extensions of positive definite integral kernels on the heisenberg group abstract we consider positive definite extensions of functions and distributions which are defined on a cylindrical neighborhood and positive definite in the heisenberg group when rotational invariance is assumed we show existence of p d extensions we also give formulas of the classical bochner type for all such extensions they are expanded in terms of an integral transform associated to the sub laplacian we further show that the p d extension problem is closely related to the theory of extensions of hermitian infinitesimal representations in the sense considered earlier by powers and fuglede these are extensions to integrable representations in generally larger so called dilation hilbert spaces

conformal field theory and elliptic cohomology abstract in this paper we use conformal field theory to construct a generalized cohomology theory which has some properties of elliptic cohomology theory which was some properties of elliptic cohomology a part of our presentation is a rigorous definition of conformal field theory following segal s axioms and some examples such as lattice theories associated with a unimodular even lattice we also include certain examples and formulate conjectures on modular forms and monstrous moonshine related to the present work

some new characters on the wire tap channel of type ii the noiseless wire tap channel of type ii with coset coding scheme was provided by ozarow and wyner in this correspondence the user is split into multiple parties who are coordinated in coding their data symbols by using the same encoder the adversary can tap not only partial transmitted symbols but also partial data symbols we are interested in the equivocation of the data symbols to this adversary who has more power than that of ozarow and wyner the generalized hamming weight of wei and the dimension length profile dlp of forney are extended to two code formats relative generalized hamming weight and relative dimension length profile rdlp upper and lower bounds of the new concepts are investigated they are useful to design a perfect secrecy coding scheme for the coordinated multiparty model under a general secrecy standard the coordinated model can provide a higher transmission rate than an uncoordinated time sharing model

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

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

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

relative generalized hamming weights of one point algebraic geometric codes security of linear ramp secret sharing schemes can be characterized by the relative generalized hamming weights of the involved codes 26 24 in this paper we elaborate on the implication of these parameters and we devise a method to estimate their value for general one point algebraic geometric codes as it is demonstrated for hermitian codes our bound is often tight furthermore for these codes the relative generalized hamming weights are often much larger than the corresponding generalized hamming weights

global anomalies on orbifolds we consider global anomalies for heterotic string theory formulated on orbifolds the vanishing of certain characteristic classes in group cohomology provides sufficient conditions for the absence of global anomalies for abelian orbifolds level matching implies these cohomology conditions so suffices for the absence of anomalies for nonabelian orbifolds level matching does not suffice and there are additional constraints we give some examples to illustrate these new constraints

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

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

relative generalized matrix weights of matrix codes for universal security on wire tap networks universal security over a network with linear network coding has been intensively studied however previous linear codes and code pairs used for this purpose were linear over a larger field than that used on the network which restricts the possible packet lengths of optimal universal secure codes does not allow to apply known list decodable rank metric codes and requires performing operations over a large field in this paper we introduce new parameters relative generalized matrix weights and relative dimension rank support profile for code pairs that are linear over the field used in the network and show that they measure the universal security performance of these code pairs for one code and non square matrices generalized matrix weights coincide with the existing delsarte generalized weights hence we prove the connection between these latter weights and secure network coding which was left open as main applications the proposed new parameters enable us to 1 obtain optimal universal secure linear codes on noiseless networks for all possible packet lengths in particular for packet lengths not considered before 2 obtain the first universal secure list decodable rank metric code pairs with polynomial sized lists based on a recent construction by guruswami et al and 3 obtain new characterizations of security equivalences of linear codes finally we show that our parameters extend relative generalized hamming weights and relative dimension length profile respectively and relative generalized rank weights and relative dimension intersection profile respectively

on the similarities between generalized rank and hamming weights and their applications to network coding rank weights and generalized rank weights have been proved to characterize error and erasure correction and information leakage in linear network coding in the same way as hamming weights and generalized hamming weights describe classical error and erasure correction and information leakage in wire tap channels of type ii and code based secret sharing although many similarities between both the cases have been established and proved in the literature many other known results in the hamming case such as bounds or characterizations of weight preserving maps have not been translated to the rank case yet or in some cases have been proved after developing a different machinery the aim of this paper is to further relate both weights and generalized weights show that the results and proofs in both cases are usually essentially the same and see the significance of these similarities in network coding some of the new results in the rank case also have new consequences in the hamming case

improved bounds on the threshold gap in ramp secret sharing in this paper we consider linear secret sharing schemes over a finite field mathbb f q where the secret is a vector in mathbb f q ell and each of the n shares is a single element of mathbb f q we obtain lower bounds on the so called threshold gap g of such schemes defined as the quantity r t where r is the smallest number such that any subset of r shares uniquely determines the secret and t is the largest number such that any subset of t shares provides no information about the secret our main result establishes a family of bounds which are tighter than previously known bounds for ell geq 2 furthermore we also provide bounds in terms of n and q on the partial reconstruction and privacy thresholds a more fine grained notion that considers the amount of information about the secret that can be contained in a set of shares of a given size finally we compare our lower bounds with known upper bounds in the asymptotic setting

representations of the heisenberg algebra on a riemann surface we formulate reflection positivity for meromorphic functions and for 1 forms on a riemann surface this construction yields representations of the heisenberg algebra on a riemann surface

relative generalized rank weight of linear codes and its applications to network coding by extending the notion of minimum rank distance this paper introduces two new relative code parameters of a linear code mathcal c 1 of length n over a field extension mathbb f q m and its subcode mathcal c 2 subsetneqq mathcal c 1 one is called the relative dimension intersection profile rdip and the other is called the relative generalized rank weight rgrw we clarify their basic properties and the relation between the rgrw and the minimum rank distance as applications of the rdip and the rgrw the security performance and the error correction capability of secure network coding guaranteed independently of the underlying network code are analyzed and clarified we propose a construction of secure network coding scheme and analyze its security performance and error correction capability as an example of applications of the rdip and the rgrw silva and kschischang showed the existence of a secure network coding in which no part of the secret message is revealed to the adversary even if any dim mathcal c 1 1 links are wiretapped which is guaranteed over any underlying network code however the explicit construction of such a scheme remained an open problem our new construction is just one instance of secure network coding that solves this open problem

unbounded elements affiliated with c algebras and non compact quantum groups the affiliation relation that allows to include unbounded elements operators into thec algebra framework is introduced investigated and applied to the quantum group theory the quantum deformation of the two fold covering of the group of motions of euclidean plane is constructed a remarkable radius quantization is discovered it is also shown that the quantumsu 1 1 group does not exist on thec algebra level for real value of the deformation parameter

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

monstrous moonshine and monstrous lie superalgebras we prove conway and norton s moonshine conjectures for the infinite dimensional representation of the monster simple group constructed by frenkel lepowsky and meurman to do this we use the no ghost theorem from string theory to construct a family of generalized kac moody superalgebras of rank 2 which are closely related to the monster and several of the other sporadic simple groups the denominator formulas of these superalgebras imply relations between the thompson functions of elements of the monster i e the traces of elements of the monster on frenkel lepowsky and meurman s representation which are the replication formulas conjectured by conway and norton these replication formulas are strong enough to verify that the thompson functions have most of the moonshine properties conjectured by conway and norton and in particular they are modular functions of genus 0 we also construct a second family of kac moody superalgebras related to elements of conway s sporadic simple group col these superalgebras have even rank between 2 and 26 for example two of the lie algebras we get have ranks 26 and 18 and one of the superalgebras has rank 10 the denominator formulas of these algebras give some new infinite product identities in the same way that the denominator formulas of the a n n e kac moody algebras give the macdonald identities

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

modular functions arising from some finite groups to each element m of the fischer griess monster group m and conjectured that these functions are hauptmoduls for certain genus zero modular groups we have found for a large number of values of n all the genus zero groups between r0 n and psl 2 r that have hauptmoduls of the above form and this provides the necessary verification that the series assigned in 21 to particular elements of m really are such hauptmoduls atkin and fong 11 have recently verified that h m really is a character of m for all n we compute thompson series for various finite groups and discuss the differences between these groups and m we find that the resulting thompson series are all hauptmoduls for suitable genus zero subgroups of psl 2 r

bounded k spherical functions on heisenberg groups abstract let hn be the 2n 1 dimensional heisenberg group and let k be a compact subgroup of aut hn the group of automorphisms of hn the pair k hn is called a gelfand pair if lk1 hn the subalgebra of elements of l1 hn that are invariant under the action of k is commutative in this case the continuous homomorphisms on lk1 hn are given by integrating against certain k invariant functions on hn these functions are the k spherical functions associated to the gelfand pair k hn in this paper we show how to compute the bounded k spherical functions on hn

dimension length profiles and trellis complexity of linear block codes this semi tutorial paper discusses the connections between the dimension length profile dlp of a linear code which is essentially the same as its generalized hamming weight hierarchy and the complexity of its minimal trellis diagram these connections are close and deep dlp duality is closely related to trellis duality the dlp of a code gives tight bounds on its state and branch complexity profiles under any coordinate ordering these bounds can often be met a maximum distance separable mds code is characterized by a certain extremal dlp from which the main properties of mds codes are easily derived the simplicity and generality of these interrelationships are emphasized

semantic security capacity for wiretap channels of type ii the secrecy capacity of the type ii wiretap channel wtc ii with a noisy main channel is currently an open problem herein its secrecy capacity is derived and shown to be equal to its semantic security ss capacity in this setting the legitimate users communicate via a discrete memoryless dm channel in the presence of an eavesdropper that has perfect access to a subset of its choosing of the transmitted symbols constrained to a fixed fraction of the blocklength the secrecy criterion is achieved simultaneously for all possible eavesdropper subset choices the ss criterion demands negligible mutual information between the message and the eavesdropper s observations even when maximized over all message distributions a key tool for the achievability proof is a novel and stronger version of wyner s soft covering lemma specifically a random codebook is shown to achieve the soft covering phenomenon with high probability the probability of failure is doubly exponentially small in the blocklength since the combined number of messages and subsets grows only exponentially with the blocklength ss for the wtc ii is established by using the union bound and invoking the stronger soft covering lemma the direct proof shows that rates up to the weak secrecy capacity of the classic wtc with a dm erasure channel ec to the eavesdropper are achievable the converse follows by establishing the capacity of this dm wiretap ec as an upper bound for the wtc ii from a broader perspective the stronger soft covering lemma constitutes a tool for showing the existence of codebooks that satisfy exponentially many constraints a beneficial ability for many other applications in information theoretic security

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

fricke lie algebras and the genus zero property in moonshine we give a new simpler proof that the canonical actions of finite groups on fricke type monstrous lie algebras yield genus zero functions in generalized monstrous moonshine using a borcherds kac moody lie algebra decomposition due to jurisich we describe a compatibility condition arising from the no ghost theorem in bosonic string theory that yields the genus zero property we give evidence for and against the conjecture that such a compatibility for symmetries of the monster lie algebra gives a characterization of the monster group

chains demi matroids and profiles it is shown that each chain of linear codes has an associated demi matroid a combinatorial structure that extends the notion of a vector matroid of a linear code these demi matroids are proven to determine important properties of the chain and it is shown that linear code chain duality is represented by demi matroid duality in a natural and yet surprising way profiles are defined for arbitrary demi matroids and thus for linear code chains which generalizes previous results in the literature and provides simple and transparent proofs thereof finally applications are given to encryption of messages through wire tap channels of type ii

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

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

new code equivalence based on relative generalized hamming weights code equivalence is a basic concept in coding theory the well known theorem by macwilliams gives a sufficient condition for code equivalence recently the macwilliams theorem has been generalized by fan liu and puig making use of the generalized hamming weights ghws in this paper we will present a further generalization of the macwilliams theorem our result extends both the macwilliams theorem and the result by fan liu and puig we will first define relative subcodes of a linear code based on the relative generalized hamming weights rghws which is a generalization of the ghws and then establish a method based on finite projective geometry to characterize relative subcodes using this method we will prove our main result

generalized hamming weights for linear codes motivated by cryptographical applications the algebraic structure of linear codes from a new perspective is studied by viewing the minimum hamming weight as a certain minimum property of one dimensional subcodes a generalized notion of higher dimensional hamming weights is obtained these weights characterize the code performance on the wire tap channel of type ii basic properties of generalized weights are derived the values of these weights for well known classes of codes are determined and lower bounds on code parameters are obtained several open problems are also listed

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

on the second relative greedy weight of 4 dimensional codes based on applications to the wire tap channel of type ii with the coset coding scheme we will define optimal codes which are described as the ones achieving upper bounds on e 2 m 2 where e 2 and m 2 are the second relative greedy weight and the second relative generalized hamming weight respectively by using the finite geometry method we will first classify all the 4 dimensional codes with respect to their k 1 dimensional subcodes for k 1 1 and k 1 2 and determine the upper bounds on e 2 m 2 for each class of codes and then obtain optimal codes by explicitly constructing the value assignments which achieve upper bounds on e 2 m 2

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

further results on the semilinear equivalence of linear codes based on relative subcodes we address an equivalent condition for a one to one semilinear mapping between two linear codes to be in fact a semimonomial transformation that is the underlying two codes are semilinearly equivalent the result in the present paper substantially improves the equivalent condition in recent literatures moreover it also generalizes the well known macwilliams theorem of code equivalence