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Manifolds of isospectral matrices and Hessenberg varieties
International Mathematics Research Notices. 2021. Vol. 2021. No. 21. P. 16671–16692.
Ayzenberg Anton, Buchstaber V.M.
We study the space Xh of Hermitian matrices having staircase form and the given simple spectrum. There is a natural action of a compact torus on this space. Using generalized Toda flow, we show that Xh is a smooth manifold and its smooth type is independent of the spectrum. Morse theory is then used to show the vanishing of odd degree cohomology, so that Xh is an equivariantly formal manifold. The equivariant and ordinary cohomology of Xh are described using GKM-theory. The main goal of this paper is to show the connection between the manifolds Xh and the semisimple Hessenberg varieties well-known in algebraic geometry. Both the spaces Xh and Hessenberg varieties form wonderful families of submanifolds in the complete flag variety. There is a certain symmetry between these families which can be generalized to other submanifolds of the flag variety.
Смирнов С. В., Миллионщиков Д. В., Уфимский математический журнал 2021 Т. 13 № 2 С. 44–73
В данной работе изучаются характеристические алгебры для систем экспоненциального типа, соответствующих вырожденным матрицам Картана. Эти системы обобщают хорошо известные в теории интегрируемых систем гиперболические уравнения синус-Гордон и Цицейки. Для таких систем, соответствующих матрицам Картана ранга 2, характеристические алгебры описаны явно в терминах образующих и соотношений, и доказано, что они имеют линейный рост. Исследуется связь между ...
Added: September 11, 2026
Смирнов С. В., Glasgow Mathematical Journal 2026 Vol. 68 No. 2 P. 299–316
In the second half of the 19th century Darboux obtained determinant formulae that provide the general solution for a linear hyperbolic second order PDE with finite Laplace series. These formulae played an important role in his study of the theory of surfaces and, in particular, in the theory of conjugate nets. During the last three ...
Added: September 11, 2026
Смирнов С. В., Journal of Physics A: Mathematical and Theoretical 2023 Vol. 56 No. 26 Article 265204
There are different methods of discretizing integrable systems. We consider semi-discrete analog of two-dimensional Toda lattices associated to the Cartan matrices of simple Lie algebras that was proposed by Habibullin in 2011. This discretization is based on the notion of Darboux integrability. Generalized Toda lattices are known to be Darboux integrable in the continuous case ...
Added: September 11, 2026
Enatskaya N., Труды Карельского научного центра РАН. Серия 10: Математическое моделирование и информационные технологии 2026 № 6 С. 132–138
In schemes S for placing r particles in n distinguishable cells, their placement in selected m cells – schemes S∗ is studied, for which the analysis is carried out by a new enumerative method (EM) in extended areas of enumerative combinatorics: finding the number of outcomes and, based on the construction of a model of ...
Added: September 11, 2026
Enatskaya N., Труды Карельского научного центра РАН. Серия 10: Математическое моделирование и информационные технологии 2026 № 6 С. 139–147
The class of particle cell arrangements under consideration is characterized by the introduction of an upper limit on the filling levels of the cells, which must be achieved in at least one cell of each outcome of each arrangement. The circuits are distinguished by the paired qualities of their constituent elements (cells and particles) according ...
Added: September 11, 2026
Podinovskiy V. V., Нелюбин А. П., Автоматика и телемеханика 2026 № 8 С. 110–123
Для многокритериальных задач принятия решений по аналогии с качественной вероятностью введены понятия полной и частичной качественной важности как бинарных отношений, обладающих постулируемыми
свойствами. Предложено новое определение отношения нестрогого предпочтения на множестве вариантов решений, порождаемое качественной
важностью. Исследованы его свойства. Указаны аналитические правила,
позволяющие попарно сравнивать варианты по предпочтительности. Проведено сравнение новых отношений предпочтения с разработанными ранее для задач, ...
Added: September 9, 2026
Podinovskiy V. V., Нелюбин А. П., Автоматика и телемеханика 2026 № 7 С. 113–126
Рассматриваются задачи принятия решений, когда предпочтения оцениваются в порядковой шкале, а возможности реализации значений
неопределенного фактора описываются качественной вероятностью (полной или только частичной). Вводятся определения отношений предпочтения и безразличий на множестве стратегий. Предлагаются простые
решающие правила, позволяющие сравнивать стратегии по предпочтительности, и приводятся иллюстративные примеры. ...
Added: September 9, 2026
Chemical Papers 2026
Benzenoid hydrocarbons are structurally regular aromatic compounds, which makes them well suited for quantitative structure–property relationship (QSPR) studies. This study presents a QSPR analysis of nineteen benzenoid hydrocarbon species using degree-based topological co-indices. We developed a Python program to compute eleven degree-based topological co-indices from molecular graphs and evaluated their ability to explain variations in ...
Added: September 8, 2026
Glutsyuk A., / Series arXiv "math". 2026.
B.Josephson (Nobel Prize, 1973) predicted a tunnelling effect for a system of two superconductors separated by a narrow dielectric (such a system is called Josephson junction): existence of a supercurrent through it and equations governing it. The overdamped Josephson junction is modeled by the family of differential equations on the 2-torus, dθdτ=1ω(cosθ+B+Acosτ), which is known as ...
Added: September 8, 2026
Glutsyuk A., / Series arXiv "math". 2026.
A planar dual billiard is a planar curve γ equipped with a family (σP)|P∈γ of projective involutions of the projective lines LP tangent to γ at P that fix P. A dual billiard is called rationally integrable, if there exists a rational function R(x,y) of two variables (called first integral) whose restriction to each tangent ...
Added: September 8, 2026
Alexandrov A., Glutsyuk A., Journal of Differential Equations 2026 Vol. 465 Article 114178
The overdamped Josephson junction in superconductivity theory can be modeled by the family of dynamical systems on the torus, which is known as the RSJ model. This family admits an equivalent description by a family of second-order differential equations: special double confluent Heun equations. In the present paper, we construct two new families of dynamical systems on ...
Added: September 8, 2026
Glutsyuk A., Inventiones Mathematicae 2026 Vol. 245 P. 977–1058
A caustic of a strictly convex planar bounded billiard is a smooth curve whose tangent lines are reflected from the billiard boundary to its tangent lines. The famous Birkhoff Conjecture, studied by many mathematicians, states that if the billiard boundary has an inner neighborhood foliated by closed caustics, then it is an ellipse. In the paper we study ...
Added: September 8, 2026
Prikhodko Artem, Kubrak D., Compositio Mathematica 2026 Vol. 162 No. 6 P. 1377–1438
In this follow-up paper we show that smooth Hodge-proper stacks over O𝐾 are ℚ𝑝-locally acyclic: namely the natural map between étale ℚ𝑝-cohomology of the algebraic and Raynaud generic fibers is an equivalence. This establishes the ℚ𝑝-case of general conjectures made in D. Kubrak and A. Prikhodko [p-adic Hodge theory for Artin stacks, Mem. Amer. Math. ...
Added: September 7, 2026
Ismailov A., Constructive Approximation 2026
The measure of the positivity set {x ∈ [0; 2π] | f (x) > 0} of a trigonometric polynomial f is bounded from below by the Motzkin density.
We generalize the bound to polynomials in several variables and almost periodic functions.
We then use these generalizations to extend known results on Taikov’s problem. ...
Added: September 7, 2026
Kelbert M., Statistics 2026 Vol. 60
We present variational representations for the weighted divergencies
and exponential error function, and discuss implications for the
statistical inference and entropic optimal transport. ...
Added: September 7, 2026
Kucheryavyy P., Математические заметки 2026 Т. 2026 № 120 С. 380–401
В работе изучаются перестановки, возникающие при упорядочивании по возрастанию дробных долей произведений элементов фиксированной целочисленной последовательности на вещественный параметр. Исследуется количество различных перестановок, которые можно получить таким образом при изменении этого параметра от нуля до единицы. ...
Added: September 7, 2026
Осипов Д.В., Математический сборник 2026 Т. 217 № 9 С. 130–146
Изучаются законы взаимности, связанные с комплексными линейными расслоениями на расслоениях на ориентируемые окружности. В частности, доказывается следующий закон взаимности. Пусть B – комплексное многообразие и πi:Mi→B – расслоение на ориентируемые окружности, где индекс i пробегает конечное множество. Пусть Li и Ni – комплексные линейные расслоения на каждом многообразии Mi. Закон взаимности утверждает, что сумма всех элементов (πi)∗(c1(Li)∪c1(Ni)), где (πi)∗ – ...
Added: September 3, 2026
Petya Pushkar, Misha Temkin, International Mathematics Research Notices 2023 Vol. 00 P. 1–67
Morse function is called strong if all its critical points have different critical values. Given such a function f and a field F Barannikov constructed a pairing of some of the critical points of f , which is now also known as barcode. With every Barannikov pair (a.k.a. bar in the barcode), we naturally associate ...
Added: January 31, 2023
Pushkar P. E., Тёмкин М. С., Доклады Российской академии наук. Математика, информатика, процессы управления (ранее - Доклады Академии Наук. Математика) 2022 Т. 507 № 1 С. 57–60
Let f be a Morse function on a manifold , such that all its critical values are pairwise distinct. Given such
a function (together with a certain choice of orientations) and a field we construct a set of non-zero elements of the field, which are called Bruhat numbers. Under certain acyclicity conditions on alternating
product of all ...
Added: January 31, 2023
Panov T., Зейникешева И. К., Труды Математического института им. В.А. Стеклова РАН 2022 Т. 317 С. 157–167
We compute the equivariant cohomology $H^*_{T_I}(Z_K)$ of moment-angle complexes $Z_K$ with respect to the action of coordinate subtori $T_I \subset T^m$. We give a criterion for the equivariant formality of $Z_K$ and obtain specifications for the cases of flag complexes and graphs. ...
Added: November 11, 2022
Ayzenberg A., Масуда М., Сато Т., Труды Математического института им. В.А. Стеклова РАН 2022 Т. 317 С. 5–26
We describe the second cohomology of a regular semisimple Hessenberg variety by generators and relations explicitly in terms of GKM theory. The cohomology of a regular semisimple Hessenberg variety becomes a module of a symmetric group Sn by the dot action introduced by Tymoczko. As an application of our explicit description, we give a formula describing the ...
Added: October 27, 2022
Petr Pushkar, Misha Tyomkin, / Series 2012.05307 "arxiv.org". 2020.
Consider the set of all rectangular n × m matrices with entries in a field. Recall that unitriangular group Tn consists of upper triangular matrices with 1’s on the diagonal. The product Tn ×Tm naturally acts on the aforementioned set: X 7→ AXB−1 . Our first observation is that each orbit of this action contains ...
Added: June 22, 2021
Soukhanov L., Journal of Geometry and Physics 2020 Vol. 149 P. 1–9
We develop the formal analogue of the Morse theory for a pair of commuting gradient-like vector fields. The resulting algebraic formalism turns out to be very similar to the algebra of the infrared of Gaiotto, Moore and Witten (see Gaiotto et al., and Kapranov et al.): from a manifold M with the pair of gradient-like ...
Added: May 10, 2020
Kurochkin S. V., Doklady Mathematics 2020 Vol. 101 No. 1 P. 62–65
It is proved that an artificial neural network with smooth activation functions and without bottlenecks
is a Morse function for almost all, with respect to the Lebesgue measure, sets of weights. ...
Added: April 20, 2020