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Abstract: This study aims to establish equivalences (in norm) of the problems of reconstructing computed tomography and computational (numerical) diameter (C(N)D), which was done in 2019 for functions of two variables. This was based on the equivalence of respective norms in the same two-dimensional Sobolev spaces proved by Frank Natterer. In this study, we prove the equivalence (in norm) of the Radon transform and the function that generated it for the case of functions of any dimension with large-scale prospects for application. PubDate: 2023-08-01

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Abstract: The problem of diffraction of harmonic shear waves on an elliptical cylindrical cavity located in a viscoelastic medium is considered. The relationship between stresses and deformations is taken into account using the integral Boltzmann–Volterra hereditary relation. The problem of a dynamic stress-strain state around an elliptical cavity in an unbounded viscoelastic medium under the action of harmonic shear waves is reduced to a plane problem (plane deformation) of viscoelasticity. The Lame equation reduces to the solution of the Mathieu equation with complex arguments. Its solution is expressed in terms of Mathieu functions. Numerical results are obtained for different frequencies of incident waves, angles of incidence of the transverse wave and the ratio of the axes of the elliptical cavity. PubDate: 2023-08-01

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Abstract: A refined geometrically nonlinear model of static and dynamic deformation is developed for a rod-strip which is attached on one of its faces to an absolutely rigid support element of finite length. This model describes the transformation of bending forms of motion of the unfixed segment into longitudinal-shear forms of motion of the fixed segment. The model is based on Timoshenko’s model for the unfixed segment, taking into account transverse shear and compression deformations, which is transformed to another model at transition from the unfixed segment to the fixed one. The main equations corresponding to the constructed model for the unfixed segment are derived with such precision and content that, in the case of static deformation, they allow identifying classical buckling modes under axial compression conditions and transverse-shear buckling modes under bending conditions, while in the case of dynamic deformation, they allow transforming the bending oscillation modes into forced and parametric longitudinal-transverse modes. To formulate linear steady-state dynamic problems, the derived equations are reduced to three uncoupled equations with exact analytical solutions. PubDate: 2023-08-01

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Abstract: Schemes of the proofs of theorems stated by S.E. Gazizova and D.V. Maklakov in their work (Lobachevsky J. Math. 42, 1969–1976 2021) are given. The theorems serve as a basis for designing supercavitating hydrofoils possessing the minimum drag coefficient for the given lift coefficient. Thus, the maximum lift-to-drag ratio is attained. PubDate: 2023-08-01

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Abstract: In an unbounded domain, for a class of equations of mixed type with a singular coefficient, the correctness of the problem is studied in which one of the internal characteristics is freed from the Gellerstedt condition and this missing local condition is replaced by an analogue of the Frankl condition on the degeneration line. The uniqueness of the solution to the stated problem is proved using the extremum principle, while the existence of a solution to the problem is proved by the method of integral equations. PubDate: 2023-08-01

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Abstract: The aim of this paper is to study the existence of infinitely many solutions for Schrödinger–Kirchhoff-type equations involving nonlocal \(p(x, \cdot )\) -fractional Laplacian \(\left\{ {\begin{array}{*{20}{l}} {M({{\sigma }_{{p(x,y)}}}(u))\mathcal{L}_{K}^{{p(x, \cdot )}}(u) = \lambda {{{\left u \right }}^{{q(x) - 2}}}u + \mu {{{\left u \right }}^{{\gamma (x) - 2}}}u\;}&{{\text{in}}\;\Omega } \\ {u(x) = 0}&{{\text{in}}\;{{\mathbb{R}}^{N}}{\kern 1pt} \backslash {\kern 1pt} \Omega ,} \end{array}} \right.\) where \({{\sigma }_{{p(x,y)}}}(u) = \int_\mathcal{Q} \frac{{{{{\left {u(x) - u(y)} \right }}^{{p(x,y)}}}}}{{p(x,y)}}K(x,y)dxdy,\) \(\mathcal{L}_{K}^{{p(x, \cdot )}}\) is a nonlocal operator with singular kernel \(K\) , \(\Omega \) is a bounded domain in \({{\mathbb{R}}^{N}}\) with Lipschitz boundary \(\partial \Omega \) , \(M:{{\mathbb{R}}^{ + }} \to \mathbb{R}\) is a continuous function, q, \(\gamma \in C(\Omega )\) and \(\lambda ,\mu \) are two parameters. Under some suitable assumptions, we show that the above problem admits infinitely many solutions by applying the Fountain Theorem and the Dual Fountain Theorem. PubDate: 2023-08-01

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Abstract: Direct and inverse problems for the equation of forced vibrations of a finite length beam with a variable stiffness coefficient at the lowest term are investigated. The direct problem is the initial–boundary value problem for this equation with boundary conditions in the form of a beam fixed at one end and free at the other. The unknown variable in the inverse problem is a multiplier in the right-hand side, which depends on the space variable \(x\) . This unknown is determined with respect to the solution of the direct problem by specifying an integral redefinition condition. The uniqueness of the solution of the direct problem is proved by the method of energy estimates. The eigenvalues and eigenfunctions of the corresponding elliptic operator are used to reduce the problems to integral equations. The method of successive approximations is used to prove existence and uniqueness theorems for solutions of these equations. PubDate: 2023-08-01

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Abstract: The problems of the oscillatory flow of a viscoelastic fluid in a flat channel for a given harmonic oscillation of the fluid flow rate are solved on the basis of the generalized Maxwell model. The transfer function of the amplitude-phase frequency characteristics is determined. These functions make it possible to evaluate the hydraulic resistance under a given law, the change in the longitudinal velocity averaged over the channel section, as well as during the flow of a viscoelastic fluid in a nonstationary flow, and allow determining the dissipation of mechanical energy in a nonstationary flow of the medium, which are important in the regulation of hydraulic and pneumatic systems. Its real part allows determining the active hydraulic resistance, and the imaginary part is reactive or inductance of the oscillatory flow. PubDate: 2023-08-01

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Abstract: We consider a convex pentagon \(D\) that has a pair of parallel and equal sides without a common vertex. We study the linear difference equation associated with this polygon. The coefficients of the equation and the free term are holomorphic in \(D\) . The solution is sought in the class of functions holomorphic outside the “half” of the \(\partial D\) boundary and vanishing at infinity. A method for its regularization is proposed and a condition for its equivalence is found. The solution is represented as a Cauchy-type integral with an unknown density. The principle of contraction mappings in a Banach space is essentially used. Applications to interpolation problems for entire functions of exponential type are indicated. PubDate: 2023-08-01

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Abstract: In a countably normed space of smooth functions on the unit circle, we consider the Riemann boundary value problem operator with smooth coefficients. The paper introduces the concept of smooth degenerate factorizations of types of plus and minus functions that are smooth on the unit circle. Criteria for the existence of such factorizations are given. An apparatus is proposed for calculating the indices of these factorizations in terms of coefficients. In terms of smooth degenerate factorizations, a criterion for the invertibility of the Riemann boundary value problem operator is obtained. This allows describing the spectrum of this operator. Relationships between the spectra of the Riemann operator in the spaces of smooth and summable functions with the same coefficients are indicated. PubDate: 2023-08-01

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Abstract: In this paper, we consider a quadratic operator on the two-dimensional simplex, which is a convex combination of two quadratic stochastic operators. It is proved that the center of the simplex is a unique fixed point of the operator and this fixed point is an attracting point. PubDate: 2023-07-01

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Abstract: In this article, we present a three-particle lattice model Hamiltonian \({{H}_{{\mu ,\lambda }}}\) , \(\mu ,\lambda > 0\) by making use nonlocal potential. The Hamiltonian under consideration acts as a tensor sum of two Friedrichs models \({{h}_{{\mu ,\lambda }}}\) which comprises a rank 2 perturbation associated with a system of three quantum particles on a d-dimensional lattice. The current study investigates the number of eigenvalues associated with the Hamiltonian. Furthermore, we provide the suitable conditions on the existence of eigenvalues localized inside, in the gap and below the bottom of the essential spectrum of \({{H}_{{\mu ,\lambda }}}\) . PubDate: 2023-07-01

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Abstract: In this paper by a local group of the semigroup C*-algebra generated by the free product of Abelian semigroups is considered. The simplest of these algebras is the Cuntz–Toeplitz algebra \(T{{O}_{n}}\) . For such algebras, an abstract version of the Fourier series by the local group is constructed. A number of properties of the “harmonics” of this series are given. PubDate: 2023-07-01

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Abstract: We consider \(A(z)\) -analytic functions in case when \(A(z)\) is anti-analytic function. This paper investigates the behavior near the boundary of the derivative of the function, \(A(z)\) -analytic inside the \(A(z)\) -lemniscate and with a bounded change of it at the boundary. Thus, this paper introduces the complex Lipschitz condition for \(A(z)\) -analytic functions and proves Fatou’s theorem for \(A(z)\) -analytic functions. PubDate: 2023-07-01

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Abstract: In the paper, we solve the problem of designing a hydrofoil in the supercavitation regime simulated by the Wu model. The distributions of velocities along the wetted part of the hydrofoil and the cavitation number are assumed to be given. We derive formulas that allow one to express the lift and drag forces in terms of integral functionals of the initial data. Thus, the formulas generalize the Kutta-Zhukovskii theorem on the lift force of a profile in continuous flow for the case of supercavitational flow. The shape of the hydrofoil is reconstructed by the methods of inverse boundary value problems. It has been rigorously proven that for lifting profiles, the lift-to-drag ratio increases monotonically with increasing cavitation numbers. PubDate: 2023-07-01

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Abstract: The paper deals with boundedness problem for maximal operators associated to hypersurfaces in the space of integrable functions with degree p. A necessary condition for boundedness is given in the space of square-integrable functions in the case one nonvanishing principal curvature. A criterion for the boundedness of the maximal operators in the space of square-integrable functions is obtained for a partial class of convex hypersurfaces. PubDate: 2023-07-01

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Abstract: The well-known theorem on the uniformalization of topological properties [1] has been strengthened and generalized to the case of a countable family of properties given by relations of open coverings. Topological properties can be linearly uniformized (the dependence between constants \(\delta \) and \(\varepsilon \) is linear). In the framework of our result, the family can be treated to be attractive in dimension n in a countable number of closed linear subspaces of a linear normed space. PubDate: 2023-07-01

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Abstract: The singular differential Bessel operator \({{B}_{{ - \gamma }}}\) with negative parameter \( - \gamma < 0\) is considered. Solutions to the singular differential Bessel equation \({{B}_{{ - \gamma }}}u + {{\lambda }^{2}}u = 0\) are represented by linearly independent functions \({{\mathbb{J}}_{\mu }}\) and \({{\mathbb{J}}_{{ - \mu }}}\) , \(\mu = \frac{{\gamma + 1}}{2}\) . Some properties of the functions \({{\mathbb{J}}_{\mu }}\) , which are expressed in terms of the properties of the Bessel–Levitan j-function, are studied. Direct and inverse Bessel \({{\mathbb{J}}_{\mu }}\) -transforms are introduced. Based on the \(\mathbb{T}\) -pseudoshift operator introduced earlier, a generalized \(\mathbb{T}\) -shift operator belonging to the Levitan class of generalized shifts commuting with the Bessel operator \({{B}_{{ - \gamma }}}\) is constructed. A fundamental solution is found for the singular differential operator \({{B}_{{ - \gamma }}}\) with a singularity at an arbitrary point on the semiaxis \([0,\infty )\) . PubDate: 2023-07-01

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Abstract: In this work, a critical circle homeomorphism with several break points is considered. The circle homeomorphism \(f\) with the irrational rotation number \(\rho \) is well known to be strictly ergodic; i.e., it has the unique \(f\) -invariant probability measure \(\mu \) . The invariant measure of critical circle homeomorphisms with a finite number of break points is proved to be singular with respect to the Lebesgue measure. PubDate: 2023-07-01

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Abstract: This paper considers three special systems of functional equations arising in the problem of embedding bimetric phenomenologically symmetric geometries of two sets of rank (3, 2) associated with complex, double, and dual numbers into a bimetric phenomenologically symmetric geometry of two sets of rank (4, 2), which is an affine group of transformations on the plane. Nondegenerate solutions of these systems are sought that cannot be easily found in general form. The set of solutions of these systems associated with a finite number of Jordan forms of second-order matrices can be found much more simply and meaningfully in the mathematical sense. The resulting solutions are directly associated with complex, double, and dual numbers. PubDate: 2023-07-01