Abstract
For the initial-boundary value problem of the dynamics of a thermoviscoelastic medium of Oldroyd type in the planar case, a nonlocal theorem regarding the existence of a weak solution is established.
For the initial-boundary value problem of the dynamics of a thermoviscoelastic medium of Oldroyd type in the planar case, a nonlocal theorem regarding the existence of a weak solution is established.
The inverse scattering method is used to integrate the Korteweg–de Vries equation with time-dependent coefficients. We derive the evolution of the scattering data of the Sturm–Liouville operator whose coefficient is a solution of the Korteweg–de Vries equation with time-dependent coefficients. An algorithm for constructing exact solutions of the Korteweg–de Vries equation with time-dependent coefficients is also proposed; we reduce it to the inverse problem of scattering theory for the Sturm–Liouville operator. Examples illustrating the stated algorithm are given.
The concepts of controlled frame and it’s dual in (n)-Hilbert space have been introduced and then some of their properties are going to be discussed. Also, we study controlled frame in tensor product of (n)-Hilbert spaces and establish a relationship between controlled frame and bounded linear operator in tensor product of (n)-Hilbert spaces. At the end, we consider the direct sum of controlled frames in (n)-Hilbert space.
An effective technique is proposed for obtaining exact formulas for estimating the area of flow regions in two-dimensional fluid flow problems with free boundaries, that allow an exact solution in terms of elliptic functions. The effectiveness of the technique is demonstrated using a specific example of the problem of capillary waves on the surface of a liquid of finite depth. This example is characterized by mirror symmetry of the flow region, but the technique can be generalized to the case of other symmetry of the flow region.
We obtain a regularized trace formula for (2m)-order differential operator perturbed by a quasi-differential perturbation and with periodic boundary conditions.
We study the Volterra integral equation of the first kind with an integral operator of order (n), a singularity, and a sufficiently smooth kernel in a certain Banach space with weight. It reduces to an integro-differential equation with two terms in the left-hand side. The first term corresponds to an equation for which an explicitly multiparameter family of solutions is constructed. For the second term we obtain an equation with an operator whose norm in an arbitrary Banach space is arbitrarily small near zero. Such splitting of the integral operator allows constructing a particular and general solutions to the integro-differential equations in the corresponding Banach space in the form of convergent series. Thus, under certain restrictions on the operator pencil corresponding to a given integral operator, a multiparameter family of solutions is constructed for the original integral equation.
We consider a family of bounded self-adjoint matrix operators (generalized Friedrichs models) acting on the direct sum of one-particle and two-particle subspaces of the Fock space. Conditions for the existence of eigenvalues of the matrix operators are obtained.
The paper investigates sufficient conditions for the absolute convergence of trigonometric Fourier series of almost-periodic functions in the sense of Besikovitch in the case when the Fourier exponents have a single limiting point at infinity. A higher-order modulus of continuity is used as a structural characteristic of the function under consideration.
We investigate the decidability of first-order logic extensions. For example, it is established in Zolotov’s works that a logic with a unary transitive closure operator for the one successor theory is decidable. We show that in a similar case, a logic with a unary partial fixed point operator is undecidable. For this purpose, we reduce the halting problem for the counter machine to the problem of truth of the underlying formula. This reduction uses only one unary nonnested partial fixed operator that is applied to a universal or existential formula.
Let (phi in mathcal{S}) with (int phi (x){kern 1pt} dx = 1), and define ({{phi }_{t}}(x) = frac{1}{{{{t}^{n}}}}phi left( {frac{x}{t}} right),) and denote the function family ({{{ {{phi }_{t}} * f(x)} }_{{t > 0}}}) by (Phi * f(x)). Let (mathcal{J}) be a subset of (mathbb{R}) (or more generally an ordered index set), and suppose that there exists a constant ({{C}_{1}}) such that (sumlimits_{t in mathcal{J}} {kern 1pt} {kern 1pt} {text{|}}{{hat {phi }}_{t}}(x){{{text{|}}}^{2}} < {{C}_{1}}) for all (x in {{mathbb{R}}^{n}}). Then
(i) There exists a constant ({{C}_{2}} > 0) such that ({text{||}}{{mathcal{V}}_{2}}(Phi * f){text{|}}{{{text{|}}}_{{{{L}^{p}}}}} leqslant {{C}_{2}}{text{||}}f{text{|}}{{{text{|}}}_{{{{H}^{p}}}}},quad frac{n}{{n + 1}} < p leqslant 1) for all (f in {{H}^{p}}({{mathbb{R}}^{n}})), (frac{n}{{n + 1}} < p leqslant 1).
(ii) The λ-jump operator ({{N}_{lambda }}(Phi * f)) satisfies ({text{||}}lambda {{[{{N}_{lambda }}(Phi * f)]}^{{1/2}}}{text{|}}{{{text{|}}}_{{{{L}^{p}}}}} leqslant {{C}_{3}}{text{||}}f{text{|}}{{{text{|}}}_{{{{H}^{p}}}}},quad frac{n}{{n + 1}} < p leqslant 1,) uniformly in (lambda > 0) for some constant ({{C}_{3}} > 0).
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