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Abstract: We prove a continuity criterion for locally bounded finite-dimensional representations of simply connected solvable Lie groups. PubDate: 2022-06-01

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Abstract: We introduce a classification of locally compact Hausdorff topological spaces with respect to the behavior of \( \sigma \) -compact subsets and, relying on this classification, we study properties of the corresponding \(C^*\) -algebras in terms of frame theory and the theory of \( {\mathscr A} \) -compact operators in Hilbert \(C^*\) -modules; some pathological examples are constructed. PubDate: 2022-06-01

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Abstract: In this paper, order estimates for the Kolmogorov widths of sets defined by restrictions on the norm in the Sobolev weighted space \(W^r_{p_1}\) and the weighted space \(L_{p_0}\) are obtained. PubDate: 2022-06-01

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Abstract: The paper is concerned with singularities of classical solutions to the eikonal equation. With this aim in view, we study the relation between the geometry of hypersurfaces and the set of singular points of its distance function from both sides of this hypersurface. We also give an algorithm for the construction of reflection caustics and compare the results and illustrations of mathematical simulation with real-world images. PubDate: 2022-06-01

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Abstract: The objective of our paper is to generalize the representation obtained in 2017 by S.Yu. Dobrokhotov, V.E. Nazaikinskii, and A.I. Shafarevich to the general case of the canonical operator on an isotropic manifold with complex germ. PubDate: 2022-06-01

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Abstract: In the paper, a boundary value problem for a singularly perturbed reaction-diffusion-advection equation is considered in a two-dimensional domain in the case of discontinuous coefficients of reaction and advection, whose discontinuity occurs on a predetermined curve lying in the domain. It is shown that this problem has a solution with a sharp internal transition layer localized near the discontinuity curve. For this solution, an asymptotic expansion in a small parameter is constructed, and also sufficient conditions are obtained for the input data of the problem under which the solution exists. The proof of the existence theorem is based on the asymptotic method of differential inequalities. It is also shown that a solution of this kind is Lyapunov asymptotically stable and locally unique. The results of the paper can be used to create mathematical models of physical phenomena at the interface between two media with different characteristics, as well as for the development of numerical-analytical methods for solving singularly perturbed problems. PubDate: 2022-06-01

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Abstract: Differential-difference operators are considered on an infinite cylinder. The objective of the paper is to present an index formula for the operators in question. We define the operator symbol as a triple consisting of an internal symbol and conormal symbols on plus and minus infinity. The conormal symbols are families of operators with a parameter and periodic coefficients. Our index formula contains three terms: the contribution of the internal symbol on the base manifold, expressed by an analog of the Atiyah–Singer integral, the contributions of the conormal symbols at infinity, described in terms of the \(\eta\) -invariant, and also the third term, which also depends on the conormal symbol. The result thus obtained generalizes the Fedosov–Schulze–Tarkhanov formula. PubDate: 2022-06-01

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Abstract: We consider certain classes of operators generated by infinite band matrices (called band operators). They can be applied to the integration via the Lax pair formalism of nonlinear dynamical systems (e.g., Volterra type lattices) by using the inverse problem theory, in particular, by the inverse spectral problem method. A key role in this method is played by the moments of the Weyl matrix of a given band operator, which are used for unique reconstruction of the latter. These band operators have a special sparse structure, namely, they contain only two nonzero diagonals. We find a description of their sparsity in terms of these moments. PubDate: 2022-06-01

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Abstract: In many physical problems describing the propagation of waves, there are dispersion effects, both spatial and temporal. In these cases, the equations describing these problems and relating the temporal frequency and spatial momenta turn out to be inhomogeneous with respect to these variables. At the same time, these relations are even not polynomial. An example with time dispersion is given by Maxwell’s equations in a situation with rapidly changing electric and magnetic fields; then the second time derivative \(t\) is replaced by a function of this derivative (a pseudodifferential operator). In the paper, asymptotic formulas are constructed for the solution of the Cauchy problem with localized initial data for the equation \(g^2\bigl(-ih\frac{\partial}{\partial t}\bigr)u=-h^2\langle \nabla,\, c^2(x)\nabla\rangle u\) with variable speed and a small parameter characterizing fast oscillations of the propagating waves. One of the main considerations in use is that the constructive asymptotics for the solution of problems of this type are represented by functions given parametrically, and the corresponding parameters are the natural coordinates on the Lagrangian manifolds defining these asymptotics. Here, in contrast to problems with spatial dispersion, the corresponding Lagrangian manifold is defined in the phase space containing the time \(t\) and the corresponding conjugate momentum coordinate, the frequency \(\omega\) . PubDate: 2022-06-01

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Abstract: We prove that every locally bounded automorphism of a connected not necessarily linear Lie central extension of a connected perfect Lie group with discrete center is continuous if and only if it is continuous on the center. PubDate: 2022-03-01 DOI: 10.1134/S1061920822010113

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Abstract: We consider the Cauchy problem for a parabolic equation with a Ё-Laplacian or a general second-order quasilinear equation with boundary conditions of the Bitsadze–Samarskii type. We prove that at least one generalized solution of such problem exists. PubDate: 2022-03-01 DOI: 10.1134/S1061920822010125

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Abstract: The phase space of a gyrostat with a fixed point and a heavy top is the Lie–Poisson space \(\textbf{e}(3)^*\cong \mathbb{R}^3\times \mathbb{R}^3\) dual to the Lie algebra \(\textbf{e}(3)\) of the Euclidean group \(E(3)\) . One has three naturally distinguished Poisson submanifolds of \(\textbf{e}(3)^*\) : (i) the dense open submanifold \(\mathbb{R}^3\times \dot{\mathbb{R}}^3\subset \textbf{e}(3)^*\) which consists of all \(4\) -dimensional symplectic leaves ( \(\vec{\Gamma}^2>0\) ); (ii) the \(5\) -dimensional Poisson submanifold of \(\mathbb{R}^3\times \dot{\mathbb{R}}^3\) defined by \(\vec{J}\cdot \vec{\Gamma} = \mu \vec{\Gamma} \) ; (iii) the \(5\) -dimensional Poisson submanifold of \(\mathbb{R}^3\times \dot{\mathbb{R}}^3\) defined by \(\vec{\Gamma}^2 = \nu^2\) , where \(\dot{\mathbb{R}}^3:= \mathbb{R}^3\backslash \{0\}\) , \((\vec{J}, \vec{\Gamma})\in \mathbb{R}^3\times \mathbb{R}^3\cong \textbf{e}(3)^*\) and \(\nu < 0 \) , \(\mu\) are some fixed real parameters. Using the \(U(2,2)\) -invariant symplectic structure of Penrose twistor space we find full and complete \(E(3)\) -equivariant symplectic realizations of these Poisson submanifolds which are \(8\) -dimensional for (i) and \(6\) -dimensional for (ii) and (iii). As a consequence of the above, Hamiltonian systems on \(\textbf{e}(3)^*\) lift to Hamiltonian systems on the above symplectic realizations. In this way, after lifting the integrable cases of a gyrostat with a fixed point and of a heavy top, we obtain a large family of integrable Hamiltonian systems on the phase spaces defined by these symplectic realizations. PubDate: 2022-03-01 DOI: 10.1134/S1061920822010095

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Abstract: The joint spectrum of the Schrödinger operator of an anisotropic Kepler problem (the component along one of the axes of the diagonal mass tensor differs from the components along the other two axes) and the angular momentum operator are considered. Using the theory of Maslov’s complex germ, series of corresponding semiclassical stationary states (with complex phases) localized in the vicinity of flat disks are constructed. The wave functions of these states, in the direction normal to the plane of the disk, have the form of Hermite–Gaussian functions; in the direction of the polar angle coordinate in the plane of the disk, they oscillate harmonically; in the radial direction in the plane of the disk, their behavior is described by the Airy function of a composed argument: inside the disk, the wave functions oscillate and, outside it, decay. The existence of such states is due to the existence of Floquet solutions of certain differential equations with periodic coefficients. PubDate: 2022-03-01 DOI: 10.1134/S1061920822010058

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Abstract: For uniformly convex asymmetric spaces, questions concerning nonempty intersections of a nested system of bounded convex closed sets are considered. Questions concerning the density of sets of points of existence and approximative uniqueness are studied in these spaces for the case of nonempty closed subsets. PubDate: 2022-03-01 DOI: 10.1134/S1061920822010137

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Abstract: Using Maslov’s complex germ in the Cauchy problem for a wave equation, we consider the asymptotics of the solution of the Cauchy problem in which the velocity depends irregularly on a small parameter. PubDate: 2022-03-01 DOI: 10.1134/S1061920822010010

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Abstract: It is proved that the graded Lie algebras of infinitesimal holomorphic automorphisms of a nondegenerate quadric of codimension \(k\) do not have weight components more than \(2k\) . It is also proved that, for \(k \leq 3\) , there are no graded components of weight greater than 2. Several questions are formulated. PubDate: 2022-03-01 DOI: 10.1134/S1061920822010022

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Abstract: In the paper, the asymptotics of eigenvalues of a perturbed Airy operator is obtained. PubDate: 2022-03-01 DOI: 10.1134/S1061920822010101

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Abstract: It was noticed recently that, given a metric space \((X,d_X)\) , the equivalence classes of metrics on the disjoint union of the two copies of \(X\) coinciding with \(d_X\) on each copy form an inverse semigroup \(M(X)\) with respect to concatenation of metrics. Now put this inverse semigroup construction in a more general context, namely, we define, for a \(C^*\) -algebra \(A\) , an inverse semigroup \(S(A)\) of Hilbert \(C^*\) - \(A\) - \(A\) -bimodules. When \(A\) is the uniform Roe algebra \(C^*_u(X)\) of a metric space \(X\) , we construct a mapping \(M(X)\to S(C^*_u(X))\) and show that this mapping is injective, but not surjective in general. This allows to define an analog of the inverse semigroup \(M(X)\) that does not depend on the choice of a metric on \(X\) within its coarse equivalence class. PubDate: 2022-03-01 DOI: 10.1134/S1061920822010071

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Abstract: Small amplitude solutions of the nonlinear shallow water equations in a one- or two-dimensional domain are considered. The amplitude is characterized by a small parameter \( \varepsilon \) . It is assumed that the basin depth is a smooth function whose gradient is nowhere zero on the set of its zeros (i.e., on the coastline in the absence of waves). A solution of the equations is understood to be a triple (time-dependent domain, free surface elevation, horizontal velocity) smoothly depending on \( \varepsilon \) and such that (i) the free surface elevation and the horizontal velocity are zero for \( \varepsilon =0\) ; (ii) the sum of the free surface elevation and the depth is positive in the domain and zero on the boundary; (iii) the free surface elevation and the horizontal velocity are smooth in the closed domain and satisfy the equations there. An asymptotic solution modulo \(O( \varepsilon ^N)\) is defined in a similar way except that the equations must be satisfied modulo \(O( \varepsilon ^N)\) . We prove that, in this setting, the nonlinear shallow water equations with small smooth initial data have an asymptotically unique asymptotic solution modulo \(O( \varepsilon ^N)\) for arbitrary \(N\) . The proof is constructive (and leads to simple explicit formulas for the leading asymptotic term). The construction uses a change of variables (depending on the unknown solution and resembling the Carrier–Greenspan transformation) that maps the unknown varying domain onto the unperturbed domain. The resulting nonlinear system is within the scope of regular perturbation theory. The zero approximation is a Cauchy problem for a linear hyperbolic system with degeneracy on the boundary, whose unique solvability in the class of smooth functions is proved by lifting the problem to a closed 3-manifold (where the spatial part of the operator turns out to be hypoelliptic). PubDate: 2022-03-01 DOI: 10.1134/S1061920822010034

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Abstract: In the paper, a boundary value problem for a singularly perturbed equation of reaction-diffusion-advection in a two-dimensional domain is considered. The key feature of the problem under consideration is the weak discontinuity of the reactive term. The discontinuity occurs on a simple closed curve known in advance that lies entirely inside the domain. It is shown that such a problem has a solution with an inner transition layer localized near the discontinuity curve. For this solution, an asymptotic expansion with respect to a small parameter is constructed, and existence theorems are proved, together with Lyapunov asymptotic stability. As a method of the proof, the asymptotic method of differential inequalities is used. An example is given illustrating the constructions carried out in the paper. PubDate: 2022-03-01 DOI: 10.1134/S1061920822010083