algorithms for solving fuzzy relational equations in a probabilistic setting abstract we discuss methods suitable for finding approximate solutions to fuzzy relational equations defined in finite universe of discourse and taking the form of either a system of equations x k r y k k 1 2 k with r being unknown fuzzy relation or a single equation x r y with x being looked for the key idea of the proposed algorithms is to incorporate some statistics concerning the solution of a single equation and utilize it in the final process of approximation of the unknown fuzzy relation set numerical considerations support the established algorithms

fuzzy modeling with tolerances abstract we consider the problem of the resolution of fuzzy equation systems a certain number of fuzzy inputs and outputs is given how can we determine a fuzzy relation which describes the behaviour of the object under consideration when attributing this problem to fuzzy equations in the classical sense it may happen that a solution does not exist therefore we introduce fuzzy tolerances as an extension of the solvability conception for this problem we get the whole solution set by means of posets moreover we consider the inverse problem how to determine tolerances to get consistency i e at least one solution for the arising equation system

numerical and applicational aspects of fuzzy relational equations we discuss numerical methods leading to a resolution of fuzzy relational equations which create a formal description of ill defined systems an applicability of the numerical approach is shown in solving some problems in fuzzy systems such as identification and control an idea of fuzzy discretization fuzzy chopping as a meaningful tool for handling various problems is also investigated

resolution of composite interval valued fuzzy relation equations abstract in this study we investigate the properties and the solution procedure of composite interval valued fuzzy relation equations with max min operator this is a generalization of existing constant valued fuzzy relation problems the study includes both theoretical development and numerical analyses it is concluded that our proposed methodology not only solves the existing fuzzy relation equations but also reveals the necessary and sufficient conditions for more general problems therefore this study facilitates the understanding of solution structure and wider applications

on the existence of minimal solutions for fuzzy equations with tolerances abstract we investigate the existence of minimal elements of solution sets for fuzzy equation problems with tolerances where the cardinality of the basic spaces of fuzzy inputs and outputs of an object need not be finite it turns out the semi continuity of applied membership functions plays a central role crisp models with fuzzy parameters are considered as well

the dual of brown representability for homotopy categories of complexes abstract we call product generator of an additive category a fixed object satisfying the property that every other object is a direct factor of a product of copies of it in this paper we start with an additive category with products and images e g a module category and we are concerned with the homotopy category of complexes with entries in that additive category we prove that the brown representability theorem is valid for the dual of the homotopy category if and only if the initial additive category has a product generator

an alternative approach to the resolution of fuzzy relation equations abstract in this paper we propose a solution procedure by means of a quasi characteristic matrix the matrix is so defined that not only the solution set can be determined but also the type of solution set such as the non empty set full set unique minimal solution and unique solution can be identified before the solution set is determined therefore it provides an insight into the structure of fuzzy relation equations

solving interval valued fuzzy relation equations solving systems of fuzzy relation equations is an important topic in fuzzy set theory this paper studies the composite interval valued fuzzy relation equations after analyzing the properties of its solution set we convert the fuzzy relation equations into a fuzzy relation inequality system and propose an efficient computational procedure to generate the whole solution set examples are included to illustrate the idea and algorithm

smashing subcategories and the telescope conjecture an algebraic approach we prove a modified version of ravenel s telescope conjecture it is shown that every smashing subcategory of the stable homotopy category is generated by a set of maps between finite spectra this result is based on a new characterization of smashing subcategories which leads in addition to a classification of these subcategories in terms of the category of finite spectra the approach presented here is purely algebraic it is based on an analysis of pure injective objects in a compactly generated triangulated category and covers therefore also situations arising in algebraic geometry and representation theory

on the fuzziness of solutions of fuzzy relation equations on finite spaces let be the set of solutions of a fuzzy relation equation defined on finite spaces in this paper we use some algorithms already introduced in our foregoing work in order to determine some relations of which have a minimal value of fuzziness measure by taking into account the fuzzy entropy concept we build a new lattice structure and some fuzzy topological results are also presented

application of fuzzy sets of type 2 to the solution of fuzzy equation systems abstract the present paper continues the authors investigation on fuzzy equation systems with tolerances we consider the situation that the outputs of an object are modelled by fuzzy sets of type 2 for special membership functions this problem can be attributed to a problem with tolerances moreover this concept enables us to define solutions for fuzzy outputs of type 1 when classical solution concepts may fail

cubature on wiener space it is well known that there is a mathematical equivalence between solving parabolic partial differential equations pdes and the integration of certain functionals on wiener space monte carlo simulation of stochastic differential equations sdes is a naive approach based on this underlying principle in finite dimensions it is well known that cubature can be a very effective approach to integration we discuss the appropriate extension of this idea to wiener space in the process we develop high order numerical schemes valid for high dimensional sdes and semi elliptic pdes

interval valued fuzzy backward reasoning the importance and efficiency of backward reasoning in nonfuzzy reasoning has been stressed for a long time especially in the case of expert systems and decision support systems the extension of this reasoning method to fuzzy theory however has never been considered in this paper the authors propose a definition of fuzzy backward reasoning based on the generalized modus ponens and show the necessity of considering interval valued fuzzy backward reasoning then the authors propose solving methods for fuzzy backward reasoning in the case of a rule with one or several conditions as well as in the case of several rules

fuzzy eigenvalues and input output analysis abstract we first generalize the perron frobenius results for nonnegative matrices to nonnegative fuzzy matrices which says that these fuzzy matrices have a dominant nonnegative fuzzy eigenvalue this result is then used to extend leontief s closed input output analysis to fuzzy economies and obtain a fuzzy model of the world s economy

the law of the euler scheme for stochastic differential equations i convergence rate of the distribution function we study the approximation problem ofef x t byef x t n where x t is the solution of a stochastic differential equation x t n is defined by the euler discretization scheme with stept n andf is a given function for smoothf s talay and tubaro have shown that the erroref x t f x t n can be expanded in powers of 1 n which permits to construct romberg extrapolation precedures to accelerate the convergence rate here we prove that the expansion exists also whenf is only supposed measurable and bounded under an additional nondegeneracy condition of hormander type for the infinitesimal generator of x t to obtain this result we use the stochastic variations calculus in the second part of this work we will consider the density of the law ofx t n and compare it to the density of the law ofx t

some adjoints in homotopy categories let r be a ring in a previous paper 11 we found a new description for the category k r proj it is equivalent to the verdier quotient k r flat s for some suitable s c k r flat in this article we show that the quotient map from k r flat to k r flat s always has a right adjoint this gives a new fully faithful embedding of k r proj into k r flat its virtue is that it generalizes to nonaffine schemes

resolution of finite fuzzy relation equations abstract general schemes for solving fuzzy relation equations with finite sets are derived in this paper the relationship between the results derived here and those of previous publications regarding this subject is also discussed

stochastic approximation finite element method analytical formulas for multidimensional diffusion process we derive an analytical weak approximation of a multidimensional diffusion process as coefficients are small or time is small our methodology combines the use of gaussian proxies to approximate the law of the diffusion and a finite element interpolation of the terminal function applied to the diffusion we call this the stochastic approximation finite element safe method we provide error bounds of our global approximation depending on the diffusion process coefficients the time horizon and the regularity of the terminal function then we give estimates of the computational cost of our algorithm this shows an improved efficiency compared to monte carlo methods in small and medium dimensions smaller than 10 which is confirmed by numerical experiments

on direct and inverse problems for fuzzy equation systems with tolerances abstract we investigate the resolution of fuzzy relational equation systems with tolerances which are a certain extension of fuzzy equations considered f i in 3 5 the extension of the concept of higashi and klir 3 enables us to describe the set of solutions to our problem for given tolerances by means of posets in a second part we investigate an inverse problem given upper lower tolerances how to determine lower upper tolerances such that the arising problem becomes consistent numerical examples are given

products of fuzzy subgroups fuzzy subgroups are different from ordinary subgroups in that one cannot tell with certainty which group elements belong and which do not of course this requires an appropriate modification of the closure property which takes the form of an inequality in this paper a study of products of fuzzy subgroups is begun some results are expected the product of fuzzy subgroups is a fuzzy subgroup products with the same but permuted factors are isomorphic and there is a natural isomorphism from any factor into the product functions called t norms are used to construct products of fuzzy subgroups different t norms yielding different products since a fuzzy subgroup must satisfy an inequality generalizing closure and the strength of that inequality is important some results relate the t norm used in forming the product to the strength of the ensuing inequality when can complicated fuzzy subgroups be expressed as products of simple ones this question is addressed for fuzzy subgroups of finite abelian groups

a rule based method to calculate exactly the widest solution sets of a max min fuzzy relational inequality abstract in this paper we propose a rule based method to calculate exactly all the widest solution sets of a fuzzy relational inequality viewed as an extension of fuzzy relational equations fuzzy relational inequalities can be applied to many fields of fuzzy logic yet they provide with more flexibility as they enable to get rid of the sensitivity of fuzzy relational equations to numerical values up to now however fuzzy relational inequalities have received few attention from researchers and few methods have been proposed to solve these inequalities the proposed analytical methods furthermore do not calculate exactly all the widest solutions sets of a fuzzy relational inequality and often lead to redundant solution sets the rule based method we propose in this paper determines easily exactly all the widest solution sets of a fuzzy relational inequality