Vol 106, No 5-6 (2019)
- Year: 2019
- Articles: 40
- URL: https://journals.rcsi.science/0001-4346/issue/view/9083
Article
Martin Integral Representation for Nonharmonic Functions and Discrete Co-Pizzetti Series
Abstract
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.
Regular Ordinary Differential Operators with Involution
Abstract
The main results of the paper are related to the study of differential operators of the form
Some Identities Involving the Cesàro Average of the Goldbach Numbers
Abstract
Let Λ(n) be the von Mangoldt function, and let rG(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 rG(n).
A Formula for the Superdifferential of the Distance Determined by the Gauge Function to the Complement of a Convex Set
Abstract
The distance determined by the Minkowski gauge function to the complement of a convex solid body in a finite-dimensional space is considered. The concavity of this distance function on a given convex set is proved, and a formula for its superdifferential at any interior point of this set is obtained. It is also proved that the distance function under consideration is directionally differentiable at the boundary points of the convex set, and formulas for its directional derivative are obtained.
Density of Sums of Shifts of a Single Function in Hardy Spaces on the Half-Plane
Abstract
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 Hp 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.
Almost-Linear Segments of Graphs of Functions
Abstract
Let f: ℝ → ℝ be a function whose graph {(x, f(x))}x∈ℝ in ℝ2 is a rectifiable curve. It is proved that, for all L < ∞ and ɛ > 0, there exist points A = (a, f(a)) and B = (b, f(b)) such that the distance between A and B is greater than L and the distances from all points (x, f(x)), a ≤ x ≤ b, to the segment AB do not exceed ε|AB|. An example of a plane rectifiable curve for which this statement is false is given. It is shown that, given a coordinate-wise nondecreasing sequence of integer points of the plane with bounded distances between adjacent points, for any r < ∞, there exists a straight line containing at least r points of this sequence.
On Singular Operators in Vanishing Generalized Variable-Exponent Morrey Spaces and Applications to Bergman-Type Spaces
Abstract
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.
Mixed Fractional Differential Equations and Generalized Operator-Valued Mittag-Leffler Functions
Abstract
We introduce the most general mixed fractional derivatives and integrals from three points of views: probability, the theory of operator semigroups, and the theory of generalized functions. The solutions to the resulting mixed fractional PDEs turned out to be representable in terms of of completely monotone functions in a certain class generalizing the usual Mittag-Leffler functions.
Aftermath of the Chernobyl Catastrophe from the Point of View of the Security Concept
Abstract
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.
Existence and Asymptotic Stability of Periodic Two-Dimensional Contrast Structures in the Problem with Weak Linear Advection
Abstract
We consider the boundary-value singularly perturbed time-periodic problem for the parabolic reaction-advection-diffusion equation in the case of a weak linear advection in a two-dimensional domain. The main result of the present paper is the justification, under certain sufficient assumptions, of the existence of a periodic solution with internal transition layer near some closed curve and the study of the Lyapunov asymptotic stability of such a solution. For this purpose, an asymptotic expansion of the solution is constructed; the justification of the existence of the solution with the constructed asymptotics is carried out by using the method of differential inequalities. The proof of Lyapunov asymptotic stability is based on the application of the so-called method of contraction barriers.
On the Unique Solvability of the Problem of the Flow of an Aqueous Solution of Polymers near a Critical Point
Abstract
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.
Asymptotic Solutions of the Cauchy Problem with Localized Initial Data for a Finite-Difference Scheme Corresponding to the One-Dimensional Wave Equation
Abstract
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.
Dynamic Properties of a Nonlinear Viscoelastic Kirchhoff-Type Equation with Acoustic Control Boundary Conditions. I
Abstract
In this paper, we consider the nonlinear viscoelastic Kirchhoff-type equation
Generalized Localization Principle for Continuous Wavelet Decompositions
Abstract
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 Lp 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.
On the Parametrization of an Algebraic Curve
Abstract
At present, a plane algebraic curve can be parametrized in the following two cases: if its genus is equal to 0 or 1 and if it has a large group of birational automorphisms. Here we propose a new polyhedron method (involving a polyhedron called a Hadamard polyhedron by the author), which allows us to divide the space ℝ2 or ℂ2 into pieces in each of which the polynomial specifying the curve is sufficiently well approximated by its truncated polynomial, which often defines the parametrized curve. This approximate parametrization in a piece can be refined by means of the Newton method. Thus, an arbitrarily exact piecewise parametrization of the original curve can be obtained.
New Examples of Locally Algebraically Integrable Bodies
Abstract
Any compact body with regular boundary in ℝN 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.
On the Volumes of Hyperbolic Simplices
Abstract
We present an explicit formula for calculating the volume of an arbitrary hyperbolic 4-simplex in terms of the coordinates of its vertices; by this formula, the volume can be expressed in terms of one-dimensional integrals of real-valued integrands over closed intervals of the real line. In addition, it is proved in the paper that the volume of a hyperbolic 5-simplex cannot be expressed as the double integral of an elementary function of the coordinates of its vertices (of edge lengths).
New Theta-Function Identities of Level 6 in the Spirit of Ramanujan
Abstract
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.
On Nonrational Fibers of del Pezzo Fibrations over Curves
Abstract
We consider threefold del Pezzo fibrations over a curve germ whose central fiber is nonrational. Under the additional assumption that the singularities of the total space are at worst ordinary double points, we apply a suitable base change and show that there is a one-to-one correspondence between such fibrations and certain nonsingular del Pezzo fibrations equipped with a cyclic group action.
Algebra of Symmetries of Three-Frequency Hyperbolic Resonance
Abstract
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.
Weak Closure of Infinite Actions of Rank 1, Joinings, and Spectrum
Abstract
It is proved that the ergodic self-joining of an infinite transformation of rank 1 is part of the weak limit of shifts of a diagonal measure. A continuous class of nonisomorphic transformations with polynomial closure is proposed. These transformations possess minimal self-joinings and certain unusual spectral properties. Thus, for example, the tensor products of the powers of transformations have both a singular and a Lebesgue spectrum, depending on the choice of the power.
Strictly Positive Definite Functions on the Space of Infinite Diagonal Matrices
Abstract
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.
Decoherence and Coherence Preservation in the Solutions of the GKSL Equation in the Theory of Open Quantum Systems
Abstract
The properties of solutions of the Gorini–Kossakowski-Sudarshan–Lindblad (GKSL) equation for the density operator (matrix) of a system that has nondegenerate energy spectrum and weakly interacts with a reservoir are considered. Conditions for the existence of solutions for which the density matrix has off-diagonal entries (“coherences”) not tending to zero at large times are derived.