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

the large n limit of the threshold values in mandelbrot s fractal percolation process the authors consider mandelbrot s fractal percolation process characterised by a density parameter p in 0 1 and an integer subdivision index n 1 for each n the process is known to have a percolation transition at density pc n in 0 1 they prove that lim pc n pc where pc is the threshold value of the ordinary square lattice site percolation model

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

phase diagram and correlation length bounds for mandelbrot aerogels the authors study a variant of the mandelbrot percolation process which is of current use as a model of aerogels the model has two parameters one of them q is the usual multiscale parameter of the mandelbrot percolation process and the other p is a bernoulli percolation parameter that is reserved for the last step of the construction they investigate the phase diagram of this model in the q p plane there are two phases a sol phase and a gel phase classified according to whether a limit of crossing probabilities vanishes or is nonzero in the sol phase they define a correlation length via the rate of decay of a rescaled connectivity function they show that this length scale diverges at the phase boundary furthermore they demonstrate that if the phase boundary is approached with q fixed and p tending up to its critical value pg q then up to logarithmic corrections the divergence is at least as fast as mod pg p mod 2dh where dh 2 mod log q log n mod can be identified as the hausdorff dimension of the background medium

membership function elicitation in possibilistic programming problems in this paper we propose an elicitation method for membership functions in the setting of possibilistic programming each possibility distribution is defined by several possible domains with different possibility degrees taking advantage of the properties of possibility and necessity measures a nested target table for the objective function and for each constraint is elicited from a decision maker to aggregate all the goals an additional nested target table is established through those nested target tables membership functions can be determined it is emphasized that the information of nested target tables is much more important than that of membership values in the setting of possibilistic programming the membership functions in possibilistic programming are not assumed to be cardinal but ordinal in this sense the decision making via possibilistic programming can be regarded as a qualitative one

on the newton method for solving fuzzy optimization problems in this article we consider optimization problems where the objectives are fuzzy functions fuzzy valued functions for this class of fuzzy optimization problems we discuss the newton method to find a non dominated solution for this purpose we use the generalized hukuhara differentiability notion which is the most general concept of existing differentiability for fuzzy functions this work improves and corrects the newton method previously proposed in the literature

an extension to possibilistic linear programming abstract this paper proposes an extension to the solution of possibilistic linear programming problems with fuzzy number parameters introduced by buckley fuzzy sets and systems31 1989 329 341 the method relies on cuts of the fuzzy number parameters to generate a succession of pairs of classical linear programs the solutions of these linear programs produce a convex possibility distribution of the optimal objective values with the corresponding levels of the decision variables the approach is illustrated with an imprecise water quality management problem

strict monotonicity for critical points in percolation and ferromagnetic models when is the numerical value of the critical point changed by an enhancement of the process or of the interaction ferromagnetic spin models independent percolation and the contact process are known to be endowed with monotonicity properties in that certain enhancements are capable of shifting the corresponding phase transition in only an obvious direction e g the addition of ferromagnetic couplings can only increase the transition temperature the question explored here is whether enhancements do indeed change the value of the critical point we present a generally applicable approach to this issue for ferromagnetic ising spin systems with pair interactions of finite range ind 2 dimensions it is shown that the critical temperaturetc is strictly monotone increasing in each coupling with the first order derivatives bounded by positive functions which are continuous on the set of fullyd dimensional interactions for independent percolation with 0 pc 1 we prove that any essential enhancement of the process has an effect on the critical probability a result with applications to the question of the existence of entanglements and to invasion percolation with trapping

on the structure of mandelbrot s percolation process and other random cantor sets we consider generalizations of mandelbrot s percolation process for the process which we call the random sierpinski carpet we show that it passes through several different phases as its parameter increases from zero to one the final section treats the percolation phase

fuzzy multiple objective programming and compromise programming with pareto optimum abstract a fuzzy multiple objective decision making approach is proposed based on the desirable features of compromise programming and the fuzzy set theory the proposed two phase approach guarantees both nondominated and balanced solutions for solving both the crisp and the fuzzy multiple objective decision making problems furthermore it is shown that the original compromise programming does not guarantee nondominated solutions if the distance parameter is assumed to be infinite and the resulting programming problem s solution is not unique

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

optimality for fuzzified mathematical programming problems a parametric approach from a conventional mathematical programming model and in accordance with which fuzzification is used several models of fuzzy mathematical programming problems can be obtained this paper deals with the study of the optimality concept for g p fuzzified mathematical programming problems an auxiliary parametric mathematical programming problem is presented which allows the above model to be solved in a straightforward way in addition some results about the g p fuzzified mathematical programming problem are obtained using the parametric mathematical programming problem

connectivity properties of mandelbrot s percolation process in 1974 mandelbrot introduced a process in 0 1 2 which he called canonical curdling and later used in this book s on fractals to generate self similar random sets with hausdorff dimension d 0 2 in this paper we will study the connectivity or percolation properties of these sets proving all of the claims he made in sect 23 of the fractal geometry of nature and a new one that he did not anticipate there is a probability pc 0 1 so that if p pc then the set is duslike i e the largest connected component is a point whereas if p pc notice the opposing sides are connected with positive probability and furthermore if we tile the plane with independent copies of the system then there is with probability one a unique unbounded connected component which intersects a positive fraction of the tiles more succinctly put the system has a first order phase transition

rees matrix covers for regular semigroups by a local submonoid of a regular semigroup s we mean a subset of the form ese where e is an idempotent of s many classes of regular semigroups can be defined in terms of properties of their local submonoids for example rectangular bands can be characterized as those regular semigroups all of whose local submonoids are trivial while completely simple semigroups are those whose local submonoids are groups we say that a regular semigroup has a property 3 locally or is a locally 5 7 semigroup if each local submonoid of s has property y in a previous paper we showed that a regular semigroup is locally inverse if and only if it is an image by a homomorphism which is one to one on local submonoids of a regular rees matrix semigroup over an inverse semigroup in this paper we extend that result to various other classes of regular semigroups in particular we show the analog of this result for locally e solid semigroups a regular semigroup is e solid if the subsemigroup generated by its idempotents is a union of groups the class of locally e solid regular semigroups is extremely extensive it includes almost all classes of regular semigroups which have been studied from a structural point of view since inverse semigroups orthodox semigroups unions of groups semigroups and their localizations all belong to this class the first section of the paper contains preliminary results which we shall require later in the paper section 2 describes a general procedure for constructing rees matrix covers for regular semigroups which is applied in subsequent sections to obtain rees matrix covers for special classes of regular semigroups these classes include locally e solid regular semigroups locally orthodox semigroups and locally sunipotent semigroups the covering theorems on these classes of regular semigroups can be obtained directly through a painstaking analaysis of special cases we have however chosen to obtain them as applications of the covering theorem for locally inverse semigroups which was proved earlier 81 by making use of recent interesting results of hall concerning the relationship between locally e solid

the computational complexity of generating random fractals we examine a number of models that generate random fractals the models are studied using the tools of computational complexity theory from the perspective of parallel computation diffusion limited aggregation and several widely used algorithms for equilibrating the ising model are shown to be highly sequential it is unlikely they can be simulated efficiently in parallel this is in contrast to mandelbrot percolation which can be simulated in constant parallel time our research helps shed light on the intrinsic complexity of these models relative to each other and to different growth processes that have been recently studied using complexity theory in addition the results may serve as a guide to simulation physics

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

evaluate fuzzy optimization problems based on biobjective programming problems abstract the embedding theorem shows that the set of fuzzy numbers can be embedded into a banach space inspired by this embedding theorem we propose a solution concept of the fuzzy optimization problem by solving a biobjective optimization problem which is transformed from the original fuzzy optimization problem using the embedding theorem we then also consider the fuzzy optimization problem with fuzzy coefficients i e the coefficients are assumed to be fuzzy numbers under a setting of core values of fuzzy numbers we show that the optimal solution of its corresponding crisp optimization problem the usual optimization problem is also an optimal solution of the original fuzzy optimization problem

on a sharp transition from area law to perimeter law in a system of random surfaces we introduce and study a phase transition which is associated with the spontaneous formation of infinite surface sheets in a bernoulli system of random plaquettes the transition is manifested by a change in the asymptotic behavior of the probability of the formation of a surface spanning a prescribed loop as such this transition offers a generalization of the bond percolation phenomenon at low plaquette densities the probability for large loops is shown to decay exponentially with the loops area whereas for high densities the decay is by a perimeter law furthermore we show that the two phases of the three dimensional plaquette system are in a precise correspondence with the two phases of the dual system of random bonds thus if a natural conjecture about the phase structure of the bond percolation model is true then there is a sharp transition in the asymptotic behavior of the surface events our analysis incorporates block variables in terms of which a non critical system is transformed into one which is close to a trivial high or low density fixed point stochastic geometric effects like those discussed here play an important role in lattice gauge theories

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

an exponential membership function for fuzzy multiple objective linear programming abstract the operator min is one of the most frequently used aggregation operators in fuzzy decision however this operator is the softest operator and no allowance is made for any compensation the product and other operators some of them may be compensatory are seldom used because of the nonlinearity of the resulting problem in this paper an exponential instead of linear membership function is proposed the advantages of using exponential membership are two fold first the resulting problems can be transformed to linear ones when the product and several other nonlinear aggregate operators are used secondly exponential representation is more realistic than the linear ones usually used for some practical applications

fuzzy approaches to multicriteria de novo programs abstract multicriteria de novo programming is a promising tool for optimal system design however there exist no algorithms for solving a general multicriteria de novo program only special cases have been discussed this paper proposes a two step fuzzy approach based on the ideal and negative ideal solutions it is shown that this approach is very efficient and is applicable to the general multicriteria de novo programming the fuzzy version of this problem is also formulated and analyzed numerical examples are given to illustrate the approaches

critical points and intermediate phases on wedges of zd the authors examine the phase structure of self avoiding walks the ising magnet and bond percolation defined on subsets of zd d or 2 which have the geometry of wedges they prove that if the cross sectional area of the wedge diverges with its width then the high temperature critical point of any of these models defined on the wedge coincides with that of the corresponding model on the full lattice for the ising magnet and bond percolation they show that there is a non trivial low temperature critical point if the cross sectional area of the wedge diverges logarithmically with its width moreover for bond percolation they show that the low temperature critical point of a logarithmic wedge may be made arbitrarily close to that of the full lattice by taking the coefficient of the logarithm large enough corollaries to these theorems include the existence of an intermediate phase for the ising magnet and bond percolation on logarithmic wedges and the existence of a first order transition for percolation models on a subclass of these wedges

possibilistic linear programming a brief review of fuzzy mathematical programming and a comparison with stochastic programming in portfolio selection problem in this paper we review some fuzzy linear programming methods and techniques from a practical point of view in the rst part the general history and the approach of fuzzy mathematical programming are introduced using a numerical example some models of fuzzy linear programming are described in the second part of the paper fuzzy mathematical programming approaches are compared to stochastic programming ones the advantages and disadvantages of fuzzy mathematical programming approaches are exemplied in the setting of an optimal portfolio selection problem finally some newly developed ideas and techniques in fuzzy mathematical programming are briey reviewed c 2000 elsevier science b v all rights reserved

optimality conditions for fuzzy constrained programming problems abstract this paper solves optimization problems where both the objective and constraints are given by fuzzy functions in order to get it we first prove that these problems are equivalent to optimization problems where the constraints functions are non fuzzy functions and we introduce a new and wider stationary point concept that generalizes all existing concepts so far this new stationary point concept is based on the gh differentiability and has many computational advantages that we describe it is well known that obtain a useful differentiability notion for fuzzy functions is a difficult task without linearity and we are in that case due to the fact that the fuzzy numbers intervals space is a nonlinear one in this direction the gh derivative for fuzzy functions is a concept that is more general than hukuhara and level wise derivatives that are usually used in fuzzy optimization so far in the sense that they can be applied to a wider number of fuzzy function classes than above concepts with this new differentiability concept we prove a necessary optimality condition for fuzzy optimization problems that is more operational and less restrictive that the few ones we can find in the literature so far moreover due to the fact that we do not have a linear space for fuzzy numbers the convex concepts and generalized convex fuzzy function notion are very restrictive also this implies that the sufficiency optimality conditions for fuzzy problems published so far are not useful

possibilistic linear programming with triangular fuzzy numbers abstract we consider a mathematical programming problem where all the parameters may be fuzzy variables specified by their possibility distribution and we define the possibility distribution of the objective function the ideas are illustrated on possibilistic linear programming

phase transitions in mandelbrot s percolation process in three dimensions we study the phase structure and transitions in three dimensional mandelbrot percolation a process which generates random fractal sets we establish the existence of three distinct phase transitions and we show that two of these transitions corresponding to percolation across the initial set by paths and sheets are discontinuous

on left quasinormal orthodox semigroups the existence of a smallest inverse congruence on an orthodox semigroup is known it is also known that a regular semigroup s is locally inverse and orthodox if and only if there exists a local isomorphism from s onto an inverse semigroup t in this paper we show the existence of a smallest r unipotent congruence on an orthodox semigroup s and give its expression in the case where s is also left quasinormal finally we prove that a regular semigroup s is left quasinormal and orthodox if and only if there exists a local isomorphism from s onto an r unipotent semigroup t

the solution and duality of imprecise network problems duality properties have been investigated by many researchers in the recent literature they are introduced in this paper for a fully fuzzified version of the minimal cost flow problem which is a basic model in network flow theory this model illustrates the least cost of the shipment of a commodity through a capacitated network in terms of the imprecisely known available supplies at certain nodes which should be transmitted to fulfil uncertain demands at other nodes first we review on the most valuable results on fuzzy duality concepts to facilitate the discussion of this paper by applying hukuhara s difference approximated and exact multiplication and wu s scalar production we exhibit the flow in network models then we use combinatorial algorithms on a reduced problem which is derived from fully fuzzified mcfp to acquire fuzzy optimal flows to give duality theorems we utilize a total order on fuzzy numbers due to the level of risk and realize optimality conditions for providing some efficient combinatorial algorithms finally we compare our results with the previous worthwhile works to demonstrate the efficiency and power of our scheme and the reasonability of our solutions in actual decision making problems

applications of fuzzy set theory to mathematical programming mathematical programming is one of the areas to which fuzzy set theory has been applied extensively primarily based on bellman and zadeh s model of decision in fuzzy environments models have been suggested which allow flexibility in constraints and fuzziness in the objective function in traditional linear and nonlinear programming in integer and fractional programming and in dynamic programming these models in turn have been used to offer computationally efficient approaches for solving vector maximum problems this paper surveys major models and theories in this area and offers some indication on future developments which can be expected

study on fuzzy optimization methods based on principal operation and inequity degree fuzzy optimization is a well known optimization problem in artificial intelligence manufacturing and management so establishing general and operable fuzzy optimization methods are important in both theory and application in this paper by distinguishing principal indices and secondary indices we give a method for comparing fuzzy information based on synthesizing effect and an operation for achieving fuzzy optimization based on a principal indices transformation further we propose an axiomatic system for fuzzy inequity degree based on the essence of constraint and give an instructive metric method for fuzzy inequity degree then by combining with genetic algorithm we give some fuzzy optimization methods based on principal operation and inequity degree denoted by bpo id fga for short finally we consider the convergence of our algorithm using the theory of markov chains and analyze its performance through two concrete examples all these indicate that bpo id fga can effectively merge decision preferences into the optimization process and that it also possesses better global convergence so it can be applied to many fuzzy optimization problems

modality constrained programming problems a unified approach to fuzzy mathematical programming problems in the setting of possibility theory abstract in this paper fuzzy mathematical programming problems are formulated based on the idea analogous with the chance constrained programming problem the difference in meaning between the ambiguity of the coefficients and that of the decision maker s preference is emphasized the constraints with fuzzy coefficients are treated as the restriction that should be satisfied properly rather than perfectly the objective functions with fuzzy coefficients are treated variously depending on the interpretations i e the optimization of the modalities the optimization of the fractile the minimization of the ambiguity and so forth the deterministic equivalent constraints and the deterministic equivalent problems are shown when the constraints and the objective functions are linear a numerical example is given to illustrate the proposed formulations

correlation length bounds for disordered ising ferromagnets thed dimensional nearest neighbor disordered ising ferromagnet h sum j ij sigma i sigma j is studied as a function of both temperature t and a disorder parameter which measures the size of fluctuations of couplingsjij 0 a finite size scaling correlation length f t is defined in terms of the magnetic response of finite samples this correlation length is shown to be equivalent in the scaling sense to the quenched average correlation length t defined as the asymptotic decay rate of the quenched average two point function furthermore the magnetic response criterion which defines f is shown to have a scale invariant property at the critical point the above results enable us to prove that the quenched correlation length satisfies c left log xi t right xi t geqq left t t c right 2 mathord left vphantom 2 d right kern nulldelimiterspace d which implies the boundv 2 d for the quenched correlation length exponent

fuzzy and semi infinite mathematical programming abstract in this paper linkups between imprecision and infinity in an optimization framework are highlighted inspired by these connections the new potential opened up by semi infinite optimization is gainfully exploited to push forward an approach for solving a fuzzy mathematical program issues regarding the viability of the approach as well as the satisfying character of solutions yielded by this way are also addressed furthermore we present a cutting plane method for solving the resulting semi infinite program we end up by some concluding remarks and indicate directions for further developments

fuzzy linear programming problems with fuzzy numbers abstract linear programming problems with fuzzy parameters are formulated by fuzzy functions the ambiguity considered here is not randomness but fuzziness which is associated with the lack of a sharp transition from membership to nonmembership parameters on constraint and objective functions are given by fuzzy numbers in this paper our object is the formulation of a fuzzy linear programming problem to obtain a reasonable solution under consideration of the ambiguity of parameters this fuzzy linear programming problem with fuzzy numbers can be regarded as a model of decision problems where human estimation is influential

generalized interval vector spaces and interval optimization this paper presents a method of endowing the generalized interval space with some different structures such as vector space topological space order relations and algebraic calculus we formulate interval optimization problems and relate them to classic multi objective optimization problems we also present a version of the mini max s theorem in the interval space context this paper presents a method for endowing the generalized interval space with some different structures such as vector spaces order relations and an algebraic calculus with these concepts we formulate interval optimization problems and relate them to classic multi objective optimization problems we also present a version of the von neumann s mini max theorem in the interval context

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