Is there a need for fuzzy logic

“Multivalued Analysis” is the theory of set-valued maps (called multifonctions) and has important applications in many different areas. Multivalued analysis is a remarkable mixture of many different parts of mathematics such as point-set topology, measure theory and nonlinear functional analysis. It is also closely related to “Nonsmooth Analysis” (Chapter 5) and in fact one of the main motivations behind the development of the theory, was in order to provide necessary analytical tools for the study of problems in nonsmooth analysis. It is not a coincidence that the development of the two fields coincide chronologically and follow parallel paths. Today multivalued analysis is a mature mathematical field with its own methods, techniques and applications that range from social and economic sciences to biological sciences and engineering. There is no doubt that a modern treatise on “Nonlinear Functional Analysis” can not afford the luxury of ignoring multivalued analysis. The omission of the theory of multifunctions will drastically limit the possible applications.

SET-Valued Analysis

Toward a generalized theory of uncertainty (GTU)--an outline

Preface 1. The approximation problem and existence of best approximations 2. The uniqueness of best approximations 3. Approximation operators and some approximating functions 4. Polynomial interpolation 5. Divided differences 6. The uniform convergence of polynomial approximations 7. The theory of minimax approximation 8. The exchange algorithm 9. The convergence of the exchange algorithm 10. Rational approximation by the exchange algorithm 11. Least squares approximation 12. Properties of orthogonal polynomials 13. Approximation of periodic functions 14. The theory of best L1 approximation 15. An example of L1 approximation and the discrete case 16. The order of convergence of polynomial approximations 17. The uniform boundedness theorem 18. Interpolation by piecewise polynomials 19. B-splines 20. Convergence properties of spline approximations 21. Knot positions and the calculation of spline approximations 22. The Peano kernel theorem 23. Natural and perfect splines 24. Optimal interpolation Appendices Index.

Approximation theory and methods, by M. J. D. Powell. Pp 339. £25 (hardcover), £8·50 (paperback). 1981. ISBN 0-521-22472-1/29514-9 (Cambridge University Press)

高等学校計算数学学報 = Numerical mathematics

Recently, fuzzy systems have become one of the hottest topics due to their applications in the area of computer science. Therefore, in this article, we are making efforts to add new useful relationships between the selected L-fuzzy (fuzzifying) systems. In particular, we establish relationships between L-fuzzy (fuzzifying) Cech closure spaces, L-fuzzy (fuzzifying) co-topological spaces and L-fuzzy (fuzzifying) approximation spaces based on reflexive L-fuzzy relations. We also show that there is a Galois correspondence between the categories of these spaces.

Categories of L-Fuzzy Čech Closure Spaces and L-Fuzzy Co-Topological Spaces

A novel theoretical background of fuzzy associative memories (FAM) is proposed. A framework of formal concept analysis is used for a new working theory of FAM. Two principal activities of FAM are formalized : data retrieval and noise reduction. It is shown that the problem of data retrieval is connected with solvability and eigen sets of a certain system of fuzzy relation equations. The differentiation of FAM models according to their ability to reduce noise is defined. It is shown how the choice of formal context determines a type of noise that can be reduced by the corresponding retrieval mechanism. Finally, we propose a fast algorithm of data retrieval.

Data Retrieval and Noise Reduction by Fuzzy Associative Memories.

Fuzzy If-Then Rules in Approximation Of Functions

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Fuzzy Methods and Approximate Reasoning in Geographical Information Systems

This paper introduces a novel construction of test spaces in the Ritz-Galerkin method. These spaces are established on the basis of linear spaces of functions that are linear combinations of polynomials and fuzzy sets of a generalized uniform fuzzy partition.

Irina Perfilieva

Papers

Categories of L-Fuzzy Čech Closure Spaces and L-Fuzzy Co-Topological Spaces

Data Retrieval and Noise Reduction by Fuzzy Associative Memories.

Fuzzy If-Then Rules in Approximation Of Functions

Fuzzy Methods and Approximate Reasoning in Geographical Information Systems

Generalized Fuzzy Partition in Galerkin Method for the Boundary Valued Problem