An infinitely long circular cylinder generally consisting of an arbitrary number of coaxial viscoelastic layers surrounded by a viscoelastic medium is considered. It is assumed that the layered package consists of thick-walled and thin-walled layers of the cylinder. When describing the motion of thin-walled elements, the equations of the theory of such shells are used, which are based on the Kirchhoff–Love hypotheses. For thin-walled layers, the initial equations are based on the linear theory of elasticity. The problem is solved by the Green–Lamb method. The displacement potentials are determined from the solutions to the Helmholtz equation. Arbitrary constants are determined from the boundary conditions that are set between the bodies. As a result, the problem is reduced to a system of inhomogeneous algebraic equations with complex coefficients, which is solved by the Gaussian elimination method with pivoting. Analytical expressions for dynamic displacements and stresses in layers and the environment are obtained in terms of special Bessel and Hankel functions. It was found that, for R_1/r_0 > 50 , the effect of a cylindrical source is decomposed as a plane wave; i.e., the curvature radius of the wave may be ignored.
The paper deals with functors acting between categories which arise in the theory of univeresal C^* -algebras. In the framework of a categorical approach to the notion of a universal C^* -algebra generated by a set of generators subject to relations, Loring introduced and studied categories called the C^* -relations. Given a set X , a C^* -relation on X is a category whose objects are functions from X to C^* -algebras and morphisms are ^* -homomorphisms of C^* -algebras making the appropriate triangle diagrams commute. Moreover, these functions and ^* -homomorphisms satisfy certain natural axioms. A C^* -relation is said to be compact if it determines a universal C^* -algebra. In this paper, it is shown that every functor between arbirary compact C^* -relations is a functor between ^* -polynomial relations on the same set X up to isomorphisms of categories. Using ideals in the algebras of involutive polynomials, we construct the factorization functors which constitute an important class of functors between C^* -relations. To study properties of the factorizatoin functors, we introduce the notion of a soft image of a functor and establish a criterion for an object to belong to a soft image.
In this study, we define the (s,t) -Fermat and (s,t) -Fermat–Lucas sequences. For these sequences, we give many features such as the characteristic equation and Binet formulas. We obtain the relations between these sequences. Then we find different representations of finding terms of these sequences. Also, we show the relationship between the positive and negative terms of the sequences and get the relationship between three consecutive terms. We present the generating functions and sum formulas of these sequences. In addition, we calculate special identities of these sequences such as Catalan identity, Melham’s identity, d’Ocagne identity, etc. We examine the relationships of (s,t) -Fermat sequence with Fibonacci, Pell, Jacobsthal, Balancing, Oresme sequences and (s,t) -Fermat–Lucas sequence with Lucas, Pell–Lucas, Jacobsthal–Lucas, Balancing–Lucas, Oresme–Lucas sequences. Moreover, we present on the application of (s,t) -Fermat and (s,t) -Fermat–Lucas sequences to hyperbolic quaternions. Furthermore, we define hyperbolic (s,t) -Fermat and (s,t) -Fermat–Lucas quaternions. For these hyperbolic quaternions, we give many properties such as Binet formulas. Finally, the terms of the (s,t) -Fermat and (s,t) -Fermat–Lucas sequences are associated with their hyperbolic quaternion values.
Issues of a priori estimation and existence of ω -periodic solutions to a system of nonlinear ordinary second-order differential equations with the main positively homogeneous nonlinear part are studied. Taking into account the explicit structure of the set of zeros of the main nonlinear part and developing the methods previously applied by the authors, new conditions are found that provide an a priori estimation of ω -periodic solutions. Under the conditions of a priori estimation, a criterion for the existence of ω -periodic solutions is formulated and proved. The proof uses the property of invariance of the existence of ω -periodic solutions under continuous variation of the main nonlinear part and applies methods for calculating the mapping degree of a vector field.
In this paper, in a certain mixed-type domain for a mixed elliptic-hyperbolic equation with different orders of degeneration and with singular coefficients, theorems of uniqueness and existence of a solution to a problem with a condition of Zhegalov type and the Nakhushev condition on internal characteristics and an analogue of the Frankl condition on the degeneration segment are proved. On the boundary of the elliptic domain, the Dirichlet condition is specified.
Let f be a measurable function defined on ℝ , and define the operator 𝒢(ϕ_n * f)(x) = sup_N | ∑_k = 0^N c_k(ϕ_n_k + 1 * f(x) - ϕ_n_k * f(x))|, where ϕ_n(x) = 1/nχ_[0,n](x) , and * stands for convolution. Let 1 < r < ∞ and r 1pt ' = r/r - 1 . Then for all 1 < ρ < ∞ , and for all (f_j) ∈L^p(w) ∩L^p(ℝ) , the weighted inequalities ( ∑_j (𝒢(ϕ_n * f_j)^ρ)^1/ρ_L^p(w)⩽C_p,ρ(w)( ∑_j | f_j|^ρ)^1/ρ_L^p(w) hold if w ∈A_p/r' and r 1pt ' ⩽ p < ∞ , or if w ∈ A_p^r' and 1 < p ⩽ r . Likewise, if w(x)^r'∈A_1 , then the weak type inequality w( {x:( ∑_j (𝒢(ϕ_n * f_j)(x))^ρ)^1/ρ > λ}) ⩽C_ρ(w)1/λ∫( ∑_j | f_j(x)|^ρ)^1/ρw(x) 1pt dx holds for all (f_j) ∈L^1(w) ∩L^1(ℝ) . In particular, the weighted inequality 𝒢(ϕ_n * f)_L^p(w)⩽C_p,ρ(w) f _L^p(w) holds for all f ∈L^p(w) ∩L^p(ℝ^n) if w ∈A_p/r' and r 1pt ' ⩽ p < ∞ , or if w ∈ A_p^r' and 1 < p ⩽ r . Likewise, if w(x)^r'∈A_1 , then the weak type inequality w({ x:𝒢(ϕ_n * f)(x) > λ} ) ⩽C_ρ(w)1/λ∫| f(x)|w(x) 1pt dx holds for all f ∈L^1(w) ∩L^1(ℝ) .
Based on the known approaches and in development of the previously obtained results, a refined mathematical model is proposed for describing the process of nonlinear deformation along the cylindrical shape of fibrous composite plates, based on the approximation of tangential displacements by a third-degree polynomial along the transverse coordinate, in which the corresponding expansion coefficients are independent, and the deflections obey a linear law. The kinematic relations for determining the strain components are constructed based on the previously proposed consistent geometrically nonlinear relations of the elasticity theory with conservation of terms, which make it possible to identify classical bending and nonclassical (in particular, transverse-shear under transverse bending conditions) modes of loss of stability of the plate. Physically nonlinear behavior of the material is taken into account in the relations for transverse tangential stresses. The issues of correctness of the constructed equations in formulating certain conditions of fastening and loading of the end sections of the plate are considered. It is shown that the solutions determined on the basis of the constructed equations, when satisfying the boundary conditions of clamping the cross-section in its vicinity, transit to a Timoshenko-model solution with a constant transverse tangential stress across the thickness. This conclusion is formulated by analyzing the exact analytical solution to the linear problem of transverse bending along the cylindrical shape of a cantilever-fixed plate.
The paper studies submanifolds that are tangent at every point to the working distribution of an affinor metric structure. This distribution is a generalization of the contact distribution of a contact metric structure, while an affinor metric structure is a generalization of a contact metric structure to manifolds of arbitrary dimension. Such submanifolds are called Legendrian submanifolds. We consider Legendrian submanifolds of general type and their particular cases: Legendrian curves and homogeneous Legendrian submanifolds.
The article studies a local problem for a third-order equation with an elliptic-hyperbolic operator in the principal part. A method is applied that does not require a special representation of the general solution of the equation under consideration. This method determines the study of an elliptic-hyperbolic equation of the second order with unknown right-hand sides, which is of interest for solving important inverse problems of mechanics and physics. Theorems of existence and uniqueness of the classical solution of the problem are proved. The proof is based on the extremum principle for a third-order equation and on the theory of singular, Fredholm integral equations.
By employing the harmonic analysis associated with the Bessel function on [0, + ∞ [ , we study two classes of generalized wavelet packets and their related generalized wavelet transformations, along with the Plancherel, Calderon, and reconstruction formulas for these transforms.
For a finite family of continuum bundles of controlled trajectory curves (trajectory–control pairs) induced in a separable Hilbert space by a given second-order nonstationary differential system with different polynomial regulators and constant delay parameters, the solvability of the problem of implementing the operator coefficients of a general (invariant) polylinear controller with the same constant delay parameters is investigated, but in the presence of which in the structure of a given differential system, the union of these dynamic bundles represents a family of its admissible solutions. This problem belongs to the type of nonstationary coefficient-operataor inverse poblems for evolutionary equations of second-order in an infinite-dimensional Hilbert space and is solved on the basis of studying the properties of continuity and semiadditivity of the nonlinear Rayleigh–Ritz functional operator. The results have applications in the general theory of adaptive systems for a class of higher-order polylinear differential models.
In this paper, an initial-boundary value problem for an inhomogeneous heat equation with a piecewise constant argument and Dirichlet boundary conditions is considered. The Fourier method is used to investigate the problem. By expanding the solution in terms of eigenfunctions, the initial-boundary value problem is reduced to the Cauchy problem for an ordinary differential equation with respect to the expansion coefficients with a piecewise continuous argument. The existence and uniqueness of the solution to this problem are proved. As a result, it is shown that the original problem has a unique solution, which is constructed in explicit form.
This paper is devoted to studying problems in the theory of approximating periodic classes of functions by trigonometric polynomials in the Hilbert space L_2. We obtain exact constants in Jackson–Stechkin type inequalities for functions f ∈ L_2^(r) whose successive derivatives f^(s) (s = 0,1, … ,r) belong to the space L_2. Additionally, we derive exact values for simultaneous approximations of a function and its successive derivatives for certain classes of functions defined by the generalized modulus of continuity of higher orders Ω_m(f^(r),t)_2. For the class of functions W_m^(r)(h), where m ∈ℕ, r ∈ℤ_ + , h ∈ (0,3π/(4n)], satisfying the constraint {Ω _m^2/m(f^(r),h) + n^2/h∫_0^h 1ptτ (h - τ )Ω _m^2/m(f^(r),τ )dτ}^m/2⩽ 1, the exact values of the Bernstein, Kolmogorov, linear, Gelfand, and projection n -widths in the space L_2 are obtained.
In this paper, we establish a new arithmetic-geometric mean type inequality for positive definite matrices. As an application, we use the new inequality to characterize operator monotone functions.
We study an iteratively regularized gradient method with an a posteriori stopping rule for solving nonlinear irregular operator equations in Hilbert spaces. An accuracy estimate in terms of the error level of input data for this method is established. We assume that the desired solution satisfies a sourcewise condition, and we use no structural conditions on the operator of the problem.
In this paper, we study inverse problems of finding the right-hand side of a one-dimensional parabolic equation using additionally specified integral conditions regarding the solution of the first initial-boundary value problem for this equation. Sufficient conditions for the given functions are found, under which the stated problems are well-posed. Using the Fourier method and integral equations, solutions to the problems are constructed in explicit form.
This paper solves an inverse problem for a convolution-type integral equation of the second kind. This problem involves finding the right-hand side of the equation from the solution to the direct problem, which is known on a certain subset of its definition. Crucially, the kernel function in the integral operator is zero in the neighborhood of zero and unknown outside its neighborhood.
We consider the problem of topological structure of limit sets of dynamical systems and construct an example of a smooth dynamical system in the space ℝ^3 whose the ω -limit set is an infinite cylinder.
We study an inhomogeneous Riemann boundary value problem with a finite index and a boundary condition on the real axis for one generalized Cauchy–Riemann equation with a strong coefficient singularity. To solve this problem, it is necessary to derive a structural formula for the general solution to the equation and conduct a complete study of the solvability of the Riemann boundary value problem for analytic function with an infinite power-order index. Based on this study, a general solution formula and a picture of the solvability of the boundary value problem for generalized analytic functions are derived.
This paper discusses the construction of an optimal interpolation formula intended for approximating functions in the Hilbert space L_2^(3)(0,1) . This space covers square-integrable functions with the third generalized derivative in the interval [0,1] . The interpolation formula is a linear combination of the function values and their first and second derivatives at equally spaced nodes in the interval [0,1] . The coefficients are determined by minimizing the norm of the error functional in the conjugate space L_2^(3)* 1pt (0,1) . This error functional is defined as the discrepancy between the function and its approximation. The key results of the study include explicit expressions for the coefficients and the norm of the error functional. The optimization problem is methodically formulated and solved, resulting in a system of linear equations for the coefficients. Analytical solutions are obtained that give a clear expression for optimal coefficients. In addition, integrating the obtained optimal interpolation formula over the interval [0,1] leads to the Euler–Maclaurin quadrature formula. The application of these results in estimating the error of the interpolation formula for functions from L_2^(3)(0,1) is demonstrated.
