spatiospectral concentration on a sphere we pose and solve the analogue of slepian s time frequency concentration problem on the surface of the unit sphere to determine an orthogonal family of strictly bandlimited functions that are optimally concentrated within a closed region of the sphere or alternatively of strictly spacelimited functions that are optimally concentrated in the spherical harmonic domain such a basis of simultaneously spatially and spectrally concentrated functions should be a useful data analysis and representation tool in a variety of geophysical and planetary applications as well as in medical imaging computer science cosmology and numerical analysis the spherical slepian functions can be found by solving either an algebraic eigenvalue problem in the spectral domain or a fredholm integral equation in the spatial domain the associated eigenvalues are a measure of the spatiospectral concentration when the concentration region is an axisymmetric polar cap the spatiospectral projection operator commutes with a sturm liouville operator this enables the eigenfunctions to be computed extremely accurately and efficiently even when their area bandwidth product or shannon number is large in the asymptotic limit of a small spatial region and a large spherical harmonic bandwidth the spherical concentration problem reduces to its planar equivalent which exhibits self similarity when the shannon number is kept invariant

multiplicity of single bump solutions for semilinear elliptic equations in multi bump domains abstract in this paper we study the decomposition of the filtration of the nehari manifold via the variation of domain shape furthermore we use this result to prove that the semilinear elliptic equation in the multi bump domain r has multiple positive solutions and we can find at least one ground state solution

spatial spectral filtering with linearly constrained minimum variance beamformers in this paper orthogonal low dimension models for broad band sources as observed by an array delay line structure are developed the models are termed broad band source representation spaces and are subspaces of the observation space they are generated from eigenstructures of source sample covariance matrices and represent sources efficiently in the 2nd order statistical sense for a single broad band source propagating in a pure delay environment and observed with an isotropic array an accurate indication of the model dimension is derived which is independent on the array configuration being a function of the observed source time bandwidth product only using these representations procedures are described for controlling the spatial and spectral response of linearly constrained minimum variance broad band array beamformers the procedures employ linear constraints to control complex gain characteristics over selected bands of frequency for multiple points of 3 d regions of source locations and do not require the use of steering delays the constraints are derived from source representation spaces and are termed eigenvector constraints simulations are presented which illustrate the effectiveness of the new linear constraints

on the relationship between monstrous moonshine and the uniqueness of the moonshine module we consider the relationship between the conjectured uniqueness of the moonshine module open image in new window and monstrous moonshine the genus zero property of the modular invariance group for each monster group thompson series we first discuss a family of possiblezn meromorphic orbifold constructions of open image in new window based on automorphisms of the leech lattice compactified bosonic string we reproduce the thompson series for all 51 non fricke classes of the monster groupm together with a new relationship between the centralisers of these classes and 51 corresponding conway group centralisers generalising a well known relationship for 5 such classes assuming that open image in new window is unique we consider meromorphic orbifoldings of open image in new window and show that monstrous moonshine holds if and onlyzr if the only meromorphic orbifoldings of open image in new window are open image in new window itself or the leech theory this constraint on the meromorphic orbifoldings of open image in new window therefore relates monstrous moonshine to the uniqueness of open image in new window in a new way

modularity in orbifold theory for vertex operator superalgebras this paper is about the orbifold theory for vertex operator superalgebras given a vertex operator superalgebra v and a finite automorphism group g of v we show that the trace functions associated to the twisted sectors are holomorphic in the upper half plane for any commuting pairs in g under the c2 cofinite condition we also establish that these functions afford a representation of the full modular group if v is c2 cofinite and g rational for any g g

mirror symmetry on kummer type k3 surfaces we investigate both geometric and conformal field theoretic aspects of mirror symmetry on n 4 4 superconformal field theories with central charge c 6 our approach enables us to determine the action of mirror symmetry on non stable singular fibers in elliptic fibrations of n orbifold limits of k3 the resulting map gives an automorphism of order 4 8 or 12 respectively on the smooth universal covering space of the moduli space we explicitly derive the geometric counterparts of the twist fields in our orbifold conformal field theories the classical mckay correspondence allows for a natural interpretation of our results

bosonic and fermionic realizations of the affine algebra g hat l_n we give an explicit description of all inequivalent heisenberg subalgebras of the affine lie algebra g hat l n mathbb c and the associated vertex operator constructions of the level one integrable highest weight representations of this algebra the construction uses multicomponent fermionic fields and yields a correspondence between bosons elements of the heisenberg subalgebra and fermions

comment on the generation number in orbifold compactifications there has been some confusion concerning the number of 1 1 forms in orbifold compactifications of the heterotic string in numerous publications in this note we point out the relevance of the underlying torus lattice on this number we answer the question when different lattices mimic the same physics and when this is not the case as a byproduct we classify all symmetriczn orbifolds with 2 2 world sheet supersymmetry obtaining also some new ones

twisted quantum affinizations and their vertex representations in this paper we generalize drinfeld s twisted quantum affine algebras to construct twisted quantum algebras for all simply laced generalized cartan matrices and present their vertex representation realizations

a construction of lower bounded generalized twisted modules for a grading restricted vertex super algebra we give a general direct and explicit construction of lower bounded generalized twisted modules satisfying a universal property for a grading restricted vertex super algebra v associated to an automorphism g of v in particular when g is the identity we obtain lower bounded generalized v modules satisfying a universal property let w be a lower bounded graded vector space equipped with a set of generating twisted fields and a set of generator twist fields satisfying a weak commutativity for generating twisted fields a generalized weak commutativity for one generating twisted field and one generator twist field and some other properties that are relatively easy to verify we first prove the convergence and commutativity of products of an arbitrary number of generating twisted fields one twist generator field and an arbitrary number of generating fields for v then using the convergence and commutativity we define a twisted vertex operator map for w and prove that w equipped with this twisted vertex operator map is a lower bounded generalized g twisted v module using this result we give an explicit construction of lower bounded generalized g twisted v modules satisfying a universal property starting from vector spaces graded by weights mathbb z 2 fermion numbers and g weights eigenvalues of g and real numbers corresponding to the lower bounds of the weights of the modules to be constructed in particular every lower bounded generalized g twisted v module every lower bounded generalized v module when g is the identity is a quotient of such a universal lower bounded generalized g twisted v module a universal lower bounded generalized v module

spectral estimation of irregularly sampled multidimensional processes by generalized prolate spheroidal sequences a nonparametric spectral estimation method is presented for bandlimited random processes that have been sampled at arbitrary points in one or more dimensions the method makes simultaneous use of several weight sequences that depend on the set of sampling point the signal band and the frequency band being analyzed these sequences are solutions to a generalized matrix eigenvalue problem and are termed generalized prolate spheroidal sequences being extensions of the familiar discrete prolate spheroidal sequences statistics of the estimator are derived and the tradeoff among bias variance and resolution is quantified the method avoids several problems typically associated with irregularly sampled data and multidimensional processes a related method is suggested that has nearly as good performance while requiring significantly fewer computations

multiple window based minimum variance spectrum estimation for multidimensional random fields spectrum analysis is viewed as the problem of estimating the power of a process contained within narrow bands this view leads naturally to consideration of filter based methods for estimating spectra multiple window based estimators where the power in a band is estimated as the average of the powers estimated at the outputs of multiple filters are considered the filters are designed using a linearly constrained minimum variance criterion commonly employed in adaptive beamforming this results in filters that automatically adjust their sidelobes to minimize leakage of energy from outside the band of interest expressions for the bias and variance of the power estimates are derived assuming the sample covariance matrix estimate is used to estimate the data covariance matrix and that the data are independent and identically gaussian distributed these expressions lead to the definition of a performance factor that indicates the degree of variance reduction obtained via multiple window processing a technique for obtaining increased suppression of energy leaking through the filter sidelobes at the expense of the response fidelity to energy in the band is presented simulations are presented to illustrate the effectiveness of the technique

vertex operators and symmetric functions an algorithm for the calculation of q dependent spin characters of the symmetric group is given with an explicit example of s4 a method for constructing the q analogue of the vertex operators is developed a 1 1 correspondence between the space v of twisted q vertex operators and the ring of q deformed symmetric functions lambda x zq q t is established and a mapping from v to lambda x zq q t is defined a number of relevant theorems are given

a relation between the domain topology and the number of minimal nodal solutions for a quasilinear elliptic problem abstract we consider the quasilinear problem div u p 2 u u p 2 u u q 2 u in u 0 on where r n is a bounded smooth domain 1 p n and p q p np n p we show that if is invariant by a non trivial orthogonal involution then for q close to p the equivariant topology of is related with the number of solutions which change sign exactly once the results complement those of castro and clapp nonlinearity 16 2003 579 590 since we consider subcritical nonlinearities and the quasilinear case without any assumption of symmetry we also extend theorem b in benci and cerami arch rational mech anal 114 1991 79 93 for the quasilinear case and prove that the topology of affects the number of positive solutions

on orbifold theory abstract let v be a simple vertex operator algebra and g a finite automorphism group of v such that v g is regular it is proved that every irreducible v g module occurs in an irreducible g twisted v module for some g g moreover the quantum dimensions of irreducible v g modules are determined and a global dimension formula for v in terms of twisted modules is obtained in particular the orbifold theory conjecture is completely solved if g is solvable

existence and multiplicity of nodal solutions for dirichlet problems in upper half strip with holes abstract in this paper we consider the existence and multiplicity of nodal solutions of semilinear elliptic equations we prove that a semilinear elliptic equation in large domains does not admit any least energy nodal sign changing solution and in an upper half strip with m holes has at least m 2 2 nodal solutions

critical point theory on partially ordered hilbert spaces we develop some abstract critical point theory in order to prove that boundary value problems like the model problem formula on a bounded domain rn 2 p 2n n 2 have infinitely many sign changing solutions uk k n which are not comparable that is uk ul and uk ul change sign for k l we also show that there are no subsolutions u such that u uk for some k and u is positive somewhere a corresponding nonexistence result applies to supersolutions related results on the existence of sign changing solutions hold for other classes of nonlinearities

on spinor vertex operator algebras and their modules abstract in this paper we first present a generator theorem and introduce an asymmetric tensor of two colored vertex operator superalgebras with the same grading and supermap then we give axioms which characterize a class of vertex operator superalgebras constructed from clifford algebras in particular we are able to give a shorter argument for the jacobi identity than the existing proofs by using elements analogous to coherent states and our generator theorem using the new tensor and the same techniques we classify twisted irreducible modules of these algebras in a constructive way

level two standard an modules abstract in this paper we study the level two standard i e irreducible integrable highest weight modules for an affine special linear lie algebra of arbitrary rank in particular we give explicit realizations of these modules in the principal picture

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

the krichever map vector bundles over algebraic curves and heisenberg algebras we study the grassmanniangr x n consisting of equivalence classes of rankn algebraic vector bundles over a riemann surfacex with an holomorphic trivialization at a fixed pointp commutative subalgebras ofgl n h h being the ring of functions holomorphic on a punctured disc aboutp define flows on the grassmannian giving rise to classes of solutions to multi component kp hierarchies these commutative subalgebras correspond to heisenberg algebras in the kac moody algebra associated togl n h one can obtain by the krichever map points ofgr x n and solutions of mckp from coveringsf y x and other geometric data conversely for every point ofgr x n and for every choice of heisenberg algebra we construct using the cotangent bundle ofgr x n an algebraic curve coveringx and other data thus inverting the krichever map we show the explicit relation between the choice of heisenberg algebra and the geometry of the covering space

twisted modules over vertex algebras on algebraic curves abstract we extend the geometric approach to vertex algebras developed by the first author to twisted modules allowing us to treat orbifold models in conformal field theory let v be a vertex algebra h a finite group of automorphisms of v and c an algebraic curve such that h aut c we show that a suitable collection of twisted v modules gives rise to a section of a certain sheaf on the quotient x c h we introduce the notion of conformal blocks for twisted modules and analyze them in the case of the heisenberg and affine kac moody vertex algebras we also give a chiral algebra interpretation of twisted modules

four 2 nodal solutions for a semilinear elliptic equation in a finite strip with a hole in this paper we study the decomposition of the filtration of the nehari manifold via the variation of domain shape we use this result to prove that the semilinear elliptic equation in a finite strip with hole has at least four 2 nodal solutions solutions with precisely two nodal domains furthermore we can describe the bump location of these solutions

minimum variance beamforming with soft response constraints soft constraints on the beamformer response to the signal are examined in the context of minimum variance beamforming a quadratic constraint on the beamformer weights is used to control the mean squared error between a desired response and the actual response in the signal direction the constraint is purposely chosen to permit distortion of the signal with the goal of achieving improved interference cancellation under the assumptions of known signal direction and spectral shape the signal to noise ratio is shown to be a nondecreasing function of the mean square distortion the distortion presented to the signal is easily computed from the beamformer weights and can be equalized after beamforming if desired properties of this beamforming method and its relationship to linear constrained beamforming are discussed simulations verify analytic results and illustrate the utility of the soft constrained approach

2 permutations of lattice vertex operator algebras higher rank abstract this paper is a continuation of our investigation on 2 permutation of lattice vertex operator algebras in particular the quantum dimensions of irreducible modules and the fusion rules are determined if the rank of the lattice is one these results have been obtained previously in 27

twisted γ lie algebras and their vertex operator representations let be a generic subgroup of the multiplicative group c of nonzero complex numbers we define a class of lie algebras associated to called twisted lie algebras which are natural generalizations of the twisted affine lie algebras starting from an arbitrary even sublattice q of zn and an arbitrary finite order isometry of zn preserving q we construct a family of twisted vertex operators acting on generalized fock spaces which afford irreducible representations for certain twisted lie algebras as an application this recovers a number of known vertex operator realizations for infinite dimensional lie algebras such as twisted affine lie algebras extended affine lie algebras of type a trigonometric lie algebras of series a and b unitary lie algebras and bc graded lie algebras

pursuing the double affine grassmannian i transversal slices via instantons on a_k singularities this article is the first in a series that describes a conjectural analog of the geometric satake isomorphism for an affine kac moody group for simplicity we only consider the untwisted and simply connected case here the usual geometric satake isomorphism for a reductive group g identifies the tensor category rep g of finite dimensional representations of the langlands dual group g with the tensor category pervg o grg of g o equivariant perverse sheaves on the affine grassmannian grg g k g o of g here k c t and o c t as a by product one gets a description of the irreducible g o equivariant intersection cohomology ic sheaves of the closures of g o orbits in grg in terms of q analogs of the weight multiplicity for finite dimensional representations of g the purpose of this article is to try to generalize the above results to the case when g is replaced by the corresponding affine kac moody group gaff we refer to the not yet constructed affine grassmannian of gaff as the double affine grassmannian more precisely in this article we construct certain varieties that should be thought of as transversal slices to various gaff o orbits inside the closure of another gaff o orbit in grgaff we present a conjecture that computes the intersection cohomology sheaf of these varieties in terms of the corresponding q analog of the weight multiplicity for the langlands dual affine group gaff and we check this conjecture in a number of cases some further constructions such as convolution of the corresponding perverse sheaves analog of the beilinson drinfeld grassmannian and so forth will be addressed in another publication

bimodules and g rationality of vertex operator algebras this paper studies the twisted representations of vertex operator algebras let v be a vertex operator algebra and g an automorphism of v of finite order t for any m n 1 tz an a g n v a g m v bimodule a g n m v is constructed the collection of these bimodules determines any admissible g twisted v module completely a verma type admissible g twisted v module is constructed naturally from any a g m v module furthermore it is shown with the help of bimodule theory that a simple vertex operator algebra v is g rational if and only if its twisted associative algebra ag v is semisimple and each irreducible admissible g twisted v module is ordinary

multiplicity of nodal solutions for elliptic problems involving non odd nonlinearities abstract in this paper we study the effect of domain shape on the number of 2 nodal solutions for the semilinear elliptic equation involving non odd nonlinearities we prove that a semilinear elliptic equation in an m bump domain possibly unbounded has m 2 2 nodal solutions and we can find a least energy nodal solution in those solutions furthermore we can describe the bump location of these solutions

a construction of the level 3 modules for the affine lie algebra and a new combinatorial identity of the rogers ramanujan type amer math soc 348 1996 pp 481 501

an extremal mathcal n 2 superconformal field theory here we provide an example of an extremal chiral mathcal n 2 superconformal field theory at c 24 the construction is based on a mathbb z 2 orbifold of the theory associated to the a 1 24 niemeier lattice the statespace is governed by representations of the sporadic group m 23

calculus of twisted vertex operators starting from an arbitrary isometry of an arbitrary even lattice twisted and shifted vertex operators are introduced under commutators these operators provide realizations of twisted affine lie algebras this construction generalizing a number of known ones is based on a self contained calculus

orbifolds of lattice vertex algebras under an isometry of order two abstract every isometry of a positive definite even lattice q can be lifted to an automorphism of the lattice vertex algebra v q an important problem in vertex algebra theory and conformal field theory is to classify the representations of the invariant subalgebra v q of v q known as an orbifold in the case when is an isometry of q of order two we classify the irreducible modules of the orbifold vertex algebra v q and identify them as submodules of twisted or untwisted v q modules the examples where q is a root lattice and is a dynkin diagram automorphism are presented in detail

realization of the level one standard _ 2 1 modules in this paper we study the level one standard or irreducible integrable highest weight modules for the affine symplectic lie algebras in particular we give concrete realizations of all level one standard modules for the affine symplectic lie algebras of even rank introduction in recent years explicit realizations of nontrivial representations of affine lie algebras have attracted the attention of many researchers because of their surprising connections with different areas of mathematics and physics among these realizations the principal realizations and the untwisted homogeneous realizations have proved to be especially useful in 1978 lepowsky and wilson 15 constructed the basic modules level one standard modules i e irreducible integrable highest weight modules for a in the principal realization in 1 o this construction was generalized to the level one standard modules for all simply laced affine lie algebras frenkel and kac 5 and segal 26 have given explicit construction of these modules in the untwisted realization the constructions of these modules in the general realization are given in 9 12 since then some higher level standard modules for certain simply laced affine lie algebras have also been constructed e g 14 17 18 19 21 22 25 however less is known regarding explicit constructions of standard modules for nonsimply laced affine lie algebras that is affine lie algebras of type b c f and g in the last section of 10 kac kazhdan lepowsky and wilson constructed reducible level one modules for the non simply laced affine lie algebras in the principal realization by taking fixed points of dynkin diagraminduced automorphisms recently in the untwisted realization the analogous representations again reducible have been constructed in 3 and 6 the level one standard modules for the affine orthogonal lie algebras b 1 have been n explicitly realized in 4 8 13 and 20 from different view points in 20 mandia has constructed the level one standard modules for the affine lie algebras f 1 and g 1 in 23 the author gave explicit constructions of the level one standard modules for the affine symplectic lie algebras of rank 3 and 4 received by the editors july 13 1989 presented to the 847th meeting of the ams pheonix arizona january 14 1989 1980 mathematics subject classification 1985 revision primary 17b65 1990 american mathematical society 0002 9947 90 1 00 25 per page

twisted ϕ coordinated modules for nonlocal vertex algebras abstract in this paper we study what we call twisted coordinated modules for nonlocal vertex algebras including weak quantum vertex algebras and we give a conceptual construction of nonlocal vertex algebras and their twisted coordinated modules as an illustrating example we study certain vertex algebras and their twisted coordinated modules associated to certain infinite dimensional heisenberg lie algebras as a reformulation and a slight extension of a result of lepowsky and a result of doyon lepowsky and milas we also give a canonical connection between twisted modules for a general 1 t z graded nonlocal vertex algebra v and twisted coordinated modules for the nonlocal vertex algebra obtained from v by zhu s change of variable theorem

lie algebra automorphisms the weyl group and tables of shift vectors the canonical lift of the weyl group in the automorphism group of a simple lie algebra is discussed a constructive algorithm is given to compute the shift vector for every weyl group conjugacy class and tables for the simply laced lie algebras of rank 8 are provided applications in physics such as the equivalence of certain orbifold and twisted wzw string models as well as the implication for the moduli space of 2 d conformal field theories are discussed

automorphisms of niemeier lattices for miyamoto s mathbb z _3 orbifold construction we classify up to conjugation all automorphisms of niemeier lattices to which we can apply miyamoto s orbifold construction using this classification we prove that the voas obtained in miyamoto symmetries integrable systems and representations springer proceedings in mathematics and statistics vol 40 springer london pp 319 344 2013 and sagaki and shimakura trans am math soc to appear are all of holomorphic non lattice voas which we can obtain by applying the mathbb z 3 orbifold construction to a niemeier lattice and its automorphism

eigenvalue beamforming using a multirank mvdr beamformer and subspace selection we derive eigenvalue beamformers to resolve an unknown signal of interest whose spatial signature lies in a known subspace but whose orientation in that subspace is otherwise unknown the unknown orientation may be fixed in which case the signal covariance is rank 1 or it may be random in which case the signal covariance is multirank we present a systematic treatment of such signal models and explain their relevance for modeling signal uncertainties we then present a multirank generalization of the mvdr beamformer the idea is to minimize the power at the output of a matrix beamformer while enforcing a data dependent distortionless constraint in the signal subspace which we design based on the type of signal we wish to resolve we show that the eigenvalues of an error covariance matrix are fundamental for resolving signals of interest signals with rank 1 covariances are resolved by the largest eigenvalues of the error covariance while signals with multirank covariances are resolved by the smallest eigenvalues thus the beamformers we design are eigenvalue beamformers which extract signal information from eigen modes of an error covariance we address the tradeoff between angular resolution of eigenvalue beamformers and the fraction of the signal power they capture

graphene lattice field theory and symmetries borrowing ideas from tight binding model we propose a board class of lattice field models that are classified by non simply laced lie algebras in the case of an 1 su n series we show that the couplings between the quantum states living at the first nearest neighbor sites of the lattice lsun are governed by the complex fundamental representations n and n of su n and the second nearest neighbor interactions are described by its adjoint n n the lattice models associated with the leading su 2 su 3 and su 4 cases are explicitly studied and their fermionic field realizations are given it is also shown that the su 2 and su 3 models describe the electronic properties of the acetylene chain and the graphene respectively it is established as well that the energy dispersion of the first nearest neighbor couplings is completely determined by the an roots through the typical dependence n 2 roots cosk with k the wave vector other features such as the so 2n extension and other application

additional symmetries of generalized integrable hierarchies the non isospectral symmetries of a general class of integrable hierarchies are found by generalizing the galilean and scaling symmetries of the korteweg de vries equation and its hierarchy the symmetries arise in a very natural way from the semi direct product structure of the virasoro algebra and the affine kac moody algebra underlying the construction of the hierarchy in particular the generators of the symmetries are shown to satisfy a subalgebra of the virasoro algebra when a tau function formalism is available the infinitesimal symmetries act directly on the tau functions as moments of virasoro currents some comments are made regarding the role of the non isospectral symmetries and the form of the string equations in matrix model formulations of quantum gravity in two dimensions and related systems

on twisted representations of vertex algebras abstract in this paper we develop a formalism for working with representations of vertex and conformal algebras by generalized fields formal power series involving non integer powers of the variable the main application of our technique is the construction of a large family of representations for the vertex superalgebra v corresponding to an integer lattice for an automorphism v v coming from a finite order automorphism we find the conditions for existence of twisted modules of v we show that the category of twisted representations of v is semisimple with finitely many isomorphism classes of simple objects

extensions of tensor products of zp orbifold models of the lattice vertex operator algebra v2ap 1 abstract let p be an odd prime and let be an order p automorphism of v 2 a p 1 which is a lift of a p cycle in the weyl group weyl a p 1 s p we study a certain extension v of a tensor product of finitely many copies of the orbifold model v 2 a p 1 and give a criterion for v that every irreducible v module is a simple current

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

cartan determinants and shapovalov forms we compute the determinant of the gram matrix of the shapovalov form on weight spaces of the basic representation of an affine kac moody algebra of ade type possibly twisted as a consequence we obtain explicit formulae for the determinants of the cartan matrices of p blocks of the symmetric group and its double cover and of the associated hecke algebras at roots of unity

twist vertex operators for twisted modules abstract we introduce and study twist vertex operators for a lower bounded generalized twisted modules for a grading restricted vertex super algebra we prove the duality property weak associativity a jacobi identity a generalized commutator formula generalized weak commutativity and convergence and commutativity for products of more than two operators involving twist vertex operators these properties of twist vertex operators play an important role in the author s recent general direct and explicit construction of lower bounded generalized twisted modules

elements of the annihilating ideal of a standard module the construction of affine lie algebras by means of vertex operators has proved to be a remarkably fruitful line of research by motivating being motivated by and linking together a disparate range of sectors of mathematics and physics vertex operators entered mathematics through the work of j lepowsky and r l wilson who gave an explicit construction of the affne algebra a l lwl their vertex operators were similar to ones used by physicists in the string model later on i frenkel v kac fk and g segal s found representations of afhne algebras using untwisted vertex operators a large amount of very interesting research has followed stimulated by this observation and by the connection between classical combinatorial identities and the weyl kac character formula k lm lpl 2 lw2 4 ma mil 21 a deeper understanding of vertex operators has been reached more recently as a result of work by borcherds b who investigated general vertex operator algebras and by frenkel lepowsky and a meurman in a series of papers culminating in the recent book flm in particular these three authors showed that the monster finite simple group is the automorphism group of a certain vertex operator algebra thus giving a natural realization of this largest among sporadic groups this also established a surprising connection between the monster and two dimensional conformal quantum field theory which is the physicists analogue of the theory of vertex operator algebras cf bpz dl fhl fj tk t zf flm

existence of positive solutions for some nonlinear elliptic equations on unbounded domains with cylindrical ends abstract in this paper we study the existence of positive solutions to nonlinear elliptic boundary value problems on unbounded domains r n with cylindrical ends for a general nonlinear term f u including f u u p 1 n 2 as a typical example u u f u u 0 x u 0 u x 0 x by using the mountain pass approach the geometry of plays an important role in our analysis

on mathbb z _p z p orbifold constructions of the moonshine vertex operator algebra for p 3 5 7 13 we consider a mathbb z p orbifold construction of the moonshine vertex operator algebra v natural we show that the vertex operator algebra obtained by the mathbb z p orbifold construction on the leech lattice vertex operator algebra v lambda and a lift of a fixed point free isometry of order p is isomorphic to the moonshine vertex operator algebra v natural we also describe the relationship between those mathbb z p orbifold constructions and the mathbb z 2 orbifold construction in a uniform manner in appendix we give a characterization of the moonshine vertex operator algebra v natural by the existence of an orthogonal pair of ising vectors

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

twisted vertex representations of quantum affine algebras recent interests in quantum groups are stimulated by their marvelous relations with quantum yang baxter equations conformal field theory invariants of links and knots and q hypergeometric series besides understanding the reason of the appearance of quantum groups in both mathematics and theoretical physics there is a natural problem of finding q deformations or quantum analogues of known structures quantum groups were first defined by drinfeld 2 and jimbo 9 also see 4 as a q deformation of the universal enveloping algebras of the kacmoody algebras in the work of trigonometric solutions of yang baxter equations in the same spirit it was shown in 13 14 that there exists a 1 1 correspondence between the integrable highest weight representations of symmetrizable kac moody algebras and those of the corresponding quantum groups where both spaces have the same dimension in the case of generic q i e q is not a root of unity moreover one can be very explicit in the case of quantum gl n to write down the irreducible highest weight representations quantum affine algebras are the quantum groups associated to affine lie algebras following drinfeld s realization 3 of q analog of loop algebras the vertex representation of untwisted simply laced quantum affine algebras was constructed in frenkel jing 6 which is a q deformation of frenkel kac 7 and segal 15 construction in the theory of affine lie algebras subsequently the same was done for the quantum affine algebra of type b in 1 in the present work we construct vertex representations of quantum affine algebras twisted by an automorphism of the dynkin diagram which generalizes certain important cases in the ordinary twisted vertex operator calculus 5

fundamental representations of the m principal realization of gl the m principal realization of gl is introduced and its fundamental representations are defined on the representation space associated with the homogeneous heisenberg subalgebra of a 1 m introduced by frenkel and kac the hirota equations associated with the representations are briefly reviewed the main results concern the reduction of the algebras to the finite rank affine lie algebras gl 1 n and a 1 n 1 where it is shown that a complete nonredundant set of realizations can be obtained this means that the associated hirota equations can be obtained directly from the reduction of the hirota equations for the fundamental m principal representations of gl without requiring the complete nonredundant set of fundamental representations of the lie algebra a 1 n 1 itself

hirota equations associated with simply laced affine lie algebras the cuspidal class e6 of e6 1 in this paper we derive hirota equations associated with the simply laced affine lie algebras g 1 where g is one of the simply laced complex lie algebras an dn e6 e7 or e8 defined by finite order automorphisms of g which we call lepowsky automorphisms in particular we investigate the hirota equations for lepowsky automorphisms of e6 defined by the cuspidal class e6 of the weyl group w e6 of e6 we also investigate the relationship between the lepowsky automorphisms of the simply laced complex lie algebras g and the conjugate canonical automorphisms defined by kac this analysis is applied to identify the canonical automorphisms for the cuspidal class e6 of e6