Some of the important concepts such as positively homogeneous, infimal convolution, right scalar multiplication, and convex hull are introduced and the corresponding theorems [r]

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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 [29] with a view to reconcile mathematical modeling and human knowledge in the engineering science. In 1992, Nanda and Kar [11] 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 [28] 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 [3]. In addition they discussed the properties of **convex** **fuzzy** optimizations. In [21], 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 [3]. In [22], Syau defined a differentiable **fuzzy** map- pings of several variables in ways that parallel the definition, proposed by Goetschel and Voxman [3], for a **fuzzy** **mapping** of one variable. Wang and Wu [24] 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.

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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.

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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.

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:

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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

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The following theorem is an extension of Theorem 3.1 of Cain and Nashed [2] for a sum of contraction and continuous mappings to a sum of certain type of asymptotically nonexpansive mappi[r]

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]

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.

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at each x ∈ X, then f : X → Y is said to be continuous on X (see [3]). 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.

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[11] Thakur S.S. & Malviya R., Generalized closed sets in **fuzzy** topology Math. Note 38(1995),137-140. [12] Yalvac T.H. **fuzzy** sets and Function on **fuzzy** spaces, J. Math. Anal. App. 126(1987), 409-423. [13] 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.

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]

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 [].

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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.

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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.

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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.

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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.

Definition 1.(see [6]) 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.

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]

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