Mathematical Notes

In this paper, we study the Martin integral representation for nonharmonic functions in discrete settings of infinite homogeneous trees. Recall that the Martin integral representation for trees is analogs to the mean-value property in Euclidean spaces. In the Euclidean case, the mean-value property for nonharmonic functions is provided by the Pizzetti (and co-Pizzetti) series. We extend the co-Pizzetti series to the discrete case. This provides us with an explicit expression for the discrete mean-value property for nonharmonic functions in discrete settings of infinite homogeneous trees.

Let Λ(n) be the von Mangoldt function, and let r_G(n):= ∑_m1+m2=n Λ (m₁)Λ(m₂) be the weighted sum for the number of Goldbach representations which also includes powers of primes. Let S̃(z): = ∑_n≥1 Λ (n)e-nz, where Λ (n) is the Von Mangoldt function, with z ∈ ℂ, Re (z) > 0. In this paper, we prove an explicit formula for S̃(z) and the Cesàro average of r_G(n).

It is proved that there exists a function defined in the closed upper half-plane for which the sums of its real shifts are dense in all Hardy spaces H_p for 2 ≤ p < ∞, as well as in the space of functions analytic in the upper half-plane, continuous on its closure, and tending to zero at infinity.

We give a proof of the boundedness of the Bergman projection in generalized variable-exponent vanishing Morrey spaces over the unit disc and the upper half-plane. To this end, we prove the boundedness of the Calderón—Zygmund operators on generalized variable-exponent vanishing Morrey spaces. We give the proof of the latter in the general context of real functions on Rⁿ, since it is new in such a setting and is of independent interest. We also study the approximation by mollified dilations and estimate the growth of functions near the boundary.

The paper deals with uncertainty relations for time and energy operators, and the aftermath of the Chernobyl catastrophe is considered as an example. The mathematical approach developed by Holevo is analyzed, which allows us to assign the corresponding observables to non-self-adjoint operators and to establish uncertainty relations for nonstandard canonical conjugate pairs.

Relations for calculating the minimal time interval in which the energy jump can be discovered are given. Based on the intensity parameter introduced by the author, which is related to a special statistics called Gentile statistics and to the polylogarithm function, properties of stable chemical elements, such as time fluctuations and the jump of specific energy in the transition from the Bose—Einstein distribution to the Fermi—Dirac distribution, are mathematically described with regard to experimental data. The obtained data are arranged in a table for 255 stable chemical elements.

The mathematical approach developed by the author of the present paper allows one to describe the “antipode” (in a certain sense) of the standard thermodynamics, i.e., the thermodynamics of nuclear matter. This field of nuclear physics is very important for the study of properties of radioactive elements and, accordingly, from the standpoint of ensuring nuclear safety.

We consider the boundary-value problem in a semibounded interval for a fourth-order equation with “double degeneracy”: the small parameter in the equation multiplies the product of the unknown function vanishing on the boundary and its highest derivative. Such a problem arises in the description of the motion of weak solutions of polymers near a critical point. For the zero value of the parameter, the solution is the classical Hiemenz solution. We prove the unique solvability of the problem for nonnegative values of the parameter not exceeding 1.

We pose the Cauchy problem with localized initial data that arises when passing from an explicit difference scheme for the wave equation to a pseudodifferential equation. The solution of the Cauchy problem for the difference scheme is compared with the asymptotics of the solution of the Cauchy problem for the pseudodifferential equation. We give a detailed study of the behavior of the asymptotic solution in the vicinity of the leading edge, where yet another version of the asymptotic solution is constructed based on vertical manifolds.

Spherically symmetric continuous wavelet decompositions are considered, and the notion of Riesz means is introduced for them. Generalized localization is proved for the decompositions under study in L_p classes without any restrictions on the wavelets. Further, generalized localization is studied for the Riesz means of wavelet decompositions of distributions from the Sobolev class with negative order of smoothness.

In the paper, systems of elementary axioms are found for classes of groupoids and ordered groupoids of binary relations with Diophantine operations that are generalized subreducts of the Tarski relation algebras.

Any compact body with regular boundary in ℝ^N defines a two-valued function on the space of affine hyperplanes: the volumes of the two parts into which these hyperplanes cut the body. This function is never algebraic if N is even and is very rarely algebraic if N is odd: all known bodies defining algebraic volume functions are ellipsoids (and have been essentially found by Archimedes for N = 3). We demonstrate a new series of locally algebraically integrable bodies with algebraic boundaries in spaces of arbitrary dimensions, that is, of bodies such that the corresponding volume functions coincide with algebraic ones in some open domains of the space of hyperplanes intersecting the body.

A nonlocal boundary-value problem for a linear ordinary differential equation with fractional discretely distributed differentiation operator is considered. The existence and uniqueness theorem for the solution of this problem is proved.

Michael Somos discovered several theta-function identities of various levels by computer and offered no proof for them. These identities highly resemble some of Ramanujan’s identities. The main focus of this paper is to prove some of these theta-function identities, in particular those of level 6 that have been discovered using computational searches. Some of the the Somos identities that we are discussing in this paper cannot be expressed in the form of P −−Q type. Furthermore, we establish certain colored partition identities for them.

The algebra of symmetries of a quantum three-frequency hyperbolic resonance oscillator is studied. It is shown that this algebra is determined by a finite set of generators with polynomial commutation relations. The irreducible representations of this algebra and the corresponding coherent states are constructed.

In this paper, using a Bochner-type theorem, we establish a characterization of strictly positive definite functions on the topological direct limit space D_∞ of real diagonal matrices relatively to the action of the product group K_∞ = U_∞ × U_∞, where U_∞ is the infinite dimensional unitary group.

The definition of a self-similar function is extended to quasi-Banach Lebesgue spaces. A sufficient condition for a function with given self-similarity parameters to lie in some such space is given.

Sets with a continuous selection from the near-best approximation operator are studied and the relationship of such sets with the radial δ-solarity property and with the metric function is discussed.