?
ON GENERALIZATIONS OF CHEBYSHEV POLYNOMIALS AND CATALAN NUMBERS
Ufa Mathematical Journal. 2021. Vol. 13. No. 2. P. 8–14.
Bychkov B., Shabat G. B.
We provide possible directions of generalizations of earlier found relations between the Chebyshev polynomials and the Catalan numbers arising in studying commuting difference operators. These generalizations are mostly related with ideas proposed by N.H. Abel in his publication in 1826, which then were reproduced by many authors in a modern language. As generalization of Chebyshev polynomials, we propose to consider polynomials with exactly two critical values well-studied in a so-called theory of dessins d'enfants. The Catalan numbers are located in the first column of the table of Harer-Zagier numbers related with the distribution by genus of orientable sewing of polygons with even number of sides. The commuting difference operators are implicitly contained in the Abel theory, who studied quasi-elliptic integrals, namely, the elliptic integrals of 3rd kind integrable in terms of logarithms. In the present work we formulate conjectures on relation between the main Abel theorem and commuting semi-infinite matrices. In the work we provide calculations supporting the conjectured relations.
Alfimov Mikhail, Kurakin Andrey, Journal of High Energy Physics 2026 Vol. 2026 Article 99
We continue studying regularization scheme dependence of the 𝒩 = 2 supersymmetric sigma models. In the present work the previous result for the four loop β-function is extended to the five loop order. Namely, we find the renormalization scheme, in which the fifth loop contribution is completely eliminated, while the fourth loop contribution is represented ...
Added: November 12, 2026
Ignatev M., Vilkin T., Mediterranean Journal of Mathematics 2026 Vol. 23 No. 2 Article 70
Trinomial varieties are affine varieties given by a system of equations consisting of polynomials with three terms. Such varieties are total coordinate spaces of normal varieties with torus action of complexity one. For an affine variety X, we consider the subgroup SAut(X) of the automorphism group generated by all algebraic subgroups isomorphic to the additive ...
Added: November 12, 2026
Maksaev A., Promyslov V., Тишутин П. С., Записки научных семинаров ПОМИ РАН 2026 Т. 554 С. 218–235
Let K be an algebraically closed field and let ℰₖ be the set of n × n matrices over K with exactly k distinct eigenvalues. Previously, the classification of linear maps T: Mₙ(K) → Mₙ(K) satisfying T(ℰₙ) ⊆ ℰₙ was obtained for invertible maps over fields of characteristic 0. In this paper, the classification is ...
Added: November 11, 2026
Barinova M., Tirskaya Karina, Владикавказский математический журнал 2026 Vol. 28 No. 3 P. 25–43
In this paper, the problem of the existence of an energy function for discrete dynamical systems defined by structurally stable diffeomorphisms is studied. For the considered class of dynamical systems, such a function is defined as a continuous function that decreases along the orbits of the system outside the non-wandering set and remains constant on ...
Added: November 11, 2026
Illarionov A., Математический сборник 2026 Т. 217 № 11 С. 56–87
Let $b(\alpha;T)$ be the number of best simultaneous Diophantine approximations with common denominator $\le T$ to the point $\alpha \in {\Bbb R}^n$. We derive asymptotic formulas for the mean value of this quantity (averaging is performed over all rational points of the form $\alpha = (k_1/N,\ldots, k_n/N)$, $N\le R$), as well as for its mathematical ...
Added: November 10, 2026
Vereshchagin N., Математический сборник 2026 Т. 217 № 10 С. 3–46
Теорема Гудмана-Штрауса утверждает, что для “почти всякой” подстановки семейство подстановочных замощений софично, т.е. может быть задано локальными правилами для некоторой раскраски плиток. Условия на подстановку, гарантирующие софичность, довольно сложные.
В настоящей работе мы предлагаем версию теоремы Гудмана-Штрауса с более простыми условиями на подстановку. Хотя эти условия достаточно ограничительны, мы показываем, что в сочетании с двумя простыми приемами (взятием достаточно ...
Added: November 10, 2026
Aleksei Samarin, Belous F., Торопов А. Г. et al., Technologies 2026 Vol. 14 No. 11 P. 1–21
Abstract
Automated interpretation of magnetic flux leakage (MFL) data is central to condition assessment of long-distance oil and gas pipelines. Yet, full-resolution panoramas are difficult to process without either suppressing small anomalies through resizing or fragmenting elongated objects at sliding-window boundaries. We propose MFL-ScrollNet, a recurrent sliding-window detector that formulates panoramic analysis as an ordered sequence ...
Added: November 8, 2026
Retinsky V., Функциональный анализ и его приложения 2026 Т. 60 № 4 С. 83–100
Приведено обобщение матричной теоремы о деревьях на случай перечисления пар остовных деревьев. Это обобщение включает в качестве частных случаев и унифицирует сразу несколько известных ранее результатов, в частности классическую теорему Кирхгофа, теорему Масбаума–Вайнтроба о перечислении 3-деревьев, а также аналог матричной теоремы о деревьях для перечисления 4-элементов Кирхгофа. ...
Added: November 5, 2026
Levashev V., Математические заметки 2026 Т. 120 № 5 С. 843–846
Пусть 𝐴 – коммутативное ассоциативное кольцо с единицей и 𝐴((𝑡)) = 𝐴[[𝑡]][𝑡−1] – кольцо рядов Лорана над 𝐴. Алгебра 𝐴((𝑡)) имеет естественную топологию, где 𝐴 – подмодули вида 𝑡𝑚𝐴[[𝑡]] образуют базу окрестностей нуля. Пусть также Ω1𝐴((𝑡))/𝐴 – модуль кэлеровых 1-дифференциалов 𝐴((𝑡)) над 𝐴. ...
Added: November 5, 2026
Алексеев О. В., Letters in Mathematical Physics 2026 Vol. 116 No. 5 Article 132
We study the mixed Hessian of the dispersionless Toda \(\tau\)-function for the one-harmonic \(s\)-fold symmetric conformal map \(f(w)=r\bigl(w+\zeta w^{1-s}\bigr)\), where \(r>0\) is the exterior conformal radius and \(\zeta>0\) is the dimensionless shape parameter. This Hessian is the susceptibility matrix generated by the inverse conformal map. Our spectral statements are formulated for its weighted symmetry-block realizations ...
Added: November 5, 2026
Levin A., Sakharova N., / Series arXiv "math". 2024. No. 2412.17795.
The modular curves serve as excellent objects for testing conjectures in arithmetic geometry. They possess a natural geometric definition in contrast with rather nontrivial structure. On the other hand, they are well-studied from the perspective of number theory. Furthermore, there is a well-developed and powerful analytic technique available. We will use the square of the ...
Added: November 4, 2026
Zhuzhoma E. V., Medvedev V., Regular and Chaotic Dynamics 2026 Vol. 31 No. 5 P. 732–742
We study Axiom A diffeomorphisms (in the sense of Smale) on a closed orientable $n$-manifold $M^n$, $n\ge 4$, whose non-wandering sets consist of codimension one orientable attractors, repellers, and isolated saddle periodic orbits with the same Morse index. Such systems combine chaotic dynamics on basic sets with regular dynamics near trivial periodic orbits. We prove ...
Added: November 3, 2026
Zhuzhoma E. V., Medvedev V., Yu B. et al., Chaos 2026 Vol. 36 No. 6 P. 063111–063118
We show that any orientable closed 3-manifold $M$ admits structurally stable non-singular flow $f^t$ whose non-wandering set $NW(f^t)$ consists of a 2-dimensional expanding attractor and finitely many repelling periodic trajectories. For $M=\mathbb{S}^3$, we prove that the set of repelling periodic trajectories can be an arbitrary link provided that this link contains the figure eight knot. ...
Added: November 3, 2026
Zhuzhoma E. V., Zhirov A. Y., Russian Journal of Nonlinear Dynamics 2026 Vol. 22 No. 2 P. 403–414
On a closed orientable surface, we consider the set of axiom A diffeomorphisms whose non-wandering sets consist of connected one-dimensional expanding attractors and contracting repellers (any attractor/repeller is locally homeomorphic to the product of segment and Cantor set). This set consists of $\Omega$-stable diffeomorphisms and structurally unstable diffeomorphisms. We classify such diffeomorphisms up to the ...
Added: November 3, 2026
Elena R. Loubenets, Sergey Kuznetsov, Quantum Information Processing 2026 Vol. 25 Article 358
The optimal implementation of quantum gates for closed $N$-qubit systems is one of the key challenges for practical realization of many quantum information processing tasks. In the present article, based on the generalized Bloch vectors formalism [J. Phys. A: Math. Theor. 54, 195301 (2021)] for a finite-dimensional quantum system, we develop a new \emph{general} model for the ...
Added: November 3, 2026
Zhuzhoma E. V., Medvedev V. S., Yu B., Mathematical notes 2025 Vol. 118 No. 2 P. 421–423
We show that given any orientable closed 3-manifold $M^3$ admits structurally stable non-singular flow $f^t$ whose non-wandering set $NW(f^t)$ consists of 2-dimensional expanding attractor and finitely many repelling periodic trajectories. For $M^3=\mathbb{S}^3$, we prove that the set of repelling periodic trajectories can be an arbitrary link provided this link contains the figure eight knot. ...
Added: November 3, 2026
Kazakov A., Kulagin N., Lerman L. et al., Chaos 2026 Vol. 36 No. 11 P. 1–24
We study perturbations of an integrable Hamiltonian system with two degrees of freedom that possesses a saddle-focus equilibrium with a homoclinic skirt—a one-parameter family of homoclinic orbits of the saddle-focus whose stable and unstable invariant manifolds coincide. The perturbations are dissipative, and we consider those that split the skirt completely (i.e., no intersections remain). We are interested ...
Added: November 2, 2026
Смирнов С. В., Дынников И. А., Успехи математических наук 2002 Т. 57 № 6 С. 183–184
Изучаются точно решаемые циклические q-цепочки Дарбу, обобщающие q-осциллятор ...
Added: November 1, 2026
Natalia Ya. Amburg, Elena M. Kreines, George B. Shabat, Sarajevo Journal of Mathematics 2024 Vol. 20 No. 1 P. 71–86
In this paper we introduce dessins d'enfants on unions of surfaces, possibly glued. We show why they are natural, discuss their relations with Belyi pairs on reducible curves and provide some examples. In particular, we provide an example of a Fried pair which degenerates to a dessin on a reducible and singular curve. ...
Added: May 26, 2025
Karpov A., Markström K., Riis S. et al., Discrete Applied Mathematics 2025 Vol. 370 P. 57–70
In this paper, we study Condorcet domains, sets of linear orders from which majority ranking produces a linear order. We introduce a new class of Condorcet domains, called coherent domains, which is natural from both a voting theoretic and combinatorial perspective. After studying the properties of these domains we introduce set-alternating schemes. This is a method for ...
Added: March 15, 2025
Дмитриев М. Д., Сибирские электронные математические известия 2023 Т. 20 № 2 С. 626–637
We give a new elementary proof of the theorem by Ambainis
et al. that for a quantum walk, the probability amplitudes of absorption
at the initial point after 4n steps are proportional to the Catalan numbers.
We also calculate the absorption probabilities at points close to the initial
one and prove a relation that connects the probability amplitudes along
the ...
Added: November 9, 2023
Avdoshin S. M., Набебин А. А., М.: ДМК Пресс, 2019.
The book contains the necessary information from the algorithm theory, graph theory, combinatorics. It is considered partially recursive functions, Turing machines, some versions of the algorithms (associative calculus, the system of substitutions, grammars, Post's productions, Marcov's normal algorithms, operator algorithms). The main types of graphs are described (multigraphs, pseudographs, Eulerian graphs, Hamiltonian graphs, trees, bipartite ...
Added: August 24, 2018