of trajectories joining a pair of critical points or more generally a pair of compact invariant sets our framework will be abstract dynamical systems we shall present applications and examples among both finite and infinite dimensional differential equations the existence of connecting orbits has attracted the attention of many researchers in gelfand 19631 a footnote on page 299 explicitly indicates that this is an interesting problem a variety of techniques have been employed to cope with the problem the tools which are employed include consideration of degree and index theories isolating blocks dimensionality of stable and unstable manifolds bifurcation type arguments fixed point theory etc consult conley 1976 conley and smoller 1975 19781 conlon 1980 foy 1964 gordon 19741 and howard and koppel 19751 some sort of differentiability and smoothness is required in any of the cited works and therefore the applicability to infinite dimensional problems is limited in some of these works the global connecting orbit was found only in a local neighborhood of some critical point in this paper we pursue a rather naive approach it would not be applicable unless some a priori demands on the global behavior of solutions are met however when the method works it is capable of handling infinite

the gross prasad conjecture and local theta correspondence we establish the fourier jacobi case of the local gross prasad conjecture for unitary groups by using local theta correspondence to relate the fourier jacobi case with the bessel case established by beuzart plessis to achieve this we prove two conjectures of prasad on the precise description of the local theta correspondence for almost equal rank unitary dual pairs in terms of the local langlands correspondence the proof uses arthur s multiplicity formula and thus is one of the first examples of a concrete application of this global reciprocity law

uniformly asymptotically zhukovskij stable orbits this article deals with zhukovskij stability of planar systems we prove that if the omega limit set of a nonclosed orbit is a stable limit cycle then the orbit is uniformly asymptotically zhukovskij stable

the existence of trajectories joining critical points abstract in this paper using the notion of an isolating block and the concept of canonical regions three existence criteria of trajectories connecting a pair of critical points of planar differential equations are given

multiplicity one theorems in the local characteristic 0 non archimedean case we consider distributions on gl n 1 which are invariant under the adjoint action of gl n we prove that such distributions are invariant by transposition this implies multiplicity at most one for restrictions from gl n 1 to gl n similar theorems are obtained for orthogonal or unitary groups

on a rankin selberg integral of the l function for widetilde mathrm sl _2 times mathrm gl _2 sl 2 gl2 we present a rankin selberg integral on the exceptional group g 2 which represents the l function for generic cuspidal representations of widetilde mathrm sl 2 times mathrm gl 2 as an application we show that certain fourier jacobi type periods on g 2 are non vanishing

local coefficients as artin factors for real groups introduction the purpose of this paper is to prove the equality of certain local coefficients of arithmetic significance which were attached to representations of quasi split real reductive algebraic groups in 27 with their corresponding artin factors attached by local class field theory 21 as a consequence we establish an identity satisfied by certain normalized intertwining operators it seems to be useful in applications of the trace formula 1 29 more precisely let g be the group of real points of a quasi split reductive algebraic group over r let a be the set of simple roots defined by a fixed minimal parabolic subgroup po m0a0u of g fix 6 c a and let p p9 be the corresponding standard parabolic subgroup of g and write p man for its langlands decomposition fix a nondegenerate character x f u i a h o be an irreducible admissible x gic banach in particular x generic unitary representation of m cf section 1 given v g a the complex dual of the lie algebra of a let i v o 0 be the continuously quasi unitarily if o is unitary induced representation indp ga e v 9 and let v v o 0 be its space section 0 then f a vfoo o now let w be the weyl group of ao in g choose w g w such that w 0 c a let n be the unipotent group opposite to n define n u c wn w where w is a representative of w in g for g f a 00 define

modelling fractal patterns via the qualitative equivalence of a nonlinear ode abstract a model of fractal patterns using the qualitative equivalence behavior of the nonlinear equation x ax 4 bx 3 cx 2 dx e where a b c d e r set of real numbers a 0 is presented from the point of view of the critical points of the equation each distinct phase portrait of the equation including the directional arrow corresponds to a fractal pattern the phase portraits when the critical points are complex valued present the more interesting cases in all a total of twenty eight 28 distinct fractal patterns can be generated via the equation up to isomorphism eighteen 18 of these arise when all the critical points are real the relevant qualitative properties are existence and uniqueness of solutions

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

local langlands correspondence for mathrm gl _ n and the exterior and symmetric square varepsilon factors let f be a p adic field that is a finite extension of qp for some prime p the local langlands correspondence llc attaches to each continuous n dimensional semisimple representation of w f the weil deligne group for f f an irreducible admissible representation of gln f such that among other things the local l and e factors of pairs are preserved this correspondence should be robust and preserve various parallel operations on the arithmetic and analytic sides such as taking the exterior square or symmetric square in this article we show that this is the case for the local arithmetic and analytic symmetric square and exterior square e factors that is that e s 2 e s 2 and e s sym2 e s sym2 the agreement of the l functions also follows by our methods but this was already known by henniart the proof is a robust deformation argument combined with local global techniques which reduces the problem to the stability of the analytic factor s 2 under highly ramified twists when is supercuspidal this last step is achieved by relating the factor to a mellin transform of a partial bessel function attached to the representation and then analyzing the asymptotics of the partial bessel function inspired in part by the theory of shalika germs for bessel integrals the stability for every irreducible admissible representation then follows from those of the corresponding arithmetic factors as a corollary

cartan matrices and morita equivalence for blocks of the symmetric groups theorem 1 the blocks of the symmetric groups with a given weight w 0 over a field of characteristic p can be collected into families according to the shapes of their cores each family consists of morita equivalent blocks the number of such families is at most flf 1 i l w 1 11 and each family contains a block of some g where n is less than or equal to p2 p 1 2 w 1 2 4 wp

from laplace to langlands via representations of orthogonal groups in the late 1960s robert langlands proposed a new and far reaching connection between the representation theory of lie groups over real and p adic fields and the structure of the galois groups of these fields 24 even though this local langlands correspondence remains largely conjectural the relation that it predicts between representation theory and number theory has profoundly changed our views of both fields moreover we now know enough about the correspondence to address and sometimes solve traditional problems in representation theory that were previously inaccessible roughly speaking the local langlands correspondence predicts that complex irreducible representations of a reductive group g over a local field k should be

on shalika periods and a theorem of jacquet martin let pi be a cuspidal automorphic representation of gl 4 bbb a with central character mu 2 it is known that pi has shalika period with respect to mu if and only if the l function l s s pi bigwedge 2 otimes mu 1 has a pole at s 1 jacquet and martin considered the analogous question for cuspidal representations pi d of the inner form gl 2 d bbb a and obtained a partial result via the relative trace formula in this paper we provide a complete solution to this problem via the method of theta correspondence and give necessary and sufficient conditions for the existence of shalika period for pi d we also resolve the analogous question in the local setting

generic transfer for general spin groups we prove langlands functoriality for the generic spectrum of general spin groups both odd and even contrary to other recent instances of functoriality our resulting automorphic representations on the general linear group are not self dual together with cases of classical groups this completes the list of cases of split reductive groups whose l groups have classical derived groups the important transfer from gsp4 to gl4 follows from our result as a special case

fractal dimension and logarithmic loss unpredictability we show that the hausdorff dimension equals the logarithmic loss unpredictability for any set of infinite sequences over a finite alphabet using computable feasible and finite state predictors this equivalence also holds for the computable feasible and finite state dimensions combining this with recent results of fortnow and lutz proc 15th ann conf on comput learning theory 2002 380 we have a tight relationship between prediction with respect to logarithmic loss and prediction with respect to absolute loss

the local theta correspondence and the local gan gross prasad conjecture for the symplectic metaplectic case we prove the local gan gross prasad conjecture for the symplectic metaplectic case under some assumptions this is the last case of the local gan gross prasad conjectures which has not been established we also prove two of prasad s conjectures on the local theta correspondence in the almost equal rank case

a new method to find homoclinic and heteroclinic orbits a new series method is provided for continuous time autonomous dynamical systems which can find exact orbits as opposed to approximate ones the method can reduce the connecting orbit problem as a boundary value problem in an infinite time domain to the initial value problem it consists of transforming time to the logarithmic scale substituting a power series around each fixed point of interest for each of the unknown functions into the system and equating the corresponding coefficients when solving for the power series coefficients additional parameters are used in order to find the intersections of the unstable manifold and the stable manifold of the equilibria this paper demonstrates how the new method allows to obtain heteroclinic and homoclinic orbits in some well known cases such as nagumo system stretch twist fold flow or mathematical pendulum

local to global extensions for gln in non zero characteristic a characterization of yf s p sym2 v and yf s p a2 v let f be a locally compact field of positive characteristic in his thesis the second author used the langlands shahidi method to define factors for the symmetric square and exterior square representations of the dual group of gl n f we prove here that such factors are characterized by local properties including a multiplicativity property with respect to parabolic induction and their role in global functional equations for l functions for this we prove that a given cuspidal representation of gl n f is the component at some place of a global automorphic representation the ramification of which is controlled at other places in fact we use an equivalent result for l adic galois representations due to o gabber and n katz and transport it via the global langlands correspondence proved by l lafforgue

uniqueness of fourier jacobi models the archimedean case we prove uniqueness of fourier jacobi models for general linear groups unitary groups symplectic groups and metaplectic groups over an archimedean local field

on the global attraction for the generalized lienard equation this paper deals with the global attraction and global weak attraction of the origin for the generalized lienard equation the problem proposed by jiang is solved by a negative answer and some new criteria for the globally asymptotically stability are obtained by the ways different from the usual filippov s conditions

a local converse theorem for mathrm sp _ 2r in this paper we prove the local converse theorem for mathrm sp 2r f over a p adic field f more precisely given two irreducible supercuspidal representations of mathrm sp 2r f with the same central character such that they are generic with the same additive character and they have the same gamma factors when twisted with generic irreducible representations of mathrm gl n f for all 1 le n le r then these two representations must be isomorphic our proof is based on the local analysis of the local integrals which define local gamma factors a key ingredient of the proof is certain partial bessel function property developed by cogdell shahidi tsai recently the same method can give the local converse theorem for mathrm u r r