




A FUZZY VERSION OF HAHNBANACH EXTENSION
THEOREM
L. ZEDAM
Abstract. In this paper, a fuzzy version of the analytic form of HahnBanach extension theorem is given. As application, the HahnBanach theorem for r fuzzy bounded linear functionals on rfuzzy normed linear spaces is obtained.





A new method for solving fuzzy differential equations
Amit Kumar, Sneh Lata
To the best of our knowledge till now there are only two analytical methods for finding the exact solution of fuzzy differential equations. In this paper, the shortcoming of one of these existing methods is pointed out. To overcome the shortcoming of the existing method, a new method, named as Mehar's method, is proposed for solving fuzzy differential equations. To show the advantage of Mehar's method over existing method the fuzzy Kolmogorov's differential equations, developed by using fuzzy Markov model of piston manufacturing system, are solved by using the existing and Mehar's method and it is shown that the results, obtained by using the existing method, may or may not be fuzzy number while the results, obtained by using Mehar's method, are always fuzzy number.





FIXED POINT THEORY FOR CYCLIC φCONTRACTIONS IN FUZZY METRIC SPACES
YONGHONG SHEN, DONG QIU AND WEI CHEN
In this paper, the notion of cyclic φcontraction in fuzzy metric spaces is introduced and a fixed point theorem for this type of mapping isestablished. Meantime, an example is provided to illustrate this theorem. The main result shows that a selfmapping on a Gcomplete fuzzy metric space has a unique fixed point if it satisfied yclic φcontraction. Afterwards, some results in connection with the fixed point are given.





REVISION OF SIGN DISTANCE METHOD FOR RANKING OF
FUZZY NUMBERS
S. ABBASBANDY, R. NURAEI AND M. GHANBARI
Recently, Abbasbandy and Asady have been proposed a modification of the distance based approach, namely \sign distance method". However, in this paper, it is shown that this method has some drawbacks, i.e., the result is not consistent with human intuition for special cases and this method can not always logically infer ranking order of the images of the fuzzy numbers. In this paper, we present a revised method which can avoid these problems for ranking fuzzy numbers. Also, we present several properties for revised sign distance method while the original method does not have some of them.





ON FUZZY CONVEX LATTICEORDERED SUBGROUPS
MAHMOOD BAKHSHI
Abstract. In this paper, the concept of fuzzy convex subgroup (resp. fuzzy convex latticeordered subgroup) of an ordered group (resp. latticeordered group) is introduced and some properties, characterizations and related results are given. Also, the fuzzy convex subgroup (resp. fuzzy convex latticeordered msubgroup) generated by a fuzzy subgroup (resp. fuzzy subsemigroup) is characterized. Furthermore, the Fundamental Homomorphism Theorem is established. Finally, it is proved that the class of all fuzzy convex latticeordered subgroups of a latticeordered group G forms a complete Heyting sublattice of
the lattice of fuzzy subgroups of G.





OPTIMAL CONTROL OF FUZZY LINEAR CONTROLLED
SYSTEM WITH FUZZY INITIAL CONDITIONS
M. NAJARIYAN, M. H. FARAHI
Abstract. In this article we found the solution of fuzzy linear controlled system with fuzzy initial conditions by using αcuts and presentation of numbers in a more compact form by moving to the field of complex numbers. Next, a fuzzy optimal control problem for a fuzzy system is considered to optimize the expected value of a fuzzy objective function. Based on Pontryagin Maximum Principle, a constructive equation for the problem is presented. In the last section, three examples are used to show that the method in effective to solve fuzzy and fuzzy optimal linear controlled systems.





FUZZY EQUATIONAL CLASSES ARE FUZZY VARIETIES
B. BUDIMIROVIC, V. BUDIMIROVIC, B. SESELJA, A. TEPAVCEVIC
In the framework of fuzzy algebras with fuzzy equalities and a complete lattice as a structure of membership values, we investigate fuzzy equational classes. They consist of special fuzzy algebras fulfilling same fuzzy identities, defined with respect to fuzzy equalities. We introduce basic notions and the corresponding operators of universal algebra: construction of fuzzy subalgebras, homomorphisms and direct products. We prove that every fuzzy equational class is closed under these three operators, which means that such a class is a fuzzy variety.





HURST EXPONENTS FOR NONPRECISE DATA
M. ALVO AND F. THEBERGE
We provide a framework for the study of statistical quantities related to the Hurst phenomenon when the data are nonprecise with finite support.





EXISTENCE OF EXTREMAL SOLUTIONS FOR IMPULSIVE
DELAY FUZZY INTEGRODIFFERENTIAL EQUATIONS IN
nDIMENSIONAL FUZZY VECTOR SPACE
Y. C. KWUN, J. S. KIM, J. S. HWANG AND J. H. PARK
Abstract. In this paper, we study the existence of extremal solutions for impulsive delay fuzzy ntegrodifferential equations in ndimensional fuzzy vector space, by using monotone method. We show that obtained result is an extension of the result of Rodr´ıguezL´opez [8] to impulsive delay fuzzy integrodifferential equations in ndimensional fuzzy vector space.





BILEVEL LINEAR PROGRAMMING WITH FUZZY
PARAMETERS
F. HAMIDI AND H. MISHMAST NEHI
Abstract. Bilevel linear programming is a decision making problem with a twolevel decentralized organization. The “leader” is in the upper level and the “follower”, in the lower. Making a decision at one level affects that at the other one. In this paper, bilevel linear programming with inexact parameters has been studied and a method is proposed to solve a fuzzy bilevel linear programming using interval bilevel linear programming.





MEETCONTINUITY ON LDIRECTED COMPLETE POSETS
SHUHUA SU, QINGGUO LI AND LANKUN GUO
Abstract. In this paper, the definition of meetcontinuity on Ldirected com plete posets (for short, Ldcpos) is introduced. As a generalization of meet continuity on crisp dcpos, meetcontinuity on Ldcpos, based in the generalized Scott topology, is characterized. In particular, it is shown that every continu ous Ldcpo is meetcontinuous and Lcontinuous retracts of meetcontinuous Ldcpos are also meetcontinuous. Then, some topological properties of meet continuity on Ldcpos are discussed. It is shown that meetcontinuity on L dcpos is a topological invariant with respect to the generalized Scott topology, and meetcontinuity on Ldcpos is hereditary with respect to generalized Scott closed subsets.





A NEW SIMILARITY MEASURE BETWEEN TYPE2 FUZZY
NUMBERS AND FUZZY RISK ANALYSIS
D. STEPHEN DINAGAR AND A. ANBALAGAN
Abstract. In this paper, we present a revised similarity measure based on ChenandChen’s similarity measure for fuzzy risk analysis. The revised similarity measure uses the corrected formulae to calculate the centre of gravity points, therefore it is more effective than the ChenandChen’s method. The revised similarity measure can overcome the drawbacks of the existing methods. We have also proposed a new similarity measure between type2 fuzzy numbers and a method for handling fuzzy risk analysis problem.





ON INTERRELATIONSHIPS BETWEEN FUZZY METRIC
STRUCTURES
A. ROLDAN, J. MARTINEZMORENO, C. ROLDAN
Abstract. Considering the increasing interest in fuzzy theory and possible applications, the concept of fuzzy metric space concept has been introduced by several authors from different perspectives. This paper interprets the theory in terms of metrics evaluated on fuzzy numbers and defines a strong Hausdorff topology. We study interrelationships between this theory and other fuzzy theories such as intuitionistic fuzzy metric spaces, Kramosil and Michalek's spaces, Kaleva and Seikkala's spaces, probabilistic metric spaces, probabilistic metric cospaces, Menger spaces and intuitionistic probabilistic metric spaces, determining their position in the framework of theses different theories.





REPRESENTATION THEOREMS OF LSUBSETS AND
LFAMILIES ON COMPLETE RESIDUATED LATTICE
HUI HAN, JINMING FANG
Abstract. In this paper, our purpose is twofold. Firstly, the tensor and residuum operations on Lnested systems are introduced under the condition of complete residuated lattice. Then we show that Lnested systems form a complete residuated lattice, which is precisely the classical isomorphic object of complete residuated power set lattice. Thus the new representation theorem of
Lsubsets on complete residuated lattice is obtained. Secondly, we introduce the concepts of L amily and the system of Lsubsets, then with the tool of the system of Lsubsets, we obtain the representation theorem of intersectionpreserving Lfamilies on complete residuated lattice.





RANDOM FUZZY SETS: A MATHEMATICAL TOOL
TO DEVELOP STATISTICAL FUZZY DATA ANALYSIS
A. BLANCOFERN´ANDEZ, M.R. CASALS, A. COLUBI, N. CORRAL, M. GARC´IAB´ARZANA, M.A. GIL, G. GONZ´ALEZRODR´IGUEZ, M.T. L´OPEZ, M.A. LUBIANO, M. MONTENEGRO, A.B. RAMOSGUAJARDO, S. DE LA ROSA DE S´AA, B. SINOVA
Abstract. Data obtained in association with many reallife random experiments from different fields cannot be perfectly/exactly quantified. Often the underlying imprecision can be suitably described in terms of fuzzy numbers/ values. For these random experiments, the scale of fuzzy numbers/values enables to capture more variability and subjectivity than that of categorical data, and more accuracy and expressiveness than that of numerical/vectorial data. On the other hand, random fuzzy numbers/sets model the random mechanisms generating experimental fuzzy data, and they are soundly formalized within the probabilistic setting.
This paper aims to review a significant part of the recent literature concerning the statistical data analysis with fuzzy data and being developed around the concept of random fuzzy numbers/sets.





AGE REPLACEMENT POLICY IN UNCERTAIN
ENVIRONMENT
KAI YAO, DAN A. RALESCU
Abstract. Age replacement policy is concerned with finding an optional time to minimize the cost, at which time the unit is replaced even if it does not fail. So far, age replacement policy involving random age has been proposed. This paper will assume the age of the unit is an uncertain variable, and find the
optimal time to replace the unit.





LINEAR OBJECTIVE FUNCTION OPTIMIZATION WITH THE
MAXPRODUCT FUZZY RELATION INEQUALITY
CONSTRAINTS
ALI ABBASI MOLAI
Abstract. In this paper, an optimization problem with a linear objective function subject to a consistent finite system of fuzzy relation inequalities us ing the maxproduct composition is studied. Since its feasible domain is non convex, traditional linear programming methods cannot be applied to solve it. We study this problem and capture some special characteristics of its feasi ble domain and optimal solutions. Some procedures are proposed to reduce and decompose the original problem into several subproblems with smaller dimensions. Combining the procedures, a new algorithm is proposed to solve the original problem. An example is also provided to show the eciency of the algorithm.





ON LACUNARY STATISTICAL LIMIT AND CLUSTER POINTS
OF SEQUENCES OF FUZZY NUMBERS
PANKAJ KUMAR, S. S. BHATIA, VIJAY KUMAR
For any lacunary sequence $\theta = (k_{r})$, we define the concepts of $S_{\theta}$limit point and $S_ \theta}$cluster point of a sequence of fuzzy numbers $X = (X_{k})$. We introduce the new sets \Lambda^{F}_{S_{\theta}}(X)$, $\Gamma^{F}_{S_{\theta}}(X)$ and prove some inclusion relaions between these and the sets $\Lambda^{F}_{S}(X)$, $\Gamma^{F}_{S}(X)$ introduced in ~\cite{Ayt:Slpsfn} by Aytar [S. ytar, Statistical limit points of sequences of fuzzy numbers, Inform. Sci. 165 (2004) 129138]. Later, we ind restriction on the lacunary sequence $\theta = (k_{r})$ for which the sets $\Lambda^{F}_{S_{\theta}} X)$ and $\Gamma^{F}_{S_{\theta}}(X)$ respectively coincides with the sets $\Lambda^{F}_{S}(X)$ and \Gamma^{F}_{S}(X)$.





A NOTE ON THE RELATIONSHIP BETWEEN HUTTON'S
QUASIUNIFORMITIES AND SHI'S QUASIUNIFORMITIES
YUELI YUE
This note studies the relationship between Hutton's quasiuniformities and Shi's quasiuniformities. It is shown that when L satis es \multiple choice principle" for coprime elements, the category of Hutton's quasiuniform spaces is a bireective full subcategory of the category of Shi's quasiuniform spaces. Especially, If the remoteneighborhood mapping defined by Shi preserves arbitrary joins, then the two categories are isomorphic to each other.





AN EMPIRICAL COMPARISON BETWEEN GRADE OF
MEMBERSHIP AND PRINCIPAL COMPONENT ANALYSIS
A. SULEMAN
Abstract. It is the purpose of this paper to contribute to the discussion ini tiated by Wachter about the arallelism between principal component (PC) and a typological grade of membership (GoM) analysis. The uthor tested empirically the close relationship between both analysis in a low dimensional framework omprising up to nine dichotomous variables and two typologies. Our contribution to the subject is also mpirical. It relies on a dataset from a survey which was especially designed to study the reward of skills in he banking sector in Portugal. The statistical data comprise thirty polythomous variables and were ecomposed in four typologies using an optimality crite rion. The empirical evidence shows a high rrelation etween the first PC scores and individual GoM scores. No correlation with the remaining PCs was found, however. In addtion to that, the first PC also proved efiective to rank individuals by skill following e particularity of data distribution meanwhile unveiled in GoM analysis.





ON APPROXIMATE CAUCHY EQUATION IN FELBIN'S TYPE
FUZZY NORMED LINEAR SPACES
ILDAR SADEQI, FRIDOUN MORADLOU AND MITRA SALEHI
Abstract. In this paper we study the HyersUlamRassias stability of Cauchy equations in Felbin's type fuzzy normed linear spaces. As a result we give an example of a fuzzy normed linear space such that the fuzzy version of the stability problem remains true, while it fails to be correct in classical analysis.
This shows how the category of fuzzy normed linear spaces di_ers from the classical normed linear spaces in general..





ALGEBRAICALLYTOPOLOGICAL SYSTEMS AND
ATTACHMENTS
ANNA FRASCELLA, COSIMO GUIDO, SERGEY A. SOLOVYOV
Abstract. The paper continues the study of the authors on relationships between topological systems of S. Vickers and attachments of C. Guido. We extend topological systems to algebraicallytopological systems. A particular instance of the latter, called attachment system, incorporates the notion of attachment, thus, making it categorically redundant in mathematics. We show that attachment systems are equipped with an internal topology, which is similar to the topology induced by locales. In particular, we provide an attachment system analogue of the wellknown categorical equivalence between sober topological spaces and spatial locales.





APPLICATION OF PREFERENCE RANKING ORGANIZATION
METHOD FOR ENRICHMENT EVALUATION METHOD IN
ENERGY PLANNING  REGIONAL LEVEL
ALI KHATAMI FIROUZABADI AND ELHAM GHAZIMATIN
Abstract. Nowadays energy is one of the most essential needs of human being and it can be considered as the basic prerequisite of social and economic development. Hence, many of the correlations and legislations of a country are affected by it. Since Iran has huge source of gas and oil, it has turned to a fossil fuel oriented county. But as oil and gas sources are nonrenewable ones and cannot be replaced, its essential for any country to focus on Renewable Energy Sources (RES). So today is the time of studying and investing on RES to be able to exploit them in the time of oil and gas crisis. In the past, the choice among alternative sources was based on cost minimization, but ranking the RES options is a complex task. The objective of this paper is determining the best renewable energy alternative for Sistan & Baluchestan province of Iran by using interval Preference Ranking Organization Method for Enrichment Evaluation (PROMETHEE) method. In the application of the proposed methodology the most appropriate renewable energy alternative is determined fuel cell and biomass for the mentioned province.





UNIVERSAL TRIPLE I METHOD FOR FUZZY REASONING
AND FUZZY CONTROLLER
Y. M. TANG, F. J. REN
Abstract. As a generalization of the triple I method, the universal triple I method is investigated from the viewpoints of both fuzzy reasoning and fuzzy controller. The universal triple I principle is put forward, which improves the previous triple I principle. Then, unified form of universal triple I method is established based on the (0,1)implication or Rimplication. Moreover, the reversibility property of universal triple I method is analyzed from expansion, reduction and other type operators, which demonstrate that its reversibility property seems fine, especially for the case employing the (0,1)implication. Lastly, we analyze the response ability of fuzzy controllers based on universal triple I method, then the practicability of triple I method is improved.





REGION MERGING STRATEGY FOR BRAIN MRI
SEGMENTATION USING DEMPSTERSHAFER THEORY
J. GHASEMI, M. R. KARAMI MOLLAEI, R. GHADERI, AND S. A. HOJJATOLESLAMI
Abstract. Detection of brain tissues using magnetic resonance imaging (MRI) is an active and challenging research area in computational neuroscience. Brain MRI artifacts lead to an uncertainty in pixel values. Therefore, brain MRI segmentation is a complicated concern which is tackled by a novel data fusion approach. The proposed algorithm has two main steps. In the rst step the brain MRI is divided to some main and ancillary cluster which is done using Fuzzy cmean (FCM). In the second one, the considering ancillary clusters are merged with main clusters employing DempsterShafer Theory. The proposed method was validated on simulated brain images from the commonly used BrainWeb dataset. The results of the proposed method are evaluated by using Dice and Tanimoto coecients which demonstrate well performance and robustness of this algorithm.





ON FUZZY NEIGHBORHOOD BASED CLUSTERING
ALGORITHM WITH LOW COMPLEXITY
G. ULUTAGAY & E. NASIBOV
Abstract. The main purpose of this paper is to achieve improvement in the speed of Fuzzy Joint Points (FJP) lgorithm. Since FJP approach is a basis for fuzzy neighborhood based clustering algorithms such as NoiseRobust FJP (NRFJP) and Fuzzy Neighborhood DBSCAN (FNDBSCAN), improving FJP algorithm ould an important achievement in terms of these FJPbased meth ods. Although FJP has many advantages uch as robustness, auto detection of the optimal number of clusters by using cluster validity, dependency rom scale, etc., it is a little bit slow. In order to eliminate this disadvantage, by im proving the FJP algorithm, we propose a novel Modified FJP algorithm, which theoretically runs approximately n / log2 n times faster nd which is less com plex than the FJP algorithm. We evaluated the performance of the Modified FJP lgorithm both analytically and experimentally.





PRESERVATION THEOREMS IN
LUKASIEWICZ MODEL THEORY
S.M. BAGHERI, M. MONIRI
Abstract. We present some model theoretic results for Lukasiewicz predicate logic by using the methods of continuous model theory developed by Chang and Keisler. We prove compactness theorem with respect to the class of all structures taking values in the Lukasiewicz BLalgebra. We also prove some appropriate preservation theorems concerning universal and inductive theories. Finally, Skolemization and Morleyization in this framework are discussed and some natural examples of fuzzy theories are presented.





A NOTE ON STRATIFIED LMFILTERS
GUNTHER JAGER
Abstract. We develop a theory of stratified LMfilters which generalizes the theory of stratified Lfilters. Our stratification condition explicitly depends on a suitable mapping between the lattices L and M. If L and M are identical and the mapping is the identity mapping, then we obtain the theory of stratified Lfilters. Based on the stratified LMfilters, a general theory of latticevalued convergence spaces can be developed.





ADAPTIVE ORDERED WEIGHTED AVERAGING FOR
ANOMALY DETECTION IN CLUSTERBASED MOBILE AD
HOC NETWORKS
M. RAHMANIMANESH AND S. JALILI
Abstract. In this paper, an anomaly detection method in clusterbased mobile ad hoc networks with ad hoc on demand distance vector (AODV) routing protocol is proposed. In the method, the required features for describing the normal behavior of AODV are de ned via step by step analysis of AODV and independent of any attack. In order to learn the normal behavior of AODV, a fuzzy averaging method is used for combining oneclass support vector machine (OCSVM), mixture of Gaussians (MoG), and selforganizing maps (SOM) oneclass classi ers and the combined model is utilized to partially detect the attacks in cluster members. The votes of cluster members are periodically transmitted to the cluster head and nal decision on attack detec tion is carried out in the cluster head. In the proposed method, an adaptive ordered weighted averaging (OWA) operator is used for aggregating the votes of cluster members in the cluster head. Since the network topology, traffic, and environmental conditions of a MANET as well as the number of nodes in each cluster dynamically change, the mere use of a xed quanti erbased weight generation approach for OWA operator is not efficient. We propose a conditionbased weight generation method for OWA operator in which the number of cluster members that participate in decision making may be varying in time and OWA weights are calculated periodically and dynamically based on the conditions of the network. Simulation results demonstrate the e ec tiveness of the proposed method in detecting rushing, RouteError fabrication, and wormhole attacks.





I_{2}CONVERGENCE OF DOUBLE SEQUENCES
OF FUZZY NUMBERS
ERDINC. DUNDAR AND OZER TALO
Abstract. In this paper, we introduce and study the concepts of I_{2}convergence, I^{*}_{2 }convergence for double sequences of fuzzy real numbers, where I_{2} denotes the ideal of subsets of N×N. Also, we study some properties and relations of them.





NEW RESULTS ON THE EXISTING FUZZY DISTANCE
MEASURES
S. ABBASBANDY, S. SALAHSHOUR
Abstract. In this paper, we investigate the properties of some recently proposed fuzzy distance measures. We find out some shortcomings for these distances and then the obtained results are illustrated by solving several examples and compared with the other fuzzy distances.






