attractors and asymptotic stability for fuzzy dynamical systems abstract in this work we study the asymptotic properties of maps on fuzzy spaces which are extensions of maps on the main results are in section 4 see theorem 21 and we give an illustrative example in the last section

numerical simulation of three dimensional nonlinear water waves we present an accurate and efficient numerical model for the simulation of fully nonlinear non breaking three dimensional surface water waves on infinite or finite depth as an extension of the work of craig and sulem 19 the numerical method is based on the reduction of the problem to a lower dimensional hamiltonian system involving surface quantities alone this is accomplished by introducing the dirichlet neumann operator which is described in terms of its taylor series expansion in homogeneous powers of the surface elevation each term in this taylor series can be computed efficiently using the fast fourier transform an important contribution of this paper is the development and implementation of a symplectic implicit scheme for the time integration of the hamiltonian equations of motion as well as detailed numerical tests on the convergence of the dirichlet neumann operator the performance of the model is illustrated by simulating the long time evolution of two dimensional steadily progressing waves as well as the development of three dimensional short crested nonlinear waves both in deep and shallow water

cones and decomposition of sub and supermartingales abstract we study sub and supermartingales with values in the class f c of all convex fuzzy subsets of a banach space having compact support the terms sub and super are understood with respect to a partial order not necessarily inclusion in this general framework orders with respect to which a doob type decomposition can hold are naturally associated to convex subcones with a special property termed f c properness decomposition theorems into the sum or difference of a martingale and a compensator are proven in the sub case the martingale need exist only in a superspace of f c

convergence of conditional expectations and strong laws of large numbers for multivalued random variables on etablit des lemmes de fatou et des theoremes de convergence de lebesgue pour des esperances conditionnelles multivaluees de variables aleatoires a valeurs dans des sous ensembles fermes d un espace de banach separable

fuzzy set valued gaussian processes and brownian motions gaussian processes and brownian motion are concepts and tools in modelling important uncertain systems in many areas in view of uncertainty complexity in many real world problems we extend these tools to the case where stochastic processes can take on fuzzy sets as values in this paper we discuss fuzzy set valued gaussian processes based on the results of s li y ogura v kreinovich limit theorems and applications of set valued and fuzzy set valued random variables kluwer academic publishers dordrecht 2002 s li y ogura h t nguyen gaussian processes and martingales for fuzzy valued variables with continuous parameter inform sci 133 2001 7 21 s li y ogura f n proske m l puri central limit theorems for generalized set valued random variables j math anal appl 285 2003 250 263 n n lyashenko on limit theorems for sums of independent compact random subsets in the euclidean space j soviet math 20 1982 2187 2196 and m l puri d a ralescu the concept of normality for fuzzy random variables ann probab 13 1985 1373 1379 we also introduce the concept of fuzzy set valued brownian motion and then prove several properties of such processes

mappin class group actions on quantum doubles we study representations of the mapping class group of the punctured torus on the double of a finite dimensional possibly non semisimple hopf algebra that arise in the construction of universal extended topological field theories we discuss how for doubles the degeneracy problem of tqft s is circumvented we find compact formulae for thes 1 matrices using the canonical non degenerate forms of hopf algebras and the bicrossed structure of doubles rather than monodromy matrices a rigorous proof of the modular relations and the computation of the projective phases is supplied using radford s relations between the canonical forms and the moduli of integrals we analyze the projectivesl 2 z action on the center ofuq sl2 forq anl 2m 1st root of unity it appears that the 3m 1 dimensional representation decomposes into anm 1 dimensional finite representation and a2m dimensional irreducible representation the latter is the tensor product of the two dimensional standard representation ofsl 2 z and the finite m dimensional representation obtained from the truncated tqft of the semisimplified representation category ofuq sl2

estimation of a fuzzy regression model using fuzzy distances regression analysis is a powerful statistical tool that has many applications in different areas the problem of regression analysis under a fuzzy environment has been treated in the literature from different points of view and considering a variety of input output data crisp or fuzzy however we realize that in general most research papers have a conflict between the solution of the fuzzy regression problem using crisp distances minimizing a real error function and the interpretation of fuzzy data as possibility distributions the main aim of this paper is to develop a methodology to solve this problem introducing a fuzzy partial order and a family of fuzzy distance measures on the whole set of fuzzy numbers the new approach allows us to obtain linear and nonlinear models that reach the lowest fuzzy error the estimation process in general can be considered easier to apply in practice and it is not limited to triangular fuzzy numbers numerical examples are provided to illustrate the usefulness and applicability of these results and comparisons with existing methodologies show that the performance of the proposed solution is very satisfactory

a high order spectral method for nonlinear water waves over moving bottom topography we present a numerical method for simulations of nonlinear surface water waves over variable bathymetry it is applicable to either two or three dimensional flows as well as to either static or moving bottom topography the method is based on the reduction of the problem to a lower dimensional hamiltonian system involving boundary quantities alone a key component of this formulation is the dirichlet neumann operator which in light of its joint analyticity properties with respect to surface and bottom deformations is computed using its taylor series representation we present new stabilized forms for the taylor terms each of which is efficiently computed by a pseudospectral method using the fast fourier transform physically relevant applications are displayed to illustrate the performance of the method comparisons with analytical solutions and laboratory experiments are provided

uncertain alternating renewal process and its application uncertain process is a sequence of uncertain variables indexed by time and space first this paper presents a kind of uncertain process known as the uncertain alternating renewal process whose alternating interarrival times are uncertain variables then it proves an uncertain alternating renewal theorem on the limit value of average working rate finally an application of the alternating renewal theorem is discussed

convergence theorem for fuzzy martingales abstract we study fuzzy set valued measures in a banach space and their relationships to fuzzy random variables our main result is a convergence theorem for fuzzy martingales our tool is a radon nikodym type theorem for fuzzy measures which are absolutely continuous with respect to probability measures

characterization of compact subsets of fuzzy sets a characterization of compact subsets is presented for the metric space of normal fuzzy convex fuzzy sets on the base space rn the metric for which is the supremum over the hausdorff distances between corresponding level sets it is shown that a closed subset is compact if and only if it is uniformly support bounded and the corresponding set of support functions is equileftcontinuous in the membership grade variable

convergence of set valued and fuzzy valued martingales the purpose of this paper is to prove the convergence theorems of set valued and fuzzy valued martingales in kuratowski mosco sense without assuming that their values are compact or of compact level sets

conformal field theory correlations in the abelian sandpile model we calculate all multipoint correlation functions of all local bond modifications in the two dimensional abelian sandpile model both at the critical point and in the model with dissipation the set of local bond modifications includes as the most physically interesting case all weakly allowed cluster variables the correlation functions show that all local bond modifications have scaling dimension 2 and can be written as linear combinations of operators in the central charge ensuremath 2 logarithmic conformal field theory in agreement with a form conjectured earlier by mahieu and ruelle in phys rev e 64 066130 2001 we find closed form expressions for the coefficients of the operators and describe methods that allow their rapid calculation we determine the fields associated with adding or removing bonds both in the bulk and along open and closed boundaries some bond defects have scaling dimension 2 while others have scaling dimension 4 we also determine the corrections to bulk probabilities for local bond modifications near open and closed boundaries

symplectic fermions and a quasi hopf algebra structure on bar u _i sl 2 abstract we consider the finite dimensional restricted quantum group u q s l 2 at q i we show that u i s l 2 does not allow for a universal r matrix even though u v v u holds for all finite dimensional representations u v of u i s l 2 we then give an explicit coassociator and a universal r matrix r such that u i s l 2 becomes a quasi triangular quasi hopf algebra our construction is motivated by the two dimensional chiral conformal field theory of symplectic fermions with central charge c 2 there a braided monoidal category sf has been computed from the factorisation and monodromy properties of conformal blocks and we prove that rep u i s l 2 r is braided monoidally equivalent to sf

separability for graph convergence of sequences of fuzzy valued random variables abstract we prove that the graph convergence in hausdorff metric or kuratowski mosco topology of a sequence of fuzzy sets follows from the convergence of the sequences of the level sets for countable dense levels as an application we give a strong law of large numbers for fuzzy valued random variables including the case when the level sets may not be bounded

hamiltonian long wave expansions for water waves over a rough bottom this paper is a study of the problem of nonlinear wave motion of the free surface of a body of fluid with a periodically varying bottom the object is to describe the character of wave propagation in a long wave asymptotic regime extending the results of r rosales g papanicolaou 1983 stud appl math 68 89 102 on periodic bottoms for two dimensional flows we take the point of view of perturbation of a hamiltonian system dependent on a small scaling parameter with the starting point being zakharov s hamiltonian v e zakharov 1968 j appl mech tech phys 9 1990 1994 for the euler equations for water waves we consider bottom topography which is periodic in horizontal variables on a short length scale with the amplitude of variation being of the same order as the fluid depth the bottom may also exhibit slow variations at the same length scale as or longer than the order of the wavelength of the surface waves we do not take up the question of random bottom variations a topic which is considered in rosales papanicolaou 1983 in the two dimensional case of waves in a channel we give an alternate derivation of the effective korteweg de vries kdv equation that is obtained in rosales papanicolaou 1983 in addition we obtain effective boussinesq equations that describe the motion of bidirectional long waves in cases in which the bottom possesses both short and long scale variations in certain cases we also obtain unidirectional equations that are similar to the kdv equation in three dimensions we obtain effective three dimensional long wave equations in a boussinesq scaling regime and again in certain cases an effective kadomtsev petviashvili kp system in the appropriate unidirectional limit the computations for these results are performed in the framework of an asymptotic analysis of multiple scale operators in the present case this involves the dirichlet neumann operator for the fluid domain which takes into account the variations in bottom topography as well as the deformations of the free surface from equilibrium

the l average value and the fuzzy expectation of a fuzzy random variable abstract the concepts of fuzzy random variable and the associated fuzzy expected value have been introduced by puri and ralescu as an extension of measurable set valued functions random sets and of the aumann integral of these functions respectively on the other hand the average function has been suggested by campos and gonzalez as an appropriate function to rank fuzzy numbers in this paper we are going to analyze some useful properties concerning the average value of the expectation of a fuzzy random variable and some practical implications of these properties are also commented on

existence and uniqueness of solution for fuzzy random differential equations with non lipschitz coefficients in this paper a class of fuzzy random differential equations with non lipschitz coefficients is studied the existence and uniqueness of solutions for fuzzy random differential equations with non lipschitz coefficients is first proved then the dependence of fuzzy random differential equations on initial values is discussed finally the non confluence property of the solution for fuzzy random differential equation is investigated our main tool is the gronwall lemma

recovering the water wave profile from pressure measurements a new method is proposed to recover the water wave surface elevation from pressure data obtained at the bottom of the fluid the new method requires the numerical solution of a nonlocal nonlinear equation relating the pressure and the surface elevation which is obtained from the euler formulation of the water wave problem without approximation from this new equation a variety of different asymptotic formulas are derived the nonlocal equation and the asymptotic formulas are compared with both numerical data and physical experiments the solvability properties of the nonlocal equation are rigorously analyzed using the implicit function theorem

set valued choquet integrals revisited this paper performs a further investigation on the set valued choquet integral given by jang and kwon fuzzy sets and systems 112 2000 233 some incorrect properties are revised it is pointed out that the definition for convergence of sequence of sets or set valued functions is unreasonable by using the usual kuratowski s convergence some new convergence theorems for the set valued choquet integral are shown meanwhile some other properties are obtained

the logarithmic triplet theory with boundary the boundary theory for the c 2 triplet model is investigated in detail in particular we show that there are four different boundary conditions that preserve the triplet algebra and check the consistency of the corresponding boundary operators by constructing their ope coefficients explicitly we also compute the correlation functions of two bulk fields in the presence of a boundary and verify that they are consistent with factorization

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

a unified approach to asymptotic behaviors for the autoregressive model with fuzzy data we propose a unified estimator for the autoregressive model with fuzzy input output variables based on the least squares method the least squares estimation is investigated in presence of a unified r distance defined on the space of fuzzy numbers we investigate asymptotic properties of the unified estimator under some simple conditions as well as a generalization which reduces to the asymptotic properties under those distances when the distances are the special cases of r distance some simulation studies are included to compare the asymptotic properties of estimators formed under several distances being the special cases of r distance

invariants of 3 manifolds and projective representations of mapping class groups via quantum groups at roots of unity an example of a finite dimensional factorizable ribbon hopf algebra is given by a quotienth u q g of the quantized universal enveloping algebrau q g at a root of unityq of odd degree the mapping class groupm g 1 of a surface of genusg with one hole projectively acts by automorphisms in theh moduleh g ifh is endowed with the coadjointh module structure there exists a projective representation of the mapping class groupm g n of a surface of genusg withn holes labeled by finite dimensionalh modulesx 1 x n in the vector space hom h x 1 x n h g an invariant of closed oriented 3 manifolds is constructed modifications of these constructions for a class of ribbon hopf algebras satisfying weaker conditions than factorizability including most ofu q g at roots of unityq of even degree are described

kazhdan lusztig dual quantum group for logarithimic extensions of virasoro minimal models we derive and study a quantum group gp q that is kazhdan lusztig dual to the w algebra wp q of the logarithmic p q conformal field theory model the algebra wp q is generated by two currents w z and w z of dimension 2p 1 2q 1 and the energy momentum tensor t z the two currents generate a vertex operator ideal r with the property that the quotient wp q r is the vertex operator algebra of the p q virasoro minimal model the number 2pq of irreducible gp q representations is the same as the number of irreducible wp q representations on which r acts nontrivially we find the center of gp q and show that the modular group representation on it is equivalent to the modular group representation on the wp q characters and pseudocharacters the factorization of the gp q ribbon element leads to a factorization of the modular group representation on the center we also find the gp q grothendieck ring which is presumably the logarithmic fusion of the p q model

almost periodic solutions of fuzzy systems we prove the existence of almost periodic solutions and asymptotically almost periodic solutions for the fuzzy functional differential equations moreover we consider uniform stable and uniformly asymptotically stable of almost periodic solutions for the fuzzy system

overview on the development of fuzzy random variables this paper presents a backward analysis on the interpretation modelling and impact of the concept of fuzzy random variable after some preliminaries the situations modelled by means of fuzzy random variables as well as the main approaches to model them are explained we also summarize briefly some of the probabilistic studies concerning this concept as well as some statistical applications

on borel measurability and large deviations for fuzzy random variables fuzzy random variables are a special case of measurable mappings taking on values in a non separable metric space due to that non separability the two main definitions in the literature turn out not to be equivalent we give an easy necessary and sufficient condition for their equivalence an intermediate result of independent value is a characterization of separable subsets of that metric space several probabilistic applications are proposed notably a generalization to this setting of the large deviations principle of bolthausen

differentials of fuzzy functions abstract in this paper the radstrom embedding theorem proc amer math soc 3 1952 165 is generalized and is used to define the concept of the differential of a fuzzy function

a banach lattice approach to convergent integrably bounded set valued martingales and their positive parts denote by cf x the set of all nonempty convex closed subsets of a separable banach space x let be a complete probability space and denote by l1 cf x the complete metric space of equivalence classes of a e equal integrably bounded cf x valued functions for any preassigned filtration i we describe the space of convergent integrably bounded cf x valued martingales in terms of the closure of i 1 l1 i cf x in l1 cf x in particular we provide a formula to calculate the join of two such martingales and the positive part of such a martingale our object is achieved by considering the more general setting of a near vector lattice s d endowed with a riesz metric d by means of radstrom s embedding theorem for such spaces a link is established between the space of convergent martingales in s and the space of convergent martingales in the radstrom completion r s of s this link provides information about the former space of martingales via known properties of measure free martingales in riesz normed vector lattices applicable to r s we also apply our general results to the spaces of convergent ck x valued martingales where ck x denotes the set of all nonempty convex compact subsets of x

on embedding problems of fuzzy number spaces iv in this paper we are going to follow the preceding papers to investigate the embedding problem of fuzzy number space ne 1 by means of a concrete embedding operator using the embedding operator some subclasses of ne 1 and metrices on ne 1 are also investigated

strong law of large numbers for t normed arithmetics we prove a strong law of large numbers for sequences of pairwise i i d fuzzy random variables addition is given by the extension principle associated with a general continuous triangular norm our result includes the slln for levelwise sums obtained when the chosen triangular norm is the minimum

stochastic integral with respect to set valued square integrable martingales in this paper we shall firstly illustrate why we should consider integral of a stochastic process with respect to a set valued square integrable martingale secondly we shall prove the representation theorem of set valued square integrable martingale thirdly we shall give the definition of stochastic integral of a stochastic process with respect to a set valued square integrable martingale and the representation theorem of this kind of integrals finally we shall prove that the stochastic integral is a set valued sub martingale

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

from non semisimple hopf algebras to correlation functions for logarithmic cft we use factorizable finite tensor categories and specifically the representation categories of factorizable ribbon hopf algebras h as a laboratory for exploring bulk correlation functions in local logarithmic conformal field theories for any ribbon hopf algebra automorphism of h we present a candidate for the space of bulk fields and endow it with a natural structure of a commutative symmetric frobenius algebra we derive an expression for the corresponding bulk partition functions as bilinear combinations of irreducible characters as a crucial ingredient this involves the cartan matrix of the category we also show how for any candidate bulk state space of the type we consider correlation functions of bulk fields for closed oriented world sheets of any genus can be constructed that are invariant under the natural action of the relevant mapping class group

central limit theorems for generalized set valued random variables abstract we give central limit theorems for generalized set valued random variables whose level sets are compact both in r d or in a banach space under milder conditions than those obtained recently by the latter two authors

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

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

the intersection operation in light of joint observables and bell inequalities in operational probability theory operational probability theory appears as a smooth extension of classical probability theory that fulfills quantum physics needs in this paper we first explain the need to extend the concept of the intersection operation in the light of the concept of joint observables then we tried to determine the basic features that characterize any possible definition of the intersection operation furthermore we tried to know if there is a relation between the definitions of the intersection operation of both fuzzy and operational probability theories these relations are used first to find a suitable suggestion to define the intersection operation second they are used besides the studies made on bell inequalities in fuzzy probability theory to find the conditions that should be satisfied by the intersection operation to ensure the possibility of the violation of bell inequalities in operational probability theory

optimal boussinesq model for shallow water waves interacting with a microstructure in this paper we consider the propagation of water waves in a long wave asymptotic regime when the bottom topography is periodic on a short length scale we perform a multiscale asymptotic analysis of the full potential theory model and of a family of reduced boussinesq systems parametrized by a free parameter that is the depth at which the velocity is evaluated we obtain explicit expressions for the coefficients of the resulting effective korteweg de vries kdv equations we show that it is possible to choose the free parameter of the reduced model so as to match the kdv limits of the full and reduced models hence the reduced model is optimal regarding the embedded linear weakly dispersive and weakly nonlinear characteristics of the underlying physical problem which has a microstructure we also discuss the impact of the rough bottom on the effective wave propagation in particular nonlinearity is enhanced and we can distinguish two regimes depending on the period of the bottom where the dispersion is either enhanced or reduced compared to the flat bottom case

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

b fuzzy probabilities abstract we treat b fuzzy probabilities in nonstandard frameworks by the methods based on boolean valued models and especially boolean powers these formal methods apart from the difficulties associated with the formal model itself give a simple and powerful theory for fuzzy probability spaces and fuzzy random variables essentially everything is reduced to a transfer principle however one can also work using classical methods of proofs we first give the necessary background on nonstandard frameworks within these frameworks we develop boolean fuzzy probabilities and fuzzy random variables

a braided monoidal category for free super bosons the chiral conformal field theory of free super bosons is generated by weight one currents whose mode algebra is the affinisation of an abelian lie super algebra h with non degenerate super symmetric pairing the mode algebras of a single free boson and of a single pair of symplectic fermions arise for even odd dimension 1 0 and 0 2 of h respectively in this paper the representations of the untwisted mode algebra of free super bosons are equipped with a tensor product a braiding and an associator in the symplectic fermion case i e if h is purely odd the braided monoidal structure is extended to representations of the z 2z twisted mode algebra the tensor product is obtained by computing spaces of vertex operators the braiding and associator are determined by explicit calculations from three and four point conformal blocks

fuzziness and randomness in an optimization framework abstract a semi infinite approach for linear programming in the presence of fuzzy random variable coefficients is described as a byproduct a way for dealing with optimization problems including both fuzzy and random data is obtained numerical examples are provided for the sake of illustration

strong laws of large numbers for independent fuzzy set valued random variables abstract in this paper we shall present strong laws of large numbers slln s for independent not necessary identically distributed fuzzy set valued random variables whose base space is a separable banach space or an euclidean space in the sense of the extended hausdorff metric d h we apply the method to the sequence of independent identically distributed fuzzy set valued random variables to give a simple proof of slln s

a convergence theorem of fuzzy valued martingales in the extended hausdorff metric h in this paper we shall give a new embedding method to prove a convergence theorem for fuzzy valued random variables in the sense of the extended hausdorff metric h without the restriction of fuzzy sets satisfying the lipschitz condition

compatibility between fuzzy set operations and level set operations applications to fuzzy difference abstract three types of binary operation between any two fuzzy sets are proposed in this paper the concept of 0 level set of fuzzy set plays an important role for defining the binary operations between two fuzzy sets according to the decomposition theorem of fuzzy sets the consistency between the membership functions and level sets should be rigorously verified when the binary operation between two fuzzy sets is defined by using their level sets the consistency is investigated thoroughly in this paper for the purpose of guaranteeing the existence of such binary operations moreover we also propose the so called hausdorff difference between any two fuzzy sets to define the difference of any two fuzzy numbers

jensen and chebyshev inequalities for pseudo integrals of set valued functions set valued functions are an important mathematical notion and play a crucial role in several practical areas at the same time pseudo analysis as a background allows extension of some classical mathematical notions to the forms that are highly applicable in some nonstandard situations this paper focuses on pseudo integration of set valued functions which is generalization of aumann s research and corresponding extensions of the jensen and chebyshev integral inequalities to the set valued case since the integral inequalities in question are widely used in various aspects of mathematics the main motivation for the presented research lies in the possibility of expanding the applicability of these inequalities by combining the properties of set valued functions with pseudo analysis

convergence of set valued sub and supermartingales in the kuratowski mosco sense the purpose of this paper is to prove some convergence theorems of closed and convex set valued sub and supermartingales in the kuratowski mosco sense to get submartingale convergence theorems we give sufficient conditions for the kudo aumann integral and hiai umegaki conditional expectation to be closed both for compact convex set valued random variables and for closed convex set valued random variables we also give an example of a bounded closed convex set valued random variable whose kudo aumann integral is not closed

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

fuzzy differential equations abstract this paper deals with fuzzy set valued mappings of a real variable whose values are normal convex upper semicontinuous and compactly supported fuzzy sets in rn we study differentiability and integrability properties of such functions and give an existence and uniqueness theorem for a solution to a fuzzy differential equation

hamiltonian long wave expansions for free surfaces and interfaces the theory of internal waves between two bodies of immiscible fluid is important both for its interest to ocean engineering and as a source of numerous interesting mathematical model equations which exhibit nonlinearity and dispersion in this paper we derive a hamiltonian formulation of the problem of a dynamic free interface with rigid lid upper boundary conditions and of a free surface and a free interface this latter situation occurring more commonly in experiment and in nature from the formulation we develop a hamiltonian perturbation theory for the long wave limits and we carry out a systematic analysis of the principal long wave scaling regimes this analysis provides a uniform treatment of the classical works of peters and stoker 28 benjamin 3 4 ono 26 and many others our considerations include the boussinesq and kdv regimes over finite depth fluids the benjamin ono regimes in the situation in which one fluid layer is infinitely deep and the intermediate long wave regimes in addition we describe a novel class of scaling regimes of the problem in which the amplitude of the interface disturbance is of the same order as the mean fluid depth and the characteristic small parameter corresponds to the slope of the interface our principal results are that we highlight the discrepancies between the case of rigid lid and of free surface upper boundary conditions which in some circumstances can be significant motivated by the recent results of choi and camassa 6 7 we also derive novel systems of nonlinear dispersive long wave equations in the large amplitude small slope regime our formulation of the dynamical free surface free interface problem is shown to be very effective for perturbation calculations and as well it holds promise as a basis for numerical simulations c 2000 wiley periodicals inc

long time dynamics of kdv solitary waves over a variable bottom we study the variable bottom generalized korteweg de vries bkdv equation tu x u f u b t x u where f is a nonlinearity and b is a small bounded and slowly varying function related to the varying depth of a channel of water many variable coefficient kdv type equations including the variable coefficient variable bottom kdv equation can be rescaled into the bkdv we study the long time behavior of solutions with initial conditions close to a stable b 0 solitary wave we prove that for long time intervals such solutions have the form of the solitary wave whose center and scale evolve according to a certain dynamical law involving the function b t x plus an h1 small fluctuation 2005 wiley periodicals inc

nonsemisimple fusion algebras and the verlinde formula we find a nonsemisimple fusion algebra open image in new window associated with each 1 p virasoro model we present a nonsemisimple generalization of the verlinde formula which allows us to derive open image in new window from modular transformations of characters

gaussian processes and martingales for fuzzy valued random variables with continuous parameter abstract the purpose of this paper is to discuss fuzzy valued gaussian processes and martingales with continuous parameter we mainly obtain a representation theorem for fuzzy valued gaussian processes after proving a new embedding theorem based on previous results about set valued and fuzzy valued martingales cf s li y ogura j fuzzy math 4 1996 905 s li y ogura fuzzy sets and systems 101 1999 453 s li y ogura ann probab 26 1998 1384 1402 s li y ogura in proc ifsa 97 vol 4 1997 pp 9 13 s li y ogura in c bertoluzza m a gil d a ralescu eds statistical modeling analysis and management of fuzzy data physica in press y ogura s li fuzzy sets and systems in press we prove optional sampling theorems for set valued and fuzzy valued martingales with continuous parameter

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

on the convergence of closed valued measurable multifunctions in this paper we study the convergence almost everywhere and in measure of sequences of closed valued multifunctions we first give a number of criteria for the convergence of sequences of closed subsets these results are used to obtain various characterizations for the convergence of measurable multifunctions in particular we are interested in the convergence properties of measurable selections

an a posteriori kam theorem for whiskered tori in hamiltonian partial differential equations with applications to some ill posed equations the goal of this paper is to develop a kam theory for tori with hyperbolic directions which applies to hamiltonian partial differential equations even to some ill posed ones the main result has an a posteriori format that is we show that if there is an approximate solution of an invariance equation which also satisfies some non degeneracy conditions then there is a true solution nearby this allows besides dealing with the quasi integrable case for the validation of numerical computations or formal perturbative expansions as well as for obtaining quasi periodic solutions in degenerate situations the a posteriori format also has other automatic consequences smooth dependence on parameters bootstrap of regularity etc we emphasize that the non degeneracy conditions required are just quantities evaluated on the approximate solution no global assumptions on the system such as twist hence they are readily verifiable in perturbation expansions we will pay attention to the quantitative relations between the sizes of the approximation and the non degeneracy conditions this will allow us to prove what experts call small twist theorems the non degeneracy conditions vanishes as the perturbation goes to zero but much slower than the error of the approximation the method of proof is based on an iterative method for solving a functional equation for the parameterization of the torus expressing that the range of the parameterization admits an evolution and is invariant we also solve functional equations for bundles which imply that are invariant under the linearization the iterative method does not use transformation theory nor action angle variables the main result does not assume that the system is close to integrable more surprisingly we do not need that the equations we study define an evolution for all initial conditions and are well posed even if the systems we study do not admit solutions for all initial conditions we show that there is a systematic way to choose initial conditions on which one can define an evolution which is quasi periodic we first develop an abstract theorem then we show how this abstract result applies to some concrete examples the examples considered in this paper are the scalar boussinesq equation and the boussinesq system both are pde models that aim to describe water waves in the long wave limit for these equations we construct small amplitude time quasi periodic solutions which are even in the spatial variable the strategy for the abstract theorem is inspired by that in fontich et al electron res announc math sci 16 9 22 2009 j differ equ 246 8 3136 3213 2009 the main part of the paper is to study infinite dimensional analogues of dichotomies which applies even to ill posed equations and which is stable under addition of unbounded perturbations this requires that we assume smoothing properties we also present very detailed bounds on the change of the splittings under perturbations

numerical simulation of gravity waves abstract we present a new spectral method to simulate numerically the waterwave problem in a channel for a fluid of finite or infinite depth it is formulated in terms of the free surface elevation and the velocity potential the numerical method is based on the reduction of this problem to a lower dimensional computation involving surface variables alone to accomplish this we describe the taylor expansion of the dirichlet neumann operator in homogeneous powers of the surface elevation each term is a concatenation of fourier multipliers with powers of and its derivatives and is valid uniformly in wavenumber these are easily calculated using the fast fourier transform the method is illustrated by computing the long time evolution of modulated wave packets and of approximations to the stokes steady wave train by imposing a surface pressure we observe surface steepening in large amplitude evolution and wake and bow wave development for flows with a close to critical froude number finally we give an example of nonlinear evolution of the distribution of energy among normal modes

logarithmic conformal field theory beyond an introduction this article aims to review a selection of central topics and examples in logarithmic conformal field theory it begins with the remarkable observation of cardy that the horizontal crossing probability of critical percolation may be computed analytically within the formalism of boundary conformal field theory cardy s derivation relies on certain implicit assumptions which are shown to lead inexorably to indecomposable modules and logarithmic singularities in correlators for this a short introduction to the fusion algorithm of nahm gaberdiel and kausch is provided while the percolation logarithmic conformal field theory is still not completely understood there are several examples for which the formalism familiar from rational conformal field theory including bulk partition functions correlation functions modular transformations fusion rules and the verlinde formula has been successfully generalized this is illustrated for three examples the singlet model related to the triplet model symplectic fermions and the fermionic bc ghost system the fractional level wess zumino witten model based on at related to the bosonic ghost system and the wess zumino witten model for the lie supergroup related to at and 1 the bershadsky polyakov algebra and the feigin semikhatov algebras these examples have been chosen because they represent the most accessible and most useful members of the three best understood families of logarithmic conformal field theories the logarithmic minimal models the fractional level wess zumino witten models and the wess zumino witten models on lie supergroups excluding in this review the emphasis lies on the representation theory of the underlying chiral algebra and the modular data pertaining to the characters of the representations each of the archetypal logarithmic conformal field theories is studied here by first determining its irreducible spectrum which turns out to be continuous as well as a selection of natural reducible but indecomposable modules this is followed by a detailed description of how to obtain character formulae for each irreducible a derivation of the action of the modular group on the characters and an application of the verlinde formula to compute the grothendieck fusion rules in each case the genuine fusion rules are known so comparisons can be made and favourable conclusions drawn in addition each example admits an infinite set of simple currents hence extended symmetry algebras may be constructed and a series of bulk modular invariants computed the spectrum of such an extended theory is typically discrete and this is how the triplet model arises for example moreover simple current technology admits a derivation of the extended algebra fusion rules from those of its continuous parent theory finally each example is concluded by a brief description of the computation of some bulk correlators a discussion of the structure of the bulk state space and remarks concerning more advanced developments and generalizations the final part gives a very short account of the theory of staggered modules the simplest class of representations that are responsible for the logarithmic singularities that distinguish logarithmic theories from their rational cousins these modules are discussed in a generality suitable to encompass all the examples met in this review and some of the very basic structure theory is proven then the important quantities known as logarithmic couplings are reviewed for virasoro staggered modules and their role as fundamentally important parameters akin to the three point constants of rational conformal field theory is discussed an appendix is also provided in order to introduce some of the necessary but perhaps unfamiliar language of homological algebra

some properties and convergence theorems of set valued choquet integrals the paper aims at discussing the set valued choquet integral which is the integral of set valued random variables with respect to capacities we mainly present representation theorems of the set valued random variable by using a sequence of choquet integrable selections and then we investigate some properties of set valued choquet integrals especially subadditive property and inequality of the metric of set valued choquet integrals we also prove fatou s lemmas lebesgue dominated convergence theorem and monotone convergence theorem of set valued choquet integrals under the weaker conditions than that in previous works

representation theorems set valued and fuzzy set valued ito integral in this paper we shall present representation theorems of set valued martingales and set valued processes of finite variation with continuous time we shall also obtain a representation theorem of a predictable set valued stochastic process we shall give a new definition of ito integral of a set valued stochastic process with respect to a brownian motion based on the work e j jung j h kim on set valued stochastic integrals stochastic anal appl 21 2 2003 401 418 we shall also discuss some properties of set valued ito integral especially the presentation theorem of set valued ito integral finally we extend some of above results to the fuzzy set valued case

limit theorems for fuzzy random variables a strong law of large numbers and a central limit theorem are proved for independent and identically distributed fuzzy random variables whose values are fuzzy sets with compact levels the proofs are based on embedding theorems as well as on probability techniques in banach space

fuzzy random variables abstract in this paper we define the concepts of fuzzy random variable and the expectation of a fuzzy random variable the new definition of expectation generalizes the integral of a set valued function we derive some properties of these new concepts by considering a suitable generalization of the hausdorff metric we derive the lebesgue dominated convergence type theorem

hukuhara derivative of the fuzzy expected value in this paper we establish sufficient conditions to guarantee the exchange of the fuzzy expected value for a fuzzy random variable depending on certain parameter and its hukuhara derivative with respect to the parameter main results are obtained for two different sets of general conditions one of them requiring measurability conditions with respect to the product space the fuzzy random variable is defined from and the second one just requiring conditions over the projections of the fuzzy random variables

braided groups and quantum fourier transform abstract we show that acting on every finite dimensional factorizable ribbon hopf algebra h there are invertible operators s t obeying the modular identities s t 3 s 2 where is a constant the class includes the finite dimensional quantum groups uq g associated to complex simple lie algebras we give the example of uq sl 2 at a root of unity in detail as well as an example relating to anyons the operator s plays the role of quantum fourier transform and acts naturally on h viewed by transmutation as a braided group h a braided cocommutative hopf algebra in a braided category it obeys s 2 s 1 where s is the antipode of h the results follow as an application of previous category theoretical constructions

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