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

The problem of solvability of a system of fuzzy relation equations with sup-* composition is considered in semilinear vector spaces. Based on the fact that a complete set of solutions is determined by minimal solutions, we focused on characterization of them. At first, sets of all minimal solutions of a single equation have been described under different assumptions on an underlying algebra. Dependently on the ordering of the support set, either necessary or sufficient conditions, or criteria of being a minimal solution have been obtained. Then minimal solutions of a system are build from minimal solutions of single equations.

/pdf/system-of-fuzzy-relation-equations-with-sup-composition-in-1dmxvt2p6o.pdf

System of fuzzy relation equations with sup-* composition in semi-linear spaces: minimal solutions

The development strategy of a complex enterprise depends on the financial performance evaluation. The latter can be made by time series analysis and forecasting of financial indicators. The paper is focused on a web-based system, which integrates fuzzy time series forecasting techniques with a web service to support enterprise managers in making plan for next period. In the paper the web service with three main components proposed: time series forecasting system using F-transform and fuzzy tendency forecasting techniques, system for linguistic description of predicted fuzzy tendency of each financial indicators and expert system for deriving the linguistic summarization of enterprise financial state and recommendation for planning for next period. As the proposed web service was successfully examined by ten Russian enterprises the example of financial performance evaluation was showed.

Web-Based System for Enterprise Performance Analysis on the Basis of Time Series Data Mining

The paper continues a study of extensional formulas and their representation by normal forms. The notion of extensional formula is extended to the case when extensionality is considered with respect to a binary reflexive relation. It has been proved that extensional formulas can be represented by normal forms of infinite type and “approximated” by normal forms of finite type. All proofs are given formally in fuzzy predicate BL-logic.

Logical approximation II

An overview of direct and inverse fuzzy transforms of three types is given and applications to data processing are considered. The construction and some important properties of fuzzy transforms are presented on the theoretical level. Three applications of $F$-transform to data processing have been chosen: compressional and reconstruction of data, removing noise and data fusion. All of them successively exploit the filtering property of the inverse fuzzy transform.

Irina Perfilieva

Papers

System of fuzzy relation equations with sup-* composition in semi-linear spaces: minimal solutions

Web-Based System for Enterprise Performance Analysis on the Basis of Time Series Data Mining

Logical approximation II

Fuzzy transforms in image compression and fusion

What is Fuzzy Modeling