new global exponential stability criteria for nonlinear delay differential systems with applications to bam neural networks we consider a nonlinear non autonomous system with time varying delaysx i t a i t x i h i t j 1mf i jt x j g i j t i 1 mwhich has a large number of applications in the theory of artificial neural networks via the m matrix method easily verifiable sufficient stability conditions for the nonlinear system and its linear version are obtained application of the main theorem requires just to check whether a matrix which is explicitly constructed using the system s parameters is an m matrix comparison with the tests obtained by gopalsamy 2007 and liu 2013 for bam neural networks illustrates novelty of the stability theorems some open problems conclude the paper

monstrous moonshine from orbifolds we consider the monster module of frenkel lepowsky and meurman as az2 orbifold of a bosonic string compactified by the leech lattice we show that the main conway and norton monstrous moonshine properties stating that the thompson series for each monster group conjugacy class has a modular invariance group of genus zero follow from an orbifold construction based on an orbifold group composed of monster group elements it is shown that a conjectured vacuum structure for the orbifold twisted sectors is sufficient to specify the modular group and the genus zero property for each thompson series it is also shown that the power map formula of conway and norton follows from the same vacuum structure finally we demonstrate the validity of the vacuum conjectures for sectors twisted by leech lattice automorphisms in many cases

cohomologie des modules de dimension finie sur la super algebre de lie osp 3 2 abstract we compute the cohomology of finite dimensional osp 3 2 modules and we give an explicit description of projective elements of the category of finite dimensional modules

homological invariants in category for the general linear superalgebra we study three related homological properties of modules in the bgg category mathcal o for basic classical lie superalgebras with specific focus on the general linear superalgebra these are the projective dimension associated variety and complexity we demonstrate connections between projective dimension and singularity of modules and blocks similarly we investigate the connection between complexity and atypicality this creates concrete tools to describe singularity and atypicality as homological and hence categorical properties of a block however we also demonstrate how two integral blocks in category mathcal o with identical global categorical characteristics of singularity and atypicality will generally still be inequivalent this principle implies that category mathcal o for mathfrak gl m vert n can contain infinitely many non equivalent blocks which we work out explicitly for mathfrak gl 3 vert 1 all of this is in sharp contrast with category mathcal o for lie algebras but also with the category of finite dimensional modules for superalgebras furthermore we characterise modules with finite projective dimension to be those with trivial associated variety we also study the associated variety of verma modules to do this we also classify the orbits in the cone of self commuting odd elements under the action of an even borel subgroup

branes and supergroups extending previous work that involved d3 branes ending on a fivebrane with theta mathrm ym neq 0 we consider a similar two sided problem this construction in case the fivebrane is of ns type is associated to the three dimensional chern simons theory of a supergroup rm u m n or rm os rm p m 2n rather than an ordinary lie group as in the one sided case by s duality we deduce a dual magnetic description of the supergroup chern simons theory a slightly different duality in the orthosymplectic case leads to a strong weak coupling duality between certain supergroup chern simons theories on mathbb r 3 and a further t duality leads to a version of khovanov homology for supergroups some cases of these statements are known in the literature we analyze how these dualities act on line and surface operators

finite dimensional representations of periplectic lie superalgebras abstract we study categories of finite dimensional modules over the periplectic lie superalgebras and obtain a bgg type reciprocity in particular we prove that these categories have only finitely many blocks we also compute the characters for irreducible modules over periplectic lie superalgebras of ranks 2 and 3 and obtain explicit description of the blocks for ranks 2 3 and 4

frame shapes and rational characters of finite groups abstract among the many moonshine properties of the monster simple group considered recently by conway and norton the leech lattice and the representation of the conway group o which it affords seem to play a particularly central role in this note we show how some of their ideas those involving so called frame shapes can be put in a somewhat more general context

supergroup osp 2 2n and super jacobi polynomials abstract the coefficients of the super jacobi polynomials of b c 1 n type are rational functions in three parameters k p q at the point 1 0 0 these coefficients may have poles let us set q 0 and consider a pair k p as the point of a 2 if we apply blow up procedure at the point 1 0 then we get a new family of polynomials this family depends on one parameter t p if t then we get euler supercharacters for lie supergroup o s p 2 2 n the supercharacters of finite dimensional irreducible modules can be also obtained by a specialization of the parameter t but in such a case the specialization depends on the highest weight of the corresponding irreducible module

beauty and the beast superconformal symmetry in a monster module frenkel lepowsky and meurman have constructed a representation of the largest sporadic simple finite group the fischer griess monster as the automorphism group of the operator product algebra of a conformal field theory with central chargec 24 in string terminology their construction corresponds to compactification on az2 asymmetric orbifold constructed from the torusr24 where is the leech lattice in this note we point out that their construction naturally embodies as well a larger algebraic structure namely a super virasoro algebra with central charge 16 with the supersymmetry generator constructed in terms of bosonic twist fields

eta quotients and theta functions abstract the jacobi triple product identity gives a closed form for many infinite product generating functions that arise naturally in combinatorics and number theory of particular interest is its application to dedekind s eta function z defined via an infinite product giving it as a certain kind of infinite sum known as a theta function using the theory of modular forms we classify all eta quotients that are theta functions

character and dimension formulae for queer lie superalgebra closed formulae are constructed for the characters and dimensions of the finite dimensional simple modules of the queer lie superalgebra mathfrak q n this is achieved by refining brundan s algorithm for computing simple mathfrak q n characters

new robust stability results for bidirectional associative memory neural networks with multiple time delays in this paper the robust stability problem is investigated for a class of bidirectional associative memory bam neural networks with multiple time delays by employing suitable lyapunov functionals and using the upper bound norm for the interconnection matrices of the neural network system some novel sufficient conditions ensuring the existence uniqueness and global robust stability of the equilibrium point are derived the obtained results impose constraint conditions on the system parameters of neural network independent of the delay parameters some numerical examples and simulation results are given to demonstrate the applicability and effectiveness of our results and to compare the results with previous robust stability results derived in the literature

finite dimensional half integer weight modules over queer lie superalgebras we give a new interpretation of representation theory of the finite dimensional half integer weight modules over the queer lie superalgebra mathfrak q n it is given in terms of the brundan s work on finite dimensional integer weight mathfrak q n modules by means of lusztig s canonical basis using this viewpoint we compute the characters of the finite dimensional half integer weight irreducible modules for a large class of irreducible modules whose highest weights are of special types i e totally connected or totally disconnected we derive closed form character formulas that are reminiscent of the kac wakimoto character formula for basic lie superalgebras

globally nilpotent differential operators and the square ising model we recall various multiple integrals with one parameter related to the isotropic square ising model and corresponding respectively to the n particle contributions of the magnetic susceptibility to the lattice form factors to the two point correlation functions and to their extensions the univariate analytic functions defined by these integrals are holonomic and even g functions they satisfy fuchsian linear differential equations with polynomial coefficients and have some arithmetic properties we recall the explicit forms found in previous work of these fuchsian equations as well as their russian doll and direct sum structures these differential operators are selected fuchsian linear differential operators and their remarkable properties have a deep geometrical origin they are all globally nilpotent or sometimes even have zero p curvature we also display miscellaneous examples of globally nilpotent operators emerging from enumerative combinatorics problems for which no integral representation is yet known focusing on the factorized parts of all these operators we find out that the global nilpotence of the factors resp p curvature nullity corresponds to a set of selected structures of algebraic geometry elliptic curves modular curves curves of genus five six and even a remarkable weight 1 modular form emerging in the three particle contribution 3 of the magnetic susceptibility of the square ising model noticeably this associated weight 1 modular form is also seen in the factors of the differential operator for another n fold integral of the ising class 3 h for the staircase polygons counting and in ap ry s study of 3 g functions naturally occur as solutions of globally nilpotent operators in the case where we do not have g functions but hamburger functions one irregular singularity at 0 or that correspond to the confluence of singularities in the scaling limit the p curvature is also found to verify new structures associated with simple deformations of the nilpotent property

kac wakimoto character formula for ortho symplectic lie superalgebras abstract we classify finite dimensional tame modules over the ortho symplectic lie superalgebras for these modules we show that their characters are given by the kac wakimoto character formula thus establishing the kac wakimoto conjecture for the ortho symplectic lie superalgebras we further relate the kac wakimoto formula to the super jacobi polynomials of sergeev and veselov and show that these polynomials up to a sign give the super characters for these tame modules

general criteria for asymptotic and exponential stabilities of neural network models with unbounded delays abstract for a family of differential equations with infinite delay we give sufficient conditions for the global asymptotic and global exponential stability of an equilibrium point this family includes most of the delayed models of neural networks of cohen grossberg type with both bounded and unbounded distributed delay for which general asymptotic and exponential stability criteria are derived as illustrations the results are applied to several concrete models studied in the literature and a comparison of results is given

tensor triangular geometry for classical lie superalgebras abstract tensor triangular geometry as introduced by balmer 3 is a powerful idea which can be used to extract the ambient geometry from a given tensor triangulated category in this paper we provide a general setting for a compactly generated tensor triangulated category which enables one to classify thick tensor ideals and the balmer spectrum for the general linear lie superalgebra g g 0 g 1 we construct a zariski space from a detecting subalgebra of g and demonstrate that this topological space governs the tensor triangular geometry for the category of finite dimensional g modules which are semisimple over g 0

primitive ideals twisting functors and star actions for classical lie superalgebras we study three related topics in representation theory of classical lie superalgebras the first one is classification of primitive ideals i e annihilator ideals of simple modules and inclusions

thick ideals in deligne s category re_p oδ abstract we describe indecomposable objects in deligne s category re p o and explain how to decompose their tensor products we then classify thick ideals in re p o as an application we classify the indecomposable summands of tensor powers of the standard representation of the orthosymplectic supergroup up to isomorphism

brief paper on nonoscillation and stability for systems of differential equations with a distributed delay for the system with a distributed delay email protected t email protected k 1 email protected h k t t d sr k t s x s 0 and a nonnegative fundamental matrix solution estimates and stability results are obtained the results are applied to integro differential equations and equations of mixed type including both distributed and concentrated delays

euler characters and super jacobi polynomials abstract we prove that euler supercharacters for orthosymplectic lie superalgebras can be obtained as a certain specialization of super jacobi polynomials a new version of weyl type formula for super schur functions and specialized super jacobi polynomials play a key role in the proof

g elliptic systems and the genus zero problem for m_ 24 toward the end of the 1970s j h conway and s p norton wrote a quite unique and influential paper 3 in which they produced incontrovertible evidence based partly on ideas of j mckay and j g thompson 13 that the fischer griess monster simple group m was intimately related to the modular function j x familiar from the theory of modular forms thus was so called moonshine brought into the world among other things it was conjectured that there is a natural infinite dimensional complex vector space v which carries a z grading into finite dimensional subspaces f v x e vx 0 v2 0 such that each vi is a cmmodule moreover the graded character chvt q j2n dimf tf of f should be the fourier expansion of j r namely q 196884 21 493 760 if the indeterminate q is interpreted as e for r in the upper half plane h there should in addition be an analogous interpretation for

on the existence of positive solutions for systems of differential equations with a distributed delay for the system with a distributed delay email protected t email protected email protected h k t t d sr k t s x s 0 sufficient conditions for positivity of the fundamental matrix are obtained in the case when the fundamental matrix is positive comparison results and sufficient positivity conditions for solutions are established

on rings of supersymmetric polynomials abstract we consider three types of rings of supersymmetric polynomials polynomial ones m n partially polynomial m n y and laurent supersymmetric rings m n for each type of rings we give their descriptions in terms of generators and relations as a corollary we get for n m an isomorphism m n y m m y 0 n m y it is also true for polynomial rings but in this case the isomorphism does not preserve the grading for each type of rings some natural basis consisting of euler supercharacters is constructed

mixed cohomology of lie superalgebras abstract supermanifolds are known to admit both differential forms and integral forms thus any appropriate super analogue of the de rham theory should take both types of forms into account however the cohomology of lie superalgebras studied so far in the literature involves only differential forms when interpreted as a de rham theory for lie supergroups thus a new cohomology theory of lie superalgebras is needed to fully incorporate differential integral forms and we investigate such a theory here this new cohomology is defined by a brst complex of lie superalgebra modules and includes the standard lie superalgebra cohomology as a special case general properties expected of a cohomology theory are established for the new cohomology and examples of the new cohomology groups are computed

super duality and irreducible characters of ortho symplectic lie superalgebras we formulate and establish a super duality which connects parabolic categories o for the ortho symplectic lie superalgebras and classical lie algebras of bcd types this provides a complete and conceptual solution of the irreducible character problem for the ortho symplectic lie superalgebras in a parabolic category o which includes all finite dimensional irreducible modules in terms of classical kazhdan lusztig polynomials

superspace conformal field theory conformal sigma models and wess zumino witten wzw models on coset superspaces provide important examples of logarithmic conformal field theories they possess many applications to problems in string and condensed matter theory we review recent results and developments including the general construction of wzw models on type i supergroups the classification of conformal sigma models and their embedding into string theory

generalised verma modules for the orthosymplectic lie superalgebra ospk 2 abstract the composition factors with their multiplicities and the submodule lattices are determined for generalised verma modules over the orthosymplectic lie superalgebra osp k 2 the results enable us to obtain explicit formulae for the formal characters and dimensions of the finite dimensional irreducible modules applying these results we also compute the first and second cohomology groups of the lie superalgebra with coefficients in finite dimensional kac modules and irreducible modules

bernstein gelfand gelfand resolutions for basic classical lie superalgebras abstract we study kostant cohomology and bernstein gelfand gelfand resolutions for finite dimensional representations of basic classical lie superalgebras for each choice of parabolic subalgebra and representation of such a lie superalgebra there is a natural definition of the boundary and coboundary operators which define co homology of the nilradical of the parabolic subalgebra we prove that complete reducibility of the homology groups is a necessary condition to have a resolution of an irreducible module in terms of generalised verma modules every such a resolution is then given by modules induced by these homology groups we also prove that if these homology groups are completely reducible a sufficient condition for the existence of this resolution is the property that these groups are isomorphic to the kernel of the kostant quabla operator which is equivalent with disjointness of the boundary and coboundary operators then we use these results to derive very explicit conditions under which bgg resolutions exist which are particularly useful for the superalgebras of type i for the unitarisable representations of gl m n and osp 2 2 n we derive conditions on the parabolic subalgebra under which the bgg resolutions exist we also apply the obtained theory to construct specific examples of bgg resolutions for osp m 2 n finally we state some results for typical modules of gl m n and osp 2 2 n

nonabelian orbifolds and the boson fermion correspondence for a positive integerl divisible by 8 there is a bosonic holomorphic vertex operator algebra voa v gamma l associated to the spin lattice l for a broad class of finite groupsg of automorphisms of v gamma l we prove the existence and uniqueness of irreducibleg twisted v gamma l modules and establish the modular invariance of the partition functionsz g h for commuting elements ing in particular for any finite group there are infinitely many holomorphic voas admittingg for which these properties hold the proof is facilitated by a boson fermion correspondence which gives a voa isomorphism between v gamma l and a certain fermionic construction and which extends work of frenkel and others

an orbifold theory of genus zero associated to the sporadic groupm24 let v gamma l be the self dual or holomorphic bosonic conformal field theory associated with the spin lattice l of rankl divisible by 24 in earlier work of the authors we showed how it is possible to establish the existence and uniqueness of irreducibleg twisted sectors for v gamma l for certain automorphismsg of v gamma l and to establish the modular invariance of the space of partition functionsz g h corresponding to commuting pairsg h of elements in certain groupsg of automorphisms of v gamma l in the present work we show that if we takel 24 andg the sporadic simple groupm24 then the corresponding orbifold has thegenus zero property that is eachz g h is either identically zero or ahauptmodul i e it generates the field of functions on the subgroup ofsl2 which fixesz g h which then necessarily has genus zero

the representation theory of the exceptional lie superalgebras f 4 and g 3 this thesis is a resolution of three related problems proposed by yu i manin and v kac for the so called exceptional lie superalgebras f 4 and g 3 the first problem posed by kac 1978 is the problem of finding character and superdimension formulae for the simple modules the second problem posed by kac 1978 is the problem of classifying all indecomposable representations the third problem posed by manin 1981 is the problem of constructing the superanalogue of borel weil bott theorem

a reduction method for atypical representations of classical lie superalgebras abstract we introduce a reduction method for studying representations of classical lie superalgebras with atypical central character we show that the atypical quotient of universal enveloping algebra has a non trivial jacobson radical the factor by this radical has a new center which is calculated for sl 1 n and psl 2 2 using this center we obtain new character formulae generalization of borel weil bott and beilinson bernstein localization theorems

character formulae for ortho symplectic lie superalgebras osp n 2 abstract the characters of finite dimensional irreducible modules of the lie superalgebra osp n 2 are completely determined in terms of those of generalized verma modules as a by product the irreducible decomposition of the tensor of any finite dimensional irreducible module with the natural module is obtained

cohomology of generalized supergrassmannians and character formulae for basic classical lie superalgebras we use geometrical methods adapted from borel weil bott theory to compute the character of every finite dimensional simple module over a basic classical lie superalgebra the answer is given by a combinatorial algorithm in terms of weight and cap diagrams which were defined by brundan and stroppel for the general linear superalgebra ams subject classification 20g05 14f05

shifted vertex operator algebras and g elliptic systems shifted vertex operator algebras and superalgebras are defined both in bosonic and in fermionic pictures their isomorphism boson fermion correspondence is shown partition functions of bosonic shifted vertex operator algebras are realized analytically and are interpreted as string path integrals over elliptic curves their modular properties follow and elliptic systems are constructed based on fermionic shifted vertex operator algebras

on kostant s theorem for the lie superalgebra q n abstract in this paper we study finite w algebras for basic superalgebras and q n associated with the regular even nilpotent coadjoint orbits we prove that this algebra satisfies the amitsur levitzki identity and therefore all its irreducible representations are finite dimensional in the case of q n we give an explicit description of the w algebra in terms of generators and relations and realize it as a quotient of the super yangian of q 1

congruence properties for a certain kind of partition functions in light of the modular equations of fifth and seventh order we derive some congruence properties for a certain kind of partition functions a n which satisfy n 0 a n qn q q k modm where k is a positive integer with 1 k 24 and m 2 3 in view of these properties we obtain many infinite families of congruences for c k n the number of generalized frobenius partitions of n with k colors and c k n the number of generalized frobenius partitions of n with k colors whose order is k under cyclic permutation of the k colors meanwhile we also apply the main theorems to some other kinds of partition functions

holomorphic eta quotients of weight 1 2 abstract we give a short proof of zagier s conjecture mersmann s theorem which states that each holomorphic eta quotient of weight 1 2 is an integral rescaling of some eta quotient from zagier s list of fourteen primitive holomorphic eta quotients in particular given any holomorphic eta quotient f of weight 1 2 this result enables us to provide a closed form expression for the coefficient of q n in the q series expansion of f for all n we also demonstrate another application of the above theorem in extending the levels of the simple resp irreducible holomorphic eta quotients