СДУ хабаршысы - 2007

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Hilbert’s Problem 13 and Related Topics in the Theory of Continuous Functions of Several Variables
(Suleyman Demirel University, 2007) F. A. Hajiyev
This paper provides an overview of Hilbert’s 13th problem and its developments concerning the representation of functions of several variables as superpositions of functions of fewer variables, particularly continuous functions. While Hilbert conjectured that certain continuous functions of three variables could not be expressed through finite superpositions of continuous functions of two variables, subsequent results by Kolmogorov and Arnol’d proved the opposite for continuous mappings on compact intervals. The study further explores generalizations of Kolmogorov’s theorem by Ostrand, Mal’cev, and others, extending these results to various metric and topological spaces, including tori, Hilbert cubes, and Cantor sets. The paper also examines the superposition problem for topological spaces (SPS(X)) and its relation to algebraic topology through homotopy groups and Post algebras. For non-abelian spaces such as the 3-sphere (S³), it is shown that the superposition problem can have negative solutions, revealing a deep connection between functional representation and homotopic structure. These investigations contribute to understanding the algebraic and topological foundations of function representation theory.