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Exact Solutions of One Nonlinear Countable-Dimensional System of Integro-Differential Equations
Журнал Средневолжского математического общества. 2023. Vol. 25. No. 1. P. 542–533.
A. E. Rassadin
In the present paper, a nonlinear countable-dimensional system of integrodifferential equations is investigated, whose vector of unknowns is a countable set of functions of two variables. These variables are interpreted as spatial coordinate and time. The nonlinearity of this system is constructed from two simultaneous convolutions: first convolution is in the sense of functional analysis and the second one is in the sense of linear space of double-sided sequences. The initial condition for this system is a doublesided sequence of functions of one variable defined on the entire real axis. The system itself can be written as a single abstract equation in the linear space of double-sided sequences. As the system may be resolved with respect to the time derivative, it may be presented as a dynamical system. The solution of this abstract equation can be interpreted as an approximation of the solution of a nonlinear integro-differential equation, whose unknown function depends not only on time, but also on two spatial variables. General representation for exact solution of system under study is obtained in the paper. Also two kinds of particular examples of exact solutions are presented. The first demonstrates oscillatory spatio-temporal behavior, and the second one shows monotone in time behavior. In the paper typical graphs of the first components of these solutions are plotted. Moreover, it is demonstrated that using some procedure one can generate countable set of new exact system’s solutions from previously found solutions. From radio engineering point of view this procedure just coincides with procedure of upsampling in digital signal processing.
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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)∗ – ...
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Basalaev A., Rarovskii A., Journal of Singularities 2026 Vol. 30 P. 61–80
Saito theory associates to an isolated singularity rich structure that plays an important role in mirror symmetry. In this note we construct Saito theory for A and D type Landau-Ginzburg orbifolds. Namely, for the pairs (f,G), where f defines an isolated singularity of A and D type and G is a group of symmetries of ...
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Rybakov M., Shkatov D., Journal of Logic and Computation 2026 Vol. 36 No. 6 Article exag026
We prove Pi-1-1-hardness, and thus lack of recursive axiomatizability, of constant-domain modal predicate logics defined by a class of Dedekind complete linear Kripke frames containing a frame with an infinitely increasing chain of worlds. The result holds even for the language with one unary predicate letter, one propositional letter, and two individual variables. ...
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Minkowski mixed volume of n subpolytopes D1,…,Dn of a polytope P⊂Rn clearly does not exceed the normalized volume n!Vol(P). Equality holds if and only if the subpolytopes are interlaced, i.e., each proper face F⊊P intersects at least dim(F)+1 of the polytopes Di. Efficiently computing mixed volumes for more general collections of subpolytopes is crucial for estimating the complexity of numerically solving polynomial systems.
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Kazaryan M., Dunin-Barkowski P., Bychkov B. et al., International Mathematics Research Notices 2026 Vol. 14 Article rnag146
We prove a recent conjecture of the fourth named author with P. Norbury that states a system of universal polynomial relations among the kappa classes on the moduli spaces of algebraic curves. The proof involves localization and materialization analysis of the spin Gromov–Witten theory of the projective line and is dictated by Z 2 -equivariant ...
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We prove that for any initial data on a genus zero spectral curve the cor responding correlation differentials of topological recursion are KP integrable. As an application we prove KP integrability of partition functions associated via ELSV-type formulas to the r-th roots of the twisted powers of the log canonical bundles ...
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Механизм происходящих в мире изменений носит эволюционный, а не экологический характер. Иначе говоря, Человечество столкнулось с кризисом развития, который имеет три независимые составляющие: кризис индустриального общества (фазовый кризис), кризис научного мышления (эпистемный кризис) и кризис формата существования разума (социосистемный кризис). Доклад посвящён аспектам этого триединого кризиса и возможным путям его преодоления, не сводящимся к первичному ...
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Devyatov R. A., Mathematical notes 2026 Vol. 119 No. 3 P. 782–786
Let G/B be a flag variety over ℂ, where G is a simple algebraic group with a simply laced Dynkin diagram, and B is a Borel subgroup. We say that the product of classes of Schubert divisors in the Chow ring is multiplicity free if it is possible to multiply it by a Schubert class ...
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We give a self-contained and simplified proof of Mukai’s classification of prime Fano threefolds of index 1 and genus g ≥ 6 with at most factorial terminal singularities, and of its extension to higher dimension. ...
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Bayer A., Kuznetsov A., Macrì E., Compositio Mathematica 2026 Vol. 162 No. 1 P. 59–99
We give a proof of Mukai’s theorem on the existence of certain exceptional vector bundles on prime Fano threefolds. To our knowledge this is the first complete proof in the literature. The result is essential for Mukai’s biregular classification of prime Fano threefolds, and for the existence of semiorthogonal decompositions in their derived categories. Our ...
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Guseva L., Novikov A., Advances in Mathematics 2026 Vol. 503 Article 111211
We prove that the Kuznetsov–Polishchuk exceptional collections on rational homogeneous spaces of the symplectic groups Sp(2n,C) are full and consist of vector bundles. To achieve this, we construct several classes of complexes, which we call generalized staircase complexes, symplectic staircase complexes and secondary staircase complexes — each of which may be of independent interest. ...
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Polishchuk A., Rains E., Journal of the Institute of Mathematics of Jussieu 2026 Vol. 25 No. 1 P. 339–373
We prove that for every relatively prime pair of integers (d,r) with r>0, there exists an exceptional pair (O,V) on any del Pezzo surface of degree 4, such that V is a bundle of rank r and degree d. As an application, we prove that every Feigin-Odesskii Poisson bracket on a projective space can be ...
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We continue the study of automorphic functions associated with a curve C over the ring k[ε]/(ε²), where k is a finite field, begun in arXiv:2303.16259. Namely, we study an example of theta-lifting in this framework and show that it can be understood in terms of the orbit decomposition of the space of automorphic functions S(SL₂(F)\SL₂(A_C)) ...
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Kuznetsov A. P., Sataev I. R., Stankevich N., Chaos 2026 Vol. 36 No. 8 Article 083131
A radio-physical system, namely, a two-mode van der Pol generator, is considered. It is shown that this system demonstrates two types of chaos: with one and two zero Lyapunov exponents. The regions of the second type of chaos are surrounded by bifurcation lines of invariant tori doubling. Quasi-periodic structures of various shapes are embedded within ...
Added: August 30, 2026
Shirokov N. A., Rozenblum G., Israel Journal of Mathematics 2026 P. 1–30
We establish that a generalized H\¨older continuous function on an (m−2)-Ahlfors regular compact set in Rm can be approximated by solutions of an elliptic equation, with the rate of approximation determined by the continuity modulus of the function ...
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E. M. Novikova, Mathematical notes 2025 Vol. 118 No. 4 P. 794–810
For each integer nonnegative n, in some Hilbert space, we introduce an integral
transform H_n. It is similar to the well-known Hankel (Fourier–Bessel) transform of nth order due
to the fact that it is related to the Fourier transform and its integral kernel is expressed in terms of
the Bessel function J_n. But, unlike the Hankel transform, intended ...
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Рассадин А. Э., Известия высших учебных заведений. Прикладная нелинейная динамика 2024 Т. 32 № 6 С. 908–920
The purpose of this work is to propose and investigate a simple and effective model of an epidemic in an animal population that takes into account migration along the plane of both diseased and healthy individuals. Within the framework of this model, the spatial migration of a population is described by introducing both diffusion and ...
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Вишняков Г. Н., Yurin A., Minaev V. et al., Известия высших учебных заведений. Прикладная нелинейная динамика 2023 Т. 31 № 4 С. 408–420
The purpose of this work is to analyze the effect of the polarimeter signal processing algorithm on the results of measurements of the optical rotation angle of the polarization plane to improve the accuracy of measurements in differential polarimetry. Methods. The paper considers the methods of polarimetry used for the analysis of optically active substances, ...
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This paper is devoted to a new method for constructing approximations to the solution of a parabolic partial differential equation. The Cauchy problem for the heat equation on a straight line with a variable heat conduction coefficient is considered. In this paper, a sequence of functions is constructed that converges to the solution of the ...
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In a number of previous papers authors have introduced quasiparticle of radio- and optical systems. We have called this quasiparticle by ‘radion’. The basis for this is the representation of Green’s function of equation of quasioptics by Feynman integral. It means that radion has quantum mechanical properties. In particular in approximation of quasioptics one can ...
Added: December 17, 2022
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V. I. Bogachev, T. I. Krasovitskii, S. V. Shaposhnikov, Doklady Mathematics 2020 Vol. 102 No. 3 P. 464–467
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В. П. Спиридонов, Теоретическая и математическая физика 2022 Т. 212 № 3 С. 403–413
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The article introduces powerful common tool for the signals analysis – basics of a spectral analysis.
It focuses on decomposition into complex sine waves (harmonic sinusoids) called the Fourier transform.
Based on qualitative review of the natural phenomenon of visible light (that is the electromagnetic field oscillation in a working range of the eyes) decomposition into a ...
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