Mathematical Notes最新文献

Abstract

We study a hyperbolic equation with an arbitrary number of potentials undergoing translation in arbitrary directions. Differential-difference equations arise in various applications that are not covered by the classical theory of differential equations. In addition, they are of considerable interest from a theoretical point of view, since the nonlocal nature of such equations gives rise to various effects that do not arise in the classical case. We find a condition on the vector of coefficients for nonlocal terms in the equation and on the vectors of potential translations that ensures the global solvability of the equation under consideration. By imposing the specified condition on the equation and using the classical Gelfand–Shilov scheme, we explicitly construct a three-parameter family of smooth global solutions to the equation under study.

Abstract

This paper continues the research on the circuit synthesis problem for a multiplexer function of logic algebra, which is a component of many integrated circuits and is also used in theoretical study. The exact value of the depth of a multiplexer with (n) select lines in the standard basis is found under the assumption that the conjunction and disjunction gates are of depth 1 and the negation gate is of depth 0; the depth equals (n+2) if (10 le n le 19). Thus, it follows from previous results that the exact depth value equals (n+2) for all positive integers (n) such that either (2 le n le 5) or (n ge 10). Moreover, for (n=1), this value equals 2, and for (6 le n le 9), it equals either (n+2) or (n+3). Similar results are also obtained for a basis consisting of all elementary conjunctions and elementary disjunctions of two variables.

Abstract

We give a new proof of the Ostapenko–Tarasov unicellularity theorem for the classical Volterra integration operator on the space (C^{(n)}[0,1]).

Abstract

Orthogonally biadditive operators preserving disjointness are studied. It is proved that, that for a Dedekind complete vector lattice (W) and order ideals (E) and (F) in (W), the set (mathfrak{N}(E,F;W)) of all orthogonally biadditive operators commuting with projections is a band in the Dedekind complete vector lattice (mathcal{OBA}_r(E,F;W)) of all regular orthogonally biadditive operators from the Cartesian product of (E) and (F) to (W). A general form of the order projection onto this band is obtained, and an operator version of the Radon–Nikodym theorem for disjointness-preserving positive orthogonally biadditive operators is proved.

Abstract

We obtain a formula describing the forward and backward wave profile for the solution of an initial–boundary value problem for the wave equation on an interval. The following combinations of boundary conditions are considered:

(i) The first-kind condition at the left endpoint of the interval and the third-kind condition at the right endpoint.

(ii) The second-kind condition at the left endpoint and the third-kind condition at the right endpoint.

(iii) The first-kind condition at the left endpoint and the attached mass condition at the right endpoint.

(iv) The second-kind condition at the left endpoint and the attached mass condition at the right endpoint.

The formula contains finitely many arithmetic operations, elementary functions, quadratures, and transformations of the independent argument of the initial data such as the multiplication by a number and taking the integer part of a number.

Abstract

An estimate of the additive energy of roots modulo a prime for sets with small doubling that has recently been obtained by Zaharescu, Kerr, Shkredov, and Shparlinskii is improved. The problem of determining the maximum cardinalities of the sets (|A+A|) and (|f(A)+f(A)|), where (f) is a polynomial of small degree and (A) is a subset of a finite field whose size is sufficiently small in comparison with the characteristic of the field, is also considered. In particular, it is proved that

(max(|A+A|,|A^4+A^4|)ge|A|^{25/24}), and (max(|A+A|,|A^5+A^5|)ge|A|^{25/24}).

Abstract

According to the Wik theorem, there exist massive Helson sets on the circle. In particular, they can be of Hausdorff dimension one. We extend Wik’s result to the multidimensional case.

Abstract

We study compactly supported solutions (u(x, t) geq 0), (x in mathbb{R}), (t geq 0), of a one-dimensional quasilinear heat transfer equation. The equation has a transport coefficient linear in (u) and a self-consistent source (alpha u+beta u^{2}) of general form. For the blow-up time of compactly supported solutions, we establish two-sided estimates functionally depending on the initial conditions (u(x, 0)).

Abstract

We consider the problem about time-periodic solutions of the quasilinear Euler–Bernoulli vibration equation for a beam subjected to tension along the horizontal axis. The boundary conditions correspond to the cases of elastically fixed, clamped, and hinged ends. The nonlinear term satisfies the nonresonance condition at infinity. Using the Schauder principle, we prove a theorem on the existence and uniqueness of a periodic solution.