No. 44 (00154) Family name : Glazunov Given name : Nikolaj Affiliation : Glushkov Institute of Cybernetics National Academy of Sci. Abbreviation : E-mail address : glanm@d105.icyb.kiev.ua,glanm@yahoo.com Title : Categorification of Fourier Transforms, Efficient Computation and Computing Intelligence Authors : Nikolaj M. Glazunov Abstract : \documentclass[12pt]{article} \begin{document} \bibliographystyle{unsrt} \bigskip \begin{center} {\Large \bf Categorification of Fourier Transforms, Efficient Computation and Computing Intelligence } \end{center} \smallskip \begin{center} {\bf Nikolaj M. Glazunov } \end{center} \smallskip \begin{center} Glushkov Institute of Cybernetics National Academy of Sci. (Kiev) \\ {\rm Email:} {\it glanm@yahoo.com } \end{center} \smallskip \begin{center} {\bf Abstract} \end{center} We investigate a category theory and homological algebra framework for various types of Fourier transforms (Fourier, Fourier-Deligne, Fourier-Cato, Fourier-Mukai) and efficient algorithms for their implementation. Overwhelming majority of the fundamental methods for Physics have set theoretic foundations. A category theory framework (by means of a category ${\cal C}$ or a functor ${\cal F}$) for a system $S,$ a process $\Pi$ or a phenomenon $\Phi$ can be thought as a collection of objects $A, B, \ldots ,$ one for each element of $S$ (respectively $\Pi,$ $\Phi$) which are combined by morphisms $f: A \rightarrow B,$ subject to the conditions (i) and (ii) below. Let ${\cal C}(A,B)$ be the class of morphisms from $A$ to $B.$ Then \\ (i) for any three (not necessary distinct) objects $A, B$ or $C$ of ${\cal C},$ there is defined a map $$ {\cal C}(A,B) \times {\cal C}(B,C) \rightarrow {\cal C}(B,C),$$ called composition which satisfied the $\it associativity$ axiom. \\ (ii) For every objects $A$ of ${\cal C}$, the set ${\cal C}(A,A)$ contains a morphism $id_{A},$ called the identity of $A.$ \\ A ${\it functor} \; {\cal F}$ from a category ${\cal C}$ to a category ${\cal K}$ is a function which maps $ Ob({\cal C}) \rightarrow Ob({\cal K}),$ and which for each pair $A, B$ of objects of ${\cal C}$ maps $ {\cal C}(A,B) \rightarrow {\cal C}({\cal F}(A),{\cal F}(B)),$ while satisfying the two conditions: \\ ${\cal F}id_{A} = id_{{\cal F}A}$ for every $A \in Ob({\cal C}),$ \\ ${\cal F}(fg) = {\cal F}(f) {\cal F}(g).$ \\ Our talk will illustrate the consequences for the category theory treatment of data representation and analysis, scientific computations, models of interaction and computing intelligence. Also we discuss several optimization and computer algebra problems concerning the approach. \begin{thebibliography}{99} \bibitem{G:IE} Glazunov N. On computational aspects of the Fourier-Mukai transform and another dualities. {Int. Conf. SYMMETRY IN NONLINEAR MATHEMATICAL PHYSICS (2003)} (Submitted) \\ Electronic abstract: http://www.imath.kiev.ua/~appmath/conf.html or mirror http://www.bgu.ac.il/~alexzh/appmath/conf.html \bibitem{Gl:CT} N.M. Glazunov, Category Theory Aspects of Complex Manifolds and Problems of Mirror Symmetry (in Russian,) {\it Problemy Programmirovaniya}. No. 3-4, Kiev, (2002) pp.104-110. \bibitem{G:MC} Glazunov N. Interval computations on Minkowski moduli space and the category theory framework for verification. SCAN2002. Int. Symposium on Scientific Computing, Computer Arithmetic, and Validated Numerics. Proceedings (Submitted) \bibitem{GK:02} Glazunov N., Kapitonova Ju. On a common context of number theory and harmonic analysis. {\it Cybernetics and System Analysis}. N 1 (2003), pp.134-147. \bibitem{G:VN} Glazunov N. On Validated Numerics, Category Theory and Computer Algebra Framework for Simulation and Computation in Theoretical Physics, {\it Nuclear Inst. and Methods in Physics, A, 2003.} P.654-656. \bibitem{G:AG} Glazunov N. On algebraic geometric and computer algebra aspects of mirror symmetry. E-print. 11p. http://arXiv.org/abs/hep-th/0112250 \bibitem{G:IA} Glazunov N. On Some Interval-Algebraic Methods for Verifications of Dynamical Systems. {\it Cybernetics and Computer Technologies. Complex Control Systems.} 109 (1997) p.15-23. \bibitem{G:KB} Glazunov N. Knowledge Base Systems and Dynamic Control Systems. {\it Cybernetics and Computer Technologies. Complex Control Systems.} 85 (1990) P.18-22. \end{thebibliography} \end{document}