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Vol 88, No 6 (2024)

Year: 2024
Articles: 9
URL: https://journals.rcsi.science/1607-0046/issue/view/17718

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Articles

Schauder's fixed point theorem and Pontryagin maximum principle

Avakov E.R., Magaril-Il'yaev G.G.

Abstract

The paper derives Pontryagin maximum principle for the general optimal control problem, where the main tool is the abstract inverse function lemma, the proof of which is essentially based on Schauder's fixed point theorem. This approach allows us to make the proof of Pontryagin maximum principle quite short and very transparent.

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Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2024;88(6):3-22

3-22

(Rus)

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Chebyshev sets composed of a union of subspaces in asymmetric normed spaces

Alimov A.R., Tsar'kov I.G.

Abstract

By definition, a Chebyshev set is a set of existence and uniqueness, that is, any point has a unique best approximant from this set. We study properties of Chebyshev sets composed of finitely or countably many planes (closed affine subspaces, possibly degenerated to points). We show that a finite union of planes is a Chebyshev set if and only if is a Chebyshev plane. Under some conditions on a space or a set, we show that a countable union of planes is never a Chebyshev set (unless this union is a Chebyshev plane itself). As a corollary, we give the following partial answer to the Efimov–Stechkin–Klee problem on convexity of Chebyshev sets in Hilbert spaces: in such spaces (and, more generally, in reflexive CLUR-spaces), an at most countable union of planes is a Chebyshev set if and only if this set is a Chebyshev plane. Results of this kind are obtained both in usual normed linear spaces and in spaces with asymmetric norm.

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Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2024;88(6):23-43

23-43

(Rus)

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Uniqueness of solutions of generalized convolution equations on the hyperbolic plane and the group $\mathrm{PSL}(2,\mathbb{R})$

Volchkov V.V., Volchkov V.V.

Abstract

The paper is devoted to the study of the uniqueness problem for convolution equations ongroups of motions of homogeneous spaces. The main results concern the case of the group of motions $G=PSL(2,\mathbb{R})$ of the hyperbolic plane $\mathbb{H}^2$ and are as follows:1) the uniqueness theorems of the John type for solutions of convolution equations on the group${G}$ are proved;2) the exact conditions for the uniqueness of the solution of the system of convolution equationson domains in $G$ are found.To prove these results, a technique is developed based on the study of generalized convolutionequations on $\mathbb{H}^2$. These equations, in turn, are investigated with the help oftransmutation operators of a special kind constructed in the work. The proposed method also allowsus to establish a number of other results related to generalized convolution equations on $\mathbb{H}^2$ and the group ${G}$.

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Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2024;88(6):44-81

44-81

(Rus)

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Cone criterion on an infinite-dimensional torus

Glyzin S.D., Kolesov A.Y.

Abstract

On the infinite-dimensional torus $\mathbb{T}^{\infty} = E/2\pi\mathbb{Z}^{\infty}$, where $E$ is an infinite-dimensional real Banach space, $\mathbb{Z}^{\infty}$is an abstract integer lattice, a special class of diffeomorphisms $\mathrm{Diff}(\mathbb{T}^{\infty})$ is considered. This class consists of the mappings $G\colon \mathbb{T}^{\infty} \to \mathbb{T}^{\infty}$ such that the differentials $DG$ and $D(G^{-1})$ are uniformly bounded and uniformly continuous on $\mathbb{T}^{\infty}$. For diffeomorphisms from $\mathrm{Diff}(\mathbb{T}^{\infty})$, we establish the validity of the so-called cone criterion, which is a classical result of finite-dimensional hyperbolic theory (that is, the hyperbolicity criterion formulated in terms of fields of invariant horizontal and vertical cones).

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Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2024;88(6):82-117

82-117

(Rus)

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On low-dimensional bases of natural fibrations for compact homogeneous spaces

Gorbatsevich V.V.

Abstract

The article gives a description of bases of dimension $\le 5$ for natural fibrations of compact homogeneous spaces.The consideration is carried out up to commensurability (generated by transitions to finite coverings).The questions of decomposability and countability of certain commensurability classes of such bases are discussed.

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Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2024;88(6):118-138

118-138

(Rus)

(JATS XML)

Superposition of layers of the cubic lattice

Grishukhin V.P.

Abstract

A cube is a Dirichlet–Voronoi cell of the integer lattice $Z^n$. A sequence of $(n+1)$-dimensional lattices $L^{n+1}(h)$ is studied. These lattices are obtained by a superposition of layers of the cubic lattice $Z^n$ at distance $h$ between layers. A sequence of quadratic forms $f_h$ corresponds to the sequence of the lattices $L^{n+1}(h)$. As $h$ changes from zero to infinity the sequence of the form $f_h$ pierces the second perfect domain of quadratic forms from one boundary to another passing some edge forms.

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Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2024;88(6):139-156

139-156

(Rus)

(JATS XML)

Residually linear abstract groupoids

Draoui K., Choulli H., Mouanis H.

Abstract

We introduce the notion of residually linear groupoids. We characterize this class in analogy with the group-theoretic setting. Various properties are proved and a relationship with residual finiteness is investigated. From a categorical point of view, our approach extends some well-known results in the theory of discrete groups, due mainly to Mal'cev and Menal. Finally, as an application, we show that the character groupoid of the Hopf algebroid of representative functions of a transitive groupoid is always residually linear.Bibliography: 24 titles.

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Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2024;88(6):157-175

157-175

(Eng)

(JATS XML)

On the Poincaré problem of the third integral of the equations of rotation of a heavy asymmetric top

Kozlov V.V.

Abstract

The Poincaré problem of the existence of the third integral of the equations of motion of a heavy asymmetric rigid body with a fixed point, which is independent of the integrals of energy and the area integral and which is represented as a series in powers of a small parameter with coefficients in the form of single-valued analytic functions on a six-dimensional phase space, is considered. The small parameter is the ratio of the distance from the center of mass to the suspension point to the characteristic size of the rigid body. This problem was formulated by Poincaré in the fifth chapter of his famous "New Methods of Celestial Mechanics". If we additionally require that the third integral is in involution with the area integral, then the answer to the Poincaré problem is negative (as was shown by the author back in 1975). In this paper, the Poincaré problem is solved in the original general formulation (without the assumption that the Poisson bracket is zero): if the body is dynamically asymmetric, then the third single-valued integral does not exist. The proof uses the Poincaré method, supplemented by some new ideas, as well as a more thorough analysis of the expansion of the perturbing function in a Fourier series in angular variables.

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Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2024;88(6):176-189

176-189

(Rus)

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On $T$-maps and ideals of antiderivatives of hypersurface singularities

Shi Q., Yau S.S., Zuo H.T.

Abstract

Mather–Yau's theorem leads to an extensive study about moduli algebras of isolated hypersurface singularities. In this paper, the Tjurina ideal is generalized as $T$-principal ideals of certain $T$-maps for Noetherian algebras. Moreover, we introduce the ideal of antiderivatives of a $T$-map, which creates many new invariants. Firstly, we compute two new invariants associated with ideals of antiderivatives for ADE singularities and conjecture a general pattern of polynomial growth of these invariants.Secondly, the language of $T$-maps is applied to generalize the well-known theorem that the Milnor number of a semi quasi-homogeneous singularity is equal to that of its principal part. Finally, we use the $T$- fullness and $T$-dependence conditions to determine whether an ideal is a $T$-principal ideal and provide a constructive way of giving a generator of a $T$-principal ideal. As a result, the problem about reconstruction of a hypersurface singularitiy from its generalized moduli algebras is solved. It generalizes the results of Rodrigues in the cases of the $0$th and $1$st moduli algebra, which inspired our solution.Bibliography: 24 titles.

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Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2024;88(6):190-226

190-226

(Eng)

(JATS XML)

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