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New Numerical Invariants of an Unfolding of a Polycycle “Tears of the Heart”
Russian Journal of Mathematical Physics. 2026. Vol. 33. No. 1. P. 89–106.
In this paper, new numerical invariants of structurally unstable vector fields in the plane
are found. One of the main tools is an improved asymptotics of sparkling saddle connections that
occur when a separatrix loop of a hyperbolic saddle breaks. Another main tool is a new topological
invariant of two arithmetic progressions, both perturbed and unperturbed, on the real line. For the
pairs of the unperturbed arithmetic progressions, we give a complete topological classification.
Keywords: векторные полябифуркацииvector fields on the spherenumeric invariantsGlobal bifurcationsчисловые инварианты
Publication based on the results of:
Shimanogov I. N., Vyalyi M., Siberian Mathematical Journal 2026 Vol. 67 No. 5 P. 1203–1212
We consider a class of Boolean algebras formed by intersections of regular languages with
a given language. In the case where such an algebra is isomorphic to the algebra of regular languages,
we prove the existence of an isomorphism that is computable using oracles for the regular realizability
problem and the infinite regular realizability problem. This result yields ...
Added: September 25, 2026
Ramazanov I., Bukh A., Shepelev Igor A., Chaos 2026 No. 36 P. 083150–083150
We investigate how stochastic Poisson impulsive forcing influences the spatiotemporal dynamics of a two-dimensional network of Hindmarsh–Rose neurons. Unlike continuous noise, impulsive forcing introduces discrete, state-dependent perturbations, making the system response highly sensitive to both the statistics and the spatial structure of the input. In most of the parameter space, stochastic impulses destabilize the initial ...
Added: September 24, 2026
Levashev V., / Series arXiv "math". 2026. No. 2609.06010.
We prove that continuous A-bilinear pairings on the ring of Laurent series that are invariant under continuous automorphisms coincide, up to a constant, with the pairing given by the residue of a differential form over any commutative associative ring with identity. ...
Added: September 24, 2026
Sokolov V., Adler V. E., Journal of Geometry and Physics 2026 Vol. 227 Article 105860
The group reduction procedure is applied to vector generalizations of the NLS, mKdV,
and KdV equations. The resulting ODE systems admit isomonodromic Lax representations
and are multicomponent generalizations of the Painlevé equations P 1, P 2, P 34, and P 4.
Some of them can be interpreted as nonautonomous deformations of well-known systems
integrable in the Liouville sense, in ...
Added: September 24, 2026
Sokolov V., BALAKHNEV M. Y., Ufa Mathematical Journal 2026 Vol. 18 No. №3 P. 85–93
A collection of miscellaneous continuous, semi-discrete, and discrete integrable
systems can be associated with each integrable evolution equation of the KdV type. We provide them for the Schwarz — KdV equation and generalize to the vector case. The existence
of these vector generalizations is a non-trivial found fact, no mathematical explanation of
which is known yet. ...
Added: September 24, 2026
Yakovlev E., Maksimov D. A., Mathematical notes 2026 Vol. 120 No. 3 P. 483–496
Smooth principal bundles whose total spaces and bases are time oriented Lorentzian manifolds and whose projections are Lorentzian submersions preserving time orientations are studied. Previously, the authors showed that the chronologicity, causality, and stable causality always lift from the base to the space of a Lorentzian bundle. For strong causality and global hyperbolicity, this holds ...
Added: September 24, 2026
Sokolov V., Shabat G. B., Tsiganov A. V., Journal of Geometry and Physics 2026 Vol. 220
We consider Novikov equations for commutative ring generated by differential operators of
orders 3,4,5. We present an explicit Hamiltonian form of these equations. Using the method
of compatible Poisson brackets, we find a separation of variables on a hyperelliptic curve
of genus 2 for the Novikov equations ...
Added: September 24, 2026
Marshakov A., Yung A., Ievlev E. et al., Physical Review D - Particles, Fields, Gravitation and Cosmology 2026 No. 114 P. 1–22
We continue the study of non-Abelian vortex string in 4D N ¼ 2 supersymmetric QCD (SQCD) as critical superstring, and extend this analysis to UðNÞ gauge theory with arbitrary even N and Nf ¼ 2N number of quarks. We introduce a special mass deformation and show that the SQCD hadron spectrum is still given by ...
Added: September 24, 2026
Demina M.V., Nechitailo V., Analysis and Mathematical Physics 2026 Vol. 16 No. 5 P. 1–25
We present a method of finding non-Liouvillian first integrals of rational two-dimensional differential systems. The method is based on the existence of two independent invariants that satisfy a linear second-order ordinary differential equation with respect to one of the variables. We call systems with this property R-integrable. These invariants are not necessarily polynomial; they can ...
Added: September 24, 2026
Bitter I., Konakov V., Mathematical notes 2026 Vol. 120 No. 4 P. 599–615
The paper provides a generalization of the local limit theorem on the convergence
of inhomogeneous Markov chains to the diffusion limit for the case in which the corresponding
coefficients of the process satisfy weak regularity conditions and coincide only asymptotically.
In particular, the drift coefficients under consideration can be unbounded with at most linear
growth, and the bounds reflect ...
Added: September 24, 2026
Pyatov P. N., Pivovarov P. A., / Series math "arxiv.org". 2026. No. 2609.06274.
We investigate a special ansats that allows for an iterative solution of the constant Yang-Baxter equation. Testing this ansatz, we construct four sequences of the constant R-matrices. In each sequence the R-matrices act on the tensor squares of vector spaces of linearly growing dimensions. Each R-matrix also depends on a single complex parameter.
By analyzing the ...
Added: September 24, 2026
Дильмухаметова Алия Мидхатовна, Напалков В. В., Муллабаева А. У., Уфимский математический журнал 2010 Т. 2 № 1 С. 52–58
В данной статье введены обобщённые пространства Фока и рассмотрены основные свойства этих пространств. Найдена операция, сопряженная к операции умножения на переменную в обобщенном пространстве Фока. Также определены собственные функции сопряженного оператора. Изучены обобщенное преобразование Лапласа и задача построения базиса для введенных пространств. ...
Added: September 21, 2026
Aleksei Samarin, Alexander Savelev, Aleksei Toropov et al., Pattern Recognition and Image Analysis 2024 Vol. 34 No. 3 P. 855–862
Tasks related to the automation of medical data processing are becoming more urgent. Particular attention is paid to systems for monitoring and analyzing human physiological parameters. Such systems often use specialized sensors to capture biomedical images, such as infrared cameras. This article describes our study of the problem of segmenting the eye pupil and iris ...
Added: September 21, 2026
Aleksei Samarin, Nazarenko A., Alexander Savelev et al., Pattern Recognition and Image Analysis 2024 Vol. 34 No. 3 P. 844–854
Improving image quality is becoming an increasingly popular task, especially when working with mobile devices. One common approach to image enhancement is the use of convolutional neural networks. However, to achieve good results, such networks must be large enough, otherwise there is a risk of unwanted artifacts. In addition, large convolutional neural networks require significant ...
Added: September 21, 2026
Aleksei Samarin, Alexander Savelev, Aleksei Toropov et al., Optical Memory and Neural Networks (Information Optics) 2024 Vol. 33 P. 424–434
This study explores the development of classifiers for microbial images, specifically focusing on streptococci captured via microscopy of live samples. Our approach uses AutoML-based techniques and automates the creation and analysis of feature spaces to produce optimal descriptors for classifying these microscopic images. This technique leverages interpretable taxonomic features based on the external geometric attributes ...
Added: September 21, 2026
Aleksei Samarin, Aleksei Toropov, Alexander Savelev et al., Pattern Recognition and Image Analysis 2024 Vol. 34 No. 4 P. 1053–1060
This paper presents a novel approach to classification in biomedical imaging, specifically targeting polyp recognition in video endoscopy snapshots. Our method leverages specialized image descriptors to enhance the accuracy and robustness of polyp recognition. By employing these specialized descriptors, we address the challenges inherent in analyzing biomedical images from open datasets. Our approach not only ...
Added: September 21, 2026
Самарин А. В., Торопов А. Г., Савельев А. Г. et al., Pattern Recognition and Image Analysis 2024 Vol. 34 No. 4 P. 1044–1052
This paper presents a study aimed at improving the detection quality of small-sized microorganisms under challenging microscopic conditions through the application of a lightweight combined image preprocessing model. We focused on the task of detecting diplococci in images obtained through dynamic sample microscopy. The proposed approach employs predefined filters for image preprocessing, combined with the ...
Added: September 21, 2026
Singapore: Springer, 2025.
Added: September 21, 2026
Dorovskiy A., / Series arXiv "math". 2026.
In this paper the structural stability of generic families of vector fields of the PC-HC class on the two-dimensional sphere is proved. A classification of these families up to moderate equivalence in neighborhoods of their large bifurcation supports is presented, based on such invariants as the configuration and the characteristic set. The realization lemma is proved. ...
Added: May 14, 2026
Kilin A. A., Ivanova T. B., Lobachevskii Journal of Mathematics 2025 No. 46 P. 1113–1138
In this paper, we address the problem of the rolling motion of a sphere with axisymmetric mass distribution on a horizontal plane. It is assumed that the sphere does not slip as it rolls in the direction of the projection of the symmetry axis onto the supporting plane. The system under consideration admits a redundant ...
Added: December 10, 2025
Kazakov A., Koryakin V., Safonov K. et al., / Series arXiv "math". 2025.
The Lorenz attractor is the first example of a robustly chaotic non-hyperbolic attractor. Each orbit of such an attractor has a positive top Lyapunov exponent, and this property persists under small perturbations despite possible bifurcations of the attractor. In this paper, we study the boundary of the Lorenz attractor existence region in the Shimizu-Morioka model. ...
Added: December 4, 2025
Никулин М. А., Попеленский Ф.Ю., Математический сборник 2025 Т. 216 № 10 С. 101–158
The paper introduces a novel integrable system within an ellipse. The interior of the ellipse is divided into subdomains by arcs of confocal quadrics, each subdomain is filled with a medium characterized by a constant coefficient of «optical» density. Upon traversing the interface between these media, the trajectory adheres to the «cosine» law of refraction. ...
Added: October 3, 2025
Karatetskaia E., Aikan Shykhmamedov, Konstantin Soldatkin et al., Regular and Chaotic Dynamics 2025 Vol. 30 No. 2 P. 306–324
We study hyperchaotic attractors characterized by three positive Lyapunov exponents in numerical experiments. In order to possess this property, periodic orbits belonging to the attractor should have a three-dimensional unstable invariant manifold. Starting with a stable fixed point we describe several bifurcation scenarios that create such periodic orbits inside the attractor. These scenarios include cascades ...
Added: May 13, 2025
http://www.shilnikov.unn.ru/en/news.html?id=20, 2020.
International Conference "ShilnikovWorkshop-2020" dedicated to the memory of the outstanding Russian mathematician Leonid Pavlovich Shilnikov (1934-2011) will be held on 17-18 December, 2020 at the Lobachevsky State University of Nizhny Novgorod. The topics of the Conference include but not restricted by the following themes of the theory of dynamical systems: bifurcations, strange attractors, conservative and ...
Added: November 1, 2021