Gui-jun Wang

Department of Mathematics, Tonghua Teacher¡¯College

Tonghua, Jilin 134002, P.R.China

**Abstract:** In this paper, we first introduce the concept of interval-valued fuzzy measures, then define and study some basic properties of the interval-valued functional. Finally, we obtain two kinds of extended interval-valued fuzzy measures in different senses respectively. Futhermore, we discuss the relation between them.

**Keywords:** Interval-valued fuzzy measure, interval-vakued functional, MIVF -algebra, extension.

**Kleene Posets and Kleene Algebras**

Zhu Degao

Department of Mathematics Central China Normal University

Wuhan, Hubei 430070, P.R.China

**Abstract:** In this paper, it will propose the notion of Kleene poset. The main results are as follows: Let X be a finite Kleene poset, but there is not nonevem-nonodd element in X, then is a Kleene algebra. And there is a central element in X iff there is not such element in .

**Keywords:** Even, odd, noneven-nonodd, central, De Morgan, Kleene.

**Extended Fuzzy Topologies. Part II, Characterizing Notions**

W.Gahler

Institute fur Mathematik, Universitat Potsdam

Postfach 601553, 14415 Potsdam, Germany

A.S.Abd-Allah

Department of Mathematics, Faculty of Science,El-Mansoura University, El-Mansourn, Egypt

A.Kandil

Department of Mathematics, Faculty of Science, Kuwait University, P.O.Box 5969, Safat, 1300 Kuwait

**Abstract:** Basic results on extended fuzzy topologies are presented obtained in applying the theory of fuzzy stacks developed in Part I and using results given in [13]. There are mainly investigated notions defined for extended fuzzy topologiess which can be used for their characterizations. They are called characterizing notions. In some cases the notions we consider are characterizing notions whicha do nor need any assumption are those of fuzzy neighborhood of a fuzzy stack, of the interior of a fuzzy stack and of the set of inner points of a fuzzy stack. The notions of open fuzzy stack and of projectively open fuzzy stack are characterizing for idempotent extended fuzzy topologies. Further characterizing notions are presented obtained by restricting to valued and superior principal fuzzy stacks.

**Keywords:** Extended fuzzy topologies, supra fuzzy topologies, fuzzy stack pretopologies, idempotent extended fuzzy topologies, stratified and strongly stratified extended fuzzy topologies, fuzzy neighborhood operators, open and projectively open fuzzy stacks, inner points of fuzzy stacks.

**Stable Fuzzy Direct Adaptive Control for a Class of Nonlinear Systems***

Tong shaocheng and Chai Tianyou

Department of Basic Mathematics, Liaoning Insititute of Teechnology,

Jin Zhou 121001, P.R.China

**Abstract:** A direct adaptive tracking control architecture is proposed for a class of continuoustime nonlinear dynamis systems, for which an explict linear parameterization of the uncertainty in the dynamic is either unknown or impossible. The architectture employs fuzzy systems to approximate the optimal controller ,developes fuzzy sliding controller to compensate for the plant uncertainties, smooth the control signals and increase robustness. Global asymptotic stablity is establishes in the Lyapunov sense, with the tracking errors converging to a neiborhood of zero. The simulation results verify the effectiveness of the proposed control algorithm.

**Keywords:** Fuzzy control, adaptive control, nonlinear systems, fuzzy sliding mode.

**On Modified Sets and Fuzzy Modified Sets**

Ranjit Biswas

Department of Mathematics, Indian Institute of Technology

Kharagpur-721 302, West Bengal, India

**Abstract:** In this paper some characterizations of modified sets are made. Fuzzy modified sets are defined and studied.

**Keywords:** Fuzzy sets, modifiers, modified boundary, weakening equality, substantiating equality, fuzzy modified sets.

**Common Fixed Fuzzy Set Theorems in Fuzzy Metric Spaces**

Ranjit Biswas

Department of Mathematics, Indian Institute of Technology

Kharagpur-721302, West Bengal, India

**Abstract:** Common or simultaneous fixed fuzzy sets of a pair of mappings are studied in fuzzy metric spaces.

**Keywords: **Fuzzy sets, fuzzy metric, fuzzy metric space, F-convergence, F-Cauchy¡¯s sequence, F-complete fuzzy metric space,F-continuity, fixed fuzzy set, common fixed fuzzy set.

**Stochastic Fuzzy Mathematical Programming and Decision**

Sandipan Gupta and M.Chakraborty

Department of Applied Mathematics, Indian School of Mines

Dhanbad-826004, India

**Abstract:** The paper deals with the development of fuzzy mathematical model for stochastic objective function with fuzzy linear constraints to solve a problem so that the probability of the event ¡°upper bound of stochastic objective function is minimum¡± would be maximum with highest degree of sarisfaction. A solution procedure has been explored and a numerical example is presented to illustrate application.

**Keywords: **Mathematical model, stochastic, fuzzy, mini operator, decision.

**Fuzzy Algebras Based on Algebraic Properties**

Godfrey C.Muganda

Department of Computer Science

North Central College, Naperville, IL 60566, USA

**Abstract:** These essence of a fuzzy set is that membership of a point in the fuzzy set is not a truefalse proposition, but is rather, a matter of degree. We suggest an algebraic approach to the constrstuction of fuzzy groups, fuzzy rings, and fuzzy modules, in which the degree of membership of a point is determined by the degree to which that point ehjoys some specified algebraic property.

**Keywords:** Fuzzy groups, fuzzy rings, fuzzy modules, fuzzy vector spaces.

**Multivalued Quasi Variational Inequalities for Fuzzy Operators**

M.Aslam Noor and Esia A.Al-Said

Mathematics Department, College of Science, King Saud University

P.O.Box 2455, Riyadh 11451, Saudi Arabia

**Abstract:** In this paper, we establish the equivalence between the multivalued quasi variational inequalities for fuzzy operators and the implicit Wiener-Hopf equations for fuzzy operators. We use this equivalence to suggest a number of new iterative algorithms fir solving the varitional inequclities for fuzzy operators. We also study the convergence criteria of these iterative algorithms. The results proved in this paper represent an improvement of previously known results.

**Keywords:** Variational inequalities, Wiener- Hopf equations, fuzzy operators, optimization.

**An Systematic Design of SMC-based Fuzzy Controller for Nonlinear Systems**

Jen-Yang Chen

Department of Electronic Engineering, China Institute of Technology and Commerce

No.245 Sec.3, Yen-Chiu-Yuan Road, Taipei, Taiwan, R.O.C.

**Abstract:** The purpose of this paper is to 1) improve the performance of sliding mode control (SMC), 2) construct the fuzzy controller (FC) that based on the concepts of SMC. The design outline rely on the following stages: First, an adaptive mechanism is proposed to approximate the equivalent control of SMC, where the mathematical model of the nonlinear system is unknown. Second, a hitting control term is added to the SMC for achieving a stable control system. Finally, a novel fuzzy controller with two fuzzy control rules will be constructed, which not only applies the SMC rational and has the property of SMC but also guarantees the stability of the fuzzy control system. A simulation example with a nonlinear plant is applied to confirm the validity of the proposed FC.

**Keywords: **Fuzzy control, sliding mode control,approximation reasoning, adaotive control.

**An Approach to Improve the Performance of the ASFC-based Fuzzy Control System**

Jen-Yang Chen

Department of Electronic Engineering, China Institute of Technology and Commerce

No. 245 Sec. 3, Yen-Chiu-Yuan Road, Taipei, Taiwan, R.O.C

**Abstract:** It is well known that fuzzy logic controllers have been successfully applied in many complex industrial process. However, the conventional design approaches deeply rely on someone who has rich experiences and expert knowledge. In may cases, it cannot be objectively elicited. Among the problems to be resolved in the design of fuzzy logic controller are a) derivation of control rules b) definition of the membership functions of each linguistic term and state space. Recently, some of these problems have been solved by machine learning, especially, one of these new algorithms is the genetic algorithm (GA) that is based on natural evolution. In this paper, we first design an adaptive state feedback contoller (ASFC), then a novel fuzzy controller based on the ASFC is constructed. Finally, the GA is applied to learn the parameter of fuzzy controller, where the fuzzy singleton functions in consequence part can be automatically adjusted under some objective criteria. A simulation example is used to confirm the design viewpoint.

**Keywords:** Genetic algorithm, fuzzy logic control, evolution, machine learning.

**A ASFC-based Fuzzy Controller Design **

Jen-Yang Chen

Department of Electronic Engineering, China Institute of Technology and Commerce

No.245 Sec.3,Yen-Chiu-Yuan Road, Taipei, Taiwan, R.O.C

**Abstract:** In this paper we present a design methodology for stabilization of a class of fuzzy control systems. Conventionally, when facing a design problem of fuzzy control system, we first specify a set of fuzzy control rules then adjust (turning, learning) it by various approaches for system specification or stability requirement. We attempt to design procedures of the fuzzy controller in this study. First, an adaptive stata feedback controller (AFSC) is used to construct an approximate optimal control by adaptive control manner, in which an adaptive law is developed toadjust the parameters of ASFC; then, a compensation part will be given to guarantee the system stability. Finally, we construct the fuzzy control rules according to the concept of ASFC control architecture. To preserve the system stability, the membership functions of fuzzy controller£¨FC£©are all well defined. Comparing with the conventional design approaches of FC, the proposed control scheme provides an effective and intuitive manner. A nonliear simulation example is applied to confirm the validity of the FC.

**Keywords:** Fuzzy sets, fuzzy control, adaptive control, approximation reasoning.

Relation Between Two Notions of Fuzzy Semi-norm

N.R.Das and Pankaja Das

Department of Mathematics Gauhati University,

Guwahati-781 014 Assam India

**Abstract:** Relation between two different notions of fuzzy seminorm, one as an absolutely convex, absorbing fuzzy subset due to katsaras [5 ] and the other as a mapping from a vector space to the set fuzzy of real numbers defined following definition of fuzzy norm due to Felbim [1] is established in this paper.

**Keywords:** Fuzzy real number, fuzzy seminorm, fuzzy norm, fuzzy topology.

**On Invariance of Clustering Algorithms**

I.Batyrshin and R.Khabibulin

Kazan State Technological Unversity

K.Marx st. 68, Kazan, 420015, Russia

E-mail. batyr@emntu.kcn.ru

**Abstract:** Theproperties of invariance of clustering algorithms to numeration of objects and to monotone transformation of similarity values between objects are considered. The problems of construction of invariant clustering algorithms and testing of existing algorithms on invariance are discussed. Invariant clustering algorithms based on the concept of fuzzy similarity relation are described. The methods of testing of clustering algorithms on the property of invariance to numeration of object on symmetric date are proposed.

**Keywords:** Cluster analysis, fuzzy similarty relation, partition, invariance, testing of algorithms, symmetric date.

**On Fuzzy Local Functions, Fuzzy Semiregularrizations and Fuzzy Mappings**

Debasis Sarkar

Department of Applied Mathematics, Calcutta University

92,A.P.C.Road, Calcuta-700 009,India

**Abstract:** In this paper some characterization theorems relating fuzzy open sets and generated fuzzy topologies by fuzzy ideals are given. We define a criterion, -interior for a fuzzy topological space with a given fuzzy ideal. With the help of this criterion, we study fuzzy semiregularization spaces associated with the generated fuzzy topologies, the original fuzzy topologies and also with the fuzzy subspaces. Finally, we discuss some fuzzy mappings between different fuzzy topological spaces and some relations are also established in connection with earlier results.

**Keywords:** Fuzzy topology, fuzzy ideal, fuzzy local function, fuzzy semiregularization, fuzzy subspace, fuzzy mappings.

**Two Generating Mappings and in Complete Lattices TVS and LFTVS***

Yan Cong-Hua

Department of Basic, Nanjing power institute,

78 Ban Cang Nanjing Jiangsu, 210042. China

**Abstract:** In this paper, we give a new decomposition theorem of L-fuzzy sets. Then we use this theorem to discuss the relations between the crisp topological vector spaces and the L-fuzzy topological vector spaces and we obtain some interesting properties.

**Keywords:** Way below; Decomposition theorem: R-neighboorhood base: L-ftvs.

**On Strong Notions in Fuzzy Topological Spaces **

Ali Kandil

Department of mathematics, Facultyof Science, Kuwait University,

P.O.Box 5969 Safat, 13060, Kuwait

Fatma Bayoumi

Department of Mathematics, Faculty of Science,

Benha Unversity,Benha, Egypt

Ail Nouh

Department of Mathematics, Faculty of Science, El-Mansoura University,

El-Mansoura, Egypt

**Abstract:** The notions of strong separation and strong continuity in fuzzy topological spaces have been introduced and studied in [14]. In this paper, we study the relation between these strong separation axioms and the fuzzy separation axioms given in [8,15]. Moreover, we introduce the notions of strong connectedness in fuzzy topological spaces. As in [14] a similar study for the strong connectedness and local strong connectedness is done. Fuzzy topological quotient spaces are studied from the point of view of strong connectedness. We also introduce examples showing that the strong notions and the analogous fuzzy notions are not equivalent in general. Some results, related to strong notions for mappings, product spaces, subspaces and sum spaces are obtained.

Keywords:
Fuzzy topologies, -level topologies, induced fuzzy topologies, fuzzy separation axioms, strong separation axioms, connected and locally connected fuzzy topological spaces, strong-connected and locally strong-conneted fuzzy topological spaces, fuzzy topological quotient spaces.

**The Seletion of Optimal Project Based on the Grey Relational Grade**

Kun-Li Wen and Fan-Hsiung Chen

Department of E.E., Chien Kuo J.College of Technology&Commerce, Chang Hua, Taiwan

John H.Wu

Department of M.E., National Centeal University, Chung-li, Taiwan.

**Abstract:** In this paper, we derive the gery relational matrix from the alobalization grey relational grade step by step, and replace the AHP method in the selection of optimal project which was further developed by the fuzzy theory and the grey theory to rewrite the same example to investigate the validity of the newly developed.

**Keywords:** Grey relational matrix, globalization grey relational grade, AHP.

**Fuzzy Rule and Its Simulations in Fuzzy Relation Equations**

Xiaozhong Li and Da Ruan

**Abstract:**Belgian Nuclear review of our previous work [2], in this paper we will present a new simplified proof to a lemma which plays an important role in proving the convergence theorem of the fuzzy perceptron. The new proof is much shorter. Moreover, we give some typical simulational results to illustrate the power of the fuzzy rule.

**Keywords:** Fuzzy relation equation, fuzzy rule, max-min operator network.