# Top PDF Convex fuzzy mapping and operations of convex fuzzy mappings ### Convex fuzzy mapping and operations of convex fuzzy mappings

Some of the important concepts such as positively homogeneous, infimal convolution, right scalar multiplication, and convex hull are introduced and the corresponding theorems [r] ### Optimizations of Convex and Generalized Convex Fuzzy Mappings in The Quotient Space of Fuzzy Numbers

T HE uncertainty includes randomness and fuzziness in the real world. Therefore, imposing the uncertainty upon the conventional optimization problems becomes an interesting research topic. The fuzzy set theory was in- troduced initially in 1965 by Zadeh  with a view to reconcile mathematical modeling and human knowledge in the engineering science. In 1992, Nanda and Kar  in- troduced the concept of convexity for fuzzy mappings and proved that a fuzzy mapping is convex if and only if its epigraph is a convex set. Yan and Xu  proposed the concepts of epigraph and convexity of the fuzzy mappings and described characteristics of the convex fuzzy mappings and quasi-convex fuzzy mappings by considering the concept of ordering due to Goetschel and Voxman . In addition they discussed the properties of convex fuzzy optimizations. In , Syau introduced the concepts of pseudo-convexity and pseudo-invexity for fuzzy mappings of one variable and investigated the relationships among them by using notion of differentiability and the results proposed by Goetschel and Voxman . In , Syau defined a differentiable fuzzy map- pings of several variables in ways that parallel the definition, proposed by Goetschel and Voxman , for a fuzzy mapping of one variable. Wang and Wu  proposed the concepts of directional derivative, differential and subdifferential of fuzzy mappings from R n into the set of fuzzy numbers and discussed the characterizations of directional derivative and differential. ### Operation and ordering of fuzzy sets, and fuzzy set-valued convex mappings

For fuzzy mathematical models using general fuzzy sets rather than fuzzy numbers or fuzzy vectors, operations (ad- dition and scalar multiplication) and orderings of fuzzy sets are needed, and the concept of fuzzy set-valued convex mappings is important. In the present paper, fundamental properties of operations, orderings, and fuzzy set-valued convex mappings for general fuzzy sets are investigated systematically. ### Fuzzy M open and Fuzzy M closed Mappings in Šostak’s Fuzzy Topological Spaces

where fo, f so, f o, fso, fpo, f o, f o, f o and f o maps are abbreviated by fuzzy open, fuzzy - semiopen, fuzzy -open, fuzzy -semiopen, fuzzy - preopen, fuzzy -open, fuzzy - open, fuzzy -open and fuzzy -open maps respectively. ### On Fuzzy *µ - Irresolute Maps and Fuzzy *µ - Homeomorphism Mappings in Fuzzy Topological Spaces

Example 5.04: Let X = Y = {a, b, c} and the fuzzy sets A, B, and C defined as follows. A= {(a, 0), (b, 0.1), (c, 0.3)}, B= {(a, 0.4), (b, 0.5), (c, 0.6)}, C = {(a, 1), (b, 0.9), (c, 0.7)}. Consider T = {0, 1, B} and  = {0, 1, A}. Then (X, T) and (Y,) are fts. Define f: X Y by f(a)=a, f(b)=b and f(c)=c. Then f is f*µ-continuous but not f-continuous (resp: not a fµ-continuous, not a f-continuous, not a fpre- continuous, not a fgµ-continuous, not a fg- continuous, not a fsg-continuous, not a fg#-s-continuous and not a fg#- continuous). As the fuzzy set C is closed fuzzy set in Y and f -1 (C) = C is not closed fuzzy set in X but *µ -closed (resp: ### Convergence theorems for generalized nonexpansive mappings in uniformly convex Banach spaces

In this paper, we prove strong and weak convergence theorems for a mapping deﬁned on a bounded, closed and convex subset of a uniformly convex Banach space, satisfying the RCSC condition. This condition was introduced by Karapınar (Dynamical Systems and Methods, 2012). We ﬁrst establish the demiclosed principle for the mapping satisfying the RCSC condition. Then, using this principle, we establish the weak and strong convergence theorems. Results in the paper extend and ### Fixed point theorems for a sum of two mappings in locally convex spaces

The following theorem is an extension of Theorem 3.1 of Cain and Nashed  for a sum of contraction and continuous mappings to a sum of certain type of asymptotically nonexpansive mappi[r] ### Fixed points and their approximations for asymptotically nonexpansive mappings in locally convex spaces

nonexpansive mappings include properly the class of nonexpansive mappings in locally convex spaces, prove a theorem on the existence of fixed points, and the convergence of the sequence [r] ### Convex combinations, barycenters and convex functions

The related problems with diﬀerent types of measures and mathematical expectations were investigated in []. The inequality in (.) under the condition in (.) was extended in []. The intention of this paper is still more to connect the quoted implication (in the extended form) with convex functions, in the discrete and integral case. We also wanted to insert the quasi-arithmetic means into this implication. at each x ∈ X, then f : X → Y is said to be continuous on X (see ). Now we examine some conditions under which the additive function found in Theorem 2.2 to be continuous. In the following theorem, we investigate fuzzy continuity of additive functions in fuzzy normed spaces. In fact, we will show that under some extra conditions on Theorem 2.2, the additive function r 7−→ A(rx) is fuzzy continuous. It follows that in such a case, A(rx) = rA(x) for all r ∈ R. ### FUZZY w- CONTINUOUS MAPPINGS

 Thakur S.S. & Malviya R., Generalized closed sets in fuzzy topology Math. Note 38(1995),137-140.  Yalvac T.H. fuzzy sets and Function on fuzzy spaces, J. Math. Anal. App. 126(1987), 409-423.  Yalvac T.H. Semi interior and semi closure of fuzzy sets,. J. math. Anal.Appl. 132 (1988) 356-364. [ 1 4 ] Zadeh L.A. fuzzy sets, Inform and control 18 (1965), 338-353. ### A Lower Bound for Continuous Convex Mappings on Normed Linear Spaces

This result can also be viewed as an estimation theorem for the continuous convex mappings defined on a normed space in terms of semi-inner product ( .,... DRAGOMIR , A characterization [r] ### An approximation of a common fixed point of nonexpansive mappings on convex metric spaces

ﬁxed point). For a bounded closed and convex subset E of a Banach space X, a mapping t : E → X is said to satisfy the conditional ﬁxed point property (CFP) if either t has no ﬁxed points, or t has a ﬁxed point in each nonempty bounded closed convex set that leaves t invariant. A set E is said to have the conditional ﬁxed point property for nonexpansive mappings (CFPP) if every nonexpansive t : E → E satisﬁes (CFP). For commuting family of nonexpansive mappings, the following is a remarkable common ﬁxed point property due to Bruck []. ### Operations on Intuitionistic Fuzzy Hypergraphs

Hypergraph is a graph in which an edge can connect more than two vertices. Hypergraphs can be applied to analyze architecture structures and to represent system partitions. The concept of hypergraphs was extended to fuzzy hypergraph . In this paper, we extend the concepts of fuzzy hypergraphs into that of intuitionistic fuzzy hypergraphs. Based on the definition of intuitionistic fuzzy graph, operations like complement, join, union, intersection, ringsum, cartesian product, composition are defined for intuitionistic fuzzy graphs. The authors further proposed to apply these operations in clustering techniques. ### Regularized Bundle Methods for Convex and Non-Convex Risks

First, in Section 2 we provide background on the cutting plane technique and on bundle meth- ods, and we describe two main existing extensions, the convex regularized bundle method (CRBM) and the non-convex bundle method (NBM). Then, we present in Section 3 our two contributions yielding our algorithm, NRBM, which is a regularized bundle method for non-convex optimization. We propose few variants of our method in Section 3.3 and we discuss in Section 4 the convergence behavior of our method both for convex risks and for non-convex risks. Finally we provide in Section 5 a number of experimental results. We investigate first artificial test problems that show that our algorithm compares well to standard non-convex bundle methods while converging much faster, suggesting our algorithm may make large scale problems practical. Second we compare our algorithms to dedicated state of the arts optimization algorithms for a number of machine learning problems, including standard problems such as learning of transductive support vector machines learning, learning of maximum margin Markov networks, learning conditional random fields, as well as less standard but difficult optimization problems related to discriminative training of com- plex graphical models for handwriting and speech recognition. ### Unsupervised Data Classification for Convex and Non Convex Classes

In this work, we propose a new method for unsup ervised data classification, which prove to be valid for convex and non convex type of classes. The method is based on three steps: split, clean and merge. The split technique is based on a neural network with evolving architecture; the network contains two layers which will make it possible to divide the data into several prototypes in an incremental way. The prototypes obtained are characterizing elements of the real classes. The clean step discards the noisy prototypes, which are non representative of the real classes. M erge step is an algorithm based on an evolving neural network, constituted of three layers. The merge technique is well adapted for regrouping the similar prototypes into end classes by means of a merge procedure after that the clean operation has been performed. ### Operations on Hesitant Fuzzy Hypergraph

In this paper, order of the HFHG, size of the HFHG, Operations like complement, join, union, intersection, ringsum, Cartesian product, composition on HFHGs are defined. Currently, the authors are working on existing clustering techniques. Further, it is proposed to apply the properties of HFHGs to develop a new clustering algorithm and the same may be checked with a numerical dataset. ### Fuzzy endpoint results for ´ Ciri´c-generalized quasicontractive fuzzy mappings

Definition 1.(see ) Let X be a space of points with generic element x and I = [0, 1]. A fuzzy set in X is a function that associates any point of X with a number in interval [0, 1]. If A is a fuzzy set in X and x ∈ X, then A(x) is called the grade of membership of x in A. ### Common fuzzy hybrid fixed point theorems for a sequence of fuzzy mappings

In this paper, we introduced g-I,-contractive type fuzzy mappings and defined the concept of the fuzz hybrid fixed point for fuzzy mappings, proved common fuzzy hybrid fixed point theore[r] ### HARMONIC MAPPINGS RELATED TO CLOSE-TO-CONVEX FUNCTIONS OF COMPLEX ORDER b

Finally, a planar harmonic mapping in the open unit disc D is a complex-valued harmonic function f , which maps D onto the some planar domain f ( D ). Since D is a simply con- nected domain, the mapping f has a canonical decomposition f = h(z) + g(z) where h(z) and g(z) are analytic in D and have the following power series expansions