on generalized convexity of fuzzy variables maps in this paper we investigate the convexity and quasiconvexity of fuzzy functions by use the interval analysis some properties of convex and quasiconvex fuzzy functions are introduced applications of fuzzy convexity and quasiconvexity to optimality are discussed

a class of convex fuzzy mappings based on the concept of ordering defined by goetschel and voxman fuzzy sets and systems 18 1996 31 in this paper we present the concepts of epigraph and convexity of the fuzzy mappings and describe characteristics of the convex fuzzy mappings and quasi convex fuzzy mappings then we discuss the properties of convex fuzzy optimizations

the homomorphic image of a fuzzy subgroup is always a fuzzy subgroup let l be a completely distributive complete lattice with universal bound 0 and 1 and x any group we will prove that the image of a fuzzy group x l under a homomorphism f x y is always a fuzzy group without any restriction on and f

fuzzy lie ideals and fuzzy lie subalgebras in this paper we have tried to apply the concepts of fuzzy sets to lie algebra to define some concepts related to fuzzy sets and to discuss their properties and the relations among them

on properties of convex fuzzy mappings the objective of this paper is to obtain some important properties about convex fuzzy mappings based on a linear ordering of fuzzy numbers proposed by goetschel and voxman firstly a new kind of fuzzy mapping termed semistrictly convex fuzzy mapping is defined through this linear ordering note that semistrict convexity does not imply convexity and the interrelationships among convex strictly convex and semistrictly convex mappings are established under certain conditions furthermore this paper obtains several characterizations for the above three classes of fuzzy mappings under the conditions of upper or lower semicontinuity finally some further characteristic properties for semistrictly convex fuzzy mappings are derived

convexity of n dimensional fuzzy number valued functions and its applications the aim of this study is to generalize convexity from vector valued maps to n dimensional fuzzy number valued functions and to investigate some relations among the convexity and generalized convexity of n dimensional fuzzy number valued functions which are based on a new ordering of n dimensional fuzzy numbers proposed in this study furthermore some criteria are obtained for the convexity and generalized convexity under the conditions of lower and upper semicontinuity respectively finally we study the local global minimum properties of the convex fuzzy number valued functions

fuzzy algebraic field extensions abstract in this paper we introduce the concept of fuzzy algebraic field extensions we determine conditions for which a fuzzy algebraic field extension has unique maximal fuzzy intermediate fields which are fuzzy separable algebraic and fuzzy purely inseparable

pseudolinear fuzzy mappings two classes of fuzzy mappings called pseudolinear and pseudolinear fuzzy mappings are introduced by relaxing the definitions of pseudo convex and pseudo invex fuzzy mappings first some characterizations of pseudolinear and pseudolinear fuzzy mappings are obtained then characterizations of the solution sets of pseudolinear and pseudolinear fuzzy programs are derived

nonconvex fuzzy mappings and the fuzzy pre variational inequality in this paper we introduce the concepts of fuzzy pseudoconvex fuzzy invex fuzzy pseudoinvex and fuzzy preinvex mapping from r n to the set of fuzzy numbers and put forward the concepts of fuzzy extended directional differential subgradient and subdifferential some properties are obtained for fuzzy pseudoconvex fuzzy invex fuzzy pseudoinvex and fuzzy preinvex mapping relations are discussed between the fuzzy pre variational inequality and the fuzzy vector optimization problem

on convex and concave fuzzy mappings abstract in this paper we introduce the concepts of convex subsets of f 0 the set of all fuzzy numbers and monotonic fuzzy mappings on r2 and show that a monotonic fuzzy mapping f on a convex set c r1 or r2 is quasi convex and quasi concave we establish two characterizations for convex fuzzy mappings and two characterizations for concave fuzzy mappings we also prove the least upper bound property

quotients with respect to similarity relations abstract the problem of the quotient w r t a similarity relation is closely related to poincare s paradox and central problems in cluster analysis amazingly this problem remains open for more than 30 years as a generalization of strong m categories we introduce the concept of m valued sets which leads to the definition of the category m set in m set we give a complete solution of the quotient problem

fuzzy linear spaces abstract nanda 1986 introduced the concept of fuzzy linear space biswas considered that nanda s definition was not rational and redefined the concept of fuzzy linear space 1989 in this paper we first point out the problems by two counter examples next we redefine the concept of fuzzy field and fuzzy linear space and give some fundamental properties

an efficient runge kutta 4 5 pair abstract a pair of explicit runge kutta formulas of orders 4 and 5 is derived it is significantly more efficient than the fehlberg and dormand prince pairs and by standard measures it is of at least as high quality there are two independent estimates of the local error the local error of the interpolant is to leading order a problem independent function of the local error at the end of the step

fuzzy weirstrass theorem and convex fuzzy mappings the convexity and continuity of fuzzy mappings are defined through a linear ordering and a metric on the set of fuzzy numbers the local global minimum property of real valued convex functions is extended to convex fuzzy mappings it is proved that a strict local minimizer of a quasiconvex fuzzy mapping is also a strict global minimizer characterizations for convex fuzzy mappings and quasiconvex fuzzy mappings are given in addition the weirstrass theorem is extended from real valued functions to fuzzy mappings

φ1 φ2 convex fuzzy mappings abstract the concepts of 1 2 convexity 1 b vexity and 1 convexity for fuzzy mappings are introduced through the so called fuzzy max order among fuzzy numbers we show that the class of 1 2 convex fuzzy mappings 1 b vex fuzzy mappings and 1 convex fuzzy mappings includes many well known classes of fuzzy mappings such as convex fuzzy mappings and preinvex fuzzy mappings as its subclasses we also give a characterization for 1 b vex fuzzy mappings by proposing the concept of 1 b vex sets in addition some basic properties of fuzzy mappings from the standpoint of convex analysis are discussed

a new bound for 3 satisfiable maxsat and its algorithmic application let f be a cnf formula with n variables and m clauses f is 3 satisfiable if for any 3 clauses in f there is a truth assignment which satisfies all of them lieberherr and specker 1982 and later yannakakis 1994 proved that in each 3 satisfiable cnf formula at least 23 of its clauses can be satisfied by a truth assignment we improve this result by showing that every 3 satisfiable cnf formula f contains a subset of variables u such that some truth assignment t will satisfy at least 23m 13m u rn clauses where m is the number of clauses of f m u is the number of clauses of f containing a variable from u n is the total number of variables in clauses not containing a variable in u and r is a positive absolute constant both u and t can be found in polynomial time we use our result to show that the following parameterized problem is fixed parameter tractable and moreover has a kernel with a linear number of variables in 3 s maxsat ae we are given a 3 satisfiable cnf formula f with m clauses and asked to determine whether there is an assignment which satisfies at least 23m k clauses where k is the parameter

fuzzy cosets and some fuzzy radicals abstract an attempt is made to formulate fuzzy versions of jacobson and prime radicals of a ring certain basic results pertaining to the notions of jacobson and prime radicals of a ring are extended to the fuzzy setting in particular the following is obtained the jacobson radical prime radical of the ring of fuzzy cosets of in r is zero where r is a commutative ring with identity and is the fuzzy jacobson radical fuzzy prime radical of r

fuzzy quotient groups abstract the operation of a group g induces through zadeh s extension principle a semigroup operation on ig we define a fuzzy quotient group of g to be a fuzzy partition of g that is a subsemigroup of ig having a group structure we show that fuzzy quotient groups of g are in one to one correspondence with normal fuzzy subgroups of g and that their associated fuzzy similarity relations are all two sided invariant we study relationships between the level structures of these structures we also give a necessary and sufficient condition for a group homomorphism to factor through fuzzy quotients

certain fuzzy ideals of rings redefined abstract an attempt is made to redefine fuzzy semiprimary fuzzy primary fuzzy prime and fuzzy maximal ideals of rings these fuzzy ideals are characterized by their level ideals fuzzy analogues of various results concerning semiprimary primary prime and maximal ideals are obtained the concept of a fuzzy nil radical of a fuzzy ideal is also formulated and discussed

robust defect control with runge kutta schemes enright numerical analysis report 122 university of manchester manchester u k 1986 implements a runge kutta method for solving the initial value problem using an alternative to the standard local error control scheme the aim is to control the defect associated with a local interpolant by sampling its value at one or more fixed points within each step however in general the quality of a sample point is problem dependent and also varies from step to step two classes of interpolant are presented for which the asymptotic behaviour of the defect is known a priori allowing optimal sample points to be chosen

fuzzy vector spaces abstract let v denote a vector space over a field f and let a denote a fuzzy subspace of v over a fuzzy subfield k of f let x be a fuzzy subset of v such that x a and let x denote the intersection of all fuzzy subspaces of v over k that contain x and are contained in a we characterize the fuzzy subspace x of a over k we use this result to introduce the concept of fuzzy freeness of a fuzzy subset x of v and characterize it in terms of linear independence in the usual sense

the relative efficiency of alternative defect control schemes for high order continuous runge kutta formulas the development of continuous runge kutta formulas for nonstiff problems has been the subject of several investigations in recent years in enright siam j numer anal 26 1989 pp 588 599 the author proposed a new error and stepsize control strategy based on controlling the size of the defect of the underlying continuous solution that is particularly appropriate for numerical methods based on this class of formulas in this investigation the author presents an analysis of the relative efficiency of existing continuous runge kutta methods the author introduces theoretical measures useful in evaluating the potential of such methods and apply these measures to several methods of orders 4 through 8 the author also presents numerical results to illustrate that the theoretical measures of potential are consistent with the observed performance of nonstiff problems over a wide range of error tolerances

adaptive stepsize based on control theory for stochastic differential equations the numerical solution of stochastic differential equations sdes has been focussed recently on the development of numerical methods with good stability and order properties these numerical implementations have been made with fixed stepsize but there are many situations when a fixed stepsize is not appropriate in the numerical solution of ordinary differential equations much work has been carried out on developing robust implementation techniques using variable stepsize it has been necessary in the deterministic case to consider the best choice for an initial stepsize as well as developing effective strategies for stepsize control the same of course must be carried out in the stochastic case in this paper proportional integral pi control is applied to a variable stepsize implementation of an embedded pair of stochastic runge kutta methods used to obtain numerical solutions of nonstiff sdes for stiff sdes the embedded pair of the balanced milstein and balanced implicit method is implemented in variable stepsize mode using a predictive controller for the stepsize change the extension of these stepsize controllers from a digital filter theory point of view via pi with derivative pid control will also be implemented the implementations show the improvement in efficiency that can be attained when using these control theory approaches compared with the regular stepsize change strategy

runge kutta triples abstract the addition of a third runge kutta rk formula to an embedded pair forming an rk triple can yield an algorithm which gives solutions of appropriate asymptotic accuracy at points within the normal step intervals in the numerical solution of the initial value problem two modes of implementation are possible and it is shown that these can be regarded as equivalent when certain conditions are satisfied an optimized rk5 4 triple based on a particularly efficient embedded pair is presented and tested the results indicate the efficiency of the dense output technique

preinvexity and φ 1 convexity of fuzzy mappings through a linear ordering the preinvexity prequasiinvexity 1 convexity and 1 quasiconvexity of fuzzy mappings are defined based on a linear ordering on the set of fuzzy numbers characterizations for these fuzzy mappings are obtained the local global minimum properties of real valued preinvex functions and 1 convex functions are extended to preinvex fuzzy mappings and 1 convex fuzzy mappings respectively it is also proved that every strict local minimizer of a prequasiinvex fuzzy mapping is a strict global minimizer and that every strict local minimizer of a 1 quasiconvex fuzzy mapping is a strict global minimizer

convex fuzzy mapping and operations of convex fuzzy mappings convex and concave fuzzy mappings are considered in this paper based on the seminal work of nanda and kar on the definition of convex fuzzy mapping the operations of convex fuzzy mapping are established some of the important concepts such as positively homogeneous infimal convolution right scalar multiplication and convex hull are introduced and the corresponding theorems are demonstrated by using the parametric representation of fuzzy numbers

on integration of fuzzy mappings abstract in this paper we introduce the concept of an integral of a fuzzy mapping this generalises the integral introduced by matloka 10 and also extends riemann stieltjes integral over a closed interval to fuzzy mappings

fuzzy field extensions abstract we introduce the concept of fuzzy field extension linear disjointness for fuzzy field extensions and the composite of two fuzzy subfields the concepts of pure inseparability and separability for fuzzy field extensions are introduced

generating properties of fuzzy algebraic structures abstract an axiomatic approach is taken to the concept of fuzzy generators for fuzzy subspaces fuzzy field extensions and fuzzy subgroups the fundamental generating properties necessary to yield fuzzy bases and uniqueness of their cardinalities for these structures are given

fuzzy hypervector spaces based on fuzzy singletons we introduce and study fuzzy hypervector spaces based on fuzzy singletons in this regard by considering the notion of fuzzy singletons we characterize a fuzzy hypervector space fuzzily spanned by a fuzzy subset then we use these results to introduce the concept of fuzzy freeness of a fuzzy subset m of a hypervector space v and finally we characterize it in terms of linear independence in the usual sense

convex fuzzy mapping with differentiability and its application in fuzzy optimization abstract the concepts of differentiability convexity generalized convexity and minimization of a fuzzy mapping are known in the literature the purpose of this present paper is to extend and generalize these concepts to fuzzy mappings of several variables using buckley feuring approach for fuzzy differentiation and derive karush kuhn tucker condition for the constrained fuzzy minimization problem

fuzzy partitions and fuzzy quotient rings abstract the cosets of a fuzzy ideal m in a ring a form another ring a m called the fuzzy quotient ring of a and m we show that a m is always a fuzzy partition of a and a fuzzy partition of a will be a fuzzy quotient ring with respect to some fuzzy ideal if and only if its associated fuzzy similarity relation is compatible with the ring operations we also discuss relationships between the two operations of the fuzzy quotient ring a m and the extensions over ia of the two operations of a by zadeh s extension principle we study relationships between the level ring structures of a fuzzy quotient ring and the level relations of its associated fuzzy similarity relations we introduce and study a new notion of the factorization of a ring homomorphism through fuzzy quotients finally a relationship between fuzzy prime ideals and fuzzy similarity relation has been established

directional derivatives and subdifferential of convex fuzzy mappings and application in convex fuzzy programming abstract in this paper we put forward the concepts of directional derivative differential and subdifferential of fuzzy mappings from r n into e 1 and discuss the characterizations of directional derivative and differential by respectively using the directional derivative and differential of two crisp functions that are determined by the fuzzy mapping and we also consider the problem of existence of directional derivative for convex fuzzy mappings and discuss the relations among directional derivative differential and subdifferential of fuzzy mappings at last we give two results of application in convex fuzzy programming

fuzzy fields and fuzzy linear spaces redefined we here redefine fuzzy fields and fuzzy linear spaces which were initially defined by nanda 5 our definitions are more suitable than the earlier

preinvex fuzzy mappings abstract in this paper we introduce the concepts of preinvex fuzzy mappings quasi preinvex fuzzy mappings and invex sets by modifying the concepts proposed by noor we also give two characterization theorems for preinvex fuzzy mappings and present some new results for preinvex fuzzy mappings strictly preinvex fuzzy mappings and quasi preinvex fuzzy mappings

fuzzy algebras over fuzzy fields redefined abstract first we point out that nanda s 1986 definition of fuzzy field was irrational hence the definition of fuzzy algebras over fuzzy fields 1990 was also irrational secondly we redefined fuzzy fields and fuzzy algebras over fuzzy fields finally we prove without any restriction that homomorphic images of fuzzy algebras form are fuzzy algebra

fourth and fifth order scaled rungs kutta algorithms for treating dense output new runge kutta algorithms are developed which determine the solution of a system of ordinary differential equations at any point within a given integration step as well as at the end of each step

on fuzzy generalized convex mappings and optimality conditions for fuzzy weakly univex mappings in this paper we first introduce the weakly invex fuzzy mappings based on weakly differentiable functions and discuss the relationships between several kinds of fuzzy generalized convex mappings then a kind of fuzzy weakly univex functions are introduced and some properties of them are investigated furthermore optimality and duality results are derived for a class of generalized convex optimization problems with fuzzy weakly univex functions

continuous numerical methods for odes with defect control abstract over the last decade several general purpose numerical methods for ordinary differential equations odes have been developed which generate a continuous piecewise polynomial approximation that is defined for all values of the independent variable in the range of interest for such methods it is possible to introduce measures of the quality of the approximate solution based on how well the piecewise polynomial satisfies the ode this leads naturally to the notion of defect control numerical methods that adopt error estimation and stepsize selection strategies in order to control the magnitude of the associated defect can be very effective and such methods are now being widely used in this paper we will review the advantages of this class of numerical methods and present examples of how they can be effectively applied we will focus on numerical methods for initial value problems ivps and boundary value problems bvps where most of the developments have been introduced but we will also discuss the implications and related developments for other classes of odes such as delay differential equations ddes and differential algebraic equations daes

free fuzzy modules and their bases abstract the notions of a basis for fuzzy modules and of a free fuzzy module are defined and the elementary properties of both are studied it is shown that a fuzzy module possesses a basis if and only if it is free results on the existence of a basis are obtained

bases of fuzzy vector spaces abstract let v be a vector space over a field f let a be a fuzzy subspace of v over f we give sufficient conditions for a to have a basis over a fuzzy subfield k of f we also show that if a has the sup property then a has a fully free basis over f but not necessarily a fully free basis over k

galois theory for fuzzy fields abstract the aim of this paper is to present a generalized form of the galois connection between the intermediate fields of a field extension and the subgroups of the corresponding galois group we prove that the fundamental theorem of galois theory accepts a generalization in the fuzzy framework

fuzzy vector spaces abstract this paper defines a basis for a fuzzy vector space and its dimension a class of fuzzy vector spaces having dimension is introduced and two standard results from crisp theory are proved for this case namely d i m e1 e2 d i m e1 d i m e2 d i m e1 e2 and rmd i m i m f d i m k e r f the product of two fuzzy vector spaces under certain conditions is also characterised

invex and generalized convex fuzzy mappings in this paper we introduce the concepts of pseudo convexity invexity and pseudo invexity for fuzzy mappings of one variable based on the notion of differentiability proposed by goetschel and voxman 4 and investigate the relationship between convex fuzzy mappings preinvex fuzzy mappings and these classes of fuzzy mappings we shall prove that pseudo convex fuzzy mappings and invex fuzzy mappings are pseudo invex and that a differentiable convex resp preinvex fuzzy mapping is pseudo convex resp invex in addition sufficient optimality conditions are obtained for pseudo convex invex and pseudo invex fuzzy mappings

some properties of weakly convex fuzzy mappings in this paper we introduce the concepts of weakly convex fuzzy sets weakly convex fuzzy mappings and weakly quasi convex fuzzy mappings and then give characterization theorems for these fuzzy sets and fuzzy mappings

a note on generalized convexity for fuzzy mappings through a linear ordering in de campos ibanez and gonzalez munoz fuzzy sets syst 29 145 154 1989 6 goestschel and voxman fuzzy sets syst 18 31 43 1986 7 the authors considered a linear ordering on the space of fuzzy intervals for each fuzzy mapping fuzzy interval valued mapping f based on the aforementioned linear ordering they introduced a real valued function t f on the domain of the fuzzy mapping f recently chalco cano et al fuzzy sets syst 231 70 83 2013 4 have studied the relationship between the generalized hukuhara differentiability of a fuzzy mapping f g differentiability for short and the differentiability of t f and some properties of local global minima this paper studies such properties for fuzzy mappings using new concepts which generalize the existing ones

differentiability and convexity of fuzzy mappings abstract goetschel and voxman 1 have introduced the notion of a derivative for fuzzy mappings of one variable in a manner different from the usual one in this paper we define a differentiable fuzzy mapping of several variables in ways that parallel the definition proposed by goetschel and voxman 1 for a fuzzy mapping of one variable and then study some basic differentiability properties of fuzzy mappings from the standpoint of convex analysis

on t fuzzy subfield and t fuzzy vector subspaces abstract in this paper we improve the definition of t fuzzy subfield and observe some of its properties the concept of t fuzzy vector subspaces of a t fuzzy subfield of a field will also be investigated and we further introduce the concept of t fuzzy subalgebra

on characterizations of preinvex fuzzy mappings several characterizations about preinvex fuzzy mapping are obtained in this paper firstly an equivalent condition of preinvex fuzzy mapping is established under certain conditions furthermore the necessary and sufficient conditions for differentiable and twice differentiable preinvex fuzzy mapping are provided by using the given equivalent condition of preinvex fuzzy mapping finally a new proof of some known important conclusions is offered these results generalize and improve some known results

blockwise adaptivity for time dependent problems based on coarse scale adjoint solutions we describe and test an adaptive algorithm for evolution problems that employs a sequence of blocks consisting of fixed though nonuniform space meshes this approach offers the advantages of adaptive mesh refinement but with reduced overhead costs associated with load balancing remeshing matrix reassembly and the solution of adjoint problems used to estimate discretization error and the effects of mesh changes a major issue with a block adaptive approach is determining block discretizations from coarse scale solution information that achieve the desired accuracy we describe several strategies for achieving this goal using adjoint based a posteriori error estimates and we demonstrate the behavior of the proposed algorithms as well as several technical issues in a set of examples

fuzzy ideals and fuzzy subalgebras of lie algebras abstract a fuzzy idea l of a lie algebra l factors it into a lie algebra l l called a fuzzy quotient lie algebra this fuzzy algebra lie algebra is always a fuzzy partition of l and a fuzzy partition of l will be a fuzzy quotient lie algebra if and only if its associated fuzzy similarity relation is compatible with the lie algebra operations relationship between the operations of the lie algebras l i t and zadeh s extension principle are studied we also study homomorphisms of lie algebras and the relationship between them and fuzzy ideals and fuzzy subalgebras of the domains and the co domains of these homomorphisms finally solvability of fuzzy ideals and studied in defining these solvable fuzzy ideals we use zadeh s extension principle

fuzzy linear and affine spaces abstract elementary properties of fuzzy linear and affine spaces are studied the concept of a basis for a fuzzy linear space is defined and an algorithm for computing a basis when the underlying space is finite dimensional is given

fuzzy subgroups as union of two fuzzy subgroups abstract the problem of writing a fuzzy subgroup of a group g as a union of two fuzzy subgroups has been discussed by dixit et al in this paper an attempt has been made to study this problem further in a broader framework of nontrivial decomposition of a fuzzy subgroup thus something more than the settlement of the question of the possibility of having nonequivalent fuzzy subgroups in this union raised by the author at the end of the section 3 of their paper has been proved

fuzzy normal subgroups and fuzzy quotients abstract in case of groups a mormal subgroup n of a group g can be defined with three equivalent approaches namely i as a kernel of some group homomorphism ii as a subgroup commuting with every element of g and iii as the subgroup n closed with respect to conjugates of elements of n in case of generalizations of this concept to fuzzy sets the elementwise third approach is found more suitable and hence is widely adopted by many authors in this paper we further study this concept of fuzzy subgroups and introduce fuzzy quotients called fuzzy quotient groups for all 0 1 two different definitions of these fuzzy quotients have been given which are shown to lead to isomorphic structures we also study the direct products of fuzzy normal subgroups min product as per the terminology of sherwood 1983 and the problem of writing a fuzzy normal subgroups of a direct product of some groups as a direct product of certain fuzzy normal subgroups has been discussed in detail

on the dimension of a fuzzy subspace abstract isomorphism between two fuzzy subspaces is defined and some properties of two isomorphic fuzzy subspaces are discussed in terms of their image sets level subspaces and dimensions the dimension of a fuzzy quotient space is also studied

fuzzy inner product spaces and fuzzy norm functions abstract we define fuzzy inner product spaces in a linear space the fuzzy inner product of an element of the linear space fuzzy norm function the fuzzy norm of an element and fuzzy orthogonality of one element to the other we prove some results on these and show that the fuzzy intersection of two fuzzy inner product spaces is also a fuzzy inner product space

generalized convex fuzzy mappings and fuzzy variational like inequality in this paper we introduce the concepts of some generalized convex fuzzy mappings from r n to the set of fuzzy numbers under the condition of upper or lower semicontinuity a new criterion be obtained for the existence of a fuzzy preinvex mapping a fuzzy preinvex mapping can be achieved when the invex subset of r n satisfies certain conditions the relations between generalized convex fuzzy mappings are discussed and some properties are obtained that relate the concepts of fuzzy pseudoinvex and fuzzy prequasiinvex mapping fuzzy invex and fuzzy preinvex mapping fuzzy preinvex and fuzzy prequasiinvex mapping we also discuss the relationship between the fuzzy variational like inequality the fuzzy weak variational like inequality and fuzzy optimization problems and an application example of a fuzzy variational like inequality the fuzzy variational like inequality representation of a fuzzy transportation equilibrium problem

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

correspondence theorem for fuzzy ideals abstract the ring r j of cosets of a fuzzy ideal j of a ring r is defined and shown to be isomorphic to a factor ring of r in a natural way if f r r is an epimorphism of rings a one to one order preserving correspondence is established between the fuzzy ideals of r and those of r which are constant on the kernel of f this generalizes the correspondence theorem of non fuzzy ideals

convexity and semicontinuity of fuzzy mappings in this paper a criterion for the convex fuzzy mapping is obtained under the condition of upper and lower semicontinuity respectively an upper lower semicontinuous fuzzy mapping is proved which convexity is equivalent to weak convexity or b vexity satisfying a special condition

fuzzy irreducible ideals in rings abstract in this paper the concept of a fuzzy set is extended by introducing the concept of a fuzzy irreducible ideal in a ring the characteristic function of an irreducible ideal and the level ideals of a fuzzy irreducible ideal are studied the algebraic nature of homomorphic images and preimages of fuzzy irreducible ideals is also examined some other fundamental results pertaining to this concept are discussed

generalization of preinvex and b vex fuzzy mappings a class of fuzzy mappings called b preinvex and strictly b preinvex fuzzy mappings is introduced by relaxing the definition of preinvexity of a fuzzy mapping similarly based on a notion of differentiability of fuzzy mappings different from the usual one the class of pseudo b vex b invex and pseudo b invex fuzzy mappings is defined as a generalization of pseudo convex invex and pseudo invex fuzzy mappings we prove that a strictly b vex or b preinvex fuzzy mapping has at most one global minimum point and that the class of b vex fuzzy mappings forms a subset of the class of quasi convex fuzzy mappings b vex resp b preinvex fuzzy mappings satisfy most of the basic properties of convex resp preinvex fuzzy mappings in addition characterizations and sufficient optimality conditions are obtained for b vex and b preinvex fuzzy mappings

fuzzy logic based generalized decision theory with imperfect information the existing decision models have been successfully applied to solving many decision problems in management business economics and other fields but nowadays arises a need to develop more realistic decision models the main drawback of the existing utility theories starting from von neumann moregnstern expected utility to the advanced non expected models is that they are designed for laboratory examples with simple well defined gambles which do not adequately enough reflect real decision situations in real life decision making problems preferences are vague and decision relevant information is imperfect as described in natural language nl vagueness of preferences and imperfect decision relevant information require using suitable utility model which would be fundamentally different to the existing precise utility models precise utility models cannot reflect vagueness of preferences vagueness of objective conditions and outcomes imprecise beliefs etc the time has come for a new generation of decision theories in this study we propose a decision theory which is capable to deal with vague preferences and imperfect information the theory discussed here is based on a fuzzy valued non expected utility model representing linguistic preference relations and imprecise beliefs

fuzzy linear spaces over valued fields abstract the concept of fuzzy linear spaces was introduced by nanda 3 the purpose of the present paper is to introduce some further concepts such as convex fuzzy sets etc in fuzzy linear spaces over any valued field

interpolation for runge kutta methods runge kutta methods provide a popular way to solve the initial value problem for a system of ordinary differential equations in contrast to the adams methods there is no natural way to approximate the solution between mesh points a way to accomplish this is proposed which is applicable to some important formulas its theoretical support is much better than that of interpolation in the popular variable order variable step adams codes

generalized convexity in fuzzy vector optimization through a linear ordering in this article we study efficiency and weakly efficiency in fuzzy vector optimization after formulating the problem we introduce two new concepts of generalized convexity for fuzzy vector mappings based on the generalized hukuhara differentiability pseudoinvexity i and pseudoinvexity ii we prove that pseudoinvexity is the necessary and sufficient condition for a stationary point to be a solution of a fuzzy vector optimization problem we give conditions to insure that a fuzzy vector mapping is invex and pseudoinvex i and ii moreover we present some examples to illustrate the results lastly we use these results to study the class of problems which have uncertainty and inaccuracies in the objective function coefficients of mathematical programming models

duality theory in fuzzy mathematical programming problems with fuzzy coefficients in this paper the notions of subgradient subdifferential and differential with respect to convex fuzzy mappings are investigated which provides the basis for the fuzzy extremum problem theory we consider the problems of minimizing or maximizing a convex fuzzy mapping over a convex set and develop the necessary and or sufficient optimality conditions furthermore the concept of saddle points and minimax theorems under fuzzy environment is discussed the results obtained are used to formulate the lagrangian dual of fuzzy programming under certain fuzzy convexity assumptions kkt conditions for fuzzy programming are derived and the perturbed convex fuzzy programming is considered finally these results are applied to fuzzy linear programming and fuzzy quadratic programming

global error estimation based on the tolerance proportionality for some adaptive runge kutta codes modern codes for the numerical solution of initial value problems ivps in odes are based in adaptive methods that for a user supplied tolerance d attempt to advance the integration selecting the size of each step so that some measure of the local error is email protected although this policy does not ensure that the global errors are under the prescribed tolerance after the early studies of stetter considerations concerning a theory for ode solvers in r burlisch r d grigorieff j schroder eds numerical treatment of differential equations proceedings of oberwolfach 1976 lecture notes in mathematics vol 631 springer berlin 1978 pp 188 200 tolerance proportionality in ode codes in r marz ed proceedings of the second conference on numerical treatment of ordinary differential equations humbold university berlin 1980 pp 109 123 and the extensions of higham global error versus tolerance for explicit runge kutta methods ima j numer anal 11 1991 457 480 the tolerance proportionality of adaptive ode solvers j comput appl math 45 1993 227 236 the reliability of standard local error control algorithms for initial value ordinary differential equations in proceedings the quality of numerical software assessment and enhancement ifip series springer berlin 1997 it has been proved that in many existing explicit runge kutta codes the global errors behave asymptotically as some rational power of d this step size policy for a given ivp determines at each grid point t n a new step size h n 1 h t n d so that h t d is a continuous function of t in this paper a study of the tolerance proportionality property under a discontinuous step size policy that does not allow to change the size of the step if the step size ratio between two consecutive steps is close to unity is carried out this theory is applied to obtain global error estimations in a few problems that have been solved with the code gauss2 s gonzalez pinto r rojas bello gauss2 a fortran 90 code for second order initial value problems based on an adaptive two stage runge kutta gauss method with this discontinuous step size policy

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

on invex fuzzy mappings and fuzzy variational like inequalities in this paper we first show the need for introducing invex fuzzy mappings after that we show that the concept of invex fuzzy mapping previously given by wu and xu are very restrictive and the examples presented there are not correct then we present more general concepts of invex and incave fuzzy mappings involving strongly generalized differentiable fuzzy mapping finally we show that the results obtained by wu and xu on the relationship between fuzzy variational like inequalities and fuzzy optimization problems are still valid using these new concepts

on fuzzy inner product spaces and fuzzy co inner product spaces abstract the notions of fuzzy co norm and fuzzy co orthogonality are introduced by formulating the notion of a fuzzy co inner product space and their basic properties are studied

on global error estimation and control for initial value problems this paper addresses global error estimation and control for initial value problems for ordinary differential equations the focus lies on a comparison between a novel approach based on the adjoint method combined with a small sample statistical initialization and the classical approach based on the first variational equation control is achieved through tolerance proportionality both approaches are found to work well and to enable estimation and control in a reliable manner

beating the random ordering is hard every ordering csp is approximation resistant we prove that assuming the unique games conjecture ugc every problem in the class of ordering constraint satisfaction problems ocsps where each constraint has constant arity is approximation resistant in other words we show that if rho is the expected fraction of constraints satisfied by a random ordering then obtaining a rho approximation for any rho rho is ug hard for the simplest ocsp the maximum acyclic subgraph mas problem this implies that obtaining a rho approximation for any constant rho 1 2 is ug hard specifically for every constant varepsilon 0 the following holds given a directed graph g that has an acyclic subgraph consisting of a fraction 1 varepsilon of its edges it is ug hard to find one with more than 1 2 varepsilon of its edges note that it is trivial to find an acyclic subgraph with 1 2 the edges by taking either the forward or backward edges in an arbitrary ordering of the vertices of g the mas problem has been well studied and beating the random ordering for mas has been a basic open problem an ocsp of arity k is specified by a subset pi subseteq s k of permutations on 1 2 dots k an instance of such an ocsp is a set v and a collection of constraints each of which is an ordered k tuple of v the objective is to find a global linear ordering of v while maximizing the number of constraints ordered as in pi a random ordering of v is expected to satisfy a rho frac pi k fraction we show that for any fixed k it is hard to obtain a rho approximation for pi ocsp for any rho rho the result is in fact stronger we show that for every lambda subseteq pi subseteq s k and an arbitrarily small varepsilon it is hard to distinguish instances where a 1 varepsilon fraction of the constraints can be ordered according to lambda from instances where at most a rho varepsilon fraction can be ordered as in pi a special case of our result is that the betweenness problem is hard to approximate beyond a factor 1 3 the results naturally generalize to ocsps which assign a payoff to the different permutations finally our results imply unconditionally that a simple semidefinite relaxation for mas does not suffice to obtain a better approximation

fuzzy preinvex functions in this paper we introduce the concept of fuzzy preinvex functions and invex sets these functions are more general than convex fuzzy functions and when differentiable are fuzzy invex some different types of fuzzy preinvexity functions are defined and their properties are studied it is proved that the minimum of the fuzzy preinvex functions on the invex sets can be characterized by variational like inequalities

prime fuzzy ideals over noncommutative rings in this paper we introduce prime fuzzy ideals over a noncommutative ring this notion of primeness is equivalent to level cuts being crisp prime ideals it also generalizes the one provided by kumbhojkar and bapat in 1993 which lacks this equivalence in a noncommutative setting semiprime fuzzy ideals over a noncommutative ring are also defined and characterized as intersection of primes this allows us to introduce the fuzzy prime radical and contribute to establish the basis of a fuzzy noncommutative ring theory

parameterized complexity of maxsat above average in maxsat we are given a cnf formula f with n variables and m clauses the task is to find a truth assignment satisfying the maximum number of clauses let r 1 r m be the number of literals in the clauses of f then asat f email protected i 1 m 1 2 r i is the expected number of clauses satisfied by a random truth assignment when the truth values to the variables are distributed uniformly and independently it is well known that in polynomial time one can find a truth assignment satisfying at least asat f clauses in the parameterized problem maxsat aa we are to decide whether there is a truth assignment satisfying at least asat f k clauses where k is the nonnegative parameter we prove that maxsat aa is para np complete and thus maxsat aa is not fixed parameter tractable unless p np this is in sharp contrast to the similar problem maxlin 2 aa which was recently proved to be fixed parameter tractable by crowston et al fsttcs 2011 in fact we consider a more refined version of maxsat aa max r n sat aa where r email protected r n for each j alon et al soda 2010 proved that if r r n is a constant then max r sat aa is fixed parameter tractable we prove that max r n sat aa is para np complete for any r n email protected email protected we also prove that assuming the exponential time hypothesis max r n sat aa is not even in xp for any integral r n email protected n where f n is any real valued unbounded strictly increasing computable function this lower bound on r n cannot be decreased much further as we prove that max r n sat aa is i in xp for any r n loglogn logloglogn and ii fixed parameter tractable for any r n email protected n where f n is any real valued unbounded strictly increasing computable function the proof uses some results on maxlin 2 aa

convex fuzzy mappings abstract in this note the concept of convexity and logarithmic convexity for fuzzy mappings are introduced and some applications to optimization are discussed

necessary and sufficient conditions for fuzzy optimality problems in this paper we define a new minimum concept for fuzzy optimization problems more general than those that exist in the literature we find necessary optimality conditions based on a new fuzzy stationary point definition and we prove that these conditions are also sufficient under new fuzzy generalized convexity notions