In this paper, we consider the one-dimensional Schrödinger equation on the semiaxis with an additional exponential potential. Using transformation operators with the asymptotics at infinity, a triangular representation of a special solution of this equation is found. An estimate is obtained with respect to the kernel of the representation.
The construction of optimal interpolation formulas is discussed. First, an exact upper bound for the error of an interpolation formula in the Sobolev space is calculated. The existence and uniqueness are proved for the optimal interpolation formula with the smallest error. An algorithm for finding the coefficients of the optimal interpolation formula is presented. This algorithm makes it possible to find the optimal coefficients.
This paper investigates conditions under which representability of each element (a) from the field (P) as the sum (a = f + g) , where ({{f}^{{{{q}_{1}}}}} = f) , ({{g}^{{{{q}_{2}}}}} = g) , and ({{q}_{1}},{{q}_{2}}) are fixed natural numbers >1, implies a similar representability of each square matrix over the field (P) . We propose a general approach to solving this problem. As an application we describe fields and commutative rings where 2 is a unit, over which each square matrix is the sum of two 4-potent matrices.
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.
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.
In this paper we consider a weakened version of the spectral synthesis for the differentiation operator in nonquasianalytic spaces of ultradifferentiable functions. We deal with the widest possible class of spaces of ultradifferentiable functions among all known ones. Namely, these are spaces of Ω‑ultradifferentiable functions which have been recently introduced and explored by Abanin. For differentiation invariant subspaces in these spaces, we establish conditions of weak spectral synthesis. As an application, we prove that a kernel of a local convolution operator admits weak spectral synthesis. We also show that a conjunction of kernels of convolution operators admits weak spectral synthesis if all generating ultradistributions have the same support equaled to {0} and there exists one generated by an ultradistribution which characteristic function is a multiplier in the corresponding space of entire functions.
We present a new logic called SPL, embedded into Solovay’s provability logic S, using a translation that embeds Visser’s formal logic FPL into Gödel–Löb’s provability logic GL. SPL is formulated as sequent and natural deduction calculi, and a the Kripke semantics is proposed for SPL.
We investigate the inverse problem of determining the time and space dependent kernel of the integral term in the (n) -dimensional integro-differential equation of heat conduction from the known solution of the Cauchy problem for this equation. First, the original problem is replaced by the equivalent problem in which an additional condition contains the unknown kernel without integral. We study the question of the uniqueness of the determining of this kernel. Next, assuming that there are two solutions ({{k}_{1}}(x,t)) and ({{k}_{2}}(x,t)) of the stated problem, an equation is formed for the difference of this solution. Further research is being conducted for the difference ({{k}_{1}}(x,t) - {{k}_{2}}(x,t)) of solutions of the problem and using the techniques of integral equations estimates. It is shown that, if the unknown kernel (k(x,t)) can be represented as (k(x,t) = sumlimits_{i = 0}^N {{a}_{i}}(x){{b}_{i}}(t)) , then ({{k}_{1}}(x,t) equiv {{k}_{2}}(x,t)) . Thus, the theorem on the uniqueness of the solution of the problem is proved.
A functional differential equation with a discrete retarded argument and a constant concentrated delay is considered. The problem of the asymptotic stability of this equation is reduced to the problem of location of the spectrum for the shift operator. Coefficient sufficient conditions for the asymptotic stability of this equation are obtained. The domain in the parameter space such that these conditions are necessary is obtained.
We consider solutions to two boundary values problems for the Poisson equation on plane domains. We prove several estimates for integrals of solutions using geometric characteristics of domains.
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