相关论文: Kolmogorov-Arnold-Moser Theorem

After carefully checking the original Russian version of Kolmogorov (1942), in the present author view, both Spalding and Wilcox misinterpretations are due to the fact that they did not notice the two different symbols, which Kolmogorov…

In this paper we develop some new KAM-technique to prove two general KAM theorems for nearly integrable hamiltonian systems without assuming any non-degeneracy condition. Many of KAM-type results (including the classical KAM theorem) are…

The discovery of high-performance thermoelectric materials requires models that are both accurate and interpretable. Traditional machine learning approaches, while effective at property prediction, often act as black boxes and provide…

For any particularly interesting theorem one proof is never enough. Instead, the first proof sets the challenge to find a more elegant method that illuminates subtle features of the math, is simpler to understand, or even avoids using…

The incompressibility method is a counting argument in the framework of algorithmic complexity that permits discovering properties that are satisfied by most objects of a class. This paper gives a preliminary insight into Kolmogorov's…

Ando established an algebraic criterion for when a complex orientation for a Morava E-theory is an $H_\infty$-map. The criterion relates such an orientation to a specific property of the formal group associated to the E-theory, namely, a…

Time series classification is a relevant step supporting decision-making processes in various domains, and deep neural models have shown promising performance in this respect. Despite significant advancements in deep learning, the…

In this paper, we present a variant of the Balog-Szemer\'edi-Gowers theorem for the Vinogradov system. We then use our result to deduce a higher degree analogue of the sum-product phenomenon.

Neural network (NN)-based transistor compact modeling has recently emerged as a transformative solution for accelerating device modeling and SPICE circuit simulations. However, conventional NN architectures, despite their widespread…

We give a new proof of the quantum version of MacMahon's Master Theorem due to Garoufalidis, Le and Zeilberger (one-parameter case) and to Konvalinka and Pak (multiparameter case) by deriving it from known facts about Koszul algebras.

The Ax-Kochen Theorem is a purely algebraic statement about the zeros of homogeneous polynomials over the p-adic numbers, but it was originally proved using techniques from mathematical logic. This document, the author's undergraduate…

This paper has been withdrawn by the author due a crucial sign error in Theorem B. We present a geometric proof of Thom conjecture, which uses Khovanov homology. Our approach doesn't use any analytic methods and is quite different from…

The results of Iv. Prodanov on abstract spectra and separative algebras were announced in the journal "Trudy Mat. Inst. Steklova", 154, 1983, 200--208, but their proofs were never written by him in the form of a manuscript, preprint or…

We survey the Kolmogorov's approach to the notion of randomness through the Kolmogorov complexity theory. The original motivation of Kolmogorov was to give up a quantitative definition of information. In this theory, an object is randomness…

The need for formal definition of the very basis of mathematics arose in the last century. The scale and complexity of mathematics, along with discovered paradoxes, revealed the danger of accumulating errors across theories. Although,…

Kolmogorov's setting for probability theory is given an original generalization to account for probabilities arising from Quantum Mechanics. The sample space has a central role in this presentation and random variables, i.e., observables,…

We investigate what can be concluded about a quantum system when sequential quantum measurements of its observable -- a prominent example of the so-called quantum stochastic process -- fulfill the Kolmogorov consistency condition and thus…

Cook's theorem is commonly expressed such as any polynomial time-verifiable problem can be reduced to the SAT problem. The proof of Cook's theorem consists in constructing a propositional formula A(w) to simulate a computation of TM, and…

In this paper we proved a new numerically explicit version of the P\'{o}lya--Vinogradov inequality. Our proof is based on the new ideas of V.A. Bykovskii and improves a recent inequality obtained by C. Pomerance.

In the following article a Ph.D. student of V.I.~Arnold gives a personal account on his teacher who unexpectedly passed away earlier this year.