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Necessary and sufficient conditions for the existence of invariant algebraic curves
Electronic Journal of Qualitative Theory of Differential Equations. 2021. Vol. 48. P. 1–22.
We present a set of conditions enabling a polynomial system of ordinary differential equations in the plane to have invariant algebraic curves. These conditions are necessary and sufficient. Our main tools include factorizations over the field of Puiseux series near infinity of bivariate polynomials generating invariant algebraic curves. The set of conditions can be algorithmically verified. This fact gives rise to a method, which is able not only to find some irreducible invariant algebraic curves, but also to perform their classification. We study in details the problem of classifying invariant algebraic curves in the most difficult case: we consider differential systems with infinite number of trajectories passing through infinity. As an example, we find necessary and sufficient conditions such that a general polynomial Liénard differential system has invariant algebraic curves. We present a set of all irreducible invariant algebraic curves for quintic Liénard differential systems with a linear damping function. It is supposed in scientific literature that the degrees of their irreducible invariant algebraic curves are bounded by 6. While we derive irreducible invariant algebraic curves of degree 9.
Абызов А. Н., Буутай П. Н., Математика и теоретические компьютерные науки 2026 Т. 4 № 2 С. 4–75
This paper is expository and methodological in nature and is devoted to the development of E.I. Zolotarev’s ideas embedded in his approach to the proof of the quadratic reciprocity law (1872). We consider extensions of Zolotarev’s approach to abstract number rings presented in the work of A. Brunyate and P.L. Clark (2015), and to finite ...
Added: July 30, 2026
Ponomarenko A., / Series Computer Science "arxiv.org". 2025.
This paper addresses the challenge of merging hierarchical navigable small world (HNSW) graphs, a critical operation for distributed systems, incremental indexing, and database compaction. We propose three algorithms for this task: Naive Graph Merge (NGM), Intra Graph Traversal Merge (IGTM), and Cross Graph Traversal Merge (CGTM). These algorithms differ in their approach to vertex selection ...
Added: July 30, 2026
Mikhaylets E. V., Razorenova A. М., Chernyshev V. L. et al., Scientific Reports 2026 Vol. 16 Article 23560
Meditation offers a naturalistic paradigm for studying introspection, yet the neural dynamics of advanced tantric practices remain largely unexplored. Buddhist Highest Yoga Tantra (BHYT) comprises a sequence of eight dissolution stages culminating in the “clear light” state. We recorded EEG during eyes-closed BHYT meditation performed in monasteries and hermitages (51 sessions from 36 male practitioners; ...
Added: July 29, 2026
Попеленский Ф. Ю., Математический сборник 2026 Т. 217 № 2 С. 108–153
In a recent paper Buchstaber and the author introduced a new structure on the cohomology of Hopf algebras in terms of the Buchstaber spectral sequence (Bss). We fully calculate this structure on the cohomology (known for a long time) of the important Hopf subalgebra A(1) of the classical Steenrod algebra A2.
As part of a demonstration ...
Added: July 28, 2026
Metlov K., Andrei B. Bogatyrëv, Annalen der Physik 2026 Vol. 538 No. 6 Article e70234
Thanks to the recent progress in bulk full three-dimensional nanoscale magnetization distribution imaging, there is a growing
interest to three-dimensional (3D) magnetization textures, promising new high information density spintronic applications.
Compared to 1D domain walls or 2D magnetic vortices/skyrmions, they are a much harder challenge to represent, analyze and
reason about. Here we build analytical representation for such ...
Added: July 28, 2026
Menovshchikov A., Ukhlov A., Rendiconti del Circolo Matematico di Palermo 2026 Vol. 75 Article 91
We consider the nonlinear Neumann eigenvalue problem in outward cuspidal domains with a weighted measure. Using composition operators on Sobolev spaces, we establish embeddings of Sobolev spaces into weighted Lebesgue spaces. These embeddings give the solvability of the Neumann spectral problem in this setting and provide estimates for the corresponding weighted Neumann eigenvalues. ...
Added: July 27, 2026
Menovshchikov A., Journal of Mathematical Sciences 2026 Vol. 298 P. 608–618
We study the set of (p, q)-eigenvalues of the p-Laplace operator with no-flux boundary conditions. We show that this set is closed and that its smallest positive element (the first nontrivial eigenvalue) admits a variational characterization. Moreover, we establish lower bounds for this eigenvalue in cuspidal domains. ...
Added: July 27, 2026
Edward R. Rzaev, Aleksandr Y. Romanov, Andrey M. Sukhov, IEEE Access 2026 Vol. 14 P. 2169–3536
This work presents a hierarchy of strictly local fault-tolerant routing algorithms for 3D mesh networks-on-chip, culminating in an algorithm that combines a live-neighbor selection rule with a bounded single-hop rollback mechanism. The proposed algorithms operate exclusively on immediate neighbor information, maintain O(1) per hop complexity, and require no global topology knowledge, additional virtual channels, or ...
Added: July 23, 2026
Singapore: Springer, 2024.
Included in the following conference series:
INdAM: INdAM Meeting: Kolmogorov Operators and their Applications Workshop
Conference proceedings info: INdAM 2022
Kolmogorov equations are a fundamental bridge between the theory of partial differential equations and that of stochastic differential equations that arise in several research fields.
This volume collects a selection of the talks given at the Cortona meeting by ...
Added: July 17, 2026
Veretennikov A., Pascucci A., Rondelli A., Stochastic Processes and their Applications 2026 Vol. 199 Article 104978
We present existence results for weak solutions to a broad class of degenerate McKean-Vlasov equations with rough coefficients, expanding upon and refining the techniques recently introduced by the third author. Under certain structural conditions, we also establish results concerning both weak and strong well-posedness. ...
Added: July 17, 2026
Veretennikov A., Ляппиева А. А., Теория вероятностей и ее применения 2026 Т. 71 № 2 С. 295–304
Установлен новый результат о сильной единственности для многомерного СДУ с невырожденной диффузией и частично нерегулярным сносом. Его можно рассматривать как комбинированный вариант на темы Ямада и Ватанабэ (1971), Звонкина (1974) и первого автора настоящей статьи (1980). ...
Added: July 17, 2026
Veretennikov A., Нуриева А. И., Доклады Российской академии наук. Математика, информатика, процессы управления (ранее - Доклады Академии Наук. Математика) 2025 Т. 525 С. 24–30
Предложено новое достаточное условие в задаче о центральной предельной теореме в схеме серий для неоднородных цепей Маркова, с возможностью того, что минимум эргодического коэффициента Маркова–Добрушина может быть ближе к нулю, чем в основном условии Добрушина. ...
Added: July 17, 2026
О частных производных модифицированных полиномов Бернштейна–Станку для функций нескольких переменных
Veretennikov A., Мазутский Н. М., Математический сборник 2025 Т. 216 № 7 С. 3–27
Целью работы является доказательство аппроксимации смешанных производных второго порядка для функции нескольких переменных в норме L1 такими же производными модифицированных полиномов Бернштейна–Станку при минимальной возможной регулярности. ...
Added: July 17, 2026
Ахмярова А. Т., Veretennikov A., Теория вероятностей и ее применения 2025 Т. 70 № 2 С. 211–227
В работе предложены новые версии слабого закона больших чисел (ЗБЧ) для слабо зависимых слагаемых (вообще говоря, разнораспределенных) как при наличии математического ожидания каждого из них, так и без такового. Одним из основных условий в первом из трех рассматриваемых случаев, в котором развиваются идеи из статьи Ю. Ш. Чау 1971 г., является равномерная интегрируемость слагаемых по Чезаро в духе работ по ЗБЧ для ...
Added: July 17, 2026
Veretennikov A., Stochastics and Dynamics 2024
A new weak existence result for degenerate multi-dimensional stochastic McKean–Vlasov equation is established under relaxed regularity conditions. ...
Added: July 17, 2026
Ахмярова А. Т., Veretennikov A., Теория вероятностей и ее применения 2024 Т. 69 № 3 С. 427–438
Предложен новый вариант усиленного закона больших чисел для попарно независимых случайных величин. Основная цель — ослабить требование существования математического ожидания каждого из слагаемых. Предположение о попарной независимости также ослаблено. ...
Added: July 17, 2026
Veretennikov A., Moscow Mathematical Journal 2024 Vol. 24 No. 1 P. 107–124
Second order recurrence are established for a d-dimensional diffusion with an additive Wiener process, with switching, and with one recurrent and one transient regime and constant switching intensities, under suitable conditions. As a corollary, the rate of convergence towards the invariant regime of order t^{−2} is claimed. The approach is based on embedded Markov chains ...
Added: July 16, 2026
Veretennikov A., Mathematics 2023 Vol. 11 No. 21 Article 4514
Positive recurrence for a single-server queueing system is established under generalized service intensity conditions, without the assumption of the existence of a service density distribution function, but with a certain integral type lower bound as a sufficient condition. Positive recurrence implies the existence of the invariant distribution and a guaranteed slow convergence to it in ...
Added: July 16, 2026
Veretennikov A., Markov Processes and Related Fields 2023 Vol. 23 No. 2 P. 259–294
The ergodic Bellman's (HJB) equation is proved for a one-dimensional controlled diffusion with switching with variable diffusion and drift coefficients both depending on control; the intensities of transitions of the discrete component are constant. Its existence and uniqueness is established. Also, the convergence of the reward iteration improvement algorithm is established to the cost constant ...
Added: July 16, 2026
On recurrence, convergence and mixing rate for generalised Wright - Fisher's diffusion with mutation
Veretennikov A., Sineokiy R., Markov Processes and Related Fields 2023 Vol. 23 No. 2 P. 241–258
Generalised one-dimensional Fisher -- Wright diffusion process with mutations is consiedered. This is a well-known model in populational genetics. The goal of the paper is an exponential recurrence of the process, which also implies exponential rate of convergence towards the invariant measure. ...
Added: July 16, 2026
Veretennikov A., Mathematics 2023 Vol. 11 No. 14 Article 3096
Polynomial recurrence bounds for a class of stochastic differential equations with a rotational symmetric gradient type drift and an additive Wiener process are established, as well as certain a priori moment inequalities for solutions. The key feature of this paper is that the approach does not use Lyapunov functions because it is not clear how ...
Added: July 16, 2026
Maria V. Demina, Varvara G. Nechitailo, Qualitative Theory of Dynamical Systems 2025 Vol. 24 No. 1 Article 29
We classify generalized Liénard differential equations, sometimes called Liénard type equations, that have integrating factors of a special form. We present first integrals of the related equations. Our families of equations contain almost all known integrable generalized Liénard equations as partial cases. We find the necessary and sufficient conditions of the Liouvillian integrability for the ...
Added: April 7, 2025
Dragovic V., Gontsov R., Goryuchkina I., Annali della Scuola Normale Superiore di Pisa, Classe di Scienze 2023 Vol. 24 No. 4 P. 2201–2213
The existence, uniqueness and convergence of formal Puiseux series solutions of non-autonomous algebraic differential equations of first order at a nonsingular point of the equation is studied, including the case where the celebrated Painleve theorem cannot be applied explicitly for the study of convergence. Several examples illustrating relationships to the Painleve theorem and lesser-known results ...
Added: November 23, 2024
M. V. Demina, D. O. Ilyukhin, Siberian Mathematical Journal 2023 Vol. 64 No. 5 P. 1145–1152
Invariant manifolds are important for describing the dynamics of systems of ODEs. In this article we classify invariant algebraic manifolds in the model of double convection given by a three-dimensional quadratic differential system. We find the explicit solutions to this system whose trajectories lie on invariant manifolds. ...
Added: October 13, 2023