abstract functional differential equations and reaction diffusion systems functional differential equations and reaction diffusion systems r h martin jr and h l smith abstract several fundamental results on the existence and behavior of solutions to semilinear functional differential equations are developed in a banach space setting the ideas are applied to reaction diffusion systems that have time delays in the nonlinear reaction terms the techniques presented here include differential inequalities invariant sets and lyapunov functions and therefore they provide for a wide range of applicability the results on inequalities and especially strict inequalities are new even in the context of semilinear equations whose nonlinear terms do not contain delays several fundamental results on the existence and behavior of solutions to semilinear functional differential equations are developed in a banach space setting the ideas are applied to reaction diffusion systems that have time delays in the nonlinear reaction terms the techniques presented here include differential inequalities invariant sets and lyapunov functions and therefore they provide for a wide range of applicability the results on inequalities and especially strict inequalities are new even in the context of semilinear equations whose nonlinear terms do not contain delays suppose q is a bounded domain in rn with aq smooth and a is the laplacian operator on q also let m be a positive integer z a positive number and f fi m a continuous bounded function from 0 xc x q x c z 0 m into rm where c t 0 is the space of continuous functions from z 0 into r the purpose of this paper is to apply abstract results for semilinear functional differential equations in banach spaces to reactiondiffusion systems with time delays having the form a tu x t diau x t i t x ut x t a xeq i 1 m rdd ai x ui x t au i x t 81 x t t a x eaq u x a 0 x x 0 5 t 0 di 0 and ca q 0 o is c and i q x 0 oo r is c2 here an is the outward normal derivative on aq and if di 0 it is assumed that no boundary conditions are specified for this i also tu x t denotes the partial with respect to t whereas ut x denotes the member of c z 0 defined by 0 u x t 0 u x t 0 m our techniques provide basic existence criteria but the main point is that they can also be effectively applied to obtain estimates for solutions especially received by the editors october 7 1987 and in revised form october 7 1988 1980 mathematics subject classification 1985 revision primary 35r10 34k30

on staggered indecomposable virasoro modules in this article certain indecomposable virasoro modules are studied specifically the virasoro mode l0 is assumed to be nondiagonalizable possessing jordan blocks of rank 2 moreover the module is further assumed to have a highest weight submodule the left module and that the quotient by this submodule yields another highest weight module the right module such modules which have been called staggered have appeared repeatedly in the logarithmic conformal field theory literature but their theory has not been explored in full generality here such a theory is developed for the virasoro algebra using rather elementary techniques the focus centers on two different but related questions typically encountered in practical studies how can one identify a given staggered module and how can one demonstrate the existence of a proposed staggered module given just the values of the highest weights of the left and right modules themselves subject to simple necessary conditions invariants are defined which together with the knowledge of the left and right modules uniquely identify a staggered module the possible values of these invariants form a vector space of dimension 0 1 or 2 and the structures of the left and right modules limit the isomorphism classes of the corresponding staggered modules to an affine subspace possibly empty the number of invariants and affine restrictions is purely determined by the structures of the left and right modules moreover in order to facilitate applications the expressions for the invariants and restrictions are given by formulas as explicit as possible they generally rely on expressions for virasoro singular vectors finally the text is liberally peppered throughout with examples illustrating the general concepts these have been carefully chosen for their physical relevance or for the novel features they exhibit

braid matrices and structure constants for minimal conformal models using the feigin fuchs representation of minimal conformal models in a form introduced recently by one of us the braid group representation matrices describing the analytic continuation properties of conformal blocks are computed in a suitable normalization their matrix elements are shown to essentially factorize into pairs of boltzmann weights of critical rsos models in a certain limit of the spectral parameter these boltzmann weights are related to quantum groupr matrices by the vertex sos transformation we show that the crossing symmetry of the four point function in left right symmetric models follows from a quantum group relation also called crossing symmetry this observation gives a simple way to evaluate the structure constants

fusion of irreducible modules in cal wlm p p prime based on symmetry principles we derive a fusion algebra generated from repeated fusions of the irreducible modules appearing in the extended logarithmic minimal model in addition to the irreducible modules themselves closure of the commutative and associative fusion algebra requires the participation of a variety of reducible yet indecomposable modules we conjecture that this fusion algebra is the same as the one obtained by application of the nahm gaberdiel kausch algorithm and find that it reproduces the known such results for and for p 1 this fusion algebra does not contain a unit requiring that the spectrum of modules is invariant under a natural notion of conjugation however introduces additional p 1 p 1 reducible yet indecomposable rank 1 modules among which the identity is found still yielding a well defined fusion algebra in this greater fusion algebra the aforementioned symmetries are generated by fusions with the three irreducible modules of conformal weights kp 1 1 k 1 2 3 we also identify polynomial fusion rings associated with our fusion algebras

nonmeromorphic operator product expansion and c2 cofiniteness for a family of w algebras we prove the existence and associativity of the nonmeromorphic operator product expansion for an infinite family of vertex operator algebras the triplet algebras using results from p z tensor product theory while doing this we also show that all these vertex operator algebras are c2 cofinite

from the representation theory of vertex operator algebras to modular tensor categories in conformal field theory two dimensional conformal quantum field theory cft has inspired an immense amount of mathematics and has interacted with mathematics in very rich ways in great part through the mathematically dynamic world of string theory one notable example of this interaction is provided by verlinde s conjecture e verlinde 1 conjectured that certain matrices formed by numbers called the fusion rules in a rational cft are diagonalized by the matrix given by a certain natural action of a fundamental modular transformation essentially a certain distinguished element of the group of two by two matrices of determinant one with integer entries his conjecture led him to the verlinde formula for the fusion rules and more generally for the dimensions of spaces of conformal blocks on riemann surfaces of arbitrary genera a great deal of progress has been achieved in interpreting and proving verlinde s physical conjecture and the verlinde formula in mathematical settings in the case of the wess zumino novikov witten models in cft which are based on affine lie algebras on the other hand moore and seiberg 2 3 showed on a physical level of rigor that the general form of the verlinde conjecture is a consequence of the axioms for rational cfts thereby providing a conceptual understanding of the conjecture in the process they formulated a cft analogue later termed modular tensor category discussed in refs 4 and 5 by i frenkel of the classical

radford drinfeld and cardy boundary states in the 1 p logarithmic conformal field models we introduce p 1 pseudocharacters in the space of 1 p model vacuum torus amplitudes to complete the distinguished basis in the 2p dimensional fusion algebra to a basis in the whole 3p 1 dimensional space of torus amplitudes and the structure constants in this basis are not necessarily non negative integer numbers we obtain a generalized verlinde formula that gives these structure constants in the context of theories with boundaries we identify the space of vacuum torus amplitudes with the space of ishibashi states then we propose 3p 1 boundary states satisfying the cardy condition

conformal algebras of two dimensional disordered systems we discuss the structure of two dimensional conformal field theories at a central charge c 0 describing critical disordered systems polymers and percolation we construct a novel extension of the c 0 virasoro algebra characterized by a number b measuring the effective number of massless degrees of freedom and by a logarithmic partner of the stress tensor it is argued to be present at a generic random critical point lacking super kac moody or other higher symmetries and is a tool to describe and classify such theories interestingly this algebra is not only consistent with but indeed naturally accommodates in general an underlying global supersymmetry polymers and percolation realize this algebra unexpectedly we find that the c 0 kac table of the degenerate fields contains two distinct theories with b 5 6 and b 5 8 which we conjecture to correspond to percolation and polymers respectively a given kac table field can be degenerate only in one of them remarkably we also find this algebra and thereby an ensuing hidden supersymmetry realized at general replica averaged critical points for which we derive an explicit formula for b

modular group representations and fusion in logarithmic conformal field theories and in the quantum group center the sl 2 representation on the center of the restricted quantum group open image in new window at the primitive 2pth root of unity is shown to be equivalent to the sl 2 representation on the extended characters of the logarithmic 1 p conformal field theory model the multiplicative jordan decomposition of the open image in new window ribbon element determines the decomposition of into a pointwise product of two commuting sl 2 representations one of which restricts to the grothendieck ring this restriction is equivalent to the sl 2 representation on the 1 p characters related to the fusion algebra via a nonsemisimple verlinde formula the grothendieck ring of open image in new window at the primitive 2pth root of unity is shown to coincide with the fusion algebra of the 1 p logarithmic conformal field theory model as a by product we derive q binomial identities implied by the fusion algebra realized in the center of open image in new window

fusion matrices generalized verlinde formulas and partition functions in cal wlm 1 p the infinite series of logarithmic minimal models is considered in the extended picture where they are denoted by as in the rational models the fusion algebra of is described by a simple graph fusion algebra the corresponding fusion matrices are mutually commuting but in general not diagonalizable nevertheless they can be simultaneously brought to jordan form by a similarity transformation the spectral decomposition of the fusion matrices is completed by a set of refined similarity matrices converting the fusion matrices into jordan canonical form consisting of jordan blocks of rank 1 2 or 3 the various similarity transformations and jordan forms are determined from the modular data this gives rise to a generalized verlinde formula for the fusion matrices its relation to the partition functions in the model is discussed in a general framework by application of a particular structure matrix and its moore penrose inverse this verlinde formula reduces to the generalized verlinde formula for the associated grothendieck ring

polynomial fusion rings of w extended logarithmic minimal models the countably infinite number of virasoro representations of the logarithmic minimal model lm p p can be reorganized into a finite number of w representations with respect to the extended virasoro algebra symmetry w using a lattice implementation of fusion we recently determined the fusion algebra of these representations and found that it closes albeit without an identity for p 1 here we provide a fusion matrix realization of this fusion algebra and identify a fusion ring isomorphic to it we also consider various extensions of it and quotients thereof and introduce and analyze commutative diagrams with morphisms between the involved fusion algebras and the corresponding quotient polynomial fusion rings one particular extension is reminiscent of the fundamental fusion algebra of lm p p and offers a natural way of introducing the missing identity for p 1 working out explicit fusion matrices is facilitated by a further enlargement based on a pair of mutual moore penrose inverses intertwining bet

virasoro representations and fusion for general augmented minimal models in this paper we present explicit results for the fusion of irreducible and higher rank representations in two logarithmically conformal models the augmented c2 3 0 model as well as the augmented yang lee model at c2 5 22 5 we analyse their spectrum of representations which is consistent with the symmetry and associativity of the fusion algebra we also describe the first few higher rank representations in detail in particular we present the first examples of consistent rank 3 indecomposable representations and describe their embedding structure knowing these two generic models we also conjecture the general representation content and fusion rules for general augmented cp q models

rationality of fuzzy reciprocal preference relations we study relationships between different transitivity and acyclicity conditions for fuzzy reciprocal relations such conditions may be treated as rationality conditions for preferences described by such kind of relations

w extended fusion algebra of critical percolation two dimensional critical percolation is the member lm 2 3 of the infinite series of yang baxter integrable logarithmic minimal models lm p p we consider the continuum scaling limit of this lattice model as a rational logarithmic conformal field theory with extended w w2 3 symmetry and use a lattice approach on a strip to study the fundamental fusion rules in this extended picture we find that the representation content of the ensuing closed fusion algebra contains 26 w indecomposable representations with eight rank1 representations 14 rank 2 representations and four rank 3 representations weidentifytheserepresentationswithsuitablelimitsofyang baxterintegrable boundary conditions on the lattice and obtain their associated w extended characters the latter decompose as finite non negative sums of w irreducible characters of which 13 are required implementation of fusion on the lattice allows us to read off the fusion rules governing the fusion algebra of the 26 representations and to construct an explicit cayley table the closure of these representations among themselves under fusion is remarkable confirmation of the proposed extended symmetry

integrable boundary conditions and w extended fusion in the logarithmic minimal models lm 1 p we consider the logarithmic minimal models lm 1 p as rational logarithmic conformal field theories with extended w symmetry to make contact with the extended picture starting from the lattice we identify 4p 2 boundary conditions as specific limits of integrable boundary conditions of the underlying yang baxter integrable lattice models specifically we identify 2p integrable boundary conditions to match the 2p known irreducible w representations these 2p extended representations naturally decompose into infinite sums of the irreducible virasoro representations r s a further 2p 2 reducible yet indecomposable w representations of rank 2 are generated by fusion and these decompose as infinite sums of indecomposable rank 2 virasoro representations the fusion rules in the extended picture are deduced from the known fusion rules for the virasoro representations of lm 1 p and are found to be in agreement with previous works the closure of the fusion algebra on a finite number of representations in the extended picture is remarkable confirmation of the consistency of the lattice approach comment 15 pages

on the structure of unitary conformal field theory ii representation theoretic approach two dimensional unitary rational conformal field theory is studied from the point of view of the representation theory of chiral algebras chiral algebras are equipped with a family of co multiplications which serve to define tensor product representations chiral vertices arise as clebsch gordan operators from tensor product representations to irreducible subrepresentations of a chiral algebra the algebra of chiral vertices is studied and shown to give rise to representations of the braid groups determined by yang baxter braid matrices chiral fusion is analyzed it is shown that the braid and fusion matrices determine invariants of knots and links connections between the representation theories of chiral algebras and of quantum groups are sketched finally it is shown how the local fields of a conformal field theory can be reconstructed from the chiral vertices of two chiral algebras

on the inverse scattering problem on branching graphs the inverse scattering problem on branching graphs is studied the definition of the schrodinger operator on such graphs is discussed the operator is defined with real potentials with finite first momentum and using special boundary conditions connecting values of the functions at the vertices it is shown that in general the scattering matrix does not determine the topology of the graph the potentials on the edges and the boundary conditions uniquely

grothendieck ring and verlinde formula for the w extended logarithmic minimal model wlm 1 p we consider the grothendieck ring of the fusion algebra of the w extended logarithmic minimal model wlm 1 p informally this is the fusion ring of w irreducible characters so it is blind to the jordan block structures associated with reducible yet indecomposable representations as in the rational models the grothendieck ring is described by a simple graph fusion algebra the 2p dimensional matrices of the regular representation are mutually commuting but not diagonalizable they are brought simultaneously to jordan form by the modular data coming from the full 3p 1 dimensional s matrix which includes transformations of the p 1 pseudo characters the spectral decomposition yields a verlinde like formula that is manifestly independent of the modular parameter tau but is in fact equivalent to the verlinde like formula recently proposed by gaberdiel and runkel involving a tau dependent s matrix comment 13 pages v2 example comments and references added

fusion algebras of logarithmic minimal models we present explicit conjectures for the chiral fusion algebras of the logarithmic minimal models lm p p considering virasoro representations with no enlarged or extended symmetry algebra the generators of fusion are countably infinite in number but the ensuing fusion rules are quasi rational in the sense that the fusion of a finite number of representations decomposes into a finite direct sum of representations the fusion rules are commutative associative and exhibit an s 2 structure but require so called kac representations which are typically reducible yet indecomposable representations of rank 1 in particular the identity of the fundamental fusion algebra p 1 is a reducible yet indecomposable kac representation of rank 1 we make detailed comparisons of our fusion rules with the results of gaberdiel and kausch for p 1 and with eberle and flohr for p p 2 5 corresponding to the logarithmic yang lee model in the latter case we confirm the appearance of indecomposable representations of rank 3 we also find that closure of a fundamental fusion algebra is achieved without the introduction of indecomposable representations of rank higher than 3 the conjectured fusion rules are supported within our lattice approach by extensive numerical studies of the associated integrable lattice models details of our lattice findings and numerical results will be presented elsewhere the agreement of our fusion rules with the previous fusion rules lends considerable support for the identification of the logarithmic minimal models lm p p with the augmented cp p minimal models defined algebraically

logarithmic torus amplitudes abstractfor the example of the logarithmic triplet theory at c 2 the chiral vacuumtorus amplitudes are analysed it is found that the space of these torus amplitudes isspanned by the characters of the irreducible representations as well as a function thatcan be associated to the logarithmic extension of the vacuum representation a fewimplications and generalisations of this result are discussed 1 introduction during the last twenty years much has been understood about the struc ture of rational conformal eld theories rational conformal eld theories are characterisedby the property that they have only nitely many irreducible highest weight representationsof the chiral algebra or vertex operator algebra and that every highest weight represen tation is completely decomposable into irreducible representations the structure of thesetheories is well understood in particular the characters of the irreducible representationstransform into one another under modular transformations 1 see also 2 and the modulars matrix determines the fusion rules via the verlinde formula 3 a general proof for thishas only recently been given in 4 on the other hand it is clear that rational conformal eld theories are rather special andit is therefore important to understand the structure of more general classes of conformal eld theories one such class are the rational logarithmic theories that possess only nitelymany indecomposable representations but for which not all highest weight representationsare completely decomposable the name logarithmic comes from the fact that their chiralcorrelation functions typically have logarithmic branch cuts the rst example of a non rational logarithmic conformal eld theory was found in 5 see also 6 and the rstrational example that shall also concern us in this paper was constructed in 7 for somerecent reviews see 8 9 10 from a physics point of view logarithmic conformal eldtheories appear naturally in various models of statistical physics for example in the theory

a modular invariant bulk theory for the c 0 triplet model a proposal for the bulk space of the logarithmic cal w 2 3 triplet model at central charge zero is made the construction is based on the idea that one may reconstruct the bulk theory of a rational conformal field theory from its boundary theory the resulting bulk space is a quotient of the direct sum of projective representations which is isomorphic as a vector space to the direct sum of tensor products of the irreducible representations with their projective covers as a consistency check of our analysis we show that the partition function of the bulk theory is modular invariant and that the boundary state analysis is compatible with the proposed annulus partition functions of this model

a crossing probability for critical percolation in two dimensions langlands et al considered two crossing probabilities and in their extensive numerical investigations of critical percolation in two dimensions cardy was able to find the exact form of by treating it as a correlation function of boundary operators in the limit of the q state potts model we extend his results to find an analogous formula for which compares very well with the numerical results