on copositive matrices abstract a criterion for copositive matrices is given and for n 3 the set of all copositive matrices is determined in terms of matrix elements copositive matrices are applied to the problem of excluding periodic solutions of certain algebraic differential equations

associative monotonic operations in fuzzy set theory abstract the properties of binary operations in a real interval are considered and used in the discussion of generalized operations on fuzzy sets on fuzzy numbers and on fuzzy probabilistic sets

quantum galois theory for compact lie groups abstract we establish a quantum galois correspondence for compact lie groups of automorphisms acting on a simple vertex operator algebra

approximation by cubic c 1 splines on arbitrary triangulations in this paper we present an efficient representation for bivariate piecewise cubicc 1 splines on arbitrary triangulations a numerical method is discussed for computing the dimension of the spaces 3 1 of these splines we consider subspaces ofs 3 1 satisfying certain boundary conditions some applications are given where piecewise cubicc 1 functions are used to solve interpolation problems and least squares approximation problems

modular categories and orbifold models in this paper we try to answer the following question given a modular tensor category with an action of a compact group g is it possible to describe in a suitable sense the quotient category g we give a full answer in the case when is the category of vector spaces in this case g turns out to be the category of representation of drinfeld s double d g this should be considered as the category theory analog of the topological identity pt g bg this implies a conjecture of dijkgraaf vafa e verlinde and h verlinde regarding so called orbifold conformal field theories if is a vertex operator algebra which has a unique irreducible module itself and g is a compact group of automorphisms of and some not too restrictive technical conditions are satisfied then g is finite and the category of representations of the algebra of invariants g is equivalent as a tensor category to the category of representations of drinfeld s double d g we also get some partial results in the non holomorphic case i e when has more than one simple module

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

integrals quantum galois extensions and the affineness criterion for quantum yetter drinfel d modules abstract in this paper we shall generalize the notion of an integral on a hopf algebra introduced by sweedler by defining the more general concept of an integral of a threetuple h a c where h is a hopf algebra coacting on an algebra a and acting on a coalgebra c we prove that there exists a total integral c hom c a of h a c if and only if any representation of h a c is injective in a functorial way as a corepresentation of c in particular the quantum integrals associated to yetter drinfel d modules are defined let now a be an h bicomodule algebra h y d a the category of quantum yetter drinfel d modules and b a a s 1 a 1 a 1 a 0 1h a the subalgebra of coinvariants of the verma structure a h y d a we shall prove the following affineness criterion if there exists h hom h a a total quantum integral and the canonical map a b a h a a b b s 1 b 1 b 1 ab 0 is surjective i e a b is a quantum homogeneous space then the induction functor b a m b h y d a is an equivalence of categories the affineness criteria proven by cline parshall and scott and independently by oberst for affine algebraic groups schemes and schneider in the noncommutative case are recovered as special cases

on principal realization of modules for the affine lie algebra a_1 1 at the critical level we present complete realization of irreducible a 1 1 modules at the critical level in the principal gradation our construction uses vertex algebraic techniques the theory of twisted modules and representations of lie conformal superalgebras we also provide an alternative z algebra approach to this construction all irreducible highest weight a 1 1 modules at the critical level are realized on the vector space m tfrac 1 2 z 1 otimes 2 where m tfrac 1 2 z 1 is the polynomial ring c alpha 1 2 alpha 3 2 explicit combinatorial bases for these modules are also given

twisted modules and g equivariantization in logarithmic conformal field theory a two dimensional chiral conformal field theory can be viewed mathematically as the representation theory of its chiral algebra a vertex operator algebra vertex operator algebras are especially well suited for studying logarithmic conformal field theory in which correlation functions have logarithmic singularities arising from non semisimple modules for the chiral algebra because of the logarithmic tensor category theory of huang lepowsky and zhang in this paper we study not necessarily semisimple or rigid braided tensor categories mathcal c of modules for the fixed point vertex operator subalgebra v g of a vertex operator super algebra v with finite automorphism group g the main results are that every v g module in mathcal c with a unital and associative v action is a direct sum of g twisted v modules for possibly several g in g that the category of all such twisted v modules is a braided g crossed super category and that the g equivariantization of this braided g crossed super category is braided tensor equivalent to the original category mathcal c of v g modules this generalizes results of kirillov and muger proved using rigidity and semisimplicity we also apply the main results to the orbifold rationality problem whether v g is strongly rational if v is strongly rational we show that v g is indeed strongly rational if v is strongly rational g is any finite automorphism group and v g is c 2 cofinite

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

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

vertex operator algebras generalized doubles and dual pairs let v be a simple vertex operator algebra and g a finite automorphism group then there is a natural right g action on the set of all inequivalent irreducible v modules let cal s be a finite set of inequivalent irreducible v modules which is closed under the action of g there is a finite dimensional semisimple associative algebra a alpha g cal s for a suitable 2 cocycle alpha naturally determined by the g action on cal s such that a alpha g cal s and the vertex operator algebra v g form a dual pair on the sum of v modules in cal s in the sense of howe in particular every irreducible v module is completely reducible v g module

quantum galois theory for finite groups dong and mason dm1 initiated a systematic research for a vertex operator algebra with a finite automorphism group which is referred to as the operator content of orbifold models by physicists dvvv the purpose of this paper is to extend one of their main results we will assume that the reader is familiar with the vertex operator algebras voa see b flm throughout this paper v denotes a simple vertex operator algebra g is a finite automorphism group of v c denotes the complex number field and z denotes rational integers let h be a subgroup of g and irr g denote the set of all irreducible cg characters in their paper dm1 they studied the sub voa v h v v h v v for all h h of h invariants and the subspace v on which g acts according to irr g especially they conjectured the following galois correspondence between sub voas of v and subgroups of g and proved it for an abelian or dihedral group g dm1 theorem 1 and later for nilpotent groups dm2 which is an origin of their title of dm1

a law of large numbers for limit order books we define a stochastic model of a two sided limit order book in terms of its key quantities best bid ask price and the standing buy sell volume density for a simple scaling of the discreteness parameters that keeps the expected volume rate over the considered price interval invariant we prove a limit theorem the limit theorem states that given regularity conditions on the random order flow the key quantities converge in probability to a tractable continuous limiting model in the limit model the buy and sell volume densities are given as the unique solution to first order linear hyperbolic pdes specified by the expected order flow parameters we calibrate order flow dynamics to market data for selected stocks and show how our model can be used to derive endogenous shape functions for models of optimal portfolio liquidation under market impact

uniform product of ag n v for an orbifold model v and g twisted zhu algebra abstract let v be a vertex operator algebra and g a finite automorphism group of v for each g g and nonnegative rational number n z g an associative algebra ag n v plays an important role in the theory of vertex operator algebras but the given product in ag n v depends on the eigenspaces of g we show that if v has no negative weights then there is a uniform definition of products on v and we introduce a g twisted zhu algebra ag n v which covers all ag n v let v be a simple vertex operator algebra with no negative weights and let s be a finite set of inequivalent irreducible twisted v modules which is closed under the action of g there is a finite dimensional semisimple associative algebra a g s for a suitable 2 cocycle naturally determined by the g action on s we show that a duality theorem of schur weyl type holds for the actions of a g s and vg on the direct sum of twisted v modules in s as an application of the theory of ag n v it follows as a natural consequence of the result that for any g g every irreducible g twisted v module is a completely reducible vg module

orbifolds of lattice vertex operator algebras at d 48 and d 72 abstract motivated by the notion of extremal vertex operator algebras we investigate cyclic orbifolds of vertex operator algebras coming from extremal even self dual lattices in d 48 and d 72 in this way we construct about one hundred new examples of holomorphic voas with a small number of low weight states

nonnegative volume matching by cubic c 1 splines on clough tocher splits we describe a numerical method to construct volume matching c1 splines satisfying the frequently occurring requirement of nonnegativity the approximated histogram is assumed to be defined on a partition of a polygonal domain into polygonal subdomains our idea consists of applying cubic c1 splines on clough tocher refinements of a triangulation that is compatible with the given partition the proposed algorithm always works successfully for nonnegative histograms

vertex operator algebras and associative algebras letvbe a vertex operator algebra we construct a sequence of associative algebrasan v n 0 1 2 such thatan v is a quotient ofan 1 v and a pair of functors between the category ofan v modules which are notan 1 v modules and the category of admissiblev modules these functors exhibit a bijection between the simple modules in each category we also show thatvis rational if and only if allan v are finite dimensional semisimple algebras

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

basic concepts for a theory of evaluation the aggregative operator abstract starting with an explication of the aggregative concept and deducing a general structure which satisfies a number of minimal requirements properties of clustering the main features of a new mathematical theory called theory of evaluation are developed the theory sheds new light on such well known concepts as membership conjunction and disjunction and seems to be a very promising tool to handle representation problems as they grow from the fields of theory of fuzzy set and its many applications of human decision making and of multicriteria analysis

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

hopf actions on vertex operator algebras ii smash product abstract the smash product v h of a hopf algebra h and an h module vertex operator algebra v are investigated a n v theory and contragredient module theory are founded for v h if h is the group algebra of a finite subgroup of aut v all the irreducible inequivalent admissible v h modules are classified we also prove the maschke theorem in vertex operator algebra version finally we prove weak duality theorems for actions and coactions in vertex operator algebras version

rank one lattice type vertex operator algebras and their automorphism groups ii e series abstract let l be the a1 root lattice and let g be a finite subgroup of au v where v vl is the associated lattice voa in this case aut v psl 2 c the fixed point sub voa vg was studied previously by the authors who found a set of generators and determined the automorphism group when g is cyclic from the a series or dihedral from the d series in the present article we obtain analogous results for the remaining possibilities for g that it belong to the e series g alt4 alt5 or sym4 for such l and g the above vgl may be rational voas

lifts of automorphisms of vertex operator algebras in simple current extensions in this article we study isomorphisms between simple current extensions of a simple voa for example we classify the isomorphism classes of simple current extensions of the voas v sqrt2e 8 and v lambda 16 where 16 is the barnes wall lattice of rank 16 moreover we consider the same simple current extension and describe the normalizer of the abelian automorphism group associated with this extension in particular we regard the moonshine module v natural as simple current extensions of five subvoas v l for 2 elementary totally even lattices l and describe corresponding five normalizers of elementary abelian 2 group in the automorphism group of v natural in terms of v l by using this description we show that three of them form a monster amalgam

block wise construction of commutative increasing monoids abstract construction of new associative commutative and increasing operations on the unit interval from a given operation is proposed it is shown that when is a t norm t conorm or a proper uninorm new t norms t conorms and proper uninorms can be obtained

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

finite element approximation of the volume matching problem optimal orderh 1 andl error bounds are obtained for a continuous piecewise linear finite element approximation of the volume matching problem this problem consists of minimising v 1 2 overv h 1 subject to the inequality constraintv 0 and a number of linear equality constraints the presence of the equality constraints leads to lagrange multipliers which in turn lead to complications with the standard error analysis for variational inequalities finally we consider an algorithm for solving the resulting algebraic problem

tannaka duals in semisimple tensor categories abstract tannaka duals of finite dimensional hopf algebras inside semisimple tensor categories are used to construct orbifold tensor categories which are shown to include the tannaka dual of the dual hopf algebras the second orbifolds are then monoidally equivalent to the initial tensor categories in a canonical fashion

on quantum galois theory the goals of the present paper are to initiate a program to systematically study and rigorously establish what a physicist might refer to as the operator content of orbifold models to explain what this might mean and to clarify the title of the paper we will assume that the reader is familiar with the algebraic formulation of 2 dimensional cft in the guise of vertex operator algebras voa see b flm and dm for more information on this point in the paper dvvv several ideas are proposed concerning the structure of a holomorphic orbifold in other words if v is a holomorphic voa and if g is a finite group of automorphisms of v then the sub voa v g of g invariants is itself a voa and the subject of dvvv is very much concerned with speculation on the nature of the v modules it turns out to be more useful at least for purpose of inductive proofs to take v to be a simple voa we will then see that v g is also simple whenever g is a finite group of automorphisms of v one consequence of our main results is the following

on the tensor structure of modules for compact orbifold vertex operator algebras suppose v g is the fixed point vertex operator subalgebra of a compact group g acting on a simple abelian intertwining algebra v we show that if all irreducible v g modules contained in v live in some braided tensor category of v g modules then they generate a tensor subcategory equivalent to the category mathrm rep g of finite dimensional representations of g with associativity and braiding isomorphisms modified by the abelian 3 cocycle defining the abelian intertwining algebra structure on v additionally we show that if the fusion rules for the irreducible v g modules contained in v agree with the dimensions of spaces of intertwiners among g modules then the irreducibles contained in v already generate a braided tensor category of v g modules these results do not require rigidity on any tensor category of v g modules and thus apply to many examples where braided tensor category structure is known to exist but rigidity is not known for example they apply when v g is c 2 cofinite but not necessarily rational when v g is both c 2 cofinite and rational and v is a vertex operator algebra we use the equivalence between mathrm rep g and the corresponding subcategory of v g modules to show that v is also rational as another application we show that a certain category of modules for the virasoro algebra at central charge 1 admits a braided tensor category structure equivalent to mathrm rep su 2 up to modification by an abelian 3 cocycle