2004
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PREFACE
PREFACE
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A NEUROFUZZY GRAPHIC OBJECT CLASSIFIER WITH MODIFIED DISTANCE MEASURE ESTIMATOR
A NEUROFUZZY GRAPHIC OBJECT CLASSIFIER WITH MODIFIED DISTANCE MEASURE ESTIMATOR
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2
The paper analyses issues leading to errors in graphic object classifiers. Thedistance measures suggested in literature and used as a basis in traditional, fuzzy, andNeuroFuzzy classifiers are found to be not suitable for classification of nonstylized orfuzzy objects in which the features of classes are much more difficult to recognize becauseof significant uncertainties in their location and graylevels. The authors suggest a neurofuzzygraphic object classifier with modified distance measure that gives betterperformance indices than systems based on traditional ordinary and cumulative distancemeasures. Simulation has shown that the quality of recognition significantly improveswhen using the suggested method.
1
The paper analyses issues leading to errors in graphic object classifiers. Thedistance measures suggested in literature and used as a basis in traditional, fuzzy, andNeuroFuzzy classifiers are found to be not suitable for classification of nonstylized orfuzzy objects in which the features of classes are much more difficult to recognize becauseof significant uncertainties in their location and graylevels. The authors suggest a neurofuzzygraphic object classifier with modified distance measure that gives betterperformance indices than systems based on traditional ordinary and cumulative distancemeasures. Simulation has shown that the quality of recognition significantly improveswhen using the suggested method.
5
15
R. A.
ALIEV
R. A.
ALIEV
MEMBER IEEE, DEPARTMENT OF COMPUTERAIDED CONTROL SYSTEMS, AZERBAIJAN
STATE OIL ACADEMY, BAKU, AZERBAIJAN
MEMBER IEEE, DEPARTMENT OF COMPUTERAIDED
Azerbaijan
raliev@iatp.az
B. G.
GUIRIMOV
B. G.
GUIRIMOV
DEPARTMENT OF COMPUTERAIDED CONTROL SYSTEMS, AZERBAIJAN STATE OIL
ACADEMY, BAKU, AZERBAIJAN
DEPARTMENT OF COMPUTERAIDED CONTROL SYSTEMS,
Azerbaijan
guirimov@hotmail.com
R. R.
ALIEV
R. R.
ALIEV
EASTERN MEDITERRANEAN UNIVERSITY, NORTH CYPRUS
EASTERN MEDITERRANEAN UNIVERSITY, NORTH CYPRUS
Azerbaijan
rashad.aliyev@emu.edu.tr
NeuroFuzzy technology
Fuzzy logic
IFTHEN rules
neural network
[[1] J. Bezdek and S. Pal (ed.), Fuzzy models for pattern recognition, New York: IEEE Press (1992).##[2] X. Ye, C.Suen and M.Cheriet, A generic system to extract and clean handwritten data from business##forms, in Prof. Int. Workshop on Frontiers in handwriting Recognition, Amsterdam (2000) 6372.##[3] R. A. Aliev and R. R. Aliev, Soft Computing and its applications, World Scientific Publishing Co.##Pte. Ltd (2001) p. 444.##[4] K. Saastamoinen, V. Könönen, and P. Luukka, A classifier based on the fuzzy similarity in the##Lukasiewicz structure with different metrics, in proceedings of IEEE International Conference on##Fuzzy Systems, FUZZIEEE’02, Vol. 1 (2002) 363367.##[5] H. Bandemer and W. Näther, Fuzzy data analysis, Theory and Decision Library, Series B:##Mathematical and Statistical Methods, Vol. 20, Cluwer Academic Publishers (1992) 6771.##[6] L. Mascarilla and C. Frélicot, Combining rejectionbased pattern classifiers, in 19th International##Conference of the North American Fuzzy Information Processing Society – NAFIPS, PeachFuzz##2000 (2000) 114118.##[7] R Aliev and B. Guirimov, Handwritten image recognition by using neural and fuzzy approaches,##Intelligent Control and Decision Making Systems, No. 1, Thematic Collected Articles, Baku,##Publishing House of Azerb. State Oil Academy (1997) 37.##[8] R. Aliev, B. Guirimov, K. Bonfig, and Steinmann, A neurofuzzy algorithm for recognition of nonstylized##images, in proceedings of Fourth International Conference on Application of Fuzzy##Systems and Soft Computing, ICAFS’2000, Siegen, Germany (2000) 238241.##[9] S. Halgamuge and M. Glesner, A fuzzy neural approach for pattern classification with generation of##rules based on supervised learning, in proceedings of Nuro Nimes 92 (1992) 165173.##[10] V. Uebele, S. Abe, and M. Lan, A neural  network based fuzzy classifier, IEEE Transactions on##Systems, Man, and Cybernetics, Vol. 23, No. 3 (1995) 353361.##[11] R. Yager, A general approach to rule aggregation in fuzzy logic control, Appl. Intelligence, 2##(1992) 333351. [12] R. Yager, On a general class of fuzzy connectives, Fuzzy Sets and Systems, 4 (1980) 235242.##]
AN AGGREGATED FUZZY RELIABILITY INDEX FOR SLOPE STABILITY ANALYSIS
AN AGGREGATED FUZZY RELIABILITY INDEX FOR SLOPE STABILITY ANALYSIS
2
2
While sophisticated analytical methods like MorgensternPrice or finite elementmethods are available for more realistic analysis of stability of slopes, assessment of the exactvalues of soil parameters is practically impossible. Uncertainty in the soil parameters arisesfrom two different sources: scatter in data and systematic error inherent in the estimate of soilproperties. Hence, stability of a slope should be expressed using a factor of safetyaccompanied by a reliability index.In this paper, the theory of fuzzy sets is used to deal simultaneously with the uncertain natureof soil parameters and the inaccuracy involved in the analysis. Soil parameters are definedusing suitable fuzzy sets and the uncertainty inherent in the value of factor of safety isassessed accordingly. It is believed that this approach accounts for the uncertainty in soilparameters more realistically compared to the conventional probabilistic approaches reportedin the literature. A computer program is developed that carries out the large amount ofcalculations required for evaluating the fuzzy factor of safety based on the concept of domaininterval analysis. An aggregated fuzzy reliability index (AFRI) is defined and assigned to thecalculated factor of safety. The proposed method is applied to a case study and the results arediscussed in details. Results from sensitivity analysis describe where the exploration effort orquality control should be concentrated. The advantage of the proposed method lies in its fastcalculation speed as well as its ease of data acquisition from experts’ opinion through fuzzysets.
1
While sophisticated analytical methods like MorgensternPrice or finite elementmethods are available for more realistic analysis of stability of slopes, assessment of the exactvalues of soil parameters is practically impossible. Uncertainty in the soil parameters arisesfrom two different sources: scatter in data and systematic error inherent in the estimate of soilproperties. Hence, stability of a slope should be expressed using a factor of safetyaccompanied by a reliability index.In this paper, the theory of fuzzy sets is used to deal simultaneously with the uncertain natureof soil parameters and the inaccuracy involved in the analysis. Soil parameters are definedusing suitable fuzzy sets and the uncertainty inherent in the value of factor of safety isassessed accordingly. It is believed that this approach accounts for the uncertainty in soilparameters more realistically compared to the conventional probabilistic approaches reportedin the literature. A computer program is developed that carries out the large amount ofcalculations required for evaluating the fuzzy factor of safety based on the concept of domaininterval analysis. An aggregated fuzzy reliability index (AFRI) is defined and assigned to thecalculated factor of safety. The proposed method is applied to a case study and the results arediscussed in details. Results from sensitivity analysis describe where the exploration effort orquality control should be concentrated. The advantage of the proposed method lies in its fastcalculation speed as well as its ease of data acquisition from experts’ opinion through fuzzysets.
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31
MEHRASHK
MEIDANI
MEHRASHK
MEIDANI
MEHRASHK MEIDANI, PHD STUDENT, CIVIL ENGINEERING DEPARTMENT, SCHOOL OF
ENGINEERING, SHIRAZ UNIVERSITY, SHIRAZ, IRAN
MEHRASHK MEIDANI, PHD STUDENT, CIVIL ENGINEERING
Iran
mehrashk@yahoo.com
GHASSEM
HABIBAGAHI
GHASSEM
HABIBAGAHI
ASSOCIATE PROF., CIVIL ENGINEERING DEPARTMENT,
SCHOOL OF ENGINEERING, SHIRAZ UNIVERSITY, SHIRAZ, IRAN
ASSOCIATE PROF., CIVIL ENGINEERING DEPARTMENT,
SCH
Iran
habibg@shirazu.ac.ir
SERAJEDIN
KATEBI
SERAJEDIN
KATEBI
DEPARTMENT OF COMPUTER SCIENCE, SCHOOL OF ENGINEERING,
SHIRAZ UNIVERSITY, SHIRAZ, IRAN
DEPARTMENT OF COMPUTER SCIENCE, SCHOOL OF
Iran
shiraz
Slope Stability
Uncertainty
Fuzzy sets
Reliability
[[1] M. Andonyadis, A. G. Altschaeffl and J. L. Chameau, “Fuzzy sets in pavement evaluation and##management: Application and interpretation.” FHWA/IN/JHRP report #8514, School of Civil##Engrg., Purdue Univ., (1985).##[2] B. M. Ayyub, Uncertainty Modeling and Analysis in Civil Engineering, CRC Press, Boca Raton,##Florida, (1998).##[3] E. N. Broomhead, The Stability of Slopes, Surrey Univ. Press, Chapman and Hall, New York,##[4] T. B. Celestino and J. M. Duncan, Simplified search for noncircular slip surface, Proc., 10th Int.##Conf. on Soil Mech. and Found. Engrg., A. A. Balkema, Rotterdam, The Netherlands, 3 (1981)##[5] R. N. Chowdhury, and D. AGrivas, Probabilistic model of progressive failure of slopes, J.##Geotech. Engrg. Div., ASCE, 108(6) (1982) 803819.##[6] R. N. Chowdhury and D. H. Xu, Slope system reliability with general slip surfaces, Soils and##Foundations, 34(3) (1994) 99105.##[7] J. T. Christian, C. C. Ladd and G. B. Baecher, Reliability applied to slope stability analysis, J.##Geotech. Engrg., ASCE, 120(12) (1994) 21802207.##[8] J. T. Christian, C. C. Ladd and G. B. Baecher, Reliability and probability in stability analysis, Proc.,##Stability and Performance of Slopes and EmbankmentsII, ASCE, New York, N.Y., Vol. 2 (1992)##10711111.##[9] W. M. Dong, H. C. Shah and F. S. Wang, Fuzzy computations in risk and decision analysis, Civ.##Engng. Syst., Vol. 2, Dec. (1985) 201208.##[10] J. M. Duncan, State of the art: limit equilibrium and finite element analysis of slopes, J. Geotech.##Engrg., ASCE, 122(7) (1996) 577596.##[11] R. B. Gilbert, S. G. Wright and E. Liedtke, Uncertainty in back analysis of slopes: Kettleman Hills##case history, J. Geotech. Engrg., ASCE, 124(12) (1998) 11671176.##[12] S. Gui, R. Zhang, J. P. Turner and X. Xue, Probabilistic slope stability analysis with stochastic soil##hydraulic conductivity, J. Geotech. Engrg., ASCE, 126(1) (2000) 19.##[13] A. M. Hassan and T. F. Wolff, Search algorithm for minimum reliability index of earth slopes, J.##Geotech. Engrg., ASCE, 125(4) (1999) 301308.##[14] C. H. Juang, J. L. Way and D. J. Elton, Model for capacity of single piles in sand using fuzzy sets, J.##Geotech. Engrg., ASCE, 117(2) (1991) 19201931.##[15] K. S. Li, and P. Lumb, Probabilistic design of slopes, Can. Geotech. J., Vol. 24 (1987) 520 535.##[16] B. K. Low and W. H. Tang, Probabilistic slope analysis using Janbu’s generalized procedure of##slices, Computers and Geotechnics, 21(2) (1997) 121142.##[17] N. R. Morgenstern and V. E. Price, The analysis of the stability of general slip surfaces,##Geotechnique, Vol. 15, (1965).##[18] T. J. Ross, Fuzzy Logic with Engineering Applications, McGraw Hill Inc., (1995).##[19] A. W. Skempton and D. J. Coats, Carsington dam failure, Failures in Earthworks, Proc. of the##symp. on Failures in Earthworks, Thomas Telford Ltd., London, (1985).##[20] S. Soulati, “A Gentic approach for determining the generalized interslice forces and the critical##noncircular slip surface for slope stability analysis”, Thesis submitted to the School of Graduate##studies in Partial Fulfillment of the Requirements for the Degree of Mater of Science, Shiraz##University, Shiraz, Iran, (2003).##[21] E. Spencer, Circular and logarithmic spiral slip surfaces, J. Soil Mech. And Found. Div., ASCE,##95(1) (1969) 929942.##[22] E. Spencer, Slip circles and critical shear planes, J. Geotech. Engrg. Div., ASCE, 107(7) (1981)##[23] W. H. Tang, M. S. Yucemen and A. H.S. Ang, Probabilitybased short term design of slopes, Can.##Geotech. J., 13(3) (1976) 201 215.##[24] E. H. Vanmarcke, Reliability of earth slopes, J.Geotech. Engrg. Div., ASCE, 103(11) (1977) 1227##[25] R. V. Whitman, Evaluating calculated risk in geotechnical engineering, J. Geotech. Engrg., ASCE,##110(2) (1984) 145188.##[26] L. A. Zadeh, Fuzzy sets, Information and Control, 8 (1965) 338353.##[27] H. J. Zimmermann, Fuzzy Set Theory and Its Applications, Kluwer Academic Publishers, (1996).##]
ON A LOSSY IMAGE COMPRESSION/RECONSTRUCTION METHOD BASED ON FUZZY RELATIONAL EQUATIONS
ON A LOSSY IMAGE COMPRESSION/RECONSTRUCTION
METHOD BASED ON FUZZY RELATIONAL EQUATIONS
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2
The pioneer work of image compression/reconstruction based onfuzzy relational equations (ICF) and the related works are introduced. TheICF regards an original image as a fuzzy relation by embedding the brightnesslevel into [0,1]. The compression/reconstruction of ICF correspond to thecomposition/solving inverse problem formulated on fuzzy relational equations.Optimizations of ICF can be consequently deduced based on fuzzy relationalcalculus, i.e., computation time reduction/improvement of reconstructed imagequality are correspond to a fast solving method/finding an approximatesolution of fuzzy relational equations, respectively. Through the experimentsusing test images extracted from Standard Image DataBAse (SIDBA), theeffectiveness of the ICF and its optimizations are shown.
1
The pioneer work of image compression/reconstruction based onfuzzy relational equations (ICF) and the related works are introduced. TheICF regards an original image as a fuzzy relation by embedding the brightnesslevel into [0,1]. The compression/reconstruction of ICF correspond to thecomposition/solving inverse problem formulated on fuzzy relational equations.Optimizations of ICF can be consequently deduced based on fuzzy relationalcalculus, i.e., computation time reduction/improvement of reconstructed imagequality are correspond to a fast solving method/finding an approximatesolution of fuzzy relational equations, respectively. Through the experimentsusing test images extracted from Standard Image DataBAse (SIDBA), theeffectiveness of the ICF and its optimizations are shown.
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42
Kaoru
Hirota
Kaoru
Hirota
Kaoru Hirota, Department
of Computational Intelligence and Systems Science, Tokyo Institute of Technology,
Yokohama, 2268502, Japan
Kaoru Hirota, Department
of Computational
Japan
hirota@hrt.dis.titech.ac.jp
Hajime
Nobuhara
Hajime
Nobuhara
Department
of Computational Intelligence and Systems Science, Tokyo Institute of Technology,
Yokohama, 2268502, Japan
Department
of Computational Intelligence
Japan
nobuhara@hrt.dis.titech.ac.jp
Kazuhiko
Kawamoto
Kazuhiko
Kawamoto
Department
of Computational Intelligence and Systems Science, Tokyo Institute of Technology,
Yokohama, 2268502, Japan
Department
of Computational Intelligence
Japan
kawa@hrt.dis.titech.ac.jp
Shinichi
Yoshida
Shinichi
Yoshida
Department
of Computational Intelligence and Systems Science, Tokyo Institute of Technology,
Yokohama, 2268502, Japan
Department
of Computational Intelligence
Japan
shin@hrt.dis.titech.ac.jp
Fuzzy relation
Fuzzy Relational Equation
Lossy Image Compression/ Reconstruction
Ordered Structure
[[1] A. DiNola, W. Pedrycz, and S. Sessa, Fuzzy Relational Structures: The StateofArt, Fuzzy##Sets and Systems, Vol. 75, No. 3(1995) 241262.##[2] A. DiNola, S. Sessa, W. Pedrycz, and E. Sanchez, Fuzzy Relation Equation and Their Applications##to Knowledge Engineering, Kluwer Academic Publishers, 1989.##[3] K. Hirota, and W. Pedrycz, Fuzzy Relational Compression, IEEE Transactions on Systems,##Man, and Cybernetics, Vol. 29 , No. 3(1999) 407415.##[4] H. Nobuhara, W. Pedrycz, and K. Hirota, Fast Solving Method of Fuzzy Relational Equation##and Its Application to Lossy Image Compression/Reconstruction, IEEE Transactions on##Fuzzy Systems, Vol. 8, No. 3(2000) 325334.##[5] H. Nobuhara, Y. Takama, and K. Hirota, Image Compression/Reconstruction Based on Various##Types of Fuzzy Relational Equations, The Transaction of The Institute of Electrical##Engineers of Japan (in Japanese), Vol. 121, No. 6 (2001) 11021113.##[6] H. Nobuhara, Y. Takama, W. Pedrycz, and K. Hirota, Lossy Image Compression and Reconstruction##Based on Fuzzy Relational Equations,Fuzzy Filters for Image Processing, Springer##(2002) 339355.##[7] H. Nobuhara, W. Pedrycz, and K. Hirota, A Digital Watermarking Algorithm using Image##Compression Method based on Fuzzy Relational Equation, IEEE International Conference on##Fuzzy Systems, Hawaii, USA, May 1217 (2002) (CDProceedings).##[8] H. Nobuhara, W. Pedrycz, and K. Hirota, Fuzzy Relational Image Compression using Nonuniform##Coders Designed by Overlap Level of Fuzzy Sets, International Conference on Fuzzy##Systems and Knowledge Discovery (FSKD’02), 2002, Singapore (CDProceedings).##[9] H. Nobuhara, and K. Hirota, Nonuniform Coders Design for Motion Compression Method by##Fuzzy Relational Equation, International Fuzzy System AssociationWorld Congress, Istanbul,##Turkey, June 29  July 2, Lecture Notes in Artificiall Intelligence, No. 2715(2003) 428435. ##[10] W. Pedrycz, Fuzzy Relational Equations with Generalized Connectives and Their Applications,##Fuzzy Sets and Systems, Vol. 10 (1983) 185201.##]
FUZZY INFORMATION AND STOCHASTICS
FUZZY INFORMATION AND STOCHASTICS
2
2
In applications there occur different forms of uncertainty. The twomost important types are randomness (stochastic variability) and imprecision(fuzziness). In modelling, the dominating concept to describe uncertainty isusing stochastic models which are based on probability. However, fuzzinessis not stochastic in nature and therefore it is not considered in probabilisticmodels.Since many years the description and analysis of fuzziness is subject of intensiveresearch. These research activities do not only deal with the fuzziness ofobserved data, but also with imprecision of informations. Especially methodsof standard statistical analysis were generalized to the situation of fuzzy observations.The present paper contains an overview about of the presentationof fuzzy information and the generalization of some basic classical statisticalconcepts to the situation of fuzzy data.
1
In applications there occur different forms of uncertainty. The twomost important types are randomness (stochastic variability) and imprecision(fuzziness). In modelling, the dominating concept to describe uncertainty isusing stochastic models which are based on probability. However, fuzzinessis not stochastic in nature and therefore it is not considered in probabilisticmodels.Since many years the description and analysis of fuzziness is subject of intensiveresearch. These research activities do not only deal with the fuzziness ofobserved data, but also with imprecision of informations. Especially methodsof standard statistical analysis were generalized to the situation of fuzzy observations.The present paper contains an overview about of the presentationof fuzzy information and the generalization of some basic classical statisticalconcepts to the situation of fuzzy data.
43
56
Reinhard
Viertl
Reinhard
Viertl
Department of Statistics and Probability Theory, Vienna University
of Technology, Wien, Austria
Department of Statistics and Probability
Australia
r.viertl@tuwien.ac.at
Dietmar
Hareter
Dietmar
Hareter
Department of Statistics and Probability Theory, Vienna University
of Technology, Wien, Austria
Department of Statistics and Probability
Australia
hareter@statistik.tuwien.ac.at
Fuzzy numbers
Fuzzy Probability Distributions
Fuzzy Random Variables
Fuzzy Stochastic Processes
Decision on Fuzzy Information
[[1] E.P. Klement, M.L. Puri, D.A. Ralescu, Law of large numbers and central limit theorem for##fuzzy random variables, Cybernetics and Systems Research 2, Proc. 7th Europ. Meet., Vienna##1984, 525529 (1984).##[2] E.P. Klement, R. Mesiar, E. Pap, Triangular Norms, Kluwer Acad. Publ., Dordrecht, 2000 .##[3] G. Klir, B. Yuan, Fuzzy Sets and Fuzzy Logic  Theory and Applications, Prentice Hall, Upper##Saddle River, New Jersey, 1995 .##[4] R. Kruse, The strong law of large numbers for fuzzy random variables, Information Science,##Vol. 28 (1982) 233241 .##[5] H. Kwakernaak, Fuzzy random variables  I. definition and theorems, Information Science,##Vol. 15 (1978) 129 .##[6] H. Kwakernaak, Fuzzy random variables  II. algorithms and examples, Information Science,##Vol. 17 (1979) 253278 .##[7] B. M¨oller, W. Graf, M. Beer, Safety assessment of structures in view of fuzzy randomness,##Computers & Structures, Vol. 81 (2003). ##[8] S. Niculescu, R. Viertl, Bernoulli’s Law of Large Numbers for Vague Data, Fuzzy Sets and##Systems, Vol. 50 (1992).##[9] M.L. Puri, D.A. Ralescu, Fuzzy Random Variables, Journal of Math. Anal. and Appl., Vol.##114 (1986) 409422 .##[10] C. R¨omer, A. Kandel, Statistical tests for fuzzy data, Fuzzy Sets and Systems, Vol. 72 (1995).##[11] J. Sickert, M. Beer, W. Graf, B. M¨oller, Fuzzy probabilistic structural analysis considering##fuzzy random functions, in A. Kiureghian, S. Madanat, J. Pestana (Eds.), Applications of##Statistics and Probability in Civil Engineering, Milpress, Rotterdam, 2003 .##[12] S.M. Taheri, Trends in Fuzzy Statistics, Austrian Journal of Statistics, Vol. 32, No. 3 (2003)##[13] R. Viertl, Statistical Methods for NonPrecise Data, CRC Press, Boca Raton, Florida, 1996##[14] R. Viertl, On the description and analysis of measurements of continuous quantities, Kybernetika,##Vol. 38 (2002).##[15] P. Filzmoser, R. Viertl, Testing Hypotheses with Fuzzy Data, The Fuzzy pvalue, to appear##in Metrika.##[16] R. Viertl, D. Hareter, Fuzzy Information and Imprecise Probability, to appear in ZAMM.##[17] W. Voß (Ed.), Taschenbuch der Statistik, Carl Hauser Verlag, M¨unchen, 2004.##[18] G. Wang, Y. Zhang, The theory of fuzzy stochastic processes, Fuzzy Sets and Systems, Vol.##51 (1992).##]
ON DEGREES OF END NODES AND CUT NODES IN FUZZY GRAPHS
ON DEGREES OF END NODES AND CUT NODES IN FUZZY
GRAPHS
2
2
The notion of strong arcs in a fuzzy graph was introduced byBhutani and Rosenfeld in [1] and fuzzy end nodes in the subsequent paper[2] using the concept of strong arcs. In Mordeson and Yao [7], the notion of“degrees” for concepts fuzzified from graph theory were defined and studied.In this note, we discuss degrees for fuzzy end nodes and study further someproperties of fuzzy end nodes and fuzzy cut nodes.
1
The notion of strong arcs in a fuzzy graph was introduced byBhutani and Rosenfeld in [1] and fuzzy end nodes in the subsequent paper[2] using the concept of strong arcs. In Mordeson and Yao [7], the notion of“degrees” for concepts fuzzified from graph theory were defined and studied.In this note, we discuss degrees for fuzzy end nodes and study further someproperties of fuzzy end nodes and fuzzy cut nodes.
57
64
Kiran R.
Bhutani
Kiran R.
Bhutani
Department of Mathematics, The Catholic University of America,
Washington, DC 20064, USA
Department of Mathematics, The Catholic University
United States
bhutani@cua.edu
John
Mordeson
John
Mordeson
Department of Mathematics and Computer Science, Creighton University,
Omaha, NB 68178, USA
Department of Mathematics and Computer Science,
United States
mordes@creighton.edu
Azriel
Rosenfeld
Azriel
Rosenfeld
Center for Automation Research, University of Maryland, College
Park, MD 20742, USA
Center for Automation Research, University
United States
ar@cfar.umd.edu
Fuzzy graph
Fuzzy End Node
Strong Arc
Fuzzy Cut Node
Weak Cut Node
[[1] K.R. Bhutani and A. Rosenfeld, Strong arcs in fuzzy graphs, Information Sciences, 152 (2003)##[2] K.R. Bhutani, A. Rosenfeld, Fuzzy end nodes in fuzzy graphs, Information Sciences, 152##(2003) 323326.##[3] S.F. Buckley, F. Harary, Distance in Graphs, AddisonWesley, Redwood City, CA, 1990.##[4] M. Delgado, J.L Verdegay, M. A. Vila, On fuzzy tree definition, European J. Operations##Research, 22 (1985) 243249.##[5] C.M. Klein, Fuzzy Shortest Paths, Fuzzy Sets and Systems, 39 (1991) 2741.##[6] J.N. Mordeson, P.S. Nair, Fuzzy Graphs and Fuzzy Hypergraphs, PhysicaVerlag, Heidelberg,##[7] J.N. Mordeson, Y. Y. Yao, it Fuzzy cycles and fuzzy trees, The Journal of Fuzzy Mathematics,##10 (2002) 189202.##[8] A. Rosenfeld, Fuzzy graphs, in L.A. Zadeh, K.S.Fu, K. Tanaka, and M. Shimura, eds., Fuzzy##Sets and their Applications to Cognitive and Decision Processes, Academic Press, New York##(1975) 77–95.##[9] M.S. Sunitha, A. Vijayakumar, A characterization of fuzzy trees, Information Sciences, 113##(1999) 293300.##]
INTUITIONISTIC FUZZY HYPER BCKIDEALS OF HYPER BCKALGEBRAS
INTUITIONISTIC FUZZY HYPER BCKIDEALS OF HYPER
BCKALGEBRAS
2
2
The intuitionistic fuzzification of (strong, weak, sweak) hyperBCKideals is introduced, and related properties are investigated. Characterizationsof an intuitionistic fuzzy hyper BCKideal are established. Using acollection of hyper BCKideals with some conditions, an intuitionistic fuzzyhyper BCKideal is built.
1
The intuitionistic fuzzification of (strong, weak, sweak) hyperBCKideals is introduced, and related properties are investigated. Characterizationsof an intuitionistic fuzzy hyper BCKideal are established. Using acollection of hyper BCKideals with some conditions, an intuitionistic fuzzyhyper BCKideal is built.
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77
Rajab Ali
Borzooei
Rajab Ali
Borzooei
Department of Mathematics, University of Sistan and Baluchestan,
Zahedan, Iran
Department of Mathematics, University of
Iran
Young Bae
Jun
Young Bae
Jun
Department of Mathematics Education, Gyeongsang National University,
Chinju (Jinju) 660701, Korea
Department of Mathematics Education, Gyeongsang
Korea
ybjun@nongae.gsnu.ac.kr
Hyper BCKalgebra
infsup property
Intuitionistic Fuzzy (Weak
sweak
Strong) Hyper BCKideal
[[1] K. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets and Systems, 20 , No. 1 (1986) 8796.##[2] K. Atanassov, New operations defined over the intuitionistic fuzzy sets, Fuzzy Sets and Systems,##61 (1994) 137142.##[3] Y. B. Jun and W. H. Shim, Fuzzy implicative hyper BCKideals of hyper BCKalgebras,##Internat. J. Math. & Math. Sci., 29 , No. 2 (2002) 63–70.##[4] Y. B. Jun and X. L. Xin, Scalar elements and hyperatoms of hyper BCKalgebras, Scientiae##Mathematicae, Vol. 2 , No. 3 (1999) 303–309.##[5] Y. B. Jun and X. L. Xin, Fuzzy hyper BCKideals of hyper BCKalgebras, Scientiae Mathematicae##Japonicae, Vol. 53 , No. 2 (2001) 353–360.##[6] Y. B. Jun, X. L. Xin, E. H. Roh and M. M. Zahedi, Strong hyper BCKideals of hyper##BCKalgebras, Math. Japonica, Vol. 51 , No. 3 (2000), 493–498.##[7] Y. B. Jun, M. M. Zahedi, X. L. Xin and R. A. Borzooei, On hyper BCKalgebras, Italian J.##of Pure and Appl. Math., 8 (2000) 127136.##[8] F. Marty, Sur une generalization de la notion de groupe, 8th Congress Math. Scandinaves,##Stockholm (1934) 4549.##[9] L. A. Zadeh, Fuzzy sets, Information and Control, 8 (1965) 338–353.##]
COUNTABLE COMPACTNESS AND THE LINDEL¨OF PROPERTY OF LFUZZY SETS
COUNTABLE COMPACTNESS AND THE LINDEL¨OF
PROPERTY OF LFUZZY SETS
2
2
In this paper, countable compactness and the Lindel¨of propertyare defined for Lfuzzy sets, where L is a complete de Morgan algebra. Theydon’t rely on the structure of the basis lattice L and no distributivity is requiredin L. A fuzzy compact Lset is countably compact and has the Lindel¨ofproperty. An Lset having the Lindel¨of property is countably compact if andonly if it is fuzzy compact. Many characterizations of countable compactnessand the Lindel¨of property are presented by means of open Lsets and closedLsets when L is a completely distributive de Morgan algebra.
1
In this paper, countable compactness and the Lindel¨of propertyare defined for Lfuzzy sets, where L is a complete de Morgan algebra. Theydon’t rely on the structure of the basis lattice L and no distributivity is requiredin L. A fuzzy compact Lset is countably compact and has the Lindel¨ofproperty. An Lset having the Lindel¨of property is countably compact if andonly if it is fuzzy compact. Many characterizations of countable compactnessand the Lindel¨of property are presented by means of open Lsets and closedLsets when L is a completely distributive de Morgan algebra.
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FuGui
Shi
FuGui
Shi
Department of Mathematics, Beijing Institute of Technology, Beijing,
100081, P.R. China
Department of Mathematics, Beijing Institute
China
fuguishi@bit.edu.cn or f.g.shi@263.net
Ltopology
Fuzzy Compactness
Countable Compactness
Lindel¨of Property
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