on the existence of periodic solutions to p laplacian rayleigh differential equation with a delay abstract in this paper the authors study the existence of periodic solutions to a p laplacian rayleigh differential equation with a delay as follows p y t f y t g y t t e t where p 1 is a constant p r r p u u p 2 u f g e c r r t t t with t 0 t 0 t and e t t e t t 0 is a constant by using mawhin s continuation theorem some new results are obtained

mirror symmetry for the landau ginzburg a model m mathbb c n w z_ 1 cdots z_ n we calculate the category of branes in the landau ginzburg a model with background m cn and superpotential w z1 zn in the form of microlocal sheaves along a natural lagrangian skeleton our arguments employ the framework of perverse schobers and our results confirm expectations from mirror symmetry

periodic solutions for p laplacian lienard equation with a deviating argument abstract by employing mawhin s continuation theorem the existence of periodic solutions of the p laplacian lienard equation with a deviating argument p x t f x t x t g x t t e t under various assumptions are obtained

existence and uniqueness of periodic solutions for a p laplacian duffing equation with a deviating argument abstract by means of mawhin s continuation theorem we study a class of p laplacian duffing type differential equations of the form p x t c x t g t x t x t t e t some new results on the existence and uniqueness of periodic solutions for the above equation are obtained it is significant that the growth degree with respect to the variables u v imposed on g t u v is allowed to be greater than p 1 so our results generalize and improve on the corresponding results in related papers

representations of semisimple lie algebras in prime characteristic and the noncommutative springer resolution we prove most of lusztig s conjectures on the canonical basis in homology of a springer ber the conjectures predict that this basis controls numerics of representations of the lie algebra of a semisimple algebraic group over an algebraically closed eld of positive characteristic we check this for almost all characteristics to this end we construct a noncommutative resolution of the nilpotent cone which is derived equivalent to the springer resolution on the one hand this noncommutative resolution is closely related to the positive characteristic derived localization equivalences obtained earlier by the present authors and rumynin on the other hand it is compatible with the t structure arising from an equivalence with the derived category of perverse sheaves on the ane ag variety of the langlands dual group this equivalence established by arkhipov and the rst author ts the framework of local geometric langlands duality the latter compatibility allows one to apply frobenius purity theorem to deduce the desired properties of the basis we expect the noncommutative counterpart of the springer resolution to be of independent interest from the perspectives of algebraic geometry and geometric langlands duality

fourier transforms and frobenius eigenvalues for finite coxeter groups abstract lusztig s classification of the unipotent characters of a finite chevalley or steinberg group involves a certain non abelian fourier transformation we construct analogous transformations for the suzuki and ree groups based on a set of axioms derived from lusztig s theory of character sheaves we also determine fourier matrices for the spetses in the sense of broue michel and the second author associated with twisted dihedral groups this completes the determination of fourier matrices for all spetses associated with finite coxeter groups we end by collecting common properties of these fourier matrices and the eigenvalues of frobenius of character sheaves and unipotent characters

character sheaves v this paper is part of a series s 131 devoted to the study of a class g of irreducible perverse sheaves called character sheaves on a connected reductive algebraic group g the numbering of chapters sections and references will continue that of s 131 this paper is a step towards the classification of character sheaves on g one of the main results is the following one under certain assumptions there is a natural surjective map with finite fibers from g to the set of all pairs 9 c up to conjugacy by the weyl group where 9 is a tame local system on the maximal torus and c is a two sided cell in the stabilizer w of 9 in the weyl group the assumptions made on g are

expressiveness of structured document query languages based on attribute grammars structured document databases can be naturally viewed as derivation trees of a context free grammar under this view the classical formalism of attribute grammars becomes a formalism for structured document query languages from this perspective we study the expressive power of bags boolean valued attribute grammars with propositional logic formulas as semantic rules and rags relation valued attribute grammars with first order logic formulas as semantic rules bags can express only unary queries rags can express queries of any arity we first show that the unary queries expressible by bags are precisely those definable in monadic second order logic we then show that the queries expressible by rags are precisely those definable by first order inductions of linear depth or equivalently those computable in linear time on a parallel machine with polynomially many processors further we show that rags that only use synthesized attributes are strictly weaker than rags that use both synthesized and inherited attributes we show that rags are more expressive than monadic second order logic for queries of any arity finally we discuss relational attribute grammars in the context of bags and rags we show that in the case of bags this does not increase the expressive power while different semantics for relational rags capture the complexity classes np conp and up coup

categorification of induced cell modules and the rough structure of generalised verma modules this paper presents categorifications of right cell modules and induced cell modules for hecke algebras of finite weyl groups fit type a we show that these categorifications depend only on the isomorphism class of the cell module not on the cell itself our main application is multiplicity formulas for parabolically induced modules over a reductive lie algebra of type a which finally determines the so called rough structure of generalised verma modules on the way we present several categorification results and give a positive answer to kostant s problem from a joseph kostant s problem goldie rank and the gelfand kirillov conjecture invent math 56 3 1980 191 213 in many cases we also present a general setup of decategorification precategorification and categorification

periodic solutions for a kind of duffing type p laplacian equation abstract based on manasevich mawhin continuation theorem and some analysis skill some new sufficient conditions for the existence of periodic solutions for duffing type p laplacian differential equation are obtained an example is provided to demonstrate effectiveness of the proposed results

the exotic robinson schensted correspondence abstract we study the action of the symplectic group on pairs of a vector and a flag considering the irreducible components of the conormal variety we obtain an exotic analogue of the robinson schensted correspondence conjecturally the resulting cells are related to exotic character sheaves

existence of periodic solutions for a lienard type p laplacian differential equation with a deviating argument by means of mawhin s continuation theorem a class of p laplacian type differential equation with a deviating argument of the form p x t f x t x t t g t x t t x e t is studied a new result related to t and the deviating argument t x is obtained it is significant that the growth degree with respect to the variable x in g t x is allowed to be greater than p 1 which could be achieved infrequently in previous papers

a categorification of morelli s theorem we prove a theorem relating torus equivariant coherent sheaves on toric varieties to polyhedrally constructible sheaves on a vector space at the level of k theory the theorem recovers morelli s description of the k theory of a smooth projective toric variety morelli in adv math 100 2 154 182 1993 specifically let x be a proper toric variety of dimension n and let m mathbb r mathrm lie t mathbb r vee cong mathbb r n be the lie algebra of the compact dual real torus t mathbb r vee cong u 1 n then there is a corresponding conical lagrangian t m and an equivalence of triangulated dg categories mathcal p mathrm erf t x cong mathit sh cc m mathbb r lambda where mathcal p mathrm erf t x is the triangulated dg category of perfect complexes of torus equivariant coherent sheaves on x and shcc m is the triangulated dg category of complex of sheaves on m with compactly supported constructible cohomology whose singular support lies in this equivalence is monoidal it intertwines the tensor product of coherent sheaves on x with the convolution product of constructible sheaves on m

periodic solutions for a higher order p laplacian neutral functional differential equation with a deviating argument abstract in this paper a higher order p laplacian neutral functional differential equation with a deviating argument p x t c t x t n m f x t x t g t x t t e t has been studied by applying mawhin s continuation degree theorem some new criteria to guarantee the existence of periodic solutions are obtained it is interesting that the power of the variable x in function g is allowed to be greater than p 1

cells in affine weyl groups and tensor categories abstract in this paper we construct a tensor or monoidal category for any two sided cell in a finite or affine weyl group the isomorphism classes of simple objects of this category correspond to the elements of the two sided cell the tensor product arises by a certain truncation procedure from the standard convolution of perverse sheaves on a flag manifold which imitates geometrically the definition of the ring j given in g lusztig j algebra 109 1987 536 548 similarly we construct for any left cell a tensor category in which the isomorphism classes of simple objects correspond to the elements in the left cell intersected with its inverse a right cell this contains as a special case the category of equivariant perverse sheaves on an affine grassmannian which as a consequence of g lusztig asterisque 101 102 1983 208 209 is a tensor category for convolution without truncation closely related to the tensor category of representations of the group dual in the sense of langlands to the group which gives rise to the affine grassmannian but our construction can be specialized in other ways and it gives a conjectural realization of the tensor category of representations of the reductive quotient of the centralizer of any unipotent element of the langlands dual group

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

the character sheaves on the group compactification abstract we give a definition of character sheaves on the group compactification which is equivalent to lusztig s definition in g lusztig parabolic character sheaves ii mosc math j 4 4 2004 869 896 we also prove some properties of the character sheaves on the group compactification

remarks on computing irreducible characters 0 1 let g be a connected reductive algebraic group defined over a finite field fq and let g fq be the finite group of all fq rational points of g we would like to present here a strategy for computing the character table of g fq under the assumption that p the characteristic of fq is sufficiently large we can assume that g has a simply connected derived group indeed in the general case we can find a connected reductive group g with simply connected derived group and a surjective homomorphism f g g whose kernel is a central torus such that g and f are defined over fq then the character table

sufficient conditions for the existence of periodic solutions to some second order differential equations with a deviating argument abstract by means of mawhin s continuation theorem we study some second order differential equations with a deviating argument x t f t x t x t t x t e t some new results on the existence of periodic solutions are obtained the interest is that we allow the degree with respect to the variables x 0 x 1 x 2 of f t x 0 x 1 x 2 to be greater than 1 and also the result theorem 3 2 is related to the deviating argument t meanwhile we give an example to demonstrate our result

homological mirror symmetry for punctured spheres we prove that the wrapped fukaya category of a punctured sphere s with an arbitrary number of points removed is equivalent to the triangulated category of singularities of a mirror landau ginzburg model proving one side of the homological mirror symmetry conjecture in this case by investigating fractional gradings on these categories we conclude that cyclic covers on the symplectic side are mirror to orbifold quotients of the landau ginzburg model

local cohomology of modules of covariants abstract let g be a connected reductive algebraic group over an algebraically closed field of characteristic zero and let w u be two finite dimensional representations of g in this paper we compute the local cohomology of u sw g provided a certain relatively weak technical condition is true

topological generation of exceptional algebraic groups abstract let g be a simple algebraic group over an algebraically closed field k and let c 1 c t be non central conjugacy classes in g in this paper we consider the problem of determining whether there exist g i c i such that g 1 g t is zariski dense in g first we establish a general result which shows that if is an irreducible subvariety of g t then the set of tuples in generating a dense subgroup of g is either empty or dense in in the special case c 1 c t by considering the dimensions of fixed point spaces we prove that this set is dense when g is an exceptional algebraic group and t 5 assuming k is not algebraic over a finite field in fact for g g 2 we only need t 4 and both of these bounds are best possible as an application we show that many faithful representations of exceptional algebraic groups are generically free we also establish new results on the topological generation of exceptional groups in the special case t 2 which have applications to random generation of finite exceptional groups of lie type in particular we prove a conjecture of liebeck and shalev on the random r s generation of exceptional groups

geometric langlands duality and representations of algebraic groups over commutative rings as such it can be viewed as a first step in the geometric langlands program the connected complex reductive groups have a combinatorial classification by their root data in the root datum the roots and the co roots appear in a symmetric manner and so the connected reductive algebraic groups come in pairs if g is a reductive group we write g for its companion and call it the dual group g the notion of the dual group itself does not appear in satake s paper but was introduced by langlands together with its various elaborations in li l2 and is a cornerstone of the langlands program it also appeared later in physics mo gno in this paper we discuss the basic relationship between g and g we begin with a reductive g and consider the affine grassmannian qx the grassmannian for the loop group of g for technical reasons we work with formal algebraic loops the affine grassmannian is an infinite dimen sional complex space we consider a certain category of sheaves the spherical perverse sheaves on r these sheaves can be multiplied using a convolution product and this leads to a rather explicit construction of a hopf algebra by what has come to be known as tannakian formalism the resulting hopf algebra turns out to be the ring of functions on g in this interpretation the spherical perverse sheaves on the affine grassman nian correspond to finite dimensional complex representations of g thus instead of defining g in terms of the classification of reductive groups we pro vide a canonical construction of g starting from g we can carry out our construction over the integers the spherical perverse sheaves are then those with integral coefficients but the grassmannian remains a complex algebraic object

characteristic cycles for the loop grassmannian and nilpotent orbits x which is a linear combination of closures of conormal bundles to submanifolds of x intuitively the microlocal multiplicities c me asurethesingularity of at in settings related to representation theory a group g acts on x is g equivariant and the microlocal multiplicities play a significant but only partially understood role in representation theory see ro sv abv and kasa e g in this paper we compute microlocal multiplicities for certain cases of interest in representation theory let g be a connected reductive group with loop group lg and let p bethesubgroup of lg consisting of loops with positive fourier coefficients then p orbits on theloop grassmannian lg p correspond to the irreducible representations l of thedual re ductivegroup g for dominant weights and of a torus of

character d modules via drinfeld center of harish chandra bimodules the category of character d modules is realized as drinfeld center of the abelian monoidal category of harish chandra bimodules tensor product of harish chandra bimodules is related to convolution of d modules via the long intertwining functor radon transform by a result of beilinson and ginzburg represent theory 3 1 31 1999 exactness property of the long intertwining functor on a cell subquotient of the harish chandra bimodules category shows that the truncated convolution category of lusztig adv math 129 85 98 1997 can be realized as a subquotient of the category of harish chandra bimodules together with the description of the truncated convolution category bezrukavnikov et al in isr j math 170 207 234 2009 this allows us to derive under a mild technical assumption a classification of irreducible character sheaves over obtained by lusztig by a different method

cyclic symmetries of an quiver representations abstract this note contains a combinatorial construction of symmetries arising in symplectic geometry partially wrapped or infinitesimal fukaya categories algebraic geometry derived categories of singularities and k theory waldhausen s s construction our specific motivation in the spirit of expectations of kontsevich is a combinatorial construction of categorical quantizations of lagrangian skeleta of symplectic manifolds the main result of this paper gives an immediate solution in the one dimensional case of ribbon graphs

modular generalized springer correspondence iii exceptional groups we complete the construction of the modular generalized springer correspondence for an arbitrary connected reductive group with a uniform proof of the disjointness of induction series that avoids the case by case arguments for classical groups used in previous papers in the series we show that the induction series containing the trivial local system on the regular nilpotent orbit is determined by the sylow subgroups of the weyl group under some assumptions we give an algorithm for determining the induction series associated to the minimal cuspidal datum with a given central character we also provide tables and other information on the modular generalized springer correspondence for quasi simple groups of exceptional type including a complete classification of cuspidal pairs in the case of good characteristic and a full determination of the correspondence in type g 2

on the existence of periodic solutions for p laplacian generalized lienard equation abstract by employing mawhin s continuation theorem the existence of periodic solutions of the p laplacian generalized lienard equation with deviating argument p x t f t x t x t t g x t t e t under various assumptions are obtained

periodic solutions for a kind of high order p laplacian differential equation with sign changing coefficient ahead of the non linear term abstract in this paper by using the theory of fourier series bernoulli number theory and the continuation theorem of coincidence degree theory we study a kind of high order p laplacian differential equation as follows p y m t m f y t y t h y t t g y t t e t some new results on the existence of periodic solutions are obtained the interesting thing is that the coefficient t is allowed to change sign but the methods used to estimate a priori bounds of periodic solutions are different from the corresponding ones used in the past

compact representation for answer sets of n ary regular queries an n ary query over trees takes an input tree t and returns a set of n tuples of the nodes of t in this paper a compact data structure is introduced for representing the answer sets of n ary queries defined by tree automata despite that the number of the elements of the answer set can be as large as t n our representation allows storing the set using only o t space several basic operations on the sets are shown to be efficiently executable on the representation

new results on the existence of periodic solutions to a p laplacian differential equation with a deviating argument by means of mawhin s continuation theorem a kind of p laplacian differential equation with a deviating argument as follows p x t f t x t x t t x t e t is studied some new results on the existence of periodic solutions are obtained the main results theorems 3 2 and 3 3 are all related to the deviating argument t meanwhile the degrees with respect to the variables x0 x1 of f t x0 x1 x2 are allowed to be grater than p 1 which is different from the corresponding conditions of known literature

query evaluation via tree decompositions a number of efficient methods for evaluating first order and monadic second order queries on finite relational structures are based on tree decompositions of structures or queries we systematically study these methods in the first part of the article we consider arbitrary formulas on tree like structures we generalize a theorem of courcelle 1990 by showing that on structures of bounded tree width a monadic second order formula with free first and second order variables can be evaluated in time linear in the structure size plus the size of the output in the second part we study tree like formulas on arbitrary structures we generalize the notions of acyclicity and bounded tree width from conjunctive queries to arbitrary first order formulas in a straightforward way and analyze the complexity of evaluating formulas of these fragments moreover we show that the acyclic and bounded tree width fragments have the same expressive power as the well known guarded fragment and the finite variable fragments of first order logic respectively

periodic solutions for a fourth order rayleigh type p laplacian delay equation abstract by applying mawhin s continuation theory and some new inequalities the existence of periodic solutions for a fourth order rayleigh type p laplacian delay equation is obtained moreover an example is given to illustrate the results

legendrian knots and constructible sheaves we study the unwrapped fukaya category of lagrangian branes ending on a legendrian knot our knots live at contact infinity in the cotangent bundle of a surface the fukaya category of which is equivalent to the category of constructible sheaves on the surface itself consequently our category can be described as constructible sheaves with singular support controlled by the front projection of the knot we use a theorem of guillermou kashiwara schapira to show that the resulting category is invariant under legendrian isotopies a subsequent article establishes its equivalence to a category of representations of the chekanov eliashberg differential graded algebra we also find two connections to topological knot theory first drawing a positive braid closure on the annulus the moduli space of rank n objects maps to the space of local systems on a circle the second page of the spectral sequence associated to the weight filtration on the pushforward of the constant sheaf is the colored by n triply graded khovanov rozansky homology second drawing a positive braid closure in the plane the number of points of our moduli spaces over a finite field with q elements recovers the lowest coefficient in a of the homfly polynomial of the braid closure

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