相关论文: Kolmogorov-Arnold-Moser Theorem
These reminiscences are about the "dark ages" of algorithmic information theory in the USSR. After a great interest in this topic in 1960s and the beginning of 1970s the number of people working in this area in the USSR decreased…
In 2004, F\'ejoz [D\'emonstration du 'th\'eor\'eme d'Arnold' sur la stabilit\'e du syst\`eme plan\'etaire (d'apr\`es M. Herman). Ergod. Th. & Dynam. Sys. 24(5) (2004), 1521-1582], completing investigations of Herman's [D\'emonstration d'un…
Kolmogorov-Arnold Networks (KANs) have gained significant attention as an alternative to traditional multilayer perceptrons, with proponents claiming superior interpretability and performance through learnable univariate activation…
Kolmogorov nonlinear averaging is complemented by a natural axiom. For this averaging, we prove a theorem on large deviations as well as establish the relationship to the tunnel canonical operator.
In this paper, we give a new proof of the classical KAM theorem on the persistence of an invariant quasi-periodic torus, whose frequency vector satisfies the Bruno-R\"ussmann condition, in real-analytic non-degenerate Hamiltonian systems…
A selfcontained proof of the KAM theorem in the Thirring model is discussed.
In this paper, we investigate Kolmogorov type theorems for small perturbations of degenerate Hamiltonian systems. These systems are index by a parameter $\xi$ as \( H(y,x,\xi) = \langle\omega(\xi),y\rangle + \varepsilon…
Kolmogorov-Arnold Networks (KANs) offer an efficient and interpretable alternative to traditional multi-layer perceptron (MLP) architectures due to their finite network topology. However, according to the results of Kolmogorov and…
We give a new proof of the KAM theorem for analytic Hamiltonians. The proof is inspired by a quantum field theory formulation of the problem and is based on a renormalization group argument treating the small denominators inductively scale…
Perturbations due to round-off errors in computer modeling are discontinuous and therefore one cannot use results like KAM theory about smooth perturbations of twist maps. We elaborate a special approximation scheme to construct two smooth…
In this paper, it is proved that the infinite KAM torus with prescribed frequency exists in a sufficiently small neighborhood of a given $ I^{0}$ for nearly integrable and analytic Hamiltonian system $ H(I,\theta) = H_{0}(I)+ \epsilon…
Recently the KAM theory has been extended to multidimensional PDEs. Nevertheless all these recent results concern PDEs on the torus, essentially because in that case the corresponding linear PDE is diagonalized in the Fourier basis and the…
Kolmogorov complexity is often used as a convenient language for counting and/or probabilistic existence proofs. However, there are some applications where Kolmogorov complexity is used in a more subtle way. We provide one (somehow)…
Data science has emerged as fourth paradigm of scientific exploration. However many machine learning models operate as black boxes offering limited insight into the reasoning behind their predictions. This lack of transparency is one of the…
In this paper, we gave a weighted compactness theory for the generalized commutators of vecotor-valued multilinear Calder\'{o}n-Zygmund operators. This was done by establishing a weighted Fr\'{e}chet-Kolmogorov theorem, which holds for…
A version of the Kolmogorov-Chentsov theorem on sample differentiability and H\"older continuity of random fields on domains of cone type is proved, and the result is generalized to manifolds.
The following is an exposition of a course of algebra that Prof. Aleksandr Aleksandrovich Zykov (1922-2013) distributed among the participants of his seminar in graph theory not far away from Odessa, Ukraine, on September, 1991. It is a…
Let $E/\mathbb{Q}$ be an elliptic curve and let $K$ be an imaginary quadratic field. Under a certain Heegner hypothesis, Kolyvagin constructed cohomology classes for $E$ using $K$-CM points and conjectured they did not all vanish.…
Kolmogorov-Arnold Networks represent a recent advancement in machine learning, with the potential to outperform traditional perceptron-based neural networks across various domains as well as provide more interpretability with the use of…
We prove Kolmogorov's law of the iterated logarithm for noncommutative martingales. The commutative case was due to Stout. The key ingredient is an exponential inequality proved recently by Junge and the author.