Kolmogorov superposition theorem and its applications

Hilbert’s 13th problem asked whether every continuous multivariate function can be written as superposition of continuous functions of 2 variables. Kolmogorov and Arnold show that every continuous multivariate function can be represented as superposition of continuous univariate functions and addition in a universal form and thus solved the problem positively. In Kolmogorov’s representation, only one univariate function (the outer function) depends on and all the other univariate functions (inner functions) are independent of the multivariate function to be represented. This greatly inspired research on representation and superposition of functions using Kolmogorov’s superposition theorem (KST). However, the numeric applications and theoretic development of KST is considerably limited due to the lack of smoothness of the univariate functions in the representation. Therefore, we investigate the properties of the outer and inner functions in detail. We show that the outer function for a given multivariate function is not unique, does not preserve the positivity of the multivariate function and has a largely degraded modulus of continuity. The structure of the set of inner functions only depends on the number of variables of the multivariate function. We show that inner functions constructed in Kolmogorov’s representation for continuous functions of a fixed number of variables can be reused by extension or projection to represent continuous functions of a different number of variables. After an investigation of the functions in KST, we combine KST with Fourier transform and write a formula regarding the change of the outer functions under different inner functions for a given multivariate function. KST is also applied to estimate the optimal cost between measures in high dimension by the optimal cost between measures in low dimension. Furthermore, we apply KST to image encryption and show that the maximal error can be obtained in the encryption schemes we suggested.Open Acces

Baire Category and Nowhere Differentiability for Feasible Real Functions ⋆

Abstract. A notion of resource-bounded Baire category is developed for the class PC[0,1] of all polynomial-time computable real-valued functions on the unit interval. The meager subsets of PC[0,1] are characterized in terms of resource-bounded Banach-Mazur games. This characterization is used to prove that, in the sense of Baire category, almost every function in PC[0,1] is nowhere differentiable. This is a complexity-theoretic extension of the analogous classical result that Banach proved for the class C[0, 1] in 1931.

The formal theory of pricing and investment for electricity.

The Thesis develops the framework of competitive equilibrium in infinite-dimensional commodity and price spaces, and applies it to the problems of electricity pricing and investment in the generating system. Alternative choices of the spaces are discussed for two different approaches to the price singularities that occur with pointed output peaks. Thermal generation costs are studied first, by using the mathematical methods of convex calculus and majorisation theory, a.k.a. rearrangement theory. Next, the thermal technology, pumped storage and hydroelectric generation are studied by duality methods of linear and convex programming. These are applied to the problems of operation and valuation of plants, and of river flows. For storage and hydro plants, both problems are approached by shadow-pricing the energy stock, and when the given electricity price is a continuous function of time, the plants' capacities, and in the case of hydro also the river flows, are shown to have definite and separate marginal values. These are used to determine the optimum investment. A short-run approach to long-run equilibrium is then developed for pricing a differentiated good such as electricity. As one tool, the Wong-Viner Envelope Theorem is extended to the case of convex but nondifferentiable costs by using the short-run profit function and the profit-imputed values of the fixed inputs, and by using the subdifferential as a multi-valued, generalised derivative. The theorem applies readily to purely thermal electricity generation. But in general the short-run approach builds on solutions to the primal-dual pair of plant operation and valuation problems, and it is this framework that is applied to the case of electricity generated by thermal, hydro and pumped-storage plants. This gives, as part of the long-run equilibrium solution, a sound method of valuing the fixed assets-in this case, the river flows and the sites suitable for reservoirs