a weakly equivalent condition of convex fuzzy sets abstract in this short communication we first introduce inner straight line g x y on the subset g in the euclidean space rn and define the concept of the weakly quasi convex fuzzy sets second we will give a weak equivalent condition that a fuzzy closed set is a convex fuzzy set

existence of nonoscillatory solutions of higher order neutral delay difference equations with variable coefficients abstract in this paper we consider the following higher order neutral delay difference equations with positive and negative coefficients m x n cx n k p n x n r q n x n l 0 n n 0 where c r m 1 k 1 r l 0 are integers and p n n n 0 and q n n n 0 are sequences of nonnegative real numbers we obtain the global results with respect to c which are some sufficient conditions for the existences of nonoscillatory solutions

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

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

existence of a global smooth solution for a degenerate goursat problem of gas dynamics the goursat problem of a mixed type equation p 0 is considered at the ends of its supports we have p 0 which means it is degenerate hyperbolic we prove the global existence of a smooth solution to the degenerate goursat problem up to a boundary where p 0 this problem comes from the expansion of a wedge of gas with constant velocity into vacuum in two dimensional pressure gradient equations in gas dynamics where p is the pressure and p 0 means vacuum

interaction of four rarefaction waves in the bi symmetric class of the pressure gradient system abstract the global existence and structure of solutions to multi dimensional pressure gradient system has some open problems in this paper we construct global classical solutions to the interaction of four planar rarefaction waves with two axes of symmetry for the pressure gradient system in two space dimensions the bi symmetric initial data is a basic type of four wave two dimensional riemann problems the solutions in this case are continuous bounded and self similar

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

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

simple waves and a characteristic decomposition of the two dimensional compressible euler equations we present a characteristic decomposition of the potential flow equation in the self similar plane the decomposition allows for a proof that any wave adjacent to a constant state is a simple wave for the adiabatic euler system this result is a generalization of the well known result on 2 d steady potential flow and a recent similar result on the pressure gradient system

interaction of rarefaction waves of the two dimensional self similar euler equations we construct classical self similar solutions to the interaction of two arbitrary planar rarefaction waves for the polytropic euler equations in two space dimensions the binary interaction represents a major type of interaction in the two dimensional riemann problems and includes in particular the classical problem of the expansion of a wedge of gas into vacuum based on the hodograph transformation the method employed here involves the phase space analysis of a second order equation and the inversion back to or development onto the physical space

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

on existence of positive solutions and bounded oscillations for neutral difference equations abstract a criterion for the existence of a positive solution for the first order difference equation xn cxn m pnxn k 0 pn 0 is established results are also obtained for the oscillation and nonoscillation of solutions of a second order difference equation 2 xn cxn m pnxn k where the pn are either of constant sign or are oscillatory some of the results are sharp

φ 1 concavity and fuzzy multiple objective decision making concave properties play a dominate role in solving both classic and fuzzy optimization problems however since fuzzy problems are generally represented by sets not crisp numbers various aggregation schemes are needed to manipulate and to combine the different elements in a fuzzy optimization problem based on these different aggregations various concavity properties can be formulated and explored in this paper the intersection aggregation and the convex combination aggregation are explored based on the email protected 1 concave fuzzy sets first the concept of f 1 convexity which covers a wider class of sets and functions is extended to fuzzy sets email protected 1 concave and email protected 1 quasiconcave fuzzy sets are then introduced and some useful aggregation and composition rules are developed based on these aggregation and composition rules and the generalized concave properties fuzzy multiple objective decision making problems are formulated and the conditions to ensure local global maximum property are discussed

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

closed and convex fuzzy sets in this paper we establish a characterization theorem for closed fuzzy sets and give two weak conditions that a closed fuzzy set is a convex fuzzy set we also present some results for convex fuzzy sets strictly convex fuzzy sets and closed fuzzy sets

on the two dimensional gas expansion for compressible euler equations we investigate the problem of two dimensional unsteady expansion of an inviscid polytropic gas which can be interpreted as the collapse of a wedge shaped dam containing water initially with a uniform velocity we model this problem by isentropic euler equations the flow is quasi stationary and using hodograph transform we describe it by a partial differential equation of second order in the state space if it is irrotational initially furthermore this equation is reduced to a linearly degenerate system of three partial differential equations with inhomogeneous source terms these properties are used to prove that the flow is globally smooth when a wedge of gas expands into a vacuum and to analyze that shocks may appear in the interaction of four planar rarefaction waves

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

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

approximative solutions to difference equations of neutral type asymptotic properties of solutions to difference equations of the form m x n u n x n k a n f x n b n are studied replacing the sequence u by its limit and the right side of the equation by zero we obtain an equation which we call the fundamental equation first we investigate the space of all solutions of the fundamental equation we show that any such solution is a sum of a polynomial sequence and a product of a geometric sequence and a periodic sequence next using a new version of the krasnoselski fixed point theorem and the iterated remainder operator we establish sufficient conditions under which a given solution of the fundamental equation is an approximative solution to the above equation our approach based on the iterated remainder operator allows us to control the degree of approximation in this paper we use o ns for a given nonpositive real s as a measure of approximation

a complete global solution to the pressure gradient equation abstract we study the domain of existence of a solution to a riemann problem for the pressure gradient equation in two space dimensions the riemann problem is the expansion of a quadrant of gas of constant state into the other three vacuum quadrants the global existence of a smooth solution was established in dai and zhang z dai t zhang existence of a global smooth solution for a degenerate goursat problem of gas dynamics arch ration mech anal 155 2000 277 298 up to the free boundary of vacuum we prove that the vacuum boundary is the coordinate axes

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

triangular norm based graded convex fuzzy sets in this paper we develop the theory of graded convex fuzzy set graded strongly convex fuzzy set graded strictly convex fuzzy set based on triangular norms in which the degree of fuzzy convexity is described by one real number we also investigate some graded properties of these graded convex fuzzy sets and present the relations between these graded convex fuzzy sets and their topological properties and give several conditions for a fuzzy set to be a graded convex fuzzy set finally we generalize the notion of global maximizer and prove an important property of graded convex fuzzy sets

the definition of convex fuzzy subset following the seminal work of zadeh on the definition of a convex fuzzy subset three new kinds of definitions of convex fuzzy subsets are proposed first by the use of the relations between fuzzy points and fuzzy sets the definition of a convex fuzzy subset is introduced the acceptable nontrivial concepts obtained in this manner are the q convex fuzzy subset and q convex fuzzy subset second by the use of the implication operators of fuzzy logic an r convex fuzzy subset is proposed relations between the convex fuzzy subset and r convex fuzzy subset are discussed finally based on the concept of falling shadow a theoretical approach of convex fuzzy subset is established and the convex fuzzy subset based on t norm is proposed

new oscillation criteria for second order linear difference equations with positive and negative coefficients the oscillation of second order neutral difference equations with positive and negative coefficients of the form d 2 x n la nx n t p nx n d q nx n s 0 l 1 is investigated we obtain many new results using the comparison between both first order and second order difference equations an example is given to show the strength of the obtained results

characteristic decomposition of compressible euler equations for a non ideal gas in two dimensions we consider a two dimensional compressible euler system for a non ideal gas and use the characteristic decomposition to establish that any pseudo steady isentropic irrotational flow adjacent to a constant state must be a simple wave further the constancy of the entropy and vorticity along the pseudo flow characteristics extends the foregoing conclusion to full euler system an attention is drawn to the fact that the result is also applicable to the shallow water system as it bears a close structural resemblance with the system under study these results are generalization of the well known theorem on reducible equations by courant and friedrichs supersonic flow and shock waves springer verlag new york 1999 and a recent result on compressible euler system for an ideal gas by li et al simple waves and a characteristic decomposition of the two dimensional compressible euler equations commun math phys 267 1 12 2006

fuzzy convexity and multiobjective convex optimization problems in this paper based on the more restrictive definition of fuzzy convexity due to ammar and metz 1 several useful composition rules are developed the advantages in using the more restrictive definition of fuzzy convexity are that local optimality implies global optimality and that any convex combination of such convex fuzzy sets is also a convex fuzzy set as shown in this paper these properties are laking in the usual convex fuzzy sets in addition to illustrate the applications in fuzzy convex optimization two examples in multiple objective programming are considered

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

some properties of m fuzzifying convexities induced by m orders abstract in this paper the notions of geometric interval spaces convex geometries and base point orders in the theory of convexity spaces are generalized to m fuzzifying convexity spaces using m fuzzifying restricted hull operators the m fuzzifying convexity spaces induced by m orders are defined conveniently then it is shown that m fuzzifying convexity spaces induced by m orders are m fuzzifying jhc and arity 2 and if the m order is strongly anti symmetric then the m fuzzifying convexity space is an m fuzzifying convex geometry and its segment operator is an m fuzzifying geometric interval operator

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

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

interaction of four rarefaction waves in the bi symmetric class of the two dimensional euler equations the global existence and structures of solutions to multi dimensional unsteady compressible euler equations are interesting and important open problems in this paper we construct global classical solutions to the interaction of four orthogonal planar rarefaction waves with two axes of symmetry for the euler equations in two space dimensions in the case where the initial rarefaction waves are large the bi symmetric initial data is a basic type of four wave two dimensional riemann problems the solutions in this case are continuous bounded and self similar and we characterize how large the rarefaction waves must be we use the methods of hodograph transformation characteristic decomposition and phase space analysis we resolve binary interactions of simple waves in the process

on fuzzy convexity and parametric fuzzy optimization abstract in this paper some different types of convexity and generalized convexity of fuzzy sets are defined and their properties are studied a general fuzzy nonlinear programming problem is formulated with application of the concept of convexity

classification of neutral difference equations of any order with respect to the asymptotic behavior of their solutions the asymptotic behavior of the solutions to a neutral difference equation of the email protected mx n q n x t n p n x s n 0 email protected n n 0 where t n is a general retarded argument s n is a general deviated argument p n n 0 and q n n 0 are sequences of real numbers d denotes the forward difference operator dx n x n 1 x n and d j denotes the jth forward difference operator d jx n email protected d j 1x n for j 2 3 m is studied examples illustrating the results are also given

asymptotic behaviour of bounded solutions of a class of higher order neutral dynamic equations in this paper necessary and sufficient conditions for the asymptotic behaviour of all bounded solutions of a class of nonlinear higher order neutral dynamic equations are stated related to various ranges of the coefficient associated with the neutral part in one of which the coefficient is allowed to oscillate and not tend to zero asymptotically the new results improve and generalize the known results stated for difference equations and differential equations and answer some problems left open for the case t r for arbitrary time scales some certain examples which the results in the literature fail to deliver an answer to are given to illustrate the applicability and significance of the new results

oscillation and nonoscillation for second order linear difference equations new oscillation and nonoscillation theorems are obtained for the second order linear difference equation 2xn 1 pnxn 0 where if pn n 1 is a real sequence with pn 0

preincavity and fuzzy decision making since almost all practical problems are fuzzy and approximate fuzzy decision making becomes one of the most important practical approaches however the resulting problems are frequently complicated and difficult to solve one effective way to overcome these difficulties is to explore the concavity or generalized concavity properties of the resulting problems in this paper we introduce and study the concept of supp preincave and supp prequasiincave fuzzy sets we give characterizations for a supp preincave fuzzy set in terms of its fuzzy hypograph and a supp prequasiincave fuzzy set in terms of its level sets furthermore we also prove that any local maximizer of a supp preincave fuzzy set is also a global maximizer and that any strictly local maximizer of a supp prequasiincave fuzzy set is also a global maximizer finally some aggregation rules for supp preincave and supp prequasiincave fuzzy sets are given and some applications to fuzzy decision making are discussed

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

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