interactive decision making for multiobjective linear fractional programming problems with block angular structure involving fuzzy numbers abstract in this paper by considering the experts vague or fuzzy understanding of the nature of the parameters in the problem formulation process multiobjective linear fractional programming problems with block angular structure involving fuzzy numbers are formulated through the use of the level sets of fuzzy numbers an extended pareto optimality concept called the pareto optimality is introduced to generate a candidate for the satisficing solution which is also pareto optimal the decision maker is asked to specify the degree and the reference objective values it is shown that the corresponding pareto optimal solution can be easily obtained by solving the minimax problems for which the dantzig wolfe decomposition method and ritter s partitioning procedure are applicable then a linear programming based interactive decision making method with decomposition procedures for deriving a satisficing solution for the decision maker efficiently from an pareto optimal solution set is presented an illustrative numerical example is provided to demonstrate the feasibility of the proposed method

vector valued variational principle in fuzzy metric space and its applications a vector valued ekeland s variational principle and its equivalents in fuzzy metric space are presented in this paper our result unifies and extends some of the previous results about single valued and vector valued ekeland s variational principle as an application we obtain a fixed point theorem in fuzzy metric space under a new contractive condition which is a new generalization of well known banach contraction principle all of our results are also new even in usual metric spaces

branch and price algorithm for fuzzy integer programming problems with block angular structure this paper deals with fuzzy integer linear programming problems with block angular structure in which the fuzzy constraints are simplified by using possibility and necessity relations this main fuzzy problem is efficiently decomposed and is solved by a branch and price algorithm in the nodes of the branch and price tree the linear relaxation of the problem is solved by applying a column generation method also the relationship between the optimal solutions of this problem under possibility and necessity relations is derived to show the validation of the proposed algorithm some results are proved in addition the application of this algorithm is illustrated on fuzzy multicommodity flow problem for this case a new branching scheme is proposed to preserve the network structure of the subproblems which are produced in the column generation method some examples are solved and their results are compared with the previous works also the results of the proposed algorithm are reported on some large scale benchmark instances

satisficing solutions and duality in interval and fuzzy linear programming in this paper we introduce a class of fuzzy linear programming problems and define the concepts of feasible and satisficing solutions the necessary tools for dealing with such problems in this way we show that the class of crisp classical lp problems can be embedded into the class of flp ones moreover for flp problems we define the concept of duality and prove the weak and strong duality theorems further we define a class of interval linear programming problems as a special subclass of flp problems and apply the previous results to this special case

gauss markov processes on hilbert spaces k it o characterised in 16 zero mean stationary gauss markov processes evolving on a class of innite dimension al spaces in this work we extend the work of it o in the case of hilbert spaces gauss markov families that are time homogenous are identied as solutions to linear stochastic dierential equations with singular coecients choosing an appropriate locally convex topology on the space of weakly sequentially continuous functions we also characterize the transition semigroup the generator and its core thus providing an innite dimension al extension of the classical result of courr ege 4 in the case of gauss markov semigroups

ergodic properties of operator semigroups in general weak topologies abstract the aim of this paper is to study ergodic properties i e properties about the limit of cesaro averages of a semigroup of bounded linear operators on a banach space x which is assumed to be continuous and locally integrable in the sense of a certain general weak topology of x then the results are applied to particular examples such as locally strongly integrable semigroups their dual semigroups and the tensor product semigroup of two c 0 semigroups

embedding problem of fuzzy number space part ii abstract using a concrete structure into which we embed the fuzzy number space e1 several necessary and sufficient conditions of fuzzy set valued functions are given by means of abstract function theory

fuzzy decision making for medical diagnosis based on fuzzy number and compositional rule of inference abstract klir and yuan 2 mentioned the medical decision making problem in chapter 17 in this paper we use interval estimation to get fuzzy number r ij and from point estimation we get r ij set r r ij and r r ij let fuzzy relation of the set p of the patients and the set s of the symptoms be q a ij and fuzzy relation of the set s of the symptoms and the set d of the diseases be r from the compositional rule of inference we have t q r therefore we can have a fuzzy decision making for medical diagnosis this paper changes r to r and use t q r to do a fuzzy decision making for medical diagnosis

statistical cluster and extreme limit points of sequences of fuzzy numbers in this paper we investigate the relations between statistical limit superior or statistical limit inferior and statistical cluster points of a statistically bounded sequence of fuzzy numbers and show that the set of statistical cluster points of a sequence of this type might be empty

on the equations ax q and sx xt q abstract in this paper we first study solutions of the functional equation ax q where a is the generator of a certain semigroup using mean ergodic theory then we apply the result to study the solvability of the operator equation sx xt q where s and t are generators of some semigroups of operators we give a characterization of the solvability using weak operator limit of an integral involving the tensor product semigroup of the two semigroups generated respectively by s and t

convergence of sequences of fuzzy numbers and fixed point theorems for increasing fuzzy mappings and application in this paper we discuss some properties of the convergence of sequences of fuzzy numbers and obtain some fixed point theorems for increasing fuzzy mappings and as applications we give two examples one is about fishery production which discusses the balance problems between the quantity of catching fish and the increase in the quantity of fish that is produced by the reproduction and growth of fish another is about the solvability of the equation of fuzzy numbers

construction of a unique self adjoint generator for a symmetric local semigroup abstract a definition is given of a symmetric local semigroup of unbounded operators p t 0 t t for some t 0 on a hilbert space h such that p t is eventually densely defined as t 0 it is shown that there exists a unique unbounded below self adjoint operator h on h such that p t is a restriction of e th as an application it is proven that h0 v is essentially self adjoint where e th0 is an lp contractive semigroup and v is multiplication by a real measurable function such that v l2 e and e v l1 for some e 0

an extension of ekeland s variational principle in fuzzy metric space and its applications abstract an extension of ekeland s variational principle in fuzzy metric space which is an essential and the most general improvement of ekeland s variational principle in fuzzy metric space up to now is established as an application we obtain caristi s coincidence theorem for set valued mappings in fuzzy metric spaces further a direct simple proof of the equivalence between the two theorems is given some applications of these results to probabilistic metric spaces are presented all these results even in usual metric space are also the latest

invariant measures and regularity properties of perturbed ornstein uhlenbeck semigroups abstract this paper deals with perturbations of the ornstein uhlenbeck operator on l 2 spaces with respect to a gaussian measure we perturb the generator of the ornstein uhlenbeck semigroup by a certain unbounded non linear drift and show various properties of the perturbed semigroup such as compactness and positivity strong feller property existence and uniqueness of an invariant measure are discussed as well

fuzzy programming approach to multi level programming problems multi level programming is characterized as mathematical programming to solve decentralized planning problems we have considered a multi level linear programming problem and applied fuzzy mathematical programming fmp approach to obtain the solution of the system we have suggested fmp method for the minimization of the objectives using linear membership functions fmp is a supervised search procedure supervised by the higher level decision maker dm the higher level dm provides the preferred values of decision variables under his control to enable the lower level dm to search for his optimum in a wider feasible space and the bounds of his objective function to direct the lower level dm to search for his solutions in the right direction

some properties of limit inferior and limit superior for sequences of fuzzy real numbers abstract the limit inferior and limit superior of a bounded sequence of fuzzy real numbers have been introduced by aytar et al 2008 1 in this paper we give a simplified expressions for limit inferior and limit superior the expressions is concise and convenient for use as a straightforward corollary of this expressions we can easily prove some properties of the limit inferior and limit superior

on fuzzy metric spaces abstract in this paper we introduce the concept of a fuzzy metric space the distance between two points in a fuzzy metric space is a non negative upper semicontinuous normal and convex fuzzy number properties of fuzzy metric spaces are studied and some fixed point theorems are proved

on w continuous cosine operator functions on donne les proprietes fondamentales des theoremes de comparaison infinitesimaux et des theoremes ergodiques

remark on lacunary statistical convergence of fuzzy numbers fatih nuray fuzzy sets and systems 99 3 1998 269 proved the inclusion relations between the sets of statistically convergent and lacunary statistically convergent sequences of fuzzy numbers in this paper we show that nuray s condition is sufficient as well as necessary

the generation of weakly integrable semigroups abstract a characterization of the generators of a class of weakly integrable semigroups on a locally convex space is proved it bears a resemblance to the hille yosida phillips theorem on the generators of strongly continuous semigroups on a banach space

topological properties of fuzzy numbers fuzzy numbers are defined and a metric is assigned to this class of fuzzy sets topological properties of the resulting metric space are studied the notion of q crisp fuzzy numbers is introduced and in particular it is shown that this class of fuzzy numbers is homeomorphic to the hilbert cube

the core of a sequence of fuzzy numbers in this paper based on level sets we define the limit inferior and limit superior of a bounded sequence of fuzzy numbers and prove some properties we extend the concept of the core of a sequence of complex numbers first introduced by knopp in 1930 to a bounded sequence of fuzzy numbers and prove that the core of a sequence of fuzzy numbers is the interval n m where n and m are extreme limit points of the sequence

a fuzzy programming approach to multiobjective multidimensional 0 1 knapsack problems abstract the objective of this paper is to present an interactive procedure for the multiobjective multidimensional 0 1 knapsack problem that takes into consideration the incorporation of fuzzy goals of the decision maker that is easy to use since it requires from the decision maker to handle only one parameter namely the aspiration level of each objective and that is fast and can treat our problem as a usual 0 1 knapsack problem using already available software namely the primal effective gradient method used primarily to solve the large scale cases to get some statistics on the behavior of the algorithm a number of randomly generated simulation of problems is solved from our numerical experience it is possible to conclude that our proposed method is a worthwhile alternative to existing methods from a practical point of view

properties of b vex fuzzy mappings and applications to fuzzy optimization convexity plays a key role in operations research and fuzzy optimization theory the concept of b vex and logarithmic b vex for fuzzy mappings is introduced by relaxing the definition of convexity of a fuzzy mapping most of the basic properties of b vex fuzzy mapping are discussed and established for the nondifferentiable case necessary and sufficient conditions for b vex fuzzy mapping are presented several important results are given for nonlinear fuzzy optimization problems assuming that objective and constraint functions are b vex fuzzy mappings

the supremum and infimum of the set of fuzzy numbers and its application abstract in this paper we prove that the bounded set of fuzzy numbers must exist supremum and infimum and give the concrete representation of supremum and infimum as an application we obtain that the continuous fuzzy valued function on a closed interval exists supremum and infimum and give the precise representation we also show that the bounded fuzzy valued function on a closed interval can define the lower and upper sums and the lower and upper integrals of riemann and riemann stieltjes by the usual way

quantum cohomology of the moduli space of stable bundles over a riemann surface we determine the quantum cohomology of the moduli space m of odd degree rank two stable vector bundles over a riemann surface of genus g 1 this work together with 10 complete the proof of the existence of an isomorphism qh m hf s1

extension principle in fuzzy optimization the paper deals with the generalization of the well known extension principle to set to set mappings multifunctions in the first part of the paper an image of a fuzzy set is defined and several properties for special classes of multifunctions are derived the second part of the paper is devoted to fuzzy optimization using the first part a new concept of solution of the fuzzy optimization problem is introduced and then compared with the usual one

elementary fuzzy calculus abstract in 2 dubois and prade introduced the notion of fuzzy numbers and defined the basic operations of addition subtraction multiplication and division a slightly modified definition of fuzzy numbers was presented in 4 and in that paper a metric was defined for this family of fuzzy sets another less restrictive definition of fuzzy numbers can be found in 1 in the present paper we consider fuzzy numbers from a somewhat different perspective basically we shall view fuzzy numbers in a topological vector space setting using the customary vector space operations together with the metric given in 4 we will define differentiation and integration of fuzzy valued functions in ways that parallel closely the corresponding definitions for real differentiation and integration

on linearly topological structure and property of fuzzy normed linear space in this paper the simplified definition of fuzzy normed linear space is introduced the different structure of fuzzy normed linear space with variable right norm r is discussed in terms of topological vector space and its properties such as compactness completeness and density are studied under more general left norm l and right norm r as application the linearly topological structure of menger pn space is obtained

a fuzzy satisficing method for large scale linear programming problems with block angular structure abstract in this paper we focus on large scale linear programming problems with block angular structure for which the dantzig wolfe decomposition method has been successfully applied by considering the vague nature of human judgements we assume that the decision maker may have a fuzzy goal for the objective function and fuzzy constraints for the coupling constraints having elicited the corresponding linear membership functions through the interaction with the decision maker if we adopt the convex fuzzy decision for combining them it is shown that under some appropriate conditions the formulated problem can be reduced to a number of independent linear subproblems and the overall satisficing solution for the decision maker is directly obtained just only solving the subproblems

on λ statistical convergence of difference sequences of fuzzy numbers in this paper the concept of strongly d l p 2 cesaro summability of a sequence of fuzzy numbers is introduced also some inclusion relations between the set of strongly d l p 2 cesaro convergent and d l 2 statistically convergent sequences of fuzzy numbers are given

the extension of the knopp core theorem to the sequences of fuzzy numbers the concept of the core of a sequence of fuzzy numbers has been introduced by aytar et al in the core of a sequence of fuzzy numbers fuzzy sets and systems 159 24 2008 3369 3379 quite recently some matrix transformations on the classical sets of sequences of fuzzy numbers have been characterized by talo and basar in determination of the duals of classical sets of sequences of fuzzy numbers and related matrix transformations comput math appl 58 2009 717 733 in this paper we have proved a core theorem for sequences of fuzzy numbers which is analogous to the famous knoop core theorem for real sequences to achieve this goal we have studied some properties of the lim sup and lim inf of sequences of fuzzy numbers also we have characterized a class of positive regular matrices of fuzzy numbers such that these matrices leave the core of any bounded sequence of fuzzy numbers invariant

some notes on the supremum and infimum of the set of fuzzy numbers abstract in this paper we give a necessary and sufficient condition under which a pair of usual functions of on 0 1 supn un supn un can be determined a fuzzy number and give a condition under which the supremum and infimum for a sequence of fuzzy numbers preserve the approximation properties for a metric d in addition we obtain the criterion that the continuous fuzzy valued function on a closed interval can attain its supremum and infimum

an interactive fuzzy satisficing method for structured multiobjective linear fractional programs with fuzzy numbers abstract in this paper by considering the experts vague or fuzzy understanding of the nature of the parameters in the problem formulation process multiobjective linear fractional programming problems with block angular structure involving fuzzy numbers are formulated using the a level sets of fuzzy numbers the corresponding nonfuzzy a multiobjective linear fractional programming problem is introduced the fuzzy goals of the decision maker for the objective functions are quantified by eliciting the corresponding membership functions including nonlinear ones through the introduction of extended pareto optimality concepts if the decision maker specifies the degree a and the reference membership values the corresponding extended pareto optimal solution can be obtained by solving the minimax problems for which the dantzig wolfe decomposition method and ritter s partitioning procedure are applicable then a linear programming based interactive fuzzy satisficing method with decomposition procedures for deriving a satisficing solution for the decision maker efficiently from an extended pareto optimal solution set is presented an illustrative numerical example is provided to demonstrate the feasibility of the proposed method

on statistically convergent sequences of fuzzy numbers abstract in this paper we study some equivalent alternative conditions for a sequence of fuzzy numbers to be statistically cauchy

on the level convergence of a sequence of fuzzy numbers abstract in this paper we first give a simplified proof of the existence theorem of supremum and infimum in fuzzy number space e 1 established by wu and wu j math anal appl 210 1997 499 and improve the expressions of the supremum and infimum as a straightforward corollary of this result we obtain a necessary and sufficient condition under which for a bounded sequence of fuzzy numbers u n the pair of functions sup n u n and sup n u n can determine a fuzzy number secondly we give a necessary and sufficient condition for a sequence of fuzzy numbers u n to be levelwise convergent in e 1 and generalize some important theorems in real number spaces to fuzzy number spaces finally we prove the existence of supremum and infimum for the level continuous fuzzy valued function on a closed interval and give a necessary and sufficient condition under which its supremum and infimum can be attained

an interactive fuzzy satisficing method for multiobjective block angular linear programming problems with fuzzy parameters abstract in this paper by considering the experts imprecise or fuzzy understanding of the nature of the parameters in the problem formulation process large scale multiobjective block angular linear programming problems involving fuzzy parameters characterized by fuzzy numbers are formulated using the level sets of fuzzy numbers the corresponding nonfuzzy programming problem is introduced the fuzzy goals of the decision maker for the objective functions are quantified by eliciting the corresponding membership functions including nonlinear ones through the introduction of an extended pareto optimality concept if the decision maker specifies the degree and the reference membership values the corresponding extended pareto optimal solution can be obtained by solving the minimax problems for which the dantzig wolfe decomposition method is applicable then a linear programming based interactive fuzzy satisficing method for deriving a satisficing solution for the decision maker efficiently from an extended pareto optimal solution set is presented along with an illustrative numerical example

a note on the completions of fuzzy metric spaces and fuzzy normed spaces in this paper we point out that the completion theorem of fuzzy metric spaces given by lee et al fuzzy sets and systems 106 1999 469 473 is incorrect and prove a new completion theorem

some properties of the space of fuzzy valued continuous functions on a compact set in this paper a generalized form of the bolzano theorem in classical analysis to fuzzy number space and a characterization of compact subsets in fuzzy number space are given some properties of the fuzzy valued continuous functions defined on a compact set k are studied completeness of the space c k e 1 of fuzzy valued continuous functions on k endowed with the supremum metric d is proved a characterization of compact subsets in the space c k e 1 d is presented which is a generalization of the arzela ascoli theorem in classical analysis

lacunary statistical convergence of sequences of fuzzy numbers the sequence x xk of fuzzy numbers is statistically convergent to the fuzzy number x0 provided that for each 0 limln the number ofk n d xk x0 0 in this paper we study a related concept of convergence in which the set k k n is replaced by k kr 1 k kr for some lacunary sequence kr also we introduce the concept of lacunary statistically cauchy sequence and show that it is equivalent to the lacunary statistical convergence in addition the inclusion relations between the sets of statistically convergent and lacunary statistically convergent sequences of fuzzy numbers are given

instanton counting and donaldson invariants for a smooth projective toric surface we determine the donaldson invariants and their wallcrossing in terms of the nekrasov partition function using the solution of the nekrasov conjecture 33 38 3 and its refinement 34 we apply this result to give a generating function for the wallcrossing of donaldson invariants of good walls of simply connected projective surfaces with b 1 in terms of modular forms this formula was proved earlier in 19 more generally for simply connected 4 manifolds with b 1 assuming the kotschick morgan conjecture and it was also derived by physical arguments in 31

formulation of fuzzy linear programming problems as four objective constrained optimization problems this paper concerns the solution of fuzzy linear programming flp problems which involve fuzzy numbers in coefficients of objective functions firstly a number of concepts of optimal solutions to flp problems are introduced and investigated then a number of theorems are developed so as to convert the flp to a multi objective optimization problem with four objective functions finally two illustrative examples are given to demonstrate the solution procedure it also shows that our method of solution includes an existing method as a special case

minimizing unconstrained fuzzy functions in this paper we introduce a definition for a fuzzy function as a possibility distribution over the set of all functions with certain properties we then examine some of the implications of this definition in particular we show that the image of a fuzzy functions is a fuzzy vector as defined herein we also show that the set of values at which a real valued fuzzy function achieves its minimum is a fuzzy number and if the fuzzy function is convex in a particular sense then the set of vectors for which the function achieves the minimum values is connected lastly we discuss minimization of an unconstrained fuzzy function and how one might proceed to defuzzify the solution

the variational principle in fuzzy metric spaces and its applications abstract in this paper we prove a fixed point theorem and a variational principle in fuzzy metric spaces applying these results to some probabilistic metric spaces we obtain a corresponding fixed point theorem and a variational principle

ergodic control of semilinear stochastic equations and the hamilton jacobi equation abstract in this paper we consider optimal control of stochastic semilinear equations with lipschitz continuous drift and cylindrical noise we show existence and uniqueness up to an additive constant of solutions to the stationary hamilton jacobi equation associated with the cost functional given by the asymptotic average per unit time cost as a consequence we find the optimizing controls given in the feedback form to obtain these results we prove also some new results on the transition semigroups of semilinear diffusion acting in the spaces of continuous function with the weighted sup norms and on the optimal control of semilinear diffusions for the discounted cost functional

kolmogorov equation associated to a stochastic kuramoto sivashinsky equation abstract this paper investigates the relation between the kolmogorov operator associated to a stochastic kuramoto sivashinsky equation and the infinitesimal generator for the corresponding transition semigroup we prove that the infinitesimal generator is the closure of kolmogorov operator in the space of continuous functions with k th polynomial growth with respect to convergence topology and the space l 2 h respectively the proof depends on various estimates on the solution invariant measure and transition semigroup as a product we also obtains smoothing properties of the transition semigroup

an interactive algorithm for large scale multiple objective programming problems with fuzzy parameters through topsis approach in this paper we extend technique for order preference by similarity ideal solution topsis for solving large scale multiple objective programming problems involving fuzzy parameters these fuzzy parameters are characterized as fuzzy numbers for such problems the pareto optimality is introduced by extending the ordinary pareto optimality on the basis of the level sets of fuzzy numbers an interactive fuzzy decision making algorithm for generating pareto optimal solution through topsis approach is provided where the decision maker dm is asked to specify the degree and the relative importance of objectives finally a numerical example is given to clarify the main results developed in the paper

on the cauchy problem for non local ornstein uhlenbeck operators abstract we study the cauchy problem involving non local ornstein uhlenbeck operators in finite and infinite dimensions we prove classical solvability without requiring that the levy measure corresponding to the large jumps part has a first finite moment moreover we determine a core of regular functions which is invariant for the associated transition markov semigroup such a core allows to characterize the marginal laws of the ornstein uhlenbeck stochastic process as unique solutions to fokker planck kolmogorov equations for measures

extensions of topsis for multi objective large scale nonlinear programming problems in this paper we focus on multi objective large scale nonlinear programming molsnlp problems with block angular structure we extend technique for order preference by similarity ideal solution topsis approach to solve molsnlp problems compromise topsis control minimizes the measure of distance providing that the closest solution should have the shortest distance from the positive ideal solution pis as well as the longest distance from the negative ideal solution nis as the measure of closeness l p metric is used thus we reduce a q dimensional objective space to a two dimensional space by a first order compromise procedure the concept of membership function of fuzzy set theory is used to represent the satisfaction level for both criteria also we get a single objective large scale nonlinear programming lsnlp problem using the max min operator for the second order compromise operation finally a numerical illustrative example is given to clarify the main results developed in the paper

compensatory operators in fuzzy linear programming with multiple objectives previously h j zimmermann 11 presented a fuzzy approach to solve the linear vectormaximum problem in the mentioned approach the author uses the minimum or the product operators to translate the semantic meaning of the and the aim of this paper is to reconsider this approach by using operators which allow some degree of compensation between aggregated membership functions firstly the c operator already defined in 12 is used to combine our fuzzy objectives it is shown that solutions obtained by this way are always efficient unfortunately the substitute problem is not generally computationally feasible we suggest then the min bounded sum operator which is a compensatory one has some desired properties and is supported by empirical arguments it allows a considerable decrease of the computational burden in the substitute problem and leads to a solution which is attractive from the stand point of efficiency

interactive decision making for large scale multiobjective linear programs with fuzzy numbers abstract in this paper by considering the experts imprecise or fuzzy understanding of the nature of the parameters in the problem formulation process large scale multiobjective block angular linear programming problems involving fuzzy numbers are formulated through the use of the level sets of fuzzy numbers an extended pareto optimality concept called the pareto optimality is introduced to generate a candidate for the satisficing solution which is also a pareto optimal decision maker is asked to specify the degree and the reference objective values it is shown that the corresponding pareto optimal solution can be easily obtained by solving the minimax problems for which the dantzig wolfe decomposition method is applicable then a linear programming based interactive decision making method for deriving a satisficing solution for the decision maker efficiently from an pareto optimal solution set is presented

diffusion semigroups in spaces of continuous functions with mixed topology abstract we study transition semigroups and kolmogorov equations corresponding to stochastic semilinear equations on a hilbert space h it is shown that the transition semigroup is strongly continuous and locally equicontinuous in the space of polynomially increasing continuous functions on h when endowed with the so called mixed topology as a result we characterize cores of certain second order differential operators in such spaces and show that they have unique extensions to generators of strongly continuous semigroups

perturbation for a class of transition semigroups on the holder space cθb loc h abstract we prove a perturbation result for the ornstein uhlenbeck generator on the holder space c b loc h h is a separable hilbert space as a consequence we obtain the existence of transition semigroups and the solution of elliptic problems associated to stochastic differential equations with non linear drifts

weakly integrable semigroups on locally convex spaces abstract a class of weakly integrable semigroups on locally convex spaces is introduced and studied the results are illustrated by examples of semigroups of unbounded operators on a banach space which include fractional powers of a closed operator and spectral local semigroups

mean ergodic theorems and linear functional equations let a be a closed linear operator with domain d a and range r a in a banach space x let ax and b be two nets of bounded operators on x such that a m r a r b d a and b a ab a i w lim aa x 0 for all x x and s lim a ax 0 for all x d a we first formulate an abstract mean ergodic theorem which contains as special cases many known or new mean ergodic theorems the theorem is then used to study the approximate solutions of the functional equation ax y one of the results says that if y a d a r a then b y converges to the unique solution x in r a the general results provide a unified approach to many specific methods for solving ax y the cases treated here include 1 a i t with t b x 2 a is the generator of a y semigroup 3 a is the generator of a strongly continuous cosine operator function 4 the operator equation a1e ea2 b

fuzzy normed space of operators and its completeness in this paper the new definition of the fuzzy norm of a linear operator from one fuzzy normed linear space into another is introduced and the boundedness of such an operator is described furthermore the space of all bounded linear operators endowed with this fuzzy norm is studied consequently its topological structure as well as completeness is given and that it can itself be made into a fuzzy normed linear space is also shown

determination of the duals of classical sets of sequences of fuzzy numbers and related matrix transformations the convergence of a series of fuzzy sets was examined via zadeh s extension principle by m stojakovic and z stojakovic m stojakovic z stojakovic addition and series of fuzzy sets fuzzy sets and systems 83 1996 341 346 since the utilization of this approach is quite difficult in practice we prefer the idea of using the sum of the series of l level sets the main purpose of the present paper is to determine the a b and c duals of the classical sets of sequences of fuzzy numbers and is to give the necessary and sufficient conditions on an infinite matrix of fuzzy numbers transforming one of the classical sets to the another one

approximation of fuzzy functions by regular fuzzy neural networks in this paper we investigate the ability of regular fuzzy neural networks to provide approximations to fuzzy functions since the operation of regular fuzzy neural networks is based on zadeh s extension principle we first present a level characterization of the zadeh s extensions of level continuous fuzzy valued functions and consider the continuity of these extensions on the basis of this we give characterizations of fuzzy functions which can be approximated by a class of four layer regular fuzzy neural networks according to supremum metric and level convergence

theta functions and hodge numbers of moduli spaces of sheaves on rational surfaces we compute generating functions for the hodge numbers of the moduli spaces of h stable rank 2 sheaves on a rational surface s in terms of theta functions for indefinite lattices if h lies in the closure of the ample cone and has self intersection 0 it follows that the generating functions are jacobi forms in particular the generating functions for the euler numbers can be expressed in terms of modular forms and their transformation behaviour is compatible with the predictions of s duality we also express the generating functions for the signatures in terms of modular forms it turns out that these generating functions are also with respect to another developing parameter the generating function for the donaldson invariants of s evaluated on all powers of the point class