The classical Szegő—Kolmogorov theorem characterizes the weights ω such that family of exponentials with positive integer frequencies spans whole weighted space \({L^2}(\mathbb{T},w)\) on circle. Kolmogorov’s probabilistic interpretation this result connects it possibility to ‘predict precisely future from past’ for stationary stochastic processes discrete time. We discuss problem whether predi...

This work (Part (I)) together with its companion (II)) develops a new framework for stochastic functional Kolmogorov equations, which are nonlinear differential equations depending on the current as well past states. Because of complexity results, it seems to be instructive divide our contributions two parts. In contrast existing literature, effort is advance knowledge by allowing delay and dep...

We present a Kolmogorov-like algorithm for the computation of normal form in neighborhood an invariant torus `isochronous' Hamiltonian systems, i.e., systems with Hamiltonians $\mathcal{H}=\mathcal{H}_0+\varepsilon \mathcal{H}_1$ where $\mathcal{H}_0$ is $N$ linear oscillators, and $\mathcal{H}_1$ expandable as polynomial series oscillators' canonical variables. This method can be regarded anal...

We survey the diverse approaches to the notion of information content: from Shannon entropy to Kolmogorov complexity. The main applications of Kolmogorov complexity are presented: namely, the mathematical notion of randomness (which goes back to the 60’s with the work of MartinLöf, Schnorr, Chaitin, Levin), and classification, which is a recent idea with provocative implementation by Vitanyi an...

We deene a (chaotic) deterministic variant of random multiplicative cascade models of turbulence. It preserves the hierarchical tree structure , thanks to the addition of innnitesimal noise. The zero-noise limit can be handled by Perron-Frobenius theory, just as the zero-diiusivity limit for the fast dynamo problem. Random multiplicative models do not possess Kolmogorov 1941 (K41) scaling becau...