ORIGINAL_ARTICLE
Cover vol. 12, no.5, October 2015
http://ijfs.usb.ac.ir/article_2642_e38b90e7cce4505fe17c4f2f1a961d07.pdf
2015-10-01T11:23:20
2017-09-23T11:23:20
0
10.22111/ijfs.2015.2642
ORIGINAL_ARTICLE
Functorial semantics of topological theories
Following the categorical approach to universal algebra through algebraic theories, proposed by F.~W.~Lawvere in his PhD thesis, this paper aims at introducing a similar setting for general topology. The cornerstone of the new framework is the notion of emph{categorically-algebraic} (emph{catalg}) emph{topological theory}, whose models induce a category of topological structures. We introduce the quasicategory of catalg topological theories and consider its functorial relationships with the quasicategory of the categories of models, in order to provide convenient means for studying topological structures via the properties of their corresponding theories.
http://ijfs.usb.ac.ir/article_2110_e9eb0f65766fb9dfb4d98bd6fea53fbc.pdf
2015-10-30T11:23:20
2017-09-23T11:23:20
1
43
10.22111/ijfs.2015.2110
Algebra
Algebraic theory
Comma category
Categorically-algebraic topology
Poslat topology
Powerset theory
Topological system
Topological theory
Variety
Sergey A.
Solovyov
sergejs.solovjovs@lumii.lv
true
1
Institute of Mathematics, Faculty of Mechanical Engineering,
Brno University of Technology, Technicka 2896/2, 616 69 Brno, Czech Republic
Institute of Mathematics, Faculty of Mechanical Engineering,
Brno University of Technology, Technicka 2896/2, 616 69 Brno, Czech Republic
Institute of Mathematics, Faculty of Mechanical Engineering,
Brno University of Technology, Technicka 2896/2, 616 69 Brno, Czech Republic
LEAD_AUTHOR
[1] J. Adamek, Introduction to coalgebra, Theory Appl. Categ., 14 (2005), 157{199.
1
[2] J. Adamek, H. Herrlich and G. E. Strecker, Abstract and concrete categories: the joy of cats,
2
Dover Publications (Mineola, New York), 2009.
3
[3] J. Adamek, J. Rosicky and E. M. Vitale, Algebraic theories. A categorical introduction to
4
general algebra, Cambridge University Press, 2011.
5
[4] J. Adamek, M. Sobral and L. Sousa, Morita equivalence of many-sorted algebraic theories,
6
J. Algebra, 297(2) (2006), 361{371.
7
[5] D. Aerts, E. Colebunders, A. van der Voorde and B. van Steirteghem, State property systems
8
and closure spaces: a study of categorical equivalence, Int. J. Theor. Phys., 38(1) (1999),
9
[6] D. Aerts, E. Colebunders, A. van der Voorde and B. van Steirteghem, On the amnestic
10
modication of the category of state property systems, Appl. Categ. Struct., 10(5) (2002),
11
[7] F. Bayoumi and S. E. Rodabaugh, Overview and comparison of localic and xed-basis topological
12
products, Fuzzy Sets Syst., 161(18) (2010), 2397{2439.
13
[8] F. Borceux, Handbook of categorical algebra. Volume 1: basic category theory, Cambridge
14
University Press, 1994.
15
[9] F. Borceux, Handbook of categorical algebra. Volume 2: categories and structures, Cambridge
16
University Press, 1994.
17
[10] F. Borceux and E. M. Vitale, On the notion of bimodel for functorial semantics, Appl. Categ.
18
Struct., 2(3)(1994), 283{295.
19
[11] A. Carboni and P. Johnstone, Connected limits, familial representability and Artin glueing,
20
Math. Struct. Comput. Sci., 5(4) (1995), 441{459.
21
[12] C. L. Chang, Fuzzy topological spaces, J. Math. Anal. Appl., 24 (1968), 182{190.
22
[13] P. M. Cohn, Universal algebra, D. Reidel Publ. Comp., 1981.
23
[14] J. T. Denniston, A. Melton and S. E. Rodabaugh, Lattice-valued topological systems, In:
24
U. Bodenhofer, B. De Baets, E. P. Klement, and S. Saminger-Platz (Eds.), Abstracts of the
25
30th Linz seminar on fuzzy set theory, Johannes Kepler Universitat, Linz, (2009), 24{31.
26
[15] J. T. Denniston, A. Melton and S. E. Rodabaugh, Lattice-valued predicate transformers and
27
interchange systems, In: P. Cintula, E. P. Klement, and L. N. Stout (Eds.), Abstracts of the
28
31st Linz seminar on fuzzy set theory, Johannes Kepler Universitat, Linz, (2010), 31{40.
29
[16] J. T. Denniston, A. Melton and S. E. Rodabaughe, Formal concept analysis and latticevalued
30
interchange systems, In: D. Dubois, M. Grabisch, R. Mesiar, and E. P. Klement
31
(Eds.), Abstracts of the 32nd Linz seminar on fuzzy set theory, Johannes Kepler Universitat,
32
Linz, (2011), 41{47.
33
[17] J. T. Denniston, A. Melton and S. E. Rodabaugh, Interweaving algebra and topology: Latticevalued
34
topological systems, Fuzzy Sets Syst., 192 (2012), 58{103.
35
[18] J. T. Denniston and S. E. Rodabaugh, Functorial relationships between lattice-valued topology
36
and topological systems, Quaest. Math., 32(2) (2009), 139{186.
37
[19] R. Diaconescu, Grothendieck institutions, Appl. Categ. Structures, 10 (2002), 338{402.
38
[20] D. Dikranjan, E. Giuli and A. Toi, Topological categories and closure operators, Quaest.
39
Math., 11(3) (1988), 323{337.
40
[21] J. J. Dukarm, Morita equivalence of algebraic theories, Colloq. Math., 55(1) (1988), 11{17.
41
[22] P. Eklund, Categorical fuzzy topology, Ph.D. thesis, Abo Akademi, 1986.
42
[23] P. Eklund, M. A. Galan and W. Gahler, Partially ordered monads for monadic topologies,
43
rough sets and Kleene algebras, Electron. Notes Theor. Comput. Sci., 225 (2009), 67{81.
44
[24] A. Frascella, C. Guido and S. Solovyov, Algebraically-topological systems and attachments,
45
Iran. J. Fuzzy Syst., 10(3) (2013), 65{102.
46
[25] W. Gahler, Monadic topology { a new concept of generalized topology, In: W. Gahler (Eds.),
47
Recent developments of general topology and its applications, international conference in
48
memory of Felix Hausdor (1868 - 1942), Math. Research, vol. 67, Akademie-Verlag, Berlin,
49
(1992), 136{149.
50
[26] W. Gahler, General topology { the monadic case, examples, applications, Acta Math. Hung.,
51
88(4) (2000), 279{290.
52
[27] J. A. Goguen, L-fuzzy sets, J. Math. Anal. Appl., 18 (1967), 145{174.
53
[28] J. A. Goguen, The fuzzy Tychono theorem, J. Math. Anal. Appl., 43 (1973), 734{742.
54
[29] G. Gratzer, Universal algebra, 2nd ed., Springer, 2008.
55
[30] C. Guido, Powerset operators based approach to fuzzy topologies on fuzzy sets, In: S. E. Rodabaugh
56
and E. P. Klement (Eds.), Topological and algebraic structures in fuzzy sets. A Handbook
57
of recent developments in the mathematics of fuzzy sets, Kluwer Academic Publishers,
58
(2003), 401{413.
59
[31] C. Guido, Fuzzy points and attachment, Fuzzy Sets Syst., 161(6) (2010), 2150{2165.
60
[32] H. Herrlich and G. E. Strecker, Category theory, 3rd ed., Sigma Series in Pure Mathematics,
61
vol. 1, Heldermann Verlag, 2007.
62
[33] D. Hofmann, Topological theories and closed objects, Adv. Math., 215(2) (2007), 789{824.
63
[34] U. Hohle, Upper semicontinuous fuzzy sets and applications, J. Math. Anal. Appl., 78 (1980),
64
[35] U. Hohle and A. P. Sostak, Axiomatic foundations of xed-basis fuzzy topology, In: U. Hohle
65
and S. E. Rodabaugh (Eds.), Mathematics of fuzzy sets: logic, topology and measure theory,
66
Kluwer Academic Publishers, (1999), 123{272.
67
[36] T. Ihringer, Allgemeine Algebra. Mit einem Anhang uber universelle Coalgebra von H. P.
68
Gumm, Heldermann Verlag, 2003.
69
[37] G. Jager, A category of L-fuzzy convergence spaces, Quaest. Math., 24(4) (2001), 501{517.
70
[38] P. Johnstone, J. Power, T. Tsujishita, H. Watanabe and J. Worrell, On the structure of
71
categories of coalgebras, Theor. Comput. Sci., 260(1-2) (2001), 87{117.
72
[39] P. T. Johnstone, Stone spaces, Cambridge University Press, 1982.
73
[40] D. Kruml and J. Paseka, Algebraic and categorical aspects of quantales, In: M. Hazewinkel
74
(Eds.), Handbook of algebra, vol. 5, Elsevier, (2008), 323{362.
75
[41] T. Kubiak and A. Sostak, Foundations of the theory of (L;M)-fuzzy topological spaces, In:
76
U. Bodenhofer, B. De Baets, E. P. Klement, and S. Saminger-Platz (Eds.), Abstracts of the
77
30th Linz seminar on fuzzy set theory, Johannes Kepler Universitat, Linz, (2009), 70{73.
78
[42] F. W. Lawvere, Functorial semantics of algebraic theories, Ph.D. thesis, Columbia University,
79
[43] R. Lowen, Fuzzy topological spaces and fuzzy compactness, J. Math. Anal. Appl., 56 (1976),
80
[44] S. Mac Lane, Categories for the working mathematician, 2nd ed., Springer-Verlag, 1998.
81
[45] E. G. Manes, Algebraic theories, Springer-Verlag, 1976.
82
[46] K. Morita, Duality for modules and its applications to the theory of rings with minimum
83
condition, Sci. Rep. Tokyo Kyoiku Daigaku, 6(A) (1958), 83{142.
84
[47] G. Preu, Semiuniform convergence spaces, Math. Jap., 41(3) (1995), 465{491.
85
[48] A. Pultr, Frames, In: M. Hazewinkel, (Eds.), Handbook of algebra, vol. 3, North-Holland
86
Elsevier, (2003), 789{858.
87
[49] S. E. Rodabaugh, Point-set lattice-theoretic topology, Fuzzy Sets Syst., 40(2) (1991), 297{
88
[50] S. E. Rodabaugh, Categorical frameworks for Stone representation theories, In: S. E. Rodabaugh,
89
E. P. Klement, and U. Hohle (Eds.), Applications of category theory to fuzzy subsets,
90
Kluwer Academic Publishers, (1992), 177{231.
91
[51] S. E. Rodabaugh, Categorical foundations of variable-basis fuzzy topology, In: U. Hohle and
92
S. E. Rodabaugh (Eds.), Mathematics of fuzzy sets: logic, topology and measure theory,
93
Kluwer Academic Publishers, (1999), 273{388.
94
[52] S. E. Rodabaugh, Relationship of algebraic theories to powerset theories and fuzzy topological
95
theories for lattice-valued mathematics, Int. J. Math. Math. Sci., 2007 (2007), 1{71.
96
[53] S. E. Rodabaugh, Functorial comparisons of bitopology with topology and the case for redundancy
97
of bitopology in lattice-valued mathematics, Appl. Gen. Topol., 9(1) (2008), 77{108.
98
[54] S. E. Rodabaugh, Relationship of algebraic theories to powersets over objects in Set and
99
SetC, Fuzzy Sets Syst., 161(3) (2010), 453{470.
100
[55] K. I. Rosenthal, Quantales and their applications, Addison Wesley Longman, 1990.
101
[56] J. J. M. M. Rutten, Universal coalgebra: a theory of systems, Theor. Comput. Sci., 249(1)
102
(2000), 3{80.
103
[57] S. Solovjovs, Categorically-algebraic topology, Abstracts of the International Conference on
104
Algebras and Lattices (Jardafest), Charles University, Prague, (2010), 20{22.
105
[58] S. Solovjovs, Lattice-valued categorically-algebraic topology, Abstracts of the 91st Peripatetic
106
Seminar on Sheaves and Logic (PSSL 91), University of Amsterdam, Amsterdam, (2010), 21.
107
[59] S. Solovjovs, Functorial semantics of topological theories, Abstracts of Applications of Algebra
108
XV, Institute of Mathematics and Computer Science of Jan D lugosz University,
109
Czestochowa, (2011), 37{41.
110
[60] S. Solovyov, Sobriety and spatiality in varieties of algebras, Fuzzy Sets Syst., 159(19) (2008),
111
2567{2585.
112
[61] S. Solovyov, Categorically-algebraic dualities, Acta Univ. M. Belii, Ser. Math., 17 (2010),
113
[62] S. Solovyov, Variable-basis topological systems versus variable-basis topological spaces, Soft
114
Comput., 14(10) (2010), 1059{1068.
115
[63] S. Solovyov, Fuzzy algebras as a framework for fuzzy topology, Fuzzy Sets Syst., 173(1)
116
(2011), 81{99.
117
[64] S. Solovyov, Categorical foundations of variety-based topology and topological systems, Fuzzy
118
Sets Syst., 192 (2012), 176{200.
119
[65] S. Solovyov, Topological systems and Artin glueing, Math. Slovaca, 62(4) (2012), 647{688.
120
[66] W. Tholen, Relative Bildzerlegungen und algebraische Kategorien, Ph.D. thesis, Westfalisch
121
Wilhelms-Universitat Munster, 1974.
122
[67] S. Vickers, Topology via logic, Cambridge University Press, 1989.
123
[68] O. Wyler, On the categories of general topology and topological algebra, Arch. Math., 22
124
(1971), 7{17.
125
[69] O. Wyler, TOP categories and categorical topology, General Topology Appl., 1 (1971), 17{28.
126
[70] L. A. Zadeh, Fuzzy sets, Inf. Control, 8 (1965), 338{365.
127
ORIGINAL_ARTICLE
CVaR Reduced Fuzzy Variables and Their Second Order Moments
Based on credibilistic value-at-risk (CVaR) of regularfuzzy variable, we introduce a new CVaR reduction method fortype-2 fuzzy variables. The reduced fuzzy variables arecharacterized by parametric possibility distributions. We establishsome useful analytical expressions for mean values and secondorder moments of common reduced fuzzy variables. The convex properties of second order moments with respect to parameters are also discussed. Finally, we take second order moment as a new risk measure, and develop a mean-moment model to optimize fuzzy portfolio selection problems. According to the analytical formulas of second order moments, the mean-moment optimization model is equivalent to parametricquadratic convex programming problems, which can be solved by general-purpose optimization software. The solution results reported in the numerical experiments demonstrate the credibility of the proposed optimization method.
http://ijfs.usb.ac.ir/article_2111_ae9c29c1bd509c95fc8928b9778558da.pdf
2015-10-30T11:23:20
2017-09-23T11:23:20
45
75
10.22111/ijfs.2015.2111
Credibilistic value-at-risk
Reduced fuzzy variable
Parametric possibility distribution
Second order moment
Xue-Jie
Bai
true
1
College of Management, Hebei University, Baoding 071002, Hebei, China and College of Science, Agricultural University of Hebei, Baoding 071001, Hebei, China
College of Management, Hebei University, Baoding 071002, Hebei, China and College of Science, Agricultural University of Hebei, Baoding 071001, Hebei, China
College of Management, Hebei University, Baoding 071002, Hebei, China and College of Science, Agricultural University of Hebei, Baoding 071001, Hebei, China
AUTHOR
Yan-Kui
Liu
true
2
College of Management, Hebei University, Baoding 071002, Hebei,
China
College of Management, Hebei University, Baoding 071002, Hebei,
China
College of Management, Hebei University, Baoding 071002, Hebei,
China
LEAD_AUTHOR
[1] J. Chachi and S. M. Taheri, A unied approach to similarity measure between intuitionistic
1
fuzzy sets, Int. J. Intell. Syst., 28 (2013), 669-685.
2
[2] S. Coupland and R. John, A fast geometric method for defuzzication of type-2 fuzzy sets,
3
IEEE T. Fuzzy Syst., 16(4) (2008), 929-941.
4
[3] D. Dubois and H. Prade, Operations in a fuzzy-valued logic, Inform. Control, 43(2) (1979),
5
[4] M. Hao and J. M. Mendel, Similarity measures for general type-2 fuzzy sets based on the
6
-plane representation, Inform. Sciences, 227 (2014), 197-215.
7
[5] B. Hu and C. Wang, On type-2 fuzzy relations and interval-valued type-2 fuzzy sets, Fuzzy
8
Set. Syst., 236 (2014), 1-32.
9
[6] C. Huang, Y. Wang, H. Chen, H. Tsai, J. Jian, A. Cheng and J. Liao, Application of cellular
10
automata and type-2 fuzzy logic to dynamic vehicle path planning, Appl. Soft Comput., 19
11
(2014), 333-342.
12
[7] C. M. Hwang, M. S. Yang, W. L. Hung and M. G. Lee, A similarity measure of intuitionistic
13
fuzzy sets based on the Sugeno integral with its application to pattern recognition, Inform.
14
Sciences, 189 (2012), 93-109.
15
[8] P. Kundu, S. Kar and M. Maiti, Fixed charge transportation problem with type-2 fuzzy vari-
16
ables, Inform. Sciences, 255 (2014), 170-186.
17
[9] B. Liu, Uncertainty theory, Springer-Verlag, Berlin, 2004.
18
[10] F. Liu, An ecient centroid type-reduction strategy for general type-2 fuzzy logic system,
19
Inform. Sciences, 178(9) (2008), 2224-2236.
20
[11] H. M. Markowitz, Portfolio selection, J. Financ., 7 (1952), 77-91.
21
[12] J. M. Mendel, Uncertain rule-based fuzzy logic system: introduction and new directions,
22
Prentice Hall, Upper Saddle River, N.J., 2001.
23
[13] J. M. Mendel and R. I. John, Type-2 fuzzy sets made simple, IEEE T. Fuzzy Syst., 10(2)
24
(2002), 117-127.
25
[14] M. Mizumoto and K. Tanaka, Some properties of fuzzy sets of type 2, Inform. Control, 31(4)
26
(1976), 312-340.
27
[15] M. Moharrer, H. Tahayori, L. Livi, A. Sadeghian and A. Rizzi, Interval type-2 fuzzy sets to
28
model linguistic label perception in online services satisfaction, Soft Comput., 19(1) (2015),
29
[16] S. C. Ngan, A type-2 linguistic set theory and its application to multi-criteria decision mak-
30
ing, Comput. Ind. Eng., 64(2) (2013), 721-730.
31
[17] A. D. Torshizi and M. H. F. Zarandi, A new cluster validity measure based on general type-2
32
fuzzy sets: application in gene expression data clustering, Knowl-Based. Syst., 64 (2014),
33
[18] P. Wang, Fuzzy contactibility and fuzzy variables, Fuzzy Set. Syst., 8(1) (1982), 81-92.
34
[19] L. A. Zadeh, The concept of a linguistic variable and its application to approximate
35
reasoning{I,II,III, Inform. Sciences, 8(3) (1975), 199-249; 8(4) (1975), 301-357; 9(1) (1975),
36
ORIGINAL_ARTICLE
Linear matrix inequality approach for synchronization of chaotic fuzzy cellular neural networks with discrete and unbounded distributed delays based on\ sampled-data control
In this paper, linear matrix inequality (LMI) approach for synchronization of chaotic fuzzy cellular neural networks (FCNNs) with discrete and unbounded distributed delays based on sampled-data controlis investigated. Lyapunov-Krasovskii functional combining with the input delay approach as well as the free-weighting matrix approach are employed to derive several sufficient criteria in terms of LMIs ensuring the delayed FCNNs to be asymptotically synchronous. The restriction such as the time-varying delay required to be differentiable or even its time-derivative assumed to be smaller than one, are removed. Instead, the time-varying delay is only assumed to be bounded. Finally, numerical examples and its simulations are provided to demonstrate the effectiveness of the derived results.
http://ijfs.usb.ac.ir/article_2112_42c61297c52cf75dfbd00eb52c889040.pdf
2015-10-30T11:23:20
2017-09-23T11:23:20
77
98
10.22111/ijfs.2015.2112
Chaos
Fuzzy cellular neural networks
Linear matrix inequality
Sampled-data control
Synchronization
P.
Balasubramaniam-pour
true
1
Department of Mathematics, Gandhigram Rural Institute - Deemed University, Gandhigram - 624 302, Tamilnadu, India
Department of Mathematics, Gandhigram Rural Institute - Deemed University, Gandhigram - 624 302, Tamilnadu, India
Department of Mathematics, Gandhigram Rural Institute - Deemed University, Gandhigram - 624 302, Tamilnadu, India
LEAD_AUTHOR
K.
Ratnavelu
true
2
Institute of Mathematical Sciences, Faculty of Science, University
of Malaya - 50603, Kuala Lumpur, Malaysia
Institute of Mathematical Sciences, Faculty of Science, University
of Malaya - 50603, Kuala Lumpur, Malaysia
Institute of Mathematical Sciences, Faculty of Science, University
of Malaya - 50603, Kuala Lumpur, Malaysia
AUTHOR
M.
Kalpana
kalpana.nitt@gmail.com
true
3
Institute of Mathematical Sciences, Faculty of Science, University of
Malaya - 50603, Kuala Lumpur, Malaysia
Institute of Mathematical Sciences, Faculty of Science, University of
Malaya - 50603, Kuala Lumpur, Malaysia
Institute of Mathematical Sciences, Faculty of Science, University of
Malaya - 50603, Kuala Lumpur, Malaysia
AUTHOR
[1] A. Arunkumar, R. Sakthivel, K. Mathiyalagan and S. Marshal Anthoni, Robust state estimation
1
for discrete-time BAM neural networks with time-varying delay, Neurocomputing, 131
2
(2014), 171-178.
3
[2] A. Arunkumar, R. Sakthivel, K. Mathiyalagan and Ju H. Park, Robust stochastic stability
4
of discrete-time fuzzy Markovian jump neural networks, ISA Transactions, 53 (2014), 1006–
5
[3] P. Balasubramaniam, M. Kalpana and R. Rakkiyappan, Linear matrix inequality approach
6
for synchronization control of fuzzy cellular neural networks with mixed time delays, Chinese
7
Physics B, 21 (2012): 048402.
8
[4] S. Boyd, L. E. Ghaoui, E. Feron and V. Balakrishnan, Linear Matrix Inequalities in Systems
9
and Control Theory (SIAM, Philadelphia, 1994).
10
[5] T. L. Carroll and L. M. Pecora, Synchronization chaotic circuits, IEEE Trans. Circuits Syst.,
11
38 (1991), 453–456.
12
[6] L. O. Chua and L. Yang, Cellular neural networks: theory, IEEE Trans. Circuits Syst., 35
13
(1988), 1257-1272.
14
[7] L. O. Chua and L. Yang, Cellular neural networks: applications, IEEE Trans. Circuits Syst.,
15
35 (1988), 1273-1290.
16
[8] X. Feng, F. Zhang and W. Wang, Global exponential synchronization of delayed fuzzy cellular
17
neural networks with impulsive effects, Chaos Solitons Fractals., 44 (2011), 9–16.
18
[9] T. Feuring, J. J. Buckley, W. M. Lippe and A. Tenhagen, Stability analysis of neural net
19
controllers using fuzzy neural networks, Fuzzy Sets and Systems, 101 (1999), 303–313.
20
[10] E. Fridman, A. Seuret and J. P. Richard, Robust sampled-data stabilization of linear systems:
21
an input delay approach, Automatica, 40 (2004), 1441–1446.
22
[11] Q. Gan and Y. Liang, Synchronization of chaotic neural networks with time delay in the
23
leakage term and parametric uncertainties based on sampled-data control, J. Franklin Inst.,
24
349 (2012), 1955–1971.
25
[12] Q. Gan, R. Xu and P. Yang, Synchronization of non-identical chaotic delayed fuzzy cellular
26
neural networks based on sliding mode control, Commun. Nonlinear Sci. Numer. Simul., 17
27
(2012), 433–443.
28
[13] Q. Gan, R. Xu and P. Yang, Exponential synchronization of stochastic fuzzy cellular neural
29
networks with time delay in the leakage term and reaction-diffusion, Commun. Nonlinear Sci.
30
Numer. Simul., 17 (2012), 1862–1870.
31
[14] K. Gu, An integral inequality in the stability problem of time-delay systems, in Proceedings
32
of the 39th IEEE Conference on Decision and Control Sydney, Australia (2000), 2805–2810.
33
[15] S. Lee, V. Kapila, M. Porfiri and A. Panda, Master-slave synchronization of continuously
34
and intermittently coupled sampled-data chaotic oscillators, Commun. Nonlinear Sci. Numer.
35
Simul., 15 (2010), 4100–4113.
36
[16] T. Li, S. Fei and Q. Zhu, Design of exponential state estimator for neural networks with
37
distributed delays, Nonlinear Anal. Real World Appl., 10 (2009), 1229–1242.
38
[17] N. Li, Y. Zhang, J. Hu and Z. Nie, Synchronization for general complex dynamical networks
39
with sampled-data, Neurocomputing, 74 (2011), 805–811.
40
[18] Z. Liu, H. Zhang and Z. Wang, Novel stability criterions of a new fuzzy cellular neural
41
networks with time-varying delays, Neurocomputing, 72 (2009), 1056–1064.
42
[19] J. Lu and D. J. Hill, Global asymptotical synchronization of chaotic Lur’e systems using
43
sampled data: a linear matrix inequality approach, IEEE Trans. Circuits Syst. II, 55 (2008),
44
586–590.
45
[20] K. Mathiyalagan, S. Hongye and R. Sakthivel, Robust stochastic stability of discrete-time
46
Markovian jump neural networks with leakage delay, Zeitschrift Fur Naturforschung Section
47
A-A Journal of Physical Sciences, 69 (2014), 70–80.
48
[21] K. Mathiyalagan, R. Sakthivel and S. Hongye, Exponential state estimation for discretetime
49
switched genetic regulatory networks with random delays, Canad. J. Phys., 92 (2014),
50
976–986.
51
[22] L. M. Pecora and T. L. Carroll, Synchronization in chaotic systems, Phys. Rev. Lett., 64
52
(1990), 821–824.
53
[23] L. M. Pecora, T. L. Carroll, G. A. Johnson, D. J. Mar and J. F. Heagy, Fundamentals of
54
synchronization in chaotic systems, concepts, and applications, Chaos, 7 (1997), 520–543.
55
[24] Y. Ping and L. Teng, Exponential synchronization of fuzzy cellular neural networks with
56
mixed delays and general boundary conditions, Commun. Nonlinear Sci. Numer. Simul., 17
57
(2012), 1003–1011.
58
[25] R. Sakthivel, R. Raja and S. Marshal Anthoni, Linear matrix inequality approach to stochastic
59
stability of uncertain delayed BAM neural networks, IMA J Appl Math., 78 (2013), 1156–
60
[26] E. N. Sanchez and J. P. Perez, Input-to-state stability (ISS) analysis for dynamic neural
61
networks, IEEE Trans. Circuits Syst. I, 46 (1999), 1395-1398.
62
[27] F. O. Souza, R. M. Palhares and P. Y. Ekel, Asymptotic stability analysis in uncertain multidelayed
63
state neural networks via Lyapunov-Krasovskii theory, Math. Comput. Modelling, 45
64
(2007), 1350–1362.
65
[28] F. O. Souza, R. M. Palhares and P. Y. Ekel, Novel stability criteria for uncertain delayed
66
Cohen-Grossberg neural networks using discretized Lyapunov functional, Chaos Solitons Fractals,
67
41 (2009), 2387–2393.
68
[29] F. O. Souza, R. M. Palhares and P. Y. Ekel, Improved asymptotic stability analysis for
69
uncertain delayed state neural networks, Chaos Solitons Fractals, 39 (2009), 240–247.
70
[30] Y. Tang and J. Fang, Robust synchronization in an array of fuzzy delayed cellular neural
71
networks with stochastically hybrid coupling, Neurocomputing, 72 (2009), 3253–3262.
72
[31] T. Yang and L. B. Yang, Global stability of fuzzy cellular neural network, IEEE Trans. Circuits
73
Syst. I, 43 (1996), 880–883.
74
[32] T. Yang, L. B. Yang, C. W. Wu and L. O. Chua, Fuzzy cellular neural networks: Theory,
75
in Proceedings of the IEEE International Workshop on Cellular Neural Networks and
76
Applications, (1996), 181–186.
77
[33] T. Yang, L. B. Yang, C. W. Wu and L. O. Chua, Fuzzy cellular neural networks: Applications,
78
in Proceedings of the IEEE International Workshop on Cellular Neural Networks and
79
Applications, (1996), 225–230.
80
[34] J. Yu, C. Hu, H. Jiang and Z. Teng, Exponential lag synchronization for delayed fuzzy cellular
81
neural networks via periodically intermittent control, Math. Comput. Simulation, 82 (2012),
82
895–908.
83
[35] F. Yu and H. Jiang, Global exponential synchronization of fuzzy cellular neural networks with
84
delays and reaction-diffusion terms, Neurocomputing, 74 (2011), 509–515.
85
[36] L. A. Zadeh, Fuzzy Sets, Information and Control, 8 (1965), 338–353.
86
[37] C. Zhang, Y. He and M. Wu, Exponential synchronization of neural networks with timevarying
87
mixed delays and sampled-data, Neurocomputing, 74 (2010), 265–273.
88
ORIGINAL_ARTICLE
Hierarchical Functional Concepts for Knowledge Transfer among Reinforcement Learning Agents
This article introduces the notions of functional space and concept as a way of knowledge representation and abstraction for Reinforcement Learning agents. These definitions are used as a tool of knowledge transfer among agents. The agents are assumed to be heterogeneous; they have different state spaces but share a same dynamic, reward and action space. In other words, the agents are assumed to have different representations of an environment while having similar actions. The learning framework is $Q$-learning. Each dimension of the functional space is the normalized expected value of an action. An unsupervisedclustering approach is used to form the functional concepts as some fuzzy areas in the functional space. The functional concepts are abstracted further in a hierarchy using the clustering approach. The hierarchical concepts are employed for knowledge transfer among agents. Properties of the proposed approach are tested in a set of case studies. The results show that the approach is very effective in transfer learning among heterogeneous agents especially in the beginning episodes of the learning.
http://ijfs.usb.ac.ir/article_2113_282ba69af4a0a581fecc56675062be1d.pdf
2015-10-30T11:23:20
2017-09-23T11:23:20
99
116
10.22111/ijfs.2015.2113
Reinforcement Learning
Transfer Learning
Heterogeneous Agents
Hierarchical Concepts
A.
Mousavi
true
1
Control and Intelligent Processing Center of Excellence, School of
Electrical and Computer Engineering, University of Tehran, Tehran, Iran
Control and Intelligent Processing Center of Excellence, School of
Electrical and Computer Engineering, University of Tehran, Tehran, Iran
Control and Intelligent Processing Center of Excellence, School of
Electrical and Computer Engineering, University of Tehran, Tehran, Iran
LEAD_AUTHOR
M.
Nili Ahmadabadi
true
2
Control and Intelligent Processing Center of Excellence,
School of Electrical and Computer Engineering, University of Tehran, Tehran, Iran
and School of Cognitive Science, Institute for Research in Fundamental Sciences
(IPM), Tehran, Iran
Control and Intelligent Processing Center of Excellence,
School of Electrical and Computer Engineering, University of Tehran, Tehran, Iran
and School of Cognitive Science, Institute for Research in Fundamental Sciences
(IPM), Tehran, Iran
Control and Intelligent Processing Center of Excellence,
School of Electrical and Computer Engineering, University of Tehran, Tehran, Iran
and School of Cognitive Science, Institute for Research in Fundamental Sciences
(IPM), Tehran, Iran
AUTHOR
H.
Vosoughpour
true
3
Control and Intelligent Processing Center of Excellence, School
of Electrical and Computer Engineering, University of Tehran, Tehran, Iran
Control and Intelligent Processing Center of Excellence, School
of Electrical and Computer Engineering, University of Tehran, Tehran, Iran
Control and Intelligent Processing Center of Excellence, School
of Electrical and Computer Engineering, University of Tehran, Tehran, Iran
AUTHOR
B. N.
Araabi
true
4
Control and Intelligent Processing Center of Excellence, School of
Electrical and Computer Engineering, University of Tehran, Tehran, Iran and School
of Cognitive Science, Institute for Research in Fundamental Sciences (IPM), Tehran,
Iran
Control and Intelligent Processing Center of Excellence, School of
Electrical and Computer Engineering, University of Tehran, Tehran, Iran and School
of Cognitive Science, Institute for Research in Fundamental Sciences (IPM), Tehran,
Iran
Control and Intelligent Processing Center of Excellence, School of
Electrical and Computer Engineering, University of Tehran, Tehran, Iran and School
of Cognitive Science, Institute for Research in Fundamental Sciences (IPM), Tehran,
Iran
AUTHOR
N.
Zaare
true
5
Control and Intelligent Processing Center of Excellence, School of
Electrical and Computer Engineering, University of Tehran, Tehran, Iran
Control and Intelligent Processing Center of Excellence, School of
Electrical and Computer Engineering, University of Tehran, Tehran, Iran
Control and Intelligent Processing Center of Excellence, School of
Electrical and Computer Engineering, University of Tehran, Tehran, Iran
AUTHOR
[1] J. S. Bruner, Actual minds, possible words, Harvard University Press, 1987.
1
[2] T. Dietterich, Hierarchical reinforcement learning with the MAXQ value function decompo-
2
sition, Journal of Articial Intelligent Research, 13 (2000), 227-303.
3
[3] K. Driessens, J. Ramon and T. Croonenborghs, Transfer learning for reinforcement learn-
4
ing through goal and policy parametrization, In ICML Workshop on Structural Knowledge
5
Transfer for Machine Learning, (2006), 1-4.
6
[4] W. Fritz, Intelligent systems and their societies, In Webpage: http://www. intelligentsys-
7
tems.com.ar/intsyst/index.htm, January (1997).
8
[5] G. L. Klir, Uncertainty and information: foundations of generalized information theory,
9
John Wiley, Hoboken, NJ (2005).
10
[6] G. L. Klir, B. Yuan, Fuzzy sets and fuzzy logic: theory and applications, Prentice Hall, 1995.
11
[7] G. Konidaris, A. Barto, Autonomous shaping: Knowledge transfer in reinforcement learning,
12
In Proceedings of the 23rd international conference on Machine learning, 2006, 489-496.
13
[8] A. Lazaric, Knowledge Transfer in Reinforcement Learning, PhD thesis, Politecnico di Mi-
14
lano, 2008.
15
[9] L. Mihalkova, T. Huynh and R. Mooney, Mapping and revising Markov Logic Networks for
16
transfer learning, In Proceedings of AAAI Conference on Articial Intelligence, (2007), 608-614.
17
[10] H. Mobahi, M. Nili Ahmadabadi, and B. Nadjar Araabi, A biologically inspired method for
18
conceptual imitation using reinforcement learning, Applied Articial Intelligence, 21 (2007),
19
[11] R. A. Mollineda, F. J. Ferri and E. Vidal, A cluster-based merging strategy for nearest pro-
20
totype classiers, In Proceedings of 15th International Conference on Pattern Recognition
21
(ICPR'00), 2 (2000), 755-758.
22
[12] G. L. Murphy, The big book of concepts, MIT Press, 2004.
23
[13] S. Pan, J. Kwok and Q. Yang, Transfer learning via dimensionality reduction, In Proceedings
24
of 23rd AAAI Conference on Articial Intelligence, (2008), 677-682.
25
[14] V. Soni, S. Singh, Using hormomorphisms to transfer options across continuous reinforce-
26
ment learning domains, In Proceedings of 21st AAAI Conference on Articial Intelligence,
27
(2006), 494-499.
28
[15] R. S. Sutton, A. G. Barto, Reinforcement Learning: An Introduction, MIT Press, Cambridge,
29
MA (1998).
30
[16] F. Tanaka, M. Yamamura, Multitask reinforcement learning on the distribution of MDPs,
31
Transactions of the Institute of Electrical Engineers of Japan, 123(5) (2003), 1004-1011.
32
[17] M. Taylor, P. Stone, Transfer learning for reinforcement learning domains: a survey, Journal
33
of Machine Learning Research, 10 (2009), 1633-1685.
34
[18] M. Taylor, G. Kuhlmann and P. Stone, Autonomous transfer for reinforcement learning, In
35
Proceedings of the 7th International Joint Conference on Autonomous Agents and Multiagent
36
systems, 1 (2008), 283-290.
37
[19] M. Taylor, P. Stone, Representation Transfer for Reinforcement Learning, In Proceedings
38
of AAAI Fall Symposium on Computational Approaches to Representation Change during
39
Learning and Development, Arlington, Virginia, (2007), 78-85.
40
[20] M. Taylor, P. Stone and Y. Liu, Value function for RL-based behavior transfer: A comparative
41
study, In Proceedings of the AAAI-05 Conference on Articial Intelligence, (2005), 880-885.
42
[21] A. Thedoridis and K. Koutroumbas, Pattern Recognition, Elsevier Academic Press, Second
43
Edition, 2003.
44
[22] L. Torrey and J. Shavlik, Transfer learning, In Soria, E., Martin, J., Magdalena, R., Martinez,
45
M., and Serrano, A., editors, Handbook of Research on Machine Learning Applications, IGI
46
Global, 2009, 242-264.
47
[23] C. J. Watkins, Learning from Delayed Rewards, Ph.D. thesis, Cambridge University, 1989.
48
[24] C. J. Watkins and P. Dayan, Q-learning, Machine Learning, 8 (1992), 279-292.
49
[25] A. Wilson, A. Fern, S. Ray and P. Tadepalli, Multitask reinforcement learning: A hierarchical
50
Bayesian approach, In Proceedings of the 24th International Conference on Machine Learning,
51
(2007), 1015-1022.
52
[26] M. Zentall, M. Galizio and T. S. Critched, Categorization, concept learning and behavior
53
analysis: an introduction, The Exprimental Analysis of Behavior, 3 (2002), 237-248.
54
ORIGINAL_ARTICLE
Non-Newtonian Fuzzy numbers and \related applications
Although there are many excellent ways presenting the principle of the classical calculus, the novel presentations probably leads most naturally to the development of the non-Newtonian calculus. The important point to note is that the non-Newtonian calculus is a self-contained system independent of any other system of calculus. Since this self-contained work is intended for a wide audience, including engineers, scientists and mathematicians. The main purpose of the present paper is to construct of fuzzy numbers with respect to the non-Newtonian calculus and is to give the necessary and sufficient conditions according to the generalization of the notion of fuzzy numbers by using the generating functions. Also we introduce the concept of non-Newtonian fuzzy distance and give some properties regarding convergence of sequences and series of fuzzy numbers with some illustrative examples.
http://ijfs.usb.ac.ir/article_2114_aafefe9d564507d2db1e4749261bbf48.pdf
2015-10-30T11:23:20
2017-09-23T11:23:20
117
137
10.22111/ijfs.2015.2114
Non-Newtonian calculus
Fuzzy level sets
Trapezoidal fuzzy numbers
Convergence of fuzzy sequences and series
Ugur
Kadak
ugurkadak@gmail.com
true
1
Department of Mathematics, Bozok University, Yozgat, Turkey
Department of Mathematics, Bozok University, Yozgat, Turkey
Department of Mathematics, Bozok University, Yozgat, Turkey
LEAD_AUTHOR
[1] A. E. Bashirov, E. Kurpnar and A. Ozyapc, Multiplicative calculus and its applications, J.
1
Math. Anal. Appl., 337(1) (2008), 36-48.
2
[2] A. F. Cakmak and F. Basar, Some new results on sequence spaces with respect to non-
3
Newtonian calculus, J. Inequal. Appl., 2012(1) (2012), 228.
4
[3] M. Grossman and R. Katz, Non-Newtonian Calculus, Lee Press, Pigeon Cove, MA, 1972.
5
[4] M. Grossman, Bigeometric Calculus, Archimedes Foundation, Rockport, Mass, USA, 1983.
6
[5] M. Grossman, The First Nonlinear System of Dierential and Integral Calculus, Mathco,
7
[6] U. Kadak and H. Efe, Matrix transformations between certain sequence spaces over the non-
8
Newtonian complex eld, Sci. World J., 2014 (2014).
9
[7] U. Kadak and M. Ozluk, Generalized Runge-Kutta method with respect to the non-Newtonian
10
calculus, Abstr. Appl. Anal., 2014 (2014).
11
[8] U. Kadak and F. Basar, On Fourier series of fuzzy-valued function, Sci. World J., 2014
12
[9] M. Matloka, Sequences of fuzzy numbers, BUSEFAL, 28 (1986), 28-37.
13
[10] E. Msrl, and Y. Gurefe, Multiplicative Adams-Bashforth-Moulton methods, Numer. Algorithms,
14
57(4) (2011), 425-439.
15
[11] M. Stojakovic and Z. Stojakovic, Series of fuzzy sets, Fuzzy Sets Syst., 160(21) (2009),
16
3115{3127.
17
[12] M. Stojakovic and Z. Stojakovic, Addition and series of fuzzy sets, Fuzzy Sets Syst., 83(3)
18
(1996), 341{346.
19
[13] O. Talo and F. Basar, Determination of the duals of classical sets of sequences of fuzzy
20
numbers and related matrix transformations, Comput. Math. Appl., 58(4) (2009), 717-733.
21
[14] O. Talo and F. Basar, Quasilinearity of the classical sets of sequences of the fuzzy numbers
22
and some applications, Taiwanese J. Math., 14(5) (2010), 1799-1819.
23
[15] S. Tekin and F. Basar, Certain sequence spaces over the non-Newtonian complex eld, Abstr.
24
Appl. Anal., 2012 (2013).
25
[16] L. A. Zadeh, Fuzzy sets, Inform. and Control, 8 (1965), 338{353.
26
ORIGINAL_ARTICLE
Order intervals in the metric space\ of fuzzy numbers
In this paper, we introduce a function in order to measure the distancebetween two order intervals of fuzzy numbers, and show that this function isa metric. We investigate some properties of this metric, and finally presentan application. We think that this study could provide a more generalframework for researchers studying on interval analysis, fuzzy analysis andfuzzy decision making.
http://ijfs.usb.ac.ir/article_2115_28bfba3b526dd2943ba3639b4db69957.pdf
2015-10-30T11:23:20
2017-09-23T11:23:20
139
147
10.22111/ijfs.2015.2115
Fuzzy number
Order interval of fuzzy numbers
Decision making
S.
Aytar
true
1
Faculty of Arts and Sciences, Department of Mathematics, Suleyman
Demirel University, Isparta, Turkey
Faculty of Arts and Sciences, Department of Mathematics, Suleyman
Demirel University, Isparta, Turkey
Faculty of Arts and Sciences, Department of Mathematics, Suleyman
Demirel University, Isparta, Turkey
LEAD_AUTHOR
[1] G. Bortolan and R. Degani, A review of some methods for ranking fuzzy subsets, Fuzzy Sets
1
and Systems, 15 (1985), 1-19.
2
[2] H. Bustince, Indicator of inclusion grade for interval-valued fuzzy sets: Application to ap-
3
proximate reasoning based on interval-valued fuzzy sets, Int. J. Approximate Reasoning, 23
4
(2000), 137-209.
5
[3] P. T. Chang and E. S. Lee, Fuzzy arithmetic and comparison of fuzzy numbers, In: M.
6
Delgado, J. Kacprzyk (Eds.), Fuzzy Optimization, Physica-Verlag, Heidelberg, (1994), 69-82.
7
[4] P. Diamond and P. Kloeden, Metric Spaces of Fuzzy Sets: Theory and Applications, World
8
Scientic, Singapore, 1994.
9
[5] J-x. Fang and H. Huang, On the level convergence of a sequence of fuzzy numbers, Fuzzy
10
Sets and Systems, 147 (2004), 417-435.
11
[6] P. Grzegorzewski, Metrics and orders in space of fuzzy numbers, Fuzzy Sets and Systems, 97
12
(1998), 83-94.
13
[7] S. Heilpern, Comparison of fuzzy numbers in decision making, Tatra Mountains Math. Publ.,
14
6 (1995), 47-53.
15
[8] M. Mizumoto and K. Tanaka, Some properties of fuzzy numbers, In: M. M. Gupta et al.
16
(Eds.) , Advances in Fuzzy Set Theory and Applications, North Holland, (1979), 153-164.
17
[9] R. E. Moore, Interval Analysis, Prentice-Hall, N.J., 1966.
18
[10] M. L. Puri and D. A. Ralescu, Fuzzy random variables, J. Math. Anal. Appl., 114 (1986),
19
[11] Q. Zhu and E. S. Lee, Comparison and ranking of fuzzy numbers, in: J. Kacprzyk, M. Fedrizzi
20
(Eds.), Fuzzy Regression Analysis, Omnitech Press, Warsaw, (1992), 21-44.
21
ORIGINAL_ARTICLE
A note on soft topological spaces
This paper demonstrates the redundancies concerning the increasing popular ``soft set" approaches to general topologies. It is shown that there is a complement preserving isomorphism (preserving arbitrary $widetilde{bigcup}$ and arbitrary $widetilde{bigcap}$) between the lattice ($mathcal{ST}_E(X,E),widetilde{subset}$) of all soft sets on $X$ with the whole parameter set $E$ as domains and the powerset lattice ($mathcal{P}(Xtimes E),subseteq$) of all subsets of $Xtimes E$. It therefore follows that soft topologies are redundant and unnecessarily complicated in theoretical sense.
http://ijfs.usb.ac.ir/article_2116_72c56938f64cab0a8312fcbb598fa53d.pdf
2015-10-30T11:23:20
2017-09-23T11:23:20
149
155
10.22111/ijfs.2015.2116
Soft set
Soft topology
Fu-Gui
Shi
fugushi@bit.edu.cn
true
1
School of Mathematics and Statistics, Beijing Institute of Technology,
5 South Zhongguancun Street, Haidian District, 100081 Beijing, P.R. China
School of Mathematics and Statistics, Beijing Institute of Technology,
5 South Zhongguancun Street, Haidian District, 100081 Beijing, P.R. China
School of Mathematics and Statistics, Beijing Institute of Technology,
5 South Zhongguancun Street, Haidian District, 100081 Beijing, P.R. China
LEAD_AUTHOR
Bin
Pang
pangbin1205@163.com
true
2
School of Mathematics and Statistics, Beijing Institute of Technology, 5
South Zhongguancun Street, Haidian District, 100081 Beijing, P. R. China
School of Mathematics and Statistics, Beijing Institute of Technology, 5
South Zhongguancun Street, Haidian District, 100081 Beijing, P. R. China
School of Mathematics and Statistics, Beijing Institute of Technology, 5
South Zhongguancun Street, Haidian District, 100081 Beijing, P. R. China
AUTHOR
[1] U. Acar, F. Koyuncu and B. Tanay, Soft sets and soft rings, Comput. Math. Appl., 59 (2010),
1
3458{3463.
2
[2] H. Aktas and N. C agan, Soft sets and soft groups, Inf. Sci., 177 (2007), 2726{2735.
3
[3] N. C agan, S. Karatas and S. Enginoglu, Soft topology, Comput. Math. Appl., 62 (2011),
4
[4] N. C agan and S. Enginoglu, Soft set theory and uni-int decision making, Eur. J. Oper. Res.,
5
207 (2010), 848{855.
6
[5] B. Chen, Soft semi-open sets and related properties in soft topological spaces, Appl. Math.
7
Inform. Sci., 7 (2013), 287{294.
8
[6] F. Feng, Y. B. Jun and X. Zhao, Soft semirings, Comput. Math. Appl., 56 (2008), 2621{2628.
9
[7] J. Gutierrez Garcia and S. E. Rodabaugh, Order-theoretic, topological, categorical redundan-
10
cies of interval-valued sets, grey sets, vague sets, interval-valued intuitionistic sets, intu-
11
itionistic fuzzy sets and topologies, Fuzzy Sets Syst., 156 (2005), 445{484.
12
[8] S. Hussain and B. Ahmad, Some properties of soft topological spaces, Comput. Math. Appl.,
13
62 (2011), 4058{4067.
14
[9] Y. B. Jun, Soft BCK/BCI-algebras, Comput. Math. Appl., 56 (2008), 1408{1413.
15
[10] Y. B. Jun and C. H. Park, Applications of soft sets in ideal theory of BCK/BCI-algebras,
16
Inf. Sci., 178 (2008), 2466{2475.
17
[11] P. K. Maji, R. Biswas and R. Roy, An application of soft sets in a decision making problem,
18
Comput. Math. Appl., 44 (2002), 1077{1083.
19
[12] P. K. Maji, R. Biswas and R. Roy, Soft set thoery, Comput. Math. Appl., 45 (2003), 555{562.
20
[13] W. K. Min, A note on soft topological spaces, Comput. Math. Appl., 62 (2011), 3524{3528.
21
[14] D. Molodtsov, Soft set theory rst results, Comput. Math. Appl., 37 (1999), 19{31.
22
[15] B. V. S. T. Sai and S. Kumar, On soft semi-open sets and soft semi-topology, Int. J. Math.
23
Arch., 4(4) (2013), 114-117.
24
[16] M. Shabir and M. Naz, On soft topological spaces, Comput. Math. Appl., 61 (2011), 1786{
25
[17] F. G. Shi and B. Pang, Redundancy of fuzzy soft topological spaces, J. Intell. Fuzzy Syst., 27
26
(2014), 1757{1760.
27
ORIGINAL_ARTICLE
Persian-translation vol. 12, no.5, October 2015
http://ijfs.usb.ac.ir/article_2643_0921b835f0d81e0b0abec67977f149f7.pdf
2015-10-29T11:23:20
2017-09-23T11:23:20
159
165
10.22111/ijfs.2015.2643