2015
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Functorial semantics of topological theories
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2
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{categoricallyalgebraic} (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.
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1
43


Sergey A.
Solovyov
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
Czech Republic
sergejs.solovjovs@lumii.lv
Algebra
Algebraic theory
Comma category
Categoricallyalgebraic topology
Poslat topology
Powerset theory
Topological system
Topological theory
Variety
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Kubiak and A. Sostak, Foundations of the theory of (L;M)fuzzy topological spaces, In:##U. Bodenhofer, B. De Baets, E. P. Klement, and S. SamingerPlatz (Eds.), Abstracts of the##30th Linz seminar on fuzzy set theory, Johannes Kepler Universitat, Linz, (2009), 70{73.##[42] F. W. Lawvere, Functorial semantics of algebraic theories, Ph.D. thesis, Columbia University,##[43] R. Lowen, Fuzzy topological spaces and fuzzy compactness, J. Math. Anal. Appl., 56 (1976),##[44] S. Mac Lane, Categories for the working mathematician, 2nd ed., SpringerVerlag, 1998.##[45] E. G. Manes, Algebraic theories, SpringerVerlag, 1976.##[46] K. Morita, Duality for modules and its applications to the theory of rings with minimum##condition, Sci. Rep. Tokyo Kyoiku Daigaku, 6(A) (1958), 83{142.##[47] G. Preu, Semiuniform convergence spaces, Math. Jap., 41(3) (1995), 465{491.##[48] A. Pultr, Frames, In: M. Hazewinkel, (Eds.), Handbook of algebra, vol. 3, NorthHolland##Elsevier, (2003), 789{858.##[49] S. E. 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Rodabaugh, Relationship of algebraic theories to powersets over objects in Set and##SetC, Fuzzy Sets Syst., 161(3) (2010), 453{470.##[55] K. I. Rosenthal, Quantales and their applications, Addison Wesley Longman, 1990.##[56] J. J. M. M. Rutten, Universal coalgebra: a theory of systems, Theor. Comput. Sci., 249(1)##(2000), 3{80.##[57] S. Solovjovs, Categoricallyalgebraic topology, Abstracts of the International Conference on##Algebras and Lattices (Jardafest), Charles University, Prague, (2010), 20{22.##[58] S. Solovjovs, Latticevalued categoricallyalgebraic topology, Abstracts of the 91st Peripatetic##Seminar on Sheaves and Logic (PSSL 91), University of Amsterdam, Amsterdam, (2010), 21.##[59] S. Solovjovs, Functorial semantics of topological theories, Abstracts of Applications of Algebra##XV, Institute of Mathematics and Computer Science of Jan D lugosz University,##Czestochowa, (2011), 37{41.##[60] S. Solovyov, Sobriety and spatiality in varieties of algebras, Fuzzy Sets Syst., 159(19) (2008),##2567{2585.##[61] S. Solovyov, Categoricallyalgebraic dualities, Acta Univ. M. Belii, Ser. Math., 17 (2010),##[62] S. Solovyov, Variablebasis topological systems versus variablebasis topological spaces, Soft##Comput., 14(10) (2010), 1059{1068.##[63] S. Solovyov, Fuzzy algebras as a framework for fuzzy topology, Fuzzy Sets Syst., 173(1)##(2011), 81{99.##[64] S. Solovyov, Categorical foundations of varietybased topology and topological systems, Fuzzy##Sets Syst., 192 (2012), 176{200.##[65] S. Solovyov, Topological systems and Artin glueing, Math. Slovaca, 62(4) (2012), 647{688.##[66] W. Tholen, Relative Bildzerlegungen und algebraische Kategorien, Ph.D. thesis, Westfalisch##WilhelmsUniversitat Munster, 1974.##[67] S. Vickers, Topology via logic, Cambridge University Press, 1989.##[68] O. Wyler, On the categories of general topology and topological algebra, Arch. 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CVaR Reduced Fuzzy Variables and Their Second Order Moments
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2
Based on credibilistic valueatrisk (CVaR) of regularfuzzy variable, we introduce a new CVaR reduction method fortype2 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 meanmoment model to optimize fuzzy portfolio selection problems. According to the analytical formulas of second order moments, the meanmoment optimization model is equivalent to parametricquadratic convex programming problems, which can be solved by generalpurpose optimization software. The solution results reported in the numerical experiments demonstrate the credibility of the proposed optimization method.
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45
75


XueJie
Bai
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,
China


YanKui
Liu
College of Management, Hebei University, Baoding 071002, Hebei,
China
College of Management, Hebei University,
China
Credibilistic valueatrisk
Reduced fuzzy variable
Parametric possibility distribution
Second order moment
[[1] J. Chachi and S. M. Taheri, A unied approach to similarity measure between intuitionistic##fuzzy sets, Int. J. Intell. Syst., 28 (2013), 669685.##[2] S. Coupland and R. John, A fast geometric method for defuzzication of type2 fuzzy sets,##IEEE T. Fuzzy Syst., 16(4) (2008), 929941.##[3] D. Dubois and H. Prade, Operations in a fuzzyvalued logic, Inform. Control, 43(2) (1979),##[4] M. Hao and J. M. Mendel, Similarity measures for general type2 fuzzy sets based on the##plane representation, Inform. Sciences, 227 (2014), 197215.##[5] B. Hu and C. Wang, On type2 fuzzy relations and intervalvalued type2 fuzzy sets, Fuzzy##Set. Syst., 236 (2014), 132.##[6] C. Huang, Y. Wang, H. Chen, H. Tsai, J. Jian, A. Cheng and J. Liao, Application of cellular##automata and type2 fuzzy logic to dynamic vehicle path planning, Appl. Soft Comput., 19##(2014), 333342.##[7] C. M. Hwang, M. S. Yang, W. L. Hung and M. G. Lee, A similarity measure of intuitionistic##fuzzy sets based on the Sugeno integral with its application to pattern recognition, Inform.##Sciences, 189 (2012), 93109.##[8] P. Kundu, S. Kar and M. Maiti, Fixed charge transportation problem with type2 fuzzy vari##ables, Inform. Sciences, 255 (2014), 170186.##[9] B. Liu, Uncertainty theory, SpringerVerlag, Berlin, 2004.##[10] F. Liu, An ecient centroid typereduction strategy for general type2 fuzzy logic system,##Inform. Sciences, 178(9) (2008), 22242236.##[11] H. M. Markowitz, Portfolio selection, J. Financ., 7 (1952), 7791.##[12] J. M. Mendel, Uncertain rulebased fuzzy logic system: introduction and new directions,##Prentice Hall, Upper Saddle River, N.J., 2001.##[13] J. M. Mendel and R. I. John, Type2 fuzzy sets made simple, IEEE T. Fuzzy Syst., 10(2)##(2002), 117127.##[14] M. Mizumoto and K. Tanaka, Some properties of fuzzy sets of type 2, Inform. Control, 31(4)##(1976), 312340.##[15] M. Moharrer, H. Tahayori, L. Livi, A. Sadeghian and A. Rizzi, Interval type2 fuzzy sets to##model linguistic label perception in online services satisfaction, Soft Comput., 19(1) (2015),##[16] S. C. Ngan, A type2 linguistic set theory and its application to multicriteria decision mak##ing, Comput. Ind. Eng., 64(2) (2013), 721730.##[17] A. D. Torshizi and M. H. F. Zarandi, A new cluster validity measure based on general type2##fuzzy sets: application in gene expression data clustering, KnowlBased. Syst., 64 (2014),##[18] P. Wang, Fuzzy contactibility and fuzzy variables, Fuzzy Set. Syst., 8(1) (1982), 8192.##[19] L. A. Zadeh, The concept of a linguistic variable and its application to approximate##reasoning{I,II,III, Inform. Sciences, 8(3) (1975), 199249; 8(4) (1975), 301357; 9(1) (1975),##]
Linear matrix inequality approach for synchronization of chaotic fuzzy cellular neural networks with discrete and unbounded distributed delays based on sampleddata control
2
2
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 sampleddata controlis investigated. LyapunovKrasovskii functional combining with the input delay approach as well as the freeweighting 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 timevarying delay required to be differentiable or even its timederivative assumed to be smaller than one, are removed. Instead, the timevarying delay is only assumed to be bounded. Finally, numerical examples and its simulations are provided to demonstrate the effectiveness of the derived results.
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77
98


P.
Balasubramaniampour
Department of Mathematics, Gandhigram Rural Institute  Deemed University, Gandhigram  624 302, Tamilnadu, India
Department of Mathematics, Gandhigram Rural
India


K.
Ratnavelu
Institute of Mathematical Sciences, Faculty of Science, University
of Malaya  50603, Kuala Lumpur, Malaysia
Institute of Mathematical Sciences, Faculty
Malaysia


M.
Kalpana
Institute of Mathematical Sciences, Faculty of Science, University of
Malaya  50603, Kuala Lumpur, Malaysia
Institute of Mathematical Sciences, Faculty
Malaysia
kalpana.nitt@gmail.com
Chaos
Fuzzy cellular neural networks
Linear matrix inequality
Sampleddata control
Synchronization
[[1] A. Arunkumar, R. Sakthivel, K. Mathiyalagan and S. Marshal Anthoni, Robust state estimation##for discretetime BAM neural networks with timevarying delay, Neurocomputing, 131##(2014), 171178.##[2] A. Arunkumar, R. Sakthivel, K. Mathiyalagan and Ju H. Park, Robust stochastic stability##of discretetime fuzzy Markovian jump neural networks, ISA Transactions, 53 (2014), 1006–##[3] P. Balasubramaniam, M. Kalpana and R. Rakkiyappan, Linear matrix inequality approach##for synchronization control of fuzzy cellular neural networks with mixed time delays, Chinese##Physics B, 21 (2012): 048402.##[4] S. Boyd, L. E. Ghaoui, E. Feron and V. Balakrishnan, Linear Matrix Inequalities in Systems##and Control Theory (SIAM, Philadelphia, 1994).##[5] T. L. Carroll and L. M. Pecora, Synchronization chaotic circuits, IEEE Trans. Circuits Syst.,##38 (1991), 453–456.##[6] L. O. Chua and L. Yang, Cellular neural networks: theory, IEEE Trans. Circuits Syst., 35##(1988), 12571272.##[7] L. O. Chua and L. Yang, Cellular neural networks: applications, IEEE Trans. Circuits Syst.,##35 (1988), 12731290.##[8] X. Feng, F. Zhang and W. Wang, Global exponential synchronization of delayed fuzzy cellular##neural networks with impulsive effects, Chaos Solitons Fractals., 44 (2011), 9–16.##[9] T. Feuring, J. J. Buckley, W. M. Lippe and A. Tenhagen, Stability analysis of neural net##controllers using fuzzy neural networks, Fuzzy Sets and Systems, 101 (1999), 303–313.##[10] E. Fridman, A. Seuret and J. P. Richard, Robust sampleddata stabilization of linear systems:##an input delay approach, Automatica, 40 (2004), 1441–1446.##[11] Q. Gan and Y. Liang, Synchronization of chaotic neural networks with time delay in the##leakage term and parametric uncertainties based on sampleddata control, J. Franklin Inst.,##349 (2012), 1955–1971.##[12] Q. Gan, R. Xu and P. Yang, Synchronization of nonidentical chaotic delayed fuzzy cellular##neural networks based on sliding mode control, Commun. Nonlinear Sci. Numer. Simul., 17##(2012), 433–443.##[13] Q. Gan, R. Xu and P. Yang, Exponential synchronization of stochastic fuzzy cellular neural##networks with time delay in the leakage term and reactiondiffusion, Commun. Nonlinear Sci.##Numer. Simul., 17 (2012), 1862–1870.##[14] K. Gu, An integral inequality in the stability problem of timedelay systems, in Proceedings##of the 39th IEEE Conference on Decision and Control Sydney, Australia (2000), 2805–2810.##[15] S. Lee, V. Kapila, M. Porfiri and A. Panda, Masterslave synchronization of continuously##and intermittently coupled sampleddata chaotic oscillators, Commun. Nonlinear Sci. Numer.##Simul., 15 (2010), 4100–4113.##[16] T. Li, S. Fei and Q. Zhu, Design of exponential state estimator for neural networks with##distributed delays, Nonlinear Anal. Real World Appl., 10 (2009), 1229–1242.##[17] N. Li, Y. Zhang, J. Hu and Z. Nie, Synchronization for general complex dynamical networks##with sampleddata, Neurocomputing, 74 (2011), 805–811.##[18] Z. Liu, H. Zhang and Z. Wang, Novel stability criterions of a new fuzzy cellular neural##networks with timevarying delays, Neurocomputing, 72 (2009), 1056–1064.##[19] J. Lu and D. J. Hill, Global asymptotical synchronization of chaotic Lur’e systems using##sampled data: a linear matrix inequality approach, IEEE Trans. Circuits Syst. II, 55 (2008),##586–590.##[20] K. Mathiyalagan, S. Hongye and R. Sakthivel, Robust stochastic stability of discretetime##Markovian jump neural networks with leakage delay, Zeitschrift Fur Naturforschung Section##AA Journal of Physical Sciences, 69 (2014), 70–80.##[21] K. Mathiyalagan, R. Sakthivel and S. Hongye, Exponential state estimation for discretetime##switched genetic regulatory networks with random delays, Canad. J. Phys., 92 (2014),##976–986.##[22] L. M. Pecora and T. L. Carroll, Synchronization in chaotic systems, Phys. Rev. Lett., 64##(1990), 821–824.##[23] L. M. Pecora, T. L. Carroll, G. A. Johnson, D. J. Mar and J. F. Heagy, Fundamentals of##synchronization in chaotic systems, concepts, and applications, Chaos, 7 (1997), 520–543.##[24] Y. Ping and L. Teng, Exponential synchronization of fuzzy cellular neural networks with##mixed delays and general boundary conditions, Commun. Nonlinear Sci. Numer. Simul., 17##(2012), 1003–1011. ##[25] R. Sakthivel, R. Raja and S. Marshal Anthoni, Linear matrix inequality approach to stochastic##stability of uncertain delayed BAM neural networks, IMA J Appl Math., 78 (2013), 1156–##[26] E. N. Sanchez and J. P. Perez, Inputtostate stability (ISS) analysis for dynamic neural##networks, IEEE Trans. Circuits Syst. I, 46 (1999), 13951398.##[27] F. O. Souza, R. M. Palhares and P. Y. Ekel, Asymptotic stability analysis in uncertain multidelayed##state neural networks via LyapunovKrasovskii theory, Math. Comput. Modelling, 45##(2007), 1350–1362.##[28] F. O. Souza, R. M. Palhares and P. Y. Ekel, Novel stability criteria for uncertain delayed##CohenGrossberg neural networks using discretized Lyapunov functional, Chaos Solitons Fractals,##41 (2009), 2387–2393.##[29] F. O. Souza, R. M. Palhares and P. Y. Ekel, Improved asymptotic stability analysis for##uncertain delayed state neural networks, Chaos Solitons Fractals, 39 (2009), 240–247.##[30] Y. Tang and J. Fang, Robust synchronization in an array of fuzzy delayed cellular neural##networks with stochastically hybrid coupling, Neurocomputing, 72 (2009), 3253–3262.##[31] T. Yang and L. B. Yang, Global stability of fuzzy cellular neural network, IEEE Trans. Circuits##Syst. I, 43 (1996), 880–883.##[32] T. Yang, L. B. Yang, C. W. Wu and L. O. Chua, Fuzzy cellular neural networks: Theory,##in Proceedings of the IEEE International Workshop on Cellular Neural Networks and##Applications, (1996), 181–186.##[33] T. Yang, L. B. Yang, C. W. Wu and L. O. Chua, Fuzzy cellular neural networks: Applications,##in Proceedings of the IEEE International Workshop on Cellular Neural Networks and##Applications, (1996), 225–230.##[34] J. Yu, C. Hu, H. Jiang and Z. Teng, Exponential lag synchronization for delayed fuzzy cellular##neural networks via periodically intermittent control, Math. Comput. Simulation, 82 (2012),##895–908.##[35] F. Yu and H. Jiang, Global exponential synchronization of fuzzy cellular neural networks with##delays and reactiondiffusion terms, Neurocomputing, 74 (2011), 509–515.##[36] L. A. Zadeh, Fuzzy Sets, Information and Control, 8 (1965), 338–353.##[37] C. Zhang, Y. He and M. Wu, Exponential synchronization of neural networks with timevarying##mixed delays and sampleddata, Neurocomputing, 74 (2010), 265–273.##]
Hierarchical Functional Concepts for Knowledge Transfer among Reinforcement Learning Agents
2
2
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.
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99
116


A.
Mousavi
Control and Intelligent Processing Center of Excellence, School of
Electrical and Computer Engineering, University of Tehran, Tehran, Iran
Control and Intelligent Processing Center
Iran


M.
Nili Ahmadabadi
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
Iran


H.
Vosoughpour
Control and Intelligent Processing Center of Excellence, School
of Electrical and Computer Engineering, University of Tehran, Tehran, Iran
Control and Intelligent Processing Center
Iran


B. N.
Araabi
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
Iran


N.
Zaare
Control and Intelligent Processing Center of Excellence, School of
Electrical and Computer Engineering, University of Tehran, Tehran, Iran
Control and Intelligent Processing Center
Iran
Reinforcement Learning
Transfer Learning
Heterogeneous Agents
Hierarchical Concepts
[[1] J. S. Bruner, Actual minds, possible words, Harvard University Press, 1987.##[2] T. Dietterich, Hierarchical reinforcement learning with the MAXQ value function decompo##sition, Journal of Articial Intelligent Research, 13 (2000), 227303.##[3] K. Driessens, J. Ramon and T. Croonenborghs, Transfer learning for reinforcement learn##ing through goal and policy parametrization, In ICML Workshop on Structural Knowledge##Transfer for Machine Learning, (2006), 14.##[4] W. Fritz, Intelligent systems and their societies, In Webpage: http://www. intelligentsys##tems.com.ar/intsyst/index.htm, January (1997).##[5] G. L. Klir, Uncertainty and information: foundations of generalized information theory,##John Wiley, Hoboken, NJ (2005).##[6] G. L. Klir, B. Yuan, Fuzzy sets and fuzzy logic: theory and applications, Prentice Hall, 1995. ##[7] G. Konidaris, A. Barto, Autonomous shaping: Knowledge transfer in reinforcement learning,##In Proceedings of the 23rd international conference on Machine learning, 2006, 489496.##[8] A. Lazaric, Knowledge Transfer in Reinforcement Learning, PhD thesis, Politecnico di Mi##lano, 2008.##[9] L. Mihalkova, T. Huynh and R. Mooney, Mapping and revising Markov Logic Networks for##transfer learning, In Proceedings of AAAI Conference on Articial Intelligence, (2007), 608614.##[10] H. Mobahi, M. Nili Ahmadabadi, and B. Nadjar Araabi, A biologically inspired method for##conceptual imitation using reinforcement learning, Applied Articial Intelligence, 21 (2007),##[11] R. A. Mollineda, F. J. Ferri and E. Vidal, A clusterbased merging strategy for nearest pro##totype classiers, In Proceedings of 15th International Conference on Pattern Recognition##(ICPR'00), 2 (2000), 755758.##[12] G. L. Murphy, The big book of concepts, MIT Press, 2004.##[13] S. Pan, J. Kwok and Q. Yang, Transfer learning via dimensionality reduction, In Proceedings##of 23rd AAAI Conference on Articial Intelligence, (2008), 677682.##[14] V. Soni, S. Singh, Using hormomorphisms to transfer options across continuous reinforce##ment learning domains, In Proceedings of 21st AAAI Conference on Articial Intelligence,##(2006), 494499.##[15] R. S. Sutton, A. G. Barto, Reinforcement Learning: An Introduction, MIT Press, Cambridge,##MA (1998).##[16] F. Tanaka, M. Yamamura, Multitask reinforcement learning on the distribution of MDPs,##Transactions of the Institute of Electrical Engineers of Japan, 123(5) (2003), 10041011.##[17] M. Taylor, P. Stone, Transfer learning for reinforcement learning domains: a survey, Journal##of Machine Learning Research, 10 (2009), 16331685.##[18] M. Taylor, G. Kuhlmann and P. Stone, Autonomous transfer for reinforcement learning, In##Proceedings of the 7th International Joint Conference on Autonomous Agents and Multiagent##systems, 1 (2008), 283290.##[19] M. Taylor, P. Stone, Representation Transfer for Reinforcement Learning, In Proceedings##of AAAI Fall Symposium on Computational Approaches to Representation Change during##Learning and Development, Arlington, Virginia, (2007), 7885.##[20] M. Taylor, P. Stone and Y. Liu, Value function for RLbased behavior transfer: A comparative##study, In Proceedings of the AAAI05 Conference on Articial Intelligence, (2005), 880885.##[21] A. Thedoridis and K. Koutroumbas, Pattern Recognition, Elsevier Academic Press, Second##Edition, 2003.##[22] L. Torrey and J. Shavlik, Transfer learning, In Soria, E., Martin, J., Magdalena, R., Martinez,##M., and Serrano, A., editors, Handbook of Research on Machine Learning Applications, IGI##Global, 2009, 242264.##[23] C. J. Watkins, Learning from Delayed Rewards, Ph.D. thesis, Cambridge University, 1989.##[24] C. J. Watkins and P. Dayan, Qlearning, Machine Learning, 8 (1992), 279292.##[25] A. Wilson, A. Fern, S. Ray and P. Tadepalli, Multitask reinforcement learning: A hierarchical##Bayesian approach, In Proceedings of the 24th International Conference on Machine Learning,##(2007), 10151022.##[26] M. Zentall, M. Galizio and T. S. Critched, Categorization, concept learning and behavior##analysis: an introduction, The Exprimental Analysis of Behavior, 3 (2002), 237248.##]
NonNewtonian Fuzzy numbers and related applications
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2
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 nonNewtonian calculus. The important point to note is that the nonNewtonian calculus is a selfcontained system independent of any other system of calculus. Since this selfcontained 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 nonNewtonian 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 nonNewtonian fuzzy distance and give some properties regarding convergence of sequences and series of fuzzy numbers with some illustrative examples.
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117
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Ugur
Kadak
Department of Mathematics, Bozok University, Yozgat, Turkey
Department of Mathematics, Bozok University,
Turkey
ugurkadak@gmail.com
NonNewtonian calculus
Fuzzy level sets
Trapezoidal fuzzy numbers
Convergence of fuzzy sequences and series
[[1] A. E. Bashirov, E. Kurpnar and A. Ozyapc, Multiplicative calculus and its applications, J.##Math. Anal. Appl., 337(1) (2008), 3648.##[2] A. F. Cakmak and F. Basar, Some new results on sequence spaces with respect to non##Newtonian calculus, J. Inequal. Appl., 2012(1) (2012), 228.##[3] M. Grossman and R. Katz, NonNewtonian Calculus, Lee Press, Pigeon Cove, MA, 1972.##[4] M. Grossman, Bigeometric Calculus, Archimedes Foundation, Rockport, Mass, USA, 1983.##[5] M. Grossman, The First Nonlinear System of Dierential and Integral Calculus, Mathco,##[6] U. Kadak and H. Efe, Matrix transformations between certain sequence spaces over the non##Newtonian complex eld, Sci. World J., 2014 (2014).##[7] U. Kadak and M. Ozluk, Generalized RungeKutta method with respect to the nonNewtonian##calculus, Abstr. Appl. Anal., 2014 (2014).##[8] U. Kadak and F. Basar, On Fourier series of fuzzyvalued function, Sci. World J., 2014##[9] M. Matloka, Sequences of fuzzy numbers, BUSEFAL, 28 (1986), 2837. ##[10] E. Msrl, and Y. Gurefe, Multiplicative AdamsBashforthMoulton methods, Numer. Algorithms,##57(4) (2011), 425439.##[11] M. Stojakovic and Z. Stojakovic, Series of fuzzy sets, Fuzzy Sets Syst., 160(21) (2009),##3115{3127.##[12] M. Stojakovic and Z. Stojakovic, Addition and series of fuzzy sets, Fuzzy Sets Syst., 83(3)##(1996), 341{346.##[13] O. Talo and F. Basar, Determination of the duals of classical sets of sequences of fuzzy##numbers and related matrix transformations, Comput. Math. Appl., 58(4) (2009), 717733.##[14] O. Talo and F. Basar, Quasilinearity of the classical sets of sequences of the fuzzy numbers##and some applications, Taiwanese J. Math., 14(5) (2010), 17991819.##[15] S. Tekin and F. Basar, Certain sequence spaces over the nonNewtonian complex eld, Abstr.##Appl. Anal., 2012 (2013).##[16] L. A. Zadeh, Fuzzy sets, Inform. and Control, 8 (1965), 338{353.##]
Order intervals in the metric space of fuzzy numbers
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2
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.
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147


S.
Aytar
Faculty of Arts and Sciences, Department of Mathematics, Suleyman
Demirel University, Isparta, Turkey
Faculty of Arts and Sciences, Department
Turkey
Fuzzy number
Order interval of fuzzy numbers
decision making
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A note on soft topological spaces
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2
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.
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149
155


FuGui
Shi
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
China
fugushi@bit.edu.cn


Bin
Pang
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
China
pangbin1205@163.com
Soft set
Soft topology
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