The Journal of Fuzzy Mathematics
Volume 15, Number 1, March 2007
CONTENT
DETAILS
Folding of Chaotic Multiple Graph and Its Fractal
M. El-Ghoul and T. Homoda
Mathematics Department, Faculty of Science
Tanta University, Tanta, Egypt
E-mail: m_elghoul2010@yahoo.com
Abstract: In this paper, we will introduce the definition of chaotic multiple graphs. The incidence matrix of this type of graph will be defined. The difference between subgraph and fractal graph will be discussed. The folding of chaotic multiple graphs will be obtained; also the limit of this floding will be obtained. Theorems governing these studies are achieved.
Keywords: Chaotic multiple graphs, fractals, folding.
Fuzzy Subspace and Local Compactness
Qingguo Li
Department of applied Mathematics, Hunan University
Changsha, 410082, Hunan, P. R. China
Abstract:
Based on crisp subsets, we introduce the notion of a fuzzy subspace, and establish the hereditary of separation axioms. Besides, we introduce a compact subspace and local compactness, and investigate their properties.
Keywords: Fuzzy subspace, m-compact, dual fuzzy point, dense
Symmetric Polygonal Fuzzy Number Space
Puyin Liu
Department of Mathematics, Beijing Normal University, Beijing, 100875, China
Department of Mathematics, National University of Defence Technology, Changsha, 410073, China
Email address: liupuyin@public.cs.hn.cn
Hongxing Li
Department of Mathematics, Beijing Normal University, Beijing, 100875, China
Abstract:
A novel class of fuzzy numbers, which are called symmetric polygonal fuzzy numbers are presented. They generalized the triangular and trapezoidal fuzzy numbers from the architectures and representing form. All polygonal fuzzy numbers constitute a metric space, whose topological properties are similar with ones of triangular of trapezoidal fuzzy number spaces. For instance, the space is a completely separable metric space; also it is locally compact; a set in the space is compact if and only if it is bounded and closed, and so on. A new fuzzy arithmetic in this novel fuzzy number space is developed. It is shown that a class of bounded fuzzy number can be approximated to any degree of accuracy by the symmetric polygonal fuzzy numbers. Finally some illustrative examples demonstrate the approximation conclusions.
Keywords: Symmetric polygonal fuzzy number, separability, compactness, fuzzy arithmetic, polygonal line operator.
Fixed Point Theorems for Pair of Fuzzy Mappings
P. Vijayaraju
Department of Mathematics College of Engineering, Guindy Anna University,
Chennai-600025, India.
E-mail:vijay@annauniv.edu
R. Mohanraj
Department of Mathematics College of Engineering, Guindy Anna University,
Chennai-600025, India.
E-mail: vrmraj@yahoo.com
Abstract:
In this paper we prove some fixed point theorems for contractive type fuzzy mappings which are generalization of Beg and Azam [1] and fuzzy extension of Kirk and Downing [5]. Using the concept ¦Á-cut of a fuzzy set, we also obtain fixed point theorems for contractive type mappings given by Park and Jeong [8]. Also we obtain simple proof of Park and Jeong [8] using the result of Beg and Azam [1].
Keywords:
Fixed point, Fuzzy set, Contraction fuzzy mapping
Separations Based on R-fuzzy Semi-open Sets
Yong Chan Kim
Department of Mathematics, Kangnung National University,
Gangneung, Gangwondo 210-702, Korea
E-mail: yck@kangnung.ac.kr
S. E. Abbas
Department of Mathematics, Faculty of Science, Sohag, Egypt
E-mail: sabbas73@yahoo.com
Abstract: We introduce r-fuzzy semi-open (closed) sets in Sostak¡¯s fuzzy topological spaces. Using them, we define the separation axiom of fuzzy topological spaces. We investigate some properties of them. Moreover, we investigate the relationship between fuzzy irresolutity and fuzzy (supra) continuity.
Keywords: Fuzzy (supra) topological spaces, fuzzy closure spaces, r-fuzzy semi-open (closed) sets, r-SRi spaces, fuzzy (supra) continuity, fuzzy irresolute maps
A Fuzzy Measure of Poverty
Mary George
Dept. of Mathematics, Mar Ivanios College, Trivandrum, Kerala, India.
P. G. Thomaskutty
Reader in Economics, Mar Ivanios College, Trivandrum, Kerala, India.
Sunny Kuriakose
Dept. of Mathematics, Union Christian College, Aluva, Kerala, India.
Abstract: The demarcating of the set of proof as well as measuring the extent of poverty in terms of classical (exact) measures is rather unrealistic as the concept of poor is non-exact. Here a fuzzy approach of measuring poverty is developed and a fuzzy poverty index is introduced.
Keywords: Fuzzy sets, poverty line, poverty index, welfare.
Solution of the Transportation Problem in Fuzzified Form
Makani Das
Department of Mathematics,Assam Engineering College
Guwahati-781013, Assam, India
Hemanta K. Baruah
Department of Statistics, Gauhati University
Gauhati-781014, Assam, India
Abstract: In this article we have discussed the closed, bounded and non-empty feasible region of the transportation problem which ensures the existence of an optimal solution to a balanced problem. For finding initial solution of the transportation problem we have preferred Vogel¡¯s Approximation Method and for Optimality Test MODI Method is taken into consideration. The fuzzified version of the problem has been discussed with the help of a numerical example.
Keywords: Triangular fuzzy numbers, Simplex method, balanced transportation problem, Vogel¡¯s Approximation Method, MODI method.
The Method of Successive Approximations of Systems of Linear Equation in Fuzzified Form
Makani Das
Department of Mathematics,Assam Engineering College
Guwahati-781013, Assam, India
Hemanta K. Baruah
Department of Statistics, Gauhati University
Gauhati-781014, Assam, India
Abstract: The approximation method for solving systems of linear equations make it possible to obtain the values of roots of the system with the specified accuracy as the limit of the sequence of some vectors. The process of constructing such a sequence is known as the iterative process. The efficiency of the application of approximation method depends on the choice of the initial vector and the rate of convergence of the process. In this topic we are going to consider vectors as fuzzy vectors. We have considered a numerical example and tried to find out solution vector X in fuzzified form using method of iteration. Discussed convergence and error approximation also
Keywords: Triangular fuzzy numbers, fuzzy normal form, dense linear system, sparse matrix, fuzzy convergence
Semi-generalized Continuous Mappings in Fuzzy Topological Spaces
M. E. El-Shafei
Mathematics Department, Faculty of Science, Mansoura University, Mansoura, Egypt.
A. Zakari
Girls College of Education in Gazan Kingdom of Saudi Arabia
Abstract:
In this paper we study the concept of semi-generalized closed fuzzy sets and generalized semi-closed fuzzy sets. Fuzzy semi-generalized continuous mappings and fuzzy generalized semi-continuous mappings are introduced and we establish some fundamental properties of these fuzzy mappings
Keywords: Semi-generalized closed fuzzy sets, generalized semi-closed fuzzy sets, Fuzzy semi-generalized continuity, fuzzy generalized semi-continuity.
N-compactness in de Morgan Topological Algebra
Xueyou Chen and Qingguo Li
College of Mathematics and Economics, Hunan University,
Changsha, Hunan 410082, P. R. CHINA
Abstract:
In this paper, Using the notion of the scale of de Morgan topological algebra (L, Q, T) (Deng [9]), N-compactness is introduced in (L, Q, T).
Keywords:
Scale, N-compactness, V(a), V*(a), ideal, ¦Á -q-net
Algebra of Fuzzy Bitopology
Qingguo Li and Zike Deng
College of Mathematics and Economics, Hunan University,
Changsha, Hunan 410082, P. R. China
E-mail: liqingguoli@yahoo.com.cn
Abstract:
We introduce an algebra ( IX , ¦Ó, ¦Ó* ) of fuzzy bitopology as a theoretic frame of studying fuzzy topology ¦Ó on X. We introduce ¦Ó-neighborhoods, ¦Ó*-dual neighborhoods, ¦Ó-properties and ¦Ó*-properties. In particular, we introduce ¦Ó-Hausdorffness with respect to ¦Ó* and investigate its properties. We also introduce ¦Ó (¦Ó*) -compactness in an algebra of fuzzy bitopology and investigate their connections with ¦Ó (¦Ó*) -convergence, ¦Ó (¦Ó*) -closedness and ¦Ó-Hausdorffness with respect to ¦Ó*. It seems that properties of compactness in classic topology remain to be true in the bitopological frame of fuzzy topology.
Keywords: ¦Ó-neighborhoods, ¦Ó*-dual neighborhoods, bitopology, ¦Ó-Hausdorffness with respect to ¦Ó*, ¦Ó-compactness
Fuzzy programming Approach to Two-person Multicriteria Bimatrix Games
Sankar Kumar Roy
Department of Mathematics, Raja Narendra Lal Khan Women¡¯s College, Gope Palace,
Paschim Midnapore, West Bengal, Pin-721102, India
E-mail: roysank@yahoo.com
Abstract:
This paper presents the study of two different solution procedures for the two-person bimatrix games. Both the procedures are useful to find the solution of the multicriteria bimatrix game. The elements of the payoff matrix of the bimatrix game are considered as crisp numbers. The first solution procedure is applied to this game on getting the probability to achieve some specified goals along with the player¡¯s strategy. The second solution procedure is applied to this game on getting the probability to achieve some specified goals along with the player¡¯s strategy by defining the fuzzy membership function defined on the pay-off matrix of the bimatrix game. Finally a numerical example is presented to illustrate the solution procedures.
Keywords: Bimatrix games, Multicriteria games, Fuzzy programming.
Function S-rough Sets and Investment Warning Estimation
Hongyu Wang
Department of Mathematics and Computer, SanMing College
SanMing, Fujian 365004, P. R. China
Shi Kai-quan
School of Math. And System Sci. Shandong University
Jinan,250100, Shan
Shdong, P. R. China
E-mail: shikq@sdu.edu.cn
Abstract:
Using function S-rough sets ( function singular rough sets), this paper gives concepts of rough system and rough dependency, presents rough dependency measure of rough systems; puts forward rough dependency order theorem, rough dependency inertia theorem and rough dependency inertia principle; and this paper also gives application of rough dependency with respect to rough system in economic system investment warning analysis.
Keywords: Function S-rough sets, rough system, rough dependency, rough dependency order theorem, rough dependency inertia principle, application.
Fuzzy Korovkin Theorems and Inequalities
George A. Anastassiou
Department of Mathematical Sciences
The University of Memphis, Memphis, TN 38152, U. S. A.
E-mail: ganastss@memphis.edu
Abstract: We introduce and study the fuzzy positive linear operators acting on fuzzy continuous functions. We prove the fuzzy Riesz representation theorem, the fuzzy Shisha-Mond type inequalities and fuzzy Korovkin type theorems regarding the fuzzy convergence of fuzzy positive linear operators to the fuzzy unit in various cases. Special attention is paid to the study of fuzzy weak convergences of finite positive measures to the unit Dirac measure. All convergence are with rates and are given via fuzzy inequalities involving the fuzzy modulus of continuity of the engaged fuzzy valued function. The assumptions for the Korovkin theorems are minimal and of natural realization, fulfilled by almost all example-fuzzy positive linear operators. The surprising fact is that the real Korovkin test functions assumptions carry over here in the fuzzy setting and they are the only enough to impose the conclusions of fuzzy Korovkin theorems. We give a lot of examples and applications to our theory, namely: to fuzzy Bernstein operators, to fuzzy Shepard operators, to fuzzy Szasz-Mirakjan and fuzzy Baskakov-type operators and to fuzzy convolution type operators.
We work in general, basically over real normed vector space domains that are compact and convex or just convex. On the way to prove our main theorems we establish a lot of other interesting and important side results.
Keywords: Fuzzy Riesz representation theorem, Fuzzy positive linear operator, Fuzzy Korovkin theory, Fuzzy Shisha-Mond inequality, Fuzzy modulus of continuity
On LI-ideal Expansions of Lattice Implication Algebras
Young Bae Jun
Department of Mathematics Education
Gyeongsang National University, Chinju 660-701, Korea
E-mail address:ybjun@gnu.ac.kr
Yang Xu
Department of Applied Mathematics Southwest Jiaotong University
Chengdu, Sichuan 610031, China. Submitted April 29, 2005
E-mail address: xuyang@swjtu.edu.cn
Abstract: The notion of LI-ideal expansions is introduced, and a generalization of a prime LI-ideal is considered. The concept of residual is defined, and related properties are investigated. Homomorphic image and inverse image of ¦Ò-prime LI-ideal are considered.
Keywords: (prime) LI-ideal; (intersection preserving, global) LI-ideal expansion; residual division.
On Fuzzy Global Smoothness Preservation
George A. Anastassiou
Department of Mathematical Sciences
The University of Memphis, Memphis, TN 38152, U. S. A.
E-mail: ganastss@memphis.edu
Abstract: Here we establish the property of global smoothness preservation for fuzzy linear operators acting on spaces of fuzzy continuous functions. Basically we transfer the property of real global smoothness preservation into the fuzzy setting, via some natural realization condition fulfilled by almost all example-fuzzy linear operators. Our derived inequalities involve fuzzy modulus of continuity and we give examples.
Keywords:
Fuzzy linear operator, fuzzy modulus of continuity, fuzzy global smoothness preservation.
Fuzzy Knowledge Spaces
Young Bae Jun
Department of Mathematics Education
Gyeongsang National University, Chinju 660-701, Korea
E-mail: ybjun@gnu.ac.kr
Abstract: Fuzzifications of knowledge structures and knowledge spaces are considered, and some fundamental properties are investigated.
Keywords: ( fuzzy ) knowledge structure, ( fuzzy ) knowledge space, quasi-ordinal ( fuzzy ) knowledge space, base.
A Fuzzy Preference Based Choice Function
Mary George
Department of Mathematics, Mar Ivanios College, Kerala, India
Sunny Kuriakose
Department of Mathematics, Union Christian College, Aluva, Kerala, India
P. G. Thomaskutty
Department of Economics, Mar Ivanios College, Kerala, India
Abstract: There are many situations in which a choice is to be made from a set of alternatives based on fuzzy preferences. Even though such choice functions are available in the literature, they lack a full sense of rationality. In this paper, a new choice function based on fuzzy strict preference relation is introduced and is checked for its rationality behavior.
Keywords:
Fuzzy preference, fuzzy ordering, choice function, domination principle
Semigroup Structure in A Set of Alternatives
Mary George
Department of Mathematics, Mar Ivanios College, Kerala, India
Sunny Kuriakose
Department of Mathematics, Union Christian College, Aluva, Kerala, India
P. G. Thomaskutty
Department of Economics, Mar Ivanios College, Kerala, India
Abstract: In this paper, we introduce a semigroup structure in a ¡®set of alternatives¡¯. This enables to define a fuzzy congruence relation of ¡®indifference¡¯ in the set and to treat the set in a better way for making exact choice.
Keywords: Fuzzy preference, set of alternatives, semigroup, indifference relation. |